Properties

Label 64.4.e.a.17.4
Level $64$
Weight $4$
Character 64.17
Analytic conductor $3.776$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,4,Mod(17,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 64.e (of order \(4\), degree \(2\), not 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: \(3.77612224037\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.4
Root \(-1.56339 - 1.24732i\) of defining polynomial
Character \(\chi\) \(=\) 64.17
Dual form 64.4.e.a.49.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.27139 + 3.27139i) q^{3} +(-12.6449 + 12.6449i) q^{5} +13.8754i q^{7} -5.59607i q^{9} +O(q^{10})\) \(q+(3.27139 + 3.27139i) q^{3} +(-12.6449 + 12.6449i) q^{5} +13.8754i q^{7} -5.59607i q^{9} +(-1.54694 + 1.54694i) q^{11} +(32.7875 + 32.7875i) q^{13} -82.7326 q^{15} +18.6531 q^{17} +(86.4042 + 86.4042i) q^{19} +(-45.3917 + 45.3917i) q^{21} -134.006i q^{23} -194.786i q^{25} +(106.634 - 106.634i) q^{27} +(-59.7949 - 59.7949i) q^{29} +31.5391 q^{31} -10.1213 q^{33} +(-175.453 - 175.453i) q^{35} +(89.1866 - 89.1866i) q^{37} +214.521i q^{39} +210.504i q^{41} +(-119.402 + 119.402i) q^{43} +(70.7617 + 70.7617i) q^{45} +182.902 q^{47} +150.474 q^{49} +(61.0213 + 61.0213i) q^{51} +(-26.1644 + 26.1644i) q^{53} -39.1219i q^{55} +565.323i q^{57} +(-441.584 + 441.584i) q^{59} +(-174.485 - 174.485i) q^{61} +77.6476 q^{63} -829.188 q^{65} +(-91.7562 - 91.7562i) q^{67} +(438.385 - 438.385i) q^{69} +348.360i q^{71} +299.436i q^{73} +(637.222 - 637.222i) q^{75} +(-21.4644 - 21.4644i) q^{77} +943.487 q^{79} +546.590 q^{81} +(-313.272 - 313.272i) q^{83} +(-235.866 + 235.866i) q^{85} -391.224i q^{87} -1412.35i q^{89} +(-454.939 + 454.939i) q^{91} +(103.177 + 103.177i) q^{93} -2185.14 q^{95} +1515.29 q^{97} +(8.65680 + 8.65680i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} - 2 q^{5} - 18 q^{11} - 2 q^{13} + 124 q^{15} - 4 q^{17} + 26 q^{19} + 52 q^{21} - 184 q^{27} - 202 q^{29} - 368 q^{31} - 4 q^{33} - 476 q^{35} - 10 q^{37} + 838 q^{43} + 194 q^{45} + 944 q^{47} + 94 q^{49} + 1500 q^{51} - 378 q^{53} - 1706 q^{59} + 910 q^{61} - 2628 q^{63} - 492 q^{65} - 1942 q^{67} + 580 q^{69} + 2954 q^{75} - 268 q^{77} + 4416 q^{79} + 482 q^{81} + 2562 q^{83} - 12 q^{85} - 3332 q^{91} - 2192 q^{93} - 6900 q^{95} - 4 q^{97} - 4958 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 3.27139 + 3.27139i 0.629578 + 0.629578i 0.947962 0.318384i \(-0.103140\pi\)
−0.318384 + 0.947962i \(0.603140\pi\)
\(4\) 0 0
\(5\) −12.6449 + 12.6449i −1.13099 + 1.13099i −0.140981 + 0.990012i \(0.545026\pi\)
−0.990012 + 0.140981i \(0.954974\pi\)
\(6\) 0 0
\(7\) 13.8754i 0.749200i 0.927187 + 0.374600i \(0.122220\pi\)
−0.927187 + 0.374600i \(0.877780\pi\)
\(8\) 0 0
\(9\) 5.59607i 0.207262i
\(10\) 0 0
\(11\) −1.54694 + 1.54694i −0.0424019 + 0.0424019i −0.727990 0.685588i \(-0.759545\pi\)
0.685588 + 0.727990i \(0.259545\pi\)
\(12\) 0 0
\(13\) 32.7875 + 32.7875i 0.699509 + 0.699509i 0.964304 0.264796i \(-0.0853046\pi\)
−0.264796 + 0.964304i \(0.585305\pi\)
\(14\) 0 0
\(15\) −82.7326 −1.42410
\(16\) 0 0
\(17\) 18.6531 0.266119 0.133060 0.991108i \(-0.457520\pi\)
0.133060 + 0.991108i \(0.457520\pi\)
\(18\) 0 0
\(19\) 86.4042 + 86.4042i 1.04329 + 1.04329i 0.999020 + 0.0442688i \(0.0140958\pi\)
0.0442688 + 0.999020i \(0.485904\pi\)
\(20\) 0 0
\(21\) −45.3917 + 45.3917i −0.471680 + 0.471680i
\(22\) 0 0
\(23\) 134.006i 1.21488i −0.794367 0.607438i \(-0.792197\pi\)
0.794367 0.607438i \(-0.207803\pi\)
\(24\) 0 0
\(25\) 194.786i 1.55829i
\(26\) 0 0
\(27\) 106.634 106.634i 0.760066 0.760066i
\(28\) 0 0
\(29\) −59.7949 59.7949i −0.382884 0.382884i 0.489256 0.872140i \(-0.337268\pi\)
−0.872140 + 0.489256i \(0.837268\pi\)
\(30\) 0 0
\(31\) 31.5391 0.182729 0.0913645 0.995818i \(-0.470877\pi\)
0.0913645 + 0.995818i \(0.470877\pi\)
\(32\) 0 0
\(33\) −10.1213 −0.0533906
\(34\) 0 0
\(35\) −175.453 175.453i −0.847340 0.847340i
\(36\) 0 0
\(37\) 89.1866 89.1866i 0.396275 0.396275i −0.480642 0.876917i \(-0.659596\pi\)
0.876917 + 0.480642i \(0.159596\pi\)
\(38\) 0 0
\(39\) 214.521i 0.880791i
\(40\) 0 0
\(41\) 210.504i 0.801834i 0.916114 + 0.400917i \(0.131308\pi\)
−0.916114 + 0.400917i \(0.868692\pi\)
\(42\) 0 0
\(43\) −119.402 + 119.402i −0.423456 + 0.423456i −0.886392 0.462936i \(-0.846796\pi\)
0.462936 + 0.886392i \(0.346796\pi\)
\(44\) 0 0
\(45\) 70.7617 + 70.7617i 0.234412 + 0.234412i
\(46\) 0 0
\(47\) 182.902 0.567638 0.283819 0.958878i \(-0.408398\pi\)
0.283819 + 0.958878i \(0.408398\pi\)
\(48\) 0 0
\(49\) 150.474 0.438699
\(50\) 0 0
\(51\) 61.0213 + 61.0213i 0.167543 + 0.167543i
\(52\) 0 0
\(53\) −26.1644 + 26.1644i −0.0678104 + 0.0678104i −0.740199 0.672388i \(-0.765269\pi\)
0.672388 + 0.740199i \(0.265269\pi\)
\(54\) 0 0
\(55\) 39.1219i 0.0959125i
\(56\) 0 0
\(57\) 565.323i 1.31366i
\(58\) 0 0
\(59\) −441.584 + 441.584i −0.974395 + 0.974395i −0.999680 0.0252856i \(-0.991950\pi\)
0.0252856 + 0.999680i \(0.491950\pi\)
\(60\) 0 0
\(61\) −174.485 174.485i −0.366238 0.366238i 0.499865 0.866103i \(-0.333383\pi\)
−0.866103 + 0.499865i \(0.833383\pi\)
\(62\) 0 0
\(63\) 77.6476 0.155281
