Embedded newform 4900.2.e.i.2549.2

• Label: 4900.2.e.i.2549.2
• Level: $4900$
• Weight: $2$
• Character: 4900.2549
• Analytic conductor: $39.127$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(39.1266969904\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.2
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	4900.2549
Dual form			4900.2.e.i.2549.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} +1.00000 q^{19} +3.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} -7.00000 q^{31} -3.00000i q^{33} +1.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +9.00000i q^{47} +3.00000 q^{51} +3.00000i q^{53} +1.00000i q^{57} -9.00000 q^{59} -1.00000 q^{61} +7.00000i q^{67} -3.00000 q^{69} -1.00000i q^{73} +13.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{87} -15.0000 q^{89} -7.00000i q^{93} +10.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} - 6 q^{11} + 2 q^{19} + 12 q^{29} - 14 q^{31} - 4 q^{39} + 12 q^{41} + 6 q^{51} - 18 q^{59} - 2 q^{61} - 6 q^{69} + 26 q^{79} + 2 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(1177\)	\(2451\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	1.00000i	0.577350i	0.957427	+	0.288675i	\(0.0932147\pi\)
−0.957427	+	0.288675i				\(0.906785\pi\)
\(5\)	0	0
			\(7\)	0	0
\(11\)	−3.00000	−0.904534				−0.452267	−	0.891883i	\(-0.649385\pi\)
			−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	2.00000i	0.554700i	0.960769	+	0.277350i	\(0.0894562\pi\)
			−0.960769	+	0.277350i	\(0.910544\pi\)
\(17\)	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	\(-0.881478\pi\)
			0.931476	−	0.363803i	\(-0.118522\pi\)
\(19\)	1.00000	0.229416	0.114708	−	0.993399i	\(-0.463407\pi\)
			0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	3.00000i	0.625543i	0.949828	+	0.312772i	\(0.101257\pi\)
			−0.949828	+	0.312772i	\(0.898743\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−7.00000	−1.25724	−0.628619	−	0.777714i	\(-0.716379\pi\)
			−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	1.00000i	0.164399i	0.996616	+	0.0821995i	\(0.0261945\pi\)
			−0.996616	+	0.0821995i	\(0.973806\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	\(-0.901344\pi\)
			0.952353	−	0.304997i	\(-0.0986555\pi\)
\(47\)	9.00000i	1.31278i	0.754420	+	0.656392i	\(0.227918\pi\)
			−0.754420	+	0.656392i	\(0.772082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.i.2549.2		2
5.2	odd	4		196.2.a.b.1.1		1
5.3	odd	4		4900.2.a.g.1.1		1
5.4	even	2	inner	4900.2.e.i.2549.1		2
7.2	even	3		700.2.r.b.249.1		4
7.4	even	3		700.2.r.b.149.2		4
7.6	odd	2		4900.2.e.h.2549.1		2
15.2	even	4		1764.2.a.a.1.1		1
20.7	even	4		784.2.a.d.1.1		1
35.2	odd	12		28.2.e.a.25.1	yes	2
35.4	even	6		700.2.r.b.149.1		4
35.9	even	6		700.2.r.b.249.2		4
35.12	even	12		196.2.e.a.165.1		2
35.13	even	4		4900.2.a.n.1.1		1
35.17	even	12		196.2.e.a.177.1		2
35.18	odd	12		700.2.i.c.401.1		2
