Properties

Label 378.4.a.p.1.1
Level $378$
Weight $4$
Character 378.1
Self dual yes
Analytic conductor $22.303$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,4,Mod(1,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)

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: yes
Analytic conductor: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{105}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.62348\) of defining polynomial
Character \(\chi\) \(=\) 378.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -19.8704 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -19.8704 q^{5} +7.00000 q^{7} +8.00000 q^{8} -39.7409 q^{10} -39.7409 q^{11} +88.6113 q^{13} +14.0000 q^{14} +16.0000 q^{16} +72.7409 q^{17} +38.0000 q^{19} -79.4817 q^{20} -79.4817 q^{22} +7.12957 q^{23} +269.834 q^{25} +177.223 q^{26} +28.0000 q^{28} +257.575 q^{29} -48.6113 q^{31} +32.0000 q^{32} +145.482 q^{34} -139.093 q^{35} -343.279 q^{37} +76.0000 q^{38} -158.963 q^{40} +217.093 q^{41} +447.445 q^{43} -158.963 q^{44} +14.2591 q^{46} +387.279 q^{47} +49.0000 q^{49} +539.668 q^{50} +354.445 q^{52} -322.093 q^{53} +789.668 q^{55} +56.0000 q^{56} +515.149 q^{58} -539.113 q^{59} +303.555 q^{61} -97.2226 q^{62} +64.0000 q^{64} -1760.74 q^{65} -365.834 q^{67} +290.963 q^{68} -278.186 q^{70} -615.389 q^{71} +123.555 q^{73} -686.558 q^{74} +152.000 q^{76} -278.186 q^{77} +399.166 q^{79} -317.927 q^{80} +434.186 q^{82} +633.133 q^{83} -1445.39 q^{85} +894.890 q^{86} -317.927 q^{88} +823.541 q^{89} +620.279 q^{91} +28.5183 q^{92} +774.558 q^{94} -755.076 q^{95} -607.448 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 9 q^{5} + 14 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} - 9 q^{5} + 14 q^{7} + 16 q^{8} - 18 q^{10} - 18 q^{11} + 85 q^{13} + 28 q^{14} + 32 q^{16} + 84 q^{17} + 76 q^{19} - 36 q^{20} - 36 q^{22} + 45 q^{23} + 263 q^{25} + 170 q^{26} + 56 q^{28} + 177 q^{29} - 5 q^{31} + 64 q^{32} + 168 q^{34} - 63 q^{35} - 41 q^{37} + 152 q^{38} - 72 q^{40} + 219 q^{41} + 526 q^{43} - 72 q^{44} + 90 q^{46} + 129 q^{47} + 98 q^{49} + 526 q^{50} + 340 q^{52} - 429 q^{53} + 1026 q^{55} + 112 q^{56} + 354 q^{58} - 156 q^{59} + 976 q^{61} - 10 q^{62} + 128 q^{64} - 1800 q^{65} - 455 q^{67} + 336 q^{68} - 126 q^{70} - 1323 q^{71} + 616 q^{73} - 82 q^{74} + 304 q^{76} - 126 q^{77} + 1075 q^{79} - 144 q^{80} + 438 q^{82} - 363 q^{83} - 1323 q^{85} + 1052 q^{86} - 144 q^{88} - 597 q^{89} + 595 q^{91} + 180 q^{92} + 258 q^{94} - 342 q^{95} + 814 q^{97} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −19.8704 −1.77726 −0.888632 0.458620i \(-0.848344\pi\)
−0.888632 + 0.458620i \(0.848344\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −39.7409 −1.25672
\(11\) −39.7409 −1.08930 −0.544651 0.838663i \(-0.683338\pi\)
−0.544651 + 0.838663i \(0.683338\pi\)
\(12\) 0 0
\(13\) 88.6113 1.89049 0.945244 0.326364i \(-0.105824\pi\)
0.945244 + 0.326364i \(0.105824\pi\)
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 72.7409 1.03778 0.518890 0.854841i \(-0.326346\pi\)
0.518890 + 0.854841i \(0.326346\pi\)
\(18\) 0 0
\(19\) 38.0000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −79.4817 −0.888632
\(21\) 0 0
\(22\) −79.4817 −0.770253
\(23\) 7.12957 0.0646356 0.0323178 0.999478i \(-0.489711\pi\)
0.0323178 + 0.999478i \(0.489711\pi\)
\(24\) 0 0
\(25\) 269.834 2.15867
\(26\) 177.223 1.33678
\(27\) 0 0
\(28\) 28.0000 0.188982
\(29\) 257.575 1.64932 0.824662 0.565625i \(-0.191365\pi\)
0.824662 + 0.565625i \(0.191365\pi\)
\(30\) 0 0
\(31\) −48.6113 −0.281640 −0.140820 0.990035i \(-0.544974\pi\)
−0.140820 + 0.990035i \(0.544974\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 145.482 0.733821
\(35\) −139.093 −0.671743
\(36\) 0 0
\(37\) −343.279 −1.52526 −0.762631 0.646834i \(-0.776093\pi\)
−0.762631 + 0.646834i \(0.776093\pi\)
\(38\) 76.0000 0.324443
\(39\) 0 0
\(40\) −158.963 −0.628358
\(41\) 217.093 0.826932 0.413466 0.910519i \(-0.364318\pi\)
0.413466 + 0.910519i \(0.364318\pi\)
\(42\) 0 0
\(43\) 447.445 1.58685 0.793427 0.608665i \(-0.208295\pi\)
0.793427 + 0.608665i \(0.208295\pi\)
\(44\) −158.963 −0.544651
\(45\) 0 0
\(46\) 14.2591 0.0457043
\(47\) 387.279 1.20192 0.600962 0.799277i \(-0.294784\pi\)
0.600962 + 0.799277i \(0.294784\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 539.668 1.52641
\(51\) 0 0
\(52\) 354.445 0.945244
\(53\) −322.093 −0.834771 −0.417386 0.908729i \(-0.637054\pi\)
−0.417386 + 0.908729i \(0.637054\pi\)
\(54\) 0 0
\(55\) 789.668 1.93598
\(56\) 56.0000 0.133631
\(57\) 0 0
\(58\) 515.149 1.16625
