Embedded newform 322.2.m.a.25.2

• Label: 322.2.m.a.25.2
• Level: $322$
• Weight: $2$
• Character: 322.25
• 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			25.2
Character	\(\chi\)	\(=\)	322.25
Dual form			322.2.m.a.219.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 + 0.998867i) q^{2} +(-1.62223 - 0.649442i) q^{3} +(-0.995472 - 0.0950560i) q^{4} +(0.0716832 + 0.295482i) q^{5} +(0.725895 - 1.58949i) q^{6} +(2.51052 + 0.835042i) q^{7} +(0.142315 - 0.989821i) q^{8} +(0.0386478 + 0.0368506i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0475819	+	0.998867i	−0.0336455	+	0.706306i
							\(3\)	−1.62223	−	0.649442i	−0.936594	−	0.374956i	−0.147384	−	0.989079i	\(-0.547085\pi\)
−0.789210	+	0.614124i	\(0.789510\pi\)
\(5\)	0.0716832	+	0.295482i	0.0320577	+	0.132144i	0.985592	−	0.169142i	\(-0.0540995\pi\)
							−0.953534	+	0.301285i	\(0.902584\pi\)
\(7\)	2.51052	+	0.835042i	0.948887	+	0.315616i
							\(11\)	0.0465679	+	0.977580i	0.0140407	+	0.294752i	0.994956	+	0.100315i	\(0.0319849\pi\)
−0.980915	+	0.194437i	\(0.937712\pi\)
\(13\)	2.34011	+	2.70063i	0.649029	+	0.749020i	0.980944	−	0.194290i	\(-0.0622401\pi\)
							−0.331915	+	0.943309i	\(0.607695\pi\)
\(17\)	3.13065	+	4.39638i	0.759294	+	1.06628i	0.995517	+	0.0945804i	\(0.0301509\pi\)
							−0.236223	+	0.971699i	\(0.575910\pi\)
\(19\)	−2.96188	+	4.15938i	−0.679502	+	0.954227i	0.320474	+	0.947257i	\(0.396158\pi\)
							−0.999976	+	0.00696956i	\(0.997782\pi\)
\(23\)	−2.57593	+	4.04532i	−0.537118	+	0.843507i
							\(29\)	0.846963	−	1.85459i	0.157277	−	0.344389i	−0.814546	−	0.580098i	\(-0.803014\pi\)
0.971824	+	0.235710i	\(0.0757414\pi\)
\(31\)	4.16891	+	3.27847i	0.748759	+	0.588831i	0.917924	−	0.396757i	\(-0.129864\pi\)
							−0.169164	+	0.985588i	\(0.554107\pi\)
\(37\)	1.14889	+	1.09546i	0.188876	+	0.180093i	0.778617	−	0.627500i	\(-0.215922\pi\)
							−0.589741	+	0.807593i	\(0.700770\pi\)
\(41\)	0.569729	+	0.167288i	0.0889768	+	0.0261259i	0.325918	−	0.945398i	\(-0.394327\pi\)
							−0.236941	+	0.971524i	\(0.576145\pi\)
\(43\)	−1.52430	−	10.6018i	−0.232454	−	1.61675i	−0.687432	−	0.726249i	\(-0.741262\pi\)
							0.454978	−	0.890503i	\(-0.349647\pi\)
\(47\)	−3.80856	−	6.59662i	−0.555536	−	0.962216i	−0.997862	−	0.0653617i	\(-0.979180\pi\)
							0.442326	−	0.896854i	\(-0.354153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.25.2	✓	160
7.2	even	3	inner	322.2.m.a.163.7	yes	160
23.12	even	11	inner	322.2.m.a.81.7	yes	160
161.58	even	33	inner	322.2.m.a.219.2	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.25.2	✓	160	1.1	even	1	trivial
322.2.m.a.81.7	yes	160	23.12	even	11	inner
322.2.m.a.163.7	yes	160	7.2	even	3	inner
322.2.m.a.219.2	yes	160	161.58	even	33	inner