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Properties

Label 392.6.i.a
Level $392$
Weight $6$
Character orbit 392.i
Analytic conductor $62.870$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,6,Mod(177,392)] 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("392.177"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-30,0,-32,0,0,0,-657] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (30 \zeta_{6} - 30) q^{3} - 32 \zeta_{6} q^{5} - 657 \zeta_{6} q^{9} + ( - 624 \zeta_{6} + 624) q^{11} - 708 q^{13} + 960 q^{15} + (934 \zeta_{6} - 934) q^{17} - 1858 \zeta_{6} q^{19} + 1120 \zeta_{6} q^{23} + \cdots - 409968 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 30 q^{3} - 32 q^{5} - 657 q^{9} + 624 q^{11} - 1416 q^{13} + 1920 q^{15} - 934 q^{17} - 1858 q^{19} + 1120 q^{23} + 2101 q^{25} + 24840 q^{27} - 2348 q^{29} - 2908 q^{31} + 18720 q^{33} + 12462 q^{37}+ \cdots - 819936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −15.0000 − 25.9808i 0 −16.0000 + 27.7128i 0 0 0 −328.500 + 568.979i 0
361.1 0 −15.0000 + 25.9808i 0 −16.0000 − 27.7128i 0 0 0 −328.500 − 568.979i 0
\(n\): e.g. 2-40 or 990-1000
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 392.6.i.a 2
7.b odd 2 1 392.6.i.f 2
7.c even 3 1 56.6.a.b ✓ 1
7.c even 3 1 inner 392.6.i.a 2
7.d odd 6 1 392.6.a.a 1
7.d odd 6 1 392.6.i.f 2
21.h odd 6 1 504.6.a.b 1
28.f even 6 1 784.6.a.n 1
28.g odd 6 1 112.6.a.a 1
56.k odd 6 1 448.6.a.p 1
56.p even 6 1 448.6.a.a 1
84.n even 6 1 1008.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.b ✓ 1 7.c even 3 1
112.6.a.a 1 28.g odd 6 1
392.6.a.a 1 7.d odd 6 1
392.6.i.a 2 1.a even 1 1 trivial
392.6.i.a 2 7.c even 3 1 inner
392.6.i.f 2 7.b odd 2 1
392.6.i.f 2 7.d odd 6 1
448.6.a.a 1 56.p even 6 1
448.6.a.p 1 56.k odd 6 1
504.6.a.b 1 21.h odd 6 1
784.6.a.n 1 28.f even 6 1
1008.6.a.h 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 30T_{3} + 900 \) acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$5$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 624T + 389376 \) Copy content Toggle raw display
$13$ \( (T + 708)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 934T + 872356 \) Copy content Toggle raw display
$19$ \( T^{2} + 1858 T + 3452164 \) Copy content Toggle raw display
$23$ \( T^{2} - 1120 T + 1254400 \) Copy content Toggle raw display
$29$ \( (T + 1174)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2908 T + 8456464 \) Copy content Toggle raw display
$37$ \( T^{2} - 12462 T + 155301444 \) Copy content Toggle raw display
$41$ \( (T - 2662)^{2} \) Copy content Toggle raw display
$43$ \( (T + 7144)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 7468 T + 55771024 \) Copy content Toggle raw display
$53$ \( T^{2} - 27274 T + 743871076 \) Copy content Toggle raw display
$59$ \( T^{2} + 2490 T + 6200100 \) Copy content Toggle raw display
$61$ \( T^{2} - 11096 T + 123121216 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1580539536 \) Copy content Toggle raw display
$71$ \( (T + 69888)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16450 T + 270602500 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 6142797376 \) Copy content Toggle raw display
$83$ \( (T - 109818)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3245125156 \) Copy content Toggle raw display
$97$ \( (T + 115946)^{2} \) Copy content Toggle raw display
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