Properties

Label 9610.2.a.bz
Level 96109610
Weight 22
Character orbit 9610.a
Self dual yes
Analytic conductor 76.73676.736
Analytic rank 00
Dimension 66
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9610,2,Mod(1,9610)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9610, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) 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("9610.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 9610=25312 9610 = 2 \cdot 5 \cdot 31^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 9610.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6,0,6,6,0,8,6,22,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 76.736236342576.7362363425
Analytic rank: 00
Dimension: 66
Coefficient field: 6.6.14623232.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x62x513x4+14x3+26x228x+1 x^{6} - 2x^{5} - 13x^{4} + 14x^{3} + 26x^{2} - 28x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 23 2^{3}
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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+q2+(β3+β1)q3+q4+q5+(β3+β1)q6+(β2+1)q7+q8+(β2+4)q9+q10β3q11+(β3+β1)q12++(β44β3+3β1)q99+O(q100) q + q^{2} + ( - \beta_{3} + \beta_1) q^{3} + q^{4} + q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - \beta_{2} + 1) q^{7} + q^{8} + (\beta_{2} + 4) q^{9} + q^{10} - \beta_{3} q^{11} + ( - \beta_{3} + \beta_1) q^{12}+ \cdots + (\beta_{4} - 4 \beta_{3} + 3 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+6q2+6q4+6q5+8q7+6q8+22q9+6q10+8q14+6q16+22q1812q19+6q20+6q25+8q28+6q32+32q33+8q35+22q3612q38++38q98+O(q100) 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 8 q^{7} + 6 q^{8} + 22 q^{9} + 6 q^{10} + 8 q^{14} + 6 q^{16} + 22 q^{18} - 12 q^{19} + 6 q^{20} + 6 q^{25} + 8 q^{28} + 6 q^{32} + 32 q^{33} + 8 q^{35} + 22 q^{36} - 12 q^{38}+ \cdots + 38 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x62x513x4+14x3+26x228x+1 x^{6} - 2x^{5} - 13x^{4} + 14x^{3} + 26x^{2} - 28x + 1 : Copy content Toggle raw display

β1\beta_{1}== (22ν5+15ν4+310ν3+95ν2513ν25)/31 ( -22\nu^{5} + 15\nu^{4} + 310\nu^{3} + 95\nu^{2} - 513\nu - 25 ) / 31 Copy content Toggle raw display
β2\beta_{2}== (28ν5+36ν4+372ν382ν2636ν+157)/31 ( -28\nu^{5} + 36\nu^{4} + 372\nu^{3} - 82\nu^{2} - 636\nu + 157 ) / 31 Copy content Toggle raw display
β3\beta_{3}== (29ν5+24ν4+403ν3+59ν2641ν+84)/31 ( -29\nu^{5} + 24\nu^{4} + 403\nu^{3} + 59\nu^{2} - 641\nu + 84 ) / 31 Copy content Toggle raw display
β4\beta_{4}== (41ν535ν4589ν346ν2+1104ν169)/31 ( 41\nu^{5} - 35\nu^{4} - 589\nu^{3} - 46\nu^{2} + 1104\nu - 169 ) / 31 Copy content Toggle raw display
β5\beta_{5}== (44ν5+30ν4+620ν3+190ν2964ν81)/31 ( -44\nu^{5} + 30\nu^{4} + 620\nu^{3} + 190\nu^{2} - 964\nu - 81 ) / 31 Copy content Toggle raw display
ν\nu== (β52β1+1)/2 ( \beta_{5} - 2\beta _1 + 1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (2β54β3+β2+11)/2 ( 2\beta_{5} - 4\beta_{3} + \beta_{2} + 11 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (12β53β49β3+β219β1+19)/2 ( 12\beta_{5} - 3\beta_{4} - 9\beta_{3} + \beta_{2} - 19\beta _1 + 19 ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (39β54β460β3+17β228β1+134)/2 ( 39\beta_{5} - 4\beta_{4} - 60\beta_{3} + 17\beta_{2} - 28\beta _1 + 134 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (181β545β4185β3+30β2243β1+381)/2 ( 181\beta_{5} - 45\beta_{4} - 185\beta_{3} + 30\beta_{2} - 243\beta _1 + 381 ) / 2 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.

