Properties

Label 245.4.a.l.1.2
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
Dimension $3$
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] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(14.4554679514\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.14360.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.861086\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.86109 q^{2} -9.53636 q^{3} -4.53636 q^{4} -5.00000 q^{5} +17.7480 q^{6} +23.3312 q^{8} +63.9421 q^{9} +O(q^{10})\) \(q-1.86109 q^{2} -9.53636 q^{3} -4.53636 q^{4} -5.00000 q^{5} +17.7480 q^{6} +23.3312 q^{8} +63.9421 q^{9} +9.30543 q^{10} -36.9807 q^{11} +43.2603 q^{12} +22.7931 q^{13} +47.6818 q^{15} -7.13061 q^{16} +135.566 q^{17} -119.002 q^{18} -6.22620 q^{19} +22.6818 q^{20} +68.8243 q^{22} -48.7397 q^{23} -222.495 q^{24} +25.0000 q^{25} -42.4199 q^{26} -352.293 q^{27} -71.1172 q^{29} -88.7399 q^{30} -124.924 q^{31} -173.379 q^{32} +352.661 q^{33} -252.300 q^{34} -290.064 q^{36} +84.9919 q^{37} +11.5875 q^{38} -217.363 q^{39} -116.656 q^{40} -92.5942 q^{41} +299.680 q^{43} +167.758 q^{44} -319.711 q^{45} +90.7088 q^{46} +72.9178 q^{47} +68.0000 q^{48} -46.5272 q^{50} -1292.81 q^{51} -103.398 q^{52} +362.685 q^{53} +655.648 q^{54} +184.904 q^{55} +59.3753 q^{57} +132.355 q^{58} +375.526 q^{59} -216.302 q^{60} -689.610 q^{61} +232.494 q^{62} +379.719 q^{64} -113.965 q^{65} -656.333 q^{66} -972.591 q^{67} -614.976 q^{68} +464.799 q^{69} +281.900 q^{71} +1491.85 q^{72} +742.980 q^{73} -158.177 q^{74} -238.409 q^{75} +28.2443 q^{76} +404.532 q^{78} +592.843 q^{79} +35.6530 q^{80} +1633.16 q^{81} +172.326 q^{82} +493.406 q^{83} -677.830 q^{85} -557.731 q^{86} +678.199 q^{87} -862.806 q^{88} -962.977 q^{89} +595.009 q^{90} +221.101 q^{92} +1191.32 q^{93} -135.706 q^{94} +31.1310 q^{95} +1653.41 q^{96} -740.748 q^{97} -2364.62 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9} + 15 q^{10} - 74 q^{11} + 152 q^{12} - 44 q^{13} + 10 q^{15} - 79 q^{16} + 52 q^{17} - 411 q^{18} - 168 q^{19} - 65 q^{20} + 184 q^{22} - 124 q^{23} - 420 q^{24} + 75 q^{25} + 446 q^{26} - 170 q^{27} + 332 q^{29} + 120 q^{30} - 320 q^{31} - 183 q^{32} + 106 q^{33} - 582 q^{34} + 181 q^{36} - 54 q^{37} + 460 q^{38} - 982 q^{39} + 75 q^{40} - 362 q^{41} - 16 q^{43} - 264 q^{44} - 405 q^{45} - 336 q^{46} + 730 q^{47} + 204 q^{48} - 75 q^{50} - 1178 q^{51} - 1202 q^{52} + 110 q^{53} + 180 q^{54} + 370 q^{55} - 956 q^{57} + 450 q^{58} + 180 q^{59} - 760 q^{60} - 1222 q^{61} - 464 q^{62} - 391 q^{64} + 220 q^{65} + 532 q^{66} + 204 q^{67} - 918 q^{68} + 716 q^{69} - 136 q^{71} + 765 q^{72} - 310 q^{73} + 502 q^{74} - 50 q^{75} - 1796 q^{76} + 3788 q^{78} - 1034 q^{79} + 395 q^{80} + 2283 q^{81} + 6 q^{82} + 1660 q^{83} - 260 q^{85} + 764 q^{86} + 1574 q^{87} - 20 q^{88} - 242 q^{89} + 2055 q^{90} + 96 q^{92} + 1376 q^{93} + 1108 q^{94} + 840 q^{95} + 3156 q^{96} - 100 q^{97} - 3488 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.86109 −0.657993 −0.328997 0.944331i \(-0.606711\pi\)
−0.328997 + 0.944331i \(0.606711\pi\)
\(3\) −9.53636 −1.83527 −0.917636 0.397421i \(-0.869905\pi\)
−0.917636 + 0.397421i \(0.869905\pi\)
\(4\) −4.53636 −0.567045
\(5\) −5.00000 −0.447214
\(6\) 17.7480 1.20760
\(7\) 0 0
\(8\) 23.3312 1.03111
\(9\) 63.9421 2.36823
\(10\) 9.30543 0.294264
\(11\) −36.9807 −1.01365 −0.506823 0.862050i \(-0.669180\pi\)
−0.506823 + 0.862050i \(0.669180\pi\)
\(12\) 43.2603 1.04068
\(13\) 22.7931 0.486282 0.243141 0.969991i \(-0.421822\pi\)
0.243141 + 0.969991i \(0.421822\pi\)
\(14\) 0 0
\(15\) 47.6818 0.820759
\(16\) −7.13061 −0.111416
\(17\) 135.566 1.93409 0.967047 0.254599i \(-0.0819436\pi\)
0.967047 + 0.254599i \(0.0819436\pi\)
\(18\) −119.002 −1.55828
\(19\) −6.22620 −0.0751784 −0.0375892 0.999293i \(-0.511968\pi\)
−0.0375892 + 0.999293i \(0.511968\pi\)
\(20\) 22.6818 0.253590
\(21\) 0 0
\(22\) 68.8243 0.666972
\(23\) −48.7397 −0.441866 −0.220933 0.975289i \(-0.570910\pi\)
−0.220933 + 0.975289i \(0.570910\pi\)
\(24\) −222.495 −1.89236
\(25\) 25.0000 0.200000
\(26\) −42.4199 −0.319970
\(27\) −352.293 −2.51107
\(28\) 0 0
\(29\) −71.1172 −0.455384 −0.227692 0.973733i \(-0.573118\pi\)
−0.227692 + 0.973733i \(0.573118\pi\)
\(30\) −88.7399 −0.540054
\(31\) −124.924 −0.723773 −0.361886 0.932222i \(-0.617867\pi\)
−0.361886 + 0.932222i \(0.617867\pi\)
\(32\) −173.379 −0.957794
\(33\) 352.661 1.86032
\(34\) −252.300 −1.27262
\(35\) 0 0
\(36\) −290.064 −1.34289
\(37\) 84.9919 0.377638 0.188819 0.982012i \(-0.439534\pi\)
0.188819 + 0.982012i \(0.439534\pi\)
\(38\) 11.5875 0.0494669
\(39\) −217.363 −0.892460
\(40\) −116.656 −0.461124
\(41\) −92.5942 −0.352702 −0.176351 0.984327i \(-0.556429\pi\)
−0.176351 + 0.984327i \(0.556429\pi\)
\(42\) 0 0
\(43\) 299.680 1.06281 0.531405 0.847118i \(-0.321664\pi\)
0.531405 + 0.847118i \(0.321664\pi\)
\(44\) 167.758 0.574782
\(45\) −319.711 −1.05910
\(46\) 90.7088 0.290745
\(47\) 72.9178 0.226301 0.113151 0.993578i \(-0.463906\pi\)
0.113151 + 0.993578i \(0.463906\pi\)
\(48\) 68.0000 0.204478
\(49\) 0 0
