Embedded newform 322.2.m.a.261.1

• Label: 322.2.m.a.261.1
• 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			261.1
Character	\(\chi\)	\(=\)	322.261
Dual form			322.2.m.a.95.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786053 + 0.618159i) q^{2} +(-1.59281 + 2.23679i) q^{3} +(0.235759 + 0.971812i) q^{4} +(-3.72233 + 0.717421i) q^{5} +(-2.63472 + 0.773624i) q^{6} +(1.75929 - 1.97609i) q^{7} +(-0.415415 + 0.909632i) q^{8} +(-1.48497 - 4.29054i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.59281	+	2.23679i	−0.919608	+	1.29141i	0.0367839	+	0.999323i	\(0.488289\pi\)
−0.956392	+	0.292086i	\(0.905651\pi\)
\(5\)	−3.72233	+	0.717421i	−1.66468	+	0.320841i	−0.932745	−	0.360536i	\(-0.882594\pi\)
							−0.731933	+	0.681376i	\(0.761382\pi\)
\(7\)	1.75929	−	1.97609i	0.664947	−	0.746890i
							\(11\)	−0.867388	+	0.682121i	−0.261527	+	0.205667i	−0.740309	−	0.672267i	\(-0.765321\pi\)
0.478782	+	0.877934i	\(0.341078\pi\)
\(13\)	−1.38819	+	0.892137i	−0.385015	+	0.247434i	−0.718803	−	0.695214i	\(-0.755310\pi\)
							0.333788	+	0.942648i	\(0.391673\pi\)
\(17\)	0.703643	+	0.670922i	0.170658	+	0.162722i	0.770595	−	0.637325i	\(-0.219959\pi\)
							−0.599937	+	0.800047i	\(0.704808\pi\)
\(19\)	−1.00289	+	0.956253i	−0.230079	+	0.219379i	−0.796345	−	0.604843i	\(-0.793236\pi\)
							0.566266	+	0.824222i	\(0.308387\pi\)
\(23\)	−4.10759	−	2.47542i	−0.856491	−	0.516161i
							\(29\)	−7.10891	+	2.08736i	−1.32009	+	0.387614i	−0.864526	−	0.502588i	\(-0.832381\pi\)
−0.455566	+	0.890202i	\(0.650563\pi\)
\(31\)	−4.64086	+	0.443148i	−0.833523	+	0.0795918i	−0.503086	−	0.864236i	\(-0.667802\pi\)
							−0.330437	+	0.943828i	\(0.607196\pi\)
\(37\)	1.99789	+	5.77252i	0.328451	+	0.948997i	0.981463	+	0.191650i	\(0.0613838\pi\)
							−0.653012	+	0.757347i	\(0.726495\pi\)
\(41\)	7.36065	+	8.49464i	1.14954	+	1.32664i	0.936935	+	0.349502i	\(0.113649\pi\)
							0.212605	+	0.977138i	\(0.431805\pi\)
\(43\)	2.32693	+	5.09527i	0.354854	+	0.777021i	0.999917	+	0.0128921i	\(0.00410379\pi\)
							−0.645063	+	0.764129i	\(0.723169\pi\)
\(47\)	−5.20301	+	9.01188i	−0.758938	+	1.31452i	0.184455	+	0.982841i	\(0.440948\pi\)
							−0.943393	+	0.331678i	\(0.892385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.261.1	yes	160
7.4	even	3	inner	322.2.m.a.123.8	yes	160
23.3	even	11	inner	322.2.m.a.233.8	yes	160
161.95	even	33	inner	322.2.m.a.95.1	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.95.1	✓	160	161.95	even	33	inner
322.2.m.a.123.8	yes	160	7.4	even	3	inner
322.2.m.a.233.8	yes	160	23.3	even	11	inner
322.2.m.a.261.1	yes	160	1.1	even	1	trivial