Properties

Label 1421.4.a.c.1.2
Level $1421$
Weight $4$
Character 1421.1
Self dual yes
Analytic conductor $83.842$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1421,4,Mod(1,1421)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1421, 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("1421.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1421.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: \(83.8417141182\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1421.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} +9.24264 q^{3} -7.82843 q^{4} -0.656854 q^{5} +3.82843 q^{6} -6.55635 q^{8} +58.4264 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} +9.24264 q^{3} -7.82843 q^{4} -0.656854 q^{5} +3.82843 q^{6} -6.55635 q^{8} +58.4264 q^{9} -0.272078 q^{10} -65.3259 q^{11} -72.3553 q^{12} +49.7696 q^{13} -6.07107 q^{15} +59.9117 q^{16} -55.4558 q^{17} +24.2010 q^{18} +64.7452 q^{19} +5.14214 q^{20} -27.0589 q^{22} +93.8823 q^{23} -60.5980 q^{24} -124.569 q^{25} +20.6152 q^{26} +290.463 q^{27} +29.0000 q^{29} -2.51472 q^{30} +236.095 q^{31} +77.2670 q^{32} -603.784 q^{33} -22.9706 q^{34} -457.387 q^{36} +76.8040 q^{37} +26.8183 q^{38} +460.002 q^{39} +4.30657 q^{40} -215.161 q^{41} +80.8305 q^{43} +511.399 q^{44} -38.3776 q^{45} +38.8873 q^{46} +357.742 q^{47} +553.742 q^{48} -51.5980 q^{50} -512.558 q^{51} -389.617 q^{52} +328.466 q^{53} +120.314 q^{54} +42.9096 q^{55} +598.416 q^{57} +12.0122 q^{58} +99.2750 q^{59} +47.5269 q^{60} +725.730 q^{61} +97.7939 q^{62} -447.288 q^{64} -32.6913 q^{65} -250.095 q^{66} +844.479 q^{67} +434.132 q^{68} +867.720 q^{69} -378.083 q^{71} -383.064 q^{72} +581.097 q^{73} +31.8133 q^{74} -1151.34 q^{75} -506.853 q^{76} +190.539 q^{78} -353.247 q^{79} -39.3532 q^{80} +1107.13 q^{81} -89.1228 q^{82} -696.510 q^{83} +36.4264 q^{85} +33.4811 q^{86} +268.037 q^{87} +428.299 q^{88} -1118.22 q^{89} -15.8965 q^{90} -734.950 q^{92} +2182.15 q^{93} +148.182 q^{94} -42.5281 q^{95} +714.151 q^{96} +805.415 q^{97} -3816.76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 10 q^{3} - 10 q^{4} + 10 q^{5} + 2 q^{6} + 18 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 10 q^{3} - 10 q^{4} + 10 q^{5} + 2 q^{6} + 18 q^{8} + 32 q^{9} - 26 q^{10} - 26 q^{11} - 74 q^{12} + 26 q^{13} + 2 q^{15} + 18 q^{16} - 60 q^{17} + 88 q^{18} + 220 q^{19} - 18 q^{20} - 122 q^{22} + 52 q^{23} - 42 q^{24} - 136 q^{25} + 78 q^{26} + 250 q^{27} + 58 q^{29} - 22 q^{30} + 294 q^{31} - 18 q^{32} - 574 q^{33} - 12 q^{34} - 400 q^{36} + 312 q^{37} - 348 q^{38} + 442 q^{39} + 266 q^{40} - 40 q^{41} - 322 q^{43} + 426 q^{44} - 320 q^{45} + 140 q^{46} + 130 q^{47} + 522 q^{48} - 24 q^{50} - 516 q^{51} - 338 q^{52} + 1002 q^{53} + 218 q^{54} + 462 q^{55} + 716 q^{57} - 58 q^{58} + 900 q^{59} + 30 q^{60} + 948 q^{61} - 42 q^{62} + 118 q^{64} - 286 q^{65} - 322 q^{66} + 320 q^{67} + 444 q^{68} + 836 q^{69} - 660 q^{71} - 1032 q^{72} - 648 q^{73} - 536 q^{74} - 1160 q^{75} - 844 q^{76} + 234 q^{78} + 258 q^{79} - 486 q^{80} + 1790 q^{81} - 512 q^{82} - 1212 q^{83} - 12 q^{85} + 1006 q^{86} + 290 q^{87} + 1394 q^{88} - 760 q^{89} + 664 q^{90} - 644 q^{92} + 2226 q^{93} + 698 q^{94} + 1612 q^{95} + 642 q^{96} - 24 q^{97} - 4856 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.146447 0.0732233 0.997316i \(-0.476671\pi\)
0.0732233 + 0.997316i \(0.476671\pi\)
\(3\) 9.24264 1.77875 0.889374 0.457181i \(-0.151141\pi\)
0.889374 + 0.457181i \(0.151141\pi\)
\(4\) −7.82843 −0.978553
\(5\) −0.656854 −0.0587508 −0.0293754 0.999568i \(-0.509352\pi\)
−0.0293754 + 0.999568i \(0.509352\pi\)
\(6\) 3.82843 0.260491
\(7\) 0 0
\(8\) −6.55635 −0.289752
\(9\) 58.4264 2.16394
\(10\) −0.272078 −0.00860386
\(11\) −65.3259 −1.79059 −0.895295 0.445473i \(-0.853036\pi\)
−0.895295 + 0.445473i \(0.853036\pi\)
\(12\) −72.3553 −1.74060
\(13\) 49.7696 1.06181 0.530907 0.847430i \(-0.321851\pi\)
0.530907 + 0.847430i \(0.321851\pi\)
\(14\) 0 0
\(15\) −6.07107 −0.104503
\(16\) 59.9117 0.936120
\(17\) −55.4558 −0.791178 −0.395589 0.918428i \(-0.629459\pi\)
−0.395589 + 0.918428i \(0.629459\pi\)
\(18\) 24.2010 0.316902
\(19\) 64.7452 0.781766 0.390883 0.920440i \(-0.372170\pi\)
0.390883 + 0.920440i \(0.372170\pi\)
\(20\) 5.14214 0.0574908
\(21\) 0 0
\(22\) −27.0589 −0.262226
\(23\) 93.8823 0.851122 0.425561 0.904930i \(-0.360077\pi\)
0.425561 + 0.904930i \(0.360077\pi\)
\(24\) −60.5980 −0.515396
\(25\) −124.569 −0.996548
\(26\) 20.6152 0.155499
\(27\) 290.463 2.07036
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) −2.51472 −0.0153041
\(31\) 236.095 1.36787 0.683935 0.729543i \(-0.260267\pi\)
0.683935 + 0.729543i \(0.260267\pi\)
\(32\) 77.2670 0.426844
\(33\) −603.784 −3.18501
\(34\) −22.9706 −0.115865
\(35\) 0 0
\(36\) −457.387 −2.11753
\(37\) 76.8040 0.341257 0.170628 0.985335i \(-0.445420\pi\)
0.170628 + 0.985335i \(0.445420\pi\)
\(38\) 26.8183 0.114487
\(39\) 460.002 1.88870
\(40\) 4.30657 0.0170232
\(41\) −215.161 −0.819575 −0.409788 0.912181i \(-0.634397\pi\)
−0.409788 + 0.912181i \(0.634397\pi\)
\(42\) 0 0
