Properties

Label 245.6.a.m.1.3
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
Analytic rank $0$
Dimension $10$
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] = [245,6,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, 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("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + 7583055 x^{2} + 5935300 x - 22888100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 7^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.72637\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.72637 q^{2} -21.0476 q^{3} -18.1142 q^{4} +25.0000 q^{5} +78.4311 q^{6} +186.744 q^{8} +200.001 q^{9} +O(q^{10})\) \(q-3.72637 q^{2} -21.0476 q^{3} -18.1142 q^{4} +25.0000 q^{5} +78.4311 q^{6} +186.744 q^{8} +200.001 q^{9} -93.1592 q^{10} -164.736 q^{11} +381.259 q^{12} +250.121 q^{13} -526.190 q^{15} -116.224 q^{16} -2178.31 q^{17} -745.276 q^{18} -18.8660 q^{19} -452.854 q^{20} +613.867 q^{22} -2625.76 q^{23} -3930.51 q^{24} +625.000 q^{25} -932.042 q^{26} +905.032 q^{27} -7253.02 q^{29} +1960.78 q^{30} -1686.54 q^{31} -5542.71 q^{32} +3467.29 q^{33} +8117.20 q^{34} -3622.85 q^{36} +738.487 q^{37} +70.3016 q^{38} -5264.44 q^{39} +4668.60 q^{40} -3352.52 q^{41} -15658.3 q^{43} +2984.06 q^{44} +5000.02 q^{45} +9784.56 q^{46} +18604.8 q^{47} +2446.23 q^{48} -2328.98 q^{50} +45848.2 q^{51} -4530.73 q^{52} +24842.7 q^{53} -3372.49 q^{54} -4118.40 q^{55} +397.083 q^{57} +27027.4 q^{58} +32607.2 q^{59} +9531.48 q^{60} -32034.5 q^{61} +6284.69 q^{62} +24373.4 q^{64} +6253.02 q^{65} -12920.4 q^{66} -47369.8 q^{67} +39458.3 q^{68} +55266.0 q^{69} -46886.4 q^{71} +37348.9 q^{72} +42017.5 q^{73} -2751.88 q^{74} -13154.7 q^{75} +341.741 q^{76} +19617.2 q^{78} +14811.9 q^{79} -2905.59 q^{80} -67648.9 q^{81} +12492.7 q^{82} +109539. q^{83} -54457.8 q^{85} +58348.7 q^{86} +152659. q^{87} -30763.4 q^{88} -13073.0 q^{89} -18631.9 q^{90} +47563.5 q^{92} +35497.7 q^{93} -69328.4 q^{94} -471.649 q^{95} +116661. q^{96} +113426. q^{97} -32947.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 58 q^{3} + 182 q^{4} + 250 q^{5} + 144 q^{6} + 270 q^{8} + 700 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 58 q^{3} + 182 q^{4} + 250 q^{5} + 144 q^{6} + 270 q^{8} + 700 q^{9} + 250 q^{10} + 794 q^{11} + 2560 q^{12} + 474 q^{13} + 1450 q^{15} + 2394 q^{16} + 802 q^{17} + 3702 q^{18} + 7292 q^{19} + 4550 q^{20} + 3948 q^{22} + 3708 q^{23} + 2092 q^{24} + 6250 q^{25} + 6576 q^{26} + 11818 q^{27} - 8866 q^{29} + 3600 q^{30} + 13292 q^{31} + 2590 q^{32} + 9854 q^{33} + 44468 q^{34} - 10690 q^{36} + 16124 q^{37} + 2180 q^{38} - 24982 q^{39} + 6750 q^{40} + 34836 q^{41} - 28604 q^{43} - 31120 q^{44} + 17500 q^{45} - 39732 q^{46} + 18106 q^{47} + 101788 q^{48} + 6250 q^{50} + 31602 q^{51} - 22480 q^{52} + 36440 q^{53} + 80836 q^{54} + 19850 q^{55} + 126988 q^{57} - 100356 q^{58} + 18644 q^{59} + 64000 q^{60} + 68120 q^{61} + 181052 q^{62} - 59358 q^{64} + 11850 q^{65} + 157780 q^{66} + 92328 q^{67} + 288540 q^{68} + 170888 q^{69} + 5044 q^{71} - 61654 q^{72} + 170160 q^{73} - 216584 q^{74} + 36250 q^{75} + 505180 q^{76} - 158008 q^{78} + 26442 q^{79} + 59850 q^{80} - 56314 q^{81} + 353948 q^{82} + 353360 q^{83} + 20050 q^{85} - 52940 q^{86} - 3190 q^{87} - 114916 q^{88} + 90704 q^{89} + 92550 q^{90} + 183520 q^{92} + 188560 q^{93} - 121388 q^{94} + 182300 q^{95} + 442220 q^{96} + 236382 q^{97} + 109024 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.72637 −0.658735 −0.329368 0.944202i \(-0.606836\pi\)
−0.329368 + 0.944202i \(0.606836\pi\)
\(3\) −21.0476 −1.35020 −0.675101 0.737725i \(-0.735900\pi\)
−0.675101 + 0.737725i \(0.735900\pi\)
\(4\) −18.1142 −0.566068
\(5\) 25.0000 0.447214
\(6\) 78.4311 0.889426
\(7\) 0 0
\(8\) 186.744 1.03162
\(9\) 200.001 0.823048
\(10\) −93.1592 −0.294595
\(11\) −164.736 −0.410494 −0.205247 0.978710i \(-0.565800\pi\)
−0.205247 + 0.978710i \(0.565800\pi\)
\(12\) 381.259 0.764306
\(13\) 250.121 0.410479 0.205240 0.978712i \(-0.434203\pi\)
0.205240 + 0.978712i \(0.434203\pi\)
\(14\) 0 0
\(15\) −526.190 −0.603829
\(16\) −116.224 −0.113500
\(17\) −2178.31 −1.82809 −0.914046 0.405611i \(-0.867059\pi\)
−0.914046 + 0.405611i \(0.867059\pi\)
\(18\) −745.276 −0.542171
\(19\) −18.8660 −0.0119893 −0.00599467 0.999982i \(-0.501908\pi\)
−0.00599467 + 0.999982i \(0.501908\pi\)
\(20\) −452.854 −0.253153
\(21\) 0 0
\(22\) 613.867 0.270407
\(23\) −2625.76 −1.03499 −0.517495 0.855686i \(-0.673135\pi\)
−0.517495 + 0.855686i \(0.673135\pi\)
\(24\) −3930.51 −1.39290
\(25\) 625.000 0.200000
\(26\) −932.042 −0.270397
\(27\) 905.032 0.238921
\(28\) 0 0
\(29\) −7253.02 −1.60149 −0.800745 0.599006i \(-0.795563\pi\)
−0.800745 + 0.599006i \(0.795563\pi\)
\(30\) 1960.78 0.397764
\(31\) −1686.54 −0.315205 −0.157603 0.987503i \(-0.550377\pi\)
−0.157603 + 0.987503i \(0.550377\pi\)
\(32\) −5542.71 −0.956858
\(33\) 3467.29 0.554250
\(34\) 8117.20 1.20423
\(35\) 0 0
\(36\) −3622.85 −0.465901
\(37\) 738.487 0.0886827 0.0443413 0.999016i \(-0.485881\pi\)
0.0443413 + 0.999016i \(0.485881\pi\)
\(38\) 70.3016 0.00789780
\(39\) −5264.44 −0.554230
\(40\) 4668.60 0.461356
