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Properties

Label 21.33.d.a
Level $21$
Weight $33$
Character orbit 21.d
Analytic conductor $136.220$
Analytic rank $0$
Dimension $42$
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(13,21)]
 
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([0, 1]))
 
N = Newforms(chi, 33, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.13");
 
S:= CuspForms(chi, 33);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 21.d (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: \(42\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 83514 q^{2} + 91789504826 q^{4} + 11178037935598 q^{7} - 379567912170246 q^{8} - 25\!\cdots\!74 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 83514 q^{2} + 91789504826 q^{4} + 11178037935598 q^{7} - 379567912170246 q^{8} - 25\!\cdots\!74 q^{9}+ \cdots + 10\!\cdots\!20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −126292. 2.48530e7i 1.16547e10 − 2.82172e11i − 3.13874e12i −2.78520e13 + 1.81299e13i −9.29482e14 −6.17673e14 3.56360e16i
13.2 −126292. − 2.48530e7i 1.16547e10 2.82172e11i 3.13874e12i −2.78520e13 − 1.81299e13i −9.29482e14 −6.17673e14 − 3.56360e16i
13.3 −123669. − 2.48530e7i 1.09989e10 − 1.25825e11i 3.07354e12i 3.32308e13 + 3.76567e11i −8.29071e14 −6.17673e14 1.55606e16i
13.4 −123669. 2.48530e7i 1.09989e10 1.25825e11i − 3.07354e12i 3.32308e13 − 3.76567e11i −8.29071e14 −6.17673e14 − 1.55606e16i
13.5 −100917. − 2.48530e7i 5.88935e9 − 1.10564e11i 2.50810e12i −2.72959e13 − 1.89568e13i −1.60901e14 −6.17673e14 1.11579e16i
13.6 −100917. 2.48530e7i 5.88935e9 1.10564e11i − 2.50810e12i −2.72959e13 + 1.89568e13i −1.60901e14 −6.17673e14 − 1.11579e16i
13.7 −89556.3 − 2.48530e7i 3.72537e9 1.29773e11i 2.22575e12i 2.63280e13 + 2.02797e13i 5.10114e13 −6.17673e14 − 1.16220e16i
13.8 −89556.3 2.48530e7i 3.72537e9 − 1.29773e11i − 2.22575e12i 2.63280e13 − 2.02797e13i 5.10114e13 −6.17673e14 1.16220e16i
13.9 −82223.4 − 2.48530e7i 2.46573e9 9.68030e10i 2.04350e12i 1.96946e13 − 2.67685e13i 1.50406e14 −6.17673e14 − 7.95947e15i
13.10 −82223.4 2.48530e7i 2.46573e9 − 9.68030e10i − 2.04350e12i 1.96946e13 + 2.67685e13i 1.50406e14 −6.17673e14 7.95947e15i
13.11 −62361.1 2.48530e7i −4.06062e8 2.64198e11i − 1.54986e12i −9.77952e12 − 3.17614e13i 2.93161e14 −6.17673e14 − 1.64757e16i
13.12 −62361.1 − 2.48530e7i −4.06062e8 − 2.64198e11i 1.54986e12i −9.77952e12 + 3.17614e13i 2.93161e14 −6.17673e14 1.64757e16i
13.13 −54893.4 2.48530e7i −1.28168e9 2.16787e11i − 1.36427e12i 1.36903e13 + 3.02821e13i 3.06121e14 −6.17673e14 − 1.19002e16i
13.14 −54893.4 − 2.48530e7i −1.28168e9 − 2.16787e11i 1.36427e12i 1.36903e13 − 3.02821e13i 3.06121e14 −6.17673e14 1.19002e16i
13.15 −54849.7 − 2.48530e7i −1.28647e9 1.94448e11i 1.36318e12i −3.02419e13 + 1.37788e13i 3.06141e14 −6.17673e14 − 1.06654e16i
13.16 −54849.7 2.48530e7i −1.28647e9 − 1.94448e11i − 1.36318e12i −3.02419e13 − 1.37788e13i 3.06141e14 −6.17673e14 1.06654e16i
13.17 −18202.6 − 2.48530e7i −3.96363e9 7.19468e10i 4.52390e11i −1.57332e13 − 2.92727e13i 1.50328e14 −6.17673e14 − 1.30962e15i
13.18 −18202.6 2.48530e7i −3.96363e9 − 7.19468e10i − 4.52390e11i −1.57332e13 + 2.92727e13i 1.50328e14 −6.17673e14 1.30962e15i
13.19 −17386.4 − 2.48530e7i −3.99268e9 2.65866e10i 4.32104e11i 1.95370e13 + 2.68837e13i 1.44092e14 −6.17673e14 − 4.62245e14i
13.20 −17386.4 2.48530e7i −3.99268e9 − 2.65866e10i − 4.32104e11i 1.95370e13 − 2.68837e13i 1.44092e14 −6.17673e14 4.62245e14i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.d.a ✓ 42
7.b odd 2 1 inner 21.33.d.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.33.d.a ✓ 42 1.a even 1 1 trivial
21.33.d.a ✓ 42 7.b odd 2 1 inner

Hecke kernels

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