# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$31.9641159291$$ Analytic rank: $$0$$ Dimension: $$179$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

## Atkin-Lehner signs

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

## 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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{179} - 22 T_{2}^{178} - 35 T_{2}^{177} + 4270 T_{2}^{176} - 18531 T_{2}^{175} + \cdots + 10\!\cdots\!20$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4003))$$.