Embedded newform 144.2.u.a.11.3

• Label: 144.2.u.a.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	144.11
Dual form			144.2.u.a.131.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20521 + 0.739910i) q^{2} +(-0.679468 - 1.59321i) q^{3} +(0.905065 - 1.78350i) q^{4} +(-2.39818 - 0.642590i) q^{5} +(1.99774 + 1.41741i) q^{6} +(-1.93190 + 3.34616i) q^{7} +(0.228833 + 2.81916i) q^{8} +(-2.07665 + 2.16507i) q^{9} +(3.36577 - 0.999982i) q^{10} +(-4.01810 + 1.07665i) q^{11} +(-3.45645 - 0.230133i) q^{12} +(3.17330 + 0.850284i) q^{13} +(-0.147504 - 5.46226i) q^{14} +(0.605703 + 4.25743i) q^{15} +(-2.36171 - 3.22836i) q^{16} -1.33161i q^{17} +(0.900838 - 4.14590i) q^{18} +(-6.09021 - 6.09021i) q^{19} +(-3.31657 + 3.69556i) q^{20} +(6.64380 + 0.804327i) q^{21} +(4.04603 - 4.27062i) q^{22} +(0.521462 - 0.301066i) q^{23} +(4.33603 - 2.28010i) q^{24} +(1.00822 + 0.582095i) q^{25} +(-4.45363 + 1.32319i) q^{26} +(4.86043 + 1.83744i) q^{27} +(4.21936 + 6.47403i) q^{28} +(0.272752 - 0.0730837i) q^{29} +(-3.88012 - 4.68293i) q^{30} +(-5.84901 + 3.37693i) q^{31} +(5.23506 + 2.14340i) q^{32} +(4.44549 + 5.67013i) q^{33} +(0.985273 + 1.60487i) q^{34} +(6.78326 - 6.78326i) q^{35} +(1.98190 + 5.66322i) q^{36} +(0.00346351 + 0.00346351i) q^{37} +(11.8462 + 2.83377i) q^{38} +(-0.801475 - 5.63348i) q^{39} +(1.26278 - 6.90789i) q^{40} +(-0.614318 - 1.06403i) q^{41} +(-8.60231 + 3.94643i) q^{42} +(-0.151042 - 0.563698i) q^{43} +(-1.71645 + 8.14070i) q^{44} +(6.37143 - 3.85780i) q^{45} +(-0.405710 + 0.748684i) q^{46} +(1.24972 - 2.16458i) q^{47} +(-3.53875 + 5.95628i) q^{48} +(-3.96450 - 6.86672i) q^{49} +(-1.64581 + 0.0444440i) q^{50} +(-2.12154 + 0.904787i) q^{51} +(4.38852 - 4.89001i) q^{52} +(-3.24883 + 3.24883i) q^{53} +(-7.21739 + 1.38178i) q^{54} +10.3280 q^{55} +(-9.87541 - 4.68063i) q^{56} +(-5.56489 + 13.8411i) q^{57} +(-0.274648 + 0.289893i) q^{58} +(1.44745 - 5.40194i) q^{59} +(8.14131 + 2.77298i) q^{60} +(0.528668 + 1.97301i) q^{61} +(4.55067 - 8.39766i) q^{62} +(-3.23278 - 11.1315i) q^{63} +(-7.89527 + 1.29023i) q^{64} +(-7.06377 - 4.07827i) q^{65} +(-9.55315 - 3.54444i) q^{66} +(-2.65385 + 9.90429i) q^{67} +(-2.37492 - 1.20519i) q^{68} +(-0.833979 - 0.626235i) q^{69} +(-3.15625 + 13.1943i) q^{70} +7.85052i q^{71} +(-6.57888 - 5.35895i) q^{72} -7.41855i q^{73} +(-0.00673695 - 0.00161157i) q^{74} +(0.242349 - 2.00182i) q^{75} +(-16.3739 + 5.34982i) q^{76} +(4.15995 - 15.5252i) q^{77} +(5.13422 + 6.19652i) q^{78} +(-0.839597 - 