Embedded newform 720.2.cq.a.173.14

• Label: 720.2.cq.a.173.14
• Level: $720$
• Weight: $2$
• Character: 720.173
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $560$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.cq (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(140\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			173.14
Character	\(\chi\)	\(=\)	720.173
Dual form			720.2.cq.a.437.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33591 + 0.464066i) q^{2} +(-1.69698 + 0.346801i) q^{3} +(1.56929 - 1.23990i) q^{4} +(-2.22855 + 0.183196i) q^{5} +(2.10606 - 1.25080i) q^{6} +(1.37032 - 0.367175i) q^{7} +(-1.52102 + 2.38464i) q^{8} +(2.75946 - 1.17703i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 6 q^{2} - 6 q^{5} - 8 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\)	\(181\)	\(271\)	\(577\)	\(641\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33591	+	0.464066i	−0.944628	+	0.328144i
							\(3\)	−1.69698	+	0.346801i	−0.979750	+	0.200226i
\(5\)	−2.22855	+	0.183196i	−0.996638	+	0.0819279i
							\(7\)	1.37032	−	0.367175i	0.517931	−	0.138779i	0.00962161	−	0.999954i	\(-0.496937\pi\)
0.508309	+	0.861175i	\(0.330271\pi\)
\(11\)	−1.61998	+	0.434073i	−0.488443	+	0.130878i	−0.494631	−	0.869103i	\(-0.664697\pi\)
							0.00618847	+	0.999981i	\(0.498030\pi\)
\(13\)	−2.65908	+	1.53522i	−0.737497	+	0.425794i	−0.821159	−	0.570700i	\(-0.806672\pi\)
							0.0836613	+	0.996494i	\(0.473339\pi\)
\(17\)	−3.37795	−	3.37795i	−0.819274	−	0.819274i	0.166729	−	0.986003i	\(-0.446679\pi\)
							−0.986003	+	0.166729i	\(0.946679\pi\)
\(19\)	2.21692	−	2.21692i	0.508597	−	0.508597i	−0.405498	−	0.914096i	\(-0.632902\pi\)
							0.914096	+	0.405498i	\(0.132902\pi\)
\(23\)	0.522854	−	1.95132i	0.109023	−	0.406878i	−0.889748	−	0.456452i	\(-0.849120\pi\)
							0.998770	+	0.0495747i	\(0.0157866\pi\)
\(29\)	4.06390	−	1.08892i	0.754648	−	0.202207i	0.139069	−	0.990283i	\(-0.455589\pi\)
							0.615579	+	0.788075i	\(0.288922\pi\)
\(31\)	−0.0472205	−	0.0817883i	−0.00848105	−	0.0146896i	0.861754	−	0.507327i	\(-0.169366\pi\)
							−0.870235	+	0.492637i	\(0.836033\pi\)
\(37\)			9.90402i			1.62821i	0.580717	+	0.814105i	\(0.302772\pi\)
							−0.580717	+	0.814105i	\(0.697228\pi\)
\(41\)	4.92727	+	8.53428i	0.769510	+	1.33283i	0.937829	+	0.347097i	\(0.112833\pi\)
							−0.168319	+	0.985733i	\(0.553834\pi\)
\(43\)	3.41722	−	5.91879i	0.521121	−	0.902607i	−0.478578	−	0.878045i	\(-0.658848\pi\)
							0.999698	−	0.0245621i	\(-0.00781914\pi\)
\(47\)	7.29462	−	1.95459i	1.06403	−	0.285106i	0.315992	−	0.948762i	\(-0.397663\pi\)
							0.748038	+	0.663656i	\(0.230996\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.cq.a.173.14	yes	560
5.2	odd	4		720.2.cm.a.317.56	yes	560
9.5	odd	6	inner	720.2.cq.a.653.61	yes	560
16.5	even	4		720.2.cm.a.533.10	yes	560
45.32	even	12		720.2.cm.a.77.10	✓	560
80.37	odd	4	inner	720.2.cq.a.677.61	yes	560
144.5	odd	12		720.2.cm.a.293.56	yes	560
720.437	even	12	inner	720.2.cq.a.437.14	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.cm.a.77.10	✓	560	45.32	even	12
720.2.cm.a.293.56	yes	560	144.5	odd	12
720.2.cm.a.317.56	yes	560	5.2	odd	4
720.2.cm.a.533.10	yes	560	16.5	even	4
720.2.cq.a.173.14	yes	560	1.1	even	1	trivial
720.2.cq.a.437.14	yes	560	720.437	even	12	inner
720.2.cq.a.653.61	yes	560	9.5	odd	6	inner
720.2.cq.a.677.61	yes	560	80.37	odd	4	inner