# Properties

 Label 784.3.c.d 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: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$7^{2}$$ Twist minimal: no (minimal twist has level 196) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{9} + ( 6 - \beta_{3} ) q^{11} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{13} + ( 18 + 2 \beta_{3} ) q^{15} -3 \beta_{2} q^{17} + ( 5 \beta_{1} - \beta_{2} ) q^{19} -26 q^{23} + ( -27 - 2 \beta_{3} ) q^{25} + ( -\beta_{1} - 2 \beta_{2} ) q^{27} + ( 8 - 2 \beta_{3} ) q^{29} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{31} + ( -\beta_{1} - 4 \beta_{2} ) q^{33} -32 q^{37} + ( -46 - 4 \beta_{3} ) q^{39} + ( -3 \beta_{1} - 5 \beta_{2} ) q^{41} + ( -10 - 7 \beta_{3} ) q^{43} + ( -7 \beta_{1} + 20 \beta_{2} ) q^{45} + ( -12 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 30 + 3 \beta_{3} ) q^{51} + ( 6 - 2 \beta_{3} ) q^{53} + ( -14 \beta_{1} + 6 \beta_{2} ) q^{55} + ( 20 + \beta_{3} ) q^{57} + ( -11 \beta_{1} - 3 \beta_{2} ) q^{59} + ( 9 \beta_{1} + 8 \beta_{2} ) q^{61} + ( 24 + 14 \beta_{3} ) q^{65} + ( -28 + 6 \beta_{3} ) q^{67} -26 \beta_{2} q^{69} + ( -56 - 4 \beta_{3} ) q^{71} + ( -11 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -2 \beta_{1} - 47 \beta_{2} ) q^{75} + ( -60 - 2 \beta_{3} ) q^{79} + ( 9 - 7 \beta_{3} ) q^{81} + ( -13 \beta_{1} - 33 \beta_{2} ) q^{83} + ( -54 - 6 \beta_{3} ) q^{85} + ( -2 \beta_{1} - 12 \beta_{2} ) q^{87} + ( 11 \beta_{1} - 25 \beta_{2} ) q^{89} + ( -84 - 8 \beta_{3} ) q^{93} + ( 62 - 12 \beta_{3} ) q^{95} + ( -22 \beta_{1} - 33 \beta_{2} ) q^{97} + ( 92 - 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 24q^{11} + 72q^{15} - 104q^{23} - 108q^{25} + 32q^{29} - 128q^{37} - 184q^{39} - 40q^{43} + 120q^{51} + 24q^{53} + 80q^{57} + 96q^{65} - 112q^{67} - 224q^{71} - 240q^{79} + 36q^{81} - 216q^{85} - 336q^{93} + 248q^{95} + 368q^{99} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 − 0.765367i − 1.84776i 1.84776i 0.765367i
0 4.46088i 0 8.47343i 0 0 0 −10.8995 0
97.2 0 0.317025i 0 5.67459i 0 0 0 8.89949 0
97.3 0 0.317025i 0 5.67459i 0 0 0 8.89949 0
97.4 0 4.46088i 0 8.47343i 0 0 0 −10.8995 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.d 4
4.b odd 2 1 196.3.b.b 4
7.b odd 2 1 inner 784.3.c.d 4
7.c even 3 2 784.3.s.g 8
7.d odd 6 2 784.3.s.g 8
12.b even 2 1 1764.3.d.e 4
28.d even 2 1 196.3.b.b 4
28.f even 6 2 196.3.h.c 8
28.g odd 6 2 196.3.h.c 8
84.h odd 2 1 1764.3.d.e 4
84.j odd 6 2 1764.3.z.k 8
84.n even 6 2 1764.3.z.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.b.b 4 4.b odd 2 1
196.3.b.b 4 28.d even 2 1
196.3.h.c 8 28.f even 6 2
196.3.h.c 8 28.g odd 6 2
784.3.c.d 4 1.a even 1 1 trivial
784.3.c.d 4 7.b odd 2 1 inner
784.3.s.g 8 7.c even 3 2
784.3.s.g 8 7.d odd 6 2
1764.3.d.e 4 12.b even 2 1
1764.3.d.e 4 84.h odd 2 1
1764.3.z.k 8 84.j odd 6 2
1764.3.z.k 8 84.n even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 20 T_{3}^{2} + 2$$ acting on $$S_{3}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$2 + 20 T^{2} + T^{4}$$
$5$ $$2312 + 104 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -62 - 12 T + T^{2} )^{2}$$
$13$ $$150152 + 776 T^{2} + T^{4}$$
$17$ $$162 + 180 T^{2} + T^{4}$$
$19$ $$45602 + 1060 T^{2} + T^{4}$$
$23$ $$( 26 + T )^{4}$$
$29$ $$( -328 - 16 T + T^{2} )^{2}$$
$31$ $$307328 + 1568 T^{2} + T^{4}$$
$37$ $$( 32 + T )^{4}$$
$41$ $$132098 + 740 T^{2} + T^{4}$$
$43$ $$( -4702 + 20 T + T^{2} )^{2}$$
$47$ $$1191968 + 6032 T^{2} + T^{4}$$
$53$ $$( -356 - 12 T + T^{2} )^{2}$$
$59$ $$334562 + 4756 T^{2} + T^{4}$$
$61$ $$2947592 + 3944 T^{2} + T^{4}$$
$67$ $$( -2744 + 56 T + T^{2} )^{2}$$
$71$ $$( 1568 + 112 T + T^{2} )^{2}$$
$73$ $$1659842 + 5284 T^{2} + T^{4}$$
$79$ $$( 3208 + 120 T + T^{2} )^{2}$$
$83$ $$102330818 + 25108 T^{2} + T^{4}$$
$89$ $$81077378 + 19540 T^{2} + T^{4}$$
$97$ $$310652738 + 35332 T^{2} + T^{4}$$