Embedded newform 76.3.g.c.7.10

• Label: 76.3.g.c.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.789173 + 1.83772i) q^{2} +(-3.65809 + 2.11200i) q^{3} +(-2.75441 + 2.90055i) q^{4} +(-1.06722 - 1.84849i) q^{5} +(-6.76812 - 5.05580i) q^{6} -1.82388i q^{7} +(-7.50411 - 2.77279i) q^{8} +(4.42107 - 7.65753i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.789173	+	1.83772i	0.394587	+	0.918859i
							\(3\)	−3.65809	+	2.11200i	−1.21936	+	0.703999i	−0.964782	−	0.263052i	\(-0.915271\pi\)
−0.254581	+	0.967051i	\(0.581938\pi\)
\(5\)	−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	\(-0.958413\pi\)
							0.991478	−	0.130277i	\(-0.0415868\pi\)
\(11\)			13.8522i			1.25929i	0.776882	+	0.629647i	\(0.216800\pi\)
							−0.776882	+	0.629647i	\(0.783200\pi\)
\(13\)	−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	0.287763	−	0.957702i	\(-0.407088\pi\)
−0.685512	+	0.728061i	\(0.740422\pi\)
\(29\)	−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	\(0.238520\pi\)
							−0.732143	+	0.681150i	\(0.761480\pi\)
\(37\)	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	0.234932	−	0.972012i	\(-0.424513\pi\)
							0.959253	+	0.282549i	\(0.0911799\pi\)
\(47\)	−77.5317	−	44.7629i	−1.64961	−	0.952403i	−0.977226	−	0.212201i	\(-0.931937\pi\)
							−0.672384	−	0.740202i	\(-0.734730\pi\)

(See \(a_n\) instead)

Twists

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

    

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