Properties

Label 825.4.a.v.1.2
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(1,825)]
 
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, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

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: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.91035289.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 285x^{2} + 286x + 19616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(13.6191\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{2} -3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} +24.2670 q^{7} +21.1771 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.56155 q^{2} -3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} +24.2670 q^{7} +21.1771 q^{8} +9.00000 q^{9} -11.0000 q^{11} +16.6847 q^{12} +84.2360 q^{13} -37.8942 q^{14} +11.4233 q^{16} -40.9641 q^{17} -14.0540 q^{18} +120.887 q^{19} -72.8010 q^{21} +17.1771 q^{22} -9.94182 q^{23} -63.5312 q^{24} -131.539 q^{26} -27.0000 q^{27} -134.962 q^{28} +196.860 q^{29} +151.123 q^{31} -187.255 q^{32} +33.0000 q^{33} +63.9675 q^{34} -50.0540 q^{36} -253.477 q^{37} -188.772 q^{38} -252.708 q^{39} -179.351 q^{41} +113.683 q^{42} -90.4851 q^{43} +61.1771 q^{44} +15.5247 q^{46} -483.434 q^{47} -34.2699 q^{48} +245.887 q^{49} +122.892 q^{51} -468.483 q^{52} +567.348 q^{53} +42.1619 q^{54} +513.904 q^{56} -362.662 q^{57} -307.407 q^{58} -491.696 q^{59} +127.008 q^{61} -235.986 q^{62} +218.403 q^{63} +201.022 q^{64} -51.5312 q^{66} +628.226 q^{67} +227.824 q^{68} +29.8255 q^{69} +309.976 q^{71} +190.594 q^{72} -1144.85 q^{73} +395.817 q^{74} -672.322 q^{76} -266.937 q^{77} +394.617 q^{78} -87.4620 q^{79} +81.0000 q^{81} +280.067 q^{82} +1040.13 q^{83} +404.887 q^{84} +141.297 q^{86} -590.579 q^{87} -232.948 q^{88} +390.723 q^{89} +2044.15 q^{91} +55.2920 q^{92} -453.368 q^{93} +754.908 q^{94} +561.764 q^{96} +165.548 q^{97} -383.966 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 12 q^{3} - 14 q^{4} - 6 q^{6} + 11 q^{7} - 6 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 12 q^{3} - 14 q^{4} - 6 q^{6} + 11 q^{7} - 6 q^{8} + 36 q^{9} - 44 q^{11} + 42 q^{12} - 25 q^{13} - 3 q^{14} - 78 q^{16} - 85 q^{17} + 18 q^{18} + 118 q^{19} - 33 q^{21} - 22 q^{22} + 10 q^{23} + 18 q^{24} - 157 q^{26} - 108 q^{27} - 47 q^{28} + 251 q^{29} - 135 q^{31} - 246 q^{32} + 132 q^{33} + 289 q^{34} - 126 q^{36} - 419 q^{37} + 76 q^{38} + 75 q^{39} - 103 q^{41} + 9 q^{42} - 15 q^{43} + 154 q^{44} - 420 q^{46} - 665 q^{47} + 234 q^{48} + 1003 q^{49} + 255 q^{51} - 57 q^{52} + 116 q^{53} - 54 q^{54} + 77 q^{56} - 354 q^{57} - 189 q^{58} + 951 q^{59} - 350 q^{61} + 213 q^{62} + 99 q^{63} + 1538 q^{64} + 66 q^{66} - 266 q^{67} + 629 q^{68} - 30 q^{69} + 1526 q^{71} - 54 q^{72} + 566 q^{73} + 1091 q^{74} - 396 q^{76} - 121 q^{77} + 471 q^{78} + 2567 q^{79} + 324 q^{81} - 9 q^{82} + 961 q^{83} + 141 q^{84} + 1837 q^{86} - 753 q^{87} + 66 q^{88} + 1053 q^{89} + 2150 q^{91} - 460 q^{92} + 405 q^{93} + 2209 q^{94} + 738 q^{96} + 172 q^{97} + 1819 q^{98} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −0.552092 −0.276046 0.961144i \(-0.589024\pi\)
−0.276046 + 0.961144i \(0.589024\pi\)
\(3\) −3.00000 −0.577350
\(4\) −5.56155 −0.695194
\(5\) 0 0
\(6\) 4.68466 0.318751
\(7\) 24.2670 1.31029 0.655147 0.755501i \(-0.272606\pi\)
0.655147 + 0.755501i \(0.272606\pi\)
\(8\) 21.1771 0.935904
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 16.6847 0.401371
\(13\) 84.2360 1.79714 0.898571 0.438828i \(-0.144606\pi\)
0.898571 + 0.438828i \(0.144606\pi\)
\(14\) −37.8942 −0.723404
\(15\) 0 0
\(16\) 11.4233 0.178489
\(17\) −40.9641 −0.584426 −0.292213 0.956353i \(-0.594392\pi\)
−0.292213 + 0.956353i \(0.594392\pi\)
\(18\) −14.0540 −0.184031
\(19\) 120.887 1.45966 0.729828 0.683630i \(-0.239600\pi\)
0.729828 + 0.683630i \(0.239600\pi\)
\(20\) 0 0
\(21\) −72.8010 −0.756499
\(22\) 17.1771 0.166462
\(23\) −9.94182 −0.0901310 −0.0450655 0.998984i \(-0.514350\pi\)
−0.0450655 + 0.998984i \(0.514350\pi\)
\(24\) −63.5312 −0.540344
\(25\) 0 0
\(26\) −131.539 −0.992189
\(27\) −27.0000 −0.192450
\(28\) −134.962 −0.910909
\(29\) 196.860 1.26055 0.630275 0.776372i \(-0.282942\pi\)
0.630275 + 0.776372i \(0.282942\pi\)
\(30\) 0 0
\(31\) 151.123 0.875562 0.437781 0.899082i \(-0.355764\pi\)
0.437781 + 0.899082i \(0.355764\pi\)
\(32\) −187.255 −1.03445
\(33\) 33.0000 0.174078
\(34\) 63.9675 0.322657
\(35\) 0 0
\(36\) −50.0540 −0.231731
\(37\) −253.477 −1.12625 −0.563126 0.826371i \(-0.690401\pi\)
−0.563126 + 0.826371i \(0.690401\pi\)
\(38\) −188.772 −0.805865
\(39\) −252.708 −1.03758
\(40\) 0 0
\(41\) −179.351 −0.683170 −0.341585 0.939851i \(-0.610964\pi\)
−0.341585 + 0.939851i \(0.610964\pi\)
\(42\) 113.683 0.417657
\(43\) −90.4851 −0.320903 −0.160452 0.987044i \(-0.551295\pi\)
−0.160452 + 0.987044i \(0.551295\pi\)
\(44\) 61.1771 0.209609
\(45\) 0 0
