# Properties

 Label 784.2.p.c Level $784$ Weight $2$ Character orbit 784.p Analytic conductor $6.260$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 −1.50000 + 0.866025i 0 0 0 1.00000 + 1.73205i 0
607.1 0 −0.500000 0.866025i 0 −1.50000 0.866025i 0 0 0 1.00000 1.73205i 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
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.p.c 2
4.b odd 2 1 784.2.p.d 2
7.b odd 2 1 112.2.p.b yes 2
7.c even 3 1 112.2.p.a 2
7.c even 3 1 784.2.f.b 2
7.d odd 6 1 784.2.f.a 2
7.d odd 6 1 784.2.p.d 2
21.c even 2 1 1008.2.cs.c 2
21.g even 6 1 7056.2.b.b 2
21.h odd 6 1 1008.2.cs.f 2
21.h odd 6 1 7056.2.b.m 2
28.d even 2 1 112.2.p.a 2
28.f even 6 1 784.2.f.b 2
28.f even 6 1 inner 784.2.p.c 2
28.g odd 6 1 112.2.p.b yes 2
28.g odd 6 1 784.2.f.a 2
56.e even 2 1 448.2.p.b 2
56.h odd 2 1 448.2.p.a 2
56.j odd 6 1 3136.2.f.b 2
56.k odd 6 1 448.2.p.a 2
56.k odd 6 1 3136.2.f.b 2
56.m even 6 1 3136.2.f.a 2
56.p even 6 1 448.2.p.b 2
56.p even 6 1 3136.2.f.a 2
84.h odd 2 1 1008.2.cs.f 2
84.j odd 6 1 7056.2.b.m 2
84.n even 6 1 1008.2.cs.c 2
84.n even 6 1 7056.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 7.c even 3 1
112.2.p.a 2 28.d even 2 1
112.2.p.b yes 2 7.b odd 2 1
112.2.p.b yes 2 28.g odd 6 1
448.2.p.a 2 56.h odd 2 1
448.2.p.a 2 56.k odd 6 1
448.2.p.b 2 56.e even 2 1
448.2.p.b 2 56.p even 6 1
784.2.f.a 2 7.d odd 6 1
784.2.f.a 2 28.g odd 6 1
784.2.f.b 2 7.c even 3 1
784.2.f.b 2 28.f even 6 1
784.2.p.c 2 1.a even 1 1 trivial
784.2.p.c 2 28.f even 6 1 inner
784.2.p.d 2 4.b odd 2 1
784.2.p.d 2 7.d odd 6 1
1008.2.cs.c 2 21.c even 2 1
1008.2.cs.c 2 84.n even 6 1
1008.2.cs.f 2 21.h odd 6 1
1008.2.cs.f 2 84.h odd 2 1
3136.2.f.a 2 56.m even 6 1
3136.2.f.a 2 56.p even 6 1
3136.2.f.b 2 56.j odd 6 1
3136.2.f.b 2 56.k odd 6 1
7056.2.b.b 2 21.g even 6 1
7056.2.b.b 2 84.n even 6 1
7056.2.b.m 2 21.h odd 6 1
7056.2.b.m 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{5}^{2} + 3 T_{5} + 3$$ $$T_{11}^{2} - 3 T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3 - 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$27 - 9 T + T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$75 + 15 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$48 + T^{2}$$
$43$ $$12 + T^{2}$$
$47$ $$9 + 3 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$75 + 15 T + T^{2}$$
$67$ $$27 + 9 T + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$3 + 3 T + T^{2}$$
$79$ $$27 - 9 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$147 - 21 T + T^{2}$$
$97$ $$48 + T^{2}$$