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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{14} \)
$3$ \( ( 1 + T )^{14} \)
$5$ \( 1 + 3 T + 34 T^{2} + 87 T^{3} + 549 T^{4} + 1149 T^{5} + 5396 T^{6} + 8301 T^{7} + 34076 T^{8} + 25703 T^{9} + 129765 T^{10} - 107288 T^{11} + 202568 T^{12} - 1634379 T^{13} - 162018 T^{14} - 8171895 T^{15} + 5064200 T^{16} - 13411000 T^{17} + 81103125 T^{18} + 80321875 T^{19} + 532437500 T^{20} + 648515625 T^{21} + 2107812500 T^{22} + 2244140625 T^{23} + 5361328125 T^{24} + 4248046875 T^{25} + 8300781250 T^{26} + 3662109375 T^{27} + 6103515625 T^{28} \)
$7$ \( 1 - 4 T + 31 T^{2} - 129 T^{3} + 592 T^{4} - 2141 T^{5} + 7785 T^{6} - 24885 T^{7} + 77505 T^{8} - 216654 T^{9} + 616823 T^{10} - 1575650 T^{11} + 4236586 T^{12} - 10566813 T^{13} + 29009354 T^{14} - 73967691 T^{15} + 207592714 T^{16} - 540447950 T^{17} + 1480992023 T^{18} - 3641303778 T^{19} + 9118385745 T^{20} - 20493867555 T^{21} + 44878975785 T^{22} - 86397072587 T^{23} + 167225347408 T^{24} - 255075149847 T^{25} + 429079903231 T^{26} - 387556041628 T^{27} + 678223072849 T^{28} \)
$11$ \( 1 - 5 T + 77 T^{2} - 255 T^{3} + 2729 T^{4} - 6593 T^{5} + 65146 T^{6} - 112950 T^{7} + 1185032 T^{8} - 1386308 T^{9} + 17737693 T^{10} - 13301606 T^{11} + 229927478 T^{12} - 117374261 T^{13} + 2659852280 T^{14} - 1291116871 T^{15} + 27821224838 T^{16} - 17704437586 T^{17} + 259697563213 T^{18} - 223266289708 T^{19} + 2099356474952 T^{20} - 2201075964450 T^{21} + 13964623661626 T^{22} - 15545949126763 T^{23} + 70783231736129 T^{24} - 72754476005805 T^{25} + 241658985007517 T^{26} - 172613560719655 T^{27} + 379749833583241 T^{28} \)
$13$ \( ( 1 - T )^{14} \)
$17$ \( 1 + 7 T + 123 T^{2} + 727 T^{3} + 7139 T^{4} + 36853 T^{5} + 264438 T^{6} + 1214718 T^{7} + 7123293 T^{8} + 29308688 T^{9} + 151804013 T^{10} + 566691726 T^{11} + 2783918143 T^{12} + 9781111041 T^{13} + 47887217988 T^{14} + 166278887697 T^{15} + 804552343327 T^{16} + 2784156449838 T^{17} + 12678822969773 T^{18} + 41614145817616 T^{19} + 171938976294717 T^{20} + 498445772189214 T^{21} + 1844655346183158 T^{22} + 4370319012543941 T^{23} + 14392180455305411 T^{24} + 24915668615649191 T^{25} + 71662535179260603 T^{26} + 69332046230341559 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 - 31 T + 642 T^{2} - 9741 T^{3} + 121456 T^{4} - 1282111 T^{5} + 11854890 T^{6} - 97421684 T^{7} + 722292215 T^{8} - 4870544446 T^{9} + 30103406600 T^{10} - 171316923644 T^{11} + 901349486328 T^{12} - 4393822430629 T^{13} + 19877998451480 T^{14} - 83482626181951 T^{15} + 325387164564408 T^{16} - 1175062779274196 T^{17} + 3923106051518600 T^{18} - 12059950232196154 T^{19} + 33980873594116415 T^{20} - 87082490093388476 T^{21} + 201338271659120490 T^{22} - 413721446887131469 T^{23} + 744654783407478256 T^{24} - 1134731611927551279 T^{25} + 1420948178040475362 T^{26} - 1303642487329968829 T^{27} + 799006685782884121 T^{28} \)
