Properties

Label 354.4.a.i
Level 354
Weight 4
Character orbit 354.a
Self dual Yes
Analytic conductor 20.887
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 354.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(20.886676142\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 3 + \beta_{1} ) q^{5} + 6 q^{6} + ( 4 + \beta_{4} ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 3 + \beta_{1} ) q^{5} + 6 q^{6} + ( 4 + \beta_{4} ) q^{7} + 8 q^{8} + 9 q^{9} + ( 6 + 2 \beta_{1} ) q^{10} + ( 11 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{11} + 12 q^{12} + ( 16 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{13} + ( 8 + 2 \beta_{4} ) q^{14} + ( 9 + 3 \beta_{1} ) q^{15} + 16 q^{16} + ( 39 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{17} + 18 q^{18} + ( 17 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + ( 12 + 4 \beta_{1} ) q^{20} + ( 12 + 3 \beta_{4} ) q^{21} + ( 22 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{22} + ( 16 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{23} + 24 q^{24} + ( 49 + \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{25} + ( 32 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} ) q^{26} + 27 q^{27} + ( 16 + 4 \beta_{4} ) q^{28} + ( 25 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} - 6 \beta_{4} ) q^{29} + ( 18 + 6 \beta_{1} ) q^{30} + ( 11 - 11 \beta_{1} - \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{31} + 32 q^{32} + ( 33 - 3 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} ) q^{33} + ( 78 - 6 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{34} + ( -19 + 8 \beta_{1} - \beta_{2} + 5 \beta_{4} + \beta_{5} ) q^{35} + 36 q^{36} + ( 1 - 13 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{37} + ( 34 - 6 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{38} + ( 48 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{5} ) q^{39} + ( 24 + 8 \beta_{1} ) q^{40} + ( 26 + 12 \beta_{1} - 8 \beta_{2} + \beta_{4} - 6 \beta_{5} ) q^{41} + ( 24 + 6 \beta_{4} ) q^{42} + ( 62 - 6 \beta_{1} - 10 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 44 - 4 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} ) q^{44} + ( 27 + 9 \beta_{1} ) q^{45} + ( 32 - 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 4 \beta_{5} ) q^{46} + ( -49 + 4 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} + 11 \beta_{4} + 5 \beta_{5} ) q^{47} + 48 q^{48} + ( -50 + 4 \beta_{1} + \beta_{2} - 7 \beta_{3} + 8 \beta_{4} - 7 \beta_{5} ) q^{49} + ( 98 + 2 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} ) q^{50} + ( 117 - 9 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} ) q^{51} + ( 64 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{5} ) q^{52} + ( -55 + 25 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 14 \beta_{4} - 6 \beta_{5} ) q^{53} + 54 q^{54} + ( -29 - 6 \beta_{1} + 9 \beta_{2} - \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{55} + ( 32 + 8 \beta_{4} ) q^{56} + ( 51 - 9 \beta_{1} - 6 