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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,8,Mod(25,72)] 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("72.25"); S:= CuspForms(chi, 8); 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, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] 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: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.9
Character \(\chi\) \(=\) 72.49
Dual form 72.8.i.b.25.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(24.3366 + 39.9340i) q^{3} +(-160.855 + 278.610i) q^{5} +(-649.814 - 1125.51i) q^{7} +(-1002.46 + 1943.72i) q^{9} +(-3161.69 - 5476.21i) q^{11} +(4522.25 - 7832.76i) q^{13} +(-15040.7 + 356.820i) q^{15} -2354.92 q^{17} +55063.5 q^{19} +(29131.9 - 53340.9i) q^{21} +(-26249.0 + 45464.6i) q^{23} +(-12686.4 - 21973.6i) q^{25} +(-102017. + 7271.56i) q^{27} +(-24332.5 - 42145.0i) q^{29} +(83691.1 - 144957. i) q^{31} +(141742. - 259532. i) q^{33} +418105. q^{35} -273838. q^{37} +(422850. - 10031.5i) q^{39} +(90519.7 - 156785. i) q^{41} +(-448720. - 777206. i) q^{43} +(-380289. - 591952. i) q^{45} +(-652359. - 1.12992e6i) q^{47} +(-432746. + 749538. i) q^{49} +(-57310.8 - 94041.4i) q^{51} -531697. q^{53} +2.03430e6 q^{55} +(1.34006e6 + 2.19891e6i) q^{57} +(-575003. + 995934. i) q^{59} +(-1.13524e6 - 1.96630e6i) q^{61} +(2.83909e6 - 134783. i) q^{63} +(1.45486e6 + 2.51988e6i) q^{65} +(-406439. + 703972. i) q^{67} +(-2.45440e6 + 58227.2i) q^{69} +1.91502e6 q^{71} -4.01043e6 q^{73} +(568748. - 1.04138e6i) q^{75} +(-4.10902e6 + 7.11703e6i) q^{77} +(917146. + 1.58854e6i) q^{79} +(-2.77314e6 - 3.89699e6i) q^{81} +(3.85397e6 + 6.67526e6i) q^{83} +(378801. - 656103. i) q^{85} +(1.09085e6 - 1.99736e6i) q^{87} -5.16070e6 q^{89} -1.17545e7 q^{91} +(7.82549e6 - 185649. i) q^{93} +(-8.85727e6 + 1.53412e7i) q^{95} +(-1.02659e6 - 1.77811e6i) q^{97} +(1.38137e7 - 655790. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 44 q^{3} - 125 q^{5} - 1245 q^{7} - 3766 q^{9} + 1699 q^{11} - 4937 q^{13} - 19349 q^{15} + 26540 q^{17} + 28976 q^{19} - 75315 q^{21} - 18239 q^{23} - 109168 q^{25} - 87680 q^{27} - 3525 q^{29}+ \cdots + 24043955 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 24.3366 + 39.9340i 0.520399 + 0.853923i
\(4\) 0 0
\(5\) −160.855 + 278.610i −0.575494 + 0.996785i 0.420494 + 0.907295i \(0.361857\pi\)
−0.995988 + 0.0894894i \(0.971476\pi\)
\(6\) 0 0
\(7\) −649.814 − 1125.51i −0.716055 − 1.24024i −0.962551 − 0.271099i \(-0.912613\pi\)
0.246497 − 0.969144i \(-0.420721\pi\)
\(8\) 0 0
\(9\) −1002.46 + 1943.72i −0.458370 + 0.888761i
\(10\) 0 0
\(11\) −3161.69 − 5476.21i −0.716217 − 1.24052i −0.962488 − 0.271324i \(-0.912539\pi\)
0.246271 − 0.969201i \(-0.420795\pi\)
\(12\) 0 0
\(13\) 4522.25 − 7832.76i 0.570890 − 0.988811i −0.425585 − 0.904919i \(-0.639932\pi\)
0.996475 − 0.0838921i \(-0.0267351\pi\)
\(14\) 0 0
\(15\) −15040.7 + 356.820i −1.15066 + 0.0272979i
\(16\) 0 0
\(17\) −2354.92 −0.116253 −0.0581266 − 0.998309i \(-0.518513\pi\)
−0.0581266 + 0.998309i \(0.518513\pi\)
\(18\) 0 0
\(19\) 55063.5 1.84173 0.920866 − 0.389880i \(-0.127483\pi\)
0.920866 + 0.389880i \(0.127483\pi\)
\(20\) 0 0
\(21\) 29131.9 − 53340.9i 0.686438 − 1.25688i
\(22\) 0 0
\(23\) −26249.0 + 45464.6i −0.449848 + 0.779159i −0.998376 − 0.0569730i \(-0.981855\pi\)
0.548528 + 0.836132i \(0.315188\pi\)
\(24\) 0 0
\(25\) −12686.4 − 21973.6i −0.162387 − 0.281262i
\(26\) 0 0
\(27\) −102017. + 7271.56i −0.997469 + 0.0710975i
\(28\) 0 0
\(29\) −24332.5 − 42145.0i −0.185265 − 0.320888i 0.758401 − 0.651788i \(-0.225981\pi\)
−0.943666 + 0.330900i \(0.892648\pi\)
\(30\) 0 0
