Properties

Label 175.8.b.c.99.2
Level $175$
Weight $8$
Character 175.99
Analytic conductor $54.667$
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] = [175,8,Mod(99,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(54.6673794597\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.8.b.c.99.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.36675i q^{2} +24.7995i q^{3} +126.132 q^{4} +33.8947 q^{6} +343.000i q^{7} -347.335i q^{8} +1571.98 q^{9} +O(q^{10})\) \(q-1.36675i q^{2} +24.7995i q^{3} +126.132 q^{4} +33.8947 q^{6} +343.000i q^{7} -347.335i q^{8} +1571.98 q^{9} -1432.37 q^{11} +3128.01i q^{12} -6136.30i q^{13} +468.795 q^{14} +15670.2 q^{16} +15858.5i q^{17} -2148.51i q^{18} +38567.5 q^{19} -8506.23 q^{21} +1957.69i q^{22} -63987.4i q^{23} +8613.73 q^{24} -8386.79 q^{26} +93220.9i q^{27} +43263.3i q^{28} -94236.6 q^{29} +275990. q^{31} -65876.1i q^{32} -35521.9i q^{33} +21674.6 q^{34} +198278. q^{36} -156532. i q^{37} -52712.1i q^{38} +152177. q^{39} -303738. q^{41} +11625.9i q^{42} +636818. i q^{43} -180667. q^{44} -87454.9 q^{46} -512021. i q^{47} +388612. i q^{48} -117649. q^{49} -393282. q^{51} -773984. i q^{52} -201249. i q^{53} +127410. q^{54} +119136. q^{56} +956454. i q^{57} +128798. i q^{58} +1.81196e6 q^{59} -982021. q^{61} -377210. i q^{62} +539191. i q^{63} +1.91575e6 q^{64} -48549.6 q^{66} +4.45336e6i q^{67} +2.00026e6i q^{68} +1.58686e6 q^{69} +725436. q^{71} -546005. i q^{72} +2.17602e6i q^{73} -213940. q^{74} +4.86459e6 q^{76} -491301. i q^{77} -207988. i q^{78} +5.21525e6 q^{79} +1.12610e6 q^{81} +415135. i q^{82} +6.07921e6i q^{83} -1.07291e6 q^{84} +870371. q^{86} -2.33702e6i q^{87} +497511. i q^{88} +1.06137e7 q^{89} +2.10475e6 q^{91} -8.07086e6i q^{92} +6.84442e6i q^{93} -699805. q^{94} +1.63369e6 q^{96} -6.64483e6i q^{97} +160797. i q^{98} -2.25166e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9} - 15812 q^{11} + 10976 q^{14} - 8640 q^{16} + 7224 q^{19} + 20580 q^{21} - 49920 q^{24} - 358656 q^{26} + 253796 q^{29} + 505536 q^{31} - 350592 q^{34} + 537120 q^{36} - 975948 q^{39} - 223840 q^{41} + 753840 q^{44} + 2102944 q^{46} - 470596 q^{49} - 2261820 q^{51} - 1944000 q^{54} + 658560 q^{56} + 2720240 q^{59} - 3627360 q^{61} + 4979968 q^{64} + 10285248 q^{66} + 11703960 q^{69} - 2989856 q^{71} + 11934048 q^{74} + 15750752 q^{76} + 15885948 q^{79} - 9551196 q^{81} - 5383728 q^{84} + 10835424 q^{86} + 35887056 q^{89} + 12223148 q^{91} + 5646208 q^{94} + 26732544 q^{96} + 6061464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.36675i − 0.120805i −0.998174 0.0604024i \(-0.980762\pi\)
0.998174 0.0604024i \(-0.0192384\pi\)
\(3\) 24.7995i 0.530296i 0.964208 + 0.265148i \(0.0854208\pi\)
−0.964208 + 0.265148i \(0.914579\pi\)
\(4\) 126.132 0.985406
\(5\) 0 0
\(6\) 33.8947 0.0640623
\(7\) 343.000i 0.377964i
\(8\) − 347.335i − 0.239847i
\(9\) 1571.98 0.718786
\(10\) 0 0
\(11\) −1432.37 −0.324474 −0.162237 0.986752i \(-0.551871\pi\)
−0.162237 + 0.986752i \(0.551871\pi\)
\(12\) 3128.01i 0.522557i
\(13\) − 6136.30i − 0.774649i −0.921943 0.387325i \(-0.873399\pi\)
0.921943 0.387325i \(-0.126601\pi\)
\(14\) 468.795 0.0456599
\(15\) 0 0
\(16\) 15670.2 0.956432
\(17\) 15858.5i 0.782871i 0.920205 + 0.391436i \(0.128021\pi\)
−0.920205 + 0.391436i \(0.871979\pi\)
\(18\) − 2148.51i − 0.0868328i
\(19\) 38567.5 1.28998 0.644991 0.764190i \(-0.276861\pi\)
0.644991 + 0.764190i \(0.276861\pi\)
\(20\) 0 0
\(21\) −8506.23 −0.200433
\(22\) 1957.69i 0.0391980i
\(23\) − 63987.4i − 1.09660i −0.836282 0.548299i \(-0.815276\pi\)
0.836282 0.548299i \(-0.184724\pi\)
\(24\) 8613.73 0.127190
\(25\) 0 0
\(26\) −8386.79 −0.0935813
\(27\) 93220.9i 0.911466i
\(28\) 43263.3i 0.372449i
\(29\) −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 \(0.187224\pi\)
\(32\) − 65876.1i − 0.355388i
\(33\) − 35521.9i − 0.172067i
\(34\) 21674.6 0.0945746
\(35\) 0 0
\(36\) 198278. 0.708296
\(37\) − 156532.i − 0.508038i −0.967199 0.254019i \(-0.918247\pi\)
0.967199 0.254019i \(-0.0817526\pi\)
\(38\) − 52712.1i − 0.155836i
\(39\) 152177. 0.410793
\(40\) 0 0
\(41\) −303738. −0.688266 −0.344133 0.938921i \(-0.611827\pi\)
−0.344133 + 0.938921i \(0.611827\pi\)
\(42\) 11625.9i 0.0242133i
\(43\) 636818.i 1.22145i 0.791843 + 0.610725i \(0.209122\pi\)
−0.791843 + 0.610725i \(0.790878\pi\)
\(44\) −180667. −0.319738
\(45\) 0 0
\(46\) −87454.9 −0.132474
\(47\) − 512021.i − 0.719358i −0.933076 0.359679i \(-0.882886\pi\)
0.933076 0.359679i \(-0.117114\pi\)
\(48\) 388612.i 0.507192i
\(49\) −117649. −0.142857
\(50\) 0 0
\(51\) −393282. −0.415154
\(52\) − 773984.i − 0.763344i
\(53\) − 201249.i − 0.185681i −0.995681 0.0928406i \(-0.970405\pi\)
0.995681 0.0928406i \(-0.0295947\pi\)
\(54\) 127410. 0.110109
\(55\) 0 0
\(56\) 119136. 0.0906535
\(57\) 956454.i 0.684072i
\(58\) 128798.i 0.0866784i
\(59\) 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.i − 0.201007i
\(63\) 539191.i 0.271676i
\(64\) 1.91575e6 0.913499
