Properties

Label 8034.2.a.bc
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} -\beta_{12} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} -\beta_{12} q^{7} + q^{8} + q^{9} -\beta_{1} q^{10} -\beta_{13} q^{11} - q^{12} - q^{13} -\beta_{12} q^{14} + \beta_{1} q^{15} + q^{16} + ( \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{17} + q^{18} + ( \beta_{6} + \beta_{10} ) q^{19} -\beta_{1} q^{20} + \beta_{12} q^{21} -\beta_{13} q^{22} + ( \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{23} - q^{24} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{25} - q^{26} - q^{27} -\beta_{12} q^{28} + ( 2 - \beta_{4} + \beta_{11} - \beta_{14} ) q^{29} + \beta_{1} q^{30} + ( \beta_{6} - \beta_{10} ) q^{31} + q^{32} + \beta_{13} q^{33} + ( \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{34} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{35} + q^{36} + ( 2 - \beta_{8} + \beta_{13} ) q^{37} + ( \beta_{6} + \beta_{10} ) q^{38} + q^{39} -\beta_{1} q^{40} + ( -\beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{41} + \beta_{12} q^{42} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{9} - 2 \beta_{12} - \beta_{14} ) q^{43} -\beta_{13} q^{44} -\beta_{1} q^{45} + ( \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{46} + ( -\beta_{4} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{47} - q^{48} + ( 2 - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{49} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{50} + ( -\beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{51} - q^{52} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{6} - \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{53} - q^{54} + ( -1 - \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{55} -\beta_{12} q^{56} + ( -\beta_{6} - \beta_{10} ) q^{57} + ( 2 - \beta_{4} + \beta_{11} - \beta_{14} ) q^{58} + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} ) q^{59} + \beta_{1} q^{60} + ( 2 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{14} ) q^{61} + ( \beta_{6} - \beta_{10} ) q^{62} -\beta_{12} q^{63} + q^{64} + \beta_{1} q^{65} + \beta_{13} q^{66} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{67} + ( \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{68} + ( -\beta_{4} + \beta_{6} - \beta_{8} + \beta_{13} - \beta_{14} ) q^{69} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{70} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{71} + q^{72} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{73} + ( 2 - \beta_{8} + \beta_{13} ) q^{74} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{75} + ( \beta_{6} + \beta_{10} ) q^{76} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{77} + q^{78} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{79} -\beta_{1} q^{80} + q^{81} + ( -\beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{82} + ( 1 + \beta_{1} - \beta_{5} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} + \beta_{12} q^{84} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{85} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{9} - 2 \beta_{12} - \beta_{14} ) q^{86} + ( -2 + \beta_{4} - \beta_{11} + \beta_{14} ) q^{87} -\beta_{13} q^{88} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{12} ) q^{89} -\beta_{1} q^{90} + \beta_{12} q^{91} + ( \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{92} + ( -\beta_{6} + \beta_{10} ) q^{93} + ( -\beta_{4} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{94} + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} ) q^{95} - q^{96} + ( 2 + \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{14} ) q^{97} + ( 2 - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{98} -\beta_{13} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(2537687267978126305 \nu^{14} - 4451260489557971801 \nu^{13} - 118607585529944247160 \nu^{12} + 201894771958174626540 \nu^{11} + 2065774167255360616712 \nu^{10} - 3389041191171673838947 \nu^{9} - 16731997871850458694196 \nu^{8} + 26184651556522014754456 \nu^{7} + 64612006487857426157383 \nu^{6} - 94142583432419588002804 \nu^{5} - 109525350370822082766017 \nu^{4} + 145425424067694665372939 \nu^{3} + 53532513774968079841586 \nu^{2} - 75481126501892671808752 \nu + 10315145682746966682944\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-3699895265701566353 \nu^{14} + 9148574385842311585 \nu^{13} + 173370522950640285632 \nu^{12} - 401147448883605874380 \nu^{11} - 3040455429814694571304 \nu^{10} + 6347593527480726095411 \nu^{9} + 25017731332666579088492 \nu^{8} - 44104795536205132068824 \nu^{7} - 100220807092977766199447 \nu^{6} + 130523696906252529491036 \nu^{5} + 189558836334691265479921 \nu^{4} - 141476191243349227852339 \nu^{3} - 143327565524240360562922 \nu^{2} + 51082916732053674000704 \nu + 31232388687353964064640\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{4}\)\(=\)\((\)\(3773756680062282389 \nu^{14} - 6621368153969086837 \nu^{13} - 173187246246377101328 \nu^{12} + 299029290976494333660 \nu^{11} + 2937458974583126918344 \nu^{10} - 4977240979565848871999 \nu^{9} - 22880252731934338749548 \nu^{8} + 37839894450753416967704 \nu^{7} + 83503603148532481698995 \nu^{6} - 132396296199577483382588 \nu^{5} - 131442940707426684242005 \nu^{4} + 201052630181593672596943 \nu^{3} + 56590561866592878591298 \nu^{2} - 105730885302223976198336 \nu + 15999879528543583420672\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-480421702545662123 \nu^{14} + 43110065350197505 \nu^{13} + 23213090564296575506 \nu^{12} - 740490576502024596 \nu^{11} - 423613518372852698680 \nu^{10} - 14892506323265156095 \nu^{9} + 3671032048907900349578 \nu^{8} + 411319014825229416136 \nu^{7} - 15679149848322058779965 \nu^{6} - 3172708855946150385058 \nu^{5} + 30948944380910040502771 \nu^{4} + 8259757592004157683545 \nu^{3} - 22217767129638255474988 \nu^{2} - 4580081011050082891228 \nu + 2332369560757484941472\)\()/ \)\(55\!\cdots\!92\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-3904290949793122853 \nu^{14} + 14888092941463260925 \nu^{13} + 169062781332764512088 \nu^{12} - 668437526636551236252 \nu^{11} - 2592166138155870798184 \nu^{10} + 10967030382881649513455 \nu^{9} + 16479152638452021523460 \nu^{8} - 80762745305159870682104 \nu^{7} - 35045824171083536765987 \nu^{6} + 263743776775326664831940 \nu^{5} - 17153063243581124000891 \nu^{4} - 329708970008761859686711 \nu^{3} + 86096965550968147850822 \nu^{2} + 85313542688545755485744 \nu - 17231425109937506795776\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{7}\)\(=\)\((\)\(1437348996203833439 \nu^{14} - 3946760310112449751 \nu^{13} - 62994684520973032856 \nu^{12} + 176671191969983900820 \nu^{11} + 984697502339295110968 \nu^{10} - 2894763636328689608669 \nu^{9} - 6497133196312926368636 \nu^{8} + 21396335118373983609512 \nu^{7} + 15463140566778105369785 \nu^{6} - 71147018611462724250524 \nu^{5} + 668266301521186841729 \nu^{4} + 96396192676307847030949 \nu^{3} - 26343781779899371656962 \nu^{2} - 36576815788443985896752 \nu + 4137311857191453969280\)\()/ \)\(14\!\cdots\!12\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-216776146664850899 \nu^{14} + 833519197187931691 \nu^{13} + 9225298037107150072 \nu^{12} - 37446683496667123780 \nu^{11} - 136909874293211051608 \nu^{10} + 615042765945550203385 \nu^{9} + 807482146801725193676 \nu^{8} - 4539938442156400956040 \nu^{7} - 1259609430605682009317 \nu^{6} + 14947524072875232440620 \nu^{5} - 2596195015386327990477 \nu^{4} - 19543885760794156101025 \nu^{3} + 5767233907927992512410 \nu^{2} + 6876707545146179250864 \nu - 1099608980575046137664\)\()/ \)\(16\!\cdots\!68\)\( \)
\(\beta_{9}\)\(=\)\((\)\(2262060993304819351 \nu^{14} - 3898546964586732887 \nu^{13} - 102677561674421271760 \nu^{12} + 172180740620193204084 \nu^{11} + 1709275636723449924632 \nu^{10} - 2762375590140558319621 \nu^{9} - 12869509104005418846724 \nu^{8} + 19631778341601514731400 \nu^{7} + 44047896643808091309313 \nu^{6} - 59268935086500033928756 \nu^{5} - 62006168022988305824471 \nu^{4} + 60676860335814627002837 \nu^{3} + 22877990863326377997014 \nu^{2} - 5496566174053900711552 \nu + 5241129959594824984640\)\()/ \)\(14\!\cdots\!12\)\( \)
