Properties

Label 175.6.b.a.99.2
Level $175$
Weight $6$
Character 175.99
Analytic conductor $28.067$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.6.b.a.99.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0000i q^{2} -14.0000i q^{3} -68.0000 q^{4} +140.000 q^{6} +49.0000i q^{7} -360.000i q^{8} +47.0000 q^{9} +O(q^{10})\) \(q+10.0000i q^{2} -14.0000i q^{3} -68.0000 q^{4} +140.000 q^{6} +49.0000i q^{7} -360.000i q^{8} +47.0000 q^{9} +232.000 q^{11} +952.000i q^{12} -140.000i q^{13} -490.000 q^{14} +1424.00 q^{16} +1722.00i q^{17} +470.000i q^{18} +98.0000 q^{19} +686.000 q^{21} +2320.00i q^{22} +1824.00i q^{23} -5040.00 q^{24} +1400.00 q^{26} -4060.00i q^{27} -3332.00i q^{28} -3418.00 q^{29} -7644.00 q^{31} +2720.00i q^{32} -3248.00i q^{33} -17220.0 q^{34} -3196.00 q^{36} +10398.0i q^{37} +980.000i q^{38} -1960.00 q^{39} -17962.0 q^{41} +6860.00i q^{42} +10880.0i q^{43} -15776.0 q^{44} -18240.0 q^{46} -9324.00i q^{47} -19936.0i q^{48} -2401.00 q^{49} +24108.0 q^{51} +9520.00i q^{52} +2262.00i q^{53} +40600.0 q^{54} +17640.0 q^{56} -1372.00i q^{57} -34180.0i q^{58} +2730.00 q^{59} +25648.0 q^{61} -76440.0i q^{62} +2303.00i q^{63} +18368.0 q^{64} +32480.0 q^{66} +48404.0i q^{67} -117096. i q^{68} +25536.0 q^{69} -58560.0 q^{71} -16920.0i q^{72} +68082.0i q^{73} -103980. q^{74} -6664.00 q^{76} +11368.0i q^{77} -19600.0i q^{78} -31784.0 q^{79} -45419.0 q^{81} -179620. i q^{82} -20538.0i q^{83} -46648.0 q^{84} -108800. q^{86} +47852.0i q^{87} -83520.0i q^{88} +50582.0 q^{89} +6860.00 q^{91} -124032. i q^{92} +107016. i q^{93} +93240.0 q^{94} +38080.0 q^{96} +58506.0i q^{97} -24010.0i q^{98} +10904.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 136 q^{4} + 280 q^{6} + 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 136 q^{4} + 280 q^{6} + 94 q^{9} + 464 q^{11} - 980 q^{14} + 2848 q^{16} + 196 q^{19} + 1372 q^{21} - 10080 q^{24} + 2800 q^{26} - 6836 q^{29} - 15288 q^{31} - 34440 q^{34} - 6392 q^{36} - 3920 q^{39} - 35924 q^{41} - 31552 q^{44} - 36480 q^{46} - 4802 q^{49} + 48216 q^{51} + 81200 q^{54} + 35280 q^{56} + 5460 q^{59} + 51296 q^{61} + 36736 q^{64} + 64960 q^{66} + 51072 q^{69} - 117120 q^{71} - 207960 q^{74} - 13328 q^{76} - 63568 q^{79} - 90838 q^{81} - 93296 q^{84} - 217600 q^{86} + 101164 q^{89} + 13720 q^{91} + 186480 q^{94} + 76160 q^{96} + 21808 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0000i 1.76777i 0.467707 + 0.883883i \(0.345080\pi\)
−0.467707 + 0.883883i \(0.654920\pi\)
\(3\) − 14.0000i − 0.898100i −0.893507 0.449050i \(-0.851762\pi\)
0.893507 0.449050i \(-0.148238\pi\)
\(4\) −68.0000 −2.12500
\(5\) 0 0
\(6\) 140.000 1.58763
\(7\) 49.0000i 0.377964i
\(8\) − 360.000i − 1.98874i
\(9\) 47.0000 0.193416
\(10\) 0 0
\(11\) 232.000 0.578104 0.289052 0.957313i \(-0.406660\pi\)
0.289052 + 0.957313i \(0.406660\pi\)
\(12\) 952.000i 1.90846i
\(13\) − 140.000i − 0.229757i −0.993380 0.114879i \(-0.963352\pi\)
0.993380 0.114879i \(-0.0366479\pi\)
\(14\) −490.000 −0.668153
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) 1722.00i 1.44514i 0.691296 + 0.722572i \(0.257040\pi\)
−0.691296 + 0.722572i \(0.742960\pi\)
\(18\) 470.000i 0.341914i
\(19\) 98.0000 0.0622791 0.0311395 0.999515i \(-0.490086\pi\)
0.0311395 + 0.999515i \(0.490086\pi\)
\(20\) 0 0
\(21\) 686.000 0.339450
\(22\) 2320.00i 1.02195i
\(23\) 1824.00i 0.718961i 0.933153 + 0.359480i \(0.117046\pi\)
−0.933153 + 0.359480i \(0.882954\pi\)
\(24\) −5040.00 −1.78609
\(25\) 0 0
\(26\) 1400.00 0.406158
\(27\) − 4060.00i − 1.07181i
\(28\) − 3332.00i − 0.803175i
\(29\) −3418.00 −0.754705 −0.377352 0.926070i \(-0.623165\pi\)
−0.377352 + 0.926070i \(0.623165\pi\)
\(30\) 0 0
\(31\) −7644.00 −1.42862 −0.714310 0.699830i \(-0.753259\pi\)
−0.714310 + 0.699830i \(0.753259\pi\)
\(32\) 2720.00i 0.469563i
\(33\) − 3248.00i − 0.519196i
\(34\) −17220.0 −2.55468
\(35\) 0 0
\(36\) −3196.00 −0.411008
\(37\) 10398.0i 1.24866i 0.781159 + 0.624332i \(0.214629\pi\)
−0.781159 + 0.624332i \(0.785371\pi\)
\(38\) 980.000i 0.110095i
\(39\) −1960.00 −0.206345
\(40\) 0 0
\(41\) −17962.0 −1.66876 −0.834382 0.551186i \(-0.814175\pi\)
−0.834382 + 0.551186i \(0.814175\pi\)
\(42\) 6860.00i 0.600069i
\(43\) 10880.0i 0.897342i 0.893697 + 0.448671i \(0.148102\pi\)
−0.893697 + 0.448671i \(0.851898\pi\)
\(44\) −15776.0 −1.22847
\(45\) 0 0
\(46\) −18240.0 −1.27096
\(47\) − 9324.00i − 0.615684i −0.951438 0.307842i \(-0.900393\pi\)
0.951438 0.307842i \(-0.0996068\pi\)
\(48\) − 19936.0i − 1.24892i
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) 24108.0 1.29788
\(52\) 9520.00i 0.488235i
\(53\) 2262.00i 0.110612i 0.998469 + 0.0553061i \(0.0176135\pi\)
−0.998469 + 0.0553061i \(0.982387\pi\)
\(54\) 40600.0 1.89471
\(55\) 0 0
\(56\) 17640.0 0.751672
\(57\) − 1372.00i − 0.0559329i
\(58\) − 34180.0i − 1.33414i
\(59\) 2730.00 0.102102 0.0510508 0.998696i \(-0.483743\pi\)
0.0510508 + 0.998696i \(0.483743\pi\)
\(60\) 0 0
\(61\) 25648.0 0.882529 0.441264 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(62\) − 76440.0i − 2.52547i
\(63\) 2303.00i 0.0731042i
\(64\) 18368.0 0.560547
\(65\) 0 0
\(66\) 32480.0 0.917817
\(67\) 48404.0i 1.31733i 0.752437 + 0.658664i \(0.228878\pi\)
