Embedded newform 605.2.k.a.56.5

• Label: 605.2.k.a.56.5
• Level: $605$
• Weight: $2$
• Character: 605.56
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			56.5
Character	\(\chi\)	\(=\)	605.56
Dual form			605.2.k.a.551.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92738 - 0.565928i) q^{2} -3.27440 q^{3} +(1.71199 + 1.10023i) q^{4} +(-0.654861 + 0.755750i) q^{5} +(6.31101 + 1.85308i) q^{6} +(-0.190974 + 0.418175i) q^{7} +(-0.0461080 - 0.0532114i) q^{8} +7.72172 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 4 q^{2} - 4 q^{3} - 24 q^{4} - 22 q^{5} - 12 q^{6} - 8 q^{7} - 12 q^{8} + 212 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92738	−	0.565928i	−1.36286	−	0.400172i	−0.483090	−	0.875571i	\(-0.660486\pi\)
							−0.879770	+	0.475399i	\(0.842304\pi\)
\(3\)	−3.27440			−1.89048			−0.945239	−	0.326379i	\(-0.894172\pi\)
							−0.945239	+	0.326379i	\(0.894172\pi\)
\(5\)	−0.654861	+	0.755750i	−0.292863	+	0.337981i
							\(7\)	−0.190974	+	0.418175i	−0.0721815	+	0.158055i	−0.942283	−	0.334817i	\(-0.891326\pi\)
0.870102	+	0.492872i	\(0.164053\pi\)
\(11\)	−2.42742	−	2.26001i	−0.731895	−	0.681417i
							\(13\)	−2.82542	−	1.81578i	−0.783629	−	0.503608i	0.0866071	−	0.996243i	\(-0.472398\pi\)
−0.870236	+	0.492635i	\(0.836034\pi\)
\(17\)	−0.791756	−	5.50679i	−0.192029	−	1.33559i	−0.826627	−	0.562750i	\(-0.809744\pi\)
							0.634598	−	0.772842i	\(-0.281166\pi\)
\(19\)	−0.601963	+	4.18674i	−0.138100	+	0.960505i	0.796458	+	0.604694i	\(0.206705\pi\)
							−0.934558	+	0.355811i	\(0.884205\pi\)
\(23\)	−0.217672	−	0.476634i	−0.0453877	−	0.0993851i	0.885580	−	0.464487i	\(-0.153761\pi\)
							−0.930967	+	0.365102i	\(0.881034\pi\)
\(29\)	0.868960	−	6.04375i	0.161362	−	1.12230i	−0.734708	−	0.678383i	\(-0.762681\pi\)
							0.896070	−	0.443913i	\(-0.146410\pi\)
\(31\)	4.59784	−	2.95485i	0.825797	−	0.530707i	−0.0581429	−	0.998308i	\(-0.518518\pi\)
							0.883940	+	0.467601i	\(0.154882\pi\)
\(37\)	1.03761	−	0.666833i	0.170583	−	0.109627i	−0.452564	−	0.891732i	\(-0.649490\pi\)
							0.623146	+	0.782105i	\(0.285854\pi\)
\(41\)	−1.51809	−	0.445751i	−0.237085	−	0.0696145i	0.161031	−	0.986949i	\(-0.448518\pi\)
							−0.398117	+	0.917335i	\(0.630336\pi\)
\(43\)	7.14558	+	8.24644i	1.08969	+	1.25757i	0.964115	+	0.265486i	\(0.0855323\pi\)
							0.125575	+	0.992084i	\(0.459922\pi\)
\(47\)	−7.97510	+	2.34170i	−1.16329	+	0.341572i	−0.805709	−	0.592311i	\(-0.798216\pi\)
							−0.357578	+	0.933883i	\(0.616397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.k.a.56.5	✓	220
121.67	even	11	inner	605.2.k.a.551.5	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.k.a.56.5	✓	220	1.1	even	1	trivial
605.2.k.a.551.5	yes	220	121.67	even	11	inner