Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,2,Mod(13,72)] 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("72.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 61.8
Root \(-0.722180 - 1.21592i\) of defining polynomial
Character \(\chi\) \(=\) 72.61
Dual form 72.2.n.b.13.8

$q$-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+(1.41411 + 0.0174668i) q^{2} +(-0.294546 - 1.70682i) q^{3} +(1.99939 + 0.0493999i) q^{4} +(-3.17262 + 1.83171i) q^{5} +(-0.386706 - 2.41877i) q^{6} +(-0.191926 + 0.332426i) q^{7} +(2.82649 + 0.104780i) q^{8} +(-2.82649 + 1.00547i) q^{9} +(-4.51841 + 2.53482i) q^{10} +(1.73849 + 1.00372i) q^{11} +(-0.504595 - 3.42715i) q^{12} +(0.397799 - 0.229669i) q^{13} +(-0.277210 + 0.466733i) q^{14} +(4.06089 + 4.87557i) q^{15} +(3.99512 + 0.197539i) q^{16} -4.08495 q^{17} +(-4.01451 + 1.37248i) q^{18} -4.72398i q^{19} +(-6.43379 + 3.50558i) q^{20} +(0.623923 + 0.229669i) q^{21} +(2.44087 + 1.44973i) q^{22} +(-2.97594 - 5.15447i) q^{23} +(-0.653689 - 4.85517i) q^{24} +(4.21034 - 7.29252i) q^{25} +(0.566541 - 0.317828i) q^{26} +(2.54870 + 4.52815i) q^{27} +(-0.400157 + 0.655168i) q^{28} +(2.03783 + 1.17654i) q^{29} +(5.65736 + 6.96551i) q^{30} +(0.592083 + 1.02552i) q^{31} +(5.64607 + 0.349123i) q^{32} +(1.20110 - 3.26293i) q^{33} +(-5.77656 - 0.0713512i) q^{34} -1.40621i q^{35} +(-5.70092 + 1.87071i) q^{36} +5.74432i q^{37} +(0.0825129 - 6.68021i) q^{38} +(-0.509175 - 0.611324i) q^{39} +(-9.15929 + 4.84488i) q^{40} +(4.75281 + 8.23212i) q^{41} +(0.878281 + 0.335675i) q^{42} +(-1.03633 - 0.598327i) q^{43} +(3.42633 + 2.09270i) q^{44} +(7.12562 - 8.36729i) q^{45} +(-4.11825 - 7.34095i) q^{46} +(3.27688 - 5.67572i) q^{47} +(-0.839581 - 6.87714i) q^{48} +(3.42633 + 5.93458i) q^{49} +(6.08124 - 10.2389i) q^{50} +(1.20321 + 6.97229i) q^{51} +(0.806700 - 0.439547i) q^{52} -7.63807i q^{53} +(3.52503 + 6.44780i) q^{54} -7.35407 q^{55} +(-0.577308 + 0.919487i) q^{56} +(-8.06300 + 1.39143i) q^{57} +(2.86116 + 1.69935i) q^{58} +(-0.603703 + 0.348548i) q^{59} +(7.87845 + 9.94878i) q^{60} +(-4.23774 - 2.44666i) q^{61} +(0.819356 + 1.46053i) q^{62} +(0.208231 - 1.13257i) q^{63} +(7.97804 + 0.592316i) q^{64} +(-0.841376 + 1.45731i) q^{65} +(1.75548 - 4.59315i) q^{66} +(-8.87932 + 5.12648i) q^{67} +(-8.16741 - 0.201796i) q^{68} +(-7.92122 + 6.59762i) q^{69} +(0.0245621 - 1.98853i) q^{70} -3.73792 q^{71} +(-8.09437 + 2.54580i) q^{72} -2.68275 q^{73} +(-0.100335 + 8.12307i) q^{74} +(-13.6872 - 5.03832i) q^{75} +(0.233364 - 9.44508i) q^{76} +(-0.667322 + 0.385279i) q^{77} +(-0.709349 - 0.873370i) q^{78} +(-5.35979 + 9.28342i) q^{79} +(-13.0368 + 6.69119i) q^{80} +(6.97804 - 5.68392i) q^{81} +(6.57719 + 11.7241i) q^{82} +(5.49039 + 3.16988i) q^{83} +(1.23612 + 0.490020i) q^{84} +(12.9600 - 7.48246i) q^{85} +(-1.45503 - 0.864199i) q^{86} +(1.40792 - 3.82477i) q^{87} +(4.80864 + 