Embedded newform 550.2.a.f.1.1

• Label: 550.2.a.f.1.1
• Level: $550$
• Weight: $2$
• Character: 550.1
• Self dual: yes
• Analytic conductor: $4.392$
• Analytic rank: $0$
• Dimension: $1$
• 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:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
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
Character	\(\chi\)	\(=\)	550.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +2.00000 q^{18} +5.00000 q^{19} -3.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} -5.00000 q^{27} -3.00000 q^{28} +5.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -7.00000 q^{34} -2.00000 q^{36} -3.00000 q^{37} -5.00000 q^{38} +6.00000 q^{39} +2.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} -6.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +7.00000 q^{51} +6.00000 q^{52} +1.00000 q^{53} +5.00000 q^{54} +3.00000 q^{56} +5.00000 q^{57} -5.00000 q^{58} -10.0000 q^{59} +7.00000 q^{61} +3.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -8.00000 q^{67} +7.00000 q^{68} +6.00000 q^{69} +7.00000 q^{71} +2.00000 q^{72} -14.0000 q^{73} +3.00000 q^{74} +5.00000 q^{76} -3.00000 q^{77} -6.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -3.00000 q^{84} +4.00000 q^{86} +5.00000 q^{87} -1.00000 q^{88} -15.0000 q^{89} -18.0000 q^{91} +6.00000 q^{92} -3.00000 q^{93} -2.00000 q^{94} -1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
			\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
0.288675	+	0.957427i				\(0.406785\pi\)
\(5\)	0	0
			\(7\)	−3.00000	−1.13389	−0.566947	−	0.823754i	\(-0.691875\pi\)
−0.566947	+	0.823754i				\(0.691875\pi\)
\(11\)	1.00000	0.301511
			\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
0.832050	+	0.554700i				\(0.187167\pi\)
\(17\)	7.00000	1.69775	0.848875	−	0.528594i	\(-0.177281\pi\)
			0.848875	+	0.528594i	\(0.177281\pi\)
\(19\)	5.00000	1.14708	0.573539	−	0.819178i	\(-0.305570\pi\)
			0.573539	+	0.819178i	\(0.305570\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	5.00000	0.928477	0.464238	−	0.885710i	\(-0.346328\pi\)
			0.464238	+	0.885710i	\(0.346328\pi\)
\(31\)	−3.00000	−0.538816	−0.269408	−	0.963026i	\(-0.586828\pi\)
			−0.269408	+	0.963026i	\(0.586828\pi\)
\(37\)	−3.00000	−0.493197	−0.246598	−	0.969118i	\(-0.579313\pi\)
			−0.246598	+	0.969118i	\(0.579313\pi\)
\(41\)	2.00000	0.312348	0.156174	−	0.987730i	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	2.00000	0.291730	0.145865	−	0.989305i	\(-0.453403\pi\)
			0.145865	+	0.989305i	\(0.453403\pi\)
\(53\)	1.00000	0.137361	0.0686803	−	0.997639i	\(-0.478121\pi\)
			0.0686803	+	0.997639i	\(0.478121\pi\)
\(59\)	−10.0000	−1.30189	−0.650945	−	0.759125i	\(-0.725627\pi\)
			−0.650945	+	0.759125i	\(0.725627\pi\)
\(61\)	7.00000	0.896258	0.448129	−	0.893969i	\(-0.352090\pi\)
			0.448129	+	0.893969i	\(0.352090\pi\)
\(67\)	−8.00000	−0.977356	−0.488678	−	0.872464i	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	7.00000	0.830747	0.415374	−	0.909651i	\(-0.363651\pi\)
			0.415374	+	0.909651i	\(0.363651\pi\)
\(73\)	−14.0000	−1.63858	−0.819288	−	0.573382i	\(-0.805631\pi\)
			−0.819288	+	0.573382i	\(0.805631\pi\)
\(79\)	10.0000	1.12509	0.562544	−	0.826767i	\(-0.309823\pi\)
			0.562544	+	0.826767i	\(0.309823\pi\)
\(83\)	6.00000	0.658586	0.329293	−	0.944228i	\(-0.393190\pi\)
			0.329293	+	0.944228i	\(0.393190\pi\)
\(89\)	−15.0000	−1.59000	−0.794998	−	0.606612i	\(-0.792528\pi\)
			−0.794998	+	0.606612i	\(0.792528\pi\)
\(97\)	12.0000	1.21842	0.609208	−	0.793011i	\(-0.291488\pi\)
			0.609208	+	0.793011i	\(0.291488\pi\)

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.f.1.1		1
3.2	odd	2		4950.2.a.bc.1.1		1
4.3	odd	2		4400.2.a.l.1.1		1
5.2	odd	4		550.2.b.a.199.1		2
5.3	odd	4		550.2.b.a.199.2		2
5.4	even	2		110.2.a.b.1.1	✓	1
11.10	odd	2		6050.2.a.bj.1.1		1
15.2	even	4		4950.2.c.m.199.2		2
15.8	even	4		4950.2.c.m.199.1		2
15.14	odd	2		990.2.a.d.1.1		1
20.3	even	4		4400.2.b.i.4049.1		2
20.7	even	4		4400.2.b.i.4049.2		2
20.19	odd	2		880.2.a.i.1.1		1
35.34	odd	2		5390.2.a.bf.1.1		1
40.19	odd	2		3520.2.a.h.1.1		1
40.29	even	2		3520.2.a.y.1.1		1
55.54	odd	2		1210.2.a.b.1.1		1
60.59	even	2		7920.2.a.d.1.1		1
220.219	even	2		9680.2.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.b.1.1	✓	1	5.4	even	2
550.2.a.f.1.1		1	1.1	even	1	trivial
550.2.b.a.199.1		2	5.2	odd	4
550.2.b.a.199.2		2	5.3	odd	4
880.2.a.i.1.1		1	20.19	odd	2
990.2.a.d.1.1		1	15.14	odd	2
1210.2.a.b.1.1		1	55.54	odd	2
3520.2.a.h.1.1		1	40.19	odd	2
3520.2.a.y.1.1		1	40.29	even	2
4400.2.a.l.1.1		1	4.3	odd	2
4400.2.b.i.4049.1		2	20.3	even	4
4400.2.b.i.4049.2		2	20.7	even	4
4950.2.a.bc.1.1		1	3.2	odd	2
4950.2.c.m.199.1		2	15.8	even	4
4950.2.c.m.199.2		2	15.2	even	4
5390.2.a.bf.1.1		1	35.34	odd	2
6050.2.a.bj.1.1		1	11.10	odd	2
7920.2.a.d.1.1		1	60.59	even	2
9680.2.a.x.1.1		1	220.219	even	2