Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,5,Mod(5,72)] 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("72.5"); S:= CuspForms(chi, 5); 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, 5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 72.j (of order \(6\), degree \(2\), 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: \(7.44263734204\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 29.12
Character \(\chi\) \(=\) 72.29
Dual form 72.5.j.a.5.12

$q$-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+(-2.83732 + 2.81950i) q^{2} +(-7.20486 - 5.39351i) q^{3} +(0.100792 - 15.9997i) q^{4} +(-21.1127 - 36.5683i) q^{5} +(35.6495 - 5.01101i) q^{6} +(-16.9047 + 29.2798i) q^{7} +(44.8252 + 45.6804i) q^{8} +(22.8201 + 77.7190i) q^{9} +(163.008 + 44.2286i) q^{10} +(28.4273 - 49.2375i) q^{11} +(-87.0206 + 114.732i) q^{12} +(-79.2879 + 45.7769i) q^{13} +(-34.5904 - 130.739i) q^{14} +(-45.1173 + 377.341i) q^{15} +(-255.980 - 3.22527i) q^{16} -122.909i q^{17} +(-283.877 - 156.173i) q^{18} +551.287i q^{19} +(-587.209 + 334.111i) q^{20} +(279.717 - 119.781i) q^{21} +(58.1680 + 219.854i) q^{22} +(595.912 - 344.050i) q^{23} +(-76.5815 - 570.886i) q^{24} +(-578.994 + 1002.85i) q^{25} +(95.8971 - 353.436i) q^{26} +(254.763 - 683.035i) q^{27} +(466.763 + 273.421i) q^{28} +(-665.451 + 1152.59i) q^{29} +(-935.903 - 1197.85i) q^{30} +(305.568 + 529.260i) q^{31} +(735.390 - 712.585i) q^{32} +(-470.378 + 201.427i) q^{33} +(346.541 + 348.731i) q^{34} +1427.62 q^{35} +(1245.78 - 357.281i) q^{36} -1555.96i q^{37} +(-1554.36 - 1564.18i) q^{38} +(818.156 + 97.8239i) q^{39} +(724.074 - 2603.62i) q^{40} +(-978.349 + 564.850i) q^{41} +(-455.923 + 1128.52i) q^{42} +(2635.89 + 1521.83i) q^{43} +(-784.919 - 459.790i) q^{44} +(2360.26 - 2475.35i) q^{45} +(-720.744 + 2656.36i) q^{46} +(-1412.72 - 815.637i) q^{47} +(1826.90 + 1403.87i) q^{48} +(628.964 + 1089.40i) q^{49} +(-1184.74 - 4477.88i) q^{50} +(-662.909 + 885.540i) q^{51} +(724.424 + 1273.19i) q^{52} -2783.44 q^{53} +(1202.98 + 2656.30i) q^{54} -2400.71 q^{55} +(-2095.27 + 540.258i) q^{56} +(2973.37 - 3971.95i) q^{57} +(-1361.65 - 5146.52i) q^{58} +(1220.18 + 2113.42i) q^{59} +(6032.79 + 759.895i) q^{60} +(-4583.29 - 2646.16i) q^{61} +(-2359.25 - 640.130i) q^{62} +(-2661.36 - 645.648i) q^{63} +(-77.4039 + 4095.27i) q^{64} +(3347.97 + 1932.95i) q^{65} +(766.690 - 1897.74i) q^{66} +(-3539.31 + 2043.42i) q^{67} +(-1966.50 - 12.3882i) q^{68} +(-6149.10 - 735.226i) q^{69} +(-4050.60 + 4025.17i) q^{70} -1182.69i q^{71} +(-2527.32 + 4526.20i) q^{72} +585.969 q^{73} +(4387.05 + 4414.77i) q^{74} +(9580.44 - 4102.56i) q^{75} +(8820.42 + 55.5652i) q^{76} +(961.108 + 1664.69i) q^{77} +(-2597.19 + 2029.24i) q^{78} +(-1960.12 + 3395.02i) q^{79} +(5286.48 + 9428.84i) q^{80} +(-5519.49 + 3547.11i) q^{81} +(1183.29 - 4361.12i) q^{82} +(-5665.44 + 9812.83i) q^{83} +(-1888.27 - 4487.45i) q^{84} +(-4494.56 + 2594.94i) q^{85} +(-11769.7 + 3113.98i) q^{86} +(11011.0 - 4715.17i) q^{87} +(3523.45 - 908.510i) q^{88} -6300.52i q^{89} +(282.452 + 13678.1i) q^{90} -3095.37i q^{91} +(-5444.63 - 9569.09i) q^{92} +(652.991 - 5461.33i) q^{93} +(6308.04 - 1668.96i) q^{94} +(20159.6 - 11639.2i) q^{95} +(-9141.72 + 1167.74i) q^{96} +(373.253 - 646.493i) q^{97} +(-4856.13 - 1317.60i) q^{98} +(4475.40 + 1085.74i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q - 3 q^{2} - q^{4} - 19 q^{6} - 2 q^{7} - 4 q^{9} + 28 q^{10} + 62 q^{12} + 852 q^{14} + 158 q^{15} - q^{16} - 506 q^{18} + 1950 q^{20} + 31 q^{22} - 6 q^{23} - 1415 q^{24} - 4752 q^{25} + 508 q^{28}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.83732 + 2.81950i −0.709330 + 0.704876i
\(3\) −7.20486 − 5.39351i −0.800540 − 0.599279i
\(4\) 0.100792 − 15.9997i 0.00629948 − 0.999980i
\(5\) −21.1127 − 36.5683i −0.844509 − 1.46273i −0.886047 − 0.463595i \(-0.846559\pi\)
0.0415382 − 0.999137i \(-0.486774\pi\)
\(6\) 35.6495 − 5.01101i 0.990265 − 0.139195i
\(7\) −16.9047 + 29.2798i −0.344993 + 0.597546i −0.985353 − 0.170529i \(-0.945452\pi\)
0.640359 + 0.768076i \(0.278786\pi\)
\(8\) 44.8252 + 45.6804i 0.700394 + 0.713757i
\(9\) 22.8201 + 77.7190i 0.281729 + 0.959494i
\(10\) 163.008 + 44.2286i 1.63008 + 0.442286i
\(11\) 28.4273 − 49.2375i 0.234936 − 0.406922i −0.724318 − 0.689466i \(-0.757845\pi\)
0.959254 + 0.282545i \(0.0911785\pi\)
\(12\) −87.0206 + 114.732i −0.604310 + 0.796749i
\(13\) −79.2879 + 45.7769i −0.469159 + 0.270869i −0.715888 − 0.698216i \(-0.753978\pi\)
0.246729 + 0.969085i \(0.420644\pi\)
\(14\) −34.5904 − 130.739i −0.176482 − 0.667035i
\(15\) −45.1173 + 377.341i −0.200521 + 1.67707i
\(16\) −255.980 − 3.22527i −0.999921 − 0.0125987i
\(17\) − 122.909i − 0.425289i −0.977130 − 0.212645i \(-0.931792\pi\)
0.977130 − 0.212645i \(-0.0682077\pi\)
\(18\) −283.877 − 156.173i −0.876164 − 0.482014i
\(19\) 551.287i 1.52711i 0.645742 + 0.763556i \(0.276548\pi\)
−0.645742 + 0.763556i \(0.723452\pi\)
\(20\) −587.209 + 334.111i −1.46802 + 0.835278i
\(21\) 279.717 − 119.781i 0.634278 − 0.271612i
\(22\) 58.1680 + 219.854i 0.120182 + 0.454243i
\(23\) 595.912 − 344.050i 1.12649 − 0.650378i 0.183439 − 0.983031i \(-0.441277\pi\)
0.943049 + 0.332653i \(0.107944\pi\)
\(24\) −76.5815 − 570.886i −0.132954 − 0.991122i
\(25\) −578.994 + 1002.85i −0.926390 + 1.60456i
\(26\) 95.8971 − 353.436i 0.141860 − 0.522835i
\(27\) 254.763 − 683.035i 0.349469 − 0.936948i
\(28\) 466.763 + 273.421i 0.595361 + 0.348751i
\(29\) −665.451 + 1152.59i −0.791261 + 1.37050i 0.133925 + 0.990991i \(0.457242\pi\)
−0.925186 + 0.379513i \(0.876091\pi\)
