Embedded newform 1089.2.e.o.364.15

• Label: 1089.2.e.o.364.15
• Level: $1089$
• Weight: $2$
• Character: 1089.364
• Analytic conductor: $8.696$
• Analytic rank: $0$
• Dimension: $36$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.69570878012\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{3})\)
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			364.15
Character	\(\chi\)	\(=\)	1089.364
Dual form			1089.2.e.o.727.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.773068 + 1.33899i) q^{2} +(1.56806 + 0.735660i) q^{3} +(-0.195269 + 0.338216i) q^{4} +(-0.296016 + 0.512715i) q^{5} +(0.227172 + 2.66833i) q^{6} +(0.360975 + 0.625227i) q^{7} +2.48845 q^{8} +(1.91761 + 2.30711i) q^{9} -0.915362 q^{10} +(-0.555005 + 0.386690i) q^{12} +(-0.787482 + 1.36396i) q^{13} +(-0.558117 + 0.966687i) q^{14} +(-0.841354 + 0.586199i) q^{15} +(2.31428 + 4.00845i) q^{16} -4.59200 q^{17} +(-1.60677 + 4.35122i) q^{18} -2.50860 q^{19} +(-0.115606 - 0.200235i) q^{20} +(0.106075 + 1.24595i) q^{21} +(2.22600 - 3.85554i) q^{23} +(3.90203 + 1.83065i) q^{24} +(2.32475 + 4.02658i) q^{25} -2.43511 q^{26} +(1.30967 + 5.02840i) q^{27} -0.281949 q^{28} +(3.48931 + 6.04366i) q^{29} +(-1.43534 - 0.673395i) q^{30} +(4.56412 - 7.90529i) q^{31} +(-1.08974 + 1.88749i) q^{32} +(-3.54993 - 6.14866i) q^{34} -0.427418 q^{35} +(-1.15475 + 0.198058i) q^{36} +2.89170 q^{37} +(-1.93932 - 3.35900i) q^{38} +(-2.23823 + 1.55945i) q^{39} +(-0.736620 + 1.27586i) q^{40} +(-0.577060 + 0.999497i) q^{41} +(-1.58631 + 1.10524i) q^{42} +(-2.10724 - 3.64985i) q^{43} +(-1.75053 + 0.300244i) q^{45} +6.88340 q^{46} +(0.113778 + 0.197069i) q^{47} +(0.680067 + 7.98800i) q^{48} +(3.23939 - 5.61079i) q^{49} +(-3.59438 + 6.22565i) q^{50} +(-7.20052 - 3.37815i) q^{51} +(-0.307542 - 0.532678i) q^{52} -5.70359 q^{53} +(-5.72053 + 5.64093i) q^{54} +(0.898268 + 1.55585i) q^{56} +(-3.93363 - 1.84548i) q^{57} +(-5.39495 + 9.34433i) q^{58} +(-3.56678 + 6.17784i) q^{59} +(-0.0339715 - 0.399026i) q^{60} +(2.27948 + 3.94817i) q^{61} +14.1135 q^{62} +(-0.750262 + 2.03175i) q^{63} +5.88733 q^{64} +(-0.466215 - 0.807507i) q^{65} +(4.04571 - 7.00738i) q^{67} +(0.896676 - 1.55309i) q^{68} +(6.32686 - 4.40814i) q^{69} +(-0.330423 - 0.572310i) q^{70} -12.3094 q^{71} +(4.77187 + 5.74113i) q^{72} -15.4833 q^{73} +(2.23548 + 3.87196i) q^{74} +(0.683144 + 8.02414i) q^{75} +(0.489852 - 0.848448i) q^{76} +(-3.81839 - 1.79141i) q^{78} +(-7.30978 - 12.6609i) q^{79} -2.74025 q^{80} +(-1.64555 + 8.84829i) q^{81} -1.78443 q^{82} +(-3.47650 - 6.02148i) q^{83} +(-0.442113 - 0.207419i) q^{84} +(1.35931 - 2.35439i) q^{85} +(3.25808 - 5.64316i) q^{86} +(1.02536 + 12.0438i) q^{87} +12.4803 q^{89} +(-1.75531 - 2.11185i) q^{90} -1.13705 q^{91} +(0.869338 + 1.50574i) q^{92} +(12.9724 - 9.03831i) q^{93} +(-0.175916 + 0.304695i) q^{94} +(0.742586 - 