Properties

Label 225.4.f.c.143.4
Level $225$
Weight $4$
Character 225.143
Analytic conductor $13.275$
Analytic rank $0$
Dimension $12$
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] = [225,4,Mod(107,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.f (of order \(4\), degree \(2\), 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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.4
Root \(-2.02004 - 2.30794i\) of defining polynomial
Character \(\chi\) \(=\) 225.143
Dual form 225.4.f.c.107.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.287902 + 0.287902i) q^{2} -7.83422i q^{4} +(-15.0557 + 15.0557i) q^{7} +(4.55871 - 4.55871i) q^{8} +O(q^{10})\) \(q+(0.287902 + 0.287902i) q^{2} -7.83422i q^{4} +(-15.0557 + 15.0557i) q^{7} +(4.55871 - 4.55871i) q^{8} -27.9992i q^{11} +(-2.49862 - 2.49862i) q^{13} -8.66913 q^{14} -60.0489 q^{16} +(-67.9104 - 67.9104i) q^{17} +95.2200i q^{19} +(8.06104 - 8.06104i) q^{22} +(-121.994 + 121.994i) q^{23} -1.43872i q^{26} +(117.950 + 117.950i) q^{28} -99.0852 q^{29} -28.7800 q^{31} +(-53.7578 - 53.7578i) q^{32} -39.1031i q^{34} +(-271.064 + 271.064i) q^{37} +(-27.4140 + 27.4140i) q^{38} -453.748i q^{41} +(-30.5679 - 30.5679i) q^{43} -219.352 q^{44} -70.2445 q^{46} +(-254.600 - 254.600i) q^{47} -110.348i q^{49} +(-19.5748 + 19.5748i) q^{52} +(224.021 - 224.021i) q^{53} +137.269i q^{56} +(-28.5268 - 28.5268i) q^{58} +483.263 q^{59} -264.372 q^{61} +(-8.28582 - 8.28582i) q^{62} +449.437i q^{64} +(498.817 - 498.817i) q^{67} +(-532.025 + 532.025i) q^{68} +609.904i q^{71} +(74.6843 + 74.6843i) q^{73} -156.080 q^{74} +745.975 q^{76} +(421.548 + 421.548i) q^{77} -406.574i q^{79} +(130.635 - 130.635i) q^{82} +(652.278 - 652.278i) q^{83} -17.6011i q^{86} +(-127.640 - 127.640i) q^{88} -139.860 q^{89} +75.2369 q^{91} +(955.726 + 955.726i) q^{92} -146.600i q^{94} +(-557.633 + 557.633i) q^{97} +(31.7693 - 31.7693i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{7} + 108 q^{13} - 648 q^{16} - 1056 q^{22} + 576 q^{28} - 1248 q^{31} - 828 q^{37} + 96 q^{43} + 672 q^{46} + 312 q^{52} + 3864 q^{58} + 96 q^{61} - 1632 q^{67} - 3972 q^{73} - 480 q^{76} + 7848 q^{82} - 7968 q^{88} + 4752 q^{91} - 2772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.287902 + 0.287902i 0.101789 + 0.101789i 0.756167 0.654378i \(-0.227070\pi\)
−0.654378 + 0.756167i \(0.727070\pi\)
\(3\) 0 0
\(4\) 7.83422i 0.979278i
\(5\) 0 0
\(6\) 0 0
\(7\) −15.0557 + 15.0557i −0.812931 + 0.812931i −0.985072 0.172141i \(-0.944931\pi\)
0.172141 + 0.985072i \(0.444931\pi\)
\(8\) 4.55871 4.55871i 0.201468 0.201468i
\(9\) 0 0
\(10\) 0 0
\(11\) 27.9992i 0.767462i −0.923445 0.383731i \(-0.874639\pi\)
0.923445 0.383731i \(-0.125361\pi\)
\(12\) 0 0
\(13\) −2.49862 2.49862i −0.0533071 0.0533071i 0.679951 0.733258i \(-0.262001\pi\)
−0.733258 + 0.679951i \(0.762001\pi\)
\(14\) −8.66913 −0.165494
\(15\) 0 0
\(16\) −60.0489 −0.938264
\(17\) −67.9104 67.9104i −0.968864 0.968864i 0.0306653 0.999530i \(-0.490237\pi\)
−0.999530 + 0.0306653i \(0.990237\pi\)
\(18\) 0 0
\(19\) 95.2200i 1.14974i 0.818246 + 0.574868i \(0.194946\pi\)
−0.818246 + 0.574868i \(0.805054\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8.06104 8.06104i 0.0781191 0.0781191i
\(23\) −121.994 + 121.994i −1.10598 + 1.10598i −0.112302 + 0.993674i \(0.535822\pi\)
−0.993674 + 0.112302i \(0.964178\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.43872i 0.0108521i
\(27\) 0 0
\(28\) 117.950 + 117.950i 0.796085 + 0.796085i
\(29\) −99.0852 −0.634471 −0.317235 0.948347i \(-0.602755\pi\)
−0.317235 + 0.948347i \(0.602755\pi\)
\(30\) 0 0
\(31\) −28.7800 −0.166743 −0.0833716 0.996519i \(-0.526569\pi\)
−0.0833716 + 0.996519i \(0.526569\pi\)
\(32\) −53.7578 53.7578i −0.296973 0.296973i
\(33\) 0 0
\(34\) 39.1031i 0.197239i
\(35\) 0 0
\(36\) 0 0
\(37\) −271.064 + 271.064i −1.20439 + 1.20439i −0.231578 + 0.972816i \(0.574389\pi\)
−0.972816 + 0.231578i \(0.925611\pi\)
\(38\) −27.4140 + 27.4140i −0.117030 + 0.117030i
\(39\) 0 0
\(40\) 0 0
\(41\) 453.748i 1.72838i −0.503166 0.864190i \(-0.667832\pi\)
0.503166 0.864190i \(-0.332168\pi\)
\(42\) 0 0
\(43\) −30.5679 30.5679i −0.108409 0.108409i 0.650822 0.759230i \(-0.274424\pi\)
−0.759230 + 0.650822i \(0.774424\pi\)
\(44\) −219.352 −0.751559
\(45\) 0 0
\(46\) −70.2445 −0.225152
\(47\) −254.600 254.600i −0.790153 0.790153i 0.191365 0.981519i \(-0.438708\pi\)
−0.981519 + 0.191365i \(0.938708\pi\)
\(48\) 0 0
\(49\) 110.348i 0.321713i
\(50\) 0 0
\(51\) 0 0
\(52\) −19.5748 + 19.5748i −0.0522025 + 0.0522025i
\(53\) 224.021 224.021i 0.580598 0.580598i −0.354470 0.935068i \(-0.615339\pi\)
0.935068 + 0.354470i \(0.115339\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 137.269i 0.327560i
\(57\) 0 0
\(58\) −28.5268 28.5268i −0.0645820 0.0645820i
\(59\) 483.263 1.06636 0.533182 0.846001i \(-0.320996\pi\)
0.533182 + 0.846001i \(0.320996\pi\)
\(60\) 0 0
\(61\) −264.372 −0.554908 −0.277454 0.960739i \(-0.589491\pi\)
−0.277454 + 0.960739i \(0.589491\pi\)
\(62\) −8.28582 8.28582i −0.0169726 0.0169726i
\(63\) 0 0
\(64\) 449.437i 0.877807i
\(65\) 0 0
\(66\) 0 0
\(67\) 498.817 498.817i 0.909556 0.909556i −0.0866805 0.996236i \(-0.527626\pi\)
0.996236 + 0.0866805i \(0.0276259\pi\)
