↑

Properties

Label 21.33.f.a
Level $21$
Weight $33$
Character orbit 21.f
Analytic conductor $136.220$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,33,Mod(10,21)]
 
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([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("21.10");
 
S:= CuspForms(chi, 33);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 21.f (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: \(136.219975799\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 41757 q^{2} - 903981141 q^{3} - 45894752413 q^{4} + 183735294405 q^{5} + 6190592189089 q^{7} + 11\!\cdots\!70 q^{8}+ \cdots + 12\!\cdots\!87 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 41757 q^{2} - 903981141 q^{3} - 45894752413 q^{4} + 183735294405 q^{5} + 6190592189089 q^{7} + 11\!\cdots\!70 q^{8}+ \cdots - 92\!\cdots\!98 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −63279.7 − 109604.i −2.15234e7 − 1.24265e7i −5.86115e9 + 1.01518e10i −1.07675e11 + 6.21661e10i 3.14538e12i −2.14746e13 + 2.53627e13i 9.40000e14 3.08837e14 + 5.34921e14i 1.36273e16 + 7.86771e15i
10.2 −60237.8 − 104335.i −2.15234e7 − 1.24265e7i −5.10969e9 + 8.85025e9i 8.13391e10 − 4.69612e10i 2.99418e12i 9.19582e12 − 3.19353e13i 7.13747e14 3.08837e14 + 5.34921e14i −9.79937e15 − 5.65767e15i
10.3 −53725.0 − 93054.5i −2.15234e7 − 1.24265e7i −3.62528e9 + 6.27916e9i 2.26301e11 − 1.30655e11i 2.67046e12i 2.05835e13 + 2.60911e13i 3.17578e14 3.08837e14 + 5.34921e14i −2.43160e16 − 1.40389e16i
10.4 −44451.2 − 76991.7i −2.15234e7 − 1.24265e7i −1.80433e9 + 3.12520e9i −1.12749e11 + 6.50959e10i 2.20949e12i −6.11642e12 + 3.26652e13i −6.10136e13 3.08837e14 + 5.34921e14i 1.00237e16 + 5.78718e15i
10.5 −43775.6 − 75821.5i −2.15234e7 − 1.24265e7i −1.68512e9 + 2.91871e9i 7.36579e10 − 4.25264e10i 2.17591e12i −3.28943e13 − 4.73175e12i −8.09616e13 3.08837e14 + 5.34921e14i −6.44883e15 − 3.72324e15i
10.6 −41812.2 − 72420.8i −2.15234e7 − 1.24265e7i −1.34903e9 + 2.33659e9i −1.45837e11 + 8.41990e10i 2.07832e12i 3.20629e13 + 8.74067e12i −1.33540e14 3.08837e14 + 5.34921e14i 1.21955e16 + 7.04108e15i
10.7 −28939.5 − 50124.7i −2.15234e7 − 1.24265e7i 4.72497e8 − 8.18388e8i 8.89464e10 − 5.13532e10i 1.43847e12i 2.74153e13 − 1.87838e13i −3.03283e14 3.08837e14 + 5.34921e14i −5.14812e15 − 2.97227e15i
10.8 −22557.9 − 39071.5i −2.15234e7 − 1.24265e7i 1.12976e9 − 1.95681e9i −1.52535e10 + 8.80662e9i 1.12127e12i −1.70986e13 − 2.84967e13i −2.95712e14 3.08837e14 + 5.34921e14i 6.88175e14 + 3.97318e14i
