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Properties

Label 1260.1.eb.a
Level 12601260
Weight 11
Character orbit 1260.eb
Analytic conductor 0.6290.629
Analytic rank 00
Dimension 88
Projective image S4S_{4}
CM/RM no
Inner twists 88

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1260,1,Mod(223,1260)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1260, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 8, 9, 6])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1260.223"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: N N == 1260=22⋅32⋅5⋅7 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1260.eb (of order 1212, degree 44, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 0.6288219159180.628821915918
Analytic rank: 00
Dimension: 88
Relative dimension: 22 over Q(ζ12)\Q(\zeta_{12})
Coefficient field: Q(ζ24)\Q(\zeta_{24})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x4+1 x^{8} - x^{4} + 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
Projective image: S4S_{4}
Projective field: Galois closure of 4.2.1134000.1

qq-expansion

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

The qq-expansion and trace form are shown below.

f(q)f(q) == q−ζ244q2+ζ243q3+ζ248q4+ζ245q5−ζ247q6+ζ24q7+q8+ζ246q9−ζ249q10+ζ2410q11+⋯−ζ244q99+O(q100) q - \zeta_{24}^{4} q^{2} + \zeta_{24}^{3} q^{3} + \zeta_{24}^{8} q^{4} + \zeta_{24}^{5} q^{5} - \zeta_{24}^{7} q^{6} + \zeta_{24} q^{7} + q^{8} + \zeta_{24}^{6} q^{9} - \zeta_{24}^{9} q^{10} + \zeta_{24}^{10} q^{11} + \cdots - \zeta_{24}^{4} q^{99} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−4q2−4q4+8q8−4q15−4q16+4q21+8q30−4q32+4q39+4q42−4q43+8q53+8q57−4q60+8q64−8q78+4q79−8q81−8q84+⋯−4q99+O(q100) 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 4 q^{15} - 4 q^{16} + 4 q^{21} + 8 q^{30} - 4 q^{32} + 4 q^{39} + 4 q^{42} - 4 q^{43} + 8 q^{53} + 8 q^{57} - 4 q^{60} + 8 q^{64} - 8 q^{78} + 4 q^{79} - 8 q^{81} - 8 q^{84}+ \cdots - 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1260Z)×\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times.

