Properties

Label 1170.4.a.x
Level 11701170
Weight 44
Character orbit 1170.a
Self dual yes
Analytic conductor 69.03269.032
Analytic rank 11
Dimension 22
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,4,Mod(1,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 1170=232513 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1170.a (trivial)

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: yes
Analytic conductor: 69.032234706769.0322347067
Analytic rank: 11
Dimension: 22
Coefficient field: Q(129)\Q(\sqrt{129})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x32 x^{2} - x - 32 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 390)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(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 β=2129\beta = 2\sqrt{129}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+2q2+4q45q5+(β+2)q7+8q810q10+(2β24)q11+13q13+(2β+4)q14+16q16+βq17+(3β34)q1920q20++(8β+354)q98+O(q100) q + 2 q^{2} + 4 q^{4} - 5 q^{5} + (\beta + 2) q^{7} + 8 q^{8} - 10 q^{10} + ( - 2 \beta - 24) q^{11} + 13 q^{13} + (2 \beta + 4) q^{14} + 16 q^{16} + \beta q^{17} + ( - 3 \beta - 34) q^{19} - 20 q^{20}+ \cdots + (8 \beta + 354) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+4q2+8q410q5+4q7+16q820q1048q11+26q13+8q14+32q1668q1940q2096q22+60q23+50q25+52q26+16q28120q29++708q98+O(q100) 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 4 q^{7} + 16 q^{8} - 20 q^{10} - 48 q^{11} + 26 q^{13} + 8 q^{14} + 32 q^{16} - 68 q^{19} - 40 q^{20} - 96 q^{22} + 60 q^{23} + 50 q^{25} + 52 q^{26} + 16 q^{28} - 120 q^{29}+ \cdots + 708 q^{98}+O(q^{100}) 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.

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}
1.1
−5.17891
6.17891
2.00000 0 4.00000 −5.00000 0 −20.7156 8.00000 0 −10.0000
1.2 2.00000 0 4.00000 −5.00000 0 24.7156 8.00000 0 −10.0000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 1 -1
55 +1 +1
1313 1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.4.a.x 2
3.b odd 2 1 390.4.a.m 2
15.d odd 2 1 1950.4.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.4.a.m 2 3.b odd 2 1
1170.4.a.x 2 1.a even 1 1 trivial
1950.4.a.y 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(1170))S_{4}^{\mathrm{new}}(\Gamma_0(1170)):

T724T7512 T_{7}^{2} - 4T_{7} - 512 Copy content Toggle raw display
T112+48T111488 T_{11}^{2} + 48T_{11} - 1488 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2)2 (T - 2)^{2} Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 (T+5)2 (T + 5)^{2} Copy content Toggle raw display
77 T24T512 T^{2} - 4T - 512 Copy content Toggle raw display
1111 T2+48T1488 T^{2} + 48T - 1488 Copy content Toggle raw display
1313 (T13)2 (T - 13)^{2} Copy content Toggle raw display
1717 T2516 T^{2} - 516 Copy content Toggle raw display
1919 T2+68T3488 T^{2} + 68T - 3488 Copy content Toggle raw display
2323 T260T12000 T^{2} - 60T - 12000 Copy content Toggle raw display
2929 T2+120T+3084 T^{2} + 120T + 3084 Copy content Toggle raw display
3131 T264T7232 T^{2} - 64T - 7232 Copy content Toggle raw display
3737 T2+476T+38068 T^{2} + 476T + 38068 Copy content Toggle raw display
4141 T2+444T+47220 T^{2} + 444T + 47220 Copy content Toggle raw display
4343 T2+320T+23536 T^{2} + 320T + 23536 Copy content Toggle raw display
4747 T2+192T197184 T^{2} + 192T - 197184 Copy content Toggle raw display
5353 T2+684T+83940 T^{2} + 684T + 83940 Copy content Toggle raw display
5959 T2+1536T+587760 T^{2} + 1536 T + 587760 Copy content Toggle raw display
6161 T21084T+285508 T^{2} - 1084 T + 285508 Copy content Toggle raw display
6767 T2+80T165584 T^{2} + 80T - 165584 Copy content Toggle raw display
7171 T2+648T144768 T^{2} + 648T - 144768 Copy content Toggle raw display
7373 T2232T618644 T^{2} - 232T - 618644 Copy content Toggle raw display
7979 T21000T+256 T^{2} - 1000T + 256 Copy content Toggle raw display
8383 T2+1440T+169584 T^{2} + 1440 T + 169584 Copy content Toggle raw display
8989 T2180T1496556 T^{2} - 180 T - 1496556 Copy content Toggle raw display
9797 T2136T144500 T^{2} - 136T - 144500 Copy content Toggle raw display
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