Properties

Label 261.12.a.a
Level $261$
Weight $12$
Character orbit 261.a
Self dual yes
Analytic conductor $200.538$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 261.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(200.537570126\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{1} ) q^{2} + ( 832 - 6 \beta_{1} + \beta_{2} ) q^{4} + ( 247 + 22 \beta_{1} + \beta_{2} - \beta_{6} ) q^{5} + ( -4504 + 121 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{9} - 2 \beta_{10} ) q^{7} + ( 13689 - 677 \beta_{1} + 8 \beta_{2} - 27 \beta_{3} + 3 \beta_{5} + 9 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{8} +O(q^{10})\) \( q + ( 3 - \beta_{1} ) q^{2} + ( 832 - 6 \beta_{1} + \beta_{2} ) q^{4} + ( 247 + 22 \beta_{1} + \beta_{2} - \beta_{6} ) q^{5} + ( -4504 + 121 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{9} - 2 \beta_{10} ) q^{7} + ( 13689 - 677 \beta_{1} + 8 \beta_{2} - 27 \beta_{3} + 3 \beta_{5} + 9 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{8} + ( -62149 - 1815 \beta_{1} - 56 \beta_{2} + 75 \beta_{3} + 7 \beta_{4} + 30 \beta_{5} + 5 \beta_{6} + \beta_{7} + 7 \beta_{8} + 4 \beta_{9} + 6 \beta_{10} ) q^{10} + ( 55432 + 1433 \beta_{1} - 42 \beta_{2} - 196 \beta_{3} - 44 \beta_{4} + 10 \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{11} + ( 138373 - 10163 \beta_{1} + 54 \beta_{2} + 271 \beta_{3} - 38 \beta_{4} + 40 \beta_{5} + 28 \beta_{6} - 25 \beta_{7} - 23 \beta_{8} - 48 \beta_{9} + 59 \beta_{10} ) q^{13} + ( -360950 + 15132 \beta_{1} - 204 \beta_{2} - 87 \beta_{3} + 79 \beta_{4} + 60 \beta_{5} - 97 \beta_{6} - 69 \beta_{7} - 23 \beta_{8} + 98 \beta_{9} + 58 \beta_{10} ) q^{14} + ( 277694 - 29552 \beta_{1} + 793 \beta_{2} + 47 \beta_{3} - 41 \beta_{4} - 202 \beta_{5} + 19 \beta_{6} + 273 \beta_{7} + 17 \beta_{8} + 348 \beta_{9} + 34 \beta_{10} ) q^{16} + ( 296264 + 35825 \beta_{1} - 362 \beta_{2} + 1066 \beta_{3} - 349 \beta_{4} - 43 \beta_{5} + 513 \beta_{6} + 145 \beta_{7} + 248 \beta_{8} - 126 \beta_{9} - \beta_{10} ) q^{17} + ( -4008188 - 42392 \beta_{1} - 168 \beta_{2} - 1577 \beta_{3} + 112 \beta_{4} + 84 \beta_{5} - 724 \beta_{6} - 388 \beta_{7} + 365 \beta_{8} - 737 \beta_{9} - 171 \beta_{10} ) q^{19} + ( 4487908 + 134996 \beta_{1} + 4151 \beta_{2} + 5270 \beta_{3} + 874 \beta_{4} - 640 \beta_{5} - 1108 \beta_{6} + 232 \beta_{7} - 586 \beta_{8} + 18 \beta_{9} - 128 \beta_{10} ) q^{20} + ( -3948940 - 41824 \beta_{1} - 5828 \beta_{2} - 10083 \beta_{3} + 802 \beta_{4} - 1664 \beta_{5} + 1681 \beta_{6} - 1546 \beta_{7} - 1213 \beta_{8} - 515 \beta_{9} - 1504 \beta_{10} ) q^{22} + ( 8058568 + 10855 \beta_{1} - 2080 \beta_{2} - 6504 \beta_{3} + 2008 \beta_{4} + 663 \beta_{5} - 866 \beta_{6} - 666 \beta_{7} - 1470 \beta_{8} - 1661 \beta_{9} - 1986 \beta_{10} ) q^{23} + ( -4043204 + 298362 \beta_{1} + 5472 \beta_{2} + 3525 \beta_{3} + 2083 \beta_{4} - 3395 \beta_{5} - 861 \beta_{6} - 606 \beta_{7} - 1187 \beta_{8} - 1464 \beta_{9} - 2456 \beta_{10} ) q^{25} + ( 29557831 - 169784 \beta_{1} + 713 \beta_{2} - 7845 \beta_{3} + 3981 \beta_{4} - 4102 \beta_{5} - 3068 \beta_{6} - 1666 \beta_{7} - 1939 \beta_{8} - 4051 \beta_{9} - 2128 \beta_{10} ) q^{26} + ( -35610386 + 577754 \beta_{1} + 4934 \beta_{2} + 28270 \beta_{3} - 956 \beta_{4} + 142 \beta_{5} - 390 \beta_{6} + 3928 \beta_{7} + 1584 \beta_{8} - 760 \beta_{9} + 3972 \beta_{10} ) q^{28} -20511149 q^{29} + ( -26611376 + 677450 \beta_{1} + 28922 \beta_{2} + 43729 \beta_{3} - 6348 \beta_{4} - 5747 \beta_{5} - 2500 \beta_{6} - 2232 \beta_{7} + 6226 \beta_{8} + 2135 \beta_{9} + 3854 \beta_{10} ) q^{31} + ( 57575293 - 521493 \beta_{1} + 27266 \beta_{2} + 48407 \beta_{3} - 12082 \beta_{4} + 6059 \beta_{5} + 7587 \beta_{6} + 9286 \beta_{7} + 2240 \beta_{8} + 21186 \beta_{9} + 12686 \beta_{10} ) q^{32} + ( -101282380 + 624889 \beta_{1} - 24695 \beta_{2} - 46491 \beta_{3} + 12387 \beta_{4} - 22288 \beta_{5} - 3382 \beta_{6} - 16302 \beta_{7} - 8689 \beta_{8} - 7855 \beta_{9} - 4728 \beta_{10} ) q^{34} + ( 119449604 - 1233151 \beta_{1} - 18358 \beta_{2} - 22340 \beta_{3} - 10940 \beta_{4} - 9875 \beta_{5} + 10108 \beta_{6} - 11740 \beta_{7} + 2280 \beta_{8} + 565 \beta_{9} + 4980 \beta_{10} ) q^{35} + ( -125450172 - 223202 \beta_{1} - 48996 \beta_{2} + 31596 \beta_{3} - 7678 \beta_{4} + 11932 \beta_{5} + 20838 \beta_{6} + 10640 \beta_{7} - 8464 \beta_{8} - 2836 \beta_{9} + 20646 \beta_{10} ) q^{37} + ( 110849486 + 4044445 \beta_{1} + 60825 \beta_{2} + 104596 \beta_{3} + 14048 \beta_{4} + 8696 \beta_{5} - 17577 \beta_{6} - 27917 \beta_{7} - 14082 \beta_{8} - 31841 \beta_{9} - 22826 \beta_{10} ) q^{38} + ( -246134287 - 6133661 \beta_{1} - 111732 \beta_{2} - 66515 \beta_{3} - 31432 \beta_{4} + 20755 \beta_{5} + 21513 \beta_{6} + 26924 \beta_{7} + 37798 \beta_{8} - 1854 \beta_{9} + 44994 \beta_{10} ) q^{40} + ( 96271118 + 3593999 \beta_{1} + 104 \beta_{2} + 35548 \beta_{3} + 50071 \beta_{4} - 7837 \beta_{5} - 27765 \beta_{6} - 79 \beta_{7} - 39898 \beta_{8} + 17918 \beta_{9} + 13733 \beta_{10} ) q^{41} + ( 6762376 + 1514809 \beta_{1} + 53152 \beta_{2} + 188322 \beta_{3} + 92512 \beta_{4} - 49636 \beta_{5} - 45820 \beta_{6} + 4284 \beta_{7} - 50139 \beta_{8} - 27729 \beta_{9} + 4285 \beta_{10} ) q^{43} + ( -5605481 + 8581164 \beta_{1} - 131387 \beta_{2} - 21136 \beta_{3} - 33369 \beta_{4} - 25033 \beta_{5} + 99923 \beta_{6} - 44244 \beta_{7} - 14361 \beta_{8} - 54689 \beta_{9} - 58842 \beta_{10} ) q^{44} + ( -7478134 - 6464054 \beta_{1} - 85470 \beta_{2} - 230015 \beta_{3} + 35853 \beta_{4} + 56828 \beta_{5} - 96955 \beta_{6} - 97941 \beta_{7} - 44079 \beta_{8} - 126838 \beta_{9} - 23126 \beta_{10} ) q^{46} + ( 165817448 - 5349636 \beta_{1} - 345658 \beta_{2} - 624506 \beta_{3} + 41978 \beta_{4} - 51103 \beta_{5} + 119538 \beta_{6} + 2610 \beta_{7} - 75233 \beta_{8} + 15888 \beta_{9} - 22503 \beta_{10} ) q^{47} + ( 426964593 - 3502586 \beta_{1} - 351534 \beta_{2} + 35288 \beta_{3} - 27040 \beta_{4} + 988 \beta_{5} + 119874 \beta_{6} - 177914 \beta_{7} - 11088 \beta_{8} + 34722 \beta_{9} + 25208 \beta_{10} ) q^{49} + ( -862902908 - 3777032 \beta_{1} - 657840 \beta_{2} - 221280 \beta_{3} - 114892 \beta_{4} + 160410 \beta_{5} + 194176 \beta_{6} - 51546 \beta_{7} + 95848 \beta_{8} - 78364 \beta_{9} + 66324 \beta_{10} ) q^{50} + ( 297665586 - 10702258 \beta_{1} - 421693 \beta_{2} - 457906 \beta_{3} - 54056 \beta_{4} + 135246 \beta_{5} + 137996 \beta_{6} - 46262 \beta_{7} + 89104 \beta_{8} - 91390 \beta_{9} - 100128 \beta_{10} ) q^{52} + ( -709246675 - 16313139 \beta_{1} + 434402 \beta_{2} + 27865 \beta_{3} - 49954 \beta_{4} + 4648 \beta_{5} - 21760 \beta_{6} + 43355 \beta_{7} - 10785 \beta_{8} + 214162 \beta_{9} + 122245 \beta_{10} ) q^{53} + ( -20339432 + 34444360 \beta_{1} - 651028 \beta_{2} + 341025 \beta_{3} + 118076 \beta_{4} - 34615 \beta_{5} + 59410 \beta_{6} + 155718 \beta_{7} + 36186 \beta_{8} + 122397 \beta_{9} - 24382 \beta_{10} ) q^{55} + ( -1024151274 + 6634910 \beta_{1} + 80206 \beta_{2} + 990280 \beta_{3} + 31462 \beta_{4} - 121382 \beta_{5} + 49408 \beta_{6} + 193530 \beta_{7} + 52726 \beta_{8} - 168908 \beta_{9} + 40248 \beta_{10} ) q^{56} + ( -61533447 + 20511149 \beta_{1} ) q^{58} + ( -106816256 - 51542383 \beta_{1} + 668322 \beta_{2} + 1027818 \beta_{3} - 202348 \beta_{4} - 247117 \beta_{5} + 538072 \beta_{6} + 234464 \beta_{7} + 269356 \beta_{8} - 12165 \beta_{9} - 65848 \beta_{10} ) q^{59} + ( -1696969736 + 63961219 \beta_{1} - 1322730 \beta_{2} - 48652 \beta_{3} + 56761 \beta_{4} + 323175 \beta_{5} - 250541 \beta_{6} + 38147 \beta_{7} + 48634 \beta_{8} - 