\(46\) 15.5247 0.0497606
\(47\) −483.434 −1.50034 −0.750172 0.661243i \(-0.770029\pi\)
−0.750172 + 0.661243i \(0.770029\pi\)
\(48\) −34.2699 −0.103051
\(49\) 245.887 0.716873
\(50\) 0 0
\(51\) 122.892 0.337419
\(52\) −468.483 −1.24936
\(53\) 567.348 1.47040 0.735200 0.677850i \(-0.237088\pi\)
0.735200 + 0.677850i \(0.237088\pi\)
\(54\) 42.1619 0.106250
\(55\) 0 0
\(56\) 513.904 1.22631
\(57\) −362.662 −0.842733
\(58\) −307.407 −0.695939
\(59\) −491.696 −1.08497 −0.542487 0.840064i \(-0.682517\pi\)
−0.542487 + 0.840064i \(0.682517\pi\)
\(60\) 0 0
\(61\) 127.008 0.266585 0.133292 0.991077i \(-0.457445\pi\)
0.133292 + 0.991077i \(0.457445\pi\)
\(62\) −235.986 −0.483391
\(63\) 218.403 0.436765
\(64\) 201.022 0.392621
\(65\) 0 0
\(66\) −51.5312 −0.0961069
\(67\) 628.226 1.14552 0.572762 0.819722i \(-0.305872\pi\)
0.572762 + 0.819722i \(0.305872\pi\)
\(68\) 227.824 0.406290
\(69\) 29.8255 0.0520372
\(70\) 0 0
\(71\) 309.976 0.518133 0.259066 0.965859i \(-0.416585\pi\)
0.259066 + 0.965859i \(0.416585\pi\)
\(72\) 190.594 0.311968
\(73\) −1144.85 −1.83554 −0.917772 0.397109i \(-0.870014\pi\)
−0.917772 + 0.397109i \(0.870014\pi\)
\(74\) 395.817 0.621795
\(75\) 0 0
\(76\) −672.322 −1.01474
\(77\) −266.937 −0.395069
\(78\) 394.617 0.572840
\(79\) −87.4620 −0.124560 −0.0622800 0.998059i \(-0.519837\pi\)
−0.0622800 + 0.998059i \(0.519837\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 280.067 0.377173
\(83\) 1040.13 1.37553 0.687764 0.725934i \(-0.258592\pi\)
0.687764 + 0.725934i \(0.258592\pi\)
\(84\) 404.887 0.525914
\(85\) 0 0
\(86\) 141.297 0.177168
\(87\) −590.579 −0.727778
\(88\) −232.948 −0.282186
\(89\) 390.723 0.465355 0.232677 0.972554i \(-0.425251\pi\)
0.232677 + 0.972554i \(0.425251\pi\)
\(90\) 0 0
\(91\) 2044.15 2.35479
\(92\) 55.2920 0.0626585
\(93\) −453.368 −0.505506
\(94\) 754.908 0.828328
\(95\) 0 0
\(96\) 561.764 0.597238
\(97\) 165.548 0.173287 0.0866433 0.996239i \(-0.472386\pi\)
0.0866433 + 0.996239i \(0.472386\pi\)
\(98\) −383.966 −0.395780
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 1510.35 1.48798 0.743990 0.668191i \(-0.232931\pi\)
0.743990 + 0.668191i \(0.232931\pi\)
\(102\) −191.903 −0.186286
\(103\) 1245.78 1.19175 0.595876 0.803076i \(-0.296805\pi\)
0.595876 + 0.803076i \(0.296805\pi\)
\(104\) 1783.87 1.68195
\(105\) 0 0
\(106\) −885.944 −0.811797
\(107\) −550.568 −0.497434 −0.248717 0.968576i \(-0.580009\pi\)
−0.248717 + 0.968576i \(0.580009\pi\)
\(108\) 150.162 0.133790
\(109\) 1380.30 1.21293 0.606463 0.795112i \(-0.292588\pi\)
0.606463 + 0.795112i \(0.292588\pi\)
\(110\) 0 0
\(111\) 760.430 0.650242
\(112\) 277.209 0.233873
\(113\) 1017.96 0.847445 0.423723 0.905792i \(-0.360723\pi\)
0.423723 + 0.905792i \(0.360723\pi\)
\(114\) 566.316 0.465267
\(115\) 0 0
\(116\) −1094.85 −0.876326
\(117\) 758.124 0.599048
\(118\) 767.810 0.599005
\(119\) −994.075 −0.765770
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −198.329 −0.147179
\(123\) 538.054 0.394429
\(124\) −840.477 −0.608686
\(125\) 0 0
\(126\) −341.048 −0.241135
\(127\) −1181.86 −0.825770 −0.412885 0.910783i \(-0.635479\pi\)
−0.412885 + 0.910783i \(0.635479\pi\)
\(128\) 1184.13 0.817683
\(129\) 271.455 0.185274
\(130\) 0 0
\(131\) −2615.26 −1.74425 −0.872124 0.489284i \(-0.837258\pi\)
−0.872124 + 0.489284i \(0.837258\pi\)
\(132\) −183.531 −0.121018
\(133\) 2933.58 1.91258
\(134\) −981.009 −0.632435
\(135\) 0 0
\(136\) −867.499 −0.546966
\(137\) −465.369 −0.290213 −0.145106 0.989416i \(-0.546352\pi\)
−0.145106 + 0.989416i \(0.546352\pi\)
\(138\) −46.5740 −0.0287293
\(139\) 2809.52 1.71439 0.857196 0.514990i \(-0.172204\pi\)
0.857196 + 0.514990i \(0.172204\pi\)
\(140\) 0 0
\(141\) 1450.30 0.866224
\(142\) −484.045 −0.286057
\(143\) −926.596 −0.541859
\(144\) 102.810 0.0594963
\(145\) 0 0
\(146\) 1787.75 1.01339
\(147\) −737.662 −0.413887
\(148\) 1409.72 0.782963
\(149\) −583.496 −0.320818 −0.160409 0.987051i \(-0.551281\pi\)
−0.160409 + 0.987051i \(0.551281\pi\)
\(150\) 0 0
\(151\) 1244.33 0.670610 0.335305 0.942110i \(-0.391161\pi\)
0.335305 + 0.942110i \(0.391161\pi\)
\(152\) 2560.04 1.36610
\(153\) −368.676 −0.194809
\(154\) 416.836 0.218114
\(155\) 0 0
\(156\) 1405.45 0.721320
\(157\) −303.991 −0.154529 −0.0772647 0.997011i \(-0.524619\pi\)
−0.0772647 + 0.997011i \(0.524619\pi\)
\(158\) 136.577 0.0687687
\(159\) −1702.04 −0.848936
\(160\) 0 0
\(161\) −241.258 −0.118098
\(162\) −126.486 −0.0613436
\(163\) 259.982 0.124929 0.0624643 0.998047i \(-0.480104\pi\)
0.0624643 + 0.998047i \(0.480104\pi\)
\(164\) 997.472 0.474936
\(165\) 0 0
\(166\) −1624.21 −0.759419
\(167\) −3468.36 −1.60712 −0.803562 0.595222i \(-0.797064\pi\)
−0.803562 + 0.595222i \(0.797064\pi\)
\(168\) −1541.71 −0.708010
\(169\) 4898.70 2.22972
\(170\) 0 0
\(171\) 1087.99 0.486552
\(172\) 503.237 0.223090
\(173\) −3864.34 −1.69827 −0.849134 0.528177i \(-0.822876\pi\)
−0.849134 + 0.528177i \(0.822876\pi\)
