# Properties

 Label 800.3.b.i Level $800$ Weight $3$ Character orbit 800.b Analytic conductor $21.798$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 800.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.7984211488$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.1827904.1 Defining polynomial: $$x^{6} + 9 x^{4} + 14 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{9}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{7} + ( -3 - \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{7} + ( -3 - \beta_{2} + \beta_{4} ) q^{9} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{11} + ( 1 - 2 \beta_{2} - \beta_{4} ) q^{13} + ( 12 + 2 \beta_{2} + 2 \beta_{4} ) q^{17} + ( -6 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{19} + ( 2 - \beta_{2} - \beta_{4} ) q^{21} + ( -2 \beta_{1} + 4 \beta_{3} + 3 \beta_{5} ) q^{23} + ( -4 \beta_{1} + 8 \beta_{3} + 2 \beta_{5} ) q^{27} + ( -6 - 4 \beta_{2} ) q^{29} + ( -4 \beta_{1} - 4 \beta_{5} ) q^{31} + ( 22 + 6 \beta_{2} ) q^{33} + ( 37 - 4 \beta_{2} - 3 \beta_{4} ) q^{37} + ( -4 \beta_{1} + 10 \beta_{3} + 4 \beta_{5} ) q^{39} + ( -10 - 3 \beta_{2} - \beta_{4} ) q^{41} + ( -\beta_{1} - 6 \beta_{5} ) q^{43} + ( 8 \beta_{1} + 12 \beta_{3} - \beta_{5} ) q^{47} + ( 13 - \beta_{2} - 7 \beta_{4} ) q^{49} + ( 12 \beta_{1} - 8 \beta_{3} - 4 \beta_{5} ) q^{51} + ( 11 - 6 \beta_{2} + 5 \beta_{4} ) q^{53} + ( 70 - 2 \beta_{2} - 8 \beta_{4} ) q^{57} + ( 6 \beta_{1} - 13 \beta_{3} + 2 \beta_{5} ) q^{59} + ( -20 + 3 \beta_{2} + 7 \beta_{4} ) q^{61} + ( 2 \beta_{1} + 4 \beta_{3} - 7 \beta_{5} ) q^{63} + ( 13 \beta_{1} + 16 \beta_{3} + 2 \beta_{5} ) q^{67} + ( 26 + 13 \beta_{2} + \beta_{4} ) q^{69} + ( 10 \beta_{3} + 8 \beta_{5} ) q^{71} + ( 10 + 6 \beta_{2} + 4 \beta_{4} ) q^{73} + ( 62 + 2 \beta_{2} + 12 \beta_{4} ) q^{77} + ( 12 \beta_{1} + 20 \beta_{3} - 4 \beta_{5} ) q^{79} + ( 33 + 13 \beta_{2} + 7 \beta_{4} ) q^{81} + ( 3 \beta_{1} + 16 \beta_{3} + 14 \beta_{5} ) q^{83} + ( -26 \beta_{1} + 24 \beta_{3} + 8 \beta_{5} ) q^{87} + ( -22 + 12 \beta_{2} + 16 \beta_{4} ) q^{89} + ( -8 \beta_{1} - 10 \beta_{3} ) q^{91} + ( 56 - 8 \beta_{4} ) q^{93} + ( 32 + 2 \beta_{2} + 6 \beta_{4} ) q^{97} + ( 34 \beta_{1} - 27 \beta_{3} + 6 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 18 q^{9} + O(q^{10})$$ $$6 q - 18 q^{9} + 80 q^{17} + 8 q^{21} - 44 q^{29} + 144 q^{33} + 208 q^{37} - 68 q^{41} + 62 q^{49} + 64 q^{53} + 400 q^{57} - 100 q^{61} + 184 q^{69} + 80 q^{73} + 400 q^{77} + 238 q^{81} - 76 q^{89} + 320 q^{93} + 208 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9 x^{4} + 14 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{4} + 20 \nu^{2} - 9$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{5} + 40 \nu^{3} + 76 \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{4} + 40 \nu^{2} + 51$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-16 \nu^{5} - 140 \nu^{3} - 194 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - \beta_{2} - 12$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + 4 \beta_{3} - 11 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{4} + 10 \beta_{2} + 69$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-10 \beta_{5} - 35 \beta_{3} + 72 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
351.1
 − 2.65109i − 1.37720i − 0.273891i 0.273891i 1.37720i 2.65109i
