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Properties

Label 21.33.b.a
Level $21$
Weight $33$
Character orbit 21.b
Analytic conductor $136.220$
Analytic rank $0$
Dimension $64$
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] = [21,33,Mod(8,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 33, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.8");
 
S:= CuspForms(chi, 33);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 21.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: \(136.219975799\)
Analytic rank: \(0\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 42774298 q^{3} - 128804556708 q^{4} + 933804622814 q^{6} - 26\!\cdots\!80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 42774298 q^{3} - 128804556708 q^{4} + 933804622814 q^{6} - 26\!\cdots\!80 q^{9}+ \cdots + 43\!\cdots\!88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 − 126039.i 1.10950e7 + 4.15923e7i −1.15908e10 8.40857e10i 5.24224e12 − 1.39840e12i 1.25609e13 9.19552e14i −1.60682e15 + 9.22932e14i 1.05980e16
8.2 − 123357.i −4.14495e7 − 1.16170e7i −1.09220e10 1.41902e11i −1.43304e12 + 5.11310e12i 1.25609e13 8.17493e14i 1.58311e15 + 9.63042e14i 1.75046e16
8.3 − 120609.i −2.70914e7 + 3.34526e7i −1.02516e10 − 1.87127e11i 4.03470e12 + 3.26748e12i −1.25609e13 7.18431e14i −3.85132e14 − 1.81256e15i −2.25692e16
8.4 − 118187.i 4.15637e7 + 1.12019e7i −9.67314e9 1.30639e11i 1.32391e12 − 4.91227e12i −1.25609e13 6.35628e14i 1.60206e15 + 9.31183e14i 1.54398e16
8.5 − 116009.i −5.83927e6 − 4.26488e7i −9.16309e9 2.63539e11i −4.94764e12 + 6.77407e11i −1.25609e13 5.64745e14i −1.78483e15 + 4.98076e14i 3.05729e16
8.6 − 110774.i −3.35919e7 − 2.69185e7i −7.97601e9 − 1.02001e11i −2.98188e12 + 3.72113e12i −1.25609e13 4.07766e14i 4.03814e14 + 1.80849e15i −1.12991e16
8.7 − 105652.i 4.26322e7 − 5.95921e6i −6.86732e9 − 1.53984e9i −6.29601e11 − 4.50417e12i 1.25609e13 2.71774e14i 1.78200e15 − 5.08109e14i −1.62687e14
8.8 − 105401.i 2.55552e7 + 3.46403e7i −6.81446e9 − 2.52354e11i 3.65113e12 − 2.69355e12i 1.25609e13 2.65558e14i −5.46883e14 + 1.77048e15i −2.65984e16
8.9 − 104647.i −4.11496e7 + 1.26386e7i −6.65603e9 − 6.51287e10i 1.32259e12 + 4.30618e12i 1.25609e13 2.47078e14i 1.53355e15 − 1.04014e15i −6.81553e15
8.10 − 98902.4i −2.83073e6 − 4.29535e7i −5.48671e9 3.77746e10i −4.24821e12 + 2.79965e11i 1.25609e13 1.17866e14i −1.83699e15 + 2.43179e14i 3.73600e15
8.11 − 91489.0i 1.27781e7 + 4.11065e7i −4.07527e9 1.28675e10i 3.76079e12 − 1.16905e12i −1.25609e13 − 2.01001e13i −1.52646e15 + 1.05052e15i 1.17723e15
8.12 − 89686.4i 4.26712e7 + 5.67388e6i −3.74869e9 − 2.24090e11i 5.08870e11 − 3.82702e12i −1.25609e13 − 4.89938e13i 1.78863e15 + 4.84222e14i −2.00979e16
8.13 − 88913.7i 3.51116e7 − 2.49038e7i −3.61067e9 1.88623e11i −2.21429e12 − 3.12190e12i 1.25609e13 − 6.08433e13i 6.12625e14 − 1.74882e15i 1.67712e16
8.14 − 86849.9i −1.75089e7 − 3.93250e7i −3.24794e9 − 2.40106e11i −3.41537e12 + 1.52065e12i 1.25609e13 − 9.09342e13i −1.23989e15 + 1.37708e15i −2.08531e16
8.15 − 82207.7i 2.56168e7 − 3.45948e7i −2.46314e9 1.01466e11i −2.84396e12 − 2.10590e12i −1.25609e13 − 1.50591e14i −5.40578e14 − 1.77242e15i 8.34128e15
8.16 − 75778.1i −4.25289e7 + 6.65652e6i −1.44736e9 1.14178e11i 5.04419e11 + 3.22276e12i −1.25609e13 − 2.15786e14i 1.76440e15 − 5.66189e14i 8.65218e15
8.17 − 74284.7i −1.81784e7 + 3.90201e7i −1.22325e9 − 7.79613e10i 2.89859e12 + 1.35038e12i 1.25609e13 − 2.28182e14i −1.19211e15 − 1.41865e15i −5.79133e15
8.18 − 70842.2i 2.43459e7 − 3.55007e7i −7.23648e8 − 1.87602e11i −2.51494e12 − 1.72472e12i −1.25609e13 − 2.53000e14i −6.67575e14 − 1.72859e15i −1.32901e16
8.19 − 67755.0i 2.90089e7 + 3.18042e7i −2.95767e8 2.21099e11i 2.15489e12 − 1.96550e12i 1.25609e13 − 2.70966e14i −1.69989e14 + 1.84521e15i 1.49806e16
8.20 − 56919.8i −3.59214e7 − 2.37208e7i 1.05511e9 2.19897e11i −1.35018e12 + 2.04464e12i 1.25609e13 − 3.04525e14i 7.27669e14 + 1.70417e15i 1.25165e16
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.64
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.33.b.a ✓ 64
3.b odd 2 1 inner 21.33.b.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.33.b.a ✓ 64 1.a even 1 1 trivial
21.33.b.a ✓ 64 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{33}^{\mathrm{new}}(21, [\chi])\).