Properties

Label 1520.2.bq.p
Level 15201520
Weight 22
Character orbit 1520.bq
Analytic conductor 12.13712.137
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(31,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1520=24519 1520 = 2^{4} \cdot 5 \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1520.bq (of order 66, degree 22, 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: 12.137261107212.1372611072
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: 8.0.1121513121.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x83x7+4x64x5+8x4+16x3+18x2+28x+13 x^{8} - 3x^{7} + 4x^{6} - 4x^{5} + 8x^{4} + 16x^{3} + 18x^{2} + 28x + 13 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
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+(β5β2)q3+(β61)q5+(2β7+β5β4+1)q7+(2β7β6+2β5+1)q9+(2β7+2β6+β5+2)q11++(7β6+β53β4+12)q99+O(q100) q + ( - \beta_{5} - \beta_{2}) q^{3} + (\beta_{6} - 1) q^{5} + (2 \beta_{7} + \beta_{5} - \beta_{4} + \cdots - 1) q^{7} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots - 1) q^{9} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 2) q^{11}+ \cdots + (7 \beta_{6} + \beta_{5} - 3 \beta_{4} + \cdots - 12) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q4q52q915q133q17+3q1912q21+12q234q2518q2712q29+42q3112q33+6q359q413q43+4q45+18q4744q49+57q99+O(q100) 8 q - 4 q^{5} - 2 q^{9} - 15 q^{13} - 3 q^{17} + 3 q^{19} - 12 q^{21} + 12 q^{23} - 4 q^{25} - 18 q^{27} - 12 q^{29} + 42 q^{31} - 12 q^{33} + 6 q^{35} - 9 q^{41} - 3 q^{43} + 4 q^{45} + 18 q^{47} - 44 q^{49}+ \cdots - 57 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x83x7+4x64x5+8x4+16x3+18x2+28x+13 x^{8} - 3x^{7} + 4x^{6} - 4x^{5} + 8x^{4} + 16x^{3} + 18x^{2} + 28x + 13 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν72ν6+2ν52ν4+6ν3+22ν2+40ν+41)/27 ( \nu^{7} - 2\nu^{6} + 2\nu^{5} - 2\nu^{4} + 6\nu^{3} + 22\nu^{2} + 40\nu + 41 ) / 27 Copy content Toggle raw display
β3\beta_{3}== (7ν7+26ν644ν5+53ν499ν313ν2100ν86)/27 ( -7\nu^{7} + 26\nu^{6} - 44\nu^{5} + 53\nu^{4} - 99\nu^{3} - 13\nu^{2} - 100\nu - 86 ) / 27 Copy content Toggle raw display
β4\beta_{4}== (3ν7+11ν620ν5+29ν450ν314ν248ν58)/9 ( -3\nu^{7} + 11\nu^{6} - 20\nu^{5} + 29\nu^{4} - 50\nu^{3} - 14\nu^{2} - 48\nu - 58 ) / 9 Copy content Toggle raw display
β5\beta_{5}== (11ν7+43ν679ν5+97ν4150ν356ν2107ν169)/27 ( -11\nu^{7} + 43\nu^{6} - 79\nu^{5} + 97\nu^{4} - 150\nu^{3} - 56\nu^{2} - 107\nu - 169 ) / 27 Copy content Toggle raw display
β6\beta_{6}== (13ν750ν6+95ν5131ν4+201ν3+58ν2+178ν+257)/27 ( 13\nu^{7} - 50\nu^{6} + 95\nu^{5} - 131\nu^{4} + 201\nu^{3} + 58\nu^{2} + 178\nu + 257 ) / 27 Copy content Toggle raw display
β7\beta_{7}== (35ν7130ν6+229ν5292ν4+477ν3+227ν2+437ν+610)/27 ( 35\nu^{7} - 130\nu^{6} + 229\nu^{5} - 292\nu^{4} + 477\nu^{3} + 227\nu^{2} + 437\nu + 610 ) / 27 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6β5β4+β3+β1 -\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 Copy content Toggle raw display
ν3\nu^{3}== β73β64β54β4+β3+β2+β1+2 -\beta_{7} - 3\beta_{6} - 4\beta_{5} - 4\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 2 Copy content Toggle raw display
ν4\nu^{4}== 3β76β611β57β4+β3+6β2+2β1+5 -3\beta_{7} - 6\beta_{6} - 11\beta_{5} - 7\beta_{4} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 5 Copy content Toggle raw display
ν5\nu^{5}== 8β76β623β511β4+2β3+20β23β11 -8\beta_{7} - 6\beta_{6} - 23\beta_{5} - 11\beta_{4} + 2\beta_{3} + 20\beta_{2} - 3\beta _1 - 1 Copy content Toggle raw display
ν6\nu^{6}== 15β7+2β629β512β43β3+51β236β126 -15\beta_{7} + 2\beta_{6} - 29\beta_{5} - 12\beta_{4} - 3\beta_{3} + 51\beta_{2} - 36\beta _1 - 26 Copy content Toggle raw display
ν7\nu^{7}== 14β7+44β6+12β5+30β436β3+95β2130β193 -14\beta_{7} + 44\beta_{6} + 12\beta_{5} + 30\beta_{4} - 36\beta_{3} + 95\beta_{2} - 130\beta _1 - 93 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1520Z)×\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times.

nn 191191 401401 11411141 12171217
χ(n)\chi(n) 1-1 β6\beta_{6} 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.

