Embedded newform 605.2.k.a.56.15

• Label: 605.2.k.a.56.15
• 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			56.15
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.a.551.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.797436 + 0.234148i) q^{2} +0.459960 q^{3} +(-1.10143 - 0.707845i) q^{4} +(-0.654861 + 0.755750i) q^{5} +(0.366789 + 0.107699i) q^{6} +(0.797120 - 1.74545i) q^{7} +(-1.80109 - 2.07857i) q^{8} -2.78844 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 4 q^{2} - 4 q^{3} - 24 q^{4} - 22 q^{5} - 12 q^{6} - 8 q^{7} - 12 q^{8} + 212 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.797436	+	0.234148i	0.563873	+	0.165568i	0.551232	−	0.834352i	\(-0.314158\pi\)
							0.0126408	+	0.999920i	\(0.495976\pi\)
\(3\)	0.459960			0.265558			0.132779	−	0.991146i	\(-0.457610\pi\)
							0.132779	+	0.991146i	\(0.457610\pi\)
\(5\)	−0.654861	+	0.755750i	−0.292863	+	0.337981i
							\(7\)	0.797120	−	1.74545i	0.301283	−	0.659718i	−0.697075	−	0.716998i	\(-0.745516\pi\)
0.998358	+	0.0572803i	\(0.0182429\pi\)
\(11\)	1.89600	−	2.72125i	0.571665	−	0.820487i
							\(13\)	0.748896	+	0.481286i	0.207706	+	0.133485i	0.640358	−	0.768076i	\(-0.278786\pi\)
−0.432652	+	0.901561i	\(0.642422\pi\)
\(17\)	0.0467294	+	0.325010i	0.0113335	+	0.0788266i	0.994703	−	0.102789i	\(-0.0327765\pi\)
							−0.983370	+	0.181615i	\(0.941867\pi\)
\(19\)	1.10763	−	7.70373i	0.254107	−	1.76736i	−0.318883	−	0.947794i	\(-0.603308\pi\)
							0.572991	−	0.819562i	\(-0.305783\pi\)
\(23\)	−3.76274	−	8.23925i	−0.784585	−	1.71800i	−0.691537	−	0.722341i	\(-0.743066\pi\)
							−0.0930483	−	0.995662i	\(-0.529661\pi\)
\(29\)	−0.0414284	+	0.288141i	−0.00769305	+	0.0535064i	−0.993306	−	0.115510i	\(-0.963150\pi\)
							0.985613	+	0.169016i	\(0.0540590\pi\)
\(31\)	−6.82467	+	4.38595i	−1.22575	+	0.787740i	−0.983223	−	0.182405i	\(-0.941612\pi\)
							−0.242524	+	0.970145i	\(0.577975\pi\)
\(37\)	6.00847	−	3.86141i	0.987786	−	0.634811i	0.0562330	−	0.998418i	\(-0.482091\pi\)
							0.931553	+	0.363606i	\(0.118455\pi\)
\(41\)	10.9211	+	3.20671i	1.70558	+	0.500804i	0.981911	−	0.189341i	\(-0.0606352\pi\)
							0.723671	+	0.690145i	\(0.242453\pi\)
\(43\)	−1.84832	−	2.13307i	−0.281866	−	0.325291i	0.597108	−	0.802161i	\(-0.296317\pi\)
							−0.878974	+	0.476870i	\(0.841771\pi\)
\(47\)	−1.14770	+	0.336995i	−0.167409	+	0.0491557i	−0.364363	−	0.931257i	\(-0.618713\pi\)
							0.196954	+	0.980413i	\(0.436895\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.a.56.15	✓	220
121.67	even	11	inner	605.2.k.a.551.15	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.a.56.15	✓	220	1.1	even	1	trivial
605.2.k.a.551.15	yes	220	121.67	even	11	inner