Properties

Label 27.11.b.c.26.3
Level $27$
Weight $11$
Character 27.26
Analytic conductor $17.155$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,11,Mod(26,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), 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: \(17.1546458222\)
Analytic rank: \(0\)
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: \( x^{4} + 188x^{2} + 756 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.3
Root \(13.5606i\) of defining polynomial
Character \(\chi\) \(=\) 27.26
Dual form 27.11.b.c.26.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+23.5425i q^{2} +469.750 q^{4} -4424.52i q^{5} -21568.0 q^{7} +35166.6i q^{8} +104164. q^{10} +242828. i q^{11} -398104. q^{13} -507765. i q^{14} -346888. q^{16} +2.02619e6i q^{17} -2.77192e6 q^{19} -2.07842e6i q^{20} -5.71678e6 q^{22} -2.47900e6i q^{23} -9.81076e6 q^{25} -9.37237e6i q^{26} -1.01315e7 q^{28} -9.13358e6i q^{29} -1.81463e7 q^{31} +2.78440e7i q^{32} -4.77017e7 q^{34} +9.54280e7i q^{35} +1.14330e7 q^{37} -6.52579e7i q^{38} +1.55595e8 q^{40} -1.37099e8i q^{41} +9.87897e7 q^{43} +1.14068e8i q^{44} +5.83620e7 q^{46} +1.41804e8i q^{47} +1.82702e8 q^{49} -2.30970e8i q^{50} -1.87009e8 q^{52} -1.39084e8i q^{53} +1.07440e9 q^{55} -7.58473e8i q^{56} +2.15027e8 q^{58} +1.06798e9i q^{59} -1.44976e9 q^{61} -4.27210e8i q^{62} -1.01073e9 q^{64} +1.76142e9i q^{65} +1.44613e9 q^{67} +9.51804e8i q^{68} -2.24662e9 q^{70} -1.33807e9i q^{71} -1.46812e9 q^{73} +2.69162e8i q^{74} -1.30211e9 q^{76} -5.23731e9i q^{77} -4.95381e9 q^{79} +1.53481e9i q^{80} +3.22766e9 q^{82} +5.60643e8i q^{83} +8.96494e9 q^{85} +2.32576e9i q^{86} -8.53944e9 q^{88} -4.73263e9i q^{89} +8.58629e9 q^{91} -1.16451e9i q^{92} -3.33843e9 q^{94} +1.22644e10i q^{95} +1.21701e10 q^{97} +4.30128e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 548 q^{4} + 20516 q^{7} - 199800 q^{10} - 262420 q^{13} - 3207800 q^{16} - 8233516 q^{19} - 21085704 q^{22} - 47203580 q^{25} - 67604212 q^{28} - 46994920 q^{31} - 78985368 q^{34} + 108086444 q^{37}+ \cdots + 12969797468 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 23.5425i 0.735704i 0.929884 + 0.367852i \(0.119907\pi\)
−0.929884 + 0.367852i \(0.880093\pi\)
\(3\) 0 0
\(4\) 469.750 0.458740
\(5\) − 4424.52i − 1.41585i −0.706289 0.707923i \(-0.749632\pi\)
0.706289 0.707923i \(-0.250368\pi\)
\(6\) 0 0
\(7\) −21568.0 −1.28327 −0.641637 0.767009i \(-0.721744\pi\)
−0.641637 + 0.767009i \(0.721744\pi\)
\(8\) 35166.6i 1.07320i
\(9\) 0 0
\(10\) 104164. 1.04164
\(11\) 242828.i 1.50777i 0.657007 + 0.753885i \(0.271822\pi\)
−0.657007 + 0.753885i \(0.728178\pi\)
\(12\) 0 0
\(13\) −398104. −1.07221 −0.536105 0.844152i \(-0.680105\pi\)
−0.536105 + 0.844152i \(0.680105\pi\)
\(14\) − 507765.i − 0.944109i
\(15\) 0 0
\(16\) −346888. −0.330818
\(17\) 2.02619e6i 1.42704i 0.700634 + 0.713520i \(0.252900\pi\)
−0.700634 + 0.713520i \(0.747100\pi\)
\(18\) 0 0
\(19\) −2.77192e6 −1.11947 −0.559735 0.828672i \(-0.689097\pi\)
−0.559735 + 0.828672i \(0.689097\pi\)
\(20\) − 2.07842e6i − 0.649505i
\(21\) 0 0
\(22\) −5.71678e6 −1.10927
\(23\) − 2.47900e6i − 0.385157i −0.981282 0.192579i \(-0.938315\pi\)
0.981282 0.192579i \(-0.0616850\pi\)
\(24\) 0 0
\(25\) −9.81076e6 −1.00462
\(26\) − 9.37237e6i − 0.788828i
\(27\) 0 0
\(28\) −1.01315e7 −0.588689
\(29\) − 9.13358e6i − 0.445298i −0.974899 0.222649i \(-0.928530\pi\)
0.974899 0.222649i \(-0.0714704\pi\)
\(30\) 0 0
\(31\) −1.81463e7 −0.633840 −0.316920 0.948452i \(-0.602649\pi\)
−0.316920 + 0.948452i \(0.602649\pi\)
\(32\) 2.78440e7i 0.829816i
\(33\) 0 0
\(34\) −4.77017e7 −1.04988
\(35\) 9.54280e7i 1.81692i
\(36\) 0 0
\(37\) 1.14330e7 0.164874 0.0824369 0.996596i \(-0.473730\pi\)
0.0824369 + 0.996596i \(0.473730\pi\)
\(38\) − 6.52579e7i − 0.823598i
\(39\) 0 0
\(40\) 1.55595e8 1.51949
\(41\) − 1.37099e8i − 1.18336i −0.806174 0.591678i \(-0.798466\pi\)
0.806174 0.591678i \(-0.201534\pi\)
\(42\) 0 0
\(43\) 9.87897e7 0.672000 0.336000 0.941862i \(-0.390926\pi\)
0.336000 + 0.941862i \(0.390926\pi\)
\(44\) 1.14068e8i 0.691674i
\(45\) 0 0
\(46\) 5.83620e7 0.283362
\(47\) 1.41804e8i 0.618302i 0.951013 + 0.309151i \(0.100045\pi\)
−0.951013 + 0.309151i \(0.899955\pi\)
\(48\) 0 0
\(49\) 1.82702e8 0.646791
\(50\) − 2.30970e8i − 0.739104i
\(51\) 0 0
\(52\) −1.87009e8 −0.491865
\(53\) − 1.39084e8i − 0.332582i −0.986077 0.166291i \(-0.946821\pi\)
0.986077 0.166291i \(-0.0531792\pi\)
\(54\) 0 0
\(55\) 1.07440e9 2.13477
\(56\) − 7.58473e8i − 1.37721i
\(57\) 0 0
\(58\) 2.15027e8 0.327608
\(59\) 1.06798e9i 1.49384i 0.664916 + 0.746918i \(0.268467\pi\)
−0.664916 + 0.746918i \(0.731533\pi\)
\(60\) 0 0
\(61\) −1.44976e9 −1.71651 −0.858256 0.513222i \(-0.828452\pi\)
−0.858256 + 0.513222i \(0.828452\pi\)
\(62\) − 4.27210e8i − 0.466318i
\(63\) 0 0
\(64\) −1.01073e9 −0.941317
\(65\) 1.76142e9i 1.51808i
\(66\) 0 0
\(67\) 1.44613e9 1.07111 0.535556 0.844500i \(-0.320102\pi\)
