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Properties

Label 392.6.i.j
Level $392$
Weight $6$
Character orbit 392.i
Analytic conductor $62.870$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,6,Mod(177,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-115})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 28x^{2} - 29x + 841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 3 \beta_1 + 3) q^{3} + (3 \beta_{3} - 3 \beta_{2} + 41 \beta_1) q^{5} + ( - 6 \beta_{3} + 6 \beta_{2} + 111 \beta_1) q^{9} + ( - 6 \beta_{3} - 170 \beta_1 - 170) q^{11} + ( - 39 \beta_{2} + 455) q^{13}+ \cdots + ( - 354 \beta_{2} + 6450) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 82 q^{5} - 222 q^{9} - 340 q^{11} + 1820 q^{13} + 3648 q^{15} - 3216 q^{17} + 674 q^{19} + 1104 q^{23} - 3322 q^{25} - 6696 q^{27} + 16128 q^{29} + 6212 q^{31} - 3120 q^{33} + 8512 q^{37}+ \cdots + 25800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 28x^{2} - 29x + 841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 28\nu^{2} - 28\nu - 841 ) / 812 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 2\nu^{2} + 114\nu + 29 ) / 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\nu^{3} + 28\nu^{2} + 1596\nu - 3335 ) / 812 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 171\beta _1 + 171 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 28\beta_{3} - 14\beta_{2} + 129 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\)

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.

comment: embeddings in the coefficient field
 
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
4.89354 + 2.24794i
−4.39354 − 3.11396i
4.89354 − 2.24794i
−4.39354 + 3.11396i
0 −7.78709 − 13.4876i 0 −48.3613 + 83.7642i 0 0 0 0.222527 − 0.385428i 0
177.2 0 10.7871 + 18.6838i 0 7.36126 − 12.7501i 0 0 0 −111.223 + 192.643i 0
361.1 0 −7.78709 + 13.4876i 0 −48.3613 − 83.7642i 0 0 0 0.222527 + 0.385428i 0
361.2 0 10.7871 − 18.6838i 0 7.36126 + 12.7501i 0 0 0 −111.223 − 192.643i 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.j 4
7.b odd 2 1 392.6.i.i 4
7.c even 3 1 56.6.a.e ✓ 2
7.c even 3 1 inner 392.6.i.j 4
7.d odd 6 1 392.6.a.d 2
7.d odd 6 1 392.6.i.i 4
21.h odd 6 1 504.6.a.i 2
28.f even 6 1 784.6.a.u 2
28.g odd 6 1 112.6.a.i 2
56.k odd 6 1 448.6.a.t 2
56.p even 6 1 448.6.a.v 2
84.n even 6 1 1008.6.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e ✓ 2 7.c even 3 1
112.6.a.i 2 28.g odd 6 1
392.6.a.d 2 7.d odd 6 1
392.6.i.i 4 7.b odd 2 1
392.6.i.i 4 7.d odd 6 1
392.6.i.j 4 1.a even 1 1 trivial
392.6.i.j 4 7.c even 3 1 inner
448.6.a.t 2 56.k odd 6 1
448.6.a.v 2 56.p even 6 1
504.6.a.i 2 21.h odd 6 1
784.6.a.u 2 28.f even 6 1
1008.6.a.bd 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6T_{3}^{3} + 372T_{3}^{2} + 2016T_{3} + 112896 \) acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + \cdots + 112896 \) Copy content Toggle raw display
$5$ \( T^{4} + 82 T^{3} + \cdots + 2027776 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 340 T^{3} + \cdots + 271590400 \) Copy content Toggle raw display
$13$ \( (T^{2} - 910 T - 317720)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 6621584683536 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 342290523136 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 2201591218176 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8064 T + 11773404)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 30994428445696 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} + 1304 T - 158165876)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10004 T - 194677376)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 69\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 89720 T + 967897600)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 52\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{2} - 35782 T - 3779136224)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + 69984 T + 1150801884)^{2} \) Copy content Toggle raw display
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