Properties

Label 115.6.a.c.1.6
Level $115$
Weight $6$
Character 115.1
Self dual yes
Analytic conductor $18.444$
Analytic rank $0$
Dimension $7$
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] = [115,6,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, 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("115.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(18.4441392785\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-8.71037\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.71037 q^{2} +18.9751 q^{3} +62.2913 q^{4} -25.0000 q^{5} +184.255 q^{6} +84.0805 q^{7} +294.140 q^{8} +117.055 q^{9} +O(q^{10})\) \(q+9.71037 q^{2} +18.9751 q^{3} +62.2913 q^{4} -25.0000 q^{5} +184.255 q^{6} +84.0805 q^{7} +294.140 q^{8} +117.055 q^{9} -242.759 q^{10} +219.102 q^{11} +1181.98 q^{12} -248.743 q^{13} +816.453 q^{14} -474.378 q^{15} +862.884 q^{16} +246.199 q^{17} +1136.64 q^{18} +31.8413 q^{19} -1557.28 q^{20} +1595.44 q^{21} +2127.56 q^{22} +529.000 q^{23} +5581.33 q^{24} +625.000 q^{25} -2415.39 q^{26} -2389.83 q^{27} +5237.49 q^{28} -5198.89 q^{29} -4606.38 q^{30} -2426.03 q^{31} -1033.55 q^{32} +4157.48 q^{33} +2390.68 q^{34} -2102.01 q^{35} +7291.48 q^{36} -1218.62 q^{37} +309.191 q^{38} -4719.93 q^{39} -7353.49 q^{40} -7062.12 q^{41} +15492.3 q^{42} +8934.89 q^{43} +13648.1 q^{44} -2926.36 q^{45} +5136.79 q^{46} +10262.7 q^{47} +16373.3 q^{48} -9737.46 q^{49} +6068.98 q^{50} +4671.65 q^{51} -15494.5 q^{52} -10911.4 q^{53} -23206.1 q^{54} -5477.54 q^{55} +24731.4 q^{56} +604.192 q^{57} -50483.1 q^{58} +3809.83 q^{59} -29549.6 q^{60} +31290.1 q^{61} -23557.6 q^{62} +9842.01 q^{63} -37648.4 q^{64} +6218.58 q^{65} +40370.6 q^{66} +7696.34 q^{67} +15336.1 q^{68} +10037.8 q^{69} -20411.3 q^{70} +71989.2 q^{71} +34430.4 q^{72} -60862.6 q^{73} -11833.2 q^{74} +11859.4 q^{75} +1983.44 q^{76} +18422.2 q^{77} -45832.2 q^{78} +53155.3 q^{79} -21572.1 q^{80} -73791.5 q^{81} -68575.8 q^{82} -56773.0 q^{83} +99381.8 q^{84} -6154.98 q^{85} +86761.1 q^{86} -98649.4 q^{87} +64446.5 q^{88} +116077. q^{89} -28416.1 q^{90} -20914.5 q^{91} +32952.1 q^{92} -46034.1 q^{93} +99655.0 q^{94} -796.032 q^{95} -19611.7 q^{96} -96388.0 q^{97} -94554.4 q^{98} +25646.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9} - 100 q^{10} + 1373 q^{11} - 285 q^{12} + 605 q^{13} + 1317 q^{14} + 75 q^{15} + 3770 q^{16} + 2505 q^{17} + 7971 q^{18} - 115 q^{19} - 4450 q^{20} + 608 q^{21} + 2977 q^{22} + 3703 q^{23} - 12447 q^{24} + 4375 q^{25} + 9379 q^{26} - 12276 q^{27} + 5777 q^{28} + 2440 q^{29} + 9525 q^{30} + 13565 q^{31} + 14086 q^{32} + 10519 q^{33} + 26997 q^{34} - 825 q^{35} + 79889 q^{36} + 9414 q^{37} + 28717 q^{38} - 21738 q^{39} - 13650 q^{40} + 13725 q^{41} + 12426 q^{42} + 76694 q^{43} + 55203 q^{44} - 11000 q^{45} + 2116 q^{46} + 59692 q^{47} - 32985 q^{48} - 53608 q^{49} + 2500 q^{50} - 24725 q^{51} + 61195 q^{52} + 49536 q^{53} - 156168 q^{54} - 34325 q^{55} - 54461 q^{56} - 7580 q^{57} - 95562 q^{58} + 44536 q^{59} + 7125 q^{60} - 49097 q^{61} - 25763 q^{62} - 3578 q^{63} - 18654 q^{64} - 15125 q^{65} - 201873 q^{66} + 788 q^{67} + 163845 q^{68} - 1587 q^{69} - 32925 q^{70} + 49521 q^{71} + 328503 q^{72} - 3760 q^{73} + 88170 q^{74} - 1875 q^{75} - 411465 q^{76} + 77728 q^{77} - 389832 q^{78} + 918 q^{79} - 94250 q^{80} + 121235 q^{81} - 227459 q^{82} + 99202 q^{83} + 336602 q^{84} - 62625 q^{85} + 24584 q^{86} - 38666 q^{87} - 201275 q^{88} - 141676 q^{89} - 199275 q^{90} - 223605 q^{91} + 94162 q^{92} + 51412 q^{93} - 354292 q^{94} + 2875 q^{95} - 592095 q^{96} + 28731 q^{97} - 149557 q^{98} + 237333 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.71037 1.71657 0.858284 0.513176i \(-0.171531\pi\)
0.858284 + 0.513176i \(0.171531\pi\)
\(3\) 18.9751 1.21725 0.608627 0.793457i \(-0.291721\pi\)
0.608627 + 0.793457i \(0.291721\pi\)
\(4\) 62.2913 1.94660
\(5\) −25.0000 −0.447214
\(6\) 184.255 2.08950
\(7\) 84.0805 0.648560 0.324280 0.945961i \(-0.394878\pi\)
0.324280 + 0.945961i \(0.394878\pi\)
\(8\) 294.140 1.62491
\(9\) 117.055 0.481706
\(10\) −242.759 −0.767672
\(11\) 219.102 0.545964 0.272982 0.962019i \(-0.411990\pi\)
0.272982 + 0.962019i \(0.411990\pi\)
\(12\) 1181.98 2.36951
\(13\) −248.743 −0.408219 −0.204109 0.978948i \(-0.565430\pi\)
−0.204109 + 0.978948i \(0.565430\pi\)
\(14\) 816.453 1.11330
\(15\) −474.378 −0.544372
\(16\) 862.884 0.842660
\(17\) 246.199 0.206616 0.103308 0.994649i \(-0.467057\pi\)
0.103308 + 0.994649i \(0.467057\pi\)
\(18\) 1136.64 0.826881
\(19\) 31.8413 0.0202352 0.0101176 0.999949i \(-0.496779\pi\)
0.0101176 + 0.999949i \(0.496779\pi\)
\(20\) −1557.28 −0.870547
\(21\) 1595.44 0.789462
\(22\) 2127.56 0.937183
\(23\) 529.000 0.208514
\(24\) 5581.33 1.97792
\(25\) 625.000 0.200000
\(26\) −2415.39 −0.700735
\(27\) −2389.83 −0.630895
\(28\) 5237.49 1.26249
\(29\) −5198.89 −1.14793 −0.573965 0.818880i \(-0.694595\pi\)
−0.573965 + 0.818880i \(0.694595\pi\)
\(30\) −4606.38 −0.934452
\(31\) −2426.03 −0.453410 −0.226705 0.973963i \(-0.572795\pi\)
−0.226705 + 0.973963i \(0.572795\pi\)
\(32\) −1033.55 −0.178425
\(33\) 4157.48 0.664576
\(34\) 2390.68 0.354670
\(35\) −2102.01 −0.290045
\(36\) 7291.48 0.937690
\(37\) −1218.62 −0.146340 −0.0731700 0.997319i \(-0.523312\pi\)
−0.0731700 + 0.997319i \(0.523312\pi\)
\(38\) 309.191 0.0347350
\(39\) −4719.93 −0.496906
\(40\) −7353.49 −0.726681
\(41\) −7062.12 −0.656108 −0.328054 0.944659i \(-0.606393\pi\)