0.535556 + 0.844500i \(0.320102\pi\)
\(68\) 9.51804e8i 0.654640i
\(69\) 0 0
\(70\) −2.24662e9 −1.33671
\(71\) − 1.33807e9i − 0.741630i −0.928707 0.370815i \(-0.879078\pi\)
0.928707 0.370815i \(-0.120922\pi\)
\(72\) 0 0
\(73\) −1.46812e9 −0.708187 −0.354094 0.935210i \(-0.615211\pi\)
−0.354094 + 0.935210i \(0.615211\pi\)
\(74\) 2.69162e8i 0.121298i
\(75\) 0 0
\(76\) −1.30211e9 −0.513545
\(77\) − 5.23731e9i − 1.93488i
\(78\) 0 0
\(79\) −4.95381e9 −1.60992 −0.804959 0.593331i \(-0.797813\pi\)
−0.804959 + 0.593331i \(0.797813\pi\)
\(80\) 1.53481e9i 0.468388i
\(81\) 0 0
\(82\) 3.22766e9 0.870600
\(83\) 5.60643e8i 0.142330i 0.997465 + 0.0711650i \(0.0226717\pi\)
−0.997465 + 0.0711650i \(0.977328\pi\)
\(84\) 0 0
\(85\) 8.96494e9 2.02047
\(86\) 2.32576e9i 0.494393i
\(87\) 0 0
\(88\) −8.53944e9 −1.61814
\(89\) − 4.73263e9i − 0.847524i −0.905774 0.423762i \(-0.860709\pi\)
0.905774 0.423762i \(-0.139291\pi\)
\(90\) 0 0
\(91\) 8.58629e9 1.37594
\(92\) − 1.16451e9i − 0.176687i
\(93\) 0 0
\(94\) −3.33843e9 −0.454887
\(95\) 1.22644e10i 1.58500i
\(96\) 0 0
\(97\) 1.21701e10 1.41722 0.708609 0.705601i \(-0.249323\pi\)
0.708609 + 0.705601i \(0.249323\pi\)
\(98\) 4.30128e9i 0.475847i
\(99\) 0 0
\(100\) −4.60860e9 −0.460860
\(101\) − 1.19970e10i − 1.14148i −0.821132 0.570738i \(-0.806657\pi\)
0.821132 0.570738i \(-0.193343\pi\)
\(102\) 0 0
\(103\) 1.17848e10 1.01657 0.508286 0.861188i \(-0.330279\pi\)
0.508286 + 0.861188i \(0.330279\pi\)
\(104\) − 1.40000e10i − 1.15070i
\(105\) 0 0
\(106\) 3.27440e9 0.244682
\(107\) 1.67650e10i 1.19532i 0.801749 + 0.597661i \(0.203903\pi\)
−0.801749 + 0.597661i \(0.796097\pi\)
\(108\) 0 0
\(109\) 1.39867e10 0.909040 0.454520 0.890736i \(-0.349811\pi\)
0.454520 + 0.890736i \(0.349811\pi\)
\(110\) 2.52940e10i 1.57056i
\(111\) 0 0
\(112\) 7.48167e9 0.424530
\(113\) 1.66178e9i 0.0901948i 0.998983 + 0.0450974i \(0.0143598\pi\)
−0.998983 + 0.0450974i \(0.985640\pi\)
\(114\) 0 0
\(115\) −1.09684e10 −0.545323
\(116\) − 4.29049e9i − 0.204276i
\(117\) 0 0
\(118\) −2.51430e10 −1.09902
\(119\) − 4.37009e10i − 1.83128i
\(120\) 0 0
\(121\) −3.30279e10 −1.27337
\(122\) − 3.41310e10i − 1.26284i
\(123\) 0 0
\(124\) −8.52421e9 −0.290768
\(125\) 1.99685e8i 0.00654329i
\(126\) 0 0
\(127\) 2.55071e9 0.0772045 0.0386022 0.999255i \(-0.487709\pi\)
0.0386022 + 0.999255i \(0.487709\pi\)
\(128\) 4.71710e9i 0.137286i
\(129\) 0 0
\(130\) −4.14682e10 −1.11686
\(131\) 1.62753e10i 0.421863i 0.977501 + 0.210932i \(0.0676498\pi\)
−0.977501 + 0.210932i \(0.932350\pi\)
\(132\) 0 0
\(133\) 5.97846e10 1.43659
\(134\) 3.40457e10i 0.788021i
\(135\) 0 0
\(136\) −7.12544e10 −1.53150
\(137\) 6.54848e10i 1.35687i 0.734661 + 0.678434i \(0.237341\pi\)
−0.734661 + 0.678434i \(0.762659\pi\)
\(138\) 0 0
\(139\) −3.01191e10 −0.580454 −0.290227 0.956958i \(-0.593731\pi\)
−0.290227 + 0.956958i \(0.593731\pi\)
\(140\) 4.48272e10i 0.833493i
\(141\) 0 0
\(142\) 3.15015e10 0.545620
\(143\) − 9.66707e10i − 1.61664i
\(144\) 0 0
\(145\) −4.04117e10 −0.630474
\(146\) − 3.45633e10i − 0.521016i
\(147\) 0 0
\(148\) 5.37065e9 0.0756342
\(149\) 9.88459e10i 1.34595i 0.739667 + 0.672973i \(0.234983\pi\)
−0.739667 + 0.672973i \(0.765017\pi\)
\(150\) 0 0
\(151\) 8.33061e10 1.06119 0.530594 0.847626i \(-0.321969\pi\)
0.530594 + 0.847626i \(0.321969\pi\)
\(152\) − 9.74790e10i − 1.20141i
\(153\) 0 0
\(154\) 1.23299e11 1.42350
\(155\) 8.02887e10i 0.897420i
\(156\) 0 0
\(157\) −7.89160e10 −0.827307 −0.413653 0.910434i \(-0.635747\pi\)
−0.413653 + 0.910434i \(0.635747\pi\)
\(158\) − 1.16625e11i − 1.18442i
\(159\) 0 0
\(160\) 1.23196e11 1.17489
\(161\) 5.34671e10i 0.494262i
\(162\) 0 0
\(163\) 1.98358e10 0.172390 0.0861951 0.996278i \(-0.472529\pi\)
0.0861951 + 0.996278i \(0.472529\pi\)
\(164\) − 6.44023e10i − 0.542853i
\(165\) 0 0
\(166\) −1.31990e10 −0.104713
\(167\) − 4.31117e10i − 0.331905i −0.986134 0.165952i \(-0.946930\pi\)
0.986134 0.165952i \(-0.0530698\pi\)
\(168\) 0 0
\(169\) 2.06281e10 0.149632
\(170\) 2.11057e11i 1.48647i
\(171\) 0 0
\(172\) 4.64064e10 0.308273
\(173\) − 1.26896e11i − 0.818875i −0.912338 0.409438i \(-0.865725\pi\)
0.912338 0.409438i \(-0.134275\pi\)
\(174\) 0 0
\(175\) 2.11598e11 1.28920
\(176\) − 8.42340e10i − 0.498797i
\(177\) 0 0
\(178\) 1.11418e11 0.623527
\(179\) − 1.71735e11i − 0.934531i −0.884117 0.467265i \(-0.845239\pi\)
0.884117 0.467265i \(-0.154761\pi\)
\(180\) 0 0
\(181\) −9.76637e10 −0.502737 −0.251368 0.967892i \(-0.580881\pi\)
−0.251368 + 0.967892i \(0.580881\pi\)
\(182\) 2.02143e11i 1.01228i
\(183\) 0 0
\(184\) 8.71782e10 0.413351
\(185\) − 5.05856e10i − 0.233436i
\(186\) 0 0
\(187\) −4.92016e11 −2.15165
\(188\) 6.66126e10i 0.283640i
\(189\) 0 0
\(190\) −2.88735e11 −1.16609
\(191\) 1.00665e11i 0.396015i 0.980201 + 0.198007i \(0.0634470\pi\)
−0.980201 + 0.198007i \(0.936553\pi\)
\(192\) 0 0
\(193\) −3.28449e11 −1.22654 −0.613270 0.789873i \(-0.710146\pi\)
−0.613270 + 0.789873i \(0.710146\pi\)
