Properties

Label 300.3.l.e.143.1
Level $300$
Weight $3$
Character 300.143
Analytic conductor $8.174$
Analytic rank $0$
Dimension $8$
CM discriminant -15
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,3,Mod(107,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.107");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.l (of order \(4\), degree \(2\), 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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 17x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 143.1
Root \(-1.72286 - 1.01575i\) of defining polynomial
Character \(\chi\) \(=\) 300.143
Dual form 300.3.l.e.107.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.72286 - 1.01575i) q^{2} +(-2.12132 - 2.12132i) q^{3} +(1.93649 + 3.50000i) q^{4} +(1.50000 + 5.80948i) q^{6} +(0.218832 - 7.99701i) q^{8} +9.00000i q^{9} +O(q^{10})\) \(q+(-1.72286 - 1.01575i) q^{2} +(-2.12132 - 2.12132i) q^{3} +(1.93649 + 3.50000i) q^{4} +(1.50000 + 5.80948i) q^{6} +(0.218832 - 7.99701i) q^{8} +9.00000i q^{9} +(3.31670 - 11.5325i) q^{12} +(-8.50000 + 13.5554i) q^{16} +(21.9089 + 21.9089i) q^{17} +(9.14178 - 15.5057i) q^{18} +30.9839 q^{19} +(-24.0416 - 24.0416i) q^{23} +(-17.4284 + 16.5000i) q^{24} +(19.0919 - 19.0919i) q^{27} -61.9677i q^{31} +(28.4133 - 14.7202i) q^{32} +(-15.4919 - 60.0000i) q^{34} +(-31.5000 + 17.4284i) q^{36} +(-53.3809 - 31.4720i) q^{38} +(17.0000 + 65.8407i) q^{46} +(-9.89949 + 9.89949i) q^{47} +(46.7867 - 10.7242i) q^{48} -49.0000i q^{49} -92.9516i q^{51} +(43.8178 - 43.8178i) q^{53} +(-52.2853 + 13.5000i) q^{54} +(-65.7267 - 65.7267i) q^{57} +118.000 q^{61} +(-62.9439 + 106.762i) q^{62} +(-63.9042 - 3.50000i) q^{64} +(-34.2548 + 119.108i) q^{68} +102.000i q^{69} +(71.9731 + 1.96949i) q^{72} +(60.0000 + 108.444i) q^{76} +123.935 q^{79} -81.0000 q^{81} +(108.894 + 108.894i) q^{83} +(37.5893 - 130.702i) q^{92} +(-131.453 + 131.453i) q^{93} +(27.1109 - 7.00000i) q^{94} +(-91.5000 - 29.0474i) q^{96} +(-49.7719 + 84.4201i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{6} - 68 q^{16} - 252 q^{36} + 136 q^{46} + 944 q^{61} + 480 q^{76} - 648 q^{81} - 732 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.72286 1.01575i −0.861430 0.507877i
\(3\) −2.12132 2.12132i −0.707107 0.707107i
\(4\) 1.93649 + 3.50000i 0.484123 + 0.875000i
\(5\) 0 0
\(6\) 1.50000 + 5.80948i 0.250000 + 0.968246i
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0.218832 7.99701i 0.0273540 0.999626i
\(9\) 9.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 3.31670 11.5325i 0.276392 0.961045i
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.50000 + 13.5554i −0.531250 + 0.847215i
\(17\) 21.9089 + 21.9089i 1.28876 + 1.28876i 0.935541 + 0.353218i \(0.114913\pi\)
0.353218 + 0.935541i \(0.385087\pi\)
\(18\) 9.14178 15.5057i 0.507877 0.861430i
\(19\) 30.9839 1.63073 0.815365 0.578947i \(-0.196536\pi\)
0.815365 + 0.578947i \(0.196536\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −24.0416 24.0416i −1.04529 1.04529i −0.998925 0.0463637i \(-0.985237\pi\)
−0.0463637 0.998925i \(-0.514763\pi\)
\(24\) −17.4284 + 16.5000i −0.726184 + 0.687500i
\(25\) 0 0
\(26\) 0 0
\(27\) 19.0919 19.0919i 0.707107 0.707107i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 61.9677i 1.99896i −0.0322581 0.999480i \(-0.510270\pi\)
0.0322581 0.999480i \(-0.489730\pi\)
\(32\) 28.4133 14.7202i 0.887915 0.460007i
\(33\) 0 0
\(34\) −15.4919 60.0000i −0.455645 1.76471i
\(35\) 0 0
\(36\) −31.5000 + 17.4284i −0.875000 + 0.484123i
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) −53.3809 31.4720i −1.40476 0.828209i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 17.0000 + 65.8407i 0.369565 + 1.43132i
\(47\) −9.89949 + 9.89949i −0.210628 + 0.210628i −0.804534 0.593907i \(-0.797585\pi\)
0.593907 + 0.804534i \(0.297585\pi\)
\(48\) 46.7867 10.7242i 0.974722 0.223421i
