Properties

Label 33.4.a.d.1.1
Level $33$
Weight $4$
Character 33.1
Self dual yes
Analytic conductor $1.947$
Analytic rank $0$
Dimension $2$
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] = [33,4,Mod(1,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, 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("33.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.37228 q^{2} +3.00000 q^{3} -2.37228 q^{4} +19.4891 q^{5} -7.11684 q^{6} +6.74456 q^{7} +24.6060 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.37228 q^{2} +3.00000 q^{3} -2.37228 q^{4} +19.4891 q^{5} -7.11684 q^{6} +6.74456 q^{7} +24.6060 q^{8} +9.00000 q^{9} -46.2337 q^{10} +11.0000 q^{11} -7.11684 q^{12} -60.9783 q^{13} -16.0000 q^{14} +58.4674 q^{15} -39.3940 q^{16} -99.1684 q^{17} -21.3505 q^{18} +24.7011 q^{19} -46.2337 q^{20} +20.2337 q^{21} -26.0951 q^{22} +112.000 q^{23} +73.8179 q^{24} +254.826 q^{25} +144.658 q^{26} +27.0000 q^{27} -16.0000 q^{28} -21.1249 q^{29} -138.701 q^{30} -318.717 q^{31} -103.394 q^{32} +33.0000 q^{33} +235.255 q^{34} +131.446 q^{35} -21.3505 q^{36} -150.380 q^{37} -58.5979 q^{38} -182.935 q^{39} +479.549 q^{40} -252.745 q^{41} -48.0000 q^{42} +214.016 q^{43} -26.0951 q^{44} +175.402 q^{45} -265.696 q^{46} +105.870 q^{47} -118.182 q^{48} -297.511 q^{49} -604.519 q^{50} -297.505 q^{51} +144.658 q^{52} +325.652 q^{53} -64.0516 q^{54} +214.380 q^{55} +165.957 q^{56} +74.1032 q^{57} +50.1143 q^{58} +196.000 q^{59} -138.701 q^{60} -402.641 q^{61} +756.087 q^{62} +60.7011 q^{63} +560.432 q^{64} -1188.41 q^{65} -78.2853 q^{66} +27.4132 q^{67} +235.255 q^{68} +336.000 q^{69} -311.826 q^{70} -300.500 q^{71} +221.454 q^{72} +427.815 q^{73} +356.745 q^{74} +764.478 q^{75} -58.5979 q^{76} +74.1902 q^{77} +433.973 q^{78} +97.5488 q^{79} -767.755 q^{80} +81.0000 q^{81} +599.581 q^{82} +1104.62 q^{83} -48.0000 q^{84} -1932.71 q^{85} -507.707 q^{86} -63.3748 q^{87} +270.666 q^{88} +463.022 q^{89} -416.103 q^{90} -411.272 q^{91} -265.696 q^{92} -956.152 q^{93} -251.152 q^{94} +481.402 q^{95} -310.182 q^{96} -1798.36 q^{97} +705.779 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9} - 58 q^{10} + 22 q^{11} + 3 q^{12} - 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} - 26 q^{17} + 9 q^{18} - 54 q^{19} - 58 q^{20} + 6 q^{21} + 11 q^{22} + 224 q^{23} + 27 q^{24} + 142 q^{25} + 94 q^{26} + 54 q^{27} - 32 q^{28} + 222 q^{29} - 174 q^{30} - 40 q^{31} - 247 q^{32} + 66 q^{33} + 482 q^{34} + 148 q^{35} + 9 q^{36} - 48 q^{37} - 324 q^{38} - 228 q^{39} + 534 q^{40} - 494 q^{41} - 96 q^{42} - 66 q^{43} + 11 q^{44} + 144 q^{45} + 112 q^{46} - 64 q^{47} - 357 q^{48} - 618 q^{49} - 985 q^{50} - 78 q^{51} + 94 q^{52} - 84 q^{53} + 27 q^{54} + 176 q^{55} + 240 q^{56} - 162 q^{57} + 870 q^{58} + 392 q^{59} - 174 q^{60} - 1104 q^{61} + 1696 q^{62} + 18 q^{63} + 713 q^{64} - 1136 q^{65} + 33 q^{66} + 928 q^{67} + 482 q^{68} + 672 q^{69} - 256 q^{70} + 456 q^{71} + 81 q^{72} - 592 q^{73} + 702 q^{74} + 426 q^{75} - 324 q^{76} + 22 q^{77} + 282 q^{78} - 230 q^{79} - 490 q^{80} + 162 q^{81} - 214 q^{82} + 348 q^{83} - 96 q^{84} - 2188 q^{85} - 1452 q^{86} + 666 q^{87} + 99 q^{88} + 972 q^{89} - 522 q^{90} - 340 q^{91} + 112 q^{92} - 120 q^{93} - 824 q^{94} + 756 q^{95} - 741 q^{96} - 1184 q^{97} - 375 q^{98} + 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37228 −0.838728 −0.419364 0.907818i \(-0.637747\pi\)
−0.419364 + 0.907818i \(0.637747\pi\)
\(3\) 3.00000 0.577350
\(4\) −2.37228 −0.296535
\(5\) 19.4891 1.74316 0.871580 0.490253i \(-0.163096\pi\)
0.871580 + 0.490253i \(0.163096\pi\)
\(6\) −7.11684 −0.484240
\(7\) 6.74456 0.364172 0.182086 0.983283i \(-0.441715\pi\)
0.182086 + 0.983283i \(0.441715\pi\)
\(8\) 24.6060 1.08744
\(9\) 9.00000 0.333333
\(10\) −46.2337 −1.46204
\(11\) 11.0000 0.301511
\(12\) −7.11684 −0.171205
\(13\) −60.9783 −1.30095 −0.650474 0.759529i \(-0.725430\pi\)
−0.650474 + 0.759529i \(0.725430\pi\)
\(14\) −16.0000 −0.305441
\(15\) 58.4674 1.00641
\(16\) −39.3940 −0.615532
\(17\) −99.1684 −1.41482 −0.707408 0.706805i \(-0.750136\pi\)
−0.707408 + 0.706805i \(0.750136\pi\)
\(18\) −21.3505 −0.279576
\(19\) 24.7011 0.298253 0.149127 0.988818i \(-0.452354\pi\)
0.149127 + 0.988818i \(0.452354\pi\)
\(20\) −46.2337 −0.516908
\(21\) 20.2337 0.210255
\(22\) −26.0951 −0.252886
\(23\) 112.000 1.01537 0.507687 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 73.8179 0.627834
\(25\) 254.826 2.03861
\(26\) 144.658 1.09114
\(27\) 27.0000 0.192450
\(28\) −16.0000 −0.107990
\(29\) −21.1249 −0.135269 −0.0676345 0.997710i \(-0.521545\pi\)
−0.0676345 + 0.997710i \(0.521545\pi\)
\(30\) −138.701 −0.844108
\(31\) −318.717 −1.84656 −0.923279 0.384130i \(-0.874502\pi\)
−0.923279 + 0.384130i \(0.874502\pi\)
\(32\) −103.394 −0.571177
\(33\) 33.0000 0.174078
\(34\) 235.255 1.18665
\(35\) 131.446 0.634810
\(36\) −21.3505 −0.0988451
\(37\) −150.380 −0.668172 −0.334086 0.942543i \(-0.608428\pi\)
−0.334086 + 0.942543i \(0.608428\pi\)
\(38\) −58.5979 −0.250153
\(39\) −182.935 −0.751103
\(40\) 479.549 1.89558
\(41\) −252.745 −0.962733 −0.481367 0.876519i \(-0.659859\pi\)
−0.481367 + 0.876519i \(0.659859\pi\)
\(42\) −48.0000 −0.176347
\(43\) 214.016 0.759004 0.379502 0.925191i \(-0.376095\pi\)
0.379502 + 0.925191i \(0.376095\pi\)
\(44\) −26.0951 −0.0894087
\(45\) 175.402 0.581053
\(46\) −265.696 −0.851623
