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Properties

Label 931.2.bf.a
Level $931$
Weight $2$
Character orbit 931.bf
Analytic conductor $7.434$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(325,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.325");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.bf (of order \(18\), degree \(6\), not 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} + 9 q^{3} - 3 q^{4} + 9 q^{5} - 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} + 9 q^{3} - 3 q^{4} + 9 q^{5} - 18 q^{8} - 3 q^{9} + 36 q^{10} + 6 q^{11} - 3 q^{12} - 30 q^{13} + 9 q^{15} + 3 q^{16} - 9 q^{17} - 36 q^{18} - 24 q^{19} - 21 q^{23} + 9 q^{24} + 27 q^{25} - 9 q^{26} + 12 q^{27} - 6 q^{29} - 6 q^{30} + 18 q^{31} - 30 q^{32} + 9 q^{33} - 36 q^{34} + 27 q^{36} + 36 q^{37} + 9 q^{38} + 12 q^{39} - 18 q^{40} + 54 q^{41} + 12 q^{43} - 63 q^{44} + 72 q^{46} + 9 q^{47} + 63 q^{48} - 63 q^{50} - 3 q^{51} - 105 q^{52} - 27 q^{53} - 9 q^{54} - 45 q^{55} - 54 q^{57} + 30 q^{58} + 84 q^{60} + 48 q^{61} - 45 q^{62} - 36 q^{64} - 153 q^{66} + 21 q^{67} - 6 q^{71} - 6 q^{72} + 6 q^{73} - 99 q^{74} - 21 q^{75} + 54 q^{76} + 3 q^{78} + 9 q^{79} - 27 q^{80} + 15 q^{81} + 54 q^{82} + 36 q^{83} - 48 q^{85} + 114 q^{86} + 9 q^{87} + 9 q^{89} - 18 q^{90} + 48 q^{92} - 84 q^{93} + 90 q^{94} - 3 q^{95} - 63 q^{96} - 27 q^{97} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1 −0.861958 − 2.36821i −0.245299 + 0.0892815i −3.33336 + 2.79702i −2.68797 + 3.20339i 0.422875 + 0.503963i 0 5.13205 + 2.96299i −2.24593 + 1.88456i 9.90323 + 3.60448i
325.2 −0.808822 − 2.22222i 2.37621 − 0.864869i −2.75199 + 2.30919i 0.682068 − 0.812857i −3.84386 − 4.58094i 0 3.26138 + 1.88296i 2.60023 − 2.18186i −2.35802 − 0.858249i
325.3 −0.677510 − 1.86144i −2.58066 + 0.939284i −1.47387 + 1.23672i 1.43966 − 1.71572i 3.49685 + 4.16738i 0 −0.130386 − 0.0752786i 3.47943 − 2.91958i −4.16909 − 1.51743i
325.4 −0.420620 − 1.15564i −0.153940 + 0.0560295i 0.373495 − 0.313400i −0.555314 + 0.661797i 0.129500 + 0.154333i 0 −2.64937 − 1.52962i −2.27758 + 1.91111i 0.998379 + 0.363380i
325.5 −0.229949 − 0.631779i 2.78157 − 1.01241i 1.18582 − 0.995022i −0.627155 + 0.747415i −1.27924 − 1.52454i 0 −2.06581 − 1.19270i 4.41404 − 3.70382i 0.616414 + 0.224356i
325.6 −0.198784 − 0.546156i −1.56536 + 0.569746i 1.27332 − 1.06844i −0.192345 + 0.229228i 0.622340 + 0.741676i 0 −1.84333 − 1.06425i −0.172381 + 0.144645i 0.163430 + 0.0594835i
325.7 0.220542 + 0.605934i 1.70919 − 0.622093i 1.21357 − 1.01831i 2.32877 − 2.77532i 0.753896 + 0.898458i 0 2.00154 + 1.15559i 0.236189 − 0.198186i 2.19525 + 0.799007i
325.8 0.255877 + 0.703015i 0.490705 − 0.178602i 1.10333 − 0.925805i −1.04177 + 1.24154i 0.251120 + 0.299273i 0 2.22898 + 1.28690i −2.08924 + 1.75308i −1.13938 − 0.414702i
325.9 0.435411 + 1.19628i −1.88185 + 0.684938i 0.290584 − 0.243829i 1.53016 − 1.82358i −1.63876 − 1.95299i 0 2.62321 + 1.51451i 0.774094 − 0.649542i 2.84776 + 1.03650i
