Embedded newform 66.6.e.a.31.2

• Label: 66.6.e.a.31.2
• Level: $66$
• Weight: $6$
• Character: 66.31
• Analytic conductor: $10.585$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 66 = 2 \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	66.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5853321077\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial:	\( x^{8} + 89x^{6} + 22551x^{4} + 4006069x^{2} + 405257161 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			31.2
Root			\(10.5047 - 7.63209i\) of defining polynomial
Character	\(\chi\)	\(=\)	66.31
Dual form			66.6.e.a.49.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23607 - 3.80423i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(-12.9443 - 9.40456i) q^{4} +(-3.34054 - 10.2811i) q^{5} +(11.1246 + 34.2380i) q^{6} +(185.412 + 134.710i) q^{7} +(-51.7771 + 37.6183i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 18 q^{3} - 32 q^{4} + 150 q^{5} - 72 q^{6} + 474 q^{7} - 128 q^{8} - 162 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.23607	−	3.80423i	0.218508	−	0.672499i
							\(3\)	−7.28115	+	5.29007i	−0.467086	+	0.339358i
\(5\)	−3.34054	−	10.2811i	−0.0597575	−	0.183915i								0.916722	−	0.399526i	\(-0.130825\pi\)
							−0.976479	+	0.215612i	\(0.930825\pi\)
\(7\)	185.412	+	134.710i	1.43019	+	1.03909i	0.989982	+	0.141196i	\(0.0450948\pi\)
							0.440207	+	0.897896i	\(0.354905\pi\)
\(11\)	−67.0175	−	395.676i	−0.166996	−	0.985958i
							\(13\)	279.202	−	859.296i	0.458206	−	1.41021i	−0.409124	−	0.912479i	\(-0.634166\pi\)
0.867330	−	0.497734i	\(-0.165834\pi\)
\(17\)	236.014	+	726.377i	0.198069	+	0.609593i	0.999927	+	0.0120786i	\(0.00384483\pi\)
							−0.801858	+	0.597514i	\(0.796155\pi\)
\(19\)	1836.70	−	1334.44i	1.16722	−	0.848038i	0.176550	−	0.984292i	\(-0.443506\pi\)
							0.990674	+	0.136254i	\(0.0435064\pi\)
\(23\)	2828.01			1.11471			0.557355	−	0.830274i	\(-0.311816\pi\)
							0.557355	+	0.830274i	\(0.311816\pi\)
\(29\)	−5640.64	−	4098.16i	−1.24547	−	0.904887i	−0.247520	−	0.968883i	\(-0.579615\pi\)
							−0.997950	+	0.0639961i	\(0.979615\pi\)
\(31\)	−2267.38	+	6978.27i	−0.423759	+	1.30420i	0.480418	+	0.877040i	\(0.340485\pi\)
							−0.904177	+	0.427157i	\(0.859515\pi\)
\(37\)	8527.79	+	6195.80i	1.02408	+	0.744035i	0.967114	−	0.254342i	\(-0.0818587\pi\)
							0.0569616	+	0.998376i	\(0.481859\pi\)
\(41\)	−4937.15	+	3587.05i	−0.458687	+	0.333256i	−0.793016	−	0.609201i	\(-0.791490\pi\)
							0.334329	+	0.942457i	\(0.391490\pi\)
\(43\)	−2029.16			−0.167358			−0.0836789	−	0.996493i	\(-0.526667\pi\)
							−0.0836789	+	0.996493i	\(0.526667\pi\)
\(47\)	16356.7	−	11883.9i	1.08007	−	0.784716i	0.102374	−	0.994746i	\(-0.467356\pi\)
							0.977695	+	0.210030i	\(0.0673561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	66.6.e.a.31.2	✓	8
3.2	odd	2		198.6.f.c.163.1		8
11.4	even	5		726.6.a.be.1.3		4
11.5	even	5	inner	66.6.e.a.49.2	yes	8
11.7	odd	10		726.6.a.bb.1.3		4
33.5	odd	10		198.6.f.c.181.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.6.e.a.31.2	✓	8	1.1	even	1	trivial
66.6.e.a.49.2	yes	8	11.5	even	5	inner
198.6.f.c.163.1		8	3.2	odd	2
198.6.f.c.181.1		8	33.5	odd	10
726.6.a.bb.1.3		4	11.7	odd	10
726.6.a.be.1.3		4	11.4	even	5