Properties

Label 144.3.q.c
Level 144144
Weight 33
Character orbit 144.q
Analytic conductor 3.9243.924
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,3,Mod(65,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 144=2432 144 = 2^{4} \cdot 3^{2}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 144.q (of order 66, degree 22, not 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: 3.923715806793.92371580679
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(2,3)\Q(\sqrt{-2}, \sqrt{-3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x42x2+4 x^{4} - 2x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 3 3
Twist minimal: no (minimal twist has level 18)
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+(β32β1+1)q3+(3β13)q5+(3β2β1)q7+(2β3+4β2+3)q9+(2β3+β2+3β16)q11+(6β3+6β2+5)q13++(6β315β2+18)q99+O(q100) q + ( - \beta_{3} - 2 \beta_1 + 1) q^{3} + ( - 3 \beta_1 - 3) q^{5} + (3 \beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} + 3) q^{9} + ( - 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 6) q^{11} + ( - 6 \beta_{3} + 6 \beta_{2} + \cdots - 5) q^{13}+ \cdots + ( - 6 \beta_{3} - 15 \beta_{2} + \cdots - 18) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q18q52q7+12q918q1110q1318q15+40q1942q2118q23+4q25+18q2938q31+54q33+128q37+102q39126q41+46q43+126q99+O(q100) 4 q - 18 q^{5} - 2 q^{7} + 12 q^{9} - 18 q^{11} - 10 q^{13} - 18 q^{15} + 40 q^{19} - 42 q^{21} - 18 q^{23} + 4 q^{25} + 18 q^{29} - 38 q^{31} + 54 q^{33} + 128 q^{37} + 102 q^{39} - 126 q^{41} + 46 q^{43}+ \cdots - 126 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x42x2+4 x^{4} - 2x^{2} + 4 : Copy content Toggle raw display

β1\beta_{1}== (ν2)/2 ( \nu^{2} ) / 2 Copy content Toggle raw display
β2\beta_{2}== (ν3+2ν)/2 ( \nu^{3} + 2\nu ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν3+4ν)/2 ( -\nu^{3} + 4\nu ) / 2 Copy content Toggle raw display
ν\nu== (β3+β2)/3 ( \beta_{3} + \beta_{2} ) / 3 Copy content Toggle raw display
ν2\nu^{2}== 2β1 2\beta_1 Copy content Toggle raw display
ν3\nu^{3}== (2β3+4β2)/3 ( -2\beta_{3} + 4\beta_{2} ) / 3 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/144Z)×\left(\mathbb{Z}/144\mathbb{Z}\right)^\times.

