Embedded newform 2100.2.k.h.1849.1

• Label: 2100.2.k.h.1849.1
• Level: $2100$
• Weight: $2$
• Character: 2100.1849
• Analytic conductor: $16.769$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.7685844245\)
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 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	2100.1849
Dual form			2100.2.k.h.1849.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -4.00000i q^{13} +2.00000i q^{17} +2.00000 q^{19} -1.00000 q^{21} -4.00000i q^{23} +1.00000i q^{27} +2.00000 q^{29} -6.00000 q^{31} -2.00000i q^{33} -6.00000i q^{37} -4.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -1.00000 q^{49} +2.00000 q^{51} -8.00000i q^{53} -2.00000i q^{57} -10.0000 q^{61} +1.00000i q^{63} -12.0000i q^{67} -4.00000 q^{69} -14.0000 q^{71} -4.00000i q^{73} -2.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -2.00000i q^{87} +14.0000 q^{89} -4.00000 q^{91} +6.00000i q^{93} +8.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 4 q^{11} + 4 q^{19} - 2 q^{21} + 4 q^{29} - 12 q^{31} - 8 q^{39} + 12 q^{41} - 2 q^{49} + 4 q^{51} - 20 q^{61} - 8 q^{69} - 28 q^{71} + 16 q^{79} + 2 q^{81} + 28 q^{89} - 8 q^{91} - 4 q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\)	\(701\)	\(1051\)	\(1177\)	\(1501\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(1\)

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
			\(3\)	− 1.00000i	− 0.577350i
\(5\)	0	0
			\(7\)	− 1.00000i	− 0.377964i
\(11\)	2.00000	0.603023				0.301511	−	0.953463i	\(-0.402509\pi\)
			0.301511	+	0.953463i	\(0.402509\pi\)
\(13\)	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	\(-0.812833\pi\)
			0.832050	−	0.554700i	\(-0.187167\pi\)
\(17\)	2.00000i	0.485071i	0.970143	+	0.242536i	\(0.0779791\pi\)
			−0.970143	+	0.242536i	\(0.922021\pi\)
\(19\)	2.00000	0.458831	0.229416	−	0.973329i	\(-0.426318\pi\)
			0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	\(-0.863071\pi\)
			0.908893	−	0.417029i	\(-0.136929\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	−6.00000	−1.07763	−0.538816	−	0.842424i	\(-0.681128\pi\)
			−0.538816	+	0.842424i	\(0.681128\pi\)
\(37\)	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	\(-0.835828\pi\)
			0.869918	−	0.493197i	\(-0.164172\pi\)
\(41\)	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	\(0.0986555\pi\)
			−0.952353	+	0.304997i	\(0.901344\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	− 8.00000i	− 1.09888i	−0.835532	−	0.549442i	\(-0.814840\pi\)
			0.835532	−	0.549442i	\(-0.185160\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−10.0000	−1.28037	−0.640184	−	0.768221i	\(-0.721142\pi\)
			−0.640184	+	0.768221i	\(0.721142\pi\)
\(67\)	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	\(-0.738112\pi\)
			0.680211	−	0.733017i	\(-0.261888\pi\)
\(71\)	−14.0000	−1.66149	−0.830747	−	0.556650i	\(-0.812086\pi\)
			−0.830747	+	0.556650i	\(0.812086\pi\)
\(73\)	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	\(-0.924791\pi\)
			0.972217	−	0.234082i	\(-0.0752085\pi\)
\(79\)	8.00000	0.900070	0.450035	−	0.893011i	\(-0.351411\pi\)
			0.450035	+	0.893011i	\(0.351411\pi\)
\(83\)	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	\(-0.771155\pi\)
			0.752506	−	0.658586i	\(-0.228845\pi\)
\(89\)	14.0000	1.48400	0.741999	−	0.670402i	\(-0.233878\pi\)
			0.741999	+	0.670402i	\(0.233878\pi\)
\(97\)	8.00000i	0.812277i	0.913812	+	0.406138i	\(0.133125\pi\)
			−0.913812	+	0.406138i	\(0.866875\pi\)

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	2100.2.k.h.1849.1		2
3.2	odd	2		6300.2.k.f.6049.1		2
5.2	odd	4		2100.2.a.h.1.1		1
5.3	odd	4		420.2.a.d.1.1	✓	1
5.4	even	2	inner	2100.2.k.h.1849.2		2
15.2	even	4		6300.2.a.u.1.1		1
15.8	even	4		1260.2.a.a.1.1		1
15.14	odd	2		6300.2.k.f.6049.2		2
20.3	even	4		1680.2.a.i.1.1		1
20.7	even	4		8400.2.a.bq.1.1		1
35.3	even	12		2940.2.q.l.961.1		2
35.13	even	4		2940.2.a.c.1.1		1
35.18	odd	12		2940.2.q.b.961.1		2
35.23	odd	12		2940.2.q.b.361.1		2
35.33	even	12		2940.2.q.l.361.1		2
40.3	even	4		6720.2.a.bs.1.1		1
40.13	odd	4		6720.2.a.e.1.1		1
60.23	odd	4		5040.2.a.r.1.1		1
105.83	odd	4		8820.2.a.s.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.a.d.1.1	✓	1	5.3	odd	4
1260.2.a.a.1.1		1	15.8	even	4
1680.2.a.i.1.1		1	20.3	even	4
2100.2.a.h.1.1		1	5.2	odd	4
2100.2.k.h.1849.1		2	1.1	even	1	trivial
2100.2.k.h.1849.2		2	5.4	even	2	inner
2940.2.a.c.1.1		1	35.13	even	4
2940.2.q.b.361.1		2	35.23	odd	12
2940.2.q.b.961.1		2	35.18	odd	12
2940.2.q.l.361.1		2	35.33	even	12
2940.2.q.l.961.1		2	35.3	even	12
5040.2.a.r.1.1		1	60.23	odd	4
6300.2.a.u.1.1		1	15.2	even	4
6300.2.k.f.6049.1		2	3.2	odd	2
6300.2.k.f.6049.2		2	15.14	odd	2
6720.2.a.e.1.1		1	40.13	odd	4
6720.2.a.bs.1.1		1	40.3	even	4
8400.2.a.bq.1.1		1	20.7	even	4
8820.2.a.s.1.1		1	105.83	odd	4