Embedded newform 845.2.u.a.456.6

• Label: 845.2.u.a.456.6
• 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			456.6
Character	\(\chi\)	\(=\)	845.456
Dual form			845.2.u.a.391.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37582 - 1.21887i) q^{2} +(-1.03338 + 2.72481i) q^{3} +(0.166162 + 1.36847i) q^{4} +(-0.970942 - 0.239316i) q^{5} +(4.74293 - 2.48928i) q^{6} +(-0.893486 + 1.29444i) q^{7} +(-0.648918 + 0.940119i) q^{8} +(-4.11117 - 3.64218i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 372 q - q^{2} - 4 q^{3} - 37 q^{4} - 31 q^{5} - 16 q^{6} + q^{7} - 9 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37582	−	1.21887i	−0.972851	−	0.861870i	0.0175851	−	0.999845i	\(-0.494402\pi\)
							−0.990436	+	0.137975i	\(0.955941\pi\)
\(3\)	−1.03338	+	2.72481i	−0.596624	+	1.57317i	0.207198	+	0.978299i	\(0.433566\pi\)
							−0.803822	+	0.594870i	\(0.797204\pi\)
\(5\)	−0.970942	−	0.239316i	−0.434218	−	0.107025i
							\(7\)	−0.893486	+	1.29444i	−0.337706	+	0.489251i	−0.954640	−	0.297764i	\(-0.903759\pi\)
0.616934	+	0.787015i	\(0.288375\pi\)
\(11\)	1.06600	−	0.944396i	0.321412	−	0.284746i	−0.486953	−	0.873428i	\(-0.661892\pi\)
							0.808364	+	0.588682i	\(0.200353\pi\)
\(13\)	−2.93499	+	2.09424i	−0.814020	+	0.580837i
							\(17\)	−0.808755	+	1.17168i	−0.196152	+	0.284175i	−0.908701	−	0.417447i	\(-0.862925\pi\)
0.712550	+	0.701622i	\(0.247540\pi\)
\(19\)	−1.48746			−0.341247			−0.170623	−	0.985336i	\(-0.554578\pi\)
							−0.170623	+	0.985336i	\(0.554578\pi\)
\(23\)	−8.32141			−1.73513			−0.867567	−	0.497321i	\(-0.834317\pi\)
							−0.867567	+	0.497321i	\(0.834317\pi\)
\(29\)	3.08304	+	2.73134i	0.572506	+	0.507196i	0.898968	−	0.438014i	\(-0.144318\pi\)
							−0.326462	+	0.945210i	\(0.605856\pi\)
\(31\)	−4.02244	+	2.11114i	−0.722452	+	0.379172i	−0.785519	−	0.618837i	\(-0.787604\pi\)
							0.0630671	+	0.998009i	\(0.479912\pi\)
\(37\)	8.42891	−	4.42383i	1.38570	−	0.727273i	0.403480	−	0.914988i	\(-0.367800\pi\)
							0.982224	+	0.187715i	\(0.0601081\pi\)
\(41\)	1.32456	−	3.49258i	0.206862	−	0.545450i	−0.790927	−	0.611910i	\(-0.790401\pi\)
							0.997789	+	0.0664603i	\(0.0211706\pi\)
\(43\)	4.88468	+	2.56368i	0.744907	+	0.390957i	0.794064	−	0.607834i	\(-0.207961\pi\)
							−0.0491574	+	0.998791i	\(0.515654\pi\)
\(47\)	1.17298	−	9.66032i	0.171096	−	1.40910i	−0.613794	−	0.789467i	\(-0.710357\pi\)
							0.784890	−	0.619636i	\(-0.212720\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.u.a.456.6	yes	372
169.53	even	13	inner	845.2.u.a.391.6	✓	372

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.u.a.391.6	✓	372	169.53	even	13	inner
845.2.u.a.456.6	yes	372	1.1	even	1	trivial