Properties

Label 441.6.a.m.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(70.7292645375\)
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 1.1
Root \(3.54138\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.08276 q^{2} +18.1655 q^{4} -79.8276 q^{5} +97.9863 q^{8} +O(q^{10})\) \(q-7.08276 q^{2} +18.1655 q^{4} -79.8276 q^{5} +97.9863 q^{8} +565.400 q^{10} -351.904 q^{11} +291.683 q^{13} -1275.31 q^{16} -370.075 q^{17} -1504.93 q^{19} -1450.11 q^{20} +2492.45 q^{22} +425.711 q^{23} +3247.45 q^{25} -2065.92 q^{26} +7783.93 q^{29} +2575.18 q^{31} +5897.16 q^{32} +2621.16 q^{34} +739.618 q^{37} +10659.0 q^{38} -7822.01 q^{40} +7029.84 q^{41} +1835.23 q^{43} -6392.51 q^{44} -3015.21 q^{46} -1532.68 q^{47} -23000.9 q^{50} +5298.57 q^{52} +9537.46 q^{53} +28091.6 q^{55} -55131.8 q^{58} -29674.1 q^{59} +46510.8 q^{61} -18239.4 q^{62} -958.246 q^{64} -23284.3 q^{65} +26746.1 q^{67} -6722.61 q^{68} +14388.8 q^{71} +70095.1 q^{73} -5238.54 q^{74} -27337.8 q^{76} -27085.8 q^{79} +101805. q^{80} -49790.7 q^{82} -79755.4 q^{83} +29542.2 q^{85} -12998.5 q^{86} -34481.7 q^{88} +43577.3 q^{89} +7733.26 q^{92} +10855.6 q^{94} +120135. q^{95} -103374. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8} + 778 q^{10} - 424 q^{11} + 924 q^{13} - 2064 q^{16} - 2346 q^{17} + 360 q^{19} - 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} + 1148 q^{26} + 7052 q^{29} - 3548 q^{31} + 8096 q^{32} - 7422 q^{34} + 11090 q^{37} + 20138 q^{38} - 15936 q^{40} + 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} - 22956 q^{47} - 29992 q^{50} + 1400 q^{52} - 3042 q^{53} + 25076 q^{55} - 58852 q^{58} - 65808 q^{59} + 42486 q^{61} - 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} + 5460 q^{68} + 2208 q^{71} + 50506 q^{73} + 47370 q^{74} - 38836 q^{76} + 9004 q^{79} + 68816 q^{80} - 67732 q^{82} - 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} - 26666 q^{89} + 10284 q^{92} - 98034 q^{94} + 198140 q^{95} - 209132 q^{97}+O(q^{100}) \) 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 \(-0.715323\pi\)
−0.626034 + 0.779796i \(0.715323\pi\)
\(3\) 0 0
\(4\) 18.1655 0.567673
\(5\) −79.8276 −1.42800 −0.714000 0.700146i \(-0.753118\pi\)
−0.714000 + 0.700146i \(0.753118\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 97.9863 0.541303
\(9\) 0 0
\(10\) 565.400 1.78795
\(11\) −351.904 −0.876884 −0.438442 0.898760i \(-0.644469\pi\)
−0.438442 + 0.898760i \(0.644469\pi\)
\(12\) 0 0
\(13\) 291.683 0.478688 0.239344 0.970935i \(-0.423068\pi\)
0.239344 + 0.970935i \(0.423068\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1275.31 −1.24542
\(17\) −370.075 −0.310576 −0.155288 0.987869i \(-0.549631\pi\)
−0.155288 + 0.987869i \(0.549631\pi\)
\(18\) 0 0
\(19\) −1504.93 −0.956381 −0.478190 0.878256i \(-0.658707\pi\)
−0.478190 + 0.878256i \(0.658707\pi\)
\(20\) −1450.11 −0.810637
\(21\) 0 0
\(22\) 2492.45 1.09792
\(23\) 425.711 0.167801 0.0839006 0.996474i \(-0.473262\pi\)
0.0839006 + 0.996474i \(0.473262\pi\)
\(24\) 0 0
\(25\) 3247.45 1.03918
\(26\) −2065.92 −0.599349
\(27\) 0 0
\(28\) 0 0
\(29\) 7783.93 1.71872 0.859358 0.511374i \(-0.170863\pi\)
0.859358 + 0.511374i \(0.170863\pi\)
\(30\) 0 0
\(31\) 2575.18 0.481285 0.240643 0.970614i \(-0.422642\pi\)
0.240643 + 0.970614i \(0.422642\pi\)
\(32\) 5897.16 1.01805
\(33\) 0 0
\(34\) 2621.16 0.388862
\(35\) 0 0
\(36\) 0 0
\(37\) 739.618 0.0888184 0.0444092 0.999013i \(-0.485859\pi\)
0.0444092 + 0.999013i \(0.485859\pi\)
\(38\) 10659.0 1.19745
\(39\) 0 0
\(40\) −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
\(45\) 0 0
\(46\) −3015.21 −0.210098
\(47\) −1532.68 −0.101206 −0.0506032 0.998719i \(-0.516114\pi\)
−0.0506032 + 0.998719i \(0.516114\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −23000.9 −1.30113
\(51\) 0 0
\(52\) 5298.57 0.271738
\(53\) 9537.46 0.466383 0.233192 0.972431i \(-0.425083\pi\)
0.233192 + 0.972431i \(0.425083\pi\)
\(54\) 0 0
\(55\) 28091.6 1.25219
\(56\) 0 0
\(57\) 0 0
\(58\) −55131.8 −2.15195
\(59\) −29674.1 −1.10981 −0.554903 0.831915i \(-0.687245\pi\)
−0.554903 + 0.831915i \(0.687245\pi\)
