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Properties

Label 476.3.x.a
Level $476$
Weight $3$
Character orbit 476.x
Analytic conductor $12.970$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [476,3,Mod(321,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("476.321");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 476.x (of order \(8\), degree \(4\), 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: \(12.9700605836\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 40 q^{11} - 16 q^{15} + 72 q^{23} + 72 q^{25} + 32 q^{35} - 256 q^{37} - 88 q^{39} - 32 q^{43} - 20 q^{49} - 120 q^{51} - 80 q^{53} + 492 q^{63} - 104 q^{65} - 144 q^{67} - 64 q^{71} + 84 q^{77}+ \cdots + 560 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1 0 −4.93903 + 2.04581i 0 2.11756 + 5.11225i 0 5.49744 + 4.33338i 0 13.8447 − 13.8447i 0
321.2 0 −4.78162 + 1.98061i 0 0.0890392 + 0.214960i 0 −6.37258 − 2.89658i 0 12.5771 − 12.5771i 0
321.3 0 −4.68527 + 1.94070i 0 −0.107047 − 0.258435i 0 6.27527 − 3.10178i 0 11.8214 − 11.8214i 0
321.4 0 −3.76134 + 1.55800i 0 −1.16212 − 2.80560i 0 −3.19109 + 6.23033i 0 5.35633 − 5.35633i 0
321.5 0 −3.43140 + 1.42133i 0 −3.65081 − 8.81383i 0 5.54157 + 4.27680i 0 3.39035 − 3.39035i 0
321.6 0 −2.58574 + 1.07105i 0 −2.55560 − 6.16976i 0 0.327431 − 6.99234i 0 −0.825039 + 0.825039i 0
321.7 0 −2.58178 + 1.06941i 0 2.86795 + 6.92385i 0 2.88122 − 6.37954i 0 −0.842001 + 0.842001i 0
321.8 0 −2.40762 + 0.997268i 0 2.93247 + 7.07962i 0 −3.61992 + 5.99134i 0 −1.56188 + 1.56188i 0
321.9 0 −2.20277 + 0.912419i 0 −1.83455 − 4.42900i 0 −6.99574 + 0.244103i 0 −2.34425 + 2.34425i 0
321.10 0 −1.57323 + 0.651655i 0 1.10724 + 2.67312i 0 3.07101 − 6.29038i 0 −4.31355 + 4.31355i 0
321.11 0 −1.14101 + 0.472620i 0 1.67386 + 4.04105i 0 −5.26535 − 4.61260i 0 −5.28544 + 5.28544i 0
321.12 0 −0.0501663 + 0.0207796i 0 0.962011 + 2.32250i 0 4.78783 + 5.10654i 0 −6.36188 + 6.36188i 0
321.13 0 0.0501663 − 0.0207796i 0 −0.962011 − 2.32250i 0 6.99637 − 0.225364i 0 −6.36188 + 6.36188i 0
321.14 0 1.14101 − 0.472620i 0 −1.67386 − 4.04105i 0 −6.98477 − 0.461559i 0 −5.28544 + 5.28544i 0
321.15 0 1.57323 − 0.651655i 0 −1.10724 − 2.67312i 0 −2.27644 + 6.61950i 0 −4.31355 + 4.31355i 0
321.16 0 2.20277 − 0.912419i 0 1.83455 + 4.42900i 0 −4.77413 − 5.11934i 0 −2.34425 + 2.34425i 0
321.17 0 2.40762 − 0.997268i 0 −2.93247 − 7.07962i 0 1.67684 − 6.79619i 0 −1.56188 + 1.56188i 0
321.18 0 2.58178 − 1.06941i 0 −2.86795 − 6.92385i 0 −2.47369 + 6.54835i 0 −0.842001 + 0.842001i 0
321.19 0 2.58574 − 1.07105i 0 2.55560 + 6.16976i 0 −4.71280 + 5.17586i 0 −0.825039 + 0.825039i 0
321.20 0 3.43140 − 1.42133i 0 3.65081 + 8.81383i 0 6.94264 + 0.894325i 0 3.39035 − 3.39035i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
17.d even 8 1 inner
119.l odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 476.3.x.a ✓ 96
7.b odd 2 1 inner 476.3.x.a ✓ 96
17.d even 8 1 inner 476.3.x.a ✓ 96
119.l odd 8 1 inner 476.3.x.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.3.x.a ✓ 96 1.a even 1 1 trivial
476.3.x.a ✓ 96 7.b odd 2 1 inner
476.3.x.a ✓ 96 17.d even 8 1 inner
476.3.x.a ✓ 96 119.l odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(476, [\chi])\).