# 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: $$x^{2} + 2$$ 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.

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

## 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 $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{2} + 8$$ $$T_{7}$$ $$T_{11}^{2} + 8$$ $$T_{13}$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

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