\(60\) 0 0
\(61\) 46510.8 1.60040 0.800201 0.599732i \(-0.204726\pi\)
0.800201 + 0.599732i \(0.204726\pi\)
\(62\) −18239.4 −0.602602
\(63\) 0 0
\(64\) −958.246 −0.0292434
\(65\) −23284.3 −0.683566
\(66\) 0 0
\(67\) 26746.1 0.727902 0.363951 0.931418i \(-0.381428\pi\)
0.363951 + 0.931418i \(0.381428\pi\)
\(68\) −6722.61 −0.176305
\(69\) 0 0
\(70\) 0 0
\(71\) 14388.8 0.338748 0.169374 0.985552i \(-0.445825\pi\)
0.169374 + 0.985552i \(0.445825\pi\)
\(72\) 0 0
\(73\) 70095.1 1.53950 0.769752 0.638343i \(-0.220380\pi\)
0.769752 + 0.638343i \(0.220380\pi\)
\(74\) −5238.54 −0.111207
\(75\) 0 0
\(76\) −27337.8 −0.542911
\(77\) 0 0
\(78\) 0 0
\(79\) −27085.8 −0.488285 −0.244143 0.969739i \(-0.578507\pi\)
−0.244143 + 0.969739i \(0.578507\pi\)
\(80\) 101805. 1.77846
\(81\) 0 0
\(82\) −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
\(87\) 0 0
\(88\) −34481.7 −0.474660
\(89\) 43577.3 0.583157 0.291579 0.956547i \(-0.405820\pi\)
0.291579 + 0.956547i \(0.405820\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7733.26 0.0952561
\(93\) 0 0
\(94\) 10855.6 0.126717
\(95\) 120135. 1.36571
\(96\) 0 0
\(97\) −103374. −1.11553 −0.557765 0.829999i \(-0.688341\pi\)
−0.557765 + 0.829999i \(0.688341\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 58991.6 0.589916
\(101\) 28700.2 0.279951 0.139975 0.990155i \(-0.455298\pi\)
0.139975 + 0.990155i \(0.455298\pi\)
\(102\) 0 0
\(103\) −29227.9 −0.271459 −0.135730 0.990746i \(-0.543338\pi\)
−0.135730 + 0.990746i \(0.543338\pi\)
\(104\) 28580.9 0.259115
\(105\) 0 0
\(106\) −67551.6 −0.583944
\(107\) −87858.4 −0.741863 −0.370932 0.928660i \(-0.620962\pi\)
−0.370932 + 0.928660i \(0.620962\pi\)
\(108\) 0 0
\(109\) 220628. 1.77867 0.889333 0.457260i \(-0.151169\pi\)
0.889333 + 0.457260i \(0.151169\pi\)
\(110\) −198966. −1.56783
\(111\) 0 0
\(112\) 0 0
\(113\) −39665.6 −0.292225 −0.146113 0.989268i \(-0.546676\pi\)
−0.146113 + 0.989268i \(0.546676\pi\)
\(114\) 0 0
\(115\) −33983.5 −0.239620
\(116\) 141399. 0.975668
\(117\) 0 0
\(118\) 210174. 1.38955
\(119\) 0 0
\(120\) 0 0
\(121\) −37214.9 −0.231075
\(122\) −329425. −2.00381
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 164917. 0.855871
\(131\) −166674. −0.848572 −0.424286 0.905528i \(-0.639475\pi\)
−0.424286 + 0.905528i \(0.639475\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −189436. −0.911382
\(135\) 0 0
\(136\) −36262.3 −0.168116
\(137\) −28259.4 −0.128636 −0.0643178 0.997929i \(-0.520487\pi\)
−0.0643178 + 0.997929i \(0.520487\pi\)
\(138\) 0 0
\(139\) −336393. −1.47676 −0.738380 0.674384i \(-0.764409\pi\)
−0.738380 + 0.674384i \(0.764409\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −101912. −0.424136
\(143\) −102644. −0.419753
\(144\) 0 0
\(145\) −621373. −2.45433
\(146\) −496467. −1.92756
\(147\) 0 0
\(148\) 13435.5 0.0504198
\(149\) 355381. 1.31138 0.655691 0.755030i \(-0.272378\pi\)
0.655691 + 0.755030i \(0.272378\pi\)
\(150\) 0 0
\(151\) −358797. −1.28058 −0.640290 0.768133i \(-0.721186\pi\)
−0.640290 + 0.768133i \(0.721186\pi\)
\(152\) −147462. −0.517692
\(153\) 0 0
\(154\) 0 0
\(155\) −205570. −0.687275
\(156\) 0 0
\(157\) −458911. −1.48586 −0.742932 0.669367i \(-0.766565\pi\)
−0.742932 + 0.669367i \(0.766565\pi\)
\(158\) 191842. 0.611366
\(159\) 0 0
\(160\) −470756. −1.45377
\(161\) 0 0
\(162\) 0 0
\(163\) 502441. 1.48121 0.740603 0.671943i \(-0.234540\pi\)
0.740603 + 0.671943i \(0.234540\pi\)
\(164\) 127701. 0.370752
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 33338.0 0.0859247
\(173\) −249160. −0.632941 −0.316470 0.948602i \(-0.602498\pi\)
−0.316470 + 0.948602i \(0.602498\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 448786. 1.09209
\(177\) 0 0
\(178\) −308648. −0.730152
\(179\) −139258. −0.324853 −0.162427 0.986721i \(-0.551932\pi\)
−0.162427 + 0.986721i \(0.551932\pi\)
\(180\) 0 0
\(181\) −306246. −0.694823 −0.347412 0.937713i \(-0.612939\pi\)
−0.347412 + 0.937713i \(0.612939\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 41713.8 0.0908312
\(185\) −59041.9 −0.126833
\(186\) 0 0
\(187\) 130231. 0.272339