10.9 −12789.0 − 22151.2i −2.15234e7 − 1.24265e7i 1.82037e9 − 3.15297e9i −1.61419e11 + 9.31954e10i 6.35692e11i −3.10598e13 − 1.18201e13i −2.02980e14 3.08837e14 + 5.34921e14i 4.12879e15 + 2.38376e15i
10.10 −12056.5 − 20882.5i −2.15234e7 − 1.24265e7i 1.85676e9 − 3.21601e9i 1.39544e11 − 8.05658e10i 5.99282e11i −2.69980e12 + 3.31231e13i −1.93109e14 3.08837e14 + 5.34921e14i −3.36483e15 − 1.94269e15i
10.11 −1548.32 − 2681.77i −2.15234e7 − 1.24265e7i 2.14269e9 − 3.71125e9i −2.02491e11 + 1.16908e11i 7.69608e10i 1.69134e13 + 2.86071e13i −2.65702e13 3.08837e14 + 5.34921e14i 6.27040e14 + 3.62022e14i
10.12 10953.8 + 18972.5i −2.15234e7 − 1.24265e7i 1.90751e9 − 3.30391e9i 2.52276e11 − 1.45652e11i − 5.44469e11i −8.24826e12 − 3.21931e13i 1.77670e14 3.08837e14 + 5.34921e14i 5.52676e15 + 3.19088e15i
10.13 12030.3 + 20837.0i −2.15234e7 − 1.24265e7i 1.85803e9 − 3.21820e9i −1.37237e11 + 7.92341e10i − 5.97977e11i 2.28256e13 − 2.41541e13i 1.92749e14 3.08837e14 + 5.34921e14i −3.30200e15 − 1.90641e15i
10.14 19284.3 + 33401.3i −2.15234e7 − 1.24265e7i 1.40372e9 − 2.43131e9i 4.63790e10 − 2.67769e10i − 9.58545e11i −3.13913e13 + 1.09093e13i 2.73929e14 3.08837e14 + 5.34921e14i 1.78877e15 + 1.03275e15i
10.15 29934.6 + 51848.3i −2.15234e7 − 1.24265e7i 3.55319e8 − 6.15430e8i 4.65852e10 − 2.68960e10i − 1.48793e12i 2.06106e13 + 2.60697e13i 2.99682e14 3.08837e14 + 5.34921e14i 2.78902e15 + 1.61024e15i
10.16 31687.4 + 54884.2i −2.15234e7 − 1.24265e7i 1.39303e8 − 2.41279e8i 4.66432e10 − 2.69295e10i − 1.57506e12i 2.97463e13 − 1.48185e13i 2.89849e14 3.08837e14 + 5.34921e14i 2.95600e15 + 1.70665e15i
10.17 45711.3 + 79174.3i −2.15234e7 − 1.24265e7i −2.03156e9 + 3.51876e9i 1.31663e10 − 7.60156e9i − 2.27213e12i −2.25706e13 − 2.43925e13i 2.11961e13 3.08837e14 + 5.34921e14i 1.20370e15 + 6.94954e14i
10.18 49855.5 + 86352.2i −2.15234e7 − 1.24265e7i −2.82365e9 + 4.89070e9i −2.35705e11 + 1.36085e11i − 2.47812e12i −5.59851e12 − 3.27580e13i −1.34842e14 3.08837e14 + 5.34921e14i −2.35024e16 − 1.35691e16i
10.19 51165.5 + 88621.3i −2.15234e7 − 1.24265e7i −3.08834e9 + 5.34916e9i 1.42679e11 − 8.23760e10i − 2.54324e12i 3.22221e13 − 8.13409e12i −1.92558e14 3.08837e14 + 5.34921e14i 1.46005e16 + 8.42962e15i
10.20 54147.0 + 93785.4i −2.15234e7 − 1.24265e7i −3.71632e9 + 6.43686e9i −1.13925e11 + 6.57746e10i − 2.69144e12i 3.84843e12 + 3.30094e13i −3.39792e14 3.08837e14 + 5.34921e14i −1.23374e16 − 7.12300e15i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.33.f.a ✓ 42
7.d odd 6 1 inner 21.33.f.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.33.f.a ✓ 42 1.a even 1 1 trivial
21.33.f.a ✓ 42 7.d odd 6 1 inner