Embedded newform 605.2.r.a.32.5

• Label: 605.2.r.a.32.5
• 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			32.5
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55047 - 2.07118i) q^{2} +(1.30510 - 1.30510i) q^{3} +(-1.32238 + 4.50362i) q^{4} +(-2.13842 + 0.653577i) q^{5} +(-4.72661 - 0.679584i) q^{6} +(-0.374253 + 1.72041i) q^{7} +(6.52992 - 2.43554i) q^{8} -0.406559i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55047	−	2.07118i	−1.09635	−	1.46455i	−0.871657	−	0.490117i	\(-0.836954\pi\)
							−0.224690	−	0.974430i	\(-0.572137\pi\)
\(3\)	1.30510	−	1.30510i	0.753498	−	0.753498i	−0.221632	−	0.975130i	\(-0.571138\pi\)
							0.975130	+	0.221632i	\(0.0711384\pi\)
\(5\)	−2.13842	+	0.653577i	−0.956330	+	0.292288i
							\(7\)	−0.374253	+	1.72041i	−0.141454	+	0.650255i	0.850743	+	0.525581i	\(0.176152\pi\)
−0.992198	+	0.124674i	\(0.960212\pi\)
\(11\)	−0.806544	−	3.21706i	−0.243182	−	0.969981i
							\(13\)	1.50728	+	2.76039i	0.418046	+	0.765593i	0.998825	−	0.0484678i	\(-0.0154338\pi\)
−0.580779	+	0.814061i	\(0.697252\pi\)
\(17\)	0.188148	+	2.63065i	0.0456326	+	0.638027i	0.967878	+	0.251421i	\(0.0808980\pi\)
							−0.922245	+	0.386606i	\(0.873647\pi\)
\(19\)	1.92685	+	2.22370i	0.442050	+	0.510153i	0.932427	−	0.361358i	\(-0.117687\pi\)
							−0.490378	+	0.871510i	\(0.663141\pi\)
\(23\)	5.60977	−	1.22033i	1.16972	−	0.254456i	0.414574	−	0.910016i	\(-0.363931\pi\)
							0.755144	+	0.655559i	\(0.227567\pi\)
\(29\)	−5.76505	−	6.65322i	−1.07054	−	1.23547i	−0.970656	−	0.240474i	\(-0.922697\pi\)
							−0.0998872	−	0.994999i	\(-0.531848\pi\)
\(31\)	5.31108	−	1.55947i	0.953898	−	0.280090i	0.232489	−	0.972599i	\(-0.425313\pi\)
							0.721409	+	0.692509i	\(0.243495\pi\)
\(37\)	5.32707	+	2.90880i	0.875765	+	0.478204i	0.853226	−	0.521542i	\(-0.174643\pi\)
							0.0225394	+	0.999746i	\(0.492825\pi\)
\(41\)	10.5507	+	1.51696i	1.64774	+	0.236909i	0.902748	−	0.430170i	\(-0.141546\pi\)
							0.744988	+	0.667078i	\(0.232455\pi\)
\(43\)	−1.67069	+	0.623135i	−0.254778	+	0.0950272i	−0.473608	−	0.880736i	\(-0.657049\pi\)
							0.218830	+	0.975763i	\(0.429776\pi\)
\(47\)	−6.32359	+	8.44733i	−0.922391	+	1.23217i	0.0499850	+	0.998750i	\(0.484083\pi\)
							−0.972376	+	0.233420i	\(0.925008\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.5	✓	1280
5.3	odd	4	inner	605.2.r.a.153.5	yes	1280
121.87	odd	22	inner	605.2.r.a.87.5	yes	1280
605.208	even	44	inner	605.2.r.a.208.5	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.5	✓	1280	1.1	even	1	trivial
605.2.r.a.87.5	yes	1280	121.87	odd	22	inner
605.2.r.a.153.5	yes	1280	5.3	odd	4	inner
605.2.r.a.208.5	yes	1280	605.208	even	44	inner