Properties

Label 975.2.b.k
Level 975975
Weight 22
Character orbit 975.b
Analytic conductor 7.7857.785
Analytic rank 00
Dimension 66
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 = [6,0,6,-8] 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: 66
Coefficient field: 6.0.559227904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x6+10x4+19x2+4 x^{6} + 10x^{4} + 19x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 1 1
Twist minimal: yes
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 a basis 1,β1,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+q3+(β21)q4+β1q6β5q7+(β5+β32β1)q8+q9+(β5+β3)q11+(β21)q12+(β5+β42β1+1)q13++(β5+β3)q99+O(q100) q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} + \beta_1 q^{6} - \beta_{5} q^{7} + ( - \beta_{5} + \beta_{3} - 2 \beta_1) q^{8} + q^{9} + (\beta_{5} + \beta_{3}) q^{11} + (\beta_{2} - 1) q^{12} + (\beta_{5} + \beta_{4} - 2 \beta_1 + 1) q^{13}+ \cdots + (\beta_{5} + \beta_{3}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+6q38q4+6q98q12+3q13+8q14+28q1612q22+4q23+26q26+6q27+16q298q36+36q38+3q39+8q4238q43+28q48+12q94+O(q100) 6 q + 6 q^{3} - 8 q^{4} + 6 q^{9} - 8 q^{12} + 3 q^{13} + 8 q^{14} + 28 q^{16} - 12 q^{22} + 4 q^{23} + 26 q^{26} + 6 q^{27} + 16 q^{29} - 8 q^{36} + 36 q^{38} + 3 q^{39} + 8 q^{42} - 38 q^{43} + 28 q^{48}+ \cdots - 12 q^{94}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6+10x4+19x2+4 x^{6} + 10x^{4} + 19x^{2} + 4 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2+3 \nu^{2} + 3 Copy content Toggle raw display
β3\beta_{3}== (ν58ν35ν)/2 ( -\nu^{5} - 8\nu^{3} - 5\nu ) / 2 Copy content Toggle raw display
β4\beta_{4}== (ν5+2ν4+10ν3+16ν2+19ν+10)/4 ( \nu^{5} + 2\nu^{4} + 10\nu^{3} + 16\nu^{2} + 19\nu + 10 ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν510ν317ν)/2 ( -\nu^{5} - 10\nu^{3} - 17\nu ) / 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β23 \beta_{2} - 3 Copy content Toggle raw display
ν3\nu^{3}== β5+β36β1 -\beta_{5} + \beta_{3} - 6\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β5+2β48β2β1+19 \beta_{5} + 2\beta_{4} - 8\beta_{2} - \beta _1 + 19 Copy content Toggle raw display
ν5\nu^{5}== 8β510β3+43β1 8\beta_{5} - 10\beta_{3} + 43\beta_1 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
2.74869i
1.48478i
0.490052i
0.490052i
1.48478i
2.74869i
2.74869i 1.00000 −5.55528 0 2.74869i 2.02107i 9.77234i 1.00000 0
376.2 1.48478i 1.00000 −0.204573 0 1.48478i 0.137780i 2.66581i 1.00000 0
376.3 0.490052i 1.00000 1.75985 0 0.490052i 3.59114i 1.84252i 1.00000 0
376.4 0.490052i 1.00000 1.75985 0 0.490052i 3.59114i 1.84252i 1.00000 0
376.5 1.48478i 1.00000 −0.204573 0 1.48478i 0.137780i 2.66581i 1.00000 0
376.6 2.74869i 1.00000 −5.55528 0 2.74869i 2.02107i 9.77234i 1.00000 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 376.6
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.k yes 6
5.b even 2 1 975.2.b.i 6
5.c odd 4 2 975.2.h.i 12
13.b even 2 1 inner 975.2.b.k yes 6
65.d even 2 1 975.2.b.i 6
65.h odd 4 2 975.2.h.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.b.i 6 5.b even 2 1
975.2.b.i 6 65.d even 2 1
975.2.b.k yes 6 1.a even 1 1 trivial
975.2.b.k yes 6 13.b even 2 1 inner
975.2.h.i 12 5.c odd 4 2
975.2.h.i 12 65.h odd 4 2

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]):

T26+10T24+19T22+4 T_{2}^{6} + 10T_{2}^{4} + 19T_{2}^{2} + 4 Copy content Toggle raw display
T17335T17+74 T_{17}^{3} - 35T_{17} + 74 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6+10T4++4 T^{6} + 10 T^{4} + \cdots + 4 Copy content Toggle raw display
33 (T1)6 (T - 1)^{6} Copy content Toggle raw display
55 T6 T^{6} Copy content Toggle raw display
77 T6+17T4++1 T^{6} + 17 T^{4} + \cdots + 1 Copy content Toggle raw display
1111 T6+54T4++36 T^{6} + 54 T^{4} + \cdots + 36 Copy content Toggle raw display
1313 T63T5++2197 T^{6} - 3 T^{5} + \cdots + 2197 Copy content Toggle raw display
1717 (T335T+74)2 (T^{3} - 35 T + 74)^{2} Copy content Toggle raw display
1919 T6+71T4++1444 T^{6} + 71 T^{4} + \cdots + 1444 Copy content Toggle raw display
2323 (T32T254T20)2 (T^{3} - 2 T^{2} - 54 T - 20)^{2} Copy content Toggle raw display
2929 (T38T2++466)2 (T^{3} - 8 T^{2} + \cdots + 466)^{2} Copy content Toggle raw display
3131 T6+109T4++29241 T^{6} + 109 T^{4} + \cdots + 29241 Copy content Toggle raw display
3737 T6+52T4++4096 T^{6} + 52 T^{4} + \cdots + 4096 Copy content Toggle raw display
4141 T6+52T4++4096 T^{6} + 52 T^{4} + \cdots + 4096 Copy content Toggle raw display
4343 (T3+19T2++178)2 (T^{3} + 19 T^{2} + \cdots + 178)^{2} Copy content Toggle raw display
4747 T6+234T4++206116 T^{6} + 234 T^{4} + \cdots + 206116 Copy content Toggle raw display
5353 (T3+2T213T24)2 (T^{3} + 2 T^{2} - 13 T - 24)^{2} Copy content Toggle raw display
5959 T6+314T4++93636 T^{6} + 314 T^{4} + \cdots + 93636 Copy content Toggle raw display
6161 (T37T2++239)2 (T^{3} - 7 T^{2} + \cdots + 239)^{2} Copy content Toggle raw display
6767 T6+45T4++729 T^{6} + 45 T^{4} + \cdots + 729 Copy content Toggle raw display
7171 T6+228T4++64 T^{6} + 228 T^{4} + \cdots + 64 Copy content Toggle raw display
7373 T6+460T4++577600 T^{6} + 460 T^{4} + \cdots + 577600 Copy content Toggle raw display
7979 (T312T2++976)2 (T^{3} - 12 T^{2} + \cdots + 976)^{2} Copy content Toggle raw display
8383 T6+362T4++427716 T^{6} + 362 T^{4} + \cdots + 427716 Copy content Toggle raw display
8989 T6+360T4++186624 T^{6} + 360 T^{4} + \cdots + 186624 Copy content Toggle raw display
9797 T6+187T4++29584 T^{6} + 187 T^{4} + \cdots + 29584 Copy content Toggle raw display
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