\(64\) 0 0
\(65\) −829.188 −1.58228
\(66\) 0 0
\(67\) −91.7562 91.7562i −0.167311 0.167311i 0.618486 0.785796i \(-0.287747\pi\)
−0.785796 + 0.618486i \(0.787747\pi\)
\(68\) 0 0
\(69\) 438.385 438.385i 0.764860 0.764860i
\(70\) 0 0
\(71\) 348.360i 0.582291i 0.956679 + 0.291146i \(0.0940364\pi\)
−0.956679 + 0.291146i \(0.905964\pi\)
\(72\) 0 0
\(73\) 299.436i 0.480087i 0.970762 + 0.240043i \(0.0771617\pi\)
−0.970762 + 0.240043i \(0.922838\pi\)
\(74\) 0 0
\(75\) 637.222 637.222i 0.981067 0.981067i
\(76\) 0 0
\(77\) −21.4644 21.4644i −0.0317675 0.0317675i
\(78\) 0 0
\(79\) 943.487 1.34368 0.671839 0.740697i \(-0.265505\pi\)
0.671839 + 0.740697i \(0.265505\pi\)
\(80\) 0 0
\(81\) 546.590 0.749781
\(82\) 0 0
\(83\) −313.272 313.272i −0.414290 0.414290i 0.468940 0.883230i \(-0.344636\pi\)
−0.883230 + 0.468940i \(0.844636\pi\)
\(84\) 0 0
\(85\) −235.866 + 235.866i −0.300979 + 0.300979i
\(86\) 0 0
\(87\) 391.224i 0.482111i
\(88\) 0 0
\(89\) 1412.35i 1.68212i −0.540942 0.841060i \(-0.681932\pi\)
0.540942 0.841060i \(-0.318068\pi\)
\(90\) 0 0
\(91\) −454.939 + 454.939i −0.524072 + 0.524072i
\(92\) 0 0
\(93\) 103.177 + 103.177i 0.115042 + 0.115042i
\(94\) 0 0
\(95\) −2185.14 −2.35990
\(96\) 0 0
\(97\) 1515.29 1.58613 0.793063 0.609140i \(-0.208485\pi\)
0.793063 + 0.609140i \(0.208485\pi\)
\(98\) 0 0
\(99\) 8.65680 + 8.65680i 0.00878830 + 0.00878830i
\(100\) 0 0
\(101\) −573.202 + 573.202i −0.564711 + 0.564711i −0.930642 0.365931i \(-0.880751\pi\)
0.365931 + 0.930642i \(0.380751\pi\)
\(102\) 0 0
\(103\) 1021.00i 0.976717i −0.872643 0.488359i \(-0.837596\pi\)
0.872643 0.488359i \(-0.162404\pi\)
\(104\) 0 0
\(105\) 1147.95i 1.06693i
\(106\) 0 0
\(107\) 1240.79 1240.79i 1.12105 1.12105i 0.129462 0.991584i \(-0.458675\pi\)
0.991584 0.129462i \(-0.0413251\pi\)
\(108\) 0 0
\(109\) −108.629 108.629i −0.0954565 0.0954565i 0.657766 0.753222i \(-0.271502\pi\)
−0.753222 + 0.657766i \(0.771502\pi\)
\(110\) 0 0
\(111\) 583.528 0.498973
\(112\) 0 0
\(113\) −1722.22 −1.43374 −0.716870 0.697207i \(-0.754426\pi\)
−0.716870 + 0.697207i \(0.754426\pi\)
\(114\) 0 0
\(115\) 1694.49 + 1694.49i 1.37402 + 1.37402i
\(116\) 0 0
\(117\) 183.481 183.481i 0.144981 0.144981i
\(118\) 0 0
\(119\) 258.818i 0.199377i
\(120\) 0 0
\(121\) 1326.21i 0.996404i
\(122\) 0 0
\(123\) −688.639 + 688.639i −0.504817 + 0.504817i
\(124\) 0 0
\(125\) 882.442 + 882.442i 0.631424 + 0.631424i
\(126\) 0 0
\(127\) −699.127 −0.488484 −0.244242 0.969714i \(-0.578539\pi\)
−0.244242 + 0.969714i \(0.578539\pi\)
\(128\) 0 0
\(129\) −781.219 −0.533198
\(130\) 0 0
\(131\) −197.970 197.970i −0.132036 0.132036i 0.638000 0.770036i \(-0.279762\pi\)
−0.770036 + 0.638000i \(0.779762\pi\)
\(132\) 0 0
\(133\) −1198.89 + 1198.89i −0.781632 + 0.781632i
\(134\) 0 0
\(135\) 2696.76i 1.71926i
\(136\) 0 0
\(137\) 271.386i 0.169242i 0.996413 + 0.0846209i \(0.0269679\pi\)
−0.996413 + 0.0846209i \(0.973032\pi\)
\(138\) 0 0
\(139\) −459.937 + 459.937i −0.280657 + 0.280657i −0.833371 0.552714i \(-0.813592\pi\)
0.552714 + 0.833371i \(0.313592\pi\)
\(140\) 0 0
\(141\) 598.343 + 598.343i 0.357373 + 0.357373i
\(142\) 0 0
\(143\) −101.441 −0.0593210
\(144\) 0 0
\(145\) 1512.20 0.866079
\(146\) 0 0
\(147\) 492.258 + 492.258i 0.276196 + 0.276196i
\(148\) 0 0
\(149\) 605.772 605.772i 0.333066 0.333066i −0.520684 0.853750i \(-0.674323\pi\)
0.853750 + 0.520684i \(0.174323\pi\)
\(150\) 0 0
\(151\) 3534.47i 1.90484i −0.304785 0.952421i \(-0.598585\pi\)
0.304785 0.952421i \(-0.401415\pi\)
\(152\) 0 0
\(153\) 104.384i 0.0551564i
\(154\) 0 0
\(155\) −398.809 + 398.809i −0.206665 + 0.206665i
\(156\) 0 0
\(157\) −1233.54 1233.54i −0.627051 0.627051i 0.320274 0.947325i \(-0.396225\pi\)
−0.947325 + 0.320274i \(0.896225\pi\)
\(158\) 0 0
\(159\) −171.187 −0.0853839
\(160\) 0 0
\(161\) 1859.38 0.910186
\(162\) 0 0
\(163\) −2569.36 2569.36i −1.23465 1.23465i −0.962158 0.272494i \(-0.912152\pi\)
−0.272494 0.962158i \(-0.587848\pi\)
\(164\) 0 0
\(165\) 127.983 127.983i 0.0603845 0.0603845i
\(166\) 0 0
\(167\) 3048.39i 1.41252i 0.707950 + 0.706262i \(0.249620\pi\)
−0.707950 + 0.706262i \(0.750380\pi\)
\(168\) 0 0
\(169\) 46.9618i 0.0213754i
\(170\) 0 0
\(171\) 483.524 483.524i 0.216234 0.216234i
\(172\) 0 0
\(173\) −1522.05 1522.05i −0.668898 0.668898i 0.288563 0.957461i \(-0.406823\pi\)
−0.957461 + 0.288563i \(0.906823\pi\)
\(174\) 0 0
\(175\) 2702.74 1.16747
\(176\) 0 0
\(177\) −2889.18 −1.22692
\(178\) 0 0
\(179\) −302.258 302.258i −0.126211 0.126211i 0.641180 0.767391i \(-0.278445\pi\)
−0.767391 + 0.641180i \(0.778445\pi\)
\(180\) 0 0
\(181\) 1696.94 1696.94i 0.696865 0.696865i −0.266868 0.963733i \(-0.585989\pi\)
0.963733 + 0.266868i \(0.0859889\pi\)
\(182\) 0 0
\(183\) 1141.62i 0.461151i
\(184\) 0 0
\(185\) 2255.51i 0.896370i
\(186\) 0 0
\(187\) −28.8552 + 28.8552i −0.0112840 + 0.0112840i
\(188\) 0 0
\(189\) 1479.59 + 1479.59i 0.569442 + 0.569442i
\(190\) 0 0
\(191\) 4035.31 1.52872 0.764358 0.644792i \(-0.223056\pi\)
0.764358 + 0.644792i \(0.223056\pi\)
\(192\) 0 0
\(193\) −886.172 −0.330508 −0.165254 0.986251i \(-0.552844\pi\)
−0.165254 + 0.986251i \(0.552844\pi\)
