Embedded newform 550.2.a.n.1.1

• Label: 550.2.a.n.1.1
• Level: $550$
• Weight: $2$
• Character: 550.1
• Self dual: yes
• Analytic conductor: $4.392$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	550.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.39177211117\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{33}) \)
Defining polynomial:	\( x^{2} - x - 8 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			\(-2.37228\) of defining polynomial
Character	\(\chi\)	\(=\)	550.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.37228 q^{3} +1.00000 q^{4} -2.37228 q^{6} +2.37228 q^{7} +1.00000 q^{8} +2.62772 q^{9} -1.00000 q^{11} -2.37228 q^{12} -2.00000 q^{13} +2.37228 q^{14} +1.00000 q^{16} +4.37228 q^{17} +2.62772 q^{18} +6.37228 q^{19} -5.62772 q^{21} -1.00000 q^{22} +8.74456 q^{23} -2.37228 q^{24} -2.00000 q^{26} +0.883156 q^{27} +2.37228 q^{28} -4.37228 q^{29} -2.37228 q^{31} +1.00000 q^{32} +2.37228 q^{33} +4.37228 q^{34} +2.62772 q^{36} -3.62772 q^{37} +6.37228 q^{38} +4.74456 q^{39} +11.4891 q^{41} -5.62772 q^{42} +4.00000 q^{43} -1.00000 q^{44} +8.74456 q^{46} +8.74456 q^{47} -2.37228 q^{48} -1.37228 q^{49} -10.3723 q^{51} -2.00000 q^{52} -13.1168 q^{53} +0.883156 q^{54} +2.37228 q^{56} -15.1168 q^{57} -4.37228 q^{58} +8.74456 q^{59} +0.372281 q^{61} -2.37228 q^{62} +6.23369 q^{63} +1.00000 q^{64} +2.37228 q^{66} -8.00000 q^{67} +4.37228 q^{68} -20.7446 q^{69} -7.11684 q^{71} +2.62772 q^{72} -7.48913 q^{73} -3.62772 q^{74} +6.37228 q^{76} -2.37228 q^{77} +4.74456 q^{78} -12.7446 q^{79} -9.97825 q^{81} +11.4891 q^{82} -8.74456 q^{83} -5.62772 q^{84} +4.00000 q^{86} +10.3723 q^{87} -1.00000 q^{88} +4.37228 q^{89} -4.74456 q^{91} +8.74456 q^{92} +5.62772 q^{93} +8.74456 q^{94} -2.37228 q^{96} +1.25544 q^{97} -1.37228 q^{98} -2.62772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 + q^3 + 2 * q^4 + q^6 - q^7 + 2 * q^8 + 11 * q^9 - 2 * q^11 + q^12 - 4 * q^13 - q^14 + 2 * q^16 + 3 * q^17 + 11 * q^18 + 7 * q^19 - 17 * q^21 - 2 * q^22 + 6 * q^23 + q^24 - 4 * q^26 + 19 * q^27 - q^28 - 3 * q^29 + q^31 + 2 * q^32 - q^33 + 3 * q^34 + 11 * q^36 - 13 * q^37 + 7 * q^38 - 2 * q^39 - 17 * q^42 + 8 * q^43 - 2 * q^44 + 6 * q^46 + 6 * q^47 + q^48 + 3 * q^49 - 15 * q^51 - 4 * q^52 - 9 * q^53 + 19 * q^54 - q^56 - 13 * q^57 - 3 * q^58 + 6 * q^59 - 5 * q^61 + q^62 - 22 * q^63 + 2 * q^64 - q^66 - 16 * q^67 + 3 * q^68 - 30 * q^69 + 3 * q^71 + 11 * q^72 + 8 * q^73 - 13 * q^74 + 7 * q^76 + q^77 - 2 * q^78 - 14 * q^79 + 26 * q^81 - 6 * q^83 - 17 * q^84 + 8 * q^86 + 15 * q^87 - 2 * q^88 + 3 * q^89 + 2 * q^91 + 6 * q^92 + 17 * q^93 + 6 * q^94 + q^96 + 14 * q^97 + 3 * q^98 - 11 * 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
			\(3\)	−2.37228	−1.36964	−0.684819	−	0.728714i	\(-0.740119\pi\)
−0.684819	+	0.728714i				\(0.740119\pi\)
\(5\)	0	0
			\(7\)	2.37228	0.896638	0.448319	−	0.893874i	\(-0.352023\pi\)
0.448319	+	0.893874i				\(0.352023\pi\)
\(11\)	−1.00000	−0.301511
			\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
−0.277350	+	0.960769i				\(0.589456\pi\)
\(17\)	4.37228	1.06043	0.530217	−	0.847862i	\(-0.322110\pi\)
