Embedded newform 700.2.i.c.501.1

• Label: 700.2.i.c.501.1
• Level: $700$
• Weight: $2$
• Character: 700.501
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			501.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	700.501
Dual form			700.2.i.c.401.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	0.500000	−	0.866025i	0.288675	−	0.500000i	−0.684819	−	0.728714i	\(-0.740119\pi\)
0.973494	+	0.228714i	\(0.0734519\pi\)
\(5\)		0			0
							\(7\)	2.00000	−	1.73205i	0.755929	−	0.654654i
\(11\)	1.50000	−	2.59808i	0.452267	−	0.783349i								−0.546259	−	0.837616i	\(-0.683949\pi\)
							0.998526	+	0.0542666i	\(0.0172821\pi\)
\(13\)	−2.00000			−0.554700			−0.277350	−	0.960769i	\(-0.589456\pi\)
							−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	\(-0.714811\pi\)
							0.988583	+	0.150675i	\(0.0481447\pi\)
\(19\)	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	\(-0.130073\pi\)
							−0.802955	+	0.596040i	\(0.796740\pi\)
\(23\)	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	\(-0.0654092\pi\)
							−0.666190	+	0.745782i	\(0.732076\pi\)
\(29\)	−6.00000			−1.11417			−0.557086	−	0.830455i	\(-0.688081\pi\)
							−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	3.50000	−	6.06218i	0.628619	−	1.08880i	−0.359211	−	0.933257i	\(-0.616954\pi\)
							0.987829	−	0.155543i	\(-0.0497126\pi\)
\(37\)	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	0.821995	−	0.569495i	\(-0.192861\pi\)
							−0.904194	+	0.427121i	\(0.859528\pi\)
\(41\)	6.00000			0.937043			0.468521	−	0.883452i	\(-0.344787\pi\)
							0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	4.00000			0.609994			0.304997	−	0.952353i	\(-0.401344\pi\)
							0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−4.50000	−	7.79423i	−0.656392	−	1.13691i	−0.981543	−	0.191243i	\(-0.938748\pi\)
							0.325150	−	0.945662i	\(-0.394585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.i.c.501.1		2
5.2	odd	4		700.2.r.b.249.1		4
5.3	odd	4		700.2.r.b.249.2		4
5.4	even	2		28.2.e.a.25.1	yes	2
7.2	even	3	inner	700.2.i.c.401.1		2
7.3	odd	6		4900.2.a.n.1.1		1
7.4	even	3		4900.2.a.g.1.1		1
15.14	odd	2		252.2.k.c.109.1		2
20.19	odd	2		112.2.i.b.81.1		2
35.2	odd	12		700.2.r.b.149.2		4
35.3	even	12		4900.2.e.h.2549.2		2
35.4	even	6		196.2.a.b.1.1		1
35.9	even	6		28.2.e.a.9.1	✓	2
35.17	even	12		4900.2.e.h.2549.1		2
35.18	odd	12		4900.2.e.i.2549.1		2
35.19	odd	6		196.2.e.a.177.1		2
35.23	odd	12		700.2.r.b.149.1		4
35.24	odd	6		196.2.a.a.1.1		1
35.32	odd	12		4900.2.e.i.2549.2		2
35.34	odd	2		196.2.e.a.165.1		2
40.19	odd	2		448.2.i.c.193.1		2
40.29	even	2		448.2.i.e.193.1		2
