Properties

Label 825.4.c.h.199.1
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not 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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{97})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 49x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-5.42443i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.h.199.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.42443i q^{2} -3.00000i q^{3} -21.4244 q^{4} -16.2733 q^{6} +7.69772i q^{7} +72.8199i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-5.42443i q^{2} -3.00000i q^{3} -21.4244 q^{4} -16.2733 q^{6} +7.69772i q^{7} +72.8199i q^{8} -9.00000 q^{9} -11.0000 q^{11} +64.2733i q^{12} +24.8489i q^{13} +41.7557 q^{14} +223.611 q^{16} +15.9420i q^{17} +48.8199i q^{18} -15.1511 q^{19} +23.0931 q^{21} +59.6687i q^{22} +17.7557i q^{23} +218.460 q^{24} +134.791 q^{26} +27.0000i q^{27} -164.919i q^{28} +128.547 q^{29} +219.395 q^{31} -630.402i q^{32} +33.0000i q^{33} +86.4763 q^{34} +192.820 q^{36} -92.0703i q^{37} +82.1863i q^{38} +74.5466 q^{39} -459.942 q^{41} -125.267i q^{42} +64.9648i q^{43} +235.669 q^{44} +96.3146 q^{46} -497.408i q^{47} -670.832i q^{48} +283.745 q^{49} +47.8260 q^{51} -532.373i q^{52} -526.919i q^{53} +146.460 q^{54} -560.547 q^{56} +45.4534i q^{57} -697.292i q^{58} +578.443 q^{59} -221.569 q^{61} -1190.09i q^{62} -69.2794i q^{63} -1630.68 q^{64} +179.006 q^{66} +860.745i q^{67} -341.548i q^{68} +53.2671 q^{69} +580.919 q^{71} -655.379i q^{72} +510.116i q^{73} -499.429 q^{74} +324.605 q^{76} -84.6749i q^{77} -404.373i q^{78} -1035.12 q^{79} +81.0000 q^{81} +2494.92i q^{82} +606.211i q^{83} -494.757 q^{84} +352.397 q^{86} -385.640i q^{87} -801.018i q^{88} +23.4411 q^{89} -191.279 q^{91} -380.406i q^{92} -658.186i q^{93} -2698.15 q^{94} -1891.20 q^{96} -719.490i q^{97} -1539.16i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 364 q^{14} + 402 q^{16} - 100 q^{19} - 144 q^{21} + 342 q^{24} + 224 q^{26} + 396 q^{29} + 720 q^{31} + 1252 q^{34} + 594 q^{36} + 180 q^{39} - 1564 q^{41} + 726 q^{44} - 836 q^{46} - 756 q^{49} - 636 q^{51} + 54 q^{54} - 2124 q^{56} + 344 q^{59} - 1556 q^{61} - 1618 q^{64} + 66 q^{66} + 804 q^{69} + 1260 q^{71} - 4716 q^{74} + 1456 q^{76} - 1304 q^{79} + 324 q^{81} + 1212 q^{84} - 2136 q^{86} + 1512 q^{89} - 56 q^{91} - 7444 q^{94} - 5142 q^{96} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.42443i − 1.91783i −0.283702 0.958913i \(-0.591563\pi\)
0.283702 0.958913i \(-0.408437\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −21.4244 −2.67805
\(5\) 0 0
\(6\) −16.2733 −1.10726
\(7\) 7.69772i 0.415638i 0.978167 + 0.207819i \(0.0666364\pi\)
−0.978167 + 0.207819i \(0.933364\pi\)
\(8\) 72.8199i 3.21821i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 64.2733i 1.54617i
\(13\) 24.8489i 0.530141i 0.964229 + 0.265071i \(0.0853952\pi\)
−0.964229 + 0.265071i \(0.914605\pi\)
\(14\) 41.7557 0.797120
\(15\) 0 0
\(16\) 223.611 3.49392
\(17\) 15.9420i 0.227441i 0.993513 + 0.113721i \(0.0362769\pi\)
−0.993513 + 0.113721i \(0.963723\pi\)
\(18\) 48.8199i 0.639275i
\(19\) −15.1511 −0.182943 −0.0914713 0.995808i \(-0.529157\pi\)
−0.0914713 + 0.995808i \(0.529157\pi\)
\(20\) 0 0
\(21\) 23.0931 0.239968
\(22\) 59.6687i 0.578246i
\(23\) 17.7557i 0.160971i 0.996756 + 0.0804853i \(0.0256470\pi\)
−0.996756 + 0.0804853i \(0.974353\pi\)
\(24\) 218.460 1.85804
\(25\) 0 0
\(26\) 134.791 1.01672
\(27\) 27.0000i 0.192450i
\(28\) − 164.919i − 1.11310i
\(29\) 128.547 0.823121 0.411560 0.911383i \(-0.364984\pi\)
0.411560 + 0.911383i \(0.364984\pi\)
\(30\) 0 0
\(31\) 219.395 1.27112 0.635558 0.772053i \(-0.280770\pi\)
0.635558 + 0.772053i \(0.280770\pi\)
\(32\) − 630.402i − 3.48251i
\(33\) 33.0000i 0.174078i
\(34\) 86.4763 0.436193
\(35\) 0 0
\(36\) 192.820 0.892685
\(37\) − 92.0703i − 0.409088i −0.978857 0.204544i \(-0.934429\pi\)
0.978857 0.204544i \(-0.0655712\pi\)
\(38\) 82.1863i 0.350852i
\(39\) 74.5466 0.306077
\(40\) 0 0
\(41\) −459.942 −1.75197 −0.875986 0.482336i \(-0.839788\pi\)
−0.875986 + 0.482336i \(0.839788\pi\)
\(42\) − 125.267i − 0.460218i
\(43\) 64.9648i 0.230396i 0.993343 + 0.115198i \(0.0367503\pi\)
−0.993343 + 0.115198i \(0.963250\pi\)
\(44\) 235.669 0.807464
\(45\) 0 0
\(46\) 96.3146 0.308713
\(47\) − 497.408i − 1.54371i −0.635799 0.771855i \(-0.719329\pi\)
0.635799 0.771855i \(-0.280671\pi\)
\(48\) − 670.832i − 2.01721i
\(49\) 283.745 0.827245
\(50\) 0 0
\(51\) 47.8260 0.131313
\(52\) − 532.373i − 1.41975i
\(53\) − 526.919i − 1.36562i −0.730596 0.682811i \(-0.760757\pi\)
0.730596 0.682811i \(-0.239243\pi\)
\(54\) 146.460 0.369086
\(55\) 0 0
\(56\) −560.547 −1.33761
\(57\) 45.4534i 0.105622i
\(58\) − 697.292i − 1.57860i
\(59\) 578.443 1.27639 0.638194 0.769876i \(-0.279682\pi\)
0.638194 + 0.769876i \(0.279682\pi\)
\(60\) 0 0
\(61\) −221.569 −0.465067 −0.232533 0.972588i \(-0.574701\pi\)
−0.232533 + 0.972588i \(0.574701\pi\)
\(62\) − 1190.09i − 2.43778i
\(63\) − 69.2794i − 0.138546i
\(64\) −1630.68 −3.18493
