# Properties

 Label 1600.2.c.g Level $1600$ Weight $2$ Character orbit 1600.c 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.c (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 800) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{3} -2 i q^{7} + 2 q^{9} +O(q^{10})$$ $$q + i q^{3} -2 i q^{7} + 2 q^{9} -5 q^{11} + 5 i q^{17} + 5 q^{19} + 2 q^{21} -6 i q^{23} + 5 i q^{27} + 4 q^{29} + 10 q^{31} -5 i q^{33} + 10 i q^{37} + 5 q^{41} + 4 i q^{43} -8 i q^{47} + 3 q^{49} -5 q^{51} -10 i q^{53} + 5 i q^{57} + 10 q^{61} -4 i q^{63} -3 i q^{67} + 6 q^{69} + 5 i q^{73} + 10 i q^{77} -10 q^{79} + q^{81} -i q^{83} + 4 i q^{87} + 9 q^{89} + 10 i q^{93} + 10 i q^{97} -10 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{9} + O(q^{10})$$ $$2q + 4q^{9} - 10q^{11} + 10q^{19} + 4q^{21} + 8q^{29} + 20q^{31} + 10q^{41} + 6q^{49} - 10q^{51} + 20q^{61} + 12q^{69} - 20q^{79} + 2q^{81} + 18q^{89} - 20q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 2.00000i 0 2.00000 0
449.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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
5.b 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.c.g 2
4.b odd 2 1 1600.2.c.j 2
5.b even 2 1 inner 1600.2.c.g 2
5.c odd 4 1 1600.2.a.g 1
5.c odd 4 1 1600.2.a.r 1
8.b even 2 1 800.2.c.d 2
8.d odd 2 1 800.2.c.c 2
20.d odd 2 1 1600.2.c.j 2
20.e even 4 1 1600.2.a.h 1
20.e even 4 1 1600.2.a.s 1
24.f even 2 1 7200.2.f.bc 2
24.h odd 2 1 7200.2.f.a 2
40.e odd 2 1 800.2.c.c 2
40.f even 2 1 800.2.c.d 2
40.i odd 4 1 800.2.a.c yes 1
40.i odd 4 1 800.2.a.h yes 1
40.k even 4 1 800.2.a.b 1
40.k even 4 1 800.2.a.g yes 1
120.i odd 2 1 7200.2.f.a 2
120.m even 2 1 7200.2.f.bc 2
120.q odd 4 1 7200.2.a.o 1
120.q odd 4 1 7200.2.a.bq 1
120.w even 4 1 7200.2.a.k 1
120.w even 4 1 7200.2.a.bm 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.b 1 40.k even 4 1
800.2.a.c yes 1 40.i odd 4 1
800.2.a.g yes 1 40.k even 4 1
800.2.a.h yes 1 40.i odd 4 1
800.2.c.c 2 8.d odd 2 1
800.2.c.c 2 40.e odd 2 1
800.2.c.d 2 8.b even 2 1
800.2.c.d 2 40.f even 2 1
1600.2.a.g 1 5.c odd 4 1
1600.2.a.h 1 20.e even 4 1
1600.2.a.r 1 5.c odd 4 1
1600.2.a.s 1 20.e even 4 1
1600.2.c.g 2 1.a even 1 1 trivial
1600.2.c.g 2 5.b even 2 1 inner
1600.2.c.j 2 4.b odd 2 1
1600.2.c.j 2 20.d odd 2 1
7200.2.a.k 1 120.w even 4 1
7200.2.a.o 1 120.q odd 4 1
7200.2.a.bm 1 120.w even 4 1
7200.2.a.bq 1 120.q odd 4 1
7200.2.f.a 2 24.h odd 2 1
7200.2.f.a 2 120.i odd 2 1
7200.2.f.bc 2 24.f even 2 1
7200.2.f.bc 2 120.m even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ $$T_{7}^{2} + 4$$ $$T_{11} + 5$$ $$T_{19} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$( -5 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -5 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$9 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$25 + T^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$1 + T^{2}$$
$89$ $$( -9 + T )^{2}$$
$97$ $$100 + T^{2}$$