Properties

Label 960.4.b.a.671.16
Level $960$
Weight $4$
Character 960.671
Analytic conductor $56.642$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,4,Mod(671,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.671");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 671.16
Root \(4.99314 + 4.99314i\) of defining polynomial
Character \(\chi\) \(=\) 960.671
Dual form 960.4.b.a.671.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.99314 + 1.43824i) q^{3} -5.00000 q^{5} -16.1856i q^{7} +(22.8629 + 14.3627i) q^{9} +15.1075i q^{11} -19.2259i q^{13} +(-24.9657 - 7.19122i) q^{15} +83.4642i q^{17} -38.2158 q^{19} +(23.2789 - 80.8171i) q^{21} +46.7087 q^{23} +25.0000 q^{25} +(93.5007 + 104.597i) q^{27} +192.661 q^{29} +132.947i q^{31} +(-21.7283 + 75.4341i) q^{33} +80.9281i q^{35} -89.0478i q^{37} +(27.6515 - 95.9975i) q^{39} -46.9335i q^{41} -8.02092 q^{43} +(-114.315 - 71.8135i) q^{45} +85.9514 q^{47} +81.0254 q^{49} +(-120.042 + 416.748i) q^{51} +372.581 q^{53} -75.5377i q^{55} +(-190.817 - 54.9636i) q^{57} -80.4063i q^{59} -154.416i q^{61} +(232.469 - 370.051i) q^{63} +96.1294i q^{65} +891.963 q^{67} +(233.223 + 67.1784i) q^{69} +1011.97 q^{71} -94.6740 q^{73} +(124.829 + 35.9561i) q^{75} +244.525 q^{77} +166.524i q^{79} +(316.425 + 656.746i) q^{81} +199.617i q^{83} -417.321i q^{85} +(961.984 + 277.094i) q^{87} +1345.95i q^{89} -311.183 q^{91} +(-191.211 + 663.824i) q^{93} +191.079 q^{95} +1240.10 q^{97} +(-216.985 + 345.402i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 80 q^{5} - 32 q^{9} + 248 q^{21} + 400 q^{25} - 192 q^{29} - 520 q^{33} + 160 q^{45} + 576 q^{49} - 960 q^{53} - 2056 q^{57} + 1992 q^{69} + 784 q^{73} - 4992 q^{77} - 3760 q^{81} + 4032 q^{93} + 1520 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 4.99314 + 1.43824i 0.960930 + 0.276790i
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 16.1856i 0.873942i −0.899476 0.436971i \(-0.856051\pi\)
0.899476 0.436971i \(-0.143949\pi\)
\(8\) 0 0
\(9\) 22.8629 + 14.3627i 0.846774 + 0.531952i
\(10\) 0 0
\(11\) 15.1075i 0.414100i 0.978330 + 0.207050i \(0.0663862\pi\)
−0.978330 + 0.207050i \(0.933614\pi\)
\(12\) 0 0
\(13\) 19.2259i 0.410177i −0.978743 0.205088i \(-0.934252\pi\)
0.978743 0.205088i \(-0.0657482\pi\)
\(14\) 0 0
\(15\) −24.9657 7.19122i −0.429741 0.123784i
\(16\) 0 0
\(17\) 83.4642i 1.19077i 0.803442 + 0.595383i \(0.203000\pi\)
−0.803442 + 0.595383i \(0.797000\pi\)
\(18\) 0 0
\(19\) −38.2158 −0.461437 −0.230718 0.973021i \(-0.574108\pi\)
−0.230718 + 0.973021i \(0.574108\pi\)
\(20\) 0 0
\(21\) 23.2789 80.8171i 0.241898 0.839797i
\(22\) 0 0
\(23\) 46.7087 0.423453 0.211727 0.977329i \(-0.432091\pi\)
0.211727 + 0.977329i \(0.432091\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 93.5007 + 104.597i 0.666452 + 0.745548i
\(28\) 0 0
\(29\) 192.661 1.23366 0.616832 0.787095i \(-0.288416\pi\)
0.616832 + 0.787095i \(0.288416\pi\)
\(30\) 0 0
\(31\) 132.947i 0.770259i 0.922862 + 0.385130i \(0.125843\pi\)
−0.922862 + 0.385130i \(0.874157\pi\)
\(32\) 0 0
\(33\) −21.7283 + 75.4341i −0.114619 + 0.397921i
\(34\) 0 0
\(35\) 80.9281i 0.390839i
\(36\) 0 0
\(37\) 89.0478i 0.395659i −0.980236 0.197829i \(-0.936611\pi\)
0.980236 0.197829i \(-0.0633892\pi\)
\(38\) 0 0
\(39\) 27.6515 95.9975i 0.113533 0.394151i
\(40\) 0 0
\(41\) 46.9335i 0.178775i −0.995997 0.0893875i \(-0.971509\pi\)
0.995997 0.0893875i \(-0.0284910\pi\)
\(42\) 0 0
\(43\) −8.02092 −0.0284460 −0.0142230 0.999899i \(-0.504527\pi\)
−0.0142230 + 0.999899i \(0.504527\pi\)
\(44\) 0 0
\(45\) −114.315 71.8135i −0.378689 0.237896i
\(46\) 0 0
\(47\) 85.9514 0.266751 0.133376 0.991066i \(-0.457418\pi\)
0.133376 + 0.991066i \(0.457418\pi\)
\(48\) 0 0
\(49\) 81.0254 0.236226
\(50\) 0 0
\(51\) −120.042 + 416.748i −0.329592 + 1.14424i
\(52\) 0 0
\(53\) 372.581 0.965622 0.482811 0.875725i \(-0.339616\pi\)
0.482811 + 0.875725i \(0.339616\pi\)
\(54\) 0 0
\(55\) 75.5377i 0.185191i
\(56\) 0 0
\(57\) −190.817 54.9636i −0.443409 0.127721i
\(58\) 0 0
\(59\) 80.4063i 0.177424i −0.996057 0.0887119i \(-0.971725\pi\)
0.996057 0.0887119i \(-0.0282750\pi\)
\(60\) 0 0
\(61\) 154.416i 0.324114i −0.986781 0.162057i \(-0.948187\pi\)
0.986781 0.162057i \(-0.0518129\pi\)
\(62\) 0 0
\(63\) 232.469 370.051i 0.464895 0.740032i
\(64\) 0 0
\(65\) 96.1294i 0.183437i
\(66\) 0 0
\(67\) 891.963 1.62643 0.813213 0.581966i \(-0.197716\pi\)
0.813213 + 0.581966i \(0.197716\pi\)
\(68\) 0 0
\(69\) 233.223 + 67.1784i 0.406909 + 0.117208i
\(70\) 0 0
\(71\) 1011.97 1.69153 0.845765 0.533556i \(-0.179145\pi\)
0.845765 + 0.533556i \(0.179145\pi\)
\(72\) 0 0
\(73\) −94.6740 −0.151791 −0.0758956 0.997116i \(-0.524182\pi\)
−0.0758956 + 0.997116i \(0.524182\pi\)
\(74\) 0 0
