Embedded newform 2646.2.h.i.361.1

• Label: 2646.2.h.i.361.1
• Level: $2646$
• Weight: $2$
• Character: 2646.361
• Analytic conductor: $21.128$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.1284163748\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	2646.361
Dual form			2646.2.h.i.667.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{8} +3.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(1.50000 - 2.59808i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(1.50000 - 2.59808i) q^{22} +6.00000 q^{23} -5.00000 q^{25} +(-1.00000 - 1.73205i) q^{26} +(3.00000 + 5.19615i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.50000 - 2.59808i) q^{34} +(2.00000 + 3.46410i) q^{37} -1.00000 q^{38} +(-4.50000 + 7.79423i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-1.50000 - 2.59808i) q^{44} +(3.00000 - 5.19615i) q^{46} +(3.00000 - 5.19615i) q^{47} +(-2.50000 + 4.33013i) q^{50} -2.00000 q^{52} +(6.00000 - 10.3923i) q^{53} +6.00000 q^{58} +(-1.50000 - 2.59808i) q^{59} +(4.00000 - 6.92820i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(-2.50000 - 4.33013i) q^{67} -3.00000 q^{68} +12.0000 q^{71} +(5.50000 - 9.52628i) q^{73} +4.00000 q^{74} +(-0.500000 + 0.866025i) q^{76} +(2.00000 - 3.46410i) q^{79} +(4.50000 + 7.79423i) q^{82} +(-6.00000 - 10.3923i) q^{83} +1.00000 q^{86} -3.00000 q^{88} +(-3.00000 - 5.19615i) q^{89} +(-3.00000 - 5.19615i) q^{92} +(-3.00000 - 5.19615i) q^{94} +(2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - q^4 - 2 * q^8 + 6 * q^11 + 2 * q^13 - q^16 + 3 * q^17 - q^19 + 3 * q^22 + 12 * q^23 - 10 * q^25 - 2 * q^26 + 6 * q^29 - 4 * q^31 + q^32 - 3 * q^34 + 4 * q^37 - 2 * q^38 - 9 * q^41 + q^43 - 3 * q^44 + 6 * q^46 + 6 * q^47 - 5 * q^50 - 4 * q^52 + 12 * q^53 + 12 * q^58 - 3 * q^59 + 8 * q^61 - 8 * q^62 + 2 * q^64 - 5 * q^67 - 6 * q^68 + 24 * q^71 + 11 * q^73 + 8 * q^74 - q^76 + 4 * q^79 + 9 * q^82 - 12 * q^83 + 2 * q^86 - 6 * q^88 - 6 * q^89 - 6 * q^92 - 6 * q^94 + 5 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	−	0.866025i	0.353553	−	0.612372i
							\(3\)		0			0
\(5\)		0			0										−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(7\)		0			0
							\(11\)	3.00000			0.904534			0.452267	−	0.891883i	\(-0.350615\pi\)
0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	\(-0.743877\pi\)
							0.970725	+	0.240192i	\(0.0772105\pi\)
\(17\)	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	\(-0.714811\pi\)
							0.988583	+	0.150675i	\(0.0481447\pi\)
\(19\)	−0.500000	−	0.866025i	−0.114708	−	0.198680i	0.802955	−	0.596040i	\(-0.203260\pi\)
							−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	6.00000			1.25109			0.625543	−	0.780189i	\(-0.284877\pi\)
							0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	3.00000	+	5.19615i	0.557086	+	0.964901i	0.997738	+	0.0672232i	\(0.0214140\pi\)
							−0.440652	+	0.897678i	\(0.645253\pi\)
\(31\)	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	\(-0.283621\pi\)
							−0.987829	+	0.155543i	\(0.950287\pi\)
