Embedded newform 230.3.k.b.3.1

• Label: 230.3.k.b.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(-5.13373 + 1.91478i) q^{3} +(0.563465 + 1.91899i) q^{4} +(4.33360 - 2.49398i) q^{5} +(-7.43487 - 2.18307i) q^{6} +(2.35485 - 10.8251i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(15.8870 - 13.7662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

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
							\(3\)	−5.13373	+	1.91478i	−1.71124	+	0.638261i	−0.998033	−	0.0626901i	\(-0.980032\pi\)
−0.713210	+	0.700951i	\(0.752759\pi\)
\(5\)	4.33360	−	2.49398i	0.866720	−	0.498795i
							\(7\)	2.35485	−	10.8251i	0.336408	−	1.54644i	−0.429246	−	0.903188i	\(-0.641221\pi\)
0.765653	−	0.643253i	\(-0.222416\pi\)
\(11\)	−1.34881	+	9.38121i	−0.122620	+	0.852837i	0.831950	+	0.554850i	\(0.187224\pi\)
							−0.954570	+	0.297987i	\(0.903685\pi\)
\(13\)	1.01429	+	4.66260i	0.0780221	+	0.358662i	0.999561	−	0.0296241i	\(-0.00943102\pi\)
							−0.921539	+	0.388286i	\(0.873067\pi\)
\(17\)	23.5612	−	12.8654i	1.38596	−	0.756789i	0.398873	−	0.917006i	\(-0.369401\pi\)
							0.987082	+	0.160217i	\(0.0512194\pi\)
\(19\)	6.37929	+	21.7259i	0.335752	+	1.14347i	0.938426	+	0.345479i	\(0.112284\pi\)
							−0.602674	+	0.797987i	\(0.705898\pi\)
\(23\)	10.3568	−	20.5362i	0.450295	−	0.892880i
							\(29\)	−3.73510	+	12.7206i	−0.128796	+	0.438640i	−0.998489	−	0.0549535i	\(-0.982499\pi\)
0.869692	+	0.493594i	\(0.164317\pi\)
\(31\)	−1.67758	−	3.67340i	−0.0541156	−	0.118497i	0.880643	−	0.473781i	\(-0.157111\pi\)
							−0.934758	+	0.355285i	\(0.884384\pi\)
\(37\)	30.6083	+	2.18915i	0.827252	+	0.0591662i	0.478542	−	0.878064i	\(-0.341165\pi\)
							0.348710	+	0.937231i	\(0.386620\pi\)
\(41\)	22.3672	−	25.8132i	0.545542	−	0.629589i	−0.414296	−	0.910142i	\(-0.635972\pi\)
							0.959839	+	0.280553i	\(0.0905177\pi\)
\(43\)	41.2511	−	15.3859i	0.959328	−	0.357811i	0.179467	−	0.983764i	\(-0.442563\pi\)
							0.779861	+	0.625953i	\(0.215290\pi\)
\(47\)	2.73010	−	2.73010i	0.0580872	−	0.0580872i	−0.677466	−	0.735554i	\(-0.736922\pi\)
							0.735554	+	0.677466i	\(0.236922\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.1	✓	240
5.2	odd	4	inner	230.3.k.b.187.12	yes	240
23.8	even	11	inner	230.3.k.b.123.12	yes	240
115.77	odd	44	inner	230.3.k.b.77.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.1	✓	240	1.1	even	1	trivial
230.3.k.b.77.1	yes	240	115.77	odd	44	inner
230.3.k.b.123.12	yes	240	23.8	even	11	inner
230.3.k.b.187.12	yes	240	5.2	odd	4	inner