Properties

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

$q$-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 \(0.408437\pi\)
−0.283702 + 0.958913i \(0.591563\pi\)
\(3\) 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 \(-0.933364\pi\)
0.978167 0.207819i \(-0.0666364\pi\)
\(8\) − 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 \(-0.914605\pi\)
0.964229 0.265071i \(-0.0853952\pi\)
\(14\) 41.7557 0.797120
\(15\) 0 0
\(16\) 223.611 3.49392
\(17\) − 15.9420i − 0.227441i −0.993513 0.113721i \(-0.963723\pi\)
0.993513 0.113721i \(-0.0362769\pi\)
\(18\) − 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 \(-0.974353\pi\)
0.996756 0.0804853i \(-0.0256470\pi\)
\(24\) 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 \(0.0655712\pi\)
−0.978857 + 0.204544i \(0.934429\pi\)
\(38\) − 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 \(-0.963250\pi\)
0.993343 0.115198i \(-0.0367503\pi\)
\(44\) 235.669 0.807464
\(45\) 0 0
\(46\) 96.3146 0.308713
\(47\) 497.408i 1.54371i 0.635799 + 0.771855i \(0.280671\pi\)
−0.635799 + 0.771855i \(0.719329\pi\)
\(48\) 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 \(0.239243\pi\)
−0.730596 + 0.682811i \(0.760757\pi\)
\(54\) 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 \(-0.712790\pi\)
0.619810 0.784752i \(-0.287210\pi\)
\(68\) 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 \(-0.865900\pi\)
0.912563 0.408935i \(-0.134100\pi\)
\(74\) −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 \(-0.868717\pi\)
0.916146 0.400845i \(-0.131283\pi\)
\(84\) −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 \(0.122894\pi\)
−0.926391 + 0.376563i \(0.877106\pi\)
\(98\) 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 \(0.0657439\pi\)
−0.978746 + 0.205075i \(0.934256\pi\)
\(104\) −1809.49 −1.70611
\(105\) 0 0
\(106\) −2858.24 −2.61902
\(107\) 1148.02i 1.03723i 0.855008 + 0.518616i \(0.173552\pi\)
−0.855008 + 0.518616i \(0.826448\pi\)
\(108\) 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 \(-0.844665\pi\)
0.883272 0.468861i \(-0.155335\pi\)
\(114\) 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 \(0.0742091\pi\)
−0.972947 + 0.231029i \(0.925791\pi\)
\(128\) − 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 \(-0.801499\pi\)
0.811776 0.583969i \(-0.198501\pi\)
\(138\) 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 \(-0.947378\pi\)
0.986366 0.164565i \(-0.0526220\pi\)
\(158\) − 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 \(-0.915399\pi\)
0.964887 0.262665i \(-0.0846013\pi\)
\(164\) 9853.99 4.69188
\(165\) 0 0
\(166\) 3288.35 1.53750
\(167\) 1123.25i 0.520479i 0.965544 + 0.260240i \(0.0838016\pi\)
−0.965544 + 0.260240i \(0.916198\pi\)
\(168\) − 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 \(-0.996782\pi\)
0.999949 0.0101106i \(-0.00321834\pi\)
\(174\) −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 \(-0.892367\pi\)
0.943373 0.331733i \(-0.107633\pi\)
\(194\) −3902.82 −1.44436
\(195\) 0 0
\(196\) −6079.08 −2.21541
\(197\) 5304.53i 1.91844i 0.282666 + 0.959218i \(0.408781\pi\)
−0.282666 + 0.959218i \(0.591219\pi\)
\(198\) 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 \(0.0663277\pi\)
−0.978368 + 0.206870i \(0.933672\pi\)
\(224\) 4852.65 1.44746
\(225\) 0 0
\(226\) 6110.06 1.79839
\(227\) − 1227.28i − 0.358843i −0.983772 0.179422i \(-0.942577\pi\)
0.983772 0.179422i \(-0.0574226\pi\)
\(228\) 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 \(0.149439\pi\)
−0.891806 + 0.452419i \(0.850561\pi\)
\(234\) −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 \(0.0132154\pi\)
−0.999138 + 0.0415055i \(0.986785\pi\)
\(258\) 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 \(-0.757141\pi\)
0.722791 0.691067i \(-0.242859\pi\)
\(264\) −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 \(0.146007\pi\)
−0.896631 + 0.442778i \(0.853993\pi\)
\(278\) 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 \(0.306110\pi\)
−0.572148 + 0.820150i \(0.693890\pi\)
