Embedded newform 605.2.o.a.34.1

• Label: 605.2.o.a.34.1
• Level: $605$
• Weight: $2$
• Character: 605.34
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.1
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.67094 + 0.384024i) q^{2} -1.90400i q^{3} +(5.06748 - 1.48795i) q^{4} +(2.16316 - 0.566331i) q^{5} +(0.731183 + 5.08549i) q^{6} +(-0.410154 + 0.638213i) q^{7} +(-8.05443 + 3.67833i) q^{8} -0.625232 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.67094	+	0.384024i	−1.88864	+	0.271546i	−0.986974	−	0.160882i	\(-0.948566\pi\)
							−0.901669	+	0.432428i	\(0.857657\pi\)
\(3\)		−	1.90400i		−	1.09928i	−0.835402	−	0.549639i	\(-0.814765\pi\)
							0.835402	−	0.549639i	\(-0.185235\pi\)
\(5\)	2.16316	−	0.566331i	0.967395	−	0.253271i
							\(7\)	−0.410154	+	0.638213i	−0.155024	+	0.241222i	−0.910075	−	0.414445i	\(-0.863976\pi\)
0.755051	+	0.655666i	\(0.227612\pi\)
\(11\)	−0.227707	−	3.30880i	−0.0686564	−	0.997640i
							\(13\)	−0.306981	−	1.04548i	−0.0851413	−	0.289965i	0.905905	−	0.423480i	\(-0.139192\pi\)
−0.991047	+	0.133515i	\(0.957373\pi\)
\(17\)	5.18168	−	4.48995i	1.25674	−	1.08897i	0.264535	−	0.964376i	\(-0.414782\pi\)
							0.992207	−	0.124597i	\(-0.0397639\pi\)
\(19\)	−4.49102	+	5.18291i	−1.03031	+	1.18904i	−0.0485697	+	0.998820i	\(0.515466\pi\)
							−0.981741	+	0.190222i	\(0.939079\pi\)
\(23\)	−1.13735	−	1.76976i	−0.237155	−	0.369020i	0.702193	−	0.711987i	\(-0.252204\pi\)
							−0.939347	+	0.342967i	\(0.888568\pi\)
\(29\)	0.510164	−	0.588761i	0.0947351	−	0.109330i	−0.706402	−	0.707810i	\(-0.749683\pi\)
							0.801138	+	0.598480i	\(0.204229\pi\)
\(31\)	−0.150040	−	0.0440557i	−0.0269480	−	0.00791265i	0.268231	−	0.963355i	\(-0.413561\pi\)
							−0.295179	+	0.955442i	\(0.595379\pi\)
\(37\)	−2.49223	+	8.48774i	−0.409719	+	1.39538i	0.453820	+	0.891094i	\(0.350061\pi\)
							−0.863539	+	0.504282i	\(0.831757\pi\)
\(41\)	−0.608451	−	4.23187i	−0.0950241	−	0.660907i	−0.980543	−	0.196304i	\(-0.937106\pi\)
							0.885519	−	0.464603i	\(-0.153803\pi\)
\(43\)	8.90370	−	4.06618i	1.35780	−	0.620087i	0.402418	−	0.915456i	\(-0.368170\pi\)
							0.955383	+	0.295369i	\(0.0954426\pi\)
\(47\)	−1.05557	−	0.151767i	−0.153970	−	0.0221376i	0.0648987	−	0.997892i	\(-0.479328\pi\)
							−0.218869	+	0.975754i	\(0.570237\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.1	✓	640
5.4	even	2	inner	605.2.o.a.34.64	yes	640
121.89	even	11	inner	605.2.o.a.89.64	yes	640
605.89	even	22	inner	605.2.o.a.89.1	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.1	✓	640	1.1	even	1	trivial
605.2.o.a.34.64	yes	640	5.4	even	2	inner
605.2.o.a.89.1	yes	640	605.89	even	22	inner
605.2.o.a.89.64	yes	640	121.89	even	11	inner