\(75\) 124.829 + 35.9561i 0.192186 + 0.0553580i
\(76\) 0 0
\(77\) 244.525 0.361899
\(78\) 0 0
\(79\) 166.524i 0.237157i 0.992945 + 0.118578i \(0.0378337\pi\)
−0.992945 + 0.118578i \(0.962166\pi\)
\(80\) 0 0
\(81\) 316.425 + 656.746i 0.434054 + 0.900887i
\(82\) 0 0
\(83\) 199.617i 0.263986i 0.991251 + 0.131993i \(0.0421376\pi\)
−0.991251 + 0.131993i \(0.957862\pi\)
\(84\) 0 0
\(85\) 417.321i 0.532527i
\(86\) 0 0
\(87\) 961.984 + 277.094i 1.18547 + 0.341466i
\(88\) 0 0
\(89\) 1345.95i 1.60303i 0.597972 + 0.801517i \(0.295973\pi\)
−0.597972 + 0.801517i \(0.704027\pi\)
\(90\) 0 0
\(91\) −311.183 −0.358471
\(92\) 0 0
\(93\) −191.211 + 663.824i −0.213200 + 0.740166i
\(94\) 0 0
\(95\) 191.079 0.206361
\(96\) 0 0
\(97\) 1240.10 1.29808 0.649038 0.760756i \(-0.275172\pi\)
0.649038 + 0.760756i \(0.275172\pi\)
\(98\) 0 0
\(99\) −216.985 + 345.402i −0.220281 + 0.350649i
\(100\) 0 0
\(101\) −1047.31 −1.03180 −0.515899 0.856649i \(-0.672542\pi\)
−0.515899 + 0.856649i \(0.672542\pi\)
\(102\) 0 0
\(103\) 1179.34i 1.12819i −0.825710 0.564095i \(-0.809225\pi\)
0.825710 0.564095i \(-0.190775\pi\)
\(104\) 0 0
\(105\) −116.394 + 404.086i −0.108180 + 0.375569i
\(106\) 0 0
\(107\) 1881.45i 1.69987i −0.526885 0.849937i \(-0.676640\pi\)
0.526885 0.849937i \(-0.323360\pi\)
\(108\) 0 0
\(109\) 1186.74i 1.04284i 0.853301 + 0.521418i \(0.174597\pi\)
−0.853301 + 0.521418i \(0.825403\pi\)
\(110\) 0 0
\(111\) 128.072 444.628i 0.109514 0.380200i
\(112\) 0 0
\(113\) 1690.98i 1.40773i 0.710333 + 0.703866i \(0.248544\pi\)
−0.710333 + 0.703866i \(0.751456\pi\)
\(114\) 0 0
\(115\) −233.543 −0.189374
\(116\) 0 0
\(117\) 276.136 439.559i 0.218194 0.347327i
\(118\) 0 0
\(119\) 1350.92 1.04066
\(120\) 0 0
\(121\) 1102.76 0.828522
\(122\) 0 0
\(123\) 67.5017 234.345i 0.0494831 0.171790i
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 295.964i 0.206792i −0.994640 0.103396i \(-0.967029\pi\)
0.994640 0.103396i \(-0.0329709\pi\)
\(128\) 0 0
\(129\) −40.0496 11.5360i −0.0273347 0.00787358i
\(130\) 0 0
\(131\) 330.510i 0.220434i −0.993908 0.110217i \(-0.964845\pi\)
0.993908 0.110217i \(-0.0351545\pi\)
\(132\) 0 0
\(133\) 618.546i 0.403269i
\(134\) 0 0
\(135\) −467.503 522.987i −0.298047 0.333419i
\(136\) 0 0
\(137\) 2370.48i 1.47828i 0.673553 + 0.739139i \(0.264767\pi\)
−0.673553 + 0.739139i \(0.735233\pi\)
\(138\) 0 0
\(139\) −2296.01 −1.40105 −0.700523 0.713630i \(-0.747050\pi\)
−0.700523 + 0.713630i \(0.747050\pi\)
\(140\) 0 0
\(141\) 429.168 + 123.619i 0.256329 + 0.0738341i
\(142\) 0 0
\(143\) 290.456 0.169854
\(144\) 0 0
\(145\) −963.305 −0.551711
\(146\) 0 0
\(147\) 404.571 + 116.534i 0.226996 + 0.0653849i
\(148\) 0 0
\(149\) −2564.14 −1.40982 −0.704908 0.709299i \(-0.749012\pi\)
−0.704908 + 0.709299i \(0.749012\pi\)
\(150\) 0 0
\(151\) 1965.29i 1.05916i 0.848260 + 0.529581i \(0.177651\pi\)
−0.848260 + 0.529581i \(0.822349\pi\)
\(152\) 0 0
\(153\) −1198.77 + 1908.23i −0.633431 + 1.00831i
\(154\) 0 0
\(155\) 664.736i 0.344470i
\(156\) 0 0
\(157\) 615.854i 0.313061i −0.987673 0.156530i \(-0.949969\pi\)
0.987673 0.156530i \(-0.0500309\pi\)
\(158\) 0 0
\(159\) 1860.35 + 535.862i 0.927895 + 0.267274i
\(160\) 0 0
\(161\) 756.009i 0.370074i
\(162\) 0 0
\(163\) 104.709 0.0503158 0.0251579 0.999683i \(-0.491991\pi\)
0.0251579 + 0.999683i \(0.491991\pi\)
\(164\) 0 0
\(165\) 108.642 377.170i 0.0512590 0.177956i
\(166\) 0 0
\(167\) 2024.38 0.938029 0.469015 0.883190i \(-0.344609\pi\)
0.469015 + 0.883190i \(0.344609\pi\)
\(168\) 0 0
\(169\) 1827.37 0.831755
\(170\) 0 0
\(171\) −873.724 548.882i −0.390733 0.245462i
\(172\) 0 0
\(173\) −2990.04 −1.31404 −0.657020 0.753873i \(-0.728183\pi\)
−0.657020 + 0.753873i \(0.728183\pi\)
\(174\) 0 0
\(175\) 404.641i 0.174788i
\(176\) 0 0
\(177\) 115.644 401.480i 0.0491092 0.170492i
\(178\) 0 0
\(179\) 563.495i 0.235294i 0.993055 + 0.117647i \(0.0375351\pi\)
−0.993055 + 0.117647i \(0.962465\pi\)
\(180\) 0 0
\(181\) 1683.92i 0.691518i 0.938323 + 0.345759i \(0.112378\pi\)
−0.938323 + 0.345759i \(0.887622\pi\)
\(182\) 0 0
\(183\) 222.088 771.021i 0.0897116 0.311451i
\(184\) 0 0
\(185\) 445.239i 0.176944i
\(186\) 0 0
\(187\) −1260.94 −0.493096
\(188\) 0 0
\(189\) 1692.98 1513.37i 0.651565 0.582441i
\(190\) 0 0
\(191\) 4690.57 1.77695 0.888476 0.458922i \(-0.151764\pi\)
0.888476 + 0.458922i \(0.151764\pi\)
\(192\) 0 0
\(193\) −3598.43 −1.34208 −0.671038 0.741423i \(-0.734151\pi\)
−0.671038 + 0.741423i \(0.734151\pi\)
\(194\) 0 0
\(195\) −138.257 + 479.988i −0.0507734 + 0.176270i
\(196\) 0 0
\(197\) −2431.31 −0.879306 −0.439653 0.898168i \(-0.644899\pi\)
−0.439653 + 0.898168i \(0.644899\pi\)
\(198\) 0 0
\(199\) 3011.78i 1.07286i −0.843944 0.536431i \(-0.819772\pi\)
0.843944 0.536431i \(-0.180228\pi\)
