Embedded newform 144.2.u.a.131.21

• Label: 144.2.u.a.131.21
• 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			131.21
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37902 + 0.313539i) q^{2} +(-0.412428 - 1.68223i) q^{3} +(1.80339 + 0.864752i) q^{4} +(0.289971 - 0.0776974i) q^{5} +(-0.0413015 - 2.44914i) q^{6} +(-0.374023 - 0.647827i) q^{7} +(2.21577 + 1.75794i) q^{8} +(-2.65981 + 1.38760i) q^{9} +(0.424236 - 0.0162292i) q^{10} +(-2.23720 - 0.599457i) q^{11} +(0.710945 - 3.39036i) q^{12} +(1.60318 - 0.429571i) q^{13} +(-0.312666 - 1.01064i) q^{14} +(-0.250297 - 0.455753i) q^{15} +(2.50441 + 3.11896i) q^{16} +6.74518i q^{17} +(-4.10299 + 1.07957i) q^{18} +(0.621335 - 0.621335i) q^{19} +(0.590118 + 0.110634i) q^{20} +(-0.935537 + 0.896375i) q^{21} +(-2.89719 - 1.52811i) q^{22} +(-6.06191 - 3.49985i) q^{23} +(2.04342 - 4.45247i) q^{24} +(-4.25208 + 2.45494i) q^{25} +(2.34550 - 0.0897272i) q^{26} +(3.43124 + 3.90212i) q^{27} +(-0.114299 - 1.49172i) q^{28} +(-5.44240 - 1.45829i) q^{29} +(-0.202268 - 0.706970i) q^{30} +(3.13647 + 1.81084i) q^{31} +(2.47571 + 5.08634i) q^{32} +(-0.0857395 + 4.01073i) q^{33} +(-2.11488 + 9.30173i) q^{34} +(-0.158790 - 0.158790i) q^{35} +(-5.99659 + 0.202306i) q^{36} +(6.74053 - 6.74053i) q^{37} +(1.05165 - 0.662020i) q^{38} +(-1.38383 - 2.51975i) q^{39} +(0.779096 + 0.337592i) q^{40} +(-1.39492 + 2.41607i) q^{41} +(-1.57117 + 0.942791i) q^{42} +(1.89907 - 7.08744i) q^{43} +(-3.51616 - 3.01568i) q^{44} +(-0.663453 + 0.609023i) q^{45} +(-7.26216 - 6.72700i) q^{46} +(0.307120 + 0.531947i) q^{47} +(4.21393 - 5.49934i) q^{48} +(3.22021 - 5.57757i) q^{49} +(-6.63342 + 2.05222i) q^{50} +(11.3470 - 2.78190i) q^{51} +(3.26263 + 0.611671i) q^{52} +(2.68523 + 2.68523i) q^{53} +(3.50828 + 6.45693i) q^{54} -0.695300 q^{55} +(0.310092 - 2.09295i) q^{56} +(-1.30149 - 0.788974i) q^{57} +(-7.04794 - 3.71741i) q^{58} +(-0.00225603 - 0.00841962i) q^{59} +(-0.0572691 - 1.03834i) q^{60} +(2.72542 - 10.1714i) q^{61} +(3.75748 + 3.48059i) q^{62} +(1.89375 + 1.20410i) q^{63} +(1.81929 + 7.79039i) q^{64} +(0.431499 - 0.249126i) q^{65} +(-1.37575 + 5.50399i) q^{66} +(2.35300 + 8.78151i) q^{67} +(-5.83291 + 12.1642i) q^{68} +(-3.38745 + 11.6410i) q^{69} +(-0.169188 - 0.268762i) q^{70} -15.9645i q^{71} +(-8.33284 - 1.60118i) q^{72} +8.17785i q^{73} +(11.4087 - 7.18190i) q^{74} +(5.88346 + 6.14050i) q^{75} +(1.65781 - 0.583207i) q^{76} +(0.448421 + 1.67353i) q^{77} +(-1.11829 - 3.90867i) q^{78} +(-7.67035 + 4.42848i) q^{79} +(0.968541 + 0.709822i) q^{80} +(5.14914 - 7.38149i) q^{81} +(-2.68115 + 2.89445i) q^{82} +(-0.353727 + 1.32013i) q^{83} +(-2.46228 + 0.807504i) q^{84} +(0.524083 + 1.95590i) q^{85} +(4.84105 - 9.17828i) q^{86} +(-0.208577 + 9.75681i) q^{87} +(-3.90332 - 5.26113i) q^{88} +15.7852 q^{89} +(-1.10587 + 0.631836i) q^{90} +(-0.877914 - 0.877914i) q^{91} +(-7.90548 - 11.5536i) q^{92} +(1.75268 - 6.02310i) q^{93} +(0.256738 + 0.829859i) q^{94} +(0.131893 - 0.228445i) q^{95} +(7.53535 - 6.26247i) q^{96} +(-4.62075 - 8.00338i) q^{97} +(6.18952 - 6.68192i) q^{98} +(6.78233 - 1.50990i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37902	+	0.313539i	0.975114	+	0.221705i
							\(3\)	−0.412428	−	1.68223i	−0.238116	−	0.971237i
\(5\)	0.289971	−	0.0776974i	0.129679	−	0.0347473i								−0.193396	−	0.981121i	\(-0.561950\pi\)
							0.323075	+	0.946373i	\(0.395283\pi\)
\(7\)	−0.374023	−	0.647827i	−0.141367	−	0.244855i	0.786644	−	0.617406i	\(-0.211817\pi\)
							−0.928012	+	0.372551i	\(0.878483\pi\)
\(11\)	−2.23720	−	0.599457i	−0.674542	−	0.180743i	−0.0947421	−	0.995502i	\(-0.530203\pi\)
							−0.579800	+	0.814759i	\(0.696869\pi\)
\(13\)	1.60318	−	0.429571i	0.444642	−	0.119142i	−0.0295491	−	0.999563i	\(-0.509407\pi\)
							0.474191	+	0.880422i	\(0.342740\pi\)
\(17\)			6.74518i			1.63595i	0.575256	+	0.817973i	\(0.304902\pi\)
							−0.575256	+	0.817973i	\(0.695098\pi\)
\(19\)	0.621335	−	0.621335i	0.142544	−	0.142544i	−0.632234	−	0.774778i	\(-0.717862\pi\)
							0.774778	+	0.632234i	\(0.217862\pi\)
\(23\)	−6.06191	−	3.49985i	−1.26400	−	0.729769i	−0.290151	−	0.956981i	\(-0.593706\pi\)
							−0.973845	+	0.227212i	\(0.927039\pi\)
\(29\)	−5.44240	−	1.45829i	−1.01063	−	0.270797i	−0.284736	−	0.958606i	\(-0.591906\pi\)
							−0.725892	+	0.687809i	\(0.758573\pi\)
\(31\)	3.13647	+	1.81084i	0.563326	+	0.325236i	0.754479	−	0.656324i	\(-0.227889\pi\)
							−0.191153	+	0.981560i	\(0.561223\pi\)
\(37\)	6.74053	−	6.74053i	1.10814	−	1.10814i	0.114740	−	0.993396i	\(-0.463396\pi\)
							0.993396	−	0.114740i	\(-0.0366036\pi\)
\(41\)	−1.39492	+	2.41607i	−0.217850	+	0.377327i	−0.954150	−	0.299328i	\(-0.903238\pi\)
							0.736301	+	0.676655i	\(0.236571\pi\)
\(43\)	1.89907	−	7.08744i	0.289606	−	1.08082i	−0.655801	−	0.754934i	\(-0.727669\pi\)
							0.945407	−	0.325891i	\(-0.105664\pi\)
\(47\)	0.307120	+	0.531947i	0.0447980	+	0.0775924i	0.887555	−	0.460702i	\(-0.152402\pi\)
							−0.842757	+	0.538294i	\(0.819069\pi\)
