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Properties

Label 931.2.v.g
Level $931$
Weight $2$
Character orbit 931.v
Analytic conductor $7.434$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(177,931)]
 
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([6, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.v (of order \(9\), 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: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 24 q^{4} - 18 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 24 q^{4} - 18 q^{8} + 18 q^{9} - 36 q^{15} + 48 q^{16} - 6 q^{18} - 48 q^{22} + 24 q^{23} + 6 q^{25} + 12 q^{29} - 180 q^{30} + 60 q^{32} + 90 q^{36} - 12 q^{37} - 24 q^{39} + 12 q^{43} - 12 q^{44} - 12 q^{46} - 42 q^{50} - 60 q^{51} + 6 q^{53} - 24 q^{57} + 60 q^{58} - 204 q^{60} - 42 q^{64} + 156 q^{65} + 72 q^{67} - 60 q^{71} + 186 q^{72} + 162 q^{74} - 96 q^{78} - 66 q^{79} - 36 q^{81} - 72 q^{85} + 12 q^{86} + 288 q^{88} - 126 q^{92} + 12 q^{93} - 108 q^{95} + 150 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1 −0.436517 − 2.47561i −0.315115 − 1.78710i −4.05873 + 1.47726i 2.79177 + 1.01612i −4.28663 + 1.56020i 0 2.91502 + 5.04896i −0.275367 + 0.100226i 1.29687 − 7.35490i
177.2 −0.436517 − 2.47561i 0.315115 + 1.78710i −4.05873 + 1.47726i −2.79177 − 1.01612i 4.28663 − 1.56020i 0 2.91502 + 5.04896i −0.275367 + 0.100226i −1.29687 + 7.35490i
177.3 −0.238364 − 1.35183i −0.217163 − 1.23159i 0.108753 − 0.0395828i −0.241269 − 0.0878147i −1.61314 + 0.587137i 0 −1.45212 − 2.51514i 1.34941 − 0.491146i −0.0612008 + 0.347087i
177.4 −0.238364 − 1.35183i 0.217163 + 1.23159i 0.108753 − 0.0395828i 0.241269 + 0.0878147i 1.61314 − 0.587137i 0 −1.45212 − 2.51514i 1.34941 − 0.491146i 0.0612008 − 0.347087i
177.5 0.0811493 + 0.460221i −0.339374 − 1.92469i 1.67417 − 0.609347i −1.39483 − 0.507677i 0.858241 − 0.312374i 0 0.883612 + 1.53046i −0.770171 + 0.280319i 0.120454 − 0.683128i
177.6 0.0811493 + 0.460221i 0.339374 + 1.92469i 1.67417 − 0.609347i 1.39483 + 0.507677i −0.858241 + 0.312374i 0 0.883612 + 1.53046i −0.770171 + 0.280319i −0.120454 + 0.683128i
177.7 0.290795 + 1.64918i −0.279390 − 1.58450i −0.755841 + 0.275104i 3.05215 + 1.11089i 2.53187 − 0.921527i 0 1.00113 + 1.73401i 0.386501 − 0.140675i −0.944510 + 5.35658i
177.8 0.290795 + 1.64918i 0.279390 + 1.58450i −0.755841 + 0.275104i −3.05215 − 1.11089i −2.53187 + 0.921527i 0 1.00113 + 1.73401i 0.386501 − 0.140675i 0.944510 − 5.35658i
177.9 0.476586 + 2.70285i −0.481629 − 2.73145i −5.19890 + 1.89225i −2.65906 − 0.967820i 7.15318 − 2.60354i 0 −4.84764 − 8.39636i −4.40978 + 1.60503i 1.34860 − 7.64831i
177.10 0.476586 + 2.70285i 0.481629 + 2.73145i −5.19890 + 1.89225i 2.65906 + 0.967820i −7.15318 + 2.60354i 0 −4.84764 − 8.39636i −4.40978 + 1.60503i −1.34860 + 7.64831i
