Embedded newform 2304.1.b.a.127.1

• Label: 2304.1.b.a.127.1
• Level: $2304$
• Weight: $1$
• Character: 2304.127
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -3, -4, 12
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2304.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 144)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\zeta_{12})\)
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.3057647616.1

Embedding invariants

Embedding label			127.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	2304.127
Dual form			2304.1.b.a.127.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.00000i q^{13} +1.00000 q^{25} -2.00000i q^{37} +1.00000 q^{49} +2.00000i q^{61} +2.00000 q^{73} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{25} + 2 q^{49} + 4 q^{73} - 4 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\)	\(1279\)	\(1793\)	\(2053\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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				1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	− 2.00000i	− 2.00000i		−	1.00000i	\(-0.5\pi\)
				−	1.00000i	\(-0.5\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	− 2.00000i	− 2.00000i		−	1.00000i	\(-0.5\pi\)
				−	1.00000i	\(-0.5\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	2.00000i	2.00000i			1.00000i	\(0.5\pi\)
					1.00000i	\(0.5\pi\)
\(67\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(79\)	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	\(0.5\pi\)
\(97\)	−2.00000	−2.00000	−1.00000			\(\pi\)
			−1.00000			\(\pi\)

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	2304.1.b.a.127.1		2
3.2	odd	2	CM	2304.1.b.a.127.1		2
4.3	odd	2	CM	2304.1.b.a.127.1		2
8.3	odd	2	inner	2304.1.b.a.127.2		2
8.5	even	2	inner	2304.1.b.a.127.2		2
12.11	even	2	RM	2304.1.b.a.127.1		2
16.3	odd	4		576.1.g.a.127.1		1
16.5	even	4		144.1.g.a.127.1	✓	1
16.11	odd	4		144.1.g.a.127.1	✓	1
16.13	even	4		576.1.g.a.127.1		1
24.5	odd	2	inner	2304.1.b.a.127.2		2
24.11	even	2	inner	2304.1.b.a.127.2		2
48.5	odd	4		144.1.g.a.127.1	✓	1
48.11	even	4		144.1.g.a.127.1	✓	1
48.29	odd	4		576.1.g.a.127.1		1
48.35	even	4		576.1.g.a.127.1		1
80.27	even	4		3600.1.j.a.1999.2		2
80.37	odd	4		3600.1.j.a.1999.2		2
80.43	even	4		3600.1.j.a.1999.1		2
80.53	odd	4		3600.1.j.a.1999.1		2
80.59	odd	4		3600.1.e.b.3151.1		1
80.69	even	4		3600.1.e.b.3151.1		1
144.5	odd	12		1296.1.o.b.703.1		2
144.11	even	12		1296.1.o.b.271.1		2
144.43	odd	12		1296.1.o.b.271.1		2
144.59	even	12		1296.1.o.b.703.1		2
144.85	even	12		1296.1.o.b.703.1		2
144.101	odd	12		1296.1.o.b.271.1		2
144.133	even	12		1296.1.o.b.271.1		2
144.139	odd	12		1296.1.o.b.703.1		2
240.53	even	4		3600.1.j.a.1999.1		2
240.59	even	4		3600.1.e.b.3151.1		1
240.107	odd	4		3600.1.j.a.1999.2		2
240.149	odd	4		3600.1.e.b.3151.1		1
240.197	even	4		3600.1.j.a.1999.2		2
240.203	odd	4		3600.1.j.a.1999.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.1.g.a.127.1	✓	1	16.5	even	4
144.1.g.a.127.1	✓	1	16.11	odd	4
144.1.g.a.127.1	✓	1	48.5	odd	4
144.1.g.a.127.1	✓	1	48.11	even	4
576.1.g.a.127.1		1	16.3	odd	4
576.1.g.a.127.1		1	16.13	even	4
576.1.g.a.127.1		1	48.29	odd	4
576.1.g.a.127.1		1	48.35	even	4
1296.1.o.b.271.1		2	144.11	even	12
1296.1.o.b.271.1		2	144.43	odd	12
1296.1.o.b.271.1		2	144.101	odd	12
1296.1.o.b.271.1		2	144.133	even	12
1296.1.o.b.703.1		2	144.5	odd	12
1296.1.o.b.703.1		2	144.59	even	12
1296.1.o.b.703.1		2	144.85	even	12
1296.1.o.b.703.1		2	144.139	odd	12
2304.1.b.a.127.1		2	1.1	even	1	trivial
2304.1.b.a.127.1		2	3.2	odd	2	CM
2304.1.b.a.127.1		2	4.3	odd	2	CM
2304.1.b.a.127.1		2	12.11	even	2	RM
2304.1.b.a.127.2		2	8.3	odd	2	inner
2304.1.b.a.127.2		2	8.5	even	2	inner
2304.1.b.a.127.2		2	24.5	odd	2	inner
2304.1.b.a.127.2		2	24.11	even	2	inner
3600.1.e.b.3151.1		1	80.59	odd	4
3600.1.e.b.3151.1		1	80.69	even	4
3600.1.e.b.3151.1		1	240.59	even	4
3600.1.e.b.3151.1		1	240.149	odd	4
3600.1.j.a.1999.1		2	80.43	even	4
3600.1.j.a.1999.1		2	80.53	odd	4
3600.1.j.a.1999.1		2	240.53	even	4
3600.1.j.a.1999.1		2	240.203	odd	4
3600.1.j.a.1999.2		2	80.27	even	4
3600.1.j.a.1999.2		2	80.37	odd	4
3600.1.j.a.1999.2		2	240.107	odd	4
3600.1.j.a.1999.2		2	240.197	even	4