# Properties

 Label 784.3.c.e Level $784$ Weight $3$ Character orbit 784.c Analytic conductor $21.362$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 784.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.3624527258$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 14) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} + 2 \beta_{3} q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} + 2 \beta_{3} q^{9} + ( 9 - \beta_{3} ) q^{11} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{13} + ( 9 - \beta_{3} ) q^{15} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + ( 15 + 3 \beta_{3} ) q^{23} + ( -2 - 4 \beta_{3} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{27} + ( 12 - 2 \beta_{3} ) q^{29} + ( 15 \beta_{1} + 7 \beta_{2} ) q^{31} + ( -12 \beta_{1} + 15 \beta_{2} ) q^{33} + ( 31 - 8 \beta_{3} ) q^{37} + ( 6 - 4 \beta_{3} ) q^{39} + ( 10 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{3} ) q^{43} + ( 6 \beta_{1} + 24 \beta_{2} ) q^{45} + ( \beta_{1} + 29 \beta_{2} ) q^{47} + ( 27 - 7 \beta_{3} ) q^{51} + ( 39 + 4 \beta_{3} ) q^{53} + ( 15 \beta_{1} - 3 \beta_{2} ) q^{55} + 3 q^{57} + ( -25 \beta_{1} + 13 \beta_{2} ) q^{59} + ( 32 \beta_{1} + 7 \beta_{2} ) q^{61} + ( 42 + 14 \beta_{3} ) q^{65} + ( -29 + 15 \beta_{3} ) q^{67} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{69} + ( 6 - 10 \beta_{3} ) q^{71} + ( -16 \beta_{1} + 53 \beta_{2} ) q^{73} + ( -10 \beta_{1} + 22 \beta_{2} ) q^{75} + ( 55 - 5 \beta_{3} ) q^{79} + ( -9 + 18 \beta_{3} ) q^{81} + ( -4 \beta_{1} - 68 \beta_{2} ) q^{83} + ( -9 + 8 \beta_{3} ) q^{85} + ( -18 \beta_{1} + 24 \beta_{2} ) q^{87} + ( -24 \beta_{1} + 63 \beta_{2} ) q^{89} + ( 69 - 8 \beta_{3} ) q^{93} + ( -15 - 3 \beta_{3} ) q^{95} + ( -26 \beta_{1} - 22 \beta_{2} ) q^{97} + ( -36 + 18 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 36 q^{11} + 36 q^{15} + 60 q^{23} - 8 q^{25} + 48 q^{29} + 124 q^{37} + 24 q^{39} + 8 q^{43} + 108 q^{51} + 156 q^{53} + 12 q^{57} + 168 q^{65} - 116 q^{67} + 24 q^{71} + 220 q^{79} - 36 q^{81} - 36 q^{85} + 276 q^{93} - 60 q^{95} - 144 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3}$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3}$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −0.707107 + 1.22474i 0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i
0 4.18154i 0 3.16693i 0 0 0 −8.48528 0
97.2 0 0.717439i 0 6.63103i 0 0 0 8.48528 0
97.3 0 0.717439i 0 6.63103i 0 0 0 8.48528 0
97.4 0 4.18154i 0 3.16693i 0 0 0 −8.48528 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.3.c.e 4
4.b odd 2 1 98.3.b.b 4
7.b odd 2 1 inner 784.3.c.e 4
7.c even 3 1 112.3.s.b 4
7.c even 3 1 784.3.s.c 4
7.d odd 6 1 112.3.s.b 4
7.d odd 6 1 784.3.s.c 4
12.b even 2 1 882.3.c.f 4
21.g even 6 1 1008.3.cg.l 4
21.h odd 6 1 1008.3.cg.l 4
28.d even 2 1 98.3.b.b 4
28.f even 6 1 14.3.d.a 4
28.f even 6 1 98.3.d.a 4
28.g odd 6 1 14.3.d.a 4
28.g odd 6 1 98.3.d.a 4
56.j odd 6 1 448.3.s.c 4
56.k odd 6 1 448.3.s.d 4
56.m even 6 1 448.3.s.d 4
56.p even 6 1 448.3.s.c 4
84.h odd 2 1 882.3.c.f 4
84.j odd 6 1 126.3.n.c 4
84.j odd 6 1 882.3.n.b 4
84.n even 6 1 126.3.n.c 4
84.n even 6 1 882.3.n.b 4
140.p odd 6 1 350.3.k.a 4
140.s even 6 1 350.3.k.a 4
140.w even 12 2 350.3.i.a 8
140.x odd 12 2 350.3.i.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 28.f even 6 1
14.3.d.a 4 28.g odd 6 1
98.3.b.b 4 4.b odd 2 1
98.3.b.b 4 28.d even 2 1
98.3.d.a 4 28.f even 6 1
98.3.d.a 4 28.g odd 6 1
112.3.s.b 4 7.c even 3 1
112.3.s.b 4 7.d odd 6 1
126.3.n.c 4 84.j odd 6 1
126.3.n.c 4 84.n even 6 1
350.3.i.a 8 140.w even 12 2
350.3.i.a 8 140.x odd 12 2
350.3.k.a 4 140.p odd 6 1
350.3.k.a 4 140.s even 6 1
448.3.s.c 4 56.j odd 6 1
448.3.s.c 4 56.p even 6 1
448.3.s.d 4 56.k odd 6 1
448.3.s.d 4 56.m even 6 1
784.3.c.e 4 1.a even 1 1 trivial
784.3.c.e 4 7.b odd 2 1 inner
784.3.s.c 4 7.c even 3 1
784.3.s.c 4 7.d odd 6 1
882.3.c.f 4 12.b even 2 1
882.3.c.f 4 84.h odd 2 1
882.3.n.b 4 84.j odd 6 1
882.3.n.b 4 84.n even 6 1
1008.3.cg.l 4 21.g even 6 1
1008.3.cg.l 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 18 T_{3}^{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 18 T^{2} + T^{4}$$
$5$ $$441 + 54 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 63 - 18 T + T^{2} )^{2}$$
$13$ $$7056 + 264 T^{2} + T^{4}$$
$17$ $$2601 + 198 T^{2} + T^{4}$$
$19$ $$9 + 18 T^{2} + T^{4}$$
$23$ $$( 63 - 30 T + T^{2} )^{2}$$
$29$ $$( 72 - 24 T + T^{2} )^{2}$$
$31$ $$1447209 + 2994 T^{2} + T^{4}$$
$37$ $$( -191 - 62 T + T^{2} )^{2}$$
$41$ $$345744 + 1224 T^{2} + T^{4}$$
$43$ $$( -68 - 4 T + T^{2} )^{2}$$
$47$ $$6335289 + 5058 T^{2} + T^{4}$$
$53$ $$( 1233 - 78 T + T^{2} )^{2}$$
$59$ $$10517049 + 8514 T^{2} + T^{4}$$
$61$ $$35964009 + 12582 T^{2} + T^{4}$$
$67$ $$( -3209 + 58 T + T^{2} )^{2}$$
$71$ $$( -1764 - 12 T + T^{2} )^{2}$$
$73$ $$47485881 + 19926 T^{2} + T^{4}$$
$79$ $$( 2575 - 110 T + T^{2} )^{2}$$
$83$ $$189778176 + 27936 T^{2} + T^{4}$$
$89$ $$71419401 + 30726 T^{2} + T^{4}$$
$97$ $$6780816 + 11016 T^{2} + T^{4}$$