−0.752437 + 0.658664i \(0.771122\pi\)
\(68\) − 117096.i − 3.07093i
\(69\) 25536.0 0.645699
\(70\) 0 0
\(71\) −58560.0 −1.37865 −0.689327 0.724450i \(-0.742094\pi\)
−0.689327 + 0.724450i \(0.742094\pi\)
\(72\) − 16920.0i − 0.384653i
\(73\) 68082.0i 1.49529i 0.664099 + 0.747645i \(0.268815\pi\)
−0.664099 + 0.747645i \(0.731185\pi\)
\(74\) −103980. −2.20735
\(75\) 0 0
\(76\) −6664.00 −0.132343
\(77\) 11368.0i 0.218503i
\(78\) − 19600.0i − 0.364770i
\(79\) −31784.0 −0.572982 −0.286491 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(80\) 0 0
\(81\) −45419.0 −0.769175
\(82\) − 179620.i − 2.94999i
\(83\) − 20538.0i − 0.327237i −0.986524 0.163619i \(-0.947683\pi\)
0.986524 0.163619i \(-0.0523167\pi\)
\(84\) −46648.0 −0.721331
\(85\) 0 0
\(86\) −108800. −1.58629
\(87\) 47852.0i 0.677801i
\(88\) − 83520.0i − 1.14970i
\(89\) 50582.0 0.676894 0.338447 0.940985i \(-0.390098\pi\)
0.338447 + 0.940985i \(0.390098\pi\)
\(90\) 0 0
\(91\) 6860.00 0.0868402
\(92\) − 124032.i − 1.52779i
\(93\) 107016.i 1.28304i
\(94\) 93240.0 1.08839
\(95\) 0 0
\(96\) 38080.0 0.421715
\(97\) 58506.0i 0.631351i 0.948867 + 0.315676i \(0.102231\pi\)
−0.948867 + 0.315676i \(0.897769\pi\)
\(98\) − 24010.0i − 0.252538i
\(99\) 10904.0 0.111814
\(100\) 0 0
\(101\) 38696.0 0.377453 0.188726 0.982030i \(-0.439564\pi\)
0.188726 + 0.982030i \(0.439564\pi\)
\(102\) 241080.i 2.29436i
\(103\) 53060.0i 0.492804i 0.969168 + 0.246402i \(0.0792483\pi\)
−0.969168 + 0.246402i \(0.920752\pi\)
\(104\) −50400.0 −0.456927
\(105\) 0 0
\(106\) −22620.0 −0.195537
\(107\) 146324.i 1.23554i 0.786360 + 0.617769i \(0.211963\pi\)
−0.786360 + 0.617769i \(0.788037\pi\)
\(108\) 276080.i 2.27759i
\(109\) −92898.0 −0.748928 −0.374464 0.927241i \(-0.622173\pi\)
−0.374464 + 0.927241i \(0.622173\pi\)
\(110\) 0 0
\(111\) 145572. 1.12143
\(112\) 69776.0i 0.525607i
\(113\) − 83354.0i − 0.614088i −0.951695 0.307044i \(-0.900660\pi\)
0.951695 0.307044i \(-0.0993398\pi\)
\(114\) 13720.0 0.0988762
\(115\) 0 0
\(116\) 232424. 1.60375
\(117\) − 6580.00i − 0.0444387i
\(118\) 27300.0i 0.180492i
\(119\) −84378.0 −0.546213
\(120\) 0 0
\(121\) −107227. −0.665795
\(122\) 256480.i 1.56011i
\(123\) 251468.i 1.49872i
\(124\) 519792. 3.03582
\(125\) 0 0
\(126\) −23030.0 −0.129231
\(127\) − 60384.0i − 0.332210i −0.986108 0.166105i \(-0.946881\pi\)
0.986108 0.166105i \(-0.0531191\pi\)
\(128\) 270720.i 1.46048i
\(129\) 152320. 0.805903
\(130\) 0 0
\(131\) −61586.0 −0.313548 −0.156774 0.987635i \(-0.550109\pi\)
−0.156774 + 0.987635i \(0.550109\pi\)
\(132\) 220864.i 1.10329i
\(133\) 4802.00i 0.0235393i
\(134\) −484040. −2.32873
\(135\) 0 0
\(136\) 619920. 2.87401
\(137\) 204462.i 0.930703i 0.885126 + 0.465352i \(0.154072\pi\)
−0.885126 + 0.465352i \(0.845928\pi\)
\(138\) 255360.i 1.14145i
\(139\) 35406.0 0.155432 0.0777159 0.996976i \(-0.475237\pi\)
0.0777159 + 0.996976i \(0.475237\pi\)
\(140\) 0 0
\(141\) −130536. −0.552946
\(142\) − 585600.i − 2.43714i
\(143\) − 32480.0i − 0.132824i
\(144\) 66928.0 0.268969
\(145\) 0 0
\(146\) −680820. −2.64332
\(147\) 33614.0i 0.128300i
\(148\) − 707064.i − 2.65341i
\(149\) 20226.0 0.0746353 0.0373177 0.999303i \(-0.488119\pi\)
0.0373177 + 0.999303i \(0.488119\pi\)
\(150\) 0 0
\(151\) 70904.0 0.253063 0.126531 0.991963i \(-0.459616\pi\)
0.126531 + 0.991963i \(0.459616\pi\)
\(152\) − 35280.0i − 0.123857i
\(153\) 80934.0i 0.279513i
\(154\) −113680. −0.386262
\(155\) 0 0
\(156\) 133280. 0.438484
\(157\) − 293524.i − 0.950374i −0.879885 0.475187i \(-0.842380\pi\)
0.879885 0.475187i \(-0.157620\pi\)
\(158\) − 317840.i − 1.01290i
\(159\) 31668.0 0.0993408
\(160\) 0 0
\(161\) −89376.0 −0.271742
\(162\) − 454190.i − 1.35972i
\(163\) 13192.0i 0.0388903i 0.999811 + 0.0194452i \(0.00618998\pi\)
−0.999811 + 0.0194452i \(0.993810\pi\)
\(164\) 1.22142e6 3.54612
\(165\) 0 0
\(166\) 205380. 0.578479
\(167\) − 493612.i − 1.36960i −0.728730 0.684801i \(-0.759889\pi\)
0.728730 0.684801i \(-0.240111\pi\)
\(168\) − 246960.i − 0.675077i
\(169\) 351693. 0.947212
\(170\) 0 0
\(171\) 4606.00 0.0120457
\(172\) − 739840.i − 1.90685i
\(173\) 240716.i 0.611490i 0.952113 + 0.305745i \(0.0989056\pi\)
−0.952113 + 0.305745i \(0.901094\pi\)
\(174\) −478520. −1.19819
\(175\) 0 0
\(176\) 330368. 0.803926
\(177\) − 38220.0i − 0.0916975i
\(178\) 505820.i 1.19659i
\(179\) −294932. −0.688001 −0.344001 0.938969i \(-0.611782\pi\)
−0.344001 + 0.938969i \(0.611782\pi\)
\(180\) 0 0
\(181\) −336980. −0.764553 −0.382277 0.924048i \(-0.624860\pi\)
−0.382277 + 0.924048i \(0.624860\pi\)
\(182\) 68600.0i 0.153513i
\(183\) − 359072.i − 0.792600i
\(184\) 656640. 1.42982
\(185\) 0 0
\(186\) −1.07016e6 −2.26812
\(187\) 399504.i 0.835444i
\(188\) 634032.i 1.30833i
\(189\) 198940. 0.405105
\(190\) 0 0
\(191\) 358264. 0.710591 0.355296 0.934754i \(-0.384380\pi\)
0.355296 + 0.934754i \(0.384380\pi\)
\(192\) − 257152.i − 0.503427i
\(193\) − 989554.i − 1.91226i −0.292948 0.956128i \(-0.594636\pi\)
0.292948 0.956128i \(-0.405364\pi\)
\(194\) −585060. −1.11608
\(195\) 0 0
\(196\) 163268. 0.303571
\(197\) 990050.i 1.81757i 0.417263 + 0.908786i \(0.362989\pi\)
−0.417263 + 0.908786i \(0.637011\pi\)
