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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 4.16
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.16

$q$-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+(-0.457884 - 0.0261828i) q^{2} +(-1.82617 + 5.62036i) q^{3} +(-7.73888 - 0.887954i) q^{4} +(-4.58861 - 8.69638i) q^{5} +(0.983329 - 2.52566i) q^{6} +(8.14222 + 3.44093i) q^{7} +(7.13558 + 1.23486i) q^{8} +(-6.41011 - 4.65722i) q^{9} +(1.87336 + 4.10208i) q^{10} +(-30.0719 - 20.6563i) q^{11} +(19.1231 - 41.8738i) q^{12} +(37.0720 - 20.9355i) q^{13} +(-3.63810 - 1.78873i) q^{14} +(57.2564 - 9.90861i) q^{15} +(57.4628 + 13.3624i) q^{16} +(50.5991 - 18.0537i) q^{17} +(2.81315 + 2.30030i) q^{18} +(3.81004 - 133.369i) q^{19} +(27.7887 + 71.3748i) q^{20} +(-34.2083 + 39.4785i) q^{21} +(13.2286 + 10.2456i) q^{22} +(124.270 + 143.415i) q^{23} +(-19.9711 + 37.8495i) q^{24} +(15.9836 - 23.3753i) q^{25} +(-17.5228 + 8.61540i) q^{26} +(-91.2049 + 66.2642i) q^{27} +(-59.9563 - 33.8589i) q^{28} +(-95.0490 - 139.005i) q^{29} +(-26.4762 + 3.03786i) q^{30} +(55.9556 - 276.152i) q^{31} +(-81.5479 - 23.9446i) q^{32} +(171.012 - 131.293i) q^{33} +(-23.6412 + 6.94168i) q^{34} +(-7.43782 - 86.5970i) q^{35} +(45.4717 + 41.7336i) q^{36} +(-278.328 + 255.447i) q^{37} +(-5.23652 + 60.9677i) q^{38} +(49.9656 + 246.590i) q^{39} +(-22.0036 - 67.7200i) q^{40} +(-131.279 - 499.437i) q^{41} +(16.6971 - 17.1809i) q^{42} +(25.4970 + 177.335i) q^{43} +(214.381 + 186.559i) q^{44} +(-11.0875 + 77.1150i) q^{45} +(-53.1462 - 68.9212i) q^{46} +(68.8342 - 56.2854i) q^{47} +(-180.038 + 298.560i) q^{48} +(-184.594 - 189.943i) q^{49} +(-7.93069 + 10.2847i) q^{50} +(9.06608 + 317.354i) q^{51} +(-305.486 + 129.099i) q^{52} +(-103.197 + 23.9975i) q^{53} +(43.4962 - 27.9533i) q^{54} +(-41.6471 + 356.300i) q^{55} +(53.8504 + 34.6075i) q^{56} +(742.624 + 264.968i) q^{57} +(39.8819 + 66.1367i) q^{58} +(-105.839 + 402.653i) q^{59} +(-451.899 + 25.8405i) q^{60} +(576.717 - 32.9779i) q^{61} +(-32.8516 + 124.980i) q^{62} +(-36.1674 - 59.9768i) q^{63} +(-407.808 - 145.506i) q^{64} +(-352.172 - 226.328i) q^{65} +(-81.7413 + 55.6393i) q^{66} +(777.748 - 499.828i) q^{67} +(-407.611 + 94.7859i) q^{68} +(-1032.98 + 436.542i) q^{69} +(1.13831 + 39.8461i) q^{70} +(-71.1561 + 92.2769i) q^{71} +(-39.9889 - 41.1476i) q^{72} +(336.795 - 558.512i) q^{73} +(134.130 - 109.678i) q^{74} +(102.189 + 132.521i) q^{75} +(-147.911 + 1028.74i) q^{76} +(-173.775 - 271.663i) q^{77} +(-16.4220 - 114.218i) q^{78} +(479.731 - 493.632i) q^{79} +(-147.470 - 561.034i) q^{80} +(-271.982 - 837.073i) q^{81} +(47.0339 + 232.121i) q^{82} +(-80.3172 + 935.117i) q^{83} +(299.789 - 275.144i) q^{84} +(-389.182 - 357.188i) q^{85} +(-7.03153 - 81.8667i) q^{86} +(954.832 - 280.364i) q^{87} +(-189.073 - 184.529i) q^{88} +(422.138 + 123.951i) q^{89} +(7.09585 - 35.0194i) q^{90} +(373.886 - 42.8994i) q^{91} +(-834.364 - 1220.22i) q^{92} +(1449.89 + 818.789i) q^{93} +(-32.9918 + 23.9699i) q^{94} +(-1177.31 + 578.844i) q^{95} +(283.497 - 414.602i) q^{96} +(320.760 - 607.908i) q^{97} +(79.5494 + 91.8049i) q^{98} +(96.5631 + 272.461i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.457884 − 0.0261828i −0.161886 − 0.00925700i −0.0240423 − 0.999711i \(-0.507654\pi\)
−0.137844 + 0.990454i \(0.544017\pi\)
\(3\) −1.82617 + 5.62036i −0.351446 + 1.08164i 0.606596 + 0.795010i \(0.292535\pi\)
−0.958042 + 0.286629i \(0.907465\pi\)
\(4\) −7.73888 − 0.887954i −0.967360 − 0.110994i
\(5\) −4.58861 − 8.69638i −0.410418 − 0.777828i 0.589126 − 0.808041i \(-0.299472\pi\)
−0.999543 + 0.0302132i \(0.990381\pi\)
\(6\) 0.983329 − 2.52566i 0.0669070 − 0.171849i
\(7\) 8.14222 + 3.44093i 0.439638 + 0.185793i 0.597842 − 0.801614i \(-0.296025\pi\)
−0.158204 + 0.987406i \(0.550570\pi\)
\(8\) 7.13558 + 1.23486i 0.315351 + 0.0545736i
\(9\) −6.41011 − 4.65722i −0.237412 − 0.172490i
\(10\) 1.87336 + 4.10208i 0.0592407 + 0.129719i
\(11\) −30.0719 − 20.6563i −0.824273 − 0.566192i
\(12\) 19.1231 − 41.8738i 0.460030 − 1.00733i
\(13\) 37.0720 − 20.9355i 0.790917 − 0.446652i −0.0426358 − 0.999091i \(-0.513576\pi\)
0.833553 + 0.552439i \(0.186303\pi\)
\(14\) −3.63810 − 1.78873i −0.0694516 − 0.0341471i
\(15\) 57.2564 − 9.90861i 0.985569 − 0.170559i
\(16\) 57.4628 + 13.3624i 0.897856 + 0.208788i
\(17\) 50.5991 − 18.0537i 0.721887 − 0.257569i 0.0505506 − 0.998722i \(-0.483902\pi\)
0.671336 + 0.741153i \(0.265721\pi\)
\(18\) 2.81315 + 2.30030i 0.0368370 + 0.0301214i
\(19\) 3.81004 − 133.369i 0.0460044 − 1.61036i −0.569337 − 0.822104i \(-0.692800\pi\)
0.615342 − 0.788261i \(-0.289018\pi\)
\(20\) 27.7887 + 71.3748i 0.310687 + 0.797994i
\(21\) −34.2083 + 39.4785i −0.355470 + 0.410234i
\(22\) 13.2286 + 10.2456i 0.128197 + 0.0992891i
\(23\) 124.270 + 143.415i 1.12661 + 1.30018i 0.948716 + 0.316129i \(0.102383\pi\)
0.177895 + 0.984050i \(0.443071\pi\)
\(24\) −19.9711 + 37.8495i −0.169858 + 0.321916i
\(25\) 15.9836 − 23.3753i 0.127869 − 0.187003i
\(26\) −17.5228 + 8.61540i −0.132173 + 0.0649853i
\(27\) −91.2049 + 66.2642i −0.650088 + 0.472317i
