Properties

Label 1620.4.i.w.541.2
Level $1620$
Weight $4$
Character 1620.541
Analytic conductor $95.583$
Analytic rank $0$
Dimension $12$
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] = [1620,4,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), 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: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.2
Root \(5.63924i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.4.i.w.1081.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.50000 - 4.33013i) q^{5} +(-8.28565 + 14.3512i) q^{7} +O(q^{10})\) \(q+(-2.50000 - 4.33013i) q^{5} +(-8.28565 + 14.3512i) q^{7} +(-36.2684 + 62.8188i) q^{11} +(29.9488 + 51.8728i) q^{13} -15.8529 q^{17} -136.646 q^{19} +(81.5503 + 141.249i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(5.79923 - 10.0446i) q^{29} +(20.5009 + 35.5086i) q^{31} +82.8565 q^{35} -242.545 q^{37} +(28.9332 + 50.1138i) q^{41} +(132.252 - 229.067i) q^{43} +(299.945 - 519.519i) q^{47} +(34.1961 + 59.2294i) q^{49} -592.285 q^{53} +362.684 q^{55} +(144.446 + 250.188i) q^{59} +(-412.690 + 714.800i) q^{61} +(149.744 - 259.364i) q^{65} +(-405.264 - 701.938i) q^{67} +966.126 q^{71} +802.840 q^{73} +(-601.015 - 1040.99i) q^{77} +(581.001 - 1006.32i) q^{79} +(251.187 - 435.069i) q^{83} +(39.6322 + 68.6450i) q^{85} -8.76580 q^{89} -992.580 q^{91} +(341.615 + 591.694i) q^{95} +(-69.1820 + 119.827i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 30 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 30 q^{5} - 12 q^{7} + 84 q^{13} - 24 q^{17} - 228 q^{19} + 30 q^{23} - 150 q^{25} + 168 q^{29} + 324 q^{31} + 120 q^{35} - 984 q^{37} + 312 q^{41} + 156 q^{43} + 462 q^{47} + 588 q^{49} - 2028 q^{53} + 1008 q^{59} - 36 q^{61} + 420 q^{65} - 144 q^{67} - 2424 q^{71} - 1800 q^{73} + 672 q^{77} + 936 q^{79} + 288 q^{83} + 60 q^{85} + 240 q^{89} + 4572 q^{91} + 570 q^{95} + 1188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −2.50000 4.33013i −0.223607 0.387298i
\(6\) 0 0
\(7\) −8.28565 + 14.3512i −0.447383 + 0.774890i −0.998215 0.0597263i \(-0.980977\pi\)
0.550832 + 0.834616i \(0.314311\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −36.2684 + 62.8188i −0.994122 + 1.72187i −0.403300 + 0.915068i \(0.632137\pi\)
−0.590822 + 0.806802i \(0.701196\pi\)
\(12\) 0 0
\(13\) 29.9488 + 51.8728i 0.638946 + 1.10669i 0.985664 + 0.168717i \(0.0539625\pi\)
−0.346719 + 0.937969i \(0.612704\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −15.8529 −0.226170 −0.113085 0.993585i \(-0.536073\pi\)
−0.113085 + 0.993585i \(0.536073\pi\)
\(18\) 0 0
\(19\) −136.646 −1.64993 −0.824966 0.565182i \(-0.808806\pi\)
−0.824966 + 0.565182i \(0.808806\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 81.5503 + 141.249i 0.739322 + 1.28054i 0.952801 + 0.303595i \(0.0981871\pi\)
−0.213479 + 0.976948i \(0.568480\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.79923 10.0446i 0.0371342 0.0643183i −0.846861 0.531814i \(-0.821510\pi\)
0.883995 + 0.467496i \(0.154844\pi\)
\(30\) 0 0
\(31\) 20.5009 + 35.5086i 0.118776 + 0.205727i 0.919283 0.393597i \(-0.128770\pi\)
−0.800507 + 0.599324i \(0.795436\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 82.8565 0.400151
\(36\) 0 0
\(37\) −242.545 −1.07768 −0.538841 0.842408i \(-0.681138\pi\)
−0.538841 + 0.842408i \(0.681138\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 28.9332 + 50.1138i 0.110210 + 0.190889i 0.915855 0.401510i \(-0.131514\pi\)
−0.805645 + 0.592399i \(0.798181\pi\)
\(42\) 0 0
\(43\) 132.252 229.067i 0.469029 0.812383i −0.530344 0.847783i \(-0.677937\pi\)
0.999373 + 0.0354000i \(0.0112705\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 299.945 519.519i 0.930882 1.61233i 0.149063 0.988828i \(-0.452374\pi\)
0.781818 0.623506i \(-0.214292\pi\)
\(48\) 0 0
\(49\) 34.1961 + 59.2294i 0.0996971 + 0.172680i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −592.285 −1.53503 −0.767516 0.641030i \(-0.778507\pi\)
−0.767516 + 0.641030i \(0.778507\pi\)
\(54\) 0 0
\(55\) 362.684 0.889170
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 144.446 + 250.188i 0.318734 + 0.552063i 0.980224 0.197890i \(-0.0634090\pi\)
−0.661490 + 0.749954i \(0.730076\pi\)
\(60\) 0 0
\(61\) −412.690 + 714.800i −0.866222 + 1.50034i −0.000392970 1.00000i \(0.500125\pi\)
−0.865829 + 0.500340i \(0.833208\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 149.744 259.364i 0.285745 0.494925i
\(66\) 0 0
\(67\) −405.264 701.938i −0.738968 1.27993i −0.952960 0.303095i \(-0.901980\pi\)
0.213992 0.976835i \(-0.431353\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 966.126 1.61490 0.807451 0.589935i \(-0.200847\pi\)
