Properties

Label 392.6.i.g
Level $392$
Weight $6$
Character orbit 392.i
Analytic conductor $62.870$
Analytic rank $1$
Dimension $4$
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] = [392,6,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: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-59})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 14x^{2} - 15x + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 13 \beta_1 - 13) q^{3} + ( - 5 \beta_{3} + 5 \beta_{2} + 31 \beta_1) q^{5} + ( - 26 \beta_{3} + 26 \beta_{2} + 103 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 13 \beta_1 - 13) q^{3} + ( - 5 \beta_{3} + 5 \beta_{2} + 31 \beta_1) q^{5} + ( - 26 \beta_{3} + 26 \beta_{2} + 103 \beta_1) q^{9} + ( - 6 \beta_{3} - 486 \beta_1 - 486) q^{11} + ( - 71 \beta_{2} - 39) q^{13} + ( - 96 \beta_{2} + 1288) q^{15} + ( - 6 \beta_{3} + 280 \beta_1 + 280) q^{17} + (37 \beta_{3} - 37 \beta_{2} - 1321 \beta_1) q^{19} + (248 \beta_{3} - 248 \beta_{2} + 1136 \beta_1) q^{23} + (310 \beta_{3} - 2261 \beta_1 - 2261) q^{25} + ( - 198 \beta_{2} + 2782) q^{27} + (14 \beta_{2} - 3904) q^{29} + ( - 130 \beta_{3} + 2722 \beta_1 + 2722) q^{31} + ( - 408 \beta_{3} + 408 \beta_{2} + 5256 \beta_1) q^{33} + ( - 650 \beta_{3} + 650 \beta_{2} + 288 \beta_1) q^{37} + (884 \beta_{3} - 12060 \beta_1 - 12060) q^{39} + (378 \beta_{2} + 8444) q^{41} + (338 \beta_{2} - 4198) q^{43} + (1321 \beta_{3} - 26203 \beta_1 - 26203) q^{45} + (1238 \beta_{3} - 1238 \beta_{2} + 2266 \beta_1) q^{47} + (358 \beta_{3} - 358 \beta_{2} - 4702 \beta_1) q^{51} + ( - 1152 \beta_{3} - 710 \beta_1 - 710) q^{53} + ( - 2244 \beta_{2} + 9756) q^{55} + (1802 \beta_{2} - 23722) q^{57} + ( - 325 \beta_{3} + 17073 \beta_1 + 17073) q^{59} + (3347 \beta_{3} - 3347 \beta_{2} - 9553 \beta_1) q^{61} + ( - 2006 \beta_{3} + 2006 \beta_{2} + 61626 \beta_1) q^{65} + (1240 \beta_{3} - 28476 \beta_1 - 28476) q^{67} + (2088 \beta_{2} - 29128) q^{69} + ( - 700 \beta_{2} - 3612) q^{71} + (1260 \beta_{3} + 64414 \beta_1 + 64414) q^{73} + ( - 6291 \beta_{3} + 6291 \beta_{2} + 84263 \beta_1) q^{75} + (1548 \beta_{3} - 1548 \beta_{2} + 26404 \beta_1) q^{79} + ( - 962 \beta_{3} - 46183 \beta_1 - 46183) q^{81} + (5209 \beta_{2} + 42243) q^{83} + (1586 \beta_{2} - 13990) q^{85} + ( - 4086 \beta_{3} + 53230 \beta_1 + 53230) q^{87} + ( - 2312 \beta_{3} + 2312 \beta_{2} + 65486 \beta_1) q^{89} + (4412 \beta_{3} - 4412 \beta_{2} - 58396 \beta_1) q^{93} + ( - 7752 \beta_{3} + 73696 \beta_1 + 73696) q^{95} + (350 \beta_{2} - 97312) q^{97} + ( - 12018 \beta_{2} + 22446) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 26 q^{3} - 62 q^{5} - 206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 26 q^{3} - 62 q^{5} - 206 q^{9} - 972 q^{11} - 156 q^{13} + 5152 q^{15} + 560 q^{17} + 2642 q^{19} - 2272 q^{23} - 4522 q^{25} + 11128 q^{27} - 15616 q^{29} + 5444 q^{31} - 10512 q^{33} - 576 q^{37} - 24120 q^{39} + 33776 q^{41} - 16792 q^{43} - 52406 q^{45} - 4532 q^{47} + 9404 q^{51} - 1420 q^{53} + 39024 q^{55} - 94888 q^{57} + 34146 q^{59} + 19106 q^{61} - 123252 q^{65} - 56952 q^{67} - 116512 q^{69} - 14448 q^{71} + 128828 q^{73} - 168526 q^{75} - 52808 q^{79} - 92366 q^{81} + 168972 q^{83} - 55960 q^{85} + 106460 q^{87} - 130972 q^{89} + 116792 q^{93} + 147392 q^{95} - 389248 q^{97} + 89784 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 14x^{2} - 15x + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 14\nu^{2} - 14\nu - 225 ) / 210 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 2\nu^{2} + 58\nu + 15 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\nu^{3} + 14\nu^{2} + 406\nu - 885 ) / 210 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 87\beta _1 + 87 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{3} - 7\beta_{2} + 66 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
