Embedded newform 76.7.l.a.23.9

• Label: 76.7.l.a.23.9
• Level: $76$
• Weight: $7$
• Character: 76.23
• Analytic conductor: $17.484$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4841103551\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(58\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.9
Character	\(\chi\)	\(=\)	76.23
Dual form			76.7.l.a.43.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.25120 + 3.37936i) q^{2} +(-38.7503 + 6.83273i) q^{3} +(41.1598 - 49.0089i) q^{4} +(178.137 - 149.475i) q^{5} +(257.896 - 180.497i) q^{6} +(-72.9496 - 42.1175i) q^{7} +(-132.840 + 494.467i) q^{8} +(769.865 - 280.208i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 6 q^{2} + 192 q^{4} - 12 q^{5} - 492 q^{6} - 3 q^{8} + 1368 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.25120	+	3.37936i	−0.906400	+	0.422420i
							\(3\)	−38.7503	+	6.83273i	−1.43520	+	0.253064i	−0.836524	−	0.547930i	\(-0.815416\pi\)
−0.598673	+	0.800994i	\(0.704305\pi\)
\(5\)	178.137	−	149.475i	1.42510	−	1.19580i	0.476557	−	0.879144i	\(-0.341885\pi\)
							0.948541	−	0.316655i	\(-0.102560\pi\)
\(7\)	−72.9496	−	42.1175i	−0.212681	−	0.122791i	0.389876	−	0.920867i	\(-0.372518\pi\)
							−0.602557	+	0.798076i	\(0.705851\pi\)
\(11\)	−332.112	+	191.745i	−0.249521	+	0.144061i	−0.619545	−	0.784961i	\(-0.712683\pi\)
							0.370024	+	0.929022i	\(0.379349\pi\)
\(13\)	−81.0045	+	459.399i	−0.0368705	+	0.209103i	−0.997677	−	0.0681153i	\(-0.978301\pi\)
							0.960807	+	0.277218i	\(0.0894125\pi\)
\(17\)	6288.38	+	2288.78i	1.27995	+	0.465863i	0.890415	−	0.455150i	\(-0.150414\pi\)
							0.389532	+	0.921013i	\(0.372637\pi\)
\(19\)	868.165	−	6803.83i	0.126573	−	0.991957i
							\(23\)	−3206.69	+	3821.58i	−0.263556	+	0.314094i	−0.881552	−	0.472088i	\(-0.843501\pi\)
0.617995	+	0.786182i	\(0.287945\pi\)
\(29\)	−29947.8	+	10900.1i	−1.22792	+	0.446927i	−0.872885	−	0.487926i	\(-0.837754\pi\)
							−0.355036	+	0.934853i	\(0.615531\pi\)
\(31\)	26370.7	+	15225.1i	0.885190	+	0.511064i	0.872366	−	0.488853i	\(-0.162585\pi\)
							0.0128236	+	0.999918i	\(0.495918\pi\)
\(37\)	−44161.8			−0.871850			−0.435925	−	0.899983i	\(-0.643579\pi\)
							−0.435925	+	0.899983i	\(0.643579\pi\)
\(41\)	−19113.9	−	108400.i	−0.277330	−	1.57282i	−0.731460	−	0.681884i	\(-0.761161\pi\)
							0.454130	−	0.890935i	\(-0.349950\pi\)
\(43\)	2054.63	+	2448.61i	0.0258421	+	0.0307975i	0.778811	−	0.627259i	\(-0.215823\pi\)
							−0.752969	+	0.658056i	\(0.771379\pi\)
\(47\)	−56216.7	−	154454.i	−0.541467	−	1.48767i	−0.844957	−	0.534834i	\(-0.820374\pi\)
							0.303491	−	0.952834i	\(-0.401848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.l.a.23.9	✓	348
4.3	odd	2	inner	76.7.l.a.23.22	yes	348
19.5	even	9	inner	76.7.l.a.43.22	yes	348
76.43	odd	18	inner	76.7.l.a.43.9	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.l.a.23.9	✓	348	1.1	even	1	trivial
76.7.l.a.23.22	yes	348	4.3	odd	2	inner
76.7.l.a.43.9	yes	348	76.43	odd	18	inner
76.7.l.a.43.22	yes	348	19.5	even	9	inner