Embedded newform 76.3.l.a.23.12

• Label: 76.3.l.a.23.12
• 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			23.12
Character	\(\chi\)	\(=\)	76.23
Dual form			76.3.l.a.43.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.531980 - 1.92795i) q^{2} +(-1.75796 + 0.309976i) q^{3} +(-3.43399 - 2.05126i) q^{4} +(6.16490 - 5.17297i) q^{5} +(-0.337581 + 3.55416i) q^{6} +(-6.99959 - 4.04121i) q^{7} +(-5.78155 + 5.52934i) q^{8} +(-5.46290 + 1.98833i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 6 q^{2} - 12 q^{5} + 12 q^{6} - 3 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.531980	−	1.92795i	0.265990	−	0.963976i
							\(3\)	−1.75796	+	0.309976i	−0.585986	+	0.103325i	−0.458778	−	0.888551i	\(-0.651713\pi\)
−0.127208	+	0.991876i	\(0.540602\pi\)
\(5\)	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	−0.0917104	−	0.995786i	\(-0.529233\pi\)
							−0.908231	+	0.418469i	\(0.862567\pi\)
\(11\)	11.1422	−	6.43297i	1.01293	−	0.584815i	0.100882	−	0.994898i	\(-0.467834\pi\)
							0.912048	+	0.410083i	\(0.134500\pi\)
\(13\)	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	−0.697894	−	0.716201i	\(-0.745879\pi\)
0.826508	+	0.562925i	\(0.190324\pi\)
\(29\)	−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	0.732115	−	0.681182i	\(-0.238534\pi\)
							−0.223863	+	0.974621i	\(0.571867\pi\)
\(37\)	−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	0.956082	−	0.293100i	\(-0.0946868\pi\)
							−0.454669	+	0.890661i	\(0.650242\pi\)
\(47\)	−25.6060	−	70.3519i	−0.544809	−	1.49685i	−0.840632	−	0.541607i	\(-0.817816\pi\)
							0.295824	−	0.955243i	\(-0.404406\pi\)

(See \(a_n\) instead)

Twists

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

    

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