Properties

Label 392.2.i.g
Level $392$
Weight $2$
Character orbit 392.i
Analytic conductor $3.130$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1) q^{5} + 5 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1) q^{5} + 5 \beta_{2} q^{9} + (4 \beta_{2} + 4) q^{11} - \beta_{3} q^{13} + 8 q^{15} - 2 \beta_1 q^{17} + (\beta_{3} + \beta_1) q^{19} + ( - 3 \beta_{2} - 3) q^{25} + 2 \beta_{3} q^{27} + 2 q^{29} - 2 \beta_1 q^{31} + (4 \beta_{3} + 4 \beta_1) q^{33} + 10 \beta_{2} q^{37} + (8 \beta_{2} + 8) q^{39} + 2 \beta_{3} q^{41} - 4 q^{43} + 5 \beta_1 q^{45} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} - 16 \beta_{2} q^{51} + ( - 6 \beta_{2} - 6) q^{53} - 4 \beta_{3} q^{55} - 8 q^{57} - \beta_1 q^{59} + ( - 5 \beta_{3} - 5 \beta_1) q^{61} - 8 \beta_{2} q^{65} + ( - 12 \beta_{2} - 12) q^{67} + ( - 3 \beta_{3} - 3 \beta_1) q^{75} + 8 \beta_{2} q^{79} + ( - \beta_{2} - 1) q^{81} - 5 \beta_{3} q^{83} - 16 q^{85} + 2 \beta_1 q^{87} - 16 \beta_{2} q^{93} + (8 \beta_{2} + 8) q^{95} + 2 \beta_{3} q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{9} + 8 q^{11} + 32 q^{15} - 6 q^{25} + 8 q^{29} - 20 q^{37} + 16 q^{39} - 16 q^{43} + 32 q^{51} - 12 q^{53} - 32 q^{57} + 16 q^{65} - 24 q^{67} - 16 q^{79} - 2 q^{81} - 64 q^{85} + 32 q^{93} + 16 q^{95} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −1.41421 2.44949i 0 −1.41421 + 2.44949i 0 0 0 −2.50000 + 4.33013i 0
177.2 0 1.41421 + 2.44949i 0 1.41421 2.44949i 0 0 0 −2.50000 + 4.33013i 0
361.1 0 −1.41421 + 2.44949i 0 −1.41421 2.44949i 0 0 0 −2.50000 4.33013i 0
361.2 0 1.41421 2.44949i 0 1.41421 + 2.44949i 0 0 0 −2.50000 4.33013i 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.i.g 4
3.b odd 2 1 3528.2.s.be 4
4.b odd 2 1 784.2.i.k 4
7.b odd 2 1 inner 392.2.i.g 4
7.c even 3 1 392.2.a.h 2
7.c even 3 1 inner 392.2.i.g 4
7.d odd 6 1 392.2.a.h 2
7.d odd 6 1 inner 392.2.i.g 4
21.c even 2 1 3528.2.s.be 4
21.g even 6 1 3528.2.a.bj 2
21.g even 6 1 3528.2.s.be 4
21.h odd 6 1 3528.2.a.bj 2
21.h odd 6 1 3528.2.s.be 4
28.d even 2 1 784.2.i.k 4
28.f even 6 1 784.2.a.n 2
28.f even 6 1 784.2.i.k 4
28.g odd 6 1 784.2.a.n 2
28.g odd 6 1 784.2.i.k 4
35.i odd 6 1 9800.2.a.bw 2
35.j even 6 1 9800.2.a.bw 2
56.j odd 6 1 3136.2.a.bt 2
56.k odd 6 1 3136.2.a.bq 2
56.m even 6 1 3136.2.a.bq 2
56.p even 6 1 3136.2.a.bt 2
84.j odd 6 1 7056.2.a.cj 2
84.n even 6 1 7056.2.a.cj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.h 2 7.c even 3 1
392.2.a.h 2 7.d odd 6 1
392.2.i.g 4 1.a even 1 1 trivial
392.2.i.g 4 7.b odd 2 1 inner
392.2.i.g 4 7.c even 3 1 inner
392.2.i.g 4 7.d odd 6 1 inner
784.2.a.n 2 28.f even 6 1
784.2.a.n 2 28.g odd 6 1
784.2.i.k 4 4.b odd 2 1
784.2.i.k 4 28.d even 2 1
784.2.i.k 4 28.f even 6 1
784.2.i.k 4 28.g odd 6 1
3136.2.a.bq 2 56.k odd 6 1
3136.2.a.bq 2 56.m even 6 1
3136.2.a.bt 2 56.j odd 6 1
3136.2.a.bt 2 56.p even 6 1
3528.2.a.bj 2 21.g even 6 1
3528.2.a.bj 2 21.h odd 6 1
3528.2.s.be 4 3.b odd 2 1
3528.2.s.be 4 21.c even 2 1
3528.2.s.be 4 21.g even 6 1
3528.2.s.be 4 21.h odd 6 1
7056.2.a.cj 2 84.j odd 6 1
7056.2.a.cj 2 84.n even 6 1
9800.2.a.bw 2 35.i odd 6 1
9800.2.a.bw 2 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
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