Properties

Label 315.8.a.c.1.2
Level $315$
Weight $8$
Character 315.1
Self dual yes
Analytic conductor $98.401$
Analytic rank $1$
Dimension $2$
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] = [315,8,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(98.4012830275\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.36675 q^{2} -126.132 q^{4} -125.000 q^{5} -343.000 q^{7} +347.335 q^{8} +O(q^{10})\) \(q-1.36675 q^{2} -126.132 q^{4} -125.000 q^{5} -343.000 q^{7} +347.335 q^{8} +170.844 q^{10} +1432.37 q^{11} -6136.30 q^{13} +468.795 q^{14} +15670.2 q^{16} +15858.5 q^{17} -38567.5 q^{19} +15766.5 q^{20} -1957.69 q^{22} +63987.4 q^{23} +15625.0 q^{25} +8386.79 q^{26} +43263.3 q^{28} -94236.6 q^{29} +275990. q^{31} -65876.1 q^{32} -21674.6 q^{34} +42875.0 q^{35} +156532. q^{37} +52712.1 q^{38} -43416.9 q^{40} +303738. q^{41} +636818. q^{43} -180667. q^{44} -87454.9 q^{46} -512021. q^{47} +117649. q^{49} -21355.5 q^{50} +773984. q^{52} +201249. q^{53} -179046. q^{55} -119136. q^{56} +128798. q^{58} +1.81196e6 q^{59} -982021. q^{61} -377210. q^{62} -1.91575e6 q^{64} +767038. q^{65} -4.45336e6 q^{67} -2.00026e6 q^{68} -58599.4 q^{70} -725436. q^{71} +2.17602e6 q^{73} -213940. q^{74} +4.86459e6 q^{76} -491301. q^{77} -5.21525e6 q^{79} -1.95877e6 q^{80} -415135. q^{82} -6.07921e6 q^{83} -1.98231e6 q^{85} -870371. q^{86} +497511. q^{88} +1.06137e7 q^{89} +2.10475e6 q^{91} -8.07086e6 q^{92} +699805. q^{94} +4.82093e6 q^{95} +6.64483e6 q^{97} -160797. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8} + 2000 q^{10} + 7906 q^{11} - 17818 q^{13} + 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 3612 q^{19} + 5000 q^{20} - 96688 q^{22} - 13844 q^{23} + 31250 q^{25} + 179328 q^{26} + 13720 q^{28} + 126898 q^{29} + 252768 q^{31} + 148224 q^{32} + 175296 q^{34} + 85750 q^{35} - 265860 q^{37} - 458800 q^{38} - 120000 q^{40} + 111920 q^{41} + 947572 q^{43} + 376920 q^{44} + 1051472 q^{46} - 271274 q^{47} + 235298 q^{49} - 250000 q^{50} - 232184 q^{52} + 1267792 q^{53} - 988250 q^{55} - 329280 q^{56} - 3107120 q^{58} + 1360120 q^{59} - 1813680 q^{61} - 37392 q^{62} - 2489984 q^{64} + 2227250 q^{65} - 2189312 q^{67} - 3159640 q^{68} - 686000 q^{70} + 1494928 q^{71} + 7169788 q^{73} + 5967024 q^{74} + 7875376 q^{76} - 2711758 q^{77} - 7942974 q^{79} + 540000 q^{80} + 2391792 q^{82} + 304712 q^{83} - 299750 q^{85} - 5417712 q^{86} + 4463680 q^{88} + 17943528 q^{89} + 6111574 q^{91} - 14774640 q^{92} - 2823104 q^{94} + 451500 q^{95} + 4258074 q^{97} - 1882384 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36675 −0.120805 −0.0604024 0.998174i \(-0.519238\pi\)
−0.0604024 + 0.998174i \(0.519238\pi\)
\(3\) 0 0
\(4\) −126.132 −0.985406
\(5\) −125.000 −0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 347.335 0.239847
\(9\) 0 0
\(10\) 170.844 0.0540256
\(11\) 1432.37 0.324474 0.162237 0.986752i \(-0.448129\pi\)
0.162237 + 0.986752i \(0.448129\pi\)
\(12\) 0 0
\(13\) −6136.30 −0.774649 −0.387325 0.921943i \(-0.626601\pi\)
−0.387325 + 0.921943i \(0.626601\pi\)
\(14\) 468.795 0.0456599
\(15\) 0 0
\(16\) 15670.2 0.956432
\(17\) 15858.5 0.782871 0.391436 0.920205i \(-0.371979\pi\)
0.391436 + 0.920205i \(0.371979\pi\)
\(18\) 0 0
\(19\) −38567.5 −1.28998 −0.644991 0.764190i \(-0.723139\pi\)
−0.644991 + 0.764190i \(0.723139\pi\)
\(20\) 15766.5 0.440687
\(21\) 0 0
\(22\) −1957.69 −0.0391980
\(23\) 63987.4 1.09660 0.548299 0.836282i \(-0.315276\pi\)
0.548299 + 0.836282i \(0.315276\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 8386.79 0.0935813
\(27\) 0 0
\(28\) 43263.3 0.372449
\(29\) −94236.6 −0.717508 −0.358754 0.933432i \(-0.616798\pi\)
−0.358754 + 0.933432i \(0.616798\pi\)
\(30\) 0 0
\(31\) 275990. 1.66390 0.831951 0.554849i \(-0.187224\pi\)
0.831951 + 0.554849i \(0.187224\pi\)
\(32\) −65876.1 −0.355388
\(33\) 0 0
\(34\) −21674.6 −0.0945746
\(35\) 42875.0 0.169031
\(36\) 0 0
\(37\) 156532. 0.508038 0.254019 0.967199i \(-0.418247\pi\)
0.254019 + 0.967199i \(0.418247\pi\)
\(38\) 52712.1 0.155836
\(39\) 0 0
\(40\) −43416.9 −0.107263
\(41\) 303738. 0.688266 0.344133 0.938921i \(-0.388173\pi\)
0.344133 + 0.938921i \(0.388173\pi\)
\(42\) 0 0
\(43\) 636818. 1.22145 0.610725 0.791843i \(-0.290878\pi\)
0.610725 + 0.791843i \(0.290878\pi\)
\(44\) −180667. −0.319738
\(45\) 0 0
\(46\) −87454.9 −0.132474
\(47\) −512021. −0.719358 −0.359679 0.933076i \(-0.617114\pi\)
−0.359679 + 0.933076i \(0.617114\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) −21355.5 −0.0241610
\(51\) 0 0
\(52\) 773984. 0.763344
\(53\) 201249. 0.185681 0.0928406 0.995681i \(-0.470405\pi\)
0.0928406 + 0.995681i \(0.470405\pi\)
\(54\) 0 0
\(55\) −179046. −0.145109
\(56\) −119136. −0.0906535
\(57\) 0 0
\(58\) 128798. 0.0866784
\(59\) 1.81196e6 1.14859 0.574296 0.818648i \(-0.305276\pi\)
