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Properties

Label 483.2.r.a
Level $483$
Weight $2$
Character orbit 483.r
Analytic conductor $3.857$
Analytic rank $0$
Dimension $320$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(34,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.r (of order \(22\), degree \(10\), 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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(32\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 4 q^{2} - 36 q^{4} + 12 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 4 q^{2} - 36 q^{4} + 12 q^{8} + 32 q^{9} - 4 q^{18} + 52 q^{23} - 68 q^{25} - 88 q^{28} + 24 q^{32} + 28 q^{35} + 14 q^{36} - 44 q^{37} + 110 q^{42} - 44 q^{43} - 154 q^{44} + 40 q^{46} - 16 q^{49} - 166 q^{50} - 44 q^{51} + 110 q^{56} - 44 q^{57} - 62 q^{58} - 84 q^{64} + 72 q^{70} - 40 q^{71} - 12 q^{72} + 22 q^{74} + 72 q^{77} - 8 q^{78} - 88 q^{79} - 32 q^{81} - 304 q^{85} + 330 q^{88} - 412 q^{92} + 24 q^{93} + 148 q^{95} - 60 q^{98} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.18339 + 1.40318i −0.989821 − 0.142315i 1.96744 − 4.30810i −1.99005 + 0.584333i 2.36086 − 1.07817i 1.49161 − 2.18520i 1.01061 + 7.02895i 0.959493 + 0.281733i 3.52514 − 4.06822i
34.2 −2.18339 + 1.40318i 0.989821 + 0.142315i 1.96744 − 4.30810i 1.99005 − 0.584333i −2.36086 + 1.07817i 2.37524 − 1.16544i 1.01061 + 7.02895i 0.959493 + 0.281733i −3.52514 + 4.06822i
34.3 −2.07156 + 1.33131i −0.989821 − 0.142315i 1.68815 − 3.69653i 3.92324 − 1.15197i 2.23994 − 1.02295i 1.23729 + 2.33861i 0.723239 + 5.03024i 0.959493 + 0.281733i −6.59362 + 7.60944i
34.4 −2.07156 + 1.33131i 0.989821 + 0.142315i 1.68815 − 3.69653i −3.92324 + 1.15197i −2.23994 + 1.02295i −2.13872 − 1.55752i 0.723239 + 5.03024i 0.959493 + 0.281733i 6.59362 − 7.60944i
34.5 −1.85697 + 1.19340i −0.989821 − 0.142315i 1.19330 − 2.61296i 1.10668 − 0.324952i 2.00791 − 0.916981i −2.23744 − 1.41204i 0.274107 + 1.90645i 0.959493 + 0.281733i −1.66728 + 1.92415i
34.6 −1.85697 + 1.19340i 0.989821 + 0.142315i 1.19330 − 2.61296i −1.10668 + 0.324952i −2.00791 + 0.916981i 1.07924 + 2.41562i 0.274107 + 1.90645i 0.959493 + 0.281733i 1.66728 − 1.92415i
34.7 −1.33182 + 0.855909i −0.989821 − 0.142315i 0.210335 − 0.460569i −4.17306 + 1.22532i 1.44007 − 0.657659i −0.894449 + 2.48997i −0.336531 − 2.34062i 0.959493 + 0.281733i 4.50901 − 5.20367i
34.8 −1.33182 + 0.855909i 0.989821 + 0.142315i 0.210335 − 0.460569i 4.17306 − 1.22532i −1.44007 + 0.657659i −2.59192 + 0.530984i −0.336531 − 2.34062i 0.959493 + 0.281733i −4.50901 + 5.20367i
34.9 −1.19713 + 0.769346i −0.989821 − 0.142315i 0.0103865 − 0.0227432i 1.85052 − 0.543361i 1.29443 − 0.591146i 1.99144 − 1.74188i −0.399972 − 2.78187i 0.959493 + 0.281733i −1.79727 + 2.07416i
