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Properties

Label 1.12.a.a
Level 11
Weight 1212
Character orbit 1.a
Self dual yes
Analytic conductor 0.7680.768
Analytic rank 00
Dimension 11
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

This is the discriminant modular form Δ=∑τ(n)qn\Delta=\sum \tau(n)q^n, where τ\tau is the Ramanujan tau function [A000594]. It is the minimal weight newform of level 11.

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1,12,Mod(1,1)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 12, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Level: N N == 1 1
Weight: k k == 12 12
Character orbit: [χ][\chi] == 1.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 0.7683431805600.768343180560
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
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)
 
f(q)f(q) == q−24q2+252q3−1472q4+4830q5−6048q6−16744q7+84480q8−113643q9−115920q10+534612q11−370944q12−577738q13+401856q14+1217160q15+⋯−60754911516q99+O(q100) q - 24 q^{2} + 252 q^{3} - 1472 q^{4} + 4830 q^{5} - 6048 q^{6} - 16744 q^{7} + 84480 q^{8} - 113643 q^{9} - 115920 q^{10} + 534612 q^{11} - 370944 q^{12} - 577738 q^{13} + 401856 q^{14} + 1217160 q^{15}+ \cdots - 60754911516 q^{99}+O(q^{100}) Copy content Toggle raw display

Expression as an eta quotient

f(z)=η(z)24=q∏n=1∞(1−qn)24f(z) = \eta(z)^{24}=q\prod_{n=1}^\infty(1 - q^{n})^{24}

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
0
−24.0000 252.000 −1472.00 4830.00 −6048.00 −16744.0 84480.0 −113643. −115920.
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.12.a.a ✓ 1
3.b odd 2 1 9.12.a.b 1
4.b odd 2 1 16.12.a.a 1
5.b even 2 1 25.12.a.b 1
5.c odd 4 2 25.12.b.b 2
7.b odd 2 1 49.12.a.a 1
7.c even 3 2 49.12.c.b 2
7.d odd 6 2 49.12.c.c 2
8.b even 2 1 64.12.a.b 1
8.d odd 2 1 64.12.a.f 1
9.c even 3 2 81.12.c.d 2
9.d odd 6 2 81.12.c.b 2
11.b odd 2 1 121.12.a.b 1
12.b even 2 1 144.12.a.d 1
13.b even 2 1 169.12.a.a 1
15.d odd 2 1 225.12.a.b 1
15.e even 4 2 225.12.b.d 2
16.e even 4 2 256.12.b.e 2
16.f odd 4 2 256.12.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a ✓ 1 1.a even 1 1 trivial
9.12.a.b 1 3.b odd 2 1
16.12.a.a 1 4.b odd 2 1
25.12.a.b 1 5.b even 2 1
25.12.b.b 2 5.c odd 4 2
49.12.a.a 1 7.b odd 2 1
49.12.c.b 2 7.c even 3 2
49.12.c.c 2 7.d odd 6 2
64.12.a.b 1 8.b even 2 1
64.12.a.f 1 8.d odd 2 1
81.12.c.b 2 9.d odd 6 2
81.12.c.d 2 9.c even 3 2
121.12.a.b 1 11.b odd 2 1
144.12.a.d 1 12.b even 2 1
169.12.a.a 1 13.b even 2 1
225.12.a.b 1 15.d odd 2 1
225.12.b.d 2 15.e even 4 2
256.12.b.c 2 16.f odd 4 2
256.12.b.e 2 16.e even 4 2

Hecke kernels

This newform subspace is the entire newspace S12new(Γ0(1))S_{12}^{\mathrm{new}}(\Gamma_0(1)).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T+24 T + 24 Copy content Toggle raw display
33 T−252 T - 252 Copy content Toggle raw display
55 T−4830 T - 4830 Copy content Toggle raw display
77 T+16744 T + 16744 Copy content Toggle raw display
1111 T−534612 T - 534612 Copy content Toggle raw display
1313 T+577738 T + 577738 Copy content Toggle raw display
1717 T+6905934 T + 6905934 Copy content Toggle raw display
1919 T−10661420 T - 10661420 Copy content Toggle raw display
2323 T−18643272 T - 18643272 Copy content Toggle raw display
2929 T−128406630 T - 128406630 Copy content Toggle raw display
3131 T+52843168 T + 52843168 Copy content Toggle raw display
3737 T+182213314 T + 182213314 Copy content Toggle raw display
4141 T−308120442 T - 308120442 Copy content Toggle raw display
4343 T+17125708 T + 17125708 Copy content Toggle raw display
4747 T−2687348496 T - 2687348496 Copy content Toggle raw display
5353 T+1596055698 T + 1596055698 Copy content Toggle raw display
5959 T+5189203740 T + 5189203740 Copy content Toggle raw display
6161 T−6956478662 T - 6956478662 Copy content Toggle raw display
6767 T+15481826884 T + 15481826884 Copy content Toggle raw display
7171 T−9791485272 T - 9791485272 Copy content Toggle raw display
7373 T−1463791322 T - 1463791322 Copy content Toggle raw display
7979 T−38116845680 T - 38116845680 Copy content Toggle raw display
8383 T+29335099668 T + 29335099668 Copy content Toggle raw display
8989 T+24992917110 T + 24992917110 Copy content Toggle raw display
9797 T−75013568546 T - 75013568546 Copy content Toggle raw display
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Additional information

q∏n≥1(1−qn)24\displaystyle q\prod_{n\geq1}(1-q^n)^{24}

η(z)24\eta(z)^{24}