Properties

Label 275.4.b.c.199.4
Level $275$
Weight $4$
Character 275.199
Analytic conductor $16.226$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,4,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(16.2255252516\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.4.b.c.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.73205i q^{2} +7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} +3.07180i q^{7} +23.3205i q^{8} -35.8564 q^{9} +O(q^{10})\) \(q+2.73205i q^{2} +7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} +3.07180i q^{7} +23.3205i q^{8} -35.8564 q^{9} -11.0000 q^{11} +4.24871i q^{12} -5.35898i q^{13} -8.39230 q^{14} -59.4256 q^{16} -41.2154i q^{17} -97.9615i q^{18} -139.923 q^{19} -24.3538 q^{21} -30.0526i q^{22} +111.354i q^{23} -184.890 q^{24} +14.6410 q^{26} -70.2154i q^{27} +1.64617i q^{28} +24.9948 q^{29} +31.4974 q^{31} +24.2102i q^{32} -87.2102i q^{33} +112.603 q^{34} -19.2154 q^{36} +13.1436i q^{37} -382.277i q^{38} +42.4871 q^{39} +261.072 q^{41} -66.5359i q^{42} +57.7128i q^{43} -5.89488 q^{44} -304.224 q^{46} -343.846i q^{47} -471.138i q^{48} +333.564 q^{49} +326.764 q^{51} -2.87187i q^{52} +342.995i q^{53} +191.832 q^{54} -71.6359 q^{56} -1109.34i q^{57} +68.2872i q^{58} -88.3693 q^{59} +738.697 q^{61} +86.0526i q^{62} -110.144i q^{63} -541.549 q^{64} +238.263 q^{66} +342.359i q^{67} -22.0873i q^{68} -882.836 q^{69} -207.364 q^{71} -836.190i q^{72} +1010.60i q^{73} -35.9090 q^{74} -74.9845 q^{76} -33.7898i q^{77} +116.077i q^{78} -1294.23 q^{79} -411.441 q^{81} +713.261i q^{82} -441.846i q^{83} -13.0512 q^{84} -157.674 q^{86} +198.164i q^{87} -256.526i q^{88} +1489.11 q^{89} +16.4617 q^{91} +59.6743i q^{92} +249.718i q^{93} +939.405 q^{94} -191.944 q^{96} +1346.42i q^{97} +911.314i q^{98} +394.420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.73205i 0.965926i 0.875641 + 0.482963i \(0.160439\pi\)
−0.875641 + 0.482963i \(0.839561\pi\)
\(3\) 7.92820i 1.52578i 0.646526 + 0.762892i \(0.276221\pi\)
−0.646526 + 0.762892i \(0.723779\pi\)
\(4\) 0.535898 0.0669873
\(5\) 0 0
\(6\) −21.6603 −1.47379
\(7\) 3.07180i 0.165861i 0.996555 + 0.0829307i \(0.0264280\pi\)
−0.996555 + 0.0829307i \(0.973572\pi\)
\(8\) 23.3205i 1.03063i
\(9\) −35.8564 −1.32802
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 4.24871i 0.102208i
\(13\) − 5.35898i − 0.114332i −0.998365 0.0571659i \(-0.981794\pi\)
0.998365 0.0571659i \(-0.0182064\pi\)
\(14\) −8.39230 −0.160210
\(15\) 0 0
\(16\) −59.4256 −0.928525
\(17\) − 41.2154i − 0.588012i −0.955804 0.294006i \(-0.905011\pi\)
0.955804 0.294006i \(-0.0949885\pi\)
\(18\) − 97.9615i − 1.28276i
\(19\) −139.923 −1.68950 −0.844751 0.535159i \(-0.820252\pi\)
−0.844751 + 0.535159i \(0.820252\pi\)
\(20\) 0 0
\(21\) −24.3538 −0.253069
\(22\) − 30.0526i − 0.291238i
\(23\) 111.354i 1.00952i 0.863261 + 0.504758i \(0.168418\pi\)
−0.863261 + 0.504758i \(0.831582\pi\)
\(24\) −184.890 −1.57252
\(25\) 0 0
\(26\) 14.6410 0.110436
\(27\) − 70.2154i − 0.500480i
\(28\) 1.64617i 0.0111106i
\(29\) 24.9948 0.160049 0.0800246 0.996793i \(-0.474500\pi\)
0.0800246 + 0.996793i \(0.474500\pi\)
\(30\) 0 0
\(31\) 31.4974 0.182487 0.0912436 0.995829i \(-0.470916\pi\)
0.0912436 + 0.995829i \(0.470916\pi\)
\(32\) 24.2102i 0.133744i
\(33\) − 87.2102i − 0.460041i
\(34\) 112.603 0.567976
\(35\) 0 0
\(36\) −19.2154 −0.0889601
\(37\) 13.1436i 0.0583998i 0.999574 + 0.0291999i \(0.00929594\pi\)
−0.999574 + 0.0291999i \(0.990704\pi\)
\(38\) − 382.277i − 1.63193i
\(39\) 42.4871 0.174446
\(40\) 0 0
\(41\) 261.072 0.994453 0.497226 0.867621i \(-0.334352\pi\)
0.497226 + 0.867621i \(0.334352\pi\)
\(42\) − 66.5359i − 0.244446i
\(43\) 57.7128i 0.204677i 0.994750 + 0.102339i \(0.0326325\pi\)
−0.994750 + 0.102339i \(0.967367\pi\)
\(44\) −5.89488 −0.0201974
\(45\) 0 0
\(46\) −304.224 −0.975118
\(47\) − 343.846i − 1.06713i −0.845759 0.533565i \(-0.820852\pi\)
0.845759 0.533565i \(-0.179148\pi\)
\(48\) − 471.138i − 1.41673i
\(49\) 333.564 0.972490
\(50\) 0 0
\(51\) 326.764 0.897179
\(52\) − 2.87187i − 0.00765879i
\(53\) 342.995i 0.888943i 0.895793 + 0.444471i \(0.146608\pi\)
−0.895793 + 0.444471i \(0.853392\pi\)
\(54\) 191.832 0.483426
\(55\) 0 0
\(56\) −71.6359 −0.170942
\(57\) − 1109.34i − 2.57782i
\(58\) 68.2872i 0.154596i
\(59\) −88.3693 −0.194995 −0.0974975 0.995236i \(-0.531084\pi\)
−0.0974975 + 0.995236i \(0.531084\pi\)
\(60\) 0 0
\(61\) 738.697 1.55050 0.775250 0.631654i \(-0.217624\pi\)
0.775250 + 0.631654i \(0.217624\pi\)
\(62\) 86.0526i 0.176269i
\(63\) − 110.144i − 0.220266i
\(64\) −541.549 −1.05771
\(65\) 0 0
\(66\) 238.263 0.444365
\(67\) 342.359i 0.624266i 0.950038 + 0.312133i \(0.101043\pi\)
−0.950038 + 0.312133i \(0.898957\pi\)
\(68\) − 22.0873i − 0.0393893i
\(69\) −882.836 −1.54030
\(70\) 0 0
\(71\) −207.364 −0.346614 −0.173307 0.984868i \(-0.555445\pi\)
−0.173307 + 0.984868i \(0.555445\pi\)
\(72\) − 836.190i − 1.36869i
\(73\) 1010.60i 1.62030i 0.586224 + 0.810149i \(0.300614\pi\)
−0.586224 + 0.810149i \(0.699386\pi\)
\(74\) −35.9090 −0.0564099
\(75\) 0 0
