Embedded newform 273.2.bt.b.145.8

• Label: 273.2.bt.b.145.8
• 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			145.8
Character	\(\chi\)	\(=\)	273.145
Dual form			273.2.bt.b.241.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.884731 - 0.884731i) q^{2} +(0.866025 + 0.500000i) q^{3} +0.434503i q^{4} +(3.68041 - 0.986163i) q^{5} +(1.20856 - 0.323834i) q^{6} +(-2.53212 - 0.767050i) q^{7} +(2.15388 + 2.15388i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.884731	−	0.884731i	0.625599	−	0.625599i	−0.321359	−	0.946958i	\(-0.604139\pi\)
							0.946958	+	0.321359i	\(0.104139\pi\)
\(3\)	0.866025	+	0.500000i	0.500000	+	0.288675i
							\(5\)	3.68041	−	0.986163i	1.64593	−	0.441025i	0.687460	−	0.726222i	\(-0.258726\pi\)
0.958469	+	0.285197i	\(0.0920590\pi\)
\(7\)	−2.53212	−	0.767050i	−0.957052	−	0.289917i
							\(11\)	−4.32142	+	1.15792i	−1.30296	+	0.349127i	−0.842568	−	0.538590i	\(-0.818957\pi\)
−0.460390	+	0.887717i	\(0.652290\pi\)
\(13\)	−3.57886	+	0.437877i	−0.992598	+	0.121445i
							\(17\)	1.20355			0.291904			0.145952	−	0.989292i	\(-0.453375\pi\)
0.145952	+	0.989292i	\(0.453375\pi\)
\(19\)	1.38832	−	5.18127i	0.318502	−	1.18866i	−0.602183	−	0.798358i	\(-0.705702\pi\)
							0.920685	−	0.390307i	\(-0.127631\pi\)
\(23\)		−	3.38143i		−	0.705077i	−0.935797	−	0.352539i	\(-0.885319\pi\)
							0.935797	−	0.352539i	\(-0.114681\pi\)
\(29\)	−3.96435	−	6.86645i	−0.736161	−	1.27507i	−0.954212	−	0.299131i	\(-0.903303\pi\)
							0.218051	−	0.975937i	\(-0.430030\pi\)
\(31\)	−1.38355	+	5.16347i	−0.248492	+	0.927386i	0.723104	+	0.690740i	\(0.242715\pi\)
							−0.971596	+	0.236646i	\(0.923952\pi\)
\(37\)	4.94238	+	4.94238i	0.812523	+	0.812523i	0.985011	−	0.172489i	\(-0.0551808\pi\)
							−0.172489	+	0.985011i	\(0.555181\pi\)
\(41\)	−0.914977	+	3.41474i	−0.142895	+	0.533293i	0.856945	+	0.515408i	\(0.172360\pi\)
							−0.999840	+	0.0178844i	\(0.994307\pi\)
\(43\)	−8.74306	−	5.04781i	−1.33330	−	0.769784i	−0.347500	−	0.937680i	\(-0.612969\pi\)
							−0.985805	+	0.167896i	\(0.946303\pi\)
\(47\)	1.33663	+	4.98837i	0.194968	+	0.727629i	0.992275	+	0.124056i	\(0.0395902\pi\)
							−0.797308	+	0.603573i	\(0.793743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.145.8	✓	40
3.2	odd	2		819.2.et.d.145.3		40
7.3	odd	6		273.2.cg.b.262.8	yes	40
13.7	odd	12		273.2.cg.b.124.8	yes	40
21.17	even	6		819.2.gh.d.262.3		40
39.20	even	12		819.2.gh.d.397.3		40
91.59	even	12	inner	273.2.bt.b.241.8	yes	40
273.59	odd	12		819.2.et.d.514.3		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.145.8	✓	40	1.1	even	1	trivial
273.2.bt.b.241.8	yes	40	91.59	even	12	inner
273.2.cg.b.124.8	yes	40	13.7	odd	12
273.2.cg.b.262.8	yes	40	7.3	odd	6
819.2.et.d.145.3		40	3.2	odd	2
819.2.et.d.514.3		40	273.59	odd	12
819.2.gh.d.262.3		40	21.17	even	6
819.2.gh.d.397.3		40	39.20	even	12