\(49\) 49.0000i 1.00000i
\(50\) 0 0
\(51\) 92.9516i 1.82258i
\(52\) 0 0
\(53\) 43.8178 43.8178i 0.826751 0.826751i −0.160315 0.987066i \(-0.551251\pi\)
0.987066 + 0.160315i \(0.0512510\pi\)
\(54\) −52.2853 + 13.5000i −0.968246 + 0.250000i
\(55\) 0 0
\(56\) 0 0
\(57\) −65.7267 65.7267i −1.15310 1.15310i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 118.000 1.93443 0.967213 0.253966i \(-0.0817352\pi\)
0.967213 + 0.253966i \(0.0817352\pi\)
\(62\) −62.9439 + 106.762i −1.01522 + 1.72196i
\(63\) 0 0
\(64\) −63.9042 3.50000i −0.998504 0.0546875i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −34.2548 + 119.108i −0.503746 + 1.75158i
\(69\) 102.000i 1.47826i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 71.9731 + 1.96949i 0.999626 + 0.0273540i
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 60.0000 + 108.444i 0.789474 + 1.42689i
\(77\) 0 0
\(78\) 0 0
\(79\) 123.935 1.56880 0.784402 0.620253i \(-0.212970\pi\)
0.784402 + 0.620253i \(0.212970\pi\)
\(80\) 0 0
\(81\) −81.0000 −1.00000
\(82\) 0 0
\(83\) 108.894 + 108.894i 1.31198 + 1.31198i 0.919953 + 0.392028i \(0.128226\pi\)
0.392028 + 0.919953i \(0.371774\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 37.5893 130.702i 0.408579 1.42068i
\(93\) −131.453 + 131.453i −1.41348 + 1.41348i
\(94\) 27.1109 7.00000i 0.288414 0.0744681i
\(95\) 0 0
\(96\) −91.5000 29.0474i −0.953125 0.302577i
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) −49.7719 + 84.4201i −0.507877 + 0.861430i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −94.4159 + 160.143i −0.925646 + 1.57003i
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −120.000 + 30.9839i −1.13208 + 0.292301i
\(107\) −74.9533 + 74.9533i −0.700498 + 0.700498i −0.964517 0.264019i \(-0.914952\pi\)
0.264019 + 0.964517i \(0.414952\pi\)
\(108\) 103.793 + 29.8503i 0.961045 + 0.276392i
\(109\) 22.0000i 0.201835i −0.994895 0.100917i \(-0.967822\pi\)
0.994895 0.100917i \(-0.0321778\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 65.7267 65.7267i 0.581652 0.581652i −0.353705 0.935357i \(-0.615078\pi\)
0.935357 + 0.353705i \(0.115078\pi\)
\(114\) 46.4758 + 180.000i 0.407682 + 1.57895i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −121.000 −1.00000
\(122\) −203.297 119.859i −1.66637 0.982450i
\(123\) 0 0
\(124\) 216.887 120.000i 1.74909 0.967742i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 106.543 + 70.9409i 0.832366 + 0.554226i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 180.000 170.411i 1.32353 1.25302i
\(137\) 109.545 + 109.545i 0.799595 + 0.799595i 0.983032 0.183437i \(-0.0587222\pi\)
−0.183437 + 0.983032i \(0.558722\pi\)
\(138\) 103.607 175.732i 0.750774 1.27342i
\(139\) −92.9516 −0.668717 −0.334358 0.942446i \(-0.608520\pi\)
−0.334358 + 0.942446i \(0.608520\pi\)
\(140\) 0 0
\(141\) 42.0000 0.297872
\(142\) 0 0
\(143\) 0 0
\(144\) −121.999 76.5000i −0.847215 0.531250i
\(145\) 0 0
\(146\) 0 0
\(147\) −103.945 + 103.945i −0.707107 + 0.707107i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 185.903i 1.23115i 0.788079 + 0.615574i \(0.211076\pi\)
−0.788079 + 0.615574i \(0.788924\pi\)
\(152\) 6.78026 247.778i 0.0446070 1.63012i
\(153\) −197.180 + 197.180i −1.28876 + 1.28876i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) −213.523 125.888i −1.35141 0.796758i
\(159\) −185.903 −1.16920
\(160\) 0 0
\(161\) 0 0
\(162\) 139.552 + 82.2760i 0.861430 + 0.507877i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −77.0000 298.220i −0.463855 1.79650i
\(167\) 179.605 179.605i 1.07548 1.07548i 0.0785713 0.996908i \(-0.474964\pi\)
0.996908 0.0785713i \(-0.0250358\pi\)
\(168\) 0 0
\(169\) 169.000i 1.00000i
\(170\) 0 0
\(171\) 278.855i 1.63073i
\(172\) 0 0
\(173\) 219.089 219.089i 1.26641 1.26641i 0.318481 0.947929i \(-0.396827\pi\)