\(76\) −74.9845 −0.113175
\(77\) − 33.7898i − 0.0500091i
\(78\) 116.077i 0.168502i
\(79\) −1294.23 −1.84319 −0.921593 0.388157i \(-0.873112\pi\)
−0.921593 + 0.388157i \(0.873112\pi\)
\(80\) 0 0
\(81\) −411.441 −0.564391
\(82\) 713.261i 0.960568i
\(83\) − 441.846i − 0.584324i −0.956369 0.292162i \(-0.905625\pi\)
0.956369 0.292162i \(-0.0943747\pi\)
\(84\) −13.0512 −0.0169524
\(85\) 0 0
\(86\) −157.674 −0.197703
\(87\) 198.164i 0.244200i
\(88\) − 256.526i − 0.310747i
\(89\) 1489.11 1.77355 0.886773 0.462205i \(-0.152942\pi\)
0.886773 + 0.462205i \(0.152942\pi\)
\(90\) 0 0
\(91\) 16.4617 0.0189633
\(92\) 59.6743i 0.0676248i
\(93\) 249.718i 0.278436i
\(94\) 939.405 1.03077
\(95\) 0 0
\(96\) −191.944 −0.204064
\(97\) 1346.42i 1.40936i 0.709526 + 0.704679i \(0.248909\pi\)
−0.709526 + 0.704679i \(0.751091\pi\)
\(98\) 911.314i 0.939353i
\(99\) 394.420 0.400412
\(100\) 0 0
\(101\) −161.461 −0.159069 −0.0795347 0.996832i \(-0.525343\pi\)
−0.0795347 + 0.996832i \(0.525343\pi\)
\(102\) 892.736i 0.866608i
\(103\) 34.7592i 0.0332517i 0.999862 + 0.0166259i \(0.00529242\pi\)
−0.999862 + 0.0166259i \(0.994708\pi\)
\(104\) 124.974 0.117834
\(105\) 0 0
\(106\) −937.079 −0.858653
\(107\) 832.179i 0.751867i 0.926647 + 0.375934i \(0.122678\pi\)
−0.926647 + 0.375934i \(0.877322\pi\)
\(108\) − 37.6283i − 0.0335258i
\(109\) −1044.26 −0.917629 −0.458815 0.888532i \(-0.651726\pi\)
−0.458815 + 0.888532i \(0.651726\pi\)
\(110\) 0 0
\(111\) −104.205 −0.0891055
\(112\) − 182.543i − 0.154007i
\(113\) − 295.082i − 0.245654i −0.992428 0.122827i \(-0.960804\pi\)
0.992428 0.122827i \(-0.0391961\pi\)
\(114\) 3030.77 2.48998
\(115\) 0 0
\(116\) 13.3947 0.0107213
\(117\) 192.154i 0.151834i
\(118\) − 241.429i − 0.188351i
\(119\) 126.605 0.0975285
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 2018.16i 1.49767i
\(123\) 2069.83i 1.51732i
\(124\) 16.8794 0.0122243
\(125\) 0 0
\(126\) 300.918 0.212761
\(127\) − 1317.60i − 0.920618i −0.887759 0.460309i \(-0.847739\pi\)
0.887759 0.460309i \(-0.152261\pi\)
\(128\) − 1285.86i − 0.887928i
\(129\) −457.559 −0.312293
\(130\) 0 0
\(131\) −1600.71 −1.06759 −0.533797 0.845612i \(-0.679235\pi\)
−0.533797 + 0.845612i \(0.679235\pi\)
\(132\) − 46.7358i − 0.0308169i
\(133\) − 429.815i − 0.280223i
\(134\) −935.342 −0.602994
\(135\) 0 0
\(136\) 961.164 0.606023
\(137\) 1611.68i 1.00507i 0.864556 + 0.502536i \(0.167600\pi\)
−0.864556 + 0.502536i \(0.832400\pi\)
\(138\) − 2411.95i − 1.48782i
\(139\) 31.8619 0.0194424 0.00972120 0.999953i \(-0.496906\pi\)
0.00972120 + 0.999953i \(0.496906\pi\)
\(140\) 0 0
\(141\) 2726.08 1.62821
\(142\) − 566.529i − 0.334803i
\(143\) 58.9488i 0.0344724i
\(144\) 2130.79 1.23310
\(145\) 0 0
\(146\) −2761.01 −1.56509
\(147\) 2644.56i 1.48381i
\(148\) 7.04363i 0.00391205i
\(149\) 2428.34 1.33515 0.667576 0.744542i \(-0.267332\pi\)
0.667576 + 0.744542i \(0.267332\pi\)
\(150\) 0 0
\(151\) −2576.68 −1.38866 −0.694328 0.719659i \(-0.744298\pi\)
−0.694328 + 0.719659i \(0.744298\pi\)
\(152\) − 3263.08i − 1.74125i
\(153\) 1477.84i 0.780889i
\(154\) 92.3154 0.0483051
\(155\) 0 0
\(156\) 22.7688 0.0116856
\(157\) 2475.94i 1.25861i 0.777158 + 0.629305i \(0.216660\pi\)
−0.777158 + 0.629305i \(0.783340\pi\)
\(158\) − 3535.89i − 1.78038i
\(159\) −2719.33 −1.35633
\(160\) 0 0
\(161\) −342.056 −0.167440
\(162\) − 1124.08i − 0.545160i
\(163\) 2725.11i 1.30949i 0.755850 + 0.654745i \(0.227224\pi\)
−0.755850 + 0.654745i \(0.772776\pi\)
\(164\) 139.908 0.0666157
\(165\) 0 0
\(166\) 1207.15 0.564414
\(167\) 2737.30i 1.26837i 0.773180 + 0.634187i \(0.218665\pi\)
−0.773180 + 0.634187i \(0.781335\pi\)
\(168\) − 567.944i − 0.260820i
\(169\) 2168.28 0.986928
\(170\) 0 0
\(171\) 5017.14 2.24368
\(172\) 30.9282i 0.0137108i
\(173\) − 2307.42i − 1.01404i −0.861933 0.507022i \(-0.830746\pi\)
0.861933 0.507022i \(-0.169254\pi\)
\(174\) −541.395 −0.235879
\(175\) 0 0
\(176\) 653.682 0.279961
\(177\) − 700.610i − 0.297520i
\(178\) 4068.33i 1.71311i
\(179\) 1312.15 0.547905 0.273953 0.961743i \(-0.411669\pi\)
0.273953 + 0.961743i \(0.411669\pi\)
\(180\) 0 0
\(181\) −803.174 −0.329831 −0.164916 0.986308i \(-0.552735\pi\)
−0.164916 + 0.986308i \(0.552735\pi\)
\(182\) 44.9742i 0.0183171i
\(183\) 5856.54i 2.36573i
\(184\) −2596.83 −1.04044
\(185\) 0 0
\(186\) −682.242 −0.268949
\(187\) 453.369i 0.177292i
\(188\) − 184.267i − 0.0714842i
\(189\) 215.687 0.0830103
\(190\) 0 0
\(191\) 1718.25 0.650932 0.325466 0.945554i \(-0.394479\pi\)
0.325466 + 0.945554i \(0.394479\pi\)
\(192\) − 4293.51i − 1.61384i
\(193\) − 1340.18i − 0.499837i −0.968267 0.249919i \(-0.919596\pi\)
0.968267 0.249919i \(-0.0804038\pi\)
\(194\) −3678.48 −1.36134
\(195\) 0 0
\(196\) 178.756 0.0651445
\(197\) − 3518.33i − 1.27244i −0.771508 0.636220i \(-0.780497\pi\)
0.771508 0.636220i \(-0.219503\pi\)
\(198\) 1077.58i 0.386768i
\(199\) −823.692 −0.293417 −0.146709 0.989180i \(-0.546868\pi\)
−0.146709 + 0.989180i \(0.546868\pi\)
\(200\) 0 0
\(201\) −2714.29 −0.952494
\(202\) − 441.121i − 0.153649i
\(203\) 76.7791i 0.0265460i
\(204\) 175.112 0.0600996
\(205\) 0 0
