Properties

Label 350.8.c.i.99.3
Level $350$
Weight $8$
Character 350.99
Analytic conductor $109.335$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,8,Mod(99,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.99");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(109.334758919\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{8761})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4381x^{2} + 4796100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.3
Root \(-46.3001i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.8.c.i.99.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000i q^{2} -44.3001i q^{3} -64.0000 q^{4} +354.401 q^{6} +343.000i q^{7} -512.000i q^{8} +224.501 q^{9} +O(q^{10})\) \(q+8.00000i q^{2} -44.3001i q^{3} -64.0000 q^{4} +354.401 q^{6} +343.000i q^{7} -512.000i q^{8} +224.501 q^{9} +8294.71 q^{11} +2835.21i q^{12} -2086.69i q^{13} -2744.00 q^{14} +4096.00 q^{16} +22276.5i q^{17} +1796.00i q^{18} -22693.4 q^{19} +15194.9 q^{21} +66357.7i q^{22} +51846.8i q^{23} -22681.7 q^{24} +16693.6 q^{26} -106830. i q^{27} -21952.0i q^{28} +69360.5 q^{29} +54654.7 q^{31} +32768.0i q^{32} -367457. i q^{33} -178212. q^{34} -14368.0 q^{36} +301929. i q^{37} -181547. i q^{38} -92440.8 q^{39} -120041. q^{41} +121559. i q^{42} +260607. i q^{43} -530862. q^{44} -414774. q^{46} -282025. i q^{47} -181453. i q^{48} -117649. q^{49} +986852. q^{51} +133548. i q^{52} -413071. i q^{53} +854638. q^{54} +175616. q^{56} +1.00532e6i q^{57} +554884. i q^{58} -961690. q^{59} -2.52312e6 q^{61} +437238. i q^{62} +77003.7i q^{63} -262144. q^{64} +2.93965e6 q^{66} -133835. i q^{67} -1.42570e6i q^{68} +2.29682e6 q^{69} -5.71568e6 q^{71} -114944. i q^{72} -28628.0i q^{73} -2.41543e6 q^{74} +1.45238e6 q^{76} +2.84509e6i q^{77} -739526. i q^{78} +7.61428e6 q^{79} -4.24159e6 q^{81} -960326. i q^{82} +2.64843e6i q^{83} -972476. q^{84} -2.08486e6 q^{86} -3.07268e6i q^{87} -4.24689e6i q^{88} +7.82855e6 q^{89} +715736. q^{91} -3.31819e6i q^{92} -2.42121e6i q^{93} +2.25620e6 q^{94} +1.45163e6 q^{96} +9.84758e6i q^{97} -941192. i q^{98} +1.86217e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 256 q^{4} - 80 q^{6} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 256 q^{4} - 80 q^{6} - 38 q^{9} + 9030 q^{11} - 10976 q^{14} + 16384 q^{16} - 87404 q^{19} - 3430 q^{21} + 5120 q^{24} + 143152 q^{26} - 18522 q^{29} + 414056 q^{31} - 600528 q^{34} + 2432 q^{36} + 491546 q^{39} + 529596 q^{41} - 577920 q^{44} + 650208 q^{46} - 470596 q^{49} + 469410 q^{51} + 176240 q^{54} + 702464 q^{56} - 3810816 q^{59} - 238636 q^{61} - 1048576 q^{64} + 8860752 q^{66} + 13712652 q^{69} - 1225344 q^{71} + 2289152 q^{74} + 5593856 q^{76} + 25241518 q^{79} - 18995596 q^{81} + 219520 q^{84} - 8402336 q^{86} + 22356660 q^{89} + 6137642 q^{91} + 8121744 q^{94} - 327680 q^{96} + 5565060 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000i 0.707107i
\(3\) − 44.3001i − 0.947284i −0.880717 0.473642i \(-0.842939\pi\)
0.880717 0.473642i \(-0.157061\pi\)
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) 354.401 0.669831
\(7\) 343.000i 0.377964i
\(8\) − 512.000i − 0.353553i
\(9\) 224.501 0.102652
\(10\) 0 0
\(11\) 8294.71 1.87900 0.939500 0.342548i \(-0.111290\pi\)
0.939500 + 0.342548i \(0.111290\pi\)
\(12\) 2835.21i 0.473642i
\(13\) − 2086.69i − 0.263425i −0.991288 0.131713i \(-0.957952\pi\)
0.991288 0.131713i \(-0.0420476\pi\)
\(14\) −2744.00 −0.267261
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 22276.5i 1.09970i 0.835262 + 0.549852i \(0.185316\pi\)
−0.835262 + 0.549852i \(0.814684\pi\)
\(18\) 1796.00i 0.0725861i
\(19\) −22693.4 −0.759035 −0.379518 0.925184i \(-0.623910\pi\)
−0.379518 + 0.925184i \(0.623910\pi\)
\(20\) 0 0
\(21\) 15194.9 0.358040
\(22\) 66357.7i 1.32865i
\(23\) 51846.8i 0.888534i 0.895894 + 0.444267i \(0.146536\pi\)
−0.895894 + 0.444267i \(0.853464\pi\)
\(24\) −22681.7 −0.334916
\(25\) 0 0
\(26\) 16693.6 0.186270
\(27\) − 106830.i − 1.04453i
\(28\) − 21952.0i − 0.188982i
\(29\) 69360.5 0.528103 0.264052 0.964509i \(-0.414941\pi\)
0.264052 + 0.964509i \(0.414941\pi\)
\(30\) 0 0
\(31\) 54654.7 0.329505 0.164752 0.986335i \(-0.447318\pi\)
0.164752 + 0.986335i \(0.447318\pi\)
\(32\) 32768.0i 0.176777i
\(33\) − 367457.i − 1.77995i
\(34\) −178212. −0.777608
\(35\) 0 0
\(36\) −14368.0 −0.0513261
\(37\) 301929.i 0.979938i 0.871740 + 0.489969i \(0.162992\pi\)
−0.871740 + 0.489969i \(0.837008\pi\)
\(38\) − 181547.i − 0.536719i
\(39\) −92440.8 −0.249538
\(40\) 0 0
\(41\) −120041. −0.272010 −0.136005 0.990708i \(-0.543426\pi\)
−0.136005 + 0.990708i \(0.543426\pi\)
\(42\) 121559.i 0.253172i
\(43\) 260607.i 0.499859i 0.968264 + 0.249929i \(0.0804074\pi\)
−0.968264 + 0.249929i \(0.919593\pi\)
\(44\) −530862. −0.939500
\(45\) 0 0
\(46\) −414774. −0.628289
\(47\) − 282025.i − 0.396228i −0.980179 0.198114i \(-0.936518\pi\)
0.980179 0.198114i \(-0.0634816\pi\)
\(48\) − 181453.i − 0.236821i
\(49\) −117649. −0.142857
\(50\) 0 0
\(51\) 986852. 1.04173
\(52\) 133548.i 0.131713i
\(53\) − 413071.i − 0.381118i −0.981676 0.190559i \(-0.938970\pi\)
0.981676 0.190559i \(-0.0610300\pi\)
\(54\) 854638. 0.738591
\(55\) 0 0
\(56\) 175616. 0.133631
\(57\) 1.00532e6i 0.719022i
\(58\) 554884.i 0.373426i
