Embedded newform 273.2.bt.b.145.4

• Label: 273.2.bt.b.145.4
• Level: $273$
• Weight: $2$
• Character: 273.145
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.bt (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			145.4
Character	\(\chi\)	\(=\)	273.145
Dual form			273.2.bt.b.241.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.507981 + 0.507981i) q^{2} +(0.866025 + 0.500000i) q^{3} +1.48391i q^{4} +(1.10096 - 0.295002i) q^{5} +(-0.693915 + 0.185934i) q^{6} +(0.718657 + 2.54628i) q^{7} +(-1.76976 - 1.76976i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.507981	+	0.507981i	−0.359197	+	0.359197i	−0.863517	−	0.504320i	\(-0.831743\pi\)
							0.504320	+	0.863517i	\(0.331743\pi\)
\(3\)	0.866025	+	0.500000i	0.500000	+	0.288675i
							\(5\)	1.10096	−	0.295002i	0.492366	−	0.131929i	−0.00408873	−	0.999992i	\(-0.501301\pi\)
0.496455	+	0.868063i	\(0.334635\pi\)
\(7\)	0.718657	+	2.54628i	0.271627	+	0.962403i
							\(11\)	1.72777	−	0.462953i	0.520941	−	0.139586i	0.0112392	−	0.999937i	\(-0.496422\pi\)
0.509702	+	0.860351i	\(0.329756\pi\)
\(13\)	−3.60058	−	0.189346i	−0.998620	−	0.0525152i
							\(17\)	3.69163			0.895351			0.447675	−	0.894196i	\(-0.352252\pi\)
0.447675	+	0.894196i	\(0.352252\pi\)
\(19\)	0.160984	−	0.600801i	0.0369323	−	0.137833i	−0.944998	−	0.327076i	\(-0.893937\pi\)
							0.981930	+	0.189243i	\(0.0606033\pi\)
\(23\)			2.26345i			0.471962i	0.971758	+	0.235981i	\(0.0758303\pi\)
							−0.971758	+	0.235981i	\(0.924170\pi\)
\(29\)	−0.0743668	−	0.128807i	−0.0138096	−	0.0239189i	0.859038	−	0.511912i	\(-0.171063\pi\)
							−0.872848	+	0.487993i	\(0.837729\pi\)
\(31\)	1.24501	−	4.64644i	0.223610	−	0.834525i	−0.759346	−	0.650687i	\(-0.774481\pi\)
							0.982956	−	0.183838i	\(-0.0588523\pi\)
\(37\)	3.17333	+	3.17333i	0.521693	+	0.521693i	0.918082	−	0.396390i	\(-0.129737\pi\)
							−0.396390	+	0.918082i	\(0.629737\pi\)
\(41\)	1.12389	−	4.19440i	0.175522	−	0.655056i	−0.820941	−	0.571014i	\(-0.806550\pi\)
							0.996462	−	0.0840421i	\(-0.0267830\pi\)
\(43\)	6.81492	+	3.93460i	1.03927	+	0.600021i	0.919625	−	0.392797i	\(-0.128492\pi\)
							0.119641	+	0.992817i	\(0.461826\pi\)
\(47\)	−0.0779824	−	0.291034i	−0.0113749	−	0.0424517i	0.960005	−	0.279982i	\(-0.0903286\pi\)
							−0.971380	+	0.237531i	\(0.923662\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.145.4	✓	40
3.2	odd	2		819.2.et.d.145.7		40
7.3	odd	6		273.2.cg.b.262.4	yes	40
13.7	odd	12		273.2.cg.b.124.4	yes	40
21.17	even	6		819.2.gh.d.262.7		40
39.20	even	12		819.2.gh.d.397.7		40
91.59	even	12	inner	273.2.bt.b.241.4	yes	40
273.59	odd	12		819.2.et.d.514.7		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.145.4	✓	40	1.1	even	1	trivial
273.2.bt.b.241.4	yes	40	91.59	even	12	inner
273.2.cg.b.124.4	yes	40	13.7	odd	12
273.2.cg.b.262.4	yes	40	7.3	odd	6
819.2.et.d.145.7		40	3.2	odd	2
819.2.et.d.514.7		40	273.59	odd	12
819.2.gh.d.262.7		40	21.17	even	6
819.2.gh.d.397.7		40	39.20	even	12