0.807451 + 0.589935i \(0.200847\pi\)
\(72\) 0 0
\(73\) 802.840 1.28720 0.643598 0.765364i \(-0.277441\pi\)
0.643598 + 0.765364i \(0.277441\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −601.015 1040.99i −0.889506 1.54067i
\(78\) 0 0
\(79\) 581.001 1006.32i 0.827440 1.43317i −0.0726002 0.997361i \(-0.523130\pi\)
0.900040 0.435807i \(-0.143537\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 251.187 435.069i 0.332186 0.575362i −0.650755 0.759288i \(-0.725547\pi\)
0.982940 + 0.183926i \(0.0588806\pi\)
\(84\) 0 0
\(85\) 39.6322 + 68.6450i 0.0505731 + 0.0875952i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.76580 −0.0104401 −0.00522007 0.999986i \(-0.501662\pi\)
−0.00522007 + 0.999986i \(0.501662\pi\)
\(90\) 0 0
\(91\) −992.580 −1.14341
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 341.615 + 591.694i 0.368936 + 0.639016i
\(96\) 0 0
\(97\) −69.1820 + 119.827i −0.0724161 + 0.125428i −0.899960 0.435973i \(-0.856404\pi\)
0.827544 + 0.561401i \(0.189738\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −662.889 + 1148.16i −0.653068 + 1.13115i 0.329306 + 0.944223i \(0.393185\pi\)
−0.982374 + 0.186924i \(0.940148\pi\)
\(102\) 0 0
\(103\) −166.772 288.857i −0.159539 0.276330i 0.775164 0.631761i \(-0.217667\pi\)
−0.934703 + 0.355431i \(0.884334\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −584.004 −0.527643 −0.263821 0.964572i \(-0.584983\pi\)
−0.263821 + 0.964572i \(0.584983\pi\)
\(108\) 0 0
\(109\) −1915.42 −1.68315 −0.841577 0.540137i \(-0.818372\pi\)
−0.841577 + 0.540137i \(0.818372\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 618.658 + 1071.55i 0.515030 + 0.892058i 0.999848 + 0.0174432i \(0.00555262\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(114\) 0 0
\(115\) 407.751 706.246i 0.330635 0.572676i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 131.351 227.507i 0.101185 0.175257i
\(120\) 0 0
\(121\) −1965.30 3403.99i −1.47656 2.55747i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1785.42 −1.24749 −0.623743 0.781629i \(-0.714389\pi\)
−0.623743 + 0.781629i \(0.714389\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 360.906 + 625.108i 0.240706 + 0.416915i 0.960916 0.276841i \(-0.0892876\pi\)
−0.720209 + 0.693757i \(0.755954\pi\)
\(132\) 0 0
\(133\) 1132.20 1961.03i 0.738152 1.27852i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −554.493 + 960.411i −0.345792 + 0.598930i −0.985497 0.169691i \(-0.945723\pi\)
0.639705 + 0.768621i \(0.279057\pi\)
\(138\) 0 0
\(139\) −218.385 378.254i −0.133260 0.230813i 0.791671 0.610947i \(-0.209211\pi\)
−0.924932 + 0.380134i \(0.875878\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4344.78 −2.54076
\(144\) 0 0
\(145\) −57.9923 −0.0332138
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −88.5585 153.388i −0.0486912 0.0843357i 0.840653 0.541575i \(-0.182172\pi\)
−0.889344 + 0.457239i \(0.848838\pi\)
\(150\) 0 0
\(151\) −414.329 + 717.639i −0.223296 + 0.386759i −0.955807 0.293996i \(-0.905015\pi\)
0.732511 + 0.680755i \(0.238348\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 102.504 177.543i 0.0531184 0.0920038i
\(156\) 0 0
\(157\) 433.262 + 750.432i 0.220242 + 0.381471i 0.954882 0.296987i \(-0.0959818\pi\)
−0.734639 + 0.678458i \(0.762648\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2702.79 −1.32304
\(162\) 0 0
\(163\) 2984.70 1.43423 0.717116 0.696954i \(-0.245462\pi\)
0.717116 + 0.696954i \(0.245462\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 979.720 + 1696.92i 0.453970 + 0.786299i 0.998628 0.0523588i \(-0.0166739\pi\)
−0.544658 + 0.838658i \(0.683341\pi\)
\(168\) 0 0
\(169\) −695.358 + 1204.40i −0.316503 + 0.548200i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 253.472 439.027i 0.111394 0.192940i −0.804939 0.593358i \(-0.797802\pi\)
0.916333 + 0.400418i \(0.131135\pi\)
\(174\) 0 0
\(175\) −207.141 358.779i −0.0894766 0.154978i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3122.26 1.30374 0.651869 0.758332i \(-0.273985\pi\)
0.651869 + 0.758332i \(0.273985\pi\)
\(180\) 0 0
\(181\) −2828.38 −1.16150 −0.580751 0.814082i \(-0.697241\pi\)
−0.580751 + 0.814082i \(0.697241\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 606.363 + 1050.25i 0.240977 + 0.417384i
\(186\) 0 0
\(187\) 574.959 995.858i 0.224840 0.389435i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2039.32 3532.20i 0.772565 1.33812i −0.163588 0.986529i \(-0.552307\pi\)
0.936153 0.351593i \(-0.114360\pi\)
\(192\) 0 0
\(193\) −1521.56 2635.42i −0.567483 0.982909i −0.996814 0.0797619i \(-0.974584\pi\)
0.429331 0.903147i \(-0.358749\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1338.21 −0.483978 −0.241989 0.970279i \(-0.577800\pi\)