\(50\) −46.5272 −0.131599
\(51\) −1292.81 −3.54959
\(52\) −103.398 −0.275744
\(53\) 362.685 0.939974 0.469987 0.882673i \(-0.344259\pi\)
0.469987 + 0.882673i \(0.344259\pi\)
\(54\) 655.648 1.65227
\(55\) 184.904 0.453316
\(56\) 0 0
\(57\) 59.3753 0.137973
\(58\) 132.355 0.299640
\(59\) 375.526 0.828633 0.414316 0.910133i \(-0.364021\pi\)
0.414316 + 0.910133i \(0.364021\pi\)
\(60\) −216.302 −0.465407
\(61\) −689.610 −1.44747 −0.723734 0.690079i \(-0.757576\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(62\) 232.494 0.476238
\(63\) 0 0
\(64\) 379.719 0.741638
\(65\) −113.965 −0.217472
\(66\) −656.333 −1.22408
\(67\) −972.591 −1.77345 −0.886723 0.462301i \(-0.847024\pi\)
−0.886723 + 0.462301i \(0.847024\pi\)
\(68\) −614.976 −1.09672
\(69\) 464.799 0.810945
\(70\) 0 0
\(71\) 281.900 0.471202 0.235601 0.971850i \(-0.424294\pi\)
0.235601 + 0.971850i \(0.424294\pi\)
\(72\) 1491.85 2.44189
\(73\) 742.980 1.19122 0.595612 0.803273i \(-0.296910\pi\)
0.595612 + 0.803273i \(0.296910\pi\)
\(74\) −158.177 −0.248483
\(75\) −238.409 −0.367055
\(76\) 28.2443 0.0426295
\(77\) 0 0
\(78\) 404.532 0.587233
\(79\) 592.843 0.844304 0.422152 0.906525i \(-0.361275\pi\)
0.422152 + 0.906525i \(0.361275\pi\)
\(80\) 35.6530 0.0498266
\(81\) 1633.16 2.24027
\(82\) 172.326 0.232076
\(83\) 493.406 0.652510 0.326255 0.945282i \(-0.394213\pi\)
0.326255 + 0.945282i \(0.394213\pi\)
\(84\) 0 0
\(85\) −677.830 −0.864953
\(86\) −557.731 −0.699322
\(87\) 678.199 0.835753
\(88\) −862.806 −1.04518
\(89\) −962.977 −1.14691 −0.573457 0.819236i \(-0.694398\pi\)
−0.573457 + 0.819236i \(0.694398\pi\)
\(90\) 595.009 0.696883
\(91\) 0 0
\(92\) 221.101 0.250558
\(93\) 1191.32 1.32832
\(94\) −135.706 −0.148905
\(95\) 31.1310 0.0336208
\(96\) 1653.41 1.75781
\(97\) −740.748 −0.775377 −0.387689 0.921790i \(-0.626726\pi\)
−0.387689 + 0.921790i \(0.626726\pi\)
\(98\) 0 0
\(99\) −2364.62 −2.40054
\(100\) −113.409 −0.113409
\(101\) −613.794 −0.604701 −0.302351 0.953197i \(-0.597771\pi\)
−0.302351 + 0.953197i \(0.597771\pi\)
\(102\) 2406.02 2.33561
\(103\) −805.493 −0.770559 −0.385280 0.922800i \(-0.625895\pi\)
−0.385280 + 0.922800i \(0.625895\pi\)
\(104\) 531.791 0.501408
\(105\) 0 0
\(106\) −674.988 −0.618497
\(107\) −1931.30 −1.74491 −0.872457 0.488691i \(-0.837474\pi\)
−0.872457 + 0.488691i \(0.837474\pi\)
\(108\) 1598.13 1.42389
\(109\) 106.462 0.0935522 0.0467761 0.998905i \(-0.485105\pi\)
0.0467761 + 0.998905i \(0.485105\pi\)
\(110\) −344.121 −0.298279
\(111\) −810.514 −0.693068
\(112\) 0 0
\(113\) 309.076 0.257305 0.128652 0.991690i \(-0.458935\pi\)
0.128652 + 0.991690i \(0.458935\pi\)
\(114\) −110.503 −0.0907852
\(115\) 243.698 0.197609
\(116\) 322.613 0.258223
\(117\) 1457.44 1.15163
\(118\) −698.887 −0.545235
\(119\) 0 0
\(120\) 1112.48 0.846289
\(121\) 36.5724 0.0274774
\(122\) 1283.42 0.952425
\(123\) 883.012 0.647305
\(124\) 566.699 0.410412
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −199.470 −0.139371 −0.0696856 0.997569i \(-0.522200\pi\)
−0.0696856 + 0.997569i \(0.522200\pi\)
\(128\) 680.345 0.469801
\(129\) −2857.86 −1.95055
\(130\) 212.100 0.143095
\(131\) −601.722 −0.401318 −0.200659 0.979661i \(-0.564308\pi\)
−0.200659 + 0.979661i \(0.564308\pi\)
\(132\) −1599.80 −1.05488
\(133\) 0 0
\(134\) 1810.08 1.16692
\(135\) 1761.47 1.12298
\(136\) 3162.92 1.99425
\(137\) 2092.21 1.30474 0.652370 0.757901i \(-0.273775\pi\)
0.652370 + 0.757901i \(0.273775\pi\)
\(138\) −865.031 −0.533597
\(139\) −834.466 −0.509198 −0.254599 0.967047i \(-0.581943\pi\)
−0.254599 + 0.967047i \(0.581943\pi\)
\(140\) 0 0
\(141\) −695.370 −0.415324
\(142\) −524.639 −0.310048
\(143\) −842.905 −0.492918
\(144\) −455.946 −0.263858
\(145\) 355.586 0.203654
\(146\) −1382.75 −0.783817
\(147\) 0 0
\(148\) −385.554 −0.214137
\(149\) −244.258 −0.134298 −0.0671491 0.997743i \(-0.521390\pi\)
−0.0671491 + 0.997743i \(0.521390\pi\)
\(150\) 443.700 0.241519
\(151\) −802.158 −0.432309 −0.216155 0.976359i \(-0.569352\pi\)
−0.216155 + 0.976359i \(0.569352\pi\)
\(152\) −145.265 −0.0775168
\(153\) 8668.38 4.58037
\(154\) 0 0
\(155\) 624.618 0.323681
\(156\) 986.037 0.506065
\(157\) −3541.38 −1.80021 −0.900105 0.435673i \(-0.856510\pi\)
−0.900105 + 0.435673i \(0.856510\pi\)
\(158\) −1103.33 −0.555546
\(159\) −3458.69 −1.72511
\(160\) 866.896 0.428339
\(161\) 0 0
\(162\) −3039.45 −1.47408
\(163\) −2214.57 −1.06416 −0.532081 0.846693i \(-0.678590\pi\)
−0.532081 + 0.846693i \(0.678590\pi\)
\(164\) 420.041 0.199998
\(165\) −1763.31 −0.831959
\(166\) −918.271 −0.429347
\(167\) 2617.07 1.21266 0.606332 0.795212i \(-0.292640\pi\)
0.606332 + 0.795212i \(0.292640\pi\)
\(168\) 0 0
\(169\) −1677.47 −0.763530
\(170\) 1261.50 0.569133
\(171\) −398.117 −0.178039
\(172\) −1359.46 −0.602661
\(173\) 1634.04 0.718114 0.359057 0.933316i \(-0.383098\pi\)
0.359057 + 0.933316i \(0.383098\pi\)
\(174\) −1262.19 −0.549920
\(175\) 0 0
\(176\) 263.695 0.112936
\(177\) −3581.15 −1.52077
\(178\) 1792.18 0.754662
\(179\) 969.160 0.404684 0.202342 0.979315i \(-0.435145\pi\)
0.202342 + 0.979315i \(0.435145\pi\)
