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

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.874820 + 0.0500240i 0.309295 + 0.0176862i 0.211047 − 0.977476i \(-0.432313\pi\)
0.0982480 + 0.995162i \(0.468676\pi\)
\(3\) 1.74018 − 5.35574i 0.334899 − 1.03071i −0.631873 − 0.775072i \(-0.717714\pi\)
0.966772 − 0.255640i \(-0.0822863\pi\)
\(4\) −7.18505 − 0.824407i −0.898131 − 0.103051i
\(5\) 9.63551 + 18.2613i 0.861826 + 1.63334i 0.770427 + 0.637528i \(0.220043\pi\)
0.0913993 + 0.995814i \(0.470866\pi\)
\(6\) 1.79026 − 4.59825i 0.121812 − 0.312871i
\(7\) 26.7923 + 11.3225i 1.44665 + 0.611359i 0.964346 − 0.264645i \(-0.0852548\pi\)
0.482303 + 0.876004i \(0.339800\pi\)
\(8\) −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 0.0834399 − 0.996513i \(-0.473409\pi\)
0.896451 + 0.443143i \(0.146137\pi\)
\(14\) 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 0.171304 − 0.985218i \(-0.445202\pi\)
0.756269 + 0.654261i \(0.227020\pi\)
\(18\) −3.19645 − 2.61373i −0.0418562 − 0.0342256i
\(19\) −1.62989 + 57.0537i −0.0196802 + 0.688896i 0.928531 + 0.371255i \(0.121072\pi\)
−0.948211 + 0.317641i \(0.897109\pi\)
\(20\) −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 0.531870 − 0.846826i \(-0.321489\pi\)
−0.913900 + 0.405940i \(0.866944\pi\)
\(24\) −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 0.813773 − 0.581183i \(-0.197410\pi\)
−0.836826 + 0.547468i \(0.815592\pi\)
\(30\) 101.220 − 11.6139i 0.616007 − 0.0706802i
\(31\) 10.3810 − 51.2324i 0.0601448 − 0.296826i −0.938708 − 0.344714i \(-0.887976\pi\)
0.998853 + 0.0478872i \(0.0152488\pi\)
\(32\) 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 −0.999171 − 0.0407165i \(-0.987036\pi\)
−0.0449369 + 0.998990i \(0.514309\pi\)
\(38\) −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 −0.989186 − 0.146664i \(-0.953146\pi\)
0.789211 − 0.614123i \(-0.210490\pi\)
\(42\) 100.029 − 102.928i 0.367496 − 0.378144i
\(43\) −7.14397 − 49.6874i −0.0253359 − 0.176215i 0.973224 − 0.229858i \(-0.0738260\pi\)
−0.998560 + 0.0536422i \(0.982917\pi\)
\(44\) 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 −0.961426 − 0.275063i \(-0.911301\pi\)
0.0786542 + 0.996902i \(0.474938\pi\)
\(48\) 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 −0.148483 − 0.988915i \(-0.547439\pi\)
0.303082 + 0.952965i \(0.401985\pi\)
\(54\) 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 −0.472753 − 0.881195i \(-0.656740\pi\)
0.844961 − 0.534828i \(-0.179624\pi\)
\(60\) −839.540 + 48.0066i −1.80640 + 0.103294i
\(61\) −780.016 + 44.6029i −1.63723 + 0.0936200i −0.851293 − 0.524691i \(-0.824181\pi\)
−0.785934 + 0.618311i \(0.787817\pi\)
\(62\) 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 −0.410342 − 0.911932i \(-0.634591\pi\)
0.659060 + 0.752090i \(0.270954\pi\)
\(68\) −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 0.174403 − 0.984674i \(-0.444201\pi\)
0.907988 − 0.418995i \(-0.137618\pi\)
\(72\) 43.8333 + 45.1034i 0.0717473 + 0.0738262i
\(73\) −456.215 + 756.547i −0.731450 + 1.21297i 0.238847 + 0.971057i \(0.423231\pi\)
−0.970297 + 0.241917i \(0.922224\pi\)
\(74\) −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 −0.466672 − 0.884431i \(-0.654547\pi\)
0.897396 + 0.441226i \(0.145456\pi\)
\(80\) 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 −0.883888 − 0.467698i \(-0.845084\pi\)
0.950696 − 0.310125i \(-0.100371\pi\)
\(84\) −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 0.246799 − 0.969067i \(-0.420621\pi\)
−0.316296 + 0.948661i \(0.602439\pi\)
\(90\) 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 −0.618526 − 0.785764i \(-0.712270\pi\)
0.997749 + 0.0670567i \(0.0213608\pi\)
\(98\) 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.4.19 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.19 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.19 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.19 yes 1280 121.91 even 55 inner