0.574296 + 0.818648i \(0.305276\pi\)
\(60\) 0 0
\(61\) −982021. −0.553945 −0.276972 0.960878i \(-0.589331\pi\)
−0.276972 + 0.960878i \(0.589331\pi\)
\(62\) −377210. −0.201007
\(63\) 0 0
\(64\) −1.91575e6 −0.913499
\(65\) 767038. 0.346434
\(66\) 0 0
\(67\) −4.45336e6 −1.80895 −0.904474 0.426528i \(-0.859736\pi\)
−0.904474 + 0.426528i \(0.859736\pi\)
\(68\) −2.00026e6 −0.771446
\(69\) 0 0
\(70\) −58599.4 −0.0204197
\(71\) −725436. −0.240544 −0.120272 0.992741i \(-0.538377\pi\)
−0.120272 + 0.992741i \(0.538377\pi\)
\(72\) 0 0
\(73\) 2.17602e6 0.654685 0.327343 0.944906i \(-0.393847\pi\)
0.327343 + 0.944906i \(0.393847\pi\)
\(74\) −213940. −0.0613735
\(75\) 0 0
\(76\) 4.86459e6 1.27116
\(77\) −491301. −0.122639
\(78\) 0 0
\(79\) −5.21525e6 −1.19009 −0.595045 0.803692i \(-0.702866\pi\)
−0.595045 + 0.803692i \(0.702866\pi\)
\(80\) −1.95877e6 −0.427729
\(81\) 0 0
\(82\) −415135. −0.0831458
\(83\) −6.07921e6 −1.16701 −0.583504 0.812110i \(-0.698319\pi\)
−0.583504 + 0.812110i \(0.698319\pi\)
\(84\) 0 0
\(85\) −1.98231e6 −0.350111
\(86\) −870371. −0.147557
\(87\) 0 0
\(88\) 497511. 0.0778239
\(89\) 1.06137e7 1.59589 0.797946 0.602729i \(-0.205920\pi\)
0.797946 + 0.602729i \(0.205920\pi\)
\(90\) 0 0
\(91\) 2.10475e6 0.292790
\(92\) −8.07086e6 −1.08059
\(93\) 0 0
\(94\) 699805. 0.0869019
\(95\) 4.82093e6 0.576897
\(96\) 0 0
\(97\) 6.64483e6 0.739236 0.369618 0.929184i \(-0.379489\pi\)
0.369618 + 0.929184i \(0.379489\pi\)
\(98\) −160797. −0.0172578
\(99\) 0 0
\(100\) −1.97081e6 −0.197081
\(101\) −1.07531e7 −1.03851 −0.519254 0.854620i \(-0.673790\pi\)
−0.519254 + 0.854620i \(0.673790\pi\)
\(102\) 0 0
\(103\) −1.05886e7 −0.954788 −0.477394 0.878689i \(-0.658419\pi\)
−0.477394 + 0.878689i \(0.658419\pi\)
\(104\) −2.13135e6 −0.185797
\(105\) 0 0
\(106\) −275057. −0.0224312
\(107\) 8.37234e6 0.660699 0.330349 0.943859i \(-0.392833\pi\)
0.330349 + 0.943859i \(0.392833\pi\)
\(108\) 0 0
\(109\) −1.95948e7 −1.44926 −0.724632 0.689136i \(-0.757990\pi\)
−0.724632 + 0.689136i \(0.757990\pi\)
\(110\) 244711. 0.0175299
\(111\) 0 0
\(112\) −5.37487e6 −0.361497
\(113\) −1.36310e7 −0.888694 −0.444347 0.895855i \(-0.646564\pi\)
−0.444347 + 0.895855i \(0.646564\pi\)
\(114\) 0 0
\(115\) −7.99843e6 −0.490413
\(116\) 1.18863e7 0.707037
\(117\) 0 0
\(118\) −2.47649e6 −0.138756
\(119\) −5.43946e6 −0.295898
\(120\) 0 0
\(121\) −1.74355e7 −0.894717
\(122\) 1.34218e6 0.0669192
\(123\) 0 0
\(124\) −3.48112e7 −1.63962
\(125\) −1.95312e6 −0.0894427
\(126\) 0 0
\(127\) 2.23763e7 0.969336 0.484668 0.874698i \(-0.338940\pi\)
0.484668 + 0.874698i \(0.338940\pi\)
\(128\) 1.10505e7 0.465743
\(129\) 0 0
\(130\) −1.04835e6 −0.0418508
\(131\) 4.53330e6 0.176183 0.0880917 0.996112i \(-0.471923\pi\)
0.0880917 + 0.996112i \(0.471923\pi\)
\(132\) 0 0
\(133\) 1.32286e7 0.487567
\(134\) 6.08663e6 0.218530
\(135\) 0 0
\(136\) 5.50821e6 0.187769
\(137\) 5.07657e7 1.68674 0.843371 0.537332i \(-0.180568\pi\)
0.843371 + 0.537332i \(0.180568\pi\)
\(138\) 0 0
\(139\) 1.05183e7 0.332195 0.166097 0.986109i \(-0.446883\pi\)
0.166097 + 0.986109i \(0.446883\pi\)
\(140\) −5.40791e6 −0.166564
\(141\) 0 0
\(142\) 991490. 0.0290589
\(143\) −8.78942e6 −0.251353
\(144\) 0 0
\(145\) 1.17796e7 0.320879
\(146\) −2.97407e6 −0.0790891
\(147\) 0 0
\(148\) −1.97437e7 −0.500624
\(149\) −5.43497e7 −1.34600 −0.673000 0.739642i \(-0.734995\pi\)
−0.673000 + 0.739642i \(0.734995\pi\)
\(150\) 0 0
\(151\) −2.23258e7 −0.527700 −0.263850 0.964564i \(-0.584992\pi\)
−0.263850 + 0.964564i \(0.584992\pi\)
\(152\) −1.33958e7 −0.309398
\(153\) 0 0
\(154\) 671486. 0.0148154
\(155\) −3.44988e7 −0.744120
\(156\) 0 0
\(157\) −4.37788e7 −0.902848 −0.451424 0.892310i \(-0.649084\pi\)
−0.451424 + 0.892310i \(0.649084\pi\)
\(158\) 7.12794e6 0.143769
\(159\) 0 0
\(160\) 8.23451e6 0.158934
\(161\) −2.19477e7 −0.414475
\(162\) 0 0
\(163\) 4.05451e7 0.733300 0.366650 0.930359i \(-0.380505\pi\)
0.366650 + 0.930359i \(0.380505\pi\)
\(164\) −3.83111e7 −0.678221
\(165\) 0 0
\(166\) 8.30876e6 0.140980
\(167\) −9.73453e7 −1.61736 −0.808682 0.588247i \(-0.799818\pi\)
−0.808682 + 0.588247i \(0.799818\pi\)
\(168\) 0 0
\(169\) −2.50943e7 −0.399919
\(170\) 2.70932e6 0.0422951
\(171\) 0 0
\(172\) −8.03231e7 −1.20362
\(173\) 5.10607e7 0.749765 0.374882 0.927072i \(-0.377683\pi\)
0.374882 + 0.927072i \(0.377683\pi\)
\(174\) 0 0
\(175\) −5.35938e6 −0.0755929
\(176\) 2.24454e7 0.310337
\(177\) 0 0
\(178\) −1.45063e7 −0.192791
\(179\) −1.45811e8 −1.90023 −0.950113 0.311907i \(-0.899032\pi\)
−0.950113 + 0.311907i \(0.899032\pi\)
\(180\) 0 0
\(181\) −6.09656e7 −0.764205 −0.382102 0.924120i \(-0.624800\pi\)
−0.382102 + 0.924120i \(0.624800\pi\)
\(182\) −2.87667e6 −0.0353704
\(183\) 0 0
\(184\) 2.22251e7 0.263015
\(185\) −1.95665e7 −0.227202
