Properties

Label 49.6.a.d.1.2
Level $49$
Weight $6$
Character 49.1
Self dual yes
Analytic conductor $7.859$
Analytic rank $1$
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] = [49,6,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(7.85880717084\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.54138\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.08276 q^{2} -10.0828 q^{3} +18.1655 q^{4} -79.8276 q^{5} -71.4138 q^{6} -97.9863 q^{8} -141.338 q^{9} +O(q^{10})\) \(q+7.08276 q^{2} -10.0828 q^{3} +18.1655 q^{4} -79.8276 q^{5} -71.4138 q^{6} -97.9863 q^{8} -141.338 q^{9} -565.400 q^{10} +351.904 q^{11} -183.159 q^{12} -291.683 q^{13} +804.883 q^{15} -1275.31 q^{16} -370.075 q^{17} -1001.06 q^{18} +1504.93 q^{19} -1450.11 q^{20} +2492.45 q^{22} -425.711 q^{23} +987.973 q^{24} +3247.45 q^{25} -2065.92 q^{26} +3875.19 q^{27} -7783.93 q^{29} +5700.80 q^{30} -2575.18 q^{31} -5897.16 q^{32} -3548.16 q^{33} -2621.16 q^{34} -2567.48 q^{36} +739.618 q^{37} +10659.0 q^{38} +2940.97 q^{39} +7822.01 q^{40} +7029.84 q^{41} +1835.23 q^{43} +6392.51 q^{44} +11282.7 q^{45} -3015.21 q^{46} -1532.68 q^{47} +12858.7 q^{48} +23000.9 q^{50} +3731.38 q^{51} -5298.57 q^{52} -9537.46 q^{53} +27447.0 q^{54} -28091.6 q^{55} -15173.8 q^{57} -55131.8 q^{58} -29674.1 q^{59} +14621.1 q^{60} -46510.8 q^{61} -18239.4 q^{62} -958.246 q^{64} +23284.3 q^{65} -25130.8 q^{66} +26746.1 q^{67} -6722.61 q^{68} +4292.34 q^{69} -14388.8 q^{71} +13849.2 q^{72} -70095.1 q^{73} +5238.54 q^{74} -32743.3 q^{75} +27337.8 q^{76} +20830.2 q^{78} -27085.8 q^{79} +101805. q^{80} -4727.49 q^{81} +49790.7 q^{82} -79755.4 q^{83} +29542.2 q^{85} +12998.5 q^{86} +78483.6 q^{87} -34481.7 q^{88} +43577.3 q^{89} +79912.5 q^{90} -7733.26 q^{92} +25964.9 q^{93} -10855.6 q^{94} -120135. q^{95} +59459.7 q^{96} +103374. q^{97} -49737.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 8 q^{3} + 12 q^{4} - 38 q^{5} - 82 q^{6} + 96 q^{8} - 380 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 8 q^{3} + 12 q^{4} - 38 q^{5} - 82 q^{6} + 96 q^{8} - 380 q^{9} - 778 q^{10} + 424 q^{11} - 196 q^{12} - 924 q^{13} + 892 q^{15} - 2064 q^{16} - 2346 q^{17} + 212 q^{18} - 360 q^{19} - 1708 q^{20} + 2126 q^{22} - 12 q^{23} + 1392 q^{24} + 1872 q^{25} + 1148 q^{26} + 2872 q^{27} - 7052 q^{29} + 5258 q^{30} + 3548 q^{31} - 8096 q^{32} - 3398 q^{33} + 7422 q^{34} - 1096 q^{36} + 11090 q^{37} + 20138 q^{38} + 1624 q^{39} + 15936 q^{40} + 3500 q^{41} - 12680 q^{43} + 5948 q^{44} + 1300 q^{45} - 5118 q^{46} - 22956 q^{47} + 11216 q^{48} + 29992 q^{50} - 384 q^{51} - 1400 q^{52} + 3042 q^{53} + 32546 q^{54} - 25076 q^{55} - 19058 q^{57} - 58852 q^{58} - 65808 q^{59} + 14084 q^{60} - 42486 q^{61} - 49362 q^{62} + 35456 q^{64} - 3164 q^{65} - 25894 q^{66} + 42312 q^{67} + 5460 q^{68} + 5154 q^{69} - 2208 q^{71} - 32448 q^{72} - 50506 q^{73} - 47370 q^{74} - 35608 q^{75} + 38836 q^{76} + 27524 q^{78} + 9004 q^{79} + 68816 q^{80} + 51178 q^{81} + 67732 q^{82} - 104328 q^{83} - 53106 q^{85} + 86776 q^{86} + 80008 q^{87} - 20496 q^{88} - 26666 q^{89} + 130652 q^{90} - 10284 q^{92} + 38718 q^{93} + 98034 q^{94} - 198140 q^{95} + 54880 q^{96} + 209132 q^{97} - 66944 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.08276 1.25207 0.626034 0.779796i \(-0.284677\pi\)
0.626034 + 0.779796i \(0.284677\pi\)
\(3\) −10.0828 −0.646810 −0.323405 0.946261i \(-0.604828\pi\)
−0.323405 + 0.946261i \(0.604828\pi\)
\(4\) 18.1655 0.567673
\(5\) −79.8276 −1.42800 −0.714000 0.700146i \(-0.753118\pi\)
−0.714000 + 0.700146i \(0.753118\pi\)
\(6\) −71.4138 −0.809849
\(7\) 0 0
\(8\) −97.9863 −0.541303
\(9\) −141.338 −0.581637
\(10\) −565.400 −1.78795
\(11\) 351.904 0.876884 0.438442 0.898760i \(-0.355531\pi\)
0.438442 + 0.898760i \(0.355531\pi\)
\(12\) −183.159 −0.367176
\(13\) −291.683 −0.478688 −0.239344 0.970935i \(-0.576932\pi\)
−0.239344 + 0.970935i \(0.576932\pi\)
\(14\) 0 0
\(15\) 804.883 0.923644
\(16\) −1275.31 −1.24542
\(17\) −370.075 −0.310576 −0.155288 0.987869i \(-0.549631\pi\)
−0.155288 + 0.987869i \(0.549631\pi\)
\(18\) −1001.06 −0.728249
\(19\) 1504.93 0.956381 0.478190 0.878256i \(-0.341293\pi\)
0.478190 + 0.878256i \(0.341293\pi\)
\(20\) −1450.11 −0.810637
\(21\) 0 0
\(22\) 2492.45 1.09792
\(23\) −425.711 −0.167801 −0.0839006 0.996474i \(-0.526738\pi\)
−0.0839006 + 0.996474i \(0.526738\pi\)
\(24\) 987.973 0.350120
\(25\) 3247.45 1.03918
\(26\) −2065.92 −0.599349
\(27\) 3875.19 1.02302
\(28\) 0 0
\(29\) −7783.93 −1.71872 −0.859358 0.511374i \(-0.829137\pi\)
−0.859358 + 0.511374i \(0.829137\pi\)
\(30\) 5700.80 1.15646
\(31\) −2575.18 −0.481285 −0.240643 0.970614i \(-0.577358\pi\)
−0.240643 + 0.970614i \(0.577358\pi\)
\(32\) −5897.16 −1.01805
\(33\) −3548.16 −0.567177
\(34\) −2621.16 −0.388862
\(35\) 0 0
\(36\) −2567.48 −0.330180
\(37\) 739.618 0.0888184 0.0444092 0.999013i \(-0.485859\pi\)
0.0444092 + 0.999013i \(0.485859\pi\)
\(38\) 10659.0 1.19745
\(39\) 2940.97 0.309620
\(40\) 7822.01 0.772981
\(41\) 7029.84 0.653109 0.326554 0.945178i \(-0.394112\pi\)
0.326554 + 0.945178i \(0.394112\pi\)
\(42\) 0 0
\(43\) 1835.23 0.151363 0.0756816 0.997132i \(-0.475887\pi\)
0.0756816 + 0.997132i \(0.475887\pi\)
\(44\) 6392.51 0.497783
