Properties

Label 230.4.a.f.1.1
Level $230$
Weight $4$
Character 230.1
Self dual yes
Analytic conductor $13.570$
Analytic rank $1$
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] = [230,4,Mod(1,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, 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("230.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.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: \(13.5704393013\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.77200\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -5.77200 q^{3} +4.00000 q^{4} +5.00000 q^{5} +11.5440 q^{6} -12.7720 q^{7} -8.00000 q^{8} +6.31601 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -5.77200 q^{3} +4.00000 q^{4} +5.00000 q^{5} +11.5440 q^{6} -12.7720 q^{7} -8.00000 q^{8} +6.31601 q^{9} -10.0000 q^{10} +51.7200 q^{11} -23.0880 q^{12} +33.9480 q^{13} +25.5440 q^{14} -28.8600 q^{15} +16.0000 q^{16} -86.4920 q^{17} -12.6320 q^{18} +59.7200 q^{19} +20.0000 q^{20} +73.7200 q^{21} -103.440 q^{22} -23.0000 q^{23} +46.1760 q^{24} +25.0000 q^{25} -67.8960 q^{26} +119.388 q^{27} -51.0880 q^{28} -64.5440 q^{29} +57.7200 q^{30} -157.072 q^{31} -32.0000 q^{32} -298.528 q^{33} +172.984 q^{34} -63.8600 q^{35} +25.2640 q^{36} -275.404 q^{37} -119.440 q^{38} -195.948 q^{39} -40.0000 q^{40} -482.128 q^{41} -147.440 q^{42} -270.984 q^{43} +206.880 q^{44} +31.5800 q^{45} +46.0000 q^{46} -320.860 q^{47} -92.3520 q^{48} -179.876 q^{49} -50.0000 q^{50} +499.232 q^{51} +135.792 q^{52} -122.596 q^{53} -238.776 q^{54} +258.600 q^{55} +102.176 q^{56} -344.704 q^{57} +129.088 q^{58} +627.060 q^{59} -115.440 q^{60} -34.7360 q^{61} +314.144 q^{62} -80.6680 q^{63} +64.0000 q^{64} +169.740 q^{65} +597.056 q^{66} -90.3479 q^{67} -345.968 q^{68} +132.756 q^{69} +127.720 q^{70} +182.712 q^{71} -50.5280 q^{72} -73.2440 q^{73} +550.808 q^{74} -144.300 q^{75} +238.880 q^{76} -660.568 q^{77} +391.896 q^{78} -283.232 q^{79} +80.0000 q^{80} -859.640 q^{81} +964.256 q^{82} +1136.32 q^{83} +294.880 q^{84} -432.460 q^{85} +541.968 q^{86} +372.548 q^{87} -413.760 q^{88} +476.104 q^{89} -63.1601 q^{90} -433.584 q^{91} -92.0000 q^{92} +906.620 q^{93} +641.720 q^{94} +298.600 q^{95} +184.704 q^{96} -1820.61 q^{97} +359.752 q^{98} +326.664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} - 17 q^{7} - 16 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} - 17 q^{7} - 16 q^{8} - 13 q^{9} - 20 q^{10} + 18 q^{11} - 12 q^{12} - 9 q^{13} + 34 q^{14} - 15 q^{15} + 32 q^{16} - 79 q^{17} + 26 q^{18} + 34 q^{19} + 40 q^{20} + 62 q^{21} - 36 q^{22} - 46 q^{23} + 24 q^{24} + 50 q^{25} + 18 q^{26} - 9 q^{27} - 68 q^{28} - 112 q^{29} + 30 q^{30} - 92 q^{31} - 64 q^{32} - 392 q^{33} + 158 q^{34} - 85 q^{35} - 52 q^{36} - 491 q^{37} - 68 q^{38} - 315 q^{39} - 80 q^{40} - 332 q^{41} - 124 q^{42} - 354 q^{43} + 72 q^{44} - 65 q^{45} + 92 q^{46} - 599 q^{47} - 48 q^{48} - 505 q^{49} - 100 q^{50} + 520 q^{51} - 36 q^{52} - 305 q^{53} + 18 q^{54} + 90 q^{55} + 136 q^{56} - 416 q^{57} + 224 q^{58} + 357 q^{59} - 60 q^{60} - 172 q^{61} + 184 q^{62} + q^{63} + 128 q^{64} - 45 q^{65} + 784 q^{66} - 531 q^{67} - 316 q^{68} + 69 q^{69} + 170 q^{70} + 1254 q^{71} + 104 q^{72} - 343 q^{73} + 982 q^{74} - 75 q^{75} + 136 q^{76} - 518 q^{77} + 630 q^{78} - 88 q^{79} + 160 q^{80} - 694 q^{81} + 664 q^{82} + 1273 q^{83} + 248 q^{84} - 395 q^{85} + 708 q^{86} + 241 q^{87} - 144 q^{88} + 1106 q^{89} + 130 q^{90} - 252 q^{91} - 184 q^{92} + 1087 q^{93} + 1198 q^{94} + 170 q^{95} + 96 q^{96} - 2240 q^{97} + 1010 q^{98} + 978 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −5.77200 −1.11082 −0.555411 0.831576i \(-0.687439\pi\)
−0.555411 + 0.831576i \(0.687439\pi\)
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) 11.5440 0.785470
\(7\) −12.7720 −0.689623 −0.344812 0.938672i \(-0.612057\pi\)
−0.344812 + 0.938672i \(0.612057\pi\)
\(8\) −8.00000 −0.353553
\(9\) 6.31601 0.233926
\(10\) −10.0000 −0.316228
\(11\) 51.7200 1.41765 0.708826 0.705383i \(-0.249225\pi\)
0.708826 + 0.705383i \(0.249225\pi\)
\(12\) −23.0880 −0.555411
\(13\) 33.9480 0.724268 0.362134 0.932126i \(-0.382048\pi\)
0.362134 + 0.932126i \(0.382048\pi\)
\(14\) 25.5440 0.487637
\(15\) −28.8600 −0.496775
\(16\) 16.0000 0.250000
\(17\) −86.4920 −1.23396 −0.616982 0.786977i \(-0.711645\pi\)
−0.616982 + 0.786977i \(0.711645\pi\)
\(18\) −12.6320 −0.165411
\(19\) 59.7200 0.721090 0.360545 0.932742i \(-0.382591\pi\)
0.360545 + 0.932742i \(0.382591\pi\)
\(20\) 20.0000 0.223607
\(21\) 73.7200 0.766049
\(22\) −103.440 −1.00243
\(23\) −23.0000 −0.208514
\(24\) 46.1760 0.392735
\(25\) 25.0000 0.200000
\(26\) −67.8960 −0.512135
\(27\) 119.388 0.850972
\(28\) −51.0880 −0.344812
\(29\) −64.5440 −0.413294 −0.206647 0.978416i \(-0.566255\pi\)
−0.206647 + 0.978416i \(0.566255\pi\)
\(30\) 57.7200 0.351273
\(31\) −157.072 −0.910031 −0.455016 0.890483i \(-0.650366\pi\)
−0.455016 + 0.890483i \(0.650366\pi\)
\(32\) −32.0000 −0.176777
\(33\) −298.528 −1.57476
\(34\) 172.984 0.872545
\(35\) −63.8600 −0.308409
\(36\) 25.2640 0.116963
\(37\) −275.404 −1.22368 −0.611840 0.790982i \(-0.709570\pi\)
−0.611840 + 0.790982i \(0.709570\pi\)
\(38\) −119.440 −0.509888
\(39\) −195.948 −0.804533
\(40\) −40.0000 −0.158114
\(41\) −482.128 −1.83648 −0.918241 0.396022i \(-0.870390\pi\)
