Embedded newform 144.2.u.a.131.11

• Label: 144.2.u.a.131.11
• Level: $144$
• Weight: $2$
• Character: 144.131
• 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			131.11
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.311077 + 1.37958i) q^{2} +(-1.36635 - 1.06447i) q^{3} +(-1.80646 - 0.858309i) q^{4} +(2.31044 - 0.619079i) q^{5} +(1.89356 - 1.55385i) q^{6} +(2.51270 + 4.35213i) q^{7} +(1.74605 - 2.22515i) q^{8} +(0.733803 + 2.90887i) q^{9} +(0.135344 + 3.38000i) q^{10} +(-1.03121 - 0.276313i) q^{11} +(1.55461 + 3.09567i) q^{12} +(4.00543 - 1.07325i) q^{13} +(-6.78573 + 2.11262i) q^{14} +(-3.81585 - 1.61352i) q^{15} +(2.52661 + 3.10101i) q^{16} -2.22468i q^{17} +(-4.24128 + 0.107453i) q^{18} +(-0.697254 + 0.697254i) q^{19} +(-4.70507 - 0.864725i) q^{20} +(1.19949 - 8.62121i) q^{21} +(0.701981 - 1.33668i) q^{22} +(-2.20833 - 1.27498i) q^{23} +(-4.75432 + 1.18170i) q^{24} +(0.624725 - 0.360685i) q^{25} +(0.234635 + 5.85966i) q^{26} +(2.09378 - 4.75564i) q^{27} +(-0.803629 - 10.0186i) q^{28} +(0.589549 + 0.157969i) q^{29} +(3.41299 - 4.76232i) q^{30} +(0.190501 + 0.109986i) q^{31} +(-5.06405 + 2.52100i) q^{32} +(1.11487 + 1.47524i) q^{33} +(3.06912 + 0.692047i) q^{34} +(8.49974 + 8.49974i) q^{35} +(1.17113 - 5.88460i) q^{36} +(-5.16341 + 5.16341i) q^{37} +(-0.745016 - 1.17882i) q^{38} +(-6.61524 - 2.79723i) q^{39} +(2.65660 - 6.22201i) q^{40} +(0.828296 - 1.43465i) q^{41} +(11.5205 + 4.33665i) q^{42} +(1.33457 - 4.98067i) q^{43} +(1.62569 + 1.38425i) q^{44} +(3.49623 + 6.26648i) q^{45} +(2.44589 - 2.64994i) q^{46} +(-5.76715 - 9.98900i) q^{47} +(-0.151291 - 6.92655i) q^{48} +(-9.12734 + 15.8090i) q^{49} +(0.303255 + 0.974056i) q^{50} +(-2.36811 + 3.03968i) q^{51} +(-8.15683 - 1.49911i) q^{52} +(-7.80379 - 7.80379i) q^{53} +(5.90944 + 4.36790i) q^{54} -2.55361 q^{55} +(14.0715 + 2.00790i) q^{56} +(1.69490 - 0.210484i) q^{57} +(-0.401326 + 0.764188i) q^{58} +(1.36607 + 5.09824i) q^{59} +(5.50829 + 6.18993i) q^{60} +(1.73861 - 6.48859i) q^{61} +(-0.210995 + 0.228597i) q^{62} +(-10.8159 + 10.5027i) q^{63} +(-1.90260 - 7.77046i) q^{64} +(8.58985 - 4.95935i) q^{65} +(-2.38201 + 1.07913i) q^{66} +(-2.20370 - 8.22431i) q^{67} +(-1.90946 + 4.01880i) q^{68} +(1.66016 + 4.09277i) q^{69} +(-14.3701 + 9.08197i) q^{70} +12.0321i q^{71} +(7.75394 + 3.44622i) q^{72} +10.3710i q^{73} +(-5.51710 - 8.72954i) q^{74} +(-1.23753 - 0.172181i) q^{75} +(1.85802 - 0.661103i) q^{76} +(-1.38858 - 5.18226i) q^{77} +(5.91684 - 8.25608i) q^{78} +(7.74084 - 