# Properties

 Label 810.2.e.l 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 30) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} - q^{8} + q^{10} -2 \zeta_{6} q^{13} -4 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 6 q^{17} -4 q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( -1 + \zeta_{6} ) q^{25} -2 q^{26} -4 q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -8 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 6 - 6 \zeta_{6} ) q^{34} + 4 q^{35} + 2 q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} -\zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} -9 \zeta_{6} q^{49} + \zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} -6 q^{53} + ( -4 + 4 \zeta_{6} ) q^{56} -6 \zeta_{6} q^{58} + ( 10 - 10 \zeta_{6} ) q^{61} -8 q^{62} + q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} -6 \zeta_{6} q^{68} + ( 4 - 4 \zeta_{6} ) q^{70} + 2 q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} + ( -8 + 8 \zeta_{6} ) q^{79} - q^{80} + 6 q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} + 6 \zeta_{6} q^{85} -4 \zeta_{6} q^{86} + 18 q^{89} -8 q^{91} -4 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} -9 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + q^{5} + 4q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + q^{5} + 4q^{7} - 2q^{8} + 2q^{10} - 2q^{13} - 4q^{14} - q^{16} + 12q^{17} - 8q^{19} + q^{20} - q^{25} - 4q^{26} - 8q^{28} + 6q^{29} - 8q^{31} + q^{32} + 6q^{34} + 8q^{35} + 4q^{37} - 4q^{38} - q^{40} + 6q^{41} + 4q^{43} - 9q^{49} + q^{50} - 2q^{52} - 12q^{53} - 4q^{56} - 6q^{58} + 10q^{61} - 16q^{62} + 2q^{64} + 2q^{65} + 4q^{67} - 6q^{68} + 4q^{70} + 4q^{73} + 2q^{74} + 4q^{76} - 8q^{79} - 2q^{80} + 12q^{82} - 12q^{83} + 6q^{85} - 4q^{86} + 36q^{89} - 16q^{91} - 4q^{95} - 2q^{97} - 18q^{98} + O(q^{100})$$

## 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 2.00000 + 3.46410i −1.00000 0 1.00000
541.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 2.00000 3.46410i −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.l 2
3.b odd 2 1 810.2.e.b 2
9.c even 3 1 30.2.a.a 1
9.c even 3 1 inner 810.2.e.l 2
9.d odd 6 1 90.2.a.c 1
9.d odd 6 1 810.2.e.b 2
36.f odd 6 1 240.2.a.b 1
36.h even 6 1 720.2.a.j 1
45.h odd 6 1 450.2.a.d 1
45.j even 6 1 150.2.a.b 1
45.k odd 12 2 150.2.c.a 2
45.l even 12 2 450.2.c.b 2
63.g even 3 1 1470.2.i.o 2
63.h even 3 1 1470.2.i.o 2
63.k odd 6 1 1470.2.i.q 2
63.l odd 6 1 1470.2.a.d 1
63.o even 6 1 4410.2.a.z 1
63.t odd 6 1 1470.2.i.q 2
72.j odd 6 1 2880.2.a.a 1
72.l even 6 1 2880.2.a.q 1
72.n even 6 1 960.2.a.e 1
72.p odd 6 1 960.2.a.p 1
99.h odd 6 1 3630.2.a.w 1
117.t even 6 1 5070.2.a.w 1
117.y odd 12 2 5070.2.b.k 2
144.v odd 12 2 3840.2.k.f 2
144.x even 12 2 3840.2.k.y 2
153.h even 6 1 8670.2.a.g 1
180.n even 6 1 3600.2.a.f 1
180.p odd 6 1 1200.2.a.k 1
180.v odd 12 2 3600.2.f.i 2
180.x even 12 2 1200.2.f.e 2
315.bg odd 6 1 7350.2.a.ct 1
360.z odd 6 1 4800.2.a.d 1
360.bk even 6 1 4800.2.a.cq 1
360.bo even 12 2 4800.2.f.w 2
360.bu odd 12 2 4800.2.f.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 9.c even 3 1
90.2.a.c 1 9.d odd 6 1
150.2.a.b 1 45.j even 6 1
150.2.c.a 2 45.k odd 12 2
240.2.a.b 1 36.f odd 6 1
450.2.a.d 1 45.h odd 6 1
450.2.c.b 2 45.l even 12 2
720.2.a.j 1 36.h even 6 1
810.2.e.b 2 3.b odd 2 1
810.2.e.b 2 9.d odd 6 1
810.2.e.l 2 1.a even 1 1 trivial
810.2.e.l 2 9.c even 3 1 inner
960.2.a.e 1 72.n even 6 1
960.2.a.p 1 72.p odd 6 1
1200.2.a.k 1 180.p odd 6 1
1200.2.f.e 2 180.x even 12 2
1470.2.a.d 1 63.l odd 6 1
1470.2.i.o 2 63.g even 3 1
1470.2.i.o 2 63.h even 3 1
1470.2.i.q 2 63.k odd 6 1
1470.2.i.q 2 63.t odd 6 1
2880.2.a.a 1 72.j odd 6 1
2880.2.a.q 1 72.l even 6 1
3600.2.a.f 1 180.n even 6 1
3600.2.f.i 2 180.v odd 12 2
3630.2.a.w 1 99.h odd 6 1
3840.2.k.f 2 144.v odd 12 2
3840.2.k.y 2 144.x even 12 2
4410.2.a.z 1 63.o even 6 1
4800.2.a.d 1 360.z odd 6 1
4800.2.a.cq 1 360.bk even 6 1
4800.2.f.p 2 360.bu odd 12 2
4800.2.f.w 2 360.bo even 12 2
5070.2.a.w 1 117.t even 6 1
5070.2.b.k 2 117.y odd 12 2
7350.2.a.ct 1 315.bg odd 6 1
8670.2.a.g 1 153.h even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$16 - 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$( -18 + T )^{2}$$
$97$ $$4 + 2 T + T^{2}$$