Embedded newform 4050.2.a.c.1.1

• Label: 4050.2.a.c.1.1
• Level: $4050$
• Weight: $2$
• Character: 4050.1
• Self dual: yes
• Analytic conductor: $32.339$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.3394128186\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4050.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	0	0
\(5\)	0	0
			\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
−0.377964	+	0.925820i				\(0.623376\pi\)
\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
			−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	−1.00000	−0.229416	−0.114708	−	0.993399i	\(-0.536593\pi\)
			−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	4.00000	0.657596	0.328798	−	0.944400i	\(-0.393356\pi\)
			0.328798	+	0.944400i	\(0.393356\pi\)
\(41\)	9.00000	1.40556	0.702782	−	0.711405i	\(-0.251941\pi\)
			0.702782	+	0.711405i	\(0.251941\pi\)
\(43\)	1.00000	0.152499	0.0762493	−	0.997089i	\(-0.475706\pi\)
			0.0762493	+	0.997089i	\(0.475706\pi\)
\(47\)	6.00000	0.875190	0.437595	−	0.899172i	\(-0.355830\pi\)
			0.437595	+	0.899172i	\(0.355830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.c.1.1		1
3.2	odd	2		4050.2.a.v.1.1		1
5.2	odd	4		4050.2.c.c.649.1		2
5.3	odd	4		4050.2.c.c.649.2		2
5.4	even	2		162.2.a.c.1.1		1
9.2	odd	6		1350.2.e.c.901.1		2
9.4	even	3		450.2.e.i.151.1		2
9.5	odd	6		1350.2.e.c.451.1		2
9.7	even	3		450.2.e.i.301.1		2
15.2	even	4		4050.2.c.r.649.2		2
15.8	even	4		4050.2.c.r.649.1		2
15.14	odd	2		162.2.a.b.1.1		1
20.19	odd	2		1296.2.a.g.1.1		1
35.34	odd	2		7938.2.a.x.1.1		1
40.19	odd	2		5184.2.a.o.1.1		1
40.29	even	2		5184.2.a.r.1.1		1
45.2	even	12		1350.2.j.a.199.2		4
45.4	even	6		18.2.c.a.7.1	✓	2
45.7	odd	12		450.2.j.e.49.1		4
45.13	odd	12		450.2.j.e.349.1		4
45.14	odd	6		54.2.c.a.19.1		2
45.22	odd	12		450.2.j.e.349.2		4
45.23	even	12		1350.2.j.a.1099.2		4
45.29	odd	6		54.2.c.a.37.1		2
45.32	even	12		1350.2.j.a.1099.1		4
45.34	even	6		18.2.c.a.13.1	yes	2
45.38	even	12		1350.2.j.a.199.1		4
45.43	odd	12		450.2.j.e.49.2		4
60.59	even	2		1296.2.a.f.1.1		1
105.104	even	2		7938.2.a.i.1.1		1
120.29	odd	2		5184.2.a.q.1.1		1
120.59	even	2		5184.2.a.p.1.1		1
180.59	even	6		432.2.i.b.289.1		2
180.79	odd	6		144.2.i.c.49.1		2
180.119	even	6		432.2.i.b.145.1		2
180.139	odd	6		144.2.i.c.97.1		2
315.4	even	6		882.2.h.c.79.1		2
315.34	odd	6		882.2.f.d.589.1		2
315.59	even	6		2646.2.h.i.667.1		2
315.74	odd	6		2646.2.e.b.1549.1		2
315.79	even	6		882.2.h.c.67.1		2
315.94	odd	6		882.2.h.b.79.1		2
315.104	even	6		2646.2.f.g.883.1		2
315.124	odd	6		882.2.h.b.67.1		2
315.139	odd	6		882.2.f.d.295.1		2
315.149	odd	6		2646.2.e.b.2125.1		2
315.164	even	6		2646.2.e.c.1549.1		2
315.184	even	6		882.2.e.i.655.1		2
