Embedded newform 87.4.g.a.16.3

• Label: 87.4.g.a.16.3
• Level: $87$
• Weight: $4$
• Character: 87.16
• Analytic conductor: $5.133$
• Analytic rank: $0$
• Dimension: $42$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	87.g (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.13316617050\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			16.3
Character	\(\chi\)	\(=\)	87.16
Dual form			87.4.g.a.49.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60781 - 1.25586i) q^{2} +(-1.87047 + 2.34549i) q^{3} +(0.235602 + 0.295435i) q^{4} +(4.97984 + 2.39816i) q^{5} +(7.82344 - 3.76757i) q^{6} +(-4.10088 + 5.14234i) q^{7} +(4.90924 + 21.5088i) q^{8} +(-2.00269 - 8.77435i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} + 21 q^{3} - 34 q^{4} - 47 q^{5} + 6 q^{6} + 20 q^{7} - 81 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(59\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60781	−	1.25586i	−0.922002	−	0.444013i	−0.0882164	−	0.996101i	\(-0.528117\pi\)
							−0.833785	+	0.552089i	\(0.813831\pi\)
\(3\)	−1.87047	+	2.34549i	−0.359972	+	0.451391i
							\(5\)	4.97984	+	2.39816i	0.445410	+	0.214498i	0.643125	−	0.765761i	\(-0.277638\pi\)
−0.197715	+	0.980260i	\(0.563352\pi\)
\(7\)	−4.10088	+	5.14234i	−0.221427	+	0.277660i	−0.880120	−	0.474751i	\(-0.842538\pi\)
							0.658693	+	0.752411i	\(0.271109\pi\)
\(11\)	10.6239	−	46.5465i	0.291203	−	1.27585i	−0.591649	−	0.806195i	\(-0.701523\pi\)
							0.882853	−	0.469650i	\(-0.155620\pi\)
\(13\)	10.3058	−	45.1525i	0.219870	−	0.963312i	−0.737704	−	0.675124i	\(-0.764090\pi\)
							0.957574	−	0.288188i	\(-0.0930528\pi\)
\(17\)	98.2544			1.40178			0.700888	−	0.713271i	\(-0.252787\pi\)
							0.700888	+	0.713271i	\(0.252787\pi\)
\(19\)	−99.7775	−	125.117i	−1.20476	−	1.51073i	−0.804092	−	0.594505i	\(-0.797348\pi\)
							−0.400673	−	0.916221i	\(-0.631223\pi\)
\(23\)	66.4952	−	32.0224i	0.602835	−	0.290310i	−0.107465	−	0.994209i	\(-0.534273\pi\)
							0.710300	+	0.703899i	\(0.248559\pi\)
\(29\)	81.0098	+	133.516i	0.518729	+	0.854939i
							\(31\)	197.792	+	95.2514i	1.14595	+	0.551860i	0.907814	−	0.419372i	\(-0.137750\pi\)
0.238135	+	0.971232i	\(0.423464\pi\)
\(37\)	−4.78145	−	20.9489i	−0.0212450	−	0.0930806i	0.963194	−	0.268808i	\(-0.0866296\pi\)
							−0.984439	+	0.175727i	\(0.943772\pi\)
\(41\)	−120.651			−0.459573			−0.229787	−	0.973241i	\(-0.573803\pi\)
							−0.229787	+	0.973241i	\(0.573803\pi\)
\(43\)	−18.9224	+	9.11253i	−0.0671078	+	0.0323174i	−0.467137	−	0.884185i	\(-0.654714\pi\)
							0.400029	+	0.916503i	\(0.369000\pi\)
\(47\)	105.335	−	461.505i	0.326910	−	1.43229i	−0.498077	−	0.867133i	\(-0.665960\pi\)
							0.824987	−	0.565152i	\(-0.191183\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.4.g.a.16.3	✓	42
29.20	even	7	inner	87.4.g.a.49.3	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.4.g.a.16.3	✓	42	1.1	even	1	trivial
87.4.g.a.49.3	yes	42	29.20	even	7	inner