Embedded newform 230.3.k.b.223.6

• Label: 230.3.k.b.223.6
• Level: $230$
• Weight: $3$
• Character: 230.223
• 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			223.6
Character	\(\chi\)	\(=\)	230.223
Dual form			230.3.k.b.197.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.100889 - 1.41061i) q^{2} +(0.265863 + 0.0578351i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(3.98700 + 3.01725i) q^{5} +(0.0547601 - 0.380865i) q^{6} +(-2.60851 + 4.77713i) q^{7} +(0.601225 + 2.76379i) q^{8} +(-8.11935 - 3.70798i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.100889	−	1.41061i	−0.0504444	−	0.705305i
							\(3\)	0.265863	+	0.0578351i	0.0886212	+	0.0192784i	0.256657	−	0.966502i	\(-0.417379\pi\)
−0.168036	+	0.985781i	\(0.553743\pi\)
\(5\)	3.98700	+	3.01725i	0.797400	+	0.603451i
							\(7\)	−2.60851	+	4.77713i	−0.372644	+	0.682447i	−0.994955	−	0.100324i	\(-0.968012\pi\)
0.622311	+	0.782770i	\(0.286194\pi\)
\(11\)	12.2267	+	14.1104i	1.11152	+	1.28276i	0.955495	+	0.295007i	\(0.0953218\pi\)
							0.156023	+	0.987753i	\(0.450133\pi\)
\(13\)	3.94921	+	7.23245i	0.303786	+	0.556342i	0.984324	−	0.176371i	\(-0.0564357\pi\)
							−0.680538	+	0.732713i	\(0.738254\pi\)
\(17\)	8.34248	+	11.1442i	0.490734	+	0.655544i	0.975731	−	0.218972i	\(-0.0702703\pi\)
							−0.484997	+	0.874516i	\(0.661179\pi\)
\(19\)	−27.0736	+	3.89260i	−1.42493	+	0.204874i	−0.811258	−	0.584688i	\(-0.801217\pi\)
							−0.613671	+	0.789562i	\(0.710308\pi\)
\(23\)	10.9564	−	20.2227i	0.476364	−	0.879248i
							\(29\)	50.8646	+	7.31323i	1.75395	+	0.252180i	0.942967	−	0.332886i	\(-0.108022\pi\)
0.810986	+	0.585066i	\(0.198931\pi\)
\(31\)	24.2824	+	15.6053i	0.783303	+	0.503398i	0.870128	−	0.492825i	\(-0.164036\pi\)
							−0.0868254	+	0.996224i	\(0.527672\pi\)
\(37\)	−18.0646	−	48.4332i	−0.488233	−	1.30900i	−0.914566	−	0.404438i	\(-0.867467\pi\)
							0.426332	−	0.904567i	\(-0.359805\pi\)
\(41\)	−8.69524	−	19.0399i	−0.212079	−	0.464388i	0.773458	−	0.633847i	\(-0.218525\pi\)
							−0.985537	+	0.169459i	\(0.945798\pi\)
\(43\)	25.6072	+	5.57051i	0.595517	+	0.129547i	0.500214	−	0.865902i	\(-0.333255\pi\)
							0.0953025	+	0.995448i	\(0.469618\pi\)
\(47\)	−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.223.6	yes	240
5.2	odd	4	inner	230.3.k.b.177.6	yes	240
23.13	even	11	inner	230.3.k.b.13.6	✓	240
115.82	odd	44	inner	230.3.k.b.197.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.6	✓	240	23.13	even	11	inner
230.3.k.b.177.6	yes	240	5.2	odd	4	inner
230.3.k.b.197.6	yes	240	115.82	odd	44	inner
230.3.k.b.223.6	yes	240	1.1	even	1	trivial