Embedded newform 4900.2.e.t.2549.1

• Label: 4900.2.e.t.2549.1
• 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			2549.1
Root			\(-0.713538i\) of defining polynomial
Character	\(\chi\)	\(=\)	4900.2549
Dual form			4900.2.e.t.2549.6

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.20440i q^{3} -7.26819 q^{9} +4.20440 q^{11} +0.204402i q^{13} +5.06379i q^{17} -1.06379 q^{19} +2.14061i q^{23} +13.6770i q^{27} +7.47259 q^{29} -8.47259 q^{31} -13.4726i q^{33} +10.6132i q^{37} +0.654985 q^{39} +10.5494 q^{41} +8.26819i q^{43} -3.26819i q^{47} +16.2264 q^{51} +5.67699i q^{53} +3.40880i q^{57} -1.20440 q^{59} +1.65498 q^{61} +12.4088i q^{67} +6.85939 q^{69} -0.591197 q^{71} -4.00000i q^{73} -6.54942 q^{79} +22.0220 q^{81} +3.88139i q^{83} -23.9452i q^{87} +9.26819 q^{89} +27.1496i q^{93} -1.33198i q^{97} -30.5584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99}+O(q^{100}) \)

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
			\(3\)	− 3.20440i	− 1.85006i	−0.379892	−	0.925031i	\(-0.624039\pi\)
0.379892	−	0.925031i				\(-0.375961\pi\)
\(5\)	0	0
			\(7\)	0	0
\(11\)	4.20440	1.26767				0.633837	−	0.773466i	\(-0.281479\pi\)
			0.633837	+	0.773466i	\(0.281479\pi\)
\(13\)	0.204402i	0.0566908i	0.999598	+	0.0283454i	\(0.00902383\pi\)
			−0.999598	+	0.0283454i	\(0.990976\pi\)
\(17\)	5.06379i	1.22815i	0.789248	+	0.614074i	\(0.210471\pi\)
			−0.789248	+	0.614074i	\(0.789529\pi\)
\(19\)	−1.06379	−0.244050	−0.122025	−	0.992527i	\(-0.538939\pi\)
			−0.122025	+	0.992527i	\(0.538939\pi\)
\(23\)	2.14061i	0.446349i	0.974779	+	0.223174i	\(0.0716419\pi\)
			−0.974779	+	0.223174i	\(0.928358\pi\)
\(29\)	7.47259	1.38763	0.693813	−	0.720156i	\(-0.255930\pi\)
			0.693813	+	0.720156i	\(0.255930\pi\)
\(31\)	−8.47259	−1.52172	−0.760861	−	0.648915i	\(-0.775223\pi\)
			−0.760861	+	0.648915i	\(0.775223\pi\)
\(37\)	10.6132i	1.74480i	0.488793	+	0.872400i	\(0.337437\pi\)
			−0.488793	+	0.872400i	\(0.662563\pi\)
\(41\)	10.5494	1.64754	0.823771	−	0.566923i	\(-0.191866\pi\)
			0.823771	+	0.566923i	\(0.191866\pi\)
\(43\)	8.26819i	1.26089i	0.776235	+	0.630444i	\(0.217127\pi\)
			−0.776235	+	0.630444i	\(0.782873\pi\)
\(47\)	− 3.26819i	− 0.476714i	−0.971178	−	0.238357i	\(-0.923391\pi\)
			0.971178	−	0.238357i	\(-0.0766089\pi\)
\(53\)	5.67699i	0.779795i	0.920858	+	0.389897i	\(0.127490\pi\)
			−0.920858	+	0.389897i	\(0.872510\pi\)
\(59\)	−1.20440	−0.156800	−0.0783999	−	0.996922i	\(-0.524981\pi\)
			−0.0783999	+	0.996922i	\(0.524981\pi\)
\(61\)	1.65498	0.211899	0.105950	−	0.994372i	\(-0.466212\pi\)
			0.105950	+	0.994372i	\(0.466212\pi\)
\(67\)	12.4088i	1.51598i	0.652268	+	0.757988i	\(0.273818\pi\)
			−0.652268	+	0.757988i	\(0.726182\pi\)
\(71\)	−0.591197	−0.0701622	−0.0350811	−	0.999384i	\(-0.511169\pi\)
			−0.0350811	+	0.999384i	\(0.511169\pi\)
\(73\)	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	\(-0.924791\pi\)
			0.972217	−	0.234082i	\(-0.0752085\pi\)
\(79\)	−6.54942	−0.736867	−0.368433	−	0.929654i	\(-0.620106\pi\)
			−0.368433	+	0.929654i	\(0.620106\pi\)
\(83\)	3.88139i	0.426038i	0.977048	+	0.213019i	\(0.0683297\pi\)
			−0.977048	+	0.213019i	\(0.931670\pi\)
\(89\)	9.26819	0.982426	0.491213	−	0.871039i	\(-0.336554\pi\)
			0.491213	+	0.871039i	\(0.336554\pi\)
\(97\)	− 1.33198i	− 0.135242i	−0.997711	−	0.0676209i	\(-0.978459\pi\)
			0.997711	−	0.0676209i	\(-0.0215408\pi\)

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.t.2549.1		6
5.2	odd	4		4900.2.a.ba.1.1		3
5.3	odd	4		4900.2.a.bc.1.3		3
5.4	even	2	inner	4900.2.e.t.2549.6		6
7.2	even	3		700.2.r.d.249.6		12
7.4	even	3		700.2.r.d.149.1		12
7.6	odd	2		4900.2.e.s.2549.6		6
35.2	odd	12		700.2.i.e.501.3	yes	6
35.4	even	6		700.2.r.d.149.6		12
35.9	even	6		700.2.r.d.249.1		12
35.13	even	4		4900.2.a.bb.1.1		3
35.18	odd	12		700.2.i.d.401.1	✓	6
35.23	odd	12		700.2.i.d.501.1	yes	6
35.27	even	4		4900.2.a.bd.1.3		3
35.32	odd	12		700.2.i.e.401.3	yes	6
35.34	odd	2		4900.2.e.s.2549.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.1	✓	6	35.18	odd	12
700.2.i.d.501.1	yes	6	35.23	odd	12
700.2.i.e.401.3	yes	6	35.32	odd	12
700.2.i.e.501.3	yes	6	35.2	odd	12
700.2.r.d.149.1		12	7.4	even	3
700.2.r.d.149.6		12	35.4	even	6
700.2.r.d.249.1		12	35.9	even	6
700.2.r.d.249.6		12	7.2	even	3
4900.2.a.ba.1.1		3	5.2	odd	4
4900.2.a.bb.1.1		3	35.13	even	4
4900.2.a.bc.1.3		3	5.3	odd	4
4900.2.a.bd.1.3		3	35.27	even	4
4900.2.e.s.2549.1		6	35.34	odd	2
4900.2.e.s.2549.6		6	7.6	odd	2
4900.2.e.t.2549.1		6	1.1	even	1	trivial
4900.2.e.t.2549.6		6	5.4	even	2	inner