Properties

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0000i q^{2} +14.0000i q^{3} -68.0000 q^{4} +140.000 q^{6} -49.0000i q^{7} +360.000i q^{8} +47.0000 q^{9} +O(q^{10})\) \(q-10.0000i q^{2} +14.0000i q^{3} -68.0000 q^{4} +140.000 q^{6} -49.0000i q^{7} +360.000i q^{8} +47.0000 q^{9} +232.000 q^{11} -952.000i q^{12} +140.000i q^{13} -490.000 q^{14} +1424.00 q^{16} -1722.00i q^{17} -470.000i q^{18} +98.0000 q^{19} +686.000 q^{21} -2320.00i q^{22} -1824.00i q^{23} -5040.00 q^{24} +1400.00 q^{26} +4060.00i q^{27} +3332.00i q^{28} -3418.00 q^{29} -7644.00 q^{31} -2720.00i q^{32} +3248.00i q^{33} -17220.0 q^{34} -3196.00 q^{36} -10398.0i q^{37} -980.000i q^{38} -1960.00 q^{39} -17962.0 q^{41} -6860.00i q^{42} -10880.0i q^{43} -15776.0 q^{44} -18240.0 q^{46} +9324.00i q^{47} +19936.0i q^{48} -2401.00 q^{49} +24108.0 q^{51} -9520.00i q^{52} -2262.00i q^{53} +40600.0 q^{54} +17640.0 q^{56} +1372.00i q^{57} +34180.0i q^{58} +2730.00 q^{59} +25648.0 q^{61} +76440.0i q^{62} -2303.00i q^{63} +18368.0 q^{64} +32480.0 q^{66} -48404.0i q^{67} +117096. i q^{68} +25536.0 q^{69} -58560.0 q^{71} +16920.0i q^{72} -68082.0i q^{73} -103980. q^{74} -6664.00 q^{76} -11368.0i q^{77} +19600.0i q^{78} -31784.0 q^{79} -45419.0 q^{81} +179620. i q^{82} +20538.0i q^{83} -46648.0 q^{84} -108800. q^{86} -47852.0i q^{87} +83520.0i q^{88} +50582.0 q^{89} +6860.00 q^{91} +124032. i q^{92} -107016. i q^{93} +93240.0 q^{94} +38080.0 q^{96} -58506.0i q^{97} +24010.0i q^{98} +10904.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 136 q^{4} + 280 q^{6} + 94 q^{9}+O(q^{10}) \) 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 \(-0.654920\pi\)
0.467707 0.883883i \(-0.345080\pi\)
\(3\) 14.0000i 0.898100i 0.893507 + 0.449050i \(0.148238\pi\)
−0.893507 + 0.449050i \(0.851762\pi\)
\(4\) −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 \(0.0366479\pi\)
−0.993380 + 0.114879i \(0.963352\pi\)
\(14\) −490.000 −0.668153
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) − 1722.00i − 1.44514i −0.691296 0.722572i \(-0.742960\pi\)
0.691296 0.722572i \(-0.257040\pi\)
\(18\) − 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 \(-0.882954\pi\)
0.933153 0.359480i \(-0.117046\pi\)
\(24\) −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 \(-0.785371\pi\)
0.781159 0.624332i \(-0.214629\pi\)
\(38\) − 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 \(-0.851898\pi\)
0.893697 0.448671i \(-0.148102\pi\)
\(44\) −15776.0 −1.22847
\(45\) 0 0
\(46\) −18240.0 −1.27096
\(47\) 9324.00i 0.615684i 0.951438 + 0.307842i \(0.0996068\pi\)
−0.951438 + 0.307842i \(0.900393\pi\)
\(48\) 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 \(-0.982387\pi\)
0.998469 0.0553061i \(-0.0176135\pi\)
\(54\) 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 \(-0.771122\pi\)
0.752437 0.658664i \(-0.228878\pi\)
\(68\) 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 \(-0.731185\pi\)
0.664099 0.747645i \(-0.268815\pi\)
\(74\) −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 \(0.0523167\pi\)
−0.986524 + 0.163619i \(0.947683\pi\)
\(84\) −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 \(-0.897769\pi\)
0.948867 0.315676i \(-0.102231\pi\)
\(98\) 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 \(-0.920752\pi\)
0.969168 0.246402i \(-0.0792483\pi\)
\(104\) −50400.0 −0.456927
\(105\) 0 0
\(106\) −22620.0 −0.195537
\(107\) − 146324.i − 1.23554i −0.786360 0.617769i \(-0.788037\pi\)
