Embedded newform 76.4.f.a.27.9

• Label: 76.4.f.a.27.9
• Level: $76$
• Weight: $4$
• Character: 76.27
• Analytic conductor: $4.484$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.48414516044\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.9
Character	\(\chi\)	\(=\)	76.27
Dual form			76.4.f.a.31.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80915 + 2.17416i) q^{2} +(-2.57005 + 4.45145i) q^{3} +(-1.45393 - 7.86677i) q^{4} +(7.41606 - 12.8450i) q^{5} +(-5.02855 - 13.6410i) q^{6} -20.1038i q^{7} +(19.7340 + 11.0711i) q^{8} +(0.289720 + 0.501809i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} + 5 q^{4} - 2 q^{5} + 21 q^{6} - 228 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80915	+	2.17416i	−0.639632	+	0.768681i
							\(3\)	−2.57005	+	4.45145i	−0.494606	+	0.856682i	−0.999981	−	0.00621762i	\(-0.998021\pi\)
0.505375	+	0.862900i	\(0.331354\pi\)
\(5\)	7.41606	−	12.8450i	0.663313	−	1.14889i	−0.316427	−	0.948617i	\(-0.602483\pi\)
							0.979740	−	0.200275i	\(-0.0641835\pi\)
\(7\)		−	20.1038i		−	1.08551i	−0.839893	−	0.542753i	\(-0.817382\pi\)
							0.839893	−	0.542753i	\(-0.182618\pi\)
\(11\)			7.47409i			0.204866i	0.994740	+	0.102433i	\(0.0326627\pi\)
							−0.994740	+	0.102433i	\(0.967337\pi\)
\(13\)	8.75417	−	5.05422i	0.186767	−	0.107830i	−0.403701	−	0.914891i	\(-0.632276\pi\)
							0.590468	+	0.807061i	\(0.298943\pi\)
\(17\)	42.5160	−	73.6398i	0.606567	−	1.05060i	−0.385235	−	0.922819i	\(-0.625880\pi\)
							0.991802	−	0.127786i	\(-0.0407871\pi\)
\(19\)	57.8226	−	59.2921i	0.698179	−	0.715923i
							\(23\)	22.1814	−	12.8065i	0.201093	−	0.116101i	−0.396072	−	0.918219i	\(-0.629627\pi\)
0.597165	+	0.802118i	\(0.296294\pi\)
\(29\)	234.929	−	135.637i	1.50432	−	0.868520i	0.504333	−	0.863509i	\(-0.331738\pi\)
							0.999987	−	0.00501089i	\(-0.00159502\pi\)
\(31\)	31.1891			0.180701			0.0903505	−	0.995910i	\(-0.471201\pi\)
							0.0903505	+	0.995910i	\(0.471201\pi\)
\(37\)			317.710i			1.41165i	0.708385	+	0.705826i	\(0.249424\pi\)
							−0.708385	+	0.705826i	\(0.750576\pi\)
\(41\)	−298.083	−	172.098i	−1.13543	−	0.655542i	−0.190137	−	0.981758i	\(-0.560893\pi\)
							−0.945296	+	0.326215i	\(0.894227\pi\)
\(43\)	−195.107	−	112.645i	−0.691943	−	0.399493i	0.112397	−	0.993663i	\(-0.464147\pi\)
							−0.804339	+	0.594170i	\(0.797481\pi\)
\(47\)	−69.5333	+	40.1451i	−0.215797	+	0.124591i	−0.604003	−	0.796982i	\(-0.706428\pi\)
							0.388205	+	0.921573i	\(0.373095\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.f.a.27.9	✓	56
4.3	odd	2	inner	76.4.f.a.27.18	yes	56
19.12	odd	6	inner	76.4.f.a.31.18	yes	56
76.31	even	6	inner	76.4.f.a.31.9	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.f.a.27.9	✓	56	1.1	even	1	trivial
76.4.f.a.27.18	yes	56	4.3	odd	2	inner
76.4.f.a.31.9	yes	56	76.31	even	6	inner
76.4.f.a.31.18	yes	56	19.12	odd	6	inner