\(186\) 0 0
\(187\) 2.27151e7 0.254021
\(188\) 6.45822e7 0.708860
\(189\) 0 0
\(190\) −6.58901e6 −0.0696920
\(191\) 1.52578e8 1.58444 0.792219 0.610237i \(-0.208926\pi\)
0.792219 + 0.610237i \(0.208926\pi\)
\(192\) 0 0
\(193\) −1.39277e8 −1.39453 −0.697267 0.716812i \(-0.745601\pi\)
−0.697267 + 0.716812i \(0.745601\pi\)
\(194\) −9.08183e6 −0.0893033
\(195\) 0 0
\(196\) −1.48393e7 −0.140772
\(197\) −6.52480e7 −0.608044 −0.304022 0.952665i \(-0.598330\pi\)
−0.304022 + 0.952665i \(0.598330\pi\)
\(198\) 0 0
\(199\) 1.93503e6 0.0174061 0.00870307 0.999962i \(-0.497230\pi\)
0.00870307 + 0.999962i \(0.497230\pi\)
\(200\) 5.42711e6 0.0479693
\(201\) 0 0
\(202\) 1.46968e7 0.125457
\(203\) 3.23232e7 0.271192
\(204\) 0 0
\(205\) −3.79673e7 −0.307802
\(206\) 1.44719e7 0.115343
\(207\) 0 0
\(208\) −9.61569e7 −0.740899
\(209\) −5.52427e7 −0.418565
\(210\) 0 0
\(211\) −5.17848e7 −0.379502 −0.189751 0.981832i \(-0.560768\pi\)
−0.189751 + 0.981832i \(0.560768\pi\)
\(212\) −2.53839e7 −0.182971
\(213\) 0 0
\(214\) −1.14429e7 −0.0798156
\(215\) −7.96023e7 −0.546249
\(216\) 0 0
\(217\) −9.46647e7 −0.628896
\(218\) 2.67812e7 0.175078
\(219\) 0 0
\(220\) 2.25834e7 0.142991
\(221\) −9.73124e7 −0.606450
\(222\) 0 0
\(223\) −1.25065e8 −0.755209 −0.377605 0.925967i \(-0.623252\pi\)
−0.377605 + 0.925967i \(0.623252\pi\)
\(224\) 2.25955e7 0.134324
\(225\) 0 0
\(226\) 1.86301e7 0.107358
\(227\) −1.92108e7 −0.109007 −0.0545036 0.998514i \(-0.517358\pi\)
−0.0545036 + 0.998514i \(0.517358\pi\)
\(228\) 0 0
\(229\) −1.05650e8 −0.581360 −0.290680 0.956820i \(-0.593882\pi\)
−0.290680 + 0.956820i \(0.593882\pi\)
\(230\) 1.09319e7 0.0592443
\(231\) 0 0
\(232\) −3.27317e7 −0.172092
\(233\) 2.31646e8 1.19972 0.599859 0.800106i \(-0.295224\pi\)
0.599859 + 0.800106i \(0.295224\pi\)
\(234\) 0 0
\(235\) 6.40026e7 0.321707
\(236\) −2.28546e8 −1.13183
\(237\) 0 0
\(238\) 7.43438e6 0.0357458
\(239\) −1.09174e8 −0.517281 −0.258641 0.965974i \(-0.583275\pi\)
−0.258641 + 0.965974i \(0.583275\pi\)
\(240\) 0 0
\(241\) −8.25277e7 −0.379787 −0.189893 0.981805i \(-0.560814\pi\)
−0.189893 + 0.981805i \(0.560814\pi\)
\(242\) 2.38300e7 0.108086
\(243\) 0 0
\(244\) 1.23864e8 0.545861
\(245\) −1.47061e7 −0.0638877
\(246\) 0 0
\(247\) 2.36662e8 0.999283
\(248\) 9.58611e7 0.399081
\(249\) 0 0
\(250\) 2.66943e6 0.0108051
\(251\) 2.40987e7 0.0961912 0.0480956 0.998843i \(-0.484685\pi\)
0.0480956 + 0.998843i \(0.484685\pi\)
\(252\) 0 0
\(253\) 9.16534e7 0.355817
\(254\) −3.05828e7 −0.117100
\(255\) 0 0
\(256\) 2.30112e8 0.857235
\(257\) −9.75049e7 −0.358311 −0.179156 0.983821i \(-0.557337\pi\)
−0.179156 + 0.983821i \(0.557337\pi\)
\(258\) 0 0
\(259\) −5.36904e7 −0.192020
\(260\) −9.67480e7 −0.341378
\(261\) 0 0
\(262\) −6.19589e6 −0.0212838
\(263\) −2.98637e8 −1.01228 −0.506138 0.862452i \(-0.668927\pi\)
−0.506138 + 0.862452i \(0.668927\pi\)
\(264\) 0 0
\(265\) −2.51561e7 −0.0830392
\(266\) −1.80803e7 −0.0589005
\(267\) 0 0
\(268\) 5.61711e8 1.78255
\(269\) 3.90722e8 1.22387 0.611934 0.790909i \(-0.290392\pi\)
0.611934 + 0.790909i \(0.290392\pi\)
\(270\) 0 0
\(271\) 2.12098e8 0.647357 0.323678 0.946167i \(-0.395080\pi\)
0.323678 + 0.946167i \(0.395080\pi\)
\(272\) 2.48505e8 0.748763
\(273\) 0 0
\(274\) −6.93841e7 −0.203767
\(275\) 2.23807e7 0.0648947
\(276\) 0 0
\(277\) 1.86723e8 0.527861 0.263930 0.964542i \(-0.414981\pi\)
0.263930 + 0.964542i \(0.414981\pi\)
\(278\) −1.43759e7 −0.0401307
\(279\) 0 0
\(280\) 1.48920e7 0.0405415
\(281\) 7.38791e8 1.98632 0.993161 0.116756i \(-0.0372496\pi\)
0.993161 + 0.116756i \(0.0372496\pi\)
\(282\) 0 0
\(283\) −3.11903e8 −0.818026 −0.409013 0.912529i \(-0.634127\pi\)
−0.409013 + 0.912529i \(0.634127\pi\)
\(284\) 9.15007e7 0.237034
\(285\) 0 0
\(286\) 1.20129e7 0.0303647
\(287\) −1.04182e8 −0.260140
\(288\) 0 0
\(289\) −1.58847e8 −0.387113
\(290\) −1.60997e7 −0.0387638
\(291\) 0 0
\(292\) −2.74466e8 −0.645131
\(293\) 5.05466e8 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(294\) 0 0
\(295\) −2.26495e8 −0.513666
\(296\) 5.43690e7 0.121851
\(297\) 0 0
\(298\) 7.42825e7 0.162603
\(299\) −3.92646e8 −0.849478
\(300\) 0 0
\(301\) −2.18429e8 −0.461665
\(302\) 3.05137e7 0.0637487
\(303\) 0 0
\(304\) −6.04359e8 −1.23378
\(305\) 1.22753e8 0.247732
\(306\) 0 0
\(307\) 4.67463e8 0.922067 0.461034 0.887383i \(-0.347479\pi\)
0.461034 + 0.887383i \(0.347479\pi\)
\(308\) 6.19688e7 0.120850
\(309\) 0 0
\(310\) 4.71512e7 0.0898933
\(311\) −1.16022e7 −0.0218714 −0.0109357 0.999940i \(-0.503481\pi\)
−0.0109357 + 0.999940i \(0.503481\pi\)
\(312\) 0 0
\(313\) 8.23197e8 1.51740 0.758698 0.651443i \(-0.225836\pi\)
0.758698 + 0.651443i \(0.225836\pi\)
\(314\) 5.98346e7 0.109068
\(315\) 0 0
\(316\) 6.57810e8 1.17272
\(317\) −3.89154e8 −0.686142 −0.343071 0.939309i \(-0.611467\pi\)
