Newform orbit 30.3.b.a

• Label: 30.3.b.a
• Level: $30$
• Weight: $3$
• Character orbit: 30.b
• Analytic conductor: $0.817$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	30.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.817440793081\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-17})\)
Defining polynomial:	\( x^{4} + 16x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + (\beta_{3} - 1) q^{6} + (\beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 8) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 4 q^{6} - 32 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 16x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 7\nu ) / 9 \)
\(\beta_{3}\)	\(=\)	\( \nu^{2} + 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

29.1

0.707107	+	2.91548i
0.707107	−	2.91548i
−0.707107	+	2.91548i
−0.707107	−	2.91548i

−1.41421

0.707107

−

2.91548i

2.00000

2.82843

−

4.12311i

−1.00000

+

4.12311i

5.83095i

−2.82843

−8.00000

−

4.12311i

−4.00000

+

5.83095i

29.2

−1.41421

0.707107

+

2.91548i

2.00000

2.82843

+

4.12311i

−1.00000

−

4.12311i

−

5.83095i

−2.82843

−8.00000

+

4.12311i

−4.00000

−

5.83095i

29.3

1.41421

−0.707107

−

2.91548i

2.00000

−2.82843

+

4.12311i

−1.00000

−

4.12311i

5.83095i

2.82843

−8.00000

+

4.12311i

−4.00000

+

5.83095i

29.4

1.41421

−0.707107

+

2.91548i

2.00000

−2.82843

−

4.12311i

−1.00000

+

4.12311i

−

5.83095i

2.82843

−8.00000

−

4.12311i

−4.00000

−

5.83095i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
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	30.3.b.a	✓	4
3.b	odd	2	1	inner	30.3.b.a	✓	4
4.b	odd	2	1		240.3.c.c		4
5.b	even	2	1	inner	30.3.b.a	✓	4
5.c	odd	4	2		150.3.d.d		4
8.b	even	2	1		960.3.c.f		4
8.d	odd	2	1		960.3.c.e		4
9.c	even	3	2		810.3.j.c		8
9.d	odd	6	2		810.3.j.c		8
12.b	even	2	1		240.3.c.c		4
15.d	odd	2	1	inner	30.3.b.a	✓	4
15.e	even	4	2		150.3.d.d		4
20.d	odd	2	1		240.3.c.c		4
20.e	even	4	2		1200.3.l.t		4
24.f	even	2	1		960.3.c.e		4
24.h	odd	2	1		960.3.c.f		4
40.e	odd	2	1		960.3.c.e		4
40.f	even	2	1		960.3.c.f		4
45.h	odd	6	2		810.3.j.c		8
45.j	even	6	2		810.3.j.c		8
60.h	even	2	1		240.3.c.c		4
60.l	odd	4	2		1200.3.l.t		4
120.i	odd	2	1		960.3.c.f		4
120.m	even	2	1		960.3.c.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.3.b.a	✓	4	1.a	even	1	1	trivial
30.3.b.a	✓	4	3.b	odd	2	1	inner
30.3.b.a	✓	4	5.b	even	2	1	inner
30.3.b.a	✓	4	15.d	odd	2	1	inner
150.3.d.d		4	5.c	odd	4	2
150.3.d.d		4	15.e	even	4	2
240.3.c.c		4	4.b	odd	2	1
240.3.c.c		4	12.b	even	2	1
240.3.c.c		4	20.d	odd	2	1
240.3.c.c		4	60.h	even	2	1
810.3.j.c		8	9.c	even	3	2
810.3.j.c		8	9.d	odd	6	2
810.3.j.c		8	45.h	odd	6	2
810.3.j.c		8	45.j	even	6	2
960.3.c.e		4	8.d	odd	2	1
960.3.c.e		4	24.f	even	2	1
960.3.c.e		4	40.e	odd	2	1
960.3.c.e		4	120.m	even	2	1
960.3.c.f		4	8.b	even	2	1
960.3.c.f		4	24.h	odd	2	1
960.3.c.f		4	40.f	even	2	1
960.3.c.f		4	120.i	odd	2	1
1200.3.l.t		4	20.e	even	4	2
1200.3.l.t		4	60.l	odd	4	2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{2} \)
$3$	\( T^{4} + 16T^{2} + 81 \)
$5$	\( T^{4} + 18T^{2} + 625 \)
$7$	\( (T^{2} + 34)^{2} \)
$11$	\( (T^{2} + 272)^{2} \)