Embedded newform 74.2.f.a.53.1

• Label: 74.2.f.a.53.1
• Level: $74$
• Weight: $2$
• Character: 74.53
• Analytic conductor: $0.591$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.590892974957\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			53.1
Root			\(-0.173648 - 0.984808i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.53
Dual form			74.2.f.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.939693 + 0.342020i) q^{2} +(-0.326352 + 0.118782i) q^{3} +(0.766044 + 0.642788i) q^{4} +(-0.0209445 + 0.118782i) q^{5} -0.347296 q^{6} +(0.233956 - 1.32683i) q^{7} +(0.500000 + 0.866025i) q^{8} +(-2.20574 + 1.85083i) q^{9} +(-0.0603074 + 0.104455i) q^{10} +(-2.26604 - 3.92490i) q^{11} +(-0.326352 - 0.118782i) q^{12} +(-0.592396 - 0.497079i) q^{13} +(0.673648 - 1.16679i) q^{14} +(-0.00727396 - 0.0412527i) q^{15} +(0.173648 + 0.984808i) q^{16} +(-2.29813 + 1.92836i) q^{17} +(-2.70574 + 0.984808i) q^{18} +(1.91875 - 0.698367i) q^{19} +(-0.0923963 + 0.0775297i) q^{20} +(0.0812519 + 0.460802i) q^{21} +(-0.786989 - 4.46324i) q^{22} +(-0.0282185 + 0.0488759i) q^{23} +(-0.266044 - 0.223238i) q^{24} +(4.68479 + 1.70513i) q^{25} +(-0.386659 - 0.669713i) q^{26} +(1.02094 - 1.76833i) q^{27} +(1.03209 - 0.866025i) q^{28} +(2.89053 + 5.00654i) q^{29} +(0.00727396 - 0.0412527i) q^{30} -3.34730 q^{31} +(-0.173648 + 0.984808i) q^{32} +(1.20574 + 1.01173i) q^{33} +(-2.81908 + 1.02606i) q^{34} +(0.152704 + 0.0555796i) q^{35} -2.87939 q^{36} +(5.60607 + 2.36051i) q^{37} +2.04189 q^{38} +(0.252374 + 0.0918566i) q^{39} +(-0.113341 + 0.0412527i) q^{40} +(-5.47565 - 4.59462i) q^{41} +(-0.0812519 + 0.460802i) q^{42} +9.31315 q^{43} +(0.786989 - 4.46324i) q^{44} +(-0.173648 - 0.300767i) q^{45} +(-0.0432332 + 0.0362770i) q^{46} +(-4.25877 + 7.37641i) q^{47} +(-0.173648 - 0.300767i) q^{48} +(4.87211 + 1.77330i) q^{49} +(3.81908 + 3.20459i) q^{50} +(0.520945 - 0.902302i) q^{51} +(-0.134285 - 0.761570i) q^{52} +(0.482926 + 2.73881i) q^{53} +(1.56418 - 1.31250i) q^{54} +(0.513671 - 0.186961i) q^{55} +(1.26604 - 0.460802i) q^{56} +(-0.543233 + 0.455827i) q^{57} +(1.00387 + 5.69323i) q^{58} +(-2.25624 - 12.7958i) q^{59} +(0.0209445 - 0.0362770i) q^{60} +(-8.82295 - 7.40333i) q^{61} +(-3.14543 - 1.14484i) q^{62} +(1.93969 + 3.35965i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(0.0714517 - 0.0599551i) q^{65} +(0.786989 + 1.36310i) q^{66} +(0.889185 - 5.04282i) q^{67} -3.00000 q^{68} +(0.00340357 - 0.0193026i) q^{69} +(0.124485 + 0.104455i) q^{70} +(-12.6236 + 4.59462i) q^{71} +(-2.70574 - 0.984808i) q^{72} -8.71688 q^{73} +(4.46064 + 4.13554i) q^{74} -1.73143 q^{75} +(1.91875 + 0.698367i) q^{76} +(-5.73783 + 2.08840i) q^{77} +(0.205737 + 