Properties

Label 2151.2.b.a.2150.5
Level $2151$
Weight $2$
Character 2151.2150
Analytic conductor $17.176$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2151,2,Mod(2150,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2151.2150");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2150.5
Character \(\chi\) \(=\) 2151.2150
Dual form 2151.2.b.a.2150.76

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.54042i q^{2} -4.45371 q^{4} +2.74390i q^{5} -3.15846i q^{7} +6.23344i q^{8} +O(q^{10})\) \(q-2.54042i q^{2} -4.45371 q^{4} +2.74390i q^{5} -3.15846i q^{7} +6.23344i q^{8} +6.97064 q^{10} -3.61764i q^{11} +6.68097i q^{13} -8.02379 q^{14} +6.92810 q^{16} +1.88970i q^{17} -1.30591i q^{19} -12.2205i q^{20} -9.19031 q^{22} -0.0247915 q^{23} -2.52897 q^{25} +16.9724 q^{26} +14.0668i q^{28} +7.25544i q^{29} +0.823399 q^{31} -5.13338i q^{32} +4.80063 q^{34} +8.66648 q^{35} -7.67446i q^{37} -3.31755 q^{38} -17.1039 q^{40} +0.294392 q^{41} +0.0550168i q^{43} +16.1119i q^{44} +0.0629807i q^{46} +8.78581 q^{47} -2.97584 q^{49} +6.42464i q^{50} -29.7551i q^{52} +8.76528 q^{53} +9.92644 q^{55} +19.6880 q^{56} +18.4318 q^{58} +13.4742 q^{59} +7.35821 q^{61} -2.09177i q^{62} +0.815284 q^{64} -18.3319 q^{65} +1.45145 q^{67} -8.41619i q^{68} -22.0165i q^{70} +15.4302i q^{71} -0.0953278i q^{73} -19.4963 q^{74} +5.81613i q^{76} -11.4262 q^{77} -12.3605i q^{79} +19.0100i q^{80} -0.747878i q^{82} +9.34663i q^{83} -5.18515 q^{85} +0.139766 q^{86} +22.5504 q^{88} +3.09409 q^{89} +21.1016 q^{91} +0.110414 q^{92} -22.3196i q^{94} +3.58328 q^{95} -12.1734i q^{97} +7.55988i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q - 80 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times\).

\(n\) \(479\) \(1441\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.54042i 1.79634i −0.439643 0.898172i \(-0.644895\pi\)
0.439643 0.898172i \(-0.355105\pi\)
\(3\) 0 0
\(4\) −4.45371 −2.22685
\(5\) 2.74390i 1.22711i 0.789653 + 0.613554i \(0.210261\pi\)
−0.789653 + 0.613554i \(0.789739\pi\)
\(6\) 0 0
\(7\) 3.15846i 1.19378i −0.802322 0.596892i \(-0.796402\pi\)
0.802322 0.596892i \(-0.203598\pi\)
\(8\) 6.23344i 2.20385i
\(9\) 0 0
\(10\) 6.97064 2.20431
\(11\) 3.61764i 1.09076i −0.838189 0.545380i \(-0.816385\pi\)
0.838189 0.545380i \(-0.183615\pi\)
\(12\) 0 0
\(13\) 6.68097i 1.85297i 0.376333 + 0.926484i \(0.377185\pi\)
−0.376333 + 0.926484i \(0.622815\pi\)
\(14\) −8.02379 −2.14445
\(15\) 0 0
\(16\) 6.92810 1.73203
\(17\) 1.88970i 0.458320i 0.973389 + 0.229160i \(0.0735979\pi\)
−0.973389 + 0.229160i \(0.926402\pi\)
\(18\) 0 0
\(19\) 1.30591i 0.299596i −0.988717 0.149798i \(-0.952138\pi\)
0.988717 0.149798i \(-0.0478623\pi\)
\(20\) 12.2205i 2.73259i
\(21\) 0 0
\(22\) −9.19031 −1.95938
\(23\) −0.0247915 −0.00516939 −0.00258469 0.999997i \(-0.500823\pi\)
−0.00258469 + 0.999997i \(0.500823\pi\)
\(24\) 0 0
\(25\) −2.52897 −0.505795
\(26\) 16.9724 3.32857
\(27\) 0 0
\(28\) 14.0668i 2.65838i
\(29\) 7.25544i 1.34730i 0.739050 + 0.673651i \(0.235275\pi\)
−0.739050 + 0.673651i \(0.764725\pi\)
\(30\) 0 0
\(31\) 0.823399 0.147887 0.0739434 0.997262i \(-0.476442\pi\)
0.0739434 + 0.997262i \(0.476442\pi\)
\(32\) 5.13338i 0.907463i
\(33\) 0 0
\(34\) 4.80063 0.823301
\(35\) 8.66648 1.46490
\(36\) 0 0
\(37\) 7.67446i 1.26167i −0.775915 0.630837i \(-0.782712\pi\)
0.775915 0.630837i \(-0.217288\pi\)
\(38\) −3.31755 −0.538177
\(39\) 0 0
\(40\) −17.1039 −2.70437
\(41\) 0.294392 0.0459763 0.0229881 0.999736i \(-0.492682\pi\)
0.0229881 + 0.999736i \(0.492682\pi\)
\(42\) 0 0
\(43\) 0.0550168i 0.00838999i 0.999991 + 0.00419499i \(0.00133531\pi\)
−0.999991 + 0.00419499i \(0.998665\pi\)
\(44\) 16.1119i 2.42896i
\(45\) 0 0
\(46\) 0.0629807i 0.00928600i
\(47\) 8.78581 1.28154 0.640771 0.767732i \(-0.278615\pi\)
0.640771 + 0.767732i \(0.278615\pi\)
\(48\) 0 0
\(49\) −2.97584 −0.425121
\(50\) 6.42464i 0.908582i
\(51\) 0 0
\(52\) 29.7551i 4.12629i
\(53\) 8.76528 1.20400 0.602002 0.798495i \(-0.294370\pi\)
0.602002 + 0.798495i \(0.294370\pi\)
\(54\) 0 0
\(55\) 9.92644 1.33848
\(56\) 19.6880 2.63093
\(57\) 0 0
\(58\) 18.4318 2.42022
\(59\) 13.4742 1.75419 0.877095 0.480317i \(-0.159478\pi\)
0.877095 + 0.480317i \(0.159478\pi\)
\(60\) 0 0
\(61\) 7.35821 0.942122 0.471061 0.882101i \(-0.343871\pi\)
0.471061 + 0.882101i \(0.343871\pi\)
\(62\) 2.09177i 0.265656i
\(63\) 0 0
\(64\) 0.815284 0.101910
\(65\) −18.3319 −2.27379
\(66\) 0 0
\(67\) 1.45145 0.177323 0.0886616 0.996062i \(-0.471741\pi\)
0.0886616 + 0.996062i \(0.471741\pi\)
\(68\) 8.41619i 1.02061i
\(69\) 0 0
