Embedded newform 605.2.w.a.2.13

• Label: 605.2.w.a.2.13
• Level: $605$
• Weight: $2$
• Character: 605.2
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $5120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.w (of order \(220\), degree \(80\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(5120\)
Relative dimension:	\(64\) over \(\Q(\zeta_{220})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{220}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	605.2
Dual form			605.2.w.a.303.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34797 - 1.42725i) q^{2} +(0.193438 - 1.22132i) q^{3} +(-0.105846 + 1.85104i) q^{4} +(2.15715 - 0.588819i) q^{5} +(-2.00388 + 1.37021i) q^{6} +(1.76243 + 2.65815i) q^{7} +(-0.219176 + 0.184504i) q^{8} +(1.39897 + 0.454553i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5120 q - 78 q^{2} - 60 q^{3} - 86 q^{5} - 156 q^{6} - 88 q^{7} - 78 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{110}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34797	−	1.42725i	−0.953158	−	1.00922i	−0.999947	−	0.0103138i	\(-0.996717\pi\)
							0.0467887	−	0.998905i	\(-0.485101\pi\)
\(3\)	0.193438	−	1.22132i	0.111681	−	0.705128i	−0.866779	−	0.498693i	\(-0.833814\pi\)
							0.978460	−	0.206436i	\(-0.0661864\pi\)
\(5\)	2.15715	−	0.588819i	0.964706	−	0.263328i
							\(7\)	1.76243	+	2.65815i	0.666135	+	1.00469i	0.998113	+	0.0614003i	\(0.0195566\pi\)
−0.331978	+	0.943287i	\(0.607716\pi\)
\(11\)	−2.81347	+	1.75625i	−0.848292	+	0.529529i
							\(13\)	0.717779	−	1.22968i	0.199076	−	0.341051i	−0.747095	−	0.664717i	\(-0.768552\pi\)
0.946171	+	0.323667i	\(0.104916\pi\)
\(17\)	3.14264	−	4.45798i	0.762202	−	1.08122i	−0.231897	−	0.972740i	\(-0.574493\pi\)
							0.994099	−	0.108478i	\(-0.0345978\pi\)
\(19\)	3.69997	+	3.80717i	0.848830	+	0.873426i	0.993238	−	0.116092i	\(-0.0370369\pi\)
							−0.144408	+	0.989518i	\(0.546128\pi\)
\(23\)	1.17600	+	3.15298i	0.245213	+	0.657442i	0.999999	+	0.00147072i	\(0.000468146\pi\)
							−0.754786	+	0.655972i	\(0.772259\pi\)
\(29\)	−3.75017	+	7.10737i	−0.696390	+	1.31981i	0.240660	+	0.970610i	\(0.422636\pi\)
							−0.937050	+	0.349196i	\(0.886455\pi\)
\(31\)	3.86656	−	3.16167i	0.694455	−	0.567853i	−0.218538	−	0.975828i	\(-0.570129\pi\)
							0.912993	+	0.407975i	\(0.133765\pi\)
\(37\)	0.649146	−	1.47688i	0.106719	−	0.242798i	−0.854027	−	0.520229i	\(-0.825847\pi\)
							0.960746	+	0.277431i	\(0.0894829\pi\)
\(41\)	−3.34931	−	2.58270i	−0.523074	−	0.403350i	0.314847	−	0.949143i	\(-0.398047\pi\)
							−0.837920	+	0.545792i	\(0.816229\pi\)
\(43\)	−1.28813	+	0.0921290i	−0.196438	+	0.0140495i	−0.169212	−	0.985580i	\(-0.554122\pi\)
							−0.0272266	+	0.999629i	\(0.508668\pi\)
\(47\)	−3.94890	−	1.87200i	−0.576006	−	0.273060i	0.118019	−	0.993011i	\(-0.462346\pi\)
							−0.694024	+	0.719952i	\(0.744164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.w.a.2.13	✓	5120
5.3	odd	4	inner	605.2.w.a.123.13	yes	5120
121.61	odd	110	inner	605.2.w.a.182.13	yes	5120
605.303	even	220	inner	605.2.w.a.303.13	yes	5120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.w.a.2.13	✓	5120	1.1	even	1	trivial
605.2.w.a.123.13	yes	5120	5.3	odd	4	inner
605.2.w.a.182.13	yes	5120	121.61	odd	110	inner
605.2.w.a.303.13	yes	5120	605.303	even	220	inner