Embedded newform 230.3.k.a.223.10

• Label: 230.3.k.a.223.10
• Level: $230$
• Weight: $3$
• Character: 230.223
• 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			223.10
Character	\(\chi\)	\(=\)	230.223
Dual form			230.3.k.a.197.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.100889 + 1.41061i) q^{2} +(3.90551 + 0.849591i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(4.80717 + 1.37517i) q^{5} +(-0.804420 + 5.59486i) q^{6} +(3.80024 - 6.95962i) q^{7} +(-0.601225 - 2.76379i) q^{8} +(6.34449 + 2.89743i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.100889	+	1.41061i	0.0504444	+	0.705305i
							\(3\)	3.90551	+	0.849591i	1.30184	+	0.283197i	0.809454	−	0.587183i	\(-0.199763\pi\)
0.492381	+	0.870380i	\(0.336127\pi\)
\(5\)	4.80717	+	1.37517i	0.961435	+	0.275034i
							\(7\)	3.80024	−	6.95962i	0.542892	−	0.994232i	−0.451877	−	0.892080i	\(-0.649245\pi\)
0.994769	−	0.102152i	\(-0.0325727\pi\)
\(11\)	11.8497	+	13.6753i	1.07725	+	1.24321i	0.968466	+	0.249146i	\(0.0801498\pi\)
							0.108783	+	0.994066i	\(0.465305\pi\)
\(13\)	−12.1081	−	22.1744i	−0.931396	−	1.70572i	−0.672934	−	0.739702i	\(-0.734966\pi\)
							−0.258461	−	0.966022i	\(-0.583215\pi\)
\(17\)	4.95478	+	6.61881i	0.291458	+	0.389342i	0.922240	−	0.386618i	\(-0.126357\pi\)
							−0.630782	+	0.775960i	\(0.717266\pi\)
\(19\)	−21.2765	+	3.05910i	−1.11982	+	0.161005i	−0.677266	−	0.735739i	\(-0.736835\pi\)
							−0.442550	+	0.896744i	\(0.645926\pi\)
\(23\)	−7.93637	+	21.5874i	−0.345060	+	0.938581i
							\(29\)	−30.8118	−	4.43007i	−1.06248	−	0.152761i	−0.411157	−	0.911565i	\(-0.634875\pi\)
−0.651319	+	0.758804i	\(0.725784\pi\)
\(31\)	−26.2664	−	16.8804i	−0.847303	−	0.544528i	0.0434297	−	0.999056i	\(-0.486172\pi\)
							−0.890732	+	0.454528i	\(0.849808\pi\)
\(37\)	−5.71309	−	15.3174i	−0.154408	−	0.413983i	0.836596	−	0.547821i	\(-0.184543\pi\)
							−0.991003	+	0.133838i	\(0.957270\pi\)
\(41\)	31.4836	+	68.9395i	0.767892	+	1.68145i	0.731238	+	0.682122i	\(0.238943\pi\)
							0.0366541	+	0.999328i	\(0.488330\pi\)
\(43\)	26.0118	+	5.65852i	0.604926	+	0.131594i	0.504589	−	0.863360i	\(-0.331644\pi\)
							0.100337	+	0.994953i	\(0.468008\pi\)
\(47\)	−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.223.10	yes	240
5.2	odd	4	inner	230.3.k.a.177.10	yes	240
23.13	even	11	inner	230.3.k.a.13.10	✓	240
115.82	odd	44	inner	230.3.k.a.197.10	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.13.10	✓	240	23.13	even	11	inner
230.3.k.a.177.10	yes	240	5.2	odd	4	inner
230.3.k.a.197.10	yes	240	115.82	odd	44	inner
230.3.k.a.223.10	yes	240	1.1	even	1	trivial