−0.918241 + 0.396022i \(0.870390\pi\)
\(42\) −147.440 −0.541678
\(43\) −270.984 −0.961039 −0.480520 0.876984i \(-0.659552\pi\)
−0.480520 + 0.876984i \(0.659552\pi\)
\(44\) 206.880 0.708826
\(45\) 31.5800 0.104615
\(46\) 46.0000 0.147442
\(47\) −320.860 −0.995792 −0.497896 0.867237i \(-0.665894\pi\)
−0.497896 + 0.867237i \(0.665894\pi\)
\(48\) −92.3520 −0.277706
\(49\) −179.876 −0.524420
\(50\) −50.0000 −0.141421
\(51\) 499.232 1.37072
\(52\) 135.792 0.362134
\(53\) −122.596 −0.317733 −0.158867 0.987300i \(-0.550784\pi\)
−0.158867 + 0.987300i \(0.550784\pi\)
\(54\) −238.776 −0.601728
\(55\) 258.600 0.633993
\(56\) 102.176 0.243819
\(57\) −344.704 −0.801003
\(58\) 129.088 0.292243
\(59\) 627.060 1.38367 0.691833 0.722058i \(-0.256803\pi\)
0.691833 + 0.722058i \(0.256803\pi\)
\(60\) −115.440 −0.248387
\(61\) −34.7360 −0.0729096 −0.0364548 0.999335i \(-0.511606\pi\)
−0.0364548 + 0.999335i \(0.511606\pi\)
\(62\) 314.144 0.643489
\(63\) −80.6680 −0.161321
\(64\) 64.0000 0.125000
\(65\) 169.740 0.323903
\(66\) 597.056 1.11352
\(67\) −90.3479 −0.164743 −0.0823713 0.996602i \(-0.526249\pi\)
−0.0823713 + 0.996602i \(0.526249\pi\)
\(68\) −345.968 −0.616982
\(69\) 132.756 0.231622
\(70\) 127.720 0.218078
\(71\) 182.712 0.305407 0.152704 0.988272i \(-0.451202\pi\)
0.152704 + 0.988272i \(0.451202\pi\)
\(72\) −50.5280 −0.0827054
\(73\) −73.2440 −0.117432 −0.0587161 0.998275i \(-0.518701\pi\)
−0.0587161 + 0.998275i \(0.518701\pi\)
\(74\) 550.808 0.865272
\(75\) −144.300 −0.222164
\(76\) 238.880 0.360545
\(77\) −660.568 −0.977646
\(78\) 391.896 0.568891
\(79\) −283.232 −0.403368 −0.201684 0.979451i \(-0.564641\pi\)
−0.201684 + 0.979451i \(0.564641\pi\)
\(80\) 80.0000 0.111803
\(81\) −859.640 −1.17920
\(82\) 964.256 1.29859
\(83\) 1136.32 1.50274 0.751372 0.659879i \(-0.229392\pi\)
0.751372 + 0.659879i \(0.229392\pi\)
\(84\) 294.880 0.383024
\(85\) −432.460 −0.551846
\(86\) 541.968 0.679557
\(87\) 372.548 0.459096
\(88\) −413.760 −0.501216
\(89\) 476.104 0.567044 0.283522 0.958966i \(-0.408497\pi\)
0.283522 + 0.958966i \(0.408497\pi\)
\(90\) −63.1601 −0.0739739
\(91\) −433.584 −0.499472
\(92\) −92.0000 −0.104257
\(93\) 906.620 1.01088
\(94\) 641.720 0.704132
\(95\) 298.600 0.322481
\(96\) 184.704 0.196367
\(97\) −1820.61 −1.90572 −0.952860 0.303412i \(-0.901874\pi\)
−0.952860 + 0.303412i \(0.901874\pi\)
\(98\) 359.752 0.370821
\(99\) 326.664 0.331626
\(100\) 100.000 0.100000
\(101\) 184.836 0.182098 0.0910488 0.995846i \(-0.470978\pi\)
0.0910488 + 0.995846i \(0.470978\pi\)
\(102\) −998.464 −0.969242
\(103\) −1249.78 −1.19557 −0.597787 0.801655i \(-0.703953\pi\)
−0.597787 + 0.801655i \(0.703953\pi\)
\(104\) −271.584 −0.256067
\(105\) 368.600 0.342587
\(106\) 245.192 0.224671
\(107\) 816.820 0.737991 0.368995 0.929431i \(-0.379702\pi\)
0.368995 + 0.929431i \(0.379702\pi\)
\(108\) 477.552 0.425486
\(109\) −259.264 −0.227826 −0.113913 0.993491i \(-0.536338\pi\)
−0.113913 + 0.993491i \(0.536338\pi\)
\(110\) −517.200 −0.448301
\(111\) 1589.63 1.35929
\(112\) −204.352 −0.172406
\(113\) 890.876 0.741651 0.370825 0.928703i \(-0.379075\pi\)
0.370825 + 0.928703i \(0.379075\pi\)
\(114\) 689.408 0.566395
\(115\) −115.000 −0.0932505
\(116\) −258.176 −0.206647
\(117\) 214.416 0.169425
\(118\) −1254.12 −0.978399
\(119\) 1104.68 0.850971
\(120\) 230.880 0.175636
\(121\) 1343.96 1.00974
\(122\) 69.4720 0.0515549
\(123\) 2782.84 2.04001
\(124\) −628.288 −0.455016
\(125\) 125.000 0.0894427
\(126\) 161.336 0.114071
\(127\) −2176.69 −1.52087 −0.760434 0.649415i \(-0.775014\pi\)
−0.760434 + 0.649415i \(0.775014\pi\)
\(128\) −128.000 −0.0883883
\(129\) 1564.12 1.06754
\(130\) −339.480 −0.229034
\(131\) −1604.85 −1.07036 −0.535178 0.844739i \(-0.679755\pi\)
−0.535178 + 0.844739i \(0.679755\pi\)
\(132\) −1194.11 −0.787380
\(133\) −762.744 −0.497281
\(134\) 180.696 0.116491
\(135\) 596.940 0.380566
\(136\) 691.936 0.436272
\(137\) −1734.60 −1.08173 −0.540865 0.841110i \(-0.681903\pi\)
−0.540865 + 0.841110i \(0.681903\pi\)
\(138\) −265.512 −0.163782
\(139\) −993.824 −0.606439 −0.303220 0.952921i \(-0.598062\pi\)
−0.303220 + 0.952921i \(0.598062\pi\)
\(140\) −255.440 −0.154204
\(141\) 1852.00 1.10615
\(142\) −365.424 −0.215955
\(143\) 1755.79 1.02676
\(144\) 101.056 0.0584815
\(145\) −322.720 −0.184831
\(146\) 146.488 0.0830372
\(147\) 1038.24 0.582537
\(148\) −1101.62 −0.611840
\(149\) −1023.08 −0.562510 −0.281255 0.959633i \(-0.590751\pi\)
−0.281255 + 0.959633i \(0.590751\pi\)
\(150\) 288.600 0.157094
\(151\) −1309.40 −0.705677 −0.352838 0.935684i \(-0.614783\pi\)
−0.352838 + 0.935684i \(0.614783\pi\)
\(152\) −477.760 −0.254944
\(153\) −546.284 −0.288657
\(154\) 1321.14 0.691300
\(155\) −785.360 −0.406978
\(156\) −783.792 −0.402267
\(157\) −1648.70 −0.838093 −0.419046 0.907965i \(-0.637636\pi\)
−0.419046 + 0.907965i \(0.637636\pi\)
\(158\) 566.464 0.285225
\(159\) 707.624 0.352945
\(160\) −160.000 −0.0790569
\(161\) 293.756 0.143796
\(162\) 1719.28 0.833824
\(163\) 3964.26 1.90494 0.952469 0.304636i \(-0.0985347\pi\)
0.952469 + 0.304636i \(0.0985347\pi\)
\(164\) −1928.51 −0.918241
\(165\) −1492.64 −0.704254
\(166\) −2272.65 −1.06260
\(167\) 3484.96 1.61482 0.807408 0.589993i \(-0.200869\pi\)
0.807408 + 0.589993i \(0.200869\pi\)
\(168\) −589.760 −0.270839
\(169\) −1044.53 −0.475436
\(170\) 864.920 0.390214
\(171\) 377.192 0.168682
