Embedded newform 5184.2.c.h.5183.4

• Label: 5184.2.c.h.5183.4
• Level: $5184$
• Weight: $2$
• Character: 5184.5183
• Analytic conductor: $41.394$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(41.3944484078\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{6} \)
Twist minimal:	no (minimal twist has level 1296)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5183.4
Root			\(-0.965926 - 0.258819i\) of defining polynomial
Character	\(\chi\)	\(=\)	5184.5183
Dual form			5184.2.c.h.5183.5

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.896575i q^{5} +1.26795i q^{7} -4.24264 q^{11} -1.00000 q^{13} -0.896575i q^{17} -4.73205i q^{19} -7.34847 q^{23} +4.19615 q^{25} +8.24504i q^{29} +6.00000i q^{31} +1.13681 q^{35} -1.19615 q^{37} +7.34847i q^{41} -8.19615i q^{43} +11.5911 q^{47} +5.39230 q^{49} -7.34847i q^{53} +3.80385i q^{55} +11.5911 q^{59} +3.19615 q^{61} +0.896575i q^{65} -10.7321i q^{67} -1.13681 q^{71} +9.19615 q^{73} -5.37945i q^{77} +4.73205i q^{79} -8.48528 q^{83} -0.803848 q^{85} -11.8313i q^{89} -1.26795i q^{91} -4.24264 q^{95} +14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{13} - 8 q^{25} + 32 q^{37} - 40 q^{49} - 16 q^{61} + 32 q^{73} - 48 q^{85} + 32 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(1217\)	\(2431\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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
\(5\)	− 0.896575i	− 0.400961i				−0.979698	−	0.200480i	\(-0.935750\pi\)
			0.979698	−	0.200480i	\(-0.0642503\pi\)
\(7\)	1.26795i	0.479240i	0.970867	+	0.239620i	\(0.0770228\pi\)
			−0.970867	+	0.239620i	\(0.922977\pi\)
\(11\)	−4.24264	−1.27920	−0.639602	−	0.768706i	\(-0.720901\pi\)
			−0.639602	+	0.768706i	\(0.720901\pi\)
\(13\)	−1.00000	−0.277350	−0.138675	−	0.990338i	\(-0.544284\pi\)
			−0.138675	+	0.990338i	\(0.544284\pi\)
\(17\)	− 0.896575i	− 0.217451i	−0.994072	−	0.108726i	\(-0.965323\pi\)
			0.994072	−	0.108726i	\(-0.0346770\pi\)
\(19\)	− 4.73205i	− 1.08561i	−0.839860	−	0.542803i	\(-0.817363\pi\)
			0.839860	−	0.542803i	\(-0.182637\pi\)
\(23\)	−7.34847	−1.53226	−0.766131	−	0.642685i	\(-0.777821\pi\)
			−0.766131	+	0.642685i	\(0.777821\pi\)
\(29\)	8.24504i	1.53107i	0.643396	+	0.765533i	\(0.277525\pi\)
			−0.643396	+	0.765533i	\(0.722475\pi\)
\(31\)	6.00000i	1.07763i	0.842424	+	0.538816i	\(0.181128\pi\)
			−0.842424	+	0.538816i	\(0.818872\pi\)
\(37\)	−1.19615	−0.196646	−0.0983231	−	0.995155i	\(-0.531348\pi\)
			−0.0983231	+	0.995155i	\(0.531348\pi\)
\(41\)	7.34847i	1.14764i	0.818982	+	0.573819i	\(0.194539\pi\)
			−0.818982	+	0.573819i	\(0.805461\pi\)
\(43\)	− 8.19615i	− 1.24990i	−0.780664	−	0.624951i	\(-0.785119\pi\)
			0.780664	−	0.624951i	\(-0.214881\pi\)
\(47\)	11.5911	1.69074	0.845369	−	0.534183i	\(-0.179381\pi\)
			0.845369	+	0.534183i	\(0.179381\pi\)
\(53\)	− 7.34847i	− 1.00939i	−0.863298	−	0.504695i	\(-0.831605\pi\)
			0.863298	−	0.504695i	\(-0.168395\pi\)
\(59\)	11.5911	1.50903	0.754517	−	0.656281i	\(-0.227871\pi\)
			0.754517	+	0.656281i	\(0.227871\pi\)
\(61\)	3.19615	0.409225	0.204613	−	0.978843i	\(-0.434407\pi\)
			0.204613	+	0.978843i	\(0.434407\pi\)
\(67\)	− 10.7321i	− 1.31113i	−0.755139	−	0.655564i	\(-0.772431\pi\)
			0.755139	−	0.655564i	\(-0.227569\pi\)
\(71\)	−1.13681	−0.134915	−0.0674574	−	0.997722i	\(-0.521489\pi\)
			−0.0674574	+	0.997722i	\(0.521489\pi\)
\(73\)	9.19615	1.07633	0.538164	−	0.842840i	\(-0.319118\pi\)
			0.538164	+	0.842840i	\(0.319118\pi\)
\(79\)	4.73205i	0.532397i	0.963918	+	0.266199i	\(0.0857677\pi\)
			−0.963918	+	0.266199i	\(0.914232\pi\)
\(83\)	−8.48528	−0.931381	−0.465690	−	0.884948i	\(-0.654194\pi\)
			−0.465690	+	0.884948i	\(0.654194\pi\)
\(89\)	− 11.8313i	− 1.25412i	−0.778971	−	0.627060i	\(-0.784258\pi\)
			0.778971	−	0.627060i	\(-0.215742\pi\)
\(97\)	14.3923	1.46132	0.730659	−	0.682743i	\(-0.239213\pi\)
			0.730659	+	0.682743i	\(0.239213\pi\)

