Embedded newform 74.4.h.a.21.9

• Label: 74.4.h.a.21.9
• Level: $74$
• Weight: $4$
• Character: 74.21
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			21.9
Character	\(\chi\)	\(=\)	74.21
Dual form			74.4.h.a.67.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.684040 - 1.87939i) q^{2} +(6.26608 - 2.28067i) q^{3} +(-3.06418 - 2.57115i) q^{4} +(17.1326 + 3.02094i) q^{5} -13.3365i q^{6} +(-3.13680 + 17.7897i) q^{7} +(-6.92820 + 4.00000i) q^{8} +(13.3792 - 11.2264i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.684040	−	1.87939i	0.241845	−	0.664463i
							\(3\)	6.26608	−	2.28067i	1.20591	−	0.438915i	0.340626	−	0.940199i	\(-0.389361\pi\)
0.865283	+	0.501284i	\(0.167139\pi\)
\(5\)	17.1326	+	3.02094i	1.53239	+	0.270201i	0.875287	−	0.483603i	\(-0.160672\pi\)
							0.657099	+	0.753804i	\(0.271783\pi\)
\(7\)	−3.13680	+	17.7897i	−0.169372	+	0.960554i	0.775070	+	0.631875i	\(0.217714\pi\)
							−0.944442	+	0.328679i	\(0.893397\pi\)
\(11\)	−22.9290	−	39.7142i	−0.628487	−	1.08857i	−0.987855	−	0.155376i	\(-0.950341\pi\)
							0.359368	−	0.933196i	\(-0.382992\pi\)
\(13\)	−38.4683	+	45.8447i	−0.820706	+	0.978080i	−0.999984	−	0.00571877i	\(-0.998180\pi\)
							0.179277	+	0.983799i	\(0.442624\pi\)
\(17\)	−40.5392	−	48.3127i	−0.578364	−	0.689268i	0.394961	−	0.918698i	\(-0.370758\pi\)
							−0.973325	+	0.229430i	\(0.926314\pi\)
\(19\)	−18.8200	−	51.7076i	−0.227242	−	0.624343i	0.772703	−	0.634767i	\(-0.218904\pi\)
							−0.999946	+	0.0104240i	\(0.996682\pi\)
\(23\)	−53.6450	−	30.9720i	−0.486338	−	0.280787i	0.236716	−	0.971579i	\(-0.423929\pi\)
							−0.723054	+	0.690792i	\(0.757262\pi\)
\(29\)	145.248	−	83.8592i	0.930068	−	0.536975i	0.0432347	−	0.999065i	\(-0.486234\pi\)
							0.886833	+	0.462090i	\(0.152900\pi\)
\(31\)			251.183i			1.45528i	0.685957	+	0.727642i	\(0.259384\pi\)
							−0.685957	+	0.727642i	\(0.740616\pi\)
\(37\)	−62.7355	+	216.142i	−0.278747	+	0.960364i
							\(41\)	33.4811	+	28.0940i	0.127534	+	0.107013i	0.704324	−	0.709879i	\(-0.251250\pi\)
−0.576790	+	0.816892i	\(0.695695\pi\)
\(43\)		−	339.827i		−	1.20519i	−0.798048	−	0.602594i	\(-0.794134\pi\)
							0.798048	−	0.602594i	\(-0.205866\pi\)
\(47\)	120.880	−	209.370i	0.375153	−	0.649784i	−0.615197	−	0.788373i	\(-0.710924\pi\)
							0.990350	+	0.138590i	\(0.0442569\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.21.9	✓	60
37.30	even	18	inner	74.4.h.a.67.9	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.21.9	✓	60	1.1	even	1	trivial
74.4.h.a.67.9	yes	60	37.30	even	18	inner