Embedded newform 869.2.u.b.45.18

• Label: 869.2.u.b.45.18
• Level: $869$
• Weight: $2$
• Character: 869.45
• Analytic conductor: $6.939$
• Analytic rank: $0$
• Dimension: $792$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	869.u (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.93899993565\)
Analytic rank:	\(0\)
Dimension:	\(792\)
Relative dimension:	\(33\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			45.18
Character	\(\chi\)	\(=\)	869.45
Dual form			869.2.u.b.309.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.191585 + 0.234649i) q^{2} +(1.52830 - 0.966439i) q^{3} +(0.381696 - 1.86967i) q^{4} +(1.83687 - 0.782616i) q^{5} +(0.519574 + 0.173459i) q^{6} +(0.123405 + 3.06227i) q^{7} +(1.04830 - 0.550192i) q^{8} +(0.115621 - 0.243666i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 792 q + 3 q^{3} + 36 q^{4} + 8 q^{6} - 9 q^{7} + 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/869\mathbb{Z}\right)^\times\).

\(n\)	\(475\)	\(793\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{32}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.191585	+	0.234649i	0.135471	+	0.165922i	0.837885	−	0.545847i	\(-0.183792\pi\)
							−0.702414	+	0.711769i	\(0.747894\pi\)
\(3\)	1.52830	−	0.966439i	0.882365	−	0.557974i	−0.0146905	−	0.999892i	\(-0.504676\pi\)
							0.897055	+	0.441918i	\(0.145702\pi\)
\(5\)	1.83687	−	0.782616i	0.821472	−	0.349996i	0.0600030	−	0.998198i	\(-0.480889\pi\)
							0.761469	+	0.648202i	\(0.224479\pi\)
\(7\)	0.123405	+	3.06227i	0.0466428	+	1.15743i	0.837857	+	0.545889i	\(0.183808\pi\)
							−0.791214	+	0.611539i	\(0.790551\pi\)
\(11\)	0.799443	−	0.600742i	0.241041	−	0.181131i
							\(13\)	4.93182	−	1.64648i	1.36784	−	0.456653i	0.464500	−	0.885573i	\(-0.346234\pi\)
0.903342	+	0.428921i	\(0.141106\pi\)
\(17\)	−0.838074	−	0.742468i	−0.203263	−	0.180075i	0.555350	−	0.831617i	\(-0.312584\pi\)
							−0.758613	+	0.651541i	\(0.774123\pi\)
\(19\)	−0.00506444	−	0.00527265i	−0.00116186	−	0.00120963i	0.720621	−	0.693329i	\(-0.243857\pi\)
							−0.721783	+	0.692119i	\(0.756677\pi\)
\(23\)	0.728877	+	1.26245i	0.151981	+	0.263239i	0.931956	−	0.362572i	\(-0.118101\pi\)
							−0.779974	+	0.625811i	\(0.784768\pi\)
\(29\)	3.07434	+	0.248186i	0.570890	+	0.0460870i	0.362541	−	0.931968i	\(-0.381909\pi\)
							0.208349	+	0.978055i	\(0.433191\pi\)
\(31\)	−9.83605	+	1.59851i	−1.76661	+	0.287102i	−0.954892	−	0.296953i	\(-0.904030\pi\)
							−0.811715	+	0.584054i	\(0.801466\pi\)
\(37\)	0.613255	−	2.11720i	0.100819	−	0.348066i	−0.894211	−	0.447646i	\(-0.852263\pi\)
							0.995030	+	0.0995796i	\(0.0317498\pi\)
\(41\)	−0.516773	+	4.25601i	−0.0807065	+	0.664677i	0.895073	+	0.445920i	\(0.147123\pi\)
							−0.975779	+	0.218757i	\(0.929800\pi\)
\(43\)	−5.02416	−	3.77541i	−0.766178	−	0.575745i	0.144188	−	0.989550i	\(-0.453943\pi\)
							−0.910366	+	0.413805i	\(0.864200\pi\)
\(47\)	−1.48300	−	5.11993i	−0.216318	−	0.746818i	−0.993123	−	0.117076i	\(-0.962648\pi\)
							0.776804	−	0.629742i	\(-0.216839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.2.u.b.45.18	✓	792
79.72	even	39	inner	869.2.u.b.309.18	yes	792

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.2.u.b.45.18	✓	792	1.1	even	1	trivial
869.2.u.b.309.18	yes	792	79.72	even	39	inner