# Properties

 Label 1200.1.l.a Level $1200$ Weight $1$ Character orbit 1200.l Self dual yes Analytic conductor $0.599$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1200.l (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 300) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.300.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.1440000.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{7} + q^{9} - q^{13} + q^{19} - q^{21} - q^{27} + q^{31} + 2q^{37} + q^{39} + q^{43} - q^{57} - q^{61} + q^{63} + q^{67} + 2q^{73} - 2q^{79} + q^{81} - q^{91} - q^{93} - q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 0
0 −1.00000 0 0 0 1.00000 0 1.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.1.l.a 1
3.b odd 2 1 CM 1200.1.l.a 1
4.b odd 2 1 300.1.g.b yes 1
5.b even 2 1 1200.1.l.b 1
5.c odd 4 2 1200.1.c.a 2
12.b even 2 1 300.1.g.b yes 1
15.d odd 2 1 1200.1.l.b 1
15.e even 4 2 1200.1.c.a 2
20.d odd 2 1 300.1.g.a 1
20.e even 4 2 300.1.b.a 2
60.h even 2 1 300.1.g.a 1
60.l odd 4 2 300.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.1.b.a 2 20.e even 4 2
300.1.b.a 2 60.l odd 4 2
300.1.g.a 1 20.d odd 2 1
300.1.g.a 1 60.h even 2 1
300.1.g.b yes 1 4.b odd 2 1
300.1.g.b yes 1 12.b even 2 1
1200.1.c.a 2 5.c odd 4 2
1200.1.c.a 2 15.e even 4 2
1200.1.l.a 1 1.a even 1 1 trivial
1200.1.l.a 1 3.b odd 2 1 CM
1200.1.l.b 1 5.b even 2 1
1200.1.l.b 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$1 + T$$
$17$ $$T$$
$19$ $$-1 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$-1 + T$$
$37$ $$-2 + T$$
$41$ $$T$$
$43$ $$-1 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$1 + T$$
$67$ $$-1 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$2 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$1 + T$$