Embedded newform 273.2.bt.b.145.1

• Label: 273.2.bt.b.145.1
• 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			145.1
Character	\(\chi\)	\(=\)	273.145
Dual form			273.2.bt.b.241.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96855 + 1.96855i) q^{2} +(0.866025 + 0.500000i) q^{3} -5.75037i q^{4} +(-1.75415 + 0.470023i) q^{5} +(-2.68909 + 0.720539i) q^{6} +(2.27503 + 1.35065i) q^{7} +(7.38279 + 7.38279i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96855	+	1.96855i	−1.39197	+	1.39197i	−0.571080	+	0.820894i	\(0.693475\pi\)
							−0.820894	+	0.571080i	\(0.806525\pi\)
\(3\)	0.866025	+	0.500000i	0.500000	+	0.288675i
							\(5\)	−1.75415	+	0.470023i	−0.784480	+	0.210201i	−0.628759	−	0.777600i	\(-0.716437\pi\)
−0.155721	+	0.987801i	\(0.549770\pi\)
\(7\)	2.27503	+	1.35065i	0.859880	+	0.510496i
							\(11\)	−4.28308	+	1.14765i	−1.29140	+	0.346029i	−0.838191	−	0.545377i	\(-0.816386\pi\)
−0.453206	+	0.891406i	\(0.649720\pi\)
\(13\)	−2.12880	+	2.91002i	−0.590424	+	0.807093i
							\(17\)	0.0643569			0.0156088			0.00780442	−	0.999970i	\(-0.497516\pi\)
0.00780442	+	0.999970i	\(0.497516\pi\)
\(19\)	−0.788412	+	2.94240i	−0.180874	+	0.675032i	0.814602	+	0.580020i	\(0.196955\pi\)
							−0.995476	+	0.0950114i	\(0.969711\pi\)
\(23\)		−	3.21686i		−	0.670761i	−0.942083	−	0.335380i	\(-0.891135\pi\)
							0.942083	−	0.335380i	\(-0.108865\pi\)
\(29\)	2.41489	+	4.18271i	0.448433	+	0.776709i	0.998284	−	0.0585536i	\(-0.0186488\pi\)
							−0.549851	+	0.835263i	\(0.685316\pi\)
\(31\)	−1.34743	+	5.02866i	−0.242005	+	0.903174i	0.732860	+	0.680379i	\(0.238185\pi\)
							−0.974865	+	0.222795i	\(0.928482\pi\)
\(37\)	−3.08432	−	3.08432i	−0.507059	−	0.507059i	0.406564	−	0.913622i	\(-0.366727\pi\)
							−0.913622	+	0.406564i	\(0.866727\pi\)
\(41\)	2.12512	−	7.93105i	0.331888	−	1.23862i	−0.575317	−	0.817931i	\(-0.695121\pi\)
							0.907204	−	0.420691i	\(-0.138212\pi\)
\(43\)	−3.22447	−	1.86165i	−0.491727	−	0.283899i	0.233564	−	0.972341i	\(-0.424961\pi\)
							−0.725291	+	0.688443i	\(0.758295\pi\)
\(47\)	1.08995	+	4.06776i	0.158986	+	0.593343i	0.998731	+	0.0503625i	\(0.0160377\pi\)
							−0.839745	+	0.542981i	\(0.817296\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.145.1	✓	40
3.2	odd	2		819.2.et.d.145.10		40
7.3	odd	6		273.2.cg.b.262.1	yes	40
13.7	odd	12		273.2.cg.b.124.1	yes	40
21.17	even	6		819.2.gh.d.262.10		40
39.20	even	12		819.2.gh.d.397.10		40
91.59	even	12	inner	273.2.bt.b.241.1	yes	40
273.59	odd	12		819.2.et.d.514.10		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.145.1	✓	40	1.1	even	1	trivial
273.2.bt.b.241.1	yes	40	91.59	even	12	inner
273.2.cg.b.124.1	yes	40	13.7	odd	12
273.2.cg.b.262.1	yes	40	7.3	odd	6
819.2.et.d.145.10		40	3.2	odd	2
819.2.et.d.514.10		40	273.59	odd	12
819.2.gh.d.262.10		40	21.17	even	6
819.2.gh.d.397.10		40	39.20	even	12