Properties

Label 1075.6.a.a.1.1
Level $1075$
Weight $6$
Character 1075.1
Self dual yes
Analytic conductor $172.413$
Analytic rank $0$
Dimension $8$
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] = [1075,6,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, 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("1075.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(172.412606299\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(10.9591\) of defining polynomial
Character \(\chi\) \(=\) 1075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.95911 q^{2} +25.1057 q^{3} +48.2657 q^{4} -224.924 q^{6} +166.001 q^{7} -145.726 q^{8} +387.294 q^{9} +O(q^{10})\) \(q-8.95911 q^{2} +25.1057 q^{3} +48.2657 q^{4} -224.924 q^{6} +166.001 q^{7} -145.726 q^{8} +387.294 q^{9} +607.827 q^{11} +1211.74 q^{12} +1039.54 q^{13} -1487.22 q^{14} -238.923 q^{16} -1439.76 q^{17} -3469.81 q^{18} -1332.62 q^{19} +4167.57 q^{21} -5445.59 q^{22} +437.244 q^{23} -3658.56 q^{24} -9313.39 q^{26} +3622.60 q^{27} +8012.16 q^{28} -87.2656 q^{29} +2654.45 q^{31} +6803.79 q^{32} +15259.9 q^{33} +12898.9 q^{34} +18693.0 q^{36} +4671.70 q^{37} +11939.1 q^{38} +26098.4 q^{39} -9012.91 q^{41} -37337.7 q^{42} +1849.00 q^{43} +29337.2 q^{44} -3917.32 q^{46} +8623.49 q^{47} -5998.32 q^{48} +10749.4 q^{49} -36146.0 q^{51} +50174.3 q^{52} +28358.3 q^{53} -32455.3 q^{54} -24190.8 q^{56} -33456.3 q^{57} +781.822 q^{58} +48066.0 q^{59} -39750.4 q^{61} -23781.5 q^{62} +64291.2 q^{63} -53310.4 q^{64} -136715. q^{66} -19233.9 q^{67} -69490.8 q^{68} +10977.3 q^{69} +17008.5 q^{71} -56439.0 q^{72} -22236.4 q^{73} -41854.3 q^{74} -64319.9 q^{76} +100900. q^{77} -233819. q^{78} +36951.1 q^{79} -3164.77 q^{81} +80747.7 q^{82} +117990. q^{83} +201151. q^{84} -16565.4 q^{86} -2190.86 q^{87} -88576.5 q^{88} +41537.6 q^{89} +172565. q^{91} +21103.9 q^{92} +66641.7 q^{93} -77258.8 q^{94} +170814. q^{96} -32593.9 q^{97} -96304.8 q^{98} +235408. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.95911 −1.58376 −0.791881 0.610675i \(-0.790898\pi\)
−0.791881 + 0.610675i \(0.790898\pi\)
\(3\) 25.1057 1.61053 0.805264 0.592916i \(-0.202023\pi\)
0.805264 + 0.592916i \(0.202023\pi\)
\(4\) 48.2657 1.50830
\(5\) 0 0
\(6\) −224.924 −2.55070
\(7\) 166.001 1.28046 0.640230 0.768183i \(-0.278839\pi\)
0.640230 + 0.768183i \(0.278839\pi\)
\(8\) −145.726 −0.805033
\(9\) 387.294 1.59380
\(10\) 0 0
\(11\) 607.827 1.51460 0.757301 0.653066i \(-0.226518\pi\)
0.757301 + 0.653066i \(0.226518\pi\)
\(12\) 1211.74 2.42917
\(13\) 1039.54 1.70602 0.853011 0.521894i \(-0.174774\pi\)
0.853011 + 0.521894i \(0.174774\pi\)
\(14\) −1487.22 −2.02794
\(15\) 0 0
\(16\) −238.923 −0.233323
\(17\) −1439.76 −1.20828 −0.604138 0.796880i \(-0.706483\pi\)
−0.604138 + 0.796880i \(0.706483\pi\)
\(18\) −3469.81 −2.52421
\(19\) −1332.62 −0.846881 −0.423440 0.905924i \(-0.639178\pi\)
−0.423440 + 0.905924i \(0.639178\pi\)
\(20\) 0 0
\(21\) 4167.57 2.06222
\(22\) −5445.59 −2.39877
\(23\) 437.244 0.172347 0.0861737 0.996280i \(-0.472536\pi\)
0.0861737 + 0.996280i \(0.472536\pi\)
\(24\) −3658.56 −1.29653
\(25\) 0 0
\(26\) −9313.39 −2.70193
\(27\) 3622.60 0.956336
\(28\) 8012.16 1.93132
\(29\) −87.2656 −0.0192685 −0.00963425 0.999954i \(-0.503067\pi\)
−0.00963425 + 0.999954i \(0.503067\pi\)
\(30\) 0 0
\(31\) 2654.45 0.496101 0.248050 0.968747i \(-0.420210\pi\)
0.248050 + 0.968747i \(0.420210\pi\)
\(32\) 6803.79 1.17456
\(33\) 15259.9 2.43931
\(34\) 12898.9 1.91362
\(35\) 0 0
\(36\) 18693.0 2.40394
\(37\) 4671.70 0.561010 0.280505 0.959853i \(-0.409498\pi\)
0.280505 + 0.959853i \(0.409498\pi\)
\(38\) 11939.1 1.34126
\(39\) 26098.4 2.74760
\(40\) 0 0
\(41\) −9012.91 −0.837347 −0.418674 0.908137i \(-0.637505\pi\)
−0.418674 + 0.908137i \(0.637505\pi\)
\(42\) −37337.7 −3.26606
\(43\) 1849.00 0.152499
\(44\) 29337.2 2.28448
\(45\) 0 0
\(46\) −3917.32 −0.272957
\(47\) 8623.49 0.569427 0.284714 0.958613i \(-0.408101\pi\)
0.284714 + 0.958613i \(0.408101\pi\)
\(48\) −5998.32 −0.375774
\(49\) 10749.4 0.639577
\(50\) 0 0
\(51\) −36146.0 −1.94596
\(52\) 50174.3 2.57320
\(53\) 28358.3 1.38672 0.693362 0.720590i \(-0.256129\pi\)
0.693362 + 0.720590i \(0.256129\pi\)
\(54\) −32455.3 −1.51461
\(55\) 0 0
\(56\) −24190.8 −1.03081
\(57\) −33456.3 −1.36393
\(58\) 781.822 0.0305167
\(59\) 48066.0 1.79766 0.898832 0.438294i \(-0.144417\pi\)
0.898832 + 0.438294i \(0.144417\pi\)
\(60\) 0 0
\(61\) −39750.4 −1.36778 −0.683891 0.729585i \(-0.739713\pi\)
−0.683891 + 0.729585i \(0.739713\pi\)
\(62\) −23781.5 −0.785706
\(63\) 64291.2 2.04080
\(64\) −53310.4 −1.62690
\(65\) 0 0
\(66\) −136715. −3.86329
\(67\) −19233.9 −0.523456 −0.261728 0.965142i \(-0.584292\pi\)
−0.261728 + 0.965142i \(0.584292\pi\)
\(68\) −69490.8 −1.82245
\(69\) 10977.3 0.277570
\(70\) 0 0
\(71\) 17008.5 0.400424 0.200212 0.979753i \(-0.435837\pi\)