\(\beta_{10}\)\(=\)\((\)\(7389528242931052763 \nu^{14} - 17619842819561824147 \nu^{13} - 335494621667721213464 \nu^{12} + 781347790842433004388 \nu^{11} + 5596799573243534456152 \nu^{10} - 12587537843796393862097 \nu^{9} - 42434297297760917373068 \nu^{8} + 89945009312798312263112 \nu^{7} + 148333866313014220649117 \nu^{6} - 276567011587519356139052 \nu^{5} - 225796791826086771929275 \nu^{4} + 305614434437315646882457 \nu^{3} + 122418852737580306892054 \nu^{2} - 64975688162446145854064 \nu - 20760519604905515508992\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{11}\)\(=\)\((\)\(8766750062989708633 \nu^{14} - 22529199141324906641 \nu^{13} - 383811818206987097848 \nu^{12} + 995941970920453001676 \nu^{11} + 6001036075358757513992 \nu^{10} - 15968681453075819058955 \nu^{9} - 39886372998134487110740 \nu^{8} + 113304212701430619470680 \nu^{7} + 99684232238791936686799 \nu^{6} - 345811571939604474832852 \nu^{5} - 22639092092040808747577 \nu^{4} + 392231796537565988390195 \nu^{3} - 156334774693565882680990 \nu^{2} - 114807959618689508407408 \nu + 54269842840724729422208\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-4417538992890062809 \nu^{14} + 10268326608661339349 \nu^{13} + 197311809783773428132 \nu^{12} - 456611053302482919612 \nu^{11} - 3196413318423942416024 \nu^{10} + 7398646108103984300491 \nu^{9} + 22849031189715444748168 \nu^{8} - 53545307775003849319096 \nu^{7} - 69555529379053943147695 \nu^{6} + 169932974581872220205488 \nu^{5} + 70235966360674007667737 \nu^{4} - 207964216939141801609367 \nu^{3} + 8344259772351224143546 \nu^{2} + 71793846729075368713576 \nu - 11807950134708017239328\)\()/ \)\(22\!\cdots\!68\)\( \)
\(\beta_{13}\)\(=\)\((\)\(7599288743075757563 \nu^{14} - 14675949069150921499 \nu^{13} - 343394734237158555296 \nu^{12} + 648617517135946752852 \nu^{11} + 5675871771319530293080 \nu^{10} - 10416374257422677241905 \nu^{9} - 42209961752351543724788 \nu^{8} + 74291750278280117421560 \nu^{7} + 141106197202116037886717 \nu^{6} - 228640259637922398414884 \nu^{5} - 190386748714222964816683 \nu^{4} + 258839300934634508381089 \nu^{3} + 69730647157650718059982 \nu^{2} - 62443733500550485578896 \nu - 2219445897514458889376\)\()/ \)\(22\!\cdots\!68\)\( \)
\(\beta_{14}\)\(=\)\((\)\(5118564908078654033 \nu^{14} - 13012426933683220879 \nu^{13} - 226696578225165112178 \nu^{12} + 578099458562396288700 \nu^{11} + 3621424480522273550704 \nu^{10} - 9348618128120933493683 \nu^{9} - 25231217957008947225314 \nu^{8} + 67395099414387524537648 \nu^{7} + 72665931767149837846055 \nu^{6} - 212442650490802176795278 \nu^{5} - 62438278530719538088105 \nu^{4} + 256098528986214242270833 \nu^{3} - 19685540454780823356992 \nu^{2} - 79244364779543304621452 \nu + 9700285797997532462848\)\()/ \)\(11\!\cdots\!84\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{14} + 2 \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(-4 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{7} - \beta_{4} + 12 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-34 \beta_{14} + 32 \beta_{13} + 5 \beta_{12} + 19 \beta_{11} + 15 \beta_{10} - 10 \beta_{9} - 19 \beta_{8} + 18 \beta_{7} + 19 \beta_{6} + 18 \beta_{5} - 16 \beta_{4} - 3 \beta_{3} + 19 \beta_{2} - 16 \beta_{1} + 85\)
\(\nu^{5}\)\(=\)\(-\beta_{14} + 8 \beta_{13} - 75 \beta_{12} - 20 \beta_{11} - 27 \beta_{10} - 22 \beta_{9} - 45 \beta_{7} - 8 \beta_{6} + \beta_{5} - 23 \beta_{4} - \beta_{3} + 4 \beta_{2} + 168 \beta_{1} - 15\)
\(\nu^{6}\)\(=\)\(-556 \beta_{14} + 502 \beta_{13} + 115 \beta_{12} + 317 \beta_{11} + 229 \beta_{10} - 96 \beta_{9} - 333 \beta_{8} + 313 \beta_{7} + 337 \beta_{6} + 302 \beta_{5} - 252 \beta_{4} - 79 \beta_{3} + 310 \beta_{2} - 247 \beta_{1} + 1198\)