35.23	odd	12		700.2.i.c.501.1		2
35.27	even	4		196.2.a.a.1.1		1
35.32	odd	12		28.2.e.a.9.1	✓	2
35.34	odd	2		4900.2.e.h.2549.2		2
40.27	even	4		3136.2.a.s.1.1		1
40.37	odd	4		3136.2.a.h.1.1		1
60.47	odd	4		7056.2.a.f.1.1		1
105.2	even	12		252.2.k.c.109.1		2
105.17	odd	12		1764.2.k.b.1549.1		2
105.32	even	12		252.2.k.c.37.1		2
105.47	odd	12		1764.2.k.b.361.1		2
105.62	odd	4		1764.2.a.j.1.1		1
140.27	odd	4		784.2.a.g.1.1		1
140.47	odd	12		784.2.i.d.753.1		2
140.67	even	12		112.2.i.b.65.1		2
140.87	odd	12		784.2.i.d.177.1		2
140.107	even	12		112.2.i.b.81.1		2
280.27	odd	4		3136.2.a.k.1.1		1
280.37	odd	12		448.2.i.e.193.1		2
280.67	even	12		448.2.i.c.65.1		2
280.107	even	12		448.2.i.c.193.1		2
280.237	even	4		3136.2.a.v.1.1		1
280.277	odd	12		448.2.i.e.65.1		2
315.2	even	12		2268.2.l.a.109.1		2
315.32	even	12		2268.2.l.a.541.1		2
315.67	odd	12		2268.2.l.h.541.1		2
315.137	even	12		2268.2.i.h.2053.1		2
315.142	odd	12		2268.2.l.h.109.1		2
315.212	even	12		2268.2.i.h.865.1		2
315.247	odd	12		2268.2.i.a.865.1		2
315.277	odd	12		2268.2.i.a.2053.1		2
420.107	odd	12		1008.2.s.p.865.1		2
420.167	even	4		7056.2.a.bw.1.1		1
420.347	odd	12		1008.2.s.p.289.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	35.32	odd	12
28.2.e.a.25.1	yes	2	35.2	odd	12
112.2.i.b.65.1		2	140.67	even	12
112.2.i.b.81.1		2	140.107	even	12
196.2.a.a.1.1		1	35.27	even	4
196.2.a.b.1.1		1	5.2	odd	4
196.2.e.a.165.1		2	35.12	even	12
196.2.e.a.177.1		2	35.17	even	12
252.2.k.c.37.1		2	105.32	even	12
252.2.k.c.109.1		2	105.2	even	12
448.2.i.c.65.1		2	280.67	even	12
448.2.i.c.193.1		2	280.107	even	12
448.2.i.e.65.1		2	280.277	odd	12
448.2.i.e.193.1		2	280.37	odd	12
700.2.i.c.401.1		2	35.18	odd	12
700.2.i.c.501.1		2	35.23	odd	12
700.2.r.b.149.1		4	35.4	even	6
700.2.r.b.149.2		4	7.4	even	3
700.2.r.b.249.1		4	7.2	even	3
700.2.r.b.249.2		4	35.9	even	6
784.2.a.d.1.1		1	20.7	even	4
784.2.a.g.1.1		1	140.27	odd	4
784.2.i.d.177.1		2	140.87	odd	12
784.2.i.d.753.1		2	140.47	odd	12
1008.2.s.p.289.1		2	420.347	odd	12
1008.2.s.p.865.1		2	420.107	odd	12
1764.2.a.a.1.1		1	15.2	even	4
1764.2.a.j.1.1		1	105.62	odd	4
1764.2.k.b.361.1		2	105.47	odd	12
1764.2.k.b.1549.1		2	105.17	odd	12
2268.2.i.a.865.1		2	315.247	odd	12
2268.2.i.a.2053.1		2	315.277	odd	12
2268.2.i.h.865.1		2	315.212	even	12
2268.2.i.h.2053.1		2	315.137	even	12
2268.2.l.a.109.1		2	315.2	even	12
2268.2.l.a.541.1		2	315.32	even	12
2268.2.l.h.109.1		2	315.142	odd	12
2268.2.l.h.541.1		2	315.67	odd	12
3136.2.a.h.1.1		1	40.37	odd	4
3136.2.a.k.1.1		1	280.27	odd	4
3136.2.a.s.1.1		1	40.27	even	4
3136.2.a.v.1.1		1	280.237	even	4
4900.2.a.g.1.1		1	5.3	odd	4
4900.2.a.n.1.1		1	35.13	even	4
4900.2.e.h.2549.1		2	7.6	odd	2
4900.2.e.h.2549.2		2	35.34	odd	2
4900.2.e.i.2549.1		2	5.4	even	2	inner
4900.2.e.i.2549.2		2	1.1	even	1	trivial
7056.2.a.f.1.1		1	60.47	odd	4
7056.2.a.bw.1.1		1	420.167	even	4