# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 4 q^{7} + 2 q^{9} +O(q^{10})$$ $$q + i q^{3} + 4 q^{7} + 2 q^{9} -3 i q^{11} - q^{17} + 7 i q^{19} + 4 i q^{21} + 4 q^{23} + 5 i q^{27} + 8 i q^{29} -4 q^{31} + 3 q^{33} -4 i q^{37} + 3 q^{41} -8 i q^{43} + 9 q^{49} -i q^{51} + 12 i q^{53} -7 q^{57} -8 i q^{59} -4 i q^{61} + 8 q^{63} -9 i q^{67} + 4 i q^{69} + 16 q^{71} + 11 q^{73} -12 i q^{77} -4 q^{79} + q^{81} -i q^{83} -8 q^{87} + 13 q^{89} -4 i q^{93} -14 q^{97} -6 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{7} + 4q^{9} + O(q^{10})$$ $$2q + 8q^{7} + 4q^{9} - 2q^{17} + 8q^{23} - 8q^{31} + 6q^{33} + 6q^{41} + 18q^{49} - 14q^{57} + 16q^{63} + 32q^{71} + 22q^{73} - 8q^{79} + 2q^{81} - 16q^{87} + 26q^{89} - 28q^{97} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 4.00000 0 2.00000 0
1601.2 0 1.00000i 0 0 0 4.00000 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
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.g yes 2
4.b odd 2 1 3200.2.d.a 2
5.b even 2 1 3200.2.d.b yes 2
5.c odd 4 1 3200.2.f.a 2
5.c odd 4 1 3200.2.f.e 2
8.b even 2 1 inner 3200.2.d.g yes 2
8.d odd 2 1 3200.2.d.a 2
16.e even 4 1 6400.2.a.b 1
16.e even 4 1 6400.2.a.q 1
16.f odd 4 1 6400.2.a.h 1
16.f odd 4 1 6400.2.a.w 1
20.d odd 2 1 3200.2.d.h yes 2
20.e even 4 1 3200.2.f.b 2
20.e even 4 1 3200.2.f.f 2
40.e odd 2 1 3200.2.d.h yes 2
40.f even 2 1 3200.2.d.b yes 2
40.i odd 4 1 3200.2.f.a 2
40.i odd 4 1 3200.2.f.e 2
40.k even 4 1 3200.2.f.b 2
40.k even 4 1 3200.2.f.f 2
80.k odd 4 1 6400.2.a.c 1
80.k odd 4 1 6400.2.a.p 1
80.q even 4 1 6400.2.a.i 1
80.q even 4 1 6400.2.a.v 1

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

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ $$T_{7} - 4$$ $$T_{11}^{2} + 9$$ $$T_{13}$$ $$T_{17} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$9 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$49 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$64 + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$64 + T^{2}$$
$61$ $$16 + T^{2}$$
$67$ $$81 + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$( -11 + T )^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$1 + T^{2}$$
$89$ $$( -13 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$