Embedded newform 78.3.l.c.37.2

• Label: 78.3.l.c.37.2
• Level: $78$
• Weight: $3$
• Character: 78.37
• 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			37.2
Root			\(5.41254 - 5.41254i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.37
Dual form			78.3.l.c.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.366025 + 1.36603i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(4.41254 + 4.41254i) q^{5} +(2.36603 - 0.633975i) q^{6} +(2.11510 + 7.89367i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{12}\right)\)

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
\(5\)	4.41254	+	4.41254i	0.882508	+	0.882508i								0.993789	−	0.111281i	\(-0.0354953\pi\)
							−0.111281	+	0.993789i	\(0.535495\pi\)
\(7\)	2.11510	+	7.89367i	0.302157	+	1.12767i	0.935365	+	0.353685i	\(0.115071\pi\)
							−0.633207	+	0.773982i	\(0.718262\pi\)
\(11\)	18.5194	+	4.96225i	1.68358	+	0.451114i	0.968721	−	0.248154i	\(-0.0798238\pi\)
							0.714860	+	0.699268i	\(0.246490\pi\)
\(13\)	−12.9213	−	1.42820i	−0.993947	−	0.109862i
							\(17\)	−19.9936	−	11.5433i	−1.17609	−	0.679018i	−0.220986	−	0.975277i	\(-0.570927\pi\)
−0.955108	+	0.296259i	\(0.904261\pi\)
\(19\)	0.417587	−	0.111892i	0.0219783	−	0.00588906i	−0.247813	−	0.968808i	\(-0.579712\pi\)
							0.269791	+	0.962919i	\(0.413045\pi\)
\(23\)	36.4688	−	21.0553i	1.58560	−	0.915447i	0.591581	−	0.806245i	\(-0.298504\pi\)
							0.994020	−	0.109202i	\(-0.0348295\pi\)
\(29\)	−3.25785	−	5.64275i	−0.112340	−	0.194578i	0.804374	−	0.594124i	\(-0.202501\pi\)
							−0.916713	+	0.399546i	\(0.869168\pi\)
\(31\)	−17.8476	−	17.8476i	−0.575730	−	0.575730i	0.357994	−	0.933724i	\(-0.383461\pi\)
							−0.933724	+	0.357994i	\(0.883461\pi\)
\(37\)	−1.37988	−	0.369738i	−0.0372940	−	0.00999291i	0.240124	−	0.970742i	\(-0.422812\pi\)
							−0.277418	+	0.960749i	\(0.589479\pi\)
\(41\)	−10.8187	+	40.3758i	−0.263870	+	0.984776i	0.699068	+	0.715055i	\(0.253598\pi\)
							−0.962938	+	0.269721i	\(0.913068\pi\)
\(43\)	−51.1299	−	29.5199i	−1.18907	−	0.686509i	−0.230973	−	0.972960i	\(-0.574191\pi\)
							−0.958095	+	0.286452i	\(0.907524\pi\)
\(47\)	15.0543	−	15.0543i	0.320303	−	0.320303i	−0.528580	−	0.848883i	\(-0.677275\pi\)
							0.848883	+	0.528580i	\(0.177275\pi\)