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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,4,Mod(4,121)] 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("121.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(110)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.g (of order \(55\), degree \(40\), 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(1280\)
Relative dimension: \(32\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label 91.5
Character \(\chi\) \(=\) 121.91
Dual form 121.4.g.a.4.5

$q$-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+(-4.44434 + 0.254137i) q^{2} +(-1.09754 - 3.37787i) q^{3} +(11.7397 - 1.34701i) q^{4} +(8.19282 - 15.5271i) q^{5} +(5.73627 + 14.7335i) q^{6} +(1.62617 - 0.687227i) q^{7} +(-16.7418 + 2.89728i) q^{8} +(11.6380 - 8.45553i) q^{9} +(-32.4657 + 71.0899i) q^{10} +(27.6665 + 23.7816i) q^{11} +(-17.4348 - 38.1769i) q^{12} +(21.2150 + 11.9806i) q^{13} +(-7.05262 + 3.46754i) q^{14} +(-61.4405 - 10.6327i) q^{15} +(-18.4072 + 4.28042i) q^{16} +(41.8552 + 14.9339i) q^{17} +(-49.5746 + 40.5369i) q^{18} +(-3.16222 - 110.692i) q^{19} +(75.2663 - 193.320i) q^{20} +(-4.10615 - 4.73875i) q^{21} +(-129.003 - 98.6623i) q^{22} +(122.134 - 140.950i) q^{23} +(28.1614 + 53.3718i) q^{24} +(-103.414 - 151.238i) q^{25} +(-97.3312 - 47.8545i) q^{26} +(-118.916 - 86.3978i) q^{27} +(18.1651 - 10.2583i) q^{28} +(-127.798 + 186.899i) q^{29} +(275.765 + 31.6411i) q^{30} +(-1.20598 - 5.95174i) q^{31} +(211.140 - 61.9962i) q^{32} +(49.9660 - 119.555i) q^{33} +(-189.814 - 55.7344i) q^{34} +(2.65230 - 30.8801i) q^{35} +(125.238 - 114.942i) q^{36} +(-45.0923 - 41.3854i) q^{37} +(42.1849 + 491.150i) q^{38} +(17.1848 - 84.8105i) q^{39} +(-92.1762 + 283.689i) q^{40} +(-68.2009 + 259.463i) q^{41} +(19.4534 + 20.0171i) q^{42} +(20.0335 - 139.336i) q^{43} +(356.832 + 241.922i) q^{44} +(-35.9417 - 249.980i) q^{45} +(-506.985 + 657.470i) q^{46} +(-304.946 - 249.353i) q^{47} +(34.6613 + 57.4793i) q^{48} +(-236.877 + 243.741i) q^{49} +(498.042 + 645.872i) q^{50} +(4.50718 - 157.772i) q^{51} +(265.196 + 112.073i) q^{52} +(127.011 + 29.5352i) q^{53} +(550.461 + 353.760i) q^{54} +(595.926 - 234.744i) q^{55} +(-25.2340 + 16.2169i) q^{56} +(-370.433 + 132.170i) q^{57} +(520.481 - 863.121i) q^{58} +(163.556 + 622.233i) q^{59} +(-735.617 - 42.0641i) q^{60} +(105.332 + 6.02309i) q^{61} +(6.87234 + 26.1451i) q^{62} +(13.1146 - 21.7481i) q^{63} +(-780.226 + 278.384i) q^{64} +(359.835 - 231.252i) q^{65} +(-191.682 + 544.042i) q^{66} +(-523.978 - 336.740i) q^{67} +(511.484 + 118.941i) q^{68} +(-610.158 - 257.855i) q^{69} +(-3.93994 + 137.916i) q^{70} +(244.868 + 317.550i) q^{71} +(-170.344 + 175.280i) q^{72} +(51.1980 + 84.9024i) q^{73} +(210.923 + 172.471i) q^{74} +(-397.362 + 515.308i) q^{75} +(-186.227 - 1295.23i) q^{76} +(61.3339 + 19.6598i) q^{77} +(-54.8218 + 381.294i) q^{78} +(-813.654 - 837.231i) q^{79} +(-84.3447 + 320.880i) q^{80} +(-41.3012 + 127.112i) q^{81} +(237.169 - 1170.47i) q^{82} +(-57.5841 - 670.439i) q^{83} +(-54.5882 - 50.1006i) q^{84} +(574.792 - 527.540i) q^{85} +(-53.6252 + 624.347i) q^{86} +(771.583 + 226.557i) q^{87} +(-532.090 - 317.989i) q^{88} +(1338.47 - 393.012i) q^{89} +(223.266 + 1101.86i) q^{90} +(42.7326 + 4.90311i) q^{91} +(1243.96 - 1819.23i) q^{92} +(-18.7806 + 10.6059i) q^{93} +(1418.66 + 1030.71i) q^{94} +(-1744.64 - 857.780i) q^{95} +(-441.148 - 645.159i) q^{96} +(158.222 + 299.865i) q^{97} +(990.821 - 1143.47i) q^{98} +(523.070 + 42.8355i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 1280 q - 37 q^{2} - 32 q^{3} + 73 q^{4} - 33 q^{5} + 73 q^{6} - 9 q^{7} - 91 q^{8} - 2748 q^{9} + 48 q^{10} + 43 q^{11} + 323 q^{12} - 287 q^{13} - 120 q^{14} + 632 q^{15} + 785 q^{16} - 13 q^{17} + 443 q^{18}+ \cdots + 13103 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{54}{55}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.44434 + 0.254137i −1.57131 + 0.0898509i −0.820881 − 0.571099i \(-0.806517\pi\)
−0.750430 + 0.660949i \(0.770154\pi\)
\(3\) −1.09754 − 3.37787i −0.211221 − 0.650071i −0.999400 − 0.0346257i \(-0.988976\pi\)
0.788179 − 0.615446i \(-0.211024\pi\)
\(4\) 11.7397 − 1.34701i 1.46747 − 0.168376i
\(5\) 8.19282 − 15.5271i 0.732788 − 1.38879i −0.181363 − 0.983416i \(-0.558051\pi\)
0.914152 − 0.405372i \(-0.132858\pi\)
\(6\) 5.73627 + 14.7335i 0.390303 + 1.00249i
\(7\) 1.62617 − 0.687227i 0.0878051 − 0.0371068i −0.344925 − 0.938630i \(-0.612096\pi\)
0.432730 + 0.901523i \(0.357550\pi\)
\(8\) −16.7418 + 2.89728i −0.739891 + 0.128043i
\(9\) 11.6380 − 8.45553i 0.431039 − 0.313168i
\(10\) −32.4657 + 71.0899i −1.02666 + 2.24806i
\(11\) 27.6665 + 23.7816i 0.758343 + 0.651855i
\(12\) −17.4348 − 38.1769i −0.419416 − 0.918393i
\(13\) 21.2150 + 11.9806i 0.452613 + 0.255602i 0.701141 − 0.713022i \(-0.252674\pi\)
−0.248528 + 0.968625i \(0.579947\pi\)
\(14\) −7.05262 + 3.46754i −0.134635 + 0.0661957i
\(15\) −61.4405 − 10.6327i −1.05759 − 0.183023i
\(16\) −18.4072 + 4.28042i −0.287613 + 0.0668816i
\(17\) 41.8552 + 14.9339i 0.597139 + 0.213059i 0.617287 − 0.786738i \(-0.288232\pi\)
−0.0201476 + 0.999797i \(0.506414\pi\)
\(18\) −49.5746 + 40.5369i −0.649158 + 0.530814i
\(19\) −3.16222 − 110.692i −0.0381822 − 1.33655i −0.762456 − 0.647040i \(-0.776007\pi\)
0.724274 − 0.689512i \(-0.242175\pi\)
\(20\) 75.2663 − 193.320i 0.841503 − 2.16138i
