Properties

Label 1728.3.o.g.1279.8
Level $1728$
Weight $3$
Character 1728.1279
Analytic conductor $47.085$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,3,Mod(127,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.o (of order \(6\), 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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + 1552 x^{8} - 3648 x^{7} + 6784 x^{6} - 9216 x^{5} + 19456 x^{4} - 30720 x^{3} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1279.8
Root \(1.63139 - 1.15696i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1279
Dual form 1728.3.o.g.127.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.07403 - 5.32438i) q^{5} +(0.511543 - 0.295340i) q^{7} +O(q^{10})\) \(q+(3.07403 - 5.32438i) q^{5} +(0.511543 - 0.295340i) q^{7} +(15.1205 - 8.72982i) q^{11} +(0.892255 - 1.54543i) q^{13} +16.9171 q^{17} -19.5058i q^{19} +(6.86778 + 3.96511i) q^{23} +(-6.39933 - 11.0840i) q^{25} +(3.17517 + 5.49956i) q^{29} +(27.6558 + 15.9671i) q^{31} -3.63153i q^{35} -58.2834 q^{37} +(2.66948 - 4.62368i) q^{41} +(33.9324 - 19.5909i) q^{43} +(9.64117 - 5.56633i) q^{47} +(-24.3255 + 42.1331i) q^{49} +35.8770 q^{53} -107.343i q^{55} +(-20.8974 - 12.0651i) q^{59} +(37.9460 + 65.7244i) q^{61} +(-5.48564 - 9.50141i) q^{65} +(31.8200 + 18.3713i) q^{67} -87.8370i q^{71} -60.0423 q^{73} +(5.15652 - 8.93136i) q^{77} +(32.1841 - 18.5815i) q^{79} +(-66.0281 + 38.1214i) q^{83} +(52.0037 - 90.0730i) q^{85} +27.5873 q^{89} -1.05407i q^{91} +(-103.856 - 59.9614i) q^{95} +(13.0585 + 22.6180i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{5} + 46 q^{13} - 12 q^{17} - 30 q^{25} + 42 q^{29} - 56 q^{37} - 84 q^{41} + 58 q^{49} - 72 q^{53} + 34 q^{61} + 30 q^{65} + 116 q^{73} - 330 q^{77} + 140 q^{85} + 384 q^{89} - 148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.07403 5.32438i 0.614806 1.06488i −0.375612 0.926777i \(-0.622568\pi\)
0.990418 0.138099i \(-0.0440991\pi\)
\(6\) 0 0
\(7\) 0.511543 0.295340i 0.0730776 0.0421914i −0.463016 0.886350i \(-0.653233\pi\)
0.536094 + 0.844159i \(0.319899\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 15.1205 8.72982i 1.37459 0.793620i 0.383088 0.923712i \(-0.374861\pi\)
0.991502 + 0.130092i \(0.0415274\pi\)
\(12\) 0 0
\(13\) 0.892255 1.54543i 0.0686350 0.118879i −0.829666 0.558261i \(-0.811469\pi\)
0.898301 + 0.439381i \(0.144802\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.9171 0.995123 0.497562 0.867429i \(-0.334229\pi\)
0.497562 + 0.867429i \(0.334229\pi\)
\(18\) 0 0
\(19\) 19.5058i 1.02662i −0.858203 0.513310i \(-0.828419\pi\)
0.858203 0.513310i \(-0.171581\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.86778 + 3.96511i 0.298599 + 0.172396i 0.641813 0.766861i \(-0.278182\pi\)
−0.343214 + 0.939257i \(0.611516\pi\)
\(24\) 0 0
\(25\) −6.39933 11.0840i −0.255973 0.443359i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.17517 + 5.49956i 0.109489 + 0.189640i 0.915563 0.402174i \(-0.131745\pi\)
−0.806075 + 0.591814i \(0.798412\pi\)
\(30\) 0 0
\(31\) 27.6558 + 15.9671i 0.892124 + 0.515068i 0.874637 0.484779i \(-0.161100\pi\)
0.0174873 + 0.999847i \(0.494433\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.63153i 0.103758i
\(36\) 0 0
\(37\) −58.2834 −1.57523 −0.787614 0.616169i \(-0.788684\pi\)
−0.787614 + 0.616169i \(0.788684\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.66948 4.62368i 0.0651093 0.112773i −0.831633 0.555325i \(-0.812594\pi\)
0.896742 + 0.442553i \(0.145927\pi\)
\(42\) 0 0
\(43\) 33.9324 19.5909i 0.789126 0.455602i −0.0505290 0.998723i \(-0.516091\pi\)
0.839655 + 0.543121i \(0.182757\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.64117 5.56633i 0.205131 0.118433i −0.393915 0.919147i \(-0.628880\pi\)
0.599047 + 0.800714i \(0.295546\pi\)
\(48\) 0 0
\(49\) −24.3255 + 42.1331i −0.496440 + 0.859859i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 35.8770 0.676925 0.338462 0.940980i \(-0.390093\pi\)
0.338462 + 0.940980i \(0.390093\pi\)
\(54\) 0 0
\(55\) 107.343i 1.95169i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −20.8974 12.0651i −0.354194 0.204494i 0.312337 0.949971i \(-0.398888\pi\)
−0.666531 + 0.745477i \(0.732222\pi\)
\(60\) 0 0
\(61\) 37.9460 + 65.7244i 0.622066 + 1.07745i 0.989100 + 0.147243i \(0.0470398\pi\)
−0.367034 + 0.930207i \(0.619627\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.48564 9.50141i −0.0843944 0.146175i
\(66\) 0 0
\(67\) 31.8200 + 18.3713i 0.474925 + 0.274198i 0.718299 0.695734i \(-0.244921\pi\)
−0.243374 + 0.969933i \(0.578254\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 87.8370i 1.23714i −0.785730 0.618570i \(-0.787712\pi\)
0.785730 0.618570i \(-0.212288\pi\)
\(72\) 0 0
\(73\) −60.0423 −0.822498 −0.411249 0.911523i \(-0.634907\pi\)