0.947929 0.318481i \(-0.103173\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 122.000 0.674033 0.337017 0.941499i \(-0.390582\pi\)
0.337017 + 0.941499i \(0.390582\pi\)
\(182\) 0 0
\(183\) −250.316 250.316i −1.36785 1.36785i
\(184\) −197.522 + 187.000i −1.07349 + 1.01630i
\(185\) 0 0
\(186\) 360.000 92.9516i 1.93548 0.499740i
\(187\) 0 0
\(188\) −53.8185 15.4779i −0.286269 0.0823295i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 128.137 + 142.986i 0.667379 + 0.744719i
\(193\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 171.500 94.8881i 0.875000 0.484123i
\(197\) 87.6356 + 87.6356i 0.444851 + 0.444851i 0.893638 0.448788i \(-0.148144\pi\)
−0.448788 + 0.893638i \(0.648144\pi\)
\(198\) 0 0
\(199\) −371.806 −1.86837 −0.934187 0.356784i \(-0.883873\pi\)
−0.934187 + 0.356784i \(0.883873\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 325.331 180.000i 1.59476 0.882353i
\(205\) 0 0
\(206\) 0 0
\(207\) 216.375 216.375i 1.04529 1.04529i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 216.887i 1.02790i 0.857820 + 0.513950i \(0.171818\pi\)
−0.857820 + 0.513950i \(0.828182\pi\)
\(212\) 238.215 + 68.5095i 1.12366 + 0.323158i
\(213\) 0 0
\(214\) 205.268 53.0000i 0.959197 0.247664i
\(215\) 0 0
\(216\) −148.500 156.856i −0.687500 0.726184i
\(217\) 0 0
\(218\) −22.3466 + 37.9029i −0.102507 + 0.173867i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −180.000 + 46.4758i −0.796460 + 0.205645i
\(227\) −94.7523 + 94.7523i −0.417411 + 0.417411i −0.884310 0.466899i \(-0.845371\pi\)
0.466899 + 0.884310i \(0.345371\pi\)
\(228\) 102.764 357.323i 0.450720 1.56720i
\(229\) 218.000i 0.951965i 0.879455 + 0.475983i \(0.157907\pi\)
−0.879455 + 0.475983i \(0.842093\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 328.634 328.634i 1.41044 1.41044i 0.653631 0.756814i \(-0.273245\pi\)
0.756814 0.653631i \(-0.226755\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −262.907 262.907i −1.10931 1.10931i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −478.000 −1.98340 −0.991701 0.128564i \(-0.958963\pi\)
−0.991701 + 0.128564i \(0.958963\pi\)
\(242\) 208.466 + 122.906i 0.861430 + 0.507877i
\(243\) 171.827 + 171.827i 0.707107 + 0.707107i
\(244\) 228.506 + 413.000i 0.936500 + 1.69262i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −495.556 13.5605i −1.99821 0.0546795i
\(249\) 462.000i 1.85542i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −111.500 230.443i −0.435547 0.900166i
\(257\) −153.362 153.362i −0.596741 0.596741i 0.342703 0.939444i \(-0.388657\pi\)
−0.939444 + 0.342703i \(0.888657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −315.370 315.370i −1.19912 1.19912i −0.974429 0.224695i \(-0.927861\pi\)
−0.224695 0.974429i \(-0.572139\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 247.871i 0.914653i 0.889299 + 0.457326i \(0.151193\pi\)
−0.889299 + 0.457326i \(0.848807\pi\)
\(272\) −483.211 + 110.759i −1.77651 + 0.407203i
\(273\) 0 0
\(274\) −77.4597 300.000i −0.282700 1.09489i
\(275\) 0 0
\(276\) −357.000 + 197.522i −1.29348 + 0.715660i
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 160.143 + 94.4159i 0.576052 + 0.339625i
\(279\) 557.710 1.99896
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −72.3601 42.6616i −0.256596 0.151282i
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 132.482 + 255.720i 0.460007 + 0.887915i
\(289\) 671.000i 2.32180i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −306.725 + 306.725i −1.04684 + 1.04684i −0.0479941 + 0.998848i \(0.515283\pi\)
−0.998848 + 0.0479941i \(0.984717\pi\)
\(294\) 284.664 73.5000i 0.968246 0.250000i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 188.832 320.285i 0.625271 1.06055i
\(303\) 0 0
\(304\) −263.363 + 420.000i −0.866325 + 1.38158i
\(305\) 0 0
\(306\) 540.000 139.427i 1.76471 0.455645i
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 240.000 + 433.774i 0.759494 + 1.37270i
