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Properties

Label 144.2.u.a.83.15
Level $144$
Weight $2$
Character 144.83
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 83.15
Character \(\chi\) \(=\) 144.83
Dual form 144.2.u.a.59.15

$q$-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+(0.417977 + 1.35103i) q^{2} +(1.68223 + 0.412428i) q^{3} +(-1.65059 + 1.12940i) q^{4} +(0.0776974 - 0.289971i) q^{5} +(0.145929 + 2.44514i) 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} +(-0.599457 - 2.23720i) q^{11} +(-3.24247 + 1.21917i) q^{12} +(-0.429571 + 1.60318i) q^{13} +(-1.03157 - 0.234541i) q^{14} +(0.250297 - 0.455753i) q^{15} +(1.44890 - 3.72836i) q^{16} -6.74518i q^{17} +(-0.762958 + 4.17348i) q^{18} +(0.621335 - 0.621335i) q^{19} +(0.199247 + 0.566375i) q^{20} +(-0.896375 + 0.935537i) q^{21} +(2.77198 - 1.74499i) q^{22} +(-6.06191 + 3.49985i) q^{23} +(-3.00242 - 3.87111i) q^{24} +(4.25208 + 2.45494i) q^{25} +(-2.34550 + 0.0897272i) q^{26} +(3.90212 + 3.43124i) q^{27} +(-0.114299 - 1.49172i) q^{28} +(-1.45829 - 5.44240i) q^{29} +(0.720357 + 0.147666i) q^{30} +(-3.13647 + 1.81084i) q^{31} +(5.64276 + 0.399142i) q^{32} +(-0.0857395 - 4.01073i) q^{33} +(9.11297 - 2.81933i) q^{34} +(0.158790 + 0.158790i) q^{35} +(-5.95741 + 0.713634i) q^{36} +(6.74053 - 6.74053i) q^{37} +(1.09915 + 0.579742i) q^{38} +(-1.38383 + 2.51975i) q^{39} +(-0.681911 + 0.505921i) q^{40} +(1.39492 + 2.41607i) q^{41} +(-1.63861 - 0.820001i) q^{42} +(-7.08744 + 1.89907i) q^{43} +(3.51616 + 3.01568i) q^{44} +(0.609023 - 0.663453i) q^{45} +(-7.26216 - 6.72700i) q^{46} +(-0.307120 + 0.531947i) q^{47} +(3.97506 - 5.67440i) q^{48} +(3.22021 + 5.57757i) q^{49} +(-1.53944 + 6.77082i) q^{50} +(2.78190 - 11.3470i) q^{51} +(-1.10159 - 3.13135i) q^{52} +(-2.68523 - 2.68523i) q^{53} +(-3.00473 + 6.70609i) q^{54} -0.695300 q^{55} +(1.96759 - 0.777926i) q^{56} +(1.30149 - 0.788974i) q^{57} +(6.74334 - 4.24499i) q^{58} +(-0.00841962 - 0.00225603i) q^{59} +(0.101591 + 1.03495i) q^{60} +(-10.1714 + 2.72542i) 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} +(5.38279 - 1.79223i) q^{66} +(-8.78151 - 2.35300i) q^{67} +(7.61803 + 11.1335i) q^{68} +(-11.6410 + 3.38745i) q^{69} +(-0.148160 + 0.280902i) q^{70} +15.9645i q^{71} +(-3.45420 - 7.75039i) q^{72} +8.17785i q^{73} +(11.9241 + 6.28930i) q^{74} +(6.14050 + 5.88346i) q^{75} +(-0.323832 + 1.72731i) q^{76} +(1.67353 + 0.448421i) q^{77} +(-3.98269 - 0.816410i) 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} +(-1.32013 + 0.353727i) q^{83} +(0.422950 - 2.55656i) q^{84} +(-1.95590 - 0.524083i) q^{85} +(-5.52810 - 8.78161i) q^{86} +(-0.208577 - 9.75681i) q^{87} +(-2.60461 + 6.01094i) q^{88} -15.7852 