# 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 $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$6.46788256372$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

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

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.5 − 0.866025i 0.5 + 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(810, [\chi])$$:

 $$T_{7}^{2} + 2 T_{7} + 4$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13}^{2} - 4 T_{13} + 16$$ $$T_{17} - 6$$ $$T_{23}$$

## Hecke characteristic polynomials

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