Embedded newform 144.2.x.e.133.11

• Label: 144.2.x.e.133.11
• Level: $144$
• Weight: $2$
• Character: 144.133
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			133.11
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.562928 + 1.29735i) q^{2} +(0.388965 + 1.68781i) q^{3} +(-1.36622 + 1.46063i) q^{4} +(0.226831 - 0.846545i) q^{5} +(-1.97072 + 1.45474i) q^{6} +(0.567074 + 0.327400i) q^{7} +(-2.66403 - 0.950239i) q^{8} +(-2.69741 + 1.31300i) q^{9} +(1.22595 - 0.182265i) q^{10} +(5.75313 - 1.54155i) q^{11} +(-2.99668 - 1.73779i) q^{12} +(-4.44367 - 1.19068i) q^{13} +(-0.105530 + 0.919995i) q^{14} +(1.51704 + 0.0535712i) q^{15} +(-0.266866 - 3.99109i) q^{16} +2.75816 q^{17} +(-3.22187 - 2.76036i) q^{18} +(-1.73499 + 1.73499i) q^{19} +(0.926585 + 1.48789i) q^{20} +(-0.332018 + 1.08446i) q^{21} +(5.23852 + 6.59603i) q^{22} +(3.50762 - 2.02512i) q^{23} +(0.567609 - 4.86599i) q^{24} +(3.66494 + 2.11595i) q^{25} +(-0.956743 - 6.43525i) q^{26} +(-3.26530 - 4.04201i) q^{27} +(-1.25296 + 0.380982i) q^{28} +(-0.662669 - 2.47312i) q^{29} +(0.784483 + 1.99828i) q^{30} +(2.08801 + 3.61654i) q^{31} +(5.02760 - 2.59291i) q^{32} +(4.83961 + 9.11059i) q^{33} +(1.55265 + 3.57830i) q^{34} +(0.405789 - 0.405789i) q^{35} +(1.76746 - 5.73377i) q^{36} +(-4.30563 - 4.30563i) q^{37} +(-3.22756 - 1.27421i) q^{38} +(0.281205 - 7.96320i) q^{39} +(-1.40870 + 2.03968i) q^{40} +(-6.15806 + 3.55536i) q^{41} +(-1.59382 + 0.179731i) q^{42} +(-0.841515 + 0.225483i) q^{43} +(-5.60844 + 10.5093i) q^{44} +(0.499657 + 2.58131i) q^{45} +(4.60183 + 3.41060i) q^{46} +(4.65521 - 8.06305i) q^{47} +(6.63240 - 2.00281i) q^{48} +(-3.28562 - 5.69086i) q^{49} +(-0.682031 + 5.94583i) q^{50} +(1.07283 + 4.65526i) q^{51} +(7.81018 - 4.86381i) q^{52} +(-7.64584 - 7.64584i) q^{53} +(3.40577 - 6.51159i) q^{54} -5.21996i q^{55} +(-1.19959 - 1.41106i) q^{56} +(-3.60318 - 2.25348i) q^{57} +(2.83546 - 2.25190i) q^{58} +(-1.83103 + 6.83351i) q^{59} +(-2.15086 + 2.14264i) q^{60} +(1.01219 + 3.77755i) q^{61} +(-3.51651 + 4.74473i) q^{62} +(-1.95951 - 0.138565i) q^{63} +(6.19409 + 5.06293i) q^{64} +(-2.01592 + 3.49168i) q^{65} +(-9.09525 + 11.4073i) q^{66} +(-11.6330 - 3.11705i) q^{67} +(-3.76827 + 4.02865i) q^{68} +(4.78237 + 5.13249i) q^{69} +(0.754880 + 0.298020i) q^{70} +4.34835i q^{71} +(8.43364 - 0.934684i) q^{72} -0.656583i q^{73} +(3.16214 - 8.00966i) q^{74} +(-2.14580 + 7.00876i) q^{75} +(-0.163790 - 4.90455i) q^{76} +(3.76715 + 1.00941i) q^{77} +(10.4893 - 4.11789i) q^{78} +(-8.16172 + 14.1365i) q^{79} +(-3.43917 - 0.679389i) q^{80} +(5.55206 - 7.08340i) q^{81} +(-8.07908 - 5.98774i) q^{82} +(-1.43847 - 5.36845i) q^{83} +(-1.13038 - 1.96657i) q^{84} +(0.625638 - 2.33491i) q^{85} +(-0.766242 - 0.964806i) q^{86} +(3.91640 - 2.08042i) q^{87} +(-16.7913 - 1.36012i) q^{88} +5.11081i q^{89} +(-3.06759 + 2.10132i) q^{90} +(-2.13006 - 2.13006i) q^{91} +(-1.83424 + 7.89010i) q^{92} +(-5.29187 + 4.93087i) q^{93} +(13.0811 + 1.50050i) q^{94} +(1.07520 + 1.86230i) q^{95} +(6.33191 + 7.47709i) q^{96} +(3.05669 - 5.29434i) q^{97} +(5.53346 - 7.46613i) q^{98} +(-13.4945 + 11.7120i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	72 * q + 4 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^5 - 28 * q^6 - 8 * q^8 - 20 * q^10 - 2 * q^11 + 8 * q^12 - 16 * q^13 - 4 * q^14 - 20 * q^15 - 10 * q^16 - 16 * q^17 + 28 * q^19 + 12 * q^20 - 16 * q^21 - 8 * q^22 - 40 * q^24 - 4 * q^26 + 8 * q^27 - 16 * q^28 + 4 * q^29 + 18 * q^30 + 28 * q^31 - 46 * q^32 - 32 * q^33 - 14 * q^34 - 16 * q^35 + 14 * q^36 + 16 * q^37 + 2 * q^38 - 10 * q^40 + 26 * q^42 - 10 * q^43 + 60 * q^44 + 40 * q^45 + 20 * q^46 - 56 * q^47 + 2 * q^48 + 4 * q^49 - 36 * q^50 - 54 * q^51 + 6 * q^52 - 8 * q^53 + 92 * q^54 + 52 * q^56 - 14 * q^58 - 14 * q^59 + 18 * q^60 - 32 * q^61 + 16 * q^62 - 108 * q^63 - 44 * q^64 - 64 * q^65 + 26 * q^66 - 18 * q^67 + 16 * q^68 + 32 * q^69 + 14 * q^70 + 114 * q^72 + 38 * q^74 + 86 * q^75 + 10 * q^76 - 36 * q^77 + 16 * q^78 + 44 * q^79 + 144 * q^80 - 44 * q^81 - 88 * q^82 + 20 * q^83 - 58 * q^84 - 8 * q^85 + 76 * q^86 - 42 * q^88 - 80 * q^91 - 68 * q^92 - 4 * q^93 + 20 * q^94 + 48 * q^95 + 94 * q^96 + 40 * q^97 + 88 * q^98 + 28 * q^99

