Properties

Label 810.2.a.l
Level $810$
Weight $2$
Character orbit 810.a
Self dual yes
Analytic conductor $6.468$
Analytic rank $0$
Dimension $2$
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] = [810,2,Mod(1,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("810.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + (\beta + 2) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{5} + (\beta + 2) q^{7} + q^{8} + q^{10} - 2 \beta q^{11} + (\beta + 2) q^{13} + (\beta + 2) q^{14} + q^{16} + 2 \beta q^{17} + ( - 4 \beta - 1) q^{19} + q^{20} - 2 \beta q^{22} + (2 \beta - 3) q^{23} + q^{25} + (\beta + 2) q^{26} + (\beta + 2) q^{28} - 4 \beta q^{29} + (2 \beta + 2) q^{31} + q^{32} + 2 \beta q^{34} + (\beta + 2) q^{35} + 8 q^{37} + ( - 4 \beta - 1) q^{38} + q^{40} + 3 \beta q^{41} + ( - 2 \beta + 2) q^{43} - 2 \beta q^{44} + (2 \beta - 3) q^{46} + ( - 2 \beta - 3) q^{47} + 4 \beta q^{49} + q^{50} + (\beta + 2) q^{52} + ( - 2 \beta - 9) q^{53} - 2 \beta q^{55} + (\beta + 2) q^{56} - 4 \beta q^{58} + 5 \beta q^{59} + ( - 6 \beta + 2) q^{61} + (2 \beta + 2) q^{62} + q^{64} + (\beta + 2) q^{65} + 8 q^{67} + 2 \beta q^{68} + (\beta + 2) q^{70} - 6 q^{71} + (6 \beta - 4) q^{73} + 8 q^{74} + ( - 4 \beta - 1) q^{76} + ( - 4 \beta - 6) q^{77} + (6 \beta - 4) q^{79} + q^{80} + 3 \beta q^{82} + (2 \beta - 12) q^{83} + 2 \beta q^{85} + ( - 2 \beta + 2) q^{86} - 2 \beta q^{88} + 12 q^{89} + (4 \beta + 7) q^{91} + (2 \beta - 3) q^{92} + ( - 2 \beta - 3) q^{94} + ( - 4 \beta - 1) q^{95} + ( - 4 \beta + 8) q^{97} + 4 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 2 q^{20} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} + 4 q^{35} + 16 q^{37} - 2 q^{38} + 2 q^{40} + 4 q^{43} - 6 q^{46} - 6 q^{47} + 2 q^{50} + 4 q^{52} - 18 q^{53} + 4 q^{56} + 4 q^{61} + 4 q^{62} + 2 q^{64} + 4 q^{65} + 16 q^{67} + 4 q^{70} - 12 q^{71} - 8 q^{73} + 16 q^{74} - 2 q^{76} - 12 q^{77} - 8 q^{79} + 2 q^{80} - 24 q^{83} + 4 q^{86} + 24 q^{89} + 14 q^{91} - 6 q^{92} - 6 q^{94} - 2 q^{95} + 16 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
1.00000 0 1.00000 1.00000 0 0.267949 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 3.73205 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

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 810.2.a.l yes 2
3.b odd 2 1 810.2.a.j 2
4.b odd 2 1 6480.2.a.bj 2
5.b even 2 1 4050.2.a.bk 2
5.c odd 4 2 4050.2.c.x 4
9.c even 3 2 810.2.e.m 4
9.d odd 6 2 810.2.e.n 4
12.b even 2 1 6480.2.a.bb 2
15.d odd 2 1 4050.2.a.bt 2
15.e even 4 2 4050.2.c.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.2.a.j 2 3.b odd 2 1
810.2.a.l yes 2 1.a even 1 1 trivial
810.2.e.m 4 9.c even 3 2
810.2.e.n 4 9.d odd 6 2
4050.2.a.bk 2 5.b even 2 1
4050.2.a.bt 2 15.d odd 2 1
4050.2.c.x 4 5.c odd 4 2
4050.2.c.z 4 15.e even 4 2
6480.2.a.bb 2 12.b even 2 1
6480.2.a.bj 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(810))\):

\( T_{7}^{2} - 4T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display
\( T_{23}^{2} + 6T_{23} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 47 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 3 \) Copy content Toggle raw display
$29$ \( T^{2} - 48 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 27 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 3 \) Copy content Toggle raw display
$53$ \( T^{2} + 18T + 69 \) Copy content Toggle raw display
$59$ \( T^{2} - 75 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$83$ \( T^{2} + 24T + 132 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 16 \) Copy content Toggle raw display
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