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Properties

Label 931.2.bj.b
Level $931$
Weight $2$
Character orbit 931.bj
Analytic conductor $7.434$
Analytic rank $0$
Dimension $72$
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(117,931)]
 
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([15, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.bj (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: \(72\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} + 6 q^{4} - 18 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} + 6 q^{4} - 18 q^{8} + 6 q^{9} + 24 q^{11} - 18 q^{15} + 30 q^{16} + 36 q^{18} + 6 q^{23} - 18 q^{25} - 42 q^{29} - 6 q^{30} - 120 q^{32} - 90 q^{36} + 72 q^{37} - 24 q^{39} - 102 q^{43} - 180 q^{44} + 72 q^{50} + 6 q^{51} - 18 q^{53} + 108 q^{57} + 48 q^{58} + 120 q^{60} + 30 q^{64} - 90 q^{65} + 54 q^{67} + 48 q^{71} + 66 q^{72} - 78 q^{78} - 60 q^{79} - 84 q^{81} + 24 q^{85} + 60 q^{86} + 18 q^{88} + 30 q^{92} - 54 q^{93} + 150 q^{95} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 −0.918226 − 2.52281i −2.25491 − 1.89209i −3.98932 + 3.34744i 0.330377 − 0.393728i −2.70287 + 7.42607i 0 7.45797 + 4.30586i 0.983654 + 5.57858i −1.29666 − 0.471945i
117.2 −0.918226 − 2.52281i 2.25491 + 1.89209i −3.98932 + 3.34744i −0.330377 + 0.393728i 2.70287 − 7.42607i 0 7.45797 + 4.30586i 0.983654 + 5.57858i 1.29666 + 0.471945i
117.3 −0.637515 − 1.75156i −0.319947 − 0.268467i −1.12944 + 0.947716i 2.10519 − 2.50887i −0.266265 + 0.731557i 0 −0.848472 − 0.489866i −0.490653 − 2.78263i −5.73653 − 2.08793i
117.4 −0.637515 − 1.75156i 0.319947 + 0.268467i −1.12944 + 0.947716i −2.10519 + 2.50887i 0.266265 − 0.731557i 0 −0.848472 − 0.489866i −0.490653 − 2.78263i 5.73653 + 2.08793i
117.5 −0.249632 − 0.685859i −1.39784 − 1.17293i 1.12400 − 0.943150i −1.26735 + 1.51037i −0.455517 + 1.25152i 0 −2.19164 − 1.26534i 0.0572544 + 0.324706i 1.35227 + 0.492186i
117.6 −0.249632 − 0.685859i 1.39784 + 1.17293i 1.12400 − 0.943150i 1.26735 − 1.51037i 0.455517 − 1.25152i 0 −2.19164 − 1.26534i 0.0572544 + 0.324706i −1.35227 − 0.492186i
117.7 0.128703 + 0.353609i −0.959799 − 0.805367i 1.42361 − 1.19455i 0.743558 − 0.886138i 0.161256 − 0.443047i 0 1.25740 + 0.725963i −0.248346 − 1.40844i 0.409045 + 0.148880i
117.8 0.128703 + 0.353609i 0.959799 + 0.805367i 1.42361 − 1.19455i −0.743558 + 0.886138i −0.161256 + 0.443047i 0 1.25740 + 0.725963i −0.248346 − 1.40844i −0.409045 − 0.148880i
117.9 0.456127 + 1.25320i −2.12943 − 1.78681i 0.169636 − 0.142342i 2.08352 − 2.48304i 1.26793 − 3.48361i 0 2.56566 + 1.48129i 0.820865 + 4.65535i 4.06209 + 1.47848i
117.10 0.456127 + 1.25320i 2.12943 + 1.78681i 0.169636 − 0.142342i −2.08352 + 2.48304i −1.26793 + 3.48361i 0 2.56566 + 1.48129i 0.820865 + 4.65535i −4.06209 − 1.47848i
