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Properties

Label 784.6.a.bm
Level $784$
Weight $6$
Character orbit 784.a
Self dual yes
Analytic conductor $125.741$
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] = [784,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(125.740914733\)
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: \( x^{5} - 167x^{3} - 387x^{2} + 1720x + 2340 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 3) q^{3} + ( - \beta_{3} - 6) q^{5} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 + 47) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 3) q^{3} + ( - \beta_{3} - 6) q^{5} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 + 47) q^{9} + ( - \beta_{4} - 2 \beta_{3} + \cdots + 72) q^{11}+ \cdots + ( - 197 \beta_{4} + 872 \beta_{3} + \cdots + 61297) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9} + 351 q^{11} + 54 q^{13} - 607 q^{15} - 111 q^{17} + 1035 q^{19} - 3639 q^{23} + 1540 q^{25} + 3607 q^{27} - 734 q^{29} + 7677 q^{31} + 7439 q^{33} + 13595 q^{37} + 1406 q^{39} - 5310 q^{41} - 764 q^{43} - 38978 q^{45} + 6675 q^{47} - 20975 q^{51} - 30753 q^{53} + 28267 q^{55} - 14389 q^{57} + 87989 q^{59} - 19899 q^{61} + 119470 q^{65} + 33067 q^{67} - 100399 q^{69} + 108720 q^{71} - 141659 q^{73} + 108788 q^{75} - 118919 q^{79} - 143851 q^{81} + 211004 q^{83} - 143379 q^{85} + 302154 q^{87} + 55861 q^{89} + 410381 q^{93} + 26279 q^{95} - 135470 q^{97} + 300154 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 167x^{3} - 387x^{2} + 1720x + 2340 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -133\nu^{4} + 490\nu^{3} + 21255\nu^{2} - 17495\nu - 229698 ) / 8068 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 91\nu^{4} + 514\nu^{3} - 21337\nu^{2} - 37287\nu + 416824 ) / 4034 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 74\nu^{4} + 152\nu^{3} - 12918\nu^{2} - 54082\nu + 104341 ) / 2017 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1485\nu^{4} + 2074\nu^{3} + 227615\nu^{2} + 176881\nu - 1166786 ) / 8068 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} - 4\beta_{3} + 2\beta_{2} + 5\beta _1 - 2 ) / 56 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -17\beta_{4} - 19\beta_{3} - 15\beta_{2} + 127\beta _1 + 3690 ) / 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -97\beta_{4} - 192\beta_{3} + 131\beta_{2} + 835\beta _1 + 6141 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3300\beta_{4} - 3925\beta_{3} - 1695\beta_{2} + 22394\beta _1 + 538504 ) / 56 \) 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
−4.45227
13.6096
−10.9009
2.94609
−1.20248
0 −18.5692 0 −57.9194 0 0 0 101.814 0
1.2 0 −14.5316 0 44.8226 0 0 0 −31.8333 0
1.3 0 6.22560 0 −9.40551 0 0 0 −204.242 0
1.4 0 14.6817 0 72.1603 0 0 0 −27.4480 0
1.5 0 25.1934 0 −80.6580 0 0 0 391.710 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 784.6.a.bm 5
4.b odd 2 1 392.6.a.i 5
7.b odd 2 1 784.6.a.bj 5
7.d odd 6 2 112.6.i.g 10
28.d even 2 1 392.6.a.l 5
28.f even 6 2 56.6.i.a ✓ 10
28.g odd 6 2 392.6.i.p 10
84.j odd 6 2 504.6.s.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.i.a ✓ 10 28.f even 6 2
112.6.i.g 10 7.d odd 6 2
392.6.a.i 5 4.b odd 2 1
392.6.a.l 5 28.d even 2 1
392.6.i.p 10 28.g odd 6 2
504.6.s.d 10 84.j odd 6 2
784.6.a.bj 5 7.b odd 2 1
784.6.a.bm 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 13T_{3}^{4} - 638T_{3}^{3} + 5718T_{3}^{2} + 90573T_{3} - 621369 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 13 T^{4} + \cdots - 621369 \) Copy content Toggle raw display
$5$ \( T^{5} + 31 T^{4} + \cdots + 142117891 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots - 16452319223691 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots - 173219419488 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 49362505838685 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots + 151493645339129 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 18\!\cdots\!17 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 36\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 12\!\cdots\!15 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 92\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 27\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 77\!\cdots\!15 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 12\!\cdots\!71 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 17\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 25\!\cdots\!45 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 16\!\cdots\!93 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 73\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 89\!\cdots\!37 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 73\!\cdots\!33 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 11\!\cdots\!52 \) Copy content Toggle raw display
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