Properties

Label 74.8.a.a.1.1
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 1.1
Root \(-1.99165\) 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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -92.6631 q^{3} +64.0000 q^{4} -162.307 q^{5} +741.305 q^{6} -829.706 q^{7} -512.000 q^{8} +6399.45 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -92.6631 q^{3} +64.0000 q^{4} -162.307 q^{5} +741.305 q^{6} -829.706 q^{7} -512.000 q^{8} +6399.45 q^{9} +1298.46 q^{10} -2491.07 q^{11} -5930.44 q^{12} +14938.3 q^{13} +6637.65 q^{14} +15039.9 q^{15} +4096.00 q^{16} +20584.9 q^{17} -51195.6 q^{18} -28580.0 q^{19} -10387.7 q^{20} +76883.2 q^{21} +19928.6 q^{22} +73036.6 q^{23} +47443.5 q^{24} -51781.3 q^{25} -119506. q^{26} -390339. q^{27} -53101.2 q^{28} -60941.6 q^{29} -120319. q^{30} -154588. q^{31} -32768.0 q^{32} +230831. q^{33} -164679. q^{34} +134667. q^{35} +409565. q^{36} +50653.0 q^{37} +228640. q^{38} -1.38423e6 q^{39} +83101.3 q^{40} +529338. q^{41} -615065. q^{42} +565222. q^{43} -159429. q^{44} -1.03868e6 q^{45} -584293. q^{46} +715209. q^{47} -379548. q^{48} -135130. q^{49} +414251. q^{50} -1.90746e6 q^{51} +956050. q^{52} -1.20158e6 q^{53} +3.12271e6 q^{54} +404319. q^{55} +424810. q^{56} +2.64831e6 q^{57} +487533. q^{58} -1.10650e6 q^{59} +962554. q^{60} -892203. q^{61} +1.23670e6 q^{62} -5.30967e6 q^{63} +262144. q^{64} -2.42459e6 q^{65} -1.84664e6 q^{66} +3.77593e6 q^{67} +1.31743e6 q^{68} -6.76780e6 q^{69} -1.07734e6 q^{70} +652761. q^{71} -3.27652e6 q^{72} -1.52330e6 q^{73} -405224. q^{74} +4.79822e6 q^{75} -1.82912e6 q^{76} +2.06686e6 q^{77} +1.10738e7 q^{78} +3.25297e6 q^{79} -664811. q^{80} +2.21744e7 q^{81} -4.23470e6 q^{82} +1.69719e6 q^{83} +4.92052e6 q^{84} -3.34108e6 q^{85} -4.52177e6 q^{86} +5.64704e6 q^{87} +1.27543e6 q^{88} +790719. q^{89} +8.30942e6 q^{90} -1.23944e7 q^{91} +4.67434e6 q^{92} +1.43246e7 q^{93} -5.72167e6 q^{94} +4.63874e6 q^{95} +3.03639e6 q^{96} -1.20273e7 q^{97} +1.08104e6 q^{98} -1.59415e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9}+O(q^{10}) \) 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
\(3\) −92.6631 −1.98145 −0.990724 0.135892i \(-0.956610\pi\)
−0.990724 + 0.135892i \(0.956610\pi\)
\(4\) 64.0000 0.500000
\(5\) −162.307 −0.580688 −0.290344 0.956922i \(-0.593770\pi\)
−0.290344 + 0.956922i \(0.593770\pi\)
\(6\) 741.305 1.40109
\(7\) −829.706 −0.914284 −0.457142 0.889394i \(-0.651127\pi\)
−0.457142 + 0.889394i \(0.651127\pi\)
\(8\) −512.000 −0.353553
\(9\) 6399.45 2.92613
\(10\) 1298.46 0.410609
\(11\) −2491.07 −0.564302 −0.282151 0.959370i \(-0.591048\pi\)
−0.282151 + 0.959370i \(0.591048\pi\)
\(12\) −5930.44 −0.990724
\(13\) 14938.3 1.88581 0.942907 0.333057i \(-0.108080\pi\)
0.942907 + 0.333057i \(0.108080\pi\)
\(14\) 6637.65 0.646497
\(15\) 15039.9 1.15060
\(16\) 4096.00 0.250000
\(17\) 20584.9 1.01620 0.508098 0.861299i \(-0.330349\pi\)
0.508098 + 0.861299i \(0.330349\pi\)
\(18\) −51195.6 −2.06909
\(19\) −28580.0 −0.955926 −0.477963 0.878380i \(-0.658625\pi\)
−0.477963 + 0.878380i \(0.658625\pi\)
\(20\) −10387.7 −0.290344
\(21\) 76883.2 1.81161
\(22\) 19928.6 0.399022
\(23\) 73036.6 1.25168 0.625840 0.779952i \(-0.284756\pi\)
0.625840 + 0.779952i \(0.284756\pi\)
\(24\) 47443.5 0.700547
\(25\) −51781.3 −0.662801
\(26\) −119506. −1.33347
\(27\) −390339. −3.81653
\(28\) −53101.2 −0.457142
\(29\) −60941.6 −0.464003 −0.232001 0.972715i \(-0.574527\pi\)
−0.232001 + 0.972715i \(0.574527\pi\)
\(30\) −120319. −0.813599
\(31\) −154588. −0.931985 −0.465992 0.884789i \(-0.654303\pi\)
−0.465992 + 0.884789i \(0.654303\pi\)
\(32\) −32768.0 −0.176777
\(33\) 230831. 1.11814
\(34\) −164679. −0.718560
\(35\) 134667. 0.530914
\(36\) 409565. 1.46307
\(37\) 50653.0 0.164399
\(38\) 228640. 0.675941
\(39\) −1.38423e6 −3.73664
\(40\) 83101.3 0.205304
\(41\) 529338. 1.19947 0.599735 0.800199i \(-0.295273\pi\)
0.599735 + 0.800199i \(0.295273\pi\)
\(42\) −615065. −1.28100
\(43\) 565222. 1.08412 0.542062 0.840338i \(-0.317644\pi\)
0.542062 + 0.840338i \(0.317644\pi\)
\(44\) −159429. −0.282151
\(45\) −1.03868e6 −1.69917
\(46\) −584293. −0.885071
\(47\) 715209. 1.00482 0.502412 0.864628i \(-0.332446\pi\)
0.502412 + 0.864628i \(0.332446\pi\)
\(48\) −379548. −0.495362
\(49\) −135130. −0.164084
\(50\) 414251. 0.468671
\(51\) −1.90746e6 −2.01354
\(52\) 956050. 0.942907
\(53\) −1.20158e6 −1.10863 −0.554317 0.832306i \(-0.687020\pi\)
−0.554317 + 0.832306i \(0.687020\pi\)
\(54\) 3.12271e6 2.69870
\(55\) 404319. 0.327684
\(56\) 424810. 0.323248
\(57\) 2.64831e6 1.89412
\(58\) 487533. 0.328100
\(59\) −1.10650e6 −0.701406 −0.350703 0.936487i \(-0.614057\pi\)
−0.350703 + 0.936487i \(0.614057\pi\)
\(60\) 962554. 0.575302
\(61\) −892203. −0.503280 −0.251640 0.967821i \(-0.580970\pi\)
−0.251640 + 0.967821i \(0.580970\pi\)
\(62\) 1.23670e6 0.659013
\(63\) −5.30967e6 −2.67532
\(64\) 262144. 0.125000
\(65\) −2.42459e6 −1.09507
\(66\) −1.84664e6 −0.790641
\(67\) 3.77593e6 1.53378 0.766888 0.641781i \(-0.221804\pi\)
0.766888 + 0.641781i \(0.221804\pi\)
\(68\) 1.31743e6 0.508098
\(69\) −6.76780e6 −2.48014
\(70\) −1.07734e6 −0.375413
\(71\) 652761. 0.216446 0.108223 0.994127i \(-0.465484\pi\)
0.108223 + 0.994127i \(0.465484\pi\)
\(72\) −3.27652e6 −1.03454
\(73\) −1.52330e6 −0.458307 −0.229153 0.973390i \(-0.573596\pi\)
−0.229153 + 0.973390i \(0.573596\pi\)
