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

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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: \(59\)
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.

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

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 −2.77566 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 have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

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