Embedded newform 201.2.m.a.10.2

• Label: 201.2.m.a.10.2
• Level: $201$
• Weight: $2$
• Character: 201.10
• 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			10.2
Character	\(\chi\)	\(=\)	201.10
Dual form			201.2.m.a.181.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.928460 - 0.885285i) q^{2} +(0.142315 + 0.989821i) q^{3} +(-0.0168551 - 0.353832i) q^{4} +(0.471271 - 1.03194i) q^{5} +(0.744140 - 1.04500i) q^{6} +(0.634488 - 2.61540i) q^{7} +(-1.97780 + 2.28250i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q + 2 q^{2} + 10 q^{3} + 6 q^{4} - 2 q^{5} + 9 q^{6} + q^{7} - 45 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.928460	−	0.885285i	−0.656520	−	0.625991i	0.286754	−	0.958004i	\(-0.407424\pi\)
							−0.943275	+	0.332013i	\(0.892272\pi\)
\(3\)	0.142315	+	0.989821i	0.0821655	+	0.571474i
							\(5\)	0.471271	−	1.03194i	0.210759	−	0.461498i	−0.774499	−	0.632576i	\(-0.781998\pi\)
0.985257	+	0.171078i	\(0.0547250\pi\)
\(7\)	0.634488	−	2.61540i	0.239814	−	0.988527i	−0.716826	−	0.697253i	\(-0.754406\pi\)
							0.956640	−	0.291274i	\(-0.0940792\pi\)
\(11\)	0.471486	+	0.662110i	0.142158	+	0.199634i	0.879516	−	0.475869i	\(-0.157866\pi\)
							−0.737358	+	0.675502i	\(0.763927\pi\)
\(13\)	1.50953	−	4.36151i	0.418669	−	1.20966i	−0.516754	−	0.856134i	\(-0.672860\pi\)
							0.935423	−	0.353531i	\(-0.115019\pi\)
\(17\)	0.224474	−	4.71229i	0.0544430	−	1.14290i	−0.792035	−	0.610475i	\(-0.790978\pi\)
							0.846478	−	0.532423i	\(-0.178718\pi\)
\(19\)	−1.16810	−	4.81496i	−0.267980	−	1.10463i	−0.933594	−	0.358331i	\(-0.883346\pi\)
							0.665615	−	0.746295i	\(-0.268169\pi\)
\(23\)	−2.43139	+	1.91206i	−0.506979	+	0.398693i	−0.838623	−	0.544712i	\(-0.816639\pi\)
							0.331644	+	0.943404i	\(0.392397\pi\)
\(29\)	−0.642806	+	1.11337i	−0.119366	+	0.206748i	−0.919517	−	0.393051i	\(-0.871420\pi\)
							0.800151	+	0.599799i	\(0.204753\pi\)
\(31\)	1.53972	+	4.44874i	0.276543	+	0.799017i	0.994670	+	0.103106i	\(0.0328781\pi\)
							−0.718128	+	0.695911i	\(0.755001\pi\)
\(37\)	1.45241	+	2.51565i	0.238775	+	0.413570i	0.960363	−	0.278752i	\(-0.0899209\pi\)
							−0.721588	+	0.692323i	\(0.756588\pi\)
\(41\)	−1.91737	+	0.988471i	−0.299442	+	0.154373i	−0.601400	−	0.798948i	\(-0.705390\pi\)
							0.301958	+	0.953321i	\(0.402360\pi\)
\(43\)	3.33374	+	2.14247i	0.508391	+	0.326723i	0.769565	−	0.638569i	\(-0.220473\pi\)
							−0.261174	+	0.965292i	\(0.584110\pi\)
\(47\)	5.47572	+	2.19215i	0.798716	+	0.319758i	0.734875	−	0.678202i	\(-0.237241\pi\)
							0.0638410	+	0.997960i	\(0.479665\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.m.a.10.2	✓	100
3.2	odd	2		603.2.z.b.10.4		100
67.47	even	33	inner	201.2.m.a.181.2	yes	100
201.47	odd	66		603.2.z.b.181.4		100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.m.a.10.2	✓	100	1.1	even	1	trivial
201.2.m.a.181.2	yes	100	67.47	even	33	inner
603.2.z.b.10.4		100	3.2	odd	2
603.2.z.b.181.4		100	201.47	odd	66