Properties

Label 5054.2.a.bl
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} - 2326 x^{3} + 570 x^{2} + 105 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} - 2326 x^{3} + 570 x^{2} + 105 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -447 \nu^{11} + 1225 \nu^{10} + 13139 \nu^{9} - 20343 \nu^{8} - 158096 \nu^{7} + 67730 \nu^{6} + 941476 \nu^{5} + 277033 \nu^{4} - 2660099 \nu^{3} - 1063803 \nu^{2} + 2788715 \nu - 530962 \)\()/162716\)
\(\beta_{3}\)\(=\)\((\)\( 839 \nu^{11} + 3252 \nu^{10} - 17654 \nu^{9} - 82762 \nu^{8} + 104083 \nu^{7} + 693188 \nu^{6} - 9086 \nu^{5} - 2103093 \nu^{4} - 941236 \nu^{3} + 1508292 \nu^{2} + 105206 \nu - 2181 \)\()/162716\)
\(\beta_{4}\)\(=\)\((\)\( -895 \nu^{11} + 7731 \nu^{10} + 18299 \nu^{9} - 193073 \nu^{8} - 119612 \nu^{7} + 1731830 \nu^{6} + 131584 \nu^{5} - 6538931 \nu^{4} + 1467335 \nu^{3} + 8749053 \nu^{2} - 4295959 \nu - 293578 \)\()/162716\)
\(\beta_{5}\)\(=\)\((\)\( -1286 \nu^{11} - 2027 \nu^{10} + 30793 \nu^{9} + 62419 \nu^{8} - 262179 \nu^{7} - 625458 \nu^{6} + 950562 \nu^{5} + 2380126 \nu^{4} - 1556147 \nu^{3} - 2734811 \nu^{2} + 1544497 \nu - 40633 \)\()/162716\)
\(\beta_{6}\)\(=\)\((\)\( 5046 \nu^{11} - 3181 \nu^{10} - 132213 \nu^{9} + 71023 \nu^{8} + 1235105 \nu^{7} - 596946 \nu^{6} - 4828586 \nu^{5} + 2366810 \nu^{4} + 6667283 \nu^{3} - 3785809 \nu^{2} - 199603 \nu + 895 \)\()/162716\)
\(\beta_{7}\)\(=\)\((\)\(-5411 \nu^{11} + 18833 \nu^{10} + 108087 \nu^{9} - 409517 \nu^{8} - 712334 \nu^{7} + 3270814 \nu^{6} + 1392036 \nu^{5} - 11448535 \nu^{4} + 2095789 \nu^{3} + 15008101 \nu^{2} - 7197671 \nu - 768532\)\()/162716\)
\(\beta_{8}\)\(=\)\((\)\( -4331 \nu^{11} + 10413 \nu^{10} + 101459 \nu^{9} - 215396 \nu^{8} - 877749 \nu^{7} + 1540074 \nu^{6} + 3329926 \nu^{5} - 4277477 \nu^{4} - 4674049 \nu^{3} + 3294301 \nu^{2} - 167712 \nu + 205005 \)\()/81358\)
\(\beta_{9}\)\(=\)\((\)\(10457 \nu^{11} - 22014 \nu^{10} - 240300 \nu^{9} + 480540 \nu^{8} + 1947439 \nu^{7} - 3867760 \nu^{6} - 6220622 \nu^{5} + 13815345 \nu^{4} + 4734210 \nu^{3} - 18793910 \nu^{2} + 5859056 \nu + 443995\)\()/162716\)
\(\beta_{10}\)\(=\)\((\)\(15220 \nu^{11} - 35795 \nu^{10} - 393225 \nu^{9} + 830535 \nu^{8} + 3775997 \nu^{7} - 6913710 \nu^{6} - 15853538 \nu^{5} + 24101664 \nu^{4} + 23736689 \nu^{3} - 29133113 \nu^{2} + 3574889 \nu + 701627\)\()/162716\)
