↑

Properties

Label 322.4.c.b
Level $322$
Weight $4$
Character orbit 322.c
Analytic conductor $18.999$
Analytic rank $0$
Dimension $24$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,4,Mod(321,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.321");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 322.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: \(18.9986150218\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{2} + 96 q^{4} + 192 q^{8} - 128 q^{9} + 384 q^{16} - 256 q^{18} + 108 q^{23} + 680 q^{25} - 56 q^{29} + 768 q^{32} + 276 q^{35} - 512 q^{36} - 1320 q^{39} + 216 q^{46} - 1404 q^{49} + 1360 q^{50}+ \cdots - 2808 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1 2.00000 − 0.390147i 4.00000 −20.6447 − 0.780294i −6.75415 − 17.2448i 8.00000 26.8478 −41.2894
321.2 2.00000 0.390147i 4.00000 −20.6447 0.780294i −6.75415 + 17.2448i 8.00000 26.8478 −41.2894
321.3 2.00000 − 8.02828i 4.00000 −13.1926 − 16.0566i −5.12912 + 17.7958i 8.00000 −37.4533 −26.3853
321.4 2.00000 8.02828i 4.00000 −13.1926 16.0566i −5.12912 − 17.7958i 8.00000 −37.4533 −26.3853
321.5 2.00000 − 4.19347i 4.00000 −14.0878 − 8.38694i 18.4618 + 1.47058i 8.00000 9.41483 −28.1757
321.6 2.00000 4.19347i 4.00000 −14.0878 8.38694i 18.4618 − 1.47058i 8.00000 9.41483 −28.1757
321.7 2.00000 − 9.03560i 4.00000 8.28684 − 18.0712i 3.85337 + 18.1150i 8.00000 −54.6420 16.5737
321.8 2.00000 9.03560i 4.00000 8.28684 18.0712i 3.85337 − 18.1150i 8.00000 −54.6420 16.5737
321.9 2.00000 − 4.65223i 4.00000 6.67343 − 9.30446i 17.8812 + 4.82311i 8.00000 5.35677 13.3469
321.10 2.00000 4.65223i 4.00000 6.67343 9.30446i 17.8812 − 4.82311i 8.00000 5.35677 13.3469
321.11 2.00000 − 2.91960i 4.00000 −2.84179 − 5.83919i 10.3029 − 15.3899i 8.00000 18.4760 −5.68359
321.12 2.00000 2.91960i 4.00000 −2.84179 5.83919i 10.3029 + 15.3899i 8.00000 18.4760 −5.68359
321.13 2.00000 − 2.91960i 4.00000 2.84179 − 5.83919i −10.3029 + 15.3899i 8.00000 18.4760 5.68359
321.14 2.00000 2.91960i 4.00000 2.84179 5.83919i −10.3029 − 15.3899i 8.00000 18.4760 5.68359
321.15 2.00000 − 4.65223i 4.00000 −6.67343 − 9.30446i −17.8812 − 4.82311i 8.00000 5.35677 −13.3469
321.16 2.00000 4.65223i 4.00000 −6.67343 9.30446i −17.8812 + 4.82311i 8.00000 5.35677 −13.3469
321.17 2.00000 − 9.03560i 4.00000 −8.28684 − 18.0712i −3.85337 − 18.1150i 8.00000 −54.6420 −16.5737
321.18 2.00000 9.03560i 4.00000 −8.28684 18.0712i −3.85337 + 18.1150i 8.00000 −54.6420 −16.5737
321.19 2.00000 − 4.19347i 4.00000 14.0878 − 8.38694i −18.4618 − 1.47058i 8.00000 9.41483 28.1757
321.20 2.00000 4.19347i 4.00000 14.0878 8.38694i −18.4618 + 1.47058i 8.00000 9.41483 28.1757
See all 24 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
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.4.c.b ✓ 24
7.b odd 2 1 inner 322.4.c.b ✓ 24
23.b odd 2 1 inner 322.4.c.b ✓ 24
161.c even 2 1 inner 322.4.c.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.c.b ✓ 24 1.a even 1 1 trivial
322.4.c.b ✓ 24 7.b odd 2 1 inner
322.4.c.b ✓ 24 23.b odd 2 1 inner
322.4.c.b ✓ 24 161.c even 2 1 inner