Properties

Label 74.8.a.a.1.4
Level $74$
Weight $8$
Character 74.1
Self dual yes
Analytic conductor $23.116$
Analytic rank $1$
Dimension $4$
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] = [74,8,Mod(1,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 74.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: \(23.1164918858\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(24.1837\) of defining polynomial
Character \(\chi\) \(=\) 74.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +55.2289 q^{3} +64.0000 q^{4} -82.2583 q^{5} -441.831 q^{6} +195.531 q^{7} -512.000 q^{8} +863.231 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +55.2289 q^{3} +64.0000 q^{4} -82.2583 q^{5} -441.831 q^{6} +195.531 q^{7} -512.000 q^{8} +863.231 q^{9} +658.066 q^{10} -3901.04 q^{11} +3534.65 q^{12} -14839.2 q^{13} -1564.25 q^{14} -4543.03 q^{15} +4096.00 q^{16} -6732.16 q^{17} -6905.84 q^{18} +36901.4 q^{19} -5264.53 q^{20} +10799.0 q^{21} +31208.3 q^{22} +46279.1 q^{23} -28277.2 q^{24} -71358.6 q^{25} +118714. q^{26} -73110.3 q^{27} +12514.0 q^{28} +60015.8 q^{29} +36344.3 q^{30} -305815. q^{31} -32768.0 q^{32} -215450. q^{33} +53857.3 q^{34} -16084.0 q^{35} +55246.8 q^{36} +50653.0 q^{37} -295211. q^{38} -819554. q^{39} +42116.2 q^{40} -670993. q^{41} -86391.6 q^{42} +220110. q^{43} -249667. q^{44} -71007.9 q^{45} -370233. q^{46} +292736. q^{47} +226218. q^{48} -785311. q^{49} +570869. q^{50} -371810. q^{51} -949710. q^{52} -715708. q^{53} +584883. q^{54} +320893. q^{55} -100112. q^{56} +2.03803e6 q^{57} -480126. q^{58} +988552. q^{59} -290754. q^{60} -3.06993e6 q^{61} +2.44652e6 q^{62} +168788. q^{63} +262144. q^{64} +1.22065e6 q^{65} +1.72360e6 q^{66} +727481. q^{67} -430859. q^{68} +2.55595e6 q^{69} +128672. q^{70} +5.23412e6 q^{71} -441974. q^{72} -2.48381e6 q^{73} -405224. q^{74} -3.94106e6 q^{75} +2.36169e6 q^{76} -762774. q^{77} +6.55643e6 q^{78} +6.57754e6 q^{79} -336930. q^{80} -5.92569e6 q^{81} +5.36794e6 q^{82} -4.65485e6 q^{83} +691133. q^{84} +553776. q^{85} -1.76088e6 q^{86} +3.31461e6 q^{87} +1.99733e6 q^{88} +1.44849e6 q^{89} +568063. q^{90} -2.90153e6 q^{91} +2.96186e6 q^{92} -1.68898e7 q^{93} -2.34188e6 q^{94} -3.03545e6 q^{95} -1.80974e6 q^{96} -597047. q^{97} +6.28249e6 q^{98} -3.36750e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9} - 888 q^{10} - 4593 q^{11} - 3392 q^{12} + 7847 q^{13} + 13328 q^{14} + 18900 q^{15} + 16384 q^{16} + 23172 q^{17} - 36872 q^{18} + 23696 q^{19} + 7104 q^{20} + 69416 q^{21} + 36744 q^{22} + 24105 q^{23} + 27136 q^{24} - 138149 q^{25} - 62776 q^{26} - 433646 q^{27} - 106624 q^{28} - 140949 q^{29} - 151200 q^{30} - 664609 q^{31} - 131072 q^{32} - 240450 q^{33} - 185376 q^{34} - 248544 q^{35} + 294976 q^{36} + 202612 q^{37} - 189568 q^{38} - 2288827 q^{39} - 56832 q^{40} - 709737 q^{41} - 555328 q^{42} - 128962 q^{43} - 293952 q^{44} - 1755342 q^{45} - 192840 q^{46} - 445842 q^{47} - 217088 q^{48} - 1602774 q^{49} + 1105192 q^{50} - 2883630 q^{51} + 502208 q^{52} - 975870 q^{53} + 3469168 q^{54} - 644145 q^{55} + 852992 q^{56} + 3494630 q^{57} + 1127592 q^{58} - 1812858 q^{59} + 1209600 q^{60} - 2955031 q^{61} + 5316872 q^{62} - 3362482 q^{63} + 1048576 q^{64} + 666 q^{65} + 1923600 q^{66} + 2737235 q^{67} + 1483008 q^{68} - 1781673 q^{69} + 1988352 q^{70} + 4958184 q^{71} - 2359808 q^{72} - 931591 q^{73} - 1620896 q^{74} + 4945810 q^{75} + 1516544 q^{76} + 4352514 q^{77} + 18310616 q^{78} + 5813561 q^{79} + 454656 q^{80} + 16394896 q^{81} + 5677896 q^{82} + 2120460 q^{83} + 4442624 q^{84} - 4845402 q^{85} + 1031696 q^{86} + 7965333 q^{87} + 2351616 q^{88} + 8833716 q^{89} + 14042736 q^{90} - 18886274 q^{91} + 1542720 q^{92} + 3024182 q^{93} + 3566736 q^{94} - 3151794 q^{95} + 1736704 q^{96} - 22666876 q^{97} + 12822192 q^{98} - 17931894 q^{99}+O(q^{100}) \) 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
\(3\) 55.2289 1.18098 0.590489 0.807046i \(-0.298935\pi\)
0.590489 + 0.807046i \(0.298935\pi\)
\(4\) 64.0000 0.500000
\(5\) −82.2583 −0.294296 −0.147148 0.989114i \(-0.547009\pi\)
−0.147148 + 0.989114i \(0.547009\pi\)
\(6\) −441.831 −0.835078
\(7\) 195.531 0.215463 0.107731 0.994180i \(-0.465641\pi\)
0.107731 + 0.994180i \(0.465641\pi\)
\(8\) −512.000 −0.353553
\(9\) 863.231 0.394710
\(10\) 658.066 0.208099
\(11\) −3901.04 −0.883702 −0.441851 0.897088i \(-0.645678\pi\)
−0.441851 + 0.897088i \(0.645678\pi\)
\(12\) 3534.65 0.590489
\(13\) −14839.2 −1.87331 −0.936655 0.350254i \(-0.886095\pi\)
−0.936655 + 0.350254i \(0.886095\pi\)
\(14\) −1564.25 −0.152355
\(15\) −4543.03 −0.347557
\(16\) 4096.00 0.250000
\(17\) −6732.16 −0.332341 −0.166170 0.986097i \(-0.553140\pi\)
−0.166170 + 0.986097i \(0.553140\pi\)
\(18\) −6905.84 −0.279102
\(19\) 36901.4 1.23426 0.617128 0.786862i \(-0.288296\pi\)
0.617128 + 0.786862i \(0.288296\pi\)
\(20\) −5264.53 −0.147148
\(21\) 10799.0 0.254457
\(22\) 31208.3 0.624872
\(23\) 46279.1 0.793118 0.396559 0.918009i \(-0.370204\pi\)
0.396559 + 0.918009i \(0.370204\pi\)
\(24\) −28277.2 −0.417539
\(25\) −71358.6 −0.913390
\(26\) 118714. 1.32463
\(27\) −73110.3 −0.714835
\(28\) 12514.0 0.107731
\(29\) 60015.8 0.456954 0.228477 0.973549i \(-0.426625\pi\)
0.228477 + 0.973549i \(0.426625\pi\)
\(30\) 36344.3 0.245760
\(31\) −305815. −1.84371 −0.921855 0.387535i \(-0.873327\pi\)
−0.921855 + 0.387535i \(0.873327\pi\)
\(32\) −32768.0 −0.176777
\(33\) −215450. −1.04363
\(34\) 53857.3 0.235000
\(35\) −16084.0 −0.0634098
\(36\) 55246.8 0.197355
\(37\) 50653.0 0.164399
\(38\) −295211. −0.872751
\(39\) −819554. −2.21234
\(40\) 42116.2 0.104049
