Properties

Label 960.3.c.i
Level 960960
Weight 33
Character orbit 960.c
Analytic conductor 26.15826.158
Analytic rank 00
Dimension 88
CM discriminant -20
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(449,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 960=2635 960 = 2^{6} \cdot 3 \cdot 5
Weight: k k == 3 3
Character orbit: [χ][\chi] == 960.c (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: 26.158105378626.1581053786
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8+7x4+1 x^{8} + 7x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 212 2^{12}
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{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 a basis 1,β1,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β5q3+5β1q5+(2β5+β4++2β2)q7+(β6β1)q9+5β3q15+(β72β6++19)q21++12β7q89+O(q100) q + \beta_{5} q^{3} + 5 \beta_1 q^{5} + ( - 2 \beta_{5} + \beta_{4} + \cdots + 2 \beta_{2}) q^{7} + (\beta_{6} - \beta_1) q^{9} + 5 \beta_{3} q^{15} + ( - \beta_{7} - 2 \beta_{6} + \cdots + 19) q^{21}+ \cdots + 12 \beta_{7} q^{89}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+152q21200q25+40q45392q49+232q69+632q81+O(q100) 8 q + 152 q^{21} - 200 q^{25} + 40 q^{45} - 392 q^{49} + 232 q^{69} + 632 q^{81}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+7x4+1 x^{8} + 7x^{4} + 1 : Copy content Toggle raw display

β1\beta_{1}== (ν6+8ν2)/3 ( \nu^{6} + 8\nu^{2} ) / 3 Copy content Toggle raw display
β2\beta_{2}== (4ν7+ν526ν3+11ν)/3 ( -4\nu^{7} + \nu^{5} - 26\nu^{3} + 11\nu ) / 3 Copy content Toggle raw display
β3\beta_{3}== (4ν7+ν5+26ν3+11ν)/3 ( 4\nu^{7} + \nu^{5} + 26\nu^{3} + 11\nu ) / 3 Copy content Toggle raw display
β4\beta_{4}== (4ν7+2ν5+29ν3+10ν)/3 ( 4\nu^{7} + 2\nu^{5} + 29\nu^{3} + 10\nu ) / 3 Copy content Toggle raw display
β5\beta_{5}== (4ν7+2ν529ν3+10ν)/3 ( -4\nu^{7} + 2\nu^{5} - 29\nu^{3} + 10\nu ) / 3 Copy content Toggle raw display
β6\beta_{6}== (8ν4+28)/3 ( 8\nu^{4} + 28 ) / 3 Copy content Toggle raw display
β7\beta_{7}== 4ν624ν2 -4\nu^{6} - 24\nu^{2} Copy content Toggle raw display
ν\nu== (β5β4+2β3+2β2)/8 ( -\beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta_{2} ) / 8 Copy content Toggle raw display
ν2\nu^{2}== (β7+12β1)/8 ( \beta_{7} + 12\beta_1 ) / 8 Copy content Toggle raw display
ν3\nu^{3}== (β5+β4β3+β2)/2 ( -\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (3β628)/8 ( 3\beta_{6} - 28 ) / 8 Copy content Toggle raw display
ν5\nu^{5}== (11β5+11β410β310β2)/8 ( 11\beta_{5} + 11\beta_{4} - 10\beta_{3} - 10\beta_{2} ) / 8 Copy content Toggle raw display
ν6\nu^{6}== β79β1 -\beta_{7} - 9\beta_1 Copy content Toggle raw display
ν7\nu^{7}== (26β526β4+29β329β2)/8 ( 26\beta_{5} - 26\beta_{4} + 29\beta_{3} - 29\beta_{2} ) / 8 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/960Z)×\left(\mathbb{Z}/960\mathbb{Z}\right)^\times.

nn 511511 577577 641641 901901
χ(n)\chi(n) 11 1-1 1-1 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.

