# Properties

 Label 819.2.c.a Level $819$ Weight $2$ Character orbit 819.c Analytic conductor $6.540$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -2 q^{4} + 3 i q^{5} + i q^{7} +O(q^{10})$$ $$q + 2 i q^{2} -2 q^{4} + 3 i q^{5} + i q^{7} -6 q^{10} + ( 2 + 3 i ) q^{13} -2 q^{14} -4 q^{16} + 2 q^{17} -i q^{19} -6 i q^{20} + q^{23} -4 q^{25} + ( -6 + 4 i ) q^{26} -2 i q^{28} -5 q^{29} -5 i q^{31} -8 i q^{32} + 4 i q^{34} -3 q^{35} + 8 i q^{37} + 2 q^{38} -10 i q^{41} + 9 q^{43} + 2 i q^{46} + 7 i q^{47} - q^{49} -8 i q^{50} + ( -4 - 6 i ) q^{52} -9 q^{53} -10 i q^{58} -4 i q^{59} -8 q^{61} + 10 q^{62} + 8 q^{64} + ( -9 + 6 i ) q^{65} -2 i q^{67} -4 q^{68} -6 i q^{70} -9 i q^{73} -16 q^{74} + 2 i q^{76} + 15 q^{79} -12 i q^{80} + 20 q^{82} + 9 i q^{83} + 6 i q^{85} + 18 i q^{86} -9 i q^{89} + ( -3 + 2 i ) q^{91} -2 q^{92} -14 q^{94} + 3 q^{95} + 13 i q^{97} -2 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + O(q^{10})$$ $$2q - 4q^{4} - 12q^{10} + 4q^{13} - 4q^{14} - 8q^{16} + 4q^{17} + 2q^{23} - 8q^{25} - 12q^{26} - 10q^{29} - 6q^{35} + 4q^{38} + 18q^{43} - 2q^{49} - 8q^{52} - 18q^{53} - 16q^{61} + 20q^{62} + 16q^{64} - 18q^{65} - 8q^{68} - 32q^{74} + 30q^{79} + 40q^{82} - 6q^{91} - 4q^{92} - 28q^{94} + 6q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 3.00000i 0 1.00000i 0 0 −6.00000
64.2 2.00000i 0 −2.00000 3.00000i 0 1.00000i 0 0 −6.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.c.a 2
3.b odd 2 1 273.2.c.a 2
12.b even 2 1 4368.2.h.e 2
13.b even 2 1 inner 819.2.c.a 2
21.c even 2 1 1911.2.c.a 2
39.d odd 2 1 273.2.c.a 2
39.f even 4 1 3549.2.a.a 1
39.f even 4 1 3549.2.a.e 1
156.h even 2 1 4368.2.h.e 2
273.g even 2 1 1911.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.a 2 3.b odd 2 1
273.2.c.a 2 39.d odd 2 1
819.2.c.a 2 1.a even 1 1 trivial
819.2.c.a 2 13.b even 2 1 inner
1911.2.c.a 2 21.c even 2 1
1911.2.c.a 2 273.g even 2 1
3549.2.a.a 1 39.f even 4 1
3549.2.a.e 1 39.f even 4 1
4368.2.h.e 2 12.b even 2 1
4368.2.h.e 2 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 - 4 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$1 + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$25 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$100 + T^{2}$$
$43$ $$( -9 + T )^{2}$$
$47$ $$49 + T^{2}$$
$53$ $$( 9 + T )^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$81 + T^{2}$$
$79$ $$( -15 + T )^{2}$$
$83$ $$81 + T^{2}$$
$89$ $$81 + T^{2}$$
$97$ $$169 + T^{2}$$