Properties

Label 308.2.n.b
Level 308308
Weight 22
Character orbit 308.n
Analytic conductor 2.4592.459
Analytic rank 00
Dimension 44
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,2,Mod(219,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.219");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 308=22711 308 = 2^{2} \cdot 7 \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 308.n (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: 2.459392382262.45939238226
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(3,7)\Q(\sqrt{-3}, \sqrt{-7})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x3x22x+4 x^{4} - x^{3} - x^{2} - 2x + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
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,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β3q2+(β22β1+1)q3+(β2+β1+1)q4+(β3β2β13)q6+(3β2+2)q7+(β3β2β1+3)q8++(8β38β28β14)q99+O(q100) q + \beta_{3} q^{2} + ( - \beta_{2} - 2 \beta_1 + 1) q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (\beta_{3} - \beta_{2} - \beta_1 - 3) q^{6} + ( - 3 \beta_{2} + 2) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{8}+ \cdots + (8 \beta_{3} - 8 \beta_{2} - 8 \beta_1 - 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+q2+3q414q6+2q7+10q8+8q94q117q12+5q14q164q1818q227q24+10q257q2612q28+11q32+14q3328q34+32q99+O(q100) 4 q + q^{2} + 3 q^{4} - 14 q^{6} + 2 q^{7} + 10 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + 5 q^{14} - q^{16} - 4 q^{18} - 18 q^{22} - 7 q^{24} + 10 q^{25} - 7 q^{26} - 12 q^{28} + 11 q^{32} + 14 q^{33} - 28 q^{34}+ \cdots - 32 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x3x22x+4 x^{4} - x^{3} - x^{2} - 2x + 4 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3+ν2ν2)/2 ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν3+ν2+ν+2)/2 ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+β2 \beta_{3} + \beta_{2} Copy content Toggle raw display
ν3\nu^{3}== β3+β2+β1+2 -\beta_{3} + \beta_{2} + \beta _1 + 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/308Z)×\left(\mathbb{Z}/308\mathbb{Z}\right)^\times.

nn 4545 5757 155155
χ(n)\chi(n) β2-\beta_{2} 1-1 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}
219.1
−0.895644 + 1.09445i
1.39564 0.228425i
−0.895644 1.09445i
1.39564 + 0.228425i
−0.895644 1.09445i 2.29129 1.32288i −0.395644 + 1.96048i 0 −3.50000 1.32288i 0.500000 + 2.59808i 2.50000 1.32288i 2.00000 3.46410i 0
219.2 1.39564 + 0.228425i −2.29129 + 1.32288i 1.89564 + 0.637600i 0 −3.50000 + 1.32288i 0.500000 + 2.59808i 2.50000 + 1.32288i 2.00000 3.46410i 0
263.1 −0.895644 + 1.09445i 2.29129 + 1.32288i −0.395644 1.96048i 0 −3.50000 + 1.32288i 0.500000 2.59808i 2.50000 + 1.32288i 2.00000 + 3.46410i 0
263.2 1.39564 0.228425i −2.29129 1.32288i 1.89564 0.637600i 0 −3.50000 1.32288i 0.500000 2.59808i 2.50000 1.32288i 2.00000 + 3.46410i 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
44.c even 2 1 inner
308.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.n.b yes 4
4.b odd 2 1 308.2.n.a 4
7.c even 3 1 inner 308.2.n.b yes 4
11.b odd 2 1 308.2.n.a 4
28.g odd 6 1 308.2.n.a 4
44.c even 2 1 inner 308.2.n.b yes 4
77.h odd 6 1 308.2.n.a 4
308.n even 6 1 inner 308.2.n.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.n.a 4 4.b odd 2 1
308.2.n.a 4 11.b odd 2 1
308.2.n.a 4 28.g odd 6 1
308.2.n.a 4 77.h odd 6 1
308.2.n.b yes 4 1.a even 1 1 trivial
308.2.n.b yes 4 7.c even 3 1 inner
308.2.n.b yes 4 44.c even 2 1 inner
308.2.n.b yes 4 308.n 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(308,[χ])S_{2}^{\mathrm{new}}(308, [\chi]):

T347T32+49 T_{3}^{4} - 7T_{3}^{2} + 49 Copy content Toggle raw display
T43+2 T_{43} + 2 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4T3T2++4 T^{4} - T^{3} - T^{2} + \cdots + 4 Copy content Toggle raw display
33 T47T2+49 T^{4} - 7T^{2} + 49 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 (T2T+7)2 (T^{2} - T + 7)^{2} Copy content Toggle raw display
1111 T4+4T3++121 T^{4} + 4 T^{3} + \cdots + 121 Copy content Toggle raw display
1313 (T2+7)2 (T^{2} + 7)^{2} Copy content Toggle raw display
1717 T428T2+784 T^{4} - 28T^{2} + 784 Copy content Toggle raw display
1919 T4 T^{4} Copy content Toggle raw display
2323 T428T2+784 T^{4} - 28T^{2} + 784 Copy content Toggle raw display
2929 (T2+63)2 (T^{2} + 63)^{2} Copy content Toggle raw display
3131 T4112T2+12544 T^{4} - 112 T^{2} + 12544 Copy content Toggle raw display
3737 (T28T+64)2 (T^{2} - 8 T + 64)^{2} Copy content Toggle raw display
4141 T4 T^{4} Copy content Toggle raw display
4343 (T+2)4 (T + 2)^{4} Copy content Toggle raw display
4747 T4 T^{4} Copy content Toggle raw display
5353 (T24T+16)2 (T^{2} - 4 T + 16)^{2} Copy content Toggle raw display
5959 T47T2+49 T^{4} - 7T^{2} + 49 Copy content Toggle raw display
6161 T47T2+49 T^{4} - 7T^{2} + 49 Copy content Toggle raw display
6767 T47T2+49 T^{4} - 7T^{2} + 49 Copy content Toggle raw display
7171 (T2+28)2 (T^{2} + 28)^{2} Copy content Toggle raw display
7373 T4112T2+12544 T^{4} - 112 T^{2} + 12544 Copy content Toggle raw display
7979 (T2+13T+169)2 (T^{2} + 13 T + 169)^{2} Copy content Toggle raw display
8383 T4 T^{4} Copy content Toggle raw display
8989 (T214T+196)2 (T^{2} - 14 T + 196)^{2} Copy content Toggle raw display
9797 (T7)4 (T - 7)^{4} Copy content Toggle raw display
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