Properties

Label 546.8.a.n.1.1
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
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] = [546,8,Mod(1,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.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: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 309949x^{4} - 14548431x^{3} + 25221499020x^{2} + 1862570808000x - 308009568384000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(457.447\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -487.447 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -487.447 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3899.58 q^{10} -6609.14 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} +13161.1 q^{15} +4096.00 q^{16} +8891.52 q^{17} -5832.00 q^{18} +7882.69 q^{19} -31196.6 q^{20} -9261.00 q^{21} +52873.2 q^{22} -107716. q^{23} +13824.0 q^{24} +159480. q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} -127254. q^{29} -105289. q^{30} +86977.9 q^{31} -32768.0 q^{32} +178447. q^{33} -71132.2 q^{34} -167194. q^{35} +46656.0 q^{36} -233049. q^{37} -63061.6 q^{38} -59319.0 q^{39} +249573. q^{40} +293448. q^{41} +74088.0 q^{42} -259605. q^{43} -422985. q^{44} -355349. q^{45} +861730. q^{46} -244441. q^{47} -110592. q^{48} +117649. q^{49} -1.27584e6 q^{50} -240071. q^{51} +140608. q^{52} -1.77201e6 q^{53} +157464. q^{54} +3.22161e6 q^{55} -175616. q^{56} -212833. q^{57} +1.01803e6 q^{58} -1.29914e6 q^{59} +842309. q^{60} -2.80338e6 q^{61} -695823. q^{62} +250047. q^{63} +262144. q^{64} -1.07092e6 q^{65} -1.42758e6 q^{66} -3.33358e6 q^{67} +569057. q^{68} +2.90834e6 q^{69} +1.33756e6 q^{70} +1.67692e6 q^{71} -373248. q^{72} -932381. q^{73} +1.86439e6 q^{74} -4.30596e6 q^{75} +504492. q^{76} -2.26694e6 q^{77} +474552. q^{78} -3.16827e6 q^{79} -1.99658e6 q^{80} +531441. q^{81} -2.34758e6 q^{82} +4.51858e6 q^{83} -592704. q^{84} -4.33415e6 q^{85} +2.07684e6 q^{86} +3.43587e6 q^{87} +3.38388e6 q^{88} -1.31198e7 q^{89} +2.84279e6 q^{90} +753571. q^{91} -6.89384e6 q^{92} -2.34840e6 q^{93} +1.95553e6 q^{94} -3.84240e6 q^{95} +884736. q^{96} -1.12922e7 q^{97} -941192. q^{98} -4.81807e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 1448 q^{10} - 6130 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} + 4887 q^{15} + 24576 q^{16} - 34610 q^{17} - 34992 q^{18} - 4085 q^{19} - 11584 q^{20} - 55566 q^{21} + 49040 q^{22} + 1515 q^{23} + 82944 q^{24} + 156609 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} - 59395 q^{29} - 39096 q^{30} + 478241 q^{31} - 196608 q^{32} + 165510 q^{33} + 276880 q^{34} - 62083 q^{35} + 279936 q^{36} + 574310 q^{37} + 32680 q^{38} - 355914 q^{39} + 92672 q^{40} + 201552 q^{41} + 444528 q^{42} + 728605 q^{43} - 392320 q^{44} - 131949 q^{45} - 12120 q^{46} + 227615 q^{47} - 663552 q^{48} + 705894 q^{49} - 1252872 q^{50} + 934470 q^{51} + 843648 q^{52} + 26321 q^{53} + 944784 q^{54} + 2115010 q^{55} - 1053696 q^{56} + 110295 q^{57} + 475160 q^{58} + 478280 q^{59} + 312768 q^{60} - 501406 q^{61} - 3825928 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 397657 q^{65} - 1324080 q^{66} - 3156366 q^{67} - 2215040 q^{68} - 40905 q^{69} + 496664 q^{70} - 2003644 q^{71} - 2239488 q^{72} + 3659111 q^{73} - 4594480 q^{74} - 4228443 q^{75} - 261440 q^{76} - 2102590 q^{77} + 2847312 q^{78} + 1131065 q^{79} - 741376 q^{80} + 3188646 q^{81} - 1612416 q^{82} - 9629297 q^{83} - 3556224 q^{84} + 895068 q^{85} - 5828840 q^{86} + 1603665 q^{87} + 3138560 q^{88} - 21977377 q^{89} + 1055592 q^{90} + 4521426 q^{91} + 96960 q^{92} - 12912507 q^{93} - 1820920 q^{94} - 19325507 q^{95} + 5308416 q^{96} - 26386649 q^{97} - 5647152 q^{98} - 4468770 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) −487.447 −1.74394 −0.871972 0.489555i \(-0.837159\pi\)
−0.871972 + 0.489555i \(0.837159\pi\)
\(6\) 216.000 0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 3899.58 1.23316
\(11\) −6609.14 −1.49717 −0.748585 0.663039i \(-0.769266\pi\)
−0.748585 + 0.663039i \(0.769266\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) 13161.1 1.00687
\(16\) 4096.00 0.250000
\(17\) 8891.52 0.438940 0.219470 0.975619i \(-0.429567\pi\)
0.219470 + 0.975619i \(0.429567\pi\)
\(18\) −5832.00 −0.235702
\(19\) 7882.69 0.263656 0.131828 0.991273i \(-0.457915\pi\)
0.131828 + 0.991273i \(0.457915\pi\)
\(20\) −31196.6 −0.871972
\(21\) −9261.00 −0.218218
\(22\) 52873.2 1.05866
\(23\) −107716. −1.84601 −0.923005 0.384788i \(-0.874274\pi\)
−0.923005 + 0.384788i \(0.874274\pi\)
\(24\) 13824.0 0.204124
\(25\) 159480. 2.04134
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) −127254. −0.968901 −0.484450 0.874819i \(-0.660980\pi\)
−0.484450 + 0.874819i \(0.660980\pi\)
\(30\) −105289. −0.711963
\(31\) 86977.9 0.524376 0.262188 0.965017i \(-0.415556\pi\)
0.262188 + 0.965017i \(0.415556\pi\)
\(32\) −32768.0 −0.176777
\(33\) 178447. 0.864391
\(34\) −71132.2 −0.310377
\(35\) −167194. −0.659149
\(36\) 46656.0 0.166667
\(37\) −233049. −0.756383 −0.378192 0.925727i \(-0.623454\pi\)
−0.378192 + 0.925727i \(0.623454\pi\)
\(38\) −63061.6 −0.186433
\(39\) −59319.0 −0.160128
\(40\) 249573. 0.616578
\(41\) 293448. 0.664947 0.332473 0.943113i \(-0.392117\pi\)
0.332473 + 0.943113i \(0.392117\pi\)
\(42\) 74088.0 0.154303
\(43\) −259605. −0.497936 −0.248968 0.968512i \(-0.580091\pi\)
−0.248968 + 0.968512i \(0.580091\pi\)
\(44\) −422985. −0.748585
\(45\) −355349. −0.581315
\(46\) 861730. 1.30533
\(47\) −244441. −0.343425 −0.171713 0.985147i \(-0.554930\pi\)