\(174\) 922.220 0.401801
\(175\) 0 0
\(176\) −125.656 −0.0538164
\(177\) 1475.09 0.626410
\(178\) −610.135 −0.256919
\(179\) 163.870 0.0684258 0.0342129 0.999415i \(-0.489108\pi\)
0.0342129 + 0.999415i \(0.489108\pi\)
\(180\) 0 0
\(181\) −3200.77 −1.31443 −0.657214 0.753704i \(-0.728265\pi\)
−0.657214 + 0.753704i \(0.728265\pi\)
\(182\) −3192.05 −1.30006
\(183\) −381.023 −0.153913
\(184\) −210.539 −0.0843539
\(185\) 0 0
\(186\) 707.958 0.279086
\(187\) 450.605 0.176211
\(188\) 2688.65 1.04303
\(189\) −655.209 −0.252166
\(190\) 0 0
\(191\) 2421.03 0.917171 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(192\) −603.065 −0.226680
\(193\) −2275.44 −0.848651 −0.424326 0.905510i \(-0.639489\pi\)
−0.424326 + 0.905510i \(0.639489\pi\)
\(194\) −258.511 −0.0956702
\(195\) 0 0
\(196\) −1367.52 −0.498366
\(197\) 1957.94 0.708107 0.354054 0.935225i \(-0.384803\pi\)
0.354054 + 0.935225i \(0.384803\pi\)
\(198\) 154.594 0.0554874
\(199\) −1975.36 −0.703665 −0.351833 0.936063i \(-0.614441\pi\)
−0.351833 + 0.936063i \(0.614441\pi\)
\(200\) 0 0
\(201\) −1884.68 −0.661368
\(202\) −2358.50 −0.821502
\(203\) 4777.19 1.65169
\(204\) −683.471 −0.234571
\(205\) 0 0
\(206\) −1945.35 −0.657958
\(207\) −89.4764 −0.0300437
\(208\) 962.252 0.320770
\(209\) −1329.76 −0.440103
\(210\) 0 0
\(211\) 1179.92 0.384973 0.192486 0.981300i \(-0.438345\pi\)
0.192486 + 0.981300i \(0.438345\pi\)
\(212\) −3155.34 −1.02221
\(213\) −929.929 −0.299144
\(214\) 859.741 0.274629
\(215\) 0 0
\(216\) −571.781 −0.180115
\(217\) 3667.29 1.14724
\(218\) −2155.41 −0.669647
\(219\) 3434.55 1.05975
\(220\) 0 0
\(221\) −3450.65 −1.05030
\(222\) −1187.45 −0.358993
\(223\) 5742.73 1.72449 0.862247 0.506489i \(-0.169057\pi\)
0.862247 + 0.506489i \(0.169057\pi\)
\(224\) −4544.11 −1.35543
\(225\) 0 0
\(226\) −1589.59 −0.467868
\(227\) 713.160 0.208520 0.104260 0.994550i \(-0.466753\pi\)
0.104260 + 0.994550i \(0.466753\pi\)
\(228\) 2016.97 0.585863
\(229\) 4438.09 1.28069 0.640343 0.768089i \(-0.278792\pi\)
0.640343 + 0.768089i \(0.278792\pi\)
\(230\) 0 0
\(231\) 800.811 0.228093
\(232\) 4168.91 1.17975
\(233\) −614.179 −0.172688 −0.0863438 0.996265i \(-0.527518\pi\)
−0.0863438 + 0.996265i \(0.527518\pi\)
\(234\) −1183.85 −0.330730
\(235\) 0 0
\(236\) 2734.60 0.754267
\(237\) 262.386 0.0719148
\(238\) 1552.30 0.422776
\(239\) 5191.00 1.40493 0.702464 0.711720i \(-0.252083\pi\)
0.702464 + 0.711720i \(0.252083\pi\)
\(240\) 0 0
\(241\) −353.708 −0.0945408 −0.0472704 0.998882i \(-0.515052\pi\)
−0.0472704 + 0.998882i \(0.515052\pi\)
\(242\) −188.948 −0.0501902
\(243\) −243.000 −0.0641500
\(244\) −706.360 −0.185328
\(245\) 0 0
\(246\) −840.200 −0.217761
\(247\) 10183.1 2.62321
\(248\) 3200.34 0.819442
\(249\) −3120.38 −0.794162
\(250\) 0 0
\(251\) 2612.91 0.657073 0.328537 0.944491i \(-0.393445\pi\)
0.328537 + 0.944491i \(0.393445\pi\)
\(252\) −1214.66 −0.303636
\(253\) 109.360 0.0271755
\(254\) 1845.53 0.455901
\(255\) 0 0
\(256\) −3457.26 −0.844057
\(257\) 2434.38 0.590866 0.295433 0.955364i \(-0.404536\pi\)
0.295433 + 0.955364i \(0.404536\pi\)
\(258\) −423.892 −0.102288
\(259\) −6151.12 −1.47572
\(260\) 0 0
\(261\) 1771.74 0.420183
\(262\) 4083.87 0.962986
\(263\) −7204.41 −1.68914 −0.844568 0.535449i \(-0.820143\pi\)
−0.844568 + 0.535449i \(0.820143\pi\)
\(264\) 698.844 0.162920
\(265\) 0 0
\(266\) −4580.93 −1.05592
\(267\) −1172.17 −0.268673
\(268\) −3493.91 −0.796361
\(269\) 5232.32 1.18595 0.592975 0.805221i \(-0.297953\pi\)
0.592975 + 0.805221i \(0.297953\pi\)
\(270\) 0 0
\(271\) 6231.13 1.39673 0.698366 0.715741i \(-0.253911\pi\)
0.698366 + 0.715741i \(0.253911\pi\)
\(272\) −467.944 −0.104314
\(273\) −6132.46 −1.35954
\(274\) 726.698 0.160224
\(275\) 0 0
\(276\) −165.876 −0.0361759
\(277\) −4940.23 −1.07159 −0.535793 0.844349i \(-0.679987\pi\)
−0.535793 + 0.844349i \(0.679987\pi\)
\(278\) −4387.22 −0.946503
\(279\) 1360.10 0.291854
\(280\) 0 0
\(281\) −1766.97 −0.375119 −0.187559 0.982253i \(-0.560058\pi\)
−0.187559 + 0.982253i \(0.560058\pi\)
\(282\) −2264.72 −0.478235
\(283\) −5304.29 −1.11416 −0.557080 0.830459i \(-0.688078\pi\)
−0.557080 + 0.830459i \(0.688078\pi\)
\(284\) −1723.95 −0.360203
\(285\) 0 0
\(286\) 1446.93 0.299156
\(287\) −4352.32 −0.895155
\(288\) −1685.29 −0.344815
\(289\) −3234.95 −0.658446
\(290\) 0 0
\(291\) −496.643 −0.100047
\(292\) 6367.15 1.27606
\(293\) 8356.39 1.66616 0.833081 0.553151i \(-0.186575\pi\)
0.833081 + 0.553151i \(0.186575\pi\)
\(294\) 1151.90 0.228504
\(295\) 0 0
\(296\) −5367.90 −1.05406
\(297\) 297.000 0.0580259
\(298\) 911.160 0.177121
\(299\) −837.459 −0.161978
\(300\) 0 0
\(301\) −2195.80 −0.420478
\(302\) −1943.09 −0.370238
\(303\) −4531.06 −0.859085
\(304\) 1380.93 0.260533
\(305\) 0 0
\(306\) 575.708 0.107552
\(307\) 8558.29 1.59103 0.795517 0.605932i \(-0.207200\pi\)
0.795517 + 0.605932i \(0.207200\pi\)
\(308\) 1484.58 0.274649
\(309\) −3737.35 −0.688059
