Properties

Label 304.4.a.c.1.2
Level $304$
Weight $4$
Character 304.1
Self dual yes
Analytic conductor $17.937$
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] = [304,4,Mod(1,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, 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("304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.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: \(17.9365806417\)
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: 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 \(\chi\) \(=\) 304.1

$q$-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 \(-0.378705\pi\)
0.371903 + 0.928272i \(0.378705\pi\)
\(6\) 0 0
\(7\) −8.08801 −0.436711 −0.218356 0.975869i \(-0.570069\pi\)
−0.218356 + 0.975869i \(0.570069\pi\)
\(8\) 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 \(-0.667309\pi\)
−0.501748 + 0.865014i \(0.667309\pi\)
\(14\) 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 \(-0.472499\pi\)
0.0862888 + 0.996270i \(0.472499\pi\)
\(24\) 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 \(-0.405670\pi\)
0.292028 + 0.956410i \(0.405670\pi\)
\(30\) 0 0
\(31\) −293.968 −1.70317 −0.851584 0.524218i \(-0.824358\pi\)
−0.851584 + 0.524218i \(0.824358\pi\)
\(32\) 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 \(-0.340994\pi\)
0.479014 + 0.877807i \(0.340994\pi\)
\(38\) 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 \(-0.554005\pi\)
−0.168850 + 0.985642i \(0.554005\pi\)
\(48\) 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 \(-0.845114\pi\)
−0.883933 + 0.467613i \(0.845114\pi\)
\(54\) 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 \(-0.608200\pi\)
−0.333412 + 0.942781i \(0.608200\pi\)
\(62\) 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 \(-0.392753\pi\)
0.330587 + 0.943775i \(0.392753\pi\)
\(72\) 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 \(-0.728213\pi\)
−0.657091 + 0.753811i \(0.728213\pi\)
\(80\) 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 \(-0.816951\pi\)
−0.839157 + 0.543889i \(0.816951\pi\)
\(102\) 0 0
\(103\) 1393.52 1.33308 0.666542 0.745468i \(-0.267774\pi\)
0.666542 + 0.745468i \(0.267774\pi\)
\(104\) 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 \(-0.376347\pi\)
0.378770 + 0.925491i \(0.376347\pi\)
\(110\) 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 \(-0.543423\pi\)
−0.135996 + 0.990709i \(0.543423\pi\)
\(128\) 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 \(-0.472681\pi\)
0.0857202 + 0.996319i \(0.472681\pi\)
\(150\) 0 0
\(151\) 1462.32 0.788093 0.394046 0.919091i \(-0.371075\pi\)
0.394046 + 0.919091i \(0.371075\pi\)
\(152\) 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 \(-0.499645\pi\)
0.00111426 + 0.999999i \(0.499645\pi\)
\(158\) 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 \(-0.566340\pi\)
−0.206908 + 0.978360i \(0.566340\pi\)
\(168\) 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 \(-0.681166\pi\)
−0.538915 + 0.842360i \(0.681166\pi\)
\(174\) 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 \(-0.673000\pi\)
−0.517132 + 0.855906i \(0.673000\pi\)
\(182\) 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 \(-0.206545\pi\)
0.796761 + 0.604294i \(0.206545\pi\)
\(192\) 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 \(-0.601565\pi\)
−0.313688 + 0.949526i \(0.601565\pi\)
\(198\) 0 0
\(199\) −380.792 −0.135646 −0.0678232 0.997697i \(-0.521605\pi\)
−0.0678232 + 0.997697i \(0.521605\pi\)
\(200\) 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 \(-0.670466\pi\)
−0.510301 + 0.859996i \(0.670466\pi\)
\(224\) 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 \(-0.601801\pi\)
−0.314393 + 0.949293i \(0.601801\pi\)
\(230\) 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 \(-0.823761\pi\)