45.4	even	6		2268.2.i.a.865.1		2
45.14	odd	6		2268.2.i.h.865.1		2
45.29	odd	6		2268.2.l.a.109.1		2
45.34	even	6		2268.2.l.h.109.1		2
60.59	even	2		1008.2.s.p.865.1		2
105.44	odd	6		252.2.k.c.37.1		2
105.59	even	6		1764.2.a.j.1.1		1
105.74	odd	6		1764.2.a.a.1.1		1
105.89	even	6		1764.2.k.b.1549.1		2
105.104	even	2		1764.2.k.b.361.1		2
140.19	even	6		784.2.i.d.177.1		2
140.39	odd	6		784.2.a.d.1.1		1
140.59	even	6		784.2.a.g.1.1		1
140.79	odd	6		112.2.i.b.65.1		2
140.139	even	2		784.2.i.d.753.1		2
280.59	even	6		3136.2.a.k.1.1		1
280.109	even	6		3136.2.a.h.1.1		1
280.149	even	6		448.2.i.e.65.1		2
280.179	odd	6		3136.2.a.s.1.1		1
280.219	odd	6		448.2.i.c.65.1		2
280.269	odd	6		3136.2.a.v.1.1		1
315.79	even	6		2268.2.i.a.2053.1		2
315.149	odd	6		2268.2.l.a.541.1		2
315.184	even	6		2268.2.l.h.541.1		2
315.254	odd	6		2268.2.i.h.2053.1		2
420.59	odd	6		7056.2.a.bw.1.1		1
420.179	even	6		7056.2.a.f.1.1		1
420.359	even	6		1008.2.s.p.289.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	35.9	even	6
28.2.e.a.25.1	yes	2	5.4	even	2
112.2.i.b.65.1		2	140.79	odd	6
112.2.i.b.81.1		2	20.19	odd	2
196.2.a.a.1.1		1	35.24	odd	6
196.2.a.b.1.1		1	35.4	even	6
196.2.e.a.165.1		2	35.34	odd	2
196.2.e.a.177.1		2	35.19	odd	6
252.2.k.c.37.1		2	105.44	odd	6
252.2.k.c.109.1		2	15.14	odd	2
448.2.i.c.65.1		2	280.219	odd	6
448.2.i.c.193.1		2	40.19	odd	2
448.2.i.e.65.1		2	280.149	even	6
448.2.i.e.193.1		2	40.29	even	2
700.2.i.c.401.1		2	7.2	even	3	inner
700.2.i.c.501.1		2	1.1	even	1	trivial
700.2.r.b.149.1		4	35.23	odd	12
700.2.r.b.149.2		4	35.2	odd	12
700.2.r.b.249.1		4	5.2	odd	4
700.2.r.b.249.2		4	5.3	odd	4
784.2.a.d.1.1		1	140.39	odd	6
784.2.a.g.1.1		1	140.59	even	6
784.2.i.d.177.1		2	140.19	even	6
784.2.i.d.753.1		2	140.139	even	2
1008.2.s.p.289.1		2	420.359	even	6
1008.2.s.p.865.1		2	60.59	even	2
1764.2.a.a.1.1		1	105.74	odd	6
1764.2.a.j.1.1		1	105.59	even	6
1764.2.k.b.361.1		2	105.104	even	2
1764.2.k.b.1549.1		2	105.89	even	6
2268.2.i.a.865.1		2	45.4	even	6
2268.2.i.a.2053.1		2	315.79	even	6
2268.2.i.h.865.1		2	45.14	odd	6
2268.2.i.h.2053.1		2	315.254	odd	6
2268.2.l.a.109.1		2	45.29	odd	6
2268.2.l.a.541.1		2	315.149	odd	6
2268.2.l.h.109.1		2	45.34	even	6
2268.2.l.h.541.1		2	315.184	even	6
3136.2.a.h.1.1		1	280.109	even	6
3136.2.a.k.1.1		1	280.59	even	6
3136.2.a.s.1.1		1	280.179	odd	6
3136.2.a.v.1.1		1	280.269	odd	6
4900.2.a.g.1.1		1	7.4	even	3
4900.2.a.n.1.1		1	7.3	odd	6
4900.2.e.h.2549.1		2	35.17	even	12
4900.2.e.h.2549.2		2	35.3	even	12
4900.2.e.i.2549.1		2	35.18	odd	12
4900.2.e.i.2549.2		2	35.32	odd	12
7056.2.a.f.1.1		1	420.179	even	6
7056.2.a.bw.1.1		1	420.59	odd	6