\(70\) 22.0165i 2.63147i
\(71\) 15.4302i 1.83123i 0.402053 + 0.915616i \(0.368297\pi\)
−0.402053 + 0.915616i \(0.631703\pi\)
\(72\) 0 0
\(73\) 0.0953278i 0.0111573i −0.999984 0.00557864i \(-0.998224\pi\)
0.999984 0.00557864i \(-0.00177574\pi\)
\(74\) −19.4963 −2.26640
\(75\) 0 0
\(76\) 5.81613i 0.667156i
\(77\) −11.4262 −1.30213
\(78\) 0 0
\(79\) 12.3605i 1.39067i −0.718688 0.695333i \(-0.755257\pi\)
0.718688 0.695333i \(-0.244743\pi\)
\(80\) 19.0100i 2.12538i
\(81\) 0 0
\(82\) 0.747878i 0.0825893i
\(83\) 9.34663i 1.02593i 0.858411 + 0.512963i \(0.171452\pi\)
−0.858411 + 0.512963i \(0.828548\pi\)
\(84\) 0 0
\(85\) −5.18515 −0.562409
\(86\) 0.139766 0.0150713
\(87\) 0 0
\(88\) 22.5504 2.40388
\(89\) 3.09409 0.327973 0.163987 0.986463i \(-0.447565\pi\)
0.163987 + 0.986463i \(0.447565\pi\)
\(90\) 0 0
\(91\) 21.1016 2.21204
\(92\) 0.110414 0.0115115
\(93\) 0 0
\(94\) 22.3196i 2.30209i
\(95\) 3.58328 0.367636
\(96\) 0 0
\(97\) 12.1734i 1.23602i −0.786169 0.618011i \(-0.787939\pi\)
0.786169 0.618011i \(-0.212061\pi\)
\(98\) 7.55988i 0.763663i
\(99\) 0 0
\(100\) 11.2633 1.12633
\(101\) 2.31914i 0.230764i −0.993321 0.115382i \(-0.963191\pi\)
0.993321 0.115382i \(-0.0368091\pi\)
\(102\) 0 0
\(103\) 3.76412i 0.370890i −0.982655 0.185445i \(-0.940627\pi\)
0.982655 0.185445i \(-0.0593726\pi\)
\(104\) −41.6454 −4.08367
\(105\) 0 0
\(106\) 22.2675i 2.16281i
\(107\) −9.64916 −0.932820 −0.466410 0.884569i \(-0.654453\pi\)
−0.466410 + 0.884569i \(0.654453\pi\)
\(108\) 0 0
\(109\) −18.4710 −1.76920 −0.884602 0.466347i \(-0.845570\pi\)
−0.884602 + 0.466347i \(0.845570\pi\)
\(110\) 25.2173i 2.40437i
\(111\) 0 0
\(112\) 21.8821i 2.06767i
\(113\) 10.1240i 0.952385i −0.879341 0.476192i \(-0.842017\pi\)
0.879341 0.476192i \(-0.157983\pi\)
\(114\) 0 0
\(115\) 0.0680254i 0.00634340i
\(116\) 32.3136i 3.00025i
\(117\) 0 0
\(118\) 34.2300i 3.15113i
\(119\) 5.96854 0.547135
\(120\) 0 0
\(121\) −2.08734 −0.189758
\(122\) 18.6929i 1.69238i
\(123\) 0 0
\(124\) −3.66718 −0.329322
\(125\) 6.78024i 0.606443i
\(126\) 0 0
\(127\) −6.13936 −0.544780 −0.272390 0.962187i \(-0.587814\pi\)
−0.272390 + 0.962187i \(0.587814\pi\)
\(128\) 12.3379i 1.09053i
\(129\) 0 0
\(130\) 46.5707i 4.08452i
\(131\) 12.1961 1.06558 0.532788 0.846248i \(-0.321144\pi\)
0.532788 + 0.846248i \(0.321144\pi\)
\(132\) 0 0
\(133\) −4.12465 −0.357653
\(134\) 3.68729i 0.318533i
\(135\) 0 0
\(136\) −11.7793 −1.01007
\(137\) −3.45116 −0.294852 −0.147426 0.989073i \(-0.547099\pi\)
−0.147426 + 0.989073i \(0.547099\pi\)
\(138\) 0 0
\(139\) 12.8269i 1.08797i 0.839096 + 0.543984i \(0.183085\pi\)
−0.839096 + 0.543984i \(0.816915\pi\)
\(140\) −38.5980 −3.26212
\(141\) 0 0
\(142\) 39.1992 3.28952
\(143\) 24.1694 2.02114
\(144\) 0 0
\(145\) −19.9082 −1.65329
\(146\) −0.242172 −0.0200423
\(147\) 0 0
\(148\) 34.1798i 2.80956i
\(149\) 7.37438 0.604133 0.302067 0.953287i \(-0.402324\pi\)
0.302067 + 0.953287i \(0.402324\pi\)
\(150\) 0 0
\(151\) 8.29877i 0.675345i −0.941264 0.337672i \(-0.890360\pi\)
0.941264 0.337672i \(-0.109640\pi\)
\(152\) 8.14029 0.660265
\(153\) 0 0
\(154\) 29.0272i 2.33908i
\(155\) 2.25932i 0.181473i
\(156\) 0 0
\(157\) 17.6696 1.41019 0.705096 0.709112i \(-0.250904\pi\)
0.705096 + 0.709112i \(0.250904\pi\)
\(158\) −31.4008 −2.49811
\(159\) 0 0
\(160\) 14.0855 1.11355
\(161\) 0.0783029i 0.00617113i
\(162\) 0 0
\(163\) 4.26672 0.334195 0.167098 0.985940i \(-0.446560\pi\)
0.167098 + 0.985940i \(0.446560\pi\)
\(164\) −1.31114 −0.102383
\(165\) 0 0
\(166\) 23.7443 1.84292
\(167\) 10.9531 0.847575 0.423787 0.905762i \(-0.360700\pi\)
0.423787 + 0.905762i \(0.360700\pi\)
\(168\) 0 0
\(169\) −31.6354 −2.43349
\(170\) 13.1724i 1.01028i
\(171\) 0 0
\(172\) 0.245029i 0.0186833i
\(173\) 4.46558 0.339512 0.169756 0.985486i \(-0.445702\pi\)
0.169756 + 0.985486i \(0.445702\pi\)
\(174\) 0 0
\(175\) 7.98765i 0.603810i
\(176\) 25.0634i 1.88923i
\(177\) 0 0
\(178\) 7.86029i 0.589153i
\(179\) −21.5478 −1.61056 −0.805281 0.592894i \(-0.797985\pi\)
−0.805281 + 0.592894i \(0.797985\pi\)
\(180\) 0 0
\(181\) 21.0683i 1.56599i −0.622026 0.782997i \(-0.713690\pi\)
0.622026 0.782997i \(-0.286310\pi\)
\(182\) 53.6067i 3.97359i
\(183\) 0 0
\(184\) 0.154536i 0.0113926i
\(185\) 21.0579 1.54821
\(186\) 0 0
\(187\) 6.83627 0.499918
\(188\) −39.1295 −2.85381
\(189\) 0 0
\(190\) 9.10301i 0.660402i
\(191\) 23.3758 1.69141 0.845706 0.533649i \(-0.179180\pi\)
0.845706 + 0.533649i \(0.179180\pi\)
\(192\) 0 0
\(193\) −7.59111 −0.546420 −0.273210 0.961954i \(-0.588085\pi\)
−0.273210 + 0.961954i \(0.588085\pi\)