−0.171713 + 0.985147i \(0.554930\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) −1.27584e6 −1.44345
\(51\) −240071. −0.253422
\(52\) 140608. 0.138675
\(53\) −1.77201e6 −1.63494 −0.817468 0.575975i \(-0.804623\pi\)
−0.817468 + 0.575975i \(0.804623\pi\)
\(54\) 157464. 0.136083
\(55\) 3.22161e6 2.61098
\(56\) −175616. −0.133631
\(57\) −212833. −0.152222
\(58\) 1.01803e6 0.685116
\(59\) −1.29914e6 −0.823519 −0.411759 0.911293i \(-0.635086\pi\)
−0.411759 + 0.911293i \(0.635086\pi\)
\(60\) 842309. 0.503434
\(61\) −2.80338e6 −1.58135 −0.790674 0.612238i \(-0.790269\pi\)
−0.790674 + 0.612238i \(0.790269\pi\)
\(62\) −695823. −0.370790
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −1.07092e6 −0.483683
\(66\) −1.42758e6 −0.611217
\(67\) −3.33358e6 −1.35410 −0.677048 0.735939i \(-0.736741\pi\)
−0.677048 + 0.735939i \(0.736741\pi\)
\(68\) 569057. 0.219470
\(69\) 2.90834e6 1.06579
\(70\) 1.33756e6 0.466089
\(71\) 1.67692e6 0.556041 0.278021 0.960575i \(-0.410322\pi\)
0.278021 + 0.960575i \(0.410322\pi\)
\(72\) −373248. −0.117851
\(73\) −932381. −0.280520 −0.140260 0.990115i \(-0.544794\pi\)
−0.140260 + 0.990115i \(0.544794\pi\)
\(74\) 1.86439e6 0.534844
\(75\) −4.30596e6 −1.17857
\(76\) 504492. 0.131828
\(77\) −2.26694e6 −0.565877
\(78\) 474552. 0.113228
\(79\) −3.16827e6 −0.722982 −0.361491 0.932376i \(-0.617732\pi\)
−0.361491 + 0.932376i \(0.617732\pi\)
\(80\) −1.99658e6 −0.435986
\(81\) 531441. 0.111111
\(82\) −2.34758e6 −0.470188
\(83\) 4.51858e6 0.867418 0.433709 0.901053i \(-0.357205\pi\)
0.433709 + 0.901053i \(0.357205\pi\)
\(84\) −592704. −0.109109
\(85\) −4.33415e6 −0.765487
\(86\) 2.07684e6 0.352094
\(87\) 3.43587e6 0.559395
\(88\) 3.38388e6 0.529329
\(89\) −1.31198e7 −1.97271 −0.986355 0.164633i \(-0.947356\pi\)
−0.986355 + 0.164633i \(0.947356\pi\)
\(90\) 2.84279e6 0.411052
\(91\) 753571. 0.104828
\(92\) −6.89384e6 −0.923005
\(93\) −2.34840e6 −0.302749
\(94\) 1.95553e6 0.242838
\(95\) −3.84240e6 −0.459801
\(96\) 884736. 0.102062
\(97\) −1.12922e7 −1.25626 −0.628130 0.778109i \(-0.716179\pi\)
−0.628130 + 0.778109i \(0.716179\pi\)
\(98\) −941192. −0.101015
\(99\) −4.81807e6 −0.499056
\(100\) 1.02067e7 1.02067
\(101\) −1.18245e7 −1.14198 −0.570989 0.820958i \(-0.693440\pi\)
−0.570989 + 0.820958i \(0.693440\pi\)
\(102\) 1.92057e6 0.179196
\(103\) 4.09889e6 0.369603 0.184802 0.982776i \(-0.440836\pi\)
0.184802 + 0.982776i \(0.440836\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) 4.51425e6 0.380560
\(106\) 1.41761e7 1.15607
\(107\) −6.61765e6 −0.522228 −0.261114 0.965308i \(-0.584090\pi\)
−0.261114 + 0.965308i \(0.584090\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 1.16847e6 0.0864222 0.0432111 0.999066i \(-0.486241\pi\)
0.0432111 + 0.999066i \(0.486241\pi\)
\(110\) −2.57729e7 −1.84624
\(111\) 6.29233e6 0.436698
\(112\) 1.40493e6 0.0944911
\(113\) 5.46121e6 0.356053 0.178026 0.984026i \(-0.443029\pi\)
0.178026 + 0.984026i \(0.443029\pi\)
\(114\) 1.70266e6 0.107637
\(115\) 5.25060e7 3.21934
\(116\) −8.14427e6 −0.484450
\(117\) 1.60161e6 0.0924500
\(118\) 1.03931e7 0.582316
\(119\) 3.04979e6 0.165904
\(120\) −6.73847e6 −0.355981
\(121\) 2.41936e7 1.24152
\(122\) 2.24270e7 1.11818
\(123\) −7.92308e6 −0.383907
\(124\) 5.56659e6 0.262188
\(125\) −3.96563e7 −1.81605
\(126\) −2.00038e6 −0.0890871
\(127\) −1.67741e6 −0.0726653 −0.0363327 0.999340i \(-0.511568\pi\)
−0.0363327 + 0.999340i \(0.511568\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 7.00933e6 0.287483
\(130\) 8.56738e6 0.342016
\(131\) 2.17219e6 0.0844207 0.0422104 0.999109i \(-0.486560\pi\)
0.0422104 + 0.999109i \(0.486560\pi\)
\(132\) 1.14206e7 0.432196
\(133\) 2.70376e6 0.0996525
\(134\) 2.66687e7 0.957490
\(135\) 9.59443e6 0.335622
\(136\) −4.55246e6 −0.155189
\(137\) −1.05432e7 −0.350308 −0.175154 0.984541i \(-0.556042\pi\)
−0.175154 + 0.984541i \(0.556042\pi\)
\(138\) −2.32667e7 −0.753630
\(139\) −4.52825e7 −1.43014 −0.715071 0.699052i \(-0.753606\pi\)
−0.715071 + 0.699052i \(0.753606\pi\)
\(140\) −1.07004e7 −0.329575
\(141\) 6.59992e6 0.198277
\(142\) −1.34153e7 −0.393181
\(143\) −1.45203e7 −0.415240
\(144\) 2.98598e6 0.0833333
\(145\) 6.20298e7 1.68971
\(146\) 7.45905e6 0.198357
\(147\) −3.17652e6 −0.0824786
\(148\) −1.49152e7 −0.378192
\(149\) 1.45929e7 0.361400 0.180700 0.983538i \(-0.442164\pi\)
0.180700 + 0.983538i \(0.442164\pi\)
\(150\) 3.44477e7 0.833375
\(151\) −4.39361e7 −1.03849 −0.519245 0.854626i \(-0.673787\pi\)
−0.519245 + 0.854626i \(0.673787\pi\)
\(152\) −4.03594e6 −0.0932163
\(153\) 6.48192e6 0.146313
\(154\) 1.81355e7 0.400135
\(155\) −4.23972e7 −0.914483
\(156\) −3.79642e6 −0.0800641
\(157\) −2.12373e7 −0.437977 −0.218989 0.975727i \(-0.570276\pi\)
−0.218989 + 0.975727i \(0.570276\pi\)
\(158\) 2.53462e7 0.511226
\(159\) 4.78443e7 0.943930
\(160\) 1.59727e7 0.308289
\(161\) −3.69467e7 −0.697726
\(162\) −4.25153e6 −0.0785674
\(163\) −1.03718e8 −1.87584 −0.937920 0.346852i \(-0.887251\pi\)
−0.937920 + 0.346852i \(0.887251\pi\)
\(164\) 1.87806e7 0.332473
\(165\) −8.69835e7 −1.50745
\(166\) −3.61486e7 −0.613357
\(167\) −4.39460e7 −0.730150 −0.365075 0.930978i \(-0.618957\pi\)
−0.365075 + 0.930978i \(0.618957\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 3.46732e7 0.541281
\(171\) 5.74648e6 0.0878852
\(172\) −1.66147e7 −0.248968
\(173\) −8.09232e7 −1.18826 −0.594130 0.804369i \(-0.702504\pi\)
−0.594130 + 0.804369i \(0.702504\pi\)
\(174\) −2.74869e7 −0.395552
\(175\) 5.47016e7 0.771555
\(176\) −2.70711e7 −0.374292
\(177\) 3.50767e7 0.475459
\(178\) 1.04959e8 1.39492
