Properties

Label 1216.4.a.g.1.2
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
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] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(71.7463225670\)
Analytic rank: \(0\)
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: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.77200\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.227998 q^{3} -8.31601 q^{5} +8.08801 q^{7} -26.9480 q^{9} +O(q^{10})\) \(q-0.227998 q^{3} -8.31601 q^{5} +8.08801 q^{7} -26.9480 q^{9} +12.7720 q^{11} +47.0360 q^{13} +1.89603 q^{15} -31.4560 q^{17} -19.0000 q^{19} -1.84405 q^{21} -19.0360 q^{23} -55.8441 q^{25} +12.3000 q^{27} -91.2120 q^{29} +293.968 q^{31} -2.91199 q^{33} -67.2599 q^{35} -215.616 q^{37} -10.7241 q^{39} -67.7200 q^{41} -308.596 q^{43} +224.100 q^{45} +108.812 q^{47} -277.584 q^{49} +7.17191 q^{51} +682.124 q^{53} -106.212 q^{55} +4.33196 q^{57} +250.300 q^{59} +317.692 q^{61} -217.956 q^{63} -391.152 q^{65} -940.444 q^{67} +4.34018 q^{69} -395.552 q^{71} +975.048 q^{73} +12.7323 q^{75} +103.300 q^{77} +922.776 q^{79} +724.792 q^{81} +1163.77 q^{83} +261.588 q^{85} +20.7962 q^{87} +685.136 q^{89} +380.428 q^{91} -67.0242 q^{93} +158.004 q^{95} +211.256 q^{97} -344.180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} + 9 q^{5} - 18 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{3} + 9 q^{5} - 18 q^{7} + 23 q^{9} + 17 q^{11} - 17 q^{13} - 150 q^{15} - 80 q^{17} - 38 q^{19} + 227 q^{21} + 73 q^{23} + 119 q^{25} - 189 q^{27} - 3 q^{29} + 212 q^{31} - 40 q^{33} - 519 q^{35} - 192 q^{37} + 551 q^{39} - 50 q^{41} - 677 q^{43} + 1089 q^{45} - 389 q^{47} + 60 q^{49} + 433 q^{51} + 1219 q^{53} - 33 q^{55} + 171 q^{57} + 287 q^{59} - 313 q^{61} - 1521 q^{63} - 1500 q^{65} - 1223 q^{67} - 803 q^{69} + 200 q^{71} + 378 q^{73} - 1521 q^{75} - 7 q^{77} + 1350 q^{79} + 1142 q^{81} + 670 q^{83} - 579 q^{85} - 753 q^{87} - 236 q^{89} + 2051 q^{91} + 652 q^{93} - 171 q^{95} + 1294 q^{97} - 133 q^{99}+O(q^{100}) \) 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 0
\(3\) −0.227998 −0.0438783 −0.0219391 0.999759i \(-0.506984\pi\)
−0.0219391 + 0.999759i \(0.506984\pi\)
\(4\) 0 0
\(5\) −8.31601 −0.743806 −0.371903 0.928272i \(-0.621295\pi\)
−0.371903 + 0.928272i \(0.621295\pi\)
\(6\) 0 0
\(7\) 8.08801 0.436711 0.218356 0.975869i \(-0.429931\pi\)
0.218356 + 0.975869i \(0.429931\pi\)
\(8\) 0 0
\(9\) −26.9480 −0.998075
\(10\) 0 0
\(11\) 12.7720 0.350082 0.175041 0.984561i \(-0.443994\pi\)
0.175041 + 0.984561i \(0.443994\pi\)
\(12\) 0 0
\(13\) 47.0360 1.00350 0.501748 0.865014i \(-0.332691\pi\)
0.501748 + 0.865014i \(0.332691\pi\)
\(14\) 0 0
\(15\) 1.89603 0.0326369
\(16\) 0 0
\(17\) −31.4560 −0.448776 −0.224388 0.974500i \(-0.572038\pi\)
−0.224388 + 0.974500i \(0.572038\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) −1.84405 −0.0191621
\(22\) 0 0
\(23\) −19.0360 −0.172578 −0.0862888 0.996270i \(-0.527501\pi\)
−0.0862888 + 0.996270i \(0.527501\pi\)
\(24\) 0 0
\(25\) −55.8441 −0.446752
\(26\) 0 0
\(27\) 12.3000 0.0876720
\(28\) 0 0
\(29\) −91.2120 −0.584057 −0.292028 0.956410i \(-0.594330\pi\)
−0.292028 + 0.956410i \(0.594330\pi\)
\(30\) 0 0
\(31\) 293.968 1.70317 0.851584 0.524218i \(-0.175642\pi\)
0.851584 + 0.524218i \(0.175642\pi\)
\(32\) 0 0
\(33\) −2.91199 −0.0153610
\(34\) 0 0
\(35\) −67.2599 −0.324829
\(36\) 0 0
\(37\) −215.616 −0.958029 −0.479014 0.877807i \(-0.659006\pi\)
−0.479014 + 0.877807i \(0.659006\pi\)
\(38\) 0 0
\(39\) −10.7241 −0.0440317
\(40\) 0 0
\(41\) −67.7200 −0.257953 −0.128977 0.991648i \(-0.541169\pi\)
−0.128977 + 0.991648i \(0.541169\pi\)
\(42\) 0 0
\(43\) −308.596 −1.09443 −0.547214 0.836992i \(-0.684312\pi\)
−0.547214 + 0.836992i \(0.684312\pi\)
\(44\) 0 0
\(45\) 224.100 0.742374
\(46\) 0 0
\(47\) 108.812 0.337700 0.168850 0.985642i \(-0.445995\pi\)
0.168850 + 0.985642i \(0.445995\pi\)
\(48\) 0 0
\(49\) −277.584 −0.809283
\(50\) 0 0
\(51\) 7.17191 0.0196915
\(52\) 0 0
\(53\) 682.124 1.76787 0.883933 0.467613i \(-0.154886\pi\)
0.883933 + 0.467613i \(0.154886\pi\)
\(54\) 0 0
\(55\) −106.212 −0.260393
\(56\) 0 0
\(57\) 4.33196 0.0100664
\(58\) 0 0
\(59\) 250.300 0.552310 0.276155 0.961113i \(-0.410940\pi\)
0.276155 + 0.961113i \(0.410940\pi\)
\(60\) 0 0
