# 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 Inner twists 1

# Related objects

## 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: $$179$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4003.2.a.c 179

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4003.2.a.c 179 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$4003$$ $$-1$$

## Hecke kernels

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

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database