Embedded newform 2368.1.k.b.1153.1

• Label: 2368.1.k.b.1153.1
• Level: $2368$
• Weight: $1$
• Character: 2368.1153
• Analytic conductor: $1.182$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $S_{4}$
• CM/RM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2368.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.18178594991\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 148)
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.0.202612.1

Embedding invariants

Embedding label			1153.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	2368.1153
Dual form			2368.1.k.b.2177.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000 q^{7} -1.00000i q^{11} +(-1.00000 - 1.00000i) q^{17} +(1.00000 + 1.00000i) q^{19} +1.00000i q^{21} +(1.00000 + 1.00000i) q^{23} -1.00000i q^{25} +1.00000i q^{27} +(1.00000 - 1.00000i) q^{29} +1.00000 q^{33} -1.00000i q^{37} +1.00000i q^{41} -1.00000 q^{47} +(1.00000 - 1.00000i) q^{51} -1.00000 q^{53} +(-1.00000 + 1.00000i) q^{57} +(-1.00000 + 1.00000i) q^{69} -1.00000 q^{71} +1.00000i q^{73} +1.00000 q^{75} -1.00000i q^{77} +(1.00000 + 1.00000i) q^{79} -1.00000 q^{81} -1.00000 q^{83} +(1.00000 + 1.00000i) q^{87} +(-1.00000 + 1.00000i) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{17} + 2 q^{19} + 2 q^{23} + 2 q^{29} + 2 q^{33} - 2 q^{47} + 2 q^{51} - 2 q^{53} - 2 q^{57} - 2 q^{69} - 2 q^{71} + 2 q^{75} + 2 q^{79} - 2 q^{81} - 2 q^{83} + 2 q^{87} - 2 q^{89}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times\).

\(n\)	\(705\)	\(1407\)	\(1925\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(1\)

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
							\(3\)			1.00000i			1.00000i	0.866025	+	0.500000i	\(0.166667\pi\)
−0.866025	+	0.500000i	\(0.833333\pi\)
\(5\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(7\)	1.00000			1.00000			0.500000	−	0.866025i	\(-0.333333\pi\)
							0.500000	+	0.866025i	\(0.333333\pi\)
\(11\)		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	−	0.500000i	\(-0.166667\pi\)
\(13\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(17\)	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	\(-0.5\pi\)
							−1.00000			\(\pi\)
\(19\)	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			\(0\)
									1.00000i	\(0.5\pi\)
\(23\)	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			\(0\)
									1.00000i	\(0.5\pi\)
\(29\)	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	\(-0.5\pi\)
							1.00000			\(0\)
\(31\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(37\)		−	1.00000i		−	1.00000i
							\(41\)			1.00000i			1.00000i	0.866025	+	0.500000i	\(0.166667\pi\)
−0.866025	+	0.500000i	\(0.833333\pi\)
\(43\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(47\)	−1.00000			−1.00000			−0.500000	−	0.866025i	\(-0.666667\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(53\)	−1.00000			−1.00000			−0.500000	−	0.866025i	\(-0.666667\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(59\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(61\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(67\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(71\)	−1.00000			−1.00000			−0.500000	−	0.866025i	\(-0.666667\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(73\)			1.00000i			1.00000i	0.866025	+	0.500000i	\(0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(79\)	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			\(0\)
									1.00000i	\(0.5\pi\)
\(83\)	−1.00000			−1.00000			−0.500000	−	0.866025i	\(-0.666667\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(89\)	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	\(0.5\pi\)
							−1.00000			\(\pi\)
\(97\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)

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	2368.1.k.b.1153.1		2
4.3	odd	2		2368.1.k.a.1153.1		2
8.3	odd	2		148.1.f.a.117.1	yes	2
8.5	even	2		592.1.k.b.561.1		2
24.11	even	2		1332.1.o.a.1153.1		2
37.31	odd	4	inner	2368.1.k.b.2177.1		2
40.3	even	4		3700.1.t.a.1449.1		2
40.19	odd	2		3700.1.j.c.1301.1		2
40.27	even	4		3700.1.t.b.1449.1		2
148.31	even	4		2368.1.k.a.2177.1		2
296.179	even	4		148.1.f.a.105.1	✓	2
296.253	odd	4		592.1.k.b.401.1		2
888.179	odd	4		1332.1.o.a.253.1		2
1480.179	even	4		3700.1.j.c.401.1		2
1480.1067	odd	4		3700.1.t.a.549.1		2
1480.1363	odd	4		3700.1.t.b.549.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.1.f.a.105.1	✓	2	296.179	even	4
148.1.f.a.117.1	yes	2	8.3	odd	2
592.1.k.b.401.1		2	296.253	odd	4
592.1.k.b.561.1		2	8.5	even	2
1332.1.o.a.253.1		2	888.179	odd	4
1332.1.o.a.1153.1		2	24.11	even	2
2368.1.k.a.1153.1		2	4.3	odd	2
2368.1.k.a.2177.1		2	148.31	even	4
2368.1.k.b.1153.1		2	1.1	even	1	trivial
2368.1.k.b.2177.1		2	37.31	odd	4	inner
3700.1.j.c.401.1		2	1480.179	even	4
3700.1.j.c.1301.1		2	40.19	odd	2
3700.1.t.a.549.1		2	1480.1067	odd	4
3700.1.t.a.1449.1		2	40.3	even	4
3700.1.t.b.549.1		2	1480.1363	odd	4
3700.1.t.b.1449.1		2	40.27	even	4