\(61\) 317.692 0.666825 0.333412 0.942781i \(-0.391800\pi\)
0.333412 + 0.942781i \(0.391800\pi\)
\(62\) 0 0
\(63\) −217.956 −0.435871
\(64\) 0 0
\(65\) −391.152 −0.746406
\(66\) 0 0
\(67\) −940.444 −1.71483 −0.857414 0.514626i \(-0.827931\pi\)
−0.857414 + 0.514626i \(0.827931\pi\)
\(68\) 0 0
\(69\) 4.34018 0.00757241
\(70\) 0 0
\(71\) −395.552 −0.661175 −0.330587 0.943775i \(-0.607247\pi\)
−0.330587 + 0.943775i \(0.607247\pi\)
\(72\) 0 0
\(73\) 975.048 1.56330 0.781649 0.623718i \(-0.214379\pi\)
0.781649 + 0.623718i \(0.214379\pi\)
\(74\) 0 0
\(75\) 12.7323 0.0196027
\(76\) 0 0
\(77\) 103.300 0.152885
\(78\) 0 0
\(79\) 922.776 1.31418 0.657091 0.753811i \(-0.271787\pi\)
0.657091 + 0.753811i \(0.271787\pi\)
\(80\) 0 0
\(81\) 724.792 0.994228
\(82\) 0 0
\(83\) 1163.77 1.53904 0.769519 0.638624i \(-0.220496\pi\)
0.769519 + 0.638624i \(0.220496\pi\)
\(84\) 0 0
\(85\) 261.588 0.333803
\(86\) 0 0
\(87\) 20.7962 0.0256274
\(88\) 0 0
\(89\) 685.136 0.816003 0.408002 0.912981i \(-0.366226\pi\)
0.408002 + 0.912981i \(0.366226\pi\)
\(90\) 0 0
\(91\) 380.428 0.438238
\(92\) 0 0
\(93\) −67.0242 −0.0747321
\(94\) 0 0
\(95\) 158.004 0.170641
\(96\) 0 0
\(97\) 211.256 0.221132 0.110566 0.993869i \(-0.464734\pi\)
0.110566 + 0.993869i \(0.464734\pi\)
\(98\) 0 0
\(99\) −344.180 −0.349408
\(100\) 0 0
\(101\) 1703.55 1.67831 0.839157 0.543889i \(-0.183049\pi\)
0.839157 + 0.543889i \(0.183049\pi\)
\(102\) 0 0
\(103\) −1393.52 −1.33308 −0.666542 0.745468i \(-0.732226\pi\)
−0.666542 + 0.745468i \(0.732226\pi\)
\(104\) 0 0
\(105\) 15.3351 0.0142529
\(106\) 0 0
\(107\) −907.996 −0.820367 −0.410184 0.912003i \(-0.634535\pi\)
−0.410184 + 0.912003i \(0.634535\pi\)
\(108\) 0 0
\(109\) −862.077 −0.757541 −0.378770 0.925491i \(-0.623653\pi\)
−0.378770 + 0.925491i \(0.623653\pi\)
\(110\) 0 0
\(111\) 49.1601 0.0420366
\(112\) 0 0
\(113\) 1502.72 1.25101 0.625505 0.780220i \(-0.284893\pi\)
0.625505 + 0.780220i \(0.284893\pi\)
\(114\) 0 0
\(115\) 158.304 0.128364
\(116\) 0 0
\(117\) −1267.53 −1.00156
\(118\) 0 0
\(119\) −254.416 −0.195986
\(120\) 0 0
\(121\) −1167.88 −0.877443
\(122\) 0 0
\(123\) 15.4400 0.0113185
\(124\) 0 0
\(125\) 1503.90 1.07610
\(126\) 0 0
\(127\) 389.280 0.271992 0.135996 0.990709i \(-0.456577\pi\)
0.135996 + 0.990709i \(0.456577\pi\)
\(128\) 0 0
\(129\) 70.3593 0.0480216
\(130\) 0 0
\(131\) 268.308 0.178948 0.0894739 0.995989i \(-0.471481\pi\)
0.0894739 + 0.995989i \(0.471481\pi\)
\(132\) 0 0
\(133\) −153.672 −0.100188
\(134\) 0 0
\(135\) −102.287 −0.0652110
\(136\) 0 0
\(137\) 2657.33 1.65716 0.828580 0.559871i \(-0.189149\pi\)
0.828580 + 0.559871i \(0.189149\pi\)
\(138\) 0 0
\(139\) 2859.92 1.74514 0.872572 0.488486i \(-0.162451\pi\)
0.872572 + 0.488486i \(0.162451\pi\)
\(140\) 0 0
\(141\) −24.8090 −0.0148177
\(142\) 0 0
\(143\) 600.744 0.351306
\(144\) 0 0
\(145\) 758.520 0.434425
\(146\) 0 0
\(147\) 63.2887 0.0355099
\(148\) 0 0
\(149\) −311.812 −0.171440 −0.0857202 0.996319i \(-0.527319\pi\)
−0.0857202 + 0.996319i \(0.527319\pi\)
\(150\) 0 0
\(151\) −1462.32 −0.788093 −0.394046 0.919091i \(-0.628925\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(152\) 0 0
\(153\) 847.677 0.447912
\(154\) 0 0
\(155\) −2444.64 −1.26683
\(156\) 0 0
\(157\) −4.38395 −0.00222852 −0.00111426 0.999999i \(-0.500355\pi\)
−0.00111426 + 0.999999i \(0.500355\pi\)
\(158\) 0 0
\(159\) −155.523 −0.0775709
\(160\) 0 0
\(161\) −153.964 −0.0753666
\(162\) 0 0
\(163\) 1777.89 0.854325 0.427162 0.904175i \(-0.359513\pi\)
0.427162 + 0.904175i \(0.359513\pi\)
\(164\) 0 0
\(165\) 24.2161 0.0114256
\(166\) 0 0
\(167\) 893.064 0.413817 0.206908 0.978360i \(-0.433660\pi\)
0.206908 + 0.978360i \(0.433660\pi\)
\(168\) 0 0
\(169\) 15.3876 0.00700391
\(170\) 0 0
\(171\) 512.012 0.228974
\(172\) 0 0
\(173\) 2452.56 1.07783 0.538915 0.842360i \(-0.318834\pi\)
0.538915 + 0.842360i \(0.318834\pi\)
\(174\) 0 0
\(175\) −451.667 −0.195102
\(176\) 0 0
\(177\) −57.0679 −0.0242344
\(178\) 0 0
\(179\) 2064.81 0.862185 0.431092 0.902308i \(-0.358128\pi\)
0.431092 + 0.902308i \(0.358128\pi\)
\(180\) 0 0
\(181\) 2518.54 1.03426 0.517132 0.855906i \(-0.327000\pi\)
0.517132 + 0.855906i \(0.327000\pi\)
\(182\) 0 0
\(183\) −72.4332 −0.0292591
\(184\) 0 0
\(185\) 1793.06 0.712588
\(186\) 0 0
\(187\) −401.756 −0.157109
\(188\) 0 0
\(189\) 99.4829 0.0382874