\(194\) −30.9255 −2.22032
\(195\) 0 0
\(196\) 13.2535 0.946682
\(197\) 13.5652i 0.966478i 0.875488 + 0.483239i \(0.160540\pi\)
−0.875488 + 0.483239i \(0.839460\pi\)
\(198\) 0 0
\(199\) 12.5598i 0.890344i 0.895445 + 0.445172i \(0.146857\pi\)
−0.895445 + 0.445172i \(0.853143\pi\)
\(200\) 15.7642i 1.11470i
\(201\) 0 0
\(202\) −5.89159 −0.414531
\(203\) 22.9160 1.60839
\(204\) 0 0
\(205\) 0.807781i 0.0564179i
\(206\) −9.56243 −0.666246
\(207\) 0 0
\(208\) 46.2865i 3.20939i
\(209\) −4.72431 −0.326787
\(210\) 0 0
\(211\) 16.3629 1.12647 0.563235 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(212\) −39.0380 −2.68114
\(213\) 0 0
\(214\) 24.5129i 1.67567i
\(215\) −0.150961 −0.0102954
\(216\) 0 0
\(217\) 2.60067i 0.176545i
\(218\) 46.9241i 3.17810i
\(219\) 0 0
\(220\) −44.2095 −2.98060
\(221\) −12.6251 −0.849253
\(222\) 0 0
\(223\) 11.0222i 0.738102i 0.929409 + 0.369051i \(0.120317\pi\)
−0.929409 + 0.369051i \(0.879683\pi\)
\(224\) −16.2136 −1.08331
\(225\) 0 0
\(226\) −25.7191 −1.71081
\(227\) −2.22510 −0.147685 −0.0738425 0.997270i \(-0.523526\pi\)
−0.0738425 + 0.997270i \(0.523526\pi\)
\(228\) 0 0
\(229\) 8.83565i 0.583876i 0.956437 + 0.291938i \(0.0943002\pi\)
−0.956437 + 0.291938i \(0.905700\pi\)
\(230\) −0.172813 −0.0113949
\(231\) 0 0
\(232\) −45.2264 −2.96926
\(233\) 19.6243 1.28563 0.642816 0.766021i \(-0.277766\pi\)
0.642816 + 0.766021i \(0.277766\pi\)
\(234\) 0 0
\(235\) 24.1074i 1.57259i
\(236\) −60.0101 −3.90633
\(237\) 0 0
\(238\) 15.1626i 0.982844i
\(239\) 13.0981 8.21211i 0.847248 0.531197i
\(240\) 0 0
\(241\) 6.96331 0.448546 0.224273 0.974526i \(-0.427999\pi\)
0.224273 + 0.974526i \(0.427999\pi\)
\(242\) 5.30271i 0.340871i
\(243\) 0 0
\(244\) −32.7713 −2.09797
\(245\) 8.16541i 0.521669i
\(246\) 0 0
\(247\) 8.72473 0.555141
\(248\) 5.13261i 0.325921i
\(249\) 0 0
\(250\) 17.2246 1.08938
\(251\) 7.70445i 0.486300i 0.969989 + 0.243150i \(0.0781808\pi\)
−0.969989 + 0.243150i \(0.921819\pi\)
\(252\) 0 0
\(253\) 0.0896868i 0.00563856i
\(254\) 15.5965i 0.978612i
\(255\) 0 0
\(256\) −29.7129 −1.85706
\(257\) 22.0120i 1.37307i 0.727097 + 0.686534i \(0.240869\pi\)
−0.727097 + 0.686534i \(0.759131\pi\)
\(258\) 0 0
\(259\) −24.2395 −1.50617
\(260\) 81.6450 5.06341
\(261\) 0 0
\(262\) 30.9831i 1.91414i
\(263\) 4.18120i 0.257824i −0.991656 0.128912i \(-0.958852\pi\)
0.991656 0.128912i \(-0.0411484\pi\)
\(264\) 0 0
\(265\) 24.0510i 1.47744i
\(266\) 10.4783i 0.642467i
\(267\) 0 0
\(268\) −6.46435 −0.394873
\(269\) 2.73513i 0.166764i 0.996518 + 0.0833818i \(0.0265721\pi\)
−0.996518 + 0.0833818i \(0.973428\pi\)
\(270\) 0 0
\(271\) −30.6712 −1.86314 −0.931571 0.363560i \(-0.881561\pi\)
−0.931571 + 0.363560i \(0.881561\pi\)
\(272\) 13.0921i 0.793823i
\(273\) 0 0
\(274\) 8.76737i 0.529656i
\(275\) 9.14892i 0.551701i
\(276\) 0 0
\(277\) 27.5314i 1.65420i 0.562053 + 0.827101i \(0.310012\pi\)
−0.562053 + 0.827101i \(0.689988\pi\)
\(278\) 32.5858 1.95437
\(279\) 0 0
\(280\) 54.0220i 3.22843i
\(281\) −30.9835 −1.84832 −0.924161 0.382002i \(-0.875235\pi\)
−0.924161 + 0.382002i \(0.875235\pi\)
\(282\) 0 0
\(283\) 13.9257 0.827798 0.413899 0.910323i \(-0.364167\pi\)
0.413899 + 0.910323i \(0.364167\pi\)
\(284\) 68.7218i 4.07789i
\(285\) 0 0
\(286\) 61.4002i 3.63067i
\(287\) 0.929824i 0.0548858i
\(288\) 0 0
\(289\) 13.4290 0.789943
\(290\) 50.5751i 2.96987i
\(291\) 0 0
\(292\) 0.424562i 0.0248456i
\(293\) 16.2221i 0.947707i 0.880604 + 0.473853i \(0.157137\pi\)
−0.880604 + 0.473853i \(0.842863\pi\)
\(294\) 0 0
\(295\) 36.9718i 2.15258i
\(296\) 47.8383 2.78054
\(297\) 0 0
\(298\) 18.7340i 1.08523i
\(299\) 0.165631i 0.00957871i
\(300\) 0 0
\(301\) 0.173768 0.0100158
\(302\) −21.0823 −1.21315
\(303\) 0 0
\(304\) 9.04746i 0.518907i
\(305\) 20.1902i 1.15609i
\(306\) 0 0
\(307\) 16.3345 0.932258 0.466129 0.884717i \(-0.345648\pi\)
0.466129 + 0.884717i \(0.345648\pi\)
\(308\) 50.8888 2.89966
\(309\) 0 0
\(310\) 5.73961 0.325988
\(311\) 5.20846i 0.295345i 0.989036 + 0.147672i \(0.0471781\pi\)
−0.989036 + 0.147672i \(0.952822\pi\)
\(312\) 0 0
\(313\) 0.279682i 0.0158086i −0.999969 0.00790429i \(-0.997484\pi\)
0.999969 0.00790429i \(-0.00251604\pi\)
\(314\) 44.8882i 2.53319i
\(315\) 0 0
\(316\) 55.0501i 3.09681i
\(317\) 26.8614 1.50869 0.754344 0.656479i \(-0.227955\pi\)
0.754344 + 0.656479i \(0.227955\pi\)
\(318\) 0 0
\(319\) 26.2476 1.46958
\(320\) 2.23706i 0.125055i
\(321\) 0 0
\(322\) 0.198922 0.0110855
\(323\) 2.46778 0.137311
\(324\) 0 0
\(325\) 16.8960i 0.937222i
\(326\) 10.8392i 0.600330i
\(327\) 0 0
\(328\) 1.83507i 0.101325i
