Embedded newform 230.3.k.a.223.6

• Label: 230.3.k.a.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.a.197.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.100889 + 1.41061i) q^{2} +(-0.429429 - 0.0934166i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(-2.14968 + 4.51429i) q^{5} +(0.0884498 - 0.615182i) q^{6} +(-0.202708 + 0.371231i) q^{7} +(-0.601225 - 2.76379i) q^{8} +(-8.01101 - 3.65850i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.429429	−	0.0934166i	−0.143143	−	0.0311389i	0.140423	−	0.990092i	\(-0.455154\pi\)
−0.283566	+	0.958953i	\(0.591517\pi\)
\(5\)	−2.14968	+	4.51429i	−0.429937	+	0.902859i
							\(7\)	−0.202708	+	0.371231i	−0.0289582	+	0.0530330i	−0.891757	−	0.452514i	\(-0.850527\pi\)
0.862799	+	0.505547i	\(0.168709\pi\)
\(11\)	−2.84602	−	3.28448i	−0.258729	−	0.298590i	0.611492	−	0.791251i	\(-0.290570\pi\)
							−0.870221	+	0.492661i	\(0.836024\pi\)
\(13\)	−0.583215	−	1.06808i	−0.0448627	−	0.0821599i	0.854286	−	0.519804i	\(-0.173995\pi\)
							−0.899148	+	0.437644i	\(0.855813\pi\)
\(17\)	−2.86575	−	3.82820i	−0.168574	−	0.225188i	0.708297	−	0.705915i	\(-0.249464\pi\)
							−0.876871	+	0.480727i	\(0.840373\pi\)
\(19\)	−8.41620	+	1.21007i	−0.442958	+	0.0636877i	−0.360185	−	0.932881i	\(-0.617287\pi\)
							−0.0827725	+	0.996568i	\(0.526377\pi\)
\(23\)	−21.7198	−	7.56627i	−0.944341	−	0.328968i
							\(29\)	−45.4973	−	6.54152i	−1.56887	−	0.225570i	−0.697642	−	0.716446i	\(-0.745768\pi\)
−0.871229	+	0.490877i	\(0.836677\pi\)
\(31\)	17.0340	+	10.9471i	0.549484	+	0.353132i	0.785738	−	0.618559i	\(-0.212283\pi\)
							−0.236254	+	0.971691i	\(0.575920\pi\)
\(37\)	−6.00855	−	16.1095i	−0.162393	−	0.435393i	0.830112	−	0.557597i	\(-0.188277\pi\)
							−0.992505	+	0.122204i	\(0.961004\pi\)
\(41\)	24.7218	+	54.1331i	0.602970	+	1.32032i	0.927280	+	0.374369i	\(0.122141\pi\)
							−0.324310	+	0.945951i	\(0.605132\pi\)
\(43\)	2.63161	+	0.572472i	0.0612003	+	0.0133133i	0.243061	−	0.970011i	\(-0.421848\pi\)
							−0.181861	+	0.983324i	\(0.558212\pi\)
\(47\)	−11.9165	+	11.9165i	−0.253543	+	0.253543i	−0.822422	−	0.568879i	\(-0.807377\pi\)
							0.568879	+	0.822422i	\(0.307377\pi\)

(See \(a_n\) instead)

Twists

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

    

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