Embedded newform 72.2.l.b.59.6

• Label: 72.2.l.b.59.6
• Level: $72$
• Weight: $2$
• Character: 72.59
• Analytic conductor: $0.575$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	72.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.574922894553\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 16*x^12 - 12*x^11 - 8*x^10 + 36*x^9 - 68*x^8 + 72*x^7 - 32*x^6 - 96*x^5 + 256*x^4 - 384*x^3 + 448*x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.6
Root			\(-1.37702 + 0.322193i\) of defining polynomial
Character	\(\chi\)	\(=\)	72.59
Dual form			72.2.l.b.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.409484 - 1.35363i) q^{2} +(-1.12774 - 1.31461i) q^{3} +(-1.66465 - 1.10858i) q^{4} +(-0.565188 + 0.978934i) q^{5} +(-2.24129 + 0.988231i) q^{6} +(3.71499 - 2.14485i) q^{7} +(-2.18226 + 1.79937i) q^{8} +(-0.456412 + 2.96508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.409484	−	1.35363i	0.289549	−	0.957163i
							\(3\)	−1.12774	−	1.31461i	−0.651100	−	0.758992i
\(5\)	−0.565188	+	0.978934i	−0.252760	+	0.437793i								−0.964285	−	0.264868i	\(-0.914672\pi\)
							0.711525	+	0.702661i	\(0.248005\pi\)
\(7\)	3.71499	−	2.14485i	1.40413	−	0.810677i	0.409320	−	0.912391i	\(-0.365766\pi\)
							0.994814	+	0.101714i	\(0.0324328\pi\)
\(11\)	1.00953	−	0.582853i	0.304385	−	0.175737i	−0.340026	−	0.940416i	\(-0.610436\pi\)
							0.644411	+	0.764679i	\(0.277103\pi\)
\(13\)	2.64466	+	1.52689i	0.733496	+	0.423484i	0.819700	−	0.572793i	\(-0.194140\pi\)
							−0.0862038	+	0.996278i	\(0.527474\pi\)
\(17\)		−	1.49654i		−	0.362963i	−0.983394	−	0.181482i	\(-0.941911\pi\)
							0.983394	−	0.181482i	\(-0.0580893\pi\)
\(19\)	−3.42378			−0.785468			−0.392734	−	0.919652i	\(-0.628471\pi\)
							−0.392734	+	0.919652i	\(0.628471\pi\)
\(23\)	−3.85938	+	6.68464i	−0.804736	+	1.39384i	0.111733	+	0.993738i	\(0.464360\pi\)
							−0.916469	+	0.400106i	\(0.868973\pi\)
\(29\)	−0.709580	−	1.22903i	−0.131766	−	0.228225i	0.792592	−	0.609753i	\(-0.208731\pi\)
							−0.924357	+	0.381528i	\(0.875398\pi\)
\(31\)	−4.66408	−	2.69281i	−0.837694	−	0.483643i	0.0187859	−	0.999824i	\(-0.494020\pi\)
							−0.856480	+	0.516181i	\(0.827353\pi\)
\(37\)			2.97201i			0.488596i	0.969700	+	0.244298i	\(0.0785575\pi\)
							−0.969700	+	0.244298i	\(0.921443\pi\)
\(41\)	−4.23339	−	2.44415i	−0.661144	−	0.381712i	0.131569	−	0.991307i	\(-0.457999\pi\)
							−0.792713	+	0.609595i	\(0.791332\pi\)
\(43\)	−1.74292	−	3.01882i	−0.265793	−	0.460366i	0.701978	−	0.712198i	\(-0.252300\pi\)
							−0.967771	+	0.251832i	\(0.918967\pi\)
\(47\)	−1.77991	−	3.08289i	−0.259627	−	0.449686i	0.706515	−	0.707698i	\(-0.250266\pi\)
							−0.966142	+	0.258011i	\(0.916933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.2.l.b.59.6	yes	16
3.2	odd	2		216.2.l.b.179.3		16
4.3	odd	2		288.2.p.b.239.5		16
8.3	odd	2	inner	72.2.l.b.59.8	yes	16
8.5	even	2		288.2.p.b.239.6		16
9.2	odd	6	inner	72.2.l.b.11.8	yes	16
9.4	even	3		648.2.f.b.323.12		16
9.5	odd	6		648.2.f.b.323.5		16
9.7	even	3		216.2.l.b.35.1		16
12.11	even	2		864.2.p.b.719.5		16
24.5	odd	2		864.2.p.b.719.4		16
24.11	even	2		216.2.l.b.179.1		16
36.7	odd	6		864.2.p.b.143.4		16
36.11	even	6		288.2.p.b.47.6		16
36.23	even	6		2592.2.f.b.1295.7		16
36.31	odd	6		2592.2.f.b.1295.9		16
72.5	odd	6		2592.2.f.b.1295.10		16
72.11	even	6	inner	72.2.l.b.11.6	✓	16
72.13	even	6		2592.2.f.b.1295.8		16
72.29	odd	6		288.2.p.b.47.5		16
72.43	odd	6		216.2.l.b.35.3		16
72.59	even	6		648.2.f.b.323.11		16
72.61	even	6		864.2.p.b.143.5		16
72.67	odd	6		648.2.f.b.323.6		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.b.11.6	✓	16	72.11	even	6	inner
72.2.l.b.11.8	yes	16	9.2	odd	6	inner
72.2.l.b.59.6	yes	16	1.1	even	1	trivial
72.2.l.b.59.8	yes	16	8.3	odd	2	inner
216.2.l.b.35.1		16	9.7	even	3
216.2.l.b.35.3		16	72.43	odd	6
216.2.l.b.179.1		16	24.11	even	2
216.2.l.b.179.3		16	3.2	odd	2
288.2.p.b.47.5		16	72.29	odd	6
288.2.p.b.47.6		16	36.11	even	6
288.2.p.b.239.5		16	4.3	odd	2
288.2.p.b.239.6		16	8.5	even	2
648.2.f.b.323.5		16	9.5	odd	6
648.2.f.b.323.6		16	72.67	odd	6
648.2.f.b.323.11		16	72.59	even	6
648.2.f.b.323.12		16	9.4	even	3
864.2.p.b.143.4		16	36.7	odd	6
864.2.p.b.143.5		16	72.61	even	6
864.2.p.b.719.4		16	24.5	odd	2
864.2.p.b.719.5		16	12.11	even	2
2592.2.f.b.1295.7		16	36.23	even	6
2592.2.f.b.1295.8		16	72.13	even	6
2592.2.f.b.1295.9		16	36.31	odd	6
2592.2.f.b.1295.10		16	72.5	odd	6