\(200\) 0 0
\(201\) 4453.70 + 1282.86i 1.56288 + 0.450179i
\(202\) 0 0
\(203\) 3118.34i 1.07815i
\(204\) 0 0
\(205\) 234.667i 0.0799506i
\(206\) 0 0
\(207\) 1067.90 + 670.863i 0.358570 + 0.225257i
\(208\) 0 0
\(209\) 577.346i 0.191081i
\(210\) 0 0
\(211\) 3830.17 1.24967 0.624833 0.780758i \(-0.285167\pi\)
0.624833 + 0.780758i \(0.285167\pi\)
\(212\) 0 0
\(213\) 5052.90 + 1455.46i 1.62544 + 0.468199i
\(214\) 0 0
\(215\) 40.1046 0.0127215
\(216\) 0 0
\(217\) 2151.84 0.673162
\(218\) 0 0
\(219\) −472.721 136.164i −0.145861 0.0420143i
\(220\) 0 0
\(221\) 1604.67 0.488425
\(222\) 0 0
\(223\) 4304.09i 1.29248i −0.763134 0.646240i \(-0.776340\pi\)
0.763134 0.646240i \(-0.223660\pi\)
\(224\) 0 0
\(225\) 571.573 + 359.068i 0.169355 + 0.106390i
\(226\) 0 0
\(227\) 2960.94i 0.865747i 0.901455 + 0.432874i \(0.142500\pi\)
−0.901455 + 0.432874i \(0.857500\pi\)
\(228\) 0 0
\(229\) 6091.99i 1.75795i −0.476870 0.878974i \(-0.658229\pi\)
0.476870 0.878974i \(-0.341771\pi\)
\(230\) 0 0
\(231\) 1220.95 + 351.687i 0.347760 + 0.100170i
\(232\) 0 0
\(233\) 5739.74i 1.61383i 0.590667 + 0.806915i \(0.298865\pi\)
−0.590667 + 0.806915i \(0.701135\pi\)
\(234\) 0 0
\(235\) −429.757 −0.119295
\(236\) 0 0
\(237\) −239.502 + 831.477i −0.0656427 + 0.227891i
\(238\) 0 0
\(239\) 385.738 0.104399 0.0521994 0.998637i \(-0.483377\pi\)
0.0521994 + 0.998637i \(0.483377\pi\)
\(240\) 0 0
\(241\) 834.546 0.223061 0.111531 0.993761i \(-0.464425\pi\)
0.111531 + 0.993761i \(0.464425\pi\)
\(242\) 0 0
\(243\) 635.395 + 3734.32i 0.167739 + 0.985831i
\(244\) 0 0
\(245\) −405.127 −0.105643
\(246\) 0 0
\(247\) 734.731i 0.189271i
\(248\) 0 0
\(249\) −287.098 + 996.717i −0.0730687 + 0.253672i
\(250\) 0 0
\(251\) 3188.54i 0.801828i −0.916116 0.400914i \(-0.868693\pi\)
0.916116 0.400914i \(-0.131307\pi\)
\(252\) 0 0
\(253\) 705.653i 0.175352i
\(254\) 0 0
\(255\) 600.209 2083.74i 0.147398 0.511722i
\(256\) 0 0
\(257\) 3900.85i 0.946803i −0.880847 0.473401i \(-0.843026\pi\)
0.880847 0.473401i \(-0.156974\pi\)
\(258\) 0 0
\(259\) −1441.29 −0.345783
\(260\) 0 0
\(261\) 4404.79 + 2767.13i 1.04464 + 0.656250i
\(262\) 0 0
\(263\) −3647.80 −0.855258 −0.427629 0.903954i \(-0.640651\pi\)
−0.427629 + 0.903954i \(0.640651\pi\)
\(264\) 0 0
\(265\) −1862.91 −0.431839
\(266\) 0 0
\(267\) −1935.80 + 6720.50i −0.443704 + 1.54040i
\(268\) 0 0
\(269\) −837.038 −0.189722 −0.0948608 0.995491i \(-0.530241\pi\)
−0.0948608 + 0.995491i \(0.530241\pi\)
\(270\) 0 0
\(271\) 6576.31i 1.47410i 0.675836 + 0.737052i \(0.263783\pi\)
−0.675836 + 0.737052i \(0.736217\pi\)
\(272\) 0 0
\(273\) −1553.78 447.557i −0.344465 0.0992211i
\(274\) 0 0
\(275\) 377.688i 0.0828199i
\(276\) 0 0
\(277\) 2414.09i 0.523641i 0.965117 + 0.261821i \(0.0843229\pi\)
−0.965117 + 0.261821i \(0.915677\pi\)
\(278\) 0 0
\(279\) −1909.48 + 3039.56i −0.409741 + 0.652236i
\(280\) 0 0
\(281\) 491.109i 0.104260i 0.998640 + 0.0521301i \(0.0166011\pi\)
−0.998640 + 0.0521301i \(0.983399\pi\)
\(282\) 0 0
\(283\) 9271.34 1.94743 0.973717 0.227761i \(-0.0731406\pi\)
0.973717 + 0.227761i \(0.0731406\pi\)
\(284\) 0 0
\(285\) 954.083 + 274.818i 0.198298 + 0.0571186i
\(286\) 0 0
\(287\) −759.648 −0.156239
\(288\) 0 0
\(289\) −2053.27 −0.417926
\(290\) 0 0
\(291\) 6192.01 + 1783.57i 1.24736 + 0.359295i
\(292\) 0 0
\(293\) 687.236 0.137026 0.0685132 0.997650i \(-0.478174\pi\)
0.0685132 + 0.997650i \(0.478174\pi\)
\(294\) 0 0
\(295\) 402.031i 0.0793463i
\(296\) 0 0
\(297\) −1580.21 + 1412.57i −0.308731 + 0.275978i
\(298\) 0 0
\(299\) 898.015i 0.173691i
\(300\) 0 0
\(301\) 129.824i 0.0248602i
\(302\) 0 0
\(303\) −5229.38 1506.29i −0.991486 0.285591i
\(304\) 0 0
\(305\) 772.081i 0.144948i
\(306\) 0 0
\(307\) −1299.44 −0.241573 −0.120787 0.992678i \(-0.538542\pi\)
−0.120787 + 0.992678i \(0.538542\pi\)
\(308\) 0 0
\(309\) 1696.17 5888.60i 0.312272 1.08411i
\(310\) 0 0
\(311\) −5227.97 −0.953219 −0.476610 0.879115i \(-0.658134\pi\)
−0.476610 + 0.879115i \(0.658134\pi\)
\(312\) 0 0
\(313\) −2260.72 −0.408255 −0.204127 0.978944i \(-0.565436\pi\)
−0.204127 + 0.978944i \(0.565436\pi\)
\(314\) 0 0
\(315\) −1162.35 + 1850.25i −0.207907 + 0.330952i
\(316\) 0 0
\(317\) 363.429 0.0643918 0.0321959 0.999482i \(-0.489750\pi\)
0.0321959 + 0.999482i \(0.489750\pi\)
\(318\) 0 0
\(319\) 2910.63i 0.510860i
\(320\) 0 0
\(321\) 2705.98 9394.33i 0.470508 1.63346i
\(322\) 0 0
\(323\) 3189.65i 0.549463i
\(324\) 0 0
\(325\) 480.647i 0.0820354i
\(326\) 0 0
\(327\) −1706.82 + 5925.56i −0.288647 + 1.00209i
\(328\) 0 0
\(329\) 1391.18i 0.233125i
\(330\) 0 0
\(331\) −601.117 −0.0998199 −0.0499100 0.998754i \(-0.515893\pi\)
−0.0499100 + 0.998754i \(0.515893\pi\)
\(332\) 0 0
\(333\) 1278.97 2035.89i 0.210471 0.335034i
\(334\) 0 0
