Embedded newform 201.2.m.a.4.5

• Label: 201.2.m.a.4.5
• Level: $201$
• Weight: $2$
• Character: 201.4
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
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			4.5
Character	\(\chi\)	\(=\)	201.4
Dual form			201.2.m.a.151.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13798 + 0.204153i) q^{2} +(-0.841254 - 0.540641i) q^{3} +(2.56544 + 0.494448i) q^{4} +(0.515825 + 3.58764i) q^{5} +(-1.68821 - 1.32763i) q^{6} +(1.76546 - 2.47924i) q^{7} +(1.26251 + 0.370705i) q^{8} +(0.415415 + 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q + 2 q^{2} + 10 q^{3} + 6 q^{4} - 2 q^{5} + 9 q^{6} + q^{7} - 45 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13798	+	0.204153i	1.51178	+	0.144358i	0.817745	−	0.575580i	\(-0.195224\pi\)
							0.694038	+	0.719938i	\(0.255830\pi\)
\(3\)	−0.841254	−	0.540641i	−0.485698	−	0.312139i
							\(5\)	0.515825	+	3.58764i	0.230684	+	1.60444i	0.695159	+	0.718856i	\(0.255334\pi\)
−0.464475	+	0.885586i	\(0.653757\pi\)
\(7\)	1.76546	−	2.47924i	0.667282	−	0.937066i	−0.332700	−	0.943033i	\(-0.607960\pi\)
							0.999982	+	0.00596620i	\(0.00189911\pi\)
\(11\)	−1.15162	+	0.905641i	−0.347225	+	0.273061i	−0.776506	−	0.630110i	\(-0.783010\pi\)
							0.429280	+	0.903171i	\(0.358767\pi\)
\(13\)	1.22204	−	5.03732i	0.338933	−	1.39710i	−0.506140	−	0.862451i	\(-0.668928\pi\)
							0.845073	−	0.534651i	\(-0.179557\pi\)
\(17\)	−6.82057	+	1.31456i	−1.65423	+	0.318827i	−0.929113	−	0.369796i	\(-0.879427\pi\)
							−0.725119	+	0.688623i	\(0.758215\pi\)
\(19\)	1.60673	+	2.25633i	0.368608	+	0.517638i	0.956668	−	0.291179i	\(-0.0940477\pi\)
							−0.588060	+	0.808817i	\(0.700108\pi\)
\(23\)	3.76901	−	1.94306i	0.785893	−	0.405156i	−0.0180691	−	0.999837i	\(-0.505752\pi\)
							0.803962	+	0.594681i	\(0.202722\pi\)
\(29\)	−3.50002	−	6.06221i	−0.649937	−	1.12572i	−0.983138	−	0.182867i	\(-0.941462\pi\)
							0.333201	−	0.942856i	\(-0.391871\pi\)
\(31\)	0.462941	+	1.90827i	0.0831467	+	0.342735i	0.998156	−	0.0606969i	\(-0.0193323\pi\)
							−0.915010	+	0.403432i	\(0.867817\pi\)
\(37\)	0.289867	−	0.502064i	0.0476538	−	0.0825388i	−0.841215	−	0.540701i	\(-0.818159\pi\)
							0.888868	+	0.458163i	\(0.151492\pi\)
\(41\)	−4.09522	+	11.8324i	−0.639565	+	1.84790i	−0.121438	+	0.992599i	\(0.538751\pi\)
							−0.518127	+	0.855304i	\(0.673371\pi\)
\(43\)	0.177389	+	0.204718i	0.0270516	+	0.0312192i	0.769114	−	0.639111i	\(-0.220698\pi\)
							−0.742063	+	0.670331i	\(0.766152\pi\)
\(47\)	−0.0289996	+	0.608777i	−0.00423003	+	0.0887993i	−0.999995	−	0.00313526i	\(-0.999002\pi\)
							0.995765	+	0.0919345i	\(0.0293050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.a.4.5	✓	100
3.2	odd	2		603.2.z.b.406.1		100
67.17	even	33	inner	201.2.m.a.151.5	yes	100
201.17	odd	66		603.2.z.b.352.1		100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.a.4.5	✓	100	1.1	even	1	trivial
201.2.m.a.151.5	yes	100	67.17	even	33	inner
603.2.z.b.352.1		100	201.17	odd	66
603.2.z.b.406.1		100	3.2	odd	2