Properties

Label 4031.2.a.b
Level 4031
Weight 2
Character orbit 4031.a
Self dual yes
Analytic conductor 32.188
Analytic rank 1
Dimension 59
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4031 = 29 \cdot 139 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4031.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1876970548\)
Analytic rank: \(1\)
Dimension: \(59\)
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.

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

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 −2.77566 2.36943 5.70431 1.60665 −6.57673 0.747628 −10.2819 2.61418 −4.45952
1.2 −2.69273 −0.478176 5.25077 0.135724 1.28760 −4.02730 −8.75344 −2.77135 −0.365467
1.3 −2.49699 −1.64487 4.23496 3.35969 4.10722 2.65574 −5.58068 −0.294408 −8.38911
1.4 −2.47917 −2.68137 4.14631 1.95848 6.64759 0.410621 −5.32107 4.18976 −4.85541
1.5 −2.43722 1.24735 3.94004 0.277998 −3.04005 4.01805 −4.72830 −1.44413 −0.677543
1.6 −2.32696 −0.944934 3.41472 −1.62617 2.19882 3.30319 −3.29200 −2.10710 3.78402
1.7 −2.24849 2.23227 3.05570 −0.764282 −5.01923 −0.688314 −2.37374 1.98301 1.71848
1.8 −2.19361 −1.71235 2.81195 −2.23633 3.75624 −3.04720 −1.78110 −0.0678453 4.90564
1.9 −2.18971 −0.875363 2.79481 −1.80760 1.91679 −0.359878 −1.74040 −2.23374 3.95812
1.10 −2.00564 2.28170 2.02257 −4.04874 −4.57625 −0.876531 −0.0452719 2.20615 8.12030
1.11 −1.84055 2.78518 1.38763 3.40108 −5.12627 −2.70641 1.12710 4.75723 −6.25987
1.12 −1.78339 0.526387 1.18048 0.564222 −0.938754 1.86157 1.46152 −2.72292 −1.00623
1.13 −1.69299 −2.95831 0.866208 −2.49874 5.00838 1.49318 1.91950 5.75160 4.23033
1.14 −1.67397 −0.759884 0.802190 2.08858 1.27203 −1.20948 2.00510 −2.42258 −3.49623
1.15 −1.54716 −2.20924 0.393702 3.10213 3.41805 −0.403081 2.48520 1.88074 −4.79950
1.16 −1.45467 0.452328 0.116052 −2.57197 −0.657986 3.66955 2.74051 −2.79540 3.74136
1.17 −1.40700 1.38079 −0.0203573 0.687727 −1.94276 −3.68772 2.84264 −1.09343 −0.967630
1.18 −1.33108 2.24902 −0.228227 −0.768051 −2.99362 2.32195 2.96595 2.05807 1.02234
1.19 −1.25853 0.116839 −0.416106 −2.24208 −0.147046 −3.42737 3.04074 −2.98635 2.82172
1.20 −1.17292 −0.112413 −0.624270 4.03608 0.131851 1.76356 3.07805 −2.98736 −4.73397
See all 59 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.59
Significant digits:
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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 4031.2.a.b 59
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4031.2.a.b 59 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)
\(139\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{59} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\).