\(43\) 80.8305 0.286664 0.143332 0.989675i \(-0.454218\pi\)
0.143332 + 0.989675i \(0.454218\pi\)
\(44\) 511.399 1.75219
\(45\) −38.3776 −0.127133
\(46\) 38.8873 0.124644
\(47\) 357.742 1.11026 0.555128 0.831765i \(-0.312669\pi\)
0.555128 + 0.831765i \(0.312669\pi\)
\(48\) 553.742 1.66512
\(49\) 0 0
\(50\) −51.5980 −0.145941
\(51\) −512.558 −1.40730
\(52\) −389.617 −1.03904
\(53\) 328.466 0.851288 0.425644 0.904891i \(-0.360048\pi\)
0.425644 + 0.904891i \(0.360048\pi\)
\(54\) 120.314 0.303197
\(55\) 42.9096 0.105199
\(56\) 0 0
\(57\) 598.416 1.39056
\(58\) 12.0122 0.0271945
\(59\) 99.2750 0.219059 0.109530 0.993984i \(-0.465066\pi\)
0.109530 + 0.993984i \(0.465066\pi\)
\(60\) 47.5269 0.102262
\(61\) 725.730 1.52328 0.761641 0.647999i \(-0.224394\pi\)
0.761641 + 0.647999i \(0.224394\pi\)
\(62\) 97.7939 0.200320
\(63\) 0 0
\(64\) −447.288 −0.873610
\(65\) −32.6913 −0.0623825
\(66\) −250.095 −0.466434
\(67\) 844.479 1.53984 0.769922 0.638138i \(-0.220295\pi\)
0.769922 + 0.638138i \(0.220295\pi\)
\(68\) 434.132 0.774209
\(69\) 867.720 1.51393
\(70\) 0 0
\(71\) −378.083 −0.631975 −0.315988 0.948763i \(-0.602336\pi\)
−0.315988 + 0.948763i \(0.602336\pi\)
\(72\) −383.064 −0.627007
\(73\) 581.097 0.931674 0.465837 0.884870i \(-0.345753\pi\)
0.465837 + 0.884870i \(0.345753\pi\)
\(74\) 31.8133 0.0499759
\(75\) −1151.34 −1.77261
\(76\) −506.853 −0.765000
\(77\) 0 0
\(78\) 190.539 0.276594
\(79\) −353.247 −0.503081 −0.251540 0.967847i \(-0.580937\pi\)
−0.251540 + 0.967847i \(0.580937\pi\)
\(80\) −39.3532 −0.0549978
\(81\) 1107.13 1.51870
\(82\) −89.1228 −0.120024
\(83\) −696.510 −0.921107 −0.460553 0.887632i \(-0.652349\pi\)
−0.460553 + 0.887632i \(0.652349\pi\)
\(84\) 0 0
\(85\) 36.4264 0.0464823
\(86\) 33.4811 0.0419809
\(87\) 268.037 0.330305
\(88\) 428.299 0.518828
\(89\) −1118.22 −1.33181 −0.665905 0.746037i \(-0.731954\pi\)
−0.665905 + 0.746037i \(0.731954\pi\)
\(90\) −15.8965 −0.0186182
\(91\) 0 0
\(92\) −734.950 −0.832868
\(93\) 2182.15 2.43310
\(94\) 148.182 0.162593
\(95\) −42.5281 −0.0459294
\(96\) 714.151 0.759248
\(97\) 805.415 0.843068 0.421534 0.906813i \(-0.361492\pi\)
0.421534 + 0.906813i \(0.361492\pi\)
\(98\) 0 0
\(99\) −3816.76 −3.87473
\(100\) 975.176 0.975176
\(101\) 1373.99 1.35363 0.676817 0.736151i \(-0.263359\pi\)
0.676817 + 0.736151i \(0.263359\pi\)
\(102\) −212.309 −0.206095
\(103\) 634.672 0.607147 0.303573 0.952808i \(-0.401820\pi\)
0.303573 + 0.952808i \(0.401820\pi\)
\(104\) −326.307 −0.307663
\(105\) 0 0
\(106\) 136.055 0.124668
\(107\) −180.956 −0.163493 −0.0817463 0.996653i \(-0.526050\pi\)
−0.0817463 + 0.996653i \(0.526050\pi\)
\(108\) −2273.87 −2.02595
\(109\) 1038.14 0.912251 0.456125 0.889916i \(-0.349237\pi\)
0.456125 + 0.889916i \(0.349237\pi\)
\(110\) 17.7737 0.0154060
\(111\) 709.872 0.607010
\(112\) 0 0
\(113\) −184.765 −0.153816 −0.0769082 0.997038i \(-0.524505\pi\)
−0.0769082 + 0.997038i \(0.524505\pi\)
\(114\) 247.872 0.203643
\(115\) −61.6670 −0.0500041
\(116\) −227.024 −0.181713
\(117\) 2907.86 2.29770
\(118\) 41.1211 0.0320805
\(119\) 0 0
\(120\) 39.8040 0.0302800
\(121\) 2936.47 2.20622
\(122\) 300.607 0.223079
\(123\) −1988.66 −1.45782
\(124\) −1848.26 −1.33853
\(125\) 163.930 0.117299
\(126\) 0 0
\(127\) 1999.58 1.39712 0.698558 0.715554i \(-0.253825\pi\)
0.698558 + 0.715554i \(0.253825\pi\)
\(128\) −803.409 −0.554781
\(129\) 747.087 0.509902
\(130\) −13.5412 −0.00913570
\(131\) −561.468 −0.374471 −0.187236 0.982315i \(-0.559953\pi\)
−0.187236 + 0.982315i \(0.559953\pi\)
\(132\) 4726.68 3.11670
\(133\) 0 0
\(134\) 349.795 0.225505
\(135\) −190.792 −0.121635
\(136\) 363.588 0.229246
\(137\) −250.489 −0.156210 −0.0781050 0.996945i \(-0.524887\pi\)
−0.0781050 + 0.996945i \(0.524887\pi\)
\(138\) 359.421 0.221710
\(139\) 242.244 0.147819 0.0739096 0.997265i \(-0.476452\pi\)
0.0739096 + 0.997265i \(0.476452\pi\)
\(140\) 0 0
\(141\) 3306.48 1.97487
\(142\) −156.607 −0.0925506
\(143\) −3251.24 −1.90128
\(144\) 3500.42 2.02571
\(145\) −19.0488 −0.0109098
\(146\) 240.698 0.136441
\(147\) 0 0
\(148\) −601.255 −0.333938
\(149\) −1632.63 −0.897652 −0.448826 0.893619i \(-0.648158\pi\)
−0.448826 + 0.893619i \(0.648158\pi\)
\(150\) −476.902 −0.259592
\(151\) −121.582 −0.0655245 −0.0327623 0.999463i \(-0.510430\pi\)
−0.0327623 + 0.999463i \(0.510430\pi\)
\(152\) −424.492 −0.226519
\(153\) −3240.09 −1.71206
\(154\) 0 0
\(155\) −155.080 −0.0803635
\(156\) −3601.09 −1.84819
\(157\) 753.163 0.382860 0.191430 0.981506i \(-0.438688\pi\)
0.191430 + 0.981506i \(0.438688\pi\)
\(158\) −146.320 −0.0736745
\(159\) 3035.89 1.51423
\(160\) −50.7532 −0.0250774
\(161\) 0 0
\(162\) 458.589 0.222408
\(163\) 537.917 0.258484 0.129242 0.991613i \(-0.458746\pi\)
0.129242 + 0.991613i \(0.458746\pi\)
\(164\) 1684.38 0.801998
\(165\) 396.598 0.187122
\(166\) −288.504 −0.134893
\(167\) −484.613 −0.224554 −0.112277 0.993677i \(-0.535814\pi\)
−0.112277 + 0.993677i \(0.535814\pi\)
\(168\) 0 0
\(169\) 280.008 0.127450
\(170\) 15.0883 0.00680718
\(171\) 3782.83 1.69170
\(172\) −632.776 −0.280516
