Embedded newform 144.2.u.a.59.8

• Label: 144.2.u.a.59.8
• Level: $144$
• Weight: $2$
• Character: 144.59
• 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			59.8
Character	\(\chi\)	\(=\)	144.59
Dual form			144.2.u.a.83.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.725676 - 1.21383i) q^{2} +(1.33407 + 1.10465i) q^{3} +(-0.946789 + 1.76170i) q^{4} +(0.178044 + 0.664471i) q^{5} +(0.372762 - 2.42096i) q^{6} +(0.645693 + 1.11837i) q^{7} +(2.82548 - 0.129179i) q^{8} +(0.559489 + 2.94737i) q^{9} +(0.677355 - 0.698307i) q^{10} +(0.860301 - 3.21069i) q^{11} +(-3.20915 + 1.30436i) q^{12} +(1.27203 + 4.74727i) q^{13} +(0.888956 - 1.59534i) q^{14} +(-0.496485 + 1.08313i) q^{15} +(-2.20718 - 3.33592i) q^{16} -5.58523i q^{17} +(3.17161 - 2.81796i) q^{18} +(-2.49649 - 2.49649i) q^{19} +(-1.33917 - 0.315453i) q^{20} +(-0.374013 + 2.20525i) q^{21} +(-4.52154 + 1.28565i) q^{22} +(-2.36529 - 1.36560i) q^{23} +(3.91208 + 2.94883i) q^{24} +(3.92031 - 2.26339i) q^{25} +(4.83933 - 4.98901i) q^{26} +(-2.50942 + 4.55004i) q^{27} +(-2.58157 + 0.0786549i) q^{28} +(-0.792277 + 2.95682i) q^{29} +(1.67503 - 0.183349i) q^{30} +(-5.28160 - 3.04933i) q^{31} +(-2.44755 + 5.09995i) q^{32} +(4.69439 - 3.33295i) q^{33} +(-6.77954 + 4.05307i) q^{34} +(-0.628165 + 0.628165i) q^{35} +(-5.72210 - 1.80488i) q^{36} +(0.507420 + 0.507420i) q^{37} +(-1.21868 + 4.84196i) q^{38} +(-3.54711 + 7.73835i) q^{39} +(0.588896 + 1.85445i) q^{40} +(-4.89892 + 8.48518i) q^{41} +(2.94823 - 1.14631i) q^{42} +(-0.949956 - 0.254540i) q^{43} +(4.84175 + 4.55544i) q^{44} +(-1.85883 + 0.896527i) q^{45} +(0.0588204 + 3.86205i) q^{46} +(-6.13774 - 10.6309i) q^{47} +(0.740492 - 6.88852i) q^{48} +(2.66616 - 4.61793i) q^{49} +(-5.59225 - 3.11611i) q^{50} +(6.16973 - 7.45109i) q^{51} +(-9.56762 - 2.25373i) q^{52} +(0.601793 - 0.601793i) q^{53} +(7.34401 - 0.255835i) q^{54} +2.28658 q^{55} +(1.96886 + 3.07653i) q^{56} +(-0.572741 - 6.08824i) q^{57} +(4.16402 - 1.18400i) q^{58} +(4.77715 - 1.28003i) q^{59} +(-1.43808 - 1.90015i) q^{60} +(10.8292 + 2.90167i) q^{61} +(0.131344 + 8.62382i) q^{62} +(-2.93500 + 2.52881i) q^{63} +(7.96663 - 0.729984i) q^{64} +(-2.92795 + 1.69045i) q^{65} +(-7.45226 - 3.27957i) q^{66} +(0.110351 - 0.0295686i) q^{67} +(9.83950 + 5.28803i) q^{68} +(-1.64695 - 4.43463i) q^{69} +(1.21833 + 0.306644i) q^{70} +0.0447904i q^{71} +(1.96156 + 8.25544i) q^{72} -13.2931i q^{73} +(0.247701 - 0.984146i) q^{74} +(7.73022 + 1.31105i) q^{75} +(6.76171 - 2.03442i) q^{76} +(4.14624 - 1.11098i) q^{77} +(11.9671 - 1.30993i) q^{78} +(-2.50052 + 