\(47\) 105.870 0.328567 0.164284 0.986413i \(-0.447469\pi\)
0.164284 + 0.986413i \(0.447469\pi\)
\(48\) −118.182 −0.355377
\(49\) −297.511 −0.867379
\(50\) −604.519 −1.70984
\(51\) −297.505 −0.816845
\(52\) 144.658 0.385777
\(53\) 325.652 0.843995 0.421998 0.906597i \(-0.361329\pi\)
0.421998 + 0.906597i \(0.361329\pi\)
\(54\) −64.0516 −0.161413
\(55\) 214.380 0.525583
\(56\) 165.957 0.396016
\(57\) 74.1032 0.172197
\(58\) 50.1143 0.113454
\(59\) 196.000 0.432492 0.216246 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) −138.701 −0.298437
\(61\) −402.641 −0.845130 −0.422565 0.906333i \(-0.638870\pi\)
−0.422565 + 0.906333i \(0.638870\pi\)
\(62\) 756.087 1.54876
\(63\) 60.7011 0.121391
\(64\) 560.432 1.09459
\(65\) −1188.41 −2.26776
\(66\) −78.2853 −0.146004
\(67\) 27.4132 0.0499860 0.0249930 0.999688i \(-0.492044\pi\)
0.0249930 + 0.999688i \(0.492044\pi\)
\(68\) 235.255 0.419543
\(69\) 336.000 0.586227
\(70\) −311.826 −0.532433
\(71\) −300.500 −0.502292 −0.251146 0.967949i \(-0.580807\pi\)
−0.251146 + 0.967949i \(0.580807\pi\)
\(72\) 221.454 0.362480
\(73\) 427.815 0.685917 0.342959 0.939351i \(-0.388571\pi\)
0.342959 + 0.939351i \(0.388571\pi\)
\(74\) 356.745 0.560415
\(75\) 764.478 1.17699
\(76\) −58.5979 −0.0884426
\(77\) 74.1902 0.109802
\(78\) 433.973 0.629971
\(79\) 97.5488 0.138925 0.0694627 0.997585i \(-0.477872\pi\)
0.0694627 + 0.997585i \(0.477872\pi\)
\(80\) −767.755 −1.07297
\(81\) 81.0000 0.111111
\(82\) 599.581 0.807472
\(83\) 1104.62 1.46082 0.730408 0.683011i \(-0.239330\pi\)
0.730408 + 0.683011i \(0.239330\pi\)
\(84\) −48.0000 −0.0623480
\(85\) −1932.71 −2.46625
\(86\) −507.707 −0.636598
\(87\) −63.3748 −0.0780976
\(88\) 270.666 0.327876
\(89\) 463.022 0.551463 0.275732 0.961235i \(-0.411080\pi\)
0.275732 + 0.961235i \(0.411080\pi\)
\(90\) −416.103 −0.487346
\(91\) −411.272 −0.473769
\(92\) −265.696 −0.301094
\(93\) −956.152 −1.06611
\(94\) −251.152 −0.275578
\(95\) 481.402 0.519903
\(96\) −310.182 −0.329769
\(97\) −1798.36 −1.88243 −0.941214 0.337810i \(-0.890314\pi\)
−0.941214 + 0.337810i \(0.890314\pi\)
\(98\) 705.779 0.727495
\(99\) 99.0000 0.100504
\(100\) −604.519 −0.604519
\(101\) 741.918 0.730927 0.365463 0.930826i \(-0.380911\pi\)
0.365463 + 0.930826i \(0.380911\pi\)
\(102\) 705.766 0.685111
\(103\) 389.837 0.372930 0.186465 0.982462i \(-0.440297\pi\)
0.186465 + 0.982462i \(0.440297\pi\)
\(104\) −1500.43 −1.41470
\(105\) 394.337 0.366508
\(106\) −772.538 −0.707882
\(107\) 1538.42 1.38995 0.694977 0.719032i \(-0.255415\pi\)
0.694977 + 0.719032i \(0.255415\pi\)
\(108\) −64.0516 −0.0570682
\(109\) 779.783 0.685226 0.342613 0.939477i \(-0.388688\pi\)
0.342613 + 0.939477i \(0.388688\pi\)
\(110\) −508.571 −0.440821
\(111\) −451.141 −0.385770
\(112\) −265.696 −0.224160
\(113\) −1514.55 −1.26086 −0.630430 0.776246i \(-0.717122\pi\)
−0.630430 + 0.776246i \(0.717122\pi\)
\(114\) −175.794 −0.144426
\(115\) 2182.78 1.76996
\(116\) 50.1143 0.0401120
\(117\) −548.804 −0.433649
\(118\) −464.967 −0.362743
\(119\) −668.848 −0.515237
\(120\) 1438.65 1.09442
\(121\) 121.000 0.0909091
\(122\) 955.179 0.708834
\(123\) −758.234 −0.555834
\(124\) 756.087 0.547569
\(125\) 2530.20 1.81046
\(126\) −144.000 −0.101814
\(127\) 2302.74 1.60894 0.804471 0.593992i \(-0.202449\pi\)
0.804471 + 0.593992i \(0.202449\pi\)
\(128\) −502.350 −0.346890
\(129\) 642.049 0.438211
\(130\) 2819.25 1.90203
\(131\) −2020.59 −1.34763 −0.673815 0.738900i \(-0.735345\pi\)
−0.673815 + 0.738900i \(0.735345\pi\)
\(132\) −78.2853 −0.0516201
\(133\) 166.598 0.108616
\(134\) −65.0319 −0.0419246
\(135\) 526.206 0.335471
\(136\) −2440.14 −1.53853
\(137\) 1475.40 0.920088 0.460044 0.887896i \(-0.347834\pi\)
0.460044 + 0.887896i \(0.347834\pi\)
\(138\) −797.087 −0.491685
\(139\) −1623.71 −0.990802 −0.495401 0.868665i \(-0.664979\pi\)
−0.495401 + 0.868665i \(0.664979\pi\)
\(140\) −311.826 −0.188244
\(141\) 317.609 0.189698
\(142\) 712.870 0.421287
\(143\) −670.761 −0.392251
\(144\) −354.546 −0.205177
\(145\) −411.707 −0.235796
\(146\) −1014.90 −0.575298
\(147\) −892.533 −0.500781
\(148\) 356.745 0.198137
\(149\) −1104.11 −0.607064 −0.303532 0.952821i \(-0.598166\pi\)
−0.303532 + 0.952821i \(0.598166\pi\)
\(150\) −1813.56 −0.987175
\(151\) 2980.07 1.60606 0.803029 0.595940i \(-0.203221\pi\)
0.803029 + 0.595940i \(0.203221\pi\)
\(152\) 607.794 0.324333
\(153\) −892.516 −0.471605
\(154\) −176.000 −0.0920941
\(155\) −6211.52 −3.21885
\(156\) 433.973 0.222728
\(157\) 2844.67 1.44605 0.723024 0.690823i \(-0.242751\pi\)
0.723024 + 0.690823i \(0.242751\pi\)
\(158\) −231.413 −0.116521
\(159\) 976.956 0.487281
\(160\) −2015.06 −0.995653
\(161\) 755.391 0.369771
\(162\) −192.155 −0.0931920
\(163\) −1528.51 −0.734492 −0.367246 0.930124i \(-0.619699\pi\)
−0.367246 + 0.930124i \(0.619699\pi\)
\(164\) 599.581 0.285484
\(165\) 643.141 0.303445
\(166\) −2620.47 −1.22523
\(167\) −383.881 −0.177878 −0.0889388 0.996037i \(-0.528348\pi\)
−0.0889388 + 0.996037i \(0.528348\pi\)
\(168\) 497.870 0.228640
\(169\) 1521.35 0.692466
\(170\) 4584.92 2.06851
\(171\) 222.310 0.0994178
\(172\) −507.707 −0.225071
\(173\) 2702.85 1.18783 0.593914 0.804529i \(-0.297582\pi\)
0.593914 + 0.804529i \(0.297582\pi\)
\(174\) 150.343 0.0655027
\(175\) 1718.69 0.742404
\(176\) −433.334 −0.185590
\(177\) 588.000 0.249699
\(178\) −1098.42 −0.462528