\(190\) 0 0
\(191\) −4206.38 −1.59352 −0.796761 0.604294i \(-0.793455\pi\)
−0.796761 + 0.604294i \(0.793455\pi\)
\(192\) 0 0
\(193\) 3245.82 1.21056 0.605282 0.796011i \(-0.293060\pi\)
0.605282 + 0.796011i \(0.293060\pi\)
\(194\) 0 0
\(195\) 89.1819 0.0327510
\(196\) 0 0
\(197\) 1734.71 0.627377 0.313688 0.949526i \(-0.398435\pi\)
0.313688 + 0.949526i \(0.398435\pi\)
\(198\) 0 0
\(199\) 380.792 0.135646 0.0678232 0.997697i \(-0.478395\pi\)
0.0678232 + 0.997697i \(0.478395\pi\)
\(200\) 0 0
\(201\) 214.420 0.0752437
\(202\) 0 0
\(203\) −737.724 −0.255064
\(204\) 0 0
\(205\) 563.160 0.191867
\(206\) 0 0
\(207\) 512.983 0.172245
\(208\) 0 0
\(209\) −242.668 −0.0803143
\(210\) 0 0
\(211\) −1010.44 −0.329675 −0.164837 0.986321i \(-0.552710\pi\)
−0.164837 + 0.986321i \(0.552710\pi\)
\(212\) 0 0
\(213\) 90.1852 0.0290112
\(214\) 0 0
\(215\) 2566.29 0.814043
\(216\) 0 0
\(217\) 2377.62 0.743793
\(218\) 0 0
\(219\) −222.309 −0.0685948
\(220\) 0 0
\(221\) −1479.57 −0.450345
\(222\) 0 0
\(223\) 3398.70 1.02060 0.510301 0.859996i \(-0.329534\pi\)
0.510301 + 0.859996i \(0.329534\pi\)
\(224\) 0 0
\(225\) 1504.89 0.445892
\(226\) 0 0
\(227\) −5760.80 −1.68439 −0.842197 0.539169i \(-0.818738\pi\)
−0.842197 + 0.539169i \(0.818738\pi\)
\(228\) 0 0
\(229\) 2179.00 0.628786 0.314393 0.949293i \(-0.398199\pi\)
0.314393 + 0.949293i \(0.398199\pi\)
\(230\) 0 0
\(231\) −23.5522 −0.00670832
\(232\) 0 0
\(233\) −2808.49 −0.789659 −0.394830 0.918754i \(-0.629196\pi\)
−0.394830 + 0.918754i \(0.629196\pi\)
\(234\) 0 0
\(235\) −904.882 −0.251183
\(236\) 0 0
\(237\) −210.391 −0.0576640
\(238\) 0 0
\(239\) 6285.67 1.70120 0.850599 0.525815i \(-0.176239\pi\)
0.850599 + 0.525815i \(0.176239\pi\)
\(240\) 0 0
\(241\) 1129.22 0.301825 0.150912 0.988547i \(-0.451779\pi\)
0.150912 + 0.988547i \(0.451779\pi\)
\(242\) 0 0
\(243\) −497.352 −0.131297
\(244\) 0 0
\(245\) 2308.39 0.601950
\(246\) 0 0
\(247\) −893.684 −0.230218
\(248\) 0 0
\(249\) −265.337 −0.0675303
\(250\) 0 0
\(251\) 2873.73 0.722661 0.361331 0.932438i \(-0.382323\pi\)
0.361331 + 0.932438i \(0.382323\pi\)
\(252\) 0 0
\(253\) −243.128 −0.0604163
\(254\) 0 0
\(255\) −59.6416 −0.0146467
\(256\) 0 0
\(257\) −3712.18 −0.901008 −0.450504 0.892774i \(-0.648756\pi\)
−0.450504 + 0.892774i \(0.648756\pi\)
\(258\) 0 0
\(259\) −1743.90 −0.418382
\(260\) 0 0
\(261\) 2457.98 0.582932
\(262\) 0 0
\(263\) 1263.04 0.296130 0.148065 0.988978i \(-0.452696\pi\)
0.148065 + 0.988978i \(0.452696\pi\)
\(264\) 0 0
\(265\) −5672.55 −1.31495
\(266\) 0 0
\(267\) −156.210 −0.0358048
\(268\) 0 0
\(269\) 5484.39 1.24308 0.621541 0.783381i \(-0.286507\pi\)
0.621541 + 0.783381i \(0.286507\pi\)
\(270\) 0 0
\(271\) −3217.66 −0.721251 −0.360625 0.932711i \(-0.617437\pi\)
−0.360625 + 0.932711i \(0.617437\pi\)
\(272\) 0 0
\(273\) −86.7368 −0.0192291
\(274\) 0 0
\(275\) −713.240 −0.156400
\(276\) 0 0
\(277\) −7668.13 −1.66330 −0.831649 0.555302i \(-0.812603\pi\)
−0.831649 + 0.555302i \(0.812603\pi\)
\(278\) 0 0
\(279\) −7921.86 −1.69989
\(280\) 0 0
\(281\) 1126.81 0.239216 0.119608 0.992821i \(-0.461836\pi\)
0.119608 + 0.992821i \(0.461836\pi\)
\(282\) 0 0
\(283\) 1502.63 0.315625 0.157813 0.987469i \(-0.449556\pi\)
0.157813 + 0.987469i \(0.449556\pi\)
\(284\) 0 0
\(285\) −36.0246 −0.00748742
\(286\) 0 0
\(287\) −547.720 −0.112651
\(288\) 0 0
\(289\) −3923.52 −0.798600
\(290\) 0 0
\(291\) −48.1659 −0.00970288
\(292\) 0 0
\(293\) 452.324 0.0901878 0.0450939 0.998983i \(-0.485641\pi\)
0.0450939 + 0.998983i \(0.485641\pi\)
\(294\) 0 0
\(295\) −2081.50 −0.410812
\(296\) 0 0
\(297\) 157.096 0.0306924
\(298\) 0 0
\(299\) −895.379 −0.173181
\(300\) 0 0
\(301\) −2495.93 −0.477950
\(302\) 0 0
\(303\) −388.407 −0.0736415
\(304\) 0 0
\(305\) −2641.93 −0.495988
\(306\) 0 0
\(307\) 2333.46 0.433803 0.216901 0.976194i \(-0.430405\pi\)
0.216901 + 0.976194i \(0.430405\pi\)
\(308\) 0 0
\(309\) 317.720 0.0584934
\(310\) 0 0
\(311\) 10476.1 1.91011 0.955055 0.296429i \(-0.0957959\pi\)
0.955055 + 0.296429i \(0.0957959\pi\)
\(312\) 0 0
\(313\) 4160.33 0.751297 0.375648 0.926762i \(-0.377420\pi\)
0.375648 + 0.926762i \(0.377420\pi\)
\(314\) 0 0
\(315\) 1812.52 0.324203
\(316\) 0 0
\(317\) 7508.56 1.33036 0.665178 0.746685i \(-0.268356\pi\)
0.665178 + 0.746685i \(0.268356\pi\)
\(318\) 0 0
\(319\) −1164.96 −0.204468
