Embedded newform 605.2.w.a.182.20

• Label: 605.2.w.a.182.20
• Level: $605$
• Weight: $2$
• Character: 605.182
• 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			182.20
Character	\(\chi\)	\(=\)	605.182
Dual form			605.2.w.a.123.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939036 - 0.886874i) q^{2} +(-0.399554 + 0.0632831i) q^{3} +(-0.0189344 - 0.331126i) q^{4} +(2.22863 + 0.182258i) q^{5} +(0.431319 + 0.294929i) q^{6} +(0.870500 + 0.577165i) q^{7} +(-1.93952 + 2.30400i) q^{8} +(-2.69753 + 0.876481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5120 q - 78 q^{2} - 60 q^{3} - 86 q^{5} - 156 q^{6} - 88 q^{7} - 78 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{109}{110}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939036	−	0.886874i	−0.663999	−	0.627114i	0.278850	−	0.960335i	\(-0.410047\pi\)
							−0.942849	+	0.333220i	\(0.891865\pi\)
\(3\)	−0.399554	+	0.0632831i	−0.230682	+	0.0365365i	−0.270705	−	0.962662i	\(-0.587257\pi\)
							0.0400229	+	0.999199i	\(0.487257\pi\)
\(5\)	2.22863	+	0.182258i	0.996673	+	0.0815085i
							\(7\)	0.870500	+	0.577165i	0.329018	+	0.218148i	0.706236	−	0.707977i	\(-0.250392\pi\)
−0.377218	+	0.926125i	\(0.623119\pi\)
\(11\)	2.79061	−	1.79234i	0.841402	−	0.540410i
							\(13\)	4.96267	−	2.89678i	1.37640	−	0.803423i	0.383632	−	0.923486i	\(-0.374673\pi\)
0.992766	+	0.120063i	\(0.0383096\pi\)
\(17\)	−2.83807	+	2.00069i	−0.688333	+	0.485238i	−0.866767	−	0.498714i	\(-0.833806\pi\)
							0.178434	+	0.983952i	\(0.442897\pi\)
\(19\)	0.0301045	−	0.0309768i	0.00690645	−	0.00710657i	−0.713670	−	0.700482i	\(-0.752968\pi\)
							0.720576	+	0.693376i	\(0.243877\pi\)
\(23\)	8.58221	+	3.20100i	1.78952	+	0.667455i	0.998446	+	0.0557279i	\(0.0177480\pi\)
							0.791069	+	0.611727i	\(0.209525\pi\)
\(29\)	−1.03974	−	1.97053i	−0.193076	−	0.365919i	0.768764	−	0.639532i	\(-0.220872\pi\)
							−0.961840	+	0.273614i	\(0.911781\pi\)
\(31\)	−4.75773	−	3.89037i	−0.854513	−	0.698732i	0.100615	−	0.994925i	\(-0.467919\pi\)
							−0.955128	+	0.296193i	\(0.904283\pi\)
\(37\)	2.53591	−	1.11463i	0.416901	−	0.183243i	−0.183363	−	0.983045i	\(-0.558699\pi\)
							0.600264	+	0.799802i	\(0.295062\pi\)
\(41\)	−1.86827	+	1.44065i	−0.291775	+	0.224992i	−0.746084	−	0.665852i	\(-0.768068\pi\)
							0.454309	+	0.890844i	\(0.349886\pi\)
\(43\)	−0.0648355	+	0.906518i	−0.00988732	+	0.138243i	−0.999998	−	0.00204631i	\(-0.999349\pi\)
							0.990111	+	0.140289i	\(0.0448032\pi\)
\(47\)	−5.24584	−	11.0658i	−0.765185	−	1.61412i	−0.788993	−	0.614402i	\(-0.789397\pi\)
							0.0238079	−	0.999717i	\(-0.492421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.w.a.182.20	yes	5120
5.3	odd	4	inner	605.2.w.a.303.20	yes	5120
121.2	odd	110	inner	605.2.w.a.2.20	✓	5120
605.123	even	220	inner	605.2.w.a.123.20	yes	5120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.w.a.2.20	✓	5120	121.2	odd	110	inner
605.2.w.a.123.20	yes	5120	605.123	even	220	inner
605.2.w.a.182.20	yes	5120	1.1	even	1	trivial
605.2.w.a.303.20	yes	5120	5.3	odd	4	inner