Properties

Label 675.4.b.n.649.6
Level $675$
Weight $4$
Character 675.649
Analytic conductor $39.826$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,4,Mod(649,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), 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: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12559936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{3} + 49x^{2} - 42x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.6
Root \(-2.05655 + 2.05655i\) of defining polynomial
Character \(\chi\) \(=\) 675.649
Dual form 675.4.b.n.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.45876i q^{2} -21.7980 q^{4} -11.8065i q^{7} -75.3201i q^{8} -56.2376 q^{11} -34.5961i q^{13} +64.4489 q^{14} +236.770 q^{16} +39.2675i q^{17} +146.561 q^{19} -306.987i q^{22} +23.5777i q^{23} +188.851 q^{26} +257.359i q^{28} -161.003 q^{29} -29.5465 q^{31} +689.908i q^{32} -214.352 q^{34} -217.688i q^{37} +800.039i q^{38} +142.290 q^{41} +468.030i q^{43} +1225.87 q^{44} -128.705 q^{46} +394.318i q^{47} +203.606 q^{49} +754.126i q^{52} +134.780i q^{53} -889.268 q^{56} -878.875i q^{58} +131.195 q^{59} +259.801 q^{61} -161.287i q^{62} -1871.88 q^{64} +445.244i q^{67} -855.954i q^{68} +560.841 q^{71} +88.6681i q^{73} +1188.30 q^{74} -3194.74 q^{76} +663.970i q^{77} -450.342 q^{79} +776.726i q^{82} -284.295i q^{83} -2554.86 q^{86} +4235.82i q^{88} -625.305 q^{89} -408.459 q^{91} -513.948i q^{92} -2152.49 q^{94} -193.261i q^{97} +1111.44i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 34 q^{4} + 10 q^{11} - 120 q^{14} + 322 q^{16} + 100 q^{19} + 370 q^{26} - 230 q^{29} - 230 q^{31} - 826 q^{34} + 1160 q^{41} + 2830 q^{44} - 570 q^{46} - 1154 q^{49} - 4380 q^{56} - 760 q^{59} - 304 q^{61}+ \cdots - 7666 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.45876i 1.92996i 0.262320 + 0.964981i \(0.415513\pi\)
−0.262320 + 0.964981i \(0.584487\pi\)
\(3\) 0 0
\(4\) −21.7980 −2.72475
\(5\) 0 0
\(6\) 0 0
\(7\) − 11.8065i − 0.637492i −0.947840 0.318746i \(-0.896738\pi\)
0.947840 0.318746i \(-0.103262\pi\)
\(8\) − 75.3201i − 3.32871i
\(9\) 0 0
\(10\) 0 0
\(11\) −56.2376 −1.54148 −0.770740 0.637150i \(-0.780113\pi\)
−0.770740 + 0.637150i \(0.780113\pi\)
\(12\) 0 0
\(13\) − 34.5961i − 0.738094i −0.929411 0.369047i \(-0.879684\pi\)
0.929411 0.369047i \(-0.120316\pi\)
\(14\) 64.4489 1.23034
\(15\) 0 0
\(16\) 236.770 3.69953
\(17\) 39.2675i 0.560221i 0.959968 + 0.280111i \(0.0903712\pi\)
−0.959968 + 0.280111i \(0.909629\pi\)
\(18\) 0 0
\(19\) 146.561 1.76965 0.884825 0.465924i \(-0.154278\pi\)
0.884825 + 0.465924i \(0.154278\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 306.987i − 2.97500i
\(23\) 23.5777i 0.213752i 0.994272 + 0.106876i \(0.0340848\pi\)
−0.994272 + 0.106876i \(0.965915\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 188.851 1.42449
\(27\) 0 0
\(28\) 257.359i 1.73701i
\(29\) −161.003 −1.03095 −0.515473 0.856906i \(-0.672384\pi\)
−0.515473 + 0.856906i \(0.672384\pi\)
\(30\) 0 0
\(31\) −29.5465 −0.171184 −0.0855921 0.996330i \(-0.527278\pi\)
−0.0855921 + 0.996330i \(0.527278\pi\)
\(32\) 689.908i 3.81124i
\(33\) 0 0
\(34\) −214.352 −1.08121
\(35\) 0 0
\(36\) 0 0
\(37\) − 217.688i − 0.967233i −0.875280 0.483617i \(-0.839323\pi\)
0.875280 0.483617i \(-0.160677\pi\)
\(38\) 800.039i 3.41536i
\(39\) 0 0
\(40\) 0 0
\(41\) 142.290 0.541999 0.270999 0.962580i \(-0.412646\pi\)
0.270999 + 0.962580i \(0.412646\pi\)
\(42\) 0 0
\(43\) 468.030i 1.65986i 0.557869 + 0.829929i \(0.311619\pi\)
−0.557869 + 0.829929i \(0.688381\pi\)
\(44\) 1225.87 4.20015
\(45\) 0 0
\(46\) −128.705 −0.412533
\(47\) 394.318i 1.22377i 0.790946 + 0.611886i \(0.209589\pi\)
−0.790946 + 0.611886i \(0.790411\pi\)
\(48\) 0 0
\(49\) 203.606 0.593604
\(50\) 0 0
\(51\) 0 0
\(52\) 754.126i 2.01112i
\(53\) 134.780i 0.349311i 0.984630 + 0.174655i \(0.0558811\pi\)
−0.984630 + 0.174655i \(0.944119\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −889.268 −2.12203
\(57\) 0 0
\(58\) − 878.875i − 1.98969i
\(59\) 131.195 0.289495 0.144747 0.989469i \(-0.453763\pi\)
0.144747 + 0.989469i \(0.453763\pi\)
\(60\) 0 0
\(61\) 259.801 0.545313 0.272657 0.962111i \(-0.412098\pi\)
0.272657 + 0.962111i \(0.412098\pi\)
\(62\) − 161.287i − 0.330379i
\(63\) 0 0
\(64\) −1871.88 −3.65602
\(65\) 0 0
\(66\) 0 0
\(67\) 445.244i 0.811869i 0.913902 + 0.405935i \(0.133054\pi\)
−0.913902 + 0.405935i \(0.866946\pi\)
\(68\) − 855.954i − 1.52647i
\(69\) 0 0
\(70\) 0 0
\(71\) 560.841 0.937459 0.468729 0.883342i \(-0.344712\pi\)
