Properties

Label 115.6.a.c.1.4
Level $115$
Weight $6$
Character 115.1
Self dual yes
Analytic conductor $18.444$
Analytic rank $0$
Dimension $7$
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] = [115,6,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(18.4441392785\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.66253\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.662532 q^{2} -4.47473 q^{3} -31.5611 q^{4} -25.0000 q^{5} +2.96465 q^{6} +91.5985 q^{7} +42.1112 q^{8} -222.977 q^{9} +O(q^{10})\) \(q-0.662532 q^{2} -4.47473 q^{3} -31.5611 q^{4} -25.0000 q^{5} +2.96465 q^{6} +91.5985 q^{7} +42.1112 q^{8} -222.977 q^{9} +16.5633 q^{10} -196.656 q^{11} +141.227 q^{12} -1148.23 q^{13} -60.6870 q^{14} +111.868 q^{15} +982.054 q^{16} +653.440 q^{17} +147.729 q^{18} +3000.86 q^{19} +789.026 q^{20} -409.879 q^{21} +130.291 q^{22} +529.000 q^{23} -188.436 q^{24} +625.000 q^{25} +760.741 q^{26} +2085.12 q^{27} -2890.94 q^{28} +7023.07 q^{29} -74.1163 q^{30} +2131.87 q^{31} -1998.20 q^{32} +879.981 q^{33} -432.925 q^{34} -2289.96 q^{35} +7037.38 q^{36} -4389.31 q^{37} -1988.17 q^{38} +5138.03 q^{39} -1052.78 q^{40} -13471.6 q^{41} +271.558 q^{42} +19142.7 q^{43} +6206.66 q^{44} +5574.42 q^{45} -350.480 q^{46} +21512.1 q^{47} -4394.42 q^{48} -8416.71 q^{49} -414.083 q^{50} -2923.97 q^{51} +36239.4 q^{52} -10580.3 q^{53} -1381.46 q^{54} +4916.39 q^{55} +3857.33 q^{56} -13428.0 q^{57} -4653.01 q^{58} +16562.2 q^{59} -3530.68 q^{60} -29662.0 q^{61} -1412.43 q^{62} -20424.3 q^{63} -30101.8 q^{64} +28705.8 q^{65} -583.016 q^{66} -18606.7 q^{67} -20623.2 q^{68} -2367.13 q^{69} +1517.17 q^{70} +21704.4 q^{71} -9389.83 q^{72} +57006.3 q^{73} +2908.06 q^{74} -2796.71 q^{75} -94710.3 q^{76} -18013.4 q^{77} -3404.11 q^{78} +26607.9 q^{79} -24551.3 q^{80} +44853.0 q^{81} +8925.36 q^{82} -26781.1 q^{83} +12936.2 q^{84} -16336.0 q^{85} -12682.7 q^{86} -31426.3 q^{87} -8281.41 q^{88} -83708.0 q^{89} -3693.23 q^{90} -105176. q^{91} -16695.8 q^{92} -9539.56 q^{93} -14252.5 q^{94} -75021.5 q^{95} +8941.41 q^{96} +27667.8 q^{97} +5576.35 q^{98} +43849.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9} - 100 q^{10} + 1373 q^{11} - 285 q^{12} + 605 q^{13} + 1317 q^{14} + 75 q^{15} + 3770 q^{16} + 2505 q^{17} + 7971 q^{18} - 115 q^{19} - 4450 q^{20} + 608 q^{21} + 2977 q^{22} + 3703 q^{23} - 12447 q^{24} + 4375 q^{25} + 9379 q^{26} - 12276 q^{27} + 5777 q^{28} + 2440 q^{29} + 9525 q^{30} + 13565 q^{31} + 14086 q^{32} + 10519 q^{33} + 26997 q^{34} - 825 q^{35} + 79889 q^{36} + 9414 q^{37} + 28717 q^{38} - 21738 q^{39} - 13650 q^{40} + 13725 q^{41} + 12426 q^{42} + 76694 q^{43} + 55203 q^{44} - 11000 q^{45} + 2116 q^{46} + 59692 q^{47} - 32985 q^{48} - 53608 q^{49} + 2500 q^{50} - 24725 q^{51} + 61195 q^{52} + 49536 q^{53} - 156168 q^{54} - 34325 q^{55} - 54461 q^{56} - 7580 q^{57} - 95562 q^{58} + 44536 q^{59} + 7125 q^{60} - 49097 q^{61} - 25763 q^{62} - 3578 q^{63} - 18654 q^{64} - 15125 q^{65} - 201873 q^{66} + 788 q^{67} + 163845 q^{68} - 1587 q^{69} - 32925 q^{70} + 49521 q^{71} + 328503 q^{72} - 3760 q^{73} + 88170 q^{74} - 1875 q^{75} - 411465 q^{76} + 77728 q^{77} - 389832 q^{78} + 918 q^{79} - 94250 q^{80} + 121235 q^{81} - 227459 q^{82} + 99202 q^{83} + 336602 q^{84} - 62625 q^{85} + 24584 q^{86} - 38666 q^{87} - 201275 q^{88} - 141676 q^{89} - 199275 q^{90} - 223605 q^{91} + 94162 q^{92} + 51412 q^{93} - 354292 q^{94} + 2875 q^{95} - 592095 q^{96} + 28731 q^{97} - 149557 q^{98} + 237333 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.662532 −0.117120 −0.0585601 0.998284i \(-0.518651\pi\)
−0.0585601 + 0.998284i \(0.518651\pi\)
\(3\) −4.47473 −0.287054 −0.143527 0.989646i \(-0.545844\pi\)
−0.143527 + 0.989646i \(0.545844\pi\)
\(4\) −31.5611 −0.986283
\(5\) −25.0000 −0.447214
\(6\) 2.96465 0.0336198
\(7\) 91.5985 0.706551 0.353275 0.935519i \(-0.385068\pi\)
0.353275 + 0.935519i \(0.385068\pi\)
\(8\) 42.1112 0.232634
\(9\) −222.977 −0.917600
\(10\) 16.5633 0.0523778
\(11\) −196.656 −0.490032 −0.245016 0.969519i \(-0.578793\pi\)
−0.245016 + 0.969519i \(0.578793\pi\)
\(12\) 141.227 0.283116
\(13\) −1148.23 −1.88439 −0.942196 0.335061i \(-0.891243\pi\)
−0.942196 + 0.335061i \(0.891243\pi\)
\(14\) −60.6870 −0.0827514
\(15\) 111.868 0.128374
\(16\) 982.054 0.959037
\(17\) 653.440 0.548382 0.274191 0.961675i \(-0.411590\pi\)
0.274191 + 0.961675i \(0.411590\pi\)
\(18\) 147.729 0.107470
\(19\) 3000.86 1.90705 0.953524 0.301317i \(-0.0974262\pi\)
0.953524 + 0.301317i \(0.0974262\pi\)
\(20\) 789.026 0.441079
\(21\) −409.879 −0.202818
\(22\) 130.291 0.0573927
\(23\) 529.000 0.208514
\(24\) −188.436 −0.0667785
\(25\) 625.000 0.200000
\(26\) 760.741 0.220701
\(27\) 2085.12 0.550455
\(28\) −2890.94 −0.696859
\(29\) 7023.07 1.55072 0.775358 0.631522i \(-0.217570\pi\)
0.775358 + 0.631522i \(0.217570\pi\)
\(30\) −74.1163 −0.0150353
\(31\) 2131.87 0.398435 0.199217 0.979955i \(-0.436160\pi\)
0.199217 + 0.979955i \(0.436160\pi\)
\(32\) −1998.20 −0.344957
\(33\) 879.981 0.140666
\(34\) −432.925 −0.0642267
\(35\) −2289.96 −0.315979
\(36\) 7037.38 0.905013
\(37\) −4389.31 −0.527098 −0.263549 0.964646i \(-0.584893\pi\)
−0.263549 + 0.964646i \(0.584893\pi\)
\(38\) −1988.17 −0.223354
\(39\) 5138.03 0.540923
\(40\) −1052.78 −0.104037
\(41\) −13471.6 −1.25158 −0.625791 0.779991i \(-0.715224\pi\)
−0.625791 + 0.779991i \(0.715224\pi\)
\(42\) 271.558 0.0237541
\(43\) 19142.7 1.57882 0.789410 0.613866i \(-0.210387\pi\)