\(65\) 0 0
\(66\) 179.006 0.333851
\(67\) 860.745i 1.56950i 0.619810 + 0.784752i \(0.287210\pi\)
−0.619810 + 0.784752i \(0.712790\pi\)
\(68\) − 341.548i − 0.609100i
\(69\) 53.2671 0.0929364
\(70\) 0 0
\(71\) 580.919 0.971020 0.485510 0.874231i \(-0.338634\pi\)
0.485510 + 0.874231i \(0.338634\pi\)
\(72\) − 655.379i − 1.07274i
\(73\) 510.116i 0.817871i 0.912563 + 0.408935i \(0.134100\pi\)
−0.912563 + 0.408935i \(0.865900\pi\)
\(74\) −499.429 −0.784560
\(75\) 0 0
\(76\) 324.605 0.489930
\(77\) − 84.6749i − 0.125319i
\(78\) − 404.373i − 0.587002i
\(79\) −1035.12 −1.47418 −0.737088 0.675797i \(-0.763800\pi\)
−0.737088 + 0.675797i \(0.763800\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 2494.92i 3.35998i
\(83\) 606.211i 0.801690i 0.916146 + 0.400845i \(0.131283\pi\)
−0.916146 + 0.400845i \(0.868717\pi\)
\(84\) −494.757 −0.642648
\(85\) 0 0
\(86\) 352.397 0.441860
\(87\) − 385.640i − 0.475229i
\(88\) − 801.018i − 0.970328i
\(89\) 23.4411 0.0279186 0.0139593 0.999903i \(-0.495556\pi\)
0.0139593 + 0.999903i \(0.495556\pi\)
\(90\) 0 0
\(91\) −191.279 −0.220347
\(92\) − 380.406i − 0.431088i
\(93\) − 658.186i − 0.733879i
\(94\) −2698.15 −2.96057
\(95\) 0 0
\(96\) −1891.20 −2.01063
\(97\) − 719.490i − 0.753126i −0.926391 0.376563i \(-0.877106\pi\)
0.926391 0.376563i \(-0.122894\pi\)
\(98\) − 1539.16i − 1.58651i
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 1871.27 1.84355 0.921774 0.387727i \(-0.126740\pi\)
0.921774 + 0.387727i \(0.126740\pi\)
\(102\) − 259.429i − 0.251836i
\(103\) − 428.745i − 0.410151i −0.978746 0.205075i \(-0.934256\pi\)
0.978746 0.205075i \(-0.0657439\pi\)
\(104\) −1809.49 −1.70611
\(105\) 0 0
\(106\) −2858.24 −2.61902
\(107\) − 1148.02i − 1.03723i −0.855008 0.518616i \(-0.826448\pi\)
0.855008 0.518616i \(-0.173552\pi\)
\(108\) − 578.460i − 0.515392i
\(109\) 1828.32 1.60662 0.803308 0.595564i \(-0.203071\pi\)
0.803308 + 0.595564i \(0.203071\pi\)
\(110\) 0 0
\(111\) −276.211 −0.236187
\(112\) 1721.29i 1.45220i
\(113\) 1126.40i 0.937722i 0.883272 + 0.468861i \(0.155335\pi\)
−0.883272 + 0.468861i \(0.844665\pi\)
\(114\) 246.559 0.202565
\(115\) 0 0
\(116\) −2754.04 −2.20436
\(117\) − 223.640i − 0.176714i
\(118\) − 3137.72i − 2.44789i
\(119\) −122.717 −0.0945332
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1201.89i 0.891916i
\(123\) 1379.83i 1.01150i
\(124\) −4700.42 −3.40412
\(125\) 0 0
\(126\) −375.801 −0.265707
\(127\) − 661.304i − 0.462057i −0.972947 0.231029i \(-0.925791\pi\)
0.972947 0.231029i \(-0.0742091\pi\)
\(128\) 3802.31i 2.62562i
\(129\) 194.895 0.133019
\(130\) 0 0
\(131\) 622.186 0.414967 0.207483 0.978239i \(-0.433473\pi\)
0.207483 + 0.978239i \(0.433473\pi\)
\(132\) − 707.006i − 0.466189i
\(133\) − 116.629i − 0.0760378i
\(134\) 4669.05 3.01003
\(135\) 0 0
\(136\) −1160.89 −0.731955
\(137\) 1872.84i 1.16794i 0.811776 + 0.583969i \(0.198501\pi\)
−0.811776 + 0.583969i \(0.801499\pi\)
\(138\) − 288.944i − 0.178236i
\(139\) 954.058 0.582174 0.291087 0.956697i \(-0.405983\pi\)
0.291087 + 0.956697i \(0.405983\pi\)
\(140\) 0 0
\(141\) −1492.22 −0.891261
\(142\) − 3151.15i − 1.86225i
\(143\) − 273.337i − 0.159844i
\(144\) −2012.50 −1.16464
\(145\) 0 0
\(146\) 2767.09 1.56853
\(147\) − 851.236i − 0.477610i
\(148\) 1972.55i 1.09556i
\(149\) 2047.01 1.12549 0.562745 0.826631i \(-0.309745\pi\)
0.562745 + 0.826631i \(0.309745\pi\)
\(150\) 0 0
\(151\) 475.863 0.256458 0.128229 0.991745i \(-0.459071\pi\)
0.128229 + 0.991745i \(0.459071\pi\)
\(152\) − 1103.30i − 0.588749i
\(153\) − 143.478i − 0.0758138i
\(154\) −459.313 −0.240341
\(155\) 0 0
\(156\) −1597.12 −0.819691
\(157\) 647.466i 0.329130i 0.986366 + 0.164565i \(0.0526220\pi\)
−0.986366 + 0.164565i \(0.947378\pi\)
\(158\) 5614.92i 2.82721i
\(159\) −1580.76 −0.788442
\(160\) 0 0
\(161\) −136.678 −0.0669054
\(162\) − 439.379i − 0.213092i
\(163\) 1093.23i 0.525329i 0.964887 + 0.262665i \(0.0846013\pi\)
−0.964887 + 0.262665i \(0.915399\pi\)
\(164\) 9853.99 4.69188
\(165\) 0 0
\(166\) 3288.35 1.53750
\(167\) − 1123.25i − 0.520479i −0.965544 0.260240i \(-0.916198\pi\)
0.965544 0.260240i \(-0.0838016\pi\)
\(168\) 1681.64i 0.772270i
\(169\) 1579.53 0.718951
\(170\) 0 0
\(171\) 136.360 0.0609809
\(172\) − 1391.83i − 0.617014i
\(173\) 46.0123i 0.0202211i 0.999949 + 0.0101106i \(0.00321834\pi\)
−0.999949 + 0.0101106i \(0.996782\pi\)
\(174\) −2091.88 −0.911406
\(175\) 0 0
\(176\) −2459.72 −1.05346
\(177\) − 1735.33i − 0.736923i
\(178\) − 127.155i − 0.0535430i
\(179\) 831.975 0.347401 0.173700 0.984799i \(-0.444428\pi\)
0.173700 + 0.984799i \(0.444428\pi\)
\(180\) 0 0
\(181\) −1810.63 −0.743553 −0.371776 0.928322i \(-0.621251\pi\)
−0.371776 + 0.928322i \(0.621251\pi\)
\(182\) 1037.58i 0.422586i
\(183\) 664.708i 0.268506i
\(184\) −1292.97 −0.518037
\(185\) 0 0
\(186\) −3570.28 −1.40745
\(187\) − 175.362i − 0.0685762i
\(188\) 10656.7i 4.13414i
\(189\) −207.838 −0.0799895
\(190\) 0 0
\(191\) 458.898 0.173847 0.0869233 0.996215i \(-0.472296\pi\)
0.0869233 + 0.996215i \(0.472296\pi\)