\(68\) −532.025 + 532.025i −0.948788 + 0.948788i
\(69\) 0 0
\(70\) 0 0
\(71\) 609.904i 1.01947i 0.860332 + 0.509735i \(0.170256\pi\)
−0.860332 + 0.509735i \(0.829744\pi\)
\(72\) 0 0
\(73\) 74.6843 + 74.6843i 0.119742 + 0.119742i 0.764438 0.644697i \(-0.223016\pi\)
−0.644697 + 0.764438i \(0.723016\pi\)
\(74\) −156.080 −0.245188
\(75\) 0 0
\(76\) 745.975 1.12591
\(77\) 421.548 + 421.548i 0.623894 + 0.623894i
\(78\) 0 0
\(79\) 406.574i 0.579027i −0.957174 0.289513i \(-0.906507\pi\)
0.957174 0.289513i \(-0.0934935\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 130.635 130.635i 0.175930 0.175930i
\(83\) 652.278 652.278i 0.862613 0.862613i −0.129028 0.991641i \(-0.541186\pi\)
0.991641 + 0.129028i \(0.0411858\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 17.6011i 0.0220695i
\(87\) 0 0
\(88\) −127.640 127.640i −0.154619 0.154619i
\(89\) −139.860 −0.166575 −0.0832873 0.996526i \(-0.526542\pi\)
−0.0832873 + 0.996526i \(0.526542\pi\)
\(90\) 0 0
\(91\) 75.2369 0.0866700
\(92\) 955.726 + 955.726i 1.08306 + 1.08306i
\(93\) 0 0
\(94\) 146.600i 0.160857i
\(95\) 0 0
\(96\) 0 0
\(97\) −557.633 + 557.633i −0.583701 + 0.583701i −0.935918 0.352217i \(-0.885428\pi\)
0.352217 + 0.935918i \(0.385428\pi\)
\(98\) 31.7693 31.7693i 0.0327468 0.0327468i
\(99\) 0 0
\(100\) 0 0
\(101\) 299.833i 0.295391i 0.989033 + 0.147695i \(0.0471855\pi\)
−0.989033 + 0.147695i \(0.952814\pi\)
\(102\) 0 0
\(103\) −577.974 577.974i −0.552907 0.552907i 0.374372 0.927279i \(-0.377858\pi\)
−0.927279 + 0.374372i \(0.877858\pi\)
\(104\) −22.7810 −0.0214794
\(105\) 0 0
\(106\) 128.992 0.118197
\(107\) 216.994 + 216.994i 0.196053 + 0.196053i 0.798305 0.602253i \(-0.205730\pi\)
−0.602253 + 0.798305i \(0.705730\pi\)
\(108\) 0 0
\(109\) 936.415i 0.822865i −0.911440 0.411432i \(-0.865029\pi\)
0.911440 0.411432i \(-0.134971\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 904.077 904.077i 0.762743 0.762743i
\(113\) 1084.00 1084.00i 0.902428 0.902428i −0.0932173 0.995646i \(-0.529715\pi\)
0.995646 + 0.0932173i \(0.0297151\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 776.256i 0.621323i
\(117\) 0 0
\(118\) 139.132 + 139.132i 0.108544 + 0.108544i
\(119\) 2044.88 1.57524
\(120\) 0 0
\(121\) 547.043 0.411001
\(122\) −76.1133 76.1133i −0.0564834 0.0564834i
\(123\) 0 0
\(124\) 225.469i 0.163288i
\(125\) 0 0
\(126\) 0 0
\(127\) 915.574 915.574i 0.639717 0.639717i −0.310769 0.950486i \(-0.600586\pi\)
0.950486 + 0.310769i \(0.100586\pi\)
\(128\) −559.457 + 559.457i −0.386324 + 0.386324i
\(129\) 0 0
\(130\) 0 0
\(131\) 629.329i 0.419731i 0.977730 + 0.209865i \(0.0673026\pi\)
−0.977730 + 0.209865i \(0.932697\pi\)
\(132\) 0 0
\(133\) −1433.60 1433.60i −0.934655 0.934655i
\(134\) 287.221 0.185165
\(135\) 0 0
\(136\) −619.167 −0.390391
\(137\) −660.931 660.931i −0.412169 0.412169i 0.470325 0.882494i \(-0.344137\pi\)
−0.882494 + 0.470325i \(0.844137\pi\)
\(138\) 0 0
\(139\) 1378.35i 0.841081i 0.907274 + 0.420540i \(0.138160\pi\)
−0.907274 + 0.420540i \(0.861840\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −175.593 + 175.593i −0.103771 + 0.103771i
\(143\) −69.9595 + 69.9595i −0.0409112 + 0.0409112i
\(144\) 0 0
\(145\) 0 0
\(146\) 43.0035i 0.0243767i
\(147\) 0 0
\(148\) 2123.57 + 2123.57i 1.17944 + 1.17944i
\(149\) −924.719 −0.508429 −0.254215 0.967148i \(-0.581817\pi\)
−0.254215 + 0.967148i \(0.581817\pi\)
\(150\) 0 0
\(151\) −401.768 −0.216526 −0.108263 0.994122i \(-0.534529\pi\)
−0.108263 + 0.994122i \(0.534529\pi\)
\(152\) 434.080 + 434.080i 0.231635 + 0.231635i
\(153\) 0 0
\(154\) 242.729i 0.127011i
\(155\) 0 0
\(156\) 0 0
\(157\) −1305.55 + 1305.55i −0.663659 + 0.663659i −0.956240 0.292582i \(-0.905486\pi\)
0.292582 + 0.956240i \(0.405486\pi\)
\(158\) 117.053 117.053i 0.0589384 0.0589384i
\(159\) 0 0
\(160\) 0 0
\(161\) 3673.40i 1.79816i
\(162\) 0 0
\(163\) 78.8393 + 78.8393i 0.0378845 + 0.0378845i 0.725795 0.687911i \(-0.241472\pi\)
−0.687911 + 0.725795i \(0.741472\pi\)
\(164\) −3554.77 −1.69256
\(165\) 0 0
\(166\) 375.585 0.175609
\(167\) −428.116 428.116i −0.198375 0.198375i 0.600928 0.799303i \(-0.294798\pi\)
−0.799303 + 0.600928i \(0.794798\pi\)
\(168\) 0 0
\(169\) 2184.51i 0.994317i
\(170\) 0 0
\(171\) 0 0
\(172\) −239.476 + 239.476i −0.106162 + 0.106162i
\(173\) −1390.57 + 1390.57i −0.611114 + 0.611114i −0.943236 0.332122i \(-0.892235\pi\)
0.332122 + 0.943236i \(0.392235\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1681.32i 0.720082i
\(177\) 0 0
\(178\) −40.2660 40.2660i −0.0169554 0.0169554i
\(179\) 110.674 0.0462131 0.0231065 0.999733i \(-0.492644\pi\)
0.0231065 + 0.999733i \(0.492644\pi\)
\(180\) 0 0
\(181\) 700.593 0.287706 0.143853 0.989599i \(-0.454051\pi\)
0.143853 + 0.989599i \(0.454051\pi\)
\(182\) 21.6609 + 21.6609i 0.00882203 + 0.00882203i
\(183\) 0 0
\(184\) 1112.27i 0.445638i
\(185\) 0 0
\(186\) 0 0
\(187\) −1901.44 + 1901.44i −0.743567 + 0.743567i
\(188\) −1994.59 + 1994.59i −0.773780 + 0.773780i
\(189\) 0 0
\(190\) 0 0
\(191\) 2214.32i 0.838862i −0.907787 0.419431i \(-0.862230\pi\)
0.907787 0.419431i \(-0.137770\pi\)
\(192\) 0 0
\(193\) 1964.16 + 1964.16i 0.732556 + 0.732556i 0.971125 0.238569i \(-0.0766785\pi\)
−0.238569 + 0.971125i \(0.576678\pi\)
\(194\) −321.087 −0.118828