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}
1.1
0.0370107
1.23688
1.14032
−1.68810
4.06531
−2.79142
1.00000 −3.36187 1.00000 1.00000 −3.36187 −3.30219 1.00000 8.30219 1.00000
1.2 1.00000 −2.33500 1.00000 1.00000 −2.33500 2.54778 1.00000 2.45222 1.00000
1.3 1.00000 −1.80155 1.00000 1.00000 −1.80155 4.75441 1.00000 0.245594 1.00000
1.4 1.00000 1.80155 1.00000 1.00000 1.80155 4.75441 1.00000 0.245594 1.00000
1.5 1.00000 2.33500 1.00000 1.00000 2.33500 2.54778 1.00000 2.45222 1.00000
1.6 1.00000 3.36187 1.00000 1.00000 3.36187 −3.30219 1.00000 8.30219 1.00000
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
55 1 -1
3131 1 -1

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9610.2.a.bz 6
31.b odd 2 1 inner 9610.2.a.bz 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9610.2.a.bz 6 1.a even 1 1 trivial
9610.2.a.bz 6 31.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(9610))S_{2}^{\mathrm{new}}(\Gamma_0(9610)):

T3620T34+116T32200 T_{3}^{6} - 20T_{3}^{4} + 116T_{3}^{2} - 200 Copy content Toggle raw display
T734T7212T7+40 T_{7}^{3} - 4T_{7}^{2} - 12T_{7} + 40 Copy content Toggle raw display
T11618T114+56T1128 T_{11}^{6} - 18T_{11}^{4} + 56T_{11}^{2} - 8 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T1)6 (T - 1)^{6} Copy content Toggle raw display
33 T620T4+200 T^{6} - 20 T^{4} + \cdots - 200 Copy content Toggle raw display
55 (T1)6 (T - 1)^{6} Copy content Toggle raw display
77 (T34T212T+40)2 (T^{3} - 4 T^{2} - 12 T + 40)^{2} Copy content Toggle raw display
1111 T618T4+8 T^{6} - 18 T^{4} + \cdots - 8 Copy content Toggle raw display
1313 T658T4+5000 T^{6} - 58 T^{4} + \cdots - 5000 Copy content Toggle raw display
1717 T6 T^{6} Copy content Toggle raw display
1919 (T3+6T2+200)2 (T^{3} + 6 T^{2} + \cdots - 200)^{2} Copy content Toggle raw display
2323 T670T4+5000 T^{6} - 70 T^{4} + \cdots - 5000 Copy content Toggle raw display
2929 T660T4+200 T^{6} - 60 T^{4} + \cdots - 200 Copy content Toggle raw display
3131 T6 T^{6} Copy content Toggle raw display
3737 T6210T4+192200 T^{6} - 210 T^{4} + \cdots - 192200 Copy content Toggle raw display
4141 (T316T2+40)2 (T^{3} - 16 T^{2} + \cdots - 40)^{2} Copy content Toggle raw display
4343 T628T4+8 T^{6} - 28 T^{4} + \cdots - 8 Copy content Toggle raw display
4747 (T3+6T2+200)2 (T^{3} + 6 T^{2} + \cdots - 200)^{2} Copy content Toggle raw display
5353 T6122T4+5000 T^{6} - 122 T^{4} + \cdots - 5000 Copy content Toggle raw display
5959 (T32T216T8)2 (T^{3} - 2 T^{2} - 16 T - 8)^{2} Copy content Toggle raw display
6161 T6340T4+1065800 T^{6} - 340 T^{4} + \cdots - 1065800 Copy content Toggle raw display
6767 (T3+4T212T40)2 (T^{3} + 4 T^{2} - 12 T - 40)^{2} Copy content Toggle raw display
7171 (T34T264T64)2 (T^{3} - 4 T^{2} - 64 T - 64)^{2} Copy content Toggle raw display
7373 T6208T4+86528 T^{6} - 208 T^{4} + \cdots - 86528 Copy content Toggle raw display
7979 T6552T4+6083072 T^{6} - 552 T^{4} + \cdots - 6083072 Copy content Toggle raw display
8383 T6396T4+773768 T^{6} - 396 T^{4} + \cdots - 773768 Copy content Toggle raw display
8989 T6470T4+1065800 T^{6} - 470 T^{4} + \cdots - 1065800 Copy content Toggle raw display
9797 (T314T2+40)2 (T^{3} - 14 T^{2} + \cdots - 40)^{2} Copy content Toggle raw display
show more
show less