Properties

Label 648.4.i.q.433.2
Level $648$
Weight $4$
Character 648.433
Analytic conductor $38.233$
Analytic rank $0$
Dimension $4$
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] = [648,4,Mod(217,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.217");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.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: \(38.2332376837\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-43})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} - 11x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.2
Root \(3.08945 - 1.20635i\) of defining polynomial
Character \(\chi\) \(=\) 648.433
Dual form 648.4.i.q.217.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(6.67891 - 11.5682i) q^{5} +(7.17891 + 12.4342i) q^{7} +O(q^{10})\) \(q+(6.67891 - 11.5682i) q^{5} +(7.17891 + 12.4342i) q^{7} +(19.5367 + 33.8386i) q^{11} +(38.3945 - 66.5013i) q^{13} +62.4313 q^{17} -39.7891 q^{19} +(-64.2524 + 111.288i) q^{23} +(-26.7156 - 46.2728i) q^{25} +(32.4680 + 56.2362i) q^{29} +(4.56873 - 7.91328i) q^{31} +191.789 q^{35} +319.505 q^{37} +(8.78908 - 15.2231i) q^{41} +(225.473 + 390.530i) q^{43} +(-290.799 - 503.678i) q^{47} +(68.4265 - 118.518i) q^{49} -329.450 q^{53} +521.936 q^{55} +(-120.927 + 209.451i) q^{59} +(-248.762 - 430.868i) q^{61} +(-512.867 - 888.312i) q^{65} +(289.032 - 500.618i) q^{67} +660.927 q^{71} +696.559 q^{73} +(-280.505 + 485.848i) q^{77} +(-365.134 - 632.430i) q^{79} +(545.239 + 944.382i) q^{83} +(416.973 - 722.218i) q^{85} +317.588 q^{89} +1102.52 q^{91} +(-265.748 + 460.288i) q^{95} +(742.827 + 1286.61i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 6 q^{7} + 10 q^{11} + 40 q^{13} + 68 q^{17} + 68 q^{19} - 98 q^{23} - 16 q^{25} - 120 q^{29} + 200 q^{31} + 540 q^{35} + 960 q^{37} - 192 q^{41} + 334 q^{43} - 300 q^{47} + 410 q^{49} + 136 q^{53} + 1588 q^{55} - 620 q^{59} - 200 q^{61} - 1370 q^{65} + 1406 q^{67} + 2780 q^{71} + 3604 q^{73} - 804 q^{77} + 334 q^{79} + 500 q^{83} + 1100 q^{85} + 180 q^{89} + 2820 q^{91} - 1222 q^{95} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 6.67891 11.5682i 0.597380 1.03469i −0.395827 0.918325i \(-0.629542\pi\)
0.993206 0.116367i \(-0.0371248\pi\)
\(6\) 0 0
\(7\) 7.17891 + 12.4342i 0.387625 + 0.671386i 0.992130 0.125216i \(-0.0399624\pi\)
−0.604505 + 0.796601i \(0.706629\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 19.5367 + 33.8386i 0.535504 + 0.927520i 0.999139 + 0.0414937i \(0.0132117\pi\)
−0.463635 + 0.886026i \(0.653455\pi\)
\(12\) 0 0
\(13\) 38.3945 66.5013i 0.819133 1.41878i −0.0871887 0.996192i \(-0.527788\pi\)
0.906322 0.422588i \(-0.138878\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 62.4313 0.890694 0.445347 0.895358i \(-0.353080\pi\)
0.445347 + 0.895358i \(0.353080\pi\)
\(18\) 0 0
\(19\) −39.7891 −0.480434 −0.240217 0.970719i \(-0.577219\pi\)
−0.240217 + 0.970719i \(0.577219\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −64.2524 + 111.288i −0.582502 + 1.00892i 0.412680 + 0.910876i \(0.364593\pi\)
−0.995182 + 0.0980467i \(0.968741\pi\)
\(24\) 0 0
\(25\) −26.7156 46.2728i −0.213725 0.370183i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 32.4680 + 56.2362i 0.207902 + 0.360097i 0.951053 0.309027i \(-0.100003\pi\)
−0.743152 + 0.669123i \(0.766670\pi\)
\(30\) 0 0
\(31\) 4.56873 7.91328i 0.0264700 0.0458473i −0.852487 0.522748i \(-0.824907\pi\)
0.878957 + 0.476901i \(0.158240\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 191.789 0.926236
\(36\) 0 0
\(37\) 319.505 1.41963 0.709814 0.704389i \(-0.248779\pi\)
0.709814 + 0.704389i \(0.248779\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.78908 15.2231i 0.0334786 0.0579867i −0.848801 0.528713i \(-0.822675\pi\)
0.882279 + 0.470726i \(0.156008\pi\)
\(42\) 0 0
\(43\) 225.473 + 390.530i 0.799634 + 1.38501i 0.919855 + 0.392260i \(0.128307\pi\)
−0.120220 + 0.992747i \(0.538360\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −290.799 503.678i −0.902496 1.56317i −0.824243 0.566236i \(-0.808399\pi\)
−0.0782529 0.996934i \(-0.524934\pi\)
\(48\) 0 0
\(49\) 68.4265 118.518i 0.199494 0.345534i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −329.450 −0.853839 −0.426919 0.904290i \(-0.640401\pi\)
−0.426919 + 0.904290i \(0.640401\pi\)
\(54\) 0 0
\(55\) 521.936 1.27960
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −120.927 + 209.451i −0.266836 + 0.462173i −0.968043 0.250785i \(-0.919311\pi\)
0.701207 + 0.712957i \(0.252645\pi\)
\(60\) 0 0
\(61\) −248.762 430.868i −0.522142 0.904377i −0.999668 0.0257594i \(-0.991800\pi\)
0.477526 0.878618i \(-0.341534\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −512.867 888.312i −0.978667 1.69510i
\(66\) 0 0