−0.411249 + 0.911523i \(0.634907\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.15652 8.93136i 0.0669678 0.115992i
\(78\) 0 0
\(79\) 32.1841 18.5815i 0.407394 0.235209i −0.282275 0.959333i \(-0.591089\pi\)
0.689669 + 0.724124i \(0.257756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −66.0281 + 38.1214i −0.795520 + 0.459294i −0.841902 0.539630i \(-0.818564\pi\)
0.0463824 + 0.998924i \(0.485231\pi\)
\(84\) 0 0
\(85\) 52.0037 90.0730i 0.611808 1.05968i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 27.5873 0.309969 0.154985 0.987917i \(-0.450467\pi\)
0.154985 + 0.987917i \(0.450467\pi\)
\(90\) 0 0
\(91\) 1.05407i 0.0115832i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −103.856 59.9614i −1.09322 0.631172i
\(96\) 0 0
\(97\) 13.0585 + 22.6180i 0.134624 + 0.233176i 0.925454 0.378861i \(-0.123684\pi\)
−0.790830 + 0.612036i \(0.790351\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.8831 22.3142i −0.127556 0.220933i 0.795173 0.606382i \(-0.207380\pi\)
−0.922729 + 0.385449i \(0.874047\pi\)
\(102\) 0 0
\(103\) −16.9947 9.81187i −0.164997 0.0952609i 0.415228 0.909717i \(-0.363702\pi\)
−0.580225 + 0.814457i \(0.697035\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 183.200i 1.71215i −0.516850 0.856076i \(-0.672895\pi\)
0.516850 0.856076i \(-0.327105\pi\)
\(108\) 0 0
\(109\) −100.841 −0.925147 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.12484 15.8047i 0.0807508 0.139865i −0.822822 0.568299i \(-0.807602\pi\)
0.903573 + 0.428435i \(0.140935\pi\)
\(114\) 0 0
\(115\) 42.2235 24.3778i 0.367161 0.211981i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.65383 4.99629i 0.0727212 0.0419856i
\(120\) 0 0
\(121\) 91.9194 159.209i 0.759664 1.31578i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 75.0146 0.600117
\(126\) 0 0
\(127\) 164.386i 1.29438i 0.762331 + 0.647188i \(0.224055\pi\)
−0.762331 + 0.647188i \(0.775945\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −123.421 71.2570i −0.942143 0.543947i −0.0515116 0.998672i \(-0.516404\pi\)
−0.890631 + 0.454726i \(0.849737\pi\)
\(132\) 0 0
\(133\) −5.76083 9.97805i −0.0433145 0.0750229i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.08176 + 5.33777i 0.0224946 + 0.0389618i 0.877054 0.480393i \(-0.159506\pi\)
−0.854559 + 0.519354i \(0.826172\pi\)
\(138\) 0 0
\(139\) −103.168 59.5642i −0.742218 0.428519i 0.0806575 0.996742i \(-0.474298\pi\)
−0.822875 + 0.568222i \(0.807631\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 31.1569i 0.217880i
\(144\) 0 0
\(145\) 39.0423 0.269257
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −103.365 + 179.034i −0.693726 + 1.20157i 0.276882 + 0.960904i \(0.410699\pi\)
−0.970608 + 0.240665i \(0.922634\pi\)
\(150\) 0 0
\(151\) 127.422 73.5670i 0.843853 0.487199i −0.0147190 0.999892i \(-0.504685\pi\)
0.858572 + 0.512693i \(0.171352\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 170.030 98.1668i 1.09697 0.633334i
\(156\) 0 0
\(157\) −31.4395 + 54.4548i −0.200251 + 0.346846i −0.948609 0.316449i \(-0.897509\pi\)
0.748358 + 0.663295i \(0.230843\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.68422 0.0290946
\(162\) 0 0
\(163\) 143.325i 0.879292i −0.898171 0.439646i \(-0.855104\pi\)
0.898171 0.439646i \(-0.144896\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −150.531 86.9089i −0.901381 0.520413i −0.0237332 0.999718i \(-0.507555\pi\)
−0.877648 + 0.479306i \(0.840889\pi\)
\(168\) 0 0
\(169\) 82.9078 + 143.600i 0.490578 + 0.849707i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −125.806 217.902i −0.727201 1.25955i −0.958062 0.286562i \(-0.907488\pi\)
0.230861 0.972987i \(-0.425846\pi\)
\(174\) 0 0
\(175\) −6.54707 3.77995i −0.0374118 0.0215997i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 96.0059i 0.536346i −0.963371 0.268173i \(-0.913580\pi\)
0.963371 0.268173i \(-0.0864199\pi\)
\(180\) 0 0
\(181\) 328.757 1.81634 0.908170 0.418603i \(-0.137480\pi\)
0.908170 + 0.418603i \(0.137480\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −179.165 + 310.323i −0.968460 + 1.67742i
\(186\) 0 0
\(187\) 255.795 147.683i 1.36789 0.789749i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.351914 0.203178i 0.00184248 0.00106376i −0.499078 0.866557i \(-0.666328\pi\)
0.500921 + 0.865493i \(0.332995\pi\)
\(192\) 0 0
\(193\) −31.2230 + 54.0798i −0.161777 + 0.280206i −0.935506 0.353311i \(-0.885056\pi\)
0.773729 + 0.633517i \(0.218389\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −207.861 −1.05513 −0.527566 0.849514i \(-0.676895\pi\)
−0.527566 + 0.849514i \(0.676895\pi\)
\(198\) 0 0
\(199\) 299.128i 1.50316i −0.659643 0.751579i \(-0.729293\pi\)
0.659643 0.751579i \(-0.270707\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.24848 + 1.87551i 0.0160024 + 0.00923896i
\(204\) 0 0