\(195\) 0 0
\(196\) −864.488 −0.315047
\(197\) −2700.70 2700.70i −0.976735 0.976735i 0.0230004 0.999735i \(-0.492678\pi\)
−0.999735 + 0.0230004i \(0.992678\pi\)
\(198\) 0 0
\(199\) 841.195i 0.299652i 0.988712 + 0.149826i \(0.0478713\pi\)
−0.988712 + 0.149826i \(0.952129\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −86.3224 + 86.3224i −0.0300674 + 0.0300674i
\(203\) 1491.80 1491.80i 0.515781 0.515781i
\(204\) 0 0
\(205\) 0 0
\(206\) 332.800i 0.112559i
\(207\) 0 0
\(208\) 150.039 + 150.039i 0.0500161 + 0.0500161i
\(209\) 2666.09 0.882379
\(210\) 0 0
\(211\) −3970.06 −1.29531 −0.647654 0.761934i \(-0.724250\pi\)
−0.647654 + 0.761934i \(0.724250\pi\)
\(212\) −1755.03 1755.03i −0.568567 0.568567i
\(213\) 0 0
\(214\) 124.946i 0.0399119i
\(215\) 0 0
\(216\) 0 0
\(217\) 433.303 433.303i 0.135551 0.135551i
\(218\) 269.596 269.596i 0.0837584 0.0837584i
\(219\) 0 0
\(220\) 0 0
\(221\) 339.365i 0.103295i
\(222\) 0 0
\(223\) −3600.34 3600.34i −1.08115 1.08115i −0.996402 0.0847496i \(-0.972991\pi\)
−0.0847496 0.996402i \(-0.527009\pi\)
\(224\) 1618.72 0.482837
\(225\) 0 0
\(226\) 624.173 0.183714
\(227\) −1734.81 1734.81i −0.507239 0.507239i 0.406439 0.913678i \(-0.366770\pi\)
−0.913678 + 0.406439i \(0.866770\pi\)
\(228\) 0 0
\(229\) 3467.74i 1.00067i 0.865831 + 0.500337i \(0.166791\pi\)
−0.865831 + 0.500337i \(0.833209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −451.700 + 451.700i −0.127826 + 0.127826i
\(233\) −1070.17 + 1070.17i −0.300899 + 0.300899i −0.841365 0.540467i \(-0.818248\pi\)
0.540467 + 0.841365i \(0.318248\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3785.99i 1.04427i
\(237\) 0 0
\(238\) 588.724 + 588.724i 0.160342 + 0.160342i
\(239\) −3909.44 −1.05808 −0.529038 0.848598i \(-0.677447\pi\)
−0.529038 + 0.848598i \(0.677447\pi\)
\(240\) 0 0
\(241\) −4787.09 −1.27952 −0.639759 0.768576i \(-0.720966\pi\)
−0.639759 + 0.768576i \(0.720966\pi\)
\(242\) 157.495 + 157.495i 0.0418353 + 0.0418353i
\(243\) 0 0
\(244\) 2071.15i 0.543409i
\(245\) 0 0
\(246\) 0 0
\(247\) 237.919 237.919i 0.0612891 0.0612891i
\(248\) −131.200 + 131.200i −0.0335935 + 0.0335935i
\(249\) 0 0
\(250\) 0 0
\(251\) 2337.42i 0.587794i 0.955837 + 0.293897i \(0.0949523\pi\)
−0.955837 + 0.293897i \(0.905048\pi\)
\(252\) 0 0
\(253\) 3415.73 + 3415.73i 0.848795 + 0.848795i
\(254\) 527.191 0.130232
\(255\) 0 0
\(256\) 3273.36 0.799160
\(257\) 977.764 + 977.764i 0.237320 + 0.237320i 0.815740 0.578419i \(-0.196330\pi\)
−0.578419 + 0.815740i \(0.696330\pi\)
\(258\) 0 0
\(259\) 8162.10i 1.95818i
\(260\) 0 0
\(261\) 0 0
\(262\) −181.185 + 181.185i −0.0427239 + 0.0427239i
\(263\) 790.690 790.690i 0.185384 0.185384i −0.608313 0.793697i \(-0.708153\pi\)
0.793697 + 0.608313i \(0.208153\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 825.474i 0.190275i
\(267\) 0 0
\(268\) −3907.85 3907.85i −0.890708 0.890708i
\(269\) −3954.49 −0.896317 −0.448159 0.893954i \(-0.647920\pi\)
−0.448159 + 0.893954i \(0.647920\pi\)
\(270\) 0 0
\(271\) 7863.30 1.76259 0.881295 0.472567i \(-0.156673\pi\)
0.881295 + 0.472567i \(0.156673\pi\)
\(272\) 4077.94 + 4077.94i 0.909050 + 0.909050i
\(273\) 0 0
\(274\) 380.567i 0.0839083i
\(275\) 0 0
\(276\) 0 0
\(277\) −908.009 + 908.009i −0.196957 + 0.196957i −0.798694 0.601737i \(-0.794475\pi\)
0.601737 + 0.798694i \(0.294475\pi\)
\(278\) −396.830 + 396.830i −0.0856126 + 0.0856126i
\(279\) 0 0
\(280\) 0 0
\(281\) 7102.70i 1.50787i 0.656949 + 0.753935i \(0.271847\pi\)
−0.656949 + 0.753935i \(0.728153\pi\)
\(282\) 0 0
\(283\) 4721.93 + 4721.93i 0.991837 + 0.991837i 0.999967 0.00813014i \(-0.00258793\pi\)
−0.00813014 + 0.999967i \(0.502588\pi\)
\(284\) 4778.13 0.998344
\(285\) 0 0
\(286\) −40.2829 −0.00832860
\(287\) 6831.49 + 6831.49i 1.40505 + 1.40505i
\(288\) 0 0
\(289\) 4310.65i 0.877396i
\(290\) 0 0
\(291\) 0 0
\(292\) 585.094 585.094i 0.117260 0.117260i
\(293\) −2078.57 + 2078.57i −0.414442 + 0.414442i −0.883283 0.468841i \(-0.844672\pi\)
0.468841 + 0.883283i \(0.344672\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2471.40i 0.485294i
\(297\) 0 0
\(298\) −266.229 266.229i −0.0517524 0.0517524i
\(299\) 609.632 0.117913
\(300\) 0 0
\(301\) 920.443 0.176257
\(302\) −115.670 115.670i −0.0220399 0.0220399i
\(303\) 0 0
\(304\) 5717.85i 1.07875i
\(305\) 0 0
\(306\) 0 0
\(307\) 5325.58 5325.58i 0.990056 0.990056i −0.00989542 0.999951i \(-0.503150\pi\)
0.999951 + 0.00989542i \(0.00314986\pi\)
\(308\) 3302.50 3302.50i 0.610966 0.610966i
\(309\) 0 0
\(310\) 0 0
\(311\) 2005.56i 0.365676i −0.983143 0.182838i \(-0.941472\pi\)
0.983143 0.182838i \(-0.0585283\pi\)
\(312\) 0 0
\(313\) −5026.49 5026.49i −0.907713 0.907713i 0.0883746 0.996087i \(-0.471833\pi\)
−0.996087 + 0.0883746i \(0.971833\pi\)
\(314\) −751.742 −0.135106
\(315\) 0 0
\(316\) −3185.19 −0.567028
\(317\) −3137.15 3137.15i −0.555836 0.555836i 0.372283 0.928119i \(-0.378575\pi\)
−0.928119 + 0.372283i \(0.878575\pi\)
\(318\) 0 0
\(319\) 2774.31i 0.486933i
\(320\) 0 0
\(321\) 0 0
\(322\) 1057.58 1057.58i 0.183033 0.183033i
\(323\) 6466.43 6466.43i 1.11394 1.11394i
\(324\) 0 0
\(325\) 0 0
\(326\) 45.3960i 0.00771243i
\(327\) 0 0
\(328\) −2068.50 2068.50i −0.348214 0.348214i
\(329\) 7666.35 1.28468
\(330\) 0 0