\(206\) −94.9639 −0.0321187
\(207\) − 3992.75i − 1.34065i
\(208\) 318.461i 0.106160i
\(209\) 1539.15 0.509404
\(210\) 0 0
\(211\) −107.343 −0.0350228 −0.0175114 0.999847i \(-0.505574\pi\)
−0.0175114 + 0.999847i \(0.505574\pi\)
\(212\) 183.810i 0.0595479i
\(213\) − 1644.03i − 0.528858i
\(214\) −2273.56 −0.726248
\(215\) 0 0
\(216\) 1637.46 0.515810
\(217\) 96.7537i 0.0302676i
\(218\) − 2852.96i − 0.886362i
\(219\) −8012.24 −2.47222
\(220\) 0 0
\(221\) −220.873 −0.0672285
\(222\) − 284.694i − 0.0860693i
\(223\) 3933.68i 1.18125i 0.806946 + 0.590625i \(0.201119\pi\)
−0.806946 + 0.590625i \(0.798881\pi\)
\(224\) −74.3689 −0.0221830
\(225\) 0 0
\(226\) 806.178 0.237284
\(227\) − 1771.90i − 0.518085i −0.965866 0.259042i \(-0.916593\pi\)
0.965866 0.259042i \(-0.0834069\pi\)
\(228\) − 594.493i − 0.172681i
\(229\) −1915.37 −0.552713 −0.276356 0.961055i \(-0.589127\pi\)
−0.276356 + 0.961055i \(0.589127\pi\)
\(230\) 0 0
\(231\) 267.892 0.0763031
\(232\) 582.892i 0.164952i
\(233\) − 4396.32i − 1.23610i −0.786137 0.618052i \(-0.787922\pi\)
0.786137 0.618052i \(-0.212078\pi\)
\(234\) −524.974 −0.146661
\(235\) 0 0
\(236\) −47.3570 −0.0130622
\(237\) − 10260.9i − 2.81230i
\(238\) 345.892i 0.0942053i
\(239\) 4084.49 1.10546 0.552728 0.833362i \(-0.313587\pi\)
0.552728 + 0.833362i \(0.313587\pi\)
\(240\) 0 0
\(241\) 3908.58 1.04471 0.522353 0.852730i \(-0.325054\pi\)
0.522353 + 0.852730i \(0.325054\pi\)
\(242\) 330.578i 0.0878114i
\(243\) − 5157.80i − 1.36162i
\(244\) 395.867 0.103864
\(245\) 0 0
\(246\) −5654.88 −1.46562
\(247\) 749.845i 0.193164i
\(248\) 734.536i 0.188077i
\(249\) 3503.05 0.891552
\(250\) 0 0
\(251\) 1094.89 0.275335 0.137667 0.990479i \(-0.456040\pi\)
0.137667 + 0.990479i \(0.456040\pi\)
\(252\) − 59.0258i − 0.0147551i
\(253\) − 1224.89i − 0.304381i
\(254\) 3599.76 0.889249
\(255\) 0 0
\(256\) −819.364 −0.200040
\(257\) 783.179i 0.190091i 0.995473 + 0.0950454i \(0.0302996\pi\)
−0.995473 + 0.0950454i \(0.969700\pi\)
\(258\) − 1250.07i − 0.301652i
\(259\) −40.3744 −0.00968628
\(260\) 0 0
\(261\) −896.225 −0.212548
\(262\) − 4373.23i − 1.03122i
\(263\) − 6180.06i − 1.44897i −0.689292 0.724484i \(-0.742078\pi\)
0.689292 0.724484i \(-0.257922\pi\)
\(264\) 2033.79 0.474132
\(265\) 0 0
\(266\) 1174.28 0.270675
\(267\) 11806.0i 2.70605i
\(268\) 183.470i 0.0418179i
\(269\) −986.965 −0.223704 −0.111852 0.993725i \(-0.535678\pi\)
−0.111852 + 0.993725i \(0.535678\pi\)
\(270\) 0 0
\(271\) 4576.99 1.02595 0.512975 0.858404i \(-0.328543\pi\)
0.512975 + 0.858404i \(0.328543\pi\)
\(272\) 2449.25i 0.545984i
\(273\) 130.512i 0.0289338i
\(274\) −4403.18 −0.970825
\(275\) 0 0
\(276\) −473.110 −0.103181
\(277\) 567.836i 0.123169i 0.998102 + 0.0615847i \(0.0196154\pi\)
−0.998102 + 0.0615847i \(0.980385\pi\)
\(278\) 87.0484i 0.0187799i
\(279\) −1129.38 −0.242346
\(280\) 0 0
\(281\) 5311.01 1.12750 0.563752 0.825944i \(-0.309357\pi\)
0.563752 + 0.825944i \(0.309357\pi\)
\(282\) 7447.79i 1.57273i
\(283\) 4728.44i 0.993204i 0.867978 + 0.496602i \(0.165419\pi\)
−0.867978 + 0.496602i \(0.834581\pi\)
\(284\) −111.126 −0.0232187
\(285\) 0 0
\(286\) −161.051 −0.0332977
\(287\) 801.960i 0.164941i
\(288\) − 868.092i − 0.177614i
\(289\) 3214.29 0.654242
\(290\) 0 0
\(291\) −10674.7 −2.15038
\(292\) 541.579i 0.108539i
\(293\) − 2328.92i − 0.464358i −0.972673 0.232179i \(-0.925415\pi\)
0.972673 0.232179i \(-0.0745854\pi\)
\(294\) −7225.08 −1.43325
\(295\) 0 0
\(296\) −306.515 −0.0601886
\(297\) 772.369i 0.150900i
\(298\) 6634.36i 1.28966i
\(299\) 596.743 0.115420
\(300\) 0 0
\(301\) −177.282 −0.0339481
\(302\) − 7039.61i − 1.34134i
\(303\) − 1280.10i − 0.242705i
\(304\) 8315.01 1.56875
\(305\) 0 0
\(306\) −4037.52 −0.754280
\(307\) − 1678.07i − 0.311962i −0.987760 0.155981i \(-0.950146\pi\)
0.987760 0.155981i \(-0.0498539\pi\)
\(308\) − 18.1079i − 0.00334997i
\(309\) −275.578 −0.0507349
\(310\) 0 0
\(311\) 3572.71 0.651413 0.325707 0.945471i \(-0.394398\pi\)
0.325707 + 0.945471i \(0.394398\pi\)
\(312\) 990.821i 0.179789i
\(313\) − 7184.36i − 1.29739i −0.761047 0.648697i \(-0.775314\pi\)
0.761047 0.648697i \(-0.224686\pi\)
\(314\) −6764.40 −1.21572
\(315\) 0 0
\(316\) −693.573 −0.123470
\(317\) − 15.7077i − 0.00278306i −0.999999 0.00139153i \(-0.999557\pi\)
0.999999 0.00139153i \(-0.000442938\pi\)
\(318\) − 7429.36i − 1.31012i
\(319\) −274.943 −0.0482566
\(320\) 0 0
\(321\) −6597.69 −1.14719
\(322\) − 934.515i − 0.161734i
\(323\) 5766.98i 0.993447i
\(324\) −220.491 −0.0378070
\(325\) 0 0
\(326\) −7445.13 −1.26487
\(327\) − 8279.08i − 1.40010i
\(328\) 6088.33i 1.02491i
\(329\) 1056.23 0.176996
\(330\) 0 0
\(331\) −1318.95 −0.219022 −0.109511 0.993986i \(-0.534928\pi\)
−0.109511 + 0.993986i \(0.534928\pi\)
\(332\) − 236.785i − 0.0391423i
\(333\) − 471.282i − 0.0775558i
\(334\) −7478.43 −1.22515
\(335\) 0 0
\(336\) 1447.24 0.234981
\(337\) − 239.183i − 0.0386621i −0.999813 0.0193310i \(-0.993846\pi\)
0.999813 0.0193310i \(-0.00615365\pi\)
\(338\) 5923.85i 0.953299i
\(339\) 2339.47 0.374816
\(340\) 0 0
\(341\) −346.472 −0.0550220
