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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [108,3,Mod(5,108)] 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("108.5"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(108, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.k (of order \(18\), degree \(6\), 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: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 65.2
Character \(\chi\) \(=\) 108.65
Dual form 108.3.k.a.5.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.95183 - 2.27824i) q^{3} +(-3.25030 + 8.93012i) q^{5} +(0.410040 - 2.32545i) q^{7} +(-1.38071 + 8.89346i) q^{9} +(4.40911 + 12.1139i) q^{11} +(-12.2099 + 10.2453i) q^{13} +(26.6889 - 10.0251i) q^{15} +(12.2018 - 7.04474i) q^{17} +(-3.29274 + 5.70319i) q^{19} +(-6.09825 + 3.60472i) q^{21} +(-25.9779 + 4.58061i) q^{23} +(-50.0315 - 41.9814i) q^{25} +(22.9563 - 14.2129i) q^{27} +(0.977913 - 1.16543i) q^{29} +(-0.620995 - 3.52184i) q^{31} +(18.9925 - 33.6893i) q^{33} +(19.4338 + 11.2201i) q^{35} +(-11.0926 - 19.2129i) q^{37} +(47.1730 + 7.81990i) q^{39} +(31.4714 + 37.5062i) q^{41} +(78.8159 - 28.6866i) q^{43} +(-74.9319 - 41.2363i) q^{45} +(-34.9622 - 6.16478i) q^{47} +(40.8053 + 14.8519i) q^{49} +(-39.8655 - 14.0485i) q^{51} +65.8880i q^{53} -122.510 q^{55} +(19.4201 - 3.63003i) q^{57} +(-17.2513 + 47.3976i) q^{59} +(-11.5793 + 65.6697i) q^{61} +(20.1152 + 6.85745i) q^{63} +(-51.8062 - 142.336i) q^{65} +(72.5027 - 60.8370i) q^{67} +(61.1402 + 50.2432i) q^{69} +(71.8787 - 41.4992i) q^{71} +(-47.8713 + 82.9155i) q^{73} +(2.00947 + 195.924i) q^{75} +(29.9782 - 5.28597i) q^{77} +(-25.9904 - 21.8086i) q^{79} +(-77.1873 - 24.5587i) q^{81} +(7.14066 - 8.50991i) q^{83} +(23.2507 + 131.861i) q^{85} +(-4.56385 + 0.0468084i) q^{87} +(-8.83930 - 5.10337i) q^{89} +(18.8185 + 32.5945i) q^{91} +(-6.81149 + 8.28880i) q^{93} +(-40.2278 - 47.9416i) q^{95} +(66.7584 - 24.2981i) q^{97} +(-113.822 + 22.4863i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 9 q^{5} + 6 q^{9} + 36 q^{11} + 45 q^{15} + 42 q^{21} - 18 q^{23} - 9 q^{25} - 18 q^{29} + 45 q^{31} - 153 q^{33} - 243 q^{35} - 123 q^{39} - 198 q^{41} + 90 q^{43} - 333 q^{45} - 243 q^{47} + 72 q^{49}+ \cdots - 945 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\)

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 0
\(3\) −1.95183 − 2.27824i −0.650610 − 0.759412i
\(4\) 0 0
\(5\) −3.25030 + 8.93012i −0.650060 + 1.78602i −0.0325374 + 0.999471i \(0.510359\pi\)
−0.617522 + 0.786553i \(0.711863\pi\)
