# Properties

 Label 784.2.p.d Level 784 Weight 2 Character orbit 784.p 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.p (of order $$6$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + ( 3 + 3 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} + ( 10 - 5 \zeta_{6} ) q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} + 5 q^{27} -6 q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + ( -2 + \zeta_{6} ) q^{33} + 5 \zeta_{6} q^{37} + ( -4 + 8 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( -2 - 2 \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + ( 6 - 3 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + 3 q^{55} + 7 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + ( -10 + 5 \zeta_{6} ) q^{61} + ( 3 + 3 \zeta_{6} ) q^{67} + ( 5 - 10 \zeta_{6} ) q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( -1 - \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} + ( -6 + 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} -9 q^{85} + ( -6 + 6 \zeta_{6} ) q^{87} + ( 14 - 7 \zeta_{6} ) q^{89} -5 \zeta_{6} q^{93} + ( -7 - 7 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 3q^{5} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} - 3q^{5} + 2q^{9} - 3q^{11} + 9q^{17} + 7q^{19} + 15q^{23} - 2q^{25} + 10q^{27} - 12q^{29} + 5q^{31} - 3q^{33} + 5q^{37} - 6q^{45} + 3q^{47} + 9q^{51} + 9q^{53} + 6q^{55} + 14q^{57} + 9q^{59} - 15q^{61} + 9q^{67} - 3q^{73} + 2q^{75} - 9q^{79} - q^{81} - 24q^{83} - 18q^{85} - 6q^{87} + 21q^{89} - 5q^{93} - 21q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 −1.50000 + 0.866025i 0 0 0 1.00000 + 1.73205i 0
607.1 0 0.500000 + 0.866025i 0 −1.50000 0.866025i 0 0 0 1.00000 1.73205i 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.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.p.d 2
4.b odd 2 1 784.2.p.c 2
7.b odd 2 1 112.2.p.a 2
7.c even 3 1 112.2.p.b yes 2
7.c even 3 1 784.2.f.a 2
7.d odd 6 1 784.2.f.b 2
7.d odd 6 1 784.2.p.c 2
21.c even 2 1 1008.2.cs.f 2
21.g even 6 1 7056.2.b.m 2
21.h odd 6 1 1008.2.cs.c 2
21.h odd 6 1 7056.2.b.b 2
28.d even 2 1 112.2.p.b yes 2
28.f even 6 1 784.2.f.a 2
28.f even 6 1 inner 784.2.p.d 2
28.g odd 6 1 112.2.p.a 2
28.g odd 6 1 784.2.f.b 2
56.e even 2 1 448.2.p.a 2
56.h odd 2 1 448.2.p.b 2
56.j odd 6 1 3136.2.f.a 2
56.k odd 6 1 448.2.p.b 2
56.k odd 6 1 3136.2.f.a 2
56.m even 6 1 3136.2.f.b 2
56.p even 6 1 448.2.p.a 2
56.p even 6 1 3136.2.f.b 2
84.h odd 2 1 1008.2.cs.c 2
84.j odd 6 1 7056.2.b.b 2
84.n even 6 1 1008.2.cs.f 2
84.n even 6 1 7056.2.b.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 7.b odd 2 1
112.2.p.a 2 28.g odd 6 1
112.2.p.b yes 2 7.c even 3 1
112.2.p.b yes 2 28.d even 2 1
448.2.p.a 2 56.e even 2 1
448.2.p.a 2 56.p even 6 1
448.2.p.b 2 56.h odd 2 1
448.2.p.b 2 56.k odd 6 1
784.2.f.a 2 7.c even 3 1
784.2.f.a 2 28.f even 6 1
784.2.f.b 2 7.d odd 6 1
784.2.f.b 2 28.g odd 6 1
784.2.p.c 2 4.b odd 2 1
784.2.p.c 2 7.d odd 6 1
784.2.p.d 2 1.a even 1 1 trivial
784.2.p.d 2 28.f even 6 1 inner
1008.2.cs.c 2 21.h odd 6 1
1008.2.cs.c 2 84.h odd 2 1
1008.2.cs.f 2 21.c even 2 1
1008.2.cs.f 2 84.n even 6 1
3136.2.f.a 2 56.j odd 6 1
3136.2.f.a 2 56.k odd 6 1
3136.2.f.b 2 56.m even 6 1
3136.2.f.b 2 56.p even 6 1
7056.2.b.b 2 21.h odd 6 1
7056.2.b.b 2 84.j odd 6 1
7056.2.b.m 2 21.g even 6 1
7056.2.b.m 2 84.n even 6 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{2} + 3 T_{5} + 3$$ $$T_{11}^{2} + 3 T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4}$$
$37$ $$1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4}$$
$41$ $$1 - 34 T^{2} + 1681 T^{4}$$
$43$ $$1 - 74 T^{2} + 1849 T^{4}$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4}$$
$59$ $$1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4}$$
$61$ $$( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$
$67$ $$1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4}$$
$71$ $$1 - 130 T^{2} + 5041 T^{4}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 21 T + 236 T^{2} - 1869 T^{3} + 7921 T^{4}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$