0.789410 + 0.613866i \(0.210387\pi\)
\(44\) 6206.66 0.483310
\(45\) 5574.42 0.410363
\(46\) −350.480 −0.0244213
\(47\) 21512.1 1.42049 0.710246 0.703953i \(-0.248584\pi\)
0.710246 + 0.703953i \(0.248584\pi\)
\(48\) −4394.42 −0.275295
\(49\) −8416.71 −0.500786
\(50\) −414.083 −0.0234241
\(51\) −2923.97 −0.157415
\(52\) 36239.4 1.85854
\(53\) −10580.3 −0.517377 −0.258688 0.965961i \(-0.583290\pi\)
−0.258688 + 0.965961i \(0.583290\pi\)
\(54\) −1381.46 −0.0644694
\(55\) 4916.39 0.219149
\(56\) 3857.33 0.164368
\(57\) −13428.0 −0.547426
\(58\) −4653.01 −0.181620
\(59\) 16562.2 0.619425 0.309713 0.950830i \(-0.399767\pi\)
0.309713 + 0.950830i \(0.399767\pi\)
\(60\) −3530.68 −0.126614
\(61\) −29662.0 −1.02065 −0.510324 0.859982i \(-0.670475\pi\)
−0.510324 + 0.859982i \(0.670475\pi\)
\(62\) −1412.43 −0.0466648
\(63\) −20424.3 −0.648331
\(64\) −30101.8 −0.918635
\(65\) 28705.8 0.842726
\(66\) −583.016 −0.0164748
\(67\) −18606.7 −0.506386 −0.253193 0.967416i \(-0.581481\pi\)
−0.253193 + 0.967416i \(0.581481\pi\)
\(68\) −20623.2 −0.540860
\(69\) −2367.13 −0.0598549
\(70\) 1517.17 0.0370075
\(71\) 21704.4 0.510977 0.255488 0.966812i \(-0.417764\pi\)
0.255488 + 0.966812i \(0.417764\pi\)
\(72\) −9389.83 −0.213465
\(73\) 57006.3 1.25203 0.626017 0.779810i \(-0.284684\pi\)
0.626017 + 0.779810i \(0.284684\pi\)
\(74\) 2908.06 0.0617339
\(75\) −2796.71 −0.0574108
\(76\) −94710.3 −1.88089
\(77\) −18013.4 −0.346233
\(78\) −3404.11 −0.0633530
\(79\) 26607.9 0.479670 0.239835 0.970814i \(-0.422907\pi\)
0.239835 + 0.970814i \(0.422907\pi\)
\(80\) −24551.3 −0.428894
\(81\) 44853.0 0.759590
\(82\) 8925.36 0.146586
\(83\) −26781.1 −0.426711 −0.213355 0.976975i \(-0.568439\pi\)
−0.213355 + 0.976975i \(0.568439\pi\)
\(84\) 12936.2 0.200036
\(85\) −16336.0 −0.245244
\(86\) −12682.7 −0.184912
\(87\) −31426.3 −0.445139
\(88\) −8281.41 −0.113998
\(89\) −83708.0 −1.12019 −0.560095 0.828428i \(-0.689235\pi\)
−0.560095 + 0.828428i \(0.689235\pi\)
\(90\) −3693.23 −0.0480618
\(91\) −105176. −1.33142
\(92\) −16695.8 −0.205654
\(93\) −9539.56 −0.114372
\(94\) −14252.5 −0.166368
\(95\) −75021.5 −0.852858
\(96\) 8941.41 0.0990212
\(97\) 27667.8 0.298569 0.149285 0.988794i \(-0.452303\pi\)
0.149285 + 0.988794i \(0.452303\pi\)
\(98\) 5576.35 0.0586522
\(99\) 43849.6 0.449654
\(100\) −19725.7 −0.197257
\(101\) −3372.26 −0.0328941 −0.0164471 0.999865i \(-0.505235\pi\)
−0.0164471 + 0.999865i \(0.505235\pi\)
\(102\) 1937.22 0.0184365
\(103\) 180827. 1.67946 0.839731 0.543002i \(-0.182713\pi\)
0.839731 + 0.543002i \(0.182713\pi\)
\(104\) −48353.5 −0.438374
\(105\) 10247.0 0.0907031
\(106\) 7009.77 0.0605953
\(107\) 61042.5 0.515434 0.257717 0.966220i \(-0.417030\pi\)
0.257717 + 0.966220i \(0.417030\pi\)
\(108\) −65808.6 −0.542904
\(109\) 64905.4 0.523257 0.261628 0.965169i \(-0.415741\pi\)
0.261628 + 0.965169i \(0.415741\pi\)
\(110\) −3257.27 −0.0256668
\(111\) 19641.0 0.151306
\(112\) 89954.6 0.677608
\(113\) −177043. −1.30432 −0.652159 0.758082i \(-0.726137\pi\)
−0.652159 + 0.758082i \(0.726137\pi\)
\(114\) 8896.51 0.0641147
\(115\) −13225.0 −0.0932505
\(116\) −221655. −1.52944
\(117\) 256029. 1.72912
\(118\) −10973.0 −0.0725472
\(119\) 59854.1 0.387460
\(120\) 4710.91 0.0298643
\(121\) −122378. −0.759868
\(122\) 19652.1 0.119539
\(123\) 60281.7 0.359271
\(124\) −67284.2 −0.392969
\(125\) −15625.0 −0.0894427
\(126\) 13531.8 0.0759327
\(127\) −60649.2 −0.333669 −0.166834 0.985985i \(-0.553355\pi\)
−0.166834 + 0.985985i \(0.553355\pi\)
\(128\) 83885.9 0.452547
\(129\) −85658.5 −0.453207
\(130\) −19018.5 −0.0987003
\(131\) 312521. 1.59111 0.795557 0.605879i \(-0.207178\pi\)
0.795557 + 0.605879i \(0.207178\pi\)
\(132\) −27773.1 −0.138736
\(133\) 274874. 1.34743
\(134\) 12327.5 0.0593080
\(135\) −52128.0 −0.246171
\(136\) 27517.2 0.127572
\(137\) 92152.8 0.419476 0.209738 0.977758i \(-0.432739\pi\)
0.209738 + 0.977758i \(0.432739\pi\)
\(138\) 1568.30 0.00701022
\(139\) −198591. −0.871812 −0.435906 0.899992i \(-0.643572\pi\)
−0.435906 + 0.899992i \(0.643572\pi\)
\(140\) 72273.6 0.311645
\(141\) −96261.0 −0.407758
\(142\) −14379.8 −0.0598457
\(143\) 225806. 0.923413
\(144\) −218975. −0.880012
\(145\) −175577. −0.693501
\(146\) −37768.5 −0.146638
\(147\) 37662.5 0.143753
\(148\) 138531. 0.519868
\(149\) 458474. 1.69180 0.845901 0.533340i \(-0.179063\pi\)
0.845901 + 0.533340i \(0.179063\pi\)
\(150\) 1852.91 0.00672397
\(151\) −78916.0 −0.281658 −0.140829 0.990034i \(-0.544977\pi\)
−0.140829 + 0.990034i \(0.544977\pi\)
\(152\) 126370. 0.443644
\(153\) −145702. −0.503195
\(154\) 11934.4 0.0405509
\(155\) −53296.8 −0.178185
\(156\) −162162. −0.533503
\(157\) 384967. 1.24645 0.623225 0.782043i \(-0.285822\pi\)
0.623225 + 0.782043i \(0.285822\pi\)
\(158\) −17628.6 −0.0561790
\(159\) 47343.8 0.148515
\(160\) 49955.1 0.154269
\(161\) 48455.6 0.147326
\(162\) −29716.6 −0.0889634
\(163\) −110303. −0.325177 −0.162589 0.986694i \(-0.551984\pi\)
−0.162589 + 0.986694i \(0.551984\pi\)
\(164\) 425177. 1.23441
\(165\) −21999.5 −0.0629076
\(166\) 17743.4 0.0499765
\(167\) 408397. 1.13316 0.566581 0.824006i \(-0.308266\pi\)
0.566581 + 0.824006i \(0.308266\pi\)
\(168\) −17260.5 −0.0471824
\(169\) 947145. 2.55094
\(170\) 10823.1 0.0287230
\(171\) −669122. −1.74991
\(172\) −604165. −1.55716
\(173\) 128548. 0.326550 0.163275 0.986581i \(-0.447794\pi\)
0.163275 + 0.986581i \(0.447794\pi\)
\(174\) 20821.0 0.0521348
\(175\) 57249.1 0.141310
