Properties

Label 310.3.c.a
Level $310$
Weight $3$
Character orbit 310.c
Analytic conductor $8.447$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [310,3,Mod(61,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.61");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 310.c (of order \(2\), degree \(1\), 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: \(8.44688819517\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} + 8 q^{7} - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} + 8 q^{7} - 96 q^{9} - 16 q^{14} + 96 q^{16} - 32 q^{18} - 56 q^{19} + 120 q^{25} + 16 q^{28} - 32 q^{31} + 248 q^{33} - 40 q^{35} - 192 q^{36} + 32 q^{38} - 104 q^{39} + 64 q^{41} - 40 q^{45} - 48 q^{47} + 552 q^{49} - 400 q^{51} - 32 q^{56} - 80 q^{59} - 40 q^{62} - 16 q^{63} + 192 q^{64} - 448 q^{66} + 568 q^{67} - 8 q^{69} + 168 q^{71} - 64 q^{72} - 112 q^{76} - 624 q^{78} + 1440 q^{81} + 256 q^{82} - 296 q^{87} - 160 q^{90} + 256 q^{93} - 432 q^{94} + 120 q^{95} - 128 q^{97} + 640 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.41421 5.88545i 2.00000 −2.23607 8.32328i 1.81095 −2.82843 −25.6385 3.16228
61.2 −1.41421 4.63999i 2.00000 2.23607 6.56194i −0.345649 −2.82843 −12.5295 −3.16228
61.3 −1.41421 2.59054i 2.00000 2.23607 3.66358i 11.2758 −2.82843 2.28909 −3.16228
61.4 −1.41421 2.35136i 2.00000 −2.23607 3.32533i 7.27991 −2.82843 3.47110 3.16228
61.5 −1.41421 1.96026i 2.00000 2.23607 2.77223i −10.7520 −2.82843 5.15737 −3.16228
61.6 −1.41421 0.304421i 2.00000 −2.23607 0.430516i −4.44058 −2.82843 8.90733 3.16228
61.7 −1.41421 0.304421i 2.00000 −2.23607 0.430516i −4.44058 −2.82843 8.90733 3.16228
61.8 −1.41421 1.96026i 2.00000 2.23607 2.77223i −10.7520 −2.82843 5.15737 −3.16228
61.9 −1.41421 2.35136i 2.00000 −2.23607 3.32533i 7.27991 −2.82843 3.47110 3.16228
61.10 −1.41421 2.59054i 2.00000 2.23607 3.66358i 11.2758 −2.82843 2.28909 −3.16228
61.11 −1.41421 4.63999i 2.00000 2.23607 6.56194i −0.345649 −2.82843 −12.5295 −3.16228
61.12 −1.41421 5.88545i 2.00000 −2.23607 8.32328i 1.81095 −2.82843 −25.6385 3.16228
61.13 1.41421 5.82697i 2.00000 2.23607 8.24058i −12.9826 2.82843 −24.9536 3.16228
61.14 1.41421 5.42696i 2.00000 −2.23607 7.67489i 7.66369 2.82843 −20.4519 −3.16228
61.15 1.41421 3.58942i 2.00000 2.23607 5.07621i 11.6269 2.82843 −3.88397 3.16228
61.16 1.41421 1.88454i 2.00000 2.23607 2.66514i −1.29462 2.82843 5.44852 3.16228
61.17 1.41421 1.46742i 2.00000 −2.23607 2.07524i −11.9745 2.82843 6.84669 −3.16228
61.18 1.41421 1.28940i 2.00000 −2.23607 1.82349i 6.13262 2.82843 7.33744 −3.16228
61.19 1.41421 1.28940i 2.00000 −2.23607 1.82349i 6.13262 2.82843 7.33744 −3.16228
61.20 1.41421 1.46742i 2.00000 −2.23607 2.07524i −11.9745 2.82843 6.84669 −3.16228
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.3.c.a 24
5.b even 2 1 1550.3.c.d 24
5.c odd 4 2 1550.3.d.c 48
31.b odd 2 1 inner 310.3.c.a 24
155.c odd 2 1 1550.3.c.d 24
155.f even 4 2 1550.3.d.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.c.a 24 1.a even 1 1 trivial
310.3.c.a 24 31.b odd 2 1 inner
1550.3.c.d 24 5.b even 2 1
1550.3.c.d 24 155.c odd 2 1
1550.3.d.c 48 5.c odd 4 2
1550.3.d.c 48 155.f even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(310, [\chi])\).