Embedded newform 700.2.be.e.443.3

• Label: 700.2.be.e.443.3
• Level: $700$
• Weight: $2$
• Character: 700.443
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			443.3
Character	\(\chi\)	\(=\)	700.443
Dual form			700.2.be.e.207.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20559 - 0.739299i) q^{2} +(2.47915 + 0.664287i) q^{3} +(0.906874 + 1.78258i) q^{4} +(-2.49773 - 2.63369i) q^{6} +(-1.41045 - 2.23844i) q^{7} +(0.224544 - 2.81950i) q^{8} +(3.10685 + 1.79374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 2 q^{2} - 16 q^{6} + 4 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20559	−	0.739299i	−0.852478	−	0.522763i
							\(3\)	2.47915	+	0.664287i	1.43134	+	0.383527i	0.889492	−	0.456950i	\(-0.151058\pi\)
0.541848	+	0.840476i	\(0.317725\pi\)
\(5\)		0			0
							\(7\)	−1.41045	−	2.23844i	−0.533101	−	0.846051i
\(11\)	1.59653	−	0.921758i	0.481372	−	0.277920i								−0.239616	−	0.970868i	\(-0.577022\pi\)
							0.720988	+	0.692947i	\(0.243688\pi\)
\(13\)	2.94578	+	2.94578i	0.817012	+	0.817012i	0.985674	−	0.168662i	\(-0.0539447\pi\)
							−0.168662	+	0.985674i	\(0.553945\pi\)
\(17\)	2.96933	+	0.795631i	0.720169	+	0.192969i	0.600247	−	0.799815i	\(-0.295069\pi\)
							0.119922	+	0.992783i	\(0.461736\pi\)
\(19\)	2.66374	−	4.61374i	0.611104	−	1.05846i	−0.379950	−	0.925007i	\(-0.624059\pi\)
							0.991055	−	0.133457i	\(-0.0426077\pi\)
\(23\)	0.654945	+	2.44429i	0.136565	+	0.509669i	0.999987	+	0.00518527i	\(0.00165053\pi\)
							−0.863421	+	0.504484i	\(0.831683\pi\)
\(29\)		−	6.35796i		−	1.18064i	−0.807168	−	0.590322i	\(-0.799001\pi\)
							0.807168	−	0.590322i	\(-0.200999\pi\)
\(31\)	−3.78330	+	2.18429i	−0.679500	+	0.392310i	−0.799667	−	0.600444i	\(-0.794991\pi\)
							0.120166	+	0.992754i	\(0.461657\pi\)
\(37\)	1.03655	+	3.86848i	0.170409	+	0.635973i	0.997288	+	0.0735946i	\(0.0234471\pi\)
							−0.826880	+	0.562379i	\(0.809886\pi\)
\(41\)	10.6462			1.66266			0.831331	−	0.555777i	\(-0.187579\pi\)
							0.831331	+	0.555777i	\(0.187579\pi\)
\(43\)	1.02386	−	1.02386i	0.156137	−	0.156137i	−0.624715	−	0.780853i	\(-0.714785\pi\)
							0.780853	+	0.624715i	\(0.214785\pi\)
\(47\)	0.785472	−	0.210467i	0.114573	−	0.0306997i	−0.201077	−	0.979575i	\(-0.564444\pi\)
							0.315650	+	0.948876i	\(0.397777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.be.e.443.3		72
4.3	odd	2	inner	700.2.be.e.443.12		72
5.2	odd	4	inner	700.2.be.e.107.14		72
5.3	odd	4		140.2.w.b.107.5	yes	72
5.4	even	2		140.2.w.b.23.16	yes	72
7.4	even	3	inner	700.2.be.e.543.17		72
20.3	even	4		140.2.w.b.107.2	yes	72
20.7	even	4	inner	700.2.be.e.107.17		72
20.19	odd	2		140.2.w.b.23.7	✓	72
28.11	odd	6	inner	700.2.be.e.543.14		72
35.3	even	12		980.2.x.m.67.7		72
35.4	even	6		140.2.w.b.123.2	yes	72
35.9	even	6		980.2.k.k.883.10		36
35.13	even	4		980.2.x.m.667.5		72
35.18	odd	12		140.2.w.b.67.7	yes	72
35.19	odd	6		980.2.k.j.883.10		36
35.23	odd	12		980.2.k.k.687.18		36
35.24	odd	6		980.2.x.m.263.2		72
35.32	odd	12	inner	700.2.be.e.207.12		72
35.33	even	12		980.2.k.j.687.18		36
35.34	odd	2		980.2.x.m.863.16		72
140.3	odd	12		980.2.x.m.67.16		72
140.19	even	6		980.2.k.j.883.18		36
140.23	even	12		980.2.k.k.687.10		36
140.39	odd	6		140.2.w.b.123.5	yes	72
140.59	even	6		980.2.x.m.263.5		72
140.67	even	12	inner	700.2.be.e.207.3		72
140.79	odd	6		980.2.k.k.883.18		36
140.83	odd	4		980.2.x.m.667.2		72
140.103	odd	12		980.2.k.j.687.10		36
140.123	even	12		140.2.w.b.67.16	yes	72
140.139	even	2		980.2.x.m.863.7		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.7	✓	72	20.19	odd	2
140.2.w.b.23.16	yes	72	5.4	even	2
140.2.w.b.67.7	yes	72	35.18	odd	12
140.2.w.b.67.16	yes	72	140.123	even	12
140.2.w.b.107.2	yes	72	20.3	even	4
140.2.w.b.107.5	yes	72	5.3	odd	4
140.2.w.b.123.2	yes	72	35.4	even	6
140.2.w.b.123.5	yes	72	140.39	odd	6
700.2.be.e.107.14		72	5.2	odd	4	inner
700.2.be.e.107.17		72	20.7	even	4	inner
700.2.be.e.207.3		72	140.67	even	12	inner
700.2.be.e.207.12		72	35.32	odd	12	inner
700.2.be.e.443.3		72	1.1	even	1	trivial
700.2.be.e.443.12		72	4.3	odd	2	inner
700.2.be.e.543.14		72	28.11	odd	6	inner
700.2.be.e.543.17		72	7.4	even	3	inner
980.2.k.j.687.10		36	140.103	odd	12
980.2.k.j.687.18		36	35.33	even	12
980.2.k.j.883.10		36	35.19	odd	6
980.2.k.j.883.18		36	140.19	even	6
980.2.k.k.687.10		36	140.23	even	12
980.2.k.k.687.18		36	35.23	odd	12
980.2.k.k.883.10		36	35.9	even	6
980.2.k.k.883.18		36	140.79	odd	6
980.2.x.m.67.7		72	35.3	even	12
980.2.x.m.67.16		72	140.3	odd	12
980.2.x.m.263.2		72	35.24	odd	6
980.2.x.m.263.5		72	140.59	even	6
980.2.x.m.667.2		72	140.83	odd	4
980.2.x.m.667.5		72	35.13	even	4
980.2.x.m.863.7		72	140.139	even	2
980.2.x.m.863.16		72	35.34	odd	2