\(172\) −1083.94 −0.480520
\(173\) 2239.51 0.984202 0.492101 0.870538i \(-0.336229\pi\)
0.492101 + 0.870538i \(0.336229\pi\)
\(174\) −745.096 −0.324630
\(175\) −319.300 −0.137925
\(176\) 827.520 0.354413
\(177\) −3619.39 −1.53701
\(178\) −952.208 −0.400961
\(179\) −2402.86 −1.00334 −0.501671 0.865058i \(-0.667281\pi\)
−0.501671 + 0.865058i \(0.667281\pi\)
\(180\) 126.320 0.0523075
\(181\) 4314.39 1.77174 0.885872 0.463929i \(-0.153561\pi\)
0.885872 + 0.463929i \(0.153561\pi\)
\(182\) 867.168 0.353180
\(183\) 200.496 0.0809896
\(184\) 184.000 0.0737210
\(185\) −1377.02 −0.547246
\(186\) −1813.24 −0.714802
\(187\) −4473.37 −1.74933
\(188\) −1283.44 −0.497896
\(189\) −1524.82 −0.586850
\(190\) −597.200 −0.228029
\(191\) −1930.75 −0.731434 −0.365717 0.930726i \(-0.619176\pi\)
−0.365717 + 0.930726i \(0.619176\pi\)
\(192\) −369.408 −0.138853
\(193\) 4302.32 1.60460 0.802299 0.596922i \(-0.203610\pi\)
0.802299 + 0.596922i \(0.203610\pi\)
\(194\) 3641.22 1.34755
\(195\) −979.740 −0.359798
\(196\) −719.504 −0.262210
\(197\) −5436.48 −1.96616 −0.983079 0.183184i \(-0.941360\pi\)
−0.983079 + 0.183184i \(0.941360\pi\)
\(198\) −653.328 −0.234495
\(199\) 1071.03 0.381525 0.190762 0.981636i \(-0.438904\pi\)
0.190762 + 0.981636i \(0.438904\pi\)
\(200\) −200.000 −0.0707107
\(201\) 521.488 0.183000
\(202\) −369.672 −0.128762
\(203\) 824.356 0.285017
\(204\) 1996.93 0.685358
\(205\) −2410.64 −0.821300
\(206\) 2499.55 0.845399
\(207\) −145.268 −0.0487770
\(208\) 543.168 0.181067
\(209\) 3088.72 1.02225
\(210\) −737.200 −0.242246
\(211\) −2350.65 −0.766946 −0.383473 0.923552i \(-0.625272\pi\)
−0.383473 + 0.923552i \(0.625272\pi\)
\(212\) −490.384 −0.158867
\(213\) −1054.61 −0.339253
\(214\) −1633.64 −0.521838
\(215\) −1354.92 −0.429790
\(216\) −955.104 −0.300864
\(217\) 2006.12 0.627579
\(218\) 518.528 0.161097
\(219\) 422.764 0.130446
\(220\) 1034.40 0.316997
\(221\) −2936.23 −0.893721
\(222\) −3179.26 −0.961163
\(223\) −2443.32 −0.733708 −0.366854 0.930279i \(-0.619565\pi\)
−0.366854 + 0.930279i \(0.619565\pi\)
\(224\) 408.704 0.121909
\(225\) 157.900 0.0467852
\(226\) −1781.75 −0.524426
\(227\) 997.432 0.291638 0.145819 0.989311i \(-0.453418\pi\)
0.145819 + 0.989311i \(0.453418\pi\)
\(228\) −1378.82 −0.400501
\(229\) −4084.76 −1.17873 −0.589364 0.807868i \(-0.700621\pi\)
−0.589364 + 0.807868i \(0.700621\pi\)
\(230\) 230.000 0.0659380
\(231\) 3812.80 1.08599
\(232\) 516.352 0.146121
\(233\) −1079.76 −0.303593 −0.151797 0.988412i \(-0.548506\pi\)
−0.151797 + 0.988412i \(0.548506\pi\)
\(234\) −428.832 −0.119802
\(235\) −1604.30 −0.445332
\(236\) 2508.24 0.691833
\(237\) 1634.82 0.448071
\(238\) −2209.35 −0.601727
\(239\) 3678.78 0.995652 0.497826 0.867277i \(-0.334132\pi\)
0.497826 + 0.867277i \(0.334132\pi\)
\(240\) −461.760 −0.124194
\(241\) −3127.15 −0.835841 −0.417920 0.908484i \(-0.637241\pi\)
−0.417920 + 0.908484i \(0.637241\pi\)
\(242\) −2687.92 −0.713992
\(243\) 1738.37 0.458915
\(244\) −138.944 −0.0364548
\(245\) −899.380 −0.234528
\(246\) −5565.69 −1.44250
\(247\) 2027.38 0.522263
\(248\) 1256.58 0.321745
\(249\) −6558.87 −1.66928
\(250\) −250.000 −0.0632456
\(251\) −1585.51 −0.398712 −0.199356 0.979927i \(-0.563885\pi\)
−0.199356 + 0.979927i \(0.563885\pi\)
\(252\) −322.672 −0.0806605
\(253\) −1189.56 −0.295601
\(254\) 4353.38 1.07542
\(255\) 2496.16 0.613002
\(256\) 256.000 0.0625000
\(257\) 7046.41 1.71028 0.855142 0.518394i \(-0.173470\pi\)
0.855142 + 0.518394i \(0.173470\pi\)
\(258\) −3128.24 −0.754867
\(259\) 3517.46 0.843878
\(260\) 678.960 0.161951
\(261\) −407.660 −0.0966802
\(262\) 3209.71 0.756856
\(263\) 4990.01 1.16995 0.584976 0.811051i \(-0.301104\pi\)
0.584976 + 0.811051i \(0.301104\pi\)
\(264\) 2388.22 0.556762
\(265\) −612.980 −0.142095
\(266\) 1525.49 0.351630
\(267\) −2748.07 −0.629885
\(268\) −361.392 −0.0823713
\(269\) 4842.06 1.09749 0.548746 0.835989i \(-0.315105\pi\)
0.548746 + 0.835989i \(0.315105\pi\)
\(270\) −1193.88 −0.269101
\(271\) −2580.28 −0.578379 −0.289189 0.957272i \(-0.593386\pi\)
−0.289189 + 0.957272i \(0.593386\pi\)
\(272\) −1383.87 −0.308491
\(273\) 2502.65 0.554825
\(274\) 3469.20 0.764898
\(275\) 1293.00 0.283530
\(276\) 531.024 0.115811
\(277\) 5444.79 1.18103 0.590516 0.807026i \(-0.298924\pi\)
0.590516 + 0.807026i \(0.298924\pi\)
\(278\) 1987.65 0.428817
\(279\) −992.068 −0.212880
\(280\) 510.880 0.109039
\(281\) −2473.91 −0.525199 −0.262599 0.964905i \(-0.584580\pi\)
−0.262599 + 0.964905i \(0.584580\pi\)
\(282\) −3704.01 −0.782165
\(283\) −1854.11 −0.389454 −0.194727 0.980858i \(-0.562382\pi\)
−0.194727 + 0.980858i \(0.562382\pi\)
\(284\) 730.847 0.152704
\(285\) −1723.52 −0.358219
\(286\) −3511.58 −0.726029
\(287\) 6157.74 1.26648
\(288\) −202.112 −0.0413527
\(289\) 2567.87 0.522668
\(290\) 645.440 0.130695
\(291\) 10508.6 2.11692
\(292\) −292.976 −0.0587161
\(293\) −4603.24 −0.917829 −0.458915 0.888480i \(-0.651762\pi\)
−0.458915 + 0.888480i \(0.651762\pi\)
\(294\) −2076.49 −0.411916
\(295\) 3135.30 0.618794
\(296\) 2203.23 0.432636
\(297\) 6174.75 1.20638
\(298\) 2046.16 0.397755
\(299\) −780.804 −0.151020
\(300\) −577.200 −0.111082
\(301\) 3461.01 0.662755
\(302\) 2618.79 0.498989
\(303\) −1066.87 −0.202278
\(304\) 955.520 0.180273
\(305\) −173.680 −0.0326062
\(306\) 1092.57 0.204111
