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Properties

Label 322.4.g.b
Level $322$
Weight $4$
Character orbit 322.g
Analytic conductor $18.999$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,4,Mod(45,322)]
 
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([1, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.45");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 322.g (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: \(18.9986150218\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} + 6 q^{3} - 96 q^{4} - 384 q^{8} + 206 q^{9} - 24 q^{12} - 384 q^{16} - 412 q^{18} - 53 q^{23} - 48 q^{24} - 748 q^{25} + 108 q^{26} - 244 q^{29} - 390 q^{31} + 768 q^{32} + 440 q^{35} - 1648 q^{36}+ \cdots + 1328 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 1.00000 + 1.73205i −8.30485 − 4.79481i −2.00000 + 3.46410i −3.30728 − 5.72837i − 19.1792i 18.4666 − 1.40941i −8.00000 32.4804 + 56.2577i 6.61455 − 11.4567i
45.2 1.00000 + 1.73205i −8.30485 − 4.79481i −2.00000 + 3.46410i 3.30728 + 5.72837i − 19.1792i −18.4666 + 1.40941i −8.00000 32.4804 + 56.2577i −6.61455 + 11.4567i
45.3 1.00000 + 1.73205i −6.35182 − 3.66723i −2.00000 + 3.46410i 10.4076 + 18.0265i − 14.6689i 18.4875 − 1.10056i −8.00000 13.3971 + 23.2044i −20.8152 + 36.0530i
45.4 1.00000 + 1.73205i −6.35182 − 3.66723i −2.00000 + 3.46410i −10.4076 − 18.0265i − 14.6689i −18.4875 + 1.10056i −8.00000 13.3971 + 23.2044i 20.8152 − 36.0530i
45.5 1.00000 + 1.73205i −4.84170 − 2.79536i −2.00000 + 3.46410i −1.72971 − 2.99594i − 11.1814i 11.5448 + 14.4817i −8.00000 2.12806 + 3.68590i 3.45941 − 5.99188i
45.6 1.00000 + 1.73205i −4.84170 − 2.79536i −2.00000 + 3.46410i 1.72971 + 2.99594i − 11.1814i −11.5448 − 14.4817i −8.00000 2.12806 + 3.68590i −3.45941 + 5.99188i
45.7 1.00000 + 1.73205i −4.14481 − 2.39301i −2.00000 + 3.46410i −6.52505 − 11.3017i − 9.57204i 5.69807 − 17.6219i −8.00000 −2.04701 − 3.54553i 13.0501 − 22.6034i
45.8 1.00000 + 1.73205i −4.14481 − 2.39301i −2.00000 + 3.46410i 6.52505 + 11.3017i − 9.57204i −5.69807 + 17.6219i −8.00000 −2.04701 − 3.54553i −13.0501 + 22.6034i
45.9 1.00000 + 1.73205i −1.92970 − 1.11411i −2.00000 + 3.46410i −9.74500 − 16.8788i − 4.45646i 13.5231 + 12.6541i −8.00000 −11.0175 − 19.0829i 19.4900 − 33.7577i
45.10 1.00000 + 1.73205i −1.92970 − 1.11411i −2.00000 + 3.46410i 9.74500 + 16.8788i − 4.45646i −13.5231 − 12.6541i −8.00000 −11.0175 − 19.0829i −19.4900 + 33.7577i
45.11 1.00000 + 1.73205i −1.25301 − 0.723424i −2.00000 + 3.46410i 3.76650 + 6.52377i − 2.89370i 12.9781 − 13.2124i −8.00000 −12.4533 − 21.5698i −7.53300 + 13.0475i
45.12 1.00000 + 1.73205i −1.25301 − 0.723424i −2.00000 + 3.46410i −3.76650 − 6.52377i − 2.89370i −12.9781 + 13.2124i −8.00000 −12.4533 − 21.5698i 7.53300 − 13.0475i
45.13 1.00000 + 1.73205i 1.60569 + 0.927044i −2.00000 + 3.46410i 0.439139 + 0.760611i 3.70818i −15.4941 + 10.1456i −8.00000 −11.7812 − 20.4056i −0.878278 + 1.52122i
45.14 1.00000 + 1.73205i 1.60569 + 0.927044i −2.00000 + 3.46410i −0.439139 − 0.760611i 3.70818i 15.4941 − 10.1456i −8.00000 −11.7812 − 20.4056i 0.878278 − 1.52122i
45.15 1.00000 + 1.73205i 2.75730 + 1.59193i −2.00000 + 3.46410i −2.64180 − 4.57574i 6.36770i −11.7394 − 14.3243i −8.00000 −8.43154 − 14.6039i 5.28361 − 9.15148i
45.16 1.00000 + 1.73205i 2.75730 + 1.59193i −2.00000 + 3.46410i 2.64180 + 4.57574i 6.36770i 11.7394 + 14.3243i −8.00000 −8.43154 − 14.6039i −5.28361 + 9.15148i
45.17 1.00000 + 1.73205i 3.71144 + 2.14280i −2.00000 + 3.46410i −8.09327 − 14.0180i 8.57121i 5.43016 + 17.7063i −8.00000 −4.31679 − 7.47690i 16.1865 − 28.0359i
45.18 1.00000 + 1.73205i 3.71144 + 2.14280i −2.00000 + 3.46410i 8.09327 + 14.0180i 8.57121i −5.43016 − 17.7063i −8.00000 −4.31679 − 7.47690i −16.1865 + 28.0359i
45.19 1.00000 + 1.73205i 5.64024 + 3.25639i −2.00000 + 3.46410i −7.10993 − 12.3148i 13.0256i 17.2059 − 6.85245i −8.00000 7.70819 + 13.3510i 14.2199 − 24.6295i
45.20 1.00000 + 1.73205i 5.64024 + 3.25639i −2.00000 + 3.46410i 7.10993 + 12.3148i 13.0256i −17.2059 + 6.85245i −8.00000 7.70819 + 13.3510i −14.2199 + 24.6295i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.4.g.b ✓ 48
7.d odd 6 1 inner 322.4.g.b ✓ 48
23.b odd 2 1 inner 322.4.g.b ✓ 48
161.g even 6 1 inner 322.4.g.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.g.b ✓ 48 1.a even 1 1 trivial
322.4.g.b ✓ 48 7.d odd 6 1 inner
322.4.g.b ✓ 48 23.b odd 2 1 inner
322.4.g.b ✓ 48 161.g even 6 1 inner