225584 \beta_{9} + 278511 \beta_{10} ) q^{61} + ( -2005835632 - 8219935 \beta_{1} - 1108755 \beta_{2} + 984172 \beta_{3} - 197195 \beta_{4} + 18380 \beta_{5} + 556237 \beta_{6} + 222444 \beta_{7} + 337832 \beta_{8} + 240212 \beta_{9} + 43320 \beta_{10} ) q^{62} + ( 1077407880 - 39278778 \beta_{1} - 295827 \beta_{2} - 244575 \beta_{3} + 97791 \beta_{4} + 76972 \beta_{5} + 136941 \beta_{6} + 395697 \beta_{7} + 320309 \beta_{8} + 458528 \beta_{9} + 326294 \beta_{10} ) q^{64} + ( -2929291497 - 17111133 \beta_{1} - 970635 \beta_{2} + 1089510 \beta_{3} + 584817 \beta_{4} + 645355 \beta_{5} + 184724 \beta_{6} + 27161 \beta_{7} - 133428 \beta_{8} + 42044 \beta_{9} - 337399 \beta_{10} ) q^{65} + ( 2496827008 - 1104490 \beta_{1} - 1070742 \beta_{2} - 2900204 \beta_{3} - 258142 \beta_{4} - 536132 \beta_{5} + 235536 \beta_{6} + 818448 \beta_{7} + 326644 \beta_{8} + 398192 \beta_{9} - 152270 \beta_{10} ) q^{67} + ( -2697457958 + 73868580 \beta_{1} - 2732430 \beta_{2} - 1784662 \beta_{3} - 650012 \beta_{4} + 512602 \beta_{5} + 372952 \beta_{6} - 260590 \beta_{7} - 212748 \beta_{8} - 153518 \beta_{9} - 225856 \beta_{10} ) q^{68} + ( 3904130402 - 90928610 \beta_{1} - 689782 \beta_{2} - 1242995 \beta_{3} - 636741 \beta_{4} - 716300 \beta_{5} + 1064325 \beta_{6} + 7177 \beta_{7} - 18311 \beta_{8} - 37772 \beta_{9} - 594458 \beta_{10} ) q^{70} + ( 1856194068 - 75282026 \beta_{1} - 3548868 \beta_{2} - 887042 \beta_{3} + 562912 \beta_{4} - 325720 \beta_{5} + 1231580 \beta_{6} - 555752 \beta_{7} + 556044 \beta_{8} + 163084 \beta_{9} - 58868 \beta_{10} ) q^{71} + ( -5250270140 + 10304308 \beta_{1} - 1030660 \beta_{2} - 2085240 \beta_{3} + 618880 \beta_{4} + 537010 \beta_{5} + 473044 \beta_{6} - 575706 \beta_{7} - 883132 \beta_{8} + 1299188 \beta_{9} - 77304 \beta_{10} ) q^{73} + ( 219434572 + 223397062 \beta_{1} - 265262 \beta_{2} - 1602682 \beta_{3} + 1019878 \beta_{4} - 1267588 \beta_{5} - 1146264 \beta_{6} - 113104 \beta_{7} - 834566 \beta_{8} - 406650 \beta_{9} - 97192 \beta_{10} ) q^{74} + ( -3023239344 - 89770230 \beta_{1} - 8004180 \beta_{2} + 159172 \beta_{3} + 717980 \beta_{4} + 197128 \beta_{5} + 56162 \beta_{6} - 857230 \beta_{7} - 868192 \beta_{8} - 1089450 \beta_{9} - 382988 \beta_{10} ) q^{76} + ( -89360190 + 140968075 \beta_{1} - 59354 \beta_{2} + 5714664 \beta_{3} - 259447 \beta_{4} + 1402923 \beta_{5} - 1822413 \beta_{6} + 612435 \beta_{7} + 51218 \beta_{8} + 693700 \beta_{9} + 1302863 \beta_{10} ) q^{77} + ( -10946876376 + 148660966 \beta_{1} + 223356 \beta_{2} - 3554956 \beta_{3} - 65890 \beta_{4} - 2140095 \beta_{5} - 907696 \beta_{6} - 377124 \beta_{7} - 353215 \beta_{8} + 299404 \beta_{9} - 1863685 \beta_{10} ) q^{79} + ( 7651084354 + 142132580 \beta_{1} + 4904921 \beta_{2} - 3623485 \beta_{3} - 248297 \beta_{4} - 961630 \beta_{5} + 647855 \beta_{6} - 211351 \beta_{7} - 423927 \beta_{8} + 894636 \beta_{9} - 298086 \beta_{10} ) q^{80} + ( -10128616390 - 54931731 \beta_{1} - 5302541 \beta_{2} + 6320293 \beta_{3} - 1679077 \beta_{4} + 2198708 \beta_{5} + 23776 \beta_{6} + 2413836 \beta_{7} + 1165275 \beta_{8} + 975671 \beta_{9} + 1605948 \beta_{10} ) q^{82} + ( 12940105572 + 161845519 \beta_{1} + 2841940 \beta_{2} + 7161682 \beta_{3} + 789874 \beta_{4} - 1488733 \beta_{5} + 256732 \beta_{6} + 523876 \beta_{7} - 735616 \beta_{8} + 2082163 \beta_{9} - 1286090 \beta_{10} ) q^{83} + ( -16517704528 + 22642929 \beta_{1} - 2815936 \beta_{2} - 8160350 \beta_{3} - 611079 \beta_{4} + 791225 \beta_{5} - 441247 \beta_{6} - 41157 \beta_{7} + 781196 \beta_{8} + 527132 \beta_{9} + 754173 \beta_{10} ) q^{85} + ( -4347175680 - 3123742 \beta_{1} - 5764394 \beta_{2} + 10978427 \beta_{3} - 2097802 \beta_{4} + 4395912 \beta_{5} + 786925 \beta_{6} + 1665880 \beta_{7} + 1929873 \beta_{8} - 942671 \beta_{9} + 