Embedded newform 2100.2.a.j.1.1

• Label: 2100.2.a.j.1.1
• Level: $2100$
• Weight: $2$
• Character: 2100.1
• Self dual: yes
• Analytic conductor: $16.769$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(16.7685844245\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 420)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2100.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +6.00000 q^{13} -2.00000 q^{17} +6.00000 q^{19} -1.00000 q^{21} -2.00000 q^{23} +1.00000 q^{27} +6.00000 q^{29} -2.00000 q^{31} -4.00000 q^{33} +4.00000 q^{37} +6.00000 q^{39} +8.00000 q^{41} +4.00000 q^{43} -4.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} +6.00000 q^{57} +4.00000 q^{59} +14.0000 q^{61} -1.00000 q^{63} -4.00000 q^{67} -2.00000 q^{69} +10.0000 q^{73} +4.00000 q^{77} +1.00000 q^{81} +16.0000 q^{83} +6.00000 q^{87} +8.00000 q^{89} -6.00000 q^{91} -2.00000 q^{93} -10.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
			\(3\)	1.00000	0.577350
\(5\)	0	0
			\(7\)	−1.00000	−0.377964
\(11\)	−4.00000	−1.20605				−0.603023	−	0.797724i	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
			0.832050	+	0.554700i	\(0.187167\pi\)
\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
			−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	6.00000	1.37649	0.688247	−	0.725476i	\(-0.258380\pi\)
			0.688247	+	0.725476i	\(0.258380\pi\)
\(23\)	−2.00000	−0.417029	−0.208514	−	0.978019i	\(-0.566863\pi\)
			−0.208514	+	0.978019i	\(0.566863\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−2.00000	−0.359211	−0.179605	−	0.983739i	\(-0.557482\pi\)
			−0.179605	+	0.983739i	\(0.557482\pi\)
\(37\)	4.00000	0.657596	0.328798	−	0.944400i	\(-0.393356\pi\)
			0.328798	+	0.944400i	\(0.393356\pi\)
\(41\)	8.00000	1.24939	0.624695	−	0.780869i	\(-0.285223\pi\)
			0.624695	+	0.780869i	\(0.285223\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−4.00000	−0.583460	−0.291730	−	0.956501i	\(-0.594231\pi\)
			−0.291730	+	0.956501i	\(0.594231\pi\)
\(53\)	−6.00000	−0.824163	−0.412082	−	0.911147i	\(-0.635198\pi\)
			−0.412082	+	0.911147i	\(0.635198\pi\)
\(59\)	4.00000	0.520756	0.260378	−	0.965507i	\(-0.416153\pi\)
			0.260378	+	0.965507i	\(0.416153\pi\)
\(61\)	14.0000	1.79252	0.896258	−	0.443533i	\(-0.146275\pi\)
			0.896258	+	0.443533i	\(0.146275\pi\)
\(67\)	−4.00000	−0.488678	−0.244339	−	0.969690i	\(-0.578571\pi\)
			−0.244339	+	0.969690i	\(0.578571\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	10.0000	1.17041	0.585206	−	0.810885i	\(-0.301014\pi\)
			0.585206	+	0.810885i	\(0.301014\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	16.0000	1.75623	0.878114	−	0.478451i	\(-0.158802\pi\)
			0.878114	+	0.478451i	\(0.158802\pi\)
\(89\)	8.00000	0.847998	0.423999	−	0.905663i	\(-0.360626\pi\)
			0.423999	+	0.905663i	\(0.360626\pi\)
\(97\)	−10.0000	−1.01535	−0.507673	−	0.861550i	\(-0.669494\pi\)
			−0.507673	+	0.861550i	\(0.669494\pi\)

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.a.j.1.1		1
3.2	odd	2		6300.2.a.n.1.1		1
4.3	odd	2		8400.2.a.bh.1.1		1
5.2	odd	4		420.2.k.a.169.1	✓	2
5.3	odd	4		420.2.k.a.169.2	yes	2
5.4	even	2		2100.2.a.e.1.1		1
15.2	even	4		1260.2.k.d.1009.1		2
15.8	even	4		1260.2.k.d.1009.2		2
15.14	odd	2		6300.2.a.bc.1.1		1
20.3	even	4		1680.2.t.a.1009.1		2
20.7	even	4		1680.2.t.a.1009.2		2
20.19	odd	2		8400.2.a.cd.1.1		1
35.2	odd	12		2940.2.bb.h.949.2		4
35.3	even	12		2940.2.bb.c.1549.1		4
35.12	even	12		2940.2.bb.c.949.1		4
35.13	even	4		2940.2.k.d.589.1		2
35.17	even	12		2940.2.bb.c.1549.2		4
35.18	odd	12		2940.2.bb.h.1549.2		4
35.23	odd	12		2940.2.bb.h.949.1		4
35.27	even	4		2940.2.k.d.589.2		2
35.32	odd	12		2940.2.bb.h.1549.1		4
35.33	even	12		2940.2.bb.c.949.2		4
60.23	odd	4		5040.2.t.o.1009.2		2
60.47	odd	4		5040.2.t.o.1009.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.k.a.169.1	✓	2	5.2	odd	4
420.2.k.a.169.2	yes	2	5.3	odd	4
1260.2.k.d.1009.1		2	15.2	even	4
1260.2.k.d.1009.2		2	15.8	even	4
1680.2.t.a.1009.1		2	20.3	even	4
1680.2.t.a.1009.2		2	20.7	even	4
2100.2.a.e.1.1		1	5.4	even	2
2100.2.a.j.1.1		1	1.1	even	1	trivial
2940.2.k.d.589.1		2	35.13	even	4
2940.2.k.d.589.2		2	35.27	even	4
2940.2.bb.c.949.1		4	35.12	even	12
2940.2.bb.c.949.2		4	35.33	even	12
2940.2.bb.c.1549.1		4	35.3	even	12
2940.2.bb.c.1549.2		4	35.17	even	12
2940.2.bb.h.949.1		4	35.23	odd	12
2940.2.bb.h.949.2		4	35.2	odd	12
2940.2.bb.h.1549.1		4	35.32	odd	12
2940.2.bb.h.1549.2		4	35.18	odd	12
5040.2.t.o.1009.1		2	60.47	odd	4
5040.2.t.o.1009.2		2	60.23	odd	4
6300.2.a.n.1.1		1	3.2	odd	2
6300.2.a.bc.1.1		1	15.14	odd	2
8400.2.a.bh.1.1		1	4.3	odd	2
8400.2.a.cd.1.1		1	20.19	odd	2