\(205\) −16.4121 28.4266i −0.0800592 0.138667i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −170.282 294.937i −0.814746 1.41118i
\(210\) 0 0
\(211\) 141.744 + 81.8360i 0.671773 + 0.387848i 0.796748 0.604311i \(-0.206552\pi\)
−0.124975 + 0.992160i \(0.539885\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 240.892i 1.12043i
\(216\) 0 0
\(217\) 18.8629 0.0869257
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.0944 26.1442i 0.0683003 0.118300i
\(222\) 0 0
\(223\) 330.681 190.919i 1.48287 0.856138i 0.483063 0.875586i \(-0.339524\pi\)
0.999811 + 0.0194478i \(0.00619081\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −51.5472 + 29.7608i −0.227080 + 0.131105i −0.609224 0.792998i \(-0.708519\pi\)
0.382144 + 0.924103i \(0.375186\pi\)
\(228\) 0 0
\(229\) −64.4366 + 111.608i −0.281383 + 0.487369i −0.971726 0.236113i \(-0.924126\pi\)
0.690343 + 0.723482i \(0.257460\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.9939 −0.0643513 −0.0321757 0.999482i \(-0.510244\pi\)
−0.0321757 + 0.999482i \(0.510244\pi\)
\(234\) 0 0
\(235\) 68.4443i 0.291252i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 315.244 + 182.006i 1.31901 + 0.761532i 0.983570 0.180529i \(-0.0577811\pi\)
0.335442 + 0.942061i \(0.391114\pi\)
\(240\) 0 0
\(241\) −40.5235 70.1888i −0.168147 0.291240i 0.769621 0.638501i \(-0.220445\pi\)
−0.937769 + 0.347261i \(0.887112\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 149.555 + 259.037i 0.610428 + 1.05729i
\(246\) 0 0
\(247\) −30.1448 17.4041i −0.122044 0.0704620i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 281.883i 1.12304i 0.827463 + 0.561520i \(0.189783\pi\)
−0.827463 + 0.561520i \(0.810217\pi\)
\(252\) 0 0
\(253\) 138.459 0.547268
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 37.6564 65.2227i 0.146523 0.253785i −0.783417 0.621496i \(-0.786525\pi\)
0.929940 + 0.367711i \(0.119858\pi\)
\(258\) 0 0
\(259\) −29.8145 + 17.2134i −0.115114 + 0.0664610i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 105.914 61.1497i 0.402716 0.232508i −0.284939 0.958546i \(-0.591973\pi\)
0.687655 + 0.726037i \(0.258640\pi\)
\(264\) 0 0
\(265\) 110.287 191.023i 0.416178 0.720841i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 280.452 1.04257 0.521287 0.853382i \(-0.325452\pi\)
0.521287 + 0.853382i \(0.325452\pi\)
\(270\) 0 0
\(271\) 81.4468i 0.300542i −0.988645 0.150271i \(-0.951985\pi\)
0.988645 0.150271i \(-0.0480146\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −193.522 111.730i −0.703716 0.406291i
\(276\) 0 0
\(277\) −224.861 389.471i −0.811774 1.40603i −0.911622 0.411031i \(-0.865169\pi\)
0.0998479 0.995003i \(-0.468164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 37.8649 + 65.5838i 0.134750 + 0.233394i 0.925502 0.378743i \(-0.123643\pi\)
−0.790752 + 0.612137i \(0.790310\pi\)
\(282\) 0 0
\(283\) 322.061 + 185.942i 1.13803 + 0.657039i 0.945941 0.324339i \(-0.105142\pi\)
0.192084 + 0.981378i \(0.438475\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.15361i 0.0109882i
\(288\) 0 0
\(289\) −2.81196 −0.00972996
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −66.3946 + 114.999i −0.226603 + 0.392488i −0.956799 0.290750i \(-0.906095\pi\)
0.730196 + 0.683237i \(0.239429\pi\)
\(294\) 0 0
\(295\) −128.479 + 74.1772i −0.435521 + 0.251448i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.2556 7.07579i 0.0409887 0.0236648i
\(300\) 0 0
\(301\) 11.5719 20.0432i 0.0384450 0.0665886i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 466.589 1.52980
\(306\) 0 0
\(307\) 336.514i 1.09614i −0.836434 0.548068i \(-0.815363\pi\)
0.836434 0.548068i \(-0.184637\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −304.206 175.634i −0.978156 0.564738i −0.0764428 0.997074i \(-0.524356\pi\)
−0.901713 + 0.432336i \(0.857690\pi\)
\(312\) 0 0
\(313\) −95.4299 165.289i −0.304888 0.528081i 0.672349 0.740235i \(-0.265286\pi\)
−0.977236 + 0.212154i \(0.931952\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −202.797 351.255i −0.639738 1.10806i −0.985490 0.169733i \(-0.945709\pi\)
0.345752 0.938326i \(-0.387624\pi\)
\(318\) 0 0
\(319\) 96.0203 + 55.4374i 0.301004 + 0.173785i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 329.981i 1.02161i
\(324\) 0 0
\(325\) −22.8393 −0.0702749
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.28792 5.69484i 0.00999368 0.0173096i
\(330\) 0 0
\(331\) −384.104 + 221.763i −1.16044 + 0.669978i −0.951408 0.307932i \(-0.900363\pi\)
−0.209027 + 0.977910i \(0.567030\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 195.631 112.948i 0.583974 0.337158i
\(336\) 0 0
\(337\) −254.239 + 440.356i −0.754420 + 1.30669i 0.191243 + 0.981543i \(0.438748\pi\)
−0.945662 + 0.325150i \(0.894585\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 557.560 1.63507
\(342\) 0 0
