↑

Properties

Label 462.2.bf.a
Level $462$
Weight $2$
Character orbit 462.bf
Analytic conductor $3.689$
Analytic rank $0$
Dimension $256$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [462,2,Mod(5,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 25, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.bf (of order \(30\), degree \(8\), 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: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(32\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 32 q^{4} + 4 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 32 q^{4} + 4 q^{7} - 12 q^{9} + 12 q^{10} + 24 q^{15} + 32 q^{16} - 8 q^{18} - 16 q^{21} - 12 q^{22} + 48 q^{25} + 6 q^{28} + 18 q^{31} - 132 q^{33} + 16 q^{36} + 4 q^{37} - 18 q^{40} - 4 q^{42} + 64 q^{43} - 48 q^{45} + 8 q^{46} + 76 q^{49} - 8 q^{51} - 88 q^{57} + 46 q^{58} - 8 q^{60} - 12 q^{63} + 64 q^{64} - 120 q^{66} - 32 q^{67} - 58 q^{70} - 12 q^{72} - 96 q^{73} - 204 q^{75} - 32 q^{78} - 4 q^{79} - 64 q^{81} + 24 q^{82} - 36 q^{84} + 232 q^{85} - 228 q^{87} - 6 q^{88} + 40 q^{91} - 2 q^{93} - 144 q^{94} + 160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.207912 + 0.978148i −1.63536 − 0.570609i −0.913545 − 0.406737i −2.60800 − 2.89648i 0.898150 − 1.48099i −2.64502 − 0.0622836i 0.587785 − 0.809017i 2.34881 + 1.86630i 3.37542 − 1.94880i
5.2 −0.207912 + 0.978148i −1.51046 + 0.847653i −0.913545 − 0.406737i −1.24203 − 1.37941i −0.515088 − 1.65369i 2.15154 + 1.53976i 0.587785 − 0.809017i 1.56297 − 2.56069i 1.60750 − 0.928093i
5.3 −0.207912 + 0.978148i −1.46408 − 0.925462i −0.913545 − 0.406737i 0.383830 + 0.426287i 1.20964 − 1.23967i −2.12466 + 1.57664i 0.587785 − 0.809017i 1.28704 + 2.70989i −0.496774 + 0.286813i
5.4 −0.207912 + 0.978148i −1.37785 + 1.04954i −0.913545 − 0.406737i 2.40395 + 2.66986i −0.740133 − 1.56595i −0.383771 + 2.61777i 0.587785 − 0.809017i 0.796937 − 2.89221i −3.11133 + 1.79633i
5.5 −0.207912 + 0.978148i −1.26644 − 1.18158i −0.913545 − 0.406737i 2.46318 + 2.73564i 1.41907 − 0.993102i 2.33823 − 1.23802i 0.587785 − 0.809017i 0.207743 + 2.99280i −3.18798 + 1.84058i
5.6 −0.207912 + 0.978148i −0.610492 + 1.62089i −0.913545 − 0.406737i −1.62461 − 1.80431i −1.45855 − 0.934154i 1.18467 − 2.36571i 0.587785 − 0.809017i −2.25460 − 1.97909i 2.10266 − 1.21397i
5.7 −0.207912 + 0.978148i −0.596483 + 1.62610i −0.913545 − 0.406737i 0.0808680 + 0.0898130i −1.46655 − 0.921534i −2.64561 − 0.0274346i 0.587785 − 0.809017i −2.28842 − 1.93988i −0.104664 + 0.0604276i
5.8 −0.207912 + 0.978148i −0.442840 − 1.67448i −0.913545 − 0.406737i −0.833125 − 0.925279i 1.72996 − 0.0850182i 0.162919 − 2.64073i 0.587785 − 0.809017i −2.60779 + 1.48306i 1.07828 − 0.622543i
5.9 −0.207912 + 0.978148i 0.263570 − 1.71188i −0.913545 − 0.406737i −1.94189 − 2.15669i 1.61967 + 0.613730i 2.26273 + 1.37115i 0.587785 − 0.809017i −2.86106 − 0.902401i 2.51330 − 1.45105i