2961260 \beta_{10} ) q^{86} + ( -16450897468 + 321289302 \beta_{1} - 7421233 \beta_{2} - 4244325 \beta_{3} - 1215299 \beta_{4} - 703912 \beta_{5} - 2581633 \beta_{6} - 2358605 \beta_{7} + 413211 \beta_{8} - 4194292 \beta_{9} - 144262 \beta_{10} ) q^{88} + ( 8769707146 + 313027197 \beta_{1} - 3813862 \beta_{2} + 14247574 \beta_{3} + 2376627 \beta_{4} - 774977 \beta_{5} - 6664931 \beta_{6} - 2232545 \beta_{7} - 1966192 \beta_{8} - 857886 \beta_{9} - 481093 \beta_{10} ) q^{89} + ( -32328870120 + 398113093 \beta_{1} + 5792756 \beta_{2} - 11818008 \beta_{3} - 554994 \beta_{4} + 3629991 \beta_{5} - 2365512 \beta_{6} + 1496664 \beta_{7} - 677528 \beta_{8} + 1911955 \beta_{9} + 445186 \beta_{10} ) q^{91} + ( 2034506096 + 101258738 \beta_{1} - 1474868 \beta_{2} + 11104634 \beta_{3} - 157934 \beta_{4} - 1775024 \beta_{5} - 3945152 \beta_{6} - 4330748 \beta_{7} - 808482 \beta_{8} - 5400158 \beta_{9} - 962384 \beta_{10} ) q^{92} + ( 15659725842 + 246136226 \beta_{1} + 1069102 \beta_{2} - 21155706 \beta_{3} - 4701791 \beta_{4} - 1803396 \beta_{5} + 4662586 \beta_{6} + 239651 \beta_{7} - 1338976 \beta_{8} + 147505 \beta_{9} - 509266 \beta_{10} ) q^{94} + ( 17830558112 - 146295916 \beta_{1} - 15045226 \beta_{2} + 14966725 \beta_{3} + 1607886 \beta_{4} + 1514150 \beta_{5} + 1322128 \beta_{6} + 1218128 \beta_{7} + 2240151 \beta_{8} - 2453213 \beta_{9} + 1064393 \beta_{10} ) q^{95} + ( -27577714358 + 179090835 \beta_{1} + 4417834 \beta_{2} + 506624 \beta_{3} + 5382735 \beta_{4} + 328513 \beta_{5} + 3333353 \beta_{6} - 2846425 \beta_{7} + 227254 \beta_{8} - 3676062 \beta_{9} + 139625 \beta_{10} ) q^{97} + ( 11224956067 + 265108623 \beta_{1} - 7343904 \beta_{2} - 667996 \beta_{3} - 6601140 \beta_{4} - 8044672 \beta_{5} + 102292 \beta_{6} + 1846132 \beta_{7} - 644092 \beta_{8} - 1257312 \beta_{9} - 6849608 \beta_{10} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 32q^{2} + 9146q^{4} + 2740q^{5} - 49432q^{7} + 150054q^{8} + O(q^{10}) \) \( 11q + 32q^{2} + 9146q^{4} + 2740q^{5} - 49432q^{7} + 150054q^{8} - 685834q^{10} + 612246q^{11} + 1510364q^{13} - 3955400q^{14} + 3024818q^{16} + 3291098q^{17} - 44121388q^{19} + 49472662q^{20} - 43435618q^{22} + 88684076q^{23} - 44195521q^{25} + 324999762q^{26} - 391274848q^{28} - 225622639q^{29} - 292235934q^{31} + 632542514q^{32} - 1113307936q^{34} + 1312820120q^{35} - 1380429338q^{37} + 1222857284q^{38} - 2713154106q^{40} + 1062067494q^{41} + 74588594q^{43} - 52891466q^{44} - 87670324q^{46} + 1821239394q^{47} + 4692522003q^{49} - 9494259926q^{50} + 3266669866q^{52} - 7818635688q^{53} - 191002682q^{55} - 11263587512q^{56} - 656356768q^{58} - 1230002712q^{59} - 18602654230q^{61} - 22075953162q^{62} + 11813658086q^{64} - 32245789334q^{65} + 27481284652q^{67} - 29588811820q^{68} + 42862666712q^{70} + 20347168516q^{71} - 57740010478q^{73} + 2640709564q^{74} - 33350650772q^{76} - 871959792q^{77} - 120245016462q^{79} + 84319695274q^{80} - 111495532412q^{82} + 142463983824q^{83} - 181628566552q^{85} - 47870165542q^{86} - 180608014462q^{88} + 96700717270q^{89} - 355162031176q^{91} + 22429477796q^{92} + 172608565078q^{94} + 195922150708q^{95} - 303190852014q^{97} + 123776497136q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2871 \)
