Embedded newform 230.3.k.b.3.11

• Label: 230.3.k.b.3.11
• Level: $230$
• Weight: $3$
• Character: 230.3
• 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			3.11
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(3.86243 - 1.44061i) q^{3} +(0.563465 + 1.91899i) q^{4} +(-0.740945 - 4.94480i) q^{5} +(5.59372 + 1.64247i) q^{6} +(1.21592 - 5.58950i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(6.04125 - 5.23477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13214	+	0.847507i	0.566068	+	0.423753i
							\(3\)	3.86243	−	1.44061i	1.28748	−	0.480204i	0.389816	−	0.920893i	\(-0.372538\pi\)
0.897660	+	0.440689i	\(0.145266\pi\)
\(5\)	−0.740945	−	4.94480i	−0.148189	−	0.988959i
							\(7\)	1.21592	−	5.58950i	0.173703	−	0.798500i	−0.805009	−	0.593262i	\(-0.797840\pi\)
0.978712	−	0.205238i	\(-0.0657967\pi\)
\(11\)	0.258948	−	1.80102i	0.0235407	−	0.163729i	−0.974660	−	0.223692i	\(-0.928189\pi\)
							0.998201	+	0.0599627i	\(0.0190982\pi\)
\(13\)	0.862944	+	3.96689i	0.0663803	+	0.305145i	0.998352	−	0.0573855i	\(-0.0182764\pi\)
							−0.931972	+	0.362531i	\(0.881913\pi\)
\(17\)	4.50514	−	2.45999i	0.265008	−	0.144705i	−0.341252	−	0.939972i	\(-0.610851\pi\)
							0.606260	+	0.795267i	\(0.292669\pi\)
\(19\)	3.90214	+	13.2895i	0.205376	+	0.699445i	0.996176	+	0.0873707i	\(0.0278465\pi\)
							−0.790800	+	0.612074i	\(0.790335\pi\)
\(23\)	14.6207	+	17.7549i	0.635682	+	0.771951i
							\(29\)	−11.8598	+	40.3908i	−0.408959	+	1.39279i	0.455568	+	0.890201i	\(0.349436\pi\)
−0.864527	+	0.502586i	\(0.832382\pi\)
\(31\)	−17.0925	−	37.4274i	−0.551371	−	1.20733i	−0.956139	−	0.292914i	\(-0.905375\pi\)
							0.404768	−	0.914420i	\(-0.367352\pi\)
\(37\)	43.8910	+	3.13914i	1.18624	+	0.0848417i	0.650477	−	0.759526i	\(-0.274569\pi\)
							0.535765	+	0.844367i	\(0.320023\pi\)
\(41\)	−28.2700	+	32.6254i	−0.689513	+	0.795741i	−0.987296	−	0.158893i	\(-0.949208\pi\)
							0.297783	+	0.954634i	\(0.403753\pi\)
\(43\)	−2.76558	+	1.03151i	−0.0643157	+	0.0239885i	−0.381416	−	0.924403i	\(-0.624563\pi\)
							0.317100	+	0.948392i	\(0.397291\pi\)
\(47\)	20.7672	−	20.7672i	0.441855	−	0.441855i	−0.450780	−	0.892635i	\(-0.648854\pi\)
							0.892635	+	0.450780i	\(0.148854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.11	✓	240
5.2	odd	4	inner	230.3.k.b.187.2	yes	240
23.8	even	11	inner	230.3.k.b.123.2	yes	240
115.77	odd	44	inner	230.3.k.b.77.11	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.11	✓	240	1.1	even	1	trivial
230.3.k.b.77.11	yes	240	115.77	odd	44	inner
230.3.k.b.123.2	yes	240	23.8	even	11	inner
230.3.k.b.187.2	yes	240	5.2	odd	4	inner