\(307\) 1076.46 0.200119 0.100060 0.994981i \(-0.468097\pi\)
0.100060 + 0.994981i \(0.468097\pi\)
\(308\) −2642.27 −0.488823
\(309\) 7213.71 1.32807
\(310\) 1570.72 0.287777
\(311\) 8141.36 1.48442 0.742209 0.670168i \(-0.233778\pi\)
0.742209 + 0.670168i \(0.233778\pi\)
\(312\) 1567.58 0.284445
\(313\) −8180.24 −1.47724 −0.738618 0.674124i \(-0.764521\pi\)
−0.738618 + 0.674124i \(0.764521\pi\)
\(314\) 3297.40 0.592621
\(315\) −403.340 −0.0721449
\(316\) −1132.93 −0.201684
\(317\) −3229.21 −0.572146 −0.286073 0.958208i \(-0.592350\pi\)
−0.286073 + 0.958208i \(0.592350\pi\)
\(318\) −1415.25 −0.249570
\(319\) −3338.22 −0.585907
\(320\) 320.000 0.0559017
\(321\) −4714.69 −0.819776
\(322\) −587.512 −0.101679
\(323\) −5165.31 −0.889800
\(324\) −3438.56 −0.589602
\(325\) 848.700 0.144854
\(326\) −7928.52 −1.34699
\(327\) 1496.47 0.253074
\(328\) 3857.03 0.649294
\(329\) 4098.02 0.686722
\(330\) 2985.28 0.497983
\(331\) −3414.79 −0.567052 −0.283526 0.958965i \(-0.591504\pi\)
−0.283526 + 0.958965i \(0.591504\pi\)
\(332\) 4545.30 0.751372
\(333\) −1739.45 −0.286251
\(334\) −6969.92 −1.14185
\(335\) −451.740 −0.0736751
\(336\) 1179.52 0.191512
\(337\) −2157.79 −0.348791 −0.174395 0.984676i \(-0.555797\pi\)
−0.174395 + 0.984676i \(0.555797\pi\)
\(338\) 2089.06 0.336184
\(339\) −5142.14 −0.823842
\(340\) −1729.84 −0.275923
\(341\) −8123.77 −1.29011
\(342\) −754.384 −0.119276
\(343\) 6678.17 1.05128
\(344\) 2167.87 0.339779
\(345\) 663.780 0.103585
\(346\) −4479.03 −0.695936
\(347\) −1920.53 −0.297116 −0.148558 0.988904i \(-0.547463\pi\)
−0.148558 + 0.988904i \(0.547463\pi\)
\(348\) 1490.19 0.229548
\(349\) 3086.66 0.473424 0.236712 0.971580i \(-0.423930\pi\)
0.236712 + 0.971580i \(0.423930\pi\)
\(350\) 638.600 0.0975275
\(351\) 4052.99 0.616332
\(352\) −1655.04 −0.250608
\(353\) −350.901 −0.0529082 −0.0264541 0.999650i \(-0.508422\pi\)
−0.0264541 + 0.999650i \(0.508422\pi\)
\(354\) 7238.79 1.08683
\(355\) 913.559 0.136582
\(356\) 1904.42 0.283522
\(357\) −6376.19 −0.945277
\(358\) 4805.72 0.709470
\(359\) −6536.65 −0.960979 −0.480489 0.877000i \(-0.659541\pi\)
−0.480489 + 0.877000i \(0.659541\pi\)
\(360\) −252.640 −0.0369870
\(361\) −3292.52 −0.480029
\(362\) −8628.77 −1.25281
\(363\) −7757.34 −1.12164
\(364\) −1734.34 −0.249736
\(365\) −366.220 −0.0525173
\(366\) −400.992 −0.0572683
\(367\) 3858.89 0.548863 0.274431 0.961607i \(-0.411510\pi\)
0.274431 + 0.961607i \(0.411510\pi\)
\(368\) −368.000 −0.0521286
\(369\) −3045.12 −0.429601
\(370\) 2754.04 0.386961
\(371\) 1565.80 0.219116
\(372\) 3626.48 0.505442
\(373\) 7132.34 0.990076 0.495038 0.868871i \(-0.335154\pi\)
0.495038 + 0.868871i \(0.335154\pi\)
\(374\) 8946.74 1.23696
\(375\) −721.500 −0.0993550
\(376\) 2566.88 0.352066
\(377\) −2191.14 −0.299336
\(378\) 3049.65 0.414966
\(379\) −10560.1 −1.43123 −0.715613 0.698497i \(-0.753853\pi\)
−0.715613 + 0.698497i \(0.753853\pi\)
\(380\) 1194.40 0.161241
\(381\) 12563.9 1.68941
\(382\) 3861.49 0.517202
\(383\) 343.459 0.0458222 0.0229111 0.999738i \(-0.492707\pi\)
0.0229111 + 0.999738i \(0.492707\pi\)
\(384\) 738.816 0.0981837
\(385\) −3302.84 −0.437217
\(386\) −8604.63 −1.13462
\(387\) −1711.54 −0.224812
\(388\) −7282.43 −0.952860
\(389\) 6173.29 0.804623 0.402312 0.915503i \(-0.368207\pi\)
0.402312 + 0.915503i \(0.368207\pi\)
\(390\) 1959.48 0.254416
\(391\) 1989.32 0.257299
\(392\) 1439.01 0.185410
\(393\) 9263.21 1.18898
\(394\) 10873.0 1.39028
\(395\) −1416.16 −0.180392
\(396\) 1306.66 0.165813
\(397\) −1490.04 −0.188371 −0.0941853 0.995555i \(-0.530025\pi\)
−0.0941853 + 0.995555i \(0.530025\pi\)
\(398\) −2142.06 −0.269779
\(399\) 4402.56 0.552390
\(400\) 400.000 0.0500000
\(401\) 2227.87 0.277443 0.138721 0.990331i \(-0.455701\pi\)
0.138721 + 0.990331i \(0.455701\pi\)
\(402\) −1042.98 −0.129400
\(403\) −5332.28 −0.659107
\(404\) 739.343 0.0910488
\(405\) −4298.20 −0.527356
\(406\) −1648.71 −0.201537
\(407\) −14243.9 −1.73475
\(408\) −3993.86 −0.484621
\(409\) −469.656 −0.0567799 −0.0283900 0.999597i \(-0.509038\pi\)
−0.0283900 + 0.999597i \(0.509038\pi\)
\(410\) 4821.28 0.580747
\(411\) 10012.1 1.20161
\(412\) −4999.11 −0.597787
\(413\) −8008.81 −0.954208
\(414\) 290.536 0.0344905
\(415\) 5681.62 0.672048
\(416\) −1086.34 −0.128034
\(417\) 5736.36 0.673647
\(418\) −6177.44 −0.722843
\(419\) −14957.6 −1.74398 −0.871991 0.489522i \(-0.837171\pi\)
−0.871991 + 0.489522i \(0.837171\pi\)
\(420\) 1474.40 0.171294
\(421\) −10830.8 −1.25382 −0.626912 0.779090i \(-0.715682\pi\)
−0.626912 + 0.779090i \(0.715682\pi\)
\(422\) 4701.31 0.542313
\(423\) −2026.55 −0.232942
\(424\) 980.768 0.112336
\(425\) −2162.30 −0.246793
\(426\) 2109.23 0.239888
\(427\) 443.648 0.0502802
\(428\) 3267.28 0.368995
\(429\) −10134.4 −1.14055
\(430\) 2709.84 0.303907
\(431\) 1774.70 0.198339 0.0991695 0.995071i \(-0.468381\pi\)
0.0991695 + 0.995071i \(0.468381\pi\)
\(432\) 1910.21 0.212743
\(433\) 4750.64 0.527254 0.263627 0.964625i \(-0.415081\pi\)
0.263627 + 0.964625i \(0.415081\pi\)
\(434\) −4012.25 −0.443765
\(435\) 1862.74 0.205314
\(436\) −1037.06 −0.113913
\(437\) −1373.56 −0.150358
\(438\) −845.529 −0.0922395
\(439\) 8283.27 0.900544 0.450272 0.892891i \(-0.351327\pi\)
0.450272 + 0.892891i \(0.351327\pi\)
\(440\) −2068.80 −0.224150
\(441\) −1136.10 −0.122675
\(442\) 5872.47 0.631956
\(443\) 13646.4 1.46357 0.731785 0.681536i \(-0.238688\pi\)
