Properties

Label 9216.2.a.i
Level 92169216
Weight 22
Character orbit 9216.a
Self dual yes
Analytic conductor 73.59073.590
Analytic rank 00
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] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 9216=21032 9216 = 2^{10} \cdot 3^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 9216.a (trivial)

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: yes
Analytic conductor: 73.590130502873.5901305028
Analytic rank: 00
Dimension: 22
Coefficient field: Q(2)\Q(\sqrt{2})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x22 x^{2} - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 1536)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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+βq5+2βq7+3βq134q17+8q19+4βq233q25βq292βq31+4q35+3βq37+4q41+8βq47+q49+5βq53++16q97+O(q100) q + \beta q^{5} + 2 \beta q^{7} + 3 \beta q^{13} - 4 q^{17} + 8 q^{19} + 4 \beta q^{23} - 3 q^{25} - \beta q^{29} - 2 \beta q^{31} + 4 q^{35} + 3 \beta q^{37} + 4 q^{41} + 8 \beta q^{47} + q^{49} + 5 \beta q^{53} + \cdots + 16 q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q8q17+16q196q25+8q35+8q41+2q4924q59+12q658q67+28q7332q8312q89+24q91+32q97+O(q100) 2 q - 8 q^{17} + 16 q^{19} - 6 q^{25} + 8 q^{35} + 8 q^{41} + 2 q^{49} - 24 q^{59} + 12 q^{65} - 8 q^{67} + 28 q^{73} - 32 q^{83} - 12 q^{89} + 24 q^{91} + 32 q^{97}+O(q^{100}) Copy content Toggle raw display

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}
1.1
−1.41421
1.41421
0 0 0 −1.41421 0 −2.82843 0 0 0
1.2 0 0 0 1.41421 0 2.82843 0 0 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
33 1 -1

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9216.2.a.i 2
3.b odd 2 1 3072.2.a.h 2
4.b odd 2 1 9216.2.a.h 2
8.b even 2 1 9216.2.a.h 2
8.d odd 2 1 inner 9216.2.a.i 2
12.b even 2 1 3072.2.a.b 2
24.f even 2 1 3072.2.a.h 2
24.h odd 2 1 3072.2.a.b 2
32.g even 8 2 4608.2.k.y 4
32.g even 8 2 4608.2.k.bb 4
32.h odd 8 2 4608.2.k.y 4
32.h odd 8 2 4608.2.k.bb 4
48.i odd 4 2 3072.2.d.c 4
48.k even 4 2 3072.2.d.c 4
96.o even 8 2 1536.2.j.b 4
96.o even 8 2 1536.2.j.c yes 4
96.p odd 8 2 1536.2.j.b 4
96.p odd 8 2 1536.2.j.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.b 4 96.o even 8 2
1536.2.j.b 4 96.p odd 8 2
1536.2.j.c yes 4 96.o even 8 2
1536.2.j.c yes 4 96.p odd 8 2
3072.2.a.b 2 12.b even 2 1
3072.2.a.b 2 24.h odd 2 1
3072.2.a.h 2 3.b odd 2 1
3072.2.a.h 2 24.f even 2 1
3072.2.d.c 4 48.i odd 4 2
3072.2.d.c 4 48.k even 4 2
4608.2.k.y 4 32.g even 8 2
4608.2.k.y 4 32.h odd 8 2
4608.2.k.bb 4 32.g even 8 2
4608.2.k.bb 4 32.h odd 8 2
9216.2.a.h 2 4.b odd 2 1
9216.2.a.h 2 8.b even 2 1
9216.2.a.i 2 1.a even 1 1 trivial
9216.2.a.i 2 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(9216))S_{2}^{\mathrm{new}}(\Gamma_0(9216)):

T522 T_{5}^{2} - 2 Copy content Toggle raw display
T728 T_{7}^{2} - 8 Copy content Toggle raw display
T11 T_{11} Copy content Toggle raw display
T13218 T_{13}^{2} - 18 Copy content Toggle raw display
T17+4 T_{17} + 4 Copy content Toggle raw display
T198 T_{19} - 8 Copy content Toggle raw display
T67+4 T_{67} + 4 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 T22 T^{2} - 2 Copy content Toggle raw display
77 T28 T^{2} - 8 Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T218 T^{2} - 18 Copy content Toggle raw display
1717 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
1919 (T8)2 (T - 8)^{2} Copy content Toggle raw display
2323 T232 T^{2} - 32 Copy content Toggle raw display
2929 T22 T^{2} - 2 Copy content Toggle raw display
3131 T28 T^{2} - 8 Copy content Toggle raw display
3737 T218 T^{2} - 18 Copy content Toggle raw display
4141 (T4)2 (T - 4)^{2} Copy content Toggle raw display
4343 T2 T^{2} Copy content Toggle raw display
4747 T2128 T^{2} - 128 Copy content Toggle raw display
5353 T250 T^{2} - 50 Copy content Toggle raw display
5959 (T+12)2 (T + 12)^{2} Copy content Toggle raw display
6161 T22 T^{2} - 2 Copy content Toggle raw display
6767 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
7171 T2128 T^{2} - 128 Copy content Toggle raw display
7373 (T14)2 (T - 14)^{2} Copy content Toggle raw display
7979 T272 T^{2} - 72 Copy content Toggle raw display
8383 (T+16)2 (T + 16)^{2} Copy content Toggle raw display
8989 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
9797 (T16)2 (T - 16)^{2} Copy content Toggle raw display
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