Properties

Label 1045.6.a.f.1.4
Level $1045$
Weight $6$
Character 1045.1
Self dual yes
Analytic conductor $167.601$
Analytic rank $0$
Dimension $38$
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] = [1045,6,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1045.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(167.601091705\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.16691 q^{2} -28.2569 q^{3} +52.0323 q^{4} +25.0000 q^{5} +259.028 q^{6} +66.4323 q^{7} -183.634 q^{8} +555.452 q^{9} +O(q^{10})\) \(q-9.16691 q^{2} -28.2569 q^{3} +52.0323 q^{4} +25.0000 q^{5} +259.028 q^{6} +66.4323 q^{7} -183.634 q^{8} +555.452 q^{9} -229.173 q^{10} -121.000 q^{11} -1470.27 q^{12} +856.238 q^{13} -608.979 q^{14} -706.422 q^{15} +18.3254 q^{16} -2251.06 q^{17} -5091.78 q^{18} -361.000 q^{19} +1300.81 q^{20} -1877.17 q^{21} +1109.20 q^{22} +2094.06 q^{23} +5188.93 q^{24} +625.000 q^{25} -7849.06 q^{26} -8828.93 q^{27} +3456.62 q^{28} -3557.08 q^{29} +6475.71 q^{30} -6117.12 q^{31} +5708.31 q^{32} +3419.08 q^{33} +20635.3 q^{34} +1660.81 q^{35} +28901.4 q^{36} +11878.1 q^{37} +3309.26 q^{38} -24194.6 q^{39} -4590.85 q^{40} -4115.68 q^{41} +17207.9 q^{42} -15128.1 q^{43} -6295.91 q^{44} +13886.3 q^{45} -19196.1 q^{46} +29205.4 q^{47} -517.818 q^{48} -12393.7 q^{49} -5729.32 q^{50} +63608.0 q^{51} +44552.0 q^{52} +8106.51 q^{53} +80934.0 q^{54} -3025.00 q^{55} -12199.2 q^{56} +10200.7 q^{57} +32607.4 q^{58} +35327.2 q^{59} -36756.8 q^{60} +13703.0 q^{61} +56075.1 q^{62} +36900.0 q^{63} -52914.0 q^{64} +21406.0 q^{65} -31342.4 q^{66} -23511.0 q^{67} -117128. q^{68} -59171.7 q^{69} -15224.5 q^{70} +22933.3 q^{71} -102000. q^{72} -53346.6 q^{73} -108885. q^{74} -17660.6 q^{75} -18783.7 q^{76} -8038.31 q^{77} +221790. q^{78} -37921.0 q^{79} +458.134 q^{80} +114503. q^{81} +37728.0 q^{82} +65371.2 q^{83} -97673.5 q^{84} -56276.6 q^{85} +138678. q^{86} +100512. q^{87} +22219.7 q^{88} +81398.5 q^{89} -127295. q^{90} +56881.9 q^{91} +108959. q^{92} +172851. q^{93} -267723. q^{94} -9025.00 q^{95} -161299. q^{96} +57974.2 q^{97} +113612. q^{98} -67209.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q + 8 q^{2} + 63 q^{3} + 616 q^{4} + 950 q^{5} + 149 q^{6} + 275 q^{7} + 264 q^{8} + 3029 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q + 8 q^{2} + 63 q^{3} + 616 q^{4} + 950 q^{5} + 149 q^{6} + 275 q^{7} + 264 q^{8} + 3029 q^{9} + 200 q^{10} - 4598 q^{11} + 2312 q^{12} + 41 q^{13} + 23 q^{14} + 1575 q^{15} + 7196 q^{16} - 2431 q^{17} - 1689 q^{18} - 13718 q^{19} + 15400 q^{20} - 1577 q^{21} - 968 q^{22} + 9284 q^{23} + 7598 q^{24} + 23750 q^{25} + 13129 q^{26} + 9228 q^{27} - 1079 q^{28} - 559 q^{29} + 3725 q^{30} + 11147 q^{31} + 11051 q^{32} - 7623 q^{33} + 40895 q^{34} + 6875 q^{35} + 55887 q^{36} + 41579 q^{37} - 2888 q^{38} + 24982 q^{39} + 6600 q^{40} + 18597 q^{41} + 61360 q^{42} + 25353 q^{43} - 74536 q^{44} + 75725 q^{45} + 1611 q^{46} + 63516 q^{47} + 187737 q^{48} + 141609 q^{49} + 5000 q^{50} + 107546 q^{51} + 60018 q^{52} + 123045 q^{53} + 256696 q^{54} - 114950 q^{55} + 157335 q^{56} - 22743 q^{57} + 218938 q^{58} + 132925 q^{59} + 57800 q^{60} - 59107 q^{61} + 166982 q^{62} + 130582 q^{63} + 313126 q^{64} + 1025 q^{65} - 18029 q^{66} + 162534 q^{67} + 182980 q^{68} + 178552 q^{69} + 575 q^{70} + 157840 q^{71} + 98630 q^{72} - 63010 q^{73} + 122683 q^{74} + 39375 q^{75} - 222376 q^{76} - 33275 q^{77} + 277272 q^{78} - 16385 q^{79} + 179900 q^{80} + 290354 q^{81} + 362302 q^{82} + 138461 q^{83} + 446870 q^{84} - 60775 q^{85} + 643902 q^{86} + 291602 q^{87} - 31944 q^{88} + 224792 q^{89} - 42225 q^{90} + 498548 q^{91} + 581088 q^{92} + 134210 q^{93} + 35864 q^{94} - 342950 q^{95} + 377376 q^{96} + 292216 q^{97} - 58230 q^{98} - 366509 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.16691 −1.62050 −0.810248 0.586087i \(-0.800668\pi\)
−0.810248 + 0.586087i \(0.800668\pi\)
\(3\) −28.2569 −1.81268 −0.906340 0.422549i \(-0.861136\pi\)
−0.906340 + 0.422549i \(0.861136\pi\)
\(4\) 52.0323 1.62601
\(5\) 25.0000 0.447214
\(6\) 259.028 2.93744
\(7\) 66.4323 0.512430 0.256215 0.966620i \(-0.417525\pi\)
0.256215 + 0.966620i \(0.417525\pi\)
\(8\) −183.634 −1.01445
\(9\) 555.452 2.28581
\(10\) −229.173 −0.724708
\(11\) −121.000 −0.301511
\(12\) −1470.27 −2.94743
\(13\) 856.238 1.40519 0.702597 0.711588i \(-0.252024\pi\)
0.702597 + 0.711588i \(0.252024\pi\)
\(14\) −608.979 −0.830390
\(15\) −706.422 −0.810655
\(16\) 18.3254 0.0178959
\(17\) −2251.06 −1.88914 −0.944572 0.328303i \(-0.893523\pi\)
−0.944572 + 0.328303i \(0.893523\pi\)
\(18\) −5091.78 −3.70415
\(19\) −361.000 −0.229416
\(20\) 1300.81 0.727173
\(21\) −1877.17 −0.928871
\(22\) 1109.20 0.488598
\(23\) 2094.06 0.825410 0.412705 0.910865i \(-0.364584\pi\)
0.412705 + 0.910865i \(0.364584\pi\)
\(24\) 5188.93 1.83886
\(25\) 625.000 0.200000
\(26\) −7849.06 −2.27711
\(27\) −8828.93 −2.33076
\(28\) 3456.62 0.833215
\(29\) −3557.08 −0.785414 −0.392707 0.919664i \(-0.628461\pi\)
−0.392707 + 0.919664i \(0.628461\pi\)
\(30\) 6475.71 1.31366
\(31\) −6117.12 −1.14325 −0.571627 0.820514i \(-0.693688\pi\)
−0.571627 + 0.820514i \(0.693688\pi\)
\(32\) 5708.31 0.985445
\(33\) 3419.08 0.546544
\(34\) 20635.3 3.06135
\(35\) 1660.81 0.229165
\(36\) 28901.4 3.71675
\(37\) 11878.1 1.42640 0.713202 0.700959i \(-0.247244\pi\)
0.713202 + 0.700959i \(0.247244\pi\)
\(38\) 3309.26 0.371767
\(39\) −24194.6 −2.54717
\(40\) −4590.85 −0.453674