\(310\) 0 0
\(311\) 6135.19 1.11863 0.559316 0.828954i \(-0.311064\pi\)
0.559316 + 0.828954i \(0.311064\pi\)
\(312\) −5351.62 −0.971076
\(313\) −4088.62 −0.738346 −0.369173 0.929361i \(-0.620359\pi\)
−0.369173 + 0.929361i \(0.620359\pi\)
\(314\) 474.698 0.0853145
\(315\) 0 0
\(316\) 486.425 0.0865934
\(317\) −7957.42 −1.40988 −0.704942 0.709265i \(-0.749027\pi\)
−0.704942 + 0.709265i \(0.749027\pi\)
\(318\) 2657.83 0.468691
\(319\) −2165.46 −0.380070
\(320\) 0 0
\(321\) 1651.70 0.287194
\(322\) 376.737 0.0652011
\(323\) −4952.04 −0.853062
\(324\) −450.486 −0.0772438
\(325\) 0 0
\(326\) −405.976 −0.0689721
\(327\) −4140.91 −0.700283
\(328\) −3798.14 −0.639382
\(329\) −11731.5 −1.96589
\(330\) 0 0
\(331\) −3113.97 −0.517097 −0.258549 0.965998i \(-0.583244\pi\)
−0.258549 + 0.965998i \(0.583244\pi\)
\(332\) −5784.72 −0.956259
\(333\) −2281.29 −0.375417
\(334\) 5416.02 0.887280
\(335\) 0 0
\(336\) −831.627 −0.135027
\(337\) 8267.16 1.33632 0.668161 0.744016i \(-0.267082\pi\)
0.668161 + 0.744016i \(0.267082\pi\)
\(338\) −7649.58 −1.23101
\(339\) −3053.87 −0.489273
\(340\) 0 0
\(341\) −1662.35 −0.263992
\(342\) −1698.95 −0.268622
\(343\) −2356.63 −0.370980
\(344\) −1916.21 −0.300335
\(345\) 0 0
\(346\) 6034.37 0.937601
\(347\) −8673.03 −1.34177 −0.670883 0.741563i \(-0.734085\pi\)
−0.670883 + 0.741563i \(0.734085\pi\)
\(348\) 3284.54 0.505947
\(349\) 3351.21 0.514000 0.257000 0.966411i \(-0.417266\pi\)
0.257000 + 0.966411i \(0.417266\pi\)
\(350\) 0 0
\(351\) −2274.37 −0.345860
\(352\) 2059.80 0.311897
\(353\) −36.5920 −0.00551727 −0.00275864 0.999996i \(-0.500878\pi\)
−0.00275864 + 0.999996i \(0.500878\pi\)
\(354\) −2303.43 −0.345836
\(355\) 0 0
\(356\) −2173.03 −0.323512
\(357\) 2982.22 0.442118
\(358\) −255.892 −0.0377773
\(359\) −12789.2 −1.88019 −0.940095 0.340912i \(-0.889264\pi\)
−0.940095 + 0.340912i \(0.889264\pi\)
\(360\) 0 0
\(361\) 7754.77 1.13060
\(362\) 4998.17 0.725686
\(363\) −363.000 −0.0524864
\(364\) −11368.7 −1.63703
\(365\) 0 0
\(366\) 594.988 0.0849741
\(367\) −1293.66 −0.184001 −0.0920005 0.995759i \(-0.529326\pi\)
−0.0920005 + 0.995759i \(0.529326\pi\)
\(368\) −113.568 −0.0160874
\(369\) −1614.16 −0.227723
\(370\) 0 0
\(371\) 13767.8 1.92666
\(372\) 2521.43 0.351425
\(373\) 8638.61 1.19917 0.599585 0.800311i \(-0.295332\pi\)
0.599585 + 0.800311i \(0.295332\pi\)
\(374\) −703.643 −0.0972848
\(375\) 0 0
\(376\) −10237.7 −1.40418
\(377\) 16582.7 2.26539
\(378\) 1023.14 0.139219
\(379\) 1853.53 0.251212 0.125606 0.992080i \(-0.459912\pi\)
0.125606 + 0.992080i \(0.459912\pi\)
\(380\) 0 0
\(381\) 3545.57 0.476758
\(382\) −3780.57 −0.506363
\(383\) −3888.44 −0.518773 −0.259387 0.965774i \(-0.583520\pi\)
−0.259387 + 0.965774i \(0.583520\pi\)
\(384\) −3552.39 −0.472090
\(385\) 0 0
\(386\) 3553.22 0.468534
\(387\) −814.366 −0.106968
\(388\) −920.701 −0.120468
\(389\) −10791.1 −1.40651 −0.703255 0.710938i \(-0.748271\pi\)
−0.703255 + 0.710938i \(0.748271\pi\)
\(390\) 0 0
\(391\) 407.257 0.0526749
\(392\) 5207.18 0.670924
\(393\) 7845.79 1.00704
\(394\) −3057.42 −0.390941
\(395\) 0 0
\(396\) 550.594 0.0698696
\(397\) 2875.92 0.363573 0.181787 0.983338i \(-0.441812\pi\)
0.181787 + 0.983338i \(0.441812\pi\)
\(398\) 3084.63 0.388488
\(399\) −8800.73 −1.10423
\(400\) 0 0
\(401\) 7615.88 0.948426 0.474213 0.880410i \(-0.342733\pi\)
0.474213 + 0.880410i \(0.342733\pi\)
\(402\) 2943.03 0.365136
\(403\) 12730.0 1.57351
\(404\) −8399.92 −1.03443
\(405\) 0 0
\(406\) −7459.84 −0.911886
\(407\) 2788.24 0.339578
\(408\) 2602.50 0.315791
\(409\) −3791.51 −0.458382 −0.229191 0.973381i \(-0.573608\pi\)
−0.229191 + 0.973381i \(0.573608\pi\)
\(410\) 0 0
\(411\) 1396.11 0.167554
\(412\) −6928.48 −0.828500
\(413\) −11932.0 −1.42163
\(414\) 139.722 0.0165869
\(415\) 0 0
\(416\) −15773.6 −1.85905
\(417\) −8428.57 −0.989805
\(418\) 2076.49 0.242978
\(419\) 8806.66 1.02681 0.513405 0.858146i \(-0.328384\pi\)
0.513405 + 0.858146i \(0.328384\pi\)
\(420\) 0 0
\(421\) −7730.71 −0.894945 −0.447473 0.894298i \(-0.647676\pi\)
−0.447473 + 0.894298i \(0.647676\pi\)
\(422\) −1842.51 −0.212541
\(423\) −4350.91 −0.500114
\(424\) 12014.8 1.37615
\(425\) 0 0
\(426\) 1452.13 0.165155
\(427\) 3082.10 0.349305
\(428\) 3062.01 0.345813
\(429\) 2779.79 0.312842
\(430\) 0 0
\(431\) 9017.97 1.00784 0.503922 0.863749i \(-0.331890\pi\)
0.503922 + 0.863749i \(0.331890\pi\)
\(432\) −308.429 −0.0343502
\(433\) 13799.1 1.53150 0.765751 0.643137i \(-0.222367\pi\)
0.765751 + 0.643137i \(0.222367\pi\)
\(434\) −5726.67 −0.633385
\(435\) 0 0
\(436\) −7676.62 −0.843219
\(437\) −1201.84 −0.131560
\(438\) −5363.24 −0.585081
\(439\) 464.472 0.0504967 0.0252483 0.999681i \(-0.491962\pi\)
0.0252483 + 0.999681i \(0.491962\pi\)
\(440\) 0 0
\(441\) 2212.99 0.238958
\(442\) 5388.37 0.579861
\(443\) −5796.17 −0.621635 −0.310818 0.950470i \(-0.600603\pi\)
−0.310818 + 0.950470i \(0.600603\pi\)
