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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,4,Mod(37,137)] 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("137.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 137.c (of order \(4\), 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: \(8.08326167079\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(33\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 37.5
Character \(\chi\) \(=\) 137.37
Dual form 137.4.c.a.100.29

$q$-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-4.52282i q^{2} +(3.03224 - 3.03224i) q^{3} -12.4559 q^{4} +(-1.60654 - 1.60654i) q^{5} +(-13.7143 - 13.7143i) q^{6} -24.8720i q^{7} +20.1532i q^{8} +8.61100i q^{9} +(-7.26611 + 7.26611i) q^{10} -14.8419i q^{11} +(-37.7693 + 37.7693i) q^{12} +(-18.3574 + 18.3574i) q^{13} -112.491 q^{14} -9.74286 q^{15} -8.49788 q^{16} +8.98410i q^{17} +38.9460 q^{18} +62.9668i q^{19} +(20.0109 + 20.0109i) q^{20} +(-75.4178 - 75.4178i) q^{21} -67.1272 q^{22} +(104.966 - 104.966i) q^{23} +(61.1094 + 61.1094i) q^{24} -119.838i q^{25} +(83.0270 + 83.0270i) q^{26} +(107.981 + 107.981i) q^{27} +309.802i q^{28} +(35.6026 + 35.6026i) q^{29} +44.0652i q^{30} +(-43.2066 + 43.2066i) q^{31} +199.660i q^{32} +(-45.0043 - 45.0043i) q^{33} +40.6335 q^{34} +(-39.9579 + 39.9579i) q^{35} -107.258i q^{36} -250.216i q^{37} +284.788 q^{38} +111.328i q^{39} +(32.3770 - 32.3770i) q^{40} +(-79.0881 - 79.0881i) q^{41} +(-341.101 + 341.101i) q^{42} +(251.988 - 251.988i) q^{43} +184.869i q^{44} +(13.8339 - 13.8339i) q^{45} +(-474.742 - 474.742i) q^{46} +(85.8932 + 85.8932i) q^{47} +(-25.7676 + 25.7676i) q^{48} -275.614 q^{49} -542.006 q^{50} +(27.2420 + 27.2420i) q^{51} +(228.657 - 228.657i) q^{52} +(-344.667 - 344.667i) q^{53} +(488.380 - 488.380i) q^{54} +(-23.8442 + 23.8442i) q^{55} +501.249 q^{56} +(190.931 + 190.931i) q^{57} +(161.024 - 161.024i) q^{58} -362.451 q^{59} +121.356 q^{60} -263.896i q^{61} +(195.416 + 195.416i) q^{62} +214.172 q^{63} +835.043 q^{64} +58.9838 q^{65} +(-203.546 + 203.546i) q^{66} +(-58.2463 - 58.2463i) q^{67} -111.905i q^{68} -636.564i q^{69} +(180.722 + 180.722i) q^{70} +(123.291 - 123.291i) q^{71} -173.539 q^{72} +849.440 q^{73} -1131.68 q^{74} +(-363.378 - 363.378i) q^{75} -784.308i q^{76} -369.147 q^{77} +503.516 q^{78} +(549.147 - 549.147i) q^{79} +(13.6522 + 13.6522i) q^{80} +422.354 q^{81} +(-357.701 + 357.701i) q^{82} +(367.298 - 367.298i) q^{83} +(939.396 + 939.396i) q^{84} +(14.4333 - 14.4333i) q^{85} +(-1139.70 - 1139.70i) q^{86} +215.912 q^{87} +299.112 q^{88} +(-880.846 + 880.846i) q^{89} +(-62.5684 - 62.5684i) q^{90} +(456.583 + 456.583i) q^{91} +(-1307.44 + 1307.44i) q^{92} +262.026i q^{93} +(388.480 - 388.480i) q^{94} +(101.159 - 101.159i) q^{95} +(605.418 + 605.418i) q^{96} +(191.893 - 191.893i) q^{97} +1246.55i q^{98} +127.804 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 66 q - 272 q^{4} - 4 q^{5} + 6 q^{6} - 174 q^{10} - 104 q^{12} - 56 q^{13} + 140 q^{14} + 276 q^{15} + 1616 q^{16} + 136 q^{18} - 18 q^{20} - 12 q^{21} + 1124 q^{22} + 170 q^{23} + 132 q^{24} + 68 q^{26}+ \cdots + 11068 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/137\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.52282i − 1.59906i −0.600627 − 0.799529i \(-0.705083\pi\)
