Embedded newform 5808.2.a.ck.1.4

• Label: 5808.2.a.ck.1.4
• Level: $5808$
• Weight: $2$
• Character: 5808.1
• Self dual: yes
• Analytic conductor: $46.377$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5808.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(46.3771134940\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial:	\( x^{4} - 7x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			\(2.52434\) of defining polynomial
Character	\(\chi\)	\(=\)	5808.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.37228 q^{5} +2.52434 q^{7} +1.00000 q^{9} +5.84096 q^{13} -3.37228 q^{15} -2.67181 q^{17} -0.939764 q^{19} -2.52434 q^{21} -2.00000 q^{23} +6.37228 q^{25} -1.00000 q^{27} +0.792287 q^{29} -1.62772 q^{31} +8.51278 q^{35} +5.00000 q^{37} -5.84096 q^{39} +10.8896 q^{41} -6.63325 q^{43} +3.37228 q^{45} +12.7446 q^{47} -0.627719 q^{49} +2.67181 q^{51} -4.11684 q^{53} +0.939764 q^{57} +6.00000 q^{59} -5.98844 q^{61} +2.52434 q^{63} +19.6974 q^{65} +1.11684 q^{67} +2.00000 q^{69} +10.7446 q^{71} -9.15759 q^{73} -6.37228 q^{75} +4.10891 q^{79} +1.00000 q^{81} -1.87953 q^{83} -9.01011 q^{85} -0.792287 q^{87} -0.627719 q^{89} +14.7446 q^{91} +1.62772 q^{93} -3.16915 q^{95} +10.4891 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 4 * q^3 + 2 * q^5 + 4 * q^9 - 2 * q^15 - 8 * q^23 + 14 * q^25 - 4 * q^27 - 18 * q^31 + 20 * q^37 + 2 * q^45 + 28 * q^47 - 14 * q^49 + 18 * q^53 + 24 * q^59 - 30 * q^67 + 8 * q^69 + 20 * q^71 - 14 * q^75 + 4 * q^81 - 14 * q^89 + 36 * q^91 + 18 * q^93 - 4 * q^97

