Embedded newform 80.6.l.a.21.11

• Label: 80.6.l.a.21.11
• Level: $80$
• Weight: $6$
• Character: 80.21
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			21.11
Character	\(\chi\)	\(=\)	80.21
Dual form			80.6.l.a.61.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.01718 - 3.98275i) q^{2} +(10.2463 - 10.2463i) q^{3} +(0.275419 + 31.9988i) q^{4} +(-17.6777 - 17.6777i) q^{5} +(-81.9696 + 0.352756i) q^{6} +107.195i q^{7} +(126.337 - 129.642i) q^{8} +33.0270i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.01718	−	3.98275i	−0.710143	−	0.704057i
							\(3\)	10.2463	−	10.2463i	0.657300	−	0.657300i	−0.297440	−	0.954740i	\(-0.596133\pi\)
0.954740	+	0.297440i	\(0.0961330\pi\)
\(5\)	−17.6777	−	17.6777i	−0.316228	−	0.316228i
							\(7\)			107.195i			0.826854i	0.910537	+	0.413427i	\(0.135668\pi\)
−0.910537	+	0.413427i	\(0.864332\pi\)
\(11\)	221.799	+	221.799i	0.552686	+	0.552686i	0.927215	−	0.374529i	\(-0.122196\pi\)
							−0.374529	+	0.927215i	\(0.622196\pi\)
\(13\)	−803.282	+	803.282i	−1.31829	+	1.31829i	−0.403154	+	0.915132i	\(0.632086\pi\)
							−0.915132	+	0.403154i	\(0.867914\pi\)
\(17\)	364.972			0.306293			0.153146	−	0.988204i	\(-0.451059\pi\)
							0.153146	+	0.988204i	\(0.451059\pi\)
\(19\)	654.936	−	654.936i	0.416212	−	0.416212i	−0.467684	−	0.883896i	\(-0.654911\pi\)
							0.883896	+	0.467684i	\(0.154911\pi\)
\(23\)			3201.73i			1.26202i	0.775776	+	0.631009i	\(0.217359\pi\)
							−0.775776	+	0.631009i	\(0.782641\pi\)
\(29\)	4482.55	−	4482.55i	0.989760	−	0.989760i	−0.0101878	−	0.999948i	\(-0.503243\pi\)
							0.999948	+	0.0101878i	\(0.00324294\pi\)
\(31\)	5318.06			0.993914			0.496957	−	0.867775i	\(-0.334451\pi\)
							0.496957	+	0.867775i	\(0.334451\pi\)
\(37\)	−4842.58	−	4842.58i	−0.581531	−	0.581531i	0.353793	−	0.935324i	\(-0.384892\pi\)
							−0.935324	+	0.353793i	\(0.884892\pi\)
\(41\)			12004.8i			1.11530i	0.830075	+	0.557652i	\(0.188298\pi\)
							−0.830075	+	0.557652i	\(0.811702\pi\)
\(43\)	196.241	+	196.241i	0.0161852	+	0.0161852i	0.715153	−	0.698968i	\(-0.246357\pi\)
							−0.698968	+	0.715153i	\(0.746357\pi\)
\(47\)	−27915.6			−1.84333			−0.921664	−	0.387990i	\(-0.873170\pi\)
							−0.921664	+	0.387990i	\(0.873170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.21.11	✓	80
4.3	odd	2		320.6.l.a.241.10		80
16.3	odd	4		320.6.l.a.81.10		80
16.13	even	4	inner	80.6.l.a.61.11	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.11	✓	80	1.1	even	1	trivial
80.6.l.a.61.11	yes	80	16.13	even	4	inner
320.6.l.a.81.10		80	16.3	odd	4
320.6.l.a.241.10		80	4.3	odd	2