Embedded newform 322.2.m.a.25.7

• Label: 322.2.m.a.25.7
• 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			25.7
Character	\(\chi\)	\(=\)	322.25
Dual form			322.2.m.a.219.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 + 0.998867i) q^{2} +(2.33543 + 0.934966i) q^{3} +(-0.995472 - 0.0950560i) q^{4} +(-0.786581 - 3.24233i) q^{5} +(-1.04503 + 2.28830i) q^{6} +(2.61565 - 0.397974i) q^{7} +(0.142315 - 0.989821i) q^{8} +(2.40888 + 2.29686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

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
							\(3\)	2.33543	+	0.934966i	1.34836	+	0.539803i	0.929693	−	0.368336i	\(-0.120072\pi\)
0.418670	+	0.908139i	\(0.362496\pi\)
\(5\)	−0.786581	−	3.24233i	−0.351770	−	1.45001i	−0.823019	−	0.568014i	\(-0.807712\pi\)
							0.471249	−	0.882000i	\(-0.343803\pi\)
\(7\)	2.61565	−	0.397974i	0.988622	−	0.150420i
							\(11\)	0.181748	+	3.81536i	0.0547991	+	1.15038i	0.844036	+	0.536286i	\(0.180173\pi\)
−0.789237	+	0.614089i	\(0.789524\pi\)
\(13\)	−0.389910	−	0.449981i	−0.108142	−	0.124802i	0.699098	−	0.715026i	\(-0.253585\pi\)
							−0.807240	+	0.590224i	\(0.799040\pi\)
\(17\)	3.66246	+	5.14321i	0.888278	+	1.24741i	0.968090	+	0.250602i	\(0.0806286\pi\)
							−0.0798119	+	0.996810i	\(0.525432\pi\)
\(19\)	2.48607	−	3.49119i	0.570343	−	0.800935i	−0.424166	−	0.905584i	\(-0.639433\pi\)
							0.994509	+	0.104650i	\(0.0333721\pi\)
\(23\)	−4.71026	−	0.901914i	−0.982157	−	0.188062i
							\(29\)	−1.63270	+	3.57512i	−0.303185	+	0.663882i	−0.998496	−	0.0548267i	\(-0.982539\pi\)
0.695311	+	0.718709i	\(0.255267\pi\)
\(31\)	−8.31732	−	6.54081i	−1.49383	−	1.17476i	−0.939538	−	0.342446i	\(-0.888745\pi\)
							−0.554297	−	0.832319i	\(-0.687013\pi\)
\(37\)	−5.12534	−	4.88700i	−0.842601	−	0.803418i	0.139804	−	0.990179i	\(-0.455353\pi\)
							−0.982405	+	0.186761i	\(0.940201\pi\)
\(41\)	−2.67708	−	0.786060i	−0.418089	−	0.122762i	0.0659203	−	0.997825i	\(-0.479002\pi\)
							−0.484009	+	0.875063i	\(0.660820\pi\)
\(43\)	0.873458	+	6.07503i	0.133201	+	0.926434i	0.941344	+	0.337448i	\(0.109564\pi\)
							−0.808143	+	0.588986i	\(0.799527\pi\)
\(47\)	−0.110894	−	0.192075i	−0.0161756	−	0.0280169i	0.857824	−	0.513943i	\(-0.171816\pi\)
							−0.874000	+	0.485926i	\(0.838482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.25.7	✓	160
7.2	even	3	inner	322.2.m.a.163.2	yes	160
23.12	even	11	inner	322.2.m.a.81.2	yes	160
161.58	even	33	inner	322.2.m.a.219.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.25.7	✓	160	1.1	even	1	trivial
322.2.m.a.81.2	yes	160	23.12	even	11	inner
322.2.m.a.163.2	yes	160	7.2	even	3	inner
322.2.m.a.219.7	yes	160	161.58	even	33	inner