# Properties

 Label 1600.2.l.a 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 16) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + i ) q^{3} -2 i q^{7} + i q^{9} +O(q^{10})$$ $$q + ( -1 + i ) q^{3} -2 i q^{7} + i q^{9} + ( -1 - i ) q^{11} + ( 1 - i ) q^{13} + 2 q^{17} + ( -3 + 3 i ) q^{19} + ( 2 + 2 i ) q^{21} + 6 i q^{23} + ( -4 - 4 i ) q^{27} + ( 3 - 3 i ) q^{29} + 8 q^{31} + 2 q^{33} + ( -3 - 3 i ) q^{37} + 2 i q^{39} + ( 5 + 5 i ) q^{43} + 8 q^{47} + 3 q^{49} + ( -2 + 2 i ) q^{51} + ( 5 + 5 i ) q^{53} -6 i q^{57} + ( 3 + 3 i ) q^{59} + ( -9 + 9 i ) q^{61} + 2 q^{63} + ( -5 + 5 i ) q^{67} + ( -6 - 6 i ) q^{69} + 10 i q^{71} + 4 i q^{73} + ( -2 + 2 i ) q^{77} + 5 q^{81} + ( -1 + i ) q^{83} + 6 i q^{87} + 4 i q^{89} + ( -2 - 2 i ) q^{91} + ( -8 + 8 i ) q^{93} + 2 q^{97} + ( 1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{11} + 2q^{13} + 4q^{17} - 6q^{19} + 4q^{21} - 8q^{27} + 6q^{29} + 16q^{31} + 4q^{33} - 6q^{37} + 10q^{43} + 16q^{47} + 6q^{49} - 4q^{51} + 10q^{53} + 6q^{59} - 18q^{61} + 4q^{63} - 10q^{67} - 12q^{69} - 4q^{77} + 10q^{81} - 2q^{83} - 4q^{91} - 16q^{93} + 4q^{97} + 2q^{99} + O(q^{100})$$

## Character values

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

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

## 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 2.00000i 0 1.00000i 0
1201.1 0 −1.00000 + 1.00000i 0 0 0 2.00000i 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.a 2
4.b odd 2 1 400.2.l.c 2
5.b even 2 1 64.2.e.a 2
5.c odd 4 1 1600.2.q.a 2
5.c odd 4 1 1600.2.q.b 2
15.d odd 2 1 576.2.k.a 2
16.e even 4 1 inner 1600.2.l.a 2
16.f odd 4 1 400.2.l.c 2
20.d odd 2 1 16.2.e.a 2
20.e even 4 1 400.2.q.a 2
20.e even 4 1 400.2.q.b 2
40.e odd 2 1 128.2.e.b 2
40.f even 2 1 128.2.e.a 2
60.h even 2 1 144.2.k.a 2
80.i odd 4 1 1600.2.q.b 2
80.j even 4 1 400.2.q.b 2
80.k odd 4 1 16.2.e.a 2
80.k odd 4 1 128.2.e.b 2
80.q even 4 1 64.2.e.a 2
80.q even 4 1 128.2.e.a 2
80.s even 4 1 400.2.q.a 2
80.t odd 4 1 1600.2.q.a 2
120.i odd 2 1 1152.2.k.a 2
120.m even 2 1 1152.2.k.b 2
140.c even 2 1 784.2.m.b 2
140.p odd 6 2 784.2.x.f 4
140.s even 6 2 784.2.x.c 4
160.y odd 8 2 1024.2.a.b 2
160.y odd 8 2 1024.2.b.e 2
160.z even 8 2 1024.2.a.e 2
160.z even 8 2 1024.2.b.b 2
240.t even 4 1 144.2.k.a 2
240.t even 4 1 1152.2.k.b 2
240.bm odd 4 1 576.2.k.a 2
240.bm odd 4 1 1152.2.k.a 2
480.bs even 8 2 9216.2.a.d 2
480.bu odd 8 2 9216.2.a.s 2
560.be even 4 1 784.2.m.b 2
560.co even 12 2 784.2.x.c 4
560.cs odd 12 2 784.2.x.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 20.d odd 2 1
16.2.e.a 2 80.k odd 4 1
64.2.e.a 2 5.b even 2 1
64.2.e.a 2 80.q even 4 1
128.2.e.a 2 40.f even 2 1
128.2.e.a 2 80.q even 4 1
128.2.e.b 2 40.e odd 2 1
128.2.e.b 2 80.k odd 4 1
144.2.k.a 2 60.h even 2 1
144.2.k.a 2 240.t even 4 1
400.2.l.c 2 4.b odd 2 1
400.2.l.c 2 16.f odd 4 1
400.2.q.a 2 20.e even 4 1
400.2.q.a 2 80.s even 4 1
400.2.q.b 2 20.e even 4 1
400.2.q.b 2 80.j even 4 1
576.2.k.a 2 15.d odd 2 1
576.2.k.a 2 240.bm odd 4 1
784.2.m.b 2 140.c even 2 1
784.2.m.b 2 560.be even 4 1
784.2.x.c 4 140.s even 6 2
784.2.x.c 4 560.co even 12 2
784.2.x.f 4 140.p odd 6 2
784.2.x.f 4 560.cs odd 12 2
1024.2.a.b 2 160.y odd 8 2
1024.2.a.e 2 160.z even 8 2
1024.2.b.b 2 160.z even 8 2
1024.2.b.e 2 160.y odd 8 2
1152.2.k.a 2 120.i odd 2 1
1152.2.k.a 2 240.bm odd 4 1
1152.2.k.b 2 120.m even 2 1
1152.2.k.b 2 240.t even 4 1
1600.2.l.a 2 1.a even 1 1 trivial
1600.2.l.a 2 16.e even 4 1 inner
1600.2.q.a 2 5.c odd 4 1
1600.2.q.a 2 80.t odd 4 1
1600.2.q.b 2 5.c odd 4 1
1600.2.q.b 2 80.i odd 4 1
9216.2.a.d 2 480.bs even 8 2
9216.2.a.s 2 480.bu odd 8 2

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} + 2$$ $$T_{7}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$2 + 2 T + T^{2}$$
$13$ $$2 - 2 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$18 + 6 T + T^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$18 - 6 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$18 + 6 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$50 - 10 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$50 - 10 T + T^{2}$$
$59$ $$18 - 6 T + T^{2}$$
$61$ $$162 + 18 T + T^{2}$$
$67$ $$50 + 10 T + T^{2}$$
$71$ $$100 + T^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$2 + 2 T + T^{2}$$
$89$ $$16 + T^{2}$$
$97$ $$( -2 + T )^{2}$$