Embedded newform 605.2.r.a.32.13

• Label: 605.2.r.a.32.13
• Level: $605$
• Weight: $2$
• Character: 605.32
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(64\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			32.13
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18351 - 1.58098i) q^{2} +(2.15121 - 2.15121i) q^{3} +(-0.535343 + 1.82321i) q^{4} +(1.95281 - 1.08928i) q^{5} +(-5.94699 - 0.855048i) q^{6} +(0.546359 - 2.51157i) q^{7} +(-0.184700 + 0.0688894i) q^{8} -6.25540i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18351	−	1.58098i	−0.836865	−	1.11792i	−0.991520	−	0.129953i	\(-0.958517\pi\)
							0.154655	−	0.987969i	\(-0.450573\pi\)
\(3\)	2.15121	−	2.15121i	1.24200	−	1.24200i	0.282832	−	0.959170i	\(-0.408726\pi\)
							0.959170	−	0.282832i	\(-0.0912738\pi\)
\(5\)	1.95281	−	1.08928i	0.873325	−	0.487139i
							\(7\)	0.546359	−	2.51157i	0.206504	−	0.949284i	−0.750874	−	0.660446i	\(-0.770367\pi\)
0.957378	−	0.288838i	\(-0.0932690\pi\)
\(11\)	−2.00171	−	2.64446i	−0.603539	−	0.797333i
							\(13\)	2.31506	+	4.23972i	0.642083	+	1.17589i	0.972688	+	0.232118i	\(0.0745656\pi\)
−0.330604	+	0.943769i	\(0.607253\pi\)
\(17\)	0.409911	+	5.73131i	0.0994180	+	1.39005i	0.765149	+	0.643854i	\(0.222665\pi\)
							−0.665731	+	0.746192i	\(0.731880\pi\)
\(19\)	0.804651	+	0.928616i	0.184599	+	0.213039i	0.840505	−	0.541804i	\(-0.182258\pi\)
							−0.655905	+	0.754843i	\(0.727713\pi\)
\(23\)	−0.685842	+	0.149196i	−0.143008	+	0.0311095i	−0.283499	−	0.958972i	\(-0.591495\pi\)
							0.140491	+	0.990082i	\(0.455132\pi\)
\(29\)	5.08028	+	5.86296i	0.943385	+	1.08872i	0.995932	+	0.0901058i	\(0.0287205\pi\)
							−0.0525473	+	0.998618i	\(0.516734\pi\)
\(31\)	1.71177	−	0.502621i	0.307443	−	0.0902734i	−0.124372	−	0.992236i	\(-0.539692\pi\)
							0.431815	+	0.901962i	\(0.357873\pi\)
\(37\)	−4.55659	−	2.48809i	−0.749099	−	0.409039i	0.0588686	−	0.998266i	\(-0.481251\pi\)
							−0.807968	+	0.589227i	\(0.799433\pi\)
\(41\)	−8.36415	−	1.20258i	−1.30626	−	0.187812i	−0.546195	−	0.837658i	\(-0.683924\pi\)
							−0.760066	+	0.649846i	\(0.774833\pi\)
\(43\)	8.32245	−	3.10411i	1.26916	−	0.473373i	0.377472	−	0.926021i	\(-0.376793\pi\)
							0.891689	+	0.452648i	\(0.149521\pi\)
\(47\)	−4.84961	+	6.47832i	−0.707388	+	0.944960i	−0.999936	−	0.0112906i	\(-0.996406\pi\)
							0.292548	+	0.956251i	\(0.405497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.13	✓	1280
5.3	odd	4	inner	605.2.r.a.153.13	yes	1280
121.87	odd	22	inner	605.2.r.a.87.13	yes	1280
605.208	even	44	inner	605.2.r.a.208.13	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.13	✓	1280	1.1	even	1	trivial
605.2.r.a.87.13	yes	1280	121.87	odd	22	inner
605.2.r.a.153.13	yes	1280	5.3	odd	4	inner
605.2.r.a.208.13	yes	1280	605.208	even	44	inner