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	5184.2.c.h.5183.4		8
3.2	odd	2	inner	5184.2.c.h.5183.6		8
4.3	odd	2	inner	5184.2.c.h.5183.3		8
8.3	odd	2		1296.2.c.g.1295.5	yes	8
8.5	even	2		1296.2.c.g.1295.6	yes	8
12.11	even	2	inner	5184.2.c.h.5183.5		8
24.5	odd	2		1296.2.c.g.1295.4	yes	8
24.11	even	2		1296.2.c.g.1295.3	✓	8
72.5	odd	6		1296.2.s.l.431.2		8
72.11	even	6		1296.2.s.l.863.3		8
72.13	even	6		1296.2.s.l.431.3		8
72.29	odd	6		1296.2.s.j.863.3		8
72.43	odd	6		1296.2.s.l.863.2		8
72.59	even	6		1296.2.s.j.431.2		8
72.61	even	6		1296.2.s.j.863.2		8
72.67	odd	6		1296.2.s.j.431.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.2.c.g.1295.3	✓	8	24.11	even	2
1296.2.c.g.1295.4	yes	8	24.5	odd	2
1296.2.c.g.1295.5	yes	8	8.3	odd	2
1296.2.c.g.1295.6	yes	8	8.5	even	2
1296.2.s.j.431.2		8	72.59	even	6
1296.2.s.j.431.3		8	72.67	odd	6
1296.2.s.j.863.2		8	72.61	even	6
1296.2.s.j.863.3		8	72.29	odd	6
1296.2.s.l.431.2		8	72.5	odd	6
1296.2.s.l.431.3		8	72.13	even	6
1296.2.s.l.863.2		8	72.43	odd	6
1296.2.s.l.863.3		8	72.11	even	6
5184.2.c.h.5183.3		8	4.3	odd	2	inner
5184.2.c.h.5183.4		8	1.1	even	1	trivial
5184.2.c.h.5183.5		8	12.11	even	2	inner
5184.2.c.h.5183.6		8	3.2	odd	2	inner