\(194\) 2.86516e11i 1.04265i
\(195\) 0 0
\(196\) 8.58244e10 0.296709
\(197\) 2.22370e11i 0.749453i 0.927135 + 0.374727i \(0.122263\pi\)
−0.927135 + 0.374727i \(0.877737\pi\)
\(198\) 0 0
\(199\) 1.76697e11 0.566191 0.283095 0.959092i \(-0.408639\pi\)
0.283095 + 0.959092i \(0.408639\pi\)
\(200\) − 3.45011e11i − 1.07816i
\(201\) 0 0
\(202\) 2.82440e11 0.839788
\(203\) 1.96993e11i 0.571439i
\(204\) 0 0
\(205\) −6.06598e11 −1.67545
\(206\) 2.77445e11i 0.747896i
\(207\) 0 0
\(208\) 1.38097e11 0.354706
\(209\) − 6.73098e11i − 1.68790i
\(210\) 0 0
\(211\) −2.67258e11 −0.639027 −0.319513 0.947582i \(-0.603519\pi\)
−0.319513 + 0.947582i \(0.603519\pi\)
\(212\) − 6.53349e10i − 0.152569i
\(213\) 0 0
\(214\) −3.94691e11 −0.879403
\(215\) − 4.37097e11i − 0.951449i
\(216\) 0 0
\(217\) 3.91379e11 0.813390
\(218\) 3.29283e11i 0.668785i
\(219\) 0 0
\(220\) 5.04697e11 0.979304
\(221\) − 8.06635e11i − 1.53009i
\(222\) 0 0
\(223\) −5.07380e11 −0.920046 −0.460023 0.887907i \(-0.652159\pi\)
−0.460023 + 0.887907i \(0.652159\pi\)
\(224\) − 6.00539e11i − 1.06488i
\(225\) 0 0
\(226\) −3.91225e10 −0.0663566
\(227\) 9.61037e11i 1.59445i 0.603682 + 0.797225i \(0.293700\pi\)
−0.603682 + 0.797225i \(0.706300\pi\)
\(228\) 0 0
\(229\) 2.08272e11 0.330715 0.165357 0.986234i \(-0.447122\pi\)
0.165357 + 0.986234i \(0.447122\pi\)
\(230\) − 2.58224e11i − 0.401197i
\(231\) 0 0
\(232\) 3.21197e11 0.477894
\(233\) 1.26797e12i 1.84642i 0.384296 + 0.923210i \(0.374444\pi\)
−0.384296 + 0.923210i \(0.625556\pi\)
\(234\) 0 0
\(235\) 6.27417e11 0.875420
\(236\) 5.01683e11i 0.685282i
\(237\) 0 0
\(238\) 1.02883e12 1.34728
\(239\) 9.98539e11i 1.28049i 0.768171 + 0.640244i \(0.221167\pi\)
−0.768171 + 0.640244i \(0.778833\pi\)
\(240\) 0 0
\(241\) 5.61004e11 0.690050 0.345025 0.938594i \(-0.387870\pi\)
0.345025 + 0.938594i \(0.387870\pi\)
\(242\) − 7.77561e11i − 0.936823i
\(243\) 0 0
\(244\) −6.81024e11 −0.787432
\(245\) − 8.08371e11i − 0.915757i
\(246\) 0 0
\(247\) 1.10351e12 1.20031
\(248\) − 6.38144e11i − 0.680237i
\(249\) 0 0
\(250\) −4.70110e9 −0.00481392
\(251\) − 1.09342e12i − 1.09753i −0.835975 0.548767i \(-0.815097\pi\)
0.835975 0.548767i \(-0.184903\pi\)
\(252\) 0 0
\(253\) 6.01971e11 0.580728
\(254\) 6.00502e10i 0.0567996i
\(255\) 0 0
\(256\) −1.14604e12 −1.04232
\(257\) 1.74563e12i 1.55699i 0.627648 + 0.778497i \(0.284018\pi\)
−0.627648 + 0.778497i \(0.715982\pi\)
\(258\) 0 0
\(259\) −2.46587e11 −0.211578
\(260\) 8.27425e11i 0.696405i
\(261\) 0 0
\(262\) −3.83161e11 −0.310367
\(263\) 3.82099e11i 0.303667i 0.988406 + 0.151833i \(0.0485177\pi\)
−0.988406 + 0.151833i \(0.951482\pi\)
\(264\) 0 0
\(265\) −6.15382e11 −0.470886
\(266\) 1.40748e12i 1.05690i
\(267\) 0 0
\(268\) 6.79321e11 0.491361
\(269\) − 1.54623e12i − 1.09777i −0.835896 0.548887i \(-0.815052\pi\)
0.835896 0.548887i \(-0.184948\pi\)
\(270\) 0 0
\(271\) 1.85954e12 1.27221 0.636104 0.771603i \(-0.280545\pi\)
0.636104 + 0.771603i \(0.280545\pi\)
\(272\) − 7.02862e11i − 0.472091i
\(273\) 0 0
\(274\) −1.54168e12 −0.998253
\(275\) − 2.38232e12i − 1.51474i
\(276\) 0 0
\(277\) −1.96780e12 −1.20665 −0.603326 0.797495i \(-0.706158\pi\)
−0.603326 + 0.797495i \(0.706158\pi\)
\(278\) − 7.09079e11i − 0.427042i
\(279\) 0 0
\(280\) −3.35588e12 −1.94992
\(281\) 3.53773e11i 0.201926i 0.994890 + 0.100963i \(0.0321924\pi\)
−0.994890 + 0.100963i \(0.967808\pi\)
\(282\) 0 0
\(283\) −1.55446e12 −0.856341 −0.428170 0.903698i \(-0.640842\pi\)
−0.428170 + 0.903698i \(0.640842\pi\)
\(284\) − 6.28558e11i − 0.340215i
\(285\) 0 0
\(286\) 2.27587e12 1.18937
\(287\) 2.95695e12i 1.51857i
\(288\) 0 0
\(289\) −2.08947e12 −1.03645
\(290\) − 9.51393e11i − 0.463842i
\(291\) 0 0
\(292\) −6.89650e11 −0.324874
\(293\) − 3.65602e12i − 1.69305i −0.532347 0.846526i \(-0.678690\pi\)
0.532347 0.846526i \(-0.321310\pi\)
\(294\) 0 0
\(295\) 4.72530e12 2.11504
\(296\) 4.02060e11i 0.176943i
\(297\) 0 0
\(298\) −2.32708e12 −0.990217
\(299\) 9.86901e11i 0.412969i
\(300\) 0 0
\(301\) −2.13069e12 −0.862360
\(302\) 1.96124e12i 0.780719i
\(303\) 0 0
\(304\) 9.61544e11 0.370341
\(305\) 6.41449e12i 2.43032i
\(306\) 0 0
\(307\) 4.14571e12 1.52022 0.760111 0.649793i \(-0.225144\pi\)
0.760111 + 0.649793i \(0.225144\pi\)
\(308\) − 2.46022e12i − 0.887607i
\(309\) 0 0
\(310\) −1.89020e12 −0.660235
\(311\) 7.41396e11i 0.254829i 0.991850 + 0.127414i \(0.0406678\pi\)
−0.991850 + 0.127414i \(0.959332\pi\)
\(312\) 0 0
\(313\) −1.19855e12 −0.398966 −0.199483 0.979901i \(-0.563926\pi\)
−0.199483 + 0.979901i \(0.563926\pi\)
\(314\) − 1.85788e12i − 0.608653i
\(315\) 0 0
\(316\) −2.32705e12 −0.738533
\(317\) 2.08772e12i 0.652191i 0.945337 + 0.326096i \(0.105733\pi\)
−0.945337 + 0.326096i \(0.894267\pi\)
\(318\) 0 0
\(319\) 2.21789e12 0.671407
\(320\) 4.47200e12i 1.33276i
\(321\) 0 0
\(322\) −1.25875e12 −0.363631
\(323\) − 5.61644e12i − 1.59753i
\(324\) 0 0
\(325\) 3.90570e12 1.07716
\(326\) 4.66986e11i 0.126828i
\(327\) 0 0
