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Properties

Label 392.6.a.j
Level $392$
Weight $6$
Character orbit 392.a
Self dual yes
Analytic conductor $62.870$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,6,Mod(1,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

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: yes
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 200x^{3} - 99x^{2} + 5803x - 3615 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 7 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1 + 16) q^{5} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 + 78) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1 + 16) q^{5} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 + 78) q^{9} + ( - \beta_{4} + 2 \beta_{2} + \cdots + 72) q^{11}+ \cdots + (51 \beta_{4} + 234 \beta_{3} + \cdots + 49911) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9} + 361 q^{11} + 342 q^{13} - 1049 q^{15} + 1809 q^{17} + 1277 q^{19} + 911 q^{23} + 3940 q^{25} - 4751 q^{27} + 5442 q^{29} + 2187 q^{31} - 5553 q^{33} - 8181 q^{37} - 3422 q^{39} + 16578 q^{41} + 6332 q^{43} + 41310 q^{45} + 16101 q^{47} - 67865 q^{51} + 16047 q^{53} + 45629 q^{55} + 22347 q^{57} + 71027 q^{59} + 31093 q^{61} - 64370 q^{65} + 47981 q^{67} + 137249 q^{69} + 22512 q^{71} + 123333 q^{73} - 45460 q^{75} - 212481 q^{79} + 52917 q^{81} + 87460 q^{83} + 222141 q^{85} + 318070 q^{87} + 129045 q^{89} - 252835 q^{93} + 300417 q^{95} + 328274 q^{97} + 249798 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 200x^{3} - 99x^{2} + 5803x - 3615 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -56\nu^{4} + 400\nu^{3} + 10484\nu^{2} - 51142\nu - 225705 ) / 3747 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 112\nu^{4} - 800\nu^{3} - 5980\nu^{2} + 87296\nu - 747630 ) / 3747 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -464\nu^{4} - 968\nu^{3} + 80444\nu^{2} + 213242\nu - 1107345 ) / 3747 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2\beta _1 + 320 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{4} - 3\beta_{3} + 52\beta_{2} + 589\beta _1 + 465 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -50\beta_{4} + 353\beta_{3} + 585\beta_{2} + 1303\beta _1 + 90895 ) / 8 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
13.2682
5.14333
0.638887
−6.72951
−12.3209
0 −27.5365 0 79.9429 0 0 0 515.257 0
1.2 0 −11.2867 0 −26.0667 0 0 0 −115.611 0
1.3 0 −2.27777 0 −50.5110 0 0 0 −237.812 0
1.4 0 12.4590 0 97.6805 0 0 0 −87.7730 0
1.5 0 23.6419 0 −20.0456 0 0 0 315.939 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.6.a.j 5
4.b odd 2 1 784.6.a.bl 5
7.b odd 2 1 392.6.a.k 5
7.c even 3 2 392.6.i.o 10
7.d odd 6 2 56.6.i.b ✓ 10
21.g even 6 2 504.6.s.b 10
28.d even 2 1 784.6.a.bk 5
28.f even 6 2 112.6.i.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.i.b ✓ 10 7.d odd 6 2
112.6.i.f 10 28.f even 6 2
392.6.a.j 5 1.a even 1 1 trivial
392.6.a.k 5 7.b odd 2 1
392.6.i.o 10 7.c even 3 2
504.6.s.b 10 21.g even 6 2
784.6.a.bk 5 28.d even 2 1
784.6.a.bl 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 5T_{3}^{4} - 790T_{3}^{3} - 1598T_{3}^{2} + 92037T_{3} + 208521 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(392))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 5 T^{4} + \cdots + 208521 \) Copy content Toggle raw display
$5$ \( T^{5} - 81 T^{4} + \cdots + 206100499 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots + 200826426843 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots - 165262458987360 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 12\!\cdots\!53 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 96608715700265 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 18\!\cdots\!17 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 32\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 27\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 13\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots + 60\!\cdots\!95 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 80\!\cdots\!95 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 43\!\cdots\!45 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 80\!\cdots\!39 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 10\!\cdots\!85 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 32\!\cdots\!71 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 70\!\cdots\!39 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 63\!\cdots\!95 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 15\!\cdots\!48 \) Copy content Toggle raw display
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