−0.850599 + 0.525815i \(0.823761\pi\)
\(240\) 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 \(-0.547304\pi\)
−0.148065 + 0.988978i \(0.547304\pi\)
\(264\) 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 \(-0.713493\pi\)
−0.621541 + 0.783381i \(0.713493\pi\)
\(270\) 0 0
\(271\) 3217.66 0.721251 0.360625 0.932711i \(-0.382563\pi\)
0.360625 + 0.932711i \(0.382563\pi\)
\(272\) 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 \(-0.187397\pi\)
0.831649 + 0.555302i \(0.187397\pi\)
\(278\) 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 \(-0.514359\pi\)
−0.0450939 + 0.998983i \(0.514359\pi\)
\(294\) 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 \(-0.904204\pi\)
−0.955055 + 0.296429i \(0.904204\pi\)
\(312\) 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 \(-0.731644\pi\)
−0.665178 + 0.746685i \(0.731644\pi\)
\(318\) 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 \(-0.519951\pi\)
−0.0626361 + 0.998036i \(0.519951\pi\)
\(350\) 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 \(-0.424137\pi\)
0.236081 + 0.971733i \(0.424137\pi\)
\(360\) 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 \(-0.694790\pi\)
−0.574466 + 0.818528i \(0.694790\pi\)
\(368\) 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 \(-0.614892\pi\)
−0.353156 + 0.935564i \(0.614892\pi\)
\(374\) 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 \(-0.344004\pi\)
0.470692 + 0.882297i \(0.344004\pi\)
\(384\) 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 \(-0.400294\pi\)
0.308138 + 0.951342i \(0.400294\pi\)
\(390\) 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 \(-0.485089\pi\)
0.0468281 + 0.998903i \(0.485089\pi\)
\(398\) 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 \(-0.375576\pi\)
0.381012 + 0.924570i \(0.375576\pi\)
\(422\) 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 \(-0.665188\pi\)
−0.495973 + 0.868338i \(0.665188\pi\)
\(432\) 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 \(-0.296427\pi\)
0.596829 + 0.802368i \(0.296427\pi\)
\(440\) 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 \(-0.261578\pi\)
0.680924 + 0.732354i \(0.261578\pi\)
\(462\) 0 0
\(463\) −7946.19 −0.797604 −0.398802 0.917037i \(-0.630574\pi\)
−0.398802 + 0.917037i \(0.630574\pi\)
\(464\) 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 \(-0.619122\pi\)
−0.365560 + 0.930788i \(0.619122\pi\)
\(480\) 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 \(-0.419973\pi\)
0.248774 + 0.968562i \(0.419973\pi\)
\(488\) 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 \(-0.174414\pi\)
0.853600 + 0.520929i \(0.174414\pi\)
\(504\) 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 \(-0.449959\pi\)
0.156561 + 0.987668i \(0.449959\pi\)
\(510\) 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 \(-0.382205\pi\)
0.361676 + 0.932304i \(0.382205\pi\)
\(542\) 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 \(-0.670728\pi\)
−0.511010 + 0.859575i \(0.670728\pi\)
\(558\) 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 \(-0.902106\pi\)
−0.953080 + 0.302719i \(0.902106\pi\)
\(600\) 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 \(-0.735589\pi\)
−0.674379 + 0.738385i \(0.735589\pi\)
\(608\) 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 \(-0.340061\pi\)
0.481587 + 0.876398i \(0.340061\pi\)
\(614\) 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 \(-0.253464\pi\)
0.699370 + 0.714759i \(0.253464\pi\)
\(632\) 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 \(-0.404617\pi\)
0.295191 + 0.955438i \(0.404617\pi\)
\(648\) 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 \(-0.599989\pi\)
−0.308985 + 0.951067i \(0.599989\pi\)
\(654\) 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 \(-0.732828\pi\)
