Embedded newform 72.2.i.a.49.1

• Label: 72.2.i.a.49.1
• Level: $72$
• Weight: $2$
• Character: 72.49
• Analytic conductor: $0.575$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	72.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.574922894553\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	72.49
Dual form			72.2.i.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +(0.500000 - 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} -3.00000 q^{9} +(-2.50000 - 4.33013i) q^{11} +(2.50000 - 4.33013i) q^{13} +(1.50000 + 0.866025i) q^{15} -2.00000 q^{17} -4.00000 q^{19} +(-4.50000 + 2.59808i) q^{21} +(0.500000 - 0.866025i) q^{23} +(2.00000 + 3.46410i) q^{25} -5.19615i q^{27} +(4.50000 + 7.79423i) q^{29} +(0.500000 - 0.866025i) q^{31} +(7.50000 - 4.33013i) q^{33} +3.00000 q^{35} -6.00000 q^{37} +(7.50000 + 4.33013i) q^{39} +(-1.50000 + 2.59808i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(-1.50000 + 2.59808i) q^{45} +(1.50000 + 2.59808i) q^{47} +(-1.00000 + 1.73205i) q^{49} -3.46410i q^{51} +2.00000 q^{53} -5.00000 q^{55} -6.92820i q^{57} +(-5.50000 + 9.52628i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(-4.50000 - 7.79423i) q^{63} +(-2.50000 - 4.33013i) q^{65} +(0.500000 - 0.866025i) q^{67} +(1.50000 + 0.866025i) q^{69} +4.00000 q^{71} -2.00000 q^{73} +(-6.00000 + 3.46410i) q^{75} +(7.50000 - 12.9904i) q^{77} +(-0.500000 - 0.866025i) q^{79} +9.00000 q^{81} +(-0.500000 - 0.866025i) q^{83} +(-1.00000 + 1.73205i) q^{85} +(-13.5000 + 7.79423i) q^{87} -18.0000 q^{89} +15.0000 q^{91} +(1.50000 + 0.866025i) q^{93} +(-2.00000 + 3.46410i) q^{95} +(6.50000 + 11.2583i) q^{97} +(7.50000 + 12.9904i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^5 + 3 * q^7 - 6 * q^9 - 5 * q^11 + 5 * q^13 + 3 * q^15 - 4 * q^17 - 8 * q^19 - 9 * q^21 + q^23 + 4 * q^25 + 9 * q^29 + q^31 + 15 * q^33 + 6 * q^35 - 12 * q^37 + 15 * q^39 - 3 * q^41 - q^43 - 3 * q^45 + 3 * q^47 - 2 * q^49 + 4 * q^53 - 10 * q^55 - 11 * q^59 - 7 * q^61 - 9 * q^63 - 5 * q^65 + q^67 + 3 * q^69 + 8 * q^71 - 4 * q^73 - 12 * q^75 + 15 * q^77 - q^79 + 18 * q^81 - q^83 - 2 * q^85 - 27 * q^87 - 36 * q^89 + 30 * q^91 + 3 * q^93 - 4 * q^95 + 13 * q^97 + 15 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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			0
							\(3\)			1.73205i			1.00000i
\(5\)	0.500000	−	0.866025i	0.223607	−	0.387298i								−0.732294	−	0.680989i	\(-0.761550\pi\)
							0.955901	+	0.293691i	\(0.0948835\pi\)
\(7\)	1.50000	+	2.59808i	0.566947	+	0.981981i	0.996866	+	0.0791130i	\(0.0252088\pi\)
							−0.429919	+	0.902867i	\(0.641458\pi\)
\(11\)	−2.50000	−	4.33013i	−0.753778	−	1.30558i	−0.945979	−	0.324227i	\(-0.894896\pi\)
							0.192201	−	0.981356i	\(-0.438437\pi\)
\(13\)	2.50000	−	4.33013i	0.693375	−	1.20096i	−0.277350	−	0.960769i	\(-0.589456\pi\)
							0.970725	−	0.240192i	\(-0.0772105\pi\)
\(17\)	−2.00000			−0.485071			−0.242536	−	0.970143i	\(-0.577979\pi\)
							−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	−4.00000			−0.917663			−0.458831	−	0.888523i	\(-0.651732\pi\)
							−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	0.500000	−	0.866025i	0.104257	−	0.180579i	−0.809177	−	0.587565i	\(-0.800087\pi\)
							0.913434	+	0.406986i	\(0.133420\pi\)
\(29\)	4.50000	+	7.79423i	0.835629	+	1.44735i	0.893517	+	0.449029i	\(0.148230\pi\)
							−0.0578882	+	0.998323i	\(0.518437\pi\)
\(31\)	0.500000	−	0.866025i	0.0898027	−	0.155543i	−0.817625	−	0.575751i	\(-0.804710\pi\)
