Embedded newform 230.3.k.b.223.10

• Label: 230.3.k.b.223.10
• 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			223.10
Character	\(\chi\)	\(=\)	230.223
Dual form			230.3.k.b.197.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.100889 - 1.41061i) q^{2} +(3.75933 + 0.817792i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(-3.90104 + 3.12760i) q^{5} +(0.774311 - 5.38545i) q^{6} +(-4.79876 + 8.78828i) q^{7} +(0.601225 + 2.76379i) q^{8} +(5.27707 + 2.40996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

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
							\(3\)	3.75933	+	0.817792i	1.25311	+	0.272597i	0.789670	−	0.613532i	\(-0.210252\pi\)
0.463439	+	0.886129i	\(0.346615\pi\)
\(5\)	−3.90104	+	3.12760i	−0.780208	+	0.625520i
							\(7\)	−4.79876	+	8.78828i	−0.685537	+	1.25547i	0.270640	+	0.962681i	\(0.412765\pi\)
−0.956178	+	0.292787i	\(0.905417\pi\)
\(11\)	4.10430	+	4.73662i	0.373118	+	0.430601i	0.910992	−	0.412424i	\(-0.135318\pi\)
							−0.537874	+	0.843025i	\(0.680772\pi\)
\(13\)	5.10454	+	9.34827i	0.392657	+	0.719097i	0.996981	−	0.0776409i	\(-0.0247388\pi\)
							−0.604325	+	0.796738i	\(0.706557\pi\)
\(17\)	5.08758	+	6.79621i	0.299269	+	0.399777i	0.924819	−	0.380407i	\(-0.124216\pi\)
							−0.625550	+	0.780184i	\(0.715125\pi\)
\(19\)	22.4818	−	3.23239i	1.18325	−	0.170126i	0.477530	−	0.878615i	\(-0.341532\pi\)
							0.705721	+	0.708490i	\(0.250623\pi\)
\(23\)	−22.4744	+	4.88877i	−0.977149	+	0.212555i
							\(29\)	−15.6900	−	2.25588i	−0.541035	−	0.0777891i	−0.133621	−	0.991033i	\(-0.542660\pi\)
−0.407414	+	0.913243i	\(0.633569\pi\)
\(31\)	−9.16240	−	5.88831i	−0.295561	−	0.189946i	0.384453	−	0.923145i	\(-0.374390\pi\)
							−0.680014	+	0.733199i	\(0.738026\pi\)
\(37\)	−3.29763	−	8.84128i	−0.0891251	−	0.238954i	0.884738	−	0.466088i	\(-0.154337\pi\)
							−0.973863	+	0.227135i	\(0.927064\pi\)
\(41\)	22.4451	+	49.1479i	0.547441	+	1.19873i	0.957966	+	0.286880i	\(0.0926181\pi\)
							−0.410525	+	0.911849i	\(0.634655\pi\)
\(43\)	62.8393	+	13.6699i	1.46138	+	0.317904i	0.871822	−	0.489822i	\(-0.162938\pi\)
							0.589558	+	0.807726i	\(0.299302\pi\)
\(47\)	31.1789	−	31.1789i	0.663381	−	0.663381i	−0.292794	−	0.956175i	\(-0.594585\pi\)
							0.956175	+	0.292794i	\(0.0945851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.223.10	yes	240
5.2	odd	4	inner	230.3.k.b.177.10	yes	240
23.13	even	11	inner	230.3.k.b.13.10	✓	240
115.82	odd	44	inner	230.3.k.b.197.10	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.10	✓	240	23.13	even	11	inner
230.3.k.b.177.10	yes	240	5.2	odd	4	inner
230.3.k.b.197.10	yes	240	115.82	odd	44	inner
230.3.k.b.223.10	yes	240	1.1	even	1	trivial