\(\nu^{7}\)\(=\)\(-31 \beta_{14} + 239 \beta_{13} - 1254 \beta_{12} - 346 \beta_{11} - 562 \beta_{10} - 432 \beta_{9} - 10 \beta_{8} - 849 \beta_{7} - 229 \beta_{6} + 18 \beta_{5} - 414 \beta_{4} - 35 \beta_{3} + 99 \beta_{2} + 2518 \beta_{1} - 180\)
\(\nu^{8}\)\(=\)\(-9097 \beta_{14} + 7967 \beta_{13} + 2088 \beta_{12} + 5193 \beta_{11} + 3576 \beta_{10} - 888 \beta_{9} - 5611 \beta_{8} + 5351 \beta_{7} + 5729 \beta_{6} + 5021 \beta_{5} - 3952 \beta_{4} - 1609 \beta_{3} + 4887 \beta_{2} - 3873 \beta_{1} + 17955\)
\(\nu^{9}\)\(=\)\(-619 \beta_{14} + 5095 \beta_{13} - 20805 \beta_{12} - 5891 \beta_{11} - 10625 \beta_{10} - 8055 \beta_{9} - 308 \beta_{8} - 15316 \beta_{7} - 5067 \beta_{6} + 61 \beta_{5} - 6877 \beta_{4} - 750 \beta_{3} + 1621 \beta_{2} + 39150 \beta_{1} - 2196\)
\(\nu^{10}\)\(=\)\(-149357 \beta_{14} + 127959 \beta_{13} + 35595 \beta_{12} + 84969 \beta_{11} + 56709 \beta_{10} - 7087 \beta_{9} - 93089 \beta_{8} + 90567 \beta_{7} + 95731 \beta_{6} + 83612 \beta_{5} - 62019 \beta_{4} - 30076 \beta_{3} + 76569 \beta_{2} - 62051 \beta_{1} + 277897\)
\(\nu^{11}\)\(=\)\(-10176 \beta_{14} + 95830 \beta_{13} - 346912 \beta_{12} - 100736 \beta_{11} - 192217 \beta_{10} - 145094 \beta_{9} - 6660 \beta_{8} - 270979 \beta_{7} - 101706 \beta_{6} - 5067 \beta_{5} - 110424 \beta_{4} - 13520 \beta_{3} + 21534 \beta_{2} + 622037 \beta_{1} - 31788\)
\(\nu^{12}\)\(=\)\(-2460611 \beta_{14} + 2074863 \beta_{13} + 595649 \beta_{12} + 1393860 \beta_{11} + 908995 \beta_{10} - 28519 \beta_{9} - 1536904 \beta_{8} + 1524931 \beta_{7} + 1589921 \beta_{6} + 1395762 \beta_{5} - 977347 \beta_{4} - 540506 \beta_{3} + 1203101 \beta_{2} - 1013593 \beta_{1} + 4386294\)
\(\nu^{13}\)\(=\)\(-143953 \beta_{14} + 1697102 \beta_{13} - 5809478 \beta_{12} - 1731649 \beta_{11} - 3399748 \beta_{10} - 2552103 \beta_{9} - 122438 \beta_{8} - 4738578 \beta_{7} - 1938971 \beta_{6} - 196689 \beta_{5} - 1745788 \beta_{4} - 226830 \beta_{3} + 231759 \beta_{2} + 10022933 \beta_{1} - 564758\)
\(\nu^{14}\)\(=\)\(-40658804 \beta_{14} + 33887265 \beta_{13} + 9929787 \beta_{12} + 22941355 \beta_{11} + 14689662 \beta_{10} + 623231 \beta_{9} - 25363596 \beta_{8} + 25613853 \beta_{7} + 26363690 \beta_{6} + 23335400 \beta_{5} - 15489614 \beta_{4} - 9506066 \beta_{3} + 19015737 \beta_{2} - 16809082 \beta_{1} + 70152806\)

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
4.06882
3.77963
2.75355
2.43337
2.05101
1.17201
0.676512
0.113262
−0.656329
−1.34829
−1.40985
−1.97610
−3.01123
−3.53778
−4.10857
1.00000 −1.00000 1.00000 −4.06882 −1.00000 4.31915 1.00000 1.00000 −4.06882
1.2 1.00000 −1.00000 1.00000 −3.77963 −1.00000 2.13919 1.00000 1.00000 −3.77963
1.3 1.00000 −1.00000 1.00000 −2.75355 −1.00000 0.824838 1.00000 1.00000 −2.75355
1.4 1.00000 −1.00000 1.00000 −2.43337 −1.00000 −2.53842 1.00000 1.00000 −2.43337
1.5 1.00000 −1.00000 1.00000 −2.05101 −1.00000 −1.63593 1.00000 1.00000 −2.05101
1.6 1.00000 −1.00000 1.00000 −1.17201 −1.00000 −4.40989 1.00000 1.00000 −1.17201
1.7 1.00000 −1.00000 1.00000 −0.676512 −1.00000 −2.72085 1.00000 1.00000 −0.676512
1.8 1.00000 −1.00000 1.00000 −0.113262 −1.00000 1.75034 1.00000 1.00000 −0.113262
1.9 1.00000 −1.00000 1.00000 0.656329 −1.00000 2.89208 1.00000 1.00000 0.656329
1.10 1.00000 −1.00000 1.00000 1.34829 −1.00000 −1.25683 1.00000 1.00000 1.34829
1.11 1.00000 −1.00000 1.00000 1.40985 −1.00000 4.86981 1.00000 1.00000 1.40985
1.12 1.00000 −1.00000 1.00000 1.97610 −1.00000 4.65524 1.00000 1.00000 1.97610
1.13 1.00000 −1.00000 1.00000 3.01123 −1.00000 −0.871929 1.00000 1.00000 3.01123
1.14 1.00000 −1.00000 1.00000 3.53778 −1.00000 −4.09818 1.00000 1.00000 3.53778
1.15 1.00000 −1.00000 1.00000 4.10857 −1.00000 1.08136 1.00000 1.00000 4.10857
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(1\)

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 8034.2.a.bc 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.bc 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{15} + \cdots\)
\(T_{7}^{15} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{15} \)
$3$ \( ( 1 + T )^{15} \)