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}
449.1
−0.437016 + 0.437016i
−0.437016 0.437016i
−1.14412 + 1.14412i
−1.14412 1.14412i
1.14412 + 1.14412i
1.14412 1.14412i
0.437016 + 0.437016i
0.437016 0.437016i
0 −2.99535 0.166925i 0 5.00000i 0 6.65841i 0 8.94427 + 1.00000i 0
449.2 0 −2.99535 + 0.166925i 0 5.00000i 0 6.65841i 0 8.94427 1.00000i 0
449.3 0 −0.166925 2.99535i 0 5.00000i 0 12.3153i 0 −8.94427 + 1.00000i 0
449.4 0 −0.166925 + 2.99535i 0 5.00000i 0 12.3153i 0 −8.94427 1.00000i 0
449.5 0 0.166925 2.99535i 0 5.00000i 0 12.3153i 0 −8.94427 1.00000i 0
449.6 0 0.166925 + 2.99535i 0 5.00000i 0 12.3153i 0 −8.94427 + 1.00000i 0
449.7 0 2.99535 0.166925i 0 5.00000i 0 6.65841i 0 8.94427 1.00000i 0
449.8 0 2.99535 + 0.166925i 0 5.00000i 0 6.65841i 0 8.94427 + 1.00000i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by Q(5)\Q(\sqrt{-5})
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.c.i 8
3.b odd 2 1 inner 960.3.c.i 8
4.b odd 2 1 inner 960.3.c.i 8
5.b even 2 1 inner 960.3.c.i 8
8.b even 2 1 480.3.c.a 8
8.d odd 2 1 480.3.c.a 8
12.b even 2 1 inner 960.3.c.i 8
15.d odd 2 1 inner 960.3.c.i 8
20.d odd 2 1 CM 960.3.c.i 8
24.f even 2 1 480.3.c.a 8
24.h odd 2 1 480.3.c.a 8
40.e odd 2 1 480.3.c.a 8
40.f even 2 1 480.3.c.a 8
60.h even 2 1 inner 960.3.c.i 8
120.i odd 2 1 480.3.c.a 8
120.m even 2 1 480.3.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.c.a 8 8.b even 2 1
480.3.c.a 8 8.d odd 2 1
480.3.c.a 8 24.f even 2 1
480.3.c.a 8 24.h odd 2 1
480.3.c.a 8 40.e odd 2 1
480.3.c.a 8 40.f even 2 1
480.3.c.a 8 120.i odd 2 1
480.3.c.a 8 120.m even 2 1
960.3.c.i 8 1.a even 1 1 trivial
960.3.c.i 8 3.b odd 2 1 inner
960.3.c.i 8 4.b odd 2 1 inner
960.3.c.i 8 5.b even 2 1 inner
960.3.c.i 8 12.b even 2 1 inner
960.3.c.i 8 15.d odd 2 1 inner
960.3.c.i 8 20.d odd 2 1 CM
960.3.c.i 8 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(960,[χ])S_{3}^{\mathrm{new}}(960, [\chi]):

T74+196T72+6724 T_{7}^{4} + 196T_{7}^{2} + 6724 Copy content Toggle raw display
T17 T_{17} Copy content Toggle raw display
T19 T_{19} Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8158T4+6561 T^{8} - 158T^{4} + 6561 Copy content Toggle raw display
55 (T2+25)4 (T^{2} + 25)^{4} Copy content Toggle raw display
77 (T4+196T2+6724)2 (T^{4} + 196 T^{2} + 6724)^{2} Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 T8 T^{8} Copy content Toggle raw display
1717 T8 T^{8} Copy content Toggle raw display
1919 T8 T^{8} Copy content Toggle raw display
2323 (T42116T2+770884)2 (T^{4} - 2116 T^{2} + 770884)^{2} Copy content Toggle raw display
2929 (T2+2880)4 (T^{2} + 2880)^{4} Copy content Toggle raw display
3131 T8 T^{8} Copy content Toggle raw display
3737 T8 T^{8} Copy content Toggle raw display
4141 (T2+3844)4 (T^{2} + 3844)^{4} Copy content Toggle raw display
4343 (T4+7396T2+4318084)2 (T^{4} + 7396 T^{2} + 4318084)^{2} Copy content Toggle raw display
4747 (T48836T2+19377604)2 (T^{4} - 8836 T^{2} + 19377604)^{2} Copy content Toggle raw display
5353 T8 T^{8} Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 (T211520)4 (T^{2} - 11520)^{4} Copy content Toggle raw display
6767 (T4+17956T2+20052484)2 (T^{4} + 17956 T^{2} + 20052484)^{2} Copy content Toggle raw display
7171 T8 T^{8} Copy content Toggle raw display
7373 T8 T^{8} Copy content Toggle raw display
7979 T8 T^{8} Copy content Toggle raw display
8383 (T427556T2+64032004)2 (T^{4} - 27556 T^{2} + 64032004)^{2} Copy content Toggle raw display
8989 (T2+11520)4 (T^{2} + 11520)^{4} Copy content Toggle raw display
9797 T8 T^{8} Copy content Toggle raw display
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