0.200212 + 0.979753i \(0.435837\pi\)
\(72\) −56439.0 −1.28306
\(73\) −22236.4 −0.488380 −0.244190 0.969727i \(-0.578522\pi\)
−0.244190 + 0.969727i \(0.578522\pi\)
\(74\) −41854.3 −0.888506
\(75\) 0 0
\(76\) −64319.9 −1.27735
\(77\) 100900. 1.93939
\(78\) −233819. −4.35154
\(79\) 36951.1 0.666131 0.333066 0.942904i \(-0.391917\pi\)
0.333066 + 0.942904i \(0.391917\pi\)
\(80\) 0 0
\(81\) −3164.77 −0.0535957
\(82\) 80747.7 1.32616
\(83\) 117990. 1.87997 0.939985 0.341216i \(-0.110839\pi\)
0.939985 + 0.341216i \(0.110839\pi\)
\(84\) 201151. 3.11045
\(85\) 0 0
\(86\) −16565.4 −0.241522
\(87\) −2190.86 −0.0310325
\(88\) −88576.5 −1.21930
\(89\) 41537.6 0.555861 0.277931 0.960601i \(-0.410351\pi\)
0.277931 + 0.960601i \(0.410351\pi\)
\(90\) 0 0
\(91\) 172565. 2.18449
\(92\) 21103.9 0.259952
\(93\) 66641.7 0.798985
\(94\) −77258.8 −0.901838
\(95\) 0 0
\(96\) 170814. 1.89166
\(97\) −32593.9 −0.351728 −0.175864 0.984415i \(-0.556272\pi\)
−0.175864 + 0.984415i \(0.556272\pi\)
\(98\) −96304.8 −1.01294
\(99\) 235408. 2.41398
\(100\) 0 0
\(101\) −7934.83 −0.0773988 −0.0386994 0.999251i \(-0.512321\pi\)
−0.0386994 + 0.999251i \(0.512321\pi\)
\(102\) 323836. 3.08194
\(103\) −132558. −1.23116 −0.615579 0.788075i \(-0.711078\pi\)
−0.615579 + 0.788075i \(0.711078\pi\)
\(104\) −151489. −1.37340
\(105\) 0 0
\(106\) −254065. −2.19624
\(107\) −120642. −1.01868 −0.509342 0.860564i \(-0.670111\pi\)
−0.509342 + 0.860564i \(0.670111\pi\)
\(108\) 174847. 1.44245
\(109\) 16769.1 0.135190 0.0675950 0.997713i \(-0.478467\pi\)
0.0675950 + 0.997713i \(0.478467\pi\)
\(110\) 0 0
\(111\) 117286. 0.903522
\(112\) −39661.5 −0.298761
\(113\) 18201.6 0.134096 0.0670478 0.997750i \(-0.478642\pi\)
0.0670478 + 0.997750i \(0.478642\pi\)
\(114\) 299739. 2.16013
\(115\) 0 0
\(116\) −4211.94 −0.0290628
\(117\) 402609. 2.71906
\(118\) −430629. −2.84707
\(119\) −239001. −1.54715
\(120\) 0 0
\(121\) 208403. 1.29402
\(122\) 356128. 2.16624
\(123\) −226275. −1.34857
\(124\) 128119. 0.748271
\(125\) 0 0
\(126\) −575993. −3.23214
\(127\) 155225. 0.853987 0.426994 0.904255i \(-0.359573\pi\)
0.426994 + 0.904255i \(0.359573\pi\)
\(128\) 259892. 1.40207
\(129\) 46420.4 0.245603
\(130\) 0 0
\(131\) 206749. 1.05261 0.526303 0.850297i \(-0.323578\pi\)
0.526303 + 0.850297i \(0.323578\pi\)
\(132\) 736530. 3.67922
\(133\) −221216. −1.08440
\(134\) 172319. 0.829030
\(135\) 0 0
\(136\) 209810. 0.972702
\(137\) 219188. 0.997736 0.498868 0.866678i \(-0.333749\pi\)
0.498868 + 0.866678i \(0.333749\pi\)
\(138\) −98347.0 −0.439606
\(139\) 83047.4 0.364577 0.182288 0.983245i \(-0.441650\pi\)
0.182288 + 0.983245i \(0.441650\pi\)
\(140\) 0 0
\(141\) 216498. 0.917079
\(142\) −152381. −0.634176
\(143\) 631863. 2.58394
\(144\) −92533.5 −0.371871
\(145\) 0 0
\(146\) 199219. 0.773478
\(147\) 269870. 1.03006
\(148\) 225483. 0.846173
\(149\) −505431. −1.86507 −0.932537 0.361074i \(-0.882410\pi\)
−0.932537 + 0.361074i \(0.882410\pi\)
\(150\) 0 0
\(151\) −222212. −0.793095 −0.396548 0.918014i \(-0.629792\pi\)
−0.396548 + 0.918014i \(0.629792\pi\)
\(152\) 194198. 0.681767
\(153\) −557609. −1.92575
\(154\) −903974. −3.07153
\(155\) 0 0
\(156\) 1.25966e6 4.14421
\(157\) −18138.1 −0.0587275 −0.0293638 0.999569i \(-0.509348\pi\)
−0.0293638 + 0.999569i \(0.509348\pi\)
\(158\) −331049. −1.05499
\(159\) 711953. 2.23336
\(160\) 0 0
\(161\) 72583.1 0.220684
\(162\) 28353.5 0.0848828
\(163\) 428372. 1.26285 0.631425 0.775437i \(-0.282470\pi\)
0.631425 + 0.775437i \(0.282470\pi\)
\(164\) −435015. −1.26297
\(165\) 0 0
\(166\) −1.05709e6 −2.97743
\(167\) −590110. −1.63735 −0.818676 0.574256i \(-0.805291\pi\)
−0.818676 + 0.574256i \(0.805291\pi\)
\(168\) −607325. −1.66015
\(169\) 709358. 1.91051
\(170\) 0 0
\(171\) −516116. −1.34976
\(172\) 89243.3 0.230014
\(173\) 194001. 0.492821 0.246410 0.969166i \(-0.420749\pi\)
0.246410 + 0.969166i \(0.420749\pi\)
\(174\) 19628.2 0.0491481
\(175\) 0 0
\(176\) −145224. −0.353392
\(177\) 1.20673e6 2.89519
\(178\) −372140. −0.880353
\(179\) −434140. −1.01274 −0.506369 0.862317i \(-0.669012\pi\)
−0.506369 + 0.862317i \(0.669012\pi\)
\(180\) 0 0
\(181\) −626829. −1.42217 −0.711087 0.703104i \(-0.751797\pi\)
−0.711087 + 0.703104i \(0.751797\pi\)
\(182\) −1.54603e6 −3.45972
\(183\) −997959. −2.20285
\(184\) −63718.1 −0.138745
\(185\) 0 0
\(186\) −597050. −1.26540
\(187\) −875122. −1.83006
\(188\) 416219. 0.858870
\(189\) 601355. 1.22455
\(190\) 0 0
\(191\) −109059. −0.216310 −0.108155 0.994134i \(-0.534494\pi\)
−0.108155 + 0.994134i \(0.534494\pi\)
\(192\) −1.33839e6 −2.62017
\(193\) 31563.2 0.0609941 0.0304971 0.999535i \(-0.490291\pi\)
0.0304971 + 0.999535i \(0.490291\pi\)
\(194\) 292012. 0.557053
\(195\) 0 0
\(196\) 518826. 0.964676
\(197\) 276843. 0.508238 0.254119 0.967173i \(-0.418214\pi\)
0.254119 + 0.967173i \(0.418214\pi\)
\(198\) −2.10905e6 −3.82316
\(199\) −777649. −1.39204 −0.696019 0.718024i \(-0.745047\pi\)