\(67\) 289.032 500.618i 0.527028 0.912839i −0.472476 0.881344i \(-0.656640\pi\)
0.999504 0.0314957i \(-0.0100271\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 660.927 1.10475 0.552377 0.833594i \(-0.313721\pi\)
0.552377 + 0.833594i \(0.313721\pi\)
\(72\) 0 0
\(73\) 696.559 1.11680 0.558398 0.829573i \(-0.311416\pi\)
0.558398 + 0.829573i \(0.311416\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −280.505 + 485.848i −0.415149 + 0.719059i
\(78\) 0 0
\(79\) −365.134 632.430i −0.520010 0.900683i −0.999729 0.0232613i \(-0.992595\pi\)
0.479720 0.877422i \(-0.340738\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 545.239 + 944.382i 0.721058 + 1.24891i 0.960576 + 0.278017i \(0.0896770\pi\)
−0.239519 + 0.970892i \(0.576990\pi\)
\(84\) 0 0
\(85\) 416.973 722.218i 0.532083 0.921594i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 317.588 0.378250 0.189125 0.981953i \(-0.439435\pi\)
0.189125 + 0.981953i \(0.439435\pi\)
\(90\) 0 0
\(91\) 1102.52 1.27006
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −265.748 + 460.288i −0.287001 + 0.497101i
\(96\) 0 0
\(97\) 742.827 + 1286.61i 0.777553 + 1.34676i 0.933348 + 0.358972i \(0.116873\pi\)
−0.155795 + 0.987789i \(0.549794\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −501.633 868.853i −0.494201 0.855982i 0.505776 0.862665i \(-0.331206\pi\)
−0.999978 + 0.00668295i \(0.997873\pi\)
\(102\) 0 0
\(103\) 858.799 1487.48i 0.821553 1.42297i −0.0829729 0.996552i \(-0.526441\pi\)
0.904526 0.426419i \(-0.140225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1006.27 0.909161 0.454581 0.890706i \(-0.349789\pi\)
0.454581 + 0.890706i \(0.349789\pi\)
\(108\) 0 0
\(109\) 1724.94 1.51577 0.757885 0.652388i \(-0.226233\pi\)
0.757885 + 0.652388i \(0.226233\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −313.509 + 543.014i −0.260995 + 0.452057i −0.966507 0.256641i \(-0.917384\pi\)
0.705511 + 0.708699i \(0.250717\pi\)
\(114\) 0 0
\(115\) 858.271 + 1486.57i 0.695950 + 1.20542i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 448.188 + 776.285i 0.345255 + 0.597999i
\(120\) 0 0
\(121\) −97.8673 + 169.511i −0.0735291 + 0.127356i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 956.002 0.684059
\(126\) 0 0
\(127\) −1990.05 −1.39046 −0.695230 0.718788i \(-0.744697\pi\)
−0.695230 + 0.718788i \(0.744697\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 78.2072 135.459i 0.0521603 0.0903442i −0.838766 0.544491i \(-0.816723\pi\)
0.890927 + 0.454147i \(0.150056\pi\)
\(132\) 0 0
\(133\) −285.642 494.747i −0.186228 0.322556i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1059.66 1835.38i −0.660822 1.14458i −0.980400 0.197017i \(-0.936875\pi\)
0.319579 0.947560i \(-0.396459\pi\)
\(138\) 0 0
\(139\) 1343.39 2326.81i 0.819745 1.41984i −0.0861255 0.996284i \(-0.527449\pi\)
0.905870 0.423555i \(-0.139218\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3000.41 1.75460
\(144\) 0 0
\(145\) 867.403 0.496785
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −775.781 + 1343.69i −0.426540 + 0.738789i −0.996563 0.0828401i \(-0.973601\pi\)
0.570023 + 0.821629i \(0.306934\pi\)
\(150\) 0 0
\(151\) 9.05458 + 15.6830i 0.00487981 + 0.00845208i 0.868455 0.495768i \(-0.165113\pi\)
−0.863575 + 0.504220i \(0.831780\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −61.0283 105.704i −0.0316252 0.0547765i
\(156\) 0 0
\(157\) −1870.75 + 3240.24i −0.950971 + 1.64713i −0.207641 + 0.978205i \(0.566579\pi\)
−0.743330 + 0.668925i \(0.766755\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1845.05 −0.903168
\(162\) 0 0
\(163\) 2608.90 1.25365 0.626826 0.779160i \(-0.284354\pi\)
0.626826 + 0.779160i \(0.284354\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −781.446 + 1353.50i −0.362097 + 0.627170i −0.988306 0.152485i \(-0.951272\pi\)
0.626209 + 0.779655i \(0.284606\pi\)
\(168\) 0 0
\(169\) −1849.78 3203.92i −0.841958 1.45831i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −751.385 1301.44i −0.330212 0.571945i 0.652341 0.757926i \(-0.273787\pi\)
−0.982553 + 0.185981i \(0.940454\pi\)
\(174\) 0 0
\(175\) 383.578 664.377i 0.165690 0.286984i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4126.50 −1.72307 −0.861534 0.507700i \(-0.830496\pi\)
−0.861534 + 0.507700i \(0.830496\pi\)
\(180\) 0 0
\(181\) 1582.33 0.649800 0.324900 0.945748i \(-0.394669\pi\)
0.324900 + 0.945748i \(0.394669\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2133.94 3696.10i 0.848057 1.46888i
\(186\) 0 0
\(187\) 1219.70 + 2112.59i 0.476970 + 0.826137i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1287.94 + 2230.78i 0.487916 + 0.845096i 0.999903 0.0138971i \(-0.00442373\pi\)
−0.511987 + 0.858993i \(0.671090\pi\)
\(192\) 0 0