−0.328054 + 0.944659i \(0.606393\pi\)
\(42\) 15492.3 1.35517
\(43\) 8934.89 0.736917 0.368458 0.929644i \(-0.379886\pi\)
0.368458 + 0.929644i \(0.379886\pi\)
\(44\) 13648.1 1.06277
\(45\) −2926.36 −0.215425
\(46\) 5136.79 0.357929
\(47\) 10262.7 0.677671 0.338835 0.940846i \(-0.389967\pi\)
0.338835 + 0.940846i \(0.389967\pi\)
\(48\) 16373.3 1.02573
\(49\) −9737.46 −0.579369
\(50\) 6068.98 0.343313
\(51\) 4671.65 0.251504
\(52\) −15494.5 −0.794640
\(53\) −10911.4 −0.533571 −0.266785 0.963756i \(-0.585961\pi\)
−0.266785 + 0.963756i \(0.585961\pi\)
\(54\) −23206.1 −1.08297
\(55\) −5477.54 −0.244162
\(56\) 24731.4 1.05385
\(57\) 604.192 0.0246313
\(58\) −50483.1 −1.97050
\(59\) 3809.83 0.142487 0.0712435 0.997459i \(-0.477303\pi\)
0.0712435 + 0.997459i \(0.477303\pi\)
\(60\) −29549.6 −1.05968
\(61\) 31290.1 1.07667 0.538334 0.842732i \(-0.319054\pi\)
0.538334 + 0.842732i \(0.319054\pi\)
\(62\) −23557.6 −0.778309
\(63\) 9842.01 0.312415
\(64\) −37648.4 −1.14894
\(65\) 6218.58 0.182561
\(66\) 40370.6 1.14079
\(67\) 7696.34 0.209458 0.104729 0.994501i \(-0.466602\pi\)
0.104729 + 0.994501i \(0.466602\pi\)
\(68\) 15336.1 0.402199
\(69\) 10037.8 0.253815
\(70\) −20411.3 −0.497882
\(71\) 71989.2 1.69481 0.847405 0.530946i \(-0.178163\pi\)
0.847405 + 0.530946i \(0.178163\pi\)
\(72\) 34430.4 0.782728
\(73\) −60862.6 −1.33673 −0.668365 0.743834i \(-0.733005\pi\)
−0.668365 + 0.743834i \(0.733005\pi\)
\(74\) −11833.2 −0.251203
\(75\) 11859.4 0.243451
\(76\) 1983.44 0.0393898
\(77\) 18422.2 0.354090
\(78\) −45832.2 −0.852972
\(79\) 53155.3 0.958250 0.479125 0.877747i \(-0.340954\pi\)
0.479125 + 0.877747i \(0.340954\pi\)
\(80\) −21572.1 −0.376849
\(81\) −73791.5 −1.24967
\(82\) −68575.8 −1.12625
\(83\) −56773.0 −0.904579 −0.452290 0.891871i \(-0.649393\pi\)
−0.452290 + 0.891871i \(0.649393\pi\)
\(84\) 99381.8 1.53677
\(85\) −6154.98 −0.0924015
\(86\) 86761.1 1.26497
\(87\) −98649.4 −1.39732
\(88\) 64446.5 0.887141
\(89\) 116077. 1.55335 0.776677 0.629900i \(-0.216904\pi\)
0.776677 + 0.629900i \(0.216904\pi\)
\(90\) −28416.1 −0.369792
\(91\) −20914.5 −0.264754
\(92\) 32952.1 0.405895
\(93\) −46034.1 −0.551915
\(94\) 99655.0 1.16327
\(95\) −796.032 −0.00904944
\(96\) −19611.7 −0.217188
\(97\) −96388.0 −1.04014 −0.520072 0.854122i \(-0.674095\pi\)
−0.520072 + 0.854122i \(0.674095\pi\)
\(98\) −94554.4 −0.994527
\(99\) 25646.8 0.262994
\(100\) 38932.1 0.389321
\(101\) −18650.2 −0.181920 −0.0909602 0.995855i \(-0.528994\pi\)
−0.0909602 + 0.995855i \(0.528994\pi\)
\(102\) 45363.5 0.431724
\(103\) −57213.0 −0.531376 −0.265688 0.964059i \(-0.585599\pi\)
−0.265688 + 0.964059i \(0.585599\pi\)
\(104\) −73165.2 −0.663317
\(105\) −39885.9 −0.353058
\(106\) −105954. −0.915910
\(107\) 109572. 0.925206 0.462603 0.886566i \(-0.346916\pi\)
0.462603 + 0.886566i \(0.346916\pi\)
\(108\) −148865. −1.22810
\(109\) −61327.9 −0.494415 −0.247208 0.968963i \(-0.579513\pi\)
−0.247208 + 0.968963i \(0.579513\pi\)
\(110\) −53188.9 −0.419121
\(111\) −23123.4 −0.178133
\(112\) 72551.7 0.546516
\(113\) −247962. −1.82679 −0.913397 0.407071i \(-0.866550\pi\)
−0.913397 + 0.407071i \(0.866550\pi\)
\(114\) 5866.93 0.0422813
\(115\) −13225.0 −0.0932505
\(116\) −323845. −2.23456
\(117\) −29116.5 −0.196641
\(118\) 36994.8 0.244589
\(119\) 20700.6 0.134003
\(120\) −139533. −0.884555
\(121\) −113045. −0.701924
\(122\) 303838. 1.84817
\(123\) −134005. −0.798650
\(124\) −151120. −0.882609
\(125\) −15625.0 −0.0894427
\(126\) 95569.6 0.536282
\(127\) 302231. 1.66276 0.831380 0.555704i \(-0.187551\pi\)
0.831380 + 0.555704i \(0.187551\pi\)
\(128\) −332507. −1.79381
\(129\) 169541. 0.897014
\(130\) 60384.7 0.313378
\(131\) 97654.9 0.497183 0.248591 0.968608i \(-0.420032\pi\)
0.248591 + 0.968608i \(0.420032\pi\)
\(132\) 258975. 1.29367
\(133\) 2677.23 0.0131237
\(134\) 74734.3 0.359549
\(135\) 59745.7 0.282145
\(136\) 72416.9 0.335732
\(137\) 403988. 1.83894 0.919470 0.393160i \(-0.128618\pi\)
0.919470 + 0.393160i \(0.128618\pi\)
\(138\) 97471.0 0.435690
\(139\) 347715. 1.52646 0.763232 0.646125i \(-0.223611\pi\)
0.763232 + 0.646125i \(0.223611\pi\)
\(140\) −130937. −0.564602
\(141\) 194737. 0.824897
\(142\) 699041. 2.90926
\(143\) −54500.0 −0.222873
\(144\) 101005. 0.405914
\(145\) 129972. 0.513370
\(146\) −590999. −2.29459
\(147\) −184769. −0.705239
\(148\) −75909.3 −0.284866
\(149\) 101121. 0.373144 0.186572 0.982441i \(-0.440262\pi\)
0.186572 + 0.982441i \(0.440262\pi\)
\(150\) 115160. 0.417899
\(151\) 15842.0 0.0565416 0.0282708 0.999600i \(-0.491000\pi\)
0.0282708 + 0.999600i \(0.491000\pi\)
\(152\) 9365.79 0.0328803
\(153\) 28818.7 0.0995282
\(154\) 178886. 0.607820
\(155\) 60650.6 0.202771
\(156\) −294010. −0.967278
\(157\) 219032. 0.709184 0.354592 0.935021i \(-0.384620\pi\)
0.354592 + 0.935021i \(0.384620\pi\)
\(158\) 516158. 1.64490
\(159\) −207045. −0.649491
\(160\) 25838.7 0.0797941
\(161\) 44478.6 0.135234
\(162\) −716543. −2.14513
\(163\) 541493. 1.59634 0.798168 0.602435i \(-0.205803\pi\)
0.798168 + 0.602435i \(0.205803\pi\)
\(164\) −439909. −1.27718
\(165\) −103937. −0.297208
\(166\) −551287. −1.55277
\(167\) 360046. 0.999003 0.499502 0.866313i \(-0.333516\pi\)
0.499502 + 0.866313i \(0.333516\pi\)
\(168\) 469281. 1.28280
\(169\) −309420. −0.833358
\(170\) −59767.1 −0.158613
\(171\) 3727.17 0.00974740
\(172\) 556566. 1.43448
\(173\) −274433. −0.697143 −0.348571 0.937282i \(-0.613333\pi\)
−0.348571 + 0.937282i \(0.613333\pi\)
