# Properties

 Label 784.4.f.f Level $784$ Weight $4$ Character orbit 784.f Analytic conductor $46.257$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.2574974445$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -5 \beta_{2} q^{5} -20 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -5 \beta_{2} q^{5} -20 q^{9} -15 \beta_{3} q^{11} -30 \beta_{2} q^{13} + 5 \beta_{3} q^{15} + 5 \beta_{2} q^{17} + 25 \beta_{1} q^{19} + 11 \beta_{3} q^{23} + 50 q^{25} -47 \beta_{1} q^{27} -168 q^{29} -85 \beta_{1} q^{31} + 105 \beta_{2} q^{33} -245 q^{37} + 30 \beta_{3} q^{39} -26 \beta_{2} q^{41} -66 \beta_{3} q^{43} + 100 \beta_{2} q^{45} -159 \beta_{1} q^{47} -5 \beta_{3} q^{51} -345 q^{53} + 225 \beta_{1} q^{55} + 175 q^{57} + 165 \beta_{1} q^{59} + 229 \beta_{2} q^{61} -450 q^{65} -97 \beta_{3} q^{67} -77 \beta_{2} q^{69} -10 \beta_{3} q^{71} + 555 \beta_{2} q^{73} + 50 \beta_{1} q^{75} -45 \beta_{3} q^{79} + 211 q^{81} -336 \beta_{1} q^{83} + 75 q^{85} -168 \beta_{1} q^{87} + 873 \beta_{2} q^{89} -595 q^{93} + 125 \beta_{3} q^{95} -250 \beta_{2} q^{97} + 300 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 80q^{9} + O(q^{10})$$ $$4q - 80q^{9} + 200q^{25} - 672q^{29} - 980q^{37} - 1380q^{53} + 700q^{57} - 1800q^{65} + 844q^{81} + 300q^{85} - 2380q^{93} + O(q^{100})$$

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 8.66025i 0 0 0 −20.0000 0
783.2 0 −2.64575 0 8.66025i 0 0 0 −20.0000 0
783.3 0 2.64575 0 8.66025i 0 0 0 −20.0000 0
783.4 0 2.64575 0 8.66025i 0 0 0 −20.0000 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 784.4.f.f 4
4.b odd 2 1 inner 784.4.f.f 4
7.b odd 2 1 inner 784.4.f.f 4
7.c even 3 1 112.4.p.e 4
7.d odd 6 1 112.4.p.e 4
28.d even 2 1 inner 784.4.f.f 4
28.f even 6 1 112.4.p.e 4
28.g odd 6 1 112.4.p.e 4
56.j odd 6 1 448.4.p.e 4
56.k odd 6 1 448.4.p.e 4
56.m even 6 1 448.4.p.e 4
56.p even 6 1 448.4.p.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.e 4 7.c even 3 1
112.4.p.e 4 7.d odd 6 1
112.4.p.e 4 28.f even 6 1
112.4.p.e 4 28.g odd 6 1
448.4.p.e 4 56.j odd 6 1
448.4.p.e 4 56.k odd 6 1
448.4.p.e 4 56.m even 6 1
448.4.p.e 4 56.p even 6 1
784.4.f.f 4 1.a even 1 1 trivial
784.4.f.f 4 4.b odd 2 1 inner
784.4.f.f 4 7.b odd 2 1 inner
784.4.f.f 4 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 7$$ acting on $$S_{4}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -7 + T^{2} )^{2}$$
$5$ $$( 75 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 4725 + T^{2} )^{2}$$
$13$ $$( 2700 + T^{2} )^{2}$$
$17$ $$( 75 + T^{2} )^{2}$$
$19$ $$( -4375 + T^{2} )^{2}$$
$23$ $$( 2541 + T^{2} )^{2}$$
$29$ $$( 168 + T )^{4}$$
$31$ $$( -50575 + T^{2} )^{2}$$
$37$ $$( 245 + T )^{4}$$
$41$ $$( 2028 + T^{2} )^{2}$$
$43$ $$( 91476 + T^{2} )^{2}$$
$47$ $$( -176967 + T^{2} )^{2}$$
$53$ $$( 345 + T )^{4}$$
$59$ $$( -190575 + T^{2} )^{2}$$
$61$ $$( 157323 + T^{2} )^{2}$$
$67$ $$( 197589 + T^{2} )^{2}$$
$71$ $$( 2100 + T^{2} )^{2}$$
$73$ $$( 924075 + T^{2} )^{2}$$
$79$ $$( 42525 + T^{2} )^{2}$$
$83$ $$( -790272 + T^{2} )^{2}$$
$89$ $$( 2286387 + T^{2} )^{2}$$
$97$ $$( 187500 + T^{2} )^{2}$$