Embedded newform 85.4.r.a.78.10

• Label: 85.4.r.a.78.10
• Level: $85$
• Weight: $4$
• Character: 85.78
• Analytic conductor: $5.015$
• Analytic rank: $0$
• Dimension: $200$
• 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			78.10
Character	\(\chi\)	\(=\)	85.78
Dual form			85.4.r.a.12.10

$q$-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 \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{16}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.16	✓	200	5.2	odd	4
85.4.o.a.63.16	yes	200	17.12	odd	16
85.4.r.a.12.10	yes	200	85.12	even	16	inner
85.4.r.a.78.10	yes	200	1.1	even	1	trivial