Embedded newform 76.5.l.a.23.15

• Label: 76.5.l.a.23.15
• Level: $76$
• Weight: $5$
• Character: 76.23
• 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			23.15
Character	\(\chi\)	\(=\)	76.23
Dual form			76.5.l.a.43.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47024 + 3.72000i) q^{2} +(-11.9688 + 2.11041i) q^{3} +(-11.6768 - 10.9386i) q^{4} +(19.6524 - 16.4903i) q^{5} +(9.74622 - 47.6266i) q^{6} +(0.422999 + 0.244219i) q^{7} +(57.8592 - 27.3552i) q^{8} +(62.6822 - 22.8144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47024	+	3.72000i	−0.367560	+	0.930000i
							\(3\)	−11.9688	+	2.11041i	−1.32986	+	0.234491i	−0.793022	−	0.609193i	\(-0.791493\pi\)
−0.536840	+	0.843684i	\(0.680382\pi\)
\(5\)	19.6524	−	16.4903i	0.786095	−	0.659612i	−0.158681	−	0.987330i	\(-0.550724\pi\)
							0.944776	+	0.327718i	\(0.106280\pi\)
\(7\)	0.422999	+	0.244219i	0.00863264	+	0.00498406i	0.504310	−	0.863523i	\(-0.331747\pi\)
							−0.495677	+	0.868507i	\(0.665080\pi\)
\(11\)	−44.7121	+	25.8145i	−0.369521	+	0.213343i	−0.673249	−	0.739415i	\(-0.735102\pi\)
							0.303728	+	0.952759i	\(0.401769\pi\)
\(13\)	−8.23706	+	46.7147i	−0.0487400	+	0.276418i	−0.999431	−	0.0337205i	\(-0.989264\pi\)
							0.950691	+	0.310139i	\(0.100375\pi\)
\(17\)	197.442	+	71.8629i	0.683190	+	0.248661i	0.660216	−	0.751075i	\(-0.270465\pi\)
							0.0229732	+	0.999736i	\(0.492687\pi\)
\(19\)	338.433	+	125.634i	0.937488	+	0.348017i
							\(23\)	117.519	−	140.054i	0.222153	−	0.264752i	−0.643443	−	0.765494i	\(-0.722495\pi\)
0.865597	+	0.500741i	\(0.166939\pi\)
\(29\)	699.726	−	254.680i	0.832017	−	0.302829i	0.109331	−	0.994005i	\(-0.465129\pi\)
							0.722686	+	0.691176i	\(0.242907\pi\)
\(31\)	917.213	+	529.553i	0.954436	+	0.551044i	0.894456	−	0.447156i	\(-0.147563\pi\)
							0.0599799	+	0.998200i	\(0.480896\pi\)
\(37\)	2476.68			1.80911			0.904557	−	0.426352i	\(-0.140202\pi\)
							0.904557	+	0.426352i	\(0.140202\pi\)
\(41\)	447.585	+	2538.38i	0.266261	+	1.51004i	0.765419	+	0.643532i	\(0.222532\pi\)
							−0.499158	+	0.866511i	\(0.666357\pi\)
\(43\)	−1201.03	−	1431.33i	−0.649557	−	0.774112i	0.336290	−	0.941759i	\(-0.390828\pi\)
							−0.985847	+	0.167646i	\(0.946383\pi\)
\(47\)	−865.351	−	2377.53i	−0.391739	−	1.07629i	−0.966207	−	0.257767i	\(-0.917013\pi\)
							0.574468	−	0.818527i	\(-0.305209\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.23.15	✓	228
4.3	odd	2	inner	76.5.l.a.23.23	yes	228
19.5	even	9	inner	76.5.l.a.43.23	yes	228
76.43	odd	18	inner	76.5.l.a.43.15	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.15	✓	228	1.1	even	1	trivial
76.5.l.a.23.23	yes	228	4.3	odd	2	inner
76.5.l.a.43.15	yes	228	76.43	odd	18	inner
76.5.l.a.43.23	yes	228	19.5	even	9	inner