Properties

Label 4033.2.a.e
Level 4033
Weight 2
Character orbit 4033.a
Self dual Yes
Analytic conductor 32.204
Analytic rank 0
Dimension 82
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: \(0\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 17q^{3} \) \(\mathstrut +\mathstrut 88q^{4} \) \(\mathstrut +\mathstrut 22q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 15q^{7} \) \(\mathstrut +\mathstrut 33q^{8} \) \(\mathstrut +\mathstrut 91q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 17q^{3} \) \(\mathstrut +\mathstrut 88q^{4} \) \(\mathstrut +\mathstrut 22q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 15q^{7} \) \(\mathstrut +\mathstrut 33q^{8} \) \(\mathstrut +\mathstrut 91q^{9} \) \(\mathstrut +\mathstrut 13q^{10} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 36q^{12} \) \(\mathstrut +\mathstrut 23q^{13} \) \(\mathstrut +\mathstrut 24q^{14} \) \(\mathstrut +\mathstrut 35q^{15} \) \(\mathstrut +\mathstrut 104q^{16} \) \(\mathstrut +\mathstrut 43q^{17} \) \(\mathstrut +\mathstrut 42q^{18} \) \(\mathstrut +\mathstrut 15q^{19} \) \(\mathstrut +\mathstrut 59q^{20} \) \(\mathstrut +\mathstrut 9q^{21} \) \(\mathstrut +\mathstrut 6q^{22} \) \(\mathstrut +\mathstrut 70q^{23} \) \(\mathstrut +\mathstrut 15q^{24} \) \(\mathstrut +\mathstrut 98q^{25} \) \(\mathstrut +\mathstrut 10q^{26} \) \(\mathstrut +\mathstrut 65q^{27} \) \(\mathstrut +\mathstrut 37q^{28} \) \(\mathstrut +\mathstrut 27q^{29} \) \(\mathstrut +\mathstrut 17q^{30} \) \(\mathstrut +\mathstrut 59q^{31} \) \(\mathstrut +\mathstrut 46q^{32} \) \(\mathstrut +\mathstrut 16q^{33} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut +\mathstrut 80q^{35} \) \(\mathstrut +\mathstrut 88q^{36} \) \(\mathstrut -\mathstrut 82q^{37} \) \(\mathstrut +\mathstrut 82q^{38} \) \(\mathstrut +\mathstrut 13q^{39} \) \(\mathstrut +\mathstrut 14q^{40} \) \(\mathstrut +\mathstrut 3q^{41} \) \(\mathstrut +\mathstrut 62q^{42} \) \(\mathstrut +\mathstrut 7q^{43} \) \(\mathstrut -\mathstrut 11q^{44} \) \(\mathstrut +\mathstrut 42q^{45} \) \(\mathstrut -\mathstrut 11q^{46} \) \(\mathstrut +\mathstrut 123q^{47} \) \(\mathstrut +\mathstrut 45q^{48} \) \(\mathstrut +\mathstrut 105q^{49} \) \(\mathstrut +\mathstrut 27q^{50} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 30q^{52} \) \(\mathstrut +\mathstrut 82q^{53} \) \(\mathstrut -\mathstrut 27q^{54} \) \(\mathstrut +\mathstrut 37q^{55} \) \(\mathstrut +\mathstrut 66q^{56} \) \(\mathstrut +\mathstrut 29q^{57} \) \(\mathstrut -\mathstrut 34q^{58} \) \(\mathstrut +\mathstrut 60q^{59} \) \(\mathstrut +\mathstrut 94q^{60} \) \(\mathstrut +\mathstrut 9q^{61} \) \(\mathstrut -\mathstrut 2q^{62} \) \(\mathstrut +\mathstrut 106q^{63} \) \(\mathstrut +\mathstrut 93q^{64} \) \(\mathstrut +\mathstrut 9q^{65} \) \(\mathstrut +\mathstrut 63q^{66} \) \(\mathstrut +\mathstrut 113q^{68} \) \(\mathstrut +\mathstrut 48q^{69} \) \(\mathstrut +\mathstrut 47q^{70} \) \(\mathstrut +\mathstrut 59q^{71} \) \(\mathstrut +\mathstrut 63q^{72} \) \(\mathstrut +\mathstrut 21q^{73} \) \(\mathstrut -\mathstrut 10q^{74} \) \(\mathstrut +\mathstrut 77q^{75} \) \(\mathstrut +\mathstrut 22q^{76} \) \(\mathstrut +\mathstrut 30q^{77} \) \(\mathstrut +\mathstrut 29q^{78} \) \(\mathstrut +\mathstrut 55q^{79} \) \(\mathstrut +\mathstrut 88q^{80} \) \(\mathstrut +\mathstrut 42q^{81} \) \(\mathstrut +\mathstrut 4q^{82} \) \(\mathstrut +\mathstrut 92q^{83} \) \(\mathstrut +\mathstrut 43q^{84} \) \(\mathstrut -\mathstrut 7q^{85} \) \(\mathstrut +\mathstrut 17q^{86} \) \(\mathstrut +\mathstrut 147q^{87} \) \(\mathstrut -\mathstrut 13q^{88} \) \(\mathstrut +\mathstrut 100q^{89} \) \(\mathstrut +\mathstrut 91q^{90} \) \(\mathstrut +\mathstrut 28q^{91} \) \(\mathstrut +\mathstrut 127q^{92} \) \(\mathstrut +\mathstrut 3q^{93} \) \(\mathstrut +\mathstrut 30q^{94} \) \(\mathstrut +\mathstrut 48q^{95} \) \(\mathstrut -\mathstrut 31q^{96} \) \(\mathstrut +\mathstrut 53q^{97} \) \(\mathstrut +\mathstrut 101q^{98} \) \(\mathstrut -\mathstrut 20q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.75814 1.25704 5.60732 1.25507 −3.46708 2.38515 −9.94947 −1.41986 −3.46166
1.2 −2.62838 −1.37635 4.90837 −2.91482 3.61757 −1.06759 −7.64429 −1.10566 7.66124
1.3 −2.58504 −2.45542 4.68242 2.00515 6.34735 −0.0308722 −6.93416 3.02908 −5.18339
1.4 −2.58335 3.13276 4.67371 4.37342 −8.09303 0.189689 −6.90713 6.81421 −11.2981
1.5 −2.53999 0.468124 4.45153 3.58258 −1.18903 0.512106 −6.22686 −2.78086 −9.09972
1.6 −2.52769 2.28955 4.38920 −2.00544 −5.78727 −4.48140 −6.03914 2.24205 5.06913
1.7 −2.40882 −0.455887 3.80243 1.02492 1.09815 3.87779 −4.34175 −2.79217 −2.46884
1.8 −2.37038 1.65069 3.61871 −2.32287 −3.91275 −2.51278 −3.83696 −0.275237 5.50608
1.9 −2.34943 −2.83343 3.51983 −1.24434 6.65694 2.59127 −3.57074 5.02830 2.92348
1.10 −2.33134 −0.785356 3.43514 0.539311 1.83093 −3.00025 −3.34579 −2.38322 −1.25731
1.11 −2.32285 2.68333 3.39564 −2.18288 −6.23298 1.55979 −3.24187 4.20026 5.07050
1.12 −2.23087 −0.522576 2.97676 0.778391 1.16580 −1.74225 −2.17903 −2.72691 −1.73649
1.13 −1.92046 1.69690 1.68818 −0.668171 −3.25884 1.48884 0.598834 −0.120514 1.28320
1.14 −1.75973 1.54621 1.09665 4.09824 −2.72090 −4.10916 1.58965 −0.609248 −7.21180
1.15 −1.75361 −2.25140 1.07516 −3.95285 3.94809 2.44850 1.62182 2.06881 6.93177
1.16 −1.67615 2.74461 0.809463 2.71622 −4.60037 4.10502 1.99551 4.53291 −4.55277
1.17 −1.62784 −2.32510 0.649851 2.52480 3.78489 −3.47012 2.19782 2.40611 −4.10996
1.18 −1.58741 −1.76709 0.519871 4.42213 2.80509 5.11364 2.34957 0.122598 −7.01973
1.19 −1.58089 0.557711 0.499209 2.89052 −0.881678 1.64472 2.37258 −2.68896 −4.56959
1.20 −1.52363 −2.34833 0.321441 0.104108 3.57798 −3.33397 2.55750 2.51466 −0.158621
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.82
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 \(T_{2}^{82} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).