\(343\) 57.6805i 0.168165i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 492.773 + 284.503i 1.42010 + 0.819893i 0.996307 0.0858678i \(-0.0273663\pi\)
0.423790 + 0.905761i \(0.360700\pi\)
\(348\) 0 0
\(349\) 206.901 + 358.363i 0.592840 + 1.02683i 0.993848 + 0.110754i \(0.0353266\pi\)
−0.401008 + 0.916074i \(0.631340\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −62.3070 107.919i −0.176507 0.305719i 0.764175 0.645009i \(-0.223147\pi\)
−0.940682 + 0.339290i \(0.889813\pi\)
\(354\) 0 0
\(355\) −467.677 270.014i −1.31740 0.760602i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 303.196i 0.844557i 0.906466 + 0.422278i \(0.138770\pi\)
−0.906466 + 0.422278i \(0.861230\pi\)
\(360\) 0 0
\(361\) −19.4752 −0.0539480
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −184.572 + 319.688i −0.505677 + 0.875858i
\(366\) 0 0
\(367\) −615.571 + 355.400i −1.67730 + 0.968392i −0.713936 + 0.700211i \(0.753089\pi\)
−0.963369 + 0.268181i \(0.913578\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 18.3527 10.5959i 0.0494681 0.0285604i
\(372\) 0 0
\(373\) −166.740 + 288.803i −0.447025 + 0.774271i −0.998191 0.0601254i \(-0.980850\pi\)
0.551166 + 0.834396i \(0.314183\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.3323 0.0300590
\(378\) 0 0
\(379\) 662.686i 1.74851i 0.485465 + 0.874256i \(0.338650\pi\)
−0.485465 + 0.874256i \(0.661350\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 69.9008 + 40.3572i 0.182509 + 0.105371i 0.588471 0.808518i \(-0.299730\pi\)
−0.405962 + 0.913890i \(0.633064\pi\)
\(384\) 0 0
\(385\) −31.7026 54.9106i −0.0823445 0.142625i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −346.006 599.301i −0.889476 1.54062i −0.840495 0.541819i \(-0.817736\pi\)
−0.0489809 0.998800i \(-0.515597\pi\)
\(390\) 0 0
\(391\) 116.183 + 67.0782i 0.297143 + 0.171556i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 228.481i 0.578432i
\(396\) 0 0
\(397\) −657.713 −1.65671 −0.828354 0.560206i \(-0.810722\pi\)
−0.828354 + 0.560206i \(0.810722\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −296.433 + 513.437i −0.739235 + 1.28039i 0.213606 + 0.976920i \(0.431479\pi\)
−0.952840 + 0.303472i \(0.901854\pi\)
\(402\) 0 0
\(403\) 49.3521 28.4935i 0.122462 0.0707034i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −881.274 + 508.804i −2.16529 + 1.25013i
\(408\) 0 0
\(409\) −161.594 + 279.889i −0.395095 + 0.684325i −0.993113 0.117157i \(-0.962622\pi\)
0.598018 + 0.801483i \(0.295955\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −14.2533 −0.0345115
\(414\) 0 0
\(415\) 468.745i 1.12951i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −222.744 128.601i −0.531608 0.306924i 0.210063 0.977688i \(-0.432633\pi\)
−0.741671 + 0.670764i \(0.765967\pi\)
\(420\) 0 0
\(421\) −41.9905 72.7297i −0.0997400 0.172755i 0.811837 0.583884i \(-0.198468\pi\)
−0.911577 + 0.411129i \(0.865134\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −108.258 187.509i −0.254725 0.441197i
\(426\) 0 0
\(427\) 38.8221 + 22.4139i 0.0909182 + 0.0524917i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 144.348i 0.334914i 0.985879 + 0.167457i \(0.0535555\pi\)
−0.985879 + 0.167457i \(0.946445\pi\)
\(432\) 0 0
\(433\) 395.353 0.913057 0.456528 0.889709i \(-0.349093\pi\)
0.456528 + 0.889709i \(0.349093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 77.3426 133.961i 0.176985 0.306548i
\(438\) 0 0
\(439\) −194.776 + 112.454i −0.443682 + 0.256160i −0.705158 0.709050i \(-0.749124\pi\)
0.261476 + 0.965210i \(0.415791\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −369.184 + 213.148i −0.833373 + 0.481148i −0.855006 0.518618i \(-0.826447\pi\)
0.0216335 + 0.999766i \(0.493113\pi\)
\(444\) 0 0
\(445\) 84.8041 146.885i 0.190571 0.330079i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 406.744 0.905888 0.452944 0.891539i \(-0.350374\pi\)
0.452944 + 0.891539i \(0.350374\pi\)
\(450\) 0 0
\(451\) 93.2163i 0.206688i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.61228 3.24025i −0.0123347 0.00712144i
\(456\) 0 0
\(457\) −159.600 276.435i −0.349234 0.604891i 0.636879 0.770963i \(-0.280225\pi\)
−0.986114 + 0.166072i \(0.946892\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 293.888 + 509.029i 0.637501 + 1.10418i 0.985979 + 0.166867i \(0.0533652\pi\)
−0.348478 + 0.937317i \(0.613302\pi\)
\(462\) 0 0
\(463\) 230.088 + 132.841i 0.496950 + 0.286914i 0.727453 0.686157i \(-0.240704\pi\)
−0.230503 + 0.973072i \(0.574037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 794.598i 1.70149i 0.525575 + 0.850747i \(0.323850\pi\)
−0.525575 + 0.850747i \(0.676150\pi\)
\(468\) 0 0
\(469\) 21.7031 0.0462752
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 342.050 592.447i 0.723149 1.25253i
\(474\) 0 0