0.600627 − 0.799529i \(-0.294917\pi\)
\(3\) 3.03224 − 3.03224i 0.583555 − 0.583555i −0.352323 − 0.935878i \(-0.614608\pi\)
0.935878 + 0.352323i \(0.114608\pi\)
\(4\) −12.4559 −1.55699
\(5\) −1.60654 − 1.60654i −0.143694 − 0.143694i 0.631600 − 0.775294i \(-0.282398\pi\)
−0.775294 + 0.631600i \(0.782398\pi\)
\(6\) −13.7143 − 13.7143i −0.933139 − 0.933139i
\(7\) − 24.8720i − 1.34296i −0.741023 − 0.671480i \(-0.765659\pi\)
0.741023 − 0.671480i \(-0.234341\pi\)
\(8\) 20.1532i 0.890654i
\(9\) 8.61100i 0.318926i
\(10\) −7.26611 + 7.26611i −0.229774 + 0.229774i
\(11\) − 14.8419i − 0.406818i −0.979094 − 0.203409i \(-0.934798\pi\)
0.979094 − 0.203409i \(-0.0652022\pi\)
\(12\) −37.7693 + 37.7693i −0.908588 + 0.908588i
\(13\) −18.3574 + 18.3574i −0.391647 + 0.391647i −0.875274 − 0.483627i \(-0.839319\pi\)
0.483627 + 0.875274i \(0.339319\pi\)
\(14\) −112.491 −2.14747
\(15\) −9.74286 −0.167706
\(16\) −8.49788 −0.132779
\(17\) 8.98410i 0.128174i 0.997944 + 0.0640872i \(0.0204136\pi\)
−0.997944 + 0.0640872i \(0.979586\pi\)
\(18\) 38.9460 0.509981
\(19\) 62.9668i 0.760294i 0.924926 + 0.380147i \(0.124127\pi\)
−0.924926 + 0.380147i \(0.875873\pi\)
\(20\) 20.0109 + 20.0109i 0.223729 + 0.223729i
\(21\) −75.4178 − 75.4178i −0.783691 − 0.783691i
\(22\) −67.1272 −0.650526
\(23\) 104.966 − 104.966i 0.951605 − 0.951605i −0.0472771 − 0.998882i \(-0.515054\pi\)
0.998882 + 0.0472771i \(0.0150544\pi\)
\(24\) 61.1094 + 61.1094i 0.519746 + 0.519746i
\(25\) − 119.838i − 0.958704i
\(26\) 83.0270 + 83.0270i 0.626267 + 0.626267i
\(27\) 107.981 + 107.981i 0.769666 + 0.769666i
\(28\) 309.802i 2.09097i
\(29\) 35.6026 + 35.6026i 0.227974 + 0.227974i 0.811846 − 0.583872i \(-0.198463\pi\)
−0.583872 + 0.811846i \(0.698463\pi\)
\(30\) 44.0652i 0.268172i
\(31\) −43.2066 + 43.2066i −0.250327 + 0.250327i −0.821105 − 0.570778i \(-0.806642\pi\)
0.570778 + 0.821105i \(0.306642\pi\)
\(32\) 199.660i 1.10298i
\(33\) −45.0043 − 45.0043i −0.237401 − 0.237401i
\(34\) 40.6335 0.204958
\(35\) −39.9579 + 39.9579i −0.192975 + 0.192975i
\(36\) − 107.258i − 0.496563i
\(37\) − 250.216i − 1.11176i −0.831262 − 0.555882i \(-0.812381\pi\)
0.831262 − 0.555882i \(-0.187619\pi\)
\(38\) 284.788 1.21575
\(39\) 111.328i 0.457096i
\(40\) 32.3770 − 32.3770i 0.127981 − 0.127981i
\(41\) −79.0881 − 79.0881i −0.301256 − 0.301256i 0.540249 − 0.841505i \(-0.318330\pi\)
−0.841505 + 0.540249i \(0.818330\pi\)
\(42\) −341.101 + 341.101i −1.25317 + 1.25317i
\(43\) 251.988 − 251.988i 0.893670 − 0.893670i −0.101196 − 0.994866i \(-0.532267\pi\)
0.994866 + 0.101196i \(0.0322671\pi\)
\(44\) 184.869i 0.633411i
\(45\) 13.8339 − 13.8339i 0.0458276 − 0.0458276i
\(46\) −474.742 − 474.742i −1.52167 − 1.52167i
\(47\) 85.8932 + 85.8932i 0.266571 + 0.266571i 0.827717 − 0.561146i \(-0.189639\pi\)
