Embedded newform 720.2.cq.a.173.18

• Label: 720.2.cq.a.173.18
• 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			173.18
Character	\(\chi\)	\(=\)	720.173
Dual form			720.2.cq.a.437.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29878 - 0.559606i) q^{2} +(1.72857 + 0.109827i) q^{3} +(1.37368 + 1.45362i) q^{4} +(1.89741 + 1.18314i) q^{5} +(-2.18357 - 1.10996i) q^{6} +(-0.405417 + 0.108631i) q^{7} +(-0.970663 - 2.65665i) q^{8} +(2.97588 + 0.379688i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 6 q^{2} - 6 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29878	−	0.559606i	−0.918379	−	0.395701i
							\(3\)	1.72857	+	0.109827i	0.997988	+	0.0634089i
\(5\)	1.89741	+	1.18314i	0.848549	+	0.529116i
							\(7\)	−0.405417	+	0.108631i	−0.153233	+	0.0410587i	−0.334620	−	0.942353i	\(-0.608608\pi\)
0.181387	+	0.983412i	\(0.441941\pi\)
\(11\)	0.943834	−	0.252900i	0.284577	−	0.0762521i	−0.113707	−	0.993514i	\(-0.536273\pi\)
							0.398284	+	0.917262i	\(0.369606\pi\)
\(13\)	0.723440	−	0.417678i	0.200646	−	0.115843i	−0.396311	−	0.918116i	\(-0.629710\pi\)
							0.596957	+	0.802273i	\(0.296376\pi\)
\(17\)	−1.77533	−	1.77533i	−0.430580	−	0.430580i	0.458246	−	0.888825i	\(-0.348478\pi\)
							−0.888825	+	0.458246i	\(0.848478\pi\)
\(19\)	2.95700	−	2.95700i	0.678382	−	0.678382i	−0.281252	−	0.959634i	\(-0.590750\pi\)
							0.959634	+	0.281252i	\(0.0907496\pi\)
\(23\)	1.61788	−	6.03800i	0.337351	−	1.25901i	−0.563948	−	0.825810i	\(-0.690718\pi\)
							0.901298	−	0.433199i	\(-0.142615\pi\)
\(29\)	−2.24304	+	0.601022i	−0.416523	+	0.111607i	−0.460993	−	0.887404i	\(-0.652507\pi\)
							0.0444703	+	0.999011i	\(0.485840\pi\)
\(31\)	3.04759	+	5.27857i	0.547363	+	0.948060i	0.998454	+	0.0555821i	\(0.0177014\pi\)
							−0.451092	+	0.892478i	\(0.648965\pi\)
\(37\)			11.0473i			1.81617i	0.418791	+	0.908083i	\(0.362454\pi\)
							−0.418791	+	0.908083i	\(0.637546\pi\)
\(41\)	−0.890466	−	1.54233i	−0.139067	−	0.240872i	0.788076	−	0.615577i	\(-0.211077\pi\)
							−0.927144	+	0.374706i	\(0.877744\pi\)
\(43\)	−0.795962	+	1.37865i	−0.121383	+	0.210242i	−0.920313	−	0.391182i	\(-0.872066\pi\)
							0.798930	+	0.601424i	\(0.205400\pi\)
\(47\)	−11.0384	+	2.95774i	−1.61012	+	0.431430i	−0.948079	−	0.318036i	\(-0.896977\pi\)
							−0.662042	+	0.749467i	\(0.730310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.cq.a.173.18	yes	560
5.2	odd	4		720.2.cm.a.317.89	yes	560
9.5	odd	6	inner	720.2.cq.a.653.31	yes	560
16.5	even	4		720.2.cm.a.533.42	yes	560
45.32	even	12		720.2.cm.a.77.42	✓	560
80.37	odd	4	inner	720.2.cq.a.677.31	yes	560
144.5	odd	12		720.2.cm.a.293.89	yes	560
720.437	even	12	inner	720.2.cq.a.437.18	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.cm.a.77.42	✓	560	45.32	even	12
720.2.cm.a.293.89	yes	560	144.5	odd	12
720.2.cm.a.317.89	yes	560	5.2	odd	4
720.2.cm.a.533.42	yes	560	16.5	even	4
720.2.cq.a.173.18	yes	560	1.1	even	1	trivial
720.2.cq.a.437.18	yes	560	720.437	even	12	inner
720.2.cq.a.653.31	yes	560	9.5	odd	6	inner
720.2.cq.a.677.31	yes	560	80.37	odd	4	inner