Embedded newform 74.5.g.b.23.2

• Label: 74.5.g.b.23.2
• Level: $74$
• Weight: $5$
• Character: 74.23
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.2
Character	\(\chi\)	\(=\)	74.23
Dual form			74.5.g.b.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.732051 - 2.73205i) q^{2} +(-9.27942 + 5.35748i) q^{3} +(-6.92820 + 4.00000i) q^{4} +(1.47068 - 5.48866i) q^{5} +(21.4299 + 21.4299i) q^{6} +(20.7504 + 35.9408i) q^{7} +(16.0000 + 16.0000i) q^{8} +(16.9051 - 29.2805i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} - 102 q^{5} + 40 q^{6} - 48 q^{7} + 448 q^{8} + 556 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.732051	−	2.73205i	−0.183013	−	0.683013i
							\(3\)	−9.27942	+	5.35748i	−1.03105	+	0.595275i	−0.917284	−	0.398233i	\(-0.869624\pi\)
−0.113763	+	0.993508i	\(0.536290\pi\)
\(5\)	1.47068	−	5.48866i	0.0588273	−	0.219546i	−0.930254	−	0.366915i	\(-0.880414\pi\)
							0.989082	+	0.147369i	\(0.0470805\pi\)
\(7\)	20.7504	+	35.9408i	0.423478	+	0.733485i	0.996277	−	0.0862105i	\(-0.0274757\pi\)
							−0.572799	+	0.819696i	\(0.694142\pi\)
\(11\)		−	234.947i		−	1.94171i	−0.239669	−	0.970855i	\(-0.577039\pi\)
							0.239669	−	0.970855i	\(-0.422961\pi\)
\(13\)	18.2333	−	68.0475i	0.107889	−	0.402648i	−0.890768	−	0.454459i	\(-0.849833\pi\)
							0.998657	+	0.0518108i	\(0.0164993\pi\)
\(17\)	414.083	−	110.953i	1.43281	−	0.383921i	0.542803	−	0.839860i	\(-0.317363\pi\)
							0.890011	+	0.455939i	\(0.150696\pi\)
\(19\)	62.9849	−	235.063i	0.174474	−	0.651144i	−0.822167	−	0.569246i	\(-0.807235\pi\)
							0.996641	−	0.0818979i	\(-0.0260982\pi\)
\(23\)	−60.2876	−	60.2876i	−0.113965	−	0.113965i	0.647824	−	0.761790i	\(-0.275679\pi\)
							−0.761790	+	0.647824i	\(0.775679\pi\)
\(29\)	120.441	−	120.441i	0.143211	−	0.143211i	−0.631866	−	0.775078i	\(-0.717711\pi\)
							0.775078	+	0.631866i	\(0.217711\pi\)
\(31\)	−633.328	+	633.328i	−0.659030	+	0.659030i	−0.955151	−	0.296120i	\(-0.904307\pi\)
							0.296120	+	0.955151i	\(0.404307\pi\)
\(37\)	1363.82	−	118.996i	0.996215	−	0.0869217i
							\(41\)	951.238	−	549.198i	0.565876	−	0.326709i	−0.189624	−	0.981857i	\(-0.560727\pi\)
0.755501	+	0.655148i	\(0.227394\pi\)
\(43\)	692.344	+	692.344i	0.374443	+	0.374443i	0.869092	−	0.494650i	\(-0.164704\pi\)
							−0.494650	+	0.869092i	\(0.664704\pi\)
\(47\)	−2409.85			−1.09092			−0.545461	−	0.838136i	\(-0.683646\pi\)
							−0.545461	+	0.838136i	\(0.683646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.b.23.2	✓	28
37.29	odd	12	inner	74.5.g.b.29.2	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.b.23.2	✓	28	1.1	even	1	trivial
74.5.g.b.29.2	yes	28	37.29	odd	12	inner