Properties

Label 784.4.a.bg
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,4,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 35x^{2} + 36x + 194 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 392)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 2 \beta_1) q^{3} + (\beta_{3} + 3 \beta_1) q^{5} + ( - 3 \beta_{2} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 2 \beta_1) q^{3} + (\beta_{3} + 3 \beta_1) q^{5} + ( - 3 \beta_{2} + 10) q^{9} + ( - 5 \beta_{2} + 31) q^{11} + (9 \beta_{3} + \beta_1) q^{13} + ( - 4 \beta_{2} + 40) q^{15} + (8 \beta_{3} + 31 \beta_1) q^{17} + (\beta_{3} - 2 \beta_1) q^{19} + (20 \beta_{2} - 32) q^{23} + ( - 5 \beta_{2} - 80) q^{25} + ( - 8 \beta_{3} + 68 \beta_1) q^{27} + (9 \beta_{2} - 63) q^{29} + (14 \beta_{3} - 64 \beta_1) q^{31} + (46 \beta_{3} + 232 \beta_1) q^{33} + (29 \beta_{2} - 91) q^{37} + ( - 10 \beta_{2} + 282) q^{39} + ( - 4 \beta_{3} - 291 \beta_1) q^{41} + (21 \beta_{2} + 305) q^{43} + (25 \beta_{3} + 135 \beta_1) q^{45} + (70 \beta_{3} + 172 \beta_1) q^{47} + ( - 39 \beta_{2} + 341) q^{51} + ( - 46 \beta_{2} - 188) q^{53} + (56 \beta_{3} + 268 \beta_1) q^{55} + (\beta_{2} + 25) q^{57} + (29 \beta_{3} - 250 \beta_1) q^{59} + ( - 73 \beta_{3} + 201 \beta_1) q^{61} + ( - 19 \beta_{2} + 275) q^{65} + ( - 18 \beta_{2} + 686) q^{67} + ( - 92 \beta_{3} - 744 \beta_1) q^{69} + (46 \beta_{2} + 430) q^{71} + ( - 50 \beta_{3} - 83 \beta_1) q^{73} + ( - 65 \beta_{3} + 10 \beta_1) q^{75} + ( - 2 \beta_{2} + 422) q^{79} + (21 \beta_{2} - 314) q^{81} + ( - 23 \beta_{3} + 106 \beta_1) q^{83} + ( - 47 \beta_{2} + 395) q^{85} + ( - 90 \beta_{3} - 432 \beta_1) q^{87} + ( - 8 \beta_{3} + 611 \beta_1) q^{89} + (50 \beta_{2} + 242) q^{93} + 20 q^{95} + (210 \beta_{3} + 177 \beta_1) q^{97} + ( - 143 \beta_{2} + 1285) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 40 q^{9} + 124 q^{11} + 160 q^{15} - 128 q^{23} - 320 q^{25} - 252 q^{29} - 364 q^{37} + 1128 q^{39} + 1220 q^{43} + 1364 q^{51} - 752 q^{53} + 100 q^{57} + 1100 q^{65} + 2744 q^{67} + 1720 q^{71} + 1688 q^{79} - 1256 q^{81} + 1580 q^{85} + 968 q^{93} + 80 q^{95} + 5140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 35x^{2} + 36x + 194 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 43\nu - 22 ) / 57 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 200\nu - 101 ) / 57 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 27\nu^{2} - 50\nu - 502 ) / 57 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + \beta_{2} + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{3} + 23\beta_{2} - 100\beta _1 + 55 ) / 2 \) Copy content Toggle raw display

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
−2.11692
3.11692
5.94534
−4.94534
0 −7.82220 0 −9.23641 0 0 0 34.1868 0
1.2 0 −3.57956 0 −2.16534 0 0 0 −14.1868 0
1.3 0 3.57956 0 2.16534 0 0 0 −14.1868 0
1.4 0 7.82220 0 9.23641 0 0 0 34.1868 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.bg 4
4.b odd 2 1 392.4.a.m 4
7.b odd 2 1 inner 784.4.a.bg 4
28.d even 2 1 392.4.a.m 4
28.f even 6 2 392.4.i.p 8
28.g odd 6 2 392.4.i.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.m 4 4.b odd 2 1
392.4.a.m 4 28.d even 2 1
392.4.i.p 8 28.f even 6 2
392.4.i.p 8 28.g odd 6 2
784.4.a.bg 4 1.a even 1 1 trivial
784.4.a.bg 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{4} - 74T_{3}^{2} + 784 \) Copy content Toggle raw display
\( T_{5}^{4} - 90T_{5}^{2} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 74T^{2} + 784 \) Copy content Toggle raw display
$5$ \( T^{4} - 90T^{2} + 400 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 62 T - 664)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 5314 T^{2} + \cdots + 6801664 \) Copy content Toggle raw display
$17$ \( T^{4} - 7076 T^{2} + 386884 \) Copy content Toggle raw display
$19$ \( T^{4} - 90T^{2} + 400 \) Copy content Toggle raw display
$23$ \( (T^{2} + 64 T - 24976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 126 T - 1296)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 32904 T^{2} + \cdots + 13778944 \) Copy content Toggle raw display
$37$ \( (T^{2} + 182 T - 46384)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 335124 T^{2} + \cdots + 27729576484 \) Copy content Toggle raw display
$43$ \( (T^{2} - 610 T + 64360)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 393576 T^{2} + \cdots + 14813810944 \) Copy content Toggle raw display
$53$ \( (T^{2} + 376 T - 102196)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 334506 T^{2} + \cdots + 12676057744 \) Copy content Toggle raw display
$61$ \( T^{4} - 572010 T^{2} + \cdots + 3645744400 \) Copy content Toggle raw display
$67$ \( (T^{2} - 1372 T + 449536)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 860 T + 47360)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 175956 T^{2} + \cdots + 5553528484 \) Copy content Toggle raw display
$79$ \( (T^{2} - 844 T + 177824)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 89610 T^{2} + \cdots + 108576400 \) Copy content Toggle raw display
$89$ \( T^{4} - 1517060 T^{2} + \cdots + 569074096900 \) Copy content Toggle raw display
$97$ \( T^{4} - 2887236 T^{2} + \cdots + 2024593185924 \) Copy content Toggle raw display
show more
show less