−0.343071 + 0.939309i \(0.611467\pi\)
\(318\) 0 0
\(319\) −1.34981e8 −0.232812
\(320\) 2.39468e8 0.408529
\(321\) 0 0
\(322\) 2.99970e7 0.0500706
\(323\) −6.11621e8 −1.00989
\(324\) 0 0
\(325\) −9.58797e7 −0.154930
\(326\) −5.54150e7 −0.0885861
\(327\) 0 0
\(328\) 1.05499e8 0.165078
\(329\) 1.75623e8 0.271892
\(330\) 0 0
\(331\) 1.48582e8 0.225199 0.112600 0.993640i \(-0.464082\pi\)
0.112600 + 0.993640i \(0.464082\pi\)
\(332\) 7.66783e8 1.14998
\(333\) 0 0
\(334\) 1.33047e8 0.195385
\(335\) 5.56670e8 0.808986
\(336\) 0 0
\(337\) 1.23379e8 0.175605 0.0878023 0.996138i \(-0.472016\pi\)
0.0878023 + 0.996138i \(0.472016\pi\)
\(338\) 3.42977e7 0.0483121
\(339\) 0 0
\(340\) 2.50033e8 0.345001
\(341\) 3.95319e8 0.539892
\(342\) 0 0
\(343\) −4.03536e7 −0.0539949
\(344\) 2.21189e8 0.292961
\(345\) 0 0
\(346\) −6.97872e7 −0.0905752
\(347\) −1.31658e9 −1.69159 −0.845793 0.533511i \(-0.820872\pi\)
−0.845793 + 0.533511i \(0.820872\pi\)
\(348\) 0 0
\(349\) 2.64521e8 0.333097 0.166549 0.986033i \(-0.446738\pi\)
0.166549 + 0.986033i \(0.446738\pi\)
\(350\) 7.32493e6 0.00913199
\(351\) 0 0
\(352\) −9.43586e7 −0.115314
\(353\) −1.30271e9 −1.57629 −0.788144 0.615490i \(-0.788958\pi\)
−0.788144 + 0.615490i \(0.788958\pi\)
\(354\) 0 0
\(355\) 9.06795e7 0.107575
\(356\) −1.33873e9 −1.57260
\(357\) 0 0
\(358\) 1.99287e8 0.229556
\(359\) 1.03262e9 1.17790 0.588952 0.808168i \(-0.299541\pi\)
0.588952 + 0.808168i \(0.299541\pi\)
\(360\) 0 0
\(361\) 5.93578e8 0.664053
\(362\) 8.33248e7 0.0923196
\(363\) 0 0
\(364\) −2.65476e8 −0.288517
\(365\) −2.72002e8 −0.292784
\(366\) 0 0
\(367\) 1.13124e9 1.19460 0.597302 0.802017i \(-0.296240\pi\)
0.597302 + 0.802017i \(0.296240\pi\)
\(368\) 1.00269e9 1.04882
\(369\) 0 0
\(370\) 2.67425e7 0.0274470
\(371\) −6.90284e7 −0.0701809
\(372\) 0 0
\(373\) 5.38130e8 0.536916 0.268458 0.963291i \(-0.413486\pi\)
0.268458 + 0.963291i \(0.413486\pi\)
\(374\) −3.10459e7 −0.0306870
\(375\) 0 0
\(376\) −1.77843e8 −0.172536
\(377\) 5.78264e8 0.555817
\(378\) 0 0
\(379\) 7.83114e8 0.738904 0.369452 0.929250i \(-0.379545\pi\)
0.369452 + 0.929250i \(0.379545\pi\)
\(380\) −6.08074e8 −0.568478
\(381\) 0 0
\(382\) −2.08536e8 −0.191408
\(383\) −8.22468e8 −0.748038 −0.374019 0.927421i \(-0.622020\pi\)
−0.374019 + 0.927421i \(0.622020\pi\)
\(384\) 0 0
\(385\) 6.14127e7 0.0548460
\(386\) 1.90357e8 0.168466
\(387\) 0 0
\(388\) −8.38126e8 −0.728448
\(389\) 1.07007e9 0.921696 0.460848 0.887479i \(-0.347545\pi\)
0.460848 + 0.887479i \(0.347545\pi\)
\(390\) 0 0
\(391\) 1.01474e9 0.858495
\(392\) 4.08636e7 0.0342638
\(393\) 0 0
\(394\) 8.91777e7 0.0734547
\(395\) 6.51906e8 0.532225
\(396\) 0 0
\(397\) −9.64552e8 −0.773676 −0.386838 0.922148i \(-0.626433\pi\)
−0.386838 + 0.922148i \(0.626433\pi\)
\(398\) −2.64471e6 −0.00210275
\(399\) 0 0
\(400\) 2.44846e8 0.191286
\(401\) 1.94810e9 1.50871 0.754357 0.656465i \(-0.227949\pi\)
0.754357 + 0.656465i \(0.227949\pi\)
\(402\) 0 0
\(403\) −1.69356e9 −1.28894
\(404\) 1.35631e9 1.02335
\(405\) 0 0
\(406\) −4.41777e7 −0.0327614
\(407\) 2.24211e8 0.164845
\(408\) 0 0
\(409\) 8.63865e8 0.624330 0.312165 0.950028i \(-0.398946\pi\)
0.312165 + 0.950028i \(0.398946\pi\)
\(410\) 5.18918e7 0.0371839
\(411\) 0 0
\(412\) 1.33556e9 0.940854
\(413\) −6.21501e8 −0.434127
\(414\) 0 0
\(415\) 7.59901e8 0.521902
\(416\) 4.04236e8 0.275301
\(417\) 0 0
\(418\) 7.55030e7 0.0505647
\(419\) −2.21337e9 −1.46996 −0.734978 0.678091i \(-0.762808\pi\)
−0.734978 + 0.678091i \(0.762808\pi\)
\(420\) 0 0
\(421\) −2.89866e9 −1.89326 −0.946631 0.322321i \(-0.895537\pi\)
−0.946631 + 0.322321i \(0.895537\pi\)
\(422\) 7.07769e7 0.0458456
\(423\) 0 0
\(424\) 6.99008e7 0.0445350
\(425\) 2.47789e8 0.156574
\(426\) 0 0
\(427\) 3.36833e8 0.209371
\(428\) −1.05602e9 −0.651057
\(429\) 0 0
\(430\) 1.08796e8 0.0659895
\(431\) 2.42056e9 1.45628 0.728142 0.685426i \(-0.240384\pi\)
0.728142 + 0.685426i \(0.240384\pi\)
\(432\) 0 0
\(433\) −2.26686e9 −1.34189 −0.670946 0.741506i \(-0.734112\pi\)
−0.670946 + 0.741506i \(0.734112\pi\)
\(434\) 1.29383e8 0.0759737
\(435\) 0 0
\(436\) 2.47153e9 1.42811
\(437\) −2.46783e9 −1.41459
\(438\) 0 0
\(439\) 1.98911e9 1.12210 0.561052 0.827780i \(-0.310397\pi\)
0.561052 + 0.827780i \(0.310397\pi\)
\(440\) −6.21888e7 −0.0348039
\(441\) 0 0
\(442\) 1.33002e8 0.0732621
\(443\) 8.78038e8 0.479844 0.239922 0.970792i \(-0.422878\pi\)
0.239922 + 0.970792i \(0.422878\pi\)
\(444\) 0 0
\(445\) −1.32672e9 −0.713705
\(446\) 1.70932e8 0.0912329
\(447\) 0 0
\(448\) 6.57101e8 0.345270
\(449\) −1.53113e8 −0.0798270 −0.0399135 0.999203i \(-0.512708\pi\)
−0.0399135 + 0.999203i \(0.512708\pi\)
\(450\) 0 0
\(451\) 4.35064e8 0.223324
\(452\) 1.71930e9 0.875724
\(453\) 0 0
\(454\) 2.62564e7 0.0131686
\(455\) −2.63094e8 −0.130940
