Embedded newform 230.3.k.b.123.8

• Label: 230.3.k.b.123.8
• Level: $230$
• Weight: $3$
• Character: 230.123
• Analytic conductor: $6.267$
• Analytic rank: $0$
• Dimension: $240$
• 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			123.8
Character	\(\chi\)	\(=\)	230.123
Dual form			230.3.k.b.187.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.847507 - 1.13214i) q^{2} +(0.546016 - 1.46393i) q^{3} +(-0.563465 + 1.91899i) q^{4} +(1.12304 - 4.87225i) q^{5} +(-2.12011 + 0.622522i) q^{6} +(12.8083 - 2.78627i) q^{7} +(2.65009 - 0.988434i) q^{8} +(4.95680 + 4.29509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.847507	−	1.13214i	−0.423753	−	0.566068i
							\(3\)	0.546016	−	1.46393i	0.182005	−	0.487975i	−0.813622	−	0.581395i	\(-0.802507\pi\)
0.995627	+	0.0934198i	\(0.0297799\pi\)
\(5\)	1.12304	−	4.87225i	0.224609	−	0.974449i
							\(7\)	12.8083	−	2.78627i	1.82975	−	0.398038i	0.840207	−	0.542267i	\(-0.182434\pi\)
0.989544	+	0.144229i	\(0.0460701\pi\)
\(11\)	1.69680	+	11.8015i	0.154254	+	1.07286i	0.908986	+	0.416828i	\(0.136858\pi\)
							−0.754731	+	0.656034i	\(0.772233\pi\)
\(13\)	−5.94049	−	1.29227i	−0.456960	−	0.0994057i	−0.0218074	−	0.999762i	\(-0.506942\pi\)
							−0.435153	+	0.900357i	\(0.643306\pi\)
\(17\)	4.79595	−	8.78312i	0.282114	−	0.516654i	−0.697744	−	0.716347i	\(-0.745813\pi\)
							0.979858	+	0.199693i	\(0.0639945\pi\)
\(19\)	1.50894	−	5.13899i	0.0794181	−	0.270473i	−0.910206	−	0.414157i	\(-0.864077\pi\)
							0.989624	+	0.143683i	\(0.0458947\pi\)
\(23\)	−2.07245	−	22.9064i	−0.0901064	−	0.995932i
							\(29\)	−1.85753	−	6.32617i	−0.0640528	−	0.218144i	0.921247	−	0.388978i	\(-0.127172\pi\)
−0.985300	+	0.170834i	\(0.945354\pi\)
\(31\)	−11.2121	+	24.5511i	−0.361681	+	0.791972i	0.638077	+	0.769973i	\(0.279730\pi\)
							−0.999758	+	0.0219988i	\(0.992997\pi\)
\(37\)	2.45503	+	34.3258i	0.0663522	+	0.927724i	0.916206	+	0.400707i	\(0.131235\pi\)
							−0.849854	+	0.527018i	\(0.823310\pi\)
\(41\)	−50.2438	−	57.9844i	−1.22546	−	1.41425i	−0.879429	−	0.476030i	\(-0.842075\pi\)
							−0.346029	−	0.938224i	\(-0.612470\pi\)
\(43\)	−13.7161	+	36.7742i	−0.318978	+	0.855214i	0.674194	+	0.738554i	\(0.264491\pi\)
							−0.993172	+	0.116659i	\(0.962781\pi\)
\(47\)	−20.6252	+	20.6252i	−0.438834	+	0.438834i	−0.891620	−	0.452785i	\(-0.850430\pi\)
							0.452785	+	0.891620i	\(0.350430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.123.8	yes	240
5.2	odd	4	inner	230.3.k.b.77.5	yes	240
23.3	even	11	inner	230.3.k.b.3.5	✓	240
115.72	odd	44	inner	230.3.k.b.187.8	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.5	✓	240	23.3	even	11	inner
230.3.k.b.77.5	yes	240	5.2	odd	4	inner
230.3.k.b.123.8	yes	240	1.1	even	1	trivial
230.3.k.b.187.8	yes	240	115.72	odd	44	inner