0.172634i) q^{78} +(0.720285 - 4.08494i) q^{79} -0.120615 q^{80} +(1.37686 - 7.80856i) q^{81} +(-3.57398 - 6.19031i) q^{82} +(-4.53596 + 3.80612i) q^{83} +(-0.233956 + 0.405223i) q^{84} +(-0.180922 - 0.313366i) q^{85} +(8.75150 + 3.18528i) q^{86} +(-1.53802 - 1.29055i) q^{87} +(2.26604 - 3.92490i) q^{88} +(1.32295 + 7.50281i) q^{89} +(-0.0603074 - 0.342020i) q^{90} +(-0.798133 + 0.669713i) q^{91} +(-0.0530334 + 0.0193026i) q^{92} +(1.09240 - 0.397600i) q^{93} +(-6.52481 + 5.47497i) q^{94} +(0.0427664 + 0.242540i) q^{95} +(-0.0603074 - 0.342020i) q^{96} +(5.12061 - 8.86916i) q^{97} +(3.97178 + 3.33272i) q^{98} +(12.2626 + 4.46324i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 3 * q^3 + 3 * q^5 + 6 * q^7 + 3 * q^8 - 3 * q^9 - 6 * q^10 - 9 * q^11 - 3 * q^12 + 3 * q^14 - 18 * q^15 - 6 * q^18 + 9 * q^19 + 3 * q^20 + 3 * q^21 + 3 * q^22 - 15 * q^23 + 3 * q^24 + 21 * q^25 - 9 * q^26 + 3 * q^27 - 3 * q^28 + 18 * q^30 - 18 * q^31 - 3 * q^33 + 3 * q^35 - 6 * q^36 + 9 * q^37 + 6 * q^38 + 18 * q^39 + 6 * q^40 + 6 * q^41 - 3 * q^42 + 12 * q^43 - 3 * q^44 + 15 * q^46 - 3 * q^47 + 6 * q^50 + 9 * q^52 - 18 * q^53 - 9 * q^54 - 18 * q^55 + 3 * q^56 + 12 * q^57 - 18 * q^58 - 6 * q^59 - 3 * q^60 - 12 * q^61 - 3 * q^62 + 6 * q^63 - 3 * q^64 - 3 * q^66 - 3 * q^67 - 18 * q^68 + 42 * q^69 - 12 * q^70 - 6 * q^71 - 6 * q^72 - 36 * q^73 + 18 * q^74 - 30 * q^75 + 9 * q^76 - 15 * q^77 - 9 * q^78 + 30 * q^79 - 12 * q^80 + 12 * q^81 - 6 * q^82 + 6 * q^83 - 6 * q^84 - 18 * q^85 + 12 * q^86 - 27 * q^87 + 9 * q^88 - 33 * q^89 - 6 * q^90 + 9 * q^91 + 12 * q^92 + 3 * q^93 - 12 * q^94 + 51 * q^95 - 6 * q^96 + 42 * q^97 + 9 * q^98 + 27 * q^99

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.939693	+	0.342020i	0.664463	+	0.241845i
							\(3\)	−0.326352	+	0.118782i	−0.188419	+	0.0685790i	−0.434507	−	0.900669i	\(-0.643077\pi\)
0.246087	+	0.969248i	\(0.420855\pi\)
\(5\)	−0.0209445	+	0.118782i	−0.00936668	+	0.0531211i	−0.989133	−	0.147024i	\(-0.953030\pi\)
							0.979766	+	0.200146i	\(0.0641415\pi\)
\(7\)	0.233956	−	1.32683i	0.0884269	−	0.501494i	−0.908137	−	0.418672i	\(-0.862496\pi\)
							0.996564	−	0.0828217i	\(-0.0263932\pi\)
\(11\)	−2.26604	−	3.92490i	−0.683238	−	1.18340i	−0.973987	−	0.226604i	\(-0.927238\pi\)
							0.290749	−	0.956799i	\(-0.406096\pi\)
\(13\)	−0.592396	−	0.497079i	−0.164301	−	0.137865i	0.556930	−	0.830559i	\(-0.311979\pi\)
							−0.721231	+	0.692694i	\(0.756424\pi\)
\(17\)	−2.29813	+	1.92836i	−0.557379	+	0.467697i	−0.877431	−	0.479703i	\(-0.840744\pi\)
							0.320051	+	0.947400i	\(0.396300\pi\)
\(19\)	1.91875	−	0.698367i	0.440191	−	0.160216i	−0.112411	−	0.993662i	\(-0.535857\pi\)
							0.552602	+	0.833445i	\(0.313635\pi\)