\(174\) −957922. −2.39860
\(175\) 52550.3 0.129712
\(176\) 189059. 0.460062
\(177\) 72291.9 0.173443
\(178\) 1.12715e6 2.66644
\(179\) 261807. 0.610730 0.305365 0.952235i \(-0.401222\pi\)
0.305365 + 0.952235i \(0.401222\pi\)
\(180\) −182287. −0.419348
\(181\) 359209. 0.814987 0.407493 0.913208i \(-0.366403\pi\)
0.407493 + 0.913208i \(0.366403\pi\)
\(182\) −203087. −0.454469
\(183\) 593732. 1.31058
\(184\) 155600. 0.338817
\(185\) 30465.4 0.0654453
\(186\) −447008. −0.947399
\(187\) 53942.6 0.112805
\(188\) 639279. 1.31916
\(189\) −200938. −0.409174
\(190\) −7729.77 −0.0155340
\(191\) 640678. 1.27074 0.635370 0.772208i \(-0.280848\pi\)
0.635370 + 0.772208i \(0.280848\pi\)
\(192\) −714383. −1.39855
\(193\) 99379.5 0.192045 0.0960226 0.995379i \(-0.469388\pi\)
0.0960226 + 0.995379i \(0.469388\pi\)
\(194\) −935963. −1.78548
\(195\) 117998. 0.222223
\(196\) −606559. −1.12780
\(197\) −378446. −0.694766 −0.347383 0.937723i \(-0.612930\pi\)
−0.347383 + 0.937723i \(0.612930\pi\)
\(198\) 249040. 0.451447
\(199\) 763816. 1.36728 0.683638 0.729821i \(-0.260397\pi\)
0.683638 + 0.729821i \(0.260397\pi\)
\(200\) 183837. 0.324982
\(201\) 146039. 0.254964
\(202\) −181101. −0.312278
\(203\) −437125. −0.744502
\(204\) 291003. 0.489579
\(205\) 176553. 0.293421
\(206\) −555559. −0.912142
\(207\) 61921.9 0.100443
\(208\) −214636. −0.343990
\(209\) 6976.48 0.0110477
\(210\) −387307. −0.606048
\(211\) 255573. 0.395192 0.197596 0.980284i \(-0.436687\pi\)
0.197596 + 0.980284i \(0.436687\pi\)
\(212\) −679687. −1.03865
\(213\) 1.36600e6 2.06301
\(214\) 1.06398e6 1.58818
\(215\) −223372. −0.329559
\(216\) −702943. −1.02515
\(217\) −203982. −0.294064
\(218\) −595516. −0.848697
\(219\) −1.15487e6 −1.62714
\(220\) −341203. −0.475287
\(221\) −61240.3 −0.0843445
\(222\) −224537. −0.305777
\(223\) −449097. −0.604752 −0.302376 0.953189i \(-0.597780\pi\)
−0.302376 + 0.953189i \(0.597780\pi\)
\(224\) −86901.2 −0.115719
\(225\) 73159.1 0.0963412
\(226\) −2.40780e6 −3.13581
\(227\) −381030. −0.490789 −0.245394 0.969423i \(-0.578917\pi\)
−0.245394 + 0.969423i \(0.578917\pi\)
\(228\) 37635.9 0.0479474
\(229\) −1.20514e6 −1.51862 −0.759311 0.650728i \(-0.774464\pi\)
−0.759311 + 0.650728i \(0.774464\pi\)
\(230\) −128420. −0.160071
\(231\) 349563. 0.431018
\(232\) −1.52920e6 −1.86528
\(233\) −792519. −0.956356 −0.478178 0.878263i \(-0.658703\pi\)
−0.478178 + 0.878263i \(0.658703\pi\)
\(234\) −282732. −0.337548
\(235\) −256569. −0.303064
\(236\) 237319. 0.277366
\(237\) 1.00863e6 1.16643
\(238\) 201010. 0.230025
\(239\) −185603. −0.210180 −0.105090 0.994463i \(-0.533513\pi\)
−0.105090 + 0.994463i \(0.533513\pi\)
\(240\) −409333. −0.458721
\(241\) −510987. −0.566718 −0.283359 0.959014i \(-0.591449\pi\)
−0.283359 + 0.959014i \(0.591449\pi\)
\(242\) −1.09771e6 −1.20490
\(243\) −819473. −0.890264
\(244\) 1.94910e6 2.09585
\(245\) 243437. 0.259102
\(246\) −1.30123e6 −1.37094
\(247\) −7920.31 −0.00826037
\(248\) −713590. −0.736749
\(249\) −1.07727e6 −1.10110
\(250\) −151725. −0.153534
\(251\) 651645. 0.652870 0.326435 0.945220i \(-0.394153\pi\)
0.326435 + 0.945220i \(0.394153\pi\)
\(252\) 613072. 0.608149
\(253\) 115905. 0.113841
\(254\) 2.93478e6 2.85424
\(255\) −116791. −0.112476
\(256\) −2.02401e6 −1.93025
\(257\) 803251. 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(258\) 1.64630e6 1.53979
\(259\) −102462. −0.0949104
\(260\) 387363. 0.355374
\(261\) −608554. −0.552965
\(262\) 948266. 0.853447
\(263\) −189088. −0.168568 −0.0842841 0.996442i \(-0.526860\pi\)
−0.0842841 + 0.996442i \(0.526860\pi\)
\(264\) 1.22288e6 1.07988
\(265\) 272786. 0.238620
\(266\) 25996.9 0.0225278
\(267\) 2.20257e6 1.89082
\(268\) 479415. 0.407732
\(269\) −898302. −0.756905 −0.378453 0.925621i \(-0.623544\pi\)
−0.378453 + 0.925621i \(0.623544\pi\)
\(270\) 580153. 0.484321
\(271\) −2.06767e6 −1.71024 −0.855122 0.518427i \(-0.826518\pi\)
−0.855122 + 0.518427i \(0.826518\pi\)
\(272\) 212441. 0.174107
\(273\) −396854. −0.322273
\(274\) 3.92288e6 3.15666
\(275\) 136938. 0.109193
\(276\) 625269. 0.494077
\(277\) 1.80297e6 1.41185 0.705927 0.708284i \(-0.250530\pi\)
0.705927 + 0.708284i \(0.250530\pi\)
\(278\) 3.37644e6 2.62028
\(279\) −283977. −0.218410
\(280\) −618286. −0.471296
\(281\) 1.05239e6 0.795081 0.397540 0.917585i \(-0.369864\pi\)
0.397540 + 0.917585i \(0.369864\pi\)
\(282\) 1.89096e6 1.41599
\(283\) −1.44912e6 −1.07557 −0.537784 0.843082i \(-0.680739\pi\)
−0.537784 + 0.843082i \(0.680739\pi\)
\(284\) 4.48430e6 3.29912
\(285\) −15104.8 −0.0110155
\(286\) −529215. −0.382576
\(287\) −593787. −0.425526
\(288\) −120981. −0.0859484
\(289\) −1.35924e6 −0.957310
\(290\) 1.26208e6 0.881234
\(291\) −1.82897e6 −1.26612
\(292\) −3.79121e6 −2.60208
\(293\) 1.44959e6 0.986450 0.493225 0.869902i \(-0.335818\pi\)
0.493225 + 0.869902i \(0.335818\pi\)
\(294\) −1.79418e6 −1.21059
\(295\) −95245.7 −0.0637221
\(296\) −358444. −0.237789
\(297\) −523615. −0.344446
\(298\) 981925. 0.640527
\(299\) −131585. −0.0851195
\(300\) 738740. 0.473902
\(301\) 751251. 0.477935
\(302\) 153832. 0.0970574
\(303\) −353890. −0.221443
\(304\) 27475.3 0.0170514
\(305\) −782252. −0.481501
\(306\) 279841. 0.170847
\(307\) 2.03959e6 1.23508 0.617542 0.786538i \(-0.288129\pi\)
0.617542 + 0.786538i \(0.288129\pi\)
\(308\) 1.14754e6 0.689274
\(309\) −1.08562e6 −0.646819
\(310\) 588940. 0.348070
\(311\) 650243. 0.381219 0.190610 0.981666i \(-0.438954\pi\)
0.190610 + 0.981666i \(0.438954\pi\)
