Embedded newform 550.2.a.n.1.2

• Label: 550.2.a.n.1.2
• 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			1.2
Root			\(3.37228\) of defining polynomial
Character	\(\chi\)	\(=\)	550.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.37228 q^{3} +1.00000 q^{4} +3.37228 q^{6} -3.37228 q^{7} +1.00000 q^{8} +8.37228 q^{9} -1.00000 q^{11} +3.37228 q^{12} -2.00000 q^{13} -3.37228 q^{14} +1.00000 q^{16} -1.37228 q^{17} +8.37228 q^{18} +0.627719 q^{19} -11.3723 q^{21} -1.00000 q^{22} -2.74456 q^{23} +3.37228 q^{24} -2.00000 q^{26} +18.1168 q^{27} -3.37228 q^{28} +1.37228 q^{29} +3.37228 q^{31} +1.00000 q^{32} -3.37228 q^{33} -1.37228 q^{34} +8.37228 q^{36} -9.37228 q^{37} +0.627719 q^{38} -6.74456 q^{39} -11.4891 q^{41} -11.3723 q^{42} +4.00000 q^{43} -1.00000 q^{44} -2.74456 q^{46} -2.74456 q^{47} +3.37228 q^{48} +4.37228 q^{49} -4.62772 q^{51} -2.00000 q^{52} +4.11684 q^{53} +18.1168 q^{54} -3.37228 q^{56} +2.11684 q^{57} +1.37228 q^{58} -2.74456 q^{59} -5.37228 q^{61} +3.37228 q^{62} -28.2337 q^{63} +1.00000 q^{64} -3.37228 q^{66} -8.00000 q^{67} -1.37228 q^{68} -9.25544 q^{69} +10.1168 q^{71} +8.37228 q^{72} +15.4891 q^{73} -9.37228 q^{74} +0.627719 q^{76} +3.37228 q^{77} -6.74456 q^{78} -1.25544 q^{79} +35.9783 q^{81} -11.4891 q^{82} +2.74456 q^{83} -11.3723 q^{84} +4.00000 q^{86} +4.62772 q^{87} -1.00000 q^{88} -1.37228 q^{89} +6.74456 q^{91} -2.74456 q^{92} +11.3723 q^{93} -2.74456 q^{94} +3.37228 q^{96} +12.7446 q^{97} +4.37228 q^{98} -8.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
			\(3\)	3.37228	1.94699	0.973494	−	0.228714i	\(-0.0734519\pi\)
0.973494	+	0.228714i				\(0.0734519\pi\)
\(5\)	0	0
			\(7\)	−3.37228	−1.27460	−0.637301	−	0.770615i	\(-0.719949\pi\)
−0.637301	+	0.770615i				\(0.719949\pi\)
\(11\)	−1.00000	−0.301511
			\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
−0.277350	+	0.960769i				\(0.589456\pi\)
\(17\)	−1.37228	−0.332827	−0.166414	−	0.986056i	\(-0.553219\pi\)
			−0.166414	+	0.986056i	\(0.553219\pi\)
\(19\)	0.627719	0.144009	0.0720043	−	0.997404i	\(-0.477060\pi\)
			0.0720043	+	0.997404i	\(0.477060\pi\)
\(23\)	−2.74456	−0.572281	−0.286140	−	0.958188i	\(-0.592372\pi\)
			−0.286140	+	0.958188i	\(0.592372\pi\)
\(29\)	1.37228	0.254826	0.127413	−	0.991850i	\(-0.459333\pi\)
			0.127413	+	0.991850i	\(0.459333\pi\)
\(31\)	3.37228	0.605680	0.302840	−	0.953041i	\(-0.402065\pi\)
			0.302840	+	0.953041i	\(0.402065\pi\)
\(37\)	−9.37228	−1.54079	−0.770397	−	0.637565i	\(-0.779942\pi\)
			−0.770397	+	0.637565i	\(0.779942\pi\)
