Embedded newform 201.3.h.b.172.8

• Label: 201.3.h.b.172.8
• Level: $201$
• Weight: $3$
• Character: 201.172
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			172.8
Character	\(\chi\)	\(=\)	201.172
Dual form			201.3.h.b.97.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.15796 + 0.668551i) q^{2} -1.73205i q^{3} +(-1.10608 - 1.91579i) q^{4} -0.749427i q^{5} +(1.15796 - 2.00565i) q^{6} +(2.89468 - 1.67125i) q^{7} -8.30629i q^{8} -3.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 34 q^{4} + 21 q^{7} - 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.15796	+	0.668551i	0.578982	+	0.334276i	0.760729	−	0.649070i	\(-0.224842\pi\)
							−0.181747	+	0.983345i	\(0.558175\pi\)
\(3\)		−	1.73205i		−	0.577350i
							\(5\)		−	0.749427i		−	0.149885i	−0.997188	−	0.0749427i	\(-0.976123\pi\)
0.997188	−	0.0749427i	\(-0.0238774\pi\)
\(7\)	2.89468	−	1.67125i	0.413526	−	0.238749i	−0.278778	−	0.960356i	\(-0.589929\pi\)
							0.692304	+	0.721606i	\(0.256596\pi\)
\(11\)	−6.87542	+	3.96953i	−0.625038	+	0.360866i	−0.778828	−	0.627238i	\(-0.784186\pi\)
							0.153790	+	0.988104i	\(0.450852\pi\)
\(13\)	−14.8848	−	8.59375i	−1.14499	−	0.661058i	−0.197325	−	0.980338i	\(-0.563225\pi\)
							−0.947660	+	0.319281i	\(0.896559\pi\)
\(17\)	9.50193	−	16.4578i	0.558937	−	0.968108i	−0.438648	−	0.898659i	\(-0.644543\pi\)
							0.997586	−	0.0694488i	\(-0.0221241\pi\)
\(19\)	14.7724	−	25.5865i	0.777493	−	1.34666i	−0.155890	−	0.987774i	\(-0.549824\pi\)
							0.933383	−	0.358883i	\(-0.116842\pi\)
\(23\)	−0.701183	+	1.21448i	−0.0304862	+	0.0528037i	−0.880866	−	0.473366i	\(-0.843039\pi\)
							0.850380	+	0.526169i	\(0.176372\pi\)
\(29\)	26.1897	+	45.3619i	0.903093	+	1.56420i	0.823457	+	0.567379i	\(0.192043\pi\)
							0.0796365	+	0.996824i	\(0.474624\pi\)
\(31\)	−16.5153	+	9.53514i	−0.532753	+	0.307585i	−0.742137	−	0.670248i	\(-0.766188\pi\)
							0.209384	+	0.977834i	\(0.432854\pi\)
\(37\)	−6.29733	+	10.9073i	−0.170198	+	0.294792i	−0.938489	−	0.345309i	\(-0.887774\pi\)
							0.768291	+	0.640101i	\(0.221107\pi\)
\(41\)	46.7236	−	26.9759i	1.13960	−	0.657949i	0.193269	−	0.981146i	\(-0.438091\pi\)
							0.946332	+	0.323197i	\(0.104758\pi\)
\(43\)		−	7.92633i		−	0.184333i	−0.995744	−	0.0921666i	\(-0.970621\pi\)
							0.995744	−	0.0921666i	\(-0.0293792\pi\)
\(47\)	16.8595	+	29.2016i	0.358714	+	0.621310i	0.987746	−	0.156069i	\(-0.0498821\pi\)
							−0.629032	+	0.777379i	\(0.716549\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.h.b.172.8	yes	24
67.30	odd	6	inner	201.3.h.b.97.8	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.h.b.97.8	✓	24	67.30	odd	6	inner
201.3.h.b.172.8	yes	24	1.1	even	1	trivial