\(188\) −27842.0 −0.0574521
\(189\) 0 0
\(190\) −850885. −1.70996
\(191\) −227494. −0.451218 −0.225609 0.974218i \(-0.572437\pi\)
−0.225609 + 0.974218i \(0.572437\pi\)
\(192\) 0 0
\(193\) −672374. −1.29933 −0.649663 0.760223i \(-0.725090\pi\)
−0.649663 + 0.760223i \(0.725090\pi\)
\(194\) 732172. 1.39672
\(195\) 0 0
\(196\) 0 0
\(197\) 1282.76 0.00235493 0.00117747 0.999999i \(-0.499625\pi\)
0.00117747 + 0.999999i \(0.499625\pi\)
\(198\) 0 0
\(199\) −368898. −0.660349 −0.330175 0.943920i \(-0.607108\pi\)
−0.330175 + 0.943920i \(0.607108\pi\)
\(200\) 318206. 0.562513
\(201\) 0 0
\(202\) −203277. −0.350517
\(203\) 0 0
\(204\) 0 0
\(205\) −561175. −0.932640
\(206\) 207014. 0.339885
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 622280. 0.928863
\(215\) −146502. −0.216147
\(216\) 0 0
\(217\) 0 0
\(218\) −1.56266e6 −2.22701
\(219\) 0 0
\(220\) 510299. 0.710834
\(221\) −107945. −0.148669
\(222\) 0 0
\(223\) 1.17328e6 1.57993 0.789967 0.613149i \(-0.210098\pi\)
0.789967 + 0.613149i \(0.210098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 280942. 0.365886
\(227\) −910159. −1.17234 −0.586168 0.810189i \(-0.699364\pi\)
−0.586168 + 0.810189i \(0.699364\pi\)
\(228\) 0 0
\(229\) −521924. −0.657686 −0.328843 0.944385i \(-0.606659\pi\)
−0.328843 + 0.944385i \(0.606659\pi\)
\(230\) 240697. 0.300020
\(231\) 0 0
\(232\) 762719. 0.930346
\(233\) 1.04279e6 1.25836 0.629182 0.777258i \(-0.283390\pi\)
0.629182 + 0.777258i \(0.283390\pi\)
\(234\) 0 0
\(235\) 122350. 0.144523
\(236\) −539045. −0.630006
\(237\) 0 0
\(238\) 0 0
\(239\) 1.53447e6 1.73766 0.868830 0.495110i \(-0.164872\pi\)
0.868830 + 0.495110i \(0.164872\pi\)
\(240\) 0 0
\(241\) 1.00758e6 1.11747 0.558735 0.829346i \(-0.311287\pi\)
0.558735 + 0.829346i \(0.311287\pi\)
\(242\) 263584. 0.289322
\(243\) 0 0
\(244\) 844893. 0.908505
\(245\) 0 0
\(246\) 0 0
\(247\) −438961. −0.457808
\(248\) 252332. 0.260521
\(249\) 0 0
\(250\) 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
\(255\) 0 0
\(256\) 1.31917e6 1.25806
\(257\) −527532. −0.498214 −0.249107 0.968476i \(-0.580137\pi\)
−0.249107 + 0.968476i \(0.580137\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −422972. −0.388042
\(261\) 0 0
\(262\) 1.18051e6 1.06247
\(263\) 352085. 0.313876 0.156938 0.987608i \(-0.449838\pi\)
0.156938 + 0.987608i \(0.449838\pi\)
\(264\) 0 0
\(265\) −761353. −0.665996
\(266\) 0 0
\(267\) 0 0
\(268\) 485856. 0.413210
\(269\) 479540. 0.404058 0.202029 0.979380i \(-0.435246\pi\)
0.202029 + 0.979380i \(0.435246\pi\)
\(270\) 0 0
\(271\) 977611. 0.808617 0.404308 0.914623i \(-0.367512\pi\)
0.404308 + 0.914623i \(0.367512\pi\)
\(272\) 471961. 0.386798
\(273\) 0 0
\(274\) 200155. 0.161061
\(275\) −1.14279e6 −0.911243
\(276\) 0 0
\(277\) 968723. 0.758578 0.379289 0.925278i \(-0.376169\pi\)
0.379289 + 0.925278i \(0.376169\pi\)
\(278\) 2.38259e6 1.84900
\(279\) 0 0
\(280\) 0 0
\(281\) 318333. 0.240501 0.120250 0.992744i \(-0.461630\pi\)
0.120250 + 0.992744i \(0.461630\pi\)
\(282\) 0 0
\(283\) −1.77210e6 −1.31529 −0.657646 0.753327i \(-0.728448\pi\)
−0.657646 + 0.753327i \(0.728448\pi\)
\(284\) 261379. 0.192298
\(285\) 0 0
\(286\) 727004. 0.525559
\(287\) 0 0
\(288\) 0 0
\(289\) −1.28290e6 −0.903543
\(290\) 4.40104e6 3.07298
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) −2.51708e6 −1.64194
\(299\) 124172. 0.0803243
\(300\) 0 0
\(301\) 0 0
\(302\) 2.54128e6 1.60337
\(303\) 0 0
\(304\) 1.91925e6 1.19110
\(305\) −3.71285e6 −2.28537
\(306\) 0 0
\(307\) 466930. 0.282752 0.141376 0.989956i \(-0.454847\pi\)
0.141376 + 0.989956i \(0.454847\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.45600e6 0.860515
\(311\) −2.43796e6 −1.42931 −0.714654 0.699478i \(-0.753416\pi\)
−0.714654 + 0.699478i \(0.753416\pi\)
\(312\) 0 0
\(313\) −2.42094e6 −1.39676 −0.698381 0.715726i \(-0.746096\pi\)
−0.698381 + 0.715726i \(0.746096\pi\)
\(314\) 3.25035e6 1.86040
\(315\) 0 0
\(316\) −492028. −0.277186
\(317\) −1.87611e6 −1.04860 −0.524301 0.851533i \(-0.675673\pi\)
−0.524301 + 0.851533i \(0.675673\pi\)
\(318\) 0 0
\(319\) −2.73919e6 −1.50711
\(320\) 76494.5 0.0417595
