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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 91.19
Character \(\chi\) \(=\) 121.91
Dual form 121.4.g.a.4.19

$q$-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+(0.874820 - 0.0500240i) q^{2} +(1.74018 + 5.35574i) q^{3} +(-7.18505 + 0.824407i) q^{4} +(9.63551 - 18.2613i) q^{5} +(1.79026 + 4.59825i) q^{6} +(26.7923 - 11.3225i) q^{7} +(-13.1517 + 2.27599i) q^{8} +(-3.81222 + 2.76974i) q^{9} +(7.51583 - 16.4574i) q^{10} +(-36.1056 + 5.23330i) q^{11} +(-16.9186 - 37.0466i) q^{12} +(45.9296 + 25.9376i) q^{13} +(22.8720 - 11.2454i) q^{14} +(114.570 + 19.8272i) q^{15} +(44.9624 - 10.4556i) q^{16} +(65.0162 + 23.1977i) q^{17} +(-3.19645 + 2.61373i) q^{18} +(-1.62989 - 57.0537i) q^{19} +(-54.1768 + 139.152i) q^{20} +(107.264 + 123.789i) q^{21} +(-31.3241 + 6.38434i) q^{22} +(-42.1394 + 48.6315i) q^{23} +(-35.0760 - 66.4764i) q^{24} +(-170.077 - 248.730i) q^{25} +(41.4776 + 20.3932i) q^{26} +(101.540 + 73.7733i) q^{27} +(-183.170 + 103.441i) q^{28} +(-3.60026 + 5.26521i) q^{29} +(101.220 + 11.6139i) q^{30} +(10.3810 + 51.2324i) q^{31} +(141.263 - 41.4786i) q^{32} +(-90.8586 - 184.265i) q^{33} +(58.0379 + 17.0415i) q^{34} +(51.3933 - 598.362i) q^{35} +(25.1076 - 23.0436i) q^{36} +(-234.989 - 215.671i) q^{37} +(-4.27992 - 49.8302i) q^{38} +(-58.9892 + 291.123i) q^{39} +(-85.1607 + 262.098i) q^{40} +(-52.4993 + 199.728i) q^{41} +(100.029 + 102.928i) q^{42} +(-7.14397 + 49.6874i) q^{43} +(255.106 - 67.3672i) q^{44} +(13.8464 + 96.3041i) q^{45} +(-34.4317 + 44.6518i) q^{46} +(-284.443 - 232.588i) q^{47} +(134.240 + 222.612i) q^{48} +(350.579 - 360.737i) q^{49} +(-161.230 - 209.086i) q^{50} +(-11.1008 + 388.578i) q^{51} +(-351.390 - 148.498i) q^{52} +(59.6511 + 13.8713i) q^{53} +(92.5199 + 59.4589i) q^{54} +(-252.329 + 709.761i) q^{55} +(-326.595 + 209.890i) q^{56} +(302.728 - 108.013i) q^{57} +(-2.88619 + 4.78621i) q^{58} +(168.680 + 641.724i) q^{59} +(-839.540 - 48.0066i) q^{60} +(-780.016 - 44.6029i) q^{61} +(11.6444 + 44.2998i) q^{62} +(-70.7778 + 117.372i) q^{63} +(-226.315 + 80.7490i) q^{64} +(916.211 - 588.813i) q^{65} +(-88.7025 - 156.654i) q^{66} +(136.402 + 87.6601i) q^{67} +(-486.269 - 113.077i) q^{68} +(-333.788 - 141.060i) q^{69} +(15.0274 - 526.029i) q^{70} +(647.548 + 839.755i) q^{71} +(43.8333 - 45.1034i) q^{72} +(-456.215 - 756.547i) q^{73} +(-216.362 - 176.918i) q^{74} +(1036.17 - 1343.73i) q^{75} +(58.7464 + 408.590i) q^{76} +(-908.098 + 549.019i) q^{77} +(-37.0418 + 