\(176\) −193126. −0.469959
\(177\) −74111.5 −0.177808
\(178\) 55459.2 0.131197
\(179\) 737569. 1.72056 0.860281 0.509821i \(-0.170288\pi\)
0.860281 + 0.509821i \(0.170288\pi\)
\(180\) −175935. −0.404734
\(181\) −614825. −1.39494 −0.697469 0.716615i \(-0.745691\pi\)
−0.697469 + 0.716615i \(0.745691\pi\)
\(182\) 69682.7 0.155936
\(183\) 132730. 0.292981
\(184\) 22276.9 0.0485075
\(185\) 109733. 0.235725
\(186\) 6320.26 0.0133953
\(187\) −128503. −0.268725
\(188\) −678946. −1.40101
\(189\) 190994. 0.388924
\(190\) 49704.2 0.0998869
\(191\) −568606. −1.12779 −0.563894 0.825847i \(-0.690698\pi\)
−0.563894 + 0.825847i \(0.690698\pi\)
\(192\) 134698. 0.263698
\(193\) −268005. −0.517905 −0.258952 0.965890i \(-0.583377\pi\)
−0.258952 + 0.965890i \(0.583377\pi\)
\(194\) −18330.8 −0.0349685
\(195\) −128451. −0.241908
\(196\) 265640. 0.493917
\(197\) 898917. 1.65027 0.825133 0.564939i \(-0.191100\pi\)
0.825133 + 0.564939i \(0.191100\pi\)
\(198\) −29051.8 −0.0526635
\(199\) −911076. −1.63088 −0.815440 0.578842i \(-0.803505\pi\)
−0.815440 + 0.578842i \(0.803505\pi\)
\(200\) 26319.5 0.0465268
\(201\) 83259.8 0.145360
\(202\) 2234.23 0.00385257
\(203\) 643303. 1.09566
\(204\) 92283.5 0.155256
\(205\) 336790. 0.559724
\(206\) −119804. −0.196699
\(207\) −117955. −0.191333
\(208\) −1.12763e6 −1.80720
\(209\) −590136. −0.934515
\(210\) −6788.94 −0.0106232
\(211\) −147387. −0.227904 −0.113952 0.993486i \(-0.536351\pi\)
−0.113952 + 0.993486i \(0.536351\pi\)
\(212\) 333924. 0.510280
\(213\) −97121.1 −0.146678
\(214\) −40442.6 −0.0603678
\(215\) −478568. −0.706070
\(216\) 87807.0 0.128054
\(217\) 195276. 0.281514
\(218\) −43001.9 −0.0612840
\(219\) −255088. −0.359401
\(220\) −155166. −0.216143
\(221\) −750301. −1.03337
\(222\) −13012.8 −0.0177210
\(223\) 302710. 0.407629 0.203814 0.979010i \(-0.434666\pi\)
0.203814 + 0.979010i \(0.434666\pi\)
\(224\) −183032. −0.243729
\(225\) −139360. −0.183520
\(226\) 117297. 0.152762
\(227\) 989967. 1.27513 0.637567 0.770395i \(-0.279941\pi\)
0.637567 + 0.770395i \(0.279941\pi\)
\(228\) 423803. 0.539917
\(229\) −321395. −0.404995 −0.202498 0.979283i \(-0.564906\pi\)
−0.202498 + 0.979283i \(0.564906\pi\)
\(230\) 8761.99 0.0109215
\(231\) 80604.9 0.0993875
\(232\) 295750. 0.360749
\(233\) −769949. −0.929121 −0.464560 0.885541i \(-0.653788\pi\)
−0.464560 + 0.885541i \(0.653788\pi\)
\(234\) −169628. −0.202515
\(235\) −537803. −0.635264
\(236\) −522721. −0.610928
\(237\) −119063. −0.137691
\(238\) −39655.3 −0.0453794
\(239\) 45691.3 0.0517414 0.0258707 0.999665i \(-0.491764\pi\)
0.0258707 + 0.999665i \(0.491764\pi\)
\(240\) 109861. 0.123116
\(241\) −31921.2 −0.0354027 −0.0177014 0.999843i \(-0.505635\pi\)
−0.0177014 + 0.999843i \(0.505635\pi\)
\(242\) 81079.1 0.0889960
\(243\) −707389. −0.768498
\(244\) 936165. 1.00665
\(245\) 210418. 0.223958
\(246\) −39938.6 −0.0420780
\(247\) −3.44568e6 −3.59363
\(248\) 89775.8 0.0926894
\(249\) 119838. 0.122489
\(250\) 10352.1 0.0104756
\(251\) 1.41375e6 1.41641 0.708205 0.706007i \(-0.249505\pi\)
0.708205 + 0.706007i \(0.249505\pi\)
\(252\) 644614. 0.639438
\(253\) −104031. −0.102179
\(254\) 40182.0 0.0390794
\(255\) 73099.2 0.0703983
\(256\) 907682. 0.865633
\(257\) 2.03450e6 1.92143 0.960717 0.277530i \(-0.0895159\pi\)
0.960717 + 0.277530i \(0.0895159\pi\)
\(258\) 56751.5 0.0530797
\(259\) −402054. −0.372422
\(260\) −905986. −0.831166
\(261\) −1.56598e6 −1.42294
\(262\) −207055. −0.186352
\(263\) 1.48823e6 1.32673 0.663363 0.748298i \(-0.269129\pi\)
0.663363 + 0.748298i \(0.269129\pi\)
\(264\) 37057.1 0.0327236
\(265\) 264507. 0.231378
\(266\) −182113. −0.157811
\(267\) 374570. 0.321555
\(268\) 587246. 0.499440
\(269\) 206275. 0.173807 0.0869033 0.996217i \(-0.472303\pi\)
0.0869033 + 0.996217i \(0.472303\pi\)
\(270\) 34536.5 0.0288316
\(271\) −1.01450e6 −0.839133 −0.419567 0.907725i \(-0.637818\pi\)
−0.419567 + 0.907725i \(0.637818\pi\)
\(272\) 641713. 0.525919
\(273\) 470636. 0.382189
\(274\) −61054.2 −0.0491291
\(275\) −122910. −0.0980064
\(276\) 74709.2 0.0590339
\(277\) 756032. 0.592026 0.296013 0.955184i \(-0.404343\pi\)
0.296013 + 0.955184i \(0.404343\pi\)
\(278\) 131573. 0.102107
\(279\) −475358. −0.365604
\(280\) −96433.2 −0.0735075
\(281\) −615482. −0.464996 −0.232498 0.972597i \(-0.574690\pi\)
−0.232498 + 0.972597i \(0.574690\pi\)
\(282\) 63776.0 0.0477567
\(283\) −1.72808e6 −1.28262 −0.641311 0.767281i \(-0.721609\pi\)
−0.641311 + 0.767281i \(0.721609\pi\)
\(284\) −685012. −0.503968
\(285\) 335701. 0.244816
\(286\) −149604. −0.108150
\(287\) −1.23398e6 −0.884305
\(288\) 445553. 0.316532
\(289\) −992873. −0.699277
\(290\) 116325. 0.0812230
\(291\) −123806. −0.0857055
\(292\) −1.79918e6 −1.23486
\(293\) 1.24616e6 0.848018 0.424009 0.905658i \(-0.360622\pi\)
0.424009 + 0.905658i \(0.360622\pi\)
\(294\) −24952.6 −0.0168364
\(295\) −414056. −0.277015
\(296\) −184839. −0.122621
\(297\) −410051. −0.269741
\(298\) −303754. −0.198144
\(299\) −607415. −0.392923
\(300\) 88267.0 0.0566233
\(301\) 1.75345e6 1.11552
\(302\) 52284.4 0.0329879
\(303\) 15090.0 0.00944239
\(304\) 2.94700e6 1.82893
\(305\) 741551. 0.456448
\(306\) 96532.2 0.0589344
\(307\) 2.94092e6 1.78089 0.890444 0.455092i \(-0.150394\pi\)
0.890444 + 0.455092i \(0.150394\pi\)
\(308\) 568521. 0.341483
\(309\) −809152. −0.482096
\(310\) 35310.9 0.0208691
\(311\) 2.07137e6 1.21438 0.607192 0.794555i \(-0.292296\pi\)
0.607192 + 0.794555i \(0.292296\pi\)
\(312\) 216369. 0.125837
