# Properties

 Label 3200.2.d.i Level $3200$ Weight $2$ Character orbit 3200.d Analytic conductor $25.552$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# 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: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 640) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - 3 \beta_1 q^{7} - 7 q^{9}+O(q^{10})$$ q + b2 * q^3 - 3*b1 * q^7 - 7 * q^9 $$q + \beta_{2} q^{3} - 3 \beta_1 q^{7} - 7 q^{9} + 3 \beta_{3} q^{21} + \beta_1 q^{23} - 4 \beta_{2} q^{27} + 2 \beta_{3} q^{29} - 12 q^{41} + \beta_{2} q^{43} + 7 \beta_1 q^{47} + 11 q^{49} + 3 \beta_{3} q^{61} + 21 \beta_1 q^{63} + 5 \beta_{2} q^{67} - \beta_{3} q^{69} + 19 q^{81} + 3 \beta_{2} q^{83} + 20 \beta_1 q^{87} + 6 q^{89}+O(q^{100})$$ q + b2 * q^3 - 3*b1 * q^7 - 7 * q^9 + 3*b3 * q^21 + b1 * q^23 - 4*b2 * q^27 + 2*b3 * q^29 - 12 * q^41 + b2 * q^43 + 7*b1 * q^47 + 11 * q^49 + 3*b3 * q^61 + 21*b1 * q^63 + 5*b2 * q^67 - b3 * q^69 + 19 * q^81 + 3*b2 * q^83 + 20*b1 * q^87 + 6 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 28 q^{9}+O(q^{10})$$ 4 * q - 28 * q^9 $$4 q - 28 q^{9} - 48 q^{41} + 44 q^{49} + 76 q^{81} + 24 q^{89}+O(q^{100})$$ 4 * q - 28 * q^9 - 48 * q^41 + 44 * q^49 + 76 * q^81 + 24 * q^89

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

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

## 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 3.16228i 0 0 0 −4.24264 0 −7.00000 0
1601.2 0 3.16228i 0 0 0 4.24264 0 −7.00000 0
1601.3 0 3.16228i 0 0 0 −4.24264 0 −7.00000 0
1601.4 0 3.16228i 0 0 0 4.24264 0 −7.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 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 3200.2.d.i 4
4.b odd 2 1 inner 3200.2.d.i 4
5.b even 2 1 inner 3200.2.d.i 4
5.c odd 4 2 640.2.f.f 4
8.b even 2 1 inner 3200.2.d.i 4
8.d odd 2 1 inner 3200.2.d.i 4
16.e even 4 2 6400.2.a.cv 4
16.f odd 4 2 6400.2.a.cv 4
20.d odd 2 1 CM 3200.2.d.i 4
20.e even 4 2 640.2.f.f 4
40.e odd 2 1 inner 3200.2.d.i 4
40.f even 2 1 inner 3200.2.d.i 4
40.i odd 4 2 640.2.f.f 4
40.k even 4 2 640.2.f.f 4
80.i odd 4 2 1280.2.c.f 4
80.j even 4 2 1280.2.c.f 4
80.k odd 4 2 6400.2.a.cv 4
80.q even 4 2 6400.2.a.cv 4
80.s even 4 2 1280.2.c.f 4
80.t odd 4 2 1280.2.c.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.f 4 5.c odd 4 2
640.2.f.f 4 20.e even 4 2
640.2.f.f 4 40.i odd 4 2
640.2.f.f 4 40.k even 4 2
1280.2.c.f 4 80.i odd 4 2
1280.2.c.f 4 80.j even 4 2
1280.2.c.f 4 80.s even 4 2
1280.2.c.f 4 80.t odd 4 2
3200.2.d.i 4 1.a even 1 1 trivial
3200.2.d.i 4 4.b odd 2 1 inner
3200.2.d.i 4 5.b even 2 1 inner
3200.2.d.i 4 8.b even 2 1 inner
3200.2.d.i 4 8.d odd 2 1 inner
3200.2.d.i 4 20.d odd 2 1 CM
3200.2.d.i 4 40.e odd 2 1 inner
3200.2.d.i 4 40.f even 2 1 inner
6400.2.a.cv 4 16.e even 4 2
6400.2.a.cv 4 16.f odd 4 2
6400.2.a.cv 4 80.k odd 4 2
6400.2.a.cv 4 80.q even 4 2

## Hecke kernels

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

 $$T_{3}^{2} + 10$$ T3^2 + 10 $$T_{7}^{2} - 18$$ T7^2 - 18 $$T_{11}$$ T11 $$T_{13}$$ T13 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 10)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 18)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 2)^{2}$$
$29$ $$(T^{2} + 80)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T + 12)^{4}$$
$43$ $$(T^{2} + 10)^{2}$$
$47$ $$(T^{2} - 98)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 180)^{2}$$
$67$ $$(T^{2} + 250)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 90)^{2}$$
$89$ $$(T - 6)^{4}$$
$97$ $$T^{4}$$