Properties

Label 112.5.k.a
Level $112$
Weight $5$
Character orbit 112.k
Analytic conductor $11.577$
Analytic rank $0$
Dimension $96$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(43,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{4} - 132 q^{6} - 200 q^{10} - 96 q^{11} + 660 q^{12} - 294 q^{14} + 642 q^{16} - 810 q^{18} + 1408 q^{19} - 2604 q^{20} - 106 q^{22} - 1152 q^{23} + 3880 q^{24} + 6828 q^{26} + 3648 q^{27} - 864 q^{29}+ \cdots - 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −3.99491 + 0.201653i −6.16471 − 6.16471i 15.9187 − 1.61118i 31.8098 + 31.8098i 25.8706 + 23.3843i −18.5203 −63.2688 + 9.64656i − 4.99279i −133.492 − 120.663i
43.2 −3.95485 + 0.599328i 1.79635 + 1.79635i 15.2816 − 4.74050i −4.04343 − 4.04343i −8.18087 − 6.02767i 18.5203 −57.5953 + 27.9066i − 74.5463i 18.4145 + 13.5678i
43.3 −3.93459 + 0.720428i −2.13007 − 2.13007i 14.9620 − 5.66917i −28.8248 − 28.8248i 9.91550 + 6.84638i −18.5203 −54.7850 + 33.0849i − 71.9256i 134.180 + 92.6475i
43.4 −3.78720 − 1.28728i −6.03112 − 6.03112i 12.6858 + 9.75038i −3.14917 − 3.14917i 15.0774 + 30.6048i 18.5203 −35.4924 − 53.2569i − 8.25109i 7.87268 + 15.9804i
43.5 −3.78615 + 1.29039i 11.4158 + 11.4158i 12.6698 − 9.77120i 8.99601 + 8.99601i −57.9529 − 28.4912i −18.5203 −35.3610 + 53.3441i 179.643i −45.6686 − 22.4519i
43.6 −3.77066 − 1.33496i 10.8491 + 10.8491i 12.4357 + 10.0674i −31.7296 − 31.7296i −26.4251 − 55.3914i 18.5203 −33.4513 − 54.5620i 154.406i 77.2836 + 161.999i
43.7 −3.74845 + 1.39612i −8.86492 − 8.86492i 12.1017 − 10.4665i 8.20272 + 8.20272i 45.6061 + 20.8532i 18.5203 −30.7502 + 56.1287i 76.1735i −42.1994 − 19.2955i
43.8 −3.71065 − 1.49368i −11.6411 − 11.6411i 11.5379 + 11.0850i −10.4274 − 10.4274i 25.8080 + 60.5840i −18.5203 −26.2555 − 58.3665i 190.030i 23.1173 + 54.2677i
43.9 −3.55330 − 1.83686i 4.91576 + 4.91576i 9.25192 + 13.0538i −5.29630 − 5.29630i −8.43764 − 26.4967i −18.5203 −8.89692 − 63.3786i − 32.6706i 9.09082 + 28.5479i
43.10 −3.35720 + 2.17468i 7.22622 + 7.22622i 6.54156 − 14.6016i −12.7239 − 12.7239i −39.9746 − 8.54517i 18.5203 9.79252 + 63.2464i 23.4366i 70.3872 + 15.0463i
43.11 −2.91721 + 2.73677i 1.88359 + 1.88359i 1.02022 − 15.9674i 6.01207 + 6.01207i −10.6498 − 0.339881i −18.5203 40.7230 + 49.3725i − 73.9042i −33.9921 − 1.08484i
43.12 −2.91660 − 2.73741i 0.212552 + 0.212552i 1.01313 + 15.9679i 14.9819 + 14.9819i −0.0380866 − 1.20177i 18.5203 40.7558 − 49.3453i − 80.9096i −2.68458 − 84.7081i
43.13 −2.74539 + 2.90910i 1.91051 + 1.91051i −0.925711 − 15.9732i 33.8528 + 33.8528i −10.8030 + 0.312775i 18.5203 49.0090 + 41.1596i − 73.6999i −191.420 + 5.54214i
43.14 −2.72871 − 2.92475i 11.3645 + 11.3645i −1.10833 + 15.9616i 25.9572 + 25.9572i 2.22795 − 64.2485i 18.5203 49.7079 − 40.3128i 177.302i 5.08878 − 146.748i
43.15 −2.35239 − 3.23516i −5.21415 − 5.21415i −4.93249 + 15.2207i 19.8037 + 19.8037i −4.60286 + 29.1343i −18.5203 60.8446 − 19.8478i − 26.6252i 17.4820 − 110.654i
43.16 −1.92494 + 3.50637i −4.29879 − 4.29879i −8.58925 − 13.4991i −23.0339 − 23.0339i 23.3480 − 6.79825i 18.5203 63.8665 − 4.13222i − 44.0408i 125.104 − 36.4266i
43.17 −1.61373 − 3.66004i 6.54673 + 6.54673i −10.7917 + 11.8126i −10.9039 − 10.9039i 13.3966 − 34.5259i −18.5203 60.6497 + 20.4357i 4.71923i −22.3128 + 57.5049i
43.18 −1.44740 + 3.72895i 8.25365 + 8.25365i −11.8101 − 10.7945i −26.0103 − 26.0103i −42.7237 + 18.8311i −18.5203 57.3462 − 28.4151i 55.2453i 134.638 − 59.3438i
43.19 −1.20482 − 3.81424i −12.1580 − 12.1580i −13.0968 + 9.19093i 7.84203 + 7.84203i −31.7254 + 61.0217i 18.5203 50.8357 + 38.8810i 214.634i 20.4632 − 39.3596i
43.20 −1.19668 + 3.81680i 10.3892 + 10.3892i −13.1359 − 9.13497i 9.65329 + 9.65329i −52.0860 + 27.2209i 18.5203 50.5858 − 39.2055i 134.871i −48.3966 + 25.2928i
See all 96 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 43.48
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.5.k.a ✓ 96
4.b odd 2 1 448.5.k.a 96
16.e even 4 1 448.5.k.a 96
16.f odd 4 1 inner 112.5.k.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.k.a ✓ 96 1.a even 1 1 trivial
112.5.k.a ✓ 96 16.f odd 4 1 inner
448.5.k.a 96 4.b odd 2 1
448.5.k.a 96 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(112, [\chi])\).