\(194\) 0 0
\(195\) −2712.59 2712.59i −0.996169 0.996169i
\(196\) 0 0
\(197\) 3270.12 3270.12i 1.18267 1.18267i 0.203624 0.979049i \(-0.434728\pi\)
0.979049 0.203624i \(-0.0652720\pi\)
\(198\) 0 0
\(199\) 222.513i 0.0792639i 0.999214 + 0.0396319i \(0.0126185\pi\)
−0.999214 + 0.0396319i \(0.987381\pi\)
\(200\) 0 0
\(201\) 600.340i 0.210670i
\(202\) 0 0
\(203\) 829.677 829.677i 0.286857 0.286857i
\(204\) 0 0
\(205\) −2661.80 2661.80i −0.906868 0.906868i
\(206\) 0 0
\(207\) −749.907 −0.251798
\(208\) 0 0
\(209\) −267.325 −0.0884748
\(210\) 0 0
\(211\) 3527.46 + 3527.46i 1.15090 + 1.15090i 0.986372 + 0.164529i \(0.0526105\pi\)
0.164529 + 0.986372i \(0.447390\pi\)
\(212\) 0 0
\(213\) −1139.62 + 1139.62i −0.366598 + 0.366598i
\(214\) 0 0
\(215\) 3019.65i 0.957852i
\(216\) 0 0
\(217\) 437.618i 0.136901i
\(218\) 0 0
\(219\) −979.571 + 979.571i −0.302252 + 0.302252i
\(220\) 0 0
\(221\) 611.587 + 611.587i 0.186153 + 0.186153i
\(222\) 0 0
\(223\) −5841.90 −1.75427 −0.877136 0.480242i \(-0.840549\pi\)
−0.877136 + 0.480242i \(0.840549\pi\)
\(224\) 0 0
\(225\) −1090.04 −0.322975
\(226\) 0 0
\(227\) 1129.54 + 1129.54i 0.330265 + 0.330265i 0.852687 0.522422i \(-0.174971\pi\)
−0.522422 + 0.852687i \(0.674971\pi\)
\(228\) 0 0
\(229\) −2905.40 + 2905.40i −0.838403 + 0.838403i −0.988649 0.150246i \(-0.951994\pi\)
0.150246 + 0.988649i \(0.451994\pi\)
\(230\) 0 0
\(231\) 140.437i 0.0400003i
\(232\) 0 0
\(233\) 734.054i 0.206393i 0.994661 + 0.103196i \(0.0329070\pi\)
−0.994661 + 0.103196i \(0.967093\pi\)
\(234\) 0 0
\(235\) −2312.78 + 2312.78i −0.641995 + 0.641995i
\(236\) 0 0
\(237\) 3086.51 + 3086.51i 0.845951 + 0.845951i
\(238\) 0 0
\(239\) 511.807 0.138519 0.0692595 0.997599i \(-0.477936\pi\)
0.0692595 + 0.997599i \(0.477936\pi\)
\(240\) 0 0
\(241\) 5920.31 1.58241 0.791204 0.611552i \(-0.209455\pi\)
0.791204 + 0.611552i \(0.209455\pi\)
\(242\) 0 0
\(243\) −1091.02 1091.02i −0.288020 0.288020i
\(244\) 0 0
\(245\) −1902.73 + 1902.73i −0.496166 + 0.496166i
\(246\) 0 0
\(247\) 5665.95i 1.45958i
\(248\) 0 0
\(249\) 2049.67i 0.521656i
\(250\) 0 0
\(251\) 309.332 309.332i 0.0777883 0.0777883i −0.667142 0.744930i \(-0.732483\pi\)
0.744930 + 0.667142i \(0.232483\pi\)
\(252\) 0 0
\(253\) 207.300 + 207.300i 0.0515131 + 0.0515131i
\(254\) 0 0
\(255\) −1543.22 −0.378980
\(256\) 0 0
\(257\) 323.723 0.0785730 0.0392865 0.999228i \(-0.487491\pi\)
0.0392865 + 0.999228i \(0.487491\pi\)
\(258\) 0 0
\(259\) 1237.50 + 1237.50i 0.296890 + 0.296890i
\(260\) 0 0
\(261\) −334.617 + 334.617i −0.0793573 + 0.0793573i
\(262\) 0 0
\(263\) 2689.15i 0.630495i 0.949009 + 0.315248i \(0.102088\pi\)
−0.949009 + 0.315248i \(0.897912\pi\)
\(264\) 0 0
\(265\) 661.691i 0.153386i
\(266\) 0 0
\(267\) 4620.34 4620.34i 1.05903 1.05903i
\(268\) 0 0
\(269\) 4703.78 + 4703.78i 1.06615 + 1.06615i 0.997651 + 0.0684995i \(0.0218212\pi\)
0.0684995 + 0.997651i \(0.478179\pi\)
\(270\) 0 0
\(271\) 2018.97 0.452561 0.226280 0.974062i \(-0.427343\pi\)
0.226280 + 0.974062i \(0.427343\pi\)
\(272\) 0 0
\(273\) −2976.56 −0.659889
\(274\) 0 0
\(275\) 301.324 + 301.324i 0.0660745 + 0.0660745i
\(276\) 0 0
\(277\) −3080.60 + 3080.60i −0.668215 + 0.668215i −0.957303 0.289088i \(-0.906648\pi\)
0.289088 + 0.957303i \(0.406648\pi\)
\(278\) 0 0
\(279\) 176.495i 0.0378728i
\(280\) 0 0
\(281\) 3893.51i 0.826575i −0.910601 0.413287i \(-0.864381\pi\)
0.910601 0.413287i \(-0.135619\pi\)
\(282\) 0 0
\(283\) −2026.38 + 2026.38i −0.425639 + 0.425639i −0.887140 0.461501i \(-0.847311\pi\)
0.461501 + 0.887140i \(0.347311\pi\)
\(284\) 0 0
\(285\) −7148.45 7148.45i −1.48575 1.48575i
\(286\) 0 0
\(287\) −2920.82 −0.600734
\(288\) 0 0
\(289\) −4565.06 −0.929180
\(290\) 0 0
\(291\) 4957.09 + 4957.09i 0.998591 + 0.998591i
\(292\) 0 0
\(293\) −1001.68 + 1001.68i −0.199724 + 0.199724i −0.799882 0.600158i \(-0.795104\pi\)
0.600158 + 0.799882i \(0.295104\pi\)
\(294\) 0 0
\(295\) 11167.6i 2.20407i
\(296\) 0 0
\(297\) 329.914i 0.0644565i
\(298\) 0 0
\(299\) 4393.72 4393.72i 0.849817 0.849817i
\(300\) 0 0
\(301\) −1656.75 1656.75i −0.317253 0.317253i
\(302\) 0 0
\(303\) −3750.33 −0.711059
\(304\) 0 0
\(305\) 4412.69 0.828425
\(306\) 0 0
\(307\) −2966.54 2966.54i −0.551497 0.551497i 0.375376 0.926873i \(-0.377514\pi\)
−0.926873 + 0.375376i \(0.877514\pi\)
\(308\) 0 0
\(309\) 3340.08 3340.08i 0.614920 0.614920i
\(310\) 0 0
\(311\) 2911.18i 0.530797i −0.964139 0.265399i \(-0.914496\pi\)
0.964139 0.265399i \(-0.0855036\pi\)
\(312\) 0 0
\(313\) 8287.74i 1.49665i −0.663333 0.748324i \(-0.730859\pi\)
0.663333 0.748324i \(-0.269141\pi\)
\(314\) 0 0
\(315\) −981.845 + 981.845i −0.175621 + 0.175621i
\(316\) 0 0
\(317\) 5742.18 + 5742.18i 1.01739 + 1.01739i 0.999846 + 0.0175452i \(0.00558510\pi\)
0.0175452 + 0.999846i \(0.494415\pi\)
\(318\) 0 0
\(319\) 184.999 0.0324700
\(320\) 0 0
\(321\) 8118.23 1.41157
\(322\) 0 0
\(323\) 1611.70 + 1611.70i 0.277639 + 0.277639i
\(324\) 0 0
\(325\) 6386.56 6386.56i 1.09004 1.09004i
\(326\) 0 0
\(327\) 710.734i 0.120195i
\(328\) 0 0
\(329\) 2537.84i 0.425275i
\(330\) 0 0
\(331\) −4499.27 + 4499.27i −0.747136 + 0.747136i −0.973940 0.226804i \(-0.927172\pi\)