257.631i) q^{78} +(302.441 + 311.204i) q^{79} +(242.303 - 921.817i) q^{80} +(-257.728 + 793.204i) q^{81} +(-35.9362 + 177.352i) q^{82} +(50.5174 + 588.163i) q^{83} +(-872.750 - 801.003i) q^{84} +(1050.09 - 963.760i) q^{85} +(-3.76412 + 43.8249i) q^{86} +(-34.4642 - 10.1196i) q^{87} +(462.939 - 151.003i) q^{88} +(-58.3509 + 17.1334i) q^{89} +(16.9307 + 83.5561i) q^{90} +(1524.24 + 174.890i) q^{91} +(262.682 - 384.160i) q^{92} +(-256.323 + 144.752i) q^{93} +(-260.471 - 189.244i) q^{94} +(-1057.58 - 519.978i) q^{95} +(467.973 + 684.388i) q^{96} +(362.287 + 686.609i) q^{97} +(288.648 - 333.117i) q^{98} +(123.148 - 119.954i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.874820 − 0.0500240i 0.309295 − 0.0176862i 0.0982480 − 0.995162i \(-0.468676\pi\)
0.211047 + 0.977476i \(0.432313\pi\)
\(3\) 1.74018 + 5.35574i 0.334899 + 1.03071i 0.966772 + 0.255640i \(0.0822863\pi\)
−0.631873 + 0.775072i \(0.717714\pi\)
\(4\) −7.18505 + 0.824407i −0.898131 + 0.103051i
\(5\) 9.63551 − 18.2613i 0.861826 − 1.63334i 0.0913993 − 0.995814i \(-0.470866\pi\)
0.770427 − 0.637528i \(-0.220043\pi\)
\(6\) 1.79026 + 4.59825i 0.121812 + 0.312871i
\(7\) 26.7923 − 11.3225i 1.44665 − 0.611359i 0.482303 − 0.876004i \(-0.339800\pi\)
0.964346 + 0.264645i \(0.0852548\pi\)
\(8\) −13.1517 + 2.27599i −0.581228 + 0.100586i
\(9\) −3.81222 + 2.76974i −0.141193 + 0.102583i
\(10\) 7.51583 − 16.4574i 0.237671 − 0.520428i
\(11\) −36.1056 + 5.23330i −0.989658 + 0.143445i
\(12\) −16.9186 − 37.0466i −0.406999 − 0.891203i
\(13\) 45.9296 + 25.9376i 0.979891 + 0.553370i 0.896451 − 0.443143i \(-0.146137\pi\)
0.0834399 + 0.996513i \(0.473409\pi\)
\(14\) 22.8720 − 11.2454i 0.436629 − 0.214676i
\(15\) 114.570 + 19.8272i 1.97213 + 0.341291i
\(16\) 44.9624 − 10.4556i 0.702537 − 0.163368i
\(17\) 65.0162 + 23.1977i 0.927573 + 0.330957i 0.756269 − 0.654261i \(-0.227020\pi\)
0.171304 + 0.985218i \(0.445202\pi\)
\(18\) −3.19645 + 2.61373i −0.0418562 + 0.0342256i
\(19\) −1.62989 − 57.0537i −0.0196802 − 0.688896i −0.948211 − 0.317641i \(-0.897109\pi\)
0.928531 − 0.371255i \(-0.121072\pi\)
\(20\) −54.1768 + 139.152i −0.605715 + 1.55577i
\(21\) 107.264 + 123.789i 1.11462 + 1.28634i
\(22\) −31.3241 + 6.38434i −0.303560 + 0.0618703i
\(23\) −42.1394 + 48.6315i −0.382029 + 0.440885i −0.913900 − 0.405940i \(-0.866944\pi\)
0.531870 + 0.846826i \(0.321489\pi\)
\(24\) −35.0760 − 66.4764i −0.298327 − 0.565393i
\(25\) −170.077 − 248.730i −1.36062 − 1.98984i
\(26\) 41.4776 + 20.3932i 0.312863 + 0.153824i
