Properties

Label 392.4.i.a
Level $392$
Weight $4$
Character orbit 392.i
Analytic conductor $23.129$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
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

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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (6 \zeta_{6} - 6) q^{3} - 8 \zeta_{6} q^{5} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (6 \zeta_{6} - 6) q^{3} - 8 \zeta_{6} q^{5} - 9 \zeta_{6} q^{9} + (56 \zeta_{6} - 56) q^{11} - 28 q^{13} + 48 q^{15} + ( - 90 \zeta_{6} + 90) q^{17} - 74 \zeta_{6} q^{19} + 96 \zeta_{6} q^{23} + ( - 61 \zeta_{6} + 61) q^{25} - 108 q^{27} - 222 q^{29} + ( - 100 \zeta_{6} + 100) q^{31} - 336 \zeta_{6} q^{33} - 58 \zeta_{6} q^{37} + ( - 168 \zeta_{6} + 168) q^{39} + 422 q^{41} + 512 q^{43} + (72 \zeta_{6} - 72) q^{45} - 148 \zeta_{6} q^{47} + 540 \zeta_{6} q^{51} + ( - 642 \zeta_{6} + 642) q^{53} + 448 q^{55} + 444 q^{57} + ( - 318 \zeta_{6} + 318) q^{59} - 720 \zeta_{6} q^{61} + 224 \zeta_{6} q^{65} + ( - 412 \zeta_{6} + 412) q^{67} - 576 q^{69} + 448 q^{71} + (994 \zeta_{6} - 994) q^{73} + 366 \zeta_{6} q^{75} + 296 \zeta_{6} q^{79} + ( - 891 \zeta_{6} + 891) q^{81} + 386 q^{83} - 720 q^{85} + ( - 1332 \zeta_{6} + 1332) q^{87} + 6 \zeta_{6} q^{89} + 600 \zeta_{6} q^{93} + (592 \zeta_{6} - 592) q^{95} - 138 q^{97} + 504 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 8 q^{5} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 8 q^{5} - 9 q^{9} - 56 q^{11} - 56 q^{13} + 96 q^{15} + 90 q^{17} - 74 q^{19} + 96 q^{23} + 61 q^{25} - 216 q^{27} - 444 q^{29} + 100 q^{31} - 336 q^{33} - 58 q^{37} + 168 q^{39} + 844 q^{41} + 1024 q^{43} - 72 q^{45} - 148 q^{47} + 540 q^{51} + 642 q^{53} + 896 q^{55} + 888 q^{57} + 318 q^{59} - 720 q^{61} + 224 q^{65} + 412 q^{67} - 1152 q^{69} + 896 q^{71} - 994 q^{73} + 366 q^{75} + 296 q^{79} + 891 q^{81} + 772 q^{83} - 1440 q^{85} + 1332 q^{87} + 6 q^{89} + 600 q^{93} - 592 q^{95} - 276 q^{97} + 1008 q^{99}+O(q^{100}) \) 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.

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
0.500000 0.866025i
0.500000 + 0.866025i
0 −3.00000 5.19615i 0 −4.00000 + 6.92820i 0 0 0 −4.50000 + 7.79423i 0
361.1 0 −3.00000 + 5.19615i 0 −4.00000 6.92820i 0 0 0 −4.50000 7.79423i 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 392.4.i.a 2
7.b odd 2 1 392.4.i.h 2
7.c even 3 1 56.4.a.b 1
7.c even 3 1 inner 392.4.i.a 2
7.d odd 6 1 392.4.a.a 1
7.d odd 6 1 392.4.i.h 2
21.h odd 6 1 504.4.a.a 1
28.f even 6 1 784.4.a.q 1
28.g odd 6 1 112.4.a.b 1
35.j even 6 1 1400.4.a.b 1
35.l odd 12 2 1400.4.g.b 2
56.k odd 6 1 448.4.a.n 1
56.p even 6 1 448.4.a.c 1
84.n even 6 1 1008.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.b 1 7.c even 3 1
112.4.a.b 1 28.g odd 6 1
392.4.a.a 1 7.d odd 6 1
392.4.i.a 2 1.a even 1 1 trivial
392.4.i.a 2 7.c even 3 1 inner
392.4.i.h 2 7.b odd 2 1
392.4.i.h 2 7.d odd 6 1
448.4.a.c 1 56.p even 6 1
448.4.a.n 1 56.k odd 6 1
504.4.a.a 1 21.h odd 6 1
784.4.a.q 1 28.f even 6 1
1008.4.a.e 1 84.n even 6 1
1400.4.a.b 1 35.j even 6 1
1400.4.g.b 2 35.l odd 12 2

Hecke kernels

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

\( T_{3}^{2} + 6T_{3} + 36 \) Copy content Toggle raw display
\( T_{5}^{2} + 8T_{5} + 64 \) 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
$5$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 56T + 3136 \) Copy content Toggle raw display
$13$ \( (T + 28)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 90T + 8100 \) Copy content Toggle raw display
$19$ \( T^{2} + 74T + 5476 \) Copy content Toggle raw display
$23$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$29$ \( (T + 222)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 100T + 10000 \) Copy content Toggle raw display
$37$ \( T^{2} + 58T + 3364 \) Copy content Toggle raw display
$41$ \( (T - 422)^{2} \) Copy content Toggle raw display
$43$ \( (T - 512)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 148T + 21904 \) Copy content Toggle raw display
$53$ \( T^{2} - 642T + 412164 \) Copy content Toggle raw display
$59$ \( T^{2} - 318T + 101124 \) Copy content Toggle raw display
$61$ \( T^{2} + 720T + 518400 \) Copy content Toggle raw display
$67$ \( T^{2} - 412T + 169744 \) Copy content Toggle raw display
$71$ \( (T - 448)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 994T + 988036 \) Copy content Toggle raw display
$79$ \( T^{2} - 296T + 87616 \) Copy content Toggle raw display
$83$ \( (T - 386)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 138)^{2} \) Copy content Toggle raw display
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