\(59\) −961690. −0.609611 −0.304806 0.952415i \(-0.598591\pi\)
−0.304806 + 0.952415i \(0.598591\pi\)
\(60\) 0 0
\(61\) −2.52312e6 −1.42326 −0.711630 0.702555i \(-0.752043\pi\)
−0.711630 + 0.702555i \(0.752043\pi\)
\(62\) 437238.i 0.232995i
\(63\) 77003.7i 0.0387989i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) 2.93965e6 1.25861
\(67\) − 133835.i − 0.0543637i −0.999631 0.0271819i \(-0.991347\pi\)
0.999631 0.0271819i \(-0.00865332\pi\)
\(68\) − 1.42570e6i − 0.549852i
\(69\) 2.29682e6 0.841695
\(70\) 0 0
\(71\) −5.71568e6 −1.89524 −0.947619 0.319404i \(-0.896517\pi\)
−0.947619 + 0.319404i \(0.896517\pi\)
\(72\) − 114944.i − 0.0362931i
\(73\) − 28628.0i − 0.00861314i −0.999991 0.00430657i \(-0.998629\pi\)
0.999991 0.00430657i \(-0.00137083\pi\)
\(74\) −2.41543e6 −0.692921
\(75\) 0 0
\(76\) 1.45238e6 0.379518
\(77\) 2.84509e6i 0.710196i
\(78\) − 739526.i − 0.176450i
\(79\) 7.61428e6 1.73754 0.868768 0.495219i \(-0.164912\pi\)
0.868768 + 0.495219i \(0.164912\pi\)
\(80\) 0 0
\(81\) −4.24159e6 −0.886810
\(82\) − 960326.i − 0.192340i
\(83\) 2.64843e6i 0.508411i 0.967150 + 0.254206i \(0.0818140\pi\)
−0.967150 + 0.254206i \(0.918186\pi\)
\(84\) −972476. −0.179020
\(85\) 0 0
\(86\) −2.08486e6 −0.353453
\(87\) − 3.07268e6i − 0.500264i
\(88\) − 4.24689e6i − 0.664327i
\(89\) 7.82855e6 1.17711 0.588554 0.808458i \(-0.299697\pi\)
0.588554 + 0.808458i \(0.299697\pi\)
\(90\) 0 0
\(91\) 715736. 0.0995653
\(92\) − 3.31819e6i − 0.444267i
\(93\) − 2.42121e6i − 0.312135i
\(94\) 2.25620e6 0.280175
\(95\) 0 0
\(96\) 1.45163e6 0.167458
\(97\) 9.84758e6i 1.09554i 0.836629 + 0.547770i \(0.184523\pi\)
−0.836629 + 0.547770i \(0.815477\pi\)
\(98\) − 941192.i − 0.101015i
\(99\) 1.86217e6 0.192884
\(100\) 0 0
\(101\) 1.89329e7 1.82848 0.914242 0.405168i \(-0.132787\pi\)
0.914242 + 0.405168i \(0.132787\pi\)
\(102\) 7.89481e6i 0.736616i
\(103\) 1.88723e7i 1.70174i 0.525374 + 0.850871i \(0.323925\pi\)
−0.525374 + 0.850871i \(0.676075\pi\)
\(104\) −1.06839e6 −0.0931348
\(105\) 0 0
\(106\) 3.30457e6 0.269491
\(107\) − 1.75676e6i − 0.138634i −0.997595 0.0693168i \(-0.977918\pi\)
0.997595 0.0693168i \(-0.0220819\pi\)
\(108\) 6.83710e6i 0.522263i
\(109\) 1.51030e7 1.11704 0.558521 0.829490i \(-0.311369\pi\)
0.558521 + 0.829490i \(0.311369\pi\)
\(110\) 0 0
\(111\) 1.33755e7 0.928280
\(112\) 1.40493e6i 0.0944911i
\(113\) 9.05559e6i 0.590394i 0.955436 + 0.295197i \(0.0953854\pi\)
−0.955436 + 0.295197i \(0.904615\pi\)
\(114\) −8.04256e6 −0.508426
\(115\) 0 0
\(116\) −4.43907e6 −0.264052
\(117\) − 468464.i − 0.0270412i
\(118\) − 7.69352e6i − 0.431060i
\(119\) −7.64084e6 −0.415649
\(120\) 0 0
\(121\) 4.93151e7 2.53064
\(122\) − 2.01850e7i − 1.00640i
\(123\) 5.31782e6i 0.257671i
\(124\) −3.49790e6 −0.164752
\(125\) 0 0
\(126\) −616029. −0.0274350
\(127\) 3.51254e7i 1.52163i 0.648971 + 0.760813i \(0.275200\pi\)
−0.648971 + 0.760813i \(0.724800\pi\)
\(128\) − 2.09715e6i − 0.0883883i
\(129\) 1.15449e7 0.473508
\(130\) 0 0
\(131\) 4.78958e7 1.86143 0.930717 0.365740i \(-0.119184\pi\)
0.930717 + 0.365740i \(0.119184\pi\)
\(132\) 2.35172e7i 0.889974i
\(133\) − 7.78384e6i − 0.286888i
\(134\) 1.07068e6 0.0384409
\(135\) 0 0
\(136\) 1.14056e7 0.388804
\(137\) − 3.21903e6i − 0.106955i −0.998569 0.0534777i \(-0.982969\pi\)
0.998569 0.0534777i \(-0.0170306\pi\)
\(138\) 1.83745e7i 0.595168i
\(139\) −2.92523e7 −0.923866 −0.461933 0.886915i \(-0.652844\pi\)
−0.461933 + 0.886915i \(0.652844\pi\)
\(140\) 0 0
\(141\) −1.24937e7 −0.375341
\(142\) − 4.57254e7i − 1.34014i
\(143\) − 1.73085e7i − 0.494976i
\(144\) 919554. 0.0256631
\(145\) 0 0
\(146\) 229024. 0.00609041
\(147\) 5.21186e6i 0.135326i
\(148\) − 1.93234e7i − 0.489969i
\(149\) −2.12164e7 −0.525436 −0.262718 0.964873i \(-0.584619\pi\)
−0.262718 + 0.964873i \(0.584619\pi\)
\(150\) 0 0
\(151\) −2.71361e7 −0.641398 −0.320699 0.947181i \(-0.603918\pi\)
−0.320699 + 0.947181i \(0.603918\pi\)
\(152\) 1.16190e7i 0.268360i
\(153\) 5.00109e6i 0.112887i
\(154\) −2.27607e7 −0.502184
\(155\) 0 0
\(156\) 5.91621e6 0.124769
\(157\) − 1.51261e7i − 0.311944i −0.987761 0.155972i \(-0.950149\pi\)
0.987761 0.155972i \(-0.0498510\pi\)
\(158\) 6.09142e7i 1.22862i
\(159\) −1.82991e7 −0.361027
\(160\) 0 0
\(161\) −1.77834e7 −0.335834
\(162\) − 3.39327e7i − 0.627070i
\(163\) 1.01081e8i 1.82816i 0.405536 + 0.914079i \(0.367085\pi\)
−0.405536 + 0.914079i \(0.632915\pi\)
\(164\) 7.68261e6 0.136005
\(165\) 0 0
\(166\) −2.11874e7 −0.359501
\(167\) − 4.83737e7i − 0.803714i −0.915702 0.401857i \(-0.868365\pi\)
0.915702 0.401857i \(-0.131635\pi\)
\(168\) − 7.77981e6i − 0.126586i
\(169\) 5.83942e7 0.930607
\(170\) 0 0
\(171\) −5.09468e6 −0.0779167
\(172\) − 1.66789e7i − 0.249929i
\(173\) − 6.02289e6i − 0.0884390i −0.999022 0.0442195i \(-0.985920\pi\)
0.999022 0.0442195i \(-0.0140801\pi\)
\(174\) 2.45814e7 0.353740
\(175\) 0 0
\(176\) 3.39751e7 0.469750
\(177\) 4.26030e7i 0.577475i
\(178\) 6.26284e7i 0.832341i
\(179\) 1.04527e7 0.136221 0.0681103 0.997678i \(-0.478303\pi\)
0.0681103 + 0.997678i \(0.478303\pi\)
\(180\) 0 0
\(181\) 1.01810e8 1.27619 0.638095 0.769958i \(-0.279723\pi\)
0.638095 + 0.769958i \(0.279723\pi\)
\(182\) 5.72589e6i 0.0704033i
\(183\) 1.11775e8i 1.34823i
\(184\) 2.65455e7 0.314144
\(185\) 0 0
\(186\) 1.93697e7 0.220712
