Embedded newform 845.2.u.a.66.18

• Label: 845.2.u.a.66.18
• Level: $845$
• Weight: $2$
• Character: 845.66
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $372$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.u (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(372\)
Relative dimension:	\(31\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			66.18
Character	\(\chi\)	\(=\)	845.66
Dual form			845.2.u.a.781.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0488709 + 0.402488i) q^{2} +(2.17777 + 1.92934i) q^{3} +(1.78228 - 0.439291i) q^{4} +(0.885456 + 0.464723i) q^{5} +(-0.670106 + 0.970816i) q^{6} +(0.162628 + 0.428815i) q^{7} +(0.551456 + 1.45407i) q^{8} +(0.658738 + 5.42520i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 372 q - q^{2} - 4 q^{3} - 37 q^{4} - 31 q^{5} - 16 q^{6} + q^{7} - 9 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{6}{13}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0488709	+	0.402488i	0.0345569	+	0.284602i	0.999735	+	0.0230091i	\(0.00732468\pi\)
							−0.965178	+	0.261593i	\(0.915752\pi\)
\(3\)	2.17777	+	1.92934i	1.25734	+	1.11390i	0.988580	+	0.150697i	\(0.0481516\pi\)
							0.268758	+	0.963208i	\(0.413387\pi\)
\(5\)	0.885456	+	0.464723i	0.395988	+	0.207831i
							\(7\)	0.162628	+	0.428815i	0.0614676	+	0.162077i	0.962232	−	0.272232i	\(-0.0877618\pi\)
−0.900764	+	0.434309i	\(0.856993\pi\)
\(11\)	−0.127220	+	1.04775i	−0.0383584	+	0.315910i	0.960883	+	0.276956i	\(0.0893256\pi\)
							−0.999241	+	0.0389537i	\(0.987598\pi\)
\(13\)	−1.92314	−	3.04984i	−0.533384	−	0.845874i
							\(17\)	−0.859905	−	2.26738i	−0.208558	−	0.549921i	0.789394	−	0.613886i	\(-0.210395\pi\)
−0.997952	+	0.0639649i	\(0.979625\pi\)
\(19\)	−4.53783			−1.04105			−0.520525	−	0.853847i	\(-0.674264\pi\)
							−0.520525	+	0.853847i	\(0.674264\pi\)
\(23\)	−0.667726			−0.139231			−0.0696153	−	0.997574i	\(-0.522177\pi\)
							−0.0696153	+	0.997574i	\(0.522177\pi\)
\(29\)	−1.05899	−	8.72159i	−0.196650	−	1.61956i	−0.672545	−	0.740057i	\(-0.734799\pi\)
							0.475894	−	0.879502i	\(-0.342124\pi\)
\(31\)	−3.56786	+	5.16894i	−0.640807	+	0.928370i	−0.999988	−	0.00489822i	\(-0.998441\pi\)
							0.359181	+	0.933268i	\(0.383056\pi\)
\(37\)	4.29187	−	6.21785i	0.705579	−	1.02221i	−0.292250	−	0.956342i	\(-0.594404\pi\)
							0.997829	−	0.0658654i	\(-0.0209808\pi\)
\(41\)	3.29176	+	2.91625i	0.514087	+	0.455442i	0.879840	−	0.475271i	\(-0.157650\pi\)
							−0.365752	+	0.930712i	\(0.619188\pi\)
\(43\)	−3.26327	−	4.72766i	−0.497644	−	0.720962i	0.491013	−	0.871152i	\(-0.336627\pi\)
							−0.988657	+	0.150190i	\(0.952011\pi\)
\(47\)	−1.75379	−	0.432271i	−0.255817	−	0.0630532i	0.109322	−	0.994006i	\(-0.465132\pi\)
							−0.365138	+	0.930953i	\(0.618978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.u.a.66.18	✓	372
169.105	even	13	inner	845.2.u.a.781.18	yes	372

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.u.a.66.18	✓	372	1.1	even	1	trivial
845.2.u.a.781.18	yes	372	169.105	even	13	inner