Embedded newform 322.2.o.a.103.1

• Label: 322.2.o.a.103.1
• Level: $322$
• Weight: $2$
• Character: 322.103
• 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.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			103.1
Character	\(\chi\)	\(=\)	322.103
Dual form			322.2.o.a.297.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 - 0.998867i) q^{2} +(-1.18055 - 2.94887i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(-0.511867 + 2.10995i) q^{5} +(-2.88936 + 1.31953i) q^{6} +(0.321409 + 2.62616i) q^{7} +(0.142315 + 0.989821i) q^{8} +(-5.13095 + 4.89235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{9}{22}\right)\)

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
							\(3\)	−1.18055	−	2.94887i	−0.681591	−	1.70253i	−0.711218	−	0.702972i	\(-0.751856\pi\)
0.0296262	−	0.999561i	\(-0.490568\pi\)
\(5\)	−0.511867	+	2.10995i	−0.228914	+	0.943597i	0.735145	+	0.677910i	\(0.237114\pi\)
							−0.964059	+	0.265687i	\(0.914401\pi\)
\(7\)	0.321409	+	2.62616i	0.121481	+	0.992594i
							\(11\)	−5.79642	−	0.276118i	−1.74769	−	0.0832526i	−0.850794	−	0.525499i	\(-0.823879\pi\)
−0.896894	+	0.442247i	\(0.854182\pi\)
\(13\)	1.60047	+	1.38681i	0.443889	+	0.384632i	0.847930	−	0.530109i	\(-0.177849\pi\)
							−0.404040	+	0.914741i	\(0.632394\pi\)
\(17\)	−1.33658	+	1.87697i	−0.324169	+	0.455232i	−0.944213	−	0.329335i	\(-0.893175\pi\)
							0.620044	+	0.784567i	\(0.287115\pi\)
\(19\)	−4.67862	−	6.57020i	−1.07335	−	1.50731i	−0.845771	−	0.533546i	\(-0.820859\pi\)
							−0.227577	−	0.973760i	\(-0.573080\pi\)
\(23\)	−4.35846	−	2.00095i	−0.908802	−	0.417227i
							\(29\)	0.445894	+	0.976372i	0.0828004	+	0.181308i	0.946504	−	0.322693i	\(-0.104588\pi\)
−0.863703	+	0.504001i	\(0.831861\pi\)
\(31\)	2.07038	+	2.63270i	0.371851	+	0.472847i	0.935581	−	0.353112i	\(-0.114877\pi\)
							−0.563730	+	0.825959i	\(0.690634\pi\)
\(37\)	−3.35496	−	3.51858i	−0.551551	−	0.578450i	0.387655	−	0.921805i	\(-0.373285\pi\)
							−0.939206	+	0.343354i	\(0.888437\pi\)
\(41\)	0.000269062	0	0.000916340i	4.20204e−5	0	0.000143108i	0.959514	−	0.281661i	\(-0.0908854\pi\)
							−0.959472	+	0.281804i	\(0.909067\pi\)
\(43\)	−5.95741	−	0.856546i	−0.908497	−	0.130622i	−0.327805	−	0.944745i	\(-0.606309\pi\)
							−0.580692	+	0.814123i	\(0.697218\pi\)
\(47\)	5.55518	+	3.20729i	0.810307	+	0.467831i	0.847062	−	0.531493i	\(-0.178369\pi\)
							−0.0367556	+	0.999324i	\(0.511702\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.103.1	✓	160
7.3	odd	6	inner	322.2.o.a.241.8	yes	160
23.21	odd	22	inner	322.2.o.a.159.8	yes	160
161.136	even	66	inner	322.2.o.a.297.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.103.1	✓	160	1.1	even	1	trivial
322.2.o.a.159.8	yes	160	23.21	odd	22	inner
322.2.o.a.241.8	yes	160	7.3	odd	6	inner
322.2.o.a.297.1	yes	160	161.136	even	66	inner