−0.667951 + 0.744206i \(0.732828\pi\)
\(662\) 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 \(-0.629759\pi\)
−0.396452 + 0.918056i \(0.629759\pi\)
\(678\) 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 \(-0.814661\pi\)
−0.835223 + 0.549912i \(0.814661\pi\)
\(702\) 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 \(-0.604242\pi\)
−0.321664 + 0.946854i \(0.604242\pi\)
\(710\) 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 \(-0.722252\pi\)
−0.642858 + 0.765985i \(0.722252\pi\)
\(720\) 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 \(-0.664174\pi\)
−0.493204 + 0.869914i \(0.664174\pi\)
\(728\) 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 \(-0.329996\pi\)
0.509052 + 0.860735i \(0.329996\pi\)
\(734\) 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 \(-0.464562\pi\)
0.111101 + 0.993809i \(0.464562\pi\)
\(744\) 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 \(-0.824769\pi\)
−0.852261 + 0.523117i \(0.824769\pi\)
\(752\) 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 \(-0.580256\pi\)
−0.249469 + 0.968383i \(0.580256\pi\)
\(758\) 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 \(-0.345935\pi\)
0.465332 + 0.885136i \(0.345935\pi\)
\(774\) 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 \(-0.549335\pi\)
−0.154370 + 0.988013i \(0.549335\pi\)
\(798\) 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 \(-0.273678\pi\)
0.652600 + 0.757703i \(0.273678\pi\)
\(822\) 0 0
\(823\) 15940.1 0.675135 0.337568 0.941301i \(-0.390396\pi\)
0.337568 + 0.941301i \(0.390396\pi\)
\(824\) 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 \(-0.642068\pi\)
−0.431649 + 0.902041i \(0.642068\pi\)
\(830\) 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 \(-0.110247\pi\)
0.940618 + 0.339468i \(0.110247\pi\)
\(840\) 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 \(-0.387603\pi\)
0.345813 + 0.938303i \(0.387603\pi\)
\(854\) 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 \(-0.439107\pi\)
0.190135 + 0.981758i \(0.439107\pi\)
\(864\) 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 \(-0.558543\pi\)
−0.182885 + 0.983134i \(0.558543\pi\)
\(878\) 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 \(-0.480175\pi\)
0.0622433 + 0.998061i \(0.480175\pi\)
\(888\) 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 \(-0.588484\pi\)
−0.274414 + 0.961612i \(0.588484\pi\)
\(912\) 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 \(-0.786214\pi\)
−0.782810 + 0.622261i \(0.786214\pi\)
\(920\) 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 \(-0.822342\pi\)
−0.848248 + 0.529599i \(0.822342\pi\)
\(942\) 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 \(-0.378996\pi\)
0.371054 + 0.928611i \(0.378996\pi\)
\(968\) 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 \(-0.300353\pi\)
0.586887 + 0.809669i \(0.300353\pi\)
\(984\) 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 \(-0.687440\pi\)
−0.555413 + 0.831575i \(0.687440\pi\)
\(992\) 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 \(-0.475929\pi\)
0.0755501 + 0.997142i \(0.475929\pi\)
\(998\) 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 304.4.a.c.1.2 2
4.3 odd 2 38.4.a.c.1.1 2
8.3 odd 2 1216.4.a.g.1.2 2
8.5 even 2 1216.4.a.p.1.1 2
12.11 even 2 342.4.a.h.1.1 2
20.3 even 4 950.4.b.i.799.1 4
20.7 even 4 950.4.b.i.799.4 4
20.19 odd 2 950.4.a.e.1.2 2
28.27 even 2 1862.4.a.e.1.2 2
76.75 even 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 4.3 odd 2
304.4.a.c.1.2 2 1.1 even 1 trivial
342.4.a.h.1.1 2 12.11 even 2
722.4.a.f.1.2 2 76.75 even 2
950.4.a.e.1.2 2 20.19 odd 2
950.4.b.i.799.1 4 20.3 even 4
950.4.b.i.799.4 4 20.7 even 4
1216.4.a.g.1.2 2 8.3 odd 2
1216.4.a.p.1.1 2 8.5 even 2
1862.4.a.e.1.2 2 28.27 even 2