Embedded newform 76.5.g.a.7.1

• Label: 76.5.g.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.99928 + 0.0760910i) q^{2} +(-4.58614 + 2.64781i) q^{3} +(15.9884 - 0.608618i) q^{4} +(-5.59416 - 9.68937i) q^{5} +(18.1398 - 10.9383i) q^{6} +39.7837i q^{7} +(-63.8958 + 3.65061i) q^{8} +(-26.4782 + 45.8616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.99928	+	0.0760910i	−0.999819	+	0.0190228i
							\(3\)	−4.58614	+	2.64781i	−0.509571	+	0.294201i	−0.732657	−	0.680598i	\(-0.761720\pi\)
0.223086	+	0.974799i	\(0.428387\pi\)
\(5\)	−5.59416	−	9.68937i	−0.223767	−	0.387575i	0.732182	−	0.681109i	\(-0.238502\pi\)
							−0.955949	+	0.293534i	\(0.905169\pi\)
\(7\)			39.7837i			0.811912i	0.913893	+	0.405956i	\(0.133061\pi\)
							−0.913893	+	0.405956i	\(0.866939\pi\)
\(11\)		−	19.2518i		−	0.159106i	−0.996831	−	0.0795530i	\(-0.974651\pi\)
							0.996831	−	0.0795530i	\(-0.0253493\pi\)
\(13\)	128.814	−	223.113i	0.762213	−	1.32019i	−0.179494	−	0.983759i	\(-0.557446\pi\)
							0.941707	−	0.336433i	\(-0.109221\pi\)
\(17\)	−140.636	−	243.589i	−0.486631	−	0.842870i	0.513251	−	0.858239i	\(-0.328441\pi\)
							−0.999882	+	0.0153689i	\(0.995108\pi\)
\(19\)	−74.3594	−	353.259i	−0.205982	−	0.978556i
							\(23\)	−240.590	−	138.905i	−0.454801	−	0.262580i	0.255054	−	0.966927i	\(-0.417907\pi\)
−0.709856	+	0.704347i	\(0.751240\pi\)
\(29\)	309.335	−	535.784i	0.367818	−	0.637080i	−0.621406	−	0.783489i	\(-0.713438\pi\)
							0.989224	+	0.146409i	\(0.0467715\pi\)
\(31\)			447.698i			0.465867i	0.972493	+	0.232934i	\(0.0748325\pi\)
							−0.972493	+	0.232934i	\(0.925168\pi\)
\(37\)	1024.23			0.748157			0.374079	−	0.927397i	\(-0.377959\pi\)
							0.374079	+	0.927397i	\(0.377959\pi\)
\(41\)	−83.3681	−	144.398i	−0.0495944	−	0.0859000i	0.840163	−	0.542335i	\(-0.182460\pi\)
							−0.889757	+	0.456435i	\(0.849126\pi\)
\(43\)	−1503.97	+	868.315i	−0.813394	+	0.469613i	−0.848133	−	0.529783i	\(-0.822273\pi\)
							0.0347389	+	0.999396i	\(0.488940\pi\)
\(47\)	−3558.01	−	2054.22i	−1.61069	−	0.929933i	−0.989210	−	0.146503i	\(-0.953198\pi\)
							−0.621480	−	0.783430i	\(-0.713468\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.1	✓	76
4.3	odd	2	inner	76.5.g.a.7.26	yes	76
19.11	even	3	inner	76.5.g.a.11.26	yes	76
76.11	odd	6	inner	76.5.g.a.11.1	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.1	✓	76	1.1	even	1	trivial
76.5.g.a.7.26	yes	76	4.3	odd	2	inner
76.5.g.a.11.1	yes	76	76.11	odd	6	inner
76.5.g.a.11.26	yes	76	19.11	even	3	inner