Properties

Label 387.6.a.e.1.7
Level $387$
Weight $6$
Character 387.1
Self dual yes
Analytic conductor $62.069$
Analytic rank $1$
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] = [387,6,Mod(1,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, 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("387.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 387.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: \(62.0685382676\)
Analytic rank: \(1\)
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: \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(3.50018\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.50018 q^{2} -25.7491 q^{4} -47.4635 q^{5} +67.4603 q^{7} -144.383 q^{8} +O(q^{10})\) \(q+2.50018 q^{2} -25.7491 q^{4} -47.4635 q^{5} +67.4603 q^{7} -144.383 q^{8} -118.667 q^{10} -81.3987 q^{11} +1058.72 q^{13} +168.663 q^{14} +462.986 q^{16} -251.378 q^{17} +1612.95 q^{19} +1222.14 q^{20} -203.512 q^{22} +32.7403 q^{23} -872.217 q^{25} +2647.00 q^{26} -1737.04 q^{28} +2583.75 q^{29} -7206.51 q^{31} +5777.81 q^{32} -628.491 q^{34} -3201.90 q^{35} -6174.39 q^{37} +4032.66 q^{38} +6852.93 q^{40} -15514.4 q^{41} +1849.00 q^{43} +2095.94 q^{44} +81.8569 q^{46} +1692.39 q^{47} -12256.1 q^{49} -2180.70 q^{50} -27261.2 q^{52} +25612.3 q^{53} +3863.47 q^{55} -9740.14 q^{56} +6459.83 q^{58} -24532.0 q^{59} +8209.11 q^{61} -18017.6 q^{62} -369.967 q^{64} -50250.7 q^{65} -12302.3 q^{67} +6472.75 q^{68} -8005.34 q^{70} -18712.0 q^{71} -12126.5 q^{73} -15437.1 q^{74} -41531.9 q^{76} -5491.18 q^{77} -52372.8 q^{79} -21974.9 q^{80} -38788.9 q^{82} -28935.9 q^{83} +11931.3 q^{85} +4622.84 q^{86} +11752.6 q^{88} -117315. q^{89} +71421.8 q^{91} -843.034 q^{92} +4231.28 q^{94} -76556.1 q^{95} -147143. q^{97} -30642.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8} - 17 q^{10} - 745 q^{11} + 1917 q^{13} - 1936 q^{14} + 5354 q^{16} - 4017 q^{17} - 2404 q^{19} - 1311 q^{20} - 5836 q^{22} - 1733 q^{23} + 7120 q^{25} + 1484 q^{26} - 15028 q^{28} - 6996 q^{29} - 4899 q^{31} + 7554 q^{32} - 27033 q^{34} - 7084 q^{35} + 1466 q^{37} - 13905 q^{38} - 93211 q^{40} - 10297 q^{41} + 18490 q^{43} + 36140 q^{44} + 17991 q^{46} - 48592 q^{47} + 29458 q^{49} - 983 q^{50} + 14232 q^{52} - 127165 q^{53} + 106672 q^{55} + 7780 q^{56} - 10305 q^{58} - 99372 q^{59} + 17408 q^{61} - 28265 q^{62} + 47202 q^{64} - 54484 q^{65} - 2021 q^{67} - 192151 q^{68} - 33194 q^{70} - 11286 q^{71} + 49892 q^{73} + 125431 q^{74} - 249803 q^{76} - 98144 q^{77} - 91524 q^{79} - 12251 q^{80} - 158909 q^{82} + 105203 q^{83} - 87212 q^{85} - 14792 q^{86} - 461824 q^{88} + 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 7259 q^{94} + 305340 q^{95} + 108383 q^{97} - 354656 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.50018 0.441974 0.220987 0.975277i \(-0.429072\pi\)
0.220987 + 0.975277i \(0.429072\pi\)
\(3\) 0 0
\(4\) −25.7491 −0.804659
\(5\) −47.4635 −0.849053 −0.424526 0.905416i \(-0.639559\pi\)
−0.424526 + 0.905416i \(0.639559\pi\)
\(6\) 0 0
\(7\) 67.4603 0.520359 0.260180 0.965560i \(-0.416218\pi\)
0.260180 + 0.965560i \(0.416218\pi\)
\(8\) −144.383 −0.797612
\(9\) 0 0
\(10\) −118.667 −0.375259
\(11\) −81.3987 −0.202832 −0.101416 0.994844i \(-0.532337\pi\)
−0.101416 + 0.994844i \(0.532337\pi\)
\(12\) 0 0
\(13\) 1058.72 1.73750 0.868749 0.495253i \(-0.164925\pi\)
0.868749 + 0.495253i \(0.164925\pi\)
\(14\) 168.663 0.229985
\(15\) 0 0
\(16\) 462.986 0.452135
\(17\) −251.378 −0.210962 −0.105481 0.994421i \(-0.533638\pi\)
−0.105481 + 0.994421i \(0.533638\pi\)
\(18\) 0 0
\(19\) 1612.95 1.02503 0.512515 0.858679i \(-0.328714\pi\)
0.512515 + 0.858679i \(0.328714\pi\)
\(20\) 1222.14 0.683198
\(21\) 0 0
\(22\) −203.512 −0.0896463
\(23\) 32.7403 0.0129052 0.00645258 0.999979i \(-0.497946\pi\)
0.00645258 + 0.999979i \(0.497946\pi\)
\(24\) 0 0
\(25\) −872.217 −0.279109
\(26\) 2647.00 0.767929
\(27\) 0 0
\(28\) −1737.04 −0.418712
\(29\) 2583.75 0.570499 0.285249 0.958453i \(-0.407924\pi\)
0.285249 + 0.958453i \(0.407924\pi\)
\(30\) 0 0
\(31\) −7206.51 −1.34685 −0.673427 0.739254i \(-0.735179\pi\)
−0.673427 + 0.739254i \(0.735179\pi\)
\(32\) 5777.81 0.997444
\(33\) 0 0
\(34\) −628.491 −0.0932399
\(35\) −3201.90 −0.441812
\(36\) 0 0
\(37\) −6174.39 −0.741463 −0.370731 0.928740i \(-0.620893\pi\)
−0.370731 + 0.928740i \(0.620893\pi\)
\(38\) 4032.66 0.453036
\(39\) 0 0
\(40\) 6852.93 0.677215
\(41\) −15514.4 −1.44137 −0.720685 0.693262i \(-0.756173\pi\)
−0.720685 + 0.693262i \(0.756173\pi\)
\(42\) 0 0
\(43\) 1849.00 0.152499
\(44\) 2095.94 0.163210
\(45\) 0 0
\(46\) 81.8569 0.00570375
\(47\) 1692.39 0.111752 0.0558760 0.998438i \(-0.482205\pi\)
0.0558760 + 0.998438i \(0.482205\pi\)
\(48\) 0 0