\(41\) −3352.52 −0.311467 −0.155733 0.987799i \(-0.549774\pi\)
−0.155733 + 0.987799i \(0.549774\pi\)
\(42\) 0 0
\(43\) −15658.3 −1.29144 −0.645720 0.763574i \(-0.723443\pi\)
−0.645720 + 0.763574i \(0.723443\pi\)
\(44\) 2984.06 0.232367
\(45\) 5000.02 0.368078
\(46\) 9784.56 0.681784
\(47\) 18604.8 1.22851 0.614257 0.789106i \(-0.289456\pi\)
0.614257 + 0.789106i \(0.289456\pi\)
\(48\) 2446.23 0.153247
\(49\) 0 0
\(50\) −2328.98 −0.131747
\(51\) 45848.2 2.46829
\(52\) −4530.73 −0.232359
\(53\) 24842.7 1.21481 0.607405 0.794392i \(-0.292211\pi\)
0.607405 + 0.794392i \(0.292211\pi\)
\(54\) −3372.49 −0.157386
\(55\) −4118.40 −0.183579
\(56\) 0 0
\(57\) 397.083 0.0161880
\(58\) 27027.4 1.05496
\(59\) 32607.2 1.21950 0.609752 0.792592i \(-0.291269\pi\)
0.609752 + 0.792592i \(0.291269\pi\)
\(60\) 9531.48 0.341808
\(61\) −32034.5 −1.10228 −0.551141 0.834412i \(-0.685808\pi\)
−0.551141 + 0.834412i \(0.685808\pi\)
\(62\) 6284.69 0.207637
\(63\) 0 0
\(64\) 24373.4 0.743816
\(65\) 6253.02 0.183572
\(66\) −12920.4 −0.365104
\(67\) −47369.8 −1.28918 −0.644592 0.764527i \(-0.722973\pi\)
−0.644592 + 0.764527i \(0.722973\pi\)
\(68\) 39458.3 1.03482
\(69\) 55266.0 1.39745
\(70\) 0 0
\(71\) −46886.4 −1.10383 −0.551913 0.833902i \(-0.686102\pi\)
−0.551913 + 0.833902i \(0.686102\pi\)
\(72\) 37348.9 0.849076
\(73\) 42017.5 0.922833 0.461417 0.887184i \(-0.347341\pi\)
0.461417 + 0.887184i \(0.347341\pi\)
\(74\) −2751.88 −0.0584184
\(75\) −13154.7 −0.270041
\(76\) 341.741 0.00678678
\(77\) 0 0
\(78\) 19617.2 0.365091
\(79\) 14811.9 0.267020 0.133510 0.991047i \(-0.457375\pi\)
0.133510 + 0.991047i \(0.457375\pi\)
\(80\) −2905.59 −0.0507586
\(81\) −67648.9 −1.14564
\(82\) 12492.7 0.205174
\(83\) 109539. 1.74531 0.872657 0.488334i \(-0.162395\pi\)
0.872657 + 0.488334i \(0.162395\pi\)
\(84\) 0 0
\(85\) −54457.8 −0.817547
\(86\) 58348.7 0.850717
\(87\) 152659. 2.16234
\(88\) −30763.4 −0.423476
\(89\) −13073.0 −0.174944 −0.0874720 0.996167i \(-0.527879\pi\)
−0.0874720 + 0.996167i \(0.527879\pi\)
\(90\) −18631.9 −0.242466
\(91\) 0 0
\(92\) 47563.5 0.585874
\(93\) 35497.7 0.425591
\(94\) −69328.4 −0.809266
\(95\) −471.649 −0.00536180
\(96\) 116661. 1.29195
\(97\) 113426. 1.22400 0.612002 0.790856i \(-0.290364\pi\)
0.612002 + 0.790856i \(0.290364\pi\)
\(98\) 0 0
\(99\) −32947.3 −0.337856
\(100\) −11321.4 −0.113214
\(101\) −200104. −1.95188 −0.975939 0.218044i \(-0.930032\pi\)
−0.975939 + 0.218044i \(0.930032\pi\)
\(102\) −170847. −1.62595
\(103\) 90658.9 0.842010 0.421005 0.907058i \(-0.361677\pi\)
0.421005 + 0.907058i \(0.361677\pi\)
\(104\) 46708.5 0.423460
\(105\) 0 0
\(106\) −92572.9 −0.800238
\(107\) −172387. −1.45561 −0.727805 0.685784i \(-0.759459\pi\)
−0.727805 + 0.685784i \(0.759459\pi\)
\(108\) −16393.9 −0.135246
\(109\) −29409.0 −0.237090 −0.118545 0.992949i \(-0.537823\pi\)
−0.118545 + 0.992949i \(0.537823\pi\)
\(110\) 15346.7 0.120930
\(111\) −15543.4 −0.119740
\(112\) 0 0
\(113\) 167768. 1.23598 0.617991 0.786185i \(-0.287947\pi\)
0.617991 + 0.786185i \(0.287947\pi\)
\(114\) −1479.68 −0.0106636
\(115\) −65644.1 −0.462861
\(116\) 131382. 0.906551
\(117\) 50024.3 0.337844
\(118\) −121506. −0.803330
\(119\) 0 0
\(120\) −98262.7 −0.622925
\(121\) −133913. −0.831495
\(122\) 119372. 0.726113
\(123\) 70562.4 0.420543
\(124\) 30550.3 0.178428
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) 109259. 0.601101 0.300551 0.953766i \(-0.402830\pi\)
0.300551 + 0.953766i \(0.402830\pi\)
\(128\) 86542.7 0.466880
\(129\) 329570. 1.74371
\(130\) −23301.1 −0.120925
\(131\) 125858. 0.640770 0.320385 0.947287i \(-0.396188\pi\)
0.320385 + 0.947287i \(0.396188\pi\)
\(132\) −62807.2 −0.313743
\(133\) 0 0
\(134\) 176517. 0.849231
\(135\) 22625.8 0.106849
\(136\) −406787. −1.88590
\(137\) 40244.5 0.183191 0.0915957 0.995796i \(-0.470803\pi\)
0.0915957 + 0.995796i \(0.470803\pi\)
\(138\) −205941. −0.920547
\(139\) 397049. 1.74304 0.871519 0.490362i \(-0.163135\pi\)
0.871519 + 0.490362i \(0.163135\pi\)
\(140\) 0 0
\(141\) −391586. −1.65874
\(142\) 174716. 0.727129
\(143\) −41203.9 −0.168499
\(144\) −23244.8 −0.0934156
\(145\) −181326. −0.716208
\(146\) −156573. −0.607903
\(147\) 0 0
\(148\) −13377.1 −0.0502004
\(149\) −243879. −0.899928 −0.449964 0.893047i \(-0.648563\pi\)
−0.449964 + 0.893047i \(0.648563\pi\)
\(150\) 49019.4 0.177885
\(151\) 487037. 1.73828 0.869139 0.494567i \(-0.164674\pi\)
0.869139 + 0.494567i \(0.164674\pi\)
\(152\) −3523.11 −0.0123685
\(153\) −435664. −1.50461
\(154\) 0 0
\(155\) −42163.6 −0.140964
\(156\) 95360.9 0.313732
\(157\) 138655. 0.448939 0.224470 0.974481i \(-0.427935\pi\)
0.224470 + 0.974481i \(0.427935\pi\)
\(158\) −55194.8 −0.175896
\(159\) −522878. −1.64024
\(160\) −138568. −0.427920
\(161\) 0 0
\(162\) 252085. 0.754674
\(163\) 329961. 0.972732 0.486366 0.873755i \(-0.338322\pi\)
0.486366 + 0.873755i \(0.338322\pi\)
\(164\) 60728.1 0.176311
\(165\) 86682.4 0.247868
\(166\) −408183. −1.14970
\(167\) −218143. −0.605273 −0.302636 0.953106i \(-0.597867\pi\)
−0.302636 + 0.953106i \(0.597867\pi\)
\(168\) 0 0
\(169\) −308733. −0.831507
\(170\) 202930. 0.538547
\(171\) −3773.21 −0.00986780
\(172\) 283638. 0.731043
