Embedded newform 975.2.db.a.67.2

• Label: 975.2.db.a.67.2
• Level: $975$
• Weight: $2$
• Character: 975.67
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $1120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.db (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(1120\)
Relative dimension:	\(70\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			67.2
Character	\(\chi\)	\(=\)	975.67
Dual form			975.2.db.a.553.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84991 + 2.05453i) q^{2} +(0.933580 - 0.358368i) q^{3} +(-0.589879 - 5.61232i) q^{4} +(-0.166732 + 2.22984i) q^{5} +(-0.990758 + 2.58101i) q^{6} +(-3.35354 - 1.93617i) q^{7} +(8.14861 + 5.92031i) q^{8} +(0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q - 4 q^{2} + 140 q^{4} - 4 q^{5} + 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\)	\(301\)	\(326\)	\(352\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(1\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84991	+	2.05453i	−1.30808	+	1.45277i	−0.497105	+	0.867690i	\(0.665604\pi\)
							−0.810975	+	0.585081i	\(0.801063\pi\)
\(3\)	0.933580	−	0.358368i	0.539003	−	0.206904i
							\(5\)	−0.166732	+	2.22984i	−0.0745647	+	0.997216i
\(7\)	−3.35354	−	1.93617i	−1.26752	−	0.731803i								−0.293002	−	0.956112i	\(-0.594654\pi\)
							−0.974518	+	0.224308i	\(0.927988\pi\)
\(11\)	−0.431861	+	0.0226329i	−0.130211	+	0.00682407i	−0.117330	−	0.993093i	\(-0.537434\pi\)
							−0.0128809	+	0.999917i	\(0.504100\pi\)
\(13\)	1.56320	−	3.24906i	0.433552	−	0.901128i
							\(17\)	−1.57482	+	4.10255i	−0.381950	+	0.995014i	0.598999	+	0.800750i	\(0.295565\pi\)
−0.980949	+	0.194265i	\(0.937768\pi\)
\(19\)	6.03217	+	2.31553i	1.38388	+	0.531220i	0.932518	−	0.361124i	\(-0.117607\pi\)
							0.451357	+	0.892343i	\(0.350940\pi\)
\(23\)	−2.29488	+	0.120269i	−0.478515	+	0.0250779i	−0.290069	−	0.957006i	\(-0.593678\pi\)
							−0.188446	+	0.982084i	\(0.560345\pi\)
\(29\)	2.17659	−	4.88871i	0.404183	−	0.907810i	−0.590711	−	0.806883i	\(-0.701153\pi\)
							0.994894	−	0.100926i	\(-0.0321807\pi\)
\(31\)	8.18563	−	1.29648i	1.47018	−	0.232854i	0.630609	−	0.776101i	\(-0.282805\pi\)
							0.839573	+	0.543247i	\(0.182805\pi\)
\(37\)	1.23596	+	5.81471i	0.203190	+	0.955933i	0.955012	+	0.296567i	\(0.0958418\pi\)
							−0.751822	+	0.659366i	\(0.770825\pi\)
\(41\)	−2.44024	+	3.75764i	−0.381101	+	0.586844i	−0.976217	−	0.216796i	\(-0.930439\pi\)
							0.595116	+	0.803640i	\(0.297106\pi\)
\(43\)	2.12046	−	0.568176i	0.323368	−	0.0866461i	−0.0934836	−	0.995621i	\(-0.529800\pi\)
							0.416851	+	0.908975i	\(0.363134\pi\)
\(47\)	6.59108	+	9.07185i	0.961408	+	1.32327i	0.946269	+	0.323380i	\(0.104819\pi\)
							0.0151390	+	0.999885i	\(0.495181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.db.a.67.2	yes	1120
13.7	odd	12		975.2.cq.a.592.2	yes	1120
25.3	odd	20		975.2.cq.a.28.2	✓	1120
325.228	even	60	inner	975.2.db.a.553.2	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.cq.a.28.2	✓	1120	25.3	odd	20
975.2.cq.a.592.2	yes	1120	13.7	odd	12
975.2.db.a.67.2	yes	1120	1.1	even	1	trivial
975.2.db.a.553.2	yes	1120	325.228	even	60	inner