0.226804 + 0.973940i \(0.427172\pi\)
\(332\) 0 0
\(333\) −499.095 499.095i −0.0821328 0.0821328i
\(334\) 0 0
\(335\) 2320.50 0.378454
\(336\) 0 0
\(337\) −5860.06 −0.947234 −0.473617 0.880731i \(-0.657052\pi\)
−0.473617 + 0.880731i \(0.657052\pi\)
\(338\) 0 0
\(339\) −5634.04 5634.04i −0.902652 0.902652i
\(340\) 0 0
\(341\) −48.7893 + 48.7893i −0.00774806 + 0.00774806i
\(342\) 0 0
\(343\) 6847.14i 1.07787i
\(344\) 0 0
\(345\) 11086.7i 1.73010i
\(346\) 0 0
\(347\) 2029.11 2029.11i 0.313915 0.313915i −0.532509 0.846424i \(-0.678751\pi\)
0.846424 + 0.532509i \(0.178751\pi\)
\(348\) 0 0
\(349\) −1943.26 1943.26i −0.298053 0.298053i 0.542198 0.840251i \(-0.317592\pi\)
−0.840251 + 0.542198i \(0.817592\pi\)
\(350\) 0 0
\(351\) 6992.54 1.06335
\(352\) 0 0
\(353\) −7548.63 −1.13817 −0.569084 0.822280i \(-0.692702\pi\)
−0.569084 + 0.822280i \(0.692702\pi\)
\(354\) 0 0
\(355\) −4404.97 4404.97i −0.658568 0.658568i
\(356\) 0 0
\(357\) −846.694 + 846.694i −0.125523 + 0.125523i
\(358\) 0 0
\(359\) 5554.15i 0.816537i 0.912862 + 0.408269i \(0.133867\pi\)
−0.912862 + 0.408269i \(0.866133\pi\)
\(360\) 0 0
\(361\) 8072.37i 1.17690i
\(362\) 0 0
\(363\) −4338.56 + 4338.56i −0.627315 + 0.627315i
\(364\) 0 0
\(365\) −3786.34 3786.34i −0.542975 0.542975i
\(366\) 0 0
\(367\) 3610.98 0.513601 0.256800 0.966464i \(-0.417332\pi\)
0.256800 + 0.966464i \(0.417332\pi\)
\(368\) 0 0
\(369\) 1177.99 0.166190
\(370\) 0 0
\(371\) −363.040 363.040i −0.0508035 0.0508035i
\(372\) 0 0
\(373\) 1215.49 1215.49i 0.168728 0.168728i −0.617692 0.786420i \(-0.711932\pi\)
0.786420 + 0.617692i \(0.211932\pi\)
\(374\) 0 0
\(375\) 5773.62i 0.795062i
\(376\) 0 0
\(377\) 3921.05i 0.535661i
\(378\) 0 0
\(379\) 7347.81 7347.81i 0.995861 0.995861i −0.00413018 0.999991i \(-0.501315\pi\)
0.999991 + 0.00413018i \(0.00131468\pi\)
\(380\) 0 0
\(381\) −2287.12 2287.12i −0.307539 0.307539i
\(382\) 0 0
\(383\) 7668.98 1.02315 0.511575 0.859238i \(-0.329062\pi\)
0.511575 + 0.859238i \(0.329062\pi\)
\(384\) 0 0
\(385\) 542.831 0.0718577
\(386\) 0 0
\(387\) 668.181 + 668.181i 0.0877663 + 0.0877663i
\(388\) 0 0
\(389\) −200.924 + 200.924i −0.0261884 + 0.0261884i −0.720080 0.693891i \(-0.755895\pi\)
0.693891 + 0.720080i \(0.255895\pi\)
\(390\) 0 0
\(391\) 2499.62i 0.323302i
\(392\) 0 0
\(393\) 1295.27i 0.166254i
\(394\) 0 0
\(395\) −11930.3 + 11930.3i −1.51969 + 1.51969i
\(396\) 0 0
\(397\) 6512.21 + 6512.21i 0.823271 + 0.823271i 0.986576 0.163305i \(-0.0522153\pi\)
−0.163305 + 0.986576i \(0.552215\pi\)
\(398\) 0 0
\(399\) −7844.07 −0.984197
\(400\) 0 0
\(401\) 5565.10 0.693036 0.346518 0.938043i \(-0.387364\pi\)
0.346518 + 0.938043i \(0.387364\pi\)
\(402\) 0 0
\(403\) 1034.09 + 1034.09i 0.127820 + 0.127820i
\(404\) 0 0
\(405\) −6911.57 + 6911.57i −0.847997 + 0.847997i
\(406\) 0 0
\(407\) 275.933i 0.0336057i
\(408\) 0 0
\(409\) 12077.6i 1.46014i 0.683370 + 0.730072i \(0.260514\pi\)
−0.683370 + 0.730072i \(0.739486\pi\)
\(410\) 0 0
\(411\) −887.810 + 887.810i −0.106551 + 0.106551i
\(412\) 0 0
\(413\) −6127.14 6127.14i −0.730017 0.730017i
\(414\) 0 0
\(415\) 7922.58 0.937119
\(416\) 0 0
\(417\) −3009.26 −0.353391
\(418\) 0 0
\(419\) 1453.03 + 1453.03i 0.169415 + 0.169415i 0.786722 0.617307i \(-0.211776\pi\)
−0.617307 + 0.786722i \(0.711776\pi\)
\(420\) 0 0
\(421\) −4822.25 + 4822.25i −0.558247 + 0.558247i −0.928808 0.370561i \(-0.879165\pi\)
0.370561 + 0.928808i \(0.379165\pi\)
\(422\) 0 0
\(423\) 1023.53i 0.117650i
\(424\) 0 0
\(425\) 3633.36i 0.414692i
\(426\) 0 0
\(427\) 2421.04 2421.04i 0.274385 0.274385i
\(428\) 0 0
\(429\) −331.852 331.852i −0.0373472 0.0373472i
\(430\) 0 0
\(431\) −12519.2 −1.39914 −0.699571 0.714563i \(-0.746626\pi\)
−0.699571 + 0.714563i \(0.746626\pi\)
\(432\) 0 0
\(433\) −2921.40 −0.324235 −0.162117 0.986771i \(-0.551832\pi\)
−0.162117 + 0.986771i \(0.551832\pi\)
\(434\) 0 0
\(435\) 4946.99 + 4946.99i 0.545264 + 0.545264i
\(436\) 0 0
\(437\) 11578.7 11578.7i 1.26747 1.26747i
\(438\) 0 0
\(439\) 1140.50i 0.123993i 0.998076 + 0.0619967i \(0.0197468\pi\)
−0.998076 + 0.0619967i \(0.980253\pi\)
\(440\) 0 0
\(441\) 842.062i 0.0909256i
\(442\) 0 0
\(443\) 1843.05 1843.05i 0.197665 0.197665i −0.601333 0.798999i \(-0.705363\pi\)
0.798999 + 0.601333i \(0.205363\pi\)
\(444\) 0 0
\(445\) 17859.0 + 17859.0i 1.90247 + 1.90247i
\(446\) 0 0
\(447\) 3963.43 0.419382
\(448\) 0 0
\(449\) 1752.13 0.184161 0.0920805 0.995752i \(-0.470648\pi\)
0.0920805 + 0.995752i \(0.470648\pi\)
\(450\) 0 0
\(451\) −325.637 325.637i −0.0339993 0.0339993i
\(452\) 0 0
\(453\) 11562.6 11562.6i 1.19925 1.19925i
\(454\) 0 0
\(455\) 11505.3i 1.18544i
\(456\) 0 0
\(457\) 12875.6i 1.31794i −0.752171 0.658968i \(-0.770993\pi\)
0.752171 0.658968i \(-0.229007\pi\)
\(458\) 0 0
\(459\) 1989.06 1989.06i 0.202268 0.202268i
\(460\) 0 0
\(461\) −13679.7 13679.7i −1.38205 1.38205i −0.840968 0.541085i \(-0.818014\pi\)
−0.541085 0.840968i \(-0.681986\pi\)
\(462\) 0 0
\(463\) −15002.4 −1.50588 −0.752938 0.658091i \(-0.771364\pi\)
−0.752938 + 0.658091i \(0.771364\pi\)
\(464\) 0 0
\(465\) −2609.32 −0.260224
\(466\) 0 0