Character values

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

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.562928	+	1.29735i	0.398050	+	0.917364i
							\(3\)	0.388965	+	1.68781i	0.224569	+	0.974458i
\(5\)	0.226831	−	0.846545i	0.101442	−	0.378587i								−0.896475	−	0.443094i	\(-0.853881\pi\)
							0.997917	+	0.0645072i	\(0.0205475\pi\)
\(7\)	0.567074	+	0.327400i	0.214334	+	0.123746i	0.603324	−	0.797496i	\(-0.293843\pi\)
							−0.388990	+	0.921242i	\(0.627176\pi\)
\(11\)	5.75313	−	1.54155i	1.73463	−	0.464794i	0.753392	−	0.657572i	\(-0.228416\pi\)
							0.981242	+	0.192778i	\(0.0617498\pi\)
\(13\)	−4.44367	−	1.19068i	−1.23245	−	0.330234i	−0.416918	−	0.908944i	\(-0.636890\pi\)
							−0.815534	+	0.578710i	\(0.803556\pi\)
\(17\)	2.75816			0.668953			0.334477	−	0.942404i	\(-0.391441\pi\)
							0.334477	+	0.942404i	\(0.391441\pi\)
\(19\)	−1.73499	+	1.73499i	−0.398034	+	0.398034i	−0.877539	−	0.479505i	\(-0.840816\pi\)
							0.479505	+	0.877539i	\(0.340816\pi\)
\(23\)	3.50762	−	2.02512i	0.731389	−	0.422268i	−0.0875410	−	0.996161i	\(-0.527901\pi\)
							0.818930	+	0.573893i	\(0.194568\pi\)
\(29\)	−0.662669	−	2.47312i	−0.123055	−	0.459246i	0.876708	−	0.481022i	\(-0.159734\pi\)
							−0.999763	+	0.0217764i	\(0.993068\pi\)
\(31\)	2.08801	+	3.61654i	0.375018	+	0.649550i	0.990330	−	0.138734i	\(-0.0443033\pi\)
							−0.615312	+	0.788284i	\(0.710970\pi\)
\(37\)	−4.30563	−	4.30563i	−0.707841	−	0.707841i	0.258240	−	0.966081i	\(-0.416858\pi\)
							−0.966081	+	0.258240i	\(0.916858\pi\)
\(41\)	−6.15806	+	3.55536i	−0.961728	+	0.555254i	−0.896704	−	0.442630i	\(-0.854046\pi\)
							−0.0650233	+	0.997884i	\(0.520712\pi\)
\(43\)	−0.841515	+	0.225483i	−0.128330	+	0.0343859i	−0.322412	−	0.946599i	\(-0.604494\pi\)
							0.194082	+	0.980985i	\(0.437827\pi\)
\(47\)	4.65521	−	8.06305i	0.679032	−	1.17612i	−0.296241	−	0.955113i	\(-0.595733\pi\)
							0.975273	−	0.221004i	\(-0.0709334\pi\)
\(53\)	−7.64584	−	7.64584i	−1.05024	−	1.05024i	−0.998669	−	0.0515677i	\(-0.983578\pi\)
							−0.0515677	−	0.998669i	\(-0.516422\pi\)
\(59\)	−1.83103	+	6.83351i	−0.238380	+	0.889647i	0.738216	+	0.674565i	\(0.235669\pi\)
							−0.976596	+	0.215082i	\(0.930998\pi\)
