Properties

Label 6003.2.a.s
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 0
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + 14484 x^{12} - 24566 x^{11} - 36791 x^{10} + 59410 x^{9} + 52109 x^{8} - 82362 x^{7} - 34967 x^{6} + 60661 x^{5} + 5201 x^{4} - 19624 x^{3} + 2768 x^{2} + 1408 x - 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 7 \nu^{2} + 5 \)
\(\beta_{5}\)\(=\)\((\)\(-9977 \nu^{19} - 284189 \nu^{18} - 706 \nu^{17} + 9004494 \nu^{16} + 5676465 \nu^{15} - 116705443 \nu^{14} - 93997884 \nu^{13} + 796777255 \nu^{12} + 679406561 \nu^{11} - 3077890179 \nu^{10} - 2551075658 \nu^{9} + 6745265900 \nu^{8} + 4995760963 \nu^{7} - 8060495321 \nu^{6} - 4647412912 \nu^{5} + 4886184991 \nu^{4} + 1615059980 \nu^{3} - 1279285860 \nu^{2} - 73689968 \nu + 67052416\)\()/3404224\)
\(\beta_{6}\)\(=\)\((\)\(59655 \nu^{19} + 83835 \nu^{18} - 1977318 \nu^{17} - 2493558 \nu^{16} + 26985321 \nu^{15} + 29771565 \nu^{14} - 195494072 \nu^{13} - 181802773 \nu^{12} + 808522329 \nu^{11} + 596421017 \nu^{10} - 1915046350 \nu^{9} - 993555752 \nu^{8} + 2487007763 \nu^{7} + 636510631 \nu^{6} - 1618911100 \nu^{5} + 152363091 \nu^{4} + 447758060 \nu^{3} - 274311304 \nu^{2} - 39969248 \nu + 42998144\)\()/6808448\)
\(\beta_{7}\)\(=\)\((\)\(32190 \nu^{19} - 101041 \nu^{18} - 1039401 \nu^{17} + 3106330 \nu^{16} + 13951816 \nu^{15} - 38936723 \nu^{14} - 101021583 \nu^{13} + 256304866 \nu^{12} + 427557079 \nu^{11} - 951947129 \nu^{10} - 1066095561 \nu^{9} + 1998865072 \nu^{8} + 1483887288 \nu^{7} - 2270424875 \nu^{6} - 990251463 \nu^{5} + 1290653668 \nu^{4} + 195615347 \nu^{3} - 326105000 \nu^{2} + 28151104 \nu + 21780992\)\()/3404224\)
\(\beta_{8}\)\(=\)\((\)\(-113117 \nu^{19} - 286133 \nu^{18} + 3436418 \nu^{17} + 9067866 \nu^{16} - 41870219 \nu^{15} - 117469803 \nu^{14} + 259419552 \nu^{13} + 800884431 \nu^{12} - 846624863 \nu^{11} - 3086463419 \nu^{10} + 1308880446 \nu^{9} + 6744972372 \nu^{8} - 477260749 \nu^{7} - 8047603685 \nu^{6} - 765017784 \nu^{5} + 4881910035 \nu^{4} + 543546028 \nu^{3} - 1275029840 \nu^{2} - 6205632 \nu + 65443584\)\()/6808448\)
\(\beta_{9}\)\(=\)\((\)\(132421 \nu^{19} + 61925 \nu^{18} - 4039074 \nu^{17} - 1758962 \nu^{16} + 49733347 \nu^{15} + 18817259 \nu^{14} - 315679888 \nu^{13} - 86908655 \nu^{12} + 1090655743 \nu^{11} + 87630643 \nu^{10} - 1976507934 \nu^{9} + 645077396 \nu^{8} + 1618257989 \nu^{7} - 2223325131 \nu^{6} - 337309496 \nu^{5} + 2337172669 \nu^{4} - 167806212 \nu^{3} - 665008808 \nu^{2} + 110604448 \nu - 8160896\)\()/6808448\)
\(\beta_{10}\)\(=\)\((\)\(-161261 \nu^{19} + 371971 \nu^{18} + 5495374 \nu^{17} - 11857542 \nu^{16} - 78622707 \nu^{15} + 155862669 \nu^{14} + 614092532 \nu^{13} - 1093593473 \nu^{12} - 2847831847 \nu^{11} + 4435330241 \nu^{10} + 7971368478 \nu^{9} - 10545189400 \nu^{8} - 13052790841 \nu^{7} + 14281125551 \nu^{6} + 11462171224 \nu^{5} - 10197457825 \nu^{4} - 4446760952 \nu^{3} + 3175929024 \nu^{2} + 397653248 \nu - 221220224\)\()/6808448\)