\(312\) −1.38832e6 −0.807426
\(313\) −716801. −0.413559 −0.206780 0.978388i \(-0.566298\pi\)
−0.206780 + 0.978388i \(0.566298\pi\)
\(314\) 2.12689e6 1.21736
\(315\) −246050. −0.139716
\(316\) 3.31111e6 1.86533
\(317\) −869542. −0.486007 −0.243004 0.970025i \(-0.578133\pi\)
−0.243004 + 0.970025i \(0.578133\pi\)
\(318\) −2.01049e6 −1.11489
\(319\) −1.13908e6 −0.626728
\(320\) 941210. 0.513821
\(321\) 2.07913e6 1.12621
\(322\) 431904. 0.232139
\(323\) 7839.30 0.00418091
\(324\) −4.59657e6 −2.43260
\(325\) −155464. −0.0816437
\(326\) 5.25810e6 2.74022
\(327\) −1.16370e6 −0.601828
\(328\) −2.07725e6 −1.06612
\(329\) 862897. 0.439510
\(330\) −1.00927e6 −0.510177
\(331\) −1.68564e6 −0.845659 −0.422829 0.906209i \(-0.638963\pi\)
−0.422829 + 0.906209i \(0.638963\pi\)
\(332\) −3.53646e6 −1.76086
\(333\) −142645. −0.0704929
\(334\) 3.49618e6 1.71486
\(335\) −192408. −0.0936725
\(336\) 1.37668e6 0.665248
\(337\) −1.64864e6 −0.790771 −0.395385 0.918515i \(-0.629389\pi\)
−0.395385 + 0.918515i \(0.629389\pi\)
\(338\) −3.00458e6 −1.43051
\(339\) −4.70511e6 −2.22367
\(340\) −383401. −0.179869
\(341\) −531546. −0.247545
\(342\) 36192.2 0.0167321
\(343\) −2.23187e6 −1.02432
\(344\) 2.62811e6 1.19742
\(345\) −250946. −0.113509
\(346\) −2.66485e6 −1.19669
\(347\) 1.65489e6 0.737810 0.368905 0.929467i \(-0.379733\pi\)
0.368905 + 0.929467i \(0.379733\pi\)
\(348\) −6.14500e6 −2.72003
\(349\) −2.77967e6 −1.22160 −0.610801 0.791784i \(-0.709153\pi\)
−0.610801 + 0.791784i \(0.709153\pi\)
\(350\) 510283. 0.222659
\(351\) 594453. 0.257543
\(352\) −226452. −0.0974135
\(353\) −2.23694e6 −0.955471 −0.477735 0.878504i \(-0.658542\pi\)
−0.477735 + 0.878504i \(0.658542\pi\)
\(354\) 701981. 0.297726
\(355\) −1.79973e6 −0.757942
\(356\) 7.23057e6 3.02376
\(357\) 392795. 0.163116
\(358\) 2.54225e6 1.04836
\(359\) 1.93471e6 0.792282 0.396141 0.918190i \(-0.370349\pi\)
0.396141 + 0.918190i \(0.370349\pi\)
\(360\) −860760. −0.350046
\(361\) −2.47509e6 −0.999591
\(362\) 3.48805e6 1.39898
\(363\) −2.14505e6 −0.854419
\(364\) −1.30279e6 −0.515372
\(365\) 1.52157e6 0.597804
\(366\) 5.76536e6 2.24970
\(367\) −3.42337e6 −1.32675 −0.663374 0.748288i \(-0.730876\pi\)
−0.663374 + 0.748288i \(0.730876\pi\)
\(368\) 456466. 0.175707
\(369\) −826654. −0.316051
\(370\) 295831. 0.112341
\(371\) −917439. −0.346053
\(372\) −2.86752e6 −1.07436
\(373\) 1.54365e6 0.574483 0.287242 0.957858i \(-0.407262\pi\)
0.287242 + 0.957858i \(0.407262\pi\)
\(374\) 523803. 0.193637
\(375\) −296486. −0.108874
\(376\) 3.01868e6 1.10115
\(377\) 1.29319e6 0.468606
\(378\) −1.95118e6 −0.702374
\(379\) 517292. 0.184985 0.0924927 0.995713i \(-0.470517\pi\)
0.0924927 + 0.995713i \(0.470517\pi\)
\(380\) −49585.9 −0.0176157
\(381\) 5.73487e6 2.02400
\(382\) 6.22123e6 2.18131
\(383\) −1.12499e6 −0.391877 −0.195939 0.980616i \(-0.562775\pi\)
−0.195939 + 0.980616i \(0.562775\pi\)
\(384\) −6.30935e6 −2.18352
\(385\) −460555. −0.158354
\(386\) 965012. 0.329659
\(387\) 1.04587e6 0.354977
\(388\) −6.00413e6 −2.02475
\(389\) 1.85887e6 0.622838 0.311419 0.950273i \(-0.399196\pi\)
0.311419 + 0.950273i \(0.399196\pi\)
\(390\) 1.14581e6 0.381461
\(391\) 130239. 0.0430824
\(392\) −2.86417e6 −0.941422
\(393\) 1.85301e6 0.605197
\(394\) −3.67485e6 −1.19261
\(395\) −1.32888e6 −0.428542
\(396\) 1.59757e6 0.511945
\(397\) 3.27369e6 1.04246 0.521232 0.853415i \(-0.325473\pi\)
0.521232 + 0.853415i \(0.325473\pi\)
\(398\) 7.41694e6 2.34702
\(399\) 50800.8 0.0159749
\(400\) 539302. 0.168532
\(401\) −92661.9 −0.0287766 −0.0143883 0.999896i \(-0.504580\pi\)
−0.0143883 + 0.999896i \(0.504580\pi\)
\(402\) 1.41809e6 0.437662
\(403\) 603457. 0.185090
\(404\) −1.16175e6 −0.354127
\(405\) 1.84479e6 0.558867
\(406\) −4.24465e6 −1.27799
\(407\) −267001. −0.0798964
\(408\) 1.37412e6 0.408671
\(409\) −4.09923e6 −1.21170 −0.605849 0.795580i \(-0.707166\pi\)
−0.605849 + 0.795580i \(0.707166\pi\)
\(410\) 1.71440e6 0.503676
\(411\) 7.66572e6 2.23846
\(412\) −3.56387e6 −1.03438
\(413\) 320332. 0.0924114
\(414\) 601284. 0.172417
\(415\) 1.41933e6 0.404540
\(416\) 257088. 0.0728364
\(417\) 6.59793e6 1.85809
\(418\) 67744.2 0.0189641
\(419\) −6.11636e6 −1.70199 −0.850996 0.525171i \(-0.824001\pi\)
−0.850996 + 0.525171i \(0.824001\pi\)
\(420\) −2.48455e6 −0.687264
\(421\) −3.46820e6 −0.953671 −0.476836 0.878992i \(-0.658216\pi\)
−0.476836 + 0.878992i \(0.658216\pi\)
\(422\) 2.48171e6 0.678374
\(423\) 1.20130e6 0.326438
\(424\) −3.20948e6 −0.867003
\(425\) 153874. 0.0413232
\(426\) 1.32644e7 3.54130
\(427\) 2.63089e6 0.698284
\(428\) 6.82535e6 1.80101
\(429\) −1.03414e6 −0.271292
\(430\) −2.16903e6 −0.565710
\(431\) −2.18327e6 −0.566127 −0.283064 0.959101i \(-0.591351\pi\)
−0.283064 + 0.959101i \(0.591351\pi\)
\(432\) −2.06214e6 −0.531630
\(433\) 4.15213e6 1.06427 0.532135 0.846660i \(-0.321390\pi\)
0.532135 + 0.846660i \(0.321390\pi\)
\(434\) −1.98074e6 −0.504780
\(435\) 2.46624e6 0.624901
\(436\) −3.82019e6 −0.962430
\(437\) 16844.0 0.00421932
\(438\) −1.12143e7 −2.79309
\(439\) −3.98317e6 −0.986432 −0.493216 0.869907i \(-0.664179\pi\)
−0.493216 + 0.869907i \(0.664179\pi\)
\(440\) −1.61116e6 −0.396741
\(441\) −1.13981e6 −0.279086
\(442\) −594666. −0.144783
\(443\) 5.08171e6 1.23027 0.615135 0.788422i \(-0.289102\pi\)
0.615135 + 0.788422i \(0.289102\pi\)
\(444\) −1.44039e6 −0.346754
\(445\) −2.90192e6 −0.694681
\(446\) −4.36089e6 −1.03810
\(447\) 1.91879e6 0.454211
\(448\) −3.16550e6 −0.745156