0.731785 + 0.681536i \(0.238688\pi\)
\(444\) 6358.53 0.679645
\(445\) 2380.52 0.253590
\(446\) 4886.64 0.518810
\(447\) 5905.22 0.624849
\(448\) −817.408 −0.0862029
\(449\) −1904.70 −0.200197 −0.100098 0.994978i \(-0.531916\pi\)
−0.100098 + 0.994978i \(0.531916\pi\)
\(450\) −315.800 −0.0330822
\(451\) −24935.7 −2.60349
\(452\) 3563.50 0.370825
\(453\) 7557.84 0.783881
\(454\) −1994.86 −0.206219
\(455\) −2167.92 −0.223371
\(456\) 2757.63 0.283197
\(457\) −10056.3 −1.02935 −0.514676 0.857385i \(-0.672088\pi\)
−0.514676 + 0.857385i \(0.672088\pi\)
\(458\) 8169.52 0.833486
\(459\) −10326.1 −1.05007
\(460\) −460.000 −0.0466252
\(461\) 9219.60 0.931453 0.465726 0.884929i \(-0.345793\pi\)
0.465726 + 0.884929i \(0.345793\pi\)
\(462\) −7625.60 −0.767911
\(463\) 2892.39 0.290326 0.145163 0.989408i \(-0.453629\pi\)
0.145163 + 0.989408i \(0.453629\pi\)
\(464\) −1032.70 −0.103323
\(465\) 4533.10 0.452081
\(466\) 2159.51 0.214673
\(467\) 16773.0 1.66201 0.831007 0.556263i \(-0.187765\pi\)
0.831007 + 0.556263i \(0.187765\pi\)
\(468\) 857.663 0.0847126
\(469\) 1153.92 0.113610
\(470\) 3208.60 0.314897
\(471\) 9516.30 0.930972
\(472\) −5016.48 −0.489200
\(473\) −14015.3 −1.36242
\(474\) −3269.63 −0.316834
\(475\) 1493.00 0.144218
\(476\) 4418.70 0.425485
\(477\) −774.317 −0.0743261
\(478\) −7357.57 −0.704032
\(479\) −1361.07 −0.129831 −0.0649153 0.997891i \(-0.520678\pi\)
−0.0649153 + 0.997891i \(0.520678\pi\)
\(480\) 923.520 0.0878182
\(481\) −9349.42 −0.886272
\(482\) 6254.31 0.591029
\(483\) −1695.56 −0.159732
\(484\) 5375.84 0.504869
\(485\) −9103.04 −0.852264
\(486\) −3476.74 −0.324502
\(487\) 10820.7 1.00684 0.503420 0.864042i \(-0.332075\pi\)
0.503420 + 0.864042i \(0.332075\pi\)
\(488\) 277.888 0.0257774
\(489\) −22881.7 −2.11605
\(490\) 1798.76 0.165836
\(491\) 18702.3 1.71899 0.859496 0.511143i \(-0.170778\pi\)
0.859496 + 0.511143i \(0.170778\pi\)
\(492\) 11131.4 1.02000
\(493\) 5582.54 0.509990
\(494\) −4054.75 −0.369295
\(495\) 1633.32 0.148308
\(496\) −2513.15 −0.227508
\(497\) −2333.60 −0.210616
\(498\) 13117.7 1.18036
\(499\) −3824.46 −0.343099 −0.171549 0.985176i \(-0.554877\pi\)
−0.171549 + 0.985176i \(0.554877\pi\)
\(500\) 500.000 0.0447214
\(501\) −20115.2 −1.79377
\(502\) 3171.03 0.281932
\(503\) −11021.2 −0.976961 −0.488481 0.872575i \(-0.662449\pi\)
−0.488481 + 0.872575i \(0.662449\pi\)
\(504\) 645.344 0.0570356
\(505\) 924.179 0.0814365
\(506\) 2379.12 0.209021
\(507\) 6029.04 0.528125
\(508\) −8706.77 −0.760434
\(509\) −16626.6 −1.44786 −0.723932 0.689872i \(-0.757667\pi\)
−0.723932 + 0.689872i \(0.757667\pi\)
\(510\) −4992.32 −0.433458
\(511\) 935.472 0.0809840
\(512\) −512.000 −0.0441942
\(513\) 7129.86 0.613627
\(514\) −14092.8 −1.20935
\(515\) −6248.88 −0.534677
\(516\) 6256.48 0.533772
\(517\) −16594.9 −1.41169
\(518\) −7034.92 −0.596712
\(519\) −12926.5 −1.09327
\(520\) −1357.92 −0.114517
\(521\) 1808.00 0.152035 0.0760174 0.997106i \(-0.475780\pi\)
0.0760174 + 0.997106i \(0.475780\pi\)
\(522\) 815.321 0.0683632
\(523\) 988.509 0.0826472 0.0413236 0.999146i \(-0.486843\pi\)
0.0413236 + 0.999146i \(0.486843\pi\)
\(524\) −6419.41 −0.535178
\(525\) 1843.00 0.153210
\(526\) −9980.02 −0.827281
\(527\) 13585.5 1.12295
\(528\) −4776.45 −0.393690
\(529\) 529.000 0.0434783
\(530\) 1225.96 0.100476
\(531\) 3960.52 0.323676
\(532\) −3050.98 −0.248640
\(533\) −16367.3 −1.33011
\(534\) 5496.15 0.445396
\(535\) 4084.10 0.330039
\(536\) 722.783 0.0582453
\(537\) 13869.3 1.11453
\(538\) −9684.12 −0.776044
\(539\) −9303.19 −0.743445
\(540\) 2387.76 0.190283
\(541\) −16521.4 −1.31296 −0.656480 0.754344i \(-0.727955\pi\)
−0.656480 + 0.754344i \(0.727955\pi\)
\(542\) 5160.55 0.408976
\(543\) −24902.6 −1.96809
\(544\) 2767.74 0.218136
\(545\) −1296.32 −0.101887
\(546\) −5005.30 −0.392320
\(547\) 14111.8 1.10307 0.551533 0.834153i \(-0.314043\pi\)
0.551533 + 0.834153i \(0.314043\pi\)
\(548\) −6938.40 −0.540865
\(549\) −219.393 −0.0170555
\(550\) −2586.00 −0.200486
\(551\) −3854.57 −0.298022
\(552\) −1062.05 −0.0818909
\(553\) 3617.44 0.278172
\(554\) −10889.6 −0.835115
\(555\) 7948.16 0.607893
\(556\) −3975.30 −0.303220
\(557\) 3746.67 0.285012 0.142506 0.989794i \(-0.454484\pi\)
0.142506 + 0.989794i \(0.454484\pi\)
\(558\) 1984.14 0.150529
\(559\) −9199.37 −0.696050
\(560\) −1021.76 −0.0771022
\(561\) 25820.3 1.94320
\(562\) 4947.81 0.371372
\(563\) 11239.5 0.841365 0.420683 0.907208i \(-0.361791\pi\)
0.420683 + 0.907208i \(0.361791\pi\)
\(564\) 7408.02 0.553074
\(565\) 4454.38 0.331676
\(566\) 3708.22 0.275385
\(567\) 10979.3 0.813207
\(568\) −1461.69 −0.107978
\(569\) 1870.13 0.137786 0.0688928 0.997624i \(-0.478053\pi\)
0.0688928 + 0.997624i \(0.478053\pi\)
\(570\) 3447.04 0.253299
\(571\) 23604.9 1.73001 0.865003 0.501766i \(-0.167316\pi\)
0.865003 + 0.501766i \(0.167316\pi\)
\(572\) 7023.17 0.513380
\(573\) 11144.3 0.812493
\(574\) −12315.5 −0.895537
\(575\) −575.000 −0.0417029
\(576\) 404.224 0.0292408
\(577\) 16572.9 1.19573 0.597866 0.801596i \(-0.296015\pi\)
0.597866 + 0.801596i \(0.296015\pi\)
\(578\) −5135.74 −0.369582
\(579\) −24833.0 −1.78242
\(580\) −1290.88 −0.0924153
\(581\) −14513.1 −1.03633
\(582\) −21017.1 −1.49689
\(583\) −6340.67 −0.450435
\(584\) 585.952 0.0415186
\(585\) 1072.08 0.0757693
\(586\) 9206.47 0.649003
