Embedded newform 144.2.u.a.11.16

• Label: 144.2.u.a.11.16
• Level: $144$
• Weight: $2$
• Character: 144.11
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	144.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			11.16
Character	\(\chi\)	\(=\)	144.11
Dual form			144.2.u.a.131.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.717593 + 1.21863i) q^{2} +(1.11494 - 1.32549i) q^{3} +(-0.970121 + 1.74896i) q^{4} +(1.20583 + 0.323102i) q^{5} +(2.41535 + 0.407536i) q^{6} +(0.140266 - 0.242948i) q^{7} +(-2.82749 + 0.0728225i) q^{8} +(-0.513832 - 2.95567i) q^{9} +(0.471556 + 1.70132i) q^{10} +(-3.07444 + 0.823794i) q^{11} +(1.23660 + 3.23586i) q^{12} +(2.76539 + 0.740984i) q^{13} +(0.396717 - 0.00340517i) q^{14} +(1.77270 - 1.23808i) q^{15} +(-2.11773 - 3.39341i) q^{16} +3.72031i q^{17} +(3.23315 - 2.74714i) q^{18} +(-4.10860 - 4.10860i) q^{19} +(-1.73490 + 1.79551i) q^{20} +(-0.165636 - 0.456792i) q^{21} +(-3.21010 - 3.15546i) q^{22} +(-1.57595 + 0.909876i) q^{23} +(-3.05595 + 3.82899i) q^{24} +(-2.98049 - 1.72078i) q^{25} +(1.08144 + 3.90172i) q^{26} +(-4.49059 - 2.61431i) q^{27} +(0.288831 + 0.481009i) q^{28} +(-3.83006 + 1.02626i) q^{29} +(2.78084 + 1.27183i) q^{30} +(8.81101 - 5.08704i) q^{31} +(2.61564 - 5.01582i) q^{32} +(-2.33588 + 4.99361i) q^{33} +(-4.53369 + 2.66967i) q^{34} +(0.247635 - 0.247635i) q^{35} +(5.66783 + 1.96868i) q^{36} +(1.76964 + 1.76964i) q^{37} +(2.05856 - 7.95516i) q^{38} +(4.06540 - 2.83934i) q^{39} +(-3.43301 - 0.825757i) q^{40} +(2.66819 + 4.62144i) q^{41} +(0.437801 - 0.529640i) q^{42} +(-1.84748 - 6.89490i) q^{43} +(1.54180 - 6.17626i) q^{44} +(0.335387 - 3.73007i) q^{45} +(-2.23969 - 1.26758i) q^{46} +(-5.48486 + 9.50006i) q^{47} +(-6.85905 - 0.976412i) q^{48} +(3.46065 + 5.99402i) q^{49} +(-0.0417747 - 4.86693i) q^{50} +(4.93123 + 4.14791i) q^{51} +(-3.97872 + 4.11772i) q^{52} +(8.58403 - 8.58403i) q^{53} +(-0.0365433 - 7.34838i) q^{54} -3.97344 q^{55} +(-0.378909 + 0.697147i) q^{56} +(-10.0267 + 0.865067i) q^{57} +(-3.99906 - 3.93099i) q^{58} +(1.44299 - 5.38532i) q^{59} +(0.445622 + 4.30146i) q^{60} +(1.66977 + 6.23168i) q^{61} +(12.5219 + 7.08695i) q^{62} +(-0.790146 - 0.289745i) q^{63} +(7.98939 - 0.411810i) q^{64} +(3.09519 + 1.78701i) q^{65} +(-7.76158 + 0.736806i) q^{66} +(-1.00652 + 3.75640i) q^{67} +(-6.50668 - 3.60915i) q^{68} +(-0.551057 + 3.10336i) q^{69} +(0.479476 + 0.124074i) q^{70} +10.6808i q^{71} +(1.66809 + 8.31970i) q^{72} +9.30419i q^{73} +(-0.886658 + 3.42643i) q^{74} +(-5.60393 + 2.03203i) q^{75} +(11.1716 - 3.19994i) q^{76} +(-0.231101 + 0.862479i) q^{77} +(6.37741 + 2.91673i) q^{78} +(8.70990 + 5.02866i) q^{79} +(-1.45721 - 4.77613i) q^{80} +(-8.47195 + 3.03744i) q^{81} +(-3.71715 + 6.56784i) q^{82} +(0.588691 + 2.19703i) q^{83} +(0.959599 + 0.153452i) q^{84} +(-1.20204 + 4.48608i) q^{85} +(7.07659 - 7.19913i) q^{86} +(-2.90998 + 6.22091i) q^{87} +(8.63296 - 2.55316i) q^{88} -1.87637 q^{89} +(4.78625 - 2.26796i) q^{90} +(0.567911 - 0.567911i) q^{91} +(-0.0624738 - 3.63897i) q^{92} +(3.08091 - 17.3506i) q^{93} +(-15.5130 + 0.133153i) q^{94} +(-3.62679 - 6.28179i) q^{95} +(-3.73212 - 9.05932i) q^{96} +(9.19070 - 15.9188i) q^{97} +(-4.82116 + 8.51852i) q^{98} +(4.01461 + 8.66374i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	88 * q - 6 * q^2 - 4 * q^3 - 2 * q^4 - 6 * q^5 + 2 * q^6 - 4 * q^7 - 8 * q^10 - 6 * q^11 - 16 * q^12 - 2 * q^13 - 6 * q^14 - 2 * q^16 - 10 * q^18 - 8 * q^19 - 48 * q^20 + 2 * q^21 - 2 * q^22 - 12 * q^23 - 16 * q^27 + 8 * q^28 - 6 * q^29 - 34 * q^30 - 6 * q^32 - 8 * q^33 + 2 * q^34 - 26 * q^36 - 8 * q^37 - 6 * q^38 - 32 * q^39 - 2 * q^40 + 48 * q^42 - 2 * q^43 + 6 * q^45 - 40 * q^46 + 42 * q^48 - 24 * q^49 + 72 * q^50 - 12 * q^51 - 2 * q^52 - 38 * q^54 - 16 * q^55 + 36 * q^56 + 16 * q^58 - 42 * q^59 + 70 * q^60 - 2 * q^61 - 44 * q^64 - 12 * q^65 + 104 * q^66 - 2 * q^67 + 96 * q^68 - 10 * q^69 - 16 * q^70 - 10 * q^72 + 78 * q^74 - 56 * q^75 - 14 * q^76 - 6 * q^77 + 12 * q^78 - 8 * q^81 - 36 * q^82 + 54 * q^83 + 158 * q^84 + 8 * q^85 + 54 * q^86 + 48 * q^87 + 22 * q^88 + 64 * q^90 + 20 * q^91 + 108 * q^92 - 34 * q^93 + 6 * q^94 - 58 * q^96 - 4 * q^97 + 46 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.717593	+	1.21863i	0.507415	+	0.861702i
							\(3\)	1.11494	−	1.32549i	0.643709	−	0.765270i
\(5\)	1.20583	+	0.323102i	0.539266	+	0.144496i								0.518164	−	0.855282i	\(-0.326616\pi\)
							0.0211020	+	0.999777i	\(0.493283\pi\)
\(7\)	0.140266	−	0.242948i	0.0530156	−	0.0918256i	−0.838300	−	0.545210i	\(-0.816450\pi\)
							0.891315	+	0.453384i	\(0.149783\pi\)
\(11\)	−3.07444	+	0.823794i	−0.926979	+	0.248383i	−0.690566	−	0.723269i	\(-0.742639\pi\)
							−0.236413	+	0.971653i	\(0.575972\pi\)
\(13\)	2.76539	+	0.740984i	0.766982	+	0.205512i	0.621038	−	0.783781i	\(-0.286711\pi\)
							0.145944	+	0.989293i	\(0.453378\pi\)
\(17\)			3.72031i			0.902309i	0.892446	+	0.451154i	\(0.148988\pi\)
							−0.892446	+	0.451154i	\(0.851012\pi\)
\(19\)	−4.10860	−	4.10860i	−0.942577	−	0.942577i	0.0558614	−	0.998439i	\(-0.482209\pi\)
							−0.998439	+	0.0558614i	\(0.982209\pi\)
\(23\)	−1.57595	+	0.909876i	−0.328609	+	0.189722i	−0.655223	−	0.755435i	\(-0.727425\pi\)
							0.326615	+	0.945158i	\(0.394092\pi\)
\(29\)	−3.83006	+	1.02626i	−0.711224	+	0.190572i	−0.596253	−	0.802797i	\(-0.703344\pi\)
							−0.114972	+	0.993369i	\(0.536678\pi\)
\(31\)	8.81101	−	5.08704i	1.58250	−	0.913659i	0.588012	−	0.808852i	\(-0.299911\pi\)
							0.994493	−	0.104807i	\(-0.0334225\pi\)
\(37\)	1.76964	+	1.76964i	0.290928	+	0.290928i	0.837447	−	0.546519i	\(-0.184047\pi\)
							−0.546519	+	0.837447i	\(0.684047\pi\)
\(41\)	2.66819	+	4.62144i	0.416701	+	0.721747i	0.995605	−	0.0936482i	\(-0.0298529\pi\)
