Newform orbit 300.3.g.c

• Label: 300.3.g.c
• Level: $300$
• Weight: $3$
• Character orbit: 300.g
• Self dual: yes
• Analytic conductor: $8.174$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.17440793081\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 3 q^{3} + 13 q^{7} + 9 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

101.1

0

0

3.00000

0

0

0

13.0000

0

9.00000

0

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	300.3.g.c	yes	1
3.b	odd	2	1	CM	300.3.g.c	yes	1
4.b	odd	2	1		1200.3.l.a		1
5.b	even	2	1		300.3.g.a	✓	1
5.c	odd	4	2		300.3.b.b		2
12.b	even	2	1		1200.3.l.a		1
15.d	odd	2	1		300.3.g.a	✓	1
15.e	even	4	2		300.3.b.b		2
20.d	odd	2	1		1200.3.l.e		1
20.e	even	4	2		1200.3.c.b		2
60.h	even	2	1		1200.3.l.e		1
60.l	odd	4	2		1200.3.c.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.3.b.b		2	5.c	odd	4	2
300.3.b.b		2	15.e	even	4	2
300.3.g.a	✓	1	5.b	even	2	1
300.3.g.a	✓	1	15.d	odd	2	1
300.3.g.c	yes	1	1.a	even	1	1	trivial
300.3.g.c	yes	1	3.b	odd	2	1	CM
1200.3.c.b		2	20.e	even	4	2
1200.3.c.b		2	60.l	odd	4	2
1200.3.l.a		1	4.b	odd	2	1
1200.3.l.a		1	12.b	even	2	1
1200.3.l.e		1	20.d	odd	2	1
1200.3.l.e		1	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\):

\( T_{7} - 13 \)

\( T_{11} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T - 3 \)
$5$	\( T \)
$7$	\( T - 13 \)
$11$	\( T \)