Embedded newform 78.3.l.c.19.1

• Label: 78.3.l.c.19.1
• Level: $78$
• Weight: $3$
• Character: 78.19
• Analytic conductor: $2.125$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	78.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.12534606201\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.1
Root			\(-4.04651 - 4.04651i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.19
Dual form			78.3.l.c.37.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-5.04651 + 5.04651i) q^{5} +(2.36603 + 0.633975i) q^{6} +(-1.34715 + 5.02764i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.366025	−	1.36603i	−0.183013	−	0.683013i
							\(3\)	−0.866025	+	1.50000i	−0.288675	+	0.500000i
\(5\)	−5.04651	+	5.04651i	−1.00930	+	1.00930i								−0.00934664	+	0.999956i	\(0.502975\pi\)
							−0.999956	+	0.00934664i	\(0.997025\pi\)
\(7\)	−1.34715	+	5.02764i	−0.192450	+	0.718235i	0.800462	+	0.599384i	\(0.204588\pi\)
							−0.992912	+	0.118851i	\(0.962079\pi\)
\(11\)	−7.32323	+	1.96225i	−0.665748	+	0.178387i	−0.575839	−	0.817563i	\(-0.695324\pi\)
							−0.0899095	+	0.995950i	\(0.528658\pi\)
\(13\)	12.9213	+	1.42820i	0.993947	+	0.109862i
							\(17\)	−13.9968	+	8.08105i	−0.823341	+	0.475356i	−0.851567	−	0.524246i	\(-0.824347\pi\)
0.0282265	+	0.999602i	\(0.491014\pi\)
\(19\)	9.87664	+	2.64644i	0.519823	+	0.139286i	0.509185	−	0.860657i	\(-0.329947\pi\)
							0.0106384	+	0.999943i	\(0.496614\pi\)
\(23\)	−8.29191	−	4.78733i	−0.360518	−	0.208145i	0.308790	−	0.951130i	\(-0.400076\pi\)
							−0.669308	+	0.742985i	\(0.733409\pi\)
\(29\)	16.5880	−	28.7312i	0.571999	−	0.990731i	−0.424362	−	0.905493i	\(-0.639501\pi\)
							0.996361	−	0.0852386i	\(-0.0271652\pi\)
\(31\)	−34.2312	+	34.2312i	−1.10423	+	1.10423i	−0.110338	+	0.993894i	\(0.535193\pi\)
							−0.993894	+	0.110338i	\(0.964807\pi\)
\(37\)	63.2267	−	16.9415i	1.70883	−	0.457879i	0.733692	−	0.679482i	\(-0.237796\pi\)
							0.975137	+	0.221603i	\(0.0711289\pi\)
\(41\)	14.0962	+	52.6079i	0.343811	+	1.28312i	0.893995	+	0.448076i	\(0.147891\pi\)
							−0.550185	+	0.835043i	\(0.685443\pi\)
\(43\)	−40.7432	+	23.5231i	−0.947515	+	0.547048i	−0.892308	−	0.451427i	\(-0.850915\pi\)
							−0.0552070	+	0.998475i	\(0.517582\pi\)
\(47\)	47.8214	+	47.8214i	1.01748	+	1.01748i	0.999845	+	0.0176317i	\(0.00561265\pi\)
							0.0176317	+	0.999845i	\(0.494387\pi\)