Embedded newform 273.2.y.b.101.7

• Label: 273.2.y.b.101.7
• 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			101.7
Character	\(\chi\)	\(=\)	273.101
Dual form			273.2.y.b.173.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.884751 + 1.53243i) q^{2} +(-1.73022 - 0.0796811i) q^{3} +(-0.565569 - 0.979594i) q^{4} +(1.27320 - 0.735080i) q^{5} +(1.65292 - 2.58094i) q^{6} +(2.56619 + 0.643937i) q^{7} -1.53745 q^{8} +(2.98730 + 0.275731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 28 q^{4} - 12 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.884751	+	1.53243i	−0.625613	+	1.08359i	0.362808	+	0.931864i	\(0.381818\pi\)
							−0.988422	+	0.151731i	\(0.951515\pi\)
\(3\)	−1.73022	−	0.0796811i	−0.998941	−	0.0460039i
							\(5\)	1.27320	−	0.735080i	0.569390	−	0.328738i	−0.187516	−	0.982262i	\(-0.560044\pi\)
0.756906	+	0.653524i	\(0.226710\pi\)
\(7\)	2.56619	+	0.643937i	0.969930	+	0.243385i
							\(11\)	4.04788			1.22048			0.610241	−	0.792216i	\(-0.291073\pi\)
0.610241	+	0.792216i	\(0.291073\pi\)
\(13\)	−3.05379	+	1.91686i	−0.846970	+	0.531640i
							\(17\)	3.11363	+	5.39297i	0.755167	+	1.30799i	0.945291	+	0.326228i	\(0.105778\pi\)
−0.190124	+	0.981760i	\(0.560889\pi\)
\(19\)	−4.05190			−0.929569			−0.464784	−	0.885424i	\(-0.653868\pi\)
							−0.464784	+	0.885424i	\(0.653868\pi\)
\(23\)	0.566709	+	0.327190i	0.118167	+	0.0682238i	0.557919	−	0.829896i	\(-0.311600\pi\)
							−0.439751	+	0.898119i	\(0.644933\pi\)
\(29\)	0.812171	−	0.468907i	0.150816	−	0.0870738i	−0.422693	−	0.906273i	\(-0.638915\pi\)
							0.573509	+	0.819199i	\(0.305582\pi\)
\(31\)	−1.65710	+	2.87019i	−0.297625	+	0.515501i	−0.975592	−	0.219591i	\(-0.929528\pi\)
							0.677967	+	0.735092i	\(0.262861\pi\)
\(37\)	8.66769	+	5.00429i	1.42496	+	0.822701i	0.996717	−	0.0809635i	\(-0.0257997\pi\)
							0.428242	+	0.903664i	\(0.359133\pi\)
\(41\)	2.97494	−	1.71758i	0.464608	−	0.268241i	−0.249372	−	0.968408i	\(-0.580224\pi\)
							0.713980	+	0.700166i	\(0.246891\pi\)
\(43\)	3.47167	−	6.01311i	0.529424	−	0.916990i	−0.469987	−	0.882674i	\(-0.655741\pi\)
							0.999411	−	0.0343164i	\(-0.0109254\pi\)
\(47\)	−7.24756	+	4.18438i	−1.05717	+	0.610355i	−0.924647	−	0.380826i	\(-0.875640\pi\)
							−0.132519	+	0.991181i	\(0.542306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.y.b.101.7	✓	64
3.2	odd	2	inner	273.2.y.b.101.26	yes	64
7.5	odd	6		273.2.br.b.257.26	yes	64
13.4	even	6		273.2.br.b.17.7	yes	64
21.5	even	6		273.2.br.b.257.7	yes	64
39.17	odd	6		273.2.br.b.17.26	yes	64
91.82	odd	6	inner	273.2.y.b.173.26	yes	64
273.173	even	6	inner	273.2.y.b.173.7	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.7	✓	64	1.1	even	1	trivial
273.2.y.b.101.26	yes	64	3.2	odd	2	inner
273.2.y.b.173.7	yes	64	273.173	even	6	inner
273.2.y.b.173.26	yes	64	91.82	odd	6	inner
273.2.br.b.17.7	yes	64	13.4	even	6
273.2.br.b.17.26	yes	64	39.17	odd	6
273.2.br.b.257.7	yes	64	21.5	even	6
273.2.br.b.257.26	yes	64	7.5	odd	6