Newform orbit 1587.1.h.c

• Label: 1587.1.h.c
• Level: $1587$
• Weight: $1$
• Character orbit: 1587.h
• Analytic conductor: $0.792$
• Analytic rank: $0$
• Dimension: $20$
• Projective image: $D_{4}$
• CM discriminant: -3
• Inner twists: $40$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1587 = 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1587.h (of order \(22\), degree \(10\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.792016175049\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{22})\)
Coefficient field:	20.0.5969915757478328440239161344.6
Defining polynomial:	\( x^{20} + 2x^{18} + 4x^{16} + 8x^{14} + 16x^{12} + 32x^{10} + 64x^{8} + 128x^{6} + 256x^{4} + 512x^{2} + 1024 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.0.36501.1
Artin image:	$C_{11}\times D_8$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{88} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b8 * q^3 + b2 * q^4 - b15 * q^7 + b16 * q^9 - b10 * q^12 + b4 * q^16 + b13 * q^19 + b1 * q^21 + b12 * q^25 - b2 * q^27 - b17 * q^28 + b18 * q^36 + b5 * q^37 + b19 * q^43 - b12 * q^48 + b8 * q^49 + (b19 + b17 + b15 + b13 + b11 + b9 + b7 + b5 + b3 + b1) * q^57 - b3 * q^61 - b9 * q^63 + b6 * q^64 - b1 * q^67 + (b18 + b16 + b14 + b12 + b10 + b8 + b6 + b4 + b2 + 1) * q^75 + b15 * q^76 - b7 * q^79 + b10 * q^81 + b3 * q^84 + b17 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} - 2 q^{4} - 2 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{25} + 2 q^{27} - 2 q^{36} + 2 q^{48} - 2 q^{49} - 2 q^{64} + 2 q^{75} - 2 q^{81}+O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 2x^{18} + 4x^{16} + 8x^{14} + 16x^{12} + 32x^{10} + 64x^{8} + 128x^{6} + 256x^{4} + 512x^{2} + 1024 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 8 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{8} ) / 16 \)
\(\beta_{9}\)	\(=\)	\( ( \nu^{9} ) / 16 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{10} ) / 32 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} ) / 32 \)
\(\beta_{12}\)	\(=\)	\( ( \nu^{12} ) / 64 \)
\(\beta_{13}\)	\(=\)	\( ( \nu^{13} ) / 64 \)
\(\beta_{14}\)	\(=\)	\( ( \nu^{14} ) / 128 \)
\(\beta_{15}\)	\(=\)	\( ( \nu^{15} ) / 128 \)
\(\beta_{16}\)	\(=\)	\( ( \nu^{16} ) / 256 \)
\(\beta_{17}\)	\(=\)	\( ( \nu^{17} ) / 256 \)
\(\beta_{18}\)	\(=\)	\( ( \nu^{18} ) / 512 \)
\(\beta_{19}\)	\(=\)	\( ( \nu^{19} ) / 512 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1587\mathbb{Z}\right)^\times\).

\(n\)	\(530\)	\(1063\)
\(\chi(n)\)	\(-1\)	\(\beta_{12}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

170.1

−1.35693	+	0.398430i
1.35693	−	0.398430i
0.926113	+	1.06879i
−0.926113	−	1.06879i
−0.201264	−	1.39982i
0.201264	+	1.39982i
−1.18971	+	0.764582i
1.18971	−	0.764582i
−0.587486	+	1.28641i
0.587486	−	1.28641i
−0.587486	−	1.28641i
0.587486	+	1.28641i
0.926113	−	1.06879i
−0.926113	+	1.06879i
−1.18971	−	0.764582i
1.18971	+	0.764582i
−0.201264	+	1.39982i
0.201264	−	1.39982i
−1.35693	−	0.398430i
1.35693	+	0.398430i

0

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170.2

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863.1

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0.142315

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995.2

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1016.1

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1313.1

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1313.2

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1457.2

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0

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1559.1

0

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1559.2

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Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
23.b	odd	2	1	inner
23.c	even	11	9	inner
23.d	odd	22	9	inner
69.c	even	2	1	inner
69.g	even	22	9	inner
69.h	odd	22	9	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1587.1.h.c		20
3.b	odd	2	1	CM	1587.1.h.c		20
23.b	odd	2	1	inner	1587.1.h.c		20
23.c	even	11	1		1587.1.b.b	✓	2
23.c	even	11	9	inner	1587.1.h.c		20
23.d	odd	22	1		1587.1.b.b	✓	2
23.d	odd	22	9	inner	1587.1.h.c		20
69.c	even	2	1	inner	1587.1.h.c		20
69.g	even	22	1		1587.1.b.b	✓	2
69.g	even	22	9	inner	1587.1.h.c		20
69.h	odd	22	1		1587.1.b.b	✓	2
69.h	odd	22	9	inner	1587.1.h.c		20

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1587.1.b.b	✓	2	23.c	even	11	1
1587.1.b.b	✓	2	23.d	odd	22	1
1587.1.b.b	✓	2	69.g	even	22	1
1587.1.b.b	✓	2	69.h	odd	22	1
1587.1.h.c		20	1.a	even	1	1	trivial
1587.1.h.c		20	3.b	odd	2	1	CM
1587.1.h.c		20	23.b	odd	2	1	inner
1587.1.h.c		20	23.c	even	11	9	inner
1587.1.h.c		20	23.d	odd	22	9	inner
1587.1.h.c		20	69.c	even	2	1	inner
1587.1.h.c		20	69.g	even	22	9	inner
1587.1.h.c		20	69.h	odd	22	9	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1587, [\chi])\):

\( T_{2} \)

T7^20 + 2*T7^18 + 4*T7^16 + 8*T7^14 + 16*T7^12 + 32*T7^10 + 64*T7^8 + 128*T7^6 + 256*T7^4 + 512*T7^2 + 1024

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{20} \)
$3$	(T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$5$	\( T^{20} \)
$7$	T^20 + 2*T^18 + 4*T^16 + 8*T^14 + 16*T^12 + 32*T^10 + 64*T^8 + 128*T^6 + 256*T^4 + 512*T^2 + 1024
$11$	\( T^{20} \)