Properties

Label 4033.2.a.c
Level 4033
Weight 2
Character orbit 4033.a
Self dual Yes
Analytic conductor 32.204
Analytic rank 1
Dimension 77
CM No

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Newspace parameters

Level: \( N \) = \( 4033 = 37 \cdot 109 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4033.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.2036671352\)
Analytic rank: \(1\)
Dimension: \(77\)
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.

\(\operatorname{Tr}(f)(q) = \) \(77q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 73q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 23q^{7} \) \(\mathstrut -\mathstrut 24q^{8} \) \(\mathstrut +\mathstrut 66q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(77q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 73q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 23q^{7} \) \(\mathstrut -\mathstrut 24q^{8} \) \(\mathstrut +\mathstrut 66q^{9} \) \(\mathstrut -\mathstrut 11q^{10} \) \(\mathstrut -\mathstrut 33q^{11} \) \(\mathstrut -\mathstrut 52q^{12} \) \(\mathstrut -\mathstrut 10q^{13} \) \(\mathstrut -\mathstrut 18q^{14} \) \(\mathstrut -\mathstrut 33q^{15} \) \(\mathstrut +\mathstrut 53q^{16} \) \(\mathstrut -\mathstrut 44q^{17} \) \(\mathstrut -\mathstrut 27q^{18} \) \(\mathstrut -\mathstrut 9q^{19} \) \(\mathstrut -\mathstrut 45q^{20} \) \(\mathstrut -\mathstrut 7q^{21} \) \(\mathstrut -\mathstrut 3q^{22} \) \(\mathstrut -\mathstrut 74q^{23} \) \(\mathstrut +\mathstrut 21q^{24} \) \(\mathstrut +\mathstrut 59q^{25} \) \(\mathstrut -\mathstrut 47q^{26} \) \(\mathstrut -\mathstrut 99q^{27} \) \(\mathstrut -\mathstrut 49q^{28} \) \(\mathstrut -\mathstrut 9q^{29} \) \(\mathstrut -\mathstrut 39q^{30} \) \(\mathstrut -\mathstrut 27q^{31} \) \(\mathstrut -\mathstrut 47q^{32} \) \(\mathstrut -\mathstrut 28q^{33} \) \(\mathstrut -\mathstrut 23q^{34} \) \(\mathstrut -\mathstrut 48q^{35} \) \(\mathstrut +\mathstrut 77q^{36} \) \(\mathstrut +\mathstrut 77q^{37} \) \(\mathstrut -\mathstrut 66q^{38} \) \(\mathstrut -\mathstrut 11q^{39} \) \(\mathstrut -\mathstrut 2q^{40} \) \(\mathstrut -\mathstrut 37q^{41} \) \(\mathstrut -\mathstrut 24q^{42} \) \(\mathstrut -\mathstrut 44q^{43} \) \(\mathstrut -\mathstrut 54q^{44} \) \(\mathstrut -\mathstrut 36q^{45} \) \(\mathstrut -\mathstrut 41q^{46} \) \(\mathstrut -\mathstrut 150q^{47} \) \(\mathstrut -\mathstrut 135q^{48} \) \(\mathstrut +\mathstrut 64q^{49} \) \(\mathstrut +\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 57q^{52} \) \(\mathstrut -\mathstrut 72q^{53} \) \(\mathstrut +\mathstrut 21q^{54} \) \(\mathstrut -\mathstrut 65q^{55} \) \(\mathstrut -\mathstrut 92q^{56} \) \(\mathstrut -\mathstrut 13q^{57} \) \(\mathstrut -\mathstrut 12q^{58} \) \(\mathstrut -\mathstrut 70q^{59} \) \(\mathstrut -\mathstrut 22q^{60} \) \(\mathstrut +\mathstrut 15q^{61} \) \(\mathstrut -\mathstrut 86q^{62} \) \(\mathstrut -\mathstrut 108q^{63} \) \(\mathstrut +\mathstrut 10q^{64} \) \(\mathstrut -\mathstrut 53q^{65} \) \(\mathstrut -\mathstrut 55q^{66} \) \(\mathstrut -\mathstrut 48q^{67} \) \(\mathstrut -\mathstrut 