3.01915i) q^{88} +7.56802 q^{89} +(10.2225 - 11.7078i) q^{90} +0.176318i q^{91} +(-5.69542 - 10.4528i) q^{92} +(1.57598 - 1.31264i) q^{93} +(4.73299 - 7.96883i) q^{94} +(8.65297 + 14.9874i) q^{95} +(-1.06713 - 9.73967i) q^{96} +(-2.98511 + 5.17036i) q^{97} +(4.74153 + 8.45196i) q^{98} +(-5.92302 - 1.08898i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - q^{4} - 7 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 16 q^{10} - 16 q^{12} + 16 q^{14} - 10 q^{15} - 9 q^{16} - 28 q^{17} + 4 q^{18} - 8 q^{20} + q^{22} - 10 q^{23} + 7 q^{24} + 2 q^{25} + 28 q^{26}+ \cdots + 66 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41411 + 0.0174668i 0.999924 + 0.0123509i
\(3\) −0.294546 − 1.70682i −0.170056 − 0.985434i
\(4\) 1.99939 + 0.0493999i 0.999695 + 0.0246999i
\(5\) −3.17262 + 1.83171i −1.41884 + 0.819167i −0.996197 − 0.0871306i \(-0.972230\pi\)
−0.422641 + 0.906297i \(0.638897\pi\)
\(6\) −0.386706 − 2.41877i −0.157872 − 0.987460i
\(7\) −0.191926 + 0.332426i −0.0725413 + 0.125645i −0.900014 − 0.435860i \(-0.856444\pi\)
0.827473 + 0.561505i \(0.189778\pi\)
\(8\) 2.82649 + 0.104780i 0.999314 + 0.0370452i
\(9\) −2.82649 + 1.00547i −0.942162 + 0.335158i
\(10\) −4.51841 + 2.53482i −1.42885 + 0.801580i
\(11\) 1.73849 + 1.00372i 0.524173 + 0.302632i 0.738640 − 0.674100i \(-0.235468\pi\)
−0.214467 + 0.976731i \(0.568801\pi\)
\(12\) −0.504595 − 3.42715i −0.145664 − 0.989334i
\(13\) 0.397799 − 0.229669i 0.110330 − 0.0636988i −0.443820 − 0.896116i \(-0.646377\pi\)
0.554149 + 0.832417i \(0.313044\pi\)
\(14\) −0.277210 + 0.466733i −0.0740876 + 0.124740i
\(15\) 4.06089 + 4.87557i 1.04852 + 1.25887i
\(16\) 3.99512 + 0.197539i 0.998780 + 0.0493848i
\(17\) −4.08495 −0.990747 −0.495373 − 0.868680i \(-0.664969\pi\)
−0.495373 + 0.868680i \(0.664969\pi\)
\(18\) −4.01451 + 1.37248i −0.946230 + 0.323496i
\(19\) − 4.72398i − 1.08376i −0.840457 − 0.541878i \(-0.817714\pi\)
0.840457 − 0.541878i \(-0.182286\pi\)
\(20\) −6.43379 + 3.50558i −1.43864 + 0.783871i
\(21\) 0.623923 + 0.229669i 0.136151 + 0.0501179i
\(22\) 2.44087 + 1.44973i 0.520396 + 0.309083i
\(23\) −2.97594 − 5.15447i −0.620525 − 1.07478i −0.989388 − 0.145298i \(-0.953586\pi\)
0.368863 − 0.929484i \(-0.379747\pi\)
\(24\) −0.653689 − 4.85517i −0.133434 − 0.991058i
\(25\) 4.21034 − 7.29252i 0.842068 − 1.45850i
\(26\) 0.566541 − 0.317828i 0.111108 − 0.0623313i
\(27\) 2.54870 + 4.52815i 0.490497 + 0.871443i
\(28\) −0.400157 + 0.655168i −0.0756226 + 0.123815i
\(29\) 2.03783 + 1.17654i 0.378416 + 0.218479i 0.677129 − 0.735864i \(-0.263224\pi\)
−0.298713 + 0.954343i \(0.596557\pi\)
\(30\) 5.65736 + 6.96551i 1.03289 + 1.27172i
\(31\) 0.592083 + 1.02552i 0.106341 + 0.184188i 0.914285 − 0.405071i \(-0.132753\pi\)
−0.807944 + 0.589259i \(0.799420\pi\)
\(32\) 5.64607 + 0.349123i 0.998094 + 0.0617169i
\(33\) 1.20110 − 3.26293i 0.209085 − 0.568003i