\(180\) 1450.32 0.600559
\(181\) −4358.20 −1.78974 −0.894869 0.446329i \(-0.852731\pi\)
−0.894869 + 0.446329i \(0.852731\pi\)
\(182\) 0 0
\(183\) 6576.37 2.65650
\(184\) −1137.16 −0.455611
\(185\) −424.960 −0.168885
\(186\) −2217.14 −0.874026
\(187\) −5013.33 −1.96049
\(188\) −330.781 −0.128323
\(189\) 0 0
\(190\) −57.9375 −0.0221223
\(191\) 2909.97 1.10240 0.551199 0.834374i \(-0.314170\pi\)
0.551199 + 0.834374i \(0.314170\pi\)
\(192\) −3621.13 −1.36111
\(193\) −3719.25 −1.38714 −0.693568 0.720391i \(-0.743962\pi\)
−0.693568 + 0.720391i \(0.743962\pi\)
\(194\) 1378.60 0.510193
\(195\) 1086.82 0.399120
\(196\) 0 0
\(197\) −1550.03 −0.560582 −0.280291 0.959915i \(-0.590431\pi\)
−0.280291 + 0.959915i \(0.590431\pi\)
\(198\) 4400.77 1.57954
\(199\) 3605.14 1.28423 0.642115 0.766608i \(-0.278057\pi\)
0.642115 + 0.766608i \(0.278057\pi\)
\(200\) 583.281 0.206221
\(201\) 9274.97 3.25476
\(202\) 1142.32 0.397889
\(203\) 0 0
\(204\) 5864.63 2.01278
\(205\) 462.971 0.157733
\(206\) 1499.09 0.507023
\(207\) −3116.52 −1.04644
\(208\) −162.529 −0.0541795
\(209\) 230.249 0.0762042
\(210\) 0 0
\(211\) −3305.27 −1.07841 −0.539204 0.842175i \(-0.681275\pi\)
−0.539204 + 0.842175i \(0.681275\pi\)
\(212\) −1645.27 −0.533007
\(213\) −2688.29 −0.864784
\(214\) 3594.31 1.14814
\(215\) −1498.40 −0.475303
\(216\) −8219.44 −2.58918
\(217\) 0 0
\(218\) −198.135 −0.0615567
\(219\) −7085.33 −2.18622
\(220\) −838.788 −0.257050
\(221\) 3089.97 0.940515
\(222\) 1508.44 0.456034
\(223\) −3451.37 −1.03642 −0.518209 0.855254i \(-0.673401\pi\)
−0.518209 + 0.855254i \(0.673401\pi\)
\(224\) 0 0
\(225\) 1598.55 0.473645
\(226\) −575.218 −0.169305
\(227\) −2047.24 −0.598591 −0.299296 0.954160i \(-0.596752\pi\)
−0.299296 + 0.954160i \(0.596752\pi\)
\(228\) −269.348 −0.0782367
\(229\) 1387.42 0.400362 0.200181 0.979759i \(-0.435847\pi\)
0.200181 + 0.979759i \(0.435847\pi\)
\(230\) −453.544 −0.130025
\(231\) 0 0
\(232\) −1659.25 −0.469549
\(233\) −374.993 −0.105436 −0.0527181 0.998609i \(-0.516788\pi\)
−0.0527181 + 0.998609i \(0.516788\pi\)
\(234\) −2712.42 −0.757762
\(235\) −364.589 −0.101205
\(236\) −1703.52 −0.469872
\(237\) −5653.56 −1.54953
\(238\) 0 0
\(239\) −5560.93 −1.50505 −0.752525 0.658564i \(-0.771164\pi\)
−0.752525 + 0.658564i \(0.771164\pi\)
\(240\) −340.000 −0.0914454
\(241\) −4706.19 −1.25789 −0.628947 0.777448i \(-0.716514\pi\)
−0.628947 + 0.777448i \(0.716514\pi\)
\(242\) −68.0644 −0.0180799
\(243\) −6062.45 −1.60044
\(244\) 3128.32 0.820779
\(245\) 0 0
\(246\) −1643.36 −0.425922
\(247\) −141.914 −0.0365579
\(248\) −2914.63 −0.746286
\(249\) −4705.29 −1.19753
\(250\) 232.636 0.0588527
\(251\) −589.085 −0.148138 −0.0740692 0.997253i \(-0.523599\pi\)
−0.0740692 + 0.997253i \(0.523599\pi\)
\(252\) 0 0
\(253\) 1802.43 0.447896
\(254\) 371.232 0.0917053
\(255\) 6464.03 1.58742
\(256\) −4303.93 −1.05076
\(257\) 4666.16 1.13256 0.566278 0.824214i \(-0.308383\pi\)
0.566278 + 0.824214i \(0.308383\pi\)
\(258\) 5318.72 1.28345
\(259\) 0 0
\(260\) 516.988 0.123316
\(261\) −4547.38 −1.07845
\(262\) 1119.86 0.264065
\(263\) −4471.96 −1.04849 −0.524246 0.851567i \(-0.675653\pi\)
−0.524246 + 0.851567i \(0.675653\pi\)
\(264\) 8228.02 1.91818
\(265\) −1813.42 −0.420369
\(266\) 0 0
\(267\) 9183.29 2.10490
\(268\) 4412.02 1.00562
\(269\) 4257.13 0.964913 0.482457 0.875920i \(-0.339745\pi\)
0.482457 + 0.875920i \(0.339745\pi\)
\(270\) −3278.24 −0.738916
\(271\) 3868.69 0.867182 0.433591 0.901110i \(-0.357246\pi\)
0.433591 + 0.901110i \(0.357246\pi\)
\(272\) −966.668 −0.215488
\(273\) 0 0
\(274\) −3893.78 −0.858510
\(275\) −924.518 −0.202729
\(276\) −2108.49 −0.459842
\(277\) 8207.30 1.78025 0.890124 0.455718i \(-0.150618\pi\)
0.890124 + 0.455718i \(0.150618\pi\)
\(278\) 1553.01 0.335049
\(279\) −7987.88 −1.71406
\(280\) 0 0
\(281\) 6471.27 1.37382 0.686910 0.726743i \(-0.258967\pi\)
0.686910 + 0.726743i \(0.258967\pi\)
\(282\) 1294.14 0.273281
\(283\) −1470.80 −0.308940 −0.154470 0.987997i \(-0.549367\pi\)
−0.154470 + 0.987997i \(0.549367\pi\)
\(284\) −1278.80 −0.267192
\(285\) −296.876 −0.0617033
\(286\) 1568.72 0.324337
\(287\) 0 0
\(288\) −11086.2 −2.26827
\(289\) 13465.1 2.74072
\(290\) −661.776 −0.134003
\(291\) 7064.04 1.42303
\(292\) −3370.42 −0.675477
\(293\) 5489.18 1.09448 0.547238 0.836977i \(-0.315679\pi\)
0.547238 + 0.836977i \(0.315679\pi\)
\(294\) 0 0
\(295\) −1877.63 −0.370576
\(296\) 1982.97 0.389384
\(297\) 13028.0 2.54533
\(298\) 454.586 0.0883673
\(299\) −1110.93 −0.214872
\(300\) 1081.51 0.208136
\(301\) 0 0
\(302\) 1492.89 0.284457
\(303\) 5853.36 1.10979
\(304\) 44.3966 0.00837605
\(305\) 3448.05 0.647328
\(306\) −16132.6 −3.01385
\(307\) −1035.35 −0.192477 −0.0962383 0.995358i \(-0.530681\pi\)
−0.0962383 + 0.995358i \(0.530681\pi\)
\(308\) 0 0
\(309\) 7681.47 1.41419
\(310\) −1162.47 −0.212980
\(311\) −2544.04 −0.463856 −0.231928 0.972733i \(-0.574503\pi\)
−0.231928 + 0.972733i \(0.574503\pi\)
\(312\) −5071.35 −0.920220
\(313\) −2599.72 −0.469473 −0.234737 0.972059i \(-0.575423\pi\)
−0.234737 + 0.972059i \(0.575423\pi\)