\(329\) 27.7496i 1.52989i
\(330\) 0 0
\(331\) 17.3353i 0.952834i −0.879219 0.476417i \(-0.841935\pi\)
0.879219 0.476417i \(-0.158065\pi\)
\(332\) 41.6272i 2.28459i
\(333\) 0 0
\(334\) 27.8254i 1.52254i
\(335\) 3.98264i 0.217595i
\(336\) 0 0
\(337\) 15.8455 0.863161 0.431580 0.902074i \(-0.357956\pi\)
0.431580 + 0.902074i \(0.357956\pi\)
\(338\) 80.3671i 4.37139i
\(339\) 0 0
\(340\) 23.0932 1.25240
\(341\) 2.97876i 0.161309i
\(342\) 0 0
\(343\) 12.7101i 0.686282i
\(344\) −0.342944 −0.0184903
\(345\) 0 0
\(346\) 11.3444i 0.609881i
\(347\) 15.3686i 0.825030i 0.910951 + 0.412515i \(0.135350\pi\)
−0.910951 + 0.412515i \(0.864650\pi\)
\(348\) 0 0
\(349\) 19.3056 1.03341 0.516703 0.856165i \(-0.327159\pi\)
0.516703 + 0.856165i \(0.327159\pi\)
\(350\) 20.2920 1.08465
\(351\) 0 0
\(352\) −18.5707 −0.989824
\(353\) −29.6340 −1.57726 −0.788629 0.614870i \(-0.789208\pi\)
−0.788629 + 0.614870i \(0.789208\pi\)
\(354\) 0 0
\(355\) −42.3390 −2.24712
\(356\) −13.7802 −0.730349
\(357\) 0 0
\(358\) 54.7405i 2.89312i
\(359\) 22.9649i 1.21204i 0.795450 + 0.606020i \(0.207235\pi\)
−0.795450 + 0.606020i \(0.792765\pi\)
\(360\) 0 0
\(361\) 17.2946 0.910242
\(362\) −53.5222 −2.81306
\(363\) 0 0
\(364\) −93.9802 −4.92590
\(365\) 0.261570 0.0136912
\(366\) 0 0
\(367\) 22.7643 1.18828 0.594142 0.804360i \(-0.297492\pi\)
0.594142 + 0.804360i \(0.297492\pi\)
\(368\) −0.171758 −0.00895352
\(369\) 0 0
\(370\) 53.4959i 2.78112i
\(371\) 27.6848i 1.43732i
\(372\) 0 0
\(373\) −23.3244 −1.20769 −0.603847 0.797100i \(-0.706366\pi\)
−0.603847 + 0.797100i \(0.706366\pi\)
\(374\) 17.3670i 0.898024i
\(375\) 0 0
\(376\) 54.7658i 2.82433i
\(377\) −48.4734 −2.49651
\(378\) 0 0
\(379\) 29.9477i 1.53831i 0.639064 + 0.769154i \(0.279322\pi\)
−0.639064 + 0.769154i \(0.720678\pi\)
\(380\) −15.9589 −0.818673
\(381\) 0 0
\(382\) 59.3842i 3.03836i
\(383\) 26.8157i 1.37022i −0.728440 0.685110i \(-0.759754\pi\)
0.728440 0.685110i \(-0.240246\pi\)
\(384\) 0 0
\(385\) 31.3522i 1.59786i
\(386\) 19.2846i 0.981559i
\(387\) 0 0
\(388\) 54.2168i 2.75244i
\(389\) 10.4488i 0.529774i −0.964279 0.264887i \(-0.914665\pi\)
0.964279 0.264887i \(-0.0853346\pi\)
\(390\) 0 0
\(391\) 0.0468486i 0.00236923i
\(392\) 18.5497i 0.936903i
\(393\) 0 0
\(394\) 34.4612 1.73613
\(395\) 33.9160 1.70650
\(396\) 0 0
\(397\) 12.1691i 0.610751i 0.952232 + 0.305375i \(0.0987820\pi\)
−0.952232 + 0.305375i \(0.901218\pi\)
\(398\) 31.9072 1.59936
\(399\) 0 0
\(400\) −17.5210 −0.876050
\(401\) 4.50204i 0.224821i −0.993662 0.112411i \(-0.964143\pi\)
0.993662 0.112411i \(-0.0358572\pi\)
\(402\) 0 0
\(403\) 5.50110i 0.274030i
\(404\) 10.3288i 0.513877i
\(405\) 0 0
\(406\) 58.2162i 2.88922i
\(407\) −27.7635 −1.37618
\(408\) 0 0
\(409\) 1.18270 0.0584808 0.0292404 0.999572i \(-0.490691\pi\)
0.0292404 + 0.999572i \(0.490691\pi\)
\(410\) 2.05210 0.101346
\(411\) 0 0
\(412\) 16.7643i 0.825918i
\(413\) 42.5576i 2.09412i
\(414\) 0 0
\(415\) −25.6462 −1.25892
\(416\) 34.2960 1.68150
\(417\) 0 0
\(418\) 12.0017i 0.587022i
\(419\) 4.13425i 0.201971i −0.994888 0.100986i \(-0.967800\pi\)
0.994888 0.100986i \(-0.0321996\pi\)
\(420\) 0 0
\(421\) 35.7666 1.74316 0.871579 0.490256i \(-0.163097\pi\)
0.871579 + 0.490256i \(0.163097\pi\)
\(422\) 41.5686i 2.02353i
\(423\) 0 0
\(424\) 54.6379i 2.65345i
\(425\) 4.77901i 0.231816i
\(426\) 0 0
\(427\) 23.2406i 1.12469i
\(428\) 42.9746 2.07725
\(429\) 0 0
\(430\) 0.383503i 0.0184941i
\(431\) 4.38847i 0.211385i −0.994399 0.105693i \(-0.966294\pi\)
0.994399 0.105693i \(-0.0337059\pi\)
\(432\) 0 0
\(433\) 33.5767i 1.61359i 0.590829 + 0.806797i \(0.298801\pi\)
−0.590829 + 0.806797i \(0.701199\pi\)
\(434\) −6.60678 −0.317135
\(435\) 0 0
\(436\) 82.2646 3.93976
\(437\) 0.0323754i 0.00154873i
\(438\) 0 0
\(439\) 1.12131 0.0535172 0.0267586 0.999642i \(-0.491481\pi\)
0.0267586 + 0.999642i \(0.491481\pi\)
\(440\) 61.8759i 2.94982i
\(441\) 0 0
\(442\) 32.0729i 1.52555i
\(443\) 34.6898i 1.64816i −0.566472 0.824081i \(-0.691692\pi\)
0.566472 0.824081i \(-0.308308\pi\)
\(444\) 0 0
\(445\) 8.48988i 0.402459i
\(446\) 28.0010 1.32589
\(447\) 0 0
\(448\) 2.57504i 0.121659i
\(449\) −27.6575 −1.30524 −0.652619 0.757686i \(-0.726330\pi\)
−0.652619 + 0.757686i \(0.726330\pi\)
\(450\) 0 0
\(451\) 1.06500i 0.0501491i
\(452\) 45.0893i 2.12082i
\(453\) 0 0
\(454\) 5.65267i 0.265293i
\(455\) 57.9005i 2.71442i
\(456\) 0 0
\(457\) −32.7132 −1.53026 −0.765130 0.643876i \(-0.777325\pi\)
−0.765130 + 0.643876i \(0.777325\pi\)
\(458\) 22.4462 1.04884
\(459\) 0 0
\(460\) 0.302965i 0.0141258i
