Embedded newform 74.5.i.b.15.1

• Label: 74.5.i.b.15.1
• Level: $74$
• Weight: $5$
• Character: 74.15
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			15.1
Character	\(\chi\)	\(=\)	74.15
Dual form			74.5.i.b.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19534 + 2.56343i) q^{2} +(-4.50230 - 12.3700i) q^{3} +(-5.14230 + 6.12836i) q^{4} +(-4.41825 - 6.30992i) q^{5} +(26.3277 - 26.3277i) q^{6} +(-6.53988 + 37.0895i) q^{7} +(-21.8564 - 5.85641i) q^{8} +(-70.6956 + 59.3207i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 114 q^{5} - 576 q^{8} - 210 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19534	+	2.56343i	0.298836	+	0.640856i
							\(3\)	−4.50230	−	12.3700i	−0.500255	−	1.37444i	−0.891026	−	0.453952i	\(-0.850014\pi\)
0.390771	−	0.920488i	\(-0.372208\pi\)
\(5\)	−4.41825	−	6.30992i	−0.176730	−	0.252397i	0.721008	−	0.692927i	\(-0.243679\pi\)
							−0.897738	+	0.440530i	\(0.854790\pi\)
\(7\)	−6.53988	+	37.0895i	−0.133467	+	0.756929i	0.842448	+	0.538778i	\(0.181114\pi\)
							−0.975915	+	0.218151i	\(0.929997\pi\)
\(11\)	−84.2588	+	48.6468i	−0.696353	+	0.402040i	−0.805988	−	0.591932i	\(-0.798365\pi\)
							0.109634	+	0.993972i	\(0.465032\pi\)
\(13\)	−299.402	+	26.1943i	−1.77161	+	0.154996i	−0.925682	−	0.378304i	\(-0.876508\pi\)
							−0.845927	+	0.533299i	\(0.820952\pi\)
\(17\)	−9.49166	+	108.490i	−0.0328431	+	0.375399i	0.961931	+	0.273294i	\(0.0881132\pi\)
							−0.994774	+	0.102105i	\(0.967442\pi\)
\(19\)	−23.9654	+	51.3940i	−0.0663862	+	0.142366i	−0.936701	−	0.350130i	\(-0.886137\pi\)
							0.870315	+	0.492496i	\(0.163915\pi\)
\(23\)	37.7334	+	140.823i	0.0713296	+	0.266206i	0.992376	−	0.123247i	\(-0.0393308\pi\)
							−0.921046	+	0.389453i	\(0.872664\pi\)
\(29\)	−7.95884	+	29.7028i	−0.00946355	+	0.0353184i	−0.970496	−	0.241117i	\(-0.922486\pi\)
							0.961033	+	0.276435i	\(0.0891531\pi\)
\(31\)	−790.334	−	790.334i	−0.822407	−	0.822407i	0.164045	−	0.986453i	\(-0.447546\pi\)
							−0.986453	+	0.164045i	\(0.947546\pi\)
\(37\)	1362.59	−	132.292i	0.995320	−	0.0966340i
							\(41\)	201.743	−	240.428i	0.120014	−	0.143027i	−0.702692	−	0.711494i	\(-0.748019\pi\)
0.822706	+	0.568467i	\(0.192463\pi\)
\(43\)	−1319.41	+	1319.41i	−0.713580	+	0.713580i	−0.967282	−	0.253702i	\(-0.918352\pi\)
							0.253702	+	0.967282i	\(0.418352\pi\)
\(47\)	−741.884	+	1284.98i	−0.335846	+	0.581702i	−0.983647	−	0.180107i	\(-0.942355\pi\)
							0.647801	+	0.761810i	\(0.275689\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.i.b.15.1	yes	72
37.5	odd	36	inner	74.5.i.b.5.1	✓	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.i.b.5.1	✓	72	37.5	odd	36	inner
74.5.i.b.15.1	yes	72	1.1	even	1	trivial