# Properties

 Label 1600.2.l.b Level $1600$ Weight $2$ Character orbit 1600.l 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.l (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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + i ) q^{3} + i q^{9} +O(q^{10})$$ $$q + ( -1 + i ) q^{3} + i q^{9} + ( 3 + 3 i ) q^{11} + ( 3 - 3 i ) q^{13} + 4 q^{17} + ( -1 + i ) q^{19} -8 i q^{23} + ( -4 - 4 i ) q^{27} + ( -3 + 3 i ) q^{29} -6 q^{33} + ( 3 + 3 i ) q^{37} + 6 i q^{39} + ( 3 + 3 i ) q^{43} -2 q^{47} + 7 q^{49} + ( -4 + 4 i ) q^{51} + ( 9 + 9 i ) q^{53} -2 i q^{57} + ( 9 + 9 i ) q^{59} + ( -5 + 5 i ) q^{61} + ( -3 + 3 i ) q^{67} + ( 8 + 8 i ) q^{69} -6 i q^{71} -6 i q^{73} + 8 q^{79} + 5 q^{81} + ( -9 + 9 i ) q^{83} -6 i q^{87} + 12 i q^{89} -12 q^{97} + ( -3 + 3 i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + O(q^{10})$$ $$2q - 2q^{3} + 6q^{11} + 6q^{13} + 8q^{17} - 2q^{19} - 8q^{27} - 6q^{29} - 12q^{33} + 6q^{37} + 6q^{43} - 4q^{47} + 14q^{49} - 8q^{51} + 18q^{53} + 18q^{59} - 10q^{61} - 6q^{67} + 16q^{69} + 16q^{79} + 10q^{81} - 18q^{83} - 24q^{97} - 6q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 − 1.00000i 1.00000i
0 −1.00000 1.00000i 0 0 0 0 0 1.00000i 0
1201.1 0 −1.00000 + 1.00000i 0 0 0 0 0 1.00000i 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
16.e 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.l.b 2
4.b odd 2 1 400.2.l.b 2
5.b even 2 1 1600.2.l.c 2
5.c odd 4 1 320.2.q.a 2
5.c odd 4 1 320.2.q.b 2
16.e even 4 1 inner 1600.2.l.b 2
16.f odd 4 1 400.2.l.b 2
20.d odd 2 1 400.2.l.a 2
20.e even 4 1 80.2.q.a 2
20.e even 4 1 80.2.q.b yes 2
40.i odd 4 1 640.2.q.a 2
40.i odd 4 1 640.2.q.d 2
40.k even 4 1 640.2.q.b 2
40.k even 4 1 640.2.q.c 2
60.l odd 4 1 720.2.bm.a 2
60.l odd 4 1 720.2.bm.b 2
80.i odd 4 1 320.2.q.b 2
80.i odd 4 1 640.2.q.d 2
80.j even 4 1 80.2.q.a 2
80.j even 4 1 640.2.q.c 2
80.k odd 4 1 400.2.l.a 2
80.q even 4 1 1600.2.l.c 2
80.s even 4 1 80.2.q.b yes 2
80.s even 4 1 640.2.q.b 2
80.t odd 4 1 320.2.q.a 2
80.t odd 4 1 640.2.q.a 2
240.z odd 4 1 720.2.bm.a 2
240.bd odd 4 1 720.2.bm.b 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} + 2$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$18 - 6 T + T^{2}$$
$13$ $$18 - 6 T + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$2 + 2 T + T^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$18 + 6 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$18 - 6 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$18 - 6 T + T^{2}$$
$47$ $$( 2 + T )^{2}$$
$53$ $$162 - 18 T + T^{2}$$
$59$ $$162 - 18 T + T^{2}$$
$61$ $$50 + 10 T + T^{2}$$
$67$ $$18 + 6 T + T^{2}$$
$71$ $$36 + T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$162 + 18 T + T^{2}$$
$89$ $$144 + T^{2}$$
$97$ $$( 12 + T )^{2}$$