−0.241989 + 0.970279i \(0.577800\pi\)
\(198\) 0 0
\(199\) 453.645 0.161598 0.0807991 0.996730i \(-0.474253\pi\)
0.0807991 + 0.996730i \(0.474253\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 96.1008 + 166.451i 0.0332264 + 0.0575498i
\(204\) 0 0
\(205\) 144.666 250.569i 0.0492874 0.0853683i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4955.93 8583.92i 1.64023 2.84097i
\(210\) 0 0
\(211\) −952.979 1650.61i −0.310928 0.538543i 0.667636 0.744488i \(-0.267306\pi\)
−0.978564 + 0.205945i \(0.933973\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1322.52 −0.419513
\(216\) 0 0
\(217\) −679.452 −0.212554
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −474.774 822.333i −0.144510 0.250299i
\(222\) 0 0
\(223\) −2227.76 + 3858.59i −0.668976 + 1.15870i 0.309215 + 0.950992i \(0.399934\pi\)
−0.978191 + 0.207708i \(0.933400\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 85.6173 148.294i 0.0250336 0.0433594i −0.853237 0.521523i \(-0.825364\pi\)
0.878271 + 0.478164i \(0.158697\pi\)
\(228\) 0 0
\(229\) −2214.97 3836.44i −0.639168 1.10707i −0.985616 0.169002i \(-0.945946\pi\)
0.346448 0.938069i \(-0.387388\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 762.810 0.214478 0.107239 0.994233i \(-0.465799\pi\)
0.107239 + 0.994233i \(0.465799\pi\)
\(234\) 0 0
\(235\) −2999.45 −0.832606
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1522.14 + 2636.43i 0.411963 + 0.713542i 0.995104 0.0988294i \(-0.0315098\pi\)
−0.583141 + 0.812371i \(0.698176\pi\)
\(240\) 0 0
\(241\) 1009.16 1747.91i 0.269733 0.467190i −0.699060 0.715063i \(-0.746398\pi\)
0.968793 + 0.247872i \(0.0797314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 170.981 296.147i 0.0445859 0.0772250i
\(246\) 0 0
\(247\) −4092.38 7088.20i −1.05422 1.82596i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3031.55 −0.762351 −0.381175 0.924503i \(-0.624481\pi\)
−0.381175 + 0.924503i \(0.624481\pi\)
\(252\) 0 0
\(253\) −11830.8 −2.93990
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2377.31 + 4117.62i 0.577013 + 0.999417i 0.995820 + 0.0913409i \(0.0291153\pi\)
−0.418806 + 0.908076i \(0.637551\pi\)
\(258\) 0 0
\(259\) 2009.65 3480.81i 0.482136 0.835084i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1982.43 + 3433.67i −0.464798 + 0.805053i −0.999192 0.0401818i \(-0.987206\pi\)
0.534395 + 0.845235i \(0.320540\pi\)
\(264\) 0 0
\(265\) 1480.71 + 2564.67i 0.343243 + 0.594515i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3989.09 −0.904161 −0.452080 0.891977i \(-0.649318\pi\)
−0.452080 + 0.891977i \(0.649318\pi\)
\(270\) 0 0
\(271\) 2684.00 0.601628 0.300814 0.953683i \(-0.402742\pi\)
0.300814 + 0.953683i \(0.402742\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −906.711 1570.47i −0.198824 0.344374i
\(276\) 0 0
\(277\) −26.7606 + 46.3507i −0.00580465 + 0.0100540i −0.868913 0.494965i \(-0.835181\pi\)
0.863108 + 0.505019i \(0.168514\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1819.03 3150.65i 0.386172 0.668869i −0.605759 0.795648i \(-0.707130\pi\)
0.991931 + 0.126779i \(0.0404638\pi\)
\(282\) 0 0
\(283\) −3663.02 6344.53i −0.769412 1.33266i −0.937882 0.346954i \(-0.887216\pi\)
0.168470 0.985707i \(-0.446117\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −958.922 −0.197224
\(288\) 0 0
\(289\) −4661.69 −0.948847
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1261.12 + 2184.32i 0.251452 + 0.435527i 0.963926 0.266172i \(-0.0857588\pi\)
−0.712474 + 0.701698i \(0.752425\pi\)
\(294\) 0 0
\(295\) 722.231 1250.94i 0.142542 0.246890i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4884.66 + 8460.48i −0.944773 + 1.63640i
\(300\) 0 0
\(301\) 2191.59 + 3795.94i 0.419671 + 0.726892i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4126.90 0.774772
\(306\) 0 0
\(307\) 640.204 0.119017 0.0595087 0.998228i \(-0.481047\pi\)
0.0595087 + 0.998228i \(0.481047\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2437.17 4221.30i −0.444371 0.769673i 0.553637 0.832758i \(-0.313239\pi\)
−0.998008 + 0.0630852i \(0.979906\pi\)
\(312\) 0 0
\(313\) 2581.99 4472.14i 0.466270 0.807604i −0.532987 0.846123i \(-0.678931\pi\)
0.999258 + 0.0385190i \(0.0122640\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3655.00 6330.64i 0.647587 1.12165i −0.336110 0.941823i \(-0.609111\pi\)
0.983697 0.179831i \(-0.0575552\pi\)
\(318\) 0 0
\(319\) 420.658 + 728.601i 0.0738318 + 0.127880i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2166.23 0.373165
\(324\) 0 0
\(325\) −1497.44 −0.255578
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4970.47 + 8609.11i 0.832921 + 1.44266i
\(330\) 0 0
