Properties

Label 925.2.b.b
Level 925925
Weight 22
Character orbit 925.b
Analytic conductor 7.3867.386
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] = [925,2,Mod(149,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("925.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 925=5237 925 = 5^{2} \cdot 37
Weight: k k == 2 2
Character orbit: [χ][\chi] == 925.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: 7.386162186977.38616218697
Analytic rank: 11
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,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 37)
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 i=1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+2iq23iq32q4+6q6+iq76q95q11+6iq122iq132q144q1612iq18+3q2110iq22+2iq23+4q26+9iq27++30q99+O(q100) q + 2 i q^{2} - 3 i q^{3} - 2 q^{4} + 6 q^{6} + i q^{7} - 6 q^{9} - 5 q^{11} + 6 i q^{12} - 2 i q^{13} - 2 q^{14} - 4 q^{16} - 12 i q^{18} + 3 q^{21} - 10 i q^{22} + 2 i q^{23} + 4 q^{26} + 9 i q^{27} + \cdots + 30 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q4q4+12q612q910q114q148q16+6q21+8q2612q298q31+24q3612q3918q41+20q448q46+12q4936q5416q59++60q99+O(q100) 2 q - 4 q^{4} + 12 q^{6} - 12 q^{9} - 10 q^{11} - 4 q^{14} - 8 q^{16} + 6 q^{21} + 8 q^{26} - 12 q^{29} - 8 q^{31} + 24 q^{36} - 12 q^{39} - 18 q^{41} + 20 q^{44} - 8 q^{46} + 12 q^{49} - 36 q^{54} - 16 q^{59}+ \cdots + 60 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/925Z)×\left(\mathbb{Z}/925\mathbb{Z}\right)^\times.

nn 7676 852852
χ(n)\chi(n) 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}
149.1
1.00000i
1.00000i
2.00000i 3.00000i −2.00000 0 6.00000 1.00000i 0 −6.00000 0
149.2 2.00000i 3.00000i −2.00000 0 6.00000 1.00000i 0 −6.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.b.b 2
5.b even 2 1 inner 925.2.b.b 2
5.c odd 4 1 37.2.a.a 1
5.c odd 4 1 925.2.a.e 1
15.e even 4 1 333.2.a.d 1
15.e even 4 1 8325.2.a.e 1
20.e even 4 1 592.2.a.e 1
35.f even 4 1 1813.2.a.a 1
40.i odd 4 1 2368.2.a.q 1
40.k even 4 1 2368.2.a.b 1
55.e even 4 1 4477.2.a.b 1
60.l odd 4 1 5328.2.a.r 1
65.h odd 4 1 6253.2.a.c 1
185.f even 4 1 1369.2.b.c 2
185.h odd 4 1 1369.2.a.e 1
185.k even 4 1 1369.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 5.c odd 4 1
333.2.a.d 1 15.e even 4 1
592.2.a.e 1 20.e even 4 1
925.2.a.e 1 5.c odd 4 1
925.2.b.b 2 1.a even 1 1 trivial
925.2.b.b 2 5.b even 2 1 inner
1369.2.a.e 1 185.h odd 4 1
1369.2.b.c 2 185.f even 4 1
1369.2.b.c 2 185.k even 4 1
1813.2.a.a 1 35.f even 4 1
2368.2.a.b 1 40.k even 4 1
2368.2.a.q 1 40.i odd 4 1
4477.2.a.b 1 55.e even 4 1
5328.2.a.r 1 60.l odd 4 1
6253.2.a.c 1 65.h odd 4 1
8325.2.a.e 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(925,[χ])S_{2}^{\mathrm{new}}(925, [\chi]):

T22+4 T_{2}^{2} + 4 Copy content Toggle raw display
T32+9 T_{3}^{2} + 9 Copy content Toggle raw display
T72+1 T_{7}^{2} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+4 T^{2} + 4 Copy content Toggle raw display
33 T2+9 T^{2} + 9 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+1 T^{2} + 1 Copy content Toggle raw display
1111 (T+5)2 (T + 5)^{2} Copy content Toggle raw display
1313 T2+4 T^{2} + 4 Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T2 T^{2} Copy content Toggle raw display
2323 T2+4 T^{2} + 4 Copy content Toggle raw display
2929 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
3131 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
3737 T2+1 T^{2} + 1 Copy content Toggle raw display
4141 (T+9)2 (T + 9)^{2} Copy content Toggle raw display
4343 T2+4 T^{2} + 4 Copy content Toggle raw display
4747 T2+81 T^{2} + 81 Copy content Toggle raw display
5353 T2+1 T^{2} + 1 Copy content Toggle raw display
5959 (T+8)2 (T + 8)^{2} Copy content Toggle raw display
6161 (T+8)2 (T + 8)^{2} Copy content Toggle raw display
6767 T2+64 T^{2} + 64 Copy content Toggle raw display
7171 (T9)2 (T - 9)^{2} Copy content Toggle raw display
7373 T2+1 T^{2} + 1 Copy content Toggle raw display
7979 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
8383 T2+225 T^{2} + 225 Copy content Toggle raw display
8989 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
9797 T2+16 T^{2} + 16 Copy content Toggle raw display
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