Embedded newform 230.3.k.b.187.8

• Label: 230.3.k.b.187.8
• Level: $230$
• Weight: $3$
• Character: 230.187
• 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			187.8
Character	\(\chi\)	\(=\)	230.187
Dual form			230.3.k.b.123.8

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{8}{11}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.5	✓	240	5.3	odd	4	inner
230.3.k.b.77.5	yes	240	23.8	even	11	inner
230.3.k.b.123.8	yes	240	115.8	odd	44	inner
230.3.k.b.187.8	yes	240	1.1	even	1	trivial