0.468729 + 0.883342i \(0.344712\pi\)
\(72\) 0 0
\(73\) 88.6681i 0.142162i 0.997471 + 0.0710809i \(0.0226448\pi\)
−0.997471 + 0.0710809i \(0.977355\pi\)
\(74\) 1188.30 1.86672
\(75\) 0 0
\(76\) −3194.74 −4.82186
\(77\) 663.970i 0.982681i
\(78\) 0 0
\(79\) −450.342 −0.641360 −0.320680 0.947188i \(-0.603911\pi\)
−0.320680 + 0.947188i \(0.603911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 776.726i 1.04604i
\(83\) − 284.295i − 0.375969i −0.982172 0.187985i \(-0.939804\pi\)
0.982172 0.187985i \(-0.0601955\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2554.86 −3.20346
\(87\) 0 0
\(88\) 4235.82i 5.13114i
\(89\) −625.305 −0.744744 −0.372372 0.928083i \(-0.621455\pi\)
−0.372372 + 0.928083i \(0.621455\pi\)
\(90\) 0 0
\(91\) −408.459 −0.470529
\(92\) − 513.948i − 0.582421i
\(93\) 0 0
\(94\) −2152.49 −2.36183
\(95\) 0 0
\(96\) 0 0
\(97\) − 193.261i − 0.202296i −0.994871 0.101148i \(-0.967748\pi\)
0.994871 0.101148i \(-0.0322516\pi\)
\(98\) 1111.44i 1.14563i
\(99\) 0 0
\(100\) 0 0
\(101\) 1374.86 1.35449 0.677245 0.735758i \(-0.263174\pi\)
0.677245 + 0.735758i \(0.263174\pi\)
\(102\) 0 0
\(103\) − 2029.60i − 1.94158i −0.239935 0.970789i \(-0.577126\pi\)
0.239935 0.970789i \(-0.422874\pi\)
\(104\) −2605.78 −2.45690
\(105\) 0 0
\(106\) −735.732 −0.674156
\(107\) − 823.062i − 0.743630i −0.928307 0.371815i \(-0.878736\pi\)
0.928307 0.371815i \(-0.121264\pi\)
\(108\) 0 0
\(109\) 829.868 0.729238 0.364619 0.931157i \(-0.381199\pi\)
0.364619 + 0.931157i \(0.381199\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 2795.43i − 2.35842i
\(113\) 1503.37i 1.25155i 0.780005 + 0.625773i \(0.215216\pi\)
−0.780005 + 0.625773i \(0.784784\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3509.54 2.80908
\(117\) 0 0
\(118\) 716.164i 0.558714i
\(119\) 463.612 0.357137
\(120\) 0 0
\(121\) 1831.67 1.37616
\(122\) 1418.19i 1.05243i
\(123\) 0 0
\(124\) 644.056 0.466435
\(125\) 0 0
\(126\) 0 0
\(127\) − 576.348i − 0.402698i −0.979520 0.201349i \(-0.935468\pi\)
0.979520 0.201349i \(-0.0645325\pi\)
\(128\) − 4698.89i − 3.24474i
\(129\) 0 0
\(130\) 0 0
\(131\) 2390.04 1.59403 0.797017 0.603957i \(-0.206410\pi\)
0.797017 + 0.603957i \(0.206410\pi\)
\(132\) 0 0
\(133\) − 1730.37i − 1.12814i
\(134\) −2430.48 −1.56688
\(135\) 0 0
\(136\) 2957.63 1.86481
\(137\) 1002.46i 0.625152i 0.949893 + 0.312576i \(0.101192\pi\)
−0.949893 + 0.312576i \(0.898808\pi\)
\(138\) 0 0
\(139\) −131.817 −0.0804356 −0.0402178 0.999191i \(-0.512805\pi\)
−0.0402178 + 0.999191i \(0.512805\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3061.50i 1.80926i
\(143\) 1945.60i 1.13776i
\(144\) 0 0
\(145\) 0 0
\(146\) −484.017 −0.274367
\(147\) 0 0
\(148\) 4745.16i 2.63547i
\(149\) −1019.49 −0.560533 −0.280267 0.959922i \(-0.590423\pi\)
−0.280267 + 0.959922i \(0.590423\pi\)
\(150\) 0 0
\(151\) 2822.38 1.52107 0.760537 0.649295i \(-0.224936\pi\)
0.760537 + 0.649295i \(0.224936\pi\)
\(152\) − 11039.0i − 5.89065i
\(153\) 0 0
\(154\) −3624.45 −1.89654
\(155\) 0 0
\(156\) 0 0
\(157\) 476.499i 0.242222i 0.992639 + 0.121111i \(0.0386456\pi\)
−0.992639 + 0.121111i \(0.961354\pi\)
\(158\) − 2458.31i − 1.23780i
\(159\) 0 0
\(160\) 0 0
\(161\) 278.371 0.136265
\(162\) 0 0
\(163\) 2242.26i 1.07747i 0.842476 + 0.538734i \(0.181097\pi\)
−0.842476 + 0.538734i \(0.818903\pi\)
\(164\) −3101.64 −1.47681
\(165\) 0 0
\(166\) 1551.90 0.725607
\(167\) 95.0390i 0.0440380i 0.999758 + 0.0220190i \(0.00700943\pi\)
−0.999758 + 0.0220190i \(0.992991\pi\)
\(168\) 0 0
\(169\) 1000.11 0.455217
\(170\) 0 0
\(171\) 0 0
\(172\) − 10202.1i − 4.52270i
\(173\) 2133.76i 0.937727i 0.883271 + 0.468864i \(0.155336\pi\)
−0.883271 + 0.468864i \(0.844664\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13315.4 −5.70275
\(177\) 0 0
\(178\) − 3413.39i − 1.43733i
\(179\) 1704.68 0.711808 0.355904 0.934523i \(-0.384173\pi\)
0.355904 + 0.934523i \(0.384173\pi\)
\(180\) 0 0
\(181\) −1360.98 −0.558902 −0.279451 0.960160i \(-0.590152\pi\)
−0.279451 + 0.960160i \(0.590152\pi\)
\(182\) − 2229.68i − 0.908103i
\(183\) 0 0
\(184\) 1775.88 0.711518
\(185\) 0 0
\(186\) 0 0
\(187\) − 2208.31i − 0.863570i
\(188\) − 8595.36i − 3.33447i
\(189\) 0 0
\(190\) 0 0
\(191\) −1096.84 −0.415522 −0.207761 0.978180i \(-0.566618\pi\)
−0.207761 + 0.978180i \(0.566618\pi\)
\(192\) 0 0
\(193\) 2867.27i 1.06938i 0.845048 + 0.534691i \(0.179572\pi\)
−0.845048 + 0.534691i \(0.820428\pi\)
\(194\) 1054.97 0.390424
\(195\) 0 0
\(196\) −4438.21 −1.61742