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} ) q^{57} + ( 50 - 6 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} - 12 \beta_{4} ) q^{58} -59 q^{59} + ( 36 + 12 \beta_{1} ) q^{60} + ( 3 + 7 \beta_{1} + 8 \beta_{2} + 9 \beta_{3} - 6 \beta_{4} + 10 \beta_{5} ) q^{61} + ( 22 - 22 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} ) q^{62} + ( 36 + 9 \beta_{4} ) q^{63} + 64 q^{64} + ( -164 + 35 \beta_{1} - \beta_{2} - 10 \beta_{3} - 17 \beta_{4} + 4 \beta_{5} ) q^{65} + ( 66 - 6 \beta_{1} - 6 \beta_{4} + 6 \beta_{5} ) q^{66} + ( 9 + 20 \beta_{1} - 11 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{67} + ( 156 - 12 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{68} + ( 48 - 12 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} ) q^{69} + ( -38 + 16 \beta_{1} - 2 \beta_{2} + 10 \beta_{4} + 2 \beta_{5} ) q^{70} + ( -152 + 5 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} ) q^{71} + 72 q^{72} + ( 35 + 12 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 41 \beta_{4} + 5 \beta_{5} ) q^{73} + ( 2 - 26 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} ) q^{74} + ( 147 + 3 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{75} + ( 68 - 12 \beta_{1} - 8 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{76} + ( -78 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 15 \beta_{4} + 2 \beta_{5} ) q^{77} + ( 96 - 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{5} ) q^{78} + ( -98 + 12 \beta_{1} - 9 \beta_{3} + 11 \beta_{4} - 14 \beta_{5} ) q^{79} + ( 48 + 16 \beta_{1} ) q^{80} + 81 q^{81} + ( 52 + 24 \beta_{1} - 16 \beta_{2} + 2 \beta_{4} - 12 \beta_{5} ) q^{82} + ( 28 - 22 \beta_{1} - 14 \beta_{2} + 18 \beta_{3} - \beta_{4} ) q^{83} + ( 48 + 12 \beta_{4} ) q^{84} + ( -28 + 86 \beta_{1} - 28 \beta_{2} + 11 \beta_{3} + 4 \beta_{4} - 22 \beta_{5} ) q^{85} + ( 124 - 12 \beta_{1} - 20 \beta_{2} + 6 \beta_{3} + 14 \beta_{4} - 4 \beta_{5} ) q^{86} + ( 75 - 9 \beta_{1} + 12 \beta_{2} - 3 \beta_{3} - 18 \beta_{4} ) q^{87} + ( 88 - 8 \beta_{1} - 8 \beta_{4} + 8 \beta_{5} ) q^{88} + ( -286 - 43 \beta_{1} + 39 \beta_{2} - \beta_{3} + 9 \beta_{4} + 16 \beta_{5} ) q^{89} + ( 54 + 18 \beta_{1} ) q^{90} + ( -76 - 26 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} + 35 \beta_{4} + 8 \beta_{5} ) q^{91} + ( 64 - 16 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} + 8 \beta_{5} ) q^{92} + ( 33 - 33 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} ) q^{93} + ( -98 + 8 \beta_{1} + 18 \beta_{2} + 8 \beta_{3} + 22 \beta_{4} + 10 \beta_{5} ) q^{94} + ( -334 - 62 \beta_{1} + 14 \beta_{2} - 25 \beta_{3} - 26 \beta_{4} - 4 \beta_{5} ) q^{95} + 96 q^{96} + ( 9 - 26 \beta_{1} + 9 \beta_{2} + 15 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} ) q^{97} + ( -100 + 8 \beta_{1} + 2 \beta_{2} - 14 \beta_{3} + 16 \beta_{4} - 14 \beta_{5} ) q^{98} + ( 99 - 9 \beta_{1} - 9 \beta_{4} + 9 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 12q^{2} + 18q^{3} + 24q^{4} + 20q^{5} + 36q^{6} + 26q^{7} + 48q^{8} + 