\(335\) −4459.81 −0.727360
\(336\) 0 0
\(337\) 4758.64 0.769198 0.384599 0.923084i \(-0.374340\pi\)
0.384599 + 0.923084i \(0.374340\pi\)
\(338\) 0 0
\(339\) −2432.03 + 8443.28i −0.389646 + 1.35273i
\(340\) 0 0
\(341\) −2008.51 −0.318964
\(342\) 0 0
\(343\) 6863.12i 1.08039i
\(344\) 0 0
\(345\) −1166.11 335.892i −0.181975 0.0524169i
\(346\) 0 0
\(347\) 8175.86i 1.26485i −0.774622 0.632425i \(-0.782060\pi\)
0.774622 0.632425i \(-0.217940\pi\)
\(348\) 0 0
\(349\) 8651.37i 1.32693i −0.748209 0.663463i \(-0.769086\pi\)
0.748209 0.663463i \(-0.230914\pi\)
\(350\) 0 0
\(351\) 2010.98 1797.63i 0.305806 0.273363i
\(352\) 0 0
\(353\) 9940.40i 1.49879i −0.662121 0.749397i \(-0.730344\pi\)
0.662121 0.749397i \(-0.269656\pi\)
\(354\) 0 0
\(355\) −5059.84 −0.756475
\(356\) 0 0
\(357\) 6745.33 + 1942.95i 1.00000 + 0.288045i
\(358\) 0 0
\(359\) −2871.55 −0.422157 −0.211079 0.977469i \(-0.567698\pi\)
−0.211079 + 0.977469i \(0.567698\pi\)
\(360\) 0 0
\(361\) −5398.56 −0.787076
\(362\) 0 0
\(363\) 5506.25 + 1586.04i 0.796152 + 0.229327i
\(364\) 0 0
\(365\) 473.370 0.0678831
\(366\) 0 0
\(367\) 1849.45i 0.263054i −0.991313 0.131527i \(-0.958012\pi\)
0.991313 0.131527i \(-0.0419879\pi\)
\(368\) 0 0
\(369\) 674.091 1073.04i 0.0950997 0.151382i
\(370\) 0 0
\(371\) 6030.46i 0.843897i
\(372\) 0 0
\(373\) 10455.0i 1.45132i 0.688055 + 0.725658i \(0.258465\pi\)
−0.688055 + 0.725658i \(0.741535\pi\)
\(374\) 0 0
\(375\) −624.143 179.780i −0.0859482 0.0247569i
\(376\) 0 0
\(377\) 3704.08i 0.506020i
\(378\) 0 0
\(379\) −6285.78 −0.851923 −0.425962 0.904741i \(-0.640064\pi\)
−0.425962 + 0.904741i \(0.640064\pi\)
\(380\) 0 0
\(381\) 425.669 1477.79i 0.0572380 0.198713i
\(382\) 0 0
\(383\) −1707.73 −0.227836 −0.113918 0.993490i \(-0.536340\pi\)
−0.113918 + 0.993490i \(0.536340\pi\)
\(384\) 0 0
\(385\) −1222.63 −0.161846
\(386\) 0 0
\(387\) −183.382 115.202i −0.0240874 0.0151319i
\(388\) 0 0
\(389\) −14594.1 −1.90218 −0.951091 0.308910i \(-0.900036\pi\)
−0.951091 + 0.308910i \(0.900036\pi\)
\(390\) 0 0
\(391\) 3898.50i 0.504234i
\(392\) 0 0
\(393\) 475.354 1650.28i 0.0610138 0.211821i
\(394\) 0 0
\(395\) 832.619i 0.106060i
\(396\) 0 0
\(397\) 3301.41i 0.417363i 0.977984 + 0.208681i \(0.0669172\pi\)
−0.977984 + 0.208681i \(0.933083\pi\)
\(398\) 0 0
\(399\) −889.620 + 3088.49i −0.111621 + 0.387513i
\(400\) 0 0
\(401\) 7988.67i 0.994850i −0.867507 0.497425i \(-0.834279\pi\)
0.867507 0.497425i \(-0.165721\pi\)
\(402\) 0 0
\(403\) 2556.03 0.315942
\(404\) 0 0
\(405\) −1582.13 3283.73i −0.194115 0.402889i
\(406\) 0 0
\(407\) 1345.29 0.163842
\(408\) 0 0
\(409\) 1717.45 0.207634 0.103817 0.994596i \(-0.466894\pi\)
0.103817 + 0.994596i \(0.466894\pi\)
\(410\) 0 0
\(411\) −3409.33 + 11836.2i −0.409172 + 1.42052i
\(412\) 0 0
\(413\) −1301.43 −0.155058
\(414\) 0 0
\(415\) 998.086i 0.118058i
\(416\) 0 0
\(417\) −11464.3 3302.23i −1.34631 0.387796i
\(418\) 0 0
\(419\) 7851.09i 0.915395i −0.889108 0.457698i \(-0.848674\pi\)
0.889108 0.457698i \(-0.151326\pi\)
\(420\) 0 0
\(421\) 5132.10i 0.594117i 0.954859 + 0.297059i \(0.0960057\pi\)
−0.954859 + 0.297059i \(0.903994\pi\)
\(422\) 0 0
\(423\) 1965.10 + 1234.49i 0.225878 + 0.141899i
\(424\) 0 0
\(425\) 2086.60i 0.238153i
\(426\) 0 0
\(427\) −2499.32 −0.283257
\(428\) 0 0
\(429\) 1450.29 + 417.746i 0.163218 + 0.0470139i
\(430\) 0 0
\(431\) −7312.72 −0.817265 −0.408633 0.912699i \(-0.633994\pi\)
−0.408633 + 0.912699i \(0.633994\pi\)
\(432\) 0 0
\(433\) −4271.64 −0.474092 −0.237046 0.971498i \(-0.576179\pi\)
−0.237046 + 0.971498i \(0.576179\pi\)
\(434\) 0 0
\(435\) −4809.92 1385.47i −0.530156 0.152708i
\(436\) 0 0
\(437\) −1785.01 −0.195397
\(438\) 0 0
\(439\) 3462.96i 0.376487i −0.982122 0.188244i \(-0.939721\pi\)
0.982122 0.188244i \(-0.0602794\pi\)
\(440\) 0 0
\(441\) 1852.48 + 1163.74i 0.200030 + 0.125661i
\(442\) 0 0
\(443\) 8898.08i 0.954312i −0.878819 0.477156i \(-0.841668\pi\)
0.878819 0.477156i \(-0.158332\pi\)
\(444\) 0 0
\(445\) 6729.73i 0.716898i
\(446\) 0 0
\(447\) −12803.1 3687.86i −1.35473 0.390223i
\(448\) 0 0
\(449\) 14845.1i 1.56032i −0.625581 0.780159i \(-0.715138\pi\)
0.625581 0.780159i \(-0.284862\pi\)
\(450\) 0 0
\(451\) 709.049 0.0740306
\(452\) 0 0
\(453\) −2826.57 + 9812.99i −0.293165 + 1.01778i
\(454\) 0 0
\(455\) 1555.91 0.160313
\(456\) 0 0
\(457\) 6834.39 0.699561 0.349780 0.936832i \(-0.386256\pi\)
0.349780 + 0.936832i \(0.386256\pi\)
\(458\) 0 0
\(459\) −8730.14 + 7803.96i −0.887773 + 0.793589i
\(460\) 0 0
\(461\) −18191.3 −1.83786 −0.918932 0.394417i \(-0.870947\pi\)
−0.918932 + 0.394417i \(0.870947\pi\)
\(462\) 0 0
\(463\) 13035.5i 1.30844i 0.756303 + 0.654222i \(0.227004\pi\)
−0.756303 + 0.654222i \(0.772996\pi\)
\(464\) 0 0
\(465\) 956.053 3319.12i 0.0953460 0.331012i