\(179\) −2777.61 −1.15982 −0.579911 0.814680i \(-0.696913\pi\)
−0.579911 + 0.814680i \(0.696913\pi\)
\(180\) −416.103 −0.172303
\(181\) −3993.61 −1.64001 −0.820007 0.572354i \(-0.806030\pi\)
−0.820007 + 0.572354i \(0.806030\pi\)
\(182\) 975.652 0.397363
\(183\) −1207.92 −0.487936
\(184\) 2755.87 1.10416
\(185\) −2930.78 −1.16473
\(186\) 2268.26 0.894177
\(187\) −1090.85 −0.426583
\(188\) −251.152 −0.0974317
\(189\) 182.103 0.0700850
\(190\) −1142.02 −0.436058
\(191\) 895.587 0.339280 0.169640 0.985506i \(-0.445740\pi\)
0.169640 + 0.985506i \(0.445740\pi\)
\(192\) 1681.30 0.631964
\(193\) 1328.24 0.495382 0.247691 0.968839i \(-0.420328\pi\)
0.247691 + 0.968839i \(0.420328\pi\)
\(194\) 4266.21 1.57885
\(195\) −3565.24 −1.30929
\(196\) 705.779 0.257208
\(197\) 154.114 0.0557370 0.0278685 0.999612i \(-0.491128\pi\)
0.0278685 + 0.999612i \(0.491128\pi\)
\(198\) −234.856 −0.0842953
\(199\) 1316.15 0.468842 0.234421 0.972135i \(-0.424681\pi\)
0.234421 + 0.972135i \(0.424681\pi\)
\(200\) 6270.24 2.21686
\(201\) 82.2397 0.0288594
\(202\) −1760.04 −0.613049
\(203\) −142.478 −0.0492612
\(204\) 705.766 0.242223
\(205\) −4925.77 −1.67820
\(206\) −924.803 −0.312787
\(207\) 1008.00 0.338458
\(208\) 2402.18 0.800775
\(209\) 271.712 0.0899268
\(210\) −935.478 −0.307401
\(211\) −1735.76 −0.566324 −0.283162 0.959072i \(-0.591383\pi\)
−0.283162 + 0.959072i \(0.591383\pi\)
\(212\) −772.538 −0.250274
\(213\) −901.499 −0.289999
\(214\) −3649.57 −1.16579
\(215\) 4170.99 1.32307
\(216\) 664.361 0.209278
\(217\) −2149.61 −0.672465
\(218\) −1849.86 −0.574718
\(219\) 1283.44 0.396014
\(220\) −508.571 −0.155854
\(221\) 6047.12 1.84060
\(222\) 1070.23 0.323556
\(223\) 1663.71 0.499597 0.249798 0.968298i \(-0.419636\pi\)
0.249798 + 0.968298i \(0.419636\pi\)
\(224\) −697.348 −0.208007
\(225\) 2293.43 0.679536
\(226\) 3592.95 1.05752
\(227\) −3658.84 −1.06980 −0.534902 0.844914i \(-0.679651\pi\)
−0.534902 + 0.844914i \(0.679651\pi\)
\(228\) −175.794 −0.0510624
\(229\) 1927.07 0.556090 0.278045 0.960568i \(-0.410314\pi\)
0.278045 + 0.960568i \(0.410314\pi\)
\(230\) −5178.17 −1.48452
\(231\) 222.571 0.0633942
\(232\) −519.800 −0.147097
\(233\) −2784.27 −0.782847 −0.391423 0.920211i \(-0.628017\pi\)
−0.391423 + 0.920211i \(0.628017\pi\)
\(234\) 1301.92 0.363714
\(235\) 2063.30 0.572745
\(236\) −464.967 −0.128249
\(237\) 292.646 0.0802086
\(238\) 1586.70 0.432144
\(239\) 1222.60 0.330892 0.165446 0.986219i \(-0.447094\pi\)
0.165446 + 0.986219i \(0.447094\pi\)
\(240\) −2303.27 −0.619480
\(241\) −2013.01 −0.538047 −0.269024 0.963134i \(-0.586701\pi\)
−0.269024 + 0.963134i \(0.586701\pi\)
\(242\) −287.046 −0.0762480
\(243\) 243.000 0.0641500
\(244\) 955.179 0.250611
\(245\) −5798.23 −1.51198
\(246\) 1798.74 0.466194
\(247\) −1506.23 −0.388012
\(248\) −7842.35 −2.00802
\(249\) 3313.86 0.843402
\(250\) −6002.33 −1.51848
\(251\) 2706.63 0.680641 0.340320 0.940310i \(-0.389464\pi\)
0.340320 + 0.940310i \(0.389464\pi\)
\(252\) −144.000 −0.0359966
\(253\) 1232.00 0.306147
\(254\) −5462.76 −1.34946
\(255\) −5798.12 −1.42389
\(256\) −3291.74 −0.803647
\(257\) 225.741 0.0547912 0.0273956 0.999625i \(-0.491279\pi\)
0.0273956 + 0.999625i \(0.491279\pi\)
\(258\) −1523.12 −0.367540
\(259\) −1014.25 −0.243330
\(260\) 2819.25 0.672471
\(261\) −190.124 −0.0450897
\(262\) 4793.40 1.13029
\(263\) 1953.42 0.457997 0.228998 0.973427i \(-0.426455\pi\)
0.228998 + 0.973427i \(0.426455\pi\)
\(264\) 811.997 0.189299
\(265\) 6346.67 1.47122
\(266\) −395.217 −0.0910989
\(267\) 1389.07 0.318387
\(268\) −65.0319 −0.0148226
\(269\) −350.848 −0.0795225 −0.0397613 0.999209i \(-0.512660\pi\)
−0.0397613 + 0.999209i \(0.512660\pi\)
\(270\) −1248.31 −0.281369
\(271\) 254.779 0.0571096 0.0285548 0.999592i \(-0.490909\pi\)
0.0285548 + 0.999592i \(0.490909\pi\)
\(272\) 3906.64 0.870864
\(273\) −1233.81 −0.273531
\(274\) −3500.07 −0.771704
\(275\) 2803.09 0.614663
\(276\) −797.087 −0.173837
\(277\) 116.478 0.0252654 0.0126327 0.999920i \(-0.495979\pi\)
0.0126327 + 0.999920i \(0.495979\pi\)
\(278\) 3851.90 0.831013
\(279\) −2868.46 −0.615519
\(280\) 3234.35 0.690319
\(281\) 8226.47 1.74644 0.873221 0.487325i \(-0.162027\pi\)
0.873221 + 0.487325i \(0.162027\pi\)
\(282\) −753.457 −0.159105
\(283\) 1561.52 0.327995 0.163997 0.986461i \(-0.447561\pi\)
0.163997 + 0.986461i \(0.447561\pi\)
\(284\) 712.870 0.148947
\(285\) 1444.21 0.300166
\(286\) 1591.23 0.328992
\(287\) −1704.65 −0.350601
\(288\) −930.546 −0.190392
\(289\) 4921.38 1.00171
\(290\) 976.684 0.197768
\(291\) −5395.07 −1.08682
\(292\) −1014.90 −0.203399
\(293\) 9486.19 1.89143 0.945715 0.324997i \(-0.105363\pi\)
0.945715 + 0.324997i \(0.105363\pi\)
\(294\) 2117.34 0.420019
\(295\) 3819.87 0.753903
\(296\) −3700.25 −0.726598
\(297\) 297.000 0.0580259
\(298\) 2619.27 0.509161
\(299\) −6829.56 −1.32095
\(300\) −1813.56 −0.349019
\(301\) 1443.45 0.276408
\(302\) −7069.56 −1.34705
\(303\) 2225.75 0.422001
\(304\) −973.074 −0.183584
\(305\) −7847.13 −1.47320
\(306\) 2117.30 0.395549
\(307\) −8443.06 −1.56961 −0.784806 0.619742i \(-0.787237\pi\)
−0.784806 + 0.619742i \(0.787237\pi\)
\(308\) −176.000 −0.0325602
\(309\) 1169.51 0.215311
\(310\) 14735.5 2.69974
\(311\) 4447.17 0.810856 0.405428 0.914127i \(-0.367123\pi\)
0.405428 + 0.914127i \(0.367123\pi\)
\(312\) −4501.29 −0.816779
