# Properties

 Label 1600.2.c.b 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} -2 i q^{7} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} -2 i q^{7} -6 q^{9} + q^{11} + 4 i q^{13} + 5 i q^{17} - q^{19} + 6 q^{21} -2 i q^{23} -9 i q^{27} -8 q^{29} -10 q^{31} + 3 i q^{33} + 6 i q^{37} -12 q^{39} -3 q^{41} -4 i q^{43} -4 i q^{47} + 3 q^{49} -15 q^{51} + 6 i q^{53} -3 i q^{57} -8 q^{59} -10 q^{61} + 12 i q^{63} -i q^{67} + 6 q^{69} + 12 q^{71} -3 i q^{73} -2 i q^{77} + 6 q^{79} + 9 q^{81} + 13 i q^{83} -24 i q^{87} + 9 q^{89} + 8 q^{91} -30 i q^{93} -14 i q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 12q^{9} + O(q^{10})$$ $$2q - 12q^{9} + 2q^{11} - 2q^{19} + 12q^{21} - 16q^{29} - 20q^{31} - 24q^{39} - 6q^{41} + 6q^{49} - 30q^{51} - 16q^{59} - 20q^{61} + 12q^{69} + 24q^{71} + 12q^{79} + 18q^{81} + 18q^{89} + 16q^{91} - 12q^{99} + O(q^{100})$$

## 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 3.00000i 0 0 0 2.00000i 0 −6.00000 0
449.2 0 3.00000i 0 0 0 2.00000i 0 −6.00000 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.b 2
4.b odd 2 1 1600.2.c.a 2
5.b even 2 1 inner 1600.2.c.b 2
5.c odd 4 1 1600.2.a.b 1
5.c odd 4 1 1600.2.a.y 1
8.b even 2 1 400.2.c.a 2
8.d odd 2 1 200.2.c.a 2
20.d odd 2 1 1600.2.c.a 2
20.e even 4 1 1600.2.a.a 1
20.e even 4 1 1600.2.a.x 1
24.f even 2 1 1800.2.f.f 2
24.h odd 2 1 3600.2.f.n 2
40.e odd 2 1 200.2.c.a 2
40.f even 2 1 400.2.c.a 2
40.i odd 4 1 400.2.a.a 1
40.i odd 4 1 400.2.a.h 1
40.k even 4 1 200.2.a.a 1
40.k even 4 1 200.2.a.e yes 1
120.i odd 2 1 3600.2.f.n 2
120.m even 2 1 1800.2.f.f 2
120.q odd 4 1 1800.2.a.h 1
120.q odd 4 1 1800.2.a.r 1
120.w even 4 1 3600.2.a.m 1
120.w even 4 1 3600.2.a.bf 1
280.y odd 4 1 9800.2.a.c 1
280.y odd 4 1 9800.2.a.bq 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.a.a 1 40.k even 4 1
200.2.a.e yes 1 40.k even 4 1
200.2.c.a 2 8.d odd 2 1
200.2.c.a 2 40.e odd 2 1
400.2.a.a 1 40.i odd 4 1
400.2.a.h 1 40.i odd 4 1
400.2.c.a 2 8.b even 2 1
400.2.c.a 2 40.f even 2 1
1600.2.a.a 1 20.e even 4 1
1600.2.a.b 1 5.c odd 4 1
1600.2.a.x 1 20.e even 4 1
1600.2.a.y 1 5.c odd 4 1
1600.2.c.a 2 4.b odd 2 1
1600.2.c.a 2 20.d odd 2 1
1600.2.c.b 2 1.a even 1 1 trivial
1600.2.c.b 2 5.b even 2 1 inner
1800.2.a.h 1 120.q odd 4 1
1800.2.a.r 1 120.q odd 4 1
1800.2.f.f 2 24.f even 2 1
1800.2.f.f 2 120.m even 2 1
3600.2.a.m 1 120.w even 4 1
3600.2.a.bf 1 120.w even 4 1
3600.2.f.n 2 24.h odd 2 1
3600.2.f.n 2 120.i odd 2 1
9800.2.a.c 1 280.y odd 4 1
9800.2.a.bq 1 280.y odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 9$$ $$T_{7}^{2} + 4$$ $$T_{11} - 1$$ $$T_{19} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$1 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$9 + T^{2}$$
$79$ $$( -6 + T )^{2}$$
$83$ $$169 + T^{2}$$
$89$ $$( -9 + T )^{2}$$
$97$ $$196 + T^{2}$$