\(187\) 1.84777e8i 2.06635i
\(188\) 1.80496e7i 0.198114i
\(189\) 3.66426e7 0.394793
\(190\) 0 0
\(191\) 1.16808e8 1.21299 0.606495 0.795088i \(-0.292575\pi\)
0.606495 + 0.795088i \(0.292575\pi\)
\(192\) 1.16130e7i 0.118411i
\(193\) 1.72514e8i 1.72732i 0.504073 + 0.863661i \(0.331834\pi\)
−0.504073 + 0.863661i \(0.668166\pi\)
\(194\) −7.87807e7 −0.774664
\(195\) 0 0
\(196\) 7.52954e6 0.0714286
\(197\) − 1.13246e8i − 1.05534i −0.849450 0.527669i \(-0.823066\pi\)
0.849450 0.527669i \(-0.176934\pi\)
\(198\) 1.48973e7i 0.136389i
\(199\) −1.87994e8 −1.69106 −0.845529 0.533930i \(-0.820715\pi\)
−0.845529 + 0.533930i \(0.820715\pi\)
\(200\) 0 0
\(201\) −5.92892e6 −0.0514979
\(202\) 1.51463e8i 1.29293i
\(203\) 2.37906e7i 0.199604i
\(204\) −6.31585e7 −0.520866
\(205\) 0 0
\(206\) −1.50978e8 −1.20331
\(207\) 1.16396e7i 0.0912101i
\(208\) − 8.54710e6i − 0.0658563i
\(209\) −1.88235e8 −1.42623
\(210\) 0 0
\(211\) −2.04724e7 −0.150031 −0.0750155 0.997182i \(-0.523901\pi\)
−0.0750155 + 0.997182i \(0.523901\pi\)
\(212\) 2.64365e7i 0.190559i
\(213\) 2.53205e8i 1.79533i
\(214\) 1.40541e7 0.0980287
\(215\) 0 0
\(216\) −5.46968e7 −0.369295
\(217\) 1.87466e7i 0.124541i
\(218\) 1.20824e8i 0.789868i
\(219\) −1.26823e6 −0.00815910
\(220\) 0 0
\(221\) 4.64843e7 0.289690
\(222\) 1.07004e8i 0.656393i
\(223\) − 1.96181e8i − 1.18465i −0.805698 0.592326i \(-0.798210\pi\)
0.805698 0.592326i \(-0.201790\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 0 0
\(226\) −7.24447e7 −0.417472
\(227\) 9.92114e7i 0.562951i 0.959568 + 0.281476i \(0.0908239\pi\)
−0.959568 + 0.281476i \(0.909176\pi\)
\(228\) − 6.43405e7i − 0.359511i
\(229\) −2.80737e7 −0.154481 −0.0772407 0.997012i \(-0.524611\pi\)
−0.0772407 + 0.997012i \(0.524611\pi\)
\(230\) 0 0
\(231\) 1.26038e8 0.672757
\(232\) − 3.55126e7i − 0.186713i
\(233\) 2.59746e7i 0.134525i 0.997735 + 0.0672626i \(0.0214265\pi\)
−0.997735 + 0.0672626i \(0.978573\pi\)
\(234\) 3.74771e6 0.0191210
\(235\) 0 0
\(236\) 6.15481e7 0.304806
\(237\) − 3.37313e8i − 1.64594i
\(238\) − 6.11267e7i − 0.293908i
\(239\) 2.89625e8 1.37228 0.686142 0.727468i \(-0.259303\pi\)
0.686142 + 0.727468i \(0.259303\pi\)
\(240\) 0 0
\(241\) −2.35224e8 −1.08248 −0.541242 0.840867i \(-0.682046\pi\)
−0.541242 + 0.840867i \(0.682046\pi\)
\(242\) 3.94521e8i 1.78944i
\(243\) − 4.57339e7i − 0.204464i
\(244\) 1.61480e8 0.711630
\(245\) 0 0
\(246\) −4.25426e7 −0.182201
\(247\) 4.73542e7i 0.199949i
\(248\) − 2.79832e7i − 0.116497i
\(249\) 1.17326e8 0.481610
\(250\) 0 0
\(251\) 3.71067e8 1.48113 0.740566 0.671984i \(-0.234558\pi\)
0.740566 + 0.671984i \(0.234558\pi\)
\(252\) − 4.92824e6i − 0.0193995i
\(253\) 4.30054e8i 1.66956i
\(254\) −2.81003e8 −1.07595
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) 2.53506e8i 0.931585i 0.884894 + 0.465792i \(0.154231\pi\)
−0.884894 + 0.465792i \(0.845769\pi\)
\(258\) 9.23595e7i 0.334821i
\(259\) −1.03562e8 −0.370382
\(260\) 0 0
\(261\) 1.55715e7 0.0542110
\(262\) 3.83166e8i 1.31623i
\(263\) − 3.39492e8i − 1.15076i −0.817887 0.575379i \(-0.804855\pi\)
0.817887 0.575379i \(-0.195145\pi\)
\(264\) −1.88138e8 −0.629307
\(265\) 0 0
\(266\) 6.22707e7 0.202861
\(267\) − 3.46806e8i − 1.11506i
\(268\) 8.56546e6i 0.0271819i
\(269\) −1.51784e8 −0.475436 −0.237718 0.971334i \(-0.576399\pi\)
−0.237718 + 0.971334i \(0.576399\pi\)
\(270\) 0 0
\(271\) 6.09610e7 0.186063 0.0930314 0.995663i \(-0.470344\pi\)
0.0930314 + 0.995663i \(0.470344\pi\)
\(272\) 9.12446e7i 0.274926i
\(273\) − 3.17072e7i − 0.0943167i
\(274\) 2.57522e7 0.0756289
\(275\) 0 0
\(276\) −1.46996e8 −0.420847
\(277\) − 2.27766e8i − 0.643888i −0.946759 0.321944i \(-0.895664\pi\)
0.946759 0.321944i \(-0.104336\pi\)
\(278\) − 2.34019e8i − 0.653272i
\(279\) 1.22700e7 0.0338244
\(280\) 0 0
\(281\) 4.86914e8 1.30912 0.654560 0.756010i \(-0.272854\pi\)
0.654560 + 0.756010i \(0.272854\pi\)
\(282\) − 9.99499e7i − 0.265406i
\(283\) 2.62185e8i 0.687631i 0.939037 + 0.343815i \(0.111719\pi\)
−0.939037 + 0.343815i \(0.888281\pi\)
\(284\) 3.65803e8 0.947619
\(285\) 0 0
\(286\) 1.38468e8 0.350001
\(287\) − 4.11740e7i − 0.102810i
\(288\) 7.35643e6i 0.0181465i
\(289\) −8.59041e7 −0.209349
\(290\) 0 0
\(291\) 4.36249e8 1.03779
\(292\) 1.83219e6i 0.00430657i
\(293\) − 7.42829e7i − 0.172525i −0.996272 0.0862625i \(-0.972508\pi\)
0.996272 0.0862625i \(-0.0274924\pi\)
\(294\) −4.16949e7 −0.0956902
\(295\) 0 0
\(296\) 1.54588e8 0.346460
\(297\) − 8.86122e8i − 1.96266i
\(298\) − 1.69731e8i − 0.371539i
\(299\) 1.08188e8 0.234062
\(300\) 0 0
\(301\) −8.93883e7 −0.188929
\(302\) − 2.17089e8i − 0.453537i
\(303\) − 8.38728e8i − 1.73210i
\(304\) −9.29522e7 −0.189759
\(305\) 0 0
\(306\) −4.00087e7 −0.0798233
\(307\) − 4.64592e8i − 0.916404i −0.888848 0.458202i \(-0.848494\pi\)
0.888848 0.458202i \(-0.151506\pi\)
\(308\) − 1.82086e8i − 0.355098i
\(309\) 8.36043e8 1.61203
\(310\) 0 0
\(311\) −6.84752e8 −1.29084 −0.645419 0.763829i \(-0.723317\pi\)
−0.645419 + 0.763829i \(0.723317\pi\)
\(312\) 4.73297e7i 0.0882252i
\(313\) 2.85953e8i 0.527096i 0.964646 + 0.263548i \(0.0848927\pi\)
−0.964646 + 0.263548i \(0.915107\pi\)
\(314\) 1.21009e8 0.220578
\(315\) 0 0
\(316\) −4.87314e8 −0.868768
\(317\) 8.17943e8i 1.44217i 0.692848 + 0.721084i \(0.256356\pi\)
