Embedded newform 845.2.u.a.66.5

• Label: 845.2.u.a.66.5
• 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			66.5
Character	\(\chi\)	\(=\)	845.66
Dual form			845.2.u.a.781.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.258552 - 2.12937i) q^{2} +(1.73405 + 1.53623i) q^{3} +(-2.52548 + 0.622476i) q^{4} +(0.885456 + 0.464723i) q^{5} +(2.82287 - 4.08963i) q^{6} +(0.677710 + 1.78698i) q^{7} +(0.457189 + 1.20551i) q^{8} +(0.285303 + 2.34968i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 372 q - q^{2} - 4 q^{3} - 37 q^{4} - 31 q^{5} - 16 q^{6} + q^{7} - 9 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.258552	−	2.12937i	−0.182824	−	1.50569i	−0.738478	−	0.674278i	\(-0.764455\pi\)
							0.555654	−	0.831414i	\(-0.312468\pi\)
\(3\)	1.73405	+	1.53623i	1.00115	+	0.886944i	0.993592	−	0.113027i	\(-0.0360546\pi\)
							0.00756140	+	0.999971i	\(0.497593\pi\)
\(5\)	0.885456	+	0.464723i	0.395988	+	0.207831i
							\(7\)	0.677710	+	1.78698i	0.256150	+	0.675413i	0.999975	+	0.00703854i	\(0.00224046\pi\)
−0.743825	+	0.668375i	\(0.766990\pi\)
\(11\)	0.717261	−	5.90718i	0.216262	−	1.78108i	−0.330512	−	0.943802i	\(-0.607221\pi\)
							0.546775	−	0.837280i	\(-0.315855\pi\)
\(13\)	2.47814	−	2.61893i	0.687313	−	0.726362i
							\(17\)	2.52248	+	6.65122i	0.611790	+	1.61316i	0.778849	+	0.627211i	\(0.215804\pi\)
−0.167059	+	0.985947i	\(0.553427\pi\)
\(19\)	−2.84908			−0.653623			−0.326812	−	0.945089i	\(-0.605974\pi\)
							−0.326812	+	0.945089i	\(0.605974\pi\)
\(23\)	7.60922			1.58663			0.793316	−	0.608810i	\(-0.208353\pi\)
							0.793316	+	0.608810i	\(0.208353\pi\)
\(29\)	0.349726	+	2.88026i	0.0649425	+	0.534850i	0.988481	+	0.151344i	\(0.0483602\pi\)
							−0.923539	+	0.383506i	\(0.874717\pi\)
\(31\)	−0.922835	+	1.33696i	−0.165746	+	0.240125i	−0.897052	−	0.441925i	\(-0.854296\pi\)
							0.731306	+	0.682050i	\(0.238911\pi\)
\(37\)	3.63860	−	5.27142i	0.598182	−	0.866616i	−0.400585	−	0.916260i	\(-0.631193\pi\)
							0.998767	+	0.0496436i	\(0.0158085\pi\)
\(41\)	−2.38053	−	2.10897i	−0.371777	−	0.329365i	0.456460	−	0.889744i	\(-0.349117\pi\)
							−0.828237	+	0.560379i	\(0.810656\pi\)
\(43\)	1.26718	+	1.83582i	0.193243	+	0.279960i	0.907613	−	0.419807i	\(-0.137902\pi\)
							−0.714371	+	0.699767i	\(0.753287\pi\)
\(47\)	−8.70474	−	2.14553i	−1.26972	−	0.312957i	−0.453788	−	0.891110i	\(-0.649928\pi\)
							−0.815929	+	0.578152i	\(0.803774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.u.a.66.5	✓	372
169.105	even	13	inner	845.2.u.a.781.5	yes	372

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.u.a.66.5	✓	372	1.1	even	1	trivial
845.2.u.a.781.5	yes	372	169.105	even	13	inner