$23$ \( 1 - 14 T + 163 T^{2} - 1459 T^{3} + 12107 T^{4} - 88663 T^{5} + 613169 T^{6} - 3888147 T^{7} + 23623554 T^{8} - 135264391 T^{9} + 749221665 T^{10} - 3960305747 T^{11} + 20420056474 T^{12} - 101542944321 T^{13} + 495218559334 T^{14} - 2335487719383 T^{15} + 10802209874746 T^{16} - 48185040023749 T^{17} + 209662939955265 T^{18} - 870608016162113 T^{19} + 3497133817729506 T^{20} - 13238461847276709 T^{21} + 48017868533765489 T^{22} - 159695598423293969 T^{23} + 501550771263648443 T^{24} - 1390149436796419493 T^{25} + 3572083782419312323 T^{26} - 7056509067110543362 T^{27} + 11592836324538749809 T^{28} \)
$29$ \( 1 - 9 T + 201 T^{2} - 1916 T^{3} + 22888 T^{4} - 202129 T^{5} + 1823927 T^{6} - 14392698 T^{7} + 109856612 T^{8} - 773276416 T^{9} + 5232404216 T^{10} - 33076055245 T^{11} + 202467959958 T^{12} - 1157495988315 T^{13} + 6456011753450 T^{14} - 33567383661135 T^{15} + 170275554324678 T^{16} - 806691911370305 T^{17} + 3700780086296696 T^{18} - 15860787786761984 T^{19} + 65345274783648452 T^{20} - 248272260252791682 T^{21} + 912412939252717847 T^{22} - 2932314908956425101 T^{23} + 9629147155775000488 T^{24} - 23376176711092368364 T^{25} + 71116771424299277241 T^{26} - 92345658416627419701 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 - 23 T + 372 T^{2} - 4265 T^{3} + 41240 T^{4} - 326033 T^{5} + 2271256 T^{6} - 13473214 T^{7} + 73083297 T^{8} - 353741572 T^{9} + 1757621474 T^{10} - 8879255438 T^{11} + 52216752422 T^{12} - 301306059209 T^{13} + 1780869905636 T^{14} - 9340487835479 T^{15} + 50180299077542 T^{16} - 264521898753458 T^{17} + 1623200341289954 T^{18} - 10127320879765372 T^{19} + 64861695107116257 T^{20} - 370683337616922754 T^{21} + 1937133886134095896 T^{22} - 8620189331910048143 T^{23} + 33801470555088233240 T^{24} - \)\(10\!\cdots\!15\)\( T^{25} + \)\(29\!\cdots\!92\)\( T^{26} - \)\(56\!\cdots\!93\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 - 12 T + 376 T^{2} - 3991 T^{3} + 69320 T^{4} - 655633 T^{5} + 8276402 T^{6} - 70179299 T^{7} + 713605277 T^{8} - 5446190073 T^{9} + 46951903950 T^{10} - 322964467504 T^{11} + 2429171482682 T^{12} - 15038861584088 T^{13} + 100348643121968 T^{14} - 556437878611256 T^{15} + 3325535759791658 T^{16} - 16359119172480112 T^{17} + 87995427258835950 T^{18} - 377660370235938861 T^{19} + 1830915904800660293 T^{20} - 6662252589948069767 T^{21} + 29070691977390672242 T^{22} - 85207205347065718741 T^{23} + \)\(33\!\cdots\!80\)\( T^{24} - \)\(71\!\cdots\!83\)\( T^{25} + \)\(24\!\cdots\!56\)\( T^{26} - \)\(29\!\cdots\!64\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 + 5 T + 232 T^{2} + 903 T^{3} + 27991 T^{4} + 86794 T^{5} + 2356178 T^{6} + 6064167 T^{7} + 157081775 T^{8} + 355737636 T^{9} + 8884986125 T^{10} + 18797724102 T^{11} + 439539202569 T^{12} + 884518962553 T^{13} + 19178916375602 T^{14} + 36265277464673 T^{15} + 738865399518489 T^{16} + 1295557942833942 T^{17} + 25106847277566125 T^{18} + 41214411059680836 T^{19} + 746154805611307775 T^{20} + 1181022440778122127 T^{21} + 18813905156499859538 T^{22} + 28414787613789451034 T^{23} + \)\(37\!