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

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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: \(48.157777459\)
Analytic rank: \(0\)
Dimension: \(133\)
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.

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

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 −2.76277 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 have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

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