0.786360 0.617769i \(-0.211963\pi\)
\(108\) − 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 \(0.0993398\pi\)
−0.951695 + 0.307044i \(0.900660\pi\)
\(114\) 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 \(0.0531191\pi\)
−0.986108 + 0.166105i \(0.946881\pi\)
\(128\) − 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 \(-0.845928\pi\)
0.885126 0.465352i \(-0.154072\pi\)
\(138\) − 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 \(0.157620\pi\)
−0.879885 + 0.475187i \(0.842380\pi\)
\(158\) 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 \(-0.993810\pi\)
0.999811 0.0194452i \(-0.00618998\pi\)
\(164\) 1.22142e6 3.54612
\(165\) 0 0
\(166\) 205380. 0.578479
\(167\) 493612.i 1.36960i 0.728730 + 0.684801i \(0.240111\pi\)
−0.728730 + 0.684801i \(0.759889\pi\)
\(168\) 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 \(-0.901094\pi\)
0.952113 0.305745i \(-0.0989056\pi\)
\(174\) −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 \(0.405364\pi\)
−0.292948 + 0.956128i \(0.594636\pi\)
\(194\) −585060. −1.11608
\(195\) 0 0
\(196\) 163268. 0.303571
\(197\) − 990050.i − 1.81757i −0.417263 0.908786i \(-0.637011\pi\)
0.417263 0.908786i \(-0.362989\pi\)
\(198\) − 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 \(0.187273\pi\)
−0.831866 + 0.554976i \(0.812727\pi\)
\(224\) −133280. −0.177478
\(225\) 0 0
\(226\) 833540. 1.08556
\(227\) 74382.0i 0.0958083i 0.998852 + 0.0479042i \(0.0152542\pi\)
−0.998852 + 0.0479042i \(0.984746\pi\)
\(228\) − 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 \(0.0382588\pi\)
−0.992785 + 0.119904i \(0.961741\pi\)
\(234\) 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 \(-0.812431\pi\)
0.831349 0.555751i \(-0.187569\pi\)
\(258\) − 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 \(-0.804831\pi\)
0.817845 0.575438i \(-0.195169\pi\)
\(264\) −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 \(-0.863212\pi\)
0.909077 0.416628i \(-0.136788\pi\)
\(278\) − 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 \(0.376941\pi\)
−0.377042 + 0.926196i \(0.623059\pi\)
\(284\) 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 \(0.228298\pi\)
−0.753637 + 0.657291i \(0.771702\pi\)
\(294\) −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 \(-0.955612\pi\)
0.990293 0.138997i \(-0.0443880\pi\)
\(308\) 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 \(0.0101974\pi\)
−0.999487 + 0.0320306i \(0.989803\pi\)
\(314\) 2.93524e6 1.68004
\(315\) 0 0
\(316\) 2.16131e6 1.21759
\(317\) − 68778.0i − 0.0384416i −0.999815 0.0192208i \(-0.993881\pi\)
0.999815 0.0192208i \(-0.00611855\pi\)
\(318\) − 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 \(0.166001\pi\)
−0.867069 + 0.498188i \(0.833999\pi\)
\(338\) − 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 \(-0.996222\pi\)
0.999930 0.0118700i \(-0.00377842\pi\)
\(348\) 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 \(-0.673162\pi\)
0.517565 0.855644i \(-0.326838\pi\)
\(354\) 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 \(0.0877264\pi\)
−0.962262 + 0.272125i \(0.912274\pi\)
\(368\) − 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 \(0.0964291\pi\)
−0.954463 + 0.298329i \(0.903571\pi\)
\(374\) −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 \(0.127019\pi\)
−0.921434 + 0.388536i \(0.872981\pi\)
\(384\) 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 \(0.0506829\pi\)
−0.987350 + 0.158553i \(0.949317\pi\)
