Properties

Label 4608.2.k.be
Level 46084608
Weight 22
Character orbit 4608.k
Analytic conductor 36.79536.795
Analytic rank 00
Dimension 88
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4608,2,Mod(1153,4608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4608, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4608.1153"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 4608=2932 4608 = 2^{9} \cdot 3^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 4608.k (of order 44, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,-8,0,0,0,0,0,0,0,-8,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 36.795065251436.7950652514
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(i)\Q(i)
Coefficient field: Q(ζ16)\Q(\zeta_{16})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8+1 x^{8} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a25]\Z[a_1, \ldots, a_{25}]
Coefficient ring index: 25 2^{5}
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

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 basis 1,β1,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β7β31)q5+(β7+2β3β2)q7+(β6+β5+β2)q11+(2β6β4+β31)q13+(β72β6++β1)q17++(2β7+4β6+2β5+4)q97+O(q100) q + (\beta_{7} - \beta_{3} - 1) q^{5} + ( - \beta_{7} + 2 \beta_{3} - \beta_{2}) q^{7} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{11} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{13} + ( - \beta_{7} - 2 \beta_{6} + \cdots + \beta_1) q^{17}+ \cdots + (2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + \cdots - 4) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q8q58q13+16q198q29+16q31+32q358q37+16q43+32q478q49+8q53+32q598q61+16q6532q6748q7932q8348q91+32q97+O(q100) 8 q - 8 q^{5} - 8 q^{13} + 16 q^{19} - 8 q^{29} + 16 q^{31} + 32 q^{35} - 8 q^{37} + 16 q^{43} + 32 q^{47} - 8 q^{49} + 8 q^{53} + 32 q^{59} - 8 q^{61} + 16 q^{65} - 32 q^{67} - 48 q^{79} - 32 q^{83} - 48 q^{91}+ \cdots - 32 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring

β1\beta_{1}== 2ζ162 2\zeta_{16}^{2} Copy content Toggle raw display
β2\beta_{2}== 2ζ163 2\zeta_{16}^{3} Copy content Toggle raw display
β3\beta_{3}== ζ164 \zeta_{16}^{4} Copy content Toggle raw display
β4\beta_{4}== 2ζ166 2\zeta_{16}^{6} Copy content Toggle raw display
β5\beta_{5}== ζ167+ζ165+ζ163+ζ16 \zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} Copy content Toggle raw display
β6\beta_{6}== ζ167ζ165ζ163+ζ16 -\zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16} Copy content Toggle raw display
β7\beta_{7}== ζ167+ζ165ζ163ζ16 -\zeta_{16}^{7} + \zeta_{16}^{5} - \zeta_{16}^{3} - \zeta_{16} Copy content Toggle raw display
ζ16\zeta_{16}== (β6+β5)/2 ( \beta_{6} + \beta_{5} ) / 2 Copy content Toggle raw display
ζ162\zeta_{16}^{2}== (β1)/2 ( \beta_1 ) / 2 Copy content Toggle raw display
ζ163\zeta_{16}^{3}== (β2)/2 ( \beta_{2} ) / 2 Copy content Toggle raw display
ζ164\zeta_{16}^{4}== β3 \beta_{3} Copy content Toggle raw display
ζ165\zeta_{16}^{5}== (β7+β5)/2 ( \beta_{7} + \beta_{5} ) / 2 Copy content Toggle raw display
ζ166\zeta_{16}^{6}== (β4)/2 ( \beta_{4} ) / 2 Copy content Toggle raw display
ζ167\zeta_{16}^{7}== (β7β6β2)/2 ( -\beta_{7} - \beta_{6} - \beta_{2} ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/4608Z)×\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times.

nn 20532053 35833583 40974097
χ(n)\chi(n) β3\beta_{3} 11 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}
1153.1
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 0.923880i
0 0 0 −2.84776 + 2.84776i 0 4.61313i 0 0 0
1153.2 0 0 0 −1.76537 + 1.76537i 0 0.917608i 0 0 0
1153.3 0 0 0 −0.234633 + 0.234633i 0 3.08239i 0 0 0
1153.4 0 0 0 0.847759 0.847759i 0 0.613126i 0 0 0
3457.1 0 0 0 −2.84776 2.84776i 0 4.61313i 0 0 0
3457.2 0 0 0 −1.76537 1.76537i 0 0.917608i 0 0 0
3457.3 0 0 0 −0.234633 0.234633i 0 3.08239i 0 0 0
3457.4 0 0 0 0.847759 + 0.847759i 0 0.613126i 0 0 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1153.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.k.be 8
3.b odd 2 1 1536.2.j.j yes 8
4.b odd 2 1 4608.2.k.bc 8
8.b even 2 1 4608.2.k.bh 8
8.d odd 2 1 4608.2.k.bj 8
12.b even 2 1 1536.2.j.i yes 8
16.e even 4 1 inner 4608.2.k.be 8
16.e even 4 1 4608.2.k.bh 8
16.f odd 4 1 4608.2.k.bc 8
16.f odd 4 1 4608.2.k.bj 8
24.f even 2 1 1536.2.j.f yes 8
24.h odd 2 1 1536.2.j.e 8
32.g even 8 1 9216.2.a.bl 4
32.g even 8 1 9216.2.a.bm 4
32.h odd 8 1 9216.2.a.z 4
32.h odd 8 1 9216.2.a.ba 4
48.i odd 4 1 1536.2.j.e 8
48.i odd 4 1 1536.2.j.j yes 8
48.k even 4 1 1536.2.j.f yes 8
48.k even 4 1 1536.2.j.i yes 8
96.o even 8 1 3072.2.a.j 4
96.o even 8 1 3072.2.a.p 4
96.o even 8 2 3072.2.d.j 8
96.p odd 8 1 3072.2.a.m 4
96.p odd 8 1 3072.2.a.s 4
96.p odd 8 2 3072.2.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.e 8 24.h odd 2 1
1536.2.j.e 8 48.i odd 4 1
1536.2.j.f yes 8 24.f even 2 1
1536.2.j.f yes 8 48.k even 4 1
1536.2.j.i yes 8 12.b even 2 1
1536.2.j.i yes 8 48.k even 4 1
1536.2.j.j yes 8 3.b odd 2 1
1536.2.j.j yes 8 48.i odd 4 1
3072.2.a.j 4 96.o even 8 1
3072.2.a.m 4 96.p odd 8 1
3072.2.a.p 4 96.o even 8 1
3072.2.a.s 4 96.p odd 8 1
3072.2.d.e 8 96.p odd 8 2
3072.2.d.j 8 96.o even 8 2
4608.2.k.bc 8 4.b odd 2 1
4608.2.k.bc 8 16.f odd 4 1
4608.2.k.be 8 1.a even 1 1 trivial
4608.2.k.be 8 16.e even 4 1 inner
4608.2.k.bh 8 8.b even 2 1
4608.2.k.bh 8 16.e even 4 1
4608.2.k.bj 8 8.d odd 2 1
4608.2.k.bj 8 16.f odd 4 1
9216.2.a.z 4 32.h odd 8 1
9216.2.a.ba 4 32.h odd 8 1
9216.2.a.bl 4 32.g even 8 1
9216.2.a.bm 4 32.g even 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(4608,[χ])S_{2}^{\mathrm{new}}(4608, [\chi]):