\(313\) 6480.75 1.17033 0.585165 0.810914i \(-0.301030\pi\)
0.585165 + 0.810914i \(0.301030\pi\)
\(314\) −6748.36 −1.21284
\(315\) 1183.01 0.211603
\(316\) −231.413 −0.0411962
\(317\) −1252.75 −0.221960 −0.110980 0.993823i \(-0.535399\pi\)
−0.110980 + 0.993823i \(0.535399\pi\)
\(318\) −2317.61 −0.408696
\(319\) −232.374 −0.0407852
\(320\) 10922.3 1.90805
\(321\) 4615.27 0.802490
\(322\) −1792.00 −0.310137
\(323\) −2449.57 −0.421974
\(324\) −192.155 −0.0329484
\(325\) −15538.8 −2.65212
\(326\) 3626.06 0.616039
\(327\) 2339.35 0.395615
\(328\) −6219.02 −1.04692
\(329\) 714.043 0.119655
\(330\) −1525.71 −0.254508
\(331\) 3801.44 0.631258 0.315629 0.948883i \(-0.397785\pi\)
0.315629 + 0.948883i \(0.397785\pi\)
\(332\) −2620.47 −0.433183
\(333\) −1353.42 −0.222724
\(334\) 910.673 0.149191
\(335\) 534.260 0.0871336
\(336\) −797.087 −0.129419
\(337\) −5456.53 −0.882007 −0.441003 0.897506i \(-0.645377\pi\)
−0.441003 + 0.897506i \(0.645377\pi\)
\(338\) −3609.06 −0.580790
\(339\) −4543.66 −0.727958
\(340\) 4584.92 0.731330
\(341\) −3505.89 −0.556758
\(342\) −527.381 −0.0833845
\(343\) −4319.97 −0.680047
\(344\) 5266.08 0.825371
\(345\) 6548.35 1.02189
\(346\) −6411.93 −0.996264
\(347\) −4240.53 −0.656033 −0.328017 0.944672i \(-0.606380\pi\)
−0.328017 + 0.944672i \(0.606380\pi\)
\(348\) 150.343 0.0231587
\(349\) −8471.53 −1.29934 −0.649672 0.760215i \(-0.725094\pi\)
−0.649672 + 0.760215i \(0.725094\pi\)
\(350\) −4077.22 −0.622675
\(351\) −1646.41 −0.250368
\(352\) −1137.33 −0.172216
\(353\) 981.282 0.147956 0.0739778 0.997260i \(-0.476431\pi\)
0.0739778 + 0.997260i \(0.476431\pi\)
\(354\) −1394.90 −0.209430
\(355\) −5856.48 −0.875576
\(356\) −1098.42 −0.163528
\(357\) −2006.54 −0.297472
\(358\) 6589.27 0.972776
\(359\) −3020.09 −0.443995 −0.221997 0.975047i \(-0.571258\pi\)
−0.221997 + 0.975047i \(0.571258\pi\)
\(360\) 4315.94 0.631861
\(361\) −6248.86 −0.911045
\(362\) 9473.96 1.37553
\(363\) 363.000 0.0524864
\(364\) 975.652 0.140489
\(365\) 8337.74 1.19566
\(366\) 2865.54 0.409246
\(367\) 8689.40 1.23592 0.617960 0.786209i \(-0.287959\pi\)
0.617960 + 0.786209i \(0.287959\pi\)
\(368\) −4412.13 −0.624995
\(369\) −2274.70 −0.320911
\(370\) 6952.64 0.976893
\(371\) 2196.38 0.307360
\(372\) 2268.26 0.316139
\(373\) 5340.53 0.741346 0.370673 0.928763i \(-0.379127\pi\)
0.370673 + 0.928763i \(0.379127\pi\)
\(374\) 2587.81 0.357787
\(375\) 7590.59 1.04527
\(376\) 2605.02 0.357297
\(377\) 1288.16 0.175978
\(378\) −432.000 −0.0587822
\(379\) −1603.49 −0.217324 −0.108662 0.994079i \(-0.534657\pi\)
−0.108662 + 0.994079i \(0.534657\pi\)
\(380\) −1142.02 −0.154170
\(381\) 6908.23 0.928923
\(382\) −2124.58 −0.284563
\(383\) −830.236 −0.110765 −0.0553826 0.998465i \(-0.517638\pi\)
−0.0553826 + 0.998465i \(0.517638\pi\)
\(384\) −1507.05 −0.200277
\(385\) 1445.90 0.191403
\(386\) −3150.96 −0.415491
\(387\) 1926.15 0.253001
\(388\) 4266.21 0.558206
\(389\) −1746.12 −0.227588 −0.113794 0.993504i \(-0.536300\pi\)
−0.113794 + 0.993504i \(0.536300\pi\)
\(390\) 8457.75 1.09814
\(391\) −11106.9 −1.43657
\(392\) −7320.54 −0.943223
\(393\) −6061.76 −0.778054
\(394\) −365.602 −0.0467482
\(395\) 1901.14 0.242169
\(396\) −234.856 −0.0298029
\(397\) −10016.2 −1.26624 −0.633119 0.774054i \(-0.718226\pi\)
−0.633119 + 0.774054i \(0.718226\pi\)
\(398\) −3122.28 −0.393231
\(399\) 499.794 0.0627092
\(400\) −10038.6 −1.25483
\(401\) 8228.38 1.02470 0.512351 0.858776i \(-0.328775\pi\)
0.512351 + 0.858776i \(0.328775\pi\)
\(402\) −195.096 −0.0242052
\(403\) 19434.8 2.40228
\(404\) −1760.04 −0.216746
\(405\) 1578.62 0.193684
\(406\) 337.999 0.0413168
\(407\) −1654.18 −0.201462
\(408\) −7320.41 −0.888270
\(409\) −12311.5 −1.48843 −0.744213 0.667943i \(-0.767175\pi\)
−0.744213 + 0.667943i \(0.767175\pi\)
\(410\) 11685.3 1.40755
\(411\) 4426.21 0.531213
\(412\) −924.803 −0.110587
\(413\) 1321.93 0.157502
\(414\) −2391.26 −0.283874
\(415\) 21528.1 2.54644
\(416\) 6304.79 0.743071
\(417\) −4871.14 −0.572040
\(418\) −644.577 −0.0754241
\(419\) 13260.4 1.54610 0.773048 0.634347i \(-0.218731\pi\)
0.773048 + 0.634347i \(0.218731\pi\)
\(420\) −935.478 −0.108683
\(421\) −6177.74 −0.715165 −0.357583 0.933881i \(-0.616399\pi\)
−0.357583 + 0.933881i \(0.616399\pi\)
\(422\) 4117.70 0.474992
\(423\) 952.826 0.109522
\(424\) 8012.98 0.917795
\(425\) −25270.7 −2.88426
\(426\) 2138.61 0.243230
\(427\) −2715.64 −0.307773
\(428\) −3649.57 −0.412170
\(429\) −2012.28 −0.226466
\(430\) −9894.76 −1.10969
\(431\) 1668.05 0.186421 0.0932103 0.995646i \(-0.470287\pi\)
0.0932103 + 0.995646i \(0.470287\pi\)
\(432\) −1063.64 −0.118459
\(433\) −731.748 −0.0812138 −0.0406069 0.999175i \(-0.512929\pi\)
−0.0406069 + 0.999175i \(0.512929\pi\)
\(434\) 5099.48 0.564015
\(435\) −1235.12 −0.136137
\(436\) −1849.86 −0.203194
\(437\) 2766.52 0.302839
\(438\) −3044.69 −0.332148
\(439\) −14248.0 −1.54902 −0.774508 0.632564i \(-0.782002\pi\)
−0.774508 + 0.632564i \(0.782002\pi\)
\(440\) 5275.04 0.571540
\(441\) −2677.60 −0.289126
\(442\) −14345.5 −1.54377
\(443\) 174.262 0.0186895 0.00934473 0.999956i \(-0.497025\pi\)
0.00934473 + 0.999956i \(0.497025\pi\)
\(444\) 1070.23 0.114394
\(445\) 9023.89 0.961288
\(446\) −3946.78 −0.419026
\(447\) −3312.34 −0.350488
\(448\) 3779.87 0.398621
\(449\) 7469.97 0.785144 0.392572 0.919721i \(-0.371585\pi\)
0.392572 + 0.919721i \(0.371585\pi\)