							−0.578904	+	0.815395i	\(0.696520\pi\)
\(43\)	−1.84748	−	6.89490i	−0.281738	−	1.05146i	−0.951190	−	0.308606i	\(-0.900138\pi\)
							0.669452	−	0.742856i	\(-0.266529\pi\)
\(47\)	−5.48486	+	9.50006i	−0.800049	+	1.38573i	0.119534	+	0.992830i	\(0.461860\pi\)
							−0.919583	+	0.392896i	\(0.871473\pi\)
\(53\)	8.58403	−	8.58403i	1.17911	−	1.17911i	0.199135	−	0.979972i	\(-0.436187\pi\)
							0.979972	−	0.199135i	\(-0.0638133\pi\)
\(59\)	1.44299	−	5.38532i	0.187862	−	0.701109i	−0.806138	−	0.591727i	\(-0.798446\pi\)
							0.994000	−	0.109382i	\(-0.0348871\pi\)
\(61\)	1.66977	+	6.23168i	0.213793	+	0.797885i	0.986588	+	0.163230i	\(0.0521911\pi\)
							−0.772796	+	0.634655i	\(0.781142\pi\)
\(67\)	−1.00652	+	3.75640i	−0.122967	+	0.458918i	−0.999759	−	0.0219514i	\(-0.993012\pi\)
							0.876792	+	0.480869i	\(0.159679\pi\)
\(71\)			10.6808i			1.26758i	0.773506	+	0.633789i	\(0.218501\pi\)
							−0.773506	+	0.633789i	\(0.781499\pi\)
\(73\)			9.30419i			1.08897i	0.838770	+	0.544487i	\(0.183275\pi\)
							−0.838770	+	0.544487i	\(0.816725\pi\)
\(79\)	8.70990	+	5.02866i	0.979940	+	0.565769i	0.902252	−	0.431209i	\(-0.141913\pi\)
							0.0776882	+	0.996978i	\(0.475246\pi\)
\(83\)	0.588691	+	2.19703i	0.0646173	+	0.241155i	0.990679	−	0.136217i	\(-0.0434945\pi\)
							−0.926062	+	0.377372i	\(0.876828\pi\)
\(89\)	−1.87637			−0.198895			−0.0994475	−	0.995043i	\(-0.531708\pi\)
							−0.0994475	+	0.995043i	\(0.531708\pi\)
\(97\)	9.19070	−	15.9188i	0.933175	−	1.61631i	0.155317	−	0.987865i	\(-0.450360\pi\)
							0.777857	−	0.628441i	\(-0.216307\pi\)

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	144.2.u.a.11.16	✓	88
3.2	odd	2		432.2.v.a.251.7		88
4.3	odd	2		576.2.y.a.335.6		88
9.4	even	3		432.2.v.a.395.1		88
9.5	odd	6	inner	144.2.u.a.59.22	yes	88
12.11	even	2		1728.2.z.a.143.7		88
16.3	odd	4	inner	144.2.u.a.83.22	yes	88
16.13	even	4		576.2.y.a.47.17		88
36.23	even	6		576.2.y.a.527.17		88
36.31	odd	6		1728.2.z.a.719.7		88
48.29	odd	4		1728.2.z.a.1007.7		88
48.35	even	4		432.2.v.a.35.1		88
144.13	even	12		1728.2.z.a.1583.7		88
144.67	odd	12		432.2.v.a.179.7		88
144.77	odd	12		576.2.y.a.239.6		88
144.131	even	12	inner	144.2.u.a.131.16	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.16	✓	88	1.1	even	1	trivial
144.2.u.a.59.22	yes	88	9.5	odd	6	inner
144.2.u.a.83.22	yes	88	16.3	odd	4	inner
144.2.u.a.131.16	yes	88	144.131	even	12	inner
432.2.v.a.35.1		88	48.35	even	4
432.2.v.a.179.7		88	144.67	odd	12
432.2.v.a.251.7		88	3.2	odd	2
432.2.v.a.395.1		88	9.4	even	3
576.2.y.a.47.17		88	16.13	even	4
576.2.y.a.239.6		88	144.77	odd	12
576.2.y.a.335.6		88	4.3	odd	2
576.2.y.a.527.17		88	36.23	even	6
1728.2.z.a.143.7		88	12.11	even	2
1728.2.z.a.719.7		88	36.31	odd	6
1728.2.z.a.1007.7		88	48.29	odd	4
1728.2.z.a.1583.7		88	144.13	even	12