Embedded newform 201.3.n.b.7.1

• Label: 201.3.n.b.7.1
• Level: $201$
• Weight: $3$
• Character: 201.7
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			7.1
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.b.115.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79502 - 3.48186i) q^{2} +(-0.487975 + 1.66189i) q^{3} +(-6.58102 + 9.24175i) q^{4} +(1.84877 + 1.60197i) q^{5} +(6.66240 - 1.28407i) q^{6} +(-4.70180 - 0.223974i) q^{7} +(28.4818 + 4.09506i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{23}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79502	−	3.48186i	−0.897512	−	1.74093i	−0.628529	−	0.777786i	\(-0.716343\pi\)
							−0.268983	−	0.963145i	\(-0.586688\pi\)
\(3\)	−0.487975	+	1.66189i	−0.162658	+	0.553964i
							\(5\)	1.84877	+	1.60197i	0.369754	+	0.320394i	0.819841	−	0.572591i	\(-0.194062\pi\)
−0.450087	+	0.892985i	\(0.648607\pi\)
\(7\)	−4.70180	−	0.223974i	−0.671686	−	0.0319963i	−0.291031	−	0.956714i	\(-0.593998\pi\)
							−0.380655	+	0.924717i	\(0.624301\pi\)
\(11\)	1.06913	−	5.54715i	0.0971933	−	0.504286i	−0.900407	−	0.435049i	\(-0.856731\pi\)
							0.997600	−	0.0692378i	\(-0.0220567\pi\)
\(13\)	4.14645	−	10.3573i	0.318958	−	0.796718i	−0.679078	−	0.734067i	\(-0.737620\pi\)
							0.998035	−	0.0626518i	\(-0.0199558\pi\)
\(17\)	−4.91013	−	6.89532i	−0.288831	−	0.405607i	0.644511	−	0.764595i	\(-0.277061\pi\)
							−0.933342	+	0.358989i	\(0.883122\pi\)
\(19\)	−0.698350	−	14.6602i	−0.0367553	−	0.771588i	−0.939735	−	0.341904i	\(-0.888928\pi\)
							0.902980	−	0.429684i	\(-0.141375\pi\)
\(23\)	21.1128	−	20.1310i	0.917949	−	0.875262i	−0.0745873	−	0.997214i	\(-0.523764\pi\)
							0.992536	+	0.121952i	\(0.0389155\pi\)
\(29\)	16.4841	+	28.5513i	0.568417	+	0.984528i	0.996723	+	0.0808938i	\(0.0257775\pi\)
							−0.428305	+	0.903634i	\(0.640889\pi\)
\(31\)	18.1809	+	45.4138i	0.586482	+	1.46496i	0.863626	+	0.504133i	\(0.168188\pi\)
							−0.277144	+	0.960828i	\(0.589388\pi\)
\(37\)	27.2543	−	47.2059i	0.736604	−	1.27584i	−0.217412	−	0.976080i	\(-0.569762\pi\)
							0.954016	−	0.299755i	\(-0.0969051\pi\)
\(41\)	−3.70886	−	38.8410i	−0.0904601	−	0.947341i	−0.920787	−	0.390065i	\(-0.872453\pi\)
							0.830327	−	0.557276i	\(-0.188153\pi\)
\(43\)	−56.9730	−	26.0187i	−1.32495	−	0.605087i	−0.377807	−	0.925884i	\(-0.623322\pi\)
							−0.947148	+	0.320798i	\(0.896049\pi\)
\(47\)	−1.34687	−	5.55186i	−0.0286567	−	0.118125i	0.955687	−	0.294385i	\(-0.0951149\pi\)
							−0.984344	+	0.176261i	\(0.943600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.7.1	✓	240
67.48	odd	66	inner	201.3.n.b.115.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.7.1	✓	240	1.1	even	1	trivial
201.3.n.b.115.1	yes	240	67.48	odd	66	inner