Properties

Label 33.4.a.c
Level 3333
Weight 44
Character orbit 33.a
Self dual yes
Analytic conductor 1.9471.947
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] = [33,4,Mod(1,33)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(33, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("33.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 33=311 33 = 3 \cdot 11
Weight: k k == 4 4
Character orbit: [χ][\chi] == 33.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 1.947063030191.94706303019
Analytic rank: 00
Dimension: 22
Coefficient field: Q(97)\Q(\sqrt{97})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x24 x^{2} - x - 24 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
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=12(1+97)\beta = \frac{1}{2}(1 + \sqrt{97}). We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+βq23q3+(β+16)q4+(2β6)q53βq6+(4β+14)q7+(9β+24)q8+9q9+(8β48)q1011q11+(3β48)q12+99q99+O(q100) q + \beta q^{2} - 3 q^{3} + (\beta + 16) q^{4} + ( - 2 \beta - 6) q^{5} - 3 \beta q^{6} + ( - 4 \beta + 14) q^{7} + (9 \beta + 24) q^{8} + 9 q^{9} + ( - 8 \beta - 48) q^{10} - 11 q^{11} + ( - 3 \beta - 48) q^{12} + \cdots - 99 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q26q3+33q414q53q6+24q7+57q8+18q9104q1022q1199q12+30q13182q14+42q15+201q16+106q17+9q18+50q19+198q99+O(q100) 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9} - 104 q^{10} - 22 q^{11} - 99 q^{12} + 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} + 106 q^{17} + 9 q^{18} + 50 q^{19}+ \cdots - 198 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
−4.42443
5.42443
−4.42443 −3.00000 11.5756 2.84886 13.2733 31.6977 −15.8199 9.00000 −12.6046
1.2 5.42443 −3.00000 21.4244 −16.8489 −16.2733 −7.69772 72.8199 9.00000 −91.3954
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
1111 +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 33.4.a.c 2
3.b odd 2 1 99.4.a.f 2
4.b odd 2 1 528.4.a.p 2
5.b even 2 1 825.4.a.l 2
5.c odd 4 2 825.4.c.h 4
7.b odd 2 1 1617.4.a.k 2
8.b even 2 1 2112.4.a.bn 2
8.d odd 2 1 2112.4.a.bg 2
11.b odd 2 1 363.4.a.i 2
12.b even 2 1 1584.4.a.bj 2
15.d odd 2 1 2475.4.a.p 2
33.d even 2 1 1089.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 1.a even 1 1 trivial
99.4.a.f 2 3.b odd 2 1
363.4.a.i 2 11.b odd 2 1
528.4.a.p 2 4.b odd 2 1
825.4.a.l 2 5.b even 2 1
825.4.c.h 4 5.c odd 4 2
1089.4.a.u 2 33.d even 2 1
1584.4.a.bj 2 12.b even 2 1
1617.4.a.k 2 7.b odd 2 1
2112.4.a.bg 2 8.d odd 2 1
2112.4.a.bn 2 8.b even 2 1
2475.4.a.p 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T22T224 T_{2}^{2} - T_{2} - 24 acting on S4new(Γ0(33))S_{4}^{\mathrm{new}}(\Gamma_0(33)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2T24 T^{2} - T - 24 Copy content Toggle raw display
33 (T+3)2 (T + 3)^{2} Copy content Toggle raw display
55 T2+14T48 T^{2} + 14T - 48 Copy content Toggle raw display
77 T224T244 T^{2} - 24T - 244 Copy content Toggle raw display
1111 (T+11)2 (T + 11)^{2} Copy content Toggle raw display
1313 T230T+128 T^{2} - 30T + 128 Copy content Toggle raw display
1717 T2106T1944 T^{2} - 106T - 1944 Copy content Toggle raw display
1919 T250T+528 T^{2} - 50T + 528 Copy content Toggle raw display
2323 T2134T+2064 T^{2} - 134T + 2064 Copy content Toggle raw display
2929 T2+198T+8928 T^{2} + 198T + 8928 Copy content Toggle raw display
3131 T2360T+30848 T^{2} - 360T + 30848 Copy content Toggle raw display
3737 T2+328T38676 T^{2} + 328T - 38676 Copy content Toggle raw display
4141 T2+782T+148128 T^{2} + 782T + 148128 Copy content Toggle raw display
4343 T2386T+20856 T^{2} - 386T + 20856 Copy content Toggle raw display
4747 T2266T115104 T^{2} - 266T - 115104 Copy content Toggle raw display
5353 T2+522T2592 T^{2} + 522T - 2592 Copy content Toggle raw display
5959 T2+172T235104 T^{2} + 172T - 235104 Copy content Toggle raw display
6161 T2+778T+123288 T^{2} + 778T + 123288 Copy content Toggle raw display
6767 T2+776T72944 T^{2} + 776T - 72944 Copy content Toggle raw display
7171 T2630T+28512 T^{2} - 630T + 28512 Copy content Toggle raw display
7373 T21296T+400892 T^{2} - 1296 T + 400892 Copy content Toggle raw display
7979 T2652T396572 T^{2} - 652T - 396572 Copy content Toggle raw display
8383 T2+324T563904 T^{2} + 324T - 563904 Copy content Toggle raw display
8989 T2+756T+17172 T^{2} + 756T + 17172 Copy content Toggle raw display
9797 T2+452T842876 T^{2} + 452T - 842876 Copy content Toggle raw display
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