\(28\) −59.9563 − 33.8589i −0.404667 − 0.228526i
\(29\) −95.0490 − 139.005i −0.608626 − 0.890087i 0.390913 − 0.920427i \(-0.372159\pi\)
−0.999540 + 0.0303400i \(0.990341\pi\)
\(30\) −26.4762 + 3.03786i −0.161129 + 0.0184878i
\(31\) 55.9556 − 276.152i 0.324191 − 1.59995i −0.404125 − 0.914704i \(-0.632424\pi\)
0.728316 − 0.685241i \(-0.240303\pi\)
\(32\) −81.5479 − 23.9446i −0.450493 − 0.132277i
\(33\) 171.012 − 131.293i 0.902103 − 0.692580i
\(34\) −23.6412 + 6.94168i −0.119248 + 0.0350144i
\(35\) −7.43782 − 86.5970i −0.0359206 − 0.418216i
\(36\) 45.4717 + 41.7336i 0.210517 + 0.193211i
\(37\) −278.328 + 255.447i −1.23667 + 1.13501i −0.250703 + 0.968064i \(0.580662\pi\)
−0.985967 + 0.166942i \(0.946611\pi\)
\(38\) −5.23652 + 60.9677i −0.0223546 + 0.260270i
\(39\) 49.9656 + 246.590i 0.205151 + 1.01246i
\(40\) −22.0036 − 67.7200i −0.0869768 − 0.267687i
\(41\) −131.279 − 499.437i −0.500056 − 1.90241i −0.424901 − 0.905240i \(-0.639691\pi\)
−0.0751554 − 0.997172i \(-0.523945\pi\)
\(42\) 16.6971 − 17.1809i 0.0613433 − 0.0631207i
\(43\) 25.4970 + 177.335i 0.0904245 + 0.628917i 0.983755 + 0.179516i \(0.0574532\pi\)
−0.893330 + 0.449400i \(0.851638\pi\)
\(44\) 214.381 + 186.559i 0.734525 + 0.639201i
\(45\) −11.0875 + 77.1150i −0.0367294 + 0.255458i
\(46\) −53.1462 − 68.9212i −0.170347 − 0.220910i
\(47\) 68.8342 − 56.2854i 0.213628 − 0.174682i −0.520144 − 0.854078i \(-0.674122\pi\)
0.733772 + 0.679396i \(0.237758\pi\)
\(48\) −180.038 + 298.560i −0.541381 + 0.897779i
\(49\) −184.594 − 189.943i −0.538175 − 0.553769i
\(50\) −7.93069 + 10.2847i −0.0224314 + 0.0290895i
\(51\) 9.06608 + 317.354i 0.0248923 + 0.871343i
\(52\) −305.486 + 129.099i −0.814678 + 0.344286i
\(53\) −103.197 + 23.9975i −0.267457 + 0.0621945i −0.358081 − 0.933691i \(-0.616569\pi\)
0.0906232 + 0.995885i \(0.471114\pi\)
\(54\) 43.4962 − 27.9533i 0.109613 − 0.0704438i
\(55\) −41.6471 + 356.300i −0.102104 + 0.873518i
\(56\) 53.8504 + 34.6075i 0.128501 + 0.0825826i
\(57\) 742.624 + 264.968i 1.72566 + 0.615716i
\(58\) 39.8819 + 66.1367i 0.0902888 + 0.149727i
\(59\) −105.839 + 402.653i −0.233543 + 0.888490i 0.742309 + 0.670058i \(0.233731\pi\)
−0.975852 + 0.218433i \(0.929906\pi\)
\(60\) −451.899 + 25.8405i −0.972331 + 0.0555999i
\(61\) 576.717 − 32.9779i 1.21051 − 0.0692195i 0.559853 − 0.828592i \(-0.310857\pi\)
0.650656 + 0.759372i \(0.274494\pi\)
\(62\) −32.8516 + 124.980i −0.0672928 + 0.256008i
\(63\) −36.1674 − 59.9768i −0.0723279 − 0.119942i
\(64\) −407.808 − 145.506i −0.796500 − 0.284191i
