# Properties

 Label 804.2.i.c Level 804 Weight 2 Character orbit 804.i Analytic conductor 6.420 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 804.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + 2 q^{5} + 3 \zeta_{6} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + 2 q^{5} + 3 \zeta_{6} q^{7} + q^{9} -\zeta_{6} q^{11} + ( 5 - 5 \zeta_{6} ) q^{13} + 2 q^{15} + ( 1 - \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{19} + 3 \zeta_{6} q^{21} + ( -7 + 7 \zeta_{6} ) q^{23} - q^{25} + q^{27} + 5 \zeta_{6} q^{29} + 3 \zeta_{6} q^{31} -\zeta_{6} q^{33} + 6 \zeta_{6} q^{35} + ( -3 + 3 \zeta_{6} ) q^{37} + ( 5 - 5 \zeta_{6} ) q^{39} -7 \zeta_{6} q^{41} + 4 q^{43} + 2 q^{45} -9 \zeta_{6} q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} + ( 1 - \zeta_{6} ) q^{51} + 2 q^{53} -2 \zeta_{6} q^{55} + ( 1 - \zeta_{6} ) q^{57} + ( 1 - \zeta_{6} ) q^{61} + 3 \zeta_{6} q^{63} + ( 10 - 10 \zeta_{6} ) q^{65} + ( 7 + 2 \zeta_{6} ) q^{67} + ( -7 + 7 \zeta_{6} ) q^{69} -5 \zeta_{6} q^{71} + ( 9 - 9 \zeta_{6} ) q^{73} - q^{75} + ( 3 - 3 \zeta_{6} ) q^{77} + 15 \zeta_{6} q^{79} + q^{81} + ( -11 + 11 \zeta_{6} ) q^{83} + ( 2 - 2 \zeta_{6} ) q^{85} + 5 \zeta_{6} q^{87} -18 q^{89} + 15 q^{91} + 3 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} + ( -15 + 15 \zeta_{6} ) q^{97} -\zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 4q^{5} + 3q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 4q^{5} + 3q^{7} + 2q^{9} - q^{11} + 5q^{13} + 4q^{15} + q^{17} + q^{19} + 3q^{21} - 7q^{23} - 2q^{25} + 2q^{27} + 5q^{29} + 3q^{31} - q^{33} + 6q^{35} - 3q^{37} + 5q^{39} - 7q^{41} + 8q^{43} + 4q^{45} - 9q^{47} - 2q^{49} + q^{51} + 4q^{53} - 2q^{55} + q^{57} + q^{61} + 3q^{63} + 10q^{65} + 16q^{67} - 7q^{69} - 5q^{71} + 9q^{73} - 2q^{75} + 3q^{77} + 15q^{79} + 2q^{81} - 11q^{83} + 2q^{85} + 5q^{87} - 36q^{89} + 30q^{91} + 3q^{93} + 2q^{95} - 15q^{97} - q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$$.

 $$n$$ $$269$$ $$337$$ $$403$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 0 2.00000 0 1.50000 2.59808i 0 1.00000 0
565.1 0 1.00000 0 2.00000 0 1.50000 + 2.59808i 0 1.00000 0
 $$n$$: 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
67.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.i.c 2
3.b odd 2 1 2412.2.l.b 2
67.c even 3 1 inner 804.2.i.c 2
201.g odd 6 1 2412.2.l.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.i.c 2 1.a even 1 1 trivial
804.2.i.c 2 67.c even 3 1 inner
2412.2.l.b 2 3.b odd 2 1
2412.2.l.b 2 201.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$:

 $$T_{5} - 2$$ $$T_{7}^{2} - 3 T_{7} + 9$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - T )^{2}$$
$5$ $$( 1 - 2 T + 5 T^{2} )^{2}$$
$7$ $$1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4}$$
$11$ $$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4}$$
$29$ $$1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4}$$
$31$ $$1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4}$$
$37$ $$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$
$41$ $$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$1 - 16 T + 67 T^{2}$$
$71$ $$1 + 5 T - 46 T^{2} + 355 T^{3} + 5041 T^{4}$$
$73$ $$1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4}$$
$79$ $$1 - 15 T + 146 T^{2} - 1185 T^{3} + 6241 T^{4}$$
$83$ $$1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 18 T + 89 T^{2} )^{2}$$
$97$ $$1 + 15 T + 128 T^{2} + 1455 T^{3} + 9409 T^{4}$$