\(449\) 6.32873e6 1.48150 0.740749 0.671782i \(-0.234471\pi\)
0.740749 + 0.671782i \(0.234471\pi\)
\(450\) 710402. 0.165376
\(451\) −1.54732e6 −0.358211
\(452\) −1.54459e7 −3.55604
\(453\) 300604. 0.0688255
\(454\) −3.69994e6 −0.842472
\(455\) 522862. 0.118402
\(456\) 177717. 0.0400236
\(457\) 2.82211e6 0.632097 0.316048 0.948743i \(-0.397644\pi\)
0.316048 + 0.948743i \(0.397644\pi\)
\(458\) −1.17024e7 −2.60682
\(459\) −588373. −0.130353
\(460\) −823802. −0.181522
\(461\) 76172.5 0.0166935 0.00834673 0.999965i \(-0.497343\pi\)
0.00834673 + 0.999965i \(0.497343\pi\)
\(462\) 3.39438e6 0.739871
\(463\) −8.27628e6 −1.79425 −0.897125 0.441777i \(-0.854348\pi\)
−0.897125 + 0.441777i \(0.854348\pi\)
\(464\) −4.48604e6 −0.967315
\(465\) 1.15085e6 0.246824
\(466\) −7.69565e6 −1.64165
\(467\) −6.55871e6 −1.39164 −0.695819 0.718217i \(-0.744958\pi\)
−0.695819 + 0.718217i \(0.744958\pi\)
\(468\) −1.81371e6 −0.382783
\(469\) 647112. 0.135846
\(470\) −2.49138e6 −0.520229
\(471\) 4.15616e6 0.863257
\(472\) 1.12062e6 0.231528
\(473\) 1.95765e6 0.402330
\(474\) 9.79414e6 2.00226
\(475\) 19900.8 0.00404703
\(476\) 1.28946e6 0.260851
\(477\) −1.27723e6 −0.257024
\(478\) −1.80228e6 −0.360788
\(479\) 6.69118e6 1.33249 0.666245 0.745733i \(-0.267901\pi\)
0.666245 + 0.745733i \(0.267901\pi\)
\(480\) 490292. 0.0971296
\(481\) 303123. 0.0597387
\(482\) −4.96187e6 −0.972810
\(483\) 843986. 0.164614
\(484\) −7.04175e6 −1.36637
\(485\) 2.40970e6 0.465167
\(486\) −7.95739e6 −1.52820
\(487\) 3.76173e6 0.718729 0.359364 0.933197i \(-0.382994\pi\)
0.359364 + 0.933197i \(0.382994\pi\)
\(488\) 9.20365e6 1.74949
\(489\) 1.02749e7 1.94314
\(490\) 2.36386e6 0.444766
\(491\) −2.46265e6 −0.460997 −0.230499 0.973073i \(-0.574036\pi\)
−0.230499 + 0.973073i \(0.574036\pi\)
\(492\) −8.34732e6 −1.55466
\(493\) −1.27996e6 −0.237181
\(494\) −76909.1 −0.0141795
\(495\) −641171. −0.117615
\(496\) −2.09338e6 −0.382070
\(497\) 6.05289e6 1.09919
\(498\) −1.04607e7 −1.89012
\(499\) 2.03788e6 0.366375 0.183188 0.983078i \(-0.441358\pi\)
0.183188 + 0.983078i \(0.441358\pi\)
\(500\) −973301. −0.174109
\(501\) 6.83191e6 1.21604
\(502\) 6.32771e6 1.12069
\(503\) 4.22596e6 0.744742 0.372371 0.928084i \(-0.378545\pi\)
0.372371 + 0.928084i \(0.378545\pi\)
\(504\) 2.89493e6 0.507646
\(505\) 466256. 0.0813572
\(506\) 1.12548e6 0.195416
\(507\) −5.87127e6 −1.01441
\(508\) 1.88264e7 3.23674
\(509\) −3.53097e6 −0.604087 −0.302044 0.953294i \(-0.597669\pi\)
−0.302044 + 0.953294i \(0.597669\pi\)
\(510\) −1.13409e6 −0.193073
\(511\) −5.11736e6 −0.866950
\(512\) −9.01370e6 −1.51960
\(513\) −76095.2 −0.0127663
\(514\) 7.79986e6 1.30220
\(515\) 1.43032e6 0.237638
\(516\) 1.05609e7 1.74613
\(517\) 2.24858e6 0.369984
\(518\) −994944. −0.162920
\(519\) −5.20740e6 −0.848599
\(520\) 1.82913e6 0.296645
\(521\) −6.66348e6 −1.07549 −0.537746 0.843107i \(-0.680724\pi\)
−0.537746 + 0.843107i \(0.680724\pi\)
\(522\) −5.90928e6 −0.949201
\(523\) −6.77352e6 −1.08283 −0.541415 0.840755i \(-0.682111\pi\)
−0.541415 + 0.840755i \(0.682111\pi\)
\(524\) 6.08305e6 0.967817
\(525\) 997148. 0.157892
\(526\) −1.83612e6 −0.289359
\(527\) −597285. −0.0936818
\(528\) 3.58742e6 0.560012
\(529\) 279841. 0.0434783
\(530\) 2.64885e6 0.409607
\(531\) 445958. 0.0686369
\(532\) 166768. 0.0255467
\(533\) 1.75665e6 0.267836
\(534\) 2.13878e7 3.24573
\(535\) −2.73929e6 −0.413765
\(536\) 2.26380e6 0.340350
\(537\) 4.96782e6 0.743413
\(538\) −8.72284e6 −1.29928
\(539\) −2.13349e6 −0.316315
\(540\) 3.72164e6 0.549224
\(541\) 5.07069e6 0.744859 0.372429 0.928061i \(-0.378525\pi\)
0.372429 + 0.928061i \(0.378525\pi\)
\(542\) −2.00778e7 −2.93575
\(543\) 6.81602e6 0.992045
\(544\) −254458. −0.0368655
\(545\) 1.53320e6 0.221109
\(546\) −3.85360e6 −0.553204
\(547\) −1.25719e7 −1.79652 −0.898259 0.439466i \(-0.855168\pi\)
−0.898259 + 0.439466i \(0.855168\pi\)
\(548\) 2.51650e7 3.57969
\(549\) 3.66264e6 0.518637
\(550\) 1.32972e6 0.187437
\(551\) −165539. −0.0232286
\(552\) 2.95252e6 0.412426
\(553\) 4.46933e6 0.621483
\(554\) 1.75075e7 2.42354
\(555\) 578085. 0.0796635
\(556\) 2.16596e7 2.97142
\(557\) −1.21618e7 −1.66097 −0.830484 0.557042i \(-0.811936\pi\)
−0.830484 + 0.557042i \(0.811936\pi\)
\(558\) −2.75752e6 −0.374916
\(559\) −2.22249e6 −0.300823
\(560\) −1.81379e6 −0.244409
\(561\) 1.02357e6 0.137312
\(562\) 1.02191e7 1.36481
\(563\) −4.47399e6 −0.594873 −0.297436 0.954742i \(-0.596132\pi\)
−0.297436 + 0.954742i \(0.596132\pi\)
\(564\) 1.21304e7 1.60575
\(565\) 6.19905e6 0.816967
\(566\) −1.40715e7 −1.84629
\(567\) −6.20443e6 −0.810483
\(568\) 2.11749e7 2.75391
\(569\) −1.20581e7 −1.56135 −0.780674 0.624938i \(-0.785124\pi\)
−0.780674 + 0.624938i \(0.785124\pi\)
\(570\) −146673. −0.0189088
\(571\) −1.18705e7 −1.52363 −0.761814 0.647796i \(-0.775691\pi\)
−0.761814 + 0.647796i \(0.775691\pi\)
\(572\) −3.39488e6 −0.433844
\(573\) 1.21569e7 1.54681
\(574\) −5.76589e6 −0.730444
\(575\) 330625. 0.0417029
\(576\) −4.40692e6 −0.553451
\(577\) 3.10897e6 0.388756 0.194378 0.980927i \(-0.437731\pi\)
0.194378 + 0.980927i \(0.437731\pi\)
\(578\) −1.31988e7 −1.64329
\(579\) 1.88574e6 0.233768
\(580\) 8.09614e6 0.999327
\(581\) −4.77351e6 −0.586674
\(582\) −1.77600e7 −2.17338
\(583\) −2.39071e6 −0.291310
\(584\) −1.79021e7 −2.17206
\(585\) 727913. 0.0879407
\(586\) 1.40760e7 1.69331
\(587\) −7.66593e6 −0.918269 −0.459134 0.888367i \(-0.651840\pi\)
−0.459134 + 0.888367i \(0.651840\pi\)
\(588\) −1.15095e7 −1.37282