\(\beta_{3}\)\(=\)\((\)\(565550727753207886291601 \nu^{10} - 12036436634048687917592128 \nu^{9} - 8679501631223090279693232390 \nu^{8} + 184357050811285113015962991580 \nu^{7} + 45484105320154672732669722806418 \nu^{6} - 930882828848581596917379957802196 \nu^{5} - 95992437020855812880527329597716284 \nu^{4} + 1730622925673756035085534196275520532 \nu^{3} + 73156959763450908017776739864204986117 \nu^{2} - 769094726496573889613325678204209667292 \nu - 20799042220528830334861868275640875459046\)\()/ \)\(14\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(804568383233756418395133 \nu^{10} - 17267803367018535072511622 \nu^{9} - 12336760382447881892656044568 \nu^{8} + 264420685272447934888564742698 \nu^{7} + 64552288847038471792600712502676 \nu^{6} - 1334535359062087490131686580269674 \nu^{5} - 135852752719164615275587072188721598 \nu^{4} + 2477728569409695037503433979951964654 \nu^{3} + 102906200407341723386076394169938910127 \nu^{2} - 1094360840468224980441703836247001677608 \nu - 29166223419008641518517735767270900815210\)\()/ \)\(37\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-9163403140657769900866107 \nu^{10} + 198633537751403329905934848 \nu^{9} + 140596166689623094462433093762 \nu^{8} - 3041665942506549913052874645172 \nu^{7} - 736417398237305496478893511100294 \nu^{6} + 15356546218183565315743456419073116 \nu^{5} + 1552588397318532196097690383647597012 \nu^{4} - 28544393468946693612023111278394674076 \nu^{3} - 1180832590628483793843189195580624803063 \nu^{2} + 12677139017446290787839141609519303240692 \nu + 336787870381508857170659774608039749532770\)\()/ \)\(14\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(925017894146270556324001 \nu^{10} - 20020628115330855489561920 \nu^{9} - 14189084609441700734893823398 \nu^{8} + 306532457196252549964930263196 \nu^{7} + 74284093661778417907203960210674 \nu^{6} - 1547182122942695217095069951118356 \nu^{5} - 156457328858201747863434643585273916 \nu^{4} + 2874126879942643802269885113303822420 \nu^{3} + 118709055269937646660049496797552834581 \nu^{2} - 1273304945394343953001065199684882896796 \nu - 33771289290772363759198711089555493152710\)\()/ \)\(14\!\cdots\!96\)\( \)
\(\beta_{7}\)\(=\)\((\)\(17924515188364112919694475 \nu^{10} - 390533907773547048878105328 \nu^{9} - 274923157022562776700670224498 \nu^{8} + 5977092590095664100881387447588 \nu^{7} + 1439054520778823019217921979158582 \nu^{6} - 30156182372896091771804617274751116 \nu^{5} - 3029909387942067083992060555638832356 \nu^{4} + 55995020279474318988140370680608573548 \nu^{3} + 2297298136347225770100641002219376169271 \nu^{2} - 24797280798141266675461305985624276255252 \nu - 653992563607077924913260367339847802624162\)\()/ \)\(14\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-10067505656708592282914281 \nu^{10} + 219233891733852304246515060 \nu^{9} + 154399929546830175885590403202 \nu^{8} - 3355681427543631266684088866112 \nu^{7} - 808057906369316743010401784537966 \nu^{6} + 16932018481560030983141642662609960 \nu^{5} + 1700796940307581694610805221167604728 \nu^{4} - 31441941084304821486089643689848324704 \nu^{3} - 1288583706240134525012238524085935159841 \nu^{2} + 13921813775931752363896357049826335499780 \nu + 366634213699662825997070617475768862963454\)\()/ \)\(74\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-21079738997064663101670405 \nu^{10} + 458644500794685761740501296 \nu^{9} + 323354430353779280539983433806 \nu^{8} - 7020628948372726028441729052476 \nu^{7} - 1692921398123508697113673323613514 \nu^{6} + 35428680220628453675122051569994452 \nu^{5} + 3566007045687488365957061058992555452 \nu^{4} - 65808228258012435057343035865902213556 \nu^{3} - 2706782697136381202012532265190577353657 \nu^{2} + 29174237010936398336754582421339819977964 \nu + 771285432236873252990289845467292253069054\)\()/ \)\(14\!\cdots\!60\)\( \)
\(\beta_{10}\)\(=\)\((\)\(2538997685966077724142287 \nu^{10} - 55249397164757927417949060 \nu^{9} - 38946640822070460985147162902 \nu^{8} + 845789085932399457365643786424 \nu^{7} + 203903709917978063752544338758090 \nu^{6} - 4268696834850637079308433925476528 \nu^{5} - 429517060878911747981686683842375824 \nu^{4} + 7930665309386432242702266640084460360 \nu^{3} + 326065648869848084054734586962683287567 \nu^{2} - 3517486448150698373256159210815439796172 \nu - 92938872384437077011075961968332563923714\)\()/ \)\(14\!\cdots\!96\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2871\)