\(456\) 0 0
\(457\) −2.39624e9 −1.17442 −0.587210 0.809435i \(-0.699774\pi\)
−0.587210 + 0.809435i \(0.699774\pi\)
\(458\) 1.44397e8 0.0702311
\(459\) 0 0
\(460\) 1.00886e9 0.483256
\(461\) −1.61913e9 −0.769713 −0.384856 0.922977i \(-0.625749\pi\)
−0.384856 + 0.922977i \(0.625749\pi\)
\(462\) 0 0
\(463\) 1.16133e9 0.543778 0.271889 0.962329i \(-0.412352\pi\)
0.271889 + 0.962329i \(0.412352\pi\)
\(464\) −1.47670e9 −0.686247
\(465\) 0 0
\(466\) −3.16602e8 −0.144932
\(467\) −2.83969e9 −1.29021 −0.645107 0.764092i \(-0.723187\pi\)
−0.645107 + 0.764092i \(0.723187\pi\)
\(468\) 0 0
\(469\) 1.52750e9 0.683718
\(470\) −8.74756e7 −0.0388637
\(471\) 0 0
\(472\) 6.29356e8 0.275486
\(473\) 9.12156e8 0.396328
\(474\) 0 0
\(475\) −6.02617e8 −0.257996
\(476\) 6.86090e8 0.291579
\(477\) 0 0
\(478\) 1.49214e8 0.0624901
\(479\) −2.38771e9 −0.992676 −0.496338 0.868129i \(-0.665322\pi\)
−0.496338 + 0.868129i \(0.665322\pi\)
\(480\) 0 0
\(481\) −9.60526e8 −0.393551
\(482\) 1.12795e8 0.0458801
\(483\) 0 0
\(484\) 2.19917e9 0.881660
\(485\) −8.30604e8 −0.330596
\(486\) 0 0
\(487\) −2.20508e9 −0.865113 −0.432556 0.901607i \(-0.642388\pi\)
−0.432556 + 0.901607i \(0.642388\pi\)
\(488\) −3.41090e8 −0.132862
\(489\) 0 0
\(490\) 2.00996e7 0.00771794
\(491\) −4.28064e8 −0.163201 −0.0816006 0.996665i \(-0.526003\pi\)
−0.0816006 + 0.996665i \(0.526003\pi\)
\(492\) 0 0
\(493\) −1.49445e9 −0.561716
\(494\) −3.23457e8 −0.120718
\(495\) 0 0
\(496\) 4.32482e9 1.59141
\(497\) 2.48824e8 0.0909171
\(498\) 0 0
\(499\) −2.95178e9 −1.06349 −0.531743 0.846906i \(-0.678463\pi\)
−0.531743 + 0.846906i \(0.678463\pi\)
\(500\) 2.46352e8 0.0881374
\(501\) 0 0
\(502\) −3.29369e7 −0.0116204
\(503\) 5.22380e9 1.83020 0.915099 0.403229i \(-0.132112\pi\)
0.915099 + 0.403229i \(0.132112\pi\)
\(504\) 0 0
\(505\) 1.34414e9 0.464435
\(506\) −1.25267e8 −0.0429844
\(507\) 0 0
\(508\) −2.82236e9 −0.955190
\(509\) 2.80532e9 0.942911 0.471455 0.881890i \(-0.343729\pi\)
0.471455 + 0.881890i \(0.343729\pi\)
\(510\) 0 0
\(511\) −7.46374e8 −0.247448
\(512\) −1.72897e9 −0.569301
\(513\) 0 0
\(514\) 1.33265e8 0.0432857
\(515\) 1.32357e9 0.426994
\(516\) 0 0
\(517\) −7.33401e8 −0.233413
\(518\) 7.33814e7 0.0231970
\(519\) 0 0
\(520\) 2.66419e8 0.0830909
\(521\) −1.40563e8 −0.0435450 −0.0217725 0.999763i \(-0.506931\pi\)
−0.0217725 + 0.999763i \(0.506931\pi\)
\(522\) 0 0
\(523\) −1.81127e9 −0.553638 −0.276819 0.960922i \(-0.589280\pi\)
−0.276819 + 0.960922i \(0.589280\pi\)
\(524\) −5.71794e8 −0.173612
\(525\) 0 0
\(526\) 4.08163e8 0.122288
\(527\) 4.37679e9 1.30262
\(528\) 0 0
\(529\) 6.89567e8 0.202526
\(530\) 3.43821e7 0.0100315
\(531\) 0 0
\(532\) −1.66855e9 −0.480452
\(533\) −1.86383e9 −0.533164
\(534\) 0 0
\(535\) −1.04654e9 −0.295474
\(536\) −1.54681e9 −0.433870
\(537\) 0 0
\(538\) −5.34019e8 −0.147849
\(539\) 1.68516e8 0.0463534
\(540\) 0 0
\(541\) −7.11633e9 −1.93226 −0.966130 0.258058i \(-0.916918\pi\)
−0.966130 + 0.258058i \(0.916918\pi\)
\(542\) −2.89885e8 −0.0782038
\(543\) 0 0
\(544\) −1.04469e9 −0.278223
\(545\) 2.44935e9 0.648130
\(546\) 0 0
\(547\) −6.02390e9 −1.57370 −0.786850 0.617144i \(-0.788290\pi\)
−0.786850 + 0.617144i \(0.788290\pi\)
\(548\) −6.40318e9 −1.66213
\(549\) 0 0
\(550\) −3.05888e7 −0.00783959
\(551\) 3.63447e9 0.925572
\(552\) 0 0
\(553\) 1.78883e9 0.449812
\(554\) −2.55204e8 −0.0637681
\(555\) 0 0
\(556\) −1.32669e9 −0.327347
\(557\) −3.55726e9 −0.872214 −0.436107 0.899895i \(-0.643643\pi\)
−0.436107 + 0.899895i \(0.643643\pi\)
\(558\) 0 0
\(559\) −3.90771e9 −0.946195
\(560\) 6.71859e8 0.161666
\(561\) 0 0
\(562\) −1.00974e9 −0.239957
\(563\) −2.51240e9 −0.593347 −0.296673 0.954979i \(-0.595877\pi\)
−0.296673 + 0.954979i \(0.595877\pi\)
\(564\) 0 0
\(565\) 1.70387e9 0.397436
\(566\) 4.26293e8 0.0988214
\(567\) 0 0
\(568\) −2.51969e8 −0.0576937
\(569\) −3.02191e9 −0.687683 −0.343841 0.939028i \(-0.611728\pi\)
−0.343841 + 0.939028i \(0.611728\pi\)
\(570\) 0 0
\(571\) 4.13151e9 0.928716 0.464358 0.885648i \(-0.346285\pi\)
0.464358 + 0.885648i \(0.346285\pi\)
\(572\) 1.10863e9 0.247685
\(573\) 0 0
\(574\) 1.42391e8 0.0314262
\(575\) 9.99804e8 0.219320
\(576\) 0 0
\(577\) −3.66048e9 −0.793274 −0.396637 0.917976i \(-0.629823\pi\)
−0.396637 + 0.917976i \(0.629823\pi\)
\(578\) 2.17105e8 0.0467651
\(579\) 0 0
\(580\) −1.48578e9 −0.316196
\(581\) 2.08517e9 0.441088
\(582\) 0 0
\(583\) 2.88262e8 0.0602487
\(584\) 7.55807e8 0.157024
\(585\) 0 0
\(586\) −6.90846e8 −0.141821
\(587\) −8.93156e9 −1.82261 −0.911305 0.411731i \(-0.864924\pi\)
−0.911305 + 0.411731i \(0.864924\pi\)
\(588\) 0 0
\(589\) −1.06442e10 −2.14640
\(590\) 3.09562e8 0.0620534
\(591\) 0 0
\(592\) 2.45288e9 0.485904
\(593\) −8.00218e9 −1.57586 −0.787929 0.615766i \(-0.788847\pi\)
−0.787929 + 0.615766i \(0.788847\pi\)