\(589\) −77247.8 −0.00917483
\(590\) −924871. −0.109383
\(591\) −7.18105e6 −0.845706
\(592\) −1.05153e6 −0.123315
\(593\) −1.04472e6 −0.122001 −0.0610004 0.998138i \(-0.519429\pi\)
−0.0610004 + 0.998138i \(0.519429\pi\)
\(594\) −5.08450e6 −0.591264
\(595\) −517514. −0.0599280
\(596\) 6.29897e6 0.726364
\(597\) 1.44935e7 1.66432
\(598\) −1.27774e6 −0.146113
\(599\) −4.05765e6 −0.462070 −0.231035 0.972945i \(-0.574211\pi\)
−0.231035 + 0.972945i \(0.574211\pi\)
\(600\) 3.48833e6 0.395585
\(601\) −8.30225e6 −0.937582 −0.468791 0.883309i \(-0.655310\pi\)
−0.468791 + 0.883309i \(0.655310\pi\)
\(602\) 7.29492e6 0.820407
\(603\) 900891. 0.100897
\(604\) 986820. 0.110064
\(605\) 2.82614e6 0.313910
\(606\) −3.43641e6 −0.380122
\(607\) 7.44097e6 0.819705 0.409853 0.912152i \(-0.365580\pi\)
0.409853 + 0.912152i \(0.365580\pi\)
\(608\) −32909.5 −0.00361046
\(609\) −8.29450e6 −0.906248
\(610\) −7.59595e6 −0.826528
\(611\) −2.55279e6 −0.276638
\(612\) 1.79516e6 0.193742
\(613\) 7.25848e6 0.780179 0.390090 0.920777i \(-0.372444\pi\)
0.390090 + 0.920777i \(0.372444\pi\)
\(614\) 1.98052e7 2.12010
\(615\) 3.35011e6 0.357167
\(616\) 5.41869e6 0.575364
\(617\) 3.29613e6 0.348571 0.174286 0.984695i \(-0.444238\pi\)
0.174286 + 0.984695i \(0.444238\pi\)
\(618\) −1.05418e7 −1.11031
\(619\) −3.57956e6 −0.375494 −0.187747 0.982217i \(-0.560119\pi\)
−0.187747 + 0.982217i \(0.560119\pi\)
\(620\) 3.77801e6 0.394715
\(621\) −1.26422e6 −0.131551
\(622\) 6.31410e6 0.654388
\(623\) 9.75980e6 1.00744
\(624\) −4.07275e6 −0.418722
\(625\) 390625. 0.0400000
\(626\) −6.96040e6 −0.709902
\(627\) 132379. 0.0134478
\(628\) 1.36438e7 1.38050
\(629\) −300023. −0.0302362
\(630\) −2.38924e6 −0.239833
\(631\) 1.45987e7 1.45963 0.729814 0.683646i \(-0.239607\pi\)
0.729814 + 0.683646i \(0.239607\pi\)
\(632\) 1.56351e7 1.55707
\(633\) 4.84952e6 0.481049
\(634\) −8.44358e6 −0.834264
\(635\) −7.55578e6 −0.743609
\(636\) −1.28971e7 −1.26430
\(637\) 2.42213e6 0.236509
\(638\) −1.10609e7 −1.07582
\(639\) 8.42666e6 0.816401
\(640\) 8.31266e6 0.802214
\(641\) 1.63500e7 1.57171 0.785857 0.618409i \(-0.212222\pi\)
0.785857 + 0.618409i \(0.212222\pi\)
\(642\) 2.01891e7 1.93322
\(643\) 3.67043e6 0.350098 0.175049 0.984560i \(-0.443992\pi\)
0.175049 + 0.984560i \(0.443992\pi\)
\(644\) 2.77063e6 0.263247
\(645\) −4.23851e6 −0.401157
\(646\) 76122.5 0.00717681
\(647\) 972667. 0.0913490 0.0456745 0.998956i \(-0.485456\pi\)
0.0456745 + 0.998956i \(0.485456\pi\)
\(648\) −2.17050e7 −2.03059
\(649\) 834739. 0.0777928
\(650\) −1.50962e6 −0.140147
\(651\) −3.87057e6 −0.357950
\(652\) 3.37303e7 3.10743
\(653\) 1.36330e7 1.25115 0.625575 0.780164i \(-0.284864\pi\)
0.625575 + 0.780164i \(0.284864\pi\)
\(654\) −1.13000e7 −1.03308
\(655\) −2.44137e6 −0.222347
\(656\) −6.09379e6 −0.552876
\(657\) −7.12425e6 −0.643911
\(658\) 8.37905e6 0.754449
\(659\) −6.41148e6 −0.575102 −0.287551 0.957765i \(-0.592841\pi\)
−0.287551 + 0.957765i \(0.592841\pi\)
\(660\) −6.47436e6 −0.578545
\(661\) −1.50140e7 −1.33657 −0.668285 0.743905i \(-0.732971\pi\)
−0.668285 + 0.743905i \(0.732971\pi\)
\(662\) −1.63682e7 −1.45163
\(663\) −1.16204e6 −0.102669
\(664\) −1.66992e7 −1.46986
\(665\) −66930.8 −0.00586911
\(666\) −1.38513e6 −0.121006
\(667\) −2.75021e6 −0.239360
\(668\) 2.24277e7 1.94466
\(669\) −8.52165e6 −0.736137
\(670\) −1.86836e6 −0.160795
\(671\) 6.85570e6 0.587822
\(672\) −1.64896e6 −0.140860
\(673\) −4.45498e6 −0.379148 −0.189574 0.981866i \(-0.560711\pi\)
−0.189574 + 0.981866i \(0.560711\pi\)
\(674\) −1.60089e7 −1.35741
\(675\) −1.49364e6 −0.126179
\(676\) −1.92742e7 −1.62222
\(677\) 2.22147e7 1.86281 0.931406 0.363981i \(-0.118583\pi\)
0.931406 + 0.363981i \(0.118583\pi\)
\(678\) −4.56883e7 −3.81708
\(679\) −8.10435e6 −0.674596
\(680\) −1.81042e6 −0.150144
\(681\) −7.23008e6 −0.597414
\(682\) −5.16151e6 −0.424928
\(683\) 1.84077e7 1.50990 0.754948 0.655785i \(-0.227662\pi\)
0.754948 + 0.655785i \(0.227662\pi\)
\(684\) 232170. 0.0189743
\(685\) −1.00997e7 −0.822399
\(686\) −2.16723e7 −1.75831
\(687\) −2.28677e7 −1.84855
\(688\) 7.70978e6 0.620970
\(689\) 2.71414e6 0.217813
\(690\) −2.43678e6 −0.194847
\(691\) −1.19728e6 −0.0953894 −0.0476947 0.998862i \(-0.515187\pi\)
−0.0476947 + 0.998862i \(0.515187\pi\)
\(692\) −1.70948e7 −1.35706
\(693\) 2.15640e6 0.170568
\(694\) 1.60696e7 1.26650
\(695\) −8.69288e6 −0.682655
\(696\) −2.90167e7 −2.27052
\(697\) −1.73869e6 −0.135563
\(698\) −2.69917e7 −2.09696
\(699\) −1.50381e7 −1.16413
\(700\) 3.27343e6 0.252498
\(701\) 1.74390e7 1.34038 0.670188 0.742191i \(-0.266213\pi\)
0.670188 + 0.742191i \(0.266213\pi\)
\(702\) 5.77236e6 0.442090
\(703\) −38802.4 −0.00296121
\(704\) −8.24883e6 −0.627279
\(705\) −4.86841e6 −0.368905
\(706\) −2.17215e7 −1.64013
\(707\) −1.56812e6 −0.117986
\(708\) 4.50315e6 0.337624
\(709\) −1.84583e7 −1.37904 −0.689520 0.724267i \(-0.742178\pi\)
−0.689520 + 0.724267i \(0.742178\pi\)
\(710\) −1.74760e7 −1.30106
\(711\) 6.22207e6 0.461595
\(712\) 3.41428e7 2.52406
\(713\) −1.28337e6 −0.0945425
\(714\) 3.81419e6 0.279999
\(715\) 1.36250e6 0.0996716
\(716\) 1.63083e7 1.18885
\(717\) −3.52184e6 −0.255842
\(718\) 1.87868e7 1.36001
\(719\) 2.12129e7 1.53030 0.765152 0.643849i \(-0.222664\pi\)
0.765152 + 0.643849i \(0.222664\pi\)
\(720\) −2.52511e6 −0.181530
\(721\) −4.81050e6 −0.344629
\(722\) −2.40340e7 −1.71586
\(723\) −9.69603e6 −0.689840
\(724\) 2.23756e7 1.58646
\(725\) −3.24930e6 −0.229586