\(65\) 0 0
\(66\) −48549.6 −0.0207865
\(67\) 4.45336e6i 1.80895i 0.426528 + 0.904474i \(0.359736\pi\)
−0.426528 + 0.904474i \(0.640264\pi\)
\(68\) 2.00026e6i 0.771446i
\(69\) 1.58686e6 0.581522
\(70\) 0 0
\(71\) 725436. 0.240544 0.120272 0.992741i \(-0.461623\pi\)
0.120272 + 0.992741i \(0.461623\pi\)
\(72\) − 546005.i − 0.172398i
\(73\) 2.17602e6i 0.654685i 0.944906 + 0.327343i \(0.106153\pi\)
−0.944906 + 0.327343i \(0.893847\pi\)
\(74\) −213940. −0.0613735
\(75\) 0 0
\(76\) 4.86459e6 1.27116
\(77\) − 491301.i − 0.122639i
\(78\) − 207988.i − 0.0496258i
\(79\) 5.21525e6 1.19009 0.595045 0.803692i \(-0.297134\pi\)
0.595045 + 0.803692i \(0.297134\pi\)
\(80\) 0 0
\(81\) 1.12610e6 0.235439
\(82\) 415135.i 0.0831458i
\(83\) 6.07921e6i 1.16701i 0.812110 + 0.583504i \(0.198319\pi\)
−0.812110 + 0.583504i \(0.801681\pi\)
\(84\) −1.07291e6 −0.197508
\(85\) 0 0
\(86\) 870371. 0.147557
\(87\) − 2.33702e6i − 0.380492i
\(88\) 497511.i 0.0778239i
\(89\) 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
\(92\) − 8.07086e6i − 1.08059i
\(93\) 6.84442e6i 0.882361i
\(94\) −699805. −0.0869019
\(95\) 0 0
\(96\) 1.63369e6 0.188461
\(97\) − 6.64483e6i − 0.739236i −0.929184 0.369618i \(-0.879489\pi\)
0.929184 0.369618i \(-0.120511\pi\)
\(98\) 160797.i 0.0172578i
\(99\) −2.25166e6 −0.233227
\(100\) 0 0
\(101\) 1.07531e7 1.03851 0.519254 0.854620i \(-0.326210\pi\)
0.519254 + 0.854620i \(0.326210\pi\)
\(102\) 537519.i 0.0501526i
\(103\) − 1.05886e7i − 0.954788i −0.878689 0.477394i \(-0.841581\pi\)
0.878689 0.477394i \(-0.158419\pi\)
\(104\) −2.13135e6 −0.185797
\(105\) 0 0
\(106\) −275057. −0.0224312
\(107\) 8.37234e6i 0.660699i 0.943859 + 0.330349i \(0.107167\pi\)
−0.943859 + 0.330349i \(0.892833\pi\)
\(108\) 1.17581e7i 0.898164i
\(109\) 1.95948e7 1.44926 0.724632 0.689136i \(-0.242010\pi\)
0.724632 + 0.689136i \(0.242010\pi\)
\(110\) 0 0
\(111\) 3.88191e6 0.269411
\(112\) 5.37487e6i 0.361497i
\(113\) 1.36310e7i 0.888694i 0.895855 + 0.444347i \(0.146564\pi\)
−0.895855 + 0.444347i \(0.853436\pi\)
\(114\) 1.30723e6 0.0826392
\(115\) 0 0
\(116\) −1.18863e7 −0.707037
\(117\) − 9.64617e6i − 0.556807i
\(118\) − 2.47649e6i − 0.138756i
\(119\) −5.43946e6 −0.295898
\(120\) 0 0
\(121\) −1.74355e7 −0.894717
\(122\) 1.34218e6i 0.0669192i
\(123\) − 7.53256e6i − 0.364985i
\(124\) 3.48112e7 1.63962
\(125\) 0 0
\(126\) 736939. 0.0328197
\(127\) − 2.23763e7i − 0.969336i −0.874698 0.484668i \(-0.838940\pi\)
0.874698 0.484668i \(-0.161060\pi\)
\(128\) − 1.10505e7i − 0.465743i
\(129\) −1.57928e7 −0.647730
\(130\) 0 0
\(131\) −4.53330e6 −0.176183 −0.0880917 0.996112i \(-0.528077\pi\)
−0.0880917 + 0.996112i \(0.528077\pi\)
\(132\) − 4.48045e6i − 0.169556i
\(133\) 1.32286e7i 0.487567i
\(134\) 6.08663e6 0.218530
\(135\) 0 0
\(136\) 5.50821e6 0.187769
\(137\) 5.07657e7i 1.68674i 0.537332 + 0.843371i \(0.319432\pi\)
−0.537332 + 0.843371i \(0.680568\pi\)
\(138\) − 2.16884e6i − 0.0702506i
\(139\) −1.05183e7 −0.332195 −0.166097 0.986109i \(-0.553117\pi\)
−0.166097 + 0.986109i \(0.553117\pi\)
\(140\) 0 0
\(141\) 1.26979e7 0.381473
\(142\) − 991490.i − 0.0290589i
\(143\) 8.78942e6i 0.251353i
\(144\) 2.46333e7 0.687470
\(145\) 0 0
\(146\) 2.97407e6 0.0790891
\(147\) − 2.91764e6i − 0.0757566i
\(148\) − 1.97437e7i − 0.500624i
\(149\) −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 \(0.584992\pi\)
\(152\) − 1.33958e7i − 0.309398i
\(153\) 2.49293e7i 0.562717i
\(154\) −671486. −0.0148154
\(155\) 0 0
\(156\) 1.91944e7 0.404798
\(157\) 4.37788e7i 0.902848i 0.892310 + 0.451424i \(0.149084\pi\)
−0.892310 + 0.451424i \(0.850916\pi\)
\(158\) − 7.12794e6i − 0.143769i
\(159\) 4.99087e6 0.0984661
\(160\) 0 0
\(161\) 2.19477e7 0.414475
\(162\) − 1.53910e6i − 0.0284422i
\(163\) 4.05451e7i 0.733300i 0.930359 + 0.366650i \(0.119495\pi\)
−0.930359 + 0.366650i \(0.880505\pi\)
\(164\) −3.83111e7 −0.678221
\(165\) 0 0
\(166\) 8.30876e6 0.140980
\(167\) − 9.73453e7i − 1.61736i −0.588247 0.808682i \(-0.700182\pi\)
0.588247 0.808682i \(-0.299818\pi\)
\(168\) 2.95451e6i 0.0480732i
\(169\) 2.50943e7 0.399919
\(170\) 0 0
\(171\) 6.06275e7 0.927221
\(172\) 8.03231e7i 1.20362i
\(173\) − 5.10607e7i − 0.749765i −0.927072 0.374882i \(-0.877683\pi\)
0.927072 0.374882i \(-0.122317\pi\)
\(174\) −3.19412e6 −0.0459652
\(175\) 0 0
\(176\) −2.24454e7 −0.310337
\(177\) 4.49356e7i 0.609094i
\(178\) − 1.45063e7i − 0.192791i
\(179\) −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 \(0.624800\pi\)
\(182\) − 2.87667e6i − 0.0353704i
\(183\) − 2.43536e7i − 0.293755i
\(184\) −2.22251e7 −0.263015
\(185\) 0 0
\(186\) 9.35462e6 0.106593
\(187\) − 2.27151e7i − 0.254021i
\(188\) − 6.45822e7i − 0.708860i
\(189\) −3.19748e7 −0.344502
\(190\) 0 0
\(191\) −1.52578e8 −1.58444 −0.792219 0.610237i \(-0.791074\pi\)
−0.792219 + 0.610237i \(0.791074\pi\)
\(192\) 4.75095e7i 0.484425i
\(193\) − 1.39277e8i − 1.39453i −0.716812 0.697267i \(-0.754399\pi\)
0.716812 0.697267i \(-0.245601\pi\)