$5$ \( 1 + T + 27 T^{2} + 26 T^{3} + 377 T^{4} + 342 T^{5} + 3655 T^{6} + 3138 T^{7} + 28216 T^{8} + 22948 T^{9} + 187699 T^{10} + 141443 T^{11} + 1119809 T^{12} + 769025 T^{13} + 6093660 T^{14} + 3911802 T^{15} + 30468300 T^{16} + 19225625 T^{17} + 139976125 T^{18} + 88401875 T^{19} + 586559375 T^{20} + 358562500 T^{21} + 2204375000 T^{22} + 1225781250 T^{23} + 7138671875 T^{24} + 3339843750 T^{25} + 18408203125 T^{26} + 6347656250 T^{27} + 32958984375 T^{28} + 6103515625 T^{29} + 30517578125 T^{30} \)
$7$ \( 1 - 5 T + 49 T^{2} - 215 T^{3} + 1236 T^{4} - 4820 T^{5} + 21339 T^{6} - 74814 T^{7} + 284031 T^{8} - 909247 T^{9} + 3106568 T^{10} - 9211198 T^{11} + 28913157 T^{12} - 80060177 T^{13} + 232747856 T^{14} - 602085894 T^{15} + 1629234992 T^{16} - 3922948673 T^{17} + 9917212851 T^{18} - 22116086398 T^{19} + 52212088376 T^{20} - 106972000303 T^{21} + 233911741833 T^{22} - 431287822014 T^{23} + 861105619773 T^{24} - 1361530700180 T^{25} + 2443975854348 T^{26} - 2975876748215 T^{27} + 4747561509943 T^{28} - 3391115364245 T^{29} + 4747561509943 T^{30} \)
$11$ \( 1 - 3 T + 65 T^{2} - 229 T^{3} + 2454 T^{4} - 9236 T^{5} + 66724 T^{6} - 253618 T^{7} + 1437743 T^{8} - 5299587 T^{9} + 25561632 T^{10} - 89155397 T^{11} + 383776013 T^{12} - 1250211022 T^{13} + 4919450306 T^{14} - 14878517304 T^{15} + 54113953366 T^{16} - 151275533662 T^{17} + 510805873303 T^{18} - 1305324167477 T^{19} + 4116726395232 T^{20} - 9388541645307 T^{21} + 28017543695053 T^{22} - 54365270681458 T^{23} + 157331701734284 T^{24} - 239558053614836 T^{25} + 700154839679394 T^{26} - 718700098269109 T^{27} + 2243976289355515 T^{28} - 1139249500749723 T^{29} + 4177248169415651 T^{30} \)
$13$ \( ( 1 + T )^{15} \)
$17$ \( 1 + 2 T + 115 T^{2} + 191 T^{3} + 6839 T^{4} + 10981 T^{5} + 281640 T^{6} + 481747 T^{7} + 8943937 T^{8} + 16838549 T^{9} + 232091030 T^{10} + 475845789 T^{11} + 5117480084 T^{12} + 10948036324 T^{13} + 98395671890 T^{14} + 205762249634 T^{15} + 1672726422130 T^{16} + 3163982497636 T^{17} + 25142179652692 T^{18} + 39743116143069 T^{19} + 329536073582710 T^{20} + 406441638347381 T^{21} + 3670043239975601 T^{22} + 3360550219929427 T^{23} + 33399089536615080 T^{24} + 22137629020830469 T^{25} + 234385498847902087 T^{26} + 111280847310884351 T^{27} + 1139026473784182755 T^{28} + 336755653118801858 T^{29} + 2862423051509815793 T^{30} \)
$19$ \( 1 - 8 T + 125 T^{2} - 796 T^{3} + 7656 T^{4} - 40392 T^{5} + 300894 T^{6} - 1396933 T^{7} + 8967100 T^{8} - 38858280 T^{9} + 229462898 T^{10} - 978014112 T^{11} + 5420517659 T^{12} - 22734357256 T^{13} + 117061484835 T^{14} - 466358709022 T^{15} + 2224168211865 T^{16} - 8207102969416 T^{17} + 37179330623081 T^{18} - 127455777089952 T^{19} + 568172852274902 T^{20} - 1828122016744680 T^{21} + 8015437270786900 T^{22} - 23724899669553253 T^{23} + 97094792135514426 T^{24} - 247646028285097992 T^{25} + 891849422124764664 T^{26} - 1761798675576664156 T^{27} + 5256622932782132375 T^{28} - 6392053486263072968 T^{29} + 15181127029874798299 T^{30} \)
$23$ \( 1 - 3 T + 118 T^{2} - 194 T^{3} + 6965 T^{4} - 717 T^{5} + 286915 T^{6} + 361421 T^{7} + 9973092 T^{8} + 21675692 T^{9} + 317713852 T^{10} + 795158967 T^{11} + 9260638682 T^{12} + 22648428516 T^{13} + 242494067671 T^{14} + 553405977852 T^{15} + 5577363556433 T^{16} + 11981018684964 T^{17} + 112674190843894 T^{18} + 222518080484247 T^{19} + 2044915327323236 T^{20} + 3208780334910188 T^{21} + 33956637426872124 T^{22} + 28303234611244301 T^{23} + 516777715863656645 T^{24} - 29702808540186333 T^{25} + 6636319963870501555 T^{26} - 4251437139811942274 T^{27} + 59476290708503151194 T^{28} - 34778508973616249427 T^{29} + \)\(26\!\cdots\!07\)\( T^{30} \)