\(179\) −7.35345e7 −0.958309 −0.479154 0.877731i \(-0.659057\pi\)
−0.479154 + 0.877731i \(0.659057\pi\)
\(180\) −2.27423e7 −0.290657
\(181\) −8.97687e7 −1.12525 −0.562626 0.826711i \(-0.690209\pi\)
−0.562626 + 0.826711i \(0.690209\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 7.56912e7 0.912991
\(184\) 5.51507e7 0.652663
\(185\) 1.13599e8 1.31909
\(186\) 1.87872e7 0.214076
\(187\) −5.87654e7 −0.657167
\(188\) −1.56443e7 −0.171713
\(189\) −6.75127e6 −0.0727393
\(190\) 3.07392e7 0.325128
\(191\) 1.23836e8 1.28597 0.642985 0.765879i \(-0.277696\pi\)
0.642985 + 0.765879i \(0.277696\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 3.79864e7 0.380345 0.190172 0.981751i \(-0.439095\pi\)
0.190172 + 0.981751i \(0.439095\pi\)
\(194\) 9.03380e7 0.888309
\(195\) 2.89149e7 0.279255
\(196\) 7.52954e6 0.0714286
\(197\) 8.33840e7 0.777053 0.388527 0.921437i \(-0.372984\pi\)
0.388527 + 0.921437i \(0.372984\pi\)
\(198\) 3.85445e7 0.352886
\(199\) 1.59684e8 1.43640 0.718199 0.695838i \(-0.244967\pi\)
0.718199 + 0.695838i \(0.244967\pi\)
\(200\) −8.16538e7 −0.721724
\(201\) 9.00067e7 0.781788
\(202\) 9.45959e7 0.807500
\(203\) −4.36482e7 −0.366210
\(204\) −1.53645e7 −0.126711
\(205\) −1.43040e8 −1.15963
\(206\) −3.27911e7 −0.261349
\(207\) −7.85252e7 −0.615337
\(208\) 8.99891e6 0.0693375
\(209\) −5.20979e7 −0.394737
\(210\) −3.61140e7 −0.269097
\(211\) −2.41498e8 −1.76980 −0.884901 0.465779i \(-0.845774\pi\)
−0.884901 + 0.465779i \(0.845774\pi\)
\(212\) −1.13409e8 −0.817468
\(213\) −4.52767e7 −0.321031
\(214\) 5.29412e7 0.369271
\(215\) 1.26544e8 0.868373
\(216\) 1.00777e7 0.0680414
\(217\) 2.98334e7 0.198196
\(218\) −9.34777e6 −0.0611097
\(219\) 2.51743e7 0.161958
\(220\) 2.06183e8 1.30549
\(221\) 1.95347e7 0.121740
\(222\) −5.03387e7 −0.308792
\(223\) 1.35473e8 0.818063 0.409031 0.912520i \(-0.365867\pi\)
0.409031 + 0.912520i \(0.365867\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 1.16261e8 0.680448
\(226\) −4.36897e7 −0.251767
\(227\) 1.35605e8 0.769460 0.384730 0.923029i \(-0.374295\pi\)
0.384730 + 0.923029i \(0.374295\pi\)
\(228\) −1.36213e7 −0.0761108
\(229\) 2.56868e8 1.41347 0.706733 0.707481i \(-0.250168\pi\)
0.706733 + 0.707481i \(0.250168\pi\)
\(230\) −4.20048e8 −2.27642
\(231\) 6.12073e7 0.326709
\(232\) 6.51542e7 0.342558
\(233\) 1.02475e8 0.530729 0.265364 0.964148i \(-0.414508\pi\)
0.265364 + 0.964148i \(0.414508\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 1.19152e8 0.598915
\(236\) −8.31449e7 −0.411759
\(237\) 8.55434e7 0.417414
\(238\) −2.43983e7 −0.117312
\(239\) 4.12642e8 1.95515 0.977577 0.210577i \(-0.0675343\pi\)
0.977577 + 0.210577i \(0.0675343\pi\)
\(240\) 5.39078e7 0.251717
\(241\) −1.55197e8 −0.714208 −0.357104 0.934065i \(-0.616236\pi\)
−0.357104 + 0.934065i \(0.616236\pi\)
\(242\) −1.93549e8 −0.877884
\(243\) −1.43489e7 −0.0641500
\(244\) −1.79416e8 −0.790674
\(245\) −5.73477e7 −0.249135
\(246\) 6.33847e7 0.271463
\(247\) 1.73183e7 0.0731249
\(248\) −4.45327e7 −0.185395
\(249\) −1.22002e8 −0.500804
\(250\) 3.17250e8 1.28414
\(251\) 8.61766e7 0.343979 0.171989 0.985099i \(-0.444981\pi\)
0.171989 + 0.985099i \(0.444981\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 7.11913e8 2.76379
\(254\) 1.34193e7 0.0513821
\(255\) 1.17022e8 0.441954
\(256\) 1.67772e7 0.0625000
\(257\) −3.02243e8 −1.11068 −0.555342 0.831622i \(-0.687413\pi\)
−0.555342 + 0.831622i \(0.687413\pi\)
\(258\) −5.60747e7 −0.203281
\(259\) −7.99359e7 −0.285886
\(260\) −6.85390e7 −0.241842
\(261\) −9.27684e7 −0.322967
\(262\) −1.73776e7 −0.0596945
\(263\) 1.59520e8 0.540719 0.270359 0.962760i \(-0.412858\pi\)
0.270359 + 0.962760i \(0.412858\pi\)
\(264\) −9.13648e7 −0.305608
\(265\) 8.63761e8 2.85124
\(266\) −2.16301e7 −0.0704649
\(267\) 3.54235e8 1.13894
\(268\) −2.13349e8 −0.677048
\(269\) 4.07599e8 1.27673 0.638366 0.769733i \(-0.279611\pi\)
0.638366 + 0.769733i \(0.279611\pi\)
\(270\) −7.67554e7 −0.237321
\(271\) 2.12138e8 0.647479 0.323740 0.946146i \(-0.395060\pi\)
0.323740 + 0.946146i \(0.395060\pi\)
\(272\) 3.64197e7 0.109735
\(273\) −2.03464e7 −0.0605228
\(274\) 8.43455e7 0.247705
\(275\) −1.05403e9 −3.05624
\(276\) 1.86134e8 0.532897
\(277\) 2.37111e8 0.670305 0.335152 0.942164i \(-0.391212\pi\)
0.335152 + 0.942164i \(0.391212\pi\)
\(278\) 3.62260e8 1.01126
\(279\) 6.34069e7 0.174792
\(280\) 8.56036e7 0.233044
\(281\) 4.12563e6 0.0110922 0.00554610 0.999985i \(-0.498235\pi\)
0.00554610 + 0.999985i \(0.498235\pi\)
\(282\) −5.27994e7 −0.140203
\(283\) 2.41524e8 0.633442 0.316721 0.948519i \(-0.397418\pi\)
0.316721 + 0.948519i \(0.397418\pi\)
\(284\) 1.07323e8 0.278021
\(285\) 1.03745e8 0.265466
\(286\) 1.16162e8 0.293619
\(287\) 1.00652e8 0.251326
\(288\) −2.38879e7 −0.0589256
\(289\) −3.31280e8 −0.807332
\(290\) −4.96238e8 −1.19481
\(291\) 3.04891e8 0.725302
\(292\) −5.96724e7 −0.140260
\(293\) 6.11484e8 1.42020 0.710099 0.704102i \(-0.248650\pi\)
0.710099 + 0.704102i \(0.248650\pi\)
\(294\) 2.54122e7 0.0583212
\(295\) 6.33262e8 1.43617
\(296\) 1.19321e8 0.267422
\(297\) 1.30088e8 0.288130
\(298\) −1.16743e8 −0.255548
\(299\) −2.36653e8 −0.511991
\(300\) −2.75581e8 −0.589285
\(301\) −8.90445e7 −0.188202
\(302\) 3.51489e8 0.734323
\(303\) 3.19261e8 0.659321
\(304\) 3.22875e7 0.0659139
\(305\) 1.36650e9 2.75778
\(306\) −5.18554e7 −0.103459
\(307\) −4.86960e8 −0.960525 −0.480263 0.877125i \(-0.659459\pi\)
−0.480263 + 0.877125i \(0.659459\pi\)
\(308\) −1.45084e8 −0.282938
\(309\) −1.10670e8 −0.213391
\(310\) 3.39177e8 0.646637
\(311\) −6.84194e8 −1.28979 −0.644893 0.764273i \(-0.723098\pi\)
