Embedded newform 76.7.g.a.7.15

• Label: 76.7.g.a.7.15
• Level: $76$
• Weight: $7$
• Character: 76.7
• Analytic conductor: $17.484$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4841103551\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.15
Character	\(\chi\)	\(=\)	76.7
Dual form			76.7.g.a.11.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.68771 + 5.62583i) q^{2} +(-33.2985 + 19.2249i) q^{3} +(0.700063 - 63.9962i) q^{4} +(-72.8107 - 126.112i) q^{5} +(81.2362 - 296.678i) q^{6} -173.537i q^{7} +(356.050 + 367.930i) q^{8} +(374.695 - 648.990i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - q^{2} - 67 q^{4} - 2 q^{5} + 33 q^{6} + 962 q^{8} + 12660 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.68771	+	5.62583i	−0.710964	+	0.703229i
							\(3\)	−33.2985	+	19.2249i	−1.23328	+	0.712034i	−0.967712	−	0.252059i	\(-0.918892\pi\)
−0.265567	+	0.964092i	\(0.585559\pi\)
\(5\)	−72.8107	−	126.112i	−0.582485	−	1.00889i	−0.995184	−	0.0980267i	\(-0.968747\pi\)
							0.412698	−	0.910868i	\(-0.364586\pi\)
\(7\)		−	173.537i		−	0.505939i	−0.967474	−	0.252969i	\(-0.918593\pi\)
							0.967474	−	0.252969i	\(-0.0814072\pi\)
\(11\)			55.9201i			0.0420136i	0.999779	+	0.0210068i	\(0.00668716\pi\)
							−0.999779	+	0.0210068i	\(0.993313\pi\)
\(13\)	239.828	−	415.394i	0.109162	−	0.189074i	−0.806269	−	0.591549i	\(-0.798517\pi\)
							0.915431	+	0.402475i	\(0.131850\pi\)
\(17\)	−911.361	−	1578.52i	−0.185500	−	0.321295i	0.758245	−	0.651970i	\(-0.226057\pi\)
							−0.943745	+	0.330674i	\(0.892724\pi\)
\(19\)	3021.08	−	6157.84i	0.440455	−	0.897775i
							\(23\)	−15509.7	−	8954.54i	−1.27474	−	0.735970i	−0.298861	−	0.954297i	\(-0.596607\pi\)
−0.975876	+	0.218327i	\(0.929940\pi\)
\(29\)	−10882.0	+	18848.1i	−0.446183	+	0.772811i	−0.998134	−	0.0610653i	\(-0.980550\pi\)
							0.551951	+	0.833877i	\(0.313884\pi\)
\(31\)			34180.9i			1.14736i	0.819080	+	0.573679i	\(0.194484\pi\)
							−0.819080	+	0.573679i	\(0.805516\pi\)
\(37\)	1864.28			0.0368049			0.0184025	−	0.999831i	\(-0.494142\pi\)
							0.0184025	+	0.999831i	\(0.494142\pi\)
\(41\)	19140.7	+	33152.7i	0.277720	+	0.481025i	0.970818	−	0.239819i	\(-0.0770879\pi\)
							−0.693098	+	0.720844i	\(0.743755\pi\)
\(43\)	−105863.	+	61120.2i	−1.33150	+	0.768740i	−0.985529	−	0.169507i	\(-0.945782\pi\)
							−0.345967	+	0.938247i	\(0.612449\pi\)
\(47\)	29630.4	+	17107.1i	0.285393	+	0.164772i	0.635862	−	0.771802i	\(-0.280645\pi\)
							−0.350469	+	0.936574i	\(0.613978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.g.a.7.15	✓	116
4.3	odd	2	inner	76.7.g.a.7.54	yes	116
19.11	even	3	inner	76.7.g.a.11.54	yes	116
76.11	odd	6	inner	76.7.g.a.11.15	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.g.a.7.15	✓	116	1.1	even	1	trivial
76.7.g.a.7.54	yes	116	4.3	odd	2	inner
76.7.g.a.11.15	yes	116	76.11	odd	6	inner
76.7.g.a.11.54	yes	116	19.11	even	3	inner