Embedded newform 2700.2.i.b.901.1

• Label: 2700.2.i.b.901.1
• Level: $2700$
• Weight: $2$
• Character: 2700.901
• Analytic conductor: $21.560$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5596085457\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			901.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	2700.901
Dual form			2700.2.i.b.1801.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{7} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\)	\(1001\)	\(1351\)	\(2377\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)

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			0
\(5\)		0			0
							\(7\)	−0.500000	−	0.866025i	−0.188982	−	0.327327i	0.755929	−	0.654654i	\(-0.227186\pi\)
−0.944911	+	0.327327i	\(0.893852\pi\)
\(11\)	1.50000	+	2.59808i	0.452267	+	0.783349i	0.998526	−	0.0542666i	\(-0.0172821\pi\)
							−0.546259	+	0.837616i	\(0.683949\pi\)
\(13\)	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	\(-0.877618\pi\)
							0.788320	+	0.615265i	\(0.210951\pi\)
\(17\)	6.00000			1.45521			0.727607	−	0.685994i	\(-0.240633\pi\)
							0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	−4.00000			−0.917663			−0.458831	−	0.888523i	\(-0.651732\pi\)
							−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	1.50000	−	2.59808i	0.312772	−	0.541736i	−0.666190	−	0.745782i	\(-0.732076\pi\)
							0.978961	+	0.204046i	\(0.0654092\pi\)
\(29\)	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	\(-0.0768152\pi\)
							−0.692480	+	0.721437i	\(0.743482\pi\)
\(31\)	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	\(-0.981558\pi\)
							0.549309	+	0.835619i	\(0.314891\pi\)
\(37\)	−2.00000			−0.328798			−0.164399	−	0.986394i	\(-0.552568\pi\)
							−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	1.50000	−	2.59808i	0.234261	−	0.405751i	−0.724797	−	0.688963i	\(-0.758066\pi\)
							0.959058	+	0.283211i	\(0.0913998\pi\)
\(43\)	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	\(-0.190961\pi\)
							−0.901629	+	0.432511i	\(0.857628\pi\)
\(47\)	4.50000	+	7.79423i	0.656392	+	1.13691i	0.981543	+	0.191243i	\(0.0612518\pi\)
							−0.325150	+	0.945662i	\(0.605415\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.i.b.901.1		2
3.2	odd	2		900.2.i.b.301.1		2
5.2	odd	4		2700.2.s.b.1549.2		4
5.3	odd	4		2700.2.s.b.1549.1		4
5.4	even	2		108.2.e.a.37.1		2
9.2	odd	6		900.2.i.b.601.1		2
9.4	even	3		8100.2.a.g.1.1		1
9.5	odd	6		8100.2.a.j.1.1		1
9.7	even	3	inner	2700.2.i.b.1801.1		2
15.2	even	4		900.2.s.b.49.2		4
15.8	even	4		900.2.s.b.49.1		4
15.14	odd	2		36.2.e.a.13.1	✓	2
20.19	odd	2		432.2.i.c.145.1		2
35.4	even	6		5292.2.i.c.1549.1		2
35.9	even	6		5292.2.l.a.361.1		2
35.19	odd	6		5292.2.l.c.361.1		2
35.24	odd	6		5292.2.i.a.1549.1		2
35.34	odd	2		5292.2.j.a.1765.1		2
40.19	odd	2		1728.2.i.c.577.1		2
40.29	even	2		1728.2.i.d.577.1		2
45.2	even	12		900.2.s.b.349.1		4
45.4	even	6		324.2.a.a.1.1		1
45.7	odd	12		2700.2.s.b.2449.1		4
45.13	odd	12		8100.2.d.c.649.1		2
45.14	odd	6		324.2.a.c.1.1		1
45.22	odd	12		8100.2.d.c.649.2		2
45.23	even	12		8100.2.d.h.649.1		2
45.29	odd	6		36.2.e.a.25.1	yes	2
45.32	even	12		8100.2.d.h.649.2		2
45.34	even	6		108.2.e.a.73.1		2
45.38	even	12		900.2.s.b.349.2		4