\(398\) − 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 \(-0.831138\pi\)
0.862557 0.505961i \(-0.168862\pi\)
\(434\) 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 \(-0.945690\pi\)
0.985480 0.169794i \(-0.0543103\pi\)
\(444\) −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 \(-0.894530\pi\)
0.945606 0.325313i \(-0.105470\pi\)
\(458\) 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 \(-0.715958\pi\)
0.627588 0.778546i \(-0.284042\pi\)
\(464\) −4.86723e6 −1.04951
\(465\) 0 0
\(466\) 1.98726e6 0.423926
\(467\) − 612570.i − 0.129976i −0.997886 0.0649881i \(-0.979299\pi\)
0.997886 0.0649881i \(-0.0207009\pi\)
\(468\) − 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 \(0.174776\pi\)
−0.853008 + 0.521898i \(0.825224\pi\)
\(488\) 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 \(0.207815\pi\)
−0.794343 + 0.607469i \(0.792185\pi\)
\(504\) 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 \(-0.855554\pi\)
0.898791 0.438377i \(-0.144446\pi\)
\(524\) 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 \(-0.883657\pi\)
0.933945 0.357418i \(-0.116343\pi\)
\(548\) 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 \(0.211213\pi\)
−0.787814 + 0.615913i \(0.788787\pi\)
\(558\) 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 \(-0.691340\pi\)
0.565561 0.824707i \(-0.308660\pi\)
\(564\) 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 \(0.0530508\pi\)
−0.986144 + 0.165894i \(0.946949\pi\)
\(578\) 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 \(-0.672493\pi\)
0.515766 0.856729i \(-0.327507\pi\)
\(588\) 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 \(0.199079\pi\)
−0.810714 + 0.585442i \(0.800921\pi\)
\(594\) 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 \(-0.875742\pi\)
0.924769 0.380530i \(-0.124258\pi\)
\(608\) − 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 \(0.174333\pi\)
−0.853734 + 0.520710i \(0.825667\pi\)
\(614\) −4.59074e6 −0.491430
\(615\) 0 0
\(616\) 4.09248e6 0.434545
\(617\) − 7.84742e6i − 0.829877i −0.909849 0.414939i \(-0.863803\pi\)
0.909849 0.414939i \(-0.136197\pi\)
\(618\) − 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 \(-0.695516\pi\)
0.576330 0.817217i \(-0.304484\pi\)
\(644\) 6.07757e6 0.577451
\(645\) 0 0
\(646\) −1.68756e6 −0.159103
\(647\) − 54964.0i − 0.00516200i −0.999997 0.00258100i \(-0.999178\pi\)
0.999997 0.00258100i \(-0.000821558\pi\)
\(648\) − 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 \(0.00708702\pi\)
−0.999752 + 0.0222627i \(0.992913\pi\)
\(654\) −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 \(-0.960237\pi\)
0.992208 0.124594i \(-0.0397629\pi\)
\(674\) 2.07729e7 1.76136
\(675\) 0 0
\(676\) −2.39151e7 −2.01282
\(677\) − 1.34992e7i − 1.13198i −0.824414 0.565988i \(-0.808495\pi\)
0.824414 0.565988i \(-0.191505\pi\)
\(678\) 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 \(0.0714830\pi\)
−0.974890 + 0.222688i \(0.928517\pi\)
\(684\) −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 \(0.181866\pi\)
−0.841172 + 0.540768i \(0.818134\pi\)
\(728\) 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 \(0.198465\pi\)
−0.811841 + 0.583878i \(0.801535\pi\)
\(734\) 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 \(0.167771\pi\)
−0.864286 + 0.503001i \(0.832229\pi\)
\(744\) 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 \(-0.886903\pi\)
0.937540 0.347877i \(-0.113097\pi\)