−0.692848 + 0.721084i \(0.743644\pi\)
\(318\) − 1.46393e8i − 0.255284i
\(319\) 5.75325e8 0.992307
\(320\) 0 0
\(321\) −7.78245e7 −0.131325
\(322\) − 1.42268e8i − 0.237471i
\(323\) − 5.05530e8i − 0.834714i
\(324\) 2.71461e8 0.443405
\(325\) 0 0
\(326\) −8.08649e8 −1.29270
\(327\) − 6.69063e8i − 1.05816i
\(328\) 6.14609e7i 0.0961702i
\(329\) 9.67346e7 0.149760
\(330\) 0 0
\(331\) 9.47073e8 1.43544 0.717720 0.696331i \(-0.245186\pi\)
0.717720 + 0.696331i \(0.245186\pi\)
\(332\) − 1.69499e8i − 0.254206i
\(333\) 6.77832e7i 0.100593i
\(334\) 3.86990e8 0.568312
\(335\) 0 0
\(336\) 6.22385e7 0.0895100
\(337\) − 1.36535e9i − 1.94330i −0.236415 0.971652i \(-0.575973\pi\)
0.236415 0.971652i \(-0.424027\pi\)
\(338\) 4.67154e8i 0.658039i
\(339\) 4.01164e8 0.559271
\(340\) 0 0
\(341\) 4.53345e8 0.619139
\(342\) − 4.07574e7i − 0.0550954i
\(343\) − 4.03536e7i − 0.0539949i
\(344\) 1.33431e8 0.176727
\(345\) 0 0
\(346\) 4.81831e7 0.0625358
\(347\) 9.66351e8i 1.24160i 0.783969 + 0.620800i \(0.213192\pi\)
−0.783969 + 0.620800i \(0.786808\pi\)
\(348\) 1.96651e8i 0.250132i
\(349\) −6.50882e8 −0.819621 −0.409811 0.912171i \(-0.634405\pi\)
−0.409811 + 0.912171i \(0.634405\pi\)
\(350\) 0 0
\(351\) −2.22921e8 −0.275154
\(352\) 2.71801e8i 0.332164i
\(353\) − 6.03836e8i − 0.730647i −0.930881 0.365323i \(-0.880958\pi\)
0.930881 0.365323i \(-0.119042\pi\)
\(354\) −3.40824e8 −0.408337
\(355\) 0 0
\(356\) −5.01027e8 −0.588554
\(357\) 3.38490e8i 0.393738i
\(358\) 8.36216e7i 0.0963225i
\(359\) −8.56904e7 −0.0977466 −0.0488733 0.998805i \(-0.515563\pi\)
−0.0488733 + 0.998805i \(0.515563\pi\)
\(360\) 0 0
\(361\) −3.78881e8 −0.423865
\(362\) 8.14480e8i 0.902403i
\(363\) − 2.18466e9i − 2.39724i
\(364\) −4.58071e7 −0.0497827
\(365\) 0 0
\(366\) −8.94197e8 −0.953343
\(367\) − 1.63658e9i − 1.72825i −0.503276 0.864126i \(-0.667872\pi\)
0.503276 0.864126i \(-0.332128\pi\)
\(368\) 2.12364e8i 0.222134i
\(369\) −2.69492e7 −0.0279225
\(370\) 0 0
\(371\) 1.41683e8 0.144049
\(372\) 1.54957e8i 0.156067i
\(373\) 1.88055e9i 1.87630i 0.346223 + 0.938152i \(0.387464\pi\)
−0.346223 + 0.938152i \(0.612536\pi\)
\(374\) −1.47822e9 −1.46113
\(375\) 0 0
\(376\) −1.44397e8 −0.140088
\(377\) − 1.44734e8i − 0.139116i
\(378\) 2.93141e8i 0.279161i
\(379\) −6.07726e8 −0.573417 −0.286708 0.958018i \(-0.592561\pi\)
−0.286708 + 0.958018i \(0.592561\pi\)
\(380\) 0 0
\(381\) 1.55606e9 1.44141
\(382\) 9.34467e8i 0.857713i
\(383\) 1.54124e8i 0.140176i 0.997541 + 0.0700882i \(0.0223281\pi\)
−0.997541 + 0.0700882i \(0.977672\pi\)
\(384\) −9.29041e7 −0.0837289
\(385\) 0 0
\(386\) −1.38011e9 −1.22140
\(387\) 5.85065e7i 0.0513116i
\(388\) − 6.30245e8i − 0.547770i
\(389\) 1.99567e9 1.71896 0.859480 0.511170i \(-0.170788\pi\)
0.859480 + 0.511170i \(0.170788\pi\)
\(390\) 0 0
\(391\) −1.15496e9 −0.977125
\(392\) 6.02363e7i 0.0505076i
\(393\) − 2.12179e9i − 1.76331i
\(394\) 9.05968e8 0.746236
\(395\) 0 0
\(396\) −1.19179e8 −0.0964419
\(397\) 6.91714e8i 0.554830i 0.960750 + 0.277415i \(0.0894777\pi\)
−0.960750 + 0.277415i \(0.910522\pi\)
\(398\) − 1.50395e9i − 1.19576i
\(399\) −3.44825e8 −0.271765
\(400\) 0 0
\(401\) −5.80549e8 −0.449608 −0.224804 0.974404i \(-0.572174\pi\)
−0.224804 + 0.974404i \(0.572174\pi\)
\(402\) − 4.74314e7i − 0.0364145i
\(403\) − 1.14048e8i − 0.0867998i
\(404\) −1.21170e9 −0.914242
\(405\) 0 0
\(406\) −1.90325e8 −0.141142
\(407\) 2.50441e9i 1.84130i
\(408\) − 5.05268e8i − 0.368308i
\(409\) 1.05158e9 0.759997 0.379998 0.924987i \(-0.375925\pi\)
0.379998 + 0.924987i \(0.375925\pi\)
\(410\) 0 0
\(411\) −1.42603e8 −0.101317
\(412\) − 1.20783e9i − 0.850871i
\(413\) − 3.29860e8i − 0.230411i
\(414\) −9.31170e7 −0.0644953
\(415\) 0 0
\(416\) 6.83768e7 0.0465674
\(417\) 1.29588e9i 0.875164i
\(418\) − 1.50588e9i − 1.00850i
\(419\) −1.96750e9 −1.30667 −0.653335 0.757069i \(-0.726631\pi\)
−0.653335 + 0.757069i \(0.726631\pi\)
\(420\) 0 0
\(421\) −2.78313e8 −0.181780 −0.0908901 0.995861i \(-0.528971\pi\)
−0.0908901 + 0.995861i \(0.528971\pi\)
\(422\) − 1.63780e8i − 0.106088i
\(423\) − 6.33148e7i − 0.0406737i
\(424\) −2.11492e8 −0.134745
\(425\) 0 0
\(426\) −2.02564e9 −1.26949
\(427\) − 8.65431e8i − 0.537941i
\(428\) 1.12432e8i 0.0693168i
\(429\) −7.66770e8 −0.468883
\(430\) 0 0
\(431\) −2.77969e9 −1.67234 −0.836172 0.548467i \(-0.815212\pi\)
−0.836172 + 0.548467i \(0.815212\pi\)
\(432\) − 4.37575e8i − 0.261131i
\(433\) − 1.40755e9i − 0.833213i −0.909087 0.416607i \(-0.863219\pi\)
0.909087 0.416607i \(-0.136781\pi\)
\(434\) −1.49972e8 −0.0880638
\(435\) 0 0
\(436\) −9.66590e8 −0.558521
\(437\) − 1.17658e9i − 0.674429i
\(438\) − 1.01458e7i − 0.00576935i
\(439\) 1.57299e9 0.887362 0.443681 0.896185i \(-0.353672\pi\)
0.443681 + 0.896185i \(0.353672\pi\)
\(440\) 0 0
\(441\) −2.64123e7 −0.0146646
\(442\) 3.71874e8i 0.204842i
\(443\) − 1.33047e9i − 0.727098i −0.931575 0.363549i \(-0.881565\pi\)
0.931575 0.363549i \(-0.118435\pi\)
\(444\) −8.56031e8 −0.464140
\(445\) 0 0
\(446\) 1.56945e9 0.837675
\(447\) 9.39888e8i 0.497737i
\(448\) − 8.99154e7i − 0.0472456i
\(449\) −2.75936e9 −1.43862 −0.719310 0.694689i \(-0.755542\pi\)
−0.719310 + 0.694689i \(0.755542\pi\)
\(450\) 0 0
\(451\) −9.95704e8 −0.511108
\(452\) − 5.79558e8i − 0.295197i
\(453\) 1.20213e9i 0.607586i
\(454\) −7.93691e8 −0.398067
