Embedded newform 322.2.m.a.261.2

• Label: 322.2.m.a.261.2
• 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			261.2
Character	\(\chi\)	\(=\)	322.261
Dual form			322.2.m.a.95.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786053 + 0.618159i) q^{2} +(-1.48844 + 2.09022i) q^{3} +(0.235759 + 0.971812i) q^{4} +(3.12787 - 0.602847i) q^{5} +(-2.46208 + 0.722932i) q^{6} +(1.79086 + 1.94752i) q^{7} +(-0.415415 + 0.909632i) q^{8} +(-1.17236 - 3.38732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.48844	+	2.09022i	−0.859350	+	1.20679i	0.117545	+	0.993068i	\(0.462498\pi\)
−0.976895	+	0.213721i	\(0.931442\pi\)
\(5\)	3.12787	−	0.602847i	1.39882	−	0.269601i	0.566578	−	0.824008i	\(-0.308267\pi\)
							0.832246	+	0.554407i	\(0.187055\pi\)
\(7\)	1.79086	+	1.94752i	0.676881	+	0.736093i
							\(11\)	2.10648	−	1.65655i	0.635127	−	0.499469i	−0.247904	−	0.968785i	\(-0.579742\pi\)
0.883031	+	0.469315i	\(0.155499\pi\)
\(13\)	−0.679232	+	0.436516i	−0.188385	+	0.121068i	−0.631434	−	0.775430i	\(-0.717533\pi\)
							0.443049	+	0.896498i	\(0.353897\pi\)
\(17\)	−3.37558	−	3.21861i	−0.818698	−	0.780627i	0.159742	−	0.987159i	\(-0.448934\pi\)
							−0.978440	+	0.206532i	\(0.933782\pi\)
\(19\)	−4.45306	+	4.24599i	−1.02160	+	0.974096i	−0.999671	−	0.0256633i	\(-0.991830\pi\)
							−0.0219320	+	0.999759i	\(0.506982\pi\)
\(23\)	−3.13472	−	3.62953i	−0.653635	−	0.756810i
							\(29\)	4.83699	−	1.42027i	0.898207	−	0.263737i	0.200138	−	0.979768i	\(-0.435861\pi\)
0.698069	+	0.716030i	\(0.254043\pi\)
\(31\)	6.14244	−	0.586532i	1.10322	−	0.105344i	0.472479	−	0.881342i	\(-0.343359\pi\)
							0.630736	+	0.775998i	\(0.282753\pi\)
\(37\)	−3.44151	−	9.94358i	−0.565780	−	1.63471i	−0.757183	−	0.653203i	\(-0.773425\pi\)
							0.191402	−	0.981512i	\(-0.438696\pi\)
\(41\)	1.02829	+	1.18671i	0.160593	+	0.185334i	0.830343	−	0.557252i	\(-0.188144\pi\)
							−0.669751	+	0.742586i	\(0.733599\pi\)
\(43\)	2.32238	+	5.08531i	0.354160	+	0.775503i	0.999928	+	0.0119694i	\(0.00381006\pi\)
							−0.645768	+	0.763534i	\(0.723463\pi\)
\(47\)	−2.49041	+	4.31352i	−0.363264	+	0.629192i	−0.988496	−	0.151247i	\(-0.951671\pi\)
							0.625232	+	0.780439i	\(0.285004\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.261.2	yes	160
7.4	even	3	inner	322.2.m.a.123.7	yes	160
23.3	even	11	inner	322.2.m.a.233.7	yes	160
161.95	even	33	inner	322.2.m.a.95.2	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.95.2	✓	160	161.95	even	33	inner
322.2.m.a.123.7	yes	160	7.4	even	3	inner
322.2.m.a.233.7	yes	160	23.3	even	11	inner
322.2.m.a.261.2	yes	160	1.1	even	1	trivial