\(321\) 0 0
\(322\) 0 0
\(323\) 556936. 0.297029
\(324\) 0 0
\(325\) 947225. 0.497445
\(326\) −3.55867e6 −1.85457
\(327\) 0 0
\(328\) 688828. 0.353530
\(329\) 0 0
\(330\) 0 0
\(331\) −1.08310e6 −0.543373 −0.271686 0.962386i \(-0.587581\pi\)
−0.271686 + 0.962386i \(0.587581\pi\)
\(332\) −1.44880e6 −0.721378
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 536650. 0.251764
\(341\) −906213. −0.422031
\(342\) 0 0
\(343\) 0 0
\(344\) 179828. 0.0819333
\(345\) 0 0
\(346\) 1.76474e6 0.792485
\(347\) 1.87051e6 0.833943 0.416972 0.908920i \(-0.363091\pi\)
0.416972 + 0.908920i \(0.363091\pi\)
\(348\) 0 0
\(349\) 1.61685e6 0.710568 0.355284 0.934758i \(-0.384384\pi\)
0.355284 + 0.934758i \(0.384384\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.07523e6 −0.892709
\(353\) −578305. −0.247013 −0.123507 0.992344i \(-0.539414\pi\)
−0.123507 + 0.992344i \(0.539414\pi\)
\(354\) 0 0
\(355\) −1.14862e6 −0.483733
\(356\) 791605. 0.331042
\(357\) 0 0
\(358\) 986330. 0.406738
\(359\) 1.96818e6 0.805988 0.402994 0.915203i \(-0.367970\pi\)
0.402994 + 0.915203i \(0.367970\pi\)
\(360\) 0 0
\(361\) −211299. −0.0853355
\(362\) 2.16907e6 0.869965
\(363\) 0 0
\(364\) 0 0
\(365\) −5.59553e6 −2.19841
\(366\) 0 0
\(367\) −2.17452e6 −0.842749 −0.421375 0.906887i \(-0.638452\pi\)
−0.421375 + 0.906887i \(0.638452\pi\)
\(368\) −542913. −0.208983
\(369\) 0 0
\(370\) 418180. 0.158803
\(371\) 0 0
\(372\) 0 0
\(373\) 1.38476e6 0.515349 0.257675 0.966232i \(-0.417044\pi\)
0.257675 + 0.966232i \(0.417044\pi\)
\(374\) −922394. −0.340987
\(375\) 0 0
\(376\) −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
\(381\) 0 0
\(382\) 1.61128e6 0.564955
\(383\) 3.28060e6 1.14276 0.571382 0.820685i \(-0.306408\pi\)
0.571382 + 0.820685i \(0.306408\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.76227e6 1.62684
\(387\) 0 0
\(388\) −1.87784e6 −0.633256
\(389\) 2.94810e6 0.987797 0.493899 0.869520i \(-0.335571\pi\)
0.493899 + 0.869520i \(0.335571\pi\)
\(390\) 0 0
\(391\) −157545. −0.0521150
\(392\) 0 0
\(393\) 0 0
\(394\) −9085.46 −0.00294854
\(395\) 2.16219e6 0.697271
\(396\) 0 0
\(397\) 69270.2 0.0220582 0.0110291 0.999939i \(-0.496489\pi\)
0.0110291 + 0.999939i \(0.496489\pi\)
\(398\) 2.61282e6 0.826802
\(399\) 0 0
\(400\) −4.14151e6 −1.29422
\(401\) 3.34786e6 1.03970 0.519848 0.854259i \(-0.325988\pi\)
0.519848 + 0.854259i \(0.325988\pi\)
\(402\) 0 0
\(403\) 751134. 0.230385
\(404\) 521354. 0.158920
\(405\) 0 0
\(406\) 0 0
\(407\) −260274. −0.0778834
\(408\) 0 0
\(409\) −2.91217e6 −0.860812 −0.430406 0.902636i \(-0.641630\pi\)
−0.430406 + 0.902636i \(0.641630\pi\)
\(410\) 3.97467e6 1.16773
\(411\) 0 0
\(412\) −530940. −0.154100
\(413\) 0 0
\(414\) 0 0
\(415\) 6.36669e6 1.81465
\(416\) 1.72010e6 0.487327
\(417\) 0 0
\(418\) −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
\(423\) 0 0
\(424\) 934541. 0.252455
\(425\) −1.20180e6 −0.322746
\(426\) 0 0
\(427\) 0 0
\(428\) −1.59599e6 −0.421135
\(429\) 0 0
\(430\) 1.03764e6 0.270630
\(431\) −7.54128e6 −1.95547 −0.977736 0.209837i \(-0.932707\pi\)
−0.977736 + 0.209837i \(0.932707\pi\)
\(432\) 0 0
\(433\) −5.83558e6 −1.49577 −0.747883 0.663830i \(-0.768930\pi\)
−0.747883 + 0.663830i \(0.768930\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00782e6 1.00970
\(437\) −640663. −0.160482
\(438\) 0 0
\(439\) −168104. −0.0416310 −0.0208155 0.999783i \(-0.506626\pi\)
−0.0208155 + 0.999783i \(0.506626\pi\)
\(440\) 2.75259e6 0.677814
\(441\) 0 0
\(442\) 764546. 0.186143
\(443\) 2.84151e6 0.687924 0.343962 0.938984i \(-0.388231\pi\)
0.343962 + 0.938984i \(0.388231\pi\)
\(444\) 0 0
\(445\) −3.47867e6 −0.832748
\(446\) −8.31005e6 −1.97818
\(447\) 0 0
\(448\) 0 0
\(449\) 1.41567e6 0.331396 0.165698 0.986177i \(-0.447012\pi\)
0.165698 + 0.986177i \(0.447012\pi\)
\(450\) 0 0
\(451\) −2.47382e6 −0.572701
\(452\) −720547. −0.165888
\(453\) 0 0
\(454\) 6.44644e6 1.46784
\(455\) 0 0
\(456\) 0 0
\(457\) 1.55727e6 0.348799 0.174399 0.984675i \(-0.444202\pi\)
0.174399 + 0.984675i \(0.444202\pi\)
\(458\) 3.69667e6 0.823468
\(459\) 0 0
