Properties

Label 4017.2.a.l
Level 4017
Weight 2
Character orbit 4017.a
Self dual Yes
Analytic conductor 32.076
Analytic rank 0
Dimension 32
CM No

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Newspace parameters

Level: \( N \) = \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4017.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0759064919\)
Analytic rank: \(0\)
Dimension: \(32\)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q + 5q^{2} + 32q^{3} + 45q^{4} + q^{5} + 5q^{6} + 11q^{7} + 12q^{8} + 32q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 5q^{2} + 32q^{3} + 45q^{4} + q^{5} + 5q^{6} + 11q^{7} + 12q^{8} + 32q^{9} + 16q^{10} + 3q^{11} + 45q^{12} - 32q^{13} + 12q^{14} + q^{15} + 75q^{16} + 10q^{17} + 5q^{18} + 4q^{20} + 11q^{21} + 27q^{22} + 53q^{23} + 12q^{24} + 67q^{25} - 5q^{26} + 32q^{27} + 32q^{28} + 6q^{29} + 16q^{30} + 12q^{31} + 19q^{32} + 3q^{33} + 25q^{34} + 16q^{35} + 45q^{36} + 36q^{37} + 8q^{38} - 32q^{39} + 36q^{40} - 19q^{41} + 12q^{42} + 43q^{43} - 23q^{44} + q^{45} + 11q^{46} + 30q^{47} + 75q^{48} + 75q^{49} + 28q^{50} + 10q^{51} - 45q^{52} + 22q^{53} + 5q^{54} + 58q^{55} + 60q^{56} + 33q^{58} - 24q^{59} + 4q^{60} + 47q^{61} - 25q^{62} + 11q^{63} + 146q^{64} - q^{65} + 27q^{66} + 34q^{67} + 58q^{68} + 53q^{69} - 35q^{70} + 18q^{71} + 12q^{72} - 2q^{73} - 20q^{74} + 67q^{75} + 24q^{76} + 39q^{77} - 5q^{78} + 39q^{79} + 2q^{80} + 32q^{81} + 64q^{82} + 17q^{83} + 32q^{84} + 35q^{85} - 13q^{86} + 6q^{87} + 55q^{88} - 48q^{89} + 16q^{90} - 11q^{91} + 39q^{92} + 12q^{93} + 58q^{94} + 59q^{95} + 19q^{96} + 42q^{97} + 16q^{98} + 3q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.75387 1.00000 5.58381 3.90153 −2.75387 0.781129 −9.86935 1.00000 −10.7443
1.2 −2.67949 1.00000 5.17965 −3.25805 −2.67949 0.886771 −8.51984 1.00000 8.72990
1.3 −2.65538 1.00000 5.05105 −1.68203 −2.65538 −3.96430 −8.10171 1.00000 4.46642
1.4 −2.55932 1.00000 4.55014 −2.63190 −2.55932 2.75923 −6.52662 1.00000 6.73589
1.5 −1.93942 1.00000 1.76133 0.142452 −1.93942 3.99599 0.462872 1.00000 −0.276273
1.6 −1.85552 1.00000 1.44297 3.38665 −1.85552 −2.21342 1.03358 1.00000 −6.28400
1.7 −1.79771 1.00000 1.23177 3.18677 −1.79771 1.45324 1.38105 1.00000 −5.72890
1.8 −1.78664 1.00000 1.19208 −1.81642 −1.78664 −4.42230 1.44346 1.00000 3.24528
1.9 −1.48812 1.00000 0.214502 −3.58678 −1.48812 0.0236485 2.65704 1.00000 5.33757
1.10 −1.42411 1.00000 0.0280857 −0.882122 −1.42411 4.13066 2.80822 1.00000 1.25624
1.11 −1.08109 1.00000 −0.831238 −0.942662 −1.08109 −2.40511 3.06083 1.00000 1.01911
1.12 −0.772884 1.00000 −1.40265 −0.606956 −0.772884 1.27278 2.62985 1.00000 0.469107
1.13 −0.365601 1.00000 −1.86634 0.232769 −0.365601 −3.73203 1.41354 1.00000 −0.0851006
1.14 −0.238219 1.00000 −1.94325 2.19571 −0.238219 5.28038 0.939357 1.00000 −0.523060
1.15 −0.163737 1.00000 −1.97319 3.16137 −0.163737 4.62100 0.650558 1.00000 −0.517632
1.16 −0.0270033 1.00000 −1.99927 −4.41470 −0.0270033 1.75798 0.107994 1.00000 0.119212
1.17 0.310907 1.00000 −1.90334 2.19968 0.310907 −3.79510 −1.21358 1.00000 0.683896
1.18 0.609784 1.00000 −1.62816 2.30670 0.609784 −1.80421 −2.21240 1.00000 1.40659
1.19 0.934190 1.00000 −1.12729 −1.38248 0.934190 −2.39473 −2.92148 1.00000 −1.29149
1.20 1.11274 1.00000 −0.761813 −3.97825 1.11274 −2.83032 −3.07318 1.00000 −4.42675
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
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Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

\(T_{2}^{32} - \cdots\)
\(T_{23}^{32} - \cdots\)