# Properties

 Label 784.4.f.c Level $784$ Weight $4$ Character orbit 784.f Analytic conductor $46.257$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.2574974445$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -26 q^{9} +O(q^{10})$$ $$q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -26 q^{9} + ( -15 + 30 \zeta_{6} ) q^{11} + ( 40 - 80 \zeta_{6} ) q^{13} + ( -3 + 6 \zeta_{6} ) q^{15} + ( -21 + 42 \zeta_{6} ) q^{17} -17 q^{19} + ( 81 - 162 \zeta_{6} ) q^{23} + 98 q^{25} -53 q^{27} + 90 q^{29} + 17 q^{31} + ( -15 + 30 \zeta_{6} ) q^{33} + 199 q^{37} + ( 40 - 80 \zeta_{6} ) q^{39} + ( -108 + 216 \zeta_{6} ) q^{41} + ( 146 - 292 \zeta_{6} ) q^{43} + ( 78 - 156 \zeta_{6} ) q^{45} + 567 q^{47} + ( -21 + 42 \zeta_{6} ) q^{51} -333 q^{53} -135 q^{55} -17 q^{57} + 801 q^{59} + ( -207 + 414 \zeta_{6} ) q^{61} + 360 q^{65} + ( 125 - 250 \zeta_{6} ) q^{67} + ( 81 - 162 \zeta_{6} ) q^{69} + ( -282 + 564 \zeta_{6} ) q^{71} + ( -233 + 466 \zeta_{6} ) q^{73} + 98 q^{75} + ( 689 - 1378 \zeta_{6} ) q^{79} + 649 q^{81} -468 q^{83} -189 q^{85} + 90 q^{87} + ( -111 + 222 \zeta_{6} ) q^{89} + 17 q^{93} + ( 51 - 102 \zeta_{6} ) q^{95} + ( 804 - 1608 \zeta_{6} ) q^{97} + ( 390 - 780 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 52q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 52q^{9} - 34q^{19} + 196q^{25} - 106q^{27} + 180q^{29} + 34q^{31} + 398q^{37} + 1134q^{47} - 666q^{53} - 270q^{55} - 34q^{57} + 1602q^{59} + 720q^{65} + 196q^{75} + 1298q^{81} - 936q^{83} - 378q^{85} + 180q^{87} + 34q^{93} + O(q^{100})$$

## 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 5.19615i 0 0 0 −26.0000 0
783.2 0 1.00000 0 5.19615i 0 0 0 −26.0000 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 784.4.f.c 2
4.b odd 2 1 784.4.f.b 2
7.b odd 2 1 784.4.f.b 2
7.c even 3 1 112.4.p.b 2
7.d odd 6 1 112.4.p.c yes 2
28.d even 2 1 inner 784.4.f.c 2
28.f even 6 1 112.4.p.b 2
28.g odd 6 1 112.4.p.c yes 2
56.j odd 6 1 448.4.p.b 2
56.k odd 6 1 448.4.p.b 2
56.m even 6 1 448.4.p.c 2
56.p even 6 1 448.4.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.b 2 7.c even 3 1
112.4.p.b 2 28.f even 6 1
112.4.p.c yes 2 7.d odd 6 1
112.4.p.c yes 2 28.g odd 6 1
448.4.p.b 2 56.j odd 6 1
448.4.p.b 2 56.k odd 6 1
448.4.p.c 2 56.m even 6 1
448.4.p.c 2 56.p even 6 1
784.4.f.b 2 4.b odd 2 1
784.4.f.b 2 7.b odd 2 1
784.4.f.c 2 1.a even 1 1 trivial
784.4.f.c 2 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{4}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$27 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$675 + T^{2}$$
$13$ $$4800 + T^{2}$$
$17$ $$1323 + T^{2}$$
$19$ $$( 17 + T )^{2}$$
$23$ $$19683 + T^{2}$$
$29$ $$( -90 + T )^{2}$$
$31$ $$( -17 + T )^{2}$$
$37$ $$( -199 + T )^{2}$$
$41$ $$34992 + T^{2}$$
$43$ $$63948 + T^{2}$$
$47$ $$( -567 + T )^{2}$$
$53$ $$( 333 + T )^{2}$$
$59$ $$( -801 + T )^{2}$$
$61$ $$128547 + T^{2}$$
$67$ $$46875 + T^{2}$$
$71$ $$238572 + T^{2}$$
$73$ $$162867 + T^{2}$$
$79$ $$1424163 + T^{2}$$
$83$ $$( 468 + T )^{2}$$
$89$ $$36963 + T^{2}$$
$97$ $$1939248 + T^{2}$$