Embedded newform 85.4.r.a.12.19

• Label: 85.4.r.a.12.19
• Level: $85$
• Weight: $4$
• Character: 85.12
• Analytic conductor: $5.015$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	85.r (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.01516235049\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			12.19
Character	\(\chi\)	\(=\)	85.12
Dual form			85.4.r.a.78.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.08910 - 2.62932i) q^{2} +(-0.533800 + 2.68359i) q^{3} +(-0.0703143 - 0.0703143i) q^{4} +(-7.46888 + 8.31960i) q^{5} +(6.47465 + 4.32623i) q^{6} +(11.5751 + 7.73424i) q^{7} +(20.7731 - 8.60449i) q^{8} +(18.0280 + 7.46745i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 72 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.08910	−	2.62932i	0.385054	−	0.929604i	−0.605917	−	0.795528i	\(-0.707194\pi\)
							0.990971	−	0.134076i	\(-0.0428065\pi\)
\(3\)	−0.533800	+	2.68359i	−0.102730	+	0.516458i	0.894816	+	0.446436i	\(0.147307\pi\)
							−0.997545	+	0.0700219i	\(0.977693\pi\)
\(5\)	−7.46888	+	8.31960i	−0.668037	+	0.744128i
							\(7\)	11.5751	+	7.73424i	0.624997	+	0.417610i	0.827337	−	0.561706i	\(-0.189855\pi\)
−0.202340	+	0.979315i	\(0.564855\pi\)
\(11\)	−15.8959	+	23.7899i	−0.435708	+	0.652083i	−0.982732	−	0.185034i	\(-0.940761\pi\)
							0.547024	+	0.837117i	\(0.315761\pi\)
\(13\)	33.3628			0.711783			0.355891	−	0.934527i	\(-0.384177\pi\)
							0.355891	+	0.934527i	\(0.384177\pi\)
\(17\)	−68.8016	+	13.3919i	−0.981578	+	0.191059i
							\(19\)	20.2580	−	8.39115i	0.244606	−	0.101319i	−0.257013	−	0.966408i	\(-0.582738\pi\)
0.501618	+	0.865089i	\(0.332738\pi\)
\(23\)	64.8278	−	12.8951i	0.587719	−	0.116905i	0.107732	−	0.994180i	\(-0.465641\pi\)
							0.479987	+	0.877275i	\(0.340641\pi\)
\(29\)	−11.8090	−	2.34895i	−0.0756162	−	0.0150410i	0.157137	−	0.987577i	\(-0.449774\pi\)
							−0.232753	+	0.972536i	\(0.574774\pi\)
\(31\)	−134.300	−	200.994i	−0.778096	−	1.16450i	−0.982618	−	0.185641i	\(-0.940564\pi\)
							0.204521	−	0.978862i	\(-0.434436\pi\)
\(37\)	211.734	+	42.1164i	0.940778	+	0.187132i	0.641578	−	0.767058i	\(-0.278280\pi\)
							0.299200	+	0.954190i	\(0.403280\pi\)
\(41\)	−296.195	+	58.9169i	−1.12824	+	0.224421i	−0.723711	−	0.690103i	\(-0.757565\pi\)
							−0.404531	+	0.914524i	\(0.632565\pi\)
\(43\)	10.7793	+	26.0235i	0.0382285	+	0.0922918i	0.941841	−	0.336060i	\(-0.109094\pi\)
							−0.903612	+	0.428352i	\(0.859094\pi\)
\(47\)		−	108.941i		−	0.338100i	−0.985607	−	0.169050i	\(-0.945930\pi\)
							0.985607	−	0.169050i	\(-0.0540699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.r.a.12.19	yes	200
5.3	odd	4		85.4.o.a.63.7	yes	200
17.10	odd	16		85.4.o.a.27.7	✓	200
85.78	even	16	inner	85.4.r.a.78.19	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.7	✓	200	17.10	odd	16
85.4.o.a.63.7	yes	200	5.3	odd	4
85.4.r.a.12.19	yes	200	1.1	even	1	trivial
85.4.r.a.78.19	yes	200	85.78	even	16	inner