Embedded newform 273.4.i.e.79.9

• Label: 273.4.i.e.79.9
• Level: $273$
• Weight: $4$
• Character: 273.79
• Analytic conductor: $16.108$
• Analytic rank: $0$
• Dimension: $26$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	273.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.1075214316\)
Analytic rank:	\(0\)
Dimension:	\(26\)
Relative dimension:	\(13\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			79.9
Character	\(\chi\)	\(=\)	273.79
Dual form			273.4.i.e.235.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.896185 - 1.55224i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(2.39370 + 4.14602i) q^{4} +(-0.238290 + 0.412731i) q^{5} -5.37711 q^{6} +(-17.3425 + 6.49897i) q^{7} +22.9198 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 3 q^{2} - 39 q^{3} - 55 q^{4} - 15 q^{5} + 18 q^{6} - 13 q^{7} - 12 q^{8} - 117 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.896185	−	1.55224i	0.316849	−	0.548799i	−0.662980	−	0.748637i	\(-0.730708\pi\)
							0.979829	+	0.199838i	\(0.0640417\pi\)
\(3\)	−1.50000	−	2.59808i	−0.288675	−	0.500000i
							\(5\)	−0.238290	+	0.412731i	−0.0213133	+	0.0369158i	−0.876485	−	0.481429i	\(-0.840118\pi\)
0.855172	+	0.518344i	\(0.173451\pi\)
\(7\)	−17.3425	+	6.49897i	−0.936409	+	0.350911i
							\(11\)	−7.96112	−	13.7891i	−0.218215	−	0.377960i	0.736047	−	0.676930i	\(-0.236690\pi\)
−0.954262	+	0.298970i	\(0.903357\pi\)
\(13\)	13.0000			0.277350
							\(17\)	27.9555	+	48.4203i	0.398835	+	0.690803i	0.993582	−	0.113110i	\(-0.0360812\pi\)
−0.594747	+	0.803913i	\(0.702748\pi\)
\(19\)	−82.1698	+	142.322i	−0.992161	+	1.71847i	−0.387854	+	0.921721i	\(0.626783\pi\)
							−0.604307	+	0.796752i	\(0.706550\pi\)
\(23\)	−47.4988	+	82.2704i	−0.430617	+	0.745851i	−0.996927	−	0.0783420i	\(-0.975037\pi\)
							0.566309	+	0.824193i	\(0.308371\pi\)
\(29\)	183.799			1.17692			0.588460	−	0.808526i	\(-0.299734\pi\)
							0.588460	+	0.808526i	\(0.299734\pi\)
\(31\)	−24.3895	−	42.2439i	−0.141306	−	0.244749i	0.786683	−	0.617358i	\(-0.211797\pi\)
							−0.927989	+	0.372608i	\(0.878463\pi\)
\(37\)	46.4892	−	80.5216i	0.206561	−	0.357775i	−0.744068	−	0.668104i	\(-0.767106\pi\)
							0.950629	+	0.310329i	\(0.100439\pi\)
\(41\)	−105.115			−0.400397			−0.200198	−	0.979755i	\(-0.564159\pi\)
							−0.200198	+	0.979755i	\(0.564159\pi\)
\(43\)	−50.2025			−0.178042			−0.0890211	−	0.996030i	\(-0.528374\pi\)
							−0.0890211	+	0.996030i	\(0.528374\pi\)
\(47\)	−170.466	+	295.255i	−0.529042	+	0.916328i	0.470384	+	0.882462i	\(0.344115\pi\)
							−0.999426	+	0.0338665i	\(0.989218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.4.i.e.79.9	✓	26
7.2	even	3		1911.4.a.bc.1.5		13
7.4	even	3	inner	273.4.i.e.235.9	yes	26
7.5	odd	6		1911.4.a.bb.1.5		13

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.4.i.e.79.9	✓	26	1.1	even	1	trivial
273.4.i.e.235.9	yes	26	7.4	even	3	inner
1911.4.a.bb.1.5		13	7.5	odd	6
1911.4.a.bc.1.5		13	7.2	even	3