4.46918i) q^{79} +(7.75734 + 5.60050i) q^{80} +(-7.92307 + 4.26907i) q^{81} +(1.72155 + 1.58899i) q^{82} +(-1.48752 + 5.55149i) q^{83} +(-9.56650 + 14.5443i) q^{84} +(-1.37725 - 5.13998i) q^{85} +(6.45606 + 3.39051i) q^{86} +(-0.637375 - 0.843399i) q^{87} +(-2.41539 + 1.81215i) q^{88} +6.56588 q^{89} +(-9.73268 + 2.87395i) q^{90} +(14.7354 + 14.7354i) q^{91} +(2.89494 + 4.19864i) q^{92} +(-0.143214 - 0.353062i) q^{93} +(15.5746 - 4.84888i) q^{94} +(-1.17930 + 2.04262i) q^{95} +(9.60277 + 1.94597i) q^{96} +(-1.51787 - 2.62902i) q^{97} +(-18.9704 - 17.5097i) q^{98} +(0.0470514 - 3.20243i) 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 \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.311077	+	1.37958i	−0.219965	+	0.975508i
							\(3\)	−1.36635	−	1.06447i	−0.788860	−	0.614573i
\(5\)	2.31044	−	0.619079i	1.03326	−	0.276861i								0.297941	−	0.954584i	\(-0.403700\pi\)
							0.735317	+	0.677724i	\(0.237033\pi\)
\(7\)	2.51270	+	4.35213i	0.949712	+	1.64495i	0.746030	+	0.665912i	\(0.231957\pi\)
							0.203681	+	0.979037i	\(0.434709\pi\)
\(11\)	−1.03121	−	0.276313i	−0.310922	−	0.0833114i	0.0999837	−	0.994989i	\(-0.468121\pi\)
							−0.410906	+	0.911678i	\(0.634788\pi\)
\(13\)	4.00543	−	1.07325i	1.11091	−	0.297666i	0.343708	−	0.939077i	\(-0.388317\pi\)
							0.767198	+	0.641411i	\(0.221650\pi\)
\(17\)		−	2.22468i		−	0.539564i	−0.962921	−	0.269782i	\(-0.913048\pi\)
							0.962921	−	0.269782i	\(-0.0869517\pi\)
\(19\)	−0.697254	+	0.697254i	−0.159961	+	0.159961i	−0.782549	−	0.622588i	\(-0.786081\pi\)
							0.622588	+	0.782549i	\(0.286081\pi\)
\(23\)	−2.20833	−	1.27498i	−0.460469	−	0.265852i	0.251773	−	0.967786i	\(-0.418987\pi\)
							−0.712241	+	0.701935i	\(0.752320\pi\)
\(29\)	0.589549	+	0.157969i	0.109477	+	0.0293342i	0.313141	−	0.949707i	\(-0.398619\pi\)
							−0.203665	+	0.979041i	\(0.565285\pi\)
\(31\)	0.190501	+	0.109986i	0.0342150	+	0.0197541i	0.517010	−	0.855979i	\(-0.327045\pi\)
							−0.482795	+	0.875733i	\(0.660378\pi\)
\(37\)	−5.16341	+	5.16341i	−0.848859	+	0.848859i	−0.989991	−	0.141132i	\(-0.954926\pi\)
							0.141132	+	0.989991i	\(0.454926\pi\)
\(41\)	0.828296	−	1.43465i	0.129358	−	0.224055i	−0.794070	−	0.607826i	\(-0.792042\pi\)
							0.923428	+	0.383772i	\(0.125375\pi\)
\(43\)	1.33457	−	4.98067i	0.203519	−	0.759545i	−0.786376	−	0.617748i	\(-0.788045\pi\)
							0.989896	−	0.141797i	\(-0.0452881\pi\)
\(47\)	−5.76715	−	9.98900i	−0.841226	−	1.45705i	−0.888859	−	0.458181i	\(-0.848501\pi\)
							0.0476333	−	0.998865i	\(-0.484832\pi\)