\(\nu^{3}\)\(=\)\(-2 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} - 4 \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + 27 \beta_{3} + \beta_{2} + 4746 \beta_{1} - 111\)
\(\nu^{4}\)\(=\)\(10 \beta_{10} + 324 \beta_{9} - 55 \beta_{8} + 225 \beta_{7} - 89 \beta_{6} - 238 \beta_{5} - 41 \beta_{4} + 371 \beta_{3} + 6895 \beta_{2} - 9356 \beta_{1} + 13621663\)
\(\nu^{5}\)\(=\)\(-28740 \beta_{10} - 32530 \beta_{9} - 51677 \beta_{8} - 38319 \beta_{7} - 81840 \beta_{6} - 33935 \beta_{5} + 11467 \beta_{4} + 175912 \beta_{3} + 10803 \beta_{2} + 26471112 \beta_{1} - 27519793\)
\(\nu^{6}\)\(=\)\(154704 \beta_{10} + 3391688 \beta_{9} - 431612 \beta_{8} + 2468940 \beta_{7} - 1134464 \beta_{6} - 2571828 \beta_{5} - 110108 \beta_{4} + 3367616 \beta_{3} + 44836183 \beta_{2} - 88076300 \beta_{1} + 75939786039\)
\(\nu^{7}\)\(=\)\(-256494054 \beta_{10} - 298313766 \beta_{9} - 383013666 \beta_{8} - 305856508 \beta_{7} - 610939499 \beta_{6} - 290285801 \beta_{5} + 131524816 \beta_{4} + 994354529 \beta_{3} + 30194683 \beta_{2} + 159206455932 \beta_{1} - 256706722429\)
\(\nu^{8}\)\(=\)\(1821509134 \beta_{10} + 27269535996 \beta_{9} - 2855699909 \beta_{8} + 20122530003 \beta_{7} - 10452712371 \beta_{6} - 20669727602 \beta_{5} + 1043359701 \beta_{4} + 25332967281 \beta_{3} + 294310414339 \beta_{2} - 715520349036 \beta_{1} + 456572329667143\)
\(\nu^{9}\)\(=\)\(-1975942874424 \beta_{10} - 2327288256370 \beta_{9} - 2713255099587 \beta_{8} - 2285739989597 \beta_{7} - 4283048928870 \beta_{6} - 2220410819223 \beta_{5} + 1114400221401 \beta_{4} + 5544822979418 \beta_{3} - 183123620537 \beta_{2} + 1001525117031748 \beta_{1} - 2076746520234869\)
\(\nu^{10}\)\(=\)\(17582847741900 \beta_{10} + 201013362531376 \beta_{9} - 18039580668670 \beta_{8} + 148670899043178 \beta_{7} - 83453288539206 \beta_{6} - 150082167079296 \beta_{5} + 17626411832470 \beta_{4} + 181739576561698 \beta_{3} + 1952786614226359 \beta_{2} - 5660625209183388 \beta_{1} + 2871559482435683511\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
81.6399
65.2743
54.9918
33.9131
30.4581
−18.9987
−20.7036
−20.7638
−56.7551
−65.3340
−82.7218
−78.6399 0 4136.23 12464.4 0 −230.276 −164218. 0 −980202.
1.2 −62.2743 0 1830.09 −1377.47 0 33050.3 13570.0 0 85780.8
1.3 −51.9918 0 655.142 −3682.79 0 −83752.5 72417.1 0 191474.
1.4 −30.9131 0 −1092.38 6165.86 0 8452.88 97078.9 0 −190606.
1.5 −27.4581 0 −1294.05 −9990.24 0 28164.0 91766.4 0 274313.
1.6 21.9987 0 −1564.06 886.321 0 44263.6 −79460.6 0 19497.9
1.7 23.7036 0 −1486.14 −6643.74 0 −61145.1 −83771.9 0 −157481.
1.8 23.7638 0 −1483.28 5297.67 0 80499.9 −83916.8 0 125893.
1.9 59.7551 0 1522.68 6379.08 0 −23005.8 −31390.8 0 381183.
1.10 68.3340 0 2621.54 −8265.40 0 −68659.4 39192.0 0 −564808.
1.11 85.7218 0 5300.23 1506.26 0 −7069.49 278787. 0 129120.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.12.a.a 11
3.b odd 2 1 29.12.a.a 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.12.a.a 11 3.b odd 2 1
261.12.a.a 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(33\!\cdots\!44\)\( T_{2}^{4} + \)\(17\!\cdots\!40\)\( T_{2}^{3} - \)\(29\!\cdots\!12\)\( T_{2}^{2} - \)\(52\!\cdots\!44\)\( T_{2} + \)\(93\!\cdots\!40\)\( \)">\(T_{2}^{11} - \cdots\) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(261))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 937413686027223040 - 52480495829254144 T - 2941272354394112 T^{2} + 170288442403840 T^{3} + 3367672715744 T^{4} - 187882927184 T^{5} - 1785306736 T^{6} + 82465976 T^{7} + 407614 T^{8} - 15325 T^{9} - 32 T^{10} + T^{11} \)
$3$ \( T^{11} \)
