Properties

Label 35.8.a.a.1.1
Level $35$
Weight $8$
Character 35.1
Self dual yes
Analytic conductor $10.933$
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] = [35,8,Mod(1,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("35.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 35.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: \(10.9334758919\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31662\) of defining polynomial
Character \(\chi\) \(=\) 35.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.36675 q^{2} +24.7995 q^{3} -126.132 q^{4} +125.000 q^{5} +33.8947 q^{6} -343.000 q^{7} -347.335 q^{8} -1571.98 q^{9} +O(q^{10})\) \(q+1.36675 q^{2} +24.7995 q^{3} -126.132 q^{4} +125.000 q^{5} +33.8947 q^{6} -343.000 q^{7} -347.335 q^{8} -1571.98 q^{9} +170.844 q^{10} -1432.37 q^{11} -3128.01 q^{12} -6136.30 q^{13} -468.795 q^{14} +3099.94 q^{15} +15670.2 q^{16} -15858.5 q^{17} -2148.51 q^{18} -38567.5 q^{19} -15766.5 q^{20} -8506.23 q^{21} -1957.69 q^{22} -63987.4 q^{23} -8613.73 q^{24} +15625.0 q^{25} -8386.79 q^{26} -93220.9 q^{27} +43263.3 q^{28} +94236.6 q^{29} +4236.84 q^{30} +275990. q^{31} +65876.1 q^{32} -35521.9 q^{33} -21674.6 q^{34} -42875.0 q^{35} +198278. q^{36} +156532. q^{37} -52712.1 q^{38} -152177. q^{39} -43416.9 q^{40} -303738. q^{41} -11625.9 q^{42} +636818. q^{43} +180667. q^{44} -196498. q^{45} -87454.9 q^{46} +512021. q^{47} +388612. q^{48} +117649. q^{49} +21355.5 q^{50} -393282. q^{51} +773984. q^{52} -201249. q^{53} -127410. q^{54} -179046. q^{55} +119136. q^{56} -956454. q^{57} +128798. q^{58} -1.81196e6 q^{59} -391001. q^{60} -982021. q^{61} +377210. q^{62} +539191. q^{63} -1.91575e6 q^{64} -767038. q^{65} -48549.6 q^{66} -4.45336e6 q^{67} +2.00026e6 q^{68} -1.58686e6 q^{69} -58599.4 q^{70} +725436. q^{71} +546005. q^{72} +2.17602e6 q^{73} +213940. q^{74} +387492. q^{75} +4.86459e6 q^{76} +491301. q^{77} -207988. q^{78} -5.21525e6 q^{79} +1.95877e6 q^{80} +1.12610e6 q^{81} -415135. q^{82} +6.07921e6 q^{83} +1.07291e6 q^{84} -1.98231e6 q^{85} +870371. q^{86} +2.33702e6 q^{87} +497511. q^{88} -1.06137e7 q^{89} -268564. q^{90} +2.10475e6 q^{91} +8.07086e6 q^{92} +6.84442e6 q^{93} +699805. q^{94} -4.82093e6 q^{95} +1.63369e6 q^{96} +6.64483e6 q^{97} +160797. q^{98} +2.25166e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9} + 2000 q^{10} - 7906 q^{11} - 7848 q^{12} - 17818 q^{13} - 5488 q^{14} - 3750 q^{15} - 4320 q^{16} - 2398 q^{17} + 9792 q^{18} - 3612 q^{19} - 5000 q^{20} + 10290 q^{21} - 96688 q^{22} + 13844 q^{23} + 24960 q^{24} + 31250 q^{25} - 179328 q^{26} - 18090 q^{27} + 13720 q^{28} - 126898 q^{29} - 96000 q^{30} + 252768 q^{31} - 148224 q^{32} + 319230 q^{33} + 175296 q^{34} - 85750 q^{35} + 268560 q^{36} - 265860 q^{37} + 458800 q^{38} + 487974 q^{39} - 120000 q^{40} - 111920 q^{41} + 263424 q^{42} + 947572 q^{43} - 376920 q^{44} - 94500 q^{45} + 1051472 q^{46} + 271274 q^{47} + 1484064 q^{48} + 235298 q^{49} + 250000 q^{50} - 1130910 q^{51} - 232184 q^{52} - 1267792 q^{53} + 972000 q^{54} - 988250 q^{55} + 329280 q^{56} - 2871996 q^{57} - 3107120 q^{58} - 1360120 q^{59} - 981000 q^{60} - 1813680 q^{61} + 37392 q^{62} + 259308 q^{63} - 2489984 q^{64} - 2227250 q^{65} + 5142624 q^{66} - 2189312 q^{67} + 3159640 q^{68} - 5851980 q^{69} - 686000 q^{70} - 1494928 q^{71} + 46080 q^{72} + 7169788 q^{73} - 5967024 q^{74} - 468750 q^{75} + 7875376 q^{76} + 2711758 q^{77} + 9159504 q^{78} - 7942974 q^{79} - 540000 q^{80} - 4775598 q^{81} + 2391792 q^{82} - 304712 q^{83} + 2691864 q^{84} - 299750 q^{85} + 5417712 q^{86} + 14455086 q^{87} + 4463680 q^{88} - 17943528 q^{89} + 1224000 q^{90} + 6111574 q^{91} + 14774640 q^{92} + 8116992 q^{93} - 2823104 q^{94} - 451500 q^{95} + 13366272 q^{96} + 4258074 q^{97} + 1882384 q^{98} - 3030732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36675 0.120805 0.0604024 0.998174i \(-0.480762\pi\)
0.0604024 + 0.998174i \(0.480762\pi\)
\(3\) 24.7995 0.530296 0.265148 0.964208i \(-0.414579\pi\)
0.265148 + 0.964208i \(0.414579\pi\)
\(4\) −126.132 −0.985406
\(5\) 125.000 0.447214
\(6\) 33.8947 0.0640623
\(7\) −343.000 −0.377964
\(8\) −347.335 −0.239847
\(9\) −1571.98 −0.718786
\(10\) 170.844 0.0540256
\(11\) −1432.37 −0.324474 −0.162237 0.986752i \(-0.551871\pi\)
−0.162237 + 0.986752i \(0.551871\pi\)
\(12\) −3128.01 −0.522557
\(13\) −6136.30 −0.774649 −0.387325 0.921943i \(-0.626601\pi\)
−0.387325 + 0.921943i \(0.626601\pi\)
\(14\) −468.795 −0.0456599
\(15\) 3099.94 0.237156
\(16\) 15670.2 0.956432
\(17\) −15858.5 −0.782871 −0.391436 0.920205i \(-0.628021\pi\)
−0.391436 + 0.920205i \(0.628021\pi\)
\(18\) −2148.51 −0.0868328
\(19\) −38567.5 −1.28998 −0.644991 0.764190i \(-0.723139\pi\)
−0.644991 + 0.764190i \(0.723139\pi\)
\(20\) −15766.5 −0.440687
\(21\) −8506.23 −0.200433
\(22\) −1957.69 −0.0391980
\(23\) −63987.4 −1.09660 −0.548299 0.836282i \(-0.684724\pi\)
−0.548299 + 0.836282i \(0.684724\pi\)
\(24\) −8613.73 −0.127190
\(25\) 15625.0 0.200000
\(26\) −8386.79 −0.0935813
\(27\) −93220.9 −0.911466
\(28\) 43263.3 0.372449
\(29\) 94236.6 0.717508 0.358754 0.933432i \(-0.383202\pi\)
0.358754 + 0.933432i \(0.383202\pi\)
\(30\) 4236.84 0.0286495
\(31\) 275990. 1.66390 0.831951 0.554849i \(-0.187224\pi\)
0.831951 + 0.554849i \(0.187224\pi\)
\(32\) 65876.1 0.355388
\(33\) −35521.9 −0.172067
\(34\) −21674.6 −0.0945746
\(35\) −42875.0 −0.169031
\(36\) 198278. 0.708296
\(37\) 156532. 0.508038 0.254019 0.967199i \(-0.418247\pi\)
0.254019 + 0.967199i \(0.418247\pi\)
\(38\) −52712.1 −0.155836
\(39\) −152177. −0.410793
\(40\) −43416.9 −0.107263
