Embedded newform 550.2.a.j.1.1

• Label: 550.2.a.j.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

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} -3.00000 q^{21} +1.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} +5.00000 q^{27} +3.00000 q^{28} +5.00000 q^{29} +7.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} -7.00000 q^{37} -5.00000 q^{38} -4.00000 q^{39} -8.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} +1.00000 q^{44} +4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -3.00000 q^{51} +4.00000 q^{52} +9.00000 q^{53} +5.00000 q^{54} +3.00000 q^{56} +5.00000 q^{57} +5.00000 q^{58} -13.0000 q^{61} +7.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -12.0000 q^{67} +3.00000 q^{68} -4.00000 q^{69} -3.00000 q^{71} -2.00000 q^{72} -6.00000 q^{73} -7.00000 q^{74} -5.00000 q^{76} +3.00000 q^{77} -4.00000 q^{78} +1.00000 q^{81} -8.00000 q^{82} +4.00000 q^{83} -3.00000 q^{84} -6.00000 q^{86} -5.00000 q^{87} +1.00000 q^{88} -15.0000 q^{89} +12.0000 q^{91} +4.00000 q^{92} -7.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} -12.0000 q^{97} +2.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	\(-0.593215\pi\)
−0.288675	+	0.957427i				\(0.593215\pi\)
\(5\)	0	0
			\(7\)	3.00000	1.13389	0.566947	−	0.823754i	\(-0.308125\pi\)
0.566947	+	0.823754i				\(0.308125\pi\)
\(11\)	1.00000	0.301511
			\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
0.554700	+	0.832050i				\(0.312833\pi\)
\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	−5.00000	−1.14708	−0.573539	−	0.819178i	\(-0.694430\pi\)
			−0.573539	+	0.819178i	\(0.694430\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	5.00000	0.928477	0.464238	−	0.885710i	\(-0.346328\pi\)
			0.464238	+	0.885710i	\(0.346328\pi\)
\(31\)	7.00000	1.25724	0.628619	−	0.777714i	\(-0.283621\pi\)
			0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	−7.00000	−1.15079	−0.575396	−	0.817875i	\(-0.695152\pi\)
			−0.575396	+	0.817875i	\(0.695152\pi\)
\(41\)	−8.00000	−1.24939	−0.624695	−	0.780869i	\(-0.714777\pi\)
			−0.624695	+	0.780869i	\(0.714777\pi\)
\(43\)	−6.00000	−0.914991	−0.457496	−	0.889212i	\(-0.651253\pi\)
			−0.457496	+	0.889212i	\(0.651253\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)
\(53\)	9.00000	1.23625	0.618123	−	0.786082i	\(-0.287894\pi\)
			0.618123	+	0.786082i	\(0.287894\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−13.0000	−1.66448	−0.832240	−	0.554416i	\(-0.812942\pi\)
			−0.832240	+	0.554416i	\(0.812942\pi\)
\(67\)	−12.0000	−1.46603	−0.733017	−	0.680211i	\(-0.761888\pi\)
			−0.733017	+	0.680211i	\(0.761888\pi\)
\(71\)	−3.00000	−0.356034	−0.178017	−	0.984027i	\(-0.556968\pi\)
			−0.178017	+	0.984027i	\(0.556968\pi\)
\(73\)	−6.00000	−0.702247	−0.351123	−	0.936329i	\(-0.614200\pi\)
			−0.351123	+	0.936329i	\(0.614200\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	4.00000	0.439057	0.219529	−	0.975606i	\(-0.429548\pi\)
			0.219529	+	0.975606i	\(0.429548\pi\)
\(89\)	−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	\(-0.708512\pi\)
			−0.609208	+	0.793011i	\(0.708512\pi\)

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.j.1.1		1
3.2	odd	2		4950.2.a.q.1.1		1
4.3	odd	2		4400.2.a.s.1.1		1
5.2	odd	4		110.2.b.a.89.2	yes	2
5.3	odd	4		110.2.b.a.89.1	✓	2
5.4	even	2		550.2.a.e.1.1		1
11.10	odd	2		6050.2.a.f.1.1		1
15.2	even	4		990.2.c.d.199.1		2
15.8	even	4		990.2.c.d.199.2		2
15.14	odd	2		4950.2.a.ba.1.1		1
20.3	even	4		880.2.b.a.529.2		2
20.7	even	4		880.2.b.a.529.1		2
20.19	odd	2		4400.2.a.k.1.1		1
55.32	even	4		1210.2.b.a.969.1		2
55.43	even	4		1210.2.b.a.969.2		2
55.54	odd	2		6050.2.a.bk.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.b.a.89.1	✓	2	5.3	odd	4
110.2.b.a.89.2	yes	2	5.2	odd	4
550.2.a.e.1.1		1	5.4	even	2
550.2.a.j.1.1		1	1.1	even	1	trivial
880.2.b.a.529.1		2	20.7	even	4
880.2.b.a.529.2		2	20.3	even	4
990.2.c.d.199.1		2	15.2	even	4
990.2.c.d.199.2		2	15.8	even	4
1210.2.b.a.969.1		2	55.32	even	4
1210.2.b.a.969.2		2	55.43	even	4
4400.2.a.k.1.1		1	20.19	odd	2
4400.2.a.s.1.1		1	4.3	odd	2
4950.2.a.q.1.1		1	3.2	odd	2
4950.2.a.ba.1.1		1	15.14	odd	2
6050.2.a.f.1.1		1	11.10	odd	2
6050.2.a.bk.1.1		1	55.54	odd	2