\(331\) 2978.02 0.494523 0.247261 0.968949i \(-0.420469\pi\)
0.247261 + 0.968949i \(0.420469\pi\)
\(332\) −5110.10 5110.10i −0.844738 0.844738i
\(333\) 0 0
\(334\) 246.511i 0.0403847i
\(335\) 0 0
\(336\) 0 0
\(337\) 5985.73 5985.73i 0.967547 0.967547i −0.0319428 0.999490i \(-0.510169\pi\)
0.999490 + 0.0319428i \(0.0101694\pi\)
\(338\) 628.926 628.926i 0.101210 0.101210i
\(339\) 0 0
\(340\) 0 0
\(341\) 805.818i 0.127969i
\(342\) 0 0
\(343\) −3502.74 3502.74i −0.551400 0.551400i
\(344\) −278.701 −0.0436818
\(345\) 0 0
\(346\) −800.693 −0.124409
\(347\) 8183.37 + 8183.37i 1.26601 + 1.26601i 0.948130 + 0.317881i \(0.102971\pi\)
0.317881 + 0.948130i \(0.397029\pi\)
\(348\) 0 0
\(349\) 1421.22i 0.217983i 0.994043 + 0.108991i \(0.0347621\pi\)
−0.994043 + 0.108991i \(0.965238\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1505.18 + 1505.18i −0.227916 + 0.227916i
\(353\) −1399.56 + 1399.56i −0.211023 + 0.211023i −0.804702 0.593679i \(-0.797675\pi\)
0.593679 + 0.804702i \(0.297675\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1095.70i 0.163123i
\(357\) 0 0
\(358\) 31.8632 + 31.8632i 0.00470397 + 0.00470397i
\(359\) 4313.98 0.634215 0.317108 0.948390i \(-0.397288\pi\)
0.317108 + 0.948390i \(0.397288\pi\)
\(360\) 0 0
\(361\) −2207.85 −0.321891
\(362\) 201.702 + 201.702i 0.0292852 + 0.0292852i
\(363\) 0 0
\(364\) 589.423i 0.0848740i
\(365\) 0 0
\(366\) 0 0
\(367\) −4533.34 + 4533.34i −0.644792 + 0.644792i −0.951730 0.306938i \(-0.900696\pi\)
0.306938 + 0.951730i \(0.400696\pi\)
\(368\) 7325.59 7325.59i 1.03770 1.03770i
\(369\) 0 0
\(370\) 0 0
\(371\) 6745.59i 0.943972i
\(372\) 0 0
\(373\) 2787.95 + 2787.95i 0.387010 + 0.387010i 0.873620 0.486609i \(-0.161766\pi\)
−0.486609 + 0.873620i \(0.661766\pi\)
\(374\) −1094.86 −0.151374
\(375\) 0 0
\(376\) −2321.29 −0.318382
\(377\) 247.576 + 247.576i 0.0338218 + 0.0338218i
\(378\) 0 0
\(379\) 8409.09i 1.13970i −0.821749 0.569849i \(-0.807002\pi\)
0.821749 0.569849i \(-0.192998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 637.508 637.508i 0.0853868 0.0853868i
\(383\) −8507.38 + 8507.38i −1.13500 + 1.13500i −0.145672 + 0.989333i \(0.546534\pi\)
−0.989333 + 0.145672i \(0.953466\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1130.97i 0.149132i
\(387\) 0 0
\(388\) 4368.62 + 4368.62i 0.571606 + 0.571606i
\(389\) −8641.23 −1.12629 −0.563146 0.826357i \(-0.690409\pi\)
−0.563146 + 0.826357i \(0.690409\pi\)
\(390\) 0 0
\(391\) 16569.3 2.14308
\(392\) −503.042 503.042i −0.0648150 0.0648150i
\(393\) 0 0
\(394\) 1555.07i 0.198841i
\(395\) 0 0
\(396\) 0 0
\(397\) −1902.21 + 1902.21i −0.240476 + 0.240476i −0.817047 0.576571i \(-0.804390\pi\)
0.576571 + 0.817047i \(0.304390\pi\)
\(398\) −242.182 + 242.182i −0.0305012 + 0.0305012i
\(399\) 0 0
\(400\) 0 0
\(401\) 13640.1i 1.69864i −0.527882 0.849318i \(-0.677014\pi\)
0.527882 0.849318i \(-0.322986\pi\)
\(402\) 0 0
\(403\) 71.9103 + 71.9103i 0.00888860 + 0.00888860i
\(404\) 2348.96 0.289270
\(405\) 0 0
\(406\) 858.982 0.105001
\(407\) 7589.57 + 7589.57i 0.924327 + 0.924327i
\(408\) 0 0
\(409\) 6395.02i 0.773138i 0.922260 + 0.386569i \(0.126340\pi\)
−0.922260 + 0.386569i \(0.873660\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4527.98 + 4527.98i −0.541450 + 0.541450i
\(413\) −7275.86 + 7275.86i −0.866880 + 0.866880i
\(414\) 0 0
\(415\) 0 0
\(416\) 268.641i 0.0316615i
\(417\) 0 0
\(418\) 767.572 + 767.572i 0.0898162 + 0.0898162i
\(419\) −8474.94 −0.988133 −0.494067 0.869424i \(-0.664490\pi\)
−0.494067 + 0.869424i \(0.664490\pi\)
\(420\) 0 0
\(421\) −9069.21 −1.04990 −0.524948 0.851134i \(-0.675915\pi\)
−0.524948 + 0.851134i \(0.675915\pi\)
\(422\) −1142.99 1142.99i −0.131848 0.131848i
\(423\) 0 0
\(424\) 2042.49i 0.233944i
\(425\) 0 0
\(426\) 0 0
\(427\) 3980.30 3980.30i 0.451102 0.451102i
\(428\) 1699.98 1699.98i 0.191990 0.191990i
\(429\) 0 0
\(430\) 0 0
\(431\) 8388.11i 0.937450i 0.883344 + 0.468725i \(0.155287\pi\)
−0.883344 + 0.468725i \(0.844713\pi\)
\(432\) 0 0
\(433\) 4132.17 + 4132.17i 0.458613 + 0.458613i 0.898200 0.439587i \(-0.144875\pi\)
−0.439587 + 0.898200i \(0.644875\pi\)
\(434\) 249.498 0.0275951
\(435\) 0 0
\(436\) −7336.08 −0.805814
\(437\) −11616.2 11616.2i −1.27158 1.27158i
\(438\) 0 0
\(439\) 10498.9i 1.14143i 0.821149 + 0.570714i \(0.193333\pi\)
−0.821149 + 0.570714i \(0.806667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −97.7038 + 97.7038i −0.0105142 + 0.0105142i
\(443\) −1465.93 + 1465.93i −0.157220 + 0.157220i −0.781333 0.624114i \(-0.785460\pi\)
0.624114 + 0.781333i \(0.285460\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2073.09i 0.220098i
\(447\) 0 0
\(448\) −6766.58 6766.58i −0.713596 0.713596i
\(449\) −6295.42 −0.661691 −0.330846 0.943685i \(-0.607334\pi\)
−0.330846 + 0.943685i \(0.607334\pi\)
\(450\) 0 0
\(451\) −12704.6 −1.32647
\(452\) −8492.32 8492.32i −0.883728 0.883728i
\(453\) 0 0
\(454\) 998.909i 0.103262i
\(455\) 0 0
\(456\) 0 0
\(457\) −9065.09 + 9065.09i −0.927893 + 0.927893i −0.997570 0.0696767i \(-0.977803\pi\)
0.0696767 + 0.997570i \(0.477803\pi\)
\(458\) −998.368 + 998.368i −0.101857 + 0.101857i
\(459\) 0 0
\(460\) 0 0
\(461\) 13305.5i 1.34424i −0.740440 0.672122i \(-0.765383\pi\)
0.740440 0.672122i \(-0.234617\pi\)