\(74\) −405224. −0.116248
\(75\) 4.79822e6 1.31331
\(76\) −1.82912e6 −0.477963
\(77\) 2.06686e6 0.515933
\(78\) 1.10738e7 2.64220
\(79\) 3.25297e6 0.742309 0.371155 0.928571i \(-0.378962\pi\)
0.371155 + 0.928571i \(0.378962\pi\)
\(80\) −664811. −0.145172
\(81\) 2.21744e7 4.63612
\(82\) −4.23470e6 −0.848154
\(83\) 1.69719e6 0.325804 0.162902 0.986642i \(-0.447915\pi\)
0.162902 + 0.986642i \(0.447915\pi\)
\(84\) 4.92052e6 0.905803
\(85\) −3.34108e6 −0.590094
\(86\) −4.52177e6 −0.766592
\(87\) 5.64704e6 0.919397
\(88\) 1.27543e6 0.199511
\(89\) 790719. 0.118893 0.0594466 0.998231i \(-0.481066\pi\)
0.0594466 + 0.998231i \(0.481066\pi\)
\(90\) 8.30942e6 1.20150
\(91\) −1.23944e7 −1.72417
\(92\) 4.67434e6 0.625840
\(93\) 1.43246e7 1.84668
\(94\) −5.72167e6 −0.710518
\(95\) 4.63874e6 0.555095
\(96\) 3.03639e6 0.350274
\(97\) −1.20273e7 −1.33804 −0.669019 0.743245i \(-0.733286\pi\)
−0.669019 + 0.743245i \(0.733286\pi\)
\(98\) 1.08104e6 0.116025
\(99\) −1.59415e7 −1.65122
\(100\) −3.31401e6 −0.331401
\(101\) 5.37844e6 0.519435 0.259718 0.965685i \(-0.416371\pi\)
0.259718 + 0.965685i \(0.416371\pi\)
\(102\) 1.52597e7 1.42379
\(103\) −1.04974e7 −0.946571 −0.473286 0.880909i \(-0.656932\pi\)
−0.473286 + 0.880909i \(0.656932\pi\)
\(104\) −7.64840e6 −0.666736
\(105\) −1.24787e7 −1.05198
\(106\) 9.61266e6 0.783923
\(107\) −1.34458e7 −1.06107 −0.530535 0.847663i \(-0.678009\pi\)
−0.530535 + 0.847663i \(0.678009\pi\)
\(108\) −2.49817e7 −1.90827
\(109\) −1.99845e7 −1.47809 −0.739043 0.673658i \(-0.764722\pi\)
−0.739043 + 0.673658i \(0.764722\pi\)
\(110\) −3.23455e6 −0.231707
\(111\) −4.69366e6 −0.325748
\(112\) −3.39848e6 −0.228571
\(113\) 1.03035e7 0.671754 0.335877 0.941906i \(-0.390967\pi\)
0.335877 + 0.941906i \(0.390967\pi\)
\(114\) −2.11865e7 −1.33934
\(115\) −1.18544e7 −0.726836
\(116\) −3.90026e6 −0.232001
\(117\) 9.55968e7 5.51814
\(118\) 8.85201e6 0.495969
\(119\) −1.70794e7 −0.929093
\(120\) −7.70043e6 −0.406800
\(121\) −1.32817e7 −0.681563
\(122\) 7.13763e6 0.355872
\(123\) −4.90501e7 −2.37669
\(124\) −9.89361e6 −0.465992
\(125\) 2.10847e7 0.965569
\(126\) 4.24773e7 1.89174
\(127\) −8.28871e6 −0.359066 −0.179533 0.983752i \(-0.557459\pi\)
−0.179533 + 0.983752i \(0.557459\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) −5.23752e7 −2.14814
\(130\) 1.93967e7 0.774331
\(131\) −1.88833e7 −0.733885 −0.366943 0.930244i \(-0.619595\pi\)
−0.366943 + 0.930244i \(0.619595\pi\)
\(132\) 1.47732e7 0.559068
\(133\) 2.37130e7 0.873988
\(134\) −3.02074e7 −1.08454
\(135\) 6.33549e7 2.21621
\(136\) −1.05395e7 −0.359280
\(137\) 3.38455e7 1.12455 0.562276 0.826950i \(-0.309926\pi\)
0.562276 + 0.826950i \(0.309926\pi\)
\(138\) 5.41424e7 1.75372
\(139\) −4.12412e7 −1.30250 −0.651252 0.758861i \(-0.725756\pi\)
−0.651252 + 0.758861i \(0.725756\pi\)
\(140\) 8.61871e6 0.265457
\(141\) −6.62735e7 −1.99101
\(142\) −5.22209e6 −0.153051
\(143\) −3.72123e7 −1.06417
\(144\) 2.62122e7 0.731533
\(145\) 9.89126e6 0.269441
\(146\) 1.21864e7 0.324072
\(147\) 1.25216e7 0.325124
\(148\) 3.24179e6 0.0821995
\(149\) −3.33745e7 −0.826538 −0.413269 0.910609i \(-0.635613\pi\)
−0.413269 + 0.910609i \(0.635613\pi\)
\(150\) −3.83858e7 −0.928647
\(151\) −733151. −0.0173290 −0.00866452 0.999962i \(-0.502758\pi\)
−0.00866452 + 0.999962i \(0.502758\pi\)
\(152\) 1.46329e7 0.337971
\(153\) 1.31732e8 2.97353
\(154\) −1.65349e7 −0.364820
\(155\) 2.50907e7 0.541193
\(156\) −8.85905e7 −1.86832
\(157\) −8.21899e7 −1.69500 −0.847500 0.530796i \(-0.821893\pi\)
−0.847500 + 0.530796i \(0.821893\pi\)
\(158\) −2.60237e7 −0.524892
\(159\) 1.11342e8 2.19670
\(160\) 5.31849e6 0.102652
\(161\) −6.05990e7 −1.14439
\(162\) −1.77395e8 −3.27823
\(163\) 4.65829e7 0.842500 0.421250 0.906945i \(-0.361592\pi\)
0.421250 + 0.906945i \(0.361592\pi\)
\(164\) 3.38776e7 0.599735
\(165\) −3.74655e7 −0.649288
\(166\) −1.35775e7 −0.230378
\(167\) −5.05560e6 −0.0839972 −0.0419986 0.999118i \(-0.513372\pi\)
−0.0419986 + 0.999118i \(0.513372\pi\)
\(168\) −3.93642e7 −0.640499
\(169\) 1.60404e8 2.55629
\(170\) 2.67287e7 0.417259
\(171\) −1.82896e8 −2.79717
\(172\) 3.61742e7 0.542062
\(173\) 8.42981e7 1.23782 0.618909 0.785463i \(-0.287575\pi\)
0.618909 + 0.785463i \(0.287575\pi\)
\(174\) −4.51763e7 −0.650112
\(175\) 4.29633e7 0.605989
\(176\) −1.02034e7 −0.141076
\(177\) 1.02532e8 1.38980
\(178\) −6.32575e6 −0.0840702
\(179\) −3.42213e7 −0.445975 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(180\) −6.64754e7 −0.849586
\(181\) −8.77181e7 −1.09955 −0.549774 0.835314i \(-0.685286\pi\)
−0.549774 + 0.835314i \(0.685286\pi\)
\(182\) 9.91551e7 1.21917
\(183\) 8.26743e7 0.997222
\(184\) −3.73948e7 −0.442536
\(185\) −8.22135e6 −0.0954646
\(186\) −1.14597e8 −1.30580
\(187\) −5.12785e7 −0.573442
\(188\) 4.57734e7 0.502412
\(189\) 3.23867e8 3.48939
\(190\) −3.71099e7 −0.392511
\(191\) −3.57361e7 −0.371099 −0.185550 0.982635i \(-0.559407\pi\)
−0.185550 + 0.982635i \(0.559407\pi\)
\(192\) −2.42911e7 −0.247681
\(193\) −1.36105e8 −1.36277 −0.681386 0.731924i \(-0.738623\pi\)
−0.681386 + 0.731924i \(0.738623\pi\)
\(194\) 9.62187e7 0.946136
\(195\) 2.24670e8 2.16982
\(196\) −8.64835e6 −0.0820421
\(197\) −9.49071e7 −0.884437 −0.442219 0.896907i \(-0.645809\pi\)
−0.442219 + 0.896907i \(0.645809\pi\)
\(198\) 1.27532e8 1.16759
\(199\) −2.06856e7 −0.186072 −0.0930362 0.995663i \(-0.529657\pi\)
