Properties

Label 354.4.a.a
Level 354
Weight 4
Character orbit 354.a
Self dual Yes
Analytic conductor 20.887
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -5 - 2 \beta ) q^{5} -6 q^{6} + ( -10 - 3 \beta ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})\) \( q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -5 - 2 \beta ) q^{5} -6 q^{6} + ( -10 - 3 \beta ) q^{7} -8 q^{8} + 9 q^{9} + ( 10 + 4 \beta ) q^{10} + ( 25 + 8 \beta ) q^{11} + 12 q^{12} + ( -9 + 26 \beta ) q^{13} + ( 20 + 6 \beta ) q^{14} + ( -15 - 6 \beta ) q^{15} + 16 q^{16} + ( -9 + 25 \beta ) q^{17} -18 q^{18} + ( -41 - 41 \beta ) q^{19} + ( -20 - 8 \beta ) q^{20} + ( -30 - 9 \beta ) q^{21} + ( -50 - 16 \beta ) q^{22} + ( -23 - 35 \beta ) q^{23} -24 q^{24} + ( -80 + 24 \beta ) q^{25} + ( 18 - 52 \beta ) q^{26} + 27 q^{27} + ( -40 - 12 \beta ) q^{28} + ( -68 + 9 \beta ) q^{29} + ( 30 + 12 \beta ) q^{30} + ( -194 + 21 \beta ) q^{31} -32 q^{32} + ( 75 + 24 \beta ) q^{33} + ( 18 - 50 \beta ) q^{34} + ( 80 + 41 \beta ) q^{35} + 36 q^{36} + ( -103 - 43 \beta ) q^{37} + ( 82 + 82 \beta ) q^{38} + ( -27 + 78 \beta ) q^{39} + ( 40 + 16 \beta ) q^{40} + ( 112 - 13 \beta ) q^{41} + ( 60 + 18 \beta ) q^{42} + ( -87 - 96 \beta ) q^{43} + ( 100 + 32 \beta ) q^{44} + ( -45 - 18 \beta ) q^{45} + ( 46 + 70 \beta ) q^{46} + ( 152 - 151 \beta ) q^{47} + 48 q^{48} + ( -198 + 69 \beta ) q^{49} + ( 160 - 48 \beta ) q^{50} + ( -27 + 75 \beta ) q^{51} + ( -36 + 104 \beta ) q^{52} + ( -143 + 26 \beta ) q^{53} -54 q^{54} + ( -205 - 106 \beta ) q^{55} + ( 80 + 24 \beta ) q^{56} + ( -123 - 123 \beta ) q^{57} + ( 136 - 18 \beta ) q^{58} -59 q^{59} + ( -60 - 24 \beta ) q^{60} + ( -504 + 93 \beta ) q^{61} + ( 388 - 42 \beta ) q^{62} + ( -90 - 27 \beta ) q^{63} + 64 q^{64} + ( -215 - 164 \beta ) q^{65} + ( -150 - 48 \beta ) q^{66} + ( -519 + 308 \beta ) q^{67} + ( -36 + 100 \beta ) q^{68} + ( -69 - 105 \beta ) q^{69} + ( -160 - 82 \beta ) q^{70} + ( -395 + 350 \beta ) q^{71} -72 q^{72} + ( -32 - 71 \beta ) q^{73} + ( 206 + 86 \beta ) q^{74} + ( -240 + 72 \beta ) q^{75} + ( -164 - 164 \beta ) q^{76} + ( -370 - 179 \beta ) q^{77} + ( 54 - 156 \beta ) q^{78} + ( -721 - 106 \beta ) q^{79} + ( -80 - 32 \beta ) q^{80} + 81 q^{81} + ( -224 + 26 \beta ) q^{82} + ( -138 + 187 \beta ) q^{83} + ( -120 - 36 \beta ) q^{84} + ( -205 - 157 \beta ) q^{85} + ( 174 + 192 \beta ) q^{86} + ( -204 + 27 \beta ) q^{87} + ( -200 - 64 \beta ) q^{88} + ( -348 + 133 \beta ) q^{89} + ( 90 + 36 \beta ) q^{90} + ( -300 - 311 \beta ) q^{91} + ( -92 - 140 \beta ) q^{92} + ( -582 + 63 \beta ) q^{93} + ( -304 + 302 \beta ) q^{94} + ( 615 + 369 \beta ) q^{95} -96 q^{96} + ( 769 - 8 \beta ) q^{97} + ( 396 - 138 \beta ) q^{98} + ( 225 + 72 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} - 23q^{7} - 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} - 23q^{7} - 16q^{8} + 18q^{9} + 24q^{10} + 58q^{11} + 24q^{12} + 8q^{13} + 46q^{14} - 36q^{15} + 32q^{16} + 7q^{17} - 36q^{18} - 123q^{19} - 48q^{20} - 69q^{21} - 116q^{22} - 81q^{23} - 48q^{24} - 136q^{25} - 16q^{26} + 54q^{27} - 92q^{28} - 127q^{29} + 72q^{30} - 367q^{31} - 64q^{32} + 174q^{33} - 14q^{34} + 201q^{35} + 72q^{36} - 249q^{37} + 246q^{38} + 24q^{39} + 96q^{40} + 211q^{41} + 138q^{42} - 270q^{43} + 232q^{44} - 108q^{45} + 162q^{46} + 153q^{47} + 96q^{48} - 327q^{49} + 272q^{50} + 21q^{51} + 32q^{52} - 260q^{53} - 108q^{54} - 516q^{55} + 184q^{56} - 369q^{57} + 254q^{58} - 118q^{59} - 144q^{60} - 915q^{61} + 734q^{62} - 207q^{63} + 128q^{64} - 594q^{65} - 348q^{66} - 730q^{67} + 28q^{68} - 243q^{69} - 402q^{70} - 440q^{71} - 144q^{72} - 135q^{73} + 498q^{74} - 408q^{75} - 492q^{76} - 919q^{77} - 48q^{78} - 1548q^{79} - 192q^{80} + 162q^{81} - 422q^{82} - 89q^{83} - 276q^{84} - 567q^{85} + 540q^{86} - 381q^{87} - 464q^{88} - 563q^{89} + 216q^{90} - 911q^{91} - 324q^{92} - 1101q^{93} - 306q^{94} + 1599q^{95} - 192q^{96} + 1530q^{97} + 654q^{98} + 522q^{99} + O(q^{100}) \)

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
2.79129
−1.79129
−2.00000 3.00000 4.00000 −10.5826 −6.00000 −18.3739 −8.00000 9.00000 21.1652
1.2 −2.00000 3.00000 4.00000 −1.41742 −6.00000 −4.62614 −8.00000 9.00000 2.83485
\(n\): 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 \( T_{5}^{2} + 12 T_{5} + 15 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\).