Embedded newform 633.2.v.b.13.2

• Label: 633.2.v.b.13.2
• Level: $633$
• Weight: $2$
• Character: 633.13
• Analytic conductor: $5.055$
• Analytic rank: $0$
• Dimension: $456$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 633 = 3 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	633.v (of order \(35\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.05453044795\)
Analytic rank:	\(0\)
Dimension:	\(456\)
Relative dimension:	\(19\) over \(\Q(\zeta_{35})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{35}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	633.13
Dual form			633.2.v.b.487.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32185 + 2.00252i) q^{2} +(-0.0448648 - 0.998993i) q^{3} +(-1.47674 - 3.45500i) q^{4} +(0.0551171 - 1.22728i) q^{5} +(2.05981 + 1.23068i) q^{6} +(1.45562 + 2.20517i) q^{7} +(4.14894 + 0.752922i) q^{8} +(-0.995974 + 0.0896393i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 3 q^{2} + 19 q^{3} + 9 q^{4} + 2 q^{5} - 3 q^{6} - 10 q^{7} - 9 q^{8} + 19 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/633\mathbb{Z}\right)^\times\).

\(n\)	\(212\)	\(424\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{24}{35}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32185	+	2.00252i	−0.934689	+	1.41599i	−0.0254701	+	0.999676i	\(0.508108\pi\)
							−0.909219	+	0.416318i	\(0.863320\pi\)
\(3\)	−0.0448648	−	0.998993i	−0.0259027	−	0.576769i
							\(5\)	0.0551171	−	1.22728i	0.0246491	−	0.548855i	−0.948236	−	0.317566i	\(-0.897134\pi\)
0.972885	−	0.231289i	\(-0.0742942\pi\)
\(7\)	1.45562	+	2.20517i	0.550173	+	0.833476i	0.998164	−	0.0605616i	\(-0.0192892\pi\)
							−0.447992	+	0.894038i	\(0.647861\pi\)
\(11\)	−0.0446006	−	0.0123090i	−0.0134476	−	0.00371130i	0.259307	−	0.965795i	\(-0.416506\pi\)
							−0.272754	+	0.962084i	\(0.587935\pi\)
\(13\)	−0.406555	+	0.425224i	−0.112758	+	0.117936i	−0.776765	−	0.629791i	\(-0.783140\pi\)
							0.664006	+	0.747727i	\(0.268855\pi\)
\(17\)	−0.314769	−	2.32372i	−0.0763428	−	0.563585i	−0.988211	−	0.153101i	\(-0.951074\pi\)
							0.911868	−	0.410484i	\(-0.134640\pi\)
\(19\)	0.561615	−	1.72847i	0.128843	−	0.396539i	−0.865738	−	0.500497i	\(-0.833151\pi\)
							0.994582	+	0.103958i	\(0.0331507\pi\)
\(23\)	−1.88198	−	5.79213i	−0.392419	−	1.20774i	−0.930953	−	0.365138i	\(-0.881022\pi\)
							0.538534	−	0.842604i	\(-0.318978\pi\)
\(29\)	−2.34855	+	2.45639i	−0.436114	+	0.456139i	−0.903926	−	0.427688i	\(-0.859328\pi\)
							0.467812	+	0.883828i	\(0.345042\pi\)
\(31\)	4.20581	−	2.02541i	0.755385	−	0.363774i	−0.0162265	−	0.999868i	\(-0.505165\pi\)
							0.771611	+	0.636094i	\(0.219451\pi\)
\(37\)	8.82329	−	2.43507i	1.45054	−	0.400324i	0.550025	−	0.835148i	\(-0.314618\pi\)
							0.900515	+	0.434824i	\(0.143190\pi\)
\(41\)	1.69342	−	3.14690i	0.264467	−	0.491463i	−0.713381	−	0.700776i	\(-0.752837\pi\)
							0.977849	+	0.209313i	\(0.0671229\pi\)
\(43\)	4.00319	−	1.92783i	0.610480	−	0.293992i	−0.102984	−	0.994683i	\(-0.532839\pi\)
							0.713465	+	0.700691i	\(0.247125\pi\)
\(47\)	4.08214	+	0.367399i	0.595441	+	0.0535907i	0.383261	−	0.923640i	\(-0.374801\pi\)
							0.212180	+	0.977231i	\(0.431944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	633.2.v.b.13.2	✓	456
211.65	even	35	inner	633.2.v.b.487.2	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
633.2.v.b.13.2	✓	456	1.1	even	1	trivial
633.2.v.b.487.2	yes	456	211.65	even	35	inner