\(53\)	2.68523	+	2.68523i	0.368844	+	0.368844i	0.867056	−	0.498211i	\(-0.166010\pi\)
							−0.498211	+	0.867056i	\(0.666010\pi\)
\(59\)	−0.00225603	−	0.00841962i	−0.000293710	−	0.00109614i	0.965779	−	0.259367i	\(-0.0835139\pi\)
							−0.966073	+	0.258271i	\(0.916847\pi\)
\(61\)	2.72542	−	10.1714i	0.348955	−	1.30232i	−0.538969	−	0.842326i	\(-0.681186\pi\)
							0.887924	−	0.459991i	\(-0.152147\pi\)
\(67\)	2.35300	+	8.78151i	0.287464	+	1.07283i	0.947020	+	0.321176i	\(0.104078\pi\)
							−0.659555	+	0.751656i	\(0.729255\pi\)
\(71\)		−	15.9645i		−	1.89463i	−0.320297	−	0.947317i	\(-0.603783\pi\)
							0.320297	−	0.947317i	\(-0.396217\pi\)
\(73\)			8.17785i			0.957145i	0.878048	+	0.478572i	\(0.158846\pi\)
							−0.878048	+	0.478572i	\(0.841154\pi\)
\(79\)	−7.67035	+	4.42848i	−0.862982	+	0.498243i	−0.865010	−	0.501755i	\(-0.832688\pi\)
							0.00202794	+	0.999998i	\(0.499354\pi\)
\(83\)	−0.353727	+	1.32013i	−0.0388266	+	0.144903i	−0.982618	−	0.185638i	\(-0.940565\pi\)
							0.943791	+	0.330541i	\(0.107231\pi\)
\(89\)	15.7852			1.67323			0.836613	−	0.547794i	\(-0.184532\pi\)
							0.836613	+	0.547794i	\(0.184532\pi\)
\(97\)	−4.62075	−	8.00338i	−0.469166	−	0.812620i	0.530212	−	0.847865i	\(-0.322112\pi\)
							−0.999379	+	0.0352448i	\(0.988779\pi\)

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.131.21	yes	88
3.2	odd	2		432.2.v.a.179.2		88
4.3	odd	2		576.2.y.a.239.12		88
9.2	odd	6	inner	144.2.u.a.83.15	yes	88
9.7	even	3		432.2.v.a.35.8		88
12.11	even	2		1728.2.z.a.1583.11		88
16.5	even	4		576.2.y.a.527.2		88
16.11	odd	4	inner	144.2.u.a.59.15	yes	88
36.7	odd	6		1728.2.z.a.1007.11		88
36.11	even	6		576.2.y.a.47.2		88
48.5	odd	4		1728.2.z.a.719.11		88
48.11	even	4		432.2.v.a.395.8		88
144.11	even	12	inner	144.2.u.a.11.21	✓	88
144.43	odd	12		432.2.v.a.251.2		88
144.101	odd	12		576.2.y.a.335.12		88
144.133	even	12		1728.2.z.a.143.11		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.21	✓	88	144.11	even	12	inner
144.2.u.a.59.15	yes	88	16.11	odd	4	inner
144.2.u.a.83.15	yes	88	9.2	odd	6	inner
144.2.u.a.131.21	yes	88	1.1	even	1	trivial
432.2.v.a.35.8		88	9.7	even	3
432.2.v.a.179.2		88	3.2	odd	2
432.2.v.a.251.2		88	144.43	odd	12
432.2.v.a.395.8		88	48.11	even	4
576.2.y.a.47.2		88	36.11	even	6
576.2.y.a.239.12		88	4.3	odd	2
576.2.y.a.335.12		88	144.101	odd	12
576.2.y.a.527.2		88	16.5	even	4
1728.2.z.a.143.11		88	144.133	even	12
1728.2.z.a.719.11		88	48.5	odd	4
1728.2.z.a.1007.11		88	36.7	odd	6
1728.2.z.a.1583.11		88	12.11	even	2