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Properties

Label 322.4.e.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 22 q^{2} - 44 q^{4} + 23 q^{5} + 43 q^{7} - 176 q^{8} - 171 q^{9} - 46 q^{10} + 48 q^{11} - 154 q^{13} - 20 q^{14} + 208 q^{15} - 176 q^{16} + 97 q^{17} + 342 q^{18} + 138 q^{19} - 184 q^{20} - 125 q^{21}+ \cdots - 5090 q^{99}+O(q^{100}) \) 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 −4.65915 − 8.06989i −2.00000 − 3.46410i 5.35680 − 9.27825i −18.6366 8.85532 − 16.2660i −8.00000 −29.9154 + 51.8151i −10.7136 − 18.5565i
93.2 1.00000 − 1.73205i −4.36891 − 7.56718i −2.00000 − 3.46410i 0.648848 − 1.12384i −17.4756 12.8195 + 13.3664i −8.00000 −24.6748 + 42.7380i −1.29770 − 2.24768i
93.3 1.00000 − 1.73205i −2.94806 − 5.10618i −2.00000 − 3.46410i −6.47227 + 11.2103i −11.7922 0.301407 + 18.5178i −8.00000 −3.88208 + 6.72395i 12.9445 + 22.4206i
93.4 1.00000 − 1.73205i −1.86913 − 3.23743i −2.00000 − 3.46410i −7.67179 + 13.2879i −7.47652 9.71607 − 15.7670i −8.00000 6.51270 − 11.2803i 15.3436 + 26.5759i
93.5 1.00000 − 1.73205i −1.69768 − 2.94047i −2.00000 − 3.46410i 1.47116 − 2.54813i −6.79072 −6.97703 − 17.1558i −8.00000 7.73576 − 13.3987i −2.94232 − 5.09625i
93.6 1.00000 − 1.73205i −0.211870 − 0.366970i −2.00000 − 3.46410i 9.01355 − 15.6119i −0.847481 −14.5555 − 11.4515i −8.00000 13.4102 − 23.2272i −18.0271 − 31.2239i
93.7 1.00000 − 1.73205i 1.13706 + 1.96945i −2.00000 − 3.46410i 7.40634 − 12.8282i 4.54825 −2.88194 + 18.2947i −8.00000 10.9142 − 18.9039i −14.8127 − 25.6563i
93.8 1.00000 − 1.73205i 1.62409 + 2.81301i −2.00000 − 3.46410i −2.83027 + 4.90218i 6.49637 17.7993 + 5.11729i −8.00000 8.22465 − 14.2455i 5.66055 + 9.80436i
93.9 1.00000 − 1.73205i 3.65246 + 6.32625i −2.00000 − 3.46410i 3.77669 − 6.54142i 14.6098 −18.3937 + 2.16118i −8.00000 −13.1810 + 22.8301i −7.55338 − 13.0828i
93.10 1.00000 − 1.73205i 4.35976 + 7.55132i −2.00000 − 3.46410i −5.10090 + 8.83502i 17.4390 −3.42555 − 18.2007i −8.00000 −24.5150 + 42.4612i 10.2018 + 17.6700i
93.11 1.00000 − 1.73205i 4.98143 + 8.62809i −2.00000 − 3.46410i 5.90184 − 10.2223i 19.9257 18.2422 + 3.19716i −8.00000 −36.1293 + 62.5777i −11.8037 − 20.4446i
277.1 1.00000 + 1.73205i −4.65915 + 8.06989i −2.00000 + 3.46410i 5.35680 + 9.27825i −18.6366 8.85532 + 16.2660i −8.00000 −29.9154 − 51.8151i −10.7136 + 18.5565i
277.2 1.00000 + 1.73205i −4.36891 + 7.56718i −2.00000 + 3.46410i 0.648848 + 1.12384i −17.4756 12.8195 − 13.3664i −8.00000 −24.6748 − 42.7380i −1.29770 + 2.24768i
277.3 1.00000 + 1.73205i −2.94806 + 5.10618i −2.00000 + 3.46410i −6.47227 − 11.2103i −11.7922 0.301407 − 18.5178i −8.00000 −3.88208 − 6.72395i 12.9445 − 22.4206i
277.4 1.00000 + 1.73205i −1.86913 + 3.23743i −2.00000 + 3.46410i −7.67179 − 13.2879i −7.47652 9.71607 + 15.7670i −8.00000 6.51270 + 11.2803i 15.3436 − 26.5759i
277.5 1.00000 + 1.73205i −1.69768 + 2.94047i −2.00000 + 3.46410i 1.47116 + 2.54813i −6.79072 −6.97703 + 17.1558i −8.00000 7.73576 + 13.3987i −2.94232 + 5.09625i
277.6 1.00000 + 1.73205i −0.211870 + 0.366970i −2.00000 + 3.46410i 9.01355 + 15.6119i −0.847481 −14.5555 + 11.4515i −8.00000 13.4102 + 23.2272i −18.0271 + 31.2239i
277.7 1.00000 + 1.73205i 1.13706 − 1.96945i −2.00000 + 3.46410i 7.40634 + 12.8282i 4.54825 −2.88194 − 18.2947i −8.00000 10.9142 + 18.9039i −14.8127 + 25.6563i
277.8 1.00000 + 1.73205i 1.62409 − 2.81301i −2.00000 + 3.46410i −2.83027 − 4.90218i 6.49637 17.7993 − 5.11729i −8.00000 8.22465 + 14.2455i 5.66055 − 9.80436i
277.9 1.00000 + 1.73205i 3.65246 − 6.32625i −2.00000 + 3.46410i 3.77669 + 6.54142i 14.6098 −18.3937 − 2.16118i −8.00000 −13.1810 − 22.8301i −7.55338 + 13.0828i
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.d ✓ 22
7.c even 3 1 inner 322.4.e.d ✓ 22
7.c even 3 1 2254.4.a.r 11
7.d odd 6 1 2254.4.a.u 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.e.d ✓ 22 1.a even 1 1 trivial
322.4.e.d ✓ 22 7.c even 3 1 inner
2254.4.a.r 11 7.c even 3 1
2254.4.a.u 11 7.d odd 6 1