Properties

Label 4.31.b.a
Level $4$
Weight $31$
Character orbit 4.b
Analytic conductor $22.806$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 31 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.8057047611\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 7 x^{13} + 7040347091761 x^{12} - 42242082550475 x^{11} + \)\(18\!\cdots\!95\)\( x^{10} - \)\(90\!\cdots\!21\)\( x^{9} + \)\(21\!\cdots\!47\)\( x^{8} - \)\(84\!\cdots\!01\)\( x^{7} + \)\(12\!\cdots\!60\)\( x^{6} - \)\(36\!\cdots\!60\)\( x^{5} + \)\(30\!\cdots\!04\)\( x^{4} - \)\(61\!\cdots\!28\)\( x^{3} + \)\(24\!\cdots\!04\)\( x^{2} - \)\(24\!\cdots\!80\)\( x + \)\(48\!\cdots\!00\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{182}\cdot 3^{19}\cdot 5^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1748 + \beta_{1} ) q^{2} + ( 9 + 30 \beta_{1} - \beta_{2} ) q^{3} + ( -58628180 - 1845 \beta_{1} + 14 \beta_{2} - \beta_{3} ) q^{4} + ( 1061747275 - 84189 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{5} + ( -32433446090 - 51676 \beta_{1} + 4394 \beta_{2} - 15 \beta_{3} - \beta_{4} - \beta_{5} ) q^{6} + ( -698284 - 2454487 \beta_{1} - 6996 \beta_{2} + 130 \beta_{3} + \beta_{5} + \beta_{7} ) q^{7} + ( -2238826483361 - 62470574 \beta_{1} - 68551 \beta_{2} + 1842 \beta_{3} + 12 \beta_{4} + 14 \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{8} + ( -51584237429141 + 634535674 \beta_{1} + 84303 \beta_{2} + 19227 \beta_{3} - 854 \beta_{4} - 37 \beta_{5} - 26 \beta_{6} - \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})\) \( q +(-1748 + \beta_{1}) q^{2} +(9 + 30 \beta_{1} - \beta_{2}) q^{3} +(-58628180 - 1845 \beta_{1} + 14 \beta_{2} - \beta_{3}) q^{4} +(1061747275 - 84189 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} + \beta_{4}) q^{5} +(-32433446090 - 51676 \beta_{1} + 4394 \beta_{2} - 15 \beta_{3} - \beta_{4} - \beta_{5}) q^{6} +(-698284 - 2454487 \beta_{1} - 6996 \beta_{2} + 130 \beta_{3} + \beta_{5} + \beta_{7}) q^{7} +(-2238826483361 - 62470574 \beta_{1} - 68551 \beta_{2} + 1842 \beta_{3} + 12 \beta_{4} + 14 \beta_{5} + 2 \beta_{6} + \beta_{8}) q^{8} +(-51584237429141 + 634535674 \beta_{1} + 84303 \beta_{2} + 19227 \beta_{3} - 854 \beta_{4} - 37 \beta_{5} - 26 \beta_{6} - \beta_{8} + \beta_{9}) q^{9} +(-91991866033229 + 913009847 \beta_{1} - 5436640 \beta_{2} + 75375 \beta_{3} - 1551 \beta_{4} + 9 \beta_{5} - 120 \beta_{6} - 23 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{13}) q^{10} +(1212546222 + 4263848226 \beta_{1} + 13290961 \beta_{2} + 83434 \beta_{3} + 193 \beta_{4} + 1198 \beta_{5} + 48 \beta_{6} + 30 \beta_{7} + 42 \beta_{8} + \beta_{9} + \beta_{11} - 3 \beta_{13}) q^{11} +(3705662592727599 - 26484029068 \beta_{1} - 30239596 \beta_{2} - 253238 \beta_{3} + 36502 \beta_{4} + 4125 \beta_{5} - 149 \beta_{6} + 975 \beta_{7} + 35 \beta_{8} - 53 \beta_{9} + 4 \beta_{11} - \beta_{12} + 8 \beta_{13}) q^{12} +(5248188478201097 + 186177698145 \beta_{1} + 23486700 \beta_{2} - 299049 \beta_{3} - 30133 \beta_{4} - 26175 \beta_{5} - 6409 \beta_{6} + 466 \beta_{7} - 871 \beta_{8} - 18 \beta_{9} + \beta_{10} + 16 \beta_{11} + 4 \beta_{12} + 16 \beta_{13}) q^{13} +(2625806519152832 + 4502548671 \beta_{1} + 734781159 \beta_{2} + 4865110 \beta_{3} + 388067 \beta_{4} - 10006 \beta_{5} - 13839 \beta_{6} - 19276 \beta_{7} - 104 \beta_{8} + 164 \beta_{9} + 4 \beta_{10} + 48 \beta_{11} - 28 \beta_{12} - 8 \beta_{13}) q^{14} +(382864122198 + 1332477766099 \beta_{1} - 5031094192 \beta_{2} + 24621874 \beta_{3} + 4018 \beta_{4} + 533875 \beta_{5} - 33260 \beta_{6} + 2795 \beta_{7} + 6964 \beta_{8} + 166 \beta_{9} + 40 \beta_{10} + 70 \beta_{11} + 96 \beta_{12} + 302 \beta_{13}) q^{15} +(-76014361712991304 - 2367533097948 \beta_{1} + 6075980832 \beta_{2} + 90445648 \beta_{3} - 9131140 \beta_{4} + 76172 \beta_{5} - 53720 \beta_{6} + 145804 \beta_{7} - 3168 \beta_{8} + 2076 \beta_{9} + 144 \beta_{10} + 80 \beta_{11} - 340 \beta_{12} - 160 \beta_{13}) q^{16} +(-315510949611366256 + 12112044526790 \beta_{1} + 1554323225 \beta_{2} + 103570657 \beta_{3} - 8193154 \beta_{4} - 3719167 \beta_{5} + 22348 \beta_{6} + 29828 \beta_{7} + 16709 \beta_{8} + 1113 \beta_{9} + 754 \beta_{10} - 224 \beta_{11} + 968 \beta_{12} - 224 \beta_{13}) q^{17} +(769529400362847554 - 50461053633869 \beta_{1} + 48438989216 \beta_{2} - 544881506 \beta_{3} + 111137090 \beta_{4} - 718062 \beta_{5} + 598768 \beta_{6} - 474926 \beta_{7} - 11646 \beta_{8} - 12860 \beta_{9} + 2432 \beta_{10} - 1280 \beta_{11} - 2240 \beta_{12} + 194 \beta_{13}) q^{18} +(19994352174902 + 69917985179254 \beta_{1} - 41498692933 \beta_{2} - 3098637850 \beta_{3} - 5166817 \beta_{4} + 15792170 \beta_{5} + 434152 \beta_{6} + 172170 \beta_{7} - 222058 \beta_{8} + 2167 \beta_{9} + 8880 \beta_{10} - 2761 \beta_{11} + 4928 \beta_{12} - 9125 \beta_{13}) q^{19} +(2234948996017120648 - 88553582168866 \beta_{1} + 176329526748 \beta_{2} - 478263370 \beta_{3} - 603535408 \beta_{4} - 3327376 \beta_{5} + 2196032 \beta_{6} - 2583792 \beta_{7} - 52528 \beta_{8} - 55856 \beta_{9} + 25536 \beta_{10} - 5696 \beta_{11} - 7536 \beta_{12} + 1152 \beta_{13}) q^{20} +(-1721936772600000162 + 172834591189758 \beta_{1} + 25658524315 \beta_{2} + 17790210765 \beta_{3} - 112141334 \beta_{4} - 34063163 \beta_{5} - 838789 \beta_{6} + 119210 \beta_{7} + 190557 \beta_{8} + 13694 \beta_{9} + 72973 \beta_{10} - 3888 \beta_{11} + 9268 \beta_{12} - 3888 \beta_{13}) q^{21} +(-4561076917613988654 - 7094698880930 \beta_{1} + 1211756604880 \beta_{2} - 234499249 \beta_{3} + 2683493823 \beta_{4} + 11827073 \beta_{5} - 11639090 \beta_{6} + 14080088 \beta_{7} - 302256 \beta_{8} + 234872 \beta_{9} + 186040 \beta_{10} + 10400 \beta_{11} - 1288 \beta_{12} - 13680 \beta_{13}) q^{22} +(-133988767406014 - 463658994592441 \beta_{1} + 3534471711160 \beta_{2} - 106464739326 \beta_{3} - 209996722 \beta_{4} - 71965689 \beta_{5} - 23628420 \beta_{6} - 2372753 \beta_{7} + 2711756 \beta_{8} + 2538 \beta_{9} + 441848 \beta_{10} + 36554 \beta_{11} - 34016 \beta_{12} + 145826 \beta_{13}) q^{23} +(-15656102616245505096 + 3690351209788224 \beta_{1} - 1154931936280 \beta_{2} + 24601421584 \beta_{3} - 9148804240 \beta_{4} + 26333376 \beta_{5} - 63342096 \beta_{6} - 8890608 \beta_{7} - 461976 \beta_{8} + 625104 \beta_{9} + 991680 \beta_{10} + 104896 \beta_{11} + 102032 \beta_{12} - 18304 \beta_{13}) q^{24} +(\)\(18\!\cdots\!31\)\( - 119091751180616 \beta_{1} + 99441794860 \beta_{2} + 536977145632 \beta_{3} + 1084397056 \beta_{4} + 937906624 \beta_{5} - 95626638 \beta_{6} - 10687052 \beta_{7} - 10741384 \beta_{8} - 449010 \beta_{9} + 2022346 \beta_{10} + 134304 \beta_{11} - 252120 \beta_{12} + 134304 \beta_{13}) q^{25} +(\)\(19\!\cdots\!67\)\( + 5582218606910187 \beta_{1} + 29478928529760 \beta_{2} - 108703523813 \beta_{3} + 3801647365 \beta_{4} + 59277117 \beta_{5} - 58280216 \beta_{6} - 119658339 \beta_{7} - 66155 \beta_{8} - 2724182 \beta_{9} + 3954432 \beta_{10} + 48640 \beta_{11} + 412800 \beta_{12} + 319509 \beta_{13}) q^{26} +(-2005365712599927 - 6898948201599476 \beta_{1} + 79887884218726 \beta_{2} - 2025306580726 \beta_{3} - 3612009351 \beta_{4} - 2502162074 \beta_{5} - 81676424 \beta_{6} - 28660282 \beta_{7} - 30883238 \beta_{8} - 688127 \beta_{9} + 7009552 \beta_{10} - 173503 \beta_{11} - 514624 \beta_{12} - 1481411 \beta_{13}) q^{27} +(\)\(16\!\cdots\!50\)\( + 2877658369020280 \beta_{1} - 76141499892168 \beta_{2} - 268431677092 \beta_{3} + 46824124260 \beta_{4} + 303995518 \beta_{5} + 55143858 \beta_{6} + 390285770 \beta_{7} - 10308350 \beta_{8} - 2419342 \beta_{9} + 11702784 \beta_{10} - 993192 \beta_{11} + 401898 \beta_{12} + 567472 \beta_{13}) q^{28} +(-\)\(37\!