Properties

Label 825.4.c.m.199.6
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

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: \(6\)
Coefficient field: 6.0.245110336.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 19x^{4} + 101x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.6
Root \(-1.12946i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.m.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.59486i q^{2} +3.00000i q^{3} -13.1127 q^{4} -13.7846 q^{6} +20.6383i q^{7} -23.4921i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+4.59486i q^{2} +3.00000i q^{3} -13.1127 q^{4} -13.7846 q^{6} +20.6383i q^{7} -23.4921i q^{8} -9.00000 q^{9} +11.0000 q^{11} -39.3381i q^{12} +15.6584i q^{13} -94.8302 q^{14} +3.04132 q^{16} +72.9507i q^{17} -41.3537i q^{18} -61.0513 q^{19} -61.9150 q^{21} +50.5434i q^{22} +13.6605i q^{23} +70.4764 q^{24} -71.9483 q^{26} -27.0000i q^{27} -270.624i q^{28} +31.4663 q^{29} -243.008 q^{31} -173.963i q^{32} +33.0000i q^{33} -335.198 q^{34} +118.014 q^{36} -65.4018i q^{37} -280.522i q^{38} -46.9753 q^{39} -109.087 q^{41} -284.491i q^{42} +121.750i q^{43} -144.240 q^{44} -62.7678 q^{46} -519.530i q^{47} +9.12396i q^{48} -82.9413 q^{49} -218.852 q^{51} -205.324i q^{52} +542.673i q^{53} +124.061 q^{54} +484.839 q^{56} -183.154i q^{57} +144.583i q^{58} -109.478 q^{59} -89.6156 q^{61} -1116.59i q^{62} -185.745i q^{63} +823.664 q^{64} -151.630 q^{66} +488.446i q^{67} -956.581i q^{68} -40.9814 q^{69} +837.423 q^{71} +211.429i q^{72} -351.216i q^{73} +300.512 q^{74} +800.547 q^{76} +227.022i q^{77} -215.845i q^{78} +831.205 q^{79} +81.0000 q^{81} -501.238i q^{82} -1389.13i q^{83} +811.873 q^{84} -559.423 q^{86} +94.3988i q^{87} -258.413i q^{88} -1523.70 q^{89} -323.164 q^{91} -179.125i q^{92} -729.025i q^{93} +2387.17 q^{94} +521.888 q^{96} -426.612i q^{97} -381.103i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 34 q^{4} - 6 q^{6} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 34 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} - 288 q^{14} + 50 q^{16} - 232 q^{19} - 36 q^{21} - 18 q^{24} + 604 q^{26} - 476 q^{29} + 184 q^{31} - 708 q^{34} + 306 q^{36} - 120 q^{39} - 92 q^{41} - 374 q^{44} - 480 q^{46} + 914 q^{49} - 192 q^{51} + 54 q^{54} + 1376 q^{56} - 2472 q^{59} + 684 q^{61} + 3838 q^{64} - 66 q^{66} + 1440 q^{69} + 3632 q^{71} + 3748 q^{74} + 792 q^{76} + 192 q^{79} + 486 q^{81} + 960 q^{84} + 376 q^{86} - 1676 q^{89} + 664 q^{91} + 6224 q^{94} - 1182 q^{96} - 594 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.59486i 1.62453i 0.583291 + 0.812263i \(0.301765\pi\)
−0.583291 + 0.812263i \(0.698235\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −13.1127 −1.63909
\(5\) 0 0
\(6\) −13.7846 −0.937921
\(7\) 20.6383i 1.11437i 0.830390 + 0.557183i \(0.188118\pi\)
−0.830390 + 0.557183i \(0.811882\pi\)
\(8\) − 23.4921i − 1.03822i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 39.3381i − 0.946328i
\(13\) 15.6584i 0.334067i 0.985951 + 0.167033i \(0.0534188\pi\)
−0.985951 + 0.167033i \(0.946581\pi\)
\(14\) −94.8302 −1.81032
\(15\) 0 0
\(16\) 3.04132 0.0475206
\(17\) 72.9507i 1.04077i 0.853931 + 0.520387i \(0.174212\pi\)
−0.853931 + 0.520387i \(0.825788\pi\)
\(18\) − 41.3537i − 0.541509i
\(19\) −61.0513 −0.737165 −0.368582 0.929595i \(-0.620157\pi\)
−0.368582 + 0.929595i \(0.620157\pi\)
\(20\) 0 0
\(21\) −61.9150 −0.643379
\(22\) 50.5434i 0.489813i
\(23\) 13.6605i 0.123844i 0.998081 + 0.0619218i \(0.0197229\pi\)
−0.998081 + 0.0619218i \(0.980277\pi\)
\(24\) 70.4764 0.599414
\(25\) 0 0
\(26\) −71.9483 −0.542701
\(27\) − 27.0000i − 0.192450i
\(28\) − 270.624i − 1.82654i
\(29\) 31.4663 0.201487 0.100744 0.994912i \(-0.467878\pi\)
0.100744 + 0.994912i \(0.467878\pi\)
\(30\) 0 0
\(31\) −243.008 −1.40792 −0.703961 0.710239i \(-0.748587\pi\)
−0.703961 + 0.710239i \(0.748587\pi\)
\(32\) − 173.963i − 0.961016i
\(33\) 33.0000i 0.174078i
\(34\) −335.198 −1.69076
\(35\) 0 0
\(36\) 118.014 0.546363
\(37\) − 65.4018i − 0.290594i −0.989388 0.145297i \(-0.953586\pi\)
0.989388 0.145297i \(-0.0464138\pi\)
\(38\) − 280.522i − 1.19754i
\(39\) −46.9753 −0.192874
\(40\) 0 0
\(41\) −109.087 −0.415524 −0.207762 0.978179i \(-0.566618\pi\)
−0.207762 + 0.978179i \(0.566618\pi\)
\(42\) − 284.491i − 1.04519i
\(43\) 121.750i 0.431783i 0.976417 + 0.215891i \(0.0692657\pi\)
−0.976417 + 0.215891i \(0.930734\pi\)
\(44\) −144.240 −0.494204
\(45\) 0 0
\(46\) −62.7678 −0.201187
\(47\) − 519.530i − 1.61237i −0.591665 0.806184i \(-0.701529\pi\)
0.591665 0.806184i \(-0.298471\pi\)
\(48\) 9.12396i 0.0274361i
\(49\) −82.9413 −0.241811
\(50\) 0 0
\(51\) −218.852 −0.600891
\(52\) − 205.324i − 0.547565i
\(53\) 542.673i 1.40645i 0.710967 + 0.703226i \(0.248258\pi\)
−0.710967 + 0.703226i \(0.751742\pi\)
\(54\) 124.061 0.312640
\(55\) 0 0
\(56\) 484.839 1.15695
\(57\) − 183.154i − 0.425602i
\(58\) 144.583i 0.327322i
\(59\) −109.478 −0.241574 −0.120787 0.992678i \(-0.538542\pi\)
−0.120787 + 0.992678i \(0.538542\pi\)