\(317\) 438.178 + 438.178i 1.38227 + 1.38227i 0.840584 + 0.541681i \(0.182212\pi\)
0.541681 + 0.840584i \(0.317788\pi\)
\(318\) 320.285 + 188.832i 1.00719 + 0.593810i
\(319\) 0 0
\(320\) 0 0
\(321\) 318.000 0.990654
\(322\) 0 0
\(323\) 678.823 + 678.823i 2.10162 + 2.10162i
\(324\) −156.856 283.500i −0.484123 0.875000i
\(325\) 0 0
\(326\) 0 0
\(327\) −46.6690 + 46.6690i −0.142719 + 0.142719i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 650.661i 1.96574i −0.184290 0.982872i \(-0.558999\pi\)
0.184290 0.982872i \(-0.441001\pi\)
\(332\) −170.257 + 592.004i −0.512823 + 1.78314i
\(333\) 0 0
\(334\) −491.869 + 127.000i −1.47266 + 0.380240i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) −171.662 + 291.163i −0.507877 + 0.861430i
\(339\) −278.855 −0.822581
\(340\) 0 0
\(341\) 0 0
\(342\) 283.248 480.428i 0.828209 1.40476i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −600.000 + 154.919i −1.73410 + 0.447744i
\(347\) −414.365 + 414.365i −1.19413 + 1.19413i −0.218239 + 0.975895i \(0.570031\pi\)
−0.975895 + 0.218239i \(0.929969\pi\)
\(348\) 0 0
\(349\) 458.000i 1.31232i −0.754621 0.656160i \(-0.772179\pi\)
0.754621 0.656160i \(-0.227821\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −460.087 + 460.087i −1.30336 + 1.30336i −0.377252 + 0.926111i \(0.623131\pi\)
−0.926111 + 0.377252i \(0.876869\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 599.000 1.65928
\(362\) −210.189 123.922i −0.580632 0.342326i
\(363\) 256.680 + 256.680i 0.707107 + 0.707107i
\(364\) 0 0
\(365\) 0 0
\(366\) 177.000 + 685.518i 0.483607 + 1.87300i
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 530.249 121.541i 1.44089 0.330275i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −714.645 205.529i −1.92109 0.552496i
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 77.0000 + 81.3327i 0.204787 + 0.216310i
\(377\) 0 0
\(378\) 0 0
\(379\) 154.919 0.408758 0.204379 0.978892i \(-0.434482\pi\)
0.204379 + 0.978892i \(0.434482\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −485.075 485.075i −1.26652 1.26652i −0.947876 0.318639i \(-0.896774\pi\)
−0.318639 0.947876i \(-0.603226\pi\)
\(384\) −75.5232 376.500i −0.196675 0.980469i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 1053.45i 2.69425i
\(392\) −391.853 10.7228i −0.999626 0.0273540i
\(393\) 0 0
\(394\) −61.9677 240.000i −0.157279 0.609137i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 640.570 + 377.663i 1.60947 + 0.948903i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −743.335 20.3408i −1.82190 0.0498548i
\(409\) 142.000i 0.347188i 0.984817 + 0.173594i \(0.0555381\pi\)
−0.984817 + 0.173594i \(0.944462\pi\)
\(410\) 0 0
\(411\) 464.758i 1.13080i
\(412\) 0 0
\(413\) 0 0
\(414\) −592.566 + 153.000i −1.43132 + 0.369565i
\(415\) 0 0
\(416\) 0 0
\(417\) 197.180 + 197.180i 0.472854 + 0.472854i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −602.000 −1.42993 −0.714964 0.699161i \(-0.753557\pi\)
−0.714964 + 0.699161i \(0.753557\pi\)
\(422\) 220.304 373.666i 0.522047 0.885464i
\(423\) −89.0955 89.0955i −0.210628 0.210628i
\(424\) −340.823 360.000i −0.803827 0.849057i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −407.483 117.190i −0.952063 0.273809i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 96.5179 + 421.080i 0.223421 + 0.974722i
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 77.0000 42.6028i 0.176606 0.0977129i
\(437\) −744.903 744.903i −1.70458 1.70458i
\(438\) 0 0
\(439\) 619.677 1.41157 0.705783 0.708428i \(-0.250595\pi\)
0.705783 + 0.708428i \(0.250595\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 400.222 + 400.222i 0.903437 + 0.903437i 0.995732 0.0922950i \(-0.0294203\pi\)
−0.0922950 + 0.995732i \(0.529420\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 357.323 + 102.764i 0.790537 + 0.227355i
\(453\) 394.360 394.360i 0.870552 0.870552i
