Properties

Label 6031.2.a.d
Level 6031
Weight 2
Character orbit 6031.a
Self dual yes
Analytic conductor 48.158
Analytic rank 0
Dimension 133
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6031 = 37 \cdot 163 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6031.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1577774590\)
Analytic rank: \(0\)
Dimension: \(133\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 133q + 14q^{2} + 8q^{3} + 142q^{4} + 34q^{5} + 20q^{6} + 8q^{7} + 42q^{8} + 177q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 133q + 14q^{2} + 8q^{3} + 142q^{4} + 34q^{5} + 20q^{6} + 8q^{7} + 42q^{8} + 177q^{9} + 9q^{10} + 23q^{11} + 24q^{12} + 23q^{13} + 31q^{14} + 9q^{15} + 168q^{16} + 98q^{17} + 38q^{18} + 29q^{19} + 83q^{20} + 26q^{21} + 2q^{22} + 34q^{23} + 75q^{24} + 177q^{25} + 67q^{26} + 32q^{27} + 32q^{28} + 91q^{29} + 12q^{30} + 24q^{31} + 88q^{32} + 27q^{33} + 23q^{34} + 66q^{35} + 232q^{36} - 133q^{37} + 26q^{38} + 28q^{39} + 41q^{40} + 132q^{41} + 13q^{42} + 11q^{43} + 65q^{44} + 107q^{45} + 20q^{46} + 10q^{47} + 27q^{48} + 229q^{49} + 78q^{50} + 19q^{51} + 71q^{52} + 7q^{53} + 43q^{54} + 41q^{55} + 67q^{56} + 45q^{57} + 25q^{58} + 97q^{59} - 42q^{60} + 65q^{61} + 24q^{62} + 39q^{63} + 200q^{64} + 60q^{65} + 35q^{66} + 25q^{67} + 227q^{68} + 120q^{69} + 37q^{70} + 26q^{71} + 93q^{72} + 55q^{73} - 14q^{74} + 5q^{75} + 34q^{76} + 21q^{77} - 2q^{78} + 50q^{79} + 162q^{80} + 341q^{81} + 66q^{82} + 30q^{83} - 89q^{84} + 30q^{85} - 12q^{86} + 80q^{87} - 85q^{88} + 225q^{89} - 86q^{90} + q^{91} + 82q^{92} + 42q^{93} - 17q^{94} + 70q^{95} + 55q^{96} + 12q^{97} + 90q^{98} + 17q^{99} + O(q^{100}) \)

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 −2.76277 2.80088 5.63291 −0.925460 −7.73820 −3.46221 −10.0369 4.84493 2.55683
1.2 −2.74334 −3.12442 5.52594 0.187024 8.57137 1.31323 −9.67287 6.76201 −0.513072
1.3 −2.65451 1.31557 5.04644 0.400622 −3.49219 1.95447 −8.08682 −1.26929 −1.06346
1.4 −2.64085 0.791428 4.97411 −0.110767 −2.09005 −4.52560 −7.85420 −2.37364 0.292520
1.5 −2.62634 −3.19656 4.89764 4.04178 8.39525 −2.68408 −7.61017 7.21801 −10.6151
1.6 −2.59785 −1.74392 4.74884 1.33654 4.53044 −2.87132 −7.14109 0.0412485 −3.47214
1.7 −2.58445 2.91776 4.67939 3.62109 −7.54082 1.31660 −6.92475 5.51335 −9.35852
1.8 −2.55998 1.31640 4.55348 3.87746 −3.36994 3.54726 −6.53684 −1.26710 −9.92621
1.9 −2.55877 −0.511653 4.54728 −3.89293 1.30920 1.47164 −6.51789 −2.73821 9.96108
1.10 −2.45235 −2.50002 4.01402 −3.16357 6.13092 −0.556288 −4.93908 3.25009 7.75817
1.11 −2.39714 0.221428 3.74630 −0.354828 −0.530796 3.04567 −4.18612 −2.95097 0.850573
1.12 −2.38561 −1.27766 3.69114 1.05833 3.04800 3.63452 −4.03440 −1.36759 −2.52477
1.13 −2.36309 −0.0862351 3.58421 −0.328445 0.203782 −0.467187 −3.74364 −2.99256 0.776147
1.14 −2.30996 −1.13672 3.33593 4.36294 2.62577 −2.53540 −3.08596 −1.70788 −10.0782
1.15 −2.29540 2.78712 3.26888 −0.0528483 −6.39757 2.45561 −2.91258 4.76805 0.121308
1.16 −2.27893 2.16909 3.19353 2.05388 −4.94321 −4.20533 −2.71998 1.70495 −4.68065
1.17 −2.27645 −1.27799 3.18222 2.39479 2.90928 2.25162 −2.69127 −1.36674 −5.45162
1.18 −2.24334 −3.44407 3.03258 −0.143577 7.72622 5.03410 −2.31642 8.86161 0.322092
1.19 −2.20408 0.816725 2.85795 −2.28441 −1.80012 0.445697 −1.89098 −2.33296 5.03501
1.20 −2.20055 −1.32438 2.84242 −2.94026 2.91437 −4.27239 −1.85380 −1.24601 6.47020
See next 80 embeddings (of 133 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.133
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 6031.2.a.d 133
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6031.2.a.d 133 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(37\) \(1\)
\(163\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{133} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database