Embedded newform 74.4.h.a.21.6

• Label: 74.4.h.a.21.6
• 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			21.6
Character	\(\chi\)	\(=\)	74.21
Dual form			74.4.h.a.67.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.684040 - 1.87939i) q^{2} +(-5.71416 + 2.07978i) q^{3} +(-3.06418 - 2.57115i) q^{4} +(5.14822 + 0.907770i) q^{5} +12.1618i q^{6} +(-3.41363 + 19.3596i) q^{7} +(-6.92820 + 4.00000i) q^{8} +(7.64290 - 6.41315i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

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
							\(3\)	−5.71416	+	2.07978i	−1.09969	+	0.400254i	−0.827202	−	0.561905i	\(-0.810069\pi\)
−0.272488	+	0.962159i	\(0.587847\pi\)
\(5\)	5.14822	+	0.907770i	0.460471	+	0.0811934i	0.399071	−	0.916920i	\(-0.369333\pi\)
							0.0613995	+	0.998113i	\(0.480444\pi\)
\(7\)	−3.41363	+	19.3596i	−0.184319	+	1.04532i	0.742509	+	0.669836i	\(0.233636\pi\)
							−0.926828	+	0.375487i	\(0.877476\pi\)
\(11\)	35.2678	+	61.0857i	0.966696	+	1.67437i	0.704989	+	0.709218i	\(0.250952\pi\)
							0.261707	+	0.965147i	\(0.415715\pi\)
\(13\)	−18.6977	+	22.2830i	−0.398908	+	0.475400i	−0.927687	−	0.373360i	\(-0.878206\pi\)
							0.528778	+	0.848760i	\(0.322650\pi\)
\(17\)	−36.7515	−	43.7988i	−0.524327	−	0.624868i	0.437271	−	0.899330i	\(-0.355945\pi\)
							−0.961598	+	0.274461i	\(0.911500\pi\)
\(19\)	−6.49433	−	17.8430i	−0.0784159	−	0.215446i	0.894290	−	0.447488i	\(-0.147681\pi\)
							−0.972706	+	0.232042i	\(0.925459\pi\)
\(23\)	25.8035	+	14.8977i	0.233930	+	0.135060i	0.612384	−	0.790560i	\(-0.290211\pi\)
							−0.378453	+	0.925620i	\(0.623544\pi\)
\(29\)	−202.057	+	116.658i	−1.29383	+	0.746992i	−0.979331	−	0.202266i	\(-0.935169\pi\)
							−0.314498	+	0.949258i	\(0.601836\pi\)
\(31\)			179.108i			1.03770i	0.854865	+	0.518851i	\(0.173640\pi\)
							−0.854865	+	0.518851i	\(0.826360\pi\)
\(37\)	210.637	−	79.2776i	0.935907	−	0.352247i
							\(41\)	241.437	+	202.590i	0.919662	+	0.771688i	0.973932	−	0.226838i	\(-0.0728388\pi\)
−0.0542704	+	0.998526i	\(0.517283\pi\)
\(43\)		−	61.9889i		−	0.219842i	−0.993940	−	0.109921i	\(-0.964940\pi\)
							0.993940	−	0.109921i	\(-0.0350598\pi\)
\(47\)	63.0581	−	109.220i	0.195702	−	0.338965i	−0.751429	−	0.659814i	\(-0.770635\pi\)
							0.947130	+	0.320849i	\(0.103968\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.21.6	✓	60
37.30	even	18	inner	74.4.h.a.67.6	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.21.6	✓	60	1.1	even	1	trivial
74.4.h.a.67.6	yes	60	37.30	even	18	inner