# Properties

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

# Related objects

## 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: $$\Q(i, \sqrt{10})$$ Defining polynomial: $$x^{4} + 25$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + 7 \beta_{2} q^{7} + 203 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + 7 \beta_{2} q^{7} + 203 q^{9} -57 \beta_{3} q^{11} + 73 \beta_{1} q^{13} -351 \beta_{1} q^{17} -216 \beta_{3} q^{19} + 280 q^{21} + 647 \beta_{2} q^{23} -446 \beta_{2} q^{27} + 4010 q^{29} + 361 \beta_{3} q^{31} + 2280 \beta_{1} q^{33} -7389 \beta_{1} q^{37} + 73 \beta_{3} q^{39} -4350 q^{41} -1965 \beta_{2} q^{43} + 951 \beta_{2} q^{47} + 14847 q^{49} -351 \beta_{3} q^{51} + 9077 \beta_{1} q^{53} + 8640 \beta_{1} q^{57} + 1558 \beta_{3} q^{59} -42130 q^{61} + 1421 \beta_{2} q^{63} + 2559 \beta_{2} q^{67} + 25880 q^{69} + 3593 \beta_{3} q^{71} -13133 \beta_{1} q^{73} -15960 \beta_{1} q^{77} + 686 \beta_{3} q^{79} + 31489 q^{81} + 15615 \beta_{2} q^{83} -4010 \beta_{2} q^{87} -30570 q^{89} -511 \beta_{3} q^{91} -14440 \beta_{1} q^{93} + 33441 \beta_{1} q^{97} -11571 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 812q^{9} + O(q^{10})$$ $$4q + 812q^{9} + 1120q^{21} + 16040q^{29} - 17400q^{41} + 59388q^{49} - 168520q^{61} + 103520q^{69} + 125956q^{81} - 122280q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/5$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 10 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{3} + 20 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} + 10 \beta_{2}$$$$)/8$$

## 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
 1.58114 + 1.58114i −1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i
0 6.32456i 0 0 0 44.2719i 0 203.000 0
449.2 0 6.32456i 0 0 0 44.2719i 0 203.000 0
449.3 0 6.32456i 0 0 0 44.2719i 0 203.000 0
449.4 0 6.32456i 0 0 0 44.2719i 0 203.000 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.i 4
4.b odd 2 1 inner 800.6.c.i 4
5.b even 2 1 inner 800.6.c.i 4
5.c odd 4 1 160.6.a.d 2
5.c odd 4 1 800.6.a.h 2
20.d odd 2 1 inner 800.6.c.i 4
20.e even 4 1 160.6.a.d 2
20.e even 4 1 800.6.a.h 2
40.i odd 4 1 320.6.a.s 2
40.k even 4 1 320.6.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.d 2 5.c odd 4 1
160.6.a.d 2 20.e even 4 1
320.6.a.s 2 40.i odd 4 1
320.6.a.s 2 40.k even 4 1
800.6.a.h 2 5.c odd 4 1
800.6.a.h 2 20.e even 4 1
800.6.c.i 4 1.a even 1 1 trivial
800.6.c.i 4 4.b odd 2 1 inner
800.6.c.i 4 5.b even 2 1 inner
800.6.c.i 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 $$S_{6}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 40$$ $$T_{11}^{2} - 519840$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 446 T^{2} + 59049 T^{4} )^{2}$$
$5$ 1
$7$ $$( 1 - 31654 T^{2} + 282475249 T^{4} )^{2}$$
$11$ $$( 1 - 197738 T^{2} + 25937424601 T^{4} )^{2}$$
$13$ $$( 1 - 721270 T^{2} + 137858491849 T^{4} )^{2}$$
$17$ $$( 1 - 2346910 T^{2} + 2015993900449 T^{4} )^{2}$$
$19$ $$( 1 - 2512762 T^{2} + 6131066257801 T^{4} )^{2}$$
$23$ $$( 1 + 3871674 T^{2} + 41426511213649 T^{4} )^{2}$$
$29$ $$( 1 - 4010 T + 20511149 T^{2} )^{4}$$
$31$ $$( 1 + 36406942 T^{2} + 819628286980801 T^{4} )^{2}$$
$37$ $$( 1 + 79701370 T^{2} + 4808584372417849 T^{4} )^{2}$$
$41$ $$( 1 + 4350 T + 115856201 T^{2} )^{4}$$
$43$ $$( 1 - 139567886 T^{2} + 21611482313284249 T^{4} )^{2}$$
$47$ $$( 1 - 422513974 T^{2} + 52599132235830049 T^{4} )^{2}$$
$53$ $$( 1 - 506823270 T^{2} + 174887470365513049 T^{4} )^{2}$$
$59$ $$( 1 + 1041470358 T^{2} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 + 42130 T + 844596301 T^{2} )^{4}$$
$67$ $$( 1 - 2438310974 T^{2} + 1822837804551761449 T^{4} )^{2}$$
$71$ $$( 1 + 1542914862 T^{2} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 - 3456240430 T^{2} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$( 1 + 6078817438 T^{2} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$( 1 + 1875047714 T^{2} + 15516041187205853449 T^{4} )^{2}$$
$89$ $$( 1 + 30570 T + 5584059449 T^{2} )^{4}$$
$97$ $$( 1 - 12701478590 T^{2} + 73742412689492826049 T^{4} )^{2}$$