Embedded newform 62.8.g.a.19.1

• Label: 62.8.g.a.19.1
• Level: $62$
• Weight: $8$
• Character: 62.19
• Analytic conductor: $19.368$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 62 = 2 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	62.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.3678715800\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			19.1
Character	\(\chi\)	\(=\)	62.19
Dual form			62.8.g.a.49.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.47214 - 4.70228i) q^{2} +(-68.5039 - 30.4999i) q^{3} +(19.7771 + 60.8676i) q^{4} +(-145.456 + 251.937i) q^{5} +(299.947 + 519.524i) q^{6} +(731.830 - 812.780i) q^{7} +(158.217 - 486.941i) q^{8} +(2299.15 + 2553.47i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 144 q^{2} - q^{3} - 1152 q^{4} + 45 q^{5} + 1072 q^{6} - 1482 q^{7} - 9216 q^{8} + 3248 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.47214	−	4.70228i	−0.572061	−	0.415627i
							\(3\)	−68.5039	−	30.4999i	−1.46484	−	0.652190i	−0.489323	−	0.872103i	\(-0.662756\pi\)
−0.975519	+	0.219913i	\(0.929423\pi\)
\(5\)	−145.456	+	251.937i	−0.520398	+	0.901356i	0.479321	+	0.877640i	\(0.340883\pi\)
							−0.999719	+	0.0237162i	\(0.992450\pi\)
\(7\)	731.830	−	812.780i	0.806431	−	0.895633i	−0.189848	−	0.981813i	\(-0.560800\pi\)
							0.996280	+	0.0861809i	\(0.0274663\pi\)
\(11\)	3005.72	+	638.886i	0.680886	+	0.144727i	0.535355	−	0.844627i	\(-0.320178\pi\)
							0.145530	+	0.989354i	\(0.453511\pi\)
\(13\)	−641.785	−	6106.18i	−0.0810192	−	0.770846i	−0.957310	−	0.289063i	\(-0.906656\pi\)
							0.876291	−	0.481783i	\(-0.160011\pi\)
\(17\)	14350.1	−	3050.21i	0.708408	−	0.150577i	0.160403	−	0.987052i	\(-0.448721\pi\)
							0.548005	+	0.836475i	\(0.315387\pi\)
\(19\)	−2255.49	+	21459.6i	−0.0754404	+	0.717768i	0.889791	+	0.456369i	\(0.150850\pi\)
							−0.965231	+	0.261398i	\(0.915816\pi\)
\(23\)	−20106.4	+	61881.1i	−0.344577	+	1.06050i	0.617233	+	0.786781i	\(0.288254\pi\)
							−0.961810	+	0.273719i	\(0.911746\pi\)
\(29\)	−148118.	−	107614.i	−1.12775	−	0.819361i	−0.142387	−	0.989811i	\(-0.545478\pi\)
							−0.985366	+	0.170451i	\(0.945478\pi\)
\(31\)	42485.7	−	160336.i	0.256140	−	0.966640i
							\(37\)	−188852.	−	327102.i	−0.612937	−	1.06164i	−0.990743	−	0.135753i	\(-0.956654\pi\)
0.377805	−	0.925885i	\(-0.376679\pi\)
\(41\)	−110330.	+	49122.2i	−0.250006	+	0.111310i	−0.527914	−	0.849298i	\(-0.677026\pi\)
							0.277908	+	0.960608i	\(0.410359\pi\)
\(43\)	54841.8	−	521785.i	0.105189	−	1.00081i	−0.806863	−	0.590738i	\(-0.798837\pi\)
							0.912053	−	0.410073i	\(-0.134497\pi\)
\(47\)	196912.	−	143065.i	0.276650	−	0.200998i	−0.440805	−	0.897603i	\(-0.645307\pi\)
							0.717455	+	0.696605i	\(0.245307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	62.8.g.a.19.1	✓	72
31.18	even	15	inner	62.8.g.a.49.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.8.g.a.19.1	✓	72	1.1	even	1	trivial
62.8.g.a.49.1	yes	72	31.18	even	15	inner