\(\beta_{11}\)\(=\)\((\)\(25258 \nu^{11} - 82415 \nu^{10} - 600369 \nu^{9} + 1855779 \nu^{8} + 5308691 \nu^{7} - 14977174 \nu^{6} - 20339674 \nu^{5} + 50488138 \nu^{4} + 24957905 \nu^{3} - 58736565 \nu^{2} + 16250289 \nu + 1189669\)\()/162716\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{7} - \beta_{6} + 7 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(9 \beta_{9} + 10 \beta_{7} - 10 \beta_{6} - 14 \beta_{5} - 4 \beta_{4} - 9 \beta_{3} + 8 \beta_{2} + 2 \beta_{1} + 38\)
\(\nu^{5}\)\(=\)\(\beta_{11} - \beta_{10} + 12 \beta_{9} + 15 \beta_{7} - 13 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + \beta_{3} + 2 \beta_{2} + 54 \beta_{1} + 33\)
\(\nu^{6}\)\(=\)\(-\beta_{11} + \beta_{10} + 80 \beta_{9} - 3 \beta_{8} + 96 \beta_{7} - 97 \beta_{6} - 151 \beta_{5} - 62 \beta_{4} - 76 \beta_{3} + 66 \beta_{2} + 33 \beta_{1} + 311\)
\(\nu^{7}\)\(=\)\(9 \beta_{11} - 6 \beta_{10} + 132 \beta_{9} - 5 \beta_{8} + 177 \beta_{7} - 164 \beta_{6} - 130 \beta_{5} - 83 \beta_{4} + 3 \beta_{3} + 41 \beta_{2} + 436 \beta_{1} + 401\)
\(\nu^{8}\)\(=\)\(-31 \beta_{11} + 36 \beta_{10} + 728 \beta_{9} - 65 \beta_{8} + 911 \beta_{7} - 960 \beta_{6} - 1518 \beta_{5} - 749 \beta_{4} - 639 \beta_{3} + 567 \beta_{2} + 407 \beta_{1} + 2668\)
\(\nu^{9}\)\(=\)\(23 \beta_{11} + 42 \beta_{10} + 1409 \beta_{9} - 131 \beta_{8} + 1918 \beta_{7} - 1979 \beta_{6} - 1945 \beta_{5} - 1247 \beta_{4} - 92 \beta_{3} + 592 \beta_{2} + 3642 \beta_{1} + 4401\)
\(\nu^{10}\)\(=\)\(-561 \beta_{11} + 692 \beta_{10} + 6765 \beta_{9} - 981 \beta_{8} + 8642 \beta_{7} - 9667 \beta_{6} - 14996 \beta_{5} - 8337 \beta_{4} - 5413 \beta_{3} + 5067 \beta_{2} + 4515 \beta_{1} + 23676\)
\(\nu^{11}\)\(=\)\(-679 \beta_{11} + 1660 \beta_{10} + 14781 \beta_{9} - 2267 \beta_{8} + 20005 \beta_{7} - 22914 \beta_{6} - 25018 \beta_{5} - 16357 \beta_{4} - 2126 \beta_{3} + 7409 \beta_{2} + 31254 \beta_{1} + 46188\)

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
3.20919
2.98161
2.93688
2.22802
0.767178
0.572050
−0.0101006
−0.113874
−1.95625
−2.25339
−2.59018
−2.77114
−1.00000 −3.20919 1.00000 2.74348 3.20919 1.00000 −1.00000 7.29887 −2.74348
1.2 −1.00000 −2.98161 1.00000 −4.31337 2.98161 1.00000 −1.00000 5.89001 4.31337
1.3 −1.00000 −2.93688 1.00000 −0.167883 2.93688 1.00000 −1.00000 5.62525 0.167883
1.4 −1.00000 −2.22802 1.00000 3.08661 2.22802 1.00000 −1.00000 1.96409 −3.08661
1.5 −1.00000 −0.767178 1.00000 2.24084 0.767178 1.00000 −1.00000 −2.41144 −2.24084
1.6 −1.00000 −0.572050 1.00000 2.21950 0.572050 1.00000 −1.00000 −2.67276 −2.21950
1.7 −1.00000 0.0101006 1.00000 −3.08958 −0.0101006 1.00000 −1.00000 −2.99990 3.08958
1.8 −1.00000 0.113874 1.00000 −3.74313 −0.113874 1.00000 −1.00000 −2.98703 3.74313