\(60\) 0 0
\(61\) −89.6156 −0.188100 −0.0940501 0.995567i \(-0.529981\pi\)
−0.0940501 + 0.995567i \(0.529981\pi\)
\(62\) − 1116.59i − 2.28721i
\(63\) − 185.745i − 0.371455i
\(64\) 823.664 1.60872
\(65\) 0 0
\(66\) −151.630 −0.282794
\(67\) 488.446i 0.890644i 0.895371 + 0.445322i \(0.146911\pi\)
−0.895371 + 0.445322i \(0.853089\pi\)
\(68\) − 956.581i − 1.70592i
\(69\) −40.9814 −0.0715011
\(70\) 0 0
\(71\) 837.423 1.39977 0.699887 0.714254i \(-0.253234\pi\)
0.699887 + 0.714254i \(0.253234\pi\)
\(72\) 211.429i 0.346072i
\(73\) − 351.216i − 0.563105i −0.959546 0.281553i \(-0.909151\pi\)
0.959546 0.281553i \(-0.0908494\pi\)
\(74\) 300.512 0.472078
\(75\) 0 0
\(76\) 800.547 1.20828
\(77\) 227.022i 0.335994i
\(78\) − 215.845i − 0.313328i
\(79\) 831.205 1.18377 0.591885 0.806022i \(-0.298384\pi\)
0.591885 + 0.806022i \(0.298384\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 501.238i − 0.675031i
\(83\) − 1389.13i − 1.83707i −0.395335 0.918537i \(-0.629371\pi\)
0.395335 0.918537i \(-0.370629\pi\)
\(84\) 811.873 1.05456
\(85\) 0 0
\(86\) −559.423 −0.701443
\(87\) 94.3988i 0.116329i
\(88\) − 258.413i − 0.313034i
\(89\) −1523.70 −1.81474 −0.907369 0.420335i \(-0.861912\pi\)
−0.907369 + 0.420335i \(0.861912\pi\)
\(90\) 0 0
\(91\) −323.164 −0.372273
\(92\) − 179.125i − 0.202990i
\(93\) − 729.025i − 0.812864i
\(94\) 2387.17 2.61933
\(95\) 0 0
\(96\) 521.888 0.554843
\(97\) − 426.612i − 0.446555i −0.974755 0.223278i \(-0.928324\pi\)
0.974755 0.223278i \(-0.0716757\pi\)
\(98\) − 381.103i − 0.392829i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 74.1387 0.0730403 0.0365202 0.999333i \(-0.488373\pi\)
0.0365202 + 0.999333i \(0.488373\pi\)
\(102\) − 1005.59i − 0.976163i
\(103\) 69.3916i 0.0663821i 0.999449 + 0.0331911i \(0.0105670\pi\)
−0.999449 + 0.0331911i \(0.989433\pi\)
\(104\) 367.850 0.346833
\(105\) 0 0
\(106\) −2493.51 −2.28482
\(107\) 1141.71i 1.03152i 0.856733 + 0.515761i \(0.172491\pi\)
−0.856733 + 0.515761i \(0.827509\pi\)
\(108\) 354.043i 0.315443i
\(109\) 2226.85 1.95682 0.978409 0.206680i \(-0.0662659\pi\)
0.978409 + 0.206680i \(0.0662659\pi\)
\(110\) 0 0
\(111\) 196.205 0.167775
\(112\) 62.7678i 0.0529554i
\(113\) 1719.76i 1.43169i 0.698257 + 0.715847i \(0.253959\pi\)
−0.698257 + 0.715847i \(0.746041\pi\)
\(114\) 841.566 0.691402
\(115\) 0 0
\(116\) −412.608 −0.330256
\(117\) − 140.926i − 0.111356i
\(118\) − 503.038i − 0.392444i
\(119\) −1505.58 −1.15980
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 411.771i − 0.305574i
\(123\) − 327.260i − 0.239903i
\(124\) 3186.49 2.30771
\(125\) 0 0
\(126\) 853.472 0.603439
\(127\) − 1601.63i − 1.11907i −0.828807 0.559534i \(-0.810980\pi\)
0.828807 0.559534i \(-0.189020\pi\)
\(128\) 2392.91i 1.65239i
\(129\) −365.249 −0.249290
\(130\) 0 0
\(131\) 2004.13 1.33665 0.668327 0.743868i \(-0.267011\pi\)
0.668327 + 0.743868i \(0.267011\pi\)
\(132\) − 432.719i − 0.285329i
\(133\) − 1260.00i − 0.821471i
\(134\) −2244.34 −1.44687
\(135\) 0 0
\(136\) 1713.77 1.08055
\(137\) − 1672.85i − 1.04322i −0.853184 0.521610i \(-0.825331\pi\)
0.853184 0.521610i \(-0.174669\pi\)
\(138\) − 188.303i − 0.116155i
\(139\) −2540.38 −1.55016 −0.775080 0.631863i \(-0.782291\pi\)
−0.775080 + 0.631863i \(0.782291\pi\)
\(140\) 0 0
\(141\) 1558.59 0.930901
\(142\) 3847.84i 2.27397i
\(143\) 172.243i 0.100725i
\(144\) −27.3719 −0.0158402
\(145\) 0 0
\(146\) 1613.79 0.914780
\(147\) − 248.824i − 0.139610i
\(148\) 857.594i 0.476310i
\(149\) −3090.68 −1.69932 −0.849658 0.527334i \(-0.823192\pi\)
−0.849658 + 0.527334i \(0.823192\pi\)
\(150\) 0 0
\(151\) 1358.74 0.732267 0.366134 0.930562i \(-0.380681\pi\)
0.366134 + 0.930562i \(0.380681\pi\)
\(152\) 1434.22i 0.765335i
\(153\) − 656.557i − 0.346925i
\(154\) −1043.13 −0.545831
\(155\) 0 0
\(156\) 615.973 0.316137
\(157\) − 1011.95i − 0.514411i −0.966357 0.257205i \(-0.917198\pi\)
0.966357 0.257205i \(-0.0828017\pi\)
\(158\) 3819.27i 1.92307i
\(159\) −1628.02 −0.812015
\(160\) 0 0
\(161\) −281.929 −0.138007
\(162\) 372.183i 0.180503i
\(163\) 2816.37i 1.35334i 0.736285 + 0.676672i \(0.236578\pi\)
−0.736285 + 0.676672i \(0.763422\pi\)
\(164\) 1430.42 0.681081
\(165\) 0 0
\(166\) 6382.87 2.98438
\(167\) 3448.89i 1.59810i 0.601262 + 0.799052i \(0.294665\pi\)
−0.601262 + 0.799052i \(0.705335\pi\)
\(168\) 1454.52i 0.667966i
\(169\) 1951.81 0.888399
\(170\) 0 0
\(171\) 549.462 0.245722
\(172\) − 1596.47i − 0.707730i
\(173\) 2287.85i 1.00545i 0.864448 + 0.502723i \(0.167668\pi\)
−0.864448 + 0.502723i \(0.832332\pi\)
\(174\) −433.749 −0.188979
\(175\) 0 0
\(176\) 33.4545 0.0143280
\(177\) − 328.435i − 0.139473i
\(178\) − 7001.17i − 2.94809i
\(179\) −3249.06 −1.35668 −0.678340 0.734748i \(-0.737300\pi\)
−0.678340 + 0.734748i \(0.737300\pi\)
\(180\) 0 0
\(181\) 1170.45 0.480655 0.240328 0.970692i \(-0.422745\pi\)
0.240328 + 0.970692i \(0.422745\pi\)
\(182\) − 1484.89i − 0.604767i
\(183\) − 268.847i − 0.108600i
\(184\) 320.913 0.128576
\(185\) 0 0
\(186\) 3349.76 1.32052
\(187\) 802.458i 0.313805i
