# Properties

 Label 300.1.b.a Level $300$ Weight $1$ Character orbit 300.b Analytic conductor $0.150$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 300.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.149719503790$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{3}$$ Projective field Galois closure of 3.1.300.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{3} -i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} -i q^{7} - q^{9} + i q^{13} + q^{19} - q^{21} + i q^{27} - q^{31} + 2 i q^{37} + q^{39} + i q^{43} -i q^{57} - q^{61} + i q^{63} -i q^{67} -2 i q^{73} -2 q^{79} + q^{81} + q^{91} + i q^{93} -i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 2q^{19} - 2q^{21} - 2q^{31} + 2q^{39} - 2q^{61} - 4q^{79} + 2q^{81} + 2q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
149.2 0 1.00000i 0 0 0 1.00000i 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 $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

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