\(467\) −9669.42 9669.42i −0.958131 0.958131i 0.0410271 0.999158i \(-0.486937\pi\)
−0.999158 + 0.0410271i \(0.986937\pi\)
\(468\) 0 0
\(469\) 1273.15 1273.15i 0.125349 0.125349i
\(470\) 0 0
\(471\) 8070.75i 0.789555i
\(472\) 0 0
\(473\) 369.416i 0.0359107i
\(474\) 0 0
\(475\) 16830.4 16830.4i 1.62575 1.62575i
\(476\) 0 0
\(477\) 146.418 + 146.418i 0.0140545 + 0.0140545i
\(478\) 0 0
\(479\) 3072.68 0.293099 0.146550 0.989203i \(-0.453183\pi\)
0.146550 + 0.989203i \(0.453183\pi\)
\(480\) 0 0
\(481\) 5848.41 0.554396
\(482\) 0 0
\(483\) 6082.76 + 6082.76i 0.573033 + 0.573033i
\(484\) 0 0
\(485\) −19160.7 + 19160.7i −1.79390 + 1.79390i
\(486\) 0 0
\(487\) 8689.64i 0.808553i 0.914637 + 0.404276i \(0.132477\pi\)
−0.914637 + 0.404276i \(0.867523\pi\)
\(488\) 0 0
\(489\) 16810.8i 1.55462i
\(490\) 0 0
\(491\) −11194.3 + 11194.3i −1.02891 + 1.02891i −0.0293379 + 0.999570i \(0.509340\pi\)
−0.999570 + 0.0293379i \(0.990660\pi\)
\(492\) 0 0
\(493\) −1115.36 1115.36i −0.101893 0.101893i
\(494\) 0 0
\(495\) −218.929 −0.0198790
\(496\) 0 0
\(497\) −4833.62 −0.436253
\(498\) 0 0
\(499\) 1632.72 + 1632.72i 0.146474 + 0.146474i 0.776541 0.630067i \(-0.216973\pi\)
−0.630067 + 0.776541i \(0.716973\pi\)
\(500\) 0 0
\(501\) −9972.47 + 9972.47i −0.889295 + 0.889295i
\(502\) 0 0
\(503\) 6901.81i 0.611802i −0.952063 0.305901i \(-0.901042\pi\)
0.952063 0.305901i \(-0.0989577\pi\)
\(504\) 0 0
\(505\) 14496.2i 1.27737i
\(506\) 0 0
\(507\) 153.630 153.630i 0.0134575 0.0134575i
\(508\) 0 0
\(509\) 92.9712 + 92.9712i 0.00809603 + 0.00809603i 0.711143 0.703047i \(-0.248178\pi\)
−0.703047 + 0.711143i \(0.748178\pi\)
\(510\) 0 0
\(511\) −4154.79 −0.359681
\(512\) 0 0
\(513\) 18427.3 1.58594
\(514\) 0 0
\(515\) 12910.4 + 12910.4i 1.10466 + 1.10466i
\(516\) 0 0
\(517\) −282.939 + 282.939i −0.0240689 + 0.0240689i
\(518\) 0 0
\(519\) 9958.43i 0.842248i
\(520\) 0 0
\(521\) 11931.3i 1.00330i 0.865071 + 0.501649i \(0.167273\pi\)
−0.865071 + 0.501649i \(0.832727\pi\)
\(522\) 0 0
\(523\) 9702.46 9702.46i 0.811203 0.811203i −0.173611 0.984814i \(-0.555544\pi\)
0.984814 + 0.173611i \(0.0555437\pi\)
\(524\) 0 0
\(525\) 8841.69 + 8841.69i 0.735015 + 0.735015i
\(526\) 0 0
\(527\) 588.301 0.0486277
\(528\) 0 0
\(529\) −5790.59 −0.475926
\(530\) 0 0
\(531\) 2471.13 + 2471.13i 0.201955 + 0.201955i
\(532\) 0 0
\(533\) −6901.89 + 6901.89i −0.560889 + 0.560889i
\(534\) 0 0
\(535\) 31379.4i 2.53579i
\(536\) 0 0
\(537\) 1977.60i 0.158920i
\(538\) 0 0
\(539\) −232.774 + 232.774i −0.0186017 + 0.0186017i
\(540\) 0 0
\(541\) −8556.67 8556.67i −0.680000 0.680000i 0.280000 0.960000i \(-0.409666\pi\)
−0.960000 + 0.280000i \(0.909666\pi\)
\(542\) 0 0
\(543\) 11102.7 0.877462
\(544\) 0 0
\(545\) 2747.20 0.215921
\(546\) 0 0
\(547\) 45.1953 + 45.1953i 0.00353274 + 0.00353274i 0.708871 0.705338i \(-0.249205\pi\)
−0.705338 + 0.708871i \(0.749205\pi\)
\(548\) 0 0
\(549\) −976.430 + 976.430i −0.0759071 + 0.0759071i
\(550\) 0 0
\(551\) 10333.1i 0.798917i
\(552\) 0 0
\(553\) 13091.2i 1.00668i
\(554\) 0 0
\(555\) −7378.64 + 7378.64i −0.564335 + 0.564335i
\(556\) 0 0
\(557\) 4279.60 + 4279.60i 0.325552 + 0.325552i 0.850892 0.525340i \(-0.176062\pi\)
−0.525340 + 0.850892i \(0.676062\pi\)
\(558\) 0 0
\(559\) −7829.77 −0.592422
\(560\) 0 0
\(561\) −188.793 −0.0142083
\(562\) 0 0
\(563\) −14593.9 14593.9i −1.09247 1.09247i −0.995265 0.0972023i \(-0.969011\pi\)
−0.0972023 0.995265i \(-0.530989\pi\)
\(564\) 0 0
\(565\) 21777.3 21777.3i 1.62155 1.62155i
\(566\) 0 0
\(567\) 7584.14i 0.561736i
\(568\) 0 0
\(569\) 21728.1i 1.60086i 0.599425 + 0.800431i \(0.295396\pi\)
−0.599425 + 0.800431i \(0.704604\pi\)
\(570\) 0 0
\(571\) −16078.0 + 16078.0i −1.17836 + 1.17836i −0.198202 + 0.980161i \(0.563510\pi\)
−0.980161 + 0.198202i \(0.936490\pi\)
\(572\) 0 0
\(573\) 13201.0 + 13201.0i 0.962446 + 0.962446i
\(574\) 0 0
\(575\) −26102.5 −1.89313
\(576\) 0 0
\(577\) −26648.2 −1.92267 −0.961335 0.275383i \(-0.911195\pi\)
−0.961335 + 0.275383i \(0.911195\pi\)
\(578\) 0 0
\(579\) −2899.01 2899.01i −0.208081 0.208081i
\(580\) 0 0
\(581\) 4346.77 4346.77i 0.310386 0.310386i
\(582\) 0 0
\(583\) 80.9495i 0.00575058i
\(584\) 0 0
\(585\) 4640.20i 0.327946i
\(586\) 0 0
\(587\) −1342.62 + 1342.62i −0.0944050 + 0.0944050i −0.752732 0.658327i \(-0.771264\pi\)
0.658327 + 0.752732i \(0.271264\pi\)
\(588\) 0 0
\(589\) 2725.11 + 2725.11i 0.190639 + 0.190639i
\(590\) 0 0
\(591\) 21395.7 1.48917
\(592\) 0 0
\(593\) 4474.79 0.309878 0.154939 0.987924i \(-0.450482\pi\)
0.154939 + 0.987924i \(0.450482\pi\)
\(594\) 0 0
\(595\) −3272.73 3272.73i −0.225494 0.225494i
\(596\) 0 0
\(597\) −727.925 + 727.925i −0.0499028 + 0.0499028i
\(598\) 0 0
\(599\) 12603.8i 0.859725i 0.902894 + 0.429863i \(0.141438\pi\)
−0.902894 + 0.429863i \(0.858562\pi\)
\(600\) 0 0
\(601\) 7220.64i 0.490077i −0.969513 0.245038i \(-0.921199\pi\)
0.969513 0.245038i \(-0.0788006\pi\)
\(602\) 0 0
\(603\) −513.474 + 513.474i −0.0346771 + 0.0346771i
\(604\) 0 0
\(605\) −16769.8 16769.8i −1.12693 1.12693i
\(606\) 0 0
\(607\) 13695.6 0.915796 0.457898 0.889005i \(-0.348603\pi\)
0.457898 + 0.889005i \(0.348603\pi\)
