Embedded newform 273.2.y.b.101.3

• Label: 273.2.y.b.101.3
• Level: $273$
• Weight: $2$
• Character: 273.101
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.3
Character	\(\chi\)	\(=\)	273.101
Dual form			273.2.y.b.173.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18406 + 2.05084i) q^{2} +(-0.641138 - 1.60902i) q^{3} +(-1.80398 - 3.12458i) q^{4} +(0.0190083 - 0.0109745i) q^{5} +(4.05899 + 0.590294i) q^{6} +(0.134389 + 2.64234i) q^{7} +3.80781 q^{8} +(-2.17788 + 2.06321i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 28 q^{4} - 12 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18406	+	2.05084i	−0.837254	+	1.45017i	0.0549283	+	0.998490i	\(0.482507\pi\)
							−0.892182	+	0.451676i	\(0.850826\pi\)
\(3\)	−0.641138	−	1.60902i	−0.370161	−	0.928968i
							\(5\)	0.0190083	−	0.0109745i	0.00850078	−	0.00490793i	−0.495744	−	0.868469i	\(-0.665104\pi\)
0.504244	+	0.863561i	\(0.331771\pi\)
\(7\)	0.134389	+	2.64234i	0.0507943	+	0.998709i
							\(11\)	−1.80919			−0.545490			−0.272745	−	0.962086i	\(-0.587932\pi\)
−0.272745	+	0.962086i	\(0.587932\pi\)
\(13\)	−2.16898	−	2.88020i	−0.601566	−	0.798823i
							\(17\)	−3.23588	−	5.60471i	−0.784817	−	1.35934i	−0.929108	−	0.369807i	\(-0.879424\pi\)
0.144292	−	0.989535i	\(-0.453910\pi\)
\(19\)	−1.75189			−0.401910			−0.200955	−	0.979600i	\(-0.564405\pi\)
							−0.200955	+	0.979600i	\(0.564405\pi\)
\(23\)	−4.93511	−	2.84929i	−1.02904	−	0.594117i	−0.112331	−	0.993671i	\(-0.535832\pi\)
							−0.916710	+	0.399554i	\(0.869165\pi\)
\(29\)	−6.52098	+	3.76489i	−1.21092	+	0.699123i	−0.962959	−	0.269648i	\(-0.913093\pi\)
							−0.247957	+	0.968771i	\(0.579759\pi\)
\(31\)	−0.347389	+	0.601695i	−0.0623929	+	0.108068i	−0.895535	−	0.444992i	\(-0.853207\pi\)
							0.833142	+	0.553060i	\(0.186540\pi\)
\(37\)	−3.24468	−	1.87332i	−0.533422	−	0.307971i	0.208987	−	0.977918i	\(-0.432983\pi\)
							−0.742409	+	0.669947i	\(0.766317\pi\)
\(41\)	−2.16550	+	1.25025i	−0.338194	+	0.195256i	−0.659473	−	0.751728i	\(-0.729221\pi\)
							0.321279	+	0.946985i	\(0.395887\pi\)
\(43\)	3.54180	−	6.13458i	0.540119	−	0.935514i	−0.458777	−	0.888551i	\(-0.651712\pi\)
							0.998897	−	0.0469630i	\(-0.0149543\pi\)
\(47\)	6.66373	−	3.84731i	0.972005	−	0.561188i	0.0721583	−	0.997393i	\(-0.477011\pi\)
							0.899847	+	0.436206i	\(0.143678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.y.b.101.3	✓	64
3.2	odd	2	inner	273.2.y.b.101.30	yes	64
7.5	odd	6		273.2.br.b.257.30	yes	64
13.4	even	6		273.2.br.b.17.3	yes	64
21.5	even	6		273.2.br.b.257.3	yes	64
39.17	odd	6		273.2.br.b.17.30	yes	64
91.82	odd	6	inner	273.2.y.b.173.30	yes	64
273.173	even	6	inner	273.2.y.b.173.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.3	✓	64	1.1	even	1	trivial
273.2.y.b.101.30	yes	64	3.2	odd	2	inner
273.2.y.b.173.3	yes	64	273.173	even	6	inner
273.2.y.b.173.30	yes	64	91.82	odd	6	inner
273.2.br.b.17.3	yes	64	13.4	even	6
273.2.br.b.17.30	yes	64	39.17	odd	6
273.2.br.b.257.3	yes	64	21.5	even	6
273.2.br.b.257.30	yes	64	7.5	odd	6