\(328\) 4.82132e12 1.26998
\(329\) − 3.05843e12i − 0.793450i
\(330\) 0 0
\(331\) 5.73492e12 1.44340 0.721701 0.692204i \(-0.243360\pi\)
0.721701 + 0.692204i \(0.243360\pi\)
\(332\) 2.63362e11i 0.0652924i
\(333\) 0 0
\(334\) 1.01496e12 0.244184
\(335\) − 6.39845e12i − 1.51653i
\(336\) 0 0
\(337\) −3.80990e11 −0.0876524 −0.0438262 0.999039i \(-0.513955\pi\)
−0.0438262 + 0.999039i \(0.513955\pi\)
\(338\) 4.85638e11i 0.110085i
\(339\) 0 0
\(340\) 4.21127e12 0.926870
\(341\) − 4.40643e12i − 0.955685i
\(342\) 0 0
\(343\) 2.15190e12 0.453264
\(344\) 3.47410e12i 0.721191i
\(345\) 0 0
\(346\) 2.98745e12 0.602450
\(347\) 3.54684e11i 0.0705009i 0.999379 + 0.0352505i \(0.0112229\pi\)
−0.999379 + 0.0352505i \(0.988777\pi\)
\(348\) 0 0
\(349\) −7.67975e12 −1.48327 −0.741634 0.670805i \(-0.765949\pi\)
−0.741634 + 0.670805i \(0.765949\pi\)
\(350\) 4.98156e12i 0.948473i
\(351\) 0 0
\(352\) −6.76130e12 −1.25117
\(353\) 6.58216e12i 1.20087i 0.799674 + 0.600434i \(0.205005\pi\)
−0.799674 + 0.600434i \(0.794995\pi\)
\(354\) 0 0
\(355\) −5.92032e12 −1.05003
\(356\) − 2.22315e12i − 0.388793i
\(357\) 0 0
\(358\) 4.04307e12 0.687538
\(359\) − 6.68773e12i − 1.12152i −0.827979 0.560760i \(-0.810509\pi\)
0.827979 0.560760i \(-0.189491\pi\)
\(360\) 0 0
\(361\) 1.55245e12 0.253211
\(362\) − 2.29925e12i − 0.369865i
\(363\) 0 0
\(364\) 4.03341e12 0.631197
\(365\) 6.49574e12i 1.00268i
\(366\) 0 0
\(367\) −9.31180e12 −1.39863 −0.699316 0.714813i \(-0.746512\pi\)
−0.699316 + 0.714813i \(0.746512\pi\)
\(368\) 8.59936e11i 0.127417i
\(369\) 0 0
\(370\) 1.19091e12 0.171740
\(371\) 2.99977e12i 0.426794i
\(372\) 0 0
\(373\) 6.18448e11 0.0856562 0.0428281 0.999082i \(-0.486363\pi\)
0.0428281 + 0.999082i \(0.486363\pi\)
\(374\) − 1.15833e13i − 1.58298i
\(375\) 0 0
\(376\) −4.98678e12 −0.663562
\(377\) 3.63611e12i 0.477453i
\(378\) 0 0
\(379\) 6.79881e11 0.0869435 0.0434717 0.999055i \(-0.486158\pi\)
0.0434717 + 0.999055i \(0.486158\pi\)
\(380\) 5.76120e12i 0.727101i
\(381\) 0 0
\(382\) −2.36991e12 −0.291350
\(383\) 5.62340e12i 0.682346i 0.940000 + 0.341173i \(0.110824\pi\)
−0.940000 + 0.341173i \(0.889176\pi\)
\(384\) 0 0
\(385\) −2.31726e13 −2.73949
\(386\) − 7.73252e12i − 0.902370i
\(387\) 0 0
\(388\) 5.71692e12 0.650135
\(389\) − 9.64393e12i − 1.08270i −0.840799 0.541348i \(-0.817914\pi\)
0.840799 0.541348i \(-0.182086\pi\)
\(390\) 0 0
\(391\) 5.02294e12 0.549635
\(392\) 6.42503e12i 0.694137i
\(393\) 0 0
\(394\) −5.23515e12 −0.551376
\(395\) 2.19182e13i 2.27940i
\(396\) 0 0
\(397\) −1.48431e11 −0.0150512 −0.00752561 0.999972i \(-0.502395\pi\)
−0.00752561 + 0.999972i \(0.502395\pi\)
\(398\) 4.15988e12i 0.416549i
\(399\) 0 0
\(400\) 3.40323e12 0.332347
\(401\) 2.15508e11i 0.0207846i 0.999946 + 0.0103923i \(0.00330803\pi\)
−0.999946 + 0.0103923i \(0.996692\pi\)
\(402\) 0 0
\(403\) 7.22411e12 0.679609
\(404\) − 5.63560e12i − 0.523640i
\(405\) 0 0
\(406\) −4.63771e12 −0.420410
\(407\) 2.77625e12i 0.248592i
\(408\) 0 0
\(409\) −1.10585e13 −0.966228 −0.483114 0.875557i \(-0.660494\pi\)
−0.483114 + 0.875557i \(0.660494\pi\)
\(410\) − 1.42809e13i − 1.23264i
\(411\) 0 0
\(412\) 5.53593e12 0.466342
\(413\) − 2.30342e13i − 1.91700i
\(414\) 0 0
\(415\) 2.48058e12 0.201517
\(416\) − 1.10848e13i − 0.889737i
\(417\) 0 0
\(418\) 1.58464e13 1.24180
\(419\) 7.75836e12i 0.600758i 0.953820 + 0.300379i \(0.0971132\pi\)
−0.953820 + 0.300379i \(0.902887\pi\)
\(420\) 0 0
\(421\) −9.44013e12 −0.713785 −0.356893 0.934145i \(-0.616164\pi\)
−0.356893 + 0.934145i \(0.616164\pi\)
\(422\) − 6.29194e12i − 0.470135i
\(423\) 0 0
\(424\) 4.89113e12 0.356928
\(425\) − 1.98785e13i − 1.43364i
\(426\) 0 0
\(427\) 3.12684e13 2.20275
\(428\) 7.87535e12i 0.548342i
\(429\) 0 0
\(430\) 1.02904e13 0.699985
\(431\) 1.56255e13i 1.05063i 0.850909 + 0.525313i \(0.176052\pi\)
−0.850909 + 0.525313i \(0.823948\pi\)
\(432\) 0 0
\(433\) −1.68288e13 −1.10564 −0.552820 0.833301i \(-0.686448\pi\)
−0.552820 + 0.833301i \(0.686448\pi\)
\(434\) 9.21405e12i 0.598414i
\(435\) 0 0
\(436\) 6.57025e12 0.417013
\(437\) 6.87159e12i 0.431172i
\(438\) 0 0
\(439\) −6.14121e12 −0.376644 −0.188322 0.982107i \(-0.560305\pi\)
−0.188322 + 0.982107i \(0.560305\pi\)
\(440\) 3.77829e13i 2.29104i
\(441\) 0 0
\(442\) 1.89902e13 1.12569
\(443\) − 1.37391e12i − 0.0805265i −0.999189 0.0402633i \(-0.987180\pi\)
0.999189 0.0402633i \(-0.0128197\pi\)
\(444\) 0 0
\(445\) −2.09396e13 −1.19996
\(446\) − 1.19450e13i − 0.676881i
\(447\) 0 0
\(448\) 2.17994e13 1.20797
\(449\) 1.04566e13i 0.573004i 0.958080 + 0.286502i \(0.0924925\pi\)
−0.958080 + 0.286502i \(0.907508\pi\)
\(450\) 0 0
\(451\) 3.32915e13 1.78423
\(452\) 7.80621e11i 0.0413759i
\(453\) 0 0
\(454\) −2.26252e13 −1.17304
\(455\) − 3.79902e13i − 1.94812i
\(456\) 0 0
\(457\) 3.39247e13 1.70190 0.850951 0.525245i \(-0.176026\pi\)
0.850951 + 0.525245i \(0.176026\pi\)
\(458\) 4.90325e12i 0.243308i
\(459\) 0 0
\(460\) −5.15240e12 −0.250162
