Embedded newform 65.4.t.a.28.13

• Label: 65.4.t.a.28.13
• Level: $65$
• Weight: $4$
• Character: 65.28
• Analytic conductor: $3.835$
• Analytic rank: $0$
• Dimension: $76$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	65.t (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			28.13
Character	\(\chi\)	\(=\)	65.28
Dual form			65.4.t.a.7.13

$q$-expansion

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

Character values

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

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.55574	−	0.898209i	0.550039	−	0.317565i	−0.199099	−	0.979979i	\(-0.563802\pi\)
							0.749138	+	0.662414i	\(0.230468\pi\)
\(3\)	1.95949	−	7.31291i	0.377104	−	1.40737i	−0.473143	−	0.880986i	\(-0.656881\pi\)
							0.850247	−	0.526384i	\(-0.176453\pi\)
\(5\)	−8.65578	−	7.07654i	−0.774197	−	0.632945i
							\(7\)	12.2976	−	21.3001i	0.664008	−	1.15010i	−0.315546	−	0.948910i	\(-0.602188\pi\)
0.979553	−	0.201185i	\(-0.0644791\pi\)
\(11\)	6.73467	−	25.1341i	0.184598	−	0.688929i	−0.810118	−	0.586267i	\(-0.800597\pi\)
							0.994716	−	0.102663i	\(-0.0327363\pi\)
\(13\)	−8.82246	+	46.0344i	−0.188224	+	0.982126i
							\(17\)	35.9722	−	9.63872i	0.513208	−	0.137514i	0.00708497	−	0.999975i	\(-0.497745\pi\)
0.506123	+	0.862461i	\(0.331078\pi\)
\(19\)	121.505	−	32.5572i	1.46712	−	0.393113i	0.565176	−	0.824971i	\(-0.308808\pi\)
							0.901942	+	0.431858i	\(0.142142\pi\)
\(23\)	106.380	+	28.5044i	0.964423	+	0.258416i	0.706472	−	0.707741i	\(-0.250286\pi\)
							0.257952	+	0.966158i	\(0.416952\pi\)
\(29\)	−179.102	+	103.405i	−1.14684	+	0.662130i	−0.948116	−	0.317925i	\(-0.897014\pi\)
							−0.198726	+	0.980055i	\(0.563681\pi\)
\(31\)	25.3577	−	25.3577i	0.146916	−	0.146916i	−0.629823	−	0.776739i	\(-0.716873\pi\)
							0.776739	+	0.629823i	\(0.216873\pi\)
\(37\)	184.824	+	320.124i	0.821212	+	1.42238i	0.904780	+	0.425879i	\(0.140035\pi\)
							−0.0835677	+	0.996502i	\(0.526631\pi\)
\(41\)	−114.474	−	30.6731i	−0.436043	−	0.116837i	0.0341185	−	0.999418i	\(-0.489138\pi\)
							−0.470161	+	0.882580i	\(0.655804\pi\)
\(43\)	−50.7091	−	189.249i	−0.179839	−	0.671167i	−0.995677	−	0.0928865i	\(-0.970391\pi\)
							0.815838	−	0.578280i	\(-0.196276\pi\)
\(47\)	−22.3546			−0.0693777			−0.0346889	−	0.999398i	\(-0.511044\pi\)
							−0.0346889	+	0.999398i	\(0.511044\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.4.o.a.2.13	✓	76	5.2	odd	4
65.4.o.a.33.13	yes	76	13.7	odd	12
65.4.t.a.7.13	yes	76	65.7	even	12	inner
65.4.t.a.28.13	yes	76	1.1	even	1	trivial