Embedded newform 273.2.br.b.17.4

• Label: 273.2.br.b.17.4
• Level: $273$
• Weight: $2$
• Character: 273.17
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.br (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			17.4
Character	\(\chi\)	\(=\)	273.17
Dual form			273.2.br.b.257.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.28808 q^{2} +(1.60560 - 0.649648i) q^{3} +3.23529 q^{4} +(3.02328 - 1.74549i) q^{5} +(-3.67374 + 1.48644i) q^{6} +(2.39720 + 1.11958i) q^{7} -2.82644 q^{8} +(2.15591 - 2.08615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{3} + 56 q^{4} - 6 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.28808			−1.61791			−0.808957	−	0.587868i	\(-0.799967\pi\)
							−0.808957	+	0.587868i	\(0.799967\pi\)
\(3\)	1.60560	−	0.649648i	0.926995	−	0.375075i
							\(5\)	3.02328	−	1.74549i	1.35205	−	0.780609i	0.363517	−	0.931588i	\(-0.381576\pi\)
0.988537	+	0.150979i	\(0.0482425\pi\)
\(7\)	2.39720	+	1.11958i	0.906055	+	0.423160i
							\(11\)	1.60796	+	2.78507i	0.484818	+	0.839730i	0.999848	−	0.0174426i	\(-0.00555243\pi\)
−0.515030	+	0.857172i	\(0.672219\pi\)
\(13\)	−3.60111	−	0.178936i	−0.998768	−	0.0496280i
							\(17\)	−2.61681			−0.634669			−0.317335	−	0.948314i	\(-0.602788\pi\)
−0.317335	+	0.948314i	\(0.602788\pi\)
\(19\)	−3.51866	+	6.09450i	−0.807236	+	1.39817i	0.107535	+	0.994201i	\(0.465704\pi\)
							−0.914771	+	0.403973i	\(0.867629\pi\)
\(23\)			1.58375i			0.330234i	0.986274	+	0.165117i	\(0.0528002\pi\)
							−0.986274	+	0.165117i	\(0.947200\pi\)
\(29\)	−4.08375	−	2.35776i	−0.758334	−	0.437824i	0.0703635	−	0.997521i	\(-0.477584\pi\)
							−0.828697	+	0.559697i	\(0.810917\pi\)
\(31\)	1.83786	−	3.18327i	0.330090	−	0.571733i	−0.652439	−	0.757841i	\(-0.726254\pi\)
							0.982529	+	0.186108i	\(0.0595876\pi\)
\(37\)			3.11103i			0.511450i	0.966750	+	0.255725i	\(0.0823141\pi\)
							−0.966750	+	0.255725i	\(0.917686\pi\)
\(41\)	−1.82354	−	1.05282i	−0.284789	−	0.164423i	0.350800	−	0.936450i	\(-0.385910\pi\)
							−0.635590	+	0.772027i	\(0.719243\pi\)
\(43\)	−3.54947	−	6.14787i	−0.541289	−	0.937541i	−0.998830	−	0.0483523i	\(-0.984603\pi\)
							0.457541	−	0.889189i	\(-0.348730\pi\)
\(47\)	−0.592823	+	0.342266i	−0.0864721	+	0.0499247i	−0.542613	−	0.839983i	\(-0.682565\pi\)
							0.456140	+	0.889908i	\(0.349231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.br.b.17.4	yes	64
3.2	odd	2	inner	273.2.br.b.17.29	yes	64
7.5	odd	6		273.2.y.b.173.29	yes	64
13.10	even	6		273.2.y.b.101.4	✓	64
21.5	even	6		273.2.y.b.173.4	yes	64
39.23	odd	6		273.2.y.b.101.29	yes	64
91.75	odd	6	inner	273.2.br.b.257.29	yes	64
273.257	even	6	inner	273.2.br.b.257.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.4	✓	64	13.10	even	6
273.2.y.b.101.29	yes	64	39.23	odd	6
273.2.y.b.173.4	yes	64	21.5	even	6
273.2.y.b.173.29	yes	64	7.5	odd	6
273.2.br.b.17.4	yes	64	1.1	even	1	trivial
273.2.br.b.17.29	yes	64	3.2	odd	2	inner
273.2.br.b.257.4	yes	64	273.257	even	6	inner
273.2.br.b.257.29	yes	64	91.75	odd	6	inner