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Properties

Label 12.20.b.a
Level $12$
Weight $20$
Character orbit 12.b
Analytic conductor $27.458$
Analytic rank $0$
Dimension $36$
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] = [12,20,Mod(11,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.11");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 12.b (of order \(2\), degree \(1\), 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: \(27.4580035868\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 47880 q^{4} + 21771144 q^{6} - 809473644 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 47880 q^{4} + 21771144 q^{6} - 809473644 q^{9} - 4162950000 q^{10} - 2268767880 q^{12} - 25913431656 q^{13} + 791689296672 q^{16} + 2852271258192 q^{18} - 1650825908424 q^{21} + 7247870602416 q^{22} + 23719435803936 q^{24} - 118879690462380 q^{25} - 7629081673968 q^{28} + 37456727138640 q^{30} + 529254701828640 q^{33} - 248241675948480 q^{34} - 11\!\cdots\!16 q^{36}+ \cdots + 12\!\cdots\!88 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −723.867 − 17.4378i 2865.95 − 33971.3i 523680. + 25245.3i − 5.49376e6i −2.66695e6 + 2.45407e7i 1.05257e8i −3.78635e8 − 2.74060e7i −1.14583e9 − 1.94720e8i −9.57989e7 + 3.97675e9i
11.2 −723.867 + 17.4378i 2865.95 + 33971.3i 523680. − 25245.3i 5.49376e6i −2.66695e6 − 2.45407e7i − 1.05257e8i −3.78635e8 + 2.74060e7i −1.14583e9 + 1.94720e8i −9.57989e7 − 3.97675e9i
11.3 −706.401 − 159.015i −32549.6 − 10138.3i 473717. + 224656.i 5.54418e6i 2.13809e7 + 1.23376e7i 1.49965e8i −2.98910e8 − 2.34025e8i 9.56692e8 + 6.59995e8i 8.81605e8 − 3.91641e9i
11.4 −706.401 + 159.015i −32549.6 + 10138.3i 473717. − 224656.i − 5.54418e6i 2.13809e7 − 1.23376e7i − 1.49965e8i −2.98910e8 + 2.34025e8i 9.56692e8 − 6.59995e8i 8.81605e8 + 3.91641e9i
11.5 −641.248 − 336.287i 33097.3 − 8174.89i 298110. + 431287.i 2.18420e6i −2.39727e7 − 5.88806e6i − 4.59862e7i −4.61264e7 − 3.76812e8i 1.02860e9 − 5.41134e8i 7.34518e8 − 1.40061e9i
11.6 −641.248 + 336.287i 33097.3 + 8174.89i 298110. − 431287.i − 2.18420e6i −2.39727e7 + 5.88806e6i 4.59862e7i −4.61264e7 + 3.76812e8i 1.02860e9 + 5.41134e8i 7.34518e8 + 1.40061e9i
11.7 −627.004 − 362.153i 8684.36 + 32967.3i 261979. + 454142.i − 6.03352e6i 6.49408e6 − 2.38157e7i 9.28886e7i 207167. − 3.79625e8i −1.01143e9 + 5.72600e8i −2.18506e9 + 3.78304e9i
11.8 −627.004 + 362.153i 8684.36 − 32967.3i 261979. − 454142.i 6.03352e6i 6.49408e6 + 2.38157e7i − 9.28886e7i 207167. + 3.79625e8i −1.01143e9 − 5.72600e8i −2.18506e9 − 3.78304e9i
11.9 −519.517 − 504.371i −19784.5 − 27763.9i 15507.7 + 524059.i − 1.34492e6i −3.72492e6 + 2.44026e7i − 1.72563e8i 2.56263e8 − 2.80079e8i −3.79408e8 + 1.09859e9i −6.78338e8 + 6.98708e8i
11.10 −519.517 + 504.371i −19784.5 + 27763.9i 15507.7 − 524059.i 1.34492e6i −3.72492e6 − 2.44026e7i 1.72563e8i 2.56263e8 + 2.80079e8i −3.79408e8 − 1.09859e9i −6.78338e8 − 6.98708e8i
11.11 −395.354 − 606.616i −27131.3 + 20643.4i −211678. + 479656.i 730760.i 2.32491e7 + 8.29684e6i − 7.02914e6i 3.74655e8 − 6.12270e7i 3.09959e8 − 1.12017e9i 4.43291e8 − 2.88909e8i
11.12 −395.354 + 606.616i −27131.3 − 20643.4i −211678. − 479656.i − 730760.i 2.32491e7 − 8.29684e6i 7.02914e6i 3.74655e8 + 6.12270e7i 3.09959e8 + 1.12017e9i 4.43291e8 + 2.88909e8i
11.13 −246.303 − 680.898i 12845.9 − 31579.2i −402957. + 335415.i 225554.i −2.46662e7 − 968699.i 9.84546e7i 3.27633e8 + 1.91759e8i −8.32227e8 − 8.11326e8i 1.53579e8 − 5.55547e7i
11.14 −246.303 + 680.898i 12845.9 + 31579.2i −402957. − 335415.i − 225554.i −2.46662e7 + 968699.i − 9.84546e7i 3.27633e8 − 1.91759e8i −8.32227e8 + 8.11326e8i 1.53579e8 + 5.55547e7i
11.15 −183.227 − 700.511i 22981.7 + 25181.4i −457144. + 256705.i 6.84881e6i 1.34289e7 − 2.07129e7i 8.10387e7i 2.63586e8 + 2.73199e8i −1.05940e8 + 1.15742e9i 4.79767e9 − 1.25489e9i
11.16 −183.227 + 700.511i 22981.7 − 25181.4i −457144. − 256705.i − 6.84881e6i 1.34289e7 + 2.07129e7i − 8.10387e7i 2.63586e8 − 2.73199e8i −1.05940e8 − 1.15742e9i 4.79767e9 + 1.25489e9i
11.17 −74.5111 − 720.233i 32706.8 + 9618.97i −513184. + 107331.i − 7.07022e6i 4.49088e6 − 2.42733e7i − 1.56654e8i 1.15541e8 + 3.61615e8i 9.77212e8 + 6.29212e8i −5.09221e9 + 5.26810e8i
11.18 −74.5111 + 720.233i 32706.8 − 9618.97i −513184. − 107331.i 7.07022e6i 4.49088e6 + 2.42733e7i 1.56654e8i 1.15541e8 − 3.61615e8i 9.77212e8 − 6.29212e8i −5.09221e9 − 5.26810e8i
11.19 74.5111 − 720.233i −32706.8 − 9618.97i −513184. − 107331.i − 7.07022e6i −9.36493e6 + 2.28398e7i 1.56654e8i −1.15541e8 + 3.61615e8i 9.77212e8 + 6.29212e8i −5.09221e9 − 5.26810e8i
11.20 74.5111 + 720.233i −32706.8 + 9618.97i −513184. + 107331.i 7.07022e6i −9.36493e6 − 2.28398e7i − 1.56654e8i −1.15541e8 − 3.61615e8i 9.77212e8 − 6.29212e8i −5.09221e9 + 5.26810e8i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.20.b.a ✓ 36
3.b odd 2 1 inner 12.20.b.a ✓ 36
4.b odd 2 1 inner 12.20.b.a ✓ 36
12.b even 2 1 inner 12.20.b.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.20.b.a ✓ 36 1.a even 1 1 trivial
12.20.b.a ✓ 36 3.b odd 2 1 inner
12.20.b.a ✓ 36 4.b odd 2 1 inner
12.20.b.a ✓ 36 12.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{20}^{\mathrm{new}}(12, [\chi])\).