\(342\) 13707.1i 2.16723i
\(343\) 2078.27i 0.327160i
\(344\) −1345.89 −0.210947
\(345\) 0 0
\(346\) 6303.98 0.979491
\(347\) − 5862.79i − 0.907006i −0.891255 0.453503i \(-0.850174\pi\)
0.891255 0.453503i \(-0.149826\pi\)
\(348\) 106.196i 0.0163583i
\(349\) −3491.73 −0.535553 −0.267776 0.963481i \(-0.586289\pi\)
−0.267776 + 0.963481i \(0.586289\pi\)
\(350\) 0 0
\(351\) −376.283 −0.0572208
\(352\) − 266.313i − 0.0403253i
\(353\) 10916.7i 1.64600i 0.568043 + 0.822999i \(0.307701\pi\)
−0.568043 + 0.822999i \(0.692299\pi\)
\(354\) 1914.10 0.287382
\(355\) 0 0
\(356\) 798.013 0.118805
\(357\) 1003.75i 0.148807i
\(358\) 3584.87i 0.529236i
\(359\) 11500.7 1.69077 0.845384 0.534160i \(-0.179372\pi\)
0.845384 + 0.534160i \(0.179372\pi\)
\(360\) 0 0
\(361\) 12719.5 1.85442
\(362\) − 2194.31i − 0.318592i
\(363\) 959.313i 0.138708i
\(364\) 8.82180 0.00127030
\(365\) 0 0
\(366\) −16000.4 −2.28512
\(367\) 6767.01i 0.962493i 0.876585 + 0.481246i \(0.159816\pi\)
−0.876585 + 0.481246i \(0.840184\pi\)
\(368\) − 6617.27i − 0.937362i
\(369\) −9361.10 −1.32065
\(370\) 0 0
\(371\) −1053.61 −0.147441
\(372\) 133.823i 0.0186517i
\(373\) 5310.22i 0.737139i 0.929600 + 0.368569i \(0.120152\pi\)
−0.929600 + 0.368569i \(0.879848\pi\)
\(374\) −1238.63 −0.171251
\(375\) 0 0
\(376\) 8018.67 1.09982
\(377\) − 133.947i − 0.0182987i
\(378\) 589.269i 0.0801818i
\(379\) 838.267 0.113612 0.0568059 0.998385i \(-0.481908\pi\)
0.0568059 + 0.998385i \(0.481908\pi\)
\(380\) 0 0
\(381\) 10446.2 1.40466
\(382\) 4694.34i 0.628752i
\(383\) 2832.16i 0.377851i 0.981991 + 0.188925i \(0.0605004\pi\)
−0.981991 + 0.188925i \(0.939500\pi\)
\(384\) 10194.5 1.35479
\(385\) 0 0
\(386\) 3661.45 0.482806
\(387\) − 2069.37i − 0.271814i
\(388\) 721.542i 0.0944091i
\(389\) −3111.25 −0.405519 −0.202759 0.979229i \(-0.564991\pi\)
−0.202759 + 0.979229i \(0.564991\pi\)
\(390\) 0 0
\(391\) 4589.49 0.593608
\(392\) 7778.88i 1.00228i
\(393\) − 12690.8i − 1.62892i
\(394\) 9612.25 1.22908
\(395\) 0 0
\(396\) 211.369 0.0268225
\(397\) 14208.7i 1.79626i 0.439728 + 0.898131i \(0.355075\pi\)
−0.439728 + 0.898131i \(0.644925\pi\)
\(398\) − 2250.37i − 0.283419i
\(399\) 3407.66 0.427560
\(400\) 0 0
\(401\) −6261.68 −0.779784 −0.389892 0.920861i \(-0.627488\pi\)
−0.389892 + 0.920861i \(0.627488\pi\)
\(402\) − 7415.58i − 0.920039i
\(403\) − 168.794i − 0.0208641i
\(404\) −86.5269 −0.0106556
\(405\) 0 0
\(406\) −209.764 −0.0256415
\(407\) − 144.580i − 0.0176082i
\(408\) 7620.30i 0.924660i
\(409\) 4192.50 0.506860 0.253430 0.967354i \(-0.418441\pi\)
0.253430 + 0.967354i \(0.418441\pi\)
\(410\) 0 0
\(411\) −12777.7 −1.53352
\(412\) 18.6274i 0.00222744i
\(413\) − 271.453i − 0.0323421i
\(414\) 10908.4 1.29497
\(415\) 0 0
\(416\) 129.742 0.0152912
\(417\) 252.608i 0.0296649i
\(418\) 4205.05i 0.492047i
\(419\) 9287.15 1.08283 0.541416 0.840755i \(-0.317888\pi\)
0.541416 + 0.840755i \(0.317888\pi\)
\(420\) 0 0
\(421\) 13146.0 1.52185 0.760923 0.648842i \(-0.224746\pi\)
0.760923 + 0.648842i \(0.224746\pi\)
\(422\) − 293.267i − 0.0338294i
\(423\) 12329.1i 1.41716i
\(424\) −7998.81 −0.916172
\(425\) 0 0
\(426\) 4491.56 0.510838
\(427\) 2269.13i 0.257168i
\(428\) 445.964i 0.0503656i
\(429\) −467.358 −0.0525974
\(430\) 0 0
\(431\) 4909.67 0.548701 0.274351 0.961630i \(-0.411537\pi\)
0.274351 + 0.961630i \(0.411537\pi\)
\(432\) 4172.59i 0.464708i
\(433\) 11743.3i 1.30334i 0.758502 + 0.651671i \(0.225932\pi\)
−0.758502 + 0.651671i \(0.774068\pi\)
\(434\) −264.336 −0.0292363
\(435\) 0 0
\(436\) −559.615 −0.0614695
\(437\) − 15581.0i − 1.70558i
\(438\) − 21889.8i − 2.38798i
\(439\) 11824.2 1.28551 0.642754 0.766073i \(-0.277792\pi\)
0.642754 + 0.766073i \(0.277792\pi\)
\(440\) 0 0
\(441\) −11960.4 −1.29148
\(442\) − 603.435i − 0.0649377i
\(443\) − 10102.1i − 1.08344i −0.840558 0.541722i \(-0.817772\pi\)
0.840558 0.541722i \(-0.182228\pi\)
\(444\) −55.8433 −0.00596894
\(445\) 0 0
\(446\) −10747.0 −1.14100
\(447\) 19252.4i 2.03715i
\(448\) − 1663.53i − 0.175434i
\(449\) 345.254 0.0362885 0.0181443 0.999835i \(-0.494224\pi\)
0.0181443 + 0.999835i \(0.494224\pi\)
\(450\) 0 0
\(451\) −2871.79 −0.299839
\(452\) − 158.134i − 0.0164557i
\(453\) − 20428.4i − 2.11879i
\(454\) 4840.93 0.500431
\(455\) 0 0
\(456\) 25870.3 2.65678
\(457\) − 10567.1i − 1.08164i −0.841138 0.540821i \(-0.818114\pi\)
0.841138 0.540821i \(-0.181886\pi\)
\(458\) − 5232.89i − 0.533879i
\(459\) −2893.95 −0.294288
\(460\) 0 0
\(461\) 4733.96 0.478270 0.239135 0.970986i \(-0.423136\pi\)
0.239135 + 0.970986i \(0.423136\pi\)
\(462\) 731.895i 0.0737031i
\(463\) − 3431.20i − 0.344409i −0.985061 0.172204i \(-0.944911\pi\)
0.985061 0.172204i \(-0.0550890\pi\)
\(464\) −1485.33 −0.148610
\(465\) 0 0
\(466\) 12011.0 1.19399
\(467\) 5116.96i 0.507034i 0.967331 + 0.253517i \(0.0815873\pi\)
−0.967331 + 0.253517i \(0.918413\pi\)
\(468\) 102.975i 0.0101710i
\(469\) −1051.66 −0.103542
\(470\) 0 0
\(471\) −19629.8 −1.92037
\(472\) − 2060.82i − 0.200968i
\(473\) − 634.841i − 0.0617125i
\(474\) 28033.2 2.71648
\(475\) 0 0
\(476\) 67.8476 0.00653317
