Newform orbit 1089.1.s.b

• Label: 1089.1.s.b
• Level: $1089$
• Weight: $1$
• Character orbit: 1089.s
• Analytic conductor: $0.543$
• Analytic rank: $0$
• Dimension: $16$
• Projective image: $S_{4}$
• CM/RM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1089.s (of order \(30\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.543481798757\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{30})\)
Coefficient field:	16.0.26873856000000000000.1
Defining polynomial:	\( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.2.107811.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b1 * q^2 - b7 * q^3 + b2 * q^4 + (b15 + b3) * q^5 - b8 * q^6 + b6 * q^7 + (-b15 + b9 + b7 + b5 - b3 - b2 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{4} ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{5} ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{6} ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{8} ) / 16 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{9} ) / 16 \)
\(\beta_{9}\)	\(=\)	\( ( \nu^{10} ) / 32 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{11} ) / 32 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{12} ) / 64 \)
\(\beta_{12}\)	\(=\)	\( ( \nu^{13} ) / 64 \)
\(\beta_{13}\)	\(=\)	\( ( \nu^{15} ) / 128 \)
\(\beta_{14}\)	\(=\)	\( ( -\nu^{13} + 32\nu^{3} ) / 64 \)
\(\beta_{15}\)	\(=\)	\( ( \nu^{14} - 32\nu^{4} ) / 128 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\)	\(244\)	\(848\)
\(\chi(n)\)	\(-\beta_{11}\)	\(-1 + \beta_{9}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

40.1

−1.40647	−	0.147826i
1.40647	+	0.147826i
−1.05097	−	0.946294i
1.05097	+	0.946294i
−0.294032	−	1.38331i
0.294032	+	1.38331i
−0.575212	−	1.29195i
0.575212	+	1.29195i
−0.294032	+	1.38331i
0.294032	−	1.38331i
−1.05097	+	0.946294i
1.05097	−	0.946294i
−0.575212	+	1.29195i
0.575212	−	1.29195i
−1.40647	+	0.147826i
1.40647	−	0.147826i

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40.2

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94.1

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94.2

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112.1

−0.294032

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403.1

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403.2

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457.1

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475.2

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481.1

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1.41421i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
99.h	odd	6	1	inner
99.m	even	15	3	inner
99.o	odd	30	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1089.1.s.b		16
3.b	odd	2	1		3267.1.w.b		16
9.c	even	3	1	inner	1089.1.s.b		16
9.d	odd	6	1		3267.1.w.b		16
11.b	odd	2	1	inner	1089.1.s.b		16
11.c	even	5	1		1089.1.h.a	✓	4
11.c	even	5	3	inner	1089.1.s.b		16
11.d	odd	10	1		1089.1.h.a	✓	4
11.d	odd	10	3	inner	1089.1.s.b		16
33.d	even	2	1		3267.1.w.b		16
33.f	even	10	1		3267.1.h.a		4
33.f	even	10	3		3267.1.w.b		16
33.h	odd	10	1		3267.1.h.a		4
33.h	odd	10	3		3267.1.w.b		16
99.g	even	6	1		3267.1.w.b		16
99.h	odd	6	1	inner	1089.1.s.b		16
99.m	even	15	1		1089.1.h.a	✓	4
99.m	even	15	3	inner	1089.1.s.b		16
99.n	odd	30	1		3267.1.h.a		4
99.n	odd	30	3		3267.1.w.b		16
99.o	odd	30	1		1089.1.h.a	✓	4
99.o	odd	30	3	inner	1089.1.s.b		16
99.p	even	30	1		3267.1.h.a		4
99.p	even	30	3		3267.1.w.b		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1089.1.h.a	✓	4	11.c	even	5	1
1089.1.h.a	✓	4	11.d	odd	10	1
1089.1.h.a	✓	4	99.m	even	15	1
1089.1.h.a	✓	4	99.o	odd	30	1
1089.1.s.b		16	1.a	even	1	1	trivial
1089.1.s.b		16	9.c	even	3	1	inner
1089.1.s.b		16	11.b	odd	2	1	inner
1089.1.s.b		16	11.c	even	5	3	inner
1089.1.s.b		16	11.d	odd	10	3	inner
1089.1.s.b		16	99.h	odd	6	1	inner
1089.1.s.b		16	99.m	even	15	3	inner
1089.1.s.b		16	99.o	odd	30	3	inner
3267.1.h.a		4	33.f	even	10	1
3267.1.h.a		4	33.h	odd	10	1
3267.1.h.a		4	99.n	odd	30	1
3267.1.h.a		4	99.p	even	30	1
3267.1.w.b		16	3.b	odd	2	1
3267.1.w.b		16	9.d	odd	6	1
3267.1.w.b		16	33.d	even	2	1
3267.1.w.b		16	33.f	even	10	3
3267.1.w.b		16	33.h	odd	10	3
3267.1.w.b		16	99.g	even	6	1
3267.1.w.b		16	99.n	odd	30	3
3267.1.w.b		16	99.p	even	30	3

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 2T_{2}^{14} - 8T_{2}^{10} - 16T_{2}^{8} - 32T_{2}^{6} + 128T_{2}^{2} + 256 \) acting on \(S_{1}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^16 + 2*T^14 - 8*T^10 - 16*T^8 - 32*T^6 + 128*T^2 + 256
$3$	(T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^2
$5$	(T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^2
$7$	T^16 + 2*T^14 - 8*T^10 - 16*T^8 - 32*T^6 + 128*T^2 + 256
$11$	\( T^{16} \)