Embedded newform 230.3.k.b.3.12

• Label: 230.3.k.b.3.12
• 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			3.12
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(5.55917 - 2.07346i) q^{3} +(0.563465 + 1.91899i) q^{4} +(-2.76117 + 4.16845i) q^{5} +(8.05101 + 2.36399i) q^{6} +(-0.783489 + 3.60164i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(19.8034 - 17.1597i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

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
							\(3\)	5.55917	−	2.07346i	1.85306	−	0.691154i	0.874551	−	0.484934i	\(-0.161156\pi\)
0.978506	−	0.206221i	\(-0.0661164\pi\)
\(5\)	−2.76117	+	4.16845i	−0.552234	+	0.833689i
							\(7\)	−0.783489	+	3.60164i	−0.111927	+	0.514520i	0.886710	+	0.462326i	\(0.152985\pi\)
−0.998637	+	0.0521939i	\(0.983379\pi\)
\(11\)	1.00857	−	7.01478i	0.0916884	−	0.637707i	−0.891214	−	0.453582i	\(-0.850146\pi\)
							0.982903	−	0.184125i	\(-0.0589450\pi\)
\(13\)	2.23291	+	10.2645i	0.171762	+	0.789578i	0.979725	+	0.200345i	\(0.0642062\pi\)
							−0.807963	+	0.589233i	\(0.799430\pi\)
\(17\)	−3.89842	+	2.12870i	−0.229319	+	0.125217i	−0.589801	−	0.807549i	\(-0.700794\pi\)
							0.360483	+	0.932766i	\(0.382612\pi\)
\(19\)	−6.00026	−	20.4350i	−0.315803	−	1.07553i	−0.952532	−	0.304439i	\(-0.901531\pi\)
							0.636729	−	0.771088i	\(-0.280287\pi\)
\(23\)	−17.0186	+	15.4716i	−0.739937	+	0.672676i
							\(29\)	12.4448	−	42.3831i	0.429131	−	1.46149i	−0.407239	−	0.913322i	\(-0.633508\pi\)
0.836370	−	0.548165i	\(-0.184673\pi\)
\(31\)	3.19850	+	7.00373i	0.103177	+	0.225927i	0.954179	−	0.299236i	\(-0.0967318\pi\)
							−0.851002	+	0.525163i	\(0.824004\pi\)
\(37\)	−51.3169	−	3.67025i	−1.38694	−	0.0991961i	−0.642194	−	0.766542i	\(-0.721976\pi\)
							−0.744748	+	0.667346i	\(0.767430\pi\)
\(41\)	−45.8661	+	52.9322i	−1.11868	+	1.29103i	−0.166318	+	0.986072i	\(0.553188\pi\)
							−0.952366	+	0.304958i	\(0.901357\pi\)
\(43\)	13.0342	−	4.86149i	0.303120	−	0.113058i	−0.193302	−	0.981139i	\(-0.561920\pi\)
							0.496422	+	0.868081i	\(0.334647\pi\)
\(47\)	8.43321	−	8.43321i	0.179430	−	0.179430i	−0.611677	−	0.791107i	\(-0.709505\pi\)
							0.791107	+	0.611677i	\(0.209505\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.12	✓	240
5.2	odd	4	inner	230.3.k.b.187.1	yes	240
23.8	even	11	inner	230.3.k.b.123.1	yes	240
115.77	odd	44	inner	230.3.k.b.77.12	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.12	✓	240	1.1	even	1	trivial
230.3.k.b.77.12	yes	240	115.77	odd	44	inner
230.3.k.b.123.1	yes	240	23.8	even	11	inner
230.3.k.b.187.1	yes	240	5.2	odd	4	inner