\(198\) 109040.i 0.197662i
\(199\) 840756. 1.50500 0.752501 0.658591i \(-0.228847\pi\)
0.752501 + 0.658591i \(0.228847\pi\)
\(200\) 0 0
\(201\) 677656. 1.18309
\(202\) 386960.i 0.667249i
\(203\) − 167482.i − 0.285252i
\(204\) −1.63934e6 −2.75800
\(205\) 0 0
\(206\) −530600. −0.871163
\(207\) 85728.0i 0.139058i
\(208\) − 199360.i − 0.319506i
\(209\) 22736.0 0.0360038
\(210\) 0 0
\(211\) 1.15073e6 1.77938 0.889689 0.456568i \(-0.150921\pi\)
0.889689 + 0.456568i \(0.150921\pi\)
\(212\) − 153816.i − 0.235051i
\(213\) 819840.i 1.23817i
\(214\) −1.46324e6 −2.18414
\(215\) 0 0
\(216\) −1.46160e6 −2.13154
\(217\) − 374556.i − 0.539967i
\(218\) − 928980.i − 1.32393i
\(219\) 953148. 1.34292
\(220\) 0 0
\(221\) 241080. 0.332032
\(222\) 1.45572e6i 1.98242i
\(223\) − 824264.i − 1.10995i −0.831866 0.554976i \(-0.812727\pi\)
0.831866 0.554976i \(-0.187273\pi\)
\(224\) −133280. −0.177478
\(225\) 0 0
\(226\) 833540. 1.08556
\(227\) − 74382.0i − 0.0958083i −0.998852 0.0479042i \(-0.984746\pi\)
0.998852 0.0479042i \(-0.0152542\pi\)
\(228\) 93296.0i 0.118857i
\(229\) −1.13196e6 −1.42640 −0.713199 0.700961i \(-0.752755\pi\)
−0.713199 + 0.700961i \(0.752755\pi\)
\(230\) 0 0
\(231\) 159152. 0.196238
\(232\) 1.23048e6i 1.50091i
\(233\) − 198726.i − 0.239809i −0.992785 0.119904i \(-0.961741\pi\)
0.992785 0.119904i \(-0.0382588\pi\)
\(234\) 65800.0 0.0785572
\(235\) 0 0
\(236\) −185640. −0.216966
\(237\) 444976.i 0.514595i
\(238\) − 843780.i − 0.965577i
\(239\) −482904. −0.546847 −0.273424 0.961894i \(-0.588156\pi\)
−0.273424 + 0.961894i \(0.588156\pi\)
\(240\) 0 0
\(241\) 805910. 0.893807 0.446904 0.894582i \(-0.352527\pi\)
0.446904 + 0.894582i \(0.352527\pi\)
\(242\) − 1.07227e6i − 1.17697i
\(243\) − 350714.i − 0.381011i
\(244\) −1.74406e6 −1.87537
\(245\) 0 0
\(246\) −2.51468e6 −2.64938
\(247\) − 13720.0i − 0.0143091i
\(248\) 2.75184e6i 2.84115i
\(249\) −287532. −0.293892
\(250\) 0 0
\(251\) 430738. 0.431548 0.215774 0.976443i \(-0.430773\pi\)
0.215774 + 0.976443i \(0.430773\pi\)
\(252\) − 156604.i − 0.155347i
\(253\) 423168.i 0.415634i
\(254\) 603840. 0.587270
\(255\) 0 0
\(256\) −2.11942e6 −2.02124
\(257\) 1.17691e6i 1.11150i 0.831349 + 0.555751i \(0.187569\pi\)
−0.831349 + 0.555751i \(0.812431\pi\)
\(258\) 1.52320e6i 1.42465i
\(259\) −509502. −0.471951
\(260\) 0 0
\(261\) −160646. −0.145972
\(262\) − 615860.i − 0.554279i
\(263\) 1.29098e6i 1.15088i 0.817845 + 0.575438i \(0.195169\pi\)
−0.817845 + 0.575438i \(0.804831\pi\)
\(264\) −1.16928e6 −1.03254
\(265\) 0 0
\(266\) −48020.0 −0.0416119
\(267\) − 708148.i − 0.607919i
\(268\) − 3.29147e6i − 2.79932i
\(269\) 1.27756e6 1.07646 0.538232 0.842797i \(-0.319093\pi\)
0.538232 + 0.842797i \(0.319093\pi\)
\(270\) 0 0
\(271\) 1.65054e6 1.36522 0.682612 0.730781i \(-0.260844\pi\)
0.682612 + 0.730781i \(0.260844\pi\)
\(272\) 2.45213e6i 2.00965i
\(273\) − 96040.0i − 0.0779912i
\(274\) −2.04462e6 −1.64527
\(275\) 0 0
\(276\) −1.73645e6 −1.37211
\(277\) 1.06409e6i 0.833257i 0.909077 + 0.416628i \(0.136788\pi\)
−0.909077 + 0.416628i \(0.863212\pi\)
\(278\) 354060.i 0.274767i
\(279\) −359268. −0.276317
\(280\) 0 0
\(281\) −22342.0 −0.0168794 −0.00843969 0.999964i \(-0.502686\pi\)
−0.00843969 + 0.999964i \(0.502686\pi\)
\(282\) − 1.30536e6i − 0.977479i
\(283\) − 2.49574e6i − 1.85239i −0.377042 0.926196i \(-0.623059\pi\)
0.377042 0.926196i \(-0.376941\pi\)
\(284\) 3.98208e6 2.92964
\(285\) 0 0
\(286\) 324800. 0.234802
\(287\) − 880138.i − 0.630734i
\(288\) 127840.i 0.0908208i
\(289\) −1.54543e6 −1.08844
\(290\) 0 0
\(291\) 819084. 0.567017
\(292\) − 4.62958e6i − 3.17749i
\(293\) − 1.93178e6i − 1.31458i −0.753637 0.657291i \(-0.771702\pi\)
0.753637 0.657291i \(-0.228298\pi\)
\(294\) −336140. −0.226805
\(295\) 0 0
\(296\) 3.74328e6 2.48326
\(297\) − 941920.i − 0.619616i
\(298\) 202260.i 0.131938i
\(299\) 255360. 0.165187
\(300\) 0 0
\(301\) −533120. −0.339163
\(302\) 709040.i 0.447356i
\(303\) − 541744.i − 0.338991i
\(304\) 139552. 0.0866068
\(305\) 0 0
\(306\) −809340. −0.494114
\(307\) 459074.i 0.277995i 0.990293 + 0.138997i \(0.0443880\pi\)
−0.990293 + 0.138997i \(0.955612\pi\)
\(308\) − 773024.i − 0.464319i
\(309\) 742840. 0.442587
\(310\) 0 0
\(311\) 667128. 0.391118 0.195559 0.980692i \(-0.437348\pi\)
0.195559 + 0.980692i \(0.437348\pi\)
\(312\) 705600.i 0.410367i
\(313\) − 111034.i − 0.0640612i −0.999487 0.0320306i \(-0.989803\pi\)
0.999487 0.0320306i \(-0.0101974\pi\)
\(314\) 2.93524e6 1.68004
\(315\) 0 0
\(316\) 2.16131e6 1.21759
\(317\) 68778.0i 0.0384416i 0.999815 + 0.0192208i \(0.00611855\pi\)
−0.999815 + 0.0192208i \(0.993881\pi\)
\(318\) 316680.i 0.175611i
\(319\) −792976. −0.436298
\(320\) 0 0
\(321\) 2.04854e6 1.10964
\(322\) − 893760.i − 0.480376i
\(323\) 168756.i 0.0900022i
\(324\) 3.08849e6 1.63450
\(325\) 0 0
\(326\) −131920. −0.0687490
\(327\) 1.30057e6i 0.672613i
\(328\) 6.46632e6i 3.31874i
\(329\) 456876. 0.232707
\(330\) 0 0
\(331\) −564448. −0.283174 −0.141587 0.989926i \(-0.545221\pi\)
−0.141587 + 0.989926i \(0.545221\pi\)
\(332\) 1.39658e6i 0.695379i
\(333\) 488706.i 0.241511i
\(334\) 4.93612e6 2.42114
\(335\) 0 0
\(336\) 976864. 0.472048
\(337\) − 2.07729e6i − 0.996376i −0.867069 0.498188i \(-0.833999\pi\)