−0.696019 + 0.718024i \(0.745047\pi\)
\(200\) 0 0
\(201\) −482879. −0.843041
\(202\) 71089.0 0.122581
\(203\) −14486.2 −0.0246725
\(204\) −1.74461e6 −2.93511
\(205\) 0 0
\(206\) 1.18761e6 1.94986
\(207\) 169342. 0.274688
\(208\) −248371. −0.398055
\(209\) −810002. −1.28269
\(210\) 0 0
\(211\) −89486.6 −0.138373 −0.0691866 0.997604i \(-0.522040\pi\)
−0.0691866 + 0.997604i \(0.522040\pi\)
\(212\) 1.36873e6 2.09160
\(213\) 427009. 0.644894
\(214\) 1.08085e6 1.61335
\(215\) 0 0
\(216\) −527908. −0.769882
\(217\) 440641. 0.635237
\(218\) −150237. −0.214109
\(219\) −558260. −0.786550
\(220\) 0 0
\(221\) −1.49669e6 −2.06135
\(222\) −1.05078e6 −1.43096
\(223\) 952284. 1.28234 0.641172 0.767398i \(-0.278449\pi\)
0.641172 + 0.767398i \(0.278449\pi\)
\(224\) 1.12944e6 1.50398
\(225\) 0 0
\(226\) −163071. −0.212376
\(227\) 465809. 0.599988 0.299994 0.953941i \(-0.403015\pi\)
0.299994 + 0.953941i \(0.403015\pi\)
\(228\) −1.61479e6 −2.05721
\(229\) 477440. 0.601631 0.300815 0.953682i \(-0.402741\pi\)
0.300815 + 0.953682i \(0.402741\pi\)
\(230\) 0 0
\(231\) 2.53316e6 3.12344
\(232\) 12716.9 0.0155118
\(233\) −743589. −0.897312 −0.448656 0.893705i \(-0.648097\pi\)
−0.448656 + 0.893705i \(0.648097\pi\)
\(234\) −3.60702e6 −4.30635
\(235\) 0 0
\(236\) 2.31994e6 2.71142
\(237\) 927682. 1.07282
\(238\) 2.14124e6 2.45032
\(239\) 1.27115e6 1.43947 0.719734 0.694250i \(-0.244264\pi\)
0.719734 + 0.694250i \(0.244264\pi\)
\(240\) 0 0
\(241\) −875960. −0.971497 −0.485749 0.874099i \(-0.661453\pi\)
−0.485749 + 0.874099i \(0.661453\pi\)
\(242\) −1.86710e6 −2.04942
\(243\) −959745. −1.04265
\(244\) −1.91858e6 −2.06303
\(245\) 0 0
\(246\) 2.02722e6 2.13582
\(247\) −1.38532e6 −1.44480
\(248\) −386823. −0.399377
\(249\) 2.96222e6 3.02774
\(250\) 0 0
\(251\) −1.19674e6 −1.19899 −0.599494 0.800380i \(-0.704631\pi\)
−0.599494 + 0.800380i \(0.704631\pi\)
\(252\) 3.10306e6 3.07815
\(253\) 265769. 0.261038
\(254\) −1.39068e6 −1.35251
\(255\) 0 0
\(256\) −622474. −0.593638
\(257\) 1.31630e6 1.24315 0.621573 0.783357i \(-0.286494\pi\)
0.621573 + 0.783357i \(0.286494\pi\)
\(258\) −415885. −0.388977
\(259\) 775507. 0.718350
\(260\) 0 0
\(261\) −33797.4 −0.0307102
\(262\) −1.85229e6 −1.66708
\(263\) −1.74612e6 −1.55663 −0.778314 0.627876i \(-0.783925\pi\)
−0.778314 + 0.627876i \(0.783925\pi\)
\(264\) −2.22377e6 −1.96372
\(265\) 0 0
\(266\) 1.98190e6 1.71743
\(267\) 1.04283e6 0.895231
\(268\) −928338. −0.789531
\(269\) −470733. −0.396637 −0.198319 0.980138i \(-0.563548\pi\)
−0.198319 + 0.980138i \(0.563548\pi\)
\(270\) 0 0
\(271\) 29459.5 0.0243670 0.0121835 0.999926i \(-0.496122\pi\)
0.0121835 + 0.999926i \(0.496122\pi\)
\(272\) 343991. 0.281919
\(273\) 4.33237e6 3.51819
\(274\) −1.96373e6 −1.58018
\(275\) 0 0
\(276\) 529828. 0.418661
\(277\) −1.01172e6 −0.792244 −0.396122 0.918198i \(-0.629644\pi\)
−0.396122 + 0.918198i \(0.629644\pi\)
\(278\) −744031. −0.577403
\(279\) 1.02805e6 0.790687
\(280\) 0 0
\(281\) −74592.9 −0.0563549 −0.0281775 0.999603i \(-0.508970\pi\)
−0.0281775 + 0.999603i \(0.508970\pi\)
\(282\) −1.93963e6 −1.45244
\(283\) 1.14789e6 0.851993 0.425996 0.904725i \(-0.359924\pi\)
0.425996 + 0.904725i \(0.359924\pi\)
\(284\) 820927. 0.603961
\(285\) 0 0
\(286\) −5.66093e6 −4.09235
\(287\) −1.49615e6 −1.07219
\(288\) 2.63507e6 1.87202
\(289\) 653038. 0.459932
\(290\) 0 0
\(291\) −818290. −0.566467
\(292\) −1.07326e6 −0.736625
\(293\) 1.30694e6 0.889376 0.444688 0.895686i \(-0.353315\pi\)
0.444688 + 0.895686i \(0.353315\pi\)
\(294\) −2.41780e6 −1.63137
\(295\) 0 0
\(296\) −680790. −0.451631
\(297\) 2.20191e6 1.44847
\(298\) 4.52821e6 2.95383
\(299\) 454535. 0.294028
\(300\) 0 0
\(301\) 306936. 0.195268
\(302\) 1.99082e6 1.25607
\(303\) −199209. −0.124653
\(304\) 318394. 0.197597
\(305\) 0 0
\(306\) 4.99568e6 3.04994
\(307\) 1.50999e6 0.914380 0.457190 0.889369i \(-0.348856\pi\)
0.457190 + 0.889369i \(0.348856\pi\)
\(308\) 4.87001e6 2.92518
\(309\) −3.32796e6 −1.98282
\(310\) 0 0
\(311\) 1.69979e6 0.996539 0.498269 0.867022i \(-0.333969\pi\)
0.498269 + 0.867022i \(0.333969\pi\)
\(312\) −3.80323e6 −2.21190
\(313\) −2.85190e6 −1.64541 −0.822703 0.568471i \(-0.807535\pi\)
−0.822703 + 0.568471i \(0.807535\pi\)
\(314\) 162501. 0.0930104
\(315\) 0 0
\(316\) 1.78347e6 1.00473
\(317\) 158403. 0.0885351 0.0442676 0.999020i \(-0.485905\pi\)
0.0442676 + 0.999020i \(0.485905\pi\)
\(318\) −6.37847e6 −3.53711
\(319\) −53042.4 −0.0291841
\(320\) 0 0
\(321\) −3.02880e6 −1.64062
\(322\) −650280. −0.349511
\(323\) 1.91865e6 1.02327
\(324\) −152750. −0.0808386
\(325\) 0 0
\(326\) −3.83783e6 −2.00006
\(327\) 421000. 0.217727
\(328\) 1.31342e6 0.674092
\(329\) 1.43151e6 0.729129
\(330\) 0 0
\(331\) 1.38726e6 0.695968 0.347984 0.937501i \(-0.386866\pi\)
0.347984 + 0.937501i \(0.386866\pi\)
\(332\) 5.69488e6 2.83557
\(333\) 1.80932e6 0.894139
