Properties

Label 4031.2.a.e
Level 4031
Weight 2
Character orbit 4031.a
Self dual Yes
Analytic conductor 32.188
Analytic rank 0
Dimension 103
CM No

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Newspace parameters

Level: \( N \) = \( 4031 = 29 \cdot 139 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4031.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1876970548\)
Analytic rank: \(0\)
Dimension: \(103\)
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.

\(\operatorname{Tr}(f)(q) = \) \(103q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 127q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 19q^{6} \) \(\mathstrut +\mathstrut 18q^{7} \) \(\mathstrut +\mathstrut 149q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(103q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 127q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 19q^{6} \) \(\mathstrut +\mathstrut 18q^{7} \) \(\mathstrut +\mathstrut 149q^{9} \) \(\mathstrut +\mathstrut 20q^{10} \) \(\mathstrut +\mathstrut 9q^{11} \) \(\mathstrut +\mathstrut 36q^{13} \) \(\mathstrut -\mathstrut 10q^{14} \) \(\mathstrut +\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 179q^{16} \) \(\mathstrut +\mathstrut 21q^{17} \) \(\mathstrut +\mathstrut 7q^{18} \) \(\mathstrut +\mathstrut 42q^{19} \) \(\mathstrut +\mathstrut 24q^{20} \) \(\mathstrut +\mathstrut 28q^{21} \) \(\mathstrut +\mathstrut 32q^{22} \) \(\mathstrut +\mathstrut 25q^{23} \) \(\mathstrut +\mathstrut 68q^{24} \) \(\mathstrut +\mathstrut 194q^{25} \) \(\mathstrut -\mathstrut 5q^{26} \) \(\mathstrut +\mathstrut 14q^{27} \) \(\mathstrut +\mathstrut 59q^{28} \) \(\mathstrut -\mathstrut 103q^{29} \) \(\mathstrut +\mathstrut 84q^{30} \) \(\mathstrut +\mathstrut 34q^{31} \) \(\mathstrut +\mathstrut 11q^{32} \) \(\mathstrut +\mathstrut 42q^{33} \) \(\mathstrut +\mathstrut 54q^{34} \) \(\mathstrut +\mathstrut 35q^{35} \) \(\mathstrut +\mathstrut 214q^{36} \) \(\mathstrut +\mathstrut 34q^{37} \) \(\mathstrut +\mathstrut 9q^{38} \) \(\mathstrut +\mathstrut 23q^{39} \) \(\mathstrut +\mathstrut 46q^{40} \) \(\mathstrut +\mathstrut 16q^{41} \) \(\mathstrut +\mathstrut 13q^{42} \) \(\mathstrut +\mathstrut 68q^{43} \) \(\mathstrut -\mathstrut 6q^{44} \) \(\mathstrut +\mathstrut 25q^{45} \) \(\mathstrut +\mathstrut 60q^{46} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 5q^{48} \) \(\mathstrut +\mathstrut 257q^{49} \) \(\mathstrut -\mathstrut 51q^{50} \) \(\mathstrut +\mathstrut 68q^{51} \) \(\mathstrut +\mathstrut 37q^{52} \) \(\mathstrut +\mathstrut 35q^{53} \) \(\mathstrut +\mathstrut 30q^{54} \) \(\mathstrut +\mathstrut 66q^{55} \) \(\mathstrut -\mathstrut 54q^{56} \) \(\mathstrut +\mathstrut 78q^{57} \) \(\mathstrut -\mathstrut q^{58} \) \(\mathstrut +\mathstrut 10q^{59} \) \(\mathstrut -\mathstrut 24q^{60} \) \(\mathstrut +\mathstrut 70q^{61} \) \(\mathstrut +\mathstrut 29q^{62} \) \(\mathstrut +\mathstrut 26q^{63} \) \(\mathstrut +\mathstrut 276q^{64} \) \(\mathstrut +\mathstrut 95q^{65} \) \(\mathstrut +\mathstrut 77q^{66} \) \(\mathstrut +\mathstrut 71q^{67} \) \(\mathstrut -\mathstrut 