Embedded newform 231.2.y.a.16.8

• Label: 231.2.y.a.16.8
• Level: $231$
• Weight: $2$
• Character: 231.16
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.y (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.84454428669\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.8
Character	\(\chi\)	\(=\)	231.16
Dual form			231.2.y.a.130.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.67076 + 0.567688i) q^{2} +(0.104528 - 0.994522i) q^{3} +(4.98362 + 2.21885i) q^{4} +(-1.97931 - 2.19824i) q^{5} +(0.843750 - 2.59679i) q^{6} +(-1.73835 + 1.99452i) q^{7} +(7.63253 + 5.54536i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{3} + 10 q^{4} - q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\)	\(155\)	\(199\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.67076	+	0.567688i	1.88852	+	0.401416i	0.998512	−	0.0545368i	\(-0.0173682\pi\)
							0.890004	+	0.455953i	\(0.150702\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	−1.97931	−	2.19824i	−0.885174	−	0.983085i	0.114773	−	0.993392i	\(-0.463386\pi\)
−0.999947	+	0.0103068i	\(0.996719\pi\)
\(7\)	−1.73835	+	1.99452i	−0.657036	+	0.753859i
							\(11\)	−2.72604	−	1.88910i	−0.821933	−	0.569584i
\(13\)	1.13875	+	3.50471i	0.315832	+	0.972031i								0.975411	+	0.220395i	\(0.0707348\pi\)
							−0.659579	+	0.751636i	\(0.729265\pi\)
\(17\)	−1.35831	+	0.288718i	−0.329439	+	0.0700244i	−0.369662	−	0.929166i	\(-0.620526\pi\)
							0.0402231	+	0.999191i	\(0.487193\pi\)
\(19\)	−1.39259	+	0.620019i	−0.319481	+	0.142242i	−0.560211	−	0.828350i	\(-0.689280\pi\)
							0.240730	+	0.970592i	\(0.422613\pi\)
\(23\)	2.58882	−	4.48397i	0.539806	−	0.934972i	−0.459108	−	0.888381i	\(-0.651831\pi\)
							0.998914	−	0.0465914i	\(-0.0148359\pi\)
\(29\)	−3.24703	+	2.35910i	−0.602958	+	0.438074i	−0.846927	−	0.531708i	\(-0.821550\pi\)
							0.243970	+	0.969783i	\(0.421550\pi\)
\(31\)	2.18796	−	2.42998i	0.392969	−	0.436437i	−0.513899	−	0.857851i	\(-0.671799\pi\)
							0.906868	+	0.421414i	\(0.138466\pi\)
\(37\)	−0.464510	−	4.41952i	−0.0763649	−	0.726564i	−0.963979	−	0.265977i	\(-0.914306\pi\)
							0.887614	−	0.460587i	\(-0.152361\pi\)
\(41\)	6.29070	+	4.57046i	0.982442	+	0.713786i	0.958253	−	0.285921i	\(-0.0922996\pi\)
							0.0241891	+	0.999707i	\(0.492300\pi\)
\(43\)	−5.34560			−0.815196			−0.407598	−	0.913161i	\(-0.633634\pi\)
							−0.407598	+	0.913161i	\(0.633634\pi\)
\(47\)	6.98930	−	3.11183i	1.01949	−	0.453908i	0.172217	−	0.985059i	\(-0.444907\pi\)
							0.847277	+	0.531151i	\(0.178240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.y.a.16.8	✓	64
3.2	odd	2		693.2.by.d.478.1		64
7.4	even	3	inner	231.2.y.a.214.1	yes	64
11.9	even	5	inner	231.2.y.a.163.1	yes	64
21.11	odd	6		693.2.by.d.676.8		64
33.20	odd	10		693.2.by.d.163.8		64
77.53	even	15	inner	231.2.y.a.130.8	yes	64
231.53	odd	30		693.2.by.d.361.1		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.y.a.16.8	✓	64	1.1	even	1	trivial
231.2.y.a.130.8	yes	64	77.53	even	15	inner
231.2.y.a.163.1	yes	64	11.9	even	5	inner
231.2.y.a.214.1	yes	64	7.4	even	3	inner
693.2.by.d.163.8		64	33.20	odd	10
693.2.by.d.361.1		64	231.53	odd	30
693.2.by.d.478.1		64	3.2	odd	2
693.2.by.d.676.8		64	21.11	odd	6