# Properties

 Label 810.2.i.b Level $810$ Weight $2$ Character orbit 810.i Analytic conductor $6.468$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 810.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.46788256372$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( -1 - 2 \zeta_{12}^{3} ) q^{10} + ( 2 - 2 \zeta_{12}^{2} ) q^{11} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{13} + 2 \zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 2 \zeta_{12}^{3} q^{17} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{20} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{22} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + 6 q^{26} + 2 \zeta_{12}^{3} q^{28} + 8 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -2 + 2 \zeta_{12}^{2} ) q^{34} + ( -2 - 4 \zeta_{12}^{3} ) q^{35} -2 \zeta_{12}^{3} q^{37} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{40} + 2 \zeta_{12}^{2} q^{41} + 4 \zeta_{12} q^{43} + 2 q^{44} + 4 q^{46} + 8 \zeta_{12} q^{47} -3 \zeta_{12}^{2} q^{49} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + 6 \zeta_{12} q^{52} -6 \zeta_{12}^{3} q^{53} + ( -4 + 2 \zeta_{12}^{3} ) q^{55} + ( -2 + 2 \zeta_{12}^{2} ) q^{56} -10 \zeta_{12}^{2} q^{59} + ( -2 + 2 \zeta_{12}^{2} ) q^{61} + 8 \zeta_{12}^{3} q^{62} - q^{64} + ( -6 - 12 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{65} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{67} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{68} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{70} -12 q^{71} -4 \zeta_{12}^{3} q^{73} + ( 2 - 2 \zeta_{12}^{2} ) q^{74} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{77} + ( 2 - \zeta_{12}^{3} ) q^{80} + 2 \zeta_{12}^{3} q^{82} -4 \zeta_{12} q^{83} + ( 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{85} + 4 \zeta_{12}^{2} q^{86} + 2 \zeta_{12} q^{88} -10 q^{89} + 12 q^{91} + 4 \zeta_{12} q^{92} + 8 \zeta_{12}^{2} q^{94} -8 \zeta_{12} q^{97} -3 \zeta_{12}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 4q^{5} + O(q^{10})$$ $$4q + 2q^{4} - 4q^{5} - 4q^{10} + 4q^{11} + 4q^{14} - 2q^{16} + 4q^{20} - 6q^{25} + 24q^{26} + 16q^{31} - 4q^{34} - 8q^{35} - 2q^{40} + 4q^{41} + 8q^{44} + 16q^{46} - 6q^{49} + 8q^{50} - 16q^{55} - 4q^{56} - 20q^{59} - 4q^{61} - 4q^{64} - 12q^{65} + 8q^{70} - 48q^{71} + 4q^{74} + 8q^{80} - 4q^{85} + 8q^{86} - 40q^{89} + 48q^{91} + 16q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.133975 + 2.23205i 0 −1.73205 + 1.00000i 1.00000i 0 −1.00000 2.00000i
109.2 0.866025 0.500000i 0 0.500000 0.866025i −1.86603 + 1.23205i 0 1.73205 1.00000i 1.00000i 0 −1.00000 + 2.00000i
379.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.133975 2.23205i 0 −1.73205 1.00000i 1.00000i 0 −1.00000 + 2.00000i
379.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.86603 1.23205i 0 1.73205 + 1.00000i 1.00000i 0 −1.00000 2.00000i
 $$n$$: 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
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.i.b 4
3.b odd 2 1 810.2.i.e 4
5.b even 2 1 inner 810.2.i.b 4
9.c even 3 1 90.2.c.a 2
9.c even 3 1 inner 810.2.i.b 4
9.d odd 6 1 30.2.c.a 2
9.d odd 6 1 810.2.i.e 4
15.d odd 2 1 810.2.i.e 4
36.f odd 6 1 720.2.f.f 2
36.h even 6 1 240.2.f.a 2
45.h odd 6 1 30.2.c.a 2
45.h odd 6 1 810.2.i.e 4
45.j even 6 1 90.2.c.a 2
45.j even 6 1 inner 810.2.i.b 4
45.k odd 12 1 450.2.a.b 1
