Embedded newform 605.2.s.b.16.13

• Label: 605.2.s.b.16.13
• Level: $605$
• Weight: $2$
• Character: 605.16
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.s (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(22\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			16.13
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.742152 + 0.0851540i) q^{2} +(0.230756 + 0.167654i) q^{3} +(-1.40448 - 0.326599i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(0.156980 + 0.144075i) q^{6} +(0.473372 + 0.487088i) q^{7} +(-2.42169 - 0.864056i) q^{8} +(-0.901911 - 2.77580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.742152	+	0.0851540i	0.524781	+	0.0602130i	0.372334	−	0.928099i	\(-0.378557\pi\)
							0.152447	+	0.988312i	\(0.451285\pi\)
\(3\)	0.230756	+	0.167654i	0.133227	+	0.0967951i	0.652403	−	0.757872i	\(-0.273761\pi\)
							−0.519176	+	0.854667i	\(0.673761\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	0.473372	+	0.487088i	0.178918	+	0.184102i	0.800658	−	0.599121i	\(-0.204483\pi\)
−0.621741	+	0.783223i	\(0.713574\pi\)
\(11\)	1.96261	−	2.67361i	0.591748	−	0.806123i
							\(13\)	3.10068	−	5.14191i	0.859975	−	1.42611i	−0.0442359	−	0.999021i	\(-0.514085\pi\)
0.904211	−	0.427087i	\(-0.140460\pi\)
\(17\)	−1.19068	+	0.973612i	−0.288782	+	0.236135i	−0.766295	−	0.642489i	\(-0.777902\pi\)
							0.477513	+	0.878624i	\(0.341538\pi\)
\(19\)	0.461091	+	0.0263661i	0.105782	+	0.00604881i	0.109899	−	0.993943i	\(-0.464947\pi\)
							−0.00411786	+	0.999992i	\(0.501311\pi\)
\(23\)	0.0315878	−	0.219698i	0.00658650	−	0.0458101i	−0.986263	−	0.165182i	\(-0.947179\pi\)
							0.992850	+	0.119372i	\(0.0380880\pi\)
\(29\)	3.59889	−	9.24368i	0.668297	−	1.71651i	−0.0288542	−	0.999584i	\(-0.509186\pi\)
							0.697151	−	0.716924i	\(-0.254451\pi\)
\(31\)	7.36953	+	3.11439i	1.32361	+	0.559361i	0.932562	−	0.361011i	\(-0.117568\pi\)
							0.391045	+	0.920372i	\(0.372114\pi\)
\(37\)	−0.0106008	+	0.123423i	−0.00174276	+	0.0202906i	−0.997151	−	0.0754253i	\(-0.975969\pi\)
							0.995409	+	0.0957159i	\(0.0305140\pi\)
\(41\)	−7.96216	+	4.49644i	−1.24348	+	0.702226i	−0.966009	−	0.258508i	\(-0.916769\pi\)
							−0.277472	+	0.960734i	\(0.589497\pi\)
\(43\)	−9.47245	+	2.78136i	−1.44454	+	0.424154i	−0.907731	−	0.419553i	\(-0.862187\pi\)
							−0.536805	+	0.843707i	\(0.680369\pi\)
\(47\)	1.96893	−	9.71705i	0.287198	−	1.41738i	−0.533396	−	0.845865i	\(-0.679085\pi\)
							0.820594	−	0.571511i	\(-0.193643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.13	✓	880
121.53	even	55	inner	605.2.s.b.416.13	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.13	✓	880	1.1	even	1	trivial
605.2.s.b.416.13	yes	880	121.53	even	55	inner