Properties

Label 155.2.e.d
Level 155155
Weight 22
Character orbit 155.e
Analytic conductor 1.2381.238
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] = [155,2,Mod(36,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 155=531 155 = 5 \cdot 31
Weight: k k == 2 2
Character orbit: [χ][\chi] == 155.e (of order 33, 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: 1.237681231331.23768123133
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: 8.0.42575625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8x7+4x6+x5+9x4x3+4x2+x+1 x^{8} - x^{7} + 4x^{6} + x^{5} + 9x^{4} - x^{3} + 4x^{2} + x + 1 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)[C3]\mathrm{SU}(2)[C_{3}]

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+(β7+β4+1)q2+(β7β4β3β1)q3+(β7+β4β2)q4+β6q5+(β5β21)q6+(β7β5β4+1)q7++(2β7+3β5+9β4++3)q99+O(q100) q + (\beta_{7} + \beta_{4} + 1) q^{2} + (\beta_{7} - \beta_{4} - \beta_{3} - \beta_1) q^{3} + (\beta_{7} + \beta_{4} - \beta_{2}) q^{4} + \beta_{6} q^{5} + ( - \beta_{5} - \beta_{2} - 1) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{7} + 3 \beta_{5} + 9 \beta_{4} + \cdots + 3) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+2q2q32q44q52q6+q7+6q85q9q10+4q11q138q14+2q156q168q17+q18+3q19+q203q218q22+7q99+O(q100) 8 q + 2 q^{2} - q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} + q^{7} + 6 q^{8} - 5 q^{9} - q^{10} + 4 q^{11} - q^{13} - 8 q^{14} + 2 q^{15} - 6 q^{16} - 8 q^{17} + q^{18} + 3 q^{19} + q^{20} - 3 q^{21} - 8 q^{22}+ \cdots - 7 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8x7+4x6+x5+9x4x3+4x2+x+1 x^{8} - x^{7} + 4x^{6} + x^{5} + 9x^{4} - x^{3} + 4x^{2} + x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (3ν7+ν6+10ν5+8ν4+46ν3+5ν2+2ν36)/25 ( 3\nu^{7} + \nu^{6} + 10\nu^{5} + 8\nu^{4} + 46\nu^{3} + 5\nu^{2} + 2\nu - 36 ) / 25 Copy content Toggle raw display
β3\beta_{3}== (4ν72ν6+5ν5+19ν4+8ν310ν239ν3)/25 ( 4\nu^{7} - 2\nu^{6} + 5\nu^{5} + 19\nu^{4} + 8\nu^{3} - 10\nu^{2} - 39\nu - 3 ) / 25 Copy content Toggle raw display
β4\beta_{4}== (6ν73ν6+20ν5+16ν4+62ν3+10ν2+4ν+8)/25 ( 6\nu^{7} - 3\nu^{6} + 20\nu^{5} + 16\nu^{4} + 62\nu^{3} + 10\nu^{2} + 4\nu + 8 ) / 25 Copy content Toggle raw display
β5\beta_{5}== (ν7+2ν65ν5+4ν48ν3+10ν2+ν+3)/5 ( -\nu^{7} + 2\nu^{6} - 5\nu^{5} + 4\nu^{4} - 8\nu^{3} + 10\nu^{2} + \nu + 3 ) / 5 Copy content Toggle raw display
β6\beta_{6}== (8ν714ν6+35ν512ν4+56ν370ν2+22ν21)/25 ( 8\nu^{7} - 14\nu^{6} + 35\nu^{5} - 12\nu^{4} + 56\nu^{3} - 70\nu^{2} + 22\nu - 21 ) / 25 Copy content Toggle raw display
β7\beta_{7}== (3ν72ν6+10ν5+8ν4+23ν3+5ν2+2ν+2)/5 ( 3\nu^{7} - 2\nu^{6} + 10\nu^{5} + 8\nu^{4} + 23\nu^{3} + 5\nu^{2} + 2\nu + 2 ) / 5 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6β5+β4β2+β12 -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta _1 - 2 Copy content Toggle raw display
ν3\nu^{3}== β7+3β4β22 -\beta_{7} + 3\beta_{4} - \beta_{2} - 2 Copy content Toggle raw display
ν4\nu^{4}== 3β6+4β5β35β1 3\beta_{6} + 4\beta_{5} - \beta_{3} - 5\beta_1 Copy content Toggle raw display
ν5\nu^{5}== 4β7+5β6+6β512β44β3+6β212β1+11 4\beta_{7} + 5\beta_{6} + 6\beta_{5} - 12\beta_{4} - 4\beta_{3} + 6\beta_{2} - 12\beta _1 + 11 Copy content Toggle raw display
ν6\nu^{6}== 6β723β4+16β2+28 6\beta_{7} - 23\beta_{4} + 16\beta_{2} + 28 Copy content Toggle raw display
ν7\nu^{7}== 23β629β5+16β3+51β1 -23\beta_{6} - 29\beta_{5} + 16\beta_{3} + 51\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/155Z)×\left(\mathbb{Z}/155\mathbb{Z}\right)^\times.

