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Properties

Label 21.30.g.b
Level $21$
Weight $30$
Character orbit 21.g
Analytic conductor $111.884$
Analytic rank $0$
Dimension $148$
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] = [21,30,Mod(5,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), 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: \(111.883889004\)
Analytic rank: \(0\)
Dimension: \(148\)
Relative dimension: \(74\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 148 q - 14348910 q^{3} + 19864223742 q^{4} + 4006780460914 q^{7} - 103459931825508 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 148 q - 14348910 q^{3} + 19864223742 q^{4} + 4006780460914 q^{7} - 103459931825508 q^{9} + 413746480644090 q^{10} - 18\!\cdots\!20 q^{12}+ \cdots - 38\!\cdots\!76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −39046.1 − 22543.3i 198812. + 8.28196e6i 7.47962e8 + 1.29551e9i −4.00749e9 + 6.94118e9i 1.78940e11 − 3.27860e11i −1.46679e12 + 1.03365e12i − 4.32404e13i −6.85513e13 + 3.29311e12i 3.12954e14 − 1.80684e14i
5.2 −37905.4 − 21884.7i −8.14580e6 + 1.50875e6i 6.89444e8 + 1.19415e9i −2.86457e9 + 4.96158e9i 3.41788e11 + 1.21078e11i 1.13993e12 − 1.38581e12i − 3.68545e13i 6.40777e13 − 2.45800e13i 2.17165e14 − 1.25380e14i
5.3 −37564.8 − 21688.1i −773024. − 8.24820e6i 6.72308e8 + 1.16447e9i −1.12400e10 + 1.94682e10i −1.49849e11 + 3.26607e11i 9.43662e11 + 1.52624e12i − 3.50368e13i −6.74352e13 + 1.27521e13i 8.44456e14 − 4.87547e14i
5.4 −36749.1 − 21217.1i 8.26359e6 + 586108.i 6.31893e8 + 1.09447e9i 8.74565e9 − 1.51479e10i −2.91244e11 − 1.96868e11i 6.87098e10 + 1.79309e12i − 3.08461e13i 6.79433e13 + 9.68672e12i −6.42789e14 + 3.71114e14i
5.5 −36583.4 − 21121.4i 7.28434e6 − 3.94573e6i 6.23793e8 + 1.08044e9i −2.67214e9 + 4.62827e9i −3.49825e11 − 9.50756e9i −4.62751e11 − 1.73371e12i − 3.00227e13i 3.74928e13 − 5.74840e13i 1.95511e14 − 1.12879e14i
5.6 −35582.8 − 20543.7i −1.20465e6 − 8.19629e6i 5.75654e8 + 9.97062e8i 1.07867e10 − 1.86831e10i −1.25517e11 + 3.16395e11i 1.73769e12 − 4.47579e11i − 2.52457e13i −6.57280e13 + 1.97474e13i −7.67639e14 + 4.43197e14i
5.7 −34757.9 − 20067.5i −7.97806e6 − 2.23181e6i 5.36973e8 + 9.30064e8i 6.76488e9 − 1.17171e10i 2.32514e11 + 2.37673e11i −1.76127e12 + 3.43261e11i − 2.15555e13i 5.86684e13 + 3.56110e13i −4.70266e14 + 2.71508e14i
5.8 −32365.9 − 18686.4i −3.76089e6 − 7.38147e6i 4.29931e8 + 7.44662e8i −1.42268e9 + 2.46416e9i −1.62090e10 + 3.09185e11i −1.63739e12 − 7.34075e11i − 1.20711e13i −4.03418e13 + 5.55218e13i 9.20928e13 − 5.31698e13i
5.9 −31842.4 − 18384.2i 4.52772e6 + 6.93759e6i 4.07523e8 + 7.05850e8i 8.06687e9 − 1.39722e10i −1.66314e10 − 3.04148e11i −1.82297e11 − 1.78513e12i − 1.02281e13i −2.76299e13 + 6.28229e13i −5.13737e14 + 2.96606e14i