\(59\) −539.113 −1.18960 −0.594801 0.803873i \(-0.702769\pi\)
−0.594801 + 0.803873i \(0.702769\pi\)
\(60\) 0 0
\(61\) 303.555 0.637151 0.318576 0.947897i \(-0.396796\pi\)
0.318576 + 0.947897i \(0.396796\pi\)
\(62\) −97.2226 −0.199150
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −1760.74 −3.35990
\(66\) 0 0
\(67\) −365.834 −0.667070 −0.333535 0.942738i \(-0.608242\pi\)
−0.333535 + 0.942738i \(0.608242\pi\)
\(68\) 290.963 0.518890
\(69\) 0 0
\(70\) −278.186 −0.474994
\(71\) −615.389 −1.02864 −0.514318 0.857599i \(-0.671955\pi\)
−0.514318 + 0.857599i \(0.671955\pi\)
\(72\) 0 0
\(73\) 123.555 0.198096 0.0990480 0.995083i \(-0.468420\pi\)
0.0990480 + 0.995083i \(0.468420\pi\)
\(74\) −686.558 −1.07852
\(75\) 0 0
\(76\) 152.000 0.229416
\(77\) −278.186 −0.411717
\(78\) 0 0
\(79\) 399.166 0.568477 0.284239 0.958754i \(-0.408259\pi\)
0.284239 + 0.958754i \(0.408259\pi\)
\(80\) −317.927 −0.444316
\(81\) 0 0
\(82\) 434.186 0.584729
\(83\) 633.133 0.837293 0.418647 0.908149i \(-0.362505\pi\)
0.418647 + 0.908149i \(0.362505\pi\)
\(84\) 0 0
\(85\) −1445.39 −1.84441
\(86\) 894.890 1.12208
\(87\) 0 0
\(88\) −317.927 −0.385126
\(89\) 823.541 0.980845 0.490422 0.871485i \(-0.336843\pi\)
0.490422 + 0.871485i \(0.336843\pi\)
\(90\) 0 0
\(91\) 620.279 0.714537
\(92\) 28.5183 0.0323178
\(93\) 0 0
\(94\) 774.558 0.849889
\(95\) −755.076 −0.815465
\(96\) 0 0
\(97\) −607.448 −0.635845 −0.317923 0.948117i \(-0.602985\pi\)
−0.317923 + 0.948117i \(0.602985\pi\)
\(98\) 98.0000 0.101015
\(99\) 0 0
\(100\) 1079.34 1.07934
\(101\) 1064.82 1.04904 0.524521 0.851397i \(-0.324244\pi\)
0.524521 + 0.851397i \(0.324244\pi\)
\(102\) 0 0
\(103\) 1494.50 1.42969 0.714844 0.699284i \(-0.246498\pi\)
0.714844 + 0.699284i \(0.246498\pi\)
\(104\) 708.890 0.668389
\(105\) 0 0
\(106\) −644.186 −0.590272
\(107\) −1733.30 −1.56602 −0.783009 0.622010i \(-0.786316\pi\)
−0.783009 + 0.622010i \(0.786316\pi\)
\(108\) 0 0
\(109\) −33.0534 −0.0290453 −0.0145227 0.999895i \(-0.504623\pi\)
−0.0145227 + 0.999895i \(0.504623\pi\)
\(110\) 1579.34 1.36894
\(111\) 0 0
\(112\) 112.000 0.0944911
\(113\) −242.963 −0.202266 −0.101133 0.994873i \(-0.532247\pi\)
−0.101133 + 0.994873i \(0.532247\pi\)
\(114\) 0 0
\(115\) −141.668 −0.114875
\(116\) 1030.30 0.824662
\(117\) 0 0
\(118\) −1078.23 −0.841176
\(119\) 509.186 0.392244
\(120\) 0 0
\(121\) 248.335 0.186578
\(122\) 607.110 0.450534
\(123\) 0 0
\(124\) −194.445 −0.140820
\(125\) −2877.91 −2.05926
\(126\) 0 0
\(127\) 1600.06 1.11797 0.558987 0.829177i \(-0.311190\pi\)
0.558987 + 0.829177i \(0.311190\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −3521.49 −2.37581
\(131\) 2470.58 1.64775 0.823875 0.566771i \(-0.191808\pi\)
0.823875 + 0.566771i \(0.191808\pi\)
\(132\) 0 0
\(133\) 266.000 0.173422
\(134\) −731.668 −0.471690
\(135\) 0 0
\(136\) 581.927 0.366910
\(137\) −241.966 −0.150895 −0.0754474 0.997150i \(-0.524038\pi\)
−0.0754474 + 0.997150i \(0.524038\pi\)
\(138\) 0 0
\(139\) 307.780 0.187810 0.0939050 0.995581i \(-0.470065\pi\)
0.0939050 + 0.995581i \(0.470065\pi\)
\(140\) −556.372 −0.335872
\(141\) 0 0
\(142\) −1230.78 −0.727356
\(143\) −3521.49 −2.05931
\(144\) 0 0
\(145\) −5118.12 −2.93129
\(146\) 247.110 0.140075
\(147\) 0 0
\(148\) −1373.12 −0.762631
\(149\) 1298.46 0.713922 0.356961 0.934119i \(-0.383813\pi\)
0.356961 + 0.934119i \(0.383813\pi\)
\(150\) 0 0
\(151\) −1430.28 −0.770823 −0.385411 0.922745i \(-0.625940\pi\)
−0.385411 + 0.922745i \(0.625940\pi\)
\(152\) 304.000 0.162221
\(153\) 0 0
\(154\) −556.372 −0.291128
\(155\) 965.927 0.500549
\(156\) 0 0
\(157\) −3641.18 −1.85094 −0.925469 0.378822i \(-0.876329\pi\)
−0.925469 + 0.378822i \(0.876329\pi\)
\(158\) 798.332 0.401974
\(159\) 0 0
\(160\) −635.854 −0.314179
\(161\) 49.9070 0.0244300
\(162\) 0 0
\(163\) −3298.45 −1.58500 −0.792498 0.609874i \(-0.791220\pi\)
−0.792498 + 0.609874i \(0.791220\pi\)
\(164\) 868.372 0.413466
\(165\) 0 0
\(166\) 1266.27 0.592056
\(167\) −1402.69 −0.649959 −0.324980 0.945721i \(-0.605357\pi\)
−0.324980 + 0.945721i \(0.605357\pi\)
\(168\) 0 0
\(169\) 5654.96 2.57395
\(170\) −2890.78 −1.30419
\(171\) 0 0
\(172\) 1789.78 0.793427
\(173\) −1683.01 −0.739635 −0.369817 0.929104i \(-0.620580\pi\)
−0.369817 + 0.929104i \(0.620580\pi\)
\(174\) 0 0
\(175\) 1888.84 0.815901
\(176\) −635.854 −0.272325
\(177\) 0 0
\(178\) 1647.08 0.693562
\(179\) 1492.08 0.623035 0.311517 0.950240i \(-0.399163\pi\)
0.311517 + 0.950240i \(0.399163\pi\)
\(180\) 0 0
\(181\) 2945.73 1.20969 0.604846 0.796343i \(-0.293235\pi\)
0.604846 + 0.796343i \(0.293235\pi\)
