Properties

Label 8034.2.a.z
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-3341759347 \nu^{13} + 518140367105 \nu^{12} - 1741789540612 \nu^{11} - 17247628357060 \nu^{10} + 62687524315698 \nu^{9} + 176417731545861 \nu^{8} - 670560099753944 \nu^{7} - 742563932810444 \nu^{6} + 2912651300509245 \nu^{5} + 1186311372117602 \nu^{4} - 5100601073725863 \nu^{3} - 235038449926693 \nu^{2} + 2735906855139534 \nu - 592135085106752\)\()/ 30448521494580 \)
\(\beta_{3}\)\(=\)\((\)\(-12670008395 \nu^{13} + 121522864993 \nu^{12} + 241219668796 \nu^{11} - 4283106154742 \nu^{10} + 1787398276572 \nu^{9} + 49208409954315 \nu^{8} - 43074872724922 \nu^{7} - 252692469808438 \nu^{6} + 223588882118493 \nu^{5} + 575804728969528 \nu^{4} - 446376121324599 \nu^{3} - 429175712907053 \nu^{2} + 305776841980152 \nu - 7510828872730\)\()/ 15224260747290 \)
\(\beta_{4}\)\(=\)\((\)\(7437129421 \nu^{13} + 8250683722 \nu^{12} - 328982202812 \nu^{11} - 355914392252 \nu^{10} + 5237503644027 \nu^{9} + 5955785366799 \nu^{8} - 36307339325161 \nu^{7} - 45172405213102 \nu^{6} + 103511839862979 \nu^{5} + 143064053475349 \nu^{4} - 94789427122899 \nu^{3} - 160392768597467 \nu^{2} + 16523568578403 \nu + 41261138165888\)\()/ 7612130373645 \)
\(\beta_{5}\)\(=\)\((\)\(16735095548 \nu^{13} - 59503252771 \nu^{12} - 534709503235 \nu^{11} + 2013117619607 \nu^{10} + 4893999455973 \nu^{9} - 20391279195918 \nu^{8} - 13884105134675 \nu^{7} + 81401298589306 \nu^{6} - 20121761021580 \nu^{5} - 121000025836105 \nu^{4} + 146667384042234 \nu^{3} + 22338499957991 \nu^{2} - 164021431732794 \nu + 67234006456384\)\()/ 7612130373645 \)
\(\beta_{6}\)\(=\)\((\)\(6829796404 \nu^{13} - 24088984468 \nu^{12} - 220735533930 \nu^{11} + 818192200816 \nu^{10} + 2167385042979 \nu^{9} - 8331580582394 \nu^{8} - 9159648834785 \nu^{7} + 33626274944798 \nu^{6} + 19719493652720 \nu^{5} - 50057127693710 \nu^{4} - 27525433033153 \nu^{3} + 16363934034728 \nu^{2} + 18990965483608 \nu - 696096892188\)\()/ 2537376791215 \)
\(\beta_{7}\)\(=\)\((\)\(101168043581 \nu^{13} - 428736219223 \nu^{12} - 3096334884004 \nu^{11} + 14443924013984 \nu^{10} + 24846996819930 \nu^{9} - 146139006849039 \nu^{8} - 37437704187596 \nu^{7} + 574313684971036 \nu^{6} - 284435428081239 \nu^{5} - 772353790449958 \nu^{4} + 1021694063716269 \nu^{3} + 18632735153039 \nu^{2} - 873824979116130 \nu + 321698456894260\)\()/ 30448521494580 \)
\(\beta_{8}\)\(=\)\((\)\(132195076697 \nu^{13} - 378058949095 \nu^{12} - 4698358621144 \nu^{11} + 13089742387748 \nu^{10} + 54759751083666 \nu^{9} - 137259949634175 \nu^{8} - 280935521431316 \nu^{7} + 563752133186128 \nu^{6} + 636877173580161 \nu^{5} - 853332240720538 \nu^{4} - 538687991427303 \nu^{3} + 267983097437951 \nu^{2} + 72415118017302 \nu + 93265012848748\)\()/ 30448521494580 \)
