Embedded newform 72.2.n.b.61.5

• Label: 72.2.n.b.61.5
• Level: $72$
• Weight: $2$
• Character: 72.61
• 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.n (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 - x^15 + x^14 + 2*x^12 - 4*x^11 - 8*x^9 + 4*x^8 - 16*x^7 - 32*x^5 + 32*x^4 + 64*x^2 - 128*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			61.5
Root			\(0.820200 - 1.15207i\) of defining polynomial
Character	\(\chi\)	\(=\)	72.61
Dual form			72.2.n.b.13.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.587625 - 1.28635i) q^{2} +(1.69028 + 0.378078i) q^{3} +(-1.30939 - 1.51178i) q^{4} +(-1.97542 + 1.14051i) q^{5} +(1.47959 - 1.95213i) q^{6} +(-0.907824 + 1.57240i) q^{7} +(-2.71411 + 0.795980i) q^{8} +(2.71411 + 1.27812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} - q^{4} - 7 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.587625	−	1.28635i	0.415513	−	0.909587i
							\(3\)	1.69028	+	0.378078i	0.975885	+	0.218283i
\(5\)	−1.97542	+	1.14051i	−0.883437	+	0.510052i								−0.871790	−	0.489880i	\(-0.837041\pi\)
							−0.0116467	+	0.999932i	\(0.503707\pi\)
\(7\)	−0.907824	+	1.57240i	−0.343125	+	0.594311i	−0.985011	−	0.172490i	\(-0.944819\pi\)
							0.641886	+	0.766800i	\(0.278152\pi\)
\(11\)	−4.24153	−	2.44885i	−1.27887	−	0.738355i	−0.302228	−	0.953236i	\(-0.597730\pi\)
							−0.976640	+	0.214880i	\(0.931064\pi\)
\(13\)	4.00895	−	2.31457i	1.11188	−	0.641946i	0.172567	−	0.984998i	\(-0.444794\pi\)
							0.939317	+	0.343052i	\(0.111461\pi\)
\(17\)	1.92788			0.467579			0.233790	−	0.972287i	\(-0.424887\pi\)
							0.233790	+	0.972287i	\(0.424887\pi\)
\(19\)			2.12907i			0.488442i	0.969720	+	0.244221i	\(0.0785322\pi\)
							−0.969720	+	0.244221i	\(0.921468\pi\)
\(23\)	−1.15765	−	2.00511i	−0.241387	−	0.418094i	0.719723	−	0.694261i	\(-0.244269\pi\)
							−0.961109	+	0.276168i	\(0.910936\pi\)
\(29\)	3.16440	+	1.82697i	0.587615	+	0.339260i	0.764154	−	0.645034i	\(-0.223157\pi\)
							−0.176539	+	0.984294i	\(0.556490\pi\)
\(31\)	−2.65800	−	4.60379i	−0.477391	−	0.826865i	0.522273	−	0.852778i	\(-0.325084\pi\)
							−0.999664	+	0.0259130i	\(0.991751\pi\)
\(37\)		−	7.98438i		−	1.31262i	−0.754489	−	0.656312i	\(-0.772115\pi\)
							0.754489	−	0.656312i	\(-0.227885\pi\)
\(41\)	−2.36240	−	4.09180i	−0.368946	−	0.639033i	0.620455	−	0.784242i	\(-0.286948\pi\)
							−0.989401	+	0.145209i	\(0.953614\pi\)
\(43\)	−2.20800	−	1.27479i	−0.336717	−	0.194404i	0.322102	−	0.946705i	\(-0.395610\pi\)
							−0.658819	+	0.752301i	\(0.728944\pi\)
\(47\)	−2.02005	+	3.49884i	−0.294655	+	0.510358i	−0.974905	−	0.222623i	\(-0.928538\pi\)
							0.680249	+	0.732981i	\(0.261871\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.2.n.b.61.5	yes	16
3.2	odd	2		216.2.n.b.181.4		16
4.3	odd	2		288.2.r.b.241.1		16
8.3	odd	2		288.2.r.b.241.8		16
8.5	even	2	inner	72.2.n.b.61.6	yes	16
9.2	odd	6		648.2.d.k.325.7		8
9.4	even	3	inner	72.2.n.b.13.6	yes	16
9.5	odd	6		216.2.n.b.37.3		16
9.7	even	3		648.2.d.j.325.2		8
12.11	even	2		864.2.r.b.721.7		16
24.5	odd	2		216.2.n.b.181.3		16
24.11	even	2		864.2.r.b.721.2		16
36.7	odd	6		2592.2.d.j.1297.2		8
36.11	even	6		2592.2.d.k.1297.7		8
36.23	even	6		864.2.r.b.145.2		16
36.31	odd	6		288.2.r.b.49.8		16
72.5	odd	6		216.2.n.b.37.4		16
72.11	even	6		2592.2.d.k.1297.2		8
72.13	even	6	inner	72.2.n.b.13.5	✓	16
72.29	odd	6		648.2.d.k.325.8		8
72.43	odd	6		2592.2.d.j.1297.7		8
72.59	even	6		864.2.r.b.145.7		16
72.61	even	6		648.2.d.j.325.1		8
72.67	odd	6		288.2.r.b.49.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.n.b.13.5	✓	16	72.13	even	6	inner
72.2.n.b.13.6	yes	16	9.4	even	3	inner
72.2.n.b.61.5	yes	16	1.1	even	1	trivial
72.2.n.b.61.6	yes	16	8.5	even	2	inner
216.2.n.b.37.3		16	9.5	odd	6
216.2.n.b.37.4		16	72.5	odd	6
216.2.n.b.181.3		16	24.5	odd	2
216.2.n.b.181.4		16	3.2	odd	2
288.2.r.b.49.1		16	72.67	odd	6
288.2.r.b.49.8		16	36.31	odd	6
288.2.r.b.241.1		16	4.3	odd	2
288.2.r.b.241.8		16	8.3	odd	2
648.2.d.j.325.1		8	72.61	even	6
648.2.d.j.325.2		8	9.7	even	3
648.2.d.k.325.7		8	9.2	odd	6
648.2.d.k.325.8		8	72.29	odd	6
864.2.r.b.145.2		16	36.23	even	6
864.2.r.b.145.7		16	72.59	even	6
864.2.r.b.721.2		16	24.11	even	2
864.2.r.b.721.7		16	12.11	even	2
2592.2.d.j.1297.2		8	36.7	odd	6
2592.2.d.j.1297.7		8	72.43	odd	6
2592.2.d.k.1297.2		8	72.11	even	6
2592.2.d.k.1297.7		8	36.11	even	6