Properties

Label 1152.1.b.a
Level 11521152
Weight 11
Character orbit 1152.b
Self dual yes
Analytic conductor 0.5750.575
Analytic rank 00
Dimension 11
Projective image D2D_{2}
CM/RM discs -4, -8, 8
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,1,Mod(703,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.703");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 1152=2732 1152 = 2^{7} \cdot 3^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1152.b (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: yes
Analytic conductor: 0.5749228945530.574922894553
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 128)
Projective image: D2D_{2}
Projective field: Galois closure of Q(ζ8)\Q(\zeta_{8})
Artin image: D4D_4
Artin field: Galois closure of 4.0.4608.1
Stark unit: Root of x4340x3+102x2340x+1x^{4} - 340x^{3} + 102x^{2} - 340x + 1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+2q17+q252q41+q492q73+2q892q97+O(q100) q + 2 q^{17} + q^{25} - 2 q^{41} + q^{49} - 2 q^{73} + 2 q^{89} - 2 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1152Z)×\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times.

nn 127127 641641 901901
χ(n)\chi(n) 1-1 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}
703.1
0
0 0 0 0 0 0 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
4.b odd 2 1 CM by Q(1)\Q(\sqrt{-1})
8.b even 2 1 RM by Q(2)\Q(\sqrt{2})
8.d odd 2 1 CM by Q(2)\Q(\sqrt{-2})

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.1.b.a 1
3.b odd 2 1 128.1.d.a 1
4.b odd 2 1 CM 1152.1.b.a 1
8.b even 2 1 RM 1152.1.b.a 1
8.d odd 2 1 CM 1152.1.b.a 1
12.b even 2 1 128.1.d.a 1
15.d odd 2 1 3200.1.g.a 1
15.e even 4 2 3200.1.e.a 2
16.e even 4 2 2304.1.g.b 1
16.f odd 4 2 2304.1.g.b 1
24.f even 2 1 128.1.d.a 1
24.h odd 2 1 128.1.d.a 1
48.i odd 4 2 256.1.c.a 1
48.k even 4 2 256.1.c.a 1
60.h even 2 1 3200.1.g.a 1
60.l odd 4 2 3200.1.e.a 2
96.o even 8 4 1024.1.f.b 2
96.p odd 8 4 1024.1.f.b 2
120.i odd 2 1 3200.1.g.a 1
120.m even 2 1 3200.1.g.a 1
120.q odd 4 2 3200.1.e.a 2
120.w even 4 2 3200.1.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 3.b odd 2 1
128.1.d.a 1 12.b even 2 1
128.1.d.a 1 24.f even 2 1
128.1.d.a 1 24.h odd 2 1
256.1.c.a 1 48.i odd 4 2
256.1.c.a 1 48.k even 4 2
1024.1.f.b 2 96.o even 8 4
1024.1.f.b 2 96.p odd 8 4
1152.1.b.a 1 1.a even 1 1 trivial
1152.1.b.a 1 4.b odd 2 1 CM
1152.1.b.a 1 8.b even 2 1 RM
1152.1.b.a 1 8.d odd 2 1 CM
2304.1.g.b 1 16.e even 4 2
2304.1.g.b 1 16.f odd 4 2
3200.1.e.a 2 15.e even 4 2
3200.1.e.a 2 60.l odd 4 2
3200.1.e.a 2 120.q odd 4 2
3200.1.e.a 2 120.w even 4 2
3200.1.g.a 1 15.d odd 2 1
3200.1.g.a 1 60.h even 2 1
3200.1.g.a 1 120.i odd 2 1
3200.1.g.a 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5 T_{5} acting on S1new(1152,[χ])S_{1}^{\mathrm{new}}(1152, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T T Copy content Toggle raw display
33 T T Copy content Toggle raw display
55 T T Copy content Toggle raw display
77 T T Copy content Toggle raw display
1111 T T Copy content Toggle raw display
1313 T T Copy content Toggle raw display
1717 T2 T - 2 Copy content Toggle raw display
1919 T T Copy content Toggle raw display
2323 T T Copy content Toggle raw display
2929 T T Copy content Toggle raw display
3131 T T Copy content Toggle raw display
3737 T T Copy content Toggle raw display
4141 T+2 T + 2 Copy content Toggle raw display
4343 T T Copy content Toggle raw display
4747 T T Copy content Toggle raw display
5353 T T Copy content Toggle raw display
5959 T T Copy content Toggle raw display
6161 T T Copy content Toggle raw display
6767 T T Copy content Toggle raw display
7171 T T Copy content Toggle raw display
7373 T+2 T + 2 Copy content Toggle raw display
7979 T T Copy content Toggle raw display
8383 T T Copy content Toggle raw display
8989 T2 T - 2 Copy content Toggle raw display
9797 T+2 T + 2 Copy content Toggle raw display
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