Embedded newform 80.6.s.a.3.11

• Label: 80.6.s.a.3.11
• Level: $80$
• Weight: $6$
• Character: 80.3
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.11
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.94622 + 2.74497i) q^{2} +14.4255 q^{3} +(16.9303 - 27.1545i) q^{4} +(-46.0646 - 31.6710i) q^{5} +(-71.3518 + 39.5976i) q^{6} +(-46.8577 + 46.8577i) q^{7} +(-9.20264 + 180.785i) q^{8} -34.9048 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.94622	+	2.74497i	−0.874377	+	0.485247i
							\(3\)	14.4255			0.925397			0.462698	−	0.886516i	\(-0.346881\pi\)
0.462698	+	0.886516i	\(0.346881\pi\)
\(5\)	−46.0646	−	31.6710i	−0.824029	−	0.566548i
							\(7\)	−46.8577	+	46.8577i	−0.361439	+	0.361439i	−0.864343	−	0.502903i	\(-0.832265\pi\)
0.502903	+	0.864343i	\(0.332265\pi\)
\(11\)	450.379	+	450.379i	1.12227	+	1.12227i	0.991400	+	0.130867i	\(0.0417760\pi\)
							0.130867	+	0.991400i	\(0.458224\pi\)
\(13\)			176.255i			0.289257i	0.989486	+	0.144629i	\(0.0461987\pi\)
							−0.989486	+	0.144629i	\(0.953801\pi\)
\(17\)	1246.06	−	1246.06i	1.04572	−	1.04572i	0.0468198	−	0.998903i	\(-0.485091\pi\)
							0.998903	−	0.0468198i	\(-0.0149087\pi\)
\(19\)	2073.88	+	2073.88i	1.31795	+	1.31795i	0.915397	+	0.402552i	\(0.131877\pi\)
							0.402552	+	0.915397i	\(0.368123\pi\)
\(23\)	2332.74	+	2332.74i	0.919491	+	0.919491i	0.996992	−	0.0775016i	\(-0.0246943\pi\)
							−0.0775016	+	0.996992i	\(0.524694\pi\)
\(29\)	−696.037	+	696.037i	−0.153687	+	0.153687i	−0.779762	−	0.626075i	\(-0.784660\pi\)
							0.626075	+	0.779762i	\(0.284660\pi\)
\(31\)			5767.43i			1.07790i	0.842338	+	0.538950i	\(0.181179\pi\)
							−0.842338	+	0.538950i	\(0.818821\pi\)
\(37\)		−	7033.57i		−	0.844639i	−0.906447	−	0.422320i	\(-0.861216\pi\)
							0.906447	−	0.422320i	\(-0.138784\pi\)
\(41\)		−	8800.26i		−	0.817591i	−0.912626	−	0.408795i	\(-0.865949\pi\)
							0.912626	−	0.408795i	\(-0.134051\pi\)
\(43\)			12534.7i			1.03381i	0.856042	+	0.516906i	\(0.172916\pi\)
							−0.856042	+	0.516906i	\(0.827084\pi\)
\(47\)	−8516.96	−	8516.96i	−0.562393	−	0.562393i	0.367594	−	0.929987i	\(-0.380182\pi\)
							−0.929987	+	0.367594i	\(0.880182\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.11	yes	116
5.2	odd	4		80.6.j.a.67.21	yes	116
16.11	odd	4		80.6.j.a.43.21	✓	116
80.27	even	4	inner	80.6.s.a.27.11	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.21	✓	116	16.11	odd	4
80.6.j.a.67.21	yes	116	5.2	odd	4
80.6.s.a.3.11	yes	116	1.1	even	1	trivial
80.6.s.a.27.11	yes	116	80.27	even	4	inner