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Properties

Label 392.4.i.n
Level $392$
Weight $4$
Character orbit 392.i
Analytic conductor $23.129$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11163123.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 49x^{2} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{5} - 4 \beta_1 + 4) q^{5} + ( - 3 \beta_{5} - 2 \beta_{4} + \cdots - 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{5} - 4 \beta_1 + 4) q^{5} + ( - 3 \beta_{5} - 2 \beta_{4} + \cdots - 15) q^{9}+ \cdots + (11 \beta_{3} - 100 \beta_{2} + 335) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 13 q^{5} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + 13 q^{5} - 50 q^{9} + 11 q^{11} - 140 q^{13} - 82 q^{15} + 97 q^{17} - 81 q^{19} - 191 q^{23} - 12 q^{25} - 230 q^{27} - 324 q^{29} + 597 q^{31} + 343 q^{33} - 217 q^{37} + 806 q^{39} - 1396 q^{41} - 616 q^{43} + 1118 q^{45} + 139 q^{47} - 1475 q^{51} - 197 q^{53} - 294 q^{55} - 2294 q^{57} + 353 q^{59} + 449 q^{61} - 906 q^{65} + 519 q^{67} - 5114 q^{69} + 448 q^{71} + 1701 q^{73} - 1132 q^{75} - 1143 q^{79} - 251 q^{81} + 2760 q^{83} - 2050 q^{85} + 1458 q^{87} + 1749 q^{89} + 601 q^{93} + 2167 q^{95} - 5204 q^{97} + 2188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 14x^{4} + 49x^{2} + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 7\nu + 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{4} - 40\nu^{2} - 57 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} + 23\nu^{3} + 6\nu^{2} + 63\nu + 27 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{5} + 4\nu^{4} - 47\nu^{3} + 40\nu^{2} - 97\nu + 57 ) / 6 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{5} + 4\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 - 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{5} - 28\beta_{4} - 7\beta_{3} + 14\beta_{2} + 58\beta _1 - 29 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{3} - 20\beta_{2} + 123 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 98\beta_{5} + 232\beta_{4} + 49\beta_{3} - 116\beta_{2} - 730\beta _1 + 365 ) / 12 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.821510i
2.13755i
− 2.95906i
− 0.821510i
− 2.13755i
2.95906i
0 −3.82512 − 6.62530i 0 7.30445 − 12.6517i 0 0 0 −15.7631 + 27.3025i 0
177.2 0 0.0691041 + 0.119692i 0 −5.04289 + 8.73453i 0 0 0 13.4904 − 23.3661i 0
177.3 0 4.25602 + 7.37164i 0 4.23844 − 7.34119i 0 0 0 −22.7274 + 39.3649i 0
361.1 0 −3.82512 + 6.62530i 0 7.30445 + 12.6517i 0 0 0 −15.7631 − 27.3025i 0
361.2 0 0.0691041 − 0.119692i 0 −5.04289 − 8.73453i 0 0 0 13.4904 + 23.3661i 0
361.3 0 4.25602 − 7.37164i 0 4.23844 + 7.34119i 0 0 0 −22.7274 − 39.3649i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.3
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.n 6
7.b odd 2 1 56.4.i.a ✓ 6
7.c even 3 1 392.4.a.j 3
7.c even 3 1 inner 392.4.i.n 6
7.d odd 6 1 56.4.i.a ✓ 6
7.d odd 6 1 392.4.a.k 3
21.c even 2 1 504.4.s.i 6
21.g even 6 1 504.4.s.i 6
28.d even 2 1 112.4.i.f 6
28.f even 6 1 112.4.i.f 6
28.f even 6 1 784.4.a.bc 3
28.g odd 6 1 784.4.a.bd 3
56.e even 2 1 448.4.i.k 6
56.h odd 2 1 448.4.i.l 6
56.j odd 6 1 448.4.i.l 6
56.m even 6 1 448.4.i.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.i.a ✓ 6 7.b odd 2 1
56.4.i.a ✓ 6 7.d odd 6 1
112.4.i.f 6 28.d even 2 1
112.4.i.f 6 28.f even 6 1
392.4.a.j 3 7.c even 3 1
392.4.a.k 3 7.d odd 6 1
392.4.i.n 6 1.a even 1 1 trivial
392.4.i.n 6 7.c even 3 1 inner
448.4.i.k 6 56.e even 2 1
448.4.i.k 6 56.m even 6 1
448.4.i.l 6 56.h odd 2 1
448.4.i.l 6 56.j odd 6 1
504.4.s.i 6 21.c even 2 1
504.4.s.i 6 21.g even 6 1
784.4.a.bc 3 28.f even 6 1
784.4.a.bd 3 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 \(S_{4}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{6} - T_{3}^{5} + 66T_{3}^{4} + 47T_{3}^{3} + 4234T_{3}^{2} - 585T_{3} + 81 \) Copy content Toggle raw display
\( T_{5}^{6} - 13T_{5}^{5} + 278T_{5}^{4} - 1081T_{5}^{3} + 28118T_{5}^{2} - 136141T_{5} + 1560001 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{6} - 13 T^{5} + \cdots + 1560001 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 11 T^{5} + \cdots + 65593801 \) Copy content Toggle raw display
$13$ \( (T^{3} + 70 T^{2} + \cdots - 17752)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 344049114249 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 15221884129 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 12196012613841 \) Copy content Toggle raw display
$29$ \( (T^{3} + 162 T^{2} + \cdots - 1324296)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 21744865333321 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 228323372199129 \) Copy content Toggle raw display
$41$ \( (T^{3} + 698 T^{2} + \cdots - 6465192)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 308 T^{2} + \cdots + 38848)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 350054034092361 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 8496286395921 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 13\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 98846286969609 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 454704744973249 \) Copy content Toggle raw display
$71$ \( (T^{3} - 224 T^{2} + \cdots + 7551488)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 52\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 88\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{3} - 1380 T^{2} + \cdots - 4797376)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{3} + 2602 T^{2} + \cdots + 528741144)^{2} \) Copy content Toggle raw display
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