Embedded newform 273.2.bn.c.230.7

• Label: 273.2.bn.c.230.7
• Level: $273$
• Weight: $2$
• Character: 273.230
• 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			230.7
Character	\(\chi\)	\(=\)	273.230
Dual form			273.2.bn.c.146.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.43804 - 0.830256i) q^{2} +(-1.23680 - 1.21257i) q^{3} +(0.378649 + 0.655839i) q^{4} -1.88889 q^{5} +(0.771823 + 2.77059i) q^{6} +(0.896028 + 2.48940i) q^{7} +2.06352i q^{8} +(0.0593312 + 2.99941i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 4 q^{7} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.43804	−	0.830256i	−1.01685	−	0.587079i	−0.103661	−	0.994613i	\(-0.533056\pi\)
							−0.913190	+	0.407533i	\(0.866389\pi\)
\(3\)	−1.23680	−	1.21257i	−0.714065	−	0.700080i
							\(5\)	−1.88889			−0.844738			−0.422369	−	0.906424i	\(-0.638801\pi\)
−0.422369	+	0.906424i	\(0.638801\pi\)
\(7\)	0.896028	+	2.48940i	0.338667	+	0.940906i
							\(11\)	−1.52869	−	0.882590i	−0.460918	−	0.266111i	0.251512	−	0.967854i	\(-0.419072\pi\)
−0.712430	+	0.701743i	\(0.752405\pi\)
\(13\)	3.51996	−	0.780953i	0.976261	−	0.216597i
							\(17\)	−1.02971	−	1.78351i	−0.249741	−	0.432565i	0.713713	−	0.700439i	\(-0.247012\pi\)
−0.963454	+	0.267874i	\(0.913679\pi\)
\(19\)	−1.72776	+	0.997525i	−0.396376	+	0.228848i	−0.684919	−	0.728619i	\(-0.740163\pi\)
							0.288543	+	0.957467i	\(0.406829\pi\)
\(23\)	4.43873	+	2.56270i	0.925538	+	0.534360i	0.885398	−	0.464834i	\(-0.153886\pi\)
							0.0401406	+	0.999194i	\(0.487219\pi\)
\(29\)	7.27489	+	4.20016i	1.35091	+	0.779950i	0.988377	−	0.152020i	\(-0.0485778\pi\)
							0.362536	+	0.931970i	\(0.381911\pi\)
\(31\)			8.13951i			1.46190i	0.682431	+	0.730950i	\(0.260923\pi\)
							−0.682431	+	0.730950i	\(0.739077\pi\)
\(37\)	2.24827	−	3.89412i	0.369614	−	0.640190i	−0.619891	−	0.784688i	\(-0.712823\pi\)
							0.989505	+	0.144498i	\(0.0461567\pi\)
\(41\)	−4.68779	+	8.11949i	−0.732109	+	1.26805i	0.223871	+	0.974619i	\(0.428131\pi\)
							−0.955980	+	0.293432i	\(0.905203\pi\)
\(43\)	2.05277	+	3.55550i	0.313044	+	0.542209i	0.979020	−	0.203765i	\(-0.0653178\pi\)
							−0.665975	+	0.745974i	\(0.731984\pi\)
\(47\)	8.61967			1.25731			0.628654	−	0.777685i	\(-0.283606\pi\)
							0.628654	+	0.777685i	\(0.283606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bn.c.230.7	yes	64
3.2	odd	2	inner	273.2.bn.c.230.25	yes	64
7.6	odd	2	inner	273.2.bn.c.230.8	yes	64
13.3	even	3	inner	273.2.bn.c.146.26	yes	64
21.20	even	2	inner	273.2.bn.c.230.26	yes	64
39.29	odd	6	inner	273.2.bn.c.146.8	yes	64
91.55	odd	6	inner	273.2.bn.c.146.25	yes	64
273.146	even	6	inner	273.2.bn.c.146.7	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bn.c.146.7	✓	64	273.146	even	6	inner
273.2.bn.c.146.8	yes	64	39.29	odd	6	inner
273.2.bn.c.146.25	yes	64	91.55	odd	6	inner
273.2.bn.c.146.26	yes	64	13.3	even	3	inner
273.2.bn.c.230.7	yes	64	1.1	even	1	trivial
273.2.bn.c.230.8	yes	64	7.6	odd	2	inner
273.2.bn.c.230.25	yes	64	3.2	odd	2	inner
273.2.bn.c.230.26	yes	64	21.20	even	2	inner