\(37\)	2.00000	+	3.46410i	0.328798	+	0.569495i	0.982274	−	0.187453i	\(-0.0600231\pi\)
							−0.653476	+	0.756948i	\(0.726690\pi\)
\(41\)	−4.50000	+	7.79423i	−0.702782	+	1.21725i	0.264704	+	0.964330i	\(0.414726\pi\)
							−0.967486	+	0.252924i	\(0.918608\pi\)
\(43\)	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	\(-0.142372\pi\)
							−0.825380	+	0.564578i	\(0.809039\pi\)
\(47\)	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	\(-0.689164\pi\)
							0.997503	+	0.0706177i	\(0.0224970\pi\)
\(53\)	6.00000	−	10.3923i	0.824163	−	1.42749i	−0.0783936	−	0.996922i	\(-0.524979\pi\)
							0.902557	−	0.430570i	\(-0.141688\pi\)
\(59\)	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	\(-0.229229\pi\)
							−0.946993	+	0.321253i	\(0.895896\pi\)
\(61\)	4.00000	−	6.92820i	0.512148	−	0.887066i	−0.487753	−	0.872982i	\(-0.662183\pi\)
							0.999901	−	0.0140840i	\(-0.00448323\pi\)
\(67\)	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	\(-0.265465\pi\)
							−0.977356	+	0.211604i	\(0.932131\pi\)
\(71\)	12.0000			1.42414			0.712069	−	0.702109i	\(-0.247758\pi\)
							0.712069	+	0.702109i	\(0.247758\pi\)
\(73\)	5.50000	−	9.52628i	0.643726	−	1.11497i	−0.340868	−	0.940111i	\(-0.610721\pi\)
							0.984594	−	0.174855i	\(-0.0559458\pi\)
\(79\)	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	\(-0.761089\pi\)
							0.956325	+	0.292306i	\(0.0944227\pi\)
\(83\)	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	\(-0.937822\pi\)
							0.322396	−	0.946605i	\(-0.395512\pi\)
\(89\)	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	\(-0.269678\pi\)
							−0.980071	+	0.198650i	\(0.936344\pi\)
\(97\)	2.50000	+	4.33013i	0.253837	+	0.439658i	0.964579	−	0.263795i	\(-0.0849741\pi\)
							−0.710742	+	0.703452i	\(0.751641\pi\)

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	2646.2.h.i.361.1		2
3.2	odd	2		882.2.h.b.67.1		2
7.2	even	3		2646.2.e.c.1549.1		2
7.3	odd	6		54.2.c.a.37.1		2
7.4	even	3		2646.2.f.g.1765.1		2
7.5	odd	6		2646.2.e.b.1549.1		2
7.6	odd	2		2646.2.h.h.361.1		2
9.2	odd	6		882.2.e.g.655.1		2
9.7	even	3		2646.2.e.c.2125.1		2
21.2	odd	6		882.2.e.g.373.1		2
21.5	even	6		882.2.e.i.373.1		2
21.11	odd	6		882.2.f.d.589.1		2
21.17	even	6		18.2.c.a.13.1	yes	2
21.20	even	2		882.2.h.c.67.1		2
28.3	even	6		432.2.i.b.145.1		2
35.3	even	12		1350.2.j.a.199.2		4
35.17	even	12		1350.2.j.a.199.1		4
35.24	odd	6		1350.2.e.c.901.1		2
56.3	even	6		1728.2.i.f.577.1		2
56.45	odd	6		1728.2.i.e.577.1		2
63.2	odd	6		882.2.h.b.79.1		2
63.4	even	3		7938.2.a.i.1.1		1
63.11	odd	6		882.2.f.d.295.1		2
63.16	even	3	inner	2646.2.h.i.667.1		2
63.20	even	6		882.2.e.i.655.1		2
63.25	even	3		2646.2.f.g.883.1		2
63.31	odd	6		162.2.a.b.1.1		1
63.32	odd	6		7938.2.a.x.1.1		1
63.34	odd	6		2646.2.e.b.2125.1		2
63.38	even	6		18.2.c.a.7.1	✓	2
63.47	even	6		882.2.h.c.79.1		2
63.52	odd	6		54.2.c.a.19.1		2
63.59	even	6		162.2.a.c.1.1		1
63.61	odd	6		2646.2.h.h.667.1		2
84.59	odd	6		144.2.i.c.49.1		2
105.17	odd	12		450.2.j.e.49.2		4
105.38	odd	12		450.2.j.e.49.1		4
105.59	even	6		450.2.e.i.301.1		2
