Embedded newform 85.4.o.a.27.23

• Label: 85.4.o.a.27.23
• Level: $85$
• Weight: $4$
• Character: 85.27
• 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.o (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			27.23
Character	\(\chi\)	\(=\)	85.27
Dual form			85.4.o.a.63.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.31285 - 1.78644i) q^{2} +(2.31120 - 0.459726i) q^{3} +(9.75244 - 9.75244i) q^{4} +(-3.82547 - 10.5055i) q^{5} +(9.14658 - 6.11155i) q^{6} +(9.68795 + 14.4990i) q^{7} +(10.3471 - 24.9802i) q^{8} +(-19.8144 + 8.20741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} + 56 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.31285	−	1.78644i	1.52482	−	0.631602i	0.546271	−	0.837608i	\(-0.316047\pi\)
							0.978551	+	0.206006i	\(0.0660467\pi\)
\(3\)	2.31120	−	0.459726i	0.444791	−	0.0884744i	0.0323867	−	0.999475i	\(-0.489689\pi\)
							0.412404	+	0.911001i	\(0.364689\pi\)
\(5\)	−3.82547	−	10.5055i	−0.342160	−	0.939642i
							\(7\)	9.68795	+	14.4990i	0.523100	+	0.782874i	0.995116	−	0.0987157i	\(-0.0314734\pi\)
−0.472016	+	0.881590i	\(0.656473\pi\)
\(11\)	8.56689	+	12.8213i	0.234820	+	0.351432i	0.930100	−	0.367306i	\(-0.119720\pi\)
							−0.695281	+	0.718738i	\(0.744720\pi\)
\(13\)		−	35.4414i		−	0.756129i	−0.925779	−	0.378065i	\(-0.876590\pi\)
							0.925779	−	0.378065i	\(-0.123410\pi\)
\(17\)	−13.4409	+	68.7920i	−0.191758	+	0.981442i
							\(19\)	101.319	+	41.9677i	1.22338	+	0.506739i	0.898482	−	0.439011i	\(-0.144671\pi\)
0.324895	+	0.945750i	\(0.394671\pi\)
\(23\)	30.6574	−	154.125i	0.277935	−	1.39727i	−0.549397	−	0.835561i	\(-0.685143\pi\)
							0.827332	−	0.561713i	\(-0.189857\pi\)
\(29\)	−224.670	+	44.6896i	−1.43862	+	0.286160i	−0.851933	−	0.523651i	\(-0.824570\pi\)
							−0.586691	+	0.809811i	\(0.699570\pi\)
\(31\)	−16.0437	+	24.0111i	−0.0929528	+	0.139114i	−0.875031	−	0.484066i	\(-0.839159\pi\)
							0.782079	+	0.623180i	\(0.214159\pi\)
\(37\)	−33.4888	−	168.360i	−0.148798	−	0.748058i	−0.981064	−	0.193682i	\(-0.937957\pi\)
							0.832266	−	0.554376i	\(-0.187043\pi\)
\(41\)	−148.320	−	29.5027i	−0.564969	−	0.112379i	−0.0956609	−	0.995414i	\(-0.530496\pi\)
							−0.469308	+	0.883035i	\(0.655496\pi\)
\(43\)	144.071	+	59.6763i	0.510946	+	0.211641i	0.623235	−	0.782035i	\(-0.285818\pi\)
							−0.112289	+	0.993676i	\(0.535818\pi\)
\(47\)	−66.9708			−0.207845			−0.103922	−	0.994585i	\(-0.533139\pi\)
							−0.103922	+	0.994585i	\(0.533139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.o.a.27.23	✓	200
5.3	odd	4		85.4.r.a.78.3	yes	200
17.12	odd	16		85.4.r.a.12.3	yes	200
85.63	even	16	inner	85.4.o.a.63.23	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.o.a.27.23	✓	200	1.1	even	1	trivial
85.4.o.a.63.23	yes	200	85.63	even	16	inner
85.4.r.a.12.3	yes	200	17.12	odd	16
85.4.r.a.78.3	yes	200	5.3	odd	4