\(41\) −303738. −0.688266 −0.344133 0.938921i \(-0.611827\pi\)
−0.344133 + 0.938921i \(0.611827\pi\)
\(42\) −11625.9 −0.0242133
\(43\) 636818. 1.22145 0.610725 0.791843i \(-0.290878\pi\)
0.610725 + 0.791843i \(0.290878\pi\)
\(44\) 180667. 0.319738
\(45\) −196498. −0.321451
\(46\) −87454.9 −0.132474
\(47\) 512021. 0.719358 0.359679 0.933076i \(-0.382886\pi\)
0.359679 + 0.933076i \(0.382886\pi\)
\(48\) 388612. 0.507192
\(49\) 117649. 0.142857
\(50\) 21355.5 0.0241610
\(51\) −393282. −0.415154
\(52\) 773984. 0.763344
\(53\) −201249. −0.185681 −0.0928406 0.995681i \(-0.529595\pi\)
−0.0928406 + 0.995681i \(0.529595\pi\)
\(54\) −127410. −0.110109
\(55\) −179046. −0.145109
\(56\) 119136. 0.0906535
\(57\) −956454. −0.684072
\(58\) 128798. 0.0866784
\(59\) −1.81196e6 −1.14859 −0.574296 0.818648i \(-0.694724\pi\)
−0.574296 + 0.818648i \(0.694724\pi\)
\(60\) −391001. −0.233695
\(61\) −982021. −0.553945 −0.276972 0.960878i \(-0.589331\pi\)
−0.276972 + 0.960878i \(0.589331\pi\)
\(62\) 377210. 0.201007
\(63\) 539191. 0.271676
\(64\) −1.91575e6 −0.913499
\(65\) −767038. −0.346434
\(66\) −48549.6 −0.0207865
\(67\) −4.45336e6 −1.80895 −0.904474 0.426528i \(-0.859736\pi\)
−0.904474 + 0.426528i \(0.859736\pi\)
\(68\) 2.00026e6 0.771446
\(69\) −1.58686e6 −0.581522
\(70\) −58599.4 −0.0204197
\(71\) 725436. 0.240544 0.120272 0.992741i \(-0.461623\pi\)
0.120272 + 0.992741i \(0.461623\pi\)
\(72\) 546005. 0.172398
\(73\) 2.17602e6 0.654685 0.327343 0.944906i \(-0.393847\pi\)
0.327343 + 0.944906i \(0.393847\pi\)
\(74\) 213940. 0.0613735
\(75\) 387492. 0.106059
\(76\) 4.86459e6 1.27116
\(77\) 491301. 0.122639
\(78\) −207988. −0.0496258
\(79\) −5.21525e6 −1.19009 −0.595045 0.803692i \(-0.702866\pi\)
−0.595045 + 0.803692i \(0.702866\pi\)
\(80\) 1.95877e6 0.427729
\(81\) 1.12610e6 0.235439
\(82\) −415135. −0.0831458
\(83\) 6.07921e6 1.16701 0.583504 0.812110i \(-0.301681\pi\)
0.583504 + 0.812110i \(0.301681\pi\)
\(84\) 1.07291e6 0.197508
\(85\) −1.98231e6 −0.350111
\(86\) 870371. 0.147557
\(87\) 2.33702e6 0.380492
\(88\) 497511. 0.0778239
\(89\) −1.06137e7 −1.59589 −0.797946 0.602729i \(-0.794080\pi\)
−0.797946 + 0.602729i \(0.794080\pi\)
\(90\) −268564. −0.0388328
\(91\) 2.10475e6 0.292790
\(92\) 8.07086e6 1.08059
\(93\) 6.84442e6 0.882361
\(94\) 699805. 0.0869019
\(95\) −4.82093e6 −0.576897
\(96\) 1.63369e6 0.188461
\(97\) 6.64483e6 0.739236 0.369618 0.929184i \(-0.379489\pi\)
0.369618 + 0.929184i \(0.379489\pi\)
\(98\) 160797. 0.0172578
\(99\) 2.25166e6 0.233227
\(100\) −1.97081e6 −0.197081
\(101\) 1.07531e7 1.03851 0.519254 0.854620i \(-0.326210\pi\)
0.519254 + 0.854620i \(0.326210\pi\)
\(102\) −537519. −0.0501526
\(103\) −1.05886e7 −0.954788 −0.477394 0.878689i \(-0.658419\pi\)
−0.477394 + 0.878689i \(0.658419\pi\)
\(104\) 2.13135e6 0.185797
\(105\) −1.06328e6 −0.0896364
\(106\) −275057. −0.0224312
\(107\) −8.37234e6 −0.660699 −0.330349 0.943859i \(-0.607167\pi\)
−0.330349 + 0.943859i \(0.607167\pi\)
\(108\) 1.17581e7 0.898164
\(109\) −1.95948e7 −1.44926 −0.724632 0.689136i \(-0.757990\pi\)
−0.724632 + 0.689136i \(0.757990\pi\)
\(110\) −244711. −0.0175299
\(111\) 3.88191e6 0.269411
\(112\) −5.37487e6 −0.361497
\(113\) 1.36310e7 0.888694 0.444347 0.895855i \(-0.353436\pi\)
0.444347 + 0.895855i \(0.353436\pi\)
\(114\) −1.30723e6 −0.0826392
\(115\) −7.99843e6 −0.490413
\(116\) −1.18863e7 −0.707037
\(117\) 9.64617e6 0.556807
\(118\) −2.47649e6 −0.138756
\(119\) 5.43946e6 0.295898
\(120\) −1.07672e6 −0.0568810
\(121\) −1.74355e7 −0.894717
\(122\) −1.34218e6 −0.0669192
\(123\) −7.53256e6 −0.364985
\(124\) −3.48112e7 −1.63962
\(125\) 1.95312e6 0.0894427
\(126\) 736939. 0.0328197
\(127\) 2.23763e7 0.969336 0.484668 0.874698i \(-0.338940\pi\)
0.484668 + 0.874698i \(0.338940\pi\)
\(128\) −1.10505e7 −0.465743
\(129\) 1.57928e7 0.647730
\(130\) −1.04835e6 −0.0418508
\(131\) −4.53330e6 −0.176183 −0.0880917 0.996112i \(-0.528077\pi\)
−0.0880917 + 0.996112i \(0.528077\pi\)
\(132\) 4.48045e6 0.169556
\(133\) 1.32286e7 0.487567
\(134\) −6.08663e6 −0.218530
\(135\) −1.16526e7 −0.407620
\(136\) 5.50821e6 0.187769
\(137\) −5.07657e7 −1.68674 −0.843371 0.537332i \(-0.819432\pi\)
−0.843371 + 0.537332i \(0.819432\pi\)
\(138\) −2.16884e6 −0.0702506
\(139\) 1.05183e7 0.332195 0.166097 0.986109i \(-0.446883\pi\)
0.166097 + 0.986109i \(0.446883\pi\)
\(140\) 5.40791e6 0.166564
\(141\) 1.26979e7 0.381473
\(142\) 991490. 0.0290589
\(143\) 8.78942e6 0.251353
\(144\) −2.46333e7 −0.687470
\(145\) 1.17796e7 0.320879
\(146\) 2.97407e6 0.0790891
\(147\) 2.91764e6 0.0757566
\(148\) −1.97437e7 −0.500624
\(149\) 5.43497e7 1.34600 0.673000 0.739642i \(-0.265005\pi\)
0.673000 + 0.739642i \(0.265005\pi\)
\(150\) 529605. 0.0128125
\(151\) −2.23258e7 −0.527700 −0.263850 0.964564i \(-0.584992\pi\)
−0.263850 + 0.964564i \(0.584992\pi\)
\(152\) 1.33958e7 0.309398
\(153\) 2.49293e7 0.562717
\(154\) 671486. 0.0148154
\(155\) 3.44988e7 0.744120
\(156\) 1.91944e7 0.404798
\(157\) −4.37788e7 −0.902848 −0.451424 0.892310i \(-0.649084\pi\)
−0.451424 + 0.892310i \(0.649084\pi\)
\(158\) −7.12794e6 −0.143769
\(159\) −4.99087e6 −0.0984661
\(160\) 8.23451e6 0.158934
\(161\) 2.19477e7 0.414475
\(162\) 1.53910e6 0.0284422
\(163\) 4.05451e7 0.733300 0.366650 0.930359i \(-0.380505\pi\)
0.366650 + 0.930359i \(0.380505\pi\)
\(164\) 3.83111e7 0.678221
\(165\) −4.44024e6 −0.0769507
\(166\) 8.30876e6 0.140980
\(167\) 9.73453e7 1.61736 0.808682 0.588247i \(-0.200182\pi\)
0.808682 + 0.588247i \(0.200182\pi\)
\(168\) 2.95451e6 0.0480732
\(169\) −2.50943e7 −0.399919
\(170\) −2.70932e6 −0.0422951
\(171\) 6.06275e7 0.927221
\(172\) −8.03231e7 −1.20362