\(334\) 5.28687e6 2.59318
\(335\) 0 0
\(336\) −995728. −0.481163
\(337\) −602990. −0.289225 −0.144612 0.989488i \(-0.546193\pi\)
−0.144612 + 0.989488i \(0.546193\pi\)
\(338\) −6.35522e6 −3.02579
\(339\) 456964. 0.215965
\(340\) 0 0
\(341\) 1.61345e6 0.751395
\(342\) 4.62394e6 2.13770
\(343\) −1.00557e6 −0.461507
\(344\) −269448. −0.122766
\(345\) 0 0
\(346\) −1.73808e6 −0.780511
\(347\) −1.38847e6 −0.619030 −0.309515 0.950895i \(-0.600167\pi\)
−0.309515 + 0.950895i \(0.600167\pi\)
\(348\) −105743. −0.0468064
\(349\) 1.90607e6 0.837676 0.418838 0.908061i \(-0.362437\pi\)
0.418838 + 0.908061i \(0.362437\pi\)
\(350\) 0 0
\(351\) 3.76585e6 1.63153
\(352\) 4.13553e6 1.77899
\(353\) 2.53782e6 1.08399 0.541993 0.840383i \(-0.317670\pi\)
0.541993 + 0.840383i \(0.317670\pi\)
\(354\) −1.08112e7 −4.58529
\(355\) 0 0
\(356\) 2.00484e6 0.838408
\(357\) −6.00028e6 −2.49173
\(358\) 3.88951e6 1.60394
\(359\) −589213. −0.241288 −0.120644 0.992696i \(-0.538496\pi\)
−0.120644 + 0.992696i \(0.538496\pi\)
\(360\) 0 0
\(361\) −700223. −0.282793
\(362\) 5.61583e6 2.25239
\(363\) 5.23209e6 2.08405
\(364\) 8.32900e6 3.29488
\(365\) 0 0
\(366\) 8.94083e6 3.48879
\(367\) −1.67673e6 −0.649826 −0.324913 0.945744i \(-0.605335\pi\)
−0.324913 + 0.945744i \(0.605335\pi\)
\(368\) −104468. −0.0402127
\(369\) −3.49065e6 −1.33457
\(370\) 0 0
\(371\) 4.70750e6 1.77564
\(372\) 3.21651e6 1.20511
\(373\) −3.12665e6 −1.16361 −0.581804 0.813329i \(-0.697653\pi\)
−0.581804 + 0.813329i \(0.697653\pi\)
\(374\) 7.84032e6 2.89838
\(375\) 0 0
\(376\) −1.25667e6 −0.458408
\(377\) −90716.4 −0.0328725
\(378\) −5.38761e6 −1.93940
\(379\) −1.98431e6 −0.709596 −0.354798 0.934943i \(-0.615450\pi\)
−0.354798 + 0.934943i \(0.615450\pi\)
\(380\) 0 0
\(381\) 3.89702e6 1.37537
\(382\) 977069. 0.342584
\(383\) −321813. −0.112100 −0.0560502 0.998428i \(-0.517851\pi\)
−0.0560502 + 0.998428i \(0.517851\pi\)
\(384\) 6.52477e6 2.25807
\(385\) 0 0
\(386\) −282778. −0.0966002
\(387\) 716107. 0.243053
\(388\) −1.57317e6 −0.530512
\(389\) −1.34564e6 −0.450874 −0.225437 0.974258i \(-0.572381\pi\)
−0.225437 + 0.974258i \(0.572381\pi\)
\(390\) 0 0
\(391\) −629525. −0.208243
\(392\) −1.56647e6 −0.514880
\(393\) 5.19057e6 1.69525
\(394\) −2.48026e6 −0.804929
\(395\) 0 0
\(396\) 1.13621e7 3.64101
\(397\) −1.40299e6 −0.446763 −0.223382 0.974731i \(-0.571710\pi\)
−0.223382 + 0.974731i \(0.571710\pi\)
\(398\) 6.96704e6 2.20466
\(399\) −5.55378e6 −1.74645
\(400\) 0 0
\(401\) 3.44258e6 1.06911 0.534556 0.845133i \(-0.320479\pi\)
0.534556 + 0.845133i \(0.320479\pi\)
\(402\) 4.32617e6 1.33518
\(403\) 2.75942e6 0.846359
\(404\) −382980. −0.116741
\(405\) 0 0
\(406\) 129783. 0.0390754
\(407\) 2.83958e6 0.849706
\(408\) 5.26743e6 1.56656
\(409\) 6.44614e6 1.90542 0.952711 0.303877i \(-0.0982811\pi\)
0.952711 + 0.303877i \(0.0982811\pi\)
\(410\) 0 0
\(411\) 5.50286e6 1.60688
\(412\) −6.39802e6 −1.85696
\(413\) 7.97902e6 2.30184
\(414\) −1.51716e6 −0.435040
\(415\) 0 0
\(416\) 7.07283e6 2.00383
\(417\) 2.08496e6 0.587161
\(418\) 7.25690e6 2.03147
\(419\) −1.77865e6 −0.494943 −0.247472 0.968895i \(-0.579600\pi\)
−0.247472 + 0.968895i \(0.579600\pi\)
\(420\) 0 0
\(421\) −3.37157e6 −0.927100 −0.463550 0.886071i \(-0.653425\pi\)
−0.463550 + 0.886071i \(0.653425\pi\)
\(422\) 801721. 0.219150
\(423\) 3.33983e6 0.907555
\(424\) −4.13255e6 −1.11636
\(425\) 0 0
\(426\) −3.82562e6 −1.02136
\(427\) −6.59861e6 −1.75139
\(428\) −5.82288e6 −1.53649
\(429\) 1.58633e7 4.16151
\(430\) 0 0
\(431\) 1.38644e6 0.359508 0.179754 0.983712i \(-0.442470\pi\)
0.179754 + 0.983712i \(0.442470\pi\)
\(432\) −865522. −0.223136
\(433\) −5.74536e6 −1.47264 −0.736322 0.676632i \(-0.763439\pi\)
−0.736322 + 0.676632i \(0.763439\pi\)
\(434\) −3.94776e6 −1.00606
\(435\) 0 0
\(436\) 809375. 0.203908
\(437\) −582681. −0.145958
\(438\) 5.00152e6 1.24571
\(439\) −5.14484e6 −1.27412 −0.637060 0.770814i \(-0.719850\pi\)
−0.637060 + 0.770814i \(0.719850\pi\)
\(440\) 0 0
\(441\) 4.16317e6 1.01936
\(442\) 1.34090e7 3.26468
\(443\) 5.65845e6 1.36990 0.684949 0.728591i \(-0.259824\pi\)
0.684949 + 0.728591i \(0.259824\pi\)
\(444\) 5.66090e6 1.36279
\(445\) 0 0
\(446\) −8.53162e6 −2.03093
\(447\) −1.26892e7 −3.00376
\(448\) −8.84958e6 −2.08318
\(449\) −956361. −0.223875 −0.111938 0.993715i \(-0.535706\pi\)
−0.111938 + 0.993715i \(0.535706\pi\)
\(450\) 0 0
\(451\) −5.47829e6 −1.26825
\(452\) 878515. 0.202257
\(453\) −5.57878e6 −1.27730
\(454\) −4.17323e6 −0.950239
\(455\) 0 0
\(456\) 4.87547e6 1.09800
\(457\) 5.50698e6 1.23345 0.616727 0.787177i \(-0.288458\pi\)
0.616727 + 0.787177i \(0.288458\pi\)
\(458\) −4.27744e6 −0.952840
\(459\) −5.21565e6 −1.15552
\(460\) 0 0
\(461\) 2.10202e6 0.460665 0.230333 0.973112i \(-0.426019\pi\)
0.230333 + 0.973112i \(0.426019\pi\)
\(462\) −2.26949e7 −4.94678
\(463\) −2.36108e6 −0.511868 −0.255934 0.966694i \(-0.582383\pi\)