\(41\)	−11.4891	−1.79430	−0.897150	−	0.441726i	\(-0.854366\pi\)
			−0.897150	+	0.441726i	\(0.854366\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−2.74456	−0.400336	−0.200168	−	0.979762i	\(-0.564149\pi\)
			−0.200168	+	0.979762i	\(0.564149\pi\)
\(53\)	4.11684	0.565492	0.282746	−	0.959195i	\(-0.408755\pi\)
			0.282746	+	0.959195i	\(0.408755\pi\)
\(59\)	−2.74456	−0.357312	−0.178656	−	0.983912i	\(-0.557175\pi\)
			−0.178656	+	0.983912i	\(0.557175\pi\)
\(61\)	−5.37228	−0.687850	−0.343925	−	0.938997i	\(-0.611757\pi\)
			−0.343925	+	0.938997i	\(0.611757\pi\)
\(67\)	−8.00000	−0.977356	−0.488678	−	0.872464i	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	10.1168	1.20065	0.600324	−	0.799757i	\(-0.295038\pi\)
			0.600324	+	0.799757i	\(0.295038\pi\)
\(73\)	15.4891	1.81286	0.906432	−	0.422351i	\(-0.138795\pi\)
			0.906432	+	0.422351i	\(0.138795\pi\)
\(79\)	−1.25544	−0.141248	−0.0706239	−	0.997503i	\(-0.522499\pi\)
			−0.0706239	+	0.997503i	\(0.522499\pi\)
\(83\)	2.74456	0.301255	0.150627	−	0.988591i	\(-0.451871\pi\)
			0.150627	+	0.988591i	\(0.451871\pi\)
\(89\)	−1.37228	−0.145462	−0.0727308	−	0.997352i	\(-0.523171\pi\)
			−0.0727308	+	0.997352i	\(0.523171\pi\)
\(97\)	12.7446	1.29401	0.647007	−	0.762484i	\(-0.276020\pi\)
			0.647007	+	0.762484i	\(0.276020\pi\)

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.1.2		2
3.2	odd	2		4950.2.a.bw.1.1		2
4.3	odd	2		4400.2.a.bl.1.1		2
5.2	odd	4		550.2.b.f.199.3		4
5.3	odd	4		550.2.b.f.199.2		4
5.4	even	2		110.2.a.d.1.1	✓	2
11.10	odd	2		6050.2.a.cb.1.2		2
15.2	even	4		4950.2.c.bc.199.1		4
15.8	even	4		4950.2.c.bc.199.4		4
15.14	odd	2		990.2.a.m.1.2		2
20.3	even	4		4400.2.b.p.4049.1		4
20.7	even	4		4400.2.b.p.4049.4		4
20.19	odd	2		880.2.a.n.1.2		2
35.34	odd	2		5390.2.a.bp.1.2		2
40.19	odd	2		3520.2.a.bj.1.1		2
40.29	even	2		3520.2.a.bq.1.2		2
55.54	odd	2		1210.2.a.r.1.1		2
60.59	even	2		7920.2.a.bq.1.1		2
220.219	even	2		9680.2.a.bt.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.1	✓	2	5.4	even	2
550.2.a.n.1.2		2	1.1	even	1	trivial
550.2.b.f.199.2		4	5.3	odd	4
550.2.b.f.199.3		4	5.2	odd	4
880.2.a.n.1.2		2	20.19	odd	2
990.2.a.m.1.2		2	15.14	odd	2
1210.2.a.r.1.1		2	55.54	odd	2
3520.2.a.bj.1.1		2	40.19	odd	2
3520.2.a.bq.1.2		2	40.29	even	2
4400.2.a.bl.1.1		2	4.3	odd	2
4400.2.b.p.4049.1		4	20.3	even	4
4400.2.b.p.4049.4		4	20.7	even	4
4950.2.a.bw.1.1		2	3.2	odd	2
4950.2.c.bc.199.1		4	15.2	even	4
4950.2.c.bc.199.4		4	15.8	even	4
5390.2.a.bp.1.2		2	35.34	odd	2
6050.2.a.cb.1.2		2	11.10	odd	2
7920.2.a.bq.1.1		2	60.59	even	2
9680.2.a.bt.1.2		2	220.219	even	2