\(461\) −19.8347 −0.923796 −0.461898 0.886933i \(-0.652831\pi\)
−0.461898 + 0.886933i \(0.652831\pi\)
\(462\) 0 0
\(463\) 40.8649i 1.89915i −0.313539 0.949575i \(-0.601515\pi\)
0.313539 0.949575i \(-0.398485\pi\)
\(464\) 50.2665i 2.33356i
\(465\) 0 0
\(466\) 49.8539i 2.30944i
\(467\) −26.7199 −1.23645 −0.618224 0.786002i \(-0.712148\pi\)
−0.618224 + 0.786002i \(0.712148\pi\)
\(468\) 0 0
\(469\) 4.58435i 0.211686i
\(470\) 61.2427 2.82492
\(471\) 0 0
\(472\) 83.9906i 3.86598i
\(473\) 0.199031 0.00915147
\(474\) 0 0
\(475\) 3.30260i 0.151534i
\(476\) −26.5822 −1.21839
\(477\) 0 0
\(478\) −20.8622 33.2747i −0.954214 1.52195i
\(479\) 9.06080i 0.413998i 0.978341 + 0.206999i \(0.0663698\pi\)
−0.978341 + 0.206999i \(0.933630\pi\)
\(480\) 0 0
\(481\) 51.2729 2.33784
\(482\) 17.6897i 0.805744i
\(483\) 0 0
\(484\) 9.29641 0.422564
\(485\) 33.4026 1.51673
\(486\) 0 0
\(487\) −33.1167 −1.50066 −0.750331 0.661062i \(-0.770106\pi\)
−0.750331 + 0.661062i \(0.770106\pi\)
\(488\) 45.8669i 2.07630i
\(489\) 0 0
\(490\) −20.7435 −0.937097
\(491\) −12.5893 −0.568149 −0.284075 0.958802i \(-0.591686\pi\)
−0.284075 + 0.958802i \(0.591686\pi\)
\(492\) 0 0
\(493\) −13.7106 −0.617496
\(494\) 22.1644i 0.997225i
\(495\) 0 0
\(496\) 5.70459 0.256144
\(497\) 48.7357 2.18610
\(498\) 0 0
\(499\) 18.3577i 0.821804i 0.911679 + 0.410902i \(0.134786\pi\)
−0.911679 + 0.410902i \(0.865214\pi\)
\(500\) 30.1972i 1.35046i
\(501\) 0 0
\(502\) 19.5725 0.873563
\(503\) 26.7645i 1.19337i 0.802475 + 0.596685i \(0.203516\pi\)
−0.802475 + 0.596685i \(0.796484\pi\)
\(504\) 0 0
\(505\) 6.36350 0.283172
\(506\) 0.227842 0.0101288
\(507\) 0 0
\(508\) 27.3429 1.21315
\(509\) 32.2824i 1.43089i −0.698668 0.715447i \(-0.746223\pi\)
0.698668 0.715447i \(-0.253777\pi\)
\(510\) 0 0
\(511\) −0.301089 −0.0133194
\(512\) 50.8072i 2.24538i
\(513\) 0 0
\(514\) 55.9195 2.46651
\(515\) 10.3284 0.455122
\(516\) 0 0
\(517\) 31.7839i 1.39786i
\(518\) 61.5783i 2.70559i
\(519\) 0 0
\(520\) 114.271i 5.01111i
\(521\) 18.3071 0.802049 0.401024 0.916067i \(-0.368654\pi\)
0.401024 + 0.916067i \(0.368654\pi\)
\(522\) 0 0
\(523\) −24.0246 −1.05052 −0.525262 0.850941i \(-0.676033\pi\)
−0.525262 + 0.850941i \(0.676033\pi\)
\(524\) −54.3178 −2.37288
\(525\) 0 0
\(526\) −10.6220 −0.463140
\(527\) 1.55598i 0.0677795i
\(528\) 0 0
\(529\) −22.9994 −0.999973
\(530\) 61.0996 2.65400
\(531\) 0 0
\(532\) 18.3700 0.796440
\(533\) 1.96682i 0.0851926i
\(534\) 0 0
\(535\) 26.4763i 1.14467i
\(536\) 9.04754i 0.390794i
\(537\) 0 0
\(538\) 6.94836 0.299565
\(539\) 10.7655i 0.463705i
\(540\) 0 0
\(541\) 10.5660i 0.454269i 0.973863 + 0.227135i \(0.0729358\pi\)
−0.973863 + 0.227135i \(0.927064\pi\)
\(542\) 77.9175i 3.34684i
\(543\) 0 0
\(544\) 9.70057 0.415908
\(545\) 50.6826i 2.17101i
\(546\) 0 0
\(547\) 0.0915418i 0.00391405i 0.999998 + 0.00195702i \(0.000622940\pi\)
−0.999998 + 0.00195702i \(0.999377\pi\)
\(548\) 15.3704 0.656593
\(549\) 0 0
\(550\) 23.2421 0.991045
\(551\) 9.47494 0.403646
\(552\) 0 0
\(553\) −39.0401 −1.66015
\(554\) 69.9412 2.97152
\(555\) 0 0
\(556\) 57.1275i 2.42275i
\(557\) 11.2511 0.476725 0.238362 0.971176i \(-0.423389\pi\)
0.238362 + 0.971176i \(0.423389\pi\)
\(558\) 0 0
\(559\) −0.367566 −0.0155464
\(560\) 60.0423 2.53725
\(561\) 0 0
\(562\) 78.7110i 3.32022i
\(563\) 7.07737i 0.298276i 0.988816 + 0.149138i \(0.0476498\pi\)
−0.988816 + 0.149138i \(0.952350\pi\)
\(564\) 0 0
\(565\) 27.7792 1.16868
\(566\) 35.3771i 1.48701i
\(567\) 0 0
\(568\) −96.1835 −4.03577
\(569\) 42.7649i 1.79280i 0.443249 + 0.896399i \(0.353826\pi\)
−0.443249 + 0.896399i \(0.646174\pi\)
\(570\) 0 0
\(571\) 1.41437 0.0591893 0.0295947 0.999562i \(-0.490578\pi\)
0.0295947 + 0.999562i \(0.490578\pi\)
\(572\) −107.643 −4.50080
\(573\) 0 0
\(574\) −2.36214 −0.0985938
\(575\) 0.0626971 0.00261465
\(576\) 0 0
\(577\) −27.9262 −1.16258 −0.581292 0.813695i \(-0.697453\pi\)
−0.581292 + 0.813695i \(0.697453\pi\)
\(578\) 34.1153i 1.41901i
\(579\) 0 0
\(580\) 88.6653 3.68163
\(581\) 29.5209 1.22473
\(582\) 0 0
\(583\) 31.7097i 1.31328i
\(584\) 0.594220 0.0245890
\(585\) 0 0
\(586\) 41.2109 1.70241
\(587\) 26.2194i 1.08219i −0.840961 0.541095i \(-0.818010\pi\)
0.840961 0.541095i \(-0.181990\pi\)
\(588\) 0 0
\(589\) 1.07528i 0.0443062i
\(590\) 93.9237 3.86678
\(591\) 0 0
\(592\) 53.1695i 2.18525i
\(593\) −24.5699 −1.00896 −0.504482 0.863422i \(-0.668317\pi\)
−0.504482 + 0.863422i \(0.668317\pi\)
\(594\) 0 0
\(595\) 16.3771i 0.671394i
\(596\) −32.8434 −1.34532
\(597\) 0 0
\(598\) −0.420773 −0.0172067
\(599\) 23.0112i 0.940211i −0.882610 0.470105i \(-0.844216\pi\)
