↑

Properties

Label 5780.2.a.n
Level $5780$
Weight $2$
Character orbit 5780.a
Self dual yes
Analytic conductor $46.154$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5780,2,Mod(1,5780)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5780, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5780.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5780.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: \(46.1535323683\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.9521152.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} - 8x^{3} + 18x^{2} + 20x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 340)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{5} - 5 \beta_{4} + 4 \beta_{3} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{11} - 8 q^{13} + 12 q^{19} - 8 q^{21} - 4 q^{23} + 6 q^{25} + 24 q^{27} - 8 q^{29} - 20 q^{31} - 24 q^{33} - 4 q^{35} - 24 q^{37} - 20 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} + 4 q^{47} + 22 q^{49} - 8 q^{53} - 8 q^{55} + 16 q^{57} - 4 q^{61} - 16 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 16 q^{73} - 24 q^{77} - 36 q^{79} + 2 q^{81} + 8 q^{83} + 28 q^{89} + 20 q^{91} - 12 q^{93} + 12 q^{95} - 32 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 10x^{4} - 8x^{3} + 18x^{2} + 20x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 4\nu^{2} - 9\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - \nu^{4} - 9\nu^{3} + \nu^{2} + 16\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{5} - 3\nu^{4} - 15\nu^{3} + 6\nu^{2} + 24\nu + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{5} + 5\nu^{4} + 22\nu^{3} - 13\nu^{2} - 35\nu - 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 4\beta_{4} - 3\beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{5} + 15\beta_{4} - 11\beta_{3} + 10\beta_{2} + 11\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{5} + 50\beta_{4} - 36\beta_{3} + 27\beta_{2} + 39\beta _1 + 37 \) 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
−1.79518
−1.65072
−1.20169
−0.111711
1.58266
3.17664
0 −1.79518 0 1.00000 0 −5.23303 0 0.222687 0
1.2 0 −1.65072 0 1.00000 0 3.17148 0 −0.275124 0
1.3 0 −1.20169 0 1.00000 0 3.60075 0 −1.55594 0
1.4 0 −0.111711 0 1.00000 0 −2.79103 0 −2.98752 0
1.5 0 1.58266 0 1.00000 0 −0.367722 0 −0.495178 0
1.6 0 3.17664 0 1.00000 0 −2.38045 0 7.09107 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(17\) \(1\)

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 5780.2.a.n 6
17.b even 2 1 5780.2.a.m 6
17.c even 4 2 5780.2.c.h 12
17.d even 8 2 340.2.o.a ✓ 12
51.g odd 8 2 3060.2.be.b 12
68.g odd 8 2 1360.2.bt.c 12
85.k odd 8 2 1700.2.m.c 12
85.m even 8 2 1700.2.o.d 12
85.n odd 8 2 1700.2.m.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
340.2.o.a ✓ 12 17.d even 8 2
1360.2.bt.c 12 68.g odd 8 2
1700.2.m.c 12 85.k odd 8 2
1700.2.m.f 12 85.n odd 8 2
1700.2.o.d 12 85.m even 8 2
3060.2.be.b 12 51.g odd 8 2
5780.2.a.m 6 17.b even 2 1
5780.2.a.n 6 1.a even 1 1 trivial
5780.2.c.h 12 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5780))\):

\( T_{3}^{6} - 10T_{3}^{4} - 8T_{3}^{3} + 18T_{3}^{2} + 20T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{6} + 4T_{7}^{5} - 24T_{7}^{4} - 84T_{7}^{3} + 122T_{7}^{2} + 452T_{7} + 146 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 10 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots + 146 \) Copy content Toggle raw display
$11$ \( T^{6} + 8 T^{5} + \cdots - 158 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + \cdots + 68 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} - 12 T^{5} + \cdots + 1088 \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + \cdots + 14866 \) Copy content Toggle raw display
$29$ \( T^{6} + 8 T^{5} + \cdots + 184 \) Copy content Toggle raw display
$31$ \( T^{6} + 20 T^{5} + \cdots + 41554 \) Copy content Toggle raw display
$37$ \( T^{6} + 24 T^{5} + \cdots - 132616 \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} + \cdots - 77888 \) Copy content Toggle raw display
$43$ \( T^{6} + 4 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$47$ \( T^{6} - 4 T^{5} + \cdots + 292 \) Copy content Toggle raw display
$53$ \( T^{6} + 8 T^{5} + \cdots + 368 \) Copy content Toggle raw display
$59$ \( T^{6} - 168 T^{4} + \cdots - 12416 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots + 544 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + \cdots + 928964 \) Copy content Toggle raw display
$71$ \( T^{6} + 16 T^{5} + \cdots - 44078 \) Copy content Toggle raw display
$73$ \( T^{6} + 16 T^{5} + \cdots + 22816 \) Copy content Toggle raw display
$79$ \( T^{6} + 36 T^{5} + \cdots - 21118 \) Copy content Toggle raw display
$83$ \( T^{6} - 8 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$89$ \( T^{6} - 28 T^{5} + \cdots + 32996 \) Copy content Toggle raw display
$97$ \( T^{6} + 32 T^{5} + \cdots - 4552 \) Copy content Toggle raw display
show more
show less