Embedded newform 869.2.l.c.315.18

• Label: 869.2.l.c.315.18
• Level: $869$
• Weight: $2$
• Character: 869.315
• Analytic conductor: $6.939$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	869.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.93899993565\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(66\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			315.18
Character	\(\chi\)	\(=\)	869.315
Dual form			869.2.l.c.789.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.959385 + 1.32048i) q^{2} +(1.91908 + 0.623546i) q^{3} +(-0.205214 - 0.631585i) q^{4} +(0.00922593 - 0.00670303i) q^{5} +(-2.66451 + 1.93588i) q^{6} +(-0.0213735 - 0.0657810i) q^{7} +(-2.07376 - 0.673806i) q^{8} +(0.866998 + 0.629911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 20 q^{2} + 48 q^{4} - 20 q^{5} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(475\)	\(793\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.959385	+	1.32048i	−0.678388	+	0.933720i	−0.999913	−	0.0131805i	\(-0.995804\pi\)
							0.321526	+	0.946901i	\(0.395804\pi\)
\(3\)	1.91908	+	0.623546i	1.10798	+	0.360005i	0.805169	−	0.593046i	\(-0.202075\pi\)
							0.302811	+	0.953051i	\(0.402075\pi\)
\(5\)	0.00922593	−	0.00670303i	0.00412596	−	0.00299769i	−0.585720	−	0.810513i	\(-0.699188\pi\)
							0.589846	+	0.807516i	\(0.299188\pi\)
\(7\)	−0.0213735	−	0.0657810i	−0.00807844	−	0.0248629i	0.946936	−	0.321422i	\(-0.104161\pi\)
							−0.955014	+	0.296559i	\(0.904161\pi\)
\(11\)	−0.561612	+	3.26873i	−0.169333	+	0.985559i
							\(13\)	−2.18394	+	3.00594i	−0.605717	+	0.833698i	−0.996217	−	0.0869054i	\(-0.972302\pi\)
0.390500	+	0.920603i	\(0.372302\pi\)
\(17\)	−1.38296	+	1.00478i	−0.335417	+	0.243695i	−0.742726	−	0.669596i	\(-0.766467\pi\)
							0.407309	+	0.913291i	\(0.366467\pi\)
\(19\)	−0.386850	−	0.125695i	−0.0887495	−	0.0288365i	0.264306	−	0.964439i	\(-0.414857\pi\)
							−0.353055	+	0.935602i	\(0.614857\pi\)
\(23\)	−0.348547			−0.0726772			−0.0363386	−	0.999340i	\(-0.511569\pi\)
							−0.0363386	+	0.999340i	\(0.511569\pi\)
\(29\)	3.11768	+	9.59523i	0.578938	+	1.78179i	0.622359	+	0.782732i	\(0.286174\pi\)
							−0.0434210	+	0.999057i	\(0.513826\pi\)
\(31\)	−5.87872	−	4.27114i	−1.05585	−	0.767120i	−0.0825336	−	0.996588i	\(-0.526301\pi\)
							−0.973316	+	0.229469i	\(0.926301\pi\)
\(37\)	0.881557	−	0.286435i	0.144927	−	0.0470897i	−0.235655	−	0.971837i	\(-0.575724\pi\)
							0.380582	+	0.924747i	\(0.375724\pi\)
\(41\)	−2.19589	+	6.75825i	−0.342940	+	1.05546i	0.619737	+	0.784809i	\(0.287239\pi\)
							−0.962677	+	0.270652i	\(0.912761\pi\)
\(43\)	−1.99762			−0.304635			−0.152317	−	0.988332i	\(-0.548674\pi\)
							−0.152317	+	0.988332i	\(0.548674\pi\)
\(47\)	3.85178	+	1.25152i	0.561840	+	0.182553i	0.576149	−	0.817345i	\(-0.304555\pi\)
							−0.0143087	+	0.999898i	\(0.504555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.2.l.c.315.18	yes	264
11.8	odd	10	inner	869.2.l.c.789.17	yes	264
79.78	odd	2	inner	869.2.l.c.315.17	✓	264
869.789	even	10	inner	869.2.l.c.789.18	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.2.l.c.315.17	✓	264	79.78	odd	2	inner
869.2.l.c.315.18	yes	264	1.1	even	1	trivial
869.2.l.c.789.17	yes	264	11.8	odd	10	inner
869.2.l.c.789.18	yes	264	869.789	even	10	inner