\(313\) 1.18809e6 0.685473 0.342736 0.939432i \(-0.388646\pi\)
0.342736 + 0.939432i \(0.388646\pi\)
\(314\) −255053. −0.145985
\(315\) 510608. 0.289942
\(316\) −839772. −0.473090
\(317\) 108896. 0.0608646 0.0304323 0.999537i \(-0.490312\pi\)
0.0304323 + 0.999537i \(0.490312\pi\)
\(318\) −31366.8 −0.0173941
\(319\) −1.38113e6 −0.759900
\(320\) 752546. 0.410826
\(321\) −273149. −0.147957
\(322\) −32103.4 −0.0172549
\(323\) 1.96088e6 1.04579
\(324\) −1.41561e6 −0.749170
\(325\) −717645. −0.376879
\(326\) 73079.6 0.0380849
\(327\) −290434. −0.150203
\(328\) −567305. −0.291160
\(329\) 1.97048e6 1.00365
\(330\) 14575.4 0.00736776
\(331\) 2.19592e6 1.10166 0.550829 0.834618i \(-0.314312\pi\)
0.550829 + 0.834618i \(0.314312\pi\)
\(332\) 845240. 0.420857
\(333\) 978713. 0.483665
\(334\) −270576. −0.132716
\(335\) 465167. 0.226463
\(336\) −402523. −0.194510
\(337\) 3635.74 0.00174389 0.000871944 1.00000i \(-0.499722\pi\)
0.000871944 1.00000i \(0.499722\pi\)
\(338\) −627514. −0.298766
\(339\) 792221. 0.374410
\(340\) 515581. 0.241880
\(341\) −419245. −0.195246
\(342\) 443315. 0.204950
\(343\) −2.31045e6 −1.06038
\(344\) 806124. 0.367287
\(345\) 59178.3 0.0267679
\(346\) −85167.1 −0.0382456
\(347\) −2.16277e6 −0.964242 −0.482121 0.876105i \(-0.660134\pi\)
−0.482121 + 0.876105i \(0.660134\pi\)
\(348\) 991848. 0.439033
\(349\) −1.65950e6 −0.729312 −0.364656 0.931142i \(-0.618813\pi\)
−0.364656 + 0.931142i \(0.618813\pi\)
\(350\) −37929.4 −0.0165503
\(351\) −2.39420e6 −1.03727
\(352\) 392958. 0.169040
\(353\) 87033.3 0.0371748 0.0185874 0.999827i \(-0.494083\pi\)
0.0185874 + 0.999827i \(0.494083\pi\)
\(354\) 49101.3 0.0208250
\(355\) −542609. −0.228516
\(356\) 2.64191e6 1.10482
\(357\) −267831. −0.111222
\(358\) −488663. −0.201513
\(359\) 1.19140e6 0.487889 0.243944 0.969789i \(-0.421559\pi\)
0.243944 + 0.969789i \(0.421559\pi\)
\(360\) 234746. 0.0954644
\(361\) 6.52906e6 2.63683
\(362\) 407342. 0.163376
\(363\) 547607. 0.218123
\(364\) 3.31948e6 1.31316
\(365\) −1.42516e6 −0.559926
\(366\) −87937.6 −0.0343141
\(367\) 1.49277e6 0.578533 0.289266 0.957249i \(-0.406589\pi\)
0.289266 + 0.957249i \(0.406589\pi\)
\(368\) 519506. 0.199973
\(369\) 3.00385e6 1.14845
\(370\) −72701.4 −0.0276082
\(371\) −969136. −0.365553
\(372\) 301078. 0.112803
\(373\) 292777. 0.108960 0.0544798 0.998515i \(-0.482650\pi\)
0.0544798 + 0.998515i \(0.482650\pi\)
\(374\) 85137.1 0.0314731
\(375\) 69917.7 0.0256749
\(376\) 905903. 0.330455
\(377\) −8.06412e6 −2.92216
\(378\) −126540. −0.0455509
\(379\) −1.06059e6 −0.379270 −0.189635 0.981855i \(-0.560731\pi\)
−0.189635 + 0.981855i \(0.560731\pi\)
\(380\) 2.36776e6 0.841159
\(381\) 271389. 0.0957809
\(382\) 376720. 0.132087
\(383\) −3.52526e6 −1.22799 −0.613993 0.789311i \(-0.710438\pi\)
−0.613993 + 0.789311i \(0.710438\pi\)
\(384\) −375367. −0.129906
\(385\) 450334. 0.154840
\(386\) 177562. 0.0606571
\(387\) −4.26838e6 −1.44873
\(388\) −873224. −0.294474
\(389\) −2.53780e6 −0.850321 −0.425161 0.905118i \(-0.639782\pi\)
−0.425161 + 0.905118i \(0.639782\pi\)
\(390\) 85102.8 0.0283323
\(391\) 345670. 0.114346
\(392\) −354438. −0.116500
\(393\) −1.39845e6 −0.456736
\(394\) −595561. −0.193280
\(395\) −665197. −0.214515
\(396\) −1.38394e6 −0.443486
\(397\) −2.75523e6 −0.877366 −0.438683 0.898642i \(-0.644555\pi\)
−0.438683 + 0.898642i \(0.644555\pi\)
\(398\) 603617. 0.191009
\(399\) −1.22999e6 −0.386784
\(400\) 613783. 0.191807
\(401\) −1.07771e6 −0.334689 −0.167345 0.985898i \(-0.553519\pi\)
−0.167345 + 0.985898i \(0.553519\pi\)
\(402\) −55162.3 −0.0170246
\(403\) −2.44789e6 −0.750807
\(404\) 106432. 0.0324429
\(405\) −1.12133e6 −0.339699
\(406\) −426209. −0.128324
\(407\) 863182. 0.258295
\(408\) −123132. −0.0366201
\(409\) −4.20349e6 −1.24252 −0.621258 0.783606i \(-0.713378\pi\)
−0.621258 + 0.783606i \(0.713378\pi\)
\(410\) −223134. −0.0655550
\(411\) −412359. −0.120412
\(412\) −5.70709e6 −1.65643
\(413\) 1.51708e6 0.437655
\(414\) 78148.8 0.0224090
\(415\) 669528. 0.190831
\(416\) 2.29440e6 0.650034
\(417\) 888642. 0.250257
\(418\) 390984. 0.109451
\(419\) 1.54943e6 0.431159 0.215579 0.976486i \(-0.430836\pi\)
0.215579 + 0.976486i \(0.430836\pi\)
\(420\) −323405. −0.0894589
\(421\) −3.46727e6 −0.953417 −0.476708 0.879061i \(-0.658170\pi\)
−0.476708 + 0.879061i \(0.658170\pi\)
\(422\) 97648.6 0.0266922
\(423\) −4.79671e6 −1.30344
\(424\) −445548. −0.120359
\(425\) 408400. 0.109676
\(426\) 64345.9 0.0171790
\(427\) −2.71700e6 −0.721140
\(428\) −1.92657e6 −0.508364
\(429\) −1.01042e6 −0.265070
\(430\) 317067. 0.0826951
\(431\) 7.13047e6 1.84895 0.924475 0.381242i \(-0.124504\pi\)
0.924475 + 0.381242i \(0.124504\pi\)
\(432\) 2.04770e6 0.527906
\(433\) 7.40722e6 1.89861 0.949304 0.314360i \(-0.101790\pi\)
0.949304 + 0.314360i \(0.101790\pi\)
\(434\) −129377. −0.0329710
\(435\) 785659. 0.199072
\(436\) −2.04848e6 −0.516079
\(437\) 1.58745e6 0.397647
\(438\) 169004. 0.0420932
\(439\) −3.50561e6 −0.868165 −0.434082 0.900873i \(-0.642927\pi\)
−0.434082 + 0.900873i \(0.642927\pi\)
\(440\) 207035. 0.0509815
\(441\) 1.87673e6 0.459521
\(442\) 497099. 0.121028
\(443\) −197545. −0.0478252 −0.0239126 0.999714i \(-0.507612\pi\)
−0.0239126 + 0.999714i \(0.507612\pi\)
\(444\) −619889. −0.149230
\(445\) 2.09270e6 0.500964
\(446\) −200555. −0.0477416
\(447\) −2.05155e6 −0.485638
\(448\) −2.75728e6 −0.649062
\(449\) 979874. 0.229379 0.114690 0.993401i \(-0.463413\pi\)
