Properties

Label 490.2.i.c
Level 490490
Weight 22
Character orbit 490.i
Analytic conductor 3.9133.913
Analytic rank 00
Dimension 88
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,2,Mod(79,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 490=2572 490 = 2 \cdot 5 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 490.i (of order 66, degree 22, 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: 3.912669699043.91266969904
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ24)\Q(\zeta_{24})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8x4+1 x^{8} - x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 32 3^{2}
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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+(β4β1)q2+(β6β5)q3+(β2+1)q4+(β7β4++β1)q5+(β7β5+β3)q6+β4q8++(6β76β5+6β3)q99+O(q100) q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{6} - \beta_{5}) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{7} - \beta_{4} + \cdots + \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{6} + \beta_{4} q^{8}+ \cdots + (6 \beta_{7} - 6 \beta_{5} + 6 \beta_{3}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+4q44q5+12q94q10+24q154q16+16q198q20+8q2616q2912q3016q31+16q34+24q3624q39+4q4048q41+12q45++4q95+O(q100) 8 q + 4 q^{4} - 4 q^{5} + 12 q^{9} - 4 q^{10} + 24 q^{15} - 4 q^{16} + 16 q^{19} - 8 q^{20} + 8 q^{26} - 16 q^{29} - 12 q^{30} - 16 q^{31} + 16 q^{34} + 24 q^{36} - 24 q^{39} + 4 q^{40} - 48 q^{41} + 12 q^{45}+ \cdots + 4 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring

β1\beta_{1}== ζ242 \zeta_{24}^{2} Copy content Toggle raw display
β2\beta_{2}== ζ244 \zeta_{24}^{4} Copy content Toggle raw display
β3\beta_{3}== ζ245+ζ24 \zeta_{24}^{5} + \zeta_{24} Copy content Toggle raw display
β4\beta_{4}== ζ246 \zeta_{24}^{6} Copy content Toggle raw display
β5\beta_{5}== ζ247+ζ243 \zeta_{24}^{7} + \zeta_{24}^{3} Copy content Toggle raw display
β6\beta_{6}== ζ245+2ζ24 -\zeta_{24}^{5} + 2\zeta_{24} Copy content Toggle raw display
β7\beta_{7}== ζ247+2ζ243 -\zeta_{24}^{7} + 2\zeta_{24}^{3} Copy content Toggle raw display
ζ24\zeta_{24}== (β6+β3)/3 ( \beta_{6} + \beta_{3} ) / 3 Copy content Toggle raw display
ζ242\zeta_{24}^{2}== β1 \beta_1 Copy content Toggle raw display
ζ243\zeta_{24}^{3}== (β7+β5)/3 ( \beta_{7} + \beta_{5} ) / 3 Copy content Toggle raw display
ζ244\zeta_{24}^{4}== β2 \beta_{2} Copy content Toggle raw display
ζ245\zeta_{24}^{5}== (β6+2β3)/3 ( -\beta_{6} + 2\beta_{3} ) / 3 Copy content Toggle raw display
ζ246\zeta_{24}^{6}== β4 \beta_{4} Copy content Toggle raw display
ζ247\zeta_{24}^{7}== (β7+2β5)/3 ( -\beta_{7} + 2\beta_{5} ) / 3 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/490Z)×\left(\mathbb{Z}/490\mathbb{Z}\right)^\times.

