Properties

Label 354.4.a.b
Level 354
Weight 4
Character orbit 354.a
Self dual Yes
Analytic conductor 20.887
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{51}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{51}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( 13 + \beta ) q^{5} -6 q^{6} -4 \beta q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})\) \( q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( 13 + \beta ) q^{5} -6 q^{6} -4 \beta q^{7} -8 q^{8} + 9 q^{9} + ( -26 - 2 \beta ) q^{10} + ( 26 + 4 \beta ) q^{11} + 12 q^{12} + ( 33 - 7 \beta ) q^{13} + 8 \beta q^{14} + ( 39 + 3 \beta ) q^{15} + 16 q^{16} + ( -32 + 6 \beta ) q^{17} -18 q^{18} + 34 q^{19} + ( 52 + 4 \beta ) q^{20} -12 \beta q^{21} + ( -52 - 8 \beta ) q^{22} + ( 58 + 6 \beta ) q^{23} -24 q^{24} + ( 95 + 26 \beta ) q^{25} + ( -66 + 14 \beta ) q^{26} + 27 q^{27} -16 \beta q^{28} + ( 175 - 9 \beta ) q^{29} + ( -78 - 6 \beta ) q^{30} + ( -121 + 17 \beta ) q^{31} -32 q^{32} + ( 78 + 12 \beta ) q^{33} + ( 64 - 12 \beta ) q^{34} + ( -204 - 52 \beta ) q^{35} + 36 q^{36} + ( -319 - 7 \beta ) q^{37} -68 q^{38} + ( 99 - 21 \beta ) q^{39} + ( -104 - 8 \beta ) q^{40} + ( -308 - 18 \beta ) q^{41} + 24 \beta q^{42} + ( 384 - 8 \beta ) q^{43} + ( 104 + 16 \beta ) q^{44} + ( 117 + 9 \beta ) q^{45} + ( -116 - 12 \beta ) q^{46} + ( 22 + 38 \beta ) q^{47} + 48 q^{48} + 473 q^{49} + ( -190 - 52 \beta ) q^{50} + ( -96 + 18 \beta ) q^{51} + ( 132 - 28 \beta ) q^{52} + ( 45 + 9 \beta ) q^{53} -54 q^{54} + ( 542 + 78 \beta ) q^{55} + 32 \beta q^{56} + 102 q^{57} + ( -350 + 18 \beta ) q^{58} + 59 q^{59} + ( 156 + 12 \beta ) q^{60} + ( 625 - 23 \beta ) q^{61} + ( 242 - 34 \beta ) q^{62} -36 \beta q^{63} + 64 q^{64} + ( 72 - 58 \beta ) q^{65} + ( -156 - 24 \beta ) q^{66} + ( 730 - 14 \beta ) q^{67} + ( -128 + 24 \beta ) q^{68} + ( 174 + 18 \beta ) q^{69} + ( 408 + 104 \beta ) q^{70} + ( 81 + 127 \beta ) q^{71} -72 q^{72} + ( 142 - 108 \beta ) q^{73} + ( 638 + 14 \beta ) q^{74} + ( 285 + 78 \beta ) q^{75} + 136 q^{76} + ( -816 - 104 \beta ) q^{77} + ( -198 + 42 \beta ) q^{78} + ( -256 + 136 \beta ) q^{79} + ( 208 + 16 \beta ) q^{80} + 81 q^{81} + ( 616 + 36 \beta ) q^{82} + ( 188 - 96 \beta ) q^{83} -48 \beta q^{84} + ( -110 + 46 \beta ) q^{85} + ( -768 + 16 \beta ) q^{86} + ( 525 - 27 \beta ) q^{87} + ( -208 - 32 \beta ) q^{88} + ( -246 + 90 \beta ) q^{89} + ( -234 - 18 \beta ) q^{90} + ( 1428 - 132 \beta ) q^{91} + ( 232 + 24 \beta ) q^{92} + ( -363 + 51 \beta ) q^{93} + ( -44 - 76 \beta ) q^{94} + ( 442 + 34 \beta ) q^{95} -96 q^{96} + ( -544 - 130 \beta ) q^{97} -946 q^{98} + ( 234 + 36 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 6q^{3} + 8q^{4} + 26q^{5} - 12q^{6} - 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 6q^{3} + 8q^{4} + 26q^{5} - 12q^{6} - 16q^{8} + 18q^{9} - 52q^{10} + 52q^{11} + 24q^{12} + 66q^{13} + 78q^{15} + 32q^{16} - 64q^{17} - 36q^{18} + 68q^{19} + 104q^{20} - 104q^{22} + 116q^{23} - 48q^{24} + 190q^{25} - 132q^{26} + 54q^{27} + 350q^{29} - 156q^{30} - 242q^{31} - 64q^{32} + 156q^{33} + 128q^{34} - 408q^{35} + 72q^{36} - 638q^{37} - 136q^{38} + 198q^{39} - 208q^{40} - 616q^{41} + 768q^{43} + 208q^{44} + 234q^{45} - 232q^{46} + 44q^{47} + 96q^{48} + 946q^{49} - 380q^{50} - 192q^{51} + 264q^{52} + 90q^{53} - 108q^{54} + 1084q^{55} + 204q^{57} - 700q^{58} + 118q^{59} + 312q^{60} + 1250q^{61} + 484q^{62} + 128q^{64} + 144q^{65} - 312q^{66} + 1460q^{67} - 256q^{68} + 348q^{69} + 816q^{70} + 162q^{71} - 144q^{72} + 284q^{73} + 1276q^{74} + 570q^{75} + 272q^{76} - 1632q^{77} - 396q^{78} - 512q^{79} + 416q^{80} + 162q^{81} + 1232q^{82} + 376q^{83} - 220q^{85} - 1536q^{86} + 1050q^{87} - 416q^{88} - 492q^{89} - 468q^{90} + 2856q^{91} + 464q^{92} - 726q^{93} - 88q^{94} + 884q^{95} - 192q^{96} - 1088q^{97} - 1892q^{98} + 468q^{99} + O(q^{100}) \)

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
−7.14143
7.14143
−2.00000 3.00000 4.00000 5.85857 −6.00000 28.5657 −8.00000 9.00000 −11.7171
1.2 −2.00000 3.00000 4.00000 20.1414 −6.00000 −28.5657 −8.00000 9.00000 −40.2829
\(n\): e.g. 2-40 or 990-1000
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 \( T_{5}^{2} - 26 T_{5} + 118 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\).