54q^{9} + O(q^{10}) \) \( 6q + 12q^{2} + 18q^{3} + 24q^{4} + 20q^{5} + 36q^{6} + 26q^{7} + 48q^{8} + 54q^{9} + 40q^{10} + 63q^{11} + 72q^{12} + 93q^{13} + 52q^{14} + 60q^{15} + 96q^{16} + 230q^{17} + 108q^{18} + 89q^{19} + 80q^{20} + 78q^{21} + 126q^{22} + 81q^{23} + 144q^{24} + 304q^{25} + 186q^{26} + 162q^{27} + 104q^{28} + 131q^{29} + 120q^{30} + 51q^{31} + 192q^{32} + 189q^{33} + 460q^{34} - 87q^{35} + 216q^{36} - 16q^{37} + 178q^{38} + 279q^{39} + 160q^{40} + 176q^{41} + 156q^{42} + 375q^{43} + 252q^{44} + 180q^{45} + 162q^{46} - 255q^{47} + 288q^{48} - 290q^{49} + 608q^{50} + 690q^{51} + 372q^{52} - 256q^{53} + 324q^{54} - 184q^{55} + 208q^{56} + 267q^{57} + 262q^{58} - 354q^{59} + 240q^{60} + 39q^{61} + 102q^{62} + 234q^{63} + 384q^{64} - 954q^{65} + 378q^{66} + 86q^{67} + 920q^{68} + 243q^{69} - 174q^{70} - 895q^{71} + 432q^{72} + 155q^{73} - 32q^{74} + 912q^{75} + 356q^{76} - 498q^{77} + 558q^{78} - 565q^{79} + 320q^{80} + 486q^{81} + 352q^{82} + 140q^{83} + 312q^{84} + q^{85} + 750q^{86} + 393q^{87} + 504q^{88} - 1769q^{89} + 360q^{90} - 422q^{91} + 324q^{92} + 153q^{93} - 510q^{94} - 2209q^{95} + 576q^{96} + 48q^{97} - 580q^{98} + 567q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 492 x^{4} + 3376 x^{3} + 13255 x^{2} - 108942 x + 106740\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -13 \nu^{5} - 193 \nu^{4} - 1329 \nu^{3} + 27671 \nu^{2} + 2400944 \nu - 5652168 \)\()/348948\)
\(\beta_{3}\)\(=\)\((\)\( 553 \nu^{5} - 7448 \nu^{4} - 312546 \nu^{3} + 5325430 \nu^{2} + 13689193 \nu - 234148170 \)\()/3315006\)
\(\beta_{4}\)\(=\)\((\)\( -7181 \nu^{5} - 8189 \nu^{4} + 3480099 \nu^{3} - 13275017 \nu^{2} - 124448624 \nu + 343017204 \)\()/6630012\)
\(\beta_{5}\)\(=\)\((\)\( -74 \nu^{5} - 353 \nu^{4} + 34935 \nu^{3} - 19199 \nu^{2} - 1502633 \nu + 1566708 \)\()/58158\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - \beta_{2} - 5 \beta_{1} + 165\)
\(\nu^{3}\)\(=\)\(13 \beta_{5} - 16 \beta_{4} - 15 \beta_{3} - 46 \beta_{2} + 414 \beta_{1} - 1327\)
\(\nu^{4}\)\(=\)\(-1137 \beta_{5} + 1623 \beta_{4} + 1803 \beta_{3} - 279 \beta_{2} - 4438 \beta_{1} + 69492\)
\(\nu^{5}\)\(=\)\(11294 \beta_{5} - 16074 \beta_{4} - 16720 \beta_{3} - 20126 \beta_{2} + 197609 \beta_{1} - 979602\)

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
−23.5704
−5.84847
1.19861
4.60566
7.93259
17.6821
2.00000 3.00000 4.00000 −20.5704 6.00000 10.6977 8.00000 9.00000 −41.1409
1.2 2.00000 3.00000 4.00000 −2.84847 6.00000 −2.00843 8.00000 9.00000 −5.69694
1.3 2.00000 3.00000 4.00000 4.19861 6.00000 31.2606 8.00000 9.00000 8.39722
1.4 2.00000 3.00000 4.00000 7.60566 6.00000 −24.7054 8.00000 9.00000 15.2113
1.5 2.00000 3.00000 4.00000 10.9326 6.00000 3.94504 8.00000 9.00000 21.8652
1.6 2.00000 3.00000 4.00000 20.6821 6.00000 6.81049 8.00000 9.00000 41.3641
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{6} - 20 T_{5}^{5} - 327 T_{5}^{4} + 8560 T_{5}^{3} - 41942 T_{5}^{2} - 46452 T_{5} + 423072 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\).