\(21\) −4.10615 − 4.73875i −0.0426683 − 0.0492419i
\(22\) −129.003 − 98.6623i −1.25016 − 0.956130i
\(23\) 122.134 − 140.950i 1.10725 − 1.27783i 0.149963 − 0.988692i \(-0.452084\pi\)
0.957286 − 0.289142i \(-0.0933701\pi\)
\(24\) 28.1614 + 53.3718i 0.239518 + 0.453936i
\(25\) −103.414 − 151.238i −0.827311 − 1.20990i
\(26\) −97.3312 − 47.8545i −0.734162 − 0.360963i
\(27\) −118.916 − 86.3978i −0.847609 − 0.615824i
\(28\) 18.1651 − 10.2583i 0.122603 − 0.0692372i
\(29\) −127.798 + 186.899i −0.818329 + 1.19677i 0.159466 + 0.987203i \(0.449023\pi\)
−0.977795 + 0.209564i \(0.932795\pi\)
\(30\) 275.765 + 31.6411i 1.67825 + 0.192561i
\(31\) −1.20598 − 5.95174i −0.00698710 − 0.0344827i 0.976437 − 0.215801i \(-0.0692363\pi\)
−0.983424 + 0.181318i \(0.941964\pi\)
\(32\) 211.140 − 61.9962i 1.16639 − 0.342484i
\(33\) 49.9660 − 119.555i 0.263575 − 0.630663i
\(34\) −189.814 − 55.7344i −0.957436 − 0.281128i
\(35\) 2.65230 − 30.8801i 0.0128091 − 0.149134i
\(36\) 125.238 − 114.942i 0.579805 − 0.532140i
\(37\) −45.0923 − 41.3854i −0.200355 − 0.183884i 0.569717 − 0.821841i \(-0.307053\pi\)
−0.770072 + 0.637957i \(0.779780\pi\)
\(38\) 42.1849 + 491.150i 0.180087 + 2.09671i
\(39\) 17.1848 − 84.8105i 0.0705584 − 0.348219i
\(40\) −92.1762 + 283.689i −0.364358 + 1.12138i
\(41\) −68.2009 + 259.463i −0.259785 + 0.988325i 0.701478 + 0.712691i \(0.252524\pi\)
−0.961263 + 0.275634i \(0.911112\pi\)
\(42\) 19.4534 + 20.0171i 0.0714697 + 0.0735405i
\(43\) 20.0335 − 139.336i 0.0710482 − 0.494151i −0.922964 − 0.384885i \(-0.874241\pi\)
0.994013 − 0.109266i \(-0.0348500\pi\)
\(44\) 356.832 + 241.922i 1.22260 + 0.828889i
\(45\) −35.9417 − 249.980i −0.119064 − 0.828107i
\(46\) −506.985 + 657.470i −1.62502 + 2.10736i
\(47\) −304.946 − 249.353i −0.946404 − 0.773871i 0.0277827 − 0.999614i \(-0.491155\pi\)
−0.974187 + 0.225743i \(0.927519\pi\)
\(48\) 34.6613 + 57.4793i 0.104228 + 0.172842i
\(49\) −236.877 + 243.741i −0.690605 + 0.710615i
\(50\) 498.042 + 645.872i 1.40867 + 1.82680i
\(51\) 4.50718 − 157.772i 0.0123751 − 0.433186i
\(52\) 265.196 + 112.073i 0.707232 + 0.298879i
\(53\) 127.011 + 29.5352i 0.329176 + 0.0765465i 0.387828 − 0.921732i \(-0.373226\pi\)
−0.0586518 + 0.998279i \(0.518680\pi\)
\(54\) 550.461 + 353.760i 1.38719 + 0.891494i
\(55\) 595.926 − 234.744i 1.46099 − 0.575506i
\(56\) −25.2340 + 16.2169i −0.0602149 + 0.0386978i
\(57\) −370.433 + 132.170i −0.860789 + 0.307129i
\(58\) 520.481 − 863.121i 1.17832 − 1.95402i
\(59\) 163.556 + 622.233i 0.360902 + 1.37301i 0.861686 + 0.507442i \(0.169409\pi\)
−0.500784 + 0.865572i \(0.666955\pi\)
\(60\) −735.617 − 42.0641i −1.58280 − 0.0905076i
\(61\) 105.332 + 6.02309i 0.221088 + 0.0126422i 0.167282 − 0.985909i \(-0.446501\pi\)
