Embedded newform 76.3.g.c.7.1

• Label: 76.3.g.c.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99453 + 0.147792i) q^{2} +(3.58197 - 2.06805i) q^{3} +(3.95632 - 0.589552i) q^{4} +(-3.72976 - 6.46014i) q^{5} +(-6.83872 + 4.65419i) q^{6} +3.06851i q^{7} +(-7.80387 + 1.76059i) q^{8} +(4.05369 - 7.02120i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99453	+	0.147792i	−0.997266	+	0.0738960i
							\(3\)	3.58197	−	2.06805i	1.19399	−	0.689351i	0.234782	−	0.972048i	\(-0.424562\pi\)
0.959209	+	0.282697i	\(0.0912291\pi\)
\(5\)	−3.72976	−	6.46014i	−0.745952	−	1.29203i	−0.949749	−	0.313014i	\(-0.898661\pi\)
							0.203796	−	0.979013i	\(-0.434672\pi\)
\(7\)			3.06851i			0.438359i	0.975685	+	0.219179i	\(0.0703380\pi\)
							−0.975685	+	0.219179i	\(0.929662\pi\)
\(11\)		−	6.31466i		−	0.574060i	−0.957922	−	0.287030i	\(-0.907332\pi\)
							0.957922	−	0.287030i	\(-0.0926679\pi\)
\(13\)	8.74659	−	15.1495i	0.672815	−	1.16535i	−0.304288	−	0.952580i	\(-0.598418\pi\)
							0.977102	−	0.212769i	\(-0.0682483\pi\)
\(17\)	10.6765	+	18.4922i	0.628030	+	1.08778i	0.987947	+	0.154795i	\(0.0494717\pi\)
							−0.359917	+	0.932984i	\(0.617195\pi\)
\(19\)	6.62587	+	17.8072i	0.348730	+	0.937223i
							\(23\)	−5.19772	−	3.00090i	−0.225988	−	0.130474i	0.382732	−	0.923859i	\(-0.374983\pi\)
−0.608720	+	0.793385i	\(0.708317\pi\)
\(29\)	18.2919	−	31.6825i	0.630755	−	1.09250i	−0.356643	−	0.934241i	\(-0.616079\pi\)
							0.987398	−	0.158258i	\(-0.0505879\pi\)
\(31\)			53.3857i			1.72212i	0.508505	+	0.861059i	\(0.330198\pi\)
							−0.508505	+	0.861059i	\(0.669802\pi\)
\(37\)	−39.2040			−1.05957			−0.529784	−	0.848133i	\(-0.677727\pi\)
							−0.529784	+	0.848133i	\(0.677727\pi\)
\(41\)	18.2738	+	31.6512i	0.445704	+	0.771981i	0.998101	−	0.0615997i	\(-0.0196202\pi\)
							−0.552397	+	0.833581i	\(0.686287\pi\)
\(43\)	31.7551	−	18.3338i	0.738491	−	0.426368i	−0.0830292	−	0.996547i	\(-0.526459\pi\)
							0.821521	+	0.570179i	\(0.193126\pi\)
\(47\)	0.0577813	+	0.0333600i	0.00122939	+	0.000709788i	0.500615	−	0.865670i	\(-0.333107\pi\)
							−0.499385	+	0.866380i	\(0.666441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.g.c.7.1	✓	28
4.3	odd	2	inner	76.3.g.c.7.11	yes	28
19.11	even	3	inner	76.3.g.c.11.11	yes	28
76.11	odd	6	inner	76.3.g.c.11.1	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.g.c.7.1	✓	28	1.1	even	1	trivial
76.3.g.c.7.11	yes	28	4.3	odd	2	inner
76.3.g.c.11.1	yes	28	76.11	odd	6	inner
76.3.g.c.11.11	yes	28	19.11	even	3	inner