Properties

Label 6031.2.a.c
Level 6031
Weight 2
Character orbit 6031.a
Self dual yes
Analytic conductor 48.158
Analytic rank 1
Dimension 110
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: \(1\)
Dimension: \(110\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 110q - 9q^{2} + 97q^{4} - 26q^{5} - 26q^{6} - 4q^{7} - 27q^{8} + 62q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 110q - 9q^{2} + 97q^{4} - 26q^{5} - 26q^{6} - 4q^{7} - 27q^{8} + 62q^{9} - 17q^{10} - 9q^{11} - 21q^{13} - 29q^{14} - 23q^{15} + 79q^{16} - 76q^{17} - 31q^{18} - 27q^{19} - 67q^{20} - 30q^{21} - 28q^{22} - 32q^{23} - 63q^{24} + 66q^{25} - 55q^{26} - 4q^{28} - 81q^{29} - 48q^{30} - 30q^{31} - 73q^{32} - 53q^{33} - 23q^{34} - 78q^{35} + 7q^{36} - 110q^{37} - 50q^{38} - 64q^{39} - 37q^{40} - 123q^{41} - 63q^{42} - 40q^{43} - 31q^{44} - 73q^{45} + 16q^{46} - 37q^{47} - 29q^{48} + 46q^{49} - 58q^{50} - 73q^{51} - 39q^{52} - 16q^{53} - 53q^{54} - 59q^{55} - 113q^{56} - 39q^{57} + 11q^{58} - 93q^{59} - 18q^{60} - 66q^{61} - 40q^{62} - 21q^{63} + 23q^{64} - 92q^{65} - 31q^{66} + q^{67} - 121q^{68} - 80q^{69} - 3q^{70} - 75q^{71} - 114q^{72} - 39q^{73} + 9q^{74} - 25q^{75} - 58q^{76} - 31q^{77} + 68q^{78} - 36q^{79} - 82q^{80} - 50q^{81} - 18q^{82} - 57q^{83} - 9q^{84} - 14q^{85} - 58q^{86} - 58q^{87} - 15q^{88} - 181q^{89} + 8q^{90} - 55q^{91} - 116q^{92} - 86q^{93} - 39q^{94} - 70q^{95} - 127q^{96} - 91q^{97} - 19q^{98} - 21q^{99} + O(q^{100}) \)

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 −2.78685 −0.498757 5.76655 1.84399 1.38996 −1.25608 −10.4968 −2.75124 −5.13894
1.2 −2.77072 1.69880 5.67692 −3.04963 −4.70691 3.28914 −10.1877 −0.114069 8.44968
1.3 −2.69148 −2.62726 5.24405 0.394241 7.07121 −0.479189 −8.73128 3.90249 −1.06109
1.4 −2.67187 1.06249 5.13890 3.00541 −2.83884 2.91292 −8.38673 −1.87111 −8.03007
1.5 −2.64482 −1.59347 4.99505 −2.42826 4.21443 4.09769 −7.92136 −0.460856 6.42231
1.6 −2.64313 2.88413 4.98613 −1.94351 −7.62312 −0.878417 −7.89274 5.31819 5.13695
1.7 −2.55998 2.69028 4.55348 −2.87781 −6.88705 3.57411 −6.53686 4.23759 7.36714
1.8 −2.51631 1.27091 4.33182 1.33976 −3.19800 0.944830 −5.86759 −1.38479 −3.37125
1.9 −2.48517 3.11043 4.17608 3.13519 −7.72995 −2.09644 −5.40793 6.67478 −7.79149
1.10 −2.45577 −2.50707 4.03080 −3.71982 6.15679 −2.33961 −4.98718 3.28541 9.13502
1.11 −2.41585 0.542785 3.83634 −2.85515 −1.31129 −2.66009 −4.43632 −2.70538 6.89763
1.12 −2.35892 −1.34501 3.56450 0.355993 3.17276 −3.31316 −3.69052 −1.19096 −0.839759
1.13 −2.33551 0.518713 3.45460 −1.17306 −1.21146 −2.50606 −3.39724 −2.73094 2.73970
1.14 −2.29692 −1.19284 3.27585 2.94374 2.73987 2.63830 −2.93052 −1.57712 −6.76154
1.15 −2.27127 −0.567339 3.15866 −3.83448 1.28858 2.52768 −2.63164 −2.67813 8.70914
1.16 −2.17795 2.23759 2.74348 1.66421 −4.87337 −3.06154 −1.61926 2.00681 −3.62457
1.17 −2.13628 −0.861354 2.56370 1.50403 1.84010 −3.07386 −1.20423 −2.25807 −3.21303
1.18 −2.12748 −2.55539 2.52618 2.02951 5.43655 −1.20737 −1.11945 3.53003 −4.31775
1.19 −2.05303 −2.17913 2.21492 2.99569 4.47382 4.05870 −0.441239 1.74862 −6.15023
1.20 −1.98962 1.97988 1.95861 0.230568 −3.93921 −0.245933 0.0823553 0.919909 −0.458744
See next 80 embeddings (of 110 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.110
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.c 110
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6031.2.a.c 110 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 \(T_{2}^{110} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database