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
			\(3\)	−1.00000	−0.577350
\(5\)	3.37228	1.50813				0.754065	−	0.656800i	\(-0.228090\pi\)
			0.754065	+	0.656800i	\(0.228090\pi\)
\(7\)	2.52434	0.954110	0.477055	−	0.878873i	\(-0.341704\pi\)
			0.477055	+	0.878873i	\(0.341704\pi\)
\(11\)	0	0
			\(13\)	5.84096	1.61999	0.809996	−	0.586436i	\(-0.199469\pi\)
0.809996	+	0.586436i				\(0.199469\pi\)
\(17\)	−2.67181	−0.648010	−0.324005	−	0.946055i	\(-0.605030\pi\)
			−0.324005	+	0.946055i	\(0.605030\pi\)
\(19\)	−0.939764	−0.215597	−0.107798	−	0.994173i	\(-0.534380\pi\)
			−0.107798	+	0.994173i	\(0.534380\pi\)
\(23\)	−2.00000	−0.417029	−0.208514	−	0.978019i	\(-0.566863\pi\)
			−0.208514	+	0.978019i	\(0.566863\pi\)
\(29\)	0.792287	0.147124	0.0735620	−	0.997291i	\(-0.476563\pi\)
			0.0735620	+	0.997291i	\(0.476563\pi\)
\(31\)	−1.62772	−0.292347	−0.146173	−	0.989259i	\(-0.546696\pi\)
			−0.146173	+	0.989259i	\(0.546696\pi\)
\(37\)	5.00000	0.821995	0.410997	−	0.911636i	\(-0.365181\pi\)
			0.410997	+	0.911636i	\(0.365181\pi\)
\(41\)	10.8896	1.70068	0.850338	−	0.526237i	\(-0.176398\pi\)
			0.850338	+	0.526237i	\(0.176398\pi\)
\(43\)	−6.63325	−1.01156	−0.505781	−	0.862662i	\(-0.668795\pi\)
			−0.505781	+	0.862662i	\(0.668795\pi\)
\(47\)	12.7446	1.85899	0.929493	−	0.368840i	\(-0.120245\pi\)
			0.929493	+	0.368840i	\(0.120245\pi\)
\(53\)	−4.11684	−0.565492	−0.282746	−	0.959195i	\(-0.591245\pi\)
			−0.282746	+	0.959195i	\(0.591245\pi\)
\(59\)	6.00000	0.781133	0.390567	−	0.920575i	\(-0.372279\pi\)
			0.390567	+	0.920575i	\(0.372279\pi\)
\(61\)	−5.98844	−0.766741	−0.383371	−	0.923595i	\(-0.625237\pi\)
			−0.383371	+	0.923595i	\(0.625237\pi\)
\(67\)	1.11684	0.136444	0.0682221	−	0.997670i	\(-0.478267\pi\)
			0.0682221	+	0.997670i	\(0.478267\pi\)
\(71\)	10.7446	1.27514	0.637572	−	0.770390i	\(-0.279939\pi\)
			0.637572	+	0.770390i	\(0.279939\pi\)
\(73\)	−9.15759	−1.07181	−0.535907	−	0.844277i	\(-0.680030\pi\)
			−0.535907	+	0.844277i	\(0.680030\pi\)
\(79\)	4.10891	0.462289	0.231144	−	0.972919i	\(-0.425753\pi\)
			0.231144	+	0.972919i	\(0.425753\pi\)
\(83\)	−1.87953	−0.206305	−0.103152	−	0.994666i	\(-0.532893\pi\)
			−0.103152	+	0.994666i	\(0.532893\pi\)
\(89\)	−0.627719	−0.0665380	−0.0332690	−	0.999446i	\(-0.510592\pi\)
			−0.0332690	+	0.999446i	\(0.510592\pi\)
\(97\)	10.4891	1.06501	0.532505	−	0.846427i	\(-0.321251\pi\)
			0.532505	+	0.846427i	\(0.321251\pi\)

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	5808.2.a.ck.1.4		4
4.3	odd	2		363.2.a.j.1.2	✓	4
11.10	odd	2	inner	5808.2.a.ck.1.3		4
12.11	even	2		1089.2.a.u.1.3		4
20.19	odd	2		9075.2.a.cv.1.3		4
44.3	odd	10		363.2.e.n.130.2		16
44.7	even	10		363.2.e.n.148.3		16
44.15	odd	10		363.2.e.n.148.2		16
44.19	even	10		363.2.e.n.130.3		16
44.27	odd	10		363.2.e.n.124.3		16
44.31	odd	10		363.2.e.n.202.3		16
44.35	even	10		363.2.e.n.202.2		16
44.39	even	10		363.2.e.n.124.2		16
44.43	even	2		363.2.a.j.1.3	yes	4
132.131	odd	2		1089.2.a.u.1.2		4
220.219	even	2		9075.2.a.cv.1.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.j.1.2	✓	4	4.3	odd	2
363.2.a.j.1.3	yes	4	44.43	even	2
363.2.e.n.124.2		16	44.39	even	10
363.2.e.n.124.3		16	44.27	odd	10
363.2.e.n.130.2		16	44.3	odd	10
363.2.e.n.130.3		16	44.19	even	10
363.2.e.n.148.2		16	44.15	odd	10
363.2.e.n.148.3		16	44.7	even	10
363.2.e.n.202.2		16	44.35	even	10
363.2.e.n.202.3		16	44.31	odd	10
1089.2.a.u.1.2		4	132.131	odd	2
1089.2.a.u.1.3		4	12.11	even	2
5808.2.a.ck.1.3		4	11.10	odd	2	inner
5808.2.a.ck.1.4		4	1.1	even	1	trivial
9075.2.a.cv.1.2		4	220.219	even	2
9075.2.a.cv.1.3		4	20.19	odd	2