0.867069 0.498188i \(-0.166001\pi\)
\(338\) 3.51693e6i 1.67445i
\(339\) −1.16696e6 −0.551512
\(340\) 0 0
\(341\) −1.77341e6 −0.825891
\(342\) 46060.0i 0.0212941i
\(343\) − 117649.i − 0.0539949i
\(344\) 3.91680e6 1.78458
\(345\) 0 0
\(346\) −2.40716e6 −1.08097
\(347\) 53248.0i 0.0237399i 0.999930 + 0.0118700i \(0.00377842\pi\)
−0.999930 + 0.0118700i \(0.996222\pi\)
\(348\) − 3.25394e6i − 1.44033i
\(349\) 2.27200e6 0.998494 0.499247 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(350\) 0 0
\(351\) −568400. −0.246256
\(352\) 631040.i 0.271456i
\(353\) 4.00645e6i 1.71129i 0.517565 + 0.855644i \(0.326838\pi\)
−0.517565 + 0.855644i \(0.673162\pi\)
\(354\) 382200. 0.162100
\(355\) 0 0
\(356\) −3.43958e6 −1.43840
\(357\) 1.18129e6i 0.490554i
\(358\) − 2.94932e6i − 1.21623i
\(359\) −73784.0 −0.0302152 −0.0151076 0.999886i \(-0.504809\pi\)
−0.0151076 + 0.999886i \(0.504809\pi\)
\(360\) 0 0
\(361\) −2.46650e6 −0.996121
\(362\) − 3.36980e6i − 1.35155i
\(363\) 1.50118e6i 0.597951i
\(364\) −466480. −0.184535
\(365\) 0 0
\(366\) 3.59072e6 1.40113
\(367\) − 1.40431e6i − 0.544250i −0.962262 0.272125i \(-0.912274\pi\)
0.962262 0.272125i \(-0.0877264\pi\)
\(368\) 2.59738e6i 0.999805i
\(369\) −844214. −0.322765
\(370\) 0 0
\(371\) −110838. −0.0418075
\(372\) − 7.27709e6i − 2.72647i
\(373\) − 1.60323e6i − 0.596657i −0.954463 0.298329i \(-0.903571\pi\)
0.954463 0.298329i \(-0.0964291\pi\)
\(374\) −3.99504e6 −1.47687
\(375\) 0 0
\(376\) −3.35664e6 −1.22443
\(377\) 478520.i 0.173399i
\(378\) 1.98940e6i 0.716131i
\(379\) 4.77012e6 1.70581 0.852906 0.522064i \(-0.174838\pi\)
0.852906 + 0.522064i \(0.174838\pi\)
\(380\) 0 0
\(381\) −845376. −0.298358
\(382\) 3.58264e6i 1.25616i
\(383\) − 2.23079e6i − 0.777072i −0.921434 0.388536i \(-0.872981\pi\)
0.921434 0.388536i \(-0.127019\pi\)
\(384\) 3.79008e6 1.31166
\(385\) 0 0
\(386\) 9.89554e6 3.38042
\(387\) 511360.i 0.173560i
\(388\) − 3.97841e6i − 1.34162i
\(389\) −4.84024e6 −1.62178 −0.810892 0.585196i \(-0.801018\pi\)
−0.810892 + 0.585196i \(0.801018\pi\)
\(390\) 0 0
\(391\) −3.14093e6 −1.03900
\(392\) 864360.i 0.284105i
\(393\) 862204.i 0.281597i
\(394\) −9.90050e6 −3.21304
\(395\) 0 0
\(396\) −741472. −0.237606
\(397\) − 995820.i − 0.317106i −0.987350 0.158553i \(-0.949317\pi\)
0.987350 0.158553i \(-0.0506829\pi\)
\(398\) 8.40756e6i 2.66049i
\(399\) 67228.0 0.0211406
\(400\) 0 0
\(401\) −3.31605e6 −1.02982 −0.514909 0.857245i \(-0.672174\pi\)
−0.514909 + 0.857245i \(0.672174\pi\)
\(402\) 6.77656e6i 2.09143i
\(403\) 1.07016e6i 0.328236i
\(404\) −2.63133e6 −0.802087
\(405\) 0 0
\(406\) 1.67482e6 0.504258
\(407\) 2.41234e6i 0.721858i
\(408\) − 8.67888e6i − 2.58115i
\(409\) −3.07273e6 −0.908274 −0.454137 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(410\) 0 0
\(411\) 2.86247e6 0.835865
\(412\) − 3.60808e6i − 1.04721i
\(413\) 133770.i 0.0385908i
\(414\) −857280. −0.245823
\(415\) 0 0
\(416\) 380800. 0.107886
\(417\) − 495684.i − 0.139593i
\(418\) 227360.i 0.0636463i
\(419\) −2.81438e6 −0.783154 −0.391577 0.920145i \(-0.628070\pi\)
−0.391577 + 0.920145i \(0.628070\pi\)
\(420\) 0 0
\(421\) 3.05802e6 0.840883 0.420441 0.907320i \(-0.361875\pi\)
0.420441 + 0.907320i \(0.361875\pi\)
\(422\) 1.15073e7i 3.14552i
\(423\) − 438228.i − 0.119083i
\(424\) 814320. 0.219979
\(425\) 0 0
\(426\) −8.19840e6 −2.18880
\(427\) 1.25675e6i 0.333565i
\(428\) − 9.95003e6i − 2.62552i
\(429\) −454720. −0.119289
\(430\) 0 0
\(431\) 1.93750e6 0.502398 0.251199 0.967936i \(-0.419175\pi\)
0.251199 + 0.967936i \(0.419175\pi\)
\(432\) − 5.78144e6i − 1.49048i
\(433\) 3.94790e6i 1.01192i 0.862557 + 0.505961i \(0.168862\pi\)
−0.862557 + 0.505961i \(0.831138\pi\)
\(434\) 3.74556e6 0.954536
\(435\) 0 0
\(436\) 6.31706e6 1.59147
\(437\) 178752.i 0.0447762i
\(438\) 9.53148e6i 2.37397i
\(439\) 7.41770e6 1.83700 0.918498 0.395426i \(-0.129403\pi\)
0.918498 + 0.395426i \(0.129403\pi\)
\(440\) 0 0
\(441\) −112847. −0.0276308
\(442\) 2.41080e6i 0.586956i
\(443\) 1.40269e6i 0.339589i 0.985480 + 0.169794i \(0.0543103\pi\)
−0.985480 + 0.169794i \(0.945690\pi\)
\(444\) −9.89890e6 −2.38303
\(445\) 0 0
\(446\) 8.24264e6 1.96214
\(447\) − 283164.i − 0.0670300i
\(448\) 900032.i 0.211867i
\(449\) 590574. 0.138248 0.0691239 0.997608i \(-0.477980\pi\)
0.0691239 + 0.997608i \(0.477980\pi\)
\(450\) 0 0
\(451\) −4.16718e6 −0.964720
\(452\) 5.66807e6i 1.30494i
\(453\) − 992656.i − 0.227276i
\(454\) 743820. 0.169367
\(455\) 0 0
\(456\) −493920. −0.111236
\(457\) 2.90484e6i 0.650627i 0.945606 + 0.325313i \(0.105470\pi\)
−0.945606 + 0.325313i \(0.894530\pi\)
\(458\) − 1.13196e7i − 2.52154i
\(459\) 6.99132e6 1.54891
\(460\) 0 0
\(461\) −922684. −0.202209 −0.101105 0.994876i \(-0.532238\pi\)
−0.101105 + 0.994876i \(0.532238\pi\)
\(462\) 1.59152e6i 0.346902i
\(463\) 7.18235e6i 1.55709i 0.627588 + 0.778546i \(0.284042\pi\)
−0.627588 + 0.778546i \(0.715958\pi\)
\(464\) −4.86723e6 −1.04951
\(465\) 0 0
\(466\) 1.98726e6 0.423926
\(467\) 612570.i 0.129976i 0.997886 + 0.0649881i \(0.0207009\pi\)
−0.997886 + 0.0649881i \(0.979299\pi\)
\(468\) 447440.i 0.0944322i
\(469\) −2.37180e6 −0.497904
\(470\) 0 0