\(173\) 3269.70 1.43694 0.718469 0.695559i \(-0.244843\pi\)
0.718469 + 0.695559i \(0.244843\pi\)
\(174\) 111.024 0.0483721
\(175\) 0 0
\(176\) −3913.79 −1.67621
\(177\) 917.563 0.389651
\(178\) −463.182 −0.195039
\(179\) −562.267 −0.234781 −0.117390 0.993086i \(-0.537453\pi\)
−0.117390 + 0.993086i \(0.537453\pi\)
\(180\) 300.437 0.124407
\(181\) 1507.32 0.618998 0.309499 0.950900i \(-0.399839\pi\)
0.309499 + 0.950900i \(0.399839\pi\)
\(182\) 0 0
\(183\) 6707.66 2.70953
\(184\) −615.525 −0.246615
\(185\) −50.4491 −0.0200491
\(186\) 903.874 0.356319
\(187\) 3622.70 1.41668
\(188\) −2800.56 −1.08645
\(189\) 0 0
\(190\) −17.6157 −0.00672621
\(191\) 4532.00 1.71688 0.858439 0.512915i \(-0.171434\pi\)
0.858439 + 0.512915i \(0.171434\pi\)
\(192\) −4134.13 −1.55393
\(193\) −2935.17 −1.09471 −0.547353 0.836902i \(-0.684364\pi\)
−0.547353 + 0.836902i \(0.684364\pi\)
\(194\) 333.614 0.123464
\(195\) −302.154 −0.110963
\(196\) 0 0
\(197\) 2682.20 0.970043 0.485022 0.874502i \(-0.338812\pi\)
0.485022 + 0.874502i \(0.338812\pi\)
\(198\) −1580.95 −0.567442
\(199\) 648.376 0.230966 0.115483 0.993309i \(-0.463158\pi\)
0.115483 + 0.993309i \(0.463158\pi\)
\(200\) 816.715 0.288752
\(201\) 7805.22 2.73899
\(202\) 569.125 0.198235
\(203\) 0 0
\(204\) 4012.53 1.37712
\(205\) 141.330 0.0481507
\(206\) 262.890 0.0889145
\(207\) 5485.20 1.84178
\(208\) 2981.78 0.993986
\(209\) −4229.54 −1.39982
\(210\) 0 0
\(211\) −4949.57 −1.61489 −0.807446 0.589941i \(-0.799151\pi\)
−0.807446 + 0.589941i \(0.799151\pi\)
\(212\) −2571.37 −0.833031
\(213\) −3494.49 −1.12412
\(214\) −74.9545 −0.0239429
\(215\) −53.0939 −0.0168417
\(216\) −1904.38 −0.599891
\(217\) 0 0
\(218\) 430.010 0.133596
\(219\) 5370.87 1.65721
\(220\) −335.915 −0.102943
\(221\) −2760.01 −0.840084
\(222\) 294.039 0.0888945
\(223\) −2216.94 −0.665729 −0.332864 0.942975i \(-0.608015\pi\)
−0.332864 + 0.942975i \(0.608015\pi\)
\(224\) 0 0
\(225\) −7278.09 −2.15647
\(226\) −76.5323 −0.0225259
\(227\) −4546.09 −1.32923 −0.664613 0.747187i \(-0.731404\pi\)
−0.664613 + 0.747187i \(0.731404\pi\)
\(228\) −4684.66 −1.36074
\(229\) −3339.05 −0.963539 −0.481770 0.876298i \(-0.660006\pi\)
−0.481770 + 0.876298i \(0.660006\pi\)
\(230\) −25.5433 −0.00732293
\(231\) 0 0
\(232\) −190.134 −0.0538057
\(233\) −3995.35 −1.12336 −0.561682 0.827353i \(-0.689846\pi\)
−0.561682 + 0.827353i \(0.689846\pi\)
\(234\) 1204.47 0.336491
\(235\) −234.984 −0.0652285
\(236\) −777.167 −0.214361
\(237\) −3264.93 −0.894853
\(238\) 0 0
\(239\) −1400.04 −0.378915 −0.189458 0.981889i \(-0.560673\pi\)
−0.189458 + 0.981889i \(0.560673\pi\)
\(240\) −363.728 −0.0978272
\(241\) 2040.94 0.545513 0.272756 0.962083i \(-0.412065\pi\)
0.272756 + 0.962083i \(0.412065\pi\)
\(242\) 1216.33 0.323093
\(243\) 2390.32 0.631026
\(244\) −5681.32 −1.49061
\(245\) 0 0
\(246\) −823.730 −0.213492
\(247\) 3222.34 0.830091
\(248\) −1547.92 −0.396344
\(249\) −6437.59 −1.63842
\(250\) 67.9021 0.0171780
\(251\) −802.648 −0.201843 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(252\) 0 0
\(253\) −6132.94 −1.52401
\(254\) 828.252 0.204603
\(255\) 336.676 0.0826803
\(256\) 3245.52 0.792364
\(257\) 4464.10 1.08351 0.541756 0.840536i \(-0.317760\pi\)
0.541756 + 0.840536i \(0.317760\pi\)
\(258\) 309.454 0.0746734
\(259\) 0 0
\(260\) 255.922 0.0610446
\(261\) 1694.37 0.401834
\(262\) −232.568 −0.0548400
\(263\) −3815.21 −0.894509 −0.447255 0.894407i \(-0.647598\pi\)
−0.447255 + 0.894407i \(0.647598\pi\)
\(264\) 3958.62 0.922864
\(265\) −215.754 −0.0500139
\(266\) 0 0
\(267\) −10335.3 −2.36895
\(268\) −6610.95 −1.50682
\(269\) −4523.98 −1.02540 −0.512699 0.858569i \(-0.671354\pi\)
−0.512699 + 0.858569i \(0.671354\pi\)
\(270\) −79.0286 −0.0178131
\(271\) −3962.65 −0.888242 −0.444121 0.895967i \(-0.646484\pi\)
−0.444121 + 0.895967i \(0.646484\pi\)
\(272\) −3322.45 −0.740637
\(273\) 0 0
\(274\) −103.756 −0.0228764
\(275\) 8137.55 1.78441
\(276\) −6792.88 −1.48146
\(277\) 2217.59 0.481019 0.240509 0.970647i \(-0.422685\pi\)
0.240509 + 0.970647i \(0.422685\pi\)
\(278\) 100.341 0.0216476
\(279\) 13794.2 2.95999
\(280\) 0 0
\(281\) −2562.96 −0.544105 −0.272053 0.962282i \(-0.587702\pi\)
−0.272053 + 0.962282i \(0.587702\pi\)
\(282\) 1369.59 0.289212
\(283\) 3869.29 0.812741 0.406370 0.913708i \(-0.366794\pi\)
0.406370 + 0.913708i \(0.366794\pi\)
\(284\) 2959.80 0.618421
\(285\) −393.072 −0.0816968
\(286\) −1346.71 −0.278435
\(287\) 0 0
\(288\) 4514.43 0.923665
\(289\) −1837.65 −0.374038
\(290\) −7.89026 −0.00159770
\(291\) 7444.17 1.49960
\(292\) −4549.07 −0.911693
\(293\) 3883.83 0.774388 0.387194 0.921998i \(-0.373444\pi\)
0.387194 + 0.921998i \(0.373444\pi\)
\(294\) 0 0
\(295\) −65.2092 −0.0128699
\(296\) −503.554 −0.0988800
\(297\) −18974.8 −3.70716
\(298\) −676.257 −0.131458
\(299\) 4672.48 0.903734
\(300\) 9013.20 1.73459
\(301\) 0 0
\(302\) −50.3609 −0.00959584
\(303\) 12699.3 2.40777
\(304\) 3878.99 0.731827
\(305\) −476.699 −0.0894941
\(306\) −1342.09 −0.250726
\(307\) 403.210 0.0749590 0.0374795 0.999297i \(-0.488067\pi\)
0.0374795 + 0.999297i \(0.488067\pi\)
