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Properties

Label 392.6.i.d
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 = [2,0,6,0,-4,0,0,0,207] 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.

\(f(q)\) \(=\) \( q + ( - 6 \zeta_{6} + 6) q^{3} - 4 \zeta_{6} q^{5} + 207 \zeta_{6} q^{9} + ( - 240 \zeta_{6} + 240) q^{11} - 744 q^{13} - 24 q^{15} + ( - 1042 \zeta_{6} + 1042) q^{17} + 986 \zeta_{6} q^{19} - 184 \zeta_{6} q^{23} + \cdots + 49680 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 4 q^{5} + 207 q^{9} + 240 q^{11} - 1488 q^{13} - 48 q^{15} + 1042 q^{17} + 986 q^{19} - 184 q^{23} + 3109 q^{25} + 5400 q^{27} - 1468 q^{29} - 5140 q^{31} - 1440 q^{33} + 6054 q^{37} - 4464 q^{39}+ \cdots + 99360 q^{99}+O(q^{100}) \) 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 3.00000 + 5.19615i 0 −2.00000 + 3.46410i 0 0 0 103.500 − 179.267i 0
361.1 0 3.00000 − 5.19615i 0 −2.00000 − 3.46410i 0 0 0 103.500 + 179.267i 0
\(n\): e.g. 2-40 or 80-90
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.d 2
7.b odd 2 1 392.6.i.c 2
7.c even 3 1 56.6.a.a ✓ 1
7.c even 3 1 inner 392.6.i.d 2
7.d odd 6 1 392.6.a.c 1
7.d odd 6 1 392.6.i.c 2
21.h odd 6 1 504.6.a.e 1
28.f even 6 1 784.6.a.e 1
28.g odd 6 1 112.6.a.f 1
56.k odd 6 1 448.6.a.g 1
56.p even 6 1 448.6.a.j 1
84.n even 6 1 1008.6.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.a ✓ 1 7.c even 3 1
112.6.a.f 1 28.g odd 6 1
392.6.a.c 1 7.d odd 6 1
392.6.i.c 2 7.b odd 2 1
392.6.i.c 2 7.d odd 6 1
392.6.i.d 2 1.a even 1 1 trivial
392.6.i.d 2 7.c even 3 1 inner
448.6.a.g 1 56.k odd 6 1
448.6.a.j 1 56.p even 6 1
504.6.a.e 1 21.h odd 6 1
784.6.a.e 1 28.f even 6 1
1008.6.a.p 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6T_{3} + 36 \) 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
$3$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 240T + 57600 \) Copy content Toggle raw display
$13$ \( (T + 744)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1042 T + 1085764 \) Copy content Toggle raw display
$19$ \( T^{2} - 986T + 972196 \) Copy content Toggle raw display
$23$ \( T^{2} + 184T + 33856 \) Copy content Toggle raw display
$29$ \( (T + 734)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5140 T + 26419600 \) Copy content Toggle raw display
$37$ \( T^{2} - 6054 T + 36650916 \) Copy content Toggle raw display
$41$ \( (T - 7598)^{2} \) Copy content Toggle raw display
$43$ \( (T - 13016)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 14668 T + 215150224 \) Copy content Toggle raw display
$53$ \( T^{2} - 14522 T + 210888484 \) Copy content Toggle raw display
$59$ \( T^{2} - 13362 T + 178543044 \) Copy content Toggle raw display
$61$ \( T^{2} + 9676 T + 93624976 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 3859391376 \) Copy content Toggle raw display
$71$ \( (T + 2112)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 28910 T + 835788100 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 10356725824 \) Copy content Toggle raw display
$83$ \( (T + 23922)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 20071522276 \) Copy content Toggle raw display
$97$ \( (T - 99982)^{2} \) Copy content Toggle raw display
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