Embedded newform 605.2.u.a.4.15

• Label: 605.2.u.a.4.15
• 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			4.15
Character	\(\chi\)	\(=\)	605.4
Dual form			605.2.u.a.454.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0908431 + 1.58866i) q^{2} +(-0.905007 - 0.294055i) q^{3} +(-0.528640 - 0.0606558i) q^{4} +(-0.647343 - 2.14031i) q^{5} +(0.549368 - 1.41104i) q^{6} +(0.322482 - 0.763085i) q^{7} +(-0.398305 + 2.30159i) q^{8} +(-1.69448 - 1.23111i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0908431	+	1.58866i	−0.0642358	+	1.12336i	0.793266	+	0.608875i	\(0.208379\pi\)
							−0.857502	+	0.514481i	\(0.827985\pi\)
\(3\)	−0.905007	−	0.294055i	−0.522506	−	0.169772i	0.0358764	−	0.999356i	\(-0.488578\pi\)
							−0.558382	+	0.829584i	\(0.688578\pi\)
\(5\)	−0.647343	−	2.14031i	−0.289500	−	0.957178i
							\(7\)	0.322482	−	0.763085i	0.121887	−	0.288419i	−0.848822	−	0.528679i	\(-0.822687\pi\)
0.970708	+	0.240260i	\(0.0772328\pi\)
\(11\)	−2.37487	+	2.31516i	−0.716051	+	0.698048i
							\(13\)	0.849799	+	1.50480i	0.235692	+	0.417356i	0.963217	−	0.268725i	\(-0.0866023\pi\)
−0.727525	+	0.686081i	\(0.759330\pi\)
\(17\)	2.57698	+	7.22248i	0.625009	+	1.75171i	0.658841	+	0.752282i	\(0.271047\pi\)
							−0.0338320	+	0.999428i	\(0.510771\pi\)
\(19\)	−0.0502574	+	1.75924i	−0.0115298	+	0.403597i	0.973246	+	0.229767i	\(0.0737963\pi\)
							−0.984776	+	0.173830i	\(0.944386\pi\)
\(23\)	−6.31516	+	5.47212i	−1.31680	+	1.14102i	−0.336898	+	0.941541i	\(0.609378\pi\)
							−0.979903	+	0.199474i	\(0.936077\pi\)
\(29\)	5.36138	+	7.84076i	0.995583	+	1.45599i	0.886925	+	0.461914i	\(0.152837\pi\)
							0.108658	+	0.994079i	\(0.465345\pi\)
\(31\)	0.982446	−	4.84856i	0.176452	−	0.870827i	−0.789827	−	0.613330i	\(-0.789829\pi\)
							0.966279	−	0.257497i	\(-0.0828978\pi\)
\(37\)	0.550291	+	0.599581i	0.0904672	+	0.0985705i	0.780324	−	0.625375i	\(-0.215054\pi\)
							−0.689857	+	0.723945i	\(0.742327\pi\)
\(41\)	−1.36443	−	5.19082i	−0.213088	−	0.810670i	−0.984654	−	0.174520i	\(-0.944163\pi\)
							0.771566	−	0.636149i	\(-0.219474\pi\)
\(43\)	−2.72530	+	0.391840i	−0.415605	+	0.0597550i	−0.346946	−	0.937885i	\(-0.612781\pi\)
							−0.0686592	+	0.997640i	\(0.521872\pi\)
\(47\)	0.504984	+	0.617569i	0.0736594	+	0.0900817i	0.809660	−	0.586899i	\(-0.199652\pi\)
							−0.736001	+	0.676981i	\(0.763288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.u.a.4.15	✓	2560
5.4	even	2	inner	605.2.u.a.4.50	yes	2560
121.91	even	55	inner	605.2.u.a.454.50	yes	2560
605.454	even	110	inner	605.2.u.a.454.15	yes	2560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.u.a.4.15	✓	2560	1.1	even	1	trivial
605.2.u.a.4.50	yes	2560	5.4	even	2	inner
605.2.u.a.454.15	yes	2560	605.454	even	110	inner
605.2.u.a.454.50	yes	2560	121.91	even	55	inner