−0.255934 + 0.966694i \(0.582383\pi\)
\(464\) 20849.8 0.00449579
\(465\) 0 0
\(466\) 6.66190e6 1.42113
\(467\) 8.56485e6 1.81730 0.908652 0.417554i \(-0.137113\pi\)
0.908652 + 0.417554i \(0.137113\pi\)
\(468\) 1.94322e7 4.10117
\(469\) −3.19285e6 −0.670264
\(470\) 0 0
\(471\) −455368. −0.0945823
\(472\) −7.00450e6 −1.44718
\(473\) 1.12387e6 0.230975
\(474\) −8.31121e6 −1.69910
\(475\) 0 0
\(476\) −1.15356e7 −2.33357
\(477\) 1.09830e7 2.21016
\(478\) −1.13884e7 −2.27977
\(479\) 3.09570e6 0.616481 0.308241 0.951308i \(-0.400260\pi\)
0.308241 + 0.951308i \(0.400260\pi\)
\(480\) 0 0
\(481\) 4.85643e6 0.957095
\(482\) 7.84782e6 1.53862
\(483\) 1.82225e6 0.355418
\(484\) 1.00587e7 1.95177
\(485\) 0 0
\(486\) 8.59846e6 1.65132
\(487\) 4.69724e6 0.897471 0.448736 0.893665i \(-0.351875\pi\)
0.448736 + 0.893665i \(0.351875\pi\)
\(488\) 5.79268e6 1.10111
\(489\) 1.07546e7 2.03386
\(490\) 0 0
\(491\) 4.40851e6 0.825255 0.412628 0.910900i \(-0.364611\pi\)
0.412628 + 0.910900i \(0.364611\pi\)
\(492\) −1.09213e7 −2.03406
\(493\) 125641. 0.0232817
\(494\) 1.24112e7 2.28821
\(495\) 0 0
\(496\) −634209. −0.115752
\(497\) 2.82343e6 0.512726
\(498\) −2.65389e7 −4.79523
\(499\) 3.52789e6 0.634255 0.317127 0.948383i \(-0.397282\pi\)
0.317127 + 0.948383i \(0.397282\pi\)
\(500\) 0 0
\(501\) −1.48151e7 −2.63700
\(502\) 1.07217e7 1.89891
\(503\) −79054.9 −0.0139319 −0.00696593 0.999976i \(-0.502217\pi\)
−0.00696593 + 0.999976i \(0.502217\pi\)
\(504\) −9.36894e6 −1.64291
\(505\) 0 0
\(506\) −2.38105e6 −0.413422
\(507\) 1.78089e7 3.07693
\(508\) 7.49203e6 1.28807
\(509\) 155909. 0.0266733 0.0133366 0.999911i \(-0.495755\pi\)
0.0133366 + 0.999911i \(0.495755\pi\)
\(510\) 0 0
\(511\) −3.69127e6 −0.625351
\(512\) −2.73974e6 −0.461885
\(513\) −4.82754e6 −0.809903
\(514\) −1.17929e7 −1.96885
\(515\) 0 0
\(516\) 2.24051e6 0.370444
\(517\) 5.24159e6 0.862456
\(518\) −6.94786e6 −1.13770
\(519\) 4.87052e6 0.793702
\(520\) 0 0
\(521\) −4.24897e6 −0.685787 −0.342893 0.939374i \(-0.611407\pi\)
−0.342893 + 0.939374i \(0.611407\pi\)
\(522\) 302795. 0.0486377
\(523\) −1.30285e6 −0.208276 −0.104138 0.994563i \(-0.533208\pi\)
−0.104138 + 0.994563i \(0.533208\pi\)
\(524\) 9.97890e6 1.58765
\(525\) 0 0
\(526\) 1.56437e7 2.46533
\(527\) −3.82176e6 −0.599427
\(528\) −3.64594e6 −0.569148
\(529\) −6.24516e6 −0.970296
\(530\) 0 0
\(531\) 1.86157e7 2.86512
\(532\) −1.06772e7 −1.63560
\(533\) −9.36932e6 −1.42853
\(534\) −9.34283e6 −1.41783
\(535\) 0 0
\(536\) 2.80289e6 0.421399
\(537\) −1.08994e7 −1.63104
\(538\) 4.21735e6 0.628179
\(539\) 6.53376e6 0.968704
\(540\) 0 0
\(541\) 1.58581e6 0.232948 0.116474 0.993194i \(-0.462841\pi\)
0.116474 + 0.993194i \(0.462841\pi\)
\(542\) −263931. −0.0385915
\(543\) −1.57370e7 −2.29045
\(544\) −9.79579e6 −1.41919
\(545\) 0 0
\(546\) −3.88142e7 −5.57197
\(547\) 5.60569e6 0.801052 0.400526 0.916285i \(-0.368827\pi\)
0.400526 + 0.916285i \(0.368827\pi\)
\(548\) 1.05793e7 1.50489
\(549\) −1.53951e7 −2.17997
\(550\) 0 0
\(551\) 116292. 0.0163181
\(552\) −1.59968e6 −0.223453
\(553\) 6.13393e6 0.852954
\(554\) 9.06408e6 1.25473
\(555\) 0 0
\(556\) 4.00834e6 0.549892
\(557\) 44346.9 0.00605655 0.00302828 0.999995i \(-0.499036\pi\)
0.00302828 + 0.999995i \(0.499036\pi\)
\(558\) −9.21043e6 −1.25226
\(559\) 1.92212e6 0.260166
\(560\) 0 0
\(561\) −2.19705e7 −2.94736
\(562\) 668286. 0.0892528
\(563\) −4.30007e6 −0.571748 −0.285874 0.958267i \(-0.592284\pi\)
−0.285874 + 0.958267i \(0.592284\pi\)
\(564\) 1.04495e7 1.38323
\(565\) 0 0
\(566\) −1.02841e7 −1.34935
\(567\) −525355. −0.0686271
\(568\) −2.47859e6 −0.322354
\(569\) 6.54339e6 0.847271 0.423635 0.905833i \(-0.360754\pi\)
0.423635 + 0.905833i \(0.360754\pi\)
\(570\) 0 0
\(571\) 8.65292e6 1.11064 0.555319 0.831637i \(-0.312596\pi\)
0.555319 + 0.831637i \(0.312596\pi\)
\(572\) 3.04973e7 3.89737
\(573\) −2.73799e6 −0.348374
\(574\) 1.34042e7 1.69809
\(575\) 0 0
\(576\) −2.06468e7 −2.59296
\(577\) −3.48407e6 −0.435660 −0.217830 0.975987i \(-0.569898\pi\)
−0.217830 + 0.975987i \(0.569898\pi\)
\(578\) −5.85064e6 −0.728423
\(579\) 792415. 0.0982328
\(580\) 0 0
\(581\) 1.95865e7 2.40723
\(582\) 7.33116e6 0.897150
\(583\) 1.72369e7 2.10033
\(584\) 3.24044e6 0.393162
\(585\) 0 0
\(586\) −1.17090e7 −1.40856
\(587\) −6.56208e6 −0.786043 −0.393021 0.919529i \(-0.628570\pi\)
−0.393021 + 0.919529i \(0.628570\pi\)
\(588\) 1.30255e7 1.55364
\(589\) −3.53737e6 −0.420138
\(590\) 0 0
\(591\) 6.95032e6 0.818532
\(592\) −1.11618e6 −0.130897
\(593\) −8.08785e6 −0.944488 −0.472244 0.881468i \(-0.656556\pi\)
−0.472244 + 0.881468i \(0.656556\pi\)
\(594\) −1.97272e7 −2.29403
\(595\) 0 0
\(596\) −2.43950e7 −2.81310
\(597\) −1.95234e7 −2.24192
\(598\) −4.07223e6 −0.465671
\(599\) −9.16029e6 −1.04314 −0.521569 0.853209i \(-0.674653\pi\)
−0.521569 + 0.853209i \(0.674653\pi\)
\(600\) 0 0