\(444\) −4229.17 −0.452044
\(445\) 0 0
\(446\) −8967.58 −0.952079
\(447\) 1750.49 0.185224
\(448\) 4878.20 0.514449
\(449\) −8955.57 −0.941291 −0.470645 0.882323i \(-0.655979\pi\)
−0.470645 + 0.882323i \(0.655979\pi\)
\(450\) 0 0
\(451\) 1972.87 0.205984
\(452\) −5661.42 −0.589139
\(453\) −3732.99 −0.387177
\(454\) −1113.64 −0.115122
\(455\) 0 0
\(456\) −7680.13 −0.788717
\(457\) 10067.8 1.03053 0.515267 0.857030i \(-0.327693\pi\)
0.515267 + 0.857030i \(0.327693\pi\)
\(458\) −6930.31 −0.707057
\(459\) 1106.03 0.112473
\(460\) 0 0
\(461\) −8293.06 −0.837844 −0.418922 0.908022i \(-0.637592\pi\)
−0.418922 + 0.908022i \(0.637592\pi\)
\(462\) −1250.51 −0.125928
\(463\) −9901.59 −0.993879 −0.496939 0.867785i \(-0.665543\pi\)
−0.496939 + 0.867785i \(0.665543\pi\)
\(464\) 2248.79 0.224994
\(465\) 0 0
\(466\) 959.073 0.0953395
\(467\) 10915.0 1.08156 0.540779 0.841165i \(-0.318130\pi\)
0.540779 + 0.841165i \(0.318130\pi\)
\(468\) −4216.35 −0.416454
\(469\) 15245.2 1.50097
\(470\) 0 0
\(471\) 911.973 0.0892176
\(472\) −10412.7 −1.01543
\(473\) 995.336 0.0967560
\(474\) −409.730 −0.0397036
\(475\) 0 0
\(476\) 5528.60 0.532359
\(477\) 5106.13 0.490134
\(478\) −8106.02 −0.775650
\(479\) 5725.15 0.546114 0.273057 0.961998i \(-0.411965\pi\)
0.273057 + 0.961998i \(0.411965\pi\)
\(480\) 0 0
\(481\) −21351.9 −2.02403
\(482\) 552.334 0.0521953
\(483\) 723.775 0.0681840
\(484\) −672.948 −0.0631995
\(485\) 0 0
\(486\) 379.457 0.0354167
\(487\) 16620.3 1.54648 0.773242 0.634111i \(-0.218634\pi\)
0.773242 + 0.634111i \(0.218634\pi\)
\(488\) 2689.65 0.249498
\(489\) −779.946 −0.0721276
\(490\) 0 0
\(491\) −9769.75 −0.897969 −0.448985 0.893540i \(-0.648214\pi\)
−0.448985 + 0.893540i \(0.648214\pi\)
\(492\) −2992.42 −0.274204
\(493\) −8064.17 −0.736698
\(494\) −15901.4 −1.44825
\(495\) 0 0
\(496\) 1726.32 0.156278
\(497\) 7522.20 0.678907
\(498\) 4872.64 0.438450
\(499\) −1886.64 −0.169254 −0.0846270 0.996413i \(-0.526970\pi\)
−0.0846270 + 0.996413i \(0.526970\pi\)
\(500\) 0 0
\(501\) 10405.1 0.927873
\(502\) −4080.20 −0.362765
\(503\) −2920.56 −0.258889 −0.129445 0.991587i \(-0.541319\pi\)
−0.129445 + 0.991587i \(0.541319\pi\)
\(504\) 4625.14 0.408770
\(505\) 0 0
\(506\) −170.771 −0.0150034
\(507\) −14696.1 −1.28733
\(508\) 6572.96 0.574070
\(509\) 7956.30 0.692842 0.346421 0.938079i \(-0.387397\pi\)
0.346421 + 0.938079i \(0.387397\pi\)
\(510\) 0 0
\(511\) −27782.1 −2.40510
\(512\) −4074.36 −0.351686
\(513\) −3263.96 −0.280911
\(514\) −3801.41 −0.326212
\(515\) 0 0
\(516\) −1509.71 −0.128801
\(517\) 5317.78 0.452371
\(518\) 9605.30 0.814735
\(519\) 11593.0 0.980496
\(520\) 0 0
\(521\) 20260.6 1.70371 0.851853 0.523780i \(-0.175479\pi\)
0.851853 + 0.523780i \(0.175479\pi\)
\(522\) −2766.66 −0.231980
\(523\) 3457.70 0.289091 0.144546 0.989498i \(-0.453828\pi\)
0.144546 + 0.989498i \(0.453828\pi\)
\(524\) 14544.9 1.21259
\(525\) 0 0
\(526\) 11250.1 0.932559
\(527\) −6190.60 −0.511702
\(528\) 376.969 0.0310709
\(529\) −12068.2 −0.991876
\(530\) 0 0
\(531\) −4425.27 −0.361658
\(532\) −16315.2 −1.32961
\(533\) −15107.8 −1.22775
\(534\) 1830.40 0.148332
\(535\) 0 0
\(536\) 13304.0 1.07210
\(537\) −491.610 −0.0395056
\(538\) −8170.55 −0.654753
\(539\) −2704.76 −0.216145
\(540\) 0 0
\(541\) −3020.10 −0.240008 −0.120004 0.992773i \(-0.538291\pi\)
−0.120004 + 0.992773i \(0.538291\pi\)
\(542\) −9730.24 −0.771125
\(543\) 9602.32 0.758885
\(544\) 7670.71 0.604557
\(545\) 0 0
\(546\) 9576.16 0.750590
\(547\) 17068.8 1.33421 0.667103 0.744965i \(-0.267534\pi\)
0.667103 + 0.744965i \(0.267534\pi\)
\(548\) 2588.17 0.201754
\(549\) 1143.07 0.0888616
\(550\) 0 0
\(551\) 23797.9 1.83997
\(552\) 631.616 0.0487018
\(553\) −2122.44 −0.163210
\(554\) 7714.42 0.591614
\(555\) 0 0
\(556\) −15625.3 −1.19184
\(557\) −4965.05 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(558\) −2123.87 −0.161130
\(559\) −7622.10 −0.576709
\(560\) 0 0
\(561\) −1351.81 −0.101736
\(562\) 2759.21 0.207100
\(563\) 18839.1 1.41025 0.705126 0.709082i \(-0.250890\pi\)
0.705126 + 0.709082i \(0.250890\pi\)
\(564\) −8065.94 −0.602194
\(565\) 0 0
\(566\) 8282.92 0.615119
\(567\) 1965.63 0.145588
\(568\) 6564.40 0.484922
\(569\) 25785.9 1.89983 0.949914 0.312513i \(-0.101171\pi\)
0.949914 + 0.312513i \(0.101171\pi\)
\(570\) 0 0
\(571\) 1732.89 0.127004 0.0635018 0.997982i \(-0.479773\pi\)
0.0635018 + 0.997982i \(0.479773\pi\)
\(572\) 5153.31 0.376697
\(573\) −7263.09 −0.529529
\(574\) 6796.38 0.494208
\(575\) 0 0
\(576\) 1809.20 0.130874
\(577\) −8872.77 −0.640170 −0.320085 0.947389i \(-0.603711\pi\)
−0.320085 + 0.947389i \(0.603711\pi\)
\(578\) 5051.54 0.363523
\(579\) 6826.32 0.489969
\(580\) 0 0
\(581\) 25240.8 1.80235
\(582\) 775.534 0.0552352
\(583\) −6240.83 −0.443343
\(584\) −24244.6 −1.71789
\(585\) 0 0
\(586\) −13048.9 −0.919875
\(587\) 11144.0 0.783582 0.391791 0.920054i \(-0.371856\pi\)
0.391791 + 0.920054i \(0.371856\pi\)