\(49\) −12256.1 −0.729226
\(50\) −2180.70 −0.123359
\(51\) 0 0
\(52\) −27261.2 −1.39809
\(53\) 25612.3 1.25245 0.626223 0.779644i \(-0.284600\pi\)
0.626223 + 0.779644i \(0.284600\pi\)
\(54\) 0 0
\(55\) 3863.47 0.172215
\(56\) −9740.14 −0.415045
\(57\) 0 0
\(58\) 6459.83 0.252146
\(59\) −24532.0 −0.917495 −0.458748 0.888567i \(-0.651702\pi\)
−0.458748 + 0.888567i \(0.651702\pi\)
\(60\) 0 0
\(61\) 8209.11 0.282470 0.141235 0.989976i \(-0.454893\pi\)
0.141235 + 0.989976i \(0.454893\pi\)
\(62\) −18017.6 −0.595275
\(63\) 0 0
\(64\) −369.967 −0.0112905
\(65\) −50250.7 −1.47523
\(66\) 0 0
\(67\) −12302.3 −0.334810 −0.167405 0.985888i \(-0.553539\pi\)
−0.167405 + 0.985888i \(0.553539\pi\)
\(68\) 6472.75 0.169753
\(69\) 0 0
\(70\) −8005.34 −0.195270
\(71\) −18712.0 −0.440530 −0.220265 0.975440i \(-0.570692\pi\)
−0.220265 + 0.975440i \(0.570692\pi\)
\(72\) 0 0
\(73\) −12126.5 −0.266334 −0.133167 0.991094i \(-0.542515\pi\)
−0.133167 + 0.991094i \(0.542515\pi\)
\(74\) −15437.1 −0.327707
\(75\) 0 0
\(76\) −41531.9 −0.824799
\(77\) −5491.18 −0.105545
\(78\) 0 0
\(79\) −52372.8 −0.944144 −0.472072 0.881560i \(-0.656494\pi\)
−0.472072 + 0.881560i \(0.656494\pi\)
\(80\) −21974.9 −0.383886
\(81\) 0 0
\(82\) −38788.9 −0.637048
\(83\) −28935.9 −0.461043 −0.230521 0.973067i \(-0.574043\pi\)
−0.230521 + 0.973067i \(0.574043\pi\)
\(84\) 0 0
\(85\) 11931.3 0.179118
\(86\) 4622.84 0.0674004
\(87\) 0 0
\(88\) 11752.6 0.161781
\(89\) −117315. −1.56992 −0.784962 0.619544i \(-0.787317\pi\)
−0.784962 + 0.619544i \(0.787317\pi\)
\(90\) 0 0
\(91\) 71421.8 0.904123
\(92\) −843.034 −0.0103843
\(93\) 0 0
\(94\) 4231.28 0.0493915
\(95\) −76556.1 −0.870304
\(96\) 0 0
\(97\) −147143. −1.58785 −0.793925 0.608016i \(-0.791966\pi\)
−0.793925 + 0.608016i \(0.791966\pi\)
\(98\) −30642.5 −0.322299
\(99\) 0 0
\(100\) 22458.8 0.224588
\(101\) 118561. 1.15648 0.578239 0.815867i \(-0.303740\pi\)
0.578239 + 0.815867i \(0.303740\pi\)
\(102\) 0 0
\(103\) 49153.5 0.456522 0.228261 0.973600i \(-0.426696\pi\)
0.228261 + 0.973600i \(0.426696\pi\)
\(104\) −152862. −1.38585
\(105\) 0 0
\(106\) 64035.4 0.553549
\(107\) 156632. 1.32258 0.661290 0.750130i \(-0.270009\pi\)
0.661290 + 0.750130i \(0.270009\pi\)
\(108\) 0 0
\(109\) 208919. 1.68427 0.842135 0.539267i \(-0.181299\pi\)
0.842135 + 0.539267i \(0.181299\pi\)
\(110\) 9659.37 0.0761144
\(111\) 0 0
\(112\) 31233.2 0.235273
\(113\) −33491.2 −0.246737 −0.123369 0.992361i \(-0.539370\pi\)
−0.123369 + 0.992361i \(0.539370\pi\)
\(114\) 0 0
\(115\) −1553.97 −0.0109572
\(116\) −66529.1 −0.459057
\(117\) 0 0
\(118\) −61334.6 −0.405509
\(119\) −16958.0 −0.109776
\(120\) 0 0
\(121\) −154425. −0.958859
\(122\) 20524.3 0.124844
\(123\) 0 0
\(124\) 185561. 1.08376
\(125\) 189722. 1.08603
\(126\) 0 0
\(127\) 72430.2 0.398483 0.199242 0.979950i \(-0.436152\pi\)
0.199242 + 0.979950i \(0.436152\pi\)
\(128\) −185815. −1.00243
\(129\) 0 0
\(130\) −125636. −0.652012
\(131\) 73042.1 0.371873 0.185937 0.982562i \(-0.440468\pi\)
0.185937 + 0.982562i \(0.440468\pi\)
\(132\) 0 0
\(133\) 108810. 0.533383
\(134\) −30757.9 −0.147977
\(135\) 0 0
\(136\) 36294.8 0.168266
\(137\) −290889. −1.32412 −0.662058 0.749453i \(-0.730317\pi\)
−0.662058 + 0.749453i \(0.730317\pi\)
\(138\) 0 0
\(139\) −375146. −1.64689 −0.823443 0.567399i \(-0.807950\pi\)
−0.823443 + 0.567399i \(0.807950\pi\)
\(140\) 82446.0 0.355508
\(141\) 0 0
\(142\) −46783.5 −0.194703
\(143\) −86178.7 −0.352420
\(144\) 0 0
\(145\) −122634. −0.484384
\(146\) −30318.3 −0.117713
\(147\) 0 0
\(148\) 158985. 0.596625
\(149\) −431136. −1.59092 −0.795460 0.606006i \(-0.792771\pi\)
−0.795460 + 0.606006i \(0.792771\pi\)
\(150\) 0 0
\(151\) 17252.4 0.0615755 0.0307878 0.999526i \(-0.490198\pi\)
0.0307878 + 0.999526i \(0.490198\pi\)
\(152\) −232883. −0.817576
\(153\) 0 0
\(154\) −13729.0 −0.0466483
\(155\) 342046. 1.14355
\(156\) 0 0
\(157\) −139497. −0.451663 −0.225832 0.974166i \(-0.572510\pi\)
−0.225832 + 0.974166i \(0.572510\pi\)
\(158\) −130942. −0.417287
\(159\) 0 0
\(160\) −274235. −0.846883
\(161\) 2208.67 0.00671532
\(162\) 0 0
\(163\) −556197. −1.63968 −0.819841 0.572591i \(-0.805938\pi\)
−0.819841 + 0.572591i \(0.805938\pi\)
\(164\) 399482. 1.15981
\(165\) 0 0
\(166\) −72345.0 −0.203769
\(167\) 239206. 0.663713 0.331857 0.943330i \(-0.392325\pi\)
0.331857 + 0.943330i \(0.392325\pi\)
\(168\) 0 0
\(169\) 749603. 2.01890
\(170\) 29830.4 0.0791656
\(171\) 0 0
\(172\) −47610.1 −0.122709
\(173\) −79563.7 −0.202116 −0.101058 0.994881i \(-0.532223\pi\)
−0.101058 + 0.994881i \(0.532223\pi\)
\(174\) 0 0
\(175\) −58840.0 −0.145237
\(176\) −37686.5 −0.0917073
\(177\) 0 0
\(178\) −293309. −0.693865
\(179\) 95838.9 0.223568 0.111784 0.993733i \(-0.464344\pi\)
