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Properties

Label 144.2.u.a.11.9
Level $144$
Weight $2$
Character 144.11
Analytic conductor $1.150$
Analytic rank $0$
Dimension $88$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,2,Mod(11,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
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 11.9
Character \(\chi\) \(=\) 144.11
Dual form 144.2.u.a.131.9

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.665270 - 1.24796i) q^{2} +(0.117756 + 1.72804i) q^{3} +(-1.11483 + 1.66047i) q^{4} +(-3.58858 - 0.961558i) q^{5} +(2.07820 - 1.29657i) q^{6} +(-1.29216 + 2.23809i) q^{7} +(2.81387 + 0.286608i) q^{8} +(-2.97227 + 0.406975i) q^{9} +(1.18739 + 5.11812i) q^{10} +(-2.02011 + 0.541286i) q^{11} +(-3.00064 - 1.73095i) q^{12} +(-0.998074 - 0.267433i) q^{13} +(3.65269 + 0.123637i) q^{14} +(1.23904 - 6.31446i) q^{15} +(-1.51431 - 3.70228i) q^{16} +4.13788i q^{17} +(2.48525 + 3.43854i) q^{18} +(2.49279 + 2.49279i) q^{19} +(5.59730 - 4.88675i) q^{20} +(-4.01968 - 1.96936i) q^{21} +(2.01942 + 2.16092i) q^{22} +(7.01915 - 4.05251i) q^{23} +(-0.163922 + 4.89624i) q^{24} +(7.62322 + 4.40127i) q^{25} +(0.330242 + 1.42348i) q^{26} +(-1.05327 - 5.08828i) q^{27} +(-2.27574 - 4.64068i) q^{28} +(-7.02439 + 1.88218i) q^{29} +(-8.70451 + 2.65455i) q^{30} +(-2.03331 + 1.17393i) q^{31} +(-3.61289 + 4.35282i) q^{32} +(-1.17324 - 3.42709i) q^{33} +(5.16393 - 2.75281i) q^{34} +(6.78909 - 6.78909i) q^{35} +(2.63781 - 5.38906i) q^{36} +(-4.75590 - 4.75590i) q^{37} +(1.45254 - 4.76930i) q^{38} +(0.344607 - 1.75621i) q^{39} +(-9.82221 - 3.73422i) q^{40} +(0.636217 + 1.10196i) q^{41} +(0.216477 + 6.32657i) q^{42} +(0.389880 + 1.45505i) q^{43} +(1.35329 - 3.95776i) q^{44} +(11.0576 + 1.39754i) q^{45} +(-9.72701 - 6.06363i) q^{46} +(-3.60814 + 6.24948i) q^{47} +(6.21938 - 3.05275i) q^{48} +(0.160635 + 0.278228i) q^{49} +(0.421123 - 12.4415i) q^{50} +(-7.15044 + 0.487260i) q^{51} +(1.55675 - 1.35913i) q^{52} +(0.546616 - 0.546616i) q^{53} +(-5.64928 + 4.69953i) q^{54} +7.76980 q^{55} +(-4.27743 + 5.92735i) q^{56} +(-4.01411 + 4.60119i) q^{57} +(7.02201 + 7.51403i) q^{58} +(-2.14469 + 8.00411i) q^{59} +(9.10364 + 9.09693i) q^{60} +(1.81511 + 6.77407i) q^{61} +(2.81773 + 1.75652i) q^{62} +(2.92980 - 7.17808i) q^{63} +(7.83571 + 1.61296i) q^{64} +(3.32452 + 1.91941i) q^{65} +(-3.49636 + 3.74411i) q^{66} +(0.751980 - 2.80643i) q^{67} +(-6.87082 - 4.61304i) q^{68} +(7.82945 + 11.6522i) q^{69} +(-12.9891 - 3.95596i) q^{70} +6.59449i q^{71} +(-8.48021 + 0.293297i) q^{72} -8.78699i q^{73} +(-2.77123 + 9.09915i) q^{74} +(-6.70790 + 13.6915i) q^{75} +(-6.91824 + 1.36016i) q^{76} +(1.39886 - 5.22061i) q^{77} +(-2.42094 + 0.738296i) q^{78} +(5.64940 + 3.26168i) q^{79} +(1.87426 + 14.7420i) q^{80} +(8.66874 - 