\(197\) 724.139i 0.261892i 0.991389 + 0.130946i \(0.0418015\pi\)
−0.991389 + 0.130946i \(0.958199\pi\)
\(198\) 0 0
\(199\) 1693.65 0.603315 0.301658 0.953416i \(-0.402460\pi\)
0.301658 + 0.953416i \(0.402460\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7505.02i 2.61411i
\(203\) 1900.88i 0.657220i
\(204\) 0 0
\(205\) 0 0
\(206\) 11079.1 3.74717
\(207\) 0 0
\(208\) − 8191.30i − 2.73060i
\(209\) −8242.22 −2.72788
\(210\) 0 0
\(211\) 947.452 0.309124 0.154562 0.987983i \(-0.450603\pi\)
0.154562 + 0.987983i \(0.450603\pi\)
\(212\) − 2937.94i − 0.951785i
\(213\) 0 0
\(214\) 4492.89 1.43518
\(215\) 0 0
\(216\) 0 0
\(217\) 348.841i 0.109129i
\(218\) 4530.05i 1.40740i
\(219\) 0 0
\(220\) 0 0
\(221\) 1358.50 0.413496
\(222\) 0 0
\(223\) − 111.866i − 0.0335923i −0.999859 0.0167961i \(-0.994653\pi\)
0.999859 0.0167961i \(-0.00534663\pi\)
\(224\) 8145.41 2.42964
\(225\) 0 0
\(226\) −8206.51 −2.41544
\(227\) 1200.70i 0.351072i 0.984473 + 0.175536i \(0.0561658\pi\)
−0.984473 + 0.175536i \(0.943834\pi\)
\(228\) 0 0
\(229\) −822.380 −0.237312 −0.118656 0.992935i \(-0.537859\pi\)
−0.118656 + 0.992935i \(0.537859\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 12126.7i 3.43172i
\(233\) 5329.21i 1.49840i 0.662341 + 0.749202i \(0.269563\pi\)
−0.662341 + 0.749202i \(0.730437\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2859.80 −0.788802
\(237\) 0 0
\(238\) 2530.75i 0.689260i
\(239\) 7085.61 1.91770 0.958850 0.283914i \(-0.0916331\pi\)
0.958850 + 0.283914i \(0.0916331\pi\)
\(240\) 0 0
\(241\) −6560.09 −1.75341 −0.876707 0.481025i \(-0.840265\pi\)
−0.876707 + 0.481025i \(0.840265\pi\)
\(242\) 9998.63i 2.65593i
\(243\) 0 0
\(244\) −5663.15 −1.48584
\(245\) 0 0
\(246\) 0 0
\(247\) − 5070.42i − 1.30617i
\(248\) 2225.45i 0.569822i
\(249\) 0 0
\(250\) 0 0
\(251\) 714.222 0.179607 0.0898033 0.995960i \(-0.471376\pi\)
0.0898033 + 0.995960i \(0.471376\pi\)
\(252\) 0 0
\(253\) − 1325.95i − 0.329494i
\(254\) 3146.14 0.777191
\(255\) 0 0
\(256\) 10675.0 2.60621
\(257\) 4396.59i 1.06713i 0.845760 + 0.533563i \(0.179147\pi\)
−0.845760 + 0.533563i \(0.820853\pi\)
\(258\) 0 0
\(259\) −2570.13 −0.616604
\(260\) 0 0
\(261\) 0 0
\(262\) 13046.6i 3.07643i
\(263\) 7550.31i 1.77024i 0.465367 + 0.885118i \(0.345922\pi\)
−0.465367 + 0.885118i \(0.654078\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 9445.68 2.17726
\(267\) 0 0
\(268\) − 9705.45i − 2.21214i
\(269\) 5536.86 1.25497 0.627487 0.778627i \(-0.284083\pi\)
0.627487 + 0.778627i \(0.284083\pi\)
\(270\) 0 0
\(271\) 3058.25 0.685518 0.342759 0.939423i \(-0.388639\pi\)
0.342759 + 0.939423i \(0.388639\pi\)
\(272\) 9297.36i 2.07256i
\(273\) 0 0
\(274\) −5472.18 −1.20652
\(275\) 0 0
\(276\) 0 0
\(277\) − 4070.19i − 0.882865i −0.897294 0.441433i \(-0.854470\pi\)
0.897294 0.441433i \(-0.145530\pi\)
\(278\) − 719.556i − 0.155238i
\(279\) 0 0
\(280\) 0 0
\(281\) 7446.19 1.58079 0.790396 0.612597i \(-0.209875\pi\)
0.790396 + 0.612597i \(0.209875\pi\)
\(282\) 0 0
\(283\) 774.651i 0.162715i 0.996685 + 0.0813573i \(0.0259255\pi\)
−0.996685 + 0.0813573i \(0.974075\pi\)
\(284\) −12225.2 −2.55434
\(285\) 0 0
\(286\) −10620.6 −2.19583
\(287\) − 1679.95i − 0.345520i
\(288\) 0 0
\(289\) 3371.06 0.686152
\(290\) 0 0
\(291\) 0 0
\(292\) − 1932.79i − 0.387356i
\(293\) − 6749.23i − 1.34571i −0.739772 0.672857i \(-0.765067\pi\)
0.739772 0.672857i \(-0.234933\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −16396.3 −3.21964
\(297\) 0 0
\(298\) − 5565.12i − 1.08181i
\(299\) 815.696 0.157769
\(300\) 0 0
\(301\) 5525.80 1.05815
\(302\) 15406.7i 2.93561i
\(303\) 0 0
\(304\) 34701.2 6.54687
\(305\) 0 0
\(306\) 0 0
\(307\) − 2204.39i − 0.409808i −0.978782 0.204904i \(-0.934312\pi\)
0.978782 0.204904i \(-0.0656883\pi\)
\(308\) − 14473.2i − 2.67756i
\(309\) 0 0
\(310\) 0 0
\(311\) −6032.98 −1.10000 −0.549998 0.835166i \(-0.685372\pi\)
−0.549998 + 0.835166i \(0.685372\pi\)
\(312\) 0 0
\(313\) 5772.96i 1.04251i 0.853400 + 0.521257i \(0.174537\pi\)
−0.853400 + 0.521257i \(0.825463\pi\)
\(314\) −2601.09 −0.467479
\(315\) 0 0
\(316\) 9816.57 1.74755
\(317\) − 3302.07i − 0.585057i −0.956257 0.292528i \(-0.905503\pi\)
0.956257 0.292528i \(-0.0944966\pi\)
\(318\) 0 0
\(319\) 9054.41 1.58918
\(320\) 0 0
\(321\) 0 0
\(322\) 1519.56i 0.262986i
\(323\) 5755.07i 0.991395i
\(324\) 0 0
\(325\) 0 0
\(326\) −12239.9 −2.07947
\(327\) 0 0
\(328\) − 10717.3i − 1.80416i
\(329\) 4655.53 0.780144
\(330\) 0 0
\(331\) 8053.15 1.33728 0.668642 0.743584i \(-0.266876\pi\)
0.668642 + 0.743584i \(0.266876\pi\)