\(450\) −5440.67 −0.569946
\(451\) −2780.19 −0.290275
\(452\) 3592.95 0.373890
\(453\) 8940.21 0.927258
\(454\) 8679.79 0.897275
\(455\) −8015.32 −0.825855
\(456\) 1823.38 0.187254
\(457\) 8762.68 0.896939 0.448469 0.893798i \(-0.351969\pi\)
0.448469 + 0.893798i \(0.351969\pi\)
\(458\) −4571.56 −0.466409
\(459\) −2677.55 −0.272282
\(460\) −5178.17 −0.524856
\(461\) 4339.67 0.438435 0.219218 0.975676i \(-0.429650\pi\)
0.219218 + 0.975676i \(0.429650\pi\)
\(462\) −528.000 −0.0531705
\(463\) −3932.49 −0.394726 −0.197363 0.980330i \(-0.563238\pi\)
−0.197363 + 0.980330i \(0.563238\pi\)
\(464\) 832.197 0.0832624
\(465\) −18634.6 −1.85840
\(466\) 6605.06 0.656596
\(467\) −8383.97 −0.830758 −0.415379 0.909648i \(-0.636351\pi\)
−0.415379 + 0.909648i \(0.636351\pi\)
\(468\) 1301.92 0.128592
\(469\) 184.890 0.0182035
\(470\) −4894.74 −0.480377
\(471\) 8534.02 0.834876
\(472\) 4822.77 0.470309
\(473\) 2354.18 0.228848
\(474\) −694.240 −0.0672732
\(475\) 6294.47 0.608022
\(476\) 1586.70 0.152786
\(477\) 2930.87 0.281332
\(478\) −2900.35 −0.277529
\(479\) 9534.10 0.909445 0.454722 0.890633i \(-0.349739\pi\)
0.454722 + 0.890633i \(0.349739\pi\)
\(480\) −6045.18 −0.574840
\(481\) 9169.93 0.869258
\(482\) 4775.43 0.451275
\(483\) 2266.17 0.213487
\(484\) −287.046 −0.0269577
\(485\) −35048.4 −3.28138
\(486\) −576.464 −0.0538044
\(487\) 4451.09 0.414164 0.207082 0.978324i \(-0.433603\pi\)
0.207082 + 0.978324i \(0.433603\pi\)
\(488\) −9907.38 −0.919029
\(489\) −4585.53 −0.424059
\(490\) 13755.0 1.26814
\(491\) −9757.40 −0.896833 −0.448417 0.893825i \(-0.648012\pi\)
−0.448417 + 0.893825i \(0.648012\pi\)
\(492\) 1798.74 0.164824
\(493\) 2094.93 0.191381
\(494\) 3573.20 0.325437
\(495\) 1929.42 0.175194
\(496\) 12555.6 1.13662
\(497\) −2026.74 −0.182921
\(498\) −7861.40 −0.707385
\(499\) −7173.69 −0.643564 −0.321782 0.946814i \(-0.604282\pi\)
−0.321782 + 0.946814i \(0.604282\pi\)
\(500\) −6002.33 −0.536865
\(501\) −1151.64 −0.102698
\(502\) −6420.88 −0.570873
\(503\) −15617.0 −1.38435 −0.692177 0.721728i \(-0.743348\pi\)
−0.692177 + 0.721728i \(0.743348\pi\)
\(504\) 1493.61 0.132005
\(505\) 14459.3 1.27412
\(506\) −2922.65 −0.256774
\(507\) 4564.04 0.399795
\(508\) −5462.76 −0.477108
\(509\) −8789.23 −0.765375 −0.382688 0.923878i \(-0.625001\pi\)
−0.382688 + 0.923878i \(0.625001\pi\)
\(510\) 13754.8 1.19426
\(511\) 2885.42 0.249792
\(512\) 11827.7 1.02093
\(513\) 666.929 0.0573989
\(514\) −535.521 −0.0459549
\(515\) 7597.58 0.650077
\(516\) −1523.12 −0.129945
\(517\) 1164.56 0.0990667
\(518\) 2406.09 0.204088
\(519\) 8108.56 0.685792
\(520\) −29242.0 −2.46606
\(521\) 13099.3 1.10152 0.550760 0.834664i \(-0.314338\pi\)
0.550760 + 0.834664i \(0.314338\pi\)
\(522\) 451.029 0.0378180
\(523\) 16824.2 1.40664 0.703318 0.710876i \(-0.251701\pi\)
0.703318 + 0.710876i \(0.251701\pi\)
\(524\) 4793.40 0.399620
\(525\) 5156.07 0.428627
\(526\) −4634.07 −0.384135
\(527\) 31606.7 2.61254
\(528\) −1300.00 −0.107150
\(529\) 377.000 0.0309855
\(530\) −15056.1 −1.23395
\(531\) 1764.00 0.144164
\(532\) −395.217 −0.0322083
\(533\) 15411.9 1.25247
\(534\) −3295.25 −0.267040
\(535\) 29982.5 2.42291
\(536\) 674.529 0.0543568
\(537\) −8332.83 −0.669624
\(538\) 832.310 0.0666978
\(539\) −3272.62 −0.261525
\(540\) −1248.31 −0.0994791
\(541\) 18863.1 1.49905 0.749527 0.661974i \(-0.230281\pi\)
0.749527 + 0.661974i \(0.230281\pi\)
\(542\) −604.407 −0.0478995
\(543\) −11980.8 −0.946862
\(544\) 10253.4 0.808110
\(545\) 15197.3 1.19446
\(546\) 2926.96 0.229418
\(547\) −12283.0 −0.960119 −0.480059 0.877236i \(-0.659385\pi\)
−0.480059 + 0.877236i \(0.659385\pi\)
\(548\) −3500.07 −0.272839
\(549\) −3623.77 −0.281710
\(550\) −6649.71 −0.515536
\(551\) −521.809 −0.0403444
\(552\) 8267.61 0.637487
\(553\) 657.924 0.0505927
\(554\) −276.320 −0.0211908
\(555\) −8792.35 −0.672458
\(556\) 3851.90 0.293808
\(557\) −9752.05 −0.741845 −0.370923 0.928664i \(-0.620958\pi\)
−0.370923 + 0.928664i \(0.620958\pi\)
\(558\) 6804.78 0.516253
\(559\) −13050.3 −0.987424
\(560\) −5178.17 −0.390746
\(561\) −3272.56 −0.246288
\(562\) −19515.5 −1.46479
\(563\) −3447.50 −0.258072 −0.129036 0.991640i \(-0.541188\pi\)
−0.129036 + 0.991640i \(0.541188\pi\)
\(564\) −753.457 −0.0562522
\(565\) −29517.3 −2.19788
\(566\) −3704.36 −0.275098
\(567\) 546.310 0.0404636
\(568\) −7394.09 −0.546213
\(569\) −3371.02 −0.248366 −0.124183 0.992259i \(-0.539631\pi\)
−0.124183 + 0.992259i \(0.539631\pi\)
\(570\) −3426.06 −0.251758
\(571\) −15852.5 −1.16183 −0.580916 0.813964i \(-0.697305\pi\)
−0.580916 + 0.813964i \(0.697305\pi\)
\(572\) 1591.23 0.116316
\(573\) 2686.76 0.195883
\(574\) 4043.91 0.294059
\(575\) 28540.5 2.06995
\(576\) 5043.89 0.364865
\(577\) 1376.35 0.0993036 0.0496518 0.998767i \(-0.484189\pi\)
0.0496518 + 0.998767i \(0.484189\pi\)
\(578\) −11674.9 −0.840159
\(579\) 3984.72 0.286009
\(580\) 976.684 0.0699217
\(581\) 7450.17 0.531988
\(582\) 12798.6 0.911547
\(583\) 3582.17 0.254474
\(584\) 10526.8 0.745894
\(585\) −10695.7 −0.755920
\(586\) −22503.9 −1.58640
\(587\) −23021.4 −1.61873 −0.809366 0.587304i \(-0.800189\pi\)
−0.809366 + 0.587304i \(0.800189\pi\)
\(588\) 2117.34 0.148499
\(589\) −7872.66 −0.550742
\(590\) −9061.80 −0.632320
\(591\) 462.343 0.0321798
\(592\) 5924.09 0.411281
\(593\) 2818.69 0.195194 0.0975968 0.995226i \(-0.468884\pi\)
