Embedded newform 460.2.j.a.47.12

• Label: 460.2.j.a.47.12
• Level: $460$
• Weight: $2$
• Character: 460.47
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.12
Character	\(\chi\)	\(=\)	460.47
Dual form			460.2.j.a.323.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22458 - 0.707387i) q^{2} +(1.89438 + 1.89438i) q^{3} +(0.999207 + 1.73251i) q^{4} +(1.67434 - 1.48208i) q^{5} +(-0.979766 - 3.65988i) q^{6} +(-2.51212 + 2.51212i) q^{7} +(0.00194187 - 2.82843i) q^{8} +4.17735i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22458	−	0.707387i	−0.865911	−	0.500198i
							\(3\)	1.89438	+	1.89438i	1.09372	+	1.09372i	0.995128	+	0.0985928i	\(0.0314341\pi\)
0.0985928	+	0.995128i	\(0.468566\pi\)
\(5\)	1.67434	−	1.48208i	0.748789	−	0.662808i
							\(7\)	−2.51212	+	2.51212i	−0.949492	+	0.949492i	−0.998784	−	0.0492927i	\(-0.984303\pi\)
0.0492927	+	0.998784i	\(0.484303\pi\)
\(11\)			4.47820i			1.35023i	0.737714	+	0.675113i	\(0.235905\pi\)
							−0.737714	+	0.675113i	\(0.764095\pi\)
\(13\)	3.24582	−	3.24582i	0.900229	−	0.900229i	−0.0952266	−	0.995456i	\(-0.530358\pi\)
							0.995456	+	0.0952266i	\(0.0303576\pi\)
\(17\)	2.33033	+	2.33033i	0.565189	+	0.565189i	0.930777	−	0.365588i	\(-0.119132\pi\)
							−0.365588	+	0.930777i	\(0.619132\pi\)
\(19\)	−3.55675			−0.815975			−0.407987	−	0.912988i	\(-0.633769\pi\)
							−0.407987	+	0.912988i	\(0.633769\pi\)
\(23\)	0.707107	+	0.707107i	0.147442	+	0.147442i
							\(29\)			7.36940i			1.36846i	0.729264	+	0.684232i	\(0.239862\pi\)
−0.729264	+	0.684232i	\(0.760138\pi\)
\(31\)		−	5.59523i		−	1.00493i	−0.864597	−	0.502466i	\(-0.832426\pi\)
							0.864597	−	0.502466i	\(-0.167574\pi\)
\(37\)	2.85832	+	2.85832i	0.469904	+	0.469904i	0.901884	−	0.431979i	\(-0.142185\pi\)
							−0.431979	+	0.901884i	\(0.642185\pi\)
\(41\)	−10.9258			−1.70632			−0.853160	−	0.521649i	\(-0.825317\pi\)
							−0.853160	+	0.521649i	\(0.825317\pi\)
\(43\)	−1.48236	−	1.48236i	−0.226058	−	0.226058i	0.584985	−	0.811044i	\(-0.301100\pi\)
							−0.811044	+	0.584985i	\(0.801100\pi\)
\(47\)	3.48091	−	3.48091i	0.507743	−	0.507743i	−0.406090	−	0.913833i	\(-0.633108\pi\)
							0.913833	+	0.406090i	\(0.133108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.j.a.47.12	✓	132
4.3	odd	2	inner	460.2.j.a.47.22	yes	132
5.3	odd	4	inner	460.2.j.a.323.22	yes	132
20.3	even	4	inner	460.2.j.a.323.12	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.j.a.47.12	✓	132	1.1	even	1	trivial
460.2.j.a.47.22	yes	132	4.3	odd	2	inner
460.2.j.a.323.12	yes	132	20.3	even	4	inner
460.2.j.a.323.22	yes	132	5.3	odd	4	inner