Embedded newform 201.3.k.a.14.14

• Label: 201.3.k.a.14.14
• 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			14.14
Character	\(\chi\)	\(=\)	201.14
Dual form			201.3.k.a.158.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.64295 + 0.750310i) q^{2} +(2.61118 - 1.47707i) q^{3} +(-0.483122 + 0.557552i) q^{4} +(6.13055 - 0.881441i) q^{5} +(-3.18178 + 4.38595i) q^{6} +(-4.82114 - 10.5568i) q^{7} +(2.41084 - 8.21055i) q^{8} +(4.63653 - 7.71379i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 440 q - 11 q^{3} + 70 q^{4} - 17 q^{6} - 30 q^{7} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.64295	+	0.750310i	−0.821475	+	0.375155i	−0.781392	−	0.624041i	\(-0.785490\pi\)
							−0.0400838	+	0.999196i	\(0.512763\pi\)
\(3\)	2.61118	−	1.47707i	0.870393	−	0.492357i
							\(5\)	6.13055	−	0.881441i	1.22611	−	0.176288i	0.501323	−	0.865260i	\(-0.332847\pi\)
0.724788	+	0.688972i	\(0.241938\pi\)
\(7\)	−4.82114	−	10.5568i	−0.688734	−	1.50812i	−0.853117	−	0.521720i	\(-0.825291\pi\)
							0.164383	−	0.986397i	\(-0.447437\pi\)
\(11\)	−11.4334	+	1.64388i	−1.03940	+	0.149443i	−0.640821	−	0.767690i	\(-0.721406\pi\)
							−0.398580	+	0.917133i	\(0.630497\pi\)
\(13\)	−9.83274	+	2.88715i	−0.756365	+	0.222089i	−0.637108	−	0.770775i	\(-0.719869\pi\)
							−0.119257	+	0.992863i	\(0.538051\pi\)
\(17\)	17.2195	−	14.9208i	1.01291	−	0.877694i	0.0203945	−	0.999792i	\(-0.493508\pi\)
							0.992518	+	0.122098i	\(0.0389623\pi\)
\(19\)	−2.44532	+	5.35450i	−0.128701	+	0.281816i	−0.963002	−	0.269493i	\(-0.913144\pi\)
							0.834301	+	0.551308i	\(0.185871\pi\)
\(23\)	2.54354	−	3.95782i	0.110589	−	0.172079i	−0.781548	−	0.623845i	\(-0.785569\pi\)
							0.892136	+	0.451766i	\(0.149206\pi\)
\(29\)		−	21.1278i		−	0.728543i	−0.931293	−	0.364272i	\(-0.881318\pi\)
							0.931293	−	0.364272i	\(-0.118682\pi\)
\(31\)	−13.1808	−	3.87025i	−0.425189	−	0.124847i	0.0621357	−	0.998068i	\(-0.480209\pi\)
							−0.487324	+	0.873221i	\(0.662027\pi\)
\(37\)	65.5228			1.77089			0.885444	−	0.464747i	\(-0.153855\pi\)
							0.885444	+	0.464747i	\(0.153855\pi\)
\(41\)	60.7731	−	52.6602i	1.48227	−	1.28439i	0.612973	−	0.790104i	\(-0.289973\pi\)
							0.869297	−	0.494290i	\(-0.164572\pi\)
\(43\)	41.9562	+	48.4200i	0.975726	+	1.12605i	0.992007	+	0.126180i	\(0.0402718\pi\)
							−0.0162815	+	0.999867i	\(0.505183\pi\)
\(47\)	−33.0140	+	51.3708i	−0.702425	+	1.09299i	0.288356	+	0.957523i	\(0.406891\pi\)
							−0.990781	+	0.135472i	\(0.956745\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.k.a.14.14	✓	440
3.2	odd	2	inner	201.3.k.a.14.31	yes	440
67.24	even	11	inner	201.3.k.a.158.31	yes	440
201.158	odd	22	inner	201.3.k.a.158.14	yes	440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.k.a.14.14	✓	440	1.1	even	1	trivial
201.3.k.a.14.31	yes	440	3.2	odd	2	inner
201.3.k.a.158.14	yes	440	201.158	odd	22	inner
201.3.k.a.158.31	yes	440	67.24	even	11	inner