Embedded newform 731.2.u.b.52.19

• Label: 731.2.u.b.52.19
• Level: $731$
• Weight: $2$
• Character: 731.52
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.u (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(29\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			52.19
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.443477 + 0.556103i) q^{2} +(1.01721 + 0.153320i) q^{3} +(0.332464 - 1.45662i) q^{4} +(2.99169 + 2.03970i) q^{5} +(0.365847 + 0.633666i) q^{6} +(0.454177 - 0.786658i) q^{7} +(2.23916 - 1.07832i) q^{8} +(-1.85551 - 0.572349i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.443477	+	0.556103i	0.313586	+	0.393224i	0.913499	−	0.406841i	\(-0.133370\pi\)
							−0.599913	+	0.800065i	\(0.704798\pi\)
\(3\)	1.01721	+	0.153320i	0.587286	+	0.0885191i	0.435962	−	0.899965i	\(-0.356408\pi\)
							0.151324	+	0.988484i	\(0.451646\pi\)
\(5\)	2.99169	+	2.03970i	1.33793	+	0.912183i	0.999584	−	0.0288405i	\(-0.00918148\pi\)
							0.338342	+	0.941023i	\(0.390134\pi\)
\(7\)	0.454177	−	0.786658i	0.171663	−	0.297329i	−0.767339	−	0.641242i	\(-0.778419\pi\)
							0.939001	+	0.343914i	\(0.111753\pi\)
\(11\)	1.36988	+	6.00183i	0.413034	+	1.80962i	0.569564	+	0.821947i	\(0.307112\pi\)
							−0.156530	+	0.987673i	\(0.550031\pi\)
\(13\)	0.126102	−	1.68272i	0.0349744	−	0.466702i	−0.952030	−	0.306006i	\(-0.901007\pi\)
							0.987004	−	0.160696i	\(-0.0513737\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	−1.18289	+	0.364873i	−0.271374	+	0.0837076i	−0.427455	−	0.904036i	\(-0.640590\pi\)
0.156082	+	0.987744i	\(0.450114\pi\)
\(23\)	1.36575	−	1.26723i	0.284779	−	0.264236i	−0.524829	−	0.851208i	\(-0.675871\pi\)
							0.809607	+	0.586972i	\(0.199680\pi\)
\(29\)	−0.274022	+	0.0413022i	−0.0508846	+	0.00766963i	−0.174435	−	0.984669i	\(-0.555810\pi\)
							0.123551	+	0.992338i	\(0.460572\pi\)
\(31\)	1.11610	−	2.84377i	0.200457	−	0.510756i	−0.794929	−	0.606702i	\(-0.792492\pi\)
							0.995387	+	0.0959454i	\(0.0305874\pi\)
\(37\)	−4.98439	−	8.63322i	−0.819429	−	1.41929i	−0.906103	−	0.423056i	\(-0.860957\pi\)
							0.0866741	−	0.996237i	\(-0.472376\pi\)
\(41\)	1.52418	+	1.91126i	0.238036	+	0.298488i	0.886473	−	0.462781i	\(-0.153148\pi\)
							−0.648437	+	0.761269i	\(0.724577\pi\)
\(43\)	−6.06789	−	2.48611i	−0.925344	−	0.379128i
							\(47\)	1.50673	−	6.60142i	0.219779	−	0.962916i	−0.737862	−	0.674952i	\(-0.764164\pi\)
0.957641	−	0.287964i	\(-0.0929785\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.19	✓	348
43.24	even	21	inner	731.2.u.b.239.19	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.19	✓	348	1.1	even	1	trivial
731.2.u.b.239.19	yes	348	43.24	even	21	inner