Properties

Label 252.3.z.a
Level 252252
Weight 33
Character orbit 252.z
Analytic conductor 6.8676.867
Analytic rank 00
Dimension 22
Inner twists 22

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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] = [252,3,Mod(73,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.73"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 252=22327 252 = 2^{2} \cdot 3^{2} \cdot 7
Weight: k k == 3 3
Character orbit: [χ][\chi] == 252.z (of order 66, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-3,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 6.866502661886.86650266188
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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)
 

Coefficients of the qq-expansion are expressed in terms of a primitive root of unity ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(ζ62)q57q7+(15ζ6+15)q11+(16ζ6+8)q13+(17ζ617)q17+(9ζ6+18)q199ζ6q23+(22ζ622)q25++(32ζ6+16)q97+O(q100) q + (\zeta_{6} - 2) q^{5} - 7 q^{7} + ( - 15 \zeta_{6} + 15) q^{11} + ( - 16 \zeta_{6} + 8) q^{13} + ( - 17 \zeta_{6} - 17) q^{17} + ( - 9 \zeta_{6} + 18) q^{19} - 9 \zeta_{6} q^{23} + (22 \zeta_{6} - 22) q^{25} + \cdots + ( - 32 \zeta_{6} + 16) q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q3q514q7+15q1151q17+27q199q2322q25+12q2921q31+21q3531q37+20q4375q47+98q4957q53+141q59141q61+27q95+O(q100) 2 q - 3 q^{5} - 14 q^{7} + 15 q^{11} - 51 q^{17} + 27 q^{19} - 9 q^{23} - 22 q^{25} + 12 q^{29} - 21 q^{31} + 21 q^{35} - 31 q^{37} + 20 q^{43} - 75 q^{47} + 98 q^{49} - 57 q^{53} + 141 q^{59} - 141 q^{61}+ \cdots - 27 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/252Z)×\left(\mathbb{Z}/252\mathbb{Z}\right)^\times.

nn 2929 7373 127127
χ(n)\chi(n) 11 ζ6\zeta_{6} 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.

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}
73.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −1.50000 + 0.866025i 0 −7.00000 0 0 0
145.1 0 0 0 −1.50000 0.866025i 0 −7.00000 0 0 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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.3.z.a 2
3.b odd 2 1 28.3.h.a 2
4.b odd 2 1 1008.3.cg.c 2
7.b odd 2 1 1764.3.z.f 2
7.c even 3 1 1764.3.d.a 2
7.c even 3 1 1764.3.z.f 2
7.d odd 6 1 inner 252.3.z.a 2
7.d odd 6 1 1764.3.d.a 2
12.b even 2 1 112.3.s.a 2
15.d odd 2 1 700.3.s.a 2
15.e even 4 2 700.3.o.a 4
21.c even 2 1 196.3.h.a 2
21.g even 6 1 28.3.h.a 2
21.g even 6 1 196.3.b.a 2
21.h odd 6 1 196.3.b.a 2
21.h odd 6 1 196.3.h.a 2
24.f even 2 1 448.3.s.b 2
24.h odd 2 1 448.3.s.a 2
28.f even 6 1 1008.3.cg.c 2
84.h odd 2 1 784.3.s.b 2
84.j odd 6 1 112.3.s.a 2
84.j odd 6 1 784.3.c.a 2
84.n even 6 1 784.3.c.a 2
84.n even 6 1 784.3.s.b 2
105.p even 6 1 700.3.s.a 2
105.w odd 12 2 700.3.o.a 4
168.ba even 6 1 448.3.s.a 2
168.be odd 6 1 448.3.s.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.h.a 2 3.b odd 2 1
28.3.h.a 2 21.g even 6 1
112.3.s.a 2 12.b even 2 1
112.3.s.a 2 84.j odd 6 1
196.3.b.a 2 21.g even 6 1
196.3.b.a 2 21.h odd 6 1
196.3.h.a 2 21.c even 2 1
196.3.h.a 2 21.h odd 6 1
252.3.z.a 2 1.a even 1 1 trivial
252.3.z.a 2 7.d odd 6 1 inner
448.3.s.a 2 24.h odd 2 1
448.3.s.a 2 168.ba even 6 1
448.3.s.b 2 24.f even 2 1
448.3.s.b 2 168.be odd 6 1
700.3.o.a 4 15.e even 4 2
700.3.o.a 4 105.w odd 12 2
700.3.s.a 2 15.d odd 2 1
700.3.s.a 2 105.p even 6 1
784.3.c.a 2 84.j odd 6 1
784.3.c.a 2 84.n even 6 1
784.3.s.b 2 84.h odd 2 1
784.3.s.b 2 84.n even 6 1
1008.3.cg.c 2 4.b odd 2 1
1008.3.cg.c 2 28.f even 6 1
1764.3.d.a 2 7.c even 3 1
1764.3.d.a 2 7.d odd 6 1
1764.3.z.f 2 7.b odd 2 1
1764.3.z.f 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(252,[χ])S_{3}^{\mathrm{new}}(252, [\chi]):

T52+3T5+3 T_{5}^{2} + 3T_{5} + 3 Copy content Toggle raw display
T11215T11+225 T_{11}^{2} - 15T_{11} + 225 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T2+3T+3 T^{2} + 3T + 3 Copy content Toggle raw display
77 (T+7)2 (T + 7)^{2} Copy content Toggle raw display
1111 T215T+225 T^{2} - 15T + 225 Copy content Toggle raw display
1313 T2+192 T^{2} + 192 Copy content Toggle raw display
1717 T2+51T+867 T^{2} + 51T + 867 Copy content Toggle raw display
1919 T227T+243 T^{2} - 27T + 243 Copy content Toggle raw display
2323 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
2929 (T6)2 (T - 6)^{2} Copy content Toggle raw display
3131 T2+21T+147 T^{2} + 21T + 147 Copy content Toggle raw display
3737 T2+31T+961 T^{2} + 31T + 961 Copy content Toggle raw display
4141 T2+3072 T^{2} + 3072 Copy content Toggle raw display
4343 (T10)2 (T - 10)^{2} Copy content Toggle raw display
4747 T2+75T+1875 T^{2} + 75T + 1875 Copy content Toggle raw display
5353 T2+57T+3249 T^{2} + 57T + 3249 Copy content Toggle raw display
5959 T2141T+6627 T^{2} - 141T + 6627 Copy content Toggle raw display
6161 T2+141T+6627 T^{2} + 141T + 6627 Copy content Toggle raw display
6767 T249T+2401 T^{2} - 49T + 2401 Copy content Toggle raw display
7171 (T126)2 (T - 126)^{2} Copy content Toggle raw display
7373 T2+45T+675 T^{2} + 45T + 675 Copy content Toggle raw display
7979 T273T+5329 T^{2} - 73T + 5329 Copy content Toggle raw display
8383 T2+192 T^{2} + 192 Copy content Toggle raw display
8989 T2+99T+3267 T^{2} + 99T + 3267 Copy content Toggle raw display
9797 T2+768 T^{2} + 768 Copy content Toggle raw display
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