\(\beta_{11}\)\(=\)\((\)\(-88710 \nu^{19} + 348943 \nu^{18} + 3305005 \nu^{17} - 10913794 \nu^{16} - 52040672 \nu^{15} + 139461381 \nu^{14} + 448941315 \nu^{13} - 938038438 \nu^{12} - 2296222593 \nu^{11} + 3571430473 \nu^{10} + 7028716107 \nu^{9} - 7739376550 \nu^{8} - 12364008590 \nu^{7} + 9238883527 \nu^{6} + 11341153205 \nu^{5} - 5750325066 \nu^{4} - 4484529451 \nu^{3} + 1641458868 \nu^{2} + 476989360 \nu - 107471168\)\()/3404224\)
\(\beta_{12}\)\(=\)\((\)\(-203145 \nu^{19} + 8703 \nu^{18} + 6311478 \nu^{17} + 38394 \nu^{16} - 80239695 \nu^{15} - 4529143 \nu^{14} + 538837948 \nu^{13} + 55520691 \nu^{12} - 2061905747 \nu^{11} - 279720755 \nu^{10} + 4537659302 \nu^{9} + 621554632 \nu^{8} - 5549815741 \nu^{7} - 467045285 \nu^{6} + 3466368864 \nu^{5} - 137492405 \nu^{4} - 896358416 \nu^{3} + 205094288 \nu^{2} + 40996096 \nu - 15271680\)\()/6808448\)
\(\beta_{13}\)\(=\)\((\)\(294337 \nu^{19} - 270767 \nu^{18} - 9616398 \nu^{17} + 8494062 \nu^{16} + 130255487 \nu^{15} - 109626129 \nu^{14} - 947801932 \nu^{13} + 753143013 \nu^{12} + 4013917123 \nu^{11} - 2978708653 \nu^{10} - 10021968214 \nu^{9} + 6853628544 \nu^{8} + 14266108021 \nu^{7} - 8841784179 \nu^{6} - 10620817160 \nu^{5} + 5854037837 \nu^{4} + 3454356288 \nu^{3} - 1635642432 \nu^{2} - 316927872 \nu + 110542464\)\()/6808448\)
\(\beta_{14}\)\(=\)\((\)\(-148537 \nu^{19} + 338681 \nu^{18} + 5041136 \nu^{17} - 10572042 \nu^{16} - 71767335 \nu^{15} + 135044723 \nu^{14} + 556857158 \nu^{13} - 910556129 \nu^{12} - 2556880073 \nu^{11} + 3490925219 \nu^{10} + 7037682580 \nu^{9} - 7663102560 \nu^{8} - 11171608865 \nu^{7} + 9303904093 \nu^{6} + 9240832926 \nu^{5} - 5836685909 \nu^{4} - 3213222030 \nu^{3} + 1626363316 \nu^{2} + 260140592 \nu - 97520832\)\()/3404224\)
\(\beta_{15}\)\(=\)\((\)\(-380439 \nu^{19} + 254997 \nu^{18} + 12633678 \nu^{17} - 8079138 \nu^{16} - 174716497 \nu^{15} + 105738795 \nu^{14} + 1306063448 \nu^{13} - 741027003 \nu^{12} - 5732647449 \nu^{11} + 3015967983 \nu^{10} + 15028081270 \nu^{9} - 7238359288 \nu^{8} - 22897375291 \nu^{7} + 9964491865 \nu^{6} + 18755286572 \nu^{5} - 7338969931 \nu^{4} - 6950567148 \nu^{3} + 2443697432 \nu^{2} + 760925792 \nu - 178908672\)\()/6808448\)
\(\beta_{16}\)\(=\)\((\)\(-202867 \nu^{19} + 156362 \nu^{18} + 6774905 \nu^{17} - 4911368 \nu^{16} - 94271295 \nu^{15} + 63450778 \nu^{14} + 709058137 \nu^{13} - 436217767 \nu^{12} - 3127379984 \nu^{11} + 1727035600 \nu^{10} + 8203441237 \nu^{9} - 3991457880 \nu^{8} - 12375653553 \nu^{7} + 5243719104 \nu^{6} + 9802374313 \nu^{5} - 3672857369 \nu^{4} - 3330475213 \nu^{3} + 1148408812 \nu^{2} + 272653424 \nu - 69421504\)\()/3404224\)