\cdots\!09\)\( + 18931611636897243 \beta_{1} + 3362906997685 \beta_{2} + 4539434238422 \beta_{3} - 15931851191 \beta_{4} - 1299050360 \beta_{5} - 149547200 \beta_{6} - 46152224 \beta_{7} + 74650408 \beta_{8} + 4707048 \beta_{9} + 17759600 \beta_{10} - 1640704 \beta_{11} + 599488 \beta_{12} - 1640704 \beta_{13}) q^{29} +(-\)\(14\!\cdots\!08\)\( - 2163567450104091 \beta_{1} + 576249775488173 \beta_{2} + 603862417002 \beta_{3} - 262558431815 \beta_{4} - 1435770410 \beta_{5} + 111760139 \beta_{6} - 75805828 \beta_{7} - 47194104 \beta_{8} + 25715660 \beta_{9} + 23937772 \beta_{10} - 1723632 \beta_{11} - 2242676 \beta_{12} - 3913176 \beta_{13}) q^{30} +(13369969937876678 + 46750130310549752 \beta_{1} - 29842152077820 \beta_{2} - 6911559295244 \beta_{3} - 15296401922 \beta_{4} + 22418402672 \beta_{5} - 2786858372 \beta_{6} + 114797016 \beta_{7} + 9117612 \beta_{8} + 4412314 \beta_{9} + 28145144 \beta_{10} - 1001350 \beta_{11} + 5413664 \beta_{12} + 10468498 \beta_{13}) q^{31} +(\)\(30\!\cdots\!08\)\( - 71563692756822384 \beta_{1} - 1138259612434880 \beta_{2} - 1475203727168 \beta_{3} + 509956383344 \beta_{4} - 2816423632 \beta_{5} + 2275856928 \beta_{6} - 3315537232 \beta_{7} - 69587136 \beta_{8} - 11554064 \beta_{9} + 22477888 \beta_{10} + 5095232 \beta_{11} - 9157840 \beta_{12} - 9604736 \beta_{13}) q^{32} +(\)\(32\!\cdots\!70\)\( + 295822370205320318 \beta_{1} + 37122338404741 \beta_{2} - 1399987016895 \beta_{3} - 53034536322 \beta_{4} - 46609147903 \beta_{5} - 9466785794 \beta_{6} + 596661176 \beta_{7} - 513266435 \beta_{8} - 44739737 \beta_{9} + 727420 \beta_{10} + 10860480 \beta_{11} + 10294768 \beta_{12} + 10860480 \beta_{13}) q^{33} +(\)\(13\!\cdots\!18\)\( - 293187151208781916 \beta_{1} + 3945156511967392 \beta_{2} - 315623231242 \beta_{3} + 91360793066 \beta_{4} + 14004964730 \beta_{5} + 11535600176 \beta_{6} + 5842107962 \beta_{7} - 34925654 \beta_{8} - 182066156 \beta_{9} - 55758464 \beta_{10} + 16304896 \beta_{11} - 9608384 \beta_{12} + 29048298 \beta_{13}) q^{34} +(-110121551488984354 - 386599666241323752 \beta_{1} - 782817175135396 \beta_{2} + 45817022076236 \beta_{3} + 61260946502 \beta_{4} + 6327676876 \beta_{5} - 17770721864 \beta_{6} - 90802884 \beta_{7} + 1048994940 \beta_{8} + 16451806 \beta_{9} - 160264272 \beta_{10} + 20482462 \beta_{11} - 4030656 \beta_{12} - 54501594 \beta_{13}) q^{35} +(-\)\(18\!\cdots\!76\)\( + 737075930510061459 \beta_{1} - 12715685354563906 \beta_{2} + 11705422948807 \beta_{3} - 2275175760992 \beta_{4} + 9545455840 \beta_{5} + 40449283200 \beta_{6} + 6308438560 \beta_{7} + 544256416 \beta_{8} + 198678432 \beta_{9} - 329689216 \beta_{10} - 8494208 \beta_{11} + 26927392 \beta_{12} + 97184000 \beta_{13}) q^{36} +(-\)\(44\!\cdots\!03\)\( - 185513596871693487 \beta_{1} - 51280757536176 \beta_{2} - 130068213096213 \beta_{3} + 278431783579 \beta_{4} + 242540979909 \beta_{5} - 59702255309 \beta_{6} - 1786487590 \beta_{7} + 1826220925 \beta_{8} + 457276230 \beta_{9} - 582488347 \beta_{10} - 36180400 \beta_{11} - 60212332 \beta_{12} - 36180400 \beta_{13}) q^{37} +(-\)\(74\!\cdots\!02\)\( - 117446004243116470 \beta_{1} + 16482388185973184 \beta_{2} - 14356985161027 \beta_{3} + 6872032148349 \beta_{4} - 58360042829 \beta_{5} + 173998562810 \beta_{6} - 35425698232 \beta_{7} + 3095906160 \beta_{8} + 813911208 \beta_{9} - 884781464 \beta_{10} - 79768352 \beta_{11} + 110407976 \beta_{12} - 130143440 \beta_{13}) q^{38} +(-1969277318093721204 - 6900088058538307585 \beta_{1} - 5078234684125884 \beta_{2} + 314725904775310 \beta_{3} + 328229221968 \beta_{4} - 725437130569 \beta_{5} - 209050565488 \beta_{6} + 5427348631 \beta_{7} - 6818433760 \beta_{8} - 270331168 \beta_{9} - 1231867360 \beta_{10} - 141826208 \beta_{11} - 128504960 \beta_{12} + 218182112 \beta_{13}) q^{39} +(\)\(17\!\cdots\!38\)\( + 2258617957171709044 \beta_{1} - 40606244275201062 \beta_{2} - 43051014528204 \beta_{3} - 8830958439944 \beta_{4} - 31464471732 \beta_{5} + 561032029524 \beta_{6} + 63085190656 \beta_{7} + 1398930090 \beta_{8} - 1360910848 \beta_{9} - 1465837568 \beta_{10} - 62502912 \beta_{11} + 136023552 \beta_{12} - 645091328 \beta_{13}) q^{40} +(\)\(10\!\cdots\!54\)\( + 6897980833399583888 \beta_{1} + 785646875316560 \beta_{2} - 407124769717044 \beta_{3} + 3757554888200 \beta_{4} + 351474617836 \beta_{5} - 596644701162 \beta_{6} + 656340652 \beta_{7} + 5429397508 \beta_{8} - 3617049786 \beta_{9} - 1472211226 \beta_{10} - 11805088 \beta_{11} - 55458920 \beta_{12} - 11805088 \beta_{13}) q^{41} +(\)\(18\!\cdots\!48\)\( - 1409490010360273476 \beta_{1} + 35434786243550528 \beta_{2} - 73539050425900 \beta_{3} + 1494306056172 \beta_{4} - 107083930548 \beta_{5} + 1186936441376 \beta_{6} + 70408798924 \beta_{7} - 17530585108 \beta_{8} - 1549210792 \beta_{9} - 963164416 \beta_{10} + 157130240 \beta_{11} - 140192640 \beta_{12} + 257706604 \beta_{13}) q^{42} +(-7799686621179933655 - 27305404747127687630 \beta_{1} - 4333639291131943 \beta_{2} + 66929742392348 \beta_{3} - 2055630567546 \beta_{4} + 2265873025092 \beta_{5} - 2022536632720 \beta_{6} - 74031634876 \beta_{7} + 28163740412 \beta_{8} + 900050774 \beta_{9} + 413758880 \beta_{10} + 496493014 \beta_{11} + 403557760 \beta_{12} - 674192770 \beta_{13}) q^{43} +(\)\(51\!\cdots\!45\)\( - 4488801365837519476 \beta_{1} + 2965750403447404 \beta_{2} + 15178893648438 \beta_{3} + 46782749810538 \beta_{4} + 302134250083 \beta_{5} + 2615543078165 \beta_{6} - 389185934415 \beta_{7} - 3538595491 \beta_{8} + 6337377717 \beta_{9} + 2850634752 \beta_{10} + 451838460 \beta_{11} - 858509439 \beta_{12} + 2859570168 \beta_{13}) q^{44} +(-\)\(11\!\cdots\!67\)\( + \)\(11\!\cdots\!81\)\( \beta_{1} + 15474520251370386 \beta_{2} + 2240268426677793 \beta_{3} - 36856510384913 \beta_{4} - 5854158665893 \beta_{5} - 5867609062859 \beta_{6} + 43703775414 \beta_{7} - 57766311037 \beta_{8} + 19417248370 \beta_{9} + 6938455203 \beta_{10} + 676391472 \beta_{11} + 1190617740 \beta_{12} + 676391472 \beta_{13}) q^{45} +(\)\(49\!\cdots\!12\)\( + 782319156307048193 \beta_{1} - 103120672297820663 \beta_{2} + 120065915896434 \beta_{3} - 105880612956363 \beta_{4} + 2253417350094 \beta_{5} + 7373939452271 \beta_{6} + 311064360652 \beta_{7} + 99634816360 \beta_{8} - 4560685860 \beta_{9} + 12247311740 \beta_{10} + 501789136 \beta_{11} - 1458373220 \beta_{12} + 708994312 \beta_{13}) q^{46} +(-52500319265177766478 - \)\(18\!\cdots\!34\)\( \beta_{1} + 113611293022097796 \beta_{2} - 4505856049007248 \beta_{3} - 20768211438494 \beta_{4} + 5421559917122 \beta_{5} - 11782576209036 \beta_{6} + 664910310394 \beta_{7} - 27940528108 \beta_{8} + 825358262 \beta_{9} + 19082793704 \beta_{10} - 381892010 \beta_{11} + 1207250272 \beta_{12} + 1309315838 \beta_{13}) q^{47} +(\)\(16\!\cdots\!52\)\( - 12861630259776600736 \beta_{1} + 609372652603020288 \beta_{2} - 1932666619725440 \beta_{3} + 151610695485344 \beta_{4} - 3531700869344 \beta_{5} + 17146878120384 \beta_{6} + 450270212384 \beta_{7} - 35615527936 \beta_{8} - 19245154656 \beta_{9} + 25931238784 \beta_{10} - 898640000 \beta_{11} - 179901408 \beta_{12} - 7736028928 \beta_{13}) q^{48} +(\)\(26\!\cdots\!89\)\( + \)\(32\!\cdots\!40\)\( \beta_{1} + 42492950249586524 \beta_{2} + 7788500996637836 \beta_{3} + 54188595326024 \beta_{4} + 10644524750156 \beta_{5} - 21807384143832 \beta_{6} - 172539847648 \beta_{7} + 183556220188 \beta_{8} - 69965194284 \beta_{9} + 31343846288 \beta_{10} - 2701526784 \beta_{11} - 1508692416 \beta_{12} - 2701526784 \beta_{13}) q^{49} +(-\)\(44\!