\(182\) 1240.56 0.505254
\(183\) 0 0
\(184\) 57.0366 0.0228521
\(185\) 6821.10 2.71080
\(186\) 0 0
\(187\) −2890.78 −1.13045
\(188\) 1549.12 0.600962
\(189\) 0 0
\(190\) −1510.15 −0.576621
\(191\) −1913.44 −0.724879 −0.362439 0.932007i \(-0.618056\pi\)
−0.362439 + 0.932007i \(0.618056\pi\)
\(192\) 0 0
\(193\) −59.3263 −0.0221264 −0.0110632 0.999939i \(-0.503522\pi\)
−0.0110632 + 0.999939i \(0.503522\pi\)
\(194\) −1214.90 −0.449611
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) 287.780 0.104079 0.0520394 0.998645i \(-0.483428\pi\)
0.0520394 + 0.998645i \(0.483428\pi\)
\(198\) 0 0
\(199\) −4014.51 −1.43006 −0.715028 0.699096i \(-0.753586\pi\)
−0.715028 + 0.699096i \(0.753586\pi\)
\(200\) 2158.67 0.763205
\(201\) 0 0
\(202\) 2129.63 0.741785
\(203\) 1803.02 0.623386
\(204\) 0 0
\(205\) −4313.73 −1.46968
\(206\) 2989.01 1.01094
\(207\) 0 0
\(208\) 1417.78 0.472622
\(209\) −1510.15 −0.499806
\(210\) 0 0
\(211\) −4810.30 −1.56945 −0.784726 0.619842i \(-0.787197\pi\)
−0.784726 + 0.619842i \(0.787197\pi\)
\(212\) −1288.37 −0.417386
\(213\) 0 0
\(214\) −3466.59 −1.10734
\(215\) −8890.93 −2.82026
\(216\) 0 0
\(217\) −340.279 −0.106450
\(218\) −66.1068 −0.0205381
\(219\) 0 0
\(220\) 3158.67 0.967989
\(221\) 6445.66 1.96191
\(222\) 0 0
\(223\) 3453.12 1.03694 0.518470 0.855096i \(-0.326502\pi\)
0.518470 + 0.855096i \(0.326502\pi\)
\(224\) 224.000 0.0668153
\(225\) 0 0
\(226\) −485.927 −0.143024
\(227\) −158.127 −0.0462345 −0.0231172 0.999733i \(-0.507359\pi\)
−0.0231172 + 0.999733i \(0.507359\pi\)
\(228\) 0 0
\(229\) −612.332 −0.176699 −0.0883495 0.996090i \(-0.528159\pi\)
−0.0883495 + 0.996090i \(0.528159\pi\)
\(230\) −283.335 −0.0812286
\(231\) 0 0
\(232\) 2060.60 0.583124
\(233\) 5206.39 1.46387 0.731936 0.681374i \(-0.238617\pi\)
0.731936 + 0.681374i \(0.238617\pi\)
\(234\) 0 0
\(235\) −7695.40 −2.13614
\(236\) −2156.45 −0.594801
\(237\) 0 0
\(238\) 1018.37 0.277358
\(239\) −647.854 −0.175340 −0.0876698 0.996150i \(-0.527942\pi\)
−0.0876698 + 0.996150i \(0.527942\pi\)
\(240\) 0 0
\(241\) 6141.67 1.64158 0.820788 0.571232i \(-0.193535\pi\)
0.820788 + 0.571232i \(0.193535\pi\)
\(242\) 496.671 0.131931
\(243\) 0 0
\(244\) 1214.22 0.318576
\(245\) −973.651 −0.253895
\(246\) 0 0
\(247\) 3367.23 0.867415
\(248\) −388.890 −0.0995748
\(249\) 0 0
\(250\) −5755.82 −1.45612
\(251\) −4779.84 −1.20199 −0.600997 0.799251i \(-0.705230\pi\)
−0.600997 + 0.799251i \(0.705230\pi\)
\(252\) 0 0
\(253\) −283.335 −0.0704077
\(254\) 3200.12 0.790526
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 1207.85 0.293165 0.146583 0.989198i \(-0.453173\pi\)
0.146583 + 0.989198i \(0.453173\pi\)
\(258\) 0 0
\(259\) −2402.95 −0.576495
\(260\) −7042.98 −1.67995
\(261\) 0 0
\(262\) 4941.16 1.16514
\(263\) 2869.66 0.672817 0.336408 0.941716i \(-0.390788\pi\)
0.336408 + 0.941716i \(0.390788\pi\)
\(264\) 0 0
\(265\) 6400.12 1.48361
\(266\) 532.000 0.122628
\(267\) 0 0
\(268\) −1463.34 −0.333535
\(269\) 5871.69 1.33087 0.665434 0.746457i \(-0.268247\pi\)
0.665434 + 0.746457i \(0.268247\pi\)
\(270\) 0 0
\(271\) 1081.96 0.242525 0.121263 0.992620i \(-0.461306\pi\)
0.121263 + 0.992620i \(0.461306\pi\)
\(272\) 1163.85 0.259445
\(273\) 0 0
\(274\) −483.933 −0.106699
\(275\) −10723.4 −2.35144
\(276\) 0 0
\(277\) 931.511 0.202054 0.101027 0.994884i \(-0.467787\pi\)
0.101027 + 0.994884i \(0.467787\pi\)
\(278\) 615.561 0.132802
\(279\) 0 0
\(280\) −1112.74 −0.237497
\(281\) −1915.50 −0.406652 −0.203326 0.979111i \(-0.565175\pi\)
−0.203326 + 0.979111i \(0.565175\pi\)
\(282\) 0 0
\(283\) 126.985 0.0266730 0.0133365 0.999911i \(-0.495755\pi\)
0.0133365 + 0.999911i \(0.495755\pi\)
\(284\) −2461.55 −0.514318
\(285\) 0 0
\(286\) −7042.98 −1.45615
\(287\) 1519.65 0.312551
\(288\) 0 0
\(289\) 378.232 0.0769859
\(290\) −10236.2 −2.07273
\(291\) 0 0
\(292\) 494.220 0.0990480
\(293\) −4129.14 −0.823300 −0.411650 0.911342i \(-0.635047\pi\)
−0.411650 + 0.911342i \(0.635047\pi\)
\(294\) 0 0
\(295\) 10712.4 2.11424
\(296\) −2746.23 −0.539262
\(297\) 0 0
\(298\) 2596.93 0.504819
\(299\) 631.761 0.122193
\(300\) 0 0
\(301\) 3132.12 0.599775
\(302\) −2860.55 −0.545054
\(303\) 0 0
\(304\) 608.000 0.114708
\(305\) −6031.76 −1.13239
\(306\) 0 0
\(307\) 1528.43 0.284144 0.142072 0.989856i \(-0.454624\pi\)
0.142072 + 0.989856i \(0.454624\pi\)
\(308\) −1112.74 −0.205859
\(309\) 0 0
\(310\) 1931.85 0.353942
\(311\) 8985.18 1.63827 0.819136 0.573599i \(-0.194453\pi\)
0.819136 + 0.573599i \(0.194453\pi\)
\(312\) 0 0
\(313\) −9218.80 −1.66478 −0.832392 0.554187i \(-0.813029\pi\)
