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

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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: \(134\)
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.

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

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 −2.78315 −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:
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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 \(T_{2}^{134} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).