\(173\) −5.10607e7 −0.749765 −0.374882 0.927072i \(-0.622317\pi\)
−0.374882 + 0.927072i \(0.622317\pi\)
\(174\) 3.19412e6 0.0459652
\(175\) −5.35938e6 −0.0755929
\(176\) −2.24454e7 −0.310337
\(177\) −4.49356e7 −0.609094
\(178\) −1.45063e7 −0.192791
\(179\) 1.45811e8 1.90023 0.950113 0.311907i \(-0.100968\pi\)
0.950113 + 0.311907i \(0.100968\pi\)
\(180\) 2.47847e7 0.316760
\(181\) −6.09656e7 −0.764205 −0.382102 0.924120i \(-0.624800\pi\)
−0.382102 + 0.924120i \(0.624800\pi\)
\(182\) 2.87667e6 0.0353704
\(183\) −2.43536e7 −0.293755
\(184\) 2.22251e7 0.263015
\(185\) 1.95665e7 0.227202
\(186\) 9.35462e6 0.106593
\(187\) 2.27151e7 0.254021
\(188\) −6.45822e7 −0.708860
\(189\) 3.19748e7 0.344502
\(190\) −6.58901e6 −0.0696920
\(191\) −1.52578e8 −1.58444 −0.792219 0.610237i \(-0.791074\pi\)
−0.792219 + 0.610237i \(0.791074\pi\)
\(192\) −4.75095e7 −0.484425
\(193\) −1.39277e8 −1.39453 −0.697267 0.716812i \(-0.745601\pi\)
−0.697267 + 0.716812i \(0.745601\pi\)
\(194\) 9.08183e6 0.0893033
\(195\) −1.90221e7 −0.183712
\(196\) −1.48393e7 −0.140772
\(197\) 6.52480e7 0.608044 0.304022 0.952665i \(-0.401670\pi\)
0.304022 + 0.952665i \(0.401670\pi\)
\(198\) 3.07745e6 0.0281750
\(199\) 1.93503e6 0.0174061 0.00870307 0.999962i \(-0.497230\pi\)
0.00870307 + 0.999962i \(0.497230\pi\)
\(200\) −5.42711e6 −0.0479693
\(201\) −1.10441e8 −0.959278
\(202\) 1.46968e7 0.125457
\(203\) −3.23232e7 −0.271192
\(204\) 4.96055e7 0.409095
\(205\) −3.79673e7 −0.307802
\(206\) −1.44719e7 −0.115343
\(207\) 1.00587e8 0.788219
\(208\) −9.61569e7 −0.740899
\(209\) 5.52427e7 0.418565
\(210\) −1.45324e6 −0.0108285
\(211\) −5.17848e7 −0.379502 −0.189751 0.981832i \(-0.560768\pi\)
−0.189751 + 0.981832i \(0.560768\pi\)
\(212\) 2.53839e7 0.182971
\(213\) 1.79904e7 0.127560
\(214\) −1.14429e7 −0.0798156
\(215\) 7.96023e7 0.546249
\(216\) 3.23789e7 0.218612
\(217\) −9.46647e7 −0.628896
\(218\) −2.67812e7 −0.175078
\(219\) 5.39642e7 0.347177
\(220\) 2.25834e7 0.142991
\(221\) 9.73124e7 0.606450
\(222\) 5.30560e6 0.0325461
\(223\) −1.25065e8 −0.755209 −0.377605 0.925967i \(-0.623252\pi\)
−0.377605 + 0.925967i \(0.623252\pi\)
\(224\) −2.25955e7 −0.134324
\(225\) −2.45623e7 −0.143757
\(226\) 1.86301e7 0.107358
\(227\) 1.92108e7 0.109007 0.0545036 0.998514i \(-0.482642\pi\)
0.0545036 + 0.998514i \(0.482642\pi\)
\(228\) 1.20639e8 0.674089
\(229\) −1.05650e8 −0.581360 −0.290680 0.956820i \(-0.593882\pi\)
−0.290680 + 0.956820i \(0.593882\pi\)
\(230\) −1.09319e7 −0.0592443
\(231\) 1.21840e7 0.0650353
\(232\) −3.27317e7 −0.172092
\(233\) −2.31646e8 −1.19972 −0.599859 0.800106i \(-0.704776\pi\)
−0.599859 + 0.800106i \(0.704776\pi\)
\(234\) 1.31839e7 0.0672649
\(235\) 6.40026e7 0.321707
\(236\) 2.28546e8 1.13183
\(237\) −1.29336e8 −0.631101
\(238\) 7.43438e6 0.0357458
\(239\) 1.09174e8 0.517281 0.258641 0.965974i \(-0.416725\pi\)
0.258641 + 0.965974i \(0.416725\pi\)
\(240\) 4.85766e7 0.226823
\(241\) −8.25277e7 −0.379787 −0.189893 0.981805i \(-0.560814\pi\)
−0.189893 + 0.981805i \(0.560814\pi\)
\(242\) −2.38300e7 −0.108086
\(243\) 2.31801e8 1.03632
\(244\) 1.23864e8 0.545861
\(245\) 1.47061e7 0.0638877
\(246\) −1.02951e7 −0.0440919
\(247\) 2.36662e8 0.999283
\(248\) −9.58611e7 −0.399081
\(249\) 1.50761e8 0.618860
\(250\) 2.66943e6 0.0108051
\(251\) −2.40987e7 −0.0961912 −0.0480956 0.998843i \(-0.515315\pi\)
−0.0480956 + 0.998843i \(0.515315\pi\)
\(252\) −6.80092e7 −0.267711
\(253\) 9.16534e7 0.355817
\(254\) 3.05828e7 0.117100
\(255\) −4.91603e7 −0.185662
\(256\) 2.30112e8 0.857235
\(257\) 9.75049e7 0.358311 0.179156 0.983821i \(-0.442663\pi\)
0.179156 + 0.983821i \(0.442663\pi\)
\(258\) 2.15848e7 0.0782489
\(259\) −5.36904e7 −0.192020
\(260\) 9.67480e7 0.341378
\(261\) −1.48139e8 −0.515735
\(262\) −6.19589e6 −0.0212838
\(263\) 2.98637e8 1.01228 0.506138 0.862452i \(-0.331073\pi\)
0.506138 + 0.862452i \(0.331073\pi\)
\(264\) 1.23380e7 0.0412697
\(265\) −2.51561e7 −0.0830392
\(266\) 1.80803e7 0.0589005
\(267\) −2.63216e8 −0.846296
\(268\) 5.61711e8 1.78255
\(269\) −3.90722e8 −1.22387 −0.611934 0.790909i \(-0.709608\pi\)
−0.611934 + 0.790909i \(0.709608\pi\)
\(270\) −1.59262e7 −0.0492424
\(271\) 2.12098e8 0.647357 0.323678 0.946167i \(-0.395080\pi\)
0.323678 + 0.946167i \(0.395080\pi\)
\(272\) −2.48505e8 −0.748763
\(273\) 5.21968e7 0.155265
\(274\) −6.93841e7 −0.203767
\(275\) −2.23807e7 −0.0648947
\(276\) 2.00153e8 0.573035
\(277\) 1.86723e8 0.527861 0.263930 0.964542i \(-0.414981\pi\)
0.263930 + 0.964542i \(0.414981\pi\)
\(278\) 1.43759e7 0.0401307
\(279\) −4.33853e8 −1.19599
\(280\) 1.48920e7 0.0405415
\(281\) −7.38791e8 −1.98632 −0.993161 0.116756i \(-0.962750\pi\)
−0.993161 + 0.116756i \(0.962750\pi\)
\(282\) 1.73548e7 0.0460838
\(283\) −3.11903e8 −0.818026 −0.409013 0.912529i \(-0.634127\pi\)
−0.409013 + 0.912529i \(0.634127\pi\)
\(284\) −9.15007e7 −0.237034
\(285\) −1.19557e8 −0.305926
\(286\) 1.20129e7 0.0303647
\(287\) 1.04182e8 0.260140
\(288\) −1.03556e8 −0.255448
\(289\) −1.58847e8 −0.387113
\(290\) 1.60997e7 0.0387638
\(291\) 1.64789e8 0.392014
\(292\) −2.74466e8 −0.645131
\(293\) −5.05466e8 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(294\) 3.98768e6 0.00915176
\(295\) −2.26495e8 −0.513666
\(296\) −5.43690e7 −0.121851
\(297\) 1.33526e8 0.295747
\(298\) 7.42825e7 0.162603
\(299\) 3.92646e8 0.849478
\(300\) −4.88752e7 −0.104511
\(301\) −2.18429e8 −0.461665
\(302\) −3.05137e7 −0.0637487
\(303\) 2.66672e8 0.550717
\(304\) −6.04359e8 −1.23378
\(305\) −1.22753e8 −0.247732
\(306\) 3.40721e7 0.0679789
\(307\) 4.67463e8 0.922067 0.461034 0.887383i \(-0.347479\pi\)
0.461034 + 0.887383i \(0.347479\pi\)