−0.832392 + 0.554187i \(0.813029\pi\)
\(314\) −7282.35 −1.30881
\(315\) 0 0
\(316\) 1596.66 0.284239
\(317\) 802.226 0.142137 0.0710686 0.997471i \(-0.477359\pi\)
0.0710686 + 0.997471i \(0.477359\pi\)
\(318\) 0 0
\(319\) −10236.2 −1.79661
\(320\) −1271.71 −0.222158
\(321\) 0 0
\(322\) 99.8140 0.0172746
\(323\) 2764.15 0.476166
\(324\) 0 0
\(325\) 23910.3 4.08094
\(326\) −6596.90 −1.12076
\(327\) 0 0
\(328\) 1736.74 0.292365
\(329\) 2710.95 0.454285
\(330\) 0 0
\(331\) −953.564 −0.158346 −0.0791731 0.996861i \(-0.525228\pi\)
−0.0791731 + 0.996861i \(0.525228\pi\)
\(332\) 2532.53 0.418647
\(333\) 0 0
\(334\) −2805.37 −0.459591
\(335\) 7269.27 1.18556
\(336\) 0 0
\(337\) 9659.00 1.56130 0.780652 0.624966i \(-0.214887\pi\)
0.780652 + 0.624966i \(0.214887\pi\)
\(338\) 11309.9 1.82005
\(339\) 0 0
\(340\) −5781.57 −0.922204
\(341\) 1931.85 0.306791
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 3579.56 0.561038
\(345\) 0 0
\(346\) −3366.02 −0.523001
\(347\) 1854.34 0.286876 0.143438 0.989659i \(-0.454184\pi\)
0.143438 + 0.989659i \(0.454184\pi\)
\(348\) 0 0
\(349\) −2643.08 −0.405389 −0.202695 0.979242i \(-0.564970\pi\)
−0.202695 + 0.979242i \(0.564970\pi\)
\(350\) 3777.67 0.576929
\(351\) 0 0
\(352\) −1271.71 −0.192563
\(353\) −3096.64 −0.466905 −0.233453 0.972368i \(-0.575002\pi\)
−0.233453 + 0.972368i \(0.575002\pi\)
\(354\) 0 0
\(355\) 12228.0 1.82816
\(356\) 3294.16 0.490422
\(357\) 0 0
\(358\) 2984.16 0.440552
\(359\) 5575.10 0.819617 0.409809 0.912172i \(-0.365595\pi\)
0.409809 + 0.912172i \(0.365595\pi\)
\(360\) 0 0
\(361\) −5415.00 −0.789474
\(362\) 5891.45 0.855381
\(363\) 0 0
\(364\) 2481.12 0.357269
\(365\) −2455.09 −0.352069
\(366\) 0 0
\(367\) 8400.17 1.19478 0.597391 0.801950i \(-0.296204\pi\)
0.597391 + 0.801950i \(0.296204\pi\)
\(368\) 114.073 0.0161589
\(369\) 0 0
\(370\) 13642.2 1.91682
\(371\) −2254.65 −0.315514
\(372\) 0 0
\(373\) 4612.62 0.640301 0.320151 0.947367i \(-0.396266\pi\)
0.320151 + 0.947367i \(0.396266\pi\)
\(374\) −5781.57 −0.799352
\(375\) 0 0
\(376\) 3098.23 0.424944
\(377\) 22824.0 3.11803
\(378\) 0 0
\(379\) 669.166 0.0906933 0.0453466 0.998971i \(-0.485561\pi\)
0.0453466 + 0.998971i \(0.485561\pi\)
\(380\) −3020.30 −0.407733
\(381\) 0 0
\(382\) −3826.88 −0.512567
\(383\) −2190.08 −0.292187 −0.146094 0.989271i \(-0.546670\pi\)
−0.146094 + 0.989271i \(0.546670\pi\)
\(384\) 0 0
\(385\) 5527.67 0.731731
\(386\) −118.653 −0.0156457
\(387\) 0 0
\(388\) −2429.79 −0.317923
\(389\) 12281.9 1.60082 0.800410 0.599452i \(-0.204615\pi\)
0.800410 + 0.599452i \(0.204615\pi\)
\(390\) 0 0
\(391\) 518.611 0.0670775
\(392\) 392.000 0.0505076
\(393\) 0 0
\(394\) 575.561 0.0735948
\(395\) −7931.60 −1.01033
\(396\) 0 0
\(397\) −11016.9 −1.39275 −0.696376 0.717677i \(-0.745206\pi\)
−0.696376 + 0.717677i \(0.745206\pi\)
\(398\) −8029.02 −1.01120
\(399\) 0 0
\(400\) 4317.34 0.539668
\(401\) −2428.73 −0.302456 −0.151228 0.988499i \(-0.548323\pi\)
−0.151228 + 0.988499i \(0.548323\pi\)
\(402\) 0 0
\(403\) −4307.51 −0.532437
\(404\) 4259.27 0.524521
\(405\) 0 0
\(406\) 3606.05 0.440801
\(407\) 13642.2 1.66147
\(408\) 0 0
\(409\) −8077.01 −0.976485 −0.488243 0.872708i \(-0.662362\pi\)
−0.488243 + 0.872708i \(0.662362\pi\)
\(410\) −8627.46 −1.03922
\(411\) 0 0
\(412\) 5978.02 0.714844
\(413\) −3773.79 −0.449627
\(414\) 0 0
\(415\) −12580.6 −1.48809
\(416\) 2835.56 0.334194
\(417\) 0 0
\(418\) −3020.30 −0.353416
\(419\) −10911.5 −1.27223 −0.636114 0.771595i \(-0.719459\pi\)
−0.636114 + 0.771595i \(0.719459\pi\)
\(420\) 0 0
\(421\) −3452.68 −0.399699 −0.199850 0.979827i \(-0.564045\pi\)
−0.199850 + 0.979827i \(0.564045\pi\)
\(422\) −9620.59 −1.10977
\(423\) 0 0
\(424\) −2576.74 −0.295136
\(425\) 19627.9 2.24022
\(426\) 0 0
\(427\) 2124.88 0.240821
\(428\) −6933.18 −0.783009
\(429\) 0 0
\(430\) −17781.9 −1.99423
\(431\) 4418.73 0.493834 0.246917 0.969037i \(-0.420582\pi\)
0.246917 + 0.969037i \(0.420582\pi\)
\(432\) 0 0
\(433\) −517.365 −0.0574203 −0.0287102 0.999588i \(-0.509140\pi\)
−0.0287102 + 0.999588i \(0.509140\pi\)
\(434\) −680.558 −0.0752715
\(435\) 0 0
\(436\) −132.214 −0.0145227
\(437\) 270.924 0.0296568
\(438\) 0 0
\(439\) 12145.5 1.32044 0.660222 0.751071i \(-0.270462\pi\)
0.660222 + 0.751071i \(0.270462\pi\)
\(440\) 6317.34 0.684471
\(441\) 0 0
\(442\) 12891.3 1.38728
\(443\) −11814.7 −1.26712 −0.633558 0.773695i \(-0.718406\pi\)
−0.633558 + 0.773695i \(0.718406\pi\)
\(444\) 0 0
\(445\) −16364.1 −1.74322
\(446\) 6906.23 0.733228
\(447\) 0 0
\(448\) 448.000 0.0472456
