# Properties

 Label 800.4.f.a Level $800$ Weight $4$ Character orbit 800.f Analytic conductor $47.202$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.2015280046$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 8) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - 4 \beta_1 q^{7} + q^{9}+O(q^{10})$$ q - b2 * q^3 - 4*b1 * q^7 + q^9 $$q - \beta_{2} q^{3} - 4 \beta_1 q^{7} + q^{9} - 3 \beta_{3} q^{11} + 10 \beta_{2} q^{13} + 7 \beta_1 q^{17} + 7 \beta_{3} q^{19} + 8 \beta_{3} q^{21} + 76 \beta_1 q^{23} + 26 \beta_{2} q^{27} - 30 \beta_{3} q^{29} - 224 q^{31} + 42 \beta_1 q^{33} - 46 \beta_{2} q^{37} - 280 q^{39} - 70 q^{41} + 83 \beta_{2} q^{43} + 168 \beta_1 q^{47} + 279 q^{49} - 14 \beta_{3} q^{51} + 6 \beta_{2} q^{53} - 98 \beta_1 q^{57} - 101 \beta_{3} q^{59} - 18 \beta_{3} q^{61} - 4 \beta_1 q^{63} + 33 \beta_{2} q^{67} - 152 \beta_{3} q^{69} + 72 q^{71} - 147 \beta_1 q^{73} - 24 \beta_{2} q^{77} - 464 q^{79} - 755 q^{81} + 103 \beta_{2} q^{83} + 420 \beta_1 q^{87} - 266 q^{89} - 80 \beta_{3} q^{91} + 224 \beta_{2} q^{93} - 497 \beta_1 q^{97} - 3 \beta_{3} q^{99}+O(q^{100})$$ q - b2 * q^3 - 4*b1 * q^7 + q^9 - 3*b3 * q^11 + 10*b2 * q^13 + 7*b1 * q^17 + 7*b3 * q^19 + 8*b3 * q^21 + 76*b1 * q^23 + 26*b2 * q^27 - 30*b3 * q^29 - 224 * q^31 + 42*b1 * q^33 - 46*b2 * q^37 - 280 * q^39 - 70 * q^41 + 83*b2 * q^43 + 168*b1 * q^47 + 279 * q^49 - 14*b3 * q^51 + 6*b2 * q^53 - 98*b1 * q^57 - 101*b3 * q^59 - 18*b3 * q^61 - 4*b1 * q^63 + 33*b2 * q^67 - 152*b3 * q^69 + 72 * q^71 - 147*b1 * q^73 - 24*b2 * q^77 - 464 * q^79 - 755 * q^81 + 103*b2 * q^83 + 420*b1 * q^87 - 266 * q^89 - 80*b3 * q^91 + 224*b2 * q^93 - 497*b1 * q^97 - 3*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} - 896 q^{31} - 1120 q^{39} - 280 q^{41} + 1116 q^{49} + 288 q^{71} - 1856 q^{79} - 3020 q^{81} - 1064 q^{89}+O(q^{100})$$ 4 * q + 4 * q^9 - 896 * q^31 - 1120 * q^39 - 280 * q^41 + 1116 * q^49 + 288 * q^71 - 1856 * q^79 - 3020 * q^81 - 1064 * q^89

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu$$ v^3 - v $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5\nu$$ -v^3 + 5*v $$\beta_{3}$$ $$=$$ $$4\nu^{2} - 6$$ 4*v^2 - 6
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 4$$ (b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 6 ) / 4$$ (b3 + 6) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{2} + 5\beta_1 ) / 4$$ (b2 + 5*b1) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.32288 + 0.500000i 1.32288 − 0.500000i −1.32288 + 0.500000i −1.32288 − 0.500000i
