Embedded newform 273.2.bt.b.136.2

• Label: 273.2.bt.b.136.2
• Level: $273$
• Weight: $2$
• Character: 273.136
• 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			136.2
Character	\(\chi\)	\(=\)	273.136
Dual form			273.2.bt.b.271.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59737 + 1.59737i) q^{2} +(-0.866025 + 0.500000i) q^{3} -3.10321i q^{4} +(0.609466 - 2.27456i) q^{5} +(0.584679 - 2.18205i) q^{6} +(1.40839 + 2.23974i) q^{7} +(1.76223 + 1.76223i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59737	+	1.59737i	−1.12951	+	1.12951i	−0.139258	+	0.990256i	\(0.544472\pi\)
							−0.990256	+	0.139258i	\(0.955528\pi\)
\(3\)	−0.866025	+	0.500000i	−0.500000	+	0.288675i
							\(5\)	0.609466	−	2.27456i	0.272561	−	1.01721i	−0.684897	−	0.728640i	\(-0.740153\pi\)
0.957458	−	0.288573i	\(-0.0931807\pi\)
\(7\)	1.40839	+	2.23974i	0.532321	+	0.846542i
							\(11\)	−1.30677	+	4.87695i	−0.394007	+	1.47045i	0.429457	+	0.903087i	\(0.358705\pi\)
−0.823465	+	0.567368i	\(0.807962\pi\)
\(13\)	−1.62322	+	3.21950i	−0.450201	+	0.892927i
							\(17\)	−5.28773			−1.28246			−0.641232	−	0.767347i	\(-0.721576\pi\)
−0.641232	+	0.767347i	\(0.721576\pi\)
\(19\)	4.08106	−	1.09352i	0.936258	−	0.250870i	0.241737	−	0.970342i	\(-0.422283\pi\)
							0.694521	+	0.719472i	\(0.255616\pi\)
\(23\)			3.98321i			0.830557i	0.909694	+	0.415278i	\(0.136316\pi\)
							−0.909694	+	0.415278i	\(0.863684\pi\)
\(29\)	0.565030	−	0.978660i	0.104923	−	0.181733i	−0.808784	−	0.588106i	\(-0.799874\pi\)
							0.913707	+	0.406374i	\(0.133207\pi\)
\(31\)	−5.23228	+	1.40198i	−0.939745	+	0.251804i	−0.696005	−	0.718037i	\(-0.745041\pi\)
							−0.243740	+	0.969841i	\(0.578374\pi\)
\(37\)	3.75894	+	3.75894i	0.617965	+	0.617965i	0.945009	−	0.327044i	\(-0.106052\pi\)
							−0.327044	+	0.945009i	\(0.606052\pi\)
\(41\)	−5.61474	+	1.50446i	−0.876875	+	0.234958i	−0.669058	−	0.743210i	\(-0.733302\pi\)
							−0.207817	+	0.978168i	\(0.566636\pi\)
\(43\)	−9.65143	+	5.57225i	−1.47183	+	0.849761i	−0.999499	−	0.0316610i	\(-0.989920\pi\)
							−0.472330	+	0.881422i	\(0.656587\pi\)
\(47\)	7.28399	+	1.95174i	1.06248	+	0.284691i	0.747400	−	0.664375i	\(-0.231302\pi\)
							0.315080	+	0.949065i	\(0.397969\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.136.2	✓	40
3.2	odd	2		819.2.et.d.136.9		40
7.5	odd	6		273.2.cg.b.19.9	yes	40
13.11	odd	12		273.2.cg.b.115.9	yes	40
21.5	even	6		819.2.gh.d.19.2		40
39.11	even	12		819.2.gh.d.388.2		40
91.89	even	12	inner	273.2.bt.b.271.2	yes	40
273.89	odd	12		819.2.et.d.271.9		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.136.2	✓	40	1.1	even	1	trivial
273.2.bt.b.271.2	yes	40	91.89	even	12	inner
273.2.cg.b.19.9	yes	40	7.5	odd	6
273.2.cg.b.115.9	yes	40	13.11	odd	12
819.2.et.d.136.9		40	3.2	odd	2
819.2.et.d.271.9		40	273.89	odd	12
819.2.gh.d.19.2		40	21.5	even	6
819.2.gh.d.388.2		40	39.11	even	12