\(449\) 15969.1 1.67846 0.839231 0.543775i \(-0.183005\pi\)
0.839231 + 0.543775i \(0.183005\pi\)
\(450\) 0 0
\(451\) −8627.46 −0.900779
\(452\) −971.854 −0.101133
\(453\) 0 0
\(454\) −316.253 −0.0326927
\(455\) −12325.2 −1.26992
\(456\) 0 0
\(457\) 12880.2 1.31840 0.659201 0.751967i \(-0.270895\pi\)
0.659201 + 0.751967i \(0.270895\pi\)
\(458\) −1224.66 −0.124945
\(459\) 0 0
\(460\) −566.671 −0.0574373
\(461\) −6764.57 −0.683421 −0.341711 0.939805i \(-0.611006\pi\)
−0.341711 + 0.939805i \(0.611006\pi\)
\(462\) 0 0
\(463\) 1121.06 0.112527 0.0562636 0.998416i \(-0.482081\pi\)
0.0562636 + 0.998416i \(0.482081\pi\)
\(464\) 4121.20 0.412331
\(465\) 0 0
\(466\) 10412.8 1.03511
\(467\) −15671.4 −1.55286 −0.776431 0.630202i \(-0.782972\pi\)
−0.776431 + 0.630202i \(0.782972\pi\)
\(468\) 0 0
\(469\) −2560.84 −0.252129
\(470\) −15390.8 −1.51048
\(471\) 0 0
\(472\) −4312.90 −0.420588
\(473\) −17781.9 −1.72856
\(474\) 0 0
\(475\) 10253.7 0.990466
\(476\) 2036.74 0.196122
\(477\) 0 0
\(478\) −1295.71 −0.123984
\(479\) −11211.7 −1.06947 −0.534734 0.845020i \(-0.679588\pi\)
−0.534734 + 0.845020i \(0.679588\pi\)
\(480\) 0 0
\(481\) −30418.4 −2.88349
\(482\) 12283.3 1.16077
\(483\) 0 0
\(484\) 993.341 0.0932890
\(485\) 12070.3 1.13007
\(486\) 0 0
\(487\) −5261.00 −0.489525 −0.244762 0.969583i \(-0.578710\pi\)
−0.244762 + 0.969583i \(0.578710\pi\)
\(488\) 2428.44 0.225267
\(489\) 0 0
\(490\) −1947.30 −0.179531
\(491\) 8754.40 0.804645 0.402322 0.915498i \(-0.368203\pi\)
0.402322 + 0.915498i \(0.368203\pi\)
\(492\) 0 0
\(493\) 18736.2 1.71164
\(494\) 6734.46 0.613355
\(495\) 0 0
\(496\) −777.780 −0.0704100
\(497\) −4307.72 −0.388788
\(498\) 0 0
\(499\) −18969.8 −1.70181 −0.850905 0.525319i \(-0.823946\pi\)
−0.850905 + 0.525319i \(0.823946\pi\)
\(500\) −11511.6 −1.02963
\(501\) 0 0
\(502\) −9559.67 −0.849938
\(503\) −3563.27 −0.315862 −0.157931 0.987450i \(-0.550482\pi\)
−0.157931 + 0.987450i \(0.550482\pi\)
\(504\) 0 0
\(505\) −21158.4 −1.86443
\(506\) −566.671 −0.0497857
\(507\) 0 0
\(508\) 6400.25 0.558987
\(509\) 722.492 0.0629153 0.0314577 0.999505i \(-0.489985\pi\)
0.0314577 + 0.999505i \(0.489985\pi\)
\(510\) 0 0
\(511\) 864.884 0.0748732
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 2415.70 0.207299
\(515\) −29696.4 −2.54094
\(516\) 0 0
\(517\) −15390.8 −1.30926
\(518\) −4805.91 −0.407644
\(519\) 0 0
\(520\) −14086.0 −1.18790
\(521\) 10687.6 0.898721 0.449361 0.893350i \(-0.351652\pi\)
0.449361 + 0.893350i \(0.351652\pi\)
\(522\) 0 0
\(523\) −4810.42 −0.402189 −0.201095 0.979572i \(-0.564450\pi\)
−0.201095 + 0.979572i \(0.564450\pi\)
\(524\) 9882.31 0.823875
\(525\) 0 0
\(526\) 5739.32 0.475753
\(527\) −3536.03 −0.292280
\(528\) 0 0
\(529\) −12116.2 −0.995822
\(530\) 12800.2 1.04907
\(531\) 0 0
\(532\) 1064.00 0.0867110
\(533\) 19236.9 1.56331
\(534\) 0 0
\(535\) 34441.3 2.78323
\(536\) −2926.67 −0.235845
\(537\) 0 0
\(538\) 11743.4 0.941065
\(539\) −1947.30 −0.155615
\(540\) 0 0
\(541\) −8090.06 −0.642918 −0.321459 0.946923i \(-0.604173\pi\)
−0.321459 + 0.946923i \(0.604173\pi\)
\(542\) 2163.92 0.171491
\(543\) 0 0
\(544\) 2327.71 0.183455
\(545\) 656.785 0.0516212
\(546\) 0 0
\(547\) −11831.9 −0.924852 −0.462426 0.886658i \(-0.653021\pi\)
−0.462426 + 0.886658i \(0.653021\pi\)
\(548\) −967.866 −0.0754474
\(549\) 0 0
\(550\) −21446.9 −1.66272
\(551\) 9787.84 0.756762
\(552\) 0 0
\(553\) 2794.16 0.214864
\(554\) 1863.02 0.142874
\(555\) 0 0
\(556\) 1231.12 0.0939050
\(557\) 15783.2 1.20064 0.600321 0.799759i \(-0.295040\pi\)
0.600321 + 0.799759i \(0.295040\pi\)
\(558\) 0 0
\(559\) 39648.7 2.99993
\(560\) −2225.49 −0.167936
\(561\) 0 0
\(562\) −3831.00 −0.287546
\(563\) −4353.81 −0.325917 −0.162958 0.986633i \(-0.552104\pi\)
−0.162958 + 0.986633i \(0.552104\pi\)
\(564\) 0 0
\(565\) 4827.79 0.359481
\(566\) 253.970 0.0188607
\(567\) 0 0
\(568\) −4923.11 −0.363678
\(569\) −16526.4 −1.21761 −0.608806 0.793319i \(-0.708351\pi\)
−0.608806 + 0.793319i \(0.708351\pi\)
\(570\) 0 0
\(571\) −24307.8 −1.78152 −0.890761 0.454471i \(-0.849828\pi\)
−0.890761 + 0.454471i \(0.849828\pi\)
\(572\) −14086.0 −1.02966
\(573\) 0 0
\(574\) 3039.30 0.221007
\(575\) 1923.80 0.139527
\(576\) 0 0
\(577\) −5474.80 −0.395007 −0.197503 0.980302i \(-0.563283\pi\)
−0.197503 + 0.980302i \(0.563283\pi\)
\(578\) 756.463 0.0544372
\(579\) 0 0
\(580\) −20472.5 −1.46564
\(581\) 4431.93 0.316467
\(582\) 0 0
\(583\) 12800.2 0.909318
\(584\) 988.439 0.0700375
\(585\) 0 0
\(586\) −8258.28 −0.582161
\(587\) −12868.2 −0.904818 −0.452409 0.891811i \(-0.649435\pi\)
−0.452409 + 0.891811i \(0.649435\pi\)
