Properties

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

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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: \(\Q(\sqrt{-3}, \sqrt{193})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 7 \beta_1) q^{3} + ( - 5 \beta_{3} - 5 \beta_{2} + \cdots - 21) q^{5}+ \cdots + ( - 14 \beta_{3} - 14 \beta_{2} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 7 \beta_1) q^{3} + ( - 5 \beta_{3} - 5 \beta_{2} + \cdots - 21) q^{5}+ \cdots + ( - 4998 \beta_{3} - 37470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{3} - 42 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{3} - 42 q^{5} + 2 q^{9} + 716 q^{11} - 1428 q^{13} - 4448 q^{15} + 1344 q^{17} + 1946 q^{19} + 1792 q^{23} - 4282 q^{25} - 3976 q^{27} - 2400 q^{29} + 6804 q^{31} - 10416 q^{33} - 14640 q^{37} - 11560 q^{39} + 15792 q^{41} + 1048 q^{43} - 26978 q^{45} + 18396 q^{47} - 30252 q^{51} - 45132 q^{53} - 84112 q^{55} - 44552 q^{57} - 22582 q^{59} + 52822 q^{61} + 47804 q^{65} - 9848 q^{67} - 61376 q^{69} - 1680 q^{71} + 122052 q^{73} + 111034 q^{75} - 31704 q^{79} + 41954 q^{81} + 73948 q^{83} - 264888 q^{85} - 219156 q^{87} + 210588 q^{89} - 219784 q^{93} - 138624 q^{95} - 88480 q^{97} - 149880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 49\nu^{2} + 4753\nu - 2304 ) / 2352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 145 ) / 49 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 97\beta _1 - 97 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 49\beta_{3} - 145 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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
3.72311 + 6.44862i
−3.22311 5.58259i
3.72311 6.44862i
−3.22311 + 5.58259i
0 −3.44622 5.96903i 0 24.2311 41.9695i 0 0 0 97.7471 169.303i 0
177.2 0 10.4462 + 18.0934i 0 −45.2311 + 78.3426i 0 0 0 −96.7471 + 167.571i 0
361.1 0 −3.44622 + 5.96903i 0 24.2311 + 41.9695i 0 0 0 97.7471 + 169.303i 0
361.2 0 10.4462 18.0934i 0 −45.2311 78.3426i 0 0 0 −96.7471 167.571i 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.k 4
7.b odd 2 1 392.6.i.h 4
7.c even 3 1 56.6.a.d 2
7.c even 3 1 inner 392.6.i.k 4
7.d odd 6 1 392.6.a.e 2
7.d odd 6 1 392.6.i.h 4
21.h odd 6 1 504.6.a.m 2
28.f even 6 1 784.6.a.q 2
28.g odd 6 1 112.6.a.j 2
56.k odd 6 1 448.6.a.r 2
56.p even 6 1 448.6.a.x 2
84.n even 6 1 1008.6.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.d 2 7.c even 3 1
112.6.a.j 2 28.g odd 6 1
392.6.a.e 2 7.d odd 6 1
392.6.i.h 4 7.b odd 2 1
392.6.i.h 4 7.d odd 6 1
392.6.i.k 4 1.a even 1 1 trivial
392.6.i.k 4 7.c even 3 1 inner
448.6.a.r 2 56.k odd 6 1
448.6.a.x 2 56.p even 6 1
504.6.a.m 2 21.h odd 6 1
784.6.a.q 2 28.f even 6 1
1008.6.a.bi 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 14T_{3}^{3} + 340T_{3}^{2} + 2016T_{3} + 20736 \) 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
$3$ \( T^{4} - 14 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$5$ \( T^{4} + 42 T^{3} + \cdots + 19219456 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 8160592896 \) Copy content Toggle raw display
$13$ \( (T^{2} + 714 T + 71672)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 12366329616 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 522046710784 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 2618493566976 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1200 T - 57176388)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 719161357689856 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{2} - 7896 T - 22118548)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 524 T - 51717888)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 520039929619456 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 212046252228864 \) Copy content Toggle raw display
$71$ \( (T^{2} + 840 T - 3181158400)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{2} - 36974 T - 3038476256)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{2} + 44240 T - 14736460932)^{2} \) Copy content Toggle raw display
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