Properties

Label 5780.2.a.i
Level $5780$
Weight $2$
Character orbit 5780.a
Self dual yes
Analytic conductor $46.154$
Analytic rank $1$
Dimension $3$
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] = [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: \(3\)
Coefficient field: 3.3.1524.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 1 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 3) q^{9} + (\beta_{2} + 3) q^{11} + ( - \beta_{2} - \beta_1) q^{13} + (\beta_1 - 1) q^{15} + ( - 2 \beta_1 - 2) q^{19} + ( - \beta_{2} - \beta_1 - 4) q^{21} + (\beta_1 - 1) q^{23} + q^{25} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{27} + (2 \beta_1 - 2) q^{29} + ( - \beta_{2} + 2 \beta_1 + 1) q^{31} + ( - \beta_{2} + 5 \beta_1 - 4) q^{33} + ( - \beta_1 - 1) q^{35} - 4 q^{37} + ( - 2 \beta_1 - 4) q^{39} + (2 \beta_1 - 4) q^{41} + ( - \beta_{2} + \beta_1 + 2) q^{43} + (\beta_{2} - \beta_1 + 3) q^{45} + (\beta_{2} - \beta_1 + 6) q^{47} + (\beta_{2} + 3 \beta_1 - 1) q^{49} + ( - 2 \beta_{2} - 8) q^{53} + (\beta_{2} + 3) q^{55} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{57} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{59} + (2 \beta_{2} - 2 \beta_1 - 2) q^{61} + ( - 3 \beta_1 + 3) q^{63} + ( - \beta_{2} - \beta_1) q^{65} + (\beta_{2} + 3 \beta_1 - 2) q^{67} + (\beta_{2} - \beta_1 + 6) q^{69} + ( - \beta_{2} - 2 \beta_1 - 5) q^{71} + (2 \beta_1 - 8) q^{73} + (\beta_1 - 1) q^{75} + ( - \beta_{2} - 5 \beta_1 - 2) q^{77} + ( - 3 \beta_{2} - 2 \beta_1 - 1) q^{79} + (\beta_{2} - 7 \beta_1 + 9) q^{81} + ( - 3 \beta_{2} + \beta_1 - 4) q^{83} + (2 \beta_{2} - 2 \beta_1 + 12) q^{87} + (\beta_{2} + \beta_1 - 6) q^{89} + (2 \beta_{2} + 4 \beta_1 + 4) q^{91} + (3 \beta_{2} - \beta_1 + 10) q^{93} + ( - 2 \beta_1 - 2) q^{95} + ( - 2 \beta_1 - 6) q^{97} + (3 \beta_{2} - 6 \beta_1 + 21) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 7 q^{9} + 8 q^{11} - 2 q^{15} - 8 q^{19} - 12 q^{21} - 2 q^{23} + 3 q^{25} - 14 q^{27} - 4 q^{29} + 6 q^{31} - 6 q^{33} - 4 q^{35} - 12 q^{37} - 14 q^{39} - 10 q^{41} + 8 q^{43} + 7 q^{45} + 16 q^{47} - q^{49} - 22 q^{53} + 8 q^{55} - 24 q^{57} + 4 q^{59} - 10 q^{61} + 6 q^{63} - 4 q^{67} + 16 q^{69} - 16 q^{71} - 22 q^{73} - 2 q^{75} - 10 q^{77} - 2 q^{79} + 19 q^{81} - 8 q^{83} + 32 q^{87} - 18 q^{89} + 14 q^{91} + 26 q^{93} - 8 q^{95} - 20 q^{97} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 5 \) 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
−2.27307
0.140435
3.13264
0 −3.27307 0 1.00000 0 1.27307 0 7.71301 0
1.2 0 −0.859565 0 1.00000 0 −1.14044 0 −2.26115 0
1.3 0 2.13264 0 1.00000 0 −4.13264 0 1.54814 0
\(n\): e.g. 2-40 or 990-1000
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.i 3
17.b even 2 1 5780.2.a.k 3
17.c even 4 2 340.2.c.a 6
51.f odd 4 2 3060.2.e.j 6
68.f odd 4 2 1360.2.c.e 6
85.f odd 4 2 1700.2.g.c 6
85.i odd 4 2 1700.2.g.b 6
85.j even 4 2 1700.2.c.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
340.2.c.a 6 17.c even 4 2
1360.2.c.e 6 68.f odd 4 2
1700.2.c.b 6 85.j even 4 2
1700.2.g.b 6 85.i odd 4 2
1700.2.g.c 6 85.f odd 4 2
3060.2.e.j 6 51.f odd 4 2
5780.2.a.i 3 1.a even 1 1 trivial
5780.2.a.k 3 17.b even 2 1

Hecke kernels

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

\( T_{3}^{3} + 2T_{3}^{2} - 6T_{3} - 6 \) Copy content Toggle raw display
\( T_{7}^{3} + 4T_{7}^{2} - 2T_{7} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 6 T - 6 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 4 T^{2} - 2 T - 6 \) Copy content Toggle raw display
$11$ \( T^{3} - 8 T^{2} + 4 T + 54 \) Copy content Toggle raw display
$13$ \( T^{3} - 24T - 4 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 8 T^{2} - 8 T - 48 \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} - 6 T - 6 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 24 T - 48 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 36 T + 214 \) Copy content Toggle raw display
$37$ \( (T + 4)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + 4 T - 72 \) Copy content Toggle raw display
$43$ \( T^{3} - 8 T^{2} - 4 T + 68 \) Copy content Toggle raw display
$47$ \( T^{3} - 16 T^{2} + 60 T - 36 \) Copy content Toggle raw display
$53$ \( T^{3} + 22 T^{2} + 92 T - 328 \) Copy content Toggle raw display
$59$ \( T^{3} - 4 T^{2} - 176 T + 1008 \) Copy content Toggle raw display
$61$ \( T^{3} + 10 T^{2} - 68 T - 456 \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 76 T - 388 \) Copy content Toggle raw display
$71$ \( T^{3} + 16 T^{2} + 40 T + 6 \) Copy content Toggle raw display
$73$ \( T^{3} + 22 T^{2} + 132 T + 168 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 180 T - 654 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} - 144 T - 924 \) Copy content Toggle raw display
$89$ \( T^{3} + 18 T^{2} + 84 T + 76 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} + 104 T + 112 \) Copy content Toggle raw display
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