0.484742i) q^{79} +(3.58930 + 9.25980i) q^{80} +(-0.375072 - 8.99218i) q^{81} +(1.52767 + 0.827840i) q^{82} +(-0.171300 - 0.639302i) q^{83} +(7.44759 - 11.1212i) q^{84} +(-0.855680 + 3.19344i) q^{85} +(0.599124 + 0.567617i) q^{86} +(-0.301764 - 0.384894i) q^{87} +(-3.95471 - 11.0813i) q^{88} +11.2223 q^{89} +(-4.82449 + 9.36375i) q^{90} +(-8.97570 + 8.97570i) q^{91} +(-0.0649933 - 1.20251i) q^{92} +(9.35438 + 7.02420i) q^{93} +(0.0954184 + 3.53345i) q^{94} +(10.6919 + 18.5189i) q^{95} +(-0.142170 - 9.79693i) q^{96} +(-3.24710 + 5.62414i) q^{97} +(9.85882 + 5.34247i) q^{98} +(6.01316 - 10.9353i) 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20521	+	0.739910i	−0.852213	+	0.523196i
							\(3\)	−0.679468	−	1.59321i	−0.392291	−	0.919841i
\(5\)	−2.39818	−	0.642590i	−1.07250	−	0.287375i								−0.320979	−	0.947086i	\(-0.604012\pi\)
							−0.751519	+	0.659711i	\(0.770679\pi\)
\(7\)	−1.93190	+	3.34616i	−0.730191	+	1.26473i	0.226610	+	0.973985i	\(0.427236\pi\)
							−0.956801	+	0.290742i	\(0.906098\pi\)
\(11\)	−4.01810	+	1.07665i	−1.21150	+	0.324621i	−0.807351	−	0.590071i	\(-0.799100\pi\)
							−0.404151	+	0.914692i	\(0.632433\pi\)
\(13\)	3.17330	+	0.850284i	0.880116	+	0.235826i	0.670457	−	0.741948i	\(-0.266098\pi\)
							0.209659	+	0.977775i	\(0.432765\pi\)
\(17\)		−	1.33161i		−	0.322963i	−0.986876	−	0.161482i	\(-0.948373\pi\)
							0.986876	−	0.161482i	\(-0.0516272\pi\)
\(19\)	−6.09021	−	6.09021i	−1.39719	−	1.39719i	−0.807981	−	0.589208i	\(-0.799440\pi\)
							−0.589208	−	0.807981i	\(-0.700560\pi\)
\(23\)	0.521462	−	0.301066i	0.108732	−	0.0627767i	−0.444648	−	0.895706i	\(-0.646671\pi\)
							0.553380	+	0.832929i	\(0.313338\pi\)
\(29\)	0.272752	−	0.0730837i	0.0506488	−	0.0135713i	−0.233406	−	0.972379i	\(-0.574987\pi\)
							0.284054	+	0.958808i	\(0.408320\pi\)
\(31\)	−5.84901	+	3.37693i	−1.05051	+	0.606514i	−0.922793	−	0.385296i	\(-0.874099\pi\)
							−0.127721	+	0.991810i	\(0.540766\pi\)
\(37\)	0.00346351	+	0.00346351i	0.000569398	+	0.000569398i	0.707391	−	0.706822i	\(-0.249872\pi\)
							−0.706822	+	0.707391i	\(0.749872\pi\)
\(41\)	−0.614318	−	1.06403i	−0.0959403	−	0.166174i	0.814060	−	0.580780i	\(-0.197252\pi\)
							−0.910001	+	0.414607i	\(0.863919\pi\)
\(43\)	−0.151042	−	0.563698i	−0.0230337	−	0.0859631i	0.953452	−	0.301544i	\(-0.0975021\pi\)
							−0.976486	+	0.215581i	\(0.930835\pi\)
\(47\)	1.24972	−	2.16458i	0.182290	−	0.315736i	−0.760370	−	0.649490i	\(-0.774982\pi\)
							0.942660	+	0.333754i	\(0.108316\pi\)
