Embedded newform 171.3.z.a.101.12

• Label: 171.3.z.a.101.12
• Level: $171$
• Weight: $3$
• Character: 171.101
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.12
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90151 + 0.335287i) q^{2} +(1.88089 + 2.33714i) q^{3} +(-0.255464 + 0.0929814i) q^{4} +(5.49536 + 6.54912i) q^{5} +(-4.36014 - 3.81345i) q^{6} +(1.56768 + 2.71530i) q^{7} +(7.14321 - 4.12414i) q^{8} +(-1.92447 + 8.79184i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90151	+	0.335287i	−0.950753	+	0.167643i	−0.627454	−	0.778654i	\(-0.715903\pi\)
							−0.323299	+	0.946297i	\(0.604792\pi\)
\(3\)	1.88089	+	2.33714i	0.626965	+	0.779048i
							\(5\)	5.49536	+	6.54912i	1.09907	+	1.30982i	0.946921	+	0.321466i	\(0.104176\pi\)
0.152151	+	0.988357i	\(0.451380\pi\)
\(7\)	1.56768	+	2.71530i	0.223954	+	0.387900i	0.956005	−	0.293350i	\(-0.0947701\pi\)
							−0.732051	+	0.681250i	\(0.761437\pi\)
\(11\)		−	8.67612i		−	0.788738i	−0.918952	−	0.394369i	\(-0.870963\pi\)
							0.918952	−	0.394369i	\(-0.129037\pi\)
\(13\)	5.85176	+	4.91021i	0.450135	+	0.377708i	0.839486	−	0.543381i	\(-0.182856\pi\)
							−0.389351	+	0.921090i	\(0.627301\pi\)
\(17\)	−13.6587	−	16.2778i	−0.803454	−	0.957519i	0.196281	−	0.980548i	\(-0.437114\pi\)
							−0.999735	+	0.0230283i	\(0.992669\pi\)
\(19\)	13.0789	+	13.7820i	0.688361	+	0.725368i
							\(23\)	−2.66589	−	7.32447i	−0.115908	−	0.318455i	0.868150	−	0.496302i	\(-0.165309\pi\)
−0.984058	+	0.177847i	\(0.943087\pi\)
\(29\)	−0.136901	−	0.376133i	−0.00472073	−	0.0129701i	0.937310	−	0.348497i	\(-0.113308\pi\)
							−0.942031	+	0.335527i	\(0.891086\pi\)
\(31\)	−36.3424			−1.17234			−0.586168	−	0.810189i	\(-0.699364\pi\)
							−0.586168	+	0.810189i	\(0.699364\pi\)
\(37\)	8.21178			0.221940			0.110970	−	0.993824i	\(-0.464604\pi\)
							0.110970	+	0.993824i	\(0.464604\pi\)
\(41\)	64.4479	−	11.3639i	1.57190	−	0.277169i	0.681317	−	0.731988i	\(-0.261407\pi\)
							0.890584	+	0.454820i	\(0.150296\pi\)
\(43\)	−24.9904	−	9.09575i	−0.581171	−	0.211529i	0.0346705	−	0.999399i	\(-0.488962\pi\)
							−0.615842	+	0.787870i	\(0.711184\pi\)
\(47\)	−4.49353	−	12.3459i	−0.0956071	−	0.262678i	0.882665	−	0.470002i	\(-0.155747\pi\)
							−0.978272	+	0.207324i	\(0.933525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.12	✓	228
9.5	odd	6		171.3.bf.a.158.12	yes	228
19.16	even	9		171.3.bf.a.92.12	yes	228
171.149	odd	18	inner	171.3.z.a.149.12	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.12	✓	228	1.1	even	1	trivial
171.3.z.a.149.12	yes	228	171.149	odd	18	inner
171.3.bf.a.92.12	yes	228	19.16	even	9
171.3.bf.a.158.12	yes	228	9.5	odd	6