\(192\) 4892.05i 1.83882i
\(193\) 1778.91i 0.663465i 0.943373 + 0.331733i \(0.107633\pi\)
−0.943373 + 0.331733i \(0.892367\pi\)
\(194\) −3902.82 −1.44436
\(195\) 0 0
\(196\) −6079.08 −2.21541
\(197\) − 5304.53i − 1.91844i −0.282666 0.959218i \(-0.591219\pi\)
0.282666 0.959218i \(-0.408781\pi\)
\(198\) − 537.018i − 0.192749i
\(199\) 5138.40 1.83041 0.915205 0.402989i \(-0.132029\pi\)
0.915205 + 0.402989i \(0.132029\pi\)
\(200\) 0 0
\(201\) 2582.24 0.906153
\(202\) − 10150.6i − 3.53560i
\(203\) 989.515i 0.342120i
\(204\) −1024.65 −0.351664
\(205\) 0 0
\(206\) −2325.70 −0.786597
\(207\) − 159.801i − 0.0536568i
\(208\) 5556.47i 1.85227i
\(209\) 166.663 0.0551593
\(210\) 0 0
\(211\) −4262.36 −1.39068 −0.695339 0.718682i \(-0.744746\pi\)
−0.695339 + 0.718682i \(0.744746\pi\)
\(212\) 11288.9i 3.65721i
\(213\) − 1742.76i − 0.560619i
\(214\) −6227.38 −1.98923
\(215\) 0 0
\(216\) −1966.14 −0.619345
\(217\) 1688.84i 0.528323i
\(218\) − 9917.58i − 3.08121i
\(219\) 1530.35 0.472198
\(220\) 0 0
\(221\) −396.141 −0.120576
\(222\) 1498.29i 0.452966i
\(223\) − 1377.80i − 0.413740i −0.978368 0.206870i \(-0.933672\pi\)
0.978368 0.206870i \(-0.0663277\pi\)
\(224\) 4852.65 1.44746
\(225\) 0 0
\(226\) 6110.06 1.79839
\(227\) 1227.28i 0.358843i 0.983772 + 0.179422i \(0.0574226\pi\)
−0.983772 + 0.179422i \(0.942577\pi\)
\(228\) − 973.814i − 0.282861i
\(229\) −3890.28 −1.12261 −0.561304 0.827610i \(-0.689700\pi\)
−0.561304 + 0.827610i \(0.689700\pi\)
\(230\) 0 0
\(231\) −254.025 −0.0723532
\(232\) 9360.74i 2.64898i
\(233\) − 3218.14i − 0.904837i −0.891806 0.452419i \(-0.850561\pi\)
0.891806 0.452419i \(-0.149439\pi\)
\(234\) −1213.12 −0.338906
\(235\) 0 0
\(236\) −12392.8 −3.41823
\(237\) 3105.35i 0.851115i
\(238\) 665.670i 0.181298i
\(239\) 428.098 0.115864 0.0579318 0.998321i \(-0.481549\pi\)
0.0579318 + 0.998321i \(0.481549\pi\)
\(240\) 0 0
\(241\) 1231.16 0.329070 0.164535 0.986371i \(-0.447388\pi\)
0.164535 + 0.986371i \(0.447388\pi\)
\(242\) − 656.356i − 0.174348i
\(243\) − 243.000i − 0.0641500i
\(244\) 4747.00 1.24547
\(245\) 0 0
\(246\) 7484.77 1.93988
\(247\) − 376.489i − 0.0969854i
\(248\) 15976.3i 4.09072i
\(249\) 1818.63 0.462856
\(250\) 0 0
\(251\) 2838.22 0.713732 0.356866 0.934156i \(-0.383845\pi\)
0.356866 + 0.934156i \(0.383845\pi\)
\(252\) 1484.27i 0.371033i
\(253\) − 195.313i − 0.0485344i
\(254\) −3587.20 −0.886145
\(255\) 0 0
\(256\) 7579.90 1.85056
\(257\) − 342.007i − 0.0830110i −0.999138 0.0415055i \(-0.986785\pi\)
0.999138 0.0415055i \(-0.0132154\pi\)
\(258\) − 1057.19i − 0.255108i
\(259\) 708.731 0.170032
\(260\) 0 0
\(261\) −1156.92 −0.274374
\(262\) − 3375.01i − 0.795834i
\(263\) 5895.00i 1.38213i 0.722791 + 0.691067i \(0.242859\pi\)
−0.722791 + 0.691067i \(0.757141\pi\)
\(264\) −2403.06 −0.560219
\(265\) 0 0
\(266\) −632.647 −0.145827
\(267\) − 70.3234i − 0.0161188i
\(268\) − 18441.0i − 4.20322i
\(269\) 2496.18 0.565779 0.282890 0.959152i \(-0.408707\pi\)
0.282890 + 0.959152i \(0.408707\pi\)
\(270\) 0 0
\(271\) −2249.68 −0.504274 −0.252137 0.967692i \(-0.581133\pi\)
−0.252137 + 0.967692i \(0.581133\pi\)
\(272\) 3564.80i 0.794662i
\(273\) 573.838i 0.127217i
\(274\) 10159.1 2.23990
\(275\) 0 0
\(276\) −1141.22 −0.248889
\(277\) − 4082.59i − 0.885556i −0.896631 0.442778i \(-0.853993\pi\)
0.896631 0.442778i \(-0.146007\pi\)
\(278\) − 5175.22i − 1.11651i
\(279\) −1974.56 −0.423705
\(280\) 0 0
\(281\) −1033.79 −0.219468 −0.109734 0.993961i \(-0.535000\pi\)
−0.109734 + 0.993961i \(0.535000\pi\)
\(282\) 8094.46i 1.70928i
\(283\) − 7809.14i − 1.64030i −0.572148 0.820150i \(-0.693890\pi\)
0.572148 0.820150i \(-0.306110\pi\)
\(284\) −12445.9 −2.60044
\(285\) 0 0
\(286\) −1482.70 −0.306552
\(287\) − 3540.50i − 0.728186i
\(288\) 5673.61i 1.16084i
\(289\) 4658.85 0.948270
\(290\) 0 0
\(291\) −2158.47 −0.434817
\(292\) − 10928.9i − 2.19030i
\(293\) − 1949.19i − 0.388645i −0.980938 0.194323i \(-0.937749\pi\)
0.980938 0.194323i \(-0.0622509\pi\)
\(294\) −4617.47 −0.915973
\(295\) 0 0
\(296\) 6704.55 1.31653
\(297\) − 297.000i − 0.0580259i
\(298\) − 11103.9i − 2.15849i
\(299\) −441.209 −0.0853371
\(300\) 0 0
\(301\) −500.081 −0.0957614
\(302\) − 2581.28i − 0.491842i
\(303\) − 5613.81i − 1.06437i
\(304\) −3387.96 −0.639187
\(305\) 0 0
\(306\) −778.286 −0.145398
\(307\) − 2364.09i − 0.439497i −0.975557 0.219748i \(-0.929476\pi\)
0.975557 0.219748i \(-0.0705236\pi\)
\(308\) 1814.11i 0.335612i
\(309\) −1286.24 −0.236801
\(310\) 0 0
\(311\) −1989.17 −0.362686 −0.181343 0.983420i \(-0.558044\pi\)
−0.181343 + 0.983420i \(0.558044\pi\)
\(312\) 5428.47i 0.985021i
\(313\) − 3878.67i − 0.700433i −0.936669 0.350216i \(-0.886108\pi\)
0.936669 0.350216i \(-0.113892\pi\)
\(314\) 3512.13 0.631214
\(315\) 0 0
\(316\) 22176.8 3.94792
\(317\) − 2913.73i − 0.516251i −0.966111 0.258126i \(-0.916895\pi\)
0.966111 0.258126i \(-0.0831048\pi\)
\(318\) 8574.71i 1.51209i
\(319\) −1414.01 −0.248180
\(320\) 0 0
\(321\) −3444.07 −0.598846
\(322\) 741.402i 0.128313i
\(323\) − 241.540i − 0.0416087i