\(45\) 11282.7 0.830578
\(46\) −3015.21 −0.210098
\(47\) −1532.68 −0.101206 −0.0506032 0.998719i \(-0.516114\pi\)
−0.0506032 + 0.998719i \(0.516114\pi\)
\(48\) 12858.7 0.805550
\(49\) 0 0
\(50\) 23000.9 1.30113
\(51\) 3731.38 0.200883
\(52\) −5298.57 −0.271738
\(53\) −9537.46 −0.466383 −0.233192 0.972431i \(-0.574917\pi\)
−0.233192 + 0.972431i \(0.574917\pi\)
\(54\) 27447.0 1.28089
\(55\) −28091.6 −1.25219
\(56\) 0 0
\(57\) −15173.8 −0.618596
\(58\) −55131.8 −2.15195
\(59\) −29674.1 −1.10981 −0.554903 0.831915i \(-0.687245\pi\)
−0.554903 + 0.831915i \(0.687245\pi\)
\(60\) 14621.1 0.524327
\(61\) −46510.8 −1.60040 −0.800201 0.599732i \(-0.795274\pi\)
−0.800201 + 0.599732i \(0.795274\pi\)
\(62\) −18239.4 −0.602602
\(63\) 0 0
\(64\) −958.246 −0.0292434
\(65\) 23284.3 0.683566
\(66\) −25130.8 −0.710143
\(67\) 26746.1 0.727902 0.363951 0.931418i \(-0.381428\pi\)
0.363951 + 0.931418i \(0.381428\pi\)
\(68\) −6722.61 −0.176305
\(69\) 4292.34 0.108535
\(70\) 0 0
\(71\) −14388.8 −0.338748 −0.169374 0.985552i \(-0.554175\pi\)
−0.169374 + 0.985552i \(0.554175\pi\)
\(72\) 13849.2 0.314842
\(73\) −70095.1 −1.53950 −0.769752 0.638343i \(-0.779620\pi\)
−0.769752 + 0.638343i \(0.779620\pi\)
\(74\) 5238.54 0.111207
\(75\) −32743.3 −0.672154
\(76\) 27337.8 0.542911
\(77\) 0 0
\(78\) 20830.2 0.387665
\(79\) −27085.8 −0.488285 −0.244143 0.969739i \(-0.578507\pi\)
−0.244143 + 0.969739i \(0.578507\pi\)
\(80\) 101805. 1.77846
\(81\) −4727.49 −0.0800604
\(82\) 49790.7 0.817736
\(83\) −79755.4 −1.27076 −0.635382 0.772198i \(-0.719157\pi\)
−0.635382 + 0.772198i \(0.719157\pi\)
\(84\) 0 0
\(85\) 29542.2 0.443502
\(86\) 12998.5 0.189517
\(87\) 78483.6 1.11168
\(88\) −34481.7 −0.474660
\(89\) 43577.3 0.583157 0.291579 0.956547i \(-0.405820\pi\)
0.291579 + 0.956547i \(0.405820\pi\)
\(90\) 79912.5 1.03994
\(91\) 0 0
\(92\) −7733.26 −0.0952561
\(93\) 25964.9 0.311300
\(94\) −10855.6 −0.126717
\(95\) −120135. −1.36571
\(96\) 59459.7 0.658483
\(97\) 103374. 1.11553 0.557765 0.829999i \(-0.311659\pi\)
0.557765 + 0.829999i \(0.311659\pi\)
\(98\) 0 0
\(99\) −49737.3 −0.510028
\(100\) 58991.6 0.589916
\(101\) 28700.2 0.279951 0.139975 0.990155i \(-0.455298\pi\)
0.139975 + 0.990155i \(0.455298\pi\)
\(102\) 26428.5 0.251520
\(103\) 29227.9 0.271459 0.135730 0.990746i \(-0.456662\pi\)
0.135730 + 0.990746i \(0.456662\pi\)
\(104\) 28580.9 0.259115
\(105\) 0 0
\(106\) −67551.6 −0.583944
\(107\) 87858.4 0.741863 0.370932 0.928660i \(-0.379038\pi\)
0.370932 + 0.928660i \(0.379038\pi\)
\(108\) 70394.8 0.580739
\(109\) 220628. 1.77867 0.889333 0.457260i \(-0.151169\pi\)
0.889333 + 0.457260i \(0.151169\pi\)
\(110\) −198966. −1.56783
\(111\) −7457.39 −0.0574486
\(112\) 0 0
\(113\) 39665.6 0.292225 0.146113 0.989268i \(-0.453324\pi\)
0.146113 + 0.989268i \(0.453324\pi\)
\(114\) −107472. −0.774524
\(115\) 33983.5 0.239620
\(116\) −141399. −0.975668
\(117\) 41225.8 0.278423
\(118\) −210174. −1.38955
\(119\) 0 0
\(120\) −78867.5 −0.499971
\(121\) −37214.9 −0.231075
\(122\) −329425. −2.00381
\(123\) −70880.2 −0.422437
\(124\) −46779.4 −0.273212
\(125\) −9774.87 −0.0559546
\(126\) 0 0
\(127\) 51740.3 0.284655 0.142328 0.989820i \(-0.454541\pi\)
0.142328 + 0.989820i \(0.454541\pi\)
\(128\) 181922. 0.981433
\(129\) −18504.2 −0.0979031
\(130\) 164917. 0.855871
\(131\) −166674. −0.848572 −0.424286 0.905528i \(-0.639475\pi\)
−0.424286 + 0.905528i \(0.639475\pi\)
\(132\) −64454.2 −0.321971
\(133\) 0 0
\(134\) 189436. 0.911382
\(135\) −309347. −1.46087
\(136\) 36262.3 0.168116
\(137\) 28259.4 0.128636 0.0643178 0.997929i \(-0.479513\pi\)
0.0643178 + 0.997929i \(0.479513\pi\)
\(138\) 30401.6 0.135894
\(139\) 336393. 1.47676 0.738380 0.674384i \(-0.235591\pi\)
0.738380 + 0.674384i \(0.235591\pi\)
\(140\) 0 0
\(141\) 15453.7 0.0654612
\(142\) −101912. −0.424136
\(143\) −102644. −0.419753
\(144\) 180250. 0.724383
\(145\) 621373. 2.45433
\(146\) −496467. −1.92756
\(147\) 0 0
\(148\) 13435.5 0.0504198
\(149\) −355381. −1.31138 −0.655691 0.755030i \(-0.727622\pi\)
−0.655691 + 0.755030i \(0.727622\pi\)
\(150\) −231913. −0.841582
\(151\) −358797. −1.28058 −0.640290 0.768133i \(-0.721186\pi\)
−0.640290 + 0.768133i \(0.721186\pi\)
\(152\) −147462. −0.517692
\(153\) 52305.7 0.180643
\(154\) 0 0
\(155\) 205570. 0.687275
\(156\) 53424.2 0.175763
\(157\) 458911. 1.48586 0.742932 0.669367i \(-0.233435\pi\)
0.742932 + 0.669367i \(0.233435\pi\)
\(158\) −191842. −0.611366
\(159\) 96164.0 0.301661
\(160\) 470756. 1.45377
\(161\) 0 0
\(162\) −33483.7 −0.100241
\(163\) 502441. 1.48121 0.740603 0.671943i \(-0.234540\pi\)
0.740603 + 0.671943i \(0.234540\pi\)
\(164\) 127701. 0.370752
\(165\) 283241. 0.809928
\(166\) −564889. −1.59108
\(167\) −676652. −1.87748 −0.938738 0.344632i \(-0.888004\pi\)
−0.938738 + 0.344632i \(0.888004\pi\)
\(168\) 0 0
\(169\) −286214. −0.770858
\(170\) 209241. 0.555295
\(171\) −212703. −0.556267
\(172\) 33338.0 0.0859247
\(173\) −249160. −0.632941 −0.316470 0.948602i \(-0.602498\pi\)
−0.316470 + 0.948602i \(0.602498\pi\)
\(174\) 555880. 1.39190
\(175\) 0 0
\(176\) −448786. −1.09209
\(177\) 299196. 0.717833
\(178\) 308648. 0.730152
\(179\) 139258. 0.324853 0.162427 0.986721i \(-0.448068\pi\)
0.162427 + 0.986721i \(0.448068\pi\)