0.882610 0.470105i \(-0.155784\pi\)
\(600\) 0 0
\(601\) 1.69938i 0.0693192i −0.999399 0.0346596i \(-0.988965\pi\)
0.999399 0.0346596i \(-0.0110347\pi\)
\(602\) 0.441444i 0.0179919i
\(603\) 0 0
\(604\) 36.9603i 1.50389i
\(605\) 5.72745i 0.232854i
\(606\) 0 0
\(607\) 27.3501i 1.11011i 0.831814 + 0.555054i \(0.187302\pi\)
−0.831814 + 0.555054i \(0.812698\pi\)
\(608\) −6.70372 −0.271872
\(609\) 0 0
\(610\) 51.2914 2.07673
\(611\) 58.6978i 2.37466i
\(612\) 0 0
\(613\) 34.5836 1.39682 0.698409 0.715699i \(-0.253892\pi\)
0.698409 + 0.715699i \(0.253892\pi\)
\(614\) 41.4964i 1.67466i
\(615\) 0 0
\(616\) 71.2243i 2.86971i
\(617\) −8.76076 −0.352695 −0.176347 0.984328i \(-0.556428\pi\)
−0.176347 + 0.984328i \(0.556428\pi\)
\(618\) 0 0
\(619\) 44.1885i 1.77609i −0.459760 0.888043i \(-0.652065\pi\)
0.459760 0.888043i \(-0.347935\pi\)
\(620\) 10.0624i 0.404114i
\(621\) 0 0
\(622\) 13.2317 0.530541
\(623\) 9.77256i 0.391529i
\(624\) 0 0
\(625\) −31.2492 −1.24997
\(626\) −0.710510 −0.0283977
\(627\) 0 0
\(628\) −78.6955 −3.14029
\(629\) 14.5025 0.578251
\(630\) 0 0
\(631\) −6.39134 −0.254435 −0.127218 0.991875i \(-0.540605\pi\)
−0.127218 + 0.991875i \(0.540605\pi\)
\(632\) 77.0484 3.06482
\(633\) 0 0
\(634\) 68.2392i 2.71012i
\(635\) 16.8458i 0.668504i
\(636\) 0 0
\(637\) 19.8815i 0.787735i
\(638\) 66.6798i 2.63988i
\(639\) 0 0
\(640\) 33.8540 1.33820
\(641\) 10.3794i 0.409963i −0.978766 0.204982i \(-0.934287\pi\)
0.978766 0.204982i \(-0.0657134\pi\)
\(642\) 0 0
\(643\) −23.6406 −0.932293 −0.466146 0.884708i \(-0.654358\pi\)
−0.466146 + 0.884708i \(0.654358\pi\)
\(644\) 0.348738i 0.0137422i
\(645\) 0 0
\(646\) 6.26918i 0.246657i
\(647\) 34.6941i 1.36397i 0.731367 + 0.681984i \(0.238882\pi\)
−0.731367 + 0.681984i \(0.761118\pi\)
\(648\) 0 0
\(649\) 48.7448i 1.91340i
\(650\) −42.9229 −1.68357
\(651\) 0 0
\(652\) −19.0027 −0.744205
\(653\) −43.4851 −1.70170 −0.850852 0.525405i \(-0.823914\pi\)
−0.850852 + 0.525405i \(0.823914\pi\)
\(654\) 0 0
\(655\) 33.4648i 1.30758i
\(656\) 2.03958 0.0796322
\(657\) 0 0
\(658\) −70.4955 −2.74820
\(659\) 8.01415 0.312187 0.156093 0.987742i \(-0.450110\pi\)
0.156093 + 0.987742i \(0.450110\pi\)
\(660\) 0 0
\(661\) 46.3416 1.80248 0.901239 0.433322i \(-0.142659\pi\)
0.901239 + 0.433322i \(0.142659\pi\)
\(662\) −44.0389 −1.71162
\(663\) 0 0
\(664\) −58.2616 −2.26099
\(665\) 11.3176i 0.438878i
\(666\) 0 0
\(667\) 0.179873i 0.00696473i
\(668\) −48.7818 −1.88743
\(669\) 0 0
\(670\) 10.1176 0.390875
\(671\) 26.6194i 1.02763i
\(672\) 0 0
\(673\) 46.6089i 1.79664i 0.439343 + 0.898320i \(0.355211\pi\)
−0.439343 + 0.898320i \(0.644789\pi\)
\(674\) 40.2542i 1.55053i
\(675\) 0 0
\(676\) 140.895 5.41903
\(677\) −10.5146 −0.404110 −0.202055 0.979374i \(-0.564762\pi\)
−0.202055 + 0.979374i \(0.564762\pi\)
\(678\) 0 0
\(679\) −38.4492 −1.47554
\(680\) 32.3213i 1.23947i
\(681\) 0 0
\(682\) −7.56729 −0.289767
\(683\) 37.6116 1.43917 0.719584 0.694405i \(-0.244332\pi\)
0.719584 + 0.694405i \(0.244332\pi\)
\(684\) 0 0
\(685\) 9.46962i 0.361816i
\(686\) −32.2890 −1.23280
\(687\) 0 0
\(688\) 0.381162i 0.0145317i
\(689\) 58.5606i 2.23098i
\(690\) 0 0
\(691\) −18.9382 −0.720442 −0.360221 0.932867i \(-0.617299\pi\)
−0.360221 + 0.932867i \(0.617299\pi\)
\(692\) −19.8884 −0.756044
\(693\) 0 0
\(694\) 39.0426 1.48204
\(695\) −35.1958 −1.33505
\(696\) 0 0
\(697\) 0.556313i 0.0210719i
\(698\) 49.0443i 1.85635i
\(699\) 0 0
\(700\) 35.5747i 1.34460i
\(701\) 7.01381 0.264908 0.132454 0.991189i \(-0.457714\pi\)
0.132454 + 0.991189i \(0.457714\pi\)
\(702\) 0 0
\(703\) −10.0221 −0.377992
\(704\) 2.94941i 0.111160i
\(705\) 0 0
\(706\) 75.2826i 2.83330i
\(707\) −7.32492 −0.275482
\(708\) 0 0
\(709\) 9.61006i 0.360913i −0.983583 0.180457i \(-0.942242\pi\)
0.983583 0.180457i \(-0.0577575\pi\)
\(710\) 107.559i 4.03660i
\(711\) 0 0
\(712\) 19.2869i 0.722805i
\(713\) −0.0204133 −0.000764484
\(714\) 0 0
\(715\) 66.3183i 2.48016i
\(716\) 95.9678 3.58649
\(717\) 0 0
\(718\) 58.3403 2.17724
\(719\) 4.04578i 0.150882i 0.997150 + 0.0754411i \(0.0240365\pi\)
−0.997150 + 0.0754411i \(0.975964\pi\)
\(720\) 0 0
\(721\) −11.8888 −0.442762
\(722\) 43.9355i 1.63511i
\(723\) 0 0
\(724\) 93.8320i 3.48724i
\(725\) 18.3488i 0.681458i
\(726\) 0 0
\(727\) −19.6221 −0.727742 −0.363871 0.931449i \(-0.618545\pi\)
−0.363871 + 0.931449i \(0.618545\pi\)
\(728\) 131.535i 4.87502i
\(729\) 0 0
\(730\) 0.664496i 0.0245941i
\(731\) −0.103965 −0.00384530
\(732\) 0 0
\(733\) 10.1672 0.375533 0.187766 0.982214i \(-0.439875\pi\)
0.187766 + 0.982214i \(0.439875\pi\)