\(460\) −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
\(465\) 0 0
\(466\) −7.38582e6 −1.57556
\(467\) 5.09090e6 1.08020 0.540098 0.841602i \(-0.318387\pi\)
0.540098 + 0.841602i \(0.318387\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −866579. −0.180952
\(471\) 0 0
\(472\) −2.90765e6 −0.600741
\(473\) −645825. −0.132728
\(474\) 0 0
\(475\) −4.88717e6 −0.993856
\(476\) 0 0
\(477\) 0 0
\(478\) −1.08683e7 −2.17567
\(479\) −8.30085e6 −1.65304 −0.826521 0.562907i \(-0.809683\pi\)
−0.826521 + 0.562907i \(0.809683\pi\)
\(480\) 0 0
\(481\) 215734. 0.0425163
\(482\) −7.13644e6 −1.39915
\(483\) 0 0
\(484\) −676028. −0.131175
\(485\) 8.25208e6 1.59298
\(486\) 0 0
\(487\) 8.63401e6 1.64964 0.824822 0.565392i \(-0.191275\pi\)
0.824822 + 0.565392i \(0.191275\pi\)
\(488\) 4.55742e6 0.866303
\(489\) 0 0
\(490\) 0 0
\(491\) −95039.5 −0.0177910 −0.00889550 0.999960i \(-0.502832\pi\)
−0.00889550 + 0.999960i \(0.502832\pi\)
\(492\) 0 0
\(493\) −2.88064e6 −0.533792
\(494\) 3.10905e6 0.573206
\(495\) 0 0
\(496\) −3.28415e6 −0.599402
\(497\) 0 0
\(498\) 0 0
\(499\) 2.14203e6 0.385101 0.192551 0.981287i \(-0.438324\pi\)
0.192551 + 0.981287i \(0.438324\pi\)
\(500\) −177566. −0.0317639
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) 939889. 0.161591
\(509\) −1.05891e7 −1.81160 −0.905802 0.423702i \(-0.860730\pi\)
−0.905802 + 0.423702i \(0.860730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3.52190e6 −0.593747
\(513\) 0 0
\(514\) 3.73639e6 0.623798
\(515\) 2.33319e6 0.387644
\(516\) 0 0
\(517\) 539357. 0.0887462
\(518\) 0 0
\(519\) 0 0
\(520\) −2.28155e6 −0.370016
\(521\) −4.54465e6 −0.733510 −0.366755 0.930318i \(-0.619531\pi\)
−0.366755 + 0.930318i \(0.619531\pi\)
\(522\) 0 0
\(523\) 5.27197e6 0.842789 0.421394 0.906877i \(-0.361541\pi\)
0.421394 + 0.906877i \(0.361541\pi\)
\(524\) −3.02772e6 −0.481711
\(525\) 0 0
\(526\) −2.49373e6 −0.392993
\(527\) −953009. −0.149476
\(528\) 0 0
\(529\) −6.25511e6 −0.971843
\(530\) 5.39248e6 0.833871
\(531\) 0 0
\(532\) 0 0
\(533\) 2.05048e6 0.312635
\(534\) 0 0
\(535\) 7.01353e6 1.05938
\(536\) 2.62075e6 0.394015
\(537\) 0 0
\(538\) −3.39647e6 −0.505908
\(539\) 0 0
\(540\) 0 0
\(541\) 5.93445e6 0.871741 0.435871 0.900009i \(-0.356441\pi\)
0.435871 + 0.900009i \(0.356441\pi\)
\(542\) −6.92418e6 −1.01244
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 8.09410e6 1.14094
\(551\) −1.17142e7 −1.64375
\(552\) 0 0
\(553\) 0 0
\(554\) −6.86124e6 −0.949791
\(555\) 0 0
\(556\) −6.11076e6 −0.838317
\(557\) 1.18224e6 0.161461 0.0807304 0.996736i \(-0.474275\pi\)
0.0807304 + 0.996736i \(0.474275\pi\)
\(558\) 0 0
\(559\) 535306. 0.0724556
\(560\) 0 0
\(561\) 0 0
\(562\) −2.25468e6 −0.301123
\(563\) 4.07741e6 0.542142 0.271071 0.962559i \(-0.412622\pi\)
0.271071 + 0.962559i \(0.412622\pi\)
\(564\) 0 0
\(565\) 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 \(-0.677562\pi\)
−0.529344 + 0.848407i \(0.677562\pi\)
\(570\) 0 0
\(571\) 3.30615e6 0.424357 0.212179 0.977231i \(-0.431944\pi\)
0.212179 + 0.977231i \(0.431944\pi\)
\(572\) −1.86458e6 −0.238282
\(573\) 0 0
\(574\) 0 0
\(575\) 1.38247e6 0.174376
\(576\) 0 0
\(577\) 7.14994e6 0.894052 0.447026 0.894521i \(-0.352483\pi\)
0.447026 + 0.894521i \(0.352483\pi\)
\(578\) 9.08648e6 1.13130
\(579\) 0 0
\(580\) −1.12876e7 −1.39325
\(581\) 0 0
\(582\) 0 0
\(583\) −3.35627e6 −0.408964
\(584\) 6.86836e6 0.833338
\(585\) 0 0
\(586\) −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
\(591\) 0 0
\(592\) −943242. −0.110616
\(593\) 6.63960e6 0.775363 0.387682 0.921793i \(-0.373276\pi\)
0.387682 + 0.921793i \(0.373276\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.45569e6 0.744435
\(597\) 0 0
\(598\) −879484. −0.100571
\(599\) 3.24191e6 0.369177 0.184588 0.982816i \(-0.440905\pi\)
0.184588 + 0.982816i \(0.440905\pi\)
\(600\) 0 0
\(601\) 5.65076e6 0.638147 0.319074 0.947730i \(-0.396628\pi\)
0.319074 + 0.947730i \(0.396628\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −6.51774e6 −0.726950
\(605\) 2.97078e6 0.329975
\(606\) 0 0
\(607\) −235674. −0.0259621 −0.0129811 0.999916i \(-0.504132\pi\)
