Properties

Label 130.3.h.b
Level 130130
Weight 33
Character orbit 130.h
Analytic conductor 3.5423.542
Analytic rank 00
Dimension 22
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] = [130,3,Mod(77,130)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(130, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("130.77"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 130=2513 130 = 2 \cdot 5 \cdot 13
Weight: k k == 3 3
Character orbit: [χ][\chi] == 130.h (of order 44, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 3.542243436683.54224343668
Analytic rank: 00
Dimension: 22
Coefficient field: Q(1)\Q(\sqrt{-1})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
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 i=1i = \sqrt{-1}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(i+1)q2+(2i2)q32iq4+5iq5+4iq6+(2i2)q7+(2i2)q8+iq9+(5i+5)q10+20iq11+(4i+4)q1213iq13+4iq14+20q99+O(q100) q + ( - i + 1) q^{2} + (2 i - 2) q^{3} - 2 i q^{4} + 5 i q^{5} + 4 i q^{6} + (2 i - 2) q^{7} + ( - 2 i - 2) q^{8} + i q^{9} + (5 i + 5) q^{10} + 20 i q^{11} + (4 i + 4) q^{12} - 13 i q^{13} + 4 i q^{14} + \cdots - 20 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+2q24q34q74q8+10q10+8q1220q158q16+34q17+2q1816q19+20q20+40q22+36q23+16q2450q2526q2640q27+40q99+O(q100) 2 q + 2 q^{2} - 4 q^{3} - 4 q^{7} - 4 q^{8} + 10 q^{10} + 8 q^{12} - 20 q^{15} - 8 q^{16} + 34 q^{17} + 2 q^{18} - 16 q^{19} + 20 q^{20} + 40 q^{22} + 36 q^{23} + 16 q^{24} - 50 q^{25} - 26 q^{26} - 40 q^{27}+ \cdots - 40 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/130Z)×\left(\mathbb{Z}/130\mathbb{Z}\right)^\times.

nn 2727 4141
χ(n)\chi(n) ii 1-1

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}
77.1
1.00000i
1.00000i
1.00000 1.00000i −2.00000 + 2.00000i 2.00000i 5.00000i 4.00000i −2.00000 + 2.00000i −2.00000 2.00000i 1.00000i 5.00000 + 5.00000i
103.1 1.00000 + 1.00000i −2.00000 2.00000i 2.00000i 5.00000i 4.00000i −2.00000 2.00000i −2.00000 + 2.00000i 1.00000i 5.00000 5.00000i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.h odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.h.b yes 2
5.b even 2 1 650.3.h.a 2
5.c odd 4 1 130.3.h.a 2
5.c odd 4 1 650.3.h.b 2
13.b even 2 1 130.3.h.a 2
65.d even 2 1 650.3.h.b 2
65.h odd 4 1 inner 130.3.h.b yes 2
65.h odd 4 1 650.3.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.h.a 2 5.c odd 4 1
130.3.h.a 2 13.b even 2 1
130.3.h.b yes 2 1.a even 1 1 trivial
130.3.h.b yes 2 65.h odd 4 1 inner
650.3.h.a 2 5.b even 2 1
650.3.h.a 2 65.h odd 4 1
650.3.h.b 2 5.c odd 4 1
650.3.h.b 2 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(130,[χ])S_{3}^{\mathrm{new}}(130, [\chi]):

T32+4T3+8 T_{3}^{2} + 4T_{3} + 8 Copy content Toggle raw display
T72+4T7+8 T_{7}^{2} + 4T_{7} + 8 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T22T+2 T^{2} - 2T + 2 Copy content Toggle raw display
33 T2+4T+8 T^{2} + 4T + 8 Copy content Toggle raw display
55 T2+25 T^{2} + 25 Copy content Toggle raw display
77 T2+4T+8 T^{2} + 4T + 8 Copy content Toggle raw display
1111 T2+400 T^{2} + 400 Copy content Toggle raw display
1313 T2+169 T^{2} + 169 Copy content Toggle raw display
1717 T234T+578 T^{2} - 34T + 578 Copy content Toggle raw display
1919 (T+8)2 (T + 8)^{2} Copy content Toggle raw display
2323 T236T+648 T^{2} - 36T + 648 Copy content Toggle raw display
2929 T2+1600 T^{2} + 1600 Copy content Toggle raw display
3131 T2+1600 T^{2} + 1600 Copy content Toggle raw display
3737 T286T+3698 T^{2} - 86T + 3698 Copy content Toggle raw display
4141 T2+100 T^{2} + 100 Copy content Toggle raw display
4343 T2+4T+8 T^{2} + 4T + 8 Copy content Toggle raw display
4747 T2+44T+968 T^{2} + 44T + 968 Copy content Toggle raw display
5353 T226T+338 T^{2} - 26T + 338 Copy content Toggle raw display
5959 (T32)2 (T - 32)^{2} Copy content Toggle raw display
6161 (T112)2 (T - 112)^{2} Copy content Toggle raw display
6767 T236T+648 T^{2} - 36T + 648 Copy content Toggle raw display
7171 T2+1600 T^{2} + 1600 Copy content Toggle raw display
7373 T214T+98 T^{2} - 14T + 98 Copy content Toggle raw display
7979 T2+14400 T^{2} + 14400 Copy content Toggle raw display
8383 T244T+968 T^{2} - 44T + 968 Copy content Toggle raw display
8989 (T82)2 (T - 82)^{2} Copy content Toggle raw display
9797 T266T+2178 T^{2} - 66T + 2178 Copy content Toggle raw display
show more
show less