0.114690 + 0.993401i \(0.463413\pi\)
\(450\) 92330.8 0.0214939
\(451\) 2.64926e6 0.613315
\(452\) 5.58768e6 1.28643
\(453\) 353128. 0.0808511
\(454\) −655885. −0.149344
\(455\) 2.62941e6 0.595429
\(456\) −565471. −0.127350
\(457\) −6.57094e6 −1.47176 −0.735880 0.677112i \(-0.763231\pi\)
−0.735880 + 0.677112i \(0.763231\pi\)
\(458\) 212934. 0.0474332
\(459\) 1.36250e6 0.301860
\(460\) 417395. 0.0919713
\(461\) 1.10170e6 0.241441 0.120721 0.992687i \(-0.461480\pi\)
0.120721 + 0.992687i \(0.461480\pi\)
\(462\) −53403.4 −0.0116403
\(463\) −1.39041e6 −0.301432 −0.150716 0.988577i \(-0.548158\pi\)
−0.150716 + 0.988577i \(0.548158\pi\)
\(464\) 6.89703e6 1.48719
\(465\) 238489. 0.0511488
\(466\) 510116. 0.108819
\(467\) 2.04924e6 0.434811 0.217406 0.976081i \(-0.430241\pi\)
0.217406 + 0.976081i \(0.430241\pi\)
\(468\) −8.08055e6 −1.70540
\(469\) −1.70434e6 −0.357787
\(470\) 356312. 0.0744022
\(471\) −1.72262e6 −0.357798
\(472\) 697456. 0.144099
\(473\) −3.76453e6 −0.773673
\(474\) 78883.1 0.0161264
\(475\) 1.87554e6 0.381410
\(476\) −1.88906e6 −0.382145
\(477\) 2.35915e6 0.474745
\(478\) −30271.9 −0.00605997
\(479\) −6.44789e6 −1.28404 −0.642020 0.766688i \(-0.721904\pi\)
−0.642020 + 0.766688i \(0.721904\pi\)
\(480\) −223535. −0.0442836
\(481\) 5.03994e6 0.993260
\(482\) 21148.8 0.00414638
\(483\) −216826. −0.0422905
\(484\) 3.86236e6 0.749445
\(485\) −691694. −0.133524
\(486\) 468668. 0.0900067
\(487\) −1.29208e6 −0.246868 −0.123434 0.992353i \(-0.539391\pi\)
−0.123434 + 0.992353i \(0.539391\pi\)
\(488\) −1.24911e6 −0.237438
\(489\) 493578. 0.0933435
\(490\) −139409. −0.0262301
\(491\) −6.75526e6 −1.26456 −0.632278 0.774741i \(-0.717880\pi\)
−0.632278 + 0.774741i \(0.717880\pi\)
\(492\) −1.90255e6 −0.354343
\(493\) 4.58915e6 0.850384
\(494\) 2.28288e6 0.420887
\(495\) −1.09624e6 −0.201091
\(496\) 2.09361e6 0.382113
\(497\) 1.98809e6 0.361031
\(498\) −79396.7 −0.0143459
\(499\) −9.20077e6 −1.65414 −0.827071 0.562097i \(-0.809995\pi\)
−0.827071 + 0.562097i \(0.809995\pi\)
\(500\) 493141. 0.0882158
\(501\) −1.82747e6 −0.325278
\(502\) −936656. −0.165890
\(503\) 5.28453e6 0.931292 0.465646 0.884971i \(-0.345822\pi\)
0.465646 + 0.884971i \(0.345822\pi\)
\(504\) −860094. −0.150824
\(505\) 84306.6 0.0147107
\(506\) 68923.8 0.0119672
\(507\) −4.23822e6 −0.732257
\(508\) 1.91415e6 0.329092
\(509\) 7.72270e6 1.32122 0.660609 0.750730i \(-0.270298\pi\)
0.660609 + 0.750730i \(0.270298\pi\)
\(510\) −48430.6 −0.00824506
\(511\) 5.22169e6 0.884625
\(512\) −3.28572e6 −0.553931
\(513\) 6.25715e6 1.04974
\(514\) −1.34792e6 −0.225039
\(515\) −4.52068e6 −0.751078
\(516\) 2.70347e6 0.446990
\(517\) −4.23048e6 −0.696087
\(518\) 266374. 0.0436181
\(519\) −575217. −0.0937375
\(520\) 1.20884e6 0.196047
\(521\) 2.51070e6 0.405230 0.202615 0.979259i \(-0.435056\pi\)
0.202615 + 0.979259i \(0.435056\pi\)
\(522\) 1.03751e6 0.166655
\(523\) 1.46743e6 0.234587 0.117294 0.993097i \(-0.462578\pi\)
0.117294 + 0.993097i \(0.462578\pi\)
\(524\) −9.86350e6 −1.56929
\(525\) −256174. −0.0405636
\(526\) −986002. −0.155387
\(527\) 1.39305e6 0.218494
\(528\) 864188. 0.134904
\(529\) 279841. 0.0434783
\(530\) −175244. −0.0270990
\(531\) −3.69299e6 −0.568384
\(532\) −8.67532e6 −1.32894
\(533\) 1.54685e7 2.35847
\(534\) −248165. −0.0376606
\(535\) −1.52606e6 −0.230509
\(536\) −783550. −0.117803
\(537\) −3.30042e6 −0.493894
\(538\) −136664. −0.0203563
\(539\) 1.65519e6 0.245401
\(540\) 1.64521e6 0.242794
\(541\) 5.92761e6 0.870736 0.435368 0.900253i \(-0.356618\pi\)
0.435368 + 0.900253i \(0.356618\pi\)
\(542\) 672142. 0.0982795
\(543\) 2.75118e6 0.400423
\(544\) −1.30570e6 −0.189168
\(545\) −1.62264e6 −0.234007
\(546\) −311811. −0.0447621
\(547\) 1.15764e7 1.65426 0.827129 0.562012i \(-0.189972\pi\)
0.827129 + 0.562012i \(0.189972\pi\)
\(548\) −2.90844e6 −0.413722
\(549\) 6.61395e6 0.936547
\(550\) 81431.7 0.0114785
\(551\) 2.10752e7 2.95729
\(552\) −99682.9 −0.0139243
\(553\) 2.43724e6 0.338911
\(554\) −500896. −0.0693383
\(555\) −491024. −0.0676659
\(556\) 6.26775e6 0.859854
\(557\) 9.53745e6 1.30255 0.651275 0.758842i \(-0.274235\pi\)
0.651275 + 0.758842i \(0.274235\pi\)
\(558\) 314940. 0.0428196
\(559\) −2.19803e7 −2.97512
\(560\) −2.24887e6 −0.303035
\(561\) 575014. 0.0771386
\(562\) 407777. 0.0544605
\(563\) −6.19086e6 −0.823152 −0.411576 0.911375i \(-0.635022\pi\)
−0.411576 + 0.911375i \(0.635022\pi\)
\(564\) 3.03810e6 0.402165
\(565\) 4.42608e6 0.583309
\(566\) 1.14491e6 0.150221
\(567\) 4.10847e6 0.536689
\(568\) 913998. 0.118871
\(569\) −439420. −0.0568982 −0.0284491 0.999595i \(-0.509057\pi\)
−0.0284491 + 0.999595i \(0.509057\pi\)
\(570\) −222413. −0.0286730
\(571\) 2.34392e6 0.300851 0.150426 0.988621i \(-0.451936\pi\)
0.150426 + 0.988621i \(0.451936\pi\)
\(572\) −7.12669e6 −0.910747
\(573\) 2.54436e6 0.323736
\(574\) 817550. 0.103570
\(575\) 330625. 0.0417029
\(576\) 6.71201e6 0.842940
\(577\) 5.86114e6 0.732896 0.366448 0.930439i \(-0.380574\pi\)
0.366448 + 0.930439i \(0.380574\pi\)
\(578\) 657811. 0.0818995
\(579\) 1.19925e6 0.148667
\(580\) 5.54139e6 0.683988
\(581\) −2.45311e6 −0.301493
\(582\) 82025.4 0.0100378
\(583\) 2.08067e6 0.253531
\(584\) 2.40061e6 0.291265
\(585\) −6.40073e6 −0.773286
\(586\) −825622. −0.0993201
\(587\) 4.61942e6 0.553340 0.276670 0.960965i \(-0.410769\pi\)
0.276670 + 0.960965i \(0.410769\pi\)
\(588\) −1.18867e6 −0.141781
\(589\) 6.39745e6 0.759834
\(590\) 274325. 0.0324441
\(591\) −4.02241e6 −0.473715