\(284\) −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 \(0.0622509\pi\)
−0.980938 + 0.194323i \(0.937749\pi\)
\(294\) −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 \(0.0705236\pi\)
−0.975557 + 0.219748i \(0.929476\pi\)
\(308\) − 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 \(0.113892\pi\)
−0.936669 + 0.350216i \(0.886108\pi\)
\(314\) 3512.13 0.631214
\(315\) 0 0
\(316\) 22176.8 3.94792
\(317\) 2913.73i 0.516251i 0.966111 + 0.258126i \(0.0831048\pi\)
−0.966111 + 0.258126i \(0.916895\pi\)
\(318\) − 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 \(0.158782\pi\)
−0.878144 + 0.478396i \(0.841218\pi\)
\(338\) 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 \(-0.770286\pi\)
0.750705 0.660638i \(-0.229714\pi\)
\(348\) − 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 \(0.00506642\pi\)
−0.999873 + 0.0159160i \(0.994934\pi\)
\(354\) −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 \(0.0233473\pi\)
−0.997311 + 0.0732821i \(0.976653\pi\)
\(368\) − 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 \(-0.774757\pi\)
0.759909 0.650029i \(-0.225243\pi\)
\(374\) −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 \(0.0247313\pi\)
−0.996983 + 0.0776176i \(0.975269\pi\)
\(384\) 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 \(-0.956158\pi\)
0.990530 0.137298i \(-0.0438417\pi\)
\(398\) 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 \(-0.898639\pi\)
0.949727 0.313080i \(-0.101361\pi\)
\(434\) 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 \(0.135878\pi\)
−0.910265 + 0.414027i \(0.864122\pi\)
\(444\) 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 \(0.193121\pi\)
−0.821529 + 0.570167i \(0.806879\pi\)
\(458\) − 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 \(-0.799031\pi\)
0.807223 0.590246i \(-0.200969\pi\)
\(464\) 28744.4 2.87592
\(465\) 0 0
\(466\) −17456.5 −1.73532
\(467\) 11854.9i 1.17469i 0.809338 + 0.587343i \(0.199826\pi\)
−0.809338 + 0.587343i \(0.800174\pi\)
\(468\) − 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 \(0.333980\pi\)
−0.498239 + 0.867040i \(0.666020\pi\)
\(488\) 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 \(-0.926685\pi\)
0.973592 0.228294i \(-0.0733146\pi\)
\(504\) 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 \(-0.961002\pi\)
0.992504 0.122208i \(-0.0389975\pi\)
\(524\) −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 \(-0.738566\pi\)
0.681256 0.732046i \(-0.261434\pi\)
\(548\) 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 \(-0.744922\pi\)
0.695736 0.718297i \(-0.255078\pi\)
\(558\) − 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 \(-0.874215\pi\)
0.922933 0.384961i \(-0.125785\pi\)
\(564\) 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 \(-0.798501\pi\)
0.806240 0.591589i \(-0.201499\pi\)
\(578\) 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 \(0.148874\pi\)
−0.892607 + 0.450836i \(0.851126\pi\)
\(588\) − 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 \(-0.801038\pi\)
0.810929 0.585144i \(-0.198962\pi\)
\(594\) −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 \(-0.795075\pi\)
0.799826 0.600232i \(-0.204925\pi\)
\(608\) − 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 \(-0.864567\pi\)
0.910843 0.412753i \(-0.135433\pi\)
\(614\) −12823.8 −0.842878
\(615\) 0 0
\(616\) 6166.01 0.403305
\(617\) − 8586.10i − 0.560232i −0.959966 0.280116i \(-0.909627\pi\)
0.959966 0.280116i \(-0.0903729\pi\)
\(618\) − 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 \(-0.790649\pi\)
0.791403 0.611294i \(-0.209351\pi\)
\(644\) 2928.26 0.179176
\(645\) 0 0
\(646\) −1310.21 −0.0797983
\(647\) − 30634.8i − 1.86148i −0.365684 0.930739i \(-0.619165\pi\)
0.365684 0.930739i \(-0.380835\pi\)
\(648\) − 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 \(0.0950494\pi\)
−0.955747 + 0.294189i \(0.904951\pi\)
\(654\) −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 \(0.328300\pi\)
−0.513630 + 0.858012i \(0.671700\pi\)