\(332\) 6197.08i 1.02442i
\(333\) 0 0
\(334\) −518.795 −0.0849916
\(335\) 0 0
\(336\) 0 0
\(337\) 3460.33i 0.559335i 0.960097 + 0.279668i \(0.0902242\pi\)
−0.960097 + 0.279668i \(0.909776\pi\)
\(338\) 5459.37i 0.878552i
\(339\) 0 0
\(340\) 0 0
\(341\) 1661.62 0.263877
\(342\) 0 0
\(343\) − 6453.51i − 1.01591i
\(344\) 35252.0 5.52518
\(345\) 0 0
\(346\) −11647.7 −1.80978
\(347\) − 9328.27i − 1.44314i −0.692344 0.721568i \(-0.743422\pi\)
0.692344 0.721568i \(-0.256578\pi\)
\(348\) 0 0
\(349\) −8899.42 −1.36497 −0.682486 0.730899i \(-0.739101\pi\)
−0.682486 + 0.730899i \(0.739101\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 38798.8i − 5.87495i
\(353\) − 3722.45i − 0.561264i −0.959816 0.280632i \(-0.909456\pi\)
0.959816 0.280632i \(-0.0905440\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 13630.4 2.02924
\(357\) 0 0
\(358\) 9305.42i 1.37376i
\(359\) −11029.3 −1.62145 −0.810727 0.585425i \(-0.800928\pi\)
−0.810727 + 0.585425i \(0.800928\pi\)
\(360\) 0 0
\(361\) 14621.1 2.13166
\(362\) − 7429.29i − 1.07866i
\(363\) 0 0
\(364\) 8903.60 1.28208
\(365\) 0 0
\(366\) 0 0
\(367\) − 4853.11i − 0.690274i −0.938552 0.345137i \(-0.887833\pi\)
0.938552 0.345137i \(-0.112167\pi\)
\(368\) 5582.49i 0.790781i
\(369\) 0 0
\(370\) 0 0
\(371\) 1591.28 0.222683
\(372\) 0 0
\(373\) 12373.8i 1.71767i 0.512254 + 0.858834i \(0.328811\pi\)
−0.512254 + 0.858834i \(0.671189\pi\)
\(374\) 12054.6 1.66666
\(375\) 0 0
\(376\) 29700.1 4.07358
\(377\) 5570.06i 0.760935i
\(378\) 0 0
\(379\) −11150.6 −1.51127 −0.755634 0.654994i \(-0.772671\pi\)
−0.755634 + 0.654994i \(0.772671\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 5987.39i − 0.801941i
\(383\) − 2199.59i − 0.293457i −0.989177 0.146728i \(-0.953126\pi\)
0.989177 0.146728i \(-0.0468743\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −15651.7 −2.06386
\(387\) 0 0
\(388\) 4212.72i 0.551207i
\(389\) −9218.92 −1.20159 −0.600794 0.799404i \(-0.705149\pi\)
−0.600794 + 0.799404i \(0.705149\pi\)
\(390\) 0 0
\(391\) −925.838 −0.119748
\(392\) − 15335.6i − 1.97594i
\(393\) 0 0
\(394\) −3952.90 −0.505442
\(395\) 0 0
\(396\) 0 0
\(397\) 1119.36i 0.141509i 0.997494 + 0.0707544i \(0.0225407\pi\)
−0.997494 + 0.0707544i \(0.977459\pi\)
\(398\) 9245.23i 1.16438i
\(399\) 0 0
\(400\) 0 0
\(401\) 12296.9 1.53137 0.765683 0.643218i \(-0.222401\pi\)
0.765683 + 0.643218i \(0.222401\pi\)
\(402\) 0 0
\(403\) 1022.19i 0.126350i
\(404\) −29969.2 −3.69065
\(405\) 0 0
\(406\) −10376.4 −1.26841
\(407\) 12242.2i 1.49097i
\(408\) 0 0
\(409\) −2500.22 −0.302269 −0.151134 0.988513i \(-0.548293\pi\)
−0.151134 + 0.988513i \(0.548293\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 44241.3i 5.29032i
\(413\) − 1548.96i − 0.184551i
\(414\) 0 0
\(415\) 0 0
\(416\) 23868.1 2.81305
\(417\) 0 0
\(418\) − 44992.3i − 5.26470i
\(419\) 8332.97 0.971581 0.485790 0.874075i \(-0.338532\pi\)
0.485790 + 0.874075i \(0.338532\pi\)
\(420\) 0 0
\(421\) 11374.2 1.31673 0.658367 0.752697i \(-0.271248\pi\)
0.658367 + 0.752697i \(0.271248\pi\)
\(422\) 5171.91i 0.596598i
\(423\) 0 0
\(424\) 10151.6 1.16275
\(425\) 0 0
\(426\) 0 0
\(427\) − 3067.35i − 0.347633i
\(428\) 17941.1i 2.02621i
\(429\) 0 0
\(430\) 0 0
\(431\) −10030.2 −1.12097 −0.560484 0.828165i \(-0.689385\pi\)
−0.560484 + 0.828165i \(0.689385\pi\)
\(432\) 0 0
\(433\) − 7609.38i − 0.844535i −0.906471 0.422267i \(-0.861234\pi\)
0.906471 0.422267i \(-0.138766\pi\)
\(434\) −1904.24 −0.210614
\(435\) 0 0
\(436\) −18089.5 −1.98699
\(437\) 3455.57i 0.378266i
\(438\) 0 0
\(439\) 12370.5 1.34490 0.672450 0.740143i \(-0.265242\pi\)
0.672450 + 0.740143i \(0.265242\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7415.72i 0.798032i
\(443\) 12084.4i 1.29605i 0.761621 + 0.648023i \(0.224404\pi\)
−0.761621 + 0.648023i \(0.775596\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 610.647 0.0648318
\(447\) 0 0
\(448\) 22100.4i 2.33068i
\(449\) 625.550 0.0657495 0.0328747 0.999459i \(-0.489534\pi\)
0.0328747 + 0.999459i \(0.489534\pi\)
\(450\) 0 0
\(451\) −8002.04 −0.835480
\(452\) − 32770.4i − 3.41016i
\(453\) 0 0
\(454\) −6554.33 −0.677555
\(455\) 0 0
\(456\) 0 0
\(457\) 1811.22i 0.185395i 0.995694 + 0.0926974i \(0.0295489\pi\)
−0.995694 + 0.0926974i \(0.970451\pi\)
\(458\) − 4489.17i − 0.458003i
\(459\) 0 0
\(460\) 0 0
\(461\) −11625.0 −1.17447 −0.587233 0.809418i \(-0.699783\pi\)
−0.587233 + 0.809418i \(0.699783\pi\)
\(462\) 0 0
\(463\) − 7291.88i − 0.731928i −0.930629 0.365964i \(-0.880739\pi\)
0.930629 0.365964i \(-0.119261\pi\)