\(6\) 0 0
\(7\) 0.410040 − 2.32545i 0.0585771 − 0.332207i −0.941410 − 0.337264i \(-0.890499\pi\)
0.999987 + 0.00505674i \(0.00160962\pi\)
\(8\) 0 0
\(9\) −1.38071 + 8.89346i −0.153413 + 0.988162i
\(10\) 0 0
\(11\) 4.40911 + 12.1139i 0.400828 + 1.10127i 0.961877 + 0.273483i \(0.0881757\pi\)
−0.561049 + 0.827783i \(0.689602\pi\)
\(12\) 0 0
\(13\) −12.2099 + 10.2453i −0.939224 + 0.788103i −0.977450 − 0.211166i \(-0.932274\pi\)
0.0382260 + 0.999269i \(0.487829\pi\)
\(14\) 0 0
\(15\) 26.6889 − 10.0251i 1.77926 − 0.668343i
\(16\) 0 0
\(17\) 12.2018 − 7.04474i 0.717755 − 0.414396i −0.0961707 − 0.995365i \(-0.530659\pi\)
0.813926 + 0.580969i \(0.197326\pi\)
\(18\) 0 0
\(19\) −3.29274 + 5.70319i −0.173302 + 0.300168i −0.939572 − 0.342350i \(-0.888777\pi\)
0.766270 + 0.642518i \(0.222110\pi\)
\(20\) 0 0
\(21\) −6.09825 + 3.60472i −0.290393 + 0.171653i
\(22\) 0 0
\(23\) −25.9779 + 4.58061i −1.12947 + 0.199157i −0.706997 − 0.707217i \(-0.749950\pi\)
−0.422477 + 0.906374i \(0.638839\pi\)
\(24\) 0 0
\(25\) −50.0315 − 41.9814i −2.00126 − 1.67926i
\(26\) 0 0
\(27\) 22.9563 − 14.2129i 0.850234 − 0.526405i
\(28\) 0 0
\(29\) 0.977913 − 1.16543i 0.0337211 − 0.0401873i −0.748920 − 0.662660i \(-0.769427\pi\)
0.782641 + 0.622473i \(0.213872\pi\)
\(30\) 0 0
\(31\) −0.620995 − 3.52184i −0.0200321 − 0.113608i 0.973152 − 0.230163i \(-0.0739259\pi\)
−0.993184 + 0.116555i \(0.962815\pi\)
\(32\) 0 0
\(33\) 18.9925 − 33.6893i 0.575531 − 1.02089i
\(34\) 0 0
\(35\) 19.4338 + 11.2201i 0.555252 + 0.320575i
\(36\) 0 0
\(37\) −11.0926 − 19.2129i −0.299800 − 0.519269i 0.676290 − 0.736635i \(-0.263587\pi\)
−0.976090 + 0.217367i \(0.930253\pi\)
\(38\) 0 0
\(39\) 47.1730 + 7.81990i 1.20956 + 0.200510i
\(40\) 0 0
\(41\) 31.4714 + 37.5062i 0.767595 + 0.914784i 0.998303 − 0.0582388i \(-0.0185485\pi\)
−0.230707 + 0.973023i \(0.574104\pi\)
\(42\) 0 0
\(43\) 78.8159 − 28.6866i 1.83293 − 0.667131i 0.840888 − 0.541210i \(-0.182033\pi\)
0.992040 − 0.125921i \(-0.0401887\pi\)
\(44\) 0 0
\(45\) −74.9319 − 41.2363i −1.66515 − 0.916363i
\(46\) 0 0
\(47\) −34.9622 − 6.16478i −0.743877 − 0.131166i −0.211151 − 0.977453i \(-0.567721\pi\)
−0.532726 + 0.846288i \(0.678832\pi\)
\(48\) 0 0
\(49\) 40.8053 + 14.8519i 0.832762 + 0.303101i
\(50\) 0 0
\(51\) −39.8655 − 14.0485i −0.781676 − 0.275461i
\(52\) 0 0
\(53\) 65.8880i 1.24317i 0.783347 + 0.621585i \(0.213511\pi\)
−0.783347 + 0.621585i \(0.786489\pi\)
\(54\) 0 0
\(55\) −122.510 −2.22745
\(56\) 0 0
\(57\) 19.4201 − 3.63003i 0.340703 − 0.0636847i
\(58\) 0 0
