Embedded newform 78.3.l.c.7.2

• Label: 78.3.l.c.7.2
• Level: $78$
• Weight: $3$
• Character: 78.7
• 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			7.2
Root			\(5.02578 - 5.02578i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.7
Dual form			78.3.l.c.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(0.866025 - 1.50000i) q^{3} +(1.73205 - 1.00000i) q^{4} +(4.02578 + 4.02578i) q^{5} +(0.633975 - 2.36603i) q^{6} +(-4.99932 - 1.33956i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36603	−	0.366025i	0.683013	−	0.183013i
							\(3\)	0.866025	−	1.50000i	0.288675	−	0.500000i
\(5\)	4.02578	+	4.02578i	0.805157	+	0.805157i								0.983896	−	0.178740i	\(-0.0572019\pi\)
							−0.178740	+	0.983896i	\(0.557202\pi\)
\(7\)	−4.99932	−	1.33956i	−0.714189	−	0.191366i	−0.116612	−	0.993178i	\(-0.537203\pi\)
							−0.597577	+	0.801811i	\(0.703870\pi\)
\(11\)	−3.41118	−	12.7307i	−0.310107	−	1.15734i	−0.928459	−	0.371435i	\(-0.878866\pi\)
							0.618352	−	0.785901i	\(-0.287801\pi\)
\(13\)	3.81310	+	12.4282i	0.293316	+	0.956016i
							\(17\)	−16.3290	+	9.42753i	−0.960527	+	0.554560i	−0.896335	−	0.443377i	\(-0.853780\pi\)
−0.0641917	+	0.997938i	\(0.520447\pi\)
\(19\)	−7.85591	+	29.3187i	−0.413469	+	1.54309i	0.374413	+	0.927262i	\(0.377844\pi\)
							−0.787883	+	0.615826i	\(0.788823\pi\)
\(23\)	−10.4840	−	6.05292i	−0.455825	−	0.263170i	0.254462	−	0.967083i	\(-0.418102\pi\)
							−0.710287	+	0.703912i	\(0.751435\pi\)
\(29\)	18.4722	−	31.9948i	0.636972	−	1.10327i	−0.349122	−	0.937077i	\(-0.613520\pi\)
							0.986094	−	0.166190i	\(-0.0531467\pi\)
\(31\)	−6.48248	−	6.48248i	−0.209112	−	0.209112i	0.594778	−	0.803890i	\(-0.297240\pi\)
							−0.803890	+	0.594778i	\(0.797240\pi\)
\(37\)	13.6094	+	50.7908i	0.367821	+	1.37273i	0.863556	+	0.504252i	\(0.168232\pi\)
							−0.495736	+	0.868473i	\(0.665102\pi\)
\(41\)	−6.38053	+	1.70966i	−0.155623	+	0.0416989i	−0.335789	−	0.941937i	\(-0.609003\pi\)
							0.180167	+	0.983636i	\(0.442336\pi\)
\(43\)	11.7826	−	6.80268i	0.274014	−	0.158202i	−0.356697	−	0.934220i	\(-0.616097\pi\)
							0.630710	+	0.776018i	\(0.282764\pi\)
\(47\)	61.6060	−	61.6060i	1.31077	−	1.31077i	0.389914	−	0.920851i	\(-0.372505\pi\)
							0.920851	−	0.389914i	\(-0.127495\pi\)