Embedded newform 76.4.k.a.3.17

• Label: 76.4.k.a.3.17
• Level: $76$
• Weight: $4$
• Character: 76.3
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48414516044\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(28\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	76.3
Dual form			76.4.k.a.51.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.339844 + 2.80794i) q^{2} +(-5.54644 + 2.01874i) q^{3} +(-7.76901 + 1.90852i) q^{4} +(2.56623 - 14.5538i) q^{5} +(-7.55341 - 14.8880i) q^{6} +(1.22017 - 0.704465i) q^{7} +(-7.99925 - 21.1663i) q^{8} +(6.00448 - 5.03835i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 6 q^{2} - 24 q^{4} - 12 q^{5} - 24 q^{6} - 9 q^{8} + 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.339844	+	2.80794i	0.120153	+	0.992755i
							\(3\)	−5.54644	+	2.01874i	−1.06741	+	0.388506i	−0.815208	−	0.579168i	\(-0.803378\pi\)
−0.252204	+	0.967674i	\(0.581155\pi\)
\(5\)	2.56623	−	14.5538i	0.229530	−	1.30173i	−0.624302	−	0.781183i	\(-0.714616\pi\)
							0.853832	−	0.520548i	\(-0.174272\pi\)
\(7\)	1.22017	−	0.704465i	0.0658829	−	0.0380375i	−0.466697	−	0.884417i	\(-0.654556\pi\)
							0.532580	+	0.846380i	\(0.321223\pi\)
\(11\)	0.803943	+	0.464157i	0.0220362	+	0.0127226i	0.510978	−	0.859594i	\(-0.329283\pi\)
							−0.488941	+	0.872317i	\(0.662617\pi\)
\(13\)	17.4777	−	48.0196i	0.372880	−	1.02448i	−0.601362	−	0.798977i	\(-0.705375\pi\)
							0.974242	−	0.225504i	\(-0.0724027\pi\)
\(17\)	−61.3858	−	51.5088i	−0.875778	−	0.734865i	0.0895282	−	0.995984i	\(-0.471464\pi\)
							−0.965307	+	0.261119i	\(0.915909\pi\)
\(19\)	−78.0122	−	27.8045i	−0.941960	−	0.335726i
							\(23\)	−127.307	+	22.4477i	−1.15415	+	0.203507i	−0.717786	−	0.696264i	\(-0.754844\pi\)
−0.436361	+	0.899771i	\(0.643733\pi\)
\(29\)	−25.0116	−	29.8077i	−0.160157	−	0.190867i	0.679998	−	0.733214i	\(-0.261980\pi\)
							−0.840155	+	0.542346i	\(0.817536\pi\)
\(31\)	−63.8093	−	110.521i	−0.369693	−	0.640328i	0.619824	−	0.784741i	\(-0.287204\pi\)
							−0.989517	+	0.144413i	\(0.953871\pi\)
\(37\)			411.728i			1.82940i	0.404136	+	0.914699i	\(0.367572\pi\)
							−0.404136	+	0.914699i	\(0.632428\pi\)
\(41\)	−72.1830	−	198.321i	−0.274953	−	0.755428i	−0.997915	−	0.0645402i	\(-0.979442\pi\)
							0.722962	−	0.690888i	\(-0.242780\pi\)
\(43\)	−41.5438	−	7.32529i	−0.147334	−	0.0259790i	0.0994946	−	0.995038i	\(-0.468277\pi\)
							−0.246829	+	0.969059i	\(0.579388\pi\)
\(47\)	127.457	+	151.898i	0.395565	+	0.471416i	0.926662	−	0.375895i	\(-0.122665\pi\)
							−0.531097	+	0.847311i	\(0.678220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.k.a.3.17	yes	168
4.3	odd	2	inner	76.4.k.a.3.1	✓	168
19.13	odd	18	inner	76.4.k.a.51.1	yes	168
76.51	even	18	inner	76.4.k.a.51.17	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.k.a.3.1	✓	168	4.3	odd	2	inner
76.4.k.a.3.17	yes	168	1.1	even	1	trivial
76.4.k.a.51.1	yes	168	19.13	odd	18	inner
76.4.k.a.51.17	yes	168	76.51	even	18	inner