Embedded newform 273.2.bt.b.136.4

• Label: 273.2.bt.b.136.4
• 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			136.4
Character	\(\chi\)	\(=\)	273.136
Dual form			273.2.bt.b.271.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.783932 + 0.783932i) q^{2} +(-0.866025 + 0.500000i) q^{3} +0.770902i q^{4} +(-0.994915 + 3.71307i) q^{5} +(0.286939 - 1.07087i) q^{6} +(0.0389254 + 2.64546i) q^{7} +(-2.17220 - 2.17220i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.783932	+	0.783932i	−0.554323	+	0.554323i	−0.927686	−	0.373362i	\(-0.878205\pi\)
							0.373362	+	0.927686i	\(0.378205\pi\)
\(3\)	−0.866025	+	0.500000i	−0.500000	+	0.288675i
							\(5\)	−0.994915	+	3.71307i	−0.444940	+	1.66054i	0.271156	+	0.962535i	\(0.412594\pi\)
−0.716096	+	0.698002i	\(0.754073\pi\)
\(7\)	0.0389254	+	2.64546i	0.0147124	+	0.999892i
							\(11\)	1.48752	−	5.55150i	0.448504	−	1.67384i	−0.258010	−	0.966142i	\(-0.583067\pi\)
0.706514	−	0.707699i	\(-0.250267\pi\)
\(13\)	2.56640	+	2.53251i	0.711791	+	0.702391i
							\(17\)	−4.62101			−1.12076			−0.560379	−	0.828236i	\(-0.689345\pi\)
−0.560379	+	0.828236i	\(0.689345\pi\)
\(19\)	1.50363	−	0.402897i	0.344957	−	0.0924310i	−0.0821808	−	0.996617i	\(-0.526189\pi\)
							0.427138	+	0.904186i	\(0.359522\pi\)
\(23\)			4.31084i			0.898871i	0.893313	+	0.449436i	\(0.148375\pi\)
							−0.893313	+	0.449436i	\(0.851625\pi\)
\(29\)	−2.38522	+	4.13133i	−0.442925	+	0.767169i	−0.997905	−	0.0646949i	\(-0.979393\pi\)
							0.554980	+	0.831864i	\(0.312726\pi\)
\(31\)	3.70936	−	0.993920i	0.666220	−	0.178513i	0.0901688	−	0.995926i	\(-0.471259\pi\)
							0.576052	+	0.817413i	\(0.304593\pi\)
\(37\)	−1.78626	−	1.78626i	−0.293659	−	0.293659i	0.544865	−	0.838524i	\(-0.316581\pi\)
							−0.838524	+	0.544865i	\(0.816581\pi\)
\(41\)	4.40726	−	1.18092i	0.688299	−	0.184429i	0.102315	−	0.994752i	\(-0.467375\pi\)
							0.585984	+	0.810323i	\(0.300708\pi\)
\(43\)	−2.65948	+	1.53545i	−0.405567	+	0.234154i	−0.688883	−	0.724872i	\(-0.741899\pi\)
							0.283316	+	0.959027i	\(0.408565\pi\)
\(47\)	7.89374	+	2.11512i	1.15142	+	0.308522i	0.783534	−	0.621349i	\(-0.213415\pi\)
							0.367886	+	0.929871i	\(0.380082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.136.4	✓	40
3.2	odd	2		819.2.et.d.136.7		40
7.5	odd	6		273.2.cg.b.19.7	yes	40
13.11	odd	12		273.2.cg.b.115.7	yes	40
21.5	even	6		819.2.gh.d.19.4		40
39.11	even	12		819.2.gh.d.388.4		40
91.89	even	12	inner	273.2.bt.b.271.4	yes	40
273.89	odd	12		819.2.et.d.271.7		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.136.4	✓	40	1.1	even	1	trivial
273.2.bt.b.271.4	yes	40	91.89	even	12	inner
273.2.cg.b.19.7	yes	40	7.5	odd	6
273.2.cg.b.115.7	yes	40	13.11	odd	12
819.2.et.d.136.7		40	3.2	odd	2
819.2.et.d.271.7		40	273.89	odd	12
819.2.gh.d.19.4		40	21.5	even	6
819.2.gh.d.388.4		40	39.11	even	12