\(188\) 6812.44i 2.64281i
\(189\) 557.235 0.214460
\(190\) 0 0
\(191\) −2760.35 −1.04572 −0.522859 0.852419i \(-0.675134\pi\)
−0.522859 + 0.852419i \(0.675134\pi\)
\(192\) 2470.99i 0.928794i
\(193\) 1250.61i 0.466430i 0.972425 + 0.233215i \(0.0749245\pi\)
−0.972425 + 0.233215i \(0.925075\pi\)
\(194\) 1960.22 0.725441
\(195\) 0 0
\(196\) 1087.58 0.396350
\(197\) − 143.991i − 0.0520756i −0.999661 0.0260378i \(-0.991711\pi\)
0.999661 0.0260378i \(-0.00828903\pi\)
\(198\) − 454.891i − 0.163271i
\(199\) −761.249 −0.271174 −0.135587 0.990765i \(-0.543292\pi\)
−0.135587 + 0.990765i \(0.543292\pi\)
\(200\) 0 0
\(201\) −1465.34 −0.514213
\(202\) 340.657i 0.118656i
\(203\) 649.411i 0.224531i
\(204\) 2869.74 0.984913
\(205\) 0 0
\(206\) −318.844 −0.107840
\(207\) − 122.944i − 0.0412812i
\(208\) 47.6223i 0.0158751i
\(209\) −671.564 −0.222263
\(210\) 0 0
\(211\) 3976.58 1.29743 0.648717 0.761029i \(-0.275306\pi\)
0.648717 + 0.761029i \(0.275306\pi\)
\(212\) − 7115.91i − 2.30530i
\(213\) 2512.27i 0.808159i
\(214\) −5245.97 −1.67573
\(215\) 0 0
\(216\) −634.287 −0.199805
\(217\) − 5015.29i − 1.56894i
\(218\) 10232.0i 3.17890i
\(219\) 1053.65 0.325109
\(220\) 0 0
\(221\) −1142.29 −0.347688
\(222\) 901.535i 0.272555i
\(223\) − 908.084i − 0.272690i −0.990661 0.136345i \(-0.956464\pi\)
0.990661 0.136345i \(-0.0435355\pi\)
\(224\) 3590.30 1.07092
\(225\) 0 0
\(226\) −7902.05 −2.32583
\(227\) − 2062.15i − 0.602951i −0.953474 0.301475i \(-0.902521\pi\)
0.953474 0.301475i \(-0.0974791\pi\)
\(228\) 2401.64i 0.697599i
\(229\) −4077.47 −1.17662 −0.588312 0.808634i \(-0.700207\pi\)
−0.588312 + 0.808634i \(0.700207\pi\)
\(230\) 0 0
\(231\) −681.065 −0.193986
\(232\) − 739.209i − 0.209187i
\(233\) − 1682.76i − 0.473138i −0.971615 0.236569i \(-0.923977\pi\)
0.971615 0.236569i \(-0.0760229\pi\)
\(234\) 647.534 0.180900
\(235\) 0 0
\(236\) 1435.56 0.395961
\(237\) 2493.62i 0.683450i
\(238\) − 6917.93i − 1.88413i
\(239\) −4024.96 −1.08934 −0.544672 0.838649i \(-0.683346\pi\)
−0.544672 + 0.838649i \(0.683346\pi\)
\(240\) 0 0
\(241\) −2784.27 −0.744194 −0.372097 0.928194i \(-0.621361\pi\)
−0.372097 + 0.928194i \(0.621361\pi\)
\(242\) 555.978i 0.147684i
\(243\) 243.000i 0.0641500i
\(244\) 1175.10 0.308313
\(245\) 0 0
\(246\) 1503.71 0.389729
\(247\) − 955.968i − 0.246262i
\(248\) 5708.78i 1.46173i
\(249\) 4167.40 1.06064
\(250\) 0 0
\(251\) −1827.60 −0.459591 −0.229796 0.973239i \(-0.573806\pi\)
−0.229796 + 0.973239i \(0.573806\pi\)
\(252\) 2435.62i 0.608848i
\(253\) 150.265i 0.0373402i
\(254\) 7359.26 1.81796
\(255\) 0 0
\(256\) −4405.79 −1.07563
\(257\) 585.171i 0.142031i 0.997475 + 0.0710155i \(0.0226240\pi\)
−0.997475 + 0.0710155i \(0.977376\pi\)
\(258\) − 1678.27i − 0.404978i
\(259\) 1349.78 0.323828
\(260\) 0 0
\(261\) −283.196 −0.0671625
\(262\) 9208.69i 2.17143i
\(263\) 238.098i 0.0558241i 0.999610 + 0.0279120i \(0.00888583\pi\)
−0.999610 + 0.0279120i \(0.991114\pi\)
\(264\) 775.240 0.180730
\(265\) 0 0
\(266\) 5789.51 1.33450
\(267\) − 4571.09i − 1.04774i
\(268\) − 6404.84i − 1.45984i
\(269\) −4618.46 −1.04681 −0.523406 0.852083i \(-0.675339\pi\)
−0.523406 + 0.852083i \(0.675339\pi\)
\(270\) 0 0
\(271\) −143.439 −0.0321525 −0.0160762 0.999871i \(-0.505117\pi\)
−0.0160762 + 0.999871i \(0.505117\pi\)
\(272\) 221.867i 0.0494582i
\(273\) − 969.493i − 0.214932i
\(274\) 7686.51 1.69474
\(275\) 0 0
\(276\) 537.376 0.117197
\(277\) − 8602.51i − 1.86597i −0.359911 0.932987i \(-0.617193\pi\)
0.359911 0.932987i \(-0.382807\pi\)
\(278\) − 11672.7i − 2.51828i
\(279\) 2187.07 0.469307
\(280\) 0 0
\(281\) −2992.81 −0.635360 −0.317680 0.948198i \(-0.602904\pi\)
−0.317680 + 0.948198i \(0.602904\pi\)
\(282\) 7161.50i 1.51227i
\(283\) 6858.89i 1.44070i 0.693610 + 0.720351i \(0.256019\pi\)
−0.693610 + 0.720351i \(0.743981\pi\)
\(284\) −10980.9 −2.29435
\(285\) 0 0
\(286\) −791.431 −0.163630
\(287\) − 2251.37i − 0.463046i
\(288\) 1565.66i 0.320339i
\(289\) −408.809 −0.0832096
\(290\) 0 0
\(291\) 1279.84 0.257819
\(292\) 4605.39i 0.922979i
\(293\) − 4049.70i − 0.807461i −0.914878 0.403731i \(-0.867713\pi\)
0.914878 0.403731i \(-0.132287\pi\)
\(294\) 1143.31 0.226800
\(295\) 0 0
\(296\) −1536.43 −0.301699
\(297\) − 297.000i − 0.0580259i
\(298\) − 14201.2i − 2.76059i
\(299\) −213.901 −0.0413720
\(300\) 0 0
\(301\) −2512.71 −0.481164
\(302\) 6243.20i 1.18959i
\(303\) 222.416i 0.0421699i
\(304\) −185.677 −0.0350305
\(305\) 0 0
\(306\) 3016.78 0.563588
\(307\) − 9572.69i − 1.77962i −0.456335 0.889808i \(-0.650838\pi\)
0.456335 0.889808i \(-0.349162\pi\)
\(308\) − 2976.87i − 0.550724i
\(309\) −208.175 −0.0383257
\(310\) 0 0
\(311\) 5396.42 0.983932 0.491966 0.870614i \(-0.336278\pi\)
0.491966 + 0.870614i \(0.336278\pi\)
\(312\) 1103.55i 0.200244i
\(313\) − 9755.04i − 1.76162i −0.473469 0.880811i \(-0.656998\pi\)
0.473469 0.880811i \(-0.343002\pi\)
\(314\) 4649.77 0.835674
\(315\) 0 0
\(316\) −10899.3 −1.94030
\(317\) − 4353.75i − 0.771391i −0.922626 0.385695i \(-0.873962\pi\)
0.922626 0.385695i \(-0.126038\pi\)