−0.0930362 + 0.995663i \(0.529657\pi\)
\(200\) 2.65120e7 0.234336
\(201\) −3.49889e8 −3.03910
\(202\) −4.30275e7 −0.367296
\(203\) 5.05636e7 0.424230
\(204\) −1.22078e8 −1.00677
\(205\) −8.59154e7 −0.696518
\(206\) 8.39796e7 0.669327
\(207\) 4.67394e8 3.66258
\(208\) 6.11872e7 0.471453
\(209\) 7.11948e7 0.539431
\(210\) 9.98296e7 0.743861
\(211\) 1.43821e8 1.05398 0.526991 0.849871i \(-0.323320\pi\)
0.526991 + 0.849871i \(0.323320\pi\)
\(212\) −7.69013e7 −0.554317
\(213\) −6.04869e7 −0.428877
\(214\) 1.07567e8 0.750290
\(215\) −9.17396e7 −0.629538
\(216\) 1.99854e8 1.34935
\(217\) 1.28262e8 0.852099
\(218\) 1.59876e8 1.04516
\(219\) 1.41154e8 0.908111
\(220\) 2.58764e7 0.163842
\(221\) 3.07503e8 1.91636
\(222\) 3.75493e7 0.230339
\(223\) −4.75735e7 −0.287275 −0.143638 0.989630i \(-0.545880\pi\)
−0.143638 + 0.989630i \(0.545880\pi\)
\(224\) 2.71878e7 0.161624
\(225\) −3.31372e8 −1.93944
\(226\) −8.24280e7 −0.475002
\(227\) 9.05773e7 0.513959 0.256980 0.966417i \(-0.417273\pi\)
0.256980 + 0.966417i \(0.417273\pi\)
\(228\) 1.69492e8 0.947058
\(229\) 1.91227e8 1.05226 0.526132 0.850403i \(-0.323642\pi\)
0.526132 + 0.850403i \(0.323642\pi\)
\(230\) 9.48350e7 0.513951
\(231\) −1.91522e8 −1.02229
\(232\) 3.12021e7 0.164050
\(233\) −2.86677e7 −0.148473 −0.0742364 0.997241i \(-0.523652\pi\)
−0.0742364 + 0.997241i \(0.523652\pi\)
\(234\) −7.64774e8 −3.90192
\(235\) −1.16084e8 −0.583490
\(236\) −7.08161e7 −0.350703
\(237\) −3.01430e8 −1.47085
\(238\) 1.36635e8 0.656968
\(239\) 2.35751e6 0.0111702 0.00558509 0.999984i \(-0.498222\pi\)
0.00558509 + 0.999984i \(0.498222\pi\)
\(240\) 6.16034e7 0.287651
\(241\) −1.43697e8 −0.661284 −0.330642 0.943756i \(-0.607265\pi\)
−0.330642 + 0.943756i \(0.607265\pi\)
\(242\) 1.06254e8 0.481938
\(243\) −1.20108e9 −5.36970
\(244\) −5.71010e7 −0.251640
\(245\) 2.19327e7 0.0952818
\(246\) 3.92401e8 1.68057
\(247\) −4.26935e8 −1.80270
\(248\) 7.91489e7 0.329506
\(249\) −1.57267e8 −0.645564
\(250\) −1.68678e8 −0.682760
\(251\) 2.31993e7 0.0926011 0.0463005 0.998928i \(-0.485257\pi\)
0.0463005 + 0.998928i \(0.485257\pi\)
\(252\) −3.39819e8 −1.33766
\(253\) −1.81940e8 −0.706326
\(254\) 6.63097e7 0.253898
\(255\) 3.09595e8 1.16924
\(256\) 1.67772e7 0.0625000
\(257\) 1.26816e8 0.466026 0.233013 0.972474i \(-0.425142\pi\)
0.233013 + 0.972474i \(0.425142\pi\)
\(258\) 4.19002e8 1.51896
\(259\) −4.20271e7 −0.150307
\(260\) −1.55174e8 −0.547535
\(261\) −3.89993e8 −1.35773
\(262\) 1.51066e8 0.518935
\(263\) 1.73549e8 0.588269 0.294135 0.955764i \(-0.404969\pi\)
0.294135 + 0.955764i \(0.404969\pi\)
\(264\) −1.18185e8 −0.395321
\(265\) 1.95026e8 0.643771
\(266\) −1.89704e8 −0.618003
\(267\) −7.32705e7 −0.235581
\(268\) 2.41659e8 0.766888
\(269\) 1.56277e8 0.489509 0.244755 0.969585i \(-0.421293\pi\)
0.244755 + 0.969585i \(0.421293\pi\)
\(270\) −5.06839e8 −1.56710
\(271\) 1.54301e8 0.470953 0.235476 0.971880i \(-0.424335\pi\)
0.235476 + 0.971880i \(0.424335\pi\)
\(272\) 8.43158e7 0.254049
\(273\) 1.14850e9 3.41635
\(274\) −2.70764e8 −0.795178
\(275\) 1.28991e8 0.374020
\(276\) −4.33139e8 −1.24007
\(277\) −4.55119e8 −1.28661 −0.643303 0.765611i \(-0.722437\pi\)
−0.643303 + 0.765611i \(0.722437\pi\)
\(278\) 3.29929e8 0.921010
\(279\) −9.89277e8 −2.72711
\(280\) −6.89497e7 −0.187706
\(281\) 3.06284e8 0.823478 0.411739 0.911302i \(-0.364922\pi\)
0.411739 + 0.911302i \(0.364922\pi\)
\(282\) 5.30188e8 1.40785
\(283\) −6.94995e8 −1.82276 −0.911379 0.411568i \(-0.864981\pi\)
−0.911379 + 0.411568i \(0.864981\pi\)
\(284\) 4.17767e7 0.108223
\(285\) −4.29840e8 −1.09989
\(286\) 2.97699e8 0.752481
\(287\) −4.39195e8 −1.09666
\(288\) −2.09697e8 −0.517272
\(289\) 1.34000e7 0.0326559
\(290\) −7.91301e7 −0.190524
\(291\) 1.11449e9 2.65125
\(292\) −9.74914e7 −0.229153
\(293\) 6.62132e7 0.153783 0.0768915 0.997039i \(-0.475501\pi\)
0.0768915 + 0.997039i \(0.475501\pi\)
\(294\) −1.00173e8 −0.229898
\(295\) 1.79593e8 0.407298
\(296\) −2.59343e7 −0.0581238
\(297\) 9.72363e8 2.15368
\(298\) 2.66996e8 0.584451
\(299\) 1.09104e9 2.36043
\(300\) 3.07086e8 0.656653
\(301\) −4.68968e8 −0.991198
\(302\) 5.86521e6 0.0122535
\(303\) −4.98383e8 −1.02923
\(304\) −1.17064e8 −0.238981
\(305\) 1.44811e8 0.292249
\(306\) −1.05386e9 −2.10260
\(307\) −4.17909e8 −0.824323 −0.412162 0.911111i \(-0.635226\pi\)
−0.412162 + 0.911111i \(0.635226\pi\)
\(308\) 1.32279e8 0.257966
\(309\) 9.72726e8 1.87558
\(310\) −2.00726e8 −0.382681
\(311\) −1.00000e9 −1.88512 −0.942560 0.334037i \(-0.891589\pi\)
−0.942560 + 0.334037i \(0.891589\pi\)
\(312\) 7.08724e8 1.32110
\(313\) −6.97386e8 −1.28549 −0.642744 0.766081i \(-0.722204\pi\)
−0.642744 + 0.766081i \(0.722204\pi\)
\(314\) 6.57519e8 1.19855
\(315\) 8.61798e8 1.55353
\(316\) 2.08190e8 0.371155
\(317\) −3.46835e8 −0.611526 −0.305763 0.952108i \(-0.598912\pi\)
−0.305763 + 0.952108i \(0.598912\pi\)
\(318\) −8.90739e8 −1.55330
\(319\) 1.51810e8 0.261838
\(320\) −4.25479e7 −0.0725860
\(321\) 1.24593e9 2.10245
\(322\) 4.84792e8 0.809207
\(323\) −5.88316e8 −0.971408
\(324\) 1.41916e9 2.31806
\(325\) −7.73524e8 −1.24992
\(326\) −3.72663e8 −0.595737
\(327\) 1.85182e9 2.92875
\(328\) −2.71021e8 −0.424077
\(329\) −5.93413e8 −0.918695
\(330\) 2.99724e8 0.459116
\(331\) −1.30141e8 −0.197249 −0.0986247 0.995125i \(-0.531444\pi\)
−0.0986247 + 0.995125i \(0.531444\pi\)
