Embedded newform 605.2.w.a.2.11

• Label: 605.2.w.a.2.11
• Level: $605$
• Weight: $2$
• Character: 605.2
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $5120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.w (of order \(220\), degree \(80\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(5120\)
Relative dimension:	\(64\) over \(\Q(\zeta_{220})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{220}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	605.2
Dual form			605.2.w.a.303.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48817 - 1.57569i) q^{2} +(-0.399974 + 2.52534i) q^{3} +(-0.153994 + 2.69305i) q^{4} +(0.336311 - 2.21063i) q^{5} +(4.57439 - 3.12789i) q^{6} +(2.61648 + 3.94627i) q^{7} +(1.15643 - 0.973491i) q^{8} +(-3.36419 - 1.09309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5120 q - 78 q^{2} - 60 q^{3} - 86 q^{5} - 156 q^{6} - 88 q^{7} - 78 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{110}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48817	−	1.57569i	−1.05229	−	1.11418i	−0.993258	−	0.115928i	\(-0.963016\pi\)
							−0.0590350	−	0.998256i	\(-0.518802\pi\)
\(3\)	−0.399974	+	2.52534i	−0.230925	+	1.45801i	0.550933	+	0.834549i	\(0.314272\pi\)
							−0.781858	+	0.623456i	\(0.785728\pi\)
\(5\)	0.336311	−	2.21063i	0.150403	−	0.988625i
							\(7\)	2.61648	+	3.94627i	0.988937	+	1.49155i	0.866565	+	0.499065i	\(0.166323\pi\)
0.122372	+	0.992484i	\(0.460950\pi\)
\(11\)	1.69012	−	2.85368i	0.509590	−	0.860417i
							\(13\)	−0.991497	+	1.69860i	−0.274992	+	0.471107i	−0.968388	−	0.249450i	\(-0.919750\pi\)
0.693396	+	0.720557i	\(0.256114\pi\)
\(17\)	3.03438	−	4.30440i	0.735944	−	1.04397i	−0.260960	−	0.965350i	\(-0.584039\pi\)
							0.996905	−	0.0786213i	\(-0.0250518\pi\)
\(19\)	−0.332523	−	0.342158i	−0.0762860	−	0.0784964i	0.677914	−	0.735141i	\(-0.262884\pi\)
							−0.754200	+	0.656645i	\(0.771975\pi\)
\(23\)	2.05293	+	5.50413i	0.428066	+	1.14769i	0.954693	+	0.297593i	\(0.0961839\pi\)
							−0.526626	+	0.850097i	\(0.676543\pi\)
\(29\)	−2.41128	+	4.56989i	−0.447764	+	0.848607i	0.552126	+	0.833761i	\(0.313817\pi\)
							−0.999890	+	0.0148463i	\(0.995274\pi\)
\(31\)	−4.02810	+	3.29376i	−0.723468	+	0.591577i	−0.921389	−	0.388641i	\(-0.872945\pi\)
							0.197921	+	0.980218i	\(0.436581\pi\)
\(37\)	0.664588	−	1.51202i	0.109258	−	0.248574i	−0.852366	−	0.522945i	\(-0.824833\pi\)
							0.961624	+	0.274371i	\(0.0884696\pi\)
\(41\)	3.53062	+	2.72252i	0.551390	+	0.425186i	0.848142	−	0.529769i	\(-0.177721\pi\)
							−0.296751	+	0.954955i	\(0.595903\pi\)
\(43\)	10.7985	−	0.772325i	1.64676	−	0.117778i	0.783113	−	0.621879i	\(-0.213631\pi\)
							0.863645	+	0.504101i	\(0.168176\pi\)
\(47\)	4.68301	+	2.22002i	0.683087	+	0.323823i	0.738127	−	0.674662i	\(-0.235710\pi\)
							−0.0550400	+	0.998484i	\(0.517529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.w.a.2.11	✓	5120
5.3	odd	4	inner	605.2.w.a.123.11	yes	5120
121.61	odd	110	inner	605.2.w.a.182.11	yes	5120
605.303	even	220	inner	605.2.w.a.303.11	yes	5120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.w.a.2.11	✓	5120	1.1	even	1	trivial
605.2.w.a.123.11	yes	5120	5.3	odd	4	inner
605.2.w.a.182.11	yes	5120	121.61	odd	110	inner
605.2.w.a.303.11	yes	5120	605.303	even	220	inner