Embedded newform 76.5.l.a.43.8

• Label: 76.5.l.a.43.8
• Level: $76$
• Weight: $5$
• Character: 76.43
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			43.8
Character	\(\chi\)	\(=\)	76.43
Dual form			76.5.l.a.23.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.15747 + 2.45569i) q^{2} +(-8.88596 - 1.56683i) q^{3} +(3.93918 - 15.5075i) q^{4} +(-25.4406 - 21.3472i) q^{5} +(31.9048 - 16.8739i) q^{6} +(18.4296 - 10.6403i) q^{7} +(25.6438 + 58.6378i) q^{8} +(0.390232 + 0.142033i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.15747	+	2.45569i	−0.789366	+	0.613922i
							\(3\)	−8.88596	−	1.56683i	−0.987329	−	0.174093i	−0.343409	−	0.939186i	\(-0.611582\pi\)
−0.643920	+	0.765093i	\(0.722693\pi\)
\(5\)	−25.4406	−	21.3472i	−1.01762	−	0.853887i	−0.0282956	−	0.999600i	\(-0.509008\pi\)
							−0.989327	+	0.145713i	\(0.953452\pi\)
\(7\)	18.4296	−	10.6403i	0.376114	−	0.217150i	−0.300012	−	0.953935i	\(-0.596991\pi\)
							0.676126	+	0.736786i	\(0.263657\pi\)
\(11\)	14.9857	+	8.65201i	0.123849	+	0.0715042i	0.560645	−	0.828057i	\(-0.310553\pi\)
							−0.436796	+	0.899561i	\(0.643887\pi\)
\(13\)	4.76034	+	26.9972i	0.0281677	+	0.159747i	0.995647	−	0.0932026i	\(-0.0297104\pi\)
							−0.967479	+	0.252950i	\(0.918599\pi\)
\(17\)	−392.828	+	142.978i	−1.35927	+	0.494732i	−0.915827	−	0.401573i	\(-0.868464\pi\)
							−0.443439	+	0.896305i	\(0.646242\pi\)
\(19\)	299.424	+	201.658i	0.829430	+	0.558611i
							\(23\)	390.606	+	465.506i	0.738386	+	0.879974i	0.996278	−	0.0861998i	\(-0.0274723\pi\)
−0.257892	+	0.966174i	\(0.583028\pi\)
\(29\)	20.3182	+	7.39523i	0.0241596	+	0.00879338i	0.354072	−	0.935218i	\(-0.384797\pi\)
							−0.329912	+	0.944012i	\(0.607019\pi\)
\(31\)	−457.709	+	264.259i	−0.476284	+	0.274983i	−0.718867	−	0.695148i	\(-0.755339\pi\)
							0.242582	+	0.970131i	\(0.422006\pi\)
\(37\)	2091.69			1.52789			0.763947	−	0.645279i	\(-0.223259\pi\)
							0.763947	+	0.645279i	\(0.223259\pi\)
\(41\)	152.801	−	866.578i	0.0908989	−	0.515514i	−0.905028	−	0.425352i	\(-0.860150\pi\)
							0.995927	−	0.0901619i	\(-0.0287385\pi\)
\(43\)	1531.51	−	1825.19i	0.828293	−	0.987121i	−0.171705	−	0.985148i	\(-0.554928\pi\)
							0.999998	−	0.00197305i	\(-0.000628041\pi\)
\(47\)	−990.180	+	2720.50i	−0.448248	+	1.23155i	0.485695	+	0.874129i	\(0.338567\pi\)
							−0.933943	+	0.357423i	\(0.883656\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.43.8	yes	228
4.3	odd	2	inner	76.5.l.a.43.2	yes	228
19.4	even	9	inner	76.5.l.a.23.2	✓	228
76.23	odd	18	inner	76.5.l.a.23.8	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.2	✓	228	19.4	even	9	inner
76.5.l.a.23.8	yes	228	76.23	odd	18	inner
76.5.l.a.43.2	yes	228	4.3	odd	2	inner
76.5.l.a.43.8	yes	228	1.1	even	1	trivial