\(455\) 0 0
\(456\) 5.14724e8 0.254213
\(457\) − 4.55675e8i − 0.223331i −0.993746 0.111665i \(-0.964381\pi\)
0.993746 0.111665i \(-0.0356185\pi\)
\(458\) − 2.24590e8i − 0.109235i
\(459\) 2.37979e9 1.14867
\(460\) 0 0
\(461\) −4.27395e8 −0.203178 −0.101589 0.994826i \(-0.532393\pi\)
−0.101589 + 0.994826i \(0.532393\pi\)
\(462\) 1.00830e9i 0.475711i
\(463\) 4.30814e8i 0.201724i 0.994900 + 0.100862i \(0.0321600\pi\)
−0.994900 + 0.100862i \(0.967840\pi\)
\(464\) 2.84100e8 0.132026
\(465\) 0 0
\(466\) −2.07797e8 −0.0951236
\(467\) − 3.54276e9i − 1.60966i −0.593508 0.804828i \(-0.702257\pi\)
0.593508 0.804828i \(-0.297743\pi\)
\(468\) 2.99817e7i 0.0135206i
\(469\) 4.59055e7 0.0205476
\(470\) 0 0
\(471\) −6.70086e8 −0.295500
\(472\) 4.92385e8i 0.215530i
\(473\) 2.16166e9i 0.939235i
\(474\) 2.69851e9 1.16386
\(475\) 0 0
\(476\) 4.89014e8 0.207825
\(477\) − 9.27346e7i − 0.0391226i
\(478\) 2.31700e9i 0.970351i
\(479\) −2.18067e9 −0.906601 −0.453301 0.891358i \(-0.649754\pi\)
−0.453301 + 0.891358i \(0.649754\pi\)
\(480\) 0 0
\(481\) 6.30033e8 0.258140
\(482\) − 1.88179e9i − 0.765432i
\(483\) 7.87808e8i 0.318131i
\(484\) −3.15617e9 −1.26532
\(485\) 0 0
\(486\) 3.65871e8 0.144578
\(487\) 8.94148e8i 0.350799i 0.984497 + 0.175399i \(0.0561217\pi\)
−0.984497 + 0.175399i \(0.943878\pi\)
\(488\) 1.29184e9i 0.503198i
\(489\) 4.47791e9 1.73179
\(490\) 0 0
\(491\) 4.17252e9 1.59079 0.795396 0.606090i \(-0.207263\pi\)
0.795396 + 0.606090i \(0.207263\pi\)
\(492\) − 3.40340e8i − 0.128836i
\(493\) 1.54511e9i 0.580758i
\(494\) −3.78834e8 −0.141385
\(495\) 0 0
\(496\) 2.23866e8 0.0823762
\(497\) − 1.96048e9i − 0.716332i
\(498\) 9.38605e8i 0.340550i
\(499\) −8.93170e8 −0.321797 −0.160899 0.986971i \(-0.551439\pi\)
−0.160899 + 0.986971i \(0.551439\pi\)
\(500\) 0 0
\(501\) −2.14296e9 −0.761346
\(502\) 2.96853e9i 1.04732i
\(503\) 2.88740e9i 1.01162i 0.862644 + 0.505812i \(0.168807\pi\)
−0.862644 + 0.505812i \(0.831193\pi\)
\(504\) 3.94259e7 0.0137175
\(505\) 0 0
\(506\) −3.44043e9 −1.18056
\(507\) − 2.58687e9i − 0.881550i
\(508\) − 2.24802e9i − 0.760813i
\(509\) 1.04446e9 0.351059 0.175529 0.984474i \(-0.443836\pi\)
0.175529 + 0.984474i \(0.443836\pi\)
\(510\) 0 0
\(511\) 9.81942e6 0.00325546
\(512\) 1.34218e8i 0.0441942i
\(513\) 2.42433e9i 0.792832i
\(514\) −2.02805e9 −0.658730
\(515\) 0 0
\(516\) −7.38876e8 −0.236754
\(517\) − 2.33932e9i − 0.744513i
\(518\) − 8.28493e8i − 0.261899i
\(519\) −2.66815e8 −0.0837769
\(520\) 0 0
\(521\) 1.58851e9 0.492106 0.246053 0.969256i \(-0.420866\pi\)
0.246053 + 0.969256i \(0.420866\pi\)
\(522\) 1.24572e8i 0.0383330i
\(523\) 4.73420e9i 1.44707i 0.690286 + 0.723537i \(0.257485\pi\)
−0.690286 + 0.723537i \(0.742515\pi\)
\(524\) −3.06533e9 −0.930717
\(525\) 0 0
\(526\) 2.71593e9 0.813709
\(527\) 1.21752e9i 0.362358i
\(528\) − 1.50510e9i − 0.444987i
\(529\) 7.16738e8 0.210507
\(530\) 0 0
\(531\) −2.15900e8 −0.0625780
\(532\) 4.98166e8i 0.143444i
\(533\) 2.50488e8i 0.0716543i
\(534\) 2.77444e9 0.788464
\(535\) 0 0
\(536\) −6.85237e7 −0.0192205
\(537\) − 4.63056e8i − 0.129040i
\(538\) − 1.21427e9i − 0.336184i
\(539\) −9.75865e8 −0.268429
\(540\) 0 0
\(541\) 4.26233e9 1.15733 0.578665 0.815566i \(-0.303574\pi\)
0.578665 + 0.815566i \(0.303574\pi\)
\(542\) 4.87688e8i 0.131566i
\(543\) − 4.51019e9i − 1.20891i
\(544\) −7.29957e8 −0.194402
\(545\) 0 0
\(546\) 2.53658e8 0.0666920
\(547\) − 5.20549e9i − 1.35990i −0.733260 0.679948i \(-0.762002\pi\)
0.733260 0.679948i \(-0.237998\pi\)
\(548\) 2.06018e8i 0.0534777i
\(549\) −5.66442e8 −0.146101
\(550\) 0 0
\(551\) −1.57402e9 −0.400849
\(552\) − 1.17597e9i − 0.297584i
\(553\) 2.61170e9i 0.656727i
\(554\) 1.82213e9 0.455298
\(555\) 0 0
\(556\) 1.87215e9 0.461933
\(557\) − 1.04344e9i − 0.255844i −0.991784 0.127922i \(-0.959169\pi\)
0.991784 0.127922i \(-0.0408307\pi\)
\(558\) 9.81601e7i 0.0239175i
\(559\) 5.43808e8 0.131675
\(560\) 0 0
\(561\) 8.18565e9 1.95742
\(562\) 3.89531e9i 0.925688i
\(563\) − 8.23842e9i − 1.94565i −0.231541 0.972825i \(-0.574377\pi\)
0.231541 0.972825i \(-0.425623\pi\)
\(564\) 7.99599e8 0.187670
\(565\) 0 0
\(566\) −2.09748e9 −0.486228
\(567\) − 1.45486e9i − 0.335183i
\(568\) 2.92643e9i 0.670068i
\(569\) −6.17425e9 −1.40505 −0.702525 0.711659i \(-0.747944\pi\)
−0.702525 + 0.711659i \(0.747944\pi\)
\(570\) 0 0
\(571\) 4.44117e9 0.998324 0.499162 0.866509i \(-0.333641\pi\)
0.499162 + 0.866509i \(0.333641\pi\)
\(572\) 1.10775e9i 0.247488i
\(573\) − 5.17462e9i − 1.14905i
\(574\) 3.29392e8 0.0726978
\(575\) 0 0
\(576\) −5.88515e7 −0.0128315
\(577\) 8.17132e9i 1.77083i 0.464801 + 0.885415i \(0.346126\pi\)
−0.464801 + 0.885415i \(0.653874\pi\)
\(578\) − 6.87233e8i − 0.148032i
\(579\) 7.64238e9 1.63626
\(580\) 0 0
\(581\) −9.08411e8 −0.192161
\(582\) 3.48999e9i 0.733827i
\(583\) − 3.42630e9i − 0.716120i
\(584\) −1.46576e7 −0.00304521
\(585\) 0 0
\(586\) 5.94263e8 0.121994
\(587\) − 6.28754e9i − 1.28306i −0.767097 0.641531i \(-0.778300\pi\)
0.767097 0.641531i \(-0.221700\pi\)
\(588\) − 3.33559e8i − 0.0676632i
\(589\) −1.24030e9 −0.250106
\(590\) 0 0
\(591\) −5.01681e9 −0.999704
\(592\) 1.23670e9i 0.244984i
\(593\) 7.01629e9i 1.38171i 0.722995 + 0.690854i \(0.242765\pi\)
−0.722995 + 0.690854i \(0.757235\pi\)
\(594\) 7.08898e9 1.38781