−0.0129811 + 0.999916i \(0.504132\pi\)
\(608\) −8.87478e6 −0.973641
\(609\) 0 0
\(610\) 2.62972e7 2.86144
\(611\) −447057. −0.0484462
\(612\) 0 0
\(613\) 788877. 0.0847926 0.0423963 0.999101i \(-0.486501\pi\)
0.0423963 + 0.999101i \(0.486501\pi\)
\(614\) −3.30716e6 −0.354025
\(615\) 0 0
\(616\) 0 0
\(617\) −1.67739e7 −1.77387 −0.886935 0.461894i \(-0.847170\pi\)
−0.886935 + 0.461894i \(0.847170\pi\)
\(618\) 0 0
\(619\) −8.22300e6 −0.862588 −0.431294 0.902211i \(-0.641943\pi\)
−0.431294 + 0.902211i \(0.641943\pi\)
\(620\) −3.73429e6 −0.390147
\(621\) 0 0
\(622\) 1.72675e7 1.78959
\(623\) 0 0
\(624\) 0 0
\(625\) −9.36798e6 −0.959281
\(626\) 1.71469e7 1.74884
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 1.32881e7 1.31292
\(635\) −4.13030e6 −0.406488
\(636\) 0 0
\(637\) 0 0
\(638\) 1.94011e7 1.88701
\(639\) 0 0
\(640\) 1.45224e7 1.40149
\(641\) 1.06761e7 1.02628 0.513141 0.858304i \(-0.328482\pi\)
0.513141 + 0.858304i \(0.328482\pi\)
\(642\) 0 0
\(643\) 3.13159e6 0.298701 0.149351 0.988784i \(-0.452282\pi\)
0.149351 + 0.988784i \(0.452282\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.94464e6 −0.371900
\(647\) −4.93457e6 −0.463435 −0.231717 0.972783i \(-0.574434\pi\)
−0.231717 + 0.972783i \(0.574434\pi\)
\(648\) 0 0
\(649\) 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 \(-0.584572\pi\)
−0.262575 + 0.964912i \(0.584572\pi\)
\(654\) 0 0
\(655\) 1.33052e7 1.21176
\(656\) −8.96523e6 −0.813395
\(657\) 0 0
\(658\) 0 0
\(659\) −362477. −0.0325137 −0.0162569 0.999868i \(-0.505175\pi\)
−0.0162569 + 0.999868i \(0.505175\pi\)
\(660\) 0 0
\(661\) 1.91211e7 1.70219 0.851096 0.525011i \(-0.175939\pi\)
0.851096 + 0.525011i \(0.175939\pi\)
\(662\) 7.67133e6 0.680339
\(663\) 0 0
\(664\) −7.81494e6 −0.687868
\(665\) 0 0
\(666\) 0 0
\(667\) 3.31370e6 0.288403
\(668\) −1.22917e7 −1.06579
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) −5.19923e6 −0.437595
\(677\) 1.16903e7 0.980291 0.490146 0.871641i \(-0.336944\pi\)
0.490146 + 0.871641i \(0.336944\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.89473e6 0.240069
\(681\) 0 0
\(682\) 6.41849e6 0.528411
\(683\) −1.83674e7 −1.50659 −0.753297 0.657681i \(-0.771538\pi\)
−0.753297 + 0.657681i \(0.771538\pi\)
\(684\) 0 0
\(685\) 2.25588e6 0.183692
\(686\) 0 0
\(687\) 0 0
\(688\) −2.34049e6 −0.188511
\(689\) 2.78191e6 0.223252
\(690\) 0 0
\(691\) −2.35611e7 −1.87716 −0.938579 0.345066i \(-0.887857\pi\)
−0.938579 + 0.345066i \(0.887857\pi\)
\(692\) −4.52612e6 −0.359303
\(693\) 0 0
\(694\) −1.32484e7 −1.04415
\(695\) 2.68535e7 2.10881
\(696\) 0 0
\(697\) −2.60157e6 −0.202840
\(698\) −1.14517e7 −0.889679
\(699\) 0 0
\(700\) 0 0
\(701\) −1.32980e7 −1.02210 −0.511048 0.859552i \(-0.670743\pi\)
−0.511048 + 0.859552i \(0.670743\pi\)
\(702\) 0 0
\(703\) −1.11307e6 −0.0849442
\(704\) 337210. 0.0256430
\(705\) 0 0
\(706\) 4.09600e6 0.309277
\(707\) 0 0
\(708\) 0 0
\(709\) −6.85353e6 −0.512034 −0.256017 0.966672i \(-0.582410\pi\)
−0.256017 + 0.966672i \(0.582410\pi\)
\(710\) 8.13540e6 0.605666
\(711\) 0 0
\(712\) 4.26998e6 0.315665
\(713\) 1.09628e6 0.0807602
\(714\) 0 0
\(715\) 8.19384e6 0.599408
\(716\) −2.52969e6 −0.184410
\(717\) 0 0
\(718\) −1.39402e7 −1.00915
\(719\) 2.65729e7 1.91698 0.958490 0.285127i \(-0.0920357\pi\)
0.958490 + 0.285127i \(0.0920357\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.49658e6 0.106846
\(723\) 0 0
\(724\) −5.56312e6 −0.394432
\(725\) 2.52779e7 1.78606
\(726\) 0 0
\(727\) −2.16991e6 −0.152267 −0.0761335 0.997098i \(-0.524258\pi\)
−0.0761335 + 0.997098i \(0.524258\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3.96318e7 2.75256
\(731\) −679174. −0.0470097
\(732\) 0 0
\(733\) 1.74653e7 1.20065 0.600324 0.799757i \(-0.295038\pi\)
0.600324 + 0.799757i \(0.295038\pi\)
\(734\) 1.54016e7 1.05518
\(735\) 0 0
\(736\) 2.51048e6 0.170829
\(737\) −9.41203e6 −0.638285
\(738\) 0 0
\(739\) −1.36461e7 −0.919172 −0.459586 0.888133i \(-0.652002\pi\)
−0.459586 + 0.888133i \(0.652002\pi\)
\(740\) −1.07253e6 −0.0719994
\(741\) 0 0
\(742\) 0 0
\(743\) −1.48965e7 −0.989944 −0.494972 0.868909i \(-0.664822\pi\)