70q^{68} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 11q^{70} \) \(\mathstrut -\mathstrut 127q^{71} \) \(\mathstrut -\mathstrut 12q^{72} \) \(\mathstrut -\mathstrut 33q^{73} \) \(\mathstrut -\mathstrut 9q^{74} \) \(\mathstrut -\mathstrut 115q^{75} \) \(\mathstrut -\mathstrut 24q^{76} \) \(\mathstrut -\mathstrut 40q^{77} \) \(\mathstrut +\mathstrut 81q^{78} \) \(\mathstrut -\mathstrut 7q^{79} \) \(\mathstrut -\mathstrut 62q^{80} \) \(\mathstrut +\mathstrut 53q^{81} \) \(\mathstrut -\mathstrut 68q^{82} \) \(\mathstrut -\mathstrut 164q^{83} \) \(\mathstrut +\mathstrut 7q^{84} \) \(\mathstrut -\mathstrut 9q^{85} \) \(\mathstrut -\mathstrut 50q^{86} \) \(\mathstrut -\mathstrut 75q^{87} \) \(\mathstrut -\mathstrut 82q^{88} \) \(\mathstrut -\mathstrut 26q^{89} \) \(\mathstrut +\mathstrut 23q^{90} \) \(\mathstrut +\mathstrut 16q^{91} \) \(\mathstrut -\mathstrut 117q^{92} \) \(\mathstrut +\mathstrut 19q^{93} \) \(\mathstrut +\mathstrut 23q^{94} \) \(\mathstrut -\mathstrut 92q^{95} \) \(\mathstrut -\mathstrut 35q^{96} \) \(\mathstrut -\mathstrut 19q^{97} \) \(\mathstrut -\mathstrut 10q^{98} \) \(\mathstrut -\mathstrut 19q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.77942 −0.824598 5.72520 1.36321 2.29191 0.958419 −10.3539 −2.32004 −3.78894
1.2 −2.66405 1.68420 5.09718 0.183970 −4.48681 1.02981 −8.25104 −0.163454 −0.490107
1.3 −2.64480 0.673502 4.99498 −3.52022 −1.78128 −3.10314 −7.92112 −2.54639 9.31030
1.4 −2.64408 −3.27443 4.99114 −3.17028 8.65784 2.13489 −7.90879 7.72189 8.38246
1.5 −2.57358 −2.79713 4.62329 0.767586 7.19864 −3.79460 −6.75123 4.82396 −1.97544
1.6 −2.57274 −3.14788 4.61898 −2.11075 8.09868 −0.361357 −6.73794 6.90917 5.43041
1.7 −2.51099 0.0536632 4.30505 −1.45353 −0.134748 3.51496 −5.78795 −2.99712 3.64980
1.8 −2.50194 −1.32701 4.25970 0.884945 3.32011 3.50232 −5.65363 −1.23903 −2.21408
1.9 −2.37004 2.31466 3.61710 2.16488 −5.48585 −1.68976 −3.83259 2.35766 −5.13086
1.10 −2.36112 2.69902 3.57489 −1.67448 −6.37271 0.928256 −3.71850 4.28471 3.95366
1.11 −2.25607 0.265455 3.08984 2.86324 −0.598885 −5.00198 −2.45875 −2.92953 −6.45966
1.12 −2.25323 −3.09773 3.07703 4.21432 6.97988 −2.71387 −2.42679 6.59591 −9.49583
1.13 −2.16010 −1.27573 2.66602 −3.85876 2.75571 4.13321 −1.43868 −1.37250 8.33530
1.14 −2.03838 1.12284 2.15500 3.13166 −2.28878 0.892972 −0.315945 −1.73922 −6.38351
1.15 −1.98064 −1.34264 1.92294 −0.618382 2.65929 −3.38656 0.152622 −1.19732 1.22479
1.16 −1.93065 −1.88783 1.72740 2.91327 3.64473 1.86456 0.526302 0.563889 −5.62450
1.17 −1.80263 2.09678 1.24948 −2.41088 −3.77972 2.66635 1.35291 1.39648 4.34594
1.18 −1.78823 2.78240 1.19778 −2.10202 −4.97558 −2.70114 1.43455 4.74173 3.75890
1.19 −1.77580 −1.89592 1.15346 0.136329 3.36677 −1.80819 1.50328 0.594515 −0.242093
1.20 −1.67758 0.712654 0.814260 2.97956 −1.19553 2.13168 1.98917 −2.49212 −4.99844
See all 77 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.77
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
\(37\) \(-1\)
\(109\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{77} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).