Embedded newform 87.4.i.a.13.2

• Label: 87.4.i.a.13.2
• Level: $87$
• Weight: $4$
• Character: 87.13
• Analytic conductor: $5.133$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	87.i (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.13316617050\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	87.13
Dual form			87.4.i.a.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22145 + 4.61288i) q^{2} +(2.34549 + 1.87047i) q^{3} +(-11.3559 - 14.2399i) q^{4} +(-10.0741 - 4.85142i) q^{5} +(-13.8386 + 6.66434i) q^{6} +(21.3564 - 26.7801i) q^{7} +(50.9810 - 11.6361i) q^{8} +(2.00269 + 8.77435i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 52 q^{4} + 28 q^{5} + 12 q^{6} - 40 q^{7} + 144 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(59\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22145	+	4.61288i	−0.785400	+	1.63090i	−0.00957188	+	0.999954i	\(0.503047\pi\)
							−0.775828	+	0.630945i	\(0.782667\pi\)
\(3\)	2.34549	+	1.87047i	0.451391	+	0.359972i
							\(5\)	−10.0741	−	4.85142i	−0.901053	−	0.433924i	−0.0747836	−	0.997200i	\(-0.523827\pi\)
−0.826269	+	0.563276i	\(0.809541\pi\)
\(7\)	21.3564	−	26.7801i	1.15314	−	1.44599i	0.279011	−	0.960288i	\(-0.409993\pi\)
							0.874126	−	0.485700i	\(-0.161435\pi\)
\(11\)	−63.1079	−	14.4040i	−1.72980	−	0.394815i	−0.762205	−	0.647336i	\(-0.775883\pi\)
							−0.967591	+	0.252521i	\(0.918740\pi\)
\(13\)	6.16044	−	26.9906i	0.131431	−	0.575835i	−0.865729	−	0.500514i	\(-0.833145\pi\)
							0.997159	−	0.0753215i	\(-0.0239983\pi\)
\(17\)		−	5.34309i		−	0.0762288i	−0.999273	−	0.0381144i	\(-0.987865\pi\)
							0.999273	−	0.0381144i	\(-0.0121351\pi\)
\(19\)	−75.4691	+	60.1846i	−0.911252	+	0.726699i	−0.962298	−	0.271997i	\(-0.912316\pi\)
							0.0510459	+	0.998696i	\(0.483745\pi\)
\(23\)	134.000	−	64.5312i	1.21483	−	0.585030i	0.286960	−	0.957942i	\(-0.407355\pi\)
							0.927866	+	0.372913i	\(0.121641\pi\)
\(29\)	−56.5979	−	145.553i	−0.362412	−	0.932018i
							\(31\)	−19.6457	+	40.7947i	−0.113822	+	0.236353i	−0.950096	−	0.311959i	\(-0.899015\pi\)
0.836274	+	0.548312i	\(0.184729\pi\)
\(37\)	18.9557	−	4.32651i	0.0842241	−	0.0192236i	−0.180201	−	0.983630i	\(-0.557675\pi\)
							0.264425	+	0.964406i	\(0.414818\pi\)
\(41\)		−	109.949i		−	0.418809i	−0.977829	−	0.209404i	\(-0.932847\pi\)
							0.977829	−	0.209404i	\(-0.0671525\pi\)
\(43\)	−118.670	−	246.421i	−0.420860	−	0.873925i	−0.998344	−	0.0575231i	\(-0.981680\pi\)
							0.577484	−	0.816402i	\(-0.304035\pi\)
\(47\)	200.679	+	45.8038i	0.622811	+	0.142152i	0.522270	−	0.852780i	\(-0.325085\pi\)
							0.100541	+	0.994933i	\(0.467943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.4.i.a.13.2	✓	96
29.9	even	14	inner	87.4.i.a.67.2	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.4.i.a.13.2	✓	96	1.1	even	1	trivial
87.4.i.a.67.2	yes	96	29.9	even	14	inner