214.1 −1.67146 − 1.40252i −1.51875 − 1.27438i 0.479417 + 2.71891i 0.226935 − 1.28702i 0.751181 + 4.26016i 0 0.830069 − 1.43772i 0.161606 + 0.916510i −2.18438 + 1.83291i
214.2 −1.67146 − 1.40252i 1.51875 + 1.27438i 0.479417 + 2.71891i −0.226935 + 1.28702i −0.751181 − 4.26016i 0 0.830069 − 1.43772i 0.161606 + 0.916510i 2.18438 − 1.83291i
214.3 −0.791970 − 0.664541i −1.48207 − 1.24361i −0.161696 − 0.917022i −0.348334 + 1.97550i 0.347328 + 1.96980i 0 −1.51518 + 2.62438i 0.129037 + 0.731803i 1.58867 − 1.33306i
214.4 −0.791970 − 0.664541i 1.48207 + 1.24361i −0.161696 − 0.917022i 0.348334 − 1.97550i −0.347328 − 1.96980i 0 −1.51518 + 2.62438i 0.129037 + 0.731803i −1.58867 + 1.33306i
214.5 0.157473 + 0.132135i −0.0390973 − 0.0328065i −0.339958 − 1.92800i 0.449719 − 2.55048i −0.00182186 − 0.0103323i 0 0.406788 − 0.704578i −0.520492 − 2.95186i 0.407827 − 0.342207i
214.6 0.157473 + 0.132135i 0.0390973 + 0.0328065i −0.339958 − 1.92800i −0.449719 + 2.55048i 0.00182186 + 0.0103323i 0 0.406788 − 0.704578i −0.520492 − 2.95186i −0.407827 + 0.342207i
214.7 1.19662 + 1.00408i −2.06445 − 1.73228i 0.0764209 + 0.433405i −0.198947 + 1.12828i −0.731009 − 4.14576i 0 1.21835 − 2.11025i 0.740214 + 4.19796i −1.37096 + 1.15037i
214.8 1.19662 + 1.00408i 2.06445 + 1.73228i 0.0764209 + 0.433405i 0.198947 − 1.12828i 0.731009 + 4.14576i 0 1.21835 − 2.11025i 0.740214 + 4.19796i 1.37096 − 1.15037i
214.9 1.87538 + 1.57363i −0.522183 − 0.438163i 0.693442 + 3.93270i 0.666145 − 3.77790i −0.289784 − 1.64345i 0 −2.44002 + 4.22625i −0.440257 − 2.49682i 7.19429 − 6.03673i
214.10 1.87538 + 1.57363i 0.522183 + 0.438163i 0.693442 + 3.93270i −0.666145 + 3.77790i 0.289784 + 1.64345i 0 −2.44002 + 4.22625i −0.440257 − 2.49682i −7.19429 + 6.03673i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
133.u even 9 1 inner
133.x odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.v.g 60
7.b odd 2 1 inner 931.2.v.g 60
7.c even 3 1 931.2.w.d ✓ 60
7.c even 3 1 931.2.x.g 60
7.d odd 6 1 931.2.w.d ✓ 60
7.d odd 6 1 931.2.x.g 60
19.e even 9 1 931.2.x.g 60
133.u even 9 1 inner 931.2.v.g 60
133.w even 9 1 931.2.w.d ✓ 60
133.x odd 18 1 inner 931.2.v.g 60
133.y odd 18 1 931.2.x.g 60
133.z odd 18 1 931.2.w.d ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.v.g 60 1.a even 1 1 trivial
931.2.v.g 60 7.b odd 2 1 inner
931.2.v.g 60 133.u even 9 1 inner
931.2.v.g 60 133.x odd 18 1 inner
931.2.w.d ✓ 60 7.c even 3 1
931.2.w.d ✓ 60 7.d odd 6 1
931.2.w.d ✓ 60 133.w even 9 1
931.2.w.d ✓ 60 133.z odd 18 1
931.2.x.g 60 7.c even 3 1
931.2.x.g 60 7.d odd 6 1
931.2.x.g 60 19.e even 9 1
931.2.x.g 60 133.y odd 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{30} + 6 T_{2}^{28} + 7 T_{2}^{27} + 3 T_{2}^{26} + 36 T_{2}^{25} + 189 T_{2}^{24} + 33 T_{2}^{23} + \cdots + 1369 \) Copy content Toggle raw display
\( T_{3}^{60} - 9 T_{3}^{58} + 9 T_{3}^{56} + 1199 T_{3}^{54} - 1797 T_{3}^{52} + 5502 T_{3}^{50} + \cdots + 423412929 \) Copy content Toggle raw display