\(318\) − 7480.52i − 1.31914i
\(319\) 346.129 0.0607508
\(320\) 0 0
\(321\) −3425.12 −0.595549
\(322\) − 1295.42i − 0.224196i
\(323\) − 4453.74i − 0.767221i
\(324\) −1062.13 −0.182121
\(325\) 0 0
\(326\) −12940.8 −2.19854
\(327\) 6680.54i 1.12977i
\(328\) 2562.68i 0.431404i
\(329\) 10722.2 1.79677
\(330\) 0 0
\(331\) 5387.64 0.894656 0.447328 0.894370i \(-0.352376\pi\)
0.447328 + 0.894370i \(0.352376\pi\)
\(332\) 18215.3i 3.01113i
\(333\) 588.616i 0.0968648i
\(334\) −15847.2 −2.59616
\(335\) 0 0
\(336\) −188.303 −0.0305738
\(337\) − 4500.27i − 0.727434i −0.931509 0.363717i \(-0.881508\pi\)
0.931509 0.363717i \(-0.118492\pi\)
\(338\) 8968.30i 1.44323i
\(339\) −5159.28 −0.826589
\(340\) 0 0
\(341\) −2673.09 −0.424504
\(342\) 2524.70i 0.399181i
\(343\) 5367.18i 0.844899i
\(344\) 2860.16 0.448283
\(345\) 0 0
\(346\) −10512.4 −1.63337
\(347\) 5906.32i 0.913740i 0.889533 + 0.456870i \(0.151030\pi\)
−0.889533 + 0.456870i \(0.848970\pi\)
\(348\) − 1237.82i − 0.190673i
\(349\) −3636.26 −0.557721 −0.278860 0.960332i \(-0.589957\pi\)
−0.278860 + 0.960332i \(0.589957\pi\)
\(350\) 0 0
\(351\) 422.778 0.0642912
\(352\) − 1913.59i − 0.289757i
\(353\) − 210.408i − 0.0317248i −0.999874 0.0158624i \(-0.994951\pi\)
0.999874 0.0158624i \(-0.00504938\pi\)
\(354\) 1509.11 0.226577
\(355\) 0 0
\(356\) 19979.8 2.97451
\(357\) − 4516.75i − 0.669612i
\(358\) − 14928.9i − 2.20396i
\(359\) −2499.68 −0.367488 −0.183744 0.982974i \(-0.558822\pi\)
−0.183744 + 0.982974i \(0.558822\pi\)
\(360\) 0 0
\(361\) −3131.74 −0.456588
\(362\) 5378.03i 0.780837i
\(363\) 363.000i 0.0524864i
\(364\) 4237.56 0.610188
\(365\) 0 0
\(366\) 1235.31 0.176423
\(367\) 5748.70i 0.817656i 0.912612 + 0.408828i \(0.134062\pi\)
−0.912612 + 0.408828i \(0.865938\pi\)
\(368\) 41.5458i 0.00588512i
\(369\) 981.781 0.138508
\(370\) 0 0
\(371\) −11199.9 −1.56730
\(372\) 9559.48i 1.33236i
\(373\) − 4467.78i − 0.620196i −0.950705 0.310098i \(-0.899638\pi\)
0.950705 0.310098i \(-0.100362\pi\)
\(374\) −3687.18 −0.509785
\(375\) 0 0
\(376\) −12204.9 −1.67398
\(377\) 492.712i 0.0673103i
\(378\) 2560.42i 0.348396i
\(379\) −7804.08 −1.05770 −0.528851 0.848715i \(-0.677377\pi\)
−0.528851 + 0.848715i \(0.677377\pi\)
\(380\) 0 0
\(381\) 4804.89 0.646094
\(382\) − 12683.4i − 1.69880i
\(383\) 11161.1i 1.48904i 0.667597 + 0.744522i \(0.267323\pi\)
−0.667597 + 0.744522i \(0.732677\pi\)
\(384\) −7178.74 −0.954007
\(385\) 0 0
\(386\) −5746.38 −0.757728
\(387\) − 1095.75i − 0.143928i
\(388\) 5594.04i 0.731944i
\(389\) −8490.24 −1.10661 −0.553306 0.832978i \(-0.686634\pi\)
−0.553306 + 0.832978i \(0.686634\pi\)
\(390\) 0 0
\(391\) −996.540 −0.128893
\(392\) 1948.47i 0.251052i
\(393\) 6012.39i 0.771717i
\(394\) 661.616 0.0845983
\(395\) 0 0
\(396\) 1298.16 0.164735
\(397\) 6019.74i 0.761013i 0.924778 + 0.380507i \(0.124250\pi\)
−0.924778 + 0.380507i \(0.875750\pi\)
\(398\) − 3497.83i − 0.440529i
\(399\) 3779.99 0.474277
\(400\) 0 0
\(401\) −10398.8 −1.29499 −0.647495 0.762069i \(-0.724184\pi\)
−0.647495 + 0.762069i \(0.724184\pi\)
\(402\) − 6733.01i − 0.835354i
\(403\) − 3805.13i − 0.470340i
\(404\) −972.158 −0.119720
\(405\) 0 0
\(406\) −2983.95 −0.364756
\(407\) − 719.420i − 0.0876175i
\(408\) 5141.30i 0.623854i
\(409\) 4733.68 0.572287 0.286144 0.958187i \(-0.407627\pi\)
0.286144 + 0.958187i \(0.407627\pi\)
\(410\) 0 0
\(411\) 5018.55 0.602304
\(412\) − 909.911i − 0.108806i
\(413\) − 2259.45i − 0.269202i
\(414\) 564.910 0.0670624
\(415\) 0 0
\(416\) 2723.98 0.321044
\(417\) − 7621.14i − 0.894985i
\(418\) − 3085.74i − 0.361073i
\(419\) 8117.57 0.946466 0.473233 0.880937i \(-0.343087\pi\)
0.473233 + 0.880937i \(0.343087\pi\)
\(420\) 0 0
\(421\) −9484.27 −1.09795 −0.548973 0.835840i \(-0.684981\pi\)
−0.548973 + 0.835840i \(0.684981\pi\)
\(422\) 18271.8i 2.10772i
\(423\) 4675.77i 0.537456i
\(424\) 12748.5 1.46020
\(425\) 0 0
\(426\) −11543.5 −1.31288
\(427\) − 1849.52i − 0.209612i
\(428\) − 14970.8i − 1.69075i
\(429\) −516.728 −0.0581536
\(430\) 0 0
\(431\) 9335.16 1.04329 0.521646 0.853162i \(-0.325318\pi\)
0.521646 + 0.853162i \(0.325318\pi\)
\(432\) − 82.1157i − 0.00914535i
\(433\) 2983.02i 0.331074i 0.986204 + 0.165537i \(0.0529357\pi\)
−0.986204 + 0.165537i \(0.947064\pi\)
\(434\) 23044.5 2.54878
\(435\) 0 0
\(436\) −29200.0 −3.20739
\(437\) − 833.988i − 0.0912931i
\(438\) 4841.36i 0.528148i
\(439\) 5232.32 0.568850 0.284425 0.958698i \(-0.408197\pi\)
0.284425 + 0.958698i \(0.408197\pi\)
\(440\) 0 0
\(441\) 746.472 0.0806038
\(442\) − 5248.68i − 0.564828i
\(443\) − 7517.71i − 0.806269i −0.915141 0.403135i \(-0.867921\pi\)
0.915141 0.403135i \(-0.132079\pi\)
\(444\) −2572.78 −0.274997
\(445\) 0 0
\(446\) 4172.52 0.442992
\(447\) − 9272.03i − 0.981101i
\(448\) 16999.1i 1.79270i
\(449\) −16070.9 −1.68916 −0.844581 0.535428i \(-0.820150\pi\)
−0.844581 + 0.535428i \(0.820150\pi\)
\(450\) 0 0
\(451\) −1199.96 −0.125285
\(452\) − 22550.7i − 2.34667i
\(453\) 4076.21i 0.422775i
\(454\) 9475.29 0.979510
\(455\) 0 0