\(734\) 57.8306i 2.13457i
\(735\) 0 0
\(736\) 0.127264i 0.00469103i
\(737\) 5.25084i 0.193417i
\(738\) 0 0
\(739\) −14.9520 −0.550017 −0.275009 0.961442i \(-0.588681\pi\)
−0.275009 + 0.961442i \(0.588681\pi\)
\(740\) −93.7859 −3.44764
\(741\) 0 0
\(742\) −70.3308 −2.58192
\(743\) 9.88641 0.362697 0.181349 0.983419i \(-0.441954\pi\)
0.181349 + 0.983419i \(0.441954\pi\)
\(744\) 0 0
\(745\) 20.2346i 0.741337i
\(746\) 59.2538i 2.16944i
\(747\) 0 0
\(748\) −30.4468 −1.11324
\(749\) 30.4765i 1.11359i
\(750\) 0 0
\(751\) −8.07548 −0.294678 −0.147339 0.989086i \(-0.547071\pi\)
−0.147339 + 0.989086i \(0.547071\pi\)
\(752\) 60.8690 2.21967
\(753\) 0 0
\(754\) 123.143i 4.48459i
\(755\) 22.7710 0.828721
\(756\) 0 0
\(757\) 3.17985 0.115573 0.0577867 0.998329i \(-0.481596\pi\)
0.0577867 + 0.998329i \(0.481596\pi\)
\(758\) 76.0795 2.76333
\(759\) 0 0
\(760\) 22.3361i 0.810217i
\(761\) 49.7358i 1.80292i 0.432860 + 0.901461i \(0.357504\pi\)
−0.432860 + 0.901461i \(0.642496\pi\)
\(762\) 0 0
\(763\) 58.3399i 2.11205i
\(764\) −104.109 −3.76653
\(765\) 0 0
\(766\) −68.1231 −2.46139
\(767\) 90.0207i 3.25046i
\(768\) 0 0
\(769\) 1.24815i 0.0450095i −0.999747 0.0225048i \(-0.992836\pi\)
0.999747 0.0225048i \(-0.00716409\pi\)
\(770\) −79.6477 −2.87030
\(771\) 0 0
\(772\) 33.8086 1.21680
\(773\) −35.7398 −1.28547 −0.642736 0.766088i \(-0.722201\pi\)
−0.642736 + 0.766088i \(0.722201\pi\)
\(774\) 0 0
\(775\) −2.08235 −0.0748004
\(776\) 75.8822 2.72401
\(777\) 0 0
\(778\) −26.5442 −0.951657
\(779\) 0.384449i 0.0137743i
\(780\) 0 0
\(781\) 55.8211 1.99744
\(782\) −0.119015 −0.00425596
\(783\) 0 0
\(784\) −20.6170 −0.736320
\(785\) 48.4837i 1.73046i
\(786\) 0 0
\(787\) 12.4044i 0.442169i −0.975255 0.221085i \(-0.929040\pi\)
0.975255 0.221085i \(-0.0709597\pi\)
\(788\) 60.4153i 2.15221i
\(789\) 0 0
\(790\) 86.1606i 3.06546i
\(791\) −31.9762 −1.13694
\(792\) 0 0
\(793\) 49.1600i 1.74572i
\(794\) 30.9146 1.09712
\(795\) 0 0
\(796\) 55.9379i 1.98267i
\(797\) 9.21427i 0.326386i 0.986594 + 0.163193i \(0.0521794\pi\)
−0.986594 + 0.163193i \(0.947821\pi\)
\(798\) 0 0
\(799\) 16.6026i 0.587357i
\(800\) 12.9822i 0.458990i
\(801\) 0 0
\(802\) −11.4371 −0.403857
\(803\) −0.344862 −0.0121699
\(804\) 0 0
\(805\) −0.214855 −0.00757265
\(806\) 13.9751 0.492251
\(807\) 0 0
\(808\) 14.4562 0.508569
\(809\) 4.93657 0.173561 0.0867804 0.996227i \(-0.472342\pi\)
0.0867804 + 0.996227i \(0.472342\pi\)
\(810\) 0 0
\(811\) 7.67772i 0.269601i −0.990873 0.134801i \(-0.956961\pi\)
0.990873 0.134801i \(-0.0430394\pi\)
\(812\) −102.061 −3.58165
\(813\) 0 0
\(814\) 70.5307i 2.47210i
\(815\) 11.7074i 0.410094i
\(816\) 0 0
\(817\) 0.0718469 0.00251360
\(818\) 3.00455i 0.105052i
\(819\) 0 0
\(820\) 3.59762i 0.125634i
\(821\) 28.3626 0.989861 0.494930 0.868933i \(-0.335194\pi\)
0.494930 + 0.868933i \(0.335194\pi\)
\(822\) 0 0
\(823\) 13.2138i 0.460605i −0.973119 0.230302i \(-0.926029\pi\)
0.973119 0.230302i \(-0.0739715\pi\)
\(824\) 23.4634 0.817387
\(825\) 0 0
\(826\) −108.114 −3.76177
\(827\) 39.3101i 1.36695i −0.729976 0.683473i \(-0.760469\pi\)
0.729976 0.683473i \(-0.239531\pi\)
\(828\) 0 0
\(829\) 37.2496i 1.29373i −0.762604 0.646866i \(-0.776079\pi\)
0.762604 0.646866i \(-0.223921\pi\)
\(830\) 65.1520i 2.26146i
\(831\) 0 0
\(832\) 5.44689i 0.188837i
\(833\) 5.62346i 0.194841i
\(834\) 0 0
\(835\) 30.0541i 1.04007i
\(836\) 21.0407 0.727707
\(837\) 0 0
\(838\) −10.5027 −0.362810
\(839\) 47.9046i 1.65385i −0.562313 0.826925i \(-0.690088\pi\)
0.562313 0.826925i \(-0.309912\pi\)
\(840\) 0 0
\(841\) −23.6415 −0.815223
\(842\) 90.8620i 3.13131i
\(843\) 0 0
\(844\) −72.8757 −2.50849
\(845\) 86.8043i 2.98616i
\(846\) 0 0
\(847\) 6.59277i 0.226530i
\(848\) 60.7268 2.08537
\(849\) 0 0
\(850\) −12.1407 −0.416421
\(851\) 0.190262i 0.00652208i
\(852\) 0 0
\(853\) −19.6700 −0.673487 −0.336743 0.941596i \(-0.609325\pi\)
−0.336743 + 0.941596i \(0.609325\pi\)
\(854\) −59.0407 −2.02033
\(855\) 0 0
\(856\) 60.1475i 2.05580i
\(857\) −9.99242 −0.341334 −0.170667 0.985329i \(-0.554592\pi\)
−0.170667 + 0.985329i \(0.554592\pi\)
\(858\) 0 0
\(859\) 16.5166 0.563541 0.281770 0.959482i \(-0.409078\pi\)
0.281770 + 0.959482i \(0.409078\pi\)
\(860\) 0.672334 0.0229264
\(861\) 0 0
\(862\) −11.1485 −0.379721
\(863\) 16.8671 0.574164 0.287082 0.957906i \(-0.407315\pi\)
0.287082 + 0.957906i \(0.407315\pi\)
\(864\) 0 0
\(865\) 12.2531i 0.416618i
\(866\) 85.2988 2.89857
\(867\) 0 0
\(868\) 11.5826i 0.393140i
\(869\) −44.7159 −1.51688
\(870\) 0 0
\(871\) 9.69711i 0.328574i
\(872\) 115.138i 3.89907i