\(27\) 101.540 + 73.7733i 0.723757 + 0.525840i
\(28\) −183.170 + 103.441i −1.23628 + 0.698159i
\(29\) −3.60026 + 5.26521i −0.0230535 + 0.0337146i −0.836826 − 0.547468i \(-0.815592\pi\)
0.813773 + 0.581183i \(0.197410\pi\)
\(30\) 101.220 + 11.6139i 0.616007 + 0.0706802i
\(31\) 10.3810 + 51.2324i 0.0601448 + 0.296826i 0.998853 − 0.0478872i \(-0.0152488\pi\)
−0.938708 + 0.344714i \(0.887976\pi\)
\(32\) 141.263 − 41.4786i 0.780376 − 0.229139i
\(33\) −90.8586 − 184.265i −0.479286 − 0.972013i
\(34\) 58.0379 + 17.0415i 0.292748 + 0.0859584i
\(35\) 51.3933 − 598.362i 0.248202 − 2.88976i
\(36\) 25.1076 − 23.0436i 0.116239 − 0.106683i
\(37\) −234.989 − 215.671i −1.04411 − 0.958273i −0.0449369 − 0.998990i \(-0.514309\pi\)
−0.999171 + 0.0407165i \(0.987036\pi\)
\(38\) −4.27992 − 49.8302i −0.0182709 − 0.212724i
\(39\) −58.9892 + 291.123i −0.242201 + 1.19531i
\(40\) −85.1607 + 262.098i −0.336627 + 1.03603i
\(41\) −52.4993 + 199.728i −0.199976 + 0.760787i 0.789211 + 0.614123i \(0.210490\pi\)
−0.989186 + 0.146664i \(0.953146\pi\)
\(42\) 100.029 + 102.928i 0.367496 + 0.378144i
\(43\) −7.14397 + 49.6874i −0.0253359 + 0.176215i −0.998560 − 0.0536422i \(-0.982917\pi\)
0.973224 + 0.229858i \(0.0738260\pi\)
\(44\) 255.106 − 67.3672i 0.874060 − 0.230818i
\(45\) 13.8464 + 96.3041i 0.0458690 + 0.319026i
\(46\) −34.4317 + 44.6518i −0.110362 + 0.143120i
\(47\) −284.443 − 232.588i −0.882772 − 0.721839i 0.0786542 − 0.996902i \(-0.474938\pi\)
−0.961426 + 0.275063i \(0.911301\pi\)
\(48\) 134.240 + 222.612i 0.403664 + 0.669402i
\(49\) 350.579 − 360.737i 1.02210 − 1.05171i
\(50\) −161.230 − 209.086i −0.456026 − 0.591385i
\(51\) −11.1008 + 388.578i −0.0304788 + 1.06690i
\(52\) −351.390 − 148.498i −0.937096 − 0.396020i
\(53\) 59.6511 + 13.8713i 0.154598 + 0.0359503i 0.303082 − 0.952965i \(-0.401985\pi\)
−0.148483 + 0.988915i \(0.547439\pi\)
\(54\) 92.5199 + 59.4589i 0.233155 + 0.149840i
\(55\) −252.329 + 709.761i −0.618618 + 1.74008i
\(56\) −326.595 + 209.890i −0.779340 + 0.500851i
\(57\) 302.728 − 108.013i 0.703462 − 0.250995i
\(58\) −2.88619 + 4.78621i −0.00653405 + 0.0108355i
\(59\) 168.680 + 641.724i 0.372207 + 1.41602i 0.844961 + 0.534828i \(0.179624\pi\)
−0.472753 + 0.881195i \(0.656740\pi\)
\(60\) −839.540 − 48.0066i −1.80640 − 0.103294i
\(61\) −780.016 − 44.6029i −1.63723 − 0.0936200i −0.785934 − 0.618311i \(-0.787817\pi\)
−0.851293 + 0.524691i \(0.824181\pi\)
\(62\) 11.6444 + 44.2998i 0.0238522 + 0.0907433i
\(63\) −70.7778 + 117.372i −0.141542 + 0.234722i