5.10 −31625.8 − 18259.2i −6.11941e6 + 5.58420e6i 3.98361e8 + 6.89981e8i 9.32010e9 − 1.61429e10i 2.95494e11 − 6.48695e10i 9.26039e11 + 1.53700e12i − 9.48930e12i 6.26388e12 − 6.83439e13i −5.89512e14 + 3.40355e14i
5.11 −31396.6 − 18126.9i 6.86877e6 + 4.63146e6i 3.88730e8 + 6.73301e8i −7.84872e9 + 1.35944e10i −1.31703e11 − 2.69921e11i 1.71129e12 + 5.39801e11i − 8.72228e12i 2.57296e13 + 6.36248e13i 4.92847e14 − 2.84545e14i
5.12 −30456.6 − 17584.2i −1.68131e6 + 8.11194e6i 3.49969e8 + 6.06165e8i −2.50471e9 + 4.33829e9i 1.93849e11 − 2.17498e11i 1.71400e12 − 5.31134e11i − 5.73483e12i −6.29768e13 − 2.72774e13i 1.52570e14 − 8.80866e13i
5.13 −28029.7 − 16182.9i 4.18466e6 − 7.14975e6i 2.55340e8 + 4.42261e8i 1.41624e9 − 2.45300e9i −2.32999e11 + 1.32685e11i −9.81846e11 + 1.50196e12i 8.47722e11i −3.36076e13 − 5.98386e13i −7.93936e13 + 4.58379e13i
5.14 −27707.6 − 15997.0i −8.25945e6 − 641753.i 2.43372e8 + 4.21533e8i −8.95842e9 + 1.55164e10i 2.18584e11 + 1.49908e11i −3.30034e10 + 1.79411e12i 1.60374e12i 6.78067e13 + 1.06011e13i 4.96433e14 − 2.86616e14i
5.15 −26433.5 − 15261.4i 7.88658e6 + 2.53620e6i 1.97386e8 + 3.41882e8i −8.95456e9 + 1.55098e10i −1.69764e11 − 1.87401e11i −1.79105e12 − 1.09705e11i 4.33726e12i 5.57658e13 + 4.00038e13i 4.73401e14 − 2.73318e14i
5.16 −26347.0 − 15211.4i −6.11550e6 + 5.58848e6i 1.94339e8 + 3.36605e8i −1.19832e10 + 2.07556e10i 2.46133e11 − 5.42139e10i −1.41864e12 − 1.09880e12i 4.50844e12i 6.16826e12 − 6.83526e13i 6.31443e14 − 3.64564e14i
5.17 −24085.1 − 13905.5i −4.42661e6 + 7.00254e6i 1.18292e8 + 2.04888e8i 5.95326e9 − 1.03114e10i 2.03989e11 − 1.07102e11i −1.70914e12 − 5.46565e11i 8.35128e12i −2.94406e13 − 6.19950e13i −2.86770e14 + 1.65567e14i
5.18 −23927.7 − 13814.7i 6.96111e6 − 4.49147e6i 1.13254e8 + 1.96162e8i −8.35112e8 + 1.44646e9i −2.28612e11 + 1.13051e10i 1.79385e12 + 4.48548e10i 8.57509e12i 2.82838e13 − 6.25312e13i 3.99646e13 − 2.30736e13i
5.19 −23237.0 − 13415.9i −5.55991e6 − 6.14148e6i 9.15355e7 + 1.58544e8i −4.01469e9 + 6.95364e9i 4.68020e10 + 2.17300e11i 4.84709e11 − 1.72770e12i 9.49307e12i −6.80528e12 + 6.82921e13i 1.86578e14 − 1.07721e14i
5.20 −22019.9 − 12713.2i −6.47130e6 − 5.17229e6i 5.48156e7 + 9.49434e7i 2.43127e9 − 4.21109e9i 7.67411e10 + 1.96164e11i 1.48330e12 + 1.00981e12i 1.08632e13i 1.51251e13 + 6.69430e13i −1.07073e14 + 6.18185e13i
See next 80 embeddings (of 148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.74
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.30.g.b ✓ 148
3.b odd 2 1 inner 21.30.g.b ✓ 148
7.d odd 6 1 inner 21.30.g.b ✓ 148
21.g even 6 1 inner 21.30.g.b ✓ 148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.g.b ✓ 148 1.a even 1 1 trivial
21.30.g.b ✓ 148 3.b odd 2 1 inner
21.30.g.b ✓ 148 7.d odd 6 1 inner
21.30.g.b ✓ 148 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{148} - 29796335615 T_{2}^{146} + \cdots + 20\!\cdots\!00 \) acting on \(S_{30}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display