\(461\) 2.28758e13i 1.09868i 0.835599 + 0.549340i \(0.185121\pi\)
−0.835599 + 0.549340i \(0.814879\pi\)
\(462\) 0 0
\(463\) −3.66690e12 −0.172343 −0.0861716 0.996280i \(-0.527463\pi\)
−0.0861716 + 0.996280i \(0.527463\pi\)
\(464\) 3.16833e12i 0.147313i
\(465\) 0 0
\(466\) −2.98513e13 −1.35842
\(467\) − 2.48699e13i − 1.11967i −0.828605 0.559834i \(-0.810865\pi\)
0.828605 0.559834i \(-0.189135\pi\)
\(468\) 0 0
\(469\) −3.11902e13 −1.37453
\(470\) 1.47710e13i 0.644050i
\(471\) 0 0
\(472\) −3.75573e13 −1.60319
\(473\) 2.39889e13i 1.01322i
\(474\) 0 0
\(475\) 2.71946e13 1.12464
\(476\) − 2.05285e13i − 0.840083i
\(477\) 0 0
\(478\) −2.35081e13 −0.942060
\(479\) 4.40756e12i 0.174792i 0.996174 + 0.0873959i \(0.0278545\pi\)
−0.996174 + 0.0873959i \(0.972145\pi\)
\(480\) 0 0
\(481\) −4.55152e12 −0.176779
\(482\) 1.32074e13i 0.507672i
\(483\) 0 0
\(484\) −1.55149e13 −0.584145
\(485\) − 5.38470e13i − 2.00656i
\(486\) 0 0
\(487\) −9.42862e12 −0.344194 −0.172097 0.985080i \(-0.555054\pi\)
−0.172097 + 0.985080i \(0.555054\pi\)
\(488\) − 5.09831e13i − 1.84216i
\(489\) 0 0
\(490\) 1.90311e13 0.673726
\(491\) 1.40743e13i 0.493195i 0.969118 + 0.246597i \(0.0793125\pi\)
−0.969118 + 0.246597i \(0.920688\pi\)
\(492\) 0 0
\(493\) 1.85064e13 0.635459
\(494\) 2.59794e13i 0.883069i
\(495\) 0 0
\(496\) 6.29473e12 0.209686
\(497\) 2.88595e13i 0.951714i
\(498\) 0 0
\(499\) −3.37708e13 −1.09154 −0.545769 0.837935i \(-0.683763\pi\)
−0.545769 + 0.837935i \(0.683763\pi\)
\(500\) 9.38021e10i 0.00300167i
\(501\) 0 0
\(502\) 2.57418e13 0.807460
\(503\) 4.24460e12i 0.131825i 0.997825 + 0.0659124i \(0.0209958\pi\)
−0.997825 + 0.0659124i \(0.979004\pi\)
\(504\) 0 0
\(505\) −5.30811e13 −1.61615
\(506\) 1.41719e13i 0.427244i
\(507\) 0 0
\(508\) 1.19819e12 0.0354168
\(509\) 2.01121e13i 0.588664i 0.955703 + 0.294332i \(0.0950972\pi\)
−0.955703 + 0.294332i \(0.904903\pi\)
\(510\) 0 0
\(511\) 3.16644e13 0.908798
\(512\) − 2.21504e13i − 0.629552i
\(513\) 0 0
\(514\) −4.10966e13 −1.14549
\(515\) − 5.21423e13i − 1.43931i
\(516\) 0 0
\(517\) −3.44341e13 −0.932257
\(518\) − 5.80527e12i − 0.155659i
\(519\) 0 0
\(520\) −6.19432e13 −1.62921
\(521\) 1.35570e12i 0.0353163i 0.999844 + 0.0176581i \(0.00562105\pi\)
−0.999844 + 0.0176581i \(0.994379\pi\)
\(522\) 0 0
\(523\) 5.72383e13 1.46278 0.731389 0.681961i \(-0.238872\pi\)
0.731389 + 0.681961i \(0.238872\pi\)
\(524\) 7.64530e12i 0.193526i
\(525\) 0 0
\(526\) −8.99557e12 −0.223409
\(527\) − 3.67679e13i − 0.904515i
\(528\) 0 0
\(529\) 3.52811e13 0.851654
\(530\) − 1.44877e13i − 0.346432i
\(531\) 0 0
\(532\) 2.80838e13 0.659019
\(533\) 5.45797e13i 1.26881i
\(534\) 0 0
\(535\) 7.41771e13 1.69239
\(536\) 5.08557e13i 1.14952i
\(537\) 0 0
\(538\) 3.64022e13 0.807637
\(539\) 4.43653e13i 0.975212i
\(540\) 0 0
\(541\) 2.34478e13 0.505961 0.252980 0.967471i \(-0.418589\pi\)
0.252980 + 0.967471i \(0.418589\pi\)
\(542\) 4.37782e13i 0.935969i
\(543\) 0 0
\(544\) −5.64174e13 −1.18418
\(545\) − 6.18845e13i − 1.28706i
\(546\) 0 0
\(547\) −1.36836e13 −0.279424 −0.139712 0.990192i \(-0.544618\pi\)
−0.139712 + 0.990192i \(0.544618\pi\)
\(548\) 3.07615e13i 0.622449i
\(549\) 0 0
\(550\) 5.60859e13 1.11440
\(551\) 2.53175e13i 0.498498i
\(552\) 0 0
\(553\) 1.06844e14 2.06597
\(554\) − 4.63269e13i − 0.887738i
\(555\) 0 0
\(556\) −1.41484e13 −0.266277
\(557\) 6.54044e13i 1.21992i 0.792432 + 0.609960i \(0.208814\pi\)
−0.792432 + 0.609960i \(0.791186\pi\)
\(558\) 0 0
\(559\) −3.93285e13 −0.720525
\(560\) − 3.31028e13i − 0.601069i
\(561\) 0 0
\(562\) −8.32871e12 −0.148558
\(563\) 3.14419e13i 0.555862i 0.960601 + 0.277931i \(0.0896486\pi\)
−0.960601 + 0.277931i \(0.910351\pi\)
\(564\) 0 0
\(565\) 7.35258e12 0.127702
\(566\) − 3.65959e13i − 0.630013i
\(567\) 0 0
\(568\) 4.70554e13 0.795917
\(569\) − 1.01433e14i − 1.70066i −0.526247 0.850332i \(-0.676401\pi\)
0.526247 0.850332i \(-0.323599\pi\)
\(570\) 0 0
\(571\) 2.86156e13 0.471435 0.235718 0.971822i \(-0.424256\pi\)
0.235718 + 0.971822i \(0.424256\pi\)
\(572\) − 4.54110e13i − 0.741619i
\(573\) 0 0
\(574\) −6.96141e13 −1.11722
\(575\) 2.43209e13i 0.386937i
\(576\) 0 0
\(577\) −1.60161e13 −0.250425 −0.125212 0.992130i \(-0.539961\pi\)
−0.125212 + 0.992130i \(0.539961\pi\)
\(578\) − 4.91913e13i − 0.762517i
\(579\) 0 0
\(580\) −1.89834e13 −0.289223
\(581\) − 1.20919e13i − 0.182648i
\(582\) 0 0
\(583\) 3.37736e13 0.501458
\(584\) − 5.16289e13i − 0.760027i
\(585\) 0 0
\(586\) 8.60719e13 1.24559
\(587\) − 1.31766e14i − 1.89066i −0.326115 0.945330i \(-0.605740\pi\)
0.326115 0.945330i \(-0.394260\pi\)
\(588\) 0 0
\(589\) 5.03000e13 0.709564
\(590\) 1.11246e14i 1.55605i
\(591\) 0 0
\(592\) −3.96597e12 −0.0545432
\(593\) 4.28136e13i 0.583859i 0.956440 + 0.291930i \(0.0942973\pi\)
−0.956440 + 0.291930i \(0.905703\pi\)
\(594\) 0 0
\(595\) −1.93356e14 −2.59282
\(596\) 4.64328e13i 0.617439i
\(597\) 0 0
\(598\) −2.32341e13 −0.303823
\(599\) 1.26813e14i 1.64448i 0.569141 + 0.822240i \(0.307276\pi\)