\(194\) −9.08183e6 −0.0893033
\(195\) 0 0
\(196\) −1.48393e7 −0.140772
\(197\) − 6.52480e7i − 0.608044i −0.952665 0.304022i \(-0.901670\pi\)
0.952665 0.304022i \(-0.0983297\pi\)
\(198\) 3.07745e6i 0.0281750i
\(199\) −1.93503e6 −0.0174061 −0.00870307 0.999962i \(-0.502770\pi\)
−0.00870307 + 0.999962i \(0.502770\pi\)
\(200\) 0 0
\(201\) −1.10441e8 −0.959278
\(202\) − 1.46968e7i − 0.125457i
\(203\) − 3.23232e7i − 0.271192i
\(204\) −4.96055e7 −0.409095
\(205\) 0 0
\(206\) −1.44719e7 −0.115343
\(207\) − 1.00587e8i − 0.788219i
\(208\) − 9.61569e7i − 0.740899i
\(209\) −5.52427e7 −0.418565
\(210\) 0 0
\(211\) −5.17848e7 −0.379502 −0.189751 0.981832i \(-0.560768\pi\)
−0.189751 + 0.981832i \(0.560768\pi\)
\(212\) − 2.53839e7i − 0.182971i
\(213\) 1.79904e7i 0.127560i
\(214\) 1.14429e7 0.0798156
\(215\) 0 0
\(216\) 3.23789e7 0.218612
\(217\) 9.46647e7i 0.628896i
\(218\) − 2.67812e7i − 0.175078i
\(219\) −5.39642e7 −0.347177
\(220\) 0 0
\(221\) 9.73124e7 0.606450
\(222\) − 5.30560e6i − 0.0325461i
\(223\) − 1.25065e8i − 0.755209i −0.925967 0.377605i \(-0.876748\pi\)
0.925967 0.377605i \(-0.123252\pi\)
\(224\) 2.25955e7 0.134324
\(225\) 0 0
\(226\) 1.86301e7 0.107358
\(227\) − 1.92108e7i − 0.109007i −0.998514 0.0545036i \(-0.982642\pi\)
0.998514 0.0545036i \(-0.0173576\pi\)
\(228\) 1.20639e8i 0.674089i
\(229\) 1.05650e8 0.581360 0.290680 0.956820i \(-0.406118\pi\)
0.290680 + 0.956820i \(0.406118\pi\)
\(230\) 0 0
\(231\) 1.21840e7 0.0650353
\(232\) 3.27317e7i 0.172092i
\(233\) − 2.31646e8i − 1.19972i −0.800106 0.599859i \(-0.795224\pi\)
0.800106 0.599859i \(-0.204776\pi\)
\(234\) −1.31839e7 −0.0672649
\(235\) 0 0
\(236\) 2.28546e8 1.13183
\(237\) 1.29336e8i 0.631101i
\(238\) 7.43438e6i 0.0357458i
\(239\) −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 \(0.560814\pi\)
\(242\) 2.38300e7i 0.108086i
\(243\) 2.31801e8i 1.03632i
\(244\) −1.23864e8 −0.545861
\(245\) 0 0
\(246\) −1.02951e7 −0.0440919
\(247\) − 2.36662e8i − 0.999283i
\(248\) − 9.58611e7i − 0.399081i
\(249\) −1.50761e8 −0.618860
\(250\) 0 0
\(251\) −2.40987e7 −0.0961912 −0.0480956 0.998843i \(-0.515315\pi\)
−0.0480956 + 0.998843i \(0.515315\pi\)
\(252\) 6.80092e7i 0.267711i
\(253\) 9.16534e7i 0.355817i
\(254\) −3.05828e7 −0.117100
\(255\) 0 0
\(256\) 2.30112e8 0.857235
\(257\) − 9.75049e7i − 0.358311i −0.983821 0.179156i \(-0.942663\pi\)
0.983821 0.179156i \(-0.0573366\pi\)
\(258\) 2.15848e7i 0.0782489i
\(259\) 5.36904e7 0.192020
\(260\) 0 0
\(261\) −1.48139e8 −0.515735
\(262\) 6.19589e6i 0.0212838i
\(263\) 2.98637e8i 1.01228i 0.862452 + 0.506138i \(0.168927\pi\)
−0.862452 + 0.506138i \(0.831073\pi\)
\(264\) −1.23380e7 −0.0412697
\(265\) 0 0
\(266\) 1.80803e7 0.0589005
\(267\) 2.63216e8i 0.846296i
\(268\) 5.61711e8i 1.78255i
\(269\) 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 \(0.395080\pi\)
\(272\) 2.48505e8i 0.748763i
\(273\) 5.21968e7i 0.155265i
\(274\) 6.93841e7 0.203767
\(275\) 0 0
\(276\) 2.00153e8 0.573035
\(277\) − 1.86723e8i − 0.527861i −0.964542 0.263930i \(-0.914981\pi\)
0.964542 0.263930i \(-0.0850189\pi\)
\(278\) 1.43759e7i 0.0401307i
\(279\) 4.33853e8 1.19599
\(280\) 0 0
\(281\) −7.38791e8 −1.98632 −0.993161 0.116756i \(-0.962750\pi\)
−0.993161 + 0.116756i \(0.962750\pi\)
\(282\) − 1.73548e7i − 0.0460838i
\(283\) − 3.11903e8i − 0.818026i −0.912529 0.409013i \(-0.865873\pi\)
0.912529 0.409013i \(-0.134127\pi\)
\(284\) 9.15007e7 0.237034
\(285\) 0 0
\(286\) 1.20129e7 0.0303647
\(287\) − 1.04182e8i − 0.260140i
\(288\) − 1.03556e8i − 0.255448i
\(289\) 1.58847e8 0.387113
\(290\) 0 0
\(291\) 1.64789e8 0.392014
\(292\) 2.74466e8i 0.645131i
\(293\) − 5.05466e8i − 1.17397i −0.809599 0.586983i \(-0.800316\pi\)
0.809599 0.586983i \(-0.199684\pi\)
\(294\) −3.98768e6 −0.00915176
\(295\) 0 0
\(296\) −5.43690e7 −0.121851
\(297\) − 1.33526e8i − 0.295747i
\(298\) 7.42825e7i 0.162603i
\(299\) −3.92646e8 −0.849478
\(300\) 0 0
\(301\) −2.18429e8 −0.461665
\(302\) 3.05137e7i 0.0637487i
\(303\) 2.66672e8i 0.550717i
\(304\) 6.04359e8 1.23378
\(305\) 0 0
\(306\) 3.40721e7 0.0679789
\(307\) − 4.67463e8i − 0.922067i −0.887383 0.461034i \(-0.847479\pi\)
0.887383 0.461034i \(-0.152521\pi\)
\(308\) − 6.19688e7i − 0.120850i
\(309\) 2.62591e8 0.506321
\(310\) 0 0
\(311\) 1.16022e7 0.0218714 0.0109357 0.999940i \(-0.496519\pi\)
0.0109357 + 0.999940i \(0.496519\pi\)
\(312\) − 5.28565e7i − 0.0985274i
\(313\) 8.23197e8i 1.51740i 0.651443 + 0.758698i \(0.274164\pi\)
−0.651443 + 0.758698i \(0.725836\pi\)
\(314\) 5.98346e7 0.109068
\(315\) 0 0
\(316\) 6.57810e8 1.17272
\(317\) − 3.89154e8i − 0.686142i −0.939309 0.343071i \(-0.888533\pi\)
0.939309 0.343071i \(-0.111467\pi\)
\(318\) − 6.82128e6i − 0.0118952i
\(319\) 1.34981e8 0.232812
\(320\) 0 0
\(321\) −2.07630e8 −0.350366
\(322\) − 2.99970e7i − 0.0500706i
\(323\) 6.11621e8i 1.00989i
\(324\) 1.42037e8 0.232003
\(325\) 0 0
\(326\) 5.54150e7 0.0885861
\(327\) 4.85940e8i 0.768539i
