Embedded newform 76.3.g.c.7.12

• Label: 76.3.g.c.7.12
• Level: $76$
• Weight: $3$
• Character: 76.7
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.12
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19692 + 1.60230i) q^{2} +(3.65809 - 2.11200i) q^{3} +(-1.13475 + 3.83567i) q^{4} +(-1.06722 - 1.84849i) q^{5} +(7.76251 + 3.33347i) q^{6} +1.82388i q^{7} +(-7.50411 + 2.77279i) q^{8} +(4.42107 - 7.65753i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19692	+	1.60230i	0.598462	+	0.801151i
							\(3\)	3.65809	−	2.11200i	1.21936	−	0.703999i	0.254581	−	0.967051i	\(-0.418062\pi\)
0.964782	+	0.263052i	\(0.0847291\pi\)
\(5\)	−1.06722	−	1.84849i	−0.213445	−	0.369698i	0.739345	−	0.673326i	\(-0.235135\pi\)
							−0.952790	+	0.303629i	\(0.901802\pi\)
\(7\)			1.82388i			0.260555i	0.991478	+	0.130277i	\(0.0415868\pi\)
							−0.991478	+	0.130277i	\(0.958413\pi\)
\(11\)		−	13.8522i		−	1.25929i	−0.776882	−	0.629647i	\(-0.783200\pi\)
							0.776882	−	0.629647i	\(-0.216800\pi\)
\(13\)	−9.95291	+	17.2389i	−0.765608	+	1.32607i	0.174316	+	0.984690i	\(0.444229\pi\)
							−0.939924	+	0.341383i	\(0.889105\pi\)
\(17\)	3.83013	+	6.63399i	0.225302	+	0.390235i	0.956410	−	0.292027i	\(-0.0943298\pi\)
							−0.731108	+	0.682262i	\(0.760996\pi\)
\(19\)	−16.7080	−	9.04663i	−0.879370	−	0.476138i
							\(23\)	9.14824	+	5.28174i	0.397749	+	0.229641i	0.685512	−	0.728061i	\(-0.259578\pi\)
−0.287763	+	0.957702i	\(0.592912\pi\)
\(29\)	−0.0147659	+	0.0255754i	−0.000509170	+	0.000881909i	−0.866280	−	0.499559i	\(-0.833495\pi\)
							0.865771	+	0.500441i	\(0.166829\pi\)
\(31\)		−	42.2313i		−	1.36230i	−0.732143	−	0.681150i	\(-0.761480\pi\)
							0.732143	−	0.681150i	\(-0.238520\pi\)
\(37\)	19.5805			0.529202			0.264601	−	0.964358i	\(-0.414760\pi\)
							0.264601	+	0.964358i	\(0.414760\pi\)
\(41\)	15.7368	+	27.2570i	0.383825	+	0.664805i	0.991605	−	0.129300i	\(-0.0412732\pi\)
							−0.607780	+	0.794105i	\(0.707940\pi\)
\(43\)	−51.3500	+	29.6469i	−1.19418	+	0.689463i	−0.959253	−	0.282549i	\(-0.908820\pi\)
							−0.234932	+	0.972012i	\(0.575487\pi\)
\(47\)	77.5317	+	44.7629i	1.64961	+	0.952403i	0.977226	+	0.212201i	\(0.0680631\pi\)
							0.672384	+	0.740202i	\(0.265270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.g.c.7.12	yes	28
4.3	odd	2	inner	76.3.g.c.7.10	✓	28
19.11	even	3	inner	76.3.g.c.11.10	yes	28
76.11	odd	6	inner	76.3.g.c.11.12	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.g.c.7.10	✓	28	4.3	odd	2	inner
76.3.g.c.7.12	yes	28	1.1	even	1	trivial
76.3.g.c.11.10	yes	28	19.11	even	3	inner
76.3.g.c.11.12	yes	28	76.11	odd	6	inner