0 5.30219i 0 0 0 0.206625i 0 −19.1132 0
351.2 0 2.75441i 0 0 0 3.84997i 0 1.41325 0
351.3 0 0.547781i 0 0 0 10.0566i 0 8.69994 0
351.4 0 0.547781i 0 0 0 10.0566i 0 8.69994 0
351.5 0 2.75441i 0 0 0 3.84997i 0 1.41325 0
351.6 0 5.30219i 0 0 0 0.206625i 0 −19.1132 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 351.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.3.b.i 6
4.b odd 2 1 inner 800.3.b.i 6
5.b even 2 1 800.3.b.h 6
5.c odd 4 1 160.3.h.a 6
5.c odd 4 1 160.3.h.b yes 6
8.b even 2 1 1600.3.b.w 6
8.d odd 2 1 1600.3.b.w 6
15.e even 4 1 1440.3.j.a 6
15.e even 4 1 1440.3.j.b 6
20.d odd 2 1 800.3.b.h 6
20.e even 4 1 160.3.h.a 6
20.e even 4 1 160.3.h.b yes 6
40.e odd 2 1 1600.3.b.v 6
40.f even 2 1 1600.3.b.v 6
40.i odd 4 1 320.3.h.f 6
40.i odd 4 1 320.3.h.g 6
40.k even 4 1 320.3.h.f 6
40.k even 4 1 320.3.h.g 6
60.l odd 4 1 1440.3.j.a 6
60.l odd 4 1 1440.3.j.b 6
80.i odd 4 1 1280.3.e.h 6
80.i odd 4 1 1280.3.e.i 6
80.j even 4 1 1280.3.e.f 6
80.j even 4 1 1280.3.e.g 6
80.s even 4 1 1280.3.e.h 6
80.s even 4 1 1280.3.e.i 6
80.t odd 4 1 1280.3.e.f 6
80.t odd 4 1 1280.3.e.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a 6 5.c odd 4 1
160.3.h.a 6 20.e even 4 1
160.3.h.b yes 6 5.c odd 4 1
160.3.h.b yes 6 20.e even 4 1
320.3.h.f 6 40.i odd 4 1
320.3.h.f 6 40.k even 4 1
320.3.h.g 6 40.i odd 4 1
320.3.h.g 6 40.k even 4 1
800.3.b.h 6 5.b even 2 1
800.3.b.h 6 20.d odd 2 1
800.3.b.i 6 1.a even 1 1 trivial
800.3.b.i 6 4.b odd 2 1 inner
1280.3.e.f 6 80.j even 4 1
1280.3.e.f 6 80.t odd 4 1
1280.3.e.g 6 80.j even 4 1
1280.3.e.g 6 80.t odd 4 1
1280.3.e.h 6 80.i odd 4 1
1280.3.e.h 6 80.s even 4 1
1280.3.e.i 6 80.i odd 4 1
1280.3.e.i 6 80.s even 4 1
1440.3.j.a 6 15.e even 4 1
1440.3.j.a 6 60.l odd 4 1
1440.3.j.b 6 15.e even 4 1
1440.3.j.b 6 60.l odd 4 1
1600.3.b.v 6 40.e odd 2 1
1600.3.b.v 6 40.f even 2 1
1600.3.b.w 6 8.b even 2 1
1600.3.b.w 6 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{6} + 36 T_{3}^{4} + 224 T_{3}^{2} + 64$$ $$T_{13}^{3} - 208 T_{13} - 832$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$64 + 224 T^{2} + 36 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$64 + 1504 T^{2} + 116 T^{4} + T^{6}$$
$11$ $$2560000 + 86784 T^{2} + 560 T^{4} + T^{6}$$
$13$ $$( -832 - 208 T + T^{3} )^{2}$$
$17$ $$( 2560 + 256 T - 40 T^{2} + T^{3} )^{2}$$
$19$ $$11505664 + 533248 T^{2} + 1712 T^{4} + T^{6}$$
$23$ $$83905600 + 645984 T^{2} + 1460 T^{4} + T^{6}$$
$29$ $$( 2120 - 948 T + 22 T^{2} + T^{3} )^{2}$$
$31$ $$419430400 + 1900544 T^{2} + 2560 T^{4} + T^{6}$$
$37$ $$( -20800 + 2704 T - 104 T^{2} + T^{3} )^{2}$$
$41$ $$( -5000 - 100 T + 34 T^{2} + T^{3} )^{2}$$
$43$ $$39438400 + 2459744 T^{2} + 4260 T^{4} + T^{6}$$
$47$ $$493550656 + 17317088 T^{2} + 8308 T^{4} + T^{6}$$
$53$ $$( 224320 - 5968 T - 32 T^{2} + T^{3} )^{2}$$
$59$ $$25416011776 + 41968384 T^{2} + 12464 T^{4} + T^{6}$$
$61$ $$( -81544 - 1732 T + 50 T^{2} + T^{3} )^{2}$$
$67$ $$613057600 + 30359904 T^{2} + 14180 T^{4} + T^{6}$$
$71$ $$14231535616 + 19738624 T^{2} + 8384 T^{4} + T^{6}$$
$73$ $$( 55808 - 1408 T - 40 T^{2} + T^{3} )^{2}$$
$79$ $$37060870144 + 159514624 T^{2} + 25856 T^{4} + T^{6}$$
$83$ $$233675560000 + 145948000 T^{2} + 24164 T^{4} + T^{6}$$
$89$ $$( -155000 - 13940 T + 38 T^{2} + T^{3} )^{2}$$
$97$ $$( -6656 + 1664 T - 104 T^{2} + T^{3} )^{2}$$