\(454\) 259.490 67.0000i 0.571564 0.147577i
\(455\) 0 0
\(456\) −540.000 + 511.234i −1.18421 + 1.12113i
\(457\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) 221.434 375.583i 0.483481 0.820051i
\(459\) 836.564 1.82258
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −900.000 + 232.379i −1.93133 + 0.498667i
\(467\) 244.659 244.659i 0.523895 0.523895i −0.394850 0.918745i \(-0.629204\pi\)
0.918745 + 0.394850i \(0.129204\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 185.903 + 720.000i 0.392201 + 1.51899i
\(475\) 0 0
\(476\) 0 0
\(477\) 394.360 + 394.360i 0.826751 + 0.826751i
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 823.527 + 485.530i 1.70856 + 1.00732i
\(483\) 0 0
\(484\) −234.315 423.500i −0.484123 0.875000i
\(485\) 0 0
\(486\) −121.500 470.567i −0.250000 0.968246i
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 25.8222 943.647i 0.0529143 1.93370i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 840.000 + 526.726i 1.69355 + 1.06195i
\(497\) 0 0
\(498\) −469.278 + 795.961i −0.942325 + 1.59832i
\(499\) 340.823 0.683011 0.341506 0.939880i \(-0.389063\pi\)
0.341506 + 0.939880i \(0.389063\pi\)
\(500\) 0 0
\(501\) −762.000 −1.52096
\(502\) 0 0
\(503\) −702.864 702.864i −1.39734 1.39734i −0.807563 0.589781i \(-0.799214\pi\)
−0.589781 0.807563i \(-0.700786\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −358.503 + 358.503i −0.707107 + 0.707107i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −41.9738 + 510.277i −0.0819801 + 0.996634i
\(513\) 591.540 591.540i 1.15310 1.15310i
\(514\) 108.444 + 420.000i 0.210980 + 0.817121i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −929.516 −1.79097
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 223.000 + 863.675i 0.423954 + 1.64197i
\(527\) 1357.65 1357.65i 2.57618 2.57618i
\(528\) 0 0
\(529\) 627.000i 1.18526i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1078.00 1.99261 0.996303 0.0859072i \(-0.0273789\pi\)
0.996303 + 0.0859072i \(0.0273789\pi\)
\(542\) 251.776 427.047i 0.464531 0.787909i
\(543\) −258.801 258.801i −0.476613 0.476613i
\(544\) 945.008 + 300.000i 1.73715 + 0.551471i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) −171.274 + 595.538i −0.312543 + 1.08675i
\(549\) 1062.00i 1.93443i
\(550\) 0 0
\(551\) 0 0
\(552\) 815.695 + 22.3209i 1.47771 + 0.0404363i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −180.000 325.331i −0.323741 0.585127i
\(557\) −657.267 657.267i −1.18001 1.18001i −0.979740 0.200272i \(-0.935817\pi\)
−0.200272 0.979740i \(-0.564183\pi\)
\(558\) −960.855 566.495i −1.72196 1.01522i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −108.894 108.894i −0.193418 0.193418i 0.603753 0.797171i \(-0.293671\pi\)
−0.797171 + 0.603753i \(0.793671\pi\)
\(564\) 81.3327 + 147.000i 0.144207 + 0.260638i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 1084.44i 1.89919i 0.313485 + 0.949593i \(0.398503\pi\)
−0.313485 + 0.949593i \(0.601497\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 31.5000 575.138i 0.0546875 0.998504i
\(577\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(578\) 681.570 1156.04i 1.17919 2.00007i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 840.000 216.887i 1.43345 0.370114i
\(587\) −603.869 + 603.869i −1.02874 + 1.02874i −0.0291633 + 0.999575i \(0.509284\pi\)
−0.999575 + 0.0291633i \(0.990716\pi\)
\(588\) −565.094 162.518i −0.961045 0.276392i
\(589\) 1920.00i 3.25976i
\(590\) 0 0
\(591\) 371.806i 0.629114i
\(592\) 0 0
\(593\) −153.362 + 153.362i −0.258621 + 0.258621i −0.824493 0.565872i \(-0.808540\pi\)
0.565872 + 0.824493i \(0.308540\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 788.720 + 788.720i 1.32114 + 1.32114i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 242.000 0.402662 0.201331 0.979523i \(-0.435473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −650.661 + 360.000i −1.07725 + 0.596026i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 880.354 456.089i 1.44795 0.750147i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1071.97 308.293i −1.75158 0.503746i