\(462\) 0 0
\(463\) 3124.86 + 3124.86i 0.313660 + 0.313660i 0.846326 0.532666i \(-0.178810\pi\)
−0.532666 + 0.846326i \(0.678810\pi\)
\(464\) 5949.95 0.595301
\(465\) 0 0
\(466\) −616.210 −0.0612562
\(467\) −4255.07 4255.07i −0.421629 0.421629i 0.464135 0.885764i \(-0.346365\pi\)
−0.885764 + 0.464135i \(0.846365\pi\)
\(468\) 0 0
\(469\) 15020.1i 1.47881i
\(470\) 0 0
\(471\) 0 0
\(472\) 2203.05 2203.05i 0.214838 0.214838i
\(473\) −855.879 + 855.879i −0.0831995 + 0.0831995i
\(474\) 0 0
\(475\) 0 0
\(476\) 16020.0i 1.54260i
\(477\) 0 0
\(478\) −1125.53 1125.53i −0.107700 0.107700i
\(479\) 10625.2 1.01352 0.506761 0.862087i \(-0.330843\pi\)
0.506761 + 0.862087i \(0.330843\pi\)
\(480\) 0 0
\(481\) 1354.57 0.128406
\(482\) −1378.21 1378.21i −0.130241 0.130241i
\(483\) 0 0
\(484\) 4285.66i 0.402485i
\(485\) 0 0
\(486\) 0 0
\(487\) 3269.03 3269.03i 0.304177 0.304177i −0.538469 0.842645i \(-0.680997\pi\)
0.842645 + 0.538469i \(0.180997\pi\)
\(488\) −1205.19 + 1205.19i −0.111796 + 0.111796i
\(489\) 0 0
\(490\) 0 0
\(491\) 18791.1i 1.72715i 0.504220 + 0.863575i \(0.331780\pi\)
−0.504220 + 0.863575i \(0.668220\pi\)
\(492\) 0 0
\(493\) 6728.92 + 6728.92i 0.614716 + 0.614716i
\(494\) 136.995 0.0124771
\(495\) 0 0
\(496\) 1728.21 0.156449
\(497\) −9182.53 9182.53i −0.828758 0.828758i
\(498\) 0 0
\(499\) 7220.67i 0.647778i −0.946095 0.323889i \(-0.895010\pi\)
0.946095 0.323889i \(-0.104990\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −672.947 + 672.947i −0.0598308 + 0.0598308i
\(503\) −2081.38 + 2081.38i −0.184501 + 0.184501i −0.793314 0.608813i \(-0.791646\pi\)
0.608813 + 0.793314i \(0.291646\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1966.79i 0.172796i
\(507\) 0 0
\(508\) −7172.81 7172.81i −0.626461 0.626461i
\(509\) −12624.1 −1.09932 −0.549659 0.835389i \(-0.685242\pi\)
−0.549659 + 0.835389i \(0.685242\pi\)
\(510\) 0 0
\(511\) −2248.85 −0.194683
\(512\) 5418.06 + 5418.06i 0.467669 + 0.467669i
\(513\) 0 0
\(514\) 563.001i 0.0483130i
\(515\) 0 0
\(516\) 0 0
\(517\) −7128.60 + 7128.60i −0.606413 + 0.606413i
\(518\) 2349.89 2349.89i 0.199321 0.199321i
\(519\) 0 0
\(520\) 0 0
\(521\) 11837.5i 0.995409i −0.867347 0.497705i \(-0.834176\pi\)
0.867347 0.497705i \(-0.165824\pi\)
\(522\) 0 0
\(523\) 5538.72 + 5538.72i 0.463081 + 0.463081i 0.899664 0.436583i \(-0.143811\pi\)
−0.436583 + 0.899664i \(0.643811\pi\)
\(524\) 4930.31 0.411033
\(525\) 0 0
\(526\) 455.282 0.0377400
\(527\) 1954.46 + 1954.46i 0.161552 + 0.161552i
\(528\) 0 0
\(529\) 17597.9i 1.44637i
\(530\) 0 0
\(531\) 0 0
\(532\) −11231.2 + 11231.2i −0.915287 + 0.915287i
\(533\) −1133.74 + 1133.74i −0.0921349 + 0.0921349i
\(534\) 0 0
\(535\) 0 0
\(536\) 4547.92i 0.366493i
\(537\) 0 0
\(538\) −1138.51 1138.51i −0.0912350 0.0912350i
\(539\) −3089.65 −0.246903
\(540\) 0 0
\(541\) 5207.24 0.413820 0.206910 0.978360i \(-0.433659\pi\)
0.206910 + 0.978360i \(0.433659\pi\)
\(542\) 2263.86 + 2263.86i 0.179412 + 0.179412i
\(543\) 0 0
\(544\) 7301.44i 0.575453i
\(545\) 0 0
\(546\) 0 0
\(547\) 3940.59 3940.59i 0.308021 0.308021i −0.536121 0.844141i \(-0.680111\pi\)
0.844141 + 0.536121i \(0.180111\pi\)
\(548\) −5177.88 + 5177.88i −0.403628 + 0.403628i
\(549\) 0 0
\(550\) 0 0
\(551\) 9434.89i 0.729473i
\(552\) 0 0
\(553\) 6121.25 + 6121.25i 0.470709 + 0.470709i
\(554\) −522.836 −0.0400959
\(555\) 0 0
\(556\) 10798.3 0.823652
\(557\) 2850.64 + 2850.64i 0.216850 + 0.216850i 0.807170 0.590320i \(-0.200998\pi\)
−0.590320 + 0.807170i \(0.700998\pi\)
\(558\) 0 0
\(559\) 152.755i 0.0115579i
\(560\) 0 0
\(561\) 0 0
\(562\) −2044.88 + 2044.88i −0.153484 + 0.153484i
\(563\) 12845.3 12845.3i 0.961569 0.961569i −0.0377198 0.999288i \(-0.512009\pi\)
0.999288 + 0.0377198i \(0.0120094\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2718.91i 0.201916i
\(567\) 0 0
\(568\) 2780.38 + 2780.38i 0.205391 + 0.205391i
\(569\) −25072.7 −1.84728 −0.923639 0.383263i \(-0.874800\pi\)
−0.923639 + 0.383263i \(0.874800\pi\)
\(570\) 0 0
\(571\) −3587.24 −0.262909 −0.131455 0.991322i \(-0.541965\pi\)
−0.131455 + 0.991322i \(0.541965\pi\)
\(572\) 548.078 + 548.078i 0.0400635 + 0.0400635i
\(573\) 0 0
\(574\) 3933.60i 0.286037i
\(575\) 0 0
\(576\) 0 0
\(577\) −760.083 + 760.083i −0.0548400 + 0.0548400i −0.733995 0.679155i \(-0.762346\pi\)
0.679155 + 0.733995i \(0.262346\pi\)
\(578\) −1241.04 + 1241.04i −0.0893091 + 0.0893091i
\(579\) 0 0
\(580\) 0 0
\(581\) 19641.0i 1.40249i
\(582\) 0 0
\(583\) −6272.43 6272.43i −0.445587 0.445587i
\(584\) 680.928 0.0482483
\(585\) 0 0
\(586\) −1196.85 −0.0843711
\(587\) 8165.43 + 8165.43i 0.574146 + 0.574146i 0.933284 0.359139i \(-0.116929\pi\)
−0.359139 + 0.933284i \(0.616929\pi\)
\(588\) 0 0
\(589\) 2740.43i 0.191711i
\(590\) 0 0
\(591\) 0 0
\(592\) 16277.1 16277.1i 1.13004 1.13004i
\(593\) −9861.32 + 9861.32i −0.682893 + 0.682893i −0.960651 0.277758i \(-0.910409\pi\)
0.277758 + 0.960651i \(0.410409\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7244.46i 0.497893i
\(597\) 0 0
\(598\) 175.514 + 175.514i 0.0120022 + 0.0120022i
\(599\) 20769.5 1.41673 0.708363 0.705848i \(-0.249434\pi\)
0.708363 + 0.705848i \(0.249434\pi\)
\(600\) 0 0
\(601\) −11642.9 −0.790222 −0.395111 0.918633i \(-0.629294\pi\)
−0.395111 + 0.918633i \(0.629294\pi\)