1.28620i) q^{95} +(-3.09733 + 2.15801i) q^{96} +(-7.09023 - 12.2806i) q^{97} +10.0171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	36 * q - 2 * q^2 + 9 * q^3 - 12 * q^4 + q^5 + q^6 + q^7 + 12 * q^8 - q^9 + 4 * q^10 - 8 * q^12 + 3 * q^13 - 5 * q^15 + 8 * q^16 + 40 * q^17 - 17 * q^18 + 6 * q^19 + 5 * q^20 + 8 * q^21 + 10 * q^23 + 57 * q^24 - 7 * q^25 + 4 * q^26 - 9 * q^27 - 38 * q^28 - 21 * q^29 - 12 * q^30 - 6 * q^31 - 9 * q^32 + 4 * q^34 + 76 * q^35 - 65 * q^36 + 14 * q^37 - 13 * q^38 - 42 * q^39 - 20 * q^41 + 18 * q^42 + 4 * q^43 + 20 * q^45 - 16 * q^46 - 7 * q^47 + 10 * q^48 - 7 * q^49 - 25 * q^50 + 4 * q^51 - 19 * q^52 + 62 * q^53 - 17 * q^54 + 57 * q^56 + 18 * q^57 - 12 * q^58 + 12 * q^59 + 11 * q^60 - 16 * q^61 + 38 * q^62 - 5 * q^63 - 32 * q^64 - 42 * q^65 + 5 * q^67 - 51 * q^68 + 31 * q^69 - 8 * q^70 + 26 * q^71 - 51 * q^72 - 9 * q^74 - 16 * q^75 - 8 * q^76 - 32 * q^78 - 2 * q^79 + 92 * q^80 + 47 * q^81 - 68 * q^82 - 36 * q^83 - 60 * q^84 + 25 * q^85 - 26 * q^86 + 60 * q^87 - 28 * q^89 - 163 * q^90 - 30 * q^91 + 15 * q^92 - 39 * q^93 + 4 * q^94 - 64 * q^95 + 21 * q^96 + 16 * q^97 + 164 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\)	\(244\)	\(848\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.773068	+	1.33899i	0.546642	+	0.946811i	0.998502	+	0.0547224i	\(0.0174274\pi\)
							−0.451860	+	0.892089i	\(0.649239\pi\)
\(3\)	1.56806	+	0.735660i	0.905318	+	0.424733i
							\(5\)	−0.296016	+	0.512715i	−0.132382	+	0.229293i	−0.924594	−	0.380953i	\(-0.875596\pi\)
0.792212	+	0.610246i	\(0.208929\pi\)
\(7\)	0.360975	+	0.625227i	0.136436	+	0.236314i	0.926145	−	0.377168i	\(-0.123102\pi\)
							−0.789709	+	0.613481i	\(0.789769\pi\)
\(11\)		0			0
							\(13\)	−0.787482	+	1.36396i	−0.218408	+	0.378294i	−0.954322	−	0.298782i	\(-0.903420\pi\)
0.735913	+	0.677076i	\(0.236753\pi\)
\(17\)	−4.59200			−1.11372			−0.556862	−	0.830605i	\(-0.687995\pi\)
							−0.556862	+	0.830605i	\(0.687995\pi\)
\(19\)	−2.50860			−0.575512			−0.287756	−	0.957704i	\(-0.592909\pi\)
							−0.287756	+	0.957704i	\(0.592909\pi\)
\(23\)	2.22600	−	3.85554i	0.464153	−	0.803937i	−0.535010	−	0.844846i	\(-0.679692\pi\)
							0.999163	+	0.0409092i	\(0.0130255\pi\)
\(29\)	3.48931	+	6.04366i	0.647949	+	1.12228i	0.983612	+	0.180298i	\(0.0577062\pi\)
							−0.335663	+	0.941982i	\(0.608960\pi\)
\(31\)	4.56412	−	7.90529i	0.819740	−	1.41983i	−0.0861334	−	0.996284i	\(-0.527451\pi\)
							0.905874	−	0.423548i	\(-0.139216\pi\)
\(37\)	2.89170			0.475392			0.237696	−	0.971340i	\(-0.423608\pi\)
							0.237696	+	0.971340i	\(0.423608\pi\)
\(41\)	−0.577060	+	0.999497i	−0.0901216	+	0.156095i	−0.907562	−	0.419918i	\(-0.862059\pi\)
							0.817441	+	0.576013i	\(0.195392\pi\)
\(43\)	−2.10724	−	3.64985i	−0.321351	−	0.556596i	0.659416	−	0.751778i	\(-0.270804\pi\)
							−0.980767	+	0.195182i	\(0.937470\pi\)