\(61\)	1.01219	+	3.77755i	0.129598	+	0.483666i	0.999962	−	0.00874306i	\(-0.00278304\pi\)
							−0.870364	+	0.492409i	\(0.836116\pi\)
\(67\)	−11.6330	−	3.11705i	−1.42120	−	0.380809i	−0.535289	−	0.844669i	\(-0.679797\pi\)
							−0.885908	+	0.463860i	\(0.846464\pi\)
\(71\)			4.34835i			0.516054i	0.966138	+	0.258027i	\(0.0830723\pi\)
							−0.966138	+	0.258027i	\(0.916928\pi\)
\(73\)		−	0.656583i		−	0.0768472i	−0.999262	−	0.0384236i	\(-0.987766\pi\)
							0.999262	−	0.0384236i	\(-0.0122336\pi\)
\(79\)	−8.16172	+	14.1365i	−0.918266	+	1.59048i	−0.116217	+	0.993224i	\(0.537077\pi\)
							−0.802048	+	0.597259i	\(0.796256\pi\)
\(83\)	−1.43847	−	5.36845i	−0.157893	−	0.589264i	−0.998840	−	0.0481470i	\(-0.984668\pi\)
							0.840948	−	0.541117i	\(-0.181998\pi\)
\(89\)			5.11081i			0.541744i	0.962615	+	0.270872i	\(0.0873121\pi\)
							−0.962615	+	0.270872i	\(0.912688\pi\)
\(97\)	3.05669	−	5.29434i	0.310360	−	0.537559i	−0.668080	−	0.744089i	\(-0.732884\pi\)
							0.978440	+	0.206530i	\(0.0662171\pi\)

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.x.e.133.11	yes	72
3.2	odd	2		432.2.y.e.181.8		72
4.3	odd	2		576.2.bb.e.241.9		72
9.4	even	3	inner	144.2.x.e.85.1	yes	72
9.5	odd	6		432.2.y.e.37.18		72
12.11	even	2		1728.2.bc.e.1585.7		72
16.3	odd	4		576.2.bb.e.529.17		72
16.13	even	4	inner	144.2.x.e.61.1	yes	72
36.23	even	6		1728.2.bc.e.1009.12		72
36.31	odd	6		576.2.bb.e.49.17		72
48.29	odd	4		432.2.y.e.397.18		72
48.35	even	4		1728.2.bc.e.721.12		72
144.13	even	12	inner	144.2.x.e.13.11	✓	72
144.67	odd	12		576.2.bb.e.337.9		72
144.77	odd	12		432.2.y.e.253.8		72
144.131	even	12		1728.2.bc.e.145.7		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.11	✓	72	144.13	even	12	inner
144.2.x.e.61.1	yes	72	16.13	even	4	inner
144.2.x.e.85.1	yes	72	9.4	even	3	inner
144.2.x.e.133.11	yes	72	1.1	even	1	trivial
432.2.y.e.37.18		72	9.5	odd	6
432.2.y.e.181.8		72	3.2	odd	2
432.2.y.e.253.8		72	144.77	odd	12
432.2.y.e.397.18		72	48.29	odd	4
576.2.bb.e.49.17		72	36.31	odd	6
576.2.bb.e.241.9		72	4.3	odd	2
576.2.bb.e.337.9		72	144.67	odd	12
576.2.bb.e.529.17		72	16.3	odd	4
1728.2.bc.e.145.7		72	144.131	even	12
1728.2.bc.e.721.12		72	48.35	even	4
1728.2.bc.e.1009.12		72	36.23	even	6
1728.2.bc.e.1585.7		72	12.11	even	2