0.111784 + 0.993733i \(0.464344\pi\)
\(180\) 0 0
\(181\) −219413. −0.497813 −0.248906 0.968528i \(-0.580071\pi\)
−0.248906 + 0.968528i \(0.580071\pi\)
\(182\) 178568. 0.399599
\(183\) 0 0
\(184\) −4727.16 −0.0102933
\(185\) 293058. 0.629541
\(186\) 0 0
\(187\) 20461.8 0.0427898
\(188\) −43577.5 −0.0899223
\(189\) 0 0
\(190\) −191404. −0.384652
\(191\) −328729. −0.652011 −0.326005 0.945368i \(-0.605703\pi\)
−0.326005 + 0.945368i \(0.605703\pi\)
\(192\) 0 0
\(193\) 575743. 1.11259 0.556295 0.830985i \(-0.312222\pi\)
0.556295 + 0.830985i \(0.312222\pi\)
\(194\) −367884. −0.701788
\(195\) 0 0
\(196\) 315584. 0.586779
\(197\) −962886. −1.76770 −0.883852 0.467768i \(-0.845058\pi\)
−0.883852 + 0.467768i \(0.845058\pi\)
\(198\) 0 0
\(199\) −760147. −1.36071 −0.680354 0.732884i \(-0.738174\pi\)
−0.680354 + 0.732884i \(0.738174\pi\)
\(200\) 125933. 0.222621
\(201\) 0 0
\(202\) 296424. 0.511133
\(203\) 174300. 0.296864
\(204\) 0 0
\(205\) 736368. 1.22380
\(206\) 122893. 0.201771
\(207\) 0 0
\(208\) 490175. 0.785584
\(209\) −131292. −0.207908
\(210\) 0 0
\(211\) −979789. −1.51505 −0.757524 0.652807i \(-0.773591\pi\)
−0.757524 + 0.652807i \(0.773591\pi\)
\(212\) −659493. −1.00779
\(213\) 0 0
\(214\) 391610. 0.584546
\(215\) −87760.0 −0.129479
\(216\) 0 0
\(217\) −486153. −0.700848
\(218\) 522336. 0.744404
\(219\) 0 0
\(220\) −99480.7 −0.138574
\(221\) −266140. −0.366547
\(222\) 0 0
\(223\) 668754. 0.900542 0.450271 0.892892i \(-0.351327\pi\)
0.450271 + 0.892892i \(0.351327\pi\)
\(224\) 389773. 0.519029
\(225\) 0 0
\(226\) −83734.2 −0.109052
\(227\) 1.40720e6 1.81256 0.906280 0.422677i \(-0.138910\pi\)
0.906280 + 0.422677i \(0.138910\pi\)
\(228\) 0 0
\(229\) 706746. 0.890583 0.445292 0.895386i \(-0.353100\pi\)
0.445292 + 0.895386i \(0.353100\pi\)
\(230\) −3885.21 −0.00484278
\(231\) 0 0
\(232\) −373050. −0.455037
\(233\) −810873. −0.978505 −0.489252 0.872142i \(-0.662730\pi\)
−0.489252 + 0.872142i \(0.662730\pi\)
\(234\) 0 0
\(235\) −80326.7 −0.0948834
\(236\) 631678. 0.738271
\(237\) 0 0
\(238\) −42398.2 −0.0485182
\(239\) −81145.7 −0.0918905 −0.0459453 0.998944i \(-0.514630\pi\)
−0.0459453 + 0.998944i \(0.514630\pi\)
\(240\) 0 0
\(241\) 538808. 0.597574 0.298787 0.954320i \(-0.403418\pi\)
0.298787 + 0.954320i \(0.403418\pi\)
\(242\) −386091. −0.423791
\(243\) 0 0
\(244\) −211377. −0.227292
\(245\) 581718. 0.619152
\(246\) 0 0
\(247\) 1.70767e6 1.78099
\(248\) 1.04050e6 1.07427
\(249\) 0 0
\(250\) 474339. 0.479998
\(251\) 228910. 0.229340 0.114670 0.993404i \(-0.463419\pi\)
0.114670 + 0.993404i \(0.463419\pi\)
\(252\) 0 0
\(253\) −2665.02 −0.00261758
\(254\) 181089. 0.176119
\(255\) 0 0
\(256\) −452733. −0.431759
\(257\) −1.42445e6 −1.34528 −0.672642 0.739968i \(-0.734840\pi\)
−0.672642 + 0.739968i \(0.734840\pi\)
\(258\) 0 0
\(259\) −416526. −0.385827
\(260\) 1.29391e6 1.18705
\(261\) 0 0
\(262\) 182618. 0.164358
\(263\) −60691.4 −0.0541050 −0.0270525 0.999634i \(-0.508612\pi\)
−0.0270525 + 0.999634i \(0.508612\pi\)
\(264\) 0 0
\(265\) −1.21565e6 −1.06339
\(266\) 272045. 0.235742
\(267\) 0 0
\(268\) 316772. 0.269408
\(269\) 1.85520e6 1.56319 0.781594 0.623788i \(-0.214407\pi\)
0.781594 + 0.623788i \(0.214407\pi\)
\(270\) 0 0
\(271\) −1.57226e6 −1.30047 −0.650236 0.759732i \(-0.725330\pi\)
−0.650236 + 0.759732i \(0.725330\pi\)
\(272\) −116385. −0.0953834
\(273\) 0 0
\(274\) −727276. −0.585225
\(275\) 70997.3 0.0566122
\(276\) 0 0
\(277\) 321513. 0.251767 0.125883 0.992045i \(-0.459823\pi\)
0.125883 + 0.992045i \(0.459823\pi\)
\(278\) −937934. −0.727881
\(279\) 0 0
\(280\) 462301. 0.352395
\(281\) 501565. 0.378932 0.189466 0.981887i \(-0.439324\pi\)
0.189466 + 0.981887i \(0.439324\pi\)
\(282\) 0 0
\(283\) 347650. 0.258034 0.129017 0.991642i \(-0.458818\pi\)
0.129017 + 0.991642i \(0.458818\pi\)
\(284\) 481818. 0.354476
\(285\) 0 0
\(286\) −215463. −0.155760
\(287\) −1.04661e6 −0.750030
\(288\) 0 0
\(289\) −1.35667e6 −0.955495
\(290\) −306606. −0.214085
\(291\) 0 0
\(292\) 312245. 0.214308
\(293\) 1.60874e6 1.09475 0.547376 0.836886i \(-0.315627\pi\)
0.547376 + 0.836886i \(0.315627\pi\)
\(294\) 0 0
\(295\) 1.16438e6 0.779002
\(296\) 891478. 0.591400
\(297\) 0 0
\(298\) −1.07792e6 −0.703145
\(299\) 34663.0 0.0224227
\(300\) 0 0
\(301\) 124734. 0.0793540
\(302\) 43134.2 0.0272148
\(303\) 0 0
\(304\) 746772. 0.463451
\(305\) −389633. −0.239832
\(306\) 0 0
\(307\) 2.50047e6 1.51417 0.757085 0.653316i \(-0.226623\pi\)
0.757085 + 0.653316i \(0.226623\pi\)
\(308\) 141393. 0.0849279
\(309\) 0 0
\(310\) 855178. 0.505420
\(311\) −2.80353e6 −1.64363 −0.821816 0.569753i \(-0.807039\pi\)
−0.821816 + 0.569753i \(0.807039\pi\)
\(312\) 0 0
\(313\) −102581. −0.0591841 −0.0295920 0.999562i \(-0.509421\pi\)