\(34\) −5.77656 − 0.0713512i −0.990671 − 0.0122366i
\(35\) − 1.40621i − 0.237693i
\(36\) −5.70092 + 1.87071i −0.950153 + 0.311785i
\(37\) 5.74432i 0.944360i 0.881502 + 0.472180i \(0.156533\pi\)
−0.881502 + 0.472180i \(0.843467\pi\)
\(38\) 0.0825129 − 6.68021i 0.0133854 − 1.08367i
\(39\) −0.509175 − 0.611324i −0.0815332 − 0.0978902i
\(40\) −9.15929 + 4.84488i −1.44821 + 0.766043i
\(41\) 4.75281 + 8.23212i 0.742265 + 1.28564i 0.951462 + 0.307767i \(0.0995817\pi\)
−0.209197 + 0.977874i \(0.567085\pi\)
\(42\) 0.878281 + 0.335675i 0.135522 + 0.0517957i
\(43\) −1.03633 − 0.598327i −0.158039 − 0.0912440i 0.418895 − 0.908035i \(-0.362418\pi\)
−0.576934 + 0.816791i \(0.695751\pi\)
\(44\) 3.42633 + 2.09270i 0.516538 + 0.315486i
\(45\) 7.12562 − 8.36729i 1.06222 − 1.24732i
\(46\) −4.11825 − 7.34095i −0.607204 − 1.08236i
\(47\) 3.27688 − 5.67572i 0.477982 − 0.827889i −0.521699 − 0.853129i \(-0.674702\pi\)
0.999681 + 0.0252403i \(0.00803510\pi\)
\(48\) −0.839581 − 6.87714i −0.121183 − 0.992630i
\(49\) 3.42633 + 5.93458i 0.489476 + 0.847796i
\(50\) 6.08124 − 10.2389i 0.860017 − 1.44799i
\(51\) 1.20321 + 6.97229i 0.168482 + 0.976316i
\(52\) 0.806700 − 0.439547i 0.111869 − 0.0609542i
\(53\) − 7.63807i − 1.04917i −0.851358 − 0.524585i \(-0.824221\pi\)
0.851358 − 0.524585i \(-0.175779\pi\)
\(54\) 3.52503 + 6.44780i 0.479696 + 0.877435i
\(55\) −7.35407 −0.991623
\(56\) −0.577308 + 0.919487i −0.0771460 + 0.122872i
\(57\) −8.06300 + 1.39143i −1.06797 + 0.184299i
\(58\) 2.86116 + 1.69935i 0.375689 + 0.223136i
\(59\) −0.603703 + 0.348548i −0.0785954 + 0.0453771i −0.538783 − 0.842445i \(-0.681116\pi\)
0.460187 + 0.887822i \(0.347782\pi\)
\(60\) 7.87845 + 9.94878i 1.01710 + 1.28438i
\(61\) −4.23774 − 2.44666i −0.542587 − 0.313263i 0.203540 − 0.979067i \(-0.434755\pi\)
−0.746127 + 0.665804i \(0.768089\pi\)
\(62\) 0.819356 + 1.46053i 0.104058 + 0.185488i
\(63\) 0.208231 − 1.13257i 0.0262346 − 0.142691i
\(64\) 7.97804 + 0.592316i 0.997255 + 0.0740395i
\(65\) −0.841376 + 1.45731i −0.104360 + 0.180757i
\(66\) 1.75548 − 4.59315i 0.216084 − 0.565377i
\(67\) −8.87932 + 5.12648i −1.08478 + 0.626299i −0.932182 − 0.361989i \(-0.882098\pi\)
−0.152599 + 0.988288i \(0.548764\pi\)
\(68\) −8.16741 − 0.201796i −0.990444 − 0.0244714i
\(69\) −7.92122 + 6.59762i −0.953603 + 0.794260i
\(70\) 0.0245621 − 1.98853i 0.00293573 − 0.237675i
\(71\) −3.73792 −0.443610 −0.221805 − 0.975091i \(-0.571195\pi\)
−0.221805 + 0.975091i \(0.571195\pi\)
\(72\) −8.09437 + 2.54580i −0.953931 + 0.300026i
\(73\) −2.68275 −0.313992 −0.156996 − 0.987599i \(-0.550181\pi\)
−0.156996 + 0.987599i \(0.550181\pi\)
\(74\) −0.100335 + 8.12307i −0.0116637 + 0.944288i
\(75\) −13.6872 − 5.03832i −1.58046 − 0.581775i
\(76\) 0.233364 − 9.44508i 0.0267687 − 1.08342i
\(77\) −0.667322 + 0.385279i −0.0760484 + 0.0439066i
\(78\) −0.709349 − 0.873370i −0.0803179 − 0.0988897i