\(601\) −1.25891e7 −1.42171 −0.710853 0.703341i \(-0.751691\pi\)
−0.710853 + 0.703341i \(0.751691\pi\)
\(602\) −2.74988e6 −0.309259
\(603\) −7.44917e6 −0.834285
\(604\) −1.07252e7 −1.19623
\(605\) 0 0
\(606\) 1.78474e6 0.197421
\(607\) 1.05315e7 1.16017 0.580083 0.814558i \(-0.303020\pi\)
0.580083 + 0.814558i \(0.303020\pi\)
\(608\) −9.06686e6 −0.994713
\(609\) −363685. −0.0397358
\(610\) 0 0
\(611\) 8.96450e6 0.971455
\(612\) −2.69134e7 −2.90462
\(613\) −2.50815e6 −0.269589 −0.134794 0.990874i \(-0.543037\pi\)
−0.134794 + 0.990874i \(0.543037\pi\)
\(614\) −1.35281e7 −1.44816
\(615\) 0 0
\(616\) −1.47038e7 −1.56127
\(617\) −1.04940e7 −1.10975 −0.554876 0.831933i \(-0.687234\pi\)
−0.554876 + 0.831933i \(0.687234\pi\)
\(618\) 2.98156e7 3.14031
\(619\) −7.31973e6 −0.767836 −0.383918 0.923367i \(-0.625426\pi\)
−0.383918 + 0.923367i \(0.625426\pi\)
\(620\) 0 0
\(621\) 1.58396e6 0.164822
\(622\) −1.52286e7 −1.57828
\(623\) 6.89529e6 0.711758
\(624\) −6.23552e6 −0.641078
\(625\) 0 0
\(626\) 2.55505e7 2.60593
\(627\) −2.03356e7 −2.06580
\(628\) −875446. −0.0885789
\(629\) −6.72610e6 −0.677855
\(630\) 0 0
\(631\) −6.30990e6 −0.630883 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(632\) −5.38476e6 −0.536258
\(633\) −2.24662e6 −0.222854
\(634\) −1.41915e6 −0.140219
\(635\) 0 0
\(636\) 3.43629e7 3.36858
\(637\) 1.11744e7 1.09113
\(638\) 475213. 0.0462207
\(639\) 6.58729e6 0.638196
\(640\) 0 0
\(641\) 176304. 0.0169479 0.00847395 0.999964i \(-0.497303\pi\)
0.00847395 + 0.999964i \(0.497303\pi\)
\(642\) 2.71354e7 2.59835
\(643\) 4.73829e6 0.451954 0.225977 0.974133i \(-0.427443\pi\)
0.225977 + 0.974133i \(0.427443\pi\)
\(644\) 3.50327e6 0.332858
\(645\) 0 0
\(646\) −1.71894e7 −1.62061
\(647\) 1.27894e7 1.20113 0.600563 0.799577i \(-0.294943\pi\)
0.600563 + 0.799577i \(0.294943\pi\)
\(648\) 461191. 0.0431463
\(649\) 2.92158e7 2.72274
\(650\) 0 0
\(651\) 1.10626e7 1.02307
\(652\) 2.06757e7 1.90476
\(653\) −2.31597e6 −0.212545 −0.106272 0.994337i \(-0.533892\pi\)
−0.106272 + 0.994337i \(0.533892\pi\)
\(654\) −3.77179e6 −0.344828
\(655\) 0 0
\(656\) 2.15339e6 0.195373
\(657\) −8.61203e6 −0.778381
\(658\) −1.28250e7 −1.15477
\(659\) −4.32880e6 −0.388288 −0.194144 0.980973i \(-0.562193\pi\)
−0.194144 + 0.980973i \(0.562193\pi\)
\(660\) 0 0
\(661\) 1.34429e7 1.19671 0.598355 0.801231i \(-0.295821\pi\)
0.598355 + 0.801231i \(0.295821\pi\)
\(662\) −1.24287e7 −1.10225
\(663\) −3.75754e7 −3.31986
\(664\) −1.71943e7 −1.51344
\(665\) 0 0
\(666\) −1.62099e7 −1.41610
\(667\) −38156.4 −0.00332088
\(668\) −2.84821e7 −2.46962
\(669\) 2.39077e7 2.06525
\(670\) 0 0
\(671\) −2.41614e7 −2.07164
\(672\) 2.83552e7 2.42220
\(673\) 1.30893e7 1.11398 0.556991 0.830518i \(-0.311956\pi\)
0.556991 + 0.830518i \(0.311956\pi\)
\(674\) 5.40225e6 0.458063
\(675\) 0 0
\(676\) 3.42377e7 2.88163
\(677\) −1.26746e6 −0.106283 −0.0531415 0.998587i \(-0.516923\pi\)
−0.0531415 + 0.998587i \(0.516923\pi\)
\(678\) −4.09399e6 −0.342037
\(679\) −5.41062e6 −0.450373
\(680\) 0 0
\(681\) 1.16944e7 0.966298
\(682\) −1.44550e7 −1.19003
\(683\) 1.77529e7 1.45619 0.728093 0.685479i \(-0.240407\pi\)
0.728093 + 0.685479i \(0.240407\pi\)
\(684\) −2.49107e7 −2.03585
\(685\) 0 0
\(686\) 9.00905e6 0.730918
\(687\) 1.19864e7 0.968944
\(688\) −441769. −0.0355815
\(689\) 2.94797e7 2.36578
\(690\) 0 0
\(691\) 5.47133e6 0.435911 0.217955 0.975959i \(-0.430061\pi\)
0.217955 + 0.975959i \(0.430061\pi\)
\(692\) 9.36360e6 0.743323
\(693\) 3.90780e7 3.09100
\(694\) 1.24394e7 0.980396
\(695\) 0 0
\(696\) 319266. 0.0249822
\(697\) 1.29764e7 1.01175
\(698\) −1.70767e7 −1.32668
\(699\) −1.86683e7 −1.44515
\(700\) 0 0
\(701\) −1.16242e7 −0.893443 −0.446721 0.894673i \(-0.647408\pi\)
−0.446721 + 0.894673i \(0.647408\pi\)
\(702\) −3.37387e7 −2.58396
\(703\) −6.22560e6 −0.475108
\(704\) −3.24035e7 −2.46411
\(705\) 0 0
\(706\) −2.27366e7 −1.71678
\(707\) −1.31719e6 −0.0991060
\(708\) 5.82437e7 4.36682
\(709\) −4.30447e6 −0.321591 −0.160795 0.986988i \(-0.551406\pi\)
−0.160795 + 0.986988i \(0.551406\pi\)
\(710\) 0 0
\(711\) 1.43110e7 1.06168
\(712\) −6.05313e6 −0.447487
\(713\) 1.16064e6 0.0855017
\(714\) 5.37572e7 3.94631
\(715\) 0 0
\(716\) −2.09541e7 −1.52752
\(717\) 3.19130e7 2.31830
\(718\) 5.27883e6 0.382143
\(719\) 1.35107e7 0.974664 0.487332 0.873217i \(-0.337970\pi\)
0.487332 + 0.873217i \(0.337970\pi\)
\(720\) 0 0
\(721\) −2.20048e7 −1.57645
\(722\) 6.27338e6 0.447877
\(723\) −2.19915e7 −1.56462
\(724\) −3.02544e7 −2.14507
\(725\) 0 0
\(726\) −4.68749e7 −3.30064
\(727\) −2.62118e7 −1.83934 −0.919668 0.392697i \(-0.871542\pi\)
−0.919668 + 0.392697i \(0.871542\pi\)
\(728\) −2.51474e7 −1.75859
\(729\) −2.33260e7 −1.62563
\(730\) 0 0
\(731\) −2.66211e6 −0.184260
\(732\) −4.81672e7 −3.32257
\(733\) 2.19136e7 1.50644 0.753222 0.657766i \(-0.228499\pi\)
0.753222 + 0.657766i \(0.228499\pi\)
\(734\) 1.50220e7 1.02917
