# Properties

 Label 800.2.d.f Level $800$ Weight $2$ Character orbit 800.d 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{2} q^{7} + q^{9}+O(q^{10})$$ q + b1 * q^3 - b2 * q^7 + q^9 $$q + \beta_1 q^{3} - \beta_{2} q^{7} + q^{9} + \beta_{3} q^{11} - 2 \beta_{2} q^{17} + \beta_{3} q^{19} - \beta_{3} q^{21} - \beta_{2} q^{23} + 4 \beta_1 q^{27} - 4 q^{31} - 2 \beta_{2} q^{33} + 6 \beta_1 q^{37} - 3 \beta_1 q^{43} - 3 \beta_{2} q^{47} - q^{49} - 2 \beta_{3} q^{51} + 4 \beta_1 q^{53} - 2 \beta_{2} q^{57} - 3 \beta_{3} q^{59} + \beta_{3} q^{61} - \beta_{2} q^{63} - 3 \beta_1 q^{67} - \beta_{3} q^{69} + 12 q^{71} + 2 \beta_{2} q^{73} - 6 \beta_1 q^{77} - 4 q^{79} - 5 q^{81} + 7 \beta_1 q^{83} - 6 q^{89} - 4 \beta_1 q^{93} + 2 \beta_{2} q^{97} + \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^3 - b2 * q^7 + q^9 + b3 * q^11 - 2*b2 * q^17 + b3 * q^19 - b3 * q^21 - b2 * q^23 + 4*b1 * q^27 - 4 * q^31 - 2*b2 * q^33 + 6*b1 * q^37 - 3*b1 * q^43 - 3*b2 * q^47 - q^49 - 2*b3 * q^51 + 4*b1 * q^53 - 2*b2 * q^57 - 3*b3 * q^59 + b3 * q^61 - b2 * q^63 - 3*b1 * q^67 - b3 * q^69 + 12 * q^71 + 2*b2 * q^73 - 6*b1 * q^77 - 4 * q^79 - 5 * q^81 + 7*b1 * q^83 - 6 * q^89 - 4*b1 * q^93 + 2*b2 * q^97 + b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} - 16 q^{31} - 4 q^{49} + 48 q^{71} - 16 q^{79} - 20 q^{81} - 24 q^{89}+O(q^{100})$$ 4 * q + 4 * q^9 - 16 * q^31 - 4 * q^49 + 48 * q^71 - 16 * q^79 - 20 * q^81 - 24 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i
0 1.41421i 0 0 0 −2.44949 0 1.00000 0
401.2 0 1.41421i 0 0 0 2.44949 0 1.00000 0
401.3 0 1.41421i 0 0 0 −2.44949 0 1.00000 0
401.4 0 1.41421i 0 0 0 2.44949 0 1.00000 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.d.f 4
3.b odd 2 1 7200.2.k.l 4
4.b odd 2 1 200.2.d.e 4
5.b even 2 1 inner 800.2.d.f 4
5.c odd 4 2 160.2.f.a 4
8.b even 2 1 inner 800.2.d.f 4
8.d odd 2 1 200.2.d.e 4
12.b even 2 1 1800.2.k.m 4
15.d odd 2 1 7200.2.k.l 4
15.e even 4 2 1440.2.d.c 4
16.e even 4 2 6400.2.a.cm 4
16.f odd 4 2 6400.2.a.co 4
20.d odd 2 1 200.2.d.e 4
20.e even 4 2 40.2.f.a 4
24.f even 2 1 1800.2.k.m 4
24.h odd 2 1 7200.2.k.l 4
40.e odd 2 1 200.2.d.e 4
40.f even 2 1 inner 800.2.d.f 4
40.i odd 4 2 160.2.f.a 4
40.k even 4 2 40.2.f.a 4
60.h even 2 1 1800.2.k.m 4
60.l odd 4 2 360.2.d.b 4
80.i odd 4 2 1280.2.c.k 4
80.j even 4 2 1280.2.c.i 4
80.k odd 4 2 6400.2.a.co 4
80.q even 4 2 6400.2.a.cm 4
80.s even 4 2 1280.2.c.i 4
80.t odd 4 2 1280.2.c.k 4
120.i odd 2 1 7200.2.k.l 4
120.m even 2 1 1800.2.k.m 4
120.q odd 4 2 360.2.d.b 4
120.w even 4 2 1440.2.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 20.e even 4 2
40.2.f.a 4 40.k even 4 2
160.2.f.a 4 5.c odd 4 2
160.2.f.a 4 40.i odd 4 2
200.2.d.e 4 4.b odd 2 1
200.2.d.e 4 8.d odd 2 1
200.2.d.e 4 20.d odd 2 1
200.2.d.e 4 40.e odd 2 1
360.2.d.b 4 60.l odd 4 2
360.2.d.b 4 120.q odd 4 2
800.2.d.f 4 1.a even 1 1 trivial
800.2.d.f 4 5.b even 2 1 inner
800.2.d.f 4 8.b even 2 1 inner
800.2.d.f 4 40.f even 2 1 inner
1280.2.c.i 4 80.j even 4 2
1280.2.c.i 4 80.s even 4 2
1280.2.c.k 4 80.i odd 4 2
1280.2.c.k 4 80.t odd 4 2
1440.2.d.c 4 15.e even 4 2
1440.2.d.c 4 120.w even 4 2
1800.2.k.m 4 12.b even 2 1
1800.2.k.m 4 24.f even 2 1
1800.2.k.m 4 60.h even 2 1
1800.2.k.m 4 120.m even 2 1
6400.2.a.cm 4 16.e even 4 2
6400.2.a.cm 4 80.q even 4 2
6400.2.a.co 4 16.f odd 4 2
6400.2.a.co 4 80.k odd 4 2
7200.2.k.l 4 3.b odd 2 1
7200.2.k.l 4 15.d odd 2 1
7200.2.k.l 4 24.h odd 2 1
7200.2.k.l 4 120.i odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 2$$ T3^2 + 2 $$T_{7}^{2} - 6$$ T7^2 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 2)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 6)^{2}$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 24)^{2}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} - 6)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T + 4)^{4}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 18)^{2}$$
$47$ $$(T^{2} - 54)^{2}$$
$53$ $$(T^{2} + 32)^{2}$$
$59$ $$(T^{2} + 108)^{2}$$
$61$ $$(T^{2} + 12)^{2}$$
$67$ $$(T^{2} + 18)^{2}$$
$71$ $$(T - 12)^{4}$$
$73$ $$(T^{2} - 24)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} + 98)^{2}$$
$89$ $$(T + 6)^{4}$$
$97$ $$(T^{2} - 24)^{2}$$