Embedded newform 67.2.g.a.10.1

• Label: 67.2.g.a.10.1
• Level: $67$
• Weight: $2$
• Character: 67.10
• Analytic conductor: $0.535$
• Analytic rank: $0$
• Dimension: $100$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	67.g (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.534997693543\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(5\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			10.1
Character	\(\chi\)	\(=\)	67.10
Dual form			67.2.g.a.47.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86173 - 1.77515i) q^{2} +(-0.395503 - 2.75079i) q^{3} +(0.219695 + 4.61197i) q^{4} +(0.754900 - 1.65300i) q^{5} +(-4.14675 + 5.82329i) q^{6} +(-0.561736 + 2.31551i) q^{7} +(4.40882 - 5.08805i) q^{8} +(-4.53192 + 1.33069i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 24 q^{2} - 14 q^{3} - 18 q^{4} - 16 q^{5} - 16 q^{6} - 24 q^{7} + 23 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86173	−	1.77515i	−1.31644	−	1.25522i	−0.945122	−	0.326717i	\(-0.894058\pi\)
							−0.371317	−	0.928506i	\(-0.621094\pi\)
\(3\)	−0.395503	−	2.75079i	−0.228344	−	1.58817i	−0.705085	−	0.709123i	\(-0.749091\pi\)
							0.476741	−	0.879044i	\(-0.341818\pi\)
\(5\)	0.754900	−	1.65300i	0.337601	−	0.739244i	−0.662349	−	0.749195i	\(-0.730440\pi\)
							0.999950	+	0.00995145i	\(0.00316770\pi\)
\(7\)	−0.561736	+	2.31551i	−0.212316	+	0.875179i	0.761559	+	0.648095i	\(0.224434\pi\)
							−0.973875	+	0.227083i	\(0.927081\pi\)
\(11\)	0.558411	+	0.784179i	0.168367	+	0.236439i	0.890135	−	0.455696i	\(-0.150610\pi\)
							−0.721768	+	0.692135i	\(0.756670\pi\)
\(13\)	2.04927	−	5.92099i	0.568366	−	1.64219i	−0.183612	−	0.982999i	\(-0.558779\pi\)
							0.751978	−	0.659188i	\(-0.229100\pi\)
\(17\)	0.0160394	−	0.336709i	0.00389013	−	0.0816638i	−0.996087	−	0.0883741i	\(-0.971833\pi\)
							0.999977	+	0.00671022i	\(0.00213595\pi\)
\(19\)	−0.0302113	−	0.124533i	−0.00693096	−	0.0285698i	0.968241	−	0.250018i	\(-0.0804367\pi\)
							−0.975172	+	0.221449i	\(0.928922\pi\)
\(23\)	−3.05448	+	2.40207i	−0.636904	+	0.500867i	−0.883610	−	0.468224i	\(-0.844894\pi\)
							0.246706	+	0.969090i	\(0.420652\pi\)
\(29\)	3.15521	−	5.46499i	0.585908	−	1.01482i	−0.408853	−	0.912600i	\(-0.634071\pi\)
							0.994762	−	0.102223i	\(-0.0325955\pi\)
\(31\)	0.950033	+	2.74494i	0.170631	+	0.493006i	0.997600	−	0.0692417i	\(-0.0220580\pi\)
							−0.826969	+	0.562248i	\(0.809937\pi\)
\(37\)	2.91909	+	5.05602i	0.479896	+	0.831205i	0.999734	−	0.0230603i	\(-0.00734098\pi\)
							−0.519838	+	0.854265i	\(0.674008\pi\)
\(41\)	5.28177	−	2.72294i	0.824873	−	0.425252i	0.00652797	−	0.999979i	\(-0.497922\pi\)
							0.818345	+	0.574727i	\(0.194892\pi\)
\(43\)	4.79837	+	3.08373i	0.731745	+	0.470264i	0.852704	−	0.522394i	\(-0.174961\pi\)
							−0.120960	+	0.992657i	\(0.538597\pi\)
\(47\)	5.05247	+	2.02271i	0.736979	+	0.295042i	0.709621	−	0.704584i	\(-0.248866\pi\)
							0.0273583	+	0.999626i	\(0.491291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.g.a.10.1	✓	100
3.2	odd	2		603.2.z.c.10.5		100
67.28	odd	66		4489.2.a.q.1.45		50
67.39	even	33		4489.2.a.p.1.6		50
67.47	even	33	inner	67.2.g.a.47.1	yes	100
201.47	odd	66		603.2.z.c.181.5		100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.g.a.10.1	✓	100	1.1	even	1	trivial
67.2.g.a.47.1	yes	100	67.47	even	33	inner
603.2.z.c.10.5		100	3.2	odd	2
603.2.z.c.181.5		100	201.47	odd	66
4489.2.a.p.1.6		50	67.39	even	33
4489.2.a.q.1.45		50	67.28	odd	66