\(41\) −670993. −1.52046 −0.760229 0.649655i \(-0.774913\pi\)
−0.760229 + 0.649655i \(0.774913\pi\)
\(42\) −86391.6 −0.179928
\(43\) 220110. 0.422182 0.211091 0.977466i \(-0.432298\pi\)
0.211091 + 0.977466i \(0.432298\pi\)
\(44\) −249667. −0.441851
\(45\) −71007.9 −0.116162
\(46\) −370233. −0.560819
\(47\) 292736. 0.411276 0.205638 0.978628i \(-0.434073\pi\)
0.205638 + 0.978628i \(0.434073\pi\)
\(48\) 226218. 0.295245
\(49\) −785311. −0.953576
\(50\) 570869. 0.645864
\(51\) −371810. −0.392487
\(52\) −949710. −0.936655
\(53\) −715708. −0.660344 −0.330172 0.943921i \(-0.607107\pi\)
−0.330172 + 0.943921i \(0.607107\pi\)
\(54\) 584883. 0.505464
\(55\) 320893. 0.260070
\(56\) −100112. −0.0761776
\(57\) 2.03803e6 1.45763
\(58\) −480126. −0.323115
\(59\) 988552. 0.626639 0.313320 0.949648i \(-0.398559\pi\)
0.313320 + 0.949648i \(0.398559\pi\)
\(60\) −290754. −0.173779
\(61\) −3.06993e6 −1.73171 −0.865854 0.500298i \(-0.833224\pi\)
−0.865854 + 0.500298i \(0.833224\pi\)
\(62\) 2.44652e6 1.30370
\(63\) 168788. 0.0850453
\(64\) 262144. 0.125000
\(65\) 1.22065e6 0.551308
\(66\) 1.72360e6 0.737960
\(67\) 727481. 0.295502 0.147751 0.989025i \(-0.452797\pi\)
0.147751 + 0.989025i \(0.452797\pi\)
\(68\) −430859. −0.166170
\(69\) 2.55595e6 0.936655
\(70\) 128672. 0.0448375
\(71\) 5.23412e6 1.73556 0.867780 0.496948i \(-0.165546\pi\)
0.867780 + 0.496948i \(0.165546\pi\)
\(72\) −441974. −0.139551
\(73\) −2.48381e6 −0.747287 −0.373644 0.927572i \(-0.621892\pi\)
−0.373644 + 0.927572i \(0.621892\pi\)
\(74\) −405224. −0.116248
\(75\) −3.94106e6 −1.07869
\(76\) 2.36169e6 0.617128
\(77\) −762774. −0.190405
\(78\) 6.55643e6 1.56436
\(79\) 6.57754e6 1.50096 0.750479 0.660895i \(-0.229823\pi\)
0.750479 + 0.660895i \(0.229823\pi\)
\(80\) −336930. −0.0735740
\(81\) −5.92569e6 −1.23891
\(82\) 5.36794e6 1.07513
\(83\) −4.65485e6 −0.893578 −0.446789 0.894639i \(-0.647433\pi\)
−0.446789 + 0.894639i \(0.647433\pi\)
\(84\) 691133. 0.127228
\(85\) 553776. 0.0978066
\(86\) −1.76088e6 −0.298528
\(87\) 3.31461e6 0.539653
\(88\) 1.99733e6 0.312436
\(89\) 1.44849e6 0.217796 0.108898 0.994053i \(-0.465268\pi\)
0.108898 + 0.994053i \(0.465268\pi\)
\(90\) 568063. 0.0821387
\(91\) −2.90153e6 −0.403628
\(92\) 2.96186e6 0.396559
\(93\) −1.68898e7 −2.17738
\(94\) −2.34188e6 −0.290816
\(95\) −3.03545e6 −0.363237
\(96\) −1.80974e6 −0.208769
\(97\) −597047. −0.0664213 −0.0332106 0.999448i \(-0.510573\pi\)
−0.0332106 + 0.999448i \(0.510573\pi\)
\(98\) 6.28249e6 0.674280
\(99\) −3.36750e6 −0.348806
\(100\) −4.56695e6 −0.456695
\(101\) 413080. 0.0398942 0.0199471 0.999801i \(-0.493650\pi\)
0.0199471 + 0.999801i \(0.493650\pi\)
\(102\) 2.97448e6 0.277530
\(103\) −1.49683e6 −0.134971 −0.0674856 0.997720i \(-0.521498\pi\)
−0.0674856 + 0.997720i \(0.521498\pi\)
\(104\) 7.59768e6 0.662315
\(105\) −888303. −0.0748857
\(106\) 5.72567e6 0.466934
\(107\) 1.54103e7 1.21609 0.608047 0.793901i \(-0.291953\pi\)
0.608047 + 0.793901i \(0.291953\pi\)
\(108\) −4.67906e6 −0.357417
\(109\) 1.80174e7 1.33260 0.666301 0.745683i \(-0.267877\pi\)
0.666301 + 0.745683i \(0.267877\pi\)
\(110\) −2.56714e6 −0.183897
\(111\) 2.79751e6 0.194152
\(112\) 800894. 0.0538657
\(113\) 3.02516e6 0.197230 0.0986151 0.995126i \(-0.468559\pi\)
0.0986151 + 0.995126i \(0.468559\pi\)
\(114\) −1.63042e7 −1.03070
\(115\) −3.80684e6 −0.233412
\(116\) 3.84101e6 0.228477
\(117\) −1.28097e7 −0.739414
\(118\) −7.90842e6 −0.443101
\(119\) −1.31635e6 −0.0716070
\(120\) 2.32603e6 0.122880
\(121\) −4.26905e6 −0.219070
\(122\) 2.45595e7 1.22450
\(123\) −3.70582e7 −1.79563
\(124\) −1.95722e7 −0.921855
\(125\) 1.22963e7 0.563103
\(126\) −1.35031e6 −0.0601361
\(127\) 4.61168e6 0.199778 0.0998888 0.994999i \(-0.468151\pi\)
0.0998888 + 0.994999i \(0.468151\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 1.21564e7 0.498588
\(130\) −9.76519e6 −0.389833
\(131\) 2.56869e7 0.998302 0.499151 0.866515i \(-0.333645\pi\)
0.499151 + 0.866515i \(0.333645\pi\)
\(132\) −1.37888e7 −0.521817
\(133\) 7.21537e6 0.265936
\(134\) −5.81985e6 −0.208951
\(135\) 6.01393e6 0.210373
\(136\) 3.44687e6 0.117500
\(137\) −1.88705e7 −0.626991 −0.313496 0.949590i \(-0.601500\pi\)
−0.313496 + 0.949590i \(0.601500\pi\)
\(138\) −2.04476e7 −0.662315
\(139\) −2.85231e7 −0.900835 −0.450417 0.892818i \(-0.648725\pi\)
−0.450417 + 0.892818i \(0.648725\pi\)
\(140\) −1.02938e6 −0.0317049
\(141\) 1.61675e7 0.485708
\(142\) −4.18730e7 −1.22723
\(143\) 5.78884e7 1.65545
\(144\) 3.53579e6 0.0986775
\(145\) −4.93679e6 −0.134480
\(146\) 1.98704e7 0.528412
\(147\) −4.33718e7 −1.12615
\(148\) 3.24179e6 0.0821995
\(149\) −6.84224e7 −1.69452 −0.847259 0.531179i \(-0.821749\pi\)
−0.847259 + 0.531179i \(0.821749\pi\)
\(150\) 3.15284e7 0.762752
\(151\) 1.77899e7 0.420488 0.210244 0.977649i \(-0.432574\pi\)
0.210244 + 0.977649i \(0.432574\pi\)
\(152\) −1.88935e7 −0.436376
\(153\) −5.81141e6 −0.131178
\(154\) 6.10219e6 0.134637
\(155\) 2.51558e7 0.542597
\(156\) −5.24515e7 −1.10617
\(157\) −6.46489e7 −1.33325 −0.666626 0.745392i \(-0.732262\pi\)
−0.666626 + 0.745392i \(0.732262\pi\)
\(158\) −5.26203e7 −1.06134
\(159\) −3.95278e7 −0.779852
\(160\) 2.69544e6 0.0520247
\(161\) 9.04900e6 0.170887
\(162\) 4.74055e7 0.876044
\(163\) −1.34057e7 −0.242455 −0.121228 0.992625i \(-0.538683\pi\)
−0.121228 + 0.992625i \(0.538683\pi\)
\(164\) −4.29435e7 −0.760229
\(165\) 1.77226e7 0.307137
\(166\) 3.72388e7 0.631855
\(167\) 1.23849e7 0.205771 0.102885 0.994693i \(-0.467193\pi\)
0.102885 + 0.994693i \(0.467193\pi\)
\(168\) −5.52906e6 −0.0899641
\(169\) 1.57454e8 2.50929
\(170\) −4.43021e6 −0.0691597