315.194	even	6		2646.2.e.c.2125.1		2
315.209	even	6		2646.2.f.g.1765.1		2
315.214	even	6		882.2.e.i.373.1		2
315.229	odd	6		882.2.e.g.655.1		2
315.254	odd	6		2646.2.h.h.361.1		2
315.284	odd	6		2646.2.h.h.667.1		2
315.299	even	6		2646.2.h.i.361.1		2
315.304	odd	6		882.2.e.g.373.1		2
360.29	odd	6		1728.2.i.e.577.1		2
360.59	even	6		1728.2.i.f.1153.1		2
360.139	odd	6		576.2.i.a.385.1		2
360.149	odd	6		1728.2.i.e.1153.1		2
360.229	even	6		576.2.i.g.385.1		2
360.259	odd	6		576.2.i.a.193.1		2
360.299	even	6		1728.2.i.f.577.1		2
360.349	even	6		576.2.i.g.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.2.c.a.7.1	✓	2	45.4	even	6
18.2.c.a.13.1	yes	2	45.34	even	6
54.2.c.a.19.1		2	45.14	odd	6
54.2.c.a.37.1		2	45.29	odd	6
144.2.i.c.49.1		2	180.79	odd	6
144.2.i.c.97.1		2	180.139	odd	6
162.2.a.b.1.1		1	15.14	odd	2
162.2.a.c.1.1		1	5.4	even	2
432.2.i.b.145.1		2	180.119	even	6
432.2.i.b.289.1		2	180.59	even	6
450.2.e.i.151.1		2	9.4	even	3
450.2.e.i.301.1		2	9.7	even	3
450.2.j.e.49.1		4	45.7	odd	12
450.2.j.e.49.2		4	45.43	odd	12
450.2.j.e.349.1		4	45.13	odd	12
450.2.j.e.349.2		4	45.22	odd	12
576.2.i.a.193.1		2	360.259	odd	6
576.2.i.a.385.1		2	360.139	odd	6
576.2.i.g.193.1		2	360.349	even	6
576.2.i.g.385.1		2	360.229	even	6
882.2.e.g.373.1		2	315.304	odd	6
882.2.e.g.655.1		2	315.229	odd	6
882.2.e.i.373.1		2	315.214	even	6
882.2.e.i.655.1		2	315.184	even	6
882.2.f.d.295.1		2	315.139	odd	6
882.2.f.d.589.1		2	315.34	odd	6
882.2.h.b.67.1		2	315.124	odd	6
882.2.h.b.79.1		2	315.94	odd	6
882.2.h.c.67.1		2	315.79	even	6
882.2.h.c.79.1		2	315.4	even	6
1296.2.a.f.1.1		1	60.59	even	2
1296.2.a.g.1.1		1	20.19	odd	2
1350.2.e.c.451.1		2	9.5	odd	6
1350.2.e.c.901.1		2	9.2	odd	6
1350.2.j.a.199.1		4	45.38	even	12
1350.2.j.a.199.2		4	45.2	even	12
1350.2.j.a.1099.1		4	45.32	even	12
1350.2.j.a.1099.2		4	45.23	even	12
1728.2.i.e.577.1		2	360.29	odd	6
1728.2.i.e.1153.1		2	360.149	odd	6
1728.2.i.f.577.1		2	360.299	even	6
1728.2.i.f.1153.1		2	360.59	even	6
2646.2.e.b.1549.1		2	315.74	odd	6
2646.2.e.b.2125.1		2	315.149	odd	6
2646.2.e.c.1549.1		2	315.164	even	6
2646.2.e.c.2125.1		2	315.194	even	6
2646.2.f.g.883.1		2	315.104	even	6
2646.2.f.g.1765.1		2	315.209	even	6
2646.2.h.h.361.1		2	315.254	odd	6
2646.2.h.h.667.1		2	315.284	odd	6
2646.2.h.i.361.1		2	315.299	even	6
2646.2.h.i.667.1		2	315.59	even	6
4050.2.a.c.1.1		1	1.1	even	1	trivial
4050.2.a.v.1.1		1	3.2	odd	2
4050.2.c.c.649.1		2	5.2	odd	4
4050.2.c.c.649.2		2	5.3	odd	4
4050.2.c.r.649.1		2	15.8	even	4
4050.2.c.r.649.2		2	15.2	even	4
5184.2.a.o.1.1		1	40.19	odd	2
5184.2.a.p.1.1		1	120.59	even	2
5184.2.a.q.1.1		1	120.29	odd	2
5184.2.a.r.1.1		1	40.29	even	2
7938.2.a.i.1.1		1	105.104	even	2
7938.2.a.x.1.1		1	35.34	odd	2