\(173\) −161771. −0.410948 −0.205474 0.978663i \(-0.565874\pi\)
−0.205474 + 0.978663i \(0.565874\pi\)
\(174\) −568862. −1.42441
\(175\) 0 0
\(176\) 19146.2 0.0465909
\(177\) −686302. −1.64658
\(178\) 48714.7 0.115242
\(179\) −139043. −0.324351 −0.162175 0.986762i \(-0.551851\pi\)
−0.162175 + 0.986762i \(0.551851\pi\)
\(180\) −90571.1 −0.208357
\(181\) −625698. −1.41961 −0.709804 0.704399i \(-0.751217\pi\)
−0.709804 + 0.704399i \(0.751217\pi\)
\(182\) 0 0
\(183\) 674248. 1.48831
\(184\) −490345. −1.06772
\(185\) 18462.2 0.0396601
\(186\) −132277. −0.280352
\(187\) 358847. 0.750421
\(188\) −337011. −0.695423
\(189\) 0 0
\(190\) 1757.54 0.00353200
\(191\) −144788. −0.287178 −0.143589 0.989637i \(-0.545864\pi\)
−0.143589 + 0.989637i \(0.545864\pi\)
\(192\) −513000. −1.00430
\(193\) 363906. 0.703227 0.351613 0.936145i \(-0.385633\pi\)
0.351613 + 0.936145i \(0.385633\pi\)
\(194\) −422667. −0.806295
\(195\) −131611. −0.247859
\(196\) 0 0
\(197\) 839307. 1.54083 0.770416 0.637541i \(-0.220048\pi\)
0.770416 + 0.637541i \(0.220048\pi\)
\(198\) 122774. 0.222558
\(199\) −214187. −0.383408 −0.191704 0.981453i \(-0.561401\pi\)
−0.191704 + 0.981453i \(0.561401\pi\)
\(200\) 116715. 0.206325
\(201\) 997020. 1.74066
\(202\) 745662. 1.28577
\(203\) 0 0
\(204\) −830502. −1.39722
\(205\) −83812.9 −0.139292
\(206\) −337829. −0.554662
\(207\) −525154. −0.851846
\(208\) −29069.9 −0.0465892
\(209\) 3107.91 0.00492155
\(210\) 0 0
\(211\) 771009. 1.19221 0.596106 0.802906i \(-0.296714\pi\)
0.596106 + 0.802906i \(0.296714\pi\)
\(212\) −450004. −0.687665
\(213\) 986844. 1.49039
\(214\) 642378. 0.958862
\(215\) −391458. −0.577550
\(216\) 169009. 0.246477
\(217\) 0 0
\(218\) 109589. 0.156180
\(219\) −884367. −1.24601
\(220\) 74601.4 0.103918
\(221\) −544841. −0.750394
\(222\) 57920.4 0.0788767
\(223\) 217469. 0.292843 0.146422 0.989222i \(-0.453224\pi\)
0.146422 + 0.989222i \(0.453224\pi\)
\(224\) 0 0
\(225\) 125000. 0.164610
\(226\) −625165. −0.814186
\(227\) −1.07247e6 −1.38140 −0.690699 0.723143i \(-0.742697\pi\)
−0.690699 + 0.723143i \(0.742697\pi\)
\(228\) −7192.83 −0.00916353
\(229\) 122096. 0.153855 0.0769277 0.997037i \(-0.475489\pi\)
0.0769277 + 0.997037i \(0.475489\pi\)
\(230\) 244614. 0.304903
\(231\) 0 0
\(232\) −1.35446e6 −1.65213
\(233\) 153610. 0.185366 0.0926831 0.995696i \(-0.470456\pi\)
0.0926831 + 0.995696i \(0.470456\pi\)
\(234\) −186409. −0.222550
\(235\) 465120. 0.549409
\(236\) −590652. −0.690322
\(237\) −311756. −0.360532
\(238\) 0 0
\(239\) 580774. 0.657677 0.328838 0.944386i \(-0.393343\pi\)
0.328838 + 0.944386i \(0.393343\pi\)
\(240\) 61155.6 0.0685344
\(241\) 25512.4 0.0282949 0.0141475 0.999900i \(-0.495497\pi\)
0.0141475 + 0.999900i \(0.495497\pi\)
\(242\) 499010. 0.547735
\(243\) 1.20392e6 1.30793
\(244\) 580278. 0.623967
\(245\) 0 0
\(246\) −262942. −0.277027
\(247\) −4718.77 −0.00492138
\(248\) −314952. −0.325173
\(249\) −2.30553e6 −2.35653
\(250\) −58224.5 −0.0589191
\(251\) 1.55324e6 1.55616 0.778080 0.628165i \(-0.216194\pi\)
0.778080 + 0.628165i \(0.216194\pi\)
\(252\) 0 0
\(253\) 432558. 0.424857
\(254\) −407139. −0.395967
\(255\) 1.14621e6 1.10385
\(256\) −1.10244e6 −1.05137
\(257\) −679193. −0.641446 −0.320723 0.947173i \(-0.603926\pi\)
−0.320723 + 0.947173i \(0.603926\pi\)
\(258\) −1.22810e6 −1.14864
\(259\) 0 0
\(260\) −113268. −0.103914
\(261\) −1.45061e6 −1.31810
\(262\) −468993. −0.422098
\(263\) 1.90847e6 1.70136 0.850680 0.525684i \(-0.176191\pi\)
0.850680 + 0.525684i \(0.176191\pi\)
\(264\) 647496. 0.571778
\(265\) 621066. 0.543279
\(266\) 0 0
\(267\) 275154. 0.236210
\(268\) 858065. 0.729765
\(269\) 965048. 0.813145 0.406573 0.913619i \(-0.366724\pi\)
0.406573 + 0.913619i \(0.366724\pi\)
\(270\) −84312.1 −0.0703851
\(271\) 1.27445e6 1.05414 0.527071 0.849822i \(-0.323290\pi\)
0.527071 + 0.849822i \(0.323290\pi\)
\(272\) 253171. 0.207488
\(273\) 0 0
\(274\) −149966. −0.120675
\(275\) −102960. −0.0820988
\(276\) −1.00110e6 −0.791049
\(277\) 187333. 0.146695 0.0733474 0.997306i \(-0.476632\pi\)
0.0733474 + 0.997306i \(0.476632\pi\)
\(278\) −1.47955e6 −1.14820
\(279\) −337310. −0.259429
\(280\) 0 0
\(281\) 2.24432e6 1.69558 0.847792 0.530329i \(-0.177931\pi\)
0.847792 + 0.530329i \(0.177931\pi\)
\(282\) 1.45919e6 1.09267
\(283\) 2.40515e6 1.78516 0.892578 0.450894i \(-0.148895\pi\)
0.892578 + 0.450894i \(0.148895\pi\)
\(284\) 849307. 0.624840
\(285\) 9927.08 0.00723951
\(286\) 153541. 0.110996
\(287\) 0 0
\(288\) −1.10855e6 −0.787540
\(289\) 3.32519e6 2.34192
\(290\) 675686. 0.471791
\(291\) −2.38734e6 −1.65265
\(292\) −761112. −0.522386
\(293\) −978243. −0.665698 −0.332849 0.942980i \(-0.608010\pi\)
−0.332849 + 0.942980i \(0.608010\pi\)
\(294\) 0 0
\(295\) 815179. 0.545379
\(296\) 137908. 0.0914872
\(297\) −149091. −0.0980757
\(298\) 908782. 0.592815
\(299\) −656758. −0.424842
\(300\) 238287. 0.152861
\(301\) 0 0
\(302\) −1.81488e6 −1.14507
\(303\) 4.21171e6 2.63543
\(304\) 2192.67 0.00136078
\(305\) −800862. −0.492956
\(306\) 1.62344e6 0.991138
\(307\) 199481. 0.120797 0.0603984 0.998174i \(-0.480763\pi\)
0.0603984 + 0.998174i \(0.480763\pi\)
\(308\) 0 0
\(309\) −1.90815e6 −1.13688
\(310\) 157117. 0.0928580
