Properties

Label 1224.1.cx.b
Level 12241224
Weight 11
Character orbit 1224.cx
Analytic conductor 0.6110.611
Analytic rank 00
Dimension 1616
Projective image D48D_{48}
CM discriminant -8
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,1,Mod(11,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 24, 8, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 1224=233217 1224 = 2^{3} \cdot 3^{2} \cdot 17
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1224.cx (of order 4848, degree 1616, 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: 0.6108555754630.610855575463
Analytic rank: 00
Dimension: 1616
Coefficient field: Q(ζ48)\Q(\zeta_{48})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x16x8+1 x^{16} - x^{8} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D48D_{48}
Projective field: Galois closure of Q[x]/(x48)\mathbb{Q}[x]/(x^{48} - \cdots)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == qζ485q2+ζ4817q3+ζ4810q4ζ4822q6ζ4815q8ζ4810q9+(ζ4821ζ484)q11ζ483q12++(ζ4814+ζ487)q99+O(q100) q - \zeta_{48}^{5} q^{2} + \zeta_{48}^{17} q^{3} + \zeta_{48}^{10} q^{4} - \zeta_{48}^{22} q^{6} - \zeta_{48}^{15} q^{8} - \zeta_{48}^{10} q^{9} + (\zeta_{48}^{21} - \zeta_{48}^{4}) q^{11} - \zeta_{48}^{3} q^{12} + \cdots + (\zeta_{48}^{14} + \zeta_{48}^{7}) q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q+8q2416q3816q438q508q54+8q578q83+O(q100) 16 q + 8 q^{24} - 16 q^{38} - 16 q^{43} - 8 q^{50} - 8 q^{54} + 8 q^{57} - 8 q^{83}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1224Z)×\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times.

nn 137137 613613 649649 919919
χ(n)\chi(n) ζ4816-\zeta_{48}^{16} 1-1 ζ4815\zeta_{48}^{15} 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}
11.1
0.130526 + 0.991445i
−0.991445 0.130526i
−0.130526 0.991445i
−0.130526 + 0.991445i
−0.991445 + 0.130526i
−0.608761 0.793353i
0.793353 + 0.608761i
0.793353 0.608761i
−0.608761 + 0.793353i
−0.793353 + 0.608761i
0.608761 + 0.793353i
0.130526 0.991445i
0.991445 + 0.130526i
0.608761 0.793353i
−0.793353 0.608761i
0.991445 0.130526i
−0.608761 0.793353i 0.793353 0.608761i −0.258819 + 0.965926i 0 −0.965926 0.258819i 0 0.923880 0.382683i 0.258819 0.965926i 0
131.1 0.793353 + 0.608761i 0.608761 0.793353i 0.258819 + 0.965926i 0 0.965926 0.258819i 0 −0.382683 + 0.923880i −0.258819 0.965926i 0
227.1 0.608761 + 0.793353i −0.793353 + 0.608761i −0.258819 + 0.965926i 0 −0.965926 0.258819i 0 −0.923880 + 0.382683i 0.258819 0.965926i 0
275.1 0.608761 0.793353i −0.793353 0.608761i −0.258819 0.965926i 0 −0.965926 + 0.258819i 0 −0.923880 0.382683i 0.258819 + 0.965926i 0
299.1 0.793353 0.608761i 0.608761 + 0.793353i 0.258819 0.965926i 0 0.965926 + 0.258819i 0 −0.382683 0.923880i −0.258819 + 0.965926i 0
347.1 −0.130526 0.991445i 0.991445 0.130526i −0.965926 + 0.258819i 0 −0.258819 0.965926i 0 0.382683 + 0.923880i 0.965926 0.258819i 0
371.1 0.991445 + 0.130526i 0.130526 0.991445i 0.965926 + 0.258819i 0 0.258819 0.965926i 0 0.923880 + 0.382683i −0.965926 0.258819i 0
419.1 0.991445 0.130526i 0.130526 + 0.991445i 0.965926 0.258819i 0 0.258819 + 0.965926i 0 0.923880 0.382683i −0.965926 + 0.258819i 0
515.1 −0.130526 + 0.991445i 0.991445 + 0.130526i −0.965926 0.258819i 0 −0.258819 + 0.965926i 0 0.382683 0.923880i 0.965926 + 0.258819i 0
635.1 −0.991445 + 0.130526i −0.130526 0.991445i 0.965926 0.258819i 0 0.258819 + 0.965926i 0 −0.923880 + 0.382683i −0.965926 + 0.258819i 0
707.1 0.130526 + 0.991445i −0.991445 + 0.130526i −0.965926 + 0.258819i 0 −0.258819 0.965926i 0 −0.382683 0.923880i 0.965926 0.258819i 0
779.1 −0.608761 + 0.793353i 0.793353 + 0.608761i −0.258819 0.965926i 0 −0.965926 + 0.258819i 0 0.923880 + 0.382683i 0.258819 + 0.965926i 0