\(608\) 0 0
\(609\) 5428.39 0.361198
\(610\) 0 0
\(611\) 5996.90 + 5996.90i 0.397068 + 0.397068i
\(612\) 0 0
\(613\) −2358.34 + 2358.34i −0.155387 + 0.155387i −0.780519 0.625132i \(-0.785045\pi\)
0.625132 + 0.780519i \(0.285045\pi\)
\(614\) 0 0
\(615\) 17415.5i 1.14189i
\(616\) 0 0
\(617\) 4186.39i 0.273157i −0.990629 0.136579i \(-0.956389\pi\)
0.990629 0.136579i \(-0.0436106\pi\)
\(618\) 0 0
\(619\) 4800.50 4800.50i 0.311710 0.311710i −0.533862 0.845572i \(-0.679260\pi\)
0.845572 + 0.533862i \(0.179260\pi\)
\(620\) 0 0
\(621\) −14289.6 14289.6i −0.923387 0.923387i
\(622\) 0 0
\(623\) 19596.9 1.26024
\(624\) 0 0
\(625\) 2031.54 0.130018
\(626\) 0 0
\(627\) −874.522 874.522i −0.0557018 0.0557018i
\(628\) 0 0
\(629\) 1663.60 1663.60i 0.105457 0.105457i
\(630\) 0 0
\(631\) 16106.3i 1.01614i −0.861316 0.508069i \(-0.830360\pi\)
0.861316 0.508069i \(-0.169640\pi\)
\(632\) 0 0
\(633\) 23079.3i 1.44917i
\(634\) 0 0
\(635\) 8840.39 8840.39i 0.552473 0.552473i
\(636\) 0 0
\(637\) 4933.66 + 4933.66i 0.306874 + 0.306874i
\(638\) 0 0
\(639\) 1949.45 0.120687
\(640\) 0 0
\(641\) −6682.21 −0.411749 −0.205875 0.978578i \(-0.566004\pi\)
−0.205875 + 0.978578i \(0.566004\pi\)
\(642\) 0 0
\(643\) 4983.47 + 4983.47i 0.305644 + 0.305644i 0.843217 0.537573i \(-0.180659\pi\)
−0.537573 + 0.843217i \(0.680659\pi\)
\(644\) 0 0
\(645\) 9878.43 9878.43i 0.603043 0.603043i
\(646\) 0 0
\(647\) 5078.45i 0.308585i 0.988025 + 0.154292i \(0.0493098\pi\)
−0.988025 + 0.154292i \(0.950690\pi\)
\(648\) 0 0
\(649\) 1366.21i 0.0826324i
\(650\) 0 0
\(651\) −1431.62 + 1431.62i −0.0861896 + 0.0861896i
\(652\) 0 0
\(653\) 6189.91 + 6189.91i 0.370949 + 0.370949i 0.867823 0.496874i \(-0.165519\pi\)
−0.496874 + 0.867823i \(0.665519\pi\)
\(654\) 0 0
\(655\) 5006.62 0.298664
\(656\) 0 0
\(657\) 1675.67 0.0995037
\(658\) 0 0
\(659\) −5751.19 5751.19i −0.339962 0.339962i 0.516391 0.856353i \(-0.327275\pi\)
−0.856353 + 0.516391i \(0.827275\pi\)
\(660\) 0 0
\(661\) −6305.38 + 6305.38i −0.371030 + 0.371030i −0.867852 0.496822i \(-0.834500\pi\)
0.496822 + 0.867852i \(0.334500\pi\)
\(662\) 0 0
\(663\) 4001.47i 0.234396i
\(664\) 0 0
\(665\) 30319.7i 1.76804i
\(666\) 0 0
\(667\) −8012.87 + 8012.87i −0.465157 + 0.465157i
\(668\) 0 0
\(669\) −19111.1 19111.1i −1.10445 1.10445i
\(670\) 0 0
\(671\) 539.837 0.0310584
\(672\) 0 0
\(673\) 14664.4 0.839925 0.419963 0.907541i \(-0.362043\pi\)
0.419963 + 0.907541i \(0.362043\pi\)
\(674\) 0 0
\(675\) −20770.9 20770.9i −1.18440 1.18440i
\(676\) 0 0
\(677\) 5795.16 5795.16i 0.328990 0.328990i −0.523212 0.852202i \(-0.675267\pi\)
0.852202 + 0.523212i \(0.175267\pi\)
\(678\) 0 0
\(679\) 21025.2i 1.18833i
\(680\) 0 0
\(681\) 7390.32i 0.415855i
\(682\) 0 0
\(683\) 13134.1 13134.1i 0.735816 0.735816i −0.235950 0.971765i \(-0.575820\pi\)
0.971765 + 0.235950i \(0.0758200\pi\)
\(684\) 0 0
\(685\) −3431.65 3431.65i −0.191411 0.191411i
\(686\) 0 0
\(687\) −19009.4 −1.05568
\(688\) 0 0
\(689\) −1715.73 −0.0948679
\(690\) 0 0
\(691\) −14324.0 14324.0i −0.788583 0.788583i 0.192679 0.981262i \(-0.438282\pi\)
−0.981262 + 0.192679i \(0.938282\pi\)
\(692\) 0 0
\(693\) −120.116 + 120.116i −0.00658419 + 0.00658419i
\(694\) 0 0
\(695\) 11631.7i 0.634843i
\(696\) 0 0
\(697\) 3926.54i 0.213383i
\(698\) 0 0
\(699\) −2401.37 + 2401.37i −0.129940 + 0.129940i
\(700\) 0 0
\(701\) 15987.2 + 15987.2i 0.861382 + 0.861382i 0.991499 0.130117i \(-0.0415351\pi\)
−0.130117 + 0.991499i \(0.541535\pi\)
\(702\) 0 0
\(703\) 15412.2 0.826859
\(704\) 0 0
\(705\) −15132.0 −0.808373
\(706\) 0 0
\(707\) −7953.40 7953.40i −0.423081 0.423081i
\(708\) 0 0
\(709\) 19580.4 19580.4i 1.03718 1.03718i 0.0378960 0.999282i \(-0.487934\pi\)
0.999282 0.0378960i \(-0.0120656\pi\)
\(710\) 0 0
\(711\) 5279.82i 0.278493i
\(712\) 0 0
\(713\) 4226.43i 0.221993i
\(714\) 0 0
\(715\) 1282.71 1282.71i 0.0670916 0.0670916i
\(716\) 0 0
\(717\) 1674.32 + 1674.32i 0.0872085 + 0.0872085i
\(718\) 0 0
\(719\) 2111.24 0.109507 0.0547537 0.998500i \(-0.482563\pi\)
0.0547537 + 0.998500i \(0.482563\pi\)
\(720\) 0 0
\(721\) 14166.7 0.731757
\(722\) 0 0
\(723\) 19367.6 + 19367.6i 0.996250 + 0.996250i
\(724\) 0 0
\(725\) −11647.2 + 11647.2i −0.596645 + 0.596645i
\(726\) 0 0
\(727\) 14763.6i 0.753164i −0.926383 0.376582i \(-0.877099\pi\)
0.926383 0.376582i \(-0.122901\pi\)
\(728\) 0 0
\(729\) 21896.2i 1.11244i
\(730\) 0 0
\(731\) −2227.21 + 2227.21i −0.112690 + 0.112690i
\(732\) 0 0
\(733\) 3419.77 + 3419.77i 0.172322 + 0.172322i 0.787999 0.615677i \(-0.211117\pi\)
−0.615677 + 0.787999i \(0.711117\pi\)
\(734\) 0 0
\(735\) −12449.1 −0.624751
\(736\) 0 0
\(737\) 283.883 0.0141886
\(738\) 0 0
\(739\) 11324.8 + 11324.8i 0.563723 + 0.563723i 0.930363 0.366640i \(-0.119492\pi\)
−0.366640 + 0.930363i \(0.619492\pi\)
\(740\) 0 0
\(741\) −18535.5 + 18535.5i −0.918919 + 0.918919i
\(742\) 0 0
\(743\) 20313.3i 1.00299i 0.865160 + 0.501495i \(0.167217\pi\)
−0.865160 + 0.501495i \(0.832783\pi\)
\(744\) 0 0
\(745\) 15319.9i 0.753391i
\(746\) 0 0
\(747\) −1753.09 + 1753.09i −0.0858665 + 0.0858665i
\(748\) 0 0
\(749\) 17216.5 + 17216.5i 0.839888 + 0.839888i
