Newform orbit 1805.1.o.b

• Label: 1805.1.o.b
• Level: $1805$
• Weight: $1$
• Character orbit: 1805.o
• Analytic conductor: $0.901$
• Analytic rank: $0$
• Dimension: $12$
• Projective image: $D_{4}$
• CM discriminant: -95
• Inner twists: $24$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1805 = 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1805.o (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.900812347803\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	12.0.101559956668416.2
Defining polynomial:	\( x^{12} + 8x^{6} + 64 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 95)
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.475.1
Artin image:	$C_9\times D_8$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{72} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b11 + b5) * q^2 + b5 * q^3 + (-b10 - b4) * q^4 - b2 * q^5 - 2*b4 * q^6 + b10 * q^9 + b1 * q^10 + b3 * q^12 + (b7 + b1) * q^13 - b7 * q^15 + (b8 + b2) * q^16 - b9 * q^18 - q^20 + b4 * q^25 + (-2*b6 - 2) * q^26 + 2*b6 * q^30 + (-b7 - b1) * q^32 + b8 * q^36 + b9 * q^37 - 2 * q^39 + (b6 + 1) * q^45 - b1 * q^48 + b6 * q^49 - b3 * q^50 - b11 * q^52 + b7 * q^53 - b5 * q^60 + (b6 + 1) * q^64 + (-b9 - b3) * q^65 - b1 * q^67 - 2*b8 * q^74 + b9 * q^75 + (-2*b11 - 2*b5) * q^78 + (-b10 - b4) * q^80 - b2 * q^81 + b11 * q^90 + 2 * q^96 + (-b11 - b5) * q^97 - b5 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{20} - 12 q^{26} - 12 q^{30} - 24 q^{39} + 6 q^{45} - 6 q^{49} + 6 q^{64} + 24 q^{96}+O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 8x^{6} + 64 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 8 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{8} ) / 16 \)
\(\beta_{9}\)	\(=\)	\( ( \nu^{9} ) / 16 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{10} ) / 32 \)
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} ) / 32 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\)	\(362\)	\(1446\)
\(\chi(n)\)	\(-1\)	\(\beta_{4} + \beta_{10}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

299.1

1.32893	+	0.483690i
−1.32893	−	0.483690i
0.245576	−	1.39273i
−0.245576	+	1.39273i
1.08335	+	0.909039i
−1.08335	−	0.909039i
1.32893	−	0.483690i
−1.32893	+	0.483690i
1.08335	−	0.909039i
−1.08335	+	0.909039i
0.245576	+	1.39273i
−0.245576	−	1.39273i

