Embedded newform 74.2.b.a.73.3

• Label: 74.2.b.a.73.3
• 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			73.3
Root			\(-2.79129i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.73
Dual form			74.2.b.a.73.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.79129 q^{3} -1.00000 q^{4} +3.79129i q^{5} -2.79129i q^{6} -2.00000 q^{7} -1.00000i q^{8} +4.79129 q^{9} -3.79129 q^{10} +3.79129 q^{11} +2.79129 q^{12} -0.791288i q^{13} -2.00000i q^{14} -10.5826i q^{15} +1.00000 q^{16} +1.58258i q^{17} +4.79129i q^{18} +7.58258i q^{19} -3.79129i q^{20} +5.58258 q^{21} +3.79129i q^{22} -0.791288i q^{23} +2.79129i q^{24} -9.37386 q^{25} +0.791288 q^{26} -5.00000 q^{27} +2.00000 q^{28} +0.791288i q^{29} +10.5826 q^{30} -5.37386i q^{31} +1.00000i q^{32} -10.5826 q^{33} -1.58258 q^{34} -7.58258i q^{35} -4.79129 q^{36} +(4.00000 - 4.58258i) q^{37} -7.58258 q^{38} +2.20871i q^{39} +3.79129 q^{40} +5.20871 q^{41} +5.58258i q^{42} +6.00000i q^{43} -3.79129 q^{44} +18.1652i q^{45} +0.791288 q^{46} +1.58258 q^{47} -2.79129 q^{48} -3.00000 q^{49} -9.37386i q^{50} -4.41742i q^{51} +0.791288i q^{52} +7.58258 q^{53} -5.00000i q^{54} +14.3739i q^{55} +2.00000i q^{56} -21.1652i q^{57} -0.791288 q^{58} +7.58258i q^{59} +10.5826i q^{60} -8.20871i q^{61} +5.37386 q^{62} -9.58258 q^{63} -1.00000 q^{64} +3.00000 q^{65} -10.5826i q^{66} -7.37386 q^{67} -1.58258i q^{68} +2.20871i q^{69} +7.58258 q^{70} +9.16515 q^{71} -4.79129i q^{72} +9.37386 q^{73} +(4.58258 + 4.00000i) q^{74} +26.1652 q^{75} -7.58258i q^{76} -7.58258 q^{77} -2.20871 q^{78} -12.7913i q^{79} +3.79129i q^{80} -0.417424 q^{81} +5.20871i q^{82} -3.16515 q^{83} -5.58258 q^{84} -6.00000 q^{85} -6.00000 q^{86} -2.20871i q^{87} -3.79129i q^{88} -6.00000i q^{89} -18.1652 q^{90} +1.58258i q^{91} +0.791288i q^{92} +15.0000i q^{93} +1.58258i q^{94} -28.7477 q^{95} -2.79129i q^{96} +4.41742i q^{97} -3.00000i q^{98} +18.1652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
							\(3\)	−2.79129			−1.61155			−0.805775	−	0.592221i	\(-0.798251\pi\)
−0.805775	+	0.592221i	\(0.798251\pi\)
\(5\)			3.79129i			1.69552i	0.530384	+	0.847758i	\(0.322048\pi\)
							−0.530384	+	0.847758i	\(0.677952\pi\)
\(7\)	−2.00000			−0.755929			−0.377964	−	0.925820i	\(-0.623376\pi\)
							−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	3.79129			1.14312			0.571558	−	0.820562i	\(-0.306339\pi\)
							0.571558	+	0.820562i	\(0.306339\pi\)
\(13\)		−	0.791288i		−	0.219464i	−0.993961	−	0.109732i	\(-0.965001\pi\)
							0.993961	−	0.109732i	\(-0.0349992\pi\)
\(17\)			1.58258i			0.383831i	0.981411	+	0.191915i	\(0.0614700\pi\)
							−0.981411	+	0.191915i	\(0.938530\pi\)
\(19\)			7.58258i			1.73956i	0.493438	+	0.869781i	\(0.335740\pi\)
							−0.493438	+	0.869781i	\(0.664260\pi\)
\(23\)		−	0.791288i		−	0.164995i	−0.996591	−	0.0824975i	\(-0.973710\pi\)
							0.996591	−	0.0824975i	\(-0.0262896\pi\)
\(29\)			0.791288i			0.146938i	0.997297	+	0.0734692i	\(0.0234071\pi\)
							−0.997297	+	0.0734692i	\(0.976593\pi\)
