Embedded newform 3600.1.j.b.1999.1

• Label: 3600.1.j.b.1999.1
• Level: $3600$
• Weight: $1$
• Character: 3600.1999
• Analytic conductor: $1.797$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{6}$
• CM discriminant: -3
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3600.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.79663404548\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.2.4320000.2

Embedding invariants

Embedding label			1999.1
Root			\(0.866025 + 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	3600.1999
Dual form			3600.1.j.b.1999.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.73205 q^{7} -1.00000i q^{13} +1.73205i q^{19} +1.73205i q^{31} +2.00000i q^{37} +1.73205 q^{43} +2.00000 q^{49} -1.00000 q^{61} -1.73205 q^{67} +2.00000i q^{73} +1.73205i q^{91} +1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{49} - 4 q^{61}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(901\)	\(2801\)	\(3151\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	0	0
			\(7\)	−1.73205	−1.73205	−0.866025	−	0.500000i	\(-0.833333\pi\)
−0.866025	+	0.500000i				\(0.833333\pi\)
\(11\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(13\)	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	\(-0.833333\pi\)
			0.866025	−	0.500000i	\(-0.166667\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	1.73205i	1.73205i	0.500000	+	0.866025i	\(0.333333\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	1.73205i	1.73205i	0.500000	+	0.866025i	\(0.333333\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(37\)	2.00000i	2.00000i			1.00000i	\(0.5\pi\)
					1.00000i	\(0.5\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	1.73205	1.73205	0.866025	−	0.500000i	\(-0.166667\pi\)
			0.866025	+	0.500000i	\(0.166667\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(61\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(67\)	−1.73205	−1.73205	−0.866025	−	0.500000i	\(-0.833333\pi\)
			−0.866025	+	0.500000i	\(0.833333\pi\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	2.00000i	2.00000i			1.00000i	\(0.5\pi\)
					1.00000i	\(0.5\pi\)
\(79\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	1.00000i	1.00000i	0.866025	+	0.500000i	\(0.166667\pi\)
			−0.866025	+	0.500000i	\(0.833333\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.1.j.b.1999.1		4
3.2	odd	2	CM	3600.1.j.b.1999.1		4
4.3	odd	2	inner	3600.1.j.b.1999.3		4
5.2	odd	4		3600.1.e.c.3151.1	✓	2
5.3	odd	4		3600.1.e.d.3151.2	yes	2
5.4	even	2	inner	3600.1.j.b.1999.4		4
12.11	even	2	inner	3600.1.j.b.1999.3		4
15.2	even	4		3600.1.e.c.3151.1	✓	2
15.8	even	4		3600.1.e.d.3151.2	yes	2
15.14	odd	2	inner	3600.1.j.b.1999.4		4
20.3	even	4		3600.1.e.d.3151.1	yes	2
20.7	even	4		3600.1.e.c.3151.2	yes	2
20.19	odd	2	inner	3600.1.j.b.1999.2		4
60.23	odd	4		3600.1.e.d.3151.1	yes	2
60.47	odd	4		3600.1.e.c.3151.2	yes	2
60.59	even	2	inner	3600.1.j.b.1999.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3600.1.e.c.3151.1	✓	2	5.2	odd	4
3600.1.e.c.3151.1	✓	2	15.2	even	4
3600.1.e.c.3151.2	yes	2	20.7	even	4
3600.1.e.c.3151.2	yes	2	60.47	odd	4
3600.1.e.d.3151.1	yes	2	20.3	even	4
3600.1.e.d.3151.1	yes	2	60.23	odd	4
3600.1.e.d.3151.2	yes	2	5.3	odd	4
3600.1.e.d.3151.2	yes	2	15.8	even	4
3600.1.j.b.1999.1		4	1.1	even	1	trivial
3600.1.j.b.1999.1		4	3.2	odd	2	CM
3600.1.j.b.1999.2		4	20.19	odd	2	inner
3600.1.j.b.1999.2		4	60.59	even	2	inner
3600.1.j.b.1999.3		4	4.3	odd	2	inner
3600.1.j.b.1999.3		4	12.11	even	2	inner
3600.1.j.b.1999.4		4	5.4	even	2	inner
3600.1.j.b.1999.4		4	15.14	odd	2	inner