\(193\) 697.649 1208.36i 0.260196 0.450673i −0.706098 0.708114i \(-0.749546\pi\)
0.966294 + 0.257441i \(0.0828794\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5047.62 −1.82552 −0.912761 0.408494i \(-0.866054\pi\)
−0.912761 + 0.408494i \(0.866054\pi\)
\(198\) 0 0
\(199\) −2441.47 −0.869704 −0.434852 0.900502i \(-0.643199\pi\)
−0.434852 + 0.900502i \(0.643199\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −466.169 + 807.429i −0.161176 + 0.279165i
\(204\) 0 0
\(205\) −117.403 203.348i −0.0399989 0.0692802i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −777.348 1346.41i −0.257274 0.445612i
\(210\) 0 0
\(211\) −2133.08 + 3694.61i −0.695959 + 1.20544i 0.273897 + 0.961759i \(0.411687\pi\)
−0.969856 + 0.243678i \(0.921646\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6023.65 1.91074
\(216\) 0 0
\(217\) 131.194 0.0410416
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2397.02 4151.76i 0.729597 1.26370i
\(222\) 0 0
\(223\) −875.574 1516.54i −0.262927 0.455404i 0.704091 0.710110i \(-0.251355\pi\)
−0.967019 + 0.254706i \(0.918021\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1100.02 + 1905.29i 0.321635 + 0.557088i 0.980825 0.194888i \(-0.0624344\pi\)
−0.659191 + 0.751976i \(0.729101\pi\)
\(228\) 0 0
\(229\) 595.645 1031.69i 0.171884 0.297711i −0.767195 0.641414i \(-0.778348\pi\)
0.939078 + 0.343703i \(0.111681\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2644.52 −0.743555 −0.371778 0.928322i \(-0.621252\pi\)
−0.371778 + 0.928322i \(0.621252\pi\)
\(234\) 0 0
\(235\) −7768.87 −2.15653
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 718.761 1244.93i 0.194530 0.336937i −0.752216 0.658917i \(-0.771015\pi\)
0.946746 + 0.321980i \(0.104348\pi\)
\(240\) 0 0
\(241\) −1633.54 2829.37i −0.436620 0.756248i 0.560806 0.827947i \(-0.310491\pi\)
−0.997426 + 0.0716990i \(0.977158\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −914.029 1583.15i −0.238348 0.412830i
\(246\) 0 0
\(247\) −1527.68 + 2646.03i −0.393539 + 0.681630i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1871.89 0.470727 0.235364 0.971907i \(-0.424372\pi\)
0.235364 + 0.971907i \(0.424372\pi\)
\(252\) 0 0
\(253\) −5021.12 −1.24773
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2345.41 + 4062.37i −0.569271 + 0.986007i 0.427367 + 0.904078i \(0.359441\pi\)
−0.996638 + 0.0819287i \(0.973892\pi\)
\(258\) 0 0
\(259\) 2293.70 + 3972.80i 0.550283 + 0.953118i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2296.69 + 3977.98i 0.538479 + 0.932673i 0.998986 + 0.0450168i \(0.0143341\pi\)
−0.460507 + 0.887656i \(0.652333\pi\)
\(264\) 0 0
\(265\) −2200.37 + 3811.15i −0.510066 + 0.883460i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −83.0074 −0.0188143 −0.00940716 0.999956i \(-0.502994\pi\)
−0.00940716 + 0.999956i \(0.502994\pi\)
\(270\) 0 0
\(271\) −3464.08 −0.776487 −0.388244 0.921557i \(-0.626918\pi\)
−0.388244 + 0.921557i \(0.626918\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1043.87 1808.04i 0.228901 0.396469i
\(276\) 0 0
\(277\) 635.350 + 1100.46i 0.137814 + 0.238701i 0.926669 0.375879i \(-0.122659\pi\)
−0.788855 + 0.614580i \(0.789326\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3825.90 + 6626.65i 0.812221 + 1.40681i 0.911306 + 0.411729i \(0.135075\pi\)
−0.0990857 + 0.995079i \(0.531592\pi\)
\(282\) 0 0
\(283\) 3.72507 6.45201i 0.000782446 0.00135524i −0.865634 0.500677i \(-0.833084\pi\)
0.866416 + 0.499322i \(0.166418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 252.384 0.0519086
\(288\) 0 0
\(289\) −1015.34 −0.206663
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1407.01 + 2437.02i −0.280541 + 0.485911i −0.971518 0.236965i \(-0.923847\pi\)
0.690977 + 0.722877i \(0.257181\pi\)
\(294\) 0 0
\(295\) 1615.31 + 2797.81i 0.318804 + 0.552185i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4933.88 + 8545.73i 0.954293 + 1.65288i
\(300\) 0 0
\(301\) −3237.30 + 5607.16i −0.619916 + 1.07373i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6645.83 −1.24767
\(306\) 0 0
\(307\) −5096.55 −0.947477 −0.473739 0.880666i \(-0.657096\pi\)
−0.473739 + 0.880666i \(0.657096\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3892.41 6741.85i 0.709705 1.22924i −0.255262 0.966872i \(-0.582162\pi\)
0.964967 0.262373i \(-0.0845050\pi\)
\(312\) 0 0
\(313\) 123.235 + 213.448i 0.0222544 + 0.0385457i 0.876938 0.480603i \(-0.159582\pi\)
−0.854684 + 0.519149i \(0.826249\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −29.6600 51.3727i −0.00525512 0.00910214i 0.863386 0.504544i \(-0.168339\pi\)
−0.868641 + 0.495442i \(0.835006\pi\)
\(318\) 0 0
\(319\) −1268.64 + 2197.34i −0.222665 + 0.385666i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2484.08 −0.427920