\(456\) −4302.67 −0.441867
\(457\) 9718.51i 0.994776i 0.867528 + 0.497388i \(0.165707\pi\)
−0.867528 + 0.497388i \(0.834293\pi\)
\(458\) − 18735.4i − 1.91146i
\(459\) 1969.67 0.200297
\(460\) 0 0
\(461\) −14538.0 −1.46877 −0.734385 0.678733i \(-0.762529\pi\)
−0.734385 + 0.678733i \(0.762529\pi\)
\(462\) − 3129.40i − 0.315136i
\(463\) 9978.17i 1.00157i 0.865573 + 0.500783i \(0.166955\pi\)
−0.865573 + 0.500783i \(0.833045\pi\)
\(464\) 95.6990 0.00957481
\(465\) 0 0
\(466\) 7732.03 0.768625
\(467\) 15188.2i 1.50498i 0.658605 + 0.752489i \(0.271147\pi\)
−0.658605 + 0.752489i \(0.728853\pi\)
\(468\) 1847.92i 0.182522i
\(469\) −10080.7 −0.992503
\(470\) 0 0
\(471\) 3035.85 0.296995
\(472\) 2571.88i 0.250806i
\(473\) 1339.25i 0.130187i
\(474\) −11457.8 −1.11028
\(475\) 0 0
\(476\) 19742.3 1.90102
\(477\) − 4884.06i − 0.468817i
\(478\) − 18494.1i − 1.76967i
\(479\) −11330.8 −1.08083 −0.540415 0.841399i \(-0.681733\pi\)
−0.540415 + 0.841399i \(0.681733\pi\)
\(480\) 0 0
\(481\) 1024.09 0.0970779
\(482\) − 12793.3i − 1.20896i
\(483\) − 845.788i − 0.0796784i
\(484\) −1586.64 −0.149008
\(485\) 0 0
\(486\) −1116.55 −0.104213
\(487\) 19086.9i 1.77599i 0.459850 + 0.887997i \(0.347903\pi\)
−0.459850 + 0.887997i \(0.652097\pi\)
\(488\) 2105.26i 0.195288i
\(489\) −8449.11 −0.781353
\(490\) 0 0
\(491\) −8112.85 −0.745677 −0.372839 0.927896i \(-0.621616\pi\)
−0.372839 + 0.927896i \(0.621616\pi\)
\(492\) 4291.27i 0.393222i
\(493\) 2295.49i 0.209703i
\(494\) 4392.53 0.400060
\(495\) 0 0
\(496\) −739.066 −0.0669053
\(497\) 17283.0i 1.55986i
\(498\) 19148.6i 1.72303i
\(499\) −18329.1 −1.64433 −0.822167 0.569246i \(-0.807235\pi\)
−0.822167 + 0.569246i \(0.807235\pi\)
\(500\) 0 0
\(501\) −10346.7 −0.922666
\(502\) − 8397.58i − 0.746618i
\(503\) − 7739.57i − 0.686064i −0.939324 0.343032i \(-0.888546\pi\)
0.939324 0.343032i \(-0.111454\pi\)
\(504\) −4363.55 −0.385651
\(505\) 0 0
\(506\) −690.446 −0.0606602
\(507\) 5855.44i 0.512918i
\(508\) 21001.7i 1.83425i
\(509\) −15914.9 −1.38589 −0.692943 0.720993i \(-0.743686\pi\)
−0.692943 + 0.720993i \(0.743686\pi\)
\(510\) 0 0
\(511\) 7248.51 0.627505
\(512\) − 1100.65i − 0.0950048i
\(513\) 1648.38i 0.141867i
\(514\) −2688.78 −0.230733
\(515\) 0 0
\(516\) 4789.40 0.408608
\(517\) − 5714.83i − 0.486147i
\(518\) 6202.07i 0.526068i
\(519\) −6863.56 −0.580495
\(520\) 0 0
\(521\) 2274.50 0.191262 0.0956312 0.995417i \(-0.469513\pi\)
0.0956312 + 0.995417i \(0.469513\pi\)
\(522\) − 1301.25i − 0.109107i
\(523\) − 10971.1i − 0.917274i −0.888624 0.458637i \(-0.848338\pi\)
0.888624 0.458637i \(-0.151662\pi\)
\(524\) −26279.6 −2.19089
\(525\) 0 0
\(526\) −1094.02 −0.0906877
\(527\) − 17727.6i − 1.46533i
\(528\) 100.364i 0.00827228i
\(529\) 11980.4 0.984663
\(530\) 0 0
\(531\) 985.306 0.0805247
\(532\) 16522.0i 1.34646i
\(533\) − 1708.13i − 0.138813i
\(534\) 21003.5 1.70208
\(535\) 0 0
\(536\) 11474.6 0.924680
\(537\) − 9747.17i − 0.783280i
\(538\) − 21221.2i − 1.70058i
\(539\) −912.355 −0.0729089
\(540\) 0 0
\(541\) 5313.05 0.422229 0.211115 0.977461i \(-0.432291\pi\)
0.211115 + 0.977461i \(0.432291\pi\)
\(542\) − 659.084i − 0.0522326i
\(543\) 3511.34i 0.277506i
\(544\) 12690.7 1.00020
\(545\) 0 0
\(546\) 4454.68 0.349162
\(547\) 20685.1i 1.61688i 0.588581 + 0.808439i \(0.299687\pi\)
−0.588581 + 0.808439i \(0.700313\pi\)
\(548\) 21935.6i 1.70993i
\(549\) 806.541 0.0627000
\(550\) 0 0
\(551\) −1921.06 −0.148529
\(552\) 962.739i 0.0742335i
\(553\) 17154.7i 1.31915i
\(554\) 39527.3 3.03132
\(555\) 0 0
\(556\) 33311.2 2.54085
\(557\) − 10853.8i − 0.825659i −0.910808 0.412830i \(-0.864540\pi\)
0.910808 0.412830i \(-0.135460\pi\)
\(558\) 10049.3i 0.762402i
\(559\) −1906.41 −0.144244
\(560\) 0 0
\(561\) −2407.37 −0.181175
\(562\) − 13751.5i − 1.03216i
\(563\) 15381.2i 1.15141i 0.817658 + 0.575704i \(0.195272\pi\)
−0.817658 + 0.575704i \(0.804728\pi\)
\(564\) −20437.3 −1.52583
\(565\) 0 0
\(566\) −31515.6 −2.34046
\(567\) 1671.71i 0.123818i
\(568\) − 19672.9i − 1.45327i
\(569\) 1348.88 0.0993814 0.0496907 0.998765i \(-0.484176\pi\)
0.0496907 + 0.998765i \(0.484176\pi\)
\(570\) 0 0
\(571\) 3463.51 0.253841 0.126920 0.991913i \(-0.459491\pi\)
0.126920 + 0.991913i \(0.459491\pi\)
\(572\) − 2258.57i − 0.165097i
\(573\) − 8281.05i − 0.603745i
\(574\) 10344.7 0.752231
\(575\) 0 0
\(576\) −7412.97 −0.536239
\(577\) 12052.6i 0.869598i 0.900528 + 0.434799i \(0.143181\pi\)
−0.900528 + 0.434799i \(0.856819\pi\)
\(578\) − 1878.42i − 0.135176i
\(579\) −3751.83 −0.269293
\(580\) 0 0
\(581\) 28669.4 2.04717
\(582\) 5880.66i 0.418834i
\(583\) 5969.41i 0.424061i
\(584\) −8250.80 −0.584624
\(585\) 0 0
\(586\) 18607.8 1.31174
\(587\) − 11133.1i − 0.782813i −0.920218 0.391407i \(-0.871989\pi\)
0.920218 0.391407i \(-0.128011\pi\)
\(588\) 3262.75i 0.228833i
\(589\) 14836.0 1.03787
\(590\) 0 0
\(591\) 431.972 0.0300659
\(592\) − 198.908i − 0.0138092i
\(593\) 7939.69i 0.549821i 0.961470 + 0.274911i \(0.0886482\pi\)
−0.961470 + 0.274911i \(0.911352\pi\)
\(594\) 1364.67 0.0942646
