Embedded newform 85.4.r.a.12.10

• Label: 85.4.r.a.12.10
• 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			12.10
Character	\(\chi\)	\(=\)	85.12
Dual form			85.4.r.a.78.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.507196 + 1.22448i) q^{2} +(0.0167418 - 0.0841667i) q^{3} +(4.41475 + 4.41475i) q^{4} +(-7.51707 + 8.27609i) q^{5} +(0.0945690 + 0.0631890i) q^{6} +(-13.0093 - 8.69252i) q^{7} +(-17.4408 + 7.22420i) q^{8} +(24.9379 + 10.3296i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 72 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.507196	+	1.22448i	−0.179321	+	0.432919i	−0.987825	−	0.155572i	\(-0.950278\pi\)
							0.808504	+	0.588491i	\(0.200278\pi\)
\(3\)	0.0167418	−	0.0841667i	0.00322196	−	0.0161979i	−0.979141	−	0.203183i	\(-0.934871\pi\)
							0.982363	+	0.186985i	\(0.0598715\pi\)
\(5\)	−7.51707	+	8.27609i	−0.672347	+	0.740236i
							\(7\)	−13.0093	−	8.69252i	−0.702435	−	0.469352i	0.152355	−	0.988326i	\(-0.451314\pi\)
−0.854790	+	0.518974i	\(0.826314\pi\)
\(11\)	−10.8595	+	16.2523i	−0.297659	+	0.445478i	−0.949909	−	0.312525i	\(-0.898825\pi\)
							0.652250	+	0.758004i	\(0.273825\pi\)
\(13\)	−56.7945			−1.21169			−0.605845	−	0.795583i	\(-0.707165\pi\)
							−0.605845	+	0.795583i	\(0.707165\pi\)
\(17\)	54.6497	+	43.8909i	0.779676	+	0.626183i
							\(19\)	−116.292	+	48.1696i	−1.40417	+	0.581624i	−0.950829	−	0.309715i	\(-0.899766\pi\)
−0.453336	+	0.891340i	\(0.649766\pi\)
\(23\)	74.7652	−	14.8717i	0.677809	−	0.134825i	0.155832	−	0.987784i	\(-0.450194\pi\)
							0.521978	+	0.852959i	\(0.325194\pi\)
\(29\)	234.820	+	46.7086i	1.50362	+	0.299089i	0.877097	−	0.480314i	\(-0.159477\pi\)
							0.626524	+	0.779403i	\(0.284477\pi\)
\(31\)	44.9217	+	67.2301i	0.260264	+	0.389512i	0.938471	−	0.345359i	\(-0.112243\pi\)
							−0.678207	+	0.734871i	\(0.737243\pi\)
\(37\)	273.464	+	54.3954i	1.21506	+	0.241691i	0.760686	−	0.649120i	\(-0.224863\pi\)
							0.454375	+	0.890811i	\(0.349863\pi\)
\(41\)	412.601	−	82.0715i	1.57165	−	0.312620i	0.669087	−	0.743184i	\(-0.266685\pi\)
							0.902560	+	0.430564i	\(0.141685\pi\)
\(43\)	88.0277	+	212.518i	0.312188	+	0.753689i	0.999623	+	0.0274434i	\(0.00873661\pi\)
							−0.687435	+	0.726246i	\(0.741263\pi\)
\(47\)		−	269.627i		−	0.836789i	−0.908265	−	0.418395i	\(-0.862593\pi\)
							0.908265	−	0.418395i	\(-0.137407\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.r.a.12.10	yes	200
5.3	odd	4		85.4.o.a.63.16	yes	200
17.10	odd	16		85.4.o.a.27.16	✓	200
85.78	even	16	inner	85.4.r.a.78.10	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.16	✓	200	17.10	odd	16
85.4.o.a.63.16	yes	200	5.3	odd	4
85.4.r.a.12.10	yes	200	1.1	even	1	trivial
85.4.r.a.78.10	yes	200	85.78	even	16	inner