Embedded newform 605.2.o.a.34.13

• Label: 605.2.o.a.34.13
• Level: $605$
• Weight: $2$
• Character: 605.34
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.13
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95165 + 0.280604i) q^{2} -2.17977i q^{3} +(1.81120 - 0.531817i) q^{4} +(0.330301 - 2.21154i) q^{5} +(0.611653 + 4.25414i) q^{6} +(-1.97410 + 3.07175i) q^{7} +(0.201476 - 0.0920112i) q^{8} -1.75139 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95165	+	0.280604i	−1.38002	+	0.198417i	−0.792047	−	0.610461i	\(-0.790984\pi\)
							−0.587976	+	0.808878i	\(0.700075\pi\)
\(3\)		−	2.17977i		−	1.25849i	−0.777207	−	0.629245i	\(-0.783364\pi\)
							0.777207	−	0.629245i	\(-0.216636\pi\)
\(5\)	0.330301	−	2.21154i	0.147715	−	0.989030i
							\(7\)	−1.97410	+	3.07175i	−0.746138	+	1.16101i	0.235798	+	0.971802i	\(0.424230\pi\)
−0.981936	+	0.189212i	\(0.939407\pi\)
\(11\)	2.43045	+	2.25675i	0.732808	+	0.680435i
							\(13\)	0.285040	+	0.970758i	0.0790560	+	0.269240i	0.989530	−	0.144326i	\(-0.0461013\pi\)
−0.910474	+	0.413566i	\(0.864283\pi\)
\(17\)	−2.93016	+	2.53900i	−0.710668	+	0.615797i	−0.933297	−	0.359106i	\(-0.883082\pi\)
							0.222629	+	0.974903i	\(0.428536\pi\)
\(19\)	−4.53028	+	5.22822i	−1.03932	+	1.19944i	−0.0597729	+	0.998212i	\(0.519038\pi\)
							−0.979545	+	0.201225i	\(0.935508\pi\)
\(23\)	3.22092	+	5.01186i	0.671609	+	1.04504i	0.995106	+	0.0988151i	\(0.0315052\pi\)
							−0.323497	+	0.946229i	\(0.604858\pi\)
\(29\)	−4.99980	+	5.77008i	−0.928440	+	1.07148i	0.0688290	+	0.997628i	\(0.478074\pi\)
							−0.997269	+	0.0738489i	\(0.976472\pi\)
\(31\)	3.82383	+	1.12278i	0.686780	+	0.201657i	0.606462	−	0.795112i	\(-0.292588\pi\)
							0.0803181	+	0.996769i	\(0.474406\pi\)
\(37\)	1.20076	−	4.08941i	0.197404	−	0.672295i	−0.799981	−	0.600025i	\(-0.795157\pi\)
							0.997385	−	0.0722705i	\(-0.0230245\pi\)
\(41\)	−0.252729	−	1.75777i	−0.0394697	−	0.274518i	0.960524	−	0.278199i	\(-0.0897374\pi\)
							−0.999993	+	0.00368109i	\(0.998828\pi\)
\(43\)	0.225254	−	0.102870i	0.0343510	−	0.0156876i	−0.398165	−	0.917314i	\(-0.630353\pi\)
							0.432516	+	0.901626i	\(0.357626\pi\)
\(47\)	7.27733	+	1.04632i	1.06151	+	0.152622i	0.650880	−	0.759181i	\(-0.274400\pi\)
							0.410628	+	0.911803i	\(0.365309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.13	✓	640
5.4	even	2	inner	605.2.o.a.34.52	yes	640
121.89	even	11	inner	605.2.o.a.89.52	yes	640
605.89	even	22	inner	605.2.o.a.89.13	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.13	✓	640	1.1	even	1	trivial
605.2.o.a.34.52	yes	640	5.4	even	2	inner
605.2.o.a.89.13	yes	640	605.89	even	22	inner
605.2.o.a.89.52	yes	640	121.89	even	11	inner