\cdots\!04\)\( + \)\(18\!\cdots\!15\)\( \beta_{1} - 1004715386974014784 \beta_{2} - 3199221019708052 \beta_{3} + 15532793274580 \beta_{4} - 11777530806668 \beta_{5} + 32700556211168 \beta_{6} - 2139584575500 \beta_{7} - 573995688748 \beta_{8} + 39313432104 \beta_{9} + 33163778304 \beta_{10} - 4256914944 \beta_{11} + 4662631296 \beta_{12} - 6681165228 \beta_{13}) q^{50} +(-\)\(27\!\cdots\!63\)\( - \)\(95\!\cdots\!40\)\( \beta_{1} + 579970015531103790 \beta_{2} - 6841753420457430 \beta_{3} - 62249710273823 \beta_{4} - 40388841397906 \beta_{5} - 47708516799376 \beta_{6} - 3938690314658 \beta_{7} - 338841679510 \beta_{8} - 11817184159 \beta_{9} + 26309904512 \beta_{10} - 4551907743 \beta_{11} - 7265276416 \beta_{12} + 1509256413 \beta_{13}) q^{51} +(-\)\(25\!\cdots\!52\)\( + \)\(19\!\cdots\!74\)\( \beta_{1} + 1385316469852391228 \beta_{2} - 1221165514020154 \beta_{3} - 527759056501744 \beta_{4} + 25032624106288 \beta_{5} + 65216941223744 \beta_{6} + 2935360974928 \beta_{7} + 42028370704 \beta_{8} + 16558619152 \beta_{9} + 9760517824 \beta_{10} - 2763871040 \beta_{11} + 10543330768 \beta_{12} + 5810380416 \beta_{13}) q^{52} +(-\)\(11\!\cdots\!39\)\( + \)\(13\!\cdots\!21\)\( \beta_{1} + 172963726658014518 \beta_{2} + 2294970489879525 \beta_{3} - 266273411787153 \beta_{4} + 37524654292911 \beta_{5} - 93105737497151 \beta_{6} - 336462695778 \beta_{7} - 301341170505 \beta_{8} + 163791247626 \beta_{9} - 24743801481 \beta_{10} + 1655076720 \beta_{11} - 11325725220 \beta_{12} + 1655076720 \beta_{13}) q^{53} +(\)\(74\!\cdots\!56\)\( + 11182275653493480386 \beta_{1} - 3143690426132480890 \beta_{2} - 3613222996172462 \beta_{3} + 1458757144900692 \beta_{4} + 35495530036046 \beta_{5} + 122461503297878 \beta_{6} + 2287502252536 \beta_{7} + 2055132030992 \beta_{8} - 107452807528 \beta_{9} - 74961050920 \beta_{10} + 10003003936 \beta_{11} + 8104470040 \beta_{12} + 18503250512 \beta_{13}) q^{54} +(-\)\(78\!\cdots\!44\)\( - \)\(27\!\cdots\!47\)\( \beta_{1} - 716080373264756900 \beta_{2} + 51087431480492002 \beta_{3} - 76459980771936 \beta_{4} + 30201632921073 \beta_{5} - 150888101919632 \beta_{6} + 15525756644113 \beta_{7} + 1252474705984 \beta_{8} + 17949612304 \beta_{9} - 145623427616 \beta_{10} + 20780721296 \beta_{11} - 2831108992 \beta_{12} - 26882659760 \beta_{13}) q^{55} +(-\)\(73\!\cdots\!80\)\( + \)\(15\!\cdots\!96\)\( \beta_{1} + 644294770774936048 \beta_{2} - 377281481170080 \beta_{3} - 1164988569046112 \beta_{4} - 113691097792384 \beta_{5} + 190134898717344 \beta_{6} - 11255918447776 \beta_{7} + 244968005360 \beta_{8} + 150517821408 \beta_{9} - 230892665216 \beta_{10} + 19278026368 \beta_{11} - 13032834976 \beta_{12} + 49255389952 \beta_{13}) q^{56} +(-\)\(12\!\cdots\!34\)\( + \)\(39\!\cdots\!54\)\( \beta_{1} + 480727993977288711 \beta_{2} - 77989464905321781 \beta_{3} + 535253296200138 \beta_{4} - 125025226104501 \beta_{5} - 241950494876238 \beta_{6} + 2163240753336 \beta_{7} - 671724929025 \beta_{8} - 178381969947 \beta_{9} - 318051571716 \beta_{10} + 23501430720 \beta_{11} + 28570513392 \beta_{12} + 23501430720 \beta_{13}) q^{57} +(\)\(20\!\cdots\!39\)\( - \)\(33\!\cdots\!97\)\( \beta_{1} + 1163862364286461728 \beta_{2} - 17036121814961913 \beta_{3} + 209577978565465 \beta_{4} - 55671833984975 \beta_{5} + 295759071505864 \beta_{6} + 15423797515025 \beta_{7} - 4535977597719 \beta_{8} + 36287817682 \beta_{9} - 401000203264 \beta_{10} + 9184557056 \beta_{11} - 50613313024 \beta_{12} + 648503017 \beta_{13}) q^{58} +(-\)\(15\!\cdots\!13\)\( - \)\(53\!\cdots\!34\)\( \beta_{1} - 5222106418014531707 \beta_{2} + 130185122975319600 \beta_{3} - 137943396268864 \beta_{4} + 210200475809288 \beta_{5} - 347715335311544 \beta_{6} - 47877494878856 \beta_{7} - 68776922752 \beta_{8} + 43296412040 \beta_{9} - 435518346736 \beta_{10} - 26322837560 \beta_{11} + 69619249600 \beta_{12} + 115135321256 \beta_{13}) q^{59} +(\)\(48\!\cdots\!18\)\( - \)\(13\!\cdots\!04\)\( \beta_{1} - 11973489839253158712 \beta_{2} - 40089878901667036 \beta_{3} + 6224585929542172 \beta_{4} + 405852416024002 \beta_{5} + 456279909333518 \beta_{6} + 5189131510 \beta_{7} - 182948644162 \beta_{8} - 705084298162 \beta_{9} - 419783727616 \beta_{10} - 26900964184 \beta_{11} - 72535158570 \beta_{12} - 239115247280 \beta_{13}) q^{60} +(-\)\(55\!\cdots\!15\)\( + \)\(68\!\cdots\!09\)\( \beta_{1} + 847002627732693692 \beta_{2} - 63709421579099401 \beta_{3} - 3496247292083061 \beta_{4} - 185278707271327 \beta_{5} - 392336610424001 \beta_{6} + 1295788460578 \beta_{7} + 5918941200281 \beta_{8} - 456225026490 \beta_{9} - 284385013271 \beta_{10} - 84000063856 \beta_{11} + 65054586788 \beta_{12} - 84000063856 \beta_{13}) q^{61} +(-\)\(50\!\cdots\!88\)\( - 75193100167568282536 \beta_{1} + 22413061624472536520 \beta_{2} + 28696853667041720 \beta_{3} - 9293547150389168 \beta_{4} - 12847351059512 \beta_{5} + 499563515477576 \beta_{6} - 46400306681696 \beta_{7} + 6502770012864 \beta_{8} + 778208738336 \beta_{9} - 26996544736 \beta_{10} - 109940279936 \beta_{11} - 14806747616 \beta_{12} - 161254712896 \beta_{13}) q^{62} +(-\)\(25\!\cdots\!86\)\( - \)\(90\!\cdots\!59\)\( \beta_{1} + 6051228720860840720 \beta_{2} - 98456216627442338 \beta_{3} - 673955263861026 \beta_{4} - 185387484924187 \beta_{5} - 438457599332116 \beta_{6} + 133465791031405 \beta_{7} - 7125046460308 \beta_{8} - 125797575190 \beta_{9} + 394670143448 \beta_{10} - 81500359094 \beta_{11} - 44297216096 \beta_{12} - 196338230494 \beta_{13}) q^{63} +(-\)\(13\!\cdots\!20\)\( + \)\(28\!\cdots\!32\)\( \beta_{1} - 34730272202656368384 \beta_{2} - 33112961855625472 \beta_{3} + 7485051846627520 \beta_{4} - 1367938241798208 \beta_{5} + 324542198698624 \beta_{6} + 79563877291456 \beta_{7} - 694125312 \beta_{8} + 955830155968 \beta_{9} + 1000779805952 \beta_{10} - 105634598656 \beta_{11} + 154473136064 \beta_{12} + 494606867968 \beta_{13}) q^{64} +(-\)\(54\!\cdots\!88\)\( + \)\(37\!\cdots\!24\)\( \beta_{1} + 579798245933764264 \beta_{2} + 515005105428836572 \beta_{3} + 10466035369455208 \beta_{4} + 872140372399548 \beta_{5} - 380817072008126 \beta_{6} - 12234314125404 \beta_{7} - 8040566407068 \beta_{8} + 3284721487530 \beta_{9} + 1717505866242 \beta_{10} + 26748645408 \beta_{11} - 252277710840 \beta_{12} + 26748645408 \beta_{13}) q^{65} +(\)\(31\!\cdots\!50\)\( + \)\(37\!\cdots\!14\)\( \beta_{1} + 51821339140437535584 \beta_{2} - 156038490829019678 \beta_{3} - 750242819005634 \beta_{4} + 198665503639662 \beta_{5} - 75216565304816 \beta_{6} - 31927690317138 \beta_{7} + 1621092378878 \beta_{8} - 2977019186372 \beta_{9} + 2586662881920 \beta_{10} + 208530533632 \beta_{11} + 338981158592 \beta_{12} + 458390763710 \beta_{13}) q^{66} +(\)\(72\!\cdots\!82\)\( + \)\(26\!\cdots\!82\)\( \beta_{1} + 41648484176560042287 \beta_{2} - 924230618287473394 \beta_{3} - 1225736766217701 \beta_{4} - 1575902458546966 \beta_{5} + 287164042459968 \beta_{6} - 299516869728934 \beta_{7} + 14300886028782 \beta_{8} - 73412161269 \beta_{9} + 3356552290656 \beta_{10} + 357871983243 \beta_{11} - 431284144512 \beta_{12} - 390296557473 \beta_{13}) q^{67} +(-\)\(11\!\cdots\!24\)\( + \)\(13\!\cdots\!38\)\( \beta_{1} - 43076590456602960900 \beta_{2} + 169245240345917662 \beta_{3} - 24423750714801120 \beta_{4} + 4248045820373600 \beta_{5} - 1175386676875648 \beta_{6} - 106317697913696 \beta_{7} - 2025657912800 \beta_{8} + 1974623866912 \beta_{9} + 4049888980352 \beta_{10} + 465288407424 \beta_{11} + 338714662816 \beta_{12} - 144432554752 \beta_{13}) q^{68} +(\)\(92\!