Embedded newform 464.2.j.a.307.15

• Label: 464.2.j.a.307.15
• Level: $464$
• Weight: $2$
• Character: 464.307
• Analytic conductor: $3.705$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	464.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.70505865379\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			307.15
Character	\(\chi\)	\(=\)	464.307
Dual form			464.2.j.a.331.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03227 + 0.966648i) q^{2} +0.0538569i q^{3} +(0.131183 - 1.99569i) q^{4} +(-0.695345 - 0.695345i) q^{5} +(-0.0520607 - 0.0555951i) q^{6} -0.265456 q^{7} +(1.79372 + 2.18691i) q^{8} +2.99710 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 8 q^{6} - 8 q^{7} + 6 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\)	\(117\)	\(175\)	\(321\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03227	+	0.966648i	−0.729929	+	0.683523i
							\(3\)			0.0538569i			0.0310943i	0.999879	+	0.0155471i	\(0.00494901\pi\)
−0.999879	+	0.0155471i	\(0.995051\pi\)
\(5\)	−0.695345	−	0.695345i	−0.310968	−	0.310968i	0.534317	−	0.845284i	\(-0.320569\pi\)
							−0.845284	+	0.534317i	\(0.820569\pi\)
\(7\)	−0.265456			−0.100333			−0.0501665	−	0.998741i	\(-0.515975\pi\)
							−0.0501665	+	0.998741i	\(0.515975\pi\)
\(11\)	−3.25761			−0.982207			−0.491104	−	0.871101i	\(-0.663406\pi\)
							−0.491104	+	0.871101i	\(0.663406\pi\)
\(13\)	−1.48814	+	1.48814i	−0.412735	+	0.412735i	−0.882690	−	0.469955i	\(-0.844270\pi\)
							0.469955	+	0.882690i	\(0.344270\pi\)
\(17\)	4.49041	−	4.49041i	1.08908	−	1.08908i	0.0934618	−	0.995623i	\(-0.470207\pi\)
							0.995623	−	0.0934618i	\(-0.0297933\pi\)
\(19\)		−	6.36294i		−	1.45976i	−0.683575	−	0.729880i	\(-0.739576\pi\)
							0.683575	−	0.729880i	\(-0.260424\pi\)
\(23\)	8.33621			1.73822			0.869110	−	0.494618i	\(-0.164692\pi\)
							0.869110	+	0.494618i	\(0.164692\pi\)
\(29\)	5.10792	+	1.70561i	0.948518	+	0.316724i
							\(31\)	−1.84021	−	1.84021i	−0.330512	−	0.330512i	0.522269	−	0.852781i	\(-0.325086\pi\)
−0.852781	+	0.522269i	\(0.825086\pi\)
\(37\)	4.00616			0.658609			0.329304	−	0.944224i	\(-0.393186\pi\)
							0.329304	+	0.944224i	\(0.393186\pi\)
\(41\)	−0.673208	+	0.673208i	−0.105137	+	0.105137i	−0.757719	−	0.652581i	\(-0.773686\pi\)
							0.652581	+	0.757719i	\(0.273686\pi\)
\(43\)	−2.71314			−0.413751			−0.206875	−	0.978367i	\(-0.566329\pi\)
							−0.206875	+	0.978367i	\(0.566329\pi\)
\(47\)	1.85028	−	1.85028i	0.269891	−	0.269891i	−0.559165	−	0.829056i	\(-0.688878\pi\)
							0.829056	+	0.559165i	\(0.188878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.2.j.a.307.15	✓	116
16.11	odd	4		464.2.t.a.75.16	yes	116
29.12	odd	4		464.2.t.a.99.16	yes	116
464.331	even	4	inner	464.2.j.a.331.15	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
464.2.j.a.307.15	✓	116	1.1	even	1	trivial
464.2.j.a.331.15	yes	116	464.331	even	4	inner
464.2.t.a.75.16	yes	116	16.11	odd	4
464.2.t.a.99.16	yes	116	29.12	odd	4