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Properties

Label 21.30.c.a
Level $21$
Weight $30$
Character orbit 21.c
Analytic conductor $111.884$
Analytic rank $0$
Dimension $76$
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(20,21)]
 
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, 1]))
 
N = Newforms(chi, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.20");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.c (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: \(111.883889004\)
Analytic rank: \(0\)
Dimension: \(76\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 20401094660 q^{4} + 1962751543872 q^{7} - 89581451225964 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 20401094660 q^{4} + 1962751543872 q^{7} - 89581451225964 q^{9} - 11\!\cdots\!48 q^{15}+ \cdots - 23\!\cdots\!24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 − 27459.9i −7.86336e6 − 2.60729e6i −2.17175e8 −2.60713e10 −7.15959e10 + 2.15927e11i −1.78867e12 − 1.43352e11i − 8.77881e12i 5.50345e13 + 4.10041e13i 7.15915e14i
20.2 27459.9i −7.86336e6 + 2.60729e6i −2.17175e8 −2.60713e10 −7.15959e10 − 2.15927e11i −1.78867e12 + 1.43352e11i 8.77881e12i 5.50345e13 − 4.10041e13i − 7.15915e14i
20.3 − 40537.3i −896293. + 8.23572e6i −1.10640e9 2.53522e10 3.33854e11 + 3.63333e10i 3.38830e11 − 1.76213e12i 2.30873e13i −6.70237e13 − 1.47632e13i − 1.02771e15i
20.4 40537.3i −896293. − 8.23572e6i −1.10640e9 2.53522e10 3.33854e11 − 3.63333e10i 3.38830e11 + 1.76213e12i − 2.30873e13i −6.70237e13 + 1.47632e13i 1.02771e15i
20.5 − 38457.7i 8.27750e6 − 336727.i −9.42122e8 −2.26488e10 −1.29498e10 − 3.18333e11i 6.92104e11 − 1.65557e12i 1.55850e13i 6.84036e13 − 5.57452e12i 8.71020e14i
20.6 38457.7i 8.27750e6 + 336727.i −9.42122e8 −2.26488e10 −1.29498e10 + 3.18333e11i 6.92104e11 + 1.65557e12i − 1.55850e13i 6.84036e13 + 5.57452e12i − 8.71020e14i
20.7 − 21897.4i −3.41396e6 + 7.54819e6i 5.73727e7 −2.19660e10 1.65286e11 + 7.47571e10i 7.76630e11 − 1.61764e12i − 1.30124e13i −4.53201e13 − 5.15385e13i 4.81000e14i
20.8 21897.4i −3.41396e6 − 7.54819e6i 5.73727e7 −2.19660e10 1.65286e11 − 7.47571e10i 7.76630e11 + 1.61764e12i 1.30124e13i −4.53201e13 + 5.15385e13i − 4.81000e14i
20.9 − 1978.96i 6.63697e6 + 4.95792e6i 5.32955e8 −2.10087e10 9.81152e9 − 1.31343e10i 1.76916e12 − 2.99987e11i − 2.11714e12i 1.94684e13 + 6.58112e13i 4.15753e13i
20.10 1978.96i 6.63697e6 − 4.95792e6i 5.32955e8 −2.10087e10 9.81152e9 + 1.31343e10i 1.76916e12 + 2.99987e11i 2.11714e12i 1.94684e13 − 6.58112e13i − 4.15753e13i
20.11 − 23469.1i −4.70121e6 − 6.82122e6i −1.39279e7 1.98757e10 −1.60088e11 + 1.10333e11i −7.54484e11 − 1.62808e12i − 1.22730e13i −2.44277e13 + 6.41359e13i − 4.66464e14i
20.12 23469.1i −4.70121e6 + 6.82122e6i −1.39279e7 1.98757e10 −1.60088e11 − 1.10333e11i −7.54484e11 + 1.62808e12i 1.22730e13i −2.44277e13 − 6.41359e13i 4.66464e14i
20.13 − 42628.6i −4.19005e6 + 7.14660e6i −1.28033e9 −1.62384e10 3.04650e11 + 1.78616e11i 1.66265e12 + 6.74901e11i 3.16925e13i −3.35174e13 − 5.98892e13i 6.92219e14i
20.14 42628.6i −4.19005e6 − 7.14660e6i −1.28033e9 −1.62384e10 3.04650e11 − 1.78616e11i 1.66265e12 − 6.74901e11i − 3.16925e13i −3.35174e13 + 5.98892e13i − 6.92219e14i
20.15 − 18452.1i −2.07416e6 + 8.02049e6i 1.96390e8 1.42232e10 1.47995e11 + 3.82726e10i −7.69210e11 + 1.62118e12i − 1.35302e13i −6.00261e13 − 3.32715e13i − 2.62448e14i
20.16 18452.1i −2.07416e6 − 8.02049e6i 1.96390e8 1.42232e10 1.47995e11 − 3.82726e10i −7.69210e11 − 1.62118e12i 1.35302e13i −6.00261e13 + 3.32715e13i 2.62448e14i
20.17 − 3217.48i 7.29657e6 − 3.92306e6i 5.26519e8 1.29709e10 −1.26224e10 − 2.34766e10i 5.86304e11 − 1.69592e12i − 3.42144e12i 3.78496e13 − 5.72498e13i − 4.17336e13i
20.18 3217.48i 7.29657e6 + 3.92306e6i 5.26519e8 1.29709e10 −1.26224e10 + 2.34766e10i 5.86304e11 + 1.69592e12i 3.42144e12i 3.78496e13 + 5.72498e13i 4.17336e13i
20.19 − 1646.15i 2.25912e6 − 7.97037e6i 5.34161e8 1.37077e10 −1.31204e10 − 3.71885e9i −1.61859e12 + 7.74652e11i − 1.76308e12i −5.84231e13 − 3.60121e13i − 2.25649e13i
20.20 1646.15i 2.25912e6 + 7.97037e6i 5.34161e8 1.37077e10 −1.31204e10 + 3.71885e9i −1.61859e12 − 7.74652e11i 1.76308e12i −5.84231e13 + 3.60121e13i 2.25649e13i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.76
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.30.c.a ✓ 76
3.b odd 2 1 inner 21.30.c.a ✓ 76
7.b odd 2 1 inner 21.30.c.a ✓ 76
21.c even 2 1 inner 21.30.c.a ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.c.a ✓ 76 1.a even 1 1 trivial
21.30.c.a ✓ 76 3.b odd 2 1 inner
21.30.c.a ✓ 76 7.b odd 2 1 inner
21.30.c.a ✓ 76 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{30}^{\mathrm{new}}(21, [\chi])\).