Embedded newform 273.2.r.b.68.16

• Label: 273.2.r.b.68.16
• 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			68.16
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.17

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.0504397i q^{2} +(-1.05075 + 1.37693i) q^{3} +1.99746 q^{4} +(-1.19579 + 2.07117i) q^{5} +(0.0694519 + 0.0529993i) q^{6} +(0.00656452 + 2.64574i) q^{7} -0.201631i q^{8} +(-0.791866 - 2.89360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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	\(-0.994323\pi\)
							0.999841	−	0.0178331i	\(-0.00567676\pi\)
\(3\)	−1.05075	+	1.37693i	−0.606648	+	0.794970i
							\(5\)	−1.19579	+	2.07117i	−0.534774	+	0.926256i	0.464400	+	0.885625i	\(0.346270\pi\)
−0.999174	+	0.0406302i	\(0.987063\pi\)
\(7\)	0.00656452	+	2.64574i	0.00248116	+	0.999997i
							\(11\)	−2.85650	−	1.64920i	−0.861267	−	0.497253i	0.00316960	−	0.999995i	\(-0.498991\pi\)
−0.864436	+	0.502742i	\(0.832324\pi\)
\(13\)	−3.26013	+	1.53997i	−0.904199	+	0.427112i
							\(17\)	6.34997			1.54009			0.770046	−	0.637988i	\(-0.220233\pi\)
0.770046	+	0.637988i	\(0.220233\pi\)
\(19\)	−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	\(0.129291\pi\)
							−0.918636	+	0.395104i	\(0.870709\pi\)
\(29\)	−2.15531	+	1.24437i	−0.400231	+	0.231073i	−0.686584	−	0.727051i	\(-0.740890\pi\)
							0.286353	+	0.958124i	\(0.407557\pi\)
\(31\)	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	0.917175	−	0.398484i	\(-0.130464\pi\)
							−0.803685	+	0.595055i	\(0.797130\pi\)
\(43\)	−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	−0.664172	−	0.747580i	\(-0.731216\pi\)
							0.979509	+	0.201400i	\(0.0645491\pi\)

(See \(a_n\) instead)

Twists

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

    

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