# Properties

 Label 784.2.f.d Level $784$ Weight $2$ Character orbit 784.f 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$2^{2}$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{5} + 4 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{5} + 4 q^{9} + \beta_{3} q^{11} + 2 \beta_{2} q^{13} + \beta_{3} q^{15} -3 \beta_{2} q^{17} + \beta_{1} q^{19} -\beta_{3} q^{23} + 2 q^{25} + \beta_{1} q^{27} -\beta_{1} q^{31} -7 \beta_{2} q^{33} + 7 q^{37} -2 \beta_{3} q^{39} -2 \beta_{2} q^{41} -2 \beta_{3} q^{43} -4 \beta_{2} q^{45} -3 \beta_{1} q^{47} + 3 \beta_{3} q^{51} + 3 q^{53} -3 \beta_{1} q^{55} + 7 q^{57} -3 \beta_{1} q^{59} + \beta_{2} q^{61} + 6 q^{65} -\beta_{3} q^{67} + 7 \beta_{2} q^{69} -2 \beta_{3} q^{71} + 3 \beta_{2} q^{73} + 2 \beta_{1} q^{75} -\beta_{3} q^{79} -5 q^{81} -9 q^{85} + \beta_{2} q^{89} -7 q^{93} + \beta_{3} q^{95} -2 \beta_{2} q^{97} + 4 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 16q^{9} + O(q^{10})$$ $$4q + 16q^{9} + 8q^{25} + 28q^{37} + 12q^{53} + 28q^{57} + 24q^{65} - 20q^{81} - 36q^{85} - 28q^{93} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/7$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{2} + 7$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 14 \nu$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{2} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{1}$$

## Character values

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

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
783.1
 1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i −1.32288 + 2.29129i
0 −2.64575 0 1.73205i 0 0 0 4.00000 0
783.2 0 −2.64575 0 1.73205i 0 0 0 4.00000 0
783.3 0 2.64575 0 1.73205i 0 0 0 4.00000 0
783.4 0 2.64575 0 1.73205i 0 0 0 4.00000 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.b odd 2 1 inner
28.d even 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 7.c even 3 1
112.2.p.c 4 7.d odd 6 1
112.2.p.c 4 28.f even 6 1
112.2.p.c 4 28.g odd 6 1
448.2.p.c 4 56.j odd 6 1
448.2.p.c 4 56.k odd 6 1
448.2.p.c 4 56.m even 6 1
448.2.p.c 4 56.p even 6 1
784.2.f.d 4 1.a even 1 1 trivial
784.2.f.d 4 4.b odd 2 1 inner
784.2.f.d 4 7.b odd 2 1 inner
784.2.f.d 4 28.d even 2 1 inner
784.2.p.g 4 7.c even 3 1
784.2.p.g 4 7.d odd 6 1
784.2.p.g 4 28.f even 6 1
784.2.p.g 4 28.g odd 6 1
1008.2.cs.q 4 21.g even 6 1
1008.2.cs.q 4 21.h odd 6 1
1008.2.cs.q 4 84.j odd 6 1
1008.2.cs.q 4 84.n even 6 1
3136.2.f.f 4 8.b even 2 1
3136.2.f.f 4 8.d odd 2 1
3136.2.f.f 4 56.e even 2 1
3136.2.f.f 4 56.h odd 2 1
7056.2.b.s 4 3.b odd 2 1
7056.2.b.s 4 12.b even 2 1
7056.2.b.s 4 21.c even 2 1
7056.2.b.s 4 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 7$$ acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$.