$5$ \( \)\(96\!\cdots\!50\)\( - \)\(97\!\cdots\!75\)\( T - \)\(69\!\cdots\!00\)\( T^{2} + \)\(60\!\cdots\!25\)\( T^{3} + \)\(90\!\cdots\!00\)\( T^{4} - \)\(57\!\cdots\!50\)\( T^{5} - 31578273416969866000 T^{6} + 18865845213698850 T^{7} + 446617757630 T^{8} - 242703127 T^{9} - 2740 T^{10} + T^{11} \)
$7$ \( -\)\(36\!\cdots\!16\)\( - \)\(16\!\cdots\!56\)\( T - \)\(66\!\cdots\!68\)\( T^{2} + \)\(31\!\cdots\!28\)\( T^{3} - \)\(37\!\cdots\!20\)\( T^{4} - \)\(72\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} + 46152539677908554560 T^{7} - 471818114692928 T^{8} - 11999796776 T^{9} + 49432 T^{10} + T^{11} \)
$11$ \( \)\(73\!\cdots\!76\)\( - \)\(30\!\cdots\!69\)\( T - \)\(92\!\cdots\!02\)\( T^{2} + \)\(72\!\cdots\!01\)\( T^{3} + \)\(32\!\cdots\!80\)\( T^{4} - \)\(13\!\cdots\!22\)\( T^{5} - \)\(27\!\cdots\!92\)\( T^{6} + \)\(73\!\cdots\!58\)\( T^{7} + 748874025379876176 T^{8} - 1490380632137 T^{9} - 612246 T^{10} + T^{11} \)
$13$ \( \)\(12\!\cdots\!62\)\( - \)\(31\!\cdots\!03\)\( T - \)\(23\!\cdots\!12\)\( T^{2} + \)\(26\!\cdots\!65\)\( T^{3} + \)\(54\!\cdots\!40\)\( T^{4} - \)\(48\!\cdots\!78\)\( T^{5} - \)\(42\!\cdots\!92\)\( T^{6} + \)\(33\!\cdots\!62\)\( T^{7} + 13701895344310761834 T^{8} - 9745795139479 T^{9} - 1510364 T^{10} + T^{11} \)
$17$ \( -\)\(18\!\cdots\!04\)\( - \)\(38\!\cdots\!52\)\( T + \)\(41\!\cdots\!92\)\( T^{2} + \)\(12\!\cdots\!32\)\( T^{3} - \)\(18\!\cdots\!28\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} - \)\(16\!\cdots\!16\)\( T^{6} + \)\(72\!\cdots\!48\)\( T^{7} + \)\(16\!\cdots\!96\)\( T^{8} - 152135401069884 T^{9} - 3291098 T^{10} + T^{11} \)
$19$ \( \)\(81\!\cdots\!72\)\( - \)\(45\!\cdots\!36\)\( T - \)\(30\!\cdots\!96\)\( T^{2} - \)\(18\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!08\)\( T^{4} + \)\(16\!\cdots\!52\)\( T^{5} - \)\(23\!\cdots\!52\)\( T^{6} - \)\(18\!\cdots\!64\)\( T^{7} - \)\(95\!\cdots\!56\)\( T^{8} + 379131733968340 T^{9} + 44121388 T^{10} + T^{11} \)
$23$ \( \)\(23\!\cdots\!08\)\( + \)\(39\!\cdots\!44\)\( T - \)\(81\!\cdots\!52\)\( T^{2} - \)\(10\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} + \)\(21\!\cdots\!72\)\( T^{5} - \)\(33\!\cdots\!40\)\( T^{6} - \)\(23\!\cdots\!28\)\( T^{7} + \)\(30\!\cdots\!52\)\( T^{8} - 1974969876252652 T^{9} - 88684076 T^{10} + T^{11} \)
$29$ \( ( 20511149 + T )^{11} \)
$31$ \( \)\(15\!\cdots\!48\)\( - \)\(40\!\cdots\!17\)\( T + \)\(20\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!17\)\( T^{3} - \)\(22\!\cdots\!32\)\( T^{4} - \)\(29\!\cdots\!10\)\( T^{5} + \)\(45\!\cdots\!72\)\( T^{6} + \)\(25\!\cdots\!18\)\( T^{7} - \)\(21\!\cdots\!60\)\( T^{8} - 85188350181927473 T^{9} + 292235934 T^{10} + T^{11} \)
$37$ \( -\)\(31\!\cdots\!88\)\( - \)\(92\!\cdots\!92\)\( T - \)\(93\!\cdots\!92\)\( T^{2} - \)\(33\!\cdots\!92\)\( T^{3} + \)\(52\!\cdots\!80\)\( T^{4} + \)\(63\!\cdots\!00\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} - \)\(26\!\cdots\!16\)\( T^{7} - \)\(77\!\cdots\!20\)\( T^{8} - 34631105003304288 T^{9} + 1380429338 T^{10} + T^{11} \)
$41$ \( \)\(36\!\cdots\!28\)\( + \)\(35\!\cdots\!80\)\( T - \)\(32\!\cdots\!64\)\( T^{2} - \)\(20\!\cdots\!32\)\( T^{3} + \)\(39\!\cdots\!32\)\( T^{4} - \)\(14\!\cdots\!16\)\( T^{5} - \)\(15\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!52\)\( T^{7} + \)\(23\!\cdots\!72\)\( T^{8} - 2036343913280571352 T^{9} - 1062067494 T^{10} + T^{11} \)
$43$ \( -\)\(14\!\cdots\!60\)\( - \)\(18\!\cdots\!29\)\( T - \)\(22\!\cdots\!46\)\( T^{2} + \)\(30\!\cdots\!29\)\( T^{3} + \)\(19\!\cdots\!40\)\( T^{4} - \)\(10\!\cdots\!66\)\( T^{5} - \)\(26\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!98\)\( T^{7} + \)\(99\!\cdots\!28\)\( T^{8} - 6328771775300000993 T^{9} - 74588594 T^{10} + T^{11} \)
