Embedded newform 460.2.q.a.11.5

• Label: 460.2.q.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	460.11
Dual form			460.2.q.a.251.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30815 - 0.537340i) q^{2} +(-0.501133 - 0.0720521i) q^{3} +(1.42253 + 1.40585i) q^{4} +(-0.281733 - 0.959493i) q^{5} +(0.616843 + 0.363534i) q^{6} +(1.51820 + 1.75210i) q^{7} +(-1.10547 - 2.60345i) q^{8} +(-2.63254 - 0.772982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30815	−	0.537340i	−0.925004	−	0.379957i
							\(3\)	−0.501133	−	0.0720521i	−0.289329	−	0.0415993i	−0.00387806	−	0.999992i	\(-0.501234\pi\)
−0.285451	+	0.958393i	\(0.592144\pi\)
\(5\)	−0.281733	−	0.959493i	−0.125995	−	0.429098i
							\(7\)	1.51820	+	1.75210i	0.573825	+	0.662230i	0.966266	−	0.257548i	\(-0.0829145\pi\)
−0.392440	+	0.919778i	\(0.628369\pi\)
\(11\)	−0.663173	−	0.426195i	−0.199954	−	0.128503i	0.436832	−	0.899543i	\(-0.356100\pi\)
							−0.636786	+	0.771040i	\(0.719737\pi\)
\(13\)	2.94377	−	3.39729i	0.816455	−	0.942240i	−0.182707	−	0.983167i	\(-0.558486\pi\)
							0.999162	+	0.0409277i	\(0.0130313\pi\)
\(17\)	3.82498	−	1.74681i	0.927694	−	0.423663i	0.106500	−	0.994313i	\(-0.466036\pi\)
							0.821194	+	0.570649i	\(0.193308\pi\)
\(19\)	−1.44661	+	3.16763i	−0.331875	+	0.726704i	−0.999847	−	0.0174820i	\(-0.994435\pi\)
							0.667973	+	0.744186i	\(0.267162\pi\)
\(23\)	4.39078	−	1.92901i	0.915540	−	0.402227i
							\(29\)	−3.63875	−	7.96774i	−0.675698	−	1.47957i	−0.867138	−	0.498068i	\(-0.834043\pi\)
0.191440	−	0.981504i	\(-0.438684\pi\)
\(31\)	2.24668	−	0.323024i	0.403516	−	0.0580168i	0.0624308	−	0.998049i	\(-0.480115\pi\)
							0.341085	+	0.940032i	\(0.389206\pi\)
\(37\)	1.65231	−	5.62725i	0.271638	−	0.925114i	−0.704816	−	0.709390i	\(-0.748971\pi\)
							0.976454	−	0.215724i	\(-0.0692113\pi\)
\(41\)	6.42271	−	1.88588i	1.00306	−	0.294525i	0.261348	−	0.965245i	\(-0.415833\pi\)
							0.741711	+	0.670720i	\(0.234015\pi\)
\(43\)	0.559450	−	3.89106i	0.0853153	−	0.593381i	−0.901652	−	0.432462i	\(-0.857645\pi\)
							0.986968	−	0.160919i	\(-0.0514459\pi\)
\(47\)		−	8.71798i		−	1.27165i	−0.771834	−	0.635824i	\(-0.780660\pi\)
							0.771834	−	0.635824i	\(-0.219340\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.q.a.11.5	✓	480
4.3	odd	2	inner	460.2.q.a.11.24	yes	480
23.21	odd	22	inner	460.2.q.a.251.24	yes	480
92.67	even	22	inner	460.2.q.a.251.5	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.q.a.11.5	✓	480	1.1	even	1	trivial
460.2.q.a.11.24	yes	480	4.3	odd	2	inner
460.2.q.a.251.5	yes	480	92.67	even	22	inner
460.2.q.a.251.24	yes	480	23.21	odd	22	inner