Embedded newform 76.5.g.a.7.9

• Label: 76.5.g.a.7.9
• 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			7.9
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.19572 - 2.40570i) q^{2} +(-7.19063 + 4.15151i) q^{3} +(4.42524 + 15.3759i) q^{4} +(-24.3962 - 42.2554i) q^{5} +(32.9665 + 4.03141i) q^{6} -41.7489i q^{7} +(22.8479 - 59.7827i) q^{8} +(-6.02993 + 10.4441i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.19572	−	2.40570i	−0.798930	−	0.601424i
							\(3\)	−7.19063	+	4.15151i	−0.798958	+	0.461279i	−0.843107	−	0.537746i	\(-0.819276\pi\)
0.0441484	+	0.999025i	\(0.485943\pi\)
\(5\)	−24.3962	−	42.2554i	−0.975847	−	1.69022i	−0.677110	−	0.735882i	\(-0.736768\pi\)
							−0.298737	−	0.954335i	\(-0.596565\pi\)
\(7\)		−	41.7489i		−	0.852019i	−0.904719	−	0.426010i	\(-0.859919\pi\)
							0.904719	−	0.426010i	\(-0.140081\pi\)
\(11\)			125.138i			1.03420i	0.855925	+	0.517100i	\(0.172988\pi\)
							−0.855925	+	0.517100i	\(0.827012\pi\)
\(13\)	−25.2283	+	43.6968i	−0.149280	+	0.258561i	−0.930962	−	0.365117i	\(-0.881029\pi\)
							0.781681	+	0.623678i	\(0.214362\pi\)
\(17\)	6.72503	+	11.6481i	0.0232700	+	0.0403048i	0.877426	−	0.479712i	\(-0.159259\pi\)
							−0.854156	+	0.520017i	\(0.825926\pi\)
\(19\)	292.291	+	211.865i	0.809671	+	0.586883i
							\(23\)	155.132	+	89.5655i	0.293255	+	0.169311i	0.639409	−	0.768867i	\(-0.279179\pi\)
−0.346154	+	0.938178i	\(0.612512\pi\)
\(29\)	578.800	−	1002.51i	0.688229	−	1.19205i	−0.284182	−	0.958770i	\(-0.591722\pi\)
							0.972410	−	0.233277i	\(-0.0749448\pi\)
\(31\)			1654.61i			1.72176i	0.508806	+	0.860881i	\(0.330087\pi\)
							−0.508806	+	0.860881i	\(0.669913\pi\)
\(37\)	−901.700			−0.658656			−0.329328	−	0.944216i	\(-0.606822\pi\)
							−0.329328	+	0.944216i	\(0.606822\pi\)
\(41\)	−267.347	−	463.058i	−0.159040	−	0.275466i	0.775483	−	0.631369i	\(-0.217507\pi\)
							−0.934523	+	0.355903i	\(0.884173\pi\)
\(43\)	−1585.27	+	915.257i	−0.857367	+	0.495001i	−0.863130	−	0.504982i	\(-0.831499\pi\)
							0.00576278	+	0.999983i	\(0.498166\pi\)
\(47\)	2719.96	+	1570.37i	1.23131	+	0.710896i	0.967303	−	0.253625i	\(-0.0816229\pi\)
							0.264006	+	0.964521i	\(0.414956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.9	✓	76
4.3	odd	2	inner	76.5.g.a.7.17	yes	76
19.11	even	3	inner	76.5.g.a.11.17	yes	76
76.11	odd	6	inner	76.5.g.a.11.9	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.9	✓	76	1.1	even	1	trivial
76.5.g.a.7.17	yes	76	4.3	odd	2	inner
76.5.g.a.11.9	yes	76	76.11	odd	6	inner
76.5.g.a.11.17	yes	76	19.11	even	3	inner