45.43	odd	12		2700.2.s.b.2449.2		4
60.59	even	2		144.2.i.a.49.1		2
105.44	odd	6		1764.2.l.c.949.1		2
105.59	even	6		1764.2.i.c.373.1		2
105.74	odd	6		1764.2.i.a.373.1		2
105.89	even	6		1764.2.l.a.949.1		2
105.104	even	2		1764.2.j.b.589.1		2
120.29	odd	2		576.2.i.f.193.1		2
120.59	even	2		576.2.i.e.193.1		2
180.59	even	6		1296.2.a.k.1.1		1
180.79	odd	6		432.2.i.c.289.1		2
180.119	even	6		144.2.i.a.97.1		2
180.139	odd	6		1296.2.a.b.1.1		1
315.34	odd	6		5292.2.j.a.3529.1		2
315.74	odd	6		1764.2.l.c.961.1		2
315.79	even	6		5292.2.i.c.2125.1		2
315.124	odd	6		5292.2.i.a.2125.1		2
315.164	even	6		1764.2.l.a.961.1		2
315.209	even	6		1764.2.j.b.1177.1		2
315.214	even	6		5292.2.l.a.3313.1		2
315.254	odd	6		1764.2.i.a.1537.1		2
315.299	even	6		1764.2.i.c.1537.1		2
315.304	odd	6		5292.2.l.c.3313.1		2
360.29	odd	6		576.2.i.f.385.1		2
360.59	even	6		5184.2.a.f.1.1		1
360.139	odd	6		5184.2.a.bb.1.1		1
360.149	odd	6		5184.2.a.e.1.1		1
360.229	even	6		5184.2.a.ba.1.1		1
360.259	odd	6		1728.2.i.c.1153.1		2
360.299	even	6		576.2.i.e.385.1		2
360.349	even	6		1728.2.i.d.1153.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.2.e.a.13.1	✓	2	15.14	odd	2
36.2.e.a.25.1	yes	2	45.29	odd	6
108.2.e.a.37.1		2	5.4	even	2
108.2.e.a.73.1		2	45.34	even	6
144.2.i.a.49.1		2	60.59	even	2
144.2.i.a.97.1		2	180.119	even	6
324.2.a.a.1.1		1	45.4	even	6
324.2.a.c.1.1		1	45.14	odd	6
432.2.i.c.145.1		2	20.19	odd	2
432.2.i.c.289.1		2	180.79	odd	6
576.2.i.e.193.1		2	120.59	even	2
576.2.i.e.385.1		2	360.299	even	6
576.2.i.f.193.1		2	120.29	odd	2
576.2.i.f.385.1		2	360.29	odd	6
900.2.i.b.301.1		2	3.2	odd	2
900.2.i.b.601.1		2	9.2	odd	6
900.2.s.b.49.1		4	15.8	even	4
900.2.s.b.49.2		4	15.2	even	4
900.2.s.b.349.1		4	45.2	even	12
900.2.s.b.349.2		4	45.38	even	12
1296.2.a.b.1.1		1	180.139	odd	6
1296.2.a.k.1.1		1	180.59	even	6
1728.2.i.c.577.1		2	40.19	odd	2
1728.2.i.c.1153.1		2	360.259	odd	6
1728.2.i.d.577.1		2	40.29	even	2
1728.2.i.d.1153.1		2	360.349	even	6
1764.2.i.a.373.1		2	105.74	odd	6
1764.2.i.a.1537.1		2	315.254	odd	6
1764.2.i.c.373.1		2	105.59	even	6
1764.2.i.c.1537.1		2	315.299	even	6
1764.2.j.b.589.1		2	105.104	even	2
1764.2.j.b.1177.1		2	315.209	even	6
1764.2.l.a.949.1		2	105.89	even	6
1764.2.l.a.961.1		2	315.164	even	6
1764.2.l.c.949.1		2	105.44	odd	6
1764.2.l.c.961.1		2	315.74	odd	6
2700.2.i.b.901.1		2	1.1	even	1	trivial
2700.2.i.b.1801.1		2	9.7	even	3	inner
2700.2.s.b.1549.1		4	5.3	odd	4
2700.2.s.b.1549.2		4	5.2	odd	4
2700.2.s.b.2449.1		4	45.7	odd	12
2700.2.s.b.2449.2		4	45.43	odd	12
5184.2.a.e.1.1		1	360.149	odd	6
5184.2.a.f.1.1		1	360.59	even	6
5184.2.a.ba.1.1		1	360.229	even	6
5184.2.a.bb.1.1		1	360.139	odd	6
5292.2.i.a.1549.1		2	35.24	odd	6
5292.2.i.a.2125.1		2	315.124	odd	6
5292.2.i.c.1549.1		2	35.4	even	6
5292.2.i.c.2125.1		2	315.79	even	6
5292.2.j.a.1765.1		2	35.34	odd	2
5292.2.j.a.3529.1		2	315.34	odd	6
5292.2.l.a.361.1		2	35.9	even	6
5292.2.l.a.3313.1		2	315.214	even	6
5292.2.l.c.361.1		2	35.19	odd	6
5292.2.l.c.3313.1		2	315.304	odd	6
8100.2.a.g.1.1		1	9.4	even	3
8100.2.a.j.1.1		1	9.5	odd	6
8100.2.d.c.649.1		2	45.13	odd	12
8100.2.d.c.649.2		2	45.22	odd	12
8100.2.d.h.649.1		2	45.23	even	12
8100.2.d.h.649.2		2	45.32	even	12