Embedded newform 845.2.u.b.456.4

• Label: 845.2.u.b.456.4
• Level: $845$
• Weight: $2$
• Character: 845.456
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $372$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.u (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(372\)
Relative dimension:	\(31\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			456.4
Character	\(\chi\)	\(=\)	845.456
Dual form			845.2.u.b.391.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79741 - 1.59237i) q^{2} +(-0.481618 + 1.26992i) q^{3} +(0.453979 + 3.73885i) q^{4} +(0.970942 + 0.239316i) q^{5} +(2.88785 - 1.51566i) q^{6} +(0.683242 - 0.989846i) q^{7} +(2.40943 - 3.49066i) q^{8} +(0.864786 + 0.766133i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 372 q + q^{2} - 29 q^{4} + 31 q^{5} + 4 q^{6} - 13 q^{7} - 3 q^{8} - 31 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{3}{13}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79741	−	1.59237i	−1.27096	−	1.12597i	−0.985877	−	0.167470i	\(-0.946440\pi\)
							−0.285083	−	0.958503i	\(-0.592021\pi\)
\(3\)	−0.481618	+	1.26992i	−0.278062	+	0.733190i	0.721122	+	0.692808i	\(0.243627\pi\)
							−0.999184	+	0.0403819i	\(0.987143\pi\)
\(5\)	0.970942	+	0.239316i	0.434218	+	0.107025i
							\(7\)	0.683242	−	0.989846i	0.258241	−	0.374127i	−0.672322	−	0.740259i	\(-0.734703\pi\)
0.930563	+	0.366132i	\(0.119318\pi\)
\(11\)	0.0189822	−	0.0168168i	0.00572335	−	0.00507045i	−0.660256	−	0.751041i	\(-0.729552\pi\)
							0.665979	+	0.745970i	\(0.268014\pi\)
\(13\)	3.44642	+	1.05934i	0.955865	+	0.293807i
							\(17\)	4.20013	−	6.08493i	1.01868	−	1.47581i	0.144927	−	0.989442i	\(-0.453705\pi\)
0.873753	−	0.486370i	\(-0.161679\pi\)
\(19\)	−5.59744			−1.28414			−0.642070	−	0.766646i	\(-0.721924\pi\)
							−0.642070	+	0.766646i	\(0.721924\pi\)
\(23\)	−0.577481			−0.120413			−0.0602065	−	0.998186i	\(-0.519176\pi\)
							−0.0602065	+	0.998186i	\(0.519176\pi\)
\(29\)	3.85135	+	3.41200i	0.715178	+	0.633592i	0.940145	−	0.340774i	\(-0.110689\pi\)
							−0.224968	+	0.974366i	\(0.572228\pi\)
\(31\)	−5.22551	+	2.74256i	−0.938530	+	0.492579i	−0.863358	−	0.504592i	\(-0.831643\pi\)
							−0.0751721	+	0.997171i	\(0.523951\pi\)
\(37\)	6.84244	−	3.59119i	1.12489	−	0.590388i	0.203519	−	0.979071i	\(-0.434762\pi\)
							0.921372	+	0.388683i	\(0.127070\pi\)
\(41\)	0.184483	−	0.486441i	0.0288114	−	0.0759693i	−0.919839	−	0.392297i	\(-0.871681\pi\)
							0.948650	+	0.316328i	\(0.102450\pi\)
\(43\)	6.48032	+	3.40114i	0.988240	+	0.518668i	0.879754	−	0.475430i	\(-0.157707\pi\)
							0.108486	+	0.994098i	\(0.465400\pi\)
\(47\)	−0.173906	+	1.43224i	−0.0253668	+	0.208914i	−0.999881	−	0.0154104i	\(-0.995095\pi\)
							0.974514	+	0.224325i	\(0.0720176\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.u.b.456.4	yes	372
169.53	even	13	inner	845.2.u.b.391.4	✓	372

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.u.b.391.4	✓	372	169.53	even	13	inner
845.2.u.b.456.4	yes	372	1.1	even	1	trivial