−0.644893 + 0.764273i \(0.723098\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) −5.66327e8 −1.04391 −0.521954 0.852974i \(-0.674797\pi\)
−0.521954 + 0.852974i \(0.674797\pi\)
\(314\) 1.69899e8 0.309697
\(315\) −1.21885e8 −0.219716
\(316\) −2.02769e8 −0.361491
\(317\) 8.09106e8 1.42659 0.713293 0.700866i \(-0.247203\pi\)
0.713293 + 0.700866i \(0.247203\pi\)
\(318\) −3.82754e8 −0.667459
\(319\) 8.41042e8 1.45061
\(320\) −1.27781e8 −0.217993
\(321\) 1.78676e8 0.301509
\(322\) 2.95573e8 0.493367
\(323\) 7.00891e7 0.115729
\(324\) 3.40122e7 0.0555556
\(325\) 3.50378e8 0.566167
\(326\) 8.29741e8 1.32642
\(327\) −3.15487e7 −0.0498959
\(328\) −1.50245e8 −0.235094
\(329\) −8.38434e7 −0.129803
\(330\) 6.95868e8 1.06593
\(331\) 1.12449e9 1.70435 0.852176 0.523255i \(-0.175283\pi\)
0.852176 + 0.523255i \(0.175283\pi\)
\(332\) 2.89189e8 0.433709
\(333\) −1.69893e8 −0.252128
\(334\) 3.51568e8 0.516294
\(335\) 1.62495e9 2.36147
\(336\) −3.79331e7 −0.0545545
\(337\) 3.87721e8 0.551841 0.275921 0.961180i \(-0.411017\pi\)
0.275921 + 0.961180i \(0.411017\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) −1.47453e8 −0.205567
\(340\) −2.77386e8 −0.382743
\(341\) −5.74850e8 −0.785080
\(342\) −4.59719e7 −0.0621442
\(343\) 4.03536e7 0.0539949
\(344\) 1.32918e8 0.176047
\(345\) −1.41766e9 −1.85869
\(346\) 6.47386e8 0.840227
\(347\) 1.45355e9 1.86757 0.933786 0.357832i \(-0.116484\pi\)
0.933786 + 0.357832i \(0.116484\pi\)
\(348\) 2.19895e8 0.279698
\(349\) −3.34462e8 −0.421170 −0.210585 0.977576i \(-0.567537\pi\)
−0.210585 + 0.977576i \(0.567537\pi\)
\(350\) −4.37613e8 −0.545572
\(351\) −4.32436e7 −0.0533761
\(352\) 2.16568e8 0.264665
\(353\) −1.12211e9 −1.35776 −0.678880 0.734249i \(-0.737534\pi\)
−0.678880 + 0.734249i \(0.737534\pi\)
\(354\) −2.80614e8 −0.336200
\(355\) −8.17408e8 −0.969705
\(356\) −8.39669e8 −0.986355
\(357\) −8.23444e7 −0.0957845
\(358\) 5.88276e8 0.677627
\(359\) 1.20452e8 0.137399 0.0686996 0.997637i \(-0.478115\pi\)
0.0686996 + 0.997637i \(0.478115\pi\)
\(360\) 1.81939e8 0.205526
\(361\) −8.31735e8 −0.930486
\(362\) 7.18150e8 0.795674
\(363\) −6.53228e8 −0.716789
\(364\) 4.82285e7 0.0524142
\(365\) 4.54487e8 0.489211
\(366\) −6.05530e8 −0.645582
\(367\) −3.56903e8 −0.376894 −0.188447 0.982083i \(-0.560345\pi\)
−0.188447 + 0.982083i \(0.560345\pi\)
\(368\) −4.41206e8 −0.461502
\(369\) 2.13923e8 0.221649
\(370\) −9.08794e8 −0.932738
\(371\) −6.07799e8 −0.617947
\(372\) −1.50298e8 −0.151374
\(373\) −3.38889e8 −0.338125 −0.169062 0.985605i \(-0.554074\pi\)
−0.169062 + 0.985605i \(0.554074\pi\)
\(374\) 4.70123e8 0.464687
\(375\) 1.07072e9 1.04849
\(376\) 1.25154e8 0.121419
\(377\) −2.79578e8 −0.268725
\(378\) 5.40102e7 0.0514344
\(379\) −1.76775e9 −1.66796 −0.833978 0.551797i \(-0.813942\pi\)
−0.833978 + 0.551797i \(0.813942\pi\)
\(380\) −2.45914e8 −0.229900
\(381\) 4.52902e7 0.0419533
\(382\) −9.90690e8 −0.909318
\(383\) −1.99915e9 −1.81824 −0.909118 0.416538i \(-0.863243\pi\)
−0.909118 + 0.416538i \(0.863243\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 1.10501e9 0.986858
\(386\) −3.03891e8 −0.268944
\(387\) −1.89252e8 −0.165979
\(388\) −7.22704e8 −0.628130
\(389\) −2.84545e8 −0.245091 −0.122546 0.992463i \(-0.539106\pi\)
−0.122546 + 0.992463i \(0.539106\pi\)
\(390\) −2.31319e8 −0.197463
\(391\) −9.57762e8 −0.810287
\(392\) −6.02363e7 −0.0505076
\(393\) −5.86492e7 −0.0487403
\(394\) −6.67072e8 −0.549460
\(395\) 1.54437e9 1.26084
\(396\) −3.08356e8 −0.249528
\(397\) 4.52749e8 0.363154 0.181577 0.983377i \(-0.441880\pi\)
0.181577 + 0.983377i \(0.441880\pi\)
\(398\) −1.27747e9 −1.01569
\(399\) −7.30016e7 −0.0575344
\(400\) 6.53230e8 0.510336
\(401\) −2.30471e7 −0.0178489 −0.00892446 0.999960i \(-0.502841\pi\)
−0.00892446 + 0.999960i \(0.502841\pi\)
\(402\) −7.20054e8 −0.552807
\(403\) 1.91090e8 0.145436
\(404\) −7.56767e8 −0.570989
\(405\) −2.59050e8 −0.193772
\(406\) 3.49186e8 0.258950
\(407\) 1.54026e9 1.13243
\(408\) 1.22916e8 0.0895982
\(409\) 1.37257e9 0.991983 0.495992 0.868327i \(-0.334805\pi\)
0.495992 + 0.868327i \(0.334805\pi\)
\(410\) 1.14432e9 0.819983
\(411\) 2.84666e8 0.202250
\(412\) 2.62329e8 0.184802
\(413\) −4.45605e8 −0.311261
\(414\) 6.28201e8 0.435109
\(415\) −2.20257e9 −1.51273
\(416\) −7.19913e7 −0.0490290
\(417\) 1.22263e9 0.825693
\(418\) 4.16783e8 0.279121
\(419\) −6.69063e7 −0.0444343 −0.0222171 0.999753i \(-0.507073\pi\)
−0.0222171 + 0.999753i \(0.507073\pi\)
\(420\) 2.88912e8 0.190280
\(421\) 1.13412e9 0.740752 0.370376 0.928882i \(-0.379229\pi\)
0.370376 + 0.928882i \(0.379229\pi\)
\(422\) 1.93198e9 1.25144
\(423\) −1.78198e8 −0.114475
\(424\) 9.07269e8 0.578037
\(425\) 1.41802e9 0.896027
\(426\) 3.62214e8 0.227003
\(427\) −9.61559e8 −0.597693
\(428\) −4.23529e8 −0.261114
\(429\) 3.92048e8 0.239739
\(430\) −1.01235e9 −0.614032
\(431\) −3.16683e9 −1.90526 −0.952629 0.304135i \(-0.901633\pi\)
−0.952629 + 0.304135i \(0.901633\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) −1.36383e9 −0.807331 −0.403666 0.914907i \(-0.632264\pi\)
−0.403666 + 0.914907i \(0.632264\pi\)
\(434\) −2.38667e8 −0.140145
\(435\) −1.67480e9 −0.975554
\(436\) 7.47822e7 0.0432111
\(437\) −8.49095e8 −0.486711
\(438\) −2.01394e8 −0.114522
\(439\) −3.17867e9 −1.79316 −0.896580 0.442881i \(-0.853956\pi\)
−0.896580 + 0.442881i \(0.853956\pi\)
\(440\) −1.64946e9 −0.923121
\(441\) 8.57661e7 0.0476190
\(442\) −1.56277e8 −0.0860832
\(443\) −1.37499e9 −0.751428 −0.375714 0.926736i \(-0.622602\pi\)
−0.375714 + 0.926736i \(0.622602\pi\)
\(444\) 4.02709e8 0.218349
