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Properties

Label 13.8.a.c
Level 1313
Weight 88
Character orbit 13.a
Self dual yes
Analytic conductor 4.0614.061
Analytic rank 00
Dimension 44
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] = [13,8,Mod(1,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 13 13
Weight: k k == 8 8
Character orbit: [χ][\chi] == 13.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 4.061005331294.06100533129
Analytic rank: 00
Dimension: 44
Coefficient field: Q[x]/(x4−⋯ )\mathbb{Q}[x]/(x^{4} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4−x3−354x2−640x+20912 x^{4} - x^{3} - 354x^{2} - 640x + 20912 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 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 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) == q+(−β1+4)q2+(β2−2β1+21)q3+(β3−2β2−4β1+63)q4+(−6β3−β2+⋯+63)q5+(11β3−4β2+⋯+391)q6+⋯+(136548β3−77786β2+⋯+1572948)q99+O(q100) q + ( - \beta_1 + 4) q^{2} + (\beta_{2} - 2 \beta_1 + 21) q^{3} + (\beta_{3} - 2 \beta_{2} - 4 \beta_1 + 63) q^{4} + ( - 6 \beta_{3} - \beta_{2} + \cdots + 63) q^{5} + (11 \beta_{3} - 4 \beta_{2} + \cdots + 391) q^{6}+ \cdots + (136548 \beta_{3} - 77786 \beta_{2} + \cdots + 1572948) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q+15q2+80q3+253q4+258q5+1579q6+1692q7+1893q8+3494q9−4495q10+1836q11−3655q12−8788q13−18285q14−29736q15−36159q16+⋯+6200852q99+O(q100) 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9} - 4495 q^{10} + 1836 q^{11} - 3655 q^{12} - 8788 q^{13} - 18285 q^{14} - 29736 q^{15} - 36159 q^{16}+ \cdots + 6200852 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4−x3−354x2−640x+20912 x^{4} - x^{3} - 354x^{2} - 640x + 20912 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3−15ν2−180ν+1956)/4 ( \nu^{3} - 15\nu^{2} - 180\nu + 1956 ) / 4 Copy content Toggle raw display
β3\beta_{3}== (ν3−13ν2−188ν+1606)/2 ( \nu^{3} - 13\nu^{2} - 188\nu + 1606 ) / 2 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3−2β2+4β1+175 \beta_{3} - 2\beta_{2} + 4\beta _1 + 175 Copy content Toggle raw display
ν3\nu^{3}== 15β3−26β2+240β1+669 15\beta_{3} - 26\beta_{2} + 240\beta _1 + 669 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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
18.6191
7.26058
−12.1994
−12.6802
−14.6191 −51.4405 85.7173 123.659 752.012 559.667 618.134 459.121 −1807.78
1.2 −3.26058 66.7551 −117.369 259.973 −217.660 1453.99 800.043 2269.24 −847.662
1.3 16.1994 71.3782 134.421 −532.467 1156.29 −6.33667 104.021 2907.85 −8625.66
1.4 16.6802 −6.69280 150.230 406.835 −111.637 −315.324 370.802 −2142.21 6786.11
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
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 13.8.a.c ✓ 4
3.b odd 2 1 117.8.a.e 4
4.b odd 2 1 208.8.a.k 4
5.b even 2 1 325.8.a.c 4
13.b even 2 1 169.8.a.c 4
13.d odd 4 2 169.8.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.a.c ✓ 4 1.a even 1 1 trivial
117.8.a.e 4 3.b odd 2 1
169.8.a.c 4 13.b even 2 1
169.8.b.c 8 13.d odd 4 2
208.8.a.k 4 4.b odd 2 1
325.8.a.c 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T24−15T23−270T22+3264T2+12880 T_{2}^{4} - 15T_{2}^{3} - 270T_{2}^{2} + 3264T_{2} + 12880 acting on S8new(Γ0(13))S_{8}^{\mathrm{new}}(\Gamma_0(13)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4−15T3+⋯+12880 T^{4} - 15 T^{3} + \cdots + 12880 Copy content Toggle raw display
33 T4−80T3+⋯+1640448 T^{4} - 80 T^{3} + \cdots + 1640448 Copy content Toggle raw display
55 T4+⋯−6964113500 T^{4} + \cdots - 6964113500 Copy content Toggle raw display
77 T4+⋯+1625961532 T^{4} + \cdots + 1625961532 Copy content Toggle raw display
1111 T4+⋯+19773464676784 T^{4} + \cdots + 19773464676784 Copy content Toggle raw display
1313 (T+2197)4 (T + 2197)^{4} Copy content Toggle raw display
1717 T4+⋯+17 ⁣⋯ ⁣84 T^{4} + \cdots + 17\!\cdots\!84 Copy content Toggle raw display
1919 T4+⋯+43 ⁣⋯ ⁣60 T^{4} + \cdots + 43\!\cdots\!60 Copy content Toggle raw display
2323 T4+⋯+20 ⁣⋯ ⁣36 T^{4} + \cdots + 20\!\cdots\!36 Copy content Toggle raw display
2929 T4+⋯−32 ⁣⋯ ⁣40 T^{4} + \cdots - 32\!\cdots\!40 Copy content Toggle raw display
3131 T4+⋯−83 ⁣⋯ ⁣08 T^{4} + \cdots - 83\!\cdots\!08 Copy content Toggle raw display
3737 T4+⋯+59 ⁣⋯ ⁣76 T^{4} + \cdots + 59\!\cdots\!76 Copy content Toggle raw display
4141 T4+⋯+19 ⁣⋯ ⁣36 T^{4} + \cdots + 19\!\cdots\!36 Copy content Toggle raw display
4343 T4+⋯+11 ⁣⋯ ⁣52 T^{4} + \cdots + 11\!\cdots\!52 Copy content Toggle raw display
4747 T4+⋯−13 ⁣⋯ ⁣60 T^{4} + \cdots - 13\!\cdots\!60 Copy content Toggle raw display
5353 T4+⋯−28 ⁣⋯ ⁣64 T^{4} + \cdots - 28\!\cdots\!64 Copy content Toggle raw display
5959 T4+⋯−44 ⁣⋯ ⁣20 T^{4} + \cdots - 44\!\cdots\!20 Copy content Toggle raw display
6161 T4+⋯+11 ⁣⋯ ⁣68 T^{4} + \cdots + 11\!\cdots\!68 Copy content Toggle raw display
6767 T4+⋯−57 ⁣⋯ ⁣08 T^{4} + \cdots - 57\!\cdots\!08 Copy content Toggle raw display
7171 T4+⋯+25 ⁣⋯ ⁣48 T^{4} + \cdots + 25\!\cdots\!48 Copy content Toggle raw display
7373 T4+⋯+69 ⁣⋯ ⁣24 T^{4} + \cdots + 69\!\cdots\!24 Copy content Toggle raw display
7979 T4+⋯+22 ⁣⋯ ⁣00 T^{4} + \cdots + 22\!\cdots\!00 Copy content Toggle raw display
8383 T4+⋯+70 ⁣⋯ ⁣76 T^{4} + \cdots + 70\!\cdots\!76 Copy content Toggle raw display
8989 T4+⋯−76 ⁣⋯ ⁣00 T^{4} + \cdots - 76\!\cdots\!00 Copy content Toggle raw display
9797 T4+⋯−44 ⁣⋯ ⁣60 T^{4} + \cdots - 44\!\cdots\!60 Copy content Toggle raw display
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