\(331\) 5867.79 10163.3i 0.974390 1.68769i 0.292456 0.956279i \(-0.405527\pi\)
0.681934 0.731414i \(-0.261139\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2026.32 + 3509.69i −0.330477 + 0.572402i
\(336\) 0 0
\(337\) −1125.51 1949.44i −0.181930 0.315112i 0.760608 0.649212i \(-0.224901\pi\)
−0.942538 + 0.334100i \(0.891568\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2974.14 −0.472313
\(342\) 0 0
\(343\) −6817.30 −1.07318
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3479.50 + 6026.66i 0.538297 + 0.932358i 0.998996 + 0.0448014i \(0.0142655\pi\)
−0.460699 + 0.887557i \(0.652401\pi\)
\(348\) 0 0
\(349\) 1462.00 2532.26i 0.224238 0.388392i −0.731852 0.681463i \(-0.761344\pi\)
0.956091 + 0.293071i \(0.0946772\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3276.24 + 5674.61i −0.493984 + 0.855606i −0.999976 0.00693255i \(-0.997793\pi\)
0.505992 + 0.862538i \(0.331127\pi\)
\(354\) 0 0
\(355\) −2415.31 4183.45i −0.361103 0.625449i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1324.66 0.194743 0.0973717 0.995248i \(-0.468956\pi\)
0.0973717 + 0.995248i \(0.468956\pi\)
\(360\) 0 0
\(361\) 11813.1 1.72228
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2007.10 3476.40i −0.287826 0.498529i
\(366\) 0 0
\(367\) 1628.11 2819.96i 0.231571 0.401092i −0.726700 0.686955i \(-0.758947\pi\)
0.958271 + 0.285863i \(0.0922802\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4907.47 8499.98i 0.686747 1.18948i
\(372\) 0 0
\(373\) 2272.33 + 3935.79i 0.315433 + 0.546347i 0.979530 0.201301i \(-0.0645168\pi\)
−0.664096 + 0.747647i \(0.731183\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 694.720 0.0949069
\(378\) 0 0
\(379\) −2484.30 −0.336701 −0.168351 0.985727i \(-0.553844\pi\)
−0.168351 + 0.985727i \(0.553844\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1414.43 2449.87i −0.188705 0.326847i 0.756114 0.654440i \(-0.227096\pi\)
−0.944819 + 0.327593i \(0.893762\pi\)
\(384\) 0 0
\(385\) −3005.07 + 5204.94i −0.397799 + 0.689009i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1295.09 2243.16i 0.168801 0.292371i −0.769198 0.639011i \(-0.779344\pi\)
0.937998 + 0.346639i \(0.112677\pi\)
\(390\) 0 0
\(391\) −1292.81 2239.21i −0.167212 0.289620i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5810.01 −0.740085
\(396\) 0 0
\(397\) 12207.4 1.54325 0.771625 0.636077i \(-0.219444\pi\)
0.771625 + 0.636077i \(0.219444\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4760.20 + 8244.91i 0.592801 + 1.02676i 0.993853 + 0.110706i \(0.0353113\pi\)
−0.401052 + 0.916055i \(0.631355\pi\)
\(402\) 0 0
\(403\) −1227.95 + 2126.88i −0.151783 + 0.262896i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8796.74 15236.4i 1.07135 1.85563i
\(408\) 0 0
\(409\) −5623.36 9739.94i −0.679846 1.17753i −0.975027 0.222087i \(-0.928713\pi\)
0.295181 0.955441i \(-0.404620\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4787.32 −0.570384
\(414\) 0 0
\(415\) −2511.87 −0.297116
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5340.99 9250.87i −0.622731 1.07860i −0.988975 0.148083i \(-0.952690\pi\)
0.366244 0.930519i \(-0.380644\pi\)
\(420\) 0 0
\(421\) −5411.99 + 9373.84i −0.626518 + 1.08516i 0.361727 + 0.932284i \(0.382187\pi\)
−0.988245 + 0.152877i \(0.951146\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 198.161 343.225i 0.0226170 0.0391738i
\(426\) 0 0
\(427\) −6838.81 11845.2i −0.775066 1.34245i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 235.904 0.0263645 0.0131823 0.999913i \(-0.495804\pi\)
0.0131823 + 0.999913i \(0.495804\pi\)
\(432\) 0 0
\(433\) −11895.5 −1.32024 −0.660118 0.751162i \(-0.729494\pi\)
−0.660118 + 0.751162i \(0.729494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11143.5 19301.1i −1.21983 2.11281i
\(438\) 0 0
\(439\) −1905.52 + 3300.46i −0.207166 + 0.358821i −0.950821 0.309742i \(-0.899757\pi\)
0.743655 + 0.668564i \(0.233091\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4016.11 + 6956.12i −0.430725 + 0.746038i −0.996936 0.0782223i \(-0.975076\pi\)
0.566210 + 0.824261i \(0.308409\pi\)
\(444\) 0 0
\(445\) 21.9145 + 37.9570i 0.00233449 + 0.00404345i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12271.3 −1.28980 −0.644898 0.764269i \(-0.723100\pi\)
−0.644898 + 0.764269i \(0.723100\pi\)
\(450\) 0 0
\(451\) −4197.45 −0.438249
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2481.45 + 4298.00i 0.255675 + 0.442842i
\(456\) 0 0
\(457\) −4972.00 + 8611.76i −0.508929 + 0.881490i 0.491018 + 0.871149i \(0.336625\pi\)
−0.999947 + 0.0103406i \(0.996708\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3374.74 5845.22i 0.340949 0.590540i −0.643661 0.765311i \(-0.722585\pi\)
