Embedded newform 230.3.k.a.223.2

• Label: 230.3.k.a.223.2
• 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			223.2
Character	\(\chi\)	\(=\)	230.223
Dual form			230.3.k.a.197.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.100889 + 1.41061i) q^{2} +(-4.70683 - 1.02391i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(4.16436 + 2.76733i) q^{5} +(0.969469 - 6.74281i) q^{6} +(-4.50592 + 8.25197i) q^{7} +(-0.601225 - 2.76379i) q^{8} +(12.9192 + 5.90000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

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
							\(3\)	−4.70683	−	1.02391i	−1.56894	−	0.341303i	−0.657736	−	0.753249i	\(-0.728486\pi\)
−0.911208	+	0.411946i	\(0.864849\pi\)
\(5\)	4.16436	+	2.76733i	0.832872	+	0.553466i
							\(7\)	−4.50592	+	8.25197i	−0.643702	+	1.17885i	0.328454	+	0.944520i	\(0.393472\pi\)
−0.972156	+	0.234333i	\(0.924709\pi\)
\(11\)	5.06204	+	5.84190i	0.460185	+	0.531082i	0.937656	−	0.347566i	\(-0.112992\pi\)
							−0.477470	+	0.878648i	\(0.658446\pi\)
\(13\)	−9.15285	−	16.7622i	−0.704066	−	1.28940i	−0.947585	−	0.319504i	\(-0.896484\pi\)
							0.243519	−	0.969896i	\(-0.421698\pi\)
\(17\)	−17.1636	−	22.9279i	−1.00962	−	1.34870i	−0.936227	−	0.351395i	\(-0.885707\pi\)
							−0.0733955	−	0.997303i	\(-0.523384\pi\)
\(19\)	−23.1417	+	3.32727i	−1.21798	+	0.175119i	−0.721182	−	0.692746i	\(-0.756401\pi\)
							−0.496800	+	0.867865i	\(0.665492\pi\)
\(23\)	−2.71233	−	22.8395i	−0.117928	−	0.993022i
							\(29\)	8.14055	+	1.17043i	0.280708	+	0.0403598i	0.281231	−	0.959640i	\(-0.409257\pi\)
−0.000522612		1.00000i	\(0.500166\pi\)
\(31\)	2.79654	+	1.79723i	0.0902110	+	0.0579751i	0.584969	−	0.811056i	\(-0.301107\pi\)
							−0.494758	+	0.869031i	\(0.664743\pi\)
\(37\)	−0.589336	−	1.58007i	−0.0159280	−	0.0427046i	0.928742	−	0.370726i	\(-0.120891\pi\)
							−0.944670	+	0.328021i	\(0.893618\pi\)
\(41\)	−23.5765	−	51.6253i	−0.575036	−	1.25915i	−0.944072	−	0.329739i	\(-0.893039\pi\)
							0.369036	−	0.929415i	\(-0.379688\pi\)
\(43\)	−49.6067	−	10.7913i	−1.15364	−	0.250960i	−0.405244	−	0.914209i	\(-0.632813\pi\)
							−0.748399	+	0.663249i	\(0.769177\pi\)
\(47\)	−21.5430	+	21.5430i	−0.458363	+	0.458363i	−0.898118	−	0.439755i	\(-0.855065\pi\)
							0.439755	+	0.898118i	\(0.355065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.a.223.2	yes	240
5.2	odd	4	inner	230.3.k.a.177.2	yes	240
23.13	even	11	inner	230.3.k.a.13.2	✓	240
115.82	odd	44	inner	230.3.k.a.197.2	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.13.2	✓	240	23.13	even	11	inner
230.3.k.a.177.2	yes	240	5.2	odd	4	inner
230.3.k.a.197.2	yes	240	115.82	odd	44	inner
230.3.k.a.223.2	yes	240	1.1	even	1	trivial