# Properties

 Label 3200.2.d.c Level $3200$ Weight $2$ Character orbit 3200.d Analytic conductor $25.552$ Analytic rank $1$ Dimension $2$ CM discriminant -8 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 5 q^{9}+O(q^{10})$$ q + b * q^3 - 5 * q^9 $$q + \beta q^{3} - 5 q^{9} - \beta q^{11} - 6 q^{17} + 3 \beta q^{19} - 2 \beta q^{27} + 8 q^{33} - 6 q^{41} - 3 \beta q^{43} - 7 q^{49} - 6 \beta q^{51} - 24 q^{57} - 5 \beta q^{59} - 3 \beta q^{67} + 2 q^{73} + q^{81} + \beta q^{83} - 18 q^{89} + 10 q^{97} + 5 \beta q^{99} +O(q^{100})$$ q + b * q^3 - 5 * q^9 - b * q^11 - 6 * q^17 + 3*b * q^19 - 2*b * q^27 + 8 * q^33 - 6 * q^41 - 3*b * q^43 - 7 * q^49 - 6*b * q^51 - 24 * q^57 - 5*b * q^59 - 3*b * q^67 + 2 * q^73 + q^81 + b * q^83 - 18 * q^89 + 10 * q^97 + 5*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{9}+O(q^{10})$$ 2 * q - 10 * q^9 $$2 q - 10 q^{9} - 12 q^{17} + 16 q^{33} - 12 q^{41} - 14 q^{49} - 48 q^{57} + 4 q^{73} + 2 q^{81} - 36 q^{89} + 20 q^{97}+O(q^{100})$$ 2 * q - 10 * q^9 - 12 * q^17 + 16 * q^33 - 12 * q^41 - 14 * q^49 - 48 * q^57 + 4 * q^73 + 2 * q^81 - 36 * q^89 + 20 * q^97

## 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
 − 1.41421i 1.41421i
0 2.82843i 0 0 0 0 0 −5.00000 0
1601.2 0 2.82843i 0 0 0 0 0 −5.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b 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.c 2
4.b odd 2 1 inner 3200.2.d.c 2
5.b even 2 1 128.2.b.a 2
5.c odd 4 2 3200.2.f.o 4
8.b even 2 1 inner 3200.2.d.c 2
8.d odd 2 1 CM 3200.2.d.c 2
15.d odd 2 1 1152.2.d.c 2
16.e even 4 2 6400.2.a.by 2
16.f odd 4 2 6400.2.a.by 2
20.d odd 2 1 128.2.b.a 2
20.e even 4 2 3200.2.f.o 4
40.e odd 2 1 128.2.b.a 2
40.f even 2 1 128.2.b.a 2
40.i odd 4 2 3200.2.f.o 4
40.k even 4 2 3200.2.f.o 4
60.h even 2 1 1152.2.d.c 2
80.k odd 4 2 256.2.a.e 2
80.q even 4 2 256.2.a.e 2
120.i odd 2 1 1152.2.d.c 2
120.m even 2 1 1152.2.d.c 2
160.y odd 8 2 1024.2.e.a 2
160.y odd 8 2 1024.2.e.f 2
160.z even 8 2 1024.2.e.a 2
160.z even 8 2 1024.2.e.f 2
240.t even 4 2 2304.2.a.t 2
240.bm odd 4 2 2304.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 5.b even 2 1
128.2.b.a 2 20.d odd 2 1
128.2.b.a 2 40.e odd 2 1
128.2.b.a 2 40.f even 2 1
256.2.a.e 2 80.k odd 4 2
256.2.a.e 2 80.q even 4 2
1024.2.e.a 2 160.y odd 8 2
1024.2.e.a 2 160.z even 8 2
1024.2.e.f 2 160.y odd 8 2
1024.2.e.f 2 160.z even 8 2
1152.2.d.c 2 15.d odd 2 1
1152.2.d.c 2 60.h even 2 1
1152.2.d.c 2 120.i odd 2 1
1152.2.d.c 2 120.m even 2 1
2304.2.a.t 2 240.t even 4 2
2304.2.a.t 2 240.bm odd 4 2
3200.2.d.c 2 1.a even 1 1 trivial
3200.2.d.c 2 4.b odd 2 1 inner
3200.2.d.c 2 8.b even 2 1 inner
3200.2.d.c 2 8.d odd 2 1 CM
3200.2.f.o 4 5.c odd 4 2
3200.2.f.o 4 20.e even 4 2
3200.2.f.o 4 40.i odd 4 2
3200.2.f.o 4 40.k even 4 2
6400.2.a.by 2 16.e even 4 2
6400.2.a.by 2 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 $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{2} + 8$$ T3^2 + 8 $$T_{7}$$ T7 $$T_{11}^{2} + 8$$ T11^2 + 8 $$T_{13}$$ T13 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 8$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8$$
$13$ $$T^{2}$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 72$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 72$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 200$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 72$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 8$$
$89$ $$(T + 18)^{2}$$
$97$ $$(T - 10)^{2}$$