\(466\) 0 0
\(467\) 16137.5i 1.59904i −0.600637 0.799522i \(-0.705086\pi\)
0.600637 0.799522i \(-0.294914\pi\)
\(468\) 0 0
\(469\) 14437.0i 1.42140i
\(470\) 0 0
\(471\) 885.749 3075.05i 0.0866521 0.300830i
\(472\) 0 0
\(473\) 121.176i 0.0117795i
\(474\) 0 0
\(475\) −955.394 −0.0922873
\(476\) 0 0
\(477\) 8518.29 + 5351.27i 0.817664 + 0.513664i
\(478\) 0 0
\(479\) −11024.0 −1.05157 −0.525783 0.850619i \(-0.676228\pi\)
−0.525783 + 0.850619i \(0.676228\pi\)
\(480\) 0 0
\(481\) −1712.02 −0.162290
\(482\) 0 0
\(483\) 1087.33 3774.86i 0.102433 0.355615i
\(484\) 0 0
\(485\) −6200.52 −0.580517
\(486\) 0 0
\(487\) 7808.76i 0.726589i 0.931674 + 0.363294i \(0.118348\pi\)
−0.931674 + 0.363294i \(0.881652\pi\)
\(488\) 0 0
\(489\) 522.829 + 150.598i 0.0483500 + 0.0139269i
\(490\) 0 0
\(491\) 5622.56i 0.516787i −0.966040 0.258393i \(-0.916807\pi\)
0.966040 0.258393i \(-0.0831931\pi\)
\(492\) 0 0
\(493\) 16080.3i 1.46901i
\(494\) 0 0
\(495\) 1084.93 1727.01i 0.0985127 0.156815i
\(496\) 0 0
\(497\) 16379.4i 1.47830i
\(498\) 0 0
\(499\) −17202.7 −1.54329 −0.771643 0.636056i \(-0.780565\pi\)
−0.771643 + 0.636056i \(0.780565\pi\)
\(500\) 0 0
\(501\) 10108.0 + 2911.55i 0.901381 + 0.259637i
\(502\) 0 0
\(503\) −10628.9 −0.942189 −0.471095 0.882083i \(-0.656141\pi\)
−0.471095 + 0.882083i \(0.656141\pi\)
\(504\) 0 0
\(505\) 5236.57 0.461434
\(506\) 0 0
\(507\) 9124.29 + 2628.20i 0.799259 + 0.230222i
\(508\) 0 0
\(509\) −1796.29 −0.156422 −0.0782112 0.996937i \(-0.524921\pi\)
−0.0782112 + 0.996937i \(0.524921\pi\)
\(510\) 0 0
\(511\) 1532.36i 0.132657i
\(512\) 0 0
\(513\) −3573.20 3997.27i −0.307526 0.344023i
\(514\) 0 0
\(515\) 5896.69i 0.504542i
\(516\) 0 0
\(517\) 1298.51i 0.110462i
\(518\) 0 0
\(519\) −14929.7 4300.41i −1.26270 0.363713i
\(520\) 0 0
\(521\) 7397.00i 0.622013i 0.950408 + 0.311006i \(0.100666\pi\)
−0.950408 + 0.311006i \(0.899334\pi\)
\(522\) 0 0
\(523\) 14977.7 1.25225 0.626126 0.779722i \(-0.284640\pi\)
0.626126 + 0.779722i \(0.284640\pi\)
\(524\) 0 0
\(525\) 581.972 2020.43i 0.0483797 0.167959i
\(526\) 0 0
\(527\) −11096.3 −0.917199
\(528\) 0 0
\(529\) −9985.30 −0.820687
\(530\) 0 0
\(531\) 1154.85 1838.32i 0.0943810 0.150238i
\(532\) 0 0
\(533\) −902.337 −0.0733294
\(534\) 0 0
\(535\) 9407.24i 0.760206i
\(536\) 0 0
\(537\) −810.443 + 2813.61i −0.0651270 + 0.226101i
\(538\) 0 0
\(539\) 1224.09i 0.0978210i
\(540\) 0 0
\(541\) 2468.07i 0.196138i 0.995180 + 0.0980691i \(0.0312666\pi\)
−0.995180 + 0.0980691i \(0.968733\pi\)
\(542\) 0 0
\(543\) −2421.89 + 8408.04i −0.191405 + 0.664500i
\(544\) 0 0
\(545\) 5933.70i 0.466371i
\(546\) 0 0
\(547\) 19532.2 1.52676 0.763380 0.645950i \(-0.223538\pi\)
0.763380 + 0.645950i \(0.223538\pi\)
\(548\) 0 0
\(549\) 2217.83 3530.40i 0.172413 0.274452i
\(550\) 0 0
\(551\) −7362.69 −0.569258
\(552\) 0 0
\(553\) 2695.29 0.207261
\(554\) 0 0
\(555\) −640.362 + 2223.14i −0.0489763 + 0.170031i
\(556\) 0 0
\(557\) −18683.3 −1.42125 −0.710625 0.703571i \(-0.751588\pi\)
−0.710625 + 0.703571i \(0.751588\pi\)
\(558\) 0 0
\(559\) 154.209i 0.0116679i
\(560\) 0 0
\(561\) −6296.04 1813.54i −0.473831 0.136484i
\(562\) 0 0
\(563\) 8115.47i 0.607507i −0.952751 0.303753i \(-0.901760\pi\)
0.952751 0.303753i \(-0.0982399\pi\)
\(564\) 0 0
\(565\) 8454.88i 0.629557i
\(566\) 0 0
\(567\) 10629.9 5121.54i 0.787323 0.379338i
\(568\) 0 0
\(569\) 18015.5i 1.32732i 0.748032 + 0.663662i \(0.230999\pi\)
−0.748032 + 0.663662i \(0.769001\pi\)
\(570\) 0 0
\(571\) 7359.90 0.539408 0.269704 0.962943i \(-0.413074\pi\)
0.269704 + 0.962943i \(0.413074\pi\)
\(572\) 0 0
\(573\) 23420.7 + 6746.19i 1.70753 + 0.491843i
\(574\) 0 0
\(575\) 1167.72 0.0846907
\(576\) 0 0
\(577\) −8867.74 −0.639807 −0.319904 0.947450i \(-0.603651\pi\)
−0.319904 + 0.947450i \(0.603651\pi\)
\(578\) 0 0
\(579\) −17967.5 5175.42i −1.28964 0.371473i
\(580\) 0 0
\(581\) 3230.93 0.230708
\(582\) 0 0
\(583\) 5628.78i 0.399863i
\(584\) 0 0
\(585\) −1380.68 + 2197.80i −0.0975795 + 0.155329i
\(586\) 0 0
\(587\) 12971.5i 0.912082i 0.889959 + 0.456041i \(0.150733\pi\)
−0.889959 + 0.456041i \(0.849267\pi\)
\(588\) 0 0
\(589\) 5080.68i 0.355426i
\(590\) 0 0
\(591\) −12139.8 3496.81i −0.844952 0.243383i
\(592\) 0 0
\(593\) 12494.6i 0.865248i −0.901575 0.432624i \(-0.857588\pi\)
0.901575 0.432624i \(-0.142412\pi\)
\(594\) 0 0
\(595\) −6754.60 −0.465398
\(596\) 0 0
\(597\) 4331.68 15038.3i 0.296958 1.03095i
\(598\) 0 0
\(599\) 12126.1 0.827146 0.413573 0.910471i \(-0.364281\pi\)
0.413573 + 0.910471i \(0.364281\pi\)
\(600\) 0 0
\(601\) −2638.14 −0.179055 −0.0895276 0.995984i \(-0.528536\pi\)
−0.0895276 + 0.995984i \(0.528536\pi\)
\(602\) 0 0
\(603\) 20392.9 + 12811.0i 1.37722 + 0.865181i
\(604\) 0 0