\cdots\!82\)\( - \)\(23\!\cdots\!50\)\( \beta_{1} - 2735910514670762119 \beta_{2} + 952117841489358783 \beta_{3} - 33895164147798258 \beta_{4} + 1094900845206055 \beta_{5} + 1603558982459945 \beta_{6} - 7691687357618 \beta_{7} - 27062354132497 \beta_{8} - 9880227609094 \beta_{9} + 4281134621711 \beta_{10} + 544502931696 \beta_{11} - 270796668868 \beta_{12} + 544502931696 \beta_{13}) q^{69} +(\)\(41\!\cdots\!32\)\( + \)\(67\!\cdots\!76\)\( \beta_{1} + 9753341725372516908 \beta_{2} + 413114238980952412 \beta_{3} + 46129026054489024 \beta_{4} - 278921105596828 \beta_{5} - 2993906850031748 \beta_{6} + 311466119197232 \beta_{7} - 47759551755360 \beta_{8} + 3641252032112 \beta_{9} + 3955273747696 \beta_{10} + 226862590784 \beta_{11} - 53934427792 \beta_{12} - 138831956448 \beta_{13}) q^{70} +(\)\(21\!\cdots\!98\)\( + \)\(73\!\cdots\!37\)\( \beta_{1} - 19791387355395527528 \beta_{2} - 770067736111773466 \beta_{3} + 2564594371522674 \beta_{4} + 2335367414067821 \beta_{5} + 3729975144574692 \beta_{6} + 393317007087173 \beta_{7} + 2249812180660 \beta_{8} + 204028103990 \beta_{9} + 2789373271240 \beta_{10} - 289140032682 \beta_{11} + 493168136672 \beta_{12} + 2969880975358 \beta_{13}) q^{71} +(-\)\(87\!\cdots\!25\)\( - \)\(20\!\cdots\!54\)\( \beta_{1} + \)\(19\!\cdots\!41\)\( \beta_{2} - 61276008996310110 \beta_{3} - 75804395153127508 \beta_{4} - 10783139870390882 \beta_{5} - 5123287045282318 \beta_{6} - 346424885068800 \beta_{7} - 6071033773319 \beta_{8} - 8621572240384 \beta_{9} + 424394895360 \beta_{10} - 229068337152 \beta_{11} - 924130698240 \beta_{12} - 2215666950144 \beta_{13}) q^{72} +(-\)\(99\!\cdots\!24\)\( - \)\(11\!\cdots\!30\)\( \beta_{1} - 14894325771281210821 \beta_{2} - 1247088384366808969 \beta_{3} + 119417159999761682 \beta_{4} - 5637739583791753 \beta_{5} + 7708440156415182 \beta_{6} + 35222001152864 \beta_{7} + 73790656677547 \beta_{8} + 14184229622949 \beta_{9} - 2899762024912 \beta_{10} - 1332994497792 \beta_{11} + 1515739650240 \beta_{12} - 1332994497792 \beta_{13}) q^{73} +(-\)\(12\!\cdots\!33\)\( - \)\(45\!\cdots\!01\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} - 580037476342610077 \beta_{3} + 27899245306407869 \beta_{4} + 783111281499189 \beta_{5} - 8933133272276056 \beta_{6} - 155482228470123 \beta_{7} + 138613373762765 \beta_{8} + 11216169456410 \beta_{9} - 7804513295616 \beta_{10} - 1547020200448 \beta_{11} - 1771612084608 \beta_{12} - 2326706048691 \beta_{13}) q^{74} +(\)\(70\!\cdots\!09\)\( + \)\(24\!\cdots\!42\)\( \beta_{1} - \)\(16\!\cdots\!59\)\( \beta_{2} + 2657101848964403644 \beta_{3} + 18536474293531758 \beta_{4} + 6034120478965004 \beta_{5} + 12961262900816744 \beta_{6} + 20666220145564 \beta_{7} - 84540317623540 \beta_{8} + 917026449974 \beta_{9} - 13284366427888 \beta_{10} - 1072323947402 \beta_{11} + 1989350397376 \beta_{12} - 5745003907426 \beta_{13}) q^{75} +(-\)\(30\!\cdots\!37\)\( - \)\(80\!\cdots\!56\)\( \beta_{1} + \)\(30\!\cdots\!24\)\( \beta_{2} + 1303771232601901250 \beta_{3} + 90077812398400030 \beta_{4} + 21422860995005881 \beta_{5} - 14522026367039697 \beta_{6} + 816684031731779 \beta_{7} + 14941138442631 \beta_{8} + 5887064094479 \beta_{9} - 19768225442816 \beta_{10} - 2569265736396 \beta_{11} - 1498989991245 \beta_{12} + 7244313877096 \beta_{13}) q^{76} +(-\)\(99\!\cdots\!14\)\( - \)\(26\!\cdots\!82\)\( \beta_{1} - 34711374741260199691 \beta_{2} - 6569092496883701021 \beta_{3} - 345331775697418858 \beta_{4} - 4366364287503445 \beta_{5} + 18387026036765925 \beta_{6} + 80957816487254 \beta_{7} + 49291138129235 \beta_{8} + 9444697560402 \beta_{9} - 25754200920461 \beta_{10} - 468202780880 \beta_{11} + 1049012709836 \beta_{12} - 468202780880 \beta_{13}) q^{77} +(\)\(73\!\cdots\!84\)\( + \)\(11\!\cdots\!77\)\( \beta_{1} - \)\(76\!\cdots\!55\)\( \beta_{2} + 4277998110749899610 \beta_{3} - 204728298744388859 \beta_{4} - 3795505907623066 \beta_{5} - 19865260886356505 \beta_{6} - 1418331839656660 \beta_{7} - 252081274366808 \beta_{8} - 54745969849412 \beta_{9} - 30761162967716 \beta_{10} + 1506964439120 \beta_{11} + 383256824444 \beta_{12} + 5987255129416 \beta_{13}) q^{78} +(\)\(10\!\cdots\!04\)\( + \)\(38\!\cdots\!74\)\( \beta_{1} - \)\(14\!\cdots\!88\)\( \beta_{2} + 7916502544438034356 \beta_{3} + 40062664713492424 \beta_{4} - 8959808819375454 \beta_{5} + 23937829209260064 \beta_{6} - 1498782186910494 \beta_{7} + 128796515453648 \beta_{8} - 171136478616 \beta_{9} - 33434915364032 \beta_{10} + 2224958166888 \beta_{11} - 2396094645504 \beta_{12} - 4690041599544 \beta_{13}) q^{79} +(-\)\(10\!\cdots\!76\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} + \)\(69\!\cdots\!28\)\( \beta_{2} + 496846971166587680 \beta_{3} + 305376287925080152 \beta_{4} - 33585694451579144 \beta_{5} - 25118337930590576 \beta_{6} + 1124995226773240 \beta_{7} + 37168369756224 \beta_{8} + 18920134021784 \beta_{9} - 31753132859488 \beta_{10} + 5937965593888 \beta_{11} + 3756029709880 \beta_{12} - 11190861359680 \beta_{13}) q^{80} +(\)\(10\!\cdots\!27\)\( - \)\(58\!\cdots\!86\)\( \beta_{1} - 76579926349344177951 \beta_{2} - 10302132722722907139 \beta_{3} + 1039992213640299942 \beta_{4} + 26446468373493117 \beta_{5} + 25563471834112446 \beta_{6} - 66109594275576 \beta_{7} - 344334803251287 \beta_{8} - 90461762300013 \beta_{9} - 25802898306588 \beta_{10} + 7328796687936 \beta_{11} - 7143008774256 \beta_{12} + 7328796687936 \beta_{13}) q^{81} +(\)\(71\!\cdots\!96\)\( + \)\(11\!\cdots\!58\)\( \beta_{1} - \)\(22\!\cdots\!88\)\( \beta_{2} - 8414699478345413812 \beta_{3} - 428688351986278412 \beta_{4} + 7028746237549972 \beta_{5} - 25959861490904096 \beta_{6} + 1572002804339476 \beta_{7} + 419318010463988 \beta_{8} + 59922996246632 \beta_{9} - 11618916309760 \beta_{10} + 3683440491008 \beta_{11} + 7924823266688 \beta_{12} - 2398676017548 \beta_{13}) q^{82} +(\)\(66\!\cdots\!65\)\( + \)\(23\!\cdots\!58\)\( \beta_{1} - \)\(13\!\cdots\!17\)\( \beta_{2} - 411031581633289624 \beta_{3} + 18857500515872332 \beta_{4} - 22686423912867120 \beta_{5} + 19006817569599944 \beta_{6} + 5233289178850048 \beta_{7} + 378385581222008 \beta_{8} - 6599238928812 \beta_{9} + 7814501729936 \beta_{10} + 1839484298900 \beta_{11} - 8438723227712 \beta_{12} + 44643741179972 \beta_{13}) q^{83} +(-\)\(19\!\cdots\!80\)\( + \)\(15\!\cdots\!64\)\( \beta_{1} - \)\(68\!\cdots\!28\)\( \beta_{2} + 274853025066221312 \beta_{3} - 34605059667063872 \beta_{4} + 38616621036480320 \beta_{5} - 17517263986150656 \beta_{6} - 3865921699280192 \beta_{7} + 31395223728064 \beta_{8} - 17483383934016 \beta_{9} + 36785456223488 \beta_{10} + 3557169077504 \beta_{11} + 8161492866240 \beta_{12} + 774979577344 \beta_{13}) q^{84} +(-\)\(11\!\cdots\!28\)\( + \)\(11\!\cdots\!32\)\( \beta_{1} + 19286768527069053571 \beta_{2} + 25695849266337036091 \beta_{3} - 2073802475598106144 \beta_{4} + 17391238487413063 \beta_{5} + 4010876743043569 \beta_{6} - 498335859295074 \beta_{7} + 253171116051567 \beta_{8} + 198656358069330 \beta_{9} + 70560867332727 \beta_{10} - 9632863787152 \beta_{11} - 4447482402340 \beta_{12} - 9632863787152 \beta_{13}) q^{85} +(\)\(29\!\cdots\!06\)\( + \)\(48\!\cdots\!76\)\( \beta_{1} + \)\(19\!\cdots\!54\)\( \beta_{2} + 33559275507063812007 \beta_{3} + 546042173571528365 \beta_{4} + 239906103167337 \beta_{5} + 2999788323311140 \beta_{6} + 6200136505799248 \beta_{7} - 129550706162848 \beta_{8} + 169631132761104 \beta_{9} + 109353647190416 \beta_{10} - 7580885949760 \beta_{11} - 279182024432 \beta_{12} - 18924146856736 \beta_{13}) q^{86} +(-\)\(83\!