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Properties

Label 121.4.g.a.4.5
Level $121$
Weight $4$
Character 121.4
Analytic conductor $7.139$
Analytic rank $0$
Dimension $1280$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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

$q$-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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{55}\right)\)

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 −0.750430 − 0.660949i \(-0.770154\pi\)
−0.820881 + 0.571099i \(0.806517\pi\)
\(3\) −1.09754 + 3.37787i −0.211221 + 0.650071i 0.788179 + 0.615446i \(0.211024\pi\)
−0.999400 + 0.0346257i \(0.988976\pi\)
\(4\) 11.7397 + 1.34701i 1.46747 + 0.168376i
\(5\) 8.19282 + 15.5271i 0.732788 + 1.38879i 0.914152 + 0.405372i \(0.132858\pi\)
−0.181363 + 0.983416i \(0.558051\pi\)
\(6\) 5.73627 − 14.7335i 0.390303 − 1.00249i
\(7\) 1.62617 + 0.687227i 0.0878051 + 0.0371068i 0.432730 − 0.901523i \(-0.357550\pi\)
−0.344925 + 0.938630i \(0.612096\pi\)
\(8\) −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 −0.248528 − 0.968625i \(-0.579947\pi\)
0.701141 + 0.713022i \(0.252674\pi\)
\(14\) −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 −0.0201476 − 0.999797i \(-0.506414\pi\)
0.617287 + 0.786738i \(0.288232\pi\)
\(18\) −49.5746 − 40.5369i −0.649158 − 0.530814i
\(19\) −3.16222 + 110.692i −0.0381822 + 1.33655i 0.724274 + 0.689512i \(0.242175\pi\)
−0.762456 + 0.647040i \(0.776007\pi\)
\(20\) 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 0.957286 + 0.289142i \(0.0933701\pi\)
0.149963 + 0.988692i \(0.452084\pi\)
\(24\) 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 −0.977795 − 0.209564i \(-0.932795\pi\)
0.159466 − 0.987203i \(-0.449023\pi\)
\(30\) 275.765 − 31.6411i 1.67825 − 0.192561i
\(31\) −1.20598 + 5.95174i −0.00698710 + 0.0344827i −0.983424 − 0.181318i \(-0.941964\pi\)
0.976437 + 0.215801i \(0.0692363\pi\)
\(32\) 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 −0.770072 − 0.637957i \(-0.779780\pi\)
0.569717 + 0.821841i \(0.307053\pi\)
\(38\) 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 −0.961263 − 0.275634i \(-0.911112\pi\)
0.701478 − 0.712691i \(-0.252524\pi\)
\(42\) 19.4534 − 20.0171i 0.0714697 − 0.0735405i
\(43\) 20.0335 + 139.336i 0.0710482 + 0.494151i 0.994013 + 0.109266i \(0.0348500\pi\)
−0.922964 + 0.384885i \(0.874241\pi\)
\(44\) 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 −0.974187 − 0.225743i \(-0.927519\pi\)
0.0277827 + 0.999614i \(0.491155\pi\)
\(48\) 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 −0.0586518 − 0.998279i \(-0.518680\pi\)
0.387828 + 0.921732i \(0.373226\pi\)
\(54\) 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 −0.500784 − 0.865572i \(-0.666955\pi\)
0.861686 − 0.507442i \(-0.169409\pi\)
\(60\) −735.617 + 42.0641i −1.58280 + 0.0905076i
\(61\) 105.332 − 6.02309i 0.221088 − 0.0126422i 0.0538060 − 0.998551i \(-0.482865\pi\)
0.167282 + 0.985909i \(0.446501\pi\)
\(62\) 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 −0.922731 − 0.385444i \(-0.874048\pi\)
−0.0327040 + 0.999465i \(0.510412\pi\)
\(68\) 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 −0.541455 − 0.840730i \(-0.682126\pi\)
0.950757 + 0.309937i \(0.100308\pi\)
\(72\) −170.344 − 175.280i −0.278822 − 0.286902i
\(73\) 51.1980 − 84.9024i 0.0820859 − 0.136124i −0.812597 − 0.582826i \(-0.801947\pi\)
0.894683 + 0.446702i \(0.147401\pi\)
\(74\) 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 −0.180825 + 0.983515i \(0.557877\pi\)
−0.977951 + 0.208837i \(0.933032\pi\)
\(80\) −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 0.854195 + 0.519952i \(0.174050\pi\)
−0.930348 + 0.366678i \(0.880495\pi\)
\(84\) −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 0.953907 − 0.300104i \(-0.0970214\pi\)
0.640229 + 0.768184i \(0.278840\pi\)
\(90\) 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 −0.787587 − 0.616203i \(-0.788670\pi\)
0.953207 + 0.302320i \(0.0977610\pi\)
\(98\) 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.4.5 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.5 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.5 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.5 yes 1280 121.91 even 55 inner