\(308\) −6.19688e7 −0.120850
\(309\) −2.62591e8 −0.506321
\(310\) 4.71512e7 0.0898933
\(311\) 1.16022e7 0.0218714 0.0109357 0.999940i \(-0.496519\pi\)
0.0109357 + 0.999940i \(0.496519\pi\)
\(312\) 5.28565e7 0.0985274
\(313\) 8.23197e8 1.51740 0.758698 0.651443i \(-0.225836\pi\)
0.758698 + 0.651443i \(0.225836\pi\)
\(314\) −5.98346e7 −0.109068
\(315\) 6.73989e7 0.121497
\(316\) 6.57810e8 1.17272
\(317\) 3.89154e8 0.686142 0.343071 0.939309i \(-0.388533\pi\)
0.343071 + 0.939309i \(0.388533\pi\)
\(318\) −6.82128e6 −0.0118952
\(319\) −1.34981e8 −0.232812
\(320\) −2.39468e8 −0.408529
\(321\) −2.07630e8 −0.350366
\(322\) 2.99970e7 0.0500706
\(323\) 6.11621e8 1.00989
\(324\) −1.42037e8 −0.232003
\(325\) −9.58797e7 −0.154930
\(326\) 5.54150e7 0.0885861
\(327\) −4.85940e8 −0.768539
\(328\) 1.05499e8 0.165078
\(329\) −1.75623e8 −0.271892
\(330\) −6.06870e6 −0.00929602
\(331\) 1.48582e8 0.225199 0.112600 0.993640i \(-0.464082\pi\)
0.112600 + 0.993640i \(0.464082\pi\)
\(332\) −7.66783e8 −1.14998
\(333\) −2.46066e8 −0.365171
\(334\) 1.33047e8 0.195385
\(335\) −5.56670e8 −0.808986
\(336\) −1.33294e8 −0.191701
\(337\) 1.23379e8 0.175605 0.0878023 0.996138i \(-0.472016\pi\)
0.0878023 + 0.996138i \(0.472016\pi\)
\(338\) −3.42977e7 −0.0483121
\(339\) 3.38041e8 0.471271
\(340\) 2.50033e8 0.345001
\(341\) −3.95319e8 −0.539892
\(342\) 8.28626e7 0.112013
\(343\) −4.03536e7 −0.0539949
\(344\) −2.21189e8 −0.292961
\(345\) −1.98357e8 −0.260064
\(346\) −6.97872e7 −0.0905752
\(347\) 1.31658e9 1.69159 0.845793 0.533511i \(-0.179128\pi\)
0.845793 + 0.533511i \(0.179128\pi\)
\(348\) −2.94773e8 −0.374939
\(349\) 2.64521e8 0.333097 0.166549 0.986033i \(-0.446738\pi\)
0.166549 + 0.986033i \(0.446738\pi\)
\(350\) −7.32493e6 −0.00913199
\(351\) 5.72032e8 0.706066
\(352\) −9.43586e7 −0.115314
\(353\) 1.30271e9 1.57629 0.788144 0.615490i \(-0.211042\pi\)
0.788144 + 0.615490i \(0.211042\pi\)
\(354\) −6.14158e7 −0.0735815
\(355\) 9.06795e7 0.107575
\(356\) 1.33873e9 1.57260
\(357\) 1.34896e8 0.156913
\(358\) 1.99287e8 0.229556
\(359\) −1.03262e9 −1.17790 −0.588952 0.808168i \(-0.700459\pi\)
−0.588952 + 0.808168i \(0.700459\pi\)
\(360\) 6.82507e7 0.0770989
\(361\) 5.93578e8 0.664053
\(362\) −8.33248e7 −0.0923196
\(363\) −4.32392e8 −0.474465
\(364\) −2.65476e8 −0.288517
\(365\) 2.72002e8 0.292784
\(366\) −3.32853e7 −0.0354870
\(367\) 1.13124e9 1.19460 0.597302 0.802017i \(-0.296240\pi\)
0.597302 + 0.802017i \(0.296240\pi\)
\(368\) −1.00269e9 −1.04882
\(369\) 4.77472e8 0.494716
\(370\) 2.67425e7 0.0274470
\(371\) 6.90284e7 0.0701809
\(372\) −8.63300e8 −0.869484
\(373\) 5.38130e8 0.536916 0.268458 0.963291i \(-0.413486\pi\)
0.268458 + 0.963291i \(0.413486\pi\)
\(374\) 3.10459e7 0.0306870
\(375\) 4.84365e7 0.0474311
\(376\) −1.77843e8 −0.172536
\(377\) −5.78264e8 −0.555817
\(378\) 4.37015e7 0.0416175
\(379\) 7.83114e8 0.738904 0.369452 0.929250i \(-0.379545\pi\)
0.369452 + 0.929250i \(0.379545\pi\)
\(380\) 6.08074e8 0.568478
\(381\) 5.54920e8 0.514035
\(382\) −2.08536e8 −0.191408
\(383\) 8.22468e8 0.748038 0.374019 0.927421i \(-0.377980\pi\)
0.374019 + 0.927421i \(0.377980\pi\)
\(384\) −2.74047e8 −0.246982
\(385\) 6.14127e7 0.0548460
\(386\) −1.90357e8 −0.168466
\(387\) −1.00107e9 −0.877961
\(388\) −8.38126e8 −0.728448
\(389\) −1.07007e9 −0.921696 −0.460848 0.887479i \(-0.652455\pi\)
−0.460848 + 0.887479i \(0.652455\pi\)
\(390\) −2.59985e7 −0.0221933
\(391\) 1.01474e9 0.858495
\(392\) −4.08636e7 −0.0342638
\(393\) −1.12424e8 −0.0934293
\(394\) 8.91777e7 0.0734547
\(395\) −6.51906e8 −0.532225
\(396\) −2.84006e8 −0.229823
\(397\) −9.64552e8 −0.773676 −0.386838 0.922148i \(-0.626433\pi\)
−0.386838 + 0.922148i \(0.626433\pi\)
\(398\) 2.64471e6 0.00210275
\(399\) 3.28064e8 0.258555
\(400\) 2.44846e8 0.191286
\(401\) −1.94810e9 −1.50871 −0.754357 0.656465i \(-0.772051\pi\)
−0.754357 + 0.656465i \(0.772051\pi\)
\(402\) −1.50945e8 −0.115885
\(403\) −1.69356e9 −1.28894
\(404\) −1.35631e9 −1.02335
\(405\) 1.40762e8 0.105292
\(406\) −4.41777e7 −0.0327614
\(407\) −2.24211e8 −0.164845
\(408\) 1.36601e8 0.0995732
\(409\) 8.63865e8 0.624330 0.312165 0.950028i \(-0.398946\pi\)
0.312165 + 0.950028i \(0.398946\pi\)
\(410\) −5.18918e7 −0.0371839
\(411\) −1.25896e9 −0.894473
\(412\) 1.33556e9 0.940854
\(413\) 6.21501e8 0.434127
\(414\) 1.37478e8 0.0952206
\(415\) 7.59901e8 0.521902
\(416\) −4.04236e8 −0.275301
\(417\) 2.60848e8 0.176162
\(418\) 7.55030e7 0.0505647
\(419\) 2.21337e9 1.46996 0.734978 0.678091i \(-0.237192\pi\)
0.734978 + 0.678091i \(0.237192\pi\)
\(420\) 1.34113e8 0.0883283
\(421\) −2.89866e9 −1.89326 −0.946631 0.322321i \(-0.895537\pi\)
−0.946631 + 0.322321i \(0.895537\pi\)
\(422\) −7.07769e7 −0.0458456
\(423\) −8.04889e8 −0.517065
\(424\) 6.99008e7 0.0445350
\(425\) −2.47789e8 −0.156574
\(426\) 2.45884e7 0.0154098
\(427\) 3.36833e8 0.209371
\(428\) 1.05602e9 0.651057
\(429\) 2.17973e8 0.133292
\(430\) 1.08796e8 0.0659895
\(431\) −2.42056e9 −1.45628 −0.728142 0.685426i \(-0.759616\pi\)
−0.728142 + 0.685426i \(0.759616\pi\)
\(432\) −1.46079e9 −0.871754
\(433\) −2.26686e9 −1.34189 −0.670946 0.741506i \(-0.734112\pi\)
−0.670946 + 0.741506i \(0.734112\pi\)
\(434\) −1.29383e8 −0.0759737
\(435\) 2.92128e8 0.170161
\(436\) 2.47153e9 1.42811
\(437\) 2.46783e9 1.41459
\(438\) 7.37555e7 0.0419407
\(439\) 1.98911e9 1.12210 0.561052 0.827780i \(-0.310397\pi\)
0.561052 + 0.827780i \(0.310397\pi\)
\(440\) 6.21888e7 0.0348039
\(441\) −1.84942e8 −0.102684
\(442\) 1.33002e8 0.0732621
\(443\) −8.78038e8 −0.479844 −0.239922 0.970792i \(-0.577122\pi\)
−0.239922 + 0.970792i \(0.577122\pi\)
\(444\) −4.89633e8 −0.265479