0.984609 + 0.174771i \(0.0559185\pi\)
\(462\) 0 0
\(463\) 4780.08 + 8279.34i 0.479804 + 0.831044i 0.999732 0.0231659i \(-0.00737460\pi\)
−0.519928 + 0.854210i \(0.674041\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6517.56 0.645818 0.322909 0.946430i \(-0.395339\pi\)
0.322909 + 0.946430i \(0.395339\pi\)
\(468\) 0 0
\(469\) 13431.5 1.32241
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9593.15 + 16615.8i 0.932545 + 1.61521i
\(474\) 0 0
\(475\) 1708.07 2958.47i 0.164993 0.285777i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2376.37 + 4115.99i −0.226678 + 0.392619i −0.956822 0.290676i \(-0.906120\pi\)
0.730143 + 0.683294i \(0.239453\pi\)
\(480\) 0 0
\(481\) −7263.94 12581.5i −0.688580 1.19266i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 691.820 0.0647710
\(486\) 0 0
\(487\) −8412.56 −0.782771 −0.391386 0.920227i \(-0.628004\pi\)
−0.391386 + 0.920227i \(0.628004\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2785.22 4824.15i −0.255999 0.443403i 0.709167 0.705040i \(-0.249071\pi\)
−0.965166 + 0.261637i \(0.915738\pi\)
\(492\) 0 0
\(493\) −91.9345 + 159.235i −0.00839863 + 0.0145469i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8004.98 + 13865.0i −0.722479 + 1.25137i
\(498\) 0 0
\(499\) 7912.32 + 13704.5i 0.709828 + 1.22946i 0.964921 + 0.262541i \(0.0845605\pi\)
−0.255093 + 0.966916i \(0.582106\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15946.4 1.41354 0.706772 0.707441i \(-0.250151\pi\)
0.706772 + 0.707441i \(0.250151\pi\)
\(504\) 0 0
\(505\) 6628.89 0.584122
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7319.34 12677.5i −0.637375 1.10397i −0.986007 0.166707i \(-0.946687\pi\)
0.348631 0.937260i \(-0.386647\pi\)
\(510\) 0 0
\(511\) −6652.05 + 11521.7i −0.575869 + 0.997435i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −833.859 + 1444.29i −0.0713480 + 0.123578i
\(516\) 0 0
\(517\) 21757.0 + 37684.3i 1.85082 + 3.20571i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13001.4 1.09329 0.546644 0.837365i \(-0.315905\pi\)
0.546644 + 0.837365i \(0.315905\pi\)
\(522\) 0 0
\(523\) −1938.88 −0.162106 −0.0810531 0.996710i \(-0.525828\pi\)
−0.0810531 + 0.996710i \(0.525828\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −324.998 562.913i −0.0268636 0.0465292i
\(528\) 0 0
\(529\) −7217.39 + 12500.9i −0.593194 + 1.02744i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1733.03 + 3001.69i −0.140836 + 0.243936i
\(534\) 0 0
\(535\) 1460.01 + 2528.81i 0.117984 + 0.204355i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4960.96 −0.396444
\(540\) 0 0
\(541\) −7574.14 −0.601918 −0.300959 0.953637i \(-0.597307\pi\)
−0.300959 + 0.953637i \(0.597307\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4788.55 + 8294.00i 0.376365 + 0.651883i
\(546\) 0 0
\(547\) 1570.94 2720.94i 0.122794 0.212686i −0.798074 0.602559i \(-0.794148\pi\)
0.920869 + 0.389873i \(0.127481\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −792.442 + 1372.55i −0.0612689 + 0.106121i
\(552\) 0 0
\(553\) 9627.94 + 16676.1i 0.740365 + 1.28235i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12191.5 −0.927412 −0.463706 0.885989i \(-0.653481\pi\)
−0.463706 + 0.885989i \(0.653481\pi\)
\(558\) 0 0
\(559\) 15843.2 1.19874
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1091.42 1890.40i −0.0817016 0.141511i 0.822279 0.569084i \(-0.192702\pi\)
−0.903981 + 0.427573i \(0.859369\pi\)
\(564\) 0 0
\(565\) 3093.29 5357.73i 0.230328 0.398941i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −892.967 + 1546.66i −0.0657911 + 0.113953i −0.897045 0.441940i \(-0.854290\pi\)
0.831254 + 0.555893i \(0.187624\pi\)
\(570\) 0 0
\(571\) −1702.15 2948.21i −0.124751 0.216075i 0.796885 0.604131i \(-0.206480\pi\)
−0.921636 + 0.388057i \(0.873146\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4077.51 −0.295729
\(576\) 0 0
\(577\) −4174.78 −0.301210 −0.150605 0.988594i \(-0.548122\pi\)
−0.150605 + 0.988594i \(0.548122\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4162.50 + 7209.66i 0.297228 + 0.514815i
\(582\) 0 0
\(583\) 21481.3 37206.6i 1.52601 2.64312i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9308.47 + 16122.8i −0.654518 + 1.13366i 0.327497 + 0.944852i \(0.393795\pi\)
−0.982014 + 0.188806i \(0.939538\pi\)
\(588\) 0 0
\(589\) −2801.36 4852.10i −0.195973 0.339435i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −24181.4 −1.67456 −0.837278 0.546777i \(-0.815855\pi\)
−0.837278 + 0.546777i \(0.815855\pi\)
\(594\) 0 0
\(595\) −1313.51 −0.0905022
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2669.72 + 4624.08i 0.182106 + 0.315417i 0.942598 0.333931i \(-0.108375\pi\)
−0.760491 + 0.649348i \(0.775042\pi\)