−0.561146 + 0.827717i \(0.689639\pi\)
\(48\) −25.7676 + 25.7676i −0.0774842 + 0.0774842i
\(49\) −275.614 −0.803540
\(50\) −542.006 −1.53302
\(51\) 27.2420 + 27.2420i 0.0747969 + 0.0747969i
\(52\) 228.657 − 228.657i 0.609790 − 0.609790i
\(53\) −344.667 − 344.667i −0.893277 − 0.893277i 0.101553 − 0.994830i \(-0.467619\pi\)
−0.994830 + 0.101553i \(0.967619\pi\)
\(54\) 488.380 − 488.380i 1.23074 − 1.23074i
\(55\) −23.8442 + 23.8442i −0.0584572 + 0.0584572i
\(56\) 501.249 1.19611
\(57\) 190.931 + 190.931i 0.443674 + 0.443674i
\(58\) 161.024 − 161.024i 0.364544 − 0.364544i
\(59\) −362.451 −0.799782 −0.399891 − 0.916563i \(-0.630952\pi\)
−0.399891 + 0.916563i \(0.630952\pi\)
\(60\) 121.356 0.261117
\(61\) − 263.896i − 0.553909i −0.960883 − 0.276954i \(-0.910675\pi\)
0.960883 − 0.276954i \(-0.0893251\pi\)
\(62\) 195.416 + 195.416i 0.400288 + 0.400288i
\(63\) 214.172 0.428305
\(64\) 835.043 1.63094
\(65\) 58.9838 0.112554
\(66\) −203.546 + 203.546i −0.379618 + 0.379618i
\(67\) −58.2463 − 58.2463i −0.106208 − 0.106208i 0.652006 − 0.758214i \(-0.273928\pi\)
−0.758214 + 0.652006i \(0.773928\pi\)
\(68\) − 111.905i − 0.199566i
\(69\) − 636.564i − 1.11063i
\(70\) 180.722 + 180.722i 0.308578 + 0.308578i
\(71\) 123.291 − 123.291i 0.206085 − 0.206085i −0.596516 − 0.802601i \(-0.703449\pi\)
0.802601 + 0.596516i \(0.203449\pi\)
\(72\) −173.539 −0.284053
\(73\) 849.440 1.36191 0.680955 − 0.732325i \(-0.261565\pi\)
0.680955 + 0.732325i \(0.261565\pi\)
\(74\) −1131.68 −1.77777
\(75\) −363.378 − 363.378i −0.559457 − 0.559457i
\(76\) − 784.308i − 1.18377i
\(77\) −369.147 −0.546340
\(78\) 503.516 0.730923
\(79\) 549.147 − 549.147i 0.782074 − 0.782074i −0.198106 − 0.980181i \(-0.563479\pi\)
0.980181 + 0.198106i \(0.0634791\pi\)
\(80\) 13.6522 + 13.6522i 0.0190796 + 0.0190796i
\(81\) 422.354 0.579360
\(82\) −357.701 + 357.701i −0.481725 + 0.481725i
\(83\) 367.298 − 367.298i 0.485738 − 0.485738i −0.421221 − 0.906958i \(-0.638398\pi\)
0.906958 + 0.421221i \(0.138398\pi\)
\(84\) 939.396 + 939.396i 1.22020 + 1.22020i
\(85\) 14.4333 − 14.4333i 0.0184178 − 0.0184178i
\(86\) −1139.70 − 1139.70i −1.42903 − 1.42903i
\(87\) 215.912 0.266071
\(88\) 299.112 0.362334
\(89\) −880.846 + 880.846i −1.04910 + 1.04910i −0.0503651 + 0.998731i \(0.516038\pi\)
−0.998731 + 0.0503651i \(0.983962\pi\)
\(90\) −62.5684 − 62.5684i −0.0732810 − 0.0732810i
\(91\) 456.583 + 456.583i 0.525967 + 0.525967i
\(92\) −1307.44 + 1307.44i −1.48164 + 1.48164i
\(93\) 262.026i 0.292159i
\(94\) 388.480 − 388.480i 0.426262 − 0.426262i
\(95\) 101.159 − 101.159i 0.109249 − 0.109249i
\(96\) 605.418 + 605.418i 0.643648 + 0.643648i
\(97\) 191.893 − 191.893i 0.200864 − 0.200864i −0.599506 − 0.800370i \(-0.704636\pi\)
0.800370 + 0.599506i \(0.204636\pi\)
\(98\) 1246.55i 1.28491i
\(99\) 127.804 0.129745
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 137.4.c.a.37.5 ✓ 66
137.100 even 4 inner 137.4.c.a.100.29 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
137.4.c.a.37.5 ✓ 66 1.1 even 1 trivial
137.4.c.a.100.29 yes 66 137.100 even 4 inner