\(471\) −4.10934e6 −0.853531
\(472\) − 982800.i − 0.203053i
\(473\) 2.52416e6i 0.518757i
\(474\) −4.44976e6 −0.909684
\(475\) 0 0
\(476\) 5.73770e6 1.16070
\(477\) 106314.i 0.0213941i
\(478\) − 4.82904e6i − 0.966699i
\(479\) −2.60330e6 −0.518424 −0.259212 0.965820i \(-0.583463\pi\)
−0.259212 + 0.965820i \(0.583463\pi\)
\(480\) 0 0
\(481\) 1.45572e6 0.286890
\(482\) 8.05910e6i 1.58004i
\(483\) 1.25126e6i 0.244051i
\(484\) 7.29144e6 1.41482
\(485\) 0 0
\(486\) 3.50714e6 0.673539
\(487\) − 5.46309e6i − 1.04380i −0.853008 0.521898i \(-0.825224\pi\)
0.853008 0.521898i \(-0.174776\pi\)
\(488\) − 9.23328e6i − 1.75512i
\(489\) 184688. 0.0349274
\(490\) 0 0
\(491\) 1.64090e6 0.307170 0.153585 0.988135i \(-0.450918\pi\)
0.153585 + 0.988135i \(0.450918\pi\)
\(492\) − 1.70998e7i − 3.18478i
\(493\) − 5.88580e6i − 1.09066i
\(494\) 137200. 0.0252951
\(495\) 0 0
\(496\) −1.08851e7 −1.98667
\(497\) − 2.86944e6i − 0.521082i
\(498\) − 2.87532e6i − 0.519533i
\(499\) −2.99796e6 −0.538983 −0.269491 0.963003i \(-0.586856\pi\)
−0.269491 + 0.963003i \(0.586856\pi\)
\(500\) 0 0
\(501\) −6.91057e6 −1.23004
\(502\) 4.30738e6i 0.762876i
\(503\) − 6.89405e6i − 1.21494i −0.794343 0.607469i \(-0.792185\pi\)
0.794343 0.607469i \(-0.207815\pi\)
\(504\) 829080. 0.145385
\(505\) 0 0
\(506\) −4.23168e6 −0.734745
\(507\) − 4.92370e6i − 0.850691i
\(508\) 4.10611e6i 0.705946i
\(509\) −2.30476e6 −0.394305 −0.197152 0.980373i \(-0.563169\pi\)
−0.197152 + 0.980373i \(0.563169\pi\)
\(510\) 0 0
\(511\) −3.33602e6 −0.565166
\(512\) − 1.25312e7i − 2.11260i
\(513\) − 397880.i − 0.0667511i
\(514\) −1.17691e7 −1.96488
\(515\) 0 0
\(516\) −1.03578e7 −1.71254
\(517\) − 2.16317e6i − 0.355929i
\(518\) − 5.09502e6i − 0.834299i
\(519\) 3.37002e6 0.549180
\(520\) 0 0
\(521\) −1.20960e7 −1.95231 −0.976155 0.217073i \(-0.930349\pi\)
−0.976155 + 0.217073i \(0.930349\pi\)
\(522\) − 1.60646e6i − 0.258044i
\(523\) 5.48443e6i 0.876753i 0.898791 + 0.438377i \(0.144446\pi\)
−0.898791 + 0.438377i \(0.855554\pi\)
\(524\) 4.18785e6 0.666289
\(525\) 0 0
\(526\) −1.29098e7 −2.03448
\(527\) − 1.31630e7i − 2.06456i
\(528\) − 4.62515e6i − 0.722007i
\(529\) 3.10937e6 0.483095
\(530\) 0 0
\(531\) 128310. 0.0197480
\(532\) − 326536.i − 0.0500210i
\(533\) 2.51468e6i 0.383411i
\(534\) 7.08148e6 1.07466
\(535\) 0 0
\(536\) 1.74254e7 2.61982
\(537\) 4.12905e6i 0.617894i
\(538\) 1.27756e7i 1.90294i
\(539\) −557032. −0.0825863
\(540\) 0 0
\(541\) −6.71799e6 −0.986839 −0.493420 0.869791i \(-0.664253\pi\)
−0.493420 + 0.869791i \(0.664253\pi\)
\(542\) 1.65054e7i 2.41340i
\(543\) 4.71772e6i 0.686646i
\(544\) −4.68384e6 −0.678586
\(545\) 0 0
\(546\) 960400. 0.137870
\(547\) 5.00235e6i 0.714835i 0.933945 + 0.357418i \(0.116343\pi\)
−0.933945 + 0.357418i \(0.883657\pi\)
\(548\) − 1.39034e7i − 1.97774i
\(549\) 1.20546e6 0.170695
\(550\) 0 0
\(551\) −334964. −0.0470023
\(552\) − 9.19296e6i − 1.28413i
\(553\) − 1.55742e6i − 0.216567i
\(554\) −1.06409e7 −1.47300
\(555\) 0 0
\(556\) −2.40761e6 −0.330293
\(557\) − 9.01961e6i − 1.23183i −0.787814 0.615913i \(-0.788787\pi\)
0.787814 0.615913i \(-0.211213\pi\)
\(558\) − 3.59268e6i − 0.488465i
\(559\) 1.52320e6 0.206171
\(560\) 0 0
\(561\) 5.59306e6 0.750312
\(562\) − 223420.i − 0.0298388i
\(563\) 1.24051e7i 1.64941i 0.565561 + 0.824707i \(0.308660\pi\)
−0.565561 + 0.824707i \(0.691340\pi\)
\(564\) 8.87645e6 1.17501
\(565\) 0 0
\(566\) 2.49574e7 3.27460
\(567\) − 2.22553e6i − 0.290721i
\(568\) 2.10816e7i 2.74178i
\(569\) −6.48804e6 −0.840103 −0.420052 0.907500i \(-0.637988\pi\)
−0.420052 + 0.907500i \(0.637988\pi\)
\(570\) 0 0
\(571\) −1.02285e7 −1.31287 −0.656435 0.754382i \(-0.727936\pi\)
−0.656435 + 0.754382i \(0.727936\pi\)
\(572\) 2.20864e6i 0.282251i
\(573\) − 5.01570e6i − 0.638182i
\(574\) 8.80138e6 1.11499
\(575\) 0 0
\(576\) 863296. 0.108419
\(577\) − 2.65338e6i − 0.331787i −0.986144 0.165894i \(-0.946949\pi\)
0.986144 0.165894i \(-0.0530508\pi\)
\(578\) − 1.54543e7i − 1.92411i
\(579\) −1.38538e7 −1.71740
\(580\) 0 0
\(581\) 1.00636e6 0.123684
\(582\) 8.19084e6i 1.00235i
\(583\) 524784.i 0.0639454i
\(584\) 2.45095e7 2.97374
\(585\) 0 0
\(586\) 1.93178e7 2.32387
\(587\) 1.43044e7i 1.71346i 0.515766 + 0.856729i \(0.327507\pi\)
−0.515766 + 0.856729i \(0.672493\pi\)
\(588\) − 2.28575e6i − 0.272638i
\(589\) −749112. −0.0889731
\(590\) 0 0
\(591\) 1.38607e7 1.63236
\(592\) 1.48068e7i 1.73642i
\(593\) − 1.00265e7i − 1.17088i −0.810714 0.585442i \(-0.800921\pi\)
0.810714 0.585442i \(-0.199079\pi\)
\(594\) 9.41920e6 1.09534
\(595\) 0 0
\(596\) −1.37537e6 −0.158600
\(597\) − 1.17706e7i − 1.35164i
\(598\) 2.55360e6i 0.292011i
\(599\) 7.52292e6 0.856681 0.428341 0.903617i \(-0.359098\pi\)
0.428341 + 0.903617i \(0.359098\pi\)
\(600\) 0 0
\(601\) 3.38625e6 0.382413 0.191207 0.981550i \(-0.438760\pi\)
0.191207 + 0.981550i \(0.438760\pi\)
\(602\) − 5.33120e6i − 0.599562i
\(603\) 2.27499e6i 0.254792i
\(604\) −4.82147e6 −0.537759
\(605\) 0 0
\(606\) 5.41744e6 0.599256
\(607\) 6.90861e6i 0.761060i 0.924769 + 0.380530i \(0.124258\pi\)
−0.924769 + 0.380530i \(0.875742\pi\)
\(608\) 266560.i 0.0292439i
\(609\) −2.34475e6 −0.256185
\(610\) 0 0
\(611\) −1.30536e6 −0.141458