\(600\) 0 0
\(601\) −4582.86 + 7937.75i −0.311046 + 0.538748i −0.978589 0.205823i \(-0.934013\pi\)
0.667543 + 0.744571i \(0.267346\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9826.49 + 17020.0i −0.660336 + 1.14374i
\(606\) 0 0
\(607\) −5715.85 9900.15i −0.382206 0.662001i 0.609171 0.793039i \(-0.291502\pi\)
−0.991377 + 0.131038i \(0.958169\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35931.9 2.37913
\(612\) 0 0
\(613\) 29975.6 1.97505 0.987523 0.157477i \(-0.0503361\pi\)
0.987523 + 0.157477i \(0.0503361\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3056.36 5293.77i −0.199424 0.345412i 0.748918 0.662663i \(-0.230574\pi\)
−0.948342 + 0.317251i \(0.897240\pi\)
\(618\) 0 0
\(619\) −693.987 + 1202.02i −0.0450625 + 0.0780505i −0.887677 0.460467i \(-0.847682\pi\)
0.842614 + 0.538517i \(0.181015\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 72.6303 125.799i 0.00467074 0.00808996i
\(624\) 0 0
\(625\) −312.500 541.266i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3845.04 0.243739
\(630\) 0 0
\(631\) 18608.3 1.17399 0.586993 0.809592i \(-0.300312\pi\)
0.586993 + 0.809592i \(0.300312\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4463.56 + 7731.12i 0.278947 + 0.483150i
\(636\) 0 0
\(637\) −2048.26 + 3547.70i −0.127402 + 0.220667i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4019.77 6962.45i 0.247693 0.429018i −0.715192 0.698928i \(-0.753661\pi\)
0.962885 + 0.269910i \(0.0869941\pi\)
\(642\) 0 0
\(643\) 13330.7 + 23089.5i 0.817592 + 1.41611i 0.907452 + 0.420157i \(0.138025\pi\)
−0.0898593 + 0.995954i \(0.528642\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6303.13 0.383001 0.191501 0.981492i \(-0.438665\pi\)
0.191501 + 0.981492i \(0.438665\pi\)
\(648\) 0 0
\(649\) −20955.4 −1.26744
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11299.2 + 19570.7i 0.677136 + 1.17283i 0.975839 + 0.218489i \(0.0701127\pi\)
−0.298703 + 0.954346i \(0.596554\pi\)
\(654\) 0 0
\(655\) 1804.53 3125.54i 0.107647 0.186450i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1198.87 + 2076.50i −0.0708667 + 0.122745i −0.899281 0.437370i \(-0.855910\pi\)
0.828415 + 0.560115i \(0.189243\pi\)
\(660\) 0 0
\(661\) 2531.81 + 4385.23i 0.148980 + 0.258042i 0.930851 0.365399i \(-0.119068\pi\)
−0.781870 + 0.623441i \(0.785734\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11322.0 −0.660223
\(666\) 0 0
\(667\) 1891.72 0.109816
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −29935.2 51849.3i −1.72226 2.98304i
\(672\) 0 0
\(673\) −3640.66 + 6305.81i −0.208525 + 0.361176i −0.951250 0.308421i \(-0.900200\pi\)
0.742725 + 0.669596i \(0.233533\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14482.1 25083.8i 0.822146 1.42400i −0.0819349 0.996638i \(-0.526110\pi\)
0.904081 0.427361i \(-0.140557\pi\)
\(678\) 0 0
\(679\) −1146.43 1985.68i −0.0647955 0.112229i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9158.42 0.513085 0.256543 0.966533i \(-0.417417\pi\)
0.256543 + 0.966533i \(0.417417\pi\)
\(684\) 0 0
\(685\) 5544.93 0.309286
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17738.2 30723.5i −0.980802 1.69880i
\(690\) 0 0
\(691\) −5014.15 + 8684.76i −0.276045 + 0.478124i −0.970398 0.241510i \(-0.922357\pi\)
0.694353 + 0.719634i \(0.255691\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1091.92 + 1891.27i −0.0595958 + 0.103223i
\(696\) 0 0
\(697\) −458.675 794.448i −0.0249262 0.0431734i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 27626.7 1.48851 0.744256 0.667894i \(-0.232804\pi\)
0.744256 + 0.667894i \(0.232804\pi\)
\(702\) 0 0
\(703\) 33142.8 1.77810
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10984.9 19026.4i −0.584343 1.01211i
\(708\) 0 0
\(709\) −4851.84 + 8403.64i −0.257003 + 0.445142i −0.965438 0.260635i \(-0.916068\pi\)
0.708435 + 0.705776i \(0.249402\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3343.70 + 5791.47i −0.175628 + 0.304197i
\(714\) 0 0
\(715\) 10861.9 + 18813.4i 0.568131 + 0.984032i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32045.4 1.66216 0.831080 0.556153i \(-0.187723\pi\)
0.831080 + 0.556153i \(0.187723\pi\)
\(720\) 0 0
\(721\) 5527.25 0.285500
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 144.981 + 251.114i 0.00742683 + 0.0128637i
\(726\) 0 0
\(727\) 3723.63 6449.51i 0.189961 0.329022i −0.755276 0.655407i \(-0.772497\pi\)
0.945237 + 0.326385i \(0.105830\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2096.58 + 3631.38i −0.106080 + 0.183736i
\(732\) 0 0
\(733\) 9072.37 + 15713.8i 0.457156 + 0.791818i 0.998809 0.0487844i \(-0.0155347\pi\)
−0.541653 + 0.840602i \(0.682201\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 58793.1 2.93850
