# Properties

 Label 800.2.f.d Level $800$ Weight $2$ Character orbit 800.f Analytic conductor $6.388$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3x^{2} + 4$$ x^4 - 3*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 200) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + 4 \beta_1 q^{7} + 4 q^{9}+O(q^{10})$$ q - b2 * q^3 + 4*b1 * q^7 + 4 * q^9 $$q - \beta_{2} q^{3} + 4 \beta_1 q^{7} + 4 q^{9} - \beta_{3} q^{11} + 3 \beta_1 q^{17} + \beta_{3} q^{19} - 4 \beta_{3} q^{21} - 4 \beta_1 q^{23} - \beta_{2} q^{27} - 4 q^{31} + 7 \beta_1 q^{33} - 4 \beta_{2} q^{37} - 5 q^{41} - 2 \beta_{2} q^{43} - 8 \beta_1 q^{47} - 9 q^{49} - 3 \beta_{3} q^{51} - 4 \beta_{2} q^{53} - 7 \beta_1 q^{57} + 2 \beta_{3} q^{59} + 4 \beta_{3} q^{61} + 16 \beta_1 q^{63} - 3 \beta_{2} q^{67} + 4 \beta_{3} q^{69} - 8 q^{71} - 7 \beta_1 q^{73} + 4 \beta_{2} q^{77} + 4 q^{79} - 5 q^{81} + 3 \beta_{2} q^{83} + q^{89} + 4 \beta_{2} q^{93} + 2 \beta_1 q^{97} - 4 \beta_{3} q^{99}+O(q^{100})$$ q - b2 * q^3 + 4*b1 * q^7 + 4 * q^9 - b3 * q^11 + 3*b1 * q^17 + b3 * q^19 - 4*b3 * q^21 - 4*b1 * q^23 - b2 * q^27 - 4 * q^31 + 7*b1 * q^33 - 4*b2 * q^37 - 5 * q^41 - 2*b2 * q^43 - 8*b1 * q^47 - 9 * q^49 - 3*b3 * q^51 - 4*b2 * q^53 - 7*b1 * q^57 + 2*b3 * q^59 + 4*b3 * q^61 + 16*b1 * q^63 - 3*b2 * q^67 + 4*b3 * q^69 - 8 * q^71 - 7*b1 * q^73 + 4*b2 * q^77 + 4 * q^79 - 5 * q^81 + 3*b2 * q^83 + q^89 + 4*b2 * q^93 + 2*b1 * q^97 - 4*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{9}+O(q^{10})$$ 4 * q + 16 * q^9 $$4 q + 16 q^{9} - 16 q^{31} - 20 q^{41} - 36 q^{49} - 32 q^{71} + 16 q^{79} - 20 q^{81} + 4 q^{89}+O(q^{100})$$ 4 * q + 16 * q^9 - 16 * q^31 - 20 * q^41 - 36 * q^49 - 32 * q^71 + 16 * q^79 - 20 * q^81 + 4 * q^89

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - \nu ) / 2$$ (v^3 - v) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu ) / 2$$ (-v^3 + 5*v) / 2 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 3$$ 2*v^2 - 3
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 3 ) / 2$$ (b3 + 3) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{2} + 5\beta_1 ) / 2$$ (b2 + 5*b1) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.32288 − 0.500000i 1.32288 + 0.500000i −1.32288 − 0.500000i −1.32288 + 0.500000i
0 −2.64575 0 0 0 4.00000i 0 4.00000 0
49.2 0 −2.64575 0 0 0 4.00000i 0 4.00000 0
49.3 0 2.64575 0 0 0 4.00000i 0 4.00000 0
49.4 0 2.64575 0 0 0 4.00000i 0 4.00000 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.f.d 4
3.b odd 2 1 7200.2.d.m 4
4.b odd 2 1 200.2.f.d 4
5.b even 2 1 inner 800.2.f.d 4
5.c odd 4 1 800.2.d.a 2
5.c odd 4 1 800.2.d.d 2
8.b even 2 1 inner 800.2.f.d 4
8.d odd 2 1 200.2.f.d 4
12.b even 2 1 1800.2.d.m 4
15.d odd 2 1 7200.2.d.m 4
