# Properties

 Label 1600.2.j.a Level $1600$ Weight $2$ Character orbit 1600.j Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.j (of order $$4$$, degree $$2$$, 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + ( -3 - 3 i ) q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} + ( -3 - 3 i ) q^{7} - q^{9} + ( 1 + i ) q^{11} + 2 q^{13} + ( -1 - i ) q^{17} + ( -3 - 3 i ) q^{19} + ( 6 - 6 i ) q^{21} + ( -1 + i ) q^{23} + 4 i q^{27} + ( 7 - 7 i ) q^{29} -2 i q^{31} + ( -2 + 2 i ) q^{33} + 6 q^{37} + 4 i q^{39} -4 i q^{41} + 4 q^{43} + ( 7 - 7 i ) q^{47} + 11 i q^{49} + ( 2 - 2 i ) q^{51} -8 i q^{53} + ( 6 - 6 i ) q^{57} + ( 3 - 3 i ) q^{59} + ( -1 - i ) q^{61} + ( 3 + 3 i ) q^{63} + 4 q^{67} + ( -2 - 2 i ) q^{69} + ( 3 + 3 i ) q^{73} -6 i q^{77} + 8 q^{79} -11 q^{81} + 2 i q^{83} + ( 14 + 14 i ) q^{87} + 6 q^{89} + ( -6 - 6 i ) q^{91} + 4 q^{93} + ( 11 + 11 i ) q^{97} + ( -1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 6q^{7} - 2q^{9} + 2q^{11} + 4q^{13} - 2q^{17} - 6q^{19} + 12q^{21} - 2q^{23} + 14q^{29} - 4q^{33} + 12q^{37} + 8q^{43} + 14q^{47} + 4q^{51} + 12q^{57} + 6q^{59} - 2q^{61} + 6q^{63} + 8q^{67} - 4q^{69} + 6q^{73} + 16q^{79} - 22q^{81} + 28q^{87} + 12q^{89} - 12q^{91} + 8q^{93} + 22q^{97} - 2q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 −3.00000 + 3.00000i 0 −1.00000 0
1007.1 0 2.00000i 0 0 0 −3.00000 3.00000i 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
80.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.j.a 2
4.b odd 2 1 400.2.j.a 2
5.b even 2 1 320.2.j.a 2
5.c odd 4 1 320.2.s.a 2
5.c odd 4 1 1600.2.s.a 2
16.e even 4 1 400.2.s.a 2
16.f odd 4 1 1600.2.s.a 2
20.d odd 2 1 80.2.j.a 2
20.e even 4 1 80.2.s.a yes 2
20.e even 4 1 400.2.s.a 2
40.e odd 2 1 640.2.j.a 2
40.f even 2 1 640.2.j.b 2
40.i odd 4 1 640.2.s.a 2
40.k even 4 1 640.2.s.b 2
60.h even 2 1 720.2.bd.a 2
60.l odd 4 1 720.2.z.d 2
80.i odd 4 1 80.2.j.a 2
80.j even 4 1 640.2.j.b 2
80.j even 4 1 inner 1600.2.j.a 2
80.k odd 4 1 320.2.s.a 2
80.k odd 4 1 640.2.s.a 2
80.q even 4 1 80.2.s.a yes 2
80.q even 4 1 640.2.s.b 2
80.s even 4 1 320.2.j.a 2
80.t odd 4 1 400.2.j.a 2
80.t odd 4 1 640.2.j.a 2
240.bb even 4 1 720.2.bd.a 2
240.bm odd 4 1 720.2.z.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.a 2 20.d odd 2 1
80.2.j.a 2 80.i odd 4 1
80.2.s.a yes 2 20.e even 4 1
80.2.s.a yes 2 80.q even 4 1
320.2.j.a 2 5.b even 2 1
320.2.j.a 2 80.s even 4 1
320.2.s.a 2 5.c odd 4 1
320.2.s.a 2 80.k odd 4 1
400.2.j.a 2 4.b odd 2 1
400.2.j.a 2 80.t odd 4 1
400.2.s.a 2 16.e even 4 1
400.2.s.a 2 20.e even 4 1
640.2.j.a 2 40.e odd 2 1
640.2.j.a 2 80.t odd 4 1
640.2.j.b 2 40.f even 2 1
640.2.j.b 2 80.j even 4 1
640.2.s.a 2 40.i odd 4 1
640.2.s.a 2 80.k odd 4 1
640.2.s.b 2 40.k even 4 1
640.2.s.b 2 80.q even 4 1
720.2.z.d 2 60.l odd 4 1
720.2.z.d 2 240.bm odd 4 1
720.2.bd.a 2 60.h even 2 1
720.2.bd.a 2 240.bb even 4 1
1600.2.j.a 2 1.a even 1 1 trivial
1600.2.j.a 2 80.j even 4 1 inner
1600.2.s.a 2 5.c odd 4 1
1600.2.s.a 2 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$18 + 6 T + T^{2}$$
$11$ $$2 - 2 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$2 + 2 T + T^{2}$$
$19$ $$18 + 6 T + T^{2}$$
$23$ $$2 + 2 T + T^{2}$$
$29$ $$98 - 14 T + T^{2}$$
$31$ $$4 + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$16 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$98 - 14 T + T^{2}$$
$53$ $$64 + T^{2}$$
$59$ $$18 - 6 T + T^{2}$$
$61$ $$2 + 2 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$18 - 6 T + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$242 - 22 T + T^{2}$$