Properties

Label 4033.2.a.f
Level 4033
Weight 2
Character orbit 4033.a
Self dual Yes
Analytic conductor 32.204
Analytic rank 0
Dimension 85
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: \(85\)
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.

\(\operatorname{Tr}(f)(q) = \) \(85q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 21q^{3} \) \(\mathstrut +\mathstrut 93q^{4} \) \(\mathstrut +\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 17q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 98q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(85q \) \(\mathstrut +\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 21q^{3} \) \(\mathstrut +\mathstrut 93q^{4} \) \(\mathstrut +\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 17q^{7} \) \(\mathstrut +\mathstrut 30q^{8} \) \(\mathstrut +\mathstrut 98q^{9} \) \(\mathstrut +\mathstrut 9q^{10} \) \(\mathstrut +\mathstrut 37q^{11} \) \(\mathstrut +\mathstrut 44q^{12} \) \(\mathstrut +\mathstrut 14q^{13} \) \(\mathstrut +\mathstrut 26q^{14} \) \(\mathstrut +\mathstrut 27q^{15} \) \(\mathstrut +\mathstrut 85q^{16} \) \(\mathstrut +\mathstrut 34q^{17} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut +\mathstrut 15q^{19} \) \(\mathstrut +\mathstrut 15q^{20} \) \(\mathstrut +\mathstrut 17q^{21} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut +\mathstrut 72q^{23} \) \(\mathstrut +\mathstrut 15q^{24} \) \(\mathstrut +\mathstrut 85q^{25} \) \(\mathstrut +\mathstrut 33q^{26} \) \(\mathstrut +\mathstrut 69q^{27} \) \(\mathstrut +\mathstrut 7q^{28} \) \(\mathstrut +\mathstrut 19q^{29} \) \(\mathstrut -\mathstrut 9q^{30} \) \(\mathstrut +\mathstrut 23q^{31} \) \(\mathstrut +\mathstrut 51q^{32} \) \(\mathstrut +\mathstrut 32q^{33} \) \(\mathstrut +\mathstrut 49q^{34} \) \(\mathstrut +\mathstrut 40q^{35} \) \(\mathstrut +\mathstrut 121q^{36} \) \(\mathstrut +\mathstrut 85q^{37} \) \(\mathstrut +\mathstrut 84q^{38} \) \(\mathstrut +\mathstrut 39q^{39} \) \(\mathstrut +\mathstrut 22q^{40} \) \(\mathstrut +\mathstrut 55q^{41} \) \(\mathstrut -\mathstrut 28q^{42} \) \(\mathstrut +\mathstrut 78q^{44} \) \(\mathstrut +\mathstrut 28q^{45} \) \(\mathstrut +\mathstrut 17q^{46} \) \(\mathstrut +\mathstrut 184q^{47} \) \(\mathstrut +\mathstrut 97q^{48} \) \(\mathstrut +\mathstrut 88q^{49} \) \(\mathstrut +\mathstrut 26q^{50} \) \(\mathstrut +\mathstrut 27q^{51} \) \(\mathstrut +\mathstrut 73q^{52} \) \(\mathstrut +\mathstrut 64q^{53} \) \(\mathstrut +\mathstrut 31q^{54} \) \(\mathstrut +\mathstrut 39q^{55} \) \(\mathstrut +\mathstrut 68q^{56} \) \(\mathstrut -\mathstrut 33q^{57} \) \(\mathstrut +\mathstrut 28q^{58} \) \(\mathstrut +\mathstrut 60q^{59} \) \(\mathstrut -\mathstrut 22q^{60} \) \(\mathstrut +\mathstrut 7q^{61} \) \(\mathstrut +\mathstrut 70q^{62} \) \(\mathstrut +\mathstrut 28q^{63} \) \(\mathstrut +\mathstrut 102q^{64} \) \(\mathstrut +\mathstrut 17q^{65} \) \(\mathstrut -\mathstrut 15q^{66} \) \(\mathstrut +\mathstrut 82q^{67} \) \(\mathstrut +\mathstrut 