\(320\) 0 0
\(321\) 207.021 0.0359963
\(322\) 0 0
\(323\) 597.664 0.102956
\(324\) 0 0
\(325\) −2626.68 −0.448314
\(326\) 0 0
\(327\) 196.552 0.0332396
\(328\) 0 0
\(329\) 880.073 0.147477
\(330\) 0 0
\(331\) −10386.8 −1.72480 −0.862400 0.506227i \(-0.831040\pi\)
−0.862400 + 0.506227i \(0.831040\pi\)
\(332\) 0 0
\(333\) 5810.43 0.956184
\(334\) 0 0
\(335\) 7820.74 1.27550
\(336\) 0 0
\(337\) 5618.29 0.908153 0.454077 0.890963i \(-0.349969\pi\)
0.454077 + 0.890963i \(0.349969\pi\)
\(338\) 0 0
\(339\) −342.617 −0.0548921
\(340\) 0 0
\(341\) 3754.56 0.596249
\(342\) 0 0
\(343\) −5019.29 −0.790135
\(344\) 0 0
\(345\) −36.0929 −0.00563240
\(346\) 0 0
\(347\) −1814.32 −0.280686 −0.140343 0.990103i \(-0.544820\pi\)
−0.140343 + 0.990103i \(0.544820\pi\)
\(348\) 0 0
\(349\) 816.757 0.125272 0.0626361 0.998036i \(-0.480049\pi\)
0.0626361 + 0.998036i \(0.480049\pi\)
\(350\) 0 0
\(351\) 578.545 0.0879785
\(352\) 0 0
\(353\) 11090.4 1.67219 0.836095 0.548585i \(-0.184833\pi\)
0.836095 + 0.548585i \(0.184833\pi\)
\(354\) 0 0
\(355\) 3289.41 0.491786
\(356\) 0 0
\(357\) 58.0064 0.00859951
\(358\) 0 0
\(359\) −3211.68 −0.472161 −0.236081 0.971733i \(-0.575863\pi\)
−0.236081 + 0.971733i \(0.575863\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 266.274 0.0385007
\(364\) 0 0
\(365\) −8108.51 −1.16279
\(366\) 0 0
\(367\) 8077.81 1.14893 0.574466 0.818528i \(-0.305210\pi\)
0.574466 + 0.818528i \(0.305210\pi\)
\(368\) 0 0
\(369\) 1824.92 0.257457
\(370\) 0 0
\(371\) 5517.02 0.772048
\(372\) 0 0
\(373\) 5088.15 0.706312 0.353156 0.935564i \(-0.385108\pi\)
0.353156 + 0.935564i \(0.385108\pi\)
\(374\) 0 0
\(375\) −342.886 −0.0472175
\(376\) 0 0
\(377\) −4290.25 −0.586099
\(378\) 0 0
\(379\) −2547.00 −0.345199 −0.172600 0.984992i \(-0.555217\pi\)
−0.172600 + 0.984992i \(0.555217\pi\)
\(380\) 0 0
\(381\) −88.7550 −0.0119345
\(382\) 0 0
\(383\) −7056.11 −0.941384 −0.470692 0.882297i \(-0.655996\pi\)
−0.470692 + 0.882297i \(0.655996\pi\)
\(384\) 0 0
\(385\) −859.044 −0.113717
\(386\) 0 0
\(387\) 8316.05 1.09232
\(388\) 0 0
\(389\) −4728.25 −0.616277 −0.308138 0.951342i \(-0.599706\pi\)
−0.308138 + 0.951342i \(0.599706\pi\)
\(390\) 0 0
\(391\) 598.797 0.0774488
\(392\) 0 0
\(393\) −61.1737 −0.00785192
\(394\) 0 0
\(395\) −7673.81 −0.977497
\(396\) 0 0
\(397\) −740.837 −0.0936563 −0.0468281 0.998903i \(-0.514911\pi\)
−0.0468281 + 0.998903i \(0.514911\pi\)
\(398\) 0 0
\(399\) 35.0370 0.00439610
\(400\) 0 0
\(401\) 1879.58 0.234070 0.117035 0.993128i \(-0.462661\pi\)
0.117035 + 0.993128i \(0.462661\pi\)
\(402\) 0 0
\(403\) 13827.1 1.70912
\(404\) 0 0
\(405\) −6027.37 −0.739513
\(406\) 0 0
\(407\) −2753.85 −0.335389
\(408\) 0 0
\(409\) −1715.45 −0.207393 −0.103697 0.994609i \(-0.533067\pi\)
−0.103697 + 0.994609i \(0.533067\pi\)
\(410\) 0 0
\(411\) −605.866 −0.0727133
\(412\) 0 0
\(413\) 2024.43 0.241200
\(414\) 0 0
\(415\) −9677.90 −1.14475
\(416\) 0 0
\(417\) −652.056 −0.0765739
\(418\) 0 0
\(419\) −2497.15 −0.291155 −0.145578 0.989347i \(-0.546504\pi\)
−0.145578 + 0.989347i \(0.546504\pi\)
\(420\) 0 0
\(421\) −6582.52 −0.762024 −0.381012 0.924570i \(-0.624424\pi\)
−0.381012 + 0.924570i \(0.624424\pi\)
\(422\) 0 0
\(423\) −2932.27 −0.337049
\(424\) 0 0
\(425\) 1756.63 0.200492
\(426\) 0 0
\(427\) 2569.50 0.291210
\(428\) 0 0
\(429\) −136.969 −0.0154147
\(430\) 0 0
\(431\) 8875.72 0.991946 0.495973 0.868338i \(-0.334812\pi\)
0.495973 + 0.868338i \(0.334812\pi\)
\(432\) 0 0
\(433\) −3636.90 −0.403645 −0.201822 0.979422i \(-0.564686\pi\)
−0.201822 + 0.979422i \(0.564686\pi\)
\(434\) 0 0
\(435\) −172.941 −0.0190618
\(436\) 0 0
\(437\) 361.684 0.0395920
\(438\) 0 0
\(439\) −10979.4 −1.19366 −0.596829 0.802368i \(-0.703573\pi\)
−0.596829 + 0.802368i \(0.703573\pi\)
\(440\) 0 0
\(441\) 7480.34 0.807725
\(442\) 0 0
\(443\) −1300.16 −0.139442 −0.0697208 0.997567i \(-0.522211\pi\)
−0.0697208 + 0.997567i \(0.522211\pi\)
\(444\) 0 0
\(445\) −5697.60 −0.606948
\(446\) 0 0
\(447\) 71.0925 0.00752250
\(448\) 0 0
\(449\) −15875.2 −1.66859 −0.834296 0.551317i \(-0.814125\pi\)
−0.834296 + 0.551317i \(0.814125\pi\)
\(450\) 0 0
\(451\) −864.920 −0.0903049
\(452\) 0 0
\(453\) 333.406 0.0345801
\(454\) 0 0
\(455\) −3163.64 −0.325964
\(456\) 0 0
\(457\) 3115.66 0.318916 0.159458 0.987205i \(-0.449025\pi\)