\(328\) 1.05499e8i 0.165078i
\(329\) 1.75623e8 0.271892
\(330\) 0 0
\(331\) 1.48582e8 0.225199 0.112600 0.993640i \(-0.464082\pi\)
0.112600 + 0.993640i \(0.464082\pi\)
\(332\) 7.66783e8i 1.14998i
\(333\) − 2.46066e8i − 0.365171i
\(334\) −1.33047e8 −0.195385
\(335\) 0 0
\(336\) −1.33294e8 −0.191701
\(337\) − 1.23379e8i − 0.175605i −0.996138 0.0878023i \(-0.972016\pi\)
0.996138 0.0878023i \(-0.0279844\pi\)
\(338\) − 3.42977e7i − 0.0483121i
\(339\) −3.38041e8 −0.471271
\(340\) 0 0
\(341\) −3.95319e8 −0.539892
\(342\) − 8.28626e7i − 0.112013i
\(343\) − 4.03536e7i − 0.0539949i
\(344\) 2.21189e8 0.292961
\(345\) 0 0
\(346\) −6.97872e7 −0.0905752
\(347\) − 1.31658e9i − 1.69159i −0.533511 0.845793i \(-0.679128\pi\)
0.533511 0.845793i \(-0.320872\pi\)
\(348\) − 2.94773e8i − 0.374939i
\(349\) −2.64521e8 −0.333097 −0.166549 0.986033i \(-0.553262\pi\)
−0.166549 + 0.986033i \(0.553262\pi\)
\(350\) 0 0
\(351\) 5.72032e8 0.706066
\(352\) 9.43586e7i 0.115314i
\(353\) 1.30271e9i 1.57629i 0.615490 + 0.788144i \(0.288958\pi\)
−0.615490 + 0.788144i \(0.711042\pi\)
\(354\) 6.14158e7 0.0735815
\(355\) 0 0
\(356\) 1.33873e9 1.57260
\(357\) − 1.34896e8i − 0.156913i
\(358\) 1.99287e8i 0.229556i
\(359\) 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
\(362\) 8.33248e7i 0.0923196i
\(363\) − 4.32392e8i − 0.474465i
\(364\) 2.65476e8 0.288517
\(365\) 0 0
\(366\) −3.32853e7 −0.0354870
\(367\) − 1.13124e9i − 1.19460i −0.802017 0.597302i \(-0.796240\pi\)
0.802017 0.597302i \(-0.203760\pi\)
\(368\) − 1.00269e9i − 1.04882i
\(369\) −4.77472e8 −0.494716
\(370\) 0 0
\(371\) 6.90284e7 0.0701809
\(372\) 8.63300e8i 0.869484i
\(373\) 5.38130e8i 0.536916i 0.963291 + 0.268458i \(0.0865140\pi\)
−0.963291 + 0.268458i \(0.913486\pi\)
\(374\) −3.10459e7 −0.0306870
\(375\) 0 0
\(376\) −1.77843e8 −0.172536
\(377\) 5.78264e8i 0.555817i
\(378\) 4.37015e7i 0.0416175i
\(379\) −7.83114e8 −0.738904 −0.369452 0.929250i \(-0.620455\pi\)
−0.369452 + 0.929250i \(0.620455\pi\)
\(380\) 0 0
\(381\) 5.54920e8 0.514035
\(382\) 2.08536e8i 0.191408i
\(383\) 8.22468e8i 0.748038i 0.927421 + 0.374019i \(0.122020\pi\)
−0.927421 + 0.374019i \(0.877980\pi\)
\(384\) 2.74047e8 0.246982
\(385\) 0 0
\(386\) −1.90357e8 −0.168466
\(387\) 1.00107e9i 0.877961i
\(388\) − 8.38126e8i − 0.728448i
\(389\) 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
\(392\) 4.08636e7i 0.0342638i
\(393\) − 1.12424e8i − 0.0934293i
\(394\) −8.91777e7 −0.0734547
\(395\) 0 0
\(396\) −2.84006e8 −0.229823
\(397\) 9.64552e8i 0.773676i 0.922148 + 0.386838i \(0.126433\pi\)
−0.922148 + 0.386838i \(0.873567\pi\)
\(398\) 2.64471e6i 0.00210275i
\(399\) −3.28064e8 −0.258555
\(400\) 0 0
\(401\) −1.94810e9 −1.50871 −0.754357 0.656465i \(-0.772051\pi\)
−0.754357 + 0.656465i \(0.772051\pi\)
\(402\) 1.50945e8i 0.115885i
\(403\) − 1.69356e9i − 1.28894i
\(404\) 1.35631e9 1.02335
\(405\) 0 0
\(406\) −4.41777e7 −0.0327614
\(407\) 2.24211e8i 0.164845i
\(408\) 1.36601e8i 0.0995732i
\(409\) −8.63865e8 −0.624330 −0.312165 0.950028i \(-0.601054\pi\)
−0.312165 + 0.950028i \(0.601054\pi\)
\(410\) 0 0
\(411\) −1.25896e9 −0.894473
\(412\) − 1.33556e9i − 0.940854i
\(413\) 6.21501e8i 0.434127i
\(414\) −1.37478e8 −0.0952206
\(415\) 0 0
\(416\) −4.04236e8 −0.275301
\(417\) − 2.60848e8i − 0.176162i
\(418\) 7.55030e7i 0.0505647i
\(419\) −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 \(0.895537\pi\)
\(422\) 7.07769e7i 0.0458456i
\(423\) − 8.04889e8i − 0.517065i
\(424\) −6.99008e7 −0.0445350
\(425\) 0 0
\(426\) 2.45884e7 0.0154098
\(427\) − 3.36833e8i − 0.209371i
\(428\) 1.05602e9i 0.651057i
\(429\) −2.17973e8 −0.133292
\(430\) 0 0
\(431\) −2.42056e9 −1.45628 −0.728142 0.685426i \(-0.759616\pi\)
−0.728142 + 0.685426i \(0.759616\pi\)
\(432\) 1.46079e9i 0.871754i
\(433\) − 2.26686e9i − 1.34189i −0.741506 0.670946i \(-0.765888\pi\)
0.741506 0.670946i \(-0.234112\pi\)
\(434\) 1.29383e8 0.0759737
\(435\) 0 0
\(436\) 2.47153e9 1.42811
\(437\) − 2.46783e9i − 1.41459i
\(438\) 7.37555e7i 0.0419407i
\(439\) −1.98911e9 −1.12210 −0.561052 0.827780i \(-0.689603\pi\)
−0.561052 + 0.827780i \(0.689603\pi\)
\(440\) 0 0
\(441\) −1.84942e8 −0.102684
\(442\) − 1.33002e8i − 0.0732621i
\(443\) − 8.78038e8i − 0.479844i −0.970792 0.239922i \(-0.922878\pi\)
0.970792 0.239922i \(-0.0771219\pi\)
\(444\) 4.89633e8 0.265479
\(445\) 0 0
\(446\) −1.70932e8 −0.0912329
\(447\) − 1.34785e9i − 0.713779i
\(448\) 6.57101e8i 0.345270i
\(449\) −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
\(452\) 1.71930e9i 0.875724i
\(453\) − 5.53668e8i − 0.279837i
\(454\) −2.62564e7 −0.0131686
\(455\) 0 0
\(456\) 3.32210e8 0.164072
\(457\) 2.39624e9i 1.17442i 0.809435 + 0.587210i \(0.199774\pi\)
−0.809435 + 0.587210i \(0.800226\pi\)
\(458\) − 1.44397e8i − 0.0702311i
\(459\) −1.47834e9 −0.713560
\(460\) 0 0
\(461\) 1.61913e9 0.769713 0.384856 0.922977i \(-0.374251\pi\)
0.384856 + 0.922977i \(0.374251\pi\)
\(462\) − 1.66525e7i − 0.00785657i