−3.07603 2.35330i
3.57603 + 1.48727i
−3.07603 + 2.35330i
3.57603 1.48727i
0 −13.1521 22.7800i 0 −48.7603 + 84.4554i 0 0 0 −224.454 + 388.765i 0
177.2 0 0.152067 + 0.263388i 0 17.7603 30.7618i 0 0 0 121.454 210.364i 0
361.1 0 −13.1521 + 22.7800i 0 −48.7603 84.4554i 0 0 0 −224.454 388.765i 0
361.2 0 0.152067 0.263388i 0 17.7603 + 30.7618i 0 0 0 121.454 + 210.364i 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.g 4
7.b odd 2 1 392.6.i.l 4
7.c even 3 1 392.6.a.f 2
7.c even 3 1 inner 392.6.i.g 4
7.d odd 6 1 56.6.a.c 2
7.d odd 6 1 392.6.i.l 4
21.g even 6 1 504.6.a.s 2
28.f even 6 1 112.6.a.k 2
28.g odd 6 1 784.6.a.p 2
56.j odd 6 1 448.6.a.z 2
56.m even 6 1 448.6.a.q 2
84.j odd 6 1 1008.6.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.c 2 7.d odd 6 1
112.6.a.k 2 28.f even 6 1
392.6.a.f 2 7.c even 3 1
392.6.i.g 4 1.a even 1 1 trivial
392.6.i.g 4 7.c even 3 1 inner
392.6.i.l 4 7.b odd 2 1
392.6.i.l 4 7.d odd 6 1
448.6.a.q 2 56.m even 6 1
448.6.a.z 2 56.j odd 6 1
504.6.a.s 2 21.g even 6 1
784.6.a.p 2 28.g odd 6 1
1008.6.a.bt 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 26T_{3}^{3} + 684T_{3}^{2} - 208T_{3} + 64 \) 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
$3$ \( T^{4} + 26 T^{3} + 684 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{4} + 62 T^{3} + 7308 T^{2} + \cdots + 11999296 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 972 T^{3} + \cdots + 52819070976 \) Copy content Toggle raw display
$13$ \( (T^{2} + 78 T - 890736)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 560 T^{3} + \cdots + 5188032784 \) Copy content Toggle raw display
$19$ \( T^{4} - 2642 T^{3} + \cdots + 2258191441984 \) Copy content Toggle raw display
$23$ \( T^{4} + 2272 T^{3} + \cdots + 92077688786944 \) Copy content Toggle raw display
$29$ \( (T^{2} + 7808 T + 15206524)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 5444 T^{3} + \cdots + 19518582624256 \) Copy content Toggle raw display
$37$ \( T^{4} + 576 T^{3} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} - 16888 T + 46010668)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8396 T - 2597984)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4532 T^{3} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + 1420 T^{3} + \cdots + 54\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{4} - 34146 T^{3} + \cdots + 74\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{4} - 19106 T^{3} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{4} + 56952 T^{3} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} + 7224 T - 73683456)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 128828 T^{3} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{4} + 52808 T^{3} + \cdots + 74\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{2} - 84486 T - 3018190488)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 130972 T^{3} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{2} + 194624 T + 9447942844)^{2} \) Copy content Toggle raw display
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