\(475\) −216.201 + 124.824i −0.455161 + 0.262787i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 572.964 330.801i 1.19617 0.690607i 0.236468 0.971639i \(-0.424010\pi\)
0.959698 + 0.281033i \(0.0906769\pi\)
\(480\) 0 0
\(481\) −52.0037 + 90.0730i −0.108116 + 0.187262i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 160.569 0.331071
\(486\) 0 0
\(487\) 57.1525i 0.117356i 0.998277 + 0.0586781i \(0.0186886\pi\)
−0.998277 + 0.0586781i \(0.981311\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 48.6600 + 28.0939i 0.0991040 + 0.0572177i 0.548733 0.835998i \(-0.315110\pi\)
−0.449629 + 0.893215i \(0.648444\pi\)
\(492\) 0 0
\(493\) 53.7147 + 93.0366i 0.108955 + 0.188715i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −25.9417 44.9324i −0.0521967 0.0904073i
\(498\) 0 0
\(499\) −522.225 301.507i −1.04654 0.604222i −0.124863 0.992174i \(-0.539849\pi\)
−0.921679 + 0.387952i \(0.873183\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 549.354i 1.09216i 0.837734 + 0.546078i \(0.183880\pi\)
−0.837734 + 0.546078i \(0.816120\pi\)
\(504\) 0 0
\(505\) −158.413 −0.313688
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −119.464 + 206.918i −0.234704 + 0.406519i −0.959187 0.282774i \(-0.908745\pi\)
0.724483 + 0.689293i \(0.242079\pi\)
\(510\) 0 0
\(511\) −30.7143 + 17.7329i −0.0601062 + 0.0347023i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −104.484 + 60.3240i −0.202882 + 0.117134i
\(516\) 0 0
\(517\) 97.1862 168.331i 0.187981 0.325593i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 567.711 1.08966 0.544828 0.838548i \(-0.316595\pi\)
0.544828 + 0.838548i \(0.316595\pi\)
\(522\) 0 0
\(523\) 941.999i 1.80114i 0.434706 + 0.900572i \(0.356852\pi\)
−0.434706 + 0.900572i \(0.643148\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 467.856 + 270.117i 0.887773 + 0.512556i
\(528\) 0 0
\(529\) −233.056 403.664i −0.440559 0.763071i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.76372 8.25100i −0.00893755 0.0154803i
\(534\) 0 0
\(535\) −975.428 563.163i −1.82323 1.05264i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 849.430i 1.57594i
\(540\) 0 0
\(541\) 242.245 0.447772 0.223886 0.974615i \(-0.428126\pi\)
0.223886 + 0.974615i \(0.428126\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −309.988 + 536.915i −0.568786 + 0.985166i
\(546\) 0 0
\(547\) 170.503 98.4402i 0.311706 0.179964i −0.335983 0.941868i \(-0.609069\pi\)
0.647690 + 0.761904i \(0.275735\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 107.273 61.9342i 0.194688 0.112403i
\(552\) 0 0
\(553\) 10.9757 19.0105i 0.0198476 0.0343770i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 958.121 1.72015 0.860073 0.510171i \(-0.170418\pi\)
0.860073 + 0.510171i \(0.170418\pi\)
\(558\) 0 0
\(559\) 69.9202i 0.125081i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 165.774 + 95.7097i 0.294448 + 0.169999i 0.639946 0.768420i \(-0.278957\pi\)
−0.345498 + 0.938419i \(0.612290\pi\)
\(564\) 0 0
\(565\) −56.1001 97.1682i −0.0992922 0.171979i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −228.215 395.280i −0.401081 0.694693i 0.592775 0.805368i \(-0.298032\pi\)
−0.993857 + 0.110675i \(0.964699\pi\)
\(570\) 0 0
\(571\) 842.764 + 486.570i 1.47594 + 0.852136i 0.999632 0.0271399i \(-0.00863995\pi\)
0.476312 + 0.879276i \(0.341973\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 101.496i 0.176515i
\(576\) 0 0
\(577\) 138.527 0.240081 0.120040 0.992769i \(-0.461698\pi\)
0.120040 + 0.992769i \(0.461698\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −22.5175 + 39.0015i −0.0387565 + 0.0671282i
\(582\) 0 0
\(583\) 542.478 313.200i 0.930494 0.537221i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 620.808 358.424i 1.05759 0.610602i 0.132829 0.991139i \(-0.457594\pi\)
0.924766 + 0.380537i \(0.124261\pi\)
\(588\) 0 0
\(589\) 311.451 539.449i 0.528779 0.915872i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −542.129 −0.914214 −0.457107 0.889412i \(-0.651114\pi\)
−0.457107 + 0.889412i \(0.651114\pi\)
\(594\) 0 0
\(595\) 61.4350i 0.103252i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 245.527 + 141.755i 0.409895 + 0.236653i 0.690744 0.723099i \(-0.257283\pi\)
−0.280850 + 0.959752i \(0.590616\pi\)
\(600\) 0 0
\(601\) 377.424 + 653.717i 0.627993 + 1.08772i 0.987954 + 0.154748i \(0.0494567\pi\)
−0.359961 + 0.932967i \(0.617210\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −565.126 978.827i −0.934093 1.61790i
\(606\) 0 0
\(607\) 77.2227 + 44.5845i 0.127220 + 0.0734506i 0.562260 0.826961i \(-0.309932\pi\)
−0.435039 + 0.900411i \(0.643266\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19.8664i 0.0325145i
\(612\) 0 0
\(613\) 316.779 0.516769 0.258385 0.966042i \(-0.416810\pi\)
0.258385 + 0.966042i \(0.416810\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −534.934 + 926.533i −0.866992 + 1.50167i −0.00193565 + 0.999998i \(0.500616\pi\)