\(311\) −641054. −0.375832 −0.187916 0.982185i \(-0.560173\pi\)
−0.187916 + 0.982185i \(0.560173\pi\)
\(312\) −983101. −0.571757
\(313\) 823954. 0.475381 0.237691 0.971341i \(-0.423610\pi\)
0.237691 + 0.971341i \(0.423610\pi\)
\(314\) −516681. −0.295732
\(315\) 0 0
\(316\) −268306. −0.151152
\(317\) 1.05147e6 0.587689 0.293845 0.955853i \(-0.405065\pi\)
0.293845 + 0.955853i \(0.405065\pi\)
\(318\) 1.94844e6 1.08048
\(319\) 1.19483e6 0.657402
\(320\) 609334. 0.332645
\(321\) 3.62833e6 1.96537
\(322\) 0 0
\(323\) 41096.0 0.0219176
\(324\) 1.22540e6 0.648510
\(325\) 156325. 0.0820959
\(326\) −1.22956e6 −0.640773
\(327\) 618988. 0.320120
\(328\) −626062. −0.321316
\(329\) 0 0
\(330\) −323011. −0.163280
\(331\) −1.72153e6 −0.863661 −0.431831 0.901955i \(-0.642132\pi\)
−0.431831 + 0.901955i \(0.642132\pi\)
\(332\) −1.98421e6 −0.987966
\(333\) 147698. 0.0729901
\(334\) 812883. 0.398714
\(335\) −1.18425e6 −0.576540
\(336\) 0 0
\(337\) −2.89933e6 −1.39067 −0.695334 0.718687i \(-0.744744\pi\)
−0.695334 + 0.718687i \(0.744744\pi\)
\(338\) 1.15045e6 0.547743
\(339\) −3.53111e6 −1.66883
\(340\) 986458. 0.462787
\(341\) 277835. 0.129390
\(342\) 14060.4 0.00650027
\(343\) 0 0
\(344\) −2.92410e6 −1.33228
\(345\) 1.38165e6 0.624957
\(346\) 602820. 0.270706
\(347\) 2.16973e6 0.967346 0.483673 0.875249i \(-0.339302\pi\)
0.483673 + 0.875249i \(0.339302\pi\)
\(348\) −2.76528e6 −1.22403
\(349\) 2.32054e6 1.01982 0.509912 0.860226i \(-0.329678\pi\)
0.509912 + 0.860226i \(0.329678\pi\)
\(350\) 0 0
\(351\) 226367. 0.0980722
\(352\) 913085. 0.392784
\(353\) 3.37510e6 1.44162 0.720808 0.693135i \(-0.243771\pi\)
0.720808 + 0.693135i \(0.243771\pi\)
\(354\) 2.55742e6 1.08466
\(355\) −1.17216e6 −0.493646
\(356\) 236806. 0.0990301
\(357\) 0 0
\(358\) 518124. 0.213661
\(359\) −1.33592e6 −0.547070 −0.273535 0.961862i \(-0.588193\pi\)
−0.273535 + 0.961862i \(0.588193\pi\)
\(360\) 933723. 0.379718
\(361\) −2.47574e6 −0.999856
\(362\) 2.33158e6 0.935146
\(363\) 2.81855e6 1.12269
\(364\) 0 0
\(365\) 1.05044e6 0.412704
\(366\) −2.51250e6 −0.980399
\(367\) −970553. −0.376144 −0.188072 0.982155i \(-0.560224\pi\)
−0.188072 + 0.982155i \(0.560224\pi\)
\(368\) 305176. 0.117471
\(369\) −670506. −0.256352
\(370\) −68796.9 −0.0261255
\(371\) 0 0
\(372\) −643011. −0.240913
\(373\) −985907. −0.366914 −0.183457 0.983028i \(-0.558729\pi\)
−0.183457 + 0.983028i \(0.558729\pi\)
\(374\) −1.33719e6 −0.494329
\(375\) −328868. −0.120766
\(376\) 3.47433e6 1.26737
\(377\) −1.81413e6 −0.657378
\(378\) 0 0
\(379\) 3.30811e6 1.18299 0.591496 0.806308i \(-0.298538\pi\)
0.591496 + 0.806308i \(0.298538\pi\)
\(380\) 8543.54 0.00303514
\(381\) −2.29964e6 −0.811609
\(382\) 539535. 0.189174
\(383\) −4.09037e6 −1.42484 −0.712419 0.701754i \(-0.752401\pi\)
−0.712419 + 0.701754i \(0.752401\pi\)
\(384\) −1.82151e6 −0.630383
\(385\) 0 0
\(386\) −1.35605e6 −0.463240
\(387\) −3.13168e6 −1.06292
\(388\) −2.05462e6 −0.692869
\(389\) −5.11644e6 −1.71433 −0.857164 0.515043i \(-0.827776\pi\)
−0.857164 + 0.515043i \(0.827776\pi\)
\(390\) 490431. 0.163274
\(391\) 5.71973e6 1.89206
\(392\) 0 0
\(393\) −2.64900e6 −0.865169
\(394\) −3.12757e6 −1.01500
\(395\) 370299. 0.119415
\(396\) 596813. 0.191250
\(397\) 5.36835e6 1.70948 0.854742 0.519054i \(-0.173716\pi\)
0.854742 + 0.519054i \(0.173716\pi\)
\(398\) 798140. 0.252564
\(399\) 0 0
\(400\) −72639.7 −0.0226999
\(401\) −3.83942e6 −1.19235 −0.596176 0.802854i \(-0.703314\pi\)
−0.596176 + 0.802854i \(0.703314\pi\)
\(402\) −3.71527e6 −1.14663
\(403\) −421840. −0.129385
\(404\) 3.62472e6 1.10490
\(405\) −1.69122e6 −0.512346
\(406\) 0 0
\(407\) −121655. −0.0364037
\(408\) 8.56187e6 2.54635
\(409\) 576841. 0.170509 0.0852546 0.996359i \(-0.472830\pi\)
0.0852546 + 0.996359i \(0.472830\pi\)
\(410\) 312318. 0.0917566
\(411\) −847049. −0.247345
\(412\) −1.64221e6 −0.476635
\(413\) 0 0
\(414\) 1.95692e6 0.561141
\(415\) 2.73847e6 0.780528
\(416\) −1.38635e6 −0.392770
\(417\) −8.35691e6 −2.35345
\(418\) −11581.2 −0.00324200
\(419\) 65975.8 0.0183590 0.00917951 0.999958i \(-0.497078\pi\)
0.00917951 + 0.999958i \(0.497078\pi\)
\(420\) 0 0
\(421\) −2.95046e6 −0.811306 −0.405653 0.914027i \(-0.632956\pi\)
−0.405653 + 0.914027i \(0.632956\pi\)
\(422\) −2.87307e6 −0.785352
\(423\) 3.72097e6 1.01113
\(424\) 4.63921e6 1.25323
\(425\) −1.36145e6 −0.365618
\(426\) −3.67735e6 −0.981772
\(427\) 0 0
\(428\) 3.12265e6 0.823974
\(429\) 867242. 0.227508
\(430\) 1.45872e6 0.380452
\(431\) 3.42138e6 0.887174 0.443587 0.896231i \(-0.353706\pi\)
0.443587 + 0.896231i \(0.353706\pi\)
\(432\) −105186. −0.0271175
\(433\) −6.00598e6 −1.53944 −0.769722 0.638379i \(-0.779605\pi\)
−0.769722 + 0.638379i \(0.779605\pi\)
\(434\) 0 0
\(435\) 3.81646e6 0.967026
\(436\) 532720. 0.134209
\(437\) 49537.6 0.0124088
\(438\) 3.29548e6 0.820792
\(439\) 58817.4 0.0145661 0.00728307 0.999973i \(-0.497682\pi\)
0.00728307 + 0.999973i \(0.497682\pi\)
\(440\) −769086. −0.189384
\(441\) 0 0
\(442\) 2.03028e6 0.494311
\(443\) −3.30577e6 −0.800320 −0.400160 0.916445i \(-0.631046\pi\)
−0.400160 + 0.916445i \(0.631046\pi\)
\(444\) 281555. 0.0677807
\(445\) −326824. −0.0782373
\(446\) −810369. −0.192906
