Embedded newform 230.3.k.a.3.2

• Label: 230.3.k.a.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.a.77.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13214 - 0.847507i) q^{2} +(-4.48436 + 1.67258i) q^{3} +(0.563465 + 1.91899i) q^{4} +(-1.64232 - 4.72258i) q^{5} +(6.49443 + 1.90694i) q^{6} +(1.22526 - 5.63243i) q^{7} +(0.988434 - 2.65009i) q^{8} +(10.5102 - 9.10714i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

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
							\(3\)	−4.48436	+	1.67258i	−1.49479	+	0.557526i	−0.958145	−	0.286284i	\(-0.907580\pi\)
−0.536641	+	0.843810i	\(0.680307\pi\)
\(5\)	−1.64232	−	4.72258i	−0.328465	−	0.944516i
							\(7\)	1.22526	−	5.63243i	0.175037	−	0.804633i	−0.802963	−	0.596029i	\(-0.796744\pi\)
0.978000	−	0.208604i	\(-0.0668919\pi\)
\(11\)	−0.308491	+	2.14560i	−0.0280447	+	0.195055i	−0.999027	−	0.0440949i	\(-0.985960\pi\)
							0.970983	+	0.239150i	\(0.0768687\pi\)
\(13\)	−1.71861	−	7.90034i	−0.132201	−	0.607718i	−0.994802	−	0.101833i	\(-0.967529\pi\)
							0.862601	−	0.505886i	\(-0.168834\pi\)
\(17\)	−16.2937	+	8.89703i	−0.958453	+	0.523355i	−0.880729	−	0.473620i	\(-0.842947\pi\)
							−0.0777236	+	0.996975i	\(0.524765\pi\)
\(19\)	4.06600	+	13.8475i	0.214000	+	0.728816i	0.994600	+	0.103785i	\(0.0330955\pi\)
							−0.780600	+	0.625031i	\(0.785086\pi\)
\(23\)	−12.8187	+	19.0966i	−0.557335	+	0.830287i
							\(29\)	2.10803	−	7.17929i	0.0726907	−	0.247562i	−0.915130	−	0.403159i	\(-0.867912\pi\)
0.987821	+	0.155598i	\(0.0497303\pi\)
\(31\)	19.6723	+	43.0763i	0.634590	+	1.38956i	0.904417	+	0.426649i	\(0.140306\pi\)
							−0.269827	+	0.962909i	\(0.586967\pi\)
\(37\)	−31.7075	−	2.26777i	−0.856961	−	0.0612910i	−0.364059	−	0.931376i	\(-0.618609\pi\)
							−0.492902	+	0.870085i	\(0.664064\pi\)
\(41\)	−6.59465	+	7.61064i	−0.160845	+	0.185625i	−0.830451	−	0.557091i	\(-0.811917\pi\)
							0.669606	+	0.742716i	\(0.266463\pi\)
\(43\)	79.6177	−	29.6959i	1.85158	−	0.690602i	0.872006	−	0.489496i	\(-0.162819\pi\)
							0.979569	−	0.201106i	\(-0.0644535\pi\)
\(47\)	−53.8367	+	53.8367i	−1.14546	+	1.14546i	−0.158028	+	0.987435i	\(0.550514\pi\)
							−0.987435	+	0.158028i	\(0.949486\pi\)

(See \(a_n\) instead)

Twists

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

    

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