−0.569141 + 0.822240i \(0.692724\pi\)
\(600\) 0 0
\(601\) −1.25899e14 −1.60565 −0.802823 0.596217i \(-0.796670\pi\)
−0.802823 + 0.596217i \(0.796670\pi\)
\(602\) − 5.01619e13i − 0.634442i
\(603\) 0 0
\(604\) 3.91330e13 0.486809
\(605\) 1.46133e14i 1.80290i
\(606\) 0 0
\(607\) −5.15751e13 −0.625889 −0.312944 0.949771i \(-0.601315\pi\)
−0.312944 + 0.949771i \(0.601315\pi\)
\(608\) − 7.71813e13i − 0.928954i
\(609\) 0 0
\(610\) −1.51013e14 −1.78799
\(611\) − 5.64529e13i − 0.662949i
\(612\) 0 0
\(613\) 6.86500e13 0.793119 0.396560 0.918009i \(-0.370204\pi\)
0.396560 + 0.918009i \(0.370204\pi\)
\(614\) 9.76005e13i 1.11843i
\(615\) 0 0
\(616\) 1.84178e14 2.07652
\(617\) − 9.31549e13i − 1.04179i −0.853621 0.520895i \(-0.825598\pi\)
0.853621 0.520895i \(-0.174402\pi\)
\(618\) 0 0
\(619\) −9.30635e13 −1.02406 −0.512031 0.858967i \(-0.671107\pi\)
−0.512031 + 0.858967i \(0.671107\pi\)
\(620\) 3.77156e13i 0.411682i
\(621\) 0 0
\(622\) −1.74543e13 −0.187478
\(623\) 1.02073e14i 1.08761i
\(624\) 0 0
\(625\) −9.49247e13 −0.995357
\(626\) − 2.82170e13i − 0.293521i
\(627\) 0 0
\(628\) −3.70707e13 −0.379519
\(629\) 2.31655e13i 0.235282i
\(630\) 0 0
\(631\) 8.95590e13 0.895288 0.447644 0.894212i \(-0.352263\pi\)
0.447644 + 0.894212i \(0.352263\pi\)
\(632\) − 1.74209e14i − 1.72776i
\(633\) 0 0
\(634\) −4.91501e13 −0.479820
\(635\) − 1.12857e13i − 0.109310i
\(636\) 0 0
\(637\) −7.27345e13 −0.693495
\(638\) 5.22147e13i 0.493957i
\(639\) 0 0
\(640\) 2.08709e13 0.194375
\(641\) − 1.07669e14i − 0.994946i −0.867480 0.497473i \(-0.834261\pi\)
0.867480 0.497473i \(-0.165739\pi\)
\(642\) 0 0
\(643\) 4.75748e13 0.432835 0.216417 0.976301i \(-0.430563\pi\)
0.216417 + 0.976301i \(0.430563\pi\)
\(644\) 2.51161e13i 0.226738i
\(645\) 0 0
\(646\) 1.32225e14 1.17531
\(647\) 1.19789e13i 0.105656i 0.998604 + 0.0528280i \(0.0168235\pi\)
−0.998604 + 0.0528280i \(0.983176\pi\)
\(648\) 0 0
\(649\) −2.59335e14 −2.25236
\(650\) 9.19500e13i 0.792474i
\(651\) 0 0
\(652\) 9.31788e12 0.0790823
\(653\) 1.60289e13i 0.135001i 0.997719 + 0.0675007i \(0.0215025\pi\)
−0.997719 + 0.0675007i \(0.978498\pi\)
\(654\) 0 0
\(655\) 7.20103e13 0.597294
\(656\) 4.75580e13i 0.391476i
\(657\) 0 0
\(658\) 7.20033e13 0.583744
\(659\) 2.23451e14i 1.79786i 0.438093 + 0.898930i \(0.355654\pi\)
−0.438093 + 0.898930i \(0.644346\pi\)
\(660\) 0 0
\(661\) 1.81957e13 0.144198 0.0720992 0.997397i \(-0.477030\pi\)
0.0720992 + 0.997397i \(0.477030\pi\)
\(662\) 1.35015e14i 1.06192i
\(663\) 0 0
\(664\) −1.97159e13 −0.152749
\(665\) − 2.64518e14i − 2.03398i
\(666\) 0 0
\(667\) −2.26422e13 −0.171510
\(668\) − 2.02517e13i − 0.152258i
\(669\) 0 0
\(670\) 1.50636e14 1.11572
\(671\) − 3.52042e14i − 2.58810i
\(672\) 0 0
\(673\) 2.39243e14 1.73286 0.866432 0.499294i \(-0.166407\pi\)
0.866432 + 0.499294i \(0.166407\pi\)
\(674\) − 8.96946e12i − 0.0644862i
\(675\) 0 0
\(676\) 9.69004e12 0.0686423
\(677\) − 2.31720e14i − 1.62937i −0.579902 0.814686i \(-0.696909\pi\)
0.579902 0.814686i \(-0.303091\pi\)
\(678\) 0 0
\(679\) −2.62485e14 −1.81868
\(680\) 3.15267e14i 2.16837i
\(681\) 0 0
\(682\) 1.03738e14 0.703101
\(683\) − 1.67561e14i − 1.12738i −0.825986 0.563690i \(-0.809381\pi\)
0.825986 0.563690i \(-0.190619\pi\)
\(684\) 0 0
\(685\) 2.89739e14 1.92112
\(686\) 5.06611e13i 0.333468i
\(687\) 0 0
\(688\) −3.42689e13 −0.222310
\(689\) 5.53701e13i 0.356598i
\(690\) 0 0
\(691\) 1.92427e14 1.22145 0.610725 0.791843i \(-0.290878\pi\)
0.610725 + 0.791843i \(0.290878\pi\)
\(692\) − 5.96094e13i − 0.375651i
\(693\) 0 0
\(694\) −8.35017e12 −0.0518678
\(695\) 1.33262e14i 0.821833i
\(696\) 0 0
\(697\) 2.77790e14 1.68870
\(698\) − 1.80801e14i − 1.09125i
\(699\) 0 0
\(700\) 9.93982e13 0.591409
\(701\) 3.26565e14i 1.92921i 0.263702 + 0.964604i \(0.415057\pi\)
−0.263702 + 0.964604i \(0.584943\pi\)
\(702\) 0 0
\(703\) −3.16913e13 −0.184571
\(704\) − 2.45434e14i − 1.41929i
\(705\) 0 0
\(706\) −1.54961e14 −0.883483
\(707\) 2.58752e14i 1.46483i
\(708\) 0 0
\(709\) −3.21622e14 −1.79521 −0.897604 0.440804i \(-0.854694\pi\)
−0.897604 + 0.440804i \(0.854694\pi\)
\(710\) − 1.39379e14i − 0.772514i
\(711\) 0 0
\(712\) 1.66431e14 0.909563
\(713\) 4.49847e13i 0.244128i
\(714\) 0 0
\(715\) −4.27721e14 −2.28892
\(716\) − 8.06724e13i − 0.428706i
\(717\) 0 0
\(718\) 1.57446e14 0.825106
\(719\) 2.74758e13i 0.142990i 0.997441 + 0.0714950i \(0.0227770\pi\)
−0.997441 + 0.0714950i \(0.977223\pi\)
\(720\) 0 0
\(721\) −2.54175e14 −1.30454
\(722\) 3.65487e13i 0.186288i
\(723\) 0 0
\(724\) −4.58775e13 −0.230625
\(725\) 8.96073e13i 0.447356i
\(726\) 0 0
\(727\) 8.42042e13 0.414631 0.207316 0.978274i \(-0.433527\pi\)
0.207316 + 0.978274i \(0.433527\pi\)
\(728\) 3.01951e14i 1.47666i
\(729\) 0 0
\(730\) −1.52926e14 −0.737679
\(731\) 2.00167e14i 0.958971i
\(732\) 0 0
\(733\) 3.62014e14 1.71082 0.855412 0.517947i \(-0.173304\pi\)
0.855412 + 0.517947i \(0.173304\pi\)
\(734\) − 2.19223e14i − 1.02898i
