Embedded newform 2738.2.a.h.1.1

• Label: 2738.2.a.h.1.1
• Level: $2738$
• Weight: $2$
• Character: 2738.1
• Self dual: yes
• Analytic conductor: $21.863$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2738 = 2 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2738.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(21.8630400734\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{21}) \)
Defining polynomial:	\( x^{2} - x - 5 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			\(-1.79129\) of defining polynomial
Character	\(\chi\)	\(=\)	2738.1

$q$-expansion

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

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	−1.79129	−1.03420	−0.517100	−	0.855925i	\(-0.672989\pi\)
−0.517100	+	0.855925i				\(0.672989\pi\)
\(5\)	−0.791288	−0.353875	−0.176937	−	0.984222i	\(-0.556619\pi\)
			−0.176937	+	0.984222i	\(0.556619\pi\)
\(7\)	−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	\(-0.461938\pi\)
			0.119291	+	0.992859i	\(0.461938\pi\)
\(13\)	3.79129	1.05151	0.525757	−	0.850635i	\(-0.323782\pi\)
			0.525757	+	0.850635i	\(0.323782\pi\)
\(17\)	−7.58258	−1.83904	−0.919522	−	0.393038i	\(-0.871424\pi\)
			−0.919522	+	0.393038i	\(0.871424\pi\)
\(19\)	−1.58258	−0.363068	−0.181534	−	0.983385i	\(-0.558106\pi\)
			−0.181534	+	0.983385i	\(0.558106\pi\)
\(23\)	3.79129	0.790538	0.395269	−	0.918565i	\(-0.370651\pi\)
			0.395269	+	0.918565i	\(0.370651\pi\)
\(29\)	3.79129	0.704024	0.352012	−	0.935995i	\(-0.385498\pi\)
			0.352012	+	0.935995i	\(0.385498\pi\)
\(31\)	−8.37386	−1.50399	−0.751995	−	0.659169i	\(-0.770908\pi\)
			−0.751995	+	0.659169i	\(0.770908\pi\)
\(37\)	0	0
			\(41\)	−9.79129	−1.52914	−0.764571	−	0.644539i	\(-0.777049\pi\)
−0.764571	+	0.644539i				\(0.777049\pi\)
\(43\)	6.00000	0.914991	0.457496	−	0.889212i	\(-0.348747\pi\)
			0.457496	+	0.889212i	\(0.348747\pi\)
\(47\)	−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	\(0.534666\pi\)
\(59\)	−1.58258	−0.206034	−0.103017	−	0.994680i	\(-0.532850\pi\)
			−0.103017	+	0.994680i	\(0.532850\pi\)
\(61\)	12.7913	1.63776	0.818878	−	0.573967i	\(-0.194596\pi\)
			0.818878	+	0.573967i	\(0.194596\pi\)
\(67\)	−6.37386	−0.778691	−0.389346	−	0.921092i	\(-0.627299\pi\)
			−0.389346	+	0.921092i	\(0.627299\pi\)
\(71\)	−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	\(-0.417608\pi\)
			0.255961	+	0.966687i	\(0.417608\pi\)
\(79\)	−8.20871	−0.923552	−0.461776	−	0.886997i	\(-0.652788\pi\)
			−0.461776	+	0.886997i	\(0.652788\pi\)
\(83\)	15.1652	1.66459	0.832296	−	0.554332i	\(-0.187026\pi\)
			0.832296	+	0.554332i	\(0.187026\pi\)
\(89\)	6.00000	0.635999	0.317999	−	0.948091i	\(-0.396989\pi\)
			0.317999	+	0.948091i	\(0.396989\pi\)
\(97\)	13.5826	1.37910	0.689551	−	0.724237i	\(-0.257808\pi\)
			0.689551	+	0.724237i	\(0.257808\pi\)

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	2738.2.a.h.1.1		2
37.6	odd	4		74.2.b.a.73.4	yes	4
37.31	odd	4		74.2.b.a.73.2	✓	4
37.36	even	2		2738.2.a.k.1.1		2
111.68	even	4		666.2.c.b.73.3		4
111.80	even	4		666.2.c.b.73.2		4
148.31	even	4		592.2.g.c.369.2		4
148.43	even	4		592.2.g.c.369.1		4
185.43	even	4		1850.2.c.h.1849.3		4
185.68	even	4		1850.2.c.g.1849.3		4
185.117	even	4		1850.2.c.g.1849.2		4
185.142	even	4		1850.2.c.h.1849.2		4
185.154	odd	4		1850.2.d.e.1701.1		4
185.179	odd	4		1850.2.d.e.1701.3		4
296.43	even	4		2368.2.g.h.961.4		4
296.117	odd	4		2368.2.g.j.961.2		4
296.179	even	4		2368.2.g.h.961.3		4
296.253	odd	4		2368.2.g.j.961.1		4
444.179	odd	4		5328.2.h.m.2737.2		4
444.191	odd	4		5328.2.h.m.2737.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.b.a.73.2	✓	4	37.31	odd	4
74.2.b.a.73.4	yes	4	37.6	odd	4
592.2.g.c.369.1		4	148.43	even	4
592.2.g.c.369.2		4	148.31	even	4
666.2.c.b.73.2		4	111.80	even	4
666.2.c.b.73.3		4	111.68	even	4
1850.2.c.g.1849.2		4	185.117	even	4
1850.2.c.g.1849.3		4	185.68	even	4
1850.2.c.h.1849.2		4	185.142	even	4
1850.2.c.h.1849.3		4	185.43	even	4
1850.2.d.e.1701.1		4	185.154	odd	4
1850.2.d.e.1701.3		4	185.179	odd	4
2368.2.g.h.961.3		4	296.179	even	4
2368.2.g.h.961.4		4	296.43	even	4
2368.2.g.j.961.1		4	296.253	odd	4
2368.2.g.j.961.2		4	296.117	odd	4
2738.2.a.h.1.1		2	1.1	even	1	trivial
2738.2.a.k.1.1		2	37.36	even	2
5328.2.h.m.2737.2		4	444.179	odd	4
5328.2.h.m.2737.3		4	444.191	odd	4