0.159458 + 0.987205i \(0.449025\pi\)
\(458\) 0 0
\(459\) −386.910 −0.0393451
\(460\) 0 0
\(461\) −13479.7 −1.36185 −0.680924 0.732354i \(-0.738422\pi\)
−0.680924 + 0.732354i \(0.738422\pi\)
\(462\) 0 0
\(463\) 7946.19 0.797604 0.398802 0.917037i \(-0.369426\pi\)
0.398802 + 0.917037i \(0.369426\pi\)
\(464\) 0 0
\(465\) 557.373 0.0555862
\(466\) 0 0
\(467\) 9148.37 0.906501 0.453250 0.891383i \(-0.350264\pi\)
0.453250 + 0.891383i \(0.350264\pi\)
\(468\) 0 0
\(469\) −7606.32 −0.748885
\(470\) 0 0
\(471\) 0.999532 9.77834e−5 0
\(472\) 0 0
\(473\) −3941.39 −0.383140
\(474\) 0 0
\(475\) 1061.04 0.102492
\(476\) 0 0
\(477\) −18381.9 −1.76446
\(478\) 0 0
\(479\) 7664.64 0.731120 0.365560 0.930788i \(-0.380878\pi\)
0.365560 + 0.930788i \(0.380878\pi\)
\(480\) 0 0
\(481\) −10141.7 −0.961378
\(482\) 0 0
\(483\) 35.1034 0.00330696
\(484\) 0 0
\(485\) −1756.80 −0.164479
\(486\) 0 0
\(487\) −5347.21 −0.497547 −0.248774 0.968562i \(-0.580027\pi\)
−0.248774 + 0.968562i \(0.580027\pi\)
\(488\) 0 0
\(489\) −405.355 −0.0374863
\(490\) 0 0
\(491\) −13647.2 −1.25436 −0.627178 0.778876i \(-0.715790\pi\)
−0.627178 + 0.778876i \(0.715790\pi\)
\(492\) 0 0
\(493\) 2869.17 0.262111
\(494\) 0 0
\(495\) 2862.20 0.259892
\(496\) 0 0
\(497\) −3199.23 −0.288743
\(498\) 0 0
\(499\) 19351.6 1.73607 0.868034 0.496504i \(-0.165383\pi\)
0.868034 + 0.496504i \(0.165383\pi\)
\(500\) 0 0
\(501\) −203.617 −0.0181576
\(502\) 0 0
\(503\) −19259.1 −1.70720 −0.853600 0.520929i \(-0.825586\pi\)
−0.853600 + 0.520929i \(0.825586\pi\)
\(504\) 0 0
\(505\) −14166.7 −1.24834
\(506\) 0 0
\(507\) −3.50834 −0.000307319 0
\(508\) 0 0
\(509\) −3595.77 −0.313123 −0.156561 0.987668i \(-0.550041\pi\)
−0.156561 + 0.987668i \(0.550041\pi\)
\(510\) 0 0
\(511\) 7886.20 0.682710
\(512\) 0 0
\(513\) −233.701 −0.0201133
\(514\) 0 0
\(515\) 11588.5 0.991556
\(516\) 0 0
\(517\) 1389.75 0.118223
\(518\) 0 0
\(519\) −559.179 −0.0472933
\(520\) 0 0
\(521\) −15211.0 −1.27909 −0.639544 0.768754i \(-0.720877\pi\)
−0.639544 + 0.768754i \(0.720877\pi\)
\(522\) 0 0
\(523\) −18307.1 −1.53062 −0.765310 0.643662i \(-0.777414\pi\)
−0.765310 + 0.643662i \(0.777414\pi\)
\(524\) 0 0
\(525\) 102.979 0.00856073
\(526\) 0 0
\(527\) −9247.06 −0.764342
\(528\) 0 0
\(529\) −11804.6 −0.970217
\(530\) 0 0
\(531\) −6745.09 −0.551247
\(532\) 0 0
\(533\) −3185.28 −0.258855
\(534\) 0 0
\(535\) 7550.90 0.610194
\(536\) 0 0
\(537\) −470.773 −0.0378312
\(538\) 0 0
\(539\) −3545.31 −0.283316
\(540\) 0 0
\(541\) −9102.17 −0.723351 −0.361676 0.932304i \(-0.617795\pi\)
−0.361676 + 0.932304i \(0.617795\pi\)
\(542\) 0 0
\(543\) −574.223 −0.0453817
\(544\) 0 0
\(545\) 7169.03 0.563464
\(546\) 0 0
\(547\) 9218.75 0.720595 0.360297 0.932837i \(-0.382675\pi\)
0.360297 + 0.932837i \(0.382675\pi\)
\(548\) 0 0
\(549\) −8561.17 −0.665541
\(550\) 0 0
\(551\) 1733.03 0.133992
\(552\) 0 0
\(553\) 7463.42 0.573918
\(554\) 0 0
\(555\) −408.815 −0.0312671
\(556\) 0 0
\(557\) 13435.1 1.02202 0.511010 0.859575i \(-0.329272\pi\)
0.511010 + 0.859575i \(0.329272\pi\)
\(558\) 0 0
\(559\) −14515.1 −1.09825
\(560\) 0 0
\(561\) 91.5996 0.00689365
\(562\) 0 0
\(563\) −11941.5 −0.893916 −0.446958 0.894555i \(-0.647493\pi\)
−0.446958 + 0.894555i \(0.647493\pi\)
\(564\) 0 0
\(565\) −12496.6 −0.930508
\(566\) 0 0
\(567\) 5862.12 0.434191
\(568\) 0 0
\(569\) −6378.91 −0.469979 −0.234989 0.971998i \(-0.575506\pi\)
−0.234989 + 0.971998i \(0.575506\pi\)
\(570\) 0 0
\(571\) −24903.9 −1.82521 −0.912605 0.408843i \(-0.865932\pi\)
−0.912605 + 0.408843i \(0.865932\pi\)
\(572\) 0 0
\(573\) 959.046 0.0699210
\(574\) 0 0
\(575\) 1063.05 0.0770995
\(576\) 0 0
\(577\) −11414.7 −0.823568 −0.411784 0.911281i \(-0.635094\pi\)
−0.411784 + 0.911281i \(0.635094\pi\)
\(578\) 0 0
\(579\) −740.040 −0.0531175
\(580\) 0 0
\(581\) 9412.57 0.672115
\(582\) 0 0
\(583\) 8712.09 0.618899
\(584\) 0 0
\(585\) 10540.8 0.744969
\(586\) 0 0
\(587\) −20732.1 −1.45776 −0.728881 0.684641i \(-0.759959\pi\)
−0.728881 + 0.684641i \(0.759959\pi\)
\(588\) 0 0
\(589\) −5585.39 −0.390734
\(590\) 0 0
\(591\) −395.511 −0.0275282
\(592\) 0 0
\(593\) 18010.5 1.24722 0.623611 0.781735i \(-0.285665\pi\)
0.623611 + 0.781735i \(0.285665\pi\)
\(594\) 0 0
\(595\) 2115.73 0.145775
\(596\) 0 0
\(597\) −86.8199 −0.00595193
\(598\) 0 0