\(41\) −4115.68 −0.382368 −0.191184 0.981554i \(-0.561233\pi\)
−0.191184 + 0.981554i \(0.561233\pi\)
\(42\) 17207.9 1.50523
\(43\) −15128.1 −1.24771 −0.623854 0.781541i \(-0.714434\pi\)
−0.623854 + 0.781541i \(0.714434\pi\)
\(44\) −6295.91 −0.490260
\(45\) 13886.3 1.02225
\(46\) −19196.1 −1.33757
\(47\) 29205.4 1.92849 0.964247 0.265003i \(-0.0853731\pi\)
0.964247 + 0.265003i \(0.0853731\pi\)
\(48\) −517.818 −0.0324395
\(49\) −12393.7 −0.737416
\(50\) −5729.32 −0.324099
\(51\) 63608.0 3.42442
\(52\) 44552.0 2.28486
\(53\) 8106.51 0.396410 0.198205 0.980161i \(-0.436489\pi\)
0.198205 + 0.980161i \(0.436489\pi\)
\(54\) 80934.0 3.77700
\(55\) −3025.00 −0.134840
\(56\) −12199.2 −0.519832
\(57\) 10200.7 0.415857
\(58\) 32607.4 1.27276
\(59\) 35327.2 1.32123 0.660615 0.750725i \(-0.270295\pi\)
0.660615 + 0.750725i \(0.270295\pi\)
\(60\) −36756.8 −1.31813
\(61\) 13703.0 0.471512 0.235756 0.971812i \(-0.424243\pi\)
0.235756 + 0.971812i \(0.424243\pi\)
\(62\) 56075.1 1.85264
\(63\) 36900.0 1.17132
\(64\) −52914.0 −1.61481
\(65\) 21406.0 0.628422
\(66\) −31342.4 −0.885672
\(67\) −23511.0 −0.639859 −0.319930 0.947441i \(-0.603659\pi\)
−0.319930 + 0.947441i \(0.603659\pi\)
\(68\) −117128. −3.07177
\(69\) −59171.7 −1.49621
\(70\) −15224.5 −0.371362
\(71\) 22933.3 0.539909 0.269954 0.962873i \(-0.412991\pi\)
0.269954 + 0.962873i \(0.412991\pi\)
\(72\) −102000. −2.31883
\(73\) −53346.6 −1.17165 −0.585827 0.810436i \(-0.699230\pi\)
−0.585827 + 0.810436i \(0.699230\pi\)
\(74\) −108885. −2.31148
\(75\) −17660.6 −0.362536
\(76\) −18783.7 −0.373032
\(77\) −8038.31 −0.154503
\(78\) 221790. 4.12768
\(79\) −37921.0 −0.683615 −0.341808 0.939770i \(-0.611039\pi\)
−0.341808 + 0.939770i \(0.611039\pi\)
\(80\) 458.134 0.00800327
\(81\) 114503. 1.93912
\(82\) 37728.0 0.619626
\(83\) 65371.2 1.04158 0.520788 0.853686i \(-0.325638\pi\)
0.520788 + 0.853686i \(0.325638\pi\)
\(84\) −97673.5 −1.51035
\(85\) −56276.6 −0.844851
\(86\) 138678. 2.02191
\(87\) 100512. 1.42370
\(88\) 22219.7 0.305867
\(89\) 81398.5 1.08929 0.544643 0.838668i \(-0.316665\pi\)
0.544643 + 0.838668i \(0.316665\pi\)
\(90\) −127295. −1.65655
\(91\) 56881.9 0.720063
\(92\) 108959. 1.34212
\(93\) 172851. 2.07235
\(94\) −267723. −3.12512
\(95\) −9025.00 −0.102598
\(96\) −161299. −1.78630
\(97\) 57974.2 0.625613 0.312806 0.949817i \(-0.398731\pi\)
0.312806 + 0.949817i \(0.398731\pi\)
\(98\) 113612. 1.19498
\(99\) −67209.7 −0.689198
\(100\) 32520.2 0.325202
\(101\) −46127.7 −0.449944 −0.224972 0.974365i \(-0.572229\pi\)
−0.224972 + 0.974365i \(0.572229\pi\)
\(102\) −583089. −5.54925
\(103\) −74583.5 −0.692708 −0.346354 0.938104i \(-0.612580\pi\)
−0.346354 + 0.938104i \(0.612580\pi\)
\(104\) −157235. −1.42549
\(105\) −46929.3 −0.415404
\(106\) −74311.7 −0.642381
\(107\) 192930. 1.62907 0.814536 0.580113i \(-0.196992\pi\)
0.814536 + 0.580113i \(0.196992\pi\)
\(108\) −459389. −3.78984
\(109\) −129162. −1.04128 −0.520639 0.853777i \(-0.674306\pi\)
−0.520639 + 0.853777i \(0.674306\pi\)
\(110\) 27729.9 0.218508
\(111\) −335638. −2.58561
\(112\) 1217.40 0.00917037
\(113\) −115751. −0.852766 −0.426383 0.904543i \(-0.640212\pi\)
−0.426383 + 0.904543i \(0.640212\pi\)
\(114\) −93509.3 −0.673896
\(115\) 52351.6 0.369135
\(116\) −185083. −1.27709
\(117\) 475599. 3.21201
\(118\) −323841. −2.14105
\(119\) −149543. −0.968054
\(120\) 129723. 0.822365
\(121\) 14641.0 0.0909091
\(122\) −125615. −0.764083
\(123\) 116296. 0.693111
\(124\) −318288. −1.85894
\(125\) 15625.0 0.0894427
\(126\) −338259. −1.89812
\(127\) 254354. 1.39936 0.699681 0.714456i \(-0.253326\pi\)
0.699681 + 0.714456i \(0.253326\pi\)
\(128\) 302392. 1.63134
\(129\) 427473. 2.26170
\(130\) −196226. −1.01836
\(131\) −124646. −0.634601 −0.317301 0.948325i \(-0.602776\pi\)
−0.317301 + 0.948325i \(0.602776\pi\)
\(132\) 177903. 0.888685
\(133\) −23982.1 −0.117559
\(134\) 215524. 1.03689
\(135\) −220723. −1.04235
\(136\) 413372. 1.91643
\(137\) −96720.4 −0.440268 −0.220134 0.975470i \(-0.570649\pi\)
−0.220134 + 0.975470i \(0.570649\pi\)
\(138\) 542422. 2.42460
\(139\) −216501. −0.950436 −0.475218 0.879868i \(-0.657631\pi\)
−0.475218 + 0.879868i \(0.657631\pi\)
\(140\) 86415.6 0.372625
\(141\) −825254. −3.49575
\(142\) −210227. −0.874920
\(143\) −103605. −0.423682
\(144\) 10178.9 0.0409066
\(145\) −88927.0 −0.351248
\(146\) 489023. 1.89866
\(147\) 350209. 1.33670
\(148\) 618044. 2.31934
\(149\) −335609. −1.23842 −0.619209 0.785226i \(-0.712547\pi\)
−0.619209 + 0.785226i \(0.712547\pi\)
\(150\) 161893. 0.587489
\(151\) 117542. 0.419520 0.209760 0.977753i \(-0.432732\pi\)
0.209760 + 0.977753i \(0.432732\pi\)
\(152\) 66291.9 0.232730
\(153\) −1.25036e6 −4.31823
\(154\) 73686.5 0.250372
\(155\) −152928. −0.511279
\(156\) −1.25890e6 −4.14172
\(157\) −355619. −1.15143 −0.575713 0.817652i \(-0.695275\pi\)
−0.575713 + 0.817652i \(0.695275\pi\)
\(158\) 347618. 1.10780
\(159\) −229065. −0.718564
\(160\) 142708. 0.440704
\(161\) 139113. 0.422965
\(162\) −1.04964e6 −3.14234
\(163\) 364837. 1.07555 0.537774 0.843089i \(-0.319265\pi\)
0.537774 + 0.843089i \(0.319265\pi\)
\(164\) −214148. −0.621734
\(165\) 85477.1 0.244422
\(166\) −599252. −1.68787
\(167\) 171652. 0.476275 0.238137 0.971231i \(-0.423463\pi\)
0.238137 + 0.971231i \(0.423463\pi\)
\(168\) 344713. 0.942289
\(169\) 361851. 0.974569
\(170\) 515882. 1.36908
\(171\) −200518. −0.524401
\(172\) −787149. −2.02878
\(173\) 245690. 0.624126 0.312063 0.950061i \(-0.398980\pi\)
