Embedded newform 230.3.k.b.3.4

• Label: 230.3.k.b.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(-2.59640 + 0.968406i) q^{3} +(0.563465 + 1.91899i) q^{4} +(4.72347 + 1.63977i) q^{5} +(-3.76021 - 1.10410i) q^{6} +(-1.02169 + 4.69661i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(-0.998275 + 0.865010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

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
							\(3\)	−2.59640	+	0.968406i	−0.865466	+	0.322802i	−0.742693	−	0.669633i	\(-0.766452\pi\)
−0.122773	+	0.992435i	\(0.539179\pi\)
\(5\)	4.72347	+	1.63977i	0.944694	+	0.327954i
							\(7\)	−1.02169	+	4.69661i	−0.145955	+	0.670945i	0.844777	+	0.535118i	\(0.179733\pi\)
−0.990733	+	0.135827i	\(0.956631\pi\)
\(11\)	1.54102	−	10.7180i	0.140093	−	0.974367i	−0.791579	−	0.611067i	\(-0.790741\pi\)
							0.931672	−	0.363300i	\(-0.118350\pi\)
\(13\)	2.60589	+	11.9791i	0.200453	+	0.921467i	0.961934	+	0.273280i	\(0.0881087\pi\)
							−0.761482	+	0.648187i	\(0.775528\pi\)
\(17\)	−17.3161	+	9.45529i	−1.01859	+	0.556194i	−0.899606	−	0.436702i	\(-0.856146\pi\)
							−0.118987	+	0.992896i	\(0.537965\pi\)
\(19\)	2.50967	+	8.54715i	0.132088	+	0.449850i	0.998801	−	0.0489571i	\(-0.0155897\pi\)
							−0.866713	+	0.498807i	\(0.833772\pi\)
\(23\)	−17.1308	+	15.3472i	−0.744816	+	0.667270i
							\(29\)	−8.37804	+	28.5330i	−0.288898	+	0.983896i	0.679331	+	0.733832i	\(0.262270\pi\)
−0.968229	+	0.250064i	\(0.919548\pi\)
\(31\)	−11.1906	−	24.5040i	−0.360987	−	0.790450i	−0.999778	−	0.0210704i	\(-0.993293\pi\)
							0.638791	−	0.769380i	\(-0.279435\pi\)
\(37\)	11.9027	+	0.851298i	0.321695	+	0.0230081i	0.231253	−	0.972894i	\(-0.425718\pi\)
							0.0904418	+	0.995902i	\(0.471172\pi\)
\(41\)	9.18899	−	10.6047i	0.224122	−	0.258650i	−0.632541	−	0.774527i	\(-0.717988\pi\)
							0.856663	+	0.515876i	\(0.172534\pi\)
\(43\)	24.4821	−	9.13135i	0.569351	−	0.212357i	−0.0482679	−	0.998834i	\(-0.515370\pi\)
							0.617619	+	0.786477i	\(0.288097\pi\)
\(47\)	33.3216	−	33.3216i	0.708969	−	0.708969i	−0.257349	−	0.966318i	\(-0.582849\pi\)
							0.966318	+	0.257349i	\(0.0828491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.4	✓	240
5.2	odd	4	inner	230.3.k.b.187.9	yes	240
23.8	even	11	inner	230.3.k.b.123.9	yes	240
115.77	odd	44	inner	230.3.k.b.77.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.4	✓	240	1.1	even	1	trivial
230.3.k.b.77.4	yes	240	115.77	odd	44	inner
230.3.k.b.123.9	yes	240	23.8	even	11	inner
230.3.k.b.187.9	yes	240	5.2	odd	4	inner