\(477\) − 12298.6i − 1.18053i
\(478\) 11159.0i 1.06779i
\(479\) −11566.9 −1.10335 −0.551675 0.834059i \(-0.686011\pi\)
−0.551675 + 0.834059i \(0.686011\pi\)
\(480\) 0 0
\(481\) 70.4363 0.00667696
\(482\) 10678.4i 1.00911i
\(483\) − 2711.89i − 0.255477i
\(484\) 64.8437 0.00608975
\(485\) 0 0
\(486\) 14091.4 1.31522
\(487\) − 18326.5i − 1.70525i −0.522527 0.852623i \(-0.675010\pi\)
0.522527 0.852623i \(-0.324990\pi\)
\(488\) 17226.8i 1.59799i
\(489\) −21605.2 −1.99800
\(490\) 0 0
\(491\) −7617.58 −0.700156 −0.350078 0.936721i \(-0.613845\pi\)
−0.350078 + 0.936721i \(0.613845\pi\)
\(492\) 1109.22i 0.101641i
\(493\) − 1030.17i − 0.0941108i
\(494\) −2048.62 −0.186582
\(495\) 0 0
\(496\) −1871.75 −0.169444
\(497\) − 636.980i − 0.0574899i
\(498\) 9570.50i 0.861173i
\(499\) −12909.1 −1.15810 −0.579050 0.815292i \(-0.696576\pi\)
−0.579050 + 0.815292i \(0.696576\pi\)
\(500\) 0 0
\(501\) −21701.8 −1.93526
\(502\) 2991.30i 0.265953i
\(503\) − 10165.7i − 0.901121i −0.892746 0.450561i \(-0.851224\pi\)
0.892746 0.450561i \(-0.148776\pi\)
\(504\) 2568.60 0.227013
\(505\) 0 0
\(506\) 3346.47 0.294009
\(507\) 17190.6i 1.50584i
\(508\) − 706.102i − 0.0616697i
\(509\) −6449.93 −0.561666 −0.280833 0.959757i \(-0.590611\pi\)
−0.280833 + 0.959757i \(0.590611\pi\)
\(510\) 0 0
\(511\) −3104.36 −0.268745
\(512\) − 12525.4i − 1.08115i
\(513\) 9824.75i 0.845562i
\(514\) −2139.68 −0.183614
\(515\) 0 0
\(516\) −245.205 −0.0209197
\(517\) 3782.31i 0.321752i
\(518\) − 110.305i − 0.00935623i
\(519\) 18293.7 1.54721
\(520\) 0 0
\(521\) −19327.4 −1.62524 −0.812620 0.582794i \(-0.801959\pi\)
−0.812620 + 0.582794i \(0.801959\pi\)
\(522\) − 2448.53i − 0.205305i
\(523\) − 6259.09i − 0.523310i −0.965161 0.261655i \(-0.915732\pi\)
0.965161 0.261655i \(-0.0842682\pi\)
\(524\) −857.819 −0.0715153
\(525\) 0 0
\(526\) 16884.2 1.39960
\(527\) − 1298.18i − 0.107305i
\(528\) 5182.52i 0.427160i
\(529\) −232.675 −0.0191235
\(530\) 0 0
\(531\) 3168.61 0.258956
\(532\) − 230.337i − 0.0187714i
\(533\) − 1399.08i − 0.113698i
\(534\) −32254.6 −2.61384
\(535\) 0 0
\(536\) −7983.99 −0.643387
\(537\) 10403.0i 0.835985i
\(538\) − 2696.44i − 0.216081i
\(539\) −3669.20 −0.293217
\(540\) 0 0
\(541\) −14008.2 −1.11323 −0.556616 0.830770i \(-0.687900\pi\)
−0.556616 + 0.830770i \(0.687900\pi\)
\(542\) 12504.6i 0.990991i
\(543\) − 6367.72i − 0.503251i
\(544\) 997.834 0.0786430
\(545\) 0 0
\(546\) −356.565 −0.0279479
\(547\) − 4949.45i − 0.386879i −0.981112 0.193440i \(-0.938036\pi\)
0.981112 0.193440i \(-0.0619644\pi\)
\(548\) 863.695i 0.0673270i
\(549\) −26487.0 −2.05909
\(550\) 0 0
\(551\) −3497.35 −0.270404
\(552\) − 20588.2i − 1.58748i
\(553\) − 3975.60i − 0.305714i
\(554\) −1551.36 −0.118973
\(555\) 0 0
\(556\) 17.0748 0.00130239
\(557\) − 3801.58i − 0.289188i −0.989491 0.144594i \(-0.953812\pi\)
0.989491 0.144594i \(-0.0461877\pi\)
\(558\) − 3085.54i − 0.234088i
\(559\) 309.282 0.0234011
\(560\) 0 0
\(561\) −3594.40 −0.270510
\(562\) 14510.0i 1.08908i
\(563\) 9900.11i 0.741101i 0.928812 + 0.370551i \(0.120831\pi\)
−0.928812 + 0.370551i \(0.879169\pi\)
\(564\) 1460.90 0.109069
\(565\) 0 0
\(566\) −12918.3 −0.959361
\(567\) − 1263.86i − 0.0936107i
\(568\) − 4835.84i − 0.357231i
\(569\) −5329.16 −0.392636 −0.196318 0.980540i \(-0.562898\pi\)
−0.196318 + 0.980540i \(0.562898\pi\)
\(570\) 0 0
\(571\) −16962.6 −1.24319 −0.621597 0.783337i \(-0.713516\pi\)
−0.621597 + 0.783337i \(0.713516\pi\)
\(572\) 31.5906i 0.00230921i
\(573\) 13622.6i 0.993181i
\(574\) −2190.99 −0.159321
\(575\) 0 0
\(576\) 19418.0 1.40466
\(577\) − 15487.0i − 1.11738i −0.829375 0.558692i \(-0.811303\pi\)
0.829375 0.558692i \(-0.188697\pi\)
\(578\) 8781.61i 0.631949i
\(579\) 10625.3 0.762643
\(580\) 0 0
\(581\) 1357.26 0.0969169
\(582\) − 29163.7i − 2.07710i
\(583\) − 3772.94i − 0.268026i
\(584\) −23567.7 −1.66993
\(585\) 0 0
\(586\) 6362.72 0.448535
\(587\) 11084.2i 0.779373i 0.920948 + 0.389686i \(0.127417\pi\)
−0.920948 + 0.389686i \(0.872583\pi\)
\(588\) 1417.22i 0.0993964i
\(589\) −4407.22 −0.308313
\(590\) 0 0
\(591\) 27894.0 1.94147
\(592\) − 781.066i − 0.0542257i
\(593\) − 4349.68i − 0.301214i −0.988594 0.150607i \(-0.951877\pi\)
0.988594 0.150607i \(-0.0481228\pi\)
\(594\) −2110.15 −0.145759
\(595\) 0 0
\(596\) 1301.34 0.0894382
\(597\) − 6530.40i − 0.447691i
\(598\) 1630.33i 0.111487i
\(599\) −13183.9 −0.899299 −0.449650 0.893205i \(-0.648451\pi\)
−0.449650 + 0.893205i \(0.648451\pi\)
\(600\) 0 0
\(601\) −18765.0 −1.27361 −0.636806 0.771024i \(-0.719745\pi\)
−0.636806 + 0.771024i \(0.719745\pi\)
\(602\) − 484.344i − 0.0327913i
\(603\) − 12275.8i − 0.829034i
\(604\) −1380.84 −0.0930223
\(605\) 0 0
\(606\) 3497.29 0.234435
\(607\) 21871.4i 1.46249i 0.682116 + 0.731244i \(0.261060\pi\)
−0.682116 + 0.731244i \(0.738940\pi\)
\(608\) − 3387.57i − 0.225961i
\(609\) −608.720 −0.0405034
\(610\) 0 0
\(611\) −1842.67 −0.122007
\(612\) 791.970i 0.0523096i
\(613\) 3527.85i 0.232445i 0.993223 + 0.116222i \(0.0370785\pi\)
−0.993223 + 0.116222i \(0.962921\pi\)
\(614\) 4584.56 0.301332
\(615\) 0 0
