Properties

Label 354.4.a.c
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: \(\Q(\sqrt{13}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( -9 - 2 \beta ) q^{5} + 6 q^{6} + ( -16 + 9 \beta ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( -9 - 2 \beta ) q^{5} + 6 q^{6} + ( -16 + 9 \beta ) q^{7} + 8 q^{8} + 9 q^{9} + ( -18 - 4 \beta ) q^{10} + ( -35 - 10 \beta ) q^{11} + 12 q^{12} + ( -41 - 20 \beta ) q^{13} + ( -32 + 18 \beta ) q^{14} + ( -27 - 6 \beta ) q^{15} + 16 q^{16} + ( -11 - 3 \beta ) q^{17} + 18 q^{18} + ( -27 - \beta ) q^{19} + ( -36 - 8 \beta ) q^{20} + ( -48 + 27 \beta ) q^{21} + ( -70 - 20 \beta ) q^{22} + ( -51 + 97 \beta ) q^{23} + 24 q^{24} + ( -32 + 40 \beta ) q^{25} + ( -82 - 40 \beta ) q^{26} + 27 q^{27} + ( -64 + 36 \beta ) q^{28} + ( -34 + 33 \beta ) q^{29} + ( -54 - 12 \beta ) q^{30} + ( 46 - 145 \beta ) q^{31} + 32 q^{32} + ( -105 - 30 \beta ) q^{33} + ( -22 - 6 \beta ) q^{34} + ( 90 - 67 \beta ) q^{35} + 36 q^{36} + ( -135 + 49 \beta ) q^{37} + ( -54 - 2 \beta ) q^{38} + ( -123 - 60 \beta ) q^{39} + ( -72 - 16 \beta ) q^{40} + ( -146 + 203 \beta ) q^{41} + ( -96 + 54 \beta ) q^{42} + ( -207 - 94 \beta ) q^{43} + ( -140 - 40 \beta ) q^{44} + ( -81 - 18 \beta ) q^{45} + ( -102 + 194 \beta ) q^{46} + ( 372 - 165 \beta ) q^{47} + 48 q^{48} + ( 156 - 207 \beta ) q^{49} + ( -64 + 80 \beta ) q^{50} + ( -33 - 9 \beta ) q^{51} + ( -164 - 80 \beta ) q^{52} + ( 325 - 218 \beta ) q^{53} + 54 q^{54} + ( 375 + 180 \beta ) q^{55} + ( -128 + 72 \beta ) q^{56} + ( -81 - 3 \beta ) q^{57} + ( -68 + 66 \beta ) q^{58} + 59 q^{59} + ( -108 - 24 \beta ) q^{60} + ( 36 + 79 \beta ) q^{61} + ( 92 - 290 \beta ) q^{62} + ( -144 + 81 \beta ) q^{63} + 64 q^{64} + ( 489 + 302 \beta ) q^{65} + ( -210 - 60 \beta ) q^{66} + ( -263 + 358 \beta ) q^{67} + ( -44 - 12 \beta ) q^{68} + ( -153 + 291 \beta ) q^{69} + ( 180 - 134 \beta ) q^{70} + ( 181 + 170 \beta ) q^{71} + 72 q^{72} + ( 36 - 121 \beta ) q^{73} + ( -270 + 98 \beta ) q^{74} + ( -96 + 120 \beta ) q^{75} + ( -108 - 4 \beta ) q^{76} + ( 290 - 245 \beta ) q^{77} + ( -246 - 120 \beta ) q^{78} + ( -261 - 142 \beta ) q^{79} + ( -144 - 32 \beta ) q^{80} + 81 q^{81} + ( -292 + 406 \beta ) q^{82} + ( 22 - 395 \beta ) q^{83} + ( -192 + 108 \beta ) q^{84} + ( 117 + 55 \beta ) q^{85} + ( -414 - 188 \beta ) q^{86} + ( -102 + 99 \beta ) q^{87} + ( -280 - 80 \beta ) q^{88} + ( -776 + 151 \beta ) q^{89} + ( -162 - 36 \beta ) q^{90} + ( 116 - 229 \beta ) q^{91} + ( -204 + 388 \beta ) q^{92} + ( 138 - 435 \beta ) q^{93} + ( 744 - 330 \beta ) q^{94} + ( 249 + 65 \beta ) q^{95} + 96 q^{96} + ( -3 + 458 \beta ) q^{97} + ( 312 - 414 \beta ) q^{98} + ( -315 - 90 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 6q^{3} + 8q^{4} - 20q^{5} + 12q^{6} - 23q^{7} + 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 4q^{2} + 6q^{3} + 8q^{4} - 20q^{5} + 12q^{6} - 23q^{7} + 16q^{8} + 18q^{9} - 40q^{10} - 80q^{11} + 24q^{12} - 102q^{13} - 46q^{14} - 60q^{15} + 32q^{16} - 25q^{17} + 36q^{18} - 55q^{19} - 80q^{20} - 69q^{21} - 160q^{22} - 5q^{23} + 48q^{24} - 24q^{25} - 204q^{26} + 54q^{27} - 92q^{28} - 35q^{29} - 120q^{30} - 53q^{31} + 64q^{32} - 240q^{33} - 50q^{34} + 113q^{35} + 72q^{36} - 221q^{37} - 110q^{38} - 306q^{39} - 160q^{40} - 89q^{41} - 138q^{42} - 508q^{43} - 320q^{44} - 180q^{45} - 10q^{46} + 579q^{47} + 96q^{48} + 105q^{49} - 48q^{50} - 75q^{51} - 408q^{52} + 432q^{53} + 108q^{54} + 930q^{55} - 184q^{56} - 165q^{57} - 70q^{58} + 118q^{59} - 240q^{60} + 151q^{61} - 106q^{62} - 207q^{63} + 128q^{64} + 1280q^{65} - 480q^{66} - 168q^{67} - 100q^{68} - 15q^{69} + 226q^{70} + 532q^{71} + 144q^{72} - 49q^{73} - 442q^{74} - 72q^{75} - 220q^{76} + 335q^{77} - 612q^{78} - 664q^{79} - 320q^{80} + 162q^{81} - 178q^{82} - 351q^{83} - 276q^{84} + 289q^{85} - 1016q^{86} - 105q^{87} - 640q^{88} - 1401q^{89} - 360q^{90} + 3q^{91} - 20q^{92} - 159q^{93} + 1158q^{94} + 563q^{95} + 192q^{96} + 452q^{97} + 210q^{98} - 720q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
2.00000 3.00000 4.00000 −13.6056 6.00000 4.72498 8.00000 9.00000 −27.2111
1.2 2.00000 3.00000 4.00000 −6.39445 6.00000 −27.7250 8.00000 9.00000 −12.7889
\(n\): 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 \( T_{5}^{2} + 20 T_{5} + 87 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\).