Embedded newform 230.3.k.a.13.10

• Label: 230.3.k.a.13.10
• Level: $230$
• Weight: $3$
• Character: 230.13
• 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			13.10
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.a.177.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41061 - 0.100889i) q^{2} +(0.849591 + 3.90551i) q^{3} +(1.97964 + 0.284630i) q^{4} +(-2.10875 - 4.53356i) q^{5} +(-0.804420 - 5.59486i) q^{6} +(6.95962 - 3.80024i) q^{7} +(-2.76379 - 0.601225i) q^{8} +(-6.34449 + 2.89743i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41061	−	0.100889i	−0.705305	−	0.0504444i
							\(3\)	0.849591	+	3.90551i	0.283197	+	1.30184i	0.870380	+	0.492381i	\(0.163873\pi\)
−0.587183	+	0.809454i	\(0.699763\pi\)
\(5\)	−2.10875	−	4.53356i	−0.421749	−	0.906713i
							\(7\)	6.95962	−	3.80024i	0.994232	−	0.542892i	0.102152	−	0.994769i	\(-0.467427\pi\)
0.892080	+	0.451877i	\(0.149245\pi\)
\(11\)	11.8497	−	13.6753i	1.07725	−	1.24321i	0.108783	−	0.994066i	\(-0.465305\pi\)
							0.968466	−	0.249146i	\(-0.0801498\pi\)
\(13\)	−22.1744	−	12.1081i	−1.70572	−	0.931396i	−0.966022	−	0.258461i	\(-0.916785\pi\)
							−0.739702	−	0.672934i	\(-0.765034\pi\)
\(17\)	−6.61881	−	4.95478i	−0.389342	−	0.291458i	0.386618	−	0.922240i	\(-0.373643\pi\)
							−0.775960	+	0.630782i	\(0.782734\pi\)
\(19\)	21.2765	+	3.05910i	1.11982	+	0.161005i	0.677266	−	0.735739i	\(-0.263165\pi\)
							0.442550	+	0.896744i	\(0.354074\pi\)
\(23\)	21.5874	−	7.93637i	0.938581	−	0.345060i
							\(29\)	30.8118	−	4.43007i	1.06248	−	0.152761i	0.411157	−	0.911565i	\(-0.365125\pi\)
0.651319	+	0.758804i	\(0.274216\pi\)
\(31\)	−26.2664	+	16.8804i	−0.847303	+	0.544528i	−0.890732	−	0.454528i	\(-0.849808\pi\)
							0.0434297	+	0.999056i	\(0.486172\pi\)
\(37\)	15.3174	+	5.71309i	0.413983	+	0.154408i	0.547821	−	0.836596i	\(-0.315457\pi\)
							−0.133838	+	0.991003i	\(0.542730\pi\)
\(41\)	31.4836	−	68.9395i	0.767892	−	1.68145i	0.0366541	−	0.999328i	\(-0.488330\pi\)
							0.731238	−	0.682122i	\(-0.238943\pi\)
\(43\)	5.65852	+	26.0118i	0.131594	+	0.604926i	0.994953	+	0.100337i	\(0.0319922\pi\)
							−0.863360	+	0.504589i	\(0.831644\pi\)
\(47\)	−32.2071	+	32.2071i	−0.685257	+	0.685257i	−0.961180	−	0.275922i	\(-0.911017\pi\)
							0.275922	+	0.961180i	\(0.411017\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.a.13.10	✓	240
5.2	odd	4	inner	230.3.k.a.197.10	yes	240
23.16	even	11	inner	230.3.k.a.223.10	yes	240
115.62	odd	44	inner	230.3.k.a.177.10	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.13.10	✓	240	1.1	even	1	trivial
230.3.k.a.177.10	yes	240	115.62	odd	44	inner
230.3.k.a.197.10	yes	240	5.2	odd	4	inner
230.3.k.a.223.10	yes	240	23.16	even	11	inner