Properties

Label 354.4.c.a
Level 354
Weight 4
Character orbit 354.c
Analytic conductor 20.887
Analytic rank 0
Dimension 30
CM No

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 354.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(20.886676142\)
Analytic rank: \(0\)
Dimension: \(30\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30q - 60q^{2} + 5q^{3} + 120q^{4} - 10q^{6} + 6q^{7} - 240q^{8} + 27q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q - 60q^{2} + 5q^{3} + 120q^{4} - 10q^{6} + 6q^{7} - 240q^{8} + 27q^{9} - 60q^{11} + 20q^{12} - 12q^{14} + 20q^{15} + 480q^{16} - 54q^{18} + 90q^{19} + 132q^{21} + 120q^{22} + 24q^{23} - 40q^{24} - 1080q^{25} - 55q^{27} + 24q^{28} - 40q^{30} - 960q^{32} + 336q^{33} + 108q^{36} - 180q^{38} + 652q^{39} - 264q^{42} - 240q^{44} - 878q^{45} - 48q^{46} + 792q^{47} + 80q^{48} + 2016q^{49} + 2160q^{50} + 650q^{51} + 110q^{54} - 48q^{56} + 846q^{57} - 480q^{59} + 80q^{60} + 887q^{63} + 1920q^{64} - 1416q^{65} - 672q^{66} - 590q^{69} - 216q^{72} - 952q^{75} + 360q^{76} + 864q^{77} - 1304q^{78} + 738q^{79} - 1217q^{81} + 876q^{83} + 528q^{84} + 1176q^{85} + 534q^{87} + 480q^{88} - 300q^{89} + 1756q^{90} + 96q^{92} + 1684q^{93} - 1584q^{94} - 160q^{96} - 4032q^{98} + 730q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1 −2.00000 −5.11454 0.917350i 4.00000 12.8770i 10.2291 + 1.83470i −4.12434 −8.00000 25.3169 + 9.38364i 25.7541i
353.2 −2.00000 −5.11454 + 0.917350i 4.00000 12.8770i 10.2291 1.83470i −4.12434 −8.00000 25.3169 9.38364i 25.7541i
353.3 −2.00000 −4.86384 1.82840i 4.00000 6.14441i 9.72768 + 3.65680i −15.5071 −8.00000 20.3139 + 17.7861i 12.2888i
353.4 −2.00000 −4.86384 + 1.82840i 4.00000 6.14441i 9.72768 3.65680i −15.5071 −8.00000 20.3139 17.7861i 12.2888i
353.5 −2.00000 −4.66792 2.28265i 4.00000 8.40010i 9.33585 + 4.56530i 31.0296 −8.00000 16.5790 + 21.3105i 16.8002i
353.6 −2.00000 −4.66792 + 2.28265i 4.00000 8.40010i 9.33585 4.56530i 31.0296 −8.00000 16.5790 21.3105i 16.8002i
353.7 −2.00000 −3.62921 3.71872i 4.00000 18.5626i 7.25841 + 7.43744i 9.26632 −8.00000 −0.657741 + 26.9920i 37.1253i
353.8 −2.00000 −3.62921 + 3.71872i 4.00000 18.5626i 7.25841 7.43744i 9.26632 −8.00000 −0.657741 26.9920i 37.1253i
353.9 −2.00000 −3.04040 4.21378i 4.00000 12.5447i 6.08081 + 8.42756i −33.4405 −8.00000 −8.51188 + 25.6232i 25.0894i
353.10 −2.00000 −3.04040 + 4.21378i 4.00000 12.5447i 6.08081 8.42756i −33.4405 −8.00000 −8.51188 25.6232i 25.0894i
353.11 −2.00000 −1.90975 4.83248i 4.00000 5.55280i 3.81949 + 9.66496i 16.8053 −8.00000 −19.7057 + 18.4576i 11.1056i
353.12 −2.00000 −1.90975 + 4.83248i 4.00000 5.55280i 3.81949 9.66496i 16.8053 −8.00000 −19.7057 18.4576i 11.1056i
353.13 −2.00000 −1.46648 4.98492i 4.00000 18.9931i 2.93296 + 9.96984i −10.1890 −8.00000 −22.6989 + 14.6206i 37.9861i
353.14 −2.00000 −1.46648 + 4.98492i 4.00000 18.9931i 2.93296 9.96984i −10.1890 −8.00000 −22.6989 14.6206i 37.9861i
353.15 −2.00000 1.05107 5.08874i 4.00000 6.49284i −2.10214 + 10.1775i −27.4926 −8.00000 −24.7905 10.6972i 12.9857i
353.16 −2.00000 1.05107 + 5.08874i 4.00000 6.49284i −2.10214 10.1775i −27.4926 −8.00000 −24.7905 + 10.6972i 12.9857i
353.17 −2.00000 1.53016 4.96574i 4.00000 1.00990i −3.06032 + 9.93149i 8.17095 −8.00000 −22.3172 15.1967i 2.01980i
353.18 −2.00000 1.53016 + 4.96574i 4.00000 1.00990i −3.06032 9.93149i 8.17095 −8.00000 −22.3172 + 15.1967i 2.01980i
353.19 −2.00000 2.99142 4.24869i 4.00000 19.3201i −5.98284 + 8.49739i 6.85976 −8.00000 −9.10280 25.4193i 38.6403i
353.20 −2.00000 2.99142 + 4.24869i 4.00000 19.3201i −5.98284 8.49739i 6.85976 −8.00000 −9.10280 + 25.4193i 38.6403i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.30
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.