\(588\) 4102.55 0.287732
\(589\) 18268.8 1.27802
\(590\) 0 0
\(591\) −5873.81 −0.408826
\(592\) −2895.54 −0.201023
\(593\) −6643.23 −0.460042 −0.230021 0.973186i \(-0.573879\pi\)
−0.230021 + 0.973186i \(0.573879\pi\)
\(594\) −463.781 −0.0320356
\(595\) 0 0
\(596\) 3245.14 0.223031
\(597\) 5926.07 0.406261
\(598\) 1307.74 0.0894270
\(599\) 26138.0 1.78292 0.891462 0.453096i \(-0.149680\pi\)
0.891462 + 0.453096i \(0.149680\pi\)
\(600\) 0 0
\(601\) 8628.54 0.585633 0.292817 0.956169i \(-0.405407\pi\)
0.292817 + 0.956169i \(0.405407\pi\)
\(602\) 3428.86 0.232143
\(603\) 5654.04 0.381841
\(604\) −6920.40 −0.466204
\(605\) 0 0
\(606\) 7075.50 0.474294
\(607\) −4043.74 −0.270396 −0.135198 0.990819i \(-0.543167\pi\)
−0.135198 + 0.990819i \(0.543167\pi\)
\(608\) −22636.7 −1.50994
\(609\) −14331.6 −0.953604
\(610\) 0 0
\(611\) −40722.6 −2.69633
\(612\) 2050.41 0.135430
\(613\) −5443.62 −0.358671 −0.179336 0.983788i \(-0.557395\pi\)
−0.179336 + 0.983788i \(0.557395\pi\)
\(614\) −13364.2 −0.878397
\(615\) 0 0
\(616\) −5652.95 −0.369746
\(617\) −234.868 −0.0153248 −0.00766241 0.999971i \(-0.502439\pi\)
−0.00766241 + 0.999971i \(0.502439\pi\)
\(618\) 5836.06 0.379872
\(619\) −4517.44 −0.293330 −0.146665 0.989186i \(-0.546854\pi\)
−0.146665 + 0.989186i \(0.546854\pi\)
\(620\) 0 0
\(621\) 268.429 0.0173457
\(622\) −9580.42 −0.617588
\(623\) 9481.68 0.609752
\(624\) −2886.76 −0.185197
\(625\) 0 0
\(626\) 6384.59 0.407635
\(627\) 3989.29 0.254094
\(628\) 1690.66 0.107428
\(629\) 10383.4 0.658211
\(630\) 0 0
\(631\) 26533.3 1.67397 0.836983 0.547229i \(-0.184317\pi\)
0.836983 + 0.547229i \(0.184317\pi\)
\(632\) −1852.19 −0.116576
\(633\) −3539.77 −0.222264
\(634\) 12425.9 0.778386
\(635\) 0 0
\(636\) 9466.01 0.590175
\(637\) 20712.6 1.28832
\(638\) 3381.47 0.209834
\(639\) 2789.79 0.172711
\(640\) 0 0
\(641\) −8233.77 −0.507355 −0.253677 0.967289i \(-0.581640\pi\)
−0.253677 + 0.967289i \(0.581640\pi\)
\(642\) −2579.22 −0.158557
\(643\) −4844.30 −0.297108 −0.148554 0.988904i \(-0.547462\pi\)
−0.148554 + 0.988904i \(0.547462\pi\)
\(644\) 1341.77 0.0821012
\(645\) 0 0
\(646\) 7732.87 0.470969
\(647\) −29856.8 −1.81421 −0.907104 0.420907i \(-0.861712\pi\)
−0.907104 + 0.420907i \(0.861712\pi\)
\(648\) 1715.34 0.103989
\(649\) 5408.66 0.327132
\(650\) 0 0
\(651\) −11001.9 −0.662362
\(652\) −1445.90 −0.0868497
\(653\) −4212.65 −0.252456 −0.126228 0.992001i \(-0.540287\pi\)
−0.126228 + 0.992001i \(0.540287\pi\)
\(654\) 6466.24 0.386621
\(655\) 0 0
\(656\) −2048.78 −0.121938
\(657\) −10303.7 −0.611848
\(658\) 18319.4 1.08535
\(659\) 14549.4 0.860039 0.430020 0.902820i \(-0.358507\pi\)
0.430020 + 0.902820i \(0.358507\pi\)
\(660\) 0 0
\(661\) 2494.86 0.146806 0.0734029 0.997302i \(-0.476614\pi\)
0.0734029 + 0.997302i \(0.476614\pi\)
\(662\) 4862.63 0.285485
\(663\) 10351.9 0.606389
\(664\) 22026.9 1.28736
\(665\) 0 0
\(666\) 3562.35 0.207265
\(667\) −1957.14 −0.113615
\(668\) 19289.5 1.11726
\(669\) −17228.2 −0.995637
\(670\) 0 0
\(671\) −1397.09 −0.0803784
\(672\) 13632.3 0.782557
\(673\) −8223.98 −0.471042 −0.235521 0.971869i \(-0.575680\pi\)
−0.235521 + 0.971869i \(0.575680\pi\)
\(674\) −12909.6 −0.737773
\(675\) 0 0
\(676\) −27244.4 −1.55009
\(677\) −9369.49 −0.531904 −0.265952 0.963986i \(-0.585686\pi\)
−0.265952 + 0.963986i \(0.585686\pi\)
\(678\) 4768.78 0.270124
\(679\) 4017.34 0.227057
\(680\) 0 0
\(681\) −2139.48 −0.120389
\(682\) 2595.85 0.145748
\(683\) −561.238 −0.0314424 −0.0157212 0.999876i \(-0.505004\pi\)
−0.0157212 + 0.999876i \(0.505004\pi\)
\(684\) −6050.90 −0.338248
\(685\) 0 0
\(686\) 3680.01 0.204815
\(687\) −13314.3 −0.739405
\(688\) −1033.64 −0.0572777
\(689\) 47791.1 2.64252
\(690\) 0 0
\(691\) 11456.5 0.630720 0.315360 0.948972i \(-0.397875\pi\)
0.315360 + 0.948972i \(0.397875\pi\)
\(692\) 21491.7 1.18063
\(693\) −2402.43 −0.131690
\(694\) 13543.4 0.740779
\(695\) 0 0
\(696\) −12506.7 −0.681130
\(697\) 7346.96 0.399263
\(698\) −5233.09 −0.283775
\(699\) 1842.54 0.0997012
\(700\) 0 0
\(701\) −2782.39 −0.149914 −0.0749569 0.997187i \(-0.523882\pi\)
−0.0749569 + 0.997187i \(0.523882\pi\)
\(702\) 3551.55 0.190947
\(703\) −30642.1 −1.64394
\(704\) −2211.24 −0.118380
\(705\) 0 0
\(706\) 57.1404 0.00304604
\(707\) 36651.8 1.94969
\(708\) −8203.79 −0.435476
\(709\) 4824.70 0.255565 0.127783 0.991802i \(-0.459214\pi\)
0.127783 + 0.991802i \(0.459214\pi\)
\(710\) 0 0
\(711\) −787.158 −0.0415200
\(712\) 8274.37 0.435527
\(713\) −1502.43 −0.0789153
\(714\) −4656.90 −0.244090
\(715\) 0 0
\(716\) −911.371 −0.0475692
\(717\) −15573.0 −0.811135
\(718\) 19971.0 1.03804
\(719\) 20717.6 1.07460 0.537298 0.843392i \(-0.319445\pi\)
0.537298 + 0.843392i \(0.319445\pi\)
\(720\) 0 0
\(721\) 30231.4 1.56155
\(722\) −12109.5 −0.624195
\(723\) 1061.12 0.0545832
\(724\) 17801.3 0.913783
\(725\) 0 0
\(726\) 566.844 0.0289773
\(727\) −22731.2 −1.15963 −0.579816 0.814747i \(-0.696876\pi\)