\(595\) 0 0
\(596\) 1.35785e9 0.262718
\(597\) 8.32816e9i 1.60191i
\(598\) 8.65507e8i 0.165507i
\(599\) 9.38103e8 0.178343 0.0891716 0.996016i \(-0.471578\pi\)
0.0891716 + 0.996016i \(0.471578\pi\)
\(600\) 0 0
\(601\) 2.95585e9 0.555421 0.277711 0.960665i \(-0.410424\pi\)
0.277711 + 0.960665i \(0.410424\pi\)
\(602\) − 7.15107e8i − 0.133593i
\(603\) − 3.00461e7i − 0.00558056i
\(604\) 1.73671e9 0.320699
\(605\) 0 0
\(606\) 6.70982e9 1.22478
\(607\) − 4.76325e9i − 0.864457i −0.901764 0.432228i \(-0.857727\pi\)
0.901764 0.432228i \(-0.142273\pi\)
\(608\) − 7.43617e8i − 0.134180i
\(609\) 1.05393e9 0.189082
\(610\) 0 0
\(611\) −5.88500e8 −0.104376
\(612\) − 3.20070e8i − 0.0564436i
\(613\) 2.32881e9i 0.408340i 0.978935 + 0.204170i \(0.0654495\pi\)
−0.978935 + 0.204170i \(0.934550\pi\)
\(614\) 3.71673e9 0.647996
\(615\) 0 0
\(616\) 1.45668e9 0.251092
\(617\) 9.82915e9i 1.68468i 0.538945 + 0.842341i \(0.318823\pi\)
−0.538945 + 0.842341i \(0.681177\pi\)
\(618\) 6.68835e9i 1.13988i
\(619\) 5.94350e8 0.100722 0.0503611 0.998731i \(-0.483963\pi\)
0.0503611 + 0.998731i \(0.483963\pi\)
\(620\) 0 0
\(621\) 5.53878e9 0.928097
\(622\) − 5.47801e9i − 0.912760i
\(623\) 2.68519e9i 0.444905i
\(624\) −3.78637e8 −0.0623846
\(625\) 0 0
\(626\) −2.28763e9 −0.372713
\(627\) 8.33884e9i 1.35104i
\(628\) 9.68068e8i 0.155972i
\(629\) −6.72592e9 −1.07764
\(630\) 0 0
\(631\) 6.06380e9 0.960820 0.480410 0.877044i \(-0.340488\pi\)
0.480410 + 0.877044i \(0.340488\pi\)
\(632\) − 3.89851e9i − 0.614312i
\(633\) 9.06931e8i 0.142122i
\(634\) −6.54354e9 −1.01977
\(635\) 0 0
\(636\) 1.17114e9 0.180513
\(637\) 2.45498e8i 0.0376322i
\(638\) 4.60260e9i 0.701667i
\(639\) −1.28317e9 −0.194550
\(640\) 0 0
\(641\) −7.74978e8 −0.116221 −0.0581107 0.998310i \(-0.518508\pi\)
−0.0581107 + 0.998310i \(0.518508\pi\)
\(642\) − 6.22596e8i − 0.0928611i
\(643\) 1.39116e9i 0.206366i 0.994662 + 0.103183i \(0.0329028\pi\)
−0.994662 + 0.103183i \(0.967097\pi\)
\(644\) 1.13814e9 0.167917
\(645\) 0 0
\(646\) 4.04424e9 0.590232
\(647\) 8.16793e9i 1.18562i 0.805341 + 0.592812i \(0.201982\pi\)
−0.805341 + 0.592812i \(0.798018\pi\)
\(648\) 2.17169e9i 0.313535i
\(649\) −7.97694e9 −1.14546
\(650\) 0 0
\(651\) 8.30475e8 0.117976
\(652\) − 6.46920e9i − 0.914079i
\(653\) 3.75842e9i 0.528213i 0.964493 + 0.264107i \(0.0850771\pi\)
−0.964493 + 0.264107i \(0.914923\pi\)
\(654\) 5.35250e9 0.748230
\(655\) 0 0
\(656\) −4.91687e8 −0.0680026
\(657\) − 6.42701e6i 0 0.000884159i
\(658\) 7.73877e8i 0.105896i
\(659\) 1.11612e10 1.51919 0.759594 0.650398i \(-0.225398\pi\)
0.759594 + 0.650398i \(0.225398\pi\)
\(660\) 0 0
\(661\) −7.25634e8 −0.0977265 −0.0488632 0.998805i \(-0.515560\pi\)
−0.0488632 + 0.998805i \(0.515560\pi\)
\(662\) 7.57658e9i 1.01501i
\(663\) − 2.05926e9i − 0.274419i
\(664\) 1.35600e9 0.179750
\(665\) 0 0
\(666\) −5.42266e8 −0.0711299
\(667\) 3.59612e9i 0.469238i
\(668\) 3.09592e9i 0.401857i
\(669\) −8.69086e9 −1.12220
\(670\) 0 0
\(671\) −2.09286e10 −2.67431
\(672\) 4.97908e8i 0.0632931i
\(673\) − 3.86734e9i − 0.489057i −0.969642 0.244528i \(-0.921367\pi\)
0.969642 0.244528i \(-0.0786331\pi\)
\(674\) 1.09228e10 1.37412
\(675\) 0 0
\(676\) −3.73723e9 −0.465304
\(677\) 6.50913e9i 0.806237i 0.915148 + 0.403118i \(0.132074\pi\)
−0.915148 + 0.403118i \(0.867926\pi\)
\(678\) 3.20931e9i 0.395465i
\(679\) −3.37772e9 −0.414076
\(680\) 0 0
\(681\) 4.39507e9 0.533275
\(682\) 3.62676e9i 0.437798i
\(683\) 5.96614e9i 0.716508i 0.933624 + 0.358254i \(0.116628\pi\)
−0.933624 + 0.358254i \(0.883372\pi\)
\(684\) 3.26060e8 0.0389584
\(685\) 0 0
\(686\) 3.22829e8 0.0381802
\(687\) 1.24367e9i 0.146338i
\(688\) 1.06745e9i 0.124965i
\(689\) −8.61953e8 −0.100396
\(690\) 0 0
\(691\) 4.47683e9 0.516176 0.258088 0.966121i \(-0.416908\pi\)
0.258088 + 0.966121i \(0.416908\pi\)
\(692\) 3.85465e8i 0.0442195i
\(693\) 6.38724e8i 0.0729032i
\(694\) −7.73081e9 −0.877944
\(695\) 0 0
\(696\) −1.57321e9 −0.176870
\(697\) − 2.67409e9i − 0.299131i
\(698\) − 5.20706e9i − 0.579560i
\(699\) 1.15068e9 0.127434
\(700\) 0 0
\(701\) −5.17249e9 −0.567135 −0.283568 0.958952i \(-0.591518\pi\)
−0.283568 + 0.958952i \(0.591518\pi\)
\(702\) − 1.78337e9i − 0.194563i
\(703\) − 6.85179e9i − 0.743808i
\(704\) −2.17441e9 −0.234875
\(705\) 0 0
\(706\) 4.83069e9 0.516645
\(707\) 6.49397e9i 0.691102i
\(708\) − 2.72659e9i − 0.288738i
\(709\) 1.92689e8 0.0203046 0.0101523 0.999948i \(-0.496768\pi\)
0.0101523 + 0.999948i \(0.496768\pi\)
\(710\) 0 0
\(711\) 1.70941e9 0.178362
\(712\) − 4.00822e9i − 0.416171i
\(713\) 2.83367e9i 0.292776i
\(714\) −2.70792e9 −0.278415
\(715\) 0 0
\(716\) −6.68973e8 −0.0681103
\(717\) − 1.28304e10i − 1.29994i
\(718\) − 6.85523e8i − 0.0691173i
\(719\) 9.94078e8 0.0997400 0.0498700 0.998756i \(-0.484119\pi\)
0.0498700 + 0.998756i \(0.484119\pi\)
\(720\) 0 0
\(721\) −6.47319e9 −0.643198
\(722\) − 3.03105e9i − 0.299718i
\(723\) 1.04204e10i 1.02542i
\(724\) −6.51584e9 −0.638095
\(725\) 0 0
\(726\) 1.74773e10 1.69510
\(727\) − 2.06212e8i − 0.0199042i −0.999950 0.00995208i \(-0.996832\pi\)
0.999950 0.00995208i \(-0.00316790\pi\)
\(728\) − 3.66457e8i − 0.0352017i
\(729\) −1.13024e10 −1.08050
\(730\) 0 0
\(731\) −5.80542e9 −0.549697
\(732\) − 7.15358e9i − 0.674116i