\(592\) −4.31053e6 −0.505506
\(593\) −1.67976e7 −1.96160 −0.980802 0.195007i \(-0.937527\pi\)
−0.980802 + 0.195007i \(0.937527\pi\)
\(594\) 271672. 0.0315921
\(595\) −1.49635e6 −0.173277
\(596\) −1.44699e7 −1.66859
\(597\) 4.07682e6 0.468151
\(598\) 402432. 0.0460193
\(599\) 7.58979e6 0.864296 0.432148 0.901803i \(-0.357756\pi\)
0.432148 + 0.901803i \(0.357756\pi\)
\(600\) −117773. −0.0133557
\(601\) 1.03810e6 0.117234 0.0586171 0.998281i \(-0.481331\pi\)
0.0586171 + 0.998281i \(0.481331\pi\)
\(602\) −1.16171e6 −0.130650
\(603\) 4.14885e6 0.464660
\(604\) 2.49067e6 0.277795
\(605\) 3.05944e6 0.339823
\(606\) −9997.59 −0.00110589
\(607\) 5.37355e6 0.591956 0.295978 0.955195i \(-0.404354\pi\)
0.295978 + 0.955195i \(0.404354\pi\)
\(608\) −5.99632e6 −0.657849
\(609\) −2.87861e6 −0.314513
\(610\) −491301. −0.0534593
\(611\) −2.47009e7 −2.67677
\(612\) 4.59851e6 0.496293
\(613\) −1.15856e6 −0.124528 −0.0622639 0.998060i \(-0.519832\pi\)
−0.0622639 + 0.998060i \(0.519832\pi\)
\(614\) −1.94845e6 −0.208578
\(615\) −1.50704e6 −0.160671
\(616\) −758565. −0.0805455
\(617\) 5.81379e6 0.614818 0.307409 0.951578i \(-0.400538\pi\)
0.307409 + 0.951578i \(0.400538\pi\)
\(618\) 536089. 0.0564633
\(619\) 8.95746e6 0.939633 0.469817 0.882764i \(-0.344320\pi\)
0.469817 + 0.882764i \(0.344320\pi\)
\(620\) 1.68210e6 0.175741
\(621\) 1.10303e6 0.114778
\(622\) −1.37235e6 −0.142229
\(623\) −7.66752e6 −0.791471
\(624\) 5.04582e6 0.518765
\(625\) 390625. 0.0400000
\(626\) −787151. −0.0802827
\(627\) 2.64070e6 0.268256
\(628\) −1.21500e7 −1.22935
\(629\) −2.86815e6 −0.289051
\(630\) −338295. −0.0339581
\(631\) 343093. 0.0343035 0.0171518 0.999853i \(-0.494540\pi\)
0.0171518 + 0.999853i \(0.494540\pi\)
\(632\) 1.12049e6 0.111587
\(633\) 659517. 0.0654209
\(634\) −72147.3 −0.00712848
\(635\) 1.51623e6 0.149221
\(636\) −1.49422e6 −0.146478
\(637\) 9.66435e6 0.943678
\(638\) 915041. 0.0889997
\(639\) −4.83957e6 −0.468872
\(640\) −2.09715e6 −0.202385
\(641\) 1.81018e7 1.74011 0.870054 0.492956i \(-0.164084\pi\)
0.870054 + 0.492956i \(0.164084\pi\)
\(642\) 180970. 0.0173288
\(643\) 4.55559e6 0.434527 0.217264 0.976113i \(-0.430287\pi\)
0.217264 + 0.976113i \(0.430287\pi\)
\(644\) −1.52931e6 −0.145305
\(645\) 2.14146e6 0.202680
\(646\) −1.29915e6 −0.122483
\(647\) −6.04974e6 −0.568167 −0.284083 0.958800i \(-0.591689\pi\)
−0.284083 + 0.958800i \(0.591689\pi\)
\(648\) 1.88882e6 0.176706
\(649\) −3.25706e6 −0.303538
\(650\) 475463. 0.0441401
\(651\) −873809. −0.0808098
\(652\) 3.48129e6 0.320717
\(653\) −2.03691e6 −0.186935 −0.0934673 0.995622i \(-0.529795\pi\)
−0.0934673 + 0.995622i \(0.529795\pi\)
\(654\) 192422. 0.0175918
\(655\) −7.81303e6 −0.711568
\(656\) −1.32298e7 −1.20031
\(657\) −1.27111e7 −1.14887
\(658\) −1.30551e6 −0.117548
\(659\) −7.42508e6 −0.666021 −0.333010 0.942923i \(-0.608064\pi\)
−0.333010 + 0.942923i \(0.608064\pi\)
\(660\) 694328. 0.0620447
\(661\) −1.32564e7 −1.18010 −0.590052 0.807365i \(-0.700893\pi\)
−0.590052 + 0.807365i \(0.700893\pi\)
\(662\) −1.45487e6 −0.129027
\(663\) 3.35739e6 0.296632
\(664\) −1.12779e6 −0.0992674
\(665\) −6.87186e6 −0.602587
\(666\) −648429. −0.0566470
\(667\) 3.71520e6 0.323346
\(668\) −1.28894e7 −1.11762
\(669\) −1.35455e6 −0.117011
\(670\) −308188. −0.0265234
\(671\) 5.83321e6 0.500151
\(672\) 819020. 0.0699635
\(673\) −1.08292e7 −0.921635 −0.460818 0.887495i \(-0.652444\pi\)
−0.460818 + 0.887495i \(0.652444\pi\)
\(674\) −2408.80 −0.000204245 0
\(675\) 1.30320e6 0.110091
\(676\) −2.98929e7 −2.51595
\(677\) −1.68467e7 −1.41268 −0.706340 0.707872i \(-0.749655\pi\)
−0.706340 + 0.707872i \(0.749655\pi\)
\(678\) −524872. −0.0438510
\(679\) 2.53433e6 0.210954
\(680\) −687929. −0.0570521
\(681\) −4.42984e6 −0.366033
\(682\) 277763. 0.0228672
\(683\) 1.53967e7 1.26292 0.631461 0.775408i \(-0.282456\pi\)
0.631461 + 0.775408i \(0.282456\pi\)
\(684\) 2.11182e7 1.72590
\(685\) −2.30382e6 −0.187595
\(686\) 1.53075e6 0.124192
\(687\) 1.43815e6 0.116256
\(688\) 1.87992e7 1.51415
\(689\) 1.21486e7 0.974941
\(690\) −39207.5 −0.00313507
\(691\) −1.81046e6 −0.144243 −0.0721213 0.997396i \(-0.522977\pi\)
−0.0721213 + 0.997396i \(0.522977\pi\)
\(692\) −4.05711e6 −0.322071
\(693\) 4.01656e6 0.317703
\(694\) 1.43290e6 0.112932
\(695\) 4.96478e6 0.389886
\(696\) −1.32340e6 −0.103554
\(697\) −8.80287e6 −0.686345
\(698\) 1.09947e6 0.0854173
\(699\) 3.44531e6 0.266708
\(700\) −1.80684e6 −0.139372
\(701\) −861882. −0.0662450 −0.0331225 0.999451i \(-0.510545\pi\)
−0.0331225 + 0.999451i \(0.510545\pi\)
\(702\) 1.58624e6 0.121486
\(703\) −1.31717e7 −1.00520
\(704\) 5.91970e6 0.450161
\(705\) 2.40653e6 0.182355
\(706\) −57662.3 −0.00435392
\(707\) −308894. −0.0232414
\(708\) 2.33904e6 0.175369
\(709\) 2.38646e7 1.78295 0.891474 0.453072i \(-0.149672\pi\)
0.891474 + 0.453072i \(0.149672\pi\)
\(710\) 359496. 0.0267638
\(711\) −5.93294e6 −0.440145
\(712\) −3.52505e6 −0.260594
\(713\) 1.12776e6 0.0830794
\(714\) 177447. 0.0130263
\(715\) −5.64516e6 −0.412963
\(716\) −2.32785e7 −1.69696
\(717\) −204456. −0.0148526
\(718\) −789340. −0.0571417
\(719\) −2.11342e7 −1.52463 −0.762315 0.647207i \(-0.775937\pi\)
−0.762315 + 0.647207i \(0.775937\pi\)
\(720\) 5.47438e6 0.393553
\(721\) 1.65635e7 1.18663
\(722\) −4.32571e6 −0.308827
\(723\) 142839. 0.0101625
\(724\) 1.94045e7 1.37580
\(725\) 4.38942e6 0.310143
\(726\) −362807. −0.0255467
\(727\) 5.46436e6 0.383445 0.191723 0.981449i \(-0.438593\pi\)
