Embedded newform 605.2.r.a.32.12

• Label: 605.2.r.a.32.12
• 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			32.12
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20651 - 1.61170i) q^{2} +(1.27231 - 1.27231i) q^{3} +(-0.578464 + 1.97007i) q^{4} +(2.15875 + 0.582930i) q^{5} +(-3.58564 - 0.515537i) q^{6} +(-1.05425 + 4.84632i) q^{7} +(0.100425 - 0.0374565i) q^{8} -0.237553i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20651	−	1.61170i	−0.853128	−	1.13965i	−0.988852	−	0.148899i	\(-0.952427\pi\)
							0.135725	−	0.990747i	\(-0.456664\pi\)
\(3\)	1.27231	−	1.27231i	0.734569	−	0.734569i	−0.236952	−	0.971521i	\(-0.576148\pi\)
							0.971521	+	0.236952i	\(0.0761484\pi\)
\(5\)	2.15875	+	0.582930i	0.965421	+	0.260694i
							\(7\)	−1.05425	+	4.84632i	−0.398470	+	1.83174i	0.141355	+	0.989959i	\(0.454854\pi\)
−0.539825	+	0.841778i	\(0.681509\pi\)
\(11\)	−2.04132	+	2.61400i	−0.615481	+	0.788152i
							\(13\)	0.771771	+	1.41339i	0.214051	+	0.392005i	0.962539	−	0.271143i	\(-0.0874018\pi\)
−0.748488	+	0.663148i	\(0.769220\pi\)
\(17\)	−0.0189566	−	0.265048i	−0.00459766	−	0.0642837i	0.994635	−	0.103444i	\(-0.0329862\pi\)
							−0.999233	+	0.0391603i	\(0.987532\pi\)
\(19\)	−1.97453	−	2.27873i	−0.452988	−	0.522776i	0.482614	−	0.875833i	\(-0.339687\pi\)
							−0.935602	+	0.353057i	\(0.885142\pi\)
\(23\)	5.45851	−	1.18743i	1.13818	−	0.247595i	0.396294	−	0.918124i	\(-0.370296\pi\)
							0.741884	+	0.670528i	\(0.233933\pi\)
\(29\)	−1.65994	−	1.91567i	−0.308243	−	0.355732i	0.580399	−	0.814332i	\(-0.302896\pi\)
							−0.888643	+	0.458600i	\(0.848351\pi\)
\(31\)	−9.48226	+	2.78424i	−1.70306	+	0.500065i	−0.981367	−	0.192144i	\(-0.938456\pi\)
							−0.721697	+	0.692209i	\(0.756638\pi\)
\(37\)	6.16833	+	3.36816i	1.01407	+	0.553722i	0.898221	−	0.439545i	\(-0.144860\pi\)
							0.115846	+	0.993267i	\(0.463042\pi\)
\(41\)	6.16502	+	0.886397i	0.962815	+	0.138432i	0.605756	−	0.795650i	\(-0.292871\pi\)
							0.357058	+	0.934082i	\(0.383780\pi\)
\(43\)	−0.254149	+	0.0947925i	−0.0387573	+	0.0144557i	−0.368768	−	0.929521i	\(-0.620220\pi\)
							0.330011	+	0.943977i	\(0.392948\pi\)
\(47\)	5.03051	−	6.71997i	0.733775	−	0.980209i	−0.266093	−	0.963947i	\(-0.585733\pi\)
							0.999868	−	0.0162612i	\(-0.00517631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.12	✓	1280
5.3	odd	4	inner	605.2.r.a.153.12	yes	1280
121.87	odd	22	inner	605.2.r.a.87.12	yes	1280
605.208	even	44	inner	605.2.r.a.208.12	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.12	✓	1280	1.1	even	1	trivial
605.2.r.a.87.12	yes	1280	121.87	odd	22	inner
605.2.r.a.153.12	yes	1280	5.3	odd	4	inner
605.2.r.a.208.12	yes	1280	605.208	even	44	inner