\(674\) −32108.2 −1.83496
\(675\) 0 0
\(676\) −33840.6 −1.92539
\(677\) − 4514.73i − 0.256300i −0.991755 0.128150i \(-0.959096\pi\)
0.991755 0.128150i \(-0.0409039\pi\)
\(678\) 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 \(0.123979\pi\)
−0.925102 + 0.379719i \(0.876021\pi\)
\(684\) −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 \(0.0138541\pi\)
−0.999053 + 0.0435101i \(0.986146\pi\)
\(728\) 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 \(-0.615127\pi\)
0.353847 0.935303i \(-0.384873\pi\)
\(734\) −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 \(-0.987676\pi\)
0.999251 0.0387080i \(-0.0123242\pi\)
\(744\) 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 \(-0.890771\pi\)
0.941698 0.336459i \(-0.109229\pi\)
\(758\) 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 \(0.0621982\pi\)
−0.980970 + 0.194160i \(0.937802\pi\)
\(774\) −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 \(-0.825637\pi\)
0.853684 0.520791i \(-0.174363\pi\)
\(788\) − 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 \(-0.980580\pi\)
0.998139 0.0609730i \(-0.0194204\pi\)
\(798\) − 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 \(-0.973886\pi\)
0.996637 0.0819461i \(-0.0261135\pi\)
\(824\) 31221.2 1.31995
\(825\) 0 0
\(826\) 24153.3 1.01743
\(827\) 7388.69i 0.310677i 0.987861 + 0.155339i \(0.0496469\pi\)
−0.987861 + 0.155339i \(0.950353\pi\)
\(828\) − 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 \(0.143700\pi\)
−0.899817 + 0.436267i \(0.856300\pi\)
\(854\) −9251.79 −0.370714
\(855\) 0 0
\(856\) 83599.0 3.33803
\(857\) − 18712.2i − 0.745852i −0.927861 0.372926i \(-0.878355\pi\)
0.927861 0.372926i \(-0.121645\pi\)
\(858\) − 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 \(-0.930993\pi\)
0.976592 0.215098i \(-0.0690071\pi\)
\(864\) 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 \(0.137664\pi\)
−0.907927 + 0.419128i \(0.862336\pi\)
\(878\) − 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 \(0.0368382\pi\)
−0.993311 + 0.115473i \(0.963162\pi\)
\(884\) 8487.09 0.322909
\(885\) 0 0
\(886\) −41881.1 −1.58806
\(887\) − 37130.2i − 1.40553i −0.711420 0.702767i \(-0.751948\pi\)
0.711420 0.702767i \(-0.248052\pi\)
\(888\) 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 \(0.00689297\pi\)
−0.999766 + 0.0216532i \(0.993107\pi\)
\(908\) 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 \(-0.740034\pi\)
0.684624 0.728896i \(-0.259966\pi\)
\(938\) − 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 \(-0.881242\pi\)
0.931206 0.364493i \(-0.118758\pi\)
\(948\) 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 \(-0.992835\pi\)
0.999747 0.0225065i \(-0.00716465\pi\)
\(954\) 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 \(0.335287\pi\)
−0.494676 + 0.869077i \(0.664713\pi\)
\(968\) − 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 \(0.0435181\pi\)
−0.990669 + 0.136291i \(0.956482\pi\)
\(978\) 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 \(0.256056\pi\)
−0.693527 + 0.720431i \(0.743944\pi\)
\(984\) −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 \(0.0277432\pi\)
−0.996204 + 0.0870474i \(0.972257\pi\)
\(998\) − 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.199.4 4
5.2 odd 4 825.4.a.l.1.1 2
5.3 odd 4 33.4.a.c.1.2 2
5.4 even 2 inner 825.4.c.h.199.1 4
15.2 even 4 2475.4.a.p.1.2 2
15.8 even 4 99.4.a.f.1.1 2
20.3 even 4 528.4.a.p.1.1 2
35.13 even 4 1617.4.a.k.1.2 2
40.3 even 4 2112.4.a.bg.1.2 2
40.13 odd 4 2112.4.a.bn.1.2 2
55.43 even 4 363.4.a.i.1.1 2
60.23 odd 4 1584.4.a.bj.1.2 2
165.98 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 5.3 odd 4
99.4.a.f.1.1 2 15.8 even 4
363.4.a.i.1.1 2 55.43 even 4
528.4.a.p.1.1 2 20.3 even 4
825.4.a.l.1.1 2 5.2 odd 4
825.4.c.h.199.1 4 5.4 even 2 inner
825.4.c.h.199.4 4 1.1 even 1 trivial
1089.4.a.u.1.2 2 165.98 odd 4
1584.4.a.bj.1.2 2 60.23 odd 4
1617.4.a.k.1.2 2 35.13 even 4
2112.4.a.bg.1.2 2 40.3 even 4
2112.4.a.bn.1.2 2 40.13 odd 4
2475.4.a.p.1.2 2 15.2 even 4