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Properties

Label 21.33.f.b
Level $21$
Weight $33$
Character orbit 21.f
Analytic conductor $136.220$
Analytic rank $0$
Dimension $44$
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(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: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 41757 q^{2} + 947027862 q^{3} - 54484687005 q^{4} + 153816227589 q^{5} - 14837835233530 q^{7} + 90590859203718 q^{8} + 13\!\cdots\!34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 41757 q^{2} + 947027862 q^{3} - 54484687005 q^{4} + 153816227589 q^{5} - 14837835233530 q^{7} + 90590859203718 q^{8} + 13\!\cdots\!34 q^{9}+ \cdots + 10\!\cdots\!22 q^{99}+O(q^{100}) \) 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 −62319.8 − 107941.i 2.15234e7 + 1.24265e7i −5.62003e9 + 9.73419e9i 2.38104e11 − 1.37469e11i − 3.09767e12i −2.66309e13 + 1.98802e13i 8.65635e14 3.08837e14 + 5.34921e14i −2.96772e16 − 1.71341e16i
10.2 −61847.9 − 107124.i 2.15234e7 + 1.24265e7i −5.50285e9 + 9.53122e9i −2.11428e11 + 1.22068e11i − 3.07422e12i 3.05188e13 + 1.31541e13i 8.30090e14 3.08837e14 + 5.34921e14i 2.61528e16 + 1.50993e16i
10.3 −55818.6 − 96680.6i 2.15234e7 + 1.24265e7i −4.08394e9 + 7.07359e9i 6.65993e10 − 3.84511e10i − 2.77452e12i 2.38881e13 − 2.31038e13i 4.32361e14 3.08837e14 + 5.34921e14i −7.43495e15 − 4.29257e15i
10.4 −47112.1 − 81600.5i 2.15234e7 + 1.24265e7i −2.29161e9 + 3.96919e9i −2.27876e10 + 1.31564e10i − 2.34176e12i −1.12756e13 + 3.12616e13i 2.71606e13 3.08837e14 + 5.34921e14i 2.14714e15 + 1.23965e15i
10.5 −46221.5 − 80058.0i 2.15234e7 + 1.24265e7i −2.12537e9 + 3.68125e9i 4.22076e10 − 2.43686e10i − 2.29749e12i −7.05515e12 − 3.24754e13i −4.08803e12 3.08837e14 + 5.34921e14i −3.90180e15 − 2.25270e15i
10.6 −43541.2 − 75415.5i 2.15234e7 + 1.24265e7i −1.64419e9 + 2.84781e9i −2.22552e11 + 1.28490e11i − 2.16426e12i −3.28458e13 − 5.05752e12i −8.76568e13 3.08837e14 + 5.34921e14i 1.93804e16 + 1.11893e16i
10.7 −31868.9 − 55198.6i 2.15234e7 + 1.24265e7i 1.16224e8 − 2.01305e8i 2.11420e11 − 1.22064e11i − 1.58408e12i 3.13088e13 + 1.11438e13i −2.88568e14 3.08837e14 + 5.34921e14i −1.34755e16 − 7.78007e15i
10.8 −22983.3 − 39808.3i 2.15234e7 + 1.24265e7i 1.09102e9 − 1.88970e9i −1.46045e11 + 8.43189e10i − 1.14241e12i 3.25404e13 − 6.74919e12i −2.97726e14 3.08837e14 + 5.34921e14i 6.71318e15 + 3.87586e15i
10.9 −16797.3 − 29093.7i 2.15234e7 + 1.24265e7i 1.58319e9 − 2.74216e9i 3.03607e10 − 1.75288e10i − 8.34926e11i −3.07326e13 − 1.26467e13i −2.50660e14 3.08837e14 + 5.34921e14i −1.01995e15 − 5.88871e14i
10.10 −13511.8 − 23403.1i 2.15234e7 + 1.24265e7i 1.78235e9 − 3.08712e9i 1.91920e11 − 1.10805e11i − 6.71617e11i −2.53769e13 + 2.14579e13i −2.12396e14 3.08837e14 + 5.34921e14i −5.18637e15 − 2.99435e15i
10.11 −330.065 − 571.689i 2.15234e7 + 1.24265e7i 2.14727e9 − 3.71917e9i 7.10415e9 − 4.10158e9i − 1.64062e10i 3.54080e12 − 3.30438e13i −5.67018e12 3.08837e14 + 5.34921e14i −4.68965e12 − 2.70757e12i
10.12 1069.08 + 1851.70i 2.15234e7 + 1.24265e7i 2.14520e9 − 3.71559e9i −5.92590e10 + 3.42132e10i 5.31397e10i 1.93737e13 + 2.70016e13i 1.83569e13 3.08837e14 + 5.34921e14i −1.26705e14 − 7.31532e13i
10.13 1621.39 + 2808.32i 2.15234e7 + 1.24265e7i 2.14223e9 − 3.71044e9i −2.12306e11 + 1.22575e11i 8.05927e10i −1.30118e13 + 3.05798e13i 2.78211e13 3.08837e14 + 5.34921e14i −6.88459e14 − 3.97482e14i
10.14 13411.6 + 23229.6i 2.15234e7 + 1.24265e7i 1.78774e9 − 3.09646e9i 2.06360e11 − 1.19142e11i 6.66637e11i 3.07924e13 − 1.25001e13i 2.11111e14 3.08837e14 + 5.34921e14i 5.53522e15 + 3.19576e15i
10.15 28224.9 + 48886.9i 2.15234e7 + 1.24265e7i 5.54196e8 − 9.59895e8i 5.12581e10 − 2.95939e10i 1.40295e12i 3.84764e12 + 3.30094e13i 3.05018e14 3.08837e14 + 5.34921e14i 2.89351e15 + 1.67057e15i
10.16 30602.9 + 53005.8i 2.15234e7 + 1.24265e7i 2.74405e8 − 4.75284e8i −1.13195e11 + 6.53534e10i 1.52115e12i 1.60242e13 − 2.91145e13i 2.96468e14 3.08837e14 + 5.34921e14i −6.92822e15 − 4.00001e15i
10.17 35721.5 + 61871.5i 2.15234e7 + 1.24265e7i −4.04571e8 + 7.00737e8i 1.65310e11 − 9.54417e10i 1.77558e12i −3.01686e13 − 1.39385e13i 2.49038e14 3.08837e14 + 5.34921e14i 1.18102e16 + 6.81865e15i
10.18 40115.2 + 69481.6i 2.15234e7 + 1.24265e7i −1.07098e9 + 1.85499e9i −8.48962e10 + 4.90149e10i 1.99397e12i −2.82716e13 + 1.74684e13i 1.72737e14 3.08837e14 + 5.34921e14i −6.81126e15 − 3.93248e15i
10.19 51660.4 + 89478.4i 2.15234e7 + 1.24265e7i −3.19010e9 + 5.52542e9i −1.28339e11 + 7.40964e10i 2.56783e12i 3.20762e13 + 8.69170e12i −2.15448e14 3.08837e14 + 5.34921e14i −1.32601e16 − 7.65570e15i
10.20 58686.3 + 101648.i 2.15234e7 + 1.24265e7i −4.74068e9 + 8.21109e9i 1.65478e11 − 9.55388e10i 2.91706e12i 8.50199e12 + 3.21270e13i −6.08739e14 3.08837e14 + 5.34921e14i 1.94226e16 + 1.12136e16i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.22
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.b ✓ 44
7.d odd 6 1 inner 21.33.f.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.33.f.b ✓ 44 1.a even 1 1 trivial
21.33.f.b ✓ 44 7.d odd 6 1 inner