\(726\) −2.08292e7 −1.46667
\(727\) 1.69692e7 1.19077 0.595383 0.803442i \(-0.297000\pi\)
0.595383 + 0.803442i \(0.297000\pi\)
\(728\) −6.15177e6 −0.430201
\(729\) 2.38175e6 0.165988
\(730\) 1.47750e7 1.02617
\(731\) 2.19976e6 0.152259
\(732\) 3.69843e7 2.55117
\(733\) 903452. 0.0621077 0.0310538 0.999518i \(-0.490114\pi\)
0.0310538 + 0.999518i \(0.490114\pi\)
\(734\) −3.32422e7 −2.27745
\(735\) 4.61923e6 0.315393
\(736\) −546747. −0.0372042
\(737\) 1.68628e6 0.114357
\(738\) −8.02711e6 −0.542523
\(739\) 386739. 0.0260499 0.0130250 0.999915i \(-0.495854\pi\)
0.0130250 + 0.999915i \(0.495854\pi\)
\(740\) 1.89773e6 0.127396
\(741\) −150289. −0.0100550
\(742\) −8.90867e6 −0.594023
\(743\) 1.90773e7 1.26779 0.633893 0.773421i \(-0.281456\pi\)
0.633893 + 0.773421i \(0.281456\pi\)
\(744\) −1.35404e7 −0.896811
\(745\) −2.52803e6 −0.166875
\(746\) 1.49894e7 0.986139
\(747\) −6.64554e6 −0.435741
\(748\) 3.36016e6 0.219586
\(749\) 9.21284e6 0.600052
\(750\) −2.87899e6 −0.186890
\(751\) 1.92271e7 1.24398 0.621990 0.783025i \(-0.286324\pi\)
0.621990 + 0.783025i \(0.286324\pi\)
\(752\) 8.85555e6 0.571046
\(753\) 1.23650e7 0.794708
\(754\) 1.25573e7 0.804394
\(755\) −396050. −0.0252862
\(756\) −1.25167e7 −0.796499
\(757\) 5.23395e6 0.331963 0.165982 0.986129i \(-0.446921\pi\)
0.165982 + 0.986129i \(0.446921\pi\)
\(758\) 5.02309e6 0.317540
\(759\) 2.19930e6 0.138574
\(760\) −234145. −0.0147045
\(761\) −1.96724e7 −1.23139 −0.615694 0.787985i \(-0.711124\pi\)
−0.615694 + 0.787985i \(0.711124\pi\)
\(762\) 5.56877e7 3.47433
\(763\) −5.15648e6 −0.320658
\(764\) 3.99087e7 2.47363
\(765\) −720468. −0.0445104
\(766\) −1.09240e7 −0.672683
\(767\) −947669. −0.0581659
\(768\) −3.84058e7 −2.34960
\(769\) −2.05543e7 −1.25339 −0.626695 0.779264i \(-0.715593\pi\)
−0.626695 + 0.779264i \(0.715593\pi\)
\(770\) −4.47215e6 −0.271825
\(771\) 1.52418e7 0.923420
\(772\) 6.19048e6 0.373836
\(773\) 2.02610e7 1.21959 0.609793 0.792561i \(-0.291253\pi\)
0.609793 + 0.792561i \(0.291253\pi\)
\(774\) 1.01558e7 0.609342
\(775\) −1.51627e6 −0.0906820
\(776\) −2.83515e7 −1.69014
\(777\) −1.94423e6 −0.115530
\(778\) 1.80503e7 1.06914
\(779\) −224867. −0.0132765
\(780\) 7.35026e6 0.432580
\(781\) 1.57729e7 0.925305
\(782\) 1.26467e6 0.0739539
\(783\) 1.24244e7 0.724223
\(784\) −8.40230e6 −0.488211
\(785\) −5.47581e6 −0.317157
\(786\) 1.79934e7 1.03886
\(787\) −2.01612e7 −1.16032 −0.580162 0.814501i \(-0.697011\pi\)
−0.580162 + 0.814501i \(0.697011\pi\)
\(788\) −2.35739e7 −1.35243
\(789\) −3.58797e6 −0.205190
\(790\) −1.29039e7 −0.735622
\(791\) −2.08488e7 −1.18479
\(792\) 7.54375e6 0.427341
\(793\) −7.78319e6 −0.439516
\(794\) 3.17887e7 1.78946
\(795\) 5.17614e6 0.290461
\(796\) 4.75791e7 2.66154
\(797\) −9.93780e6 −0.554172 −0.277086 0.960845i \(-0.589369\pi\)
−0.277086 + 0.960845i \(0.589369\pi\)
\(798\) 493294. 0.0274220
\(799\) 2.52668e6 0.140018
\(800\) −645967. −0.0356850
\(801\) 1.35873e7 0.748260
\(802\) −899781. −0.0493970
\(803\) −1.33351e7 −0.729806
\(804\) 9.09695e6 0.496313
\(805\) −1.11197e6 −0.0604786
\(806\) 5.85979e6 0.317720
\(807\) −1.70454e7 −0.921345
\(808\) −5.48578e6 −0.295604
\(809\) 3.48213e7 1.87057 0.935284 0.353898i \(-0.115144\pi\)
0.935284 + 0.353898i \(0.115144\pi\)
\(810\) 1.79136e7 0.959333
\(811\) 7.77806e6 0.415259 0.207630 0.978208i \(-0.433425\pi\)
0.207630 + 0.978208i \(0.433425\pi\)
\(812\) −2.72291e7 −1.44925
\(813\) −3.92342e7 −2.08180
\(814\) −2.59268e6 −0.137147
\(815\) −1.35373e7 −0.713903
\(816\) 4.03109e6 0.211933
\(817\) 284499. 0.0149116
\(818\) −3.98051e7 −2.07996
\(819\) −2.44813e6 −0.127534
\(820\) 1.09977e7 0.571173
\(821\) 2.03211e7 1.05218 0.526089 0.850430i \(-0.323658\pi\)
0.526089 + 0.850430i \(0.323658\pi\)
\(822\) 7.44370e7 3.84246
\(823\) 2.14456e7 1.10367 0.551834 0.833954i \(-0.313928\pi\)
0.551834 + 0.833954i \(0.313928\pi\)
\(824\) −1.68286e7 −0.863436
\(825\) 2.59842e6 0.132915
\(826\) 3.11055e6 0.158630
\(827\) 2.88309e7 1.46587 0.732934 0.680300i \(-0.238150\pi\)
0.732934 + 0.680300i \(0.238150\pi\)
\(828\) 3.85719e6 0.195522
\(829\) −5.70528e6 −0.288330 −0.144165 0.989554i \(-0.546050\pi\)
−0.144165 + 0.989554i \(0.546050\pi\)
\(830\) 1.37822e7 0.694421
\(831\) 3.42116e7 1.71859
\(832\) 9.36479e6 0.469018
\(833\) −2.39735e6 −0.119707
\(834\) 6.40683e7 3.18954
\(835\) −9.00115e6 −0.446768
\(836\) 434574. 0.0215054
\(837\) 5.79778e6 0.286054
\(838\) −5.93921e7 −2.92159
\(839\) −952844. −0.0467323 −0.0233661 0.999727i \(-0.507438\pi\)
−0.0233661 + 0.999727i \(0.507438\pi\)
\(840\) −1.17320e7 −0.573687
\(841\) 6.51728e6 0.317743
\(842\) −3.36775e7 −1.63704
\(843\) 1.99692e7 0.967815
\(844\) 1.59200e7 0.769282
\(845\) 7.73550e6 0.372689
\(846\) 1.16651e7 0.560353
\(847\) −9.50493e6 −0.455240
\(848\) −9.41529e6 −0.449619
\(849\) −2.74972e7 −1.30924
\(850\) 1.49418e6 0.0709341
\(851\) −644649. −0.0305140
\(852\) 8.50900e7 4.01587
\(853\) −4.02321e7 −1.89321 −0.946607 0.322389i \(-0.895514\pi\)
−0.946607 + 0.322389i \(0.895514\pi\)
\(854\) 2.55469e7 1.19865
\(855\) −93179.2 −0.00435917
\(856\) 3.22293e7 1.50337
\(857\) −2.86224e7 −1.33123 −0.665617 0.746294i \(-0.731831\pi\)
−0.665617 + 0.746294i \(0.731831\pi\)
\(858\) −1.00419e7 −0.465692
\(859\) −3.61703e7 −1.67251 −0.836256 0.548339i \(-0.815260\pi\)
−0.836256 + 0.548339i \(0.815260\pi\)
\(860\) −1.39142e7 −0.641521
\(861\) −1.12672e7 −0.517973
\(862\) −2.12003e7 −0.971795
\(863\) 6.70416e6 0.306420 0.153210 0.988194i \(-0.451039\pi\)
