Embedded newform 201.2.m.b.10.3

• Label: 201.2.m.b.10.3
• Level: $201$
• Weight: $2$
• Character: 201.10
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $120$
• 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:	\(120\)
Relative dimension:	\(6\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			10.3
Character	\(\chi\)	\(=\)	201.10
Dual form			201.2.m.b.181.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.601743 - 0.573760i) q^{2} +(-0.142315 - 0.989821i) q^{3} +(-0.0622707 - 1.30722i) q^{4} +(1.55758 - 3.41063i) q^{5} +(-0.482283 + 0.677272i) q^{6} +(-0.343913 + 1.41763i) q^{7} +(-1.80152 + 2.07906i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 2 q^{2} - 12 q^{3} + 8 q^{4} + 2 q^{5} - 9 q^{6} + 5 q^{7} - 33 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{8}{33}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.601743	−	0.573760i	−0.425496	−	0.405710i	0.446842	−	0.894613i	\(-0.352549\pi\)
							−0.872338	+	0.488903i	\(0.837397\pi\)
\(3\)	−0.142315	−	0.989821i	−0.0821655	−	0.571474i
							\(5\)	1.55758	−	3.41063i	0.696573	−	1.52528i	−0.147505	−	0.989061i	\(-0.547124\pi\)
0.844078	−	0.536220i	\(-0.180148\pi\)
\(7\)	−0.343913	+	1.41763i	−0.129987	+	0.535813i	0.869161	+	0.494530i	\(0.164660\pi\)
							−0.999147	+	0.0412830i	\(0.986855\pi\)
\(11\)	0.668031	+	0.938118i	0.201419	+	0.282853i	0.902914	−	0.429821i	\(-0.141423\pi\)
							−0.701495	+	0.712674i	\(0.747484\pi\)
\(13\)	−0.687727	+	1.98706i	−0.190741	+	0.551110i	−0.999304	−	0.0372994i	\(-0.988124\pi\)
							0.808563	+	0.588410i	\(0.200246\pi\)
\(17\)	0.301334	−	6.32578i	0.0730843	−	1.53423i	−0.603202	−	0.797588i	\(-0.706109\pi\)
							0.676286	−	0.736639i	\(-0.263588\pi\)
\(19\)	0.875128	+	3.60733i	0.200768	+	0.827577i	0.979700	+	0.200469i	\(0.0642466\pi\)
							−0.778932	+	0.627108i	\(0.784238\pi\)
\(23\)	6.12248	−	4.81477i	1.27663	−	1.00395i	0.277561	−	0.960708i	\(-0.410474\pi\)
							0.999065	−	0.0432415i	\(-0.0137685\pi\)
\(29\)	−2.21965	+	3.84455i	−0.412179	+	0.713915i	−0.995128	−	0.0985937i	\(-0.968566\pi\)
							0.582949	+	0.812509i	\(0.301899\pi\)
\(31\)	0.663127	+	1.91598i	0.119101	+	0.344120i	0.988756	−	0.149541i	\(-0.0477796\pi\)
							−0.869654	+	0.493661i	\(0.835658\pi\)
\(37\)	−1.98952	−	3.44595i	−0.327075	−	0.566511i	0.654855	−	0.755755i	\(-0.272730\pi\)
							−0.981930	+	0.189244i	\(0.939396\pi\)
\(41\)	0.225976	−	0.116499i	0.0352915	−	0.0181940i	−0.440490	−	0.897757i	\(-0.645195\pi\)
							0.475782	+	0.879563i	\(0.342165\pi\)
\(43\)	6.55555	+	4.21300i	0.999712	+	0.642476i	0.934711	−	0.355409i	\(-0.115658\pi\)
							0.0650016	+	0.997885i	\(0.479295\pi\)
\(47\)	−6.93031	−	2.77448i	−1.01089	−	0.404699i	−0.193696	−	0.981062i	\(-0.562048\pi\)
							−0.817194	+	0.576362i	\(0.804472\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.b.10.3	✓	120
3.2	odd	2		603.2.z.d.10.4		120
67.47	even	33	inner	201.2.m.b.181.3	yes	120
201.47	odd	66		603.2.z.d.181.4		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.b.10.3	✓	120	1.1	even	1	trivial
201.2.m.b.181.3	yes	120	67.47	even	33	inner
603.2.z.d.10.4		120	3.2	odd	2
603.2.z.d.181.4		120	201.47	odd	66