\(599\) 27944.7 1.90616 0.953080 0.302719i \(-0.0978942\pi\)
0.953080 + 0.302719i \(0.0978942\pi\)
\(600\) 0 0
\(601\) −11598.1 −0.787179 −0.393590 0.919286i \(-0.628767\pi\)
−0.393590 + 0.919286i \(0.628767\pi\)
\(602\) 0 0
\(603\) 25343.1 1.71153
\(604\) 0 0
\(605\) 9712.06 0.652647
\(606\) 0 0
\(607\) 20170.5 1.34876 0.674379 0.738385i \(-0.264411\pi\)
0.674379 + 0.738385i \(0.264411\pi\)
\(608\) 0 0
\(609\) 168.200 0.0111918
\(610\) 0 0
\(611\) 5118.09 0.338880
\(612\) 0 0
\(613\) −14618.3 −0.963174 −0.481587 0.876398i \(-0.659939\pi\)
−0.481587 + 0.876398i \(0.659939\pi\)
\(614\) 0 0
\(615\) −128.399 −0.00841880
\(616\) 0 0
\(617\) −17538.1 −1.14434 −0.572171 0.820134i \(-0.693899\pi\)
−0.572171 + 0.820134i \(0.693899\pi\)
\(618\) 0 0
\(619\) 8815.75 0.572431 0.286216 0.958165i \(-0.407603\pi\)
0.286216 + 0.958165i \(0.407603\pi\)
\(620\) 0 0
\(621\) −234.144 −0.0151302
\(622\) 0 0
\(623\) 5541.39 0.356358
\(624\) 0 0
\(625\) −5525.94 −0.353660
\(626\) 0 0
\(627\) 55.3279 0.00352405
\(628\) 0 0
\(629\) 6782.42 0.429941
\(630\) 0 0
\(631\) −22170.8 −1.39874 −0.699370 0.714759i \(-0.746536\pi\)
−0.699370 + 0.714759i \(0.746536\pi\)
\(632\) 0 0
\(633\) 230.378 0.0144655
\(634\) 0 0
\(635\) −3237.25 −0.202309
\(636\) 0 0
\(637\) −13056.5 −0.812112
\(638\) 0 0
\(639\) 10659.3 0.659902
\(640\) 0 0
\(641\) −22067.7 −1.35978 −0.679891 0.733313i \(-0.737973\pi\)
−0.679891 + 0.733313i \(0.737973\pi\)
\(642\) 0 0
\(643\) 11795.4 0.723428 0.361714 0.932289i \(-0.382192\pi\)
0.361714 + 0.932289i \(0.382192\pi\)
\(644\) 0 0
\(645\) −585.108 −0.0357188
\(646\) 0 0
\(647\) −9716.04 −0.590382 −0.295191 0.955438i \(-0.595383\pi\)
−0.295191 + 0.955438i \(0.595383\pi\)
\(648\) 0 0
\(649\) 3196.83 0.193354
\(650\) 0 0
\(651\) −542.092 −0.0326363
\(652\) 0 0
\(653\) 10311.9 0.617969 0.308985 0.951067i \(-0.400011\pi\)
0.308985 + 0.951067i \(0.400011\pi\)
\(654\) 0 0
\(655\) −2231.25 −0.133102
\(656\) 0 0
\(657\) −26275.6 −1.56029
\(658\) 0 0
\(659\) −4019.80 −0.237616 −0.118808 0.992917i \(-0.537907\pi\)
−0.118808 + 0.992917i \(0.537907\pi\)
\(660\) 0 0
\(661\) 22702.6 1.33590 0.667951 0.744206i \(-0.267172\pi\)
0.667951 + 0.744206i \(0.267172\pi\)
\(662\) 0 0
\(663\) 337.338 0.0197604
\(664\) 0 0
\(665\) 1277.94 0.0745208
\(666\) 0 0
\(667\) 1736.31 0.100795
\(668\) 0 0
\(669\) −774.898 −0.0447822
\(670\) 0 0
\(671\) 4057.57 0.233443
\(672\) 0 0
\(673\) 11132.8 0.637652 0.318826 0.947813i \(-0.396711\pi\)
0.318826 + 0.947813i \(0.396711\pi\)
\(674\) 0 0
\(675\) −686.884 −0.0391677
\(676\) 0 0
\(677\) 13967.0 0.792903 0.396452 0.918056i \(-0.370241\pi\)
0.396452 + 0.918056i \(0.370241\pi\)
\(678\) 0 0
\(679\) 1708.64 0.0965707
\(680\) 0 0
\(681\) 1313.45 0.0739083
\(682\) 0 0
\(683\) −1173.88 −0.0657648 −0.0328824 0.999459i \(-0.510469\pi\)
−0.0328824 + 0.999459i \(0.510469\pi\)
\(684\) 0 0
\(685\) −22098.4 −1.23261
\(686\) 0 0
\(687\) −496.807 −0.0275901
\(688\) 0 0
\(689\) 32084.4 1.77405
\(690\) 0 0
\(691\) −8713.33 −0.479697 −0.239849 0.970810i \(-0.577098\pi\)
−0.239849 + 0.970810i \(0.577098\pi\)
\(692\) 0 0
\(693\) −2783.73 −0.152590
\(694\) 0 0
\(695\) −23783.1 −1.29805
\(696\) 0 0
\(697\) 2130.20 0.115763
\(698\) 0 0
\(699\) 640.331 0.0346489
\(700\) 0 0
\(701\) 31003.4 1.67045 0.835223 0.549912i \(-0.185339\pi\)
0.835223 + 0.549912i \(0.185339\pi\)
\(702\) 0 0
\(703\) 4096.70 0.219787
\(704\) 0 0
\(705\) 206.311 0.0110215
\(706\) 0 0
\(707\) 13778.3 0.732939
\(708\) 0 0
\(709\) 12145.1 0.643328 0.321664 0.946854i \(-0.395758\pi\)
0.321664 + 0.946854i \(0.395758\pi\)
\(710\) 0 0
\(711\) −24867.0 −1.31165
\(712\) 0 0
\(713\) −5595.98 −0.293929
\(714\) 0 0
\(715\) −4995.79 −0.261304
\(716\) 0 0
\(717\) −1433.12 −0.0746456
\(718\) 0 0
\(719\) 24787.8 1.28572 0.642858 0.765985i \(-0.277748\pi\)
0.642858 + 0.765985i \(0.277748\pi\)
\(720\) 0 0
\(721\) −11270.8 −0.582173
\(722\) 0 0
\(723\) −257.461 −0.0132435
\(724\) 0 0
\(725\) 5093.65 0.260929
\(726\) 0 0
\(727\) 19335.6 0.986409 0.493204 0.869914i \(-0.335826\pi\)
0.493204 + 0.869914i \(0.335826\pi\)
\(728\) 0 0
\(729\) −19456.0 −0.988467
\(730\) 0 0
\(731\) 9707.19 0.491154
\(732\) 0 0
\(733\) −20204.5 −1.01810 −0.509052 0.860735i \(-0.670004\pi\)
−0.509052 + 0.860735i \(0.670004\pi\)
\(734\) 0 0
\(735\) −526.309 −0.0264125
\(736\) 0 0
