Embedded newform 74.2.b.a.73.2

• Label: 74.2.b.a.73.2
• Level: $74$
• Weight: $2$
• Character: 74.73
• Analytic conductor: $0.591$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	74.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.590892974957\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{21})\)
Defining polynomial:	\( x^{4} + 11x^{2} + 25 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.2
Root			\(-1.79129i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.73
Dual form			74.2.b.a.73.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.79129 q^{3} -1.00000 q^{4} +0.791288i q^{5} -1.79129i q^{6} -2.00000 q^{7} +1.00000i q^{8} +0.208712 q^{9} +0.791288 q^{10} -0.791288 q^{11} -1.79129 q^{12} -3.79129i q^{13} +2.00000i q^{14} +1.41742i q^{15} +1.00000 q^{16} +7.58258i q^{17} -0.208712i q^{18} +1.58258i q^{19} -0.791288i q^{20} -3.58258 q^{21} +0.791288i q^{22} -3.79129i q^{23} +1.79129i q^{24} +4.37386 q^{25} -3.79129 q^{26} -5.00000 q^{27} +2.00000 q^{28} +3.79129i q^{29} +1.41742 q^{30} -8.37386i q^{31} -1.00000i q^{32} -1.41742 q^{33} +7.58258 q^{34} -1.58258i q^{35} -0.208712 q^{36} +(4.00000 - 4.58258i) q^{37} +1.58258 q^{38} -6.79129i q^{39} -0.791288 q^{40} +9.79129 q^{41} +3.58258i q^{42} -6.00000i q^{43} +0.791288 q^{44} +0.165151i q^{45} -3.79129 q^{46} -7.58258 q^{47} +1.79129 q^{48} -3.00000 q^{49} -4.37386i q^{50} +13.5826i q^{51} +3.79129i q^{52} -1.58258 q^{53} +5.00000i q^{54} -0.626136i q^{55} -2.00000i q^{56} +2.83485i q^{57} +3.79129 q^{58} +1.58258i q^{59} -1.41742i q^{60} +12.7913i q^{61} -8.37386 q^{62} -0.417424 q^{63} -1.00000 q^{64} +3.00000 q^{65} +1.41742i q^{66} +6.37386 q^{67} -7.58258i q^{68} -6.79129i q^{69} -1.58258 q^{70} -9.16515 q^{71} +0.208712i q^{72} -4.37386 q^{73} +(-4.58258 - 4.00000i) q^{74} +7.83485 q^{75} -1.58258i q^{76} +1.58258 q^{77} -6.79129 q^{78} +8.20871i q^{79} +0.791288i q^{80} -9.58258 q^{81} -9.79129i q^{82} +15.1652 q^{83} +3.58258 q^{84} -6.00000 q^{85} -6.00000 q^{86} +6.79129i q^{87} -0.791288i q^{88} +6.00000i q^{89} +0.165151 q^{90} +7.58258i q^{91} +3.79129i q^{92} -15.0000i q^{93} +7.58258i q^{94} -1.25227 q^{95} -1.79129i q^{96} -13.5826i q^{97} +3.00000i q^{98} -0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 2 * q^3 - 4 * q^4 - 8 * q^7 + 10 * q^9 - 6 * q^10 + 6 * q^11 + 2 * q^12 + 4 * q^16 + 4 * q^21 - 10 * q^25 - 6 * q^26 - 20 * q^27 + 8 * q^28 + 24 * q^30 - 24 * q^33 + 12 * q^34 - 10 * q^36 + 16 * q^37 - 12 * q^38 + 6 * q^40 + 30 * q^41 - 6 * q^44 - 6 * q^46 - 12 * q^47 - 2 * q^48 - 12 * q^49 + 12 * q^53 + 6 * q^58 - 6 * q^62 - 20 * q^63 - 4 * q^64 + 12 * q^65 - 2 * q^67 + 12 * q^70 + 10 * q^73 + 68 * q^75 - 12 * q^77 - 18 * q^78 - 20 * q^81 + 24 * q^83 - 4 * q^84 - 24 * q^85 - 24 * q^86 - 36 * q^90 - 60 * q^95 + 36 * q^99