5.10 −0.207912 + 0.978148i 0.665114 − 1.59926i −0.913545 − 0.406737i 1.17526 + 1.30525i 1.42602 + 0.983084i −1.84250 − 1.89873i 0.587785 − 0.809017i −2.11525 − 2.12738i −1.52108 + 0.878197i
5.11 −0.207912 + 0.978148i 0.915662 + 1.47023i −0.913545 − 0.406737i −2.41026 − 2.67687i −1.62847 + 0.589975i −0.0504795 + 2.64527i 0.587785 − 0.809017i −1.32313 + 2.69246i 3.11949 − 1.80104i
5.12 −0.207912 + 0.978148i 0.940283 + 1.45460i −0.913545 − 0.406737i 1.15142 + 1.27878i −1.61831 + 0.617307i 2.35020 − 1.21514i 0.587785 − 0.809017i −1.23174 + 2.73548i −1.49023 + 0.860384i
5.13 −0.207912 + 0.978148i 1.30084 − 1.14360i −0.913545 − 0.406737i 1.92888 + 2.14224i 0.848146 + 1.51018i 2.60946 + 0.436733i 0.587785 − 0.809017i 0.384375 − 2.97527i −2.49647 + 1.44134i
5.14 −0.207912 + 0.978148i 1.38379 + 1.04170i −0.913545 − 0.406737i 1.51673 + 1.68449i −1.30664 + 1.13697i −2.61122 − 0.426069i 0.587785 − 0.809017i 0.829732 + 2.88298i −1.96303 + 1.13336i
5.15 −0.207912 + 0.978148i 1.70274 − 0.317274i −0.913545 − 0.406737i −1.03302 − 1.14729i −0.0436800 + 1.73150i −0.907001 + 2.48543i 0.587785 − 0.809017i 2.79867 − 1.08047i 1.33700 − 0.771915i
5.16 −0.207912 + 0.978148i 1.73200 + 0.0134239i −0.913545 − 0.406737i −1.28642 − 1.42871i −0.373233 + 1.69136i 0.715130 − 2.54727i 0.587785 − 0.809017i 2.99964 + 0.0465004i 1.66495 − 0.961262i
5.17 0.207912 − 0.978148i −1.71826 + 0.218129i −0.913545 − 0.406737i 2.60800 + 2.89648i −0.143884 + 1.72606i −2.64502 − 0.0622836i −0.587785 + 0.809017i 2.90484 − 0.749604i 3.37542 − 1.94880i
5.18 0.207912 − 0.978148i −1.62450 + 0.600840i −0.913545 − 0.406737i −0.383830 − 0.426287i 0.249958 + 1.71392i −2.12466 + 1.57664i −0.587785 + 0.809017i 2.27798 − 1.95213i −0.496774 + 0.286813i
5.19 0.207912 − 0.978148i −1.48443 + 0.892451i −0.913545 − 0.406737i −2.46318 − 2.73564i 0.564318 + 1.63754i 2.33823 − 1.23802i −0.587785 + 0.809017i 1.40706 − 2.64956i −3.18798 + 1.84058i
5.20 0.207912 − 0.978148i −1.30121 − 1.14317i −0.913545 − 0.406737i 1.24203 + 1.37941i −1.38873 + 1.03510i 2.15154 + 1.53976i −0.587785 + 0.809017i 0.386315 + 2.97502i 1.60750 − 0.928093i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
11.c even 5 1 inner
21.g even 6 1 inner
33.h odd 10 1 inner
77.p odd 30 1 inner
231.bc even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.bf.a ✓ 256
3.b odd 2 1 inner 462.2.bf.a ✓ 256
7.d odd 6 1 inner 462.2.bf.a ✓ 256
11.c even 5 1 inner 462.2.bf.a ✓ 256
21.g even 6 1 inner 462.2.bf.a ✓ 256
33.h odd 10 1 inner 462.2.bf.a ✓ 256
77.p odd 30 1 inner 462.2.bf.a ✓ 256
231.bc even 30 1 inner 462.2.bf.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.bf.a ✓ 256 1.a even 1 1 trivial
462.2.bf.a ✓ 256 3.b odd 2 1 inner
462.2.bf.a ✓ 256 7.d odd 6 1 inner
462.2.bf.a ✓ 256 11.c even 5 1 inner
462.2.bf.a ✓ 256 21.g even 6 1 inner
462.2.bf.a ✓ 256 33.h odd 10 1 inner
462.2.bf.a ✓ 256 77.p odd 30 1 inner
462.2.bf.a ✓ 256 231.bc even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(462, [\chi])\).