nn 101101 197197
χ(n)\chi(n) β2-\beta_{2} 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}
79.1
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.866025 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i −1.30701 + 1.81431i 2.44949 0 1.00000i 1.50000 2.59808i 2.03906 0.917738i
79.2 −0.866025 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i 2.03906 + 0.917738i −2.44949 0 1.00000i 1.50000 2.59808i −1.30701 1.81431i
79.3 0.866025 + 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i −1.81431 1.30701i −2.44949 0 1.00000i 1.50000 2.59808i −0.917738 2.03906i
79.4 0.866025 + 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i −0.917738 + 2.03906i 2.44949 0 1.00000i 1.50000 2.59808i −1.81431 + 1.30701i
459.1 −0.866025 + 0.500000i −2.12132 1.22474i 0.500000 0.866025i −1.30701 1.81431i 2.44949 0 1.00000i 1.50000 + 2.59808i 2.03906 + 0.917738i
459.2 −0.866025 + 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i 2.03906 0.917738i −2.44949 0 1.00000i 1.50000 + 2.59808i −1.30701 + 1.81431i
459.3 0.866025 0.500000i −2.12132 1.22474i 0.500000 0.866025i −1.81431 + 1.30701i −2.44949 0 1.00000i 1.50000 + 2.59808i −0.917738 + 2.03906i
459.4 0.866025 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i −0.917738 2.03906i 2.44949 0 1.00000i 1.50000 + 2.59808i −1.81431 1.30701i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.i.c 8
5.b even 2 1 inner 490.2.i.c 8
7.b odd 2 1 490.2.i.f 8
7.c even 3 1 70.2.c.a 4
7.c even 3 1 inner 490.2.i.c 8
7.d odd 6 1 490.2.c.e 4
7.d odd 6 1 490.2.i.f 8
21.h odd 6 1 630.2.g.g 4
28.g odd 6 1 560.2.g.e 4
35.c odd 2 1 490.2.i.f 8
35.i odd 6 1 490.2.c.e 4
35.i odd 6 1 490.2.i.f 8
35.j even 6 1 70.2.c.a 4
35.j even 6 1 inner 490.2.i.c 8
35.k even 12 1 2450.2.a.bl 2
35.k even 12 1 2450.2.a.bq 2
35.l odd 12 1 350.2.a.g 2
35.l odd 12 1 350.2.a.h 2
56.k odd 6 1 2240.2.g.i 4
56.p even 6 1 2240.2.g.j 4
84.n even 6 1 5040.2.t.t 4
105.o odd 6 1 630.2.g.g 4
105.x even 12 1 3150.2.a.bs 2
105.x even 12 1 3150.2.a.bt 2
140.p odd 6 1 560.2.g.e 4
140.w even 12 1 2800.2.a.bl 2
140.w even 12 1 2800.2.a.bm 2
280.bf even 6 1 2240.2.g.j 4
280.bi odd 6 1 2240.2.g.i 4
420.ba even 6 1 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 7.c even 3 1
70.2.c.a 4 35.j even 6 1
350.2.a.g 2 35.l odd 12 1
350.2.a.h 2 35.l odd 12 1
490.2.c.e 4 7.d odd 6 1
490.2.c.e 4 35.i odd 6 1
490.2.i.c 8 1.a even 1 1 trivial
490.2.i.c 8 5.b even 2 1 inner
490.2.i.c 8 7.c even 3 1 inner
490.2.i.c 8 35.j even 6 1 inner
490.2.i.f 8 7.b odd 2 1
490.2.i.f 8 7.d odd 6 1
490.2.i.f 8 35.c odd 2 1
490.2.i.f 8 35.i odd 6 1
560.2.g.e 4 28.g odd 6 1
560.2.g.e 4 140.p odd 6 1
630.2.g.g 4 21.h odd 6 1
630.2.g.g 4 105.o odd 6 1
2240.2.g.i 4 56.k odd 6 1
2240.2.g.i 4 280.bi odd 6 1
2240.2.g.j 4 56.p even 6 1
2240.2.g.j 4 280.bf even 6 1
2450.2.a.bl 2 35.k even 12 1
2450.2.a.bq 2 35.k even 12 1
2800.2.a.bl 2 140.w even 12 1
2800.2.a.bm 2 140.w even 12 1
3150.2.a.bs 2 105.x even 12 1
3150.2.a.bt 2 105.x even 12 1
5040.2.t.t 4 84.n even 6 1
5040.2.t.t 4 420.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(490,[χ])S_{2}^{\mathrm{new}}(490, [\chi]):

T346T32+36 T_{3}^{4} - 6T_{3}^{2} + 36 Copy content Toggle raw display
T114+24T112+576 T_{11}^{4} + 24T_{11}^{2} + 576 Copy content Toggle raw display
T1948T193+54T19280T19+100 T_{19}^{4} - 8T_{19}^{3} + 54T_{19}^{2} - 80T_{19} + 100 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4T2+1)2 (T^{4} - T^{2} + 1)^{2} Copy content Toggle raw display
33 (T46T2+36)2 (T^{4} - 6 T^{2} + 36)^{2} Copy content Toggle raw display
55 T8+4T7++625 T^{8} + 4 T^{7} + \cdots + 625 Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 (T4+24T2+576)2 (T^{4} + 24 T^{2} + 576)^{2} Copy content Toggle raw display
1313 (T4+20T2+4)2 (T^{4} + 20 T^{2} + 4)^{2} Copy content Toggle raw display
1717 (T44T2+16)2 (T^{4} - 4 T^{2} + 16)^{2} Copy content Toggle raw display
1919 (T48T3++100)2 (T^{4} - 8 T^{3} + \cdots + 100)^{2} Copy content Toggle raw display
2323 T856T6++160000 T^{8} - 56 T^{6} + \cdots + 160000 Copy content Toggle raw display
2929 (T2+4T20)4 (T^{2} + 4 T - 20)^{4} Copy content Toggle raw display
3131 (T4+8T3+72T2++64)2 (T^{4} + 8 T^{3} + 72 T^{2} + \cdots + 64)^{2} Copy content Toggle raw display
3737 (T44T2+16)2 (T^{4} - 4 T^{2} + 16)^{2} Copy content Toggle raw display
4141 (T2+12T+12)4 (T^{2} + 12 T + 12)^{4} Copy content Toggle raw display
4343 (T4+80T2+64)2 (T^{4} + 80 T^{2} + 64)^{2} Copy content Toggle raw display
4747 T880T6++4096 T^{8} - 80 T^{6} + \cdots + 4096 Copy content Toggle raw display
5353 T8120T6++20736 T^{8} - 120 T^{6} + \cdots + 20736 Copy content Toggle raw display
5959 (T4+8T3++100)2 (T^{4} + 8 T^{3} + \cdots + 100)^{2} Copy content Toggle raw display
6161 (T4+12T3++900)2 (T^{4} + 12 T^{3} + \cdots + 900)^{2} Copy content Toggle raw display
6767 (T464T2+4096)2 (T^{4} - 64 T^{2} + 4096)^{2} Copy content Toggle raw display
7171 (T2+12T+12)4 (T^{2} + 12 T + 12)^{4} Copy content Toggle raw display
7373 T856T6++160000 T^{8} - 56 T^{6} + \cdots + 160000 Copy content Toggle raw display
7979 (T44T3++400)2 (T^{4} - 4 T^{3} + \cdots + 400)^{2} Copy content Toggle raw display
8383 (T2+6)4 (T^{2} + 6)^{4} Copy content Toggle raw display
8989 (T2+10T+100)4 (T^{2} + 10 T + 100)^{4} Copy content Toggle raw display
9797 (T4+264T2+3600)2 (T^{4} + 264 T^{2} + 3600)^{2} Copy content Toggle raw display
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