\(314\) 6590.82 1.18453
\(315\) 0 0
\(316\) −2689.35 −0.478758
\(317\) −2725.13 −0.482835 −0.241417 0.970421i \(-0.577612\pi\)
−0.241417 + 0.970421i \(0.577612\pi\)
\(318\) 6436.93 1.13511
\(319\) 2629.96 0.461598
\(320\) −1898.59 −0.331671
\(321\) 18417.6 3.20239
\(322\) 0 0
\(323\) −844.061 −0.145402
\(324\) −7408.58 −1.27033
\(325\) 569.827 0.0972564
\(326\) 4121.50 0.700212
\(327\) −1015.26 −0.171694
\(328\) −2160.34 −0.363673
\(329\) 0 0
\(330\) 3281.66 0.547423
\(331\) −5178.12 −0.859865 −0.429933 0.902861i \(-0.641463\pi\)
−0.429933 + 0.902861i \(0.641463\pi\)
\(332\) −2238.26 −0.370002
\(333\) 5434.56 0.894331
\(334\) −4870.59 −0.797924
\(335\) 4862.95 0.793109
\(336\) 0 0
\(337\) −3656.07 −0.590975 −0.295488 0.955347i \(-0.595482\pi\)
−0.295488 + 0.955347i \(0.595482\pi\)
\(338\) 3121.93 0.502398
\(339\) −2947.46 −0.472225
\(340\) 3074.88 0.490467
\(341\) 4619.77 0.733649
\(342\) 740.929 0.117149
\(343\) 0 0
\(344\) 6991.92 1.09587
\(345\) −2323.99 −0.362666
\(346\) −3041.09 −0.472514
\(347\) −1673.15 −0.258846 −0.129423 0.991589i \(-0.541312\pi\)
−0.129423 + 0.991589i \(0.541312\pi\)
\(348\) −3076.55 −0.473910
\(349\) −777.313 −0.119222 −0.0596112 0.998222i \(-0.518986\pi\)
−0.0596112 + 0.998222i \(0.518986\pi\)
\(350\) 0 0
\(351\) −8029.85 −1.22109
\(352\) 6411.69 0.970864
\(353\) −4422.92 −0.666879 −0.333439 0.942772i \(-0.608209\pi\)
−0.333439 + 0.942772i \(0.608209\pi\)
\(354\) 6664.83 1.00065
\(355\) −1409.50 −0.210728
\(356\) 4368.41 0.650351
\(357\) 0 0
\(358\) −1803.69 −0.266279
\(359\) −962.163 −0.141451 −0.0707256 0.997496i \(-0.522531\pi\)
−0.0707256 + 0.997496i \(0.522531\pi\)
\(360\) −7459.24 −1.09205
\(361\) −6820.23 −0.994348
\(362\) 8110.99 1.17764
\(363\) −348.767 −0.0504285
\(364\) 0 0
\(365\) −3714.90 −0.532731
\(366\) −12239.2 −1.74796
\(367\) 7282.68 1.03584 0.517919 0.855430i \(-0.326707\pi\)
0.517919 + 0.855430i \(0.326707\pi\)
\(368\) 347.543 0.0492308
\(369\) −5920.67 −0.835279
\(370\) 790.887 0.111125
\(371\) 0 0
\(372\) −5404.24 −0.753217
\(373\) −1149.24 −0.159532 −0.0797658 0.996814i \(-0.525417\pi\)
−0.0797658 + 0.996814i \(0.525417\pi\)
\(374\) 9330.23 1.28999
\(375\) 1192.04 0.164152
\(376\) 1701.26 0.233340
\(377\) −1620.98 −0.221445
\(378\) 0 0
\(379\) −10452.6 −1.41666 −0.708332 0.705879i \(-0.750552\pi\)
−0.708332 + 0.705879i \(0.750552\pi\)
\(380\) −141.221 −0.0190645
\(381\) 1902.22 0.255784
\(382\) −5415.71 −0.725371
\(383\) 13469.3 1.79699 0.898496 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(384\) −6488.01 −0.862214
\(385\) 0 0
\(386\) 6921.84 0.912727
\(387\) 19162.2 2.51698
\(388\) 3360.30 0.439674
\(389\) 10635.5 1.38622 0.693110 0.720832i \(-0.256240\pi\)
0.693110 + 0.720832i \(0.256240\pi\)
\(390\) −2022.66 −0.262619
\(391\) −6607.44 −0.854611
\(392\) 0 0
\(393\) 5738.23 0.736528
\(394\) 2884.73 0.368860
\(395\) −2964.21 −0.377584
\(396\) 10726.8 1.36121
\(397\) −1031.93 −0.130456 −0.0652282 0.997870i \(-0.520778\pi\)
−0.0652282 + 0.997870i \(0.520778\pi\)
\(398\) −6709.48 −0.845015
\(399\) 0 0
\(400\) −178.265 −0.0222831
\(401\) −6468.95 −0.805596 −0.402798 0.915289i \(-0.631962\pi\)
−0.402798 + 0.915289i \(0.631962\pi\)
\(402\) −17261.5 −2.14161
\(403\) −2847.40 −0.351958
\(404\) 2784.39 0.342893
\(405\) −8165.78 −1.00188
\(406\) 0 0
\(407\) −3143.06 −0.382791
\(408\) −30162.8 −3.66000
\(409\) 8652.18 1.04602 0.523011 0.852326i \(-0.324809\pi\)
0.523011 + 0.852326i \(0.324809\pi\)
\(410\) −861.630 −0.103787
\(411\) −19952.0 −2.39455
\(412\) 3654.01 0.436942
\(413\) 0 0
\(414\) 5800.11 0.688550
\(415\) −2467.03 −0.291811
\(416\) −3951.85 −0.465758
\(417\) 7957.77 0.934517
\(418\) −428.514 −0.0501419
\(419\) −7303.41 −0.851539 −0.425770 0.904832i \(-0.639997\pi\)
−0.425770 + 0.904832i \(0.639997\pi\)
\(420\) 0 0
\(421\) −11599.8 −1.34285 −0.671425 0.741072i \(-0.734317\pi\)
−0.671425 + 0.741072i \(0.734317\pi\)
\(422\) 6151.39 0.709585
\(423\) 4662.52 0.535933
\(424\) 8461.89 0.969212
\(425\) 3389.15 0.386819
\(426\) 5003.15 0.569022
\(427\) 0 0
\(428\) 8761.06 0.989444
\(429\) 8038.24 0.904638
\(430\) 2788.66 0.312746
\(431\) −1506.87 −0.168407 −0.0842034 0.996449i \(-0.526835\pi\)
−0.0842034 + 0.996449i \(0.526835\pi\)
\(432\) 2512.06 0.279772
\(433\) −2112.02 −0.234405 −0.117203 0.993108i \(-0.537393\pi\)
−0.117203 + 0.993108i \(0.537393\pi\)
\(434\) 0 0
\(435\) −3390.99 −0.373760
\(436\) −482.949 −0.0530483
\(437\) 303.463 0.0332188
\(438\) 13186.4 1.43852
\(439\) 3492.88 0.379740 0.189870 0.981809i \(-0.439193\pi\)
0.189870 + 0.981809i \(0.439193\pi\)
\(440\) 4314.03 0.467417
\(441\) 0 0
\(442\) −5750.70 −0.618853
\(443\) 974.674 0.104533 0.0522666 0.998633i \(-0.483355\pi\)
0.0522666 + 0.998633i \(0.483355\pi\)
\(444\) 3676.78 0.393000
\(445\) 4814.88 0.512916
\(446\) 6423.30 0.681956
\(447\) 2329.34 0.246474
\(448\) 0 0
\(449\) 6113.63 0.642584 0.321292 0.946980i \(-0.395883\pi\)
0.321292 + 0.946980i \(0.395883\pi\)
\(450\) −2975.04 −0.311655
\(451\) 3424.20 0.357515
\(452\) −1402.08 −0.145903
\(453\) 7649.66 0.793405
\(454\) 3810.09 0.393869
