Properties

Label 4033.2.a.d
Level 4033
Weight 2
Character orbit 4033.a
Self dual Yes
Analytic conductor 32.204
Analytic rank 1
Dimension 79
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: \(79\)
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.

\(\operatorname{Tr}(f)(q) = \) \(79q \) \(\mathstrut -\mathstrut 11q^{2} \) \(\mathstrut -\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 79q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 14q^{6} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 76q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(79q \) \(\mathstrut -\mathstrut 11q^{2} \) \(\mathstrut -\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 79q^{4} \) \(\mathstrut -\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 14q^{6} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut -\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 76q^{9} \) \(\mathstrut -\mathstrut 13q^{10} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut 40q^{12} \) \(\mathstrut -\mathstrut 18q^{13} \) \(\mathstrut -\mathstrut 42q^{14} \) \(\mathstrut -\mathstrut 49q^{15} \) \(\mathstrut +\mathstrut 83q^{16} \) \(\mathstrut -\mathstrut 62q^{17} \) \(\mathstrut -\mathstrut 33q^{18} \) \(\mathstrut -\mathstrut 25q^{19} \) \(\mathstrut -\mathstrut 39q^{20} \) \(\mathstrut -\mathstrut 15q^{21} \) \(\mathstrut -\mathstrut 31q^{22} \) \(\mathstrut -\mathstrut 94q^{23} \) \(\mathstrut -\mathstrut 39q^{24} \) \(\mathstrut +\mathstrut 71q^{25} \) \(\mathstrut -\mathstrut 35q^{26} \) \(\mathstrut -\mathstrut 47q^{27} \) \(\mathstrut -\mathstrut 13q^{28} \) \(\mathstrut -\mathstrut 17q^{29} \) \(\mathstrut +\mathstrut 15q^{30} \) \(\mathstrut -\mathstrut 37q^{31} \) \(\mathstrut -\mathstrut 105q^{32} \) \(\mathstrut -\mathstrut 60q^{33} \) \(\mathstrut +\mathstrut 9q^{34} \) \(\mathstrut -\mathstrut 60q^{35} \) \(\mathstrut +\mathstrut 43q^{36} \) \(\mathstrut -\mathstrut 79q^{37} \) \(\mathstrut -\mathstrut 80q^{38} \) \(\mathstrut -\mathstrut 41q^{39} \) \(\mathstrut -\mathstrut 64q^{40} \) \(\mathstrut -\mathstrut 37q^{41} \) \(\mathstrut -\mathstrut 30q^{42} \) \(\mathstrut -\mathstrut 20q^{43} \) \(\mathstrut -\mathstrut 14q^{44} \) \(\mathstrut -\mathstrut 8q^{45} \) \(\mathstrut +\mathstrut 61q^{46} \) \(\mathstrut -\mathstrut 148q^{47} \) \(\mathstrut -\mathstrut 39q^{48} \) \(\mathstrut +\mathstrut 82q^{49} \) \(\mathstrut -\mathstrut 90q^{50} \) \(\mathstrut -\mathstrut 45q^{51} \) \(\mathstrut -\mathstrut 27q^{52} \) \(\mathstrut -\mathstrut 70q^{53} \) \(\mathstrut -\mathstrut 41q^{54} \) \(\mathstrut -\mathstrut 105q^{55} \) \(\mathstrut -\mathstrut 68q^{56} \) \(\mathstrut -\mathstrut 31q^{57} \) \(\mathstrut -\mathstrut 14q^{58} \) \(\mathstrut -\mathstrut 96q^{59} \) \(\mathstrut -\mathstrut 74q^{60} \) \(\mathstrut -\mathstrut 21q^{61} \) \(\mathstrut +\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 60q^{63} \) \(\mathstrut +\mathstrut 132q^{64} \) \(\mathstrut -\mathstrut 15q^{65} \) \(\mathstrut +\mathstrut 71q^{66} \) \(\mathstrut -\mathstrut 44q^{67} \) \(\mathstrut -\mathstrut 166q^{68} \) \(\mathstrut -\mathstrut 72q^{69} \) \(\mathstrut -\mathstrut 9q^{70} \) \(\mathstrut -\mathstrut 55q^{71} \) \(\mathstrut -\mathstrut 126q^{72} \) \(\mathstrut -\mathstrut 27q^{73} \) \(\mathstrut +\mathstrut 11q^{74} \) \(\mathstrut -\mathstrut 39q^{75} \) \(\mathstrut -\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 104q^{77} \) \(\mathstrut -\mathstrut 47q^{78} \) \(\mathstrut -\mathstrut 49q^{79} \) \(\mathstrut -\mathstrut 82q^{80} \) \(\mathstrut +\mathstrut 55q^{81} \) \(\mathstrut +\mathstrut 20q^{82} \) \(\mathstrut -\mathstrut 52q^{83} \) \(\mathstrut -\mathstrut 29q^{84} \) \(\mathstrut +\mathstrut 3q^{85} \) \(\mathstrut -\mathstrut 32q^{86} \) \(\mathstrut -\mathstrut 113q^{87} \) \(\mathstrut +\mathstrut 14q^{88} \) \(\mathstrut -\mathstrut 68q^{89} \) \(\mathstrut -\mathstrut 39q^{90} \) \(\mathstrut -\mathstrut 30q^{91} \) \(\mathstrut -\mathstrut 179q^{92} \) \(\mathstrut -\mathstrut 53q^{93} \) \(\mathstrut -\mathstrut 33q^{94} \) \(\mathstrut -\mathstrut 86q^{95} \) \(\mathstrut +\mathstrut 33q^{96} \) \(\mathstrut -\mathstrut 57q^{97} \) \(\mathstrut -\mathstrut 116q^{98} \) \(\mathstrut +\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.81086 −2.68213 5.90091 2.99603 7.53909 4.86255 −10.9649 4.19384 −8.42140
1.2 −2.77527 1.86757 5.70214 1.50597 −5.18302 −3.04702 −10.2744 0.487817 −4.17947
1.3 −2.76342 −1.75117 5.63647 −3.34867 4.83922 −0.176382 −10.0491 0.0666133 9.25378
1.4 −2.75248 1.04725 5.57617 −2.70262 −2.88255 3.33593 −9.84336 −1.90326 7.43892
1.5 −2.66315 1.52081 5.09239 1.90830 −4.05016 −2.83326 −8.23552 −0.687131 −5.08211
1.6 −2.65601 −2.08930 5.05439 −0.388260 5.54922 −5.15918 −8.11249 1.36520 1.03122
1.7 −2.60105 3.31359 4.76544 −1.17110 −8.61880 2.01510 −7.19305 7.97988 3.04608
1.8 −2.50943 1.55716 4.29722 −3.66304 −3.90758 4.73862 −5.76470 −0.575254 9.19213
1.9 −2.50108 −2.74712 4.25541 3.75941 6.87078 −1.77571 −5.64098 4.54668 −9.40258
1.10 −2.32027 1.35424 3.38367 2.79503 −3.14221 4.55894 −3.21050 −1.16603 −6.48525
1.11 −2.27855 −2.02568 3.19181 −1.32967 4.61563 2.20899 −2.71560 1.10340 3.02972
1.12 −2.25482 −0.982243 3.08420 2.20296 2.21478 0.164703 −2.44467 −2.03520 −4.96728
1.13 −2.24777 −0.430015 3.05245 −2.54923 0.966572 −1.16281 −2.36567 −2.81509 5.73008
1.14 −2.13414 −1.44031 2.55454 3.76689 3.07382 1.36855 −1.18346 −0.925504 −8.03907
1.15 −2.08653 2.88703 2.35363 −4.18679 −6.02388 −1.99666 −0.737856 5.33492 8.73588
1.16 −2.06826 0.566483 2.27770 0.827262 −1.17163 0.908209 −0.574346 −2.67910 −1.71099
1.17 −1.97055 −1.02182 1.88308 −1.99471 2.01355 −4.59826 0.230406 −1.95588 3.93068
1.18 −1.80025 1.41167 1.24090 0.213559 −2.54136 −2.54704 1.36657 −1.00719 −0.384459
1.19 −1.79445 −3.46032 1.22004 0.638539 6.20936 −3.14453 1.39960 8.97380 −1.14582
1.20 −1.74629 2.61915 1.04954 −0.0450252 −4.57381 −1.64580 1.65978 3.85995 0.0786272
See all 79 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.79
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 \(T_{2}^{79} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).