2.41928i) q^{81} +(0.951950 - 1.52708i) q^{82} +(2.06791 + 7.71754i) q^{83} +(7.75132 - 4.47904i) q^{84} +(3.97882 - 14.8491i) q^{85} +(1.55648 - 1.45456i) q^{86} +(-4.07965 - 11.9168i) q^{87} +(-5.83945 + 0.944128i) q^{88} -14.1938 q^{89} +(-5.61218 - 14.7292i) q^{90} +(1.88821 - 1.88821i) q^{91} +(-1.09610 + 16.1729i) q^{92} +(-2.26805 - 3.37542i) q^{93} +(10.1995 + 0.345234i) q^{94} +(-6.54863 - 11.3426i) q^{95} +(-7.94730 - 5.73066i) q^{96} +(6.54551 - 11.3372i) q^{97} +(0.240353 - 0.385564i) q^{98} +(5.78401 - 2.43098i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7} - 8 q^{10} - 6 q^{11} - 16 q^{12} - 2 q^{13} - 6 q^{14} - 2 q^{16} - 10 q^{18} - 8 q^{19} - 48 q^{20} + 2 q^{21} - 2 q^{22} - 12 q^{23} - 16 q^{27}+ \cdots + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.665270 − 1.24796i −0.470417 − 0.882444i
\(3\) 0.117756 + 1.72804i 0.0679864 + 0.997686i
\(4\) −1.11483 + 1.66047i −0.557415 + 0.830234i
\(5\) −3.58858 − 0.961558i −1.60486 − 0.430022i −0.658357 − 0.752706i \(-0.728748\pi\)
−0.946507 + 0.322684i \(0.895415\pi\)
\(6\) 2.07820 − 1.29657i 0.848420 − 0.529323i
\(7\) −1.29216 + 2.23809i −0.488391 + 0.845919i −0.999911 − 0.0133531i \(-0.995749\pi\)
0.511520 + 0.859272i \(0.329083\pi\)
\(8\) 2.81387 + 0.286608i 0.994853 + 0.101331i
\(9\) −2.97227 + 0.406975i −0.990756 + 0.135658i
\(10\) 1.18739 + 5.11812i 0.375485 + 1.61849i
\(11\) −2.02011 + 0.541286i −0.609085 + 0.163204i −0.550161 − 0.835059i \(-0.685434\pi\)
−0.0589240 + 0.998262i \(0.518767\pi\)
\(12\) −3.00064 − 1.73095i −0.866210 − 0.499681i
\(13\) −0.998074 − 0.267433i −0.276816 − 0.0741726i 0.117740 − 0.993044i \(-0.462435\pi\)
−0.394556 + 0.918872i \(0.629102\pi\)
\(14\) 3.65269 + 0.123637i 0.976224 + 0.0330433i
\(15\) 1.23904 − 6.31446i 0.319918 − 1.63039i
\(16\) −1.51431 − 3.70228i −0.378577 − 0.925570i
\(17\) 4.13788i 1.00358i 0.864988 + 0.501792i \(0.167326\pi\)
−0.864988 + 0.501792i \(0.832674\pi\)
\(18\) 2.48525 + 3.43854i 0.585779 + 0.810471i
\(19\) 2.49279 + 2.49279i 0.571886 + 0.571886i 0.932655 − 0.360769i \(-0.117486\pi\)
−0.360769 + 0.932655i \(0.617486\pi\)
\(20\) 5.59730 − 4.88675i 1.25159 − 1.09271i
\(21\) −4.01968 − 1.96936i −0.877165 − 0.429750i
\(22\) 2.01942 + 2.16092i 0.430542 + 0.460709i
\(23\) 7.01915 − 4.05251i 1.46359 − 0.845006i 0.464418 − 0.885616i \(-0.346264\pi\)
0.999175 + 0.0406102i \(0.0129302\pi\)
\(24\) −0.163922 + 4.89624i −0.0334604 + 0.999440i
\(25\) 7.62322 + 4.40127i 1.52464 + 0.880253i
\(26\) 0.330242 + 1.42348i 0.0647658 + 0.279167i
\(27\) −1.05327 − 5.08828i −0.202702 − 0.979240i
\(28\) −2.27574 − 4.64068i −0.430074 − 0.877007i
\(29\) −7.02439 + 1.88218i −1.30440 + 0.349512i −0.843111 − 0.537740i \(-0.819278\pi\)
−0.461286 + 0.887252i \(0.652612\pi\)
\(30\) −8.70451 + 2.65455i −1.58922 + 0.484652i
