Embedded newform 460.2.q.a.11.2

• Label: 460.2.q.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	460.11
Dual form			460.2.q.a.251.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41004 + 0.108605i) q^{2} +(-2.31335 - 0.332609i) q^{3} +(1.97641 - 0.306274i) q^{4} +(-0.281733 - 0.959493i) q^{5} +(3.29803 + 0.217751i) q^{6} +(-1.52622 - 1.76136i) q^{7} +(-2.75355 + 0.646505i) q^{8} +(2.36248 + 0.693685i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41004	+	0.108605i	−0.997047	+	0.0767953i
							\(3\)	−2.31335	−	0.332609i	−1.33561	−	0.192032i	−0.562761	−	0.826619i	\(-0.690261\pi\)
−0.772851	+	0.634587i	\(0.781170\pi\)
\(5\)	−0.281733	−	0.959493i	−0.125995	−	0.429098i
							\(7\)	−1.52622	−	1.76136i	−0.576858	−	0.665730i	0.390068	−	0.920786i	\(-0.372451\pi\)
−0.966926	+	0.255056i	\(0.917906\pi\)
\(11\)	−0.980273	−	0.629983i	−0.295563	−	0.189947i	0.384452	−	0.923145i	\(-0.374390\pi\)
							−0.680015	+	0.733198i	\(0.738027\pi\)
\(13\)	2.14199	−	2.47199i	0.594081	−	0.685606i	−0.376490	−	0.926421i	\(-0.622869\pi\)
							0.970571	+	0.240815i	\(0.0774146\pi\)
\(17\)	−0.352578	+	0.161017i	−0.0855127	+	0.0390523i	−0.457712	−	0.889100i	\(-0.651331\pi\)
							0.372199	+	0.928153i	\(0.378604\pi\)
\(19\)	0.159288	−	0.348792i	0.0365432	−	0.0800184i	−0.890480	−	0.455022i	\(-0.849631\pi\)
							0.927023	+	0.375004i	\(0.122359\pi\)
\(23\)	−4.72599	−	0.815490i	−0.985437	−	0.170041i
							\(29\)	2.50706	+	5.48970i	0.465549	+	1.01941i	0.986187	+	0.165634i	\(0.0529669\pi\)
−0.520638	+	0.853778i	\(0.674306\pi\)
\(31\)	−4.23497	+	0.608897i	−0.760623	+	0.109361i	−0.511702	−	0.859163i	\(-0.670985\pi\)
							−0.248921	+	0.968524i	\(0.580076\pi\)
\(37\)	−1.15575	+	3.93613i	−0.190005	+	0.647096i	0.808294	+	0.588780i	\(0.200391\pi\)
							−0.998298	+	0.0583165i	\(0.981427\pi\)
\(41\)	−9.31928	+	2.73639i	−1.45543	+	0.427352i	−0.911332	−	0.411672i	\(-0.864945\pi\)
							−0.544095	+	0.839024i	\(0.683127\pi\)
\(43\)	−1.16496	+	8.10246i	−0.177654	+	1.23561i	0.684517	+	0.728997i	\(0.260013\pi\)
							−0.862171	+	0.506617i	\(0.830896\pi\)
\(47\)			3.41098i			0.497543i	0.968562	+	0.248771i	\(0.0800268\pi\)
							−0.968562	+	0.248771i	\(0.919973\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.q.a.11.2	✓	480
4.3	odd	2	inner	460.2.q.a.11.15	yes	480
23.21	odd	22	inner	460.2.q.a.251.15	yes	480
92.67	even	22	inner	460.2.q.a.251.2	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.q.a.11.2	✓	480	1.1	even	1	trivial
460.2.q.a.11.15	yes	480	4.3	odd	2	inner
460.2.q.a.251.2	yes	480	92.67	even	22	inner
460.2.q.a.251.15	yes	480	23.21	odd	22	inner