1.44368i) q^{79} +(1.82364 - 2.06055i) q^{80} +(-8.37394 + 3.29804i) q^{81} +(13.8546 - 0.211011i) q^{82} +(3.79568 + 1.01705i) q^{83} +(-3.53089 - 2.74681i) q^{84} +(3.71122 - 0.994419i) q^{85} +(0.380391 + 1.33780i) q^{86} +(-4.32321 + 3.06941i) q^{87} +(2.01601 - 9.18285i) q^{88} -12.7362 q^{89} +(2.43714 + 1.60572i) q^{90} +(-4.48788 + 4.48788i) q^{91} +(4.64521 - 2.87400i) q^{92} +(-3.67758 - 9.90236i) q^{93} +(-8.45011 + 15.1648i) q^{94} +(1.21436 - 2.10333i) q^{95} +(-8.89888 + 4.10000i) q^{96} +(4.41066 + 7.63949i) q^{97} +(-7.54017 + 0.114840i) q^{98} +(9.94440 + 0.739279i) 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 \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

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

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.725676	−	1.21383i	−0.513130	−	0.858311i
							\(3\)	1.33407	+	1.10465i	0.770226	+	0.637771i
\(5\)	0.178044	+	0.664471i	0.0796239	+	0.297161i								0.994242	−	0.107158i	\(-0.0341752\pi\)
							−0.914618	+	0.404319i	\(0.867509\pi\)
\(7\)	0.645693	+	1.11837i	0.244049	+	0.422705i	0.961864	−	0.273529i	\(-0.0881909\pi\)
							−0.717815	+	0.696234i	\(0.754858\pi\)
\(11\)	0.860301	−	3.21069i	0.259390	−	0.968058i	−0.706205	−	0.708008i	\(-0.749594\pi\)
							0.965595	−	0.260051i	\(-0.0837392\pi\)
\(13\)	1.27203	+	4.74727i	0.352797	+	1.31666i	0.883234	+	0.468933i	\(0.155361\pi\)
							−0.530437	+	0.847725i	\(0.677972\pi\)
\(17\)		−	5.58523i		−	1.35462i	−0.735699	−	0.677308i	\(-0.763146\pi\)
							0.735699	−	0.677308i	\(-0.236854\pi\)
\(19\)	−2.49649	−	2.49649i	−0.572733	−	0.572733i	0.360158	−	0.932891i	\(-0.382723\pi\)
							−0.932891	+	0.360158i	\(0.882723\pi\)
\(23\)	−2.36529	−	1.36560i	−0.493197	−	0.284747i	0.232703	−	0.972548i	\(-0.425243\pi\)
							−0.725900	+	0.687801i	\(0.758576\pi\)
\(29\)	−0.792277	+	2.95682i	−0.147122	+	0.549067i	0.852530	+	0.522679i	\(0.175067\pi\)
							−0.999652	+	0.0263884i	\(0.991599\pi\)
\(31\)	−5.28160	−	3.04933i	−0.948604	−	0.547677i	−0.0559568	−	0.998433i	\(-0.517821\pi\)
							−0.892647	+	0.450757i	\(0.851154\pi\)
\(37\)	0.507420	+	0.507420i	0.0834193	+	0.0834193i	0.747585	−	0.664166i	\(-0.231213\pi\)
							−0.664166	+	0.747585i	\(0.731213\pi\)
\(41\)	−4.89892	+	8.48518i	−0.765083	+	1.32516i	0.175120	+	0.984547i	\(0.443969\pi\)
							−0.940203	+	0.340616i	\(0.889365\pi\)
\(43\)	−0.949956	−	0.254540i	−0.144867	−	0.0388170i	0.185657	−	0.982615i	\(-0.440559\pi\)
							−0.330524	+	0.943798i	\(0.607225\pi\)
\(47\)	−6.13774	−	10.6309i	−0.895281	−	1.55067i	−0.833456	−	0.552586i	\(-0.813641\pi\)