\(588\) 0 0
\(589\) −1847.23 −0.129225
\(590\) 21424.8 1.49499
\(591\) 0 0
\(592\) −5492.46 −0.381316
\(593\) −16734.6 −1.15886 −0.579432 0.815020i \(-0.696726\pi\)
−0.579432 + 0.815020i \(0.696726\pi\)
\(594\) 0 0
\(595\) −10117.7 −0.697121
\(596\) 5193.86 0.356961
\(597\) 0 0
\(598\) 1263.52 0.0864034
\(599\) −7640.15 −0.521149 −0.260574 0.965454i \(-0.583912\pi\)
−0.260574 + 0.965454i \(0.583912\pi\)
\(600\) 0 0
\(601\) −6068.66 −0.411890 −0.205945 0.978564i \(-0.566027\pi\)
−0.205945 + 0.978564i \(0.566027\pi\)
\(602\) 6264.23 0.424105
\(603\) 0 0
\(604\) −5721.10 −0.385411
\(605\) −4934.53 −0.331599
\(606\) 0 0
\(607\) 11318.1 0.756814 0.378407 0.925639i \(-0.376472\pi\)
0.378407 + 0.925639i \(0.376472\pi\)
\(608\) 1216.00 0.0811107
\(609\) 0 0
\(610\) −12063.5 −0.800718
\(611\) 34317.3 2.27222
\(612\) 0 0
\(613\) 16973.1 1.11833 0.559167 0.829055i \(-0.311121\pi\)
0.559167 + 0.829055i \(0.311121\pi\)
\(614\) 3056.87 0.200920
\(615\) 0 0
\(616\) −2225.49 −0.145564
\(617\) 13351.7 0.871182 0.435591 0.900145i \(-0.356539\pi\)
0.435591 + 0.900145i \(0.356539\pi\)
\(618\) 0 0
\(619\) −7258.04 −0.471284 −0.235642 0.971840i \(-0.575719\pi\)
−0.235642 + 0.971840i \(0.575719\pi\)
\(620\) 3863.71 0.250275
\(621\) 0 0
\(622\) 17970.4 1.15843
\(623\) 5764.79 0.370724
\(624\) 0 0
\(625\) 23456.1 1.50119
\(626\) −18437.6 −1.17718
\(627\) 0 0
\(628\) −14564.7 −0.925469
\(629\) −24970.4 −1.58289
\(630\) 0 0
\(631\) −9332.14 −0.588758 −0.294379 0.955689i \(-0.595113\pi\)
−0.294379 + 0.955689i \(0.595113\pi\)
\(632\) 3193.33 0.200987
\(633\) 0 0
\(634\) 1604.45 0.100506
\(635\) −31793.9 −1.98693
\(636\) 0 0
\(637\) 4341.95 0.270070
\(638\) −20472.5 −1.27040
\(639\) 0 0
\(640\) −2543.41 −0.157090
\(641\) −9908.84 −0.610571 −0.305285 0.952261i \(-0.598752\pi\)
−0.305285 + 0.952261i \(0.598752\pi\)
\(642\) 0 0
\(643\) −7468.14 −0.458032 −0.229016 0.973423i \(-0.573551\pi\)
−0.229016 + 0.973423i \(0.573551\pi\)
\(644\) 199.628 0.0122150
\(645\) 0 0
\(646\) 5528.30 0.336700
\(647\) 4863.05 0.295496 0.147748 0.989025i \(-0.452798\pi\)
0.147748 + 0.989025i \(0.452798\pi\)
\(648\) 0 0
\(649\) 21424.8 1.29584
\(650\) 47820.6 2.88566
\(651\) 0 0
\(652\) −13193.8 −0.792498
\(653\) 1144.48 0.0685863 0.0342931 0.999412i \(-0.489082\pi\)
0.0342931 + 0.999412i \(0.489082\pi\)
\(654\) 0 0
\(655\) −49091.4 −2.92849
\(656\) 3473.49 0.206733
\(657\) 0 0
\(658\) 5421.91 0.321228
\(659\) −20783.3 −1.22853 −0.614267 0.789098i \(-0.710548\pi\)
−0.614267 + 0.789098i \(0.710548\pi\)
\(660\) 0 0
\(661\) 6071.70 0.357280 0.178640 0.983915i \(-0.442830\pi\)
0.178640 + 0.983915i \(0.442830\pi\)
\(662\) −1907.13 −0.111968
\(663\) 0 0
\(664\) 5065.06 0.296028
\(665\) −5285.53 −0.308217
\(666\) 0 0
\(667\) 1836.40 0.106605
\(668\) −5610.75 −0.324980
\(669\) 0 0
\(670\) 14538.5 0.838318
\(671\) −12063.5 −0.694050
\(672\) 0 0
\(673\) −25493.0 −1.46015 −0.730077 0.683365i \(-0.760516\pi\)
−0.730077 + 0.683365i \(0.760516\pi\)
\(674\) 19318.0 1.10401
\(675\) 0 0
\(676\) 22619.8 1.28697
\(677\) −14911.3 −0.846511 −0.423256 0.906010i \(-0.639113\pi\)
−0.423256 + 0.906010i \(0.639113\pi\)
\(678\) 0 0
\(679\) −4252.14 −0.240327
\(680\) −11563.1 −0.652097
\(681\) 0 0
\(682\) 3863.71 0.216934
\(683\) 7005.90 0.392494 0.196247 0.980555i \(-0.437125\pi\)
0.196247 + 0.980555i \(0.437125\pi\)
\(684\) 0 0
\(685\) 4807.98 0.268180
\(686\) 686.000 0.0381802
\(687\) 0 0
\(688\) 7159.12 0.396714
\(689\) −28541.1 −1.57813
\(690\) 0 0
\(691\) 2772.02 0.152609 0.0763045 0.997085i \(-0.475688\pi\)
0.0763045 + 0.997085i \(0.475688\pi\)
\(692\) −6732.04 −0.369817
\(693\) 0 0
\(694\) 3708.68 0.202852
\(695\) −6115.73 −0.333788
\(696\) 0 0
\(697\) 15791.5 0.858173
\(698\) −5286.16 −0.286654
\(699\) 0 0
\(700\) 7555.35 0.407950
\(701\) −9933.72 −0.535223 −0.267612 0.963527i \(-0.586234\pi\)
−0.267612 + 0.963527i \(0.586234\pi\)
\(702\) 0 0
\(703\) −13044.6 −0.699838
\(704\) −2543.41 −0.136163
\(705\) 0 0
\(706\) −6193.28 −0.330152
\(707\) 7453.72 0.396501
\(708\) 0 0
\(709\) −13314.4 −0.705266 −0.352633 0.935762i \(-0.614714\pi\)
−0.352633 + 0.935762i \(0.614714\pi\)
\(710\) 24456.1 1.29270
\(711\) 0 0
\(712\) 6588.33 0.346781
\(713\) −346.578 −0.0182040
\(714\) 0 0
\(715\) 69973.5 3.65994
\(716\) 5968.32 0.311517
\(717\) 0 0
\(718\) 11150.2 0.579557
\(719\) −17436.0 −0.904386 −0.452193 0.891920i \(-0.649358\pi\)
−0.452193 + 0.891920i \(0.649358\pi\)
\(720\) 0 0
\(721\) 10461.5 0.540371
\(722\) −10830.0 −0.558242
\(723\) 0 0
\(724\) 11782.9 0.604846
\(725\) 69502.4 3.56035