\(308\) 0 0
\(309\) 5866.05 1.07996
\(310\) −64.2364 −0.0117690
\(311\) 4838.71 0.882244 0.441122 0.897447i \(-0.354581\pi\)
0.441122 + 0.897447i \(0.354581\pi\)
\(312\) −3015.93 −0.547255
\(313\) 8544.28 1.54298 0.771488 0.636244i \(-0.219513\pi\)
0.771488 + 0.636244i \(0.219513\pi\)
\(314\) 311.970 0.0560685
\(315\) 0 0
\(316\) 2765.37 0.492291
\(317\) −1773.06 −0.314148 −0.157074 0.987587i \(-0.550206\pi\)
−0.157074 + 0.987587i \(0.550206\pi\)
\(318\) 1257.51 0.221753
\(319\) −1894.45 −0.332504
\(320\) 293.803 0.0513253
\(321\) −1672.51 −0.290812
\(322\) 0 0
\(323\) −3590.50 −0.618516
\(324\) −8667.10 −1.48613
\(325\) −6199.72 −1.05815
\(326\) 222.813 0.0378541
\(327\) 9595.11 1.62266
\(328\) 1410.67 0.237474
\(329\) 0 0
\(330\) 164.276 0.0274034
\(331\) 801.875 0.133157 0.0665786 0.997781i \(-0.478792\pi\)
0.0665786 + 0.997781i \(0.478792\pi\)
\(332\) 5452.58 0.901352
\(333\) 4487.38 0.738460
\(334\) −200.733 −0.0328851
\(335\) −554.700 −0.0904671
\(336\) 0 0
\(337\) 8193.23 1.32437 0.662186 0.749339i \(-0.269629\pi\)
0.662186 + 0.749339i \(0.269629\pi\)
\(338\) 115.983 0.0186647
\(339\) −1707.72 −0.273601
\(340\) −285.161 −0.0454854
\(341\) −15423.1 −2.44930
\(342\) 1566.90 0.247743
\(343\) 0 0
\(344\) −529.953 −0.0830615
\(345\) −569.966 −0.0889447
\(346\) 1354.35 0.210435
\(347\) −10914.9 −1.68860 −0.844301 0.535869i \(-0.819984\pi\)
−0.844301 + 0.535869i \(0.819984\pi\)
\(348\) −2098.30 −0.323221
\(349\) −6697.83 −1.02730 −0.513649 0.858001i \(-0.671706\pi\)
−0.513649 + 0.858001i \(0.671706\pi\)
\(350\) 0 0
\(351\) 14456.2 2.19833
\(352\) −5047.54 −0.764303
\(353\) 3764.83 0.567654 0.283827 0.958875i \(-0.408396\pi\)
0.283827 + 0.958875i \(0.408396\pi\)
\(354\) 380.067 0.0570631
\(355\) 248.346 0.0371291
\(356\) 8753.90 1.30325
\(357\) 0 0
\(358\) −232.898 −0.0343829
\(359\) 6577.13 0.966930 0.483465 0.875364i \(-0.339378\pi\)
0.483465 + 0.875364i \(0.339378\pi\)
\(360\) 251.617 0.0368372
\(361\) −2667.06 −0.388841
\(362\) 624.354 0.0906501
\(363\) 27140.8 3.92430
\(364\) 0 0
\(365\) −381.696 −0.0547366
\(366\) 2778.40 0.396802
\(367\) −2274.27 −0.323477 −0.161738 0.986834i \(-0.551710\pi\)
−0.161738 + 0.986834i \(0.551710\pi\)
\(368\) 5624.64 0.796752
\(369\) −12571.1 −1.77351
\(370\) −20.8967 −0.00293613
\(371\) 0 0
\(372\) −17082.8 −2.38091
\(373\) 1284.94 0.178369 0.0891844 0.996015i \(-0.471574\pi\)
0.0891844 + 0.996015i \(0.471574\pi\)
\(374\) 1500.57 0.207467
\(375\) 1515.15 0.208645
\(376\) −2345.48 −0.321700
\(377\) 1443.32 0.197174
\(378\) 0 0
\(379\) 174.785 0.0236890 0.0118445 0.999930i \(-0.496230\pi\)
0.0118445 + 0.999930i \(0.496230\pi\)
\(380\) 332.928 0.0449444
\(381\) 18481.4 2.48512
\(382\) 1877.22 0.251431
\(383\) 5558.62 0.741599 0.370799 0.928713i \(-0.379084\pi\)
0.370799 + 0.928713i \(0.379084\pi\)
\(384\) −7425.62 −0.986816
\(385\) 0 0
\(386\) −1215.79 −0.160316
\(387\) 4722.64 0.620323
\(388\) −6305.14 −0.824987
\(389\) −2556.05 −0.333154 −0.166577 0.986028i \(-0.553271\pi\)
−0.166577 + 0.986028i \(0.553271\pi\)
\(390\) −125.156 −0.0162501
\(391\) −5206.32 −0.673388
\(392\) 0 0
\(393\) −5189.45 −0.666089
\(394\) 1111.00 0.142060
\(395\) 232.032 0.0295564
\(396\) 29879.2 3.79163
\(397\) −5927.27 −0.749323 −0.374662 0.927162i \(-0.622241\pi\)
−0.374662 + 0.927162i \(0.622241\pi\)
\(398\) 268.566 0.0338241
\(399\) 0 0
\(400\) −7463.11 −0.932889
\(401\) 4747.99 0.591280 0.295640 0.955299i \(-0.404467\pi\)
0.295640 + 0.955299i \(0.404467\pi\)
\(402\) 3233.03 0.401116
\(403\) 11750.4 1.45243
\(404\) −10756.2 −1.32460
\(405\) −727.224 −0.0892249
\(406\) 0 0
\(407\) −5017.29 −0.611052
\(408\) 3360.51 0.407770
\(409\) −5200.19 −0.628686 −0.314343 0.949309i \(-0.601784\pi\)
−0.314343 + 0.949309i \(0.601784\pi\)
\(410\) 58.5407 0.00705151
\(411\) −2315.18 −0.277858
\(412\) −4968.48 −0.594125
\(413\) 0 0
\(414\) 2272.05 0.269722
\(415\) 457.505 0.0541158
\(416\) 3845.55 0.453229
\(417\) 2238.97 0.262933
\(418\) −1751.93 −0.204999
\(419\) 6425.59 0.749189 0.374595 0.927189i \(-0.377782\pi\)
0.374595 + 0.927189i \(0.377782\pi\)
\(420\) 0 0
\(421\) 10037.6 1.16201 0.581003 0.813902i \(-0.302661\pi\)
0.581003 + 0.813902i \(0.302661\pi\)
\(422\) −2050.18 −0.236496
\(423\) 20901.6 2.40253
\(424\) −2153.54 −0.246663
\(425\) 6908.05 0.788447
\(426\) −1447.46 −0.164624
\(427\) 0 0
\(428\) 1416.60 0.159986
\(429\) −30050.1 −3.38189
\(430\) −21.9922 −0.00246641
\(431\) 16646.8 1.86044 0.930218 0.367006i \(-0.119617\pi\)
0.930218 + 0.367006i \(0.119617\pi\)
\(432\) 17402.1 1.93810
\(433\) −15089.1 −1.67468 −0.837340 0.546682i \(-0.815891\pi\)
−0.837340 + 0.546682i \(0.815891\pi\)
\(434\) 0 0
\(435\) −176.061 −0.0194057
\(436\) −8126.97 −0.892686
\(437\) 6078.42 0.665378
\(438\) 2224.69 0.242693
\(439\) 3777.24 0.410656 0.205328 0.978693i \(-0.434174\pi\)
0.205328 + 0.978693i \(0.434174\pi\)
\(440\) −281.330 −0.0304816
\(441\) 0 0
\(442\) −1143.23 −0.123027
\(443\) 7992.65 0.857206 0.428603 0.903493i \(-0.359006\pi\)
0.428603 + 0.903493i \(0.359006\pi\)
\(444\) −5557.18 −0.593991
