# Properties

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

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## 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q -7 q^{3} + ( -9 + 18 \zeta_{6} ) q^{5} + 22 q^{9} +O(q^{10})$$ $$q -7 q^{3} + ( -9 + 18 \zeta_{6} ) q^{5} + 22 q^{9} + ( 7 - 14 \zeta_{6} ) q^{11} + ( -40 + 80 \zeta_{6} ) q^{13} + ( 63 - 126 \zeta_{6} ) q^{15} + ( -19 + 38 \zeta_{6} ) q^{17} + 119 q^{19} + ( -77 + 154 \zeta_{6} ) q^{23} -118 q^{25} + 35 q^{27} + 210 q^{29} + 301 q^{31} + ( -49 + 98 \zeta_{6} ) q^{33} -77 q^{37} + ( 280 - 560 \zeta_{6} ) q^{39} + ( 68 - 136 \zeta_{6} ) q^{41} + ( 70 - 140 \zeta_{6} ) q^{43} + ( -198 + 396 \zeta_{6} ) q^{45} -357 q^{47} + ( 133 - 266 \zeta_{6} ) q^{51} + 327 q^{53} + 189 q^{55} -833 q^{57} + 609 q^{59} + ( -397 + 794 \zeta_{6} ) q^{61} -1080 q^{65} + ( 91 - 182 \zeta_{6} ) q^{67} + ( 539 - 1078 \zeta_{6} ) q^{69} + ( -126 + 252 \zeta_{6} ) q^{71} + ( 33 - 66 \zeta_{6} ) q^{73} + 826 q^{75} + ( -469 + 938 \zeta_{6} ) q^{79} -839 q^{81} -588 q^{83} -513 q^{85} -1470 q^{87} + ( -425 + 850 \zeta_{6} ) q^{89} -2107 q^{93} + ( -1071 + 2142 \zeta_{6} ) q^{95} + ( 180 - 360 \zeta_{6} ) q^{97} + ( 154 - 308 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 14q^{3} + 44q^{9} + O(q^{10})$$ $$2q - 14q^{3} + 44q^{9} + 238q^{19} - 236q^{25} + 70q^{27} + 420q^{29} + 602q^{31} - 154q^{37} - 714q^{47} + 654q^{53} + 378q^{55} - 1666q^{57} + 1218q^{59} - 2160q^{65} + 1652q^{75} - 1678q^{81} - 1176q^{83} - 1026q^{85} - 2940q^{87} - 4214q^{93} + O(q^{100})$$

## 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 −7.00000 0 15.5885i 0 0 0 22.0000 0
783.2 0 −7.00000 0 15.5885i 0 0 0 22.0000 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.a 2
4.b odd 2 1 784.4.f.d 2
7.b odd 2 1 784.4.f.d 2
7.c even 3 1 112.4.p.d yes 2
7.d odd 6 1 112.4.p.a 2
28.d even 2 1 inner 784.4.f.a 2
28.f even 6 1 112.4.p.d yes 2
28.g odd 6 1 112.4.p.a 2
56.j odd 6 1 448.4.p.d 2
56.k odd 6 1 448.4.p.d 2
56.m even 6 1 448.4.p.a 2
56.p even 6 1 448.4.p.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 7$$ acting on $$S_{4}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 7 + T )^{2}$$
$5$ $$243 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$147 + T^{2}$$
$13$ $$4800 + T^{2}$$
$17$ $$1083 + T^{2}$$
$19$ $$( -119 + T )^{2}$$
$23$ $$17787 + T^{2}$$
$29$ $$( -210 + T )^{2}$$
$31$ $$( -301 + T )^{2}$$
$37$ $$( 77 + T )^{2}$$
$41$ $$13872 + T^{2}$$
$43$ $$14700 + T^{2}$$
$47$ $$( 357 + T )^{2}$$
$53$ $$( -327 + T )^{2}$$
$59$ $$( -609 + T )^{2}$$
$61$ $$472827 + T^{2}$$
$67$ $$24843 + T^{2}$$
$71$ $$47628 + T^{2}$$
$73$ $$3267 + T^{2}$$
$79$ $$659883 + T^{2}$$
$83$ $$( 588 + T )^{2}$$
$89$ $$541875 + T^{2}$$
$97$ $$97200 + T^{2}$$
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