\(750\) 0 0
\(751\) −28755.3 −1.39720 −0.698598 0.715514i \(-0.746192\pi\)
−0.698598 + 0.715514i \(0.746192\pi\)
\(752\) 0 0
\(753\) 2023.89 0.0979477
\(754\) 0 0
\(755\) 44693.0 + 44693.0i 2.15436 + 2.15436i
\(756\) 0 0
\(757\) −23006.0 + 23006.0i −1.10458 + 1.10458i −0.110730 + 0.993850i \(0.535319\pi\)
−0.993850 + 0.110730i \(0.964681\pi\)
\(758\) 0 0
\(759\) 1356.31i 0.0648631i
\(760\) 0 0
\(761\) 9298.53i 0.442932i 0.975168 + 0.221466i \(0.0710843\pi\)
−0.975168 + 0.221466i \(0.928916\pi\)
\(762\) 0 0
\(763\) 1507.27 1507.27i 0.0715160 0.0715160i
\(764\) 0 0
\(765\) 1319.92 + 1319.92i 0.0623815 + 0.0623815i
\(766\) 0 0
\(767\) −28956.8 −1.36319
\(768\) 0 0
\(769\) −20402.0 −0.956717 −0.478358 0.878165i \(-0.658768\pi\)
−0.478358 + 0.878165i \(0.658768\pi\)
\(770\) 0 0
\(771\) 1059.02 + 1059.02i 0.0494679 + 0.0494679i
\(772\) 0 0
\(773\) 7337.03 7337.03i 0.341390 0.341390i −0.515500 0.856890i \(-0.672394\pi\)
0.856890 + 0.515500i \(0.172394\pi\)
\(774\) 0 0
\(775\) 6143.40i 0.284745i
\(776\) 0 0
\(777\) 8096.67i 0.373831i
\(778\) 0 0
\(779\) −18188.4 + 18188.4i −0.836544 + 0.836544i
\(780\) 0 0
\(781\) −538.893 538.893i −0.0246903 0.0246903i
\(782\) 0 0
\(783\) −12752.4 −0.582034
\(784\) 0 0
\(785\) 31195.9 1.41838
\(786\) 0 0
\(787\) 11928.6 + 11928.6i 0.540292 + 0.540292i 0.923615 0.383323i \(-0.125220\pi\)
−0.383323 + 0.923615i \(0.625220\pi\)
\(788\) 0 0
\(789\) −8797.25 + 8797.25i −0.396946 + 0.396946i
\(790\) 0 0
\(791\) 23896.4i 1.07416i
\(792\) 0 0
\(793\) 11441.8i 0.512373i
\(794\) 0 0
\(795\) 2164.65 2164.65i 0.0965686 0.0965686i
\(796\) 0 0
\(797\) −6576.18 6576.18i −0.292271 0.292271i 0.545706 0.837977i \(-0.316262\pi\)
−0.837977 + 0.545706i \(0.816262\pi\)
\(798\) 0 0
\(799\) 3411.68 0.151060
\(800\) 0 0
\(801\) −7903.61 −0.348639
\(802\) 0 0
\(803\) −463.211 463.211i −0.0203566 0.0203566i
\(804\) 0 0
\(805\) −23511.7 + 23511.7i −1.02941 + 1.02941i
\(806\) 0 0
\(807\) 30775.8i 1.34245i
\(808\) 0 0
\(809\) 29320.9i 1.27425i −0.770760 0.637126i \(-0.780123\pi\)
0.770760 0.637126i \(-0.219877\pi\)
\(810\) 0 0
\(811\) 14487.9 14487.9i 0.627297 0.627297i −0.320090 0.947387i \(-0.603713\pi\)
0.947387 + 0.320090i \(0.103713\pi\)
\(812\) 0 0
\(813\) 6604.84 + 6604.84i 0.284922 + 0.284922i
\(814\) 0 0
\(815\) 64978.7 2.79276
\(816\) 0 0
\(817\) −20633.6 −0.883574
\(818\) 0 0
\(819\) 2545.87 + 2545.87i 0.108620 + 0.108620i
\(820\) 0 0
\(821\) 20259.2 20259.2i 0.861208 0.861208i −0.130270 0.991479i \(-0.541585\pi\)
0.991479 + 0.130270i \(0.0415845\pi\)
\(822\) 0 0
\(823\) 24605.9i 1.04217i −0.853504 0.521086i \(-0.825527\pi\)
0.853504 0.521086i \(-0.174473\pi\)
\(824\) 0 0
\(825\) 1971.49i 0.0831982i
\(826\) 0 0
\(827\) 24095.8 24095.8i 1.01317 1.01317i 0.0132601 0.999912i \(-0.495779\pi\)
0.999912 0.0132601i \(-0.00422096\pi\)
\(828\) 0 0
\(829\) −914.616 914.616i −0.0383184 0.0383184i 0.687688 0.726006i \(-0.258626\pi\)
−0.726006 + 0.687688i \(0.758626\pi\)
\(830\) 0 0
\(831\) −20155.7 −0.841388
\(832\) 0 0
\(833\) 2806.80 0.116746
\(834\) 0 0
\(835\) −38546.6 38546.6i −1.59756 1.59756i
\(836\) 0 0
\(837\) 3363.16 3363.16i 0.138886 0.138886i
\(838\) 0 0
\(839\) 28847.5i 1.18704i 0.804819 + 0.593521i \(0.202263\pi\)
−0.804819 + 0.593521i \(0.797737\pi\)
\(840\) 0 0
\(841\) 17238.1i 0.706800i
\(842\) 0 0
\(843\) 12737.2 12737.2i 0.520394 0.520394i
\(844\) 0 0
\(845\) 593.827 + 593.827i 0.0241755 + 0.0241755i
\(846\) 0 0
\(847\) −18401.7 −0.746506
\(848\) 0 0
\(849\) −13258.2 −0.535947
\(850\) 0 0
\(851\) −11951.5 11951.5i −0.481426 0.481426i
\(852\) 0 0
\(853\) 41.5562 41.5562i 0.00166806 0.00166806i −0.706272 0.707940i \(-0.749625\pi\)
0.707940 + 0.706272i \(0.249625\pi\)
\(854\) 0 0
\(855\) 12228.2i 0.489118i
\(856\) 0 0
\(857\) 20953.6i 0.835194i 0.908632 + 0.417597i \(0.137128\pi\)
−0.908632 + 0.417597i \(0.862872\pi\)
\(858\) 0 0
\(859\) −29316.3 + 29316.3i −1.16444 + 1.16444i −0.180953 + 0.983492i \(0.557918\pi\)
−0.983492 + 0.180953i \(0.942082\pi\)
\(860\) 0 0
\(861\) −9555.13 9555.13i −0.378209 0.378209i
\(862\) 0 0
\(863\) 3389.59 0.133700 0.0668499 0.997763i \(-0.478705\pi\)
0.0668499 + 0.997763i \(0.478705\pi\)
\(864\) 0 0
\(865\) 38492.3 1.51304
\(866\) 0 0
\(867\) −14934.1 14934.1i −0.584992 0.584992i
\(868\) 0 0
\(869\) −1459.52 + 1459.52i −0.0569745 + 0.0569745i
\(870\) 0 0
\(871\) 6016.91i 0.234070i
\(872\) 0 0
\(873\) 8479.66i 0.328743i
\(874\) 0 0
\(875\) −12244.2 + 12244.2i −0.473063 + 0.473063i
\(876\) 0 0
\(877\) −13912.5 13912.5i −0.535681 0.535681i 0.386576 0.922257i \(-0.373658\pi\)
−0.922257 + 0.386576i \(0.873658\pi\)
\(878\) 0 0
\(879\) −6553.79 −0.251484
\(880\) 0 0
\(881\) −1497.48 −0.0572660 −0.0286330 0.999590i \(-0.509115\pi\)
−0.0286330 + 0.999590i \(0.509115\pi\)
\(882\) 0 0
\(883\) 4143.81 + 4143.81i 0.157928 + 0.157928i 0.781648 0.623720i \(-0.214379\pi\)
−0.623720 + 0.781648i \(0.714379\pi\)
\(884\) 0 0
\(885\) 36533.4 36533.4i 1.38763 1.38763i
\(886\) 0 0
\(887\) 18058.0i 0.683573i −0.939778 0.341787i \(-0.888968\pi\)
0.939778 0.341787i \(-0.111032\pi\)
\(888\) 0 0
\(889\) 9700.66i 0.365973i
\(890\) 0 0
