Properties

Label 7600.2.a.o
Level 76007600
Weight 22
Character orbit 7600.a
Self dual yes
Analytic conductor 60.68660.686
Analytic rank 11
Dimension 11
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7600,2,Mod(1,7600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 7600=245219 7600 = 2^{4} \cdot 5^{2} \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 7600.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: 60.686305536260.6863055362
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 152)
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)
 
f(q)f(q) == q+q3+3q72q92q11q13+5q17q19+3q21q235q273q294q312q332q37q398q418q438q47+2q49++4q99+O(q100) q + q^{3} + 3 q^{7} - 2 q^{9} - 2 q^{11} - q^{13} + 5 q^{17} - q^{19} + 3 q^{21} - q^{23} - 5 q^{27} - 3 q^{29} - 4 q^{31} - 2 q^{33} - 2 q^{37} - q^{39} - 8 q^{41} - 8 q^{43} - 8 q^{47} + 2 q^{49}+ \cdots + 4 q^{99}+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
0
0 1.00000 0 0 0 3.00000 0 −2.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
55 +1 +1
1919 +1 +1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7600.2.a.o 1
4.b odd 2 1 3800.2.a.d 1
5.b even 2 1 304.2.a.b 1
15.d odd 2 1 2736.2.a.k 1
20.d odd 2 1 152.2.a.b 1
20.e even 4 2 3800.2.d.f 2
40.e odd 2 1 1216.2.a.f 1
40.f even 2 1 1216.2.a.l 1
60.h even 2 1 1368.2.a.g 1
95.d odd 2 1 5776.2.a.l 1
140.c even 2 1 7448.2.a.g 1
380.d even 2 1 2888.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.b 1 20.d odd 2 1
304.2.a.b 1 5.b even 2 1
1216.2.a.f 1 40.e odd 2 1
1216.2.a.l 1 40.f even 2 1
1368.2.a.g 1 60.h even 2 1
2736.2.a.k 1 15.d odd 2 1
2888.2.a.b 1 380.d even 2 1
3800.2.a.d 1 4.b odd 2 1
3800.2.d.f 2 20.e even 4 2
5776.2.a.l 1 95.d odd 2 1
7448.2.a.g 1 140.c even 2 1
7600.2.a.o 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(7600))S_{2}^{\mathrm{new}}(\Gamma_0(7600)):

T31 T_{3} - 1 Copy content Toggle raw display
T73 T_{7} - 3 Copy content Toggle raw display
T11+2 T_{11} + 2 Copy content Toggle raw display
T13+1 T_{13} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T T Copy content Toggle raw display
33 T1 T - 1 Copy content Toggle raw display
55 T T Copy content Toggle raw display
77 T3 T - 3 Copy content Toggle raw display
1111 T+2 T + 2 Copy content Toggle raw display
1313 T+1 T + 1 Copy content Toggle raw display
1717 T5 T - 5 Copy content Toggle raw display
1919 T+1 T + 1 Copy content Toggle raw display
2323 T+1 T + 1 Copy content Toggle raw display
2929 T+3 T + 3 Copy content Toggle raw display
3131 T+4 T + 4 Copy content Toggle raw display
3737 T+2 T + 2 Copy content Toggle raw display
4141 T+8 T + 8 Copy content Toggle raw display
4343 T+8 T + 8 Copy content Toggle raw display
4747 T+8 T + 8 Copy content Toggle raw display
5353 T+9 T + 9 Copy content Toggle raw display
5959 T+1 T + 1 Copy content Toggle raw display
6161 T14 T - 14 Copy content Toggle raw display
6767 T13 T - 13 Copy content Toggle raw display
7171 T+10 T + 10 Copy content Toggle raw display
7373 T+9 T + 9 Copy content Toggle raw display
7979 T10 T - 10 Copy content Toggle raw display
8383 T10 T - 10 Copy content Toggle raw display
8989 T+12 T + 12 Copy content Toggle raw display
9797 T+14 T + 14 Copy content Toggle raw display
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