\cdots\!82\)\( - \)\(28\!\cdots\!41\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} - 44788449718815815438 \beta_{3} - 101698274316044534 \beta_{4} - 422321766897365 \beta_{5} - 24123468110696972 \beta_{6} - 12931487438422429 \beta_{7} - 949136277660764 \beta_{8} + 821351526238 \beta_{9} + 149224254099688 \beta_{10} - 5425375593986 \beta_{11} + 6246727120224 \beta_{12} - 69906713820154 \beta_{13}) q^{87} +(-\)\(53\!\cdots\!52\)\( - \)\(36\!\cdots\!60\)\( \beta_{1} - \)\(32\!\cdots\!48\)\( \beta_{2} - 3004159386360983312 \beta_{3} - 1463146894953343856 \beta_{4} - 7613034992144576 \beta_{5} + 51023134692101648 \beta_{6} - 3009502055628560 \beta_{7} - 208324260267880 \beta_{8} - 87415847915984 \beta_{9} + 183459057466944 \beta_{10} - 32886522247616 \beta_{11} - 13535374734992 \beta_{12} + 48144721577856 \beta_{13}) q^{88} +(\)\(70\!\cdots\!80\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + \)\(16\!\cdots\!91\)\( \beta_{2} + 42365009738268139431 \beta_{3} + 3533537833744964594 \beta_{4} - 88709842435425113 \beta_{5} - 74157258449989914 \beta_{6} + 329441753865616 \beta_{7} + 859719975642427 \beta_{8} - 213524169121923 \beta_{9} + 210162964020104 \beta_{10} - 12196800546688 \beta_{11} + 26856856563232 \beta_{12} - 12196800546688 \beta_{13}) q^{89} +(\)\(12\!\cdots\!87\)\( - \)\(90\!\cdots\!17\)\( \beta_{1} + \)\(75\!\cdots\!28\)\( \beta_{2} - \)\(10\!\cdots\!61\)\( \beta_{3} + 2539571263601948861 \beta_{4} - 19088030246066763 \beta_{5} + 119736405786088744 \beta_{6} - 10017965984473067 \beta_{7} - 1873070526451635 \beta_{8} - 588406931351270 \beta_{9} + 213760523862272 \beta_{10} - 5727724539392 \beta_{11} - 29644183763072 \beta_{12} + 48132069003725 \beta_{13}) q^{90} +(-\)\(64\!\cdots\!06\)\( - \)\(22\!\cdots\!08\)\( \beta_{1} - \)\(34\!\cdots\!72\)\( \beta_{2} - 49167967887692321740 \beta_{3} - 263066526825829174 \beta_{4} + 149000562170078244 \beta_{5} - 158490477156026568 \beta_{6} + 21884223124086228 \beta_{7} - 1660951308968924 \beta_{8} + 20051378334498 \beta_{9} + 200475286132016 \beta_{10} - 18227424849438 \beta_{11} + 38278803183936 \beta_{12} - 71934610463142 \beta_{13}) q^{91} +(-\)\(13\!\cdots\!46\)\( + \)\(28\!\cdots\!80\)\( \beta_{1} - \)\(40\!\cdots\!60\)\( \beta_{2} - 7044764280412187276 \beta_{3} - 1267599792336534324 \beta_{4} - 140076272322096982 \beta_{5} + 204641514815384934 \beta_{6} + 12731191803984238 \beta_{7} - 620108099554858 \beta_{8} + 100445486043302 \beta_{9} + 144715182124544 \beta_{10} + 31404230190280 \beta_{11} - 41388218016050 \beta_{12} - 138693106577008 \beta_{13}) q^{92} +(-\)\(87\!\cdots\!92\)\( + \)\(53\!\cdots\!96\)\( \beta_{1} + \)\(68\!\cdots\!88\)\( \beta_{2} + 37491673287886782516 \beta_{3} - 6758152144022012728 \beta_{4} - 110192853604819372 \beta_{5} - 280100537375882756 \beta_{6} + 1499472259909192 \beta_{7} - 2315281302119052 \beta_{8} - 41882932851416 \beta_{9} + 62299049812868 \beta_{10} + 42464977123392 \beta_{11} + 20939751812624 \beta_{12} + 42464977123392 \beta_{13}) q^{93} +(\)\(19\!\cdots\!32\)\( + \)\(32\!\cdots\!14\)\( \beta_{1} + \)\(18\!\cdots\!98\)\( \beta_{2} + \)\(18\!\cdots\!12\)\( \beta_{3} - 705841981194411066 \beta_{4} + 37950946994145484 \beta_{5} + 324844088253091498 \beta_{6} - 23344465760603128 \beta_{7} + 4848488431162864 \beta_{8} + 278690584319848 \beta_{9} - 64875658445528 \beta_{10} + 8163966184928 \beta_{11} - 14157919498264 \beta_{12} - 39640135543376 \beta_{13}) q^{94} +(-\)\(20\!\cdots\!72\)\( - \)\(70\!\cdots\!61\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2} + \)\(10\!\cdots\!10\)\( \beta_{3} - 244960549706192280 \beta_{4} + 51662114384474467 \beta_{5} - 402157437335237808 \beta_{6} - 19539539068366013 \beta_{7} + 6007532863374608 \beta_{8} + 38673827932984 \beta_{9} - 229896050107744 \beta_{10} + 47248890057144 \beta_{11} - 8575062124160 \beta_{12} + 406261721469656 \beta_{13}) q^{95} +(-\)\(28\!\cdots\!44\)\( + \)\(11\!\cdots\!28\)\( \beta_{1} - \)\(69\!\cdots\!64\)\( \beta_{2} + 6566795347145581056 \beta_{3} + 4745988928670996096 \beta_{4} + 523809147950024832 \beta_{5} + 401207395853146880 \beta_{6} + 13546982464646272 \beta_{7} + 879752009715200 \beta_{8} + 570003904873088 \beta_{9} - 419769607866880 \beta_{10} + 77431660002816 \beta_{11} + 47827322394752 \beta_{12} + 154516240008192 \beta_{13}) q^{96} +(-\)\(73\!\cdots\!32\)\( + \)\(56\!\cdots\!74\)\( \beta_{1} + \)\(68\!\cdots\!53\)\( \beta_{2} - \)\(13\!\cdots\!91\)\( \beta_{3} + 8616285493530604742 \beta_{4} + 266589706786438061 \beta_{5} - 481432192989759588 \beta_{6} - 1836411799529548 \beta_{7} + 567919304649121 \beta_{8} + 847967897366757 \beta_{9} - 637885285848886 \beta_{10} - 21291684668256 \beta_{11} - 72604018320600 \beta_{12} - 21291684668256 \beta_{13}) q^{97} +(\)\(34\!\cdots\!16\)\( + \)\(32\!\cdots\!81\)\( \beta_{1} - \)\(69\!\cdots\!92\)\( \beta_{2} - \)\(38\!\cdots\!92\)\( \beta_{3} - 8388684744291685432 \beta_{4} - 87924864625922296 \beta_{5} + 477710871218241472 \beta_{6} + 50683295500848648 \beta_{7} - 7521935691211064 \beta_{8} + 1609726061890704 \beta_{9} - 843207865827840 \beta_{10} + 70513405271040 \beta_{11} + 95377116360960 \beta_{12} - 74885612839992 \beta_{13}) q^{98} +(-\)\(32\!\cdots\!37\)\( - \)\(11\!\cdots\!78\)\( \beta_{1} - \)\(73\!\cdots\!51\)\( \beta_{2} + \)\(29\!\cdots\!76\)\( \beta_{3} - 22126703498322744 \beta_{4} - 616035429745141832 \beta_{5} - 503905113401764472 \beta_{6} - 15694563451760344 \beta_{7} + 2275284282300112 \beta_{8} - 86817595478960 \beta_{9} - 1043128630909040 \beta_{10} + 72385795604624 \beta_{11} - 159203391083584 \beta_{12} - 361733216611760 \beta_{13}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 24476q^{2} - 820787056q^{4} + 14864798540q^{5} - 454068012192q^{6} - 31343321296064q^{8} - 722181861644898q^{9} + O(q^{10}) \) \( 14q - 24476q^{2} - 820787056q^{4} + 14864798540q^{5} - 454068012192q^{6} - 31343321296064q^{8} - 722181861644898q^{9} - 1287889809124760q^{10} + 51879382052868480q^{12} + 73473894124935532q^{13} + 36761277666527808q^{14} - 1064191557393108736q^{16} - 4417201733411195812q^{17} + 10773613739927673156q^{18} + 31289641216539815840q^{20} - 24107806000815293184q^{21} - 63855041197225323360q^{22} - \)\(21\!\cdots\!92\)\(q^{24} + \)\(25\!\cdots\!30\)\(q^{25} + \)\(26\!\cdots\!20\)\(q^{26} + \)\(22\!\cdots\!40\)\(q^{28} - \)\(51\!\cdots\!12\)\(q^{29} - \)\(19\!\cdots\!60\)\(q^{30} + \)\(42\!\cdots\!64\)\(q^{32} + \)\(45\!\cdots\!00\)\(q^{33} + \)\(18\!\cdots\!80\)\(q^{34} - \)\(26\!\cdots\!28\)\(q^{36} - \)\(62\!\cdots\!92\)\(q^{37} - \)\(10\!\cdots\!20\)\(q^{38} + \)\(24\!\cdots\!80\)\(q^{40} + \)\(14\!\cdots\!28\)\(q^{41} + \)\(26\!\cdots\!00\)\(q^{42} + \)\(72\!\cdots\!00\)\(q^{44} - \)\(15\!\cdots\!00\)\(q^{45} + \)\(69\!\cdots\!08\)\(q^{46} + \)\(22\!\cdots\!20\)\(q^{48} + \)\(36\!\cdots\!02\)\(q^{49} - \)\(62\!\cdots\!40\)\(q^{50} - \)\(35\!\cdots\!32\)\(q^{52} - \)\(16\!\cdots\!68\)\(q^{53} + \)\(10\!\cdots\!16\)\(q^{54} - \)\(10\!\cdots\!92\)\(q^{56} - \)\(17\!\cdots\!20\)\(q^{57} + \)\(29\!\cdots\!52\)\(q^{58} + \)\(67\!\cdots\!60\)\(q^{60} - \)\(78\!\cdots\!72\)\(q^{61} - \)\(70\!\cdots\!00\)\(q^{62} - \)\(18\!\cdots\!56\)\(q^{64} - \)\(76\!\cdots\!00\)\(q^{65} + \)\(43\!\cdots\!00\)\(q^{66} - \)\(15\!\cdots\!28\)\(q^{68} + \)\(12\!\cdots\!16\)\(q^{69} + \)\(57\!\cdots\!00\)\(q^{70} - \)\(12\!\cdots\!56\)\(q^{72} - \)\(13\!\cdots\!08\)\(q^{73} - \)\(16\!\cdots\!20\)\(q^{74} - \)\(43\!\cdots\!00\)\(q^{76} - \)\(13\!