Embedded newform 201.3.k.a.14.13

• Label: 201.3.k.a.14.13
• Level: $201$
• Weight: $3$
• Character: 201.14
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $440$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(440\)
Relative dimension:	\(44\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.13
Character	\(\chi\)	\(=\)	201.14
Dual form			201.3.k.a.158.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74004 + 0.794650i) q^{2} +(2.75393 + 1.18990i) q^{3} +(-0.223170 + 0.257552i) q^{4} +(4.42297 - 0.635927i) q^{5} +(-5.73751 + 0.117936i) q^{6} +(2.38025 + 5.21203i) q^{7} +(2.33937 - 7.96717i) q^{8} +(6.16827 + 6.55381i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 440 q - 11 q^{3} + 70 q^{4} - 17 q^{6} - 30 q^{7} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74004	+	0.794650i	−0.870020	+	0.397325i	−0.799843	−	0.600210i	\(-0.795084\pi\)
							−0.0701776	+	0.997535i	\(0.522357\pi\)
\(3\)	2.75393	+	1.18990i	0.917977	+	0.396633i
							\(5\)	4.42297	−	0.635927i	0.884593	−	0.127185i	0.314980	−	0.949098i	\(-0.398002\pi\)
0.569613	+	0.821913i	\(0.307093\pi\)
\(7\)	2.38025	+	5.21203i	0.340036	+	0.744575i	0.999977	−	0.00674196i	\(-0.00214605\pi\)
							−0.659941	+	0.751317i	\(0.729419\pi\)
\(11\)	−0.664170	+	0.0954933i	−0.0603791	+	0.00868121i	−0.172438	−	0.985020i	\(-0.555165\pi\)
							0.112059	+	0.993702i	\(0.464255\pi\)
\(13\)	11.2732	−	3.31010i	0.867167	−	0.254623i	0.182257	−	0.983251i	\(-0.441660\pi\)
							0.684910	+	0.728628i	\(0.259841\pi\)
\(17\)	−7.26143	+	6.29206i	−0.427143	+	0.370121i	−0.841740	−	0.539884i	\(-0.818468\pi\)
							0.414597	+	0.910005i	\(0.363923\pi\)
\(19\)	5.78310	−	12.6632i	0.304374	−	0.666485i	−0.694205	−	0.719777i	\(-0.744244\pi\)
							0.998579	+	0.0532917i	\(0.0169713\pi\)
\(23\)	−7.45379	+	11.5983i	−0.324078	+	0.504275i	−0.964619	−	0.263649i	\(-0.915074\pi\)
							0.640541	+	0.767924i	\(0.278710\pi\)
\(29\)			15.5461i			0.536074i	0.963409	+	0.268037i	\(0.0863749\pi\)
							−0.963409	+	0.268037i	\(0.913625\pi\)
\(31\)	−1.02467	−	0.300871i	−0.0330540	−	0.00970552i	0.265164	−	0.964203i	\(-0.414574\pi\)
							−0.298218	+	0.954498i	\(0.596392\pi\)
\(37\)	−9.62772			−0.260209			−0.130104	−	0.991500i	\(-0.541531\pi\)
							−0.130104	+	0.991500i	\(0.541531\pi\)
\(41\)	−29.9496	+	25.9515i	−0.730478	+	0.632963i	−0.938547	−	0.345151i	\(-0.887828\pi\)
							0.208069	+	0.978114i	\(0.433282\pi\)
\(43\)	27.6694	+	31.9322i	0.643474	+	0.742609i	0.979985	−	0.199071i	\(-0.0637924\pi\)
							−0.336511	+	0.941680i	\(0.609247\pi\)
\(47\)	−4.11570	+	6.40415i	−0.0875680	+	0.136258i	−0.882274	−	0.470736i	\(-0.843989\pi\)
							0.794706	+	0.606994i	\(0.207625\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.k.a.14.13	✓	440
3.2	odd	2	inner	201.3.k.a.14.32	yes	440
67.24	even	11	inner	201.3.k.a.158.32	yes	440
201.158	odd	22	inner	201.3.k.a.158.13	yes	440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.k.a.14.13	✓	440	1.1	even	1	trivial
201.3.k.a.14.32	yes	440	3.2	odd	2	inner
201.3.k.a.158.13	yes	440	201.158	odd	22	inner
201.3.k.a.158.32	yes	440	67.24	even	11	inner