\(332\) 1.08620e8 0.162902
\(333\) 3.24152e8 0.481053
\(334\) 4.04448e7 0.0593950
\(335\) −6.12861e8 −0.890646
\(336\) 3.14913e8 0.452902
\(337\) 9.96083e8 1.41772 0.708860 0.705349i \(-0.249210\pi\)
0.708860 + 0.705349i \(0.249210\pi\)
\(338\) −1.28323e9 −1.80757
\(339\) −9.54754e8 −1.33104
\(340\) −2.13829e8 −0.295047
\(341\) 3.85089e8 0.525921
\(342\) 1.46317e9 1.97789
\(343\) 7.95417e8 1.06430
\(344\) −2.89393e8 −0.383296
\(345\) 1.09846e9 1.44019
\(346\) −6.74385e8 −0.875269
\(347\) −1.34875e9 −1.73292 −0.866461 0.499244i \(-0.833611\pi\)
−0.866461 + 0.499244i \(0.833611\pi\)
\(348\) 3.61410e8 0.459699
\(349\) −1.32724e9 −1.67133 −0.835664 0.549241i \(-0.814917\pi\)
−0.835664 + 0.549241i \(0.814917\pi\)
\(350\) −3.43706e8 −0.428499
\(351\) −5.83099e9 −7.19727
\(352\) 8.16275e7 0.0997555
\(353\) 1.17151e9 1.41754 0.708768 0.705441i \(-0.249251\pi\)
0.708768 + 0.705441i \(0.249251\pi\)
\(354\) −8.20255e8 −0.982737
\(355\) −1.05948e8 −0.125688
\(356\) 5.06060e7 0.0594466
\(357\) 1.58263e9 1.84095
\(358\) 2.73770e8 0.315352
\(359\) 9.75337e8 1.11256 0.556281 0.830994i \(-0.312228\pi\)
0.556281 + 0.830994i \(0.312228\pi\)
\(360\) 5.31803e8 0.600748
\(361\) −7.70574e7 −0.0862063
\(362\) 7.01744e8 0.777497
\(363\) 1.23073e9 1.35048
\(364\) −7.93240e8 −0.862085
\(365\) 2.47243e8 0.266133
\(366\) −6.61395e8 −0.705142
\(367\) 7.36459e8 0.777710 0.388855 0.921299i \(-0.372871\pi\)
0.388855 + 0.921299i \(0.372871\pi\)
\(368\) 2.99158e8 0.312920
\(369\) 3.38747e9 3.50981
\(370\) 6.57708e7 0.0675036
\(371\) 9.96961e8 1.01361
\(372\) 9.16773e8 0.923340
\(373\) 1.10444e9 1.10195 0.550976 0.834521i \(-0.314256\pi\)
0.550976 + 0.834521i \(0.314256\pi\)
\(374\) 4.10228e8 0.405485
\(375\) −1.95378e9 −1.91322
\(376\) −3.66187e8 −0.355259
\(377\) −9.10362e8 −0.875023
\(378\) −2.59093e9 −2.46737
\(379\) −1.61968e8 −0.152824 −0.0764119 0.997076i \(-0.524346\pi\)
−0.0764119 + 0.997076i \(0.524346\pi\)
\(380\) 2.96879e8 0.277547
\(381\) 7.68057e8 0.711469
\(382\) 2.85889e8 0.262407
\(383\) −5.68832e8 −0.517355 −0.258677 0.965964i \(-0.583287\pi\)
−0.258677 + 0.965964i \(0.583287\pi\)
\(384\) 1.94329e8 0.175137
\(385\) −3.35466e8 −0.299596
\(386\) 1.08884e9 0.963626
\(387\) 3.61711e9 3.17229
\(388\) −7.69750e8 −0.669019
\(389\) −1.72728e9 −1.48778 −0.743889 0.668303i \(-0.767021\pi\)
−0.743889 + 0.668303i \(0.767021\pi\)
\(390\) −1.79736e9 −1.53430
\(391\) 1.50345e9 1.27195
\(392\) 6.91868e7 0.0580125
\(393\) 1.74978e9 1.45415
\(394\) 7.59257e8 0.625392
\(395\) −5.27980e8 −0.431050
\(396\) −1.02026e9 −0.825612
\(397\) −9.48140e8 −0.760511 −0.380256 0.924881i \(-0.624164\pi\)
−0.380256 + 0.924881i \(0.624164\pi\)
\(398\) 1.65485e8 0.131573
\(399\) −2.19732e9 −1.73176
\(400\) −2.12096e8 −0.165700
\(401\) 3.21964e8 0.249346 0.124673 0.992198i \(-0.460212\pi\)
0.124673 + 0.992198i \(0.460212\pi\)
\(402\) 2.79911e9 2.14897
\(403\) −2.30927e9 −1.75755
\(404\) 3.44220e8 0.259718
\(405\) −3.59907e9 −2.69214
\(406\) −4.04509e8 −0.299976
\(407\) −1.26180e8 −0.0927707
\(408\) 9.76621e8 0.711894
\(409\) 6.10954e8 0.441547 0.220774 0.975325i \(-0.429142\pi\)
0.220774 + 0.975325i \(0.429142\pi\)
\(410\) 6.87323e8 0.492513
\(411\) −3.13623e9 −2.22824
\(412\) −6.71837e8 −0.473286
\(413\) 9.18071e8 0.641285
\(414\) −3.73916e9 −2.58984
\(415\) −2.75466e8 −0.189191
\(416\) −4.89497e8 −0.333368
\(417\) 3.82154e9 2.58084
\(418\) −5.69558e8 −0.381435
\(419\) −7.32708e8 −0.486611 −0.243305 0.969950i \(-0.578232\pi\)
−0.243305 + 0.969950i \(0.578232\pi\)
\(420\) −7.98637e8 −0.525989
\(421\) 3.74946e8 0.244896 0.122448 0.992475i \(-0.460926\pi\)
0.122448 + 0.992475i \(0.460926\pi\)
\(422\) −1.15057e9 −0.745277
\(423\) 4.57694e9 2.94025
\(424\) 6.15210e8 0.391961
\(425\) −1.06591e9 −0.673536
\(426\) 4.83895e8 0.303262
\(427\) 7.40267e8 0.460141
\(428\) −8.60532e8 −0.530535
\(429\) 3.44821e9 2.10859
\(430\) 7.33917e8 0.445151
\(431\) −4.63081e8 −0.278604 −0.139302 0.990250i \(-0.544486\pi\)
−0.139302 + 0.990250i \(0.544486\pi\)
\(432\) −1.59883e9 −0.954133
\(433\) −2.04983e9 −1.21342 −0.606709 0.794924i \(-0.707511\pi\)
−0.606709 + 0.794924i \(0.707511\pi\)
\(434\) −1.02610e9 −0.602525
\(435\) −9.16555e8 −0.533883
\(436\) −1.27901e9 −0.739043
\(437\) −2.08738e9 −1.19651
\(438\) −1.12923e9 −0.642131
\(439\) −3.54372e8 −0.199909 −0.0999547 0.994992i \(-0.531870\pi\)
−0.0999547 + 0.994992i \(0.531870\pi\)
\(440\) −2.07011e8 −0.115854
\(441\) −8.64761e8 −0.480132
\(442\) −2.46002e9 −1.35507
\(443\) 1.19336e9 0.652166 0.326083 0.945341i \(-0.394271\pi\)
0.326083 + 0.945341i \(0.394271\pi\)
\(444\) −3.00395e8 −0.162874
\(445\) −1.28339e8 −0.0690399
\(446\) 3.80588e8 0.203134
\(447\) 3.09258e9 1.63774
\(448\) −2.17503e8 −0.114286
\(449\) 3.36121e9 1.75240 0.876201 0.481946i \(-0.160070\pi\)
0.876201 + 0.481946i \(0.160070\pi\)
\(450\) 2.65098e9 1.37139
\(451\) −1.31862e9 −0.676864
\(452\) 6.59424e8 0.335877
\(453\) 6.79361e7 0.0343366
\(454\) −7.24618e8 −0.363424
\(455\) 2.01170e9 1.00121
\(456\) −1.35593e9 −0.669671
\(457\) −4.06865e8 −0.199409 −0.0997044 0.995017i \(-0.531790\pi\)
−0.0997044 + 0.995017i \(0.531790\pi\)
\(458\) −1.52981e9 −0.744063
\(459\) −8.03510e9 −3.87835
\(460\) −7.58680e8 −0.363418
\(461\) −3.53394e9 −1.67999 −0.839994 0.542596i \(-0.817441\pi\)
−0.839994 + 0.542596i \(0.817441\pi\)
\(462\) 1.53217e9 0.722871