\(171\) 3.18544e7 0.487173
\(172\) 1.40870e7 0.211091
\(173\) 4.83378e7 0.709783 0.354891 0.934908i \(-0.384518\pi\)
0.354891 + 0.934908i \(0.384518\pi\)
\(174\) −2.65168e7 −0.381592
\(175\) −1.39528e7 −0.196801
\(176\) −1.59787e7 −0.220926
\(177\) 5.45966e7 0.740047
\(178\) −1.15879e7 −0.154005
\(179\) 8.54494e7 1.11359 0.556793 0.830652i \(-0.312032\pi\)
0.556793 + 0.830652i \(0.312032\pi\)
\(180\) −4.54450e6 −0.0580808
\(181\) −2.36467e7 −0.296412 −0.148206 0.988956i \(-0.547350\pi\)
−0.148206 + 0.988956i \(0.547350\pi\)
\(182\) 2.32122e7 0.285408
\(183\) −1.69549e8 −2.04511
\(184\) −2.36949e7 −0.280410
\(185\) −4.16663e6 −0.0483820
\(186\) 1.35119e8 1.53964
\(187\) 2.62624e7 0.293690
\(188\) 1.87351e7 0.205638
\(189\) −1.42953e7 −0.154020
\(190\) 2.42836e7 0.256847
\(191\) 9.64062e7 1.00113 0.500563 0.865700i \(-0.333127\pi\)
0.500563 + 0.865700i \(0.333127\pi\)
\(192\) 1.44779e7 0.147622
\(193\) 7.87097e7 0.788093 0.394047 0.919090i \(-0.371075\pi\)
0.394047 + 0.919090i \(0.371075\pi\)
\(194\) 4.77637e6 0.0469669
\(195\) 6.74151e7 0.651082
\(196\) −5.02599e7 −0.476788
\(197\) 1.55205e8 1.44635 0.723175 0.690665i \(-0.242682\pi\)
0.723175 + 0.690665i \(0.242682\pi\)
\(198\) 2.69400e7 0.246643
\(199\) −1.64147e8 −1.47655 −0.738273 0.674502i \(-0.764358\pi\)
−0.738273 + 0.674502i \(0.764358\pi\)
\(200\) 3.65356e7 0.322932
\(201\) 4.01780e7 0.348981
\(202\) −3.30464e6 −0.0282094
\(203\) 1.17349e7 0.0984566
\(204\) −2.37958e7 −0.196244
\(205\) 5.51947e7 0.447465
\(206\) 1.19746e7 0.0954391
\(207\) 3.99496e7 0.313052
\(208\) −6.07815e7 −0.468327
\(209\) −1.43954e8 −1.09072
\(210\) 7.10642e6 0.0529522
\(211\) −1.22343e8 −0.896580 −0.448290 0.893888i \(-0.647967\pi\)
−0.448290 + 0.893888i \(0.647967\pi\)
\(212\) −4.58053e7 −0.330172
\(213\) 2.89075e8 2.04966
\(214\) −1.23282e8 −0.859909
\(215\) −1.81059e7 −0.124247
\(216\) 3.74325e7 0.252732
\(217\) −5.97962e7 −0.397251
\(218\) −1.44140e8 −0.942292
\(219\) −1.37178e8 −0.882530
\(220\) 2.05371e7 0.130035
\(221\) 9.99001e7 0.622577
\(222\) −2.23801e7 −0.137286
\(223\) −9.24807e7 −0.558450 −0.279225 0.960226i \(-0.590078\pi\)
−0.279225 + 0.960226i \(0.590078\pi\)
\(224\) −6.40715e6 −0.0380888
\(225\) −6.15989e7 −0.360524
\(226\) −2.42013e7 −0.139463
\(227\) −1.36652e8 −0.775399 −0.387699 0.921786i \(-0.626730\pi\)
−0.387699 + 0.921786i \(0.626730\pi\)
\(228\) 1.30434e8 0.728815
\(229\) 1.47748e8 0.813011 0.406505 0.913648i \(-0.366747\pi\)
0.406505 + 0.913648i \(0.366747\pi\)
\(230\) 3.04547e7 0.165047
\(231\) −4.21272e7 −0.224864
\(232\) −3.07281e7 −0.161558
\(233\) 2.49263e8 1.29096 0.645480 0.763777i \(-0.276657\pi\)
0.645480 + 0.763777i \(0.276657\pi\)
\(234\) 1.02477e8 0.522844
\(235\) −2.40799e7 −0.121037
\(236\) 6.32673e7 0.313320
\(237\) 3.63270e8 1.77260
\(238\) 1.05308e7 0.0506338
\(239\) 3.06876e8 1.45402 0.727010 0.686627i \(-0.240909\pi\)
0.727010 + 0.686627i \(0.240909\pi\)
\(240\) −1.86083e7 −0.0868893
\(241\) −2.86171e7 −0.131694 −0.0658470 0.997830i \(-0.520975\pi\)
−0.0658470 + 0.997830i \(0.520975\pi\)
\(242\) 3.41524e7 0.154906
\(243\) −1.67377e8 −0.748296
\(244\) −1.96476e8 −0.865854
\(245\) 6.45983e7 0.280634
\(246\) 2.96465e8 1.26970
\(247\) −5.47589e8 −2.31214
\(248\) 1.56577e8 0.651850
\(249\) −2.57082e8 −1.05530
\(250\) −9.83701e7 −0.398174
\(251\) −1.53399e8 −0.612300 −0.306150 0.951983i \(-0.599041\pi\)
−0.306150 + 0.951983i \(0.599041\pi\)
\(252\) 1.08024e7 0.0425226
\(253\) −1.80537e8 −0.700880
\(254\) −3.68935e7 −0.141264
\(255\) 3.05844e7 0.115507
\(256\) 1.67772e7 0.0625000
\(257\) −3.85881e8 −1.41804 −0.709018 0.705190i \(-0.750862\pi\)
−0.709018 + 0.705190i \(0.750862\pi\)
\(258\) −9.72514e7 −0.352555
\(259\) 9.90422e6 0.0354219
\(260\) 7.81215e7 0.275654
\(261\) 5.18075e7 0.180364
\(262\) −2.05495e8 −0.705906
\(263\) −4.54495e8 −1.54058 −0.770289 0.637695i \(-0.779888\pi\)
−0.770289 + 0.637695i \(0.779888\pi\)
\(264\) 1.10310e8 0.368980
\(265\) 5.88729e7 0.194337
\(266\) −5.77229e7 −0.188045
\(267\) 7.99985e7 0.257213
\(268\) 4.65588e7 0.147751
\(269\) −2.71270e8 −0.849705 −0.424852 0.905263i \(-0.639674\pi\)
−0.424852 + 0.905263i \(0.639674\pi\)
\(270\) −4.81114e7 −0.148756
\(271\) 2.26702e8 0.691932 0.345966 0.938247i \(-0.387551\pi\)
0.345966 + 0.938247i \(0.387551\pi\)
\(272\) −2.75749e7 −0.0830852
\(273\) −1.60248e8 −0.476676
\(274\) 1.50964e8 0.443350
\(275\) 2.78373e8 0.807165
\(276\) 1.63581e8 0.468328
\(277\) −2.31320e8 −0.653933 −0.326967 0.945036i \(-0.606026\pi\)
−0.326967 + 0.945036i \(0.606026\pi\)
\(278\) 2.28185e8 0.636987
\(279\) −2.63989e8 −0.727731
\(280\) 8.23502e6 0.0224188
\(281\) −4.14589e8 −1.11467 −0.557334 0.830289i \(-0.688176\pi\)
−0.557334 + 0.830289i \(0.688176\pi\)
\(282\) −1.29340e8 −0.343447
\(283\) −4.52047e8 −1.18558 −0.592790 0.805357i \(-0.701973\pi\)
−0.592790 + 0.805357i \(0.701973\pi\)
\(284\) 3.34984e8 0.867780
\(285\) −1.67644e8 −0.428975
\(286\) −4.63107e8 −1.17058
\(287\) −1.31200e8 −0.327602
\(288\) −2.82863e7 −0.0697755
\(289\) −3.65017e8 −0.889550
\(290\) 3.94944e7 0.0950916
\(291\) −3.29742e7 −0.0784421
\(292\) −1.58964e8 −0.373644
\(293\) 6.51503e8 1.51314 0.756572 0.653911i \(-0.226873\pi\)
0.756572 + 0.653911i \(0.226873\pi\)
\(294\) 3.46975e8 0.796310
\(295\) −8.13166e7 −0.184417
\(296\) −2.59343e7 −0.0581238
\(297\) 2.85206e8 0.631701
\(298\) 5.47379e8 1.19821
\(299\) −6.86747e8 −1.48576
\(300\) −2.52228e8 −0.539347
\(301\) 4.30383e7 0.0909645
\(302\) −1.42319e8 −0.297330
\(303\) 2.28140e7 0.0471141
\(304\) 1.51148e8 0.308564
\(305\) 2.52527e8 0.509635
\(306\) 4.64913e7 0.0927570