\(447\) 5.13305e6 1.21509
\(448\) 0 0
\(449\) 6.71550e6 1.57204 0.786018 0.618203i \(-0.212139\pi\)
0.786018 + 0.618203i \(0.212139\pi\)
\(450\) −465798. −0.108434
\(451\) 552280. 0.127855
\(452\) −3.03897e6 −0.699650
\(453\) −1.02509e7 −2.34703
\(454\) 3.99640e6 0.909975
\(455\) 0 0
\(456\) 74152.9 0.0167000
\(457\) −8.13981e6 −1.82316 −0.911578 0.411126i \(-0.865135\pi\)
−0.911578 + 0.411126i \(0.865135\pi\)
\(458\) −454975. −0.101350
\(459\) −1.97144e6 −0.436770
\(460\) 1.18909e6 0.262011
\(461\) −6.80207e6 −1.49070 −0.745348 0.666676i \(-0.767717\pi\)
−0.745348 + 0.666676i \(0.767717\pi\)
\(462\) 0 0
\(463\) −1.62848e6 −0.353045 −0.176523 0.984297i \(-0.556485\pi\)
−0.176523 + 0.984297i \(0.556485\pi\)
\(464\) 842972. 0.181768
\(465\) 887442. 0.190330
\(466\) −572409. −0.122107
\(467\) −4.85435e6 −1.03000 −0.515002 0.857189i \(-0.672209\pi\)
−0.515002 + 0.857189i \(0.672209\pi\)
\(468\) −906149. −0.191243
\(469\) 0 0
\(470\) −1.73321e6 −0.361915
\(471\) −2.91836e6 −0.606159
\(472\) 6.08919e6 1.25807
\(473\) 2.57949e6 0.530128
\(474\) 1.16172e6 0.237495
\(475\) −11791.2 −0.00239787
\(476\) 0 0
\(477\) 4.96855e6 0.999847
\(478\) −2.16418e6 −0.433235
\(479\) −1.79890e6 −0.358235 −0.179118 0.983828i \(-0.557324\pi\)
−0.179118 + 0.983828i \(0.557324\pi\)
\(480\) 2.91652e6 0.577779
\(481\) 184711. 0.0364024
\(482\) −95068.6 −0.0186389
\(483\) 0 0
\(484\) 2.42572e6 0.470682
\(485\) 2.83565e6 0.547391
\(486\) −4.48626e6 −0.861577
\(487\) 5.00914e6 0.957063 0.478531 0.878070i \(-0.341169\pi\)
0.478531 + 0.878070i \(0.341169\pi\)
\(488\) −5.98224e6 −1.13714
\(489\) −6.94487e6 −1.31339
\(490\) 0 0
\(491\) −439989. −0.0823641 −0.0411820 0.999152i \(-0.513112\pi\)
−0.0411820 + 0.999152i \(0.513112\pi\)
\(492\) −1.27818e6 −0.238056
\(493\) 1.57993e7 2.92767
\(494\) 17583.9 0.00324188
\(495\) −823683. −0.151094
\(496\) 196016. 0.0357757
\(497\) 0 0
\(498\) 8.59126e6 1.55233
\(499\) −178880. −0.0321597 −0.0160798 0.999871i \(-0.505119\pi\)
−0.0160798 + 0.999871i \(0.505119\pi\)
\(500\) −283034. −0.0506306
\(501\) 4.59139e6 0.817241
\(502\) −5.78795e6 −1.02510
\(503\) −447148. −0.0788009 −0.0394004 0.999224i \(-0.512545\pi\)
−0.0394004 + 0.999224i \(0.512545\pi\)
\(504\) 0 0
\(505\) −5.00260e6 −0.872906
\(506\) −1.61187e6 −0.279868
\(507\) 6.49807e6 1.12270
\(508\) −1.97914e6 −0.340264
\(509\) −98185.6 −0.0167978 −0.00839892 0.999965i \(-0.502673\pi\)
−0.00839892 + 0.999965i \(0.502673\pi\)
\(510\) −4.27118e6 −0.727148
\(511\) 0 0
\(512\) 1.33872e6 0.225692
\(513\) −17074.3 −0.00286451
\(514\) 2.53092e6 0.422543
\(515\) 2.26647e6 0.376558
\(516\) −5.96989e6 −0.987056
\(517\) −3.06488e6 −0.504298
\(518\) 0 0
\(519\) 3.40490e6 0.554863
\(520\) 1.16771e6 0.189377
\(521\) −1.02358e7 −1.65207 −0.826036 0.563618i \(-0.809409\pi\)
−0.826036 + 0.563618i \(0.809409\pi\)
\(522\) 5.40551e6 0.868281
\(523\) 7.82929e6 1.25161 0.625804 0.779981i \(-0.284771\pi\)
0.625804 + 0.779981i \(0.284771\pi\)
\(524\) −2.27981e6 −0.362719
\(525\) 0 0
\(526\) −7.11167e6 −1.12075
\(527\) 3.67382e6 0.576224
\(528\) −402981. −0.0629072
\(529\) 458288. 0.0712032
\(530\) −2.31432e6 −0.357877
\(531\) 6.52146e6 1.00371
\(532\) 0 0
\(533\) −838534. −0.127851
\(534\) −1.02533e6 −0.155600
\(535\) −4.30967e6 −0.650968
\(536\) −8.84603e6 −1.32995
\(537\) 2.92651e6 0.437939
\(538\) −3.59612e6 −0.535647
\(539\) 0 0
\(540\) −409848. −0.0604837
\(541\) −5.83527e6 −0.857172 −0.428586 0.903501i \(-0.640988\pi\)
−0.428586 + 0.903501i \(0.640988\pi\)
\(542\) −4.74906e6 −0.694400
\(543\) 1.31694e7 1.91676
\(544\) 1.20738e7 1.74922
\(545\) −735225. −0.106030
\(546\) 0 0
\(547\) 1.82111e6 0.260237 0.130118 0.991498i \(-0.458464\pi\)
0.130118 + 0.991498i \(0.458464\pi\)
\(548\) −728995. −0.103699
\(549\) −6.40692e6 −0.907232
\(550\) 383667. 0.0540814
\(551\) 136835. 0.0192008
\(552\) 1.03206e7 1.44164
\(553\) 0 0
\(554\) −698072. −0.0966331
\(555\) −388584. −0.0535492
\(556\) −7.19221e6 −0.986677
\(557\) −1.09965e7 −1.50182 −0.750911 0.660404i \(-0.770385\pi\)
−0.750911 + 0.660404i \(0.770385\pi\)
\(558\) 1.25694e6 0.170895
\(559\) −3.91647e6 −0.530109
\(560\) 0 0
\(561\) −7.55285e6 −1.01322
\(562\) −8.36317e6 −1.11694
\(563\) 2.09557e6 0.278632 0.139316 0.990248i \(-0.455510\pi\)
0.139316 + 0.990248i \(0.455510\pi\)
\(564\) 7.09326e6 0.938962
\(565\) 4.19419e6 0.552748
\(566\) −8.96248e6 −1.17594
\(567\) 0 0
\(568\) −8.75574e6 −1.13873
\(569\) 6.24664e6 0.808846 0.404423 0.914572i \(-0.367472\pi\)
0.404423 + 0.914572i \(0.367472\pi\)
\(570\) −36992.0 −0.00476892
\(571\) 8.34847e6 1.07156 0.535780 0.844358i \(-0.320018\pi\)
0.535780 + 0.844358i \(0.320018\pi\)
\(572\) 746374. 0.0953820
\(573\) 3.04745e6 0.387748
\(574\) 0 0
\(575\) −1.64110e6 −0.206998
\(576\) 4.87469e6 0.612196
\(577\) −3.77464e6 −0.471993 −0.235997 0.971754i \(-0.575835\pi\)
−0.235997 + 0.971754i \(0.575835\pi\)
\(578\) −1.23909e7 −1.54270
\(579\) −7.65933e6 −0.949499
\(580\) 3.28456e6 0.405422
\(581\) 0 0
\(582\) 8.89612e6 1.08866
\(583\) −4.09248e6 −0.498672
\(584\) 7.84652e6 0.952017
\(585\) 1.25061e6 0.151088
\(586\) 3.64529e6 0.438519
\(587\) −1.53838e6 −0.184275 −0.0921377 0.995746i \(-0.529370\pi\)