\cdots\!20\)\(q^{77} + \)\(10\!\cdots\!00\)\(q^{78} - \)\(14\!\cdots\!80\)\(q^{80} + \)\(14\!\cdots\!30\)\(q^{81} + \)\(10\!\cdots\!88\)\(q^{82} - \)\(27\!\cdots\!84\)\(q^{84} - \)\(16\!\cdots\!80\)\(q^{85} + \)\(40\!\cdots\!08\)\(q^{86} - \)\(74\!\cdots\!40\)\(q^{88} + \)\(98\!\cdots\!88\)\(q^{89} + \)\(18\!\cdots\!40\)\(q^{90} - \)\(18\!\cdots\!00\)\(q^{92} - \)\(12\!\cdots\!20\)\(q^{93} + \)\(27\!\cdots\!08\)\(q^{94} - \)\(40\!\cdots\!92\)\(q^{96} - \)\(10\!\cdots\!32\)\(q^{97} + \)\(47\!\cdots\!36\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 7 x^{13} + 7040347091761 x^{12} - 42242082550475 x^{11} + \)\(18\!\cdots\!95\)\( x^{10} - \)\(90\!\cdots\!21\)\( x^{9} + \)\(21\!\cdots\!47\)\( x^{8} - \)\(84\!\cdots\!01\)\( x^{7} + \)\(12\!\cdots\!60\)\( x^{6} - \)\(36\!\cdots\!60\)\( x^{5} + \)\(30\!\cdots\!04\)\( x^{4} - \)\(61\!\cdots\!28\)\( x^{3} + \)\(24\!\cdots\!04\)\( x^{2} - \)\(24\!\cdots\!80\)\( x + \)\(48\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(28\!\cdots\!75\)\( \nu^{13} - \)\(56\!\cdots\!76\)\( \nu^{12} + \)\(17\!\cdots\!31\)\( \nu^{11} - \)\(35\!\cdots\!56\)\( \nu^{10} + \)\(36\!\cdots\!25\)\( \nu^{9} - \)\(73\!\cdots\!96\)\( \nu^{8} + \)\(29\!\cdots\!25\)\( \nu^{7} - \)\(58\!\cdots\!08\)\( \nu^{6} + \)\(90\!\cdots\!28\)\( \nu^{5} - \)\(17\!\cdots\!40\)\( \nu^{4} + \)\(51\!\cdots\!08\)\( \nu^{3} - \)\(96\!\cdots\!36\)\( \nu^{2} - \)\(20\!\cdots\!80\)\( \nu - \)\(75\!\cdots\!00\)\(\)\()/ \)\(57\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(28\!\cdots\!75\)\( \nu^{13} - \)\(56\!\cdots\!76\)\( \nu^{12} + \)\(17\!\cdots\!31\)\( \nu^{11} - \)\(35\!\cdots\!56\)\( \nu^{10} + \)\(36\!\cdots\!25\)\( \nu^{9} - \)\(73\!\cdots\!96\)\( \nu^{8} + \)\(29\!\cdots\!25\)\( \nu^{7} - \)\(58\!\cdots\!08\)\( \nu^{6} + \)\(90\!\cdots\!28\)\( \nu^{5} - \)\(17\!\cdots\!40\)\( \nu^{4} + \)\(51\!\cdots\!08\)\( \nu^{3} - \)\(96\!\cdots\!36\)\( \nu^{2} + \)\(28\!\cdots\!60\)\( \nu - \)\(75\!\cdots\!60\)\(\)\()/ \)\(19\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(15\!\cdots\!11\)\( \nu^{13} - \)\(87\!\cdots\!84\)\( \nu^{12} + \)\(98\!\cdots\!59\)\( \nu^{11} - \)\(54\!\cdots\!88\)\( \nu^{10} + \)\(20\!\cdots\!01\)\( \nu^{9} - \)\(11\!\cdots\!68\)\( \nu^{8} + \)\(17\!\cdots\!53\)\( \nu^{7} - \)\(94\!\cdots\!32\)\( \nu^{6} + \)\(60\!\cdots\!84\)\( \nu^{5} - \)\(29\!\cdots\!68\)\( \nu^{4} + \)\(59\!\cdots\!60\)\( \nu^{3} - \)\(18\!\cdots\!88\)\( \nu^{2} + \)\(21\!\cdots\!60\)\( \nu - \)\(15\!\cdots\!00\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(11\!\cdots\!97\)\( \nu^{13} + \)\(89\!\cdots\!24\)\( \nu^{12} + \)\(69\!\cdots\!41\)\( \nu^{11} + \)\(56\!\cdots\!56\)\( \nu^{10} + \)\(14\!\cdots\!27\)\( \nu^{9} + \)\(11\!\cdots\!56\)\( \nu^{8} + \)\(12\!\cdots\!91\)\( \nu^{7} + \)\(10\!\cdots\!72\)\( \nu^{6} + \)\(37\!\cdots\!52\)\( \nu^{5} + \)\(34\!\cdots\!44\)\( \nu^{4} + \)\(26\!\cdots\!04\)\( \nu^{3} + \)\(31\!\cdots\!56\)\( \nu^{2} + \)\(37\!\cdots\!80\)\( \nu + \)\(77\!\cdots\!00\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(49\!\cdots\!51\)\( \nu^{13} - \)\(19\!\cdots\!16\)\( \nu^{12} - \)\(30\!\cdots\!79\)\( \nu^{11} - \)\(12\!\cdots\!12\)\( \nu^{10} - \)\(63\!\cdots\!81\)\( \nu^{9} - \)\(25\!\cdots\!32\)\( \nu^{8} - \)\(51\!\cdots\!93\)\( \nu^{7} - \)\(21\!\cdots\!08\)\( \nu^{6} - \)\(15\!\cdots\!84\)\( \nu^{5} - \)\(69\!\cdots\!92\)\( \nu^{4} - \)\(92\!\cdots\!00\)\( \nu^{3} - \)\(58\!\cdots\!92\)\( \nu^{2} - \)\(10\!\cdots\!60\)\( \nu - \)\(12\!\cdots\!00\)\(\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(38\!\cdots\!13\)\( \nu^{13} + \)\(51\!\cdots\!88\)\( \nu^{12} + \)\(23\!\cdots\!37\)\( \nu^{11} + \)\(31\!\cdots\!36\)\( \nu^{10} + \)\(49\!\cdots\!83\)\( \nu^{9} + \)\(64\!\cdots\!76\)\( \nu^{8} + \)\(40\!\cdots\!59\)\( \nu^{7} + \)\(50\!\cdots\!24\)\( \nu^{6} + \)\(12\!\cdots\!52\)\( \nu^{5} + \)\(13\!\cdots\!56\)\( \nu^{4} + \)\(73\!\cdots\!20\)\( \nu^{3} + \)\(41\!\cdots\!76\)\( \nu^{2} - \)\(28\!\cdots\!20\)\( \nu - \)\(97\!\cdots\!00\)\(\)\()/ \)\(47\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(24\!\cdots\!37\)\( \nu^{13} + \)\(40\!\cdots\!00\)\( \nu^{12} + \)\(15\!\cdots\!25\)\( \nu^{11} + \)\(25\!\cdots\!92\)\( \nu^{10} + \)\(34\!\cdots\!47\)\( \nu^{9} + \)\(53\!\cdots\!52\)\( \nu^{8} + \)\(31\!\cdots\!91\)\( \nu^{7} + \)\(44\!\cdots\!60\)\( \nu^{6} + \)\(12\!\cdots\!84\)\( \nu^{5} + \)\(14\!\cdots\!24\)\( \nu^{4} + \)\(17\!\cdots\!36\)\( \nu^{3} + \)\(12\!\cdots\!92\)\( \nu^{2} + \)\(56\!\cdots\!60\)\( \nu + \)\(24\!\cdots\!00\)\(\)\()/ \)\(57\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(90\!\cdots\!79\)\( \nu^{13} + \)\(10\!\cdots\!56\)\( \nu^{12} - \)\(57\!\cdots\!11\)\( \nu^{11} + \)\(66\!\cdots\!12\)\( \nu^{10} - \)\(12\!\cdots\!49\)\( \nu^{9} + \)\(14\!\cdots\!72\)\( \nu^{8} - \)\(10\!\cdots\!17\)\( \nu^{7} + \)\(11\!\cdots\!28\)\( \nu^{6} - \)\(34\!\cdots\!16\)\( \nu^{5} + \)\(38\!\cdots\!32\)\( \nu^{4} - \)\(28\!\cdots\!00\)\( \nu^{3} + \)\(32\!\cdots\!12\)\( \nu^{2} - \)\(28\!\cdots\!40\)\( \nu + \)\(57\!\cdots\!00\)\(\)\()/ \)\(32\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(32\!\cdots\!37\)\( \nu^{13} + \)\(38\!\cdots\!16\)\( \nu^{12} - \)\(20\!\cdots\!01\)\( \nu^{11} + \)\(23\!\cdots\!04\)\( \nu^{10} - \)\(42\!\cdots\!07\)\( \nu^{9} + \)\(50\!\cdots\!24\)\( \nu^{8} - \)\(35\!\cdots\!91\)\( \nu^{7} + \)\(42\!\cdots\!08\)\( \nu^{6} - \)\(11\!\cdots\!32\)\( \nu^{5} + \)\(13\!\cdots\!96\)\( \nu^{4} - \)\(96\!\cdots\!84\)\( \nu^{3} + \)\(11\!\cdots\!84\)\( \nu^{2} - \)\(16\!\cdots\!80\)\( \nu + \)\(37\!\cdots\!00\)\(\)\()/ \)\(57\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(20\!\cdots\!27\)\( \nu^{13} - \)\(44\!\cdots\!28\)\( \nu^{12} - \)\(13\!\cdots\!07\)\( \nu^{11} - \)\(27\!\cdots\!80\)\( \nu^{10} - \)\(27\!\cdots\!37\)\( \nu^{9} - \)\(57\!\cdots\!40\)\( \nu^{8} - \)\(23\!\cdots\!21\)\( \nu^{7} - \)\(46\!\cdots\!84\)\( \nu^{6} - \)\(78\!\cdots\!40\)\( \nu^{5} - \)\(13\!\cdots\!24\)\( \nu^{4} - \)\(71\!\cdots\!52\)\( \nu^{3} - \)\(36\!\cdots\!60\)\( \nu^{2} - \)\(21\!\cdots\!00\)\( \nu + \)\(16\!\cdots\!00\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(20\!\cdots\!65\)\( \nu^{13} - \)\(30\!\cdots\!16\)\( \nu^{12} - \)\(12\!\cdots\!69\)\( \nu^{11} - \)\(19\!\cdots\!76\)\( \nu^{10} - \)\(26\!\cdots\!95\)\( \nu^{9} - \)\(40\!\cdots\!36\)\( \nu^{8} - \)\(21\!\cdots\!75\)\( \nu^{7} - \)\(34\!\cdots\!08\)\( \nu^{6} - \)\(59\!\cdots\!12\)\( \nu^{5} - \)\(11\!\cdots\!80\)\( \nu^{4} - \)\(73\!\cdots\!12\)\( \nu^{3} - \)\(98\!\cdots\!76\)\( \nu^{2} + \)\(44\!\cdots\!20\)\( \nu - \)\(16\!\cdots\!00\)\(\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(11\!\cdots\!67\)\( \nu^{13} + \)\(25\!\cdots\!24\)\( \nu^{12} - \)\(69\!\cdots\!79\)\( \nu^{11} + \)\(16\!\cdots\!72\)\( \nu^{10} - \)\(14\!\cdots\!97\)\( \nu^{9} + \)\(34\!\cdots\!52\)\( \nu^{8} - \)\(11\!\cdots\!81\)\( \nu^{7} + \)\(28\!\cdots\!12\)\( \nu^{6} - \)\(36\!\cdots\!16\)\( \nu^{5} + \)\(92\!\cdots\!36\)\( \nu^{4} - \)\(24\!\cdots\!08\)\( \nu^{3} + \)\(77\!\cdots\!92\)\( \nu^{2} - \)\(25\!\cdots\!40\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(96\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(43\!\cdots\!29\)\( \nu^{13} - \)\(53\!\cdots\!33\)\( \nu^{12} - \)\(27\!\cdots\!27\)\( \nu^{11} - \)\(33\!\cdots\!77\)\( \nu^{10} - \)\(59\!\cdots\!49\)\( \nu^{9} - \)\(71\!\cdots\!07\)\( \nu^{8} - \)\(51\!\cdots\!77\)\( \nu^{7} - \)\(60\!\cdots\!59\)\( \nu^{6} - \)\(18\!\cdots\!34\)\( \nu^{5} - \)\(19\!\cdots\!28\)\( \nu^{4} - \)\(21\!\cdots\!08\)\( \nu^{3} - \)\(17\!\cdots\!12\)\( \nu^{2} - \)\(81\!\cdots\!60\)\( \nu - \)\(30\!\cdots\!00\)\(\)\()/ \)\(36\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 30 \beta_{1} - 1\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9} - \beta_{8} - 26 \beta_{6} - 37 \beta_{5} - 854 \beta_{4} + 19227 \beta_{3} + 84319 \beta_{2} + 634535194 \beta_{1} - 257475369523870\)\()/256\)
