# Properties

 Label 200.1.e.a Level 200 Weight 1 Character orbit 200.e Analytic conductor 0.100 Analytic rank 0 Dimension 2 Projective image $$D_{3}$$ CM discriminant -8 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 200.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0998130025266$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{3}$$ Projective field Galois closure of 3.1.200.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + i q^{8} +O(q^{10})$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + i q^{8} - q^{11} + i q^{12} + q^{16} + i q^{17} + q^{19} + i q^{22} + q^{24} -i q^{27} -i q^{32} + i q^{33} + q^{34} -i q^{38} - q^{41} + 2 i q^{43} + q^{44} -i q^{48} - q^{49} + q^{51} - q^{54} -i q^{57} -2 q^{59} - q^{64} + q^{66} + i q^{67} -i q^{68} -i q^{73} - q^{76} - q^{81} + i q^{82} -i q^{83} + 2 q^{86} -i q^{88} + q^{89} - q^{96} -2 i q^{97} + i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{11} + 2q^{16} + 2q^{19} + 2q^{24} + 2q^{34} - 2q^{41} + 2q^{44} - 2q^{49} + 2q^{51} - 2q^{54} - 4q^{59} - 2q^{64} + 2q^{66} - 2q^{76} - 2q^{81} + 4q^{86} + 2q^{89} - 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$177$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i 0 0
99.2 1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
5.b even 2 1 inner
40.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.1.e.a 2
3.b odd 2 1 1800.1.p.a 2
4.b odd 2 1 800.1.e.a 2
5.b even 2 1 inner 200.1.e.a 2
5.c odd 4 1 200.1.g.a 1
5.c odd 4 1 200.1.g.b yes 1
8.b even 2 1 800.1.e.a 2
8.d odd 2 1 CM 200.1.e.a 2
15.d odd 2 1 1800.1.p.a 2
15.e even 4 1 1800.1.g.a 1
15.e even 4 1 1800.1.g.b 1
20.d odd 2 1 800.1.e.a 2
20.e even 4 1 800.1.g.a 1
20.e even 4 1 800.1.g.b 1
24.f even 2 1 1800.1.p.a 2
40.e odd 2 1 inner 200.1.e.a 2
40.f even 2 1 800.1.e.a 2
40.i odd 4 1 800.1.g.a 1
40.i odd 4 1 800.1.g.b 1
40.k even 4 1 200.1.g.a 1
40.k even 4 1 200.1.g.b yes 1
120.m even 2 1 1800.1.p.a 2
120.q odd 4 1 1800.1.g.a 1
120.q odd 4 1 1800.1.g.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.1.e.a 2 1.a even 1 1 trivial
200.1.e.a 2 5.b even 2 1 inner
200.1.e.a 2 8.d odd 2 1 CM
200.1.e.a 2 40.e odd 2 1 inner
200.1.g.a 1 5.c odd 4 1
200.1.g.a 1 40.k even 4 1
200.1.g.b yes 1 5.c odd 4 1
200.1.g.b yes 1 40.k even 4 1
800.1.e.a 2 4.b odd 2 1
800.1.e.a 2 8.b even 2 1
800.1.e.a 2 20.d odd 2 1
800.1.e.a 2 40.f even 2 1
800.1.g.a 1 20.e even 4 1
800.1.g.a 1 40.i odd 4 1
800.1.g.b 1 20.e even 4 1
800.1.g.b 1 40.i odd 4 1
1800.1.g.a 1 15.e even 4 1
1800.1.g.a 1 120.q odd 4 1
1800.1.g.b 1 15.e even 4 1
1800.1.g.b 1 120.q odd 4 1
1800.1.p.a 2 3.b odd 2 1
1800.1.p.a 2 15.d odd 2 1
1800.1.p.a 2 24.f even 2 1
1800.1.p.a 2 120.m even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(200, [\chi])$$.