1.9 −1.00000 1.95625 1.00000 −2.38328 −1.95625 1.00000 −1.00000 0.826915 2.38328
1.10 −1.00000 2.25339 1.00000 1.19857 −2.25339 1.00000 −1.00000 2.07775 −1.19857
1.11 −1.00000 2.59018 1.00000 3.85497 −2.59018 1.00000 −1.00000 3.70904 −3.85497
1.12 −1.00000 2.77114 1.00000 1.35329 −2.77114 1.00000 −1.00000 4.67919 −1.35329
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

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 5054.2.a.bl 12
19.b odd 2 1 5054.2.a.bm 12
19.e even 9 2 266.2.u.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.d 24 19.e even 9 2
5054.2.a.bl 12 1.a even 1 1 trivial
5054.2.a.bm 12 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\(T_{3}^{12} + \cdots\)
\(T_{5}^{12} - \cdots\)
\(T_{13}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( 1 - 105 T + 570 T^{2} + 2326 T^{3} + 1005 T^{4} - 1911 T^{5} - 814 T^{6} + 555 T^{7} + 213 T^{8} - 68 T^{9} - 24 T^{10} + 3 T^{11} + T^{12} \)
$5$ \( -5256 - 19872 T + 61866 T^{2} - 39705 T^{3} - 11682 T^{4} + 17952 T^{5} - 1847 T^{6} - 2571 T^{7} + 561 T^{8} + 149 T^{9} - 42 T^{10} - 3 T^{11} + T^{12} \)
$7$ \( ( -1 + T )^{12} \)
$11$ \( 56832 - 407808 T + 1024704 T^{2} - 1019616 T^{3} + 227280 T^{4} + 180156 T^{5} - 57907 T^{6} - 11457 T^{7} + 3885 T^{8} + 311 T^{9} - 105 T^{10} - 3 T^{11} + T^{12} \)
$13$ \( 2483704 + 2784300 T - 973740 T^{2} - 1350445 T^{3} + 212895 T^{4} + 235059 T^{5} - 33436 T^{6} - 17346 T^{7} + 2613 T^{8} + 545 T^{9} - 87 T^{10} - 6 T^{11} + T^{12} \)
$17$ \( 4198848 + 1938240 T - 2832624 T^{2} - 877824 T^{3} + 742656 T^{4} + 126840 T^{5} - 85819 T^{6} - 8226 T^{7} + 4644 T^{8} + 253 T^{9} - 114 T^{10} - 3 T^{11} + T^{12} \)
$19$ \( T^{12} \)
$23$ \( -20544 + 148608 T + 202752 T^{2} - 746976 T^{3} - 314796 T^{4} + 433500 T^{5} + 29365 T^{6} - 44574 T^{7} + 1149 T^{8} + 1378 T^{9} - 90 T^{10} - 12 T^{11} + T^{12} \)
$29$ \( -3319296 - 499968 T + 6757632 T^{2} + 3637248 T^{3} - 1185888 T^{4} - 987696 T^{5} - 45608 T^{6} + 57420 T^{7} + 6702 T^{8} - 1087 T^{9} - 159 T^{10} + 6 T^{11} + T^{12} \)
$31$ \( -32997376 + 63111168 T - 20318208 T^{2} - 22541824 T^{3} + 13820160 T^{4} - 81024 T^{5} - 937600 T^{6} + 33288 T^{7} + 23772 T^{8} - 609 T^{9} - 258 T^{10} + 3 T^{11} + T^{12} \)
$37$ \( -7091712 + 57853440 T - 29144448 T^{2} - 52622784 T^{3} - 10980576 T^{4} + 3228480 T^{5} + 1224640 T^{6} + 27588 T^{7} - 29358 T^{8} - 3049 T^{9} + 102 T^{10} + 27 T^{11} + T^{12} \)