−0.0295920 + 0.999562i \(0.509421\pi\)
\(314\) −348767. −0.199623
\(315\) 0 0
\(316\) 1.34855e6 0.759714
\(317\) 263420. 0.147232 0.0736158 0.997287i \(-0.476546\pi\)
0.0736158 + 0.997287i \(0.476546\pi\)
\(318\) 0 0
\(319\) −210313. −0.115715
\(320\) 17559.9 0.00958623
\(321\) 0 0
\(322\) 5522.09 0.00296800
\(323\) −405459. −0.216243
\(324\) 0 0
\(325\) −923437. −0.484952
\(326\) −1.39059e6 −0.724697
\(327\) 0 0
\(328\) 2.24002e6 1.14965
\(329\) 114169. 0.0581512
\(330\) 0 0
\(331\) −2.31706e6 −1.16243 −0.581216 0.813749i \(-0.697423\pi\)
−0.581216 + 0.813749i \(0.697423\pi\)
\(332\) 745072. 0.370982
\(333\) 0 0
\(334\) 598058. 0.293344
\(335\) 583909. 0.284271
\(336\) 0 0
\(337\) 1.40113e6 0.672055 0.336028 0.941852i \(-0.390916\pi\)
0.336028 + 0.941852i \(0.390916\pi\)
\(338\) 1.87414e6 0.892301
\(339\) 0 0
\(340\) −307219. −0.144129
\(341\) 586600. 0.273185
\(342\) 0 0
\(343\) −1.96061e6 −0.899819
\(344\) −266965. −0.121635
\(345\) 0 0
\(346\) −198924. −0.0893298
\(347\) 3.64231e6 1.62388 0.811940 0.583741i \(-0.198412\pi\)
0.811940 + 0.583741i \(0.198412\pi\)
\(348\) 0 0
\(349\) 2.59872e6 1.14208 0.571039 0.820923i \(-0.306540\pi\)
0.571039 + 0.820923i \(0.306540\pi\)
\(350\) −147111. −0.0641910
\(351\) 0 0
\(352\) −470306. −0.202313
\(353\) 4.35884e6 1.86180 0.930902 0.365269i \(-0.119023\pi\)
0.930902 + 0.365269i \(0.119023\pi\)
\(354\) 0 0
\(355\) 888139. 0.374033
\(356\) 3.02075e6 1.26325
\(357\) 0 0
\(358\) 239615. 0.0988112
\(359\) −3.30615e6 −1.35390 −0.676949 0.736029i \(-0.736698\pi\)
−0.676949 + 0.736029i \(0.736698\pi\)
\(360\) 0 0
\(361\) 125500. 0.0506845
\(362\) −548573. −0.220020
\(363\) 0 0
\(364\) −1.83905e6 −0.727511
\(365\) 575564. 0.226132
\(366\) 0 0
\(367\) −548156. −0.212441 −0.106221 0.994343i \(-0.533875\pi\)
−0.106221 + 0.994343i \(0.533875\pi\)
\(368\) 15158.3 0.00583488
\(369\) 0 0
\(370\) 732698. 0.278241
\(371\) 1.72781e6 0.651722
\(372\) 0 0
\(373\) −549721. −0.204583 −0.102292 0.994754i \(-0.532617\pi\)
−0.102292 + 0.994754i \(0.532617\pi\)
\(374\) 51158.3 0.0189120
\(375\) 0 0
\(376\) −244353. −0.0891348
\(377\) 2.73547e6 0.991240
\(378\) 0 0
\(379\) −2.24483e6 −0.802760 −0.401380 0.915912i \(-0.631469\pi\)
−0.401380 + 0.915912i \(0.631469\pi\)
\(380\) 1.97125e6 0.700298
\(381\) 0 0
\(382\) −821883. −0.288172
\(383\) −5.10313e6 −1.77762 −0.888812 0.458272i \(-0.848469\pi\)
−0.888812 + 0.458272i \(0.848469\pi\)
\(384\) 0 0
\(385\) 260631. 0.0896135
\(386\) 1.43946e6 0.491736
\(387\) 0 0
\(388\) 3.78879e6 1.27768
\(389\) −4.70768e6 −1.57737 −0.788683 0.614800i \(-0.789237\pi\)
−0.788683 + 0.614800i \(0.789237\pi\)
\(390\) 0 0
\(391\) −8230.20 −0.00272250
\(392\) 1.76958e6 0.581640
\(393\) 0 0
\(394\) −2.40739e6 −0.781279
\(395\) 2.48580e6 0.801628
\(396\) 0 0
\(397\) 3.81755e6 1.21565 0.607825 0.794071i \(-0.292042\pi\)
0.607825 + 0.794071i \(0.292042\pi\)
\(398\) −1.90051e6 −0.601397
\(399\) 0 0
\(400\) −403824. −0.126195
\(401\) −1.31578e6 −0.408622 −0.204311 0.978906i \(-0.565495\pi\)
−0.204311 + 0.978906i \(0.565495\pi\)
\(402\) 0 0
\(403\) −7.62970e6 −2.34016
\(404\) −3.05283e6 −0.930571
\(405\) 0 0
\(406\) 435782. 0.131206
\(407\) 502587. 0.150392
\(408\) 0 0
\(409\) 3.92071e6 1.15893 0.579464 0.814998i \(-0.303262\pi\)
0.579464 + 0.814998i \(0.303262\pi\)
\(410\) 1.84105e6 0.540888
\(411\) 0 0
\(412\) −1.26566e6 −0.367344
\(413\) −1.65494e6 −0.477427
\(414\) 0 0
\(415\) 1.37340e6 0.391450
\(416\) 6.11711e6 1.73306
\(417\) 0 0
\(418\) −328253. −0.0918901
\(419\) −2.92842e6 −0.814890 −0.407445 0.913230i \(-0.633580\pi\)
−0.407445 + 0.913230i \(0.633580\pi\)
\(420\) 0 0
\(421\) 4.43337e6 1.21907 0.609535 0.792759i \(-0.291356\pi\)
0.609535 + 0.792759i \(0.291356\pi\)
\(422\) −2.44965e6 −0.669612
\(423\) 0 0
\(424\) −3.69799e6 −0.998966
\(425\) 219256. 0.0588816
\(426\) 0 0
\(427\) 553789. 0.146986
\(428\) −4.03314e6 −1.06423
\(429\) 0 0
\(430\) −219416. −0.0572265
\(431\) −4.27188e6 −1.10771 −0.553854 0.832614i \(-0.686844\pi\)
−0.553854 + 0.832614i \(0.686844\pi\)
\(432\) 0 0
\(433\) −1.65833e6 −0.425061 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(434\) −1.21547e6 −0.309757
\(435\) 0 0
\(436\) −5.37947e6 −1.35526
\(437\) 52808.5 0.0132282
\(438\) 0 0
\(439\) −1.44252e6 −0.357240 −0.178620 0.983918i \(-0.557163\pi\)
−0.178620 + 0.983918i \(0.557163\pi\)
\(440\) −557820. −0.137361
\(441\) 0 0
\(442\) −665398. −0.162004
\(443\) −5.80251e6 −1.40477 −0.702387 0.711795i \(-0.747882\pi\)
−0.702387 + 0.711795i \(0.747882\pi\)
\(444\) 0 0
\(445\) 5.56818e6 1.33295
\(446\) 1.67201e6 0.398016
\(447\) 0 0
\(448\) −24958.1 −0.00587511
\(449\) 1.86453e6 0.436470 0.218235 0.975896i \(-0.429970\pi\)
0.218235 + 0.975896i \(0.429970\pi\)
\(450\) 0 0