\(464\) −38120.6 −3.81402
\(465\) 0 0
\(466\) −29090.9 −2.89186
\(467\) − 11637.6i − 1.15316i −0.817042 0.576579i \(-0.804387\pi\)
0.817042 0.576579i \(-0.195613\pi\)
\(468\) 0 0
\(469\) 5256.78 0.517560
\(470\) 0 0
\(471\) 0 0
\(472\) − 9881.66i − 0.963644i
\(473\) − 26320.9i − 2.55864i
\(474\) 0 0
\(475\) 0 0
\(476\) −10105.8 −0.973109
\(477\) 0 0
\(478\) 38678.6i 3.70109i
\(479\) −12041.4 −1.14862 −0.574309 0.818639i \(-0.694729\pi\)
−0.574309 + 0.818639i \(0.694729\pi\)
\(480\) 0 0
\(481\) −7531.14 −0.713909
\(482\) − 35810.0i − 3.38402i
\(483\) 0 0
\(484\) −39926.7 −3.74969
\(485\) 0 0
\(486\) 0 0
\(487\) 7037.81i 0.654853i 0.944877 + 0.327427i \(0.106181\pi\)
−0.944877 + 0.327427i \(0.893819\pi\)
\(488\) − 19568.2i − 1.81519i
\(489\) 0 0
\(490\) 0 0
\(491\) 6603.07 0.606909 0.303454 0.952846i \(-0.401860\pi\)
0.303454 + 0.952846i \(0.401860\pi\)
\(492\) 0 0
\(493\) − 6322.17i − 0.577558i
\(494\) 27678.2 2.52085
\(495\) 0 0
\(496\) −6995.72 −0.633301
\(497\) − 6621.58i − 0.597623i
\(498\) 0 0
\(499\) 36.7047 0.00329284 0.00164642 0.999999i \(-0.499476\pi\)
0.00164642 + 0.999999i \(0.499476\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3898.76i 0.346634i
\(503\) − 21242.5i − 1.88302i −0.336990 0.941508i \(-0.609409\pi\)
0.336990 0.941508i \(-0.390591\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 7238.06 0.635911
\(507\) 0 0
\(508\) 12563.2i 1.09725i
\(509\) 11280.4 0.982309 0.491155 0.871072i \(-0.336575\pi\)
0.491155 + 0.871072i \(0.336575\pi\)
\(510\) 0 0
\(511\) 1046.86 0.0906270
\(512\) 20681.3i 1.78514i
\(513\) 0 0
\(514\) −23999.9 −2.05951
\(515\) 0 0
\(516\) 0 0
\(517\) − 22175.5i − 1.88642i
\(518\) − 14029.7i − 1.19002i
\(519\) 0 0
\(520\) 0 0
\(521\) −10239.3 −0.861023 −0.430511 0.902585i \(-0.641667\pi\)
−0.430511 + 0.902585i \(0.641667\pi\)
\(522\) 0 0
\(523\) 2822.00i 0.235942i 0.993017 + 0.117971i \(0.0376389\pi\)
−0.993017 + 0.117971i \(0.962361\pi\)
\(524\) −52098.1 −4.34335
\(525\) 0 0
\(526\) −41215.3 −3.41649
\(527\) − 1160.22i − 0.0959010i
\(528\) 0 0
\(529\) 11611.1 0.954310
\(530\) 0 0
\(531\) 0 0
\(532\) 37718.7i 3.07390i
\(533\) − 4922.67i − 0.400046i
\(534\) 0 0
\(535\) 0 0
\(536\) 33535.8 2.70248
\(537\) 0 0
\(538\) 30224.4i 2.42205i
\(539\) −11450.3 −0.915028
\(540\) 0 0
\(541\) −9409.63 −0.747785 −0.373892 0.927472i \(-0.621977\pi\)
−0.373892 + 0.927472i \(0.621977\pi\)
\(542\) 16694.2i 1.32302i
\(543\) 0 0
\(544\) −27091.0 −2.13514
\(545\) 0 0
\(546\) 0 0
\(547\) − 3836.43i − 0.299879i −0.988695 0.149940i \(-0.952092\pi\)
0.988695 0.149940i \(-0.0479079\pi\)
\(548\) − 21851.6i − 1.70339i
\(549\) 0 0
\(550\) 0 0
\(551\) −23596.7 −1.82441
\(552\) 0 0
\(553\) 5316.97i 0.408862i
\(554\) 22218.2 1.70390
\(555\) 0 0
\(556\) 2873.35 0.219167
\(557\) 6145.92i 0.467524i 0.972294 + 0.233762i \(0.0751036\pi\)
−0.972294 + 0.233762i \(0.924896\pi\)
\(558\) 0 0
\(559\) 16192.0 1.22513
\(560\) 0 0
\(561\) 0 0
\(562\) 40646.9i 3.05087i
\(563\) − 13247.9i − 0.991707i −0.868406 0.495854i \(-0.834855\pi\)
0.868406 0.495854i \(-0.165145\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4228.63 −0.314033
\(567\) 0 0
\(568\) − 42242.6i − 3.12053i
\(569\) −6544.89 −0.482208 −0.241104 0.970499i \(-0.577509\pi\)
−0.241104 + 0.970499i \(0.577509\pi\)
\(570\) 0 0
\(571\) 20362.1 1.49234 0.746170 0.665755i \(-0.231890\pi\)
0.746170 + 0.665755i \(0.231890\pi\)
\(572\) − 42410.2i − 3.10011i
\(573\) 0 0
\(574\) 9170.43 0.666840
\(575\) 0 0
\(576\) 0 0
\(577\) − 26247.4i − 1.89375i −0.321600 0.946876i \(-0.604221\pi\)
0.321600 0.946876i \(-0.395779\pi\)
\(578\) 18401.8i 1.32425i
\(579\) 0 0
\(580\) 0 0
\(581\) −3356.54 −0.239677
\(582\) 0 0
\(583\) − 7579.71i − 0.538455i
\(584\) 6678.49 0.473215
\(585\) 0 0
\(586\) 36842.4 2.59718
\(587\) − 14098.2i − 0.991301i −0.868522 0.495650i \(-0.834930\pi\)
0.868522 0.495650i \(-0.165070\pi\)
\(588\) 0 0
\(589\) −4330.36 −0.302936
\(590\) 0 0
\(591\) 0 0
\(592\) − 51541.9i − 3.57831i
\(593\) − 3476.71i − 0.240761i −0.992728 0.120380i \(-0.961589\pi\)
0.992728 0.120380i \(-0.0384115\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22222.8 1.52732
\(597\) 0 0
\(598\) 4452.69i 0.304488i
\(599\) 16179.0 1.10360 0.551798 0.833978i \(-0.313942\pi\)
0.551798 + 0.833978i \(0.313942\pi\)
\(600\) 0 0
\(601\) 8112.16 0.550586 0.275293 0.961360i \(-0.411225\pi\)
0.275293 + 0.961360i \(0.411225\pi\)
\(602\) 30164.0i 2.04218i
\(603\) 0 0
\(604\) −61522.3 −4.14455
\(605\) 0 0
\(606\) 0 0