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −240.998 240.998i −0.390596 0.390596i 0.484304 0.874900i \(-0.339073\pi\)
−0.874900 + 0.484304i \(0.839073\pi\)
\(618\) 0 0
\(619\) −1022.47 −1.65181 −0.825903 0.563813i \(-0.809334\pi\)
−0.825903 + 0.563813i \(0.809334\pi\)
\(620\) 0 0
\(621\) −918.000 −1.47826
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1239.35i 1.96411i 0.188590 + 0.982056i \(0.439608\pi\)
−0.188590 + 0.982056i \(0.560392\pi\)
\(632\) 27.1210 991.113i 0.0429130 1.56822i
\(633\) 460.087 460.087i 0.726836 0.726836i
\(634\) −309.839 1200.00i −0.488705 1.89274i
\(635\) 0 0
\(636\) −360.000 650.661i −0.566038 1.02305i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −547.869 323.009i −0.853379 0.503130i
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −480.000 1859.03i −0.743034 2.87776i
\(647\) −499.217 + 499.217i −0.771588 + 0.771588i −0.978384 0.206796i \(-0.933696\pi\)
0.206796 + 0.978384i \(0.433696\pi\)
\(648\) −17.7254 + 647.758i −0.0273540 + 0.999626i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −481.996 + 481.996i −0.738125 + 0.738125i −0.972215 0.234090i \(-0.924789\pi\)
0.234090 + 0.972215i \(0.424789\pi\)
\(654\) 127.808 33.0000i 0.195426 0.0504587i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −838.000 −1.26778 −0.633888 0.773425i \(-0.718542\pi\)
−0.633888 + 0.773425i \(0.718542\pi\)
\(662\) −660.911 + 1121.00i −0.998355 + 1.69335i
\(663\) 0 0
\(664\) 894.659 847.000i 1.34738 1.27560i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 976.422 + 280.814i 1.46171 + 0.420380i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 591.500 327.267i 0.875000 0.484123i
\(677\) 920.174 + 920.174i 1.35919 + 1.35919i 0.874913 + 0.484281i \(0.160919\pi\)
0.484281 + 0.874913i \(0.339081\pi\)
\(678\) 480.428 + 283.248i 0.708595 + 0.417769i
\(679\) 0 0
\(680\) 0 0
\(681\) 402.000 0.590308
\(682\) 0 0
\(683\) 60.8112 + 60.8112i 0.0890354 + 0.0890354i 0.750222 0.661186i \(-0.229947\pi\)
−0.661186 + 0.750222i \(0.729947\pi\)
\(684\) −975.992 + 540.000i −1.42689 + 0.789474i
\(685\) 0 0
\(686\) 0 0
\(687\) 462.448 462.448i 0.673141 0.673141i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 402.790i 0.582909i −0.956585 0.291455i \(-0.905861\pi\)
0.956585 0.291455i \(-0.0941392\pi\)
\(692\) 1191.08 + 342.548i 1.72121 + 0.495011i
\(693\) 0 0
\(694\) 1134.78 293.000i 1.63514 0.422190i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −465.215 + 789.070i −0.666497 + 1.13047i
\(699\) −1394.27 −1.99467
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1260.00 325.331i 1.78470 0.460808i
\(707\) 0 0
\(708\) 0 0
\(709\) 742.000i 1.04654i −0.852166 0.523272i \(-0.824711\pi\)
0.852166 0.523272i \(-0.175289\pi\)
\(710\) 0 0
\(711\) 1115.42i 1.56880i
\(712\) 0 0
\(713\) −1489.81 + 1489.81i −2.08949 + 2.08949i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1031.99 608.436i −1.42935 0.842709i
\(723\) 1013.99 + 1013.99i 1.40248 + 1.40248i
\(724\) 236.252 + 427.000i 0.326315 + 0.589779i
\(725\) 0 0
\(726\) −181.500 702.946i −0.250000 0.968246i
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 729.000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 391.371 1360.84i 0.534660 1.85907i
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1037.00 329.204i −1.40897 0.447287i
\(737\) 0 0
\(738\) 0 0
\(739\) −216.887 −0.293487 −0.146744 0.989175i \(-0.546879\pi\)
−0.146744 + 0.989175i \(0.546879\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 363.453 + 363.453i 0.489169 + 0.489169i 0.908044 0.418875i \(-0.137575\pi\)
−0.418875 + 0.908044i \(0.637575\pi\)
\(744\) 1022.47 + 1080.00i 1.37428 + 1.45161i