92q^{68} \) \(\mathstrut +\mathstrut 22q^{69} \) \(\mathstrut -\mathstrut 41q^{70} \) \(\mathstrut +\mathstrut 113q^{71} \) \(\mathstrut -\mathstrut 19q^{73} \) \(\mathstrut +\mathstrut 11q^{74} \) \(\mathstrut +\mathstrut 45q^{75} \) \(\mathstrut +\mathstrut 34q^{76} \) \(\mathstrut +\mathstrut 64q^{77} \) \(\mathstrut +\mathstrut 29q^{78} \) \(\mathstrut +\mathstrut 23q^{79} \) \(\mathstrut +\mathstrut 54q^{80} \) \(\mathstrut +\mathstrut 149q^{81} \) \(\mathstrut +\mathstrut 4q^{82} \) \(\mathstrut +\mathstrut 100q^{83} \) \(\mathstrut -\mathstrut 49q^{84} \) \(\mathstrut -\mathstrut 5q^{85} \) \(\mathstrut -\mathstrut 24q^{86} \) \(\mathstrut +\mathstrut 65q^{87} \) \(\mathstrut +\mathstrut 14q^{88} \) \(\mathstrut +\mathstrut 84q^{89} \) \(\mathstrut -\mathstrut 21q^{90} \) \(\mathstrut +\mathstrut 32q^{91} \) \(\mathstrut +\mathstrut 95q^{92} \) \(\mathstrut +\mathstrut 19q^{93} \) \(\mathstrut -\mathstrut 47q^{94} \) \(\mathstrut +\mathstrut 102q^{95} \) \(\mathstrut +\mathstrut 29q^{96} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 26q^{98} \) \(\mathstrut +\mathstrut 107q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.76864 3.12213 5.66539 −1.61337 −8.64407 −3.34335 −10.1482 6.74771 4.46684
1.2 −2.68964 −0.345637 5.23416 3.86855 0.929638 −2.16399 −8.69874 −2.88054 −10.4050
1.3 −2.66672 −1.32046 5.11137 0.282486 3.52128 −2.11170 −8.29715 −1.25639 −0.753309
1.4 −2.62495 2.49737 4.89034 0.915854 −6.55547 2.77253 −7.58698 3.23688 −2.40407
1.5 −2.49324 −1.70202 4.21624 −1.58478 4.24353 2.99306 −5.52560 −0.103143 3.95124
1.6 −2.42751 0.768452 3.89281 −1.80408 −1.86543 −1.49541 −4.59481 −2.40948 4.37943
1.7 −2.36110 2.47312 3.57478 1.50101 −5.83928 4.30200 −3.71820 3.11634 −3.54403
1.8 −2.31084 0.814794 3.33999 −0.287417 −1.88286 −2.20640 −3.09651 −2.33611 0.664174
1.9 −2.27621 −2.60153 3.18111 2.30412 5.92161 1.78008 −2.68845 3.76795 −5.24464
1.10 −2.26706 −2.69699 3.13957 −3.90758 6.11425 −4.79951 −2.58348 4.27377 8.85872
1.11 −2.15921 −0.135675 2.66221 −3.14294 0.292951 0.215452 −1.42985 −2.98159 6.78628
1.12 −2.11247 2.70421 2.46253 2.42834 −5.71257 1.36575 −0.977087 4.31276 −5.12979
1.13 −2.03156 3.25300 2.12723 1.52354 −6.60865 −4.26037 −0.258468 7.58200 −3.09515
1.14 −1.99373 −2.36120 1.97494 −0.776635 4.70759 −0.440394 0.0499623 2.57527 1.54840
1.15 −1.96262 −1.03474 1.85187 3.48986 2.03080 −2.58926 0.290726 −1.92931 −6.84926
1.16 −1.92648 −3.23973 1.71134 1.34988 6.24129 3.96616 0.556094 7.49587 −2.60052
1.17 −1.88550 3.14155 1.55513 −2.47668 −5.92341 4.66231 0.838810 6.86935 4.66980
1.18 −1.82466 0.0328358 1.32938 −0.181740 −0.0599140 4.41961 1.22366 −2.99892 0.331613
1.19 −1.67574 1.31893 0.808091 −4.46158 −2.21019 0.153470 1.99732 −1.26041 7.47643
1.20 −1.66354 1.93600 0.767368 −2.27919 −3.22061 0.280246 2.05053 0.748091 3.79153
See all 85 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.85
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 \(T_{2}^{85} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).