\(463\) −1.05179e9 −0.492488 −0.246244 0.969208i \(-0.579196\pi\)
−0.246244 + 0.969208i \(0.579196\pi\)
\(464\) −2.49617e8 −0.116001
\(465\) −2.32498e9 −1.07234
\(466\) 2.29341e8 0.104986
\(467\) −1.82295e9 −0.828259 −0.414130 0.910218i \(-0.635914\pi\)
−0.414130 + 0.910218i \(0.635914\pi\)
\(468\) 6.11820e9 2.75907
\(469\) −3.13291e9 −1.40231
\(470\) 9.28669e8 0.412590
\(471\) 7.61597e9 3.35855
\(472\) 5.66528e8 0.247985
\(473\) −1.40801e9 −0.611774
\(474\) 2.41144e9 1.04005
\(475\) 1.47991e9 0.633589
\(476\) −1.09308e9 −0.464546
\(477\) −7.68947e9 −3.24401
\(478\) −1.88600e7 −0.00789851
\(479\) 3.30582e9 1.37437 0.687187 0.726480i \(-0.258845\pi\)
0.687187 + 0.726480i \(0.258845\pi\)
\(480\) −4.92827e8 −0.203400
\(481\) 7.56669e8 0.310026
\(482\) 1.14958e9 0.467598
\(483\) 5.61529e9 2.26755
\(484\) −8.50031e8 −0.340781
\(485\) 1.95212e9 0.776983
\(486\) 9.60864e9 3.79695
\(487\) 3.05374e9 1.19806 0.599032 0.800725i \(-0.295552\pi\)
0.599032 + 0.800725i \(0.295552\pi\)
\(488\) 4.56808e8 0.177936
\(489\) −4.31652e9 −1.66937
\(490\) −1.75461e8 −0.0673744
\(491\) 3.58565e9 1.36704 0.683522 0.729930i \(-0.260447\pi\)
0.683522 + 0.729930i \(0.260447\pi\)
\(492\) −3.13921e9 −1.18834
\(493\) −1.25448e9 −0.471518
\(494\) 3.41548e9 1.27470
\(495\) 2.58742e9 0.958846
\(496\) −6.33191e8 −0.232996
\(497\) −5.41600e8 −0.197894
\(498\) 1.25813e9 0.456483
\(499\) 4.42952e9 1.59590 0.797948 0.602726i \(-0.205919\pi\)
0.797948 + 0.602726i \(0.205919\pi\)
\(500\) 1.34942e9 0.482785
\(501\) 4.68467e8 0.166436
\(502\) −1.85594e8 −0.0654788
\(503\) −5.41281e8 −0.189642 −0.0948211 0.995494i \(-0.530228\pi\)
−0.0948211 + 0.995494i \(0.530228\pi\)
\(504\) 2.71855e9 0.945868
\(505\) −8.72960e8 −0.301630
\(506\) 1.45552e9 0.499448
\(507\) −1.48635e10 −5.06516
\(508\) −5.30477e8 −0.179533
\(509\) −3.16819e9 −1.06487 −0.532437 0.846469i \(-0.678724\pi\)
−0.532437 + 0.846469i \(0.678724\pi\)
\(510\) −2.47676e9 −0.826777
\(511\) 1.26389e9 0.419023
\(512\) −1.34218e8 −0.0441942
\(513\) 1.11559e10 3.64832
\(514\) −1.01453e9 −0.329530
\(515\) 1.70381e9 0.549663
\(516\) −3.35201e9 −1.07407
\(517\) −1.78164e9 −0.567025
\(518\) 3.36217e8 0.106283
\(519\) −7.81133e9 −2.45267
\(520\) 1.24139e9 0.387166
\(521\) −2.92545e9 −0.906276 −0.453138 0.891440i \(-0.649695\pi\)
−0.453138 + 0.891440i \(0.649695\pi\)
\(522\) 3.11994e9 0.960063
\(523\) 2.78256e8 0.0850529 0.0425264 0.999095i \(-0.486459\pi\)
0.0425264 + 0.999095i \(0.486459\pi\)
\(524\) −1.20853e9 −0.366943
\(525\) −3.98111e9 −1.20073
\(526\) −1.38839e9 −0.415969
\(527\) −3.18217e9 −0.947080
\(528\) 9.45482e8 0.279534
\(529\) 1.92952e9 0.566703
\(530\) −1.56021e9 −0.455215
\(531\) −7.08100e9 −2.05241
\(532\) 1.51763e9 0.436994
\(533\) 7.90740e9 2.26198
\(534\) 5.86164e8 0.166581
\(535\) 2.18235e9 0.616151
\(536\) −1.93328e9 −0.542272
\(537\) 3.17105e9 0.883676
\(538\) −1.25021e9 −0.346135
\(539\) 3.36620e8 0.0925931
\(540\) 4.05471e9 1.10811
\(541\) 5.70898e9 1.55013 0.775065 0.631882i \(-0.217717\pi\)
0.775065 + 0.631882i \(0.217717\pi\)
\(542\) −1.23441e9 −0.333014
\(543\) 8.12823e9 2.17870
\(544\) −6.74526e8 −0.179640
\(545\) 3.24362e9 0.858307
\(546\) −9.18802e9 −2.41573
\(547\) −4.03987e9 −1.05539 −0.527693 0.849435i \(-0.676943\pi\)
−0.527693 + 0.849435i \(0.676943\pi\)
\(548\) 2.16611e9 0.562276
\(549\) −5.70961e9 −1.47266
\(550\) −1.03193e9 −0.264472
\(551\) 1.74171e9 0.443552
\(552\) 3.46511e9 0.876861
\(553\) −2.69901e9 −0.678682
\(554\) 3.64095e9 0.909768
\(555\) 7.61816e8 0.189158
\(556\) −2.63944e9 −0.651252
\(557\) −3.56678e9 −0.874546 −0.437273 0.899329i \(-0.644056\pi\)
−0.437273 + 0.899329i \(0.644056\pi\)
\(558\) 7.91421e9 1.92836
\(559\) 8.44344e9 2.04446
\(560\) 5.51598e8 0.132729
\(561\) 4.75163e9 1.13625
\(562\) −2.45027e9 −0.582287
\(563\) −2.36473e9 −0.558473 −0.279237 0.960222i \(-0.590081\pi\)
−0.279237 + 0.960222i \(0.590081\pi\)
\(564\) −4.24150e9 −0.995504
\(565\) −1.67233e9 −0.390079
\(566\) 5.55996e9 1.28888
\(567\) −1.83983e10 −4.23873
\(568\) −3.34214e8 −0.0765253
\(569\) −4.24062e9 −0.965020 −0.482510 0.875890i \(-0.660275\pi\)
−0.482510 + 0.875890i \(0.660275\pi\)
\(570\) 3.43872e9 0.777740
\(571\) 2.50010e9 0.561992 0.280996 0.959709i \(-0.409335\pi\)
0.280996 + 0.959709i \(0.409335\pi\)
\(572\) −2.38159e9 −0.532085
\(573\) 3.31142e9 0.735314
\(574\) 3.51356e9 0.775453
\(575\) −3.78193e9 −0.829615
\(576\) 1.67758e9 0.365767
\(577\) 7.62148e9 1.65167 0.825837 0.563909i \(-0.190703\pi\)
0.825837 + 0.563909i \(0.190703\pi\)
\(578\) −1.07200e8 −0.0230912
\(579\) 1.26119e10 2.70026
\(580\) 6.33041e8 0.134720
\(581\) −1.40817e9 −0.297878
\(582\) −8.91592e9 −1.87472
\(583\) 2.99323e9 0.625605
\(584\) 7.79931e8 0.162036
\(585\) −1.55161e10 −3.20432
\(586\) −5.29706e8 −0.108741
\(587\) −5.60882e8 −0.114456 −0.0572279 0.998361i \(-0.518226\pi\)
−0.0572279 + 0.998361i \(0.518226\pi\)
\(588\) 8.01383e8 0.162562
\(589\) 4.41811e9 0.890908
\(590\) −1.43675e9 −0.288004
\(591\) 8.79439e9 1.75247
\(592\) 2.07475e8 0.0410997
\(593\) 1.90793e8 0.0375725 0.0187863 0.999824i \(-0.494020\pi\)
0.0187863 + 0.999824i \(0.494020\pi\)
\(594\) −7.77890e9 −1.52288
\(595\) 2.77212e9 0.539513
\(596\) −2.13597e9 −0.413269
\(597\) 1.91679e9 0.368693
\(598\) −8.72833e9 −1.66908
\(599\) −6.24289e9 −1.18684 −0.593419 0.804894i \(-0.702222\pi\)
−0.593419 + 0.804894i \(0.702222\pi\)