\(180\) 204956. 0.471497
\(181\) 306246. 0.694823 0.347412 0.937713i \(-0.387061\pi\)
0.347412 + 0.937713i \(0.387061\pi\)
\(182\) 0 0
\(183\) 468957. 1.03516
\(184\) 41713.8 0.0908312
\(185\) −59041.9 −0.126833
\(186\) 183903. 0.389768
\(187\) −130231. −0.272339
\(188\) −27842.0 −0.0574521
\(189\) 0 0
\(190\) −850885. −1.70996
\(191\) 227494. 0.451218 0.225609 0.974218i \(-0.427563\pi\)
0.225609 + 0.974218i \(0.427563\pi\)
\(192\) 9661.77 0.0189149
\(193\) −672374. −1.29933 −0.649663 0.760223i \(-0.725090\pi\)
−0.649663 + 0.760223i \(0.725090\pi\)
\(194\) 732172. 1.39672
\(195\) −234770. −0.442137
\(196\) 0 0
\(197\) −1282.76 −0.00235493 −0.00117747 0.999999i \(-0.500375\pi\)
−0.00117747 + 0.999999i \(0.500375\pi\)
\(198\) −352278. −0.638590
\(199\) 368898. 0.660349 0.330175 0.943920i \(-0.392892\pi\)
0.330175 + 0.943920i \(0.392892\pi\)
\(200\) −318206. −0.562513
\(201\) −269674. −0.470814
\(202\) 203277. 0.350517
\(203\) 0 0
\(204\) 67782.5 0.114036
\(205\) −561175. −0.932640
\(206\) 207014. 0.339885
\(207\) 60169.0 0.0975994
\(208\) 371986. 0.596167
\(209\) 529589. 0.838635
\(210\) 0 0
\(211\) 502168. 0.776503 0.388251 0.921553i \(-0.373079\pi\)
0.388251 + 0.921553i \(0.373079\pi\)
\(212\) −173253. −0.264753
\(213\) 145078. 0.219106
\(214\) 622280. 0.928863
\(215\) −146502. −0.216147
\(216\) −379715. −0.553763
\(217\) 0 0
\(218\) 1.56266e6 2.22701
\(219\) 706753. 0.995765
\(220\) −510299. −0.710834
\(221\) 107945. 0.148669
\(222\) −52818.9 −0.0719295
\(223\) −1.17328e6 −1.57993 −0.789967 0.613149i \(-0.789902\pi\)
−0.789967 + 0.613149i \(0.789902\pi\)
\(224\) 0 0
\(225\) −458988. −0.604428
\(226\) 280942. 0.365886
\(227\) −910159. −1.17234 −0.586168 0.810189i \(-0.699364\pi\)
−0.586168 + 0.810189i \(0.699364\pi\)
\(228\) −275640. −0.351160
\(229\) 521924. 0.657686 0.328843 0.944385i \(-0.393341\pi\)
0.328843 + 0.944385i \(0.393341\pi\)
\(230\) 240697. 0.300020
\(231\) 0 0
\(232\) 762719. 0.930346
\(233\) −1.04279e6 −1.25836 −0.629182 0.777258i \(-0.716610\pi\)
−0.629182 + 0.777258i \(0.716610\pi\)
\(234\) 291993. 0.348604
\(235\) 122350. 0.144523
\(236\) −539045. −0.630006
\(237\) 273100. 0.315828
\(238\) 0 0
\(239\) −1.53447e6 −1.73766 −0.868830 0.495110i \(-0.835128\pi\)
−0.868830 + 0.495110i \(0.835128\pi\)
\(240\) −1.02648e6 −1.15033
\(241\) −1.00758e6 −1.11747 −0.558735 0.829346i \(-0.688713\pi\)
−0.558735 + 0.829346i \(0.688713\pi\)
\(242\) −263584. −0.289322
\(243\) −894004. −0.971234
\(244\) −844893. −0.908505
\(245\) 0 0
\(246\) −502028. −0.528920
\(247\) −438961. −0.457808
\(248\) 252332. 0.260521
\(249\) 804155. 0.821942
\(250\) −69233.1 −0.0700590
\(251\) 8511.89 0.00852789 0.00426394 0.999991i \(-0.498643\pi\)
0.00426394 + 0.999991i \(0.498643\pi\)
\(252\) 0 0
\(253\) −149809. −0.147142
\(254\) 366464. 0.356408
\(255\) −297867. −0.286862
\(256\) 1.31917e6 1.25806
\(257\) −527532. −0.498214 −0.249107 0.968476i \(-0.580137\pi\)
−0.249107 + 0.968476i \(0.580137\pi\)
\(258\) −131061. −0.122581
\(259\) 0 0
\(260\) 422972. 0.388042
\(261\) 1.10016e6 0.999670
\(262\) −1.18051e6 −1.06247
\(263\) −352085. −0.313876 −0.156938 0.987608i \(-0.550162\pi\)
−0.156938 + 0.987608i \(0.550162\pi\)
\(264\) 347671. 0.307014
\(265\) 761353. 0.665996
\(266\) 0 0
\(267\) −439380. −0.377192
\(268\) 485856. 0.413210
\(269\) 479540. 0.404058 0.202029 0.979380i \(-0.435246\pi\)
0.202029 + 0.979380i \(0.435246\pi\)
\(270\) −2.19103e6 −1.82911
\(271\) −977611. −0.808617 −0.404308 0.914623i \(-0.632488\pi\)
−0.404308 + 0.914623i \(0.632488\pi\)
\(272\) 471961. 0.386798
\(273\) 0 0
\(274\) 200155. 0.161061
\(275\) 1.14279e6 0.911243
\(276\) 77972.6 0.0616126
\(277\) 968723. 0.758578 0.379289 0.925278i \(-0.376169\pi\)
0.379289 + 0.925278i \(0.376169\pi\)
\(278\) 2.38259e6 1.84900
\(279\) 363970. 0.279934
\(280\) 0 0
\(281\) −318333. −0.240501 −0.120250 0.992744i \(-0.538370\pi\)
−0.120250 + 0.992744i \(0.538370\pi\)
\(282\) 109455. 0.0819619
\(283\) 1.77210e6 1.31529 0.657646 0.753327i \(-0.271552\pi\)
0.657646 + 0.753327i \(0.271552\pi\)
\(284\) −261379. −0.192298
\(285\) 1.21129e6 0.883356
\(286\) −727004. −0.525559
\(287\) 0 0
\(288\) 833492. 0.592134
\(289\) −1.28290e6 −0.903543
\(290\) 4.40104e6 3.07298
\(291\) −1.04229e6 −0.721535
\(292\) −1.27331e6 −0.873934
\(293\) 1.64148e6 1.11703 0.558516 0.829494i \(-0.311371\pi\)
0.558516 + 0.829494i \(0.311371\pi\)
\(294\) 0 0
\(295\) 2.36881e6 1.58480
\(296\) −72472.4 −0.0480777
\(297\) 1.36369e6 0.897068
\(298\) −2.51708e6 −1.64194
\(299\) 124172. 0.0803243
\(300\) −594799. −0.381563
\(301\) 0 0
\(302\) −2.54128e6 −1.60337
\(303\) −289377. −0.181075
\(304\) −1.91925e6 −1.19110
\(305\) 3.71285e6 2.28537
\(306\) 370469. 0.226177
\(307\) −466930. −0.282752 −0.141376 0.989956i \(-0.545153\pi\)
−0.141376 + 0.989956i \(0.545153\pi\)
\(308\) 0 0
\(309\) −294698. −0.175582
\(310\) 1.45600e6 0.860515
\(311\) −2.43796e6 −1.42931 −0.714654 0.699478i \(-0.753416\pi\)
−0.714654 + 0.699478i \(0.753416\pi\)
\(312\) −288174. −0.167598
\(313\) 2.42094e6 1.39676 0.698381 0.715726i \(-0.253904\pi\)
0.698381 + 0.715726i \(0.253904\pi\)
\(314\) 3.25035e6 1.86040
\(315\) 0 0
\(316\) −492028. −0.277186
\(317\) 1.87611e6 1.04860 0.524301 0.851533i \(-0.324327\pi\)