\(587\) −27505.5 −1.93403 −0.967015 0.254721i \(-0.918017\pi\)
−0.967015 + 0.254721i \(0.918017\pi\)
\(588\) 4152.98 0.291269
\(589\) −9380.35 −0.656215
\(590\) −6270.60 −0.437554
\(591\) 31379.4 2.18405
\(592\) −4406.46 −0.305920
\(593\) 21998.3 1.52338 0.761689 0.647943i \(-0.224370\pi\)
0.761689 + 0.647943i \(0.224370\pi\)
\(594\) −12349.5 −0.853041
\(595\) 5523.38 0.380566
\(596\) −4092.32 −0.281255
\(597\) −6182.00 −0.423806
\(598\) 1561.61 0.106788
\(599\) −13628.2 −0.929605 −0.464802 0.885415i \(-0.653875\pi\)
−0.464802 + 0.885415i \(0.653875\pi\)
\(600\) 1154.40 0.0785470
\(601\) −14029.2 −0.952182 −0.476091 0.879396i \(-0.657947\pi\)
−0.476091 + 0.879396i \(0.657947\pi\)
\(602\) −6922.02 −0.468638
\(603\) −570.638 −0.0385376
\(604\) −5237.59 −0.352838
\(605\) 6719.80 0.451568
\(606\) 2133.75 0.143032
\(607\) −3276.29 −0.219079 −0.109539 0.993982i \(-0.534938\pi\)
−0.109539 + 0.993982i \(0.534938\pi\)
\(608\) −1911.04 −0.127472
\(609\) −4758.19 −0.316603
\(610\) 347.360 0.0230560
\(611\) −10892.6 −0.721221
\(612\) −2185.14 −0.144328
\(613\) 5419.49 0.357082 0.178541 0.983932i \(-0.442862\pi\)
0.178541 + 0.983932i \(0.442862\pi\)
\(614\) −2152.91 −0.141506
\(615\) 13914.2 0.912318
\(616\) 5284.55 0.345650
\(617\) 2820.18 0.184013 0.0920067 0.995758i \(-0.470672\pi\)
0.0920067 + 0.995758i \(0.470672\pi\)
\(618\) −14427.4 −0.939088
\(619\) 22656.3 1.47113 0.735567 0.677452i \(-0.236916\pi\)
0.735567 + 0.677452i \(0.236916\pi\)
\(620\) −3141.44 −0.203489
\(621\) −2745.93 −0.177440
\(622\) −16282.7 −1.04964
\(623\) −6080.80 −0.391047
\(624\) −3135.17 −0.201133
\(625\) 625.000 0.0400000
\(626\) 16360.5 1.04456
\(627\) −17828.1 −1.13554
\(628\) −6594.80 −0.419046
\(629\) 23820.2 1.50998
\(630\) 806.680 0.0510142
\(631\) −4336.70 −0.273600 −0.136800 0.990599i \(-0.543682\pi\)
−0.136800 + 0.990599i \(0.543682\pi\)
\(632\) 2265.86 0.142612
\(633\) 13568.0 0.851941
\(634\) 6458.41 0.404568
\(635\) −10883.5 −0.680153
\(636\) 2830.50 0.176472
\(637\) −6106.43 −0.379821
\(638\) 6676.43 0.414299
\(639\) 1154.01 0.0714427
\(640\) −640.000 −0.0395285
\(641\) 27123.4 1.67131 0.835654 0.549255i \(-0.185089\pi\)
0.835654 + 0.549255i \(0.185089\pi\)
\(642\) 9429.38 0.579670
\(643\) 20845.2 1.27847 0.639234 0.769012i \(-0.279252\pi\)
0.639234 + 0.769012i \(0.279252\pi\)
\(644\) 1175.02 0.0718982
\(645\) 7820.60 0.477420
\(646\) 10330.6 0.629183
\(647\) −3035.99 −0.184478 −0.0922388 0.995737i \(-0.529402\pi\)
−0.0922388 + 0.995737i \(0.529402\pi\)
\(648\) 6877.12 0.416912
\(649\) 32431.6 1.96156
\(650\) −1697.40 −0.102427
\(651\) −11579.4 −0.697129
\(652\) 15857.0 0.952469
\(653\) 3794.77 0.227413 0.113707 0.993514i \(-0.463728\pi\)
0.113707 + 0.993514i \(0.463728\pi\)
\(654\) −2992.94 −0.178950
\(655\) −8024.26 −0.478678
\(656\) −7714.05 −0.459121
\(657\) −462.609 −0.0274705
\(658\) −8196.05 −0.485586
\(659\) 22291.8 1.31770 0.658849 0.752275i \(-0.271043\pi\)
0.658849 + 0.752275i \(0.271043\pi\)
\(660\) −5970.56 −0.352127
\(661\) 30545.5 1.79740 0.898700 0.438565i \(-0.144513\pi\)
0.898700 + 0.438565i \(0.144513\pi\)
\(662\) 6829.59 0.400966
\(663\) 16947.9 0.992765
\(664\) −9090.59 −0.531300
\(665\) −3813.72 −0.222391
\(666\) 3478.91 0.202410
\(667\) 1484.51 0.0861777
\(668\) 13939.8 0.807408
\(669\) 14102.9 0.815019
\(670\) 903.479 0.0520962
\(671\) −1796.55 −0.103360
\(672\) −2359.04 −0.135420
\(673\) 482.978 0.0276633 0.0138317 0.999904i \(-0.495597\pi\)
0.0138317 + 0.999904i \(0.495597\pi\)
\(674\) 4315.59 0.246632
\(675\) 2984.70 0.170194
\(676\) −4178.13 −0.237718
\(677\) 33458.8 1.89945 0.949725 0.313085i \(-0.101363\pi\)
0.949725 + 0.313085i \(0.101363\pi\)
\(678\) 10284.3 0.582545
\(679\) 23252.8 1.31423
\(680\) 3459.68 0.195107
\(681\) −5757.18 −0.323958
\(682\) 16247.5 0.912244
\(683\) −12633.5 −0.707770 −0.353885 0.935289i \(-0.615140\pi\)
−0.353885 + 0.935289i \(0.615140\pi\)
\(684\) 1508.77 0.0843409
\(685\) −8673.00 −0.483764
\(686\) −13356.3 −0.743364
\(687\) 23577.2 1.30936
\(688\) −4335.74 −0.240260
\(689\) −4161.89 −0.230124
\(690\) −1327.56 −0.0732455
\(691\) −15067.1 −0.829491 −0.414745 0.909938i \(-0.636129\pi\)
−0.414745 + 0.909938i \(0.636129\pi\)
\(692\) 8958.05 0.492101
\(693\) −4172.15 −0.228697
\(694\) 3841.06 0.210093
\(695\) −4969.12 −0.271208
\(696\) −2980.38 −0.162315
\(697\) 41700.2 2.26615
\(698\) −6173.32 −0.334762
\(699\) 6232.36 0.337238
\(700\) −1277.20 −0.0689623
\(701\) −16517.0 −0.889927 −0.444964 0.895549i \(-0.646783\pi\)
−0.444964 + 0.895549i \(0.646783\pi\)
\(702\) −8105.98 −0.435812
\(703\) −16447.1 −0.882383
\(704\) 3310.08 0.177207
\(705\) 9260.02 0.494685
\(706\) 701.802 0.0374117
\(707\) −2360.72 −0.125579
\(708\) −14477.6 −0.768503
\(709\) −3206.99 −0.169874 −0.0849372 0.996386i \(-0.527069\pi\)
−0.0849372 + 0.996386i \(0.527069\pi\)
\(710\) −1827.12 −0.0965782
\(711\) −1788.90 −0.0943584
\(712\) −3808.83 −0.200480
\(713\) 3612.66 0.189755
\(714\) 12752.4 0.668412
\(715\) 8778.96 0.459181
\(716\) −9611.44 −0.501671
\(717\) −21233.9 −1.10599
\(718\) 13073.3 0.679515
\(719\) −26482.4 −1.37361 −0.686806 0.726841i \(-0.740988\pi\)
−0.686806 + 0.726841i \(0.740988\pi\)
\(720\) 505.280 0.0261537
\(721\) 15962.1 0.824496
\(722\) 6585.04 0.339432
\(723\) 18049.9 0.928471
\(724\) 17257.5 0.885872
\(725\) −1613.60 −0.0826588
\(726\) 15514.7 0.793118
