# Properties

 Label 3200.2.d.r Level $3200$ Weight $2$ Character orbit 3200.d Analytic conductor $25.552$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + 4x^{2} + 9$$ x^4 + 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} - \beta_1 q^{7} - 2 q^{9}+O(q^{10})$$ q + b3 * q^3 - b1 * q^7 - 2 * q^9 $$q + \beta_{3} q^{3} - \beta_1 q^{7} - 2 q^{9} - \beta_{3} q^{11} - \beta_{2} q^{13} + 5 q^{17} + \beta_{3} q^{19} + \beta_{2} q^{21} + 2 \beta_1 q^{23} + \beta_{3} q^{27} - \beta_{2} q^{29} + 5 q^{33} - \beta_{2} q^{37} - 5 \beta_1 q^{39} + 3 q^{41} - 4 \beta_{3} q^{43} - \beta_1 q^{47} + q^{49} + 5 \beta_{3} q^{51} - 2 \beta_{2} q^{53} - 5 q^{57} - 4 \beta_{3} q^{59} + \beta_{2} q^{61} + 2 \beta_1 q^{63} - 5 \beta_{3} q^{67} - 2 \beta_{2} q^{69} - 5 \beta_1 q^{71} - 15 q^{73} - \beta_{2} q^{77} - 5 \beta_1 q^{79} - 11 q^{81} + 3 \beta_{3} q^{83} - 5 \beta_1 q^{87} + q^{89} - 8 \beta_{3} q^{91} - 10 q^{97} + 2 \beta_{3} q^{99}+O(q^{100})$$ q + b3 * q^3 - b1 * q^7 - 2 * q^9 - b3 * q^11 - b2 * q^13 + 5 * q^17 + b3 * q^19 + b2 * q^21 + 2*b1 * q^23 + b3 * q^27 - b2 * q^29 + 5 * q^33 - b2 * q^37 - 5*b1 * q^39 + 3 * q^41 - 4*b3 * q^43 - b1 * q^47 + q^49 + 5*b3 * q^51 - 2*b2 * q^53 - 5 * q^57 - 4*b3 * q^59 + b2 * q^61 + 2*b1 * q^63 - 5*b3 * q^67 - 2*b2 * q^69 - 5*b1 * q^71 - 15 * q^73 - b2 * q^77 - 5*b1 * q^79 - 11 * q^81 + 3*b3 * q^83 - 5*b1 * q^87 + q^89 - 8*b3 * q^91 - 10 * q^97 + 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{9}+O(q^{10})$$ 4 * q - 8 * q^9 $$4 q - 8 q^{9} + 20 q^{17} + 20 q^{33} + 12 q^{41} + 4 q^{49} - 20 q^{57} - 60 q^{73} - 44 q^{81} + 4 q^{89} - 40 q^{97}+O(q^{100})$$ 4 * q - 8 * q^9 + 20 * q^17 + 20 * q^33 + 12 * q^41 + 4 * q^49 - 20 * q^57 - 60 * q^73 - 44 * q^81 + 4 * q^89 - 40 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} + 2\nu ) / 3$$ (2*v^3 + 2*v) / 3 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 14\nu ) / 3$$ (2*v^3 + 14*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 4$$ (b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 7\beta_1 ) / 4$$ (-b2 + 7*b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1601.1
 −0.707107 + 1.58114i 0.707107 − 1.58114i −0.707107 − 1.58114i 0.707107 + 1.58114i
0 2.23607i 0 0 0 −2.82843 0 −2.00000 0
1601.2 0 2.23607i 0 0 0 2.82843 0 −2.00000 0
1601.3 0 2.23607i 0 0 0 −2.82843 0 −2.00000 0
1601.4 0 2.23607i 0 0 0 2.82843 0 −2.00000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.d.r yes 4
4.b odd 2 1 inner 3200.2.d.r yes 4
5.b even 2 1 3200.2.d.m 4
5.c odd 4 2 3200.2.f.r 8
8.b even 2 1 inner 3200.2.d.r yes 4
8.d odd 2 1 inner 3200.2.d.r yes 4
16.e even 4 2 6400.2.a.cs 4
16.f odd 4 2 6400.2.a.cs 4
20.d odd 2 1 3200.2.d.m 4
20.e even 4 2 3200.2.f.r 8
40.e odd 2 1 3200.2.d.m 4
40.f even 2 1 3200.2.d.m 4
40.i odd 4 2 3200.2.f.r 8
40.k even 4 2 3200.2.f.r 8
80.k odd 4 2 6400.2.a.cp 4
80.q even 4 2 6400.2.a.cp 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.m 4 5.b even 2 1
3200.2.d.m 4 20.d odd 2 1
3200.2.d.m 4 40.e odd 2 1
3200.2.d.m 4 40.f even 2 1
3200.2.d.r yes 4 1.a even 1 1 trivial
3200.2.d.r yes 4 4.b odd 2 1 inner
3200.2.d.r yes 4 8.b even 2 1 inner
3200.2.d.r yes 4 8.d odd 2 1 inner
3200.2.f.r 8 5.c odd 4 2
3200.2.f.r 8 20.e even 4 2
3200.2.f.r 8 40.i odd 4 2
3200.2.f.r 8 40.k even 4 2
6400.2.a.cp 4 80.k odd 4 2
6400.2.a.cp 4 80.q even 4 2
6400.2.a.cs 4 16.e even 4 2
6400.2.a.cs 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{2} + 5$$ T3^2 + 5 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{2} + 5$$ T11^2 + 5 $$T_{13}^{2} + 40$$ T13^2 + 40 $$T_{17} - 5$$ T17 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 5)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 8)^{2}$$
$11$ $$(T^{2} + 5)^{2}$$
$13$ $$(T^{2} + 40)^{2}$$
$17$ $$(T - 5)^{4}$$
$19$ $$(T^{2} + 5)^{2}$$
$23$ $$(T^{2} - 32)^{2}$$
$29$ $$(T^{2} + 40)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} + 40)^{2}$$
$41$ $$(T - 3)^{4}$$
$43$ $$(T^{2} + 80)^{2}$$
$47$ $$(T^{2} - 8)^{2}$$
$53$ $$(T^{2} + 160)^{2}$$
$59$ $$(T^{2} + 80)^{2}$$
$61$ $$(T^{2} + 40)^{2}$$
$67$ $$(T^{2} + 125)^{2}$$
$71$ $$(T^{2} - 200)^{2}$$
$73$ $$(T + 15)^{4}$$
$79$ $$(T^{2} - 200)^{2}$$
$83$ $$(T^{2} + 45)^{2}$$
$89$ $$(T - 1)^{4}$$
$97$ $$(T + 10)^{4}$$