0.312063 + 0.950061i \(0.398980\pi\)
\(174\) −921385. −2.30711
\(175\) 41520.2 0.102486
\(176\) −2217.37 −0.00539581
\(177\) −998236. −2.39497
\(178\) −746173. −1.76518
\(179\) −348150. −0.812146 −0.406073 0.913841i \(-0.633102\pi\)
−0.406073 + 0.913841i \(0.633102\pi\)
\(180\) 722536. 1.66218
\(181\) −62340.8 −0.141441 −0.0707206 0.997496i \(-0.522530\pi\)
−0.0707206 + 0.997496i \(0.522530\pi\)
\(182\) −521431. −1.16686
\(183\) −387205. −0.854700
\(184\) −384541. −0.837334
\(185\) 296952. 0.637907
\(186\) −1.58451e6 −3.35824
\(187\) 272379. 0.569599
\(188\) 1.51962e6 3.13575
\(189\) −586526. −1.19435
\(190\) 82731.4 0.166259
\(191\) 604468. 1.19892 0.599459 0.800405i \(-0.295382\pi\)
0.599459 + 0.800405i \(0.295382\pi\)
\(192\) 1.49518e6 2.92713
\(193\) 53159.1 0.102727 0.0513635 0.998680i \(-0.483643\pi\)
0.0513635 + 0.998680i \(0.483643\pi\)
\(194\) −531444. −1.01380
\(195\) −604866. −1.13913
\(196\) −644875. −1.19904
\(197\) −960324. −1.76300 −0.881500 0.472184i \(-0.843466\pi\)
−0.881500 + 0.472184i \(0.843466\pi\)
\(198\) 616105. 1.11684
\(199\) −132861. −0.237829 −0.118914 0.992905i \(-0.537941\pi\)
−0.118914 + 0.992905i \(0.537941\pi\)
\(200\) −114771. −0.202889
\(201\) 664349. 1.15986
\(202\) 422849. 0.729133
\(203\) −236305. −0.402469
\(204\) 3.30967e6 5.56813
\(205\) −102892. −0.171000
\(206\) 683701. 1.12253
\(207\) 1.16315e6 1.88673
\(208\) 15690.9 0.0251471
\(209\) 43681.0 0.0691714
\(210\) 430196. 0.673160
\(211\) −412287. −0.637519 −0.318760 0.947836i \(-0.603266\pi\)
−0.318760 + 0.947836i \(0.603266\pi\)
\(212\) 421800. 0.644566
\(213\) −648023. −0.978682
\(214\) −1.76857e6 −2.63991
\(215\) −378202. −0.557992
\(216\) 1.62129e6 2.36443
\(217\) −406374. −0.585837
\(218\) 1.18401e6 1.68739
\(219\) 1.50741e6 2.12383
\(220\) −157398. −0.219251
\(221\) −1.92745e6 −2.65461
\(222\) 3.07676e6 4.18998
\(223\) 1.17182e6 1.57796 0.788982 0.614416i \(-0.210608\pi\)
0.788982 + 0.614416i \(0.210608\pi\)
\(224\) 379216. 0.504971
\(225\) 347158. 0.457162
\(226\) 1.06108e6 1.38190
\(227\) 626009. 0.806336 0.403168 0.915126i \(-0.367909\pi\)
0.403168 + 0.915126i \(0.367909\pi\)
\(228\) 530768. 0.676188
\(229\) 403204. 0.508084 0.254042 0.967193i \(-0.418240\pi\)
0.254042 + 0.967193i \(0.418240\pi\)
\(230\) −479902. −0.598182
\(231\) 227138. 0.280065
\(232\) 653201. 0.796759
\(233\) 518044. 0.625139 0.312570 0.949895i \(-0.398810\pi\)
0.312570 + 0.949895i \(0.398810\pi\)
\(234\) −4.35978e6 −5.20505
\(235\) 730135. 0.862449
\(236\) 1.83815e6 2.14833
\(237\) 1.07153e6 1.23918
\(238\) 1.37085e6 1.56873
\(239\) 1.34862e6 1.52720 0.763598 0.645691i \(-0.223431\pi\)
0.763598 + 0.645691i \(0.223431\pi\)
\(240\) −12945.4 −0.0145074
\(241\) 613835. 0.680784 0.340392 0.940284i \(-0.389440\pi\)
0.340392 + 0.940284i \(0.389440\pi\)
\(242\) −134213. −0.147318
\(243\) −1.09007e6 −1.18424
\(244\) 713000. 0.766682
\(245\) −309844. −0.329782
\(246\) −1.06608e6 −1.12318
\(247\) −309102. −0.322374
\(248\) 1.12331e6 1.15977
\(249\) −1.84719e6 −1.88805
\(250\) −143233. −0.144942
\(251\) −689394. −0.690690 −0.345345 0.938476i \(-0.612238\pi\)
−0.345345 + 0.938476i \(0.612238\pi\)
\(252\) 1.91999e6 1.90457
\(253\) −253382. −0.248871
\(254\) −2.33164e6 −2.26766
\(255\) 1.59020e6 1.53145
\(256\) −1.07875e6 −1.02878
\(257\) 564460. 0.533089 0.266545 0.963823i \(-0.414118\pi\)
0.266545 + 0.963823i \(0.414118\pi\)
\(258\) −3.91861e6 −3.66507
\(259\) 789089. 0.730931
\(260\) 1.11380e6 1.02182
\(261\) −1.97579e6 −1.79531
\(262\) 1.14262e6 1.02837
\(263\) −1.73353e6 −1.54541 −0.772703 0.634768i \(-0.781096\pi\)
−0.772703 + 0.634768i \(0.781096\pi\)
\(264\) −627861. −0.554439
\(265\) 202663. 0.177280
\(266\) 219841. 0.190505
\(267\) −2.30007e6 −1.97453
\(268\) −1.22333e6 −1.04042
\(269\) 570898. 0.481036 0.240518 0.970645i \(-0.422683\pi\)
0.240518 + 0.970645i \(0.422683\pi\)
\(270\) 2.02335e6 1.68912
\(271\) 689745. 0.570513 0.285256 0.958451i \(-0.407921\pi\)
0.285256 + 0.958451i \(0.407921\pi\)
\(272\) −41251.5 −0.0338079
\(273\) −1.60730e6 −1.30524
\(274\) 886627. 0.713452
\(275\) −75625.0 −0.0603023
\(276\) −3.07884e6 −2.43284
\(277\) −1.22302e6 −0.957710 −0.478855 0.877894i \(-0.658948\pi\)
−0.478855 + 0.877894i \(0.658948\pi\)
\(278\) 1.98465e6 1.54018
\(279\) −3.39777e6 −2.61326
\(280\) −304981. −0.232476
\(281\) 2.38326e6 1.80055 0.900275 0.435322i \(-0.143366\pi\)
0.900275 + 0.435322i \(0.143366\pi\)
\(282\) 7.56503e6 5.66484
\(283\) −270350. −0.200659 −0.100330 0.994954i \(-0.531990\pi\)
−0.100330 + 0.994954i \(0.531990\pi\)
\(284\) 1.19327e6 0.877896
\(285\) 255018. 0.185977
\(286\) 949736. 0.686575
\(287\) −273414. −0.195937
\(288\) 3.17069e6 2.25254
\(289\) 3.64742e6 2.56887
\(290\) 815186. 0.569196
\(291\) −1.63817e6 −1.13404
\(292\) −2.77574e6 −1.90512
\(293\) −1.92919e6 −1.31282 −0.656412 0.754403i \(-0.727927\pi\)
−0.656412 + 0.754403i \(0.727927\pi\)
\(294\) −3.21033e6 −2.16612
\(295\) 883179. 0.590872
\(296\) −2.18122e6 −1.44701
\(297\) 1.06830e6 0.702752
\(298\) 3.07649e6 2.00685
\(299\) 1.79302e6 1.15986
\(300\) −918919. −0.589487
\(301\) −1.00499e6 −0.639363
\(302\) −1.07750e6 −0.679830
\(303\) 1.30343e6 0.815605
\(304\) −6615.46 −0.00410559
\(305\) 342576. 0.210866
\(306\) 1.14619e7 6.99767
\(307\) −1.51935e6 −0.920050 −0.460025 0.887906i \(-0.652160\pi\)
−0.460025 + 0.887906i \(0.652160\pi\)
\(308\) −418252. −0.251224
\(309\) 2.10750e6 1.25566
\(310\) 1.40188e6 0.828525