\(726\) 0 0
\(727\) 5380.39 0.274481 0.137240 0.990538i \(-0.456177\pi\)
0.137240 + 0.990538i \(0.456177\pi\)
\(728\) 4962.23 0.252627
\(729\) 0 0
\(730\) −4910.18 −0.248950
\(731\) 32547.5 1.64680
\(732\) 0 0
\(733\) −37200.7 −1.87454 −0.937271 0.348600i \(-0.886657\pi\)
−0.937271 + 0.348600i \(0.886657\pi\)
\(734\) 16800.3 0.844839
\(735\) 0 0
\(736\) 228.146 0.0114261
\(737\) 14538.5 0.726641
\(738\) 0 0
\(739\) 17236.5 0.857988 0.428994 0.903307i \(-0.358868\pi\)
0.428994 + 0.903307i \(0.358868\pi\)
\(740\) 27284.4 1.35540
\(741\) 0 0
\(742\) −4509.30 −0.223102
\(743\) −16450.2 −0.812246 −0.406123 0.913818i \(-0.633120\pi\)
−0.406123 + 0.913818i \(0.633120\pi\)
\(744\) 0 0
\(745\) −25801.1 −1.26883
\(746\) 9225.24 0.452761
\(747\) 0 0
\(748\) −11563.1 −0.565227
\(749\) −12133.1 −0.591900
\(750\) 0 0
\(751\) −29439.6 −1.43045 −0.715223 0.698897i \(-0.753675\pi\)
−0.715223 + 0.698897i \(0.753675\pi\)
\(752\) 6196.46 0.300481
\(753\) 0 0
\(754\) 45648.0 2.20478
\(755\) 28420.2 1.36996
\(756\) 0 0
\(757\) 12924.0 0.620518 0.310259 0.950652i \(-0.399584\pi\)
0.310259 + 0.950652i \(0.399584\pi\)
\(758\) 1338.33 0.0641298
\(759\) 0 0
\(760\) −6040.61 −0.288310
\(761\) −12513.7 −0.596087 −0.298043 0.954552i \(-0.596334\pi\)
−0.298043 + 0.954552i \(0.596334\pi\)
\(762\) 0 0
\(763\) −231.374 −0.0109781
\(764\) −7653.77 −0.362439
\(765\) 0 0
\(766\) −4380.16 −0.206608
\(767\) −47771.5 −2.24893
\(768\) 0 0
\(769\) −12823.8 −0.601351 −0.300675 0.953727i \(-0.597212\pi\)
−0.300675 + 0.953727i \(0.597212\pi\)
\(770\) 11055.3 0.517412
\(771\) 0 0
\(772\) −237.305 −0.0110632
\(773\) −8348.45 −0.388451 −0.194226 0.980957i \(-0.562219\pi\)
−0.194226 + 0.980957i \(0.562219\pi\)
\(774\) 0 0
\(775\) −13117.0 −0.607968
\(776\) −4859.59 −0.224805
\(777\) 0 0
\(778\) 24563.9 1.13195
\(779\) 8249.53 0.379423
\(780\) 0 0
\(781\) 24456.1 1.12050
\(782\) 1037.22 0.0474310
\(783\) 0 0
\(784\) 784.000 0.0357143
\(785\) 72351.7 3.28961
\(786\) 0 0
\(787\) −23593.9 −1.06865 −0.534327 0.845278i \(-0.679435\pi\)
−0.534327 + 0.845278i \(0.679435\pi\)
\(788\) 1151.12 0.0520394
\(789\) 0 0
\(790\) −15863.2 −0.714415
\(791\) −1700.74 −0.0764494
\(792\) 0 0
\(793\) 26898.4 1.20453
\(794\) −22033.8 −0.984825
\(795\) 0 0
\(796\) −16058.0 −0.715028
\(797\) 25753.3 1.14458 0.572289 0.820052i \(-0.306055\pi\)
0.572289 + 0.820052i \(0.306055\pi\)
\(798\) 0 0
\(799\) 28171.0 1.24733
\(800\) 8634.68 0.381603
\(801\) 0 0
\(802\) −4857.46 −0.213869
\(803\) −4910.18 −0.215786
\(804\) 0 0
\(805\) −991.674 −0.0434185
\(806\) −8615.02 −0.376490
\(807\) 0 0
\(808\) 8518.54 0.370892
\(809\) 15768.2 0.685266 0.342633 0.939469i \(-0.388681\pi\)
0.342633 + 0.939469i \(0.388681\pi\)
\(810\) 0 0
\(811\) −18577.0 −0.804351 −0.402175 0.915563i \(-0.631746\pi\)
−0.402175 + 0.915563i \(0.631746\pi\)
\(812\) 7212.09 0.311693
\(813\) 0 0
\(814\) 27284.4 1.17484
\(815\) 65541.6 2.81696
\(816\) 0 0
\(817\) 17002.9 0.728099
\(818\) −16154.0 −0.690479
\(819\) 0 0
\(820\) −17254.9 −0.734839
\(821\) 25406.7 1.08002 0.540012 0.841657i \(-0.318420\pi\)
0.540012 + 0.841657i \(0.318420\pi\)
\(822\) 0 0
\(823\) 43664.6 1.84940 0.924698 0.380702i \(-0.124318\pi\)
0.924698 + 0.380702i \(0.124318\pi\)
\(824\) 11956.0 0.505471
\(825\) 0 0
\(826\) −7547.58 −0.317934
\(827\) −23206.4 −0.975773 −0.487887 0.872907i \(-0.662232\pi\)
−0.487887 + 0.872907i \(0.662232\pi\)
\(828\) 0 0
\(829\) 10247.3 0.429316 0.214658 0.976689i \(-0.431136\pi\)
0.214658 + 0.976689i \(0.431136\pi\)
\(830\) −25161.2 −1.05224
\(831\) 0 0
\(832\) 5671.12 0.236311
\(833\) 3564.30 0.148254
\(834\) 0 0
\(835\) 27872.0 1.15515
\(836\) −6040.61 −0.249903
\(837\) 0 0
\(838\) −21823.1 −0.899600
\(839\) 12322.0 0.507035 0.253518 0.967331i \(-0.418412\pi\)
0.253518 + 0.967331i \(0.418412\pi\)
\(840\) 0 0
\(841\) 41955.7 1.72027
\(842\) −6905.37 −0.282630
\(843\) 0 0
\(844\) −19241.2 −0.784726
\(845\) −112366. −4.57458
\(846\) 0 0
\(847\) 1738.35 0.0705199
\(848\) −5153.49 −0.208693
\(849\) 0 0
\(850\) 39255.9 1.58408
\(851\) −2447.43 −0.0985863
\(852\) 0 0
\(853\) 38665.8 1.55204 0.776020 0.630708i \(-0.217235\pi\)
0.776020 + 0.630708i \(0.217235\pi\)
\(854\) 4249.77 0.170286
\(855\) 0 0
\(856\) −13866.4 −0.553671
\(857\) −6516.94 −0.259760 −0.129880 0.991530i \(-0.541459\pi\)
−0.129880 + 0.991530i \(0.541459\pi\)
\(858\) 0 0
\(859\) 45946.7 1.82501 0.912504 0.409068i \(-0.134146\pi\)
0.912504 + 0.409068i \(0.134146\pi\)
\(860\) −35563.7 −1.41013
\(861\) 0 0
\(862\) 8837.45 0.349193
\(863\) −25825.2 −1.01866 −0.509328 0.860573i \(-0.670106\pi\)