45.k odd 12 1 450.2.a.f 1
45.l even 12 1 150.2.a.a 1
45.l even 12 1 150.2.a.c 1
63.i even 6 1 1470.2.n.a 4
63.j odd 6 1 1470.2.n.h 4
63.n odd 6 1 1470.2.n.h 4
63.o even 6 1 1470.2.g.g 2
63.s even 6 1 1470.2.n.a 4
72.j odd 6 1 960.2.f.h 2
72.l even 6 1 960.2.f.i 2
72.n even 6 1 2880.2.f.e 2
72.p odd 6 1 2880.2.f.c 2
144.u even 12 1 3840.2.d.j 2
144.u even 12 1 3840.2.d.x 2
144.w odd 12 1 3840.2.d.g 2
144.w odd 12 1 3840.2.d.y 2
180.n even 6 1 240.2.f.a 2
180.p odd 6 1 720.2.f.f 2
180.v odd 12 1 1200.2.a.g 1
180.v odd 12 1 1200.2.a.m 1
180.x even 12 1 3600.2.a.o 1
180.x even 12 1 3600.2.a.bg 1
315.u even 6 1 1470.2.n.a 4
315.v odd 6 1 1470.2.n.h 4
315.z even 6 1 1470.2.g.g 2
315.bq even 6 1 1470.2.n.a 4
315.br odd 6 1 1470.2.n.h 4
315.cf odd 12 1 7350.2.a.bg 1
315.cf odd 12 1 7350.2.a.cc 1
360.z odd 6 1 2880.2.f.c 2
360.bd even 6 1 960.2.f.i 2
360.bh odd 6 1 960.2.f.h 2
360.bk even 6 1 2880.2.f.e 2
360.br even 12 1 4800.2.a.l 1
360.br even 12 1 4800.2.a.cg 1
360.bt odd 12 1 4800.2.a.m 1
360.bt odd 12 1 4800.2.a.cj 1
720.ch odd 12 1 3840.2.d.g 2
720.ch odd 12 1 3840.2.d.y 2
720.da even 12 1 3840.2.d.j 2
720.da even 12 1 3840.2.d.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 9.d odd 6 1
30.2.c.a 2 45.h odd 6 1
90.2.c.a 2 9.c even 3 1
90.2.c.a 2 45.j even 6 1
150.2.a.a 1 45.l even 12 1
150.2.a.c 1 45.l even 12 1
240.2.f.a 2 36.h even 6 1
240.2.f.a 2 180.n even 6 1
450.2.a.b 1 45.k odd 12 1
450.2.a.f 1 45.k odd 12 1
720.2.f.f 2 36.f odd 6 1
720.2.f.f 2 180.p odd 6 1
810.2.i.b 4 1.a even 1 1 trivial
810.2.i.b 4 5.b even 2 1 inner
810.2.i.b 4 9.c even 3 1 inner
810.2.i.b 4 45.j even 6 1 inner
810.2.i.e 4 3.b odd 2 1
810.2.i.e 4 9.d odd 6 1
810.2.i.e 4 15.d odd 2 1
810.2.i.e 4 45.h odd 6 1
960.2.f.h 2 72.j odd 6 1
960.2.f.h 2 360.bh odd 6 1
960.2.f.i 2 72.l even 6 1
960.2.f.i 2 360.bd even 6 1
1200.2.a.g 1 180.v odd 12 1
1200.2.a.m 1 180.v odd 12 1
1470.2.g.g 2 63.o even 6 1
1470.2.g.g 2 315.z even 6 1
1470.2.n.a 4 63.i even 6 1
1470.2.n.a 4 63.s even 6 1
1470.2.n.a 4 315.u even 6 1
1470.2.n.a 4 315.bq even 6 1
1470.2.n.h 4 63.j odd 6 1
1470.2.n.h 4 63.n odd 6 1
1470.2.n.h 4 315.v odd 6 1
1470.2.n.h 4 315.br odd 6 1
2880.2.f.c 2 72.p odd 6 1
2880.2.f.c 2 360.z odd 6 1
2880.2.f.e 2 72.n even 6 1
2880.2.f.e 2 360.bk even 6 1
3600.2.a.o 1 180.x even 12 1
3600.2.a.bg 1 180.x even 12 1
3840.2.d.g 2 144.w odd 12 1
3840.2.d.g 2 720.ch odd 12 1
3840.2.d.j 2 144.u even 12 1
3840.2.d.j 2 720.da even 12 1
3840.2.d.x 2 144.u even 12 1
3840.2.d.x 2 720.da even 12 1
3840.2.d.y 2 144.w odd 12 1
3840.2.d.y 2 720.ch odd 12 1
4800.2.a.l 1 360.br even 12 1
4800.2.a.m 1 360.bt odd 12 1
4800.2.a.cg 1 360.br even 12 1
4800.2.a.cj 1 360.bt odd 12 1
7350.2.a.bg 1 315.cf odd 12 1
7350.2.a.cc 1 315.cf odd 12 1

## Hecke kernels

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

 $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ $$T_{11}^{2} - 2 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ 1
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$
$11$ $$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8}$$
$47$ $$1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 130 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 79 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{4}$$
$97$ $$( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} )$$