\cdots\!91\)\( T^{24} + \)\(49\!\cdots\!23\)\( T^{25} + \)\(52\!\cdots\!92\)\( T^{26} + \)\(46\!\cdots\!05\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( 1 + 2 T + 364 T^{2} + 680 T^{3} + 66871 T^{4} + 109964 T^{5} + 8178187 T^{6} + 11596709 T^{7} + 742336047 T^{8} + 907283845 T^{9} + 52880438636 T^{10} + 56375649615 T^{11} + 3047263202649 T^{12} + 2889825087241 T^{13} + 144238454924186 T^{14} + 124262478751363 T^{15} + 5634389661698001 T^{16} + 4482258773939805 T^{17} + 180787696489195436 T^{18} + 133378385412503335 T^{19} + 4692575657446527303 T^{20} + 3152201333792046863 T^{21} + 95588287563672889387 T^{22} + 55267093979023003652 T^{23} + \)\(14\!\cdots\!79\)\( T^{24} + \)\(63\!\cdots\!60\)\( T^{25} + \)\(14\!\cdots\!64\)\( T^{26} + \)\(34\!\cdots\!86\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 + 11 T + 397 T^{2} + 4309 T^{3} + 82643 T^{4} + 827019 T^{5} + 11547735 T^{6} + 104704066 T^{7} + 1188810483 T^{8} + 9771070409 T^{9} + 95017480563 T^{10} + 708893639374 T^{11} + 6077161764777 T^{12} + 41137288674466 T^{13} + 315981100890954 T^{14} + 1933452567699902 T^{15} + 13424450338392393 T^{16} + 73599464320726802 T^{17} + 463654994571140403 T^{18} + 2240946211349597863 T^{19} + 12814444181629493907 T^{20} + 53045500642083902558 T^{21} + \)\(27\!\cdots\!35\)\( T^{22} + \)\(92\!\cdots\!73\)\( T^{23} + \)\(43\!\cdots\!07\)\( T^{24} + \)\(10\!\cdots\!27\)\( T^{25} + \)\(46\!\cdots\!77\)\( T^{26} + \)\(60\!\cdots\!97\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( 1 - 6 T + 412 T^{2} - 1749 T^{3} + 83252 T^{4} - 247756 T^{5} + 11201644 T^{6} - 22303687 T^{7} + 1133664124 T^{8} - 1409415786 T^{9} + 92045778336 T^{10} - 67796356263 T^{11} + 6221456621063 T^{12} - 2954360400877 T^{13} + 356868631966448 T^{14} - 156581101246481 T^{15} + 17476071648565967 T^{16} - 10093318131366651 T^{17} + 726285465090419616 T^{18} - 589411329468252498 T^{19} + 25126941043327435996 T^{20} - 26200389578337679019 T^{21} + \)\(69\!\cdots\!84\)\( T^{22} - \)\(81\!\cdots\!48\)\( T^{23} + \)\(14\!\cdots\!48\)\( T^{24} - \)\(16\!\cdots\!53\)\( T^{25} + \)\(20\!\cdots\!92\)\( T^{26} - \)\(15\!\cdots\!38\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \)
$59$ \( 1 + 4 T + 435 T^{2} + 547 T^{3} + 87910 T^{4} - 121965 T^{5} + 11584809 T^{6} - 43984765 T^{7} + 1192198149 T^{8} - 6391994138 T^{9} + 105987857061 T^{10} - 600490468346 T^{11} + 8219291912012 T^{12} - 42863794863453 T^{13} + 534923792078606 T^{14} - 2528963896943727 T^{15} + 28611355145713772 T^{16} - 123328131898433134 T^{17} + 1284293125624536021 T^{18} - 4569791928321759262 T^{19} + 50287554130632430509 T^{20} - \)\(10\!\cdots\!35\)\( T^{21} + \)\(17\!\cdots\!89\)\( T^{22} - \)\(10\!\cdots\!35\)\( T^{23} + \)\(44\!\cdots\!10\)\( T^{24} + \)\(16\!\cdots\!73\)\( T^{25} + \)\(77\!\cdots\!35\)\( T^{26} + \)\(41\!\cdots\!16\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \)
