# Properties

 Label 1600.2.o.d Level $1600$ Weight $2$ Character orbit 1600.o 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.o (of order $$4$$, degree $$2$$, 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 320) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - i ) q^{3} + ( 1 - i ) q^{7} + i q^{9} +O(q^{10})$$ $$q + ( 1 - i ) q^{3} + ( 1 - i ) q^{7} + i q^{9} -4 q^{11} + ( 3 + 3 i ) q^{13} + ( 3 + 3 i ) q^{17} + 6 i q^{19} -2 i q^{21} + ( 3 + 3 i ) q^{23} + ( 4 + 4 i ) q^{27} + 2 q^{29} -6 i q^{31} + ( -4 + 4 i ) q^{33} + ( 3 - 3 i ) q^{37} + 6 q^{39} + 6 q^{41} + ( -3 + 3 i ) q^{43} + ( 9 - 9 i ) q^{47} + 5 i q^{49} + 6 q^{51} + ( -5 - 5 i ) q^{53} + ( 6 + 6 i ) q^{57} -10 i q^{59} + 12 i q^{61} + ( 1 + i ) q^{63} + ( -9 - 9 i ) q^{67} + 6 q^{69} -6 i q^{71} + ( -5 + 5 i ) q^{73} + ( -4 + 4 i ) q^{77} + 5 q^{81} + ( -3 + 3 i ) q^{83} + ( 2 - 2 i ) q^{87} + 6 q^{91} + ( -6 - 6 i ) q^{93} + ( 7 + 7 i ) q^{97} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{7} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{7} - 8q^{11} + 6q^{13} + 6q^{17} + 6q^{23} + 8q^{27} + 4q^{29} - 8q^{33} + 6q^{37} + 12q^{39} + 12q^{41} - 6q^{43} + 18q^{47} + 12q^{51} - 10q^{53} + 12q^{57} + 2q^{63} - 18q^{67} + 12q^{69} - 10q^{73} - 8q^{77} + 10q^{81} - 6q^{83} + 4q^{87} + 12q^{91} - 12q^{93} + 14q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
543.1
 − 1.00000i 1.00000i
0 1.00000 + 1.00000i 0 0 0 1.00000 + 1.00000i 0 1.00000i 0
607.1 0 1.00000 1.00000i 0 0 0 1.00000 1.00000i 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
40.k 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.o.d 2
4.b odd 2 1 1600.2.o.a 2
5.b even 2 1 320.2.o.a 2
5.c odd 4 1 320.2.o.b yes 2
5.c odd 4 1 1600.2.o.c 2
8.b even 2 1 1600.2.o.b 2
8.d odd 2 1 1600.2.o.c 2
20.d odd 2 1 320.2.o.c yes 2
20.e even 4 1 320.2.o.d yes 2
20.e even 4 1 1600.2.o.b 2
40.e odd 2 1 320.2.o.b yes 2
40.f even 2 1 320.2.o.d yes 2
40.i odd 4 1 320.2.o.c yes 2
40.i odd 4 1 1600.2.o.a 2
40.k even 4 1 320.2.o.a 2
40.k even 4 1 inner 1600.2.o.d 2
80.i odd 4 1 1280.2.n.d 2
80.j even 4 1 1280.2.n.c 2
80.k odd 4 1 1280.2.n.d 2
80.k odd 4 1 1280.2.n.i 2
80.q even 4 1 1280.2.n.c 2
80.q even 4 1 1280.2.n.j 2
80.s even 4 1 1280.2.n.j 2
80.t odd 4 1 1280.2.n.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.o.a 2 5.b even 2 1
320.2.o.a 2 40.k even 4 1
320.2.o.b yes 2 5.c odd 4 1
320.2.o.b yes 2 40.e odd 2 1
320.2.o.c yes 2 20.d odd 2 1
320.2.o.c yes 2 40.i odd 4 1
320.2.o.d yes 2 20.e even 4 1
320.2.o.d yes 2 40.f even 2 1
1280.2.n.c 2 80.j even 4 1
1280.2.n.c 2 80.q even 4 1
1280.2.n.d 2 80.i odd 4 1
1280.2.n.d 2 80.k odd 4 1
1280.2.n.i 2 80.k odd 4 1
1280.2.n.i 2 80.t odd 4 1
1280.2.n.j 2 80.q even 4 1
1280.2.n.j 2 80.s even 4 1
1600.2.o.a 2 4.b odd 2 1
1600.2.o.a 2 40.i odd 4 1
1600.2.o.b 2 8.b even 2 1
1600.2.o.b 2 20.e even 4 1
1600.2.o.c 2 5.c odd 4 1
1600.2.o.c 2 8.d odd 2 1
1600.2.o.d 2 1.a even 1 1 trivial
1600.2.o.d 2 40.k even 4 1 inner

## Hecke kernels

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

 $$T_{3}^{2} - 2 T_{3} + 2$$ $$T_{7}^{2} - 2 T_{7} + 2$$ $$T_{79}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$2 - 2 T + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$18 - 6 T + T^{2}$$
$17$ $$18 - 6 T + T^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$18 - 6 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$36 + T^{2}$$
$37$ $$18 - 6 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$18 + 6 T + T^{2}$$
$47$ $$162 - 18 T + T^{2}$$
$53$ $$50 + 10 T + T^{2}$$
$59$ $$100 + T^{2}$$
$61$ $$144 + T^{2}$$
$67$ $$162 + 18 T + T^{2}$$
$71$ $$36 + T^{2}$$
$73$ $$50 + 10 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$18 + 6 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$98 - 14 T + T^{2}$$