\(463\) 1.16133e9i 0.543778i 0.962329 + 0.271889i \(0.0876483\pi\)
−0.962329 + 0.271889i \(0.912352\pi\)
\(464\) −1.47670e9 −0.686247
\(465\) 0 0
\(466\) −3.16602e8 −0.144932
\(467\) − 2.83969e9i − 1.29021i −0.764092 0.645107i \(-0.776813\pi\)
0.764092 0.645107i \(-0.223187\pi\)
\(468\) − 1.21669e9i − 0.548681i
\(469\) −1.52750e9 −0.683718
\(470\) 0 0
\(471\) −1.08569e9 −0.478777
\(472\) − 6.29356e8i − 0.275486i
\(473\) − 9.12156e8i − 0.396328i
\(474\) 1.76769e8 0.0762400
\(475\) 0 0
\(476\) −6.86090e8 −0.291579
\(477\) − 3.16360e8i − 0.133465i
\(478\) 1.49214e8i 0.0624901i
\(479\) −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
\(482\) 1.12795e8i 0.0458801i
\(483\) 5.44292e8i 0.219794i
\(484\) −2.19917e9 −0.881660
\(485\) 0 0
\(486\) 3.16814e8 0.125192
\(487\) 2.20508e9i 0.865113i 0.901607 + 0.432556i \(0.142388\pi\)
−0.901607 + 0.432556i \(0.857612\pi\)
\(488\) 3.41090e8i 0.132862i
\(489\) −1.00550e9 −0.388866
\(490\) 0 0
\(491\) 4.28064e8 0.163201 0.0816006 0.996665i \(-0.473997\pi\)
0.0816006 + 0.996665i \(0.473997\pi\)
\(492\) − 9.50097e8i − 0.359658i
\(493\) − 1.49445e9i − 0.561716i
\(494\) −3.23457e8 −0.120718
\(495\) 0 0
\(496\) 4.32482e9 1.59141
\(497\) 2.48824e8i 0.0909171i
\(498\) 2.06053e8i 0.0747613i
\(499\) 2.95178e9 1.06349 0.531743 0.846906i \(-0.321537\pi\)
0.531743 + 0.846906i \(0.321537\pi\)
\(500\) 0 0
\(501\) 2.41412e9 0.857681
\(502\) 3.29369e7i 0.0116204i
\(503\) − 5.22380e9i − 1.83020i −0.403229 0.915099i \(-0.632112\pi\)
0.403229 0.915099i \(-0.367888\pi\)
\(504\) 1.87280e8 0.0651605
\(505\) 0 0
\(506\) 1.25267e8 0.0429844
\(507\) 6.22326e8i 0.212075i
\(508\) − 2.82236e9i − 0.955190i
\(509\) 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
\(512\) − 1.72897e9i − 0.569301i
\(513\) 3.59530e9i 1.17577i
\(514\) −1.33265e8 −0.0432857
\(515\) 0 0
\(516\) −1.99197e9 −0.638277
\(517\) 7.33401e8i 0.233413i
\(518\) − 7.33814e7i − 0.0231970i
\(519\) 1.26628e9 0.397597
\(520\) 0 0
\(521\) 1.40563e8 0.0435450 0.0217725 0.999763i \(-0.493069\pi\)
0.0217725 + 0.999763i \(0.493069\pi\)
\(522\) 2.02468e8i 0.0623032i
\(523\) − 1.81127e9i − 0.553638i −0.960922 0.276819i \(-0.910720\pi\)
0.960922 0.276819i \(-0.0892803\pi\)
\(524\) −5.71794e8 −0.173612
\(525\) 0 0
\(526\) 4.08163e8 0.122288
\(527\) 4.37679e9i 1.30262i
\(528\) − 5.56635e8i − 0.164570i
\(529\) −6.89567e8 −0.202526
\(530\) 0 0
\(531\) 2.84837e9 0.825592
\(532\) 1.66855e9i 0.480452i
\(533\) 1.86383e9i 0.533164i
\(534\) 3.59750e8 0.102237
\(535\) 0 0
\(536\) 1.54681e9 0.433870
\(537\) − 3.61604e9i − 1.00768i
\(538\) − 5.34019e8i − 0.147849i
\(539\) 1.68516e8 0.0463534
\(540\) 0 0
\(541\) −7.11633e9 −1.93226 −0.966130 0.258058i \(-0.916918\pi\)
−0.966130 + 0.258058i \(0.916918\pi\)
\(542\) − 2.89885e8i − 0.0782038i
\(543\) − 1.51192e9i − 0.405255i
\(544\) 1.04469e9 0.278223
\(545\) 0 0
\(546\) 7.13400e7 0.0187568
\(547\) 6.02390e9i 1.57370i 0.617144 + 0.786850i \(0.288290\pi\)
−0.617144 + 0.786850i \(0.711710\pi\)
\(548\) 6.40318e9i 1.66213i
\(549\) −1.54372e9 −0.398168
\(550\) 0 0
\(551\) −3.63447e9 −0.925572
\(552\) − 5.51171e8i − 0.139476i
\(553\) 1.78883e9i 0.449812i
\(554\) −2.55204e8 −0.0637681
\(555\) 0 0
\(556\) −1.32669e9 −0.327347
\(557\) − 3.55726e9i − 0.872214i −0.899895 0.436107i \(-0.856357\pi\)
0.899895 0.436107i \(-0.143643\pi\)
\(558\) − 5.92968e8i − 0.144481i
\(559\) 3.90771e9 0.946195
\(560\) 0 0
\(561\) 5.63324e8 0.134706
\(562\) 1.00974e9i 0.239957i
\(563\) 2.51240e9i 0.593347i 0.954979 + 0.296673i \(0.0958773\pi\)
−0.954979 + 0.296673i \(0.904123\pi\)
\(564\) 1.60161e9 0.375906
\(565\) 0 0
\(566\) −4.26293e8 −0.0988214
\(567\) 3.86252e8i 0.0889877i
\(568\) − 2.51969e8i − 0.0576937i
\(569\) −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 \(0.346285\pi\)
\(572\) 1.10863e9i 0.247685i
\(573\) − 3.78386e9i − 0.840221i
\(574\) −1.42391e8 −0.0314262
\(575\) 0 0
\(576\) 3.01152e9 0.656610
\(577\) 3.66048e9i 0.793274i 0.917976 + 0.396637i \(0.129823\pi\)
−0.917976 + 0.396637i \(0.870177\pi\)
\(578\) − 2.17105e8i − 0.0467651i
\(579\) 3.45400e9 0.739516
\(580\) 0 0
\(581\) −2.08517e9 −0.441088
\(582\) − 2.25225e8i − 0.0473572i
\(583\) 2.88262e8i 0.0602487i
\(584\) 7.55807e8 0.157024
\(585\) 0 0
\(586\) −6.90846e8 −0.141821
\(587\) − 8.93156e9i − 1.82261i −0.411731 0.911305i \(-0.635076\pi\)
0.411731 0.911305i \(-0.364924\pi\)
\(588\) − 3.68007e8i − 0.0746510i
\(589\) 1.06442e10 2.14640
\(590\) 0 0
\(591\) 1.61812e9 0.322444
\(592\) − 2.45288e9i − 0.485904i
\(593\) 8.00218e9i 1.57586i 0.615766 + 0.787929i \(0.288847\pi\)
−0.615766 + 0.787929i \(0.711153\pi\)
\(594\) −1.82497e8 −0.0357276
\(595\) 0 0
\(596\) −6.85524e9 −1.32636
\(597\) − 4.79879e7i − 0.00923041i
\(598\) 5.36649e8i 0.102621i
\(599\) −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 \(0.230323\pi\)
\(602\) 2.98537e8i 0.0557713i
\(603\) 7.00062e9i 1.30025i
\(604\) −2.81599e9 −0.519999
\(605\) 0 0
\(606\) 3.64474e8 0.0665293