\(595\) 0 0
\(596\) 40527.1 2.78533
\(597\) − 2283.75i − 0.156562i
\(598\) − 982.846i − 0.0672100i
\(599\) 19474.7 1.32840 0.664202 0.747553i \(-0.268771\pi\)
0.664202 + 0.747553i \(0.268771\pi\)
\(600\) 0 0
\(601\) −19946.1 −1.35377 −0.676887 0.736087i \(-0.736671\pi\)
−0.676887 + 0.736087i \(0.736671\pi\)
\(602\) − 11545.6i − 0.781664i
\(603\) − 4396.01i − 0.296881i
\(604\) −17816.7 −1.20025
\(605\) 0 0
\(606\) −1021.97 −0.0685061
\(607\) 1427.44i 0.0954496i 0.998861 + 0.0477248i \(0.0151970\pi\)
−0.998861 + 0.0477248i \(0.984803\pi\)
\(608\) 10620.6i 0.708427i
\(609\) −1948.23 −0.129633
\(610\) 0 0
\(611\) 8135.03 0.538638
\(612\) 8609.23i 0.568640i
\(613\) 8029.40i 0.529045i 0.964380 + 0.264522i \(0.0852143\pi\)
−0.964380 + 0.264522i \(0.914786\pi\)
\(614\) 43985.1 2.89103
\(615\) 0 0
\(616\) 5333.22 0.348834
\(617\) 20795.5i 1.35688i 0.734655 + 0.678440i \(0.237344\pi\)
−0.734655 + 0.678440i \(0.762656\pi\)
\(618\) − 956.533i − 0.0622612i
\(619\) −1677.43 −0.108920 −0.0544602 0.998516i \(-0.517344\pi\)
−0.0544602 + 0.998516i \(0.517344\pi\)
\(620\) 0 0
\(621\) 368.832 0.0238337
\(622\) 24795.8i 1.59842i
\(623\) − 31446.6i − 2.02228i
\(624\) −142.867 −0.00916547
\(625\) 0 0
\(626\) 44823.0 2.86180
\(627\) − 2014.69i − 0.128324i
\(628\) 13269.4i 0.843165i
\(629\) 4771.11 0.302443
\(630\) 0 0
\(631\) −25225.2 −1.59144 −0.795719 0.605666i \(-0.792907\pi\)
−0.795719 + 0.605666i \(0.792907\pi\)
\(632\) − 19526.8i − 1.22901i
\(633\) 11929.7i 0.749074i
\(634\) 20004.8 1.25315
\(635\) 0 0
\(636\) 21347.7 1.33096
\(637\) − 1298.73i − 0.0807812i
\(638\) 1590.41i 0.0986912i
\(639\) −7536.81 −0.466591
\(640\) 0 0
\(641\) 15165.3 0.934468 0.467234 0.884134i \(-0.345251\pi\)
0.467234 + 0.884134i \(0.345251\pi\)
\(642\) − 15737.9i − 0.967486i
\(643\) − 27156.1i − 1.66553i −0.553630 0.832763i \(-0.686758\pi\)
0.553630 0.832763i \(-0.313242\pi\)
\(644\) 3696.85 0.226206
\(645\) 0 0
\(646\) 20464.3 1.24637
\(647\) 29154.9i 1.77156i 0.464110 + 0.885778i \(0.346374\pi\)
−0.464110 + 0.885778i \(0.653626\pi\)
\(648\) − 1902.86i − 0.115357i
\(649\) −1204.26 −0.0728374
\(650\) 0 0
\(651\) 15045.9 0.905828
\(652\) − 36930.2i − 2.21825i
\(653\) 19141.7i 1.14713i 0.819161 + 0.573564i \(0.194440\pi\)
−0.819161 + 0.573564i \(0.805560\pi\)
\(654\) −30696.1 −1.83534
\(655\) 0 0
\(656\) −331.768 −0.0197460
\(657\) 3160.94i 0.187702i
\(658\) 49267.2i 2.91890i
\(659\) 24939.6 1.47422 0.737110 0.675773i \(-0.236190\pi\)
0.737110 + 0.675773i \(0.236190\pi\)
\(660\) 0 0
\(661\) 22617.7 1.33090 0.665452 0.746440i \(-0.268239\pi\)
0.665452 + 0.746440i \(0.268239\pi\)
\(662\) 24755.4i 1.45339i
\(663\) − 3426.88i − 0.200738i
\(664\) −32633.7 −1.90728
\(665\) 0 0
\(666\) −2704.61 −0.157359
\(667\) 429.843i 0.0249529i
\(668\) − 45224.3i − 2.61943i
\(669\) 2724.25 0.157438
\(670\) 0 0
\(671\) −985.772 −0.0567143
\(672\) 10770.9i 0.618298i
\(673\) 13855.8i 0.793615i 0.917902 + 0.396807i \(0.129882\pi\)
−0.917902 + 0.396807i \(0.870118\pi\)
\(674\) 20678.1 1.18174
\(675\) 0 0
\(676\) −25593.5 −1.45616
\(677\) − 24992.8i − 1.41884i −0.704787 0.709419i \(-0.748957\pi\)
0.704787 0.709419i \(-0.251043\pi\)
\(678\) − 23706.1i − 1.34282i
\(679\) 8804.57 0.497626
\(680\) 0 0
\(681\) 6186.46 0.348114
\(682\) − 12282.5i − 0.689619i
\(683\) 14420.5i 0.807887i 0.914784 + 0.403943i \(0.132361\pi\)
−0.914784 + 0.403943i \(0.867639\pi\)
\(684\) −7204.93 −0.402759
\(685\) 0 0
\(686\) −24661.4 −1.37256
\(687\) − 12232.4i − 0.679324i
\(688\) 370.280i 0.0205186i
\(689\) −8497.42 −0.469849
\(690\) 0 0
\(691\) 30552.4 1.68201 0.841005 0.541027i \(-0.181964\pi\)
0.841005 + 0.541027i \(0.181964\pi\)
\(692\) − 29999.9i − 1.64801i
\(693\) − 2043.20i − 0.111998i
\(694\) −27138.7 −1.48440
\(695\) 0 0
\(696\) 2217.63 0.120774
\(697\) − 7957.96i − 0.432467i
\(698\) − 16708.1i − 0.906032i
\(699\) 5048.27 0.273166
\(700\) 0 0
\(701\) −9151.47 −0.493076 −0.246538 0.969133i \(-0.579293\pi\)
−0.246538 + 0.969133i \(0.579293\pi\)
\(702\) 1942.60i 0.104443i
\(703\) 3992.86i 0.214216i
\(704\) 9060.30 0.485047
\(705\) 0 0
\(706\) 966.793 0.0515378
\(707\) 1530.10i 0.0813937i
\(708\) 4306.67i 0.228608i
\(709\) 6261.96 0.331697 0.165848 0.986151i \(-0.446964\pi\)
0.165848 + 0.986151i \(0.446964\pi\)
\(710\) 0 0
\(711\) −7480.85 −0.394590
\(712\) 35794.9i 1.88409i
\(713\) − 3319.60i − 0.174362i
\(714\) 20753.8 1.08780
\(715\) 0 0
\(716\) 42603.9 2.22372
\(717\) − 12074.9i − 0.628933i
\(718\) − 11485.7i − 0.596995i
\(719\) 18228.7 0.945500 0.472750 0.881197i \(-0.343261\pi\)
0.472750 + 0.881197i \(0.343261\pi\)
\(720\) 0 0
\(721\) −1432.13 −0.0739740
\(722\) − 14389.9i − 0.741740i
\(723\) − 8352.81i − 0.429660i
\(724\) −15347.7 −0.787836
\(725\) 0 0
\(726\) −1667.93 −0.0852655
\(727\) − 7233.66i − 0.369026i −0.982830 0.184513i \(-0.940929\pi\)
0.982830 0.184513i \(-0.0590707\pi\)
\(728\) 7591.81i 0.386499i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −8881.73 −0.449388
\(732\) 3525.31i 0.178004i