\(445\) 734.507 0.0782449
\(446\) −918.288 −0.0974937
\(447\) −15089.8 −1.59670
\(448\) 0 0
\(449\) 6433.54 0.676209 0.338104 0.941109i \(-0.390214\pi\)
0.338104 + 0.941109i \(0.390214\pi\)
\(450\) −3014.68 −0.315808
\(451\) 14055.6 1.46752
\(452\) 1446.42 0.150518
\(453\) −1123.74 −0.116552
\(454\) −1883.05 −0.194661
\(455\) 0 0
\(456\) −3923.43 −0.402919
\(457\) 6975.18 0.713972 0.356986 0.934110i \(-0.383804\pi\)
0.356986 + 0.934110i \(0.383804\pi\)
\(458\) −1383.08 −0.141107
\(459\) −16107.9 −1.63802
\(460\) 482.755 0.0489317
\(461\) 14758.9 1.49109 0.745543 0.666458i \(-0.232190\pi\)
0.745543 + 0.666458i \(0.232190\pi\)
\(462\) 0 0
\(463\) −18951.2 −1.90224 −0.951121 0.308818i \(-0.900067\pi\)
−0.951121 + 0.308818i \(0.900067\pi\)
\(464\) 1737.44 0.173833
\(465\) −1433.35 −0.142946
\(466\) −1654.93 −0.164513
\(467\) −12442.4 −1.23290 −0.616449 0.787395i \(-0.711429\pi\)
−0.616449 + 0.787395i \(0.711429\pi\)
\(468\) −22763.9 −2.24843
\(469\) 0 0
\(470\) −97.3338 −0.00955249
\(471\) 6961.22 0.681010
\(472\) −650.882 −0.0634730
\(473\) −5280.33 −0.513297
\(474\) −1352.38 −0.131048
\(475\) −8065.21 −0.779068
\(476\) 0 0
\(477\) 19191.1 1.84214
\(478\) −579.914 −0.0554909
\(479\) −12947.5 −1.23504 −0.617522 0.786554i \(-0.711863\pi\)
−0.617522 + 0.786554i \(0.711863\pi\)
\(480\) −469.093 −0.0446064
\(481\) 3822.50 0.362352
\(482\) 845.385 0.0798885
\(483\) 0 0
\(484\) −22988.0 −2.15890
\(485\) −529.041 −0.0495309
\(486\) 990.104 0.0924116
\(487\) 9844.72 0.916030 0.458015 0.888944i \(-0.348561\pi\)
0.458015 + 0.888944i \(0.348561\pi\)
\(488\) −4758.14 −0.441375
\(489\) 4971.77 0.459778
\(490\) 0 0
\(491\) −6809.50 −0.625883 −0.312941 0.949772i \(-0.601314\pi\)
−0.312941 + 0.949772i \(0.601314\pi\)
\(492\) 15568.1 1.42655
\(493\) −1608.22 −0.146918
\(494\) 1334.74 0.121564
\(495\) 2507.05 0.227644
\(496\) 14144.9 1.28049
\(497\) 0 0
\(498\) −2666.54 −0.239940
\(499\) −15953.8 −1.43124 −0.715622 0.698488i \(-0.753857\pi\)
−0.715622 + 0.698488i \(0.753857\pi\)
\(500\) −1283.32 −0.114783
\(501\) −4479.11 −0.399424
\(502\) −332.468 −0.0295593
\(503\) 14582.7 1.29267 0.646334 0.763054i \(-0.276301\pi\)
0.646334 + 0.763054i \(0.276301\pi\)
\(504\) 0 0
\(505\) −902.511 −0.0795272
\(506\) −2540.35 −0.223186
\(507\) 2588.02 0.226702
\(508\) −15653.5 −1.36715
\(509\) −20906.4 −1.82055 −0.910273 0.414008i \(-0.864129\pi\)
−0.910273 + 0.414008i \(0.864129\pi\)
\(510\) 139.456 0.0121083
\(511\) 0 0
\(512\) 7771.61 0.670820
\(513\) 18806.1 1.61854
\(514\) 1849.09 0.158677
\(515\) −416.887 −0.0356704
\(516\) −5848.52 −0.498967
\(517\) −23369.8 −1.98802
\(518\) 0 0
\(519\) 30220.6 2.55595
\(520\) 214.336 0.0180755
\(521\) 15131.7 1.27242 0.636212 0.771515i \(-0.280500\pi\)
0.636212 + 0.771515i \(0.280500\pi\)
\(522\) 701.829 0.0588472
\(523\) −12146.9 −1.01558 −0.507790 0.861481i \(-0.669537\pi\)
−0.507790 + 0.861481i \(0.669537\pi\)
\(524\) 4395.41 0.366440
\(525\) 0 0
\(526\) −1580.31 −0.130998
\(527\) −13092.9 −1.08223
\(528\) −36173.7 −2.98155
\(529\) −3353.12 −0.275592
\(530\) −89.3683 −0.00732436
\(531\) 5800.28 0.474032
\(532\) 0 0
\(533\) −10708.5 −0.870237
\(534\) −4281.02 −0.346925
\(535\) 118.862 0.00960532
\(536\) −5536.70 −0.446174
\(537\) −5196.83 −0.417616
\(538\) −1873.89 −0.150166
\(539\) 0 0
\(540\) 1493.60 0.119027
\(541\) −22291.8 −1.77153 −0.885767 0.464130i \(-0.846367\pi\)
−0.885767 + 0.464130i \(0.846367\pi\)
\(542\) −1641.38 −0.130080
\(543\) 13931.7 1.10104
\(544\) −4284.91 −0.337709
\(545\) −681.904 −0.0535955
\(546\) 0 0
\(547\) −15439.4 −1.20684 −0.603421 0.797423i \(-0.706196\pi\)
−0.603421 + 0.797423i \(0.706196\pi\)
\(548\) 1960.94 0.152860
\(549\) 42401.8 3.29629
\(550\) 3370.68 0.261321
\(551\) 1877.61 0.145170
\(552\) −5689.07 −0.438665
\(553\) 0 0
\(554\) 918.557 0.0704436
\(555\) −466.283 −0.0356623
\(556\) −1896.39 −0.144649
\(557\) 2336.99 0.177776 0.0888881 0.996042i \(-0.471669\pi\)
0.0888881 + 0.996042i \(0.471669\pi\)
\(558\) 5713.75 0.433481
\(559\) 4022.90 0.304384
\(560\) 0 0
\(561\) 33483.3 2.51991
\(562\) −1061.61 −0.0796824
\(563\) −19833.3 −1.48468 −0.742340 0.670023i \(-0.766284\pi\)
−0.742340 + 0.670023i \(0.766284\pi\)
\(564\) −25884.6 −1.93251
\(565\) 121.364 0.00903685
\(566\) 1602.71 0.119023
\(567\) 0 0
\(568\) 2478.85 0.183116
\(569\) 11063.7 0.815141 0.407571 0.913174i \(-0.366376\pi\)
0.407571 + 0.913174i \(0.366376\pi\)
\(570\) −162.816 −0.0119642
\(571\) −665.827 −0.0487986 −0.0243993 0.999702i \(-0.507767\pi\)
−0.0243993 + 0.999702i \(0.507767\pi\)
\(572\) 25452.1 1.86050
\(573\) 41887.6 3.05389
\(574\) 0 0
\(575\) −11694.8 −0.848184
\(576\) −26133.5 −1.89044
\(577\) 7165.21 0.516970 0.258485 0.966015i \(-0.416777\pi\)
0.258485 + 0.966015i \(0.416777\pi\)
\(578\) −761.179 −0.0547766
\(579\) −27128.7 −1.94720
\(580\) 149.122 0.0106758
\(581\) 0 0
\(582\) 3083.47 0.219612
\(583\) −21457.3 −1.52431
\(584\) −3809.87 −0.269955
\(585\) −1910.04 −0.134992
\(586\) 1608.73 0.113406
\(587\) 10375.2 0.729525 0.364763 0.931101i \(-0.381150\pi\)