\(600\) −2.45669e9 −0.464324
\(601\) −8.29739e9 −1.55913 −0.779563 0.626324i \(-0.784559\pi\)
−0.779563 + 0.626324i \(0.784559\pi\)
\(602\) 3.75174e9 0.700883
\(603\) 2.41639e10 4.48803
\(604\) −4.69217e7 −0.00866452
\(605\) 2.15572e9 0.395776
\(606\) 3.98706e9 0.727778
\(607\) −1.53780e9 −0.279087 −0.139544 0.990216i \(-0.544564\pi\)
−0.139544 + 0.990216i \(0.544564\pi\)
\(608\) 9.36508e8 0.168985
\(609\) −4.68538e9 −0.840590
\(610\) −1.15849e9 −0.206651
\(611\) 1.06840e10 1.89491
\(612\) 8.43086e9 1.48676
\(613\) 1.96256e9 0.344122 0.172061 0.985086i \(-0.444957\pi\)
0.172061 + 0.985086i \(0.444957\pi\)
\(614\) 3.34327e9 0.582885
\(615\) 7.96119e9 1.38011
\(616\) −1.05823e9 −0.182410
\(617\) 8.98380e9 1.53979 0.769896 0.638169i \(-0.220308\pi\)
0.769896 + 0.638169i \(0.220308\pi\)
\(618\) −7.78181e9 −1.32624
\(619\) 1.00564e9 0.170422 0.0852108 0.996363i \(-0.472844\pi\)
0.0852108 + 0.996363i \(0.472844\pi\)
\(620\) 1.60581e9 0.270596
\(621\) −2.85090e10 −4.77708
\(622\) 8.00001e9 1.33298
\(623\) −6.56064e8 −0.108702
\(624\) −5.66979e9 −0.934160
\(625\) 6.23209e8 0.102107
\(626\) 5.57909e9 0.908978
\(627\) −6.59713e9 −1.06885
\(628\) −5.26015e9 −0.847500
\(629\) 1.04269e9 0.167062
\(630\) −6.89438e9 −1.09851
\(631\) −5.39559e9 −0.854941 −0.427470 0.904029i \(-0.640595\pi\)
−0.427470 + 0.904029i \(0.640595\pi\)
\(632\) −1.66552e9 −0.262446
\(633\) −1.33269e10 −2.08841
\(634\) 2.77468e9 0.432414
\(635\) 1.34532e9 0.208505
\(636\) 7.12591e9 1.09835
\(637\) −2.01862e9 −0.309432
\(638\) −1.21448e9 −0.185147
\(639\) 4.17732e9 0.633351
\(640\) 3.40383e8 0.0513261
\(641\) 3.66480e9 0.549601 0.274800 0.961501i \(-0.411388\pi\)
0.274800 + 0.961501i \(0.411388\pi\)
\(642\) −9.96745e9 −1.48666
\(643\) −3.47973e9 −0.516187 −0.258093 0.966120i \(-0.583094\pi\)
−0.258093 + 0.966120i \(0.583094\pi\)
\(644\) −3.87833e9 −0.572196
\(645\) 8.50088e9 1.24740
\(646\) 4.70653e9 0.686890
\(647\) −9.32462e9 −1.35352 −0.676762 0.736202i \(-0.736618\pi\)
−0.676762 + 0.736202i \(0.736618\pi\)
\(648\) −1.13533e10 −1.63912
\(649\) 2.75637e9 0.395805
\(650\) 6.18819e9 0.883826
\(651\) −1.18852e10 −1.68839
\(652\) 2.98130e9 0.421250
\(653\) 1.21479e10 1.70728 0.853640 0.520864i \(-0.174390\pi\)
0.853640 + 0.520864i \(0.174390\pi\)
\(654\) −1.48146e10 −2.07094
\(655\) 3.06490e9 0.426159
\(656\) 2.16817e9 0.299868
\(657\) −9.74831e9 −1.34107
\(658\) 4.74731e9 0.649616
\(659\) −7.70738e9 −1.04908 −0.524539 0.851386i \(-0.675762\pi\)
−0.524539 + 0.851386i \(0.675762\pi\)
\(660\) −2.39779e9 −0.324644
\(661\) 2.21755e9 0.298654 0.149327 0.988788i \(-0.452289\pi\)
0.149327 + 0.988788i \(0.452289\pi\)
\(662\) 1.04113e9 0.139476
\(663\) −2.84942e10 −3.79716
\(664\) −8.68960e8 −0.115189
\(665\) −3.84879e9 −0.507514
\(666\) −2.59321e9 −0.340156
\(667\) −4.45097e9 −0.580783
\(668\) −3.23558e8 −0.0419986
\(669\) 4.40831e9 0.569220
\(670\) 4.90289e9 0.629782
\(671\) 2.22254e9 0.284002
\(672\) −2.51931e9 −0.320250
\(673\) −1.10936e10 −1.40287 −0.701437 0.712731i \(-0.747458\pi\)
−0.701437 + 0.712731i \(0.747458\pi\)
\(674\) −7.96866e9 −1.00248
\(675\) 2.02123e10 2.52960
\(676\) 1.02658e10 1.27815
\(677\) −6.44746e9 −0.798598 −0.399299 0.916821i \(-0.630747\pi\)
−0.399299 + 0.916821i \(0.630747\pi\)
\(678\) 7.63803e9 0.941191
\(679\) 9.97916e9 1.22335
\(680\) 1.71063e9 0.208630
\(681\) −8.39317e9 −1.01838
\(682\) −3.08071e9 −0.371883
\(683\) −1.68247e9 −0.202057 −0.101029 0.994884i \(-0.532213\pi\)
−0.101029 + 0.994884i \(0.532213\pi\)
\(684\) −1.17054e10 −1.39858
\(685\) −5.49338e9 −0.653014
\(686\) −6.36334e9 −0.752577
\(687\) −1.77197e10 −2.08500
\(688\) 2.31515e9 0.271031
\(689\) −1.79496e10 −2.09068
\(690\) −8.78771e9 −1.01837
\(691\) 4.16940e9 0.480729 0.240365 0.970683i \(-0.422733\pi\)
0.240365 + 0.970683i \(0.422733\pi\)
\(692\) 5.39508e9 0.618909
\(693\) 1.32268e10 1.50969
\(694\) 1.07900e10 1.22536
\(695\) 6.69374e9 0.756349
\(696\) −2.89128e9 −0.325056
\(697\) 1.08964e10 1.21890
\(698\) 1.06180e10 1.18181
\(699\) 2.65644e9 0.294191
\(700\) 2.74965e9 0.302994
\(701\) 7.00636e9 0.768209 0.384105 0.923290i \(-0.374510\pi\)
0.384105 + 0.923290i \(0.374510\pi\)
\(702\) 4.66479e10 5.08924
\(703\) −1.44766e9 −0.157153
\(704\) −6.53020e8 −0.0705378
\(705\) 1.07567e10 1.15615
\(706\) −9.37208e9 −1.00235
\(707\) −4.46252e9 −0.474911
\(708\) 6.56204e9 0.694900
\(709\) 2.63358e9 0.277514 0.138757 0.990326i \(-0.455689\pi\)
0.138757 + 0.990326i \(0.455689\pi\)
\(710\) 8.47584e8 0.0888747
\(711\) 2.08172e10 2.17210
\(712\) −4.04848e8 −0.0420351
\(713\) −1.12906e10 −1.16655
\(714\) −1.26611e10 −1.30175
\(715\) 6.03983e9 0.617951
\(716\) −2.19016e9 −0.222988
\(717\) −2.18454e8 −0.0221331
\(718\) −7.80269e9 −0.786700
\(719\) −1.37240e10 −1.37699 −0.688493 0.725243i \(-0.741727\pi\)
−0.688493 + 0.725243i \(0.741727\pi\)
\(720\) −4.25443e9 −0.424793
\(721\) 8.70980e9 0.865435
\(722\) 6.16459e8 0.0609571
\(723\) 1.33154e10 1.31030
\(724\) −5.61396e9 −0.549774
\(725\) 3.15564e9 0.307542
\(726\) −9.84581e9 −0.954934
\(727\) 7.19432e9 0.694415 0.347208 0.937788i \(-0.387130\pi\)
0.347208 + 0.937788i \(0.387130\pi\)
\(728\) 6.34592e9 0.609586
\(729\) 6.28004e10 6.00366
\(730\) −1.97795e9 −0.188185
\(731\) 1.16350e10 1.10168
\(732\) 5.29116e9 0.498611
\(733\) 2.44649e9 0.229446 0.114723 0.993398i \(-0.463402\pi\)
0.114723 + 0.993398i \(0.463402\pi\)