\(607\) − 5.42119e9i − 0.983863i −0.870634 0.491931i \(-0.836291\pi\)
0.870634 0.491931i \(-0.163709\pi\)
\(608\) − 2.54067e9i − 0.458444i
\(609\) 8.01598e8 0.143812
\(610\) 0 0
\(611\) −3.14191e9 −0.557250
\(612\) 3.14438e9i 0.554505i
\(613\) 8.21824e9i 1.44101i 0.693450 + 0.720505i \(0.256090\pi\)
−0.693450 + 0.720505i \(0.743910\pi\)
\(614\) −6.38905e8 −0.111390
\(615\) 0 0
\(616\) −1.70646e8 −0.0294147
\(617\) − 8.15621e9i − 1.39795i −0.715148 0.698973i \(-0.753641\pi\)
0.715148 0.698973i \(-0.246359\pi\)
\(618\) − 3.58897e8i − 0.0611660i
\(619\) 6.46052e9 1.09484 0.547420 0.836858i \(-0.315610\pi\)
0.547420 + 0.836858i \(0.315610\pi\)
\(620\) 0 0
\(621\) 5.96497e9 0.999511
\(622\) − 1.58573e7i − 0.00264218i
\(623\) 3.64051e9i 0.603191i
\(624\) 2.38464e9 0.392896
\(625\) 0 0
\(626\) 1.12511e9 0.183309
\(627\) − 1.36999e9i − 0.221963i
\(628\) 5.52190e9i 0.889672i
\(629\) 2.48236e9 0.397729
\(630\) 0 0
\(631\) −8.82660e9 −1.39859 −0.699295 0.714833i \(-0.746503\pi\)
−0.699295 + 0.714833i \(0.746503\pi\)
\(632\) − 1.81144e9i − 0.285439i
\(633\) − 1.28424e9i − 0.201248i
\(634\) −5.31876e8 −0.0828893
\(635\) 0 0
\(636\) 6.29509e8 0.0970291
\(637\) 7.21930e8i 0.110664i
\(638\) − 1.84486e8i − 0.0281249i
\(639\) 1.14037e9 0.172900
\(640\) 0 0
\(641\) 8.54151e9 1.28095 0.640474 0.767980i \(-0.278738\pi\)
0.640474 + 0.767980i \(0.278738\pi\)
\(642\) 2.83778e8i 0.0423259i
\(643\) − 1.20342e10i − 1.78517i −0.450878 0.892585i \(-0.648889\pi\)
0.450878 0.892585i \(-0.351111\pi\)
\(644\) 2.76831e9 0.408426
\(645\) 0 0
\(646\) 8.35934e8 0.122000
\(647\) − 1.89174e8i − 0.0274598i −0.999906 0.0137299i \(-0.995630\pi\)
0.999906 0.0137299i \(-0.00437050\pi\)
\(648\) − 3.91133e8i − 0.0564693i
\(649\) −2.59539e9 −0.372688
\(650\) 0 0
\(651\) −2.34764e9 −0.333501
\(652\) 5.11403e9i 0.722598i
\(653\) 8.70977e9i 1.22408i 0.790826 + 0.612041i \(0.209651\pi\)
−0.790826 + 0.612041i \(0.790349\pi\)
\(654\) 6.64159e8 0.0928432
\(655\) 0 0
\(656\) −4.75963e9 −0.658279
\(657\) 3.42067e9i 0.470579i
\(658\) − 2.40033e8i − 0.0328458i
\(659\) 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 \(0.307173\pi\)
\(662\) − 2.03074e8i − 0.0272052i
\(663\) 2.41330e9i 0.321598i
\(664\) 2.11152e9 0.279903
\(665\) 0 0
\(666\) −3.36310e8 −0.0441144
\(667\) 6.02996e9i 0.786817i
\(668\) − 1.22784e10i − 1.59376i
\(669\) 3.10154e9 0.400485
\(670\) 0 0
\(671\) 1.40661e9 0.179740
\(672\) 5.60357e8i 0.0712316i
\(673\) 4.78543e9i 0.605157i 0.953124 + 0.302578i \(0.0978474\pi\)
−0.953124 + 0.302578i \(0.902153\pi\)
\(674\) −1.68628e8 −0.0212139
\(675\) 0 0
\(676\) 3.16520e9 0.394083
\(677\) 1.29662e10i 1.60603i 0.595958 + 0.803015i \(0.296772\pi\)
−0.595958 + 0.803015i \(0.703228\pi\)
\(678\) 4.62018e8i 0.0569318i
\(679\) 2.27918e9 0.279405
\(680\) 0 0
\(681\) 4.76419e8 0.0578061
\(682\) 5.40302e8i 0.0652216i
\(683\) − 9.15988e9i − 1.10006i −0.835144 0.550031i \(-0.814616\pi\)
0.835144 0.550031i \(-0.185384\pi\)
\(684\) 7.64706e9 0.913689
\(685\) 0 0
\(686\) −5.51533e7 −0.00652285
\(687\) 2.62007e9i 0.308293i
\(688\) 9.97905e9i 1.16823i
\(689\) −1.23492e9 −0.143838
\(690\) 0 0
\(691\) 1.05298e10 1.21407 0.607037 0.794673i \(-0.292358\pi\)
0.607037 + 0.794673i \(0.292358\pi\)
\(692\) − 6.44038e9i − 0.738823i
\(693\) − 7.72318e8i − 0.0881515i
\(694\) −1.79944e9 −0.204352
\(695\) 0 0
\(696\) −8.11729e8 −0.0912596
\(697\) − 4.81683e9i − 0.538824i
\(698\) 3.61534e8i 0.0402398i
\(699\) 5.74470e9 0.636205
\(700\) 0 0
\(701\) 1.27411e9 0.139699 0.0698497 0.997558i \(-0.477748\pi\)
0.0698497 + 0.997558i \(0.477748\pi\)
\(702\) − 7.81825e8i − 0.0852962i
\(703\) − 6.03703e9i − 0.655360i
\(704\) −2.74405e9 −0.296406
\(705\) 0 0
\(706\) 1.78048e9 0.190423
\(707\) 3.68832e9i 0.392519i
\(708\) 5.66782e9i 0.600205i
\(709\) 7.17795e9 0.756378 0.378189 0.925728i \(-0.376547\pi\)
0.378189 + 0.925728i \(0.376547\pi\)
\(710\) 0 0
\(711\) 8.19829e9 0.855421
\(712\) − 3.68653e9i − 0.382769i
\(713\) − 1.76599e10i − 1.82463i
\(714\) −1.84369e8 −0.0189559
\(715\) 0 0
\(716\) −1.83914e10 −1.87249
\(717\) − 2.70746e9i − 0.274312i
\(718\) − 1.41133e9i − 0.142296i
\(719\) −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
\(722\) − 8.11273e8i − 0.0802208i
\(723\) − 2.04665e9i − 0.201400i
\(724\) −7.68971e9 −0.753052
\(725\) 0 0
\(726\) −5.90971e8 −0.0573176
\(727\) 4.67874e9i 0.451605i 0.974173 + 0.225802i \(0.0725004\pi\)
−0.974173 + 0.225802i \(0.927500\pi\)
\(728\) − 7.31054e8i − 0.0702246i
\(729\) −3.28577e9 −0.314116
\(730\) 0 0
\(731\) −1.00990e10 −0.956238
\(732\) − 3.07177e9i − 0.289468i
\(733\) 1.28552e9i 0.120563i 0.998181 + 0.0602817i \(0.0191999\pi\)
−0.998181 + 0.0602817i \(0.980800\pi\)
\(734\) −1.54612e9 −0.144314
\(735\) 0 0
\(736\) −4.21524e9 −0.389718
\(737\) − 6.37884e9i − 0.586956i
\(738\) 6.52585e8i 0.0597641i
\(739\) 5.26720e9 0.480091 0.240046 0.970762i \(-0.422838\pi\)
0.240046 + 0.970762i \(0.422838\pi\)
\(740\) 0 0
\(741\) 5.86909e9 0.529916