\(616\) 787.994 0.0515409
\(617\) − 22728.1i − 1.48298i −0.670963 0.741490i \(-0.734119\pi\)
0.670963 0.741490i \(-0.265881\pi\)
\(618\) − 752.893i − 0.0490062i
\(619\) 21443.3 1.39237 0.696187 0.717861i \(-0.254879\pi\)
0.696187 + 0.717861i \(0.254879\pi\)
\(620\) 0 0
\(621\) 7818.75 0.505243
\(622\) 9760.81i 0.629217i
\(623\) 4574.25i 0.294163i
\(624\) −2524.82 −0.161977
\(625\) 0 0
\(626\) 19628.0 1.25319
\(627\) 12202.7i 0.777240i
\(628\) 1326.85i 0.0843109i
\(629\) 541.718 0.0343398
\(630\) 0 0
\(631\) 21532.0 1.35844 0.679219 0.733936i \(-0.262319\pi\)
0.679219 + 0.733936i \(0.262319\pi\)
\(632\) − 30182.0i − 1.89964i
\(633\) − 851.038i − 0.0534372i
\(634\) 42.9141 0.00268823
\(635\) 0 0
\(636\) −1457.29 −0.0908572
\(637\) − 1787.56i − 0.111187i
\(638\) − 751.159i − 0.0466123i
\(639\) 7435.33 0.460309
\(640\) 0 0
\(641\) 20148.3 1.24151 0.620756 0.784004i \(-0.286826\pi\)
0.620756 + 0.784004i \(0.286826\pi\)
\(642\) − 18025.2i − 1.10810i
\(643\) − 28869.7i − 1.77062i −0.465000 0.885310i \(-0.653946\pi\)
0.465000 0.885310i \(-0.346054\pi\)
\(644\) −183.307 −0.0112163
\(645\) 0 0
\(646\) −15755.7 −0.959597
\(647\) − 1590.02i − 0.0966155i −0.998833 0.0483077i \(-0.984617\pi\)
0.998833 0.0483077i \(-0.0153828\pi\)
\(648\) − 9595.02i − 0.581679i
\(649\) 972.062 0.0587932
\(650\) 0 0
\(651\) −767.083 −0.0461818
\(652\) 1460.38i 0.0877192i
\(653\) − 20028.1i − 1.20024i −0.799909 0.600122i \(-0.795119\pi\)
0.799909 0.600122i \(-0.204881\pi\)
\(654\) 22618.9 1.35240
\(655\) 0 0
\(656\) −15514.4 −0.923375
\(657\) − 36236.5i − 2.15178i
\(658\) 2885.66i 0.170965i
\(659\) 10520.7 0.621897 0.310948 0.950427i \(-0.399353\pi\)
0.310948 + 0.950427i \(0.399353\pi\)
\(660\) 0 0
\(661\) 3295.83 0.193938 0.0969690 0.995287i \(-0.469085\pi\)
0.0969690 + 0.995287i \(0.469085\pi\)
\(662\) − 3603.45i − 0.211559i
\(663\) − 1751.12i − 0.102576i
\(664\) 10304.1 0.602222
\(665\) 0 0
\(666\) 1287.57 0.0749132
\(667\) 2783.27i 0.161572i
\(668\) 1466.91i 0.0849649i
\(669\) −31187.0 −1.80233
\(670\) 0 0
\(671\) −8125.67 −0.467493
\(672\) − 589.612i − 0.0338464i
\(673\) 1187.64i 0.0680239i 0.999421 + 0.0340119i \(0.0108284\pi\)
−0.999421 + 0.0340119i \(0.989172\pi\)
\(674\) 653.460 0.0373447
\(675\) 0 0
\(676\) 1161.98 0.0661117
\(677\) 13221.4i 0.750574i 0.926909 + 0.375287i \(0.122456\pi\)
−0.926909 + 0.375287i \(0.877544\pi\)
\(678\) 6391.55i 0.362044i
\(679\) −4135.91 −0.233758
\(680\) 0 0
\(681\) 14048.0 0.790485
\(682\) − 946.578i − 0.0531471i
\(683\) 13831.4i 0.774882i 0.921894 + 0.387441i \(0.126641\pi\)
−0.921894 + 0.387441i \(0.873359\pi\)
\(684\) 2688.68 0.150298
\(685\) 0 0
\(686\) −5677.93 −0.316012
\(687\) − 15185.4i − 0.843320i
\(688\) − 3429.62i − 0.190048i
\(689\) 1838.10 0.101635
\(690\) 0 0
\(691\) −9817.07 −0.540462 −0.270231 0.962796i \(-0.587100\pi\)
−0.270231 + 0.962796i \(0.587100\pi\)
\(692\) − 1236.54i − 0.0679280i
\(693\) 1211.58i 0.0664128i
\(694\) 16017.4 0.876101
\(695\) 0 0
\(696\) −4621.29 −0.251680
\(697\) − 10760.2i − 0.584750i
\(698\) − 9539.58i − 0.517304i
\(699\) 34854.9 1.88603
\(700\) 0 0
\(701\) 29949.8 1.61368 0.806838 0.590773i \(-0.201177\pi\)
0.806838 + 0.590773i \(0.201177\pi\)
\(702\) − 1028.02i − 0.0552711i
\(703\) − 1839.09i − 0.0986667i
\(704\) 5957.03 0.318912
\(705\) 0 0
\(706\) −29825.0 −1.58991
\(707\) − 495.976i − 0.0263835i
\(708\) − 375.456i − 0.0199301i
\(709\) −11307.5 −0.598959 −0.299479 0.954103i \(-0.596813\pi\)
−0.299479 + 0.954103i \(0.596813\pi\)
\(710\) 0 0
\(711\) 46406.3 2.44778
\(712\) 34726.9i 1.82787i
\(713\) 3507.36i 0.184224i
\(714\) −2742.30 −0.143737
\(715\) 0 0
\(716\) 703.181 0.0367027
\(717\) 32382.7i 1.68669i
\(718\) 31420.6i 1.63316i
\(719\) 32623.4 1.69214 0.846070 0.533071i \(-0.178962\pi\)
0.846070 + 0.533071i \(0.178962\pi\)
\(720\) 0 0
\(721\) −106.773 −0.00551518
\(722\) 34750.2i 1.79123i
\(723\) 30988.0i 1.59399i
\(724\) −430.420 −0.0220945
\(725\) 0 0
\(726\) −2620.89 −0.133981
\(727\) − 502.545i − 0.0256373i −0.999918 0.0128187i \(-0.995920\pi\)
0.999918 0.0128187i \(-0.00408042\pi\)
\(728\) 383.895i 0.0195441i
\(729\) 29783.2 1.51314
\(730\) 0 0
\(731\) 2378.66 0.120353
\(732\) 3138.51i 0.158474i
\(733\) − 8631.37i − 0.434935i −0.976068 0.217467i \(-0.930220\pi\)
0.976068 0.217467i \(-0.0697796\pi\)
\(734\) −18487.8 −0.929697
\(735\) 0 0
\(736\) −2695.90 −0.135017
\(737\) − 3765.95i − 0.188223i
\(738\) − 25575.0i − 1.27565i
\(739\) 18357.5 0.913792 0.456896 0.889520i \(-0.348961\pi\)
0.456896 + 0.889520i \(0.348961\pi\)
\(740\) 0 0
\(741\) −5944.93 −0.294726
\(742\) − 2878.52i − 0.142417i
\(743\) − 11182.6i − 0.552155i −0.961135 0.276078i \(-0.910965\pi\)
0.961135 0.276078i \(-0.0890347\pi\)
\(744\) −5823.55 −0.286965
\(745\) 0 0
\(746\) −14507.8 −0.712021
\(747\) 15843.0i 0.775991i
\(748\) 242.960i 0.0118763i
\(749\) −2556.29 −0.124706
\(750\) 0 0
\(751\) 16733.4 0.813063 0.406531 0.913637i \(-0.366738\pi\)
0.406531 + 0.913637i \(0.366738\pi\)
\(752\) 20433.3i 0.990857i
\(753\) 8680.53i 0.420101i
\(754\) 365.950 0.0176752
\(755\) 0 0