\(79\) −5.35979 + 9.28342i −0.603023 + 1.04447i 0.389337 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123372i \(0.960629\pi\)
\(80\) −13.0368 + 6.69119i −1.45756 + 0.748098i
\(81\) 6.97804 − 5.68392i 0.775338 − 0.631546i
\(82\) 6.57719 + 11.7241i 0.726329 + 1.29471i
\(83\) 5.49039 + 3.16988i 0.602648 + 0.347939i 0.770083 − 0.637944i \(-0.220215\pi\)
−0.167434 + 0.985883i \(0.553548\pi\)
\(84\) 1.23612 + 0.490020i 0.134872 + 0.0534656i
\(85\) 12.9600 − 7.48246i 1.40571 − 0.811586i
\(86\) −1.45503 − 0.864199i −0.156900 − 0.0931890i
\(87\) 1.40792 − 3.82477i 0.150945 − 0.410058i
\(88\) 4.80864 + 3.01915i 0.512603 + 0.321842i
\(89\) 7.56802 0.802208 0.401104 − 0.916032i \(-0.368627\pi\)
0.401104 + 0.916032i \(0.368627\pi\)
\(90\) 10.2225 − 11.7078i 1.07755 − 1.23411i
\(91\) 0.176318i 0.0184832i
\(92\) −5.69542 − 10.4528i −0.593789 − 1.08978i
\(93\) 1.57598 − 1.31264i 0.163422 − 0.136115i
\(94\) 4.73299 − 7.96883i 0.488171 − 0.821922i
\(95\) 8.65297 + 14.9874i 0.887776 + 1.53767i
\(96\) −1.06713 − 9.73967i −0.108914 − 0.994051i
\(97\) −2.98511 + 5.17036i −0.303092 + 0.524971i −0.976835 − 0.213995i \(-0.931352\pi\)
0.673743 + 0.738966i \(0.264686\pi\)
\(98\) 4.74153 + 8.45196i 0.478967 + 0.853777i
\(99\) −5.92302 − 1.08898i −0.595286 − 0.109447i
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 72.2.n.b.61.8 yes 16
3.2 odd 2 216.2.n.b.181.1 16
4.3 odd 2 288.2.r.b.241.5 16
8.3 odd 2 288.2.r.b.241.4 16
8.5 even 2 inner 72.2.n.b.61.3 yes 16
9.2 odd 6 648.2.d.k.325.6 8
9.4 even 3 inner 72.2.n.b.13.3 ✓ 16
9.5 odd 6 216.2.n.b.37.6 16
9.7 even 3 648.2.d.j.325.3 8
12.11 even 2 864.2.r.b.721.8 16
24.5 odd 2 216.2.n.b.181.6 16
24.11 even 2 864.2.r.b.721.1 16
36.7 odd 6 2592.2.d.j.1297.1 8
36.11 even 6 2592.2.d.k.1297.8 8
36.23 even 6 864.2.r.b.145.1 16
36.31 odd 6 288.2.r.b.49.4 16
72.5 odd 6 216.2.n.b.37.1 16
72.11 even 6 2592.2.d.k.1297.1 8
72.13 even 6 inner 72.2.n.b.13.8 yes 16
72.29 odd 6 648.2.d.k.325.5 8
72.43 odd 6 2592.2.d.j.1297.8 8
72.59 even 6 864.2.r.b.145.8 16
72.61 even 6 648.2.d.j.325.4 8
72.67 odd 6 288.2.r.b.49.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.n.b.13.3 ✓ 16 9.4 even 3 inner
72.2.n.b.13.8 yes 16 72.13 even 6 inner
72.2.n.b.61.3 yes 16 8.5 even 2 inner
72.2.n.b.61.8 yes 16 1.1 even 1 trivial
216.2.n.b.37.1 16 72.5 odd 6
216.2.n.b.37.6 16 9.5 odd 6
216.2.n.b.181.1 16 3.2 odd 2
216.2.n.b.181.6 16 24.5 odd 2
288.2.r.b.49.4 16 36.31 odd 6
288.2.r.b.49.5 16 72.67 odd 6
288.2.r.b.241.4 16 8.3 odd 2
288.2.r.b.241.5 16 4.3 odd 2
648.2.d.j.325.3 8 9.7 even 3
648.2.d.j.325.4 8 72.61 even 6
648.2.d.k.325.5 8 72.29 odd 6
648.2.d.k.325.6 8 9.2 odd 6
864.2.r.b.145.1 16 36.23 even 6
864.2.r.b.145.8 16 72.59 even 6
864.2.r.b.721.1 16 24.11 even 2
864.2.r.b.721.8 16 12.11 even 2
2592.2.d.j.1297.1 8 36.7 odd 6
2592.2.d.j.1297.8 8 72.43 odd 6
2592.2.d.k.1297.1 8 72.11 even 6
2592.2.d.k.1297.8 8 36.11 even 6