# Properties

 Label 784.2.i.n Level $784$ Weight $2$ Character orbit 784.i 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 392) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} -\beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{5} -\beta_{2} q^{9} + ( 6 + 6 \beta_{2} ) q^{11} + 4 \beta_{3} q^{13} -4 q^{15} -\beta_{1} q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} -4 \beta_{2} q^{23} + ( -3 - 3 \beta_{2} ) q^{25} -4 \beta_{3} q^{27} -6 q^{29} + 2 \beta_{1} q^{31} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{33} + 2 \beta_{2} q^{37} + ( -8 - 8 \beta_{2} ) q^{39} + \beta_{3} q^{41} -10 q^{43} + 2 \beta_{1} q^{45} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} -2 \beta_{2} q^{51} + ( 2 + 2 \beta_{2} ) q^{53} + 12 \beta_{3} q^{55} -6 q^{57} + \beta_{1} q^{59} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{61} -16 \beta_{2} q^{65} + ( 4 + 4 \beta_{2} ) q^{67} -4 \beta_{3} q^{69} + 12 q^{71} + 7 \beta_{1} q^{73} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{75} + 4 \beta_{2} q^{79} + ( 5 + 5 \beta_{2} ) q^{81} + \beta_{3} q^{83} + 4 q^{85} -6 \beta_{1} q^{87} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{89} + 4 \beta_{2} q^{93} + ( -12 - 12 \beta_{2} ) q^{95} -9 \beta_{3} q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} + 12q^{11} - 16q^{15} + 8q^{23} - 6q^{25} - 24q^{29} - 4q^{37} - 16q^{39} - 40q^{43} + 4q^{51} + 4q^{53} - 24q^{57} + 32q^{65} + 8q^{67} + 48q^{71} - 8q^{79} + 10q^{81} + 16q^{85} - 8q^{93} - 24q^{95} + 24q^{99} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.707107 1.22474i 0 1.41421 2.44949i 0 0 0 0.500000 0.866025i 0
177.2 0 0.707107 + 1.22474i 0 −1.41421 + 2.44949i 0 0 0 0.500000 0.866025i 0
753.1 0 −0.707107 + 1.22474i 0 1.41421 + 2.44949i 0 0 0 0.500000 + 0.866025i 0
753.2 0 0.707107 1.22474i 0 −1.41421 2.44949i 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.b odd 2 1 inner
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 784.2.i.n 4
4.b odd 2 1 392.2.i.h 4
7.b odd 2 1 inner 784.2.i.n 4
7.c even 3 1 784.2.a.k 2
7.c even 3 1 inner 784.2.i.n 4
7.d odd 6 1 784.2.a.k 2
7.d odd 6 1 inner 784.2.i.n 4
12.b even 2 1 3528.2.s.bj 4
21.g even 6 1 7056.2.a.ct 2
21.h odd 6 1 7056.2.a.ct 2
28.d even 2 1 392.2.i.h 4
28.f even 6 1 392.2.a.g 2
28.f even 6 1 392.2.i.h 4
28.g odd 6 1 392.2.a.g 2
28.g odd 6 1 392.2.i.h 4
56.j odd 6 1 3136.2.a.bp 2
56.k odd 6 1 3136.2.a.bk 2
56.m even 6 1 3136.2.a.bk 2
56.p even 6 1 3136.2.a.bp 2
84.h odd 2 1 3528.2.s.bj 4
84.j odd 6 1 3528.2.a.be 2
84.j odd 6 1 3528.2.s.bj 4
84.n even 6 1 3528.2.a.be 2
84.n even 6 1 3528.2.s.bj 4
140.p odd 6 1 9800.2.a.bv 2
140.s even 6 1 9800.2.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 28.f even 6 1
392.2.a.g 2 28.g odd 6 1
392.2.i.h 4 4.b odd 2 1
392.2.i.h 4 28.d even 2 1
392.2.i.h 4 28.f even 6 1
392.2.i.h 4 28.g odd 6 1
784.2.a.k 2 7.c even 3 1
784.2.a.k 2 7.d odd 6 1
784.2.i.n 4 1.a even 1 1 trivial
784.2.i.n 4 7.b odd 2 1 inner
784.2.i.n 4 7.c even 3 1 inner
784.2.i.n 4 7.d odd 6 1 inner
3136.2.a.bk 2 56.k odd 6 1
3136.2.a.bk 2 56.m even 6 1
3136.2.a.bp 2 56.j odd 6 1
3136.2.a.bp 2 56.p even 6 1
3528.2.a.be 2 84.j odd 6 1
3528.2.a.be 2 84.n even 6 1
3528.2.s.bj 4 12.b even 2 1
3528.2.s.bj 4 84.h odd 2 1
3528.2.s.bj 4 84.j odd 6 1
3528.2.s.bj 4 84.n even 6 1
7056.2.a.ct 2 21.g even 6 1
7056.2.a.ct 2 21.h odd 6 1
9800.2.a.bv 2 140.p odd 6 1
9800.2.a.bv 2 140.s even 6 1

## Hecke kernels

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

 $$T_{3}^{4} + 2 T_{3}^{2} + 4$$ $$T_{5}^{4} + 8 T_{5}^{2} + 64$$ $$T_{11}^{2} - 6 T_{11} + 36$$