\(53\)	−3.24883	+	3.24883i	−0.446262	+	0.446262i	−0.894110	−	0.447848i	\(-0.852191\pi\)
							0.447848	+	0.894110i	\(0.352191\pi\)
\(59\)	1.44745	−	5.40194i	0.188441	−	0.703273i	−0.805426	−	0.592696i	\(-0.798064\pi\)
							0.993868	−	0.110577i	\(-0.0352698\pi\)
\(61\)	0.528668	+	1.97301i	0.0676890	+	0.252619i	0.991476	−	0.130287i	\(-0.0415900\pi\)
							−0.923787	+	0.382906i	\(0.874923\pi\)
\(67\)	−2.65385	+	9.90429i	−0.324219	+	1.21000i	0.590876	+	0.806763i	\(0.298782\pi\)
							−0.915095	+	0.403239i	\(0.867884\pi\)
\(71\)			7.85052i			0.931685i	0.884868	+	0.465843i	\(0.154249\pi\)
							−0.884868	+	0.465843i	\(0.845751\pi\)
\(73\)		−	7.41855i		−	0.868275i	−0.900847	−	0.434138i	\(-0.857053\pi\)
							0.900847	−	0.434138i	\(-0.142947\pi\)
\(79\)	−0.839597	−	0.484742i	−0.0944620	−	0.0545377i	0.452025	−	0.892005i	\(-0.350702\pi\)
							−0.546487	+	0.837468i	\(0.684035\pi\)
\(83\)	−0.171300	−	0.639302i	−0.0188027	−	0.0701725i	0.955887	−	0.293735i	\(-0.0948982\pi\)
							−0.974690	+	0.223562i	\(0.928231\pi\)
\(89\)	11.2223			1.18956			0.594781	−	0.803887i	\(-0.297239\pi\)
							0.594781	+	0.803887i	\(0.297239\pi\)
\(97\)	−3.24710	+	5.62414i	−0.329693	+	0.571045i	−0.982451	−	0.186521i	\(-0.940279\pi\)
							0.652758	+	0.757567i	\(0.273612\pi\)

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.11.3	✓	88
3.2	odd	2		432.2.v.a.251.20		88
4.3	odd	2		576.2.y.a.335.14		88
9.4	even	3		432.2.v.a.395.11		88
9.5	odd	6	inner	144.2.u.a.59.12	yes	88
12.11	even	2		1728.2.z.a.143.17		88
16.3	odd	4	inner	144.2.u.a.83.12	yes	88
16.13	even	4		576.2.y.a.47.19		88
36.23	even	6		576.2.y.a.527.19		88
36.31	odd	6		1728.2.z.a.719.17		88
48.29	odd	4		1728.2.z.a.1007.17		88
48.35	even	4		432.2.v.a.35.11		88
144.13	even	12		1728.2.z.a.1583.17		88
144.67	odd	12		432.2.v.a.179.20		88
144.77	odd	12		576.2.y.a.239.14		88
144.131	even	12	inner	144.2.u.a.131.3	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.3	✓	88	1.1	even	1	trivial
144.2.u.a.59.12	yes	88	9.5	odd	6	inner
144.2.u.a.83.12	yes	88	16.3	odd	4	inner
144.2.u.a.131.3	yes	88	144.131	even	12	inner
432.2.v.a.35.11		88	48.35	even	4
432.2.v.a.179.20		88	144.67	odd	12
432.2.v.a.251.20		88	3.2	odd	2
432.2.v.a.395.11		88	9.4	even	3
576.2.y.a.47.19		88	16.13	even	4
576.2.y.a.239.14		88	144.77	odd	12
576.2.y.a.335.14		88	4.3	odd	2
576.2.y.a.527.19		88	36.23	even	6
1728.2.z.a.143.17		88	12.11	even	2
1728.2.z.a.719.17		88	36.31	odd	6
1728.2.z.a.1007.17		88	48.29	odd	4
1728.2.z.a.1583.17		88	144.13	even	12