$61$ \( 1 - 12 T + 482 T^{2} - 4638 T^{3} + 106787 T^{4} - 828032 T^{5} + 14480297 T^{6} - 87816605 T^{7} + 1344767383 T^{8} - 5842521801 T^{9} + 91879935358 T^{10} - 230541494581 T^{11} + 5135016712645 T^{12} - 4901001983979 T^{13} + 289194053480718 T^{14} - 298961121022719 T^{15} + 19107397187752045 T^{16} - 52328538981489961 T^{17} + 1272154976057146078 T^{18} - 4934572301636458101 T^{19} + 69282919000622267263 T^{20} - \)\(27\!\cdots\!05\)\( T^{21} + \)\(27\!\cdots\!57\)\( T^{22} - \)\(96\!\cdots\!12\)\( T^{23} + \)\(76\!\cdots\!87\)\( T^{24} - \)\(20\!\cdots\!18\)\( T^{25} + \)\(12\!\cdots\!22\)\( T^{26} - \)\(19\!\cdots\!72\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 - 24 T + 728 T^{2} - 11067 T^{3} + 196045 T^{4} - 2144049 T^{5} + 28074428 T^{6} - 224629985 T^{7} + 2399433248 T^{8} - 12472021562 T^{9} + 120741266923 T^{10} - 110918142082 T^{11} + 2706198176098 T^{12} + 37496687932729 T^{13} + 246324937482 T^{14} + 2512278091492843 T^{15} + 12148123612503922 T^{16} - 33360073167008566 T^{17} + 2433071879458670683 T^{18} - 16838789445901557134 T^{19} + \)\(21\!\cdots\!12\)\( T^{20} - \)\(13\!\cdots\!55\)\( T^{21} + \)\(11\!\cdots\!48\)\( T^{22} - \)\(58\!\cdots\!03\)\( T^{23} + \)\(35\!\cdots\!05\)\( T^{24} - \)\(13\!\cdots\!61\)\( T^{25} + \)\(59\!\cdots\!08\)\( T^{26} - \)\(13\!\cdots\!88\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \)
$71$ \( 1 - 20 T + 625 T^{2} - 8542 T^{3} + 149291 T^{4} - 1431774 T^{5} + 17687299 T^{6} - 106070439 T^{7} + 1019982579 T^{8} - 522497538 T^{9} + 10266084344 T^{10} + 578764158604 T^{11} - 2595505604495 T^{12} + 61931023687227 T^{13} - 244091435636824 T^{14} + 4397102681793117 T^{15} - 13083943752259295 T^{16} + 207146058770116244 T^{17} + 260878460468822264 T^{18} - 942705393884837838 T^{19} + \)\(13\!\cdots\!59\)\( T^{20} - \)\(96\!\cdots\!49\)\( T^{21} + \)\(11\!\cdots\!39\)\( T^{22} - \)\(65\!\cdots\!94\)\( T^{23} + \)\(48\!\cdots\!91\)\( T^{24} - \)\(19\!\cdots\!82\)\( T^{25} + \)\(10\!\cdots\!25\)\( T^{26} - \)\(23\!\cdots\!20\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 - 2 T + 443 T^{2} - 1375 T^{3} + 93120 T^{4} - 417438 T^{5} + 12418477 T^{6} - 75514184 T^{7} + 1185969174 T^{8} - 9498474967 T^{9} + 87804401592 T^{10} - 927760695467 T^{11} + 5594072409883 T^{12} - 76465537821639 T^{13} + 372417748110468 T^{14} - 5581984260979647 T^{15} + 29810811872266507 T^{16} - 360914682468485939 T^{17} + 2493490557270399672 T^{18} - 19691018630909312431 T^{19} + \)\(17\!\cdots\!86\)\( T^{20} - \)\(83\!\cdots\!48\)\( T^{21} + \)\(10\!\cdots\!37\)\( T^{22} - \)\(24\!\cdots\!94\)\( T^{23} + \)\(40\!\cdots\!80\)\( T^{24} - \)\(43\!\cdots\!75\)\( T^{25} + \)\(10\!\cdots\!03\)\( T^{26} - \)\(33\!