Embedded newform 76.3.l.a.23.7

• Label: 76.3.l.a.23.7
• 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			23.7
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.831743 + 1.81885i) q^{2} +(1.75796 - 0.309976i) q^{3} +(-2.61641 - 3.02563i) q^{4} +(6.16490 - 5.17297i) q^{5} +(-0.898372 + 3.45528i) q^{6} +(6.99959 + 4.04121i) q^{7} +(7.67933 - 2.24230i) q^{8} +(-5.46290 + 1.98833i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.831743	+	1.81885i	−0.415872	+	0.909423i
							\(3\)	1.75796	−	0.309976i	0.585986	−	0.103325i	0.127208	−	0.991876i	\(-0.459398\pi\)
0.458778	+	0.888551i	\(0.348287\pi\)
\(5\)	6.16490	−	5.17297i	1.23298	−	1.03459i	0.234941	−	0.972010i	\(-0.424510\pi\)
							0.998040	−	0.0625838i	\(-0.0199341\pi\)
\(7\)	6.99959	+	4.04121i	0.999941	+	0.577316i	0.908231	−	0.418469i	\(-0.137433\pi\)
							0.0917104	+	0.995786i	\(0.470767\pi\)
\(11\)	−11.1422	+	6.43297i	−1.01293	+	0.584815i	−0.912048	−	0.410083i	\(-0.865500\pi\)
							−0.100882	+	0.994898i	\(0.532166\pi\)
\(13\)	1.24625	−	7.06781i	0.0958651	−	0.543678i	−0.898614	−	0.438741i	\(-0.855425\pi\)
							0.994479	−	0.104937i	\(-0.0334642\pi\)
\(17\)	25.0534	+	9.11869i	1.47373	+	0.536394i	0.949111	−	0.314943i	\(-0.101985\pi\)
							0.524619	+	0.851337i	\(0.324208\pi\)
\(19\)	−13.2465	−	13.6210i	−0.697184	−	0.716893i
							\(23\)	−2.95812	+	3.52535i	−0.128614	+	0.153276i	−0.826508	−	0.562925i	\(-0.809676\pi\)
0.697894	+	0.716201i	\(0.254121\pi\)
\(29\)	−28.8625	+	10.5051i	−0.995259	+	0.362245i	−0.787755	−	0.615989i	\(-0.788756\pi\)
							−0.207505	+	0.978234i	\(0.566534\pi\)
\(31\)	−15.7558	−	9.09661i	−0.508251	−	0.293439i	0.223863	−	0.974621i	\(-0.428133\pi\)
							−0.732115	+	0.681182i	\(0.761466\pi\)
\(37\)	−36.4512			−0.985168			−0.492584	−	0.870265i	\(-0.663947\pi\)
							−0.492584	+	0.870265i	\(0.663947\pi\)
\(41\)	1.77249	+	10.0523i	0.0432314	+	0.245178i	0.998764	−	0.0497087i	\(-0.0158293\pi\)
							−0.955532	+	0.294886i	\(0.904718\pi\)
\(43\)	−21.5608	−	25.6951i	−0.501413	−	0.597561i	0.454669	−	0.890661i	\(-0.349758\pi\)
							−0.956082	+	0.293100i	\(0.905313\pi\)
\(47\)	25.6060	+	70.3519i	0.544809	+	1.49685i	0.840632	+	0.541607i	\(0.182184\pi\)
							−0.295824	+	0.955243i	\(0.595594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.l.a.23.7	✓	108
4.3	odd	2	inner	76.3.l.a.23.12	yes	108
19.5	even	9	inner	76.3.l.a.43.12	yes	108
76.43	odd	18	inner	76.3.l.a.43.7	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.l.a.23.7	✓	108	1.1	even	1	trivial
76.3.l.a.23.12	yes	108	4.3	odd	2	inner
76.3.l.a.43.7	yes	108	76.43	odd	18	inner
76.3.l.a.43.12	yes	108	19.5	even	9	inner