\(31\)		−	5.37386i		−	0.965174i	−0.875848	−	0.482587i	\(-0.839697\pi\)
							0.875848	−	0.482587i	\(-0.160303\pi\)
\(37\)	4.00000	−	4.58258i	0.657596	−	0.753371i
							\(41\)	5.20871			0.813464			0.406732	−	0.913547i	\(-0.366668\pi\)
0.406732	+	0.913547i	\(0.366668\pi\)
\(43\)			6.00000i			0.914991i	0.889212	+	0.457496i	\(0.151253\pi\)
							−0.889212	+	0.457496i	\(0.848747\pi\)
\(47\)	1.58258			0.230842			0.115421	−	0.993317i	\(-0.463178\pi\)
							0.115421	+	0.993317i	\(0.463178\pi\)
\(53\)	7.58258			1.04155			0.520773	−	0.853695i	\(-0.325644\pi\)
							0.520773	+	0.853695i	\(0.325644\pi\)
\(59\)			7.58258i			0.987167i	0.869698	+	0.493584i	\(0.164313\pi\)
							−0.869698	+	0.493584i	\(0.835687\pi\)
\(61\)		−	8.20871i		−	1.05102i	−0.850788	−	0.525509i	\(-0.823875\pi\)
							0.850788	−	0.525509i	\(-0.176125\pi\)
\(67\)	−7.37386			−0.900861			−0.450430	−	0.892812i	\(-0.648729\pi\)
							−0.450430	+	0.892812i	\(0.648729\pi\)
\(71\)	9.16515			1.08770			0.543852	−	0.839181i	\(-0.316965\pi\)
							0.543852	+	0.839181i	\(0.316965\pi\)
\(73\)	9.37386			1.09713			0.548564	−	0.836109i	\(-0.315175\pi\)
							0.548564	+	0.836109i	\(0.315175\pi\)
\(79\)		−	12.7913i		−	1.43913i	−0.694424	−	0.719566i	\(-0.744341\pi\)
							0.694424	−	0.719566i	\(-0.255659\pi\)
\(83\)	−3.16515			−0.347421			−0.173710	−	0.984797i	\(-0.555576\pi\)
							−0.173710	+	0.984797i	\(0.555576\pi\)
\(89\)		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	\(-0.896989\pi\)
							0.948091	−	0.317999i	\(-0.103011\pi\)
\(97\)			4.41742i			0.448521i	0.974529	+	0.224261i	\(0.0719967\pi\)
							−0.974529	+	0.224261i	\(0.928003\pi\)

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.73.3	yes	4
3.2	odd	2		666.2.c.b.73.1		4
4.3	odd	2		592.2.g.c.369.4		4
5.2	odd	4		1850.2.c.g.1849.4		4
5.3	odd	4		1850.2.c.h.1849.1		4
5.4	even	2		1850.2.d.e.1701.2		4
8.3	odd	2		2368.2.g.h.961.1		4
8.5	even	2		2368.2.g.j.961.3		4
12.11	even	2		5328.2.h.m.2737.1		4
37.6	odd	4		2738.2.a.k.1.2		2
37.31	odd	4		2738.2.a.h.1.2		2
37.36	even	2	inner	74.2.b.a.73.1	✓	4
111.110	odd	2		666.2.c.b.73.4		4
148.147	odd	2		592.2.g.c.369.3		4
185.73	odd	4		1850.2.c.g.1849.1		4
185.147	odd	4		1850.2.c.h.1849.4		4
185.184	even	2		1850.2.d.e.1701.4		4
296.147	odd	2		2368.2.g.h.961.2		4
296.221	even	2		2368.2.g.j.961.4		4
444.443	even	2		5328.2.h.m.2737.4		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.b.a.73.1	✓	4	37.36	even	2	inner
74.2.b.a.73.3	yes	4	1.1	even	1	trivial
592.2.g.c.369.3		4	148.147	odd	2
592.2.g.c.369.4		4	4.3	odd	2
666.2.c.b.73.1		4	3.2	odd	2
666.2.c.b.73.4		4	111.110	odd	2
1850.2.c.g.1849.1		4	185.73	odd	4
1850.2.c.g.1849.4		4	5.2	odd	4
1850.2.c.h.1849.1		4	5.3	odd	4
1850.2.c.h.1849.4		4	185.147	odd	4
1850.2.d.e.1701.2		4	5.4	even	2
1850.2.d.e.1701.4		4	185.184	even	2
2368.2.g.h.961.1		4	8.3	odd	2
2368.2.g.h.961.2		4	296.147	odd	2
2368.2.g.j.961.3		4	8.5	even	2
2368.2.g.j.961.4		4	296.221	even	2
2738.2.a.h.1.2		2	37.31	odd	4
2738.2.a.k.1.2		2	37.6	odd	4
5328.2.h.m.2737.1		4	12.11	even	2
5328.2.h.m.2737.4		4	444.443	even	2