\(756\) 115.587 0.00556064
\(757\) − 24402.4i − 1.17163i −0.810446 0.585813i \(-0.800775\pi\)
0.810446 0.585813i \(-0.199225\pi\)
\(758\) 2290.19i 0.109741i
\(759\) 9711.19 0.464419
\(760\) 0 0
\(761\) 8469.33 0.403434 0.201717 0.979444i \(-0.435348\pi\)
0.201717 + 0.979444i \(0.435348\pi\)
\(762\) 28539.7i 1.35680i
\(763\) − 3207.74i − 0.152199i
\(764\) 920.805 0.0436042
\(765\) 0 0
\(766\) −7737.62 −0.364976
\(767\) 473.570i 0.0222941i
\(768\) − 6496.08i − 0.305218i
\(769\) −32834.7 −1.53973 −0.769864 0.638208i \(-0.779676\pi\)
−0.769864 + 0.638208i \(0.779676\pi\)
\(770\) 0 0
\(771\) −6209.20 −0.290038
\(772\) − 718.202i − 0.0334827i
\(773\) 35571.4i 1.65513i 0.561371 + 0.827564i \(0.310274\pi\)
−0.561371 + 0.827564i \(0.689726\pi\)
\(774\) 5653.64 0.262553
\(775\) 0 0
\(776\) −31399.1 −1.45253
\(777\) − 320.097i − 0.0147792i
\(778\) − 8500.11i − 0.391701i
\(779\) −36530.0 −1.68013
\(780\) 0 0
\(781\) 2281.01 0.104508
\(782\) 12538.7i 0.573381i
\(783\) − 1755.02i − 0.0801014i
\(784\) −19822.3 −0.902982
\(785\) 0 0
\(786\) 34671.8 1.57341
\(787\) 15729.6i 0.712452i 0.934400 + 0.356226i \(0.115937\pi\)
−0.934400 + 0.356226i \(0.884063\pi\)
\(788\) − 1885.47i − 0.0852373i
\(789\) 48996.7 2.21081
\(790\) 0 0
\(791\) 906.431 0.0407446
\(792\) 9198.09i 0.412676i
\(793\) − 3958.67i − 0.177272i
\(794\) −38819.0 −1.73506
\(795\) 0 0
\(796\) −441.415 −0.0196552
\(797\) 7888.07i 0.350577i 0.984517 + 0.175288i \(0.0560858\pi\)
−0.984517 + 0.175288i \(0.943914\pi\)
\(798\) 9309.91i 0.412991i
\(799\) −14171.8 −0.627485
\(800\) 0 0
\(801\) −53394.2 −2.35530
\(802\) − 17107.2i − 0.753214i
\(803\) − 11116.6i − 0.488538i
\(804\) −1454.58 −0.0638050
\(805\) 0 0
\(806\) 461.154 0.0201532
\(807\) − 7824.86i − 0.341323i
\(808\) − 3765.36i − 0.163942i
\(809\) −5896.97 −0.256275 −0.128138 0.991756i \(-0.540900\pi\)
−0.128138 + 0.991756i \(0.540900\pi\)
\(810\) 0 0
\(811\) 14197.9 0.614744 0.307372 0.951589i \(-0.400550\pi\)
0.307372 + 0.951589i \(0.400550\pi\)
\(812\) 41.1458i 0.00177824i
\(813\) 36287.3i 1.56538i
\(814\) 394.999 0.0170082
\(815\) 0 0
\(816\) −19418.2 −0.833053
\(817\) − 8075.35i − 0.345803i
\(818\) 11454.1i 0.489589i
\(819\) −590.258 −0.0251835
\(820\) 0 0
\(821\) −19841.7 −0.843459 −0.421729 0.906722i \(-0.638577\pi\)
−0.421729 + 0.906722i \(0.638577\pi\)
\(822\) − 34909.3i − 1.48127i
\(823\) 28202.2i 1.19449i 0.802058 + 0.597246i \(0.203738\pi\)
−0.802058 + 0.597246i \(0.796262\pi\)
\(824\) −810.602 −0.0342702
\(825\) 0 0
\(826\) 741.622 0.0312401
\(827\) 34031.0i 1.43092i 0.698651 + 0.715462i \(0.253784\pi\)
−0.698651 + 0.715462i \(0.746216\pi\)
\(828\) − 2139.71i − 0.0898067i
\(829\) −4931.55 −0.206610 −0.103305 0.994650i \(-0.532942\pi\)
−0.103305 + 0.994650i \(0.532942\pi\)
\(830\) 0 0
\(831\) −4501.92 −0.187930
\(832\) 2902.15i 0.120930i
\(833\) − 13748.0i − 0.571836i
\(834\) −690.138 −0.0286541
\(835\) 0 0
\(836\) 824.830 0.0341236
\(837\) − 2211.60i − 0.0913312i
\(838\) 25373.0i 1.04594i
\(839\) 38189.8 1.57146 0.785731 0.618568i \(-0.212287\pi\)
0.785731 + 0.618568i \(0.212287\pi\)
\(840\) 0 0
\(841\) −23764.3 −0.974384
\(842\) 35915.6i 1.46999i
\(843\) 42106.8i 1.72033i
\(844\) −57.5250 −0.00234608
\(845\) 0 0
\(846\) −33683.7 −1.36888
\(847\) 371.687i 0.0150783i
\(848\) − 20382.7i − 0.825406i
\(849\) −37488.0 −1.51541
\(850\) 0 0
\(851\) −1463.59 −0.0589556
\(852\) − 881.030i − 0.0354268i
\(853\) − 42966.8i − 1.72469i −0.506325 0.862343i \(-0.668997\pi\)
0.506325 0.862343i \(-0.331003\pi\)
\(854\) −6199.37 −0.248405
\(855\) 0 0
\(856\) −19406.8 −0.774898
\(857\) − 17281.5i − 0.688828i −0.938818 0.344414i \(-0.888078\pi\)
0.938818 0.344414i \(-0.111922\pi\)
\(858\) − 1276.85i − 0.0508052i
\(859\) −9316.75 −0.370062 −0.185031 0.982733i \(-0.559239\pi\)
−0.185031 + 0.982733i \(0.559239\pi\)
\(860\) 0 0
\(861\) −6358.10 −0.251665
\(862\) 13413.5i 0.530005i
\(863\) 9647.65i 0.380544i 0.981731 + 0.190272i \(0.0609370\pi\)
−0.981731 + 0.190272i \(0.939063\pi\)
\(864\) 1699.93 0.0669361
\(865\) 0 0
\(866\) −32083.3 −1.25893
\(867\) 25483.6i 0.998232i
\(868\) 51.8501i 0.00202754i
\(869\) 14236.5 0.555742
\(870\) 0 0
\(871\) 1834.70 0.0713735
\(872\) − 24352.6i − 0.945737i
\(873\) − 48277.6i − 1.87165i
\(874\) 42568.0 1.64746
\(875\) 0 0
\(876\) −4293.75 −0.165608
\(877\) 19728.7i 0.759624i 0.925064 + 0.379812i \(0.124011\pi\)
−0.925064 + 0.379812i \(0.875989\pi\)
\(878\) 32304.3i 1.24171i
\(879\) 18464.1 0.708509
\(880\) 0 0
\(881\) 19473.9 0.744712 0.372356 0.928090i \(-0.378550\pi\)
0.372356 + 0.928090i \(0.378550\pi\)
\(882\) − 32676.4i − 1.24748i
\(883\) − 49092.4i − 1.87100i −0.353329 0.935499i \(-0.614950\pi\)
0.353329 0.935499i \(-0.385050\pi\)
\(884\) −118.365 −0.00450346
\(885\) 0 0
\(886\) 27599.5 1.04653
\(887\) 9292.86i 0.351774i 0.984410 + 0.175887i \(0.0562793\pi\)
−0.984410 + 0.175887i \(0.943721\pi\)
\(888\) − 2430.12i − 0.0918348i
\(889\) 4047.41 0.152695
\(890\) 0 0
\(891\) 4525.85 0.170170
\(892\) 2108.05i 0.0791288i
\(893\) 48112.0i 1.80292i
\(894\) −52598.5 −1.96774
\(895\) 0 0
