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Properties

Label 462.2.i.g
Level $462$
Weight $2$
Character orbit 462.i
Analytic conductor $3.689$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [462,2,Mod(67,462)] 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("462.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(462, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3,3,-3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.21870000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_1) q^{5} - q^{6} + (\beta_{3} + \beta_{2} + 1) q^{7} + q^{8} + \beta_{2} q^{9} + ( - \beta_{3} + \beta_1 - 1) q^{10}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 6 q^{6} + 6 q^{8} - 3 q^{9} + 3 q^{11} + 3 q^{12} - 3 q^{14} - 3 q^{16} - 3 q^{17} - 3 q^{18} + 9 q^{19} - 3 q^{21} - 6 q^{22} - 3 q^{23} + 3 q^{24} - 15 q^{25} - 6 q^{27}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 3.23735i
0.500000 − 3.05087i
0.500000 + 0.679547i
0.500000 − 3.23735i
0.500000 + 3.05087i
0.500000 − 0.679547i
−0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i −2.20942 − 3.82682i −1.00000 2.20942 − 1.45550i 1.00000 −0.500000 − 0.866025i −2.20942 + 3.82682i
67.2 −0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 0.806615 + 1.39710i −1.00000 −0.806615 − 2.51980i 1.00000 −0.500000 − 0.866025i 0.806615 − 1.39710i
67.3 −0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.40280 + 2.42972i −1.00000 −1.40280 + 2.24325i 1.00000 −0.500000 − 0.866025i 1.40280 − 2.42972i
331.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i −2.20942 + 3.82682i −1.00000 2.20942 + 1.45550i 1.00000 −0.500000 + 0.866025i −2.20942 − 3.82682i
331.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 0.806615 − 1.39710i −1.00000 −0.806615 + 2.51980i 1.00000 −0.500000 + 0.866025i 0.806615 + 1.39710i
331.3 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 1.40280 − 2.42972i −1.00000 −1.40280 − 2.24325i 1.00000 −0.500000 + 0.866025i 1.40280 + 2.42972i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.i.g ✓ 6
3.b odd 2 1 1386.2.k.v 6
7.c even 3 1 inner 462.2.i.g ✓ 6
7.c even 3 1 3234.2.a.bf 3
7.d odd 6 1 3234.2.a.bh 3
21.g even 6 1 9702.2.a.dw 3
21.h odd 6 1 1386.2.k.v 6
21.h odd 6 1 9702.2.a.dv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g ✓ 6 1.a even 1 1 trivial
462.2.i.g ✓ 6 7.c even 3 1 inner
1386.2.k.v 6 3.b odd 2 1
1386.2.k.v 6 21.h odd 6 1
3234.2.a.bf 3 7.c even 3 1
3234.2.a.bh 3 7.d odd 6 1
9702.2.a.dv 3 21.h odd 6 1
9702.2.a.dw 3 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{6} + 15T_{5}^{4} - 40T_{5}^{3} + 225T_{5}^{2} - 300T_{5} + 400 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 15 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{4} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 19321 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + \cdots + 784 \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + \cdots + 7921 \) Copy content Toggle raw display
$29$ \( (T^{3} - 9 T^{2} + \cdots + 348)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + \cdots + 36864 \) Copy content Toggle raw display
$37$ \( T^{6} + 21 T^{5} + \cdots + 51984 \) Copy content Toggle raw display
$41$ \( (T^{3} - 12 T^{2} + \cdots + 306)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 3 T^{2} + \cdots + 404)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T + 64)^{3} \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} + \cdots + 2304 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + \cdots + 9604 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 44944 \) Copy content Toggle raw display
$71$ \( (T^{3} - 9 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 120 T^{4} + \cdots + 230400 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( (T^{3} - 18 T^{2} + \cdots + 164)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 12 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$97$ \( (T - 7)^{6} \) Copy content Toggle raw display
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