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.00000i		−	0.707107i
							\(3\)	1.79129			1.03420			0.517100	−	0.855925i	\(-0.327011\pi\)
0.517100	+	0.855925i	\(0.327011\pi\)
\(5\)			0.791288i			0.353875i	0.984222	+	0.176937i	\(0.0566190\pi\)
							−0.984222	+	0.176937i	\(0.943381\pi\)
\(7\)	−2.00000			−0.755929			−0.377964	−	0.925820i	\(-0.623376\pi\)
							−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	−0.791288			−0.238582			−0.119291	−	0.992859i	\(-0.538062\pi\)
							−0.119291	+	0.992859i	\(0.538062\pi\)
\(13\)		−	3.79129i		−	1.05151i	−0.850635	−	0.525757i	\(-0.823782\pi\)
							0.850635	−	0.525757i	\(-0.176218\pi\)
\(17\)			7.58258i			1.83904i	0.393038	+	0.919522i	\(0.371424\pi\)
							−0.393038	+	0.919522i	\(0.628576\pi\)
\(19\)			1.58258i			0.363068i	0.983385	+	0.181534i	\(0.0581062\pi\)
							−0.983385	+	0.181534i	\(0.941894\pi\)
\(23\)		−	3.79129i		−	0.790538i	−0.918565	−	0.395269i	\(-0.870651\pi\)
							0.918565	−	0.395269i	\(-0.129349\pi\)
\(29\)			3.79129i			0.704024i	0.935995	+	0.352012i	\(0.114502\pi\)
							−0.935995	+	0.352012i	\(0.885498\pi\)
\(31\)		−	8.37386i		−	1.50399i	−0.659169	−	0.751995i	\(-0.729092\pi\)
							0.659169	−	0.751995i	\(-0.270908\pi\)
\(37\)	4.00000	−	4.58258i	0.657596	−	0.753371i
							\(41\)	9.79129			1.52914			0.764571	−	0.644539i	\(-0.222951\pi\)
0.764571	+	0.644539i	\(0.222951\pi\)
\(43\)		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	\(-0.848747\pi\)
							0.889212	−	0.457496i	\(-0.151253\pi\)
\(47\)	−7.58258			−1.10603			−0.553016	−	0.833171i	\(-0.686523\pi\)
							−0.553016	+	0.833171i	\(0.686523\pi\)
\(53\)	−1.58258			−0.217383			−0.108692	−	0.994076i	\(-0.534666\pi\)
							−0.108692	+	0.994076i	\(0.534666\pi\)
\(59\)			1.58258i			0.206034i	0.994680	+	0.103017i	\(0.0328496\pi\)
							−0.994680	+	0.103017i	\(0.967150\pi\)
\(61\)			12.7913i			1.63776i	0.573967	+	0.818878i	\(0.305404\pi\)
							−0.573967	+	0.818878i	\(0.694596\pi\)
\(67\)	6.37386			0.778691			0.389346	−	0.921092i	\(-0.372701\pi\)
							0.389346	+	0.921092i	\(0.372701\pi\)
\(71\)	−9.16515			−1.08770			−0.543852	−	0.839181i	\(-0.683035\pi\)
							−0.543852	+	0.839181i	\(0.683035\pi\)
\(73\)	−4.37386			−0.511922			−0.255961	−	0.966687i	\(-0.582392\pi\)
							−0.255961	+	0.966687i	\(0.582392\pi\)
\(79\)			8.20871i			0.923552i	0.886997	+	0.461776i	\(0.152788\pi\)
							−0.886997	+	0.461776i	\(0.847212\pi\)
\(83\)	15.1652			1.66459			0.832296	−	0.554332i	\(-0.187026\pi\)
							0.832296	+	0.554332i	\(0.187026\pi\)
\(89\)			6.00000i			0.635999i	0.948091	+	0.317999i	\(0.103011\pi\)
							−0.948091	+	0.317999i	\(0.896989\pi\)
\(97\)		−	13.5826i		−	1.37910i	−0.724237	−	0.689551i	\(-0.757808\pi\)
							0.724237	−	0.689551i	\(-0.242192\pi\)

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.b.a.73.2	✓	4
3.2	odd	2		666.2.c.b.73.3		4
4.3	odd	2		592.2.g.c.369.2		4
5.2	odd	4		1850.2.c.h.1849.2		4
5.3	odd	4		1850.2.c.g.1849.3		4
5.4	even	2		1850.2.d.e.1701.3		4
8.3	odd	2		2368.2.g.h.961.3		4
8.5	even	2		2368.2.g.j.961.1		4
12.11	even	2		5328.2.h.m.2737.2		4
37.6	odd	4		2738.2.a.h.1.1		2
37.31	odd	4		2738.2.a.k.1.1		2
37.36	even	2	inner	74.2.b.a.73.4	yes	4
111.110	odd	2		666.2.c.b.73.2		4
148.147	odd	2		592.2.g.c.369.1		4
185.73	odd	4		1850.2.c.h.1849.3		4
185.147	odd	4		1850.2.c.g.1849.2		4
185.184	even	2		1850.2.d.e.1701.1		4
296.147	odd	2		2368.2.g.h.961.4		4
296.221	even	2		2368.2.g.j.961.2		4
444.443	even	2		5328.2.h.m.2737.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.b.a.73.2	✓	4	1.1	even	1	trivial
74.2.b.a.73.4	yes	4	37.36	even	2	inner
592.2.g.c.369.1		4	148.147	odd	2
592.2.g.c.369.2		4	4.3	odd	2
666.2.c.b.73.2		4	111.110	odd	2
666.2.c.b.73.3		4	3.2	odd	2
1850.2.c.g.1849.2		4	185.147	odd	4
1850.2.c.g.1849.3		4	5.3	odd	4
1850.2.c.h.1849.2		4	5.2	odd	4
1850.2.c.h.1849.3		4	185.73	odd	4
1850.2.d.e.1701.1		4	185.184	even	2
1850.2.d.e.1701.3		4	5.4	even	2
2368.2.g.h.961.3		4	8.3	odd	2
2368.2.g.h.961.4		4	296.147	odd	2
2368.2.g.j.961.1		4	8.5	even	2
2368.2.g.j.961.2		4	296.221	even	2
2738.2.a.h.1.1		2	37.6	odd	4
2738.2.a.k.1.1		2	37.31	odd	4
5328.2.h.m.2737.2		4	12.11	even	2
5328.2.h.m.2737.3		4	444.443	even	2