−0.0921377 + 0.995746i \(0.529370\pi\)
\(588\) 0 0
\(589\) 31818.3 0.00377910
\(590\) −3.03766e6 −0.359260
\(591\) −1.76654e7 −2.08044
\(592\) −85829.6 −0.0100654
\(593\) 2.65317e6 0.309834 0.154917 0.987928i \(-0.450489\pi\)
0.154917 + 0.987928i \(0.450489\pi\)
\(594\) 555570. 0.0646060
\(595\) 0 0
\(596\) 4.41766e6 0.509420
\(597\) 4.50812e6 0.517678
\(598\) 2.44732e6 0.279858
\(599\) −1.43136e7 −1.62998 −0.814992 0.579472i \(-0.803259\pi\)
−0.814992 + 0.579472i \(0.803259\pi\)
\(600\) −2.45657e6 −0.278580
\(601\) −3.27781e6 −0.370167 −0.185084 0.982723i \(-0.559256\pi\)
−0.185084 + 0.982723i \(0.559256\pi\)
\(602\) 0 0
\(603\) −9.47399e6 −1.06106
\(604\) −8.82226e6 −0.983983
\(605\) −3.34783e6 −0.371856
\(606\) −1.56944e7 −1.73605
\(607\) 7.19646e6 0.792770 0.396385 0.918084i \(-0.370264\pi\)
0.396385 + 0.918084i \(0.370264\pi\)
\(608\) 104569. 0.0114721
\(609\) 0 0
\(610\) 2.98431e6 0.324727
\(611\) 4.65345e6 0.504280
\(612\) 7.89169e6 0.851709
\(613\) 1.05406e7 1.13296 0.566478 0.824077i \(-0.308306\pi\)
0.566478 + 0.824077i \(0.308306\pi\)
\(614\) −743340. −0.0795732
\(615\) 1.76406e6 0.188073
\(616\) 0 0
\(617\) 1.88228e6 0.199054 0.0995272 0.995035i \(-0.468267\pi\)
0.0995272 + 0.995035i \(0.468267\pi\)
\(618\) 7.11047e6 0.748906
\(619\) 4.38233e6 0.459705 0.229852 0.973226i \(-0.426176\pi\)
0.229852 + 0.973226i \(0.426176\pi\)
\(620\) 763759. 0.0797952
\(621\) −2.37640e6 −0.247281
\(622\) 2.38880e6 0.247574
\(623\) 0 0
\(624\) 611852. 0.0629049
\(625\) 390625. 0.0400000
\(626\) −3.07036e6 −0.313151
\(627\) −65413.9 −0.00664509
\(628\) −2.51163e6 −0.254130
\(629\) −1.60866e6 −0.162120
\(630\) 0 0
\(631\) −1.59096e7 −1.59069 −0.795345 0.606157i \(-0.792710\pi\)
−0.795345 + 0.606157i \(0.792710\pi\)
\(632\) 2.76604e6 0.275465
\(633\) −1.62279e7 −1.60973
\(634\) −3.91816e6 −0.387132
\(635\) 2.73147e6 0.268821
\(636\) 9.47149e6 0.928487
\(637\) 0 0
\(638\) −4.45239e6 −0.433054
\(639\) −9.37730e6 −0.908502
\(640\) 2.16357e6 0.208795
\(641\) 5.45547e6 0.524430 0.262215 0.965009i \(-0.415547\pi\)
0.262215 + 0.965009i \(0.415547\pi\)
\(642\) −1.35205e7 −1.29466
\(643\) 45853.5 0.00437366 0.00218683 0.999998i \(-0.499304\pi\)
0.00218683 + 0.999998i \(0.499304\pi\)
\(644\) 0 0
\(645\) 8.23925e6 0.779809
\(646\) −153139. −0.0144379
\(647\) −1.75738e6 −0.165046 −0.0825230 0.996589i \(-0.526298\pi\)
−0.0825230 + 0.996589i \(0.526298\pi\)
\(648\) −1.26330e7 −1.18187
\(649\) −5.37158e6 −0.500599
\(650\) −582526. −0.0540794
\(651\) 0 0
\(652\) −5.97696e6 −0.550632
\(653\) 6.36106e6 0.583776 0.291888 0.956452i \(-0.405716\pi\)
0.291888 + 0.956452i \(0.405716\pi\)
\(654\) −2.30658e6 −0.210875
\(655\) 3.14645e6 0.286561
\(656\) 389642. 0.0353513
\(657\) 8.40353e6 0.759536
\(658\) 0 0
\(659\) −9.78759e6 −0.877935 −0.438967 0.898503i \(-0.644656\pi\)
−0.438967 + 0.898503i \(0.644656\pi\)
\(660\) −1.57018e6 −0.140310
\(661\) 1.43971e7 1.28165 0.640827 0.767686i \(-0.278592\pi\)
0.640827 + 0.767686i \(0.278592\pi\)
\(662\) 6.41504e6 0.568924
\(663\) 1.14676e7 1.01318
\(664\) 2.04557e7 1.80051
\(665\) 0 0
\(666\) −550377. −0.0480811
\(667\) 1.90447e7 1.65752
\(668\) 3.95149e6 0.342625
\(669\) −4.57719e6 −0.395398
\(670\) 4.41294e6 0.379788
\(671\) 5.27723e6 0.452481
\(672\) 0 0
\(673\) 7.07466e6 0.602099 0.301050 0.953608i \(-0.402663\pi\)
0.301050 + 0.953608i \(0.402663\pi\)
\(674\) 1.08040e7 0.916082
\(675\) 565645. 0.0477842
\(676\) 5.59243e6 0.470689
\(677\) 1.29681e7 1.08744 0.543720 0.839267i \(-0.317015\pi\)
0.543720 + 0.839267i \(0.317015\pi\)
\(678\) 1.31582e7 1.09932
\(679\) 0 0
\(680\) −1.01697e7 −0.843402
\(681\) 2.25728e7 1.86517
\(682\) −1.03531e6 −0.0852337
\(683\) 1.23817e6 0.101561 0.0507806 0.998710i \(-0.483829\pi\)
0.0507806 + 0.998710i \(0.483829\pi\)
\(684\) 68348.5 0.00558584
\(685\) 1.00611e6 0.0819256
\(686\) 0 0
\(687\) −2.56983e6 −0.207736
\(688\) 1.81987e6 0.146578
\(689\) 6.21366e6 0.498654
\(690\) −5.14853e6 −0.411681
\(691\) 1.48745e7 1.18508 0.592541 0.805541i \(-0.298125\pi\)
0.592541 + 0.805541i \(0.298125\pi\)
\(692\) 2.93036e6 0.232624
\(693\) 0 0
\(694\) −8.08521e6 −0.637225
\(695\) 9.92622e6 0.779510
\(696\) 2.85081e7 2.23072
\(697\) 7.30283e6 0.569389
\(698\) −8.64719e6 −0.671795
\(699\) −3.23313e6 −0.250282
\(700\) 0 0
\(701\) 1.89434e7 1.45601 0.728004 0.685573i \(-0.240448\pi\)
0.728004 + 0.685573i \(0.240448\pi\)
\(702\) −843528. −0.0646036
\(703\) −13932.3 −0.00106325
\(704\) −4.01517e6 −0.305332
\(705\) −9.78965e6 −0.741813
\(706\) −1.25769e7 −0.949643
\(707\) 0 0
\(708\) 1.24318e7 0.932074
\(709\) 1.10355e7 0.824472 0.412236 0.911077i \(-0.364748\pi\)
0.412236 + 0.911077i \(0.364748\pi\)
\(710\) 4.36790e6 0.325182
\(711\) 2.96240e6 0.219771
\(712\) −2.44130e6 −0.180476
\(713\) 4.42846e6 0.326234
\(714\) 0 0
\(715\) −1.03010e6 −0.0753552
\(716\) 2.51864e6 0.183605
\(717\) −1.22239e7 −0.887997
\(718\) 4.97812e6 0.360375
\(719\) 4.65186e6 0.335587 0.167793 0.985822i \(-0.446336\pi\)
0.167793 + 0.985822i \(0.446336\pi\)
\(720\) −581120. −0.0417767
\(721\) 0 0
\(722\) 9.22553e6 0.658641
\(723\) −536974. −0.0382039
\(724\) 1.13340e7 0.803594
\(725\) −4.53314e6 −0.320298