34.10 −1.19713 + 0.769346i 0.989821 + 0.142315i 0.0103865 − 0.0227432i −1.85052 + 0.543361i −1.29443 + 0.591146i 2.00756 − 1.72328i −0.399972 − 2.78187i 0.959493 + 0.281733i 1.79727 − 2.07416i
34.11 −0.818407 + 0.525958i −0.989821 − 0.142315i −0.437672 + 0.958368i −0.929338 + 0.272878i 0.884929 − 0.404133i 2.61509 + 0.401643i −0.422768 − 2.94041i 0.959493 + 0.281733i 0.617054 − 0.712119i
34.12 −0.818407 + 0.525958i 0.989821 + 0.142315i −0.437672 + 0.958368i 0.929338 − 0.272878i −0.884929 + 0.404133i −0.0253891 − 2.64563i −0.422768 − 2.94041i 0.959493 + 0.281733i −0.617054 + 0.712119i
34.13 −0.795109 + 0.510986i −0.989821 − 0.142315i −0.459738 + 1.00669i 1.49456 − 0.438843i 0.859737 − 0.392629i −0.275939 + 2.63132i −0.417877 − 2.90640i 0.959493 + 0.281733i −0.964098 + 1.11263i
34.14 −0.795109 + 0.510986i 0.989821 + 0.142315i −0.459738 + 1.00669i −1.49456 + 0.438843i −0.859737 + 0.392629i −2.64381 − 0.101346i −0.417877 − 2.90640i 0.959493 + 0.281733i 0.964098 − 1.11263i
34.15 −0.129024 + 0.0829186i −0.989821 − 0.142315i −0.821058 + 1.79787i −3.74200 + 1.09875i 0.139511 − 0.0637126i −0.158342 − 2.64101i −0.0867944 − 0.603668i 0.959493 + 0.281733i 0.391701 − 0.452047i
34.16 −0.129024 + 0.0829186i 0.989821 + 0.142315i −0.821058 + 1.79787i 3.74200 − 1.09875i −0.139511 + 0.0637126i 2.59159 + 0.532585i −0.0867944 − 0.603668i 0.959493 + 0.281733i −0.391701 + 0.452047i
34.17 0.0475988 − 0.0305899i −0.989821 − 0.142315i −0.829500 + 1.81635i 1.34994 − 0.396378i −0.0514677 + 0.0235045i −1.67496 − 2.04805i 0.0321834 + 0.223840i 0.959493 + 0.281733i 0.0521303 − 0.0601616i
34.18 0.0475988 − 0.0305899i 0.989821 + 0.142315i −0.829500 + 1.81635i −1.34994 + 0.396378i 0.0514677 − 0.0235045i 1.78883 + 1.94938i 0.0321834 + 0.223840i 0.959493 + 0.281733i −0.0521303 + 0.0601616i
34.19 0.227214 − 0.146021i −0.989821 − 0.142315i −0.800526 + 1.75291i −1.03996 + 0.305361i −0.245682 + 0.112199i −2.17465 + 1.50695i 0.150947 + 1.04986i 0.959493 + 0.281733i −0.191705 + 0.221239i
34.20 0.227214 − 0.146021i 0.989821 + 0.142315i −0.800526 + 1.75291i 1.03996 − 0.305361i 0.245682 − 0.112199i −1.80110 + 1.93805i 0.150947 + 1.04986i 0.959493 + 0.281733i 0.191705 − 0.221239i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.d odd 22 1 inner
161.k even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.r.a ✓ 320
7.b odd 2 1 inner 483.2.r.a ✓ 320
23.d odd 22 1 inner 483.2.r.a ✓ 320
161.k even 22 1 inner 483.2.r.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.r.a ✓ 320 1.a even 1 1 trivial
483.2.r.a ✓ 320 7.b odd 2 1 inner
483.2.r.a ✓ 320 23.d odd 22 1 inner
483.2.r.a ✓ 320 161.k even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(483, [\chi])\).