\(\beta_{9}\)\(=\)\((\)\(37454660035 \nu^{13} - 65377606061 \nu^{12} - 1543899870542 \nu^{11} + 2517262146709 \nu^{10} + 22989677068086 \nu^{9} - 31415177667555 \nu^{8} - 162281355426796 \nu^{7} + 170536386071396 \nu^{6} + 566623348152159 \nu^{5} - 440040849178856 \nu^{4} - 910338490594662 \nu^{3} + 529906004653921 \nu^{2} + 498400052484456 \nu - 240724229362315\)\()/ 7612130373645 \)
\(\beta_{10}\)\(=\)\((\)\(61385752497 \nu^{13} - 224068680263 \nu^{12} - 2016688298880 \nu^{11} + 7666899083920 \nu^{10} + 20353862368234 \nu^{9} - 80361026660595 \nu^{8} - 88069692253324 \nu^{7} + 350770178370856 \nu^{6} + 171611111684913 \nu^{5} - 659138629686466 \nu^{4} - 144254361560619 \nu^{3} + 482165035857659 \nu^{2} + 23740354322750 \nu - 113491089496292\)\()/ 10149507164860 \)
\(\beta_{11}\)\(=\)\((\)\(61385752497 \nu^{13} - 224068680263 \nu^{12} - 2016688298880 \nu^{11} + 7666899083920 \nu^{10} + 20353862368234 \nu^{9} - 80361026660595 \nu^{8} - 88069692253324 \nu^{7} + 350770178370856 \nu^{6} + 171611111684913 \nu^{5} - 659138629686466 \nu^{4} - 144254361560619 \nu^{3} + 472015528692799 \nu^{2} + 23740354322750 \nu - 52594046507132\)\()/ 10149507164860 \)
\(\beta_{12}\)\(=\)\((\)\(47799908567 \nu^{13} - 114887884225 \nu^{12} - 1783176710104 \nu^{11} + 4177793518358 \nu^{10} + 22844790904641 \nu^{9} - 47813470495515 \nu^{8} - 135044548880321 \nu^{7} + 231962318420908 \nu^{6} + 378111654620451 \nu^{5} - 496358883141133 \nu^{4} - 434168909972403 \nu^{3} + 413593508460356 \nu^{2} + 106258840381992 \nu - 91513305290492\)\()/ 7612130373645 \)
\(\beta_{13}\)\(=\)\((\)\(-120974609621 \nu^{13} + 303334971739 \nu^{12} + 4434352237832 \nu^{11} - 10647593900116 \nu^{10} - 55483675391710 \nu^{9} + 114757728296943 \nu^{8} + 325219663165232 \nu^{7} - 513376507176776 \nu^{6} - 932125820605101 \nu^{5} + 1007892134519590 \nu^{4} + 1197390391324439 \nu^{3} - 853786741692003 \nu^{2} - 506871319390622 \nu + 280944669298364\)\()/ 10149507164860 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + \beta_{10} + 6\)
\(\nu^{3}\)\(=\)\(3 \beta_{13} + 3 \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 10 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{13} - 4 \beta_{12} - 19 \beta_{11} + 16 \beta_{10} + \beta_{9} + 4 \beta_{8} - 6 \beta_{7} + 3 \beta_{6} + 8 \beta_{5} - 4 \beta_{3} - 4 \beta_{1} + 71\)
\(\nu^{5}\)\(=\)\(60 \beta_{13} + 61 \beta_{12} + 53 \beta_{11} + 7 \beta_{10} + 21 \beta_{9} - 41 \beta_{8} + 41 \beta_{7} + \beta_{6} - 49 \beta_{5} + 46 \beta_{4} + 39 \beta_{3} - 2 \beta_{2} + 137 \beta_{1} - 54\)
\(\nu^{6}\)\(=\)\(-59 \beta_{13} - 126 \beta_{12} - 354 \beta_{11} + 250 \beta_{10} + 15 \beta_{9} + 101 \beta_{8} - 148 \beta_{7} + 64 \beta_{6} + 212 \beta_{5} - 30 \beta_{4} - 105 \beta_{3} - 151 \beta_{1} + 1064\)
\(\nu^{7}\)\(=\)\(1048 \beta_{13} + 1103 \beta_{12} + 1135 \beta_{11} - 76 \beta_{10} + 365 \beta_{9} - 772 \beta_{8} + 788 \beta_{7} + 3 \beta_{6} - 981 \beta_{5} + 844 \beta_{4} + 701 \beta_{3} - 47 \beta_{2} + 2167 \beta_{1} - 1613\)
