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Properties

Label 115.5.f.a
Level $115$
Weight $5$
Character orbit 115.f
Analytic conductor $11.888$
Analytic rank $0$
Dimension $88$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,5,Mod(47,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.47");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 115.f (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(11.8875457546\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 48 q^{5} + 160 q^{7} + 180 q^{8} + 16 q^{10} + 312 q^{11} - 720 q^{12} - 760 q^{13} - 60 q^{15} - 4528 q^{16} + 420 q^{17} + 980 q^{18} + 1344 q^{20} + 1976 q^{21} + 720 q^{22} - 1176 q^{25} + 1872 q^{26}+ \cdots - 22320 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −5.45879 − 5.45879i 0.249588 − 0.249588i 43.5967i 18.7380 + 16.5495i −2.72490 19.4401 + 19.4401i 150.645 − 150.645i 80.8754i −11.9468 − 192.627i
47.2 −5.33319 − 5.33319i −7.23554 + 7.23554i 40.8858i −24.4965 − 4.99232i 77.1770 29.3451 + 29.3451i 132.721 − 132.721i − 23.7062i 104.019 + 157.269i
47.3 −4.97771 − 4.97771i 7.01798 − 7.01798i 33.5552i −23.4643 + 8.62722i −69.8670 −45.7084 − 45.7084i 87.3849 − 87.3849i − 17.5042i 159.742 + 73.8545i
47.4 −4.97086 − 4.97086i 2.56662 − 2.56662i 33.4190i 6.66462 − 24.0953i −25.5166 17.2314 + 17.2314i 86.5873 − 86.5873i 67.8250i −152.903 + 86.6455i
47.5 −4.73356 − 4.73356i −10.2373 + 10.2373i 28.8132i 11.3791 − 22.2602i 96.9182 −46.2071 − 46.2071i 60.6523 − 60.6523i − 128.606i −159.234 + 51.5063i
47.6 −4.33895 − 4.33895i −0.812388 + 0.812388i 21.6530i 24.9780 + 1.04909i 7.04982 −60.2203 − 60.2203i 24.5280 − 24.5280i 79.6801i −103.826 − 112.930i
47.7 −4.30136 − 4.30136i 10.0667 − 10.0667i 21.0033i 7.80587 + 23.7501i −86.6013 36.0694 + 36.0694i 21.5211 − 21.5211i − 121.679i 68.5819 − 135.734i
47.8 −4.20795 − 4.20795i 9.21285 − 9.21285i 19.4137i −8.64878 − 23.4563i −77.5344 16.5187 + 16.5187i 14.3646 − 14.3646i − 88.7533i −62.3093 + 135.097i
47.9 −4.08601 − 4.08601i −10.8237 + 10.8237i 17.3909i 16.5685 + 18.7212i 88.4518 35.7857 + 35.7857i 5.68317 − 5.68317i − 153.307i 8.79606 − 144.194i
47.10 −3.82088 − 3.82088i −3.62979 + 3.62979i 13.1983i −11.2115 + 22.3451i 27.7380 −9.87428 − 9.87428i −10.7050 + 10.7050i 54.6493i 128.216 − 42.5403i
47.11 −3.55253 − 3.55253i −5.80472 + 5.80472i 9.24088i −21.4504 − 12.8406i 41.2428 2.81474 + 2.81474i −24.0119 + 24.0119i 13.6105i 30.5864 + 121.820i
47.12 −2.95338 − 2.95338i 0.0137449 − 0.0137449i 1.44491i 21.7668 − 12.2965i −0.0811878 52.4860 + 52.4860i −42.9867 + 42.9867i 80.9996i −100.602 − 27.9694i
47.13 −2.86409 − 2.86409i 9.27176 − 9.27176i 0.406037i 24.7873 − 3.25454i −53.1103 −31.1339 − 31.1339i −44.6625 + 44.6625i − 90.9310i −80.3142 − 61.6717i
47.14 −2.39187 − 2.39187i 3.09239 − 3.09239i − 4.55787i −24.9722 + 1.17796i −14.7932 52.6653 + 52.6653i −49.1719 + 49.1719i 61.8743i 62.5480 + 56.9129i
47.15 −2.19710 − 2.19710i −7.89548 + 7.89548i − 6.34549i 12.9495 − 21.3848i 34.6943 18.7701 + 18.7701i −49.0953 + 49.0953i − 43.6771i −75.4360 + 18.5332i
47.16 −2.04029 − 2.04029i 2.48006 − 2.48006i − 7.67444i −14.0131 − 20.7035i −10.1201 −49.5729 − 49.5729i −48.3027 + 48.3027i 68.6986i −13.6504 + 70.8318i
47.17 −1.82478 − 1.82478i 3.91359 − 3.91359i − 9.34035i −0.467365 + 24.9956i −14.2829 −5.47905 − 5.47905i −46.2406 + 46.2406i 50.3677i 46.4644 − 44.7587i
47.18 −1.60759 − 1.60759i −11.9434 + 11.9434i − 10.8313i −22.7256 + 10.4187i 38.4002 −60.8173 − 60.8173i −43.1337 + 43.1337i − 204.290i 53.2823 + 19.7844i
47.19 −1.05609 − 1.05609i 11.5301 − 11.5301i − 13.7694i −24.8276 + 2.93067i −24.3536 12.8452 + 12.8452i −31.4390 + 31.4390i − 184.888i 29.3151 + 23.1251i
47.20 −0.148728 − 0.148728i 10.5922 − 10.5922i − 15.9558i 9.50761 − 23.1215i −3.15072 0.0244290 + 0.0244290i −4.75271 + 4.75271i − 143.391i −4.85286 + 2.02477i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.44
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.5.f.a ✓ 88
5.c odd 4 1 inner 115.5.f.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.5.f.a ✓ 88 1.a even 1 1 trivial
115.5.f.a ✓ 88 5.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(115, [\chi])\).