0.524301 + 0.851533i \(0.324327\pi\)
\(318\) 681107. 0.377700
\(319\) −2.73919e6 −1.50711
\(320\) 76494.5 0.0417595
\(321\) −885855. −0.479844
\(322\) 0 0
\(323\) −556936. −0.297029
\(324\) −85877.3 −0.0454481
\(325\) −947225. −0.497445
\(326\) 3.55867e6 1.85457
\(327\) −2.22454e6 −1.15046
\(328\) −688828. −0.353530
\(329\) 0 0
\(330\) 2.00613e6 1.01408
\(331\) −1.08310e6 −0.543373 −0.271686 0.962386i \(-0.587581\pi\)
−0.271686 + 0.962386i \(0.587581\pi\)
\(332\) −1.44880e6 −0.721378
\(333\) −104536. −0.0516601
\(334\) −4.79257e6 −2.35073
\(335\) −2.13507e6 −1.03944
\(336\) 0 0
\(337\) −2.59465e6 −1.24453 −0.622263 0.782809i \(-0.713786\pi\)
−0.622263 + 0.782809i \(0.713786\pi\)
\(338\) −2.02719e6 −0.965166
\(339\) −399939. −0.189014
\(340\) 536650. 0.251764
\(341\) −906213. −0.422031
\(342\) −1.50652e6 −0.696484
\(343\) 0 0
\(344\) −179828. −0.0819333
\(345\) −342647. −0.154989
\(346\) −1.76474e6 −0.792485
\(347\) −1.87051e6 −0.833943 −0.416972 0.908920i \(-0.636909\pi\)
−0.416972 + 0.908920i \(0.636909\pi\)
\(348\) 1.42569e6 0.631071
\(349\) −1.61685e6 −0.710568 −0.355284 0.934758i \(-0.615616\pi\)
−0.355284 + 0.934758i \(0.615616\pi\)
\(350\) 0 0
\(351\) −1.13033e6 −0.489706
\(352\) −2.07523e6 −0.892709
\(353\) −578305. −0.247013 −0.123507 0.992344i \(-0.539414\pi\)
−0.123507 + 0.992344i \(0.539414\pi\)
\(354\) 2.11914e6 0.898775
\(355\) 1.14862e6 0.483733
\(356\) 791605. 0.331042
\(357\) 0 0
\(358\) 986330. 0.406738
\(359\) −1.96818e6 −0.805988 −0.402994 0.915203i \(-0.632030\pi\)
−0.402994 + 0.915203i \(0.632030\pi\)
\(360\) −1.10555e6 −0.449594
\(361\) −211299. −0.0853355
\(362\) 2.16907e6 0.869965
\(363\) 375229. 0.149462
\(364\) 0 0
\(365\) 5.59553e6 2.19841
\(366\) 3.32151e6 1.29608
\(367\) 2.17452e6 0.842749 0.421375 0.906887i \(-0.361548\pi\)
0.421375 + 0.906887i \(0.361548\pi\)
\(368\) 542913. 0.208983
\(369\) −993583. −0.379873
\(370\) −418180. −0.158803
\(371\) 0 0
\(372\) 471666. 0.176716
\(373\) 1.38476e6 0.515349 0.257675 0.966232i \(-0.417044\pi\)
0.257675 + 0.966232i \(0.417044\pi\)
\(374\) −922394. −0.340987
\(375\) 98557.7 0.0361920
\(376\) 150182. 0.0547833
\(377\) 2.27044e6 0.822728
\(378\) 0 0
\(379\) 3.37190e6 1.20580 0.602902 0.797815i \(-0.294011\pi\)
0.602902 + 0.797815i \(0.294011\pi\)
\(380\) −2.18231e6 −0.775277
\(381\) −521685. −0.184118
\(382\) 1.61128e6 0.564955
\(383\) 3.28060e6 1.14276 0.571382 0.820685i \(-0.306408\pi\)
0.571382 + 0.820685i \(0.306408\pi\)
\(384\) −1.83428e6 −0.634800
\(385\) 0 0
\(386\) −4.76227e6 −1.62684
\(387\) −259388. −0.0880385
\(388\) 1.87784e6 0.633256
\(389\) −2.94810e6 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(390\) −1.66282e6 −0.553585
\(391\) 157545. 0.0521150
\(392\) 0 0
\(393\) 1.68053e6 0.548865
\(394\) −9085.46 −0.00294854
\(395\) 2.16219e6 0.697271
\(396\) −903504. −0.289529
\(397\) −69270.2 −0.0220582 −0.0110291 0.999939i \(-0.503511\pi\)
−0.0110291 + 0.999939i \(0.503511\pi\)
\(398\) 2.61282e6 0.826802
\(399\) 0 0
\(400\) −4.14151e6 −1.29422
\(401\) −3.34786e6 −1.03970 −0.519848 0.854259i \(-0.674012\pi\)
−0.519848 + 0.854259i \(0.674012\pi\)
\(402\) −1.91004e6 −0.589491
\(403\) 751134. 0.230385
\(404\) 521354. 0.158920
\(405\) 377384. 0.114326
\(406\) 0 0
\(407\) 260274. 0.0778834
\(408\) −365624. −0.108739
\(409\) 2.91217e6 0.860812 0.430406 0.902636i \(-0.358370\pi\)
0.430406 + 0.902636i \(0.358370\pi\)
\(410\) −3.97467e6 −1.16773
\(411\) −284933. −0.0832028
\(412\) 530940. 0.154100
\(413\) 0 0
\(414\) 426163. 0.122201
\(415\) 6.36669e6 1.81465
\(416\) 1.72010e6 0.487327
\(417\) −3.39177e6 −0.955183
\(418\) 3.75095e6 1.05003
\(419\) −4.62361e6 −1.28661 −0.643304 0.765611i \(-0.722437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(420\) 0 0
\(421\) −2.63042e6 −0.723303 −0.361652 0.932313i \(-0.617787\pi\)
−0.361652 + 0.932313i \(0.617787\pi\)
\(422\) 3.55674e6 0.972234
\(423\) 216626. 0.0588654
\(424\) 934541. 0.252455
\(425\) −1.20180e6 −0.322746
\(426\) 1.02756e6 0.274335
\(427\) 0 0
\(428\) 1.59599e6 0.421135
\(429\) 1.03494e6 0.271500
\(430\) −1.03764e6 −0.270630
\(431\) 7.54128e6 1.95547 0.977736 0.209837i \(-0.0672932\pi\)
0.977736 + 0.209837i \(0.0672932\pi\)
\(432\) −4.94207e6 −1.27409
\(433\) 5.83558e6 1.49577 0.747883 0.663830i \(-0.231070\pi\)
0.747883 + 0.663830i \(0.231070\pi\)
\(434\) 0 0
\(435\) −6.26516e6 −1.58748
\(436\) 4.00782e6 1.00970
\(437\) −640663. −0.160482
\(438\) 5.00576e6 1.24677
\(439\) 168104. 0.0416310 0.0208155 0.999783i \(-0.493374\pi\)
0.0208155 + 0.999783i \(0.493374\pi\)
\(440\) 2.75259e6 0.677814
\(441\) 0 0
\(442\) 764546. 0.186143
\(443\) −2.84151e6 −0.687924 −0.343962 0.938984i \(-0.611769\pi\)
−0.343962 + 0.938984i \(0.611769\pi\)
\(444\) −135467. −0.0326120
\(445\) −3.47867e6 −0.832748
\(446\) −8.31005e6 −1.97818
\(447\) 3.58323e6 0.848214
\(448\) 0 0
\(449\) −1.41567e6 −0.331396 −0.165698 0.986177i \(-0.552988\pi\)
−0.165698 + 0.986177i \(0.552988\pi\)
\(450\) −3.25090e6 −0.756785
\(451\) 2.47382e6 0.572701
\(452\) 720547. 0.165888
\(453\) 3.61767e6 0.828291
\(454\) −6.44644e6 −1.46784
\(455\) 0 0
\(456\) 1.48682e6 0.334848
\(457\) 1.55727e6 0.348799 0.174399 0.984675i \(-0.444202\pi\)
0.174399 + 0.984675i \(0.444202\pi\)