\(727\) 3270.55 0.166848 0.0834238 0.996514i \(-0.473414\pi\)
0.0834238 + 0.996514i \(0.473414\pi\)
\(728\) 3468.67 0.176590
\(729\) 13176.4 0.669432
\(730\) 732.440 0.0371353
\(731\) 23438.0 1.18589
\(732\) 801.985 0.0404948
\(733\) −25182.1 −1.26892 −0.634462 0.772954i \(-0.718778\pi\)
−0.634462 + 0.772954i \(0.718778\pi\)
\(734\) −7717.79 −0.388105
\(735\) 5191.22 0.260519
\(736\) 736.000 0.0368605
\(737\) −4672.80 −0.233548
\(738\) 6090.25 0.303774
\(739\) −19406.3 −0.965999 −0.483000 0.875621i \(-0.660453\pi\)
−0.483000 + 0.875621i \(0.660453\pi\)
\(740\) −5508.08 −0.273623
\(741\) −11702.0 −0.580141
\(742\) −3131.59 −0.154939
\(743\) −11547.0 −0.570146 −0.285073 0.958506i \(-0.592018\pi\)
−0.285073 + 0.958506i \(0.592018\pi\)
\(744\) −7252.96 −0.357401
\(745\) −5115.40 −0.251562
\(746\) −14264.7 −0.700090
\(747\) 7177.03 0.351531
\(748\) −17893.5 −0.874666
\(749\) −10432.4 −0.508936
\(750\) 1443.00 0.0702546
\(751\) −30113.3 −1.46318 −0.731591 0.681744i \(-0.761222\pi\)
−0.731591 + 0.681744i \(0.761222\pi\)
\(752\) −5133.76 −0.248948
\(753\) 9151.58 0.442898
\(754\) 4382.28 0.211662
\(755\) −6546.98 −0.315588
\(756\) −6099.30 −0.293425
\(757\) −26192.3 −1.25756 −0.628782 0.777582i \(-0.716446\pi\)
−0.628782 + 0.777582i \(0.716446\pi\)
\(758\) 21120.2 1.01203
\(759\) 6866.15 0.328360
\(760\) −2388.80 −0.114014
\(761\) −5051.36 −0.240620 −0.120310 0.992736i \(-0.538389\pi\)
−0.120310 + 0.992736i \(0.538389\pi\)
\(762\) −25127.7 −1.19460
\(763\) 3311.32 0.157114
\(764\) −7722.98 −0.365717
\(765\) −2731.42 −0.129091
\(766\) −686.917 −0.0324012
\(767\) 21287.5 1.00215
\(768\) −1477.63 −0.0694264
\(769\) 41142.8 1.92932 0.964661 0.263494i \(-0.0848749\pi\)
0.964661 + 0.263494i \(0.0848749\pi\)
\(770\) 6605.68 0.309159
\(771\) −40671.9 −1.89982
\(772\) 17209.3 0.802299
\(773\) 3087.82 0.143675 0.0718376 0.997416i \(-0.477114\pi\)
0.0718376 + 0.997416i \(0.477114\pi\)
\(774\) 3423.07 0.158966
\(775\) −3926.80 −0.182006
\(776\) 14564.9 0.673773
\(777\) −20302.8 −0.937398
\(778\) −12346.6 −0.568954
\(779\) −28792.7 −1.32427
\(780\) −3918.96 −0.179899
\(781\) 9449.86 0.432961
\(782\) −3978.63 −0.181938
\(783\) −7705.78 −0.351701
\(784\) −2878.02 −0.131105
\(785\) −8243.50 −0.374807
\(786\) −18526.4 −0.840732
\(787\) 31166.9 1.41167 0.705833 0.708378i \(-0.250573\pi\)
0.705833 + 0.708378i \(0.250573\pi\)
\(788\) −21745.9 −0.983079
\(789\) −28802.4 −1.29961
\(790\) 2832.32 0.127556
\(791\) −11378.3 −0.511460
\(792\) −2613.31 −0.117247
\(793\) −1179.22 −0.0528061
\(794\) 2980.09 0.133198
\(795\) 3538.12 0.157842
\(796\) 4284.13 0.190762
\(797\) 27495.3 1.22200 0.611000 0.791631i \(-0.290768\pi\)
0.611000 + 0.791631i \(0.290768\pi\)
\(798\) −8805.12 −0.390599
\(799\) 27751.8 1.22877
\(800\) −800.000 −0.0353553
\(801\) 3007.08 0.132646
\(802\) −4455.74 −0.196182
\(803\) −3788.18 −0.166478
\(804\) 2085.95 0.0914999
\(805\) 1468.78 0.0643077
\(806\) 10664.6 0.466059
\(807\) −27948.4 −1.21912
\(808\) −1478.69 −0.0643812
\(809\) −1773.85 −0.0770893 −0.0385447 0.999257i \(-0.512272\pi\)
−0.0385447 + 0.999257i \(0.512272\pi\)
\(810\) 8596.40 0.372897
\(811\) −17.8982 −0.000774956 0 −0.000387478 1.00000i \(-0.500123\pi\)
−0.000387478 1.00000i \(0.500123\pi\)
\(812\) 3297.42 0.142509
\(813\) 14893.4 0.642476
\(814\) 28487.8 1.22665
\(815\) 19821.3 0.851914
\(816\) 7987.71 0.342679
\(817\) −16183.2 −0.692996
\(818\) 939.311 0.0401495
\(819\) −2738.52 −0.116840
\(820\) −9642.56 −0.410650
\(821\) −15459.5 −0.657173 −0.328587 0.944474i \(-0.606572\pi\)
−0.328587 + 0.944474i \(0.606572\pi\)
\(822\) −20024.2 −0.849666
\(823\) −29737.4 −1.25951 −0.629757 0.776792i \(-0.716845\pi\)
−0.629757 + 0.776792i \(0.716845\pi\)
\(824\) 9998.21 0.422699
\(825\) −7463.20 −0.314952
\(826\) 16017.6 0.674727
\(827\) −33405.8 −1.40463 −0.702317 0.711864i \(-0.747851\pi\)
−0.702317 + 0.711864i \(0.747851\pi\)
\(828\) −581.073 −0.0243885
\(829\) 45802.5 1.91892 0.959460 0.281844i \(-0.0909459\pi\)
0.959460 + 0.281844i \(0.0909459\pi\)
\(830\) −11363.2 −0.475210
\(831\) −31427.3 −1.31192
\(832\) 2172.67 0.0905335
\(833\) 15557.8 0.647115
\(834\) −11472.7 −0.476340
\(835\) 17424.8 0.722168
\(836\) 12354.9 0.511127
\(837\) −18752.5 −0.774411
\(838\) 29915.3 1.23318
\(839\) 5331.64 0.219391 0.109695 0.993965i \(-0.465013\pi\)
0.109695 + 0.993965i \(0.465013\pi\)
\(840\) −2948.80 −0.121123
\(841\) −20223.1 −0.829188
\(842\) 21661.6 0.886588
\(843\) 14279.4 0.583402
\(844\) −9402.61 −0.383473
\(845\) −5222.66 −0.212621
\(846\) 4053.11 0.164715
\(847\) −17165.1 −0.696338
\(848\) −1961.54 −0.0794333
\(849\) 10701.9 0.432614
\(850\) 4324.60 0.174509
\(851\) 6334.29 0.255155
\(852\) −4218.45 −0.169626
\(853\) −5792.08 −0.232494 −0.116247 0.993220i \(-0.537086\pi\)
−0.116247 + 0.993220i \(0.537086\pi\)
\(854\) −887.296 −0.0355534
\(855\) 1885.96 0.0754368
\(856\) −6534.56 −0.260919
\(857\) −22502.2 −0.896918 −0.448459 0.893803i \(-0.648027\pi\)
−0.448459 + 0.893803i \(0.648027\pi\)
\(858\) 20268.9 0.806489
\(859\) 92.1371 0.00365969 0.00182985 0.999998i \(-0.499418\pi\)
0.00182985 + 0.999998i \(0.499418\pi\)
\(860\) −5419.68 −0.214895
\(861\) −35542.5 −1.40684
\(862\) −3549.39 −0.140247
\(863\) −33877.4 −1.33627 −0.668134 0.744041i \(-0.732907\pi\)
−0.668134 + 0.744041i \(0.732907\pi\)
\(864\) −3820.42 −0.150432
