# Properties

 Label 804.2.o.c Level 804 Weight 2 Character orbit 804.o Analytic conductor 6.420 Analytic rank 0 Dimension 4 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.o (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} ) q^{3} -2 q^{5} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} + ( -1 + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{3} ) q^{3} -2 q^{5} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} + ( -1 + 2 \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( 3 + 3 \beta_{2} ) q^{13} + ( -2 - 2 \beta_{3} ) q^{15} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{17} + ( -3 + 3 \beta_{2} ) q^{19} + ( 2 - \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{21} + ( -3 + 3 \beta_{1} - 3 \beta_{2} ) q^{23} - q^{25} + ( -5 + \beta_{3} ) q^{27} + ( 2 - 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{29} + ( -2 + \beta_{2} ) q^{31} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{33} + ( -4 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{35} + ( -5 - 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{37} + ( 3 - 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{39} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( 4 - 8 \beta_{2} + 4 \beta_{3} ) q^{43} + ( 2 - 4 \beta_{3} ) q^{45} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{49} + ( 5 - 2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{51} + 6 q^{53} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 3 + 4 \beta_{1} + 3 \beta_{2} ) q^{61} + ( -2 + 4 \beta_{1} - 7 \beta_{2} ) q^{63} + ( -6 - 6 \beta_{2} ) q^{65} + ( 9 - 2 \beta_{2} ) q^{67} + ( -9 + 6 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{69} + ( 10 + 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{71} + ( 1 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{73} + ( -1 - \beta_{3} ) q^{75} + ( -5 + 5 \beta_{1} - 5 \beta_{2} ) q^{77} + ( -6 + 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{79} + ( -7 - 4 \beta_{3} ) q^{81} + ( -1 - 7 \beta_{1} - \beta_{2} ) q^{83} + ( 2 + 6 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 2 - 4 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} ) q^{87} + ( -6 + 12 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 9 - 12 \beta_{1} + 6 \beta_{3} ) q^{91} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 6 - 6 \beta_{2} ) q^{95} + ( 3 + 6 \beta_{1} + 3 \beta_{2} ) q^{97} + ( -8 + \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 8q^{5} + 6q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 8q^{5} + 6q^{7} - 4q^{9} - 2q^{11} + 18q^{13} - 8q^{15} - 6q^{17} - 6q^{19} - 2q^{21} - 18q^{23} - 4q^{25} - 20q^{27} + 6q^{29} - 6q^{31} - 14q^{33} - 12q^{35} - 10q^{37} + 18q^{39} - 2q^{41} + 8q^{45} - 6q^{47} + 8q^{49} + 6q^{51} + 24q^{53} + 4q^{55} - 6q^{57} + 18q^{61} - 22q^{63} - 36q^{65} + 32q^{67} - 30q^{69} + 30q^{71} + 2q^{73} - 4q^{75} - 30q^{77} - 18q^{79} - 28q^{81} - 6q^{83} + 12q^{85} - 14q^{87} + 36q^{91} - 6q^{93} + 12q^{95} + 18q^{97} - 22q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$269$$ $$337$$ $$403$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
365.1
 1.22474 − 0.707107i −1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 1.00000 1.41421i 0 −2.00000 0 −0.949490 0.548188i 0 −1.00000 2.82843i 0
365.2 0 1.00000 + 1.41421i 0 −2.00000 0 3.94949 + 2.28024i 0 −1.00000 + 2.82843i 0
641.1 0 1.00000 1.41421i 0 −2.00000 0 3.94949 2.28024i 0 −1.00000 2.82843i 0
641.2 0 1.00000 + 1.41421i 0 −2.00000 0 −0.949490 + 0.548188i 0 −1.00000 + 2.82843i 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
201.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.o.c yes 4
3.b odd 2 1 804.2.o.b 4
67.d odd 6 1 804.2.o.b 4
201.f even 6 1 inner 804.2.o.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.o.b 4 3.b odd 2 1
804.2.o.b 4 67.d odd 6 1
804.2.o.c yes 4 1.a even 1 1 trivial
804.2.o.c yes 4 201.f even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - 2 T + 3 T^{2} )^{2}$$
$5$ $$( 1 + 2 T + 5 T^{2} )^{4}$$
$7$ $$1 - 6 T + 21 T^{2} - 54 T^{3} + 116 T^{4} - 378 T^{5} + 1029 T^{6} - 2058 T^{7} + 2401 T^{8}$$
$11$ $$1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 110 T^{5} - 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$( 1 - 7 T + 13 T^{2} )^{2}( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$1 + 6 T + 31 T^{2} + 114 T^{3} + 276 T^{4} + 1938 T^{5} + 8959 T^{6} + 29478 T^{7} + 83521 T^{8}$$
$19$ $$( 1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 18 T + 163 T^{2} + 990 T^{3} + 4980 T^{4} + 22770 T^{5} + 86227 T^{6} + 219006 T^{7} + 279841 T^{8}$$
$29$ $$1 - 6 T + 23 T^{2} - 66 T^{3} - 372 T^{4} - 1914 T^{5} + 19343 T^{6} - 146334 T^{7} + 707281 T^{8}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$1 + 10 T + 25 T^{2} + 10 T^{3} + 556 T^{4} + 370 T^{5} + 34225 T^{6} + 506530 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 2 T - 73 T^{2} - 10 T^{3} + 4084 T^{4} - 410 T^{5} - 122713 T^{6} + 137842 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 12 T^{2} - 2410 T^{4} - 22188 T^{6} + 3418801 T^{8}$$
$47$ $$1 + 6 T + 107 T^{2} + 570 T^{3} + 7380 T^{4} + 26790 T^{5} + 236363 T^{6} + 622938 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{4}$$
$59$ $$1 - 196 T^{2} + 16182 T^{4} - 682276 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 18 T + 225 T^{2} - 2106 T^{3} + 16556 T^{4} - 128466 T^{5} + 837225 T^{6} - 4085658 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 - 16 T + 67 T^{2} )^{2}$$
$71$ $$1 - 30 T + 499 T^{2} - 5970 T^{3} + 55860 T^{4} - 423870 T^{5} + 2515459 T^{6} - 10737330 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 2 T - 119 T^{2} + 46 T^{3} + 9508 T^{4} + 3358 T^{5} - 634151 T^{6} - 778034 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 18 T + 261 T^{2} + 2754 T^{3} + 25700 T^{4} + 217566 T^{5} + 1628901 T^{6} + 8874702 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 6 T + 83 T^{2} + 426 T^{3} - 852 T^{4} + 35358 T^{5} + 571787 T^{6} + 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 76 T^{2} + 3462 T^{4} - 601996 T^{6} + 62742241 T^{8}$$
$97$ $$1 - 18 T + 257 T^{2} - 2682 T^{3} + 23268 T^{4} - 260154 T^{5} + 2418113 T^{6} - 16428114 T^{7} + 88529281 T^{8}$$