Embedded newform 605.2.u.a.4.11

• Label: 605.2.u.a.4.11
• Level: $605$
• Weight: $2$
• Character: 605.4
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $2560$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.u (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(2560\)
Relative dimension:	\(64\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

Embedding invariants

Embedding label			4.11
Character	\(\chi\)	\(=\)	605.4
Dual form			605.2.u.a.454.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.120780 + 2.11219i) q^{2} +(1.16323 + 0.377956i) q^{3} +(-2.45981 - 0.282237i) q^{4} +(2.04224 + 0.910624i) q^{5} +(-0.938810 + 2.41131i) q^{6} +(-1.54461 + 3.65497i) q^{7} +(0.171705 - 0.992189i) q^{8} +(-1.21680 - 0.884057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.120780	+	2.11219i	−0.0854041	+	1.49355i	0.619922	+	0.784663i	\(0.287164\pi\)
							−0.705326	+	0.708883i	\(0.749199\pi\)
\(3\)	1.16323	+	0.377956i	0.671590	+	0.218213i	0.624910	−	0.780697i	\(-0.285136\pi\)
							0.0466806	+	0.998910i	\(0.485136\pi\)
\(5\)	2.04224	+	0.910624i	0.913320	+	0.407244i
							\(7\)	−1.54461	+	3.65497i	−0.583806	+	1.38145i	0.317496	+	0.948260i	\(0.397158\pi\)
−0.901302	+	0.433191i	\(0.857388\pi\)
\(11\)	−1.83505	−	2.76271i	−0.553290	−	0.832989i
							\(13\)	−1.99229	−	3.52788i	−0.552561	−	0.978459i	−0.996595	−	0.0824488i	\(-0.973726\pi\)
0.444034	−	0.896010i	\(-0.353547\pi\)
\(17\)	2.50368	+	7.01705i	0.607232	+	1.70189i	0.707308	+	0.706906i	\(0.249909\pi\)
							−0.100076	+	0.994980i	\(0.531909\pi\)
\(19\)	−0.169346	+	5.92787i	−0.0388506	+	1.35995i	0.713255	+	0.700904i	\(0.247220\pi\)
							−0.752106	+	0.659042i	\(0.770962\pi\)
\(23\)	3.86211	−	3.34653i	0.805305	−	0.697801i	−0.151524	−	0.988454i	\(-0.548418\pi\)
							0.956829	+	0.290653i	\(0.0938725\pi\)
\(29\)	1.55392	+	2.27253i	0.288555	+	0.421999i	0.942327	−	0.334693i	\(-0.108633\pi\)
							−0.653772	+	0.756692i	\(0.726814\pi\)
\(31\)	−0.356242	+	1.75812i	−0.0639830	+	0.315768i	−0.999273	−	0.0381126i	\(-0.987865\pi\)
							0.935290	+	0.353881i	\(0.115138\pi\)
\(37\)	5.00138	+	5.44937i	0.822222	+	0.895870i	0.996040	−	0.0889076i	\(-0.0283376\pi\)
							−0.173818	+	0.984778i	\(0.555610\pi\)
\(41\)	−0.970512	−	3.69221i	−0.151569	−	0.576627i	−0.998957	−	0.0456537i	\(-0.985463\pi\)
							0.847389	−	0.530973i	\(-0.178173\pi\)
\(43\)	8.02879	−	1.15437i	1.22438	−	0.176039i	0.500359	−	0.865818i	\(-0.333201\pi\)
							0.724020	+	0.689779i	\(0.242292\pi\)
\(47\)	0.310735	+	0.380013i	0.0453254	+	0.0554307i	0.796308	−	0.604891i	\(-0.206783\pi\)
							−0.750983	+	0.660322i	\(0.770420\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.u.a.4.11	✓	2560
5.4	even	2	inner	605.2.u.a.4.54	yes	2560
121.91	even	55	inner	605.2.u.a.454.54	yes	2560
605.454	even	110	inner	605.2.u.a.454.11	yes	2560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.u.a.4.11	✓	2560	1.1	even	1	trivial
605.2.u.a.4.54	yes	2560	5.4	even	2	inner
605.2.u.a.454.11	yes	2560	605.454	even	110	inner
605.2.u.a.454.54	yes	2560	121.91	even	55	inner