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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,3,Mod(65,432)] 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("432.65"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 13])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 432.bc (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0,0,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.7711474204\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 65.5
Character \(\chi\) \(=\) 432.65
Dual form 432.3.bc.b.113.5

$q$-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 \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{13}{18}\right)\)

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
\(3\) 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 −0.999987 − 0.00505674i \(-0.998390\pi\)
0.941410 + 0.337264i \(0.109501\pi\)
\(8\) 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 \(-0.911824\pi\)
0.561049 − 0.827783i \(-0.310398\pi\)
\(12\) 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 −0.766270 − 0.642518i \(-0.777890\pi\)
0.939572 + 0.342350i \(0.111223\pi\)
\(20\) 0 0
\(21\) −6.09825 + 3.60472i −0.290393 + 0.171653i
\(22\) 0 0
\(23\) 25.9779 − 4.58061i 1.12947 − 0.199157i 0.422477 − 0.906374i \(-0.361161\pi\)
0.706997 + 0.707217i \(0.250050\pi\)
\(24\) 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 0.993184 − 0.116555i \(-0.0371852\pi\)
−0.973152 + 0.230163i \(0.926074\pi\)
\(32\) 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 \(0.817967\pi\)
−0.992040 + 0.125921i \(0.959811\pi\)
\(44\) 0 0
\(45\) −74.9319 − 41.2363i −1.66515 − 0.916363i
\(46\) 0 0
\(47\) 34.9622 + 6.16478i 0.743877 + 0.131166i 0.532726 − 0.846288i \(-0.321168\pi\)
0.211151 + 0.977453i \(0.432279\pi\)
\(48\) 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 \(-0.748300\pi\)
0.995715 − 0.0924754i \(-0.0294780\pi\)
\(60\) 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 −0.996096 − 0.0882774i \(-0.971864\pi\)
−0.0860340 + 0.996292i \(0.527419\pi\)
\(68\) 0 0
\(69\) 61.1402 + 50.2432i 0.886089 + 0.728163i
\(70\) 0 0
\(71\) −71.8787 + 41.4992i −1.01238 + 0.584496i −0.911887 − 0.410442i \(-0.865374\pi\)
−0.100490 + 0.994938i \(0.532041\pi\)
\(72\) 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 0.792289 − 0.610146i \(-0.208889\pi\)
−0.463296 + 0.886203i \(0.653333\pi\)
\(80\) 0 0
\(81\) −77.1873 − 24.5587i −0.952929 − 0.303193i
\(82\) 0 0
\(83\) −7.14066 + 8.50991i −0.0860321 + 0.102529i −0.807343 − 0.590082i \(-0.799095\pi\)
0.721311 + 0.692611i \(0.243540\pi\)
\(84\) 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 432.3.bc.b.65.5 36
4.3 odd 2 108.3.k.a.65.2 yes 36
12.11 even 2 324.3.k.a.197.6 36
27.5 odd 18 inner 432.3.bc.b.113.5 36
108.7 odd 18 2916.3.c.b.1457.1 36
108.47 even 18 2916.3.c.b.1457.36 36
108.59 even 18 108.3.k.a.5.2 ✓ 36
108.103 odd 18 324.3.k.a.125.6 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.5.2 ✓ 36 108.59 even 18
108.3.k.a.65.2 yes 36 4.3 odd 2
324.3.k.a.125.6 36 108.103 odd 18
324.3.k.a.197.6 36 12.11 even 2
432.3.bc.b.65.5 36 1.1 even 1 trivial
432.3.bc.b.113.5 36 27.5 odd 18 inner
2916.3.c.b.1457.1 36 108.7 odd 18
2916.3.c.b.1457.36 36 108.47 even 18