\(455\) 0 0
\(456\) 1385.30 0.142264
\(457\) −1553.51 −0.159015 −0.0795077 0.996834i \(-0.525335\pi\)
−0.0795077 + 0.996834i \(0.525335\pi\)
\(458\) −2582.10 −0.263436
\(459\) −47759.0 −4.85664
\(460\) −1105.50 −0.112053
\(461\) −9419.28 −0.951626 −0.475813 0.879546i \(-0.657846\pi\)
−0.475813 + 0.879546i \(0.657846\pi\)
\(462\) 0 0
\(463\) −11458.4 −1.15014 −0.575070 0.818104i \(-0.695025\pi\)
−0.575070 + 0.818104i \(0.695025\pi\)
\(464\) 507.109 0.0507369
\(465\) −5956.59 −0.594043
\(466\) 697.895 0.0693763
\(467\) −2121.08 −0.210175 −0.105087 0.994463i \(-0.533512\pi\)
−0.105087 + 0.994463i \(0.533512\pi\)
\(468\) −6611.46 −0.653023
\(469\) 0 0
\(470\) 678.532 0.0665922
\(471\) 33771.9 3.30388
\(472\) 8761.49 0.854407
\(473\) −11082.4 −1.07731
\(474\) 10521.8 1.01958
\(475\) −155.655 −0.0150357
\(476\) 0 0
\(477\) 23190.8 2.22607
\(478\) 10349.4 0.990313
\(479\) −17261.2 −1.64653 −0.823263 0.567660i \(-0.807849\pi\)
−0.823263 + 0.567660i \(0.807849\pi\)
\(480\) −8267.03 −0.786118
\(481\) 1937.23 0.183638
\(482\) 8758.63 0.827686
\(483\) 0 0
\(484\) −165.905 −0.0155809
\(485\) 3703.74 0.346759
\(486\) 11282.7 1.05308
\(487\) −18516.8 −1.72295 −0.861473 0.507804i \(-0.830457\pi\)
−0.861473 + 0.507804i \(0.830457\pi\)
\(488\) −16089.5 −1.49249
\(489\) 21118.9 1.95303
\(490\) 0 0
\(491\) 7914.77 0.727472 0.363736 0.931502i \(-0.381501\pi\)
0.363736 + 0.931502i \(0.381501\pi\)
\(492\) −4005.66 −0.367051
\(493\) −9641.07 −0.880755
\(494\) 264.115 0.0240549
\(495\) 11823.1 1.07356
\(496\) 890.782 0.0806397
\(497\) 0 0
\(498\) 8756.96 0.787969
\(499\) 2388.91 0.214314 0.107157 0.994242i \(-0.465825\pi\)
0.107157 + 0.994242i \(0.465825\pi\)
\(500\) 567.045 0.0507180
\(501\) −24957.3 −2.22557
\(502\) 1096.34 0.0974740
\(503\) −18073.0 −1.60206 −0.801030 0.598624i \(-0.795715\pi\)
−0.801030 + 0.598624i \(0.795715\pi\)
\(504\) 0 0
\(505\) 3068.97 0.270431
\(506\) −3354.47 −0.294713
\(507\) 15997.0 1.40129
\(508\) 904.869 0.0790296
\(509\) −18671.7 −1.62595 −0.812975 0.582299i \(-0.802153\pi\)
−0.812975 + 0.582299i \(0.802153\pi\)
\(510\) −12030.1 −1.04451
\(511\) 0 0
\(512\) 2567.23 0.221595
\(513\) 2193.45 0.188778
\(514\) −8684.12 −0.745214
\(515\) 4027.47 0.344605
\(516\) 12964.3 1.10605
\(517\) −2696.55 −0.229389
\(518\) 0 0
\(519\) −15582.8 −1.31793
\(520\) −2658.96 −0.224236
\(521\) 3526.04 0.296504 0.148252 0.988950i \(-0.452635\pi\)
0.148252 + 0.988950i \(0.452635\pi\)
\(522\) 8463.07 0.709614
\(523\) 10966.3 0.916865 0.458433 0.888729i \(-0.348411\pi\)
0.458433 + 0.888729i \(0.348411\pi\)
\(524\) 2729.62 0.227565
\(525\) 0 0
\(526\) 8322.71 0.689900
\(527\) −16935.4 −1.39984
\(528\) −2514.69 −0.207268
\(529\) −9791.44 −0.804754
\(530\) 3374.94 0.276600
\(531\) 24011.9 1.96239
\(532\) 0 0
\(533\) −2110.51 −0.171513
\(534\) −17090.9 −1.38501
\(535\) 9656.50 0.780349
\(536\) −22691.8 −1.82861
\(537\) −9242.26 −0.742705
\(538\) −7922.88 −0.634907
\(539\) 0 0
\(540\) −7990.64 −0.636782
\(541\) 11349.8 0.901971 0.450985 0.892531i \(-0.351073\pi\)
0.450985 + 0.892531i \(0.351073\pi\)
\(542\) −7199.97 −0.570600
\(543\) 41561.4 3.28466
\(544\) −23504.3 −1.85246
\(545\) −532.309 −0.0418378
\(546\) 0 0
\(547\) 11206.1 0.875940 0.437970 0.898989i \(-0.355698\pi\)
0.437970 + 0.898989i \(0.355698\pi\)
\(548\) −9491.00 −0.739845
\(549\) −44095.1 −3.42793
\(550\) 1720.61 0.133394
\(551\) 442.790 0.0342350
\(552\) 10844.3 0.836170
\(553\) 0 0
\(554\) −15274.5 −1.17139
\(555\) 4052.57 0.309949
\(556\) 3785.44 0.288738
\(557\) 12631.0 0.960847 0.480424 0.877037i \(-0.340483\pi\)
0.480424 + 0.877037i \(0.340483\pi\)
\(558\) 14866.1 1.12784
\(559\) 6830.65 0.516826
\(560\) 0 0
\(561\) 47808.9 3.59803
\(562\) −12043.6 −0.903964
\(563\) −7000.69 −0.524057 −0.262028 0.965060i \(-0.584391\pi\)
−0.262028 + 0.965060i \(0.584391\pi\)
\(564\) 3154.45 0.235508
\(565\) −1545.38 −0.115070
\(566\) 2737.29 0.203281
\(567\) 0 0
\(568\) 6577.07 0.485858
\(569\) −8659.25 −0.637987 −0.318994 0.947757i \(-0.603345\pi\)
−0.318994 + 0.947757i \(0.603345\pi\)
\(570\) 552.513 0.0406004
\(571\) 25631.9 1.87857 0.939283 0.343145i \(-0.111492\pi\)
0.939283 + 0.343145i \(0.111492\pi\)
\(572\) 3823.72 0.279506
\(573\) −27750.5 −2.02320
\(574\) 0 0
\(575\) −1218.49 −0.0883733
\(576\) 24280.0 1.75637
\(577\) −2546.85 −0.183755 −0.0918775 0.995770i \(-0.529287\pi\)
−0.0918775 + 0.995770i \(0.529287\pi\)
\(578\) −25059.8 −1.80337
\(579\) 35468.1 2.54577
\(580\) −1613.06 −0.115481
\(581\) 0 0
\(582\) −13146.8 −0.936344
\(583\) −13412.3 −0.952800
\(584\) 17334.7 1.22828
\(585\) −7287.19 −0.515023
\(586\) −10215.8 −0.720158
\(587\) −17798.8 −1.25150 −0.625752 0.780022i \(-0.715208\pi\)
−0.625752 + 0.780022i \(0.715208\pi\)
\(588\) 0 0
\(589\) 777.800 0.0544121
\(590\) 3494.43 0.243836
\(591\) 14781.6 1.02882
\(592\) −606.044 −0.0420748
\(593\) −3191.29 −0.220996 −0.110498 0.993876i \(-0.535245\pi\)
−0.110498 + 0.993876i \(0.535245\pi\)
\(594\) −24246.3 −1.67481
\(595\) 0 0
\(596\) 1108.04 0.0761531
\(597\) −34379.9 −2.35691
\(598\) 2067.53 0.141384
