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Properties

Label 1792.2.b.i
Level 17921792
Weight 22
Character orbit 1792.b
Analytic conductor 14.30914.309
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] = [1792,2,Mod(897,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.897");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1792=28⋅7 1792 = 2^{8} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1792.b (of order 22, degree 11, not 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: 14.309192042214.3091920422
Analytic rank: 00
Dimension: 22
Coefficient field: Q(−1)\Q(\sqrt{-1})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 56)
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=2i\beta = 2i. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−βq5+q7+3q9+2βq11+βq13−6q17+4βq19+q25+3βq29+8q31−βq35+βq37−2q41+2βq43−3βq45+⋯+6βq99+O(q100) q - \beta q^{5} + q^{7} + 3 q^{9} + 2 \beta q^{11} + \beta q^{13} - 6 q^{17} + 4 \beta q^{19} + q^{25} + 3 \beta q^{29} + 8 q^{31} - \beta q^{35} + \beta q^{37} - 2 q^{41} + 2 \beta q^{43} - 3 \beta q^{45} + \cdots + 6 \beta q^{99} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q+2q7+6q9−12q17+2q25+16q31−4q41−16q47+2q49+16q55+6q63+8q65+16q71−20q73+32q79+18q81+12q89+32q95−12q97+O(q100) 2 q + 2 q^{7} + 6 q^{9} - 12 q^{17} + 2 q^{25} + 16 q^{31} - 4 q^{41} - 16 q^{47} + 2 q^{49} + 16 q^{55} + 6 q^{63} + 8 q^{65} + 16 q^{71} - 20 q^{73} + 32 q^{79} + 18 q^{81} + 12 q^{89} + 32 q^{95} - 12 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1792Z)×\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times.

nn 10231023 10251025 15411541
χ(n)\chi(n) 11 11 −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}
897.1
1.00000i
− 1.00000i
0 0 0 − 2.00000i 0 1.00000 0 3.00000 0
897.2 0 0 0 2.00000i 0 1.00000 0 3.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.b.i 2
4.b odd 2 1 1792.2.b.d 2
8.b even 2 1 inner 1792.2.b.i 2
8.d odd 2 1 1792.2.b.d 2
16.e even 4 1 56.2.a.a ✓ 1
16.e even 4 1 448.2.a.d 1
16.f odd 4 1 112.2.a.b 1
16.f odd 4 1 448.2.a.e 1
48.i odd 4 1 504.2.a.c 1
48.i odd 4 1 4032.2.a.bb 1
48.k even 4 1 1008.2.a.d 1
48.k even 4 1 4032.2.a.bk 1
80.i odd 4 1 1400.2.g.g 2
80.j even 4 1 2800.2.g.p 2
80.k odd 4 1 2800.2.a.p 1
80.q even 4 1 1400.2.a.g 1
80.s even 4 1 2800.2.g.p 2
80.t odd 4 1 1400.2.g.g 2
112.j even 4 1 784.2.a.e 1
112.j even 4 1 3136.2.a.p 1
112.l odd 4 1 392.2.a.d 1
112.l odd 4 1 3136.2.a.q 1
112.u odd 12 2 784.2.i.e 2
112.v even 12 2 784.2.i.g 2
112.w even 12 2 392.2.i.c 2
112.x odd 12 2 392.2.i.d 2
176.l odd 4 1 6776.2.a.g 1
208.p even 4 1 9464.2.a.c 1
336.v odd 4 1 7056.2.a.bo 1
336.y even 4 1 3528.2.a.x 1
336.bo even 12 2 3528.2.s.e 2
336.bt odd 12 2 3528.2.s.t 2
560.bf odd 4 1 9800.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a ✓ 1 16.e even 4 1
112.2.a.b 1 16.f odd 4 1
392.2.a.d 1 112.l odd 4 1
392.2.i.c 2 112.w even 12 2
392.2.i.d 2 112.x odd 12 2
448.2.a.d 1 16.e even 4 1
448.2.a.e 1 16.f odd 4 1
504.2.a.c 1 48.i odd 4 1
784.2.a.e 1 112.j even 4 1
784.2.i.e 2 112.u odd 12 2
784.2.i.g 2 112.v even 12 2
1008.2.a.d 1 48.k even 4 1
1400.2.a.g 1 80.q even 4 1
1400.2.g.g 2 80.i odd 4 1
1400.2.g.g 2 80.t odd 4 1
1792.2.b.d 2 4.b odd 2 1
1792.2.b.d 2 8.d odd 2 1
1792.2.b.i 2 1.a even 1 1 trivial
1792.2.b.i 2 8.b even 2 1 inner
2800.2.a.p 1 80.k odd 4 1
2800.2.g.p 2 80.j even 4 1
2800.2.g.p 2 80.s even 4 1
3136.2.a.p 1 112.j even 4 1
3136.2.a.q 1 112.l odd 4 1
3528.2.a.x 1 336.y even 4 1
3528.2.s.e 2 336.bo even 12 2
3528.2.s.t 2 336.bt odd 12 2
4032.2.a.bb 1 48.i odd 4 1
4032.2.a.bk 1 48.k even 4 1
6776.2.a.g 1 176.l odd 4 1
7056.2.a.bo 1 336.v odd 4 1
9464.2.a.c 1 208.p even 4 1
9800.2.a.u 1 560.bf odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1792,[χ])S_{2}^{\mathrm{new}}(1792, [\chi]):

T3 T_{3} Copy content Toggle raw display
T52+4 T_{5}^{2} + 4 Copy content Toggle raw display
T112+16 T_{11}^{2} + 16 Copy content Toggle raw display
T23 T_{23} Copy content Toggle raw display
T31−8 T_{31} - 8 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+4 T^{2} + 4 Copy content Toggle raw display
77 (T−1)2 (T - 1)^{2} Copy content Toggle raw display
1111 T2+16 T^{2} + 16 Copy content Toggle raw display
1313 T2+4 T^{2} + 4 Copy content Toggle raw display
1717 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
1919 T2+64 T^{2} + 64 Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2+36 T^{2} + 36 Copy content Toggle raw display
3131 (T−8)2 (T - 8)^{2} Copy content Toggle raw display
3737 T2+4 T^{2} + 4 Copy content Toggle raw display
4141 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
4343 T2+16 T^{2} + 16 Copy content Toggle raw display
4747 (T+8)2 (T + 8)^{2} Copy content Toggle raw display
5353 T2+36 T^{2} + 36 Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 T2+36 T^{2} + 36 Copy content Toggle raw display
6767 T2+16 T^{2} + 16 Copy content Toggle raw display
7171 (T−8)2 (T - 8)^{2} Copy content Toggle raw display
7373 (T+10)2 (T + 10)^{2} Copy content Toggle raw display
7979 (T−16)2 (T - 16)^{2} Copy content Toggle raw display
8383 T2+64 T^{2} + 64 Copy content Toggle raw display
8989 (T−6)2 (T - 6)^{2} Copy content Toggle raw display
9797 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
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