\(458\) 3.69667e6 0.823468
\(459\) −1.43411e6 −0.317725
\(460\) 617328. 0.136026
\(461\) −4.45345e6 −0.975987 −0.487994 0.872847i \(-0.662271\pi\)
−0.487994 + 0.872847i \(0.662271\pi\)
\(462\) 0 0
\(463\) 4.92263e6 1.06720 0.533599 0.845738i \(-0.320839\pi\)
0.533599 + 0.845738i \(0.320839\pi\)
\(464\) 9.92693e6 2.14052
\(465\) −2.07271e6 −0.444536
\(466\) −7.38582e6 −1.57556
\(467\) 5.09090e6 1.08020 0.540098 0.841602i \(-0.318387\pi\)
0.540098 + 0.841602i \(0.318387\pi\)
\(468\) 748889. 0.158053
\(469\) 0 0
\(470\) 866579. 0.180952
\(471\) −4.62709e6 −0.961071
\(472\) 2.90765e6 0.600741
\(473\) 645825. 0.132728
\(474\) 1.93430e6 0.395437
\(475\) 4.88717e6 0.993856
\(476\) 0 0
\(477\) 1.34800e6 0.271266
\(478\) −1.08683e7 −2.17567
\(479\) −8.30085e6 −1.65304 −0.826521 0.562907i \(-0.809683\pi\)
−0.826521 + 0.562907i \(0.809683\pi\)
\(480\) −4.74652e6 −0.940313
\(481\) −215734. −0.0425163
\(482\) −7.13644e6 −1.39915
\(483\) 0 0
\(484\) −676028. −0.131175
\(485\) −8.25208e6 −1.59298
\(486\) −6.33202e6 −1.21605
\(487\) 8.63401e6 1.64964 0.824822 0.565392i \(-0.191275\pi\)
0.824822 + 0.565392i \(0.191275\pi\)
\(488\) 4.55742e6 0.866303
\(489\) −5.06599e6 −0.958059
\(490\) 0 0
\(491\) 95039.5 0.0177910 0.00889550 0.999960i \(-0.497168\pi\)
0.00889550 + 0.999960i \(0.497168\pi\)
\(492\) −1.28758e6 −0.239806
\(493\) 2.88064e6 0.533792
\(494\) −3.10905e6 −0.573206
\(495\) 3.97041e6 0.728320
\(496\) 3.28415e6 0.599402
\(497\) 0 0
\(498\) 5.69564e6 1.02913
\(499\) 2.14203e6 0.385101 0.192551 0.981287i \(-0.438324\pi\)
0.192551 + 0.981287i \(0.438324\pi\)
\(500\) −177566. −0.0317639
\(501\) 6.82252e6 1.21437
\(502\) 60287.7 0.0106775
\(503\) 5.24794e6 0.924844 0.462422 0.886660i \(-0.346981\pi\)
0.462422 + 0.886660i \(0.346981\pi\)
\(504\) 0 0
\(505\) −2.29107e6 −0.399769
\(506\) −1.06106e6 −0.184232
\(507\) 2.88583e6 0.498598
\(508\) 939889. 0.161591
\(509\) −1.05891e7 −1.81160 −0.905802 0.423702i \(-0.860730\pi\)
−0.905802 + 0.423702i \(0.860730\pi\)
\(510\) −2.10972e6 −0.359170
\(511\) 0 0
\(512\) 3.52190e6 0.593747
\(513\) 5.83187e6 0.978395
\(514\) −3.73639e6 −0.623798
\(515\) −2.33319e6 −0.387644
\(516\) −336139. −0.0555769
\(517\) −539357. −0.0887462
\(518\) 0 0
\(519\) 2.51222e6 0.409392
\(520\) −2.28155e6 −0.370016
\(521\) −4.54465e6 −0.733510 −0.366755 0.930318i \(-0.619531\pi\)
−0.366755 + 0.930318i \(0.619531\pi\)
\(522\) 7.79221e6 1.25165
\(523\) −5.27197e6 −0.842789 −0.421394 0.906877i \(-0.638459\pi\)
−0.421394 + 0.906877i \(0.638459\pi\)
\(524\) −3.02772e6 −0.481711
\(525\) 0 0
\(526\) −2.49373e6 −0.392993
\(527\) 953009. 0.149476
\(528\) 4.52501e6 0.706373
\(529\) −6.25511e6 −0.971843
\(530\) 5.39248e6 0.833871
\(531\) 4.19407e6 0.645504
\(532\) 0 0
\(533\) −2.05048e6 −0.312635
\(534\) −3.11202e6 −0.472269
\(535\) −7.01353e6 −1.05938
\(536\) −2.62075e6 −0.394015
\(537\) −1.40410e6 −0.210118
\(538\) 3.39647e6 0.505908
\(539\) 0 0
\(540\) −5.61945e6 −0.829296
\(541\) 5.93445e6 0.871741 0.435871 0.900009i \(-0.356441\pi\)
0.435871 + 0.900009i \(0.356441\pi\)
\(542\) −6.92418e6 −1.01244
\(543\) −3.08781e6 −0.449418
\(544\) 2.18239e6 0.316181
\(545\) −1.76122e7 −2.53994
\(546\) 0 0
\(547\) −8.82017e6 −1.26040 −0.630200 0.776433i \(-0.717027\pi\)
−0.630200 + 0.776433i \(0.717027\pi\)
\(548\) 513347. 0.0730230
\(549\) 6.57374e6 0.930854
\(550\) 8.09410e6 1.14094
\(551\) −1.17142e7 −1.64375
\(552\) −420590. −0.0587505
\(553\) 0 0
\(554\) 6.86124e6 0.949791
\(555\) 595306. 0.0820366
\(556\) 6.11076e6 0.838317
\(557\) −1.18224e6 −0.161461 −0.0807304 0.996736i \(-0.525725\pi\)
−0.0807304 + 0.996736i \(0.525725\pi\)
\(558\) 2.57791e6 0.350496
\(559\) −535306. −0.0724556
\(560\) 0 0
\(561\) 1.31309e6 0.176151
\(562\) −2.25468e6 −0.301123
\(563\) 4.07741e6 0.542142 0.271071 0.962559i \(-0.412622\pi\)
0.271071 + 0.962559i \(0.412622\pi\)
\(564\) 280724. 0.0371606
\(565\) −3.16641e6 −0.417298
\(566\) 1.25514e7 1.64684
\(567\) 0 0
\(568\) 1.40990e6 0.183366
\(569\) 8.17615e6 1.05869 0.529344 0.848407i \(-0.322438\pi\)
0.529344 + 0.848407i \(0.322438\pi\)
\(570\) 8.57927e6 1.10602
\(571\) 3.30615e6 0.424357 0.212179 0.977231i \(-0.431944\pi\)
0.212179 + 0.977231i \(0.431944\pi\)
\(572\) −1.86458e6 −0.238282
\(573\) −2.29377e6 −0.291852
\(574\) 0 0
\(575\) −1.38247e6 −0.174376
\(576\) 135437. 0.0170090
\(577\) −7.14994e6 −0.894052 −0.447026 0.894521i \(-0.647517\pi\)
−0.447026 + 0.894521i \(0.647517\pi\)
\(578\) −9.08648e6 −1.13130
\(579\) 6.77939e6 0.840416
\(580\) 1.12876e7 1.39325
\(581\) 0 0
\(582\) −7.38232e6 −0.903411
\(583\) −3.35627e6 −0.408964
\(584\) 6.86836e6 0.833338
\(585\) −3.29096e6 −0.397588
\(586\) 1.16262e7 1.39860
\(587\) 9.69191e6 1.16095 0.580476 0.814277i \(-0.302867\pi\)
0.580476 + 0.814277i \(0.302867\pi\)
\(588\) 0 0
\(589\) −3.87545e6 −0.460292
\(590\) 1.67777e7 1.98428
\(591\) 12933.7 0.00152319
\(592\) −943242. −0.110616
\(593\) 6.63960e6 0.775363 0.387682 0.921793i \(-0.373276\pi\)
0.387682 + 0.921793i \(0.373276\pi\)
\(594\) 9.65871e6 1.12319
\(595\) 0 0
\(596\) −6.45569e6 −0.744435
\(597\) −3.71951e6 −0.427120
\(598\) 879484. 0.100571
\(599\) −3.24191e6 −0.369177 −0.184588 0.982816i \(-0.559095\pi\)
−0.184588 + 0.982816i \(0.559095\pi\)