−0.865056 + 0.501675i \(0.832717\pi\)
\(618\) 0 0
\(619\) −578.542 + 334.021i −0.934640 + 0.539615i −0.888276 0.459310i \(-0.848097\pi\)
−0.0463638 + 0.998925i \(0.514763\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.1121 8.14762i 0.0226518 0.0130780i
\(624\) 0 0
\(625\) 390.580 676.505i 0.624929 1.08241i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −985.986 −1.56755
\(630\) 0 0
\(631\) 150.631i 0.238718i 0.992851 + 0.119359i \(0.0380839\pi\)
−0.992851 + 0.119359i \(0.961916\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 875.252 + 505.327i 1.37835 + 0.795790i
\(636\) 0 0
\(637\) 43.4092 + 75.1869i 0.0681463 + 0.118033i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −351.521 608.852i −0.548395 0.949847i −0.998385 0.0568139i \(-0.981906\pi\)
0.449990 0.893034i \(-0.351427\pi\)
\(642\) 0 0
\(643\) −742.057 428.427i −1.15405 0.666293i −0.204182 0.978933i \(-0.565454\pi\)
−0.949872 + 0.312639i \(0.898787\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 156.257i 0.241510i −0.992682 0.120755i \(-0.961468\pi\)
0.992682 0.120755i \(-0.0385316\pi\)
\(648\) 0 0
\(649\) −421.306 −0.649162
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −441.773 + 765.173i −0.676528 + 1.17178i 0.299492 + 0.954099i \(0.403183\pi\)
−0.976020 + 0.217682i \(0.930151\pi\)
\(654\) 0 0
\(655\) −758.798 + 438.092i −1.15847 + 0.668843i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −379.533 + 219.123i −0.575922 + 0.332509i −0.759511 0.650494i \(-0.774562\pi\)
0.183589 + 0.983003i \(0.441228\pi\)
\(660\) 0 0
\(661\) −233.924 + 405.168i −0.353894 + 0.612963i −0.986928 0.161161i \(-0.948476\pi\)
0.633034 + 0.774124i \(0.281809\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −70.8359 −0.106520
\(666\) 0 0
\(667\) 50.3597i 0.0755018i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1147.52 + 662.524i 1.71017 + 0.987368i
\(672\) 0 0
\(673\) 273.302 + 473.372i 0.406094 + 0.703376i 0.994448 0.105227i \(-0.0335571\pi\)
−0.588354 + 0.808604i \(0.700224\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 227.606 + 394.225i 0.336198 + 0.582312i 0.983714 0.179740i \(-0.0575255\pi\)
−0.647516 + 0.762052i \(0.724192\pi\)
\(678\) 0 0
\(679\) 13.3600 + 7.71341i 0.0196760 + 0.0113600i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 123.214i 0.180400i 0.995924 + 0.0902002i \(0.0287507\pi\)
−0.995924 + 0.0902002i \(0.971249\pi\)
\(684\) 0 0
\(685\) 37.8937 0.0553193
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 32.0115 55.4455i 0.0464607 0.0804724i
\(690\) 0 0
\(691\) 163.326 94.2965i 0.236362 0.136464i −0.377141 0.926156i \(-0.623093\pi\)
0.613504 + 0.789692i \(0.289760\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −634.285 + 366.204i −0.912640 + 0.526913i
\(696\) 0 0
\(697\) 45.1599 78.2192i 0.0647918 0.112223i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −810.064 −1.15558 −0.577792 0.816184i \(-0.696085\pi\)
−0.577792 + 0.816184i \(0.696085\pi\)
\(702\) 0 0
\(703\) 1136.86i 1.61716i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.1806 7.60980i −0.0186429 0.0107635i
\(708\) 0 0
\(709\) 651.819 + 1128.98i 0.919349 + 1.59236i 0.800406 + 0.599459i \(0.204618\pi\)
0.118944 + 0.992901i \(0.462049\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 126.623 + 219.317i 0.177592 + 0.307598i
\(714\) 0 0
\(715\) −165.891 95.7772i −0.232015 0.133954i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 788.981i 1.09733i 0.836042 + 0.548666i \(0.184864\pi\)
−0.836042 + 0.548666i \(0.815136\pi\)
\(720\) 0 0
\(721\) −11.5913 −0.0160768
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 40.6380 70.3870i 0.0560524 0.0970856i
\(726\) 0 0
\(727\) −232.676 + 134.335i −0.320049 + 0.184780i −0.651414 0.758722i \(-0.725824\pi\)
0.331365 + 0.943502i \(0.392491\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 574.038 331.421i 0.785277 0.453380i
\(732\) 0 0
\(733\) 36.8343 63.7989i 0.0502514 0.0870380i −0.839806 0.542887i \(-0.817331\pi\)
0.890057 + 0.455849i \(0.150664\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 641.512 0.870437
\(738\) 0 0
\(739\) 448.249i 0.606562i −0.952901 0.303281i \(-0.901918\pi\)
0.952901 0.303281i \(-0.0980820\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −656.602 379.089i −0.883718 0.510215i −0.0118352 0.999930i \(-0.503767\pi\)
−0.871882 + 0.489715i \(0.837101\pi\)
\(744\) 0 0
\(745\) 635.496 + 1100.71i 0.853015 + 1.47746i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −54.1063 93.7149i −0.0722381 0.125120i
\(750\) 0 0
\(751\) 1141.58 + 659.091i 1.52008 + 0.877618i 0.999720 + 0.0236697i \(0.00753501\pi\)
0.520358 + 0.853948i \(0.325798\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 904.589i 1.19813i
\(756\) 0 0
