Embedded newform 74.6.h.a.3.14

• Label: 74.6.h.a.3.14
• Level: $74$
• Weight: $6$
• Character: 74.3
• Analytic conductor: $11.868$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8684026662\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	74.3
Dual form			74.6.h.a.25.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57115 - 3.06418i) q^{2} +(10.7924 - 9.05588i) q^{3} +(-2.77837 - 15.7569i) q^{4} +(17.1692 - 47.1719i) q^{5} -56.3538i q^{6} +(-205.267 - 74.7112i) q^{7} +(-55.4256 - 32.0000i) q^{8} +(-7.73005 + 43.8393i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 6 q^{3} + 90 q^{5} - 228 q^{7} - 186 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.57115	−	3.06418i	0.454519	−	0.541675i
							\(3\)	10.7924	−	9.05588i	0.692331	−	0.580935i	−0.227249	−	0.973837i	\(-0.572973\pi\)
0.919581	+	0.392902i	\(0.128529\pi\)
\(5\)	17.1692	−	47.1719i	0.307131	−	0.843837i	−0.686081	−	0.727525i	\(-0.740671\pi\)
							0.993213	−	0.116312i	\(-0.0371072\pi\)
\(7\)	−205.267	−	74.7112i	−1.58334	−	0.576290i	−0.607415	−	0.794385i	\(-0.707794\pi\)
							−0.975928	+	0.218095i	\(0.930016\pi\)
\(11\)	−42.0813	+	72.8869i	−0.104859	+	0.181622i	−0.913681	−	0.406433i	\(-0.866773\pi\)
							0.808821	+	0.588054i	\(0.200106\pi\)
\(13\)	483.834	−	85.3131i	0.794033	−	0.140009i	0.238104	−	0.971240i	\(-0.423474\pi\)
							0.555928	+	0.831230i	\(0.312363\pi\)
\(17\)	1276.12	+	225.014i	1.07095	+	0.188837i	0.681210	−	0.732088i	\(-0.261454\pi\)
							0.389739	+	0.920925i	\(0.372565\pi\)
\(19\)	−1757.34	−	2094.32i	−1.11679	−	1.33094i	−0.937834	−	0.347084i	\(-0.887172\pi\)
							−0.178958	−	0.983857i	\(-0.557273\pi\)
\(23\)	518.470	−	299.339i	0.204364	−	0.117989i	−0.394326	−	0.918971i	\(-0.629022\pi\)
							0.598689	+	0.800981i	\(0.295688\pi\)
\(29\)	−3286.16	−	1897.26i	−0.725594	−	0.418922i	0.0912144	−	0.995831i	\(-0.470925\pi\)
							−0.816808	+	0.576910i	\(0.804258\pi\)
\(31\)		−	6933.26i		−	1.29579i	−0.761731	−	0.647893i	\(-0.775650\pi\)
							0.761731	−	0.647893i	\(-0.224350\pi\)
\(37\)	7518.70	−	3579.55i	0.902897	−	0.429857i
							\(41\)	−1485.00	−	8421.87i	−0.137965	−	0.782436i	−0.972749	−	0.231860i	\(-0.925519\pi\)
0.834785	−	0.550577i	\(-0.185592\pi\)
\(43\)		−	11964.8i		−	0.986812i	−0.869799	−	0.493406i	\(-0.835752\pi\)
							0.869799	−	0.493406i	\(-0.164248\pi\)
\(47\)	7749.02	+	13421.7i	0.511684	+	0.886263i	0.999908	+	0.0135449i	\(0.00431161\pi\)
							−0.488224	+	0.872718i	\(0.662355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.h.a.3.14	✓	96
37.25	even	18	inner	74.6.h.a.25.14	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.h.a.3.14	✓	96	1.1	even	1	trivial
74.6.h.a.25.14	yes	96	37.25	even	18	inner