Embedded newform 1350.2.e.c.451.1

• Label: 1350.2.e.c.451.1
• Level: $1350$
• Weight: $2$
• Character: 1350.451
• Analytic conductor: $10.780$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.7798042729\)
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			451.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1350.451
Dual form			1350.2.e.c.901.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{7} +1.00000 q^{8} +(-1.50000 + 2.59808i) q^{11} +(1.00000 + 1.73205i) q^{13} +(1.00000 + 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} -3.00000 q^{17} -1.00000 q^{19} +(-1.50000 - 2.59808i) q^{22} +(3.00000 + 5.19615i) q^{23} -2.00000 q^{26} -2.00000 q^{28} +(3.00000 - 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(1.50000 - 2.59808i) q^{34} +4.00000 q^{37} +(0.500000 - 0.866025i) q^{38} +(4.50000 + 7.79423i) q^{41} +(-0.500000 + 0.866025i) q^{43} +3.00000 q^{44} -6.00000 q^{46} +(3.00000 - 5.19615i) q^{47} +(1.50000 + 2.59808i) q^{49} +(1.00000 - 1.73205i) q^{52} +12.0000 q^{53} +(1.00000 - 1.73205i) q^{56} +(3.00000 + 5.19615i) q^{58} +(1.50000 + 2.59808i) q^{59} +(-4.00000 + 6.92820i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(2.50000 + 4.33013i) q^{67} +(1.50000 + 2.59808i) q^{68} +12.0000 q^{71} -11.0000 q^{73} +(-2.00000 + 3.46410i) q^{74} +(0.500000 + 0.866025i) q^{76} +(3.00000 + 5.19615i) q^{77} +(2.00000 - 3.46410i) q^{79} -9.00000 q^{82} +(-6.00000 + 10.3923i) q^{83} +(-0.500000 - 0.866025i) q^{86} +(-1.50000 + 2.59808i) q^{88} -6.00000 q^{89} +4.00000 q^{91} +(3.00000 - 5.19615i) q^{92} +(3.00000 + 5.19615i) q^{94} +(2.50000 - 4.33013i) q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 - q^4 + 2 * q^7 + 2 * q^8 - 3 * q^11 + 2 * q^13 + 2 * q^14 - q^16 - 6 * q^17 - 2 * q^19 - 3 * q^22 + 6 * q^23 - 4 * q^26 - 4 * q^28 + 6 * q^29 + 4 * q^31 - q^32 + 3 * q^34 + 8 * q^37 + q^38 + 9 * q^41 - q^43 + 6 * q^44 - 12 * q^46 + 6 * q^47 + 3 * q^49 + 2 * q^52 + 24 * q^53 + 2 * q^56 + 6 * q^58 + 3 * q^59 - 8 * q^61 - 8 * q^62 + 2 * q^64 + 5 * q^67 + 3 * q^68 + 24 * q^71 - 22 * q^73 - 4 * q^74 + q^76 + 6 * q^77 + 4 * q^79 - 18 * q^82 - 12 * q^83 - q^86 - 3 * q^88 - 12 * q^89 + 8 * q^91 + 6 * q^92 + 6 * q^94 + 5 * q^97 - 6 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\)	\(1001\)	\(1027\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.500000	+	0.866025i	−0.353553	+	0.612372i
							\(3\)		0			0
\(5\)		0			0
							\(7\)	1.00000	−	1.73205i	0.377964	−	0.654654i	−0.612801	−	0.790237i	\(-0.709957\pi\)
0.990766	+	0.135583i	\(0.0432908\pi\)
\(11\)	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	\(-0.982718\pi\)
							0.546259	+	0.837616i	\(0.316051\pi\)
\(13\)	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	\(-0.0772105\pi\)
							−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	−3.00000			−0.727607			−0.363803	−	0.931476i	\(-0.618522\pi\)
							−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	−1.00000			−0.229416			−0.114708	−	0.993399i	\(-0.536593\pi\)
							−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	\(0.0484560\pi\)
							−0.362892	+	0.931831i	\(0.618211\pi\)
\(29\)	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	\(-0.645253\pi\)
							0.997738	−	0.0672232i	\(-0.0214140\pi\)
\(31\)	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	\(-0.0497126\pi\)
							−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	4.00000			0.657596			0.328798	−	0.944400i	\(-0.393356\pi\)
							0.328798	+	0.944400i	\(0.393356\pi\)
\(41\)	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	\(0.0813924\pi\)