\(758\) − 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 \(-0.810662\pi\)
0.828247 0.560363i \(-0.189338\pi\)
\(774\) −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 \(0.272546\pi\)
−0.655291 + 0.755377i \(0.727454\pi\)
\(788\) 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 \(-0.909712\pi\)
0.960041 0.279860i \(-0.0902881\pi\)
\(798\) − 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 \(-0.848541\pi\)
0.888916 0.458071i \(-0.151459\pi\)
\(824\) 1.91016e7 0.980058
\(825\) 0 0
\(826\) −1.33770e6 −0.0682195
\(827\) 1.62921e7i 0.828350i 0.910197 + 0.414175i \(0.135930\pi\)
−0.910197 + 0.414175i \(0.864070\pi\)
\(828\) 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 \(0.179328\pi\)
−0.845458 + 0.534042i \(0.820672\pi\)
\(854\) −1.25675e7 −0.589664
\(855\) 0 0
\(856\) 5.26766e7 2.45716
\(857\) 2.52900e7i 1.17624i 0.808774 + 0.588120i \(0.200132\pi\)
−0.808774 + 0.588120i \(0.799868\pi\)
\(858\) 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 \(-0.544033\pi\)
0.137892 0.990447i \(-0.455967\pi\)
\(864\) 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 \(0.303736\pi\)
−0.578248 + 0.815861i \(0.696264\pi\)
\(878\) − 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 \(-0.748028\pi\)
0.702712 0.711474i \(-0.251972\pi\)
\(884\) −1.63934e7 −0.705569
\(885\) 0 0
\(886\) −1.40269e7 −0.600313
\(887\) − 1.61099e7i − 0.687517i −0.939058 0.343758i \(-0.888300\pi\)
0.939058 0.343758i \(-0.111700\pi\)
\(888\) 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 \(-0.641592\pi\)
0.430300 0.902686i \(-0.358408\pi\)
\(908\) − 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 \(-0.793310\pi\)
0.796485 0.604658i \(-0.206690\pi\)
\(938\) 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 \(-0.735056\pi\)
0.673142 0.739513i \(-0.264944\pi\)
\(948\) 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 \(0.0382376\pi\)
−0.992793 + 0.119838i \(0.961762\pi\)
\(954\) −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 \(-0.984725\pi\)
0.998849 0.0479707i \(-0.0152754\pi\)
\(968\) − 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 \(-0.959347\pi\)
0.991855 0.127370i \(-0.0406534\pi\)
\(978\) − 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 \(0.0305035\pi\)
−0.995412 + 0.0956829i \(0.969497\pi\)
\(984\) 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 \(0.0738853\pi\)
−0.973182 + 0.230039i \(0.926115\pi\)
\(998\) 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.99.1 2
5.2 odd 4 175.6.a.b.1.1 1
5.3 odd 4 7.6.a.a.1.1 1
5.4 even 2 inner 175.6.b.a.99.2 2
15.8 even 4 63.6.a.e.1.1 1
20.3 even 4 112.6.a.g.1.1 1
35.3 even 12 49.6.c.b.30.1 2
35.13 even 4 49.6.a.a.1.1 1
35.18 odd 12 49.6.c.c.30.1 2
35.23 odd 12 49.6.c.c.18.1 2
35.33 even 12 49.6.c.b.18.1 2
40.3 even 4 448.6.a.c.1.1 1
40.13 odd 4 448.6.a.m.1.1 1
55.43 even 4 847.6.a.b.1.1 1
60.23 odd 4 1008.6.a.y.1.1 1
105.83 odd 4 441.6.a.k.1.1 1
140.83 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 5.3 odd 4
49.6.a.a.1.1 1 35.13 even 4
49.6.c.b.18.1 2 35.33 even 12
49.6.c.b.30.1 2 35.3 even 12
49.6.c.c.18.1 2 35.23 odd 12
49.6.c.c.30.1 2 35.18 odd 12
63.6.a.e.1.1 1 15.8 even 4
112.6.a.g.1.1 1 20.3 even 4
175.6.a.b.1.1 1 5.2 odd 4
175.6.b.a.99.1 2 1.1 even 1 trivial
175.6.b.a.99.2 2 5.4 even 2 inner
441.6.a.k.1.1 1 105.83 odd 4
448.6.a.c.1.1 1 40.3 even 4
448.6.a.m.1.1 1 40.13 odd 4
784.6.a.c.1.1 1 140.83 odd 4
847.6.a.b.1.1 1 55.43 even 4
1008.6.a.y.1.1 1 60.23 odd 4