\(\nu^{3}\)\(=\)\((\)\(1481411 \beta_{13} + 514624 \beta_{12} + 173503 \beta_{11} - 7009552 \beta_{10} + 688151 \beta_{9} + 30883214 \beta_{8} + 28660282 \beta_{7} + 81675800 \beta_{6} + 2502161186 \beta_{5} + 3611988855 \beta_{4} + 2025307042174 \beta_{3} - 491670146384560 \beta_{2} + 19252431356128832 \beta_{1} - 468002778268567\)\()/4096\)
\(\nu^{4}\)\(=\)\((\)\(1832211023272 \beta_{13} - 1785748076572 \beta_{12} + 1832200560008 \beta_{11} - 6450780653063 \beta_{10} - 177033784140878 \beta_{9} + 68334895323973 \beta_{8} - 16527169286638 \beta_{7} + 10405745687782663 \beta_{6} + 12325116026292829 \beta_{5} + 391871352412690494 \beta_{4} - 5544518575814097427 \beta_{3} - 32166845030246611065 \beta_{2} - 245151629067851769047294 \beta_{1} + 31712401328843084445352459762\)\()/16384\)
\(\nu^{5}\)\(=\)\((\)\(-175891237964564420195 \beta_{13} - 70120483351473374080 \beta_{12} - 23631311331552959455 \beta_{11} + 681867695654635504360 \beta_{10} - 93755336287758821095 \beta_{9} - 3752482540876937553270 \beta_{8} - 9020073593656627258528 \beta_{7} - 45595621737143233529626 \beta_{6} - 345949731472766393007168 \beta_{5} - 395378799364644183202721 \beta_{4} - 200147785510284623813988826 \beta_{3} + 37213965650671490527090865750 \beta_{2} - 2525173451368884095742245608660 \beta_{1} - 103177536317049301415113121293\)\()/131072\)
\(\nu^{6}\)\(=\)\((\)\(-314296427615509451285185912 \beta_{13} + 565138824498393461067919540 \beta_{12} - 314292773377270257336005912 \beta_{11} + 1764814034298070729536816365 \beta_{10} + 30272179110125627985006376028 \beta_{9} - 10731570290097588513872472409 \beta_{8} + 11094542828200810381019559658 \beta_{7} - 2457822729076517736583995489265 \beta_{6} - 3132023655053440959601912538809 \beta_{5} - 89632383038023146852659511825450 \beta_{4} + 1156450346713150387350231899232847 \beta_{3} + 7581394729376887967694363753476621 \beta_{2} + 58031753069024097548595634900772982290 \beta_{1} - 4819141842045086816163539279861815195017830\)\()/1048576\)
\(\nu^{7}\)\(=\)\((\)\(\)\(16\!\cdots\!92\)\( \beta_{13} + \)\(76\!\cdots\!80\)\( \beta_{12} + \)\(28\!\cdots\!40\)\( \beta_{11} - \)\(58\!\cdots\!20\)\( \beta_{10} + \)\(10\!\cdots\!12\)\( \beta_{9} + \)\(39\!\cdots\!16\)\( \beta_{8} + \)\(11\!\cdots\!00\)\( \beta_{7} + \)\(68\!\cdots\!95\)\( \beta_{6} + \)\(38\!\cdots\!64\)\( \beta_{5} + \)\(37\!\cdots\!87\)\( \beta_{4} + \)\(17\!\cdots\!60\)\( \beta_{3} - \)\(30\!\cdots\!15\)\( \beta_{2} + \)\(29\!\cdots\!45\)\( \beta_{1} + \)\(17\!\cdots\!44\)\(\)\()/4194304\)
\(\nu^{8}\)\(=\)\((\)\(\)\(66\!\cdots\!88\)\( \beta_{13} - \)\(20\!\cdots\!12\)\( \beta_{12} + \)\(66\!\cdots\!12\)\( \beta_{11} - \)\(57\!\cdots\!14\)\( \beta_{10} - \)\(81\!\cdots\!24\)\( \beta_{9} + \)\(37\!\cdots\!30\)\( \beta_{8} - \)\(52\!\cdots\!60\)\( \beta_{7} + \)\(80\!\cdots\!63\)\( \beta_{6} + \)\(10\!\cdots\!70\)\( \beta_{5} + \)\(28\!\cdots\!85\)\( \beta_{4} - \)\(34\!\cdots\!78\)\( \beta_{3} - \)\(24\!\cdots\!23\)\( \beta_{2} - \)\(19\!\cdots\!57\)\( \beta_{1} + \)\(12\!\cdots\!48\)\(\)\()/1048576\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(23\!\cdots\!80\)\( \beta_{13} - \)\(12\!\cdots\!72\)\( \beta_{12} - \)\(48\!\cdots\!80\)\( \beta_{11} + \)\(77\!\cdots\!80\)\( \beta_{10} - \)\(16\!\cdots\!52\)\( \beta_{9} - \)\(60\!\cdots\!80\)\( \beta_{8} - \)\(18\!\cdots\!88\)\( \beta_{7} - \)\(12\!\cdots\!39\)\( \beta_{6} - \)\(60\!\cdots\!32\)\( \beta_{5} - \)\(53\!\cdots\!03\)\( \beta_{4} - \)\(23\!\cdots\!48\)\( \beta_{3} + \)\(41\!\cdots\!99\)\( \beta_{2} - \)\(48\!\cdots\!65\)\( \beta_{1} - \)\(27\!\cdots\!20\)\(\)\()/2097152\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(13\!\cdots\!36\)\( \beta_{13} + \)\(68\!\cdots\!68\)\( \beta_{12} - \)\(13\!\cdots\!36\)\( \beta_{11} + \)\(17\!\cdots\!71\)\( \beta_{10} + \)\(22\!\cdots\!84\)\( \beta_{9} - \)\(12\!\cdots\!35\)\( \beta_{8} + \)\(19\!\cdots\!10\)\( \beta_{7} - \)\(25\!\cdots\!71\)\( \beta_{6} - \)\(34\!\cdots\!35\)\( \beta_{5} - \)\(85\!\cdots\!54\)\( \beta_{4} + \)\(99\!\cdots\!33\)\( \beta_{3} + \)\(76\!\cdots\!95\)\( \beta_{2} + \)\(59\!\cdots\!10\)\( \beta_{1} - \)\(33\!\cdots\!82\)\(\)\()/1048576\)
\(\nu^{11}\)\(=\)\((\)\(\)\(12\!\cdots\!68\)\( \beta_{13} + \)\(74\!\cdots\!48\)\( \beta_{12} + \)\(30\!\cdots\!20\)\( \beta_{11} - \)\(41\!\cdots\!84\)\( \beta_{10} + \)\(10\!\cdots\!12\)\( \beta_{9} + \)\(36\!\cdots\!56\)\( \beta_{8} + \)\(11\!\cdots\!08\)\( \beta_{7} + \)\(82\!\cdots\!45\)\( \beta_{6} + \)\(37\!\cdots\!68\)\( \beta_{5} + \)\(30\!\cdots\!21\)\( \beta_{4} + \)\(12\!\cdots\!04\)\( \beta_{3} - \)\(22\!\cdots\!65\)\( \beta_{2} + \)\(30\!\cdots\!95\)\( \beta_{1} + \)\(13\!\cdots\!80\)\(\)\()/4194304\)
\(\nu^{12}\)\(=\)\((\)\(\)\(24\!\cdots\!44\)\( \beta_{13} - \)\(21\!\cdots\!52\)\( \beta_{12} + \)\(24\!\cdots\!84\)\( \beta_{11} - \)\(51\!\cdots\!04\)\( \beta_{10} - \)\(62\!\cdots\!64\)\( \beta_{9} + \)\(40\!\cdots\!04\)\( \beta_{8} - \)\(64\!\cdots\!08\)\( \beta_{7} + \)\(75\!\cdots\!05\)\( \beta_{6} + \)\(10\!\cdots\!92\)\( \beta_{5} + \)\(25\!\cdots\!53\)\( \beta_{4} - \)\(28\!\cdots\!44\)\( \beta_{3} - \)\(23\!\cdots\!89\)\( \beta_{2} - \)\(17\!\cdots\!81\)\( \beta_{1} + \)\(92\!\cdots\!76\)\(\)\()/1048576\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(11\!\cdots\!35\)\( \beta_{13} - \)\(69\!\cdots\!52\)\( \beta_{12} - \)\(29\!\cdots\!63\)\( \beta_{11} + \)\(35\!\cdots\!06\)\( \beta_{10} - \)\(98\!\cdots\!91\)\( \beta_{9} - \)\(33\!\cdots\!84\)\( \beta_{8} - \)\(10\!\cdots\!40\)\( \beta_{7} - \)\(80\!\cdots\!04\)\( \beta_{6} - \)\(34\!\cdots\!94\)\( \beta_{5} - \)\(27\!\cdots\!21\)\( \beta_{4} - \)\(10\!\cdots\!96\)\( \beta_{3} + \)\(19\!\cdots\!44\)\( \beta_{2} - \)\(28\!\cdots\!72\)\( \beta_{1} - \)\(73\!\cdots\!57\)\(\)\()/131072\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 287382.i
0.500000 + 287382.i
0.500000 + 1.69534e6i
0.500000 1.69534e6i
0.500000 872651.i
0.500000 + 872651.i
0.500000 + 868866.i
0.500000 868866.i
0.500000 + 174763.i
0.500000 174763.i
0.500000 1.39431e6i
0.500000 + 1.39431e6i
0.500000 + 769735.i
0.500000 769735.i
−31548.7 8855.74i 4.59811e6i 9.16894e8 + 5.58773e8i −1.46052e10 4.07197e10 1.45064e11i 9.18461e12i −2.39784e13 2.57483e13i 1.84749e14 4.60774e14 + 1.29340e14i
3.2 −31548.7 + 8855.74i 4.59811e6i 9.16894e8 5.58773e8i −1.46052e10 4.07197e10 + 1.45064e11i 9.18461e12i −2.39784e13 + 2.57483e13i 1.84749e14 4.60774e14 1.29340e14i
3.3 −26662.2 19049.1i 2.71254e7i 3.48003e8 + 1.01578e9i 3.98373e10 −5.16715e11 + 7.23222e11i 1.87458e12i 1.00713e13 3.37122e13i −5.29895e14 −1.06215e15 7.58866e14i
3.4 −26662.2 + 19049.1i 2.71254e7i 3.48003e8 1.01578e9i 3.98373e10 −5.16715e11 7.23222e11i 1.87458e12i 1.00713e13 + 3.37122e13i −5.29895e14 −1.06215e15 + 7.58866e14i
3.5 −16991.5 28018.4i 1.39624e7i −4.96318e8 + 9.52150e8i 8.38288e9 3.91204e11 2.37243e11i 4.67445e12i 3.51109e13 2.27243e12i 1.09422e13 −1.42438e14 2.34875e14i
3.6 −16991.5 + 28018.4i 1.39624e7i −4.96318e8 9.52150e8i 8.38288e9 3.91204e11 + 2.37243e11i 4.67445e12i 3.51109e13 + 2.27243e12i 1.09422e13 −1.42438e14 + 2.34875e14i
3.7 −713.932 32760.2i 1.39019e7i −1.07272e9 + 4.67771e7i −5.42656e10 −4.55428e11 + 9.92498e9i 5.92817e11i 2.29828e12 + 3.51092e13i 1.26296e13 3.87420e13 + 1.77775e15i