\(445\) 6.39523e9 3.44030
\(446\) −1.08379e9 −0.578458
\(447\) −3.94007e8 −0.208654
\(448\) 8.99154e7 0.0472456
\(449\) 5.84627e8 0.304801 0.152401 0.988319i \(-0.451300\pi\)
0.152401 + 0.988319i \(0.451300\pi\)
\(450\) −9.30087e8 −0.481149
\(451\) −1.93944e9 −0.995538
\(452\) 3.49517e8 0.178026
\(453\) 1.18627e9 0.599572
\(454\) −1.08484e9 −0.544091
\(455\) −3.67326e8 −0.182815
\(456\) 1.08970e8 0.0538185
\(457\) 2.47958e9 1.21527 0.607634 0.794217i \(-0.292119\pi\)
0.607634 + 0.794217i \(0.292119\pi\)
\(458\) −2.05494e9 −0.999471
\(459\) −1.75012e8 −0.0844740
\(460\) 3.36039e9 1.60967
\(461\) 1.16093e9 0.551890 0.275945 0.961173i \(-0.411009\pi\)
0.275945 + 0.961173i \(0.411009\pi\)
\(462\) −4.89658e8 −0.231018
\(463\) 2.66646e9 1.24854 0.624268 0.781210i \(-0.285397\pi\)
0.624268 + 0.781210i \(0.285397\pi\)
\(464\) −5.21234e8 −0.242225
\(465\) 1.14472e9 0.527977
\(466\) −8.19801e8 −0.375282
\(467\) −4.30406e8 −0.195555 −0.0977776 0.995208i \(-0.531173\pi\)
−0.0977776 + 0.995208i \(0.531173\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) −1.14342e9 −0.511800
\(470\) −9.53219e8 −0.423497
\(471\) 5.73408e8 0.252866
\(472\) 6.65159e8 0.291158
\(473\) 1.71577e9 0.745494
\(474\) −6.84347e8 −0.295156
\(475\) 1.25713e9 0.538212
\(476\) 1.95187e8 0.0829518
\(477\) −1.29179e9 −0.544978
\(478\) −3.30114e9 −1.38250
\(479\) 1.60185e9 0.665960 0.332980 0.942934i \(-0.391946\pi\)
0.332980 + 0.942934i \(0.391946\pi\)
\(480\) −4.31262e8 −0.177991
\(481\) −5.12009e8 −0.209783
\(482\) 1.24158e9 0.505022
\(483\) 9.97561e8 0.402832
\(484\) 1.54839e9 0.620758
\(485\) 5.50438e9 2.19085
\(486\) 1.14791e8 0.0453609
\(487\) 2.62672e9 1.03054 0.515268 0.857029i \(-0.327692\pi\)
0.515268 + 0.857029i \(0.327692\pi\)
\(488\) 1.43533e9 0.559091
\(489\) 2.80037e9 1.08302
\(490\) 4.58782e8 0.176165
\(491\) 4.97690e9 1.89746 0.948732 0.316083i \(-0.102368\pi\)
0.948732 + 0.316083i \(0.102368\pi\)
\(492\) −5.07077e8 −0.191954
\(493\) −1.13148e9 −0.425289
\(494\) −1.38546e8 −0.0517071
\(495\) 2.34855e9 0.870327
\(496\) 3.56261e8 0.131094
\(497\) 5.75182e8 0.210164
\(498\) 9.76012e8 0.354122
\(499\) −2.98769e9 −1.07642 −0.538212 0.842809i \(-0.680900\pi\)
−0.538212 + 0.842809i \(0.680900\pi\)
\(500\) −2.53800e9 −0.908023
\(501\) 1.18654e9 0.421552
\(502\) −6.89413e8 −0.243230
\(503\) −3.10846e9 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 5.76382e9 1.99155
\(506\) −5.69530e9 −1.95429
\(507\) −1.30324e8 −0.0444116
\(508\) −1.07354e8 −0.0363327
\(509\) −1.21043e9 −0.406845 −0.203422 0.979091i \(-0.565206\pi\)
−0.203422 + 0.979091i \(0.565206\pi\)
\(510\) −9.36176e8 −0.312509
\(511\) −3.19807e8 −0.106026
\(512\) −1.34218e8 −0.0441942
\(513\) −1.55155e8 −0.0507406
\(514\) 2.41795e9 0.785372
\(515\) −1.99799e9 −0.644568
\(516\) 4.48597e8 0.143742
\(517\) 1.61555e9 0.514166
\(518\) 6.39487e8 0.202152
\(519\) 2.18493e9 0.686043
\(520\) 5.48312e8 0.171008
\(521\) −7.31274e8 −0.226542 −0.113271 0.993564i \(-0.536133\pi\)
−0.113271 + 0.993564i \(0.536133\pi\)
\(522\) 7.42147e8 0.228372
\(523\) −6.22512e9 −1.90280 −0.951398 0.307965i \(-0.900352\pi\)
−0.951398 + 0.307965i \(0.900352\pi\)
\(524\) 1.39020e8 0.0422104
\(525\) −1.47694e9 −0.445458
\(526\) −1.27616e9 −0.382346
\(527\) 7.73366e8 0.230170
\(528\) 7.30919e8 0.216098
\(529\) 8.19797e9 2.40775
\(530\) −6.91009e9 −2.01613
\(531\) −9.47072e8 −0.274506
\(532\) 1.73041e8 0.0498262
\(533\) 6.44704e8 0.184423
\(534\) −2.83388e9 −0.805355
\(535\) 3.22575e9 0.910737
\(536\) 1.70679e9 0.478745
\(537\) 1.98543e9 0.553280
\(538\) −3.26079e9 −0.902786
\(539\) −7.77559e8 −0.213881
\(540\) 6.14043e8 0.167811
\(541\) −2.23903e9 −0.607953 −0.303976 0.952680i \(-0.598314\pi\)
−0.303976 + 0.952680i \(0.598314\pi\)
\(542\) −1.69710e9 −0.457837
\(543\) 2.42376e9 0.649665
\(544\) −2.91357e8 −0.0775943
\(545\) −5.69568e8 −0.150716
\(546\) 1.62771e8 0.0427960
\(547\) −2.84122e9 −0.742248 −0.371124 0.928583i \(-0.621027\pi\)
−0.371124 + 0.928583i \(0.621027\pi\)
\(548\) −6.74764e8 −0.175154
\(549\) −2.04366e9 −0.527116
\(550\) 8.43221e9 2.16109
\(551\) −1.00311e9 −0.255456
\(552\) −1.48907e9 −0.376815
\(553\) −1.08672e9 −0.273262
\(554\) −1.89689e9 −0.473977
\(555\) −3.06718e9 −0.761577
\(556\) −2.89808e9 −0.715071
\(557\) −3.15591e9 −0.773804 −0.386902 0.922121i \(-0.626455\pi\)
−0.386902 + 0.922121i \(0.626455\pi\)
\(558\) −5.07255e8 −0.123597
\(559\) −5.70352e8 −0.138103
\(560\) −6.84829e8 −0.164787
\(561\) 1.58666e9 0.379416
\(562\) −3.30050e7 −0.00784337
\(563\) 4.29568e9 1.01450 0.507250 0.861799i \(-0.330662\pi\)
0.507250 + 0.861799i \(0.330662\pi\)
\(564\) 4.22395e8 0.0991384
\(565\) −2.66205e9 −0.620936
\(566\) −1.93219e9 −0.447911
\(567\) 1.82284e8 0.0419961
\(568\) −8.58581e8 −0.196590
\(569\) −6.87856e9 −1.56532 −0.782662 0.622447i \(-0.786139\pi\)
−0.782662 + 0.622447i \(0.786139\pi\)
\(570\) −8.29958e8 −0.187713
\(571\) 3.19999e9 0.719319 0.359660 0.933084i \(-0.382893\pi\)
0.359660 + 0.933084i \(0.382893\pi\)
\(572\) −9.29299e8 −0.207620
\(573\) −3.34358e9 −0.742455
\(574\) −8.05220e8 −0.177715
\(575\) −1.71786e10 −3.76834
\(576\) 1.91103e8 0.0416667
\(577\) −6.66011e9 −1.44333 −0.721666 0.692242i \(-0.756623\pi\)
−0.721666 + 0.692242i \(0.756623\pi\)
\(578\) 2.65024e9 0.570870
\(579\) −1.02563e9 −0.219592
\(580\) 3.96991e9 0.844855
\(581\) 1.54987e9 0.327853
\(582\) −2.43912e9 −0.512866
\(583\) 1.17115e10 2.44777
\(584\) 4.77379e8 0.0991787
\(585\) −7.80702e8 −0.161228
\(586\) −4.89188e9 −1.00423
\(587\) −2.95382e9 −0.602769 −0.301384 0.953503i \(-0.597449\pi\)