\(734\) −5.89168e9 −0.549924
\(735\) −2.03235e9 −0.188796
\(736\) −2.39326e9 −0.221268
\(737\) −9.40611e9 −0.865513
\(738\) −2.70998e10 −2.48181
\(739\) −4.23228e9 −0.385761 −0.192881 0.981222i \(-0.561783\pi\)
−0.192881 + 0.981222i \(0.561783\pi\)
\(740\) −5.26167e8 −0.0477323
\(741\) 3.95612e10 3.57195
\(742\) −7.97569e9 −0.716728
\(743\) 1.35161e10 1.20890 0.604450 0.796643i \(-0.293393\pi\)
0.604450 + 0.796643i \(0.293393\pi\)
\(744\) −7.33418e9 −0.652900
\(745\) 5.41692e9 0.479961
\(746\) −8.83554e9 −0.779197
\(747\) 1.08611e10 0.953346
\(748\) −3.28182e9 −0.286721
\(749\) 1.11561e10 0.970119
\(750\) 1.56302e10 1.35285
\(751\) −1.99922e10 −1.72235 −0.861175 0.508308i \(-0.830271\pi\)
−0.861175 + 0.508308i \(0.830271\pi\)
\(752\) 2.92949e9 0.251206
\(753\) −2.14972e9 −0.183484
\(754\) 7.28290e9 0.618735
\(755\) 1.18996e8 0.0100628
\(756\) 2.07275e10 1.74470
\(757\) 6.53816e9 0.547798 0.273899 0.961759i \(-0.411687\pi\)
0.273899 + 0.961759i \(0.411687\pi\)
\(758\) 1.29574e9 0.108063
\(759\) 1.68591e10 1.39955
\(760\) −2.37503e9 −0.196256
\(761\) −1.50769e10 −1.24013 −0.620064 0.784551i \(-0.712894\pi\)
−0.620064 + 0.784551i \(0.712894\pi\)
\(762\) −6.14446e9 −0.503085
\(763\) 1.65812e10 1.35139
\(764\) −2.28711e9 −0.185550
\(765\) −2.13811e10 −1.72669
\(766\) 4.55066e9 0.365825
\(767\) −1.65292e10 −1.32272
\(768\) −1.55463e9 −0.123840
\(769\) −4.34803e9 −0.344787 −0.172393 0.985028i \(-0.555150\pi\)
−0.172393 + 0.985028i \(0.555150\pi\)
\(770\) 2.68373e9 0.211846
\(771\) −1.17512e10 −0.923405
\(772\) −8.71072e9 −0.681386
\(773\) −8.80910e9 −0.685967 −0.342983 0.939341i \(-0.611437\pi\)
−0.342983 + 0.939341i \(0.611437\pi\)
\(774\) −2.89369e10 −2.24315
\(775\) 8.00476e9 0.617721
\(776\) 6.15800e9 0.473068
\(777\) 3.89436e9 0.297826
\(778\) 1.38182e10 1.05202
\(779\) −1.51285e10 −1.14660
\(780\) 1.43789e10 1.08491
\(781\) −1.62608e9 −0.122141
\(782\) −1.20276e10 −0.899407
\(783\) 2.37879e10 1.77088
\(784\) −5.53494e8 −0.0410211
\(785\) 1.33400e10 0.984266
\(786\) −1.39983e10 −1.02824
\(787\) −2.46888e10 −1.80547 −0.902733 0.430202i \(-0.858442\pi\)
−0.902733 + 0.430202i \(0.858442\pi\)
\(788\) −6.07406e9 −0.442219
\(789\) −1.60816e10 −1.16562
\(790\) 4.22384e9 0.304799
\(791\) −8.54888e9 −0.614174
\(792\) 8.16205e9 0.583796
\(793\) −1.33280e10 −0.949092
\(794\) 7.58512e9 0.537763
\(795\) −1.80717e10 −1.27560
\(796\) −1.32388e9 −0.0930362
\(797\) −2.28771e10 −1.60065 −0.800326 0.599565i \(-0.795340\pi\)
−0.800326 + 0.599565i \(0.795340\pi\)
\(798\) 1.75785e10 1.22454
\(799\) 1.47225e10 1.02110
\(800\) 1.69677e9 0.117168
\(801\) 5.06017e9 0.347897
\(802\) −2.57571e9 −0.176314
\(803\) 3.79466e9 0.258624
\(804\) −2.23929e10 −1.51955
\(805\) 9.83565e9 0.664535
\(806\) 1.84742e10 1.24278
\(807\) −1.44811e10 −0.969937
\(808\) −2.75376e9 −0.183648
\(809\) −2.69945e10 −1.79248 −0.896242 0.443566i \(-0.853713\pi\)
−0.896242 + 0.443566i \(0.853713\pi\)
\(810\) 2.87926e10 1.90363
\(811\) −6.28429e9 −0.413698 −0.206849 0.978373i \(-0.566321\pi\)
−0.206849 + 0.978373i \(0.566321\pi\)
\(812\) 3.23607e9 0.212115
\(813\) −1.42980e10 −0.933168
\(814\) 1.00944e9 0.0655988
\(815\) −7.56074e9 −0.489230
\(816\) −7.81297e9 −0.503385
\(817\) −1.61540e10 −1.03634
\(818\) −4.88763e9 −0.312221
\(819\) −7.93173e10 −5.04515
\(820\) −5.49859e9 −0.348259
\(821\) −2.20355e10 −1.38970 −0.694851 0.719154i \(-0.744530\pi\)
−0.694851 + 0.719154i \(0.744530\pi\)
\(822\) 2.50899e10 1.57560
\(823\) 2.99068e10 1.87012 0.935062 0.354485i \(-0.115344\pi\)
0.935062 + 0.354485i \(0.115344\pi\)
\(824\) 5.37469e9 0.334664
\(825\) −1.19527e10 −0.741101
\(826\) −7.34457e9 −0.453457
\(827\) 3.20833e9 0.197247 0.0986235 0.995125i \(-0.468556\pi\)
0.0986235 + 0.995125i \(0.468556\pi\)
\(828\) 2.99132e10 1.83129
\(829\) −1.33414e10 −0.813321 −0.406660 0.913579i \(-0.633307\pi\)
−0.406660 + 0.913579i \(0.633307\pi\)
\(830\) 2.20373e9 0.133778
\(831\) 4.21728e10 2.54934
\(832\) 3.91598e9 0.235727
\(833\) −2.78165e9 −0.166742
\(834\) −3.05723e10 −1.82493
\(835\) 8.20560e8 0.0487762
\(836\) 4.55646e9 0.269716
\(837\) 6.03416e10 3.55695
\(838\) 5.86166e9 0.344086
\(839\) 2.28049e9 0.133310 0.0666549 0.997776i \(-0.478767\pi\)
0.0666549 + 0.997776i \(0.478767\pi\)
\(840\) 6.38909e9 0.371931
\(841\) −1.35360e10 −0.784701
\(842\) −2.99957e9 −0.173167
\(843\) −2.83812e10 −1.63168
\(844\) 9.20453e9 0.526991
\(845\) −2.60347e10 −1.48441
\(846\) −3.66156e10 −2.07907
\(847\) 1.10199e10 0.623142
\(848\) −4.92168e9 −0.277159
\(849\) 6.44004e10 3.61170
\(850\) 8.52731e9 0.476262
\(851\) 3.69952e9 0.205775
\(852\) −3.87116e9 −0.214439
\(853\) −6.93868e8 −0.0382785 −0.0191393 0.999817i \(-0.506093\pi\)
−0.0191393 + 0.999817i \(0.506093\pi\)
\(854\) −5.92213e9 −0.325369
\(855\) 2.96854e10 1.62428
\(856\) 6.88426e9 0.375145
\(857\) −6.73153e9 −0.365326 −0.182663 0.983176i \(-0.558472\pi\)
−0.182663 + 0.983176i \(0.558472\pi\)
\(858\) −2.75857e10 −1.49100
\(859\) 3.87789e9 0.208747 0.104373 0.994538i \(-0.466716\pi\)
0.104373 + 0.994538i \(0.466716\pi\)
\(860\) −5.87133e9 −0.314769
\(861\) 4.06972e10 2.17297
\(862\) 3.70465e9 0.197002
\(863\) 5.08731e9 0.269433 0.134716 0.990884i \(-0.456988\pi\)
0.134716 + 0.990884i \(0.456988\pi\)
\(864\) 1.27906e10 0.674674
\(865\) −1.36822e10 −0.718786
\(866\) 1.63987e10 0.858016
\(867\) −1.24168e9 −0.0647059
