Embedded newform 161.2.m.a.100.3

• Label: 161.2.m.a.100.3
• Level: $161$
• Weight: $2$
• Character: 161.100
• Analytic conductor: $1.286$
• Analytic rank: $0$
• Dimension: $280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	161.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.28559147254\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(14\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			100.3
Character	\(\chi\)	\(=\)	161.100
Dual form			161.2.m.a.95.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50843 - 1.18624i) q^{2} +(-0.564492 + 0.792718i) q^{3} +(0.396672 + 1.63510i) q^{4} +(-0.812668 + 0.156629i) q^{5} +(1.79185 - 0.526135i) q^{6} +(-1.33836 - 2.28228i) q^{7} +(-0.253079 + 0.554166i) q^{8} +(0.671453 + 1.94004i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 11 q^{2} - 9 q^{3} + q^{4} - 11 q^{5} - 52 q^{6} - 20 q^{7} - 32 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50843	−	1.18624i	−1.06662	−	0.838800i	−0.0791913	−	0.996859i	\(-0.525234\pi\)
							−0.987430	+	0.158059i	\(0.949476\pi\)
\(3\)	−0.564492	+	0.792718i	−0.325910	+	0.457676i	−0.944727	−	0.327859i	\(-0.893673\pi\)
							0.618817	+	0.785535i	\(0.287612\pi\)
\(5\)	−0.812668	+	0.156629i	−0.363436	+	0.0700466i	−0.367700	−	0.929944i	\(-0.619855\pi\)
							0.00426412	+	0.999991i	\(0.498643\pi\)
\(7\)	−1.33836	−	2.28228i	−0.505852	−	0.862620i
							\(11\)	−3.74569	+	2.94565i	−1.12937	+	0.888146i	−0.994708	−	0.102746i	\(-0.967237\pi\)
−0.134662	+	0.990892i	\(0.542995\pi\)
\(13\)	−3.75437	+	2.41279i	−1.04127	+	0.669187i	−0.945302	−	0.326197i	\(-0.894233\pi\)
							−0.0959731	+	0.995384i	\(0.530596\pi\)
\(17\)	4.06851	+	3.87931i	0.986757	+	0.940871i	0.998254	−	0.0590626i	\(-0.0188112\pi\)
							−0.0114969	+	0.999934i	\(0.503660\pi\)
\(19\)	0.735584	−	0.701378i	0.168755	−	0.160907i	−0.600996	−	0.799252i	\(-0.705229\pi\)
							0.769750	+	0.638345i	\(0.220381\pi\)
\(23\)	−4.76191	−	0.569381i	−0.992927	−	0.118724i
							\(29\)	−6.21460	+	1.82477i	−1.15402	+	0.338851i	−0.802107	−	0.597181i	\(-0.796288\pi\)
−0.351915	+	0.936032i	\(0.614469\pi\)
\(31\)	9.48157	−	0.905381i	1.70294	−	0.162611i	0.802226	−	0.597021i	\(-0.203649\pi\)
							0.900715	+	0.434410i	\(0.143043\pi\)
\(37\)	−3.19562	−	9.23314i	−0.525357	−	1.51792i	−0.825712	−	0.564091i	\(-0.809227\pi\)
							0.300356	−	0.953827i	\(-0.402895\pi\)
\(41\)	1.65752	+	1.91288i	0.258862	+	0.298742i	0.870272	−	0.492571i	\(-0.163943\pi\)
							−0.611410	+	0.791314i	\(0.709398\pi\)
\(43\)	−1.75200	−	3.83634i	−0.267177	−	0.585037i	0.727726	−	0.685868i	\(-0.240577\pi\)
							−0.994904	+	0.100831i	\(0.967850\pi\)
\(47\)	1.59695	−	2.76601i	0.232940	−	0.403463i	−0.725732	−	0.687977i	\(-0.758499\pi\)
							0.958672	+	0.284514i	\(0.0918322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.m.a.100.3	yes	280
7.4	even	3	inner	161.2.m.a.123.12	yes	280
23.3	even	11	inner	161.2.m.a.72.12	✓	280
161.95	even	33	inner	161.2.m.a.95.3	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.m.a.72.12	✓	280	23.3	even	11	inner
161.2.m.a.95.3	yes	280	161.95	even	33	inner
161.2.m.a.100.3	yes	280	1.1	even	1	trivial
161.2.m.a.123.12	yes	280	7.4	even	3	inner