\(\nu^{8}\)\(=\)\(-1728 \beta_{13} - 2966 \beta_{12} - 6601 \beta_{11} + 4024 \beta_{10} + 101 \beta_{9} + 2173 \beta_{8} - 3046 \beta_{7} + 1168 \beta_{6} + 4462 \beta_{5} - 1070 \beta_{4} - 2280 \beta_{3} + 16 \beta_{2} - 3982 \beta_{1} + 17581\)
\(\nu^{9}\)\(=\)\(18247 \beta_{13} + 19912 \beta_{12} + 23085 \beta_{11} - 4313 \beta_{10} + 6128 \beta_{9} - 14338 \beta_{8} + 15014 \beta_{7} - 420 \beta_{6} - 18991 \beta_{5} + 14884 \beta_{4} + 12767 \beta_{3} - 873 \beta_{2} + 36799 \beta_{1} - 38962\)
\(\nu^{10}\)\(=\)\(-41245 \beta_{13} - 63259 \beta_{12} - 123644 \beta_{11} + 67177 \beta_{10} - 1642 \beta_{9} + 44844 \beta_{8} - 60274 \beta_{7} + 20510 \beta_{6} + 88164 \beta_{5} - 27624 \beta_{4} - 46650 \beta_{3} + 676 \beta_{2} - 91210 \beta_{1} + 305954\)
\(\nu^{11}\)\(=\)\(323677 \beta_{13} + 365063 \beta_{12} + 460293 \beta_{11} - 120204 \beta_{10} + 102976 \beta_{9} - 266446 \beta_{8} + 285976 \beta_{7} - 17338 \beta_{6} - 366787 \beta_{5} + 263343 \beta_{4} + 237094 \beta_{3} - 15430 \beta_{2} + 650857 \beta_{1} - 854144\)
\(\nu^{12}\)\(=\)\(-900094 \beta_{13} - 1291247 \beta_{12} - 2328995 \beta_{11} + 1159985 \beta_{10} - 89596 \beta_{9} + 907064 \beta_{8} - 1178554 \beta_{7} + 357528 \beta_{6} + 1706092 \beta_{5} - 629847 \beta_{4} - 929026 \beta_{3} + 19530 \beta_{2} - 1948105 \beta_{1} + 5498570\)
\(\nu^{13}\)\(=\)\(5856451 \beta_{13} + 6791797 \beta_{12} + 9079144 \beta_{11} - 2797826 \beta_{10} + 1753751 \beta_{9} - 4976458 \beta_{8} + 5457293 \beta_{7} - 489879 \beta_{6} - 7096317 \beta_{5} + 4725157 \beta_{4} + 4464975 \beta_{3} - 271490 \beta_{2} + 11817884 \beta_{1} - 17797008\)

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
3.68342
3.65422
3.21497
2.05100
1.52247
0.937159
0.664239
0.343018
−1.02849
−1.34598
−1.59859
−2.28948
−2.41893
−4.38902
−1.00000 −1.00000 1.00000 −3.68342 1.00000 5.03243 −1.00000 1.00000 3.68342
1.2 −1.00000 −1.00000 1.00000 −3.65422 1.00000 −2.98447 −1.00000 1.00000 3.65422
1.3 −1.00000 −1.00000 1.00000 −3.21497 1.00000 −1.40491 −1.00000 1.00000 3.21497
1.4 −1.00000 −1.00000 1.00000 −2.05100 1.00000 5.03739 −1.00000 1.00000 2.05100
1.5 −1.00000 −1.00000 1.00000 −1.52247 1.00000 −3.40750 −1.00000 1.00000 1.52247
1.6 −1.00000 −1.00000 1.00000 −0.937159 1.00000 −0.792704 −1.00000 1.00000 0.937159
1.7 −1.00000 −1.00000 1.00000 −0.664239 1.00000 2.36162 −1.00000 1.00000 0.664239
1.8 −1.00000 −1.00000 1.00000 −0.343018 1.00000 2.39676 −1.00000 1.00000 0.343018
1.9 −1.00000 −1.00000 1.00000 1.02849 1.00000 −4.96874 −1.00000 1.00000 −1.02849
1.10 −1.00000 −1.00000 1.00000 1.34598 1.00000 −3.24510 −1.00000 1.00000 −1.34598
1.11 −1.00000 −1.00000 1.00000 1.59859 1.00000 4.24832 −1.00000 1.00000 −1.59859
