# Properties

 Label 800.4.c.l Level $800$ Weight $4$ Character orbit 800.c Analytic conductor $47.202$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.2015280046$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + 7 \beta_{2} q^{7} + 7 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + 7 \beta_{2} q^{7} + 7 q^{9} -\beta_{3} q^{11} -31 \beta_{1} q^{13} + 23 \beta_{1} q^{17} -12 \beta_{3} q^{19} -140 q^{21} + 43 \beta_{2} q^{23} + 34 \beta_{2} q^{27} + 90 q^{29} -17 \beta_{3} q^{31} -20 \beta_{1} q^{33} + 107 \beta_{1} q^{37} + 31 \beta_{3} q^{39} -10 q^{41} -15 \beta_{2} q^{43} -89 \beta_{2} q^{47} -637 q^{49} -23 \beta_{3} q^{51} -339 \beta_{1} q^{53} -240 \beta_{1} q^{57} + 46 \beta_{3} q^{59} + 250 q^{61} + 49 \beta_{2} q^{63} -11 \beta_{2} q^{67} -860 q^{69} -41 \beta_{3} q^{71} + 261 \beta_{1} q^{73} -140 \beta_{1} q^{77} -98 \beta_{3} q^{79} -491 q^{81} + 85 \beta_{2} q^{83} + 90 \beta_{2} q^{87} -970 q^{89} + 217 \beta_{3} q^{91} -340 \beta_{1} q^{93} + 467 \beta_{1} q^{97} -7 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 28q^{9} + O(q^{10})$$ $$4q + 28q^{9} - 560q^{21} + 360q^{29} - 40q^{41} - 2548q^{49} + 1000q^{61} - 3440q^{69} - 1964q^{81} - 3880q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 12$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1}$$$$)/2$$

## 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
 − 0.618034i − 1.61803i 0.618034i 1.61803i
0 4.47214i 0 0 0 31.3050i 0 7.00000 0
449.2 0 4.47214i 0 0 0 31.3050i 0 7.00000 0
449.3 0 4.47214i 0 0 0 31.3050i 0 7.00000 0
449.4 0 4.47214i 0 0 0 31.3050i 0 7.00000 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 800.4.c.l 4
4.b odd 2 1 inner 800.4.c.l 4
5.b even 2 1 inner 800.4.c.l 4
5.c odd 4 1 160.4.a.d 2
5.c odd 4 1 800.4.a.o 2
15.e even 4 1 1440.4.a.bb 2
20.d odd 2 1 inner 800.4.c.l 4
20.e even 4 1 160.4.a.d 2
20.e even 4 1 800.4.a.o 2
40.i odd 4 1 320.4.a.q 2
40.i odd 4 1 1600.4.a.cg 2
40.k even 4 1 320.4.a.q 2
40.k even 4 1 1600.4.a.cg 2
60.l odd 4 1 1440.4.a.bb 2
80.i odd 4 1 1280.4.d.v 4
80.j even 4 1 1280.4.d.v 4
80.s even 4 1 1280.4.d.v 4
80.t odd 4 1 1280.4.d.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 5.c odd 4 1
160.4.a.d 2 20.e even 4 1
320.4.a.q 2 40.i odd 4 1
320.4.a.q 2 40.k even 4 1
800.4.a.o 2 5.c odd 4 1
800.4.a.o 2 20.e even 4 1
800.4.c.l 4 1.a even 1 1 trivial
800.4.c.l 4 4.b odd 2 1 inner
800.4.c.l 4 5.b even 2 1 inner
800.4.c.l 4 20.d odd 2 1 inner
1280.4.d.v 4 80.i odd 4 1
1280.4.d.v 4 80.j even 4 1
1280.4.d.v 4 80.s even 4 1
1280.4.d.v 4 80.t odd 4 1
1440.4.a.bb 2 15.e even 4 1
1440.4.a.bb 2 60.l odd 4 1
1600.4.a.cg 2 40.i odd 4 1
1600.4.a.cg 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 20$$ $$T_{11}^{2} - 80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 20 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 980 + T^{2} )^{2}$$
$11$ $$( -80 + T^{2} )^{2}$$
$13$ $$( 3844 + T^{2} )^{2}$$
$17$ $$( 2116 + T^{2} )^{2}$$
$19$ $$( -11520 + T^{2} )^{2}$$
$23$ $$( 36980 + T^{2} )^{2}$$
$29$ $$( -90 + T )^{4}$$
$31$ $$( -23120 + T^{2} )^{2}$$
$37$ $$( 45796 + T^{2} )^{2}$$
$41$ $$( 10 + T )^{4}$$
$43$ $$( 4500 + T^{2} )^{2}$$
$47$ $$( 158420 + T^{2} )^{2}$$
$53$ $$( 459684 + T^{2} )^{2}$$
$59$ $$( -169280 + T^{2} )^{2}$$
$61$ $$( -250 + T )^{4}$$
$67$ $$( 2420 + T^{2} )^{2}$$
$71$ $$( -134480 + T^{2} )^{2}$$
$73$ $$( 272484 + T^{2} )^{2}$$
$79$ $$( -768320 + T^{2} )^{2}$$
$83$ $$( 144500 + T^{2} )^{2}$$
$89$ $$( 970 + T )^{4}$$
$97$ $$( 872356 + T^{2} )^{2}$$