\(65\) −352.172 − 226.328i −0.672025 − 0.431884i
\(66\) −81.7413 + 55.6393i −0.152449 + 0.103769i
\(67\) 777.748 − 499.828i 1.41816 − 0.911399i 0.418169 − 0.908369i \(-0.362672\pi\)
0.999995 − 0.00303019i \(-0.000964542\pi\)
\(68\) −407.611 + 94.7859i −0.726913 + 0.169036i
\(69\) −1032.98 + 436.542i −1.80227 + 0.761644i
\(70\) 1.13831 + 39.8461i 0.00194363 + 0.0680360i
\(71\) −71.1561 + 92.2769i −0.118939 + 0.154243i −0.847608 − 0.530624i \(-0.821958\pi\)
0.728668 + 0.684867i \(0.240140\pi\)
\(72\) −39.9889 − 41.1476i −0.0654546 − 0.0673512i
\(73\) 336.795 − 558.512i 0.539985 − 0.895465i −0.459986 − 0.887926i \(-0.652146\pi\)
0.999972 − 0.00753881i \(-0.00239970\pi\)
\(74\) 134.130 − 109.678i 0.210707 − 0.172294i
\(75\) 102.189 + 132.521i 0.157330 + 0.204030i
\(76\) −147.911 + 1028.74i −0.223244 + 1.55270i
\(77\) −173.775 − 271.663i −0.257188 − 0.402064i
\(78\) −16.4220 − 114.218i −0.0238388 − 0.165803i
\(79\) 479.731 − 493.632i 0.683215 − 0.703012i −0.283470 − 0.958981i \(-0.591486\pi\)
0.966685 + 0.255969i \(0.0823946\pi\)
\(80\) −147.470 − 561.034i −0.206095 − 0.784068i
\(81\) −271.982 − 837.073i −0.373089 − 1.14825i
\(82\) 47.0339 + 232.121i 0.0633417 + 0.312604i
\(83\) −80.3172 + 935.117i −0.106216 + 1.23666i 0.728139 + 0.685429i \(0.240385\pi\)
−0.834356 + 0.551226i \(0.814160\pi\)
\(84\) 299.789 − 275.144i 0.389401 − 0.357389i
\(85\) −389.182 − 357.188i −0.496620 − 0.455793i
\(86\) −7.03153 − 81.8667i −0.00881663 − 0.102650i
\(87\) 954.832 − 280.364i 1.17665 − 0.345496i
\(88\) −189.073 − 184.529i −0.229036 − 0.223533i
\(89\) 422.138 + 123.951i 0.502770 + 0.147627i 0.523276 − 0.852163i \(-0.324710\pi\)
−0.0205060 + 0.999790i \(0.506528\pi\)
\(90\) 7.09585 − 35.0194i 0.00831076 − 0.0410152i
\(91\) 373.886 − 42.8994i 0.430702 − 0.0494185i
\(92\) −834.364 − 1220.22i −0.945526 − 1.38279i
\(93\) 1449.89 + 818.789i 1.61663 + 0.912951i
\(94\) −32.9918 + 23.9699i −0.0362005 + 0.0263012i
\(95\) −1177.31 + 578.844i −1.27147 + 0.625139i
\(96\) 283.497 − 414.602i 0.301399 − 0.440782i
\(97\) 320.760 − 607.908i 0.335755 − 0.636326i −0.657346 − 0.753589i \(-0.728321\pi\)
0.993101 + 0.117262i \(0.0374119\pi\)
\(98\) 79.5494 + 91.8049i 0.0819969 + 0.0946295i
\(99\) 96.5631 + 272.461i 0.0980298 + 0.276599i
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.4.16 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.16 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.16 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.16 yes 1280 121.91 even 55 inner