# Properties

 Label 784.2.p.g Level $784$ Weight $2$ Character orbit 784.p Analytic conductor $6.260$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + 4 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + 4 \beta_{2} q^{9} + ( -\beta_{1} - 2 \beta_{3} ) q^{11} + ( 2 + 4 \beta_{2} ) q^{13} + ( 2 \beta_{1} + \beta_{3} ) q^{15} + ( -3 + 3 \beta_{2} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{3} ) q^{23} + ( -2 - 2 \beta_{2} ) q^{25} + \beta_{3} q^{27} -\beta_{1} q^{31} + ( 14 + 7 \beta_{2} ) q^{33} + 7 \beta_{2} q^{37} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{39} + ( -2 - 4 \beta_{2} ) q^{41} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{2} ) q^{45} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{47} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{51} + ( -3 - 3 \beta_{2} ) q^{53} -3 \beta_{3} q^{55} + 7 q^{57} -3 \beta_{1} q^{59} + ( -2 - \beta_{2} ) q^{61} + 6 \beta_{2} q^{65} + ( \beta_{1} + 2 \beta_{3} ) q^{67} + ( 7 + 14 \beta_{2} ) q^{69} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( 3 - 3 \beta_{2} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{75} + ( \beta_{1} - \beta_{3} ) q^{79} + ( 5 + 5 \beta_{2} ) q^{81} -9 q^{85} + ( -2 - \beta_{2} ) q^{89} -7 \beta_{2} q^{93} + ( -\beta_{1} - 2 \beta_{3} ) q^{95} + ( -2 - 4 \beta_{2} ) q^{97} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} - 8q^{9} + O(q^{10})$$ $$4q + 6q^{5} - 8q^{9} - 18q^{17} - 4q^{25} + 42q^{33} - 14q^{37} - 24q^{45} - 6q^{53} + 28q^{57} - 6q^{61} - 12q^{65} + 18q^{73} + 10q^{81} - 36q^{85} - 6q^{89} + 14q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{2}$$

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0 −1.32288 + 2.29129i 0 1.50000 0.866025i 0 0 0 −2.00000 3.46410i 0
31.2 0 1.32288 2.29129i 0 1.50000 0.866025i 0 0 0 −2.00000 3.46410i 0
607.1 0 −1.32288 2.29129i 0 1.50000 + 0.866025i 0 0 0 −2.00000 + 3.46410i 0
607.2 0 1.32288 + 2.29129i 0 1.50000 + 0.866025i 0 0 0 −2.00000 + 3.46410i 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
4.b odd 2 1 inner
7.d odd 6 1 inner
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.g 4
4.b odd 2 1 inner 784.2.p.g 4
7.b odd 2 1 112.2.p.c 4
7.c even 3 1 112.2.p.c 4
7.c even 3 1 784.2.f.d 4
7.d odd 6 1 784.2.f.d 4
7.d odd 6 1 inner 784.2.p.g 4
21.c even 2 1 1008.2.cs.q 4
21.g even 6 1 7056.2.b.s 4
21.h odd 6 1 1008.2.cs.q 4
21.h odd 6 1 7056.2.b.s 4
28.d even 2 1 112.2.p.c 4
28.f even 6 1 784.2.f.d 4
28.f even 6 1 inner 784.2.p.g 4
28.g odd 6 1 112.2.p.c 4
28.g odd 6 1 784.2.f.d 4
56.e even 2 1 448.2.p.c 4
56.h odd 2 1 448.2.p.c 4
56.j odd 6 1 3136.2.f.f 4
56.k odd 6 1 448.2.p.c 4
56.k odd 6 1 3136.2.f.f 4
56.m even 6 1 3136.2.f.f 4
56.p even 6 1 448.2.p.c 4
56.p even 6 1 3136.2.f.f 4
84.h odd 2 1 1008.2.cs.q 4
84.j odd 6 1 7056.2.b.s 4
84.n even 6 1 1008.2.cs.q 4
84.n even 6 1 7056.2.b.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 7.b odd 2 1
112.2.p.c 4 7.c even 3 1
112.2.p.c 4 28.d even 2 1
112.2.p.c 4 28.g odd 6 1
448.2.p.c 4 56.e even 2 1
448.2.p.c 4 56.h odd 2 1
448.2.p.c 4 56.k odd 6 1
448.2.p.c 4 56.p even 6 1
784.2.f.d 4 7.c even 3 1
784.2.f.d 4 7.d odd 6 1
784.2.f.d 4 28.f even 6 1
784.2.f.d 4 28.g odd 6 1
784.2.p.g 4 1.a even 1 1 trivial
784.2.p.g 4 4.b odd 2 1 inner
784.2.p.g 4 7.d odd 6 1 inner
784.2.p.g 4 28.f even 6 1 inner
1008.2.cs.q 4 21.c even 2 1
1008.2.cs.q 4 21.h odd 6 1
1008.2.cs.q 4 84.h odd 2 1
1008.2.cs.q 4 84.n even 6 1
3136.2.f.f 4 56.j odd 6 1
3136.2.f.f 4 56.k odd 6 1
3136.2.f.f 4 56.m even 6 1
3136.2.f.f 4 56.p even 6 1
7056.2.b.s 4 21.g even 6 1
7056.2.b.s 4 21.h odd 6 1
7056.2.b.s 4 84.j odd 6 1
7056.2.b.s 4 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 $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{4} + 7 T_{3}^{2} + 49$$ $$T_{5}^{2} - 3 T_{5} + 3$$ $$T_{11}^{4} - 21 T_{11}^{2} + 441$$