0.364763 + 0.931101i \(0.381150\pi\)
\(588\) 0 0
\(589\) 15286.0 1.06936
\(590\) −27.0105 −0.00188476
\(591\) 24790.6 1.72546
\(592\) 4601.46 0.319457
\(593\) −18931.5 −1.31100 −0.655501 0.755194i \(-0.727542\pi\)
−0.655501 + 0.755194i \(0.727542\pi\)
\(594\) −7859.60 −0.542901
\(595\) 0 0
\(596\) 12780.9 0.878401
\(597\) 5992.71 0.410829
\(598\) 1935.40 0.132349
\(599\) −12244.2 −0.835199 −0.417600 0.908631i \(-0.637129\pi\)
−0.417600 + 0.908631i \(0.637129\pi\)
\(600\) 7548.60 0.513617
\(601\) −15596.9 −1.05859 −0.529293 0.848439i \(-0.677543\pi\)
−0.529293 + 0.848439i \(0.677543\pi\)
\(602\) 0 0
\(603\) 49339.9 3.33213
\(604\) 951.796 0.0641192
\(605\) −1928.84 −0.129617
\(606\) 5260.22 0.352610
\(607\) 10155.5 0.679076 0.339538 0.940592i \(-0.389729\pi\)
0.339538 + 0.940592i \(0.389729\pi\)
\(608\) 5002.67 0.333692
\(609\) 0 0
\(610\) −197.455 −0.0131061
\(611\) 17804.7 1.17889
\(612\) 25364.8 1.67534
\(613\) −6227.67 −0.410331 −0.205166 0.978727i \(-0.565773\pi\)
−0.205166 + 0.978727i \(0.565773\pi\)
\(614\) 167.015 0.0109775
\(615\) 1306.26 0.0856479
\(616\) 0 0
\(617\) 14357.5 0.936808 0.468404 0.883514i \(-0.344829\pi\)
0.468404 + 0.883514i \(0.344829\pi\)
\(618\) 2429.80 0.158156
\(619\) 13220.6 0.858453 0.429227 0.903197i \(-0.358786\pi\)
0.429227 + 0.903197i \(0.358786\pi\)
\(620\) 1214.03 0.0786400
\(621\) 27269.3 1.76213
\(622\) 2004.26 0.129202
\(623\) 0 0
\(624\) 27559.5 1.76805
\(625\) 15463.4 0.989657
\(626\) 3539.16 0.225963
\(627\) −39092.1 −2.48993
\(628\) −5896.08 −0.374649
\(629\) −4259.23 −0.269995
\(630\) 0 0
\(631\) 1828.97 0.115389 0.0576943 0.998334i \(-0.481625\pi\)
0.0576943 + 0.998334i \(0.481625\pi\)
\(632\) 2316.01 0.145769
\(633\) −45747.1 −2.87249
\(634\) −734.426 −0.0460059
\(635\) −1313.43 −0.0820817
\(636\) −23766.3 −1.48175
\(637\) 0 0
\(638\) −784.707 −0.0486941
\(639\) −22090.0 −1.36756
\(640\) 527.723 0.0325939
\(641\) −22644.1 −1.39530 −0.697651 0.716437i \(-0.745772\pi\)
−0.697651 + 0.716437i \(0.745772\pi\)
\(642\) −692.778 −0.0425884
\(643\) −22728.4 −1.39397 −0.696983 0.717088i \(-0.745475\pi\)
−0.696983 + 0.717088i \(0.745475\pi\)
\(644\) 0 0
\(645\) −490.728 −0.0299572
\(646\) −1487.23 −0.0905796
\(647\) 5844.85 0.355154 0.177577 0.984107i \(-0.443174\pi\)
0.177577 + 0.984107i \(0.443174\pi\)
\(648\) −7258.74 −0.440047
\(649\) −6485.23 −0.392246
\(650\) −2568.01 −0.154962
\(651\) 0 0
\(652\) −4211.04 −0.252941
\(653\) 15174.1 0.909355 0.454677 0.890656i \(-0.349755\pi\)
0.454677 + 0.890656i \(0.349755\pi\)
\(654\) 3974.43 0.237634
\(655\) 368.803 0.0220005
\(656\) −12890.7 −0.767221
\(657\) 33951.4 2.01609
\(658\) 0 0
\(659\) −27857.0 −1.64667 −0.823333 0.567558i \(-0.807888\pi\)
−0.823333 + 0.567558i \(0.807888\pi\)
\(660\) −3104.74 −0.183109
\(661\) 4966.64 0.292254 0.146127 0.989266i \(-0.453319\pi\)
0.146127 + 0.989266i \(0.453319\pi\)
\(662\) 332.148 0.0195004
\(663\) −25509.8 −1.49430
\(664\) 4566.56 0.266893
\(665\) 0 0
\(666\) 1858.74 0.108145
\(667\) 2722.59 0.158049
\(668\) 3793.76 0.219738
\(669\) −20490.4 −1.18416
\(670\) −229.764 −0.0132486
\(671\) −47409.0 −2.72758
\(672\) 0 0
\(673\) −2338.02 −0.133914 −0.0669569 0.997756i \(-0.521329\pi\)
−0.0669569 + 0.997756i \(0.521329\pi\)
\(674\) 3393.75 0.193950
\(675\) −36182.6 −2.06321
\(676\) −2192.03 −0.124717
\(677\) −6342.30 −0.360051 −0.180025 0.983662i \(-0.557618\pi\)
−0.180025 + 0.983662i \(0.557618\pi\)
\(678\) −707.361 −0.0400679
\(679\) 0 0
\(680\) −238.824 −0.0134684
\(681\) −42017.9 −2.36436
\(682\) −6388.48 −0.358691
\(683\) −30366.6 −1.70124 −0.850620 0.525780i \(-0.823773\pi\)
−0.850620 + 0.525780i \(0.823773\pi\)
\(684\) −29613.6 −1.65542
\(685\) 164.535 0.00917746
\(686\) 0 0
\(687\) −30861.6 −1.71389
\(688\) 4842.69 0.268352
\(689\) 16347.6 0.903910
\(690\) −236.087 −0.0130256
\(691\) 11826.5 0.651089 0.325545 0.945527i \(-0.394452\pi\)
0.325545 + 0.945527i \(0.394452\pi\)
\(692\) −25596.6 −1.40612
\(693\) 0 0
\(694\) −4521.12 −0.247290
\(695\) −159.119 −0.00868450
\(696\) −1757.34 −0.0957067
\(697\) 11932.0 0.648429
\(698\) −2774.33 −0.150444
\(699\) −36927.6 −1.99818
\(700\) 0 0
\(701\) −2776.33 −0.149587 −0.0747936 0.997199i \(-0.523830\pi\)
−0.0747936 + 0.997199i \(0.523830\pi\)
\(702\) 5987.96 0.321939
\(703\) 4972.69 0.266783
\(704\) 29219.5 1.56428
\(705\) −2171.88 −0.116025
\(706\) 1559.45 0.0831310
\(707\) 0 0
\(708\) −7183.08 −0.381295
\(709\) −15962.7 −0.845543 −0.422772 0.906236i \(-0.638943\pi\)
−0.422772 + 0.906236i \(0.638943\pi\)
\(710\) 102.868 0.00543742
\(711\) −20638.9 −1.08864
\(712\) 7331.44 0.385895
\(713\) 22165.2 1.16422
\(714\) 0 0
\(715\) 2135.59 0.111702
\(716\) 4401.66 0.229746
\(717\) −12940.0 −0.673994
\(718\) 2724.34 0.141604
\(719\) 20832.9 1.08058 0.540289 0.841480i \(-0.318315\pi\)
0.540289 + 0.841480i \(0.318315\pi\)
\(720\) −2299.27 −0.119012
\(721\) 0 0
\(722\) −1104.73 −0.0569445
\(723\) 18863.7 0.970329
\(724\) −11800.0 −0.605722
\(725\) −3612.49 −0.185054
\(726\) 11242.1 0.574700
\(727\) −4452.04 −0.227121 −0.113561 0.993531i \(-0.536226\pi\)
