# Properties

 Label 800.2.o.f Level $800$ Weight $2$ Character orbit 800.o Analytic conductor $6.388$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.o (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} + ( -3 + \beta_{1} - \beta_{3} ) q^{11} + ( -2 + 2 \beta_{2} - 3 \beta_{3} ) q^{17} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{19} + ( -5 + 5 \beta_{2} + 3 \beta_{3} ) q^{27} -\beta_{1} q^{33} + ( 3 + 4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -6 - 6 \beta_{2} ) q^{43} -7 \beta_{2} q^{49} + ( 5 + \beta_{1} - \beta_{3} ) q^{51} + ( -10 + 10 \beta_{2} + 7 \beta_{3} ) q^{57} -6 \beta_{2} q^{59} + ( -3 + 3 \beta_{2} - 7 \beta_{3} ) q^{67} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{73} + ( -13 - 2 \beta_{1} + 2 \beta_{3} ) q^{81} + ( 1 - 9 \beta_{1} + \beta_{2} ) q^{83} + ( 2 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 12 - 12 \beta_{2} ) q^{97} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + O(q^{10})$$ $$4q + 4q^{3} - 12q^{11} - 8q^{17} - 20q^{27} + 12q^{41} - 24q^{43} + 20q^{51} - 40q^{57} - 12q^{67} - 24q^{73} - 52q^{81} + 4q^{83} + 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
0 −0.224745 0.224745i 0 0 0 0 0 2.89898i 0
143.2 0 2.22474 + 2.22474i 0 0 0 0 0 6.89898i 0
207.1 0 −0.224745 + 0.224745i 0 0 0 0 0 2.89898i 0
207.2 0 2.22474 2.22474i 0 0 0 0 0 6.89898i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
5.c odd 4 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.o.f 4
4.b odd 2 1 200.2.k.f yes 4
5.b even 2 1 800.2.o.e 4
5.c odd 4 1 800.2.o.e 4
5.c odd 4 1 inner 800.2.o.f 4
8.b even 2 1 200.2.k.f yes 4
8.d odd 2 1 CM 800.2.o.f 4
20.d odd 2 1 200.2.k.e 4
20.e even 4 1 200.2.k.e 4
20.e even 4 1 200.2.k.f yes 4
40.e odd 2 1 800.2.o.e 4
40.f even 2 1 200.2.k.e 4
40.i odd 4 1 200.2.k.e 4
40.i odd 4 1 200.2.k.f yes 4
40.k even 4 1 800.2.o.e 4
40.k even 4 1 inner 800.2.o.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.e 4 20.d odd 2 1
200.2.k.e 4 20.e even 4 1
200.2.k.e 4 40.f even 2 1
200.2.k.e 4 40.i odd 4 1
200.2.k.f yes 4 4.b odd 2 1
200.2.k.f yes 4 8.b even 2 1
200.2.k.f yes 4 20.e even 4 1
200.2.k.f yes 4 40.i odd 4 1
800.2.o.e 4 5.b even 2 1
800.2.o.e 4 5.c odd 4 1
800.2.o.e 4 40.e odd 2 1
800.2.o.e 4 40.k even 4 1
800.2.o.f 4 1.a even 1 1 trivial
800.2.o.f 4 5.c odd 4 1 inner
800.2.o.f 4 8.d odd 2 1 CM
800.2.o.f 4 40.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{4} - 4 T_{3}^{3} + 8 T_{3}^{2} + 4 T_{3} + 1$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + 4 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 3 + 6 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$361 - 152 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$2809 + 110 T^{2} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -87 - 6 T + T^{2} )^{2}$$
$43$ $$( 72 + 12 T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$16641 - 1548 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$4761 + 1656 T + 288 T^{2} + 24 T^{3} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$58081 + 964 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$89$ $$3249 + 210 T^{2} + T^{4}$$
$97$ $$( 288 - 24 T + T^{2} )^{2}$$