$29$ \( 1 - 26 T + 492 T^{2} - 6878 T^{3} + 82755 T^{4} - 859175 T^{5} + 8093356 T^{6} - 69368335 T^{7} + 553881683 T^{8} - 4125364715 T^{9} + 29034620693 T^{10} - 193116487897 T^{11} + 1223332281964 T^{12} - 7373929885144 T^{13} + 42510010099452 T^{14} - 233928549023228 T^{15} + 1232790292884108 T^{16} - 6201475033406104 T^{17} + 29835851024819996 T^{18} - 136587622676278057 T^{19} + 595533431192606257 T^{20} - 2453863140112518515 T^{21} + 9554390521570748047 T^{22} - 34701260756826989935 T^{23} + \)\(11\!\cdots\!64\)\( T^{24} - \)\(36\!\cdots\!75\)\( T^{25} + \)\(10\!\cdots\!95\)\( T^{26} - \)\(24\!\cdots\!98\)\( T^{27} + \)\(50\!\cdots\!88\)\( T^{28} - \)\(77\!\cdots\!06\)\( T^{29} + \)\(86\!\cdots\!49\)\( T^{30} \)
$31$ \( 1 + 171 T^{2} - 248 T^{3} + 15492 T^{4} - 35834 T^{5} + 1030848 T^{6} - 2695925 T^{7} + 55435144 T^{8} - 147303946 T^{9} + 2501133760 T^{10} - 6533582356 T^{11} + 97825130111 T^{12} - 246577142588 T^{13} + 3388706218081 T^{14} - 8138732000126 T^{15} + 105049892760511 T^{16} - 236960634027068 T^{17} + 2914308451136801 T^{18} - 6033900510995476 T^{19} + 71605336086237760 T^{20} - 130732794300825226 T^{21} + 1525165725059716984 T^{22} - 2299330270113127925 T^{23} + 27255231625083379008 T^{24} - 29370560035670023034 T^{25} + \)\(39\!\cdots\!52\)\( T^{26} - \)\(19\!\cdots\!28\)\( T^{27} + \)\(41\!\cdots\!61\)\( T^{28} + \)\(23\!\cdots\!51\)\( T^{30} \)
$37$ \( 1 - 25 T + 637 T^{2} - 10733 T^{3} + 168993 T^{4} - 2195276 T^{5} + 26543409 T^{6} - 283850250 T^{7} + 2843128342 T^{8} - 25957673047 T^{9} + 223327355434 T^{10} - 1778267126925 T^{11} + 13400103358884 T^{12} - 94198114875268 T^{13} + 628189236579148 T^{14} - 3921599955222616 T^{15} + 23243001753428476 T^{16} - 128957219264241892 T^{17} + 678755435437551252 T^{18} - 3332758896864884925 T^{19} + 15486402532139012338 T^{20} - 66600287252875398223 T^{21} + \)\(26\!\cdots\!86\)\( T^{22} - \)\(99\!\cdots\!50\)\( T^{23} + \)\(34\!\cdots\!93\)\( T^{24} - \)\(10\!\cdots\!24\)\( T^{25} + \)\(30\!\cdots\!09\)\( T^{26} - \)\(70\!\cdots\!73\)\( T^{27} + \)\(15\!\cdots\!89\)\( T^{28} - \)\(22\!\cdots\!25\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} \)
$41$ \( 1 + T + 274 T^{2} + 411 T^{3} + 39122 T^{4} + 64747 T^{5} + 3941083 T^{6} + 6497047 T^{7} + 311528655 T^{8} + 502102102 T^{9} + 20299643915 T^{10} + 31599706679 T^{11} + 1120377565252 T^{12} + 1656373409302 T^{13} + 53116320707002 T^{14} + 73564007527758 T^{15} + 2177769148987082 T^{16} + 2784363701036662 T^{17} + 77217542174733092 T^{18} + 89293218744957719 T^{19} + 2351839625644666915 T^{20} + 2385037324125214582 T^{21} + 60671536997649560055 T^{22} + 51878434505084905687 T^{23} + \)\(12\!\cdots\!63\)\( T^{24} + \)\(86\!\cdots\!47\)\( T^{25} + \)\(21\!\cdots\!02\)\( T^{26} + \)\(92\!\cdots\!91\)\( T^{27} + \)\(25\!\cdots\!54\)\( T^{28} + \)\(37\!\cdots\!61\)\( T^{29} + \)\(15\!\cdots\!01\)\( T^{30} \)
$43$ \( 1 - 10 T + 404 T^{2} - 3980 T^{3} + 82520 T^{4} - 779132 T^{5} + 11211210 T^{6} - 99954811 T^{7} + 1127769859 T^{8} - 9402978727 T^{9} + 88728742215 T^{10} - 686546499866 T^{11} + 5629398959891 T^{12} - 40134130897355 T^{13} + 292917365393374 T^{14} - 1908635853771102 T^{15} + 12595446711915082 T^{16} - 74208008029209395 T^{17} + 447576623104053737 T^{18} - 2347165860288380666 T^{19} + 13043874242375521245 T^{20} - 59439642275390858623 T^{21} + \)\(30\!\cdots\!13\)\( T^{22} - \)\(11\!\cdots\!11\)\( T^{23} + \)\(56\!\cdots\!30\)\( T^{24} - \)\(16\!\cdots\!68\)\( T^{25} + \)\(76\!\cdots\!40\)\( T^{26} - \)\(15\!\cdots\!80\)\( T^{27} + \)\(69\!\cdots\!72\)\( T^{28} - \)\(73\!\cdots\!90\)\( T^{29} + \)\(31\!\cdots\!07\)\( T^{30} \)
$47$ \( 1 + 3 T + 235 T^{2} + 855 T^{3} + 31094 T^{4} + 103386 T^{5} + 2822261 T^{6} + 7409638 T^{7} + 190245714 T^{8} + 275690210 T^{9} + 10065437023 T^{10} - 2792300935 T^{11} + 442737001354 T^{12} - 1062516621220 T^{13} + 18503391215176 T^{14} - 70326469147170 T^{15} + 869659387113272 T^{16} - 2347099216274980 T^{17} + 45966283691576342 T^{18} - 13625537818801735 T^{19} + 2308457724497994161 T^{20} + 2971724137687229090 T^{21} + 96382877281391445582 T^{22} + \)\(17\!\cdots\!18\)\( T^{23} + \)\(31\!\cdots\!87\)\( T^{24} + \)\(54\!\cdots\!14\)\( T^{25} + \)\(76\!\cdots\!82\)\( T^{26} + \)\(99\!\cdots\!55\)\( T^{27} + \)\(12\!\cdots\!45\)\( T^{28} + \)\(77\!\cdots\!07\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \)
