Properties

Label 4003.2.a.c
Level 4003
Weight 2
Character orbit 4003.a
Self dual Yes
Analytic conductor 31.964
Analytic rank 0
Dimension 179
CM No

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Newspace parameters

Level: \( N \) = \( 4003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9641159291\)
Analytic rank: \(0\)
Dimension: \(179\)
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.

\(\operatorname{Tr}(f)(q) = \) \(179q \) \(\mathstrut +\mathstrut 22q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 196q^{4} \) \(\mathstrut +\mathstrut 61q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 21q^{7} \) \(\mathstrut +\mathstrut 60q^{8} \) \(\mathstrut +\mathstrut 221q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(179q \) \(\mathstrut +\mathstrut 22q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 196q^{4} \) \(\mathstrut +\mathstrut 61q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 21q^{7} \) \(\mathstrut +\mathstrut 60q^{8} \) \(\mathstrut +\mathstrut 221q^{9} \) \(\mathstrut +\mathstrut 9q^{10} \) \(\mathstrut +\mathstrut 46q^{11} \) \(\mathstrut +\mathstrut 33q^{12} \) \(\mathstrut +\mathstrut 47q^{13} \) \(\mathstrut +\mathstrut 22q^{14} \) \(\mathstrut +\mathstrut 36q^{15} \) \(\mathstrut +\mathstrut 222q^{16} \) \(\mathstrut +\mathstrut 103q^{17} \) \(\mathstrut +\mathstrut 43q^{18} \) \(\mathstrut +\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut 102q^{20} \) \(\mathstrut +\mathstrut 50q^{21} \) \(\mathstrut +\mathstrut 39q^{22} \) \(\mathstrut +\mathstrut 121q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut +\mathstrut 246q^{25} \) \(\mathstrut +\mathstrut 52q^{26} \) \(\mathstrut +\mathstrut 49q^{27} \) \(\mathstrut +\mathstrut 41q^{28} \) \(\mathstrut +\mathstrut 138q^{29} \) \(\mathstrut +\mathstrut 28q^{30} \) \(\mathstrut +\mathstrut 5q^{31} \) \(\mathstrut +\mathstrut 137q^{32} \) \(\mathstrut +\mathstrut 63q^{33} \) \(\mathstrut +\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 72q^{35} \) \(\mathstrut +\mathstrut 279q^{36} \) \(\mathstrut +\mathstrut 118q^{37} \) \(\mathstrut +\mathstrut 123q^{38} \) \(\mathstrut +\mathstrut q^{39} \) \(\mathstrut +\mathstrut 9q^{40} \) \(\mathstrut +\mathstrut 50q^{41} \) \(\mathstrut +\mathstrut 48q^{42} \) \(\mathstrut +\mathstrut 48q^{43} \) \(\mathstrut +\mathstrut 108q^{44} \) \(\mathstrut +\mathstrut 158q^{45} \) \(\mathstrut +\mathstrut 13q^{46} \) \(\mathstrut +\mathstrut 85q^{47} \) \(\mathstrut +\mathstrut 50q^{48} \) \(\mathstrut +\mathstrut 230q^{49} \) \(\mathstrut +\mathstrut 78q^{50} \) \(\mathstrut +\mathstrut 15q^{51} \) \(\mathstrut +\mathstrut 41q^{52} \) \(\mathstrut +\mathstrut 399q^{53} \) \(\mathstrut -\mathstrut 5q^{54} \) \(\mathstrut +\mathstrut 24q^{55} \) \(\mathstrut +\mathstrut 53q^{56} \) \(\mathstrut +\mathstrut 45q^{57} \) \(\mathstrut +\mathstrut 27q^{58} \) \(\mathstrut +\mathstrut 48q^{59} \) \(\mathstrut +\mathstrut 66q^{60} \) \(\mathstrut +\mathstrut 46q^{61} \) \(\mathstrut +\mathstrut 81q^{62} \) \(\mathstrut +\mathstrut 78q^{63} \) \(\mathstrut +\mathstrut 252q^{64} \) \(\mathstrut +\mathstrut 153q^{65} \) \(\mathstrut +\mathstrut 6q^{66} \) \(\mathstrut +\mathstrut 70q^{67} \) \(\mathstrut +\mathstrut 240q^{68} \) \(\mathstrut +\mathstrut 120q^{69} \) \(\mathstrut -\mathstrut 31q^{70} \) \(\mathstrut +\mathstrut 86q^{71} \) \(\mathstrut +\mathstrut 89q^{72} \) \(\mathstrut +\mathstrut 45q^{73} \) \(\mathstrut +\mathstrut 68q^{74} \) \(\mathstrut +\mathstrut 17q^{75} \) \(\mathstrut -\mathstrut 13q^{76} \) \(\mathstrut +\mathstrut 362q^{77} \) \(\mathstrut +\mathstrut 69q^{78} \) \(\mathstrut +\mathstrut 31q^{79} \) \(\mathstrut +\mathstrut 169q^{80} \) \(\mathstrut +\mathstrut 303q^{81} \) \(\mathstrut +\mathstrut 25q^{82} \) \(\mathstrut +\mathstrut 106q^{83} \) \(\mathstrut +\mathstrut 13q^{84} \) \(\mathstrut +\mathstrut 115q^{85} \) \(\mathstrut +\mathstrut 95q^{86} \) \(\mathstrut +\mathstrut 32q^{87} \) \(\mathstrut +\mathstrut 83q^{88} \) \(\mathstrut +\mathstrut 105q^{89} \) \(\mathstrut -\mathstrut 38q^{90} \) \(\mathstrut +\mathstrut 3q^{91} \) \(\mathstrut +\mathstrut 310q^{92} \) \(\mathstrut +\mathstrut 298q^{93} \) \(\mathstrut -\mathstrut 17q^{94} \) \(\mathstrut +\mathstrut 102q^{95} \) \(\mathstrut -\mathstrut 82q^{96} \) \(\mathstrut +\mathstrut 34q^{97} \) \(\mathstrut +\mathstrut 81q^{98} \) \(\mathstrut +\mathstrut 58q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.79615 2.84308 5.81846 −1.26517 −7.94968 1.49582 −10.6770 5.08309 3.53762
1.2 −2.74344 −0.797814 5.52649 −1.01783 2.18876 −0.805682 −9.67473 −2.36349 2.79236
1.3 −2.71994 −1.21777 5.39810 4.02685 3.31228 −3.20136 −9.24264 −1.51703 −10.9528
1.4 −2.64748 2.89261 5.00915 3.62616 −7.65814 1.08314 −7.96667 5.36721 −9.60018
1.5 −2.63071 2.57872 4.92061 −1.88131 −6.78385 −5.19574 −7.68328 3.64979 4.94917
1.6 −2.62123 2.13484 4.87085 4.03687 −5.59591 2.86410 −7.52515 1.55755 −10.5816
1.7 −2.61690 −2.40767 4.84815 1.76120 6.30063 −1.48829 −7.45330 2.79688 −4.60888
1.8 −2.61424 −1.76762 4.83424 0.635426 4.62098 1.21621 −7.40938 0.124480 −1.66116
1.9 −2.60379 0.316007 4.77975 1.57805 −0.822817 2.80041 −7.23789 −2.90014 −4.10892
1.10 −2.54699 −0.919443 4.48717 −0.00498874 2.34182 −0.132134 −6.33481 −2.15462 0.0127063
1.11 −2.53975 −3.24477 4.45032 3.59171 8.24090 4.74281 −6.22319 7.52854 −9.12202
1.12 −2.46969 0.639879 4.09938 −1.75762 −1.58030 −0.0776251 −5.18482 −2.59056 4.34079
1.13 −2.45370 −0.415301 4.02066 −3.18084 1.01903 −2.85630 −4.95811 −2.82752 7.80484
1.14 −2.40245 0.531316 3.77174 3.68505 −1.27646 −3.72446 −4.25652 −2.71770 −8.85314
1.15 −2.39320 3.24043 3.72738 2.56572 −7.75497 −2.79627 −4.13396 7.50035 −6.14026
1.16 −2.38397 1.62509 3.68332 −0.985677 −3.87416 −1.38154 −4.01299 −0.359094 2.34983
1.17 −2.34837 −3.35198 3.51485 −2.33127 7.87170 −2.55818 −3.55742 8.23579 5.47468
1.18 −2.34196 −1.89472 3.48479 −4.06412 4.43736 3.16270 −3.47733 0.589962 9.51803
1.19 −2.33012 −0.602166 3.42946 1.32408 1.40312 −1.36511 −3.33081 −2.63740 −3.08526
1.20 −2.32611 2.50152 3.41079 0.271197 −5.81882 2.22103 −3.28166 3.25762 −0.630834
See next 80 embeddings (of 179 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.179
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(4003\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{179} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\).