−0.494972 + 0.868909i \(0.664822\pi\)
\(744\) 0 0
\(745\) −2.83692e7 −1.87265
\(746\) −9.80791e6 −0.645252
\(747\) 0 0
\(748\) 2.36571e6 0.154599
\(749\) 0 0
\(750\) 0 0
\(751\) −2.53463e7 −1.63989 −0.819944 0.572443i \(-0.805996\pi\)
−0.819944 + 0.572443i \(0.805996\pi\)
\(752\) 1.95465e6 0.126044
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) 1.17715e7 0.739264
\(761\) 579829. 0.0362943 0.0181471 0.999835i \(-0.494223\pi\)
0.0181471 + 0.999835i \(0.494223\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −4.13255e6 −0.256144
\(765\) 0 0
\(766\) −2.32357e7 −1.43082
\(767\) −8.65541e6 −0.531250
\(768\) 0 0
\(769\) −1.52438e7 −0.929562 −0.464781 0.885426i \(-0.653867\pi\)
−0.464781 + 0.885426i \(0.653867\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.22140e7 −0.737591
\(773\) −1.94926e7 −1.17333 −0.586665 0.809830i \(-0.699559\pi\)
−0.586665 + 0.809830i \(0.699559\pi\)
\(774\) 0 0
\(775\) 8.36275e6 0.500144
\(776\) −1.01292e7 −0.603839
\(777\) 0 0
\(778\) −2.08807e7 −1.23679
\(779\) −1.05794e7 −0.624621
\(780\) 0 0
\(781\) −5.06345e6 −0.297043
\(782\) 1.11585e6 0.0652515
\(783\) 0 0
\(784\) 0 0
\(785\) 3.66337e7 2.12181
\(786\) 0 0
\(787\) −1.36277e7 −0.784305 −0.392153 0.919900i \(-0.628270\pi\)
−0.392153 + 0.919900i \(0.628270\pi\)
\(788\) 23302.0 0.00133683
\(789\) 0 0
\(790\) −1.53143e7 −0.873031
\(791\) 0 0
\(792\) 0 0
\(793\) 1.35664e7 0.766093
\(794\) −490624. −0.0276184
\(795\) 0 0
\(796\) −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
\(801\) 0 0
\(802\) −2.37121e7 −1.30177
\(803\) −2.46667e7 −1.34997
\(804\) 0 0
\(805\) 0 0
\(806\) −5.32010e6 −0.288458
\(807\) 0 0
\(808\) 2.81223e6 0.151538
\(809\) −2.98780e7 −1.60502 −0.802509 0.596640i \(-0.796502\pi\)
−0.802509 + 0.596640i \(0.796502\pi\)
\(810\) 0 0
\(811\) 1.02643e6 0.0547995 0.0273998 0.999625i \(-0.491277\pi\)
0.0273998 + 0.999625i \(0.491277\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.84346e6 0.0975153
\(815\) −4.01086e7 −2.11516
\(816\) 0 0
\(817\) −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 \(-0.403719\pi\)
0.297883 + 0.954602i \(0.403719\pi\)
\(822\) 0 0
\(823\) −2.51210e7 −1.29282 −0.646408 0.762992i \(-0.723730\pi\)
−0.646408 + 0.762992i \(0.723730\pi\)
\(824\) −2.86393e6 −0.146942
\(825\) 0 0
\(826\) 0 0
\(827\) 2.14447e6 0.109032 0.0545162 0.998513i \(-0.482638\pi\)
0.0545162 + 0.998513i \(0.482638\pi\)
\(828\) 0 0
\(829\) −818065. −0.0413430 −0.0206715 0.999786i \(-0.506580\pi\)
−0.0206715 + 0.999786i \(0.506580\pi\)
\(830\) −4.50937e7 −2.27207
\(831\) 0 0
\(832\) −279504. −0.0139984
\(833\) 0 0
\(834\) 0 0
\(835\) 5.40155e7 2.68104
\(836\) 9.62025e6 0.476070
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 9.12215e6 0.440799
\(845\) 2.28478e7 1.10079
\(846\) 0 0
\(847\) 0 0
\(848\) −1.21632e7 −0.580844
\(849\) 0 0
\(850\) 8.51207e6 0.404099
\(851\) 314863. 0.0149038
\(852\) 0 0
\(853\) 1.89000e7 0.889386 0.444693 0.895683i \(-0.353313\pi\)
0.444693 + 0.895683i \(0.353313\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.60892e6 −0.401573
\(857\) 3.72286e7 1.73151 0.865753 0.500471i \(-0.166840\pi\)
0.865753 + 0.500471i \(0.166840\pi\)
\(858\) 0 0
\(859\) −3.02064e6 −0.139674 −0.0698371 0.997558i \(-0.522248\pi\)
−0.0698371 + 0.997558i \(0.522248\pi\)
\(860\) −2.66129e6 −0.122700
\(861\) 0 0
\(862\) 5.34131e7 2.44838
\(863\) 2.71843e7 1.24248 0.621242 0.783619i \(-0.286629\pi\)
0.621242 + 0.783619i \(0.286629\pi\)
\(864\) 0 0
\(865\) 1.98899e7 0.903839
\(866\) 4.13320e7 1.87280
\(867\) 0 0
\(868\) 0 0
\(869\) 9.53158e6 0.428169
\(870\) 0 0
\(871\) 7.80136e6 0.348438
\(872\) 2.16185e7 0.962797
\(873\) 0 0
\(874\) 4.53766e6 0.200934
\(875\) 0 0
\(876\) 0 0
\(877\) 1.73868e7 0.763344 0.381672 0.924298i \(-0.375348\pi\)
0.381672 + 0.924298i \(0.375348\pi\)
\(878\) 1.19064e6 0.0521248
\(879\) 0 0
\(880\) −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
\(885\) 0 0
\(886\) −2.01258e7 −0.861327
\(887\) −1.47028e7 −0.627465 −0.313733 0.949511i \(-0.601580\pi\)