\(445\) −1.32672e9 −0.713705
\(446\) −1.70932e8 −0.0912329
\(447\) 1.34785e9 0.713779
\(448\) 6.57101e8 0.345270
\(449\) 1.53113e8 0.0798270 0.0399135 0.999203i \(-0.487292\pi\)
0.0399135 + 0.999203i \(0.487292\pi\)
\(450\) −3.35705e7 −0.0173666
\(451\) 4.35064e8 0.223324
\(452\) −1.71930e9 −0.875724
\(453\) −5.53668e8 −0.279837
\(454\) 2.62564e7 0.0131686
\(455\) 2.63094e8 0.130940
\(456\) 3.32210e8 0.164072
\(457\) −2.39624e9 −1.17442 −0.587210 0.809435i \(-0.699774\pi\)
−0.587210 + 0.809435i \(0.699774\pi\)
\(458\) −1.44397e8 −0.0702311
\(459\) 1.47834e9 0.713560
\(460\) 1.00886e9 0.483256
\(461\) 1.61913e9 0.769713 0.384856 0.922977i \(-0.374251\pi\)
0.384856 + 0.922977i \(0.374251\pi\)
\(462\) 1.66525e7 0.00785657
\(463\) 1.16133e9 0.543778 0.271889 0.962329i \(-0.412352\pi\)
0.271889 + 0.962329i \(0.412352\pi\)
\(464\) 1.47670e9 0.686247
\(465\) 8.55553e8 0.394604
\(466\) −3.16602e8 −0.144932
\(467\) 2.83969e9 1.29021 0.645107 0.764092i \(-0.276813\pi\)
0.645107 + 0.764092i \(0.276813\pi\)
\(468\) −1.21669e9 −0.548681
\(469\) 1.52750e9 0.683718
\(470\) 8.74756e7 0.0388637
\(471\) −1.08569e9 −0.478777
\(472\) 6.29356e8 0.275486
\(473\) −9.12156e8 −0.396328
\(474\) −1.76769e8 −0.0762400
\(475\) −6.02617e8 −0.257996
\(476\) −6.86090e8 −0.291579
\(477\) 3.16360e8 0.133465
\(478\) 1.49214e8 0.0624901
\(479\) 2.38771e9 0.992676 0.496338 0.868129i \(-0.334678\pi\)
0.496338 + 0.868129i \(0.334678\pi\)
\(480\) 2.04212e8 0.0842823
\(481\) −9.60526e8 −0.393551
\(482\) −1.12795e8 −0.0458801
\(483\) 5.44292e8 0.219794
\(484\) 2.19917e9 0.881660
\(485\) 8.30604e8 0.330596
\(486\) 3.16814e8 0.125192
\(487\) −2.20508e9 −0.865113 −0.432556 0.901607i \(-0.642388\pi\)
−0.432556 + 0.901607i \(0.642388\pi\)
\(488\) 3.41090e8 0.132862
\(489\) 1.00550e9 0.388866
\(490\) 2.00996e7 0.00771794
\(491\) 4.28064e8 0.163201 0.0816006 0.996665i \(-0.473997\pi\)
0.0816006 + 0.996665i \(0.473997\pi\)
\(492\) 9.50097e8 0.359658
\(493\) −1.49445e9 −0.561716
\(494\) 3.23457e8 0.120718
\(495\) 2.81457e8 0.104302
\(496\) 4.32482e9 1.59141
\(497\) −2.48824e8 −0.0909171
\(498\) 2.06053e8 0.0747613
\(499\) −2.95178e9 −1.06349 −0.531743 0.846906i \(-0.678463\pi\)
−0.531743 + 0.846906i \(0.678463\pi\)
\(500\) −2.46352e8 −0.0881374
\(501\) 2.41412e9 0.857681
\(502\) −3.29369e7 −0.0116204
\(503\) −5.22380e9 −1.83020 −0.915099 0.403229i \(-0.867888\pi\)
−0.915099 + 0.403229i \(0.867888\pi\)
\(504\) −1.87280e8 −0.0651605
\(505\) 1.34414e9 0.464435
\(506\) 1.25267e8 0.0429844
\(507\) −6.22326e8 −0.212075
\(508\) −2.82236e9 −0.955190
\(509\) −2.80532e9 −0.942911 −0.471455 0.881890i \(-0.656271\pi\)
−0.471455 + 0.881890i \(0.656271\pi\)
\(510\) −6.71899e7 −0.0224289
\(511\) −7.46374e8 −0.247448
\(512\) 1.72897e9 0.569301
\(513\) 3.59530e9 1.17577
\(514\) 1.33265e8 0.0432857
\(515\) −1.32357e9 −0.426994
\(516\) −1.99197e9 −0.638277
\(517\) −7.33401e8 −0.233413
\(518\) −7.33814e7 −0.0231970
\(519\) −1.26628e9 −0.397597
\(520\) 2.66419e8 0.0830909
\(521\) 1.40563e8 0.0435450 0.0217725 0.999763i \(-0.493069\pi\)
0.0217725 + 0.999763i \(0.493069\pi\)
\(522\) −2.02468e8 −0.0623032
\(523\) −1.81127e9 −0.553638 −0.276819 0.960922i \(-0.589280\pi\)
−0.276819 + 0.960922i \(0.589280\pi\)
\(524\) 5.71794e8 0.173612
\(525\) −1.32910e8 −0.0400866
\(526\) 4.08163e8 0.122288
\(527\) −4.37679e9 −1.30262
\(528\) −5.56635e8 −0.164570
\(529\) 6.89567e8 0.202526
\(530\) −3.43821e7 −0.0100315
\(531\) 2.84837e9 0.825592
\(532\) −1.66855e9 −0.480452
\(533\) 1.86383e9 0.533164
\(534\) −3.59750e8 −0.102237
\(535\) −1.04654e9 −0.295474
\(536\) 1.54681e9 0.433870
\(537\) 3.61604e9 1.00768
\(538\) −5.34019e8 −0.147849
\(539\) −1.68516e8 −0.0463534
\(540\) 1.46977e9 0.401671
\(541\) −7.11633e9 −1.93226 −0.966130 0.258058i \(-0.916918\pi\)
−0.966130 + 0.258058i \(0.916918\pi\)
\(542\) 2.89885e8 0.0782038
\(543\) −1.51192e9 −0.405255
\(544\) −1.04469e9 −0.278223
\(545\) −2.44935e9 −0.648130
\(546\) 7.13400e7 0.0187568
\(547\) −6.02390e9 −1.57370 −0.786850 0.617144i \(-0.788290\pi\)
−0.786850 + 0.617144i \(0.788290\pi\)
\(548\) 6.40318e9 1.66213
\(549\) 1.54372e9 0.398168
\(550\) −3.05888e7 −0.00783959
\(551\) −3.63447e9 −0.925572
\(552\) 5.51171e8 0.139476
\(553\) 1.78883e9 0.449812
\(554\) 2.55204e8 0.0637681
\(555\) 4.85239e8 0.120484
\(556\) −1.32669e9 −0.327347
\(557\) 3.55726e9 0.872214 0.436107 0.899895i \(-0.356357\pi\)
0.436107 + 0.899895i \(0.356357\pi\)
\(558\) −5.92968e8 −0.144481
\(559\) −3.90771e9 −0.946195
\(560\) −6.71859e8 −0.161666
\(561\) 5.63324e8 0.134706
\(562\) −1.00974e9 −0.239957
\(563\) 2.51240e9 0.593347 0.296673 0.954979i \(-0.404123\pi\)
0.296673 + 0.954979i \(0.404123\pi\)
\(564\) −1.60161e9 −0.375906
\(565\) 1.70387e9 0.397436
\(566\) −4.26293e8 −0.0988214
\(567\) −3.86252e8 −0.0889877
\(568\) −2.51969e8 −0.0576937
\(569\) 3.02191e9 0.687683 0.343841 0.939028i \(-0.388272\pi\)
0.343841 + 0.939028i \(0.388272\pi\)
\(570\) −1.63404e8 −0.0369574
\(571\) 4.13151e9 0.928716 0.464358 0.885648i \(-0.346285\pi\)
0.464358 + 0.885648i \(0.346285\pi\)
\(572\) −1.10863e9 −0.247685
\(573\) −3.78386e9 −0.840221
\(574\) 1.42391e8 0.0314262
\(575\) −9.99804e8 −0.219320
\(576\) 3.01152e9 0.656610
\(577\) −3.66048e9 −0.793274 −0.396637 0.917976i \(-0.629823\pi\)
−0.396637 + 0.917976i \(0.629823\pi\)
\(578\) −2.17105e8 −0.0467651
\(579\) −3.45400e9 −0.739516
\(580\) −1.48578e9 −0.316196
\(581\) −2.08517e9 −0.441088
\(582\) 2.25225e8 0.0473572
\(583\) 2.88262e8 0.0602487
\(584\) −7.55807e8 −0.157024
\(585\) 1.20577e9 0.249012
\(586\) −6.90846e8 −0.141821
\(587\) 8.93156e9 1.82261 0.911305 0.411731i \(-0.135076\pi\)
