Embedded newform 230.3.k.b.77.5

• Label: 230.3.k.b.77.5
• Level: $230$
• Weight: $3$
• Character: 230.77
• 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			77.5
Character	\(\chi\)	\(=\)	230.77
Dual form			230.3.k.b.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 - 0.847507i) q^{2} +(-1.46393 - 0.546016i) q^{3} +(0.563465 - 1.91899i) q^{4} +(-4.98248 + 0.418219i) q^{5} +(-2.12011 + 0.622522i) q^{6} +(2.78627 + 12.8083i) q^{7} +(-0.988434 - 2.65009i) q^{8} +(-4.95680 - 4.29509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.13214	−	0.847507i	0.566068	−	0.423753i
							\(3\)	−1.46393	−	0.546016i	−0.487975	−	0.182005i	0.0934198	−	0.995627i	\(-0.470220\pi\)
−0.581395	+	0.813622i	\(0.697493\pi\)
\(5\)	−4.98248	+	0.418219i	−0.996496	+	0.0836438i
							\(7\)	2.78627	+	12.8083i	0.398038	+	1.82975i	0.542267	+	0.840207i	\(0.317566\pi\)
−0.144229	+	0.989544i	\(0.546070\pi\)
\(11\)	1.69680	+	11.8015i	0.154254	+	1.07286i	0.908986	+	0.416828i	\(0.136858\pi\)
							−0.754731	+	0.656034i	\(0.772233\pi\)
\(13\)	−1.29227	+	5.94049i	−0.0994057	+	0.456960i	0.900357	+	0.435153i	\(0.143306\pi\)
							−0.999762	+	0.0218074i	\(0.993058\pi\)
\(17\)	8.78312	+	4.79595i	0.516654	+	0.282114i	0.716347	−	0.697744i	\(-0.245813\pi\)
							−0.199693	+	0.979858i	\(0.563995\pi\)
\(19\)	−1.50894	+	5.13899i	−0.0794181	+	0.270473i	−0.989624	−	0.143683i	\(-0.954105\pi\)
							0.910206	+	0.414157i	\(0.135923\pi\)
\(23\)	−22.9064	+	2.07245i	−0.995932	+	0.0901064i
							\(29\)	1.85753	+	6.32617i	0.0640528	+	0.218144i	0.985300	−	0.170834i	\(-0.0546463\pi\)
−0.921247	+	0.388978i	\(0.872828\pi\)
\(31\)	−11.2121	+	24.5511i	−0.361681	+	0.791972i	0.638077	+	0.769973i	\(0.279730\pi\)
							−0.999758	+	0.0219988i	\(0.992997\pi\)
\(37\)	−34.3258	+	2.45503i	−0.927724	+	0.0663522i	−0.527018	−	0.849854i	\(-0.676690\pi\)
							−0.400707	+	0.916206i	\(0.631235\pi\)
\(41\)	−50.2438	−	57.9844i	−1.22546	−	1.41425i	−0.879429	−	0.476030i	\(-0.842075\pi\)
							−0.346029	−	0.938224i	\(-0.612470\pi\)
\(43\)	36.7742	+	13.7161i	0.855214	+	0.318978i	0.738554	−	0.674194i	\(-0.235509\pi\)
							0.116659	+	0.993172i	\(0.462781\pi\)
\(47\)	−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.77.5	yes	240
5.3	odd	4	inner	230.3.k.b.123.8	yes	240
23.3	even	11	inner	230.3.k.b.187.8	yes	240
115.3	odd	44	inner	230.3.k.b.3.5	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.5	✓	240	115.3	odd	44	inner
230.3.k.b.77.5	yes	240	1.1	even	1	trivial
230.3.k.b.123.8	yes	240	5.3	odd	4	inner
230.3.k.b.187.8	yes	240	23.3	even	11	inner