\(451\) 1.26285e6 0.292355
\(452\) 862368. 0.198539
\(453\) 0 0
\(454\) 3.51827e6 0.801105
\(455\) −3.38993e6 −0.767648
\(456\) 0 0
\(457\) 3.61109e6 0.808812 0.404406 0.914580i \(-0.367478\pi\)
0.404406 + 0.914580i \(0.367478\pi\)
\(458\) 1.76699e6 0.393615
\(459\) 0 0
\(460\) 40013.3 0.00881679
\(461\) −5.82273e6 −1.27607 −0.638034 0.770008i \(-0.720252\pi\)
−0.638034 + 0.770008i \(0.720252\pi\)
\(462\) 0 0
\(463\) 759569. 0.164670 0.0823351 0.996605i \(-0.473762\pi\)
0.0823351 + 0.996605i \(0.473762\pi\)
\(464\) 1.19624e6 0.257942
\(465\) 0 0
\(466\) −2.02733e6 −0.432474
\(467\) −3.30533e6 −0.701330 −0.350665 0.936501i \(-0.614044\pi\)
−0.350665 + 0.936501i \(0.614044\pi\)
\(468\) 0 0
\(469\) −829915. −0.174221
\(470\) −200831. −0.0419360
\(471\) 0 0
\(472\) 3.54202e6 0.731805
\(473\) −150506. −0.0309315
\(474\) 0 0
\(475\) −1.40684e6 −0.286095
\(476\) 436654. 0.0883324
\(477\) 0 0
\(478\) −202879. −0.0406132
\(479\) 7.63918e6 1.52128 0.760638 0.649176i \(-0.224886\pi\)
0.760638 + 0.649176i \(0.224886\pi\)
\(480\) 0 0
\(481\) −6.53697e6 −1.28829
\(482\) 1.34712e6 0.264112
\(483\) 0 0
\(484\) 3.97631e6 0.771555
\(485\) 6.98391e6 1.34817
\(486\) 0 0
\(487\) −4.03922e6 −0.771747 −0.385873 0.922552i \(-0.626100\pi\)
−0.385873 + 0.922552i \(0.626100\pi\)
\(488\) −1.18526e6 −0.225301
\(489\) 0 0
\(490\) 1.45440e6 0.273649
\(491\) −7.62584e6 −1.42753 −0.713763 0.700387i \(-0.753011\pi\)
−0.713763 + 0.700387i \(0.753011\pi\)
\(492\) 0 0
\(493\) −649497. −0.120354
\(494\) 4.26948e6 0.787149
\(495\) 0 0
\(496\) −3.33651e6 −0.608960
\(497\) −1.26232e6 −0.229234
\(498\) 0 0
\(499\) 9.55168e6 1.71723 0.858615 0.512621i \(-0.171325\pi\)
0.858615 + 0.512621i \(0.171325\pi\)
\(500\) −4.88516e6 −0.873885
\(501\) 0 0
\(502\) 572316. 0.101362
\(503\) 4.69119e6 0.826730 0.413365 0.910566i \(-0.364353\pi\)
0.413365 + 0.910566i \(0.364353\pi\)
\(504\) 0 0
\(505\) −5.62731e6 −0.981911
\(506\) −6663.04 −0.00115690
\(507\) 0 0
\(508\) −1.86501e6 −0.320643
\(509\) −2.37623e6 −0.406531 −0.203265 0.979124i \(-0.565155\pi\)
−0.203265 + 0.979124i \(0.565155\pi\)
\(510\) 0 0
\(511\) −818054. −0.138589
\(512\) 4.81417e6 0.811608
\(513\) 0 0
\(514\) −3.56138e6 −0.594580
\(515\) −2.33300e6 −0.387611
\(516\) 0 0
\(517\) −137758. −0.0226668
\(518\) −1.04139e6 −0.170526
\(519\) 0 0
\(520\) 7.25536e6 1.17666
\(521\) −6.31208e6 −1.01878 −0.509388 0.860537i \(-0.670128\pi\)
−0.509388 + 0.860537i \(0.670128\pi\)
\(522\) 0 0
\(523\) −1.03668e7 −1.65725 −0.828627 0.559801i \(-0.810878\pi\)
−0.828627 + 0.559801i \(0.810878\pi\)
\(524\) −1.88077e6 −0.299231
\(525\) 0 0
\(526\) −151739. −0.0239130
\(527\) 1.81156e6 0.284136
\(528\) 0 0
\(529\) −6.43527e6 −0.999833
\(530\) −3.03935e6 −0.469992
\(531\) 0 0
\(532\) −2.80176e6 −0.429192
\(533\) −1.64255e7 −2.50438
\(534\) 0 0
\(535\) −7.43432e6 −1.12294
\(536\) 1.77624e6 0.267049
\(537\) 0 0
\(538\) 4.63835e6 0.690888
\(539\) 997631. 0.147910
\(540\) 0 0
\(541\) −5.71736e6 −0.839851 −0.419926 0.907559i \(-0.637944\pi\)
−0.419926 + 0.907559i \(0.637944\pi\)
\(542\) −3.93093e6 −0.574775
\(543\) 0 0
\(544\) −1.45242e6 −0.210423
\(545\) −9.91603e6 −1.43003
\(546\) 0 0
\(547\) 4.37335e6 0.624950 0.312475 0.949926i \(-0.398842\pi\)
0.312475 + 0.949926i \(0.398842\pi\)
\(548\) 7.49013e6 1.06546
\(549\) 0 0
\(550\) 177506. 0.0250211
\(551\) 4.16744e6 0.584778
\(552\) 0 0
\(553\) −3.53309e6 −0.491294
\(554\) 803841. 0.111274
\(555\) 0 0
\(556\) 9.65967e6 1.32518
\(557\) 5.52441e6 0.754481 0.377240 0.926115i \(-0.376873\pi\)
0.377240 + 0.926115i \(0.376873\pi\)
\(558\) 0 0
\(559\) 1.95758e6 0.264966
\(560\) −1.48244e6 −0.199759
\(561\) 0 0
\(562\) 1.25400e6 0.167478
\(563\) −1.38558e7 −1.84230 −0.921149 0.389211i \(-0.872748\pi\)
−0.921149 + 0.389211i \(0.872748\pi\)
\(564\) 0 0
\(565\) 1.58961e6 0.209493
\(566\) 869189. 0.114044
\(567\) 0 0
\(568\) 2.70171e6 0.351372
\(569\) −8.96028e6 −1.16022 −0.580111 0.814538i \(-0.696991\pi\)
−0.580111 + 0.814538i \(0.696991\pi\)
\(570\) 0 0
\(571\) 1.86281e6 0.239099 0.119550 0.992828i \(-0.461855\pi\)
0.119550 + 0.992828i \(0.461855\pi\)
\(572\) 2.21902e6 0.283578
\(573\) 0 0
\(574\) −2.61671e6 −0.331494
\(575\) −28556.7 −0.00360195
\(576\) 0 0
\(577\) −8.48683e6 −1.06122 −0.530610 0.847616i \(-0.678037\pi\)
−0.530610 + 0.847616i \(0.678037\pi\)
\(578\) −3.39191e6 −0.422304
\(579\) 0 0
\(580\) 3.15770e6 0.389764
\(581\) −1.95202e6 −0.239908
\(582\) 0 0
\(583\) −2.08481e6 −0.254036
\(584\) 1.75086e6 0.212431
\(585\) 0 0
\(586\) 4.02214e6 0.483852
\(587\) 9.06080e6 1.08535 0.542677 0.839942i \(-0.317411\pi\)
0.542677 + 0.839942i \(0.317411\pi\)
\(588\) 0 0
\(589\) −1.16237e7 −1.38056
\(590\) 2.91115e6 0.344299
\(591\) 0 0
\(592\) −2.85866e6 −0.335241
