Embedded newform 447.2.i.a.100.1

• Label: 447.2.i.a.100.1
• Level: $447$
• Weight: $2$
• Character: 447.100
• Analytic conductor: $3.569$
• Analytic rank: $0$
• Dimension: $864$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 447 = 3 \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	447.i (of order \(74\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.56931297035\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(24\) over \(\Q(\zeta_{74})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{74}]$

Embedding invariants

Embedding label			100.1
Character	\(\chi\)	\(=\)	447.100
Dual form			447.2.i.a.76.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.659648 - 2.53341i) q^{2} +(0.169001 + 0.985616i) q^{3} +(-4.23700 + 2.36693i) q^{4} +(0.0880200 - 0.408414i) q^{5} +(2.38549 - 1.07831i) q^{6} +(-0.128891 - 3.03419i) q^{7} +(5.16850 + 5.39272i) q^{8} +(-0.942877 + 0.333140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q + 18 q^{4} + 8 q^{5} + 2 q^{6} - 4 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/447\mathbb{Z}\right)^\times\).

\(n\)	\(151\)	\(299\)
\(\chi(n)\)	\(e\left(\frac{31}{74}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.659648	−	2.53341i	−0.466442	−	1.79139i	−0.599604	−	0.800297i	\(-0.704675\pi\)
							0.133162	−	0.991094i	\(-0.457487\pi\)
\(3\)	0.169001	+	0.985616i	0.0975727	+	0.569046i
							\(5\)	0.0880200	−	0.408414i	0.0393637	−	0.182648i	−0.953598	−	0.301082i	\(-0.902652\pi\)
0.992962	+	0.118434i	\(0.0377873\pi\)
\(7\)	−0.128891	−	3.03419i	−0.0487162	−	1.14682i	−0.842528	−	0.538653i	\(-0.818933\pi\)
							0.793812	−	0.608164i	\(-0.208094\pi\)
\(11\)	0.319426	−	0.571799i	0.0963106	−	0.172404i	−0.820593	−	0.571514i	\(-0.806356\pi\)
							0.916903	+	0.399110i	\(0.130681\pi\)
\(13\)	−2.98963	−	2.21112i	−0.829174	−	0.613254i	0.0948915	−	0.995488i	\(-0.469750\pi\)
							−0.924066	+	0.382234i	\(0.875155\pi\)
\(17\)	−2.57189	+	0.440994i	−0.623774	+	0.106957i	−0.472204	−	0.881490i	\(-0.656541\pi\)
							−0.151571	+	0.988446i	\(0.548433\pi\)
\(19\)	−5.12947	−	3.46691i	−1.17678	−	0.795365i	−0.194156	−	0.980971i	\(-0.562197\pi\)
							−0.982624	+	0.185606i	\(0.940575\pi\)
\(23\)	−6.08568	+	4.50094i	−1.26895	+	0.938512i	−0.999698	−	0.0245577i	\(-0.992182\pi\)
							−0.269253	+	0.963069i	\(0.586777\pi\)
\(29\)	3.31236	+	3.76357i	0.615089	+	0.698877i	0.971925	−	0.235290i	\(-0.0756040\pi\)
							−0.356836	+	0.934167i	\(0.616145\pi\)
\(31\)	−1.89508	−	2.56231i	−0.340366	−	0.460205i	0.600183	−	0.799862i	\(-0.295094\pi\)
							−0.940549	+	0.339657i	\(0.889689\pi\)
\(37\)	−7.84723	−	4.38373i	−1.29008	−	0.720681i	−0.316403	−	0.948625i	\(-0.602475\pi\)
							−0.973674	+	0.227944i	\(0.926800\pi\)
\(41\)	−8.63578	−	5.31726i	−1.34868	−	0.830417i	−0.354185	−	0.935176i	\(-0.615242\pi\)
							−0.994498	+	0.104759i	\(0.966593\pi\)
\(43\)	3.06397	−	0.392356i	0.467251	−	0.0598338i	0.110181	−	0.993912i	\(-0.464857\pi\)
							0.357070	+	0.934078i	\(0.383776\pi\)
\(47\)	−1.98984	+	1.34490i	−0.290248	+	0.196173i	−0.696441	−	0.717614i	\(-0.745234\pi\)
							0.406194	+	0.913787i	\(0.366856\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	447.2.i.a.100.1	yes	864
149.76	even	74	inner	447.2.i.a.76.1	✓	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
447.2.i.a.76.1	✓	864	149.76	even	74	inner
447.2.i.a.100.1	yes	864	1.1	even	1	trivial