Embedded newform 201.3.k.a.14.12

• Label: 201.3.k.a.14.12
• 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			14.12
Character	\(\chi\)	\(=\)	201.14
Dual form			201.3.k.a.158.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74808 + 0.798320i) q^{2} +(-2.32890 - 1.89108i) q^{3} +(-0.200985 + 0.231949i) q^{4} +(1.59523 - 0.229359i) q^{5} +(5.58080 + 1.44655i) q^{6} +(-2.44056 - 5.34409i) q^{7} +(2.33184 - 7.94150i) q^{8} +(1.84760 + 8.80831i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 440 q - 11 q^{3} + 70 q^{4} - 17 q^{6} - 30 q^{7} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74808	+	0.798320i	−0.874038	+	0.399160i	−0.801351	−	0.598195i	\(-0.795885\pi\)
							−0.0726877	+	0.997355i	\(0.523158\pi\)
\(3\)	−2.32890	−	1.89108i	−0.776302	−	0.630362i
							\(5\)	1.59523	−	0.229359i	0.319046	−	0.0458719i	0.0190684	−	0.999818i	\(-0.493930\pi\)
0.299977	+	0.953946i	\(0.403021\pi\)
\(7\)	−2.44056	−	5.34409i	−0.348652	−	0.763441i	−0.999989	−	0.00465279i	\(-0.998519\pi\)
							0.651337	−	0.758788i	\(-0.274208\pi\)
\(11\)	8.23231	−	1.18363i	0.748392	−	0.107602i	0.242443	−	0.970166i	\(-0.422051\pi\)
							0.505949	+	0.862563i	\(0.331142\pi\)
\(13\)	−15.9697	+	4.68914i	−1.22844	+	0.360703i	−0.830661	−	0.556778i	\(-0.812037\pi\)
							−0.397780	+	0.917481i	\(0.630219\pi\)
\(17\)	11.9813	−	10.3818i	0.704781	−	0.610696i	−0.226924	−	0.973912i	\(-0.572867\pi\)
							0.931705	+	0.363216i	\(0.118321\pi\)
\(19\)	−8.67957	+	19.0056i	−0.456820	+	1.00030i	0.531381	+	0.847133i	\(0.321673\pi\)
							−0.988201	+	0.153163i	\(0.951054\pi\)
\(23\)	−24.3581	+	37.9020i	−1.05905	+	1.64791i	−0.359808	+	0.933026i	\(0.617158\pi\)
							−0.699241	+	0.714886i	\(0.746479\pi\)
\(29\)			34.3515i			1.18453i	0.805742	+	0.592267i	\(0.201767\pi\)
							−0.805742	+	0.592267i	\(0.798233\pi\)
\(31\)	44.7487	+	13.1394i	1.44351	+	0.423852i	0.907390	−	0.420291i	\(-0.138072\pi\)
							0.536118	+	0.844143i	\(0.319890\pi\)
\(37\)	−57.4094			−1.55160			−0.775802	−	0.630976i	\(-0.782654\pi\)
							−0.775802	+	0.630976i	\(0.782654\pi\)
\(41\)	−44.0024	+	38.1283i	−1.07323	+	0.929959i	−0.997740	−	0.0671875i	\(-0.978597\pi\)
							−0.0754897	+	0.997147i	\(0.524052\pi\)
\(43\)	−6.15708	−	7.10565i	−0.143188	−	0.165248i	0.679626	−	0.733559i	\(-0.262142\pi\)
							−0.822813	+	0.568312i	\(0.807597\pi\)
\(47\)	30.9961	−	48.2309i	0.659492	−	1.02619i	−0.336918	−	0.941534i	\(-0.609384\pi\)
							0.996410	−	0.0846561i	\(-0.0269792\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.k.a.14.12	✓	440
3.2	odd	2	inner	201.3.k.a.14.33	yes	440
67.24	even	11	inner	201.3.k.a.158.33	yes	440
201.158	odd	22	inner	201.3.k.a.158.12	yes	440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.k.a.14.12	✓	440	1.1	even	1	trivial
201.3.k.a.14.33	yes	440	3.2	odd	2	inner
201.3.k.a.158.12	yes	440	201.158	odd	22	inner
201.3.k.a.158.33	yes	440	67.24	even	11	inner