\(324\) 0 0
\(325\) −4102.94 −0.700277
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4175.23 7231.71i 0.699660 1.21185i
\(330\) 0 0
\(331\) −2204.55 3818.39i −0.366082 0.634072i 0.622867 0.782327i \(-0.285968\pi\)
−0.988949 + 0.148255i \(0.952634\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3860.84 6687.17i −0.629672 1.09062i
\(336\) 0 0
\(337\) −3016.15 + 5224.12i −0.487537 + 0.844440i −0.999897 0.0143313i \(-0.995438\pi\)
0.512360 + 0.858771i \(0.328771\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 357.032 0.0566991
\(342\) 0 0
\(343\) 6889.64 1.08456
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3120.39 + 5404.68i −0.482742 + 0.836133i −0.999804 0.0198147i \(-0.993692\pi\)
0.517062 + 0.855948i \(0.327026\pi\)
\(348\) 0 0
\(349\) −2817.31 4879.72i −0.432111 0.748439i 0.564943 0.825130i \(-0.308898\pi\)
−0.997055 + 0.0766906i \(0.975565\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1720.66 2980.27i −0.259437 0.449359i 0.706654 0.707559i \(-0.250204\pi\)
−0.966091 + 0.258201i \(0.916870\pi\)
\(354\) 0 0
\(355\) 4414.27 7645.74i 0.659958 1.14308i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7808.79 −1.14800 −0.574000 0.818855i \(-0.694609\pi\)
−0.574000 + 0.818855i \(0.694609\pi\)
\(360\) 0 0
\(361\) −5275.83 −0.769183
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4652.26 8057.94i 0.667151 1.15554i
\(366\) 0 0
\(367\) −6804.06 11785.0i −0.967763 1.67621i −0.702001 0.712176i \(-0.747710\pi\)
−0.265763 0.964038i \(-0.585624\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2365.09 4096.46i −0.330969 0.573255i
\(372\) 0 0
\(373\) 1874.08 3246.01i 0.260151 0.450595i −0.706131 0.708082i \(-0.749561\pi\)
0.966282 + 0.257486i \(0.0828942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4986.37 0.681197
\(378\) 0 0
\(379\) −10210.3 −1.38382 −0.691909 0.721985i \(-0.743230\pi\)
−0.691909 + 0.721985i \(0.743230\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3064.75 5308.30i 0.408880 0.708202i −0.585884 0.810395i \(-0.699253\pi\)
0.994765 + 0.102193i \(0.0325860\pi\)
\(384\) 0 0
\(385\) 3746.93 + 6489.87i 0.496003 + 0.859103i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2432.99 4214.05i −0.317114 0.549257i 0.662771 0.748822i \(-0.269380\pi\)
−0.979885 + 0.199565i \(0.936047\pi\)
\(390\) 0 0
\(391\) −4011.36 + 6947.87i −0.518831 + 0.898642i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9754.78 −1.24257
\(396\) 0 0
\(397\) −438.311 −0.0554110 −0.0277055 0.999616i \(-0.508820\pi\)
−0.0277055 + 0.999616i \(0.508820\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5876.83 + 10179.0i −0.731857 + 1.26761i 0.224231 + 0.974536i \(0.428013\pi\)
−0.956089 + 0.293078i \(0.905320\pi\)
\(402\) 0 0
\(403\) −350.829 607.653i −0.0433648 0.0751101i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6242.08 + 10811.6i 0.760217 + 1.31673i
\(408\) 0 0
\(409\) −5623.24 + 9739.75i −0.679833 + 1.17750i 0.295198 + 0.955436i \(0.404614\pi\)
−0.975031 + 0.222069i \(0.928719\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3472.48 −0.413728
\(414\) 0 0
\(415\) 14566.4 1.72298
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6158.18 + 10666.3i −0.718012 + 1.24363i 0.243774 + 0.969832i \(0.421614\pi\)
−0.961786 + 0.273801i \(0.911719\pi\)
\(420\) 0 0
\(421\) 2351.30 + 4072.57i 0.272198 + 0.471461i 0.969424 0.245390i \(-0.0789160\pi\)
−0.697226 + 0.716851i \(0.745583\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1667.89 2888.87i −0.190364 0.329720i
\(426\) 0 0
\(427\) 3571.68 6186.32i 0.404790 0.701118i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2204.89 0.246417 0.123208 0.992381i \(-0.460682\pi\)
0.123208 + 0.992381i \(0.460682\pi\)
\(432\) 0 0
\(433\) 9426.46 1.04620 0.523102 0.852270i \(-0.324775\pi\)
0.523102 + 0.852270i \(0.324775\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2556.54 4428.06i 0.279854 0.484721i
\(438\) 0 0
\(439\) 3842.29 + 6655.05i 0.417728 + 0.723527i 0.995711 0.0925227i \(-0.0294931\pi\)
−0.577982 + 0.816049i \(0.696160\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6197.15 10733.8i −0.664640 1.15119i −0.979383 0.202013i \(-0.935252\pi\)
0.314743 0.949177i \(-0.398082\pi\)
\(444\) 0 0
\(445\) 2121.14 3673.92i 0.225959 0.391372i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14568.7 1.53127 0.765635 0.643275i \(-0.222425\pi\)
0.765635 + 0.643275i \(0.222425\pi\)
\(450\) 0 0
\(451\) 686.840 0.0717118
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7363.65 12754.2i 0.758711 1.31413i
\(456\) 0 0
\(457\) 323.024 + 559.493i 0.0330643 + 0.0572691i 0.882084 0.471092i \(-0.156140\pi\)