\(311\) −2.51623e6 −1.47519 −0.737597 0.675241i \(-0.764040\pi\)
−0.737597 + 0.675241i \(0.764040\pi\)
\(312\) 4.44296e6 2.58396
\(313\) −545024. −0.314452 −0.157226 0.987563i \(-0.550255\pi\)
−0.157226 + 0.987563i \(0.550255\pi\)
\(314\) 3.25993e6 1.86588
\(315\) 922499. 0.523829
\(316\) −1.97311e6 −1.11156
\(317\) 133688. 0.0747214 0.0373607 0.999302i \(-0.488105\pi\)
0.0373607 + 0.999302i \(0.488105\pi\)
\(318\) 2.09982e6 1.16443
\(319\) 430406. 0.236811
\(320\) −1.32285e6 −0.722163
\(321\) −5.45160e6 −2.95299
\(322\) −1.27524e6 −0.685413
\(323\) 812634. 0.433400
\(324\) 5.95786e6 3.15303
\(325\) 535149. 0.281039
\(326\) −3.34443e6 −1.74292
\(327\) 3.64970e6 1.88751
\(328\) 755779. 0.387891
\(329\) 1.94018e6 0.988218
\(330\) −783561. −0.396085
\(331\) −3.12617e6 −1.56835 −0.784175 0.620539i \(-0.786914\pi\)
−0.784175 + 0.620539i \(0.786914\pi\)
\(332\) 3.40141e6 1.69361
\(333\) 6.59771e6 3.26049
\(334\) −1.57352e6 −0.771802
\(335\) −587776. −0.286154
\(336\) −34399.8 −0.0166229
\(337\) 2.05851e6 0.987365 0.493683 0.869642i \(-0.335650\pi\)
0.493683 + 0.869642i \(0.335650\pi\)
\(338\) −3.31705e6 −1.57929
\(339\) 3.27077e6 1.54579
\(340\) −2.92820e6 −1.37374
\(341\) 740172. 0.344704
\(342\) 1.83813e6 0.849790
\(343\) −1.93987e6 −0.890303
\(344\) 2.77803e6 1.26573
\(345\) −1.47929e6 −0.669123
\(346\) −2.25222e6 −1.01139
\(347\) 3.98161e6 1.77515 0.887574 0.460665i \(-0.152389\pi\)
0.887574 + 0.460665i \(0.152389\pi\)
\(348\) 5.22987e6 2.31496
\(349\) −2.24411e6 −0.986237 −0.493119 0.869962i \(-0.664143\pi\)
−0.493119 + 0.869962i \(0.664143\pi\)
\(350\) −380612. −0.166078
\(351\) −7.55966e6 −3.27518
\(352\) −690705. −0.297123
\(353\) 307957. 0.131539 0.0657694 0.997835i \(-0.479050\pi\)
0.0657694 + 0.997835i \(0.479050\pi\)
\(354\) 9.15074e6 3.88104
\(355\) 573332. 0.241455
\(356\) 4.23535e6 1.77119
\(357\) 4.22563e6 1.75477
\(358\) 3.19146e6 1.31608
\(359\) −1.11335e6 −0.455929 −0.227964 0.973669i \(-0.573207\pi\)
−0.227964 + 0.973669i \(0.573207\pi\)
\(360\) −2.55000e6 −1.03701
\(361\) 130321. 0.0526316
\(362\) 571473. 0.229205
\(363\) −413709. −0.164789
\(364\) 2.95969e6 1.17083
\(365\) −1.33366e6 −0.523979
\(366\) 3.54948e6 1.38504
\(367\) −25188.1 −0.00976180 −0.00488090 0.999988i \(-0.501554\pi\)
−0.00488090 + 0.999988i \(0.501554\pi\)
\(368\) 38374.4 0.0147714
\(369\) −2.28606e6 −0.874021
\(370\) −2.72214e6 −1.03373
\(371\) 538534. 0.203132
\(372\) 8.99382e6 3.36967
\(373\) 4.83284e6 1.79858 0.899292 0.437349i \(-0.144082\pi\)
0.899292 + 0.437349i \(0.144082\pi\)
\(374\) −2.49687e6 −0.923032
\(375\) −441514. −0.162131
\(376\) −5.36311e6 −1.95635
\(377\) −3.04571e6 −1.10366
\(378\) 5.37663e6 1.93544
\(379\) 2.02328e6 0.723534 0.361767 0.932269i \(-0.382174\pi\)
0.361767 + 0.932269i \(0.382174\pi\)
\(380\) −469591. −0.166825
\(381\) −7.18726e6 −2.53659
\(382\) −5.54110e6 −1.94284
\(383\) −2.05417e6 −0.715549 −0.357775 0.933808i \(-0.616464\pi\)
−0.357775 + 0.933808i \(0.616464\pi\)
\(384\) −8.54465e6 −2.95710
\(385\) −200958. −0.0690960
\(386\) −487305. −0.166469
\(387\) −8.40293e6 −2.85202
\(388\) 3.01653e6 1.01725
\(389\) 2.43820e6 0.816950 0.408475 0.912770i \(-0.366061\pi\)
0.408475 + 0.912770i \(0.366061\pi\)
\(390\) 5.54475e6 1.84595
\(391\) −4.71386e6 −1.55932
\(392\) 2.27592e6 0.748068
\(393\) 3.52212e6 1.15033
\(394\) 8.80321e6 2.85694
\(395\) −948024. −0.305722
\(396\) −3.49707e6 −1.12064
\(397\) −3.58215e6 −1.14069 −0.570345 0.821405i \(-0.693191\pi\)
−0.570345 + 0.821405i \(0.693191\pi\)
\(398\) 1.21792e6 0.385401
\(399\) 677659. 0.213098
\(400\) 11453.4 0.00357917
\(401\) −2.46752e6 −0.766301 −0.383150 0.923686i \(-0.625161\pi\)
−0.383150 + 0.923686i \(0.625161\pi\)
\(402\) −6.09003e6 −1.87955
\(403\) −5.23771e6 −1.60649
\(404\) −2.40013e6 −0.731613
\(405\) 2.86258e6 0.867201
\(406\) 2.16619e6 0.652200
\(407\) −1.43725e6 −0.430077
\(408\) −1.16806e7 −3.47388
\(409\) −4.63675e6 −1.37058 −0.685292 0.728269i \(-0.740325\pi\)
−0.685292 + 0.728269i \(0.740325\pi\)
\(410\) 943201. 0.277105
\(411\) 2.73302e6 0.798064
\(412\) −3.88075e6 −1.12635
\(413\) 2.34686e6 0.677038
\(414\) −1.06625e7 −3.05744
\(415\) 1.63428e6 0.465807
\(416\) 4.88767e6 1.38474
\(417\) 6.11765e6 1.72284
\(418\) −400420. −0.112092
\(419\) 3.35416e6 0.933359 0.466680 0.884426i \(-0.345450\pi\)
0.466680 + 0.884426i \(0.345450\pi\)
\(420\) −2.44184e6 −0.675450
\(421\) −6.02799e6 −1.65755 −0.828777 0.559580i \(-0.810963\pi\)
−0.828777 + 0.559580i \(0.810963\pi\)
\(422\) 3.77940e6 1.03310
\(423\) 1.62222e7 4.40817
\(424\) −1.48863e6 −0.402136
\(425\) −1.40691e6 −0.377829
\(426\) 5.94037e6 1.58595
\(427\) 910324. 0.241616
\(428\) 1.00386e7 2.64889
\(429\) 2.92755e6 0.768000
\(430\) 3.46695e6 0.904224
\(431\) 2.57138e6 0.666766 0.333383 0.942792i \(-0.391810\pi\)
0.333383 + 0.942792i \(0.391810\pi\)
\(432\) −161793. −0.0417110
\(433\) −2.38192e6 −0.610532 −0.305266 0.952267i \(-0.598745\pi\)
−0.305266 + 0.952267i \(0.598745\pi\)
\(434\) 3.72520e6 0.949347
\(435\) 2.51280e6 0.636700
\(436\) −6.72057e6 −1.69313
\(437\) −755956. −0.189362
\(438\) −1.38183e7 −3.44167
\(439\) −4.98108e6 −1.23357 −0.616783 0.787134i \(-0.711564\pi\)
−0.616783 + 0.787134i \(0.711564\pi\)
\(440\) 555493. 0.136788
\(441\) −6.88413e6 −1.68559
\(442\) 1.76687e7 4.30179
\(443\) −943151. −0.228335 −0.114167 0.993462i \(-0.536420\pi\)
−0.114167 + 0.993462i \(0.536420\pi\)
\(444\) −1.74640e7 −4.20423
\(445\) 2.03496e6 0.487143