\cdots\!66\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 59 T + 2407 T^{2} - 71445 T^{3} + 1755844 T^{4} - 36460999 T^{5} + 665505041 T^{6} - 10797401048 T^{7} + 158556521206 T^{8} - 2121765780844 T^{9} + 26133042315189 T^{10} - 297453653405510 T^{11} + 3146959336917445 T^{12} - 31001951260452789 T^{13} + 285206931233742566 T^{14} - 2449154149575770331 T^{15} + 19640173221701774245 T^{16} - \)\(14\!\cdots\!90\)\( T^{17} + \)\(10\!\cdots\!09\)\( T^{18} - \)\(65\!\cdots\!56\)\( T^{19} + \)\(38\!\cdots\!26\)\( T^{20} - \)\(20\!\cdots\!32\)\( T^{21} + \)\(10\!\cdots\!01\)\( T^{22} - \)\(43\!\cdots\!81\)\( T^{23} + \)\(16\!\cdots\!44\)\( T^{24} - \)\(53\!\cdots\!55\)\( T^{25} + \)\(14\!\cdots\!87\)\( T^{26} - \)\(27\!\cdots\!01\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 - T + 578 T^{2} - 451 T^{3} + 171663 T^{4} - 58266 T^{5} + 34522926 T^{6} + 7365824 T^{7} + 5258965667 T^{8} + 3990734318 T^{9} + 646035334504 T^{10} + 762906296233 T^{11} + 66697995009421 T^{12} + 91565920089015 T^{13} + 5936164767077296 T^{14} + 7599971367388245 T^{15} + 459482487619901269 T^{16} + 436219902404178371 T^{17} + 30659752282233207784 T^{18} + 15719664674016886474 T^{19} + \)\(17\!\cdots\!23\)\( T^{20} + \)\(19\!\cdots\!48\)\( T^{21} + \)\(77\!\cdots\!66\)\( T^{22} - \)\(10\!\cdots\!98\)\( T^{23} + \)\(26\!\cdots\!87\)\( T^{24} - \)\(58\!\cdots\!17\)\( T^{25} + \)\(61\!\cdots\!58\)\( T^{26} - \)\(88\!\cdots\!63\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( 1 - 6 T + 314 T^{2} - 1552 T^{3} + 53306 T^{4} - 187401 T^{5} + 6529520 T^{6} - 13363945 T^{7} + 697212845 T^{8} - 486604536 T^{9} + 64791377804 T^{10} + 53496142742 T^{11} + 5464351022744 T^{12} + 12805961052930 T^{13} + 458759171432148 T^{14} + 1139730533710770 T^{15} + 43283124451155224 T^{16} + 37713122252684998 T^{17} + 4065156240900618764 T^{18} - 2717228657177060664 T^{19} + \)\(34\!\cdots\!45\)\( T^{20} - \)\(59\!\cdots\!05\)\( T^{21} + \)\(25\!\cdots\!20\)\( T^{22} - \)\(65\!\cdots\!09\)\( T^{23} + \)\(16\!\cdots\!06\)\( T^{24} - \)\(43\!\cdots\!28\)\( T^{25} + \)\(77\!\cdots\!94\)\( T^{26} - \)\(13\!\cdots\!14\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \)
$97$ \( 1 + 6 T + 539 T^{2} + 4360 T^{3} + 176400 T^{4} + 1470473 T^{5} + 41966887 T^{6} + 343070551 T^{7} + 7783476451 T^{8} + 61109403984 T^{9} + 1174953427935 T^{10} + 8706554053390 T^{11} + 147557628968604 T^{12} + 1018584151007508 T^{13} + 15576640897061022 T^{14} + 98802662647728276 T^{15} + 1388369730965595036 T^{16} + 7946236807569611470 T^{17} + \)\(10\!\cdots\!35\)\( T^{18} + \)\(52\!\cdots\!88\)\( T^{19} + \)\(64\!\cdots\!79\)\( T^{20} + \)\(27\!\cdots\!63\)\( T^{21} + \)\(32\!\cdots\!07\)\( T^{22} + \)\(11\!\cdots\!41\)\( T^{23} + \)\(13\!\cdots\!00\)\( T^{24} + \)\(31\!\cdots\!80\)\( T^{25} + \)\(37\!\cdots\!99\)\( T^{26} + \)\(40\!\cdots\!62\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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