\(745\) 0 0
\(746\) 0 0
\(747\) −980.050 + 980.050i −1.31198 + 1.31198i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 433.774i 0.577595i 0.957390 + 0.288798i \(0.0932555\pi\)
−0.957390 + 0.288798i \(0.906745\pi\)
\(752\) −50.0463 218.338i −0.0665510 0.290343i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) −266.904 157.360i −0.352116 0.207599i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 343.000 + 1328.43i 0.447781 + 1.73425i
\(767\) 0 0
\(768\) −252.315 + 725.370i −0.328535 + 0.944492i
\(769\) 578.000i 0.751625i 0.926696 + 0.375813i \(0.122636\pi\)
−0.926696 + 0.375813i \(0.877364\pi\)
\(770\) 0 0
\(771\) 650.661i 0.843919i
\(772\) 0 0
\(773\) −175.271 + 175.271i −0.226742 + 0.226742i −0.811330 0.584588i \(-0.801256\pi\)
0.584588 + 0.811330i \(0.301256\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −1070.05 + 1814.95i −1.36835 + 2.32091i
\(783\) 0 0
\(784\) 664.217 + 416.500i 0.847215 + 0.531250i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) −137.019 + 476.430i −0.173882 + 0.604607i
\(789\) 1338.00i 1.69582i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −720.000 1301.32i −0.904523 1.63483i
\(797\) −963.992 963.992i −1.20953 1.20953i −0.971181 0.238345i \(-0.923395\pi\)
−0.238345 0.971181i \(-0.576605\pi\)
\(798\) 0 0
\(799\) −433.774 −0.542896
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 1208.37i 1.48998i 0.667078 + 0.744988i \(0.267545\pi\)
−0.667078 + 0.744988i \(0.732455\pi\)
\(812\) 0 0
\(813\) 525.814 525.814i 0.646757 0.646757i
\(814\) 0 0
\(815\) 0 0
\(816\) 1260.00 + 790.089i 1.54412 + 0.968246i
\(817\) 0 0
\(818\) 144.237 244.646i 0.176329 0.299078i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −472.079 + 800.713i −0.574306 + 0.974103i
\(823\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −264.458 + 264.458i −0.319780 + 0.319780i −0.848683 0.528903i \(-0.822604\pi\)
0.528903 + 0.848683i \(0.322604\pi\)
\(828\) 1176.32 + 338.304i 1.42068 + 0.408579i
\(829\) 502.000i 0.605549i 0.953062 + 0.302774i \(0.0979129\pi\)
−0.953062 + 0.302774i \(0.902087\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1073.54 1073.54i 1.28876 1.28876i
\(834\) −139.427 540.000i −0.167179 0.647482i
\(835\) 0 0
\(836\) 0 0
\(837\) −1183.08 1183.08i −1.41348 1.41348i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −841.000 −1.00000
\(842\) 1037.16 + 611.483i 1.23178 + 0.726227i
\(843\) 0 0
\(844\) −759.105 + 420.000i −0.899413 + 0.497630i
\(845\) 0 0
\(846\) 63.0000 + 243.998i 0.0744681 + 0.288414i
\(847\) 0 0
\(848\) 221.518 + 966.421i 0.261224 + 1.13965i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 583.000 + 615.804i 0.681075 + 0.719398i
\(857\) 284.816 + 284.816i 0.332340 + 0.332340i 0.853475 0.521134i \(-0.174491\pi\)
−0.521134 + 0.853475i \(0.674491\pi\)
\(858\) 0 0
\(859\) 1704.11 1.98383 0.991917 0.126892i \(-0.0405001\pi\)
0.991917 + 0.126892i \(0.0405001\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 193.747 + 193.747i 0.224504 + 0.224504i 0.810392 0.585888i \(-0.199254\pi\)
−0.585888 + 0.810392i \(0.699254\pi\)
\(864\) 261.426 823.500i 0.302577 0.953125i
\(865\) 0 0
\(866\) 0 0
\(867\) 1423.41 1423.41i 1.64176 1.64176i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −175.934 4.81430i −0.201759 0.00552099i
\(873\) 0 0
\(874\) 526.726 + 2040.00i 0.602661 + 2.33410i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(878\) −1067.62 629.439i −1.21596 0.716901i
\(879\) 1301.32 1.48046
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −759.781 447.947i −0.861430 0.507877i
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −283.000 1096.05i −0.319413 1.23708i
\(887\) −1197.84 + 1197.84i −1.35044 + 1.35044i −0.465269 + 0.885169i \(0.654042\pi\)