$41$ \( 1993061184 - 1389839616 T + 32194080 T^{2} + 170312640 T^{3} - 31323660 T^{4} - 5590944 T^{5} + 1956017 T^{6} - 37065 T^{7} - 38115 T^{8} + 3767 T^{9} + 87 T^{10} - 27 T^{11} + T^{12} \)
$43$ \( -21992896 + 46324224 T - 24208032 T^{2} - 11645776 T^{3} + 16778196 T^{4} - 6437484 T^{5} + 814205 T^{6} + 95658 T^{7} - 35829 T^{8} + 2352 T^{9} + 195 T^{10} - 30 T^{11} + T^{12} \)
$47$ \( 6744576 - 48072960 T + 26509824 T^{2} + 9674688 T^{3} - 7498272 T^{4} - 191664 T^{5} + 660136 T^{6} - 61176 T^{7} - 18492 T^{8} + 3283 T^{9} - 30 T^{10} - 21 T^{11} + T^{12} \)
$53$ \( 332453376 - 656024832 T - 44737920 T^{2} + 204379776 T^{3} + 28773792 T^{4} - 12119904 T^{5} - 1831672 T^{6} + 253992 T^{7} + 39552 T^{8} - 2105 T^{9} - 339 T^{10} + 6 T^{11} + T^{12} \)
$59$ \( -11454423 - 20924055 T - 283536 T^{2} + 11070456 T^{3} - 297066 T^{4} - 2116008 T^{5} + 398245 T^{6} + 87000 T^{7} - 33246 T^{8} + 2783 T^{9} + 117 T^{10} - 27 T^{11} + T^{12} \)
$61$ \( 531001664 + 395034528 T - 272932944 T^{2} - 229013416 T^{3} + 2953764 T^{4} + 15658794 T^{5} - 100501 T^{6} - 416340 T^{7} + 11049 T^{8} + 4656 T^{9} - 204 T^{10} - 18 T^{11} + T^{12} \)
$67$ \( 7338176 - 32211456 T + 50848176 T^{2} - 36082992 T^{3} + 10041444 T^{4} + 825672 T^{5} - 907067 T^{6} + 61785 T^{7} + 25785 T^{8} - 1799 T^{9} - 285 T^{10} + 9 T^{11} + T^{12} \)
$71$ \( 127877837592 + 72066556188 T + 4111442136 T^{2} - 4804204227 T^{3} - 1044178662 T^{4} - 3000972 T^{5} + 19345369 T^{6} + 1610751 T^{7} - 66417 T^{8} - 12963 T^{9} - 234 T^{10} + 27 T^{11} + T^{12} \)
$73$ \( 3241283264 + 5910814272 T - 4151072880 T^{2} + 102148688 T^{3} + 236495604 T^{4} - 14676876 T^{5} - 5595805 T^{6} + 330645 T^{7} + 67527 T^{8} - 2901 T^{9} - 411 T^{10} + 9 T^{11} + T^{12} \)
$79$ \( -979192 - 13061652 T - 34553340 T^{2} - 21589185 T^{3} + 10137825 T^{4} + 4793193 T^{5} - 1071902 T^{6} - 225579 T^{7} + 31074 T^{8} + 3043 T^{9} - 306 T^{10} - 12 T^{11} + T^{12} \)
$83$ \( -6892045047 - 8570809395 T - 2497046094 T^{2} + 430316958 T^{3} + 233477460 T^{4} + 409416 T^{5} - 6846227 T^{6} - 248514 T^{7} + 83880 T^{8} + 2913 T^{9} - 471 T^{10} - 9 T^{11} + T^{12} \)
$89$ \( -6220575552 - 594619200 T + 2138932800 T^{2} - 42147312 T^{3} - 217418820 T^{4} + 27514416 T^{5} + 4734179 T^{6} - 911832 T^{7} - 14457 T^{8} + 8816 T^{9} - 231 T^{10} - 24 T^{11} + T^{12} \)
$97$ \( -6948749312 - 2686992384 T + 3766081536 T^{2} + 371656192 T^{3} - 669920256 T^{4} + 116527680 T^{5} + 3177312 T^{6} - 1995552 T^{7} + 64584 T^{8} + 10651 T^{9} - 525 T^{10} - 18 T^{11} + T^{12} \)
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