Embedded newform 7350.2.a.cw.1.1

• Label: 7350.2.a.cw.1.1
• Level: $7350$
• Weight: $2$
• Character: 7350.1
• Self dual: yes
• Analytic conductor: $58.690$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7350.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$58.6900454856$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7350.1

$q$-expansion

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} +1.00000 q^{18} +4.00000 q^{19} +3.00000 q^{22} +1.00000 q^{24} -4.00000 q^{26} +1.00000 q^{27} +9.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +1.00000 q^{36} -8.00000 q^{37} +4.00000 q^{38} -4.00000 q^{39} +10.0000 q^{43} +3.00000 q^{44} -6.00000 q^{47} +1.00000 q^{48} -4.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} +4.00000 q^{57} +9.00000 q^{58} -3.00000 q^{59} +10.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.00000 q^{66} +10.0000 q^{67} -6.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -8.00000 q^{74} +4.00000 q^{76} -4.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -9.00000 q^{83} +10.0000 q^{86} +9.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +1.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} -1.00000 q^{97} +3.00000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$
$2$	1.00000	0.707107
			$3$	1.00000	0.577350
$5$	0	0
			$7$	0	0
$11$	3.00000	0.904534				0.452267	−	0.891883i	$-0.350615\pi$
			0.452267	+	0.891883i	$0.350615\pi$
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
			−0.554700	+	0.832050i	$0.687167\pi$
$17$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
			0.458831	+	0.888523i	$0.348268\pi$
$23$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
			0.835629	+	0.549294i	$0.185103\pi$
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
			0.0898027	+	0.995960i	$0.471376\pi$
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
			−0.657596	+	0.753371i	$0.728427\pi$
$41$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
			0.762493	+	0.646997i	$0.223975\pi$
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
			−0.437595	+	0.899172i	$0.644170\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.cw.1.1		1
5.4	even	2		294.2.a.a.1.1		1
7.3	odd	6		1050.2.i.e.751.1		2
7.5	odd	6		1050.2.i.e.151.1		2
7.6	odd	2		7350.2.a.ce.1.1		1
15.14	odd	2		882.2.a.k.1.1		1
20.19	odd	2		2352.2.a.n.1.1		1
35.3	even	12		1050.2.o.b.499.2		4
35.4	even	6		294.2.e.f.79.1		2
35.9	even	6		294.2.e.f.67.1		2
35.12	even	12		1050.2.o.b.949.2		4
35.17	even	12		1050.2.o.b.499.1		4
35.19	odd	6		42.2.e.b.25.1	✓	2
35.24	odd	6		42.2.e.b.37.1	yes	2
35.33	even	12		1050.2.o.b.949.1		4
35.34	odd	2		294.2.a.d.1.1		1
40.19	odd	2		9408.2.a.bm.1.1		1
40.29	even	2		9408.2.a.db.1.1		1
60.59	even	2		7056.2.a.bz.1.1		1
105.44	odd	6		882.2.g.b.361.1		2
105.59	even	6		126.2.g.b.37.1		2
105.74	odd	6		882.2.g.b.667.1		2
105.89	even	6		126.2.g.b.109.1		2
105.104	even	2		882.2.a.g.1.1		1
140.19	even	6		336.2.q.d.193.1		2
140.39	odd	6		2352.2.q.m.961.1		2
140.59	even	6		336.2.q.d.289.1		2
140.79	odd	6		2352.2.q.m.1537.1		2
140.139	even	2		2352.2.a.m.1.1		1
280.19	even	6		1344.2.q.j.193.1		2
280.59	even	6		1344.2.q.j.961.1		2
280.69	odd	2		9408.2.a.d.1.1		1
280.139	even	2		9408.2.a.bu.1.1		1
280.229	odd	6		1344.2.q.v.193.1		2
280.269	odd	6		1344.2.q.v.961.1		2
315.59	even	6		1134.2.h.a.541.1		2
315.94	odd	6		1134.2.h.p.541.1		2
315.124	odd	6		1134.2.h.p.109.1		2
315.164	even	6		1134.2.e.p.919.1		2
315.194	even	6		1134.2.e.p.865.1		2
315.229	odd	6		1134.2.e.a.865.1		2
315.299	even	6		1134.2.h.a.109.1		2
315.304	odd	6		1134.2.e.a.919.1		2
420.59	odd	6		1008.2.s.n.289.1		2
420.299	odd	6		1008.2.s.n.865.1		2
420.419	odd	2		7056.2.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.e.b.25.1	✓	2	35.19	odd	6
42.2.e.b.37.1	yes	2	35.24	odd	6
126.2.g.b.37.1		2	105.59	even	6
126.2.g.b.109.1		2	105.89	even	6
294.2.a.a.1.1		1	5.4	even	2
294.2.a.d.1.1		1	35.34	odd	2
294.2.e.f.67.1		2	35.9	even	6
294.2.e.f.79.1		2	35.4	even	6
336.2.q.d.193.1		2	140.19	even	6
336.2.q.d.289.1		2	140.59	even	6
882.2.a.g.1.1		1	105.104	even	2
882.2.a.k.1.1		1	15.14	odd	2
882.2.g.b.361.1		2	105.44	odd	6
882.2.g.b.667.1		2	105.74	odd	6
1008.2.s.n.289.1		2	420.59	odd	6
1008.2.s.n.865.1		2	420.299	odd	6
1050.2.i.e.151.1		2	7.5	odd	6
1050.2.i.e.751.1		2	7.3	odd	6
1050.2.o.b.499.1		4	35.17	even	12
1050.2.o.b.499.2		4	35.3	even	12
1050.2.o.b.949.1		4	35.33	even	12
1050.2.o.b.949.2		4	35.12	even	12
1134.2.e.a.865.1		2	315.229	odd	6
1134.2.e.a.919.1		2	315.304	odd	6
1134.2.e.p.865.1		2	315.194	even	6
1134.2.e.p.919.1		2	315.164	even	6
1134.2.h.a.109.1		2	315.299	even	6
1134.2.h.a.541.1		2	315.59	even	6
1134.2.h.p.109.1		2	315.124	odd	6
1134.2.h.p.541.1		2	315.94	odd	6
1344.2.q.j.193.1		2	280.19	even	6
1344.2.q.j.961.1		2	280.59	even	6
1344.2.q.v.193.1		2	280.229	odd	6
1344.2.q.v.961.1		2	280.269	odd	6
2352.2.a.m.1.1		1	140.139	even	2
2352.2.a.n.1.1		1	20.19	odd	2
2352.2.q.m.961.1		2	140.39	odd	6
2352.2.q.m.1537.1		2	140.79	odd	6
7056.2.a.g.1.1		1	420.419	odd	2
7056.2.a.bz.1.1		1	60.59	even	2
7350.2.a.ce.1.1		1	7.6	odd	2
7350.2.a.cw.1.1		1	1.1	even	1	trivial
9408.2.a.d.1.1		1	280.69	odd	2
9408.2.a.bm.1.1		1	40.19	odd	2
9408.2.a.bu.1.1		1	280.139	even	2
9408.2.a.db.1.1		1	40.29	even	2