\(868\) 8.20879e9 0.426050
\(869\) −8.10337e9 −0.418887
\(870\) 7.33244e9 0.377512
\(871\) 5.64059e10 2.89242
\(872\) 1.02320e10 0.522582
\(873\) −7.69684e10 −3.91528
\(874\) 1.66991e10 0.846062
\(875\) −1.74941e10 −0.882805
\(876\) 9.03386e9 0.454055
\(877\) 1.60299e10 0.802476 0.401238 0.915974i \(-0.368580\pi\)
0.401238 + 0.915974i \(0.368580\pi\)
\(878\) 2.83497e9 0.141357
\(879\) −6.13553e9 −0.304713
\(880\) 1.65609e9 0.0819209
\(881\) −3.22718e10 −1.59004 −0.795018 0.606586i \(-0.792539\pi\)
−0.795018 + 0.606586i \(0.792539\pi\)
\(882\) 6.91809e9 0.339505
\(883\) −1.30984e9 −0.0640261 −0.0320130 0.999487i \(-0.510192\pi\)
−0.0320130 + 0.999487i \(0.510192\pi\)
\(884\) 1.96802e10 0.958179
\(885\) −1.66417e10 −0.807040
\(886\) −9.54688e9 −0.461151
\(887\) −1.02440e10 −0.492876 −0.246438 0.969159i \(-0.579260\pi\)
−0.246438 + 0.969159i \(0.579260\pi\)
\(888\) 2.40316e9 0.115169
\(889\) 6.87719e9 0.328288
\(890\) 1.02672e9 0.0488186
\(891\) −5.52381e10 −2.61617
\(892\) −3.04470e9 −0.143638
\(893\) −2.04406e10 −0.960538
\(894\) −2.47407e10 −1.15806
\(895\) 5.55436e9 0.258973
\(896\) 1.74002e9 0.0808121
\(897\) −1.01099e11 −4.67708
\(898\) −2.68897e10 −1.23914
\(899\) 9.42081e9 0.432444
\(900\) −2.12078e10 −0.969722
\(901\) −2.47345e10 −1.12659
\(902\) 1.05490e10 0.478615
\(903\) 4.34560e10 1.96401
\(904\) −5.27539e9 −0.237501
\(905\) 1.42373e10 0.638494
\(906\) −5.43489e8 −0.0242796
\(907\) −5.59296e9 −0.248895 −0.124447 0.992226i \(-0.539716\pi\)
−0.124447 + 0.992226i \(0.539716\pi\)
\(908\) 5.79695e9 0.256980
\(909\) 3.44191e10 1.51994
\(910\) −1.60936e10 −0.707959
\(911\) −4.22317e10 −1.85065 −0.925324 0.379177i \(-0.876207\pi\)
−0.925324 + 0.379177i \(0.876207\pi\)
\(912\) 1.08475e10 0.473529
\(913\) −4.22782e9 −0.183852
\(914\) 3.25492e9 0.141003
\(915\) −1.34186e10 −0.579075
\(916\) 1.22385e10 0.526132
\(917\) 1.56676e10 0.670980
\(918\) 6.42808e10 2.74241
\(919\) 2.34854e10 0.998146 0.499073 0.866560i \(-0.333674\pi\)
0.499073 + 0.866560i \(0.333674\pi\)
\(920\) 6.06944e9 0.256975
\(921\) 3.87248e10 1.63335
\(922\) 2.82715e10 1.18793
\(923\) 9.75113e9 0.408177
\(924\) −1.22574e10 −0.511147
\(925\) −2.62288e9 −0.108964
\(926\) 8.41431e9 0.348241
\(927\) −6.71779e10 −2.76979
\(928\) 1.99693e9 0.0820249
\(929\) 2.42044e10 0.990463 0.495232 0.868761i \(-0.335083\pi\)
0.495232 + 0.868761i \(0.335083\pi\)
\(930\) 1.85999e10 0.758262
\(931\) 3.86202e9 0.156852
\(932\) −1.83473e9 −0.0742364
\(933\) 9.26632e10 3.73527
\(934\) 1.45836e10 0.585668
\(935\) 8.32288e9 0.332991
\(936\) −4.89456e10 −1.95096
\(937\) 4.02922e9 0.160005 0.0800023 0.996795i \(-0.474507\pi\)
0.0800023 + 0.996795i \(0.474507\pi\)
\(938\) 2.50633e10 0.991581
\(939\) 6.46220e10 2.54713
\(940\) −7.42935e9 −0.291745
\(941\) −2.81815e10 −1.10255 −0.551277 0.834322i \(-0.685859\pi\)
−0.551277 + 0.834322i \(0.685859\pi\)
\(942\) −6.09278e10 −2.37485
\(943\) 3.86611e10 1.50135
\(944\) −4.53223e9 −0.175352
\(945\) −5.25659e10 −2.02625
\(946\) 1.12641e10 0.432590
\(947\) 2.87539e10 1.10020 0.550100 0.835099i \(-0.314590\pi\)
0.550100 + 0.835099i \(0.314590\pi\)
\(948\) −1.92915e10 −0.735423
\(949\) −2.27555e10 −0.864281
\(950\) −1.18393e10 −0.448015
\(951\) 3.21388e10 1.21171
\(952\) 8.74467e9 0.328484
\(953\) −2.06957e10 −0.774559 −0.387280 0.921962i \(-0.626585\pi\)
−0.387280 + 0.921962i \(0.626585\pi\)
\(954\) 6.15158e10 2.29386
\(955\) 5.80023e9 0.215493
\(956\) 1.50880e8 0.00558509
\(957\) −1.40672e10 −0.518818
\(958\) −2.64466e10 −0.971829
\(959\) −2.80818e10 −1.02816
\(960\) 3.94262e9 0.143825
\(961\) −3.61527e9 −0.131404
\(962\) −6.05335e9 −0.219221
\(963\) −8.60459e10 −3.10483
\(964\) −9.19660e9 −0.330642
\(965\) 2.20908e10 0.791346
\(966\) −4.49223e10 −1.60340
\(967\) −1.30646e10 −0.464627 −0.232314 0.972641i \(-0.574630\pi\)
−0.232314 + 0.972641i \(0.574630\pi\)
\(968\) 6.80025e9 0.240969
\(969\) 5.45152e10 1.92479
\(970\) −1.56170e10 −0.549410
\(971\) 3.18275e10 1.11567 0.557835 0.829952i \(-0.311632\pi\)
0.557835 + 0.829952i \(0.311632\pi\)
\(972\) −7.68691e10 −2.68485
\(973\) 3.42181e10 1.19086
\(974\) −2.44299e10 −0.847160
\(975\) 7.16771e10 2.47665
\(976\) −3.65446e9 −0.125820
\(977\) −2.26551e10 −0.777203 −0.388601 0.921406i \(-0.627042\pi\)
−0.388601 + 0.921406i \(0.627042\pi\)
\(978\) 3.45321e10 1.18042
\(979\) −1.96974e9 −0.0670917
\(980\) 1.40369e9 0.0476409
\(981\) −1.27890e11 −4.32508
\(982\) −2.86852e10 −0.966646
\(983\) 2.41520e10 0.810991 0.405496 0.914097i \(-0.367099\pi\)
0.405496 + 0.914097i \(0.367099\pi\)
\(984\) 2.51136e10 0.840286
\(985\) 1.54041e10 0.513582
\(986\) 1.00358e10 0.333414
\(987\) 5.49875e10 1.82035
\(988\) −2.73239e10 −0.901349
\(989\) 4.12819e10 1.35698
\(990\) −2.06994e10 −0.678007
\(991\) −3.64240e10 −1.18886 −0.594429 0.804148i \(-0.702622\pi\)
−0.594429 + 0.804148i \(0.702622\pi\)
\(992\) 5.06553e9 0.164753
\(993\) 1.20593e10 0.390839
\(994\) 4.33280e9 0.139932
\(995\) 3.35742e9 0.108050
\(996\) −1.00651e10 −0.322782
\(997\) 1.72705e10 0.551914 0.275957 0.961170i \(-0.411005\pi\)
0.275957 + 0.961170i \(0.411005\pi\)
\(998\) −3.54361e10 −1.12847
\(999\) −1.97718e10 −0.627434
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.1.1 4
4.3 odd 2 592.8.a.b.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.8.a.a.1.1 4 1.1 even 1 trivial
592.8.a.b.1.4 4 4.3 odd 2