−0.509328 + 0.860573i \(0.670106\pi\)
\(864\) 0 0
\(865\) 33442.1 1.31453
\(866\) −1034.73 −0.0406023
\(867\) 0 0
\(868\) −1361.12 −0.0532250
\(869\) −15863.2 −0.619243
\(870\) 0 0
\(871\) −32417.0 −1.26109
\(872\) −264.427 −0.0102691
\(873\) 0 0
\(874\) 541.848 0.0209706
\(875\) −20145.4 −0.778329
\(876\) 0 0
\(877\) 36784.9 1.41635 0.708174 0.706038i \(-0.249519\pi\)
0.708174 + 0.706038i \(0.249519\pi\)
\(878\) 24291.1 0.933695
\(879\) 0 0
\(880\) 12634.7 0.483994
\(881\) −41594.5 −1.59064 −0.795321 0.606189i \(-0.792698\pi\)
−0.795321 + 0.606189i \(0.792698\pi\)
\(882\) 0 0
\(883\) −22923.9 −0.873670 −0.436835 0.899542i \(-0.643901\pi\)
−0.436835 + 0.899542i \(0.643901\pi\)
\(884\) 25782.6 0.980955
\(885\) 0 0
\(886\) −23629.4 −0.895986
\(887\) 2603.19 0.0985418 0.0492709 0.998785i \(-0.484310\pi\)
0.0492709 + 0.998785i \(0.484310\pi\)
\(888\) 0 0
\(889\) 11200.4 0.422554
\(890\) −32728.2 −1.23264
\(891\) 0 0
\(892\) 13812.5 0.518470
\(893\) 14716.6 0.551481
\(894\) 0 0
\(895\) −29648.2 −1.10730
\(896\) 896.000 0.0334077
\(897\) 0 0
\(898\) 31938.3 1.18685
\(899\) −12521.0 −0.464516
\(900\) 0 0
\(901\) −23429.3 −0.866308
\(902\) −17254.9 −0.636947
\(903\) 0 0
\(904\) −1943.71 −0.0715119
\(905\) −58532.9 −2.14994
\(906\) 0 0
\(907\) 29167.9 1.06781 0.533905 0.845544i \(-0.320724\pi\)
0.533905 + 0.845544i \(0.320724\pi\)
\(908\) −632.506 −0.0231172
\(909\) 0 0
\(910\) −24650.4 −0.897971
\(911\) 9798.14 0.356341 0.178171 0.984000i \(-0.442982\pi\)
0.178171 + 0.984000i \(0.442982\pi\)
\(912\) 0 0
\(913\) −25161.2 −0.912065
\(914\) 25760.4 0.932251
\(915\) 0 0
\(916\) −2449.33 −0.0883495
\(917\) 17294.0 0.622791
\(918\) 0 0
\(919\) −49986.6 −1.79424 −0.897120 0.441786i \(-0.854345\pi\)
−0.897120 + 0.441786i \(0.854345\pi\)
\(920\) −1133.34 −0.0406143
\(921\) 0 0
\(922\) −13529.1 −0.483252
\(923\) −54530.4 −1.94463
\(924\) 0 0
\(925\) −92628.3 −3.29254
\(926\) 2242.12 0.0795687
\(927\) 0 0
\(928\) 8242.39 0.291562
\(929\) 50651.4 1.78883 0.894413 0.447242i \(-0.147594\pi\)
0.894413 + 0.447242i \(0.147594\pi\)
\(930\) 0 0
\(931\) 1862.00 0.0655474
\(932\) 20825.6 0.731936
\(933\) 0 0
\(934\) −31342.8 −1.09804
\(935\) 57441.1 2.00912
\(936\) 0 0
\(937\) 29457.1 1.02702 0.513511 0.858083i \(-0.328344\pi\)
0.513511 + 0.858083i \(0.328344\pi\)
\(938\) −5121.67 −0.178282
\(939\) 0 0
\(940\) −30781.6 −1.06807
\(941\) −12285.0 −0.425590 −0.212795 0.977097i \(-0.568257\pi\)
−0.212795 + 0.977097i \(0.568257\pi\)
\(942\) 0 0
\(943\) 1547.78 0.0534493
\(944\) −8625.80 −0.297400
\(945\) 0 0
\(946\) −35563.7 −1.22228
\(947\) −10735.0 −0.368363 −0.184182 0.982892i \(-0.558963\pi\)
−0.184182 + 0.982892i \(0.558963\pi\)
\(948\) 0 0
\(949\) 10948.4 0.374498
\(950\) 20507.4 0.700365
\(951\) 0 0
\(952\) 4073.49 0.138679
\(953\) 101.551 0.00345178 0.00172589 0.999999i \(-0.499451\pi\)
0.00172589 + 0.999999i \(0.499451\pi\)
\(954\) 0 0
\(955\) 38020.9 1.28830
\(956\) −2591.41 −0.0876698
\(957\) 0 0
\(958\) −22423.4 −0.756228
\(959\) −1693.76 −0.0570329
\(960\) 0 0
\(961\) −27427.9 −0.920679
\(962\) −60836.8 −2.03894
\(963\) 0 0
\(964\) 24566.7 0.820788
\(965\) 1178.84 0.0393245
\(966\) 0 0
\(967\) 5751.74 0.191276 0.0956378 0.995416i \(-0.469511\pi\)
0.0956378 + 0.995416i \(0.469511\pi\)
\(968\) 1986.68 0.0659653
\(969\) 0 0
\(970\) 24140.5 0.799077
\(971\) 1143.79 0.0378024 0.0189012 0.999821i \(-0.493983\pi\)
0.0189012 + 0.999821i \(0.493983\pi\)
\(972\) 0 0
\(973\) 2154.46 0.0709855
\(974\) −10522.0 −0.346146
\(975\) 0 0
\(976\) 4856.88 0.159288
\(977\) −16323.6 −0.534532 −0.267266 0.963623i \(-0.586120\pi\)
−0.267266 + 0.963623i \(0.586120\pi\)
\(978\) 0 0
\(979\) −32728.2 −1.06844
\(980\) −3894.60 −0.126947
\(981\) 0 0
\(982\) 17508.8 0.568970
\(983\) −19016.4 −0.617017 −0.308509 0.951222i \(-0.599830\pi\)
−0.308509 + 0.951222i \(0.599830\pi\)
\(984\) 0 0
\(985\) −5718.32 −0.184975
\(986\) 37472.4 1.21031
\(987\) 0 0
\(988\) 13468.9 0.433708
\(989\) 3190.09 0.102567
\(990\) 0 0
\(991\) −53795.4 −1.72439 −0.862193 0.506580i \(-0.830909\pi\)
−0.862193 + 0.506580i \(0.830909\pi\)
\(992\) −1555.56 −0.0497874
\(993\) 0 0
\(994\) −8615.44 −0.274915
\(995\) 79770.0 2.54159
\(996\) 0 0
\(997\) 29029.7 0.922145 0.461073 0.887362i \(-0.347465\pi\)
0.461073 + 0.887362i \(0.347465\pi\)
\(998\) −37939.5 −1.20336
\(999\) 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 378.4.a.p.1.1 yes 2
3.2 odd 2 378.4.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.4.a.o.1.2 2 3.2 odd 2
378.4.a.p.1.1 yes 2 1.1 even 1 trivial