\(602\) 264.997 + 264.997i 0.0179410 + 0.0179410i
\(603\) 0 0
\(604\) 3147.54i 0.212039i
\(605\) 0 0
\(606\) 0 0
\(607\) 17174.6 17174.6i 1.14843 1.14843i 0.161565 0.986862i \(-0.448346\pi\)
0.986862 0.161565i \(-0.0516540\pi\)
\(608\) 5118.82 5118.82i 0.341440 0.341440i
\(609\) 0 0
\(610\) 0 0
\(611\) 1272.30i 0.0842416i
\(612\) 0 0
\(613\) −11694.1 11694.1i −0.770504 0.770504i 0.207691 0.978195i \(-0.433405\pi\)
−0.978195 + 0.207691i \(0.933405\pi\)
\(614\) 3066.49 0.201553
\(615\) 0 0
\(616\) 3843.43 0.251390
\(617\) 5586.69 + 5586.69i 0.364525 + 0.364525i 0.865476 0.500951i \(-0.167016\pi\)
−0.500951 + 0.865476i \(0.667016\pi\)
\(618\) 0 0
\(619\) 16137.1i 1.04783i 0.851771 + 0.523914i \(0.175529\pi\)
−0.851771 + 0.523914i \(0.824471\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 577.406 577.406i 0.0372217 0.0372217i
\(623\) 2105.69 2105.69i 0.135414 0.135414i
\(624\) 0 0
\(625\) 0 0
\(626\) 2894.27i 0.184790i
\(627\) 0 0
\(628\) 10228.0 + 10228.0i 0.649906 + 0.649906i
\(629\) 36816.1 2.33379
\(630\) 0 0
\(631\) 25292.3 1.59567 0.797837 0.602874i \(-0.205978\pi\)
0.797837 + 0.602874i \(0.205978\pi\)
\(632\) −1853.45 1853.45i −0.116656 0.116656i
\(633\) 0 0
\(634\) 1806.38i 0.113156i
\(635\) 0 0
\(636\) 0 0
\(637\) −275.717 + 275.717i −0.0171496 + 0.0171496i
\(638\) −798.729 + 798.729i −0.0495643 + 0.0495643i
\(639\) 0 0
\(640\) 0 0
\(641\) 8057.35i 0.496484i 0.968698 + 0.248242i \(0.0798528\pi\)
−0.968698 + 0.248242i \(0.920147\pi\)
\(642\) 0 0
\(643\) −15024.4 15024.4i −0.921471 0.921471i 0.0756629 0.997133i \(-0.475893\pi\)
−0.997133 + 0.0756629i \(0.975893\pi\)
\(644\) −28778.2 −1.76090
\(645\) 0 0
\(646\) 3723.40 0.226773
\(647\) 2396.69 + 2396.69i 0.145631 + 0.145631i 0.776163 0.630532i \(-0.217163\pi\)
−0.630532 + 0.776163i \(0.717163\pi\)
\(648\) 0 0
\(649\) 13531.0i 0.818394i
\(650\) 0 0
\(651\) 0 0
\(652\) 617.645 617.645i 0.0370995 0.0370995i
\(653\) 10852.1 10852.1i 0.650343 0.650343i −0.302732 0.953076i \(-0.597899\pi\)
0.953076 + 0.302732i \(0.0978988\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 27247.1i 1.62168i
\(657\) 0 0
\(658\) 2207.16 + 2207.16i 0.130766 + 0.130766i
\(659\) 21164.7 1.25108 0.625538 0.780193i \(-0.284879\pi\)
0.625538 + 0.780193i \(0.284879\pi\)
\(660\) 0 0
\(661\) −6399.33 −0.376558 −0.188279 0.982116i \(-0.560291\pi\)
−0.188279 + 0.982116i \(0.560291\pi\)
\(662\) 857.380 + 857.380i 0.0503369 + 0.0503369i
\(663\) 0 0
\(664\) 5947.09i 0.347578i
\(665\) 0 0
\(666\) 0 0
\(667\) 12087.8 12087.8i 0.701710 0.701710i
\(668\) −3353.96 + 3353.96i −0.194264 + 0.194264i
\(669\) 0 0
\(670\) 0 0
\(671\) 7402.22i 0.425871i
\(672\) 0 0
\(673\) 1618.29 + 1618.29i 0.0926903 + 0.0926903i 0.751932 0.659241i \(-0.229122\pi\)
−0.659241 + 0.751932i \(0.729122\pi\)
\(674\) 3446.61 0.196971
\(675\) 0 0
\(676\) −17114.0 −0.973713
\(677\) 16119.8 + 16119.8i 0.915118 + 0.915118i 0.996669 0.0815511i \(-0.0259874\pi\)
−0.0815511 + 0.996669i \(0.525987\pi\)
\(678\) 0 0
\(679\) 16791.1i 0.949017i
\(680\) 0 0
\(681\) 0 0
\(682\) −231.997 + 231.997i −0.0130258 + 0.0130258i
\(683\) 23612.1 23612.1i 1.32283 1.32283i 0.411351 0.911477i \(-0.365057\pi\)
0.911477 0.411351i \(-0.134943\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2016.89i 0.112253i
\(687\) 0 0
\(688\) 1835.57 + 1835.57i 0.101716 + 0.101716i
\(689\) −1119.49 −0.0619000
\(690\) 0 0
\(691\) 12036.1 0.662624 0.331312 0.943521i \(-0.392509\pi\)
0.331312 + 0.943521i \(0.392509\pi\)
\(692\) 10894.0 + 10894.0i 0.598451 + 0.598451i
\(693\) 0 0
\(694\) 4712.02i 0.257732i
\(695\) 0 0
\(696\) 0 0
\(697\) −30814.2 + 30814.2i −1.67457 + 1.67457i
\(698\) −409.171 + 409.171i −0.0221882 + 0.0221882i
\(699\) 0 0
\(700\) 0 0
\(701\) 33366.3i 1.79776i −0.438200 0.898878i \(-0.644384\pi\)
0.438200 0.898878i \(-0.355616\pi\)
\(702\) 0 0
\(703\) −25810.7 25810.7i −1.38473 1.38473i
\(704\) 12583.9 0.673684
\(705\) 0 0
\(706\) −805.875 −0.0429596
\(707\) −4514.19 4514.19i −0.240132 0.240132i
\(708\) 0 0
\(709\) 4987.22i 0.264173i −0.991238 0.132087i \(-0.957832\pi\)
0.991238 0.132087i \(-0.0421677\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −637.581 + 637.581i −0.0335595 + 0.0335595i
\(713\) 3510.98 3510.98i 0.184414 0.184414i
\(714\) 0 0
\(715\) 0 0
\(716\) 867.043i 0.0452555i
\(717\) 0 0
\(718\) 1242.00 + 1242.00i 0.0645560 + 0.0645560i
\(719\) −33618.4 −1.74375 −0.871873 0.489732i \(-0.837095\pi\)
−0.871873 + 0.489732i \(0.837095\pi\)
\(720\) 0 0
\(721\) 17403.6 0.898951
\(722\) −635.644 635.644i −0.0327649 0.0327649i
\(723\) 0 0
\(724\) 5488.61i 0.281744i
\(725\) 0 0
\(726\) 0 0
\(727\) 10057.9 10057.9i 0.513106 0.513106i −0.402371 0.915477i \(-0.631814\pi\)
0.915477 + 0.402371i \(0.131814\pi\)
\(728\) 342.983 342.983i 0.0174613 0.0174613i
\(729\) 0 0
\(730\) 0 0
\(731\) 4151.76i 0.210066i
\(732\) 0 0
\(733\) 4209.55 + 4209.55i 0.212119 + 0.212119i 0.805167 0.593048i \(-0.202076\pi\)
−0.593048 + 0.805167i \(0.702076\pi\)
\(734\) −2610.32 −0.131265
\(735\) 0 0
\(736\) 13116.2 0.656890
\(737\) −13966.5 13966.5i −0.698050 0.698050i
\(738\) 0 0
\(739\) 4750.38i 0.236462i −0.992986 0.118231i \(-0.962278\pi\)