923.1 −0.793353 0.608761i −0.608761 + 0.793353i 0.258819 + 0.965926i 0 0.965926 0.258819i 0 0.382683 0.923880i −0.258819 0.965926i 0
947.1 0.130526 0.991445i −0.991445 0.130526i −0.965926 0.258819i 0 −0.258819 + 0.965926i 0 −0.382683 + 0.923880i 0.965926 + 0.258819i 0
1091.1 −0.991445 0.130526i −0.130526 + 0.991445i 0.965926 + 0.258819i 0 0.258819 0.965926i 0 −0.923880 0.382683i −0.965926 0.258819i 0
1163.1 −0.793353 + 0.608761i −0.608761 0.793353i 0.258819 0.965926i 0 0.965926 + 0.258819i 0 0.382683 + 0.923880i −0.258819 + 0.965926i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by Q(2)\Q(\sqrt{-2})
153.s even 48 1 inner
1224.cx odd 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.cx.b yes 16
3.b odd 2 1 3672.1.dv.b 16
8.d odd 2 1 CM 1224.1.cx.b yes 16
9.c even 3 1 3672.1.dv.a 16
9.d odd 6 1 1224.1.cx.a 16
17.e odd 16 1 1224.1.cx.a 16
24.f even 2 1 3672.1.dv.b 16
51.i even 16 1 3672.1.dv.a 16
72.l even 6 1 1224.1.cx.a 16
72.p odd 6 1 3672.1.dv.a 16
136.s even 16 1 1224.1.cx.a 16
153.s even 48 1 inner 1224.1.cx.b yes 16
153.t odd 48 1 3672.1.dv.b 16
408.bg odd 16 1 3672.1.dv.a 16
1224.cx odd 48 1 inner 1224.1.cx.b yes 16
1224.da even 48 1 3672.1.dv.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.cx.a 16 9.d odd 6 1
1224.1.cx.a 16 17.e odd 16 1
1224.1.cx.a 16 72.l even 6 1
1224.1.cx.a 16 136.s even 16 1
1224.1.cx.b yes 16 1.a even 1 1 trivial
1224.1.cx.b yes 16 8.d odd 2 1 CM
1224.1.cx.b yes 16 153.s even 48 1 inner
1224.1.cx.b yes 16 1224.cx odd 48 1 inner
3672.1.dv.a 16 9.c even 3 1
3672.1.dv.a 16 51.i even 16 1
3672.1.dv.a 16 72.p odd 6 1
3672.1.dv.a 16 408.bg odd 16 1
3672.1.dv.b 16 3.b odd 2 1
3672.1.dv.b 16 24.f even 2 1
3672.1.dv.b 16 153.t odd 48 1
3672.1.dv.b 16 1224.da even 48 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T11164T1114+10T1112+8T111116T1110+48T119+21T118++1 T_{11}^{16} - 4 T_{11}^{14} + 10 T_{11}^{12} + 8 T_{11}^{11} - 16 T_{11}^{10} + 48 T_{11}^{9} + 21 T_{11}^{8} + \cdots + 1 acting on S1new(1224,[χ])S_{1}^{\mathrm{new}}(1224, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16T8+1 T^{16} - T^{8} + 1 Copy content Toggle raw display
33 T16T8+1 T^{16} - T^{8} + 1 Copy content Toggle raw display
55 T16 T^{16} Copy content Toggle raw display
77 T16 T^{16} Copy content Toggle raw display
1111 T164T14++1 T^{16} - 4 T^{14} + \cdots + 1 Copy content Toggle raw display
1313 T16 T^{16} Copy content Toggle raw display
1717 T16T8+1 T^{16} - T^{8} + 1 Copy content Toggle raw display
1919 T16+194T8+1 T^{16} + 194T^{8} + 1 Copy content Toggle raw display
2323 T16 T^{16} Copy content Toggle raw display
2929 T16 T^{16} Copy content Toggle raw display
3131 T16 T^{16} Copy content Toggle raw display
3737 T16 T^{16} Copy content Toggle raw display
4141 T16+4T12++1 T^{16} + 4 T^{12} + \cdots + 1 Copy content Toggle raw display
4343 (T8+8T7+28T6++1)2 (T^{8} + 8 T^{7} + 28 T^{6} + \cdots + 1)^{2} Copy content Toggle raw display
4747 T16 T^{16} Copy content Toggle raw display
5353 T16 T^{16} Copy content Toggle raw display
5959 (T8+4T64T5++1)2 (T^{8} + 4 T^{6} - 4 T^{5} + \cdots + 1)^{2} Copy content Toggle raw display
6161 T16 T^{16} Copy content Toggle raw display
6767 T168T14++1 T^{16} - 8 T^{14} + \cdots + 1 Copy content Toggle raw display
7171 T16 T^{16} Copy content Toggle raw display
7373 T168T13++1 T^{16} - 8 T^{13} + \cdots + 1 Copy content Toggle raw display
7979 T16 T^{16} Copy content Toggle raw display
8383 (T8+4T7+10T6++4)2 (T^{8} + 4 T^{7} + 10 T^{6} + \cdots + 4)^{2} Copy content Toggle raw display
8989 (T8+12T4+4)2 (T^{8} + 12 T^{4} + 4)^{2} Copy content Toggle raw display
9797 T16+8T14++1 T^{16} + 8 T^{14} + \cdots + 1 Copy content Toggle raw display
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