0.0975968 + 0.995226i \(0.468884\pi\)
\(594\) −704.568 −0.0486679
\(595\) −13035.3 −0.898140
\(596\) 2619.27 0.180016
\(597\) 3948.46 0.270686
\(598\) 16201.6 1.10792
\(599\) −17691.4 −1.20676 −0.603380 0.797454i \(-0.706180\pi\)
−0.603380 + 0.797454i \(0.706180\pi\)
\(600\) 18810.7 1.27991
\(601\) −24516.1 −1.66394 −0.831972 0.554817i \(-0.812788\pi\)
−0.831972 + 0.554817i \(0.812788\pi\)
\(602\) −3424.26 −0.231831
\(603\) 246.719 0.0166620
\(604\) −7069.56 −0.476252
\(605\) 2358.18 0.158469
\(606\) −5280.12 −0.353944
\(607\) 4288.59 0.286768 0.143384 0.989667i \(-0.454202\pi\)
0.143384 + 0.989667i \(0.454202\pi\)
\(608\) −2553.94 −0.170355
\(609\) −427.435 −0.0284410
\(610\) 18615.6 1.23561
\(611\) −6455.74 −0.427449
\(612\) 2117.30 0.139848
\(613\) −8124.07 −0.535283 −0.267641 0.963519i \(-0.586244\pi\)
−0.267641 + 0.963519i \(0.586244\pi\)
\(614\) 20029.3 1.31648
\(615\) −14777.3 −0.968908
\(616\) 1825.52 0.119403
\(617\) −14923.1 −0.973714 −0.486857 0.873482i \(-0.661857\pi\)
−0.486857 + 0.873482i \(0.661857\pi\)
\(618\) −2774.41 −0.180588
\(619\) 3627.68 0.235556 0.117778 0.993040i \(-0.462423\pi\)
0.117778 + 0.993040i \(0.462423\pi\)
\(620\) 14735.5 0.954501
\(621\) 3024.00 0.195409
\(622\) −10549.9 −0.680087
\(623\) 3122.88 0.200827
\(624\) 7206.54 0.462328
\(625\) 17458.0 1.11731
\(626\) −15374.2 −0.981589
\(627\) 815.135 0.0519192
\(628\) −6748.36 −0.428804
\(629\) 14913.0 0.945341
\(630\) −2806.43 −0.177478
\(631\) 12576.5 0.793445 0.396723 0.917939i \(-0.370147\pi\)
0.396723 + 0.917939i \(0.370147\pi\)
\(632\) 2400.28 0.151073
\(633\) −5207.27 −0.326967
\(634\) 2971.87 0.186164
\(635\) 44878.5 2.80464
\(636\) −2317.61 −0.144496
\(637\) 18141.7 1.12841
\(638\) 551.257 0.0342077
\(639\) −2704.50 −0.167431
\(640\) −9790.36 −0.604685
\(641\) −7292.77 −0.449371 −0.224686 0.974431i \(-0.572136\pi\)
−0.224686 + 0.974431i \(0.572136\pi\)
\(642\) −10948.7 −0.673071
\(643\) −12946.3 −0.794016 −0.397008 0.917815i \(-0.629951\pi\)
−0.397008 + 0.917815i \(0.629951\pi\)
\(644\) −1792.00 −0.109650
\(645\) 12513.0 0.763872
\(646\) 5811.06 0.353921
\(647\) 21973.3 1.33518 0.667589 0.744530i \(-0.267327\pi\)
0.667589 + 0.744530i \(0.267327\pi\)
\(648\) 1993.08 0.120827
\(649\) 2156.00 0.130401
\(650\) 36862.5 2.22441
\(651\) −6448.83 −0.388248
\(652\) 3626.06 0.217803
\(653\) −27495.6 −1.64776 −0.823880 0.566764i \(-0.808195\pi\)
−0.823880 + 0.566764i \(0.808195\pi\)
\(654\) −5549.59 −0.331814
\(655\) −39379.5 −2.34913
\(656\) 9956.63 0.592593
\(657\) 3850.33 0.228639
\(658\) −1693.91 −0.100358
\(659\) 5156.28 0.304796 0.152398 0.988319i \(-0.451301\pi\)
0.152398 + 0.988319i \(0.451301\pi\)
\(660\) −1525.71 −0.0899822
\(661\) 9328.59 0.548926 0.274463 0.961598i \(-0.411500\pi\)
0.274463 + 0.961598i \(0.411500\pi\)
\(662\) −9018.10 −0.529454
\(663\) 18141.4 1.06267
\(664\) 27180.2 1.58855
\(665\) 3246.85 0.189334
\(666\) 3210.70 0.186805
\(667\) −2365.99 −0.137349
\(668\) 910.673 0.0527470
\(669\) 4991.12 0.288442
\(670\) −1267.42 −0.0730814
\(671\) −4429.06 −0.254816
\(672\) −2092.04 −0.120093
\(673\) 22182.4 1.27053 0.635267 0.772293i \(-0.280890\pi\)
0.635267 + 0.772293i \(0.280890\pi\)
\(674\) 12944.4 0.739764
\(675\) 6880.30 0.392330
\(676\) −3609.06 −0.205340
\(677\) −13507.1 −0.766797 −0.383399 0.923583i \(-0.625246\pi\)
−0.383399 + 0.923583i \(0.625246\pi\)
\(678\) 10778.8 0.610559
\(679\) −12129.1 −0.685528
\(680\) −47556.1 −2.68190
\(681\) −10976.5 −0.617652
\(682\) 8316.96 0.466969
\(683\) −19465.6 −1.09053 −0.545264 0.838264i \(-0.683571\pi\)
−0.545264 + 0.838264i \(0.683571\pi\)
\(684\) −527.381 −0.0294809
\(685\) 28754.3 1.60386
\(686\) 10248.2 0.570375
\(687\) 5781.22 0.321059
\(688\) −8430.96 −0.467191
\(689\) −19857.7 −1.09799
\(690\) −15534.5 −0.857086
\(691\) −19971.8 −1.09951 −0.549757 0.835324i \(-0.685280\pi\)
−0.549757 + 0.835324i \(0.685280\pi\)
\(692\) −6411.93 −0.352233
\(693\) 667.712 0.0366007
\(694\) 10059.7 0.550234
\(695\) −31644.7 −1.72713
\(696\) −1559.40 −0.0849265
\(697\) 25064.3 1.36209
\(698\) 20096.9 1.08980
\(699\) −8352.80 −0.451977
\(700\) −4077.22 −0.220149
\(701\) −14180.1 −0.764018 −0.382009 0.924159i \(-0.624768\pi\)
−0.382009 + 0.924159i \(0.624768\pi\)
\(702\) 3905.75 0.209990
\(703\) −3714.56 −0.199285
\(704\) 6164.75 0.330032
\(705\) 6189.91 0.330675
\(706\) −2327.88 −0.124095
\(707\) 5003.91 0.266183
\(708\) −1394.90 −0.0740446
\(709\) 15870.6 0.840667 0.420334 0.907370i \(-0.361913\pi\)
0.420334 + 0.907370i \(0.361913\pi\)
\(710\) 13893.2 0.734370
\(711\) 877.939 0.0463084
\(712\) 11393.1 0.599683
\(713\) −35696.3 −1.87495
\(714\) 4760.09 0.249498
\(715\) −13072.5 −0.683756
\(716\) 6589.27 0.343928
\(717\) 3667.79 0.191041
\(718\) 7164.50 0.372391
\(719\) −6040.05 −0.313290 −0.156645 0.987655i \(-0.550068\pi\)
−0.156645 + 0.987655i \(0.550068\pi\)
\(720\) −6909.80 −0.357657
\(721\) 2629.28 0.135811
\(722\) 14824.0 0.764119
\(723\) −6039.03 −0.310642
\(724\) 9473.96 0.486322
\(725\) −5383.18 −0.275761
\(726\) −861.138 −0.0440218
\(727\) 37252.3 1.90043 0.950213 0.311601i \(-0.100865\pi\)
0.950213 + 0.311601i \(0.100865\pi\)
\(728\) −10119.7 −0.515196
\(729\) 729.000 0.0370370
\(730\) −19779.5 −1.00284
\(731\) −21223.7 −1.07385
\(732\) 2865.54 0.144690
\(733\) 6125.52 0.308665 0.154332 0.988019i \(-0.450677\pi\)
0.154332 + 0.988019i \(0.450677\pi\)