\(59\) −17.2513 + 47.3976i −0.292395 + 0.803349i 0.703320 + 0.710874i \(0.251700\pi\)
−0.995715 + 0.0924754i \(0.970522\pi\)
\(60\) 0 0
\(61\) −11.5793 + 65.6697i −0.189825 + 1.07655i 0.729772 + 0.683690i \(0.239626\pi\)
−0.919597 + 0.392862i \(0.871485\pi\)
\(62\) 0 0
\(63\) 20.1152 + 6.85745i 0.319288 + 0.108848i
\(64\) 0 0
\(65\) −51.8062 − 142.336i −0.797019 − 2.18979i
\(66\) 0 0
\(67\) 72.5027 − 60.8370i 1.08213 − 0.908015i 0.0860340 − 0.996292i \(-0.472581\pi\)
0.996096 + 0.0882774i \(0.0281362\pi\)
\(68\) 0 0
\(69\) 61.1402 + 50.2432i 0.886089 + 0.728163i
\(70\) 0 0
\(71\) 71.8787 − 41.4992i 1.01238 − 0.584496i 0.100490 − 0.994938i \(-0.467959\pi\)
0.911887 + 0.410442i \(0.134626\pi\)
\(72\) 0 0
\(73\) −47.8713 + 82.9155i −0.655771 + 1.13583i 0.325929 + 0.945394i \(0.394323\pi\)
−0.981700 + 0.190434i \(0.939010\pi\)
\(74\) 0 0
\(75\) 2.00947 + 195.924i 0.0267929 + 2.61232i
\(76\) 0 0
\(77\) 29.9782 − 5.28597i 0.389328 − 0.0686490i
\(78\) 0 0
\(79\) −25.9904 − 21.8086i −0.328993 − 0.276058i 0.463296 − 0.886203i \(-0.346667\pi\)
−0.792289 + 0.610146i \(0.791111\pi\)
\(80\) 0 0
\(81\) −77.1873 − 24.5587i −0.952929 − 0.303193i
\(82\) 0 0
\(83\) 7.14066 − 8.50991i 0.0860321 − 0.102529i −0.721311 − 0.692611i \(-0.756460\pi\)
0.807343 + 0.590082i \(0.200905\pi\)
\(84\) 0 0
\(85\) 23.2507 + 131.861i 0.273538 + 1.55131i
\(86\) 0 0
\(87\) −4.56385 + 0.0468084i −0.0524580 + 0.000538028i
\(88\) 0 0
\(89\) −8.83930 − 5.10337i −0.0993180 − 0.0573412i 0.449518 − 0.893271i \(-0.351596\pi\)
−0.548836 + 0.835930i \(0.684929\pi\)
\(90\) 0 0
\(91\) 18.8185 + 32.5945i 0.206796 + 0.358182i
\(92\) 0 0
\(93\) −6.81149 + 8.28880i −0.0732419 + 0.0891269i
\(94\) 0 0
\(95\) −40.2278 − 47.9416i −0.423450 − 0.504648i
\(96\) 0 0
\(97\) 66.7584 − 24.2981i 0.688231 − 0.250496i 0.0258533 − 0.999666i \(-0.491770\pi\)
0.662378 + 0.749170i \(0.269547\pi\)
\(98\) 0 0
\(99\) −113.822 + 22.4863i −1.14972 + 0.227135i
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 108.3.k.a.65.2 yes 36
3.2 odd 2 324.3.k.a.197.6 36
4.3 odd 2 432.3.bc.b.65.5 36
27.5 odd 18 inner 108.3.k.a.5.2 ✓ 36
27.7 even 9 2916.3.c.b.1457.1 36
27.20 odd 18 2916.3.c.b.1457.36 36
27.22 even 9 324.3.k.a.125.6 36
108.59 even 18 432.3.bc.b.113.5 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.5.2 ✓ 36 27.5 odd 18 inner
108.3.k.a.65.2 yes 36 1.1 even 1 trivial
324.3.k.a.125.6 36 27.22 even 9
324.3.k.a.197.6 36 3.2 odd 2
432.3.bc.b.65.5 36 4.3 odd 2
432.3.bc.b.113.5 36 108.59 even 18
2916.3.c.b.1457.1 36 27.7 even 9
2916.3.c.b.1457.36 36 27.20 odd 18