\(\beta_{17}\)\(=\)\((\)\(-423491 \nu^{19} + 589261 \nu^{18} + 14300798 \nu^{17} - 18316426 \nu^{16} - 202063709 \nu^{15} + 232872267 \nu^{14} + 1551417160 \nu^{13} - 1562401159 \nu^{12} - 7027134489 \nu^{11} + 5963688187 \nu^{10} + 19037717322 \nu^{9} - 13077406708 \nu^{8} - 29747755059 \nu^{7} + 16061786213 \nu^{6} + 24293520488 \nu^{5} - 10588454955 \nu^{4} - 8318188780 \nu^{3} + 3293324216 \nu^{2} + 584394208 \nu - 197504512\)\()/6808448\)
\(\beta_{18}\)\(=\)\((\)\(-595707 \nu^{19} + 195465 \nu^{18} + 19683950 \nu^{17} - 6229090 \nu^{16} - 270259309 \nu^{15} + 82762231 \nu^{14} + 1999221168 \nu^{13} - 597242311 \nu^{12} - 8642079917 \nu^{11} + 2553180667 \nu^{10} + 22150191358 \nu^{9} - 6590508888 \nu^{8} - 32604889607 \nu^{7} + 9966835125 \nu^{6} + 25189860044 \nu^{5} - 8106916351 \nu^{4} - 8224828108 \nu^{3} + 2852863952 \nu^{2} + 555519872 \nu - 174423552\)\()/6808448\)
\(\beta_{19}\)\(=\)\((\)\(975995 \nu^{19} - 593997 \nu^{18} - 31897570 \nu^{17} + 18413026 \nu^{16} + 432558821 \nu^{15} - 235226483 \nu^{14} - 3156059852 \nu^{13} + 1607414239 \nu^{12} + 13439770617 \nu^{11} - 6389315479 \nu^{10} - 33902595298 \nu^{9} + 15053725344 \nu^{8} + 49088306735 \nu^{7} - 20480764017 \nu^{6} - 37324091800 \nu^{5} + 14898925983 \nu^{4} + 12074218624 \nu^{3} - 4763008568 \nu^{2} - 845754080 \nu + 286605824\)\()/6808448\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 7 \beta_{2} + 23\)
\(\nu^{5}\)\(=\)\(\beta_{19} + \beta_{18} - \beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{6} + \beta_{4} + 9 \beta_{3} + 37 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{19} + 2 \beta_{18} - \beta_{17} - \beta_{15} + 2 \beta_{14} - \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{7} + 2 \beta_{5} + 11 \beta_{4} + 47 \beta_{2} - 3 \beta_{1} + 149\)
\(\nu^{7}\)\(=\)\(15 \beta_{19} + 17 \beta_{18} - 13 \beta_{17} + 12 \beta_{16} + 12 \beta_{15} + 13 \beta_{14} - 2 \beta_{13} + 12 \beta_{12} - 13 \beta_{11} + \beta_{10} + 14 \beta_{9} - 14 \beta_{8} - 3 \beta_{7} + 16 \beta_{6} + 2 \beta_{5} + 13 \beta_{4} + 70 \beta_{3} - 2 \beta_{2} + 236 \beta_{1} - 2\)
\(\nu^{8}\)\(=\)\(15 \beta_{19} + 29 \beta_{18} - 16 \beta_{17} + 5 \beta_{16} - 16 \beta_{15} + 31 \beta_{14} + \beta_{13} - 15 \beta_{12} - \beta_{11} + 15 \beta_{10} + 15 \beta_{9} - 2 \beta_{8} - 28 \beta_{7} + 2 \beta_{6} + 31 \beta_{5} + 97 \beta_{4} - \beta_{3} + 322 \beta_{2} - 47 \beta_{1} + 1019\)
\(\nu^{9}\)\(=\)\(163 \beta_{19} + 195 \beta_{18} - 125 \beta_{17} + 109 \beta_{16} + 107 \beta_{15} + 128 \beta_{14} - 38 \beta_{13} + 110 \beta_{12} - 126 \beta_{11} + 20 \beta_{10} + 144 \beta_{9} - 139 \beta_{8} - 49 \beta_{7} + 177 \beta_{6} + 35 \beta_{5} + 128 \beta_{4} + 527 \beta_{3} - 38 \beta_{2} + 1550 \beta_{1} - 32\)
