Embedded newform 78.3.l.c.7.1

• Label: 78.3.l.c.7.1
• 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			7.1
Root			\(-5.39181 + 5.39181i\) of defining polynomial
Character	\(\chi\)	\(=\)	78.7
Dual form			78.3.l.c.67.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(0.866025 - 1.50000i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-6.39181 - 6.39181i) q^{5} +(0.633975 - 2.36603i) q^{6} +(9.23137 + 2.47354i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9}+O(q^{10}) \)

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
\(5\)	−6.39181	−	6.39181i	−1.27836	−	1.27836i								−0.941584	−	0.336778i	\(-0.890663\pi\)
							−0.336778	−	0.941584i	\(-0.609337\pi\)
\(7\)	9.23137	+	2.47354i	1.31877	+	0.353363i	0.848516	−	0.529170i	\(-0.177496\pi\)
							0.470252	+	0.882532i	\(0.344163\pi\)
\(11\)	4.21503	+	15.7307i	0.383184	+	1.43006i	0.841009	+	0.541021i	\(0.181962\pi\)
							−0.457825	+	0.889042i	\(0.651371\pi\)
\(13\)	−3.81310	+	12.4282i	−0.293316	+	0.956016i
							\(17\)	8.31934	−	4.80317i	0.489373	−	0.282540i	−0.234941	−	0.972010i	\(-0.575490\pi\)
0.724314	+	0.689470i	\(0.242157\pi\)
\(19\)	2.56168	−	9.56033i	0.134825	−	0.503175i	−0.865173	−	0.501473i	\(-0.832792\pi\)
							0.999999	−	0.00170204i	\(-0.000541777\pi\)
\(23\)	−23.6929	−	13.6791i	−1.03013	−	0.594745i	−0.113107	−	0.993583i	\(-0.536080\pi\)
							−0.917021	+	0.398838i	\(0.869414\pi\)
\(29\)	−13.8023	+	23.9063i	−0.475942	+	0.824356i	−0.999620	−	0.0275606i	\(-0.991226\pi\)
							0.523678	+	0.851916i	\(0.324559\pi\)
\(31\)	11.5613	+	11.5613i	0.372946	+	0.372946i	0.868549	−	0.495603i	\(-0.165053\pi\)
							−0.495603	+	0.868549i	\(0.665053\pi\)
\(37\)	−5.45615	−	20.3626i	−0.147464	−	0.550342i	−0.999633	−	0.0270763i	\(-0.991380\pi\)
							0.852170	−	0.523265i	\(-0.175286\pi\)
\(41\)	39.1030	−	10.4776i	0.953731	−	0.255551i	0.251786	−	0.967783i	\(-0.418982\pi\)
							0.701945	+	0.712232i	\(0.252315\pi\)
\(43\)	−30.9095	+	17.8456i	−0.718826	+	0.415014i	−0.814320	−	0.580416i	\(-0.802890\pi\)
							0.0954945	+	0.995430i	\(0.469557\pi\)
\(47\)	25.5184	−	25.5184i	0.542944	−	0.542944i	−0.381447	−	0.924391i	\(-0.624574\pi\)
							0.924391	+	0.381447i	\(0.124574\pi\)