\(600\) 3.20839e6 0.363839
\(601\) −5.65076e6 −0.638147 −0.319074 0.947730i \(-0.603372\pi\)
−0.319074 + 0.947730i \(0.603372\pi\)
\(602\) 0 0
\(603\) −3.78023e6 −0.423375
\(604\) −6.51774e6 −0.726950
\(605\) 2.97078e6 0.329975
\(606\) −2.04959e6 −0.226718
\(607\) 235674. 0.0259621 0.0129811 0.999916i \(-0.495868\pi\)
0.0129811 + 0.999916i \(0.495868\pi\)
\(608\) −8.87478e6 −0.973641
\(609\) 0 0
\(610\) 2.62972e7 2.86144
\(611\) 447057. 0.0484462
\(612\) 950160. 0.102546
\(613\) 788877. 0.0847926 0.0423963 0.999101i \(-0.486501\pi\)
0.0423963 + 0.999101i \(0.486501\pi\)
\(614\) −3.30716e6 −0.354025
\(615\) 5.65820e6 0.603240
\(616\) 0 0
\(617\) 1.67739e7 1.77387 0.886935 0.461894i \(-0.152830\pi\)
0.886935 + 0.461894i \(0.152830\pi\)
\(618\) −2.08728e6 −0.219841
\(619\) 8.22300e6 0.862588 0.431294 0.902211i \(-0.358057\pi\)
0.431294 + 0.902211i \(0.358057\pi\)
\(620\) 3.73429e6 0.390147
\(621\) −1.64971e6 −0.171664
\(622\) −1.72675e7 −1.78959
\(623\) 0 0
\(624\) −3.75065e6 −0.385607
\(625\) −9.36798e6 −0.959281
\(626\) 1.71469e7 1.74884
\(627\) −5.33972e6 −0.542437
\(628\) 8.33635e6 0.843484
\(629\) −273714. −0.0275849
\(630\) 0 0
\(631\) −5.94507e6 −0.594406 −0.297203 0.954814i \(-0.596054\pi\)
−0.297203 + 0.954814i \(0.596054\pi\)
\(632\) 2.65404e6 0.264310
\(633\) −5.06324e6 −0.502249
\(634\) 1.32881e7 1.31292
\(635\) −4.13030e6 −0.406488
\(636\) 1.74687e6 0.171245
\(637\) 0 0
\(638\) −1.94011e7 −1.88701
\(639\) 2.03368e6 0.197029
\(640\) −1.45224e7 −1.40149
\(641\) −1.06761e7 −1.02628 −0.513141 0.858304i \(-0.671518\pi\)
−0.513141 + 0.858304i \(0.671518\pi\)
\(642\) −6.27430e6 −0.600797
\(643\) −3.13159e6 −0.298701 −0.149351 0.988784i \(-0.547718\pi\)
−0.149351 + 0.988784i \(0.547718\pi\)
\(644\) 0 0
\(645\) 1.47715e6 0.139806
\(646\) −3.94464e6 −0.371900
\(647\) −4.93457e6 −0.463435 −0.231717 0.972783i \(-0.574434\pi\)
−0.231717 + 0.972783i \(0.574434\pi\)
\(648\) 463229. 0.0433369
\(649\) −1.04424e7 −0.973170
\(650\) −6.70897e6 −0.622834
\(651\) 0 0
\(652\) 9.12710e6 0.840841
\(653\) 5.72224e6 0.525150 0.262575 0.964912i \(-0.415428\pi\)
0.262575 + 0.964912i \(0.415428\pi\)
\(654\) −1.57559e7 −1.44045
\(655\) 1.33052e7 1.21176
\(656\) −8.96523e6 −0.813395
\(657\) 9.90710e6 0.895433
\(658\) 0 0
\(659\) 362477. 0.0325137 0.0162569 0.999868i \(-0.494825\pi\)
0.0162569 + 0.999868i \(0.494825\pi\)
\(660\) 5.14522e6 0.459774
\(661\) −1.91211e7 −1.70219 −0.851096 0.525011i \(-0.824061\pi\)
−0.851096 + 0.525011i \(0.824061\pi\)
\(662\) −7.67133e6 −0.680339
\(663\) −1.08838e6 −0.0961604
\(664\) 7.81494e6 0.687868
\(665\) 0 0
\(666\) −740404. −0.0646819
\(667\) 3.31370e6 0.288403
\(668\) −1.22917e7 −1.06579
\(669\) 1.18299e7 1.02192
\(670\) −1.51222e7 −1.30145
\(671\) −1.63673e7 −1.40337
\(672\) 0 0
\(673\) −573374. −0.0487978 −0.0243989 0.999702i \(-0.507767\pi\)
−0.0243989 + 0.999702i \(0.507767\pi\)
\(674\) −1.83773e7 −1.55823
\(675\) 1.25845e7 1.06310
\(676\) −5.19923e6 −0.437595
\(677\) 1.16903e7 0.980291 0.490146 0.871641i \(-0.336944\pi\)
0.490146 + 0.871641i \(0.336944\pi\)
\(678\) −2.83267e6 −0.236659
\(679\) 0 0
\(680\) −2.89473e6 −0.240069
\(681\) 9.17691e6 0.758279
\(682\) −6.41849e6 −0.528411
\(683\) 1.83674e7 1.50659 0.753297 0.657681i \(-0.228462\pi\)
0.753297 + 0.657681i \(0.228462\pi\)
\(684\) −3.86386e6 −0.315778
\(685\) −2.25588e6 −0.183692
\(686\) 0 0
\(687\) −5.26244e6 −0.425398
\(688\) −2.34049e6 −0.188511
\(689\) 2.78191e6 0.223252
\(690\) −2.42689e6 −0.194056
\(691\) 2.35611e7 1.87716 0.938579 0.345066i \(-0.112143\pi\)
0.938579 + 0.345066i \(0.112143\pi\)
\(692\) −4.52612e6 −0.359303
\(693\) 0 0
\(694\) −1.32484e7 −1.04415
\(695\) −2.68535e7 −2.10881
\(696\) −7.69031e6 −0.601757
\(697\) −2.60157e6 −0.202840
\(698\) −1.14517e7 −0.889679
\(699\) 1.05142e7 0.813921
\(700\) 0 0
\(701\) 1.32980e7 1.02210 0.511048 0.859552i \(-0.329257\pi\)
0.511048 + 0.859552i \(0.329257\pi\)
\(702\) −8.00582e6 −0.613145
\(703\) 1.11307e6 0.0849442
\(704\) −337210. −0.0256430
\(705\) −1.23363e6 −0.0934786
\(706\) −4.09600e6 −0.309277
\(707\) 0 0
\(708\) 5.43506e6 0.407494
\(709\) −6.85353e6 −0.512034 −0.256017 0.966672i \(-0.582410\pi\)
−0.256017 + 0.966672i \(0.582410\pi\)
\(710\) 8.13540e6 0.605666
\(711\) 3.82825e6 0.284005
\(712\) −4.26998e6 −0.315665
\(713\) 1.09628e6 0.0807602
\(714\) 0 0
\(715\) 8.19384e6 0.599408
\(716\) 2.52969e6 0.184410
\(717\) 1.54717e7 1.12394
\(718\) −1.39402e7 −1.00915
\(719\) 2.65729e7 1.91698 0.958490 0.285127i \(-0.0920357\pi\)
0.958490 + 0.285127i \(0.0920357\pi\)
\(720\) −1.43889e7 −1.03442
\(721\) 0 0
\(722\) −1.49658e6 −0.106846
\(723\) 1.01592e7 0.722791
\(724\) 5.56312e6 0.394432
\(725\) −2.52779e7 −1.78606
\(726\) 2.65766e6 0.187136
\(727\) 2.16991e6 0.152267 0.0761335 0.997098i \(-0.475742\pi\)
0.0761335 + 0.997098i \(0.475742\pi\)
\(728\) 0 0
\(729\) 1.01628e7 0.708264
\(730\) 3.96318e7 2.75256
\(731\) −679174. −0.0470097
\(732\) 8.51886e6 0.587629
\(733\) −1.74653e7 −1.20065 −0.600324 0.799757i \(-0.704962\pi\)
−0.600324 + 0.799757i \(0.704962\pi\)
\(734\) 1.54016e7 1.05518
\(735\) 0 0
\(736\) 2.51048e6 0.170829
\(737\) 9.41203e6 0.638285
\(738\) −7.03731e6 −0.475626
\(739\) −1.36461e7 −0.919172 −0.459586 0.888133i \(-0.652002\pi\)