\(446\) −1.07419e7 −2.55709
\(447\) 9.48325e6 2.24486
\(448\) −3.51520e6 −0.827474
\(449\) −358465. −0.0839133 −0.0419567 0.999119i \(-0.513359\pi\)
−0.0419567 + 0.999119i \(0.513359\pi\)
\(450\) −3.18236e6 −0.740830
\(451\) 497997. 0.115288
\(452\) −6.02280e6 −1.38660
\(453\) −3.32139e6 −0.760455
\(454\) −5.73857e6 −1.30667
\(455\) 1.42205e6 0.322022
\(456\) −1.87320e6 −0.421865
\(457\) −4.58896e6 −1.02784 −0.513918 0.857839i \(-0.671806\pi\)
−0.513918 + 0.857839i \(0.671806\pi\)
\(458\) −3.69613e6 −0.823349
\(459\) 1.98745e7 4.40315
\(460\) 2.72397e6 0.600216
\(461\) 6.80506e6 1.49135 0.745675 0.666310i \(-0.232127\pi\)
0.745675 + 0.666310i \(0.232127\pi\)
\(462\) −2.08215e6 −0.453845
\(463\) 5.76634e6 1.25011 0.625054 0.780581i \(-0.285077\pi\)
0.625054 + 0.780581i \(0.285077\pi\)
\(464\) −65184.8 −0.0140557
\(465\) 4.32127e6 0.926785
\(466\) −4.74886e6 −1.01304
\(467\) −2.25580e6 −0.478640 −0.239320 0.970941i \(-0.576925\pi\)
−0.239320 + 0.970941i \(0.576925\pi\)
\(468\) 2.47465e7 5.22275
\(469\) −1.56189e6 −0.327883
\(470\) −6.69308e6 −1.39760
\(471\) 1.00487e7 2.08717
\(472\) −6.48727e6 −1.34032
\(473\) 1.83050e6 0.376198
\(474\) −9.82261e6 −2.00808
\(475\) −225625. −0.0458831
\(476\) −7.78108e6 −1.57406
\(477\) 4.50278e6 0.906118
\(478\) −1.23627e7 −2.47482
\(479\) 5.82530e6 1.16006 0.580029 0.814596i \(-0.303041\pi\)
0.580029 + 0.814596i \(0.303041\pi\)
\(480\) −4.03248e6 −0.798856
\(481\) 1.01705e7 2.00437
\(482\) −5.62697e6 −1.10321
\(483\) −3.93091e6 −0.766700
\(484\) 761805. 0.147819
\(485\) 1.44936e6 0.279782
\(486\) 9.99262e6 1.91906
\(487\) −2.83168e6 −0.541030 −0.270515 0.962716i \(-0.587194\pi\)
−0.270515 + 0.962716i \(0.587194\pi\)
\(488\) −2.51635e6 −0.478323
\(489\) −1.03092e7 −1.94963
\(490\) 2.84031e6 0.534411
\(491\) −5.25571e6 −0.983846 −0.491923 0.870639i \(-0.663706\pi\)
−0.491923 + 0.870639i \(0.663706\pi\)
\(492\) 6.05116e6 1.12700
\(493\) 8.00721e6 1.48376
\(494\) 2.83351e6 0.522405
\(495\) −1.68024e6 −0.308219
\(496\) −112098. −0.0204595
\(497\) 1.52351e6 0.276665
\(498\) 1.69330e7 3.05957
\(499\) −4.86665e6 −0.874941 −0.437471 0.899233i \(-0.644126\pi\)
−0.437471 + 0.899233i \(0.644126\pi\)
\(500\) 813004. 0.145435
\(501\) −4.85035e6 −0.863334
\(502\) 6.31962e6 1.11926
\(503\) −5.00425e6 −0.881899 −0.440949 0.897532i \(-0.645358\pi\)
−0.440949 + 0.897532i \(0.645358\pi\)
\(504\) −6.77609e6 −1.18824
\(505\) −1.15319e6 −0.201221
\(506\) 2.32273e6 0.403294
\(507\) −1.02248e7 −1.76658
\(508\) 1.32346e7 2.27537
\(509\) −76974.8 −0.0131690 −0.00658452 0.999978i \(-0.502096\pi\)
−0.00658452 + 0.999978i \(0.502096\pi\)
\(510\) −1.45772e7 −2.48170
\(511\) −3.54393e6 −0.600390
\(512\) 212292. 0.0357898
\(513\) 3.18724e6 0.534714
\(514\) −5.17435e6 −0.863870
\(515\) −1.86459e6 −0.309788
\(516\) 2.22424e7 3.67754
\(517\) −3.53385e6 −0.581463
\(518\) −7.23351e6 −1.18447
\(519\) −6.94243e6 −1.13134
\(520\) −3.93086e6 −0.637499
\(521\) 6.10975e6 0.986119 0.493059 0.869996i \(-0.335878\pi\)
0.493059 + 0.869996i \(0.335878\pi\)
\(522\) 1.81119e7 2.90929
\(523\) 7.63002e6 1.21975 0.609876 0.792497i \(-0.291219\pi\)
0.609876 + 0.792497i \(0.291219\pi\)
\(524\) −6.48563e6 −1.03187
\(525\) −1.17323e6 −0.185774
\(526\) 1.58911e7 2.50433
\(527\) 1.37700e7 2.15977
\(528\) 62656.0 0.00978087
\(529\) −2.05125e6 −0.318698
\(530\) −1.85779e6 −0.287281
\(531\) 1.96225e7 3.02008
\(532\) −1.24784e6 −0.191153
\(533\) −3.52400e6 −0.537301
\(534\) 2.10845e7 3.19971
\(535\) 4.82325e6 0.728543
\(536\) 4.31743e6 0.649102
\(537\) 9.83764e6 1.47216
\(538\) −5.23337e6 −0.779518
\(539\) 1.49964e6 0.222339
\(540\) −1.14847e7 −1.69487
\(541\) 1.73668e6 0.255110 0.127555 0.991832i \(-0.459287\pi\)
0.127555 + 0.991832i \(0.459287\pi\)
\(542\) −6.32283e6 −0.924514
\(543\) 1.76156e6 0.256388
\(544\) −1.28498e7 −1.86165
\(545\) −3.22904e6 −0.465674
\(546\) 1.47340e7 2.11514
\(547\) −7.51122e6 −1.07335 −0.536676 0.843788i \(-0.680320\pi\)
−0.536676 + 0.843788i \(0.680320\pi\)
\(548\) −5.03258e6 −0.715879
\(549\) 7.61138e6 1.07779
\(550\) 693248. 0.0977196
\(551\) 1.28411e6 0.180186
\(552\) 1.08659e7 1.51782
\(553\) −2.51918e6 −0.350305
\(554\) 1.12113e7 1.55196
\(555\) −8.39095e6 −1.15632
\(556\) −1.12650e7 −1.54542
\(557\) −1.34726e7 −1.83998 −0.919990 0.391942i \(-0.871803\pi\)
−0.919990 + 0.391942i \(0.871803\pi\)
\(558\) 3.11470e7 4.23478
\(559\) −1.29532e7 −1.75327
\(560\) 30434.9 0.00410111
\(561\) −7.69657e6 −1.03250
\(562\) −2.18471e7 −2.91778
\(563\) 4.28186e6 0.569327 0.284663 0.958628i \(-0.408118\pi\)
0.284663 + 0.958628i \(0.408118\pi\)
\(564\) −4.29398e7 −5.68411
\(565\) −2.89378e6 −0.381368
\(566\) 2.47827e6 0.325168
\(567\) 7.60671e6 0.993663
\(568\) −4.21133e6 −0.547708
\(569\) −1.24720e7 −1.61494 −0.807471 0.589907i \(-0.799164\pi\)
−0.807471 + 0.589907i \(0.799164\pi\)
\(570\) −2.33773e6 −0.301375
\(571\) 1.26377e7 1.62210 0.811049 0.584978i \(-0.198897\pi\)
0.811049 + 0.584978i \(0.198897\pi\)
\(572\) −5.39079e6 −0.688910
\(573\) −1.70804e7 −2.17326
\(574\) 2.50636e6 0.317515
\(575\) 1.30879e6 0.165082
\(576\) −2.93912e7 −3.69114
\(577\) 1.20353e6 0.150494 0.0752470 0.997165i \(-0.476025\pi\)
0.0752470 + 0.997165i \(0.476025\pi\)
\(578\) −3.34356e7 −4.16284
\(579\) −1.50211e6 −0.186211
\(580\) −4.62707e6 −0.571132
\(581\) 4.34276e6 0.533735
\(582\) 1.50170e7 1.83770
\(583\) −980888. −0.119522
\(584\) 9.79625e6 1.18858
\(585\) 1.18900e7 1.43645
\(586\) 1.76847e7 2.12743
