Embedded newform 85.4.r.a.12.16

• Label: 85.4.r.a.12.16
• 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			12.16
Character	\(\chi\)	\(=\)	85.12
Dual form			85.4.r.a.78.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.665575 - 1.60684i) q^{2} +(1.08698 - 5.46463i) q^{3} +(3.51791 + 3.51791i) q^{4} +(-8.49133 - 7.27305i) q^{5} +(-8.05732 - 5.38373i) q^{6} +(-22.3480 - 14.9324i) q^{7} +(20.8489 - 8.63589i) q^{8} +(-3.73585 - 1.54744i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 72 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.665575	−	1.60684i	0.235316	−	0.568104i	−0.761471	−	0.648199i	\(-0.775522\pi\)
							0.996787	+	0.0800951i	\(0.0255224\pi\)
\(3\)	1.08698	−	5.46463i	0.209190	−	1.05167i	−0.723317	−	0.690516i	\(-0.757384\pi\)
							0.932507	−	0.361152i	\(-0.117616\pi\)
\(5\)	−8.49133	−	7.27305i	−0.759488	−	0.650521i
							\(7\)	−22.3480	−	14.9324i	−1.20668	−	0.806275i	−0.221058	−	0.975261i	\(-0.570951\pi\)
−0.985619	+	0.168985i	\(0.945951\pi\)
\(11\)	1.88345	−	2.81879i	0.0516257	−	0.0772634i	−0.804759	−	0.593602i	\(-0.797705\pi\)
							0.856385	+	0.516339i	\(0.172705\pi\)
\(13\)	13.2554			0.282798			0.141399	−	0.989953i	\(-0.454840\pi\)
							0.141399	+	0.989953i	\(0.454840\pi\)
\(17\)	−40.7266	−	57.0469i	−0.581038	−	0.813876i
							\(19\)	0.580355	−	0.240391i	0.00700750	−	0.00290260i	−0.379177	−	0.925324i	\(-0.623793\pi\)
0.386184	+	0.922422i	\(0.373793\pi\)
\(23\)	77.3802	−	15.3919i	0.701517	−	0.139540i	0.168571	−	0.985690i	\(-0.446085\pi\)
							0.532946	+	0.846149i	\(0.321085\pi\)
\(29\)	235.331	+	46.8103i	1.50689	+	0.299740i	0.878343	−	0.478030i	\(-0.158649\pi\)
							0.628549	+	0.777770i	\(0.283649\pi\)
\(31\)	21.7921	+	32.6141i	0.126257	+	0.188957i	0.889213	−	0.457493i	\(-0.151253\pi\)
							−0.762956	+	0.646451i	\(0.776253\pi\)
\(37\)	96.9894	+	19.2924i	0.430945	+	0.0857202i	0.405796	−	0.913963i	\(-0.366994\pi\)
							0.0251483	+	0.999684i	\(0.491994\pi\)
\(41\)	42.5331	−	8.46036i	0.162014	−	0.0322265i	−0.113417	−	0.993547i	\(-0.536180\pi\)
							0.275431	+	0.961321i	\(0.411180\pi\)
\(43\)	−156.058	−	376.758i	−0.553457	−	1.33616i	−0.914867	−	0.403756i	\(-0.867704\pi\)
							0.361409	−	0.932407i	\(-0.382296\pi\)
\(47\)			342.958i			1.06437i	0.846627	+	0.532187i	\(0.178630\pi\)
							−0.846627	+	0.532187i	\(0.821370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.r.a.12.16	yes	200
5.3	odd	4		85.4.o.a.63.10	yes	200
17.10	odd	16		85.4.o.a.27.10	✓	200
85.78	even	16	inner	85.4.r.a.78.16	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.10	✓	200	17.10	odd	16
85.4.o.a.63.10	yes	200	5.3	odd	4
85.4.r.a.12.16	yes	200	1.1	even	1	trivial
85.4.r.a.78.16	yes	200	85.78	even	16	inner