Embedded newform 322.2.m.b.95.7

• Label: 322.2.m.b.95.7
• Level: $322$
• Weight: $2$
• Character: 322.95
• Analytic conductor: $2.571$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			95.7
Character	\(\chi\)	\(=\)	322.95
Dual form			322.2.m.b.261.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.786053 + 0.618159i) q^{2} +(1.20508 + 1.69230i) q^{3} +(0.235759 - 0.971812i) q^{4} +(2.58003 + 0.497261i) q^{5} +(-1.99337 - 0.585307i) q^{6} +(-2.50532 + 0.850515i) q^{7} +(0.415415 + 0.909632i) q^{8} +(-0.430459 + 1.24373i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.786053	+	0.618159i	−0.555823	+	0.437104i
							\(3\)	1.20508	+	1.69230i	0.695755	+	0.977051i	0.999703	+	0.0243628i	\(0.00775567\pi\)
−0.303948	+	0.952689i	\(0.598305\pi\)
\(5\)	2.58003	+	0.497261i	1.15383	+	0.222382i	0.730055	−	0.683388i	\(-0.239494\pi\)
							0.423770	+	0.905770i	\(0.360706\pi\)
\(7\)	−2.50532	+	0.850515i	−0.946922	+	0.321464i
							\(11\)	4.29995	+	3.38152i	1.29648	+	1.01957i	0.997873	+	0.0651908i	\(0.0207656\pi\)
0.298611	+	0.954375i	\(0.403477\pi\)
\(13\)	−2.71427	−	1.74436i	−0.752803	−	0.483797i	0.107104	−	0.994248i	\(-0.465842\pi\)
							−0.859907	+	0.510451i	\(0.829479\pi\)
\(17\)	−0.255497	+	0.243616i	−0.0619670	+	0.0590854i	−0.720430	−	0.693528i	\(-0.756055\pi\)
							0.658463	+	0.752613i	\(0.271207\pi\)
\(19\)	2.34066	+	2.23181i	0.536984	+	0.512013i	0.909327	−	0.416082i	\(-0.136597\pi\)
							−0.372343	+	0.928095i	\(0.621446\pi\)
\(23\)	−1.10278	−	4.66732i	−0.229945	−	0.973204i
							\(29\)	−5.85271	−	1.71851i	−1.08682	−	0.319119i	−0.311216	−	0.950339i	\(-0.600736\pi\)
−0.775604	+	0.631220i	\(0.782555\pi\)
\(31\)	1.99558	+	0.190554i	0.358416	+	0.0342246i	0.272711	−	0.962096i	\(-0.412080\pi\)
							0.0857050	+	0.996321i	\(0.472686\pi\)
\(37\)	−2.61219	+	7.54744i	−0.429442	+	1.24079i	0.498115	+	0.867111i	\(0.334026\pi\)
							−0.927557	+	0.373681i	\(0.878096\pi\)
\(41\)	−5.65371	+	6.52473i	−0.882962	+	1.01899i	0.116705	+	0.993167i	\(0.462767\pi\)
							−0.999666	+	0.0258253i	\(0.991779\pi\)
\(43\)	1.89201	−	4.14293i	0.288529	−	0.631791i	−0.708754	−	0.705456i	\(-0.750742\pi\)
							0.997283	+	0.0736650i	\(0.0234695\pi\)
\(47\)	−0.134519	−	0.232994i	−0.0196216	−	0.0339856i	0.856048	−	0.516897i	\(-0.172913\pi\)
							−0.875669	+	0.482911i	\(0.839579\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.b.95.7	✓	160
7.2	even	3	inner	322.2.m.b.233.2	yes	160
23.8	even	11	inner	322.2.m.b.123.2	yes	160
161.100	even	33	inner	322.2.m.b.261.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.b.95.7	✓	160	1.1	even	1	trivial
322.2.m.b.123.2	yes	160	23.8	even	11	inner
322.2.m.b.233.2	yes	160	7.2	even	3	inner
322.2.m.b.261.7	yes	160	161.100	even	33	inner