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}
31.1
−0.212319 + 1.23705i
−0.731824 0.0280585i
2.26669 + 1.18132i
0.177457 1.52428i
−0.212319 1.23705i
−0.731824 + 0.0280585i
2.26669 1.18132i
0.177457 + 1.52428i
0 −1.17747 2.03944i 0 −0.500000 0.866025i 0 3.23155i 0 −1.27288 + 2.20470i 0
31.2 0 −0.341612 0.591690i 0 −0.500000 0.866025i 0 4.79806i 0 1.26660 2.19382i 0
31.3 0 0.110293 + 0.191033i 0 −0.500000 0.866025i 0 1.12206i 0 1.47567 2.55594i 0
31.4 0 1.40879 + 2.44010i 0 −0.500000 0.866025i 0 3.90855i 0 −2.46939 + 4.27711i 0
1471.1 0 −1.17747 + 2.03944i 0 −0.500000 + 0.866025i 0 3.23155i 0 −1.27288 2.20470i 0
1471.2 0 −0.341612 + 0.591690i 0 −0.500000 + 0.866025i 0 4.79806i 0 1.26660 + 2.19382i 0
1471.3 0 0.110293 0.191033i 0 −0.500000 + 0.866025i 0 1.12206i 0 1.47567 + 2.55594i 0
1471.4 0 1.40879 2.44010i 0 −0.500000 + 0.866025i 0 3.90855i 0 −2.46939 4.27711i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.bq.p yes 8
4.b odd 2 1 1520.2.bq.o 8
19.d odd 6 1 1520.2.bq.o 8
76.f even 6 1 inner 1520.2.bq.p yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.bq.o 8 4.b odd 2 1
1520.2.bq.o 8 19.d odd 6 1
1520.2.bq.p yes 8 1.a even 1 1 trivial
1520.2.bq.p yes 8 76.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1520,[χ])S_{2}^{\mathrm{new}}(1520, [\chi]):

T38+7T36+6T35+48T34+21T33+16T323T3+1 T_{3}^{8} + 7T_{3}^{6} + 6T_{3}^{5} + 48T_{3}^{4} + 21T_{3}^{3} + 16T_{3}^{2} - 3T_{3} + 1 Copy content Toggle raw display
T78+50T76+813T74+4619T72+4624 T_{7}^{8} + 50T_{7}^{6} + 813T_{7}^{4} + 4619T_{7}^{2} + 4624 Copy content Toggle raw display
T118+50T116+327T114+569T112+169 T_{11}^{8} + 50T_{11}^{6} + 327T_{11}^{4} + 569T_{11}^{2} + 169 Copy content Toggle raw display
T138+15T137+88T136+195T13596T134897T133+547T132+5520T13+6400 T_{13}^{8} + 15T_{13}^{7} + 88T_{13}^{6} + 195T_{13}^{5} - 96T_{13}^{4} - 897T_{13}^{3} + 547T_{13}^{2} + 5520T_{13} + 6400 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8+7T6++1 T^{8} + 7 T^{6} + \cdots + 1 Copy content Toggle raw display
55 (T2+T+1)4 (T^{2} + T + 1)^{4} Copy content Toggle raw display
77 T8+50T6++4624 T^{8} + 50 T^{6} + \cdots + 4624 Copy content Toggle raw display
1111 T8+50T6++169 T^{8} + 50 T^{6} + \cdots + 169 Copy content Toggle raw display
1313 T8+15T7++6400 T^{8} + 15 T^{7} + \cdots + 6400 Copy content Toggle raw display
1717 T8+3T7++8100 T^{8} + 3 T^{7} + \cdots + 8100 Copy content Toggle raw display
1919 T83T7++130321 T^{8} - 3 T^{7} + \cdots + 130321 Copy content Toggle raw display
2323 T812T7++28900 T^{8} - 12 T^{7} + \cdots + 28900 Copy content Toggle raw display
2929 T8+12T7++5184 T^{8} + 12 T^{7} + \cdots + 5184 Copy content Toggle raw display
3131 (T421T3++252)2 (T^{4} - 21 T^{3} + \cdots + 252)^{2} Copy content Toggle raw display
3737 T8+154T6++64 T^{8} + 154 T^{6} + \cdots + 64 Copy content Toggle raw display
4141 T8+9T7++164025 T^{8} + 9 T^{7} + \cdots + 164025 Copy content Toggle raw display
4343 T8+3T7++396900 T^{8} + 3 T^{7} + \cdots + 396900 Copy content Toggle raw display
4747 T818T7++4 T^{8} - 18 T^{7} + \cdots + 4 Copy content Toggle raw display
5353 T815T7++2676496 T^{8} - 15 T^{7} + \cdots + 2676496 Copy content Toggle raw display
5959 T815T7++337561 T^{8} - 15 T^{7} + \cdots + 337561 Copy content Toggle raw display
6161 T825T7++62500 T^{8} - 25 T^{7} + \cdots + 62500 Copy content Toggle raw display
6767 T827T7++279023616 T^{8} - 27 T^{7} + \cdots + 279023616 Copy content Toggle raw display
7171 T8+21T7++321269776 T^{8} + 21 T^{7} + \cdots + 321269776 Copy content Toggle raw display
7373 T821T7++3143529 T^{8} - 21 T^{7} + \cdots + 3143529 Copy content Toggle raw display
7979 T8+6T7++1327104 T^{8} + 6 T^{7} + \cdots + 1327104 Copy content Toggle raw display
8383 T8+414T6++21986721 T^{8} + 414 T^{6} + \cdots + 21986721 Copy content Toggle raw display
8989 T830T7++82944 T^{8} - 30 T^{7} + \cdots + 82944 Copy content Toggle raw display
9797 T818T7++52780225 T^{8} - 18 T^{7} + \cdots + 52780225 Copy content Toggle raw display
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