\(307\) −5.71722e8 −1.12772 −0.563859 0.825871i \(-0.690684\pi\)
−0.563859 + 0.825871i \(0.690684\pi\)
\(308\) −4.88175e7 −0.0952025
\(309\) −8.26681e7 −0.159398
\(310\) −2.01246e8 −0.383674
\(311\) 2.12508e8 0.400602 0.200301 0.979734i \(-0.435808\pi\)
0.200301 + 0.979734i \(0.435808\pi\)
\(312\) 4.19612e8 0.782180
\(313\) 5.38948e8 0.993440 0.496720 0.867911i \(-0.334538\pi\)
0.496720 + 0.867911i \(0.334538\pi\)
\(314\) 5.17191e8 0.942752
\(315\) −1.38842e7 −0.0250285
\(316\) 4.20962e8 0.750479
\(317\) 7.71803e8 1.36081 0.680407 0.732834i \(-0.261803\pi\)
0.680407 + 0.732834i \(0.261803\pi\)
\(318\) 3.16222e8 0.551439
\(319\) −2.34124e8 −0.403811
\(320\) −2.15635e7 −0.0367870
\(321\) 8.51093e8 1.43618
\(322\) −7.23920e7 −0.120836
\(323\) −2.48426e8 −0.410194
\(324\) −3.79244e8 −0.619457
\(325\) 1.05891e9 1.71106
\(326\) 1.07245e8 0.171442
\(327\) 9.95083e8 1.57377
\(328\) 3.43548e8 0.537563
\(329\) 5.72388e7 0.0886146
\(330\) −1.41780e8 −0.217179
\(331\) 2.94395e8 0.446204 0.223102 0.974795i \(-0.428382\pi\)
0.223102 + 0.974795i \(0.428382\pi\)
\(332\) −2.97910e8 −0.446789
\(333\) 4.37252e7 0.0648899
\(334\) −9.90789e7 −0.145502
\(335\) −5.98413e7 −0.0869650
\(336\) 4.42325e7 0.0636142
\(337\) −2.10884e8 −0.300151 −0.150075 0.988675i \(-0.547952\pi\)
−0.150075 + 0.988675i \(0.547952\pi\)
\(338\) −1.25963e9 −1.77433
\(339\) 1.67076e8 0.232925
\(340\) 3.54417e7 0.0489033
\(341\) 1.19300e9 1.62929
\(342\) −2.54836e8 −0.344484
\(343\) −3.14581e8 −0.420923
\(344\) −1.12696e8 −0.149264
\(345\) −2.10248e8 −0.275654
\(346\) −3.86702e8 −0.501892
\(347\) 1.38481e9 1.77925 0.889623 0.456696i \(-0.150967\pi\)
0.889623 + 0.456696i \(0.150967\pi\)
\(348\) 2.12135e8 0.269826
\(349\) −1.34732e9 −1.69661 −0.848305 0.529507i \(-0.822377\pi\)
−0.848305 + 0.529507i \(0.822377\pi\)
\(350\) 1.11622e8 0.139160
\(351\) 1.08490e9 1.33911
\(352\) 1.27829e8 0.156218
\(353\) 2.94781e8 0.356687 0.178344 0.983968i \(-0.442926\pi\)
0.178344 + 0.983968i \(0.442926\pi\)
\(354\) −4.36773e8 −0.523292
\(355\) −4.30550e8 −0.510769
\(356\) 9.27034e7 0.108898
\(357\) −7.27003e7 −0.0845663
\(358\) −6.83595e8 −0.787424
\(359\) 8.58811e8 0.979642 0.489821 0.871823i \(-0.337062\pi\)
0.489821 + 0.871823i \(0.337062\pi\)
\(360\) 3.63560e7 0.0410693
\(361\) 4.67844e8 0.523390
\(362\) 1.89174e8 0.209595
\(363\) −2.35775e8 −0.258717
\(364\) −1.85698e8 −0.201814
\(365\) 2.04314e8 0.219924
\(366\) 1.35639e9 1.44611
\(367\) −1.83485e8 −0.193762 −0.0968812 0.995296i \(-0.530887\pi\)
−0.0968812 + 0.995296i \(0.530887\pi\)
\(368\) 1.89559e8 0.198280
\(369\) −5.79221e8 −0.600140
\(370\) 3.33330e7 0.0342112
\(371\) −1.39943e8 −0.142280
\(372\) −1.08095e9 −1.08869
\(373\) 8.02534e8 0.800724 0.400362 0.916357i \(-0.368884\pi\)
0.400362 + 0.916357i \(0.368884\pi\)
\(374\) −2.10100e8 −0.207670
\(375\) 6.79109e8 0.665013
\(376\) −1.49881e8 −0.145408
\(377\) −8.90588e8 −0.856016
\(378\) 1.14363e8 0.108909
\(379\) −2.01554e9 −1.90175 −0.950876 0.309572i \(-0.899814\pi\)
−0.950876 + 0.309572i \(0.899814\pi\)
\(380\) −1.94269e8 −0.181619
\(381\) 2.54698e8 0.235933
\(382\) −7.71250e8 −0.707902
\(383\) −9.40060e8 −0.854988 −0.427494 0.904018i \(-0.640603\pi\)
−0.427494 + 0.904018i \(0.640603\pi\)
\(384\) −1.15823e8 −0.104385
\(385\) 6.27445e7 0.0560354
\(386\) −6.29677e8 −0.557266
\(387\) 1.90006e8 0.166639
\(388\) −3.82110e7 −0.0332106
\(389\) −1.24350e9 −1.07108 −0.535539 0.844511i \(-0.679891\pi\)
−0.535539 + 0.844511i \(0.679891\pi\)
\(390\) −5.39321e8 −0.460385
\(391\) −3.11559e8 −0.263585
\(392\) 4.02079e8 0.337140
\(393\) 1.41866e9 1.17897
\(394\) −1.24164e9 −1.02272
\(395\) −5.41057e8 −0.441726
\(396\) −2.15520e8 −0.174403
\(397\) −2.28435e9 −1.83230 −0.916148 0.400841i \(-0.868718\pi\)
−0.916148 + 0.400841i \(0.868718\pi\)
\(398\) 1.31318e9 1.04408
\(399\) 3.98497e8 0.314065
\(400\) −2.92285e8 −0.228347
\(401\) −1.47181e9 −1.13985 −0.569925 0.821697i \(-0.693028\pi\)
−0.569925 + 0.821697i \(0.693028\pi\)
\(402\) −3.21424e8 −0.246767
\(403\) 4.53806e9 3.45384
\(404\) 2.64371e7 0.0199471
\(405\) 4.87437e8 0.364608
\(406\) −9.38795e7 −0.0696193
\(407\) −1.97599e8 −0.145280
\(408\) 1.90367e8 0.138765
\(409\) −4.00328e8 −0.289324 −0.144662 0.989481i \(-0.546209\pi\)
−0.144662 + 0.989481i \(0.546209\pi\)
\(410\) −4.41558e8 −0.316405
\(411\) −1.04220e9 −0.740463
\(412\) −9.57969e7 −0.0674856
\(413\) 1.93292e8 0.135017
\(414\) −3.19597e8 −0.221361
\(415\) 3.82900e8 0.262977
\(416\) 4.86252e8 0.331157
\(417\) −1.57530e9 −1.06387
\(418\) 1.15163e9 0.771253
\(419\) −1.38722e9 −0.921292 −0.460646 0.887584i \(-0.652382\pi\)
−0.460646 + 0.887584i \(0.652382\pi\)
\(420\) −5.68514e7 −0.0374428
\(421\) 1.19793e9 0.782428 0.391214 0.920300i \(-0.372055\pi\)
0.391214 + 0.920300i \(0.372055\pi\)
\(422\) 9.78741e8 0.633978
\(423\) 2.52698e8 0.162335
\(424\) 3.66443e8 0.233467
\(425\) 4.80398e8 0.303557
\(426\) −2.31260e9 −1.44933
\(427\) −6.00266e8 −0.373118
\(428\) 9.86258e8 0.608047
\(429\) 3.19711e9 1.95505
\(430\) 1.44847e8 0.0878556
\(431\) −1.17438e9 −0.706542 −0.353271 0.935521i \(-0.614931\pi\)
−0.353271 + 0.935521i \(0.614931\pi\)
\(432\) −2.99460e8 −0.178709
\(433\) 7.67669e8 0.454429 0.227215 0.973845i \(-0.427038\pi\)
0.227215 + 0.973845i \(0.427038\pi\)
\(434\) 4.78370e8 0.280899
\(435\) −2.72654e8 −0.158818
\(436\) 1.15312e9 0.666301
\(437\) 1.70777e9 0.978911
\(438\) 1.09742e9 0.624043
\(439\) −6.04257e8 −0.340875 −0.170438 0.985368i \(-0.554518\pi\)
−0.170438 + 0.985368i \(0.554518\pi\)
\(440\) −1.64297e8 −0.0919487
\(441\) −6.77904e8 −0.376386
\(442\) −7.99201e8 −0.440228