\(30\) −935.903 − 1197.85i −1.03989 − 1.33094i
\(31\) 305.568 + 529.260i 0.317969 + 0.550739i 0.980064 − 0.198681i \(-0.0636658\pi\)
−0.662095 + 0.749420i \(0.730332\pi\)
\(32\) 735.390 − 712.585i 0.718155 − 0.695883i
\(33\) −470.378 + 201.427i −0.431935 + 0.184965i
\(34\) 346.541 + 348.731i 0.299776 + 0.301671i
\(35\) 1427.62 1.16540
\(36\) 1245.78 − 357.281i 0.961250 − 0.275680i
\(37\) − 1555.96i − 1.13657i −0.822832 − 0.568285i \(-0.807607\pi\)
0.822832 − 0.568285i \(-0.192393\pi\)
\(38\) −1554.36 − 1564.18i −1.07642 − 1.08323i
\(39\) 818.156 + 97.8239i 0.537907 + 0.0643155i
\(40\) 724.074 − 2603.62i 0.452546 − 1.62726i
\(41\) −978.349 + 564.850i −0.582004 + 0.336020i −0.761929 − 0.647660i \(-0.775748\pi\)
0.179925 + 0.983680i \(0.442414\pi\)
\(42\) −455.923 + 1128.52i −0.258460 + 0.639750i
\(43\) 2635.89 + 1521.83i 1.42558 + 0.823057i 0.996768 − 0.0803375i \(-0.0255998\pi\)
0.428810 + 0.903395i \(0.358933\pi\)
\(44\) −784.919 − 459.790i −0.405433 − 0.237495i
\(45\) 2360.26 − 2475.35i 1.16556 − 1.22240i
\(46\) −720.744 + 2656.36i −0.340616 + 1.25537i
\(47\) −1412.72 − 815.637i −0.639531 − 0.369233i 0.144903 − 0.989446i \(-0.453713\pi\)
−0.784434 + 0.620212i \(0.787046\pi\)
\(48\) 1826.90 + 1403.87i 0.792927 + 0.609317i
\(49\) 628.964 + 1089.40i 0.261959 + 0.453726i
\(50\) −1184.74 − 4477.88i −0.473896 − 1.79115i
\(51\) −662.909 + 885.540i −0.254867 + 0.340461i
\(52\) 724.424 + 1273.19i 0.267908 + 0.470856i
\(53\) −2783.44 −0.990901 −0.495451 − 0.868636i \(-0.664997\pi\)
−0.495451 + 0.868636i \(0.664997\pi\)
\(54\) 1202.98 + 2656.30i 0.412543 + 0.910938i
\(55\) −2400.71 −0.793623
\(56\) −2095.27 + 540.258i −0.668134 + 0.172276i
\(57\) 2973.37 − 3971.95i 0.915166 − 1.22251i
\(58\) −1361.65 − 5146.52i −0.404770 − 1.52988i
\(59\) 1220.18 + 2113.42i 0.350526 + 0.607129i 0.986342 − 0.164712i \(-0.0526695\pi\)
−0.635816 + 0.771841i \(0.719336\pi\)
\(60\) 6032.79 + 759.895i 1.67578 + 0.211082i
\(61\) −4583.29 − 2646.16i −1.23174 − 0.711143i −0.264344 − 0.964429i \(-0.585155\pi\)
−0.967391 + 0.253286i \(0.918489\pi\)
\(62\) −2359.25 − 640.130i −0.613748 − 0.166527i
\(63\) −2661.36 − 645.648i −0.670537 − 0.162673i
\(64\) −77.4039 + 4095.27i −0.0188974 + 0.999821i
\(65\) 3347.97 + 1932.95i 0.792418 + 0.457503i
\(66\) 766.690 − 1897.74i 0.176008 − 0.435662i
\(67\) −3539.31 + 2043.42i −0.788441 + 0.455207i −0.839413 − 0.543493i \(-0.817101\pi\)
0.0509724 + 0.998700i \(0.483768\pi\)
\(68\) −1966.50 − 12.3882i −0.425281 − 0.00267910i
\(69\) −6149.10 − 735.226i −1.29156 − 0.154427i
\(70\) −4050.60 + 4025.17i −0.826654 + 0.821463i
\(71\) − 1182.69i − 0.234615i −0.993096 − 0.117307i \(-0.962574\pi\)
0.993096 − 0.117307i \(-0.0374262\pi\)
\(72\) −2527.32 + 4526.20i −0.487524 + 0.873110i
