Embedded newform 322.2.m.b.261.4

• Label: 322.2.m.b.261.4
• 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			261.4
Character	\(\chi\)	\(=\)	322.261
Dual form			322.2.m.b.95.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.786053 - 0.618159i) q^{2} +(-0.0197625 + 0.0277526i) q^{3} +(0.235759 + 0.971812i) q^{4} +(3.71493 - 0.715993i) q^{5} +(0.0326899 - 0.00959862i) q^{6} +(0.765630 + 2.53255i) q^{7} +(0.415415 - 0.909632i) q^{8} +(0.980824 + 2.83391i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 2 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} - 11 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.0197625	+	0.0277526i	−0.0114099	+	0.0160230i	−0.820242	−	0.572017i	\(-0.806161\pi\)
0.808832	+	0.588040i	\(0.200100\pi\)
\(5\)	3.71493	−	0.715993i	1.66137	−	0.320202i	0.729769	−	0.683694i	\(-0.239628\pi\)
							0.931597	+	0.363492i	\(0.118416\pi\)
\(7\)	0.765630	+	2.53255i	0.289381	+	0.957214i
							\(11\)	−4.60525	+	3.62161i	−1.38853	+	1.09196i	−0.404389	+	0.914587i	\(0.632516\pi\)
−0.984145	+	0.177368i	\(0.943242\pi\)
\(13\)	1.18727	−	0.763009i	0.329288	−	0.211621i	−0.365541	−	0.930795i	\(-0.619116\pi\)
							0.694830	+	0.719174i	\(0.255480\pi\)
\(17\)	−2.31441	−	2.20678i	−0.561327	−	0.535224i	0.355413	−	0.934709i	\(-0.384340\pi\)
							−0.916739	+	0.399486i	\(0.869189\pi\)
\(19\)	1.93459	−	1.84463i	0.443825	−	0.423186i	−0.434926	−	0.900466i	\(-0.643226\pi\)
							0.878751	+	0.477280i	\(0.158377\pi\)
\(23\)	−2.45258	+	4.12127i	−0.511399	+	0.859343i
							\(29\)	4.98434	−	1.46353i	0.925569	−	0.271772i	0.215988	−	0.976396i	\(-0.430703\pi\)
0.709580	+	0.704624i	\(0.248884\pi\)
\(31\)	6.89186	−	0.658093i	1.23782	−	0.118197i	0.544456	−	0.838789i	\(-0.316736\pi\)
							0.693359	+	0.720592i	\(0.256130\pi\)
\(37\)	0.106158	+	0.306723i	0.0174522	+	0.0504249i	0.953391	−	0.301739i	\(-0.0975671\pi\)
							−0.935938	+	0.352164i	\(0.885446\pi\)
\(41\)	−6.54967	−	7.55872i	−1.02289	−	1.18047i	−0.983437	−	0.181253i	\(-0.941985\pi\)
							−0.0394503	−	0.999222i	\(-0.512561\pi\)
\(43\)	−4.88584	−	10.6985i	−0.745084	−	1.63151i	−0.775007	−	0.631953i	\(-0.782253\pi\)
							0.0299229	−	0.999552i	\(-0.490474\pi\)
\(47\)	4.21347	−	7.29795i	0.614598	−	1.06452i	−0.375856	−	0.926678i	\(-0.622651\pi\)
							0.990455	−	0.137838i	\(-0.0440152\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.b.261.4	yes	160
7.4	even	3	inner	322.2.m.b.123.5	yes	160
23.3	even	11	inner	322.2.m.b.233.5	yes	160
161.95	even	33	inner	322.2.m.b.95.4	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.b.95.4	✓	160	161.95	even	33	inner
322.2.m.b.123.5	yes	160	7.4	even	3	inner
322.2.m.b.233.5	yes	160	23.3	even	11	inner
322.2.m.b.261.4	yes	160	1.1	even	1	trivial