# Properties

 Label 805.2.c.a Level 805 Weight 2 Character orbit 805.c Analytic conductor 6.428 Analytic rank 0 Dimension 2 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ = $$805 = 5 \cdot 7 \cdot 23$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 805.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.42795736271$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 q^{4} + ( 2 - i ) q^{5} + i q^{7} + 2 q^{9} +O(q^{10})$$ $$q + i q^{3} + 2 q^{4} + ( 2 - i ) q^{5} + i q^{7} + 2 q^{9} - q^{11} + 2 i q^{12} + i q^{13} + ( 1 + 2 i ) q^{15} + 4 q^{16} -i q^{17} -2 q^{19} + ( 4 - 2 i ) q^{20} - q^{21} + i q^{23} + ( 3 - 4 i ) q^{25} + 5 i q^{27} + 2 i q^{28} -7 q^{29} + 4 q^{31} -i q^{33} + ( 1 + 2 i ) q^{35} + 4 q^{36} -8 i q^{37} - q^{39} -6 q^{41} -8 i q^{43} -2 q^{44} + ( 4 - 2 i ) q^{45} + 7 i q^{47} + 4 i q^{48} - q^{49} + q^{51} + 2 i q^{52} + 4 i q^{53} + ( -2 + i ) q^{55} -2 i q^{57} + 4 q^{59} + ( 2 + 4 i ) q^{60} -10 q^{61} + 2 i q^{63} + 8 q^{64} + ( 1 + 2 i ) q^{65} + 14 i q^{67} -2 i q^{68} - q^{69} + 2 i q^{73} + ( 4 + 3 i ) q^{75} -4 q^{76} -i q^{77} + 15 q^{79} + ( 8 - 4 i ) q^{80} + q^{81} -8 i q^{83} -2 q^{84} + ( -1 - 2 i ) q^{85} -7 i q^{87} -6 q^{89} - q^{91} + 2 i q^{92} + 4 i q^{93} + ( -4 + 2 i ) q^{95} + 7 i q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{4} + 4q^{5} + 4q^{9} + O(q^{10})$$ $$2q + 4q^{4} + 4q^{5} + 4q^{9} - 2q^{11} + 2q^{15} + 8q^{16} - 4q^{19} + 8q^{20} - 2q^{21} + 6q^{25} - 14q^{29} + 8q^{31} + 2q^{35} + 8q^{36} - 2q^{39} - 12q^{41} - 4q^{44} + 8q^{45} - 2q^{49} + 2q^{51} - 4q^{55} + 8q^{59} + 4q^{60} - 20q^{61} + 16q^{64} + 2q^{65} - 2q^{69} + 8q^{75} - 8q^{76} + 30q^{79} + 16q^{80} + 2q^{81} - 4q^{84} - 2q^{85} - 12q^{89} - 2q^{91} - 8q^{95} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/805\mathbb{Z}\right)^\times$$.

 $$n$$ $$162$$ $$281$$ $$346$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
484.1
 − 1.00000i 1.00000i
0 1.00000i 2.00000 2.00000 + 1.00000i 0 1.00000i 0 2.00000 0
484.2 0 1.00000i 2.00000 2.00000 1.00000i 0 1.00000i 0 2.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.c.a 2
5.b even 2 1 inner 805.2.c.a 2
5.c odd 4 1 4025.2.a.c 1
5.c odd 4 1 4025.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.c.a 2 1.a even 1 1 trivial
805.2.c.a 2 5.b even 2 1 inner
4025.2.a.c 1 5.c odd 4 1
4025.2.a.d 1 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(805, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}$$
$3$ $$1 - 5 T^{2} + 9 T^{4}$$
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + T + 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 - 33 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( 1 + 7 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$1 - 45 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$
$59$ $$( 1 - 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 10 T + 61 T^{2} )^{2}$$
$67$ $$1 + 62 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 15 T + 79 T^{2} )^{2}$$
$83$ $$1 - 102 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 145 T^{2} + 9409 T^{4}$$
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