Embedded newform 5328.2.h.g.2737.3

• Label: 5328.2.h.g.2737.3
• Level: $5328$
• Weight: $2$
• Character: 5328.2737
• Analytic conductor: $42.544$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(42.5442941969\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial:	\( x^{4} + 6x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 444)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2737.3
Root			\(-0.874032i\) of defining polynomial
Character	\(\chi\)	\(=\)	5328.2737
Dual form			5328.2.h.g.2737.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.540182i q^{5} +1.23607 q^{7} +2.47214 q^{11} -4.57649i q^{13} +3.36861i q^{17} -1.74806i q^{19} -8.61280i q^{23} +4.70820 q^{25} +7.94510i q^{29} -6.32456i q^{31} +0.667701i q^{35} +(-2.23607 - 5.65685i) q^{37} -2.00000 q^{41} -2.82843i q^{43} +4.00000 q^{47} -5.47214 q^{49} -10.9443 q^{53} +1.33540i q^{55} -2.28825i q^{59} +5.65685i q^{61} +2.47214 q^{65} +7.70820 q^{67} -8.94427 q^{71} +3.23607 q^{73} +3.05573 q^{77} -9.56564i q^{79} +1.52786 q^{83} -1.81966 q^{85} -8.61280i q^{89} -5.65685i q^{91} +0.944272 q^{95} +13.7295i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 8 q^{11} - 8 q^{25} - 8 q^{41} + 16 q^{47} - 4 q^{49} - 8 q^{53} - 8 q^{65} + 4 q^{67} + 4 q^{73} + 48 q^{77} + 24 q^{83} - 52 q^{85} - 32 q^{95}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times\).

\(n\)	\(1297\)	\(1333\)	\(1999\)	\(2369\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)

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
\(5\)			0.540182i			0.241577i								0.992678	+	0.120788i	\(0.0385422\pi\)
							−0.992678	+	0.120788i	\(0.961458\pi\)
\(7\)	1.23607			0.467190			0.233595	−	0.972334i	\(-0.424951\pi\)
							0.233595	+	0.972334i	\(0.424951\pi\)
\(11\)	2.47214			0.745377			0.372689	−	0.927957i	\(-0.378436\pi\)
							0.372689	+	0.927957i	\(0.378436\pi\)
\(13\)		−	4.57649i		−	1.26929i	−0.772804	−	0.634645i	\(-0.781146\pi\)
							0.772804	−	0.634645i	\(-0.218854\pi\)
\(17\)			3.36861i			0.817008i	0.912757	+	0.408504i	\(0.133949\pi\)
							−0.912757	+	0.408504i	\(0.866051\pi\)
\(19\)		−	1.74806i		−	0.401033i	−0.979690	−	0.200517i	\(-0.935738\pi\)
							0.979690	−	0.200517i	\(-0.0642621\pi\)
\(23\)		−	8.61280i		−	1.79589i	−0.440104	−	0.897947i	\(-0.645059\pi\)
							0.440104	−	0.897947i	\(-0.354941\pi\)
\(29\)			7.94510i			1.47537i	0.675146	+	0.737684i	\(0.264081\pi\)
							−0.675146	+	0.737684i	\(0.735919\pi\)
\(31\)		−	6.32456i		−	1.13592i	−0.823055	−	0.567962i	\(-0.807732\pi\)
							0.823055	−	0.567962i	\(-0.192268\pi\)
\(37\)	−2.23607	−	5.65685i	−0.367607	−	0.929981i
							\(41\)	−2.00000			−0.312348			−0.156174	−	0.987730i	\(-0.549916\pi\)
−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)		−	2.82843i		−	0.431331i	−0.976467	−	0.215666i	\(-0.930808\pi\)
							0.976467	−	0.215666i	\(-0.0691921\pi\)
\(47\)	4.00000			0.583460			0.291730	−	0.956501i	\(-0.405769\pi\)
							0.291730	+	0.956501i	\(0.405769\pi\)
\(53\)	−10.9443			−1.50331			−0.751656	−	0.659556i	\(-0.770744\pi\)
							−0.751656	+	0.659556i	\(0.770744\pi\)
\(59\)		−	2.28825i		−	0.297904i	−0.988844	−	0.148952i	\(-0.952410\pi\)
							0.988844	−	0.148952i	\(-0.0475900\pi\)
\(61\)			5.65685i			0.724286i	0.932123	+	0.362143i	\(0.117955\pi\)
							−0.932123	+	0.362143i	\(0.882045\pi\)
\(67\)	7.70820			0.941707			0.470853	−	0.882211i	\(-0.343946\pi\)
							0.470853	+	0.882211i	\(0.343946\pi\)
\(71\)	−8.94427			−1.06149			−0.530745	−	0.847532i	\(-0.678088\pi\)
							−0.530745	+	0.847532i	\(0.678088\pi\)
\(73\)	3.23607			0.378753			0.189377	−	0.981905i	\(-0.439353\pi\)
							0.189377	+	0.981905i	\(0.439353\pi\)
\(79\)		−	9.56564i		−	1.07622i	−0.842875	−	0.538110i	\(-0.819139\pi\)
							0.842875	−	0.538110i	\(-0.180861\pi\)
\(83\)	1.52786			0.167705			0.0838524	−	0.996478i	\(-0.473278\pi\)
							0.0838524	+	0.996478i	\(0.473278\pi\)
\(89\)		−	8.61280i		−	0.912955i	−0.889735	−	0.456478i	\(-0.849111\pi\)
							0.889735	−	0.456478i	\(-0.150889\pi\)
\(97\)			13.7295i			1.39402i	0.717063	+	0.697008i	\(0.245486\pi\)
							−0.717063	+	0.697008i	\(0.754514\pi\)

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	5328.2.h.g.2737.3		4
3.2	odd	2		1776.2.h.f.961.2		4
4.3	odd	2		1332.2.e.e.73.3		4
12.11	even	2		444.2.e.a.73.2	✓	4
37.36	even	2	inner	5328.2.h.g.2737.2		4
111.110	odd	2		1776.2.h.f.961.3		4
148.147	odd	2		1332.2.e.e.73.2		4
444.443	even	2		444.2.e.a.73.3	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
444.2.e.a.73.2	✓	4	12.11	even	2
444.2.e.a.73.3	yes	4	444.443	even	2
1332.2.e.e.73.2		4	148.147	odd	2
1332.2.e.e.73.3		4	4.3	odd	2
1776.2.h.f.961.2		4	3.2	odd	2
1776.2.h.f.961.3		4	111.110	odd	2
5328.2.h.g.2737.2		4	37.36	even	2	inner
5328.2.h.g.2737.3		4	1.1	even	1	trivial