							0.907428	+	0.420208i	\(0.138043\pi\)
\(37\)	−6.00000			−0.986394			−0.493197	−	0.869918i	\(-0.664172\pi\)
							−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−1.50000	+	2.59808i	−0.234261	+	0.405751i	−0.959058	−	0.283211i	\(-0.908600\pi\)
							0.724797	+	0.688963i	\(0.241934\pi\)
\(43\)	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	\(-0.190961\pi\)
							−0.901629	+	0.432511i	\(0.857628\pi\)
\(47\)	1.50000	+	2.59808i	0.218797	+	0.378968i	0.954441	−	0.298401i	\(-0.0964533\pi\)
							−0.735643	+	0.677369i	\(0.763120\pi\)
\(53\)	2.00000			0.274721			0.137361	−	0.990521i	\(-0.456138\pi\)
							0.137361	+	0.990521i	\(0.456138\pi\)
\(59\)	−5.50000	+	9.52628i	−0.716039	+	1.24022i	0.246518	+	0.969138i	\(0.420713\pi\)
							−0.962557	+	0.271078i	\(0.912620\pi\)
\(61\)	−3.50000	−	6.06218i	−0.448129	−	0.776182i	0.550135	−	0.835076i	\(-0.314576\pi\)
							−0.998264	+	0.0588933i	\(0.981243\pi\)
\(67\)	0.500000	−	0.866025i	0.0610847	−	0.105802i	−0.833866	−	0.551967i	\(-0.813877\pi\)
							0.894951	+	0.446165i	\(0.147211\pi\)
\(71\)	4.00000			0.474713			0.237356	−	0.971423i	\(-0.423719\pi\)
							0.237356	+	0.971423i	\(0.423719\pi\)
\(73\)	−2.00000			−0.234082			−0.117041	−	0.993127i	\(-0.537341\pi\)
							−0.117041	+	0.993127i	\(0.537341\pi\)
\(79\)	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	\(-0.184582\pi\)
							−0.892781	+	0.450490i	\(0.851249\pi\)
\(83\)	−0.500000	−	0.866025i	−0.0548821	−	0.0950586i	0.837279	−	0.546776i	\(-0.184145\pi\)
							−0.892161	+	0.451717i	\(0.850812\pi\)
\(89\)	−18.0000			−1.90800			−0.953998	−	0.299813i	\(-0.903076\pi\)
							−0.953998	+	0.299813i	\(0.903076\pi\)
\(97\)	6.50000	+	11.2583i	0.659975	+	1.14311i	0.980622	+	0.195911i	\(0.0627665\pi\)
							−0.320647	+	0.947199i	\(0.603900\pi\)

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	72.2.i.a.49.1	yes	2
3.2	odd	2		216.2.i.a.145.1		2
4.3	odd	2		144.2.i.b.49.1		2
8.3	odd	2		576.2.i.c.193.1		2
8.5	even	2		576.2.i.d.193.1		2
9.2	odd	6		216.2.i.a.73.1		2
9.4	even	3		648.2.a.a.1.1		1
9.5	odd	6		648.2.a.c.1.1		1
9.7	even	3	inner	72.2.i.a.25.1	✓	2
12.11	even	2		432.2.i.a.145.1		2
24.5	odd	2		1728.2.i.h.577.1		2
24.11	even	2		1728.2.i.g.577.1		2
36.7	odd	6		144.2.i.b.97.1		2
36.11	even	6		432.2.i.a.289.1		2
36.23	even	6		1296.2.a.i.1.1		1
36.31	odd	6		1296.2.a.e.1.1		1
72.5	odd	6		5184.2.a.i.1.1		1
72.11	even	6		1728.2.i.g.1153.1		2
72.13	even	6		5184.2.a.s.1.1		1
72.29	odd	6		1728.2.i.h.1153.1		2
72.43	odd	6		576.2.i.c.385.1		2
72.59	even	6		5184.2.a.n.1.1		1
72.61	even	6		576.2.i.d.385.1		2
72.67	odd	6		5184.2.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.i.a.25.1	✓	2	9.7	even	3	inner
72.2.i.a.49.1	yes	2	1.1	even	1	trivial
144.2.i.b.49.1		2	4.3	odd	2
144.2.i.b.97.1		2	36.7	odd	6
216.2.i.a.73.1		2	9.2	odd	6
216.2.i.a.145.1		2	3.2	odd	2
432.2.i.a.145.1		2	12.11	even	2
432.2.i.a.289.1		2	36.11	even	6
576.2.i.c.193.1		2	8.3	odd	2
576.2.i.c.385.1		2	72.43	odd	6
576.2.i.d.193.1		2	8.5	even	2
576.2.i.d.385.1		2	72.61	even	6
648.2.a.a.1.1		1	9.4	even	3
648.2.a.c.1.1		1	9.5	odd	6
1296.2.a.e.1.1		1	36.31	odd	6
1296.2.a.i.1.1		1	36.23	even	6
1728.2.i.g.577.1		2	24.11	even	2
1728.2.i.g.1153.1		2	72.11	even	6
1728.2.i.h.577.1		2	24.5	odd	2
1728.2.i.h.1153.1		2	72.29	odd	6
5184.2.a.i.1.1		1	72.5	odd	6
5184.2.a.n.1.1		1	72.59	even	6
5184.2.a.s.1.1		1	72.13	even	6
5184.2.a.x.1.1		1	72.67	odd	6