Embedded newform 76.3.l.a.23.15

• Label: 76.3.l.a.23.15
• Level: $76$
• Weight: $3$
• Character: 76.23
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) 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			76.3.l.a.43.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40142 - 1.42690i) q^{2} +(-5.33798 + 0.941229i) q^{3} +(-0.0720694 - 3.99935i) q^{4} +(-5.57927 + 4.68156i) q^{5} +(-6.13769 + 8.93580i) q^{6} +(-3.83135 - 2.21203i) q^{7} +(-5.80766 - 5.50192i) q^{8} +(19.1508 - 6.97034i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40142	−	1.42690i	0.700708	−	0.713448i
							\(3\)	−5.33798	+	0.941229i	−1.77933	+	0.313743i	−0.964128	−	0.265439i	\(-0.914483\pi\)
−0.815198	+	0.579182i	\(0.803372\pi\)
\(5\)	−5.57927	+	4.68156i	−1.11585	+	0.936313i	−0.998388	−	0.0567641i	\(-0.981922\pi\)
							−0.117466	+	0.993077i	\(0.537477\pi\)
\(7\)	−3.83135	−	2.21203i	−0.547336	−	0.316005i	0.200711	−	0.979651i	\(-0.435675\pi\)
							−0.748047	+	0.663646i	\(0.769008\pi\)
\(11\)	−5.20089	+	3.00274i	−0.472808	+	0.272976i	−0.717415	−	0.696646i	\(-0.754675\pi\)
							0.244606	+	0.969622i	\(0.421341\pi\)
\(13\)	1.78484	−	10.1223i	0.137295	−	0.778641i	−0.835938	−	0.548824i	\(-0.815076\pi\)
							0.973234	−	0.229818i	\(-0.0738130\pi\)
\(17\)	−19.9602	−	7.26493i	−1.17413	−	0.427349i	−0.320006	−	0.947415i	\(-0.603685\pi\)
							−0.854126	+	0.520066i	\(0.825907\pi\)
\(19\)	8.81643	+	16.8306i	0.464023	+	0.885823i
							\(23\)	6.75219	−	8.04695i	0.293574	−	0.349867i	−0.599016	−	0.800737i	\(-0.704442\pi\)
0.892590	+	0.450869i	\(0.148886\pi\)
\(29\)	−23.6489	+	8.60751i	−0.815480	+	0.296811i	−0.715886	−	0.698218i	\(-0.753977\pi\)
							−0.0995947	+	0.995028i	\(0.531755\pi\)
\(31\)	−6.21616	−	3.58890i	−0.200521	−	0.115771i	0.396377	−	0.918088i	\(-0.370267\pi\)
							−0.596899	+	0.802317i	\(0.703601\pi\)
\(37\)	10.9458			0.295831			0.147916	−	0.989000i	\(-0.452744\pi\)
							0.147916	+	0.989000i	\(0.452744\pi\)
\(41\)	10.4370	+	59.1911i	0.254561	+	1.44368i	0.797198	+	0.603718i	\(0.206315\pi\)
							−0.542637	+	0.839967i	\(0.682574\pi\)
\(43\)	−42.6907	−	50.8768i	−0.992807	−	1.18318i	−0.983070	−	0.183228i	\(-0.941345\pi\)
							−0.00973637	−	0.999953i	\(-0.503099\pi\)
\(47\)	2.74884	+	7.55237i	0.0584859	+	0.160689i	0.965495	−	0.260423i	\(-0.0838621\pi\)
							−0.907009	+	0.421112i	\(0.861640\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.15	yes	108
4.3	odd	2	inner	76.3.l.a.23.11	✓	108
19.5	even	9	inner	76.3.l.a.43.11	yes	108
76.43	odd	18	inner	76.3.l.a.43.15	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.11	✓	108	4.3	odd	2	inner
76.3.l.a.23.15	yes	108	1.1	even	1	trivial
76.3.l.a.43.11	yes	108	19.5	even	9	inner
76.3.l.a.43.15	yes	108	76.43	odd	18	inner