Properties

Label 810.2.a.k
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: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + \beta q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{5} + \beta q^{7} + q^{8} + q^{10} + (\beta + 1) q^{11} - 2 \beta q^{13} + \beta q^{14} + q^{16} + ( - \beta + 5) q^{17} + ( - \beta + 1) q^{19} + q^{20} + (\beta + 1) q^{22} + ( - \beta + 2) q^{23} + q^{25} - 2 \beta q^{26} + \beta q^{28} + ( - \beta + 2) q^{29} + (2 \beta - 2) q^{31} + q^{32} + ( - \beta + 5) q^{34} + \beta q^{35} - 4 q^{37} + ( - \beta + 1) q^{38} + q^{40} + 3 q^{41} + (\beta - 9) q^{43} + (\beta + 1) q^{44} + ( - \beta + 2) q^{46} + (\beta + 4) q^{47} + (\beta + 1) q^{49} + q^{50} - 2 \beta q^{52} + (4 \beta - 2) q^{53} + (\beta + 1) q^{55} + \beta q^{56} + ( - \beta + 2) q^{58} + ( - \beta - 1) q^{59} + ( - 3 \beta + 2) q^{61} + (2 \beta - 2) q^{62} + q^{64} - 2 \beta q^{65} - 7 q^{67} + ( - \beta + 5) q^{68} + \beta q^{70} + 6 q^{71} + (3 \beta - 7) q^{73} - 4 q^{74} + ( - \beta + 1) q^{76} + (2 \beta + 8) q^{77} + 2 q^{79} + q^{80} + 3 q^{82} + ( - \beta - 4) q^{83} + ( - \beta + 5) q^{85} + (\beta - 9) q^{86} + (\beta + 1) q^{88} + ( - 3 \beta - 6) q^{89} + ( - 2 \beta - 16) q^{91} + ( - \beta + 2) q^{92} + (\beta + 4) q^{94} + ( - \beta + 1) q^{95} + ( - \beta - 5) q^{97} + (\beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} + 3 q^{11} - 2 q^{13} + q^{14} + 2 q^{16} + 9 q^{17} + q^{19} + 2 q^{20} + 3 q^{22} + 3 q^{23} + 2 q^{25} - 2 q^{26} + q^{28} + 3 q^{29} - 2 q^{31} + 2 q^{32} + 9 q^{34} + q^{35} - 8 q^{37} + q^{38} + 2 q^{40} + 6 q^{41} - 17 q^{43} + 3 q^{44} + 3 q^{46} + 9 q^{47} + 3 q^{49} + 2 q^{50} - 2 q^{52} + 3 q^{55} + q^{56} + 3 q^{58} - 3 q^{59} + q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{65} - 14 q^{67} + 9 q^{68} + q^{70} + 12 q^{71} - 11 q^{73} - 8 q^{74} + q^{76} + 18 q^{77} + 4 q^{79} + 2 q^{80} + 6 q^{82} - 9 q^{83} + 9 q^{85} - 17 q^{86} + 3 q^{88} - 15 q^{89} - 34 q^{91} + 3 q^{92} + 9 q^{94} + q^{95} - 11 q^{97} + 3 q^{98}+O(q^{100}) \) 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
−2.37228
3.37228
1.00000 0 1.00000 1.00000 0 −2.37228 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 3.37228 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.k 2
3.b odd 2 1 810.2.a.i 2
4.b odd 2 1 6480.2.a.bn 2
5.b even 2 1 4050.2.a.bo 2
5.c odd 4 2 4050.2.c.ba 4
9.c even 3 2 270.2.e.c 4
9.d odd 6 2 90.2.e.c 4
12.b even 2 1 6480.2.a.be 2
15.d odd 2 1 4050.2.a.bw 2
15.e even 4 2 4050.2.c.v 4
36.f odd 6 2 2160.2.q.f 4
36.h even 6 2 720.2.q.f 4
45.h odd 6 2 450.2.e.j 4
45.j even 6 2 1350.2.e.l 4
45.k odd 12 4 1350.2.j.f 8
45.l even 12 4 450.2.j.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 9.d odd 6 2
270.2.e.c 4 9.c even 3 2
450.2.e.j 4 45.h odd 6 2
450.2.j.g 8 45.l even 12 4
720.2.q.f 4 36.h even 6 2
810.2.a.i 2 3.b odd 2 1
810.2.a.k 2 1.a even 1 1 trivial
1350.2.e.l 4 45.j even 6 2
1350.2.j.f 8 45.k odd 12 4
2160.2.q.f 4 36.f odd 6 2
4050.2.a.bo 2 5.b even 2 1
4050.2.a.bw 2 15.d odd 2 1
4050.2.c.v 4 15.e even 4 2
4050.2.c.ba 4 5.c odd 4 2
6480.2.a.be 2 12.b even 2 1
6480.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(810))\):

\( T_{7}^{2} - T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} - 9T_{17} + 12 \) Copy content Toggle raw display
\( T_{23}^{2} - 3T_{23} - 6 \) 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
$7$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 17T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 132 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$61$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$67$ \( (T + 7)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11T - 44 \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 15T - 18 \) Copy content Toggle raw display
$97$ \( T^{2} + 11T + 22 \) Copy content Toggle raw display
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