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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 $1$

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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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

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

Display 100 coefficients 10 coefficients

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:

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\). Copy content Toggle raw display