q^{89} +(1.15091 + 0.545504i) q^{90} +(-0.877914 - 0.877914i) q^{91} +(6.05300 - 12.6232i) q^{92} +(-6.02310 + 1.75268i) q^{93} +(-0.847048 - 0.192588i) q^{94} +(-0.131893 - 0.228445i) q^{95} +(9.32780 + 2.99868i) q^{96} +(-4.62075 + 8.00338i) q^{97} +(-6.18952 + 6.68192i) q^{98} +(1.50990 - 6.78233i) 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 \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.417977 + 1.35103i 0.295554 + 0.955326i
\(3\) 1.68223 + 0.412428i 0.971237 + 0.238116i
\(4\) −1.65059 + 1.12940i −0.825295 + 0.564701i
\(5\) 0.0776974 − 0.289971i 0.0347473 − 0.129679i −0.946373 − 0.323075i \(-0.895283\pi\)
0.981121 + 0.193396i \(0.0619502\pi\)
\(6\) 0.145929 + 2.44514i 0.0595753 + 0.998224i
\(7\) −0.374023 + 0.647827i −0.141367 + 0.244855i −0.928012 − 0.372551i \(-0.878483\pi\)
0.786644 + 0.617406i \(0.211817\pi\)
\(8\) −2.21577 − 1.75794i −0.783394 − 0.621526i
\(9\) 2.65981 + 1.38760i 0.886602 + 0.462533i
\(10\) 0.424236 − 0.0162292i 0.134155 − 0.00513211i
\(11\) −0.599457 − 2.23720i −0.180743 − 0.674542i −0.995502 − 0.0947421i \(-0.969797\pi\)
0.814759 − 0.579800i \(-0.196869\pi\)
\(12\) −3.24247 + 1.21917i −0.936021 + 0.351943i
\(13\) −0.429571 + 1.60318i −0.119142 + 0.444642i −0.999563 − 0.0295491i \(-0.990593\pi\)
0.880422 + 0.474191i \(0.157260\pi\)
\(14\) −1.03157 − 0.234541i −0.275699 − 0.0626838i
\(15\) 0.250297 − 0.455753i 0.0646264 − 0.117675i
\(16\) 1.44890 − 3.72836i 0.362225 − 0.932091i
\(17\) − 6.74518i − 1.63595i −0.575256 − 0.817973i \(-0.695098\pi\)
0.575256 − 0.817973i \(-0.304902\pi\)
\(18\) −0.762958 + 4.17348i −0.179831 + 0.983698i
\(19\) 0.621335 − 0.621335i 0.142544 − 0.142544i −0.632234 − 0.774778i \(-0.717862\pi\)
0.774778 + 0.632234i \(0.217862\pi\)
\(20\) 0.199247 + 0.566375i 0.0445530 + 0.126645i
\(21\) −0.896375 + 0.935537i −0.195605 + 0.204151i
\(22\) 2.77198 − 1.74499i 0.590988 − 0.372032i
\(23\) −6.06191 + 3.49985i −1.26400 + 0.729769i −0.973845 − 0.227212i \(-0.927039\pi\)
−0.290151 + 0.956981i \(0.593706\pi\)
\(24\) −3.00242 − 3.87111i −0.612866 − 0.790187i
\(25\) 4.25208 + 2.45494i 0.850416 + 0.490988i
\(26\) −2.34550 + 0.0897272i −0.459991 + 0.0175970i
\(27\) 3.90212 + 3.43124i 0.750964 + 0.660343i
\(28\) −0.114299 − 1.49172i −0.0216004 − 0.281908i
\(29\) −1.45829 − 5.44240i −0.270797 − 1.01063i −0.958606 − 0.284736i \(-0.908094\pi\)
0.687809 − 0.725892i \(-0.258573\pi\)
\(30\) 0.720357 + 0.147666i 0.131519 + 0.0269600i
\(31\) −3.13647 + 1.81084i −0.563326 + 0.325236i −0.754479 − 0.656324i \(-0.772111\pi\)
0.191153 + 0.981560i \(0.438777\pi\)
\(32\) 5.64276 + 0.399142i 0.997508 + 0.0705590i
\(33\) −0.0857395 − 4.01073i −0.0149253 − 0.698178i
\(34\) 9.11297 − 2.81933i 1.56286 − 0.483511i