\(31\) 83691.1 − 144957.i 0.504561 − 0.873925i −0.495425 − 0.868651i \(-0.664988\pi\)
0.999986 − 0.00527429i \(-0.00167887\pi\)
\(32\) 0 0
\(33\) 141742. − 259532.i 0.686594 − 1.25716i
\(34\) 0 0
\(35\) 418105. 1.64834
\(36\) 0 0
\(37\) −273838. −0.888766 −0.444383 − 0.895837i \(-0.646577\pi\)
−0.444383 + 0.895837i \(0.646577\pi\)
\(38\) 0 0
\(39\) 422850. − 10031.5i 1.14146 − 0.0270795i
\(40\) 0 0
\(41\) 90519.7 − 156785.i 0.205116 − 0.355271i −0.745054 − 0.667005i \(-0.767576\pi\)
0.950170 + 0.311733i \(0.100909\pi\)
\(42\) 0 0
\(43\) −448720. − 777206.i −0.860668 − 1.49072i −0.871285 − 0.490778i \(-0.836713\pi\)
0.0106162 − 0.999944i \(-0.496621\pi\)
\(44\) 0 0
\(45\) −380289. − 591952.i −0.622115 − 0.968373i
\(46\) 0 0
\(47\) −652359. − 1.12992e6i −0.916525 − 1.58747i −0.804653 − 0.593745i \(-0.797649\pi\)
−0.111872 − 0.993723i \(-0.535685\pi\)
\(48\) 0 0
\(49\) −432746. + 749538.i −0.525468 + 0.910138i
\(50\) 0 0
\(51\) −57310.8 − 94041.4i −0.0604980 − 0.0992713i
\(52\) 0 0
\(53\) −531697. −0.490568 −0.245284 − 0.969451i \(-0.578881\pi\)
−0.245284 + 0.969451i \(0.578881\pi\)
\(54\) 0 0
\(55\) 2.03430e6 1.64871
\(56\) 0 0
\(57\) 1.34006e6 + 2.19891e6i 0.958435 + 1.57270i
\(58\) 0 0
\(59\) −575003. + 995934.i −0.364492 + 0.631318i −0.988694 − 0.149944i \(-0.952091\pi\)
0.624203 + 0.781262i \(0.285424\pi\)
\(60\) 0 0
\(61\) −1.13524e6 − 1.96630e6i −0.640374 − 1.10916i −0.985349 − 0.170549i \(-0.945446\pi\)
0.344975 − 0.938612i \(-0.387887\pi\)
\(62\) 0 0
\(63\) 2.83909e6 − 134783.i 1.43050 − 0.0679114i
\(64\) 0 0
\(65\) 1.45486e6 + 2.51988e6i 0.657088 + 1.13811i
\(66\) 0 0
\(67\) −406439. + 703972.i −0.165095 + 0.285952i −0.936689 − 0.350163i \(-0.886126\pi\)
0.771594 + 0.636115i \(0.219460\pi\)
\(68\) 0 0
\(69\) −2.45440e6 + 58227.2i −0.899443 + 0.0213380i
\(70\) 0 0
\(71\) 1.91502e6 0.634994 0.317497 − 0.948259i \(-0.397158\pi\)
0.317497 + 0.948259i \(0.397158\pi\)
\(72\) 0 0
\(73\) −4.01043e6 −1.20659 −0.603297 − 0.797516i \(-0.706147\pi\)
−0.603297 + 0.797516i \(0.706147\pi\)
\(74\) 0 0
\(75\) 568748. − 1.04138e6i 0.155670 − 0.285034i
\(76\) 0 0
\(77\) −4.10902e6 + 7.11703e6i −1.02570 + 1.77657i
\(78\) 0 0
\(79\) 917146. + 1.58854e6i 0.209288 + 0.362497i 0.951490 − 0.307679i \(-0.0995522\pi\)
−0.742203 + 0.670175i \(0.766219\pi\)
\(80\) 0 0
\(81\) −2.77314e6 − 3.89699e6i −0.579794 − 0.814763i
\(82\) 0 0
\(83\) 3.85397e6 + 6.67526e6i 0.739834 + 1.28143i 0.952570 + 0.304321i \(0.0984295\pi\)
−0.212735 + 0.977110i \(0.568237\pi\)
\(84\) 0 0
\(85\) 378801. − 656103.i 0.0669030 − 0.115879i
\(86\) 0 0
\(87\) 1.09085e6 − 1.99736e6i 0.177602 − 0.325192i
\(88\) 0 0
\(89\) −5.16070e6 −0.775968 −0.387984 − 0.921666i \(-0.626828\pi\)
−0.387984 + 0.921666i \(0.626828\pi\)
\(90\) 0 0
\(91\) −1.17545e7 −1.63515
\(92\) 0 0
\(93\) 7.82549e6 − 185649.i 1.00884 − 0.0239333i
\(94\) 0 0
\(95\) −8.85727e6 + 1.53412e7i −1.05991 + 1.83581i
\(96\) 0 0
\(97\) −1.02659e6 − 1.77811e6i −0.114208 − 0.197814i 0.803255 − 0.595635i \(-0.203100\pi\)
−0.917463 + 0.397821i \(0.869766\pi\)
\(98\) 0 0
\(99\) 1.38137e7 − 655790.i 1.43082 − 0.0679268i
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 72.8.i.b.49.9 yes 22
3.2 odd 2 216.8.i.b.145.9 22
4.3 odd 2 144.8.i.f.49.3 22
9.2 odd 6 216.8.i.b.73.9 22
9.7 even 3 inner 72.8.i.b.25.9 ✓ 22
12.11 even 2 432.8.i.f.145.9 22
36.7 odd 6 144.8.i.f.97.3 22
36.11 even 6 432.8.i.f.289.9 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.i.b.25.9 ✓ 22 9.7 even 3 inner
72.8.i.b.49.9 yes 22 1.1 even 1 trivial
144.8.i.f.49.3 22 4.3 odd 2
144.8.i.f.97.3 22 36.7 odd 6
216.8.i.b.73.9 22 9.2 odd 6
216.8.i.b.145.9 22 3.2 odd 2
432.8.i.f.145.9 22 12.11 even 2
432.8.i.f.289.9 22 36.11 even 6