Embedded newform 605.2.o.a.34.16

• Label: 605.2.o.a.34.16
• 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			34.16
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.51095 + 0.217241i) q^{2} +3.08747i q^{3} +(0.316777 - 0.0930142i) q^{4} +(0.837700 + 2.07322i) q^{5} +(-0.670726 - 4.66500i) q^{6} +(-1.71177 + 2.66356i) q^{7} +(2.31865 - 1.05889i) q^{8} -6.53248 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.51095	+	0.217241i	−1.06840	+	0.153613i	−0.654009	−	0.756486i	\(-0.726914\pi\)
							−0.414391	+	0.910099i	\(0.636005\pi\)
\(3\)			3.08747i			1.78255i	0.453461	+	0.891276i	\(0.350189\pi\)
							−0.453461	+	0.891276i	\(0.649811\pi\)
\(5\)	0.837700	+	2.07322i	0.374631	+	0.927174i
							\(7\)	−1.71177	+	2.66356i	−0.646987	+	1.00673i	0.350547	+	0.936545i	\(0.385996\pi\)
−0.997534	+	0.0701864i	\(0.977641\pi\)
\(11\)	−2.66285	−	1.97718i	−0.802880	−	0.596141i
							\(13\)	−1.23297	−	4.19912i	−0.341966	−	1.16463i	−0.933569	−	0.358398i	\(-0.883323\pi\)
0.591603	−	0.806229i	\(-0.298495\pi\)
\(17\)	−0.407571	+	0.353162i	−0.0988505	+	0.0856544i	−0.702884	−	0.711304i	\(-0.748105\pi\)
							0.604034	+	0.796959i	\(0.293559\pi\)
\(19\)	3.76887	−	4.34951i	0.864638	−	0.997845i	−0.135337	−	0.990800i	\(-0.543212\pi\)
							0.999975	−	0.00704561i	\(-0.00224271\pi\)
\(23\)	3.33102	+	5.18316i	0.694565	+	1.08076i	0.992026	+	0.126031i	\(0.0402237\pi\)
							−0.297461	+	0.954734i	\(0.596140\pi\)
\(29\)	−4.55939	+	5.26181i	−0.846657	+	0.977094i	−0.999939	−	0.0110887i	\(-0.996470\pi\)
							0.153282	+	0.988182i	\(0.451016\pi\)
\(31\)	1.67465	+	0.491721i	0.300776	+	0.0883157i	0.428638	−	0.903476i	\(-0.358993\pi\)
							−0.127863	+	0.991792i	\(0.540812\pi\)
\(37\)	−3.27770	+	11.1628i	−0.538851	+	1.83516i	0.0110646	+	0.999939i	\(0.496478\pi\)
							−0.549916	+	0.835220i	\(0.685340\pi\)
\(41\)	−1.06867	−	7.43275i	−0.166898	−	1.16080i	−0.885248	−	0.465119i	\(-0.846012\pi\)
							0.718350	−	0.695681i	\(-0.244897\pi\)
\(43\)	0.812682	−	0.371140i	0.123933	−	0.0565983i	−0.352484	−	0.935818i	\(-0.614663\pi\)
							0.476416	+	0.879220i	\(0.341935\pi\)
\(47\)	−0.759325	−	0.109174i	−0.110759	−	0.0159247i	0.0867125	−	0.996233i	\(-0.472364\pi\)
							−0.197471	+	0.980309i	\(0.563273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.16	✓	640
5.4	even	2	inner	605.2.o.a.34.49	yes	640
121.89	even	11	inner	605.2.o.a.89.49	yes	640
605.89	even	22	inner	605.2.o.a.89.16	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.16	✓	640	1.1	even	1	trivial
605.2.o.a.34.49	yes	640	5.4	even	2	inner
605.2.o.a.89.16	yes	640	605.89	even	22	inner
605.2.o.a.89.49	yes	640	121.89	even	11	inner