\(35\) 0.158790 + 0.158790i 0.0268404 + 0.0268404i
\(36\) −5.95741 + 0.713634i −0.992902 + 0.118939i
\(37\) 6.74053 − 6.74053i 1.10814 − 1.10814i 0.114740 − 0.993396i \(-0.463396\pi\)
0.993396 − 0.114740i \(-0.0366036\pi\)
\(38\) 1.09915 + 0.579742i 0.178306 + 0.0940465i
\(39\) −1.38383 + 2.51975i −0.221591 + 0.403483i
\(40\) −0.681911 + 0.505921i −0.107820 + 0.0799932i
\(41\) 1.39492 + 2.41607i 0.217850 + 0.377327i 0.954150 − 0.299328i \(-0.0967624\pi\)
−0.736301 + 0.676655i \(0.763429\pi\)
\(42\) −1.63861 − 0.820001i −0.252843 − 0.126529i
\(43\) −7.08744 + 1.89907i −1.08082 + 0.289606i −0.754934 − 0.655801i \(-0.772331\pi\)
−0.325891 + 0.945407i \(0.605664\pi\)
\(44\) 3.51616 + 3.01568i 0.530081 + 0.454631i
\(45\) 0.609023 − 0.663453i 0.0907878 − 0.0989017i
\(46\) −7.26216 − 6.72700i −1.07075 − 0.991842i
\(47\) −0.307120 + 0.531947i −0.0447980 + 0.0775924i −0.887555 − 0.460702i \(-0.847598\pi\)
0.842757 + 0.538294i \(0.180931\pi\)
\(48\) 3.97506 − 5.67440i 0.573751 − 0.819030i
\(49\) 3.22021 + 5.57757i 0.460031 + 0.796796i
\(50\) −1.53944 + 6.77082i −0.217709 + 0.957538i
\(51\) 2.78190 − 11.3470i 0.389544 − 1.58889i
\(52\) −1.10159 − 3.13135i −0.152763 − 0.434241i
\(53\) −2.68523 − 2.68523i −0.368844 − 0.368844i 0.498211 − 0.867056i \(-0.333990\pi\)
−0.867056 + 0.498211i \(0.833990\pi\)
\(54\) −3.00473 + 6.70609i −0.408892 + 0.912583i
\(55\) −0.695300 −0.0937542
\(56\) 1.96759 − 0.777926i 0.262930 − 0.103955i
\(57\) 1.30149 − 0.788974i 0.172386 − 0.104502i
\(58\) 6.74334 − 4.24499i 0.885444 − 0.557395i
\(59\) −0.00841962 − 0.00225603i −0.00109614 − 0.000293710i 0.258271 − 0.966073i \(-0.416847\pi\)
−0.259367 + 0.965779i \(0.583514\pi\)
\(60\) 0.101591 + 1.03495i 0.0131153 + 0.133611i
\(61\) −10.1714 + 2.72542i −1.30232 + 0.348955i −0.842326 − 0.538969i \(-0.818814\pi\)
−0.459991 + 0.887924i \(0.652147\pi\)
\(62\) −3.75748 − 3.48059i −0.477200 − 0.442035i
\(63\) −1.89375 + 1.20410i −0.238590 + 0.151702i
\(64\) 1.81929 + 7.79039i 0.227411 + 0.973799i
\(65\) 0.431499 + 0.249126i 0.0535208 + 0.0309003i
\(66\) 5.38279 − 1.79223i 0.662576 − 0.220608i
\(67\) −8.78151 − 2.35300i −1.07283 − 0.287464i −0.321176 − 0.947020i \(-0.604078\pi\)
−0.751656 + 0.659555i \(0.770745\pi\)
\(68\) 7.61803 + 11.1335i 0.923821 + 1.35014i
\(69\) −11.6410 + 3.38745i −1.40141 + 0.407801i
\(70\) −0.148160 + 0.280902i −0.0177086 + 0.0335742i
\(71\) 15.9645i 1.89463i 0.320297 + 0.947317i \(0.396217\pi\)
−0.320297 + 0.947317i \(0.603783\pi\)
\(72\) −3.45420 − 7.75039i −0.407082 − 0.913392i
\(73\) 8.17785i 0.957145i 0.878048 + 0.478572i \(0.158846\pi\)
−0.878048 + 0.478572i \(0.841154\pi\)
\(74\) 11.9241 + 6.28930i 1.38615 + 0.731116i
