Properties

Label 810.2.e.e
Level $810$
Weight $2$
Character orbit 810.e
Analytic conductor $6.468$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,2,Mod(271,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.e (of order \(3\), degree \(2\), not minimal)

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: no
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + q^{8} - q^{10} + ( - 6 \zeta_{6} + 6) q^{11} + 4 \zeta_{6} q^{13} - 2 \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 6 q^{17} - 4 q^{19} + ( - \zeta_{6} + 1) q^{20} + 6 \zeta_{6} q^{22} + (\zeta_{6} - 1) q^{25} - 4 q^{26} + 2 q^{28} + (6 \zeta_{6} - 6) q^{29} + 4 \zeta_{6} q^{31} - \zeta_{6} q^{32} + (6 \zeta_{6} - 6) q^{34} - 2 q^{35} + 8 q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + \zeta_{6} q^{40} + (8 \zeta_{6} - 8) q^{43} - 6 q^{44} + 3 \zeta_{6} q^{49} - \zeta_{6} q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + 6 q^{53} + 6 q^{55} + (2 \zeta_{6} - 2) q^{56} - 6 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + (2 \zeta_{6} - 2) q^{61} - 4 q^{62} + q^{64} + (4 \zeta_{6} - 4) q^{65} + 4 \zeta_{6} q^{67} - 6 \zeta_{6} q^{68} + ( - 2 \zeta_{6} + 2) q^{70} + 12 q^{71} - 10 q^{73} + (8 \zeta_{6} - 8) q^{74} + 4 \zeta_{6} q^{76} + 12 \zeta_{6} q^{77} + ( - 4 \zeta_{6} + 4) q^{79} - q^{80} + ( - 12 \zeta_{6} + 12) q^{83} + 6 \zeta_{6} q^{85} - 8 \zeta_{6} q^{86} + ( - 6 \zeta_{6} + 6) q^{88} - 12 q^{89} - 8 q^{91} - 4 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} - 3 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} + 6 q^{11} + 4 q^{13} - 2 q^{14} - q^{16} + 12 q^{17} - 8 q^{19} + q^{20} + 6 q^{22} - q^{25} - 8 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 6 q^{34} - 4 q^{35} + 16 q^{37} + 4 q^{38} + q^{40} - 8 q^{43} - 12 q^{44} + 3 q^{49} - q^{50} + 4 q^{52} + 12 q^{53} + 12 q^{55} - 2 q^{56} - 6 q^{58} + 6 q^{59} - 2 q^{61} - 8 q^{62} + 2 q^{64} - 4 q^{65} + 4 q^{67} - 6 q^{68} + 2 q^{70} + 24 q^{71} - 20 q^{73} - 8 q^{74} + 4 q^{76} + 12 q^{77} + 4 q^{79} - 2 q^{80} + 12 q^{83} + 6 q^{85} - 8 q^{86} + 6 q^{88} - 24 q^{89} - 16 q^{91} - 4 q^{95} - 2 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 −1.00000 1.73205i 1.00000 0 −1.00000
541.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.e.e 2
3.b odd 2 1 810.2.e.h 2
9.c even 3 1 90.2.a.b yes 1
9.c even 3 1 inner 810.2.e.e 2
9.d odd 6 1 90.2.a.a 1
9.d odd 6 1 810.2.e.h 2
36.f odd 6 1 720.2.a.b 1
36.h even 6 1 720.2.a.g 1
45.h odd 6 1 450.2.a.e 1
45.j even 6 1 450.2.a.a 1
45.k odd 12 2 450.2.c.a 2
45.l even 12 2 450.2.c.d 2
63.l odd 6 1 4410.2.a.bf 1
63.o even 6 1 4410.2.a.k 1
72.j odd 6 1 2880.2.a.k 1
72.l even 6 1 2880.2.a.h 1
72.n even 6 1 2880.2.a.bf 1
72.p odd 6 1 2880.2.a.u 1
180.n even 6 1 3600.2.a.ba 1
180.p odd 6 1 3600.2.a.bj 1
180.v odd 12 2 3600.2.f.a 2
180.x even 12 2 3600.2.f.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.a.a 1 9.d odd 6 1
90.2.a.b yes 1 9.c even 3 1
450.2.a.a 1 45.j even 6 1
450.2.a.e 1 45.h odd 6 1
450.2.c.a 2 45.k odd 12 2
450.2.c.d 2 45.l even 12 2
720.2.a.b 1 36.f odd 6 1
720.2.a.g 1 36.h even 6 1
810.2.e.e 2 1.a even 1 1 trivial
810.2.e.e 2 9.c even 3 1 inner
810.2.e.h 2 3.b odd 2 1
810.2.e.h 2 9.d odd 6 1
2880.2.a.h 1 72.l even 6 1
2880.2.a.k 1 72.j odd 6 1
2880.2.a.u 1 72.p odd 6 1
2880.2.a.bf 1 72.n even 6 1
3600.2.a.ba 1 180.n even 6 1
3600.2.a.bj 1 180.p odd 6 1
3600.2.f.a 2 180.v odd 12 2
3600.2.f.u 2 180.x even 12 2
4410.2.a.k 1 63.o even 6 1
4410.2.a.bf 1 63.l odd 6 1

Hecke kernels

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

\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 16 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( (T + 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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