Embedded newform 161.2.m.a.100.14

• Label: 161.2.m.a.100.14
• 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			100.14
Character	\(\chi\)	\(=\)	161.100
Dual form			161.2.m.a.95.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.92438 + 1.51335i) q^{2} +(-0.261853 + 0.367721i) q^{3} +(0.941492 + 3.88088i) q^{4} +(-1.46274 + 0.281919i) q^{5} +(-1.06040 + 0.311360i) q^{6} +(-2.01990 - 1.70881i) q^{7} +(-2.02734 + 4.43926i) q^{8} +(0.914552 + 2.64242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 11 q^{2} - 9 q^{3} + q^{4} - 11 q^{5} - 52 q^{6} - 20 q^{7} - 32 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.92438	+	1.51335i	1.36074	+	1.07010i	0.989955	+	0.141380i	\(0.0451539\pi\)
							0.370786	+	0.928719i	\(0.379088\pi\)
\(3\)	−0.261853	+	0.367721i	−0.151181	+	0.212304i	−0.883220	−	0.468958i	\(-0.844629\pi\)
							0.732039	+	0.681263i	\(0.238569\pi\)
\(5\)	−1.46274	+	0.281919i	−0.654156	+	0.126078i	−0.505521	−	0.862814i	\(-0.668700\pi\)
							−0.148634	+	0.988892i	\(0.547488\pi\)
\(7\)	−2.01990	−	1.70881i	−0.763449	−	0.645868i
							\(11\)	4.03256	−	3.17124i	1.21586	−	0.956164i	0.216066	−	0.976379i	\(-0.430677\pi\)
0.999796	+	0.0202143i	\(0.00643484\pi\)
\(13\)	5.59553	−	3.59603i	1.55192	−	0.997358i	0.567126	−	0.823631i	\(-0.308055\pi\)
							0.984794	−	0.173728i	\(-0.0555813\pi\)
\(17\)	−1.75038	−	1.66899i	−0.424530	−	0.404788i	0.447467	−	0.894301i	\(-0.352326\pi\)
							−0.871996	+	0.489512i	\(0.837175\pi\)
\(19\)	−4.76581	+	4.54419i	−1.09335	+	1.04251i	−0.0944567	+	0.995529i	\(0.530111\pi\)
							−0.998896	+	0.0469805i	\(0.985040\pi\)
\(23\)	−4.13118	−	2.43585i	−0.861411	−	0.507909i
							\(29\)	−3.87635	+	1.13820i	−0.719820	+	0.211358i	−0.621071	−	0.783754i	\(-0.713302\pi\)
−0.0987486	+	0.995112i	\(0.531484\pi\)
\(31\)	−1.22113	+	0.116604i	−0.219321	+	0.0209426i	−0.204138	−	0.978942i	\(-0.565439\pi\)
							−0.0151831	+	0.999885i	\(0.504833\pi\)
\(37\)	−0.215287	−	0.622030i	−0.0353929	−	0.102261i	0.925920	−	0.377719i	\(-0.123292\pi\)
							−0.961313	+	0.275458i	\(0.911170\pi\)
\(41\)	−0.187892	−	0.216838i	−0.0293437	−	0.0338645i	0.740888	−	0.671629i	\(-0.234405\pi\)
							−0.770232	+	0.637764i	\(0.779859\pi\)
\(43\)	−0.936001	−	2.04956i	−0.142739	−	0.312555i	0.824738	−	0.565516i	\(-0.191323\pi\)
							−0.967476	+	0.252961i	\(0.918596\pi\)
\(47\)	3.09368	−	5.35841i	0.451259	−	0.781604i	−0.547205	−	0.836999i	\(-0.684308\pi\)
							0.998465	+	0.0553943i	\(0.0176416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.m.a.100.14	yes	280
7.4	even	3	inner	161.2.m.a.123.1	yes	280
23.3	even	11	inner	161.2.m.a.72.1	✓	280
161.95	even	33	inner	161.2.m.a.95.14	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.m.a.72.1	✓	280	23.3	even	11	inner
161.2.m.a.95.14	yes	280	161.95	even	33	inner
161.2.m.a.100.14	yes	280	1.1	even	1	trivial
161.2.m.a.123.1	yes	280	7.4	even	3	inner