$47$ \( -\)\(21\!\cdots\!68\)\( - \)\(70\!\cdots\!89\)\( T + \)\(23\!\cdots\!62\)\( T^{2} + \)\(78\!\cdots\!73\)\( T^{3} - \)\(13\!\cdots\!24\)\( T^{4} - \)\(11\!\cdots\!26\)\( T^{5} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(57\!\cdots\!10\)\( T^{7} + \)\(14\!\cdots\!96\)\( T^{8} - 12243512776017368689 T^{9} - 1821239394 T^{10} + T^{11} \)
$53$ \( -\)\(17\!\cdots\!82\)\( + \)\(13\!\cdots\!45\)\( T + \)\(41\!\cdots\!20\)\( T^{2} - \)\(19\!\cdots\!35\)\( T^{3} + \)\(32\!\cdots\!36\)\( T^{4} + \)\(19\!\cdots\!14\)\( T^{5} + \)\(13\!\cdots\!44\)\( T^{6} - \)\(69\!\cdots\!94\)\( T^{7} - \)\(27\!\cdots\!70\)\( T^{8} - 14495590711615911727 T^{9} + 7818635688 T^{10} + T^{11} \)
$59$ \( -\)\(21\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!76\)\( T^{3} - \)\(99\!\cdots\!44\)\( T^{4} - \)\(21\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!48\)\( T^{7} - \)\(92\!\cdots\!88\)\( T^{8} - \)\(18\!\cdots\!16\)\( T^{9} + 1230002712 T^{10} + T^{11} \)
$61$ \( \)\(21\!\cdots\!20\)\( - \)\(25\!\cdots\!52\)\( T + \)\(26\!\cdots\!44\)\( T^{2} + \)\(60\!\cdots\!16\)\( T^{3} - \)\(17\!\cdots\!00\)\( T^{4} - \)\(25\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!32\)\( T^{6} + \)\(48\!\cdots\!48\)\( T^{7} - \)\(23\!\cdots\!12\)\( T^{8} - 84689162462461737068 T^{9} + 18602654230 T^{10} + T^{11} \)
$67$ \( -\)\(21\!\cdots\!80\)\( - \)\(41\!\cdots\!92\)\( T + \)\(15\!\cdots\!48\)\( T^{2} - \)\(93\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!60\)\( T^{4} + \)\(67\!\cdots\!16\)\( T^{5} - \)\(11\!\cdots\!24\)\( T^{6} - \)\(21\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!00\)\( T^{8} - \)\(24\!\cdots\!72\)\( T^{9} - 27481284652 T^{10} + T^{11} \)
$71$ \( -\)\(26\!\cdots\!92\)\( + \)\(13\!\cdots\!36\)\( T - \)\(17\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!32\)\( T^{3} + \)\(40\!\cdots\!28\)\( T^{4} - \)\(86\!\cdots\!28\)\( T^{5} - \)\(21\!\cdots\!76\)\( T^{6} + \)\(85\!\cdots\!68\)\( T^{7} + \)\(37\!\cdots\!60\)\( T^{8} - \)\(17\!\cdots\!96\)\( T^{9} - 20347168516 T^{10} + T^{11} \)
$73$ \( -\)\(11\!\cdots\!48\)\( + \)\(56\!\cdots\!88\)\( T + \)\(59\!\cdots\!16\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} - \)\(32\!\cdots\!60\)\( T^{4} + \)\(31\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!12\)\( T^{6} - \)\(72\!\cdots\!44\)\( T^{7} - \)\(60\!\cdots\!48\)\( T^{8} - \)\(15\!\cdots\!84\)\( T^{9} + 57740010478 T^{10} + T^{11} \)
$79$ \( -\)\(12\!\cdots\!80\)\( + \)\(10\!\cdots\!39\)\( T - \)\(17\!\cdots\!58\)\( T^{2} - \)\(12\!\cdots\!07\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(95\!\cdots\!90\)\( T^{5} - \)\(11\!\cdots\!24\)\( T^{6} - \)\(14\!\cdots\!30\)\( T^{7} - \)\(30\!\cdots\!44\)\( T^{8} + \)\(18\!\cdots\!67\)\( T^{9} + 120245016462 T^{10} + T^{11} \)
$83$ \( \)\(45\!\cdots\!64\)\( + \)\(15\!\cdots\!84\)\( T - \)\(65\!\cdots\!60\)\( T^{2} - \)\(24\!\cdots\!12\)\( T^{3} + \)\(96\!\cdots\!60\)\( T^{4} + \)\(42\!\cdots\!88\)\( T^{5} - \)\(22\!\cdots\!40\)\( T^{6} - \)\(77\!\cdots\!08\)\( T^{7} + \)\(26\!\cdots\!64\)\( T^{8} + \)\(33\!\cdots\!80\)\( T^{9} - 142463983824 T^{10} + T^{11} \)
$89$ \( \)\(45\!\cdots\!40\)\( + \)\(17\!\cdots\!68\)\( T - \)\(24\!\cdots\!12\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(65\!\cdots\!04\)\( T^{4} + \)\(47\!\cdots\!68\)\( T^{5} - \)\(55\!\cdots\!12\)\( T^{6} + \)\(30\!\cdots\!32\)\( T^{7} + \)\(13\!\cdots\!24\)\( T^{8} - \)\(11\!\cdots\!48\)\( T^{9} - 96700717270 T^{10} + T^{11} \)
$97$ \( -\)\(18\!\cdots\!04\)\( - \)\(43\!\cdots\!56\)\( T - \)\(14\!\cdots\!20\)\( T^{2} + \)\(90\!\cdots\!92\)\( T^{3} + \)\(61\!\cdots\!28\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} - \)\(80\!\cdots\!24\)\( T^{6} - \)\(39\!\cdots\!92\)\( T^{7} - \)\(29\!\cdots\!28\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + 303190852014 T^{10} + T^{11} \)
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