−0.579816 + 0.814747i \(0.696876\pi\)
\(728\) 43289.2 2.20385
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 3706.64 0.187544
\(732\) 2119.08 0.106999
\(733\) 24311.8 1.22507 0.612536 0.790443i \(-0.290149\pi\)
0.612536 + 0.790443i \(0.290149\pi\)
\(734\) 2020.11 0.101586
\(735\) 0 0
\(736\) 1861.65 0.0932357
\(737\) −6910.49 −0.345388
\(738\) 2520.60 0.125724
\(739\) 16033.9 0.798126 0.399063 0.916924i \(-0.369335\pi\)
0.399063 + 0.916924i \(0.369335\pi\)
\(740\) 0 0
\(741\) −30549.2 −1.51451
\(742\) −21499.2 −1.06369
\(743\) 19512.2 0.963435 0.481718 0.876326i \(-0.340013\pi\)
0.481718 + 0.876326i \(0.340013\pi\)
\(744\) −9601.01 −0.473105
\(745\) 0 0
\(746\) −13489.7 −0.662053
\(747\) 9361.15 0.458509
\(748\) −2506.06 −0.122501
\(749\) −13360.6 −0.651785
\(750\) 0 0
\(751\) 8393.65 0.407841 0.203920 0.978987i \(-0.434632\pi\)
0.203920 + 0.978987i \(0.434632\pi\)
\(752\) −5522.41 −0.267795
\(753\) −7838.73 −0.379361
\(754\) −25894.7 −1.25070
\(755\) 0 0
\(756\) 3643.98 0.175305
\(757\) −20208.6 −0.970270 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(758\) −2894.38 −0.138692
\(759\) −328.080 −0.0156898
\(760\) 0 0
\(761\) 3529.19 0.168112 0.0840558 0.996461i \(-0.473213\pi\)
0.0840558 + 0.996461i \(0.473213\pi\)
\(762\) −5536.59 −0.263215
\(763\) 33495.8 1.58929
\(764\) −13464.7 −0.637612
\(765\) 0 0
\(766\) 6072.01 0.286411
\(767\) −41418.5 −1.94985
\(768\) 10371.8 0.487317
\(769\) −26692.6 −1.25171 −0.625853 0.779941i \(-0.715249\pi\)
−0.625853 + 0.779941i \(0.715249\pi\)
\(770\) 0 0
\(771\) −7303.14 −0.341136
\(772\) 12655.0 0.589977
\(773\) −32009.0 −1.48937 −0.744684 0.667417i \(-0.767400\pi\)
−0.744684 + 0.667417i \(0.767400\pi\)
\(774\) 1271.67 0.0590561
\(775\) 0 0
\(776\) 3505.81 0.162180
\(777\) 18453.4 0.852008
\(778\) 16850.9 0.776523
\(779\) −21681.3 −0.997194
\(780\) 0 0
\(781\) −3409.74 −0.156223
\(782\) −635.954 −0.0290814
\(783\) −5315.21 −0.242593
\(784\) 2808.84 0.127954
\(785\) 0 0
\(786\) −12251.6 −0.555980
\(787\) −8180.18 −0.370511 −0.185255 0.982690i \(-0.559311\pi\)
−0.185255 + 0.982690i \(0.559311\pi\)
\(788\) −10889.2 −0.492272
\(789\) 21613.2 0.975223
\(790\) 0 0
\(791\) 24702.8 1.11040
\(792\) −2096.53 −0.0940619
\(793\) 10698.6 0.479091
\(794\) −4490.91 −0.200726
\(795\) 0 0
\(796\) 10986.1 0.489184
\(797\) 25471.0 1.13203 0.566015 0.824395i \(-0.308484\pi\)
0.566015 + 0.824395i \(0.308484\pi\)
\(798\) 13742.8 0.609636
\(799\) 19803.4 0.876840
\(800\) 0 0
\(801\) 3516.51 0.155118
\(802\) −11892.6 −0.523619
\(803\) 12593.4 0.553437
\(804\) 10481.7 0.459779
\(805\) 0 0
\(806\) −19878.5 −0.868723
\(807\) −15697.0 −0.684708
\(808\) 31984.9 1.39261
\(809\) 12381.1 0.538067 0.269034 0.963131i \(-0.413296\pi\)
0.269034 + 0.963131i \(0.413296\pi\)
\(810\) 0 0
\(811\) 29129.5 1.26125 0.630627 0.776086i \(-0.282798\pi\)
0.630627 + 0.776086i \(0.282798\pi\)
\(812\) −26568.6 −1.14825
\(813\) −18693.4 −0.806403
\(814\) −4353.99 −0.187478
\(815\) 0 0
\(816\) 1403.83 0.0602255
\(817\) −10938.5 −0.468409
\(818\) 5920.64 0.253069
\(819\) 18397.4 0.784929
\(820\) 0 0
\(821\) −5279.00 −0.224407 −0.112204 0.993685i \(-0.535791\pi\)
−0.112204 + 0.993685i \(0.535791\pi\)
\(822\) −2180.09 −0.0925055
\(823\) −19991.7 −0.846738 −0.423369 0.905957i \(-0.639153\pi\)
−0.423369 + 0.905957i \(0.639153\pi\)
\(824\) 26382.0 1.11537
\(825\) 0 0
\(826\) 18632.4 0.784874
\(827\) −9918.83 −0.417063 −0.208532 0.978016i \(-0.566868\pi\)
−0.208532 + 0.978016i \(0.566868\pi\)
\(828\) 497.628 0.0208862
\(829\) −5671.76 −0.237622 −0.118811 0.992917i \(-0.537908\pi\)
−0.118811 + 0.992917i \(0.537908\pi\)
\(830\) 0 0
\(831\) 14820.7 0.618681
\(832\) 16933.3 0.705595
\(833\) −10072.5 −0.418959
\(834\) 13161.7 0.546464
\(835\) 0 0
\(836\) 7395.54 0.305957
\(837\) −4080.31 −0.168502
\(838\) −13752.1 −0.566894
\(839\) −29160.1 −1.19990 −0.599951 0.800036i \(-0.704813\pi\)
−0.599951 + 0.800036i \(0.704813\pi\)
\(840\) 0 0
\(841\) 14364.7 0.588984
\(842\) 12071.9 0.494092
\(843\) 5300.90 0.216575
\(844\) −6562.21 −0.267631
\(845\) 0 0
\(846\) 6794.17 0.276109
\(847\) 2936.31 0.119118
\(848\) 6480.98 0.262450
\(849\) 15912.9 0.643260
\(850\) 0 0
\(851\) 2520.02 0.101510
\(852\) 5171.85 0.207963
\(853\) −18361.2 −0.737017 −0.368508 0.929624i \(-0.620131\pi\)
−0.368508 + 0.929624i \(0.620131\pi\)
\(854\) −4812.86 −0.192849
\(855\) 0 0
\(856\) −11659.4 −0.465550
\(857\) −1755.28 −0.0699642 −0.0349821 0.999388i \(-0.511137\pi\)
−0.0349821 + 0.999388i \(0.511137\pi\)
\(858\) −4340.78 −0.172718
\(859\) −31317.3 −1.24392 −0.621962 0.783047i \(-0.713664\pi\)
−0.621962 + 0.783047i \(0.713664\pi\)
\(860\) 0 0
\(861\) 13057.0 0.516818
\(862\) −14082.0 −0.556422
\(863\) 42862.3 1.69067 0.845336 0.534235i \(-0.179400\pi\)
0.845336 + 0.534235i \(0.179400\pi\)
\(864\) 5055.88 0.199079
\(865\) 0 0
\(866\) −21548.0 −0.845530
\(867\) 9704.84 0.380154