\(734\) −20613.7 −1.03660
\(735\) −17394.7 −0.872942
\(736\) −11580.1 −0.579958
\(737\) 301.546 0.0150713
\(738\) 5396.23 0.269157
\(739\) −5225.97 −0.260136 −0.130068 0.991505i \(-0.541520\pi\)
−0.130068 + 0.991505i \(0.541520\pi\)
\(740\) 6952.64 0.345384
\(741\) −4518.68 −0.224019
\(742\) −5210.43 −0.257791
\(743\) 7062.96 0.348741 0.174371 0.984680i \(-0.444211\pi\)
0.174371 + 0.984680i \(0.444211\pi\)
\(744\) −23527.0 −1.15933
\(745\) −21518.2 −1.05821
\(746\) −12669.2 −0.621788
\(747\) 9941.57 0.486939
\(748\) 2587.81 0.126497
\(749\) 10376.0 0.506182
\(750\) −18007.0 −0.876697
\(751\) −20755.4 −1.00849 −0.504244 0.863561i \(-0.668229\pi\)
−0.504244 + 0.863561i \(0.668229\pi\)
\(752\) −4170.63 −0.202243
\(753\) 8119.89 0.392968
\(754\) −3055.88 −0.147598
\(755\) 58079.0 2.79962
\(756\) −432.000 −0.0207827
\(757\) 31182.9 1.49717 0.748587 0.663037i \(-0.230733\pi\)
0.748587 + 0.663037i \(0.230733\pi\)
\(758\) 3803.93 0.182276
\(759\) 3696.00 0.176754
\(760\) 11845.4 0.565364
\(761\) 32047.8 1.52659 0.763293 0.646052i \(-0.223581\pi\)
0.763293 + 0.646052i \(0.223581\pi\)
\(762\) −16388.3 −0.779114
\(763\) 5259.29 0.249540
\(764\) −2124.58 −0.100608
\(765\) −17394.4 −0.822084
\(766\) 1969.55 0.0929019
\(767\) −11951.7 −0.562650
\(768\) −9875.22 −0.463986
\(769\) 2215.88 0.103910 0.0519548 0.998649i \(-0.483455\pi\)
0.0519548 + 0.998649i \(0.483455\pi\)
\(770\) −3430.09 −0.160535
\(771\) 677.223 0.0316337
\(772\) −3150.96 −0.146898
\(773\) 5300.56 0.246634 0.123317 0.992367i \(-0.460647\pi\)
0.123317 + 0.992367i \(0.460647\pi\)
\(774\) −4569.36 −0.212199
\(775\) −81217.4 −3.76441
\(776\) −44250.3 −2.04703
\(777\) −3042.75 −0.140487
\(778\) 4142.29 0.190885
\(779\) −6243.06 −0.287138
\(780\) 8457.75 0.388251
\(781\) −3305.50 −0.151447
\(782\) 26348.6 1.20489
\(783\) −570.373 −0.0260325
\(784\) 11720.2 0.533899
\(785\) 55440.2 2.52069
\(786\) 14380.2 0.652576
\(787\) 34374.1 1.55693 0.778466 0.627687i \(-0.215998\pi\)
0.778466 + 0.627687i \(0.215998\pi\)
\(788\) −365.602 −0.0165280
\(789\) 5860.27 0.264425
\(790\) −4510.04 −0.203114
\(791\) −10215.0 −0.459170
\(792\) 2435.99 0.109292
\(793\) 24552.4 1.09947
\(794\) 23761.2 1.06203
\(795\) 19040.0 0.849409
\(796\) −3122.28 −0.139028
\(797\) 29867.1 1.32741 0.663707 0.747993i \(-0.268982\pi\)
0.663707 + 0.747993i \(0.268982\pi\)
\(798\) −1185.65 −0.0525960
\(799\) −10498.9 −0.464862
\(800\) −26347.5 −1.16441
\(801\) 4167.20 0.183821
\(802\) −19520.0 −0.859447
\(803\) 4705.96 0.206812
\(804\) −195.096 −0.00855783
\(805\) 14721.9 0.644570
\(806\) −46104.9 −2.01486
\(807\) −1052.54 −0.0459124
\(808\) 18255.6 0.794839
\(809\) −15857.2 −0.689133 −0.344566 0.938762i \(-0.611974\pi\)
−0.344566 + 0.938762i \(0.611974\pi\)
\(810\) −3744.93 −0.162449
\(811\) 36122.6 1.56404 0.782020 0.623253i \(-0.214189\pi\)
0.782020 + 0.623253i \(0.214189\pi\)
\(812\) 337.999 0.0146077
\(813\) 764.337 0.0329723
\(814\) 3924.19 0.168971
\(815\) −29789.3 −1.28034
\(816\) 11719.9 0.502794
\(817\) 5286.43 0.226375
\(818\) 29206.4 1.24838
\(819\) −3701.44 −0.157923
\(820\) 11685.3 0.497645
\(821\) 25779.6 1.09588 0.547938 0.836519i \(-0.315413\pi\)
0.547938 + 0.836519i \(0.315413\pi\)
\(822\) −10500.2 −0.445544
\(823\) −22130.2 −0.937315 −0.468658 0.883380i \(-0.655262\pi\)
−0.468658 + 0.883380i \(0.655262\pi\)
\(824\) 9592.32 0.405539
\(825\) 8409.26 0.354876
\(826\) −3136.00 −0.132101
\(827\) 18288.6 0.768994 0.384497 0.923126i \(-0.374375\pi\)
0.384497 + 0.923126i \(0.374375\pi\)
\(828\) −2391.26 −0.100365
\(829\) −34956.6 −1.46453 −0.732263 0.681022i \(-0.761536\pi\)
−0.732263 + 0.681022i \(0.761536\pi\)
\(830\) −51070.6 −2.13577
\(831\) 349.435 0.0145870
\(832\) −34174.2 −1.42401
\(833\) 29503.7 1.22718
\(834\) 11555.7 0.479786
\(835\) −7481.50 −0.310069
\(836\) −644.577 −0.0266664
\(837\) −8605.37 −0.355370
\(838\) −31457.5 −1.29675
\(839\) −14507.5 −0.596965 −0.298482 0.954415i \(-0.596480\pi\)
−0.298482 + 0.954415i \(0.596480\pi\)
\(840\) 9703.04 0.398556
\(841\) −23942.7 −0.981702
\(842\) 14655.3 0.599829
\(843\) 24679.4 1.00831
\(844\) 4117.70 0.167935
\(845\) 29649.7 1.20708
\(846\) −2260.37 −0.0918595
\(847\) 816.092 0.0331066
\(848\) −12828.7 −0.519506
\(849\) 4684.55 0.189368
\(850\) 59949.2 2.41911
\(851\) −16842.6 −0.678445
\(852\) 2138.61 0.0859948
\(853\) −44146.9 −1.77205 −0.886027 0.463634i \(-0.846545\pi\)
−0.886027 + 0.463634i \(0.846545\pi\)
\(854\) 6442.26 0.258138
\(855\) 4332.62 0.173301
\(856\) 37854.4 1.51149
\(857\) −47679.3 −1.90046 −0.950230 0.311549i \(-0.899152\pi\)
−0.950230 + 0.311549i \(0.899152\pi\)
\(858\) 4773.70 0.189943
\(859\) 7525.09 0.298897 0.149449 0.988769i \(-0.452250\pi\)
0.149449 + 0.988769i \(0.452250\pi\)
\(860\) −9894.76 −0.392335
\(861\) −5113.95 −0.202419
\(862\) −3957.09 −0.156356
\(863\) 45816.1 1.80718 0.903591 0.428396i \(-0.140921\pi\)
0.903591 + 0.428396i \(0.140921\pi\)
\(864\) −2791.64 −0.109923
\(865\) 52676.2 2.07057
\(866\) 1735.91 0.0681163
\(867\) 14764.1 0.578335
\(868\) 5099.48 0.199410
\(869\) 1073.04 0.0418876
\(870\) 2930.05 0.114182
\(871\) −1671.61 −0.0650292
\(872\) 19187.3 0.745142
\(873\) −16185.2 −0.627476
\(874\) −6562.96 −0.253999
\(875\) 17065.1 0.659319
\(876\) −3044.69 −0.117432
\(877\) −34168.3 −1.31560 −0.657801 0.753192i \(-0.728513\pi\)