\(612\) − 5.50351e6i − 0.593966i
\(613\) − 9.68896e6i − 1.04142i −0.853734 0.520710i \(-0.825667\pi\)
0.853734 0.520710i \(-0.174333\pi\)
\(614\) −4.59074e6 −0.491430
\(615\) 0 0
\(616\) 4.09248e6 0.434545
\(617\) 7.84742e6i 0.829877i 0.909849 + 0.414939i \(0.136197\pi\)
−0.909849 + 0.414939i \(0.863803\pi\)
\(618\) 7.42840e6i 0.782391i
\(619\) 1.01972e7 1.06968 0.534840 0.844953i \(-0.320372\pi\)
0.534840 + 0.844953i \(0.320372\pi\)
\(620\) 0 0
\(621\) 7.40544e6 0.770587
\(622\) 6.67128e6i 0.691406i
\(623\) 2.47852e6i 0.255842i
\(624\) −2.79104e6 −0.286949
\(625\) 0 0
\(626\) 1.11034e6 0.113245
\(627\) − 318304.i − 0.0323350i
\(628\) 1.99596e7i 2.01954i
\(629\) −1.79054e7 −1.80450
\(630\) 0 0
\(631\) −8.36258e6 −0.836116 −0.418058 0.908420i \(-0.637289\pi\)
−0.418058 + 0.908420i \(0.637289\pi\)
\(632\) 1.14422e7i 1.13951i
\(633\) − 1.61102e7i − 1.59806i
\(634\) −687780. −0.0679558
\(635\) 0 0
\(636\) −2.15342e6 −0.211099
\(637\) 336140.i 0.0328225i
\(638\) − 7.92976e6i − 0.771273i
\(639\) −2.75232e6 −0.266653
\(640\) 0 0
\(641\) 1.10283e6 0.106014 0.0530070 0.998594i \(-0.483119\pi\)
0.0530070 + 0.998594i \(0.483119\pi\)
\(642\) 2.04854e7i 1.96158i
\(643\) 1.71354e7i 1.63443i 0.576330 + 0.817217i \(0.304484\pi\)
−0.576330 + 0.817217i \(0.695516\pi\)
\(644\) 6.07757e6 0.577451
\(645\) 0 0
\(646\) −1.68756e6 −0.159103
\(647\) 54964.0i 0.00516200i 0.999997 + 0.00258100i \(0.000821558\pi\)
−0.999997 + 0.00258100i \(0.999178\pi\)
\(648\) 1.63508e7i 1.52969i
\(649\) 633360. 0.0590254
\(650\) 0 0
\(651\) −5.24378e6 −0.484945
\(652\) − 897056.i − 0.0826420i
\(653\) − 485166.i − 0.0445254i −0.999752 0.0222627i \(-0.992913\pi\)
0.999752 0.0222627i \(-0.00708702\pi\)
\(654\) −1.30057e7 −1.18902
\(655\) 0 0
\(656\) −2.55779e7 −2.32063
\(657\) 3.19985e6i 0.289212i
\(658\) 4.56876e6i 0.411371i
\(659\) 2.72136e6 0.244103 0.122051 0.992524i \(-0.461053\pi\)
0.122051 + 0.992524i \(0.461053\pi\)
\(660\) 0 0
\(661\) −2.14525e6 −0.190974 −0.0954869 0.995431i \(-0.530441\pi\)
−0.0954869 + 0.995431i \(0.530441\pi\)
\(662\) − 5.64448e6i − 0.500586i
\(663\) − 3.37512e6i − 0.298198i
\(664\) −7.39368e6 −0.650789
\(665\) 0 0
\(666\) −4.88706e6 −0.426935
\(667\) − 6.23443e6i − 0.542603i
\(668\) 3.35656e7i 2.91041i
\(669\) −1.15397e7 −0.996848
\(670\) 0 0
\(671\) 5.95034e6 0.510194
\(672\) 1.86592e6i 0.159393i
\(673\) 2.92796e6i 0.249188i 0.992208 + 0.124594i \(0.0397629\pi\)
−0.992208 + 0.124594i \(0.960237\pi\)
\(674\) 2.07729e7 1.76136
\(675\) 0 0
\(676\) −2.39151e7 −2.01282
\(677\) 1.34992e7i 1.13198i 0.824414 + 0.565988i \(0.191505\pi\)
−0.824414 + 0.565988i \(0.808495\pi\)
\(678\) − 1.16696e7i − 0.974945i
\(679\) −2.86679e6 −0.238628
\(680\) 0 0
\(681\) −1.04135e6 −0.0860455
\(682\) − 1.77341e7i − 1.45998i
\(683\) − 5.42972e6i − 0.445375i −0.974890 0.222688i \(-0.928517\pi\)
0.974890 0.222688i \(-0.0714830\pi\)
\(684\) −313208. −0.0255972
\(685\) 0 0
\(686\) 1.17649e6 0.0954504
\(687\) 1.58474e7i 1.28105i
\(688\) 1.54931e7i 1.24787i
\(689\) 316680. 0.0254140
\(690\) 0 0
\(691\) 2.08280e7 1.65940 0.829702 0.558207i \(-0.188510\pi\)
0.829702 + 0.558207i \(0.188510\pi\)
\(692\) − 1.63687e7i − 1.29942i
\(693\) 534296.i 0.0422619i
\(694\) −532480. −0.0419667
\(695\) 0 0
\(696\) 1.72267e7 1.34797
\(697\) − 3.09306e7i − 2.41160i
\(698\) 2.27200e7i 1.76510i
\(699\) −2.78216e6 −0.215372
\(700\) 0 0
\(701\) 2.35141e7 1.80731 0.903655 0.428261i \(-0.140874\pi\)
0.903655 + 0.428261i \(0.140874\pi\)
\(702\) − 5.68400e6i − 0.435323i
\(703\) 1.01900e6i 0.0777656i
\(704\) 4.26138e6 0.324055
\(705\) 0 0
\(706\) −4.00645e7 −3.02516
\(707\) 1.89610e6i 0.142664i
\(708\) 2.59896e6i 0.194857i
\(709\) 1.95747e7 1.46244 0.731221 0.682140i \(-0.238951\pi\)
0.731221 + 0.682140i \(0.238951\pi\)
\(710\) 0 0
\(711\) −1.49385e6 −0.110824
\(712\) − 1.82095e7i − 1.34617i
\(713\) − 1.39427e7i − 1.02712i
\(714\) −1.18129e7 −0.867185
\(715\) 0 0
\(716\) 2.00554e7 1.46200
\(717\) 6.76066e6i 0.491124i
\(718\) − 737840.i − 0.0534135i
\(719\) 2.61152e7 1.88396 0.941978 0.335674i \(-0.108964\pi\)
0.941978 + 0.335674i \(0.108964\pi\)
\(720\) 0 0
\(721\) −2.59994e6 −0.186262
\(722\) − 2.46650e7i − 1.76091i
\(723\) − 1.12827e7i − 0.802729i
\(724\) 2.29146e7 1.62468
\(725\) 0 0
\(726\) −1.50118e7 −1.05704
\(727\) − 1.54126e7i − 1.08154i −0.841172 0.540768i \(-0.818134\pi\)
0.841172 0.540768i \(-0.181866\pi\)
\(728\) − 2.46960e6i − 0.172702i
\(729\) −1.59468e7 −1.11136
\(730\) 0 0
\(731\) −1.87354e7 −1.29679
\(732\) 2.44169e7i 1.68427i
\(733\) − 1.69868e7i − 1.16776i −0.811841 0.583878i \(-0.801535\pi\)
0.811841 0.583878i \(-0.198465\pi\)
\(734\) 1.40431e7 0.962107
\(735\) 0 0
\(736\) −4.96128e6 −0.337597
\(737\) 1.12297e7i 0.761554i
\(738\) − 8.44214e6i − 0.570574i
\(739\) −2.01511e6 −0.135734 −0.0678669 0.997694i \(-0.521619\pi\)
−0.0678669 + 0.997694i \(0.521619\pi\)
\(740\) 0 0
\(741\) −192080. −0.0128510
\(742\) − 1.10838e6i − 0.0739059i
\(743\) − 1.51381e7i − 1.00600i −0.864286 0.503001i \(-0.832229\pi\)
0.864286 0.503001i \(-0.167771\pi\)
\(744\) 3.85258e7 2.55164
\(745\) 0 0
\(746\) 1.60323e7 1.05475
\(747\) − 965286.i − 0.0632928i
\(748\) − 2.71663e7i − 1.77532i