21q^{68} \) \(\mathstrut -\mathstrut 20q^{69} \) \(\mathstrut +\mathstrut 48q^{70} \) \(\mathstrut +\mathstrut 32q^{71} \) \(\mathstrut +\mathstrut 32q^{72} \) \(\mathstrut +\mathstrut 94q^{73} \) \(\mathstrut +\mathstrut 35q^{74} \) \(\mathstrut +\mathstrut 7q^{75} \) \(\mathstrut +\mathstrut 134q^{76} \) \(\mathstrut +\mathstrut 17q^{77} \) \(\mathstrut +\mathstrut 58q^{78} \) \(\mathstrut +\mathstrut 110q^{79} \) \(\mathstrut +\mathstrut 78q^{80} \) \(\mathstrut +\mathstrut 267q^{81} \) \(\mathstrut -\mathstrut 71q^{82} \) \(\mathstrut +\mathstrut 35q^{83} \) \(\mathstrut +\mathstrut 96q^{84} \) \(\mathstrut +\mathstrut 71q^{85} \) \(\mathstrut +\mathstrut 33q^{86} \) \(\mathstrut -\mathstrut 2q^{87} \) \(\mathstrut +\mathstrut 100q^{88} \) \(\mathstrut +\mathstrut 22q^{89} \) \(\mathstrut -\mathstrut 134q^{90} \) \(\mathstrut +\mathstrut 108q^{91} \) \(\mathstrut -\mathstrut 11q^{92} \) \(\mathstrut +\mathstrut 78q^{93} \) \(\mathstrut +\mathstrut 90q^{94} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 177q^{96} \) \(\mathstrut +\mathstrut 44q^{97} \) \(\mathstrut -\mathstrut 18q^{98} \) \(\mathstrut +\mathstrut 83q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.80660 −2.96734 5.87699 4.35928 8.32812 −3.03197 −10.8811 5.80510 −12.2348
1.2 −2.80086 −1.28750 5.84483 −1.26767 3.60610 −2.24450 −10.7688 −1.34235 3.55056
1.3 −2.75205 2.78767 5.57380 −3.50027 −7.67183 2.04758 −9.83529 4.77113 9.63294
1.4 −2.71060 −0.763863 5.34738 1.53299 2.07053 3.02090 −9.07341 −2.41651 −4.15533
1.5 −2.68510 −0.634380 5.20976 1.13239 1.70337 4.26000 −8.61852 −2.59756 −3.04058
1.6 −2.63961 1.15663 4.96757 −1.20001 −3.05306 −2.18899 −7.83323 −1.66221 3.16755
1.7 −2.62778 2.50964 4.90520 −0.626861 −6.59477 4.88608 −7.63422 3.29828 1.64725
1.8 −2.59576 −0.589470 4.73795 2.97822 1.53012 −2.97611 −7.10705 −2.65253 −7.73074
1.9 −2.54769 −2.43102 4.49072 −4.16655 6.19349 −1.47765 −6.34558 2.90987 10.6151
1.10 −2.50341 −3.31200 4.26704 −1.06782 8.29127 3.43750 −5.67533 7.96931 2.67318
1.11 −2.41155 2.97695 3.81557 2.55399 −7.17907 0.984137 −4.37835 5.86223 −6.15908
1.12 −2.38730 −1.73906 3.69919 0.845395 4.15166 −4.68935 −4.05648 0.0243354 −2.01821
1.13 −2.35149 2.74768 3.52949 4.05795 −6.46113 3.42169 −3.59657 4.54974 −9.54222
1.14 −2.35028 1.87633 3.52380 −3.27377 −4.40989 −2.87617 −3.58134 0.520602 7.69425
1.15 −2.34575 −2.20403 3.50256 −4.15589 5.17012 5.11824 −3.52465 1.85776 9.74869
1.16 −2.20982 −3.43245 2.88331 3.18051 7.58510 4.63810 −1.95196 8.78169 −7.02836
1.17 −2.11397 0.494442 2.46887 0.938762 −1.04524 1.45468 −0.991174 −2.75553 −1.98451
1.18 −2.09508 −0.524599 2.38936 −2.74536 1.09908 −3.42586 −0.815732 −2.72480 5.75174
1.19 −2.06403 0.822287 2.26023 −2.41369 −1.69723 3.09601 −0.537124 −2.32384 4.98194
1.20 −2.04048 0.724757 2.16355 −3.93757 −1.47885 3.69332 −0.333725 −2.47473 8.03452
See next 80 embeddings (of 103 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.103
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
\(29\) \(1\)
\(139\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{103} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\).