0.191723 + 0.981449i \(0.438593\pi\)
\(728\) −4.42911e6 −0.309733
\(729\) −7.73391e6 −0.538989
\(730\) 944213. 0.0655787
\(731\) 1.25086e7 0.865797
\(732\) −4.18909e6 −0.288963
\(733\) −5.11833e6 −0.351858 −0.175929 0.984403i \(-0.556293\pi\)
−0.175929 + 0.984403i \(0.556293\pi\)
\(734\) −989009. −0.0677579
\(735\) −941563. −0.0642882
\(736\) −1.05705e6 −0.0719284
\(737\) 3.65910e6 0.248145
\(738\) −1.99015e6 −0.134507
\(739\) −7.07135e6 −0.476312 −0.238156 0.971227i \(-0.576543\pi\)
−0.238156 + 0.971227i \(0.576543\pi\)
\(740\) −3.46328e6 −0.232492
\(741\) 1.54185e7 1.03157
\(742\) 642084. 0.0428136
\(743\) −2.94959e6 −0.196015 −0.0980077 0.995186i \(-0.531247\pi\)
−0.0980077 + 0.995186i \(0.531247\pi\)
\(744\) −401723. −0.0266069
\(745\) −1.14619e7 −0.756597
\(746\) −193974. −0.0127614
\(747\) 5.97157e6 0.391550
\(748\) 4.05568e6 0.265039
\(749\) 5.59140e6 0.364180
\(750\) −46322.7 −0.00300705
\(751\) −2.21927e7 −1.43585 −0.717927 0.696118i \(-0.754909\pi\)
−0.717927 + 0.696118i \(0.754909\pi\)
\(752\) 2.11261e7 1.36230
\(753\) −6.32616e6 −0.406586
\(754\) 5.34274e6 0.342244
\(755\) 1.97290e6 0.125961
\(756\) −6.02797e6 −0.383589
\(757\) 1.60910e7 1.02057 0.510285 0.860005i \(-0.329540\pi\)
0.510285 + 0.860005i \(0.329540\pi\)
\(758\) 702674. 0.0444202
\(759\) 465510. 0.0293308
\(760\) −3.15925e6 −0.198404
\(761\) −2.99467e7 −1.87451 −0.937255 0.348644i \(-0.886642\pi\)
−0.937255 + 0.348644i \(0.886642\pi\)
\(762\) −179804. −0.0112179
\(763\) 5.94524e6 0.369707
\(764\) 1.79458e7 1.11232
\(765\) 3.64255e6 0.225036
\(766\) 2.33560e6 0.143822
\(767\) −1.90173e7 −1.16724
\(768\) −4.06163e6 −0.248483
\(769\) −2.61805e7 −1.59648 −0.798239 0.602341i \(-0.794235\pi\)
−0.798239 + 0.602341i \(0.794235\pi\)
\(770\) −298361. −0.0181349
\(771\) −9.10385e6 −0.551555
\(772\) 8.45852e6 0.510801
\(773\) 2.50929e7 1.51043 0.755216 0.655476i \(-0.227532\pi\)
0.755216 + 0.655476i \(0.227532\pi\)
\(774\) 2.82794e6 0.169675
\(775\) 1.33242e6 0.0796869
\(776\) 1.16512e6 0.0694573
\(777\) 1.79908e6 0.106905
\(778\) 1.68137e6 0.0995899
\(779\) −4.04263e7 −2.38683
\(780\) 4.05404e6 0.238590
\(781\) −4.26828e6 −0.250395
\(782\) −229017. −0.0133922
\(783\) 1.46439e7 0.853599
\(784\) −8.26566e6 −0.480272
\(785\) −9.62418e6 −0.557429
\(786\) 926517. 0.0534930
\(787\) 1.69454e7 0.975247 0.487623 0.873054i \(-0.337864\pi\)
0.487623 + 0.873054i \(0.337864\pi\)
\(788\) −2.83708e7 −1.62763
\(789\) −6.65944e6 −0.380842
\(790\) 440714. 0.0251240
\(791\) −1.62169e7 −0.921567
\(792\) 1.84656e6 0.104605
\(793\) 3.40589e7 1.92330
\(794\) 1.82543e6 0.102757
\(795\) −1.18360e6 −0.0664180
\(796\) 2.87545e7 1.60851
\(797\) 2.53659e7 1.41450 0.707251 0.706962i \(-0.249935\pi\)
0.707251 + 0.706962i \(0.249935\pi\)
\(798\) 814907. 0.0453003
\(799\) 1.40569e7 0.778973
\(800\) −1.24888e6 −0.0689913
\(801\) 1.86649e7 1.02789
\(802\) 714019. 0.0391989
\(803\) −1.12106e7 −0.613537
\(804\) −2.62777e6 −0.143366
\(805\) −1.21139e6 −0.0658862
\(806\) 1.62180e6 0.0879348
\(807\) −923025. −0.0498919
\(808\) −142010. −0.00765229
\(809\) 5.89962e6 0.316922 0.158461 0.987365i \(-0.449347\pi\)
0.158461 + 0.987365i \(0.449347\pi\)
\(810\) 742914. 0.0397856
\(811\) −2.20181e7 −1.17552 −0.587758 0.809037i \(-0.699989\pi\)
−0.587758 + 0.809037i \(0.699989\pi\)
\(812\) −2.03033e7 −1.08063
\(813\) 4.53964e6 0.240877
\(814\) −571886. −0.0302516
\(815\) 2.75759e6 0.145424
\(816\) −2.87149e6 −0.150967
\(817\) 5.74446e7 3.01089
\(818\) 2.78495e6 0.145524
\(819\) 2.34519e7 1.22171
\(820\) −1.06294e7 −0.552046
\(821\) −2.61484e7 −1.35390 −0.676952 0.736028i \(-0.736699\pi\)
−0.676952 + 0.736028i \(0.736699\pi\)
\(822\) 273201. 0.0141027
\(823\) 1.02919e7 0.529661 0.264830 0.964295i \(-0.414684\pi\)
0.264830 + 0.964295i \(0.414684\pi\)
\(824\) 7.61485e6 0.390700
\(825\) 549988. 0.0281331
\(826\) −1.00511e6 −0.0512583
\(827\) −2.13133e7 −1.08364 −0.541822 0.840493i \(-0.682265\pi\)
−0.541822 + 0.840493i \(0.682265\pi\)
\(828\) 3.72278e6 0.188708
\(829\) −9.73422e6 −0.491943 −0.245971 0.969277i \(-0.579107\pi\)
−0.245971 + 0.969277i \(0.579107\pi\)
\(830\) −443584. −0.0223502
\(831\) −3.38304e6 −0.169943
\(832\) 3.45639e7 1.73107
\(833\) −5.49982e6 −0.274622
\(834\) −588754. −0.0293102
\(835\) −1.02099e7 −0.506765
\(836\) 1.86253e7 0.921696
\(837\) 4.44521e6 0.219320
\(838\) −1.02655e6 −0.0504974
\(839\) −250114. −0.0122668 −0.00613342 0.999981i \(-0.501952\pi\)
−0.00613342 + 0.999981i \(0.501952\pi\)
\(840\) 431512. 0.0211006
\(841\) 2.88124e7 1.40472
\(842\) 2.29718e6 0.111664
\(843\) 2.75411e6 0.133479
\(844\) 4.65169e6 0.224778
\(845\) −2.36786e7 −1.14081
\(846\) 3.17797e6 0.152660
\(847\) −1.12096e7 −0.536885
\(848\) −1.03904e7 −0.496183
\(849\) 7.73270e6 0.368182
\(850\) −270578. −0.0128453
\(851\) −2.32194e6 −0.109908
\(852\) 3.06525e6 0.144666
\(853\) 2.79614e7 1.31579 0.657895 0.753110i \(-0.271447\pi\)
0.657895 + 0.753110i \(0.271447\pi\)
\(854\) 1.80010e6 0.0844601
\(855\) 1.67281e7 0.782582
\(856\) 2.57058e6 0.119907
\(857\) 1.71141e7 0.795980 0.397990 0.917390i \(-0.369708\pi\)
0.397990 + 0.917390i \(0.369708\pi\)
\(858\) 669438. 0.0310450
\(859\) 6.77393e6 0.313226 0.156613 0.987660i \(-0.449942\pi\)
0.156613 + 0.987660i \(0.449942\pi\)
\(860\) 1.51041e7 0.696385
\(861\) 5.52171e6 0.253843
\(862\) −4.72417e6 −0.216550
\(863\) −3.61430e6 −0.165195 −0.0825976 0.996583i \(-0.526322\pi\)
−0.0825976 + 0.996583i \(0.526322\pi\)
\(864\) −4.16649e6 −0.189883
