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Properties

Label 430.2.r.a
Level $430$
Weight $2$
Character orbit 430.r
Analytic conductor $3.434$
Analytic rank $0$
Dimension $264$
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] = [430,2,Mod(27,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([7, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.r (of order \(28\), degree \(12\), 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: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(22\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q - 8 q^{6} - 40 q^{13} + 44 q^{16} - 8 q^{17} - 16 q^{21} - 24 q^{23} - 8 q^{25} + 48 q^{31} - 56 q^{33} + 32 q^{35} + 256 q^{36} - 100 q^{38} + 40 q^{41} - 168 q^{43} + 48 q^{47} - 280 q^{51} - 16 q^{52} + 24 q^{53} - 8 q^{56} - 40 q^{57} - 16 q^{60} - 84 q^{62} + 84 q^{65} + 144 q^{66} + 40 q^{67} + 20 q^{68} - 252 q^{73} - 56 q^{77} - 56 q^{78} + 48 q^{81} - 56 q^{82} - 44 q^{83} + 20 q^{86} - 144 q^{87} + 4 q^{90} + 24 q^{92} + 28 q^{95} + 8 q^{96} - 60 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −0.846724 − 0.532032i −2.43675 + 1.53111i 0.433884 + 0.900969i 0.235258 − 2.22366i 2.87785 −1.03661 + 1.03661i 0.111964 − 0.993712i 2.29179 − 4.75895i −1.38226 + 1.75766i
27.2 −0.846724 − 0.532032i −2.36912 + 1.48862i 0.433884 + 0.900969i 1.52080 + 1.63926i 2.79798 2.44635 − 2.44635i 0.111964 − 0.993712i 2.09509 − 4.35051i −0.415555 − 2.19711i
27.3 −0.846724 − 0.532032i −1.58096 + 0.993385i 0.433884 + 0.900969i −2.11822 + 0.716352i 1.86715 −0.600259 + 0.600259i 0.111964 − 0.993712i 0.210980 − 0.438105i 2.17467 + 0.520406i
27.4 −0.846724 − 0.532032i −1.25844 + 0.790731i 0.433884 + 0.900969i 1.73201 − 1.41427i 1.48625 1.74142 − 1.74142i 0.111964 − 0.993712i −0.343232 + 0.712728i −2.21897 + 0.276012i
27.5 −0.846724 − 0.532032i −0.0459382 + 0.0288649i 0.433884 + 0.900969i −1.90179 − 1.17609i 0.0542540 −0.499604 + 0.499604i 0.111964 − 0.993712i −1.30037 + 2.70025i 0.984578 + 2.00764i
27.6 −0.846724 − 0.532032i 0.296314 − 0.186186i 0.433884 + 0.900969i −0.486637 + 2.18247i −0.349954 0.0102848 − 0.0102848i 0.111964 − 0.993712i −1.24851 + 2.59257i 1.57319 − 1.58905i
27.7 −0.846724 − 0.532032i 0.306612 − 0.192657i 0.433884 + 0.900969i 1.17132 − 1.90473i −0.362116 −3.59081 + 3.59081i 0.111964 − 0.993712i −1.24476 + 2.58476i −2.00516 + 0.989606i
27.8 −0.846724 − 0.532032i 1.28668 − 0.808477i 0.433884 + 0.900969i 2.21966 + 0.270414i −1.51960 2.16897 − 2.16897i 0.111964 − 0.993712i −0.299731 + 0.622397i −1.73557 − 1.40989i
27.9 −0.846724 − 0.532032i 1.93498 − 1.21583i 0.433884 + 0.900969i 0.121846 − 2.23275i −2.28525 1.40881 − 1.40881i 0.111964 − 0.993712i 0.964257 − 2.00230i −1.29106 + 1.82569i
27.10 −0.846724 − 0.532032i 1.99342 − 1.25255i 0.433884 + 0.900969i 1.05240 + 1.97293i −2.35427 −2.68327 + 2.68327i 0.111964 − 0.993712i 1.10318 − 2.29079i 0.158565 − 2.23044i
27.11 −0.846724 − 0.532032i 2.25003 − 1.41379i 0.433884 + 0.900969i −2.02218 + 0.954350i −2.65734 2.89927 − 2.89927i 0.111964 − 0.993712i 1.76219 − 3.65923i 2.21997 + 0.267794i
27.12 0.846724 + 0.532032i −2.83739 + 1.78285i 0.433884 + 0.900969i −0.608113 − 2.15179i −3.35102 1.61190 − 1.61190i −0.111964 + 0.993712i 3.57059 − 7.41440i 0.629917 − 2.14551i
27.13 0.846724 + 0.532032i −1.90831 + 1.19907i 0.433884 + 0.900969i 2.05629 + 0.878446i −2.25376 2.27342 − 2.27342i −0.111964 + 0.993712i 0.902233 − 1.87351i 1.27375 + 1.83781i
27.14 0.846724 + 0.532032i −1.65275 + 1.03849i 0.433884 + 0.900969i −1.53740 + 1.62370i −1.95193 0.349245 − 0.349245i −0.111964 + 0.993712i 0.351458 − 0.729810i −2.16561 + 0.556884i
27.15 0.846724 + 0.532032i −1.05896 + 0.665389i 0.433884 + 0.900969i −1.40404 − 1.74031i −1.25066 −1.58379 + 1.58379i −0.111964 + 0.993712i −0.622995 + 1.29366i −0.262931 − 2.22056i
27.16 0.846724 + 0.532032i −0.527328 + 0.331342i 0.433884 + 0.900969i 1.70699 + 1.44436i −0.622786 −1.95690 + 1.95690i −0.111964 + 0.993712i −1.13336 + 2.35345i 0.676901 + 2.13115i
27.17 0.846724 + 0.532032i −0.126989 + 0.0797926i 0.433884 + 0.900969i 1.11185 − 1.94005i −0.149977 −0.830650 + 0.830650i −0.111964 + 0.993712i −1.29189 + 2.68264i 1.97360 − 1.05114i
27.18 0.846724 + 0.532032i 0.919405 − 0.577701i 0.433884 + 0.900969i −1.69116 + 1.46286i 1.08584 −2.01843 + 2.01843i −0.111964 + 0.993712i −0.790083 + 1.64062i −2.21024 + 0.338890i
27.19 0.846724 + 0.532032i 0.969909 − 0.609434i 0.433884 + 0.900969i 1.59901 − 1.56306i 1.14548 2.66216 − 2.66216i −0.111964 + 0.993712i −0.732338 + 1.52072i 2.18552 − 0.472759i
27.20 0.846724 + 0.532032i 1.04232 − 0.654933i 0.433884 + 0.900969i −0.0867089 + 2.23439i 1.23100 2.75433 − 2.75433i −0.111964 + 0.993712i −0.644158 + 1.33761i −1.26218 + 1.84578i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.f odd 14 1 inner
215.r even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.r.a ✓ 264
5.c odd 4 1 inner 430.2.r.a ✓ 264
43.f odd 14 1 inner 430.2.r.a ✓ 264
215.r even 28 1 inner 430.2.r.a ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.r.a ✓ 264 1.a even 1 1 trivial
430.2.r.a ✓ 264 5.c odd 4 1 inner
430.2.r.a ✓ 264 43.f odd 14 1 inner
430.2.r.a ✓ 264 215.r even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(430, [\chi])\).