\(593\) −1.56414e7 −1.82658 −0.913290 0.407310i \(-0.866467\pi\)
−0.913290 + 0.407310i \(0.866467\pi\)
\(594\) 0 0
\(595\) 804887. 0.0932058
\(596\) 1.11013e7 1.28015
\(597\) 0 0
\(598\) 86663.8 0.00991025
\(599\) 1.40306e6 0.159775 0.0798877 0.996804i \(-0.474544\pi\)
0.0798877 + 0.996804i \(0.474544\pi\)
\(600\) 0 0
\(601\) −1.02325e6 −0.115557 −0.0577786 0.998329i \(-0.518402\pi\)
−0.0577786 + 0.998329i \(0.518402\pi\)
\(602\) 311858. 0.0350724
\(603\) 0 0
\(604\) −444235. −0.0495473
\(605\) 7.32956e6 0.814122
\(606\) 0 0
\(607\) 6.79951e6 0.749042 0.374521 0.927219i \(-0.377807\pi\)
0.374521 + 0.927219i \(0.377807\pi\)
\(608\) 9.31931e6 1.02241
\(609\) 0 0
\(610\) −974154. −0.105999
\(611\) 1.79177e6 0.194169
\(612\) 0 0
\(613\) 3.70194e6 0.397904 0.198952 0.980009i \(-0.436246\pi\)
0.198952 + 0.980009i \(0.436246\pi\)
\(614\) 6.25162e6 0.669224
\(615\) 0 0
\(616\) 792834. 0.0841842
\(617\) −1.14193e7 −1.20761 −0.603806 0.797131i \(-0.706350\pi\)
−0.603806 + 0.797131i \(0.706350\pi\)
\(618\) 0 0
\(619\) −8.12725e6 −0.852545 −0.426272 0.904595i \(-0.640173\pi\)
−0.426272 + 0.904595i \(0.640173\pi\)
\(620\) −8.80737e6 −0.920168
\(621\) 0 0
\(622\) −7.00934e6 −0.726443
\(623\) −7.91410e6 −0.816924
\(624\) 0 0
\(625\) −6.27919e6 −0.642989
\(626\) −256470. −0.0261578
\(627\) 0 0
\(628\) 3.59191e6 0.363435
\(629\) 1.55210e6 0.156421
\(630\) 0 0
\(631\) 8.36884e6 0.836743 0.418371 0.908276i \(-0.362601\pi\)
0.418371 + 0.908276i \(0.362601\pi\)
\(632\) 7.56176e6 0.753061
\(633\) 0 0
\(634\) 658599. 0.0650725
\(635\) −3.43779e6 −0.338334
\(636\) 0 0
\(637\) −1.29758e7 −1.26703
\(638\) −525822. −0.0511431
\(639\) 0 0
\(640\) 8.81943e6 0.851120
\(641\) 1.04858e7 1.00799 0.503996 0.863706i \(-0.331863\pi\)
0.503996 + 0.863706i \(0.331863\pi\)
\(642\) 0 0
\(643\) 1.22946e7 1.17270 0.586349 0.810058i \(-0.300565\pi\)
0.586349 + 0.810058i \(0.300565\pi\)
\(644\) −56871.3 −0.00540354
\(645\) 0 0
\(646\) −1.01372e6 −0.0955736
\(647\) 1.39774e7 1.31270 0.656349 0.754457i \(-0.272100\pi\)
0.656349 + 0.754457i \(0.272100\pi\)
\(648\) 0 0
\(649\) 1.99688e6 0.186097
\(650\) −2.30876e6 −0.214336
\(651\) 0 0
\(652\) 1.43216e7 1.31939
\(653\) 5.00675e6 0.459487 0.229743 0.973251i \(-0.426211\pi\)
0.229743 + 0.973251i \(0.426211\pi\)
\(654\) 0 0
\(655\) −3.46683e6 −0.315740
\(656\) −7.18296e6 −0.651694
\(657\) 0 0
\(658\) 285443. 0.0257013
\(659\) −5.50845e6 −0.494101 −0.247050 0.969003i \(-0.579461\pi\)
−0.247050 + 0.969003i \(0.579461\pi\)
\(660\) 0 0
\(661\) −6.69305e6 −0.595827 −0.297914 0.954593i \(-0.596291\pi\)
−0.297914 + 0.954593i \(0.596291\pi\)
\(662\) −5.79308e6 −0.513765
\(663\) 0 0
\(664\) 4.17786e6 0.367733
\(665\) −5.16450e6 −0.452870
\(666\) 0 0
\(667\) 84592.7 0.00736238
\(668\) −6.15933e6 −0.534063
\(669\) 0 0
\(670\) 1.45988e6 0.125641
\(671\) −668211. −0.0572938
\(672\) 0 0
\(673\) −7.54961e6 −0.642521 −0.321260 0.946991i \(-0.604106\pi\)
−0.321260 + 0.946991i \(0.604106\pi\)
\(674\) 3.50309e6 0.297031
\(675\) 0 0
\(676\) −1.93016e7 −1.62453
\(677\) 1.51658e6 0.127173 0.0635863 0.997976i \(-0.479746\pi\)
0.0635863 + 0.997976i \(0.479746\pi\)
\(678\) 0 0
\(679\) −9.92629e6 −0.826252
\(680\) −1.72268e6 −0.142867
\(681\) 0 0
\(682\) 1.46661e6 0.120741
\(683\) 2.30637e7 1.89181 0.945904 0.324448i \(-0.105178\pi\)
0.945904 + 0.324448i \(0.105178\pi\)
\(684\) 0 0
\(685\) 1.38066e7 1.12424
\(686\) −4.90187e6 −0.397696
\(687\) 0 0
\(688\) 856062. 0.0689499
\(689\) 2.71164e7 2.17612
\(690\) 0 0
\(691\) −1.55625e7 −1.23990 −0.619948 0.784643i \(-0.712846\pi\)
−0.619948 + 0.784643i \(0.712846\pi\)
\(692\) 2.04869e6 0.162634
\(693\) 0 0
\(694\) 9.10645e6 0.717713
\(695\) 1.78058e7 1.39829
\(696\) 0 0
\(697\) 3.89998e6 0.304075
\(698\) 6.49727e6 0.504769
\(699\) 0 0
\(700\) 1.51508e6 0.116866
\(701\) −1.48227e6 −0.113929 −0.0569643 0.998376i \(-0.518142\pi\)
−0.0569643 + 0.998376i \(0.518142\pi\)
\(702\) 0 0
\(703\) −9.95896e6 −0.760021
\(704\) 30114.8 0.00229007
\(705\) 0 0
\(706\) 1.08979e7 0.822869
\(707\) 7.99814e6 0.601784
\(708\) 0 0
\(709\) 1.23095e7 0.919652 0.459826 0.888009i \(-0.347912\pi\)
0.459826 + 0.888009i \(0.347912\pi\)
\(710\) 2.22051e6 0.165313
\(711\) 0 0
\(712\) 1.69383e7 1.25219
\(713\) −235944. −0.0173814
\(714\) 0 0
\(715\) 4.09034e6 0.299223
\(716\) −2.46776e6 −0.179896
\(717\) 0 0
\(718\) −8.26598e6 −0.598388
\(719\) 2.07641e7 1.49793 0.748963 0.662612i \(-0.230552\pi\)
0.748963 + 0.662612i \(0.230552\pi\)
\(720\) 0 0
\(721\) 3.31591e6 0.237555
\(722\) 313773. 0.0224012
\(723\) 0 0
\(724\) 5.64968e6 0.400569
\(725\) −2.25359e6 −0.159232
\(726\) 0 0
\(727\) −7.15312e6 −0.501949 −0.250974 0.967994i \(-0.580751\pi\)
−0.250974 + 0.967994i \(0.580751\pi\)