\(443\) −1.49397e9 −0.816451 −0.408225 0.912881i \(-0.633852\pi\)
−0.408225 + 0.912881i \(0.633852\pi\)
\(444\) 1.79041e8 0.0970758
\(445\) −1.19150e8 −0.0640966
\(446\) 7.39846e8 0.394884
\(447\) −3.77889e9 −2.00119
\(448\) 5.12572e7 0.0269328
\(449\) −2.37104e9 −1.23616 −0.618082 0.786113i \(-0.712090\pi\)
−0.618082 + 0.786113i \(0.712090\pi\)
\(450\) 4.92791e8 0.254929
\(451\) 2.61757e9 1.34363
\(452\) 1.93610e8 0.0986151
\(453\) 9.82515e8 0.496587
\(454\) 1.09322e9 0.548290
\(455\) 2.38675e8 0.118786
\(456\) −1.04347e9 −0.515350
\(457\) −7.29815e8 −0.357689 −0.178845 0.983877i \(-0.557236\pi\)
−0.178845 + 0.983877i \(0.557236\pi\)
\(458\) −1.18198e9 −0.574886
\(459\) 4.92191e8 0.237569
\(460\) −2.43638e8 −0.116706
\(461\) 2.98314e9 1.41814 0.709072 0.705136i \(-0.249114\pi\)
0.709072 + 0.705136i \(0.249114\pi\)
\(462\) 3.37017e8 0.159003
\(463\) 7.61541e8 0.356583 0.178291 0.983978i \(-0.442943\pi\)
0.178291 + 0.983978i \(0.442943\pi\)
\(464\) 2.45825e8 0.114239
\(465\) 1.38933e9 0.640795
\(466\) −1.99411e9 −0.912847
\(467\) −1.83218e9 −0.832452 −0.416226 0.909261i \(-0.636648\pi\)
−0.416226 + 0.909261i \(0.636648\pi\)
\(468\) −8.19819e8 −0.369707
\(469\) 1.42245e8 0.0636696
\(470\) 1.92639e8 0.0855860
\(471\) −3.57049e9 −1.57454
\(472\) −5.06139e8 −0.221550
\(473\) −8.58657e8 −0.373083
\(474\) −2.90616e9 −1.25342
\(475\) −2.63323e9 −1.12736
\(476\) −8.42461e7 −0.0358035
\(477\) −6.17821e8 −0.260644
\(478\) −2.45501e9 −1.02815
\(479\) 3.61205e9 1.50169 0.750843 0.660481i \(-0.229648\pi\)
0.750843 + 0.660481i \(0.229648\pi\)
\(480\) 1.48866e8 0.0614400
\(481\) −7.51651e8 −0.307970
\(482\) 2.28937e8 0.0931218
\(483\) 4.99766e8 0.201814
\(484\) −2.73219e8 −0.109535
\(485\) 4.91120e7 0.0195475
\(486\) 1.33901e9 0.529125
\(487\) 2.18474e9 0.857132 0.428566 0.903511i \(-0.359019\pi\)
0.428566 + 0.903511i \(0.359019\pi\)
\(488\) 1.57181e9 0.612251
\(489\) −7.40380e8 −0.286334
\(490\) −5.16786e8 −0.198438
\(491\) −7.61117e8 −0.290179 −0.145090 0.989419i \(-0.546347\pi\)
−0.145090 + 0.989419i \(0.546347\pi\)
\(492\) −2.37172e9 −0.897814
\(493\) −4.04036e8 −0.151864
\(494\) 4.38071e9 1.63493
\(495\) 2.77005e8 0.102652
\(496\) −1.25262e9 −0.460928
\(497\) 1.02343e9 0.373949
\(498\) 2.05666e9 0.746207
\(499\) −4.35203e9 −1.56798 −0.783989 0.620775i \(-0.786818\pi\)
−0.783989 + 0.620775i \(0.786818\pi\)
\(500\) 7.86961e8 0.281552
\(501\) 6.84002e8 0.243011
\(502\) 1.22719e9 0.432962
\(503\) −4.76391e9 −1.66907 −0.834537 0.550952i \(-0.814265\pi\)
−0.834537 + 0.550952i \(0.814265\pi\)
\(504\) −8.64196e7 −0.0300680
\(505\) −3.39792e7 −0.0117407
\(506\) 1.44429e9 0.495597
\(507\) 8.69601e9 2.96341
\(508\) 2.95148e8 0.0998888
\(509\) −5.39076e8 −0.181192 −0.0905958 0.995888i \(-0.528877\pi\)
−0.0905958 + 0.995888i \(0.528877\pi\)
\(510\) −2.44676e8 −0.0816761
\(511\) −4.85661e8 −0.161013
\(512\) −1.34218e8 −0.0441942
\(513\) −2.69788e9 −0.882289
\(514\) 3.08705e9 1.00270
\(515\) 1.23126e8 0.0397215
\(516\) 7.78011e8 0.249294
\(517\) −1.14197e9 −0.363445
\(518\) −7.92338e7 −0.0250470
\(519\) 2.66964e9 0.838238
\(520\) −6.24972e8 −0.194917
\(521\) −3.23894e9 −1.00339 −0.501696 0.865044i \(-0.667290\pi\)
−0.501696 + 0.865044i \(0.667290\pi\)
\(522\) −4.14460e8 −0.127537
\(523\) 1.31356e9 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(524\) 1.64396e9 0.499151
\(525\) −7.70598e8 −0.232418
\(526\) 3.63596e9 1.08935
\(527\) 2.05880e9 0.612740
\(528\) −8.82484e8 −0.260908
\(529\) −1.26307e9 −0.370964
\(530\) −4.70983e8 −0.137417
\(531\) 8.53348e8 0.247341
\(532\) 4.61784e8 0.132968
\(533\) 9.95701e9 2.84829
\(534\) −6.39988e8 −0.181877
\(535\) −1.26762e9 −0.357892
\(536\) −3.72470e8 −0.104476
\(537\) 4.71928e9 1.31512
\(538\) 2.17016e9 0.600832
\(539\) 3.06353e9 0.842677
\(540\) 3.84891e8 0.105187
\(541\) −1.68280e9 −0.456922 −0.228461 0.973553i \(-0.573369\pi\)
−0.228461 + 0.973553i \(0.573369\pi\)
\(542\) −1.81362e9 −0.489270
\(543\) −1.30598e9 −0.350057
\(544\) 2.20600e8 0.0587501
\(545\) −1.48208e9 −0.392180
\(546\) 1.28198e9 0.337061
\(547\) −3.91304e8 −0.102225 −0.0511127 0.998693i \(-0.516277\pi\)
−0.0511127 + 0.998693i \(0.516277\pi\)
\(548\) −1.20771e9 −0.313496
\(549\) −2.65006e9 −0.683522
\(550\) −2.22698e9 −0.570752
\(551\) 2.21467e9 0.563999
\(552\) −1.30864e9 −0.331158
\(553\) 1.28611e9 0.323400
\(554\) 1.85056e9 0.462401
\(555\) −2.30118e8 −0.0571381
\(556\) −1.82548e9 −0.450417
\(557\) −1.91820e9 −0.470328 −0.235164 0.971956i \(-0.575563\pi\)
−0.235164 + 0.971956i \(0.575563\pi\)
\(558\) 2.11191e9 0.514583
\(559\) −3.26626e9 −0.790878
\(560\) −6.58802e7 −0.0158525
\(561\) 1.45045e9 0.346842
\(562\) 3.31671e9 0.788189
\(563\) −3.86396e9 −0.912544 −0.456272 0.889840i \(-0.650816\pi\)
−0.456272 + 0.889840i \(0.650816\pi\)
\(564\) 1.03472e9 0.242854
\(565\) −2.48844e8 −0.0580441
\(566\) 3.61637e9 0.838331
\(567\) −1.15865e9 −0.266940
\(568\) −2.67987e9 −0.613613
\(569\) 5.70371e8 0.129797 0.0648985 0.997892i \(-0.479328\pi\)
0.0648985 + 0.997892i \(0.479328\pi\)
\(570\) 1.34116e9 0.303331
\(571\) −1.41942e9 −0.319070 −0.159535 0.987192i \(-0.550999\pi\)
−0.159535 + 0.987192i \(0.550999\pi\)
\(572\) 3.70486e9 0.827724
\(573\) 5.32441e9 1.18231
\(574\) 1.04960e9 0.231650
\(575\) −3.30241e9 −0.724426
\(576\) 2.26291e8 0.0493387
\(577\) 4.28493e9 0.928599 0.464299 0.885678i \(-0.346306\pi\)
0.464299 + 0.885678i \(0.346306\pi\)
\(578\) 2.92013e9 0.629007
\(579\) 4.34705e9 0.930721
\(580\) −3.15955e8 −0.0672399
\(581\) −9.10167e8 −0.192533
\(582\) 2.63794e8 0.0554669
\(583\) 2.79201e9 0.583548
\(584\) 1.27171e9 0.264206