\(75\) 6.14050 + 5.88346i 0.709044 + 0.679363i
\(76\) −0.323832 + 1.72731i −0.0371461 + 0.198136i
\(77\) 1.67353 + 0.448421i 0.190717 + 0.0511023i
\(78\) −3.98269 − 0.816410i −0.450950 − 0.0924402i
\(79\) 7.67035 + 4.42848i 0.862982 + 0.498243i 0.865010 − 0.501755i \(-0.167312\pi\)
−0.00202794 + 0.999998i \(0.500646\pi\)
\(80\) −0.968541 − 0.709822i −0.108286 − 0.0793605i
\(81\) 5.14914 + 7.38149i 0.572126 + 0.820166i
\(82\) −2.68115 + 2.89445i −0.296084 + 0.319638i
\(83\) −1.32013 + 0.353727i −0.144903 + 0.0388266i −0.330541 − 0.943791i \(-0.607231\pi\)
0.185638 + 0.982618i \(0.440565\pi\)
\(84\) 0.422950 − 2.55656i 0.0461477 − 0.278943i
\(85\) −1.95590 − 0.524083i −0.212148 − 0.0568448i
\(86\) −5.52810 − 8.78161i −0.596111 − 0.946946i
\(87\) −0.208577 − 9.75681i −0.0223618 − 1.04604i
\(88\) −2.60461 + 6.01094i −0.277653 + 0.640768i
\(89\) −15.7852 −1.67323 −0.836613 − 0.547794i \(-0.815468\pi\)
−0.836613 + 0.547794i \(0.815468\pi\)
\(90\) 1.15091 + 0.545504i 0.121316 + 0.0575011i
\(91\) −0.877914 − 0.877914i −0.0920304 − 0.0920304i
\(92\) 6.05300 − 12.6232i 0.631069 − 1.31606i
\(93\) −6.02310 + 1.75268i −0.624567 + 0.181745i
\(94\) −0.847048 − 0.192588i −0.0873663 − 0.0198639i
\(95\) −0.131893 − 0.228445i −0.0135319 − 0.0234380i
\(96\) 9.32780 + 2.99868i 0.952015 + 0.306052i
\(97\) −4.62075 + 8.00338i −0.469166 + 0.812620i −0.999379 − 0.0352448i \(-0.988779\pi\)
0.530212 + 0.847865i \(0.322112\pi\)
\(98\) −6.18952 + 6.68192i −0.625236 + 0.674976i
\(99\) 1.50990 − 6.78233i 0.151751 − 0.681650i
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.83.15 yes 88
3.2 odd 2 432.2.v.a.35.8 88
4.3 odd 2 576.2.y.a.47.2 88
9.4 even 3 432.2.v.a.179.2 88
9.5 odd 6 inner 144.2.u.a.131.21 yes 88
12.11 even 2 1728.2.z.a.1007.11 88
16.5 even 4 576.2.y.a.335.12 88
16.11 odd 4 inner 144.2.u.a.11.21 ✓ 88
36.23 even 6 576.2.y.a.239.12 88
36.31 odd 6 1728.2.z.a.1583.11 88
48.5 odd 4 1728.2.z.a.143.11 88
48.11 even 4 432.2.v.a.251.2 88
144.5 odd 12 576.2.y.a.527.2 88
144.59 even 12 inner 144.2.u.a.59.15 yes 88
144.85 even 12 1728.2.z.a.719.11 88
144.139 odd 12 432.2.v.a.395.8 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.u.a.11.21 ✓ 88 16.11 odd 4 inner
144.2.u.a.59.15 yes 88 144.59 even 12 inner
144.2.u.a.83.15 yes 88 1.1 even 1 trivial
144.2.u.a.131.21 yes 88 9.5 odd 6 inner
432.2.v.a.35.8 88 3.2 odd 2
432.2.v.a.179.2 88 9.4 even 3
432.2.v.a.251.2 88 48.11 even 4
432.2.v.a.395.8 88 144.139 odd 12
576.2.y.a.47.2 88 4.3 odd 2
576.2.y.a.239.12 88 36.23 even 6
576.2.y.a.335.12 88 16.5 even 4
576.2.y.a.527.2 88 144.5 odd 12
1728.2.z.a.143.11 88 48.5 odd 4
1728.2.z.a.719.11 88 144.85 even 12
1728.2.z.a.1007.11 88 12.11 even 2
1728.2.z.a.1583.11 88 36.31 odd 6