\(587\) 690137. 0.0826685 0.0413342 0.999145i \(-0.486839\pi\)
0.0413342 + 0.999145i \(0.486839\pi\)
\(588\) 1.82222e7 2.17349
\(589\) 2.20828e6 0.262280
\(590\) −8.09602e6 −0.957506
\(591\) 2.71358e7 3.19576
\(592\) 217670. 0.0255267
\(593\) −3.51525e6 −0.410506 −0.205253 0.978709i \(-0.565802\pi\)
−0.205253 + 0.978709i \(0.565802\pi\)
\(594\) −9.79301e6 −1.13881
\(595\) −3.73858e6 −0.432927
\(596\) −1.74625e7 −2.01368
\(597\) 3.75423e6 0.431108
\(598\) −1.64364e7 −1.87955
\(599\) 8.04943e6 0.916638 0.458319 0.888788i \(-0.348452\pi\)
0.458319 + 0.888788i \(0.348452\pi\)
\(600\) 3.24308e6 0.367773
\(601\) −3.27785e6 −0.370171 −0.185085 0.982722i \(-0.559256\pi\)
−0.185085 + 0.982722i \(0.559256\pi\)
\(602\) 9.21269e6 1.03608
\(603\) −1.30592e7 −1.46260
\(604\) 6.11600e6 0.682143
\(605\) 366025. 0.0406558
\(606\) −1.19484e7 −1.32168
\(607\) −678553. −0.0747502 −0.0373751 0.999301i \(-0.511900\pi\)
−0.0373751 + 0.999301i \(0.511900\pi\)
\(608\) −2.06070e6 −0.226077
\(609\) 6.67724e6 0.729548
\(610\) −3.14036e6 −0.341708
\(611\) 2.50068e7 2.70991
\(612\) −6.50589e7 −7.02148
\(613\) −1.28758e7 −1.38395 −0.691977 0.721920i \(-0.743260\pi\)
−0.691977 + 0.721920i \(0.743260\pi\)
\(614\) 1.39277e7 1.49094
\(615\) 2.90741e6 0.309969
\(616\) 1.47611e6 0.156735
\(617\) −1.72087e7 −1.81985 −0.909924 0.414776i \(-0.863860\pi\)
−0.909924 + 0.414776i \(0.863860\pi\)
\(618\) −1.93193e7 −2.03479
\(619\) 6.64082e6 0.696619 0.348309 0.937380i \(-0.386756\pi\)
0.348309 + 0.937380i \(0.386756\pi\)
\(620\) −7.95719e6 −0.831344
\(621\) −1.84883e7 −1.92384
\(622\) 2.30661e7 2.39055
\(623\) 5.40749e6 0.558182
\(624\) −443375. −0.0455837
\(625\) 390625. 0.0400000
\(626\) 4.99619e6 0.509569
\(627\) −1.23429e6 −0.125386
\(628\) −1.85037e7 −1.87223
\(629\) −2.67383e7 −2.69468
\(630\) −8.45647e6 −0.848863
\(631\) 5.84026e6 0.583927 0.291964 0.956429i \(-0.405691\pi\)
0.291964 + 0.956429i \(0.405691\pi\)
\(632\) 6.96359e6 0.693490
\(633\) 1.16499e7 1.15562
\(634\) −1.22551e6 −0.121086
\(635\) 6.35886e6 0.625813
\(636\) −1.19188e7 −1.16839
\(637\) −1.06120e7 −1.03621
\(638\) −3.94550e6 −0.383752
\(639\) 1.27383e7 1.23413
\(640\) 7.55980e6 0.729558
\(641\) −2.49949e6 −0.240274 −0.120137 0.992757i \(-0.538333\pi\)
−0.120137 + 0.992757i \(0.538333\pi\)
\(642\) 4.99744e7 4.78531
\(643\) 7.25773e6 0.692267 0.346134 0.938185i \(-0.387494\pi\)
0.346134 + 0.938185i \(0.387494\pi\)
\(644\) 7.23839e6 0.687744
\(645\) 1.06868e7 1.01146
\(646\) −7.44934e6 −0.702322
\(647\) 1.21058e7 1.13692 0.568462 0.822709i \(-0.307539\pi\)
0.568462 + 0.822709i \(0.307539\pi\)
\(648\) −2.10267e7 −1.96713
\(649\) −4.27459e6 −0.398366
\(650\) −4.90566e6 −0.455422
\(651\) 1.14829e7 1.06194
\(652\) 1.89833e7 1.74885
\(653\) 2.08897e7 1.91712 0.958561 0.284888i \(-0.0919563\pi\)
0.958561 + 0.284888i \(0.0919563\pi\)
\(654\) −3.34565e7 −3.05870
\(655\) −3.11616e6 −0.283802
\(656\) −75421.3 −0.00684280
\(657\) −2.96315e7 −2.67818
\(658\) −1.77855e7 −1.60140
\(659\) −4.46750e6 −0.400729 −0.200365 0.979721i \(-0.564213\pi\)
−0.200365 + 0.979721i \(0.564213\pi\)
\(660\) 4.44757e6 0.397432
\(661\) 1.70197e6 0.151512 0.0757560 0.997126i \(-0.475863\pi\)
0.0757560 + 0.997126i \(0.475863\pi\)
\(662\) 2.86574e7 2.54151
\(663\) 5.44636e7 4.81197
\(664\) −1.20044e7 −1.05662
\(665\) −599551. −0.0525742
\(666\) −6.04806e7 −5.28361
\(667\) −7.44874e6 −0.648289
\(668\) 8.93144e6 0.774427
\(669\) −3.31119e7 −2.86035
\(670\) 5.38809e6 0.463711
\(671\) −1.65807e6 −0.142166
\(672\) −1.07155e7 −0.915351
\(673\) 1.82947e7 1.55699 0.778496 0.627649i \(-0.215983\pi\)
0.778496 + 0.627649i \(0.215983\pi\)
\(674\) −1.88702e7 −1.60002
\(675\) −5.51808e6 −0.466153
\(676\) 1.88279e7 1.58466
\(677\) 5.79481e6 0.485923 0.242961 0.970036i \(-0.421881\pi\)
0.242961 + 0.970036i \(0.421881\pi\)
\(678\) −2.99829e7 −2.50495
\(679\) 3.85136e6 0.320582
\(680\) 1.03343e7 0.857055
\(681\) −1.76891e7 −1.46163
\(682\) −6.78509e6 −0.558592
\(683\) −5.53455e6 −0.453973 −0.226987 0.973898i \(-0.572887\pi\)
−0.226987 + 0.973898i \(0.572887\pi\)
\(684\) −1.04334e7 −0.852681
\(685\) −2.41801e6 −0.196894
\(686\) 1.77826e7 1.44273
\(687\) −1.13933e7 −0.920994
\(688\) −277228. −0.0223288
\(689\) 6.94110e6 0.557032
\(690\) 1.35605e7 1.08431
\(691\) 8.62709e6 0.687336 0.343668 0.939091i \(-0.388331\pi\)
0.343668 + 0.939091i \(0.388331\pi\)
\(692\) 1.27838e7 1.01483
\(693\) −4.46489e6 −0.353165
\(694\) −3.64990e7 −2.87662
\(695\) −5.41252e6 −0.425048
\(696\) −1.84574e7 −1.44427
\(697\) 9.26464e6 0.722348
\(698\) 2.05716e7 1.59819
\(699\) −1.46383e7 −1.13318
\(700\) 2.16039e6 0.166643
\(701\) 2.57068e7 1.97584 0.987922 0.154952i \(-0.0495224\pi\)
0.987922 + 0.154952i \(0.0495224\pi\)
\(702\) 6.92988e7 5.30741
\(703\) −4.28799e6 −0.327239
\(704\) 6.40259e6 0.486882
\(705\) −2.06313e7 −1.56334
\(706\) −2.82302e6 −0.213158
\(707\) −3.06437e6 −0.230565
\(708\) −5.19405e7 −3.89424
\(709\) 5.87521e6 0.438943 0.219471 0.975619i \(-0.429567\pi\)
0.219471 + 0.975619i \(0.429567\pi\)
\(710\) −5.25568e6 −0.391276
\(711\) −2.10633e7 −1.56262
\(712\) −1.49476e7 −1.10502
\(713\) −1.28096e7 −0.943654
\(714\) −3.87360e7 −2.84360
\(715\) −2.59012e6 −0.189476
\(716\) −1.81150e7 −1.32056
\(717\) −3.81078e7 −2.76832
\(718\) 1.02060e7 0.738831
\(719\) −2.20444e7 −1.59029 −0.795143 0.606422i \(-0.792604\pi\)
−0.795143 + 0.606422i \(0.792604\pi\)
\(720\) 254472. 0.0182940
\(721\) −4.95476e6 −0.354964
\(722\) −1.19464e6 −0.0852893