\(733\) 13444.8i 0.677485i 0.940879 + 0.338743i \(0.110002\pi\)
−0.940879 + 0.338743i \(0.889998\pi\)
\(734\) −26414.4 −1.32830
\(735\) 0 0
\(736\) 2376.41 0.119016
\(737\) 5372.90i 0.268539i
\(738\) 4511.14i 0.225010i
\(739\) −18490.9 −0.920432 −0.460216 0.887807i \(-0.652228\pi\)
−0.460216 + 0.887807i \(0.652228\pi\)
\(740\) 0 0
\(741\) 2867.90 0.142180
\(742\) − 51461.8i − 2.54612i
\(743\) 25160.9i 1.24235i 0.783674 + 0.621173i \(0.213343\pi\)
−0.783674 + 0.621173i \(0.786657\pi\)
\(744\) −17126.3 −0.843927
\(745\) 0 0
\(746\) 20528.8 1.00752
\(747\) 12502.2i 0.612358i
\(748\) − 10522.4i − 0.514354i
\(749\) −23562.9 −1.14949
\(750\) 0 0
\(751\) −13419.5 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(752\) − 1580.06i − 0.0766207i
\(753\) − 5482.81i − 0.265345i
\(754\) −2263.94 −0.109347
\(755\) 0 0
\(756\) −7306.86 −0.351518
\(757\) − 7014.90i − 0.336804i −0.985718 0.168402i \(-0.946139\pi\)
0.985718 0.168402i \(-0.0538607\pi\)
\(758\) − 35858.6i − 1.71826i
\(759\) −450.795 −0.0215584
\(760\) 0 0
\(761\) −30156.9 −1.43651 −0.718256 0.695779i \(-0.755059\pi\)
−0.718256 + 0.695779i \(0.755059\pi\)
\(762\) 22077.8i 1.04960i
\(763\) 45958.4i 2.18061i
\(764\) 36195.7 1.71402
\(765\) 0 0
\(766\) −51283.5 −2.41899
\(767\) − 1714.26i − 0.0807019i
\(768\) − 13217.4i − 0.621017i
\(769\) −11292.2 −0.529530 −0.264765 0.964313i \(-0.585294\pi\)
−0.264765 + 0.964313i \(0.585294\pi\)
\(770\) 0 0
\(771\) −1755.51 −0.0820016
\(772\) − 16398.9i − 0.764519i
\(773\) − 8524.10i − 0.396624i −0.980139 0.198312i \(-0.936454\pi\)
0.980139 0.198312i \(-0.0635460\pi\)
\(774\) 5034.80 0.233814
\(775\) 0 0
\(776\) −10022.0 −0.463621
\(777\) 4049.35i 0.186962i
\(778\) − 39011.4i − 1.79772i
\(779\) 6659.89 0.306310
\(780\) 0 0
\(781\) 9211.66 0.422047
\(782\) − 4578.96i − 0.209390i
\(783\) − 849.589i − 0.0387763i
\(784\) −252.251 −0.0114910
\(785\) 0 0
\(786\) −27626.1 −1.25368
\(787\) − 14983.9i − 0.678676i −0.940665 0.339338i \(-0.889797\pi\)
0.940665 0.339338i \(-0.110203\pi\)
\(788\) 1888.10i 0.0853565i
\(789\) −714.293 −0.0322300
\(790\) 0 0
\(791\) −35493.0 −1.59543
\(792\) 2325.72i 0.104345i
\(793\) − 1403.24i − 0.0628380i
\(794\) −27659.9 −1.23629
\(795\) 0 0
\(796\) 9982.04 0.444477
\(797\) 37172.3i 1.65208i 0.563610 + 0.826041i \(0.309412\pi\)
−0.563610 + 0.826041i \(0.690588\pi\)
\(798\) 17368.5i 0.770475i
\(799\) 37900.1 1.67811
\(800\) 0 0
\(801\) 13713.3 0.604913
\(802\) − 47781.0i − 2.10375i
\(803\) − 3863.37i − 0.169783i
\(804\) 19214.5 0.842841
\(805\) 0 0
\(806\) 17484.0 0.764080
\(807\) − 13855.4i − 0.604378i
\(808\) − 1741.68i − 0.0758316i
\(809\) −23797.1 −1.03419 −0.517096 0.855928i \(-0.672987\pi\)
−0.517096 + 0.855928i \(0.672987\pi\)
\(810\) 0 0
\(811\) 8988.35 0.389178 0.194589 0.980885i \(-0.437663\pi\)
0.194589 + 0.980885i \(0.437663\pi\)
\(812\) − 8515.54i − 0.368026i
\(813\) − 430.318i − 0.0185633i
\(814\) 3305.63 0.142337
\(815\) 0 0
\(816\) −665.600 −0.0285547
\(817\) − 7432.98i − 0.318295i
\(818\) 21750.6i 0.929696i
\(819\) 2908.48 0.124091
\(820\) 0 0
\(821\) −25156.8 −1.06940 −0.534702 0.845041i \(-0.679576\pi\)
−0.534702 + 0.845041i \(0.679576\pi\)
\(822\) 23059.5i 0.978459i
\(823\) − 1318.51i − 0.0558447i −0.999610 0.0279224i \(-0.991111\pi\)
0.999610 0.0279224i \(-0.00888912\pi\)
\(824\) 1630.16 0.0689189
\(825\) 0 0
\(826\) 10381.9 0.437326
\(827\) 124.982i 0.00525519i 0.999997 + 0.00262760i \(0.000836391\pi\)
−0.999997 + 0.00262760i \(0.999164\pi\)
\(828\) 1612.13i 0.0676635i
\(829\) 8886.80 0.372318 0.186159 0.982520i \(-0.440396\pi\)
0.186159 + 0.982520i \(0.440396\pi\)
\(830\) 0 0
\(831\) 25807.5 1.07732
\(832\) 12897.3i 0.537419i
\(833\) − 6050.63i − 0.251671i
\(834\) 35018.0 1.45393
\(835\) 0 0
\(836\) 8806.02 0.364309
\(837\) 6561.22i 0.270955i
\(838\) 37299.1i 1.53756i
\(839\) −2995.21 −0.123249 −0.0616247 0.998099i \(-0.519628\pi\)
−0.0616247 + 0.998099i \(0.519628\pi\)
\(840\) 0 0
\(841\) −23398.9 −0.959403
\(842\) − 43578.8i − 1.78364i
\(843\) − 8978.43i − 0.366825i
\(844\) −52143.6 −2.12661
\(845\) 0 0
\(846\) −21484.5 −0.873111
\(847\) 2497.24i 0.101306i
\(848\) 1650.44i 0.0668355i
\(849\) −20576.7 −0.831789
\(850\) 0 0
\(851\) 893.418 0.0359882
\(852\) − 32942.7i − 1.32464i
\(853\) − 18130.5i − 0.727757i −0.931446 0.363878i \(-0.881452\pi\)
0.931446 0.363878i \(-0.118548\pi\)
\(854\) 8498.27 0.340521
\(855\) 0 0
\(856\) 26821.1 1.07094
\(857\) 26394.1i 1.05205i 0.850470 + 0.526024i \(0.176318\pi\)
−0.850470 + 0.526024i \(0.823682\pi\)
\(858\) − 2374.29i − 0.0944720i
\(859\) 29456.2 1.17000 0.585002 0.811032i \(-0.301094\pi\)
0.585002 + 0.811032i \(0.301094\pi\)
\(860\) 0 0
\(861\) 6754.12 0.267340
\(862\) 42893.7i 1.69486i
\(863\) 762.616i 0.0300808i 0.999887 + 0.0150404i \(0.00478769\pi\)
−0.999887 + 0.0150404i \(0.995212\pi\)
\(864\) −4696.99 −0.184948
\(865\) 0 0
\(866\) −13706.6 −0.537838
\(867\) − 1226.43i − 0.0480411i
\(868\) 65764.0i 2.57163i
\(869\) 9143.26 0.356920
\(870\) 0 0
\(871\) −7648.29 −0.297535
\(872\) − 52313.3i − 2.03160i