0.992986 0.118231i \(-0.0377223\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1942.07 + 1942.07i −0.0960857 + 0.0960857i
\(743\) −16266.1 + 16266.1i −0.803159 + 0.803159i −0.983588 0.180429i \(-0.942251\pi\)
0.180429 + 0.983588i \(0.442251\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1605.32i 0.0787866i
\(747\) 0 0
\(748\) 14896.3 + 14896.3i 0.728159 + 0.728159i
\(749\) −6534.00 −0.318754
\(750\) 0 0
\(751\) −32690.2 −1.58839 −0.794197 0.607660i \(-0.792108\pi\)
−0.794197 + 0.607660i \(0.792108\pi\)
\(752\) 15288.4 + 15288.4i 0.741372 + 0.741372i
\(753\) 0 0
\(754\) 142.555i 0.00688536i
\(755\) 0 0
\(756\) 0 0
\(757\) 6722.97 6722.97i 0.322788 0.322788i −0.527048 0.849836i \(-0.676701\pi\)
0.849836 + 0.527048i \(0.176701\pi\)
\(758\) 2420.99 2420.99i 0.116008 0.116008i
\(759\) 0 0
\(760\) 0 0
\(761\) 7001.94i 0.333535i −0.985996 0.166767i \(-0.946667\pi\)
0.985996 0.166767i \(-0.0533329\pi\)
\(762\) 0 0
\(763\) 14098.4 + 14098.4i 0.668932 + 0.668932i
\(764\) −17347.5 −0.821480
\(765\) 0 0
\(766\) −4898.58 −0.231061
\(767\) −1207.49 1207.49i −0.0568448 0.0568448i
\(768\) 0 0
\(769\) 25176.4i 1.18060i 0.807183 + 0.590301i \(0.200991\pi\)
−0.807183 + 0.590301i \(0.799009\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15387.7 15387.7i 0.717376 0.717376i
\(773\) −7322.13 + 7322.13i −0.340697 + 0.340697i −0.856629 0.515932i \(-0.827446\pi\)
0.515932 + 0.856629i \(0.327446\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5084.17i 0.235195i
\(777\) 0 0
\(778\) −2487.83 2487.83i −0.114644 0.114644i
\(779\) 43205.9 1.98718
\(780\) 0 0
\(781\) 17076.9 0.782405
\(782\) 4770.33 + 4770.33i 0.218142 + 0.218142i
\(783\) 0 0
\(784\) 6626.25i 0.301852i
\(785\) 0 0
\(786\) 0 0
\(787\) −16794.5 + 16794.5i −0.760683 + 0.760683i −0.976446 0.215763i \(-0.930776\pi\)
0.215763 + 0.976446i \(0.430776\pi\)
\(788\) −21157.9 + 21157.9i −0.956495 + 0.956495i
\(789\) 0 0
\(790\) 0 0
\(791\) 32640.8i 1.46722i
\(792\) 0 0
\(793\) 660.565 + 660.565i 0.0295805 + 0.0295805i
\(794\) −1095.30 −0.0489556
\(795\) 0 0
\(796\) 6590.11 0.293443
\(797\) −18950.9 18950.9i −0.842251 0.842251i 0.146900 0.989151i \(-0.453070\pi\)
−0.989151 + 0.146900i \(0.953070\pi\)
\(798\) 0 0
\(799\) 34580.0i 1.53110i
\(800\) 0 0
\(801\) 0 0
\(802\) 3927.01 3927.01i 0.172902 0.172902i
\(803\) 2091.10 2091.10i 0.0918972 0.0918972i
\(804\) 0 0
\(805\) 0 0
\(806\) 41.4062i 0.00180952i
\(807\) 0 0
\(808\) 1366.85 + 1366.85i 0.0595118 + 0.0595118i
\(809\) 3967.32 0.172415 0.0862074 0.996277i \(-0.472525\pi\)
0.0862074 + 0.996277i \(0.472525\pi\)
\(810\) 0 0
\(811\) −23204.4 −1.00471 −0.502353 0.864663i \(-0.667532\pi\)
−0.502353 + 0.864663i \(0.667532\pi\)
\(812\) −11687.1 11687.1i −0.505093 0.505093i
\(813\) 0 0
\(814\) 4370.11i 0.188172i
\(815\) 0 0
\(816\) 0 0
\(817\) 2910.68 2910.68i 0.124641 0.124641i
\(818\) −1841.14 + 1841.14i −0.0786968 + 0.0786968i
\(819\) 0 0
\(820\) 0 0
\(821\) 5783.29i 0.245844i −0.992416 0.122922i \(-0.960773\pi\)
0.992416 0.122922i \(-0.0392265\pi\)
\(822\) 0 0
\(823\) −2021.92 2021.92i −0.0856375 0.0856375i 0.662990 0.748628i \(-0.269287\pi\)
−0.748628 + 0.662990i \(0.769287\pi\)
\(824\) −5269.62 −0.222786
\(825\) 0 0
\(826\) −4189.47 −0.176477
\(827\) −18025.3 18025.3i −0.757922 0.757922i 0.218022 0.975944i \(-0.430039\pi\)
−0.975944 + 0.218022i \(0.930039\pi\)
\(828\) 0 0
\(829\) 42400.5i 1.77640i −0.459462 0.888198i \(-0.651958\pi\)
0.459462 0.888198i \(-0.348042\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1122.97 1122.97i 0.0467933 0.0467933i
\(833\) −7493.75 + 7493.75i −0.311696 + 0.311696i
\(834\) 0 0
\(835\) 0 0
\(836\) 20886.7i 0.864094i
\(837\) 0 0
\(838\) −2439.95 2439.95i −0.100581 0.100581i
\(839\) −38073.9 −1.56670 −0.783348 0.621584i \(-0.786490\pi\)
−0.783348 + 0.621584i \(0.786490\pi\)
\(840\) 0 0
\(841\) −14571.1 −0.597447
\(842\) −2611.04 2611.04i −0.106868 0.106868i
\(843\) 0 0
\(844\) 31102.3i 1.26847i
\(845\) 0 0
\(846\) 0 0
\(847\) −8236.11 + 8236.11i −0.334116 + 0.334116i
\(848\) −13452.2 + 13452.2i −0.544754 + 0.544754i
\(849\) 0 0
\(850\) 0 0
\(851\) 66136.1i 2.66406i
\(852\) 0 0
\(853\) 26447.7 + 26447.7i 1.06161 + 1.06161i 0.997973 + 0.0636372i \(0.0202701\pi\)
0.0636372 + 0.997973i \(0.479730\pi\)
\(854\) 2291.88 0.0918342
\(855\) 0 0
\(856\) 1978.43 0.0789967
\(857\) −18997.5 18997.5i −0.757227 0.757227i 0.218590 0.975817i \(-0.429854\pi\)
−0.975817 + 0.218590i \(0.929854\pi\)
\(858\) 0 0
\(859\) 29326.5i 1.16485i −0.812885 0.582425i \(-0.802104\pi\)
0.812885 0.582425i \(-0.197896\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2414.95 + 2414.95i −0.0954219 + 0.0954219i
\(863\) 23070.8 23070.8i 0.910011 0.910011i −0.0862616 0.996273i \(-0.527492\pi\)
0.996273 + 0.0862616i \(0.0274921\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2379.32i 0.0933633i
\(867\) 0 0
\(868\) −3394.59 3394.59i −0.132742 0.132742i
\(869\) −11383.8 −0.444381
\(870\) 0 0
\(871\) −2492.71 −0.0969716
\(872\) −4268.84 4268.84i −0.165781 0.165781i
\(873\) 0 0
\(874\) 6688.68i 0.258865i
\(875\) 0 0
\(876\) 0 0
\(877\) −20252.6 + 20252.6i −0.779795 + 0.779795i −0.979796 0.200001i \(-0.935906\pi\)
0.200001 + 0.979796i \(0.435906\pi\)
\(878\) −3022.67 + 3022.67i −0.116185 + 0.116185i
\(879\) 0 0
\(880\) 0 0