\(728\) −1.03121e7 −0.721140
\(729\) 0 0
\(730\) 1.43901e6 0.0999443
\(731\) −464798. −0.0321715
\(732\) 0 0
\(733\) 1.65135e7 1.13521 0.567607 0.823299i \(-0.307869\pi\)
0.567607 + 0.823299i \(0.307869\pi\)
\(734\) −1.37049e6 −0.0938936
\(735\) 0 0
\(736\) 189168. 0.0128722
\(737\) 1.00139e6 0.0679100
\(738\) 0 0
\(739\) −7.25575e6 −0.488733 −0.244366 0.969683i \(-0.578580\pi\)
−0.244366 + 0.969683i \(0.578580\pi\)
\(740\) −7.54597e6 −0.506566
\(741\) 0 0
\(742\) 4.31985e6 0.288044
\(743\) 1.59618e7 1.06074 0.530372 0.847765i \(-0.322052\pi\)
0.530372 + 0.847765i \(0.322052\pi\)
\(744\) 0 0
\(745\) 2.04632e7 1.35077
\(746\) −1.37440e6 −0.0904205
\(747\) 0 0
\(748\) −526874. −0.0344312
\(749\) 1.05665e7 0.688217
\(750\) 0 0
\(751\) 2.56869e7 1.66192 0.830962 0.556329i \(-0.187791\pi\)
0.830962 + 0.556329i \(0.187791\pi\)
\(752\) 783553. 0.0505270
\(753\) 0 0
\(754\) 6.83918e6 0.438102
\(755\) −818861. −0.0522809
\(756\) 0 0
\(757\) 7.97201e6 0.505624 0.252812 0.967515i \(-0.418645\pi\)
0.252812 + 0.967515i \(0.418645\pi\)
\(758\) −5.61249e6 −0.354799
\(759\) 0 0
\(760\) 1.10534e7 0.694165
\(761\) −2.21020e7 −1.38347 −0.691736 0.722150i \(-0.743154\pi\)
−0.691736 + 0.722150i \(0.743154\pi\)
\(762\) 0 0
\(763\) 1.40937e7 0.876425
\(764\) 8.46448e6 0.524646
\(765\) 0 0
\(766\) −1.27588e7 −0.785664
\(767\) −2.59727e7 −1.59415
\(768\) 0 0
\(769\) −2.64726e7 −1.61428 −0.807142 0.590357i \(-0.798987\pi\)
−0.807142 + 0.590357i \(0.798987\pi\)
\(770\) 651624. 0.0396068
\(771\) 0 0
\(772\) −1.48249e7 −0.895256
\(773\) −2.48033e7 −1.49301 −0.746503 0.665382i \(-0.768268\pi\)
−0.746503 + 0.665382i \(0.768268\pi\)
\(774\) 0 0
\(775\) 6.28564e6 0.375920
\(776\) 2.12449e7 1.26649
\(777\) 0 0
\(778\) −1.17701e7 −0.697155
\(779\) −2.50239e7 −1.47745
\(780\) 0 0
\(781\) 1.52314e6 0.0893534
\(782\) −20577.0 −0.00120328
\(783\) 0 0
\(784\) −5.67441e6 −0.329709
\(785\) 6.62100e6 0.383486
\(786\) 0 0
\(787\) 8.09631e6 0.465962 0.232981 0.972481i \(-0.425152\pi\)
0.232981 + 0.972481i \(0.425152\pi\)
\(788\) 2.47934e7 1.42240
\(789\) 0 0
\(790\) 6.21495e6 0.354299
\(791\) −2.25933e6 −0.128392
\(792\) 0 0
\(793\) 8.69118e6 0.490790
\(794\) 9.54457e6 0.537286
\(795\) 0 0
\(796\) 1.95731e7 1.09491
\(797\) −1.30377e7 −0.727034 −0.363517 0.931587i \(-0.618424\pi\)
−0.363517 + 0.931587i \(0.618424\pi\)
\(798\) 0 0
\(799\) −425429. −0.0235755
\(800\) −5.03951e6 −0.278396
\(801\) 0 0
\(802\) −3.28968e6 −0.180600
\(803\) 987077. 0.0540210
\(804\) 0 0
\(805\) −104831. −0.00570166
\(806\) −1.90756e7 −1.03429
\(807\) 0 0
\(808\) −1.71182e7 −0.922422
\(809\) 3.29258e6 0.176874 0.0884371 0.996082i \(-0.471813\pi\)
0.0884371 + 0.996082i \(0.471813\pi\)
\(810\) 0 0
\(811\) 8.58147e6 0.458152 0.229076 0.973409i \(-0.426430\pi\)
0.229076 + 0.973409i \(0.426430\pi\)
\(812\) −4.48807e6 −0.238874
\(813\) 0 0
\(814\) 1.25656e6 0.0664694
\(815\) 2.63991e7 1.39218
\(816\) 0 0
\(817\) 2.98234e6 0.156315
\(818\) 9.80249e6 0.512216
\(819\) 0 0
\(820\) −1.89608e7 −0.984741
\(821\) 7.34065e6 0.380081 0.190041 0.981776i \(-0.439138\pi\)
0.190041 + 0.981776i \(0.439138\pi\)
\(822\) 0 0
\(823\) 3.32707e7 1.71223 0.856116 0.516783i \(-0.172871\pi\)
0.856116 + 0.516783i \(0.172871\pi\)
\(824\) −7.09694e6 −0.364127
\(825\) 0 0
\(826\) −4.13765e6 −0.211010
\(827\) 3.77399e7 1.91883 0.959417 0.281992i \(-0.0909951\pi\)
0.959417 + 0.281992i \(0.0909951\pi\)
\(828\) 0 0
\(829\) −2.81993e7 −1.42512 −0.712560 0.701611i \(-0.752465\pi\)
−0.712560 + 0.701611i \(0.752465\pi\)
\(830\) 3.43374e6 0.173011
\(831\) 0 0
\(832\) −391693. −0.0196172
\(833\) 3.08092e6 0.153839
\(834\) 0 0
\(835\) −1.13535e7 −0.563528
\(836\) 3.38064e6 0.167295
\(837\) 0 0
\(838\) −7.32159e6 −0.360160
\(839\) 9.00386e6 0.441594 0.220797 0.975320i \(-0.429134\pi\)
0.220797 + 0.975320i \(0.429134\pi\)
\(840\) 0 0
\(841\) −1.38354e7 −0.674531
\(842\) 1.10842e7 0.538797
\(843\) 0 0
\(844\) 2.52287e7 1.21910
\(845\) −3.55788e7 −1.71415
\(846\) 0 0
\(847\) −1.04176e7 −0.498951
\(848\) 1.18581e7 0.566275
\(849\) 0 0
\(850\) 548180. 0.0260241
\(851\) −202152. −0.00956870
\(852\) 0 0
\(853\) 2.76660e7 1.30189 0.650944 0.759126i \(-0.274373\pi\)
0.650944 + 0.759126i \(0.274373\pi\)
\(854\) 1.38457e6 0.0649638
\(855\) 0 0
\(856\) −2.26151e7 −1.05491
\(857\) 3.77780e7 1.75706 0.878531 0.477686i \(-0.158524\pi\)
0.878531 + 0.477686i \(0.158524\pi\)
\(858\) 0 0
\(859\) 2.02066e7 0.934352 0.467176 0.884164i \(-0.345271\pi\)
0.467176 + 0.884164i \(0.345271\pi\)
\(860\) 2.25974e6 0.104187
\(861\) 0 0
\(862\) −1.06805e7 −0.489578
\(863\) 992399. 0.0453586 0.0226793 0.999743i \(-0.492780\pi\)
0.0226793 + 0.999743i \(0.492780\pi\)
\(864\) 0 0
\(865\) 3.77637e6 0.171607
\(866\) −4.14613e6 −0.187866
