# Properties

 Label 800.6.c.b Level 800 Weight 6 Character orbit 800.c Analytic conductor 128.307 Analytic rank 0 Dimension 2 CM no Inner twists 2

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 i q^{3} -208 i q^{7} + 179 q^{9} +O(q^{10})$$ $$q + 8 i q^{3} -208 i q^{7} + 179 q^{9} + 536 q^{11} + 694 i q^{13} + 1278 i q^{17} + 1112 q^{19} + 1664 q^{21} -3216 i q^{23} + 3376 i q^{27} -2918 q^{29} + 2624 q^{31} + 4288 i q^{33} + 9458 i q^{37} -5552 q^{39} + 170 q^{41} + 19928 i q^{43} + 32 i q^{47} -26457 q^{49} -10224 q^{51} -22178 i q^{53} + 8896 i q^{57} + 41480 q^{59} + 15462 q^{61} -37232 i q^{63} -20744 i q^{67} + 25728 q^{69} -28592 q^{71} -53670 i q^{73} -111488 i q^{77} -69152 q^{79} + 16489 q^{81} + 37800 i q^{83} -23344 i q^{87} + 126806 q^{89} + 144352 q^{91} + 20992 i q^{93} -62290 i q^{97} + 95944 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 358q^{9} + O(q^{10})$$ $$2q + 358q^{9} + 1072q^{11} + 2224q^{19} + 3328q^{21} - 5836q^{29} + 5248q^{31} - 11104q^{39} + 340q^{41} - 52914q^{49} - 20448q^{51} + 82960q^{59} + 30924q^{61} + 51456q^{69} - 57184q^{71} - 138304q^{79} + 32978q^{81} + 253612q^{89} + 288704q^{91} + 191888q^{99} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 8.00000i 0 0 0 208.000i 0 179.000 0
449.2 0 8.00000i 0 0 0 208.000i 0 179.000 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
5.b even 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.b 2
4.b odd 2 1 800.6.c.a 2
5.b even 2 1 inner 800.6.c.b 2
5.c odd 4 1 32.6.a.c yes 1
5.c odd 4 1 800.6.a.a 1
15.e even 4 1 288.6.a.e 1
20.d odd 2 1 800.6.c.a 2
20.e even 4 1 32.6.a.a 1
20.e even 4 1 800.6.a.e 1
40.i odd 4 1 64.6.a.c 1
40.k even 4 1 64.6.a.e 1
60.l odd 4 1 288.6.a.d 1
80.i odd 4 1 256.6.b.b 2
80.j even 4 1 256.6.b.h 2
80.s even 4 1 256.6.b.h 2
80.t odd 4 1 256.6.b.b 2
120.q odd 4 1 576.6.a.u 1
120.w even 4 1 576.6.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.a 1 20.e even 4 1
32.6.a.c yes 1 5.c odd 4 1
64.6.a.c 1 40.i odd 4 1
64.6.a.e 1 40.k even 4 1
256.6.b.b 2 80.i odd 4 1
256.6.b.b 2 80.t odd 4 1
256.6.b.h 2 80.j even 4 1
256.6.b.h 2 80.s even 4 1
288.6.a.d 1 60.l odd 4 1
288.6.a.e 1 15.e even 4 1
576.6.a.u 1 120.q odd 4 1
576.6.a.v 1 120.w even 4 1
800.6.a.a 1 5.c odd 4 1
800.6.a.e 1 20.e even 4 1
800.6.c.a 2 4.b odd 2 1
800.6.c.a 2 20.d odd 2 1
800.6.c.b 2 1.a even 1 1 trivial
800.6.c.b 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{2} + 64$$ $$T_{11} - 536$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 422 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 + 9650 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 536 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 260950 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 1206430 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 1112 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 2530030 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 2918 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 2624 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 49234150 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 170 T + 115856201 T^{2} )^{2}$$
$43$ $$1 + 103108298 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 458688990 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 344527302 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 41480 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 15462 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 2269936678 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 28592 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 1265674286 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 69152 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 6449241286 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 126806 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 13294636414 T^{2} + 73742412689492826049 T^{4}$$