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Properties

Label 322.4.e.c
Level $322$
Weight $4$
Character orbit 322.e
Analytic conductor $18.999$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,4,Mod(93,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.93");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\), 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: \(18.9986150218\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 22 q^{2} - 44 q^{4} - 17 q^{5} + 11 q^{7} - 176 q^{8} - 59 q^{9} + 34 q^{10} + 6 q^{13} + 44 q^{14} - 288 q^{15} - 176 q^{16} - 47 q^{17} + 118 q^{18} + 138 q^{19} + 136 q^{20} + 97 q^{21} + 253 q^{23}+ \cdots + 7050 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 1.00000 − 1.73205i −4.32590 − 7.49269i −2.00000 − 3.46410i −4.14453 + 7.17853i −17.3036 −17.4586 − 6.18033i −8.00000 −23.9269 + 41.4426i 8.28906 + 14.3571i
93.2 1.00000 − 1.73205i −3.46396 − 5.99975i −2.00000 − 3.46410i 9.19292 − 15.9226i −13.8558 −6.38650 + 17.3843i −8.00000 −10.4980 + 18.1830i −18.3858 − 31.8452i
93.3 1.00000 − 1.73205i −2.34129 − 4.05524i −2.00000 − 3.46410i −9.91510 + 17.1735i −9.36518 11.3676 − 14.6212i −8.00000 2.53668 − 4.39366i 19.8302 + 34.3469i
93.4 1.00000 − 1.73205i −1.81101 − 3.13677i −2.00000 − 3.46410i 5.84292 − 10.1202i −7.24405 18.4157 + 1.96493i −8.00000 6.94046 − 12.0212i −11.6858 − 20.2405i
93.5 1.00000 − 1.73205i −1.58834 − 2.75108i −2.00000 − 3.46410i 0.0130321 − 0.0225723i −6.35334 −17.5995 + 5.76708i −8.00000 8.45438 − 14.6434i −0.0260643 − 0.0451447i
93.6 1.00000 − 1.73205i −0.982658 − 1.70201i −2.00000 − 3.46410i −2.18935 + 3.79207i −3.93063 13.1963 + 12.9945i −8.00000 11.5688 − 20.0377i 4.37870 + 7.58413i
93.7 1.00000 − 1.73205i 1.92946 + 3.34193i −2.00000 − 3.46410i 7.63693 − 13.2276i 7.71785 14.1857 − 11.9066i −8.00000 6.05435 − 10.4864i −15.2739 − 26.4551i
93.8 1.00000 − 1.73205i 2.02449 + 3.50652i −2.00000 − 3.46410i −2.86575 + 4.96362i 8.09796 −15.1732 − 10.6194i −8.00000 5.30289 − 9.18487i 5.73149 + 9.92723i
93.9 1.00000 − 1.73205i 2.72170 + 4.71412i −2.00000 − 3.46410i −3.53343 + 6.12007i 10.8868 6.47254 − 17.3524i −8.00000 −1.31529 + 2.27814i 7.06685 + 12.2401i
93.10 1.00000 − 1.73205i 3.70035 + 6.40919i −2.00000 − 3.46410i 1.93015 − 3.34311i 14.8014 −9.96340 + 15.6119i −8.00000 −13.8851 + 24.0498i −3.86030 − 6.68623i
93.11 1.00000 − 1.73205i 4.13716 + 7.16578i −2.00000 − 3.46410i −10.4678 + 18.1308i 16.5487 8.44343 + 16.4836i −8.00000 −20.7322 + 35.9093i 20.9356 + 36.2615i
277.1 1.00000 + 1.73205i −4.32590 + 7.49269i −2.00000 + 3.46410i −4.14453 − 7.17853i −17.3036 −17.4586 + 6.18033i −8.00000 −23.9269 − 41.4426i 8.28906 − 14.3571i
277.2 1.00000 + 1.73205i −3.46396 + 5.99975i −2.00000 + 3.46410i 9.19292 + 15.9226i −13.8558 −6.38650 − 17.3843i −8.00000 −10.4980 − 18.1830i −18.3858 + 31.8452i
277.3 1.00000 + 1.73205i −2.34129 + 4.05524i −2.00000 + 3.46410i −9.91510 − 17.1735i −9.36518 11.3676 + 14.6212i −8.00000 2.53668 + 4.39366i 19.8302 − 34.3469i
277.4 1.00000 + 1.73205i −1.81101 + 3.13677i −2.00000 + 3.46410i 5.84292 + 10.1202i −7.24405 18.4157 − 1.96493i −8.00000 6.94046 + 12.0212i −11.6858 + 20.2405i
277.5 1.00000 + 1.73205i −1.58834 + 2.75108i −2.00000 + 3.46410i 0.0130321 + 0.0225723i −6.35334 −17.5995 − 5.76708i −8.00000 8.45438 + 14.6434i −0.0260643 + 0.0451447i
277.6 1.00000 + 1.73205i −0.982658 + 1.70201i −2.00000 + 3.46410i −2.18935 − 3.79207i −3.93063 13.1963 − 12.9945i −8.00000 11.5688 + 20.0377i 4.37870 − 7.58413i
277.7 1.00000 + 1.73205i 1.92946 − 3.34193i −2.00000 + 3.46410i 7.63693 + 13.2276i 7.71785 14.1857 + 11.9066i −8.00000 6.05435 + 10.4864i −15.2739 + 26.4551i
277.8 1.00000 + 1.73205i 2.02449 − 3.50652i −2.00000 + 3.46410i −2.86575 − 4.96362i 8.09796 −15.1732 + 10.6194i −8.00000 5.30289 + 9.18487i 5.73149 − 9.92723i
277.9 1.00000 + 1.73205i 2.72170 − 4.71412i −2.00000 + 3.46410i −3.53343 − 6.12007i 10.8868 6.47254 + 17.3524i −8.00000 −1.31529 − 2.27814i 7.06685 − 12.2401i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.4.e.c ✓ 22
7.c even 3 1 inner 322.4.e.c ✓ 22
7.c even 3 1 2254.4.a.t 11
7.d odd 6 1 2254.4.a.s 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.e.c ✓ 22 1.a even 1 1 trivial
322.4.e.c ✓ 22 7.c even 3 1 inner
2254.4.a.s 11 7.d odd 6 1
2254.4.a.t 11 7.c even 3 1