\(742\) − 9.43446e7i − 0.00847819i
\(743\) 4.15012e9i 0.371193i 0.982626 + 0.185596i \(0.0594217\pi\)
−0.982626 + 0.185596i \(0.940578\pi\)
\(744\) 2.37731e9 0.211631
\(745\) 0 0
\(746\) 7.35489e8 0.0648620
\(747\) 9.55643e9i 0.838829i
\(748\) − 2.86511e9i − 0.250314i
\(749\) −2.87171e9 −0.249721
\(750\) 0 0
\(751\) −6.37970e9 −0.549618 −0.274809 0.961499i \(-0.588615\pi\)
−0.274809 + 0.961499i \(0.588615\pi\)
\(752\) − 8.02346e9i − 0.688017i
\(753\) − 5.97635e8i − 0.0510098i
\(754\) 7.90343e8 0.0671453
\(755\) 0 0
\(756\) −4.03304e9 −0.339474
\(757\) 1.19658e10i 1.00255i 0.865289 + 0.501274i \(0.167135\pi\)
−0.865289 + 0.501274i \(0.832865\pi\)
\(758\) 1.07032e9i 0.0892631i
\(759\) −2.27296e9 −0.188688
\(760\) 0 0
\(761\) −2.00959e10 −1.65296 −0.826479 0.562967i \(-0.809660\pi\)
−0.826479 + 0.562967i \(0.809660\pi\)
\(762\) − 7.58437e8i − 0.0620979i
\(763\) 6.72100e9i 0.547770i
\(764\) −1.92450e10 −1.56131
\(765\) 0 0
\(766\) 1.12411e9 0.0903665
\(767\) − 1.11187e10i − 0.889756i
\(768\) 5.70667e9i 0.454588i
\(769\) −2.46683e10 −1.95613 −0.978064 0.208304i \(-0.933206\pi\)
−0.978064 + 0.208304i \(0.933206\pi\)
\(770\) 0 0
\(771\) 2.41807e9 0.190011
\(772\) − 1.75673e10i − 1.37418i
\(773\) 8.88824e9i 0.692130i 0.938211 + 0.346065i \(0.112482\pi\)
−0.938211 + 0.346065i \(0.887518\pi\)
\(774\) 1.36821e9 0.106062
\(775\) 0 0
\(776\) −2.30798e9 −0.177303
\(777\) 1.33149e9i 0.101828i
\(778\) − 1.46252e9i − 0.111345i
\(779\) −1.17144e10 −0.887850
\(780\) 0 0
\(781\) −1.03909e9 −0.0780502
\(782\) − 1.38690e9i − 0.103710i
\(783\) − 8.78483e9i − 0.653984i
\(784\) −1.84358e9 −0.136633
\(785\) 0 0
\(786\) −1.53655e8 −0.0112867
\(787\) 4.65006e9i 0.340053i 0.985439 + 0.170027i \(0.0543854\pi\)
−0.985439 + 0.170027i \(0.945615\pi\)
\(788\) − 8.22986e9i − 0.599171i
\(789\) −7.40606e9 −0.536806
\(790\) 0 0
\(791\) −4.67542e9 −0.335895
\(792\) 7.82079e8i 0.0559387i
\(793\) 6.02598e9i 0.429113i
\(794\) 1.31830e9 0.0934638
\(795\) 0 0
\(796\) −2.44070e8 −0.0171521
\(797\) − 1.42890e10i − 0.999762i −0.866094 0.499881i \(-0.833377\pi\)
0.866094 0.499881i \(-0.166623\pi\)
\(798\) 4.48381e8i 0.0312347i
\(799\) 8.11987e9 0.563165
\(800\) 0 0
\(801\) 1.66846e10 1.14711
\(802\) 2.66257e9i 0.182260i
\(803\) − 3.11685e9i − 0.212428i
\(804\) −1.39302e10 −0.945279
\(805\) 0 0
\(806\) −2.31467e9 −0.155710
\(807\) 9.68971e9i 0.649013i
\(808\) − 3.73494e9i − 0.249083i
\(809\) 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 \(0.217173\pi\)
\(812\) − 4.07698e9i − 0.267235i
\(813\) 5.25992e9i 0.343291i
\(814\) 3.06440e8 0.0199141
\(815\) 0 0
\(816\) −6.16280e9 −0.397066
\(817\) 2.45605e10i 1.57565i
\(818\) 1.18069e9i 0.0754221i
\(819\) 3.30864e9 0.210453
\(820\) 0 0
\(821\) −2.86630e10 −1.80768 −0.903838 0.427875i \(-0.859262\pi\)
−0.903838 + 0.427875i \(0.859262\pi\)
\(822\) 1.72069e9i 0.108057i
\(823\) − 2.76897e10i − 1.73148i −0.500490 0.865742i \(-0.666847\pi\)
0.500490 0.865742i \(-0.333153\pi\)
\(824\) −3.67778e9 −0.229003
\(825\) 0 0
\(826\) 8.49437e8 0.0524447
\(827\) 1.27176e10i 0.781873i 0.920418 + 0.390936i \(0.127849\pi\)
−0.920418 + 0.390936i \(0.872151\pi\)
\(828\) − 1.26873e10i − 0.776716i
\(829\) 1.50770e10 0.919127 0.459563 0.888145i \(-0.348006\pi\)
0.459563 + 0.888145i \(0.348006\pi\)
\(830\) 0 0
\(831\) 4.63064e9 0.279923
\(832\) − 1.17556e10i − 0.707641i
\(833\) − 1.86573e9i − 0.111839i
\(834\) −3.56514e8 −0.0212812
\(835\) 0 0
\(836\) −6.96787e9 −0.412457
\(837\) 2.57281e10i 1.51659i
\(838\) 3.02512e9i 0.177578i
\(839\) −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
\(842\) 3.96175e9i 0.228715i
\(843\) − 1.83216e10i − 1.05334i
\(844\) −6.53172e9 −0.373963
\(845\) 0 0
\(846\) −1.10008e9 −0.0624639
\(847\) − 5.98038e9i − 0.338171i
\(848\) − 3.15361e9i − 0.177591i
\(849\) 7.73504e9 0.433796
\(850\) 0 0
\(851\) −1.00161e10 −0.557114
\(852\) 2.26917e9i 0.125698i
\(853\) − 1.13971e9i − 0.0628740i −0.999506 0.0314370i \(-0.989992\pi\)
0.999506 0.0314370i \(-0.0100084\pi\)
\(854\) −4.60367e8 −0.0252931
\(855\) 0 0
\(856\) 2.90801e9 0.158466
\(857\) − 7.79419e9i − 0.422998i −0.977378 0.211499i \(-0.932166\pi\)
0.977378 0.211499i \(-0.0678345\pi\)
\(858\) 2.97915e8i 0.0161023i
\(859\) −1.27280e10 −0.685147 −0.342573 0.939491i \(-0.611299\pi\)
−0.342573 + 0.939491i \(0.611299\pi\)
\(860\) 0 0
\(861\) 2.58367e9 0.137951
\(862\) 3.30831e9i 0.175926i
\(863\) 2.53204e9i 0.134101i 0.997750 + 0.0670507i \(0.0213589\pi\)
−0.997750 + 0.0670507i \(0.978641\pi\)
\(864\) 6.14103e9 0.323924
\(865\) 0 0
\(866\) −3.09823e9 −0.162107
\(867\) 3.93933e9i 0.205284i
\(868\) 1.19402e10i 0.619718i
\(869\) −7.47014e9 −0.386153
\(870\) 0 0
\(871\) 2.73272e10 1.40130
\(872\) − 6.80595e9i − 0.347601i
\(873\) − 1.04456e10i − 0.531352i
\(874\) −3.37291e9 −0.170889
\(875\) 0 0
\(876\) −6.80661e9 −0.342110
\(877\) − 5.00988e9i − 0.250800i −0.992106 0.125400i \(-0.959979\pi\)
0.992106 0.125400i \(-0.0400215\pi\)