1.12 −1.00000 −1.00000 1.00000 2.28948 1.00000 −1.19378 −1.00000 1.00000 −2.28948
1.13 −1.00000 −1.00000 1.00000 2.41893 1.00000 −0.257176 −1.00000 1.00000 −2.41893
1.14 −1.00000 −1.00000 1.00000 4.38902 1.00000 3.17787 −1.00000 1.00000 −4.38902
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.z 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.z 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{14} + \cdots\)
\(T_{7}^{14} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{14} \)
$3$ \( ( 1 + T )^{14} \)
$5$ \( 1552 + 6316 T + 1118 T^{2} - 15763 T^{3} - 7015 T^{4} + 14618 T^{5} + 6361 T^{6} - 6204 T^{7} - 2404 T^{8} + 1239 T^{9} + 434 T^{10} - 108 T^{11} - 36 T^{12} + 3 T^{13} + T^{14} \)
$7$ \( 108608 + 657008 T + 964156 T^{2} + 49673 T^{3} - 640804 T^{4} - 170741 T^{5} + 171823 T^{6} + 45906 T^{7} - 23659 T^{8} - 4948 T^{9} + 1761 T^{10} + 235 T^{11} - 67 T^{12} - 4 T^{13} + T^{14} \)
$11$ \( 1849152 + 3061120 T - 8125040 T^{2} + 2804704 T^{3} + 2370440 T^{4} - 1417614 T^{5} - 164591 T^{6} + 220317 T^{7} - 11436 T^{8} - 15063 T^{9} + 1882 T^{10} + 460 T^{11} - 77 T^{12} - 5 T^{13} + T^{14} \)
$13$ \( ( -1 + T )^{14} \)
$17$ \( -57600 - 562496 T + 4012608 T^{2} - 6423120 T^{3} + 366160 T^{4} + 3702132 T^{5} + 244396 T^{6} - 544255 T^{7} - 61384 T^{8} + 32399 T^{9} + 4300 T^{10} - 820 T^{11} - 115 T^{12} + 7 T^{13} + T^{14} \)
$19$ \( 177664 - 88832 T - 724416 T^{2} + 536184 T^{3} + 687896 T^{4} - 826510 T^{5} + 70333 T^{6} + 263377 T^{7} - 146992 T^{8} + 26343 T^{9} + 2877 T^{10} - 2084 T^{11} + 376 T^{12} - 31 T^{13} + T^{14} \)
$23$ \( 2497011840 - 1856116368 T - 1217259168 T^{2} + 906424103 T^{3} + 139161270 T^{4} - 121803940 T^{5} - 4885935 T^{6} + 7078138 T^{7} - 70253 T^{8} - 200926 T^{9} + 7852 T^{10} + 2727 T^{11} - 159 T^{12} - 14 T^{13} + T^{14} \)
$29$ \( 946837836 + 1698705900 T + 707534247 T^{2} - 253516496 T^{3} - 207985431 T^{4} - 1670255 T^{5} + 19309274 T^{6} + 1705463 T^{7} - 807069 T^{8} - 82910 T^{9} + 17697 T^{10} + 1477 T^{11} - 205 T^{12} - 9 T^{13} + T^{14} \)
$31$ \( 6181087360 + 2288189792 T - 4002433360 T^{2} - 646585376 T^{3} + 850043216 T^{4} + 28022064 T^{5} - 72777841 T^{6} + 4038841 T^{7} + 2535314 T^{8} - 308363 T^{9} - 23147 T^{10} + 5004 T^{11} - 62 T^{12} - 23 T^{13} + T^{14} \)
$37$ \( 106231104 - 272968864 T + 28672992 T^{2} + 153492576 T^{3} - 23418076 T^{4} - 31060364 T^{5} + 3162283 T^{6} + 2567030 T^{7} - 212952 T^{8} - 99116 T^{9} + 7789 T^{10} + 1781 T^{11} - 142 T^{12} - 12 T^{13} + T^{14} \)
$41$ \( 228866240 + 2023585176 T + 3170369868 T^{2} - 3245414710 T^{3} - 873089151 T^{4} + 333937444 T^{5} + 71036346 T^{6} - 12931707 T^{7} - 2533974 T^{8} + 225866 T^{9} + 43284 T^{10} - 1762 T^{11} - 342 T^{12} + 5 T^{13} + T^{14} \)
$43$ \( 157075968 - 693779200 T + 286288544 T^{2} + 322420448 T^{3} - 195616688 T^{4} - 12135748 T^{5} + 20382602 T^{6} - 443463 T^{7} - 928869 T^{8} + 28694 T^{9} + 21420 T^{10} - 438 T^{11} - 238 T^{12} + 2 T^{13} + T^{14} \)
