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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\).