Embedded newform 76.5.g.a.7.12

• Label: 76.5.g.a.7.12
• Level: $76$
• Weight: $5$
• Character: 76.7
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) 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			76.5.g.a.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.44419 + 3.16638i) q^{2} +(7.49168 - 4.32533i) q^{3} +(-4.05190 - 15.4784i) q^{4} +(5.46741 + 9.46983i) q^{5} +(-4.61545 + 34.2934i) q^{6} -37.0844i q^{7} +(58.9142 + 25.0023i) q^{8} +(-3.08311 + 5.34010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.44419	+	3.16638i	−0.611047	+	0.791595i
							\(3\)	7.49168	−	4.32533i	0.832409	−	0.480592i	−0.0222675	−	0.999752i	\(-0.507089\pi\)
0.854677	+	0.519160i	\(0.173755\pi\)
\(5\)	5.46741	+	9.46983i	0.218696	+	0.378793i	0.954410	−	0.298500i	\(-0.0964863\pi\)
							−0.735713	+	0.677293i	\(0.763153\pi\)
\(7\)		−	37.0844i		−	0.756824i	−0.925637	−	0.378412i	\(-0.876470\pi\)
							0.925637	−	0.378412i	\(-0.123530\pi\)
\(11\)		−	197.000i		−	1.62810i	−0.580795	−	0.814050i	\(-0.697258\pi\)
							0.580795	−	0.814050i	\(-0.302742\pi\)
\(13\)	104.805	−	181.527i	0.620147	−	1.07413i	−0.369311	−	0.929306i	\(-0.620406\pi\)
							0.989458	−	0.144820i	\(-0.0462604\pi\)
\(17\)	182.154	+	315.499i	0.630289	+	1.09169i	0.987493	+	0.157666i	\(0.0503969\pi\)
							−0.357204	+	0.934027i	\(0.616270\pi\)
\(19\)	154.149	−	326.434i	0.427006	−	0.904249i
							\(23\)	425.313	+	245.554i	0.803994	+	0.464186i	0.844866	−	0.534979i	\(-0.179680\pi\)
−0.0408721	+	0.999164i	\(0.513014\pi\)
\(29\)	432.994	−	749.967i	0.514856	−	0.891757i	−0.484995	−	0.874517i	\(-0.661179\pi\)
							0.999851	−	0.0172399i	\(-0.00548791\pi\)
\(31\)			486.754i			0.506508i	0.967400	+	0.253254i	\(0.0815008\pi\)
							−0.967400	+	0.253254i	\(0.918499\pi\)
\(37\)	−2121.09			−1.54938			−0.774688	−	0.632344i	\(-0.782093\pi\)
							−0.774688	+	0.632344i	\(0.782093\pi\)
\(41\)	−33.4460	−	57.9302i	−0.0198965	−	0.0344618i	0.855906	−	0.517132i	\(-0.173000\pi\)
							−0.875802	+	0.482670i	\(0.839667\pi\)
\(43\)	−2364.35	+	1365.06i	−1.27872	+	0.738269i	−0.976613	−	0.215005i	\(-0.931023\pi\)
							−0.302106	+	0.953274i	\(0.597690\pi\)
\(47\)	−748.297	−	432.030i	−0.338749	−	0.195577i	0.320970	−	0.947090i	\(-0.395991\pi\)
							−0.659719	+	0.751513i	\(0.729325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.12	✓	76
4.3	odd	2	inner	76.5.g.a.7.38	yes	76
19.11	even	3	inner	76.5.g.a.11.38	yes	76
76.11	odd	6	inner	76.5.g.a.11.12	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.12	✓	76	1.1	even	1	trivial
76.5.g.a.7.38	yes	76	4.3	odd	2	inner
76.5.g.a.11.12	yes	76	76.11	odd	6	inner
76.5.g.a.11.38	yes	76	19.11	even	3	inner