$53$ \( 1 - 13 T + 583 T^{2} - 6905 T^{3} + 167825 T^{4} - 1800869 T^{5} + 31358538 T^{6} - 305078001 T^{7} + 4237425741 T^{8} - 37424788657 T^{9} + 437822971446 T^{10} - 3511297865535 T^{11} + 35696212815386 T^{12} - 259551311750413 T^{13} + 2337210583998580 T^{14} - 15345611632581678 T^{15} + 123872160951924740 T^{16} - 729079634706910117 T^{17} + 5314345075316221522 T^{18} - 27705829093344472335 T^{19} + \)\(18\!\cdots\!78\)\( T^{20} - \)\(82\!\cdots\!53\)\( T^{21} + \)\(49\!\cdots\!17\)\( T^{22} - \)\(18\!\cdots\!61\)\( T^{23} + \)\(10\!\cdots\!54\)\( T^{24} - \)\(31\!\cdots\!81\)\( T^{25} + \)\(15\!\cdots\!25\)\( T^{26} - \)\(33\!\cdots\!05\)\( T^{27} + \)\(15\!\cdots\!59\)\( T^{28} - \)\(17\!\cdots\!97\)\( T^{29} + \)\(73\!\cdots\!57\)\( T^{30} \)
$59$ \( 1 - 28 T + 928 T^{2} - 16863 T^{3} + 327017 T^{4} - 4457648 T^{5} + 64017165 T^{6} - 701394844 T^{7} + 8203485572 T^{8} - 75531633997 T^{9} + 764634024596 T^{10} - 6152604967098 T^{11} + 56753724151169 T^{12} - 416284794875227 T^{13} + 3649391031165450 T^{14} - 25404790897055438 T^{15} + 215314070838761550 T^{16} - 1449087370960665187 T^{17} + 11656023112442938051 T^{18} - 74553335476719588378 T^{19} + \)\(54\!\cdots\!04\)\( T^{20} - \)\(31\!\cdots\!77\)\( T^{21} + \)\(20\!\cdots\!68\)\( T^{22} - \)\(10\!\cdots\!24\)\( T^{23} + \)\(55\!\cdots\!35\)\( T^{24} - \)\(22\!\cdots\!48\)\( T^{25} + \)\(98\!\cdots\!03\)\( T^{26} - \)\(30\!\cdots\!03\)\( T^{27} + \)\(97\!\cdots\!12\)\( T^{28} - \)\(17\!\cdots\!08\)\( T^{29} + \)\(36\!\cdots\!99\)\( T^{30} \)
$61$ \( 1 - 22 T + 644 T^{2} - 11036 T^{3} + 196600 T^{4} - 2765294 T^{5} + 38169654 T^{6} - 459020073 T^{7} + 5341068529 T^{8} - 56570472501 T^{9} + 578065233341 T^{10} - 5503410184670 T^{11} + 50560085586603 T^{12} - 438296586915695 T^{13} + 3669732916010368 T^{14} - 29157690683995002 T^{15} + 223853707876632448 T^{16} - 1630901599913301095 T^{17} + 11476178786532735543 T^{18} - 76199342374721457470 T^{19} + \)\(48\!\cdots\!41\)\( T^{20} - \)\(29\!\cdots\!61\)\( T^{21} + \)\(16\!\cdots\!09\)\( T^{22} - \)\(87\!\cdots\!13\)\( T^{23} + \)\(44\!\cdots\!14\)\( T^{24} - \)\(19\!\cdots\!94\)\( T^{25} + \)\(85\!\cdots\!00\)\( T^{26} - \)\(29\!\cdots\!56\)\( T^{27} + \)\(10\!\cdots\!64\)\( T^{28} - \)\(21\!\cdots\!02\)\( T^{29} + \)\(60\!\cdots\!01\)\( T^{30} \)
$67$ \( 1 - 29 T + 899 T^{2} - 16475 T^{3} + 306532 T^{4} - 4232197 T^{5} + 59579872 T^{6} - 673192766 T^{7} + 7916505077 T^{8} - 77617882877 T^{9} + 813861050007 T^{10} - 7257358894729 T^{11} + 70681960181205 T^{12} - 588519867282757 T^{13} + 5387864066877213 T^{14} - 42062755847947284 T^{15} + 360986892480773271 T^{16} - 2641865684232296173 T^{17} + 21258518389979759415 T^{18} - \)\(14\!\cdots\!09\)\( T^{19} + \)\(10\!\cdots\!49\)\( T^{20} - \)\(70\!\cdots\!13\)\( T^{21} + \)\(47\!\cdots\!71\)\( T^{22} - \)\(27\!\cdots\!06\)\( T^{23} + \)\(16\!\cdots\!84\)\( T^{24} - \)\(77\!\cdots\!53\)\( T^{25} + \)\(37\!\cdots\!56\)\( T^{26} - \)\(13\!\cdots\!75\)\( T^{27} + \)\(49\!\cdots\!13\)\( T^{28} - \)\(10\!\cdots\!41\)\( T^{29} + \)\(24\!\cdots\!43\)\( T^{30} \)
$71$ \( 1 - 18 T + 786 T^{2} - 12246 T^{3} + 300544 T^{4} - 4154590 T^{5} + 74405584 T^{6} - 923967185 T^{7} + 13335847710 T^{8} - 149752123327 T^{9} + 1831680494665 T^{10} - 18652367064118 T^{11} + 199025978213365 T^{12} - 1837570173972345 T^{13} + 17411928699927097 T^{14} - 145299914209922134 T^{15} + 1236246937694823887 T^{16} - 9263191246994591145 T^{17} + 71233586888323680515 T^{18} - \)\(47\!\cdots\!58\)\( T^{19} + \)\(33\!\cdots\!15\)\( T^{20} - \)\(19\!\cdots\!67\)\( T^{21} + \)\(12\!\cdots\!10\)\( T^{22} - \)\(59\!\cdots\!85\)\( T^{23} + \)\(34\!\cdots\!04\)\( T^{24} - \)\(13\!\cdots\!90\)\( T^{25} + \)\(69\!\cdots\!24\)\( T^{26} - \)\(20\!\cdots\!86\)\( T^{27} + \)\(91\!\cdots\!46\)\( T^{28} - \)\(14\!\cdots\!58\)\( T^{29} + \)\(58\!\cdots\!51\)\( T^{30} \)