\(324\) −1735.38 −0.297562
\(325\) 0 0
\(326\) 5930.17 1.00749
\(327\) − 5484.95i − 0.927580i
\(328\) − 33492.9i − 5.63822i
\(329\) 3828.90 0.641624
\(330\) 0 0
\(331\) 8104.46 1.34580 0.672902 0.739731i \(-0.265047\pi\)
0.672902 + 0.739731i \(0.265047\pi\)
\(332\) − 12987.7i − 2.14697i
\(333\) 828.633i 0.136363i
\(334\) −6093.02 −0.998189
\(335\) 0 0
\(336\) 5163.88 0.838430
\(337\) − 5919.19i − 0.956792i −0.878144 0.478396i \(-0.841218\pi\)
0.878144 0.478396i \(-0.158782\pi\)
\(338\) − 8568.07i − 1.37882i
\(339\) 3379.19 0.541394
\(340\) 0 0
\(341\) −2413.35 −0.383256
\(342\) − 739.677i − 0.116951i
\(343\) 4824.51i 0.759472i
\(344\) −4730.73 −0.741465
\(345\) 0 0
\(346\) 249.590 0.0387805
\(347\) 8540.59i 1.32128i 0.750705 + 0.660638i \(0.229714\pi\)
−0.750705 + 0.660638i \(0.770286\pi\)
\(348\) 8262.11i 1.27269i
\(349\) −937.337 −0.143767 −0.0718833 0.997413i \(-0.522901\pi\)
−0.0718833 + 0.997413i \(0.522901\pi\)
\(350\) 0 0
\(351\) −670.919 −0.102026
\(352\) 6934.42i 1.05002i
\(353\) − 211.118i − 0.0318319i −0.999873 0.0159160i \(-0.994934\pi\)
0.999873 0.0159160i \(-0.00506642\pi\)
\(354\) −9413.17 −1.41329
\(355\) 0 0
\(356\) −502.213 −0.0747675
\(357\) 368.151i 0.0545788i
\(358\) − 4512.99i − 0.666254i
\(359\) 1376.31 0.202337 0.101169 0.994869i \(-0.467742\pi\)
0.101169 + 0.994869i \(0.467742\pi\)
\(360\) 0 0
\(361\) −6629.44 −0.966532
\(362\) 9821.63i 1.42600i
\(363\) − 363.000i − 0.0524864i
\(364\) 4098.05 0.590100
\(365\) 0 0
\(366\) 3605.66 0.514948
\(367\) − 1030.45i − 0.146564i −0.997311 0.0732821i \(-0.976653\pi\)
0.997311 0.0732821i \(-0.0233473\pi\)
\(368\) 3970.37i 0.562418i
\(369\) 4139.48 0.583991
\(370\) 0 0
\(371\) 4056.07 0.567603
\(372\) 14101.3i 1.96537i
\(373\) 9365.39i 1.30006i 0.759909 + 0.650029i \(0.225243\pi\)
−0.759909 + 0.650029i \(0.774757\pi\)
\(374\) −951.239 −0.131517
\(375\) 0 0
\(376\) 36221.2 4.96799
\(377\) 3194.24i 0.436370i
\(378\) 1127.40i 0.153406i
\(379\) 7120.23 0.965017 0.482509 0.875891i \(-0.339726\pi\)
0.482509 + 0.875891i \(0.339726\pi\)
\(380\) 0 0
\(381\) −1983.91 −0.266769
\(382\) − 2489.26i − 0.333407i
\(383\) − 1163.56i − 0.155235i −0.996983 0.0776176i \(-0.975269\pi\)
0.996983 0.0776176i \(-0.0247313\pi\)
\(384\) 11406.9 1.51590
\(385\) 0 0
\(386\) 9649.57 1.27241
\(387\) − 584.684i − 0.0767988i
\(388\) 15414.7i 2.01691i
\(389\) 10958.9 1.42838 0.714188 0.699954i \(-0.246796\pi\)
0.714188 + 0.699954i \(0.246796\pi\)
\(390\) 0 0
\(391\) −283.062 −0.0366114
\(392\) 20662.3i 2.66225i
\(393\) − 1866.56i − 0.239581i
\(394\) −28774.0 −3.67923
\(395\) 0 0
\(396\) −2121.02 −0.269155
\(397\) 2172.09i 0.274595i 0.990530 + 0.137298i \(0.0438417\pi\)
−0.990530 + 0.137298i \(0.956158\pi\)
\(398\) − 27872.9i − 3.51041i
\(399\) −349.888 −0.0439005
\(400\) 0 0
\(401\) 7830.71 0.975180 0.487590 0.873073i \(-0.337876\pi\)
0.487590 + 0.873073i \(0.337876\pi\)
\(402\) − 14007.2i − 1.73784i
\(403\) 5451.73i 0.673870i
\(404\) −40090.9 −4.93712
\(405\) 0 0
\(406\) 5367.55 0.656126
\(407\) 1012.77i 0.123345i
\(408\) 3482.68i 0.422594i
\(409\) 10731.2 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(410\) 0 0
\(411\) 5618.52 0.674309
\(412\) 9185.62i 1.09841i
\(413\) 4452.69i 0.530515i
\(414\) −866.831 −0.102904
\(415\) 0 0
\(416\) 15664.8 1.84622
\(417\) − 2862.17i − 0.336118i
\(418\) − 904.049i − 0.105786i
\(419\) 7315.88 0.852994 0.426497 0.904489i \(-0.359748\pi\)
0.426497 + 0.904489i \(0.359748\pi\)
\(420\) 0 0
\(421\) −12495.7 −1.44657 −0.723284 0.690551i \(-0.757368\pi\)
−0.723284 + 0.690551i \(0.757368\pi\)
\(422\) 23120.9i 2.66708i
\(423\) 4476.67i 0.514570i
\(424\) 38370.2 4.39486
\(425\) 0 0
\(426\) −9453.46 −1.07517
\(427\) − 1705.58i − 0.193299i
\(428\) 24595.8i 2.77776i
\(429\) −820.012 −0.0922857
\(430\) 0 0
\(431\) 6075.01 0.678939 0.339470 0.940617i \(-0.389752\pi\)
0.339470 + 0.940617i \(0.389752\pi\)
\(432\) 6037.49i 0.672405i
\(433\) 5641.79i 0.626160i 0.949727 + 0.313080i \(0.101361\pi\)
−0.949727 + 0.313080i \(0.898639\pi\)
\(434\) 9161.01 1.01323
\(435\) 0 0
\(436\) −39170.7 −4.30260
\(437\) − 269.019i − 0.0294484i
\(438\) − 8301.26i − 0.905593i
\(439\) −10897.0 −1.18470 −0.592351 0.805680i \(-0.701800\pi\)
−0.592351 + 0.805680i \(0.701800\pi\)
\(440\) 0 0
\(441\) −2553.71 −0.275748
\(442\) 2148.84i 0.231244i
\(443\) − 7720.83i − 0.828054i −0.910265 0.414027i \(-0.864122\pi\)
0.910265 0.414027i \(-0.135878\pi\)
\(444\) 5917.66 0.632522
\(445\) 0 0
\(446\) −7473.76 −0.793481
\(447\) − 6141.04i − 0.649802i
\(448\) − 12552.5i − 1.32378i
\(449\) 7473.86 0.785553 0.392776 0.919634i \(-0.371515\pi\)
0.392776 + 0.919634i \(0.371515\pi\)
\(450\) 0 0
\(451\) 5059.36 0.528240
\(452\) − 24132.4i − 2.51127i
\(453\) − 1427.59i − 0.148066i
\(454\) 6657.29 0.688198
\(455\) 0 0
\(456\) −3309.91 −0.339914
\(457\) − 11140.5i − 1.14033i −0.821529 0.570167i \(-0.806879\pi\)
0.821529 0.570167i \(-0.193121\pi\)
\(458\) 21102.6i 2.15296i
\(459\) −430.434 −0.0437711
\(460\) 0 0
\(461\) 14328.8 1.44763 0.723817 0.689992i \(-0.242386\pi\)