\(873\) 3839.51i 0.148852i
\(874\) 3832.06 0.148308
\(875\) 0 0
\(876\) −13816.2 −0.532882
\(877\) 44767.2i 1.72369i 0.507168 + 0.861847i \(0.330692\pi\)
−0.507168 + 0.861847i \(0.669308\pi\)
\(878\) 24041.8i 0.924112i
\(879\) 12149.1 0.466188
\(880\) 0 0
\(881\) −32057.9 −1.22595 −0.612973 0.790104i \(-0.710027\pi\)
−0.612973 + 0.790104i \(0.710027\pi\)
\(882\) 3429.93i 0.130943i
\(883\) − 7078.95i − 0.269791i −0.990860 0.134896i \(-0.956930\pi\)
0.990860 0.134896i \(-0.0430699\pi\)
\(884\) 14978.6 0.569891
\(885\) 0 0
\(886\) 34542.8 1.30981
\(887\) − 25148.1i − 0.951964i −0.879455 0.475982i \(-0.842093\pi\)
0.879455 0.475982i \(-0.157907\pi\)
\(888\) − 4609.28i − 0.174186i
\(889\) 33055.0 1.24705
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 11907.4i 0.446963i
\(893\) 31718.0i 1.18858i
\(894\) 42603.6 1.59382
\(895\) 0 0
\(896\) −49385.8 −1.84137
\(897\) − 641.704i − 0.0238862i
\(898\) − 73843.6i − 2.74409i
\(899\) −7646.56 −0.283679
\(900\) 0 0
\(901\) −39588.4 −1.46380
\(902\) − 5513.62i − 0.203529i
\(903\) − 7538.14i − 0.277800i
\(904\) 40400.8 1.48641
\(905\) 0 0
\(906\) −18729.6 −0.686809
\(907\) 1269.76i 0.0464848i 0.999730 + 0.0232424i \(0.00739895\pi\)
−0.999730 + 0.0232424i \(0.992601\pi\)
\(908\) 27040.4i 0.988289i
\(909\) −667.248 −0.0243468
\(910\) 0 0
\(911\) −33783.1 −1.22863 −0.614316 0.789060i \(-0.710568\pi\)
−0.614316 + 0.789060i \(0.710568\pi\)
\(912\) − 557.030i − 0.0202249i
\(913\) − 15280.5i − 0.553899i
\(914\) −44655.1 −1.61604
\(915\) 0 0
\(916\) 53466.6 1.92859
\(917\) 41361.9i 1.48952i
\(918\) 9050.35i 0.325388i
\(919\) −39262.5 −1.40930 −0.704652 0.709553i \(-0.748897\pi\)
−0.704652 + 0.709553i \(0.748897\pi\)
\(920\) 0 0
\(921\) 28718.1 1.02746
\(922\) − 66800.1i − 2.38606i
\(923\) 13112.7i 0.467618i
\(924\) 8930.61 0.317960
\(925\) 0 0
\(926\) −45848.3 −1.62707
\(927\) − 624.525i − 0.0221274i
\(928\) − 5473.95i − 0.193633i
\(929\) 21175.0 0.747825 0.373913 0.927464i \(-0.378016\pi\)
0.373913 + 0.927464i \(0.378016\pi\)
\(930\) 0 0
\(931\) 5063.68 0.178255
\(932\) 22065.5i 0.775515i
\(933\) 16189.3i 0.568073i
\(934\) −69787.5 −2.44488
\(935\) 0 0
\(936\) −3310.65 −0.115611
\(937\) − 5135.11i − 0.179036i −0.995985 0.0895180i \(-0.971467\pi\)
0.995985 0.0895180i \(-0.0285327\pi\)
\(938\) − 46319.4i − 1.61235i
\(939\) 29265.1 1.01707
\(940\) 0 0
\(941\) −9702.77 −0.336133 −0.168067 0.985776i \(-0.553752\pi\)
−0.168067 + 0.985776i \(0.553752\pi\)
\(942\) 13949.3i 0.482477i
\(943\) − 1490.18i − 0.0514600i
\(944\) −332.959 −0.0114798
\(945\) 0 0
\(946\) −6153.65 −0.211493
\(947\) − 699.579i − 0.0240055i −0.999928 0.0120028i \(-0.996179\pi\)
0.999928 0.0120028i \(-0.00382069\pi\)
\(948\) − 32698.0i − 1.12024i
\(949\) 5499.49 0.188115
\(950\) 0 0
\(951\) 13061.2 0.445363
\(952\) 35369.3i 1.20412i
\(953\) − 42039.3i − 1.42895i −0.699663 0.714473i \(-0.746667\pi\)
0.699663 0.714473i \(-0.253333\pi\)
\(954\) 22441.6 0.761606
\(955\) 0 0
\(956\) 52778.1 1.78553
\(957\) 1038.39i 0.0350745i
\(958\) − 52063.4i − 1.75584i
\(959\) 34524.9 1.16253
\(960\) 0 0
\(961\) 29262.0 0.982243
\(962\) 4705.55i 0.157706i
\(963\) − 10275.3i − 0.343841i
\(964\) 36509.3 1.21980
\(965\) 0 0
\(966\) 3886.27 0.129440
\(967\) − 32794.8i − 1.09060i −0.838242 0.545299i \(-0.816416\pi\)
0.838242 0.545299i \(-0.183584\pi\)
\(968\) − 2842.55i − 0.0943832i
\(969\) 13361.2 0.442955
\(970\) 0 0
\(971\) −3322.53 −0.109810 −0.0549048 0.998492i \(-0.517486\pi\)
−0.0549048 + 0.998492i \(0.517486\pi\)
\(972\) − 3186.39i − 0.105148i
\(973\) − 52429.3i − 1.72745i
\(974\) −87701.4 −2.88515
\(975\) 0 0
\(976\) −272.550 −0.00893864
\(977\) 22192.5i 0.726716i 0.931650 + 0.363358i \(0.118370\pi\)
−0.931650 + 0.363358i \(0.881630\pi\)
\(978\) − 38822.4i − 1.26933i
\(979\) −16760.7 −0.547164
\(980\) 0 0
\(981\) −20041.6 −0.652272
\(982\) − 37277.4i − 1.21137i
\(983\) − 7383.09i − 0.239556i −0.992801 0.119778i \(-0.961782\pi\)
0.992801 0.119778i \(-0.0382183\pi\)
\(984\) −7688.04 −0.249071
\(985\) 0 0
\(986\) −10547.4 −0.340668
\(987\) 32166.7i 1.03736i
\(988\) 12535.3i 0.403645i
\(989\) −1663.16 −0.0534735
\(990\) 0 0
\(991\) 46260.8 1.48287 0.741434 0.671026i \(-0.234146\pi\)
0.741434 + 0.671026i \(0.234146\pi\)
\(992\) 42274.3i 1.35304i
\(993\) 16162.9i 0.516530i
\(994\) −79413.1 −2.53403
\(995\) 0 0
\(996\) −54645.9 −1.73847
\(997\) 41196.8i 1.30864i 0.756217 + 0.654320i \(0.227045\pi\)
−0.756217 + 0.654320i \(0.772955\pi\)
\(998\) − 84219.5i − 2.67127i
\(999\) −1765.85 −0.0559249
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.m.199.6 6
5.2 odd 4 825.4.a.p.1.1 3
5.3 odd 4 165.4.a.g.1.3 3
5.4 even 2 inner 825.4.c.m.199.1 6
15.2 even 4 2475.4.a.z.1.3 3
15.8 even 4 495.4.a.i.1.1 3
55.43 even 4 1815.4.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 5.3 odd 4
495.4.a.i.1.1 3 15.8 even 4
825.4.a.p.1.1 3 5.2 odd 4
825.4.c.m.199.1 6 5.4 even 2 inner
825.4.c.m.199.6 6 1.1 even 1 trivial
1815.4.a.q.1.1 3 55.43 even 4
2475.4.a.z.1.3 3 15.2 even 4