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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [650,2,Mod(101,650)] 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("650.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,4,0,0] 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots + 1) q^{3} + \beta_{6} q^{4} + ( - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{7} - \beta_{2}) q^{8}+ \cdots + (3 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 4 q^{4} - 6 q^{7} + 2 q^{9} + 12 q^{11} - 4 q^{12} + 10 q^{13} - 4 q^{16} + 6 q^{17} + 6 q^{19} - 6 q^{22} + 6 q^{26} + 4 q^{27} - 6 q^{28} - 12 q^{29} - 24 q^{33} - 2 q^{36} + 6 q^{37}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 10\nu^{2} - 16\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 18\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} + 2\nu^{5} + 7\nu^{4} - 8\nu^{3} - 9\nu^{2} + 28\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} - 3\beta_{6} - 5\beta_{5} + \beta_{4} - 4\beta_{3} + 3\beta_{2} + 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -6\beta_{7} - 8\beta_{6} - 2\beta_{5} + 4\beta_{4} + 2\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.665665 + 1.24775i
1.20036 − 0.747754i
−1.27597 − 0.609843i
1.40994 + 0.109843i
0.665665 − 1.24775i
1.20036 + 0.747754i
−1.27597 + 0.609843i
1.40994 − 0.109843i
−0.866025 + 0.500000i −1.24775 − 2.16117i 0.500000 − 0.866025i 0 2.16117 + 1.24775i −0.286941 − 0.165665i 1.00000i −1.61378 + 2.79515i 0
101.2 −0.866025 + 0.500000i 0.747754 + 1.29515i 0.500000 − 0.866025i 0 −1.29515 − 0.747754i −1.21306 − 0.700360i 1.00000i 0.381728 − 0.661173i 0
101.3 0.866025 − 0.500000i −0.609843 − 1.05628i 0.500000 − 0.866025i 0 −1.05628 − 0.609843i −3.07606 − 1.77597i − 1.00000i 0.756182 − 1.30975i 0
101.4 0.866025 − 0.500000i 0.109843 + 0.190254i 0.500000 − 0.866025i 0 0.190254 + 0.109843i 1.57606 + 0.909941i − 1.00000i 1.47587 − 2.55628i 0
251.1 −0.866025 − 0.500000i −1.24775 + 2.16117i 0.500000 + 0.866025i 0 2.16117 − 1.24775i −0.286941 + 0.165665i − 1.00000i −1.61378 − 2.79515i 0
251.2 −0.866025 − 0.500000i 0.747754 − 1.29515i 0.500000 + 0.866025i 0 −1.29515 + 0.747754i −1.21306 + 0.700360i − 1.00000i 0.381728 + 0.661173i 0
251.3 0.866025 + 0.500000i −0.609843 + 1.05628i 0.500000 + 0.866025i 0 −1.05628 + 0.609843i −3.07606 + 1.77597i 1.00000i 0.756182 + 1.30975i 0
251.4 0.866025 + 0.500000i 0.109843 − 0.190254i 0.500000 + 0.866025i 0 0.190254 − 0.109843i 1.57606 − 0.909941i 1.00000i 1.47587 + 2.55628i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 101.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.m.b ✓ 8
5.b even 2 1 650.2.m.d yes 8
5.c odd 4 1 650.2.n.c 8
5.c odd 4 1 650.2.n.f 8
13.e even 6 1 inner 650.2.m.b ✓ 8
13.f odd 12 1 8450.2.a.ck 4
13.f odd 12 1 8450.2.a.co 4
65.l even 6 1 650.2.m.d yes 8
65.r odd 12 1 650.2.n.c 8
65.r odd 12 1 650.2.n.f 8
65.s odd 12 1 8450.2.a.ch 4
65.s odd 12 1 8450.2.a.cl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.m.b ✓ 8 1.a even 1 1 trivial
650.2.m.b ✓ 8 13.e even 6 1 inner
650.2.m.d yes 8 5.b even 2 1
650.2.m.d yes 8 65.l even 6 1
650.2.n.c 8 5.c odd 4 1
650.2.n.c 8 65.r odd 12 1
650.2.n.f 8 5.c odd 4 1
650.2.n.f 8 65.r odd 12 1
8450.2.a.ch 4 65.s odd 12 1
8450.2.a.ck 4 13.f odd 12 1
8450.2.a.cl 4 65.s odd 12 1
8450.2.a.co 4 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} + 7T_{3}^{6} + 2T_{3}^{5} + 16T_{3}^{4} + 8T_{3}^{3} + 19T_{3}^{2} - 4T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + \cdots + 42849 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 84681 \) Copy content Toggle raw display
$23$ \( T^{8} + 42 T^{6} + \cdots + 1521 \) Copy content Toggle raw display
$29$ \( T^{8} + 12 T^{7} + \cdots + 576081 \) Copy content Toggle raw display
$31$ \( T^{8} + 120 T^{6} + \cdots + 389376 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + \cdots + 558009 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( T^{8} + 138 T^{6} + \cdots + 281961 \) Copy content Toggle raw display
$53$ \( (T^{4} - 6 T^{3} + \cdots - 507)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 30 T^{7} + \cdots + 178929 \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{7} + \cdots + 5031049 \) Copy content Toggle raw display
$67$ \( T^{8} - 30 T^{7} + \cdots + 8071281 \) Copy content Toggle raw display
$71$ \( T^{8} - 180 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$73$ \( T^{8} + 192 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 120 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$89$ \( T^{8} + 6 T^{7} + \cdots + 11881809 \) Copy content Toggle raw display
$97$ \( T^{8} + 18 T^{7} + \cdots + 7469289 \) Copy content Toggle raw display
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