\(726\) −1.05029e7 −0.739553
\(727\) −1.87498e7 −1.31571 −0.657857 0.753143i \(-0.728537\pi\)
−0.657857 + 0.753143i \(0.728537\pi\)
\(728\) 0 0
\(729\) −8.90098e6 −0.620325
\(730\) −3.91432e6 −0.271862
\(731\) 3.41087e7 2.36087
\(732\) −1.22134e7 −0.842482
\(733\) 1.08016e7 0.742557 0.371279 0.928521i \(-0.378919\pi\)
0.371279 + 0.928521i \(0.378919\pi\)
\(734\) 3.61664e6 0.247779
\(735\) 0 0
\(736\) 1.45539e7 0.990338
\(737\) 7.80351e6 0.529202
\(738\) 2.49855e6 0.168868
\(739\) 1.72048e7 1.15888 0.579442 0.815014i \(-0.303271\pi\)
0.579442 + 0.815014i \(0.303271\pi\)
\(740\) −334427. −0.0224503
\(741\) 99318.7 0.00664486
\(742\) 0 0
\(743\) 6.25027e6 0.415362 0.207681 0.978197i \(-0.433408\pi\)
0.207681 + 0.978197i \(0.433408\pi\)
\(744\) 6.62897e6 0.439050
\(745\) −6.09696e6 −0.402460
\(746\) 3.67385e6 0.241699
\(747\) 2.19079e7 1.43648
\(748\) −6.50021e6 −0.424789
\(749\) 0 0
\(750\) 1.22549e6 0.0795527
\(751\) −1.71447e7 −1.10925 −0.554626 0.832099i \(-0.687139\pi\)
−0.554626 + 0.832099i \(0.687139\pi\)
\(752\) −2.16232e6 −0.139436
\(753\) −3.26919e7 −2.10113
\(754\) 6.76012e6 0.433038
\(755\) 1.21759e7 0.777382
\(756\) 0 0
\(757\) −2.56731e7 −1.62831 −0.814157 0.580645i \(-0.802800\pi\)
−0.814157 + 0.580645i \(0.802800\pi\)
\(758\) −1.23272e7 −0.779279
\(759\) −9.10429e6 −0.573643
\(760\) −88077.7 −0.00553136
\(761\) −198469. −0.0124231 −0.00621156 0.999981i \(-0.501977\pi\)
−0.00621156 + 0.999981i \(0.501977\pi\)
\(762\) 8.56930e6 0.534635
\(763\) 0 0
\(764\) 2.62272e6 0.162562
\(765\) −1.08916e7 −0.672881
\(766\) 1.52422e7 0.938591
\(767\) 8.15573e6 0.500581
\(768\) 2.32036e7 1.41956
\(769\) 3.50740e6 0.213880 0.106940 0.994265i \(-0.465895\pi\)
0.106940 + 0.994265i \(0.465895\pi\)
\(770\) 0 0
\(771\) 1.42954e7 0.866083
\(772\) −6.59185e6 −0.398074
\(773\) −1.80013e7 −1.08356 −0.541782 0.840519i \(-0.682250\pi\)
−0.541782 + 0.840519i \(0.682250\pi\)
\(774\) 1.16698e7 0.700181
\(775\) −1.05409e6 −0.0630410
\(776\) 2.11816e7 1.26271
\(777\) 0 0
\(778\) 1.90658e7 1.12929
\(779\) 63248.5 0.00373428
\(780\) 2.38402e6 0.140305
\(781\) 7.72387e6 0.453114
\(782\) −2.13138e7 −1.24636
\(783\) −6.56422e6 −0.382630
\(784\) 0 0
\(785\) 3.46639e6 0.200772
\(786\) 9.87116e6 0.569917
\(787\) 1.79041e7 1.03042 0.515211 0.857063i \(-0.327714\pi\)
0.515211 + 0.857063i \(0.327714\pi\)
\(788\) −1.52034e7 −0.872216
\(789\) −4.01687e7 −2.29718
\(790\) −1.37987e6 −0.0786630
\(791\) 0 0
\(792\) −6.15271e6 −0.348541
\(793\) −8.01249e6 −0.452464
\(794\) −2.00045e7 −1.12610
\(795\) −1.30719e7 −0.733537
\(796\) 3.87982e6 0.217035
\(797\) −7.45389e6 −0.415659 −0.207829 0.978165i \(-0.566640\pi\)
−0.207829 + 0.978165i \(0.566640\pi\)
\(798\) 0 0
\(799\) −4.05271e7 −2.24584
\(800\) −3.46420e6 −0.191372
\(801\) −2.61460e6 −0.143987
\(802\) 1.43071e7 0.785444
\(803\) −6.92180e6 −0.378818
\(804\) −1.80602e7 −0.985331
\(805\) 0 0
\(806\) 1.57193e6 0.0852306
\(807\) −2.03119e7 −1.09791
\(808\) −3.73682e7 −2.01360
\(809\) −7.85717e6 −0.422080 −0.211040 0.977477i \(-0.567685\pi\)
−0.211040 + 0.977477i \(0.567685\pi\)
\(810\) 6.30212e6 0.337500
\(811\) −550490. −0.0293899 −0.0146949 0.999892i \(-0.504678\pi\)
−0.0146949 + 0.999892i \(0.504678\pi\)
\(812\) 0 0
\(813\) −2.68240e7 −1.42330
\(814\) 453333. 0.0239804
\(815\) 8.24901e6 0.435019
\(816\) −5.32864e6 −0.280150
\(817\) 295410. 0.0154835
\(818\) −2.14952e6 −0.112320
\(819\) 0 0
\(820\) 1.51820e6 0.0788487
\(821\) 2.66311e7 1.37890 0.689448 0.724335i \(-0.257853\pi\)
0.689448 + 0.724335i \(0.257853\pi\)
\(822\) 3.15642e6 0.162935
\(823\) 1.47532e7 0.759253 0.379627 0.925140i \(-0.376052\pi\)
0.379627 + 0.925140i \(0.376052\pi\)
\(824\) 1.69300e7 0.868638
\(825\) 2.16706e6 0.110850
\(826\) 0 0
\(827\) −2.54872e7 −1.29586 −0.647931 0.761699i \(-0.724365\pi\)
−0.647931 + 0.761699i \(0.724365\pi\)
\(828\) 9.51273e6 0.482202
\(829\) 941474. 0.0475797 0.0237899 0.999717i \(-0.492427\pi\)
0.0237899 + 0.999717i \(0.492427\pi\)
\(830\) −1.02046e7 −0.514161
\(831\) −3.94291e6 −0.198068
\(832\) 6.09628e6 0.305321
\(833\) 0 0
\(834\) 3.11410e7 1.55030
\(835\) −5.45359e6 −0.270686
\(836\) −56297.1 −0.00278593
\(837\) −1.52638e6 −0.0753092
\(838\) −245850. −0.0120937
\(839\) 2.99251e7 1.46768 0.733839 0.679323i \(-0.237727\pi\)
0.733839 + 0.679323i \(0.237727\pi\)
\(840\) 0 0
\(841\) 3.20952e7 1.56477
\(842\) 1.09945e7 0.534436
\(843\) −4.72376e7 −2.28938
\(844\) −1.39662e7 −0.674873
\(845\) −7.71832e6 −0.371861
\(846\) −1.38657e7 −0.666065
\(847\) 0 0
\(848\) −2.88730e6 −0.137880
\(849\) −5.06226e7 −2.41032
\(850\) 5.07325e6 0.240846
\(851\) −1.93909e6 −0.0917856
\(852\) −1.78759e7 −0.843661
\(853\) 1.25646e7 0.591254 0.295627 0.955303i \(-0.404471\pi\)
0.295627 + 0.955303i \(0.404471\pi\)
\(854\) 0 0
\(855\) −94330.2 −0.00441301
\(856\) −3.21922e7 −1.50164
\(857\) −2.20590e7 −1.02597 −0.512985 0.858398i \(-0.671460\pi\)
−0.512985 + 0.858398i \(0.671460\pi\)
\(858\) −3.23167e6 −0.149868
\(859\) −1.44949e6 −0.0670243 −0.0335122 0.999438i \(-0.510669\pi\)
−0.0335122 + 0.999438i \(0.510669\pi\)
\(860\) 7.09094e6 0.326932
\(861\) 0 0
\(862\) −1.27493e7 −0.584413
\(863\) −3.08530e7 −1.41016 −0.705082 0.709125i \(-0.749090\pi\)
