Properties

Label 392.4.i.k
Level $392$
Weight $4$
Character orbit 392.i
Analytic conductor $23.129$
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] = [392,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.177");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(23.1287487223\)
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: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (10 \beta_{3} + 10 \beta_1) q^{5} - 25 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (10 \beta_{3} + 10 \beta_1) q^{5} - 25 \beta_{2} q^{9} + ( - 54 \beta_{2} - 54) q^{11} - 16 \beta_{3} q^{13} - 20 q^{15} - 23 \beta_1 q^{17} + ( - 45 \beta_{3} - 45 \beta_1) q^{19} + 28 \beta_{2} q^{23} + ( - 75 \beta_{2} - 75) q^{25} - 52 \beta_{3} q^{27} + 282 q^{29} + 194 \beta_1 q^{31} + ( - 54 \beta_{3} - 54 \beta_1) q^{33} + 146 \beta_{2} q^{37} + (32 \beta_{2} + 32) q^{39} - 241 \beta_{3} q^{41} + 10 q^{43} + 250 \beta_1 q^{45} + ( - 358 \beta_{3} - 358 \beta_1) q^{47} - 46 \beta_{2} q^{51} + ( - 598 \beta_{2} - 598) q^{53} - 540 \beta_{3} q^{55} + 90 q^{57} - 407 \beta_1 q^{59} + (330 \beta_{3} + 330 \beta_1) q^{61} + 320 \beta_{2} q^{65} + ( - 916 \beta_{2} - 916) q^{67} + 28 \beta_{3} q^{69} + 420 q^{71} + 497 \beta_1 q^{73} + ( - 75 \beta_{3} - 75 \beta_1) q^{75} - 292 \beta_{2} q^{79} + ( - 571 \beta_{2} - 571) q^{81} - 815 \beta_{3} q^{83} + 460 q^{85} + 282 \beta_1 q^{87} + (315 \beta_{3} + 315 \beta_1) q^{89} + 388 \beta_{2} q^{93} + (900 \beta_{2} + 900) q^{95} + 417 \beta_{3} q^{97} - 1350 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 50 q^{9} - 108 q^{11} - 80 q^{15} - 56 q^{23} - 150 q^{25} + 1128 q^{29} - 292 q^{37} + 64 q^{39} + 40 q^{43} + 92 q^{51} - 1196 q^{53} + 360 q^{57} - 640 q^{65} - 1832 q^{67} + 1680 q^{71} + 584 q^{79} - 1142 q^{81} + 1840 q^{85} - 776 q^{93} + 1800 q^{95} - 5400 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) 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 −0.707107 1.22474i 0 7.07107 12.2474i 0 0 0 12.5000 21.6506i 0
177.2 0 0.707107 + 1.22474i 0 −7.07107 + 12.2474i 0 0 0 12.5000 21.6506i 0
361.1 0 −0.707107 + 1.22474i 0 7.07107 + 12.2474i 0 0 0 12.5000 + 21.6506i 0
361.2 0 0.707107 1.22474i 0 −7.07107 12.2474i 0 0 0 12.5000 + 21.6506i 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 392.4.i.k 4
7.b odd 2 1 inner 392.4.i.k 4
7.c even 3 1 392.4.a.f 2
7.c even 3 1 inner 392.4.i.k 4
7.d odd 6 1 392.4.a.f 2
7.d odd 6 1 inner 392.4.i.k 4
28.f even 6 1 784.4.a.v 2
28.g odd 6 1 784.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.f 2 7.c even 3 1
392.4.a.f 2 7.d odd 6 1
392.4.i.k 4 1.a even 1 1 trivial
392.4.i.k 4 7.b odd 2 1 inner
392.4.i.k 4 7.c even 3 1 inner
392.4.i.k 4 7.d odd 6 1 inner
784.4.a.v 2 28.f even 6 1
784.4.a.v 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 200T_{5}^{2} + 40000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 54 T + 2916)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 512)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 1058 T^{2} + \cdots + 1119364 \) Copy content Toggle raw display
$19$ \( T^{4} + 4050 T^{2} + \cdots + 16402500 \) Copy content Toggle raw display
$23$ \( (T^{2} + 28 T + 784)^{2} \) Copy content Toggle raw display
$29$ \( (T - 282)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 75272 T^{2} + \cdots + 5665873984 \) Copy content Toggle raw display
$37$ \( (T^{2} + 146 T + 21316)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 116162)^{2} \) Copy content Toggle raw display
$43$ \( (T - 10)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 256328 T^{2} + \cdots + 65704043584 \) Copy content Toggle raw display
$53$ \( (T^{2} + 598 T + 357604)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 331298 T^{2} + \cdots + 109758364804 \) Copy content Toggle raw display
$61$ \( T^{4} + 217800 T^{2} + \cdots + 47436840000 \) Copy content Toggle raw display
$67$ \( (T^{2} + 916 T + 839056)^{2} \) Copy content Toggle raw display
$71$ \( (T - 420)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 494018 T^{2} + \cdots + 244053784324 \) Copy content Toggle raw display
$79$ \( (T^{2} - 292 T + 85264)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1328450)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 198450 T^{2} + \cdots + 39382402500 \) Copy content Toggle raw display
$97$ \( (T^{2} - 347778)^{2} \) Copy content Toggle raw display
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