# Properties

 Label 800.6.c.h Level 800 Weight 6 Character orbit 800.c Analytic conductor 128.307 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$128.307055850$$ 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.

 $$f(q)$$ $$=$$ $$q -3 \beta_{2} q^{3} + 31 \beta_{2} q^{7} + 63 q^{9} +O(q^{10})$$ $$q -3 \beta_{2} q^{3} + 31 \beta_{2} q^{7} + 63 q^{9} -29 \beta_{3} q^{11} + 77 \beta_{1} q^{13} -89 \beta_{1} q^{17} + 108 \beta_{3} q^{19} + 1860 q^{21} -589 \beta_{2} q^{23} -918 \beta_{2} q^{27} -4110 q^{29} -353 \beta_{3} q^{31} + 1740 \beta_{1} q^{33} -3721 \beta_{1} q^{37} + 231 \beta_{3} q^{39} + 7270 q^{41} + 4005 \beta_{2} q^{43} -1657 \beta_{2} q^{47} -2413 q^{49} -267 \beta_{3} q^{51} + 16113 \beta_{1} q^{53} -6480 \beta_{1} q^{57} + 3806 \beta_{3} q^{59} + 26770 q^{61} + 1953 \beta_{2} q^{63} + 11137 \beta_{2} q^{67} -35340 q^{69} -6049 \beta_{3} q^{71} -9267 \beta_{1} q^{73} -17980 \beta_{1} q^{77} -9698 \beta_{3} q^{79} -39771 q^{81} + 17585 \beta_{2} q^{83} + 12330 \beta_{2} q^{87} + 107590 q^{89} -2387 \beta_{3} q^{91} + 21180 \beta_{1} q^{93} + 54419 \beta_{1} q^{97} -1827 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 252q^{9} + O(q^{10})$$ $$4q + 252q^{9} + 7440q^{21} - 16440q^{29} + 29080q^{41} - 9652q^{49} + 107080q^{61} - 141360q^{69} - 159084q^{81} + 430360q^{89} + O(q^{100})$$

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 13.4164i 0 0 0 138.636i 0 63.0000 0
449.2 0 13.4164i 0 0 0 138.636i 0 63.0000 0
449.3 0 13.4164i 0 0 0 138.636i 0 63.0000 0
449.4 0 13.4164i 0 0 0 138.636i 0 63.0000 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 800.6.c.h 4
4.b odd 2 1 inner 800.6.c.h 4
5.b even 2 1 inner 800.6.c.h 4
5.c odd 4 1 160.6.a.b 2
5.c odd 4 1 800.6.a.i 2
20.d odd 2 1 inner 800.6.c.h 4
20.e even 4 1 160.6.a.b 2
20.e even 4 1 800.6.a.i 2
40.i odd 4 1 320.6.a.t 2
40.k even 4 1 320.6.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.b 2 5.c odd 4 1
160.6.a.b 2 20.e even 4 1
320.6.a.t 2 40.i odd 4 1
320.6.a.t 2 40.k even 4 1
800.6.a.i 2 5.c odd 4 1
800.6.a.i 2 20.e even 4 1
800.6.c.h 4 1.a even 1 1 trivial
800.6.c.h 4 4.b odd 2 1 inner
800.6.c.h 4 5.b even 2 1 inner
800.6.c.h 4 20.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 180$$ $$T_{11}^{2} - 67280$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 306 T^{2} + 59049 T^{4} )^{2}$$
$5$ 1
$7$ $$( 1 - 14394 T^{2} + 282475249 T^{4} )^{2}$$
$11$ $$( 1 + 254822 T^{2} + 25937424601 T^{4} )^{2}$$
$13$ $$( 1 - 718870 T^{2} + 137858491849 T^{4} )^{2}$$
$17$ $$( 1 - 2808030 T^{2} + 2015993900449 T^{4} )^{2}$$
$19$ $$( 1 + 4019078 T^{2} + 6131066257801 T^{4} )^{2}$$
$23$ $$( 1 - 5934266 T^{2} + 41426511213649 T^{4} )^{2}$$
$29$ $$( 1 + 4110 T + 20511149 T^{2} )^{4}$$
$31$ $$( 1 + 47289582 T^{2} + 819628286980801 T^{4} )^{2}$$
$37$ $$( 1 - 83304550 T^{2} + 4808584372417849 T^{4} )^{2}$$
$41$ $$( 1 - 7270 T + 115856201 T^{2} )^{4}$$
$43$ $$( 1 + 26783614 T^{2} + 21611482313284249 T^{4} )^{2}$$
$47$ $$( 1 - 403777034 T^{2} + 52599132235830049 T^{4} )^{2}$$
$53$ $$( 1 + 202124090 T^{2} + 174887470365513049 T^{4} )^{2}$$
$59$ $$( 1 + 270997718 T^{2} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 - 26770 T + 844596301 T^{2} )^{4}$$
$67$ $$( 1 - 219594834 T^{2} + 1822837804551761449 T^{4} )^{2}$$
$71$ $$( 1 + 681226622 T^{2} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 - 3802634030 T^{2} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$( 1 - 1369983522 T^{2} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$( 1 - 1693436786 T^{2} + 15516041187205853449 T^{4} )^{2}$$
$89$ $$( 1 - 107590 T + 5584059449 T^{2} )^{4}$$
$97$ $$( 1 - 5328970270 T^{2} + 73742412689492826049 T^{4} )^{2}$$
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