\(868\) −20395.9 −0.797558
\(869\) 962.082 0.0375563
\(870\) 0 0
\(871\) 52919.3 2.05867
\(872\) 29230.8 1.13518
\(873\) 1489.93 0.0577622
\(874\) 1876.74 0.0726335
\(875\) 0 0
\(876\) −19101.4 −0.736733
\(877\) −38306.6 −1.47494 −0.737471 0.675379i \(-0.763980\pi\)
−0.737471 + 0.675379i \(0.763980\pi\)
\(878\) −725.298 −0.0278788
\(879\) −25069.2 −0.961959
\(880\) 0 0
\(881\) −12994.5 −0.496931 −0.248466 0.968641i \(-0.579926\pi\)
−0.248466 + 0.968641i \(0.579926\pi\)
\(882\) −3455.69 −0.131927
\(883\) 4256.48 0.162222 0.0811110 0.996705i \(-0.474153\pi\)
0.0811110 + 0.996705i \(0.474153\pi\)
\(884\) 19191.0 0.730160
\(885\) 0 0
\(886\) 9051.03 0.343200
\(887\) −38188.8 −1.44561 −0.722804 0.691053i \(-0.757147\pi\)
−0.722804 + 0.691053i \(0.757147\pi\)
\(888\) 16103.7 0.608563
\(889\) −28680.1 −1.08200
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) −31938.5 −1.19886
\(893\) −58441.1 −2.18999
\(894\) −2733.48 −0.102261
\(895\) 0 0
\(896\) 28735.3 1.07141
\(897\) 2512.38 0.0935182
\(898\) 13984.6 0.519679
\(899\) 29750.0 1.10369
\(900\) 0 0
\(901\) −23240.9 −0.859341
\(902\) −3080.73 −0.113722
\(903\) 6587.40 0.242763
\(904\) 21557.3 0.793127
\(905\) 0 0
\(906\) 5829.26 0.213757
\(907\) 31257.8 1.14432 0.572161 0.820142i \(-0.306105\pi\)
0.572161 + 0.820142i \(0.306105\pi\)
\(908\) −3966.28 −0.144962
\(909\) 13593.2 0.495993
\(910\) 0 0
\(911\) −35781.4 −1.30131 −0.650653 0.759375i \(-0.725505\pi\)
−0.650653 + 0.759375i \(0.725505\pi\)
\(912\) −4142.80 −0.150419
\(913\) −11441.4 −0.414737
\(914\) −15721.5 −0.568950
\(915\) 0 0
\(916\) −24682.7 −0.890326
\(917\) −63464.6 −2.28548
\(918\) −1727.12 −0.0620954
\(919\) 31489.1 1.13028 0.565140 0.824995i \(-0.308822\pi\)
0.565140 + 0.824995i \(0.308822\pi\)
\(920\) 0 0
\(921\) −25674.9 −0.918583
\(922\) 12950.0 0.462567
\(923\) 26111.2 0.931159
\(924\) −4453.75 −0.158569
\(925\) 0 0
\(926\) 15461.9 0.548713
\(927\) 11212.0 0.397251
\(928\) −36862.9 −1.30397
\(929\) −40466.1 −1.42912 −0.714559 0.699575i \(-0.753373\pi\)
−0.714559 + 0.699575i \(0.753373\pi\)
\(930\) 0 0
\(931\) 29724.7 1.04639
\(932\) 3415.79 0.120051
\(933\) −18405.6 −0.645843
\(934\) −17044.4 −0.597120
\(935\) 0 0
\(936\) 16054.8 0.560651
\(937\) −39041.1 −1.36117 −0.680586 0.732668i \(-0.738275\pi\)
−0.680586 + 0.732668i \(0.738275\pi\)
\(938\) −23806.1 −0.828676
\(939\) 12265.9 0.426284
\(940\) 0 0
\(941\) 10842.0 0.375598 0.187799 0.982207i \(-0.439865\pi\)
0.187799 + 0.982207i \(0.439865\pi\)
\(942\) −1424.09 −0.0492563
\(943\) 1783.08 0.0615748
\(944\) −5616.79 −0.193656
\(945\) 0 0
\(946\) −1554.27 −0.0534182
\(947\) 17810.1 0.611142 0.305571 0.952169i \(-0.401153\pi\)
0.305571 + 0.952169i \(0.401153\pi\)
\(948\) −1459.27 −0.0499947
\(949\) −96437.6 −3.29873
\(950\) 0 0
\(951\) 23872.3 0.813997
\(952\) −21051.6 −0.716687
\(953\) 45621.7 1.55071 0.775357 0.631523i \(-0.217570\pi\)
0.775357 + 0.631523i \(0.217570\pi\)
\(954\) −7973.49 −0.270599
\(955\) 0 0
\(956\) −28870.0 −0.976697
\(957\) 6496.37 0.219433
\(958\) −8940.12 −0.301506
\(959\) −11293.1 −0.380264
\(960\) 0 0
\(961\) −6952.93 −0.233390
\(962\) 33342.0 1.11745
\(963\) −4955.11 −0.165811
\(964\) 1967.17 0.0657242
\(965\) 0 0
\(966\) −1130.21 −0.0376439
\(967\) 41111.6 1.36718 0.683588 0.729868i \(-0.260419\pi\)
0.683588 + 0.729868i \(0.260419\pi\)
\(968\) 2562.43 0.0850821
\(969\) 14856.1 0.492515
\(970\) 0 0
\(971\) 22030.1 0.728093 0.364046 0.931381i \(-0.381395\pi\)
0.364046 + 0.931381i \(0.381395\pi\)
\(972\) 1351.46 0.0445967
\(973\) 68178.7 2.24636
\(974\) −25953.5 −0.853802
\(975\) 0 0
\(976\) 1450.85 0.0475825
\(977\) 34593.0 1.13278 0.566391 0.824137i \(-0.308339\pi\)
0.566391 + 0.824137i \(0.308339\pi\)
\(978\) 1217.93 0.0398211
\(979\) −4297.95 −0.140310
\(980\) 0 0
\(981\) 12422.7 0.404309
\(982\) 15256.0 0.495762
\(983\) 20638.9 0.669663 0.334831 0.942278i \(-0.391321\pi\)
0.334831 + 0.942278i \(0.391321\pi\)
\(984\) 11394.4 0.369147
\(985\) 0 0
\(986\) 12592.6 0.406725
\(987\) 35194.5 1.13501
\(988\) −56633.7 −1.82364
\(989\) 899.586 0.0289233
\(990\) 0 0
\(991\) −35229.7 −1.12927 −0.564636 0.825340i \(-0.690983\pi\)
−0.564636 + 0.825340i \(0.690983\pi\)
\(992\) −28298.4 −0.905722
\(993\) 9341.91 0.298546
\(994\) −11746.3 −0.374819
\(995\) 0 0
\(996\) 17354.2 0.552096
\(997\) −29366.2 −0.932834 −0.466417 0.884565i \(-0.654455\pi\)
−0.466417 + 0.884565i \(0.654455\pi\)
\(998\) 2946.09 0.0934439
\(999\) 6843.87 0.216747
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.a.v.1.2 yes 4
3.2 odd 2 2475.4.a.bb.1.4 4
5.2 odd 4 825.4.c.q.199.4 8
5.3 odd 4 825.4.c.q.199.5 8
5.4 even 2 825.4.a.u.1.3 4
15.14 odd 2 2475.4.a.bd.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.u.1.3 4 5.4 even 2
825.4.a.v.1.2 yes 4 1.1 even 1 trivial
825.4.c.q.199.4 8 5.2 odd 4
825.4.c.q.199.5 8 5.3 odd 4
2475.4.a.bb.1.4 4 3.2 odd 2
2475.4.a.bd.1.1 4 15.14 odd 2