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Properties

Label 363.6.a.c
Level $363$
Weight $6$
Character orbit 363.a
Self dual yes
Analytic conductor $58.219$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [363,6,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,-9,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(58.2193265921\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 9 q^{3} - 28 q^{4} + 46 q^{5} - 18 q^{6} - 148 q^{7} - 120 q^{8} + 81 q^{9} + 92 q^{10} + 252 q^{12} - 574 q^{13} - 296 q^{14} - 414 q^{15} + 656 q^{16} + 722 q^{17} + 162 q^{18} - 2160 q^{19}+ \cdots + 10194 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 −9.00000 −28.0000 46.0000 −18.0000 −148.000 −120.000 81.0000 92.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( -1 \)

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 363.6.a.c 1
3.b odd 2 1 1089.6.a.d 1
11.b odd 2 1 33.6.a.a ✓ 1
33.d even 2 1 99.6.a.b 1
44.c even 2 1 528.6.a.i 1
55.d odd 2 1 825.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a ✓ 1 11.b odd 2 1
99.6.a.b 1 33.d even 2 1
363.6.a.c 1 1.a even 1 1 trivial
528.6.a.i 1 44.c even 2 1
825.6.a.b 1 55.d odd 2 1
1089.6.a.d 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T - 46 \) Copy content Toggle raw display
$7$ \( T + 148 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 574 \) Copy content Toggle raw display
$17$ \( T - 722 \) Copy content Toggle raw display
$19$ \( T + 2160 \) Copy content Toggle raw display
$23$ \( T + 2536 \) Copy content Toggle raw display
$29$ \( T + 4650 \) Copy content Toggle raw display
$31$ \( T - 5032 \) Copy content Toggle raw display
$37$ \( T - 8118 \) Copy content Toggle raw display
$41$ \( T - 5138 \) Copy content Toggle raw display
$43$ \( T + 8304 \) Copy content Toggle raw display
$47$ \( T - 24728 \) Copy content Toggle raw display
$53$ \( T + 28746 \) Copy content Toggle raw display
$59$ \( T + 5860 \) Copy content Toggle raw display
$61$ \( T - 53658 \) Copy content Toggle raw display
$67$ \( T - 30908 \) Copy content Toggle raw display
$71$ \( T + 69648 \) Copy content Toggle raw display
$73$ \( T - 18446 \) Copy content Toggle raw display
$79$ \( T - 25300 \) Copy content Toggle raw display
$83$ \( T - 17556 \) Copy content Toggle raw display
$89$ \( T - 132570 \) Copy content Toggle raw display
$97$ \( T - 70658 \) Copy content Toggle raw display
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