\(607\) 21139.2i 1.41353i 0.707449 + 0.706765i \(0.249846\pi\)
−0.707449 + 0.706765i \(0.750154\pi\)
\(608\) 101113.i 6.74456i
\(609\) 0 0
\(610\) 0 0
\(611\) 13641.9 0.903258
\(612\) 0 0
\(613\) 11440.3i 0.753785i 0.926257 + 0.376892i \(0.123007\pi\)
−0.926257 + 0.376892i \(0.876993\pi\)
\(614\) 12033.2 0.790915
\(615\) 0 0
\(616\) 50010.3 3.27106
\(617\) 21566.2i 1.40717i 0.710612 + 0.703584i \(0.248418\pi\)
−0.710612 + 0.703584i \(0.751582\pi\)
\(618\) 0 0
\(619\) 15198.3 0.986866 0.493433 0.869784i \(-0.335742\pi\)
0.493433 + 0.869784i \(0.335742\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 32932.6i − 2.12295i
\(623\) 7382.68i 0.474769i
\(624\) 0 0
\(625\) 0 0
\(626\) −31513.2 −2.01201
\(627\) 0 0
\(628\) − 10386.7i − 0.659994i
\(629\) 8548.05 0.541865
\(630\) 0 0
\(631\) −4929.66 −0.311009 −0.155504 0.987835i \(-0.549700\pi\)
−0.155504 + 0.987835i \(0.549700\pi\)
\(632\) 33919.8i 2.13490i
\(633\) 0 0
\(634\) 18025.2 1.12914
\(635\) 0 0
\(636\) 0 0
\(637\) − 7043.97i − 0.438135i
\(638\) 49425.8i 3.06706i
\(639\) 0 0
\(640\) 0 0
\(641\) −7535.73 −0.464342 −0.232171 0.972675i \(-0.574583\pi\)
−0.232171 + 0.972675i \(0.574583\pi\)
\(642\) 0 0
\(643\) 15997.2i 0.981135i 0.871403 + 0.490568i \(0.163211\pi\)
−0.871403 + 0.490568i \(0.836789\pi\)
\(644\) −6067.93 −0.371289
\(645\) 0 0
\(646\) −31415.5 −1.91336
\(647\) 6020.46i 0.365825i 0.983129 + 0.182913i \(0.0585525\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(648\) 0 0
\(649\) −7378.12 −0.446250
\(650\) 0 0
\(651\) 0 0
\(652\) − 48876.8i − 2.93583i
\(653\) − 10948.7i − 0.656133i −0.944655 0.328067i \(-0.893603\pi\)
0.944655 0.328067i \(-0.106397\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 33690.0 2.00514
\(657\) 0 0
\(658\) 25413.4i 1.50565i
\(659\) 12338.3 0.729334 0.364667 0.931138i \(-0.381183\pi\)
0.364667 + 0.931138i \(0.381183\pi\)
\(660\) 0 0
\(661\) 20016.8 1.17786 0.588928 0.808186i \(-0.299550\pi\)
0.588928 + 0.808186i \(0.299550\pi\)
\(662\) 43960.2i 2.58091i
\(663\) 0 0
\(664\) −21413.1 −1.25149
\(665\) 0 0
\(666\) 0 0
\(667\) − 3796.08i − 0.220367i
\(668\) − 2071.66i − 0.119993i
\(669\) 0 0
\(670\) 0 0
\(671\) −14610.6 −0.840589
\(672\) 0 0
\(673\) 8419.22i 0.482225i 0.970497 + 0.241112i \(0.0775122\pi\)
−0.970497 + 0.241112i \(0.922488\pi\)
\(674\) −18889.1 −1.07950
\(675\) 0 0
\(676\) −21800.5 −1.24036
\(677\) − 25707.1i − 1.45938i −0.683776 0.729692i \(-0.739663\pi\)
0.683776 0.729692i \(-0.260337\pi\)
\(678\) 0 0
\(679\) −2281.74 −0.128962
\(680\) 0 0
\(681\) 0 0
\(682\) 9070.41i 0.509272i
\(683\) − 19624.3i − 1.09942i −0.835357 0.549708i \(-0.814739\pi\)
0.835357 0.549708i \(-0.185261\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 35228.2 1.96067
\(687\) 0 0
\(688\) 110815.i 6.14069i
\(689\) 4662.86 0.257824
\(690\) 0 0
\(691\) 7273.23 0.400415 0.200207 0.979754i \(-0.435838\pi\)
0.200207 + 0.979754i \(0.435838\pi\)
\(692\) − 46511.8i − 2.55508i
\(693\) 0 0
\(694\) 50920.8 2.78520
\(695\) 0 0
\(696\) 0 0
\(697\) 5587.37i 0.303639i
\(698\) − 48579.8i − 2.63434i
\(699\) 0 0
\(700\) 0 0
\(701\) −17644.3 −0.950664 −0.475332 0.879807i \(-0.657672\pi\)
−0.475332 + 0.879807i \(0.657672\pi\)
\(702\) 0 0
\(703\) − 31904.5i − 1.71166i
\(704\) 105270. 5.63568
\(705\) 0 0
\(706\) 20320.0 1.08322
\(707\) − 16232.3i − 0.863476i
\(708\) 0 0
\(709\) −24304.4 −1.28741 −0.643703 0.765276i \(-0.722603\pi\)
−0.643703 + 0.765276i \(0.722603\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 47098.1i 2.47904i
\(713\) − 696.639i − 0.0365909i
\(714\) 0 0
\(715\) 0 0
\(716\) −37158.6 −1.93950
\(717\) 0 0
\(718\) − 60206.0i − 3.12934i
\(719\) −15170.2 −0.786863 −0.393431 0.919354i \(-0.628712\pi\)
−0.393431 + 0.919354i \(0.628712\pi\)
\(720\) 0 0
\(721\) −23962.5 −1.23774
\(722\) 79812.8i 4.11402i
\(723\) 0 0
\(724\) 29666.8 1.52287
\(725\) 0 0
\(726\) 0 0
\(727\) − 17487.0i − 0.892102i −0.895008 0.446051i \(-0.852830\pi\)
0.895008 0.446051i \(-0.147170\pi\)
\(728\) 30765.2i 1.56625i
\(729\) 0 0
\(730\) 0 0
\(731\) −18378.4 −0.929888
\(732\) 0 0
\(733\) − 18698.0i − 0.942194i −0.882082 0.471097i \(-0.843858\pi\)
0.882082 0.471097i \(-0.156142\pi\)
\(734\) 26492.0 1.33220
\(735\) 0 0
\(736\) −16266.5 −0.814660
\(737\) − 25039.5i − 1.25148i
\(738\) 0 0
\(739\) 35250.3 1.75467 0.877336 0.479876i \(-0.159318\pi\)
0.877336 + 0.479876i \(0.159318\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8686.43i 0.429769i
\(743\) − 11133.6i − 0.549731i −0.961483 0.274866i \(-0.911367\pi\)