\(599\) 20511.7 1.39914 0.699571 0.714563i \(-0.253374\pi\)
0.699571 + 0.714563i \(0.253374\pi\)
\(600\) −5562.38 −0.378472
\(601\) −22802.2 −1.54762 −0.773810 0.633418i \(-0.781651\pi\)
−0.773810 + 0.633418i \(0.781651\pi\)
\(602\) 0 0
\(603\) −62189.5 −4.19992
\(604\) 3638.87 0.245139
\(605\) −182.862 −0.0122883
\(606\) −10893.6 −0.730236
\(607\) −3130.15 −0.209306 −0.104653 0.994509i \(-0.533373\pi\)
−0.104653 + 0.994509i \(0.533373\pi\)
\(608\) 1079.49 0.0720054
\(609\) 0 0
\(610\) −6417.12 −0.425937
\(611\) 1662.02 0.110046
\(612\) −39322.9 −2.59727
\(613\) −12736.1 −0.839162 −0.419581 0.907718i \(-0.637823\pi\)
−0.419581 + 0.907718i \(0.637823\pi\)
\(614\) 1926.87 0.126648
\(615\) −4415.06 −0.289484
\(616\) 0 0
\(617\) −16662.4 −1.08720 −0.543600 0.839345i \(-0.682939\pi\)
−0.543600 + 0.839345i \(0.682939\pi\)
\(618\) −14295.9 −0.930526
\(619\) 9967.73 0.647232 0.323616 0.946188i \(-0.395101\pi\)
0.323616 + 0.946188i \(0.395101\pi\)
\(620\) −2833.49 −0.183542
\(621\) 17170.7 1.10956
\(622\) 4734.68 0.305215
\(623\) 0 0
\(624\) 1549.93 0.0994341
\(625\) 625.000 0.0400000
\(626\) 4838.31 0.308910
\(627\) −2195.74 −0.139856
\(628\) 16065.0 1.02080
\(629\) 11522.0 0.730386
\(630\) 0 0
\(631\) −6243.76 −0.393914 −0.196957 0.980412i \(-0.563106\pi\)
−0.196957 + 0.980412i \(0.563106\pi\)
\(632\) 13831.8 0.870566
\(633\) 31520.2 1.97917
\(634\) 5071.70 0.317702
\(635\) 997.352 0.0623287
\(636\) 15689.9 0.978214
\(637\) 0 0
\(638\) −4894.59 −0.303728
\(639\) 18025.3 1.11591
\(640\) −3401.72 −0.210102
\(641\) −25915.9 −1.59690 −0.798452 0.602059i \(-0.794347\pi\)
−0.798452 + 0.602059i \(0.794347\pi\)
\(642\) −34276.7 −2.10715
\(643\) −11833.7 −0.725778 −0.362889 0.931832i \(-0.618210\pi\)
−0.362889 + 0.931832i \(0.618210\pi\)
\(644\) 0 0
\(645\) 14289.3 0.872311
\(646\) 1570.87 0.0956735
\(647\) −7268.54 −0.441662 −0.220831 0.975312i \(-0.570877\pi\)
−0.220831 + 0.975312i \(0.570877\pi\)
\(648\) 38103.6 2.30995
\(649\) −13887.2 −0.839940
\(650\) −1060.50 −0.0639941
\(651\) 0 0
\(652\) 10046.1 0.603427
\(653\) −30455.8 −1.82515 −0.912577 0.408904i \(-0.865911\pi\)
−0.912577 + 0.408904i \(0.865911\pi\)
\(654\) 1889.48 0.112973
\(655\) 3008.61 0.179475
\(656\) 660.253 0.0392966
\(657\) 47507.7 2.82109
\(658\) 0 0
\(659\) 16170.2 0.955843 0.477922 0.878403i \(-0.341390\pi\)
0.477922 + 0.878403i \(0.341390\pi\)
\(660\) 7998.99 0.471758
\(661\) −10331.9 −0.607962 −0.303981 0.952678i \(-0.598316\pi\)
−0.303981 + 0.952678i \(0.598316\pi\)
\(662\) 9636.93 0.565786
\(663\) −29467.0 −1.72610
\(664\) 11511.8 0.672806
\(665\) 0 0
\(666\) −10114.2 −0.588464
\(667\) 3466.23 0.201219
\(668\) −11872.0 −0.687634
\(669\) 32913.5 1.90211
\(670\) −9050.38 −0.521861
\(671\) 25502.3 1.46722
\(672\) 0 0
\(673\) 18387.1 1.05315 0.526576 0.850128i \(-0.323475\pi\)
0.526576 + 0.850128i \(0.323475\pi\)
\(674\) 6804.25 0.388858
\(675\) −8807.33 −0.502214
\(676\) 7609.63 0.432955
\(677\) 1795.92 0.101954 0.0509770 0.998700i \(-0.483766\pi\)
0.0509770 + 0.998700i \(0.483766\pi\)
\(678\) 5485.48 0.310721
\(679\) 0 0
\(680\) −15814.6 −0.891857
\(681\) 19523.2 1.09858
\(682\) −8597.78 −0.482736
\(683\) −5203.86 −0.291537 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(684\) 1806.00 0.100956
\(685\) −10461.0 −0.583497
\(686\) 0 0
\(687\) −13230.9 −0.734774
\(688\) −2136.90 −0.118414
\(689\) 8266.71 0.457092
\(690\) 4325.16 0.238632
\(691\) 8903.56 0.490170 0.245085 0.969502i \(-0.421184\pi\)
0.245085 + 0.969502i \(0.421184\pi\)
\(692\) −7412.58 −0.407202
\(693\) 0 0
\(694\) 3113.88 0.170319
\(695\) 4172.33 0.227720
\(696\) 15823.2 0.861750
\(697\) −12552.6 −0.682159
\(698\) 1446.65 0.0784476
\(699\) 3576.07 0.193504
\(700\) 0 0
\(701\) −8343.11 −0.449522 −0.224761 0.974414i \(-0.572160\pi\)
−0.224761 + 0.974414i \(0.572160\pi\)
\(702\) 14944.2 0.803468
\(703\) −529.177 −0.0283902
\(704\) −14042.3 −0.751758
\(705\) 3476.85 0.185739
\(706\) 8231.43 0.438802
\(707\) 0 0
\(708\) 16245.4 0.862343
\(709\) −28590.0 −1.51441 −0.757206 0.653176i \(-0.773436\pi\)
−0.757206 + 0.653176i \(0.773436\pi\)
\(710\) 2623.20 0.138658
\(711\) 37907.6 1.99950
\(712\) −22467.4 −1.18259
\(713\) 6088.74 0.319811
\(714\) 0 0
\(715\) 4214.52 0.220439
\(716\) −4396.46 −0.229474
\(717\) 53031.0 2.76218
\(718\) 1790.67 0.0930740
\(719\) −26730.4 −1.38647 −0.693237 0.720710i \(-0.743816\pi\)
−0.693237 + 0.720710i \(0.743816\pi\)
\(720\) 2279.73 0.118001
\(721\) 0 0
\(722\) 12693.0 0.654275
\(723\) 44879.9 2.30858
\(724\) 19770.4 1.01486
\(725\) −1777.93 −0.0910768
\(726\) 649.086 0.0331816
\(727\) 8903.62 0.454219 0.227109 0.973869i \(-0.427073\pi\)
0.227109 + 0.973869i \(0.427073\pi\)
\(728\) 0 0
\(729\) 13718.4 0.696969
\(730\) 6913.75 0.350534
\(731\) 40626.5 2.05557
\(732\) −29832.8 −1.50635
\(733\) 6022.56 0.303477 0.151738 0.988421i \(-0.451513\pi\)
0.151738 + 0.988421i \(0.451513\pi\)
\(734\) −13553.7 −0.681575
\(735\) 0 0
\(736\) 8450.45 0.423217
\(737\) 35967.1 1.79765
\(738\) 11018.9 0.549608
\(739\) 14078.5 0.700795 0.350398 0.936601i \(-0.386046\pi\)