−0.849020 + 0.528361i \(0.822807\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9500.49 + 16455.3i 0.959831 + 1.66248i 0.722904 + 0.690948i \(0.242807\pi\)
0.236927 + 0.971528i \(0.423860\pi\)
\(462\) 0 0
\(463\) −6832.86 + 11834.9i −0.685853 + 1.18793i 0.287315 + 0.957836i \(0.407237\pi\)
−0.973168 + 0.230096i \(0.926096\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3762.83 −0.372855 −0.186427 0.982469i \(-0.559691\pi\)
−0.186427 + 0.982469i \(0.559691\pi\)
\(468\) 0 0
\(469\) 8299.74 0.817156
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8810.00 + 15259.4i −0.856415 + 1.48335i
\(474\) 0 0
\(475\) 1062.99 + 1841.15i 0.102681 + 0.177848i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3381.94 5857.68i −0.322598 0.558757i 0.658425 0.752646i \(-0.271223\pi\)
−0.981023 + 0.193890i \(0.937890\pi\)
\(480\) 0 0
\(481\) 12267.2 21247.5i 1.16286 2.01414i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 19845.1 1.85798
\(486\) 0 0
\(487\) −1447.72 −0.134708 −0.0673538 0.997729i \(-0.521456\pi\)
−0.0673538 + 0.997729i \(0.521456\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3919.14 + 6788.15i −0.360220 + 0.623920i −0.987997 0.154474i \(-0.950632\pi\)
0.627777 + 0.778394i \(0.283965\pi\)
\(492\) 0 0
\(493\) 2027.02 + 3510.90i 0.185177 + 0.320736i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4744.73 + 8218.12i 0.428230 + 0.741716i
\(498\) 0 0
\(499\) 2122.67 3676.57i 0.190428 0.329832i −0.754964 0.655766i \(-0.772346\pi\)
0.945392 + 0.325935i \(0.105679\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 17914.9 1.58804 0.794021 0.607891i \(-0.207984\pi\)
0.794021 + 0.607891i \(0.207984\pi\)
\(504\) 0 0
\(505\) −13401.4 −1.18090
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7260.14 + 12574.9i −0.632220 + 1.09504i 0.354877 + 0.934913i \(0.384523\pi\)
−0.987097 + 0.160124i \(0.948810\pi\)
\(510\) 0 0
\(511\) 5000.54 + 8661.18i 0.432898 + 0.749801i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −11471.7 19869.5i −0.981558 1.70011i
\(516\) 0 0
\(517\) 11362.5 19680.4i 0.966581 1.67417i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3372.49 −0.283592 −0.141796 0.989896i \(-0.545288\pi\)
−0.141796 + 0.989896i \(0.545288\pi\)
\(522\) 0 0
\(523\) 4339.40 0.362808 0.181404 0.983409i \(-0.441936\pi\)
0.181404 + 0.983409i \(0.441936\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 285.232 494.036i 0.0235766 0.0408359i
\(528\) 0 0
\(529\) −2173.23 3764.15i −0.178617 0.309373i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −674.906 1168.97i −0.0548469 0.0949977i
\(534\) 0 0
\(535\) 6720.82 11640.8i 0.543115 0.940702i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5347.32 0.427320
\(540\) 0 0
\(541\) −3831.98 −0.304528 −0.152264 0.988340i \(-0.548656\pi\)
−0.152264 + 0.988340i \(0.548656\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11520.7 19954.4i 0.905490 1.56835i
\(546\) 0 0
\(547\) −6701.34 11607.1i −0.523818 0.907280i −0.999616 0.0277247i \(-0.991174\pi\)
0.475797 0.879555i \(-0.342160\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1291.87 2237.59i −0.0998831 0.173003i
\(552\) 0 0
\(553\) 5242.52 9080.32i 0.403137 0.698254i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14810.0 −1.12661 −0.563304 0.826249i \(-0.690470\pi\)
−0.563304 + 0.826249i \(0.690470\pi\)
\(558\) 0 0
\(559\) 34627.7 2.62003
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6116.61 + 10594.3i −0.457877 + 0.793066i −0.998849 0.0479751i \(-0.984723\pi\)
0.540972 + 0.841041i \(0.318057\pi\)
\(564\) 0 0
\(565\) 4187.80 + 7253.49i 0.311827 + 0.540100i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4855.84 + 8410.56i 0.357763 + 0.619664i 0.987587 0.157074i \(-0.0502061\pi\)
−0.629823 + 0.776738i \(0.716873\pi\)
\(570\) 0 0
\(571\) −4264.81 + 7386.86i −0.312568 + 0.541384i −0.978918 0.204255i \(-0.934523\pi\)
0.666349 + 0.745640i \(0.267856\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6866.17 0.497981
\(576\) 0 0
\(577\) 15314.0 1.10491 0.552453 0.833544i \(-0.313692\pi\)
0.552453 + 0.833544i \(0.313692\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7828.44 + 13559.3i −0.558999 + 0.968215i
\(582\) 0 0
\(583\) −6436.38 11148.1i −0.457234 0.791953i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9837.53 17039.1i −0.691717 1.19809i −0.971275 0.237961i \(-0.923521\pi\)
0.279557 0.960129i \(-0.409812\pi\)
\(588\) 0 0
\(589\) −181.786 + 314.862i −0.0127171 + 0.0220266i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15635.9 1.08278 0.541391 0.840771i \(-0.317898\pi\)
0.541391 + 0.840771i \(0.317898\pi\)
\(594\) 0 0
\(595\) 11973.6 0.824994
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 242.594 420.186i 0.0165478 0.0286616i −0.857633 0.514262i \(-0.828066\pi\)