\(\nu^{10}\)\(=\)\(161 \beta_{19} + 298 \beta_{18} - 177 \beta_{17} + 101 \beta_{16} - 176 \beta_{15} + 341 \beta_{14} + 23 \beta_{13} - 159 \beta_{12} - 25 \beta_{11} + 163 \beta_{10} + 163 \beta_{9} - 39 \beta_{8} - 280 \beta_{7} + 39 \beta_{6} + 339 \beta_{5} + 800 \beta_{4} - 22 \beta_{3} + 2262 \beta_{2} - 522 \beta_{1} + 7189\)
\(\nu^{11}\)\(=\)\(1554 \beta_{19} + 1914 \beta_{18} - 1080 \beta_{17} + 900 \beta_{16} + 862 \beta_{15} + 1144 \beta_{14} - 476 \beta_{13} + 920 \beta_{12} - 1108 \beta_{11} + 262 \beta_{10} + 1326 \beta_{9} - 1218 \beta_{8} - 544 \beta_{7} + 1704 \beta_{6} + 414 \beta_{5} + 1145 \beta_{4} + 3940 \beta_{3} - 472 \beta_{2} + 10431 \beta_{1} - 361\)
\(\nu^{12}\)\(=\)\(1517 \beta_{19} + 2679 \beta_{18} - 1689 \beta_{17} + 1337 \beta_{16} - 1661 \beta_{15} + 3273 \beta_{14} + 336 \beta_{13} - 1479 \beta_{12} - 377 \beta_{11} + 1564 \beta_{10} + 1559 \beta_{9} - 487 \beta_{8} - 2472 \beta_{7} + 517 \beta_{6} + 3224 \beta_{5} + 6419 \beta_{4} - 306 \beta_{3} + 16243 \beta_{2} - 5047 \beta_{1} + 51776\)
\(\nu^{13}\)\(=\)\(13817 \beta_{19} + 17346 \beta_{18} - 8891 \beta_{17} + 7126 \beta_{16} + 6667 \beta_{15} + 9750 \beta_{14} - 4998 \beta_{13} + 7381 \beta_{12} - 9352 \beta_{11} + 2857 \beta_{10} + 11597 \beta_{9} - 10098 \beta_{8} - 5144 \beta_{7} + 15336 \beta_{6} + 4158 \beta_{5} + 9812 \beta_{4} + 29497 \beta_{3} - 4892 \beta_{2} + 71651 \beta_{1} - 3564\)
\(\nu^{14}\)\(=\)\(13384 \beta_{19} + 22530 \beta_{18} - 14956 \beta_{17} + 14755 \beta_{16} - 14479 \beta_{15} + 29274 \beta_{14} + 4010 \beta_{13} - 12905 \beta_{12} - 4542 \beta_{11} + 14095 \beta_{10} + 13955 \beta_{9} - 5009 \beta_{8} - 20601 \beta_{7} + 5797 \beta_{6} + 28526 \beta_{5} + 50833 \beta_{4} - 3497 \beta_{3} + 118747 \beta_{2} - 45393 \beta_{1} + 378519\)
\(\nu^{15}\)\(=\)\(117785 \beta_{19} + 150054 \beta_{18} - 71434 \beta_{17} + 55290 \beta_{16} + 50746 \beta_{15} + 80782 \beta_{14} - 47767 \beta_{13} + 57941 \beta_{12} - 77366 \beta_{11} + 28192 \beta_{10} + 98627 \beta_{9} - 81521 \beta_{8} - 44718 \beta_{7} + 132887 \beta_{6} + 38293 \beta_{5} + 82225 \beta_{4} + 221776 \beta_{3} - 46055 \beta_{2} + 500887 \beta_{1} - 32922\)
\(\nu^{16}\)\(=\)\(113673 \beta_{19} + 182464 \beta_{18} - 126943 \beta_{17} + 147497 \beta_{16} - 120512 \beta_{15} + 251097 \beta_{14} + 42710 \beta_{13} - 108603 \beta_{12} - 48321 \beta_{11} + 122482 \beta_{10} + 120137 \beta_{9} - 46421 \beta_{8} - 166712 \beta_{7} + 59163 \beta_{6} + 241951 \beta_{5} + 399854 \beta_{4} - 35925 \beta_{3} + 880615 \beta_{2} - 391163 \beta_{1} + 2799099\)
