# Properties

 Label 800.2.o.h Level $800$ Weight $2$ Character orbit 800.o Analytic conductor $6.388$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.3317760000.5 Defining polynomial: $$x^{8} - 7 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{6} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{6} q^{7} + q^{11} -\beta_{4} q^{13} -3 \beta_{1} q^{17} -3 \beta_{3} q^{19} + \beta_{7} q^{21} -3 \beta_{1} q^{27} + \beta_{5} q^{29} -\beta_{2} q^{33} + \beta_{6} q^{37} + \beta_{5} q^{39} - q^{41} + 2 \beta_{2} q^{43} + \beta_{6} q^{47} -13 \beta_{3} q^{49} + 9 q^{51} + 2 \beta_{4} q^{53} + 3 \beta_{1} q^{57} -4 \beta_{3} q^{59} -\beta_{7} q^{61} + 3 \beta_{1} q^{67} + \beta_{7} q^{71} + \beta_{2} q^{73} + \beta_{6} q^{77} + \beta_{5} q^{79} + 9 q^{81} -\beta_{2} q^{83} -3 \beta_{6} q^{87} + 13 \beta_{3} q^{89} -20 q^{91} -4 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{11} - 8q^{41} + 72q^{51} + 72q^{81} - 160q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + \nu^{3}$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 5 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{2}$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - \nu$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 11 \nu^{2}$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 13 \nu^{3}$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$4 \nu^{4} - 14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + 2 \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} + 6 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} + 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{4} + 2 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{5} + 22 \beta_{3}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{6} + 26 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{3}$$

## 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
 −0.178197 + 1.40294i 1.40294 − 0.178197i −1.40294 + 0.178197i 0.178197 − 1.40294i −0.178197 − 1.40294i 1.40294 + 0.178197i −1.40294 − 0.178197i 0.178197 + 1.40294i
0 −1.22474 1.22474i 0 0 0 −3.16228 3.16228i 0 0 0
143.2 0 −1.22474 1.22474i 0 0 0 3.16228 + 3.16228i 0 0 0
143.3 0 1.22474 + 1.22474i 0 0 0 −3.16228 3.16228i 0 0 0
143.4 0 1.22474 + 1.22474i 0 0 0 3.16228 + 3.16228i 0 0 0
207.1 0 −1.22474 + 1.22474i 0 0 0 −3.16228 + 3.16228i 0 0 0
207.2 0 −1.22474 + 1.22474i 0 0 0 3.16228 3.16228i 0 0 0
207.3 0 1.22474 1.22474i 0 0 0 −3.16228 + 3.16228i 0 0 0
207.4 0 1.22474 1.22474i 0 0 0 3.16228 3.16228i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 207.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} + 9$$ $$T_{7}^{4} + 400$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 400 + T^{4} )^{2}$$
$11$ $$( -1 + T )^{8}$$
$13$ $$( 400 + T^{4} )^{2}$$
$17$ $$( 729 + T^{4} )^{2}$$
$19$ $$( 9 + T^{2} )^{4}$$
$23$ $$T^{8}$$
$29$ $$( -60 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( 400 + T^{4} )^{2}$$
$41$ $$( 1 + T )^{8}$$
$43$ $$( 144 + T^{4} )^{2}$$
$47$ $$( 400 + T^{4} )^{2}$$
$53$ $$( 6400 + T^{4} )^{2}$$
$59$ $$( 16 + T^{2} )^{4}$$
$61$ $$( 60 + T^{2} )^{4}$$
$67$ $$( 729 + T^{4} )^{2}$$
$71$ $$( 60 + T^{2} )^{4}$$
$73$ $$( 9 + T^{4} )^{2}$$
$79$ $$( -60 + T^{2} )^{4}$$
$83$ $$( 9 + T^{4} )^{2}$$
$89$ $$( 169 + T^{2} )^{4}$$
$97$ $$( 2304 + T^{4} )^{2}$$