0.961483 0.274866i \(-0.0886334\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −67545.5 −3.31503
\(747\) 0 0
\(748\) 48136.8i 2.35301i
\(749\) −9717.49 −0.474058
\(750\) 0 0
\(751\) 17197.6 0.835619 0.417809 0.908535i \(-0.362798\pi\)
0.417809 + 0.908535i \(0.362798\pi\)
\(752\) 93362.7i 4.52738i
\(753\) 0 0
\(754\) −30405.6 −1.46858
\(755\) 0 0
\(756\) 0 0
\(757\) 804.647i 0.0386333i 0.999813 + 0.0193166i \(0.00614906\pi\)
−0.999813 + 0.0193166i \(0.993851\pi\)
\(758\) − 60868.7i − 2.91669i
\(759\) 0 0
\(760\) 0 0
\(761\) 26208.9 1.24845 0.624225 0.781245i \(-0.285415\pi\)
0.624225 + 0.781245i \(0.285415\pi\)
\(762\) 0 0
\(763\) − 9797.85i − 0.464883i
\(764\) 23909.0 1.13219
\(765\) 0 0
\(766\) 12007.0 0.566360
\(767\) − 4538.85i − 0.213674i
\(768\) 0 0
\(769\) −36544.2 −1.71367 −0.856837 0.515587i \(-0.827574\pi\)
−0.856837 + 0.515587i \(0.827574\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 62500.8i − 2.91380i
\(773\) 42387.4i 1.97228i 0.165923 + 0.986139i \(0.446940\pi\)
−0.165923 + 0.986139i \(0.553060\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −14556.5 −0.673385
\(777\) 0 0
\(778\) − 50323.8i − 2.31902i
\(779\) 20854.1 0.959148
\(780\) 0 0
\(781\) −31540.4 −1.44507
\(782\) − 5053.92i − 0.231110i
\(783\) 0 0
\(784\) 48207.8 2.19606
\(785\) 0 0
\(786\) 0 0
\(787\) 26849.4i 1.21611i 0.793896 + 0.608054i \(0.208050\pi\)
−0.793896 + 0.608054i \(0.791950\pi\)
\(788\) − 15784.8i − 0.713592i
\(789\) 0 0
\(790\) 0 0
\(791\) 17749.5 0.797851
\(792\) 0 0
\(793\) − 8988.09i − 0.402492i
\(794\) −6110.31 −0.273107
\(795\) 0 0
\(796\) −36918.3 −1.64389
\(797\) − 31307.0i − 1.39141i −0.718328 0.695704i \(-0.755092\pi\)
0.718328 0.695704i \(-0.244908\pi\)
\(798\) 0 0
\(799\) −15483.9 −0.685583
\(800\) 0 0
\(801\) 0 0
\(802\) 67125.7i 2.95548i
\(803\) − 4986.48i − 0.219140i
\(804\) 0 0
\(805\) 0 0
\(806\) −5579.90 −0.243851
\(807\) 0 0
\(808\) − 103554.i − 4.50870i
\(809\) 10011.9 0.435106 0.217553 0.976049i \(-0.430193\pi\)
0.217553 + 0.976049i \(0.430193\pi\)
\(810\) 0 0
\(811\) 4603.68 0.199331 0.0996653 0.995021i \(-0.468223\pi\)
0.0996653 + 0.995021i \(0.468223\pi\)
\(812\) − 41435.5i − 1.79076i
\(813\) 0 0
\(814\) −66827.4 −2.87752
\(815\) 0 0
\(816\) 0 0
\(817\) 68594.8i 2.93737i
\(818\) − 13648.1i − 0.583367i
\(819\) 0 0
\(820\) 0 0
\(821\) −35429.0 −1.50607 −0.753033 0.657983i \(-0.771410\pi\)
−0.753033 + 0.657983i \(0.771410\pi\)
\(822\) 0 0
\(823\) 28297.6i 1.19853i 0.800550 + 0.599265i \(0.204541\pi\)
−0.800550 + 0.599265i \(0.795459\pi\)
\(824\) −152870. −6.46295
\(825\) 0 0
\(826\) 8455.40 0.356176
\(827\) 41059.9i 1.72647i 0.504800 + 0.863236i \(0.331566\pi\)
−0.504800 + 0.863236i \(0.668434\pi\)
\(828\) 0 0
\(829\) −9030.68 −0.378345 −0.189173 0.981944i \(-0.560581\pi\)
−0.189173 + 0.981944i \(0.560581\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 64759.8i 2.69849i
\(833\) 7995.10i 0.332550i
\(834\) 0 0
\(835\) 0 0
\(836\) 179664. 7.43280
\(837\) 0 0
\(838\) 45487.7i 1.87511i
\(839\) −5227.81 −0.215118 −0.107559 0.994199i \(-0.534303\pi\)
−0.107559 + 0.994199i \(0.534303\pi\)
\(840\) 0 0
\(841\) 1532.87 0.0628508
\(842\) 62089.0i 2.54125i
\(843\) 0 0
\(844\) −20652.6 −0.842288
\(845\) 0 0
\(846\) 0 0
\(847\) − 21625.6i − 0.877290i
\(848\) 31911.8i 1.29228i
\(849\) 0 0
\(850\) 0 0
\(851\) 5132.58 0.206748
\(852\) 0 0
\(853\) 30002.9i 1.20432i 0.798377 + 0.602158i \(0.205692\pi\)
−0.798377 + 0.602158i \(0.794308\pi\)
\(854\) 16743.9 0.670918
\(855\) 0 0
\(856\) −61993.1 −2.47533
\(857\) 15671.7i 0.624662i 0.949973 + 0.312331i \(0.101110\pi\)
−0.949973 + 0.312331i \(0.898890\pi\)
\(858\) 0 0
\(859\) 31306.8 1.24351 0.621755 0.783212i \(-0.286420\pi\)
0.621755 + 0.783212i \(0.286420\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 54752.3i − 2.16342i
\(863\) − 13212.3i − 0.521150i −0.965454 0.260575i \(-0.916088\pi\)
0.965454 0.260575i \(-0.0839121\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 41537.8 1.62992
\(867\) 0 0
\(868\) − 7604.05i − 0.297348i
\(869\) 25326.2 0.988643
\(870\) 0 0
\(871\) 15403.7 0.599236
\(872\) − 62505.7i − 2.42742i
\(873\) 0 0
\(874\) −18863.1 −0.730039
\(875\) 0 0
\(876\) 0 0
\(877\) 15440.6i 0.594519i 0.954797 + 0.297260i \(0.0960727\pi\)
−0.954797 + 0.297260i \(0.903927\pi\)
\(878\) 67527.5i 2.59561i
\(879\) 0 0
\(880\) 0 0
\(881\) −27563.1 −1.05406 −0.527029 0.849847i \(-0.676694\pi\)
−0.527029 + 0.849847i \(0.676694\pi\)
\(882\) 0 0
\(883\) − 19897.7i − 0.758337i −0.925328 0.379169i \(-0.876210\pi\)
