Properties

Label 650.4.b.l
Level 650650
Weight 44
Character orbit 650.b
Analytic conductor 38.35138.351
Analytic rank 00
Dimension 44
Inner twists 22

Related objects

Downloads

Learn more

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(599,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.599"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: N N == 650=25213 650 = 2 \cdot 5^{2} \cdot 13
Weight: k k == 4 4
Character orbit: [χ][\chi] == 650.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-16,0,-8,0,0,28,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 38.351241503738.3512415037
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,19)\Q(i, \sqrt{19})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x49x2+25 x^{4} - 9x^{2} + 25 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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 a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q2β1q2+(β3β1)q34q4+(2β22)q6+6β3q7+8β1q8+(2β2+7)q9+(7β225)q11+(4β3+4β1)q12++(β2+91)q99+O(q100) q - 2 \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} - 4 q^{4} + (2 \beta_{2} - 2) q^{6} + 6 \beta_{3} q^{7} + 8 \beta_1 q^{8} + (2 \beta_{2} + 7) q^{9} + (7 \beta_{2} - 25) q^{11} + ( - 4 \beta_{3} + 4 \beta_1) q^{12}+ \cdots + ( - \beta_{2} + 91) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q16q48q6+28q9100q11+64q16+188q19456q21+32q24104q26+520q29804q31352q34112q3652q391072q41+400q44++364q99+O(q100) 4 q - 16 q^{4} - 8 q^{6} + 28 q^{9} - 100 q^{11} + 64 q^{16} + 188 q^{19} - 456 q^{21} + 32 q^{24} - 104 q^{26} + 520 q^{29} - 804 q^{31} - 352 q^{34} - 112 q^{36} - 52 q^{39} - 1072 q^{41} + 400 q^{44}+ \cdots + 364 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x49x2+25 x^{4} - 9x^{2} + 25 : Copy content Toggle raw display

β1\beta_{1}== (ν34ν)/5 ( \nu^{3} - 4\nu ) / 5 Copy content Toggle raw display
β2\beta_{2}== (ν3+14ν)/5 ( -\nu^{3} + 14\nu ) / 5 Copy content Toggle raw display
β3\beta_{3}== 2ν29 2\nu^{2} - 9 Copy content Toggle raw display
ν\nu== (β2+β1)/2 ( \beta_{2} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β3+9)/2 ( \beta_{3} + 9 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== 2β2+7β1 2\beta_{2} + 7\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/650Z)×\left(\mathbb{Z}/650\mathbb{Z}\right)^\times.

nn 2727 301301
χ(n)\chi(n) 1-1 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}
599.1
−2.17945 + 0.500000i
2.17945 + 0.500000i
2.17945 0.500000i
−2.17945 0.500000i
2.00000i 5.35890i −4.00000 0 −10.7178 26.1534i 8.00000i −1.71780 0
599.2 2.00000i 3.35890i −4.00000 0 6.71780 26.1534i 8.00000i 15.7178 0
599.3 2.00000i 3.35890i −4.00000 0 6.71780 26.1534i 8.00000i 15.7178 0
599.4 2.00000i 5.35890i −4.00000 0 −10.7178 26.1534i 8.00000i −1.71780 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.4.b.l 4
5.b even 2 1 inner 650.4.b.l 4
5.c odd 4 1 130.4.a.d 2
5.c odd 4 1 650.4.a.r 2
15.e even 4 1 1170.4.a.ba 2
20.e even 4 1 1040.4.a.i 2
65.h odd 4 1 1690.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.4.a.d 2 5.c odd 4 1
650.4.a.r 2 5.c odd 4 1
650.4.b.l 4 1.a even 1 1 trivial
650.4.b.l 4 5.b even 2 1 inner
1040.4.a.i 2 20.e even 4 1
1170.4.a.ba 2 15.e even 4 1
1690.4.a.s 2 65.h odd 4 1

Hecke kernels

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

T34+40T32+324 T_{3}^{4} + 40T_{3}^{2} + 324 Copy content Toggle raw display
T72+684 T_{7}^{2} + 684 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
33 T4+40T2+324 T^{4} + 40T^{2} + 324 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 (T2+684)2 (T^{2} + 684)^{2} Copy content Toggle raw display
1111 (T2+50T306)2 (T^{2} + 50 T - 306)^{2} Copy content Toggle raw display
1313 (T2+169)2 (T^{2} + 169)^{2} Copy content Toggle raw display
1717 T4+11320T2+3196944 T^{4} + 11320 T^{2} + 3196944 Copy content Toggle raw display
1919 (T294T2066)2 (T^{2} - 94 T - 2066)^{2} Copy content Toggle raw display
2323 T4+7784T2+12602500 T^{4} + 7784 T^{2} + 12602500 Copy content Toggle raw display
2929 (T2260T+7704)2 (T^{2} - 260 T + 7704)^{2} Copy content Toggle raw display
3131 (T2+402T+40382)2 (T^{2} + 402 T + 40382)^{2} Copy content Toggle raw display
3737 T4+90112T2+37748736 T^{4} + 90112 T^{2} + 37748736 Copy content Toggle raw display
4141 (T2+536T+49860)2 (T^{2} + 536 T + 49860)^{2} Copy content Toggle raw display
4343 T4++22361613444 T^{4} + \cdots + 22361613444 Copy content Toggle raw display
4747 T4++19453554576 T^{4} + \cdots + 19453554576 Copy content Toggle raw display
5353 T4++71513456400 T^{4} + \cdots + 71513456400 Copy content Toggle raw display
5959 (T2330T+24014)2 (T^{2} - 330 T + 24014)^{2} Copy content Toggle raw display
6161 (T2620T+13336)2 (T^{2} - 620 T + 13336)^{2} Copy content Toggle raw display
6767 T4++57237691536 T^{4} + \cdots + 57237691536 Copy content Toggle raw display
7171 (T2+734T+131478)2 (T^{2} + 734 T + 131478)^{2} Copy content Toggle raw display
7373 T4++210717721600 T^{4} + \cdots + 210717721600 Copy content Toggle raw display
7979 (T2924T51112)2 (T^{2} - 924 T - 51112)^{2} Copy content Toggle raw display
8383 T4++1759390016400 T^{4} + \cdots + 1759390016400 Copy content Toggle raw display
8989 (T2748T+71476)2 (T^{2} - 748 T + 71476)^{2} Copy content Toggle raw display
9797 T4++79625552400 T^{4} + \cdots + 79625552400 Copy content Toggle raw display
show more
show less