Embedded newform 633.2.k.a.23.18

• Label: 633.2.k.a.23.18
• Level: $633$
• Weight: $2$
• Character: 633.23
• Analytic conductor: $5.055$
• Analytic rank: $0$
• Dimension: $272$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 633 = 3 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	633.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.05453044795\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(68\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			23.18
Character	\(\chi\)	\(=\)	633.23
Dual form			633.2.k.a.578.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.504007 + 1.55117i) q^{2} +(0.690180 - 1.58860i) q^{3} +(-0.534085 - 0.388035i) q^{4} +(0.00566708 - 0.00780006i) q^{5} +(2.11634 + 1.87126i) q^{6} +(-3.86122 + 1.25459i) q^{7} +(-1.76792 + 1.28447i) q^{8} +(-2.04730 - 2.19284i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 10 q^{3} - 74 q^{4} + 9 q^{6} - 30 q^{7} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(212\)	\(424\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.504007	+	1.55117i	−0.356387	+	1.09685i	0.598814	+	0.800888i	\(0.295639\pi\)
							−0.955201	+	0.295958i	\(0.904361\pi\)
\(3\)	0.690180	−	1.58860i	0.398476	−	0.917179i
							\(5\)	0.00566708	−	0.00780006i	0.00253439	−	0.00348829i	−0.807748	−	0.589528i	\(-0.799314\pi\)
0.810282	+	0.586040i	\(0.199314\pi\)
\(7\)	−3.86122	+	1.25459i	−1.45941	+	0.474190i	−0.927888	−	0.372858i	\(-0.878378\pi\)
							−0.531517	+	0.847048i	\(0.678378\pi\)
\(11\)	2.34483	−	3.22738i	0.706992	−	0.973090i	−0.292865	−	0.956154i	\(-0.594609\pi\)
							0.999857	−	0.0169366i	\(-0.00539134\pi\)
\(13\)	−1.74363	−	5.36634i	−0.483596	−	1.48835i	−0.834005	−	0.551757i	\(-0.813957\pi\)
							0.350409	−	0.936597i	\(-0.386043\pi\)
\(17\)	−1.66140	−	5.11326i	−0.402949	−	1.24015i	−0.922596	−	0.385768i	\(-0.873937\pi\)
							0.519647	−	0.854381i	\(-0.326063\pi\)
\(19\)	2.48071	+	7.63485i	0.569114	+	1.75155i	0.655399	+	0.755283i	\(0.272501\pi\)
							−0.0862842	+	0.996271i	\(0.527499\pi\)
\(23\)	1.92118	−	5.91277i	0.400593	−	1.23290i	−0.523927	−	0.851763i	\(-0.675533\pi\)
							0.924520	−	0.381135i	\(-0.124467\pi\)
\(29\)	−1.84851	−	5.68913i	−0.343260	−	1.05644i	−0.962509	−	0.271250i	\(-0.912563\pi\)
							0.619249	−	0.785194i	\(-0.287437\pi\)
\(31\)		−	7.08491i		−	1.27249i	−0.771488	−	0.636244i	\(-0.780487\pi\)
							0.771488	−	0.636244i	\(-0.219513\pi\)
\(37\)	−1.58764	+	1.15349i	−0.261006	+	0.189632i	−0.710590	−	0.703606i	\(-0.751572\pi\)
							0.449584	+	0.893238i	\(0.351572\pi\)
\(41\)	−0.878794	−	0.638481i	−0.137245	−	0.0997140i	0.517045	−	0.855958i	\(-0.327032\pi\)
							−0.654289	+	0.756244i	\(0.727032\pi\)
\(43\)	−8.17961			−1.24738			−0.623689	−	0.781672i	\(-0.714367\pi\)
							−0.623689	+	0.781672i	\(0.714367\pi\)
\(47\)	1.84336	−	0.598945i	0.268882	−	0.0873652i	−0.171473	−	0.985189i	\(-0.554853\pi\)
							0.440355	+	0.897824i	\(0.354853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	633.2.k.a.23.18	✓	272
3.2	odd	2	inner	633.2.k.a.23.51	yes	272
211.156	odd	10	inner	633.2.k.a.578.51	yes	272
633.578	even	10	inner	633.2.k.a.578.18	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
633.2.k.a.23.18	✓	272	1.1	even	1	trivial
633.2.k.a.23.51	yes	272	3.2	odd	2	inner
633.2.k.a.578.18	yes	272	633.578	even	10	inner
633.2.k.a.578.51	yes	272	211.156	odd	10	inner