Properties

Label 1815.2.c.f
Level 18151815
Weight 22
Character orbit 1815.c
Analytic conductor 14.49314.493
Analytic rank 00
Dimension 88
Inner twists 44

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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] = [1815,2,Mod(364,1815)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1815, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1815.364"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1815=35112 1815 = 3 \cdot 5 \cdot 11^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1815.c (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-4,16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 14.492847966914.4928479669
Analytic rank: 00
Dimension: 88
Coefficient field: 8.0.49787136.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8+3x6+5x4+12x2+16 x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 24 2^{4}
Twist minimal: yes
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,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β6q2β3q3+β4q4+(β3+2)q5β7q62β6q7+(β6β2)q8q9+(β7+2β6)q10++(13β64β2)q98+O(q100) q + \beta_{6} q^{2} - \beta_{3} q^{3} + \beta_{4} q^{4} + ( - \beta_{3} + 2) q^{5} - \beta_{7} q^{6} - 2 \beta_{6} q^{7} + ( - \beta_{6} - \beta_{2}) q^{8} - q^{9} + ( - \beta_{7} + 2 \beta_{6}) q^{10}+ \cdots + ( - 13 \beta_{6} - 4 \beta_{2}) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q4q4+16q58q9+40q148q15+4q168q20+24q25+40q2616q31+44q34+4q3616q4524q4924q568q59+4q60+64q64+80q91+O(q100) 8 q - 4 q^{4} + 16 q^{5} - 8 q^{9} + 40 q^{14} - 8 q^{15} + 4 q^{16} - 8 q^{20} + 24 q^{25} + 40 q^{26} - 16 q^{31} + 44 q^{34} + 4 q^{36} - 16 q^{45} - 24 q^{49} - 24 q^{56} - 8 q^{59} + 4 q^{60} + 64 q^{64}+ \cdots - 80 q^{91}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+3x6+5x4+12x2+16 x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 : Copy content Toggle raw display

β1\beta_{1}== (ν7+5ν5+15ν3+42ν)/20 ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 20 Copy content Toggle raw display
β2\beta_{2}== (ν6+5ν4+15ν2+8)/20 ( -\nu^{6} + 5\nu^{4} + 15\nu^{2} + 8 ) / 20 Copy content Toggle raw display
β3\beta_{3}== (3ν75ν5+5ν316ν)/40 ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 Copy content Toggle raw display
β4\beta_{4}== (ν63ν4ν28)/4 ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 8 ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν73ν55ν34ν)/8 ( -\nu^{7} - 3\nu^{5} - 5\nu^{3} - 4\nu ) / 8 Copy content Toggle raw display
β6\beta_{6}== (7ν6+5ν4+15ν2+44)/20 ( 7\nu^{6} + 5\nu^{4} + 15\nu^{2} + 44 ) / 20 Copy content Toggle raw display
β7\beta_{7}== (ν7ν53ν36ν)/4 ( -\nu^{7} - \nu^{5} - 3\nu^{3} - 6\nu ) / 4 Copy content Toggle raw display
ν\nu== (β5β3+β1)/2 ( \beta_{5} - \beta_{3} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β6+β4+2β21)/2 ( \beta_{6} + \beta_{4} + 2\beta_{2} - 1 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (β7β5+5β3)/2 ( -\beta_{7} - \beta_{5} + 5\beta_{3} ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (2β63β4+β22)/2 ( -2\beta_{6} - 3\beta_{4} + \beta_{2} - 2 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (5β76β56β3+β1)/2 ( 5\beta_{7} - 6\beta_{5} - 6\beta_{3} + \beta_1 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== (5β65β29)/2 ( 5\beta_{6} - 5\beta_{2} - 9 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (10β7+3β53β37β1)/2 ( -10\beta_{7} + 3\beta_{5} - 3\beta_{3} - 7\beta_1 ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1815Z)×\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times.