\(605\) −5513.81 −0.370526
\(606\) 0 0
\(607\) 6958.98i 0.465331i −0.972557 0.232666i \(-0.925255\pi\)
0.972557 0.232666i \(-0.0747448\pi\)
\(608\) 0 0
\(609\) 4484.93 15570.3i 0.298421 1.03603i
\(610\) 0 0
\(611\) 1652.49i 0.109415i
\(612\) 0 0
\(613\) 15348.6i 1.01129i 0.862741 + 0.505646i \(0.168746\pi\)
−0.862741 + 0.505646i \(0.831254\pi\)
\(614\) 0 0
\(615\) −337.509 + 1171.73i −0.0221295 + 0.0768270i
\(616\) 0 0
\(617\) 25707.3i 1.67737i 0.544615 + 0.838686i \(0.316676\pi\)
−0.544615 + 0.838686i \(0.683324\pi\)
\(618\) 0 0
\(619\) −27355.0 −1.77624 −0.888118 0.459615i \(-0.847987\pi\)
−0.888118 + 0.459615i \(0.847987\pi\)
\(620\) 0 0
\(621\) 4367.29 + 4885.61i 0.282212 + 0.315705i
\(622\) 0 0
\(623\) 21785.0 1.40096
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 830.364 2882.77i 0.0528892 0.183615i
\(628\) 0 0
\(629\) 7432.30 0.471137
\(630\) 0 0
\(631\) 20107.2i 1.26855i −0.773108 0.634275i \(-0.781299\pi\)
0.773108 0.634275i \(-0.218701\pi\)
\(632\) 0 0
\(633\) 19124.6 + 5508.71i 1.20084 + 0.345895i
\(634\) 0 0
\(635\) 1479.82i 0.0924802i
\(636\) 0 0
\(637\) 1557.78i 0.0968943i
\(638\) 0 0
\(639\) 23136.6 + 14534.6i 1.43234 + 0.899813i
\(640\) 0 0
\(641\) 7562.85i 0.466013i −0.972475 0.233007i \(-0.925144\pi\)
0.972475 0.233007i \(-0.0748564\pi\)
\(642\) 0 0
\(643\) 6839.33 0.419466 0.209733 0.977759i \(-0.432740\pi\)
0.209733 + 0.977759i \(0.432740\pi\)
\(644\) 0 0
\(645\) 200.248 + 57.6802i 0.0122244 + 0.00352117i
\(646\) 0 0
\(647\) −2348.57 −0.142708 −0.0713539 0.997451i \(-0.522732\pi\)
−0.0713539 + 0.997451i \(0.522732\pi\)
\(648\) 0 0
\(649\) 1214.74 0.0734711
\(650\) 0 0
\(651\) 10744.4 + 3094.86i 0.646862 + 0.186324i
\(652\) 0 0
\(653\) 3786.26 0.226903 0.113452 0.993544i \(-0.463809\pi\)
0.113452 + 0.993544i \(0.463809\pi\)
\(654\) 0 0
\(655\) 1652.55i 0.0985809i
\(656\) 0 0
\(657\) −2164.52 1359.77i −0.128533 0.0807456i
\(658\) 0 0
\(659\) 13627.8i 0.805562i 0.915296 + 0.402781i \(0.131956\pi\)
−0.915296 + 0.402781i \(0.868044\pi\)
\(660\) 0 0
\(661\) 15895.8i 0.935363i 0.883897 + 0.467681i \(0.154911\pi\)
−0.883897 + 0.467681i \(0.845089\pi\)
\(662\) 0 0
\(663\) 8012.35 + 2307.91i 0.469342 + 0.135191i
\(664\) 0 0
\(665\) 3092.73i 0.180347i
\(666\) 0 0
\(667\) 8998.94 0.522399
\(668\) 0 0
\(669\) 6190.33 21490.9i 0.357746 1.24198i
\(670\) 0 0
\(671\) 2332.85 0.134215
\(672\) 0 0
\(673\) 31175.0 1.78560 0.892800 0.450453i \(-0.148738\pi\)
0.892800 + 0.450453i \(0.148738\pi\)
\(674\) 0 0
\(675\) 2337.52 + 2614.94i 0.133290 + 0.149110i
\(676\) 0 0
\(677\) −6503.22 −0.369186 −0.184593 0.982815i \(-0.559097\pi\)
−0.184593 + 0.982815i \(0.559097\pi\)
\(678\) 0 0
\(679\) 20071.9i 1.13444i
\(680\) 0 0
\(681\) −4258.55 + 14784.4i −0.239630 + 0.831923i
\(682\) 0 0
\(683\) 8888.28i 0.497951i −0.968510 0.248975i \(-0.919906\pi\)
0.968510 0.248975i \(-0.0800939\pi\)
\(684\) 0 0
\(685\) 11852.4i 0.661106i
\(686\) 0 0
\(687\) 8761.77 30418.2i 0.486583 1.68927i
\(688\) 0 0
\(689\) 7163.20i 0.396076i
\(690\) 0 0
\(691\) 16298.8 0.897301 0.448651 0.893707i \(-0.351905\pi\)
0.448651 + 0.893707i \(0.351905\pi\)
\(692\) 0 0
\(693\) 5590.55 + 3512.04i 0.306447 + 0.192513i
\(694\) 0 0
\(695\) 11480.1 0.626567
\(696\) 0 0
\(697\) 3917.26 0.212879
\(698\) 0 0
\(699\) −8255.14 + 28659.3i −0.446692 + 1.55078i
\(700\) 0 0
\(701\) 24267.7 1.30753 0.653764 0.756698i \(-0.273189\pi\)
0.653764 + 0.756698i \(0.273189\pi\)
\(702\) 0 0
\(703\) 3403.03i 0.182571i
\(704\) 0 0
\(705\) −2145.84 618.095i −0.114634 0.0330196i
\(706\) 0 0
\(707\) 16951.4i 0.901731i
\(708\) 0 0
\(709\) 36767.6i 1.94758i −0.227447 0.973790i \(-0.573038\pi\)
0.227447 0.973790i \(-0.426962\pi\)
\(710\) 0 0
\(711\) −2391.73 + 3807.22i −0.126156 + 0.200818i
\(712\) 0 0
\(713\) 6209.79i 0.326169i
\(714\) 0 0
\(715\) −1452.28 −0.0759610
\(716\) 0 0
\(717\) 1926.04 + 554.785i 0.100320 + 0.0288965i
\(718\) 0 0
\(719\) 14128.1 0.732806 0.366403 0.930456i \(-0.380589\pi\)
0.366403 + 0.930456i \(0.380589\pi\)
\(720\) 0 0
\(721\) −19088.3 −0.985972
\(722\) 0 0
\(723\) 4167.00 + 1200.28i 0.214347 + 0.0617412i
\(724\) 0 0
\(725\) 4816.53 0.246733
\(726\) 0 0
\(727\) 30461.0i 1.55397i 0.629518 + 0.776986i \(0.283252\pi\)
−0.629518 + 0.776986i \(0.716748\pi\)
\(728\) 0 0
\(729\) −2198.25 + 19559.9i −0.111683 + 0.993744i
\(730\) 0 0
\(731\) 669.460i 0.0338726i
\(732\) 0 0
\(733\) 15628.1i 0.787498i 0.919218 + 0.393749i \(0.128822\pi\)
−0.919218 + 0.393749i \(0.871178\pi\)
\(734\) 0 0
\(735\) −2022.86 582.671i −0.101516 0.0292410i
\(736\) 0 0
\(737\) 13475.4i 0.673502i
\(738\) 0 0
\(739\) −27037.9 −1.34588 −0.672940 0.739697i \(-0.734969\pi\)
−0.672940 + 0.739697i \(0.734969\pi\)
\(740\) 0 0
\(741\) −1056.72 + 3668.62i −0.0523882 + 0.181876i