\(865\) −3.21370e6 −0.146038
\(866\) −4.90752e6 −0.222365
\(867\) 4.44284e6 0.200730
\(868\) −6.16313e6 −0.277653
\(869\) −5.23259e6 −0.235054
\(870\) −520524. −0.0233154
\(871\) 2.13648e7 0.954230
\(872\) 2.73325e6 0.121727
\(873\) −6.16927e6 −0.273967
\(874\) −1.05174e6 −0.0465725
\(875\) −1.43123e6 −0.0631958
\(876\) 8.05084e6 0.354471
\(877\) −3.05086e7 −1.33944 −0.669721 0.742613i \(-0.733586\pi\)
−0.669721 + 0.742613i \(0.733586\pi\)
\(878\) 2.32258e6 0.101680
\(879\) −5.57623e6 −0.243427
\(880\) 4.82816e6 0.210172
\(881\) −1.71201e7 −0.743133 −0.371566 0.928406i \(-0.621179\pi\)
−0.371566 + 0.928406i \(0.621179\pi\)
\(882\) −1.24340e6 −0.0538193
\(883\) −1.31582e7 −0.567931 −0.283966 0.958834i \(-0.591650\pi\)
−0.283966 + 0.958834i \(0.591650\pi\)
\(884\) 2.36803e7 1.01919
\(885\) 1.85279e6 0.0795184
\(886\) 130880. 0.00560130
\(887\) −3.48528e7 −1.48740 −0.743701 0.668513i \(-0.766931\pi\)
−0.743701 + 0.668513i \(0.766931\pi\)
\(888\) 827105. 0.0351988
\(889\) −5.55537e6 −0.235754
\(890\) −1.38648e6 −0.0586731
\(891\) −8.82060e6 −0.372223
\(892\) −9.55385e6 −0.402037
\(893\) 6.45549e7 2.70895
\(894\) 1.35922e6 0.0568781
\(895\) −1.84392e7 −0.769458
\(896\) 7.68382e6 0.319748
\(897\) 2.71802e6 0.112790
\(898\) −649198. −0.0268650
\(899\) 1.49723e7 0.617859
\(900\) 4.39836e6 0.181003
\(901\) −6.91357e6 −0.283720
\(902\) −1.75522e6 −0.0718316
\(903\) −7.84619e6 −0.320214
\(904\) −7.45552e6 −0.303429
\(905\) 1.53706e7 0.623836
\(906\) −233958. −0.00946931
\(907\) 2.10620e7 0.850124 0.425062 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(908\) −3.12444e7 −1.25764
\(909\) 751937. 0.0301836
\(910\) −1.74207e6 −0.0697368
\(911\) 1.03915e7 0.414842 0.207421 0.978252i \(-0.433493\pi\)
0.207421 + 0.978252i \(0.433493\pi\)
\(912\) −1.31870e7 −0.525001
\(913\) 5.26666e6 0.209102
\(914\) 4.35346e6 0.172373
\(915\) −3.31824e6 −0.131025
\(916\) 1.01436e7 0.399440
\(917\) 2.86265e7 1.12420
\(918\) −902701. −0.0353539
\(919\) 2.44332e7 0.954315 0.477158 0.878818i \(-0.341667\pi\)
0.477158 + 0.878818i \(0.341667\pi\)
\(920\) −556921. −0.0216932
\(921\) −1.31598e7 −0.511211
\(922\) −729912. −0.0282776
\(923\) −2.49217e7 −0.962881
\(924\) −2.54398e6 −0.0980241
\(925\) −2.74332e6 −0.105420
\(926\) 921189. 0.0353038
\(927\) −4.03202e7 −1.54107
\(928\) −1.40335e7 −0.534929
\(929\) −4.96917e7 −1.88906 −0.944528 0.328430i \(-0.893481\pi\)
−0.944528 + 0.328430i \(0.893481\pi\)
\(930\) −158007. −0.00599057
\(931\) −2.52574e7 −0.955024
\(932\) 2.43004e7 0.916376
\(933\) −9.26880e6 −0.348594
\(934\) −1.35769e6 −0.0509252
\(935\) 3.21257e6 0.120177
\(936\) 1.07817e7 0.402252
\(937\) −4.31900e7 −1.60707 −0.803534 0.595258i \(-0.797050\pi\)
−0.803534 + 0.595258i \(0.797050\pi\)
\(938\) 1.12918e6 0.0419041
\(939\) −5.31640e6 −0.196768
\(940\) 1.69736e7 0.626550
\(941\) 6.09014e6 0.224209 0.112105 0.993696i \(-0.464241\pi\)
0.112105 + 0.993696i \(0.464241\pi\)
\(942\) 1.14129e6 0.0419054
\(943\) −7.12647e6 −0.260973
\(944\) 1.62650e7 0.594051
\(945\) −4.77485e6 −0.173932
\(946\) 2.49412e6 0.0906128
\(947\) 3.57186e7 1.29425 0.647127 0.762382i \(-0.275970\pi\)
0.647127 + 0.762382i \(0.275970\pi\)
\(948\) 3.75775e6 0.135802
\(949\) −6.54565e7 −2.35932
\(950\) −1.24260e6 −0.0446708
\(951\) −487282. −0.0174714
\(952\) 2.52053e6 0.0901363
\(953\) −1.94149e7 −0.692472 −0.346236 0.938147i \(-0.612540\pi\)
−0.346236 + 0.938147i \(0.612540\pi\)
\(954\) −1.56302e6 −0.0556022
\(955\) 1.42151e7 0.504363
\(956\) −1.44206e6 −0.0510317
\(957\) 6.18017e6 0.218132
\(958\) 4.27194e6 0.150387
\(959\) 8.44106e6 0.296381
\(960\) −3.36744e6 −0.117929
\(961\) −2.40843e7 −0.841250
\(962\) −3.33913e6 −0.116331
\(963\) −1.36111e7 −0.472962
\(964\) 1.00747e6 0.0349171
\(965\) 6.70013e6 0.231614
\(966\) 143654. 0.00495308
\(967\) 1.49642e7 0.514622 0.257311 0.966329i \(-0.417164\pi\)
0.257311 + 0.966329i \(0.417164\pi\)
\(968\) −5.15347e6 −0.176771
\(969\) −8.77441e6 −0.300199
\(970\) 458270. 0.0156384
\(971\) 6.58853e6 0.224254 0.112127 0.993694i \(-0.464234\pi\)
0.112127 + 0.993694i \(0.464234\pi\)
\(972\) 2.23259e7 0.757957
\(973\) −1.81907e7 −0.615980
\(974\) 856042. 0.0289133
\(975\) 3.21127e6 0.108185
\(976\) −2.91297e7 −0.978840
\(977\) −1.95844e6 −0.0656408 −0.0328204 0.999461i \(-0.510449\pi\)
−0.0328204 + 0.999461i \(0.510449\pi\)
\(978\) −327012. −0.0109324
\(979\) 1.64616e7 0.548929
\(980\) −6.64101e6 −0.220886
\(981\) −1.44724e7 −0.480140
\(982\) 4.47558e6 0.148105
\(983\) 3.34298e7 1.10344 0.551722 0.834028i \(-0.313971\pi\)
0.551722 + 0.834028i \(0.313971\pi\)
\(984\) 2.53854e6 0.0835787
\(985\) −2.24729e7 −0.738021
\(986\) −3.04046e6 −0.0995973
\(987\) −8.81736e6 −0.288102
\(988\) 1.08749e8 3.54433
\(989\) 1.01265e7 0.329207
\(990\) 726295. 0.0235519
\(991\) 2.04616e7 0.661843 0.330921 0.943658i \(-0.392640\pi\)
0.330921 + 0.943658i \(0.392640\pi\)
\(992\) −4.25991e6 −0.137443
\(993\) −9.82616e6 −0.316235
\(994\) −1.31717e6 −0.0422840
\(995\) 2.27769e7 0.729352
\(996\) −3.78222e6 −0.120809
\(997\) −4.83696e7 −1.54111 −0.770557 0.637372i \(-0.780022\pi\)
−0.770557 + 0.637372i \(0.780022\pi\)
\(998\) 6.09581e6 0.193734
\(999\) −9.15223e6 −0.290144
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 115.6.a.c.1.4 7
3.2 odd 2 1035.6.a.b.1.4 7
5.4 even 2 575.6.a.d.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.6.a.c.1.4 7 1.1 even 1 trivial
575.6.a.d.1.4 7 5.4 even 2
1035.6.a.b.1.4 7 3.2 odd 2