3.8 −713.932 + 32760.2i 1.39019e7i −1.07272e9 4.67771e7i −5.42656e10 −4.55428e11 9.92498e9i 5.92817e11i 2.29828e12 3.51092e13i 1.26296e13 3.87420e13 1.77775e15i
3.9 10610.7 31002.5i 2.79620e6i −8.48568e8 6.57916e8i 4.95390e10 −8.66893e10 2.96697e10i 4.58323e12i −2.94010e13 + 1.93268e13i 1.98072e14 5.25643e14 1.53583e15i
3.10 10610.7 + 31002.5i 2.79620e6i −8.48568e8 + 6.57916e8i 4.95390e10 −8.66893e10 + 2.96697e10i 4.58323e12i −2.94010e13 1.93268e13i 1.98072e14 5.25643e14 + 1.53583e15i
3.11 22243.3 24061.9i 2.23090e7i −8.42119e7 1.07043e9i −2.29401e10 5.36798e11 + 4.96227e11i 1.19922e12i −2.76299e13 2.17837e13i −2.91801e14 −5.10263e14 + 5.51982e14i
3.12 22243.3 + 24061.9i 2.23090e7i −8.42119e7 + 1.07043e9i −2.29401e10 5.36798e11 4.96227e11i 1.19922e12i −2.76299e13 + 2.17837e13i −2.91801e14 −5.10263e14 5.51982e14i
3.13 30824.3 11117.8i 1.23158e7i 8.26531e8 6.85396e8i 1.48412e9 −1.36924e11 3.79624e11i 2.60240e12i 1.78571e13 3.03161e13i 5.42133e13 4.57469e13 1.65001e13i
3.14 30824.3 + 11117.8i 1.23158e7i 8.26531e8 + 6.85396e8i 1.48412e9 −1.36924e11 + 3.79624e11i 2.60240e12i 1.78571e13 + 3.03161e13i 5.42133e13 4.57469e13 + 1.65001e13i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.31.b.a 14
3.b odd 2 1 36.31.d.c 14
4.b odd 2 1 inner 4.31.b.a 14
12.b even 2 1 36.31.d.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.31.b.a 14 1.a even 1 1 trivial
4.31.b.a 14 4.b odd 2 1 inner
36.31.d.c 14 3.b odd 2 1
36.31.d.c 14 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{31}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(16\!\cdots\!24\)\( + \)\(37\!\cdots\!76\)\( T + \)\(10\!\cdots\!84\)\( T^{2} + \)\(30\!\cdots\!12\)\( T^{3} + \)\(92\!\cdots\!48\)\( T^{4} + \)\(97\!\cdots\!04\)\( T^{5} + \)\(56\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!16\)\( T^{7} + \)\(52\!\cdots\!92\)\( T^{8} + \)\(84\!\cdots\!04\)\( T^{9} + 743860952814845952 T^{10} + 22936390643712 T^{11} + 709930816 T^{12} + 24476 T^{13} + T^{14} \)
$3$ \( \)\(34\!\cdots\!00\)\( + \)\(67\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{4} + \)\(51\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{8} + \)\(11\!\cdots\!16\)\( T^{10} + 1802328855484992 T^{12} + T^{14} \)
$5$ \( ( \)\(44\!\cdots\!00\)\( - \)\(31\!\cdots\!00\)\( T + \)\(55\!\cdots\!00\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!00\)\( T^{4} - \)\(38\!\cdots\!20\)\( T^{5} - 7432399270 T^{6} + T^{7} )^{2} \)
$7$ \( \)\(46\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{4} + \)\(64\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{8} + \)\(56\!\cdots\!16\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} + T^{14} \)
$11$ \( \)\(42\!\cdots\!00\)\( + \)\(74\!\cdots\!00\)\( T^{2} + \)\(43\!\cdots\!00\)\( T^{4} + \)\(97\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{8} + \)\(53\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{12} + T^{14} \)
$13$ \( ( -\)\(24\!\cdots\!00\)\( - \)\(18\!\cdots\!40\)\( T + \)\(32\!\cdots\!64\)\( T^{2} + \)\(22\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!32\)\( T^{4} - \)\(85\!\cdots\!12\)\( T^{5} - 36736947062467766 T^{6} + T^{7} )^{2} \)
$17$ \( ( \)\(10\!\cdots\!00\)\( + \)\(39\!\cdots\!80\)\( T + \)\(46\!\cdots\!56\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{3} - \)\(66\!\cdots\!92\)\( T^{4} - \)\(26\!\cdots\!72\)\( T^{5} + 2208600866705597906 T^{6} + T^{7} )^{2} \)
$19$ \( \)\(33\!\cdots\!00\)\( + \)\(35\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(73\!\cdots\!00\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{10} + \)\(21\!\cdots\!00\)\( T^{12} + T^{14} \)
$23$ \( \)\(19\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{6} + \)\(50\!\cdots\!00\)\( T^{8} + \)\(72\!\cdots\!16\)\( T^{10} + \)\(45\!\cdots\!92\)\( T^{12} + T^{14} \)
$29$ \( ( -\)\(41\!\cdots\!72\)\( - \)\(59\!\cdots\!08\)\( T + \)\(61\!\cdots\!32\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} - \)\(87\!\cdots\!40\)\( T^{4} - \)\(24\!\cdots\!96\)\( T^{5} + \)\(25\!\cdots\!06\)\( T^{6} + T^{7} )^{2} \)
$31$ \( \)\(12\!\cdots\!00\)\( + \)\(33\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{8} + \)\(43\!\cdots\!00\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{12} + T^{14} \)
$37$ \( ( -\)\(86\!\cdots\!00\)\( + \)\(26\!\cdots\!40\)\( T + \)\(26\!\cdots\!16\)\( T^{2} + \)\(57\!\cdots\!16\)\( T^{3} - \)\(91\!\cdots\!12\)\( T^{4} - \)\(29\!\cdots\!32\)\( T^{5} + \)\(31\!\cdots\!46\)\( T^{6} + T^{7} )^{2} \)
$41$ \( ( -\)\(15\!\cdots\!28\)\( - \)\(39\!\cdots\!52\)\( T + \)\(69\!\cdots\!28\)\( T^{2} + \)\(71\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(44\!\cdots\!16\)\( T^{5} - \)\(74\!\cdots\!14\)\( T^{6} + T^{7} )^{2} \)
$43$ \( \)\(15\!\cdots\!00\)\( + \)\(72\!\cdots\!00\)\( T^{2} + \)\(73\!\cdots\!00\)\( T^{4} + \)\(32\!\cdots\!00\)\( T^{6} + \)\(74\!\cdots\!00\)\( T^{8} + \)\(86\!\cdots\!16\)\( T^{10} + \)\(48\!\cdots\!92\)\( T^{12} + T^{14} \)
$47$ \( \)\(27\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{4} + \)\(49\!\cdots\!00\)\( T^{6} + \)\(65\!\cdots\!00\)\( T^{8} + \)\(38\!\cdots\!16\)\( T^{10} + \)\(10\!\cdots\!92\)\( T^{12} + T^{14} \)
$53$ \( ( -\)\(41\!\cdots\!00\)\( - \)\(10\!\cdots\!20\)\( T + \)\(53\!\cdots\!44\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} - \)\(15\!\cdots\!28\)\( T^{4} - \)\(21\!\cdots\!12\)\( T^{5} + \)\(81\!\cdots\!34\)\( T^{6} + T^{7} )^{2} \)
$59$ \( \)\(87\!\cdots\!00\)\( + \)\(47\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{8} + \)\(68\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} + T^{14} \)
$61$ \( ( -\)\(18\!\cdots\!28\)\( - \)\(48\!\cdots\!52\)\( T + \)\(55\!\cdots\!28\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} - \)\(29\!\cdots\!80\)\( T^{4} - \)\(73\!\cdots\!16\)\( T^{5} + \)\(39\!\cdots\!86\)\( T^{6} + T^{7} )^{2} \)
$67$ \( \)\(17\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{4} + \)\(20\!\cdots\!00\)\( T^{6} + \)\(55\!\cdots\!00\)\( T^{8} + \)\(71\!\cdots\!16\)\( T^{10} + \)\(43\!\cdots\!92\)\( T^{12} + T^{14} \)
$71$ \( \)\(76\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T^{2} + \)\(56\!\cdots\!00\)\( T^{4} + \)\(73\!\cdots\!00\)\( T^{6} + \)\(43\!\cdots\!00\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!00\)\( T^{12} + T^{14} \)
$73$ \( ( -\)\(32\!\cdots\!00\)\( - \)\(90\!\cdots\!60\)\( T + \)\(60\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!36\)\( T^{3} - \)\(20\!\cdots\!08\)\( T^{4} - \)\(27\!\cdots\!32\)\( T^{5} + \)\(69\!\cdots\!54\)\( T^{6} + T^{7} )^{2} \)
$79$ \( \)\(32\!\cdots\!00\)\( + \)\(63\!\cdots\!00\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{4} + \)\(85\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!00\)\( T^{8} + \)\(62\!\cdots\!00\)\( T^{10} + \)\(42\!\cdots\!00\)\( T^{12} + T^{14} \)
$83$ \( \)\(14\!\cdots\!00\)\( + \)\(82\!\cdots\!00\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(70\!\cdots\!00\)\( T^{8} + \)\(17\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!92\)\( T^{12} + T^{14} \)
$89$ \( ( \)\(26\!\cdots\!28\)\( + \)\(88\!\cdots\!92\)\( T - \)\(36\!\cdots\!68\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(10\!\cdots\!96\)\( T^{5} - \)\(49\!\cdots\!94\)\( T^{6} + T^{7} )^{2} \)
$97$ \( ( \)\(22\!\cdots\!00\)\( + \)\(33\!\cdots\!20\)\( T + \)\(13\!\cdots\!96\)\( T^{2} + \)\(30\!\cdots\!36\)\( T^{3} - \)\(71\!\cdots\!92\)\( T^{4} - \)\(93\!\cdots\!12\)\( T^{5} + \)\(51\!\cdots\!66\)\( T^{6} + T^{7} )^{2} \)
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