Newform orbit 300.9.g.d

• Label: 300.9.g.d
• Level: $300$
• Weight: $9$
• Character orbit: 300.g
• Analytic conductor: $122.214$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(122.213583018\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-110}) \)
Defining polynomial:	\( x^{2} + 110 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2\cdot 3 \)
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{-110}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta + 51) q^{3} + 3094 q^{7} + (102 \beta - 1359) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 102 q^{3} + 6188 q^{7} - 2718 q^{9}+O(q^{10}) \)

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

	−	10.4881i
		10.4881i

0

51.0000

−

62.9285i

0

0

0

3094.00

0

−1359.00

−

6418.71i

0

101.2

0

51.0000

+

62.9285i

0

0

0

3094.00

0

−1359.00

+

6418.71i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.9.g.d		2
3.b	odd	2	1	inner	300.9.g.d		2
5.b	even	2	1		12.9.c.b	✓	2
5.c	odd	4	2		300.9.b.c		4
15.d	odd	2	1		12.9.c.b	✓	2
15.e	even	4	2		300.9.b.c		4
20.d	odd	2	1		48.9.e.c		2
40.e	odd	2	1		192.9.e.d		2
40.f	even	2	1		192.9.e.g		2
45.h	odd	6	2		324.9.g.f		4
45.j	even	6	2		324.9.g.f		4
60.h	even	2	1		48.9.e.c		2
120.i	odd	2	1		192.9.e.g		2
120.m	even	2	1		192.9.e.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.9.c.b	✓	2	5.b	even	2	1
12.9.c.b	✓	2	15.d	odd	2	1
48.9.e.c		2	20.d	odd	2	1
48.9.e.c		2	60.h	even	2	1
192.9.e.d		2	40.e	odd	2	1
192.9.e.d		2	120.m	even	2	1
192.9.e.g		2	40.f	even	2	1
192.9.e.g		2	120.i	odd	2	1
300.9.b.c		4	5.c	odd	4	2
300.9.b.c		4	15.e	even	4	2
300.9.g.d		2	1.a	even	1	1	trivial
300.9.g.d		2	3.b	odd	2	1	inner
324.9.g.f		4	45.h	odd	6	2
324.9.g.f		4	45.j	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 3094 \) acting on \(S_{9}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 102T + 6561 \)
$5$	\( T^{2} \)
$7$	\( (T - 3094)^{2} \)
$11$	\( T^{2} + 1283040 \)