Embedded newform 4900.2.e.s.2549.3

• Label: 4900.2.e.s.2549.3
• Level: $4900$
• Weight: $2$
• Character: 4900.2549
• Analytic conductor: $39.127$
• Analytic rank: $0$
• Dimension: $6$
• 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:	\(6\)
Coefficient field:	6.0.4227136.2
Defining polynomial:	\( x^{6} + 9x^{4} + 22x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 700)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.3
Root			\(-2.19869i\) of defining polynomial
Character	\(\chi\)	\(=\)	4900.2549
Dual form			4900.2.e.s.2549.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.364448i q^{3} +2.86718 q^{9} +1.36445 q^{11} -2.63555i q^{13} -2.23163i q^{17} -6.23163 q^{19} -6.59607i q^{23} -2.13828i q^{27} -5.50273 q^{29} -4.50273 q^{31} -0.497270i q^{33} -2.09334i q^{37} -0.960522 q^{39} -9.32497 q^{41} +1.86718i q^{43} +6.86718i q^{47} -0.813312 q^{51} +10.1383i q^{53} +2.27110i q^{57} -1.63555 q^{59} -0.0394782 q^{61} -6.72890i q^{67} -2.40393 q^{69} -6.27110 q^{71} -4.00000i q^{73} -5.32497 q^{79} +7.82224 q^{81} -14.7738i q^{83} +2.00546i q^{87} +0.867178 q^{89} +1.64101i q^{93} +16.0988i q^{97} +3.91211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) \)

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 \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
			\(3\)	− 0.364448i	− 0.210414i	−0.994450	−	0.105207i	\(-0.966449\pi\)
0.994450	−	0.105207i				\(-0.0335505\pi\)
\(5\)	0	0
			\(7\)	0	0
\(11\)	1.36445	0.411397				0.205698	−	0.978615i	\(-0.434053\pi\)
			0.205698	+	0.978615i	\(0.434053\pi\)
\(13\)	− 2.63555i	− 0.730971i	−0.930817	−	0.365485i	\(-0.880903\pi\)
			0.930817	−	0.365485i	\(-0.119097\pi\)
\(17\)	− 2.23163i	− 0.541249i	−0.962685	−	0.270624i	\(-0.912770\pi\)
			0.962685	−	0.270624i	\(-0.0872301\pi\)
\(19\)	−6.23163	−1.42963	−0.714816	−	0.699312i	\(-0.753490\pi\)
			−0.714816	+	0.699312i	\(0.753490\pi\)
\(23\)	− 6.59607i	− 1.37538i	−0.726006	−	0.687688i	\(-0.758626\pi\)
			0.726006	−	0.687688i	\(-0.241374\pi\)
\(29\)	−5.50273	−1.02183	−0.510916	−	0.859631i	\(-0.670694\pi\)
			−0.510916	+	0.859631i	\(0.670694\pi\)
\(31\)	−4.50273	−0.808714	−0.404357	−	0.914601i	\(-0.632505\pi\)
			−0.404357	+	0.914601i	\(0.632505\pi\)
\(37\)	− 2.09334i	− 0.344144i	−0.985084	−	0.172072i	\(-0.944954\pi\)
			0.985084	−	0.172072i	\(-0.0550461\pi\)
\(41\)	−9.32497	−1.45632	−0.728158	−	0.685410i	\(-0.759623\pi\)
			−0.728158	+	0.685410i	\(0.759623\pi\)
\(43\)	1.86718i	0.284742i	0.989813	+	0.142371i	\(0.0454726\pi\)
			−0.989813	+	0.142371i	\(0.954527\pi\)
\(47\)	6.86718i	1.00168i	0.865540	+	0.500840i	\(0.166976\pi\)
			−0.865540	+	0.500840i	\(0.833024\pi\)
\(53\)	10.1383i	1.39260i	0.717751	+	0.696300i	\(0.245172\pi\)
			−0.717751	+	0.696300i	\(0.754828\pi\)
\(59\)	−1.63555	−0.212931	−0.106465	−	0.994316i	\(-0.533953\pi\)
			−0.106465	+	0.994316i	\(0.533953\pi\)
\(61\)	−0.0394782	−0.00505467	−0.00252733	−	0.999997i	\(-0.500804\pi\)
			−0.00252733	+	0.999997i	\(0.500804\pi\)
\(67\)	− 6.72890i	− 0.822065i	−0.911621	−	0.411033i	\(-0.865168\pi\)
			0.911621	−	0.411033i	\(-0.134832\pi\)
\(71\)	−6.27110	−0.744243	−0.372122	−	0.928184i	\(-0.621370\pi\)
			−0.372122	+	0.928184i	\(0.621370\pi\)
\(73\)	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	\(-0.924791\pi\)
			0.972217	−	0.234082i	\(-0.0752085\pi\)
\(79\)	−5.32497	−0.599106	−0.299553	−	0.954080i	\(-0.596838\pi\)
			−0.299553	+	0.954080i	\(0.596838\pi\)
\(83\)	− 14.7738i	− 1.62164i	−0.585296	−	0.810819i	\(-0.699022\pi\)
			0.585296	−	0.810819i	\(-0.300978\pi\)
\(89\)	0.867178	0.0919206	0.0459603	−	0.998943i	\(-0.485365\pi\)
			0.0459603	+	0.998943i	\(0.485365\pi\)
\(97\)	16.0988i	1.63459i	0.576222	+	0.817293i	\(0.304526\pi\)
			−0.576222	+	0.817293i	\(0.695474\pi\)

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	4900.2.e.s.2549.3		6
5.2	odd	4		4900.2.a.bb.1.2		3
5.3	odd	4		4900.2.a.bd.1.2		3
5.4	even	2	inner	4900.2.e.s.2549.4		6
7.3	odd	6		700.2.r.d.149.4		12
7.5	odd	6		700.2.r.d.249.3		12
7.6	odd	2		4900.2.e.t.2549.4		6
35.3	even	12		700.2.i.e.401.2	yes	6
35.12	even	12		700.2.i.d.501.2	yes	6
35.13	even	4		4900.2.a.ba.1.2		3
35.17	even	12		700.2.i.d.401.2	✓	6
35.19	odd	6		700.2.r.d.249.4		12
35.24	odd	6		700.2.r.d.149.3		12
35.27	even	4		4900.2.a.bc.1.2		3
35.33	even	12		700.2.i.e.501.2	yes	6
35.34	odd	2		4900.2.e.t.2549.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.2	✓	6	35.17	even	12
700.2.i.d.501.2	yes	6	35.12	even	12
700.2.i.e.401.2	yes	6	35.3	even	12
700.2.i.e.501.2	yes	6	35.33	even	12
700.2.r.d.149.3		12	35.24	odd	6
700.2.r.d.149.4		12	7.3	odd	6
700.2.r.d.249.3		12	7.5	odd	6
700.2.r.d.249.4		12	35.19	odd	6
4900.2.a.ba.1.2		3	35.13	even	4
4900.2.a.bb.1.2		3	5.2	odd	4
4900.2.a.bc.1.2		3	35.27	even	4
4900.2.a.bd.1.2		3	5.3	odd	4
4900.2.e.s.2549.3		6	1.1	even	1	trivial
4900.2.e.s.2549.4		6	5.4	even	2	inner
4900.2.e.t.2549.3		6	35.34	odd	2
4900.2.e.t.2549.4		6	7.6	odd	2