0.723817 + 0.689992i \(0.242386\pi\)
\(462\) 1377.94i 0.138761i
\(463\) 11760.7i 1.18049i 0.807223 + 0.590246i \(0.200969\pi\)
−0.807223 + 0.590246i \(0.799031\pi\)
\(464\) 28744.4 2.87592
\(465\) 0 0
\(466\) −17456.5 −1.73532
\(467\) − 11854.9i − 1.17469i −0.809338 0.587343i \(-0.800174\pi\)
0.809338 0.587343i \(-0.199826\pi\)
\(468\) 4791.35i 0.473249i
\(469\) −6625.77 −0.652345
\(470\) 0 0
\(471\) 1942.40 0.190023
\(472\) 42122.1i 4.10769i
\(473\) − 714.613i − 0.0694671i
\(474\) 16844.8 1.63229
\(475\) 0 0
\(476\) 2629.14 0.253165
\(477\) 4742.27i 0.455207i
\(478\) − 2322.19i − 0.222206i
\(479\) 1324.68 0.126359 0.0631796 0.998002i \(-0.479876\pi\)
0.0631796 + 0.998002i \(0.479876\pi\)
\(480\) 0 0
\(481\) 2287.84 0.216874
\(482\) − 6678.34i − 0.631100i
\(483\) 410.035i 0.0386278i
\(484\) −2592.36 −0.243459
\(485\) 0 0
\(486\) −1318.14 −0.123029
\(487\) − 18636.4i − 1.73408i −0.498239 0.867040i \(-0.666020\pi\)
0.498239 0.867040i \(-0.333980\pi\)
\(488\) − 16134.7i − 1.49668i
\(489\) 3279.70 0.303299
\(490\) 0 0
\(491\) 124.552 0.0114480 0.00572398 0.999984i \(-0.498178\pi\)
0.00572398 + 0.999984i \(0.498178\pi\)
\(492\) − 29562.0i − 2.70886i
\(493\) 2049.29i 0.187212i
\(494\) −2042.24 −0.186001
\(495\) 0 0
\(496\) 49059.2 4.44117
\(497\) 4471.75i 0.403592i
\(498\) − 9865.04i − 0.887677i
\(499\) −10230.2 −0.917768 −0.458884 0.888496i \(-0.651751\pi\)
−0.458884 + 0.888496i \(0.651751\pi\)
\(500\) 0 0
\(501\) −3369.76 −0.300499
\(502\) − 15395.7i − 1.36881i
\(503\) 5150.81i 0.456587i 0.973592 + 0.228294i \(0.0733146\pi\)
−0.973592 + 0.228294i \(0.926685\pi\)
\(504\) 5044.92 0.445870
\(505\) 0 0
\(506\) −1059.46 −0.0930806
\(507\) − 4738.60i − 0.415086i
\(508\) 14168.1i 1.23741i
\(509\) 22.7715 0.00198296 0.000991481 1.00000i \(-0.499684\pi\)
0.000991481 1.00000i \(0.499684\pi\)
\(510\) 0 0
\(511\) −3926.73 −0.339938
\(512\) − 10698.1i − 0.923429i
\(513\) − 409.081i − 0.0352073i
\(514\) −1855.19 −0.159201
\(515\) 0 0
\(516\) −4175.50 −0.356233
\(517\) 5471.49i 0.465446i
\(518\) − 3844.46i − 0.326093i
\(519\) 138.037 0.0116747
\(520\) 0 0
\(521\) 21521.7 1.80976 0.904879 0.425669i \(-0.139961\pi\)
0.904879 + 0.425669i \(0.139961\pi\)
\(522\) 6275.63i 0.526201i
\(523\) 2923.36i 0.244416i 0.992504 + 0.122208i \(0.0389975\pi\)
−0.992504 + 0.122208i \(0.961002\pi\)
\(524\) −13330.0 −1.11130
\(525\) 0 0
\(526\) 31977.0 2.65069
\(527\) 3497.60i 0.289104i
\(528\) 7379.15i 0.608213i
\(529\) 11851.7 0.974088
\(530\) 0 0
\(531\) −5205.99 −0.425462
\(532\) 2498.71i 0.203633i
\(533\) − 11429.0i − 0.928792i
\(534\) −381.464 −0.0309130
\(535\) 0 0
\(536\) −62679.3 −5.05100
\(537\) − 2495.93i − 0.200572i
\(538\) − 13540.3i − 1.08507i
\(539\) −3121.20 −0.249424
\(540\) 0 0
\(541\) 21272.8 1.69056 0.845278 0.534327i \(-0.179435\pi\)
0.845278 + 0.534327i \(0.179435\pi\)
\(542\) 12203.2i 0.967108i
\(543\) 5431.89i 0.429290i
\(544\) 10049.9 0.792067
\(545\) 0 0
\(546\) 3112.75 0.243980
\(547\) 18730.5i 1.46409i 0.681256 + 0.732046i \(0.261434\pi\)
−0.681256 + 0.732046i \(0.738566\pi\)
\(548\) − 40124.5i − 3.12780i
\(549\) 1994.12 0.155022
\(550\) 0 0
\(551\) −1947.63 −0.150584
\(552\) 3878.91i 0.299089i
\(553\) − 7968.04i − 0.612723i
\(554\) −22145.7 −1.69834
\(555\) 0 0
\(556\) −20440.1 −1.55909
\(557\) 18885.0i 1.43659i 0.695736 + 0.718297i \(0.255078\pi\)
−0.695736 + 0.718297i \(0.744922\pi\)
\(558\) 10710.9i 0.812593i
\(559\) −1614.30 −0.122143
\(560\) 0 0
\(561\) −526.086 −0.0395925
\(562\) 5607.71i 0.420902i
\(563\) 10285.1i 0.769922i 0.922933 + 0.384961i \(0.125785\pi\)
−0.922933 + 0.384961i \(0.874215\pi\)
\(564\) 31970.0 2.38685
\(565\) 0 0
\(566\) −42360.1 −3.14581
\(567\) 623.515i 0.0461820i
\(568\) 42302.5i 3.12495i
\(569\) −18008.8 −1.32683 −0.663415 0.748251i \(-0.730894\pi\)
−0.663415 + 0.748251i \(0.730894\pi\)
\(570\) 0 0
\(571\) −7010.79 −0.513822 −0.256911 0.966435i \(-0.582705\pi\)
−0.256911 + 0.966435i \(0.582705\pi\)
\(572\) 5856.10i 0.428070i
\(573\) − 1376.69i − 0.100370i
\(574\) −19205.2 −1.39653
\(575\) 0 0
\(576\) 14676.1 1.06164
\(577\) 16398.9i 1.18318i 0.806240 + 0.591589i \(0.201499\pi\)
−0.806240 + 0.591589i \(0.798501\pi\)
\(578\) − 25271.6i − 1.81862i
\(579\) 5336.73 0.383052
\(580\) 0 0
\(581\) −4666.44 −0.333213
\(582\) 11708.5i 0.833903i
\(583\) 5796.11i 0.411750i
\(584\) −37146.6 −2.63208
\(585\) 0 0
\(586\) −10573.3 −0.745354
\(587\) − 12823.5i − 0.901671i −0.892607 0.450836i \(-0.851126\pi\)
0.892607 0.450836i \(-0.148874\pi\)
\(588\) 18237.2i 1.27907i
\(589\) −3324.09 −0.232541
\(590\) 0 0
\(591\) −15913.6 −1.10761
\(592\) − 20587.9i − 1.42932i
\(593\) 16899.5i 1.17029i 0.810929 + 0.585144i \(0.198962\pi\)
−0.810929 + 0.585144i \(0.801038\pi\)
\(594\) −1611.06 −0.111284
\(595\) 0 0
\(596\) −43856.1 −3.01412
\(597\) − 15415.2i − 1.05679i
\(598\) 2393.31i 0.163662i
\(599\) 15074.9 1.02829 0.514143 0.857704i \(-0.328110\pi\)
0.514143 + 0.857704i \(0.328110\pi\)
\(600\) 0 0
\(601\) −11418.8 −0.775014 −0.387507 0.921867i \(-0.626664\pi\)