\(757\) −587.874 −0.776583 −0.388292 0.921537i \(-0.626935\pi\)
−0.388292 + 0.921537i \(0.626935\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 188.496 326.485i 0.247695 0.429021i −0.715191 0.698929i \(-0.753660\pi\)
0.962886 + 0.269908i \(0.0869934\pi\)
\(762\) 0 0
\(763\) −51.5845 + 29.7823i −0.0676075 + 0.0390332i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −37.2917 + 21.5304i −0.0486202 + 0.0280709i
\(768\) 0 0
\(769\) 643.939 1115.34i 0.837372 1.45037i −0.0547122 0.998502i \(-0.517424\pi\)
0.892084 0.451869i \(-0.149243\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 778.578 1.00722 0.503608 0.863932i \(-0.332006\pi\)
0.503608 + 0.863932i \(0.332006\pi\)
\(774\) 0 0
\(775\) 408.715i 0.527375i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −90.1884 52.0703i −0.115775 0.0668425i
\(780\) 0 0
\(781\) −766.801 1328.14i −0.981819 1.70056i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 193.292 + 334.791i 0.246232 + 0.426486i
\(786\) 0 0
\(787\) 390.283 + 225.330i 0.495913 + 0.286315i 0.727024 0.686612i \(-0.240903\pi\)
−0.231111 + 0.972927i \(0.574236\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.7797i 0.0136280i
\(792\) 0 0
\(793\) 135.430 0.170782
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −182.891 + 316.776i −0.229474 + 0.397461i −0.957652 0.287927i \(-0.907034\pi\)
0.728178 + 0.685388i \(0.240367\pi\)
\(798\) 0 0
\(799\) 163.101 94.1662i 0.204131 0.117855i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −907.869 + 524.158i −1.13060 + 0.652750i
\(804\) 0 0
\(805\) 14.3994 24.9406i 0.0178875 0.0309821i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1167.70 −1.44339 −0.721695 0.692212i \(-0.756636\pi\)
−0.721695 + 0.692212i \(0.756636\pi\)
\(810\) 0 0
\(811\) 810.121i 0.998916i −0.866338 0.499458i \(-0.833533\pi\)
0.866338 0.499458i \(-0.166467\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −763.114 440.584i −0.936336 0.540594i
\(816\) 0 0
\(817\) −382.135 661.878i −0.467730 0.810132i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −280.513 485.862i −0.341672 0.591793i 0.643071 0.765806i \(-0.277660\pi\)
−0.984743 + 0.174013i \(0.944327\pi\)
\(822\) 0 0
\(823\) −1016.04 586.612i −1.23456 0.712773i −0.266583 0.963812i \(-0.585895\pi\)
−0.967977 + 0.251039i \(0.919228\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 267.739i 0.323747i 0.986811 + 0.161874i \(0.0517537\pi\)
−0.986811 + 0.161874i \(0.948246\pi\)
\(828\) 0 0
\(829\) 432.474 0.521682 0.260841 0.965382i \(-0.416000\pi\)
0.260841 + 0.965382i \(0.416000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −411.518 + 712.769i −0.494019 + 0.855665i
\(834\) 0 0
\(835\) −925.472 + 534.321i −1.10835 + 0.639906i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 459.103 265.063i 0.547202 0.315927i −0.200790 0.979634i \(-0.564351\pi\)
0.747993 + 0.663707i \(0.231018\pi\)
\(840\) 0 0
\(841\) 400.337 693.403i 0.476024 0.824499i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1019.44 1.20644
\(846\) 0 0
\(847\) 108.590i 0.128205i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −400.278 231.100i −0.470361 0.271563i
\(852\) 0 0
\(853\) −88.3868 153.090i −0.103619 0.179473i 0.809554 0.587045i \(-0.199709\pi\)
−0.913173 + 0.407572i \(0.866376\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 194.859 + 337.505i 0.227373 + 0.393821i 0.957029 0.289993i \(-0.0936531\pi\)
−0.729656 + 0.683815i \(0.760320\pi\)
\(858\) 0 0
\(859\) 503.279 + 290.568i 0.585889 + 0.338263i 0.763470 0.645843i \(-0.223494\pi\)
−0.177581 + 0.984106i \(0.556827\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 827.326i 0.958663i 0.877634 + 0.479331i \(0.159121\pi\)
−0.877634 + 0.479331i \(0.840879\pi\)
\(864\) 0 0
\(865\) −1546.92 −1.78835
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 324.426 561.923i 0.373333 0.646632i
\(870\) 0 0
\(871\) 56.7831 32.7837i 0.0651930 0.0376392i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 38.3732 22.1548i 0.0438551 0.0253198i
\(876\) 0 0
\(877\) 279.815 484.653i 0.319059 0.552626i −0.661233 0.750181i \(-0.729967\pi\)
0.980292 + 0.197554i \(0.0632999\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 957.127 1.08641 0.543205 0.839600i \(-0.317211\pi\)
0.543205 + 0.839600i \(0.317211\pi\)
\(882\) 0 0
\(883\) 625.252i 0.708100i 0.935227 + 0.354050i \(0.115196\pi\)
−0.935227 + 0.354050i \(0.884804\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −921.187 531.847i −1.03854 0.599602i −0.119122 0.992880i \(-0.538008\pi\)
−0.919420 + 0.393277i \(0.871341\pi\)
\(888\) 0 0
\(889\) 48.5496 + 84.0904i 0.0546115 + 0.0945899i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −108.576 188.059i −0.121585 0.210592i
\(894\) 0 0
\(895\) −511.172 295.125i −0.571142 0.329749i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 202.793i 0.225577i
