Embedded newform 605.2.k.a.56.8

• Label: 605.2.k.a.56.8
• Level: $605$
• Weight: $2$
• Character: 605.56
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			56.8
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.a.551.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12455 - 0.330198i) q^{2} -2.26851 q^{3} +(-0.526923 - 0.338633i) q^{4} +(-0.654861 + 0.755750i) q^{5} +(2.55105 + 0.749057i) q^{6} +(1.97574 - 4.32626i) q^{7} +(2.01576 + 2.32632i) q^{8} +2.14613 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 4 q^{2} - 4 q^{3} - 24 q^{4} - 22 q^{5} - 12 q^{6} - 8 q^{7} - 12 q^{8} + 212 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12455	−	0.330198i	−0.795178	−	0.233485i	−0.141182	−	0.989984i	\(-0.545090\pi\)
							−0.653996	+	0.756498i	\(0.726908\pi\)
\(3\)	−2.26851			−1.30972			−0.654862	−	0.755749i	\(-0.727273\pi\)
							−0.654862	+	0.755749i	\(0.727273\pi\)
\(5\)	−0.654861	+	0.755750i	−0.292863	+	0.337981i
							\(7\)	1.97574	−	4.32626i	0.746758	−	1.63517i	−0.0253493	−	0.999679i	\(-0.508070\pi\)
0.772107	−	0.635493i	\(-0.219203\pi\)
\(11\)	−2.24762	+	2.43889i	−0.677683	+	0.735354i
							\(13\)	3.33966	+	2.14627i	0.926254	+	0.595267i	0.914466	−	0.404663i	\(-0.132611\pi\)
0.0117880	+	0.999931i	\(0.496248\pi\)
\(17\)	1.16501	+	8.10284i	0.282557	+	1.96523i	0.260517	+	0.965469i	\(0.416107\pi\)
							0.0220401	+	0.999757i	\(0.492984\pi\)
\(19\)	0.463974	−	3.22701i	0.106443	−	0.740328i	−0.864779	−	0.502152i	\(-0.832542\pi\)
							0.971222	−	0.238175i	\(-0.0765493\pi\)
\(23\)	−2.80895	−	6.15074i	−0.585706	−	1.28252i	−0.938002	−	0.346629i	\(-0.887326\pi\)
							0.352296	−	0.935889i	\(-0.385401\pi\)
\(29\)	0.395397	−	2.75005i	0.0734234	−	0.510671i	−0.919609	−	0.392834i	\(-0.871495\pi\)
							0.993033	−	0.117837i	\(-0.0375961\pi\)
\(31\)	2.38203	−	1.53084i	0.427825	−	0.274947i	−0.308958	−	0.951076i	\(-0.599980\pi\)
							0.736783	+	0.676129i	\(0.236344\pi\)
\(37\)	7.24071	−	4.65332i	1.19037	−	0.765001i	0.213102	−	0.977030i	\(-0.431643\pi\)
							0.977263	+	0.212029i	\(0.0680070\pi\)
\(41\)	−8.20199	−	2.40832i	−1.28094	−	0.376117i	−0.430689	−	0.902501i	\(-0.641729\pi\)
							−0.850247	+	0.526384i	\(0.823547\pi\)
\(43\)	−1.61116	−	1.85938i	−0.245700	−	0.283553i	0.619482	−	0.785011i	\(-0.287343\pi\)
							−0.865182	+	0.501458i	\(0.832797\pi\)
\(47\)	2.65074	−	0.778327i	0.386650	−	0.113531i	−0.0826297	−	0.996580i	\(-0.526332\pi\)
							0.469279	+	0.883050i	\(0.344514\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.a.56.8	✓	220
121.67	even	11	inner	605.2.k.a.551.8	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.a.56.8	✓	220	1.1	even	1	trivial
605.2.k.a.551.8	yes	220	121.67	even	11	inner