\(733\) 8.04851e9i 0.754834i 0.926043 + 0.377417i \(0.123188\pi\)
−0.926043 + 0.377417i \(0.876812\pi\)
\(734\) 1.30927e10 1.22206
\(735\) 0 0
\(736\) −1.69891e9 −0.157072
\(737\) − 1.11013e9i − 0.102149i
\(738\) − 2.15594e8i − 0.0197442i
\(739\) 5.35129e9 0.487756 0.243878 0.969806i \(-0.421580\pi\)
0.243878 + 0.969806i \(0.421580\pi\)
\(740\) 0 0
\(741\) 2.09780e9 0.189409
\(742\) 1.13347e9i 0.101858i
\(743\) 1.53558e10i 1.37345i 0.726919 + 0.686723i \(0.240952\pi\)
−0.726919 + 0.686723i \(0.759048\pi\)
\(744\) −1.23966e9 −0.110356
\(745\) 0 0
\(746\) −1.50444e10 −1.32675
\(747\) 5.94574e8i 0.0521896i
\(748\) − 1.18257e10i − 1.03317i
\(749\) 6.02568e8 0.0523986
\(750\) 0 0
\(751\) 2.05007e9 0.176615 0.0883077 0.996093i \(-0.471854\pi\)
0.0883077 + 0.996093i \(0.471854\pi\)
\(752\) − 1.15517e9i − 0.0990570i
\(753\) − 1.64383e10i − 1.40305i
\(754\) 1.15787e9 0.0983697
\(755\) 0 0
\(756\) −2.34513e9 −0.197397
\(757\) − 1.05302e10i − 0.882270i −0.897441 0.441135i \(-0.854576\pi\)
0.897441 0.441135i \(-0.145424\pi\)
\(758\) − 4.86181e9i − 0.405467i
\(759\) 1.90514e10 1.58155
\(760\) 0 0
\(761\) 1.69743e10 1.39620 0.698098 0.716002i \(-0.254030\pi\)
0.698098 + 0.716002i \(0.254030\pi\)
\(762\) 1.24485e10i 1.01923i
\(763\) 5.18032e9i 0.422202i
\(764\) −7.47573e9 −0.606495
\(765\) 0 0
\(766\) −1.23299e9 −0.0991198
\(767\) 2.00675e9i 0.160587i
\(768\) − 7.43232e8i − 0.0592053i
\(769\) −3.02600e9 −0.239953 −0.119977 0.992777i \(-0.538282\pi\)
−0.119977 + 0.992777i \(0.538282\pi\)
\(770\) 0 0
\(771\) 1.12303e10 0.882476
\(772\) − 1.10409e10i − 0.863661i
\(773\) 1.38384e10i 1.07760i 0.842434 + 0.538799i \(0.181122\pi\)
−0.842434 + 0.538799i \(0.818878\pi\)
\(774\) −4.68052e8 −0.0362828
\(775\) 0 0
\(776\) 5.04196e9 0.387332
\(777\) 4.58779e9i 0.350857i
\(778\) 1.59654e10i 1.21549i
\(779\) 2.72413e9 0.206465
\(780\) 0 0
\(781\) −4.74099e10 −3.56115
\(782\) − 9.23972e9i − 0.690932i
\(783\) − 7.40976e9i − 0.551617i
\(784\) −4.81890e8 −0.0357143
\(785\) 0 0
\(786\) 1.69743e10 1.24685
\(787\) − 9.83270e9i − 0.719054i −0.933135 0.359527i \(-0.882938\pi\)
0.933135 0.359527i \(-0.117062\pi\)
\(788\) 7.24775e9i 0.527669i
\(789\) −1.50395e10 −1.09009
\(790\) 0 0
\(791\) −3.10607e9 −0.223148
\(792\) − 9.53430e8i − 0.0681947i
\(793\) 5.26499e9i 0.374922i
\(794\) −5.53371e9 −0.392324
\(795\) 0 0
\(796\) 1.20316e10 0.845529
\(797\) 3.16521e9i 0.221462i 0.993850 + 0.110731i \(0.0353191\pi\)
−0.993850 + 0.110731i \(0.964681\pi\)
\(798\) − 2.75860e9i − 0.192167i
\(799\) 6.28253e9 0.435733
\(800\) 0 0
\(801\) 1.75751e9 0.120833
\(802\) − 4.64439e9i − 0.317921i
\(803\) − 2.37461e8i − 0.0161841i
\(804\) 3.79451e8 0.0257489
\(805\) 0 0
\(806\) 9.12381e8 0.0613767
\(807\) 6.72403e9i 0.450373i
\(808\) − 9.69362e9i − 0.646467i
\(809\) −2.18461e10 −1.45062 −0.725311 0.688422i \(-0.758304\pi\)
−0.725311 + 0.688422i \(0.758304\pi\)
\(810\) 0 0
\(811\) 1.11360e10 0.733090 0.366545 0.930400i \(-0.380541\pi\)
0.366545 + 0.930400i \(0.380541\pi\)
\(812\) − 1.52260e9i − 0.0998022i
\(813\) − 2.70058e9i − 0.176254i
\(814\) −2.00353e10 −1.30200
\(815\) 0 0
\(816\) 4.04214e9 0.260433
\(817\) − 5.91407e9i − 0.379410i
\(818\) 8.41266e9i 0.537399i
\(819\) 1.60683e8 0.0102206
\(820\) 0 0
\(821\) 2.08012e9 0.131186 0.0655931 0.997846i \(-0.479106\pi\)
0.0655931 + 0.997846i \(0.479106\pi\)
\(822\) − 1.14083e9i − 0.0716421i
\(823\) − 1.89553e10i − 1.18531i −0.805458 0.592653i \(-0.798081\pi\)
0.805458 0.592653i \(-0.201919\pi\)
\(824\) 9.66260e9 0.601657
\(825\) 0 0
\(826\) 2.63888e9 0.162925
\(827\) 2.12105e10i 1.30401i 0.758215 + 0.652005i \(0.226072\pi\)
−0.758215 + 0.652005i \(0.773928\pi\)
\(828\) − 7.44936e8i − 0.0456050i
\(829\) 1.14471e10 0.697840 0.348920 0.937152i \(-0.386548\pi\)
0.348920 + 0.937152i \(0.386548\pi\)
\(830\) 0 0
\(831\) −1.00901e10 −0.609945
\(832\) 5.47014e8i 0.0329281i
\(833\) − 2.62081e9i − 0.157101i
\(834\) −1.03671e10 −0.618834
\(835\) 0 0
\(836\) 1.20471e10 0.713114
\(837\) − 5.83875e9i − 0.344176i
\(838\) − 1.57400e10i − 0.923956i
\(839\) −3.31864e10 −1.93996 −0.969981 0.243179i \(-0.921810\pi\)
−0.969981 + 0.243179i \(0.921810\pi\)
\(840\) 0 0
\(841\) −1.24390e10 −0.721107
\(842\) − 2.22651e9i − 0.128538i
\(843\) − 2.15703e10i − 1.24011i
\(844\) 1.31024e9 0.0750155
\(845\) 0 0
\(846\) 5.06518e8 0.0287606
\(847\) 1.69151e10i 0.956494i
\(848\) − 1.69194e9i − 0.0952794i
\(849\) 1.16148e10 0.651382
\(850\) 0 0
\(851\) −1.56540e10 −0.870709
\(852\) − 1.62051e10i − 0.897664i
\(853\) − 2.28601e10i − 1.26112i −0.776140 0.630560i \(-0.782825\pi\)
0.776140 0.630560i \(-0.217175\pi\)
\(854\) 6.92345e9 0.380382
\(855\) 0 0
\(856\) −8.99460e8 −0.0490144
\(857\) 2.93946e10i 1.59527i 0.603140 + 0.797635i \(0.293916\pi\)
−0.603140 + 0.797635i \(0.706084\pi\)
\(858\) − 6.13416e9i − 0.331550i
\(859\) −1.12866e10 −0.607558 −0.303779 0.952743i \(-0.598248\pi\)
−0.303779 + 0.952743i \(0.598248\pi\)
\(860\) 0 0
\(861\) −1.82401e9 −0.0973905
\(862\) − 2.22375e10i − 1.18253i
\(863\) − 2.36165e10i − 1.25077i −0.780315 0.625387i \(-0.784941\pi\)
0.780315 0.625387i \(-0.215059\pi\)
\(864\) 3.50060e9 0.184648
\(865\) 0 0
\(866\) 1.12604e10 0.589171
\(867\) 3.80556e9i 0.198313i
\(868\) − 1.19978e9i − 0.0622705i
\(869\) 6.31583e10 3.26483
\(870\) 0 0