117.11 0.780851 + 2.14537i −0.0352593 − 0.0295861i −2.46080 + 2.06486i −1.23328 + 1.46977i 0.0359409 − 0.0987467i 0 −2.39703 − 1.38393i −0.520577 − 2.95234i −4.11621 − 1.49818i
117.12 0.780851 + 2.14537i 0.0352593 + 0.0295861i −2.46080 + 2.06486i 1.23328 − 1.46977i −0.0359409 + 0.0987467i 0 −2.39703 − 1.38393i −0.520577 − 2.95234i 4.11621 + 1.49818i
129.1 −2.59411 + 0.457412i −1.14360 − 0.416236i 4.64080 − 1.68911i −1.24336 + 3.41610i 3.15701 + 0.556666i 0 −6.70368 + 3.87037i −1.16357 − 0.976351i 1.66284 − 9.43046i
129.2 −2.59411 + 0.457412i 1.14360 + 0.416236i 4.64080 − 1.68911i 1.24336 − 3.41610i −3.15701 − 0.556666i 0 −6.70368 + 3.87037i −1.16357 − 0.976351i −1.66284 + 9.43046i
129.3 −1.61839 + 0.285366i −2.14915 − 0.782226i 0.658369 − 0.239627i 0.289709 − 0.795969i 3.70138 + 0.652654i 0 1.84926 − 1.06767i 1.70883 + 1.43388i −0.241720 + 1.37086i
129.4 −1.61839 + 0.285366i 2.14915 + 0.782226i 0.658369 − 0.239627i −0.289709 + 0.795969i −3.70138 − 0.652654i 0 1.84926 − 1.06767i 1.70883 + 1.43388i 0.241720 − 1.37086i
129.5 0.0355693 − 0.00627183i −2.99270 − 1.08925i −1.87816 + 0.683594i −0.874743 + 2.40334i −0.113280 − 0.0199743i 0 −0.125076 + 0.0722125i 5.47164 + 4.59125i −0.0160407 + 0.0909712i
129.6 0.0355693 − 0.00627183i 2.99270 + 1.08925i −1.87816 + 0.683594i 0.874743 − 2.40334i 0.113280 + 0.0199743i 0 −0.125076 + 0.0722125i 5.47164 + 4.59125i 0.0160407 − 0.0909712i
129.7 0.543162 − 0.0957740i −0.574904 − 0.209248i −1.59353 + 0.579999i 0.314774 − 0.864834i −0.332306 − 0.0585945i 0 −1.76529 + 1.01919i −2.01140 − 1.68777i 0.0881444 − 0.499892i
129.8 0.543162 − 0.0957740i 0.574904 + 0.209248i −1.59353 + 0.579999i −0.314774 + 0.864834i 0.332306 + 0.0585945i 0 −1.76529 + 1.01919i −2.01140 − 1.68777i −0.0881444 + 0.499892i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
133.be odd 18 1 inner
133.bf 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.bj.b 72
7.b odd 2 1 inner 931.2.bj.b 72
7.c even 3 1 133.2.ba.a ✓ 72
7.c even 3 1 931.2.bf.b 72
7.d odd 6 1 133.2.ba.a ✓ 72
7.d odd 6 1 931.2.bf.b 72
19.f odd 18 1 931.2.bf.b 72
133.ba even 18 1 931.2.bf.b 72
133.bb even 18 1 133.2.ba.a ✓ 72
133.bd odd 18 1 133.2.ba.a ✓ 72
133.be odd 18 1 inner 931.2.bj.b 72
133.bf even 18 1 inner 931.2.bj.b 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.ba.a ✓ 72 7.c even 3 1
133.2.ba.a ✓ 72 7.d odd 6 1
133.2.ba.a ✓ 72 133.bb even 18 1
133.2.ba.a ✓ 72 133.bd odd 18 1
931.2.bf.b 72 7.c even 3 1
931.2.bf.b 72 7.d odd 6 1
931.2.bf.b 72 19.f odd 18 1
931.2.bf.b 72 133.ba even 18 1
931.2.bj.b 72 1.a even 1 1 trivial
931.2.bj.b 72 7.b odd 2 1 inner
931.2.bj.b 72 133.be odd 18 1 inner
931.2.bj.b 72 133.bf even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{35} + 3 T_{2}^{34} + 3 T_{2}^{33} - 24 T_{2}^{32} + 111 T_{2}^{31} - 481 T_{2}^{30} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\). Copy content Toggle raw display