0.153210 + 0.988194i \(0.451039\pi\)
\(864\) 2.47000e6 0.112567
\(865\) 6.86083e6 0.311772
\(866\) 4.03188e7 1.82689
\(867\) −2.57918e7 −1.16529
\(868\) −1.27063e7 −0.572425
\(869\) 1.16464e7 0.523170
\(870\) 2.39481e7 1.07269
\(871\) −1.91441e6 −0.0855047
\(872\) −1.80390e7 −0.803379
\(873\) −1.12827e7 −0.501044
\(874\) 163562. 0.00724275
\(875\) −1.31376e6 −0.0580090
\(876\) −7.19386e7 −3.16739
\(877\) 2.69344e7 1.18252 0.591261 0.806480i \(-0.298630\pi\)
0.591261 + 0.806480i \(0.298630\pi\)
\(878\) −3.86780e7 −1.69328
\(879\) 2.75061e7 1.20076
\(880\) −4.72648e6 −0.205746
\(881\) 8.29228e6 0.359943 0.179972 0.983672i \(-0.442399\pi\)
0.179972 + 0.983672i \(0.442399\pi\)
\(882\) −1.10680e7 −0.479069
\(883\) 2.48054e7 1.07064 0.535322 0.844648i \(-0.320190\pi\)
0.535322 + 0.844648i \(0.320190\pi\)
\(884\) −3.81474e6 −0.164185
\(885\) −1.80730e6 −0.0775660
\(886\) 4.93452e7 2.11184
\(887\) 3.33079e6 0.142147 0.0710736 0.997471i \(-0.477357\pi\)
0.0710736 + 0.997471i \(0.477357\pi\)
\(888\) −6.80151e6 −0.289450
\(889\) 2.54118e7 1.07840
\(890\) −2.81787e7 −1.19247
\(891\) −1.61678e7 −0.682272
\(892\) −2.79748e7 −1.17721
\(893\) 326779. 0.0137128
\(894\) 1.86321e7 0.779684
\(895\) −6.54518e6 −0.273127
\(896\) −2.79573e7 −1.16339
\(897\) −2.49684e6 −0.103612
\(898\) 6.14543e7 2.54309
\(899\) 1.26126e7 0.520483
\(900\) 4.55718e6 0.187538
\(901\) −2.68638e6 −0.110244
\(902\) −1.50251e7 −0.614894
\(903\) 1.42551e7 0.581768
\(904\) −7.29355e7 −2.96837
\(905\) −8.98022e6 −0.364473
\(906\) 2.91898e6 0.118144
\(907\) −3.87276e7 −1.56316 −0.781578 0.623807i \(-0.785585\pi\)
−0.781578 + 0.623807i \(0.785585\pi\)
\(908\) −2.37348e7 −0.955371
\(909\) −2.18310e6 −0.0876321
\(910\) 5.07718e6 0.203245
\(911\) 1.56365e7 0.624229 0.312114 0.950045i \(-0.398963\pi\)
0.312114 + 0.950045i \(0.398963\pi\)
\(912\) 521347. 0.0207558
\(913\) −1.24391e7 −0.493868
\(914\) 2.74037e7 1.08504
\(915\) −1.48433e7 −0.586108
\(916\) −7.50698e7 −2.95615
\(917\) 8.21088e6 0.322453
\(918\) −5.71332e6 −0.223760
\(919\) 6.11633e6 0.238892 0.119446 0.992841i \(-0.461888\pi\)
0.119446 + 0.992841i \(0.461888\pi\)
\(920\) −3.89000e6 −0.151523
\(921\) 3.87014e7 1.50341
\(922\) 739664. 0.0286554
\(923\) −1.79068e7 −0.691853
\(924\) 2.17747e7 0.839021
\(925\) −761636. −0.0292680
\(926\) −8.03658e7 −3.07995
\(927\) −6.69704e6 −0.255967
\(928\) 5.37330e6 0.204819
\(929\) −1.46888e7 −0.558403 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(930\) 1.11752e7 0.423690
\(931\) −310053. −0.0117236
\(932\) −4.93670e7 −1.86165
\(933\) 1.23384e7 0.464040
\(934\) −6.36875e7 −2.38884
\(935\) −1.34857e6 −0.0504479
\(936\) −8.56433e6 −0.319524
\(937\) 1.85682e7 0.690910 0.345455 0.938435i \(-0.387725\pi\)
0.345455 + 0.938435i \(0.387725\pi\)
\(938\) 6.28370e6 0.233189
\(939\) −1.36014e7 −0.503406
\(940\) −1.59820e7 −0.589944
\(941\) −5.08610e7 −1.87245 −0.936227 0.351397i \(-0.885707\pi\)
−0.936227 + 0.351397i \(0.885707\pi\)
\(942\) 4.03579e7 1.48184
\(943\) −3.73586e6 −0.136808
\(944\) 3.28744e6 0.120068
\(945\) 5.02345e6 0.182988
\(946\) 1.90095e7 0.690626
\(947\) −2.18050e7 −0.790098 −0.395049 0.918660i \(-0.629272\pi\)
−0.395049 + 0.918660i \(0.629272\pi\)
\(948\) 6.28287e7 2.27058
\(949\) 1.51392e7 0.545678
\(950\) 193244. 0.00694700
\(951\) −1.64997e7 −0.591594
\(952\) 6.08885e6 0.217743
\(953\) 2.63434e7 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(954\) −1.24024e7 −0.441199
\(955\) −1.60170e7 −0.568292
\(956\) −1.15615e7 −0.409137
\(957\) −2.16142e7 −0.762887
\(958\) 6.49738e7 2.28731
\(959\) 3.39676e7 1.19266
\(960\) 1.78596e7 0.625450
\(961\) −2.27436e7 −0.794419
\(962\) 2.94344e6 0.102546
\(963\) 1.28259e7 0.445677
\(964\) −3.18301e7 −1.10318
\(965\) −2.48449e6 −0.0858852
\(966\) 8.19542e6 0.282571
\(967\) −6.88107e6 −0.236641 −0.118320 0.992975i \(-0.537751\pi\)
−0.118320 + 0.992975i \(0.537751\pi\)
\(968\) −3.32512e7 −1.14056
\(969\) 148752. 0.00508923
\(970\) 2.33991e7 0.798490
\(971\) −8.39882e6 −0.285871 −0.142936 0.989732i \(-0.545654\pi\)
−0.142936 + 0.989732i \(0.545654\pi\)
\(972\) −5.10460e7 −1.73299
\(973\) 2.92361e7 0.990004
\(974\) 3.65278e7 1.23375
\(975\) −2.94995e6 −0.0993811
\(976\) 2.69997e7 0.907265
\(977\) 3.21445e7 1.07738 0.538692 0.842503i \(-0.318919\pi\)
0.538692 + 0.842503i \(0.318919\pi\)
\(978\) 9.97730e7 3.33554
\(979\) 2.54326e7 0.848075
\(980\) 1.51640e7 0.504368
\(981\) −7.17871e6 −0.238163
\(982\) −2.39132e7 −0.791333
\(983\) −1.00759e7 −0.332583 −0.166292 0.986077i \(-0.553179\pi\)
−0.166292 + 0.986077i \(0.553179\pi\)
\(984\) −3.94160e7 −1.29773
\(985\) 9.46115e6 0.310709
\(986\) −1.24289e7 −0.407137
\(987\) 1.63736e7 0.534996
\(988\) −493366. −0.0160797
\(989\) 4.72656e6 0.153658
\(990\) −6.22601e6 −0.201893
\(991\) −1.13729e7 −0.367864 −0.183932 0.982939i \(-0.558883\pi\)
−0.183932 + 0.982939i \(0.558883\pi\)
\(992\) 2.50741e6 0.0808996
\(993\) −3.19852e7 −1.02938
\(994\) 5.87758e7 1.88683
\(995\) −1.90954e7 −0.611465
\(996\) −6.71048e7 −2.14341
\(997\) −2.73242e7 −0.870582 −0.435291 0.900290i \(-0.643354\pi\)
−0.435291 + 0.900290i \(0.643354\pi\)
\(998\) 1.97885e7 0.628908
\(999\) 2.91229e6 0.0923252
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 115.6.a.c.1.6 7
3.2 odd 2 1035.6.a.b.1.2 7
5.4 even 2 575.6.a.d.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.6.a.c.1.6 7 1.1 even 1 trivial
575.6.a.d.1.2 7 5.4 even 2
1035.6.a.b.1.2 7 3.2 odd 2