−0.113561 + 0.993531i \(0.536226\pi\)
\(728\) 0 0
\(729\) −7799.67 −0.396264
\(730\) −158.104 −0.00801599
\(731\) −4482.52 −0.226802
\(732\) −52510.4 −2.65142
\(733\) 12107.2 0.610082 0.305041 0.952339i \(-0.401330\pi\)
0.305041 + 0.952339i \(0.401330\pi\)
\(734\) −942.034 −0.0473721
\(735\) 0 0
\(736\) 7254.00 0.363296
\(737\) −55166.4 −2.75723
\(738\) −5207.12 −0.259725
\(739\) −4506.27 −0.224311 −0.112156 0.993691i \(-0.535775\pi\)
−0.112156 + 0.993691i \(0.535775\pi\)
\(740\) 394.937 0.0196191
\(741\) 29782.9 1.47652
\(742\) 0 0
\(743\) 1177.19 0.0581253 0.0290626 0.999578i \(-0.490748\pi\)
0.0290626 + 0.999578i \(0.490748\pi\)
\(744\) −14306.9 −0.704996
\(745\) 1072.40 0.0527378
\(746\) 532.239 0.0261215
\(747\) −40694.6 −1.99322
\(748\) −28360.1 −1.38629
\(749\) 0 0
\(750\) 627.595 0.0305554
\(751\) 27631.6 1.34260 0.671300 0.741186i \(-0.265736\pi\)
0.671300 + 0.741186i \(0.265736\pi\)
\(752\) 21432.9 1.03933
\(753\) −7418.59 −0.359028
\(754\) 597.841 0.0288755
\(755\) 79.8616 0.00384962
\(756\) 0 0
\(757\) 11336.6 0.544300 0.272150 0.962255i \(-0.412265\pi\)
0.272150 + 0.962255i \(0.412265\pi\)
\(758\) 72.3984 0.00346917
\(759\) −56684.6 −2.71083
\(760\) 278.829 0.0133082
\(761\) −4356.58 −0.207524 −0.103762 0.994602i \(-0.533088\pi\)
−0.103762 + 0.994602i \(0.533088\pi\)
\(762\) 7655.23 0.363937
\(763\) 0 0
\(764\) −35478.4 −1.68006
\(765\) 2128.26 0.100585
\(766\) 2302.46 0.108605
\(767\) 4940.87 0.232601
\(768\) 29997.2 1.40942
\(769\) 21718.1 1.01843 0.509217 0.860638i \(-0.329935\pi\)
0.509217 + 0.860638i \(0.329935\pi\)
\(770\) 0 0
\(771\) 41260.0 1.92729
\(772\) 22977.8 1.07123
\(773\) −22688.4 −1.05568 −0.527842 0.849343i \(-0.676999\pi\)
−0.527842 + 0.849343i \(0.676999\pi\)
\(774\) 1956.18 0.0908442
\(775\) −29410.1 −1.36315
\(776\) −5280.58 −0.244281
\(777\) 0 0
\(778\) −1058.75 −0.0487893
\(779\) −13930.7 −0.640716
\(780\) 2365.39 0.108583
\(781\) 24698.6 1.13161
\(782\) −2156.53 −0.0986155
\(783\) 8423.43 0.384456
\(784\) 0 0
\(785\) −494.718 −0.0224933
\(786\) −2149.54 −0.0975465
\(787\) 32890.9 1.48975 0.744875 0.667204i \(-0.232509\pi\)
0.744875 + 0.667204i \(0.232509\pi\)
\(788\) −20997.4 −0.949239
\(789\) −35262.6 −1.59111
\(790\) 96.1107 0.00432844
\(791\) 0 0
\(792\) 25024.0 1.12271
\(793\) 36119.3 1.61744
\(794\) −2455.16 −0.109736
\(795\) −1994.14 −0.0889620
\(796\) −5075.76 −0.226012
\(797\) −30404.1 −1.35128 −0.675638 0.737233i \(-0.736132\pi\)
−0.675638 + 0.737233i \(0.736132\pi\)
\(798\) 0 0
\(799\) −19838.9 −0.878410
\(800\) −9625.04 −0.425371
\(801\) −65333.5 −2.88196
\(802\) 1966.68 0.0865909
\(803\) −37960.7 −1.66825
\(804\) −61102.6 −2.68025
\(805\) 0 0
\(806\) 4867.16 0.212703
\(807\) −41813.5 −1.82392
\(808\) −9008.36 −0.392219
\(809\) −37889.3 −1.64662 −0.823311 0.567591i \(-0.807876\pi\)
−0.823311 + 0.567591i \(0.807876\pi\)
\(810\) −301.226 −0.0130667
\(811\) 8123.23 0.351720 0.175860 0.984415i \(-0.443729\pi\)
0.175860 + 0.984415i \(0.443729\pi\)
\(812\) 0 0
\(813\) −36625.3 −1.57996
\(814\) −2078.23 −0.0894864
\(815\) −353.333 −0.0151862
\(816\) −30708.2 −1.31741
\(817\) 5233.39 0.224104
\(818\) −2153.99 −0.0920690
\(819\) 0 0
\(820\) −1106.39 −0.0471180
\(821\) 13226.8 0.562264 0.281132 0.959669i \(-0.409290\pi\)
0.281132 + 0.959669i \(0.409290\pi\)
\(822\) −958.981 −0.0406914
\(823\) −29575.2 −1.25265 −0.626323 0.779563i \(-0.715441\pi\)
−0.626323 + 0.779563i \(0.715441\pi\)
\(824\) −4161.13 −0.175922
\(825\) 75212.5 3.17401
\(826\) 0 0
\(827\) −36661.2 −1.54152 −0.770758 0.637128i \(-0.780122\pi\)
−0.770758 + 0.637128i \(0.780122\pi\)
\(828\) −42940.5 −1.80228
\(829\) −11277.0 −0.472455 −0.236228 0.971698i \(-0.575911\pi\)
−0.236228 + 0.971698i \(0.575911\pi\)
\(830\) 189.505 0.00792507
\(831\) 20496.4 0.855611
\(832\) −22261.3 −0.927612
\(833\) 0 0
\(834\) 927.413 0.0385056
\(835\) 318.320 0.0131927
\(836\) 33110.6 1.36980
\(837\) 68577.0 2.83198
\(838\) 2661.56 0.109716
\(839\) 18965.7 0.780417 0.390208 0.920727i \(-0.372403\pi\)
0.390208 + 0.920727i \(0.372403\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 4157.72 0.170172
\(843\) −23688.5 −0.967825
\(844\) 38747.3 1.58026
\(845\) −183.925 −0.00748781
\(846\) 8657.72 0.351842
\(847\) 0 0
\(848\) 19678.9 0.796908
\(849\) 35762.5 1.44566
\(850\) 2861.41 0.115465
\(851\) 7210.54 0.290451
\(852\) 27356.3 1.10002
\(853\) −8067.23 −0.323818 −0.161909 0.986806i \(-0.551765\pi\)
−0.161909 + 0.986806i \(0.551765\pi\)
\(854\) 0 0
\(855\) −2484.77 −0.0993886
\(856\) 1186.41 0.0473724
\(857\) −15281.7 −0.609118 −0.304559 0.952493i \(-0.598509\pi\)
−0.304559 + 0.952493i \(0.598509\pi\)
\(858\) −12447.1 −0.495266
\(859\) 36789.1 1.46127 0.730634 0.682770i \(-0.239225\pi\)
0.730634 + 0.682770i \(0.239225\pi\)
\(860\) 415.641 0.0164805
\(861\) 0 0
\(862\) 6895.33 0.272455
\(863\) 40907.8 1.61358 0.806788 0.590840i \(-0.201204\pi\)
0.806788 + 0.590840i \(0.201204\pi\)
\(864\) 22443.2 0.883719
\(865\) −2147.71 −0.0844213
\(866\) −6250.12 −0.245251
\(867\) −16984.7 −0.665319
\(868\) 0 0
\(869\) 23076.2 0.900812