0 −5.29150 0 0 0 8.00000i 0 1.00000 0
49.2 0 −5.29150 0 0 0 8.00000i 0 1.00000 0
49.3 0 5.29150 0 0 0 8.00000i 0 1.00000 0
49.4 0 5.29150 0 0 0 8.00000i 0 1.00000 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 800.4.f.a 4
4.b odd 2 1 200.4.f.a 4
5.b even 2 1 inner 800.4.f.a 4
5.c odd 4 1 32.4.b.a 2
5.c odd 4 1 800.4.d.a 2
8.b even 2 1 inner 800.4.f.a 4
8.d odd 2 1 200.4.f.a 4
15.e even 4 1 288.4.d.a 2
20.d odd 2 1 200.4.f.a 4
20.e even 4 1 8.4.b.a 2
20.e even 4 1 200.4.d.a 2
40.e odd 2 1 200.4.f.a 4
40.f even 2 1 inner 800.4.f.a 4
40.i odd 4 1 32.4.b.a 2
40.i odd 4 1 800.4.d.a 2
40.k even 4 1 8.4.b.a 2
40.k even 4 1 200.4.d.a 2
60.l odd 4 1 72.4.d.b 2
80.i odd 4 1 256.4.a.j 2
80.j even 4 1 256.4.a.l 2
80.s even 4 1 256.4.a.l 2
80.t odd 4 1 256.4.a.j 2
120.q odd 4 1 72.4.d.b 2
120.w even 4 1 288.4.d.a 2
240.z odd 4 1 2304.4.a.bn 2
240.bb even 4 1 2304.4.a.v 2
240.bd odd 4 1 2304.4.a.bn 2
240.bf even 4 1 2304.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 20.e even 4 1
8.4.b.a 2 40.k even 4 1
32.4.b.a 2 5.c odd 4 1
32.4.b.a 2 40.i odd 4 1
72.4.d.b 2 60.l odd 4 1
72.4.d.b 2 120.q odd 4 1
200.4.d.a 2 20.e even 4 1
200.4.d.a 2 40.k even 4 1
200.4.f.a 4 4.b odd 2 1
200.4.f.a 4 8.d odd 2 1
200.4.f.a 4 20.d odd 2 1
200.4.f.a 4 40.e odd 2 1
256.4.a.j 2 80.i odd 4 1
256.4.a.j 2 80.t odd 4 1
256.4.a.l 2 80.j even 4 1
256.4.a.l 2 80.s even 4 1
288.4.d.a 2 15.e even 4 1
288.4.d.a 2 120.w even 4 1
800.4.d.a 2 5.c odd 4 1
800.4.d.a 2 40.i odd 4 1
800.4.f.a 4 1.a even 1 1 trivial
800.4.f.a 4 5.b even 2 1 inner
800.4.f.a 4 8.b even 2 1 inner
800.4.f.a 4 40.f even 2 1 inner
2304.4.a.v 2 240.bb even 4 1
2304.4.a.v 2 240.bf even 4 1
2304.4.a.bn 2 240.z odd 4 1
2304.4.a.bn 2 240.bd odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 28$$ acting on $$S_{4}^{\mathrm{new}}(800, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 28)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 64)^{2}$$
$11$ $$(T^{2} + 252)^{2}$$
$13$ $$(T^{2} - 2800)^{2}$$
$17$ $$(T^{2} + 196)^{2}$$
$19$ $$(T^{2} + 1372)^{2}$$
$23$ $$(T^{2} + 23104)^{2}$$
$29$ $$(T^{2} + 25200)^{2}$$
$31$ $$(T + 224)^{4}$$
$37$ $$(T^{2} - 59248)^{2}$$
$41$ $$(T + 70)^{4}$$
$43$ $$(T^{2} - 192892)^{2}$$
$47$ $$(T^{2} + 112896)^{2}$$
$53$ $$(T^{2} - 1008)^{2}$$
$59$ $$(T^{2} + 285628)^{2}$$
$61$ $$(T^{2} + 9072)^{2}$$
$67$ $$(T^{2} - 30492)^{2}$$
$71$ $$(T - 72)^{4}$$
$73$ $$(T^{2} + 86436)^{2}$$
$79$ $$(T + 464)^{4}$$
$83$ $$(T^{2} - 297052)^{2}$$
$89$ $$(T + 266)^{4}$$
$97$ $$(T^{2} + 988036)^{2}$$