Embedded newform 74.5.g.b.23.3

• Label: 74.5.g.b.23.3
• 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			23.3
Character	\(\chi\)	\(=\)	74.23
Dual form			74.5.g.b.29.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.732051 - 2.73205i) q^{2} +(-7.91112 + 4.56749i) q^{3} +(-6.92820 + 4.00000i) q^{4} +(9.46674 - 35.3304i) q^{5} +(18.2700 + 18.2700i) q^{6} +(-10.9589 - 18.9814i) q^{7} +(16.0000 + 16.0000i) q^{8} +(1.22390 - 2.11986i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} - 102 q^{5} + 40 q^{6} - 48 q^{7} + 448 q^{8} + 556 q^{9}+O(q^{10}) \)

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
							\(3\)	−7.91112	+	4.56749i	−0.879014	+	0.507499i	−0.870333	−	0.492464i	\(-0.836096\pi\)
−0.00868054	+	0.999962i	\(0.502763\pi\)
\(5\)	9.46674	−	35.3304i	0.378670	−	1.41321i	−0.469239	−	0.883071i	\(-0.655472\pi\)
							0.847908	−	0.530143i	\(-0.177862\pi\)
\(7\)	−10.9589	−	18.9814i	−0.223652	−	0.387376i	0.732262	−	0.681023i	\(-0.238465\pi\)
							−0.955914	+	0.293646i	\(0.905131\pi\)
\(11\)			135.072i			1.11630i	0.829740	+	0.558150i	\(0.188489\pi\)
							−0.829740	+	0.558150i	\(0.811511\pi\)
\(13\)	−72.3767	+	270.113i	−0.428264	+	1.59830i	0.328425	+	0.944530i	\(0.393482\pi\)
							−0.756690	+	0.653774i	\(0.773185\pi\)
\(17\)	−450.123	+	120.610i	−1.55752	+	0.417336i	−0.931877	−	0.362775i	\(-0.881829\pi\)
							−0.625642	+	0.780111i	\(0.715163\pi\)
\(19\)	103.764	−	387.254i	0.287436	−	1.07272i	−0.659605	−	0.751612i	\(-0.729277\pi\)
							0.947041	−	0.321112i	\(-0.104057\pi\)
\(23\)	716.848	+	716.848i	1.35510	+	1.35510i	0.879852	+	0.475248i	\(0.157642\pi\)
							0.475248	+	0.879852i	\(0.342358\pi\)
\(29\)	−380.347	+	380.347i	−0.452256	+	0.452256i	−0.896103	−	0.443847i	\(-0.853613\pi\)
							0.443847	+	0.896103i	\(0.353613\pi\)
\(31\)	−275.960	+	275.960i	−0.287159	+	0.287159i	−0.835956	−	0.548797i	\(-0.815086\pi\)
							0.548797	+	0.835956i	\(0.315086\pi\)
\(37\)	−1353.25	−	207.033i	−0.988499	−	0.151229i
							\(41\)	−1387.67	+	801.172i	−0.825503	+	0.476604i	−0.852310	−	0.523036i	\(-0.824799\pi\)
0.0268076	+	0.999641i	\(0.491466\pi\)
\(43\)	1661.17	+	1661.17i	0.898413	+	0.898413i	0.995296	−	0.0968830i	\(-0.0308873\pi\)
							−0.0968830	+	0.995296i	\(0.530887\pi\)
\(47\)	−1226.78			−0.555356			−0.277678	−	0.960674i	\(-0.589565\pi\)
							−0.277678	+	0.960674i	\(0.589565\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.b.23.3	✓	28
37.29	odd	12	inner	74.5.g.b.29.3	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.b.23.3	✓	28	1.1	even	1	trivial
74.5.g.b.29.3	yes	28	37.29	odd	12	inner