0.925328 0.379169i \(-0.123790\pi\)
\(884\) −29612.6 −1.12667
\(885\) 0 0
\(886\) −65965.9 −2.50132
\(887\) 40061.6i 1.51650i 0.651963 + 0.758251i \(0.273946\pi\)
−0.651963 + 0.758251i \(0.726054\pi\)
\(888\) 0 0
\(889\) −6804.66 −0.256716
\(890\) 0 0
\(891\) 0 0
\(892\) 2438.45i 0.0915306i
\(893\) 57791.6i 2.16565i
\(894\) 0 0
\(895\) 0 0
\(896\) −55477.5 −2.06850
\(897\) 0 0
\(898\) 3414.72i 0.126894i
\(899\) 4757.07 0.176482
\(900\) 0 0
\(901\) −5292.47 −0.195691
\(902\) − 43681.2i − 1.61244i
\(903\) 0 0
\(904\) 113234. 4.16603
\(905\) 0 0
\(906\) 0 0
\(907\) 27839.6i 1.01918i 0.860417 + 0.509591i \(0.170203\pi\)
−0.860417 + 0.509591i \(0.829797\pi\)
\(908\) − 26172.9i − 0.956584i
\(909\) 0 0
\(910\) 0 0
\(911\) 22251.0 0.809232 0.404616 0.914487i \(-0.367405\pi\)
0.404616 + 0.914487i \(0.367405\pi\)
\(912\) 0 0
\(913\) 15988.1i 0.579549i
\(914\) −9887.02 −0.357805
\(915\) 0 0
\(916\) 17926.3 0.646616
\(917\) − 28218.0i − 1.01618i
\(918\) 0 0
\(919\) −29480.5 −1.05819 −0.529093 0.848564i \(-0.677468\pi\)
−0.529093 + 0.848564i \(0.677468\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 63457.9i − 2.26668i
\(923\) − 19402.9i − 0.691933i
\(924\) 0 0
\(925\) 0 0
\(926\) 39804.6 1.41259
\(927\) 0 0
\(928\) − 111077.i − 3.92919i
\(929\) −42351.8 −1.49571 −0.747857 0.663860i \(-0.768917\pi\)
−0.747857 + 0.663860i \(0.768917\pi\)
\(930\) 0 0
\(931\) 29840.7 1.05047
\(932\) − 116166.i − 4.08278i
\(933\) 0 0
\(934\) 63526.9 2.22555
\(935\) 0 0
\(936\) 0 0
\(937\) − 35930.6i − 1.25272i −0.779532 0.626362i \(-0.784543\pi\)
0.779532 0.626362i \(-0.215457\pi\)
\(938\) 28695.5i 0.998872i
\(939\) 0 0
\(940\) 0 0
\(941\) −21564.0 −0.747041 −0.373521 0.927622i \(-0.621849\pi\)
−0.373521 + 0.927622i \(0.621849\pi\)
\(942\) 0 0
\(943\) 3354.87i 0.115853i
\(944\) 31063.1 1.07099
\(945\) 0 0
\(946\) 143679. 4.93807
\(947\) − 16632.4i − 0.570729i −0.958419 0.285364i \(-0.907885\pi\)
0.958419 0.285364i \(-0.0921146\pi\)
\(948\) 0 0
\(949\) 3067.57 0.104929
\(950\) 0 0
\(951\) 0 0
\(952\) − 34919.3i − 1.18880i
\(953\) − 39557.8i − 1.34460i −0.740279 0.672299i \(-0.765307\pi\)
0.740279 0.672299i \(-0.234693\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −154452. −5.22526
\(957\) 0 0
\(958\) − 65731.4i − 2.21679i
\(959\) 11835.5 0.398529
\(960\) 0 0
\(961\) −28918.0 −0.970696
\(962\) − 41110.6i − 1.37782i
\(963\) 0 0
\(964\) 142997. 4.77762
\(965\) 0 0
\(966\) 0 0
\(967\) − 10666.2i − 0.354707i −0.984147 0.177354i \(-0.943246\pi\)
0.984147 0.177354i \(-0.0567536\pi\)
\(968\) − 137961.i − 4.58083i
\(969\) 0 0
\(970\) 0 0
\(971\) 31821.8 1.05171 0.525855 0.850574i \(-0.323746\pi\)
0.525855 + 0.850574i \(0.323746\pi\)
\(972\) 0 0
\(973\) 1556.30i 0.0512771i
\(974\) −38417.7 −1.26384
\(975\) 0 0
\(976\) 61513.0 2.01740
\(977\) 11126.5i 0.364347i 0.983266 + 0.182173i \(0.0583132\pi\)
−0.983266 + 0.182173i \(0.941687\pi\)
\(978\) 0 0
\(979\) 35165.7 1.14801
\(980\) 0 0
\(981\) 0 0
\(982\) 36044.5i 1.17131i
\(983\) 991.225i 0.0321619i 0.999871 + 0.0160810i \(0.00511895\pi\)
−0.999871 + 0.0160810i \(0.994881\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 34511.2 1.11467
\(987\) 0 0
\(988\) 110525.i 3.55898i
\(989\) −11035.1 −0.354798
\(990\) 0 0
\(991\) −48714.9 −1.56153 −0.780767 0.624822i \(-0.785171\pi\)
−0.780767 + 0.624822i \(0.785171\pi\)
\(992\) − 20384.4i − 0.652424i
\(993\) 0 0
\(994\) 36145.6 1.15339
\(995\) 0 0
\(996\) 0 0
\(997\) − 42207.2i − 1.34074i −0.742028 0.670369i \(-0.766136\pi\)
0.742028 0.670369i \(-0.233864\pi\)
\(998\) 200.362i 0.00635505i
\(999\) 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 675.4.b.n.649.6 6
3.2 odd 2 675.4.b.m.649.1 6
5.2 odd 4 675.4.a.p.1.1 3
5.3 odd 4 135.4.a.h.1.3 yes 3
5.4 even 2 inner 675.4.b.n.649.1 6
15.2 even 4 675.4.a.s.1.3 3
15.8 even 4 135.4.a.e.1.1 3
15.14 odd 2 675.4.b.m.649.6 6
20.3 even 4 2160.4.a.bq.1.2 3
45.13 odd 12 405.4.e.q.136.1 6
45.23 even 12 405.4.e.v.136.3 6
45.38 even 12 405.4.e.v.271.3 6
45.43 odd 12 405.4.e.q.271.1 6
60.23 odd 4 2160.4.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.e.1.1 3 15.8 even 4
135.4.a.h.1.3 yes 3 5.3 odd 4
405.4.e.q.136.1 6 45.13 odd 12
405.4.e.q.271.1 6 45.43 odd 12
405.4.e.v.136.3 6 45.23 even 12
405.4.e.v.271.3 6 45.38 even 12
675.4.a.p.1.1 3 5.2 odd 4
675.4.a.s.1.3 3 15.2 even 4
675.4.b.m.649.1 6 3.2 odd 2
675.4.b.m.649.6 6 15.14 odd 2
675.4.b.n.649.1 6 5.4 even 2 inner
675.4.b.n.649.6 6 1.1 even 1 trivial
2160.4.a.bi.1.2 3 60.23 odd 4
2160.4.a.bq.1.2 3 20.3 even 4