Embedded newform 230.3.i.a.19.3

• Label: 230.3.i.a.19.3
• Level: $230$
• Weight: $3$
• Character: 230.19
• Analytic conductor: $6.267$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			19.3
Character	\(\chi\)	\(=\)	230.19
Dual form			230.3.i.a.109.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 + 0.587486i) q^{2} +(-0.859820 - 2.92828i) q^{3} +(1.30972 - 1.51150i) q^{4} +(2.44584 - 4.36095i) q^{5} +(2.82640 + 3.26184i) q^{6} +(1.96650 + 13.6773i) q^{7} +(-0.796860 + 2.71386i) q^{8} +(-0.264236 + 0.169814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28641	+	0.587486i	−0.643207	+	0.293743i
							\(3\)	−0.859820	−	2.92828i	−0.286607	−	0.976092i	−0.969401	−	0.245482i	\(-0.921054\pi\)
0.682795	−	0.730610i	\(-0.260764\pi\)
\(5\)	2.44584	−	4.36095i	0.489168	−	0.872190i
							\(7\)	1.96650	+	13.6773i	0.280929	+	1.95390i	0.299560	+	0.954078i	\(0.403160\pi\)
−0.0186309	+	0.999826i	\(0.505931\pi\)
\(11\)	6.64888	+	3.03644i	0.604444	+	0.276040i	0.694038	−	0.719938i	\(-0.255830\pi\)
							−0.0895944	+	0.995978i	\(0.528557\pi\)
\(13\)	21.6520	+	3.11308i	1.66554	+	0.239468i	0.909692	−	0.415283i	\(-0.136317\pi\)
							0.755845	+	0.654751i	\(0.227227\pi\)
\(17\)	−17.9452	−	20.7099i	−1.05560	−	1.21823i	−0.975168	−	0.221467i	\(-0.928916\pi\)
							−0.0804317	−	0.996760i	\(-0.525630\pi\)
\(19\)	3.89374	+	3.37394i	0.204934	+	0.177576i	0.751281	−	0.659983i	\(-0.229437\pi\)
							−0.546347	+	0.837559i	\(0.683982\pi\)
\(23\)	9.09563	−	21.1251i	0.395462	−	0.918482i
							\(29\)	18.1204	+	20.9120i	0.624840	+	0.721104i	0.976619	−	0.214980i	\(-0.0689685\pi\)
−0.351779	+	0.936083i	\(0.614423\pi\)
\(31\)	14.0578	+	4.12775i	0.453478	+	0.133153i	0.500492	−	0.865741i	\(-0.333152\pi\)
							−0.0470141	+	0.998894i	\(0.514971\pi\)
\(37\)	25.9243	−	16.6605i	0.700656	−	0.450284i	−0.141204	−	0.989981i	\(-0.545097\pi\)
							0.841860	+	0.539696i	\(0.181461\pi\)
\(41\)	17.5368	+	11.2702i	0.427726	+	0.274883i	0.736742	−	0.676174i	\(-0.236363\pi\)
							−0.309016	+	0.951057i	\(0.600000\pi\)
\(43\)	−5.99830	+	1.76126i	−0.139495	+	0.0409596i	−0.350735	−	0.936475i	\(-0.614068\pi\)
							0.211239	+	0.977434i	\(0.432250\pi\)
\(47\)			47.2251i			1.00479i	0.864638	+	0.502395i	\(0.167548\pi\)
							−0.864638	+	0.502395i	\(0.832452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.19.3	✓	240
5.4	even	2	inner	230.3.i.a.19.22	yes	240
23.17	odd	22	inner	230.3.i.a.109.22	yes	240
115.109	odd	22	inner	230.3.i.a.109.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.3	✓	240	1.1	even	1	trivial
230.3.i.a.19.22	yes	240	5.4	even	2	inner
230.3.i.a.109.3	yes	240	115.109	odd	22	inner
230.3.i.a.109.22	yes	240	23.17	odd	22	inner