0.874181 + 0.485601i \(0.161399\pi\)
\(600\) 0 0
\(601\) 7398.12 + 12813.9i 0.502123 + 0.869702i 0.999997 + 0.00245282i \(0.000780759\pi\)
−0.497874 + 0.867249i \(0.665886\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1307.29 + 2264.30i 0.0878496 + 0.152160i
\(606\) 0 0
\(607\) −1773.99 + 3072.64i −0.118623 + 0.205461i −0.919222 0.393739i \(-0.871181\pi\)
0.800599 + 0.599200i \(0.204515\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −44660.3 −2.95706
\(612\) 0 0
\(613\) 20450.9 1.34748 0.673740 0.738968i \(-0.264687\pi\)
0.673740 + 0.738968i \(0.264687\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7652.60 13254.7i 0.499323 0.864853i −0.500677 0.865634i \(-0.666915\pi\)
1.00000 0.000781625i \(0.000248799\pi\)
\(618\) 0 0
\(619\) 2075.59 + 3595.03i 0.134774 + 0.233435i 0.925511 0.378720i \(-0.123636\pi\)
−0.790737 + 0.612156i \(0.790302\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2279.93 + 3948.96i 0.146619 + 0.253951i
\(624\) 0 0
\(625\) 9724.50 16843.3i 0.622368 1.07797i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 19947.1 1.26446
\(630\) 0 0
\(631\) 25954.4 1.63745 0.818724 0.574187i \(-0.194682\pi\)
0.818724 + 0.574187i \(0.194682\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13291.4 + 23021.3i −0.830632 + 1.43870i
\(636\) 0 0
\(637\) −5254.41 9100.91i −0.326825 0.566077i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8630.66 14948.7i −0.531811 0.921123i −0.999310 0.0371297i \(-0.988179\pi\)
0.467500 0.883993i \(-0.345155\pi\)
\(642\) 0 0
\(643\) −1430.18 + 2477.15i −0.0877152 + 0.151927i −0.906545 0.422109i \(-0.861290\pi\)
0.818830 + 0.574036i \(0.194623\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28384.9 1.72477 0.862384 0.506255i \(-0.168971\pi\)
0.862384 + 0.506255i \(0.168971\pi\)
\(648\) 0 0
\(649\) −9450.04 −0.571566
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5518.22 + 9557.84i −0.330696 + 0.572783i −0.982649 0.185477i \(-0.940617\pi\)
0.651952 + 0.758260i \(0.273950\pi\)
\(654\) 0 0
\(655\) −1044.68 1809.43i −0.0623190 0.107940i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10647.5 18441.9i −0.629387 1.09013i −0.987675 0.156519i \(-0.949973\pi\)
0.358288 0.933611i \(-0.383361\pi\)
\(660\) 0 0
\(661\) −6162.37 + 10673.5i −0.362615 + 0.628067i −0.988390 0.151936i \(-0.951449\pi\)
0.625776 + 0.780003i \(0.284783\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7631.11 −0.444995
\(666\) 0 0
\(667\) −8344.58 −0.484413
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9719.98 16835.5i 0.559219 0.968595i
\(672\) 0 0
\(673\) −1525.47 2642.19i −0.0873738 0.151336i 0.819027 0.573756i \(-0.194514\pi\)
−0.906400 + 0.422420i \(0.861181\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −842.733 1459.66i −0.0478418 0.0828643i 0.841113 0.540860i \(-0.181901\pi\)
−0.888955 + 0.457995i \(0.848568\pi\)
\(678\) 0 0
\(679\) −10665.4 + 18473.0i −0.602797 + 1.04408i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −34715.5 −1.94488 −0.972440 0.233155i \(-0.925095\pi\)
−0.972440 + 0.233155i \(0.925095\pi\)
\(684\) 0 0
\(685\) −28309.4 −1.57905
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12649.1 + 21908.9i −0.699408 + 1.21141i
\(690\) 0 0
\(691\) 4147.68 + 7184.00i 0.228343 + 0.395502i 0.957317 0.289039i \(-0.0933358\pi\)
−0.728974 + 0.684542i \(0.760002\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17944.7 31081.1i −0.979398 1.69637i
\(696\) 0 0
\(697\) 548.714 950.400i 0.0298192 0.0516484i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23420.5 1.26188 0.630942 0.775830i \(-0.282669\pi\)
0.630942 + 0.775830i \(0.282669\pi\)
\(702\) 0 0
\(703\) −12712.8 −0.682037
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7202.35 12474.8i 0.383129 0.663599i
\(708\) 0 0
\(709\) 2641.33 + 4574.92i 0.139912 + 0.242334i 0.927463 0.373915i \(-0.121985\pi\)
−0.787551 + 0.616249i \(0.788651\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 587.104 + 1016.89i 0.0308376 + 0.0534123i
\(714\) 0 0
\(715\) 20039.5 34709.4i 1.04816 1.81547i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 714.975 0.0370849 0.0185425 0.999828i \(-0.494097\pi\)
0.0185425 + 0.999828i \(0.494097\pi\)
\(720\) 0 0
\(721\) 24660.9 1.27382
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1734.81 3004.77i 0.0888677 0.153923i
\(726\) 0 0
\(727\) −12897.7 22339.5i −0.657979 1.13965i −0.981138 0.193308i \(-0.938078\pi\)
0.323159 0.946345i \(-0.395255\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14076.5 + 24381.3i 0.712230 + 1.23362i
\(732\) 0 0
\(733\) −1815.78 + 3145.02i −0.0914969 + 0.158477i −0.908141 0.418664i \(-0.862498\pi\)
