Properties

Label 6031.2.a.e
Level 6031
Weight 2
Character orbit 6031.a
Self dual yes
Analytic conductor 48.158
Analytic rank 0
Dimension 134
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: \(134\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 134q + 9q^{2} + 7q^{3} + 149q^{4} + 22q^{5} + 20q^{6} + 11q^{7} + 27q^{8} + 181q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 134q + 9q^{2} + 7q^{3} + 149q^{4} + 22q^{5} + 20q^{6} + 11q^{7} + 27q^{8} + 181q^{9} + 15q^{10} + 20q^{11} + 28q^{12} + 11q^{13} + 17q^{14} - 13q^{15} + 143q^{16} + 76q^{17} + 23q^{18} + 15q^{19} + 67q^{20} + 63q^{21} + 2q^{22} + 22q^{23} + 33q^{24} + 160q^{25} + 65q^{26} + 31q^{27} + 10q^{28} + 73q^{29} + 20q^{30} + 10q^{31} + 53q^{32} + 72q^{33} - 7q^{34} + 52q^{35} + 201q^{36} + 134q^{37} + 70q^{38} + 6q^{39} + 11q^{40} + 182q^{41} - 15q^{42} + 12q^{43} + 33q^{44} + 29q^{45} + 24q^{46} + 80q^{47} + 21q^{48} + 229q^{49} + 37q^{50} + 57q^{51} - 15q^{52} + 75q^{53} + 95q^{54} - 9q^{55} + 39q^{56} + 19q^{57} - 21q^{58} + 91q^{59} + 62q^{60} + 58q^{61} + 108q^{62} + 9q^{63} + 167q^{64} + 76q^{65} + 105q^{66} - 17q^{67} + 109q^{68} + 48q^{69} - 55q^{70} + 56q^{71} + 48q^{72} + 54q^{73} + 9q^{74} + 28q^{75} + 82q^{76} + 156q^{77} + 16q^{78} - 2q^{79} + 98q^{80} + 270q^{81} - 42q^{82} + 130q^{83} + 229q^{84} + 22q^{85} + 72q^{86} + 22q^{87} + 61q^{88} + 157q^{89} + 176q^{90} + 31q^{91} - 18q^{92} + 36q^{93} + 83q^{94} + 98q^{95} + 111q^{96} + 35q^{97} + 53q^{98} + 55q^{99} + O(q^{100}) \)

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 −2.78315 −1.65667 5.74594 3.71472 4.61077 4.59606 −10.4255 −0.255436 −10.3386
1.2 −2.77108 1.62557 5.67890 −0.123356 −4.50458 −1.34606 −10.1945 −0.357536 0.341829
1.3 −2.71257 −2.51910 5.35803 −0.492593 6.83324 −2.96732 −9.10890 3.34588 1.33619
1.4 −2.70867 −0.769737 5.33687 −2.82113 2.08496 −1.98436 −9.03847 −2.40750 7.64150
1.5 −2.62103 −1.46003 4.86979 3.30025 3.82679 −2.17387 −7.52179 −0.868300 −8.65005
1.6 −2.56681 3.22691 4.58852 1.32695 −8.28287 4.31369 −6.64424 7.41296 −3.40604
1.7 −2.54672 −0.772659 4.48577 3.37038 1.96774 0.335508 −6.33055 −2.40300 −8.58341
1.8 −2.53107 −0.695993 4.40632 1.03970 1.76161 0.140699 −6.09057 −2.51559 −2.63157
1.9 −2.49038 1.16097 4.20199 0.527322 −2.89126 0.382261 −5.48378 −1.65215 −1.31323
1.10 −2.47900 1.64888 4.14545 −0.897925 −4.08758 2.76964 −5.31858 −0.281188 2.22596
1.11 −2.47262 1.83929 4.11384 −2.27118 −4.54786 −1.50415 −5.22672 0.382983 5.61575
1.12 −2.43815 −2.94184 3.94457 −1.35496 7.17265 −0.834817 −4.74114 5.65444 3.30360
1.13 −2.41121 3.43364 3.81394 −3.49586 −8.27924 1.03348 −4.37378 8.78990 8.42926
1.14 −2.39957 0.225532 3.75794 −1.91995 −0.541181 −3.36005 −4.21829 −2.94914 4.60705
1.15 −2.39066 −3.44347 3.71525 −1.21628 8.23217 −4.84430 −4.10059 8.85751 2.90771
1.16 −2.25933 −1.97048 3.10458 −1.05665 4.45197 1.22634 −2.49561 0.882799 2.38733
1.17 −2.23037 2.48760 2.97457 3.37158 −5.54828 −3.17676 −2.17365 3.18815 −7.51989
1.18 −2.21168 1.70673 2.89152 4.27910 −3.77473 1.99718 −1.97176 −0.0870880 −9.46398
1.19 −2.16741 3.17923 2.69765 1.60787 −6.89069 1.52618 −1.51209 7.10752 −3.48491
1.20 −2.07881 1.94163 2.32147 −0.214355 −4.03630 −5.01421 −0.668275 0.769946 0.445604
See next 80 embeddings (of 134 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.134
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.e 134
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6031.2.a.e 134 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 \(T_{2}^{134} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).