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
Inner twists 1

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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: \(103\)
Coefficient ring index: multiple of None
Twist minimal: yes
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.

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

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 −2.80660 −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:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4031.2.a.e 103
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4031.2.a.e 103 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)
\(139\) \(-1\)

Hecke kernels

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

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database