0.350398 + 0.936601i \(0.386046\pi\)
\(740\) 1927.77 0.0957651
\(741\) 1353.35 0.0670937
\(742\) 0 0
\(743\) 31431.4 1.55196 0.775980 0.630757i \(-0.217256\pi\)
0.775980 + 0.630757i \(0.217256\pi\)
\(744\) 27794.9 1.36964
\(745\) 1221.29 0.0600600
\(746\) 2138.83 0.104971
\(747\) 31549.4 1.54529
\(748\) 22742.2 1.11168
\(749\) 0 0
\(750\) −2218.50 −0.108011
\(751\) −2463.12 −0.119681 −0.0598405 0.998208i \(-0.519059\pi\)
−0.0598405 + 0.998208i \(0.519059\pi\)
\(752\) −519.948 −0.0252135
\(753\) 5617.73 0.271874
\(754\) 3016.79 0.145709
\(755\) 4010.79 0.193335
\(756\) 0 0
\(757\) 37987.2 1.82387 0.911935 0.410335i \(-0.134588\pi\)
0.911935 + 0.410335i \(0.134588\pi\)
\(758\) 19453.3 0.932156
\(759\) −17188.6 −0.822011
\(760\) 726.325 0.0346666
\(761\) −18691.9 −0.890384 −0.445192 0.895435i \(-0.646865\pi\)
−0.445192 + 0.895435i \(0.646865\pi\)
\(762\) −3540.20 −0.168304
\(763\) 0 0
\(764\) −13200.7 −0.625109
\(765\) −43341.9 −2.04840
\(766\) −25067.5 −1.18241
\(767\) 8559.40 0.402949
\(768\) 41043.8 1.92844
\(769\) 27250.5 1.27786 0.638932 0.769263i \(-0.279376\pi\)
0.638932 + 0.769263i \(0.279376\pi\)
\(770\) 0 0
\(771\) −44498.1 −2.07855
\(772\) 16871.8 0.786568
\(773\) 1092.67 0.0508417 0.0254209 0.999677i \(-0.491907\pi\)
0.0254209 + 0.999677i \(0.491907\pi\)
\(774\) −35662.5 −1.65615
\(775\) −3123.09 −0.144755
\(776\) −17282.6 −0.799495
\(777\) 0 0
\(778\) −19793.5 −0.912124
\(779\) 576.511 0.0265156
\(780\) −4930.18 −0.226319
\(781\) −10424.8 −0.477631
\(782\) 12297.0 0.562328
\(783\) 25054.1 1.14350
\(784\) 0 0
\(785\) 17706.9 0.805078
\(786\) −10679.3 −0.484631
\(787\) 12639.1 0.572474 0.286237 0.958159i \(-0.407596\pi\)
0.286237 + 0.958159i \(0.407596\pi\)
\(788\) 7031.47 0.317875
\(789\) 42646.3 1.92427
\(790\) 5516.66 0.248448
\(791\) 0 0
\(792\) −55169.6 −2.47521
\(793\) −15718.4 −0.703878
\(794\) 1920.51 0.0858394
\(795\) 17293.5 0.771492
\(796\) −16354.2 −0.728216
\(797\) −8666.66 −0.385180 −0.192590 0.981279i \(-0.561689\pi\)
−0.192590 + 0.981279i \(0.561689\pi\)
\(798\) 0 0
\(799\) 9885.18 0.437688
\(800\) −4334.48 −0.191559
\(801\) −61574.8 −2.71615
\(802\) 12039.3 0.530077
\(803\) −27475.9 −1.20748
\(804\) −42074.6 −1.84559
\(805\) 0 0
\(806\) 5299.25 0.231586
\(807\) −40597.5 −1.77088
\(808\) −14320.6 −0.623511
\(809\) 3555.20 0.154504 0.0772522 0.997012i \(-0.475385\pi\)
0.0772522 + 0.997012i \(0.475385\pi\)
\(810\) 15197.2 0.659230
\(811\) −21940.0 −0.949961 −0.474981 0.879996i \(-0.657545\pi\)
−0.474981 + 0.879996i \(0.657545\pi\)
\(812\) 0 0
\(813\) −36893.2 −1.59152
\(814\) 5849.51 0.251874
\(815\) 11072.8 0.475908
\(816\) 9218.49 0.395480
\(817\) −1865.87 −0.0799003
\(818\) −16102.5 −0.688275
\(819\) 0 0
\(820\) −2100.20 −0.0894418
\(821\) −29572.7 −1.25712 −0.628560 0.777761i \(-0.716355\pi\)
−0.628560 + 0.777761i \(0.716355\pi\)
\(822\) 37132.5 1.57560
\(823\) −19314.7 −0.818067 −0.409034 0.912519i \(-0.634134\pi\)
−0.409034 + 0.912519i \(0.634134\pi\)
\(824\) −18793.2 −0.794528
\(825\) 8816.53 0.372063
\(826\) 0 0
\(827\) 21107.8 0.887535 0.443768 0.896142i \(-0.353642\pi\)
0.443768 + 0.896142i \(0.353642\pi\)
\(828\) 14137.6 0.593378
\(829\) −10799.8 −0.452463 −0.226231 0.974074i \(-0.572641\pi\)
−0.226231 + 0.974074i \(0.572641\pi\)
\(830\) 4591.35 0.192010
\(831\) −78267.7 −3.26724
\(832\) 8654.96 0.360645
\(833\) 0 0
\(834\) −14810.1 −0.614906
\(835\) −13085.3 −0.542319
\(836\) −1044.49 −0.0432112
\(837\) 44009.8 1.81744
\(838\) 13592.3 0.560307
\(839\) 11829.3 0.486761 0.243381 0.969931i \(-0.421744\pi\)
0.243381 + 0.969931i \(0.421744\pi\)
\(840\) 0 0
\(841\) −19331.3 −0.792626
\(842\) 21588.2 0.883587
\(843\) −61712.3 −2.52133
\(844\) 14993.9 0.611506
\(845\) 8387.37 0.341461
\(846\) −8677.35 −0.352640
\(847\) 0 0
\(848\) −2586.16 −0.104728
\(849\) 14026.1 0.566989
\(850\) −6307.50 −0.254524
\(851\) −4142.48 −0.166865
\(852\) 12195.1 0.490371
\(853\) 3426.91 0.137556 0.0687779 0.997632i \(-0.478090\pi\)
0.0687779 + 0.997632i \(0.478090\pi\)
\(854\) 0 0
\(855\) 1990.58 0.0796216
\(856\) −45059.6 −1.79919
\(857\) −7308.16 −0.291298 −0.145649 0.989336i \(-0.546527\pi\)
−0.145649 + 0.989336i \(0.546527\pi\)
\(858\) −14959.9 −0.595246
\(859\) 22539.8 0.895282 0.447641 0.894213i \(-0.352264\pi\)
0.447641 + 0.894213i \(0.352264\pi\)
\(860\) 6797.29 0.269518
\(861\) 0 0
\(862\) 2804.41 0.110811
\(863\) 1690.95 0.0666983 0.0333491 0.999444i \(-0.489383\pi\)
0.0333491 + 0.999444i \(0.489383\pi\)
\(864\) 61080.3 2.40509
\(865\) −8170.19 −0.321150
\(866\) 3930.66 0.154237
\(867\) −128408. −5.02996
\(868\) 0 0
\(869\) −21923.7 −0.855825
\(870\) 6310.93 0.245932
\(871\) −22168.4 −0.862395
\(872\) 2483.89 0.0964621
\(873\) −47365.0 −1.83627
\(874\) −564.771 −0.0218577
\(875\) 0 0
\(876\) 32141.6 1.23968
\(877\) 33021.5 1.27145 0.635723 0.771917i \(-0.280702\pi\)
0.635723 + 0.771917i \(0.280702\pi\)
\(878\) −6500.54 −0.249866
\(879\) −52346.8 −2.00866
\(880\) −1318.47 −0.0505065
\(881\) 29413.5 1.12482 0.562410 0.826859i \(-0.309874\pi\)
0.562410 + 0.826859i \(0.309874\pi\)
\(882\) 0 0
