Embedded newform 2700.2.d.g.649.2

• Label: 2700.2.d.g.649.2
• Level: $2700$
• Weight: $2$
• Character: 2700.649
• Analytic conductor: $21.560$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5596085457\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			649.2
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	2700.649
Dual form			2700.2.d.g.649.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+5.00000i q^{7} +7.00000i q^{13} +1.00000 q^{19} -4.00000 q^{31} -1.00000i q^{37} -8.00000i q^{43} -18.0000 q^{49} -13.0000 q^{61} +11.0000i q^{67} -17.0000i q^{73} +13.0000 q^{79} -35.0000 q^{91} +5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{19} - 8 q^{31} - 36 q^{49} - 26 q^{61} + 26 q^{79} - 70 q^{91}+O(q^{100}) \)

Character values

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

\(n\)	\(1001\)	\(1351\)	\(2377\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	0	0
			\(7\)	5.00000i	1.88982i	0.327327	+	0.944911i	\(0.393852\pi\)
−0.327327	+	0.944911i				\(0.606148\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	7.00000i	1.94145i	0.240192	+	0.970725i	\(0.422790\pi\)
			−0.240192	+	0.970725i	\(0.577210\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	1.00000	0.229416	0.114708	−	0.993399i	\(-0.463407\pi\)
			0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	\(-0.973806\pi\)
			0.996616	−	0.0821995i	\(-0.0261945\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	\(-0.791172\pi\)
			0.792406	−	0.609994i	\(-0.208828\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−13.0000	−1.66448	−0.832240	−	0.554416i	\(-0.812942\pi\)
			−0.832240	+	0.554416i	\(0.812942\pi\)
\(67\)	11.0000i	1.34386i	0.740613	+	0.671932i	\(0.234535\pi\)
			−0.740613	+	0.671932i	\(0.765465\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	− 17.0000i	− 1.98970i	−0.101361	−	0.994850i	\(-0.532320\pi\)
			0.101361	−	0.994850i	\(-0.467680\pi\)
\(79\)	13.0000	1.46261	0.731307	−	0.682048i	\(-0.238911\pi\)
			0.731307	+	0.682048i	\(0.238911\pi\)
\(83\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	5.00000i	0.507673i	0.967247	+	0.253837i	\(0.0816925\pi\)
			−0.967247	+	0.253837i	\(0.918307\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.d.g.649.2		2
3.2	odd	2	CM	2700.2.d.g.649.2		2
5.2	odd	4		2700.2.a.b.1.1		1
5.3	odd	4		108.2.a.a.1.1	✓	1
5.4	even	2	inner	2700.2.d.g.649.1		2
15.2	even	4		2700.2.a.b.1.1		1
15.8	even	4		108.2.a.a.1.1	✓	1
15.14	odd	2	inner	2700.2.d.g.649.1		2
20.3	even	4		432.2.a.d.1.1		1
35.13	even	4		5292.2.a.j.1.1		1
40.3	even	4		1728.2.a.m.1.1		1
40.13	odd	4		1728.2.a.p.1.1		1
45.13	odd	12		324.2.e.b.217.1		2
45.23	even	12		324.2.e.b.217.1		2
45.38	even	12		324.2.e.b.109.1		2
45.43	odd	12		324.2.e.b.109.1		2
60.23	odd	4		432.2.a.d.1.1		1
105.83	odd	4		5292.2.a.j.1.1		1
120.53	even	4		1728.2.a.p.1.1		1
120.83	odd	4		1728.2.a.m.1.1		1
180.23	odd	12		1296.2.i.j.865.1		2
180.43	even	12		1296.2.i.j.433.1		2
180.83	odd	12		1296.2.i.j.433.1		2
180.103	even	12		1296.2.i.j.865.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.2.a.a.1.1	✓	1	5.3	odd	4
108.2.a.a.1.1	✓	1	15.8	even	4
324.2.e.b.109.1		2	45.38	even	12
324.2.e.b.109.1		2	45.43	odd	12
324.2.e.b.217.1		2	45.13	odd	12
324.2.e.b.217.1		2	45.23	even	12
432.2.a.d.1.1		1	20.3	even	4
432.2.a.d.1.1		1	60.23	odd	4
1296.2.i.j.433.1		2	180.43	even	12
1296.2.i.j.433.1		2	180.83	odd	12
1296.2.i.j.865.1		2	180.23	odd	12
1296.2.i.j.865.1		2	180.103	even	12
1728.2.a.m.1.1		1	40.3	even	4
1728.2.a.m.1.1		1	120.83	odd	4
1728.2.a.p.1.1		1	40.13	odd	4
1728.2.a.p.1.1		1	120.53	even	4
2700.2.a.b.1.1		1	5.2	odd	4
2700.2.a.b.1.1		1	15.2	even	4
2700.2.d.g.649.1		2	5.4	even	2	inner
2700.2.d.g.649.1		2	15.14	odd	2	inner
2700.2.d.g.649.2		2	1.1	even	1	trivial
2700.2.d.g.649.2		2	3.2	odd	2	CM
5292.2.a.j.1.1		1	35.13	even	4
5292.2.a.j.1.1		1	105.83	odd	4