Embedded newform 230.3.k.b.3.2

• Label: 230.3.k.b.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(-4.87737 + 1.81916i) q^{3} +(0.563465 + 1.91899i) q^{4} +(-2.81617 - 4.13149i) q^{5} +(-7.06360 - 2.07406i) q^{6} +(-2.62119 + 12.0494i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(13.6776 - 11.8517i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

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
							\(3\)	−4.87737	+	1.81916i	−1.62579	+	0.606388i	−0.986672	−	0.162721i	\(-0.947973\pi\)
−0.639118	+	0.769109i	\(0.720700\pi\)
\(5\)	−2.81617	−	4.13149i	−0.563234	−	0.826298i
							\(7\)	−2.62119	+	12.0494i	−0.374455	+	1.72134i	0.275439	+	0.961318i	\(0.411177\pi\)
−0.649894	+	0.760024i	\(0.725187\pi\)
\(11\)	1.27021	−	8.83452i	0.115474	−	0.803138i	−0.846967	−	0.531646i	\(-0.821574\pi\)
							0.962441	−	0.271492i	\(-0.0875171\pi\)
\(13\)	−4.40805	−	20.2635i	−0.339081	−	1.55873i	−0.758952	−	0.651146i	\(-0.774288\pi\)
							0.419871	−	0.907584i	\(-0.362075\pi\)
\(17\)	15.7392	−	8.59427i	0.925837	−	0.505545i	0.0557505	−	0.998445i	\(-0.482245\pi\)
							0.870086	+	0.492900i	\(0.164063\pi\)
\(19\)	−4.53467	−	15.4437i	−0.238667	−	0.812825i	−0.988503	−	0.151201i	\(-0.951686\pi\)
							0.749836	−	0.661624i	\(-0.230132\pi\)
\(23\)	−14.1509	+	18.1315i	−0.615256	+	0.788327i
							\(29\)	1.73946	−	5.92405i	0.0599814	−	0.204278i	−0.924050	−	0.382271i	\(-0.875142\pi\)
0.984032	+	0.177993i	\(0.0569604\pi\)
\(31\)	−16.0692	−	35.1867i	−0.518363	−	1.13506i	−0.970056	−	0.242883i	\(-0.921907\pi\)
							0.451693	−	0.892173i	\(-0.350820\pi\)
\(37\)	−0.998437	−	0.0714096i	−0.0269848	−	0.00192999i	0.0578403	−	0.998326i	\(-0.481579\pi\)
							−0.0848250	+	0.996396i	\(0.527033\pi\)
\(41\)	−0.330784	+	0.381745i	−0.00806789	+	0.00931085i	−0.759769	−	0.650193i	\(-0.774688\pi\)
							0.751701	+	0.659504i	\(0.229234\pi\)
\(43\)	−37.6883	+	14.0570i	−0.876472	+	0.326907i	−0.747118	−	0.664691i	\(-0.768563\pi\)
							−0.129354	+	0.991599i	\(0.541290\pi\)
\(47\)	38.1396	−	38.1396i	0.811480	−	0.811480i	−0.173376	−	0.984856i	\(-0.555468\pi\)
							0.984856	+	0.173376i	\(0.0554675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.2	✓	240
5.2	odd	4	inner	230.3.k.b.187.11	yes	240
23.8	even	11	inner	230.3.k.b.123.11	yes	240
115.77	odd	44	inner	230.3.k.b.77.2	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.2	✓	240	1.1	even	1	trivial
230.3.k.b.77.2	yes	240	115.77	odd	44	inner
230.3.k.b.123.11	yes	240	23.8	even	11	inner
230.3.k.b.187.11	yes	240	5.2	odd	4	inner