0.911305 + 0.411731i \(0.135076\pi\)
\(588\) −3.68007e8 −0.0746510
\(589\) −1.06442e10 −2.14640
\(590\) −3.09562e8 −0.0620534
\(591\) 1.61812e9 0.322444
\(592\) 2.45288e9 0.485904
\(593\) 8.00218e9 1.57586 0.787929 0.615766i \(-0.211153\pi\)
0.787929 + 0.615766i \(0.211153\pi\)
\(594\) 1.82497e8 0.0357276
\(595\) 6.79932e8 0.132329
\(596\) −6.85524e9 −1.32636
\(597\) 4.79879e7 0.00923041
\(598\) 5.36649e8 0.102621
\(599\) 6.37081e9 1.21116 0.605579 0.795785i \(-0.292941\pi\)
0.605579 + 0.795785i \(0.292941\pi\)
\(600\) −1.34590e8 −0.0254379
\(601\) 7.97677e9 1.49888 0.749439 0.662073i \(-0.230323\pi\)
0.749439 + 0.662073i \(0.230323\pi\)
\(602\) −2.98537e8 −0.0557713
\(603\) 7.00062e9 1.30025
\(604\) 2.81599e9 0.519999
\(605\) −2.17944e9 −0.400130
\(606\) 3.64474e8 0.0665293
\(607\) 5.42119e9 0.983863 0.491931 0.870634i \(-0.336291\pi\)
0.491931 + 0.870634i \(0.336291\pi\)
\(608\) −2.54067e9 −0.458444
\(609\) −8.01598e8 −0.143812
\(610\) −1.67772e8 −0.0299272
\(611\) −3.14191e9 −0.557250
\(612\) −3.14438e9 −0.554505
\(613\) 8.21824e9 1.44101 0.720505 0.693450i \(-0.243910\pi\)
0.720505 + 0.693450i \(0.243910\pi\)
\(614\) 6.38905e8 0.111390
\(615\) −9.41570e8 −0.163226
\(616\) −1.70646e8 −0.0294147
\(617\) 8.15621e9 1.39795 0.698973 0.715148i \(-0.253641\pi\)
0.698973 + 0.715148i \(0.253641\pi\)
\(618\) −3.58897e8 −0.0611660
\(619\) −6.46052e9 −1.09484 −0.547420 0.836858i \(-0.684390\pi\)
−0.547420 + 0.836858i \(0.684390\pi\)
\(620\) −4.35140e9 −0.733260
\(621\) 5.96497e9 0.999511
\(622\) 1.58573e7 0.00264218
\(623\) 3.64051e9 0.603191
\(624\) −2.38464e9 −0.392896
\(625\) 2.44141e8 0.0400000
\(626\) 1.12511e9 0.183309
\(627\) 1.36999e9 0.221963
\(628\) 5.52190e9 0.889672
\(629\) −2.48236e9 −0.397729
\(630\) 9.21174e7 0.0146774
\(631\) −8.82660e9 −1.39859 −0.699295 0.714833i \(-0.746503\pi\)
−0.699295 + 0.714833i \(0.746503\pi\)
\(632\) 1.81144e9 0.285439
\(633\) −1.28424e9 −0.201248
\(634\) 5.31876e8 0.0828893
\(635\) 2.79703e9 0.433500
\(636\) 6.29509e8 0.0970291
\(637\) −7.21930e8 −0.110664
\(638\) −1.84486e8 −0.0281249
\(639\) −1.14037e9 −0.172900
\(640\) −1.38131e9 −0.208287
\(641\) 8.54151e9 1.28095 0.640474 0.767980i \(-0.278738\pi\)
0.640474 + 0.767980i \(0.278738\pi\)
\(642\) −2.83778e8 −0.0423259
\(643\) −1.20342e10 −1.78517 −0.892585 0.450878i \(-0.851111\pi\)
−0.892585 + 0.450878i \(0.851111\pi\)
\(644\) −2.76831e9 −0.408426
\(645\) 1.97410e9 0.289674
\(646\) 8.35934e8 0.122000
\(647\) 1.89174e8 0.0274598 0.0137299 0.999906i \(-0.495630\pi\)
0.0137299 + 0.999906i \(0.495630\pi\)
\(648\) −3.91133e8 −0.0564693
\(649\) 2.59539e9 0.372688
\(650\) −1.31044e8 −0.0187163
\(651\) −2.34764e9 −0.333501
\(652\) −5.11403e9 −0.722598
\(653\) 8.70977e9 1.22408 0.612041 0.790826i \(-0.290349\pi\)
0.612041 + 0.790826i \(0.290349\pi\)
\(654\) −6.64159e8 −0.0928432
\(655\) −5.66662e8 −0.0787916
\(656\) −4.75963e9 −0.658279
\(657\) −3.42067e9 −0.470579
\(658\) −2.40033e8 −0.0328458
\(659\) −7.48288e8 −0.101852 −0.0509260 0.998702i \(-0.516217\pi\)
−0.0509260 + 0.998702i \(0.516217\pi\)
\(660\) 5.60057e8 0.0758277
\(661\) 8.45586e9 1.13881 0.569407 0.822056i \(-0.307173\pi\)
0.569407 + 0.822056i \(0.307173\pi\)
\(662\) 2.03074e8 0.0272052
\(663\) 2.41330e9 0.321598
\(664\) −2.11152e9 −0.279903
\(665\) 1.65358e9 0.218047
\(666\) −3.36310e8 −0.0441144
\(667\) −6.02996e9 −0.786817
\(668\) −1.22784e10 −1.59376
\(669\) −3.10154e9 −0.400485
\(670\) −7.60829e8 −0.0977294
\(671\) 1.40661e9 0.179740
\(672\) −5.60357e8 −0.0712316
\(673\) 4.78543e9 0.605157 0.302578 0.953124i \(-0.402153\pi\)
0.302578 + 0.953124i \(0.402153\pi\)
\(674\) 1.68628e8 0.0212139
\(675\) −1.45658e9 −0.182293
\(676\) 3.16520e9 0.394083
\(677\) −1.29662e10 −1.60603 −0.803015 0.595958i \(-0.796772\pi\)
−0.803015 + 0.595958i \(0.796772\pi\)
\(678\) 4.62018e8 0.0569318
\(679\) −2.27918e9 −0.279405
\(680\) 6.88526e8 0.0839729
\(681\) 4.76419e8 0.0578061
\(682\) −5.40302e8 −0.0652216
\(683\) −9.15988e9 −1.10006 −0.550031 0.835144i \(-0.685384\pi\)
−0.550031 + 0.835144i \(0.685384\pi\)
\(684\) −7.64706e9 −0.913689
\(685\) −6.34572e9 −0.754334
\(686\) −5.51533e7 −0.00652285
\(687\) −2.62007e9 −0.308293
\(688\) 9.97905e9 1.16823
\(689\) 1.23492e9 0.143838
\(690\) −2.71105e8 −0.0314170
\(691\) 1.05298e10 1.21407 0.607037 0.794673i \(-0.292358\pi\)
0.607037 + 0.794673i \(0.292358\pi\)
\(692\) 6.44038e9 0.738823
\(693\) −7.72318e8 −0.0881515
\(694\) 1.79944e9 0.204352
\(695\) 1.31478e9 0.148562
\(696\) −8.11729e8 −0.0912596
\(697\) 4.81683e9 0.538824
\(698\) 3.61534e8 0.0402398
\(699\) −5.74470e9 −0.636205
\(700\) 6.75989e8 0.0744897
\(701\) 1.27411e9 0.139699 0.0698497 0.997558i \(-0.477748\pi\)
0.0698497 + 0.997558i \(0.477748\pi\)
\(702\) 7.81825e8 0.0852962
\(703\) −6.03703e9 −0.655360
\(704\) 2.74405e9 0.296406
\(705\) 1.58723e9 0.170600
\(706\) 1.78048e9 0.190423
\(707\) −3.68832e9 −0.392519
\(708\) 5.66782e9 0.600205
\(709\) −7.17795e9 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(710\) 1.23936e8 0.0129955
\(711\) 8.19829e9 0.855421
\(712\) 3.68653e9 0.382769
\(713\) −1.76599e10 −1.82463
\(714\) 1.84369e8 0.0189559
\(715\) 1.09868e9 0.112409
\(716\) −1.83914e10 −1.87249
\(717\) 2.70746e9 0.274312
\(718\) −1.41133e9 −0.142296
\(719\) 1.18502e10 1.18898 0.594488 0.804104i \(-0.297355\pi\)
0.594488 + 0.804104i \(0.297355\pi\)
\(720\) −3.07916e9 −0.307446
\(721\) 3.63188e9 0.360876
\(722\) 8.11273e8 0.0802208
\(723\) −2.04665e9 −0.201400
\(724\) 7.68971e9 0.753052
\(725\) 1.47245e9 0.143502
\(726\) −5.90971e8 −0.0573176
\(727\) −4.67874e9 −0.451605 −0.225802 0.974173i \(-0.572500\pi\)
