Properties

Label 354.4.a.g
Level 354
Weight 4
Character orbit 354.a
Self dual Yes
Analytic conductor 20.887
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: 3.3.47277.1
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( -6 + \beta_{1} + \beta_{2} ) q^{5} -6 q^{6} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})\) \( q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( -6 + \beta_{1} + \beta_{2} ) q^{5} -6 q^{6} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{7} + 8 q^{8} + 9 q^{9} + ( -12 + 2 \beta_{1} + 2 \beta_{2} ) q^{10} + ( -3 + 6 \beta_{1} - 2 \beta_{2} ) q^{11} -12 q^{12} + ( -15 - 4 \beta_{1} + 4 \beta_{2} ) q^{13} + ( 4 - 6 \beta_{1} - 4 \beta_{2} ) q^{14} + ( 18 - 3 \beta_{1} - 3 \beta_{2} ) q^{15} + 16 q^{16} + ( -32 - 2 \beta_{1} - 9 \beta_{2} ) q^{17} + 18 q^{18} + ( -57 - 5 \beta_{1} + 6 \beta_{2} ) q^{19} + ( -24 + 4 \beta_{1} + 4 \beta_{2} ) q^{20} + ( -6 + 9 \beta_{1} + 6 \beta_{2} ) q^{21} + ( -6 + 12 \beta_{1} - 4 \beta_{2} ) q^{22} + ( -59 + 13 \beta_{1} + 12 \beta_{2} ) q^{23} -24 q^{24} + ( -23 - 9 \beta_{1} - 17 \beta_{2} ) q^{25} + ( -30 - 8 \beta_{1} + 8 \beta_{2} ) q^{26} -27 q^{27} + ( 8 - 12 \beta_{1} - 8 \beta_{2} ) q^{28} + ( -117 + 14 \beta_{1} + 9 \beta_{2} ) q^{29} + ( 36 - 6 \beta_{1} - 6 \beta_{2} ) q^{30} + ( -89 - 30 \beta_{1} - 5 \beta_{2} ) q^{31} + 32 q^{32} + ( 9 - 18 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -64 - 4 \beta_{1} - 18 \beta_{2} ) q^{34} + ( -163 + 8 \beta_{1} + 23 \beta_{2} ) q^{35} + 36 q^{36} + ( -74 + 6 \beta_{1} - 23 \beta_{2} ) q^{37} + ( -114 - 10 \beta_{1} + 12 \beta_{2} ) q^{38} + ( 45 + 12 \beta_{1} - 12 \beta_{2} ) q^{39} + ( -48 + 8 \beta_{1} + 8 \beta_{2} ) q^{40} + ( -260 - 5 \beta_{1} + 12 \beta_{2} ) q^{41} + ( -12 + 18 \beta_{1} + 12 \beta_{2} ) q^{42} + ( 7 + 30 \beta_{1} + 20 \beta_{2} ) q^{43} + ( -12 + 24 \beta_{1} - 8 \beta_{2} ) q^{44} + ( -54 + 9 \beta_{1} + 9 \beta_{2} ) q^{45} + ( -118 + 26 \beta_{1} + 24 \beta_{2} ) q^{46} + ( -239 + 26 \beta_{1} + 17 \beta_{2} ) q^{47} -48 q^{48} + ( 33 + 26 \beta_{1} - 21 \beta_{2} ) q^{49} + ( -46 - 18 \beta_{1} - 34 \beta_{2} ) q^{50} + ( 96 + 6 \beta_{1} + 27 \beta_{2} ) q^{51} + ( -60 - 16 \beta_{1} + 16 \beta_{2} ) q^{52} + ( -78 - 9 \beta_{1} - 9 \beta_{2} ) q^{53} -54 q^{54} + ( 38 + 3 \beta_{1} + 27 \beta_{2} ) q^{55} + ( 16 - 24 \beta_{1} - 16 \beta_{2} ) q^{56} + ( 171 + 15 \beta_{1} - 18 \beta_{2} ) q^{57} + ( -234 + 28 \beta_{1} + 18 \beta_{2} ) q^{58} -59 q^{59} + ( 72 - 12 \beta_{1} - 12 \beta_{2} ) q^{60} + ( 205 - 26 \beta_{1} - 49 \beta_{2} ) q^{61} + ( -178 - 60 \beta_{1} - 10 \beta_{2} ) q^{62} + ( 18 - 27 \beta_{1} - 18 \beta_{2} ) q^{63} + 64 q^{64} + ( 202 - 27 \beta_{1} - 67 \beta_{2} ) q^{65} + ( 18 - 36 \beta_{1} + 12 \beta_{2} ) q^{66} + ( -48 - 43 \beta_{1} - 13 \beta_{2} ) q^{67} + ( -128 - 8 \beta_{1} - 36 \beta_{2} ) q^{68} + ( 177 - 39 \beta_{1} - 36 \beta_{2} ) q^{69} + ( -326 + 16 \beta_{1} + 46 \beta_{2} ) q^{70} + ( 327 - 34 \beta_{1} - 2 \beta_{2} ) q^{71} + 72 q^{72} + ( -21 - 32 \beta_{1} + 33 \beta_{2} ) q^{73} + ( -148 + 12 \beta_{1} - 46 \beta_{2} ) q^{74} + ( 69 + 27 \beta_{1} + 51 \beta_{2} ) q^{75} + ( -228 - 20 \beta_{1} + 24 \beta_{2} ) q^{76} + ( -264 - 67 \beta_{1} - 56 \beta_{2} ) q^{77} + ( 90 + 24 \beta_{1} - 24 \beta_{2} ) q^{78} + ( 185 - 32 \beta_{1} - 106 \beta_{2} ) q^{79} + ( -96 + 16 \beta_{1} + 16 \beta_{2} ) q^{80} + 81 q^{81} + ( -520 - 10 \beta_{1} + 24 \beta_{2} ) q^{82} + ( 22 + 153 \beta_{1} + 126 \beta_{2} ) q^{83} + ( -24 + 36 \beta_{1} + 24 \beta_{2} ) q^{84} + ( -269 - 5 \beta_{1} + 74 \beta_{2} ) q^{85} + ( 14 + 60 \beta_{1} + 40 \beta_{2} ) q^{86} + ( 351 - 42 \beta_{1} - 27 \beta_{2} ) q^{87} + ( -24 + 48 \beta_{1} - 16 \beta_{2} ) q^{88} + ( -463 - 158 \beta_{1} - 15 \beta_{2} ) q^{89} + ( -108 + 18 \beta_{1} + 18 \beta_{2} ) q^{90} + ( -74 + 101 \beta_{1} + 114 \beta_{2} ) q^{91} + ( -236 + 52 \beta_{1} + 48 \beta_{2} ) q^{92} + ( 267 + 90 \beta_{1} + 15 \beta_{2} ) q^{93} + ( -478 + 52 \beta_{1} + 34 \beta_{2} ) q^{94} + ( 529 - 75 \beta_{1} - 134 \beta_{2} ) q^{95} -96 q^{96} + ( 320 - 91 \beta_{1} - 29 \beta_{2} ) q^{97} + ( 66 + 52 \beta_{1} - 42 \beta_{2} ) q^{98} + ( -27 + 54 \beta_{1} - 18 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 6q^{2} - 9q^{3} + 12q^{4} - 18q^{5} - 18q^{6} + 6q^{7} + 24q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + 6q^{2} - 9q^{3} + 12q^{4} - 18q^{5} - 18q^{6} + 6q^{7} + 24q^{8} + 27q^{9} - 36q^{10} - 9q^{11} - 36q^{12} - 45q^{13} + 12q^{14} + 54q^{15} + 48q^{16} - 96q^{17} + 54q^{18} - 171q^{19} - 72q^{20} - 18q^{21} - 18q^{22} - 177q^{23} - 72q^{24} - 69q^{25} - 90q^{26} - 81q^{27} + 24q^{28} - 351q^{29} + 108q^{30} - 267q^{31} + 96q^{32} + 27q^{33} - 192q^{34} - 489q^{35} + 108q^{36} - 222q^{37} - 342q^{38} + 135q^{39} - 144q^{40} - 780q^{41} - 36q^{42} + 21q^{43} - 36q^{44} - 162q^{45} - 354q^{46} - 717q^{47} - 144q^{48} + 99q^{49} - 138q^{50} + 288q^{51} - 180q^{52} - 234q^{53} - 162q^{54} + 114q^{55} + 48q^{56} + 513q^{57} - 702q^{58} - 177q^{59} + 216q^{60} + 615q^{61} - 534q^{62} + 54q^{63} + 192q^{64} + 606q^{65} + 54q^{66} - 144q^{67} - 384q^{68} + 531q^{69} - 978q^{70} + 981q^{71} + 216q^{72} - 63q^{73} - 444q^{74} + 207q^{75} - 684q^{76} - 792q^{77} + 270q^{78} + 555q^{79} - 288q^{80} + 243q^{81} - 1560q^{82} + 66q^{83} - 72q^{84} - 807q^{85} + 42q^{86} + 1053q^{87} - 72q^{88} - 1389q^{89} - 324q^{90} - 222q^{91} - 708q^{92} + 801q^{93} - 1434q^{94} + 1587q^{95} - 288q^{96} + 960q^{97} + 198q^{98} - 81q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 48 x - 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 2 \nu - 32 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2} + 2 \beta_{1} + 32\)

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
−0.523828
−6.65142
7.17525
2.00000 −3.00000 4.00000 −16.7498 −6.00000 24.0234 8.00000 9.00000 −33.4996
1.2 2.00000 −3.00000 4.00000 −4.13667 −6.00000 4.92477 8.00000 9.00000 −8.27334
1.3 2.00000 −3.00000 4.00000 2.88648 −6.00000 −22.9482 8.00000 9.00000 5.77297
\(n\): 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 \( T_{5}^{3} + 18 T_{5}^{2} + 9 T_{5} - 200 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\).