Embedded newform 111.2.g.a.68.7

• Label: 111.2.g.a.68.7
• Level: $111$
• Weight: $2$
• Character: 111.68
• Analytic conductor: $0.886$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	\( x^{20} + 51x^{16} + 751x^{12} + 3533x^{8} + 2532x^{4} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			68.7
Root			\(0.651715 - 0.651715i\) of defining polynomial
Character	\(\chi\)	\(=\)	111.68
Dual form			111.2.g.a.80.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.651715 - 0.651715i) q^{2} +(-0.901905 + 1.47870i) q^{3} +1.15054i q^{4} +(0.457049 + 0.457049i) q^{5} +(0.375908 + 1.55148i) q^{6} +0.624804 q^{7} +(2.05325 + 2.05325i) q^{8} +(-1.37313 - 2.66730i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.651715	−	0.651715i	0.460832	−	0.460832i	−0.438096	−	0.898928i	\(-0.644347\pi\)
							0.898928	+	0.438096i	\(0.144347\pi\)
\(3\)	−0.901905	+	1.47870i	−0.520715	+	0.853730i
							\(5\)	0.457049	+	0.457049i	0.204399	+	0.204399i	0.801882	−	0.597483i	\(-0.203832\pi\)
−0.597483	+	0.801882i	\(0.703832\pi\)
\(7\)	0.624804			0.236154			0.118077	−	0.993004i	\(-0.462327\pi\)
							0.118077	+	0.993004i	\(0.462327\pi\)
\(11\)	0.161990			0.0488417			0.0244208	−	0.999702i	\(-0.492226\pi\)
							0.0244208	+	0.999702i	\(0.492226\pi\)
\(13\)	1.85184	−	1.85184i	0.513607	−	0.513607i	−0.402022	−	0.915630i	\(-0.631693\pi\)
							0.915630	+	0.402022i	\(0.131693\pi\)
\(17\)	−4.72554	−	4.72554i	−1.14611	−	1.14611i	−0.987311	−	0.158801i	\(-0.949237\pi\)
							−0.158801	−	0.987311i	\(-0.550763\pi\)
\(19\)	1.70130	−	1.70130i	0.390305	−	0.390305i	−0.484491	−	0.874796i	\(-0.660995\pi\)
							0.874796	+	0.484491i	\(0.160995\pi\)
\(23\)	2.22245	+	2.22245i	0.463412	+	0.463412i	0.899772	−	0.436360i	\(-0.143733\pi\)
							−0.436360	+	0.899772i	\(0.643733\pi\)
\(29\)	−4.91597	+	4.91597i	−0.912873	+	0.912873i	−0.996497	−	0.0836246i	\(-0.973350\pi\)
							0.0836246	+	0.996497i	\(0.473350\pi\)
\(31\)	0.0805365	+	0.0805365i	0.0144648	+	0.0144648i	0.714302	−	0.699837i	\(-0.246744\pi\)
							−0.699837	+	0.714302i	\(0.746744\pi\)
\(37\)	5.72871	−	2.04496i	0.941794	−	0.336190i
							\(41\)	6.38631			0.997374			0.498687	−	0.866782i	\(-0.333816\pi\)
0.498687	+	0.866782i	\(0.333816\pi\)
\(43\)	−3.72871	+	3.72871i	−0.568623	+	0.568623i	−0.931743	−	0.363120i	\(-0.881712\pi\)
							0.363120	+	0.931743i	\(0.381712\pi\)
\(47\)		−	6.30948i		−	0.920332i	−0.887833	−	0.460166i	\(-0.847790\pi\)
							0.887833	−	0.460166i	\(-0.152210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.g.a.68.7	yes	20
3.2	odd	2	inner	111.2.g.a.68.4	✓	20
37.6	odd	4	inner	111.2.g.a.80.4	yes	20
111.80	even	4	inner	111.2.g.a.80.7	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.g.a.68.4	✓	20	3.2	odd	2	inner
111.2.g.a.68.7	yes	20	1.1	even	1	trivial
111.2.g.a.80.4	yes	20	37.6	odd	4	inner
111.2.g.a.80.7	yes	20	111.80	even	4	inner