325.10 0.791178 + 2.17374i 2.32656 − 0.846797i −2.56712 + 2.15407i −0.499788 + 0.595624i 3.68144 + 4.38737i 0 −2.70677 − 1.56275i 2.39767 − 2.01188i −1.69016 − 0.615167i
325.11 0.820989 + 2.25565i −0.225025 + 0.0819025i −2.88184 + 2.41815i 0.602739 − 0.718316i −0.369487 − 0.440337i 0 −3.66282 − 2.11473i −2.25420 + 1.89150i 2.11511 + 0.769837i
509.1 −1.58722 − 1.89157i 0.450080 − 0.377662i −0.711488 + 4.03505i 1.97213 − 0.347740i −1.42875 − 0.251927i 0 4.48496 − 2.58939i −0.461001 + 2.61447i −3.78797 − 3.17848i
509.2 −1.31618 − 1.56856i 2.21400 − 1.85776i −0.380756 + 2.15938i −2.33956 + 0.412527i −5.82801 − 1.02764i 0 0.341691 − 0.197275i 0.929550 − 5.27174i 3.72634 + 3.12677i
509.3 −0.783961 − 0.934288i −1.18122 + 0.991162i 0.0889966 − 0.504725i −3.48153 + 0.613887i 1.85206 + 0.326568i 0 −2.65378 + 1.53216i −0.108064 + 0.612860i 3.30293 + 2.77149i
509.4 −0.777661 − 0.926781i −1.46113 + 1.22604i 0.0931311 − 0.528173i 1.16366 − 0.205185i 2.27253 + 0.400709i 0 −2.65741 + 1.53425i 0.110799 − 0.628375i −1.09510 − 0.918896i
509.5 −0.259595 − 0.309374i 1.15901 − 0.972521i 0.318974 − 1.80899i 4.03471 − 0.711429i −0.601745 − 0.106104i 0 −1.34196 + 0.774783i −0.123448 + 0.700108i −1.26749 − 1.06355i
509.6 0.0147569 + 0.0175866i 1.20481 − 1.01096i 0.347205 − 1.96910i −1.47726 + 0.260481i 0.0355586 + 0.00626995i 0 0.0795172 − 0.0459093i −0.0914061 + 0.518390i −0.0263808 − 0.0221361i
509.7 0.455090 + 0.542355i −1.85621 + 1.55754i 0.260254 − 1.47598i 1.67567 − 0.295466i −1.68948 − 0.297901i 0 2.14522 − 1.23854i 0.498621 − 2.82782i 0.922829 + 0.774345i
509.8 0.606100 + 0.722322i 0.152885 − 0.128286i 0.192905 − 1.09402i −2.31432 + 0.408077i 0.185327 + 0.0326782i 0 2.54034 − 1.46667i −0.514028 + 2.91520i −1.69747 − 1.42435i
509.9 1.20450 + 1.43547i 2.10108 − 1.76302i −0.262448 + 1.48842i 2.46054 − 0.433859i 5.06151 + 0.892480i 0 0.792944 − 0.457807i 0.785375 − 4.45408i 3.58650 + 3.00943i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 325.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.bb even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.bf.a 66
7.b odd 2 1 133.2.bb.a ✓ 66
7.c even 3 1 133.2.bf.a yes 66
7.c even 3 1 931.2.be.a 66
7.d odd 6 1 931.2.be.b 66
7.d odd 6 1 931.2.bj.a 66
19.f odd 18 1 931.2.bj.a 66
133.ba even 18 1 133.2.bf.a yes 66
133.bb even 18 1 inner 931.2.bf.a 66
133.bd odd 18 1 133.2.bb.a ✓ 66
133.be odd 18 1 931.2.be.b 66
133.bf even 18 1 931.2.be.a 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.bb.a ✓ 66 7.b odd 2 1
133.2.bb.a ✓ 66 133.bd odd 18 1
133.2.bf.a yes 66 7.c even 3 1
133.2.bf.a yes 66 133.ba even 18 1
931.2.be.a 66 7.c even 3 1
931.2.be.a 66 133.bf even 18 1
931.2.be.b 66 7.d odd 6 1
931.2.be.b 66 133.be odd 18 1
931.2.bf.a 66 1.a even 1 1 trivial
931.2.bf.a 66 133.bb even 18 1 inner
931.2.bj.a 66 7.d odd 6 1
931.2.bj.a 66 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 3 T_{2}^{65} + 6 T_{2}^{64} + 15 T_{2}^{63} + 24 T_{2}^{62} + 96 T_{2}^{61} - 125 T_{2}^{60} + \cdots + 243 \) acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\). Copy content Toggle raw display