Embedded newform 605.2.u.a.454.18

• Label: 605.2.u.a.454.18
• Level: $605$
• Weight: $2$
• Character: 605.454
• 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			454.18
Character	\(\chi\)	\(=\)	605.454
Dual form			605.2.u.a.4.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0843987 - 1.47596i) q^{2} +(-1.88212 + 0.611538i) q^{3} +(-0.184383 + 0.0211560i) q^{4} +(-1.88871 - 1.19699i) q^{5} +(1.06146 + 2.72633i) q^{6} +(1.72780 + 4.08846i) q^{7} +(-0.457404 - 2.64309i) q^{8} +(0.741350 - 0.538623i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{54}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0843987	−	1.47596i	−0.0596789	−	1.04366i	−0.880975	−	0.473162i	\(-0.843112\pi\)
							0.821296	−	0.570502i	\(-0.193251\pi\)
\(3\)	−1.88212	+	0.611538i	−1.08664	+	0.353072i	−0.796948	−	0.604047i	\(-0.793554\pi\)
							−0.289695	+	0.957119i	\(0.593554\pi\)
\(5\)	−1.88871	−	1.19699i	−0.844656	−	0.535310i
							\(7\)	1.72780	+	4.08846i	0.653046	+	1.54529i	0.828008	+	0.560717i	\(0.189474\pi\)
−0.174962	+	0.984575i	\(0.555980\pi\)
\(11\)	3.04141	−	1.32280i	0.917020	−	0.398840i
							\(13\)	1.63135	−	2.88875i	0.452456	−	0.801196i	−0.546895	−	0.837201i	\(-0.684190\pi\)
0.999352	+	0.0360049i	\(0.0114632\pi\)
\(17\)	0.256166	−	0.717954i	0.0621293	−	0.174129i	−0.906747	−	0.421675i	\(-0.861442\pi\)
							0.968876	+	0.247546i	\(0.0796241\pi\)
\(19\)	−0.0639439	−	2.23833i	−0.0146697	−	0.513507i	−0.973390	−	0.229156i	\(-0.926403\pi\)
							0.958720	−	0.284352i	\(-0.0917783\pi\)
\(23\)	−4.43018	−	3.83878i	−0.923757	−	0.800440i	0.0564518	−	0.998405i	\(-0.482021\pi\)
							−0.980209	+	0.197965i	\(0.936567\pi\)
\(29\)	−1.24879	+	1.82630i	−0.231895	+	0.339136i	−0.923815	−	0.382838i	\(-0.874947\pi\)
							0.691920	+	0.721974i	\(0.256765\pi\)
\(31\)	−2.09084	−	10.3187i	−0.375526	−	1.85330i	−0.506958	−	0.861971i	\(-0.669230\pi\)
							0.131432	−	0.991325i	\(-0.458043\pi\)
\(37\)	−2.93836	+	3.20156i	−0.483064	+	0.526333i	−0.929668	−	0.368399i	\(-0.879906\pi\)
							0.446604	+	0.894732i	\(0.352633\pi\)
\(41\)	1.34688	−	5.12405i	0.210347	−	0.800242i	−0.775325	−	0.631563i	\(-0.782414\pi\)
							0.985671	−	0.168678i	\(-0.0539500\pi\)
\(43\)	8.07712	+	1.16131i	1.23175	+	0.177099i	0.727286	−	0.686334i	\(-0.240781\pi\)
							0.504463	+	0.863433i	\(0.331690\pi\)
\(47\)	−1.73868	+	2.12632i	−0.253613	+	0.310156i	−0.885256	−	0.465104i	\(-0.846017\pi\)
							0.631643	+	0.775259i	\(0.282381\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.u.a.454.18	yes	2560
5.4	even	2	inner	605.2.u.a.454.47	yes	2560
121.4	even	55	inner	605.2.u.a.4.47	yes	2560
605.4	even	110	inner	605.2.u.a.4.18	✓	2560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.u.a.4.18	✓	2560	605.4	even	110	inner
605.2.u.a.4.47	yes	2560	121.4	even	55	inner
605.2.u.a.454.18	yes	2560	1.1	even	1	trivial
605.2.u.a.454.47	yes	2560	5.4	even	2	inner