nn 3737 6565 127127
χ(n)\chi(n) 11 1β11 - \beta_{1} 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}
65.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0 −2.44949 + 1.73205i 0 −4.50000 + 2.59808i 0 3.17423 5.49794i 0 3.00000 8.48528i 0
65.2 0 2.44949 + 1.73205i 0 −4.50000 + 2.59808i 0 −4.17423 + 7.22999i 0 3.00000 + 8.48528i 0
113.1 0 −2.44949 1.73205i 0 −4.50000 2.59808i 0 3.17423 + 5.49794i 0 3.00000 + 8.48528i 0
113.2 0 2.44949 1.73205i 0 −4.50000 2.59808i 0 −4.17423 7.22999i 0 3.00000 8.48528i 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
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.q.c 4
3.b odd 2 1 432.3.q.d 4
4.b odd 2 1 18.3.d.a 4
8.b even 2 1 576.3.q.e 4
8.d odd 2 1 576.3.q.f 4
9.c even 3 1 432.3.q.d 4
9.c even 3 1 1296.3.e.g 4
9.d odd 6 1 inner 144.3.q.c 4
9.d odd 6 1 1296.3.e.g 4
12.b even 2 1 54.3.d.a 4
20.d odd 2 1 450.3.i.b 4
20.e even 4 2 450.3.k.a 8
24.f even 2 1 1728.3.q.d 4
24.h odd 2 1 1728.3.q.c 4
36.f odd 6 1 54.3.d.a 4
36.f odd 6 1 162.3.b.a 4
36.h even 6 1 18.3.d.a 4
36.h even 6 1 162.3.b.a 4
60.h even 2 1 1350.3.i.b 4
60.l odd 4 2 1350.3.k.a 8
72.j odd 6 1 576.3.q.e 4
72.l even 6 1 576.3.q.f 4
72.n even 6 1 1728.3.q.c 4
72.p odd 6 1 1728.3.q.d 4
180.n even 6 1 450.3.i.b 4
180.p odd 6 1 1350.3.i.b 4
180.v odd 12 2 450.3.k.a 8
180.x even 12 2 1350.3.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 4.b odd 2 1
18.3.d.a 4 36.h even 6 1
54.3.d.a 4 12.b even 2 1
54.3.d.a 4 36.f odd 6 1
144.3.q.c 4 1.a even 1 1 trivial
144.3.q.c 4 9.d odd 6 1 inner
162.3.b.a 4 36.f odd 6 1
162.3.b.a 4 36.h even 6 1
432.3.q.d 4 3.b odd 2 1
432.3.q.d 4 9.c even 3 1
450.3.i.b 4 20.d odd 2 1
450.3.i.b 4 180.n even 6 1
450.3.k.a 8 20.e even 4 2
450.3.k.a 8 180.v odd 12 2
576.3.q.e 4 8.b even 2 1
576.3.q.e 4 72.j odd 6 1
576.3.q.f 4 8.d odd 2 1
576.3.q.f 4 72.l even 6 1
1296.3.e.g 4 9.c even 3 1
1296.3.e.g 4 9.d odd 6 1
1350.3.i.b 4 60.h even 2 1
1350.3.i.b 4 180.p odd 6 1
1350.3.k.a 8 60.l odd 4 2
1350.3.k.a 8 180.x even 12 2
1728.3.q.c 4 24.h odd 2 1
1728.3.q.c 4 72.n even 6 1
1728.3.q.d 4 24.f even 2 1
1728.3.q.d 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T52+9T5+27 T_{5}^{2} + 9T_{5} + 27 acting on S3new(144,[χ])S_{3}^{\mathrm{new}}(144, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T46T2+81 T^{4} - 6T^{2} + 81 Copy content Toggle raw display
55 (T2+9T+27)2 (T^{2} + 9 T + 27)^{2} Copy content Toggle raw display
77 T4+2T3++2809 T^{4} + 2 T^{3} + \cdots + 2809 Copy content Toggle raw display
1111 T4+18T3++81 T^{4} + 18 T^{3} + \cdots + 81 Copy content Toggle raw display
1313 T4+10T3++36481 T^{4} + 10 T^{3} + \cdots + 36481 Copy content Toggle raw display
1717 T4+360T2+1296 T^{4} + 360T^{2} + 1296 Copy content Toggle raw display
1919 (T220T116)2 (T^{2} - 20 T - 116)^{2} Copy content Toggle raw display
2323 T4+18T3++81 T^{4} + 18 T^{3} + \cdots + 81 Copy content Toggle raw display
2929 T418T3++2025 T^{4} - 18 T^{3} + \cdots + 2025 Copy content Toggle raw display
3131 T4+38T3++15625 T^{4} + 38 T^{3} + \cdots + 15625 Copy content Toggle raw display
3737 (T264T+808)2 (T^{2} - 64 T + 808)^{2} Copy content Toggle raw display
4141 T4+126T3++455625 T^{4} + 126 T^{3} + \cdots + 455625 Copy content Toggle raw display
4343 T446T3++1849 T^{4} - 46 T^{3} + \cdots + 1849 Copy content Toggle raw display
4747 T4+54T3++408321 T^{4} + 54 T^{3} + \cdots + 408321 Copy content Toggle raw display
5353 T4+9000T2+810000 T^{4} + 9000 T^{2} + 810000 Copy content Toggle raw display
5959 T4+126T3++2954961 T^{4} + 126 T^{3} + \cdots + 2954961 Copy content Toggle raw display
6161 T462T3++966289 T^{4} - 62 T^{3} + \cdots + 966289 Copy content Toggle raw display
6767 T4106T3++5396329 T^{4} - 106 T^{3} + \cdots + 5396329 Copy content Toggle raw display
7171 T4+7704T2+2396304 T^{4} + 7704 T^{2} + 2396304 Copy content Toggle raw display
7373 (T2+104T+760)2 (T^{2} + 104 T + 760)^{2} Copy content Toggle raw display
7979 T4+14T3++1692601 T^{4} + 14 T^{3} + \cdots + 1692601 Copy content Toggle raw display
8383 T4378T3++131262849 T^{4} - 378 T^{3} + \cdots + 131262849 Copy content Toggle raw display
8989 T4+22824T2+36144144 T^{4} + 22824 T^{2} + 36144144 Copy content Toggle raw display
9797 T414T3++110986225 T^{4} - 14 T^{3} + \cdots + 110986225 Copy content Toggle raw display
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