\(585\) 1.05370e9 0.217607
\(586\) −5.21203e9 −1.06995
\(587\) −8.43305e9 −1.72088 −0.860441 0.509550i \(-0.829812\pi\)
−0.860441 + 0.509550i \(0.829812\pi\)
\(588\) −2.77580e9 −0.563076
\(589\) −1.12850e10 −2.27561
\(590\) 6.50533e8 0.130403
\(591\) 8.57179e9 1.70811
\(592\) 2.07475e8 0.0410997
\(593\) −2.18440e9 −0.430170 −0.215085 0.976595i \(-0.569003\pi\)
−0.215085 + 0.976595i \(0.569003\pi\)
\(594\) −2.28165e9 −0.446680
\(595\) 1.08280e8 0.0210737
\(596\) −4.37903e9 −0.847259
\(597\) −9.06565e9 −1.74377
\(598\) 5.49397e9 1.05059
\(599\) −6.26686e9 −1.19140 −0.595698 0.803209i \(-0.703124\pi\)
−0.595698 + 0.803209i \(0.703124\pi\)
\(600\) 2.01782e9 0.381376
\(601\) −6.89136e9 −1.29492 −0.647462 0.762098i \(-0.724170\pi\)
−0.647462 + 0.762098i \(0.724170\pi\)
\(602\) −3.44306e8 −0.0643216
\(603\) 6.27984e8 0.116637
\(604\) 1.13855e9 0.210244
\(605\) 3.51165e8 0.0644714
\(606\) −1.82512e8 −0.0333147
\(607\) 1.06049e10 1.92462 0.962312 0.271949i \(-0.0876680\pi\)
0.962312 + 0.271949i \(0.0876680\pi\)
\(608\) −1.20919e9 −0.218188
\(609\) 6.48108e8 0.116275
\(610\) −2.02022e9 −0.360366
\(611\) −4.34397e9 −0.770446
\(612\) −3.71930e8 −0.0655891
\(613\) 5.15559e9 0.903997 0.451998 0.892019i \(-0.350711\pi\)
0.451998 + 0.892019i \(0.350711\pi\)
\(614\) 4.57378e9 0.797417
\(615\) 3.04834e9 0.528446
\(616\) 3.90540e8 0.0673183
\(617\) −6.22489e9 −1.06692 −0.533462 0.845824i \(-0.679109\pi\)
−0.533462 + 0.845824i \(0.679109\pi\)
\(618\) 6.61345e8 0.112712
\(619\) 3.06576e8 0.0519542 0.0259771 0.999663i \(-0.491730\pi\)
0.0259771 + 0.999663i \(0.491730\pi\)
\(620\) 1.60997e9 0.271298
\(621\) −3.38348e9 −0.566948
\(622\) −1.70006e9 −0.283268
\(623\) 2.83225e8 0.0469270
\(624\) −3.35689e9 −0.553084
\(625\) 4.56342e9 0.747671
\(626\) −4.31158e9 −0.702468
\(627\) −7.95042e9 −1.28811
\(628\) −4.13753e9 −0.666626
\(629\) −3.41004e8 −0.0546365
\(630\) 1.11074e8 0.0176978
\(631\) −6.18111e9 −0.979409 −0.489704 0.871889i \(-0.662895\pi\)
−0.489704 + 0.871889i \(0.662895\pi\)
\(632\) −3.36770e9 −0.530669
\(633\) −6.75684e9 −1.05884
\(634\) −6.17442e9 −0.962241
\(635\) −3.79349e8 −0.0587937
\(636\) −2.52978e9 −0.389926
\(637\) 1.16534e10 1.78634
\(638\) 1.87299e9 0.285538
\(639\) 4.51826e9 0.685043
\(640\) 1.72508e8 0.0260123
\(641\) 1.47764e8 0.0221598 0.0110799 0.999939i \(-0.496473\pi\)
0.0110799 + 0.999939i \(0.496473\pi\)
\(642\) −6.80874e9 −1.01553
\(643\) 4.08393e9 0.605814 0.302907 0.953020i \(-0.402043\pi\)
0.302907 + 0.953020i \(0.402043\pi\)
\(644\) 5.79136e8 0.0854437
\(645\) −9.99966e8 −0.146733
\(646\) 1.98741e9 0.290051
\(647\) 5.02073e9 0.728790 0.364395 0.931245i \(-0.381276\pi\)
0.364395 + 0.931245i \(0.381276\pi\)
\(648\) 3.03395e9 0.438022
\(649\) −3.85638e9 −0.553763
\(650\) −8.47125e9 −1.20990
\(651\) −3.30248e9 −0.469145
\(652\) −8.57962e8 −0.121228
\(653\) −1.79275e9 −0.251955 −0.125978 0.992033i \(-0.540207\pi\)
−0.125978 + 0.992033i \(0.540207\pi\)
\(654\) −7.96067e9 −1.11283
\(655\) −2.11296e9 −0.293796
\(656\) −2.74839e9 −0.380114
\(657\) −2.14410e9 −0.294962
\(658\) −4.57911e8 −0.0626600
\(659\) −6.24023e9 −0.849379 −0.424689 0.905339i \(-0.639617\pi\)
−0.424689 + 0.905339i \(0.639617\pi\)
\(660\) 1.13424e9 0.153569
\(661\) −3.34819e8 −0.0450925 −0.0225463 0.999746i \(-0.507177\pi\)
−0.0225463 + 0.999746i \(0.507177\pi\)
\(662\) −2.35516e9 −0.315514
\(663\) 5.51737e9 0.735250
\(664\) 2.38328e9 0.315928
\(665\) −5.93524e8 −0.0782640
\(666\) −3.49802e8 −0.0458841
\(667\) 2.77748e9 0.362418
\(668\) 7.92631e8 0.102885
\(669\) −5.10761e9 −0.659517
\(670\) 4.78731e8 0.0614935
\(671\) 1.19759e10 1.53031
\(672\) −3.53860e8 −0.0449820
\(673\) −2.61406e9 −0.330569 −0.165285 0.986246i \(-0.552854\pi\)
−0.165285 + 0.986246i \(0.552854\pi\)
\(674\) 1.68707e9 0.212238
\(675\) 5.21705e9 0.652923
\(676\) 1.00771e10 1.25464
\(677\) −1.47546e9 −0.182754 −0.0913768 0.995816i \(-0.529127\pi\)
−0.0913768 + 0.995816i \(0.529127\pi\)
\(678\) −1.33661e9 −0.164703
\(679\) −1.16741e8 −0.0143113
\(680\) −2.83533e8 −0.0345798
\(681\) −7.54713e9 −0.915729
\(682\) −9.54397e9 −1.15208
\(683\) 1.03256e10 1.24006 0.620031 0.784578i \(-0.287120\pi\)
0.620031 + 0.784578i \(0.287120\pi\)
\(684\) 2.03868e9 0.243587
\(685\) 1.55226e9 0.184521
\(686\) 2.51664e9 0.297637
\(687\) 8.15994e9 0.960148
\(688\) 9.01570e8 0.105546
\(689\) 1.06206e10 1.23703
\(690\) 1.68198e9 0.194917
\(691\) 1.26802e10 1.46202 0.731011 0.682365i \(-0.239049\pi\)
0.731011 + 0.682365i \(0.239049\pi\)
\(692\) 3.09362e9 0.354891
\(693\) −6.58450e8 −0.0751547
\(694\) −1.10785e10 −1.25812
\(695\) 2.34626e9 0.265112
\(696\) −1.69708e9 −0.190796
\(697\) 4.51723e9 0.505310
\(698\) 1.07786e10 1.19968
\(699\) 1.37665e10 1.52460
\(700\) −8.92979e8 −0.0984007
\(701\) −1.40075e10 −1.53585 −0.767923 0.640543i \(-0.778709\pi\)
−0.767923 + 0.640543i \(0.778709\pi\)
\(702\) −8.67920e9 −0.946891
\(703\) 1.86917e9 0.202911
\(704\) −1.02263e9 −0.110463
\(705\) −1.32991e9 −0.142942
\(706\) −2.35825e9 −0.252216
\(707\) 8.07699e7 0.00859570
\(708\) 3.49418e9 0.370024
\(709\) −1.16059e10 −1.22297 −0.611486 0.791255i \(-0.709428\pi\)
−0.611486 + 0.791255i \(0.709428\pi\)
\(710\) 3.44440e9 0.361168
\(711\) 5.67793e9 0.592443
\(712\) −7.41627e8 −0.0770027
\(713\) −1.41528e10 −1.46228
\(714\) 5.81603e8 0.0597974
\(715\) −4.76180e9 −0.487192
\(716\) 5.46876e9 0.556793
\(717\) 1.69484e10 1.71717
\(718\) −6.87049e9 −0.692711
\(719\) 5.37751e9 0.539548 0.269774 0.962924i \(-0.413051\pi\)
0.269774 + 0.962924i \(0.413051\pi\)
\(720\) −2.90848e8 −0.0290404
\(721\) −2.92676e8 −0.0290813
\(722\) −3.74275e9 −0.370093