nn 281281 631631 757757 10811081
χ(n)\chi(n) ζ248\zeta_{24}^{8} −1-1 −ζ246-\zeta_{24}^{6} −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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
223.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.500000 + 0.866025i −0.707107 − 0.707107i −0.500000 − 0.866025i 0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i 1.00000 1.00000i −0.707107 + 0.707107i
223.2 −0.500000 + 0.866025i 0.707107 + 0.707107i −0.500000 − 0.866025i −0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 1.00000 1.00000i 0.707107 − 0.707107i
643.1 −0.500000 − 0.866025i −0.707107 − 0.707107i −0.500000 + 0.866025i −0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 1.00000 1.00000i −0.707107 + 0.707107i
643.2 −0.500000 − 0.866025i 0.707107 + 0.707107i −0.500000 + 0.866025i 0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i 1.00000 1.00000i 0.707107 − 0.707107i
727.1 −0.500000 + 0.866025i −0.707107 + 0.707107i −0.500000 − 0.866025i −0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 1.00000 − 1.00000i −0.707107 − 0.707107i
727.2 −0.500000 + 0.866025i 0.707107 − 0.707107i −0.500000 − 0.866025i 0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i 1.00000 − 1.00000i 0.707107 + 0.707107i
1147.1 −0.500000 − 0.866025i −0.707107 + 0.707107i −0.500000 + 0.866025i 0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i 1.00000 − 1.00000i −0.707107 − 0.707107i
1147.2 −0.500000 − 0.866025i 0.707107 − 0.707107i −0.500000 + 0.866025i −0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 1.00000 − 1.00000i 0.707107 + 0.707107i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.c even 3 1 inner
20.e even 4 1 inner
63.l odd 6 1 inner
140.j odd 4 1 inner
180.x even 12 1 inner
1260.eb odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.eb.a ✓ 8
3.b odd 2 1 3780.1.ee.b 8
4.b odd 2 1 1260.1.eb.b yes 8
5.c odd 4 1 1260.1.eb.b yes 8
7.b odd 2 1 inner 1260.1.eb.a ✓ 8
9.c even 3 1 inner 1260.1.eb.a ✓ 8
9.d odd 6 1 3780.1.ee.b 8
12.b even 2 1 3780.1.ee.a 8
15.e even 4 1 3780.1.ee.a 8
20.e even 4 1 inner 1260.1.eb.a ✓ 8
21.c even 2 1 3780.1.ee.b 8
28.d even 2 1 1260.1.eb.b yes 8
35.f even 4 1 1260.1.eb.b yes 8
36.f odd 6 1 1260.1.eb.b yes 8
36.h even 6 1 3780.1.ee.a 8
45.k odd 12 1 1260.1.eb.b yes 8
45.l even 12 1 3780.1.ee.a 8
60.l odd 4 1 3780.1.ee.b 8
63.l odd 6 1 inner 1260.1.eb.a ✓ 8
63.o even 6 1 3780.1.ee.b 8
84.h odd 2 1 3780.1.ee.a 8
105.k odd 4 1 3780.1.ee.a 8
140.j odd 4 1 inner 1260.1.eb.a ✓ 8
180.v odd 12 1 3780.1.ee.b 8
180.x even 12 1 inner 1260.1.eb.a ✓ 8
252.s odd 6 1 3780.1.ee.a 8
252.bi even 6 1 1260.1.eb.b yes 8
315.cb even 12 1 1260.1.eb.b yes 8
315.cf odd 12 1 3780.1.ee.a 8
420.w even 4 1 3780.1.ee.b 8
1260.do even 12 1 3780.1.ee.b 8
1260.eb odd 12 1 inner 1260.1.eb.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.eb.a ✓ 8 1.a even 1 1 trivial
1260.1.eb.a ✓ 8 7.b odd 2 1 inner
1260.1.eb.a ✓ 8 9.c even 3 1 inner
1260.1.eb.a ✓ 8 20.e even 4 1 inner
1260.1.eb.a ✓ 8 63.l odd 6 1 inner
1260.1.eb.a ✓ 8 140.j odd 4 1 inner
1260.1.eb.a ✓ 8 180.x even 12 1 inner
1260.1.eb.a ✓ 8 1260.eb odd 12 1 inner
1260.1.eb.b yes 8 4.b odd 2 1
1260.1.eb.b yes 8 5.c odd 4 1
1260.1.eb.b yes 8 28.d even 2 1
1260.1.eb.b yes 8 35.f even 4 1
1260.1.eb.b yes 8 36.f odd 6 1
1260.1.eb.b yes 8 45.k odd 12 1
1260.1.eb.b yes 8 252.bi even 6 1
1260.1.eb.b yes 8 315.cb even 12 1
3780.1.ee.a 8 12.b even 2 1
3780.1.ee.a 8 15.e even 4 1
3780.1.ee.a 8 36.h even 6 1
3780.1.ee.a 8 45.l even 12 1
3780.1.ee.a 8 84.h odd 2 1
3780.1.ee.a 8 105.k odd 4 1
3780.1.ee.a 8 252.s odd 6 1
3780.1.ee.a 8 315.cf odd 12 1
3780.1.ee.b 8 3.b odd 2 1
3780.1.ee.b 8 9.d odd 6 1
3780.1.ee.b 8 21.c even 2 1
3780.1.ee.b 8 60.l odd 4 1
3780.1.ee.b 8 63.o even 6 1
3780.1.ee.b 8 180.v odd 12 1
3780.1.ee.b 8 420.w even 4 1
3780.1.ee.b 8 1260.do even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T434+2T433+2T432+4T43+4 T_{43}^{4} + 2T_{43}^{3} + 2T_{43}^{2} + 4T_{43} + 4 acting on S1new(1260,[χ])S_{1}^{\mathrm{new}}(1260, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+T+1)4 (T^{2} + T + 1)^{4} Copy content Toggle raw display
33 (T4+1)2 (T^{4} + 1)^{2} Copy content Toggle raw display
55 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
77 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
1111 (T4−T2+1)2 (T^{4} - T^{2} + 1)^{2} Copy content Toggle raw display
1313 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
1717 (T4+1)2 (T^{4} + 1)^{2} Copy content Toggle raw display
1919 (T2+2)4 (T^{2} + 2)^{4} Copy content Toggle raw display
2323 T8 T^{8} Copy content Toggle raw display
2929 T8 T^{8} Copy content Toggle raw display
3131 T8 T^{8} Copy content Toggle raw display
3737 T8 T^{8} Copy content Toggle raw display
4141 T8 T^{8} Copy content Toggle raw display
4343 (T4+2T3+2T2+⋯+4)2 (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} Copy content Toggle raw display
4747 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
5353 (T2−2T+2)4 (T^{2} - 2 T + 2)^{4} Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 (T4−2T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
6767 T8 T^{8} Copy content Toggle raw display
7171 (T2+1)4 (T^{2} + 1)^{4} Copy content Toggle raw display
7373 (T4+1)2 (T^{4} + 1)^{2} Copy content Toggle raw display
7979 (T2−T+1)4 (T^{2} - T + 1)^{4} Copy content Toggle raw display
8383 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
8989 T8 T^{8} Copy content Toggle raw display
9797 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
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