\(867\) 0 0
\(868\) 1.25180e7 0.563944
\(869\) 4.26308e6 0.191502
\(870\) 0 0
\(871\) −1.30247e7 −0.581732
\(872\) −3.01644e7 −1.34339
\(873\) 0 0
\(874\) 132031. 0.00584651
\(875\) 1.27987e7 0.565126
\(876\) 0 0
\(877\) 3.48750e7 1.53114 0.765572 0.643351i \(-0.222456\pi\)
0.765572 + 0.643351i \(0.222456\pi\)
\(878\) −3.60656e6 −0.157891
\(879\) 0 0
\(880\) 1.78873e6 0.0778643
\(881\) −2.68921e7 −1.16731 −0.583654 0.812002i \(-0.698378\pi\)
−0.583654 + 0.812002i \(0.698378\pi\)
\(882\) 0 0
\(883\) −2.40336e6 −0.103733 −0.0518666 0.998654i \(-0.516517\pi\)
−0.0518666 + 0.998654i \(0.516517\pi\)
\(884\) 6.85286e6 0.294945
\(885\) 0 0
\(886\) −1.45073e7 −0.620874
\(887\) 2.56525e7 1.09477 0.547383 0.836882i \(-0.315624\pi\)
0.547383 + 0.836882i \(0.315624\pi\)
\(888\) 0 0
\(889\) 4.88616e6 0.207355
\(890\) 1.39215e7 0.589128
\(891\) 0 0
\(892\) −1.72198e7 −0.724629
\(893\) 2.72973e6 0.114549
\(894\) 0 0
\(895\) −4.54885e6 −0.189821
\(896\) −1.25351e7 −0.521626
\(897\) 0 0
\(898\) 4.66167e6 0.192908
\(899\) −1.86198e7 −0.768379
\(900\) 0 0
\(901\) −6.43837e6 −0.264219
\(902\) 3.15736e6 0.129214
\(903\) 0 0
\(904\) 4.83557e6 0.196801
\(905\) 1.04141e7 0.422669
\(906\) 0 0
\(907\) −2.68686e7 −1.08450 −0.542248 0.840219i \(-0.682426\pi\)
−0.542248 + 0.840219i \(0.682426\pi\)
\(908\) −3.62342e7 −1.45849
\(909\) 0 0
\(910\) −8.47544e6 −0.339280
\(911\) −1.69356e7 −0.676091 −0.338045 0.941130i \(-0.609766\pi\)
−0.338045 + 0.941130i \(0.609766\pi\)
\(912\) 0 0
\(913\) 2.35534e6 0.0935141
\(914\) 9.02838e6 0.357474
\(915\) 0 0
\(916\) −1.81981e7 −0.716616
\(917\) 4.92744e6 0.193508
\(918\) 0 0
\(919\) −2.20166e7 −0.859927 −0.429964 0.902846i \(-0.641474\pi\)
−0.429964 + 0.902846i \(0.641474\pi\)
\(920\) 224367. 0.00873957
\(921\) 0 0
\(922\) −1.45579e7 −0.563989
\(923\) −1.98109e7 −0.765420
\(924\) 0 0
\(925\) 5.38540e6 0.206949
\(926\) 1.89906e6 0.0727799
\(927\) 0 0
\(928\) 1.49284e7 0.569041
\(929\) 8.11263e6 0.308406 0.154203 0.988039i \(-0.450719\pi\)
0.154203 + 0.988039i \(0.450719\pi\)
\(930\) 0 0
\(931\) −1.97685e7 −0.747478
\(932\) 2.08792e7 0.787362
\(933\) 0 0
\(934\) −8.26392e6 −0.309970
\(935\) −971190. −0.0363308
\(936\) 0 0
\(937\) −1.80064e7 −0.670004 −0.335002 0.942217i \(-0.608737\pi\)
−0.335002 + 0.942217i \(0.608737\pi\)
\(938\) −2.07494e6 −0.0770013
\(939\) 0 0
\(940\) 2.06834e6 0.0763488
\(941\) −2.90643e7 −1.07000 −0.535002 0.844851i \(-0.679689\pi\)
−0.535002 + 0.844851i \(0.679689\pi\)
\(942\) 0 0
\(943\) −507947. −0.0186011
\(944\) −1.13580e7 −0.414832
\(945\) 0 0
\(946\) −376293. −0.0136709
\(947\) 1.97188e7 0.714504 0.357252 0.934008i \(-0.383714\pi\)
0.357252 + 0.934008i \(0.383714\pi\)
\(948\) 0 0
\(949\) −1.28386e7 −0.462755
\(950\) −3.51736e6 −0.126447
\(951\) 0 0
\(952\) 2.44846e6 0.0875588
\(953\) 5.34568e6 0.190665 0.0953325 0.995445i \(-0.469609\pi\)
0.0953325 + 0.995445i \(0.469609\pi\)
\(954\) 0 0
\(955\) 1.56026e7 0.553592
\(956\) 2.08943e6 0.0739405
\(957\) 0 0
\(958\) 1.90993e7 0.672364
\(959\) −1.96235e7 −0.689016
\(960\) 0 0
\(961\) 2.33046e7 0.814017
\(962\) −1.63436e7 −0.569391
\(963\) 0 0
\(964\) −1.38738e7 −0.480843
\(965\) −2.73268e7 −0.944648
\(966\) 0 0
\(967\) −605934. −0.0208381 −0.0104191 0.999946i \(-0.503317\pi\)
−0.0104191 + 0.999946i \(0.503317\pi\)
\(968\) 2.22964e7 0.764798
\(969\) 0 0
\(970\) 1.74610e7 0.595855
\(971\) −2.31358e7 −0.787475 −0.393737 0.919223i \(-0.628818\pi\)
−0.393737 + 0.919223i \(0.628818\pi\)
\(972\) 0 0
\(973\) −2.53075e7 −0.856972
\(974\) −1.00988e7 −0.341092
\(975\) 0 0
\(976\) 3.80071e6 0.127714
\(977\) −2.22967e6 −0.0747315 −0.0373658 0.999302i \(-0.511897\pi\)
−0.0373658 + 0.999302i \(0.511897\pi\)
\(978\) 0 0
\(979\) 9.54928e6 0.318430
\(980\) −1.49787e7 −0.498206
\(981\) 0 0
\(982\) −1.90660e7 −0.630929
\(983\) 2.21182e6 0.0730071 0.0365036 0.999334i \(-0.488378\pi\)
0.0365036 + 0.999334i \(0.488378\pi\)
\(984\) 0 0
\(985\) 4.57019e7 1.50087
\(986\) −1.62386e6 −0.0531932
\(987\) 0 0
\(988\) −4.39708e7 −1.43309
\(989\) 60536.9 0.00196802
\(990\) 0 0
\(991\) −1.66837e7 −0.539645 −0.269823 0.962910i \(-0.586965\pi\)
−0.269823 + 0.962910i \(0.586965\pi\)
\(992\) −4.16379e7 −1.34341
\(993\) 0 0
\(994\) −3.15603e6 −0.101315
\(995\) 3.60792e7 1.15531
\(996\) 0 0
\(997\) 1.01549e7 0.323547 0.161774 0.986828i \(-0.448279\pi\)
0.161774 + 0.986828i \(0.448279\pi\)
\(998\) 2.38810e7 0.758971
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 387.6.a.e.1.7 10
3.2 odd 2 43.6.a.b.1.4 10
12.11 even 2 688.6.a.h.1.4 10
15.14 odd 2 1075.6.a.b.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.b.1.4 10 3.2 odd 2
387.6.a.e.1.7 10 1.1 even 1 trivial
688.6.a.h.1.4 10 12.11 even 2
1075.6.a.b.1.7 10 15.14 odd 2