\(594\) 0 0
\(595\) 6.79932e8 0.132329
\(596\) 6.85524e9 1.32636
\(597\) 0 0
\(598\) 5.36649e8 0.102621
\(599\) −6.37081e9 −1.21116 −0.605579 0.795785i \(-0.707059\pi\)
−0.605579 + 0.795785i \(0.707059\pi\)
\(600\) 0 0
\(601\) 7.97677e9 1.49888 0.749439 0.662073i \(-0.230323\pi\)
0.749439 + 0.662073i \(0.230323\pi\)
\(602\) 2.98537e8 0.0557713
\(603\) 0 0
\(604\) 2.81599e9 0.519999
\(605\) 2.17944e9 0.400130
\(606\) 0 0
\(607\) 5.42119e9 0.983863 0.491931 0.870634i \(-0.336291\pi\)
0.491931 + 0.870634i \(0.336291\pi\)
\(608\) 2.54067e9 0.458444
\(609\) 0 0
\(610\) −1.67772e8 −0.0299272
\(611\) 3.14191e9 0.557250
\(612\) 0 0
\(613\) 8.21824e9 1.44101 0.720505 0.693450i \(-0.243910\pi\)
0.720505 + 0.693450i \(0.243910\pi\)
\(614\) −6.38905e8 −0.111390
\(615\) 0 0
\(616\) −1.70646e8 −0.0294147
\(617\) −8.15621e9 −1.39795 −0.698973 0.715148i \(-0.746359\pi\)
−0.698973 + 0.715148i \(0.746359\pi\)
\(618\) 0 0
\(619\) −6.46052e9 −1.09484 −0.547420 0.836858i \(-0.684390\pi\)
−0.547420 + 0.836858i \(0.684390\pi\)
\(620\) 4.35140e9 0.733260
\(621\) 0 0
\(622\) 1.58573e7 0.00264218
\(623\) −3.64051e9 −0.603191
\(624\) 0 0
\(625\) 2.44141e8 0.0400000
\(626\) −1.12511e9 −0.183309
\(627\) 0 0
\(628\) 5.52190e9 0.889672
\(629\) 2.48236e9 0.397729
\(630\) 0 0
\(631\) −8.82660e9 −1.39859 −0.699295 0.714833i \(-0.746503\pi\)
−0.699295 + 0.714833i \(0.746503\pi\)
\(632\) −1.81144e9 −0.285439
\(633\) 0 0
\(634\) 5.31876e8 0.0828893
\(635\) −2.79703e9 −0.433500
\(636\) 0 0
\(637\) −7.21930e8 −0.110664
\(638\) 1.84486e8 0.0281249
\(639\) 0 0
\(640\) −1.38131e9 −0.208287
\(641\) −8.54151e9 −1.28095 −0.640474 0.767980i \(-0.721262\pi\)
−0.640474 + 0.767980i \(0.721262\pi\)
\(642\) 0 0
\(643\) −1.20342e10 −1.78517 −0.892585 0.450878i \(-0.851111\pi\)
−0.892585 + 0.450878i \(0.851111\pi\)
\(644\) 2.76831e9 0.408426
\(645\) 0 0
\(646\) 8.35934e8 0.122000
\(647\) −1.89174e8 −0.0274598 −0.0137299 0.999906i \(-0.504370\pi\)
−0.0137299 + 0.999906i \(0.504370\pi\)
\(648\) 0 0
\(649\) 2.59539e9 0.372688
\(650\) 1.31044e8 0.0187163
\(651\) 0 0
\(652\) −5.11403e9 −0.722598
\(653\) −8.70977e9 −1.22408 −0.612041 0.790826i \(-0.709651\pi\)
−0.612041 + 0.790826i \(0.709651\pi\)
\(654\) 0 0
\(655\) −5.66662e8 −0.0787916
\(656\) 4.75963e9 0.658279
\(657\) 0 0
\(658\) −2.40033e8 −0.0328458
\(659\) 7.48288e8 0.101852 0.0509260 0.998702i \(-0.483783\pi\)
0.0509260 + 0.998702i \(0.483783\pi\)
\(660\) 0 0
\(661\) 8.45586e9 1.13881 0.569407 0.822056i \(-0.307173\pi\)
0.569407 + 0.822056i \(0.307173\pi\)
\(662\) −2.03074e8 −0.0272052
\(663\) 0 0
\(664\) −2.11152e9 −0.279903
\(665\) −1.65358e9 −0.218047
\(666\) 0 0
\(667\) −6.02996e9 −0.786817
\(668\) 1.22784e10 1.59376
\(669\) 0 0
\(670\) −7.60829e8 −0.0977294
\(671\) −1.40661e9 −0.179740
\(672\) 0 0
\(673\) 4.78543e9 0.605157 0.302578 0.953124i \(-0.402153\pi\)
0.302578 + 0.953124i \(0.402153\pi\)
\(674\) −1.68628e8 −0.0212139
\(675\) 0 0
\(676\) 3.16520e9 0.394083
\(677\) 1.29662e10 1.60603 0.803015 0.595958i \(-0.203228\pi\)
0.803015 + 0.595958i \(0.203228\pi\)
\(678\) 0 0
\(679\) −2.27918e9 −0.279405
\(680\) −6.88526e8 −0.0839729
\(681\) 0 0
\(682\) −5.40302e8 −0.0652216
\(683\) 9.15988e9 1.10006 0.550031 0.835144i \(-0.314616\pi\)
0.550031 + 0.835144i \(0.314616\pi\)
\(684\) 0 0
\(685\) −6.34572e9 −0.754334
\(686\) 5.51533e7 0.00652285
\(687\) 0 0
\(688\) 9.97905e9 1.16823
\(689\) −1.23492e9 −0.143838
\(690\) 0 0
\(691\) 1.05298e10 1.21407 0.607037 0.794673i \(-0.292358\pi\)
0.607037 + 0.794673i \(0.292358\pi\)
\(692\) −6.44038e9 −0.738823
\(693\) 0 0
\(694\) 1.79944e9 0.204352
\(695\) −1.31478e9 −0.148562
\(696\) 0 0
\(697\) 4.81683e9 0.538824
\(698\) −3.61534e8 −0.0402398
\(699\) 0 0
\(700\) 6.75989e8 0.0744897
\(701\) −1.27411e9 −0.139699 −0.0698497 0.997558i \(-0.522252\pi\)
−0.0698497 + 0.997558i \(0.522252\pi\)
\(702\) 0 0
\(703\) −6.03703e9 −0.655360
\(704\) −2.74405e9 −0.296406
\(705\) 0 0
\(706\) 1.78048e9 0.190423
\(707\) 3.68832e9 0.392519
\(708\) 0 0
\(709\) −7.17795e9 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(710\) −1.23936e8 −0.0129955
\(711\) 0 0
\(712\) 3.68653e9 0.382769
\(713\) 1.76599e10 1.82463
\(714\) 0 0
\(715\) 1.09868e9 0.112409
\(716\) 1.83914e10 1.87249
\(717\) 0 0
\(718\) −1.41133e9 −0.142296
\(719\) −1.18502e10 −1.18898 −0.594488 0.804104i \(-0.702645\pi\)
−0.594488 + 0.804104i \(0.702645\pi\)
\(720\) 0 0
\(721\) 3.63188e9 0.360876
\(722\) −8.11273e8 −0.0802208
\(723\) 0 0
\(724\) 7.68971e9 0.753052
\(725\) −1.47245e9 −0.143502
\(726\) 0 0
\(727\) −4.67874e9 −0.451605 −0.225802 0.974173i \(-0.572500\pi\)
−0.225802 + 0.974173i \(0.572500\pi\)
\(728\) 7.31054e8 0.0702246
\(729\) 0 0
\(730\) 3.71759e8 0.0353697
\(731\) 1.00990e10 0.956238
\(732\) 0 0
\(733\) 1.28552e9 0.120563 0.0602817 0.998181i \(-0.480800\pi\)
