# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{3} + ( -3 + 3 i ) q^{7} -i q^{9} +O(q^{10})$$ $$q + ( 1 + i ) q^{3} + ( -3 + 3 i ) q^{7} -i q^{9} + 2 i q^{11} + ( -3 + 3 i ) q^{13} + ( -1 - i ) q^{17} -4 q^{19} -6 q^{21} + ( -1 - i ) q^{23} + ( 4 - 4 i ) q^{27} + 10 i q^{31} + ( -2 + 2 i ) q^{33} + ( 1 + i ) q^{37} -6 q^{39} -10 q^{41} + ( 5 + 5 i ) q^{43} + ( -3 + 3 i ) q^{47} -11 i q^{49} -2 i q^{51} + ( 5 - 5 i ) q^{53} + ( -4 - 4 i ) q^{57} + 12 q^{59} + 2 q^{61} + ( 3 + 3 i ) q^{63} + ( -1 + i ) q^{67} -2 i q^{69} + 2 i q^{71} + ( -1 + i ) q^{73} + ( -6 - 6 i ) q^{77} -8 q^{79} + 5 q^{81} + ( 5 + 5 i ) q^{83} + 16 i q^{89} -18 i q^{91} + ( -10 + 10 i ) q^{93} + ( 3 + 3 i ) q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 6q^{7} + O(q^{10})$$ $$2q + 2q^{3} - 6q^{7} - 6q^{13} - 2q^{17} - 8q^{19} - 12q^{21} - 2q^{23} + 8q^{27} - 4q^{33} + 2q^{37} - 12q^{39} - 20q^{41} + 10q^{43} - 6q^{47} + 10q^{53} - 8q^{57} + 24q^{59} + 4q^{61} + 6q^{63} - 2q^{67} - 2q^{73} - 12q^{77} - 16q^{79} + 10q^{81} + 10q^{83} - 20q^{93} + 6q^{97} + 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
543.1
 − 1.00000i 1.00000i
0 1.00000 1.00000i 0 0 0 −3.00000 3.00000i 0 1.00000i 0
607.1 0 1.00000 + 1.00000i 0 0 0 −3.00000 + 3.00000i 0 1.00000i 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
20.e 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.n.f 2
4.b odd 2 1 800.2.n.e 2
5.b even 2 1 160.2.n.c 2
5.c odd 4 1 160.2.n.d yes 2
5.c odd 4 1 800.2.n.e 2
8.b even 2 1 1600.2.n.c 2
8.d odd 2 1 1600.2.n.m 2
15.d odd 2 1 1440.2.x.f 2
15.e even 4 1 1440.2.x.a 2
20.d odd 2 1 160.2.n.d yes 2
20.e even 4 1 160.2.n.c 2
20.e even 4 1 inner 800.2.n.f 2
40.e odd 2 1 320.2.n.b 2
40.f even 2 1 320.2.n.g 2
40.i odd 4 1 320.2.n.b 2
40.i odd 4 1 1600.2.n.m 2
40.k even 4 1 320.2.n.g 2
40.k even 4 1 1600.2.n.c 2
60.h even 2 1 1440.2.x.a 2
60.l odd 4 1 1440.2.x.f 2
80.i odd 4 1 1280.2.o.f 2
80.j even 4 1 1280.2.o.c 2
80.k odd 4 1 1280.2.o.f 2
80.k odd 4 1 1280.2.o.l 2
80.q even 4 1 1280.2.o.c 2
80.q even 4 1 1280.2.o.m 2
80.s even 4 1 1280.2.o.m 2
80.t odd 4 1 1280.2.o.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.c 2 5.b even 2 1
160.2.n.c 2 20.e even 4 1
160.2.n.d yes 2 5.c odd 4 1
160.2.n.d yes 2 20.d odd 2 1
320.2.n.b 2 40.e odd 2 1
320.2.n.b 2 40.i odd 4 1
320.2.n.g 2 40.f even 2 1
320.2.n.g 2 40.k even 4 1
800.2.n.e 2 4.b odd 2 1
800.2.n.e 2 5.c odd 4 1
800.2.n.f 2 1.a even 1 1 trivial
800.2.n.f 2 20.e even 4 1 inner
1280.2.o.c 2 80.j even 4 1
1280.2.o.c 2 80.q even 4 1
1280.2.o.f 2 80.i odd 4 1
1280.2.o.f 2 80.k odd 4 1
1280.2.o.l 2 80.k odd 4 1
1280.2.o.l 2 80.t odd 4 1
1280.2.o.m 2 80.q even 4 1
1280.2.o.m 2 80.s even 4 1
1440.2.x.a 2 15.e even 4 1
1440.2.x.a 2 60.h even 2 1
1440.2.x.f 2 15.d odd 2 1
1440.2.x.f 2 60.l odd 4 1
1600.2.n.c 2 8.b even 2 1
1600.2.n.c 2 40.k even 4 1
1600.2.n.m 2 8.d odd 2 1
1600.2.n.m 2 40.i odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} - 2 T_{3} + 2$$ $$T_{7}^{2} + 6 T_{7} + 18$$ $$T_{11}^{2} + 4$$ $$T_{13}^{2} + 6 T_{13} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$18 + 6 T + T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$18 + 6 T + T^{2}$$
$17$ $$2 + 2 T + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$2 + 2 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$100 + T^{2}$$
$37$ $$2 - 2 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$50 - 10 T + T^{2}$$
$47$ $$18 + 6 T + T^{2}$$
$53$ $$50 - 10 T + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$2 + 2 T + T^{2}$$
$71$ $$4 + T^{2}$$
$73$ $$2 + 2 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$50 - 10 T + T^{2}$$
$89$ $$256 + T^{2}$$
$97$ $$18 - 6 T + T^{2}$$
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