Embedded newform 230.3.i.a.109.3

• Label: 230.3.i.a.109.3
• Level: $230$
• Weight: $3$
• Character: 230.109
• 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			109.3
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.3

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{22}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.3	✓	240	115.19	odd	22	inner
230.3.i.a.19.22	yes	240	23.19	odd	22	inner
230.3.i.a.109.3	yes	240	1.1	even	1	trivial
230.3.i.a.109.22	yes	240	5.4	even	2	inner