−1.32893

+

0.483690i

−0.245576

+

1.39273i

0.766044

−

0.642788i

−0.766044

−

0.642788i

−0.347296

−

1.96962i

0

0

−0.939693

−

0.342020i

1.32893

+

0.483690i

299.2

1.32893

−

0.483690i

0.245576

−

1.39273i

0.766044

−

0.642788i

−0.766044

−

0.642788i

−0.347296

−

1.96962i

0

0

−0.939693

−

0.342020i

−1.32893

−

0.483690i

694.1

−0.245576

−

1.39273i

1.08335

−

0.909039i

−0.939693

+

0.342020i

0.939693

+

0.342020i

−1.53209

−

1.28558i

0

0

0.173648

−

0.984808i

0.245576

−

1.39273i

694.2

0.245576

+

1.39273i

−1.08335

+

0.909039i

−0.939693

+

0.342020i

0.939693

+

0.342020i

−1.53209

−

1.28558i

0

0

0.173648

−

0.984808i

−0.245576

+

1.39273i

849.1

−1.08335

+

0.909039i

−1.32893

−

0.483690i

0.173648

−

0.984808i

−0.173648

−

0.984808i

1.87939

−

0.684040i

0

0

0.766044

+

0.642788i

1.08335

+

0.909039i

849.2

1.08335

−

0.909039i

1.32893

+

0.483690i

0.173648

−

0.984808i

−0.173648

−

0.984808i

1.87939

−

0.684040i

0

0

0.766044

+

0.642788i

−1.08335

−

0.909039i

984.1

−1.32893

−

0.483690i

−0.245576

−

1.39273i

0.766044

+

0.642788i

−0.766044

+

0.642788i

−0.347296

+

1.96962i

0

0

−0.939693

+

0.342020i

1.32893

−

0.483690i

984.2

1.32893

+

0.483690i

0.245576

+

1.39273i

0.766044

+

0.642788i

−0.766044

+

0.642788i

−0.347296

+

1.96962i

0

0

−0.939693

+

0.342020i

−1.32893

+

0.483690i

1029.1

−1.08335

−

0.909039i

−1.32893

+

0.483690i

0.173648

+

0.984808i

−0.173648

+

0.984808i

1.87939

+

0.684040i

0

0

0.766044

−

0.642788i

1.08335

−

0.909039i

1029.2

1.08335

+

0.909039i

1.32893

−

0.483690i

0.173648

+

0.984808i

−0.173648

+

0.984808i

1.87939

+

0.684040i

0

0

0.766044

−

0.642788i

−1.08335

+

0.909039i

1199.1

−0.245576

+

1.39273i

1.08335

+

0.909039i

−0.939693

−

0.342020i

0.939693

−

0.342020i

−1.53209

+

1.28558i

0

0

0.173648

+

0.984808i

0.245576

+

1.39273i

1199.2

0.245576

−

1.39273i

−1.08335

−

0.909039i

−0.939693

−

0.342020i

0.939693

−

0.342020i

−1.53209

+

1.28558i

0

0

0.173648

+

0.984808i

−0.245576

−

1.39273i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
95.d	odd	2	1	CM by \(\Q(\sqrt{-95}) \)
5.b	even	2	1	inner
19.b	odd	2	1	inner
19.c	even	3	2	inner
19.d	odd	6	2	inner
19.e	even	9	3	inner
19.f	odd	18	3	inner
95.h	odd	6	2	inner
95.i	even	6	2	inner
95.o	odd	18	3	inner
95.p	even	18	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1805.1.o.b		12
5.b	even	2	1	inner	1805.1.o.b		12
19.b	odd	2	1	inner	1805.1.o.b		12
19.c	even	3	2	inner	1805.1.o.b		12
19.d	odd	6	2	inner	1805.1.o.b		12
19.e	even	9	1		95.1.d.b	✓	2
19.e	even	9	2		1805.1.h.b		4
19.e	even	9	3	inner	1805.1.o.b		12
19.f	odd	18	1		95.1.d.b	✓	2
19.f	odd	18	2		1805.1.h.b		4
19.f	odd	18	3	inner	1805.1.o.b		12
57.j	even	18	1		855.1.g.c		2
57.l	odd	18	1		855.1.g.c		2
76.k	even	18	1		1520.1.m.b		2
76.l	odd	18	1		1520.1.m.b		2
95.d	odd	2	1	CM	1805.1.o.b		12
95.h	odd	6	2	inner	1805.1.o.b		12
95.i	even	6	2	inner	1805.1.o.b		12
95.o	odd	18	1		95.1.d.b	✓	2
95.o	odd	18	2		1805.1.h.b		4
95.o	odd	18	3	inner	1805.1.o.b		12
95.p	even	18	1		95.1.d.b	✓	2
95.p	even	18	2		1805.1.h.b		4
95.p	even	18	3	inner	1805.1.o.b		12
95.q	odd	36	2		475.1.c.b		2
95.r	even	36	2		475.1.c.b		2
285.bd	odd	18	1		855.1.g.c		2
285.bf	even	18	1		855.1.g.c		2
380.ba	odd	18	1		1520.1.m.b		2
380.bb	even	18	1		1520.1.m.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.1.d.b	✓	2	19.e	even	9	1
95.1.d.b	✓	2	19.f	odd	18	1
95.1.d.b	✓	2	95.o	odd	18	1
95.1.d.b	✓	2	95.p	even	18	1
475.1.c.b		2	95.q	odd	36	2
475.1.c.b		2	95.r	even	36	2
855.1.g.c		2	57.j	even	18	1
855.1.g.c		2	57.l	odd	18	1
855.1.g.c		2	285.bd	odd	18	1
855.1.g.c		2	285.bf	even	18	1
1520.1.m.b		2	76.k	even	18	1
1520.1.m.b		2	76.l	odd	18	1
1520.1.m.b		2	380.ba	odd	18	1
1520.1.m.b		2	380.bb	even	18	1
1805.1.h.b		4	19.e	even	9	2
1805.1.h.b		4	19.f	odd	18	2
1805.1.h.b		4	95.o	odd	18	2
1805.1.h.b		4	95.p	even	18	2
1805.1.o.b		12	1.a	even	1	1	trivial
1805.1.o.b		12	5.b	even	2	1	inner
1805.1.o.b		12	19.b	odd	2	1	inner
1805.1.o.b		12	19.c	even	3	2	inner
1805.1.o.b		12	19.d	odd	6	2	inner
1805.1.o.b		12	19.e	even	9	3	inner
1805.1.o.b		12	19.f	odd	18	3	inner
1805.1.o.b		12	95.d	odd	2	1	CM
1805.1.o.b		12	95.h	odd	6	2	inner
1805.1.o.b		12	95.i	even	6	2	inner
1805.1.o.b		12	95.o	odd	18	3	inner
1805.1.o.b		12	95.p	even	18	3	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8T_{2}^{6} + 64 \) acting on \(S_{1}^{\mathrm{new}}(1805, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{12} + 8T^{6} + 64 \)
$3$	\( T^{12} + 8T^{6} + 64 \)
$5$	\( (T^{6} - T^{3} + 1)^{2} \)
$7$	\( T^{12} \)
$11$	\( T^{12} \)