\(742\) 0 0
\(743\) 1697.21 0.0838017 0.0419009 0.999122i \(-0.486659\pi\)
0.0419009 + 0.999122i \(0.486659\pi\)
\(744\) 0 0
\(745\) 12820.7 0.630489
\(746\) 0 0
\(747\) −2867.04 + 4563.83i −0.140428 + 0.223537i
\(748\) 0 0
\(749\) −30452.4 −1.48559
\(750\) 0 0
\(751\) 29710.1i 1.44359i −0.692107 0.721795i \(-0.743317\pi\)
0.692107 0.721795i \(-0.256683\pi\)
\(752\) 0 0
\(753\) 4585.90 15920.8i 0.221938 0.770501i
\(754\) 0 0
\(755\) 9826.47i 0.473671i
\(756\) 0 0
\(757\) 24486.7i 1.17567i −0.808980 0.587836i \(-0.799980\pi\)
0.808980 0.587836i \(-0.200020\pi\)
\(758\) 0 0
\(759\) −1014.90 + 3523.42i −0.0485357 + 0.168501i
\(760\) 0 0
\(761\) 27354.7i 1.30303i 0.758635 + 0.651515i \(0.225866\pi\)
−0.758635 + 0.651515i \(0.774134\pi\)
\(762\) 0 0
\(763\) 19208.1 0.911378
\(764\) 0 0
\(765\) 5993.86 9541.17i 0.283279 0.450930i
\(766\) 0 0
\(767\) −1545.88 −0.0727751
\(768\) 0 0
\(769\) −19052.5 −0.893432 −0.446716 0.894676i \(-0.647407\pi\)
−0.446716 + 0.894676i \(0.647407\pi\)
\(770\) 0 0
\(771\) 5610.37 19477.5i 0.262066 0.909811i
\(772\) 0 0
\(773\) 4302.43 0.200191 0.100095 0.994978i \(-0.468085\pi\)
0.100095 + 0.994978i \(0.468085\pi\)
\(774\) 0 0
\(775\) 3323.68i 0.154052i
\(776\) 0 0
\(777\) −7196.59 2072.93i −0.332273 0.0957092i
\(778\) 0 0
\(779\) 1793.60i 0.0824933i
\(780\) 0 0
\(781\) 15288.4i 0.700462i
\(782\) 0 0
\(783\) 18013.9 + 20151.9i 0.822178 + 0.919755i
\(784\) 0 0
\(785\) 3079.27i 0.140005i
\(786\) 0 0
\(787\) −18936.4 −0.857701 −0.428851 0.903375i \(-0.641081\pi\)
−0.428851 + 0.903375i \(0.641081\pi\)
\(788\) 0 0
\(789\) −18214.0 5246.42i −0.821844 0.236727i
\(790\) 0 0
\(791\) 27369.5 1.23028
\(792\) 0 0
\(793\) −2968.78 −0.132944
\(794\) 0 0
\(795\) −9301.75 2679.31i −0.414967 0.119529i
\(796\) 0 0
\(797\) 26499.5 1.17774 0.588871 0.808227i \(-0.299572\pi\)
0.588871 + 0.808227i \(0.299572\pi\)
\(798\) 0 0
\(799\) 7173.87i 0.317638i
\(800\) 0 0
\(801\) −19331.4 + 30772.2i −0.852737 + 1.35741i
\(802\) 0 0
\(803\) 1430.29i 0.0628566i
\(804\) 0 0
\(805\) 3780.05i 0.165502i
\(806\) 0 0
\(807\) −4179.45 1203.86i −0.182309 0.0525131i
\(808\) 0 0
\(809\) 17982.5i 0.781496i 0.920498 + 0.390748i \(0.127784\pi\)
−0.920498 + 0.390748i \(0.872216\pi\)
\(810\) 0 0
\(811\) 25868.5 1.12005 0.560027 0.828474i \(-0.310791\pi\)
0.560027 + 0.828474i \(0.310791\pi\)
\(812\) 0 0
\(813\) −9458.33 + 32836.4i −0.408018 + 1.41651i
\(814\) 0 0
\(815\) −523.547 −0.0225019
\(816\) 0 0
\(817\) 306.526 0.0131260
\(818\) 0 0
\(819\) −7114.55 4469.43i −0.303544 0.190689i
\(820\) 0 0
\(821\) 9280.65 0.394515 0.197258 0.980352i \(-0.436796\pi\)
0.197258 + 0.980352i \(0.436796\pi\)
\(822\) 0 0
\(823\) 260.648i 0.0110396i −0.999985 0.00551981i \(-0.998243\pi\)
0.999985 0.00551981i \(-0.00175702\pi\)
\(824\) 0 0
\(825\) −543.208 + 1885.85i −0.0229237 + 0.0795842i
\(826\) 0 0
\(827\) 29070.2i 1.22233i −0.791502 0.611166i \(-0.790701\pi\)
0.791502 0.611166i \(-0.209299\pi\)
\(828\) 0 0
\(829\) 33065.3i 1.38529i −0.721279 0.692644i \(-0.756446\pi\)
0.721279 0.692644i \(-0.243554\pi\)
\(830\) 0 0
\(831\) −3472.05 + 12053.9i −0.144939 + 0.503183i
\(832\) 0 0
\(833\) 6762.72i 0.281290i
\(834\) 0 0
\(835\) −10121.9 −0.419499
\(836\) 0 0
\(837\) −13905.9 + 12430.7i −0.574265 + 0.513341i
\(838\) 0 0
\(839\) −32192.6 −1.32469 −0.662344 0.749200i \(-0.730438\pi\)
−0.662344 + 0.749200i \(0.730438\pi\)
\(840\) 0 0
\(841\) 12729.3 0.521927
\(842\) 0 0
\(843\) −706.335 + 2452.18i −0.0288582 + 0.100187i
\(844\) 0 0
\(845\) −9136.83 −0.371972
\(846\) 0 0
\(847\) 17848.9i 0.724080i
\(848\) 0 0
\(849\) 46293.1 + 13334.4i 1.87135 + 0.539030i
\(850\) 0 0
\(851\) 4159.30i 0.167543i
\(852\) 0 0
\(853\) 6772.88i 0.271863i 0.990718 + 0.135931i \(0.0434027\pi\)
−0.990718 + 0.135931i \(0.956597\pi\)
\(854\) 0 0
\(855\) 4368.62 + 2744.41i 0.174741 + 0.109774i
\(856\) 0 0
\(857\) 20970.9i 0.835883i 0.908474 + 0.417942i \(0.137248\pi\)
−0.908474 + 0.417942i \(0.862752\pi\)
\(858\) 0 0
\(859\) −15056.3 −0.598039 −0.299019 0.954247i \(-0.596660\pi\)
−0.299019 + 0.954247i \(0.596660\pi\)
\(860\) 0 0
\(861\) −3793.03 1092.56i −0.150135 0.0432454i
\(862\) 0 0
\(863\) 31506.7 1.24276 0.621378 0.783511i \(-0.286573\pi\)
0.621378 + 0.783511i \(0.286573\pi\)
\(864\) 0 0
\(865\) 14950.2 0.587656
\(866\) 0 0
\(867\) −10252.3 2953.10i −0.401598 0.115678i
\(868\) 0 0
\(869\) −2515.77 −0.0982066
\(870\) 0 0
\(871\) 17148.8i 0.667122i
\(872\) 0 0
\(873\) 28352.4 + 17811.2i 1.09918 + 0.690514i
\(874\) 0 0
\(875\) 2023.20i 0.0781677i
\(876\) 0 0
\(877\) 27344.6i 1.05286i 0.850218 + 0.526431i \(0.176470\pi\)
−0.850218 + 0.526431i \(0.823530\pi\)
\(878\) 0 0
\(879\) 3431.46 + 988.412i 0.131673 + 0.0379276i
\(880\) 0 0