\(738\) 0 0
\(739\) 25920.2 1.29024 0.645121 0.764081i \(-0.276807\pi\)
0.645121 + 0.764081i \(0.276807\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4656.55 + 8065.38i 0.229922 + 0.398237i 0.957785 0.287486i \(-0.0928195\pi\)
−0.727863 + 0.685723i \(0.759486\pi\)
\(744\) 0 0
\(745\) −442.792 + 766.939i −0.0217754 + 0.0377161i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4838.85 8381.13i 0.236058 0.408865i
\(750\) 0 0
\(751\) 2492.14 + 4316.52i 0.121091 + 0.209736i 0.920198 0.391453i \(-0.128027\pi\)
−0.799107 + 0.601189i \(0.794694\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4143.29 0.199722
\(756\) 0 0
\(757\) −10452.6 −0.501857 −0.250928 0.968006i \(-0.580736\pi\)
−0.250928 + 0.968006i \(0.580736\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4702.97 8145.78i −0.224024 0.388022i 0.732002 0.681303i \(-0.238586\pi\)
−0.956026 + 0.293281i \(0.905253\pi\)
\(762\) 0 0
\(763\) 15870.5 27488.5i 0.753014 1.30426i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8651.98 + 14985.7i −0.407307 + 0.705477i
\(768\) 0 0
\(769\) 11784.1 + 20410.7i 0.552596 + 0.957125i 0.998086 + 0.0618381i \(0.0196962\pi\)
−0.445490 + 0.895287i \(0.646970\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3172.70 −0.147625 −0.0738125 0.997272i \(-0.523517\pi\)
−0.0738125 + 0.997272i \(0.523517\pi\)
\(774\) 0 0
\(775\) −1025.04 −0.0475106
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3953.60 6847.84i −0.181839 0.314954i
\(780\) 0 0
\(781\) −35039.9 + 60690.8i −1.60541 + 2.78065i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2166.31 3752.16i 0.0984954 0.170599i
\(786\) 0 0
\(787\) −7721.19 13373.5i −0.349721 0.605735i 0.636479 0.771294i \(-0.280390\pi\)
−0.986200 + 0.165559i \(0.947057\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20503.9 −0.921663
\(792\) 0 0
\(793\) −49438.2 −2.21388
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4984.27 + 8633.02i 0.221521 + 0.383685i 0.955270 0.295735i \(-0.0955646\pi\)
−0.733749 + 0.679420i \(0.762231\pi\)
\(798\) 0 0
\(799\) −4754.99 + 8235.88i −0.210537 + 0.364661i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −29117.7 + 50433.4i −1.27963 + 2.21638i
\(804\) 0 0
\(805\) 6756.97 + 11703.4i 0.295841 + 0.512411i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10439.0 −0.453667 −0.226833 0.973934i \(-0.572837\pi\)
−0.226833 + 0.973934i \(0.572837\pi\)
\(810\) 0 0
\(811\) 17472.9 0.756544 0.378272 0.925694i \(-0.376518\pi\)
0.378272 + 0.925694i \(0.376518\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7461.75 12924.1i −0.320704 0.555476i
\(816\) 0 0
\(817\) −18071.7 + 31301.1i −0.773867 + 1.34038i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9750.89 16889.0i 0.414505 0.717943i −0.580872 0.813995i \(-0.697288\pi\)
0.995376 + 0.0960521i \(0.0306216\pi\)
\(822\) 0 0
\(823\) 8296.21 + 14369.5i 0.351383 + 0.608612i 0.986492 0.163810i \(-0.0523783\pi\)
−0.635109 + 0.772422i \(0.719045\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32449.4 −1.36442 −0.682210 0.731156i \(-0.738981\pi\)
−0.682210 + 0.731156i \(0.738981\pi\)
\(828\) 0 0
\(829\) 7197.10 0.301527 0.150763 0.988570i \(-0.451827\pi\)
0.150763 + 0.988570i \(0.451827\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −542.107 938.956i −0.0225485 0.0390551i
\(834\) 0 0
\(835\) 4898.60 8484.62i 0.203022 0.351644i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7715.97 + 13364.4i −0.317503 + 0.549931i −0.979966 0.199163i \(-0.936178\pi\)
0.662464 + 0.749094i \(0.269511\pi\)
\(840\) 0 0
\(841\) 12127.2 + 21005.0i 0.497242 + 0.861249i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6953.58 0.283089
\(846\) 0 0
\(847\) 65135.0 2.64235
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −19779.6 34259.3i −0.796753 1.38002i
\(852\) 0 0
\(853\) −3957.37 + 6854.36i −0.158848 + 0.275134i −0.934454 0.356085i \(-0.884111\pi\)
0.775605 + 0.631218i \(0.217445\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13406.4 23220.6i 0.534369 0.925554i −0.464825 0.885403i \(-0.653883\pi\)
0.999194 0.0401510i \(-0.0127839\pi\)
\(858\) 0 0
\(859\) −809.265 1401.69i −0.0321441 0.0556752i 0.849506 0.527579i \(-0.176900\pi\)
−0.881650 + 0.471904i \(0.843567\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16678.5 −0.657869 −0.328935 0.944353i \(-0.606690\pi\)
−0.328935 + 0.944353i \(0.606690\pi\)
\(864\) 0 0
\(865\) −2534.72 −0.0996338
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 42144.0 + 72995.6i 1.64515 + 2.84949i
\(870\) 0 0
\(871\) 24274.3 42044.3i 0.944321 1.63561i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1035.71 + 1793.90i −0.0400151 + 0.0693083i