−0.705082 + 0.709125i \(0.749090\pi\)
\(864\) −5.01634e6 −0.228614
\(865\) −4.04429e6 −0.183781
\(866\) 2.23805e7 1.01409
\(867\) −6.99872e7 −3.16206
\(868\) 0 0
\(869\) −2.44006e6 −0.109610
\(870\) −1.42216e7 −0.637014
\(871\) −1.18482e7 −0.529183
\(872\) −5.49195e6 −0.244588
\(873\) 2.26853e7 1.00741
\(874\) −184595. −0.00817414
\(875\) 0 0
\(876\) 1.60196e7 0.705327
\(877\) 3.59009e7 1.57618 0.788090 0.615560i \(-0.211070\pi\)
0.788090 + 0.615560i \(0.211070\pi\)
\(878\) −219175. −0.00959523
\(879\) 2.05896e7 0.898828
\(880\) 478655. 0.0208361
\(881\) −4.08291e6 −0.177227 −0.0886135 0.996066i \(-0.528244\pi\)
−0.0886135 + 0.996066i \(0.528244\pi\)
\(882\) 0 0
\(883\) 3.66369e6 0.158131 0.0790655 0.996869i \(-0.474806\pi\)
0.0790655 + 0.996869i \(0.474806\pi\)
\(884\) 9.86934e6 0.424774
\(885\) −1.71576e7 −0.736372
\(886\) 1.23185e7 0.527199
\(887\) −2.52478e7 −1.07749 −0.538746 0.842468i \(-0.681102\pi\)
−0.538746 + 0.842468i \(0.681102\pi\)
\(888\) −2.90263e6 −0.123526
\(889\) 0 0
\(890\) 1.21787e6 0.0515377
\(891\) 1.11442e7 0.470278
\(892\) −3.93927e6 −0.165769
\(893\) −350998. −0.0147291
\(894\) −1.91277e7 −0.800420
\(895\) −3.47606e6 −0.145054
\(896\) 0 0
\(897\) 1.38232e7 0.573623
\(898\) −2.50244e7 −1.03556
\(899\) 1.22325e7 0.504798
\(900\) −2.26428e6 −0.0931802
\(901\) −5.41151e7 −2.22078
\(902\) −2.05800e6 −0.0842227
\(903\) 0 0
\(904\) 3.13296e7 1.27507
\(905\) −1.56425e7 −0.634868
\(906\) 3.81988e7 1.54607
\(907\) 3.59572e7 1.45134 0.725668 0.688045i \(-0.241531\pi\)
0.725668 + 0.688045i \(0.241531\pi\)
\(908\) 1.94268e7 0.781964
\(909\) −4.00210e7 −1.60649
\(910\) 0 0
\(911\) −6.81358e6 −0.272006 −0.136003 0.990708i \(-0.543426\pi\)
−0.136003 + 0.990708i \(0.543426\pi\)
\(912\) −46150.4 −0.00183734
\(913\) −1.80450e7 −0.716441
\(914\) 3.03320e7 1.20098
\(915\) 1.68562e7 0.665590
\(916\) −2.21167e6 −0.0870926
\(917\) 0 0
\(918\) 7.34633e6 0.287716
\(919\) 1.37705e7 0.537848 0.268924 0.963161i \(-0.413332\pi\)
0.268924 + 0.963161i \(0.413332\pi\)
\(920\) −1.22586e7 −0.477499
\(921\) −4.19859e6 −0.163100
\(922\) 2.53470e7 0.981974
\(923\) −1.17272e7 −0.453098
\(924\) 0 0
\(925\) 461555. 0.0177365
\(926\) 6.06832e6 0.232563
\(927\) 1.81318e7 0.693015
\(928\) 4.02014e7 1.53240
\(929\) −5.04155e7 −1.91657 −0.958285 0.285814i \(-0.907736\pi\)
−0.958285 + 0.285814i \(0.907736\pi\)
\(930\) −3.30694e6 −0.125377
\(931\) 0 0
\(932\) −2.78252e6 −0.104930
\(933\) 1.34926e7 0.507449
\(934\) 1.80891e7 0.678500
\(935\) 8.97116e6 0.335598
\(936\) 9.34173e6 0.348528
\(937\) −7.60181e6 −0.282858 −0.141429 0.989948i \(-0.545170\pi\)
−0.141429 + 0.989948i \(0.545170\pi\)
\(938\) 0 0
\(939\) −1.73422e7 −0.641861
\(940\) −8.42526e6 −0.311002
\(941\) 1.02579e6 0.0377647 0.0188824 0.999822i \(-0.493989\pi\)
0.0188824 + 0.999822i \(0.493989\pi\)
\(942\) 1.08749e7 0.399299
\(943\) 8.80292e6 0.322365
\(944\) −3.78972e6 −0.138413
\(945\) 0 0
\(946\) −9.61214e6 −0.349214
\(947\) 3.46475e6 0.125544 0.0627722 0.998028i \(-0.480006\pi\)
0.0627722 + 0.998028i \(0.480006\pi\)
\(948\) 5.64719e6 0.204085
\(949\) 1.05095e7 0.378804
\(950\) 43938.5 0.00157956
\(951\) −2.21309e7 −0.793500
\(952\) 0 0
\(953\) −728447. −0.0259816 −0.0129908 0.999916i \(-0.504135\pi\)
−0.0129908 + 0.999916i \(0.504135\pi\)
\(954\) −1.85146e7 −0.658634
\(955\) −3.61971e6 −0.128430
\(956\) −1.05202e7 −0.372290
\(957\) −2.51484e7 −0.887626
\(958\) 6.70337e6 0.235982
\(959\) 0 0
\(960\) −1.28250e7 −0.449138
\(961\) −2.57847e7 −0.900646
\(962\) −688301. −0.0239795
\(963\) −3.44775e7 −1.19804
\(964\) −462136. −0.0160168
\(965\) 9.09764e6 0.314493
\(966\) 0 0
\(967\) 5.01305e7 1.72399 0.861997 0.506913i \(-0.169213\pi\)
0.861997 + 0.506913i \(0.169213\pi\)
\(968\) −2.50074e7 −0.857790
\(969\) −864971. −0.0295932
\(970\) −1.05667e7 −0.360586
\(971\) −2.12323e7 −0.722684 −0.361342 0.932433i \(-0.617681\pi\)
−0.361342 + 0.932433i \(0.617681\pi\)
\(972\) −2.18081e7 −0.740374
\(973\) 0 0
\(974\) −1.86659e7 −0.630451
\(975\) −3.29027e6 −0.110846
\(976\) 3.72316e6 0.125109
\(977\) −5.47337e6 −0.183450 −0.0917251 0.995784i \(-0.529238\pi\)
−0.0917251 + 0.995784i \(0.529238\pi\)
\(978\) 2.58792e7 0.865173
\(979\) 2.15359e6 0.0718135
\(980\) 0 0
\(981\) −5.88182e6 −0.195137
\(982\) 1.63956e6 0.0542561
\(983\) 1.74099e7 0.574663 0.287331 0.957831i \(-0.407232\pi\)
0.287331 + 0.957831i \(0.407232\pi\)
\(984\) 1.31771e7 0.433842
\(985\) 2.09827e7 0.689081
\(986\) −5.88742e7 −1.92856
\(987\) 0 0
\(988\) 85476.6 0.00278583
\(989\) 4.11151e7 1.33663
\(990\) 3.06935e6 0.0995309
\(991\) 2.81682e7 0.911119 0.455559 0.890205i \(-0.349439\pi\)
0.455559 + 0.890205i \(0.349439\pi\)
\(992\) 9.34803e6 0.301607
\(993\) 3.62339e7 1.16612
\(994\) 0 0
\(995\) −5.35468e6 −0.171465
\(996\) 4.17628e7 1.33395
\(997\) 2.73631e7 0.871822 0.435911 0.899990i \(-0.356426\pi\)
0.435911 + 0.899990i \(0.356426\pi\)
\(998\) 666575. 0.0211847
\(999\) 668355. 0.0211882
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 245.6.a.m.1.3 yes 10
7.6 odd 2 245.6.a.l.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.6.a.l.1.3 10 7.6 odd 2
245.6.a.m.1.3 yes 10 1.1 even 1 trivial