\(900\) 0 0
\(901\) 606.935 0.673624
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1010.61 1750.43i 1.11670 1.93418i
\(906\) 0 0
\(907\) −207.207 + 119.631i −0.228453 + 0.131898i −0.609858 0.792510i \(-0.708774\pi\)
0.381405 + 0.924408i \(0.375440\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −175.804 + 101.501i −0.192980 + 0.111417i −0.593377 0.804925i \(-0.702206\pi\)
0.400397 + 0.916342i \(0.368872\pi\)
\(912\) 0 0
\(913\) −665.585 + 1152.83i −0.729009 + 1.26268i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −84.1801 −0.0917994
\(918\) 0 0
\(919\) 878.708i 0.956156i −0.878317 0.478078i \(-0.841334\pi\)
0.878317 0.478078i \(-0.158666\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −135.746 78.3730i −0.147070 0.0849111i
\(924\) 0 0
\(925\) 372.975 + 646.011i 0.403216 + 0.698391i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −300.259 520.064i −0.323207 0.559810i 0.657941 0.753069i \(-0.271428\pi\)
−0.981148 + 0.193259i \(0.938094\pi\)
\(930\) 0 0
\(931\) 821.838 + 474.489i 0.882748 + 0.509655i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1815.93i 1.94217i
\(936\) 0 0
\(937\) −184.325 −0.196718 −0.0983589 0.995151i \(-0.531359\pi\)
−0.0983589 + 0.995151i \(0.531359\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −377.587 + 653.999i −0.401261 + 0.695005i −0.993878 0.110479i \(-0.964761\pi\)
0.592617 + 0.805484i \(0.298095\pi\)
\(942\) 0 0
\(943\) 36.6668 21.1696i 0.0388831 0.0224492i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −769.965 + 444.539i −0.813057 + 0.469419i −0.848016 0.529970i \(-0.822203\pi\)
0.0349595 + 0.999389i \(0.488870\pi\)
\(948\) 0 0
\(949\) −53.5731 + 92.7913i −0.0564521 + 0.0977779i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −15.5920 −0.0163610 −0.00818050 0.999967i \(-0.502604\pi\)
−0.00818050 + 0.999967i \(0.502604\pi\)
\(954\) 0 0
\(955\) 2.49830i 0.00261602i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.15291 + 1.82033i 0.00328770 + 0.00189816i
\(960\) 0 0
\(961\) 29.3970 + 50.9172i 0.0305900 + 0.0529835i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 191.961 + 332.486i 0.198923 + 0.344545i
\(966\) 0 0
\(967\) 847.921 + 489.548i 0.876858 + 0.506254i 0.869621 0.493720i \(-0.164363\pi\)
0.00723669 + 0.999974i \(0.497696\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 67.3838i 0.0693963i −0.999398 0.0346982i \(-0.988953\pi\)
0.999398 0.0346982i \(-0.0110470\pi\)
\(972\) 0 0
\(973\) −70.3667 −0.0723193
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −353.710 + 612.644i −0.362037 + 0.627067i −0.988296 0.152549i \(-0.951252\pi\)
0.626259 + 0.779615i \(0.284585\pi\)
\(978\) 0 0
\(979\) 417.133 240.832i 0.426081 0.245998i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 338.829 195.623i 0.344689 0.199006i −0.317655 0.948206i \(-0.602895\pi\)
0.662343 + 0.749200i \(0.269562\pi\)
\(984\) 0 0
\(985\) −638.971 + 1106.73i −0.648701 + 1.12358i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 310.720 0.314176
\(990\) 0 0
\(991\) 104.988i 0.105941i −0.998596 0.0529706i \(-0.983131\pi\)
0.998596 0.0529706i \(-0.0168690\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1592.67 919.530i −1.60068 0.924151i
\(996\) 0 0
\(997\) 39.0028 + 67.5547i 0.0391201 + 0.0677580i 0.884923 0.465738i \(-0.154211\pi\)
−0.845802 + 0.533496i \(0.820878\pi\)
\(998\) 0 0
\(999\) 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 1728.3.o.g.1279.8 16
3.2 odd 2 576.3.o.g.319.7 16
4.3 odd 2 inner 1728.3.o.g.1279.7 16
8.3 odd 2 108.3.f.c.91.6 16
8.5 even 2 108.3.f.c.91.5 16
9.2 odd 6 576.3.o.g.511.2 16
9.7 even 3 inner 1728.3.o.g.127.7 16
12.11 even 2 576.3.o.g.319.2 16
24.5 odd 2 36.3.f.c.31.4 yes 16
24.11 even 2 36.3.f.c.31.3 yes 16
36.7 odd 6 inner 1728.3.o.g.127.8 16
36.11 even 6 576.3.o.g.511.7 16
72.5 odd 6 324.3.d.i.163.8 8
72.11 even 6 36.3.f.c.7.4 yes 16
72.13 even 6 324.3.d.g.163.1 8
72.29 odd 6 36.3.f.c.7.3 16
72.43 odd 6 108.3.f.c.19.5 16
72.59 even 6 324.3.d.i.163.7 8
72.61 even 6 108.3.f.c.19.6 16
72.67 odd 6 324.3.d.g.163.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.c.7.3 16 72.29 odd 6
36.3.f.c.7.4 yes 16 72.11 even 6
36.3.f.c.31.3 yes 16 24.11 even 2
36.3.f.c.31.4 yes 16 24.5 odd 2
108.3.f.c.19.5 16 72.43 odd 6
108.3.f.c.19.6 16 72.61 even 6
108.3.f.c.91.5 16 8.5 even 2
108.3.f.c.91.6 16 8.3 odd 2
324.3.d.g.163.1 8 72.13 even 6
324.3.d.g.163.2 8 72.67 odd 6
324.3.d.i.163.7 8 72.59 even 6
324.3.d.i.163.8 8 72.5 odd 6
576.3.o.g.319.2 16 12.11 even 2
576.3.o.g.319.7 16 3.2 odd 2
576.3.o.g.511.2 16 9.2 odd 6
576.3.o.g.511.7 16 36.11 even 6
1728.3.o.g.127.7 16 9.7 even 3 inner
1728.3.o.g.127.8 16 36.7 odd 6 inner
1728.3.o.g.1279.7 16 4.3 odd 2 inner
1728.3.o.g.1279.8 16 1.1 even 1 trivial