Properties

Label 4013.2.a.b
Level 4013
Weight 2
Character orbit 4013.a
Self dual Yes
Analytic conductor 32.044
Analytic rank 1
Dimension 157
CM No

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Newspace parameters

Level: \( N \) = \( 4013 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4013.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0439663311\)
Analytic rank: \(1\)
Dimension: \(157\)
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.

\(\operatorname{Tr}(f)(q) = \) \(157q \) \(\mathstrut -\mathstrut 15q^{2} \) \(\mathstrut -\mathstrut 51q^{3} \) \(\mathstrut +\mathstrut 137q^{4} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 144q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(157q \) \(\mathstrut -\mathstrut 15q^{2} \) \(\mathstrut -\mathstrut 51q^{3} \) \(\mathstrut +\mathstrut 137q^{4} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 144q^{9} \) \(\mathstrut -\mathstrut 61q^{10} \) \(\mathstrut -\mathstrut 27q^{11} \) \(\mathstrut -\mathstrut 93q^{12} \) \(\mathstrut -\mathstrut 97q^{13} \) \(\mathstrut -\mathstrut 12q^{14} \) \(\mathstrut -\mathstrut 36q^{15} \) \(\mathstrut +\mathstrut 105q^{16} \) \(\mathstrut -\mathstrut 45q^{17} \) \(\mathstrut -\mathstrut 68q^{18} \) \(\mathstrut -\mathstrut 128q^{19} \) \(\mathstrut -\mathstrut 30q^{20} \) \(\mathstrut -\mathstrut 26q^{21} \) \(\mathstrut -\mathstrut 68q^{22} \) \(\mathstrut -\mathstrut 41q^{23} \) \(\mathstrut -\mathstrut 40q^{24} \) \(\mathstrut +\mathstrut 102q^{25} \) \(\mathstrut -\mathstrut 5q^{26} \) \(\mathstrut -\mathstrut 189q^{27} \) \(\mathstrut -\mathstrut 115q^{28} \) \(\mathstrut -\mathstrut 26q^{29} \) \(\mathstrut -\mathstrut 12q^{30} \) \(\mathstrut -\mathstrut 88q^{31} \) \(\mathstrut -\mathstrut 89q^{32} \) \(\mathstrut -\mathstrut 52q^{33} \) \(\mathstrut -\mathstrut 61q^{34} \) \(\mathstrut -\mathstrut 87q^{35} \) \(\mathstrut +\mathstrut 110q^{36} \) \(\mathstrut -\mathstrut 62q^{37} \) \(\mathstrut -\mathstrut 37q^{38} \) \(\mathstrut -\mathstrut 20q^{39} \) \(\mathstrut -\mathstrut 161q^{40} \) \(\mathstrut -\mathstrut 34q^{41} \) \(\mathstrut -\mathstrut 53q^{42} \) \(\mathstrut -\mathstrut 254q^{43} \) \(\mathstrut -\mathstrut 19q^{44} \) \(\mathstrut -\mathstrut 46q^{45} \) \(\mathstrut -\mathstrut 52q^{46} \) \(\mathstrut -\mathstrut 76q^{47} \) \(\mathstrut -\mathstrut 162q^{48} \) \(\mathstrut +\mathstrut 96q^{49} \) \(\mathstrut -\mathstrut 54q^{50} \) \(\mathstrut -\mathstrut 76q^{51} \) \(\mathstrut -\mathstrut 259q^{52} \) \(\mathstrut -\mathstrut 48q^{53} \) \(\mathstrut -\mathstrut 12q^{54} \) \(\mathstrut -\mathstrut 194q^{55} \) \(\mathstrut -\mathstrut 10q^{56} \) \(\mathstrut -\mathstrut 30q^{57} \) \(\mathstrut -\mathstrut 52q^{58} \) \(\mathstrut -\mathstrut 64q^{59} \) \(\mathstrut -\mathstrut 31q^{60} \) \(\mathstrut -\mathstrut 107q^{61} \) \(\mathstrut -\mathstrut 51q^{62} \) \(\mathstrut -\mathstrut 106q^{63} \) \(\mathstrut +\mathstrut 54q^{64} \) \(\mathstrut -\mathstrut 17q^{65} \) \(\mathstrut -\mathstrut 13q^{66} \) \(\mathstrut -\mathstrut 193q^{67} \) \(\mathstrut -\mathstrut 118q^{68} \) \(\mathstrut -\mathstrut 55q^{69} \) \(\mathstrut -\mathstrut 86q^{70} \) \(\mathstrut -\mathstrut 11q^{71} \) \(\mathstrut -\mathstrut 172q^{72} \) \(\mathstrut -\mathstrut 173q^{73} \) \(\mathstrut -\mathstrut 11q^{74} \) \(\mathstrut -\mathstrut 209q^{75} \) \(\mathstrut -\mathstrut 213q^{76} \) \(\mathstrut -\mathstrut 84q^{77} \) \(\mathstrut -\mathstrut 30q^{78} \) \(\mathstrut -\mathstrut 111q^{79} \) \(\mathstrut -\mathstrut 6q^{80} \) \(\mathstrut +\mathstrut 157q^{81} \) \(\mathstrut -\mathstrut 117q^{82} \) \(\mathstrut -\mathstrut 154q^{83} \) \(\mathstrut -\mathstrut 6q^{84} \) \(\mathstrut -\mathstrut 91q^{85} \) \(\mathstrut +\mathstrut 28q^{86} \) \(\mathstrut -\mathstrut 165q^{87} \) \(\mathstrut -\mathstrut 165q^{88} \) \(\mathstrut -\mathstrut 32q^{89} \) \(\mathstrut -\mathstrut 103q^{90} \) \(\mathstrut -\mathstrut 200q^{91} \) \(\mathstrut -\mathstrut 86q^{92} \) \(\mathstrut -\mathstrut 39q^{93} \) \(\mathstrut -\mathstrut 118q^{94} \) \(\mathstrut -\mathstrut 22q^{95} \) \(\mathstrut -\mathstrut 28q^{96} \) \(\mathstrut -\mathstrut 151q^{97} \) \(\mathstrut -\mathstrut 38q^{98} \) \(\mathstrut -\mathstrut 91q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81982 1.03159 5.95140 1.63220 −2.90891 3.00127 −11.1422 −1.93581 −4.60252
1.2 −2.78653 −0.812844 5.76476 −1.14246 2.26502 −4.07959 −10.4906 −2.33929 3.18350
1.3 −2.78090 −3.27207 5.73341 3.80445 9.09929 2.37189 −10.3822 7.70642 −10.5798
1.4 −2.68953 2.61392 5.23356 1.08630 −7.03022 −3.47806 −8.69675 3.83259 −2.92162
1.5 −2.65659 1.81764 5.05746 3.63888 −4.82872 −3.82226 −8.12240 0.303819 −9.66700
1.6 −2.65066 −3.08601 5.02601 −3.86295 8.17996 −0.380070 −8.02093 6.52343 10.2394
1.7 −2.61349 −2.30050 4.83033 −0.685238 6.01232 −4.02398 −7.39704 2.29228 1.79086
1.8 −2.60727 1.83175 4.79788 −1.15472 −4.77588 −1.01890 −7.29484 0.355310 3.01066
1.9 −2.54071 0.214630 4.45522 0.391559 −0.545314 2.42547 −6.23801 −2.95393 −0.994840
1.10 −2.53912 −1.94061 4.44715 3.91571 4.92744 −2.75166 −6.21361 0.765960 −9.94248
1.11 −2.52958 −1.28339 4.39880 −1.06771 3.24644 0.845401 −6.06796 −1.35291 2.70087
1.12 −2.51870 −2.55292 4.34385 0.976255 6.43005 1.70978 −5.90347 3.51742 −2.45890
1.13 −2.51500 −3.29902 4.32520 −0.150822 8.29703 −3.52495 −5.84787 7.88356 0.379318
1.14 −2.47196 3.17720 4.11056 1.49907 −7.85391 0.578951 −5.21722 7.09463 −3.70563
1.15 −2.42617 1.73131 3.88629 −0.987697 −4.20045 1.01261 −4.57644 −0.00256059 2.39632
1.16 −2.39096 −0.986021 3.71671 −2.27233 2.35754 −0.314319 −4.10458 −2.02776 5.43305
1.17 −2.35914 −1.36085 3.56554 2.69250 3.21043 1.19672 −3.69333 −1.14810 −6.35199
1.18 −2.34257 −1.48585 3.48762 0.440014 3.48071 1.63946 −3.48485 −0.792243 −1.03076
1.19 −2.28387 2.83481 3.21604 −3.24199 −6.47432 0.379335 −2.77728 5.03614 7.40427
1.20 −2.28352 −2.81571 3.21448 2.96925 6.42974 −4.34900 −2.77330 4.92822 −6.78035
See next 80 embeddings (of 157 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.157
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(4013\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{157} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4013))\).