$73$ \( 1 - 23 T + 714 T^{2} - 13064 T^{3} + 256630 T^{4} - 3875083 T^{5} + 59775379 T^{6} - 779527221 T^{7} + 10167980773 T^{8} - 117350588257 T^{9} + 1344637744889 T^{10} - 13953210591736 T^{11} + 143306121500168 T^{12} - 1350456422408969 T^{13} + 12579042289107098 T^{14} - 108104577844055246 T^{15} + 918270087104818154 T^{16} - 7196582275017395801 T^{17} + 55748517467630854856 T^{18} - \)\(39\!\cdots\!76\)\( T^{19} + \)\(27\!\cdots\!77\)\( T^{20} - \)\(17\!\cdots\!73\)\( T^{21} + \)\(11\!\cdots\!81\)\( T^{22} - \)\(62\!\cdots\!01\)\( T^{23} + \)\(35\!\cdots\!27\)\( T^{24} - \)\(16\!\cdots\!67\)\( T^{25} + \)\(80\!\cdots\!10\)\( T^{26} - \)\(29\!\cdots\!44\)\( T^{27} + \)\(11\!\cdots\!62\)\( T^{28} - \)\(28\!\cdots\!07\)\( T^{29} + \)\(89\!\cdots\!57\)\( T^{30} \)
$79$ \( 1 - 27 T + 910 T^{2} - 18404 T^{3} + 380561 T^{4} - 6259264 T^{5} + 100308501 T^{6} - 1406356045 T^{7} + 18991686269 T^{8} - 233831106522 T^{9} + 2768926692491 T^{10} - 30504586984480 T^{11} + 323420177010880 T^{12} - 3222825558556267 T^{13} + 30922538909939857 T^{14} - 280074029752122718 T^{15} + 2442880573885248703 T^{16} - 20113654310949662347 T^{17} + \)\(15\!\cdots\!20\)\( T^{18} - \)\(11\!\cdots\!80\)\( T^{19} + \)\(85\!\cdots\!09\)\( T^{20} - \)\(56\!\cdots\!62\)\( T^{21} + \)\(36\!\cdots\!71\)\( T^{22} - \)\(21\!\cdots\!45\)\( T^{23} + \)\(12\!\cdots\!19\)\( T^{24} - \)\(59\!\cdots\!64\)\( T^{25} + \)\(28\!\cdots\!19\)\( T^{26} - \)\(10\!\cdots\!64\)\( T^{27} + \)\(42\!\cdots\!90\)\( T^{28} - \)\(99\!\cdots\!87\)\( T^{29} + \)\(29\!\cdots\!99\)\( T^{30} \)
$83$ \( 1 - 7 T + 803 T^{2} - 6078 T^{3} + 320913 T^{4} - 2495885 T^{5} + 84851115 T^{6} - 653557372 T^{7} + 16575535401 T^{8} - 123233920374 T^{9} + 2528376058709 T^{10} - 17803860830837 T^{11} + 310540724025273 T^{12} - 2037881282757994 T^{13} + 31235525263958377 T^{14} - 187948737106099098 T^{15} + 2592548596908545291 T^{16} - 14038964156919820666 T^{17} + \)\(17\!\cdots\!51\)\( T^{18} - \)\(84\!\cdots\!77\)\( T^{19} + \)\(99\!\cdots\!87\)\( T^{20} - \)\(40\!\cdots\!06\)\( T^{21} + \)\(44\!\cdots\!27\)\( T^{22} - \)\(14\!\cdots\!52\)\( T^{23} + \)\(15\!\cdots\!45\)\( T^{24} - \)\(38\!\cdots\!65\)\( T^{25} + \)\(41\!\cdots\!71\)\( T^{26} - \)\(64\!\cdots\!58\)\( T^{27} + \)\(71\!\cdots\!89\)\( T^{28} - \)\(51\!\cdots\!03\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} \)
$89$ \( 1 - 35 T + 1232 T^{2} - 27319 T^{3} + 587810 T^{4} - 9969660 T^{5} + 165094388 T^{6} - 2331344325 T^{7} + 32445012640 T^{8} - 399770682470 T^{9} + 4895813405688 T^{10} - 54196917574773 T^{11} + 600693274693670 T^{12} - 6091283663324139 T^{13} + 62174942885772339 T^{14} - 584045746436932254 T^{15} + 5533569916833738171 T^{16} - 48249057897190505019 T^{17} + \)\(42\!\cdots\!30\)\( T^{18} - \)\(34\!\cdots\!93\)\( T^{19} + \)\(27\!\cdots\!12\)\( T^{20} - \)\(19\!\cdots\!70\)\( T^{21} + \)\(14\!\cdots\!60\)\( T^{22} - \)\(91\!\cdots\!25\)\( T^{23} + \)\(57\!\cdots\!92\)\( T^{24} - \)\(31\!\cdots\!60\)\( T^{25} + \)\(16\!\cdots\!90\)\( T^{26} - \)\(67\!\cdots\!99\)\( T^{27} + \)\(27\!\cdots\!08\)\( T^{28} - \)\(68\!\cdots\!35\)\( T^{29} + \)\(17\!\cdots\!49\)\( T^{30} \)
$97$ \( 1 - 19 T + 1013 T^{2} - 16060 T^{3} + 496672 T^{4} - 6871349 T^{5} + 158516831 T^{6} - 1956969685 T^{7} + 36984580873 T^{8} - 412417120207 T^{9} + 6688966113246 T^{10} - 67778641389360 T^{11} + 968687130938725 T^{12} - 8935899536293225 T^{13} + 114351538528897631 T^{14} - 958724353660235134 T^{15} + 11092099237303070207 T^{16} - 84077878736982954025 T^{17} + \)\(88\!\cdots\!25\)\( T^{18} - \)\(60\!\cdots\!60\)\( T^{19} + \)\(57\!\cdots\!22\)\( T^{20} - \)\(34\!\cdots\!03\)\( T^{21} + \)\(29\!\cdots\!49\)\( T^{22} - \)\(15\!\cdots\!85\)\( T^{23} + \)\(12\!\cdots\!27\)\( T^{24} - \)\(50\!\cdots\!01\)\( T^{25} + \)\(35\!\cdots\!16\)\( T^{26} - \)\(11\!\cdots\!60\)\( T^{27} + \)\(68\!\cdots\!01\)\( T^{28} - \)\(12\!\cdots\!11\)\( T^{29} + \)\(63\!\cdots\!93\)\( T^{30} \)
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