							−0.0618250	−	0.998087i	\(-0.519692\pi\)
\(53\)	0.601793	−	0.601793i	0.0826626	−	0.0826626i	−0.664567	−	0.747229i	\(-0.731384\pi\)
							0.747229	+	0.664567i	\(0.231384\pi\)
\(59\)	4.77715	−	1.28003i	0.621932	−	0.166646i	0.0659263	−	0.997824i	\(-0.479000\pi\)
							0.556006	+	0.831178i	\(0.312333\pi\)
\(61\)	10.8292	+	2.90167i	1.38653	+	0.371520i	0.873490	−	0.486842i	\(-0.161851\pi\)
							0.513043	+	0.858363i	\(0.328518\pi\)
\(67\)	0.110351	−	0.0295686i	0.0134816	−	0.00361238i	−0.252072	−	0.967708i	\(-0.581112\pi\)
							0.265554	+	0.964096i	\(0.414445\pi\)
\(71\)			0.0447904i			0.00531565i	0.999996	+	0.00265782i	\(0.000846013\pi\)
							−0.999996	+	0.00265782i	\(0.999154\pi\)
\(73\)		−	13.2931i		−	1.55585i	−0.628360	−	0.777923i	\(-0.716274\pi\)
							0.628360	−	0.777923i	\(-0.283726\pi\)
\(79\)	−2.50052	+	1.44368i	−0.281331	+	0.162426i	−0.634026	−	0.773312i	\(-0.718599\pi\)
							0.352695	+	0.935738i	\(0.385265\pi\)
\(83\)	3.79568	+	1.01705i	0.416630	+	0.111636i	0.461043	−	0.887378i	\(-0.347475\pi\)
							−0.0444135	+	0.999013i	\(0.514142\pi\)
\(89\)	−12.7362			−1.35003			−0.675017	−	0.737802i	\(-0.735864\pi\)
							−0.675017	+	0.737802i	\(0.735864\pi\)
\(97\)	4.41066	+	7.63949i	0.447835	+	0.775673i	0.998245	−	0.0592215i	\(-0.0188618\pi\)
							−0.550410	+	0.834895i	\(0.685529\pi\)

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.59.8	yes	88
3.2	odd	2		432.2.v.a.395.15		88
4.3	odd	2		576.2.y.a.527.5		88
9.2	odd	6	inner	144.2.u.a.11.15	✓	88
9.7	even	3		432.2.v.a.251.8		88
12.11	even	2		1728.2.z.a.719.10		88
16.3	odd	4	inner	144.2.u.a.131.15	yes	88
16.13	even	4		576.2.y.a.239.7		88
36.7	odd	6		1728.2.z.a.143.10		88
36.11	even	6		576.2.y.a.335.7		88
48.29	odd	4		1728.2.z.a.1583.10		88
48.35	even	4		432.2.v.a.179.8		88
144.29	odd	12		576.2.y.a.47.5		88
144.61	even	12		1728.2.z.a.1007.10		88
144.83	even	12	inner	144.2.u.a.83.8	yes	88
144.115	odd	12		432.2.v.a.35.15		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.15	✓	88	9.2	odd	6	inner
144.2.u.a.59.8	yes	88	1.1	even	1	trivial
144.2.u.a.83.8	yes	88	144.83	even	12	inner
144.2.u.a.131.15	yes	88	16.3	odd	4	inner
432.2.v.a.35.15		88	144.115	odd	12
432.2.v.a.179.8		88	48.35	even	4
432.2.v.a.251.8		88	9.7	even	3
432.2.v.a.395.15		88	3.2	odd	2
576.2.y.a.47.5		88	144.29	odd	12
576.2.y.a.239.7		88	16.13	even	4
576.2.y.a.335.7		88	36.11	even	6
576.2.y.a.527.5		88	4.3	odd	2
1728.2.z.a.143.10		88	36.7	odd	6
1728.2.z.a.719.10		88	12.11	even	2
1728.2.z.a.1007.10		88	144.61	even	12
1728.2.z.a.1583.10		88	48.29	odd	4