			0.530217	+	0.847862i	\(0.322110\pi\)
\(19\)	6.37228	1.46190	0.730951	−	0.682430i	\(-0.239077\pi\)
			0.730951	+	0.682430i	\(0.239077\pi\)
\(23\)	8.74456	1.82337	0.911684	−	0.410893i	\(-0.134783\pi\)
			0.911684	+	0.410893i	\(0.134783\pi\)
\(29\)	−4.37228	−0.811912	−0.405956	−	0.913893i	\(-0.633061\pi\)
			−0.405956	+	0.913893i	\(0.633061\pi\)
\(31\)	−2.37228	−0.426074	−0.213037	−	0.977044i	\(-0.568336\pi\)
			−0.213037	+	0.977044i	\(0.568336\pi\)
\(37\)	−3.62772	−0.596393	−0.298197	−	0.954504i	\(-0.596385\pi\)
			−0.298197	+	0.954504i	\(0.596385\pi\)
\(41\)	11.4891	1.79430	0.897150	−	0.441726i	\(-0.145634\pi\)
			0.897150	+	0.441726i	\(0.145634\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	8.74456	1.27553	0.637763	−	0.770233i	\(-0.279860\pi\)
			0.637763	+	0.770233i	\(0.279860\pi\)
\(53\)	−13.1168	−1.80174	−0.900869	−	0.434092i	\(-0.857069\pi\)
			−0.900869	+	0.434092i	\(0.857069\pi\)
\(59\)	8.74456	1.13845	0.569223	−	0.822183i	\(-0.307244\pi\)
			0.569223	+	0.822183i	\(0.307244\pi\)
\(61\)	0.372281	0.0476657	0.0238329	−	0.999716i	\(-0.492413\pi\)
			0.0238329	+	0.999716i	\(0.492413\pi\)
\(67\)	−8.00000	−0.977356	−0.488678	−	0.872464i	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	−7.11684	−0.844614	−0.422307	−	0.906453i	\(-0.638780\pi\)
			−0.422307	+	0.906453i	\(0.638780\pi\)
\(73\)	−7.48913	−0.876536	−0.438268	−	0.898844i	\(-0.644408\pi\)
			−0.438268	+	0.898844i	\(0.644408\pi\)
\(79\)	−12.7446	−1.43388	−0.716938	−	0.697137i	\(-0.754457\pi\)
			−0.716938	+	0.697137i	\(0.754457\pi\)
\(83\)	−8.74456	−0.959840	−0.479920	−	0.877312i	\(-0.659334\pi\)
			−0.479920	+	0.877312i	\(0.659334\pi\)
\(89\)	4.37228	0.463461	0.231730	−	0.972780i	\(-0.425561\pi\)
			0.231730	+	0.972780i	\(0.425561\pi\)
\(97\)	1.25544	0.127470	0.0637352	−	0.997967i	\(-0.479699\pi\)
			0.0637352	+	0.997967i	\(0.479699\pi\)

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	550.2.a.n.1.1		2
3.2	odd	2		4950.2.a.bw.1.2		2
4.3	odd	2		4400.2.a.bl.1.2		2
5.2	odd	4		550.2.b.f.199.4		4
5.3	odd	4		550.2.b.f.199.1		4
5.4	even	2		110.2.a.d.1.2	✓	2
11.10	odd	2		6050.2.a.cb.1.1		2
15.2	even	4		4950.2.c.bc.199.2		4
15.8	even	4		4950.2.c.bc.199.3		4
15.14	odd	2		990.2.a.m.1.1		2
20.3	even	4		4400.2.b.p.4049.3		4
20.7	even	4		4400.2.b.p.4049.2		4
20.19	odd	2		880.2.a.n.1.1		2
35.34	odd	2		5390.2.a.bp.1.1		2
40.19	odd	2		3520.2.a.bj.1.2		2
40.29	even	2		3520.2.a.bq.1.1		2
55.54	odd	2		1210.2.a.r.1.2		2
60.59	even	2		7920.2.a.bq.1.2		2
220.219	even	2		9680.2.a.bt.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.2	✓	2	5.4	even	2
550.2.a.n.1.1		2	1.1	even	1	trivial
550.2.b.f.199.1		4	5.3	odd	4
550.2.b.f.199.4		4	5.2	odd	4
880.2.a.n.1.1		2	20.19	odd	2
990.2.a.m.1.1		2	15.14	odd	2
1210.2.a.r.1.2		2	55.54	odd	2
3520.2.a.bj.1.2		2	40.19	odd	2
3520.2.a.bq.1.1		2	40.29	even	2
4400.2.a.bl.1.2		2	4.3	odd	2
4400.2.b.p.4049.2		4	20.7	even	4
4400.2.b.p.4049.3		4	20.3	even	4
4950.2.a.bw.1.2		2	3.2	odd	2
4950.2.c.bc.199.2		4	15.2	even	4
4950.2.c.bc.199.3		4	15.8	even	4
5390.2.a.bp.1.1		2	35.34	odd	2
6050.2.a.cb.1.1		2	11.10	odd	2
7920.2.a.bq.1.2		2	60.59	even	2
9680.2.a.bt.1.1		2	220.219	even	2