\(31\) −2.03331 + 1.17393i −0.365194 + 0.210845i −0.671357 − 0.741134i \(-0.734288\pi\)
0.306163 + 0.951979i \(0.400955\pi\)
\(32\) −3.61289 + 4.35282i −0.638675 + 0.769477i
\(33\) −1.17324 − 3.42709i −0.204236 − 0.596580i
\(34\) 5.16393 − 2.75281i 0.885607 − 0.472103i
\(35\) 6.78909 − 6.78909i 1.14757 − 1.14757i
\(36\) 2.63781 − 5.38906i 0.439634 − 0.898177i
\(37\) −4.75590 − 4.75590i −0.781865 − 0.781865i 0.198280 − 0.980145i \(-0.436464\pi\)
−0.980145 + 0.198280i \(0.936464\pi\)
\(38\) 1.45254 − 4.76930i 0.235632 − 0.773682i
\(39\) 0.344607 − 1.75621i 0.0551813 − 0.281218i
\(40\) −9.82221 − 3.73422i −1.55303 − 0.590432i
\(41\) 0.636217 + 1.10196i 0.0993604 + 0.172097i 0.911420 − 0.411477i \(-0.134987\pi\)
−0.812060 + 0.583574i \(0.801654\pi\)
\(42\) 0.216477 + 6.32657i 0.0334031 + 0.976211i
\(43\) 0.389880 + 1.45505i 0.0594562 + 0.221894i 0.989261 − 0.146159i \(-0.0466912\pi\)
−0.929805 + 0.368053i \(0.880025\pi\)
\(44\) 1.35329 − 3.95776i 0.204016 − 0.596655i
\(45\) 11.0576 + 1.39754i 1.64836 + 0.208334i
\(46\) −9.72701 − 6.06363i −1.43417 − 0.894034i
\(47\) −3.60814 + 6.24948i −0.526301 + 0.911580i 0.473230 + 0.880939i \(0.343088\pi\)
−0.999530 + 0.0306407i \(0.990245\pi\)
\(48\) 6.21938 − 3.05275i 0.897690 − 0.440627i
\(49\) 0.160635 + 0.278228i 0.0229479 + 0.0397469i
\(50\) 0.421123 − 12.4415i 0.0595557 − 1.75950i
\(51\) −7.15044 + 0.487260i −1.00126 + 0.0682301i
\(52\) 1.55675 − 1.35913i 0.215882 − 0.188477i
\(53\) 0.546616 − 0.546616i 0.0750835 − 0.0750835i −0.668568 − 0.743651i \(-0.733092\pi\)
0.743651 + 0.668568i \(0.233092\pi\)
\(54\) −5.64928 + 4.69953i −0.768770 + 0.639525i
\(55\) 7.76980 1.04768
\(56\) −4.27743 + 5.92735i −0.571596 + 0.792075i
\(57\) −4.01411 + 4.60119i −0.531682 + 0.609443i
\(58\) 7.02201 + 7.51403i 0.922036 + 0.986641i
\(59\) −2.14469 + 8.00411i −0.279215 + 1.04205i 0.673748 + 0.738961i \(0.264683\pi\)
−0.952963 + 0.303085i \(0.901983\pi\)
\(60\) 9.10364 + 9.09693i 1.17527 + 1.17441i
\(61\) 1.81511 + 6.77407i 0.232401 + 0.867331i 0.979303 + 0.202398i \(0.0648734\pi\)
−0.746903 + 0.664933i \(0.768460\pi\)
\(62\) 2.81773 + 1.75652i 0.357852 + 0.223078i
\(63\) 2.92980 − 7.17808i 0.369121 − 0.904353i
\(64\) 7.83571 + 1.61296i 0.979464 + 0.201620i
\(65\) 3.32452 + 1.91941i 0.412356 + 0.238074i
\(66\) −3.49636 + 3.74411i −0.430372 + 0.460868i
\(67\) 0.751980 − 2.80643i 0.0918690 − 0.342860i −0.904657 − 0.426140i \(-0.859873\pi\)
0.996526 + 0.0832804i \(0.0265397\pi\)
\(68\) −6.87082 − 4.61304i −0.833209 − 0.559413i
\(69\) 7.82945 + 11.6522i 0.942555 + 1.40276i
\(70\) −12.9891 − 3.95596i −1.55250 − 0.472828i
\(71\) 6.59449i 0.782622i 0.920258 + 0.391311i \(0.127978\pi\)
−0.920258 + 0.391311i \(0.872022\pi\)
\(72\) −8.48021 + 0.293297i −0.999402 + 0.0345653i
\(73\) − 8.78699i − 1.02844i −0.857658 − 0.514220i \(-0.828082\pi\)