−0.225802 + 0.974173i \(0.572500\pi\)
\(728\) −7.31054e8 −0.0702246
\(729\) 3.28577e9 0.314116
\(730\) 3.71759e8 0.0353697
\(731\) −1.00990e10 −0.956238
\(732\) 3.07177e9 0.289468
\(733\) 1.28552e9 0.120563 0.0602817 0.998181i \(-0.480800\pi\)
0.0602817 + 0.998181i \(0.480800\pi\)
\(734\) 1.54612e9 0.144314
\(735\) 3.64705e8 0.0338794
\(736\) −4.21524e9 −0.389718
\(737\) 6.37884e9 0.586956
\(738\) 6.52585e8 0.0597641
\(739\) −5.26720e9 −0.480091 −0.240046 0.970762i \(-0.577162\pi\)
−0.240046 + 0.970762i \(0.577162\pi\)
\(740\) −2.46796e9 −0.223886
\(741\) 5.86909e9 0.529916
\(742\) 9.43446e7 0.00847819
\(743\) 4.15012e9 0.371193 0.185596 0.982626i \(-0.440578\pi\)
0.185596 + 0.982626i \(0.440578\pi\)
\(744\) −2.37731e9 −0.211631
\(745\) 6.79371e9 0.601950
\(746\) 7.35489e8 0.0648620
\(747\) −9.55643e9 −0.838829
\(748\) −2.86511e9 −0.250314
\(749\) 2.87171e9 0.249721
\(750\) 6.62006e7 0.00572991
\(751\) −6.37970e9 −0.549618 −0.274809 0.961499i \(-0.588615\pi\)
−0.274809 + 0.961499i \(0.588615\pi\)
\(752\) 8.02346e9 0.688017
\(753\) −5.97635e8 −0.0510098
\(754\) −7.90343e8 −0.0671453
\(755\) −2.79072e9 −0.235995
\(756\) −4.03304e9 −0.339474
\(757\) −1.19658e10 −1.00255 −0.501274 0.865289i \(-0.667135\pi\)
−0.501274 + 0.865289i \(0.667135\pi\)
\(758\) 1.07032e9 0.0892631
\(759\) 2.27296e9 0.188688
\(760\) 1.67448e9 0.138367
\(761\) −2.00959e10 −1.65296 −0.826479 0.562967i \(-0.809660\pi\)
−0.826479 + 0.562967i \(0.809660\pi\)
\(762\) 7.58437e8 0.0620979
\(763\) 6.72100e9 0.547770
\(764\) 1.92450e10 1.56131
\(765\) 3.11616e9 0.251655
\(766\) 1.12411e9 0.0903665
\(767\) 1.11187e10 0.889756
\(768\) 5.70667e9 0.454588
\(769\) 2.46683e10 1.95613 0.978064 0.208304i \(-0.0667944\pi\)
0.978064 + 0.208304i \(0.0667944\pi\)
\(770\) 8.39358e7 0.00662567
\(771\) 2.41807e9 0.190011
\(772\) 1.75673e10 1.37418
\(773\) 8.88824e9 0.692130 0.346065 0.938211i \(-0.387518\pi\)
0.346065 + 0.938211i \(0.387518\pi\)
\(774\) −1.36821e9 −0.106062
\(775\) 4.31235e9 0.332781
\(776\) −2.30798e9 −0.177303
\(777\) −1.33149e9 −0.101828
\(778\) −1.46252e9 −0.111345
\(779\) 1.17144e10 0.887850
\(780\) 2.39930e9 0.181031
\(781\) −1.03909e9 −0.0780502
\(782\) 1.38690e9 0.103710
\(783\) −8.78483e9 −0.653984
\(784\) 1.84358e9 0.136633
\(785\) −5.47235e9 −0.403766
\(786\) −1.53655e8 −0.0112867
\(787\) −4.65006e9 −0.340053 −0.170027 0.985439i \(-0.554385\pi\)
−0.170027 + 0.985439i \(0.554385\pi\)
\(788\) −8.22986e9 −0.599171
\(789\) 7.40606e9 0.536806
\(790\) −8.90993e8 −0.0642953
\(791\) −4.67542e9 −0.335895
\(792\) −7.82079e8 −0.0559387
\(793\) 6.02598e9 0.429113
\(794\) −1.31830e9 −0.0934638
\(795\) −6.23859e8 −0.0440354
\(796\) −2.44070e8 −0.0171521
\(797\) 1.42890e10 0.999762 0.499881 0.866094i \(-0.333377\pi\)
0.499881 + 0.866094i \(0.333377\pi\)
\(798\) 4.48381e8 0.0312347
\(799\) −8.11987e9 −0.563165
\(800\) 1.02931e9 0.0710776
\(801\) 1.66846e10 1.14711
\(802\) −2.66257e9 −0.182260
\(803\) −3.11685e9 −0.212428
\(804\) 1.39302e10 0.945279
\(805\) 2.74346e9 0.185359
\(806\) −2.31467e9 −0.155710
\(807\) −9.68971e9 −0.649013
\(808\) −3.73494e9 −0.249083
\(809\) −4.92320e9 −0.326909 −0.163455 0.986551i \(-0.552264\pi\)
−0.163455 + 0.986551i \(0.552264\pi\)
\(810\) 1.92387e8 0.0127197
\(811\) 2.35801e10 1.55229 0.776145 0.630555i \(-0.217173\pi\)
0.776145 + 0.630555i \(0.217173\pi\)
\(812\) 4.07698e9 0.267235
\(813\) 5.25992e9 0.343291
\(814\) −3.06440e8 −0.0199141
\(815\) 5.06813e9 0.327942
\(816\) −6.16280e9 −0.397066
\(817\) −2.45605e10 −1.57565
\(818\) 1.18069e9 0.0754221
\(819\) −3.30864e9 −0.210453
\(820\) 4.78889e9 0.303310
\(821\) −2.86630e10 −1.80768 −0.903838 0.427875i \(-0.859262\pi\)
−0.903838 + 0.427875i \(0.859262\pi\)
\(822\) −1.72069e9 −0.108057
\(823\) −2.76897e10 −1.73148 −0.865742 0.500490i \(-0.833153\pi\)
−0.865742 + 0.500490i \(0.833153\pi\)
\(824\) 3.67778e9 0.229003
\(825\) −5.55030e8 −0.0344134
\(826\) 8.49437e8 0.0524447
\(827\) −1.27176e10 −0.781873 −0.390936 0.920418i \(-0.627849\pi\)
−0.390936 + 0.920418i \(0.627849\pi\)
\(828\) −1.26873e10 −0.776716
\(829\) −1.50770e10 −0.919127 −0.459563 0.888145i \(-0.651994\pi\)
−0.459563 + 0.888145i \(0.651994\pi\)
\(830\) 1.03860e9 0.0630483
\(831\) 4.63064e9 0.279923
\(832\) 1.17556e10 0.707641
\(833\) −1.86573e9 −0.111839
\(834\) 3.56514e8 0.0212812
\(835\) 1.21682e10 0.723307
\(836\) −6.96787e9 −0.412457
\(837\) −2.57281e10 −1.51659
\(838\) 3.02512e9 0.177578
\(839\) 4.59511e9 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(840\) 3.69314e8 0.0214990
\(841\) −8.36934e9 −0.485182
\(842\) −3.96175e9 −0.228715
\(843\) −1.83216e10 −1.05334
\(844\) 6.53172e9 0.373963
\(845\) −3.13679e9 −0.178849
\(846\) −1.10008e9 −0.0624639
\(847\) 5.98038e9 0.338171
\(848\) −3.15361e9 −0.177591
\(849\) −7.73504e9 −0.433796
\(850\) −3.38665e8 −0.0189149
\(851\) −1.00161e10 −0.557114
\(852\) −2.26917e9 −0.125698
\(853\) −1.13971e9 −0.0628740 −0.0314370 0.999506i \(-0.510008\pi\)
−0.0314370 + 0.999506i \(0.510008\pi\)
\(854\) 4.60367e8 0.0252931
\(855\) 7.57844e9 0.414666
\(856\) 2.90801e9 0.158466
\(857\) 7.79419e9 0.422998 0.211499 0.977378i \(-0.432166\pi\)
0.211499 + 0.977378i \(0.432166\pi\)
\(858\) 2.97915e8 0.0161023
\(859\) 1.27280e10 0.685147 0.342573 0.939491i \(-0.388701\pi\)
0.342573 + 0.939491i \(0.388701\pi\)
\(860\) −1.00404e10 −0.538277
\(861\) 2.58367e9 0.137951
\(862\) −3.30831e9 −0.175926
\(863\) 2.53204e9 0.134101 0.0670507 0.997750i \(-0.478641\pi\)
0.0670507 + 0.997750i \(0.478641\pi\)
\(864\) −6.14103e9 −0.323924
\(865\) −6.38258e9 −0.335305
\(866\) −3.09823e9 −0.162107
\(867\) −3.93933e9 −0.205284