\(749\) −7.16988e6 −0.466989
\(750\) 0 0
\(751\) 7.21401e6 0.466742 0.233371 0.972388i \(-0.425024\pi\)
0.233371 + 0.972388i \(0.425024\pi\)
\(752\) − 1.32774e7i − 0.856185i
\(753\) − 6.03033e6i − 0.387573i
\(754\) −4.78520e6 −0.306529
\(755\) 0 0
\(756\) −1.35279e7 −0.860848
\(757\) 1.09697e7i 0.695755i 0.937540 + 0.347877i \(0.113097\pi\)
−0.937540 + 0.347877i \(0.886903\pi\)
\(758\) 4.77012e7i 3.01548i
\(759\) 5.92435e6 0.373281
\(760\) 0 0
\(761\) 1.92442e7 1.20459 0.602293 0.798275i \(-0.294254\pi\)
0.602293 + 0.798275i \(0.294254\pi\)
\(762\) − 8.45376e6i − 0.527427i
\(763\) − 4.55200e6i − 0.283068i
\(764\) −2.43620e7 −1.51001
\(765\) 0 0
\(766\) 2.23079e7 1.37368
\(767\) − 382200.i − 0.0234586i
\(768\) 2.96719e7i 1.81528i
\(769\) −8.21185e6 −0.500755 −0.250378 0.968148i \(-0.580555\pi\)
−0.250378 + 0.968148i \(0.580555\pi\)
\(770\) 0 0
\(771\) 1.64767e7 0.998241
\(772\) 6.72897e7i 4.06355i
\(773\) 1.86187e7i 1.12073i 0.828247 + 0.560363i \(0.189338\pi\)
−0.828247 + 0.560363i \(0.810662\pi\)
\(774\) −5.11360e6 −0.306813
\(775\) 0 0
\(776\) 2.10622e7 1.25559
\(777\) 7.13303e6i 0.423859i
\(778\) − 4.84024e7i − 2.86694i
\(779\) −1.76028e6 −0.103929
\(780\) 0 0
\(781\) −1.35859e7 −0.797006
\(782\) − 3.14093e7i − 1.83671i
\(783\) 1.38771e7i 0.808898i
\(784\) −3.41902e6 −0.198661
\(785\) 0 0
\(786\) −8.62204e6 −0.497799
\(787\) − 2.62501e7i − 1.51075i −0.655291 0.755377i \(-0.727454\pi\)
0.655291 0.755377i \(-0.272546\pi\)
\(788\) − 6.73234e7i − 3.86234i
\(789\) 1.80737e7 1.03360
\(790\) 0 0
\(791\) 4.08435e6 0.232103
\(792\) − 3.92544e6i − 0.222370i
\(793\) − 3.59072e6i − 0.202768i
\(794\) 9.95820e6 0.560570
\(795\) 0 0
\(796\) −5.71714e7 −3.19813
\(797\) 1.00373e7i 0.559720i 0.960041 + 0.279860i \(0.0902881\pi\)
−0.960041 + 0.279860i \(0.909712\pi\)
\(798\) 672280.i 0.0373717i
\(799\) 1.60559e7 0.889751
\(800\) 0 0
\(801\) 2.37735e6 0.130922
\(802\) − 3.31605e7i − 1.82048i
\(803\) 1.57950e7i 0.864433i
\(804\) −4.60806e7 −2.51407
\(805\) 0 0
\(806\) −1.07016e7 −0.580245
\(807\) − 1.78858e7i − 0.966772i
\(808\) − 1.39306e7i − 0.750655i
\(809\) −1.40884e7 −0.756816 −0.378408 0.925639i \(-0.623528\pi\)
−0.378408 + 0.925639i \(0.623528\pi\)
\(810\) 0 0
\(811\) 1.81433e7 0.968646 0.484323 0.874889i \(-0.339066\pi\)
0.484323 + 0.874889i \(0.339066\pi\)
\(812\) 1.13888e7i 0.606160i
\(813\) − 2.31076e7i − 1.22611i
\(814\) −2.41234e7 −1.27608
\(815\) 0 0
\(816\) 3.43298e7 1.80487
\(817\) 1.06624e6i 0.0558856i
\(818\) − 3.07273e7i − 1.60562i
\(819\) 322420. 0.0167962
\(820\) 0 0
\(821\) −2.13669e7 −1.10633 −0.553164 0.833072i \(-0.686580\pi\)
−0.553164 + 0.833072i \(0.686580\pi\)
\(822\) 2.86247e7i 1.47761i
\(823\) 1.78017e7i 0.916142i 0.888916 + 0.458071i \(0.151459\pi\)
−0.888916 + 0.458071i \(0.848541\pi\)
\(824\) 1.91016e7 0.980058
\(825\) 0 0
\(826\) −1.33770e6 −0.0682195
\(827\) − 1.62921e7i − 0.828350i −0.910197 0.414175i \(-0.864070\pi\)
0.910197 0.414175i \(-0.135930\pi\)
\(828\) − 5.82950e6i − 0.295499i
\(829\) 2.08499e6 0.105370 0.0526851 0.998611i \(-0.483222\pi\)
0.0526851 + 0.998611i \(0.483222\pi\)
\(830\) 0 0
\(831\) 1.48973e7 0.748348
\(832\) − 2.57152e6i − 0.128790i
\(833\) − 4.13452e6i − 0.206449i
\(834\) 4.95684e6 0.246769
\(835\) 0 0
\(836\) −1.54605e6 −0.0765081
\(837\) 3.10346e7i 1.53120i
\(838\) − 2.81438e7i − 1.38443i
\(839\) 2.27850e7 1.11749 0.558745 0.829340i \(-0.311283\pi\)
0.558745 + 0.829340i \(0.311283\pi\)
\(840\) 0 0
\(841\) −8.82842e6 −0.430421
\(842\) 3.05802e7i 1.48648i
\(843\) 312788.i 0.0151594i
\(844\) −7.82498e7 −3.78118
\(845\) 0 0
\(846\) 4.38228e6 0.210511
\(847\) − 5.25412e6i − 0.251647i
\(848\) 3.22109e6i 0.153820i
\(849\) −3.49403e7 −1.66363
\(850\) 0 0
\(851\) −1.89660e7 −0.897740
\(852\) − 5.57491e7i − 2.63111i
\(853\) − 2.26975e7i − 1.06808i −0.845458 0.534042i \(-0.820672\pi\)
0.845458 0.534042i \(-0.179328\pi\)
\(854\) −1.25675e7 −0.589664
\(855\) 0 0
\(856\) 5.26766e7 2.45716
\(857\) − 2.52900e7i − 1.17624i −0.808774 0.588120i \(-0.799868\pi\)
0.808774 0.588120i \(-0.200132\pi\)
\(858\) − 4.54720e6i − 0.210875i
\(859\) 1.03947e7 0.480652 0.240326 0.970692i \(-0.422746\pi\)
0.240326 + 0.970692i \(0.422746\pi\)
\(860\) 0 0
\(861\) −1.23219e7 −0.566462
\(862\) 1.93750e7i 0.888122i
\(863\) 4.33399e7i 1.98089i 0.137892 + 0.990447i \(0.455967\pi\)
−0.137892 + 0.990447i \(0.544033\pi\)
\(864\) 1.10432e7 0.503281
\(865\) 0 0
\(866\) −3.94790e7 −1.78884
\(867\) 2.16360e7i 0.977527i
\(868\) 2.54698e7i 1.14743i
\(869\) −7.37389e6 −0.331243
\(870\) 0 0
\(871\) 6.77656e6 0.302666
\(872\) 3.34433e7i 1.48942i
\(873\) 2.74978e6i 0.122113i
\(874\) −1.78752e6 −0.0791539
\(875\) 0 0
\(876\) −6.48141e7 −2.85370
\(877\) − 3.71659e7i − 1.63172i −0.578248 0.815861i \(-0.696264\pi\)
0.578248 0.815861i \(-0.303736\pi\)
\(878\) 7.41770e7i 3.24738i
\(879\) −2.70449e7 −1.18063
\(880\) 0 0
\(881\) 9.04785e6 0.392740 0.196370 0.980530i \(-0.437085\pi\)
0.196370 + 0.980530i \(0.437085\pi\)
\(882\) − 1.12847e6i − 0.0488448i
\(883\) 3.29679e7i 1.42295i 0.702712 + 0.711474i \(0.251972\pi\)
−0.702712 + 0.711474i \(0.748028\pi\)
\(884\) −1.63934e7 −0.705569
\(885\) 0 0
\(886\) −1.40269e7 −0.600313