\(891\) −845.544 + 845.544i −0.0317921 + 0.0317921i
\(892\) 0 0
\(893\) 15803.5 + 15803.5i 0.592211 + 0.592211i
\(894\) 0 0
\(895\) 7644.03 0.285488
\(896\) 0 0
\(897\) 28747.1 1.07005
\(898\) 0 0
\(899\) −1885.88 1885.88i −0.0699640 0.0699640i
\(900\) 0 0
\(901\) −488.045 + 488.045i −0.0180457 + 0.0180457i
\(902\) 0 0
\(903\) 10839.7i 0.399472i
\(904\) 0 0
\(905\) 42915.2i 1.57630i
\(906\) 0 0
\(907\) 21423.5 21423.5i 0.784295 0.784295i −0.196257 0.980552i \(-0.562879\pi\)
0.980552 + 0.196257i \(0.0628788\pi\)
\(908\) 0 0
\(909\) 3207.68 + 3207.68i 0.117043 + 0.117043i
\(910\) 0 0
\(911\) 31977.7 1.16297 0.581487 0.813556i \(-0.302471\pi\)
0.581487 + 0.813556i \(0.302471\pi\)
\(912\) 0 0
\(913\) 969.228 0.0351334
\(914\) 0 0
\(915\) 14435.6 + 14435.6i 0.521559 + 0.521559i
\(916\) 0 0
\(917\) 2746.91 2746.91i 0.0989215 0.0989215i
\(918\) 0 0
\(919\) 40696.7i 1.46078i −0.683029 0.730391i \(-0.739338\pi\)
0.683029 0.730391i \(-0.260662\pi\)
\(920\) 0 0
\(921\) 19409.4i 0.694421i
\(922\) 0 0
\(923\) −11421.8 + 11421.8i −0.407318 + 0.407318i
\(924\) 0 0
\(925\) −17372.4 17372.4i −0.617513 0.617513i
\(926\) 0 0
\(927\) −5713.57 −0.202436
\(928\) 0 0
\(929\) −11467.5 −0.404989 −0.202495 0.979283i \(-0.564905\pi\)
−0.202495 + 0.979283i \(0.564905\pi\)
\(930\) 0 0
\(931\) 13001.6 + 13001.6i 0.457690 + 0.457690i
\(932\) 0 0
\(933\) 9523.60 9523.60i 0.334179 0.334179i
\(934\) 0 0
\(935\) 729.742i 0.0255242i
\(936\) 0 0
\(937\) 14100.2i 0.491603i 0.969320 + 0.245802i \(0.0790512\pi\)
−0.969320 + 0.245802i \(0.920949\pi\)
\(938\) 0 0
\(939\) 27112.4 27112.4i 0.942257 0.942257i
\(940\) 0 0
\(941\) −21058.7 21058.7i −0.729538 0.729538i 0.240989 0.970528i \(-0.422528\pi\)
−0.970528 + 0.240989i \(0.922528\pi\)
\(942\) 0 0
\(943\) 28208.8 0.974129
\(944\) 0 0
\(945\) −37418.5 −1.28807
\(946\) 0 0
\(947\) −24998.9 24998.9i −0.857820 0.857820i 0.133261 0.991081i \(-0.457455\pi\)
−0.991081 + 0.133261i \(0.957455\pi\)
\(948\) 0 0
\(949\) −9817.75 + 9817.75i −0.335825 + 0.335825i
\(950\) 0 0
\(951\) 37569.8i 1.28106i
\(952\) 0 0
\(953\) 6456.01i 0.219445i −0.993962 0.109722i \(-0.965004\pi\)
0.993962 0.109722i \(-0.0349961\pi\)
\(954\) 0 0
\(955\) −51026.0 + 51026.0i −1.72897 + 1.72897i
\(956\) 0 0
\(957\) 605.202 + 605.202i 0.0204424 + 0.0204424i
\(958\) 0 0
\(959\) −3765.59 −0.126796
\(960\) 0 0
\(961\) −28796.3 −0.966610
\(962\) 0 0
\(963\) −6943.57 6943.57i −0.232350 0.232350i
\(964\) 0 0
\(965\) 11205.5 11205.5i 0.373802 0.373802i
\(966\) 0 0
\(967\) 15099.9i 0.502153i 0.967967 + 0.251076i \(0.0807845\pi\)
−0.967967 + 0.251076i \(0.919215\pi\)
\(968\) 0 0
\(969\) 10545.0i 0.349591i
\(970\) 0 0
\(971\) 6223.12 6223.12i 0.205674 0.205674i −0.596752 0.802426i \(-0.703542\pi\)
0.802426 + 0.596752i \(0.203542\pi\)
\(972\) 0 0
\(973\) −6381.80 6381.80i −0.210268 0.210268i
\(974\) 0 0
\(975\) 41785.8 1.37253
\(976\) 0 0
\(977\) −34900.0 −1.14284 −0.571418 0.820659i \(-0.693607\pi\)
−0.571418 + 0.820659i \(0.693607\pi\)
\(978\) 0 0
\(979\) 2184.82 + 2184.82i 0.0713251 + 0.0713251i
\(980\) 0 0
\(981\) −607.895 + 607.895i −0.0197845 + 0.0197845i
\(982\) 0 0
\(983\) 21221.5i 0.688567i 0.938866 + 0.344283i \(0.111878\pi\)
−0.938866 + 0.344283i \(0.888122\pi\)
\(984\) 0 0
\(985\) 82700.7i 2.67519i
\(986\) 0 0
\(987\) −8302.24 + 8302.24i −0.267744 + 0.267744i
\(988\) 0 0
\(989\) 16000.6 + 16000.6i 0.514447 + 0.514447i
\(990\) 0 0
\(991\) −23985.3 −0.768838 −0.384419 0.923159i \(-0.625598\pi\)
−0.384419 + 0.923159i \(0.625598\pi\)
\(992\) 0 0
\(993\) −29437.7 −0.940761
\(994\) 0 0
\(995\) −2813.65 2813.65i −0.0896469 0.0896469i
\(996\) 0 0
\(997\) 10763.8 10763.8i 0.341919 0.341919i −0.515169 0.857088i \(-0.672271\pi\)
0.857088 + 0.515169i \(0.172271\pi\)
\(998\) 0 0
\(999\) 19020.7i 0.602391i
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 64.4.e.a.17.4 10
3.2 odd 2 576.4.k.a.145.5 10
4.3 odd 2 16.4.e.a.13.5 yes 10
8.3 odd 2 128.4.e.b.33.4 10
8.5 even 2 128.4.e.a.33.2 10
12.11 even 2 144.4.k.a.109.1 10
16.3 odd 4 128.4.e.b.97.4 10
16.5 even 4 inner 64.4.e.a.49.4 10
16.11 odd 4 16.4.e.a.5.5 10
16.13 even 4 128.4.e.a.97.2 10
32.3 odd 8 1024.4.b.j.513.3 10
32.5 even 8 1024.4.a.m.1.3 10
32.11 odd 8 1024.4.a.n.1.3 10
32.13 even 8 1024.4.b.k.513.3 10
32.19 odd 8 1024.4.b.j.513.8 10
32.21 even 8 1024.4.a.m.1.8 10
32.27 odd 8 1024.4.a.n.1.8 10
32.29 even 8 1024.4.b.k.513.8 10
48.5 odd 4 576.4.k.a.433.5 10
48.11 even 4 144.4.k.a.37.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.5 10 16.11 odd 4
16.4.e.a.13.5 yes 10 4.3 odd 2
64.4.e.a.17.4 10 1.1 even 1 trivial
64.4.e.a.49.4 10 16.5 even 4 inner
128.4.e.a.33.2 10 8.5 even 2
128.4.e.a.97.2 10 16.13 even 4
128.4.e.b.33.4 10 8.3 odd 2
128.4.e.b.97.4 10 16.3 odd 4
144.4.k.a.37.1 10 48.11 even 4
144.4.k.a.109.1 10 12.11 even 2
576.4.k.a.145.5 10 3.2 odd 2
576.4.k.a.433.5 10 48.5 odd 4
1024.4.a.m.1.3 10 32.5 even 8
1024.4.a.m.1.8 10 32.21 even 8
1024.4.a.n.1.3 10 32.11 odd 8
1024.4.a.n.1.8 10 32.27 odd 8
1024.4.b.j.513.3 10 32.3 odd 8
1024.4.b.j.513.8 10 32.19 odd 8
1024.4.b.k.513.3 10 32.13 even 8
1024.4.b.k.513.8 10 32.29 even 8