\(878\) 2.71862e9i 0.135556i
\(879\) 1.25353e10 0.622550
\(880\) 0 0
\(881\) 9.46900e9 0.466539 0.233270 0.972412i \(-0.425058\pi\)
0.233270 + 0.972412i \(0.425058\pi\)
\(882\) 2.52770e8i 0.0124047i
\(883\) 1.11146e10i 0.543289i 0.962398 + 0.271644i \(0.0875675\pi\)
−0.962398 + 0.271644i \(0.912433\pi\)
\(884\) 1.22742e10 0.597600
\(885\) 0 0
\(886\) −1.20006e9 −0.0579675
\(887\) 7.27986e9i 0.350260i 0.984545 + 0.175130i \(0.0560346\pi\)
−0.984545 + 0.175130i \(0.943965\pi\)
\(888\) − 1.34832e9i − 0.0646173i
\(889\) 7.67506e9 0.366375
\(890\) 0 0
\(891\) −1.61298e9 −0.0763938
\(892\) − 1.57746e10i − 0.744188i
\(893\) − 1.97473e10i − 0.927959i
\(894\) −1.84217e9 −0.0862280
\(895\) 0 0
\(896\) 3.79032e9 0.176034
\(897\) − 9.73743e9i − 0.450475i
\(898\) 2.09267e8i 0.00964348i
\(899\) −2.60084e10 −1.19386
\(900\) 0 0
\(901\) 3.19150e9 0.145365
\(902\) − 5.94624e8i − 0.0269786i
\(903\) − 5.41692e9i − 0.244819i
\(904\) 4.73451e9 0.213150
\(905\) 0 0
\(906\) −7.56726e8 −0.0338057
\(907\) 1.39503e10i 0.620809i 0.950605 + 0.310405i \(0.100465\pi\)
−0.950605 + 0.310405i \(0.899535\pi\)
\(908\) − 2.42310e9i − 0.107416i
\(909\) 1.69038e10 0.746465
\(910\) 0 0
\(911\) 2.98148e8 0.0130653 0.00653263 0.999979i \(-0.497921\pi\)
0.00653263 + 0.999979i \(0.497921\pi\)
\(912\) 1.49878e10i 0.654268i
\(913\) − 8.70765e9i − 0.378663i
\(914\) 3.27506e9 0.141876
\(915\) 0 0
\(916\) 1.33259e10 0.572876
\(917\) − 1.55492e9i − 0.0665910i
\(918\) 2.02053e9i 0.0862015i
\(919\) −2.67202e10 −1.13563 −0.567814 0.823157i \(-0.692211\pi\)
−0.567814 + 0.823157i \(0.692211\pi\)
\(920\) 0 0
\(921\) 1.15928e10 0.488969
\(922\) − 2.21295e9i − 0.0929850i
\(923\) − 4.45149e9i − 0.186337i
\(924\) 1.53680e9 0.0640861
\(925\) 0 0
\(926\) 1.58725e9 0.0656910
\(927\) − 1.66451e10i − 0.686288i
\(928\) 6.20794e9i 0.254994i
\(929\) 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
\(932\) − 2.92180e10i − 1.18221i
\(933\) 2.87728e8i 0.0115983i
\(934\) −3.88115e9 −0.155864
\(935\) 0 0
\(936\) −3.35045e9 −0.133548
\(937\) − 1.28088e10i − 0.508649i −0.967119 0.254325i \(-0.918147\pi\)
0.967119 0.254325i \(-0.0818531\pi\)
\(938\) 2.08772e9i 0.0825965i
\(939\) −2.04149e10 −0.804669
\(940\) 0 0
\(941\) −1.20663e10 −0.472073 −0.236037 0.971744i \(-0.575849\pi\)
−0.236037 + 0.971744i \(0.575849\pi\)
\(942\) 1.48387e9i 0.0578386i
\(943\) 1.94354e10i 0.754751i
\(944\) 2.83937e10 1.09855
\(945\) 0 0
\(946\) −1.24669e9 −0.0478784
\(947\) − 8.36023e9i − 0.319885i −0.987126 0.159942i \(-0.948869\pi\)
0.987126 0.159942i \(-0.0511308\pi\)
\(948\) 1.63133e10i 0.621890i
\(949\) 1.33527e10 0.507151
\(950\) 0 0
\(951\) 9.65082e9 0.363859
\(952\) 1.88931e9i 0.0709700i
\(953\) 4.49530e10i 1.68242i 0.540710 + 0.841209i \(0.318156\pi\)
−0.540710 + 0.841209i \(0.681844\pi\)
\(954\) −4.32386e8 −0.0161232
\(955\) 0 0
\(956\) −1.37703e10 −0.509732
\(957\) 3.34747e9i 0.123459i
\(958\) 3.26341e9i 0.119920i
\(959\) −1.74126e10 −0.637528
\(960\) 0 0
\(961\) 4.86580e10 1.76857
\(962\) 1.31280e9i 0.0475429i
\(963\) 1.31612e10i 0.474901i
\(964\) −1.04094e10 −0.374244
\(965\) 0 0
\(966\) 7.43911e8 0.0265522
\(967\) − 1.34247e8i − 0.00477432i −0.999997 0.00238716i \(-0.999240\pi\)
0.999997 0.00238716i \(-0.000759858\pi\)
\(968\) 6.05596e9i 0.214595i
\(969\) −1.51679e10 −0.535541
\(970\) 0 0
\(971\) −3.00377e10 −1.05293 −0.526465 0.850197i \(-0.676483\pi\)
−0.526465 + 0.850197i \(0.676483\pi\)
\(972\) 2.92375e10i 1.02119i
\(973\) − 3.60777e9i − 0.125558i
\(974\) 3.01379e9 0.104510
\(975\) 0 0
\(976\) −1.53884e10 −0.529810
\(977\) − 4.52860e10i − 1.55358i −0.629761 0.776789i \(-0.716847\pi\)
0.629761 0.776789i \(-0.283153\pi\)
\(978\) 1.37426e9i 0.0469769i
\(979\) −1.52028e10 −0.517825
\(980\) 0 0
\(981\) 3.08027e10 1.04171
\(982\) − 5.85057e8i − 0.0197155i
\(983\) 4.61443e10i 1.54946i 0.632290 + 0.774731i \(0.282115\pi\)
−0.632290 + 0.774731i \(0.717885\pi\)
\(984\) −2.61632e9 −0.0875404
\(985\) 0 0
\(986\) −2.04254e9 −0.0678580
\(987\) 4.35537e9i 0.144183i
\(988\) − 2.98506e10i − 0.984700i
\(989\) 4.07484e10 1.33944
\(990\) 0 0
\(991\) −1.05400e10 −0.344018 −0.172009 0.985095i \(-0.555026\pi\)
−0.172009 + 0.985095i \(0.555026\pi\)
\(992\) − 1.81812e10i − 0.591331i
\(993\) 3.68475e9i 0.119422i
\(994\) 3.40081e8 0.0109832
\(995\) 0 0
\(996\) −1.90158e10 −0.609828
\(997\) 5.00734e10i 1.60020i 0.599868 + 0.800099i \(0.295220\pi\)
−0.599868 + 0.800099i \(0.704780\pi\)
\(998\) − 4.03435e9i − 0.128474i
\(999\) 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 175.8.b.c.99.2 4
5.2 odd 4 35.8.a.a.1.1 2
5.3 odd 4 175.8.a.b.1.2 2
5.4 even 2 inner 175.8.b.c.99.3 4
15.2 even 4 315.8.a.c.1.2 2
20.7 even 4 560.8.a.i.1.1 2
35.27 even 4 245.8.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.1 2 5.2 odd 4
175.8.a.b.1.2 2 5.3 odd 4
175.8.b.c.99.2 4 1.1 even 1 trivial
175.8.b.c.99.3 4 5.4 even 2 inner
245.8.a.b.1.1 2 35.27 even 4
315.8.a.c.1.2 2 15.2 even 4
560.8.a.i.1.1 2 20.7 even 4