Properties

Label 4608.2.c.a
Level 46084608
Weight 22
Character orbit 4608.c
Analytic conductor 36.79536.795
Analytic rank 11
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4608,2,Mod(4607,4608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4608.4607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 4608=2932 4608 = 2^{9} \cdot 3^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 4608.c (of order 22, degree 11, 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: 36.795065251436.7950652514
Analytic rank: 11
Dimension: 22
Coefficient field: Q(2)\Q(\sqrt{-2})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+2 x^{2} + 2 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of β=2\beta = \sqrt{-2}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+3βq74q116q13+3βq172βq19+6q23+5q25+6βq293βq316q37βq412βq436q4711q49+6βq53+12q97+O(q100) q + 3 \beta q^{7} - 4 q^{11} - 6 q^{13} + 3 \beta q^{17} - 2 \beta q^{19} + 6 q^{23} + 5 q^{25} + 6 \beta q^{29} - 3 \beta q^{31} - 6 q^{37} - \beta q^{41} - 2 \beta q^{43} - 6 q^{47} - 11 q^{49} + 6 \beta q^{53} + \cdots - 12 q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q8q1112q13+12q23+10q2512q3712q4722q49+8q59+12q6112q71+12q7332q8324q97+O(q100) 2 q - 8 q^{11} - 12 q^{13} + 12 q^{23} + 10 q^{25} - 12 q^{37} - 12 q^{47} - 22 q^{49} + 8 q^{59} + 12 q^{61} - 12 q^{71} + 12 q^{73} - 32 q^{83} - 24 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/4608Z)×\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times.

nn 20532053 35833583 40974097
χ(n)\chi(n) 11 1-1 1-1

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
4607.1
1.41421i
1.41421i
0 0 0 0 0 4.24264i 0 0 0
4607.2 0 0 0 0 0 4.24264i 0 0 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.c.a 2
3.b odd 2 1 4608.2.c.g yes 2
4.b odd 2 1 4608.2.c.g yes 2
8.b even 2 1 4608.2.c.h yes 2
8.d odd 2 1 4608.2.c.b yes 2
12.b even 2 1 inner 4608.2.c.a 2
16.e even 4 2 4608.2.f.i 4
16.f odd 4 2 4608.2.f.k 4
24.f even 2 1 4608.2.c.h yes 2
24.h odd 2 1 4608.2.c.b yes 2
48.i odd 4 2 4608.2.f.k 4
48.k even 4 2 4608.2.f.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.c.a 2 1.a even 1 1 trivial
4608.2.c.a 2 12.b even 2 1 inner
4608.2.c.b yes 2 8.d odd 2 1
4608.2.c.b yes 2 24.h odd 2 1
4608.2.c.g yes 2 3.b odd 2 1
4608.2.c.g yes 2 4.b odd 2 1
4608.2.c.h yes 2 8.b even 2 1
4608.2.c.h yes 2 24.f even 2 1
4608.2.f.i 4 16.e even 4 2
4608.2.f.i 4 48.k even 4 2
4608.2.f.k 4 16.f odd 4 2
4608.2.f.k 4 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(4608,[χ])S_{2}^{\mathrm{new}}(4608, [\chi]):

T5 T_{5} Copy content Toggle raw display
T72+18 T_{7}^{2} + 18 Copy content Toggle raw display
T11+4 T_{11} + 4 Copy content Toggle raw display
T13+6 T_{13} + 6 Copy content Toggle raw display
T236 T_{23} - 6 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+18 T^{2} + 18 Copy content Toggle raw display
1111 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
1313 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
1717 T2+18 T^{2} + 18 Copy content Toggle raw display
1919 T2+8 T^{2} + 8 Copy content Toggle raw display
2323 (T6)2 (T - 6)^{2} Copy content Toggle raw display
2929 T2+72 T^{2} + 72 Copy content Toggle raw display
3131 T2+18 T^{2} + 18 Copy content Toggle raw display
3737 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
4141 T2+2 T^{2} + 2 Copy content Toggle raw display
4343 T2+8 T^{2} + 8 Copy content Toggle raw display
4747 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
5353 T2+72 T^{2} + 72 Copy content Toggle raw display
5959 (T4)2 (T - 4)^{2} Copy content Toggle raw display
6161 (T6)2 (T - 6)^{2} Copy content Toggle raw display
6767 T2+128 T^{2} + 128 Copy content Toggle raw display
7171 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
7373 (T6)2 (T - 6)^{2} Copy content Toggle raw display
7979 T2+18 T^{2} + 18 Copy content Toggle raw display
8383 (T+16)2 (T + 16)^{2} Copy content Toggle raw display
8989 T2+162 T^{2} + 162 Copy content Toggle raw display
9797 (T+12)2 (T + 12)^{2} Copy content Toggle raw display
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