							−0.264704	+	0.964330i	\(0.585274\pi\)
\(43\)	−0.500000	+	0.866025i	−0.0762493	+	0.132068i	−0.901629	−	0.432511i	\(-0.857628\pi\)
							0.825380	+	0.564578i	\(0.190961\pi\)
\(47\)	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	\(-0.689164\pi\)
							0.997503	+	0.0706177i	\(0.0224970\pi\)
\(53\)	12.0000			1.64833			0.824163	−	0.566352i	\(-0.191646\pi\)
							0.824163	+	0.566352i	\(0.191646\pi\)
\(59\)	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	\(-0.104104\pi\)
							−0.751710	+	0.659494i	\(0.770771\pi\)
\(61\)	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	\(0.337817\pi\)
							−0.999901	+	0.0140840i	\(0.995517\pi\)
\(67\)	2.50000	+	4.33013i	0.305424	+	0.529009i	0.977356	−	0.211604i	\(-0.0678686\pi\)
							−0.671932	+	0.740613i	\(0.734535\pi\)
\(71\)	12.0000			1.42414			0.712069	−	0.702109i	\(-0.247758\pi\)
							0.712069	+	0.702109i	\(0.247758\pi\)
\(73\)	−11.0000			−1.28745			−0.643726	−	0.765256i	\(-0.722612\pi\)
							−0.643726	+	0.765256i	\(0.722612\pi\)
\(79\)	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	0.322396	+	0.946605i	\(0.395512\pi\)
							−0.980982	+	0.194099i	\(0.937822\pi\)
\(89\)	−6.00000			−0.635999			−0.317999	−	0.948091i	\(-0.603011\pi\)
							−0.317999	+	0.948091i	\(0.603011\pi\)
\(97\)	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	\(-0.751641\pi\)
							0.964579	+	0.263795i	\(0.0849741\pi\)

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.2.c.a.7.1	✓	2	15.14	odd	2
18.2.c.a.13.1	yes	2	45.14	odd	6
54.2.c.a.19.1		2	5.4	even	2
54.2.c.a.37.1		2	45.4	even	6
144.2.i.c.49.1		2	180.59	even	6
144.2.i.c.97.1		2	60.59	even	2
162.2.a.b.1.1		1	45.34	even	6
162.2.a.c.1.1		1	45.29	odd	6
432.2.i.b.145.1		2	180.139	odd	6
432.2.i.b.289.1		2	20.19	odd	2
450.2.e.i.151.1		2	3.2	odd	2
450.2.e.i.301.1		2	9.5	odd	6
450.2.j.e.49.1		4	45.32	even	12
450.2.j.e.49.2		4	45.23	even	12
450.2.j.e.349.1		4	15.8	even	4
450.2.j.e.349.2		4	15.2	even	4
576.2.i.a.193.1		2	360.59	even	6
576.2.i.a.385.1		2	120.59	even	2
576.2.i.g.193.1		2	360.149	odd	6
576.2.i.g.385.1		2	120.29	odd	2
882.2.e.g.373.1		2	315.59	even	6
882.2.e.g.655.1		2	105.89	even	6
882.2.e.i.373.1		2	315.284	odd	6
882.2.e.i.655.1		2	105.44	odd	6
882.2.f.d.295.1		2	105.104	even	2
882.2.f.d.589.1		2	315.104	even	6
882.2.h.b.67.1		2	315.194	even	6
882.2.h.b.79.1		2	105.59	even	6
882.2.h.c.67.1		2	315.149	odd	6
882.2.h.c.79.1		2	105.74	odd	6
1296.2.a.f.1.1		1	180.79	odd	6
1296.2.a.g.1.1		1	180.119	even	6
1350.2.e.c.451.1		2	1.1	even	1	trivial
1350.2.e.c.901.1		2	9.4	even	3	inner
1350.2.j.a.199.1		4	45.13	odd	12
1350.2.j.a.199.2		4	45.22	odd	12
1350.2.j.a.1099.1		4	5.2	odd	4
1350.2.j.a.1099.2		4	5.3	odd	4
1728.2.i.e.577.1		2	360.229	even	6
1728.2.i.e.1153.1		2	40.29	even	2
1728.2.i.f.577.1		2	360.139	odd	6
1728.2.i.f.1153.1		2	40.19	odd	2
2646.2.e.b.1549.1		2	315.4	even	6
2646.2.e.b.2125.1		2	35.9	even	6
2646.2.e.c.1549.1		2	315.94	odd	6
2646.2.e.c.2125.1		2	35.19	odd	6
2646.2.f.g.883.1		2	35.34	odd	2
2646.2.f.g.1765.1		2	315.139	odd	6
2646.2.h.h.361.1		2	315.184	even	6
2646.2.h.h.667.1		2	35.4	even	6
2646.2.h.i.361.1		2	315.229	odd	6
2646.2.h.i.667.1		2	35.24	odd	6
4050.2.a.c.1.1		1	9.2	odd	6
4050.2.a.v.1.1		1	9.7	even	3
4050.2.c.c.649.1		2	45.2	even	12
4050.2.c.c.649.2		2	45.38	even	12
4050.2.c.r.649.1		2	45.43	odd	12
4050.2.c.r.649.2		2	45.7	odd	12
5184.2.a.o.1.1		1	360.299	even	6
5184.2.a.p.1.1		1	360.259	odd	6
5184.2.a.q.1.1		1	360.349	even	6
5184.2.a.r.1.1		1	360.29	odd	6
7938.2.a.i.1.1		1	315.34	odd	6
7938.2.a.x.1.1		1	315.209	even	6