−0.459586 + 0.888133i \(0.652002\pi\)
\(740\) −1.07253e6 −0.0719994
\(741\) 4.42594e6 0.296114
\(742\) 0 0
\(743\) 1.48965e7 0.989944 0.494972 0.868909i \(-0.335178\pi\)
0.494972 + 0.868909i \(0.335178\pi\)
\(744\) −2.54420e6 −0.168508
\(745\) 2.83692e7 1.87265
\(746\) 9.80791e6 0.645252
\(747\) 1.12725e7 0.739124
\(748\) −2.36571e6 −0.154599
\(749\) 0 0
\(750\) 698061. 0.0453148
\(751\) −2.53463e7 −1.63989 −0.819944 0.572443i \(-0.805996\pi\)
−0.819944 + 0.572443i \(0.805996\pi\)
\(752\) 1.95465e6 0.126044
\(753\) −85823.3 −0.00551592
\(754\) 1.60810e7 1.03011
\(755\) 2.86419e7 1.82867
\(756\) 0 0
\(757\) −2.66725e7 −1.69170 −0.845852 0.533417i \(-0.820908\pi\)
−0.845852 + 0.533417i \(0.820908\pi\)
\(758\) 2.38824e7 1.50975
\(759\) 1.51049e6 0.0951729
\(760\) 1.17715e7 0.739264
\(761\) 579829. 0.0362943 0.0181471 0.999835i \(-0.494223\pi\)
0.0181471 + 0.999835i \(0.494223\pi\)
\(762\) −3.69497e6 −0.230528
\(763\) 0 0
\(764\) 4.13255e6 0.256144
\(765\) −4.17544e6 −0.257958
\(766\) 2.32357e7 1.43082
\(767\) 8.65541e6 0.531250
\(768\) −1.33009e7 −0.813727
\(769\) 1.52438e7 0.929562 0.464781 0.885426i \(-0.346133\pi\)
0.464781 + 0.885426i \(0.346133\pi\)
\(770\) 0 0
\(771\) 5.31898e6 0.322250
\(772\) −1.22140e7 −0.737591
\(773\) −1.94926e7 −1.17333 −0.586665 0.809830i \(-0.699559\pi\)
−0.586665 + 0.809830i \(0.699559\pi\)
\(774\) −1.83718e6 −0.110230
\(775\) −8.36275e6 −0.500144
\(776\) −1.01292e7 −0.603839
\(777\) 0 0
\(778\) −2.08807e7 −1.23679
\(779\) 1.05794e7 0.624621
\(780\) −4.26473e6 −0.250989
\(781\) −5.06345e6 −0.297043
\(782\) 1.11585e6 0.0652515
\(783\) −3.01642e7 −1.75828
\(784\) 0 0
\(785\) −3.66337e7 −2.12181
\(786\) 1.19028e7 0.687215
\(787\) 1.36277e7 0.784305 0.392153 0.919900i \(-0.371730\pi\)
0.392153 + 0.919900i \(0.371730\pi\)
\(788\) −23302.0 −0.00133683
\(789\) 3.54999e6 0.203018
\(790\) 1.53143e7 0.873031
\(791\) 0 0
\(792\) 4.87357e6 0.276080
\(793\) 1.35664e7 0.766093
\(794\) −490624. −0.0276184
\(795\) −7.67654e6 −0.430772
\(796\) 6.70123e6 0.374862
\(797\) 3.62853e6 0.202341 0.101171 0.994869i \(-0.467741\pi\)
0.101171 + 0.994869i \(0.467741\pi\)
\(798\) 0 0
\(799\) 567208. 0.0314323
\(800\) −1.91507e7 −1.05794
\(801\) −6.15913e6 −0.339186
\(802\) −2.37121e7 −1.30177
\(803\) −2.46667e7 −1.34997
\(804\) −4.89877e6 −0.267268
\(805\) 0 0
\(806\) 5.32010e6 0.288458
\(807\) −4.83509e6 −0.261349
\(808\) −2.81223e6 −0.151538
\(809\) 2.98780e7 1.60502 0.802509 0.596640i \(-0.203498\pi\)
0.802509 + 0.596640i \(0.203498\pi\)
\(810\) 2.67292e6 0.143144
\(811\) −1.02643e6 −0.0547995 −0.0273998 0.999625i \(-0.508723\pi\)
−0.0273998 + 0.999625i \(0.508723\pi\)
\(812\) 0 0
\(813\) 9.85702e6 0.523021
\(814\) 1.84346e6 0.0975153
\(815\) −4.01086e7 −2.11516
\(816\) −4.75867e6 −0.250184
\(817\) 2.76189e6 0.144761
\(818\) 2.06262e7 1.07779
\(819\) 0 0
\(820\) −1.01940e7 −0.529434
\(821\) −1.15062e7 −0.595766 −0.297883 0.954602i \(-0.596281\pi\)
−0.297883 + 0.954602i \(0.596281\pi\)
\(822\) −2.01811e6 −0.104175
\(823\) −2.51210e7 −1.29282 −0.646408 0.762992i \(-0.723730\pi\)
−0.646408 + 0.762992i \(0.723730\pi\)
\(824\) −2.86393e6 −0.146942
\(825\) −1.15225e7 −0.589401
\(826\) 0 0
\(827\) −2.14447e6 −0.109032 −0.0545162 0.998513i \(-0.517362\pi\)
−0.0545162 + 0.998513i \(0.517362\pi\)
\(828\) 1.09300e6 0.0554045
\(829\) 818065. 0.0413430 0.0206715 0.999786i \(-0.493420\pi\)
0.0206715 + 0.999786i \(0.493420\pi\)
\(830\) 4.50937e7 2.27207
\(831\) −9.76741e6 −0.490656
\(832\) 279504. 0.0139984
\(833\) 0 0
\(834\) −2.40231e7 −1.19595
\(835\) 5.40155e7 2.68104
\(836\) 9.62025e6 0.476070
\(837\) −9.97929e6 −0.492364
\(838\) −3.27480e7 −1.61092
\(839\) −2.11279e7 −1.03622 −0.518110 0.855314i \(-0.673364\pi\)
−0.518110 + 0.855314i \(0.673364\pi\)
\(840\) 0 0
\(841\) 4.00785e7 1.95398
\(842\) −1.86307e7 −0.905624
\(843\) 3.20968e6 0.155558
\(844\) 9.12215e6 0.440799
\(845\) 2.28478e7 1.10079
\(846\) 1.53431e6 0.0737035
\(847\) 0 0
\(848\) 1.21632e7 0.580844
\(849\) −1.78677e7 −0.850744
\(850\) −8.51207e6 −0.404099
\(851\) −314863. −0.0149038
\(852\) 2.63543e6 0.124380
\(853\) −1.89000e7 −0.889386 −0.444693 0.895683i \(-0.646687\pi\)
−0.444693 + 0.895683i \(0.646687\pi\)
\(854\) 0 0
\(855\) 1.69796e7 0.794349
\(856\) −8.60892e6 −0.401573
\(857\) 3.72286e7 1.73151 0.865753 0.500471i \(-0.166840\pi\)
0.865753 + 0.500471i \(0.166840\pi\)
\(858\) 7.33021e6 0.339937
\(859\) 3.02064e6 0.139674 0.0698371 0.997558i \(-0.477752\pi\)
0.0698371 + 0.997558i \(0.477752\pi\)
\(860\) −2.66129e6 −0.122700
\(861\) 0 0
\(862\) 5.34131e7 2.44838
\(863\) −2.71843e7 −1.24248 −0.621242 0.783619i \(-0.713371\pi\)
−0.621242 + 0.783619i \(0.713371\pi\)
\(864\) −2.28526e7 −1.04148
\(865\) 1.98899e7 0.903839
\(866\) 4.13320e7 1.87280
\(867\) 1.29352e7 0.584420
\(868\) 0 0
\(869\) −9.53158e6 −0.428169
\(870\) −4.43746e7 −1.98763
\(871\) −7.80136e6 −0.348438
\(872\) −2.16185e7 −0.962797
\(873\) −1.46106e7 −0.648834
\(874\) −4.53766e6 −0.200934
\(875\) 0 0
\(876\) 1.28385e7 0.565269
\(877\) 1.73868e7 0.763344 0.381672 0.924298i \(-0.375348\pi\)
0.381672 + 0.924298i \(0.375348\pi\)
\(878\) 1.19064e6 0.0521248
\(879\) −1.65506e7 −0.722507
\(880\) 3.58255e7 1.55950