0.0602817 + 0.998181i \(0.480800\pi\)
\(734\) −1.54612e9 −0.144314
\(735\) 0 0
\(736\) −4.21524e9 −0.389718
\(737\) −6.37884e9 −0.586956
\(738\) 0 0
\(739\) −5.26720e9 −0.480091 −0.240046 0.970762i \(-0.577162\pi\)
−0.240046 + 0.970762i \(0.577162\pi\)
\(740\) 2.46796e9 0.223886
\(741\) 0 0
\(742\) 9.43446e7 0.00847819
\(743\) −4.15012e9 −0.371193 −0.185596 0.982626i \(-0.559422\pi\)
−0.185596 + 0.982626i \(0.559422\pi\)
\(744\) 0 0
\(745\) 6.79371e9 0.601950
\(746\) −7.35489e8 −0.0648620
\(747\) 0 0
\(748\) −2.86511e9 −0.250314
\(749\) −2.87171e9 −0.249721
\(750\) 0 0
\(751\) −6.37970e9 −0.549618 −0.274809 0.961499i \(-0.588615\pi\)
−0.274809 + 0.961499i \(0.588615\pi\)
\(752\) −8.02346e9 −0.688017
\(753\) 0 0
\(754\) −7.90343e8 −0.0671453
\(755\) 2.79072e9 0.235995
\(756\) 0 0
\(757\) −1.19658e10 −1.00255 −0.501274 0.865289i \(-0.667135\pi\)
−0.501274 + 0.865289i \(0.667135\pi\)
\(758\) −1.07032e9 −0.0892631
\(759\) 0 0
\(760\) 1.67448e9 0.138367
\(761\) 2.00959e10 1.65296 0.826479 0.562967i \(-0.190340\pi\)
0.826479 + 0.562967i \(0.190340\pi\)
\(762\) 0 0
\(763\) 6.72100e9 0.547770
\(764\) −1.92450e10 −1.56131
\(765\) 0 0
\(766\) 1.12411e9 0.0903665
\(767\) −1.11187e10 −0.889756
\(768\) 0 0
\(769\) 2.46683e10 1.95613 0.978064 0.208304i \(-0.0667944\pi\)
0.978064 + 0.208304i \(0.0667944\pi\)
\(770\) −8.39358e7 −0.00662567
\(771\) 0 0
\(772\) 1.75673e10 1.37418
\(773\) −8.88824e9 −0.692130 −0.346065 0.938211i \(-0.612482\pi\)
−0.346065 + 0.938211i \(0.612482\pi\)
\(774\) 0 0
\(775\) 4.31235e9 0.332781
\(776\) 2.30798e9 0.177303
\(777\) 0 0
\(778\) −1.46252e9 −0.111345
\(779\) −1.17144e10 −0.887850
\(780\) 0 0
\(781\) −1.03909e9 −0.0780502
\(782\) −1.38690e9 −0.103710
\(783\) 0 0
\(784\) 1.84358e9 0.136633
\(785\) 5.47235e9 0.403766
\(786\) 0 0
\(787\) −4.65006e9 −0.340053 −0.170027 0.985439i \(-0.554385\pi\)
−0.170027 + 0.985439i \(0.554385\pi\)
\(788\) 8.22986e9 0.599171
\(789\) 0 0
\(790\) −8.90993e8 −0.0642953
\(791\) 4.67542e9 0.335895
\(792\) 0 0
\(793\) 6.02598e9 0.429113
\(794\) 1.31830e9 0.0934638
\(795\) 0 0
\(796\) −2.44070e8 −0.0171521
\(797\) −1.42890e10 −0.999762 −0.499881 0.866094i \(-0.666623\pi\)
−0.499881 + 0.866094i \(0.666623\pi\)
\(798\) 0 0
\(799\) −8.11987e9 −0.563165
\(800\) −1.02931e9 −0.0710776
\(801\) 0 0
\(802\) −2.66257e9 −0.182260
\(803\) 3.11685e9 0.212428
\(804\) 0 0
\(805\) 2.74346e9 0.185359
\(806\) 2.31467e9 0.155710
\(807\) 0 0
\(808\) −3.73494e9 −0.249083
\(809\) 4.92320e9 0.326909 0.163455 0.986551i \(-0.447736\pi\)
0.163455 + 0.986551i \(0.447736\pi\)
\(810\) 0 0
\(811\) 2.35801e10 1.55229 0.776145 0.630555i \(-0.217173\pi\)
0.776145 + 0.630555i \(0.217173\pi\)
\(812\) −4.07698e9 −0.267235
\(813\) 0 0
\(814\) −3.06440e8 −0.0199141
\(815\) −5.06813e9 −0.327942
\(816\) 0 0
\(817\) −2.45605e10 −1.57565
\(818\) −1.18069e9 −0.0754221
\(819\) 0 0
\(820\) 4.78889e9 0.303310
\(821\) 2.86630e10 1.80768 0.903838 0.427875i \(-0.140738\pi\)
0.903838 + 0.427875i \(0.140738\pi\)
\(822\) 0 0
\(823\) −2.76897e10 −1.73148 −0.865742 0.500490i \(-0.833153\pi\)
−0.865742 + 0.500490i \(0.833153\pi\)
\(824\) −3.67778e9 −0.229003
\(825\) 0 0
\(826\) 8.49437e8 0.0524447
\(827\) 1.27176e10 0.781873 0.390936 0.920418i \(-0.372151\pi\)
0.390936 + 0.920418i \(0.372151\pi\)
\(828\) 0 0
\(829\) −1.50770e10 −0.919127 −0.459563 0.888145i \(-0.651994\pi\)
−0.459563 + 0.888145i \(0.651994\pi\)
\(830\) −1.03860e9 −0.0630483
\(831\) 0 0
\(832\) 1.17556e10 0.707641
\(833\) 1.86573e9 0.111839
\(834\) 0 0
\(835\) 1.21682e10 0.723307
\(836\) 6.96787e9 0.412457
\(837\) 0 0
\(838\) 3.02512e9 0.177578
\(839\) −4.59511e9 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(840\) 0 0
\(841\) −8.36934e9 −0.485182
\(842\) 3.96175e9 0.228715
\(843\) 0 0
\(844\) 6.53172e9 0.373963
\(845\) 3.13679e9 0.178849
\(846\) 0 0
\(847\) 5.98038e9 0.338171
\(848\) 3.15361e9 0.177591
\(849\) 0 0
\(850\) −3.38665e8 −0.0189149
\(851\) 1.00161e10 0.557114
\(852\) 0 0
\(853\) −1.13971e9 −0.0628740 −0.0314370 0.999506i \(-0.510008\pi\)
−0.0314370 + 0.999506i \(0.510008\pi\)
\(854\) −4.60367e8 −0.0252931
\(855\) 0 0
\(856\) 2.90801e9 0.158466
\(857\) −7.79419e9 −0.422998 −0.211499 0.977378i \(-0.567834\pi\)
−0.211499 + 0.977378i \(0.567834\pi\)
\(858\) 0 0
\(859\) 1.27280e10 0.685147 0.342573 0.939491i \(-0.388701\pi\)
0.342573 + 0.939491i \(0.388701\pi\)
\(860\) 1.00404e10 0.538277
\(861\) 0 0
\(862\) −3.30831e9 −0.175926
\(863\) −2.53204e9 −0.134101 −0.0670507 0.997750i \(-0.521359\pi\)
−0.0670507 + 0.997750i \(0.521359\pi\)
\(864\) 0 0
\(865\) −6.38258e9 −0.335305
\(866\) 3.09823e9 0.162107
\(867\) 0 0
\(868\) 1.19402e10 0.619718
\(869\) −7.47014e9 −0.386153
\(870\) 0 0
\(871\) 2.73272e10 1.40130
\(872\) −6.80595e9 −0.347601
