Embedded newform 2738.2.a.a.1.1

• Label: 2738.2.a.a.1.1
• Level: $2738$
• Weight: $2$
• Character: 2738.1
• Self dual: yes
• Analytic conductor: $21.863$
• Analytic rank: $1$
• Dimension: $1$
• 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:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
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
Character	\(\chi\)	\(=\)	2738.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -6.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -6.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +3.00000 q^{20} +8.00000 q^{21} +6.00000 q^{22} +6.00000 q^{23} +2.00000 q^{24} +4.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -4.00000 q^{28} +3.00000 q^{29} +6.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} +12.0000 q^{33} -3.00000 q^{34} -12.0000 q^{35} +1.00000 q^{36} -2.00000 q^{38} -4.00000 q^{39} -3.00000 q^{40} +3.00000 q^{41} -8.00000 q^{42} -4.00000 q^{43} -6.00000 q^{44} +3.00000 q^{45} -6.00000 q^{46} -6.00000 q^{47} -2.00000 q^{48} +9.00000 q^{49} -4.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} -18.0000 q^{55} +4.00000 q^{56} -4.00000 q^{57} -3.00000 q^{58} -6.00000 q^{60} -1.00000 q^{61} -2.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -12.0000 q^{66} +2.00000 q^{67} +3.00000 q^{68} -12.0000 q^{69} +12.0000 q^{70} -12.0000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -8.00000 q^{75} +2.00000 q^{76} +24.0000 q^{77} +4.00000 q^{78} +14.0000 q^{79} +3.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} +6.00000 q^{83} +8.00000 q^{84} +9.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +6.00000 q^{88} +3.00000 q^{89} -3.00000 q^{90} -8.00000 q^{91} +6.00000 q^{92} -4.00000 q^{93} +6.00000 q^{94} +6.00000 q^{95} +2.00000 q^{96} -13.0000 q^{97} -9.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
			\(3\)	−2.00000	−1.15470	−0.577350	−	0.816497i	\(-0.695913\pi\)
−0.577350	+	0.816497i				\(0.695913\pi\)
\(5\)	3.00000	1.34164	0.670820	−	0.741620i	\(-0.265942\pi\)
			0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	−4.00000	−1.51186	−0.755929	−	0.654654i	\(-0.772814\pi\)
			−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	−6.00000	−1.80907	−0.904534	−	0.426401i	\(-0.859781\pi\)
			−0.904534	+	0.426401i	\(0.859781\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	2.00000	0.458831	0.229416	−	0.973329i	\(-0.426318\pi\)
			0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	3.00000	0.557086	0.278543	−	0.960424i	\(-0.410149\pi\)
			0.278543	+	0.960424i	\(0.410149\pi\)
\(31\)	2.00000	0.359211	0.179605	−	0.983739i	\(-0.442518\pi\)
			0.179605	+	0.983739i	\(0.442518\pi\)
\(37\)	0	0
			\(41\)	3.00000	0.468521	0.234261	−	0.972174i	\(-0.424733\pi\)
0.234261	+	0.972174i				\(0.424733\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)
\(53\)	−6.00000	−0.824163	−0.412082	−	0.911147i	\(-0.635198\pi\)
			−0.412082	+	0.911147i	\(0.635198\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−1.00000	−0.128037	−0.0640184	−	0.997949i	\(-0.520392\pi\)
			−0.0640184	+	0.997949i	\(0.520392\pi\)
\(67\)	2.00000	0.244339	0.122169	−	0.992509i	\(-0.461015\pi\)
			0.122169	+	0.992509i	\(0.461015\pi\)
\(71\)	−12.0000	−1.42414	−0.712069	−	0.702109i	\(-0.752242\pi\)
			−0.712069	+	0.702109i	\(0.752242\pi\)
\(73\)	−10.0000	−1.17041	−0.585206	−	0.810885i	\(-0.698986\pi\)
			−0.585206	+	0.810885i	\(0.698986\pi\)
\(79\)	14.0000	1.57512	0.787562	−	0.616236i	\(-0.211343\pi\)
			0.787562	+	0.616236i	\(0.211343\pi\)
\(83\)	6.00000	0.658586	0.329293	−	0.944228i	\(-0.393190\pi\)
			0.329293	+	0.944228i	\(0.393190\pi\)
\(89\)	3.00000	0.317999	0.159000	−	0.987279i	\(-0.449173\pi\)
			0.159000	+	0.987279i	\(0.449173\pi\)
\(97\)	−13.0000	−1.31995	−0.659975	−	0.751288i	\(-0.729433\pi\)
			−0.659975	+	0.751288i	\(0.729433\pi\)

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.a.1.1		1
37.10	even	3		74.2.c.b.63.1	yes	2
37.26	even	3		74.2.c.b.47.1	✓	2
37.36	even	2		2738.2.a.c.1.1		1
111.26	odd	6		666.2.f.d.343.1		2
111.47	odd	6		666.2.f.d.433.1		2
148.47	odd	6		592.2.i.a.433.1		2
148.63	odd	6		592.2.i.a.417.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.c.b.47.1	✓	2	37.26	even	3
74.2.c.b.63.1	yes	2	37.10	even	3
592.2.i.a.417.1		2	148.63	odd	6
592.2.i.a.433.1		2	148.47	odd	6
666.2.f.d.343.1		2	111.26	odd	6
666.2.f.d.433.1		2	111.47	odd	6
2738.2.a.a.1.1		1	1.1	even	1	trivial
2738.2.a.c.1.1		1	37.36	even	2