\(735\) 0 0
\(736\) 6.90254e13 0.319610
\(737\) 3.51162e14i 1.61499i
\(738\) 0 0
\(739\) −2.44206e14 −1.10799 −0.553993 0.832522i \(-0.686896\pi\)
−0.553993 + 0.832522i \(0.686896\pi\)
\(740\) − 2.37625e13i − 0.107086i
\(741\) 0 0
\(742\) −7.06222e13 −0.313994
\(743\) − 6.89118e13i − 0.304333i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486250\pi\)
\(744\) 0 0
\(745\) 4.37346e14 1.90565
\(746\) 1.45598e13i 0.0630176i
\(747\) 0 0
\(748\) −2.31124e14 −0.987047
\(749\) − 3.61587e14i − 1.53392i
\(750\) 0 0
\(751\) −2.44541e14 −1.02365 −0.511825 0.859090i \(-0.671030\pi\)
−0.511825 + 0.859090i \(0.671030\pi\)
\(752\) − 4.91902e13i − 0.204545i
\(753\) 0 0
\(754\) −8.56032e13 −0.351264
\(755\) − 3.68589e14i − 1.50248i
\(756\) 0 0
\(757\) −3.72001e13 −0.149646 −0.0748230 0.997197i \(-0.523839\pi\)
−0.0748230 + 0.997197i \(0.523839\pi\)
\(758\) 1.60061e13i 0.0639646i
\(759\) 0 0
\(760\) −4.31298e14 −1.70102
\(761\) − 5.57377e13i − 0.218386i −0.994021 0.109193i \(-0.965173\pi\)
0.994021 0.109193i \(-0.0348267\pi\)
\(762\) 0 0
\(763\) −3.01665e14 −1.16655
\(764\) 4.72873e13i 0.181668i
\(765\) 0 0
\(766\) −1.32389e14 −0.502005
\(767\) − 4.25167e14i − 1.60171i
\(768\) 0 0
\(769\) 4.18923e13 0.155777 0.0778883 0.996962i \(-0.475182\pi\)
0.0778883 + 0.996962i \(0.475182\pi\)
\(770\) − 5.45541e14i − 2.01546i
\(771\) 0 0
\(772\) −1.54289e14 −0.562663
\(773\) 4.44998e13i 0.161236i 0.996745 + 0.0806178i \(0.0256893\pi\)
−0.996745 + 0.0806178i \(0.974311\pi\)
\(774\) 0 0
\(775\) 1.78029e14 0.636769
\(776\) 4.27983e14i 1.52096i
\(777\) 0 0
\(778\) 2.27043e14 0.796543
\(779\) 3.80027e14i 1.32473i
\(780\) 0 0
\(781\) 3.24921e14 1.11821
\(782\) 1.18253e14i 0.404369i
\(783\) 0 0
\(784\) −6.33773e13 −0.213970
\(785\) 3.49165e14i 1.17134i
\(786\) 0 0
\(787\) −5.29172e12 −0.0175276 −0.00876381 0.999962i \(-0.502790\pi\)
−0.00876381 + 0.999962i \(0.502790\pi\)
\(788\) 1.04458e14i 0.343804i
\(789\) 0 0
\(790\) −5.16010e14 −1.67696
\(791\) − 3.58412e13i − 0.115745i
\(792\) 0 0
\(793\) 5.77155e14 1.84046
\(794\) − 3.49444e12i − 0.0110732i
\(795\) 0 0
\(796\) 8.30031e13 0.259734
\(797\) 2.29589e14i 0.713936i 0.934117 + 0.356968i \(0.116190\pi\)
−0.934117 + 0.356968i \(0.883810\pi\)
\(798\) 0 0
\(799\) −2.87323e14 −0.882342
\(800\) − 2.73171e14i − 0.833651i
\(801\) 0 0
\(802\) −5.07360e12 −0.0152913
\(803\) − 3.56501e14i − 1.06778i
\(804\) 0 0
\(805\) 2.36566e14 0.699799
\(806\) 1.70074e14i 0.499991i
\(807\) 0 0
\(808\) 4.21895e14 1.22503
\(809\) − 2.21639e14i − 0.639594i −0.947486 0.319797i \(-0.896385\pi\)
0.947486 0.319797i \(-0.103615\pi\)
\(810\) 0 0
\(811\) 2.70111e14 0.769907 0.384953 0.922936i \(-0.374218\pi\)
0.384953 + 0.922936i \(0.374218\pi\)
\(812\) 9.25373e13i 0.262142i
\(813\) 0 0
\(814\) −6.53600e13 −0.182890
\(815\) − 8.77641e13i − 0.244078i
\(816\) 0 0
\(817\) −2.73837e14 −0.752283
\(818\) − 2.60345e14i − 0.710858i
\(819\) 0 0
\(820\) −2.84949e14 −0.768596
\(821\) 1.46064e14i 0.391586i 0.980645 + 0.195793i \(0.0627281\pi\)
−0.980645 + 0.195793i \(0.937272\pi\)
\(822\) 0 0
\(823\) −3.82425e13 −0.101286 −0.0506428 0.998717i \(-0.516127\pi\)
−0.0506428 + 0.998717i \(0.516127\pi\)
\(824\) 4.14434e14i 1.09099i
\(825\) 0 0
\(826\) 5.42283e14 1.41035
\(827\) 3.49746e13i 0.0904119i 0.998978 + 0.0452060i \(0.0143944\pi\)
−0.998978 + 0.0452060i \(0.985606\pi\)
\(828\) 0 0
\(829\) 6.43554e14 1.64366 0.821831 0.569732i \(-0.192953\pi\)
0.821831 + 0.569732i \(0.192953\pi\)
\(830\) 5.83991e13i 0.148257i
\(831\) 0 0
\(832\) 4.02376e14 1.00929
\(833\) 3.70191e14i 0.922997i
\(834\) 0 0
\(835\) −1.90749e14 −0.469926
\(836\) − 3.16188e14i − 0.774308i
\(837\) 0 0
\(838\) −1.82651e14 −0.441980
\(839\) 7.76705e13i 0.186830i 0.995627 + 0.0934149i \(0.0297783\pi\)
−0.995627 + 0.0934149i \(0.970222\pi\)
\(840\) 0 0
\(841\) 3.37285e14 0.801710
\(842\) − 2.22245e14i − 0.525135i
\(843\) 0 0
\(844\) −1.25545e14 −0.293147
\(845\) − 9.12695e13i − 0.211857i
\(846\) 0 0
\(847\) 7.12346e14 1.63408
\(848\) 4.82467e13i 0.110024i
\(849\) 0 0
\(850\) 4.67990e14 1.05473
\(851\) − 2.83425e13i − 0.0635023i
\(852\) 0 0
\(853\) −3.53687e14 −0.783202 −0.391601 0.920135i \(-0.628079\pi\)
−0.391601 + 0.920135i \(0.628079\pi\)
\(854\) 7.36136e14i 1.62057i
\(855\) 0 0
\(856\) −5.89569e14 −1.28282
\(857\) 5.53847e14i 1.19808i 0.800719 + 0.599040i \(0.204451\pi\)
−0.800719 + 0.599040i \(0.795549\pi\)
\(858\) 0 0
\(859\) 4.82091e14 1.03077 0.515387 0.856958i \(-0.327648\pi\)
0.515387 + 0.856958i \(0.327648\pi\)
\(860\) − 2.05326e14i − 0.436467i
\(861\) 0 0
\(862\) −3.67864e14 −0.772950
\(863\) 2.02122e14i 0.422240i 0.977460 + 0.211120i \(0.0677111\pi\)
−0.977460 + 0.211120i \(0.932289\pi\)
\(864\) 0 0
\(865\) −5.61454e14 −1.15940
\(866\) − 3.96193e14i − 0.813424i
\(867\) 0 0
\(868\) 1.83850e14 0.373134
\(869\) − 1.20292e15i − 2.42739i
\(870\) 0 0
\(871\) −5.75712e14 −1.14846
\(872\) 4.91866e14i 0.975583i
\(873\) 0 0