\(\nu^{17}\)\(=\)\(977165 \beta_{19} + 1260560 \beta_{18} - 566566 \beta_{17} + 424748 \beta_{16} + 384398 \beta_{15} + 656953 \beta_{14} - 431490 \beta_{13} + 449294 \beta_{12} - 632839 \beta_{11} + 261711 \beta_{10} + 824566 \beta_{9} - 649433 \beta_{8} - 370153 \beta_{7} + 1124485 \beta_{6} + 334904 \beta_{5} + 680038 \beta_{4} + 1675901 \beta_{3} - 409578 \beta_{2} + 3555127 \beta_{1} - 292518\)
\(\nu^{18}\)\(=\)\(942530 \beta_{19} + 1444290 \beta_{18} - 1050191 \beta_{17} + 1387506 \beta_{16} - 974521 \beta_{15} + 2096617 \beta_{14} + 423211 \beta_{13} - 893741 \beta_{12} - 475861 \beta_{11} + 1039783 \beta_{10} + 1009051 \beta_{9} - 404469 \beta_{8} - 1328183 \beta_{7} + 568587 \beta_{6} + 1998264 \beta_{5} + 3134246 \beta_{4} - 345522 \beta_{3} + 6604802 \beta_{2} - 3279990 \beta_{1} + 20887382\)
\(\nu^{19}\)\(=\)\(7960315 \beta_{19} + 10386058 \beta_{18} - 4462429 \beta_{17} + 3248175 \beta_{16} + 2912926 \beta_{15} + 5273968 \beta_{14} - 3758411 \beta_{13} + 3459400 \beta_{12} - 5139162 \beta_{11} + 2333897 \beta_{10} + 6815068 \beta_{9} - 5138883 \beta_{8} - 2972779 \beta_{7} + 9364341 \beta_{6} + 2833876 \beta_{5} + 5575805 \beta_{4} + 12727477 \beta_{3} - 3513325 \beta_{2} + 25568807 \beta_{1} - 2533741\)

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
2.80377
2.62822
2.45061
2.44527
2.23669
1.74461
1.20801
0.762638
0.603983
0.458217
0.194668
−0.302778
−0.852674
−1.19744
−1.41510
−1.94926
−1.97506
−2.40422
−2.63931
−2.80084
−2.80377 0 5.86110 −2.52828 0 −2.67086 −10.8256 0 7.08870
1.2 −2.62822 0 4.90753 4.03060 0 −1.30880 −7.64164 0 −10.5933
1.3 −2.45061 0 4.00547 −1.21123 0 0.971463 −4.91461 0 2.96824
1.4 −2.44527 0 3.97935 1.74520 0 5.10283 −4.84005 0 −4.26748
1.5 −2.23669 0 3.00280 −1.57347 0 −1.14044 −2.24296 0 3.51938
1.6 −1.74461 0 1.04367 −0.892571 0 3.89655 1.66842 0 1.55719
1.7 −1.20801 0 −0.540713 −2.72981 0 −1.05258 3.06921 0 3.29764
1.8 −0.762638 0 −1.41838 4.10409 0 −2.37592 2.60699 0 −3.12993
1.9 −0.603983 0 −1.63520 2.98438 0 4.14035 2.19560 0 −1.80252
1.10 −0.458217 0 −1.79004 0.388274 0 −4.22376 1.73666 0 −0.177914
1.11 −0.194668 0 −1.96210 −4.10517 0 4.22277 0.771295 0 0.799144
1.12 0.302778 0 −1.90833 −3.34986 0 0.428154 −1.18336 0 −1.01426
1.13 0.852674 0 −1.27295 0.278169 0 2.70488 −2.79076 0 0.237188
1.14 1.19744 0 −0.566132 2.06209 0 −5.04192 −3.07279 0 2.46923
1.15 1.41510 0 0.00250280 −0.0661489 0 1.68271 −2.82665 0 −0.0936071
1.16 1.94926 0 1.79960 −3.69296 0 −2.45473 −0.390627 0 −7.19852
1.17 1.97506 0 1.90086 3.88400 0 2.64382 −0.195801 0 7.67114
1.18 2.40422 0 3.78028 −2.94625 0 2.89217 4.28018 0 −7.08343
1.19 2.63931 0 4.96597 1.13791 0 1.99715 7.82811 0 3.00330
1.20 2.80084 0 5.84471 3.48103 0 −1.41384 10.7684 0 9.74980
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{20} + \cdots\)
\(T_{5}^{20} - \cdots\)