\(868\) 1.19402e10 0.619718
\(869\) 7.47014e9 0.386153
\(870\) 3.99265e8 0.0205563
\(871\) 2.73272e10 1.40130
\(872\) 6.80595e9 0.347601
\(873\) −1.04456e10 −0.531352
\(874\) 3.37291e9 0.170889
\(875\) −6.69922e8 −0.0338062
\(876\) −6.80661e9 −0.342110
\(877\) 5.00988e9 0.250800 0.125400 0.992106i \(-0.459979\pi\)
0.125400 + 0.992106i \(0.459979\pi\)
\(878\) 2.71862e9 0.135556
\(879\) −1.25353e10 −0.622550
\(880\) −2.80568e9 −0.138787
\(881\) 9.46900e9 0.466539 0.233270 0.972412i \(-0.425058\pi\)
0.233270 + 0.972412i \(0.425058\pi\)
\(882\) −2.52770e8 −0.0124047
\(883\) 1.11146e10 0.543289 0.271644 0.962398i \(-0.412433\pi\)
0.271644 + 0.962398i \(0.412433\pi\)
\(884\) −1.22742e10 −0.597600
\(885\) −5.61696e9 −0.272395
\(886\) −1.20006e9 −0.0579675
\(887\) −7.27986e9 −0.350260 −0.175130 0.984545i \(-0.556035\pi\)
−0.175130 + 0.984545i \(0.556035\pi\)
\(888\) −1.34832e9 −0.0646173
\(889\) −7.67506e9 −0.366375
\(890\) −1.81329e9 −0.0862190
\(891\) −1.61298e9 −0.0763938
\(892\) 1.57746e10 0.744188
\(893\) −1.97473e10 −0.927959
\(894\) 1.84217e9 0.0862280
\(895\) 1.82264e10 0.849807
\(896\) 3.79032e9 0.176034
\(897\) 9.73743e9 0.450475
\(898\) 2.09267e8 0.00964348
\(899\) 2.60084e10 1.19386
\(900\) 3.09809e9 0.141659
\(901\) 3.19150e9 0.145365
\(902\) 5.94624e8 0.0269786
\(903\) −5.41692e9 −0.244819
\(904\) −4.73451e9 −0.213150
\(905\) −7.62070e9 −0.341763
\(906\) −7.56726e8 −0.0338057
\(907\) −1.39503e10 −0.620809 −0.310405 0.950605i \(-0.600465\pi\)
−0.310405 + 0.950605i \(0.600465\pi\)
\(908\) −2.42310e9 −0.107416
\(909\) −1.69038e10 −0.746465
\(910\) 3.59584e8 0.0158181
\(911\) 2.98148e8 0.0130653 0.00653263 0.999979i \(-0.497921\pi\)
0.00653263 + 0.999979i \(0.497921\pi\)
\(912\) −1.49878e10 −0.654268
\(913\) −8.70765e9 −0.378663
\(914\) −3.27506e9 −0.141876
\(915\) −3.04420e9 −0.131371
\(916\) 1.33259e10 0.572876
\(917\) 1.55492e9 0.0665910
\(918\) 2.02053e9 0.0862015
\(919\) 2.67202e10 1.13563 0.567814 0.823157i \(-0.307789\pi\)
0.567814 + 0.823157i \(0.307789\pi\)
\(920\) 2.77813e9 0.117624
\(921\) 1.15928e10 0.488969
\(922\) 2.21295e9 0.0929850
\(923\) −4.45149e9 −0.186337
\(924\) −1.53680e9 −0.0640861
\(925\) 2.44581e9 0.101608
\(926\) 1.58725e9 0.0656910
\(927\) 1.66451e10 0.686288
\(928\) 6.20794e9 0.254994
\(929\) −3.66336e10 −1.49908 −0.749540 0.661959i \(-0.769725\pi\)
−0.749540 + 0.661959i \(0.769725\pi\)
\(930\) 1.16933e9 0.0476701
\(931\) −4.53742e9 −0.184283
\(932\) 2.92180e10 1.18221
\(933\) 2.87728e8 0.0115983
\(934\) 3.88115e9 0.155864
\(935\) 2.83939e9 0.113602
\(936\) −3.35045e9 −0.133548
\(937\) 1.28088e10 0.508649 0.254325 0.967119i \(-0.418147\pi\)
0.254325 + 0.967119i \(0.418147\pi\)
\(938\) 2.08772e9 0.0825965
\(939\) 2.04149e10 0.804669
\(940\) −8.07278e9 −0.317012
\(941\) −1.20663e10 −0.472073 −0.236037 0.971744i \(-0.575849\pi\)
−0.236037 + 0.971744i \(0.575849\pi\)
\(942\) −1.48387e9 −0.0578386
\(943\) 1.94354e10 0.754751
\(944\) −2.83937e10 −1.09855
\(945\) 3.99685e9 0.154066
\(946\) −1.24669e9 −0.0478784
\(947\) 8.36023e9 0.319885 0.159942 0.987126i \(-0.448869\pi\)
0.159942 + 0.987126i \(0.448869\pi\)
\(948\) 1.63133e10 0.621890
\(949\) −1.33527e10 −0.507151
\(950\) −8.23627e8 −0.0311672
\(951\) 9.65082e9 0.363859
\(952\) −1.88931e9 −0.0709700
\(953\) 4.49530e10 1.68242 0.841209 0.540710i \(-0.181844\pi\)
0.841209 + 0.540710i \(0.181844\pi\)
\(954\) 4.32386e8 0.0161232
\(955\) −1.90722e10 −0.708582
\(956\) −1.37703e10 −0.509732
\(957\) −3.34747e9 −0.123459
\(958\) 3.26341e9 0.119920
\(959\) 1.74126e10 0.637528
\(960\) −5.93869e9 −0.216641
\(961\) 4.86580e10 1.76857
\(962\) −1.31280e9 −0.0475429
\(963\) 1.31612e10 0.474901
\(964\) 1.04094e10 0.374244
\(965\) −1.74096e10 −0.623654
\(966\) 7.43911e8 0.0265522
\(967\) 1.34247e8 0.00477432 0.00238716 0.999997i \(-0.499240\pi\)
0.00238716 + 0.999997i \(0.499240\pi\)
\(968\) 6.05596e9 0.214595
\(969\) 1.51679e10 0.535541
\(970\) 1.13523e9 0.0399376
\(971\) −3.00377e10 −1.05293 −0.526465 0.850197i \(-0.676483\pi\)
−0.526465 + 0.850197i \(0.676483\pi\)
\(972\) −2.92375e10 −1.02119
\(973\) −3.60777e9 −0.125558
\(974\) −3.01379e9 −0.104510
\(975\) −2.37777e9 −0.0821587
\(976\) −1.53884e10 −0.529810
\(977\) 4.52860e10 1.55358 0.776789 0.629761i \(-0.216847\pi\)
0.776789 + 0.629761i \(0.216847\pi\)
\(978\) 1.37426e9 0.0469769
\(979\) 1.52028e10 0.517825
\(980\) −1.85491e9 −0.0629553
\(981\) 3.08027e10 1.04171
\(982\) 5.85057e8 0.0197155
\(983\) 4.61443e10 1.54946 0.774731 0.632290i \(-0.217885\pi\)
0.774731 + 0.632290i \(0.217885\pi\)
\(984\) 2.61632e9 0.0875404
\(985\) 8.15600e9 0.271926
\(986\) −2.04254e9 −0.0678580
\(987\) −4.35537e9 −0.144183
\(988\) −2.98506e10 −0.984700
\(989\) −4.07484e10 −1.33944
\(990\) 3.84682e8 0.0126002
\(991\) −1.05400e10 −0.344018 −0.172009 0.985095i \(-0.555026\pi\)
−0.172009 + 0.985095i \(0.555026\pi\)
\(992\) 1.81812e10 0.591331
\(993\) 3.68475e9 0.119422
\(994\) −3.40081e8 −0.0109832
\(995\) 2.41879e8 0.00778427
\(996\) −1.90158e10 −0.609828
\(997\) −5.00734e10 −1.60020 −0.800099 0.599868i \(-0.795220\pi\)
−0.800099 + 0.599868i \(0.795220\pi\)
\(998\) −4.03435e9 −0.128474
\(999\) −1.45920e10 −0.463059
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 35.8.a.a.1.1 2
3.2 odd 2 315.8.a.c.1.2 2
4.3 odd 2 560.8.a.i.1.1 2
5.2 odd 4 175.8.b.c.99.3 4
5.3 odd 4 175.8.b.c.99.2 4
5.4 even 2 175.8.a.b.1.2 2
7.6 odd 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 1.1 even 1 trivial
175.8.a.b.1.2 2 5.4 even 2
175.8.b.c.99.2 4 5.3 odd 4
175.8.b.c.99.3 4 5.2 odd 4
245.8.a.b.1.1 2 7.6 odd 2
315.8.a.c.1.2 2 3.2 odd 2
560.8.a.i.1.1 2 4.3 odd 2