# Properties

 Label 784.2.f.a Level $784$ Weight $2$ Character orbit 784.f Analytic conductor $6.260$ Analytic rank $1$ 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
783.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 0 1.73205i 0 0 0 −2.00000 0
783.2 0 −1.00000 0 1.73205i 0 0 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
28.d even 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 7.d odd 6 1
112.2.p.a 2 28.g odd 6 1
112.2.p.b yes 2 7.c even 3 1
112.2.p.b yes 2 28.f even 6 1
448.2.p.a 2 56.m even 6 1
448.2.p.a 2 56.p even 6 1
448.2.p.b 2 56.j odd 6 1
448.2.p.b 2 56.k odd 6 1
784.2.f.a 2 1.a even 1 1 trivial
784.2.f.a 2 28.d even 2 1 inner
784.2.f.b 2 4.b odd 2 1
784.2.f.b 2 7.b odd 2 1
784.2.p.c 2 7.d odd 6 1
784.2.p.c 2 28.g odd 6 1
784.2.p.d 2 7.c even 3 1
784.2.p.d 2 28.f even 6 1
1008.2.cs.c 2 21.h odd 6 1
1008.2.cs.c 2 84.j odd 6 1
1008.2.cs.f 2 21.g even 6 1
1008.2.cs.f 2 84.n even 6 1
3136.2.f.a 2 8.d odd 2 1
3136.2.f.a 2 56.h odd 2 1
3136.2.f.b 2 8.b even 2 1
3136.2.f.b 2 56.e even 2 1
7056.2.b.b 2 3.b odd 2 1
7056.2.b.b 2 84.h odd 2 1
7056.2.b.m 2 12.b even 2 1
7056.2.b.m 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

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