\(723\) −1.73451e7 −1.23404
\(724\) −3.24374e6 −0.229985
\(725\) −2.22317e6 −0.157083
\(726\) 3.79244e6 0.267040
\(727\) 1.69303e7 1.18803 0.594015 0.804454i \(-0.297542\pi\)
0.594015 + 0.804454i \(0.297542\pi\)
\(728\) −1.04455e7 −0.730464
\(729\) 2.97786e6 0.207532
\(730\) 1.22256e7 0.849107
\(731\) 3.40543e7 2.35710
\(732\) −2.01472e7 −1.38975
\(733\) 2.50996e7 1.72547 0.862733 0.505661i \(-0.168751\pi\)
0.862733 + 0.505661i \(0.168751\pi\)
\(734\) 230897. 0.0158190
\(735\) 8.75522e6 0.597790
\(736\) 1.19535e7 0.813397
\(737\) 2.84483e6 0.192925
\(738\) 2.09561e7 1.41635
\(739\) 2.29213e7 1.54393 0.771967 0.635662i \(-0.219273\pi\)
0.771967 + 0.635662i \(0.219273\pi\)
\(740\) 1.54511e7 1.03724
\(741\) 8.73426e6 0.584360
\(742\) −4.93670e6 −0.329175
\(743\) 2.72611e7 1.81164 0.905820 0.423662i \(-0.139255\pi\)
0.905820 + 0.423662i \(0.139255\pi\)
\(744\) −3.17413e7 −2.10229
\(745\) −8.39021e6 −0.553837
\(746\) −4.43023e7 −2.91460
\(747\) 3.63106e7 2.38085
\(748\) 1.41725e7 0.926172
\(749\) 1.28168e7 0.834785
\(750\) 4.04732e6 0.262733
\(751\) −3.04081e7 −1.96739 −0.983693 0.179854i \(-0.942437\pi\)
−0.983693 + 0.179854i \(0.942437\pi\)
\(752\) 535200. 0.0345121
\(753\) 1.94801e7 1.25200
\(754\) 2.79197e7 1.78847
\(755\) 2.93856e6 0.187615
\(756\) −3.05183e7 −1.94203
\(757\) −2.16756e6 −0.137478 −0.0687388 0.997635i \(-0.521898\pi\)
−0.0687388 + 0.997635i \(0.521898\pi\)
\(758\) −1.85473e7 −1.17248
\(759\) 7.15977e6 0.451123
\(760\) 1.65730e6 0.104080
\(761\) −1.27130e7 −0.795765 −0.397882 0.917436i \(-0.630255\pi\)
−0.397882 + 0.917436i \(0.630255\pi\)
\(762\) 6.58850e7 4.11054
\(763\) −8.58050e6 −0.533582
\(764\) 3.14518e7 1.94945
\(765\) −3.12589e7 −1.93117
\(766\) 1.88304e7 1.15955
\(767\) 3.02485e7 1.85658
\(768\) 3.04822e7 1.86485
\(769\) 8.41597e6 0.513202 0.256601 0.966517i \(-0.417397\pi\)
0.256601 + 0.966517i \(0.417397\pi\)
\(770\) 1.84216e6 0.111970
\(771\) −1.59499e7 −0.966321
\(772\) 2.76599e6 0.167035
\(773\) 1.79031e7 1.07765 0.538827 0.842416i \(-0.318868\pi\)
0.538827 + 0.842416i \(0.318868\pi\)
\(774\) 7.70289e7 4.62170
\(775\) −3.82320e6 −0.228651
\(776\) −1.06460e7 −0.634650
\(777\) −2.22972e7 −1.32495
\(778\) −2.23508e7 −1.32386
\(779\) 1.48576e6 0.0877212
\(780\) −3.14725e7 −1.85223
\(781\) −2.77493e6 −0.162789
\(782\) 4.32116e7 2.52687
\(783\) 3.14052e7 1.83061
\(784\) −227120. −0.0131967
\(785\) −8.89047e6 −0.514933
\(786\) −3.22869e7 −1.86410
\(787\) 2.59120e7 1.49130 0.745648 0.666340i \(-0.232140\pi\)
0.745648 + 0.666340i \(0.232140\pi\)
\(788\) −4.99679e7 −2.86665
\(789\) 4.89843e7 2.80133
\(790\) 8.69046e6 0.495421
\(791\) −7.68962e6 −0.436982
\(792\) 1.23420e7 0.699153
\(793\) 1.17331e7 0.662565
\(794\) 3.28373e7 1.84849
\(795\) −5.72662e6 −0.321352
\(796\) −6.91305e6 −0.386712
\(797\) 1.67109e7 0.931866 0.465933 0.884820i \(-0.345719\pi\)
0.465933 + 0.884820i \(0.345719\pi\)
\(798\) −6.21204e6 −0.345324
\(799\) −6.57432e7 −3.64321
\(800\) 3.56769e6 0.197089
\(801\) 4.52130e7 2.48990
\(802\) 2.26195e7 1.24179
\(803\) 6.45493e6 0.353267
\(804\) 3.45676e7 1.88594
\(805\) 3.47783e6 0.189156
\(806\) 4.80136e7 2.60332
\(807\) −1.61318e7 −0.871965
\(808\) 8.47063e6 0.456444
\(809\) −1.13038e7 −0.607229 −0.303614 0.952795i \(-0.598193\pi\)
−0.303614 + 0.952795i \(0.598193\pi\)
\(810\) −2.62410e7 −1.40530
\(811\) −3.69748e7 −1.97403 −0.987015 0.160630i \(-0.948647\pi\)
−0.987015 + 0.160630i \(0.948647\pi\)
\(812\) −1.22955e7 −0.654418
\(813\) −1.94901e7 −1.03416
\(814\) 1.31751e7 0.696938
\(815\) 9.12093e6 0.481000
\(816\) 1.16564e6 0.0612829
\(817\) 5.46124e6 0.286244
\(818\) 4.25047e7 2.22103
\(819\) 3.15951e7 1.64593
\(820\) −5.35370e6 −0.278048
\(821\) −3.01143e7 −1.55924 −0.779622 0.626250i \(-0.784589\pi\)
−0.779622 + 0.626250i \(0.784589\pi\)
\(822\) −2.50533e7 −1.29326
\(823\) −2.11678e7 −1.08937 −0.544685 0.838641i \(-0.683351\pi\)
−0.544685 + 0.838641i \(0.683351\pi\)
\(824\) 1.36961e7 0.702714
\(825\) 2.13693e6 0.109309
\(826\) −2.15135e7 −1.09714
\(827\) 2.42006e7 1.23044 0.615222 0.788354i \(-0.289066\pi\)
0.615222 + 0.788354i \(0.289066\pi\)
\(828\) 6.05214e7 3.06784
\(829\) 2.15825e7 1.09073 0.545364 0.838199i \(-0.316391\pi\)
0.545364 + 0.838199i \(0.316391\pi\)
\(830\) −1.49813e7 −0.754839
\(831\) 3.45587e7 1.73602
\(832\) −4.53069e7 −2.26911
\(833\) 2.78991e7 1.39309
\(834\) −5.60799e7 −2.79185
\(835\) 4.29130e6 0.212997
\(836\) 2.27282e6 0.112473
\(837\) 5.40076e7 2.66466
\(838\) −3.07473e7 −1.51251
\(839\) −1.13438e6 −0.0556355 −0.0278178 0.999613i \(-0.508856\pi\)
−0.0278178 + 0.999613i \(0.508856\pi\)
\(840\) 8.61782e6 0.421404
\(841\) −7.85834e6 −0.383125
\(842\) 5.52581e7 2.68606
\(843\) −6.73434e7 −3.26382
\(844\) −2.14522e7 −1.03661
\(845\) 9.04627e6 0.435840
\(846\) −1.48707e8 −7.14343
\(847\) 972635. 0.0465845
\(848\) 148555. 0.00709409
\(849\) 7.63924e6 0.363732
\(850\) 1.28971e7 0.612270
\(851\) 2.48735e7 1.17737
\(852\) −3.37181e7 −1.59135
\(853\) −6.38506e6 −0.300464 −0.150232 0.988651i \(-0.548002\pi\)
−0.150232 + 0.988651i \(0.548002\pi\)
\(854\) −8.34486e6 −0.391539
\(855\) −5.01295e6 −0.234519
\(856\) −3.54285e7 −1.65260
\(857\) −3.31421e7 −1.54144 −0.770722 0.637172i \(-0.780104\pi\)
−0.770722 + 0.637172i \(0.780104\pi\)
\(858\) −2.68366e7 −1.24454
\(859\) 2.29603e7 1.06168 0.530840 0.847472i \(-0.321876\pi\)
0.530840 + 0.847472i \(0.321876\pi\)
\(860\) −1.96787e7 −0.907300
\(861\) 7.72583e6 0.355171