\(53\)	−7.80379	−	7.80379i	−1.07193	−	1.07193i	−0.997204	−	0.0747298i	\(-0.976191\pi\)
							−0.0747298	−	0.997204i	\(-0.523809\pi\)
\(59\)	1.36607	+	5.09824i	0.177847	+	0.663735i	0.996049	+	0.0888039i	\(0.0283045\pi\)
							−0.818202	+	0.574931i	\(0.805029\pi\)
\(61\)	1.73861	−	6.48859i	0.222606	−	0.830779i	−0.760743	−	0.649053i	\(-0.775165\pi\)
							0.983349	−	0.181725i	\(-0.0581681\pi\)
\(67\)	−2.20370	−	8.22431i	−0.269224	−	1.00476i	−0.959614	−	0.281321i	\(-0.909227\pi\)
							0.690389	−	0.723438i	\(-0.257439\pi\)
\(71\)			12.0321i			1.42795i	0.700171	+	0.713975i	\(0.253107\pi\)
							−0.700171	+	0.713975i	\(0.746893\pi\)
\(73\)			10.3710i			1.21383i	0.794766	+	0.606917i	\(0.207594\pi\)
							−0.794766	+	0.606917i	\(0.792406\pi\)
\(79\)	7.74084	−	4.46918i	0.870913	−	0.502822i	0.00326134	−	0.999995i	\(-0.498962\pi\)
							0.867651	+	0.497173i	\(0.165629\pi\)
\(83\)	−1.48752	+	5.55149i	−0.163276	+	0.609355i	0.834978	+	0.550284i	\(0.185480\pi\)
							−0.998254	+	0.0590710i	\(0.981186\pi\)
\(89\)	6.56588			0.695982			0.347991	−	0.937498i	\(-0.386864\pi\)
							0.347991	+	0.937498i	\(0.386864\pi\)
\(97\)	−1.51787	−	2.62902i	−0.154116	−	0.266936i	0.778621	−	0.627495i	\(-0.215920\pi\)
							−0.932737	+	0.360558i	\(0.882586\pi\)

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.131.11	yes	88
3.2	odd	2		432.2.v.a.179.12		88
4.3	odd	2		576.2.y.a.239.17		88
9.2	odd	6	inner	144.2.u.a.83.3	yes	88
9.7	even	3		432.2.v.a.35.20		88
12.11	even	2		1728.2.z.a.1583.4		88
16.5	even	4		576.2.y.a.527.6		88
16.11	odd	4	inner	144.2.u.a.59.3	yes	88
36.7	odd	6		1728.2.z.a.1007.4		88
36.11	even	6		576.2.y.a.47.6		88
48.5	odd	4		1728.2.z.a.719.4		88
48.11	even	4		432.2.v.a.395.20		88
144.11	even	12	inner	144.2.u.a.11.11	✓	88
144.43	odd	12		432.2.v.a.251.12		88
144.101	odd	12		576.2.y.a.335.17		88
144.133	even	12		1728.2.z.a.143.4		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.11	✓	88	144.11	even	12	inner
144.2.u.a.59.3	yes	88	16.11	odd	4	inner
144.2.u.a.83.3	yes	88	9.2	odd	6	inner
144.2.u.a.131.11	yes	88	1.1	even	1	trivial
432.2.v.a.35.20		88	9.7	even	3
432.2.v.a.179.12		88	3.2	odd	2
432.2.v.a.251.12		88	144.43	odd	12
432.2.v.a.395.20		88	48.11	even	4
576.2.y.a.47.6		88	36.11	even	6
576.2.y.a.239.17		88	4.3	odd	2
576.2.y.a.335.17		88	144.101	odd	12
576.2.y.a.527.6		88	16.5	even	4
1728.2.z.a.143.4		88	144.133	even	12
1728.2.z.a.719.4		88	48.5	odd	4
1728.2.z.a.1007.4		88	36.7	odd	6
1728.2.z.a.1583.4		88	12.11	even	2