\(881\) 9957.73i 0.380799i 0.981707 + 0.190400i \(0.0609784\pi\)
−0.981707 + 0.190400i \(0.939022\pi\)
\(882\) 0 0
\(883\) 41428.0 1.57889 0.789447 0.613819i \(-0.210368\pi\)
0.789447 + 0.613819i \(0.210368\pi\)
\(884\) 0 0
\(885\) −578.219 + 2007.40i −0.0219623 + 0.0762463i
\(886\) 0 0
\(887\) 32204.1 1.21906 0.609531 0.792762i \(-0.291358\pi\)
0.609531 + 0.792762i \(0.291358\pi\)
\(888\) 0 0
\(889\) −4790.37 −0.180724
\(890\) 0 0
\(891\) −9921.82 + 4780.41i −0.373057 + 0.179742i
\(892\) 0 0
\(893\) −3284.70 −0.123089
\(894\) 0 0
\(895\) 2817.47i 0.105227i
\(896\) 0 0
\(897\) 1291.56 4483.92i 0.0480759 0.166905i
\(898\) 0 0
\(899\) 25613.8i 0.950241i
\(900\) 0 0
\(901\) 31097.2i 1.14983i
\(902\) 0 0
\(903\) −186.718 + 648.228i −0.00688105 + 0.0238889i
\(904\) 0 0
\(905\) 8419.59i 0.309256i
\(906\) 0 0
\(907\) −13255.9 −0.485288 −0.242644 0.970115i \(-0.578015\pi\)
−0.242644 + 0.970115i \(0.578015\pi\)
\(908\) 0 0
\(909\) −23944.6 15042.3i −0.873700 0.548867i
\(910\) 0 0
\(911\) 15092.0 0.548868 0.274434 0.961606i \(-0.411509\pi\)
0.274434 + 0.961606i \(0.411509\pi\)
\(912\) 0 0
\(913\) −3015.72 −0.109316
\(914\) 0 0
\(915\) −1110.44 + 3855.11i −0.0401202 + 0.139285i
\(916\) 0 0
\(917\) −5349.51 −0.192646
\(918\) 0 0
\(919\) 47179.7i 1.69349i −0.532000 0.846744i \(-0.678559\pi\)
0.532000 0.846744i \(-0.321441\pi\)
\(920\) 0 0
\(921\) −6488.29 1868.91i −0.232135 0.0668651i
\(922\) 0 0
\(923\) 19456.0i 0.693826i
\(924\) 0 0
\(925\) 2226.19i 0.0791317i
\(926\) 0 0
\(927\) 16938.5 26963.1i 0.600143 0.955322i
\(928\) 0 0
\(929\) 33295.7i 1.17589i 0.808902 + 0.587943i \(0.200062\pi\)
−0.808902 + 0.587943i \(0.799938\pi\)
\(930\) 0 0
\(931\) −3096.45 −0.109003
\(932\) 0 0
\(933\) −26104.0 7519.10i −0.915977 0.263842i
\(934\) 0 0
\(935\) 6304.69 0.220519
\(936\) 0 0
\(937\) 6092.97 0.212432 0.106216 0.994343i \(-0.466127\pi\)
0.106216 + 0.994343i \(0.466127\pi\)
\(938\) 0 0
\(939\) −11288.1 3251.47i −0.392304 0.113001i
\(940\) 0 0
\(941\) −1728.53 −0.0598814 −0.0299407 0.999552i \(-0.509532\pi\)
−0.0299407 + 0.999552i \(0.509532\pi\)
\(942\) 0 0
\(943\) 2192.20i 0.0757029i
\(944\) 0 0
\(945\) −8464.88 + 7566.84i −0.291389 + 0.260475i
\(946\) 0 0
\(947\) 4971.37i 0.170589i 0.996356 + 0.0852945i \(0.0271831\pi\)
−0.996356 + 0.0852945i \(0.972817\pi\)
\(948\) 0 0
\(949\) 1820.19i 0.0622612i
\(950\) 0 0
\(951\) 1814.65 + 522.699i 0.0618760 + 0.0178230i
\(952\) 0 0
\(953\) 13671.0i 0.464688i 0.972634 + 0.232344i \(0.0746395\pi\)
−0.972634 + 0.232344i \(0.925361\pi\)
\(954\) 0 0
\(955\) −23452.9 −0.794677
\(956\) 0 0
\(957\) −4186.20 + 14533.2i −0.141401 + 0.490901i
\(958\) 0 0
\(959\) 38367.7 1.29193
\(960\) 0 0
\(961\) 12116.0 0.406701
\(962\) 0 0
\(963\) 27022.7 43015.4i 0.904251 1.43941i
\(964\) 0 0
\(965\) 17992.2 0.600195
\(966\) 0 0
\(967\) 39411.1i 1.31063i 0.755357 + 0.655314i \(0.227464\pi\)
−0.755357 + 0.655314i \(0.772536\pi\)
\(968\) 0 0
\(969\) 4587.49 15926.4i 0.152086 0.527996i
\(970\) 0 0
\(971\) 42243.7i 1.39615i −0.716023 0.698076i \(-0.754040\pi\)
0.716023 0.698076i \(-0.245960\pi\)
\(972\) 0 0
\(973\) 37162.4i 1.22443i
\(974\) 0 0
\(975\) 691.287 2399.94i 0.0227066 0.0788303i
\(976\) 0 0
\(977\) 4702.62i 0.153992i 0.997031 + 0.0769960i \(0.0245329\pi\)
−0.997031 + 0.0769960i \(0.975467\pi\)
\(978\) 0 0
\(979\) −20333.9 −0.663815
\(980\) 0 0
\(981\) −17044.8 + 27132.3i −0.554739 + 0.883047i
\(982\) 0 0
\(983\) 41600.6 1.34980 0.674900 0.737910i \(-0.264187\pi\)
0.674900 + 0.737910i \(0.264187\pi\)
\(984\) 0 0
\(985\) 12156.5 0.393238
\(986\) 0 0
\(987\) 2000.85 6946.35i 0.0645267 0.224017i
\(988\) 0 0
\(989\) −374.647 −0.0120456
\(990\) 0 0
\(991\) 1313.39i 0.0421003i −0.999778 0.0210501i \(-0.993299\pi\)
0.999778 0.0210501i \(-0.00670096\pi\)
\(992\) 0 0
\(993\) −3001.46 864.553i −0.0959200 0.0276292i
\(994\) 0 0
\(995\) 15058.9i 0.479799i
\(996\) 0 0
\(997\) 34690.2i 1.10195i 0.834520 + 0.550977i \(0.185745\pi\)
−0.834520 + 0.550977i \(0.814255\pi\)
\(998\) 0 0
\(999\) 9314.17 8326.03i 0.294982 0.263688i
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 960.4.b.a.671.16 yes 16
3.2 odd 2 960.4.b.b.671.15 yes 16
4.3 odd 2 inner 960.4.b.a.671.1 16
8.3 odd 2 960.4.b.b.671.16 yes 16
8.5 even 2 960.4.b.b.671.1 yes 16
12.11 even 2 960.4.b.b.671.2 yes 16
24.5 odd 2 inner 960.4.b.a.671.2 yes 16
24.11 even 2 inner 960.4.b.a.671.15 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.4.b.a.671.1 16 4.3 odd 2 inner
960.4.b.a.671.2 yes 16 24.5 odd 2 inner
960.4.b.a.671.15 yes 16 24.11 even 2 inner
960.4.b.a.671.16 yes 16 1.1 even 1 trivial
960.4.b.b.671.1 yes 16 8.5 even 2
960.4.b.b.671.2 yes 16 12.11 even 2
960.4.b.b.671.15 yes 16 3.2 odd 2
960.4.b.b.671.16 yes 16 8.3 odd 2