\(64\) −226.315 + 80.7490i −0.442021 + 0.157713i
\(65\) 916.211 − 588.813i 1.74834 − 1.12359i
\(66\) −88.7025 − 156.654i −0.165432 − 0.292162i
\(67\) 136.402 + 87.6601i 0.248718 + 0.159842i 0.659060 − 0.752090i \(-0.270954\pi\)
−0.410342 + 0.911932i \(0.634591\pi\)
\(68\) −486.269 − 113.077i −0.867188 − 0.201656i
\(69\) −333.788 − 141.060i −0.582367 − 0.246110i
\(70\) 15.0274 − 526.029i 0.0256589 − 0.898179i
\(71\) 647.548 + 839.755i 1.08239 + 1.40367i 0.907988 + 0.418995i \(0.137618\pi\)
0.174403 + 0.984674i \(0.444201\pi\)
\(72\) 43.8333 − 45.1034i 0.0717473 − 0.0738262i
\(73\) −456.215 − 756.547i −0.731450 − 1.21297i −0.970297 − 0.241917i \(-0.922224\pi\)
0.238847 − 0.971057i \(-0.423231\pi\)
\(74\) −216.362 − 176.918i −0.339886 − 0.277923i
\(75\) 1036.17 − 1343.73i 1.59529 − 2.06880i
\(76\) 58.7464 + 408.590i 0.0886667 + 0.616691i
\(77\) −908.098 + 549.019i −1.34399 + 0.812552i
\(78\) −37.0418 + 257.631i −0.0537712 + 0.373987i
\(79\) 302.441 + 311.204i 0.430724 + 0.443205i 0.897396 − 0.441226i \(-0.145456\pi\)
−0.466672 + 0.884431i \(0.654547\pi\)
\(80\) 242.303 − 921.817i 0.338629 − 1.28828i
\(81\) −257.728 + 793.204i −0.353536 + 1.08807i
\(82\) −35.9362 + 177.352i −0.0483962 + 0.238845i
\(83\) 50.5174 + 588.163i 0.0668073 + 0.777823i 0.950696 + 0.310125i \(0.100371\pi\)
−0.883888 + 0.467698i \(0.845084\pi\)
\(84\) −872.750 − 801.003i −1.13363 − 1.04044i
\(85\) 1050.09 − 963.760i 1.33997 − 1.22982i
\(86\) −3.76412 + 43.8249i −0.00471972 + 0.0549507i
\(87\) −34.4642 − 10.1196i −0.0424707 − 0.0124705i
\(88\) 462.939 − 151.003i 0.560789 − 0.182920i
\(89\) −58.3509 + 17.1334i −0.0694965 + 0.0204060i −0.316296 − 0.948661i \(-0.602439\pi\)
0.246799 + 0.969067i \(0.420621\pi\)
\(90\) 16.9307 + 83.5561i 0.0198294 + 0.0978621i
\(91\) 1524.24 + 174.890i 1.75587 + 0.201467i
\(92\) 262.682 − 384.160i 0.297679 − 0.435341i
\(93\) −256.323 + 144.752i −0.285800 + 0.161399i
\(94\) −260.471 − 189.244i −0.285804 − 0.207649i
\(95\) −1057.58 − 519.978i −1.14216 − 0.561564i
\(96\) 467.973 + 684.388i 0.497523 + 0.727605i
\(97\) 362.287 + 686.609i 0.379223 + 0.718708i 0.997749 − 0.0670567i \(-0.0213608\pi\)
−0.618526 + 0.785764i \(0.712270\pi\)
\(98\) 288.648 − 333.117i 0.297529 − 0.343367i
\(99\) 123.148 − 119.954i 0.125018 − 0.121776i
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.91.19 yes 1280
121.4 even 55 inner 121.4.g.a.4.19 ✓ 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.19 ✓ 1280 121.4 even 55 inner
121.4.g.a.91.19 yes 1280 1.1 even 1 trivial