nn 3232 9696
χ(n)\chi(n) 11 β6\beta_{6}

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}
36.1
0.368820 + 0.638815i
1.04765 + 1.81458i
−0.677837 1.17405i
−0.238630 0.413319i
0.368820 0.638815i
1.04765 1.81458i
−0.677837 + 1.17405i
−0.238630 + 0.413319i
−1.35567 −0.440197 + 0.762443i −0.162147 −0.500000 0.866025i 0.596764 1.03362i 1.77460 3.07370i 2.93117 1.11245 + 1.92683i 0.677837 + 1.17405i
36.2 −0.477260 1.35666 2.34981i −1.77222 −0.500000 0.866025i −0.647481 + 1.12147i 0.0911485 0.157874i 1.80033 −2.18107 3.77773i 0.238630 + 0.413319i
36.3 0.737640 −1.48685 + 2.57531i −1.45589 −0.500000 0.866025i −1.09676 + 1.89965i −0.965584 + 1.67244i −2.54920 −2.92147 5.06014i −0.368820 0.638815i
36.4 2.09529 0.0703870 0.121914i 2.39026 −0.500000 0.866025i 0.147481 0.255445i −0.400166 + 0.693107i 0.817703 1.49009 + 2.58091i −1.04765 1.81458i
56.1 −1.35567 −0.440197 0.762443i −0.162147 −0.500000 + 0.866025i 0.596764 + 1.03362i 1.77460 + 3.07370i 2.93117 1.11245 1.92683i 0.677837 1.17405i
56.2 −0.477260 1.35666 + 2.34981i −1.77222 −0.500000 + 0.866025i −0.647481 1.12147i 0.0911485 + 0.157874i 1.80033 −2.18107 + 3.77773i 0.238630 0.413319i
56.3 0.737640 −1.48685 2.57531i −1.45589 −0.500000 + 0.866025i −1.09676 1.89965i −0.965584 1.67244i −2.54920 −2.92147 + 5.06014i −0.368820 + 0.638815i
56.4 2.09529 0.0703870 + 0.121914i 2.39026 −0.500000 + 0.866025i 0.147481 + 0.255445i −0.400166 0.693107i 0.817703 1.49009 2.58091i −1.04765 + 1.81458i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.2.e.d 8
5.b even 2 1 775.2.e.f 8
5.c odd 4 2 775.2.o.f 16
31.c even 3 1 inner 155.2.e.d 8
31.c even 3 1 4805.2.a.o 4
31.e odd 6 1 4805.2.a.m 4
155.j even 6 1 775.2.e.f 8
155.o odd 12 2 775.2.o.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.e.d 8 1.a even 1 1 trivial
155.2.e.d 8 31.c even 3 1 inner
775.2.e.f 8 5.b even 2 1
775.2.e.f 8 155.j even 6 1
775.2.o.f 16 5.c odd 4 2
775.2.o.f 16 155.o odd 12 2
4805.2.a.m 4 31.e odd 6 1
4805.2.a.o 4 31.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T24T233T22+T2+1 T_{2}^{4} - T_{2}^{3} - 3T_{2}^{2} + T_{2} + 1 acting on S2new(155,[χ])S_{2}^{\mathrm{new}}(155, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4T33T2++1)2 (T^{4} - T^{3} - 3 T^{2} + \cdots + 1)^{2} Copy content Toggle raw display
33 T8+T7+9T6++1 T^{8} + T^{7} + 9 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 T8T7+9T6++1 T^{8} - T^{7} + 9 T^{6} + \cdots + 1 Copy content Toggle raw display
1111 T84T7++6241 T^{8} - 4 T^{7} + \cdots + 6241 Copy content Toggle raw display
1313 T8+T7+11T6++1 T^{8} + T^{7} + 11 T^{6} + \cdots + 1 Copy content Toggle raw display
1717 T8+8T7++6241 T^{8} + 8 T^{7} + \cdots + 6241 Copy content Toggle raw display
1919 T83T7++841 T^{8} - 3 T^{7} + \cdots + 841 Copy content Toggle raw display
2323 (T419T3++211)2 (T^{4} - 19 T^{3} + \cdots + 211)^{2} Copy content Toggle raw display
2929 (T419T3+1421)2 (T^{4} - 19 T^{3} + \cdots - 1421)^{2} Copy content Toggle raw display
3131 T8+5T7++923521 T^{8} + 5 T^{7} + \cdots + 923521 Copy content Toggle raw display
3737 T86T7++546121 T^{8} - 6 T^{7} + \cdots + 546121 Copy content Toggle raw display
4141 T8+4T7++2401 T^{8} + 4 T^{7} + \cdots + 2401 Copy content Toggle raw display
4343 T8+11T7++121 T^{8} + 11 T^{7} + \cdots + 121 Copy content Toggle raw display
4747 (T4+4T3++271)2 (T^{4} + 4 T^{3} + \cdots + 271)^{2} Copy content Toggle raw display
5353 T8+7T7++4289041 T^{8} + 7 T^{7} + \cdots + 4289041 Copy content Toggle raw display
5959 (T4+9T3++1681)2 (T^{4} + 9 T^{3} + \cdots + 1681)^{2} Copy content Toggle raw display
6161 (T4+5T3++589)2 (T^{4} + 5 T^{3} + \cdots + 589)^{2} Copy content Toggle raw display
6767 T89T7++6561 T^{8} - 9 T^{7} + \cdots + 6561 Copy content Toggle raw display
7171 T8+12T7++183358681 T^{8} + 12 T^{7} + \cdots + 183358681 Copy content Toggle raw display
7373 T8+4T7++430336 T^{8} + 4 T^{7} + \cdots + 430336 Copy content Toggle raw display
7979 T814T7++11566801 T^{8} - 14 T^{7} + \cdots + 11566801 Copy content Toggle raw display
8383 T8+18T7++495018001 T^{8} + 18 T^{7} + \cdots + 495018001 Copy content Toggle raw display
8989 (T4+T338T2++121)2 (T^{4} + T^{3} - 38 T^{2} + \cdots + 121)^{2} Copy content Toggle raw display
9797 (T47T3++121)2 (T^{4} - 7 T^{3} + \cdots + 121)^{2} Copy content Toggle raw display
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