Properties

Label 384.4.a.o
Level 384384
Weight 44
Character orbit 384.a
Self dual yes
Analytic conductor 22.65722.657
Analytic rank 00
Dimension 22
CM no
Inner twists 11

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [384,4,Mod(1,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 384=273 384 = 2^{7} \cdot 3
Weight: k k == 4 4
Character orbit: [χ][\chi] == 384.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 22.656733442222.6567334422
Analytic rank: 00
Dimension: 22
Coefficient field: Q(15)\Q(\sqrt{15})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x215 x^{2} - 15 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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 β=415\beta = 4\sqrt{15}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+3q3+(β+4)q5+(β+2)q7+9q9+(2β+20)q11+2q13+(3β+12)q15+(6β+14)q17+(2β+56)q19+(3β+6)q21+(6β28)q23++(18β+180)q99+O(q100) q + 3 q^{3} + (\beta + 4) q^{5} + (\beta + 2) q^{7} + 9 q^{9} + (2 \beta + 20) q^{11} + 2 q^{13} + (3 \beta + 12) q^{15} + ( - 6 \beta + 14) q^{17} + ( - 2 \beta + 56) q^{19} + (3 \beta + 6) q^{21} + ( - 6 \beta - 28) q^{23}+ \cdots + (18 \beta + 180) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+6q3+8q5+4q7+18q9+40q11+4q13+24q15+28q17+112q19+12q2156q23+262q25+54q27144q29+276q31+120q33+496q35++360q99+O(q100) 2 q + 6 q^{3} + 8 q^{5} + 4 q^{7} + 18 q^{9} + 40 q^{11} + 4 q^{13} + 24 q^{15} + 28 q^{17} + 112 q^{19} + 12 q^{21} - 56 q^{23} + 262 q^{25} + 54 q^{27} - 144 q^{29} + 276 q^{31} + 120 q^{33} + 496 q^{35}+ \cdots + 360 q^{99}+O(q^{100}) Copy content Toggle raw display

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}
1.1
−3.87298
3.87298
0 3.00000 0 −11.4919 0 −13.4919 0 9.00000 0
1.2 0 3.00000 0 19.4919 0 17.4919 0 9.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.a.o yes 2
3.b odd 2 1 1152.4.a.p 2
4.b odd 2 1 384.4.a.k yes 2
8.b even 2 1 384.4.a.j 2
8.d odd 2 1 384.4.a.n yes 2
12.b even 2 1 1152.4.a.o 2
16.e even 4 2 768.4.d.q 4
16.f odd 4 2 768.4.d.t 4
24.f even 2 1 1152.4.a.u 2
24.h odd 2 1 1152.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.a.j 2 8.b even 2 1
384.4.a.k yes 2 4.b odd 2 1
384.4.a.n yes 2 8.d odd 2 1
384.4.a.o yes 2 1.a even 1 1 trivial
768.4.d.q 4 16.e even 4 2
768.4.d.t 4 16.f odd 4 2
1152.4.a.o 2 12.b even 2 1
1152.4.a.p 2 3.b odd 2 1
1152.4.a.u 2 24.f even 2 1
1152.4.a.v 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(384))S_{4}^{\mathrm{new}}(\Gamma_0(384)):

T528T5224 T_{5}^{2} - 8T_{5} - 224 Copy content Toggle raw display
T724T7236 T_{7}^{2} - 4T_{7} - 236 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 (T3)2 (T - 3)^{2} Copy content Toggle raw display
55 T28T224 T^{2} - 8T - 224 Copy content Toggle raw display
77 T24T236 T^{2} - 4T - 236 Copy content Toggle raw display
1111 T240T560 T^{2} - 40T - 560 Copy content Toggle raw display
1313 (T2)2 (T - 2)^{2} Copy content Toggle raw display
1717 T228T8444 T^{2} - 28T - 8444 Copy content Toggle raw display
1919 T2112T+2176 T^{2} - 112T + 2176 Copy content Toggle raw display
2323 T2+56T7856 T^{2} + 56T - 7856 Copy content Toggle raw display
2929 T2+144T35376 T^{2} + 144T - 35376 Copy content Toggle raw display
3131 T2276T+7284 T^{2} - 276T + 7284 Copy content Toggle raw display
3737 T2+580T+83140 T^{2} + 580T + 83140 Copy content Toggle raw display
4141 T2588T+85476 T^{2} - 588T + 85476 Copy content Toggle raw display
4343 T2+96T113856 T^{2} + 96T - 113856 Copy content Toggle raw display
4747 T2712T+125776 T^{2} - 712T + 125776 Copy content Toggle raw display
5353 T2+1104T+304464 T^{2} + 1104 T + 304464 Copy content Toggle raw display
5959 T2536T66416 T^{2} - 536T - 66416 Copy content Toggle raw display
6161 T2+452T111164 T^{2} + 452T - 111164 Copy content Toggle raw display
6767 T2296T730736 T^{2} - 296T - 730736 Copy content Toggle raw display
7171 T2216T+3024 T^{2} - 216T + 3024 Copy content Toggle raw display
7373 T2364T615836 T^{2} - 364T - 615836 Copy content Toggle raw display
7979 T21076T+235444 T^{2} - 1076 T + 235444 Copy content Toggle raw display
8383 T2648T817584 T^{2} - 648T - 817584 Copy content Toggle raw display
8989 T22308T+1235716 T^{2} - 2308 T + 1235716 Copy content Toggle raw display
9797 T2+1900T+806500 T^{2} + 1900 T + 806500 Copy content Toggle raw display
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