\(735\) 0 0
\(736\) 2.97492e6 0.202433
\(737\) −1.16909e7 −0.792827
\(738\) 3.12731e7 2.11364
\(739\) −2.72080e6 −0.183267 −0.0916337 0.995793i \(-0.529209\pi\)
−0.0916337 + 0.995793i \(0.529209\pi\)
\(740\) 0 0
\(741\) −3.47793e7 −2.32689
\(742\) −4.21751e7 −2.81220
\(743\) 5.79913e6 0.385381 0.192691 0.981260i \(-0.438279\pi\)
0.192691 + 0.981260i \(0.438279\pi\)
\(744\) −9.71146e6 −0.643209
\(745\) 0 0
\(746\) 2.80120e7 1.84288
\(747\) 4.56969e7 2.99630
\(748\) −4.22384e7 −2.76028
\(749\) −2.00267e7 −1.30438
\(750\) 0 0
\(751\) −2.22708e6 −0.144091 −0.0720455 0.997401i \(-0.522953\pi\)
−0.0720455 + 0.997401i \(0.522953\pi\)
\(752\) −2.06035e6 −0.132861
\(753\) −3.00449e7 −1.93100
\(754\) 812739. 0.0520622
\(755\) 0 0
\(756\) 2.90248e7 1.84699
\(757\) −1.68706e7 −1.07002 −0.535009 0.844846i \(-0.679692\pi\)
−0.535009 + 0.844846i \(0.679692\pi\)
\(758\) 1.77776e7 1.12383
\(759\) 6.67231e6 0.420409
\(760\) 0 0
\(761\) −6.73683e6 −0.421691 −0.210845 0.977519i \(-0.567622\pi\)
−0.210845 + 0.977519i \(0.567622\pi\)
\(762\) −3.49138e7 −2.17826
\(763\) 2.78370e6 0.173105
\(764\) −5.26380e6 −0.326261
\(765\) 0 0
\(766\) 2.88316e6 0.177540
\(767\) 4.99668e7 3.06685
\(768\) −1.56276e7 −0.956071
\(769\) 2.99965e7 1.82917 0.914587 0.404390i \(-0.132516\pi\)
0.914587 + 0.404390i \(0.132516\pi\)
\(770\) 0 0
\(771\) 3.30466e7 2.00212
\(772\) 1.52342e6 0.0919977
\(773\) −2.03964e7 −1.22773 −0.613867 0.789409i \(-0.710387\pi\)
−0.613867 + 0.789409i \(0.710387\pi\)
\(774\) −6.41568e6 −0.384938
\(775\) 0 0
\(776\) 4.74979e6 0.283152
\(777\) 1.94696e7 1.15692
\(778\) 1.20558e7 0.714078
\(779\) 1.20108e7 0.709133
\(780\) 0 0
\(781\) 1.03382e7 0.606482
\(782\) 5.63999e6 0.329808
\(783\) −316128. −0.0184272
\(784\) −2.56827e6 −0.149228
\(785\) 0 0
\(786\) −4.65029e7 −2.68488
\(787\) 9.96285e6 0.573386 0.286693 0.958023i \(-0.407444\pi\)
0.286693 + 0.958023i \(0.407444\pi\)
\(788\) 1.33620e7 0.766578
\(789\) −4.38375e7 −2.50699
\(790\) 0 0
\(791\) 3.02149e6 0.171704
\(792\) −3.43052e7 −1.94333
\(793\) −4.13223e7 −2.33346
\(794\) 1.25695e7 0.707567
\(795\) 0 0
\(796\) −3.75338e7 −2.09961
\(797\) 4.33007e6 0.241462 0.120731 0.992685i \(-0.461476\pi\)
0.120731 + 0.992685i \(0.461476\pi\)
\(798\) 4.97570e7 2.76597
\(799\) −1.24157e7 −0.688026
\(800\) 0 0
\(801\) 1.60873e7 0.885933
\(802\) −3.08425e7 −1.69322
\(803\) −1.35159e7 −0.739701
\(804\) −2.33065e7 −1.27156
\(805\) 0 0
\(806\) −2.47219e7 −1.34043
\(807\) −1.18181e7 −0.638796
\(808\) 1.15631e6 0.0623085
\(809\) 4.65045e6 0.249818 0.124909 0.992168i \(-0.460136\pi\)
0.124909 + 0.992168i \(0.460136\pi\)
\(810\) 0 0
\(811\) −2.23521e7 −1.19334 −0.596672 0.802486i \(-0.703510\pi\)
−0.596672 + 0.802486i \(0.703510\pi\)
\(812\) −699186. −0.0372137
\(813\) 739599. 0.0392437
\(814\) −2.54402e7 −1.34573
\(815\) 0 0
\(816\) 8.63611e6 0.454039
\(817\) −2.46401e6 −0.129148
\(818\) −5.77517e7 −3.01774
\(819\) 6.68336e7 3.48165
\(820\) 0 0
\(821\) 6.86450e6 0.355427 0.177714 0.984082i \(-0.443130\pi\)
0.177714 + 0.984082i \(0.443130\pi\)
\(822\) −4.93008e7 −2.54492
\(823\) 3.78159e7 1.94614 0.973072 0.230502i \(-0.0740369\pi\)
0.973072 + 0.230502i \(0.0740369\pi\)
\(824\) 1.93173e7 0.991123
\(825\) 0 0
\(826\) −7.14849e7 −3.64556
\(827\) −3.11812e7 −1.58536 −0.792681 0.609637i \(-0.791315\pi\)
−0.792681 + 0.609637i \(0.791315\pi\)
\(828\) 8.17342e6 0.414313
\(829\) −7.56161e6 −0.382145 −0.191072 0.981576i \(-0.561196\pi\)
−0.191072 + 0.981576i \(0.561196\pi\)
\(830\) 0 0
\(831\) −2.53998e7 −1.27593
\(832\) −5.54185e7 −2.77553
\(833\) −1.54765e7 −0.772786
\(834\) −1.86794e7 −0.929924
\(835\) 0 0
\(836\) −3.90954e7 −1.93468
\(837\) 9.61600e6 0.474439
\(838\) 1.59351e7 0.783873
\(839\) 1.20663e7 0.591792 0.295896 0.955220i \(-0.404382\pi\)
0.295896 + 0.955220i \(0.404382\pi\)
\(840\) 0 0
\(841\) −2.05035e7 −0.999629
\(842\) 3.02062e7 1.46831
\(843\) −1.87270e6 −0.0907612
\(844\) −4.31914e6 −0.208709
\(845\) 0 0
\(846\) −2.99219e7 −1.43735
\(847\) 3.45951e7 1.65694
\(848\) −6.77544e6 −0.323555
\(849\) 2.88186e7 1.37216
\(850\) 0 0
\(851\) 2.04267e6 0.0966886
\(852\) 2.06099e7 0.972696
\(853\) −2.73307e7 −1.28611 −0.643055 0.765820i \(-0.722333\pi\)
−0.643055 + 0.765820i \(0.722333\pi\)
\(854\) 5.91177e7 2.77378
\(855\) 0 0
\(856\) 1.75808e7 0.820074
\(857\) 3.10213e7 1.44280 0.721402 0.692516i \(-0.243498\pi\)
0.721402 + 0.692516i \(0.243498\pi\)
\(858\) −1.42121e8 −6.59085
\(859\) 2.98878e7 1.38201 0.691006 0.722849i \(-0.257168\pi\)
0.691006 + 0.722849i \(0.257168\pi\)
\(860\) 0 0
\(861\) −3.75619e7 −1.72679
\(862\) −1.24213e7 −0.569375
\(863\) 2.37019e7 1.08332 0.541660 0.840598i \(-0.317796\pi\)
0.541660 + 0.840598i \(0.317796\pi\)
\(864\) 2.46474e7 1.12328
\(865\) 0 0
\(866\) 5.14733e7 2.33232
\(867\) 1.63949e7 0.740734
\(868\) 2.12679e7 0.958131
\(869\) 2.24599e7 1.00892
\(870\) 0 0