\(873\) 0 0
\(874\) 0.0822470 0.00278205
\(875\) 21.4151 0.723962
\(876\) 0 0
\(877\) −15.7423 −0.531579 −0.265790 0.964031i \(-0.585633\pi\)
−0.265790 + 0.964031i \(0.585633\pi\)
\(878\) 2.84859i 0.0961353i
\(879\) 0 0
\(880\) 68.7714 2.31828
\(881\) −24.3002 −0.818695 −0.409348 0.912379i \(-0.634244\pi\)
−0.409348 + 0.912379i \(0.634244\pi\)
\(882\) 0 0
\(883\) −36.8222 −1.23917 −0.619583 0.784931i \(-0.712698\pi\)
−0.619583 + 0.784931i \(0.712698\pi\)
\(884\) 56.2283 1.89116
\(885\) 0 0
\(886\) −88.1265 −2.96067
\(887\) 4.70714i 0.158050i −0.996873 0.0790252i \(-0.974819\pi\)
0.996873 0.0790252i \(-0.0251807\pi\)
\(888\) 0 0
\(889\) 19.3909i 0.650349i
\(890\) 21.5678 0.722955
\(891\) 0 0
\(892\) 49.0897i 1.64364i
\(893\) 11.4735i 0.383945i
\(894\) 0 0
\(895\) 59.1251i 1.97633i
\(896\) −38.9688 −1.30186
\(897\) 0 0
\(898\) 70.2615i 2.34466i
\(899\) 5.97412i 0.199248i
\(900\) 0 0
\(901\) 16.5638i 0.551819i
\(902\) −2.70555 −0.0900851
\(903\) 0 0
\(904\) 63.1073 2.09892
\(905\) 57.8092 1.92164
\(906\) 0 0
\(907\) 50.7684i 1.68574i 0.538118 + 0.842869i \(0.319135\pi\)
−0.538118 + 0.842869i \(0.680865\pi\)
\(908\) 9.90994 0.328873
\(909\) 0 0
\(910\) 147.091 4.87603
\(911\) 28.6351 0.948722 0.474361 0.880330i \(-0.342679\pi\)
0.474361 + 0.880330i \(0.342679\pi\)
\(912\) 0 0
\(913\) 33.8128 1.11904
\(914\) 83.1052i 2.74887i
\(915\) 0 0
\(916\) 39.3514i 1.30021i
\(917\) 38.5208i 1.27207i
\(918\) 0 0
\(919\) 10.5541 0.348146 0.174073 0.984733i \(-0.444307\pi\)
0.174073 + 0.984733i \(0.444307\pi\)
\(920\) 0.424032 0.0139799
\(921\) 0 0
\(922\) 50.3885i 1.65946i
\(923\) −103.089 −3.39322
\(924\) 0 0
\(925\) 19.4085i 0.638148i
\(926\) −103.814 −3.41153
\(927\) 0 0
\(928\) 37.2450 1.22263
\(929\) 16.5948 0.544457 0.272228 0.962233i \(-0.412239\pi\)
0.272228 + 0.962233i \(0.412239\pi\)
\(930\) 0 0
\(931\) 3.88618i 0.127364i
\(932\) −87.4010 −2.86291
\(933\) 0 0
\(934\) 67.8796i 2.22109i
\(935\) 18.7580i 0.613453i
\(936\) 0 0
\(937\) 51.6131 1.68613 0.843063 0.537815i \(-0.180750\pi\)
0.843063 + 0.537815i \(0.180750\pi\)
\(938\) −11.6461 −0.380260
\(939\) 0 0
\(940\) 107.367i 3.50193i
\(941\) 47.3870 1.54477 0.772386 0.635154i \(-0.219063\pi\)
0.772386 + 0.635154i \(0.219063\pi\)
\(942\) 0 0
\(943\) −0.00729842 −0.000237669
\(944\) 93.3506 3.03830
\(945\) 0 0
\(946\) 0.505622i 0.0164392i
\(947\) 49.7804 1.61765 0.808823 0.588052i \(-0.200105\pi\)
0.808823 + 0.588052i \(0.200105\pi\)
\(948\) 0 0
\(949\) 0.636883 0.0206741
\(950\) 8.38999 0.272207
\(951\) 0 0
\(952\) 37.2045i 1.20581i
\(953\) −7.13021 −0.230970 −0.115485 0.993309i \(-0.536842\pi\)
−0.115485 + 0.993309i \(0.536842\pi\)
\(954\) 0 0
\(955\) 64.1407i 2.07555i
\(956\) −58.3353 + 36.5744i −1.88670 + 1.18290i
\(957\) 0 0
\(958\) 23.0182 0.743684
\(959\) 10.9003i 0.351990i
\(960\) 0 0
\(961\) −30.3220 −0.978130
\(962\) 130.254i 4.19957i
\(963\) 0 0
\(964\) −31.0126 −0.998847
\(965\) 20.8292i 0.670517i
\(966\) 0 0
\(967\) −29.2478 −0.940545 −0.470272 0.882521i \(-0.655844\pi\)
−0.470272 + 0.882521i \(0.655844\pi\)
\(968\) 13.0113i 0.418199i
\(969\) 0 0
\(970\) 84.8565i 2.72458i
\(971\) 21.0708i 0.676193i 0.941111 + 0.338096i \(0.109783\pi\)
−0.941111 + 0.338096i \(0.890217\pi\)
\(972\) 0 0
\(973\) 40.5134 1.29880
\(974\) 84.1302i 2.69571i
\(975\) 0 0
\(976\) 50.9784 1.63178
\(977\) −36.7623 −1.17613 −0.588065 0.808814i \(-0.700110\pi\)
−0.588065 + 0.808814i \(0.700110\pi\)
\(978\) 0 0
\(979\) 11.1933i 0.357740i
\(980\) 36.3664i 1.16168i
\(981\) 0 0
\(982\) 31.9822i 1.02059i
\(983\) 37.1739i 1.18566i −0.805327 0.592831i \(-0.798010\pi\)
0.805327 0.592831i \(-0.201990\pi\)
\(984\) 0 0
\(985\) −37.2214 −1.18597
\(986\) 34.8307i 1.10924i
\(987\) 0 0
\(988\) −38.8574 −1.23622
\(989\) 0.00136395i 4.33711e-5i
\(990\) 0 0
\(991\) 0.00579448i 0.000184068i 1.00000 9.20339e-5i \(2.92953e-5\pi\)
−1.00000 9.20339e-5i \(0.999971\pi\)
\(992\) 4.22682i 0.134202i
\(993\) 0 0
\(994\) 123.809i 3.92698i
\(995\) −34.4629 −1.09255
\(996\) 0 0
\(997\) 13.0063i 0.411913i −0.978561 0.205956i \(-0.933969\pi\)
0.978561 0.205956i \(-0.0660305\pi\)
\(998\) 46.6362 1.47624
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2151.2.b.a.2150.5 80
3.2 odd 2 inner 2151.2.b.a.2150.75 yes 80
239.238 odd 2 inner 2151.2.b.a.2150.6 yes 80
717.716 even 2 inner 2151.2.b.a.2150.76 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.b.a.2150.5 80 1.1 even 1 trivial
2151.2.b.a.2150.6 yes 80 239.238 odd 2 inner
2151.2.b.a.2150.75 yes 80 3.2 odd 2 inner
2151.2.b.a.2150.76 yes 80 717.716 even 2 inner