Embedded newform 273.2.r.b.68.17

• Label: 273.2.r.b.68.17
• Level: $273$
• Weight: $2$
• Character: 273.68
• 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.r (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			68.17
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.0504397i q^{2} +(0.667082 - 1.59844i) q^{3} +1.99746 q^{4} +(1.19579 - 2.07117i) q^{5} +(0.0806247 + 0.0336474i) q^{6} +(0.00656452 + 2.64574i) q^{7} +0.201631i q^{8} +(-2.11000 - 2.13258i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			0.0504397i			0.0356663i	0.999841	+	0.0178331i	\(0.00567676\pi\)
							−0.999841	+	0.0178331i	\(0.994323\pi\)
\(3\)	0.667082	−	1.59844i	0.385140	−	0.922858i
							\(5\)	1.19579	−	2.07117i	0.534774	−	0.926256i	−0.464400	−	0.885625i	\(-0.653730\pi\)
0.999174	−	0.0406302i	\(-0.0129366\pi\)
\(7\)	0.00656452	+	2.64574i	0.00248116	+	0.999997i
							\(11\)	2.85650	+	1.64920i	0.861267	+	0.497253i	0.864436	−	0.502742i	\(-0.167676\pi\)
−0.00316960	+	0.999995i	\(0.501009\pi\)
\(13\)	−3.26013	+	1.53997i	−0.904199	+	0.427112i
							\(17\)	−6.34997			−1.54009			−0.770046	−	0.637988i	\(-0.779767\pi\)
−0.770046	+	0.637988i	\(0.779767\pi\)
\(19\)	−4.30999	+	2.48837i	−0.988778	+	0.570871i	−0.904909	−	0.425605i	\(-0.860061\pi\)
							−0.0838695	+	0.996477i	\(0.526728\pi\)
\(23\)		−	3.78971i		−	0.790208i	−0.918636	−	0.395104i	\(-0.870709\pi\)
							0.918636	−	0.395104i	\(-0.129291\pi\)
\(29\)	2.15531	−	1.24437i	0.400231	−	0.231073i	−0.286353	−	0.958124i	\(-0.592443\pi\)
							0.686584	+	0.727051i	\(0.259110\pi\)
\(31\)	4.40944	−	2.54579i	0.791960	−	0.457238i	−0.0486923	−	0.998814i	\(-0.515505\pi\)
							0.840652	+	0.541576i	\(0.182172\pi\)
\(37\)	5.24612			0.862457			0.431228	−	0.902243i	\(-0.358080\pi\)
							0.431228	+	0.902243i	\(0.358080\pi\)
\(41\)	−0.726691	−	1.25866i	−0.113490	−	0.196570i	0.803685	−	0.595055i	\(-0.202870\pi\)
							−0.917175	+	0.398484i	\(0.869536\pi\)
\(43\)	−0.219074	+	0.379447i	−0.0334085	+	0.0578652i	−0.882246	−	0.470788i	\(-0.843970\pi\)
							0.848838	+	0.528653i	\(0.177303\pi\)
\(47\)	−2.16184	+	3.74442i	−0.315337	+	0.546180i	−0.979509	−	0.201400i	\(-0.935451\pi\)
							0.664172	+	0.747580i	\(0.268784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.68.17	yes	64
3.2	odd	2	inner	273.2.r.b.68.16	✓	64
7.3	odd	6		273.2.bf.b.185.16	yes	64
13.9	even	3		273.2.bf.b.152.17	yes	64
21.17	even	6		273.2.bf.b.185.17	yes	64
39.35	odd	6		273.2.bf.b.152.16	yes	64
91.87	odd	6	inner	273.2.r.b.269.17	yes	64
273.269	even	6	inner	273.2.r.b.269.16	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.16	✓	64	3.2	odd	2	inner
273.2.r.b.68.17	yes	64	1.1	even	1	trivial
273.2.r.b.269.16	yes	64	273.269	even	6	inner
273.2.r.b.269.17	yes	64	91.87	odd	6	inner
273.2.bf.b.152.16	yes	64	39.35	odd	6
273.2.bf.b.152.17	yes	64	13.9	even	3
273.2.bf.b.185.16	yes	64	7.3	odd	6
273.2.bf.b.185.17	yes	64	21.17	even	6