Embedded newform 230.3.k.b.13.7

• Label: 230.3.k.b.13.7
• 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			13.7
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.b.177.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41061 + 0.100889i) q^{2} +(0.161368 + 0.741796i) q^{3} +(1.97964 + 0.284630i) q^{4} +(4.91560 - 0.914803i) q^{5} +(0.152788 + 1.06267i) q^{6} +(2.14811 - 1.17296i) q^{7} +(2.76379 + 0.601225i) q^{8} +(7.66247 - 3.49933i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	0.161368	+	0.741796i	0.0537893	+	0.247265i	0.996189	−	0.0872244i	\(-0.0277997\pi\)
−0.942399	+	0.334490i	\(0.891436\pi\)
\(5\)	4.91560	−	0.914803i	0.983120	−	0.182961i
							\(7\)	2.14811	−	1.17296i	0.306873	−	0.167565i	−0.318432	−	0.947946i	\(-0.603156\pi\)
0.625305	+	0.780380i	\(0.284975\pi\)
\(11\)	−7.11457	+	8.21065i	−0.646779	+	0.746422i	−0.980558	−	0.196229i	\(-0.937130\pi\)
							0.333779	+	0.942651i	\(0.391676\pi\)
\(13\)	−18.1470	−	9.90900i	−1.39592	−	0.762231i	−0.407391	−	0.913254i	\(-0.633561\pi\)
							−0.988530	+	0.151023i	\(0.951743\pi\)
\(17\)	5.70029	+	4.26718i	0.335311	+	0.251011i	0.753644	−	0.657283i	\(-0.228294\pi\)
							−0.418333	+	0.908294i	\(0.637385\pi\)
\(19\)	22.8352	+	3.28320i	1.20185	+	0.172800i	0.714012	−	0.700134i	\(-0.246876\pi\)
							0.487839	+	0.872934i	\(0.337785\pi\)
\(23\)	21.0019	+	9.37664i	0.913125	+	0.407680i
							\(29\)	−44.1820	+	6.35242i	−1.52352	+	0.219049i	−0.852632	−	0.522512i	\(-0.824995\pi\)
−0.670886	+	0.741561i	\(0.734086\pi\)
\(31\)	−10.2239	+	6.57049i	−0.329803	+	0.211951i	−0.695054	−	0.718958i	\(-0.744619\pi\)
							0.365251	+	0.930909i	\(0.380983\pi\)
\(37\)	−42.4332	−	15.8268i	−1.14684	−	0.427751i	−0.297048	−	0.954862i	\(-0.596002\pi\)
							−0.849796	+	0.527111i	\(0.823275\pi\)
\(41\)	−5.99857	+	13.1350i	−0.146307	+	0.320367i	−0.968570	−	0.248740i	\(-0.919984\pi\)
							0.822264	+	0.569107i	\(0.192711\pi\)
\(43\)	−6.53521	−	30.0419i	−0.151982	−	0.698648i	−0.988563	−	0.150806i	\(-0.951813\pi\)
							0.836582	−	0.547842i	\(-0.184550\pi\)
\(47\)	−26.8848	+	26.8848i	−0.572018	+	0.572018i	−0.932692	−	0.360674i	\(-0.882547\pi\)
							0.360674	+	0.932692i	\(0.382547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.13.7	✓	240
5.2	odd	4	inner	230.3.k.b.197.7	yes	240
23.16	even	11	inner	230.3.k.b.223.7	yes	240
115.62	odd	44	inner	230.3.k.b.177.7	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.7	✓	240	1.1	even	1	trivial
230.3.k.b.177.7	yes	240	115.62	odd	44	inner
230.3.k.b.197.7	yes	240	5.2	odd	4	inner
230.3.k.b.223.7	yes	240	23.16	even	11	inner