\(73\) 585.969 0.109959 0.0549793 − 0.998487i \(-0.482491\pi\)
0.0549793 + 0.998487i \(0.482491\pi\)
\(74\) 4387.05 + 4414.77i 0.801141 + 0.806203i
\(75\) 9580.44 − 4102.56i 1.70319 − 0.729345i
\(76\) 8820.42 + 55.5652i 1.52708 + 0.00962001i
\(77\) 961.108 + 1664.69i 0.162103 + 0.280771i
\(78\) −2597.19 + 2029.24i −0.426888 + 0.333537i
\(79\) −1960.12 + 3395.02i −0.314071 + 0.543986i −0.979240 − 0.202707i \(-0.935026\pi\)
0.665169 + 0.746693i \(0.268360\pi\)
\(80\) 5286.48 + 9428.84i 0.826013 + 1.47326i
\(81\) −5519.49 + 3547.11i −0.841257 + 0.540635i
\(82\) 1183.29 − 4361.12i 0.175981 − 0.648590i
\(83\) −5665.44 + 9812.83i −0.822390 + 1.42442i 0.0815082 + 0.996673i \(0.474026\pi\)
−0.903898 + 0.427748i \(0.859307\pi\)
\(84\) −1888.27 − 4487.45i −0.267611 − 0.635976i
\(85\) −4494.56 + 2594.94i −0.622085 + 0.359161i
\(86\) −11769.7 + 3113.98i −1.59136 + 0.421036i
\(87\) 11011.0 − 4715.17i 1.45475 − 0.622958i
\(88\) 3523.45 − 908.510i 0.454991 − 0.117318i
\(89\) − 6300.52i − 0.795419i −0.917511 − 0.397710i \(-0.869805\pi\)
0.917511 − 0.397710i \(-0.130195\pi\)
\(90\) 282.452 + 13678.1i 0.0348706 + 1.68866i
\(91\) − 3095.37i − 0.373792i
\(92\) −5444.63 − 9569.09i −0.643269 − 1.13056i
\(93\) 652.991 − 5461.33i 0.0754990 − 0.631441i
\(94\) 6308.04 − 1668.96i 0.713903 − 0.188882i
\(95\) 20159.6 − 11639.2i 2.23376 − 1.28966i
\(96\) −9141.72 + 1167.74i −0.991940 + 0.126708i
\(97\) 373.253 − 646.493i 0.0396697 − 0.0687100i −0.845509 − 0.533961i \(-0.820703\pi\)
0.885179 + 0.465251i \(0.154036\pi\)
\(98\) −4856.13 − 1317.60i −0.505636 − 0.137193i
\(99\) 4475.40 + 1085.74i 0.456627 + 0.110778i
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 72.5.j.a.29.12 yes 92
3.2 odd 2 216.5.j.a.197.35 92
4.3 odd 2 288.5.n.a.209.36 92
8.3 odd 2 288.5.n.a.209.11 92
8.5 even 2 inner 72.5.j.a.29.27 yes 92
9.4 even 3 216.5.j.a.125.20 92
9.5 odd 6 inner 72.5.j.a.5.27 yes 92
12.11 even 2 864.5.n.a.305.45 92
24.5 odd 2 216.5.j.a.197.20 92
24.11 even 2 864.5.n.a.305.2 92
36.23 even 6 288.5.n.a.113.11 92
36.31 odd 6 864.5.n.a.17.2 92
72.5 odd 6 inner 72.5.j.a.5.12 ✓ 92
72.13 even 6 216.5.j.a.125.35 92
72.59 even 6 288.5.n.a.113.36 92
72.67 odd 6 864.5.n.a.17.45 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.5.j.a.5.12 ✓ 92 72.5 odd 6 inner
72.5.j.a.5.27 yes 92 9.5 odd 6 inner
72.5.j.a.29.12 yes 92 1.1 even 1 trivial
72.5.j.a.29.27 yes 92 8.5 even 2 inner
216.5.j.a.125.20 92 9.4 even 3
216.5.j.a.125.35 92 72.13 even 6
216.5.j.a.197.20 92 24.5 odd 2
216.5.j.a.197.35 92 3.2 odd 2
288.5.n.a.113.11 92 36.23 even 6
288.5.n.a.113.36 92 72.59 even 6
288.5.n.a.209.11 92 8.3 odd 2
288.5.n.a.209.36 92 4.3 odd 2
864.5.n.a.17.2 92 36.31 odd 6
864.5.n.a.17.45 92 72.67 odd 6
864.5.n.a.305.2 92 24.11 even 2
864.5.n.a.305.45 92 12.11 even 2