Embedded newform 85.4.r.a.12.4

• Label: 85.4.r.a.12.4
• 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			12.4
Character	\(\chi\)	\(=\)	85.12
Dual form			85.4.r.a.78.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56343 + 3.77446i) q^{2} +(0.501843 - 2.52294i) q^{3} +(-6.14535 - 6.14535i) q^{4} +(9.04488 + 6.57192i) q^{5} +(8.73812 + 5.83862i) q^{6} +(4.58649 + 3.06459i) q^{7} +(2.60756 - 1.08009i) q^{8} +(18.8314 + 7.80021i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 72 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56343	+	3.77446i	−0.552756	+	1.33447i	0.362644	+	0.931928i	\(0.381874\pi\)
							−0.915400	+	0.402544i	\(0.868126\pi\)
\(3\)	0.501843	−	2.52294i	0.0965798	−	0.485539i	−0.901975	−	0.431789i	\(-0.857882\pi\)
							0.998554	−	0.0537504i	\(-0.0171175\pi\)
\(5\)	9.04488	+	6.57192i	0.808999	+	0.587811i
							\(7\)	4.58649	+	3.06459i	0.247647	+	0.165472i	0.673199	−	0.739462i	\(-0.264920\pi\)
−0.425552	+	0.904934i	\(0.639920\pi\)
\(11\)	−30.9106	+	46.2610i	−0.847264	+	1.26802i	0.114301	+	0.993446i	\(0.463537\pi\)
							−0.961566	+	0.274575i	\(0.911463\pi\)
\(13\)	4.43864			0.0946968			0.0473484	−	0.998878i	\(-0.484923\pi\)
							0.0473484	+	0.998878i	\(0.484923\pi\)
\(17\)	−53.8536	+	44.8641i	−0.768319	+	0.640068i
							\(19\)	59.3917	−	24.6008i	0.717126	−	0.297043i	0.00587619	−	0.999983i	\(-0.498130\pi\)
0.711250	+	0.702939i	\(0.248130\pi\)
\(23\)	−150.412	+	29.9188i	−1.36361	+	0.271239i	−0.822047	−	0.569420i	\(-0.807168\pi\)
							−0.541564	+	0.840659i	\(0.682168\pi\)
\(29\)	169.794	+	33.7741i	1.08724	+	0.216265i	0.706000	−	0.708212i	\(-0.250498\pi\)
							0.381238	+	0.924477i	\(0.375498\pi\)
\(31\)	82.7109	+	123.786i	0.479204	+	0.717179i	0.989772	−	0.142658i	\(-0.0455650\pi\)
							−0.510568	+	0.859837i	\(0.670565\pi\)
\(37\)	9.46736	+	1.88318i	0.0420655	+	0.00836735i	0.216078	−	0.976376i	\(-0.430673\pi\)
							−0.174013	+	0.984743i	\(0.555673\pi\)
\(41\)	390.933	−	77.7614i	1.48911	−	0.296202i	0.617565	−	0.786520i	\(-0.288119\pi\)
							0.871544	+	0.490318i	\(0.163119\pi\)
\(43\)	−46.0546	−	111.186i	−0.163332	−	0.394317i	0.820932	−	0.571027i	\(-0.193455\pi\)
							−0.984263	+	0.176709i	\(0.943455\pi\)
\(47\)		−	183.241i		−	0.568692i	−0.958722	−	0.284346i	\(-0.908224\pi\)
							0.958722	−	0.284346i	\(-0.0917764\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.r.a.12.4	yes	200
5.3	odd	4		85.4.o.a.63.22	yes	200
17.10	odd	16		85.4.o.a.27.22	✓	200
85.78	even	16	inner	85.4.r.a.78.4	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.22	✓	200	17.10	odd	16
85.4.o.a.63.22	yes	200	5.3	odd	4
85.4.r.a.12.4	yes	200	1.1	even	1	trivial
85.4.r.a.78.4	yes	200	85.78	even	16	inner