168.59	odd	6		576.2.i.a.193.1		2
168.101	even	6		576.2.i.g.193.1		2
252.31	even	6		1296.2.a.f.1.1		1
252.59	odd	6		1296.2.a.g.1.1		1
252.115	even	6		432.2.i.b.289.1		2
252.227	odd	6		144.2.i.c.97.1		2
315.38	odd	12		450.2.j.e.349.2		4
315.52	even	12		1350.2.j.a.1099.2		4
315.59	even	6		4050.2.a.c.1.1		1
315.94	odd	6		4050.2.a.v.1.1		1
315.122	odd	12		4050.2.c.c.649.2		2
315.157	even	12		4050.2.c.r.649.1		2
315.164	even	6		450.2.e.i.151.1		2
315.178	even	12		1350.2.j.a.1099.1		4
315.227	odd	12		450.2.j.e.349.1		4
315.248	odd	12		4050.2.c.c.649.1		2
315.283	even	12		4050.2.c.r.649.2		2
315.304	odd	6		1350.2.e.c.451.1		2
504.59	odd	6		5184.2.a.o.1.1		1
504.101	even	6		576.2.i.g.385.1		2
504.115	even	6		1728.2.i.f.1153.1		2
504.157	odd	6		5184.2.a.q.1.1		1
504.227	odd	6		576.2.i.a.385.1		2
504.283	even	6		5184.2.a.p.1.1		1
504.437	even	6		5184.2.a.r.1.1		1
504.493	odd	6		1728.2.i.e.1153.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.2.c.a.7.1	✓	2	63.38	even	6
18.2.c.a.13.1	yes	2	21.17	even	6
54.2.c.a.19.1		2	63.52	odd	6
54.2.c.a.37.1		2	7.3	odd	6
144.2.i.c.49.1		2	84.59	odd	6
144.2.i.c.97.1		2	252.227	odd	6
162.2.a.b.1.1		1	63.31	odd	6
162.2.a.c.1.1		1	63.59	even	6
432.2.i.b.145.1		2	28.3	even	6
432.2.i.b.289.1		2	252.115	even	6
450.2.e.i.151.1		2	315.164	even	6
450.2.e.i.301.1		2	105.59	even	6
450.2.j.e.49.1		4	105.38	odd	12
450.2.j.e.49.2		4	105.17	odd	12
450.2.j.e.349.1		4	315.227	odd	12
450.2.j.e.349.2		4	315.38	odd	12
576.2.i.a.193.1		2	168.59	odd	6
576.2.i.a.385.1		2	504.227	odd	6
576.2.i.g.193.1		2	168.101	even	6
576.2.i.g.385.1		2	504.101	even	6
882.2.e.g.373.1		2	21.2	odd	6
882.2.e.g.655.1		2	9.2	odd	6
882.2.e.i.373.1		2	21.5	even	6
882.2.e.i.655.1		2	63.20	even	6
882.2.f.d.295.1		2	63.11	odd	6
882.2.f.d.589.1		2	21.11	odd	6
882.2.h.b.67.1		2	3.2	odd	2
882.2.h.b.79.1		2	63.2	odd	6
882.2.h.c.67.1		2	21.20	even	2
882.2.h.c.79.1		2	63.47	even	6
1296.2.a.f.1.1		1	252.31	even	6
1296.2.a.g.1.1		1	252.59	odd	6
1350.2.e.c.451.1		2	315.304	odd	6
1350.2.e.c.901.1		2	35.24	odd	6
1350.2.j.a.199.1		4	35.17	even	12
1350.2.j.a.199.2		4	35.3	even	12
1350.2.j.a.1099.1		4	315.178	even	12
1350.2.j.a.1099.2		4	315.52	even	12
1728.2.i.e.577.1		2	56.45	odd	6
1728.2.i.e.1153.1		2	504.493	odd	6
1728.2.i.f.577.1		2	56.3	even	6
1728.2.i.f.1153.1		2	504.115	even	6
2646.2.e.b.1549.1		2	7.5	odd	6
2646.2.e.b.2125.1		2	63.34	odd	6
2646.2.e.c.1549.1		2	7.2	even	3
2646.2.e.c.2125.1		2	9.7	even	3
2646.2.f.g.883.1		2	63.25	even	3
2646.2.f.g.1765.1		2	7.4	even	3
2646.2.h.h.361.1		2	7.6	odd	2
2646.2.h.h.667.1		2	63.61	odd	6
2646.2.h.i.361.1		2	1.1	even	1	trivial
2646.2.h.i.667.1		2	63.16	even	3	inner
4050.2.a.c.1.1		1	315.59	even	6
4050.2.a.v.1.1		1	315.94	odd	6
4050.2.c.c.649.1		2	315.248	odd	12
4050.2.c.c.649.2		2	315.122	odd	12
4050.2.c.r.649.1		2	315.157	even	12
4050.2.c.r.649.2		2	315.283	even	12
5184.2.a.o.1.1		1	504.59	odd	6
5184.2.a.p.1.1		1	504.283	even	6
5184.2.a.q.1.1		1	504.157	odd	6
5184.2.a.r.1.1		1	504.437	even	6
7938.2.a.i.1.1		1	63.4	even	3
7938.2.a.x.1.1		1	63.32	odd	6