Properties

Label 120.2.d.a
Level 120120
Weight 22
Character orbit 120.d
Analytic conductor 0.9580.958
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,2,Mod(109,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 120=2335 120 = 2^{3} \cdot 3 \cdot 5
Weight: k k == 2 2
Character orbit: [χ][\chi] == 120.d (of order 22, degree 11, 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: 0.9582048242550.958204824255
Analytic rank: 00
Dimension: 66
Coefficient field: 6.0.839056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6+6x4+8x2+1 x^{6} + 6x^{4} + 8x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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 a basis 1,β1,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β3q2+q3+(β2β1)q4+(β3+β2)q5+β3q6+(β4+β3β1)q7+(β5+β4++β1)q8++(β5β3+β1)q99+O(q100) q + \beta_{3} q^{2} + q^{3} + ( - \beta_{2} - \beta_1) q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + \beta_{3} q^{6} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{8}+ \cdots + (\beta_{5} - \beta_{3} + \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6qq2+6q3+q4q6q8+6q9+q10+q128q1310q14+q16q1813q20+10q22q24+2q2514q26+6q272q28++21q98+O(q100) 6 q - q^{2} + 6 q^{3} + q^{4} - q^{6} - q^{8} + 6 q^{9} + q^{10} + q^{12} - 8 q^{13} - 10 q^{14} + q^{16} - q^{18} - 13 q^{20} + 10 q^{22} - q^{24} + 2 q^{25} - 14 q^{26} + 6 q^{27} - 2 q^{28}+ \cdots + 21 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6+6x4+8x2+1 x^{6} + 6x^{4} + 8x^{2} + 1 : Copy content Toggle raw display

β1\beta_{1}== ν3+4ν \nu^{3} + 4\nu Copy content Toggle raw display
β2\beta_{2}== (ν5+ν4+5ν3+3ν2+3ν1)/2 ( \nu^{5} + \nu^{4} + 5\nu^{3} + 3\nu^{2} + 3\nu - 1 ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν5+ν45ν3+5ν25ν+3)/2 ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 5\nu^{2} - 5\nu + 3 ) / 2 Copy content Toggle raw display
β4\beta_{4}== (ν5+ν47ν3+5ν29ν+3)/2 ( -\nu^{5} + \nu^{4} - 7\nu^{3} + 5\nu^{2} - 9\nu + 3 ) / 2 Copy content Toggle raw display
β5\beta_{5}== (3ν5+ν4+17ν3+5ν2+19ν+3)/2 ( 3\nu^{5} + \nu^{4} + 17\nu^{3} + 5\nu^{2} + 19\nu + 3 ) / 2 Copy content Toggle raw display
ν\nu== (β4β3+β1)/2 ( \beta_{4} - \beta_{3} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β5+β32β2β14)/2 ( \beta_{5} + \beta_{3} - 2\beta_{2} - \beta _1 - 4 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== 2β4+2β3β1 -2\beta_{4} + 2\beta_{3} - \beta_1 Copy content Toggle raw display
ν4\nu^{4}== (4β5+β43β3+10β2+5β1+14)/2 ( -4\beta_{5} + \beta_{4} - 3\beta_{3} + 10\beta_{2} + 5\beta _1 + 14 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (β5+16β417β3+5β1)/2 ( \beta_{5} + 16\beta_{4} - 17\beta_{3} + 5\beta_1 ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/120Z)×\left(\mathbb{Z}/120\mathbb{Z}\right)^\times.