−0.387507 + 0.921867i \(0.626664\pi\)
\(602\) 2712.65i 0.183654i
\(603\) − 7746.71i − 0.523168i
\(604\) −10195.1 −0.686809
\(605\) 0 0
\(606\) −30451.7 −2.04128
\(607\) 17952.8i 1.20046i 0.799826 + 0.600232i \(0.204925\pi\)
−0.799826 + 0.600232i \(0.795075\pi\)
\(608\) 9551.30i 0.637100i
\(609\) 2968.54 0.197523
\(610\) 0 0
\(611\) 12360.0 0.818384
\(612\) 3073.94i 0.203033i
\(613\) 12528.9i 0.825507i 0.910843 + 0.412753i \(0.135433\pi\)
−0.910843 + 0.412753i \(0.864567\pi\)
\(614\) −12823.8 −0.842878
\(615\) 0 0
\(616\) 6166.01 0.403305
\(617\) 8586.10i 0.560232i 0.959966 + 0.280116i \(0.0903729\pi\)
−0.959966 + 0.280116i \(0.909627\pi\)
\(618\) 6977.09i 0.454142i
\(619\) −18415.4 −1.19576 −0.597882 0.801584i \(-0.703991\pi\)
−0.597882 + 0.801584i \(0.703991\pi\)
\(620\) 0 0
\(621\) −479.404 −0.0309788
\(622\) 10790.1i 0.695569i
\(623\) 180.443i 0.0116040i
\(624\) 16669.4 1.06941
\(625\) 0 0
\(626\) −21039.6 −1.34331
\(627\) − 499.988i − 0.0318462i
\(628\) − 13871.6i − 0.881427i
\(629\) 1467.79 0.0930436
\(630\) 0 0
\(631\) 2374.38 0.149798 0.0748989 0.997191i \(-0.476137\pi\)
0.0748989 + 0.997191i \(0.476137\pi\)
\(632\) − 75377.1i − 4.74421i
\(633\) 12787.1i 0.802909i
\(634\) −15805.3 −0.990080
\(635\) 0 0
\(636\) 33866.8 2.11149
\(637\) 7050.74i 0.438557i
\(638\) 7670.21i 0.475966i
\(639\) −5228.27 −0.323673
\(640\) 0 0
\(641\) 11086.0 0.683104 0.341552 0.939863i \(-0.389047\pi\)
0.341552 + 0.939863i \(0.389047\pi\)
\(642\) 18682.1i 1.14848i
\(643\) 19934.1i 1.22259i 0.791403 + 0.611294i \(0.209351\pi\)
−0.791403 + 0.611294i \(0.790649\pi\)
\(644\) 2928.26 0.179176
\(645\) 0 0
\(646\) −1310.21 −0.0797983
\(647\) 30634.8i 1.86148i 0.365684 + 0.930739i \(0.380835\pi\)
−0.365684 + 0.930739i \(0.619165\pi\)
\(648\) 5898.41i 0.357579i
\(649\) −6362.87 −0.384845
\(650\) 0 0
\(651\) 5066.53 0.305028
\(652\) − 23421.9i − 1.40686i
\(653\) − 9818.07i − 0.588378i −0.955747 0.294189i \(-0.904951\pi\)
0.955747 0.294189i \(-0.0950494\pi\)
\(654\) −29752.7 −1.77894
\(655\) 0 0
\(656\) −102848. −6.12125
\(657\) − 4591.04i − 0.272624i
\(658\) − 20769.6i − 1.23052i
\(659\) −16478.5 −0.974070 −0.487035 0.873383i \(-0.661922\pi\)
−0.487035 + 0.873383i \(0.661922\pi\)
\(660\) 0 0
\(661\) 2958.12 0.174066 0.0870328 0.996205i \(-0.472262\pi\)
0.0870328 + 0.996205i \(0.472262\pi\)
\(662\) − 43962.1i − 2.58102i
\(663\) 1188.42i 0.0696146i
\(664\) −44144.2 −2.58001
\(665\) 0 0
\(666\) 4494.86 0.261520
\(667\) 2282.44i 0.132498i
\(668\) 24065.1i 1.39387i
\(669\) −4133.39 −0.238873
\(670\) 0 0
\(671\) 2437.26 0.140223
\(672\) − 14558.0i − 0.835692i
\(673\) − 29960.3i − 1.71602i −0.513630 0.858012i \(-0.671700\pi\)
0.513630 0.858012i \(-0.328300\pi\)
\(674\) −32108.2 −1.83496
\(675\) 0 0
\(676\) −33840.6 −1.92539
\(677\) 4514.73i 0.256300i 0.991755 + 0.128150i \(0.0409039\pi\)
−0.991755 + 0.128150i \(0.959096\pi\)
\(678\) − 18330.2i − 1.03830i
\(679\) 5538.43 0.313027
\(680\) 0 0
\(681\) 3681.84 0.207178
\(682\) 13091.0i 0.735018i
\(683\) − 13555.7i − 0.759438i −0.925102 0.379719i \(-0.876021\pi\)
0.925102 0.379719i \(-0.123979\pi\)
\(684\) −2921.44 −0.163310
\(685\) 0 0
\(686\) 26170.2 1.45653
\(687\) 11670.8i 0.648137i
\(688\) 14526.8i 0.804986i
\(689\) 13093.3 0.723972
\(690\) 0 0
\(691\) −11471.3 −0.631535 −0.315768 0.948837i \(-0.602262\pi\)
−0.315768 + 0.948837i \(0.602262\pi\)
\(692\) − 985.787i − 0.0541532i
\(693\) 762.074i 0.0417731i
\(694\) 46327.8 2.53398
\(695\) 0 0
\(696\) 28082.2 1.52939
\(697\) − 7332.40i − 0.398471i
\(698\) 5084.52i 0.275719i
\(699\) −9654.41 −0.522408
\(700\) 0 0
\(701\) 22229.0 1.19769 0.598843 0.800866i \(-0.295627\pi\)
0.598843 + 0.800866i \(0.295627\pi\)
\(702\) 3639.35i 0.195667i
\(703\) 1394.97i 0.0748397i
\(704\) 17937.5 0.960292
\(705\) 0 0
\(706\) −1145.19 −0.0610480
\(707\) 14404.5i 0.766248i
\(708\) 37178.4i 1.97352i
\(709\) 15081.2 0.798851 0.399426 0.916766i \(-0.369210\pi\)
0.399426 + 0.916766i \(0.369210\pi\)
\(710\) 0 0
\(711\) 9316.06 0.491392
\(712\) 1706.98i 0.0898480i
\(713\) 3895.52i 0.204612i
\(714\) 1997.01 0.104673
\(715\) 0 0
\(716\) −17824.6 −0.930358
\(717\) − 1284.30i − 0.0668938i
\(718\) − 7465.71i − 0.388047i
\(719\) 7399.80 0.383819 0.191910 0.981413i \(-0.438532\pi\)
0.191910 + 0.981413i \(0.438532\pi\)
\(720\) 0 0
\(721\) 3300.36 0.170474
\(722\) 35960.9i 1.85364i
\(723\) − 3693.48i − 0.189989i
\(724\) 38791.7 1.99127
\(725\) 0 0
\(726\) −1969.07 −0.100660
\(727\) − 1705.77i − 0.0870202i −0.999053 0.0435101i \(-0.986146\pi\)
0.999053 0.0435101i \(-0.0138541\pi\)
\(728\) − 13928.9i − 0.709122i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −1035.67 −0.0524017
\(732\) − 14241.0i − 0.719074i
\(733\) 37122.6i 1.87061i 0.353847 + 0.935303i \(0.384873\pi\)
−0.353847 + 0.935303i \(0.615127\pi\)
\(734\) −5589.60 −0.281084
\(735\) 0 0
\(736\) 11193.2 0.560581
\(737\) − 9468.20i − 0.473223i
\(738\) − 22454.3i − 1.11999i
\(739\) −34256.3 −1.70520 −0.852598 0.522568i \(-0.824974\pi\)
−0.852598 + 0.522568i \(0.824974\pi\)
\(740\) 0 0
