Embedded newform 473.2.v.a.4.16

• Label: 473.2.v.a.4.16
• Level: $473$
• Weight: $2$
• Character: 473.4
• Analytic conductor: $3.777$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• 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			4.16
Character	\(\chi\)	\(=\)	473.4
Dual form			473.2.v.a.355.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.739140 + 0.203990i) q^{2} +(1.12386 - 0.421792i) q^{3} +(-1.21218 + 0.724245i) q^{4} +(-2.05879 + 0.373616i) q^{5} +(-0.744649 + 0.541019i) q^{6} +(1.02720 + 3.16140i) q^{7} +(1.80801 - 1.89103i) q^{8} +(-1.17406 + 1.02575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 21 q^{2} - 19 q^{3} + 21 q^{4} - 19 q^{5} - 38 q^{6} - 6 q^{7} - 63 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(89\)	\(431\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.739140	+	0.203990i	−0.522651	+	0.144242i	−0.517400	−	0.855744i	\(-0.673100\pi\)
							−0.00525087	+	0.999986i	\(0.501671\pi\)
\(3\)	1.12386	−	0.421792i	0.648861	−	0.243522i	−0.00517297	−	0.999987i	\(-0.501647\pi\)
							0.654034	+	0.756465i	\(0.273075\pi\)
\(5\)	−2.05879	+	0.373616i	−0.920719	+	0.167086i	−0.618167	−	0.786047i	\(-0.712124\pi\)
							−0.302552	+	0.953133i	\(0.597839\pi\)
\(7\)	1.02720	+	3.16140i	0.388246	+	1.19490i	0.934098	+	0.357017i	\(0.116206\pi\)
							−0.545852	+	0.837882i	\(0.683794\pi\)
\(11\)	−1.81066	−	2.77876i	−0.545935	−	0.837827i
							\(13\)	−1.24846	−	2.32002i	−0.346260	−	0.643459i	0.646589	−	0.762839i	\(-0.276195\pi\)
−0.992849	+	0.119380i	\(0.961909\pi\)
\(17\)	1.35876	−	2.52499i	0.329547	−	0.612400i	−0.660872	−	0.750499i	\(-0.729813\pi\)
							0.990419	+	0.138098i	\(0.0440990\pi\)
\(19\)	−6.48721	−	0.583860i	−1.48827	−	0.133947i	−0.684554	−	0.728962i	\(-0.740003\pi\)
							−0.803714	+	0.595015i	\(0.797146\pi\)
\(23\)	0.781717	−	3.42493i	0.162999	−	0.714146i	−0.825685	−	0.564132i	\(-0.809211\pi\)
							0.988684	−	0.150014i	\(-0.0479320\pi\)
\(29\)	−7.52397	−	2.82379i	−1.39717	−	0.524366i	−0.464651	−	0.885494i	\(-0.653820\pi\)
							−0.932516	+	0.361128i	\(0.882392\pi\)
\(31\)	−8.83317	+	2.43780i	−1.58648	+	0.437842i	−0.944402	−	0.328794i	\(-0.893358\pi\)
							−0.642082	+	0.766636i	\(0.721929\pi\)
\(37\)	1.08352	+	3.33474i	0.178130	+	0.548227i	0.999763	−	0.0217899i	\(-0.00693650\pi\)
							−0.821633	+	0.570017i	\(0.806936\pi\)
\(41\)	5.71497	+	8.65781i	0.892528	+	1.35212i	0.935604	+	0.353050i	\(0.114856\pi\)
							−0.0430763	+	0.999072i	\(0.513716\pi\)
\(43\)	6.54479	−	0.407106i	0.998071	−	0.0620831i
							\(47\)	9.50386	+	0.855363i	1.38628	+	0.124768i	0.757510	−	0.652823i	\(-0.226416\pi\)
0.628771	+	0.777591i	\(0.283558\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
473.2.v.a.4.16	✓	1008	1.1	even	1	trivial
473.2.v.a.47.16	yes	1008	11.3	even	5	inner
473.2.v.a.312.16	yes	1008	43.11	even	7	inner
473.2.v.a.355.16	yes	1008	473.355	even	35	inner