\(896\) 3949.89 0.147273
\(897\) 4731.10i 0.176106i
\(898\) 943.252i 0.0350520i
\(899\) 787.273 0.0292069
\(900\) 0 0
\(901\) 14136.7 0.522709
\(902\) − 7845.88i − 0.289622i
\(903\) − 1405.53i − 0.0517974i
\(904\) 6881.46 0.253179
\(905\) 0 0
\(906\) 55811.5 2.04659
\(907\) 37688.7i 1.37975i 0.723928 + 0.689875i \(0.242335\pi\)
−0.723928 + 0.689875i \(0.757665\pi\)
\(908\) − 949.559i − 0.0347051i
\(909\) 5789.42 0.211246
\(910\) 0 0
\(911\) 33049.6 1.20196 0.600979 0.799265i \(-0.294778\pi\)
0.600979 + 0.799265i \(0.294778\pi\)
\(912\) 65923.1i 2.39357i
\(913\) 4860.31i 0.176180i
\(914\) 28870.0 1.04479
\(915\) 0 0
\(916\) −1026.44 −0.0370247
\(917\) − 4917.06i − 0.177073i
\(918\) − 7906.43i − 0.284260i
\(919\) 23148.0 0.830883 0.415442 0.909620i \(-0.363627\pi\)
0.415442 + 0.909620i \(0.363627\pi\)
\(920\) 0 0
\(921\) 13304.1 0.475986
\(922\) 12933.4i 0.461973i
\(923\) 1111.26i 0.0396290i
\(924\) 143.563 0.00511134
\(925\) 0 0
\(926\) 9374.21 0.332673
\(927\) − 1246.34i − 0.0441588i
\(928\) 605.131i 0.0214056i
\(929\) 23177.9 0.818561 0.409280 0.912409i \(-0.365780\pi\)
0.409280 + 0.912409i \(0.365780\pi\)
\(930\) 0 0
\(931\) −46673.3 −1.64302
\(932\) − 2355.98i − 0.0828033i
\(933\) 28325.1i 0.993916i
\(934\) −13979.8 −0.489757
\(935\) 0 0
\(936\) −4481.13 −0.156485
\(937\) − 34574.7i − 1.20545i −0.797950 0.602724i \(-0.794082\pi\)
0.797950 0.602724i \(-0.205918\pi\)
\(938\) − 2873.18i − 0.100014i
\(939\) 56959.1 1.97954
\(940\) 0 0
\(941\) 41831.2 1.44916 0.724578 0.689192i \(-0.242034\pi\)
0.724578 + 0.689192i \(0.242034\pi\)
\(942\) − 53629.6i − 1.85493i
\(943\) 29071.3i 1.00392i
\(944\) 5251.40 0.181058
\(945\) 0 0
\(946\) 1734.42 0.0596097
\(947\) 27231.2i 0.934419i 0.884147 + 0.467209i \(0.154741\pi\)
−0.884147 + 0.467209i \(0.845259\pi\)
\(948\) − 5498.79i − 0.188389i
\(949\) 5415.79 0.185252
\(950\) 0 0
\(951\) 124.534 0.00424635
\(952\) 2952.50i 0.100516i
\(953\) − 40939.4i − 1.39156i −0.718254 0.695781i \(-0.755058\pi\)
0.718254 0.695781i \(-0.244942\pi\)
\(954\) 33600.3 1.14030
\(955\) 0 0
\(956\) 2188.87 0.0740515
\(957\) − 2179.81i − 0.0736292i
\(958\) − 31601.3i − 1.06575i
\(959\) −4950.74 −0.166703
\(960\) 0 0
\(961\) −28798.9 −0.966698
\(962\) 192.436i 0.00644945i
\(963\) − 29839.0i − 0.998491i
\(964\) 2094.60 0.0699820
\(965\) 0 0
\(966\) 7409.03 0.246772
\(967\) − 46173.1i − 1.53550i −0.640750 0.767750i \(-0.721376\pi\)
0.640750 0.767750i \(-0.278624\pi\)
\(968\) 2821.78i 0.0936937i
\(969\) −45721.8 −1.51579
\(970\) 0 0
\(971\) −5153.91 −0.170337 −0.0851683 0.996367i \(-0.527143\pi\)
−0.0851683 + 0.996367i \(0.527143\pi\)
\(972\) − 2764.06i − 0.0912111i
\(973\) 97.8734i 0.00322474i
\(974\) 50069.0 1.64714
\(975\) 0 0
\(976\) −43897.6 −1.43968
\(977\) 9692.13i 0.317378i 0.987329 + 0.158689i \(0.0507268\pi\)
−0.987329 + 0.158689i \(0.949273\pi\)
\(978\) − 59026.5i − 1.92992i
\(979\) −16380.2 −0.534744
\(980\) 0 0
\(981\) 37443.3 1.21863
\(982\) − 20811.6i − 0.676299i
\(983\) − 32915.7i − 1.06800i −0.845483 0.534002i \(-0.820687\pi\)
0.845483 0.534002i \(-0.179313\pi\)
\(984\) −48269.5 −1.56380
\(985\) 0 0
\(986\) 2814.48 0.0909041
\(987\) 8373.97i 0.270057i
\(988\) 401.841i 0.0129395i
\(989\) −6426.54 −0.206625
\(990\) 0 0
\(991\) 29477.9 0.944901 0.472451 0.881357i \(-0.343370\pi\)
0.472451 + 0.881357i \(0.343370\pi\)
\(992\) 762.560i 0.0244066i
\(993\) − 10456.9i − 0.334180i
\(994\) 1740.26 0.0555310
\(995\) 0 0
\(996\) 1877.28 0.0597227
\(997\) − 31944.4i − 1.01473i −0.861731 0.507366i \(-0.830619\pi\)
0.861731 0.507366i \(-0.169381\pi\)
\(998\) − 35268.4i − 1.11864i
\(999\) 922.883 0.0292279
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 275.4.b.c.199.4 4
5.2 odd 4 275.4.a.b.1.1 2
5.3 odd 4 11.4.a.a.1.2 2
5.4 even 2 inner 275.4.b.c.199.1 4
15.2 even 4 2475.4.a.q.1.2 2
15.8 even 4 99.4.a.c.1.1 2
20.3 even 4 176.4.a.i.1.2 2
35.13 even 4 539.4.a.e.1.2 2
40.3 even 4 704.4.a.n.1.1 2
40.13 odd 4 704.4.a.p.1.2 2
55.3 odd 20 121.4.c.c.9.2 8
55.8 even 20 121.4.c.f.9.1 8
55.13 even 20 121.4.c.f.81.2 8
55.18 even 20 121.4.c.f.27.1 8
55.28 even 20 121.4.c.f.3.2 8
55.38 odd 20 121.4.c.c.3.1 8
55.43 even 4 121.4.a.c.1.1 2
55.48 odd 20 121.4.c.c.27.2 8
55.53 odd 20 121.4.c.c.81.1 8
60.23 odd 4 1584.4.a.bc.1.1 2
65.38 odd 4 1859.4.a.a.1.1 2
165.98 odd 4 1089.4.a.v.1.2 2
220.43 odd 4 1936.4.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 5.3 odd 4
99.4.a.c.1.1 2 15.8 even 4
121.4.a.c.1.1 2 55.43 even 4
121.4.c.c.3.1 8 55.38 odd 20
121.4.c.c.9.2 8 55.3 odd 20
121.4.c.c.27.2 8 55.48 odd 20
121.4.c.c.81.1 8 55.53 odd 20
121.4.c.f.3.2 8 55.28 even 20
121.4.c.f.9.1 8 55.8 even 20
121.4.c.f.27.1 8 55.18 even 20
121.4.c.f.81.2 8 55.13 even 20
176.4.a.i.1.2 2 20.3 even 4
275.4.a.b.1.1 2 5.2 odd 4
275.4.b.c.199.1 4 5.4 even 2 inner
275.4.b.c.199.4 4 1.1 even 1 trivial
539.4.a.e.1.2 2 35.13 even 4
704.4.a.n.1.1 2 40.3 even 4
704.4.a.p.1.2 2 40.13 odd 4
1089.4.a.v.1.2 2 165.98 odd 4
1584.4.a.bc.1.1 2 60.23 odd 4
1859.4.a.a.1.1 2 65.38 odd 4
1936.4.a.w.1.2 2 220.43 odd 4
2475.4.a.q.1.2 2 15.2 even 4