# Properties

 Label 784.2.i.i Level $784$ Weight $2$ Character orbit 784.i Analytic conductor $6.260$ Analytic rank $0$ 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.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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{9} + 4 q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 q^{27} -6 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{37} + ( 8 - 8 \zeta_{6} ) q^{39} -6 q^{41} -8 q^{43} + 12 \zeta_{6} q^{47} -12 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} -4 q^{57} + ( 6 - 6 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + ( -4 + 4 \zeta_{6} ) q^{67} + ( 2 - 2 \zeta_{6} ) q^{73} -10 \zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -6 q^{83} + ( -12 + 12 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + 10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - q^{9} + O(q^{10})$$ $$2q + 2q^{3} - q^{9} + 8q^{13} + 6q^{17} - 2q^{19} + 5q^{25} + 8q^{27} - 12q^{29} + 4q^{31} - 2q^{37} + 8q^{39} - 12q^{41} - 16q^{43} + 12q^{47} - 12q^{51} - 6q^{53} - 8q^{57} + 6q^{59} + 8q^{61} - 4q^{67} + 2q^{73} - 10q^{75} + 8q^{79} + 11q^{81} - 12q^{83} - 12q^{87} - 6q^{89} - 8q^{93} + 20q^{97} + O(q^{100})$$

## 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 1.00000 + 1.73205i 0 0 0 0 0 −0.500000 + 0.866025i 0
753.1 0 1.00000 1.73205i 0 0 0 0 0 −0.500000 0.866025i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.i.i 2
4.b odd 2 1 98.2.c.a 2
7.b odd 2 1 784.2.i.c 2
7.c even 3 1 784.2.a.b 1
7.c even 3 1 inner 784.2.i.i 2
7.d odd 6 1 112.2.a.c 1
7.d odd 6 1 784.2.i.c 2
12.b even 2 1 882.2.g.d 2
21.g even 6 1 1008.2.a.h 1
21.h odd 6 1 7056.2.a.bd 1
28.d even 2 1 98.2.c.b 2
28.f even 6 1 14.2.a.a 1
28.f even 6 1 98.2.c.b 2
28.g odd 6 1 98.2.a.a 1
28.g odd 6 1 98.2.c.a 2
35.i odd 6 1 2800.2.a.g 1
35.k even 12 2 2800.2.g.h 2
56.j odd 6 1 448.2.a.a 1
56.k odd 6 1 3136.2.a.e 1
56.m even 6 1 448.2.a.g 1
56.p even 6 1 3136.2.a.z 1
84.h odd 2 1 882.2.g.c 2
84.j odd 6 1 126.2.a.b 1
84.j odd 6 1 882.2.g.c 2
84.n even 6 1 882.2.a.i 1
84.n even 6 1 882.2.g.d 2
112.v even 12 2 1792.2.b.c 2
112.x odd 12 2 1792.2.b.g 2
140.p odd 6 1 2450.2.a.t 1
140.s even 6 1 350.2.a.f 1
140.w even 12 2 2450.2.c.c 2
140.x odd 12 2 350.2.c.d 2
168.ba even 6 1 4032.2.a.r 1
168.be odd 6 1 4032.2.a.w 1
252.n even 6 1 1134.2.f.l 2
252.r odd 6 1 1134.2.f.f 2
252.bj even 6 1 1134.2.f.l 2
252.bn odd 6 1 1134.2.f.f 2
308.m odd 6 1 1694.2.a.e 1
364.x even 6 1 2366.2.a.j 1
364.bw odd 12 2 2366.2.d.b 2
420.be odd 6 1 3150.2.a.i 1
420.br even 12 2 3150.2.g.j 2
476.q even 6 1 4046.2.a.f 1
532.bh odd 6 1 5054.2.a.c 1
644.j odd 6 1 7406.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 28.f even 6 1
98.2.a.a 1 28.g odd 6 1
98.2.c.a 2 4.b odd 2 1
98.2.c.a 2 28.g odd 6 1
98.2.c.b 2 28.d even 2 1
98.2.c.b 2 28.f even 6 1
112.2.a.c 1 7.d odd 6 1
126.2.a.b 1 84.j odd 6 1
350.2.a.f 1 140.s even 6 1
350.2.c.d 2 140.x odd 12 2
448.2.a.a 1 56.j odd 6 1
448.2.a.g 1 56.m even 6 1
784.2.a.b 1 7.c even 3 1
784.2.i.c 2 7.b odd 2 1
784.2.i.c 2 7.d odd 6 1
784.2.i.i 2 1.a even 1 1 trivial
784.2.i.i 2 7.c even 3 1 inner
882.2.a.i 1 84.n even 6 1
882.2.g.c 2 84.h odd 2 1
882.2.g.c 2 84.j odd 6 1
882.2.g.d 2 12.b even 2 1
882.2.g.d 2 84.n even 6 1
1008.2.a.h 1 21.g even 6 1
1134.2.f.f 2 252.r odd 6 1
1134.2.f.f 2 252.bn odd 6 1
1134.2.f.l 2 252.n even 6 1
1134.2.f.l 2 252.bj even 6 1
1694.2.a.e 1 308.m odd 6 1
1792.2.b.c 2 112.v even 12 2
1792.2.b.g 2 112.x odd 12 2
2366.2.a.j 1 364.x even 6 1
2366.2.d.b 2 364.bw odd 12 2
2450.2.a.t 1 140.p odd 6 1
2450.2.c.c 2 140.w even 12 2
2800.2.a.g 1 35.i odd 6 1
2800.2.g.h 2 35.k even 12 2
3136.2.a.e 1 56.k odd 6 1
3136.2.a.z 1 56.p even 6 1
3150.2.a.i 1 420.be odd 6 1
3150.2.g.j 2 420.br even 12 2
4032.2.a.r 1 168.ba even 6 1
4032.2.a.w 1 168.be odd 6 1
4046.2.a.f 1 476.q even 6 1
5054.2.a.c 1 532.bh odd 6 1
7056.2.a.bd 1 21.h odd 6 1
7406.2.a.a 1 644.j odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$144 - 12 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 - 2 T + T^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$