T58+8T57+32T56+48T55+24T5432T53+128T52+64T5+16 T_{5}^{8} + 8T_{5}^{7} + 32T_{5}^{6} + 48T_{5}^{5} + 24T_{5}^{4} - 32T_{5}^{3} + 128T_{5}^{2} + 64T_{5} + 16 Copy content Toggle raw display
T78+32T76+240T74+256T72+64 T_{7}^{8} + 32T_{7}^{6} + 240T_{7}^{4} + 256T_{7}^{2} + 64 Copy content Toggle raw display
T118+192T114+1024 T_{11}^{8} + 192T_{11}^{4} + 1024 Copy content Toggle raw display
T138+8T137+32T136+48T135+1160T134+9120T133+36992T132+51136T13+35344 T_{13}^{8} + 8T_{13}^{7} + 32T_{13}^{6} + 48T_{13}^{5} + 1160T_{13}^{4} + 9120T_{13}^{3} + 36992T_{13}^{2} + 51136T_{13} + 35344 Copy content Toggle raw display
T19816T197+128T196512T195+1088T194512T193+1024 T_{19}^{8} - 16T_{19}^{7} + 128T_{19}^{6} - 512T_{19}^{5} + 1088T_{19}^{4} - 512T_{19}^{3} + 1024 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 T8+8T7++16 T^{8} + 8 T^{7} + \cdots + 16 Copy content Toggle raw display
77 T8+32T6++64 T^{8} + 32 T^{6} + \cdots + 64 Copy content Toggle raw display
1111 T8+192T4+1024 T^{8} + 192T^{4} + 1024 Copy content Toggle raw display
1313 T8+8T7++35344 T^{8} + 8 T^{7} + \cdots + 35344 Copy content Toggle raw display
1717 (T432T2+32)2 (T^{4} - 32 T^{2} + \cdots - 32)^{2} Copy content Toggle raw display
1919 T816T7++1024 T^{8} - 16 T^{7} + \cdots + 1024 Copy content Toggle raw display
2323 (T2+16)4 (T^{2} + 16)^{4} Copy content Toggle raw display
2929 T8+8T7++4624 T^{8} + 8 T^{7} + \cdots + 4624 Copy content Toggle raw display
3131 (T48T3++248)2 (T^{4} - 8 T^{3} + \cdots + 248)^{2} Copy content Toggle raw display
3737 T8+8T7++150544 T^{8} + 8 T^{7} + \cdots + 150544 Copy content Toggle raw display
4141 T8+192T6++984064 T^{8} + 192 T^{6} + \cdots + 984064 Copy content Toggle raw display
4343 T816T7++984064 T^{8} - 16 T^{7} + \cdots + 984064 Copy content Toggle raw display
4747 (T28T16)4 (T^{2} - 8 T - 16)^{4} Copy content Toggle raw display
5353 T88T7++35344 T^{8} - 8 T^{7} + \cdots + 35344 Copy content Toggle raw display
5959 T832T7++18939904 T^{8} - 32 T^{7} + \cdots + 18939904 Copy content Toggle raw display
6161 T8+8T7++16 T^{8} + 8 T^{7} + \cdots + 16 Copy content Toggle raw display
6767 T8+32T7++62980096 T^{8} + 32 T^{7} + \cdots + 62980096 Copy content Toggle raw display
7171 T8+256T6++4734976 T^{8} + 256 T^{6} + \cdots + 4734976 Copy content Toggle raw display
7373 T8+304T6++2408704 T^{8} + 304 T^{6} + \cdots + 2408704 Copy content Toggle raw display
7979 (T4+24T3+7688)2 (T^{4} + 24 T^{3} + \cdots - 7688)^{2} Copy content Toggle raw display
8383 T8+32T7++295936 T^{8} + 32 T^{7} + \cdots + 295936 Copy content Toggle raw display
8989 T8+400T6++73984 T^{8} + 400 T^{6} + \cdots + 73984 Copy content Toggle raw display
9797 (T4+16T3+256)2 (T^{4} + 16 T^{3} + \cdots - 256)^{2} Copy content Toggle raw display
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