\(723\) −1.58049e9 −0.155528
\(724\) −1.51339e9 −0.148206
\(725\) −4.28264e9 −0.417377
\(726\) 1.88620e9 0.182940
\(727\) 1.17805e10 1.13708 0.568541 0.822655i \(-0.307508\pi\)
0.568541 + 0.822655i \(0.307508\pi\)
\(728\) 1.48558e9 0.142704
\(729\) 3.71544e9 0.355192
\(730\) −1.63451e9 −0.155510
\(731\) −1.48182e9 −0.140308
\(732\) −1.08511e10 −1.02255
\(733\) 1.88637e10 1.76914 0.884572 0.466403i \(-0.154450\pi\)
0.884572 + 0.466403i \(0.154450\pi\)
\(734\) 1.46788e9 0.137011
\(735\) 3.56769e9 0.331422
\(736\) −1.51647e9 −0.140205
\(737\) −2.83793e9 −0.261135
\(738\) 4.63377e9 0.424363
\(739\) 1.27884e10 1.16563 0.582815 0.812605i \(-0.301951\pi\)
0.582815 + 0.812605i \(0.301951\pi\)
\(740\) −2.66664e8 −0.0241910
\(741\) −3.02427e10 −2.73059
\(742\) 1.11954e9 0.100607
\(743\) 1.45516e10 1.30152 0.650758 0.759286i \(-0.274451\pi\)
0.650758 + 0.759286i \(0.274451\pi\)
\(744\) 8.64759e9 0.769821
\(745\) 5.62831e9 0.498690
\(746\) −6.42027e9 −0.566197
\(747\) −4.01821e9 −0.352704
\(748\) 1.68080e9 0.146845
\(749\) 3.01318e9 0.262023
\(750\) −5.43287e9 −0.470235
\(751\) 5.73628e9 0.494186 0.247093 0.968992i \(-0.420525\pi\)
0.247093 + 0.968992i \(0.420525\pi\)
\(752\) 1.19904e9 0.102819
\(753\) −8.47206e9 −0.723113
\(754\) 7.12470e9 0.605295
\(755\) −1.46336e9 −0.123748
\(756\) −9.14901e8 −0.0770101
\(757\) 8.37480e9 0.701680 0.350840 0.936435i \(-0.385896\pi\)
0.350840 + 0.936435i \(0.385896\pi\)
\(758\) 1.61243e10 1.34474
\(759\) −9.97085e9 −0.827725
\(760\) 1.55415e9 0.128424
\(761\) −4.44829e9 −0.365887 −0.182943 0.983123i \(-0.558563\pi\)
−0.182943 + 0.983123i \(0.558563\pi\)
\(762\) −2.03759e9 −0.166830
\(763\) 3.52297e9 0.287126
\(764\) 6.17000e9 0.500563
\(765\) 4.78037e8 0.0386052
\(766\) 7.52048e9 0.604568
\(767\) −1.46693e10 −1.17389
\(768\) 9.26587e8 0.0738111
\(769\) 1.26976e10 1.00689 0.503443 0.864029i \(-0.332066\pi\)
0.503443 + 0.864029i \(0.332066\pi\)
\(770\) −5.01956e8 −0.0396230
\(771\) −2.13118e10 −1.67467
\(772\) 5.03742e9 0.394047
\(773\) −1.24267e10 −0.967667 −0.483834 0.875160i \(-0.660756\pi\)
−0.483834 + 0.875160i \(0.660756\pi\)
\(774\) −1.52004e9 −0.117832
\(775\) 2.18225e10 1.68403
\(776\) 3.05688e8 0.0234835
\(777\) 5.46999e8 0.0418324
\(778\) 9.94797e9 0.757366
\(779\) −2.47606e10 −1.87664
\(780\) 4.31457e9 0.325541
\(781\) −2.04185e10 −1.53372
\(782\) 2.49247e9 0.186383
\(783\) −4.38777e9 −0.326647
\(784\) −3.21663e9 −0.238394
\(785\) 5.31791e9 0.392371
\(786\) −1.13493e10 −0.833660
\(787\) 1.93687e10 1.41641 0.708206 0.706006i \(-0.249505\pi\)
0.708206 + 0.706006i \(0.249505\pi\)
\(788\) 9.93310e9 0.723175
\(789\) −2.51012e10 −1.81939
\(790\) 4.32845e9 0.312347
\(791\) 5.91512e8 0.0424958
\(792\) 1.72416e9 0.123322
\(793\) 4.55554e10 3.24402
\(794\) 1.82748e10 1.29563
\(795\) 3.25149e9 0.229508
\(796\) −1.05054e10 −0.738273
\(797\) 1.20656e10 0.844199 0.422099 0.906550i \(-0.361293\pi\)
0.422099 + 0.906550i \(0.361293\pi\)
\(798\) −3.18797e9 −0.222078
\(799\) −1.97074e9 −0.136684
\(800\) 2.33828e9 0.161466
\(801\) 1.25038e9 0.0859664
\(802\) 1.17745e10 0.805995
\(803\) 9.68943e9 0.660380
\(804\) 2.57139e9 0.174490
\(805\) −7.44355e8 −0.0502915
\(806\) −3.63044e10 −2.44223
\(807\) −1.49819e10 −1.00348
\(808\) −2.11497e8 −0.0141047
\(809\) −2.52731e6 −0.000167818 0 −8.39090e−5 1.00000i \(-0.500027\pi\)
−8.39090e−5 1.00000i \(0.500027\pi\)
\(810\) −3.89949e9 −0.257817
\(811\) −1.17598e10 −0.774150 −0.387075 0.922048i \(-0.626515\pi\)
−0.387075 + 0.922048i \(0.626515\pi\)
\(812\) 7.51036e8 0.0492283
\(813\) 1.25205e10 0.817157
\(814\) 1.58080e9 0.102728
\(815\) 1.10273e9 0.0713536
\(816\) −1.52293e9 −0.0981218
\(817\) 8.12237e9 0.521081
\(818\) 3.20263e9 0.204583
\(819\) −2.50469e9 −0.159316
\(820\) 3.53246e9 0.223732
\(821\) −3.04383e10 −1.91964 −0.959820 0.280615i \(-0.909461\pi\)
−0.959820 + 0.280615i \(0.909461\pi\)
\(822\) 8.33758e9 0.523587
\(823\) −2.22031e10 −1.38840 −0.694200 0.719782i \(-0.744242\pi\)
−0.694200 + 0.719782i \(0.744242\pi\)
\(824\) 7.66376e8 0.0477196
\(825\) 1.53742e10 0.953244
\(826\) −1.54634e9 −0.0954717
\(827\) −1.43976e9 −0.0885156 −0.0442578 0.999020i \(-0.514092\pi\)
−0.0442578 + 0.999020i \(0.514092\pi\)
\(828\) 2.55677e9 0.156526
\(829\) −5.46818e8 −0.0333351 −0.0166675 0.999861i \(-0.505306\pi\)
−0.0166675 + 0.999861i \(0.505306\pi\)
\(830\) −3.06320e9 −0.185953
\(831\) −1.27755e10 −0.772281
\(832\) −3.89001e9 −0.234164
\(833\) 5.28684e9 0.316912
\(834\) 1.26024e10 0.752267
\(835\) −1.01876e9 −0.0605575
\(836\) −9.21305e9 −0.545358
\(837\) 2.23582e10 1.31795
\(838\) 1.10978e10 0.651451
\(839\) −1.24569e10 −0.728187 −0.364093 0.931362i \(-0.618621\pi\)
−0.364093 + 0.931362i \(0.618621\pi\)
\(840\) 4.54811e8 0.0264761
\(841\) −1.36480e10 −0.791193
\(842\) −9.58344e9 −0.553260
\(843\) −2.28973e10 −1.31640
\(844\) −7.82992e9 −0.448290
\(845\) −1.29519e10 −0.738474
\(846\) −2.02159e9 −0.114788
\(847\) −8.34731e8 −0.0472014
\(848\) −2.93154e9 −0.165086
\(849\) −2.49660e10 −1.40014
\(850\) −3.84318e9 −0.214647
\(851\) 2.34418e9 0.130388
\(852\) 1.85008e10 1.02483
\(853\) 3.24413e10 1.78968 0.894842 0.446383i \(-0.147288\pi\)
0.894842 + 0.446383i \(0.147288\pi\)
\(854\) 4.80213e9 0.263834
\(855\) −2.62029e9 −0.143373
\(856\) −7.89006e9 −0.429954
\(857\) 1.13661e10 0.616851 0.308425 0.951249i \(-0.400198\pi\)
0.308425 + 0.951249i \(0.400198\pi\)
\(858\) −2.55769e10 −1.38243
\(859\) −1.56465e10 −0.842249 −0.421125 0.907003i \(-0.638364\pi\)
−0.421125 + 0.907003i \(0.638364\pi\)
\(860\) −1.15877e9 −0.0621233
\(861\) −7.24602e9 −0.386891
