Embedded newform 322.2.m.a.261.6

• Label: 322.2.m.a.261.6
• Level: $322$
• Weight: $2$
• Character: 322.261
• 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			261.6
Character	\(\chi\)	\(=\)	322.261
Dual form			322.2.m.a.95.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786053 + 0.618159i) q^{2} +(1.00269 - 1.40808i) q^{3} +(0.235759 + 0.971812i) q^{4} +(1.51728 - 0.292431i) q^{5} +(1.65859 - 0.487005i) q^{6} +(-0.430333 + 2.61052i) q^{7} +(-0.415415 + 0.909632i) q^{8} +(0.00389841 + 0.0112637i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{11}\right)\)

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
							\(3\)	1.00269	−	1.40808i	0.578903	−	0.812956i	−0.416459	−	0.909155i	\(-0.636729\pi\)
0.995362	+	0.0961985i	\(0.0306684\pi\)
\(5\)	1.51728	−	0.292431i	0.678546	−	0.130779i	0.161677	−	0.986844i	\(-0.448310\pi\)
							0.516869	+	0.856065i	\(0.327097\pi\)
\(7\)	−0.430333	+	2.61052i	−0.162651	+	0.986684i
							\(11\)	0.138232	−	0.108707i	0.0416786	−	0.0327765i	−0.597102	−	0.802165i	\(-0.703681\pi\)
0.638781	+	0.769389i	\(0.279439\pi\)
\(13\)	1.09944	−	0.706568i	0.304930	−	0.195967i	−0.379222	−	0.925306i	\(-0.623809\pi\)
							0.684153	+	0.729339i	\(0.260172\pi\)
\(17\)	−4.12149	−	3.92983i	−0.999608	−	0.953124i	−0.000707076	−	1.00000i	\(-0.500225\pi\)
							−0.998901	+	0.0468756i	\(0.985074\pi\)
\(19\)	4.19253	−	3.99757i	0.961833	−	0.917105i	−0.0347759	−	0.999395i	\(-0.511072\pi\)
							0.996608	+	0.0822897i	\(0.0262233\pi\)
\(23\)	0.158445	−	4.79321i	0.0330380	−	0.999454i
							\(29\)	−7.53939	+	2.21376i	−1.40003	+	0.411086i	−0.892694	−	0.450663i	\(-0.851188\pi\)
−0.507335	+	0.861749i	\(0.669369\pi\)
\(31\)	−6.47069	+	0.617876i	−1.16217	+	0.110974i	−0.658266	−	0.752785i	\(-0.728710\pi\)
							−0.503905	+	0.863759i	\(0.668104\pi\)
\(37\)	0.938333	+	2.71113i	0.154261	+	0.445708i	0.995478	−	0.0949894i	\(-0.0302817\pi\)
							−0.841217	+	0.540697i	\(0.818161\pi\)
\(41\)	3.48787	+	4.02522i	0.544714	+	0.628633i	0.959643	−	0.281220i	\(-0.0907392\pi\)
							−0.414929	+	0.909854i	\(0.636194\pi\)
\(43\)	−1.77186	−	3.87983i	−0.270206	−	0.591668i	0.725079	−	0.688666i	\(-0.241803\pi\)
							−0.995285	+	0.0969978i	\(0.969076\pi\)
\(47\)	−1.31876	+	2.28416i	−0.192361	+	0.333179i	−0.946032	−	0.324072i	\(-0.894948\pi\)
							0.753671	+	0.657252i	\(0.228281\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.261.6	yes	160
7.4	even	3	inner	322.2.m.a.123.3	yes	160
23.3	even	11	inner	322.2.m.a.233.3	yes	160
161.95	even	33	inner	322.2.m.a.95.6	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.95.6	✓	160	161.95	even	33	inner
322.2.m.a.123.3	yes	160	7.4	even	3	inner
322.2.m.a.233.3	yes	160	23.3	even	11	inner
322.2.m.a.261.6	yes	160	1.1	even	1	trivial