Embedded newform 19.5.f.a.10.4

• Label: 19.5.f.a.10.4
• Level: $19$
• Weight: $5$
• Character: 19.10
• Analytic conductor: $1.964$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.96402929859\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			10.4
Character	\(\chi\)	\(=\)	19.10
Dual form			19.5.f.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.301706 + 0.0531989i) q^{2} +(-9.51563 - 11.3403i) q^{3} +(-14.9469 + 5.44022i) q^{4} +(-0.554323 - 0.201757i) q^{5} +(3.47421 + 2.91521i) q^{6} +(33.9928 - 58.8772i) q^{7} +(8.46520 - 4.88738i) q^{8} +(-23.9894 + 136.050i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 6 q^{2} - 18 q^{3} - 48 q^{4} - 6 q^{5} - 48 q^{7} - 9 q^{8} + 246 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.301706	+	0.0531989i	−0.0754265	+	0.0132997i	−0.211234	−	0.977436i	\(-0.567748\pi\)
							0.135808	+	0.990735i	\(0.456637\pi\)
\(3\)	−9.51563	−	11.3403i	−1.05729	−	1.26003i	−0.964426	−	0.264353i	\(-0.914842\pi\)
							−0.0928659	−	0.995679i	\(-0.529603\pi\)
\(5\)	−0.554323	−	0.201757i	−0.0221729	−	0.00807028i	0.330910	−	0.943662i	\(-0.392644\pi\)
							−0.353083	+	0.935592i	\(0.614867\pi\)
\(7\)	33.9928	−	58.8772i	0.693730	−	1.20158i	−0.276877	−	0.960905i	\(-0.589299\pi\)
							0.970607	−	0.240671i	\(-0.0773673\pi\)
\(11\)	−45.0933	−	78.1039i	−0.372672	−	0.645487i	0.617304	−	0.786725i	\(-0.288225\pi\)
							−0.989976	+	0.141238i	\(0.954892\pi\)
\(13\)	−53.1527	+	63.3449i	−0.314513	+	0.374822i	−0.900022	−	0.435844i	\(-0.856450\pi\)
							0.585509	+	0.810666i	\(0.300894\pi\)
\(17\)	−47.2962	−	268.230i	−0.163655	−	0.928132i	−0.950441	−	0.310906i	\(-0.899368\pi\)
							0.786786	−	0.617226i	\(-0.211744\pi\)
\(19\)	−315.632	+	175.206i	−0.874328	+	0.485336i
							\(23\)	878.173	−	319.629i	1.66006	−	0.604213i	0.669690	−	0.742641i	\(-0.266427\pi\)
0.990373	+	0.138428i	\(0.0442048\pi\)
\(29\)	1279.69	+	225.644i	1.52163	+	0.268305i	0.871074	−	0.491152i	\(-0.163424\pi\)
							0.650558	+	0.759457i	\(0.274535\pi\)
\(31\)	−87.0170	−	50.2393i	−0.0905483	−	0.0522781i	0.454042	−	0.890980i	\(-0.349982\pi\)
							−0.544590	+	0.838702i	\(0.683315\pi\)
\(37\)			1632.28i			1.19231i	0.802868	+	0.596156i	\(0.203306\pi\)
							−0.802868	+	0.596156i	\(0.796694\pi\)
\(41\)	−1488.06	−	1773.41i	−0.885225	−	1.05497i	−0.998116	−	0.0613630i	\(-0.980455\pi\)
							0.112890	−	0.993607i	\(-0.463989\pi\)
\(43\)	501.193	+	182.419i	0.271062	+	0.0986584i	0.473975	−	0.880538i	\(-0.342819\pi\)
							−0.202913	+	0.979197i	\(0.565041\pi\)
\(47\)	216.920	−	1230.21i	0.0981982	−	0.556910i	−0.895522	−	0.445017i	\(-0.853198\pi\)
							0.993720	−	0.111893i	\(-0.0356912\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.5.f.a.10.4	yes	36
19.2	odd	18	inner	19.5.f.a.2.4	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.5.f.a.2.4	✓	36	19.2	odd	18	inner
19.5.f.a.10.4	yes	36	1.1	even	1	trivial