\(741\) −1129.47 −0.0559945
\(742\) − 22001.9i − 1.08856i
\(743\) 1567.88i 0.0774160i 0.999251 + 0.0387080i \(0.0123242\pi\)
−0.999251 + 0.0387080i \(0.987676\pi\)
\(744\) 47929.0 2.36178
\(745\) 0 0
\(746\) 50801.9 2.49328
\(747\) − 5455.90i − 0.267230i
\(748\) 3757.03i 0.183651i
\(749\) 8837.17 0.431112
\(750\) 0 0
\(751\) −955.613 −0.0464325 −0.0232163 0.999730i \(-0.507391\pi\)
−0.0232163 + 0.999730i \(0.507391\pi\)
\(752\) − 111226.i − 5.39360i
\(753\) − 8514.65i − 0.412073i
\(754\) 17326.9 0.836881
\(755\) 0 0
\(756\) 4452.82 0.214216
\(757\) 14015.4i 0.672918i 0.941698 + 0.336459i \(0.109229\pi\)
−0.941698 + 0.336459i \(0.890771\pi\)
\(758\) − 38623.2i − 1.85073i
\(759\) −585.938 −0.0280214
\(760\) 0 0
\(761\) −36271.0 −1.72776 −0.863879 0.503699i \(-0.831972\pi\)
−0.863879 + 0.503699i \(0.831972\pi\)
\(762\) 10761.6i 0.511616i
\(763\) 14073.9i 0.667770i
\(764\) −9831.63 −0.465571
\(765\) 0 0
\(766\) −6311.64 −0.297714
\(767\) 14373.6i 0.676665i
\(768\) − 22739.7i − 1.06842i
\(769\) 18163.6 0.851749 0.425874 0.904782i \(-0.359967\pi\)
0.425874 + 0.904782i \(0.359967\pi\)
\(770\) 0 0
\(771\) −1026.02 −0.0479264
\(772\) − 38112.1i − 1.77680i
\(773\) − 8345.65i − 0.388321i −0.980970 0.194160i \(-0.937802\pi\)
0.980970 0.194160i \(-0.0621982\pi\)
\(774\) −3171.57 −0.147287
\(775\) 0 0
\(776\) 52393.2 2.42372
\(777\) − 2126.19i − 0.0981683i
\(778\) − 59445.7i − 2.73937i
\(779\) 6968.65 0.320510
\(780\) 0 0
\(781\) −6390.11 −0.292774
\(782\) 1535.45i 0.0702142i
\(783\) 3470.76i 0.158410i
\(784\) 63448.5 2.89033
\(785\) 0 0
\(786\) −10125.0 −0.459475
\(787\) 22996.2i 1.04158i 0.853684 + 0.520791i \(0.174363\pi\)
−0.853684 + 0.520791i \(0.825637\pi\)
\(788\) 113647.i 5.13768i
\(789\) 17685.0 0.797975
\(790\) 0 0
\(791\) −8670.69 −0.389752
\(792\) 7209.17i 0.323443i
\(793\) − 5505.75i − 0.246551i
\(794\) 11782.4 0.526626
\(795\) 0 0
\(796\) −110087. −4.90194
\(797\) 2743.82i 0.121946i 0.998139 + 0.0609730i \(0.0194204\pi\)
−0.998139 + 0.0609730i \(0.980580\pi\)
\(798\) 1897.94i 0.0841934i
\(799\) 7929.68 0.351104
\(800\) 0 0
\(801\) −210.970 −0.00930619
\(802\) − 42477.1i − 1.87022i
\(803\) − 5611.28i − 0.246597i
\(804\) −55322.9 −2.42673
\(805\) 0 0
\(806\) 29572.5 1.29237
\(807\) − 7488.53i − 0.326653i
\(808\) 136266.i 5.93293i
\(809\) −41241.7 −1.79231 −0.896156 0.443738i \(-0.853652\pi\)
−0.896156 + 0.443738i \(0.853652\pi\)
\(810\) 0 0
\(811\) 12832.9 0.555641 0.277820 0.960633i \(-0.410388\pi\)
0.277820 + 0.960633i \(0.410388\pi\)
\(812\) − 21199.8i − 0.916215i
\(813\) 6749.03i 0.291142i
\(814\) 5493.72 0.236554
\(815\) 0 0
\(816\) 10694.4 0.458798
\(817\) − 984.292i − 0.0421493i
\(818\) − 58210.4i − 2.48812i
\(819\) 1721.51 0.0734488
\(820\) 0 0
\(821\) 16368.5 0.695817 0.347908 0.937529i \(-0.386892\pi\)
0.347908 + 0.937529i \(0.386892\pi\)
\(822\) − 30477.3i − 1.29321i
\(823\) 3869.53i 0.163892i 0.996637 + 0.0819461i \(0.0261135\pi\)
−0.996637 + 0.0819461i \(0.973886\pi\)
\(824\) 31221.2 1.31995
\(825\) 0 0
\(826\) 24153.3 1.01743
\(827\) − 7388.69i − 0.310677i −0.987861 0.155339i \(-0.950353\pi\)
0.987861 0.155339i \(-0.0496469\pi\)
\(828\) 3423.65i 0.143696i
\(829\) −23990.1 −1.00508 −0.502539 0.864554i \(-0.667601\pi\)
−0.502539 + 0.864554i \(0.667601\pi\)
\(830\) 0 0
\(831\) −12247.8 −0.511276
\(832\) − 40520.6i − 1.68846i
\(833\) 4523.47i 0.188150i
\(834\) −15525.7 −0.644616
\(835\) 0 0
\(836\) −3570.65 −0.147720
\(837\) 5923.68i 0.244626i
\(838\) − 39684.5i − 1.63589i
\(839\) 18228.3 0.750074 0.375037 0.927010i \(-0.377630\pi\)
0.375037 + 0.927010i \(0.377630\pi\)
\(840\) 0 0
\(841\) −7864.78 −0.322472
\(842\) 67782.3i 2.77427i
\(843\) 3101.36i 0.126710i
\(844\) 91318.7 3.72431
\(845\) 0 0
\(846\) 24283.4 0.986855
\(847\) 931.424i 0.0377852i
\(848\) − 117825.i − 4.77137i
\(849\) −23427.4 −0.947028
\(850\) 0 0
\(851\) 1634.77 0.0658511
\(852\) 37337.6i 1.50137i
\(853\) − 21737.3i − 0.872534i −0.899817 0.436267i \(-0.856300\pi\)
0.899817 0.436267i \(-0.143700\pi\)
\(854\) −9251.79 −0.370714
\(855\) 0 0
\(856\) 83599.0 3.33803
\(857\) 18712.2i 0.745852i 0.927861 + 0.372926i \(0.121645\pi\)
−0.927861 + 0.372926i \(0.878355\pi\)
\(858\) 4448.10i 0.176988i
\(859\) −30527.6 −1.21256 −0.606279 0.795252i \(-0.707339\pi\)
−0.606279 + 0.795252i \(0.707339\pi\)
\(860\) 0 0
\(861\) −10621.5 −0.420418
\(862\) − 32953.4i − 1.30209i
\(863\) 10906.4i 0.430196i 0.976592 + 0.215098i \(0.0690071\pi\)
−0.976592 + 0.215098i \(0.930993\pi\)
\(864\) 17020.8 0.670209
\(865\) 0 0
\(866\) 30603.5 1.20087
\(867\) − 13976.6i − 0.547484i
\(868\) − 36182.5i − 1.41488i
\(869\) 11386.3 0.444481
\(870\) 0 0
\(871\) −21388.5 −0.832058
\(872\) 133138.i 5.17043i
\(873\) 6475.41i 0.251042i
\(874\) −1459.28 −0.0564768
\(875\) 0 0
\(876\) −32786.8 −1.26457
\(877\) − 21770.9i − 0.838256i −0.907927 0.419128i \(-0.862336\pi\)
0.907927 0.419128i \(-0.137664\pi\)
\(878\) 59109.8i 2.27205i
\(879\) −5847.58 −0.224384
\(880\) 0 0
\(881\) 47206.9 1.80527 0.902634 0.430409i \(-0.141631\pi\)
