# Properties

 Label 784.2.i.f Level $784$ Weight $2$ Character orbit 784.i Analytic conductor $6.260$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.i (of order $$3$$, 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 3 \zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{11} + 8 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + 2 q^{29} + 6 \zeta_{6} q^{37} + 12 q^{43} + ( 10 - 10 \zeta_{6} ) q^{53} + ( 4 - 4 \zeta_{6} ) q^{67} -16 q^{71} + 8 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{9} + O(q^{10})$$ $$2 q + 3 q^{9} + 4 q^{11} + 8 q^{23} + 5 q^{25} + 4 q^{29} + 6 q^{37} + 24 q^{43} + 10 q^{53} + 4 q^{67} - 32 q^{71} + 8 q^{79} - 9 q^{81} + 24 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 0 0 1.50000 2.59808i 0
753.1 0 0 0 0 0 0 0 1.50000 + 2.59808i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.i.f 2
4.b odd 2 1 49.2.c.a 2
7.b odd 2 1 CM 784.2.i.f 2
7.c even 3 1 784.2.a.f 1
7.c even 3 1 inner 784.2.i.f 2
7.d odd 6 1 784.2.a.f 1
7.d odd 6 1 inner 784.2.i.f 2
12.b even 2 1 441.2.e.d 2
21.g even 6 1 7056.2.a.bg 1
21.h odd 6 1 7056.2.a.bg 1
28.d even 2 1 49.2.c.a 2
28.f even 6 1 49.2.a.a 1
28.f even 6 1 49.2.c.a 2
28.g odd 6 1 49.2.a.a 1
28.g odd 6 1 49.2.c.a 2
56.j odd 6 1 3136.2.a.o 1
56.k odd 6 1 3136.2.a.n 1
56.m even 6 1 3136.2.a.n 1
56.p even 6 1 3136.2.a.o 1
84.h odd 2 1 441.2.e.d 2
84.j odd 6 1 441.2.a.c 1
84.j odd 6 1 441.2.e.d 2
84.n even 6 1 441.2.a.c 1
84.n even 6 1 441.2.e.d 2
140.p odd 6 1 1225.2.a.c 1
140.s even 6 1 1225.2.a.c 1
140.w even 12 2 1225.2.b.c 2
140.x odd 12 2 1225.2.b.c 2
308.m odd 6 1 5929.2.a.c 1
308.n even 6 1 5929.2.a.c 1
364.x even 6 1 8281.2.a.d 1
364.bl odd 6 1 8281.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 28.f even 6 1
49.2.a.a 1 28.g odd 6 1
49.2.c.a 2 4.b odd 2 1
49.2.c.a 2 28.d even 2 1
49.2.c.a 2 28.f even 6 1
49.2.c.a 2 28.g odd 6 1
441.2.a.c 1 84.j odd 6 1
441.2.a.c 1 84.n even 6 1
441.2.e.d 2 12.b even 2 1
441.2.e.d 2 84.h odd 2 1
441.2.e.d 2 84.j odd 6 1
441.2.e.d 2 84.n even 6 1
784.2.a.f 1 7.c even 3 1
784.2.a.f 1 7.d odd 6 1
784.2.i.f 2 1.a even 1 1 trivial
784.2.i.f 2 7.b odd 2 1 CM
784.2.i.f 2 7.c even 3 1 inner
784.2.i.f 2 7.d odd 6 1 inner
1225.2.a.c 1 140.p odd 6 1
1225.2.a.c 1 140.s even 6 1
1225.2.b.c 2 140.w even 12 2
1225.2.b.c 2 140.x odd 12 2
3136.2.a.n 1 56.k odd 6 1
3136.2.a.n 1 56.m even 6 1
3136.2.a.o 1 56.j odd 6 1
3136.2.a.o 1 56.p even 6 1
5929.2.a.c 1 308.m odd 6 1
5929.2.a.c 1 308.n even 6 1
7056.2.a.bg 1 21.g even 6 1
7056.2.a.bg 1 21.h odd 6 1
8281.2.a.d 1 364.x even 6 1
8281.2.a.d 1 364.bl odd 6 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{5}$$ $$T_{11}^{2} - 4 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$64 - 8 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$36 - 6 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -12 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$100 - 10 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( 16 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$