\(887\) 1.61099e7i 0.687517i 0.939058 + 0.343758i \(0.111700\pi\)
−0.939058 + 0.343758i \(0.888300\pi\)
\(888\) − 5.24059e7i − 2.23022i
\(889\) 2.95882e6 0.125564
\(890\) 0 0
\(891\) −1.05372e7 −0.444663
\(892\) 5.60500e7i 2.35865i
\(893\) − 913752.i − 0.0383442i
\(894\) 2.83164e6 0.118493
\(895\) 0 0
\(896\) −1.32653e7 −0.552009
\(897\) − 3.57504e6i − 0.148354i
\(898\) 5.90574e6i 0.244390i
\(899\) 2.61272e7 1.07819
\(900\) 0 0
\(901\) −3.89516e6 −0.159850
\(902\) − 4.16718e7i − 1.70540i
\(903\) 7.46368e6i 0.304603i
\(904\) −3.00074e7 −1.22126
\(905\) 0 0
\(906\) 9.92656e6 0.401771
\(907\) 4.47286e7i 1.80537i 0.430300 + 0.902686i \(0.358408\pi\)
−0.430300 + 0.902686i \(0.641592\pi\)
\(908\) 5.05798e6i 0.203593i
\(909\) 1.81871e6 0.0730053
\(910\) 0 0
\(911\) −6.60518e6 −0.263687 −0.131844 0.991271i \(-0.542090\pi\)
−0.131844 + 0.991271i \(0.542090\pi\)
\(912\) − 1.95373e6i − 0.0777816i
\(913\) − 4.76482e6i − 0.189177i
\(914\) −2.90484e7 −1.15016
\(915\) 0 0
\(916\) 7.69730e7 3.03110
\(917\) − 3.01771e6i − 0.118510i
\(918\) 6.99132e7i 2.73812i
\(919\) 3.08930e7 1.20662 0.603311 0.797506i \(-0.293848\pi\)
0.603311 + 0.797506i \(0.293848\pi\)
\(920\) 0 0
\(921\) 6.42704e6 0.249667
\(922\) − 9.22684e6i − 0.357459i
\(923\) 8.19840e6i 0.316756i
\(924\) −1.08223e7 −0.417005
\(925\) 0 0
\(926\) −7.18235e7 −2.75258
\(927\) 2.49382e6i 0.0953160i
\(928\) − 9.29696e6i − 0.354381i
\(929\) 4.87215e6 0.185217 0.0926087 0.995703i \(-0.470479\pi\)
0.0926087 + 0.995703i \(0.470479\pi\)
\(930\) 0 0
\(931\) −235298. −0.00889701
\(932\) 1.35134e7i 0.509593i
\(933\) − 9.33979e6i − 0.351264i
\(934\) −6.12570e6 −0.229767
\(935\) 0 0
\(936\) −2.36880e6 −0.0883769
\(937\) 3.25004e7i 1.20932i 0.796485 + 0.604658i \(0.206690\pi\)
−0.796485 + 0.604658i \(0.793310\pi\)
\(938\) − 2.37180e7i − 0.880177i
\(939\) −1.55448e6 −0.0575334
\(940\) 0 0
\(941\) −2.64040e6 −0.0972066 −0.0486033 0.998818i \(-0.515477\pi\)
−0.0486033 + 0.998818i \(0.515477\pi\)
\(942\) − 4.10934e7i − 1.50884i
\(943\) − 3.27627e7i − 1.19978i
\(944\) 3.88752e6 0.141985
\(945\) 0 0
\(946\) −2.52416e7 −0.917042
\(947\) 4.08179e7i 1.47903i 0.673142 + 0.739513i \(0.264944\pi\)
−0.673142 + 0.739513i \(0.735056\pi\)
\(948\) − 3.02584e7i − 1.09351i
\(949\) 9.53148e6 0.343554
\(950\) 0 0
\(951\) 962892. 0.0345244
\(952\) 3.03761e7i 1.08627i
\(953\) − 6.71983e6i − 0.239677i −0.992793 0.119838i \(-0.961762\pi\)
0.992793 0.119838i \(-0.0382376\pi\)
\(954\) −1.06314e6 −0.0378198
\(955\) 0 0
\(956\) 3.28375e7 1.16205
\(957\) 1.11017e7i 0.391840i
\(958\) − 2.60330e7i − 0.916454i
\(959\) −1.00186e7 −0.351773
\(960\) 0 0
\(961\) 2.98016e7 1.04095
\(962\) 1.45572e7i 0.507154i
\(963\) 6.87723e6i 0.238972i
\(964\) −5.48019e7 −1.89934
\(965\) 0 0
\(966\) −1.25126e7 −0.431426
\(967\) 2.78979e6i 0.0959413i 0.998849 + 0.0479707i \(0.0152754\pi\)
−0.998849 + 0.0479707i \(0.984725\pi\)
\(968\) 3.86017e7i 1.32409i
\(969\) 2.36258e6 0.0808310
\(970\) 0 0
\(971\) 3.33594e7 1.13545 0.567727 0.823217i \(-0.307823\pi\)
0.567727 + 0.823217i \(0.307823\pi\)
\(972\) 2.38486e7i 0.809648i
\(973\) 1.73489e6i 0.0587477i
\(974\) 5.46309e7 1.84519
\(975\) 0 0
\(976\) 3.65228e7 1.22727
\(977\) 7.60033e6i 0.254739i 0.991855 + 0.127370i \(0.0406534\pi\)
−0.991855 + 0.127370i \(0.959347\pi\)
\(978\) 1.84688e6i 0.0617435i
\(979\) 1.17350e7 0.391316
\(980\) 0 0
\(981\) −4.36621e6 −0.144854
\(982\) 1.64090e7i 0.543004i
\(983\) − 5.79760e6i − 0.191366i −0.995412 0.0956829i \(-0.969497\pi\)
0.995412 0.0956829i \(-0.0305035\pi\)
\(984\) 9.05285e7 2.98056
\(985\) 0 0
\(986\) 5.88580e7 1.92803
\(987\) − 6.39626e6i − 0.208994i
\(988\) 932960.i 0.0304068i
\(989\) −1.98451e7 −0.645153
\(990\) 0 0
\(991\) 1.26825e7 0.410224 0.205112 0.978739i \(-0.434244\pi\)
0.205112 + 0.978739i \(0.434244\pi\)
\(992\) − 2.07917e7i − 0.670827i
\(993\) 7.90227e6i 0.254319i
\(994\) 2.86944e7 0.921152
\(995\) 0 0
\(996\) 1.95522e7 0.624521
\(997\) − 1.44400e7i − 0.460077i −0.973182 0.230039i \(-0.926115\pi\)
0.973182 0.230039i \(-0.0738853\pi\)
\(998\) − 2.99796e7i − 0.952796i
\(999\) 4.22159e7 1.33833
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 175.6.b.a.99.2 2
5.2 odd 4 7.6.a.a.1.1 1
5.3 odd 4 175.6.a.b.1.1 1
5.4 even 2 inner 175.6.b.a.99.1 2
15.2 even 4 63.6.a.e.1.1 1
20.7 even 4 112.6.a.g.1.1 1
35.2 odd 12 49.6.c.c.18.1 2
35.12 even 12 49.6.c.b.18.1 2
35.17 even 12 49.6.c.b.30.1 2
35.27 even 4 49.6.a.a.1.1 1
35.32 odd 12 49.6.c.c.30.1 2
40.27 even 4 448.6.a.c.1.1 1
40.37 odd 4 448.6.a.m.1.1 1
55.32 even 4 847.6.a.b.1.1 1
60.47 odd 4 1008.6.a.y.1.1 1
105.62 odd 4 441.6.a.k.1.1 1
140.27 odd 4 784.6.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 5.2 odd 4
49.6.a.a.1.1 1 35.27 even 4
49.6.c.b.18.1 2 35.12 even 12
49.6.c.b.30.1 2 35.17 even 12
49.6.c.c.18.1 2 35.2 odd 12
49.6.c.c.30.1 2 35.32 odd 12
63.6.a.e.1.1 1 15.2 even 4
112.6.a.g.1.1 1 20.7 even 4
175.6.a.b.1.1 1 5.3 odd 4
175.6.b.a.99.1 2 5.4 even 2 inner
175.6.b.a.99.2 2 1.1 even 1 trivial
441.6.a.k.1.1 1 105.62 odd 4
448.6.a.c.1.1 1 40.27 even 4
448.6.a.m.1.1 1 40.37 odd 4
784.6.a.c.1.1 1 140.27 odd 4
847.6.a.b.1.1 1 55.32 even 4
1008.6.a.y.1.1 1 60.47 odd 4