Embedded newform 605.2.o.a.89.55

• Label: 605.2.o.a.89.55
• Level: $605$
• Weight: $2$
• Character: 605.89
• 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			89.55
Character	\(\chi\)	\(=\)	605.89
Dual form			605.2.o.a.34.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.18609 + 0.314312i) q^{2} -1.37149i q^{3} +(2.76121 + 0.810764i) q^{4} +(1.33367 - 1.79480i) q^{5} +(0.431075 - 2.99819i) q^{6} +(2.66241 + 4.14280i) q^{7} +(1.76344 + 0.805337i) q^{8} +1.11903 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.18609	+	0.314312i	1.54580	+	0.222252i	0.861806	−	0.507237i	\(-0.169333\pi\)
							0.683992	+	0.729490i	\(0.260242\pi\)
\(3\)		−	1.37149i		−	0.791828i	−0.918288	−	0.395914i	\(-0.870428\pi\)
							0.918288	−	0.395914i	\(-0.129572\pi\)
\(5\)	1.33367	−	1.79480i	0.596437	−	0.802660i
							\(7\)	2.66241	+	4.14280i	1.00630	+	1.56583i	0.810967	+	0.585092i	\(0.198942\pi\)
0.195331	+	0.980737i	\(0.437422\pi\)
\(11\)	−3.26615	−	0.576408i	−0.984782	−	0.173793i
							\(13\)	−0.872682	+	2.97208i	−0.242039	+	0.824308i	0.745442	+	0.666570i	\(0.232238\pi\)
−0.987481	+	0.157738i	\(0.949580\pi\)
\(17\)	−4.57600	−	3.96513i	−1.10984	−	0.961685i	−0.110362	−	0.993891i	\(-0.535201\pi\)
							−0.999481	+	0.0322068i	\(0.989746\pi\)
\(19\)	3.01019	+	3.47395i	0.690586	+	0.796979i	0.987448	−	0.157941i	\(-0.0504857\pi\)
							−0.296862	+	0.954920i	\(0.595940\pi\)
\(23\)	1.45862	−	2.26965i	0.304143	−	0.473255i	−0.655217	−	0.755440i	\(-0.727423\pi\)
							0.959360	+	0.282185i	\(0.0910593\pi\)
\(29\)	−1.70768	−	1.97077i	−0.317109	−	0.365963i	0.574709	−	0.818358i	\(-0.305115\pi\)
							−0.891818	+	0.452395i	\(0.850570\pi\)
\(31\)	−9.06548	+	2.66186i	−1.62821	+	0.478085i	−0.963208	−	0.268758i	\(-0.913387\pi\)
							−0.665000	+	0.746843i	\(0.731569\pi\)
\(37\)	1.37866	+	4.69528i	0.226650	+	0.771899i	0.991772	+	0.128014i	\(0.0408603\pi\)
							−0.765122	+	0.643885i	\(0.777322\pi\)
\(41\)	−0.0332730	+	0.231419i	−0.00519637	+	0.0361415i	−0.992255	−	0.124220i	\(-0.960357\pi\)
							0.987058	+	0.160362i	\(0.0512661\pi\)
\(43\)	2.50506	+	1.14402i	0.382019	+	0.174462i	0.597162	−	0.802120i	\(-0.296295\pi\)
							−0.215144	+	0.976582i	\(0.569022\pi\)
\(47\)	7.64201	−	1.09875i	1.11470	−	0.160270i	0.439741	−	0.898124i	\(-0.355070\pi\)
							0.674960	+	0.737855i	\(0.264161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.89.55	yes	640
5.4	even	2	inner	605.2.o.a.89.10	yes	640
121.34	even	11	inner	605.2.o.a.34.10	✓	640
605.34	even	22	inner	605.2.o.a.34.55	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.10	✓	640	121.34	even	11	inner
605.2.o.a.34.55	yes	640	605.34	even	22	inner
605.2.o.a.89.10	yes	640	5.4	even	2	inner
605.2.o.a.89.55	yes	640	1.1	even	1	trivial