0.857658 − 0.514220i \(-0.171918\pi\)
\(74\) −2.77123 + 9.09915i −0.322149 + 1.05776i
\(75\) −6.70790 + 13.6915i −0.774562 + 1.58096i
\(76\) −6.91824 + 1.36016i −0.793577 + 0.156021i
\(77\) 1.39886 − 5.22061i 0.159415 − 0.594944i
\(78\) −2.42094 + 0.738296i −0.274118 + 0.0835955i
\(79\) 5.64940 + 3.26168i 0.635607 + 0.366968i 0.782920 − 0.622122i \(-0.213729\pi\)
−0.147313 + 0.989090i \(0.547063\pi\)
\(80\) 1.87426 + 14.7420i 0.209549 + 1.64821i
\(81\) 8.66874 − 2.41928i 0.963194 − 0.268808i
\(82\) 0.951950 − 1.52708i 0.105125 − 0.168637i
\(83\) 2.06791 + 7.71754i 0.226982 + 0.847110i 0.981601 + 0.190946i \(0.0611554\pi\)
−0.754618 + 0.656164i \(0.772178\pi\)
\(84\) 7.75132 − 4.47904i 0.845739 − 0.488703i
\(85\) 3.97882 − 14.8491i 0.431563 − 1.61062i
\(86\) 1.55648 − 1.45456i 0.167839 − 0.156849i
\(87\) −4.07965 − 11.9168i −0.437385 − 1.27762i
\(88\) −5.83945 + 0.944128i −0.622487 + 0.100644i
\(89\) −14.1938 −1.50454 −0.752272 − 0.658852i \(-0.771042\pi\)
−0.752272 + 0.658852i \(0.771042\pi\)
\(90\) −5.61218 − 14.7292i −0.591576 − 1.55259i
\(91\) 1.88821 − 1.88821i 0.197939 − 0.197939i
\(92\) −1.09610 + 16.1729i −0.114277 + 1.68614i
\(93\) −2.26805 − 3.37542i −0.235185 − 0.350015i
\(94\) 10.1995 + 0.345234i 1.05200 + 0.0356082i
\(95\) −6.54863 − 11.3426i −0.671875 − 1.16372i
\(96\) −7.94730 − 5.73066i −0.811118 − 0.584883i
\(97\) 6.54551 − 11.3372i 0.664596 − 1.15111i −0.314799 − 0.949158i \(-0.601937\pi\)
0.979395 − 0.201955i \(-0.0647295\pi\)
\(98\) 0.240353 − 0.385564i 0.0242793 − 0.0389478i
\(99\) 5.78401 − 2.43098i 0.581314 − 0.244322i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 144.2.u.a.11.9 ✓ 88
3.2 odd 2 432.2.v.a.251.14 88
4.3 odd 2 576.2.y.a.335.10 88
9.4 even 3 432.2.v.a.395.21 88
9.5 odd 6 inner 144.2.u.a.59.2 yes 88
12.11 even 2 1728.2.z.a.143.21 88
16.3 odd 4 inner 144.2.u.a.83.2 yes 88
16.13 even 4 576.2.y.a.47.1 88
36.23 even 6 576.2.y.a.527.1 88
36.31 odd 6 1728.2.z.a.719.21 88
48.29 odd 4 1728.2.z.a.1007.21 88
48.35 even 4 432.2.v.a.35.21 88
144.13 even 12 1728.2.z.a.1583.21 88
144.67 odd 12 432.2.v.a.179.14 88
144.77 odd 12 576.2.y.a.239.10 88
144.131 even 12 inner 144.2.u.a.131.9 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.u.a.11.9 ✓ 88 1.1 even 1 trivial
144.2.u.a.59.2 yes 88 9.5 odd 6 inner
144.2.u.a.83.2 yes 88 16.3 odd 4 inner
144.2.u.a.131.9 yes 88 144.131 even 12 inner
432.2.v.a.35.21 88 48.35 even 4
432.2.v.a.179.14 88 144.67 odd 12
432.2.v.a.251.14 88 3.2 odd 2
432.2.v.a.395.21 88 9.4 even 3
576.2.y.a.47.1 88 16.13 even 4
576.2.y.a.239.10 88 144.77 odd 12
576.2.y.a.335.10 88 4.3 odd 2
576.2.y.a.527.1 88 36.23 even 6
1728.2.z.a.143.21 88 12.11 even 2
1728.2.z.a.719.21 88 36.31 odd 6
1728.2.z.a.1007.21 88 48.29 odd 4
1728.2.z.a.1583.21 88 144.13 even 12