Properties

Label 975.2.b.e
Level 975975
Weight 22
Character orbit 975.b
Analytic conductor 7.7857.785
Analytic rank 00
Dimension 22
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [975,2,Mod(376,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.376"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 975=35213 975 = 3 \cdot 5^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 975.b (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 7.785414197077.78541419707
Analytic rank: 00
Dimension: 22
Coefficient field: Q(1)\Q(\sqrt{-1})
Copy content comment:defining polynomial
 
Copy content 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,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 3 3
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of β=3i\beta = 3i. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+q3+2q4βq7+q9βq11+2q12+(β2)q13+4q163q17βq21+3q23+q272βq286q292βq31βq33+βq99+O(q100) q + q^{3} + 2 q^{4} - \beta q^{7} + q^{9} - \beta q^{11} + 2 q^{12} + ( - \beta - 2) q^{13} + 4 q^{16} - 3 q^{17} - \beta q^{21} + 3 q^{23} + q^{27} - 2 \beta q^{28} - 6 q^{29} - 2 \beta q^{31} - \beta q^{33} + \cdots - \beta q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+2q3+4q4+2q9+4q124q13+8q166q17+6q23+2q2712q29+4q364q39+20q43+8q484q496q518q52+6q53+2q61++12q92+O(q100) 2 q + 2 q^{3} + 4 q^{4} + 2 q^{9} + 4 q^{12} - 4 q^{13} + 8 q^{16} - 6 q^{17} + 6 q^{23} + 2 q^{27} - 12 q^{29} + 4 q^{36} - 4 q^{39} + 20 q^{43} + 8 q^{48} - 4 q^{49} - 6 q^{51} - 8 q^{52} + 6 q^{53} + 2 q^{61}+ \cdots + 12 q^{92}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/975Z)×\left(\mathbb{Z}/975\mathbb{Z}\right)^\times.

nn 301301 326326 352352
χ(n)\chi(n) 1-1 11 11

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
376.1
1.00000i
1.00000i
0 1.00000 2.00000 0 0 3.00000i 0 1.00000 0
376.2 0 1.00000 2.00000 0 0 3.00000i 0 1.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.b.e 2
5.b even 2 1 195.2.b.a 2
5.c odd 4 1 975.2.h.b 2
5.c odd 4 1 975.2.h.c 2
13.b even 2 1 inner 975.2.b.e 2
15.d odd 2 1 585.2.b.d 2
20.d odd 2 1 3120.2.g.j 2
65.d even 2 1 195.2.b.a 2
65.g odd 4 1 2535.2.a.f 1
65.g odd 4 1 2535.2.a.g 1
65.h odd 4 1 975.2.h.b 2
65.h odd 4 1 975.2.h.c 2
195.e odd 2 1 585.2.b.d 2
195.n even 4 1 7605.2.a.i 1
195.n even 4 1 7605.2.a.n 1
260.g odd 2 1 3120.2.g.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.a 2 5.b even 2 1
195.2.b.a 2 65.d even 2 1
585.2.b.d 2 15.d odd 2 1
585.2.b.d 2 195.e odd 2 1
975.2.b.e 2 1.a even 1 1 trivial
975.2.b.e 2 13.b even 2 1 inner
975.2.h.b 2 5.c odd 4 1
975.2.h.b 2 65.h odd 4 1
975.2.h.c 2 5.c odd 4 1
975.2.h.c 2 65.h odd 4 1
2535.2.a.f 1 65.g odd 4 1
2535.2.a.g 1 65.g odd 4 1
3120.2.g.j 2 20.d odd 2 1
3120.2.g.j 2 260.g odd 2 1
7605.2.a.i 1 195.n even 4 1
7605.2.a.n 1 195.n 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(975,[χ])S_{2}^{\mathrm{new}}(975, [\chi]):

T2 T_{2} Copy content Toggle raw display
T17+3 T_{17} + 3 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 (T1)2 (T - 1)^{2} Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+9 T^{2} + 9 Copy content Toggle raw display
1111 T2+9 T^{2} + 9 Copy content Toggle raw display
1313 T2+4T+13 T^{2} + 4T + 13 Copy content Toggle raw display
1717 (T+3)2 (T + 3)^{2} Copy content Toggle raw display
1919 T2 T^{2} Copy content Toggle raw display
2323 (T3)2 (T - 3)^{2} Copy content Toggle raw display
2929 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
3131 T2+36 T^{2} + 36 Copy content Toggle raw display
3737 T2+81 T^{2} + 81 Copy content Toggle raw display
4141 T2+9 T^{2} + 9 Copy content Toggle raw display
4343 (T10)2 (T - 10)^{2} Copy content Toggle raw display
4747 T2+144 T^{2} + 144 Copy content Toggle raw display
5353 (T3)2 (T - 3)^{2} Copy content Toggle raw display
5959 T2+144 T^{2} + 144 Copy content Toggle raw display
6161 (T1)2 (T - 1)^{2} Copy content Toggle raw display
6767 T2 T^{2} Copy content Toggle raw display
7171 T2+81 T^{2} + 81 Copy content Toggle raw display
7373 T2+36 T^{2} + 36 Copy content Toggle raw display
7979 (T1)2 (T - 1)^{2} Copy content Toggle raw display
8383 T2+36 T^{2} + 36 Copy content Toggle raw display
8989 T2+225 T^{2} + 225 Copy content Toggle raw display
9797 T2+81 T^{2} + 81 Copy content Toggle raw display
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