\(865\) 11197.6 0.440149
\(866\) −9501.27 −0.372825
\(867\) −14821.7 −0.580592
\(868\) 8024.50 0.313789
\(869\) −14648.8 −0.571836
\(870\) −3725.48 −0.145179
\(871\) −3067.13 −0.119318
\(872\) 2074.11 0.0805485
\(873\) −11499.0 −0.445798
\(874\) 2747.12 0.106319
\(875\) −1596.50 −0.0616818
\(876\) 1691.06 0.0652232
\(877\) 31434.6 1.21034 0.605172 0.796094i \(-0.293104\pi\)
0.605172 + 0.796094i \(0.293104\pi\)
\(878\) −16566.5 −0.636781
\(879\) 26569.9 1.01955
\(880\) 4137.60 0.158498
\(881\) −15812.0 −0.604676 −0.302338 0.953201i \(-0.597767\pi\)
−0.302338 + 0.953201i \(0.597767\pi\)
\(882\) 2272.20 0.0867447
\(883\) 786.986 0.0299934 0.0149967 0.999888i \(-0.495226\pi\)
0.0149967 + 0.999888i \(0.495226\pi\)
\(884\) −11744.9 −0.446861
\(885\) −18097.0 −0.687370
\(886\) −27292.9 −1.03490
\(887\) −52830.3 −1.99985 −0.999925 0.0122640i \(-0.996096\pi\)
−0.999925 + 0.0122640i \(0.996096\pi\)
\(888\) −12717.1 −0.480582
\(889\) 27800.7 1.04883
\(890\) −4761.04 −0.179315
\(891\) −44460.6 −1.67170
\(892\) −9773.28 −0.366854
\(893\) −19161.8 −0.718056
\(894\) −11810.4 −0.441835
\(895\) −12014.3 −0.448708
\(896\) 1634.82 0.0609547
\(897\) 4506.80 0.167757
\(898\) 3809.40 0.141560
\(899\) 10138.1 0.376110
\(900\) 631.601 0.0233926
\(901\) 10603.6 0.392071
\(902\) 49871.4 1.84095
\(903\) −19976.9 −0.736203
\(904\) −7127.01 −0.262213
\(905\) 21571.9 0.792348
\(906\) −15115.7 −0.554288
\(907\) −31587.2 −1.15638 −0.578189 0.815903i \(-0.696240\pi\)
−0.578189 + 0.815903i \(0.696240\pi\)
\(908\) 3989.73 0.145819
\(909\) 1167.42 0.0425974
\(910\) 4335.84 0.157947
\(911\) 5713.89 0.207804 0.103902 0.994588i \(-0.466867\pi\)
0.103902 + 0.994588i \(0.466867\pi\)
\(912\) −5515.26 −0.200251
\(913\) 58770.7 2.13037
\(914\) 20112.6 0.727861
\(915\) 1002.48 0.0362197
\(916\) −16339.0 −0.589364
\(917\) 20497.2 0.738142
\(918\) 20652.2 0.742511
\(919\) −52580.9 −1.88736 −0.943680 0.330860i \(-0.892661\pi\)
−0.943680 + 0.330860i \(0.892661\pi\)
\(920\) 920.000 0.0329690
\(921\) −6213.30 −0.222297
\(922\) −18439.2 −0.658636
\(923\) 6202.70 0.221197
\(924\) 15251.2 0.542995
\(925\) −6885.10 −0.244736
\(926\) −5784.79 −0.205291
\(927\) −7893.60 −0.279676
\(928\) 2065.41 0.0730607
\(929\) 51129.0 1.80569 0.902845 0.429965i \(-0.141474\pi\)
0.902845 + 0.429965i \(0.141474\pi\)
\(930\) −9066.20 −0.319669
\(931\) −10742.2 −0.378154
\(932\) −4319.03 −0.151797
\(933\) −46991.9 −1.64893
\(934\) −33545.9 −1.17522
\(935\) −22366.8 −0.782325
\(936\) −1715.33 −0.0599009
\(937\) −5328.37 −0.185774 −0.0928870 0.995677i \(-0.529610\pi\)
−0.0928870 + 0.995677i \(0.529610\pi\)
\(938\) −2307.85 −0.0803346
\(939\) 47216.4 1.64095
\(940\) −6417.20 −0.222666
\(941\) 41912.5 1.45198 0.725988 0.687708i \(-0.241383\pi\)
0.725988 + 0.687708i \(0.241383\pi\)
\(942\) −19032.6 −0.658297
\(943\) 11088.9 0.382933
\(944\) 10033.0 0.345916
\(945\) −7624.12 −0.262447
\(946\) 28030.6 0.963376
\(947\) −32175.8 −1.10409 −0.552045 0.833814i \(-0.686152\pi\)
−0.552045 + 0.833814i \(0.686152\pi\)
\(948\) 6539.26 0.224035
\(949\) −2486.49 −0.0850525
\(950\) −2986.00 −0.101978
\(951\) 18639.0 0.635553
\(952\) −8837.41 −0.300864
\(953\) −22469.7 −0.763763 −0.381882 0.924211i \(-0.624724\pi\)
−0.381882 + 0.924211i \(0.624724\pi\)
\(954\) 1548.63 0.0525565
\(955\) −9653.73 −0.327107
\(956\) 14715.1 0.497826
\(957\) 19268.2 0.650838
\(958\) 2722.14 0.0918042
\(959\) 22154.3 0.745986
\(960\) −1847.04 −0.0620969
\(961\) −5119.37 −0.171843
\(962\) 18698.8 0.626689
\(963\) 5159.04 0.172635
\(964\) −12508.6 −0.417920
\(965\) 21511.6 0.717598
\(966\) 3391.12 0.112948
\(967\) 15967.7 0.531009 0.265504 0.964110i \(-0.414462\pi\)
0.265504 + 0.964110i \(0.414462\pi\)
\(968\) −10751.7 −0.356996
\(969\) 29814.2 0.988409
\(970\) 18206.1 0.602641
\(971\) 58395.0 1.92995 0.964977 0.262334i \(-0.0844923\pi\)
0.964977 + 0.262334i \(0.0844923\pi\)
\(972\) 6953.47 0.229457
\(973\) 12693.1 0.418215
\(974\) −21641.3 −0.711944
\(975\) −4898.70 −0.160907
\(976\) −555.776 −0.0182274
\(977\) 32548.6 1.06584 0.532918 0.846167i \(-0.321095\pi\)
0.532918 + 0.846167i \(0.321095\pi\)
\(978\) 45763.4 1.49627
\(979\) 24624.1 0.803871
\(980\) −3597.52 −0.117264
\(981\) −1637.51 −0.0532944
\(982\) −37404.7 −1.21551
\(983\) −19691.8 −0.638933 −0.319466 0.947598i \(-0.603504\pi\)
−0.319466 + 0.947598i \(0.603504\pi\)
\(984\) −22262.8 −0.721251
\(985\) −27182.4 −0.879292
\(986\) −11165.1 −0.360617
\(987\) −23653.8 −0.762826
\(988\) 8109.50 0.261131
\(989\) 6232.63 0.200390
\(990\) −3266.64 −0.104869
\(991\) 31329.5 1.00425 0.502127 0.864794i \(-0.332551\pi\)
0.502127 + 0.864794i \(0.332551\pi\)
\(992\) 5026.31 0.160872
\(993\) 19710.2 0.629893
\(994\) 4667.19 0.148928
\(995\) 5355.16 0.170623
\(996\) −26235.5 −0.834641
\(997\) 10050.3 0.319254 0.159627 0.987177i \(-0.448971\pi\)
0.159627 + 0.987177i \(0.448971\pi\)
\(998\) 7648.91 0.242607
\(999\) −32879.9 −1.04132
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 230.4.a.f.1.1 2
3.2 odd 2 2070.4.a.s.1.1 2
4.3 odd 2 1840.4.a.i.1.2 2
5.2 odd 4 1150.4.b.k.599.2 4
5.3 odd 4 1150.4.b.k.599.3 4
5.4 even 2 1150.4.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.f.1.1 2 1.1 even 1 trivial
1150.4.a.l.1.2 2 5.4 even 2
1150.4.b.k.599.2 4 5.2 odd 4
1150.4.b.k.599.3 4 5.3 odd 4
1840.4.a.i.1.2 2 4.3 odd 2
2070.4.a.s.1.1 2 3.2 odd 2