\(873\) 0 0
\(874\) 3.37291e9 0.170889
\(875\) 6.69922e8 0.0338062
\(876\) 0 0
\(877\) 5.00988e9 0.250800 0.125400 0.992106i \(-0.459979\pi\)
0.125400 + 0.992106i \(0.459979\pi\)
\(878\) −2.71862e9 −0.135556
\(879\) 0 0
\(880\) −2.80568e9 −0.138787
\(881\) −9.46900e9 −0.466539 −0.233270 0.972412i \(-0.574942\pi\)
−0.233270 + 0.972412i \(0.574942\pi\)
\(882\) 0 0
\(883\) 1.11146e10 0.543289 0.271644 0.962398i \(-0.412433\pi\)
0.271644 + 0.962398i \(0.412433\pi\)
\(884\) 1.22742e10 0.597600
\(885\) 0 0
\(886\) −1.20006e9 −0.0579675
\(887\) 7.27986e9 0.350260 0.175130 0.984545i \(-0.443965\pi\)
0.175130 + 0.984545i \(0.443965\pi\)
\(888\) 0 0
\(889\) −7.67506e9 −0.366375
\(890\) 1.81329e9 0.0862190
\(891\) 0 0
\(892\) 1.57746e10 0.744188
\(893\) 1.97473e10 0.927959
\(894\) 0 0
\(895\) 1.82264e10 0.849807
\(896\) −3.79032e9 −0.176034
\(897\) 0 0
\(898\) 2.09267e8 0.00964348
\(899\) −2.60084e10 −1.19386
\(900\) 0 0
\(901\) 3.19150e9 0.145365
\(902\) −5.94624e8 −0.0269786
\(903\) 0 0
\(904\) −4.73451e9 −0.213150
\(905\) 7.62070e9 0.341763
\(906\) 0 0
\(907\) −1.39503e10 −0.620809 −0.310405 0.950605i \(-0.600465\pi\)
−0.310405 + 0.950605i \(0.600465\pi\)
\(908\) 2.42310e9 0.107416
\(909\) 0 0
\(910\) 3.59584e8 0.0158181
\(911\) −2.98148e8 −0.0130653 −0.00653263 0.999979i \(-0.502079\pi\)
−0.00653263 + 0.999979i \(0.502079\pi\)
\(912\) 0 0
\(913\) −8.70765e9 −0.378663
\(914\) 3.27506e9 0.141876
\(915\) 0 0
\(916\) 1.33259e10 0.572876
\(917\) −1.55492e9 −0.0665910
\(918\) 0 0
\(919\) 2.67202e10 1.13563 0.567814 0.823157i \(-0.307789\pi\)
0.567814 + 0.823157i \(0.307789\pi\)
\(920\) −2.77813e9 −0.117624
\(921\) 0 0
\(922\) 2.21295e9 0.0929850
\(923\) 4.45149e9 0.186337
\(924\) 0 0
\(925\) 2.44581e9 0.101608
\(926\) −1.58725e9 −0.0656910
\(927\) 0 0
\(928\) 6.20794e9 0.254994
\(929\) 3.66336e10 1.49908 0.749540 0.661959i \(-0.230275\pi\)
0.749540 + 0.661959i \(0.230275\pi\)
\(930\) 0 0
\(931\) −4.53742e9 −0.184283
\(932\) −2.92180e10 −1.18221
\(933\) 0 0
\(934\) 3.88115e9 0.155864
\(935\) −2.83939e9 −0.113602
\(936\) 0 0
\(937\) 1.28088e10 0.508649 0.254325 0.967119i \(-0.418147\pi\)
0.254325 + 0.967119i \(0.418147\pi\)
\(938\) −2.08772e9 −0.0825965
\(939\) 0 0
\(940\) −8.07278e9 −0.317012
\(941\) 1.20663e10 0.472073 0.236037 0.971744i \(-0.424151\pi\)
0.236037 + 0.971744i \(0.424151\pi\)
\(942\) 0 0
\(943\) 1.94354e10 0.754751
\(944\) 2.83937e10 1.09855
\(945\) 0 0
\(946\) −1.24669e9 −0.0478784
\(947\) −8.36023e9 −0.319885 −0.159942 0.987126i \(-0.551131\pi\)
−0.159942 + 0.987126i \(0.551131\pi\)
\(948\) 0 0
\(949\) −1.33527e10 −0.507151
\(950\) 8.23627e8 0.0311672
\(951\) 0 0
\(952\) −1.88931e9 −0.0709700
\(953\) −4.49530e10 −1.68242 −0.841209 0.540710i \(-0.818156\pi\)
−0.841209 + 0.540710i \(0.818156\pi\)
\(954\) 0 0
\(955\) −1.90722e10 −0.708582
\(956\) 1.37703e10 0.509732
\(957\) 0 0
\(958\) 3.26341e9 0.119920
\(959\) −1.74126e10 −0.637528
\(960\) 0 0
\(961\) 4.86580e10 1.76857
\(962\) 1.31280e9 0.0475429
\(963\) 0 0
\(964\) 1.04094e10 0.374244
\(965\) 1.74096e10 0.623654
\(966\) 0 0
\(967\) 1.34247e8 0.00477432 0.00238716 0.999997i \(-0.499240\pi\)
0.00238716 + 0.999997i \(0.499240\pi\)
\(968\) −6.05596e9 −0.214595
\(969\) 0 0
\(970\) 1.13523e9 0.0399376
\(971\) 3.00377e10 1.05293 0.526465 0.850197i \(-0.323517\pi\)
0.526465 + 0.850197i \(0.323517\pi\)
\(972\) 0 0
\(973\) −3.60777e9 −0.125558
\(974\) 3.01379e9 0.104510
\(975\) 0 0
\(976\) −1.53884e10 −0.529810
\(977\) −4.52860e10 −1.55358 −0.776789 0.629761i \(-0.783153\pi\)
−0.776789 + 0.629761i \(0.783153\pi\)
\(978\) 0 0
\(979\) 1.52028e10 0.517825
\(980\) 1.85491e9 0.0629553
\(981\) 0 0
\(982\) 5.85057e8 0.0197155
\(983\) −4.61443e10 −1.54946 −0.774731 0.632290i \(-0.782115\pi\)
−0.774731 + 0.632290i \(0.782115\pi\)
\(984\) 0 0
\(985\) 8.15600e9 0.271926
\(986\) 2.04254e9 0.0678580
\(987\) 0 0
\(988\) −2.98506e10 −0.984700
\(989\) 4.07484e10 1.33944
\(990\) 0 0
\(991\) −1.05400e10 −0.344018 −0.172009 0.985095i \(-0.555026\pi\)
−0.172009 + 0.985095i \(0.555026\pi\)
\(992\) −1.81812e10 −0.591331
\(993\) 0 0
\(994\) −3.40081e8 −0.0109832
\(995\) −2.41879e8 −0.00778427
\(996\) 0 0
\(997\) −5.00734e10 −1.60020 −0.800099 0.599868i \(-0.795220\pi\)
−0.800099 + 0.599868i \(0.795220\pi\)
\(998\) 4.03435e9 0.128474
\(999\) 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 315.8.a.c.1.2 2
3.2 odd 2 35.8.a.a.1.1 2
12.11 even 2 560.8.a.i.1.1 2
15.2 even 4 175.8.b.c.99.3 4
15.8 even 4 175.8.b.c.99.2 4
15.14 odd 2 175.8.a.b.1.2 2
21.20 even 2 245.8.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.1 2 3.2 odd 2
175.8.a.b.1.2 2 15.14 odd 2
175.8.b.c.99.2 4 15.8 even 4
175.8.b.c.99.3 4 15.2 even 4
245.8.a.b.1.1 2 21.20 even 2
315.8.a.c.1.2 2 1.1 even 1 trivial
560.8.a.i.1.1 2 12.11 even 2