Embedded newform 460.2.q.a.11.4

• Label: 460.2.q.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	460.11
Dual form			460.2.q.a.251.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31872 - 0.510869i) q^{2} +(0.994266 + 0.142954i) q^{3} +(1.47803 + 1.34738i) q^{4} +(-0.281733 - 0.959493i) q^{5} +(-1.23812 - 0.696455i) q^{6} +(-3.13881 - 3.62238i) q^{7} +(-1.26076 - 2.53189i) q^{8} +(-1.91035 - 0.560929i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31872	−	0.510869i	−0.932473	−	0.361239i
							\(3\)	0.994266	+	0.142954i	0.574040	+	0.0825345i	0.423220	−	0.906027i	\(-0.360900\pi\)
0.150820	+	0.988561i	\(0.451809\pi\)
\(5\)	−0.281733	−	0.959493i	−0.125995	−	0.429098i
							\(7\)	−3.13881	−	3.62238i	−1.18636	−	1.36913i	−0.913377	−	0.407114i	\(-0.866535\pi\)
−0.272983	−	0.962019i	\(-0.588010\pi\)
\(11\)	1.37669	+	0.884744i	0.415087	+	0.266760i	0.731473	−	0.681870i	\(-0.238833\pi\)
							−0.316386	+	0.948631i	\(0.602470\pi\)
\(13\)	−2.96837	+	3.42568i	−0.823278	+	0.950114i	−0.999413	−	0.0342522i	\(-0.989095\pi\)
							0.176135	+	0.984366i	\(0.443640\pi\)
\(17\)	−6.59084	+	3.00994i	−1.59851	+	0.730017i	−0.997596	−	0.0692943i	\(-0.977925\pi\)
							−0.600918	+	0.799311i	\(0.705198\pi\)
\(19\)	0.394119	−	0.862999i	0.0904170	−	0.197986i	−0.859021	−	0.511941i	\(-0.828927\pi\)
							0.949438	+	0.313955i	\(0.101654\pi\)
\(23\)	−0.128403	+	4.79411i	−0.0267738	+	0.999642i
							\(29\)	−1.78941	−	3.91826i	−0.332285	−	0.727603i	0.667571	−	0.744546i	\(-0.267334\pi\)
−0.999856	+	0.0169431i	\(0.994607\pi\)
\(31\)	7.18144	−	1.03254i	1.28982	−	0.185449i	0.536951	−	0.843613i	\(-0.319576\pi\)
							0.752874	+	0.658165i	\(0.228667\pi\)
\(37\)	2.21357	−	7.53871i	0.363908	−	1.23936i	−0.550593	−	0.834774i	\(-0.685598\pi\)
							0.914501	−	0.404583i	\(-0.132583\pi\)
\(41\)	−11.7429	+	3.44801i	−1.83393	+	0.538489i	−0.999912	−	0.0132539i	\(-0.995781\pi\)
							−0.834014	+	0.551743i	\(0.813963\pi\)
\(43\)	0.687892	−	4.78439i	0.104903	−	0.729613i	−0.867691	−	0.497103i	\(-0.834397\pi\)
							0.972594	−	0.232510i	\(-0.0746939\pi\)
\(47\)		−	4.66250i		−	0.680095i	−0.940408	−	0.340048i	\(-0.889557\pi\)
							0.940408	−	0.340048i	\(-0.110443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.q.a.11.4	✓	480
4.3	odd	2	inner	460.2.q.a.11.23	yes	480
23.21	odd	22	inner	460.2.q.a.251.23	yes	480
92.67	even	22	inner	460.2.q.a.251.4	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.q.a.11.4	✓	480	1.1	even	1	trivial
460.2.q.a.11.23	yes	480	4.3	odd	2	inner
460.2.q.a.251.4	yes	480	92.67	even	22	inner
460.2.q.a.251.23	yes	480	23.21	odd	22	inner