0.0538060 + 0.998551i \(0.482865\pi\)
\(62\) 6.87234 + 26.1451i 0.0140772 + 0.0535553i
\(63\) 13.1146 − 21.7481i 0.0262268 − 0.0434922i
\(64\) −780.226 + 278.384i −1.52388 + 0.543719i
\(65\) 359.835 − 231.252i 0.686647 − 0.441281i
\(66\) −191.682 + 544.042i −0.357492 + 1.01465i
\(67\) −523.978 − 336.740i −0.955435 − 0.614021i −0.0327040 − 0.999465i \(-0.510412\pi\)
−0.922731 + 0.385444i \(0.874048\pi\)
\(68\) 511.484 + 118.941i 0.912156 + 0.212113i
\(69\) −610.158 − 257.855i −1.06456 − 0.449886i
\(70\) −3.93994 + 137.916i −0.00672733 + 0.235487i
\(71\) 244.868 + 317.550i 0.409303 + 0.530793i 0.950757 − 0.309937i \(-0.100308\pi\)
−0.541455 + 0.840730i \(0.682126\pi\)
\(72\) −170.344 + 175.280i −0.278822 + 0.286902i
\(73\) 51.1980 + 84.9024i 0.0820859 + 0.136124i 0.894683 − 0.446702i \(-0.147401\pi\)
−0.812597 + 0.582826i \(0.801947\pi\)
\(74\) 210.923 + 172.471i 0.331342 + 0.270937i
\(75\) −397.362 + 515.308i −0.611778 + 0.793368i
\(76\) −186.227 − 1295.23i −0.281074 − 1.95492i
\(77\) 61.3339 + 19.6598i 0.0907747 + 0.0290966i
\(78\) −54.8218 + 381.294i −0.0795814 + 0.553501i
\(79\) −813.654 − 837.231i −1.15878 − 1.19235i −0.977951 − 0.208837i \(-0.933032\pi\)
−0.180825 − 0.983515i \(-0.557877\pi\)
\(80\) −84.3447 + 320.880i −0.117875 + 0.448444i
\(81\) −41.3012 + 127.112i −0.0566547 + 0.174365i
\(82\) 237.169 − 1170.47i 0.319402 − 1.57631i
\(83\) −57.5841 − 670.439i −0.0761527 − 0.886630i −0.930348 − 0.366678i \(-0.880495\pi\)
0.854195 − 0.519952i \(-0.174050\pi\)
\(84\) −54.5882 − 50.1006i −0.0709055 − 0.0650764i
\(85\) 574.792 − 527.540i 0.733470 − 0.673173i
\(86\) −53.6252 + 624.347i −0.0672390 + 0.782849i
\(87\) 771.583 + 226.557i 0.950833 + 0.279190i
\(88\) −532.090 − 317.989i −0.644557 − 0.385201i
\(89\) 1338.47 − 393.012i 1.59414 − 0.468080i 0.640229 − 0.768184i \(-0.278840\pi\)
0.953907 + 0.300104i \(0.0970214\pi\)
\(90\) 223.266 + 1101.86i 0.261493 + 1.29052i
\(91\) 42.7326 + 4.90311i 0.0492263 + 0.00564820i
\(92\) 1243.96 − 1819.23i 1.40969 − 2.06161i
\(93\) −18.7806 + 10.6059i −0.0209404 + 0.0118256i
\(94\) 1418.66 + 1030.71i 1.55663 + 1.13096i
\(95\) −1744.64 − 857.780i −1.88417 − 0.926383i
\(96\) −441.148 − 645.159i −0.469005 − 0.685899i
\(97\) 158.222 + 299.865i 0.165619 + 0.313883i 0.953207 − 0.302320i \(-0.0977610\pi\)
−0.787587 + 0.616203i \(0.788670\pi\)
\(98\) 990.821 − 1143.47i 1.02131 − 1.17865i
\(99\) 523.070 + 42.8355i 0.531016 + 0.0434862i
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 121.4.g.a.91.5 yes 1280
121.4 even 55 inner 121.4.g.a.4.5 ✓ 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.5 ✓ 1280 121.4 even 55 inner
121.4.g.a.91.5 yes 1280 1.1 even 1 trivial