0.816644 + 0.577141i \(0.195832\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22587.0 1.12890
\(738\) 0 0
\(739\) −24758.3 −1.23241 −0.616203 0.787588i \(-0.711330\pi\)
−0.616203 + 0.787588i \(0.711330\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4422.66 7660.28i 0.218374 0.378235i −0.735937 0.677050i \(-0.763258\pi\)
0.954311 + 0.298815i \(0.0965914\pi\)
\(744\) 0 0
\(745\) 10362.7 + 17948.8i 0.509612 + 0.882675i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7223.96 + 12512.3i 0.352413 + 0.610398i
\(750\) 0 0
\(751\) −830.199 + 1437.95i −0.0403387 + 0.0698688i −0.885490 0.464659i \(-0.846177\pi\)
0.845151 + 0.534527i \(0.179510\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 241.899 0.0116604
\(756\) 0 0
\(757\) 19937.2 0.957239 0.478619 0.878022i \(-0.341137\pi\)
0.478619 + 0.878022i \(0.341137\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4221.87 7312.50i 0.201107 0.348328i −0.747778 0.663949i \(-0.768879\pi\)
0.948886 + 0.315620i \(0.102213\pi\)
\(762\) 0 0
\(763\) 12383.2 + 21448.3i 0.587550 + 1.01767i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9285.84 + 16083.5i 0.437148 + 0.757162i
\(768\) 0 0
\(769\) −8553.93 + 14815.8i −0.401122 + 0.694763i −0.993862 0.110631i \(-0.964713\pi\)
0.592740 + 0.805394i \(0.298046\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9821.05 0.456971 0.228486 0.973547i \(-0.426623\pi\)
0.228486 + 0.973547i \(0.426623\pi\)
\(774\) 0 0
\(775\) −488.226 −0.0226292
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −349.710 + 605.715i −0.0160843 + 0.0278588i
\(780\) 0 0
\(781\) 12912.3 + 22364.8i 0.591600 + 1.02468i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24989.2 + 43282.6i 1.13618 + 1.96792i
\(786\) 0 0
\(787\) 2622.78 4542.80i 0.118796 0.205760i −0.800495 0.599339i \(-0.795430\pi\)
0.919291 + 0.393579i \(0.128763\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9002.62 −0.404673
\(792\) 0 0
\(793\) −38204.4 −1.71082
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4609.26 + 7983.47i −0.204854 + 0.354817i −0.950086 0.311988i \(-0.899005\pi\)
0.745232 + 0.666805i \(0.232339\pi\)
\(798\) 0 0
\(799\) −18154.9 31445.2i −0.803848 1.39231i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13608.5 + 23570.6i 0.598049 + 1.03585i
\(804\) 0 0
\(805\) −12322.9 + 21343.9i −0.539534 + 0.934501i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16397.4 −0.712611 −0.356305 0.934370i \(-0.615964\pi\)
−0.356305 + 0.934370i \(0.615964\pi\)
\(810\) 0 0
\(811\) 8468.88 0.366686 0.183343 0.983049i \(-0.441308\pi\)
0.183343 + 0.983049i \(0.441308\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 17424.6 30180.3i 0.748906 1.29714i
\(816\) 0 0
\(817\) −8971.35 15538.8i −0.384171 0.665404i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2616.73 4532.31i −0.111236 0.192666i 0.805033 0.593230i \(-0.202148\pi\)
−0.916269 + 0.400564i \(0.868814\pi\)
\(822\) 0 0
\(823\) 1562.34 2706.05i 0.0661722 0.114614i −0.831041 0.556211i \(-0.812255\pi\)
0.897213 + 0.441597i \(0.145588\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11037.6 −0.464103 −0.232052 0.972703i \(-0.574544\pi\)
−0.232052 + 0.972703i \(0.574544\pi\)
\(828\) 0 0
\(829\) −23941.9 −1.00306 −0.501529 0.865141i \(-0.667229\pi\)
−0.501529 + 0.865141i \(0.667229\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4271.96 7399.25i 0.177688 0.307765i
\(834\) 0 0
\(835\) 10438.4 + 18079.9i 0.432618 + 0.749317i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10629.7 + 18411.2i 0.437399 + 0.757597i 0.997488 0.0708352i \(-0.0225664\pi\)
−0.560089 + 0.828432i \(0.689233\pi\)
\(840\) 0 0
\(841\) 10086.2 17469.7i 0.413554 0.716296i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −49418.1 −2.01187
\(846\) 0 0
\(847\) −2810.32 −0.114007
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −20528.9 + 35557.2i −0.826936 + 1.43230i
\(852\) 0 0
\(853\) −4414.37 7645.91i −0.177192 0.306906i 0.763725 0.645541i \(-0.223368\pi\)
−0.940918 + 0.338635i \(0.890035\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −19007.4 32921.8i −0.757621 1.31224i −0.944061 0.329771i \(-0.893028\pi\)
0.186440 0.982466i \(-0.440305\pi\)
\(858\) 0 0
\(859\) −2458.81 + 4258.78i −0.0976640 + 0.169159i −0.910717 0.413030i \(-0.864470\pi\)
0.813053 + 0.582189i \(0.197804\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20474.1 0.807585 0.403792 0.914851i \(-0.367692\pi\)
0.403792 + 0.914851i \(0.367692\pi\)
\(864\) 0 0
\(865\) −20073.7 −0.789049
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14267.0 24711.2i 0.556934 0.964639i
\(870\) 0 0
\(871\) −22194.5 38442.0i −0.863412 1.49547i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6863.05 + 11887.2i 0.265158 + 0.459268i
\(876\) 0 0
