Embedded newform 845.2.z.a.138.24

• Label: 845.2.z.a.138.24
• Level: $845$
• Weight: $2$
• Character: 845.138
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $2136$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.z (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(2136\)
Relative dimension:	\(89\) over \(\Q(\zeta_{52})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{52}]$

Embedding invariants

Embedding label			138.24
Character	\(\chi\)	\(=\)	845.138
Dual form			845.2.z.a.447.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.492370 - 1.29827i) q^{2} +(-0.995448 + 0.182423i) q^{3} +(0.0539350 - 0.0477822i) q^{4} +(1.93594 - 1.11899i) q^{5} +(0.726964 + 1.20255i) q^{6} +(-0.569280 + 1.08467i) q^{7} +(-2.54751 - 1.33704i) q^{8} +(-1.84741 + 0.700630i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2136 q - 24 q^{2} - 22 q^{3} - 174 q^{4} - 26 q^{5} - 48 q^{6} - 26 q^{7} - 20 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{33}{52}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.492370	−	1.29827i	−0.348158	−	0.918018i	−0.988675	−	0.150074i	\(-0.952049\pi\)
							0.640516	−	0.767944i	\(-0.278720\pi\)
\(3\)	−0.995448	+	0.182423i	−0.574722	+	0.105322i	−0.459752	−	0.888047i	\(-0.652062\pi\)
							−0.114970	+	0.993369i	\(0.536677\pi\)
\(5\)	1.93594	−	1.11899i	0.865778	−	0.500428i
							\(7\)	−0.569280	+	1.08467i	−0.215168	+	0.409968i	−0.968986	−	0.247115i	\(-0.920517\pi\)
0.753819	+	0.657083i	\(0.228210\pi\)
\(11\)	1.99319	+	4.42869i	0.600970	+	1.33530i	0.921617	+	0.388102i	\(0.126869\pi\)
							−0.320647	+	0.947199i	\(0.603900\pi\)
\(13\)	−3.51839	−	0.787999i	−0.975825	−	0.218551i
							\(17\)	−0.845076	−	2.71194i	−0.204961	−	0.657743i	−0.998770	−	0.0495764i	\(-0.984213\pi\)
0.793809	−	0.608167i	\(-0.208095\pi\)
\(19\)	−6.04943	+	6.04943i	−1.38783	+	1.38783i	−0.557979	+	0.829855i	\(0.688423\pi\)
							−0.829855	+	0.557979i	\(0.811577\pi\)
\(23\)	−5.02376	+	5.02376i	−1.04753	+	1.04753i	−0.0487127	+	0.998813i	\(0.515512\pi\)
							−0.998813	+	0.0487127i	\(0.984488\pi\)
\(29\)	−3.01578	+	1.14373i	−0.560016	+	0.212386i	−0.618339	−	0.785912i	\(-0.712194\pi\)
							0.0583227	+	0.998298i	\(0.481425\pi\)
\(31\)	−3.17919	−	5.25903i	−0.571000	−	0.944550i	−0.999165	−	0.0408491i	\(-0.986994\pi\)
							0.428165	−	0.903700i	\(-0.359160\pi\)
\(37\)	2.25542	−	9.15059i	0.370788	−	1.50435i	−0.428579	−	0.903504i	\(-0.640986\pi\)
							0.799367	−	0.600843i	\(-0.205168\pi\)
\(41\)	−11.3449	+	2.07903i	−1.77178	+	0.324690i	−0.964220	−	0.265102i	\(-0.914594\pi\)
							−0.807555	+	0.589792i	\(0.799210\pi\)
\(43\)	−0.174539	+	0.288723i	−0.0266170	+	0.0440299i	−0.868807	−	0.495152i	\(-0.835112\pi\)
							0.842190	+	0.539182i	\(0.181266\pi\)
\(47\)	−2.80817	+	3.16977i	−0.409613	+	0.462358i	−0.916723	−	0.399524i	\(-0.869176\pi\)
							0.507109	+	0.861882i	\(0.330714\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.z.a.138.24	✓	2136
5.2	odd	4		845.2.be.a.307.66	yes	2136
169.109	odd	52		845.2.be.a.278.66	yes	2136
845.447	even	52	inner	845.2.z.a.447.24	yes	2136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.z.a.138.24	✓	2136	1.1	even	1	trivial
845.2.z.a.447.24	yes	2136	845.447	even	52	inner
845.2.be.a.278.66	yes	2136	169.109	odd	52
845.2.be.a.307.66	yes	2136	5.2	odd	4