\(23\)	−0.0282185	+	0.0488759i	−0.00588396	+	0.0101913i	−0.868952	−	0.494896i	\(-0.835206\pi\)
							0.863068	+	0.505087i	\(0.168540\pi\)
\(29\)	2.89053	+	5.00654i	0.536758	+	0.929692i	0.999076	+	0.0429778i	\(0.0136845\pi\)
							−0.462318	+	0.886714i	\(0.652982\pi\)
\(31\)	−3.34730			−0.601192			−0.300596	−	0.953752i	\(-0.597186\pi\)
							−0.300596	+	0.953752i	\(0.597186\pi\)
\(37\)	5.60607	+	2.36051i	0.921632	+	0.388066i
							\(41\)	−5.47565	−	4.59462i	−0.855153	−	0.717559i	0.105765	−	0.994391i	\(-0.466271\pi\)
−0.960918	+	0.276832i	\(0.910715\pi\)
\(43\)	9.31315			1.42024			0.710121	−	0.704080i	\(-0.248640\pi\)
							0.710121	+	0.704080i	\(0.248640\pi\)
\(47\)	−4.25877	+	7.37641i	−0.621206	+	1.07596i	0.368056	+	0.929804i	\(0.380023\pi\)
							−0.989262	+	0.146156i	\(0.953310\pi\)
\(53\)	0.482926	+	2.73881i	0.0663350	+	0.376204i	0.999844	+	0.0176510i	\(0.00561877\pi\)
							−0.933509	+	0.358553i	\(0.883270\pi\)
\(59\)	−2.25624	−	12.7958i	−0.293738	−	1.66587i	−0.672288	−	0.740290i	\(-0.734688\pi\)
							0.378550	−	0.925581i	\(-0.376423\pi\)
\(61\)	−8.82295	−	7.40333i	−1.12966	−	0.947900i	−0.130611	−	0.991434i	\(-0.541694\pi\)
							−0.999052	+	0.0435341i	\(0.986138\pi\)
\(67\)	0.889185	−	5.04282i	0.108631	−	0.616079i	−0.881076	−	0.472974i	\(-0.843180\pi\)
							0.989708	−	0.143104i	\(-0.0457085\pi\)
\(71\)	−12.6236	+	4.59462i	−1.49815	+	0.545281i	−0.955580	−	0.294732i	\(-0.904770\pi\)
							−0.542567	+	0.840013i	\(0.682547\pi\)
\(73\)	−8.71688			−1.02023			−0.510117	−	0.860105i	\(-0.670398\pi\)
							−0.510117	+	0.860105i	\(0.670398\pi\)
\(79\)	0.720285	−	4.08494i	0.0810384	−	0.459592i	−0.917103	−	0.398650i	\(-0.869479\pi\)
							0.998141	−	0.0609412i	\(-0.0194102\pi\)
\(83\)	−4.53596	+	3.80612i	−0.497886	+	0.417776i	−0.856843	−	0.515578i	\(-0.827577\pi\)
							0.358956	+	0.933354i	\(0.383133\pi\)
\(89\)	1.32295	+	7.50281i	0.140232	+	0.795297i	0.971072	+	0.238785i	\(0.0767492\pi\)
							−0.830840	+	0.556511i	\(0.812140\pi\)
\(97\)	5.12061	−	8.86916i	0.519920	−	0.900527i	−0.479812	−	0.877371i	\(-0.659295\pi\)
							0.999732	−	0.0231560i	\(-0.00737146\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.2.f.a.53.1	yes	6
3.2	odd	2		666.2.x.c.127.1		6
4.3	odd	2		592.2.bc.b.497.1		6
37.7	even	9	inner	74.2.f.a.7.1	✓	6
37.9	even	9		2738.2.a.m.1.2		3
37.28	even	18		2738.2.a.p.1.2		3
111.44	odd	18		666.2.x.c.451.1		6
148.7	odd	18		592.2.bc.b.81.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.f.a.7.1	✓	6	37.7	even	9	inner
74.2.f.a.53.1	yes	6	1.1	even	1	trivial
592.2.bc.b.81.1		6	148.7	odd	18
592.2.bc.b.497.1		6	4.3	odd	2
666.2.x.c.127.1		6	3.2	odd	2
666.2.x.c.451.1		6	111.44	odd	18
2738.2.a.m.1.2		3	37.9	even	9
2738.2.a.p.1.2		3	37.28	even	18