\(870\) −72.9268 −0.00284190
\(871\) 42029.4 1.63503
\(872\) −6806.38 −0.264327
\(873\) 47057.5 1.82435
\(874\) 2517.76 0.0974424
\(875\) 0 0
\(876\) −42045.4 −1.62167
\(877\) 2391.60 0.0920852 0.0460426 0.998939i \(-0.485339\pi\)
0.0460426 + 0.998939i \(0.485339\pi\)
\(878\) 1564.59 0.0601392
\(879\) 35896.8 1.37744
\(880\) 2570.79 0.0984786
\(881\) −5487.72 −0.209859 −0.104930 0.994480i \(-0.533462\pi\)
−0.104930 + 0.994480i \(0.533462\pi\)
\(882\) 0 0
\(883\) 170.008 0.00647931 0.00323966 0.999995i \(-0.498969\pi\)
0.00323966 + 0.999995i \(0.498969\pi\)
\(884\) 21606.6 0.822067
\(885\) −602.705 −0.0228923
\(886\) 3310.66 0.125535
\(887\) −25867.3 −0.979188 −0.489594 0.871950i \(-0.662855\pi\)
−0.489594 + 0.871950i \(0.662855\pi\)
\(888\) −4654.17 −0.175883
\(889\) 0 0
\(890\) 304.243 0.0114587
\(891\) −72324.4 −2.71937
\(892\) 17355.2 0.651451
\(893\) 23162.1 0.867961
\(894\) −6250.40 −0.233831
\(895\) 369.327 0.0137936
\(896\) 0 0
\(897\) 43186.0 1.60751
\(898\) 2664.86 0.0990285
\(899\) 6846.77 0.254007
\(900\) 56976.0 2.11022
\(901\) −18215.4 −0.673520
\(902\) 5822.03 0.214914
\(903\) 0 0
\(904\) 1211.39 0.0445687
\(905\) −990.093 −0.0363666
\(906\) −465.468 −0.0170686
\(907\) 11411.5 0.417765 0.208882 0.977941i \(-0.433017\pi\)
0.208882 + 0.977941i \(0.433017\pi\)
\(908\) 35588.7 1.30072
\(909\) 80277.3 2.92919
\(910\) 0 0
\(911\) −38718.5 −1.40813 −0.704063 0.710138i \(-0.748633\pi\)
−0.704063 + 0.710138i \(0.748633\pi\)
\(912\) 35852.1 1.30174
\(913\) 45500.1 1.64933
\(914\) 2889.21 0.104559
\(915\) −4405.96 −0.159187
\(916\) 26139.5 0.942875
\(917\) 0 0
\(918\) −6672.10 −0.239882
\(919\) −48465.3 −1.73963 −0.869817 0.493374i \(-0.835763\pi\)
−0.869817 + 0.493374i \(0.835763\pi\)
\(920\) 404.310 0.0144888
\(921\) 3726.72 0.133333
\(922\) 6113.34 0.218364
\(923\) −18817.0 −0.671040
\(924\) 0 0
\(925\) −9567.37 −0.340079
\(926\) −7849.85 −0.278577
\(927\) 37081.6 1.31383
\(928\) 2240.74 0.0792630
\(929\) −10560.1 −0.372946 −0.186473 0.982460i \(-0.559706\pi\)
−0.186473 + 0.982460i \(0.559706\pi\)
\(930\) −593.714 −0.0209340
\(931\) 0 0
\(932\) 31277.3 1.09927
\(933\) 44722.4 1.56929
\(934\) −5153.79 −0.180554
\(935\) −2379.59 −0.0832309
\(936\) −19064.9 −0.665765
\(937\) 23025.0 0.802769 0.401384 0.915910i \(-0.368529\pi\)
0.401384 + 0.915910i \(0.368529\pi\)
\(938\) 0 0
\(939\) 78971.7 2.74456
\(940\) 1839.56 0.0638296
\(941\) −40778.1 −1.41268 −0.706338 0.707874i \(-0.749654\pi\)
−0.706338 + 0.707874i \(0.749654\pi\)
\(942\) 2883.43 0.0997317
\(943\) −20199.8 −0.697558
\(944\) 5947.74 0.205066
\(945\) 0 0
\(946\) −2187.18 −0.0751707
\(947\) 36129.8 1.23977 0.619884 0.784693i \(-0.287179\pi\)
0.619884 + 0.784693i \(0.287179\pi\)
\(948\) 25559.3 0.875662
\(949\) 28920.9 0.989265
\(950\) −3340.72 −0.114092
\(951\) −16387.8 −0.558790
\(952\) 0 0
\(953\) 20831.4 0.708075 0.354037 0.935231i \(-0.384809\pi\)
0.354037 + 0.935231i \(0.384809\pi\)
\(954\) 7949.21 0.269775
\(955\) −2976.86 −0.100868
\(956\) 10960.1 0.370789
\(957\) −17509.7 −0.591441
\(958\) −5363.02 −0.180868
\(959\) 0 0
\(960\) 2715.52 0.0912948
\(961\) 25950.1 0.871071
\(962\) 1583.33 0.0530652
\(963\) −10572.6 −0.353788
\(964\) −15977.4 −0.533813
\(965\) 1927.98 0.0643149
\(966\) 0 0
\(967\) 49242.5 1.63757 0.818785 0.574100i \(-0.194648\pi\)
0.818785 + 0.574100i \(0.194648\pi\)
\(968\) −19252.5 −0.639256
\(969\) −33185.7 −1.10018
\(970\) −219.136 −0.00725363
\(971\) 2352.05 0.0777351 0.0388675 0.999244i \(-0.487625\pi\)
0.0388675 + 0.999244i \(0.487625\pi\)
\(972\) −18712.5 −0.617493
\(973\) 0 0
\(974\) 4077.82 0.134150
\(975\) −57301.8 −1.88218
\(976\) 43479.7 1.42598
\(977\) 18768.3 0.614588 0.307294 0.951615i \(-0.400577\pi\)
0.307294 + 0.951615i \(0.400577\pi\)
\(978\) 2059.38 0.0673329
\(979\) 73048.7 2.38473
\(980\) 0 0
\(981\) 60654.5 1.97406
\(982\) −2820.59 −0.0916584
\(983\) 49014.5 1.59036 0.795179 0.606375i \(-0.207377\pi\)
0.795179 + 0.606375i \(0.207377\pi\)
\(984\) 13038.4 0.422406
\(985\) −1761.81 −0.0569908
\(986\) −666.146 −0.0215156
\(987\) 0 0
\(988\) −25225.8 −0.812288
\(989\) 7588.55 0.243986
\(990\) 1038.46 0.0333377
\(991\) −48860.6 −1.56620 −0.783102 0.621893i \(-0.786364\pi\)
−0.783102 + 0.621893i \(0.786364\pi\)
\(992\) 18242.4 0.583868
\(993\) 7411.44 0.236853
\(994\) 0 0
\(995\) −425.888 −0.0135694
\(996\) 50396.2 1.60328
\(997\) −2934.57 −0.0932184 −0.0466092 0.998913i \(-0.514842\pi\)
−0.0466092 + 0.998913i \(0.514842\pi\)
\(998\) −6608.29 −0.209601
\(999\) 22308.7 0.706524
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 1421.4.a.c.1.2 2
7.6 odd 2 29.4.a.a.1.2 2
21.20 even 2 261.4.a.b.1.1 2
28.27 even 2 464.4.a.f.1.2 2
35.34 odd 2 725.4.a.b.1.1 2
56.13 odd 2 1856.4.a.n.1.2 2
56.27 even 2 1856.4.a.h.1.1 2
203.202 odd 2 841.4.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.a.1.2 2 7.6 odd 2
261.4.a.b.1.1 2 21.20 even 2
464.4.a.f.1.2 2 28.27 even 2
725.4.a.b.1.1 2 35.34 odd 2
841.4.a.a.1.1 2 203.202 odd 2
1421.4.a.c.1.2 2 1.1 even 1 trivial
1856.4.a.h.1.1 2 56.27 even 2
1856.4.a.n.1.2 2 56.13 odd 2