\(883\) 22413.5 0.854218 0.427109 0.904200i \(-0.359532\pi\)
0.427109 + 0.904200i \(0.359532\pi\)
\(884\) −14017.2 −0.533314
\(885\) 17905.8 0.680108
\(886\) −1813.95 −0.0687821
\(887\) 49069.5 1.85749 0.928744 0.370723i \(-0.120890\pi\)
0.928744 + 0.370723i \(0.120890\pi\)
\(888\) −18910.3 −0.714626
\(889\) 0 0
\(890\) −8960.92 −0.337495
\(891\) −60395.3 −2.27084
\(892\) 15656.7 0.587695
\(893\) −454.001 −0.0170130
\(894\) −4335.10 −0.162178
\(895\) −4845.80 −0.180980
\(896\) 0 0
\(897\) 10594.2 0.394348
\(898\) −11378.0 −0.422816
\(899\) 8884.22 0.329594
\(900\) −7251.61 −0.268578
\(901\) 49167.8 1.81800
\(902\) −6372.73 −0.235243
\(903\) 0 0
\(904\) 7211.13 0.265308
\(905\) 21791.0 0.800395
\(906\) −14236.7 −0.522055
\(907\) −42854.4 −1.56886 −0.784431 0.620216i \(-0.787045\pi\)
−0.784431 + 0.620216i \(0.787045\pi\)
\(908\) 9287.02 0.339428
\(909\) −39247.3 −1.43207
\(910\) 0 0
\(911\) −48846.7 −1.77647 −0.888234 0.459391i \(-0.848068\pi\)
−0.888234 + 0.459391i \(0.848068\pi\)
\(912\) −423.382 −0.0153723
\(913\) −18246.5 −0.661414
\(914\) 2891.21 0.104631
\(915\) −32881.9 −1.18802
\(916\) −6293.81 −0.227023
\(917\) 0 0
\(918\) 88883.6 3.19564
\(919\) −22400.9 −0.804067 −0.402033 0.915625i \(-0.631696\pi\)
−0.402033 + 0.915625i \(0.631696\pi\)
\(920\) 5685.79 0.203755
\(921\) 9873.43 0.353247
\(922\) 17530.1 0.626164
\(923\) 6425.36 0.229137
\(924\) 0 0
\(925\) 2124.80 0.0755275
\(926\) 21325.0 0.756785
\(927\) −51504.9 −1.82486
\(928\) 12330.2 0.436164
\(929\) −2298.05 −0.0811587 −0.0405794 0.999176i \(-0.512920\pi\)
−0.0405794 + 0.999176i \(0.512920\pi\)
\(930\) 11085.7 0.390876
\(931\) 0 0
\(932\) 1701.10 0.0597870
\(933\) 24260.9 0.851303
\(934\) 3947.51 0.138294
\(935\) 25066.6 0.876756
\(936\) 34003.9 1.18745
\(937\) 47163.3 1.64435 0.822176 0.569233i \(-0.192760\pi\)
0.822176 + 0.569233i \(0.192760\pi\)
\(938\) 0 0
\(939\) 24791.9 0.861611
\(940\) 1653.91 0.0573877
\(941\) −40457.0 −1.40155 −0.700776 0.713382i \(-0.747163\pi\)
−0.700776 + 0.713382i \(0.747163\pi\)
\(942\) −62852.4 −2.17393
\(943\) 4513.01 0.155847
\(944\) −2677.73 −0.0923227
\(945\) 0 0
\(946\) 20625.3 0.708865
\(947\) −6363.14 −0.218347 −0.109173 0.994023i \(-0.534820\pi\)
−0.109173 + 0.994023i \(0.534820\pi\)
\(948\) 25646.6 0.878652
\(949\) 16934.8 0.579270
\(950\) 289.688 0.00989337
\(951\) 25987.8 0.886133
\(952\) 0 0
\(953\) −48268.8 −1.64069 −0.820346 0.571867i \(-0.806219\pi\)
−0.820346 + 0.571867i \(0.806219\pi\)
\(954\) −43160.2 −1.46474
\(955\) −14549.9 −0.493008
\(956\) 25226.4 0.853430
\(957\) −25080.3 −0.847158
\(958\) 32124.7 1.08340
\(959\) 0 0
\(960\) 18105.7 0.608706
\(961\) −14185.1 −0.476153
\(962\) −3605.35 −0.120833
\(963\) −123491. −4.13235
\(964\) 21349.0 0.713282
\(965\) 18596.2 0.620346
\(966\) 0 0
\(967\) −26795.3 −0.891084 −0.445542 0.895261i \(-0.646989\pi\)
−0.445542 + 0.895261i \(0.646989\pi\)
\(968\) 853.280 0.0283321
\(969\) 8049.27 0.266852
\(970\) −6892.98 −0.228165
\(971\) −21583.8 −0.713345 −0.356672 0.934230i \(-0.616089\pi\)
−0.356672 + 0.934230i \(0.616089\pi\)
\(972\) 27501.4 0.907519
\(973\) 0 0
\(974\) 34461.3 1.13369
\(975\) −5434.08 −0.178492
\(976\) 4917.34 0.161271
\(977\) 51106.3 1.67353 0.836763 0.547565i \(-0.184445\pi\)
0.836763 + 0.547565i \(0.184445\pi\)
\(978\) −39304.1 −1.28508
\(979\) 35611.6 1.16256
\(980\) 0 0
\(981\) 6807.39 0.221553
\(982\) −14730.1 −0.478672
\(983\) 41585.7 1.34932 0.674658 0.738131i \(-0.264291\pi\)
0.674658 + 0.738131i \(0.264291\pi\)
\(984\) 20601.8 0.667439
\(985\) 7750.13 0.250700
\(986\) 17942.9 0.579531
\(987\) 0 0
\(988\) 643.775 0.0207300
\(989\) −14606.3 −0.469620
\(990\) −22003.8 −0.706392
\(991\) −29082.5 −0.932227 −0.466114 0.884725i \(-0.654346\pi\)
−0.466114 + 0.884725i \(0.654346\pi\)
\(992\) 21659.2 0.693226
\(993\) 49380.4 1.57809
\(994\) 0 0
\(995\) −18025.7 −0.574325
\(996\) 21344.9 0.679055
\(997\) −34503.0 −1.09601 −0.548004 0.836476i \(-0.684612\pi\)
−0.548004 + 0.836476i \(0.684612\pi\)
\(998\) −4445.98 −0.141017
\(999\) −29942.1 −0.948274
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 245.4.a.l.1.2 3
3.2 odd 2 2205.4.a.bm.1.2 3
5.4 even 2 1225.4.a.y.1.2 3
7.2 even 3 245.4.e.n.116.2 6
7.3 odd 6 245.4.e.m.226.2 6
7.4 even 3 245.4.e.n.226.2 6
7.5 odd 6 245.4.e.m.116.2 6
7.6 odd 2 35.4.a.c.1.2 3
21.20 even 2 315.4.a.p.1.2 3
28.27 even 2 560.4.a.u.1.1 3
35.13 even 4 175.4.b.e.99.4 6
35.27 even 4 175.4.b.e.99.3 6
35.34 odd 2 175.4.a.f.1.2 3
56.13 odd 2 2240.4.a.bt.1.1 3
56.27 even 2 2240.4.a.bv.1.3 3
105.104 even 2 1575.4.a.ba.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.2 3 7.6 odd 2
175.4.a.f.1.2 3 35.34 odd 2
175.4.b.e.99.3 6 35.27 even 4
175.4.b.e.99.4 6 35.13 even 4
245.4.a.l.1.2 3 1.1 even 1 trivial
245.4.e.m.116.2 6 7.5 odd 6
245.4.e.m.226.2 6 7.3 odd 6
245.4.e.n.116.2 6 7.2 even 3
245.4.e.n.226.2 6 7.4 even 3
315.4.a.p.1.2 3 21.20 even 2
560.4.a.u.1.1 3 28.27 even 2
1225.4.a.y.1.2 3 5.4 even 2
1575.4.a.ba.1.2 3 105.104 even 2
2205.4.a.bm.1.2 3 3.2 odd 2
2240.4.a.bt.1.1 3 56.13 odd 2
2240.4.a.bv.1.3 3 56.27 even 2