−0.301384 + 0.953503i \(0.597449\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) 6.85620e8 0.138255
\(590\) −5.06610e9 −1.01553
\(591\) −2.25137e9 −0.448632
\(592\) −9.54570e8 −0.189096
\(593\) −4.54892e9 −0.895813 −0.447906 0.894080i \(-0.647830\pi\)
−0.447906 + 0.894080i \(0.647830\pi\)
\(594\) −1.04070e9 −0.203739
\(595\) −1.48661e9 −0.289327
\(596\) 9.33942e8 0.180700
\(597\) −4.31146e9 −0.829305
\(598\) 1.89322e9 0.362032
\(599\) −8.80898e9 −1.67468 −0.837340 0.546683i \(-0.815890\pi\)
−0.837340 + 0.546683i \(0.815890\pi\)
\(600\) 2.20465e9 0.416688
\(601\) −4.78732e9 −0.899563 −0.449781 0.893139i \(-0.648498\pi\)
−0.449781 + 0.893139i \(0.648498\pi\)
\(602\) 7.12356e8 0.133079
\(603\) −2.43018e9 −0.451365
\(604\) −2.81191e9 −0.519245
\(605\) −1.17931e10 −2.16513
\(606\) −2.55409e9 −0.466210
\(607\) −5.37360e9 −0.975226 −0.487613 0.873060i \(-0.662132\pi\)
−0.487613 + 0.873060i \(0.662132\pi\)
\(608\) −2.58300e8 −0.0466082
\(609\) 1.17850e9 0.211432
\(610\) −1.09320e10 −1.95005
\(611\) −5.37038e8 −0.0952491
\(612\) 4.14843e8 0.0731566
\(613\) 1.02097e9 0.179020 0.0895100 0.995986i \(-0.471470\pi\)
0.0895100 + 0.995986i \(0.471470\pi\)
\(614\) 3.89568e9 0.679194
\(615\) 3.86209e9 0.669513
\(616\) 1.16067e9 0.200068
\(617\) 4.36863e9 0.748768 0.374384 0.927274i \(-0.377854\pi\)
0.374384 + 0.927274i \(0.377854\pi\)
\(618\) 8.85360e8 0.150890
\(619\) 9.55909e9 1.61994 0.809970 0.586471i \(-0.199483\pi\)
0.809970 + 0.586471i \(0.199483\pi\)
\(620\) −2.71342e9 −0.457242
\(621\) 2.12018e9 0.355265
\(622\) 5.47355e9 0.912017
\(623\) −4.50010e9 −0.745614
\(624\) −2.42971e8 −0.0400320
\(625\) 6.87098e9 1.12574
\(626\) 4.53062e9 0.738154
\(627\) 1.40664e9 0.227902
\(628\) −1.35919e9 −0.218989
\(629\) −2.07216e9 −0.332007
\(630\) 9.75078e8 0.155363
\(631\) 3.76791e9 0.597032 0.298516 0.954405i \(-0.403508\pi\)
0.298516 + 0.954405i \(0.403508\pi\)
\(632\) 1.62216e9 0.255613
\(633\) 6.52044e9 1.02180
\(634\) −6.47284e9 −1.00875
\(635\) 8.17651e8 0.126724
\(636\) 3.06203e9 0.471965
\(637\) 2.58475e8 0.0396214
\(638\) −6.72834e9 −1.02574
\(639\) 1.22247e9 0.185347
\(640\) 1.02225e9 0.154144
\(641\) −1.10760e10 −1.66104 −0.830519 0.556991i \(-0.811956\pi\)
−0.830519 + 0.556991i \(0.811956\pi\)
\(642\) −1.42941e9 −0.213199
\(643\) 1.71600e9 0.254553 0.127276 0.991867i \(-0.459376\pi\)
0.127276 + 0.991867i \(0.459376\pi\)
\(644\) −2.36459e9 −0.348863
\(645\) −3.41668e9 −0.501355
\(646\) −5.60713e8 −0.0818327
\(647\) 3.74804e9 0.544050 0.272025 0.962290i \(-0.412307\pi\)
0.272025 + 0.962290i \(0.412307\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) 8.58620e9 1.23295
\(650\) −2.80302e9 −0.400340
\(651\) −8.05502e8 −0.114428
\(652\) −6.63793e9 −0.937920
\(653\) −2.89195e9 −0.406438 −0.203219 0.979133i \(-0.565140\pi\)
−0.203219 + 0.979133i \(0.565140\pi\)
\(654\) 2.52390e8 0.0352817
\(655\) −1.05883e9 −0.147225
\(656\) 1.20196e9 0.166237
\(657\) −6.79706e8 −0.0935066
\(658\) 6.70747e8 0.0917843
\(659\) 5.98887e7 0.00815166 0.00407583 0.999992i \(-0.498703\pi\)
0.00407583 + 0.999992i \(0.498703\pi\)
\(660\) −5.56694e9 −0.753725
\(661\) 2.69737e9 0.363275 0.181637 0.983366i \(-0.441860\pi\)
0.181637 + 0.983366i \(0.441860\pi\)
\(662\) −8.99596e9 −1.20516
\(663\) −5.27436e8 −0.0702866
\(664\) −2.31351e9 −0.306678
\(665\) −1.31794e9 −0.173788
\(666\) 1.35914e9 0.178281
\(667\) 1.37074e10 1.78860
\(668\) −2.81255e9 −0.365075
\(669\) −3.65778e9 −0.472309
\(670\) −1.29996e10 −1.66981
\(671\) 1.85279e10 2.36754
\(672\) 3.03464e8 0.0385758
\(673\) −1.57258e9 −0.198866 −0.0994329 0.995044i \(-0.531703\pi\)
−0.0994329 + 0.995044i \(0.531703\pi\)
\(674\) −3.10177e9 −0.390211
\(675\) −3.13904e9 −0.392857
\(676\) 3.08916e8 0.0384615
\(677\) −6.48212e9 −0.802891 −0.401446 0.915883i \(-0.631492\pi\)
−0.401446 + 0.915883i \(0.631492\pi\)
\(678\) 1.17962e9 0.145358
\(679\) −3.87324e9 −0.474821
\(680\) 2.21908e9 0.270640
\(681\) −3.66134e9 −0.444248
\(682\) 4.59880e9 0.555135
\(683\) 7.54875e9 0.906573 0.453286 0.891365i \(-0.350252\pi\)
0.453286 + 0.891365i \(0.350252\pi\)
\(684\) 3.67775e8 0.0439426
\(685\) 5.13925e9 0.610918
\(686\) −3.22829e8 −0.0381802
\(687\) −6.93543e9 −0.816065
\(688\) −1.06334e9 −0.124484
\(689\) −3.89310e9 −0.453449
\(690\) 1.13413e10 1.31429
\(691\) −1.97717e9 −0.227966 −0.113983 0.993483i \(-0.536361\pi\)
−0.113983 + 0.993483i \(0.536361\pi\)
\(692\) −5.17908e9 −0.594130
\(693\) −1.65260e9 −0.188626
\(694\) −1.16284e10 −1.32057
\(695\) 2.20729e10 2.49409
\(696\) −1.75916e9 −0.197776
\(697\) 2.60919e9 0.291872
\(698\) 2.67569e9 0.297812
\(699\) −2.76683e9 −0.306416
\(700\) 3.50090e9 0.385778
\(701\) 9.57044e9 1.04935 0.524673 0.851304i \(-0.324188\pi\)
0.524673 + 0.851304i \(0.324188\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) −1.83706e9 −0.199425
\(704\) −1.73255e9 −0.187146
\(705\) −3.21711e9 −0.345784
\(706\) 8.97686e9 0.960082
\(707\) −4.05580e9 −0.431627
\(708\) 2.24491e9 0.237729
\(709\) −1.34890e10 −1.42141 −0.710704 0.703492i \(-0.751623\pi\)
−0.710704 + 0.703492i \(0.751623\pi\)
\(710\) 6.53927e9 0.685685
\(711\) −2.30967e9 −0.240994
\(712\) 6.71735e9 0.697458
\(713\) −9.36894e9 −0.968004
\(714\) 6.58755e8 0.0677299
\(715\) 7.07788e9 0.724156
\(716\) −4.70621e9 −0.479154
\(717\) −1.11413e10 −1.12881
\(718\) −9.63618e8 −0.0971560
\(719\) −7.99651e9 −0.802323 −0.401161 0.916007i \(-0.631393\pi\)
−0.401161 + 0.916007i \(0.631393\pi\)
\(720\) −1.45551e9 −0.145329
\(721\) 1.40592e9 0.139697
\(722\) 6.65388e9 0.657953
\(723\) 4.19033e9 0.412348