\(862\) 9.39503e9 0.499601
\(863\) −2.74534e9 −0.145398 −0.0726990 0.997354i \(-0.523161\pi\)
−0.0726990 + 0.997354i \(0.523161\pi\)
\(864\) 2.39568e9 0.126366
\(865\) −3.97618e9 −0.208886
\(866\) −6.14135e9 −0.321330
\(867\) −2.01595e10 −1.05054
\(868\) −3.82696e9 −0.198625
\(869\) −2.56592e10 −1.32640
\(870\) 2.18123e9 0.112301
\(871\) −1.07953e10 −0.553566
\(872\) −9.22493e9 −0.471146
\(873\) −5.15389e8 −0.0262171
\(874\) −1.36621e10 −0.692195
\(875\) 2.40430e9 0.121328
\(876\) −8.77938e9 −0.441265
\(877\) −3.57403e10 −1.78920 −0.894600 0.446867i \(-0.852540\pi\)
−0.894600 + 0.446867i \(0.852540\pi\)
\(878\) 4.83405e9 0.241035
\(879\) 3.59818e10 1.78699
\(880\) 1.31438e9 0.0650176
\(881\) 1.17526e10 0.579051 0.289525 0.957170i \(-0.406503\pi\)
0.289525 + 0.957170i \(0.406503\pi\)
\(882\) 5.42323e9 0.266145
\(883\) 8.73602e9 0.427023 0.213511 0.976941i \(-0.431510\pi\)
0.213511 + 0.976941i \(0.431510\pi\)
\(884\) 6.39361e9 0.311288
\(885\) −4.49102e9 −0.217793
\(886\) 1.19518e10 0.577318
\(887\) 4.61304e9 0.221950 0.110975 0.993823i \(-0.464603\pi\)
0.110975 + 0.993823i \(0.464603\pi\)
\(888\) −1.43232e9 −0.0686430
\(889\) 9.01727e8 0.0430446
\(890\) 9.53203e8 0.0453232
\(891\) 2.31163e10 1.09483
\(892\) −5.91877e9 −0.279225
\(893\) 1.08024e10 0.507620
\(894\) 3.02311e10 1.41506
\(895\) −7.02892e9 −0.327724
\(896\) −4.10058e8 −0.0190444
\(897\) −3.79282e10 −1.75464
\(898\) 1.89683e10 0.874100
\(899\) −1.83537e10 −0.842491
\(900\) −3.94233e9 −0.180262
\(901\) 4.81827e9 0.219459
\(902\) −2.09406e10 −0.950091
\(903\) 2.37696e9 0.107427
\(904\) −1.54888e9 −0.0697314
\(905\) 1.94514e9 0.0872330
\(906\) −7.86012e9 −0.351140
\(907\) 1.84894e10 0.822806 0.411403 0.911454i \(-0.365039\pi\)
0.411403 + 0.911454i \(0.365039\pi\)
\(908\) −8.74572e9 −0.387699
\(909\) 3.56583e8 0.0157466
\(910\) −1.90940e9 −0.0839946
\(911\) 2.98023e9 0.130598 0.0652988 0.997866i \(-0.479200\pi\)
0.0652988 + 0.997866i \(0.479200\pi\)
\(912\) 8.34775e9 0.364408
\(913\) 1.81588e10 0.789657
\(914\) 5.83852e9 0.252925
\(915\) 1.39468e10 0.601868
\(916\) 9.45585e9 0.406505
\(917\) 5.02258e9 0.215097
\(918\) −3.93753e9 −0.167986
\(919\) 1.43439e10 0.609624 0.304812 0.952413i \(-0.401406\pi\)
0.304812 + 0.952413i \(0.401406\pi\)
\(920\) 1.94910e9 0.0825234
\(921\) −3.15756e10 −1.33181
\(922\) −2.38651e10 −1.00278
\(923\) −7.76704e10 −3.25124
\(924\) −2.69614e9 −0.112432
\(925\) −3.61453e9 −0.150160
\(926\) −6.09233e9 −0.252142
\(927\) −1.29211e9 −0.0532745
\(928\) −1.96660e9 −0.0807788
\(929\) 4.83590e10 1.97889 0.989447 0.144897i \(-0.0462852\pi\)
0.989447 + 0.144897i \(0.0462852\pi\)
\(930\) −1.11146e10 −0.453111
\(931\) −2.89791e10 −1.17696
\(932\) 1.59529e10 0.645480
\(933\) 1.17366e10 0.473102
\(934\) 1.46575e10 0.588633
\(935\) −2.16030e9 −0.0864319
\(936\) 6.55855e9 0.261422
\(937\) −5.46114e9 −0.216868 −0.108434 0.994104i \(-0.534584\pi\)
−0.108434 + 0.994104i \(0.534584\pi\)
\(938\) −1.13796e9 −0.0450212
\(939\) 2.97655e10 1.17323
\(940\) −1.54111e9 −0.0605184
\(941\) 4.34968e10 1.70174 0.850871 0.525374i \(-0.176075\pi\)
0.850871 + 0.525374i \(0.176075\pi\)
\(942\) 2.85639e10 1.11337
\(943\) −3.10530e10 −1.20590
\(944\) 4.04911e9 0.156660
\(945\) 1.17591e9 0.0453275
\(946\) 6.86926e9 0.263810
\(947\) −1.81345e10 −0.693873 −0.346937 0.937889i \(-0.612778\pi\)
−0.346937 + 0.937889i \(0.612778\pi\)
\(948\) 2.32493e10 0.886299
\(949\) 3.68578e10 1.39990
\(950\) 2.10659e10 0.797162
\(951\) 4.26258e10 1.60709
\(952\) 6.73969e8 0.0253169
\(953\) 1.00536e10 0.376265 0.188133 0.982144i \(-0.439756\pi\)
0.188133 + 0.982144i \(0.439756\pi\)
\(954\) 4.94257e9 0.184303
\(955\) −7.93021e9 −0.294627
\(956\) 1.96401e10 0.727010
\(957\) −1.29304e10 −0.476893
\(958\) −2.88964e10 −1.06185
\(959\) −3.68977e9 −0.135093
\(960\) −1.19093e9 −0.0434447
\(961\) 6.60101e10 2.39927
\(962\) 6.01321e9 0.217768
\(963\) 1.33026e10 0.480005
\(964\) −1.83150e9 −0.0658470
\(965\) −6.47452e9 −0.231933
\(966\) −3.99813e9 −0.142704
\(967\) 1.37918e9 0.0490489 0.0245244 0.999699i \(-0.492193\pi\)
0.0245244 + 0.999699i \(0.492193\pi\)
\(968\) 2.18575e9 0.0774529
\(969\) −1.37203e10 −0.484430
\(970\) −3.92896e8 −0.0138222
\(971\) −5.41405e10 −1.89782 −0.948911 0.315545i \(-0.897813\pi\)
−0.948911 + 0.315545i \(0.897813\pi\)
\(972\) −1.07121e10 −0.374148
\(973\) −5.57715e9 −0.194096
\(974\) −1.74779e10 −0.606084
\(975\) 5.84822e10 2.02073
\(976\) −1.25744e10 −0.432927
\(977\) −5.40754e10 −1.85511 −0.927554 0.373690i \(-0.878092\pi\)
−0.927554 + 0.373690i \(0.878092\pi\)
\(978\) 5.92304e9 0.202469
\(979\) −5.65062e9 −0.192467
\(980\) 4.13429e9 0.140317
\(981\) 1.55532e10 0.525991
\(982\) 6.08894e9 0.205188
\(983\) 4.72172e10 1.58549 0.792743 0.609556i \(-0.208652\pi\)
0.792743 + 0.609556i \(0.208652\pi\)
\(984\) 1.89738e10 0.634850
\(985\) −1.27669e10 −0.425655
\(986\) 3.23229e9 0.107384
\(987\) 3.16124e9 0.104652
\(988\) −3.50457e10 −1.15607
\(989\) 1.01865e10 0.334840
\(990\) −2.21604e9 −0.0725861
\(991\) −3.70899e10 −1.21059 −0.605297 0.796000i \(-0.706946\pi\)
−0.605297 + 0.796000i \(0.706946\pi\)
\(992\) 1.00209e10 0.325925
\(993\) 1.62591e10 0.526957
\(994\) −8.18746e9 −0.264422
\(995\) 1.35024e10 0.434542
\(996\) −1.64533e10 −0.527648
\(997\) −1.44459e10 −0.461650 −0.230825 0.972995i \(-0.574142\pi\)
−0.230825 + 0.972995i \(0.574142\pi\)
\(998\) 3.48162e10 1.10873
\(999\) −3.70326e9 −0.117518
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 74.8.a.a.1.4 4
4.3 odd 2 592.8.a.b.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.8.a.a.1.4 4 1.1 even 1 trivial
592.8.a.b.1.1 4 4.3 odd 2