−0.885169 + 0.465269i \(0.845958\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −306.725 + 306.725i −0.343477 + 0.343477i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 1920.00 2.13097
\(902\) 0 0
\(903\) 0 0
\(904\) −511.234 540.000i −0.565524 0.597345i
\(905\) 0 0
\(906\) −1080.00 + 278.855i −1.19205 + 0.307787i
\(907\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) −515.120 148.146i −0.567313 0.163156i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 1449.63 332.278i 1.58951 0.364339i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −763.000 + 422.155i −0.832969 + 0.460868i
\(917\) 0 0
\(918\) −1441.28 849.743i −1.57003 0.925646i
\(919\) −1301.32 −1.41602 −0.708010 0.706202i \(-0.750407\pi\)
−0.708010 + 0.706202i \(0.750407\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 1518.21i 1.63073i
\(932\) 1786.61 + 513.821i 1.91697 + 0.551310i
\(933\) 0 0
\(934\) −670.026 + 173.000i −0.717373 + 0.185225i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1112.99 1112.99i 1.17528 1.17528i 0.194342 0.980934i \(-0.437743\pi\)
0.980934 0.194342i \(-0.0622571\pi\)
\(948\) 411.057 1429.29i 0.433604 1.50769i
\(949\) 0 0
\(950\) 0 0
\(951\) 1859.03i 1.95482i
\(952\) 0 0
\(953\) 854.447 854.447i 0.896587 0.896587i −0.0985458 0.995133i \(-0.531419\pi\)
0.995133 + 0.0985458i \(0.0314191\pi\)
\(954\) −278.855 1080.00i −0.292301 1.13208i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2879.00 −2.99584
\(962\) 0 0
\(963\) −674.580 674.580i −0.700498 0.700498i
\(964\) −925.643 1673.00i −0.960211 1.73548i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) −26.4787 + 967.638i −0.0273540 + 0.999626i
\(969\) 2880.00i 2.97214i
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −268.653 + 934.136i −0.276392 + 0.961045i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1003.00 + 1599.54i −1.02766 + 1.63888i
\(977\) −197.180 197.180i −0.201822 0.201822i 0.598958 0.800780i \(-0.295582\pi\)
−0.800780 + 0.598958i \(0.795582\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 198.000 0.201835
\(982\) 0 0
\(983\) −1381.69 1381.69i −1.40558 1.40558i −0.780798 0.624783i \(-0.785187\pi\)
−0.624783 0.780798i \(-0.714813\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1735.10i 1.75085i −0.483350 0.875427i \(-0.660580\pi\)
0.483350 0.875427i \(-0.339420\pi\)
\(992\) −912.179 1760.71i −0.919535 1.77491i
\(993\) −1380.26 + 1380.26i −1.38999 + 1.38999i
\(994\) 0 0
\(995\) 0 0
\(996\) 1617.00 894.659i 1.62349 0.898252i
\(997\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) −587.189 346.192i −0.588366 0.346885i
\(999\) 0 0
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 300.3.l.e.143.1 yes 8
3.2 odd 2 inner 300.3.l.e.143.4 yes 8
4.3 odd 2 inner 300.3.l.e.143.2 yes 8
5.2 odd 4 inner 300.3.l.e.107.3 yes 8
5.3 odd 4 inner 300.3.l.e.107.2 yes 8
5.4 even 2 inner 300.3.l.e.143.4 yes 8
12.11 even 2 inner 300.3.l.e.143.3 yes 8
15.2 even 4 inner 300.3.l.e.107.2 yes 8
15.8 even 4 inner 300.3.l.e.107.3 yes 8
15.14 odd 2 CM 300.3.l.e.143.1 yes 8
20.3 even 4 inner 300.3.l.e.107.1 8
20.7 even 4 inner 300.3.l.e.107.4 yes 8
20.19 odd 2 inner 300.3.l.e.143.3 yes 8
60.23 odd 4 inner 300.3.l.e.107.4 yes 8
60.47 odd 4 inner 300.3.l.e.107.1 8
60.59 even 2 inner 300.3.l.e.143.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.l.e.107.1 8 20.3 even 4 inner
300.3.l.e.107.1 8 60.47 odd 4 inner
300.3.l.e.107.2 yes 8 5.3 odd 4 inner
300.3.l.e.107.2 yes 8 15.2 even 4 inner
300.3.l.e.107.3 yes 8 5.2 odd 4 inner
300.3.l.e.107.3 yes 8 15.8 even 4 inner
300.3.l.e.107.4 yes 8 20.7 even 4 inner
300.3.l.e.107.4 yes 8 60.23 odd 4 inner
300.3.l.e.143.1 yes 8 1.1 even 1 trivial
300.3.l.e.143.1 yes 8 15.14 odd 2 CM
300.3.l.e.143.2 yes 8 4.3 odd 2 inner
300.3.l.e.143.2 yes 8 60.59 even 2 inner
300.3.l.e.143.3 yes 8 12.11 even 2 inner
300.3.l.e.143.3 yes 8 20.19 odd 2 inner
300.3.l.e.143.4 yes 8 3.2 odd 2 inner
300.3.l.e.143.4 yes 8 5.4 even 2 inner