\(47\)	0.113778	+	0.197069i	0.0165962	+	0.0287454i	0.874204	−	0.485558i	\(-0.161384\pi\)
							−0.857608	+	0.514304i	\(0.828050\pi\)
\(53\)	−5.70359			−0.783449			−0.391724	−	0.920083i	\(-0.628121\pi\)
							−0.391724	+	0.920083i	\(0.628121\pi\)
\(59\)	−3.56678	+	6.17784i	−0.464355	+	0.804286i	−0.999172	−	0.0406816i	\(-0.987047\pi\)
							0.534817	+	0.844968i	\(0.320380\pi\)
\(61\)	2.27948	+	3.94817i	0.291857	+	0.505512i	0.974249	−	0.225475i	\(-0.0723934\pi\)
							−0.682392	+	0.730987i	\(0.739060\pi\)
\(67\)	4.04571	−	7.00738i	0.494262	−	0.856087i	−0.505716	−	0.862700i	\(-0.668772\pi\)
							0.999978	+	0.00661279i	\(0.00210493\pi\)
\(71\)	−12.3094			−1.46086			−0.730429	−	0.682989i	\(-0.760680\pi\)
							−0.730429	+	0.682989i	\(0.760680\pi\)
\(73\)	−15.4833			−1.81219			−0.906093	−	0.423078i	\(-0.860950\pi\)
							−0.906093	+	0.423078i	\(0.860950\pi\)
\(79\)	−7.30978	−	12.6609i	−0.822414	−	1.42446i	−0.903880	−	0.427787i	\(-0.859293\pi\)
							0.0814654	−	0.996676i	\(-0.474040\pi\)
\(83\)	−3.47650	−	6.02148i	−0.381596	−	0.660943i	0.609695	−	0.792636i	\(-0.291292\pi\)
							−0.991291	+	0.131693i	\(0.957959\pi\)
\(89\)	12.4803			1.32291			0.661453	−	0.749986i	\(-0.269940\pi\)
							0.661453	+	0.749986i	\(0.269940\pi\)
\(97\)	−7.09023	−	12.2806i	−0.719904	−	1.24691i	−0.961037	−	0.276419i	\(-0.910852\pi\)
							0.241133	−	0.970492i	\(-0.422481\pi\)

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	1089.2.e.o.364.15		36
9.4	even	3		9801.2.a.co.1.4		18
9.5	odd	6		9801.2.a.cn.1.15		18
9.7	even	3	inner	1089.2.e.o.727.15		36
11.2	odd	10		99.2.m.b.4.2	✓	72
11.6	odd	10		99.2.m.b.58.8	yes	72
11.10	odd	2		1089.2.e.p.364.4		36
33.2	even	10		297.2.n.b.37.8		72
33.17	even	10		297.2.n.b.91.2		72
99.2	even	30		297.2.n.b.235.2		72
99.13	odd	30		891.2.f.f.730.2		36
99.32	even	6		9801.2.a.cp.1.4		18
99.43	odd	6		1089.2.e.p.727.4		36
99.50	even	30		891.2.f.e.487.8		36
99.61	odd	30		99.2.m.b.25.2	yes	72
99.68	even	30		891.2.f.e.730.8		36
99.76	odd	6		9801.2.a.cm.1.15		18
99.79	odd	30		99.2.m.b.70.8	yes	72
99.83	even	30		297.2.n.b.289.8		72
99.94	odd	30		891.2.f.f.487.2		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.b.4.2	✓	72	11.2	odd	10
99.2.m.b.25.2	yes	72	99.61	odd	30
99.2.m.b.58.8	yes	72	11.6	odd	10
99.2.m.b.70.8	yes	72	99.79	odd	30
297.2.n.b.37.8		72	33.2	even	10
297.2.n.b.91.2		72	33.17	even	10
297.2.n.b.235.2		72	99.2	even	30
297.2.n.b.289.8		72	99.83	even	30
891.2.f.e.487.8		36	99.50	even	30
891.2.f.e.730.8		36	99.68	even	30
891.2.f.f.487.2		36	99.94	odd	30
891.2.f.f.730.2		36	99.13	odd	30
1089.2.e.o.364.15		36	1.1	even	1	trivial
1089.2.e.o.727.15		36	9.7	even	3	inner
1089.2.e.p.364.4		36	11.10	odd	2
1089.2.e.p.727.4		36	99.43	odd	6
9801.2.a.cm.1.15		18	99.76	odd	6
9801.2.a.cn.1.15		18	9.5	odd	6
9801.2.a.co.1.4		18	9.4	even	3
9801.2.a.cp.1.4		18	99.32	even	6