Embedded newform 460.2.q.a.11.1

• Label: 460.2.q.a.11.1
• Level: $460$
• Weight: $2$
• Character: 460.11
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.q (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(48\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	460.11
Dual form			460.2.q.a.251.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41401 - 0.0241273i) q^{2} +(0.0682998 + 0.00982003i) q^{3} +(1.99884 + 0.0682323i) q^{4} +(0.281733 + 0.959493i) q^{5} +(-0.0963395 - 0.0155335i) q^{6} +(0.126885 + 0.146433i) q^{7} +(-2.82472 - 0.144707i) q^{8} +(-2.87391 - 0.843856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41401	−	0.0241273i	−0.999854	−	0.0170605i
							\(3\)	0.0682998	+	0.00982003i	0.0394329	+	0.00566959i	0.162003	−	0.986790i	\(-0.448205\pi\)
−0.122570	+	0.992460i	\(0.539114\pi\)
\(5\)	0.281733	+	0.959493i	0.125995	+	0.429098i
							\(7\)	0.126885	+	0.146433i	0.0479580	+	0.0553465i	0.779222	−	0.626748i	\(-0.215615\pi\)
−0.731264	+	0.682095i	\(0.761069\pi\)
\(11\)	−4.69685	−	3.01848i	−1.41615	−	0.910106i	−1.00000		0.000811050i	\(-0.999742\pi\)
							−0.416153	−	0.909295i	\(-0.636622\pi\)
\(13\)	−2.93656	+	3.38898i	−0.814456	+	0.939933i	−0.999080	−	0.0428764i	\(-0.986348\pi\)
							0.184624	+	0.982809i	\(0.440893\pi\)
\(17\)	−1.38160	+	0.630954i	−0.335087	+	0.153029i	−0.575851	−	0.817555i	\(-0.695329\pi\)
							0.240764	+	0.970584i	\(0.422602\pi\)
\(19\)	1.48392	−	3.24933i	0.340434	−	0.745447i	−0.659547	−	0.751664i	\(-0.729252\pi\)
							0.999981	+	0.00621698i	\(0.00197894\pi\)
\(23\)	1.35679	−	4.59991i	0.282910	−	0.959147i
							\(29\)	−2.01846	−	4.41981i	−0.374819	−	0.820738i	−0.999214	−	0.0396302i	\(-0.987382\pi\)
0.624396	−	0.781108i	\(-0.285345\pi\)
\(31\)	−3.65864	+	0.526032i	−0.657110	+	0.0944782i	−0.462801	−	0.886462i	\(-0.653156\pi\)
							−0.194309	+	0.980940i	\(0.562247\pi\)
\(37\)	−3.27232	+	11.1445i	−0.537966	+	1.83214i	0.0163945	+	0.999866i	\(0.494781\pi\)
							−0.554360	+	0.832277i	\(0.687037\pi\)
\(41\)	−3.68947	+	1.08333i	−0.576198	+	0.169187i	−0.556832	−	0.830625i	\(-0.687983\pi\)
							−0.0193667	+	0.999812i	\(0.506165\pi\)
\(43\)	0.361594	−	2.51494i	0.0551425	−	0.383525i	−0.943497	−	0.331381i	\(-0.892486\pi\)
							0.998640	−	0.0521437i	\(-0.0166054\pi\)
\(47\)			0.798714i			0.116504i	0.998302	+	0.0582522i	\(0.0185528\pi\)
							−0.998302	+	0.0582522i	\(0.981447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.q.a.11.1	✓	480
4.3	odd	2	inner	460.2.q.a.11.17	yes	480
23.21	odd	22	inner	460.2.q.a.251.17	yes	480
92.67	even	22	inner	460.2.q.a.251.1	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.q.a.11.1	✓	480	1.1	even	1	trivial
460.2.q.a.11.17	yes	480	4.3	odd	2	inner
460.2.q.a.251.1	yes	480	92.67	even	22	inner
460.2.q.a.251.17	yes	480	23.21	odd	22	inner