\(876\) 0 0
\(877\) 1064.16 + 1843.17i 0.0409738 + 0.0709686i 0.885785 0.464096i \(-0.153621\pi\)
−0.844811 + 0.535064i \(0.820287\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2379.31 −0.0909886 −0.0454943 0.998965i \(-0.514486\pi\)
−0.0454943 + 0.998965i \(0.514486\pi\)
\(882\) 0 0
\(883\) −36926.5 −1.40733 −0.703666 0.710531i \(-0.748455\pi\)
−0.703666 + 0.710531i \(0.748455\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2390.70 4140.81i −0.0904981 0.156747i 0.817223 0.576322i \(-0.195513\pi\)
−0.907721 + 0.419575i \(0.862179\pi\)
\(888\) 0 0
\(889\) 14793.4 25622.9i 0.558104 0.966665i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −40986.2 + 70990.2i −1.53589 + 2.66024i
\(894\) 0 0
\(895\) −7805.66 13519.8i −0.291525 0.504935i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 475.558 0.0176426
\(900\) 0 0
\(901\) 9389.43 0.347178
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7070.95 + 12247.2i 0.259720 + 0.449847i
\(906\) 0 0
\(907\) −24032.4 + 41625.3i −0.879804 + 1.52387i −0.0282484 + 0.999601i \(0.508993\pi\)
−0.851556 + 0.524264i \(0.824340\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12817.2 22200.1i 0.466140 0.807378i −0.533112 0.846044i \(-0.678978\pi\)
0.999252 + 0.0386666i \(0.0123110\pi\)
\(912\) 0 0
\(913\) 18220.3 + 31558.6i 0.660466 + 1.14396i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −11961.4 −0.430751
\(918\) 0 0
\(919\) −9651.68 −0.346441 −0.173221 0.984883i \(-0.555417\pi\)
−0.173221 + 0.984883i \(0.555417\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 28934.3 + 50115.6i 1.03183 + 1.78719i
\(924\) 0 0
\(925\) 3031.82 5251.26i 0.107768 0.186660i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 166.997 289.248i 0.00589775 0.0102152i −0.863061 0.505099i \(-0.831456\pi\)
0.868959 + 0.494884i \(0.164789\pi\)
\(930\) 0 0
\(931\) −4672.76 8093.45i −0.164493 0.284911i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5749.59 −0.201103
\(936\) 0 0
\(937\) 11567.3 0.403295 0.201648 0.979458i \(-0.435370\pi\)
0.201648 + 0.979458i \(0.435370\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −10130.8 17547.1i −0.350963 0.607886i 0.635455 0.772138i \(-0.280812\pi\)
−0.986418 + 0.164252i \(0.947479\pi\)
\(942\) 0 0
\(943\) −4719.02 + 8173.59i −0.162961 + 0.282257i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13676.7 23688.8i 0.469307 0.812863i −0.530077 0.847949i \(-0.677837\pi\)
0.999384 + 0.0350859i \(0.0111705\pi\)
\(948\) 0 0
\(949\) 24044.1 + 41645.5i 0.822448 + 1.42452i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −45432.0 −1.54427 −0.772133 0.635460i \(-0.780810\pi\)
−0.772133 + 0.635460i \(0.780810\pi\)
\(954\) 0 0
\(955\) −20393.2 −0.691003
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9188.67 15915.2i −0.309403 0.535902i
\(960\) 0 0
\(961\) 14054.9 24343.8i 0.471784 0.817154i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7607.79 + 13177.1i −0.253786 + 0.439570i
\(966\) 0 0
\(967\) 3567.09 + 6178.39i 0.118625 + 0.205464i 0.919223 0.393738i \(-0.128818\pi\)
−0.800598 + 0.599202i \(0.795485\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −31719.4 −1.04833 −0.524163 0.851618i \(-0.675622\pi\)
−0.524163 + 0.851618i \(0.675622\pi\)
\(972\) 0 0
\(973\) 7237.84 0.238473
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8785.15 15216.3i −0.287678 0.498274i 0.685577 0.728000i \(-0.259550\pi\)
−0.973255 + 0.229727i \(0.926217\pi\)
\(978\) 0 0
\(979\) 317.922 550.656i 0.0103788 0.0179766i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −28993.6 + 50218.5i −0.940746 + 1.62942i −0.176694 + 0.984266i \(0.556540\pi\)
−0.764052 + 0.645154i \(0.776793\pi\)
\(984\) 0 0
\(985\) 3345.53 + 5794.63i 0.108221 + 0.187444i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 43140.8 1.38705
\(990\) 0 0
\(991\) 20741.2 0.664851 0.332425 0.943130i \(-0.392133\pi\)
0.332425 + 0.943130i \(0.392133\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1134.11 1964.34i −0.0361344 0.0625867i
\(996\) 0 0
\(997\) 4960.70 8592.19i 0.157580 0.272936i −0.776416 0.630221i \(-0.782964\pi\)
0.933995 + 0.357285i \(0.116298\pi\)
\(998\) 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 1620.4.i.w.541.2 12
3.2 odd 2 1620.4.i.x.541.2 12
9.2 odd 6 1620.4.a.i.1.5 6
9.4 even 3 inner 1620.4.i.w.1081.2 12
9.5 odd 6 1620.4.i.x.1081.2 12
9.7 even 3 1620.4.a.j.1.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.4.a.i.1.5 6 9.2 odd 6
1620.4.a.j.1.5 yes 6 9.7 even 3
1620.4.i.w.541.2 12 1.1 even 1 trivial
1620.4.i.w.1081.2 12 9.4 even 3 inner
1620.4.i.x.541.2 12 3.2 odd 2
1620.4.i.x.1081.2 12 9.5 odd 6