Embedded newform 65.4.o.a.2.13

• Label: 65.4.o.a.2.13
• Level: $65$
• Weight: $4$
• Character: 65.2
• Analytic conductor: $3.835$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	65.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.83512415037\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(19\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	65.2
Dual form			65.4.o.a.33.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.898209 + 1.55574i) q^{2} +(-7.31291 - 1.95949i) q^{3} +(2.38644 - 4.13344i) q^{4} +(7.07654 + 8.65578i) q^{5} +(-3.52006 - 13.1370i) q^{6} +(21.3001 + 12.2976i) q^{7} +22.9454 q^{8} +(26.2564 + 15.1591i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 6 q^{2} - 2 q^{3} - 136 q^{4} + 10 q^{5} - 8 q^{6} - 6 q^{7} + 120 q^{8} + 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.898209	+	1.55574i	0.317565	+	0.550039i	0.979979	−	0.199099i	\(-0.0638015\pi\)
							−0.662414	+	0.749138i	\(0.730468\pi\)
\(3\)	−7.31291	−	1.95949i	−1.40737	−	0.377104i	−0.526384	−	0.850247i	\(-0.676453\pi\)
							−0.880986	+	0.473143i	\(0.843119\pi\)
\(5\)	7.07654	+	8.65578i	0.632945	+	0.774197i
							\(7\)	21.3001	+	12.2976i	1.15010	+	0.664008i	0.948910	−	0.315546i	\(-0.102188\pi\)
0.201185	+	0.979553i	\(0.435521\pi\)
\(11\)	6.73467	−	25.1341i	0.184598	−	0.688929i	−0.810118	−	0.586267i	\(-0.800597\pi\)
							0.994716	−	0.102663i	\(-0.0327363\pi\)
\(13\)	46.0344	+	8.82246i	0.982126	+	0.188224i
							\(17\)	9.63872	+	35.9722i	0.137514	+	0.513208i	0.999975	+	0.00708497i	\(0.00225523\pi\)
−0.862461	+	0.506123i	\(0.831078\pi\)
\(19\)	−121.505	+	32.5572i	−1.46712	+	0.393113i	−0.901942	−	0.431858i	\(-0.857858\pi\)
							−0.565176	+	0.824971i	\(0.691192\pi\)
\(23\)	28.5044	−	106.380i	0.258416	−	0.964423i	−0.707741	−	0.706472i	\(-0.750286\pi\)
							0.966158	−	0.257952i	\(-0.0830475\pi\)
\(29\)	179.102	−	103.405i	1.14684	−	0.662130i	0.198726	−	0.980055i	\(-0.436319\pi\)
							0.948116	+	0.317925i	\(0.102986\pi\)
\(31\)	25.3577	−	25.3577i	0.146916	−	0.146916i	−0.629823	−	0.776739i	\(-0.716873\pi\)
							0.776739	+	0.629823i	\(0.216873\pi\)
\(37\)	−320.124	+	184.824i	−1.42238	+	0.821212i	−0.996502	−	0.0835677i	\(-0.973369\pi\)
							−0.425879	+	0.904780i	\(0.640035\pi\)
\(41\)	−114.474	−	30.6731i	−0.436043	−	0.116837i	0.0341185	−	0.999418i	\(-0.489138\pi\)
							−0.470161	+	0.882580i	\(0.655804\pi\)
\(43\)	−189.249	+	50.7091i	−0.671167	+	0.179839i	−0.578280	−	0.815838i	\(-0.696276\pi\)
							−0.0928865	+	0.995677i	\(0.529609\pi\)
\(47\)		−	22.3546i		−	0.0693777i	−0.999398	−	0.0346889i	\(-0.988956\pi\)
							0.999398	−	0.0346889i	\(-0.0110440\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.4.o.a.2.13	✓	76
5.3	odd	4		65.4.t.a.28.13	yes	76
13.7	odd	12		65.4.t.a.7.13	yes	76
65.33	even	12	inner	65.4.o.a.33.13	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.4.o.a.2.13	✓	76	1.1	even	1	trivial
65.4.o.a.33.13	yes	76	65.33	even	12	inner
65.4.t.a.7.13	yes	76	13.7	odd	12
65.4.t.a.28.13	yes	76	5.3	odd	4