Embedded newform 322.2.m.a.261.3

• Label: 322.2.m.a.261.3
• 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			261.3
Character	\(\chi\)	\(=\)	322.261
Dual form			322.2.m.a.95.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786053 + 0.618159i) q^{2} +(-1.23036 + 1.72780i) q^{3} +(0.235759 + 0.971812i) q^{4} +(0.437942 - 0.0844064i) q^{5} +(-2.03518 + 0.597584i) q^{6} +(-2.59636 + 0.508835i) q^{7} +(-0.415415 + 0.909632i) q^{8} +(-0.490303 - 1.41664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.23036	+	1.72780i	−0.710349	+	0.997546i	0.288843	+	0.957376i	\(0.406729\pi\)
−0.999193	+	0.0401699i	\(0.987210\pi\)
\(5\)	0.437942	−	0.0844064i	0.195854	−	0.0377477i	−0.0903809	−	0.995907i	\(-0.528808\pi\)
							0.286235	+	0.958160i	\(0.407596\pi\)
\(7\)	−2.59636	+	0.508835i	−0.981332	+	0.192321i
							\(11\)	−4.44203	+	3.49325i	−1.33932	+	1.05326i	−0.345958	+	0.938250i	\(0.612446\pi\)
−0.993365	+	0.115006i	\(0.963311\pi\)
\(13\)	3.05723	−	1.96476i	0.847922	−	0.544926i	−0.0430042	−	0.999075i	\(-0.513693\pi\)
							0.890926	+	0.454149i	\(0.150057\pi\)
\(17\)	−0.108260	−	0.103226i	−0.0262569	−	0.0250359i	0.676837	−	0.736133i	\(-0.263350\pi\)
							−0.703094	+	0.711097i	\(0.748199\pi\)
\(19\)	0.909344	−	0.867058i	0.208618	−	0.198917i	−0.578565	−	0.815636i	\(-0.696387\pi\)
							0.787183	+	0.616719i	\(0.211539\pi\)
\(23\)	2.97333	+	3.76288i	0.619982	+	0.784616i
							\(29\)	1.83316	−	0.538265i	0.340409	−	0.0999532i	−0.107060	−	0.994253i	\(-0.534144\pi\)
0.447469	+	0.894299i	\(0.352325\pi\)
\(31\)	2.84947	−	0.272091i	0.511779	−	0.0488690i	0.164028	−	0.986456i	\(-0.447551\pi\)
							0.347752	+	0.937587i	\(0.386945\pi\)
\(37\)	2.88003	+	8.32131i	0.473474	+	1.36801i	0.888781	+	0.458333i	\(0.151553\pi\)
							−0.415306	+	0.909682i	\(0.636326\pi\)
\(41\)	3.33855	+	3.85289i	0.521393	+	0.601720i	0.953979	−	0.299873i	\(-0.0969442\pi\)
							−0.432586	+	0.901593i	\(0.642399\pi\)
\(43\)	0.201117	+	0.440385i	0.0306700	+	0.0671580i	0.924347	−	0.381553i	\(-0.124611\pi\)
							−0.893677	+	0.448711i	\(0.851883\pi\)
\(47\)	3.16503	−	5.48200i	0.461667	−	0.799631i	−0.537377	−	0.843342i	\(-0.680585\pi\)
							0.999044	+	0.0437108i	\(0.0139180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.261.3	yes	160
7.4	even	3	inner	322.2.m.a.123.6	yes	160
23.3	even	11	inner	322.2.m.a.233.6	yes	160
161.95	even	33	inner	322.2.m.a.95.3	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.95.3	✓	160	161.95	even	33	inner
322.2.m.a.123.6	yes	160	7.4	even	3	inner
322.2.m.a.233.6	yes	160	23.3	even	11	inner
322.2.m.a.261.3	yes	160	1.1	even	1	trivial