−0.657801 + 0.753192i \(0.728513\pi\)
\(878\) 33800.2 1.29920
\(879\) 28458.6 1.09202
\(880\) −8445.31 −0.323513
\(881\) −100.796 −0.00385460 −0.00192730 0.999998i \(-0.500613\pi\)
−0.00192730 + 0.999998i \(0.500613\pi\)
\(882\) 6352.02 0.242498
\(883\) −12346.1 −0.470531 −0.235266 0.971931i \(-0.575596\pi\)
−0.235266 + 0.971931i \(0.575596\pi\)
\(884\) −14345.5 −0.545803
\(885\) 11459.6 0.435266
\(886\) −413.398 −0.0156754
\(887\) −37345.1 −1.41367 −0.706834 0.707379i \(-0.749877\pi\)
−0.706834 + 0.707379i \(0.749877\pi\)
\(888\) −11100.8 −0.419501
\(889\) 15531.0 0.585932
\(890\) −21407.2 −0.806260
\(891\) 891.000 0.0335013
\(892\) −3946.78 −0.148148
\(893\) 2615.09 0.0979962
\(894\) 7857.80 0.293964
\(895\) −54133.2 −2.02176
\(896\) −3388.13 −0.126328
\(897\) −20488.7 −0.762651
\(898\) −17720.9 −0.658523
\(899\) 6732.88 0.249782
\(900\) −5440.67 −0.201506
\(901\) −32294.4 −1.19410
\(902\) 6595.39 0.243462
\(903\) 4330.34 0.159584
\(904\) −37267.1 −1.37111
\(905\) −77831.9 −2.85881
\(906\) −21208.7 −0.777717
\(907\) 10308.6 0.377390 0.188695 0.982036i \(-0.439574\pi\)
0.188695 + 0.982036i \(0.439574\pi\)
\(908\) 8679.79 0.317235
\(909\) 6677.26 0.243642
\(910\) 19014.6 0.692668
\(911\) −22590.8 −0.821589 −0.410795 0.911728i \(-0.634749\pi\)
−0.410795 + 0.911728i \(0.634749\pi\)
\(912\) −2919.22 −0.105992
\(913\) 12150.8 0.440453
\(914\) −20787.5 −0.752288
\(915\) −23541.4 −0.850551
\(916\) −4571.56 −0.164900
\(917\) −13628.0 −0.490769
\(918\) 6351.90 0.228370
\(919\) 47712.1 1.71260 0.856299 0.516481i \(-0.172758\pi\)
0.856299 + 0.516481i \(0.172758\pi\)
\(920\) 53709.5 1.92473
\(921\) −25329.2 −0.906216
\(922\) −10294.9 −0.367728
\(923\) 18323.9 0.653456
\(924\) −528.000 −0.0187986
\(925\) −38320.8 −1.36214
\(926\) 9328.96 0.331068
\(927\) 3508.53 0.124310
\(928\) 2184.19 0.0772625
\(929\) 10714.3 0.378391 0.189196 0.981939i \(-0.439412\pi\)
0.189196 + 0.981939i \(0.439412\pi\)
\(930\) 44206.4 1.55869
\(931\) −7348.84 −0.258699
\(932\) 6605.06 0.232142
\(933\) 13341.5 0.468148
\(934\) 19889.1 0.696780
\(935\) −21259.8 −0.743603
\(936\) −13503.9 −0.471568
\(937\) −14719.5 −0.513197 −0.256599 0.966518i \(-0.582602\pi\)
−0.256599 + 0.966518i \(0.582602\pi\)
\(938\) −438.612 −0.0152678
\(939\) 19442.2 0.675691
\(940\) −4894.74 −0.169839
\(941\) 3694.44 0.127987 0.0639933 0.997950i \(-0.479616\pi\)
0.0639933 + 0.997950i \(0.479616\pi\)
\(942\) −20245.1 −0.700234
\(943\) −28307.4 −0.977535
\(944\) −7721.23 −0.266213
\(945\) 3549.03 0.122169
\(946\) −5584.77 −0.191941
\(947\) 36416.7 1.24961 0.624806 0.780780i \(-0.285178\pi\)
0.624806 + 0.780780i \(0.285178\pi\)
\(948\) −694.240 −0.0237847
\(949\) −26087.4 −0.892342
\(950\) −14932.3 −0.509965
\(951\) −3758.25 −0.128149
\(952\) −16457.6 −0.560289
\(953\) 20779.4 0.706306 0.353153 0.935566i \(-0.385110\pi\)
0.353153 + 0.935566i \(0.385110\pi\)
\(954\) −6952.84 −0.235961
\(955\) 17454.2 0.591419
\(956\) −2900.35 −0.0981212
\(957\) −697.123 −0.0235473
\(958\) −22617.6 −0.762777
\(959\) 9950.94 0.335071
\(960\) 32767.0 1.10161
\(961\) 71789.7 2.40978
\(962\) −21753.7 −0.729071
\(963\) 13845.8 0.463318
\(964\) 4775.43 0.159550
\(965\) 25886.2 0.863531
\(966\) −5376.00 −0.179058
\(967\) 56812.8 1.88932 0.944662 0.328044i \(-0.106389\pi\)
0.944662 + 0.328044i \(0.106389\pi\)
\(968\) 2977.32 0.0988582
\(969\) −7348.70 −0.243627
\(970\) 83144.7 2.75218
\(971\) −26459.7 −0.874493 −0.437247 0.899342i \(-0.644046\pi\)
−0.437247 + 0.899342i \(0.644046\pi\)
\(972\) −576.464 −0.0190227
\(973\) −10951.2 −0.360822
\(974\) −10559.2 −0.347371
\(975\) −46616.5 −1.53120
\(976\) 15861.7 0.520204
\(977\) −21009.2 −0.687967 −0.343984 0.938976i \(-0.611776\pi\)
−0.343984 + 0.938976i \(0.611776\pi\)
\(978\) 10878.2 0.355670
\(979\) 5093.24 0.166272
\(980\) 13755.0 0.448355
\(981\) 7018.04 0.228409
\(982\) 23147.3 0.752199
\(983\) −9076.80 −0.294512 −0.147256 0.989098i \(-0.547044\pi\)
−0.147256 + 0.989098i \(0.547044\pi\)
\(984\) −18657.1 −0.604437
\(985\) 3003.55 0.0971585
\(986\) −4969.76 −0.160517
\(987\) 2142.13 0.0690828
\(988\) 3573.20 0.115059
\(989\) 23969.8 0.770673
\(990\) −4577.14 −0.146940
\(991\) 2629.24 0.0842791 0.0421395 0.999112i \(-0.486583\pi\)
0.0421395 + 0.999112i \(0.486583\pi\)
\(992\) 32953.5 1.05471
\(993\) 11404.3 0.364457
\(994\) 4808.00 0.153421
\(995\) 25650.6 0.817267
\(996\) −7861.40 −0.250098
\(997\) −50423.9 −1.60175 −0.800874 0.598833i \(-0.795631\pi\)
−0.800874 + 0.598833i \(0.795631\pi\)
\(998\) 17018.0 0.539775
\(999\) −4060.27 −0.128590
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 33.4.a.d.1.1 2
3.2 odd 2 99.4.a.e.1.2 2
4.3 odd 2 528.4.a.o.1.2 2
5.2 odd 4 825.4.c.i.199.2 4
5.3 odd 4 825.4.c.i.199.3 4
5.4 even 2 825.4.a.k.1.2 2
7.6 odd 2 1617.4.a.j.1.1 2
8.3 odd 2 2112.4.a.bh.1.1 2
8.5 even 2 2112.4.a.ba.1.1 2
11.10 odd 2 363.4.a.j.1.2 2
12.11 even 2 1584.4.a.x.1.1 2
15.14 odd 2 2475.4.a.o.1.1 2
33.32 even 2 1089.4.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.1 2 1.1 even 1 trivial
99.4.a.e.1.2 2 3.2 odd 2
363.4.a.j.1.2 2 11.10 odd 2
528.4.a.o.1.2 2 4.3 odd 2
825.4.a.k.1.2 2 5.4 even 2
825.4.c.i.199.2 4 5.2 odd 4
825.4.c.i.199.3 4 5.3 odd 4
1089.4.a.t.1.1 2 33.32 even 2
1584.4.a.x.1.1 2 12.11 even 2
1617.4.a.j.1.1 2 7.6 odd 2
2112.4.a.ba.1.1 2 8.5 even 2
2112.4.a.bh.1.1 2 8.3 odd 2
2475.4.a.o.1.1 2 15.14 odd 2