# Properties

 Label 1600.2.f.b Level $1600$ Weight $2$ Character orbit 1600.f Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 64) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + q^{9} +O(q^{10})$$ $$q + 2 q^{3} + q^{9} -6 i q^{11} -6 i q^{17} -2 i q^{19} -4 q^{27} -12 i q^{33} -6 q^{41} + 10 q^{43} + 7 q^{49} -12 i q^{51} -4 i q^{57} + 6 i q^{59} + 14 q^{67} -2 i q^{73} -11 q^{81} + 18 q^{83} -18 q^{89} + 10 i q^{97} -6 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 2q^{9} - 8q^{27} - 12q^{41} + 20q^{43} + 14q^{49} + 28q^{67} - 22q^{81} + 36q^{83} - 36q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 1.00000i − 1.00000i
0 2.00000 0 0 0 0 0 1.00000 0
1249.2 0 2.00000 0 0 0 0 0 1.00000 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 $$\Q(\sqrt{-2})$$
20.d 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 1600.2.f.b 2
4.b odd 2 1 1600.2.f.a 2
5.b even 2 1 1600.2.f.a 2
5.c odd 4 1 64.2.b.a 2
5.c odd 4 1 1600.2.d.a 2
8.b even 2 1 1600.2.f.a 2
8.d odd 2 1 CM 1600.2.f.b 2
15.e even 4 1 576.2.d.a 2
20.d odd 2 1 inner 1600.2.f.b 2
20.e even 4 1 64.2.b.a 2
20.e even 4 1 1600.2.d.a 2
35.f even 4 1 3136.2.b.b 2
40.e odd 2 1 1600.2.f.a 2
40.f even 2 1 inner 1600.2.f.b 2
40.i odd 4 1 64.2.b.a 2
40.i odd 4 1 1600.2.d.a 2
40.k even 4 1 64.2.b.a 2
40.k even 4 1 1600.2.d.a 2
60.l odd 4 1 576.2.d.a 2
80.i odd 4 1 256.2.a.a 1
80.i odd 4 1 6400.2.a.a 1
80.j even 4 1 256.2.a.a 1
80.j even 4 1 6400.2.a.a 1
80.s even 4 1 256.2.a.d 1
80.s even 4 1 6400.2.a.x 1
80.t odd 4 1 256.2.a.d 1
80.t odd 4 1 6400.2.a.x 1
120.q odd 4 1 576.2.d.a 2
120.w even 4 1 576.2.d.a 2
140.j odd 4 1 3136.2.b.b 2
160.u even 8 2 1024.2.e.l 4
160.v odd 8 2 1024.2.e.l 4
160.ba even 8 2 1024.2.e.l 4
160.bb odd 8 2 1024.2.e.l 4
240.z odd 4 1 2304.2.a.h 1
240.bb even 4 1 2304.2.a.i 1
240.bd odd 4 1 2304.2.a.i 1
240.bf even 4 1 2304.2.a.h 1
280.s even 4 1 3136.2.b.b 2
280.y odd 4 1 3136.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 5.c odd 4 1
64.2.b.a 2 20.e even 4 1
64.2.b.a 2 40.i odd 4 1
64.2.b.a 2 40.k even 4 1
256.2.a.a 1 80.i odd 4 1
256.2.a.a 1 80.j even 4 1
256.2.a.d 1 80.s even 4 1
256.2.a.d 1 80.t odd 4 1
576.2.d.a 2 15.e even 4 1
576.2.d.a 2 60.l odd 4 1
576.2.d.a 2 120.q odd 4 1
576.2.d.a 2 120.w even 4 1
1024.2.e.l 4 160.u even 8 2
1024.2.e.l 4 160.v odd 8 2
1024.2.e.l 4 160.ba even 8 2
1024.2.e.l 4 160.bb odd 8 2
1600.2.d.a 2 5.c odd 4 1
1600.2.d.a 2 20.e even 4 1
1600.2.d.a 2 40.i odd 4 1
1600.2.d.a 2 40.k even 4 1
1600.2.f.a 2 4.b odd 2 1
1600.2.f.a 2 5.b even 2 1
1600.2.f.a 2 8.b even 2 1
1600.2.f.a 2 40.e odd 2 1
1600.2.f.b 2 1.a even 1 1 trivial
1600.2.f.b 2 8.d odd 2 1 CM
1600.2.f.b 2 20.d odd 2 1 inner
1600.2.f.b 2 40.f even 2 1 inner
2304.2.a.h 1 240.z odd 4 1
2304.2.a.h 1 240.bf even 4 1
2304.2.a.i 1 240.bb even 4 1
2304.2.a.i 1 240.bd odd 4 1
3136.2.b.b 2 35.f even 4 1
3136.2.b.b 2 140.j odd 4 1
3136.2.b.b 2 280.s even 4 1
3136.2.b.b 2 280.y odd 4 1
6400.2.a.a 1 80.i odd 4 1
6400.2.a.a 1 80.j even 4 1
6400.2.a.x 1 80.s even 4 1
6400.2.a.x 1 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3} - 2$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -2 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$36 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -10 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$( -14 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -18 + T )^{2}$$
$89$ $$( 18 + T )^{2}$$
$97$ $$100 + T^{2}$$