Embedded newform 230.3.k.b.13.6

• Label: 230.3.k.b.13.6
• Level: $230$
• Weight: $3$
• Character: 230.13
• 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.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			13.6
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.b.177.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41061 + 0.100889i) q^{2} +(0.0578351 + 0.265863i) q^{3} +(1.97964 + 0.284630i) q^{4} +(-0.330642 - 4.98906i) q^{5} +(0.0547601 + 0.380865i) q^{6} +(-4.77713 + 2.60851i) q^{7} +(2.76379 + 0.601225i) q^{8} +(8.11935 - 3.70798i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41061	+	0.100889i	0.705305	+	0.0504444i
							\(3\)	0.0578351	+	0.265863i	0.0192784	+	0.0886212i	0.985781	−	0.168036i	\(-0.0537425\pi\)
−0.966502	+	0.256657i	\(0.917379\pi\)
\(5\)	−0.330642	−	4.98906i	−0.0661285	−	0.997811i
							\(7\)	−4.77713	+	2.60851i	−0.682447	+	0.372644i	−0.782770	−	0.622311i	\(-0.786194\pi\)
0.100324	+	0.994955i	\(0.468012\pi\)
\(11\)	12.2267	−	14.1104i	1.11152	−	1.28276i	0.156023	−	0.987753i	\(-0.450133\pi\)
							0.955495	−	0.295007i	\(-0.0953218\pi\)
\(13\)	7.23245	+	3.94921i	0.556342	+	0.303786i	0.732713	−	0.680538i	\(-0.238254\pi\)
							−0.176371	+	0.984324i	\(0.556436\pi\)
\(17\)	−11.1442	−	8.34248i	−0.655544	−	0.490734i	0.218972	−	0.975731i	\(-0.429730\pi\)
							−0.874516	+	0.484997i	\(0.838821\pi\)
\(19\)	27.0736	+	3.89260i	1.42493	+	0.204874i	0.811258	−	0.584688i	\(-0.198783\pi\)
							0.613671	+	0.789562i	\(0.289692\pi\)
\(23\)	−20.2227	+	10.9564i	−0.879248	+	0.476364i
							\(29\)	−50.8646	+	7.31323i	−1.75395	+	0.252180i	−0.942967	−	0.332886i	\(-0.891978\pi\)
−0.810986	+	0.585066i	\(0.801069\pi\)
\(31\)	24.2824	−	15.6053i	0.783303	−	0.503398i	−0.0868254	−	0.996224i	\(-0.527672\pi\)
							0.870128	+	0.492825i	\(0.164036\pi\)
\(37\)	48.4332	+	18.0646i	1.30900	+	0.488233i	0.904567	−	0.426332i	\(-0.140195\pi\)
							0.404438	+	0.914566i	\(0.367467\pi\)
\(41\)	−8.69524	+	19.0399i	−0.212079	+	0.464388i	−0.985537	−	0.169459i	\(-0.945798\pi\)
							0.773458	+	0.633847i	\(0.218525\pi\)
\(43\)	5.57051	+	25.6072i	0.129547	+	0.595517i	0.995448	+	0.0953025i	\(0.0303818\pi\)
							−0.865902	+	0.500214i	\(0.833255\pi\)
\(47\)	−19.9708	+	19.9708i	−0.424911	+	0.424911i	−0.886890	−	0.461980i	\(-0.847139\pi\)
							0.461980	+	0.886890i	\(0.347139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.13.6	✓	240
5.2	odd	4	inner	230.3.k.b.197.6	yes	240
23.16	even	11	inner	230.3.k.b.223.6	yes	240
115.62	odd	44	inner	230.3.k.b.177.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.6	✓	240	1.1	even	1	trivial
230.3.k.b.177.6	yes	240	115.62	odd	44	inner
230.3.k.b.197.6	yes	240	5.2	odd	4	inner
230.3.k.b.223.6	yes	240	23.16	even	11	inner