nn 3131 4141 6161 9797
χ(n)\chi(n) 11 11 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}
109.1
1.32132i
1.32132i
2.02852i
2.02852i
0.373087i
0.373087i
−1.34067 0.450129i 1.00000 1.59477 + 1.20695i −0.254102 + 2.22158i −1.34067 0.450129i 2.64265i −1.59477 2.33596i 1.00000 1.34067 2.86402i
109.2 −1.34067 + 0.450129i 1.00000 1.59477 1.20695i −0.254102 2.22158i −1.34067 + 0.450129i 2.64265i −1.59477 + 2.33596i 1.00000 1.34067 + 2.86402i
109.3 −0.321037 1.37729i 1.00000 −1.79387 + 0.884323i 2.11491 + 0.726062i −0.321037 1.37729i 4.05705i 1.79387 + 2.18678i 1.00000 0.321037 3.14594i
109.4 −0.321037 + 1.37729i 1.00000 −1.79387 0.884323i 2.11491 0.726062i −0.321037 + 1.37729i 4.05705i 1.79387 2.18678i 1.00000 0.321037 + 3.14594i
109.5 1.16170 0.806504i 1.00000 0.699104 1.87383i −1.86081 + 1.23992i 1.16170 0.806504i 0.746175i −0.699104 2.74067i 1.00000 −1.16170 + 2.94116i
109.6 1.16170 + 0.806504i 1.00000 0.699104 + 1.87383i −1.86081 1.23992i 1.16170 + 0.806504i 0.746175i −0.699104 + 2.74067i 1.00000 −1.16170 2.94116i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.d.a 6
3.b odd 2 1 360.2.d.f 6
4.b odd 2 1 480.2.d.a 6
5.b even 2 1 120.2.d.b yes 6
5.c odd 4 2 600.2.k.f 12
8.b even 2 1 120.2.d.b yes 6
8.d odd 2 1 480.2.d.b 6
12.b even 2 1 1440.2.d.e 6
15.d odd 2 1 360.2.d.e 6
15.e even 4 2 1800.2.k.u 12
16.e even 4 2 3840.2.f.l 12
16.f odd 4 2 3840.2.f.m 12
20.d odd 2 1 480.2.d.b 6
20.e even 4 2 2400.2.k.f 12
24.f even 2 1 1440.2.d.f 6
24.h odd 2 1 360.2.d.e 6
40.e odd 2 1 480.2.d.a 6
40.f even 2 1 inner 120.2.d.a 6
40.i odd 4 2 600.2.k.f 12
40.k even 4 2 2400.2.k.f 12
60.h even 2 1 1440.2.d.f 6
60.l odd 4 2 7200.2.k.u 12
80.k odd 4 2 3840.2.f.m 12
80.q even 4 2 3840.2.f.l 12
120.i odd 2 1 360.2.d.f 6
120.m even 2 1 1440.2.d.e 6
120.q odd 4 2 7200.2.k.u 12
120.w even 4 2 1800.2.k.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 1.a even 1 1 trivial
120.2.d.a 6 40.f even 2 1 inner
120.2.d.b yes 6 5.b even 2 1
120.2.d.b yes 6 8.b even 2 1
360.2.d.e 6 15.d odd 2 1
360.2.d.e 6 24.h odd 2 1
360.2.d.f 6 3.b odd 2 1
360.2.d.f 6 120.i odd 2 1
480.2.d.a 6 4.b odd 2 1
480.2.d.a 6 40.e odd 2 1
480.2.d.b 6 8.d odd 2 1
480.2.d.b 6 20.d odd 2 1
600.2.k.f 12 5.c odd 4 2
600.2.k.f 12 40.i odd 4 2
1440.2.d.e 6 12.b even 2 1
1440.2.d.e 6 120.m even 2 1
1440.2.d.f 6 24.f even 2 1
1440.2.d.f 6 60.h even 2 1
1800.2.k.u 12 15.e even 4 2
1800.2.k.u 12 120.w even 4 2
2400.2.k.f 12 20.e even 4 2
2400.2.k.f 12 40.k even 4 2
3840.2.f.l 12 16.e even 4 2
3840.2.f.l 12 80.q even 4 2
3840.2.f.m 12 16.f odd 4 2
3840.2.f.m 12 80.k odd 4 2
7200.2.k.u 12 60.l odd 4 2
7200.2.k.u 12 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T133+4T13216T1356 T_{13}^{3} + 4T_{13}^{2} - 16T_{13} - 56 acting on S2new(120,[χ])S_{2}^{\mathrm{new}}(120, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6+T5+4T+8 T^{6} + T^{5} + 4T + 8 Copy content Toggle raw display
33 (T1)6 (T - 1)^{6} Copy content Toggle raw display
55 T6T4++125 T^{6} - T^{4} + \cdots + 125 Copy content Toggle raw display
77 T6+24T4++64 T^{6} + 24 T^{4} + \cdots + 64 Copy content Toggle raw display
1111 T6+32T4++64 T^{6} + 32 T^{4} + \cdots + 64 Copy content Toggle raw display
1313 (T3+4T216T56)2 (T^{3} + 4 T^{2} - 16 T - 56)^{2} Copy content Toggle raw display
1717 T6+36T4++1024 T^{6} + 36 T^{4} + \cdots + 1024 Copy content Toggle raw display
1919 T6+60T4++1024 T^{6} + 60 T^{4} + \cdots + 1024 Copy content Toggle raw display
2323 T6+92T4++16384 T^{6} + 92 T^{4} + \cdots + 16384 Copy content Toggle raw display
2929 T6+108T4++12544 T^{6} + 108 T^{4} + \cdots + 12544 Copy content Toggle raw display
3131 (T3+8T24T64)2 (T^{3} + 8 T^{2} - 4 T - 64)^{2} Copy content Toggle raw display
3737 (T3+8T28)2 (T^{3} + 8 T^{2} - 8)^{2} Copy content Toggle raw display
4141 (T3+2T2100T+56)2 (T^{3} + 2 T^{2} - 100 T + 56)^{2} Copy content Toggle raw display
4343 (T364T+64)2 (T^{3} - 64 T + 64)^{2} Copy content Toggle raw display
4747 T6+60T4++1024 T^{6} + 60 T^{4} + \cdots + 1024 Copy content Toggle raw display
5353 (T312T2+32T+8)2 (T^{3} - 12 T^{2} + 32 T + 8)^{2} Copy content Toggle raw display
5959 T6+176T4++179776 T^{6} + 176 T^{4} + \cdots + 179776 Copy content Toggle raw display
6161 T6+176T4++65536 T^{6} + 176 T^{4} + \cdots + 65536 Copy content Toggle raw display
6767 (T364T64)2 (T^{3} - 64 T - 64)^{2} Copy content Toggle raw display
7171 (T38T2++128)2 (T^{3} - 8 T^{2} + \cdots + 128)^{2} Copy content Toggle raw display
7373 T6+384T4++16384 T^{6} + 384 T^{4} + \cdots + 16384 Copy content Toggle raw display
7979 (T38T24T+64)2 (T^{3} - 8 T^{2} - 4 T + 64)^{2} Copy content Toggle raw display
8383 (T3+8T2+448)2 (T^{3} + 8 T^{2} + \cdots - 448)^{2} Copy content Toggle raw display
8989 (T3+10T2+1384)2 (T^{3} + 10 T^{2} + \cdots - 1384)^{2} Copy content Toggle raw display
9797 T6+336T4++262144 T^{6} + 336 T^{4} + \cdots + 262144 Copy content Toggle raw display
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