Properties

Label 650.4.a.m
Level 650650
Weight 44
Character orbit 650.a
Self dual yes
Analytic conductor 38.35138.351
Analytic rank 00
Dimension 22
CM no
Inner twists 11

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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] = [650,4,Mod(1,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, 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("650.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 650=25213 650 = 2 \cdot 5^{2} \cdot 13
Weight: k k == 4 4
Character orbit: [χ][\chi] == 650.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,7,8,0,-14,5,-16,23] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 38.351241503738.3512415037
Analytic rank: 00
Dimension: 22
Coefficient field: Q(105)\Q(\sqrt{105})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x26 x^{2} - x - 26 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
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+105)\beta = \frac{1}{2}(1 + \sqrt{105}). We also show the integral qq-expansion of the trace form.

f(q)f(q) == q2q2+(β+4)q3+4q4+(2β8)q6+(β+3)q78q8+(7β+15)q9+(5β23)q11+(4β+16)q1213q13+(2β6)q14++(121β+565)q99+O(q100) q - 2 q^{2} + ( - \beta + 4) q^{3} + 4 q^{4} + (2 \beta - 8) q^{6} + ( - \beta + 3) q^{7} - 8 q^{8} + ( - 7 \beta + 15) q^{9} + ( - 5 \beta - 23) q^{11} + ( - 4 \beta + 16) q^{12} - 13 q^{13} + (2 \beta - 6) q^{14}+ \cdots + (121 \beta + 565) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q4q2+7q3+8q414q6+5q716q8+23q951q11+28q1226q1310q14+32q16+63q1746q18+28q19+70q21+102q22+9q23++1251q99+O(q100) 2 q - 4 q^{2} + 7 q^{3} + 8 q^{4} - 14 q^{6} + 5 q^{7} - 16 q^{8} + 23 q^{9} - 51 q^{11} + 28 q^{12} - 26 q^{13} - 10 q^{14} + 32 q^{16} + 63 q^{17} - 46 q^{18} + 28 q^{19} + 70 q^{21} + 102 q^{22} + 9 q^{23}+ \cdots + 1251 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
5.62348
−4.62348
−2.00000 −1.62348 4.00000 0 3.24695 −2.62348 −8.00000 −24.3643 0
1.2 −2.00000 8.62348 4.00000 0 −17.2470 7.62348 −8.00000 47.3643 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
55 +1 +1
1313 +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 650.4.a.m 2
5.b even 2 1 650.4.a.p yes 2
5.c odd 4 2 650.4.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.4.a.m 2 1.a even 1 1 trivial
650.4.a.p yes 2 5.b even 2 1
650.4.b.i 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(650))S_{4}^{\mathrm{new}}(\Gamma_0(650)):

T327T314 T_{3}^{2} - 7T_{3} - 14 Copy content Toggle raw display
T725T720 T_{7}^{2} - 5T_{7} - 20 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
33 T27T14 T^{2} - 7T - 14 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T25T20 T^{2} - 5T - 20 Copy content Toggle raw display
1111 T2+51T6 T^{2} + 51T - 6 Copy content Toggle raw display
1313 (T+13)2 (T + 13)^{2} Copy content Toggle raw display
1717 T263T294 T^{2} - 63T - 294 Copy content Toggle raw display
1919 T228T3584 T^{2} - 28T - 3584 Copy content Toggle raw display
2323 T29T7566 T^{2} - 9T - 7566 Copy content Toggle raw display
2929 T2+198T+8121 T^{2} + 198T + 8121 Copy content Toggle raw display
3131 T2175T+4480 T^{2} - 175T + 4480 Copy content Toggle raw display
3737 T2632T+96076 T^{2} - 632T + 96076 Copy content Toggle raw display
4141 T2210T1680 T^{2} - 210T - 1680 Copy content Toggle raw display
4343 T223T7454 T^{2} - 23T - 7454 Copy content Toggle raw display
4747 T2231T78036 T^{2} - 231T - 78036 Copy content Toggle raw display
5353 T2+186T11931 T^{2} + 186T - 11931 Copy content Toggle raw display
5959 T263T294 T^{2} - 63T - 294 Copy content Toggle raw display
6161 T2+182T213899 T^{2} + 182T - 213899 Copy content Toggle raw display
6767 T2695T77930 T^{2} - 695T - 77930 Copy content Toggle raw display
7171 T2390T909600 T^{2} - 390T - 909600 Copy content Toggle raw display
7373 T21526T+453544 T^{2} - 1526 T + 453544 Copy content Toggle raw display
7979 T2+863T+5356 T^{2} + 863T + 5356 Copy content Toggle raw display
8383 T2315T557970 T^{2} - 315T - 557970 Copy content Toggle raw display
8989 T2+1638T+170856 T^{2} + 1638 T + 170856 Copy content Toggle raw display
9797 T298T829304 T^{2} - 98T - 829304 Copy content Toggle raw display
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