\(724\) −5.74520e9 −0.562626
\(725\) −2.02945e10 −1.97786
\(726\) 5.22582e9 0.506847
\(727\) 4.80961e9 0.464237 0.232118 0.972688i \(-0.425434\pi\)
0.232118 + 0.972688i \(0.425434\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −3.63589e9 −0.345924
\(731\) −2.30828e9 −0.218564
\(732\) 4.84424e9 0.456496
\(733\) 1.74677e10 1.63822 0.819109 0.573638i \(-0.194468\pi\)
0.819109 + 0.573638i \(0.194468\pi\)
\(734\) 2.85522e9 0.266504
\(735\) 1.54839e9 0.143838
\(736\) 3.52965e9 0.326331
\(737\) 2.20321e10 2.02731
\(738\) −1.71139e9 −0.156729
\(739\) 8.05884e9 0.734543 0.367271 0.930114i \(-0.380292\pi\)
0.367271 + 0.930114i \(0.380292\pi\)
\(740\) 7.27035e9 0.659545
\(741\) −4.67594e8 −0.0422187
\(742\) 4.86239e9 0.436955
\(743\) −1.74429e10 −1.56012 −0.780058 0.625707i \(-0.784810\pi\)
−0.780058 + 0.625707i \(0.784810\pi\)
\(744\) 1.20238e9 0.107038
\(745\) −7.11325e9 −0.630262
\(746\) 2.71111e9 0.239090
\(747\) 3.29404e9 0.289139
\(748\) −3.76098e9 −0.328583
\(749\) −2.26985e9 −0.197384
\(750\) −8.56576e9 −0.741398
\(751\) 9.80161e9 0.844418 0.422209 0.906498i \(-0.361255\pi\)
0.422209 + 0.906498i \(0.361255\pi\)
\(752\) −1.00123e9 −0.0858563
\(753\) −2.32677e9 −0.198596
\(754\) 2.23662e9 0.190017
\(755\) 2.14165e10 1.81107
\(756\) −4.32081e8 −0.0363696
\(757\) −9.93192e9 −0.832142 −0.416071 0.909332i \(-0.636593\pi\)
−0.416071 + 0.909332i \(0.636593\pi\)
\(758\) 1.41420e10 1.17942
\(759\) −1.92216e10 −1.59567
\(760\) 1.96731e9 0.162564
\(761\) 1.50372e9 0.123686 0.0618432 0.998086i \(-0.480302\pi\)
0.0618432 + 0.998086i \(0.480302\pi\)
\(762\) −3.62321e8 −0.0296655
\(763\) 4.00786e8 0.0326645
\(764\) 7.92552e9 0.642985
\(765\) −3.15959e9 −0.255162
\(766\) 1.59932e10 1.28569
\(767\) −2.85421e9 −0.228403
\(768\) −4.52985e8 −0.0360844
\(769\) 2.10157e9 0.166649 0.0833244 0.996522i \(-0.473446\pi\)
0.0833244 + 0.996522i \(0.473446\pi\)
\(770\) −8.84010e9 −0.697814
\(771\) 8.16057e9 0.641254
\(772\) 2.43113e9 0.190172
\(773\) −1.88721e10 −1.46958 −0.734788 0.678297i \(-0.762718\pi\)
−0.734788 + 0.678297i \(0.762718\pi\)
\(774\) 1.51402e9 0.117365
\(775\) 1.38712e10 1.07043
\(776\) 5.78163e9 0.444155
\(777\) 2.15827e9 0.165056
\(778\) 2.27636e9 0.173306
\(779\) 2.31316e9 0.175317
\(780\) 1.85055e9 0.139627
\(781\) −1.10830e10 −0.832488
\(782\) 7.66209e9 0.572959
\(783\) 2.50475e9 0.186465
\(784\) 4.81890e8 0.0357143
\(785\) 1.03521e10 0.763808
\(786\) 4.69194e8 0.0344646
\(787\) 1.27674e10 0.933668 0.466834 0.884345i \(-0.345395\pi\)
0.466834 + 0.884345i \(0.345395\pi\)
\(788\) 5.33657e9 0.388527
\(789\) −4.30705e9 −0.312184
\(790\) −1.23549e10 −0.891550
\(791\) 1.87320e9 0.134575
\(792\) 2.46685e9 0.176443
\(793\) −6.15902e9 −0.438587
\(794\) −3.62199e9 −0.256788
\(795\) −2.33216e10 −1.64616
\(796\) 1.02198e10 0.718199
\(797\) 1.13285e10 0.792629 0.396315 0.918115i \(-0.370289\pi\)
0.396315 + 0.918115i \(0.370289\pi\)
\(798\) 5.84013e8 0.0406830
\(799\) −2.17346e9 −0.150743
\(800\) −5.22584e9 −0.360862
\(801\) −9.56436e9 −0.657570
\(802\) 1.84377e8 0.0126211
\(803\) 6.16224e9 0.419986
\(804\) 5.76043e9 0.390894
\(805\) 1.80096e10 1.21680
\(806\) −1.52872e9 −0.102839
\(807\) −1.10052e10 −0.737122
\(808\) 6.05414e9 0.403750
\(809\) −1.20927e9 −0.0802975 −0.0401487 0.999194i \(-0.512783\pi\)
−0.0401487 + 0.999194i \(0.512783\pi\)
\(810\) 2.07240e9 0.137017
\(811\) −1.99159e10 −1.31107 −0.655535 0.755164i \(-0.727557\pi\)
−0.655535 + 0.755164i \(0.727557\pi\)
\(812\) −2.79349e9 −0.183105
\(813\) −5.72772e9 −0.373822
\(814\) −1.23221e10 −0.800751
\(815\) 5.05569e10 3.27136
\(816\) −9.83331e8 −0.0633555
\(817\) −2.04639e9 −0.131284
\(818\) −1.09806e10 −0.701438
\(819\) 5.49353e8 0.0349428
\(820\) −9.15457e9 −0.579815
\(821\) 1.79789e9 0.113387 0.0566934 0.998392i \(-0.481944\pi\)
0.0566934 + 0.998392i \(0.481944\pi\)
\(822\) −2.27733e9 −0.143013
\(823\) 2.96870e10 1.85638 0.928191 0.372103i \(-0.121363\pi\)
0.928191 + 0.372103i \(0.121363\pi\)
\(824\) −2.09863e9 −0.130675
\(825\) 2.84587e10 1.76452
\(826\) 3.56484e9 0.220095
\(827\) −1.82607e10 −1.12266 −0.561329 0.827593i \(-0.689710\pi\)
−0.561329 + 0.827593i \(0.689710\pi\)
\(828\) −5.02561e9 −0.307668
\(829\) −1.11037e9 −0.0676904 −0.0338452 0.999427i \(-0.510775\pi\)
−0.0338452 + 0.999427i \(0.510775\pi\)
\(830\) 1.76205e10 1.06966
\(831\) −6.40199e9 −0.387001
\(832\) 5.75930e8 0.0346688
\(833\) 1.04608e9 0.0627057
\(834\) −9.78103e9 −0.583853
\(835\) 2.14214e10 1.27334
\(836\) −3.33426e9 −0.197369
\(837\) −1.71199e9 −0.100916
\(838\) 5.35251e8 0.0314198
\(839\) −9.11686e9 −0.532940 −0.266470 0.963843i \(-0.585857\pi\)
−0.266470 + 0.963843i \(0.585857\pi\)
\(840\) −2.31130e9 −0.134548
\(841\) −1.05623e9 −0.0612309
\(842\) −9.07299e9 −0.523791
\(843\) −1.11392e8 −0.00640409
\(844\) −1.54559e10 −0.884901
\(845\) −2.35282e9 −0.134150
\(846\) 1.42558e9 0.0809461
\(847\) 8.29841e9 0.469249
\(848\) −7.25815e9 −0.408734
\(849\) −6.52114e9 −0.365718
\(850\) −1.13442e10 −0.633587
\(851\) 2.51032e10 1.39629
\(852\) −2.89771e9 −0.160515
\(853\) 2.39883e10 1.32336 0.661680 0.749787i \(-0.269844\pi\)
0.661680 + 0.749787i \(0.269844\pi\)
\(854\) 7.69247e9 0.422633
\(855\) −2.80111e9 −0.153267
\(856\) 3.38823e9 0.184636
\(857\) −4.74952e9 −0.257761 −0.128880 0.991660i \(-0.541138\pi\)
−0.128880 + 0.991660i \(0.541138\pi\)
\(858\) −3.13638e9 −0.169521
\(859\) 2.55278e10 1.37416 0.687079 0.726583i \(-0.258893\pi\)
0.687079 + 0.726583i \(0.258893\pi\)
\(860\) 8.09880e9 0.434186
\(861\) −2.71762e9 −0.145103
\(862\) 2.53346e10 1.34722
