Embedded newform 273.2.bt.a.145.6

• Label: 273.2.bt.a.145.6
• Level: $273$
• Weight: $2$
• Character: 273.145
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $36$
• 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:	\(36\)
Relative dimension:	\(9\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			145.6
Character	\(\chi\)	\(=\)	273.145
Dual form			273.2.bt.a.241.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.430820 - 0.430820i) q^{2} +(-0.866025 - 0.500000i) q^{3} +1.62879i q^{4} +(1.97745 - 0.529856i) q^{5} +(-0.588511 + 0.157691i) q^{6} +(-1.23433 + 2.34018i) q^{7} +(1.56335 + 1.56335i) q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{7} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.430820	−	0.430820i	0.304636	−	0.304636i	−0.538189	−	0.842824i	\(-0.680891\pi\)
							0.842824	+	0.538189i	\(0.180891\pi\)
\(3\)	−0.866025	−	0.500000i	−0.500000	−	0.288675i
							\(5\)	1.97745	−	0.529856i	0.884342	−	0.236959i	0.212062	−	0.977256i	\(-0.431982\pi\)
0.672279	+	0.740297i	\(0.265315\pi\)
\(7\)	−1.23433	+	2.34018i	−0.466534	+	0.884503i
							\(11\)	−0.0981532	+	0.0263001i	−0.0295943	+	0.00792977i	−0.273586	−	0.961848i	\(-0.588210\pi\)
0.243992	+	0.969777i	\(0.421543\pi\)
\(13\)	3.09469	+	1.85010i	0.858313	+	0.513126i
							\(17\)	3.63007			0.880422			0.440211	−	0.897894i	\(-0.354904\pi\)
0.440211	+	0.897894i	\(0.354904\pi\)
\(19\)	0.374251	−	1.39672i	0.0858590	−	0.320430i	−0.909616	−	0.415449i	\(-0.863624\pi\)
							0.995475	+	0.0950189i	\(0.0302911\pi\)
\(23\)		−	4.80660i		−	1.00224i	−0.865376	−	0.501122i	\(-0.832921\pi\)
							0.865376	−	0.501122i	\(-0.167079\pi\)
\(29\)	3.79375	+	6.57097i	0.704482	+	1.22020i	0.966878	+	0.255238i	\(0.0821538\pi\)
							−0.262397	+	0.964960i	\(0.584513\pi\)
\(31\)	1.73499	−	6.47506i	0.311613	−	1.16296i	−0.615489	−	0.788146i	\(-0.711041\pi\)
							0.927102	−	0.374810i	\(-0.122292\pi\)
\(37\)	−2.15422	−	2.15422i	−0.354152	−	0.354152i	0.507500	−	0.861652i	\(-0.330570\pi\)
							−0.861652	+	0.507500i	\(0.830570\pi\)
\(41\)	−0.872020	+	3.25442i	−0.136187	+	0.508256i	0.863804	+	0.503829i	\(0.168076\pi\)
							−0.999990	+	0.00442675i	\(0.998591\pi\)
\(43\)	−7.21579	−	4.16604i	−1.10040	−	0.635315i	−0.164073	−	0.986448i	\(-0.552463\pi\)
							−0.936326	+	0.351133i	\(0.885796\pi\)
\(47\)	−0.529965	−	1.97786i	−0.0773034	−	0.288500i	0.916442	−	0.400167i	\(-0.131048\pi\)
							−0.993746	+	0.111667i	\(0.964381\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.a.145.6	✓	36
3.2	odd	2		819.2.et.c.145.4		36
7.3	odd	6		273.2.cg.a.262.6	yes	36
13.7	odd	12		273.2.cg.a.124.6	yes	36
21.17	even	6		819.2.gh.c.262.4		36
39.20	even	12		819.2.gh.c.397.4		36
91.59	even	12	inner	273.2.bt.a.241.6	yes	36
273.59	odd	12		819.2.et.c.514.4		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.a.145.6	✓	36	1.1	even	1	trivial
273.2.bt.a.241.6	yes	36	91.59	even	12	inner
273.2.cg.a.124.6	yes	36	13.7	odd	12
273.2.cg.a.262.6	yes	36	7.3	odd	6
819.2.et.c.145.4		36	3.2	odd	2
819.2.et.c.514.4		36	273.59	odd	12
819.2.gh.c.262.4		36	21.17	even	6
819.2.gh.c.397.4		36	39.20	even	12