$47$ \( -159375400 + 930393616 T - 1941589872 T^{2} + 1747875853 T^{3} - 602471680 T^{4} - 29296365 T^{5} + 54646010 T^{6} - 4467675 T^{7} - 1740763 T^{8} + 178701 T^{9} + 28828 T^{10} - 2412 T^{11} - 261 T^{12} + 11 T^{13} + T^{14} \)
$53$ \( 9892473600 - 3825576896 T - 8923879488 T^{2} + 4449051808 T^{3} + 509438056 T^{4} - 491999248 T^{5} + 19108574 T^{6} + 19055393 T^{7} - 1691454 T^{8} - 323599 T^{9} + 37513 T^{10} + 2385 T^{11} - 330 T^{12} - 6 T^{13} + T^{14} \)
$59$ \( 6276618880 - 4167748592 T - 9828582320 T^{2} - 2615882247 T^{3} + 1076021466 T^{4} + 481670059 T^{5} - 8877595 T^{6} - 23597138 T^{7} - 1642991 T^{8} + 428092 T^{9} + 47967 T^{10} - 2521 T^{11} - 391 T^{12} + 4 T^{13} + T^{14} \)
$61$ \( -672637440 - 13879896832 T - 3850418720 T^{2} + 5498955616 T^{3} + 584911584 T^{4} - 713222548 T^{5} + 3122342 T^{6} + 32329483 T^{7} - 1475595 T^{8} - 618314 T^{9} + 40480 T^{10} + 4878 T^{11} - 372 T^{12} - 12 T^{13} + T^{14} \)
$67$ \( -12800870040 + 71675235776 T - 1400890692 T^{2} - 21380526709 T^{3} + 4264298585 T^{4} + 1292071245 T^{5} - 445432379 T^{6} + 8378662 T^{7} + 10035616 T^{8} - 990510 T^{9} - 43614 T^{10} + 9837 T^{11} - 210 T^{12} - 24 T^{13} + T^{14} \)
$71$ \( -240723840 + 12800144 T + 1989329140 T^{2} - 3098885762 T^{3} + 1355048649 T^{4} + 231761661 T^{5} - 268981454 T^{6} + 30865699 T^{7} + 6663129 T^{8} - 1313772 T^{9} + 4948 T^{10} + 9918 T^{11} - 369 T^{12} - 20 T^{13} + T^{14} \)
$73$ \( 2427482624 - 8728168596 T + 4612396486 T^{2} + 6767863665 T^{3} - 4729822073 T^{4} - 122743546 T^{5} + 349470788 T^{6} - 2288614 T^{7} - 9772355 T^{8} - 6083 T^{9} + 115385 T^{10} + 523 T^{11} - 579 T^{12} - 2 T^{13} + T^{14} \)
$79$ \( -3471863808 + 7493339392 T - 4988797568 T^{2} - 108292496 T^{3} + 1557652696 T^{4} - 644335236 T^{5} + 31017472 T^{6} + 45217417 T^{7} - 13957211 T^{8} + 1690471 T^{9} - 45435 T^{10} - 10852 T^{11} + 1301 T^{12} - 59 T^{13} + T^{14} \)
$83$ \( -547859593152 - 14183948256 T + 166694194680 T^{2} - 2122897865 T^{3} - 15900876970 T^{4} + 92572503 T^{5} + 663810190 T^{6} + 3383982 T^{7} - 13013166 T^{8} - 94288 T^{9} + 126428 T^{10} + 628 T^{11} - 584 T^{12} - T^{13} + T^{14} \)
$89$ \( -22829917834624 - 4056419240928 T + 3295802787888 T^{2} + 444629478896 T^{3} - 177931564960 T^{4} - 16507668264 T^{5} + 4480352577 T^{6} + 255685992 T^{7} - 54647834 T^{8} - 1757183 T^{9} + 327871 T^{10} + 5390 T^{11} - 932 T^{12} - 6 T^{13} + T^{14} \)
$97$ \( -9472848370240 - 7167536613504 T + 2556560838432 T^{2} + 538608195784 T^{3} - 180280963064 T^{4} - 8000732514 T^{5} + 4239703089 T^{6} + 10108254 T^{7} - 46944089 T^{8} + 487863 T^{9} + 273497 T^{10} - 3206 T^{11} - 819 T^{12} + 6 T^{13} + T^{14} \)
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