\(881\) −8.14472e6 −0.353538 −0.176769 0.984252i \(-0.556565\pi\)
−0.176769 + 0.984252i \(0.556565\pi\)
\(882\) 0 0
\(883\) −3.10298e7 −1.33930 −0.669649 0.742678i \(-0.733555\pi\)
−0.669649 + 0.742678i \(0.733555\pi\)
\(884\) 1.96087e6 0.0843953
\(885\) −2.38841e7 −1.02507
\(886\) −2.01258e7 −0.861327
\(887\) −1.47028e7 −0.627465 −0.313733 0.949511i \(-0.601580\pi\)
−0.313733 + 0.949511i \(0.601580\pi\)
\(888\) 730722. 0.0310971
\(889\) 0 0
\(890\) −2.46386e7 −1.04266
\(891\) −1.66362e6 −0.0702037
\(892\) −2.13132e7 −0.896885
\(893\) −2.30657e6 −0.0967918
\(894\) 2.53791e7 1.06202
\(895\) −1.11166e7 −0.463890
\(896\) 0 0
\(897\) −1.25200e6 −0.0519545
\(898\) −1.00269e7 −0.414930
\(899\) 2.00450e7 0.827193
\(900\) −8.33775e6 −0.343117
\(901\) 3.52958e6 0.144848
\(902\) 1.75215e7 0.717060
\(903\) 0 0
\(904\) −3.88669e6 −0.158183
\(905\) −2.44469e7 −0.992208
\(906\) 2.56231e7 1.03708
\(907\) 1.30940e7 0.528512 0.264256 0.964453i \(-0.414874\pi\)
0.264256 + 0.964453i \(0.414874\pi\)
\(908\) −1.65335e7 −0.665504
\(909\) −4.05643e6 −0.162830
\(910\) 0 0
\(911\) 2.68695e7 1.07266 0.536332 0.844007i \(-0.319809\pi\)
0.536332 + 0.844007i \(0.319809\pi\)
\(912\) 1.93513e7 0.770412
\(913\) −2.80662e7 −1.11431
\(914\) 1.10298e7 0.436719
\(915\) −3.74357e7 −1.47820
\(916\) 9.48103e6 0.373351
\(917\) 0 0
\(918\) −1.01575e7 −0.397813
\(919\) 1.27317e6 0.0497277 0.0248638 0.999691i \(-0.492085\pi\)
0.0248638 + 0.999691i \(0.492085\pi\)
\(920\) −3.32991e6 −0.129707
\(921\) 4.70795e6 0.182887
\(922\) −3.15427e7 −1.22200
\(923\) 4.19695e6 0.162155
\(924\) 0 0
\(925\) 2.40187e6 0.0922986
\(926\) 3.48658e7 1.33620
\(927\) −4.13101e6 −0.157891
\(928\) 4.59031e7 1.74973
\(929\) −9.31705e6 −0.354192 −0.177096 0.984194i \(-0.556670\pi\)
−0.177096 + 0.984194i \(0.556670\pi\)
\(930\) −1.46805e7 −0.556589
\(931\) 0 0
\(932\) −1.89428e7 −0.714338
\(933\) 2.45814e7 0.924490
\(934\) 3.60577e7 1.35248
\(935\) 1.03960e7 0.388900
\(936\) −4.03956e6 −0.150711
\(937\) 1.18158e7 0.439657 0.219829 0.975539i \(-0.429450\pi\)
0.219829 + 0.975539i \(0.429450\pi\)
\(938\) 0 0
\(939\) −2.44097e7 −0.903439
\(940\) 2.22256e6 0.0820416
\(941\) −2.53529e7 −0.933371 −0.466685 0.884423i \(-0.654552\pi\)
−0.466685 + 0.884423i \(0.654552\pi\)
\(942\) −3.27725e7 −1.20333
\(943\) −2.99268e6 −0.109592
\(944\) 3.78436e7 1.38217
\(945\) 0 0
\(946\) 4.57422e6 0.166184
\(947\) 2.64941e7 0.960008 0.480004 0.877266i \(-0.340635\pi\)
0.480004 + 0.877266i \(0.340635\pi\)
\(948\) 4.96100e6 0.179287
\(949\) 2.04455e7 0.736941
\(950\) 3.46147e7 1.24437
\(951\) −1.89164e7 −0.678246
\(952\) 0 0
\(953\) −1.88335e7 −0.671735 −0.335868 0.941909i \(-0.609030\pi\)
−0.335868 + 0.941909i \(0.609030\pi\)
\(954\) 9.54760e6 0.339643
\(955\) −1.81603e7 −0.644339
\(956\) −2.78745e7 −0.986423
\(957\) 2.76186e7 0.974816
\(958\) −5.87929e7 −2.06972
\(959\) 0 0
\(960\) −771276. −0.0270105
\(961\) −2.19976e7 −0.768365
\(962\) −1.52799e6 −0.0532332
\(963\) −1.24177e7 −0.431495
\(964\) −1.83032e7 −0.634357
\(965\) 5.36740e7 1.85544
\(966\) 0 0
\(967\) −3.14956e7 −1.08314 −0.541569 0.840656i \(-0.682169\pi\)
−0.541569 + 0.840656i \(0.682169\pi\)
\(968\) 3.64655e6 0.125082
\(969\) 5.61545e6 0.192121
\(970\) −5.84475e7 −1.99451
\(971\) −6.85669e6 −0.233381 −0.116691 0.993168i \(-0.537229\pi\)
−0.116691 + 0.993168i \(0.537229\pi\)
\(972\) −1.62401e7 −0.551343
\(973\) 0 0
\(974\) 6.11527e7 2.06547
\(975\) 9.55064e6 0.321752
\(976\) 5.93157e7 1.99317
\(977\) 2.81471e7 0.943402 0.471701 0.881759i \(-0.343640\pi\)
0.471701 + 0.881759i \(0.343640\pi\)
\(978\) −3.58812e7 −1.19955
\(979\) 1.53350e7 0.511361
\(980\) 0 0
\(981\) −3.11831e7 −1.03454
\(982\) 673142. 0.0222755
\(983\) 2.34916e7 0.775406 0.387703 0.921784i \(-0.373269\pi\)
0.387703 + 0.921784i \(0.373269\pi\)
\(984\) 6.94529e6 0.228666
\(985\) 102399. 0.00336285
\(986\) 2.04029e7 0.668343
\(987\) 0 0
\(988\) −7.97395e6 −0.259885
\(989\) −781278. −0.0253989
\(990\) 2.81215e7 0.911906
\(991\) −2.14412e6 −0.0693530 −0.0346765 0.999399i \(-0.511040\pi\)
−0.0346765 + 0.999399i \(0.511040\pi\)
\(992\) 1.51862e7 0.489971
\(993\) 1.09206e7 0.351459
\(994\) 0 0
\(995\) −2.94483e7 −0.942979
\(996\) 1.46079e7 0.466594
\(997\) 2.50872e7 0.799307 0.399654 0.916666i \(-0.369130\pi\)
0.399654 + 0.916666i \(0.369130\pi\)
\(998\) 1.51715e7 0.482173
\(999\) 2.86616e6 0.0908628
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 49.6.a.d.1.2 2
3.2 odd 2 441.6.a.n.1.1 2
4.3 odd 2 784.6.a.ba.1.2 2
7.2 even 3 7.6.c.a.4.1 yes 4
7.3 odd 6 49.6.c.f.30.1 4
7.4 even 3 7.6.c.a.2.1 4
7.5 odd 6 49.6.c.f.18.1 4
7.6 odd 2 49.6.a.e.1.2 2
21.2 odd 6 63.6.e.d.46.2 4
21.11 odd 6 63.6.e.d.37.2 4
21.20 even 2 441.6.a.m.1.1 2
28.11 odd 6 112.6.i.c.65.1 4
28.23 odd 6 112.6.i.c.81.1 4
28.27 even 2 784.6.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.1 4 7.4 even 3
7.6.c.a.4.1 yes 4 7.2 even 3
49.6.a.d.1.2 2 1.1 even 1 trivial
49.6.a.e.1.2 2 7.6 odd 2
49.6.c.f.18.1 4 7.5 odd 6
49.6.c.f.30.1 4 7.3 odd 6
63.6.e.d.37.2 4 21.11 odd 6
63.6.e.d.46.2 4 21.2 odd 6
112.6.i.c.65.1 4 28.11 odd 6
112.6.i.c.81.1 4 28.23 odd 6
441.6.a.m.1.1 2 21.20 even 2
441.6.a.n.1.1 2 3.2 odd 2
784.6.a.t.1.1 2 28.27 even 2
784.6.a.ba.1.2 2 4.3 odd 2