Embedded newform 273.2.bn.c.146.12

• Label: 273.2.bn.c.146.12
• Level: $273$
• Weight: $2$
• Character: 273.146
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.bn (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			146.12
Character	\(\chi\)	\(=\)	273.146
Dual form			273.2.bn.c.230.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.828363 + 0.478255i) q^{2} +(1.28843 + 1.15756i) q^{3} +(-0.542544 + 0.939713i) q^{4} -3.79208 q^{5} +(-1.62090 - 0.342680i) q^{6} +(2.36163 + 1.19277i) q^{7} -2.95092i q^{8} +(0.320110 + 2.98287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 4 q^{7} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.828363	+	0.478255i	−0.585741	+	0.338178i	−0.763412	−	0.645912i	\(-0.776477\pi\)
							0.177671	+	0.984090i	\(0.443144\pi\)
\(3\)	1.28843	+	1.15756i	0.743876	+	0.668318i
							\(5\)	−3.79208			−1.69587			−0.847935	−	0.530101i	\(-0.822154\pi\)
−0.847935	+	0.530101i	\(0.822154\pi\)
\(7\)	2.36163	+	1.19277i	0.892612	+	0.450825i
							\(11\)	−0.918580	+	0.530343i	−0.276962	+	0.159904i	−0.632047	−	0.774930i	\(-0.717785\pi\)
0.355085	+	0.934834i	\(0.384452\pi\)
\(13\)	−3.56834	−	0.516671i	−0.989679	−	0.143299i
							\(17\)	−2.87769	+	4.98430i	−0.697942	+	1.20887i	0.271237	+	0.962513i	\(0.412567\pi\)
−0.969179	+	0.246358i	\(0.920766\pi\)
\(19\)	−5.33218	−	3.07853i	−1.22329	−	0.706264i	−0.257668	−	0.966233i	\(-0.582954\pi\)
							−0.965617	+	0.259969i	\(0.916288\pi\)
\(23\)	4.94616	−	2.85567i	1.03135	−	0.595448i	0.113976	−	0.993484i	\(-0.463641\pi\)
							0.917370	+	0.398036i	\(0.130308\pi\)
\(29\)	−0.971709	+	0.561016i	−0.180442	+	0.104178i	−0.587500	−	0.809224i	\(-0.699888\pi\)
							0.407058	+	0.913402i	\(0.366555\pi\)
\(31\)			2.62050i			0.470656i	0.971916	+	0.235328i	\(0.0756165\pi\)
							−0.971916	+	0.235328i	\(0.924384\pi\)
\(37\)	1.05765	+	1.83191i	0.173877	+	0.301164i	0.939772	−	0.341802i	\(-0.111037\pi\)
							−0.765895	+	0.642966i	\(0.777704\pi\)
\(41\)	2.79696	+	4.84447i	0.436811	+	0.756580i	0.997442	−	0.0714867i	\(-0.0227743\pi\)
							−0.560630	+	0.828066i	\(0.689441\pi\)
\(43\)	−3.65756	+	6.33508i	−0.557772	+	0.966090i	0.439910	+	0.898042i	\(0.355010\pi\)
							−0.997682	+	0.0680480i	\(0.978323\pi\)
\(47\)	7.08267			1.03311			0.516557	−	0.856253i	\(-0.327214\pi\)
							0.516557	+	0.856253i	\(0.327214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bn.c.146.12	yes	64
3.2	odd	2	inner	273.2.bn.c.146.21	yes	64
7.6	odd	2	inner	273.2.bn.c.146.11	✓	64
13.9	even	3	inner	273.2.bn.c.230.22	yes	64
21.20	even	2	inner	273.2.bn.c.146.22	yes	64
39.35	odd	6	inner	273.2.bn.c.230.11	yes	64
91.48	odd	6	inner	273.2.bn.c.230.21	yes	64
273.230	even	6	inner	273.2.bn.c.230.12	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bn.c.146.11	✓	64	7.6	odd	2	inner
273.2.bn.c.146.12	yes	64	1.1	even	1	trivial
273.2.bn.c.146.21	yes	64	3.2	odd	2	inner
273.2.bn.c.146.22	yes	64	21.20	even	2	inner
273.2.bn.c.230.11	yes	64	39.35	odd	6	inner
273.2.bn.c.230.12	yes	64	273.230	even	6	inner
273.2.bn.c.230.21	yes	64	91.48	odd	6	inner
273.2.bn.c.230.22	yes	64	13.9	even	3	inner