\(863\) 8.21920e9 0.435303 0.217651 0.976027i \(-0.430160\pi\)
0.217651 + 0.976027i \(0.430160\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 3.94458e10 2.07226
\(866\) 1.09106e10 0.570869
\(867\) 8.94455e9 0.466113
\(868\) 1.90934e9 0.0990978
\(869\) 2.09396e10 1.08243
\(870\) 1.33984e10 0.689821
\(871\) −7.32388e9 −0.375559
\(872\) −5.98257e8 −0.0305549
\(873\) −8.23205e9 −0.418753
\(874\) 6.79276e9 0.344157
\(875\) −1.36021e10 −0.686401
\(876\) 1.61115e9 0.0809791
\(877\) −3.69625e10 −1.85039 −0.925194 0.379493i \(-0.876098\pi\)
−0.925194 + 0.379493i \(0.876098\pi\)
\(878\) 2.54293e10 1.26796
\(879\) −1.65101e10 −0.819951
\(880\) 1.31957e10 0.652745
\(881\) −2.66580e10 −1.31345 −0.656723 0.754132i \(-0.728058\pi\)
−0.656723 + 0.754132i \(0.728058\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −2.02249e10 −0.988607 −0.494304 0.869289i \(-0.664577\pi\)
−0.494304 + 0.869289i \(0.664577\pi\)
\(884\) 1.25022e9 0.0608700
\(885\) −1.70981e10 −0.829174
\(886\) 1.09999e10 0.531340
\(887\) 3.02784e9 0.145680 0.0728400 0.997344i \(-0.476794\pi\)
0.0728400 + 0.997344i \(0.476794\pi\)
\(888\) −3.22167e9 −0.154396
\(889\) −5.75353e8 −0.0274649
\(890\) −5.11618e10 −2.43266
\(891\) −3.51237e9 −0.166352
\(892\) 8.67029e9 0.409031
\(893\) −1.92686e9 −0.0905460
\(894\) 3.15206e9 0.147541
\(895\) 3.58442e10 1.67124
\(896\) −7.19323e8 −0.0334077
\(897\) 6.38962e9 0.295598
\(898\) −4.67702e9 −0.215527
\(899\) −1.10683e10 −0.508069
\(900\) 7.44070e9 0.340224
\(901\) −1.57559e10 −0.717638
\(902\) 1.55155e10 0.703952
\(903\) 2.40420e9 0.108659
\(904\) −2.79614e9 −0.125884
\(905\) 4.37575e10 1.96238
\(906\) −9.49019e9 −0.423961
\(907\) 3.44478e10 1.53298 0.766490 0.642257i \(-0.222002\pi\)
0.766490 + 0.642257i \(0.222002\pi\)
\(908\) 8.67874e9 0.384730
\(909\) −8.62005e9 −0.380659
\(910\) 2.93861e9 0.129270
\(911\) −2.62762e10 −1.15146 −0.575729 0.817641i \(-0.695282\pi\)
−0.575729 + 0.817641i \(0.695282\pi\)
\(912\) −8.71763e8 −0.0380554
\(913\) −2.98639e10 −1.29867
\(914\) −1.98367e10 −0.859324
\(915\) −3.68955e10 −1.59221
\(916\) 1.64395e10 0.706733
\(917\) 7.45062e8 0.0319080
\(918\) 1.40009e9 0.0597321
\(919\) 1.02393e10 0.435178 0.217589 0.976041i \(-0.430181\pi\)
0.217589 + 0.976041i \(0.430181\pi\)
\(920\) −2.68831e10 −1.13821
\(921\) 1.31479e10 0.554560
\(922\) −9.28744e9 −0.390245
\(923\) 3.68418e9 0.154218
\(924\) 3.91727e9 0.163355
\(925\) −3.71667e10 −1.54404
\(926\) −2.13317e10 −0.882848
\(927\) 2.98809e9 0.123201
\(928\) 4.16987e9 0.171279
\(929\) 2.65627e10 1.08697 0.543485 0.839419i \(-0.317105\pi\)
0.543485 + 0.839419i \(0.317105\pi\)
\(930\) −9.15779e9 −0.373336
\(931\) 9.27391e8 0.0376651
\(932\) 6.55841e9 0.265364
\(933\) 1.84732e10 0.744659
\(934\) 3.44325e9 0.138278
\(935\) 2.86450e10 1.14606
\(936\) −8.20026e8 −0.0326860
\(937\) 3.40056e10 1.35040 0.675199 0.737636i \(-0.264058\pi\)
0.675199 + 0.737636i \(0.264058\pi\)
\(938\) 9.14735e9 0.361897
\(939\) 1.52908e10 0.602700
\(940\) 7.62575e9 0.299457
\(941\) −2.47895e10 −0.969848 −0.484924 0.874556i \(-0.661153\pi\)
−0.484924 + 0.874556i \(0.661153\pi\)
\(942\) −4.58727e9 −0.178803
\(943\) −3.16091e10 −1.22750
\(944\) −5.32127e9 −0.205880
\(945\) 3.29089e9 0.126853
\(946\) −1.37261e10 −0.527144
\(947\) 2.16151e10 0.827052 0.413526 0.910492i \(-0.364297\pi\)
0.413526 + 0.910492i \(0.364297\pi\)
\(948\) 5.47478e9 0.208707
\(949\) −2.04844e9 −0.0778022
\(950\) −1.00571e10 −0.380573
\(951\) −2.18459e10 −0.823639
\(952\) −1.56149e9 −0.0586558
\(953\) 1.62128e10 0.606782 0.303391 0.952866i \(-0.401881\pi\)
0.303391 + 0.952866i \(0.401881\pi\)
\(954\) 1.03344e10 0.385358
\(955\) −6.03637e10 −2.24266
\(956\) 2.64091e10 0.977577
\(957\) −2.27081e10 −0.837509
\(958\) −1.28148e10 −0.470905
\(959\) −3.61632e9 −0.132404
\(960\) 3.45010e9 0.125858
\(961\) −1.99475e10 −0.725030
\(962\) 4.09607e9 0.148339
\(963\) −4.82426e9 −0.174076
\(964\) −9.93264e9 −0.357104
\(965\) −1.85164e10 −0.663300
\(966\) −7.98048e9 −0.284845
\(967\) −3.00338e10 −1.06811 −0.534057 0.845449i \(-0.679333\pi\)
−0.534057 + 0.845449i \(0.679333\pi\)
\(968\) −1.23871e10 −0.438942
\(969\) −1.89241e9 −0.0668161
\(970\) −4.40350e10 −1.54916
\(971\) −1.32906e10 −0.465884 −0.232942 0.972491i \(-0.574835\pi\)
−0.232942 + 0.972491i \(0.574835\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) −1.55319e10 −0.540543
\(974\) −2.10138e10 −0.728699
\(975\) −9.46019e9 −0.326877
\(976\) −1.14826e10 −0.395337
\(977\) 1.99798e10 0.685424 0.342712 0.939440i \(-0.388654\pi\)
0.342712 + 0.939440i \(0.388654\pi\)
\(978\) −2.24030e10 −0.765808
\(979\) 8.67109e10 2.95348
\(980\) −3.67025e9 −0.124567
\(981\) 8.51816e8 0.0288074
\(982\) −3.98152e10 −1.34171
\(983\) 5.18069e9 0.173960 0.0869802 0.996210i \(-0.472278\pi\)
0.0869802 + 0.996210i \(0.472278\pi\)
\(984\) 4.05662e9 0.135732
\(985\) −4.06453e10 −1.35514
\(986\) 9.05187e9 0.300725
\(987\) 2.26377e9 0.0749416
\(988\) 1.10837e9 0.0365625
\(989\) 2.79637e10 0.919194
\(990\) −1.87884e10 −0.615414
\(991\) 5.16222e10 1.68492 0.842459 0.538761i \(-0.181107\pi\)
0.842459 + 0.538761i \(0.181107\pi\)
\(992\) −2.85009e9 −0.0926975
\(993\) −3.03613e10 −0.984008
\(994\) −4.60146e9 −0.148608
\(995\) −7.78374e10 −2.50500
\(996\) −7.80810e9 −0.250402
\(997\) −1.99089e10 −0.636229 −0.318115 0.948052i \(-0.603050\pi\)
−0.318115 + 0.948052i \(0.603050\pi\)
\(998\) 2.39015e10 0.761147
\(999\) 4.58711e9 0.145566
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 546.8.a.n.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.n.1.1 6 1.1 even 1 trivial