\(862\) −2.35716e7 −1.08049
\(863\) −1.01302e7 −0.463010 −0.231505 0.972834i \(-0.574365\pi\)
−0.231505 + 0.972834i \(0.574365\pi\)
\(864\) −5.03982e7 −2.29684
\(865\) 6.14225e6 0.279118
\(866\) 2.18349e7 0.989364
\(867\) −1.03065e8 −4.65654
\(868\) −2.11446e7 −0.952576
\(869\) 4.58844e6 0.206118
\(870\) −2.30346e7 −1.03177
\(871\) −2.01310e7 −0.899126
\(872\) 2.37185e7 1.05632
\(873\) 3.22019e7 1.43003
\(874\) 6.92979e6 0.306861
\(875\) 1.03800e6 0.0458331
\(876\) 7.84339e7 3.45337
\(877\) −2.12082e7 −0.931119 −0.465559 0.885017i \(-0.654147\pi\)
−0.465559 + 0.885017i \(0.654147\pi\)
\(878\) 4.56611e7 1.99899
\(879\) 5.45130e7 2.37973
\(880\) −55434.2 −0.00241308
\(881\) 4.48994e7 1.94895 0.974475 0.224494i \(-0.0720731\pi\)
0.974475 + 0.224494i \(0.0720731\pi\)
\(882\) 6.31063e7 2.73150
\(883\) 4.21540e6 0.181943 0.0909717 0.995853i \(-0.471003\pi\)
0.0909717 + 0.995853i \(0.471003\pi\)
\(884\) −1.00289e8 −4.31643
\(885\) −2.49559e7 −1.07106
\(886\) 8.64578e6 0.370015
\(887\) 1.47395e7 0.629035 0.314518 0.949252i \(-0.398157\pi\)
0.314518 + 0.949252i \(0.398157\pi\)
\(888\) 6.16346e7 2.62296
\(889\) 1.68973e7 0.717074
\(890\) −1.86543e7 −0.789414
\(891\) −1.38549e7 −0.584667
\(892\) 6.09723e7 2.56578
\(893\) −1.05431e7 −0.442427
\(894\) −8.69322e7 −3.63778
\(895\) −8.70375e6 −0.363203
\(896\) 2.00886e7 0.835948
\(897\) −5.06651e7 −2.10246
\(898\) 3.28602e6 0.135981
\(899\) 2.17591e7 0.897927
\(900\) 1.80634e7 0.743350
\(901\) −1.82483e7 −0.748875
\(902\) −4.56509e6 −0.186824
\(903\) 2.83980e7 1.15896
\(904\) 2.12559e7 0.865084
\(905\) −1.55852e6 −0.0632545
\(906\) 3.04468e7 1.23232
\(907\) −6.63616e6 −0.267854 −0.133927 0.990991i \(-0.542759\pi\)
−0.133927 + 0.990991i \(0.542759\pi\)
\(908\) 3.25727e7 1.31111
\(909\) −2.56217e7 −1.02849
\(910\) −1.30358e7 −0.521835
\(911\) 2.09681e7 0.837074 0.418537 0.908200i \(-0.362543\pi\)
0.418537 + 0.908200i \(0.362543\pi\)
\(912\) 186932. 0.00744213
\(913\) −7.90992e6 −0.314047
\(914\) 4.20666e7 1.66560
\(915\) −9.68013e6 −0.382233
\(916\) 2.09796e7 0.826149
\(917\) −8.28054e6 −0.325188
\(918\) −1.82187e8 −7.13529
\(919\) 8.30744e6 0.324473 0.162236 0.986752i \(-0.448129\pi\)
0.162236 + 0.986752i \(0.448129\pi\)
\(920\) −9.61353e6 −0.374467
\(921\) 4.29321e7 1.66776
\(922\) −6.23814e7 −2.41673
\(923\) 1.96363e7 0.758676
\(924\) 1.18185e7 0.455388
\(925\) 7.42381e6 0.285281
\(926\) −5.28595e7 −2.02580
\(927\) −4.14276e7 −1.58340
\(928\) −2.03049e7 −0.773982
\(929\) −2.68907e7 −1.02226 −0.511132 0.859502i \(-0.670774\pi\)
−0.511132 + 0.859502i \(0.670774\pi\)
\(930\) −3.96127e7 −1.50185
\(931\) 4.47414e6 0.169175
\(932\) 2.69550e7 1.01648
\(933\) 7.11008e7 2.67406
\(934\) 2.06788e7 0.775635
\(935\) 6.80946e6 0.254732
\(936\) −8.73363e7 −3.25840
\(937\) −1.95375e7 −0.726975 −0.363488 0.931599i \(-0.618414\pi\)
−0.363488 + 0.931599i \(0.618414\pi\)
\(938\) 1.43177e7 0.531333
\(939\) 1.54007e7 0.570002
\(940\) 3.79906e7 1.40235
\(941\) 3.68525e7 1.35673 0.678364 0.734726i \(-0.262689\pi\)
0.678364 + 0.734726i \(0.262689\pi\)
\(942\) −9.21154e7 −3.38225
\(943\) −8.61848e6 −0.315611
\(944\) 647383. 0.0236446
\(945\) −1.46631e7 −0.534131
\(946\) −1.67800e7 −0.609628
\(947\) −151048. −0.00547318 −0.00273659 0.999996i \(-0.500871\pi\)
−0.00273659 + 0.999996i \(0.500871\pi\)
\(948\) 5.57541e7 2.01491
\(949\) −4.56774e7 −1.64640
\(950\) 2.06828e6 0.0743535
\(951\) −3.77761e6 −0.135446
\(952\) 2.74613e7 0.982037
\(953\) 1.04441e7 0.372509 0.186255 0.982501i \(-0.440365\pi\)
0.186255 + 0.982501i \(0.440365\pi\)
\(954\) −4.12766e7 −1.46836
\(955\) 1.51117e7 0.536173
\(956\) 7.01718e7 2.48324
\(957\) −1.21620e7 −0.429263
\(958\) −5.34001e7 −1.87987
\(959\) −6.42536e6 −0.225606
\(960\) 3.73796e7 1.30905
\(961\) 8.79001e6 0.307030
\(962\) −9.32319e7 −3.24808
\(963\) 1.07163e8 3.72375
\(964\) 3.19393e7 1.10696
\(965\) 1.32898e6 0.0459409
\(966\) 3.60343e7 1.24243
\(967\) −2.27014e7 −0.780704 −0.390352 0.920666i \(-0.627647\pi\)
−0.390352 + 0.920666i \(0.627647\pi\)
\(968\) −2.68859e6 −0.0922223
\(969\) −2.29625e7 −0.785615
\(970\) −1.32861e7 −0.453387
\(971\) 2.51537e7 0.856158 0.428079 0.903741i \(-0.359191\pi\)
0.428079 + 0.903741i \(0.359191\pi\)
\(972\) −5.67191e7 −1.92559
\(973\) −1.43827e7 −0.487031
\(974\) 2.59577e7 0.876737
\(975\) −1.51216e7 −0.509433
\(976\) 251113. 0.00843810
\(977\) 3.04134e7 1.01936 0.509681 0.860363i \(-0.329763\pi\)
0.509681 + 0.860363i \(0.329763\pi\)
\(978\) 9.45032e7 3.15936
\(979\) −9.84922e6 −0.328432
\(980\) −1.61219e7 −0.536229
\(981\) −7.17430e7 −2.38017
\(982\) 4.81786e7 1.59432
\(983\) −6.43832e6 −0.212515 −0.106257 0.994339i \(-0.533887\pi\)
−0.106257 + 0.994339i \(0.533887\pi\)
\(984\) −2.13560e7 −0.703123
\(985\) −2.40081e7 −0.788437
\(986\) −7.34014e7 −2.40443
\(987\) −5.48235e7 −1.79132
\(988\) −1.60833e7 −0.524182
\(989\) −3.16792e7 −1.02987
\(990\) 1.54026e7 0.499467
\(991\) 2.37565e7 0.768418 0.384209 0.923246i \(-0.374474\pi\)
0.384209 + 0.923246i \(0.374474\pi\)
\(992\) −3.49184e7 −1.12661
\(993\) 8.83360e7 2.84292
\(994\) −1.39659e7 −0.448335
\(995\) −3.32152e6 −0.106360
\(996\) −9.61134e7 −3.06998
\(997\) 4.00676e7 1.27660 0.638301 0.769787i \(-0.279638\pi\)
0.638301 + 0.769787i \(0.279638\pi\)
\(998\) 4.46122e7 1.41784
\(999\) −1.04871e8 −3.32461
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 1045.6.a.f.1.4 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.6.a.f.1.4 38 1.1 even 1 trivial