Properties

Label 4003.2.a.b
Level 4003
Weight 2
Character orbit 4003.a
Self dual Yes
Analytic conductor 31.964
Analytic rank 1
Dimension 152
CM No

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Newspace parameters

Level: \( N \) = \( 4003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9641159291\)
Analytic rank: \(1\)
Dimension: \(152\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 152q - 22q^{2} - 18q^{3} + 138q^{4} - 59q^{5} - 17q^{6} - 19q^{7} - 66q^{8} + 106q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 152q - 22q^{2} - 18q^{3} + 138q^{4} - 59q^{5} - 17q^{6} - 19q^{7} - 66q^{8} + 106q^{9} - 15q^{10} - 40q^{11} - 53q^{12} - 59q^{13} - 36q^{14} - 40q^{15} + 118q^{16} - 93q^{17} - 59q^{18} - 16q^{19} - 108q^{20} - 62q^{21} - 37q^{22} - 107q^{23} - 31q^{24} + 101q^{25} - 64q^{26} - 63q^{27} - 53q^{28} - 124q^{29} - 68q^{30} - 15q^{31} - 129q^{32} - 49q^{33} - 76q^{35} + 45q^{36} - 98q^{37} - 125q^{38} - 47q^{39} - 7q^{40} - 56q^{41} - 84q^{42} - 62q^{43} - 114q^{44} - 142q^{45} - 3q^{46} - 111q^{47} - 92q^{48} + 117q^{49} - 64q^{50} - 21q^{51} - 85q^{52} - 347q^{53} + 3q^{54} - 16q^{55} - 73q^{56} - 115q^{57} - 29q^{58} - 50q^{59} - 54q^{60} - 62q^{61} - 55q^{62} - 70q^{63} + 64q^{64} - 147q^{65} + 34q^{66} - 86q^{67} - 174q^{68} - 104q^{69} - 7q^{70} - 86q^{71} - 139q^{72} - 27q^{73} - 52q^{74} - 49q^{75} - 11q^{76} - 346q^{77} - 59q^{78} - 17q^{79} - 149q^{80} - 8q^{81} - 31q^{82} - 106q^{83} - 51q^{84} - 69q^{85} - 85q^{86} - 32q^{87} - 113q^{88} - 59q^{89} + 10q^{90} - 9q^{91} - 314q^{92} - 230q^{93} + 7q^{94} - 74q^{95} - 54q^{96} - 60q^{97} - 77q^{98} - 96q^{99} + O(q^{100}) \)

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 −2.81629 1.48630 5.93147 2.03150 −4.18584 −2.55919 −11.0722 −0.790918 −5.72129
1.2 −2.78616 −2.78181 5.76271 −2.35951 7.75059 2.45659 −10.4835 4.73848 6.57398
1.3 −2.74603 −0.470182 5.54067 −1.60967 1.29113 −2.65994 −9.72280 −2.77893 4.42020
1.4 −2.71591 −1.21901 5.37616 2.34595 3.31073 3.38769 −9.16936 −1.51401 −6.37139
1.5 −2.71344 −2.64650 5.36273 −1.98460 7.18110 −4.86510 −9.12456 4.00395 5.38509
1.6 −2.70039 3.13054 5.29213 −0.910228 −8.45368 0.335007 −8.89004 6.80026 2.45797
1.7 −2.68690 0.420690 5.21946 −4.00007 −1.13035 −1.97783 −8.65037 −2.82302 10.7478
1.8 −2.64207 1.02501 4.98054 2.32673 −2.70814 −0.493904 −7.87479 −1.94936 −6.14739
1.9 −2.60205 1.48054 4.77066 1.31197 −3.85243 3.44490 −7.20939 −0.808005 −3.41380
1.10 −2.59568 2.00123 4.73753 −2.55331 −5.19456 4.30357 −7.10575 1.00494 6.62756
1.11 −2.57418 1.68195 4.62641 −3.90988 −4.32965 1.64141 −6.76085 −0.171031 10.0647
1.12 −2.54789 −3.05680 4.49172 1.69745 7.78837 −1.95167 −6.34863 6.34402 −4.32491
1.13 −2.45287 −0.154053 4.01656 −1.18015 0.377872 4.09823 −4.94635 −2.97627 2.89476
1.14 −2.44543 −1.15615 3.98012 −3.94364 2.82729 3.67660 −4.84225 −1.66331 9.64388
1.15 −2.43441 1.77953 3.92633 −0.360928 −4.33209 −1.53438 −4.68946 0.166714 0.878646
1.16 −2.42958 −1.44088 3.90288 0.473210 3.50074 −4.22120 −4.62321 −0.923870 −1.14970
1.17 −2.41802 1.98760 3.84680 1.18517 −4.80604 −3.98093 −4.46560 0.950540 −2.86577
1.18 −2.37503 −2.00580 3.64079 0.380635 4.76385 3.69515 −3.89692 1.02324 −0.904022
1.19 −2.36790 −2.62074 3.60697 3.23023 6.20567 1.16162 −3.80516 3.86830 −7.64888
1.20 −2.36704 −1.77609 3.60290 −2.33381 4.20407 1.88575 −3.79413 0.154480 5.52424
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.152
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
\(4003\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{152} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\).