nn 727727 12111211 16961696
χ(n)\chi(n) 1-1 11 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}
364.1
0.228425 1.39564i
−0.228425 + 1.39564i
1.09445 + 0.895644i
−1.09445 0.895644i
−1.09445 + 0.895644i
1.09445 0.895644i
−0.228425 1.39564i
0.228425 + 1.39564i
2.18890i 1.00000i −2.79129 2.00000 1.00000i −2.18890 4.37780i 1.73205i −1.00000 −2.18890 4.37780i
364.2 2.18890i 1.00000i −2.79129 2.00000 + 1.00000i 2.18890 4.37780i 1.73205i −1.00000 2.18890 4.37780i
364.3 0.456850i 1.00000i 1.79129 2.00000 1.00000i −0.456850 0.913701i 1.73205i −1.00000 −0.456850 0.913701i
364.4 0.456850i 1.00000i 1.79129 2.00000 + 1.00000i 0.456850 0.913701i 1.73205i −1.00000 0.456850 0.913701i
364.5 0.456850i 1.00000i 1.79129 2.00000 1.00000i 0.456850 0.913701i 1.73205i −1.00000 0.456850 + 0.913701i
364.6 0.456850i 1.00000i 1.79129 2.00000 + 1.00000i −0.456850 0.913701i 1.73205i −1.00000 −0.456850 + 0.913701i
364.7 2.18890i 1.00000i −2.79129 2.00000 1.00000i 2.18890 4.37780i 1.73205i −1.00000 2.18890 + 4.37780i
364.8 2.18890i 1.00000i −2.79129 2.00000 + 1.00000i −2.18890 4.37780i 1.73205i −1.00000 −2.18890 + 4.37780i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.c.f 8
5.b even 2 1 inner 1815.2.c.f 8
5.c odd 4 1 9075.2.a.cs 4
5.c odd 4 1 9075.2.a.cz 4
11.b odd 2 1 inner 1815.2.c.f 8
55.d odd 2 1 inner 1815.2.c.f 8
55.e even 4 1 9075.2.a.cs 4
55.e even 4 1 9075.2.a.cz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.f 8 1.a even 1 1 trivial
1815.2.c.f 8 5.b even 2 1 inner
1815.2.c.f 8 11.b odd 2 1 inner
1815.2.c.f 8 55.d odd 2 1 inner
9075.2.a.cs 4 5.c odd 4 1
9075.2.a.cs 4 55.e even 4 1
9075.2.a.cz 4 5.c odd 4 1
9075.2.a.cz 4 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1815,[χ])S_{2}^{\mathrm{new}}(1815, [\chi]):

T24+5T22+1 T_{2}^{4} + 5T_{2}^{2} + 1 Copy content Toggle raw display
T19228 T_{19}^{2} - 28 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4+5T2+1)2 (T^{4} + 5 T^{2} + 1)^{2} Copy content Toggle raw display
33 (T2+1)4 (T^{2} + 1)^{4} Copy content Toggle raw display
55 (T24T+5)4 (T^{2} - 4 T + 5)^{4} Copy content Toggle raw display
77 (T4+20T2+16)2 (T^{4} + 20 T^{2} + 16)^{2} Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 (T4+20T2+16)2 (T^{4} + 20 T^{2} + 16)^{2} Copy content Toggle raw display
1717 (T4+62T2+625)2 (T^{4} + 62 T^{2} + 625)^{2} Copy content Toggle raw display
1919 (T228)4 (T^{2} - 28)^{4} Copy content Toggle raw display
2323 (T4+74T2+25)2 (T^{4} + 74 T^{2} + 25)^{2} Copy content Toggle raw display
2929 (T420T2+16)2 (T^{4} - 20 T^{2} + 16)^{2} Copy content Toggle raw display
3131 (T2+4T17)4 (T^{2} + 4 T - 17)^{4} Copy content Toggle raw display
3737 (T4+92T2+16)2 (T^{4} + 92 T^{2} + 16)^{2} Copy content Toggle raw display
4141 (T248)4 (T^{2} - 48)^{4} Copy content Toggle raw display
4343 (T4+20T2+16)2 (T^{4} + 20 T^{2} + 16)^{2} Copy content Toggle raw display
4747 (T4+50T2+289)2 (T^{4} + 50 T^{2} + 289)^{2} Copy content Toggle raw display
5353 (T2+25)4 (T^{2} + 25)^{4} Copy content Toggle raw display
5959 (T2+2T20)4 (T^{2} + 2 T - 20)^{4} Copy content Toggle raw display
6161 (T275)4 (T^{2} - 75)^{4} Copy content Toggle raw display
6767 (T4+380T2+35344)2 (T^{4} + 380 T^{2} + 35344)^{2} Copy content Toggle raw display
7171 (T284)4 (T^{2} - 84)^{4} Copy content Toggle raw display
7373 (T4+272T2+6400)2 (T^{4} + 272 T^{2} + 6400)^{2} Copy content Toggle raw display
7979 (T27)4 (T^{2} - 7)^{4} Copy content Toggle raw display
8383 (T4+152T2+400)2 (T^{4} + 152 T^{2} + 400)^{2} Copy content Toggle raw display
8989 (T2+6T180)4 (T^{2} + 6 T - 180)^{4} Copy content Toggle raw display
9797 (T4+44T2+400)2 (T^{4} + 44 T^{2} + 400)^{2} Copy content Toggle raw display
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