# Properties

 Label 800.2.c.f Level $800$ Weight $2$ Character orbit 800.c Analytic conductor $6.388$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 160) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{2} q^{7} - 5 q^{9}+O(q^{10})$$ q + b2 * q^3 + b2 * q^7 - 5 * q^9 $$q + \beta_{2} q^{3} + \beta_{2} q^{7} - 5 q^{9} + \beta_{3} q^{11} + \beta_1 q^{13} + \beta_1 q^{17} - 8 q^{21} + \beta_{2} q^{23} - 2 \beta_{2} q^{27} - 6 q^{29} - \beta_{3} q^{31} + 8 \beta_1 q^{33} - 5 \beta_1 q^{37} - \beta_{3} q^{39} + 2 q^{41} - 3 \beta_{2} q^{43} + \beta_{2} q^{47} - q^{49} - \beta_{3} q^{51} - 3 \beta_1 q^{53} + 2 \beta_{3} q^{59} - 2 q^{61} - 5 \beta_{2} q^{63} + \beta_{2} q^{67} - 8 q^{69} - \beta_{3} q^{71} + 3 \beta_1 q^{73} + 8 \beta_1 q^{77} + 2 \beta_{3} q^{79} + q^{81} + \beta_{2} q^{83} - 6 \beta_{2} q^{87} - 10 q^{89} - \beta_{3} q^{91} - 8 \beta_1 q^{93} + \beta_1 q^{97} - 5 \beta_{3} q^{99}+O(q^{100})$$ q + b2 * q^3 + b2 * q^7 - 5 * q^9 + b3 * q^11 + b1 * q^13 + b1 * q^17 - 8 * q^21 + b2 * q^23 - 2*b2 * q^27 - 6 * q^29 - b3 * q^31 + 8*b1 * q^33 - 5*b1 * q^37 - b3 * q^39 + 2 * q^41 - 3*b2 * q^43 + b2 * q^47 - q^49 - b3 * q^51 - 3*b1 * q^53 + 2*b3 * q^59 - 2 * q^61 - 5*b2 * q^63 + b2 * q^67 - 8 * q^69 - b3 * q^71 + 3*b1 * q^73 + 8*b1 * q^77 + 2*b3 * q^79 + q^81 + b2 * q^83 - 6*b2 * q^87 - 10 * q^89 - b3 * q^91 - 8*b1 * q^93 + b1 * q^97 - 5*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{9}+O(q^{10})$$ 4 * q - 20 * q^9 $$4 q - 20 q^{9} - 32 q^{21} - 24 q^{29} + 8 q^{41} - 4 q^{49} - 8 q^{61} - 32 q^{69} + 4 q^{81} - 40 q^{89}+O(q^{100})$$ 4 * q - 20 * q^9 - 32 * q^21 - 24 * q^29 + 8 * q^41 - 4 * q^49 - 8 * q^61 - 32 * q^69 + 4 * q^81 - 40 * q^89

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v $$\beta_{3}$$ $$=$$ $$-4\zeta_{8}^{3} + 4\zeta_{8}$$ -4*v^3 + 4*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} ) / 8$$ (b3 + 2*b2) / 8 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 8$$ (-b3 + 2*b2) / 8

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.2 0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.3 0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.4 0 2.82843i 0 0 0 2.82843i 0 −5.00000 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
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.c.f 4
3.b odd 2 1 7200.2.f.bh 4
4.b odd 2 1 inner 800.2.c.f 4
5.b even 2 1 inner 800.2.c.f 4
5.c odd 4 1 160.2.a.c 2
5.c odd 4 1 800.2.a.m 2
8.b even 2 1 1600.2.c.n 4
8.d odd 2 1 1600.2.c.n 4
12.b even 2 1 7200.2.f.bh 4
15.d odd 2 1 7200.2.f.bh 4
15.e even 4 1 1440.2.a.o 2
15.e even 4 1 7200.2.a.cm 2
20.d odd 2 1 inner 800.2.c.f 4
20.e even 4 1 160.2.a.c 2
20.e even 4 1 800.2.a.m 2
35.f even 4 1 7840.2.a.bf 2
40.e odd 2 1 1600.2.c.n 4
40.f even 2 1 1600.2.c.n 4
40.i odd 4 1 320.2.a.g 2
40.i odd 4 1 1600.2.a.bc 2
40.k even 4 1 320.2.a.g 2
40.k even 4 1 1600.2.a.bc 2
60.h even 2 1 7200.2.f.bh 4
60.l odd 4 1 1440.2.a.o 2
60.l odd 4 1 7200.2.a.cm 2
80.i odd 4 1 1280.2.d.l 4
80.j even 4 1 1280.2.d.l 4
80.s even 4 1 1280.2.d.l 4
80.t odd 4 1 1280.2.d.l 4
120.q odd 4 1 2880.2.a.bk 2
120.w even 4 1 2880.2.a.bk 2
140.j odd 4 1 7840.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 5.c odd 4 1
160.2.a.c 2 20.e even 4 1
320.2.a.g 2 40.i odd 4 1
320.2.a.g 2 40.k even 4 1
800.2.a.m 2 5.c odd 4 1
800.2.a.m 2 20.e even 4 1
800.2.c.f 4 1.a even 1 1 trivial
800.2.c.f 4 4.b odd 2 1 inner
800.2.c.f 4 5.b even 2 1 inner
800.2.c.f 4 20.d odd 2 1 inner
1280.2.d.l 4 80.i odd 4 1
1280.2.d.l 4 80.j even 4 1
1280.2.d.l 4 80.s even 4 1
1280.2.d.l 4 80.t odd 4 1
1440.2.a.o 2 15.e even 4 1
1440.2.a.o 2 60.l odd 4 1
1600.2.a.bc 2 40.i odd 4 1
1600.2.a.bc 2 40.k even 4 1
1600.2.c.n 4 8.b even 2 1
1600.2.c.n 4 8.d odd 2 1
1600.2.c.n 4 40.e odd 2 1
1600.2.c.n 4 40.f even 2 1
2880.2.a.bk 2 120.q odd 4 1
2880.2.a.bk 2 120.w even 4 1
7200.2.a.cm 2 15.e even 4 1
7200.2.a.cm 2 60.l odd 4 1
7200.2.f.bh 4 3.b odd 2 1
7200.2.f.bh 4 12.b even 2 1
7200.2.f.bh 4 15.d odd 2 1
7200.2.f.bh 4 60.h even 2 1
7840.2.a.bf 2 35.f even 4 1
7840.2.a.bf 2 140.j odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 8$$ T3^2 + 8 $$T_{11}^{2} - 32$$ T11^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 8)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} - 32)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$(T^{2} + 4)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 8)^{2}$$
$29$ $$(T + 6)^{4}$$
$31$ $$(T^{2} - 32)^{2}$$
$37$ $$(T^{2} + 100)^{2}$$
$41$ $$(T - 2)^{4}$$
$43$ $$(T^{2} + 72)^{2}$$
$47$ $$(T^{2} + 8)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 128)^{2}$$
$61$ $$(T + 2)^{4}$$
$67$ $$(T^{2} + 8)^{2}$$
$71$ $$(T^{2} - 32)^{2}$$
$73$ $$(T^{2} + 36)^{2}$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$(T^{2} + 8)^{2}$$
$89$ $$(T + 10)^{4}$$
$97$ $$(T^{2} + 4)^{2}$$