\(871\) −2.79274e8 −0.0143208
\(872\) − 7.73272e9i − 0.394934i
\(873\) 2.21079e9i 0.112460i
\(874\) 9.41264e9 0.476893
\(875\) 0 0
\(876\) 8.11664e7 0.00407955
\(877\) − 3.02456e9i − 0.151413i −0.997130 0.0757065i \(-0.975879\pi\)
0.997130 0.0757065i \(-0.0241212\pi\)
\(878\) 1.25839e10i 0.627459i
\(879\) −3.29074e9 −0.163430
\(880\) 0 0
\(881\) −2.09728e10 −1.03333 −0.516666 0.856187i \(-0.672827\pi\)
−0.516666 + 0.856187i \(0.672827\pi\)
\(882\) − 2.11298e8i − 0.0103694i
\(883\) 1.76578e10i 0.863124i 0.902083 + 0.431562i \(0.142037\pi\)
−0.902083 + 0.431562i \(0.857963\pi\)
\(884\) −2.97499e9 −0.144845
\(885\) 0 0
\(886\) 1.06438e10 0.514136
\(887\) − 3.58732e10i − 1.72599i −0.505215 0.862993i \(-0.668587\pi\)
0.505215 0.862993i \(-0.331413\pi\)
\(888\) − 6.84825e9i − 0.328197i
\(889\) −1.20480e10 −0.575121
\(890\) 0 0
\(891\) −3.51827e10 −1.66632
\(892\) 1.25556e10i 0.592326i
\(893\) 6.40011e9i 0.300751i
\(894\) −7.51911e9 −0.351953
\(895\) 0 0
\(896\) 7.19323e8 0.0334077
\(897\) − 4.79276e9i − 0.221724i
\(898\) − 2.20749e10i − 1.01726i
\(899\) 3.79087e9 0.174013
\(900\) 0 0
\(901\) 9.20177e9 0.419117
\(902\) − 7.96563e9i − 0.361408i
\(903\) 3.95991e9i 0.178969i
\(904\) 4.63646e9 0.208736
\(905\) 0 0
\(906\) −9.61704e9 −0.429628
\(907\) − 1.65563e10i − 0.736780i −0.929671 0.368390i \(-0.879909\pi\)
0.929671 0.368390i \(-0.120091\pi\)
\(908\) − 6.34953e9i − 0.281476i
\(909\) 4.25044e9 0.187698
\(910\) 0 0
\(911\) 2.04661e9 0.0896852 0.0448426 0.998994i \(-0.485721\pi\)
0.0448426 + 0.998994i \(0.485721\pi\)
\(912\) 4.11779e9i 0.179756i
\(913\) 2.19680e10i 0.955305i
\(914\) 3.64540e9 0.157919
\(915\) 0 0
\(916\) 1.79672e9 0.0772407
\(917\) 1.64283e10i 0.703556i
\(918\) 1.90383e10i 0.812232i
\(919\) −3.62621e10 −1.54116 −0.770582 0.637341i \(-0.780034\pi\)
−0.770582 + 0.637341i \(0.780034\pi\)
\(920\) 0 0
\(921\) −2.05815e10 −0.868096
\(922\) − 3.41916e9i − 0.143669i
\(923\) 1.19269e10i 0.499253i
\(924\) −8.06641e9 −0.336379
\(925\) 0 0
\(926\) −3.44652e9 −0.142640
\(927\) 4.23683e9i 0.174688i
\(928\) 2.27280e9i 0.0933564i
\(929\) 1.05595e10 0.432105 0.216053 0.976382i \(-0.430682\pi\)
0.216053 + 0.976382i \(0.430682\pi\)
\(930\) 0 0
\(931\) 2.66986e9 0.108434
\(932\) − 1.66238e9i − 0.0672626i
\(933\) 3.03346e10i 1.22279i
\(934\) 2.83421e10 1.13820
\(935\) 0 0
\(936\) −2.39854e8 −0.00956050
\(937\) − 5.66531e9i − 0.224975i −0.993653 0.112488i \(-0.964118\pi\)
0.993653 0.112488i \(-0.0358819\pi\)
\(938\) 3.67244e8i 0.0145293i
\(939\) 1.26678e10 0.499310
\(940\) 0 0
\(941\) −3.76376e10 −1.47251 −0.736255 0.676704i \(-0.763408\pi\)
−0.736255 + 0.676704i \(0.763408\pi\)
\(942\) − 5.36069e9i − 0.208950i
\(943\) − 6.22373e9i − 0.241690i
\(944\) −3.93908e9 −0.152403
\(945\) 0 0
\(946\) −1.72933e10 −0.664139
\(947\) 3.56031e10i 1.36227i 0.732159 + 0.681134i \(0.238513\pi\)
−0.732159 + 0.681134i \(0.761487\pi\)
\(948\) 2.15881e10i 0.822970i
\(949\) −5.97380e7 −0.00226892
\(950\) 0 0
\(951\) 3.62350e10 1.36614
\(952\) 3.91211e9i 0.146954i
\(953\) − 3.27959e10i − 1.22742i −0.789530 0.613712i \(-0.789675\pi\)
0.789530 0.613712i \(-0.210325\pi\)
\(954\) 7.41877e8 0.0276638
\(955\) 0 0
\(956\) −1.85360e10 −0.686142
\(957\) − 2.54870e10i − 0.939997i
\(958\) − 1.74454e10i − 0.641064i
\(959\) 1.10413e9 0.0404253
\(960\) 0 0
\(961\) −2.45255e10 −0.891427
\(962\) 5.04027e9i 0.182533i
\(963\) − 3.94393e8i − 0.0142311i
\(964\) 1.50543e10 0.541242
\(965\) 0 0
\(966\) −6.30247e9 −0.224952
\(967\) − 1.21805e10i − 0.433184i −0.976262 0.216592i \(-0.930506\pi\)
0.976262 0.216592i \(-0.0694941\pi\)
\(968\) − 2.52493e10i − 0.894718i
\(969\) −2.23950e10 −0.790712
\(970\) 0 0
\(971\) 2.37312e10 0.831865 0.415933 0.909395i \(-0.363455\pi\)
0.415933 + 0.909395i \(0.363455\pi\)
\(972\) 2.92697e9i 0.102232i
\(973\) − 1.00336e10i − 0.349188i
\(974\) −7.15318e9 −0.248052
\(975\) 0 0
\(976\) −1.03347e10 −0.355815
\(977\) 3.44206e9i 0.118083i 0.998256 + 0.0590416i \(0.0188044\pi\)
−0.998256 + 0.0590416i \(0.981196\pi\)
\(978\) 3.58233e10i 1.22456i
\(979\) 6.49356e10 2.21179
\(980\) 0 0
\(981\) 3.39062e9 0.114667
\(982\) 3.33802e10i 1.12486i
\(983\) − 5.25354e10i − 1.76407i −0.471187 0.882033i \(-0.656174\pi\)
0.471187 0.882033i \(-0.343826\pi\)
\(984\) 2.72272e9 0.0911005
\(985\) 0 0
\(986\) −1.23609e10 −0.410658
\(987\) − 4.28535e9i − 0.141865i
\(988\) − 3.03067e9i − 0.0999745i
\(989\) −1.35117e10 −0.444142
\(990\) 0 0
\(991\) 5.03399e10 1.64306 0.821531 0.570163i \(-0.193120\pi\)
0.821531 + 0.570163i \(0.193120\pi\)
\(992\) 1.79092e9i 0.0582487i
\(993\) − 4.19554e10i − 1.35977i
\(994\) 1.56838e10 0.506524
\(995\) 0 0
\(996\) −7.50884e9 −0.240805
\(997\) − 4.80693e9i − 0.153615i −0.997046 0.0768077i \(-0.975527\pi\)
0.997046 0.0768077i \(-0.0244727\pi\)
\(998\) − 7.14536e9i − 0.227545i
\(999\) 3.22550e10 1.02357
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 350.8.c.i.99.3 4
5.2 odd 4 350.8.a.l.1.1 2
5.3 odd 4 70.8.a.g.1.2 2
5.4 even 2 inner 350.8.c.i.99.2 4
20.3 even 4 560.8.a.h.1.1 2
35.13 even 4 490.8.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.8.a.g.1.2 2 5.3 odd 4
350.8.a.l.1.1 2 5.2 odd 4
350.8.c.i.99.2 4 5.4 even 2 inner
350.8.c.i.99.3 4 1.1 even 1 trivial
490.8.a.k.1.1 2 35.13 even 4
560.8.a.h.1.1 2 20.3 even 4