\(874\) −1.61775e14 −0.317215
\(875\) − 4.30681e12i − 0.00839683i
\(876\) 0 0
\(877\) −2.83664e14 −0.546771 −0.273386 0.961905i \(-0.588143\pi\)
−0.273386 + 0.961905i \(0.588143\pi\)
\(878\) − 1.44580e14i − 0.277099i
\(879\) 0 0
\(880\) −3.72695e14 −0.706221
\(881\) − 7.92783e14i − 1.49374i −0.664970 0.746870i \(-0.731556\pi\)
0.664970 0.746870i \(-0.268444\pi\)
\(882\) 0 0
\(883\) 5.58732e14 1.04088 0.520439 0.853899i \(-0.325768\pi\)
0.520439 + 0.853899i \(0.325768\pi\)
\(884\) − 3.78917e14i − 0.701911i
\(885\) 0 0
\(886\) 3.23453e13 0.0592437
\(887\) 5.60792e13i 0.102137i 0.998695 + 0.0510685i \(0.0162627\pi\)
−0.998695 + 0.0510685i \(0.983737\pi\)
\(888\) 0 0
\(889\) −5.50137e13 −0.0990745
\(890\) − 4.92971e14i − 0.882818i
\(891\) 0 0
\(892\) −2.38342e14 −0.422062
\(893\) − 3.93070e14i − 0.692170i
\(894\) 0 0
\(895\) −7.59845e14 −1.32315
\(896\) − 1.01738e14i − 0.176175i
\(897\) 0 0
\(898\) −2.46174e14 −0.421561
\(899\) 1.65741e14i 0.282248i
\(900\) 0 0
\(901\) 2.81812e14 0.474609
\(902\) 7.83766e14i 1.31266i
\(903\) 0 0
\(904\) −5.84392e13 −0.0967971
\(905\) 4.32115e14i 0.711798i
\(906\) 0 0
\(907\) −1.02359e14 −0.166759 −0.0833795 0.996518i \(-0.526571\pi\)
−0.0833795 + 0.996518i \(0.526571\pi\)
\(908\) 4.51447e14i 0.731438i
\(909\) 0 0
\(910\) 8.94386e14 1.43324
\(911\) 2.03072e14i 0.323637i 0.986821 + 0.161818i \(0.0517358\pi\)
−0.986821 + 0.161818i \(0.948264\pi\)
\(912\) 0 0
\(913\) −1.36140e14 −0.214601
\(914\) 7.98672e14i 1.25210i
\(915\) 0 0
\(916\) 9.78357e13 0.151712
\(917\) − 3.51025e14i − 0.541366i
\(918\) 0 0
\(919\) −1.21378e15 −1.85167 −0.925835 0.377929i \(-0.876636\pi\)
−0.925835 + 0.377929i \(0.876636\pi\)
\(920\) − 3.85722e14i − 0.585241i
\(921\) 0 0
\(922\) −5.38554e14 −0.808304
\(923\) 5.32691e14i 0.795182i
\(924\) 0 0
\(925\) −1.12166e14 −0.165636
\(926\) − 8.63281e13i − 0.126794i
\(927\) 0 0
\(928\) 2.54315e14 0.369516
\(929\) − 7.35885e14i − 1.06348i −0.846906 0.531742i \(-0.821537\pi\)
0.846906 0.531742i \(-0.178463\pi\)
\(930\) 0 0
\(931\) −5.06436e14 −0.724063
\(932\) 5.95630e14i 0.847026i
\(933\) 0 0
\(934\) 5.85499e14 0.823744
\(935\) 2.17694e15i 3.04640i
\(936\) 0 0
\(937\) −9.68431e14 −1.34082 −0.670410 0.741991i \(-0.733882\pi\)
−0.670410 + 0.741991i \(0.733882\pi\)
\(938\) − 7.34296e14i − 1.01125i
\(939\) 0 0
\(940\) 2.94729e14 0.401590
\(941\) − 5.71203e14i − 0.774180i −0.922042 0.387090i \(-0.873480\pi\)
0.922042 0.387090i \(-0.126520\pi\)
\(942\) 0 0
\(943\) −3.39869e14 −0.455778
\(944\) − 3.70469e14i − 0.494188i
\(945\) 0 0
\(946\) −5.64759e14 −0.745431
\(947\) − 5.06129e14i − 0.664525i −0.943187 0.332263i \(-0.892188\pi\)
0.943187 0.332263i \(-0.107812\pi\)
\(948\) 0 0
\(949\) 5.84465e14 0.759325
\(950\) 6.40229e14i 0.827404i
\(951\) 0 0
\(952\) 1.53681e15 1.96533
\(953\) − 1.56038e14i − 0.198503i −0.995062 0.0992515i \(-0.968355\pi\)
0.995062 0.0992515i \(-0.0316448\pi\)
\(954\) 0 0
\(955\) 4.45394e14 0.560696
\(956\) 4.69063e14i 0.587411i
\(957\) 0 0
\(958\) −1.03765e14 −0.128595
\(959\) − 1.41237e15i − 1.74123i
\(960\) 0 0
\(961\) −4.90340e14 −0.598247
\(962\) − 1.07154e14i − 0.130057i
\(963\) 0 0
\(964\) 2.63531e14 0.316553
\(965\) 1.45323e15i 1.73659i
\(966\) 0 0
\(967\) −8.20480e14 −0.970366 −0.485183 0.874413i \(-0.661247\pi\)
−0.485183 + 0.874413i \(0.661247\pi\)
\(968\) − 1.16148e15i − 1.36658i
\(969\) 0 0
\(970\) 1.26769e15 1.47624
\(971\) − 3.49738e14i − 0.405179i −0.979264 0.202590i \(-0.935064\pi\)
0.979264 0.202590i \(-0.0649357\pi\)
\(972\) 0 0
\(973\) 6.49608e14 0.744881
\(974\) − 2.21974e14i − 0.253225i
\(975\) 0 0
\(976\) 5.02904e14 0.567853
\(977\) 4.03993e13i 0.0453838i 0.999743 + 0.0226919i \(0.00722368\pi\)
−0.999743 + 0.0226919i \(0.992776\pi\)
\(978\) 0 0
\(979\) 1.14921e15 1.27787
\(980\) − 3.79732e14i − 0.420094i
\(981\) 0 0
\(982\) −3.31344e14 −0.362845
\(983\) 3.08888e14i 0.336537i 0.985741 + 0.168269i \(0.0538176\pi\)
−0.985741 + 0.168269i \(0.946182\pi\)
\(984\) 0 0
\(985\) 9.83880e14 1.06111
\(986\) 4.35687e14i 0.467509i
\(987\) 0 0
\(988\) 5.18373e14 0.550628
\(989\) − 2.44900e14i − 0.258826i
\(990\) 0 0
\(991\) 1.06886e15 1.11828 0.559140 0.829073i \(-0.311131\pi\)
0.559140 + 0.829073i \(0.311131\pi\)
\(992\) − 5.05266e14i − 0.525971i
\(993\) 0 0
\(994\) −6.79425e14 −0.700179
\(995\) − 7.81798e14i − 0.801639i
\(996\) 0 0
\(997\) 9.25262e14 0.939267 0.469633 0.882862i \(-0.344386\pi\)
0.469633 + 0.882862i \(0.344386\pi\)
\(998\) − 7.95051e14i − 0.803049i
\(999\) 0 0
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 27.11.b.c.26.3 yes 4
3.2 odd 2 inner 27.11.b.c.26.2 4
9.2 odd 6 81.11.d.e.53.2 8
9.4 even 3 81.11.d.e.26.2 8
9.5 odd 6 81.11.d.e.26.3 8
9.7 even 3 81.11.d.e.53.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.11.b.c.26.2 4 3.2 odd 2 inner
27.11.b.c.26.3 yes 4 1.1 even 1 trivial
81.11.d.e.26.2 8 9.4 even 3
81.11.d.e.26.3 8 9.5 odd 6
81.11.d.e.53.2 8 9.2 odd 6
81.11.d.e.53.3 8 9.7 even 3