Embedded newform 322.2.m.b.179.3

• Label: 322.2.m.b.179.3
• Level: $322$
• Weight: $2$
• Character: 322.179
• 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			179.3
Character	\(\chi\)	\(=\)	322.179
Dual form			322.2.m.b.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.235759 + 0.971812i) q^{2} +(-0.526198 - 1.52035i) q^{3} +(-0.888835 + 0.458227i) q^{4} +(-1.39502 + 0.558482i) q^{5} +(1.35344 - 0.869801i) q^{6} +(-1.65886 + 2.06112i) q^{7} +(-0.654861 - 0.755750i) q^{8} +(0.323581 - 0.254467i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.235759	+	0.971812i	0.166707	+	0.687175i
							\(3\)	−0.526198	−	1.52035i	−0.303801	−	0.877774i	−0.988802	−	0.149233i	\(-0.952320\pi\)
0.685002	−	0.728542i	\(-0.259802\pi\)
\(5\)	−1.39502	+	0.558482i	−0.623872	+	0.249761i	−0.661990	−	0.749512i	\(-0.730288\pi\)
							0.0381179	+	0.999273i	\(0.487864\pi\)
\(7\)	−1.65886	+	2.06112i	−0.626988	+	0.779029i
							\(11\)	−1.00077	+	4.12525i	−0.301745	+	1.24381i	0.595889	+	0.803067i	\(0.296800\pi\)
−0.897634	+	0.440742i	\(0.854715\pi\)
\(13\)	−2.57194	+	5.63176i	−0.713327	+	1.56197i	0.109698	+	0.993965i	\(0.465012\pi\)
							−0.823026	+	0.568004i	\(0.807716\pi\)
\(17\)	−0.0683106	−	1.43402i	−0.0165677	−	0.347800i	−0.991897	−	0.127045i	\(-0.959451\pi\)
							0.975329	−	0.220755i	\(-0.0708522\pi\)
\(19\)	0.0833982	−	1.75074i	0.0191329	−	0.401648i	−0.968905	−	0.247434i	\(-0.920413\pi\)
							0.988038	−	0.154214i	\(-0.0492844\pi\)
\(23\)	−0.394450	+	4.77958i	−0.0822484	+	0.996612i
							\(29\)	1.35187	−	0.868793i	0.251036	−	0.161331i	−0.409071	−	0.912502i	\(-0.634147\pi\)
0.660107	+	0.751172i	\(0.270511\pi\)
\(31\)	−0.694218	+	0.133800i	−0.124685	+	0.0240311i	−0.251212	−	0.967932i	\(-0.580829\pi\)
							0.126527	+	0.991963i	\(0.459617\pi\)
\(37\)	4.69304	−	3.69065i	0.771531	−	0.606739i	−0.152832	−	0.988252i	\(-0.548839\pi\)
							0.924363	+	0.381514i	\(0.124597\pi\)
\(41\)	0.620949	−	4.31880i	0.0969760	−	0.674483i	−0.882111	−	0.471041i	\(-0.843878\pi\)
							0.979087	−	0.203441i	\(-0.0652126\pi\)
\(43\)	−6.27143	+	7.23762i	−0.956384	+	1.10373i	0.0381457	+	0.999272i	\(0.487855\pi\)
							−0.994530	+	0.104454i	\(0.966691\pi\)
\(47\)	−2.01036	−	3.48205i	−0.293241	−	0.507909i	0.681333	−	0.731974i	\(-0.261401\pi\)
							−0.974574	+	0.224065i	\(0.928067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.b.179.3	yes	160
7.2	even	3	inner	322.2.m.b.317.6	yes	160
23.9	even	11	inner	322.2.m.b.193.6	yes	160
161.9	even	33	inner	322.2.m.b.9.3	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.b.9.3	✓	160	161.9	even	33	inner
322.2.m.b.179.3	yes	160	1.1	even	1	trivial
322.2.m.b.193.6	yes	160	23.9	even	11	inner
322.2.m.b.317.6	yes	160	7.2	even	3	inner