Embedded newform 473.2.v.a.47.16

• Label: 473.2.v.a.47.16
• Level: $473$
• Weight: $2$
• Character: 473.47
• Analytic conductor: $3.777$
• Analytic rank: $0$
• Dimension: $1008$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 473 = 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	473.v (of order \(35\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.77692401561\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(42\) over \(\Q(\zeta_{35})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{35}]$

Embedding invariants

Embedding label			47.16
Character	\(\chi\)	\(=\)	473.47
Dual form			473.2.v.a.312.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0344011 + 0.766000i) q^{2} +(-0.661299 + 1.00183i) q^{3} +(1.40638 + 0.126576i) q^{4} +(-0.991531 - 1.84257i) q^{5} +(-0.744649 - 0.541019i) q^{6} +(1.02720 - 3.16140i) q^{7} +(-0.351191 + 2.59259i) q^{8} +(0.612737 + 1.43357i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 21 q^{2} - 19 q^{3} + 21 q^{4} - 19 q^{5} - 38 q^{6} - 6 q^{7} - 63 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/473\mathbb{Z}\right)^\times\).

\(n\)	\(89\)	\(431\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0344011	+	0.766000i	−0.0243252	+	0.541644i	0.949421	+	0.314007i	\(0.101671\pi\)
							−0.973746	+	0.227637i	\(0.926900\pi\)
\(3\)	−0.661299	+	1.00183i	−0.381801	+	0.578404i	−0.973764	−	0.227561i	\(-0.926925\pi\)
							0.591962	+	0.805966i	\(0.298353\pi\)
\(5\)	−0.991531	−	1.84257i	−0.443426	−	0.824024i	0.556551	−	0.830814i	\(-0.312124\pi\)
							−0.999977	+	0.00678978i	\(0.997839\pi\)
\(7\)	1.02720	−	3.16140i	0.388246	−	1.19490i	−0.545852	−	0.837882i	\(-0.683794\pi\)
							0.934098	−	0.357017i	\(-0.116206\pi\)
\(11\)	2.83701	−	1.71796i	0.855390	−	0.517984i
							\(13\)	−2.59227	+	0.470427i	−0.718966	+	0.130473i	−0.525696	−	0.850673i	\(-0.676195\pi\)
−0.193270	+	0.981146i	\(0.561909\pi\)
\(17\)	2.82129	+	0.511988i	0.684263	+	0.124175i	0.509546	−	0.860443i	\(-0.329813\pi\)
							0.174717	+	0.984619i	\(0.444099\pi\)
\(19\)	5.59145	−	3.34074i	1.28277	−	0.766417i	0.300478	−	0.953789i	\(-0.402854\pi\)
							0.982289	+	0.187372i	\(0.0599968\pi\)
\(23\)	0.781717	−	3.42493i	0.162999	−	0.714146i	−0.825685	−	0.564132i	\(-0.809211\pi\)
							0.988684	−	0.150014i	\(-0.0479320\pi\)
\(29\)	4.42724	+	6.70698i	0.822117	+	1.24545i	0.966687	+	0.255960i	\(0.0823916\pi\)
							−0.144570	+	0.989495i	\(0.546180\pi\)
\(31\)	−0.411114	+	9.15416i	−0.0738382	+	1.64414i	0.530700	+	0.847560i	\(0.321929\pi\)
							−0.604538	+	0.796576i	\(0.706642\pi\)
\(37\)	1.08352	−	3.33474i	0.178130	−	0.548227i	−0.821633	−	0.570017i	\(-0.806936\pi\)
							0.999763	+	0.0217899i	\(0.00693650\pi\)
\(41\)	−9.71244	−	3.64514i	−1.51683	−	0.569275i	−0.552390	−	0.833586i	\(-0.686284\pi\)
							−0.964438	+	0.264311i	\(0.914856\pi\)
\(43\)	6.54479	−	0.407106i	0.998071	−	0.0620831i
							\(47\)	−8.19156	+	4.89423i	−1.19486	+	0.713896i	−0.965743	−	0.259502i	\(-0.916442\pi\)
−0.229119	+	0.973399i	\(0.573584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	473.2.v.a.47.16	yes	1008
11.4	even	5	inner	473.2.v.a.4.16	✓	1008
43.11	even	7	inner	473.2.v.a.355.16	yes	1008
473.312	even	35	inner	473.2.v.a.312.16	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
473.2.v.a.4.16	✓	1008	11.4	even	5	inner
473.2.v.a.47.16	yes	1008	1.1	even	1	trivial
473.2.v.a.312.16	yes	1008	473.312	even	35	inner
473.2.v.a.355.16	yes	1008	43.11	even	7	inner