\(881\) 15547.7i 0.594569i 0.954789 + 0.297284i \(0.0960809\pi\)
−0.954789 + 0.297284i \(0.903919\pi\)
\(882\) 0 0
\(883\) −17599.2 17599.2i −0.670737 0.670737i 0.287149 0.957886i \(-0.407293\pi\)
−0.957886 + 0.287149i \(0.907293\pi\)
\(884\) 2658.66 0.101154
\(885\) 0 0
\(886\) −844.087 −0.0320064
\(887\) 9099.97 + 9099.97i 0.344472 + 0.344472i 0.858046 0.513573i \(-0.171679\pi\)
−0.513573 + 0.858046i \(0.671679\pi\)
\(888\) 0 0
\(889\) 27569.2i 1.04009i
\(890\) 0 0
\(891\) 0 0
\(892\) −28205.9 + 28205.9i −1.05875 + 1.05875i
\(893\) 24243.0 24243.0i 0.908467 0.908467i
\(894\) 0 0
\(895\) 0 0
\(896\) 16846.0i 0.628109i
\(897\) 0 0
\(898\) −1812.47 1812.47i −0.0673528 0.0673528i
\(899\) 2851.67 0.105794
\(900\) 0 0
\(901\) −30426.8 −1.12504
\(902\) −3657.68 3657.68i −0.135019 0.135019i
\(903\) 0 0
\(904\) 9883.30i 0.363621i
\(905\) 0 0
\(906\) 0 0
\(907\) −28827.2 + 28827.2i −1.05534 + 1.05534i −0.0569624 + 0.998376i \(0.518142\pi\)
−0.998376 + 0.0569624i \(0.981858\pi\)
\(908\) −13590.9 + 13590.9i −0.496728 + 0.496728i
\(909\) 0 0
\(910\) 0 0
\(911\) 32922.1i 1.19732i −0.801003 0.598660i \(-0.795700\pi\)
0.801003 0.598660i \(-0.204300\pi\)
\(912\) 0 0
\(913\) −18263.3 18263.3i −0.662023 0.662023i
\(914\) −5219.72 −0.188898
\(915\) 0 0
\(916\) 27167.0 0.979938
\(917\) −9474.99 9474.99i −0.341212 0.341212i
\(918\) 0 0
\(919\) 28643.9i 1.02815i −0.857744 0.514077i \(-0.828135\pi\)
0.857744 0.514077i \(-0.171865\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3830.67 3830.67i 0.136829 0.136829i
\(923\) 1523.92 1523.92i 0.0543450 0.0543450i
\(924\) 0 0
\(925\) 0 0
\(926\) 1799.31i 0.0638541i
\(927\) 0 0
\(928\) 5326.61 + 5326.61i 0.188421 + 0.188421i
\(929\) 12080.7 0.426648 0.213324 0.976981i \(-0.431571\pi\)
0.213324 + 0.976981i \(0.431571\pi\)
\(930\) 0 0
\(931\) 10507.3 0.369885
\(932\) 8383.98 + 8383.98i 0.294664 + 0.294664i
\(933\) 0 0
\(934\) 2450.08i 0.0858343i
\(935\) 0 0
\(936\) 0 0
\(937\) 18170.0 18170.0i 0.633500 0.633500i −0.315444 0.948944i \(-0.602154\pi\)
0.948944 + 0.315444i \(0.102154\pi\)
\(938\) −4324.31 + 4324.31i −0.150526 + 0.150526i
\(939\) 0 0
\(940\) 0 0
\(941\) 7693.26i 0.266518i 0.991081 + 0.133259i \(0.0425442\pi\)
−0.991081 + 0.133259i \(0.957456\pi\)
\(942\) 0 0
\(943\) 55354.4 + 55354.4i 1.91155 + 1.91155i
\(944\) −29019.4 −1.00053
\(945\) 0 0
\(946\) −492.819 −0.0169375
\(947\) 9792.71 + 9792.71i 0.336030 + 0.336030i 0.854871 0.518841i \(-0.173636\pi\)
−0.518841 + 0.854871i \(0.673636\pi\)
\(948\) 0 0
\(949\) 373.216i 0.0127662i
\(950\) 0 0
\(951\) 0 0
\(952\) 9321.99 9321.99i 0.317361 0.317361i
\(953\) −22620.8 + 22620.8i −0.768899 + 0.768899i −0.977913 0.209014i \(-0.932975\pi\)
0.209014 + 0.977913i \(0.432975\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 30627.4i 1.03615i
\(957\) 0 0
\(958\) 3059.01 + 3059.01i 0.103165 + 0.103165i
\(959\) 19901.5 0.670130
\(960\) 0 0
\(961\) −28962.7 −0.972197
\(962\) 389.984 + 389.984i 0.0130702 + 0.0130702i
\(963\) 0 0
\(964\) 37503.2i 1.25300i
\(965\) 0 0
\(966\) 0 0
\(967\) 4426.36 4426.36i 0.147200 0.147200i −0.629666 0.776866i \(-0.716808\pi\)
0.776866 + 0.629666i \(0.216808\pi\)
\(968\) 2493.81 2493.81i 0.0828037 0.0828037i
\(969\) 0 0
\(970\) 0 0
\(971\) 36943.3i 1.22097i 0.792026 + 0.610487i \(0.209026\pi\)
−0.792026 + 0.610487i \(0.790974\pi\)
\(972\) 0 0
\(973\) −20752.0 20752.0i −0.683741 0.683741i
\(974\) 1882.32 0.0619236
\(975\) 0 0
\(976\) 15875.2 0.520650
\(977\) −14789.2 14789.2i −0.484287 0.484287i 0.422211 0.906498i \(-0.361254\pi\)
−0.906498 + 0.422211i \(0.861254\pi\)
\(978\) 0 0
\(979\) 3915.98i 0.127840i
\(980\) 0 0
\(981\) 0 0
\(982\) −5410.00 + 5410.00i −0.175805 + 0.175805i
\(983\) −12134.8 + 12134.8i −0.393733 + 0.393733i −0.876015 0.482283i \(-0.839808\pi\)
0.482283 + 0.876015i \(0.339808\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3874.54i 0.125142i
\(987\) 0 0
\(988\) −1863.91 1863.91i −0.0600190 0.0600190i
\(989\) 7458.19 0.239794
\(990\) 0 0
\(991\) 29327.6 0.940084 0.470042 0.882644i \(-0.344239\pi\)
0.470042 + 0.882644i \(0.344239\pi\)
\(992\) 1547.15 + 1547.15i 0.0495182 + 0.0495182i
\(993\) 0 0
\(994\) 5287.34i 0.168717i
\(995\) 0 0
\(996\) 0 0
\(997\) 44483.8 44483.8i 1.41306 1.41306i 0.677912 0.735143i \(-0.262885\pi\)
0.735143 0.677912i \(-0.237115\pi\)
\(998\) 2078.84 2078.84i 0.0659365 0.0659365i
\(999\) 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 225.4.f.c.143.4 12
3.2 odd 2 inner 225.4.f.c.143.3 12
5.2 odd 4 inner 225.4.f.c.107.3 12
5.3 odd 4 45.4.f.a.17.4 yes 12
5.4 even 2 45.4.f.a.8.3 12
15.2 even 4 inner 225.4.f.c.107.4 12
15.8 even 4 45.4.f.a.17.3 yes 12
15.14 odd 2 45.4.f.a.8.4 yes 12
20.3 even 4 720.4.w.d.17.6 12
20.19 odd 2 720.4.w.d.593.1 12
60.23 odd 4 720.4.w.d.17.1 12
60.59 even 2 720.4.w.d.593.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.f.a.8.3 12 5.4 even 2
45.4.f.a.8.4 yes 12 15.14 odd 2
45.4.f.a.17.3 yes 12 15.8 even 4
45.4.f.a.17.4 yes 12 5.3 odd 4
225.4.f.c.107.3 12 5.2 odd 4 inner
225.4.f.c.107.4 12 15.2 even 4 inner
225.4.f.c.143.3 12 3.2 odd 2 inner
225.4.f.c.143.4 12 1.1 even 1 trivial
720.4.w.d.17.1 12 60.23 odd 4
720.4.w.d.17.6 12 20.3 even 4
720.4.w.d.593.1 12 20.19 odd 2
720.4.w.d.593.6 12 60.59 even 2