Properties

Label 4027.2.a.b
Level 4027
Weight 2
Character orbit 4027.a
Self dual Yes
Analytic conductor 32.156
Analytic rank 1
Dimension 159
CM No

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Newspace parameters

Level: \( N \) = \( 4027 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.155756894\)
Analytic rank: \(1\)
Dimension: \(159\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 159q - 22q^{2} - 19q^{3} + 148q^{4} - 70q^{5} - 23q^{6} - 19q^{7} - 66q^{8} + 126q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 159q - 22q^{2} - 19q^{3} + 148q^{4} - 70q^{5} - 23q^{6} - 19q^{7} - 66q^{8} + 126q^{9} - 23q^{10} - 33q^{11} - 57q^{12} - 90q^{13} - 28q^{14} - 22q^{15} + 130q^{16} - 145q^{17} - 50q^{18} - 28q^{19} - 121q^{20} - 69q^{21} - 26q^{22} - 79q^{23} - 62q^{24} + 123q^{25} - 40q^{26} - 70q^{27} - 43q^{28} - 109q^{29} - 43q^{30} - 21q^{31} - 139q^{32} - 83q^{33} - 93q^{35} + 75q^{36} - 65q^{37} - 122q^{38} - 18q^{39} - 43q^{40} - 71q^{41} - 88q^{42} - 72q^{43} - 79q^{44} - 181q^{45} - 11q^{46} - 114q^{47} - 118q^{48} + 118q^{49} - 77q^{50} - 29q^{51} - 169q^{52} - 220q^{53} - 80q^{54} - 37q^{55} - 72q^{56} - 90q^{57} - 8q^{58} - 60q^{59} - 42q^{60} - 108q^{61} - 152q^{62} - 65q^{63} + 114q^{64} - 81q^{65} - 40q^{66} - 50q^{67} - 319q^{68} - 103q^{69} + 4q^{70} - 7q^{71} - 129q^{72} - 94q^{73} - 79q^{74} - 59q^{75} - 46q^{76} - 329q^{77} + 8q^{78} - 18q^{79} - 190q^{80} + 59q^{81} - 56q^{82} - 201q^{83} - 71q^{84} - 26q^{85} - 52q^{86} - 126q^{87} - 66q^{88} - 114q^{89} - 33q^{90} - 30q^{91} - 204q^{92} - 125q^{93} + 9q^{94} - 84q^{95} - 88q^{96} - 56q^{97} - 110q^{98} - 46q^{99} + O(q^{100}) \)

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 −2.82139 1.96833 5.96026 2.22438 −5.55342 0.144940 −11.1735 0.874306 −6.27587
1.2 −2.78842 −1.28715 5.77530 −4.20295 3.58911 −2.09011 −10.5271 −1.34326 11.7196
1.3 −2.73932 −1.57093 5.50386 0.430202 4.30328 4.35305 −9.59817 −0.532177 −1.17846
1.4 −2.73389 −3.04946 5.47414 −1.23983 8.33688 −1.91443 −9.49790 6.29922 3.38954
1.5 −2.72666 0.318995 5.43469 −1.22389 −0.869792 −4.59347 −9.36524 −2.89824 3.33713
1.6 −2.70494 2.29108 5.31668 −3.64034 −6.19723 2.95034 −8.97141 2.24905 9.84688
1.7 −2.69495 1.90961 5.26276 −0.957907 −5.14631 0.102731 −8.79297 0.646618 2.58151
1.8 −2.64324 2.50355 4.98674 1.52675 −6.61749 −4.42325 −7.89468 3.26775 −4.03556
1.9 −2.63499 −1.76075 4.94319 1.57936 4.63957 3.18044 −7.75529 0.100248 −4.16161
1.10 −2.59596 −2.97309 4.73903 −2.99704 7.71803 3.05793 −7.11041 5.83925 7.78021
1.11 −2.59401 −2.55671 4.72888 1.35267 6.63213 −3.48116 −7.07874 3.53677 −3.50885
1.12 −2.58046 0.749522 4.65878 −1.56103 −1.93411 3.58823 −6.86087 −2.43822 4.02819
1.13 −2.57109 −0.605717 4.61051 3.41025 1.55735 −4.02437 −6.71185 −2.63311 −8.76807
1.14 −2.54982 3.26459 4.50158 −1.30834 −8.32411 3.58480 −6.37858 7.65753 3.33603
1.15 −2.50942 −0.724063 4.29717 −3.46864 1.81698 2.45320 −5.76455 −2.47573 8.70427
1.16 −2.48250 −1.21941 4.16279 0.264971 3.02717 −2.37151 −5.36913 −1.51305 −0.657790
1.17 −2.46215 −2.74752 4.06218 4.08202 6.76481 −0.414263 −5.07738 4.54887 −10.0505
1.18 −2.44784 0.695688 3.99191 2.35878 −1.70293 2.06757 −4.87586 −2.51602 −5.77391
1.19 −2.37585 0.673958 3.64466 −1.12762 −1.60122 1.18808 −3.90747 −2.54578 2.67904
1.20 −2.29049 2.02403 3.24633 −4.01248 −4.63602 1.58118 −2.85471 1.09671 9.19054
See next 80 embeddings (of 159 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.159
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
\(4027\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{159} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4027))\).