−0.313733 + 0.949511i \(0.601580\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.46386e7 1.04266
\(891\) 0 0
\(892\) 2.13132e7 0.896885
\(893\) 2.30657e6 0.0967918
\(894\) 0 0
\(895\) 1.11166e7 0.463890
\(896\) 0 0
\(897\) 0 0
\(898\) −1.00269e7 −0.414930
\(899\) 2.00450e7 0.827193
\(900\) 0 0
\(901\) −3.52958e6 −0.144848
\(902\) 1.75215e7 0.717060
\(903\) 0 0
\(904\) −3.88669e6 −0.158183
\(905\) 2.44469e7 0.992208
\(906\) 0 0
\(907\) 1.30940e7 0.528512 0.264256 0.964453i \(-0.414874\pi\)
0.264256 + 0.964453i \(0.414874\pi\)
\(908\) −1.65335e7 −0.665504
\(909\) 0 0
\(910\) 0 0
\(911\) −2.68695e7 −1.07266 −0.536332 0.844007i \(-0.680191\pi\)
−0.536332 + 0.844007i \(0.680191\pi\)
\(912\) 0 0
\(913\) 2.80662e7 1.11431
\(914\) −1.10298e7 −0.436719
\(915\) 0 0
\(916\) −9.48103e6 −0.373351
\(917\) 0 0
\(918\) 0 0
\(919\) 1.27317e6 0.0497277 0.0248638 0.999691i \(-0.492085\pi\)
0.0248638 + 0.999691i \(0.492085\pi\)
\(920\) −3.32991e6 −0.129707
\(921\) 0 0
\(922\) 3.15427e7 1.22200
\(923\) 4.19695e6 0.162155
\(924\) 0 0
\(925\) 2.40187e6 0.0922986
\(926\) −3.48658e7 −1.33620
\(927\) 0 0
\(928\) 4.59031e7 1.74973
\(929\) −9.31705e6 −0.354192 −0.177096 0.984194i \(-0.556670\pi\)
−0.177096 + 0.984194i \(0.556670\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.89428e7 0.714338
\(933\) 0 0
\(934\) −3.60577e7 −1.35248
\(935\) −1.03960e7 −0.388900
\(936\) 0 0
\(937\) −1.18158e7 −0.439657 −0.219829 0.975539i \(-0.570550\pi\)
−0.219829 + 0.975539i \(0.570550\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2.22256e6 0.0820416
\(941\) −2.53529e7 −0.933371 −0.466685 0.884423i \(-0.654552\pi\)
−0.466685 + 0.884423i \(0.654552\pi\)
\(942\) 0 0
\(943\) 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 \(-0.659365\pi\)
−0.480004 + 0.877266i \(0.659365\pi\)
\(948\) 0 0
\(949\) 2.04455e7 0.736941
\(950\) 3.46147e7 1.24437
\(951\) 0 0
\(952\) 0 0
\(953\) 1.88335e7 0.671735 0.335868 0.941909i \(-0.390970\pi\)
0.335868 + 0.941909i \(0.390970\pi\)
\(954\) 0 0
\(955\) 1.81603e7 0.644339
\(956\) 2.78745e7 0.986423
\(957\) 0 0
\(958\) 5.87929e7 2.06972
\(959\) 0 0
\(960\) 0 0
\(961\) −2.19976e7 −0.768365
\(962\) −1.52799e6 −0.0532332
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) −5.84475e7 −1.99451
\(971\) −6.85669e6 −0.233381 −0.116691 0.993168i \(-0.537229\pi\)
−0.116691 + 0.993168i \(0.537229\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −6.11527e7 −2.06547
\(975\) 0 0
\(976\) −5.93157e7 −1.99317
\(977\) −2.81471e7 −0.943402 −0.471701 0.881759i \(-0.656360\pi\)
−0.471701 + 0.881759i \(0.656360\pi\)
\(978\) 0 0
\(979\) −1.53350e7 −0.511361
\(980\) 0 0
\(981\) 0 0
\(982\) 673142. 0.0222755
\(983\) 2.34916e7 0.775406 0.387703 0.921784i \(-0.373269\pi\)
0.387703 + 0.921784i \(0.373269\pi\)
\(984\) 0 0
\(985\) −102399. −0.00336285
\(986\) 2.04029e7 0.668343
\(987\) 0 0
\(988\) −7.97395e6 −0.259885
\(989\) 781278. 0.0253989
\(990\) 0 0
\(991\) −2.14412e6 −0.0693530 −0.0346765 0.999399i \(-0.511040\pi\)
−0.0346765 + 0.999399i \(0.511040\pi\)
\(992\) 1.51862e7 0.489971
\(993\) 0 0
\(994\) 0 0
\(995\) 2.94483e7 0.942979
\(996\) 0 0
\(997\) −2.50872e7 −0.799307 −0.399654 0.916666i \(-0.630870\pi\)
−0.399654 + 0.916666i \(0.630870\pi\)
\(998\) −1.51715e7 −0.482173
\(999\) 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 441.6.a.m.1.1 2
3.2 odd 2 49.6.a.e.1.2 2
7.3 odd 6 63.6.e.d.37.2 4
7.5 odd 6 63.6.e.d.46.2 4
7.6 odd 2 441.6.a.n.1.1 2
12.11 even 2 784.6.a.t.1.1 2
21.2 odd 6 49.6.c.f.18.1 4
21.5 even 6 7.6.c.a.4.1 yes 4
21.11 odd 6 49.6.c.f.30.1 4
21.17 even 6 7.6.c.a.2.1 4
21.20 even 2 49.6.a.d.1.2 2
84.47 odd 6 112.6.i.c.81.1 4
84.59 odd 6 112.6.i.c.65.1 4
84.83 odd 2 784.6.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.1 4 21.17 even 6
7.6.c.a.4.1 yes 4 21.5 even 6
49.6.a.d.1.2 2 21.20 even 2
49.6.a.e.1.2 2 3.2 odd 2
49.6.c.f.18.1 4 21.2 odd 6
49.6.c.f.30.1 4 21.11 odd 6
63.6.e.d.37.2 4 7.3 odd 6
63.6.e.d.46.2 4 7.5 odd 6
112.6.i.c.65.1 4 84.59 odd 6
112.6.i.c.81.1 4 84.47 odd 6
441.6.a.m.1.1 2 1.1 even 1 trivial
441.6.a.n.1.1 2 7.6 odd 2
784.6.a.t.1.1 2 12.11 even 2
784.6.a.ba.1.2 2 84.83 odd 2