# Properties

 Label 816.2.c.c Level $816$ Weight $2$ Character orbit 816.c Analytic conductor $6.516$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$816 = 2^{4} \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 816.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.51579280494$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} -4 i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} -4 i q^{7} - q^{9} + 4 i q^{11} + 2 q^{13} + ( 1 - 4 i ) q^{17} + 4 q^{19} + 4 q^{21} -4 i q^{23} + 5 q^{25} -i q^{27} -4 i q^{31} -4 q^{33} -8 i q^{37} + 2 i q^{39} + 8 i q^{41} + 4 q^{43} + 8 q^{47} -9 q^{49} + ( 4 + i ) q^{51} + 6 q^{53} + 4 i q^{57} + 12 q^{59} + 8 i q^{61} + 4 i q^{63} -12 q^{67} + 4 q^{69} + 12 i q^{71} + 5 i q^{75} + 16 q^{77} -4 i q^{79} + q^{81} -12 q^{83} -10 q^{89} -8 i q^{91} + 4 q^{93} -16 i q^{97} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{13} + 2q^{17} + 8q^{19} + 8q^{21} + 10q^{25} - 8q^{33} + 8q^{43} + 16q^{47} - 18q^{49} + 8q^{51} + 12q^{53} + 24q^{59} - 24q^{67} + 8q^{69} + 32q^{77} + 2q^{81} - 24q^{83} - 20q^{89} + 8q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$511$$ $$545$$ $$613$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
577.2 0 1.00000i 0 0 0 4.00000i 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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.c.c 2
3.b odd 2 1 2448.2.c.j 2
4.b odd 2 1 51.2.d.b 2
8.b even 2 1 3264.2.c.d 2
8.d odd 2 1 3264.2.c.e 2
12.b even 2 1 153.2.d.a 2
17.b even 2 1 inner 816.2.c.c 2
20.d odd 2 1 1275.2.g.a 2
20.e even 4 1 1275.2.d.b 2
20.e even 4 1 1275.2.d.d 2
51.c odd 2 1 2448.2.c.j 2
68.d odd 2 1 51.2.d.b 2
68.f odd 4 1 867.2.a.a 1
68.f odd 4 1 867.2.a.b 1
68.g odd 8 4 867.2.e.d 4
68.i even 16 8 867.2.h.d 8
136.e odd 2 1 3264.2.c.e 2
136.h even 2 1 3264.2.c.d 2
204.h even 2 1 153.2.d.a 2
204.l even 4 1 2601.2.a.i 1
204.l even 4 1 2601.2.a.j 1
340.d odd 2 1 1275.2.g.a 2
340.r even 4 1 1275.2.d.b 2
340.r even 4 1 1275.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 4.b odd 2 1
51.2.d.b 2 68.d odd 2 1
153.2.d.a 2 12.b even 2 1
153.2.d.a 2 204.h even 2 1
816.2.c.c 2 1.a even 1 1 trivial
816.2.c.c 2 17.b even 2 1 inner
867.2.a.a 1 68.f odd 4 1
867.2.a.b 1 68.f odd 4 1
867.2.e.d 4 68.g odd 8 4
867.2.h.d 8 68.i even 16 8
1275.2.d.b 2 20.e even 4 1
1275.2.d.b 2 340.r even 4 1
1275.2.d.d 2 20.e even 4 1
1275.2.d.d 2 340.r even 4 1
1275.2.g.a 2 20.d odd 2 1
1275.2.g.a 2 340.d odd 2 1
2448.2.c.j 2 3.b odd 2 1
2448.2.c.j 2 51.c odd 2 1
2601.2.a.i 1 204.l even 4 1
2601.2.a.j 1 204.l even 4 1
3264.2.c.d 2 8.b even 2 1
3264.2.c.d 2 136.h even 2 1
3264.2.c.e 2 8.d odd 2 1
3264.2.c.e 2 136.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(816, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$17 - 2 T + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$16 + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$256 + T^{2}$$