0.902634 + 0.430409i \(0.141631\pi\)
\(882\) 13852.4i 0.528837i
\(883\) − 6059.68i − 0.230945i −0.993311 0.115473i \(-0.963162\pi\)
0.993311 0.115473i \(-0.0368382\pi\)
\(884\) 8487.09 0.322909
\(885\) 0 0
\(886\) −41881.1 −1.58806
\(887\) 37130.2i 1.40553i 0.711420 + 0.702767i \(0.248052\pi\)
−0.711420 + 0.702767i \(0.751948\pi\)
\(888\) − 20113.6i − 0.760101i
\(889\) 5090.53 0.192048
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 29518.5i 1.10802i
\(893\) 7536.30i 0.282410i
\(894\) −33311.6 −1.24621
\(895\) 0 0
\(896\) −29269.1 −1.09131
\(897\) 1323.63i 0.0492694i
\(898\) − 40541.4i − 1.50655i
\(899\) 28202.5 1.04628
\(900\) 0 0
\(901\) 8400.15 0.310599
\(902\) − 27444.1i − 1.01307i
\(903\) 1500.24i 0.0552879i
\(904\) −82024.1 −3.01779
\(905\) 0 0
\(906\) −7743.85 −0.283965
\(907\) − 1182.94i − 0.0433064i −0.999766 0.0216532i \(-0.993107\pi\)
0.999766 0.0216532i \(-0.00689297\pi\)
\(908\) − 26293.8i − 0.961001i
\(909\) −16841.4 −0.614516
\(910\) 0 0
\(911\) 37676.4 1.37022 0.685112 0.728438i \(-0.259753\pi\)
0.685112 + 0.728438i \(0.259753\pi\)
\(912\) 10163.9i 0.369035i
\(913\) − 6668.32i − 0.241719i
\(914\) −60431.1 −2.18696
\(915\) 0 0
\(916\) 83347.1 3.00640
\(917\) 4789.41i 0.172476i
\(918\) 2334.86i 0.0839454i
\(919\) −8697.82 −0.312203 −0.156101 0.987741i \(-0.549893\pi\)
−0.156101 + 0.987741i \(0.549893\pi\)
\(920\) 0 0
\(921\) −7092.26 −0.253744
\(922\) − 77725.7i − 2.77631i
\(923\) 14435.2i 0.514778i
\(924\) 5442.33 0.193766
\(925\) 0 0
\(926\) 63795.3 2.26398
\(927\) 3858.71i 0.136717i
\(928\) − 81036.0i − 2.86653i
\(929\) 17247.5 0.609119 0.304559 0.952493i \(-0.401491\pi\)
0.304559 + 0.952493i \(0.401491\pi\)
\(930\) 0 0
\(931\) −4299.06 −0.151338
\(932\) 68946.7i 2.42320i
\(933\) 5967.51i 0.209397i
\(934\) −64306.0 −2.25284
\(935\) 0 0
\(936\) 16285.4 0.568702
\(937\) 41812.4i 1.45779i 0.684624 + 0.728896i \(0.259966\pi\)
−0.684624 + 0.728896i \(0.740034\pi\)
\(938\) 35941.0i 1.25108i
\(939\) −11636.0 −0.404395
\(940\) 0 0
\(941\) 37655.9 1.30451 0.652257 0.757998i \(-0.273822\pi\)
0.652257 + 0.757998i \(0.273822\pi\)
\(942\) − 10536.4i − 0.364431i
\(943\) − 8166.60i − 0.282016i
\(944\) 129346. 4.45959
\(945\) 0 0
\(946\) −3876.37 −0.133226
\(947\) 21244.4i 0.728986i 0.931206 + 0.364493i \(0.118758\pi\)
−0.931206 + 0.364493i \(0.881242\pi\)
\(948\) − 66530.4i − 2.27933i
\(949\) −12675.8 −0.433587
\(950\) 0 0
\(951\) −8741.20 −0.298058
\(952\) − 8936.24i − 0.304228i
\(953\) 1324.27i 0.0450130i 0.999747 + 0.0225065i \(0.00716465\pi\)
−0.999747 + 0.0225065i \(0.992835\pi\)
\(954\) 25724.1 0.873007
\(955\) 0 0
\(956\) −9171.77 −0.310289
\(957\) 4242.04i 0.143287i
\(958\) − 7185.62i − 0.242335i
\(959\) −14416.6 −0.485439
\(960\) 0 0
\(961\) 18343.4 0.615735
\(962\) − 12410.2i − 0.415927i
\(963\) 10332.2i 0.345744i
\(964\) −26376.9 −0.881268
\(965\) 0 0
\(966\) 2224.21 0.0740815
\(967\) − 52267.1i − 1.73815i −0.494676 0.869077i \(-0.664713\pi\)
0.494676 0.869077i \(-0.335287\pi\)
\(968\) 8811.20i 0.292565i
\(969\) −724.619 −0.0240228
\(970\) 0 0
\(971\) −52489.8 −1.73479 −0.867394 0.497622i \(-0.834207\pi\)
−0.867394 + 0.497622i \(0.834207\pi\)
\(972\) 5206.14i 0.171797i
\(973\) 7344.07i 0.241973i
\(974\) −101092. −3.32566
\(975\) 0 0
\(976\) −49545.3 −1.62490
\(977\) − 8324.11i − 0.272581i −0.990669 0.136291i \(-0.956482\pi\)
0.990669 0.136291i \(-0.0435181\pi\)
\(978\) − 17790.5i − 0.581675i
\(979\) −257.852 −0.00841777
\(980\) 0 0
\(981\) −16454.9 −0.535539
\(982\) − 675.623i − 0.0219552i
\(983\) − 44407.1i − 1.44086i −0.693527 0.720431i \(-0.743944\pi\)
0.693527 0.720431i \(-0.256056\pi\)
\(984\) −100479. −3.25523
\(985\) 0 0
\(986\) 11116.2 0.359039
\(987\) − 11486.7i − 0.370442i
\(988\) 8066.05i 0.259732i
\(989\) −1153.50 −0.0370870
\(990\) 0 0
\(991\) −45124.7 −1.44645 −0.723226 0.690612i \(-0.757341\pi\)
−0.723226 + 0.690612i \(0.757341\pi\)
\(992\) − 138307.i − 4.42667i
\(993\) − 24313.4i − 0.777001i
\(994\) 24256.7 0.774020
\(995\) 0 0
\(996\) −38963.2 −1.23955
\(997\) − 5480.61i − 0.174095i −0.996204 0.0870474i \(-0.972257\pi\)
0.996204 0.0870474i \(-0.0277432\pi\)
\(998\) 55492.9i 1.76012i
\(999\) 2485.90 0.0787291
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 825.4.c.h.199.1 4
5.2 odd 4 33.4.a.c.1.2 2
5.3 odd 4 825.4.a.l.1.1 2
5.4 even 2 inner 825.4.c.h.199.4 4
15.2 even 4 99.4.a.f.1.1 2
15.8 even 4 2475.4.a.p.1.2 2
20.7 even 4 528.4.a.p.1.1 2
35.27 even 4 1617.4.a.k.1.2 2
40.27 even 4 2112.4.a.bg.1.2 2
40.37 odd 4 2112.4.a.bn.1.2 2
55.32 even 4 363.4.a.i.1.1 2
60.47 odd 4 1584.4.a.bj.1.2 2
165.32 odd 4 1089.4.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 5.2 odd 4
99.4.a.f.1.1 2 15.2 even 4
363.4.a.i.1.1 2 55.32 even 4
528.4.a.p.1.1 2 20.7 even 4
825.4.a.l.1.1 2 5.3 odd 4
825.4.c.h.199.1 4 1.1 even 1 trivial
825.4.c.h.199.4 4 5.4 even 2 inner
1089.4.a.u.1.2 2 165.32 odd 4
1584.4.a.bj.1.2 2 60.47 odd 4
1617.4.a.k.1.2 2 35.27 even 4
2112.4.a.bg.1.2 2 40.27 even 4
2112.4.a.bn.1.2 2 40.37 odd 4
2475.4.a.p.1.2 2 15.8 even 4