Embedded newform 87.4.i.a.13.4

• Label: 87.4.i.a.13.4
• Level: $87$
• Weight: $4$
• Character: 87.13
• Analytic conductor: $5.133$
• Analytic rank: $0$
• Dimension: $96$
• 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			13.4
Character	\(\chi\)	\(=\)	87.13
Dual form			87.4.i.a.67.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78908 + 3.71506i) q^{2} +(-2.34549 - 1.87047i) q^{3} +(-5.61296 - 7.03842i) q^{4} +(16.3624 + 7.87972i) q^{5} +(11.1452 - 5.36724i) q^{6} +(21.8101 - 27.3490i) q^{7} +(4.03000 - 0.919821i) q^{8} +(2.00269 + 8.77435i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 52 q^{4} + 28 q^{5} + 12 q^{6} - 40 q^{7} + 144 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78908	+	3.71506i	−0.632535	+	1.31347i	0.300533	+	0.953772i	\(0.402836\pi\)
							−0.933068	+	0.359701i	\(0.882879\pi\)
\(3\)	−2.34549	−	1.87047i	−0.451391	−	0.359972i
							\(5\)	16.3624	+	7.87972i	1.46350	+	0.704784i	0.984880	−	0.173239i	\(-0.0554232\pi\)
0.478619	+	0.878023i	\(0.341137\pi\)
\(7\)	21.8101	−	27.3490i	1.17764	−	1.47671i	0.331731	−	0.943374i	\(-0.392367\pi\)
							0.845904	−	0.533335i	\(-0.179061\pi\)
\(11\)	27.2729	+	6.22487i	0.747554	+	0.170624i	0.579295	−	0.815118i	\(-0.303328\pi\)
							0.168260	+	0.985743i	\(0.446185\pi\)
\(13\)	−13.2952	+	58.2501i	−0.283648	+	1.24274i	0.609429	+	0.792841i	\(0.291399\pi\)
							−0.893077	+	0.449904i	\(0.851458\pi\)
\(17\)			32.6214i			0.465403i	0.972548	+	0.232701i	\(0.0747565\pi\)
							−0.972548	+	0.232701i	\(0.925244\pi\)
\(19\)	−26.1132	+	20.8246i	−0.315305	+	0.251447i	−0.768335	−	0.640048i	\(-0.778914\pi\)
							0.453030	+	0.891495i	\(0.350343\pi\)
\(23\)	48.6743	−	23.4403i	0.441274	−	0.212506i	−0.200036	−	0.979789i	\(-0.564106\pi\)
							0.641309	+	0.767282i	\(0.278392\pi\)
\(29\)	123.560	−	95.5087i	0.791191	−	0.611569i
							\(31\)	−17.0233	+	35.3493i	−0.0986283	+	0.204804i	−0.944445	−	0.328670i	\(-0.893400\pi\)
0.845816	+	0.533474i	\(0.179114\pi\)
\(37\)	51.2753	−	11.7032i	0.227827	−	0.0520001i	−0.107083	−	0.994250i	\(-0.534151\pi\)
							0.334910	+	0.942250i	\(0.391294\pi\)
\(41\)			347.570i			1.32394i	0.749532	+	0.661968i	\(0.230278\pi\)
							−0.749532	+	0.661968i	\(0.769722\pi\)
\(43\)	46.7802	+	97.1400i	0.165905	+	0.344505i	0.967302	−	0.253628i	\(-0.0816238\pi\)
							−0.801397	+	0.598133i	\(0.795910\pi\)
\(47\)	−411.684	−	93.9641i	−1.27766	−	0.291619i	−0.470740	−	0.882272i	\(-0.656013\pi\)
							−0.806925	+	0.590653i	\(0.798870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.4.i.a.13.4	✓	96
29.9	even	14	inner	87.4.i.a.67.4	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.4.i.a.13.4	✓	96	1.1	even	1	trivial
87.4.i.a.67.4	yes	96	29.9	even	14	inner