\(877\) −7140.36 + 12367.5i −0.274929 + 0.476191i −0.970117 0.242637i \(-0.921988\pi\)
0.695188 + 0.718828i \(0.255321\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12505.2 −0.478217 −0.239109 0.970993i \(-0.576855\pi\)
−0.239109 + 0.970993i \(0.576855\pi\)
\(882\) 0 0
\(883\) −34255.9 −1.30555 −0.652776 0.757551i \(-0.726396\pi\)
−0.652776 + 0.757551i \(0.726396\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12186.2 + 21107.2i −0.461300 + 0.798995i −0.999026 0.0441244i \(-0.985950\pi\)
0.537726 + 0.843120i \(0.319284\pi\)
\(888\) 0 0
\(889\) −14286.4 24744.7i −0.538976 0.933534i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11570.6 + 20040.9i 0.433590 + 0.750999i
\(894\) 0 0
\(895\) −27560.5 + 47736.2i −1.02933 + 1.78284i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 593.350 0.0220126
\(900\) 0 0
\(901\) −20568.0 −0.760510
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10568.2 18304.7i 0.388177 0.672343i
\(906\) 0 0
\(907\) 21418.0 + 37097.1i 0.784095 + 1.35809i 0.929538 + 0.368726i \(0.120206\pi\)
−0.145443 + 0.989367i \(0.546461\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18072.6 31302.7i −0.657271 1.13843i −0.981319 0.192386i \(-0.938378\pi\)
0.324049 0.946040i \(-0.394956\pi\)
\(912\) 0 0
\(913\) −21304.4 + 36900.3i −0.772258 + 1.33759i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2245.77 0.0808744
\(918\) 0 0
\(919\) −23283.2 −0.835738 −0.417869 0.908507i \(-0.637223\pi\)
−0.417869 + 0.908507i \(0.637223\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 25376.0 43952.5i 0.904941 1.56740i
\(924\) 0 0
\(925\) −8535.77 14784.4i −0.303410 0.525522i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19595.2 33939.9i −0.692033 1.19864i −0.971171 0.238385i \(-0.923382\pi\)
0.279138 0.960251i \(-0.409951\pi\)
\(930\) 0 0
\(931\) −2722.63 + 4715.73i −0.0958438 + 0.166006i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 32585.1 1.13973
\(936\) 0 0
\(937\) −36892.8 −1.28627 −0.643135 0.765753i \(-0.722366\pi\)
−0.643135 + 0.765753i \(0.722366\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 13302.9 23041.3i 0.460853 0.798220i −0.538151 0.842848i \(-0.680877\pi\)
0.999004 + 0.0446282i \(0.0142103\pi\)
\(942\) 0 0
\(943\) 1129.44 + 1956.25i 0.0390027 + 0.0675547i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6420.67 + 11120.9i 0.220321 + 0.381607i 0.954905 0.296910i \(-0.0959563\pi\)
−0.734585 + 0.678517i \(0.762623\pi\)
\(948\) 0 0
\(949\) 26744.1 46322.1i 0.914804 1.58449i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −33977.5 −1.15492 −0.577460 0.816419i \(-0.695956\pi\)
−0.577460 + 0.816419i \(0.695956\pi\)
\(954\) 0 0
\(955\) 34408.1 1.16589
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 15214.4 26352.0i 0.512301 0.887332i
\(960\) 0 0
\(961\) 14853.8 + 25727.5i 0.498599 + 0.863598i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −9319.07 16141.1i −0.310872 0.538446i
\(966\) 0 0
\(967\) −24671.4 + 42732.2i −0.820455 + 1.42107i 0.0848894 + 0.996390i \(0.472946\pi\)
−0.905344 + 0.424679i \(0.860387\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −43375.7 −1.43356 −0.716782 0.697297i \(-0.754386\pi\)
−0.716782 + 0.697297i \(0.754386\pi\)
\(972\) 0 0
\(973\) 38576.2 1.27101
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19847.3 + 34376.5i −0.649919 + 1.12569i 0.333223 + 0.942848i \(0.391864\pi\)
−0.983142 + 0.182845i \(0.941470\pi\)
\(978\) 0 0
\(979\) 6204.62 + 10746.7i 0.202554 + 0.350834i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4529.45 + 7845.23i 0.146965 + 0.254551i 0.930104 0.367295i \(-0.119716\pi\)
−0.783139 + 0.621847i \(0.786383\pi\)
\(984\) 0 0
\(985\) −33712.6 + 58391.9i −1.09053 + 1.88885i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −57948.6 −1.86315
\(990\) 0 0
\(991\) −33006.3 −1.05800 −0.529001 0.848621i \(-0.677433\pi\)
−0.529001 + 0.848621i \(0.677433\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16306.3 + 28243.4i −0.519544 + 0.899876i
\(996\) 0 0
\(997\) −22162.2 38386.1i −0.703996 1.21936i −0.967053 0.254577i \(-0.918064\pi\)
0.263056 0.964781i \(-0.415270\pi\)
\(998\) 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 648.4.i.q.433.2 4
3.2 odd 2 648.4.i.p.433.1 4
9.2 odd 6 648.4.i.p.217.1 4
9.4 even 3 648.4.a.d.1.1 2
9.5 odd 6 648.4.a.e.1.2 yes 2
9.7 even 3 inner 648.4.i.q.217.2 4
36.23 even 6 1296.4.a.p.1.2 2
36.31 odd 6 1296.4.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
648.4.a.d.1.1 2 9.4 even 3
648.4.a.e.1.2 yes 2 9.5 odd 6
648.4.i.p.217.1 4 9.2 odd 6
648.4.i.p.433.1 4 3.2 odd 2
648.4.i.q.217.2 4 9.7 even 3 inner
648.4.i.q.433.2 4 1.1 even 1 trivial
1296.4.a.n.1.1 2 36.31 odd 6
1296.4.a.p.1.2 2 36.23 even 6