\(737\) −12011.4 −0.600331
\(738\) 0 0
\(739\) −15643.7 −0.778706 −0.389353 0.921089i \(-0.627301\pi\)
−0.389353 + 0.921089i \(0.627301\pi\)
\(740\) 0 0
\(741\) 203.758 0.0101016
\(742\) 0 0
\(743\) −4500.20 −0.222202 −0.111101 0.993809i \(-0.535438\pi\)
−0.111101 + 0.993809i \(0.535438\pi\)
\(744\) 0 0
\(745\) 2593.03 0.127518
\(746\) 0 0
\(747\) −31361.2 −1.53608
\(748\) 0 0
\(749\) −7343.88 −0.358264
\(750\) 0 0
\(751\) 35080.2 1.70452 0.852261 0.523117i \(-0.175231\pi\)
0.852261 + 0.523117i \(0.175231\pi\)
\(752\) 0 0
\(753\) −655.204 −0.0317091
\(754\) 0 0
\(755\) 12160.7 0.586188
\(756\) 0 0
\(757\) 10391.8 0.498938 0.249469 0.968383i \(-0.419744\pi\)
0.249469 + 0.968383i \(0.419744\pi\)
\(758\) 0 0
\(759\) 55.4328 0.00265096
\(760\) 0 0
\(761\) 11810.5 0.562590 0.281295 0.959621i \(-0.409236\pi\)
0.281295 + 0.959621i \(0.409236\pi\)
\(762\) 0 0
\(763\) −6972.48 −0.330827
\(764\) 0 0
\(765\) −7049.28 −0.333160
\(766\) 0 0
\(767\) 11773.1 0.554241
\(768\) 0 0
\(769\) 35125.5 1.64715 0.823574 0.567209i \(-0.191977\pi\)
0.823574 + 0.567209i \(0.191977\pi\)
\(770\) 0 0
\(771\) 846.369 0.0395347
\(772\) 0 0
\(773\) −20001.5 −0.930665 −0.465332 0.885136i \(-0.654065\pi\)
−0.465332 + 0.885136i \(0.654065\pi\)
\(774\) 0 0
\(775\) −16416.4 −0.760895
\(776\) 0 0
\(777\) 397.607 0.0183579
\(778\) 0 0
\(779\) 1286.68 0.0591786
\(780\) 0 0
\(781\) −5051.99 −0.231465
\(782\) 0 0
\(783\) −1121.91 −0.0512055
\(784\) 0 0
\(785\) 36.4569 0.00165758
\(786\) 0 0
\(787\) −13593.3 −0.615690 −0.307845 0.951437i \(-0.599608\pi\)
−0.307845 + 0.951437i \(0.599608\pi\)
\(788\) 0 0
\(789\) −287.970 −0.0129937
\(790\) 0 0
\(791\) 12154.0 0.546330
\(792\) 0 0
\(793\) 14943.0 0.669156
\(794\) 0 0
\(795\) 1293.33 0.0576977
\(796\) 0 0
\(797\) 6946.75 0.308741 0.154370 0.988013i \(-0.450665\pi\)
0.154370 + 0.988013i \(0.450665\pi\)
\(798\) 0 0
\(799\) −3422.79 −0.151552
\(800\) 0 0
\(801\) −18463.1 −0.814432
\(802\) 0 0
\(803\) 12453.3 0.547283
\(804\) 0 0
\(805\) 1280.36 0.0560581
\(806\) 0 0
\(807\) −1250.43 −0.0545443
\(808\) 0 0
\(809\) 24987.2 1.08591 0.542955 0.839762i \(-0.317305\pi\)
0.542955 + 0.839762i \(0.317305\pi\)
\(810\) 0 0
\(811\) 23172.5 1.00332 0.501662 0.865064i \(-0.332722\pi\)
0.501662 + 0.865064i \(0.332722\pi\)
\(812\) 0 0
\(813\) 733.621 0.0316472
\(814\) 0 0
\(815\) −14784.9 −0.635452
\(816\) 0 0
\(817\) 5863.32 0.251079
\(818\) 0 0
\(819\) −10251.8 −0.437394
\(820\) 0 0
\(821\) −30703.8 −1.30520 −0.652600 0.757703i \(-0.726322\pi\)
−0.652600 + 0.757703i \(0.726322\pi\)
\(822\) 0 0
\(823\) −15940.1 −0.675135 −0.337568 0.941301i \(-0.609604\pi\)
−0.337568 + 0.941301i \(0.609604\pi\)
\(824\) 0 0
\(825\) 162.617 0.00686256
\(826\) 0 0
\(827\) 6662.20 0.280130 0.140065 0.990142i \(-0.455269\pi\)
0.140065 + 0.990142i \(0.455269\pi\)
\(828\) 0 0
\(829\) 20606.0 0.863299 0.431649 0.902041i \(-0.357932\pi\)
0.431649 + 0.902041i \(0.357932\pi\)
\(830\) 0 0
\(831\) 1748.32 0.0729826
\(832\) 0 0
\(833\) 8731.69 0.363187
\(834\) 0 0
\(835\) −7426.73 −0.307799
\(836\) 0 0
\(837\) 3615.82 0.149320
\(838\) 0 0
\(839\) −45717.9 −1.88124 −0.940618 0.339468i \(-0.889753\pi\)
−0.940618 + 0.339468i \(0.889753\pi\)
\(840\) 0 0
\(841\) −16069.4 −0.658878
\(842\) 0 0
\(843\) −256.910 −0.0104964
\(844\) 0 0
\(845\) −127.963 −0.00520955
\(846\) 0 0
\(847\) −9445.79 −0.383189
\(848\) 0 0
\(849\) −342.597 −0.0138491
\(850\) 0 0
\(851\) 4104.47 0.165334
\(852\) 0 0
\(853\) −17230.4 −0.691626 −0.345813 0.938303i \(-0.612397\pi\)
−0.345813 + 0.938303i \(0.612397\pi\)
\(854\) 0 0
\(855\) −4257.90 −0.170312
\(856\) 0 0
\(857\) 44484.4 1.77311 0.886557 0.462619i \(-0.153090\pi\)
0.886557 + 0.462619i \(0.153090\pi\)
\(858\) 0 0
\(859\) 23213.4 0.922039 0.461019 0.887390i \(-0.347484\pi\)
0.461019 + 0.887390i \(0.347484\pi\)
\(860\) 0 0
\(861\) 124.879 0.00494294
\(862\) 0 0
\(863\) −9640.68 −0.380270 −0.190135 0.981758i \(-0.560893\pi\)
−0.190135 + 0.981758i \(0.560893\pi\)
\(864\) 0 0
\(865\) −20395.5 −0.801697
\(866\) 0 0
\(867\) 894.555 0.0350412
\(868\) 0 0
\(869\) 11785.7 0.460072
\(870\) 0 0
\(871\) −44234.8 −1.72082
\(872\) 0 0
\(873\) −5692.93 −0.220706
\(874\) 0 0
\(875\) 12163.6 0.469947
\(876\) 0 0
\(877\) 9499.62 0.365769 0.182885 0.983134i \(-0.441457\pi\)
0.182885 + 0.983134i \(0.441457\pi\)