15.e even 4 1 7200.2.k.b 2
15.e even 4 1 7200.2.k.i 2
20.d odd 2 1 200.2.f.d 4
20.e even 4 1 200.2.d.b 2
20.e even 4 1 200.2.d.c yes 2
24.f even 2 1 1800.2.d.m 4
24.h odd 2 1 7200.2.d.m 4
40.e odd 2 1 200.2.f.d 4
40.f even 2 1 inner 800.2.f.d 4
40.i odd 4 1 800.2.d.a 2
40.i odd 4 1 800.2.d.d 2
40.k even 4 1 200.2.d.b 2
40.k even 4 1 200.2.d.c yes 2
60.h even 2 1 1800.2.d.m 4
60.l odd 4 1 1800.2.k.d 2
60.l odd 4 1 1800.2.k.f 2
80.i odd 4 1 6400.2.a.bh 2
80.i odd 4 1 6400.2.a.cb 2
80.j even 4 1 6400.2.a.bg 2
80.j even 4 1 6400.2.a.cc 2
80.s even 4 1 6400.2.a.bg 2
80.s even 4 1 6400.2.a.cc 2
80.t odd 4 1 6400.2.a.bh 2
80.t odd 4 1 6400.2.a.cb 2
120.i odd 2 1 7200.2.d.m 4
120.m even 2 1 1800.2.d.m 4
120.q odd 4 1 1800.2.k.d 2
120.q odd 4 1 1800.2.k.f 2
120.w even 4 1 7200.2.k.b 2
120.w even 4 1 7200.2.k.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b 2 20.e even 4 1
200.2.d.b 2 40.k even 4 1
200.2.d.c yes 2 20.e even 4 1
200.2.d.c yes 2 40.k even 4 1
200.2.f.d 4 4.b odd 2 1
200.2.f.d 4 8.d odd 2 1
200.2.f.d 4 20.d odd 2 1
200.2.f.d 4 40.e odd 2 1
800.2.d.a 2 5.c odd 4 1
800.2.d.a 2 40.i odd 4 1
800.2.d.d 2 5.c odd 4 1
800.2.d.d 2 40.i odd 4 1
800.2.f.d 4 1.a even 1 1 trivial
800.2.f.d 4 5.b even 2 1 inner
800.2.f.d 4 8.b even 2 1 inner
800.2.f.d 4 40.f even 2 1 inner
1800.2.d.m 4 12.b even 2 1
1800.2.d.m 4 24.f even 2 1
1800.2.d.m 4 60.h even 2 1
1800.2.d.m 4 120.m even 2 1
1800.2.k.d 2 60.l odd 4 1
1800.2.k.d 2 120.q odd 4 1
1800.2.k.f 2 60.l odd 4 1
1800.2.k.f 2 120.q odd 4 1
6400.2.a.bg 2 80.j even 4 1
6400.2.a.bg 2 80.s even 4 1
6400.2.a.bh 2 80.i odd 4 1
6400.2.a.bh 2 80.t odd 4 1
6400.2.a.cb 2 80.i odd 4 1
6400.2.a.cb 2 80.t odd 4 1
6400.2.a.cc 2 80.j even 4 1
6400.2.a.cc 2 80.s even 4 1
7200.2.d.m 4 3.b odd 2 1
7200.2.d.m 4 15.d odd 2 1
7200.2.d.m 4 24.h odd 2 1
7200.2.d.m 4 120.i odd 2 1
7200.2.k.b 2 15.e even 4 1
7200.2.k.b 2 120.w even 4 1
7200.2.k.i 2 15.e even 4 1
7200.2.k.i 2 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 7$$ acting on $$S_{2}^{\mathrm{new}}(800, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 7)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 16)^{2}$$
$11$ $$(T^{2} + 7)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 9)^{2}$$
$19$ $$(T^{2} + 7)^{2}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T + 4)^{4}$$
$37$ $$(T^{2} - 112)^{2}$$
$41$ $$(T + 5)^{4}$$
$43$ $$(T^{2} - 28)^{2}$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$(T^{2} - 112)^{2}$$
$59$ $$(T^{2} + 28)^{2}$$
$61$ $$(T^{2} + 112)^{2}$$
$67$ $$(T^{2} - 63)^{2}$$
$71$ $$(T + 8)^{4}$$
$73$ $$(T^{2} + 49)^{2}$$
$79$ $$(T - 4)^{4}$$
$83$ $$(T^{2} - 63)^{2}$$
$89$ $$(T - 1)^{4}$$
$97$ $$(T^{2} + 4)^{2}$$