# Properties

 Label 784.3.s.a Level 784 Weight 3 Character orbit 784.s Analytic conductor 21.362 Analytic rank 0 Dimension 2 CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 784.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.3624527258$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q -9 \zeta_{6} q^{9} + ( -6 + 6 \zeta_{6} ) q^{11} + 18 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -54 q^{29} + 38 \zeta_{6} q^{37} -58 q^{43} + ( 6 - 6 \zeta_{6} ) q^{53} + ( -118 + 118 \zeta_{6} ) q^{67} -114 q^{71} -94 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} + 54 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{9} + O(q^{10})$$ $$2q - 9q^{9} - 6q^{11} + 18q^{23} - 25q^{25} - 108q^{29} + 38q^{37} - 116q^{43} + 6q^{53} - 118q^{67} - 228q^{71} - 94q^{79} - 81q^{81} + 108q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 0 0 −4.50000 7.79423i 0
705.1 0 0 0 0 0 0 0 −4.50000 + 7.79423i 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 784.3.s.a 2
4.b odd 2 1 49.3.d.a 2
7.b odd 2 1 CM 784.3.s.a 2
7.c even 3 1 112.3.c.a 1
7.c even 3 1 inner 784.3.s.a 2
7.d odd 6 1 112.3.c.a 1
7.d odd 6 1 inner 784.3.s.a 2
12.b even 2 1 441.3.m.a 2
21.g even 6 1 1008.3.f.a 1
21.h odd 6 1 1008.3.f.a 1
28.d even 2 1 49.3.d.a 2
28.f even 6 1 7.3.b.a 1
28.f even 6 1 49.3.d.a 2
28.g odd 6 1 7.3.b.a 1
28.g odd 6 1 49.3.d.a 2
56.j odd 6 1 448.3.c.b 1
56.k odd 6 1 448.3.c.a 1
56.m even 6 1 448.3.c.a 1
56.p even 6 1 448.3.c.b 1
84.h odd 2 1 441.3.m.a 2
84.j odd 6 1 63.3.d.a 1
84.j odd 6 1 441.3.m.a 2
84.n even 6 1 63.3.d.a 1
84.n even 6 1 441.3.m.a 2
140.p odd 6 1 175.3.d.a 1
140.s even 6 1 175.3.d.a 1
140.w even 12 2 175.3.c.a 2
140.x odd 12 2 175.3.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 28.f even 6 1
7.3.b.a 1 28.g odd 6 1
49.3.d.a 2 4.b odd 2 1
49.3.d.a 2 28.d even 2 1
49.3.d.a 2 28.f even 6 1
49.3.d.a 2 28.g odd 6 1
63.3.d.a 1 84.j odd 6 1
63.3.d.a 1 84.n even 6 1
112.3.c.a 1 7.c even 3 1
112.3.c.a 1 7.d odd 6 1
175.3.c.a 2 140.w even 12 2
175.3.c.a 2 140.x odd 12 2
175.3.d.a 1 140.p odd 6 1
175.3.d.a 1 140.s even 6 1
441.3.m.a 2 12.b even 2 1
441.3.m.a 2 84.h odd 2 1
441.3.m.a 2 84.j odd 6 1
441.3.m.a 2 84.n even 6 1
448.3.c.a 1 56.k odd 6 1
448.3.c.a 1 56.m even 6 1
448.3.c.b 1 56.j odd 6 1
448.3.c.b 1 56.p even 6 1
784.3.s.a 2 1.a even 1 1 trivial
784.3.s.a 2 7.b odd 2 1 CM
784.3.s.a 2 7.c even 3 1 inner
784.3.s.a 2 7.d odd 6 1 inner
1008.3.f.a 1 21.g even 6 1
1008.3.f.a 1 21.h odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 3 T + 9 T^{2} )( 1 + 3 T + 9 T^{2} )$$
$5$ $$( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} )$$
$7$ 1
$11$ $$1 + 6 T - 85 T^{2} + 726 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - 13 T )^{2}( 1 + 13 T )^{2}$$
$17$ $$( 1 - 17 T + 289 T^{2} )( 1 + 17 T + 289 T^{2} )$$
$19$ $$( 1 - 19 T + 361 T^{2} )( 1 + 19 T + 361 T^{2} )$$
$23$ $$1 - 18 T - 205 T^{2} - 9522 T^{3} + 279841 T^{4}$$
$29$ $$( 1 + 54 T + 841 T^{2} )^{2}$$
$31$ $$( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} )$$
$37$ $$1 - 38 T + 75 T^{2} - 52022 T^{3} + 1874161 T^{4}$$
$41$ $$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$
$43$ $$( 1 + 58 T + 1849 T^{2} )^{2}$$
$47$ $$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$
$53$ $$1 - 6 T - 2773 T^{2} - 16854 T^{3} + 7890481 T^{4}$$
$59$ $$( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} )$$
$61$ $$( 1 - 61 T + 3721 T^{2} )( 1 + 61 T + 3721 T^{2} )$$
$67$ $$1 + 118 T + 9435 T^{2} + 529702 T^{3} + 20151121 T^{4}$$
$71$ $$( 1 + 114 T + 5041 T^{2} )^{2}$$
$73$ $$( 1 - 73 T + 5329 T^{2} )( 1 + 73 T + 5329 T^{2} )$$
$79$ $$1 + 94 T + 2595 T^{2} + 586654 T^{3} + 38950081 T^{4}$$
$83$ $$( 1 - 83 T )^{2}( 1 + 83 T )^{2}$$
$89$ $$( 1 - 89 T + 7921 T^{2} )( 1 + 89 T + 7921 T^{2} )$$
$97$ $$( 1 - 97 T )^{2}( 1 + 97 T )^{2}$$