\(878\) 0 0
\(879\) −103.129 −0.00395729
\(880\) 0 0
\(881\) −8252.54 −0.315590 −0.157795 0.987472i \(-0.550439\pi\)
−0.157795 + 0.987472i \(0.550439\pi\)
\(882\) 0 0
\(883\) 34768.9 1.32510 0.662552 0.749016i \(-0.269474\pi\)
0.662552 + 0.749016i \(0.269474\pi\)
\(884\) 0 0
\(885\) 474.577 0.0180257
\(886\) 0 0
\(887\) −3288.58 −0.124487 −0.0622433 0.998061i \(-0.519825\pi\)
−0.0622433 + 0.998061i \(0.519825\pi\)
\(888\) 0 0
\(889\) 3148.50 0.118782
\(890\) 0 0
\(891\) 9257.05 0.348061
\(892\) 0 0
\(893\) −2067.43 −0.0774736
\(894\) 0 0
\(895\) −17171.0 −0.641298
\(896\) 0 0
\(897\) 204.145 0.00759888
\(898\) 0 0
\(899\) −26813.4 −0.994747
\(900\) 0 0
\(901\) −21456.9 −0.793377
\(902\) 0 0
\(903\) 569.067 0.0209716
\(904\) 0 0
\(905\) −20944.2 −0.769292
\(906\) 0 0
\(907\) −37686.1 −1.37966 −0.689828 0.723973i \(-0.742314\pi\)
−0.689828 + 0.723973i \(0.742314\pi\)
\(908\) 0 0
\(909\) −45907.4 −1.67508
\(910\) 0 0
\(911\) 15090.9 0.548828 0.274414 0.961612i \(-0.411516\pi\)
0.274414 + 0.961612i \(0.411516\pi\)
\(912\) 0 0
\(913\) 14863.7 0.538790
\(914\) 0 0
\(915\) 602.355 0.0217631
\(916\) 0 0
\(917\) 2170.08 0.0781485
\(918\) 0 0
\(919\) 43617.4 1.56562 0.782810 0.622261i \(-0.213786\pi\)
0.782810 + 0.622261i \(0.213786\pi\)
\(920\) 0 0
\(921\) −532.024 −0.0190345
\(922\) 0 0
\(923\) −18605.2 −0.663486
\(924\) 0 0
\(925\) 12040.9 0.428002
\(926\) 0 0
\(927\) 37552.6 1.33052
\(928\) 0 0
\(929\) 32446.2 1.14588 0.572942 0.819596i \(-0.305802\pi\)
0.572942 + 0.819596i \(0.305802\pi\)
\(930\) 0 0
\(931\) 5274.10 0.185662
\(932\) 0 0
\(933\) −2388.53 −0.0838123
\(934\) 0 0
\(935\) 3341.01 0.116858
\(936\) 0 0
\(937\) −28355.4 −0.988614 −0.494307 0.869287i \(-0.664578\pi\)
−0.494307 + 0.869287i \(0.664578\pi\)
\(938\) 0 0
\(939\) −948.548 −0.0329656
\(940\) 0 0
\(941\) 48970.8 1.69650 0.848248 0.529599i \(-0.177658\pi\)
0.848248 + 0.529599i \(0.177658\pi\)
\(942\) 0 0
\(943\) 1289.12 0.0445170
\(944\) 0 0
\(945\) −827.300 −0.0284784
\(946\) 0 0
\(947\) 9198.84 0.315652 0.157826 0.987467i \(-0.449552\pi\)
0.157826 + 0.987467i \(0.449552\pi\)
\(948\) 0 0
\(949\) 45862.4 1.56876
\(950\) 0 0
\(951\) −1711.94 −0.0583737
\(952\) 0 0
\(953\) 28428.9 0.966321 0.483160 0.875532i \(-0.339489\pi\)
0.483160 + 0.875532i \(0.339489\pi\)
\(954\) 0 0
\(955\) 34980.3 1.18527
\(956\) 0 0
\(957\) 265.609 0.00897170
\(958\) 0 0
\(959\) 21492.5 0.723700
\(960\) 0 0
\(961\) 56626.2 1.90078
\(962\) 0 0
\(963\) 24468.7 0.818788
\(964\) 0 0
\(965\) −26992.2 −0.900426
\(966\) 0 0
\(967\) −22315.5 −0.742109 −0.371054 0.928611i \(-0.621004\pi\)
−0.371054 + 0.928611i \(0.621004\pi\)
\(968\) 0 0
\(969\) −136.266 −0.00451755
\(970\) 0 0
\(971\) −208.410 −0.00688795 −0.00344398 0.999994i \(-0.501096\pi\)
−0.00344398 + 0.999994i \(0.501096\pi\)
\(972\) 0 0
\(973\) 23131.0 0.762124
\(974\) 0 0
\(975\) 598.879 0.0196712
\(976\) 0 0
\(977\) 35744.3 1.17048 0.585241 0.810860i \(-0.301000\pi\)
0.585241 + 0.810860i \(0.301000\pi\)
\(978\) 0 0
\(979\) 8750.56 0.285668
\(980\) 0 0
\(981\) 23231.3 0.756082
\(982\) 0 0
\(983\) −36175.5 −1.17377 −0.586887 0.809669i \(-0.699647\pi\)
−0.586887 + 0.809669i \(0.699647\pi\)
\(984\) 0 0
\(985\) −14425.9 −0.466647
\(986\) 0 0
\(987\) −200.655 −0.00647104
\(988\) 0 0
\(989\) 5874.44 0.188874
\(990\) 0 0
\(991\) 34654.3 1.11083 0.555413 0.831575i \(-0.312560\pi\)
0.555413 + 0.831575i \(0.312560\pi\)
\(992\) 0 0
\(993\) 2368.17 0.0756812
\(994\) 0 0
\(995\) −3166.67 −0.100895
\(996\) 0 0
\(997\) −4756.72 −0.151100 −0.0755501 0.997142i \(-0.524071\pi\)
−0.0755501 + 0.997142i \(0.524071\pi\)
\(998\) 0 0
\(999\) −2652.09 −0.0839923
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 1216.4.a.g.1.2 2
4.3 odd 2 1216.4.a.p.1.1 2
8.3 odd 2 304.4.a.c.1.2 2
8.5 even 2 38.4.a.c.1.1 2
24.5 odd 2 342.4.a.h.1.1 2
40.13 odd 4 950.4.b.i.799.1 4
40.29 even 2 950.4.a.e.1.2 2
40.37 odd 4 950.4.b.i.799.4 4
56.13 odd 2 1862.4.a.e.1.2 2
152.37 odd 2 722.4.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.4.a.c.1.1 2 8.5 even 2
304.4.a.c.1.2 2 8.3 odd 2
342.4.a.h.1.1 2 24.5 odd 2
722.4.a.f.1.2 2 152.37 odd 2
950.4.a.e.1.2 2 40.29 even 2
950.4.b.i.799.1 4 40.13 odd 4
950.4.b.i.799.4 4 40.37 odd 4
1216.4.a.g.1.2 2 1.1 even 1 trivial
1216.4.a.p.1.1 2 4.3 odd 2
1862.4.a.e.1.2 2 56.13 odd 2