Embedded newform 2880.2.o.a.2879.2

• Label: 2880.2.o.a.2879.2
• Level: $2880$
• Weight: $2$
• Character: 2880.2879
• Analytic conductor: $22.997$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -4
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.9969157821\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{8})\)
Defining polynomial:	\( x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2879.2
Root			\(-0.707107 + 0.707107i\) of defining polynomial
Character	\(\chi\)	\(=\)	2880.2879
Dual form			2880.2.o.a.2879.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 2.12132i) q^{5} +6.00000i q^{13} +7.07107 q^{17} +(-4.00000 - 3.00000i) q^{25} +4.24264i q^{29} -12.0000i q^{37} +12.7279i q^{41} -7.00000 q^{49} +7.07107 q^{53} -10.0000 q^{61} +(-12.7279 - 4.24264i) q^{65} +6.00000i q^{73} +(-5.00000 + 15.0000i) q^{85} +4.24264i q^{89} +18.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{25} - 28 q^{49} - 40 q^{61} - 20 q^{85}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(641\)	\(901\)	\(2431\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

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
\(5\)	−0.707107	+	2.12132i	−0.316228	+	0.948683i
							\(7\)		0			0			−	1.00000i	\(-0.5\pi\)
		1.00000i	\(0.5\pi\)
\(11\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(13\)			6.00000i			1.66410i	0.554700	+	0.832050i	\(0.312833\pi\)
							−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)	7.07107			1.71499			0.857493	−	0.514496i	\(-0.172021\pi\)
							0.857493	+	0.514496i	\(0.172021\pi\)
\(19\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(23\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(29\)			4.24264i			0.787839i	0.919145	+	0.393919i	\(0.128881\pi\)
							−0.919145	+	0.393919i	\(0.871119\pi\)
\(31\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(37\)		−	12.0000i		−	1.97279i	−0.164399	−	0.986394i	\(-0.552568\pi\)
							0.164399	−	0.986394i	\(-0.447432\pi\)
\(41\)			12.7279i			1.98777i	0.110432	+	0.993884i	\(0.464777\pi\)
							−0.110432	+	0.993884i	\(0.535223\pi\)
\(43\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(47\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(53\)	7.07107			0.971286			0.485643	−	0.874157i	\(-0.338586\pi\)
							0.485643	+	0.874157i	\(0.338586\pi\)
\(59\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(61\)	−10.0000			−1.28037			−0.640184	−	0.768221i	\(-0.721142\pi\)
							−0.640184	+	0.768221i	\(0.721142\pi\)
\(67\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)			6.00000i			0.702247i	0.936329	+	0.351123i	\(0.114200\pi\)
							−0.936329	+	0.351123i	\(0.885800\pi\)
\(79\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(83\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(89\)			4.24264i			0.449719i	0.974391	+	0.224860i	\(0.0721923\pi\)
							−0.974391	+	0.224860i	\(0.927808\pi\)
\(97\)			18.0000i			1.82762i	0.406138	+	0.913812i	\(0.366875\pi\)
							−0.406138	+	0.913812i	\(0.633125\pi\)

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	2880.2.o.a.2879.2		4
3.2	odd	2	inner	2880.2.o.a.2879.3		4
4.3	odd	2	CM	2880.2.o.a.2879.2		4
5.4	even	2	inner	2880.2.o.a.2879.4		4
8.3	odd	2		180.2.h.a.179.1	✓	4
8.5	even	2		180.2.h.a.179.1	✓	4
12.11	even	2	inner	2880.2.o.a.2879.3		4
15.14	odd	2	inner	2880.2.o.a.2879.1		4
20.19	odd	2	inner	2880.2.o.a.2879.4		4
24.5	odd	2		180.2.h.a.179.4	yes	4
24.11	even	2		180.2.h.a.179.4	yes	4
40.3	even	4		900.2.e.c.251.2		2
40.13	odd	4		900.2.e.c.251.2		2
40.19	odd	2		180.2.h.a.179.3	yes	4
40.27	even	4		900.2.e.a.251.1		2
40.29	even	2		180.2.h.a.179.3	yes	4
40.37	odd	4		900.2.e.a.251.1		2
60.59	even	2	inner	2880.2.o.a.2879.1		4
120.29	odd	2		180.2.h.a.179.2	yes	4
120.53	even	4		900.2.e.c.251.1		2
120.59	even	2		180.2.h.a.179.2	yes	4
120.77	even	4		900.2.e.a.251.2		2
120.83	odd	4		900.2.e.c.251.1		2
120.107	odd	4		900.2.e.a.251.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.h.a.179.1	✓	4	8.3	odd	2
180.2.h.a.179.1	✓	4	8.5	even	2
180.2.h.a.179.2	yes	4	120.29	odd	2
180.2.h.a.179.2	yes	4	120.59	even	2
180.2.h.a.179.3	yes	4	40.19	odd	2
180.2.h.a.179.3	yes	4	40.29	even	2
180.2.h.a.179.4	yes	4	24.5	odd	2
180.2.h.a.179.4	yes	4	24.11	even	2
900.2.e.a.251.1		2	40.27	even	4
900.2.e.a.251.1		2	40.37	odd	4
900.2.e.a.251.2		2	120.77	even	4
900.2.e.a.251.2		2	120.107	odd	4
900.2.e.c.251.1		2	120.53	even	4
900.2.e.c.251.1		2	120.83	odd	4
900.2.e.c.251.2		2	40.3	even	4
900.2.e.c.251.2		2	40.13	odd	4
2880.2.o.a.2879.1		4	15.14	odd	2	inner
2880.2.o.a.2879.1		4	60.59	even	2	inner
2880.2.o.a.2879.2		4	1.1	even	1	trivial
2880.2.o.a.2879.2		4	4.3	odd	2	CM
2880.2.o.a.2879.3		4	3.2	odd	2	inner
2880.2.o.a.2879.3		4	12.11	even	2	inner
2880.2.o.a.2879.4		4	5.4	even	2	inner
2880.2.o.a.2879.4		4	20.19	odd	2	inner