\(871\) −1.99945e7 −0.893027
\(872\) −2.44371e6 −0.108832
\(873\) −1.26234e7 −0.560584
\(874\) 5.22030e6 0.231162
\(875\) 0 0
\(876\) −2.69448e7 −1.18636
\(877\) −3.85645e7 −1.69312 −0.846562 0.532290i \(-0.821332\pi\)
−0.846562 + 0.532290i \(0.821332\pi\)
\(878\) 4.60932e7 2.01790
\(879\) 3.28115e7 1.43237
\(880\) 0 0
\(881\) −3.45951e7 −1.50167 −0.750836 0.660489i \(-0.770349\pi\)
−0.750836 + 0.660489i \(0.770349\pi\)
\(882\) −3.72983e7 −1.61442
\(883\) 8.71550e6 0.376175 0.188088 0.982152i \(-0.439771\pi\)
0.188088 + 0.982152i \(0.439771\pi\)
\(884\) −7.22388e7 −3.10914
\(885\) 0 0
\(886\) −5.06947e7 −2.16959
\(887\) −4.58909e6 −0.195847 −0.0979237 0.995194i \(-0.531220\pi\)
−0.0979237 + 0.995194i \(0.531220\pi\)
\(888\) −1.70917e7 −0.727365
\(889\) 2.57675e7 1.09350
\(890\) 0 0
\(891\) −1.92363e6 −0.0811761
\(892\) 4.59627e7 1.93416
\(893\) −1.14918e7 −0.482237
\(894\) 1.13684e8 4.75723
\(895\) 0 0
\(896\) 4.31424e7 1.79529
\(897\) 1.14114e7 0.473541
\(898\) 8.56815e6 0.354565
\(899\) −231642. −0.00955912
\(900\) 0 0
\(901\) −4.08290e7 −1.67555
\(902\) 4.90807e7 2.00860
\(903\) 7.70583e6 0.314485
\(904\) −2.65246e6 −0.107951
\(905\) 0 0
\(906\) 4.99809e7 2.02294
\(907\) −1.10880e7 −0.447544 −0.223772 0.974642i \(-0.571837\pi\)
−0.223772 + 0.974642i \(0.571837\pi\)
\(908\) 2.24826e7 0.904965
\(909\) −3.07311e6 −0.123358
\(910\) 0 0
\(911\) 3.54367e7 1.41468 0.707339 0.706875i \(-0.249896\pi\)
0.707339 + 0.706875i \(0.249896\pi\)
\(912\) 7.99348e6 0.318236
\(913\) 7.17177e7 2.84740
\(914\) −4.93377e7 −1.95350
\(915\) 0 0
\(916\) 2.30440e7 0.907442
\(917\) 3.43206e7 1.34782
\(918\) 4.67276e7 1.83007
\(919\) 3.32990e7 1.30060 0.650298 0.759679i \(-0.274644\pi\)
0.650298 + 0.759679i \(0.274644\pi\)
\(920\) 0 0
\(921\) 3.79092e7 1.47264
\(922\) −1.88323e7 −0.729584
\(923\) 1.76811e7 0.683131
\(924\) 1.22265e8 4.71109
\(925\) 0 0
\(926\) 2.11532e7 0.810677
\(927\) −5.13390e7 −1.96222
\(928\) −593736. −0.0226320
\(929\) 3.43816e7 1.30703 0.653517 0.756912i \(-0.273293\pi\)
0.653517 + 0.756912i \(0.273293\pi\)
\(930\) 0 0
\(931\) −1.43248e7 −0.541645
\(932\) −3.58899e7 −1.35342
\(933\) 4.26743e7 1.60495
\(934\) −7.67335e7 −2.87818
\(935\) 0 0
\(936\) −5.86708e7 −2.18893
\(937\) −3.81989e7 −1.42135 −0.710677 0.703518i \(-0.751611\pi\)
−0.710677 + 0.703518i \(0.751611\pi\)
\(938\) 2.86051e7 1.06154
\(939\) −7.15988e7 −2.64997
\(940\) 0 0
\(941\) 2.92661e7 1.07744 0.538718 0.842486i \(-0.318909\pi\)
0.538718 + 0.842486i \(0.318909\pi\)
\(942\) 4.07969e6 0.149796
\(943\) −3.94085e6 −0.144315
\(944\) −1.14841e7 −0.419437
\(945\) 0 0
\(946\) −1.00689e7 −0.365809
\(947\) −4.49447e6 −0.162856 −0.0814280 0.996679i \(-0.525948\pi\)
−0.0814280 + 0.996679i \(0.525948\pi\)
\(948\) 4.47753e7 1.61814
\(949\) −2.31157e7 −0.833186
\(950\) 0 0
\(951\) 3.97681e6 0.142588
\(952\) 3.48288e7 1.24551
\(953\) 7.96508e6 0.284091 0.142046 0.989860i \(-0.454632\pi\)
0.142046 + 0.989860i \(0.454632\pi\)
\(954\) −9.83978e7 −3.50038
\(955\) 0 0
\(956\) 6.13529e7 2.17115
\(957\) −1.33166e6 −0.0470018
\(958\) −2.77347e7 −0.976360
\(959\) 3.63855e7 1.27756
\(960\) 0 0
\(961\) −2.15831e7 −0.753884
\(962\) −4.35094e7 −1.51581
\(963\) −4.67240e7 −1.62358
\(964\) −4.22788e7 −1.46531
\(965\) 0 0
\(966\) −1.63257e7 −0.562897
\(967\) −3.91114e6 −0.134505 −0.0672523 0.997736i \(-0.521423\pi\)
−0.0672523 + 0.997736i \(0.521423\pi\)
\(968\) −3.03698e7 −1.04173
\(969\) 4.81689e7 1.64800
\(970\) 0 0
\(971\) −2.98964e7 −1.01759 −0.508793 0.860889i \(-0.669908\pi\)
−0.508793 + 0.860889i \(0.669908\pi\)
\(972\) −4.63228e7 −1.57264
\(973\) 1.37860e7 0.466826
\(974\) −4.20831e7 −1.42138
\(975\) 0 0
\(976\) 9.49728e6 0.319135
\(977\) 4.36187e7 1.46196 0.730981 0.682398i \(-0.239063\pi\)
0.730981 + 0.682398i \(0.239063\pi\)
\(978\) −9.63513e7 −3.22115
\(979\) 2.52477e7 0.841909
\(980\) 0 0
\(981\) 6.49459e6 0.215466
\(982\) −3.94964e7 −1.30701
\(983\) 1.95377e7 0.644895 0.322448 0.946587i \(-0.395494\pi\)
0.322448 + 0.946587i \(0.395494\pi\)
\(984\) 3.29743e7 1.08564
\(985\) 0 0
\(986\) −1.12563e6 −0.0368727
\(987\) 3.59390e7 1.17428
\(988\) −6.68633e7 −2.17919
\(989\) 808465. 0.0262827
\(990\) 0 0
\(991\) 3.45462e7 1.11742 0.558709 0.829364i \(-0.311297\pi\)
0.558709 + 0.829364i \(0.311297\pi\)
\(992\) 1.80603e7 0.582701
\(993\) 3.48282e7 1.12088
\(994\) −2.52954e7 −0.812037
\(995\) 0 0
\(996\) 1.42974e8 4.56676
\(997\) −3.34457e7 −1.06562 −0.532810 0.846235i \(-0.678864\pi\)
−0.532810 + 0.846235i \(0.678864\pi\)
\(998\) −3.16068e7 −1.00451
\(999\) 1.69237e7 0.536514
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 1075.6.a.a.1.1 8
5.4 even 2 43.6.a.a.1.8 8
15.14 odd 2 387.6.a.c.1.1 8
20.19 odd 2 688.6.a.e.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.8 8 5.4 even 2
387.6.a.c.1.1 8 15.14 odd 2
688.6.a.e.1.7 8 20.19 odd 2
1075.6.a.a.1.1 8 1.1 even 1 trivial