Embedded newform 87.4.i.a.13.3

• Label: 87.4.i.a.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	87.13
Dual form			87.4.i.a.67.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93516 + 4.01839i) q^{2} +(2.34549 + 1.87047i) q^{3} +(-7.41472 - 9.29777i) q^{4} +(11.6028 + 5.58764i) q^{5} +(-12.0552 + 5.80547i) q^{6} +(-13.8901 + 17.4176i) q^{7} +(16.9247 - 3.86294i) q^{8} +(2.00269 + 8.77435i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 52 q^{4} + 28 q^{5} + 12 q^{6} - 40 q^{7} + 144 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93516	+	4.01839i	−0.684181	+	1.42072i	0.212109	+	0.977246i	\(0.431967\pi\)
							−0.896289	+	0.443470i	\(0.853747\pi\)
\(3\)	2.34549	+	1.87047i	0.451391	+	0.359972i
							\(5\)	11.6028	+	5.58764i	1.03779	+	0.499773i	0.873595	−	0.486654i	\(-0.161783\pi\)
0.164196	+	0.986428i	\(0.447497\pi\)
\(7\)	−13.8901	+	17.4176i	−0.749993	+	0.940462i	−0.999611	−	0.0278827i	\(-0.991124\pi\)
							0.249618	+	0.968344i	\(0.419695\pi\)
\(11\)	−4.91728	−	1.12234i	−0.134783	−	0.0307634i	0.154597	−	0.987978i	\(-0.450592\pi\)
							−0.289380	+	0.957214i	\(0.593449\pi\)
\(13\)	−8.44940	+	37.0192i	−0.180265	+	0.789792i	0.801238	+	0.598345i	\(0.204175\pi\)
							−0.981503	+	0.191446i	\(0.938682\pi\)
\(17\)		−	72.4406i		−	1.03350i	−0.856138	−	0.516748i	\(-0.827143\pi\)
							0.856138	−	0.516748i	\(-0.172857\pi\)
\(19\)	11.2217	−	8.94903i	0.135497	−	0.108055i	−0.553396	−	0.832918i	\(-0.686668\pi\)
							0.688893	+	0.724863i	\(0.258097\pi\)
\(23\)	−0.469985	+	0.226333i	−0.00426081	+	0.00205190i	−0.436013	−	0.899940i	\(-0.643610\pi\)
							0.431752	+	0.901992i	\(0.357896\pi\)
\(29\)	117.592	+	102.768i	0.752974	+	0.658051i
							\(31\)	−135.134	+	280.609i	−0.782930	+	1.62577i	−0.00292751	+	0.999996i	\(0.500932\pi\)
−0.780003	+	0.625776i	\(0.784782\pi\)
\(37\)	83.9218	−	19.1546i	0.372883	−	0.0851080i	−0.0319728	−	0.999489i	\(-0.510179\pi\)
							0.404855	+	0.914381i	\(0.367322\pi\)
\(41\)		−	319.879i		−	1.21846i	−0.792995	−	0.609228i	\(-0.791479\pi\)
							0.792995	−	0.609228i	\(-0.208521\pi\)
\(43\)	90.6315	+	188.198i	0.321423	+	0.667441i	0.997596	−	0.0693018i	\(-0.0220771\pi\)
							−0.676173	+	0.736743i	\(0.736363\pi\)
\(47\)	591.634	+	135.037i	1.83614	+	0.419088i	0.992951	−	0.118524i	\(-0.0378163\pi\)
							0.843192	+	0.537612i	\(0.180673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.4.i.a.13.3	✓	96
29.9	even	14	inner	87.4.i.a.67.3	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.4.i.a.13.3	✓	96	1.1	even	1	trivial
87.4.i.a.67.3	yes	96	29.9	even	14	inner