Properties

Label 350.6.c.g.99.1
Level $350$
Weight $6$
Character 350.99
Analytic conductor $56.134$
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] = [350,6,Mod(99,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, 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("350.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 350.c (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: \(56.1343369345\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.6.c.g.99.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -49.0000i q^{7} +64.0000i q^{8} +162.000 q^{9} +O(q^{10})\) \(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -49.0000i q^{7} +64.0000i q^{8} +162.000 q^{9} -187.000 q^{11} -144.000i q^{12} -627.000i q^{13} -196.000 q^{14} +256.000 q^{16} +1813.00i q^{17} -648.000i q^{18} -258.000 q^{19} +441.000 q^{21} +748.000i q^{22} -2970.00i q^{23} -576.000 q^{24} -2508.00 q^{26} +3645.00i q^{27} +784.000i q^{28} -1299.00 q^{29} +1916.00 q^{31} -1024.00i q^{32} -1683.00i q^{33} +7252.00 q^{34} -2592.00 q^{36} +6578.00i q^{37} +1032.00i q^{38} +5643.00 q^{39} +6676.00 q^{41} -1764.00i q^{42} -3178.00i q^{43} +2992.00 q^{44} -11880.0 q^{46} -22001.0i q^{47} +2304.00i q^{48} -2401.00 q^{49} -16317.0 q^{51} +10032.0i q^{52} -26168.0i q^{53} +14580.0 q^{54} +3136.00 q^{56} -2322.00i q^{57} +5196.00i q^{58} -3932.00 q^{59} -48740.0 q^{61} -7664.00i q^{62} -7938.00i q^{63} -4096.00 q^{64} -6732.00 q^{66} -44832.0i q^{67} -29008.0i q^{68} +26730.0 q^{69} +63736.0 q^{71} +10368.0i q^{72} -60470.0i q^{73} +26312.0 q^{74} +4128.00 q^{76} +9163.00i q^{77} -22572.0i q^{78} +43721.0 q^{79} +6561.00 q^{81} -26704.0i q^{82} -97276.0i q^{83} -7056.00 q^{84} -12712.0 q^{86} -11691.0i q^{87} -11968.0i q^{88} -45560.0 q^{89} -30723.0 q^{91} +47520.0i q^{92} +17244.0i q^{93} -88004.0 q^{94} +9216.00 q^{96} -57295.0i q^{97} +9604.00i q^{98} -30294.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 72 q^{6} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 72 q^{6} + 324 q^{9} - 374 q^{11} - 392 q^{14} + 512 q^{16} - 516 q^{19} + 882 q^{21} - 1152 q^{24} - 5016 q^{26} - 2598 q^{29} + 3832 q^{31} + 14504 q^{34} - 5184 q^{36} + 11286 q^{39} + 13352 q^{41} + 5984 q^{44} - 23760 q^{46} - 4802 q^{49} - 32634 q^{51} + 29160 q^{54} + 6272 q^{56} - 7864 q^{59} - 97480 q^{61} - 8192 q^{64} - 13464 q^{66} + 53460 q^{69} + 127472 q^{71} + 52624 q^{74} + 8256 q^{76} + 87442 q^{79} + 13122 q^{81} - 14112 q^{84} - 25424 q^{86} - 91120 q^{89} - 61446 q^{91} - 176008 q^{94} + 18432 q^{96} - 60588 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) 9.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 36.0000 0.408248
\(7\) − 49.0000i − 0.377964i
\(8\) 64.0000i 0.353553i
\(9\) 162.000 0.666667
\(10\) 0 0
\(11\) −187.000 −0.465972 −0.232986 0.972480i \(-0.574850\pi\)
−0.232986 + 0.972480i \(0.574850\pi\)
\(12\) − 144.000i − 0.288675i
\(13\) − 627.000i − 1.02899i −0.857495 0.514493i \(-0.827980\pi\)
0.857495 0.514493i \(-0.172020\pi\)
\(14\) −196.000 −0.267261
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1813.00i 1.52151i 0.649038 + 0.760756i \(0.275172\pi\)
−0.649038 + 0.760756i \(0.724828\pi\)
\(18\) − 648.000i − 0.471405i
\(19\) −258.000 −0.163959 −0.0819796 0.996634i \(-0.526124\pi\)
−0.0819796 + 0.996634i \(0.526124\pi\)
\(20\) 0 0
\(21\) 441.000 0.218218
\(22\) 748.000i 0.329492i
\(23\) − 2970.00i − 1.17068i −0.810789 0.585338i \(-0.800962\pi\)
0.810789 0.585338i \(-0.199038\pi\)
\(24\) −576.000 −0.204124
\(25\) 0 0
\(26\) −2508.00 −0.727602
\(27\) 3645.00i 0.962250i
\(28\) 784.000i 0.188982i
\(29\) −1299.00 −0.286823 −0.143412 0.989663i \(-0.545807\pi\)
−0.143412 + 0.989663i \(0.545807\pi\)
\(30\) 0 0
\(31\) 1916.00 0.358089 0.179045 0.983841i \(-0.442699\pi\)
0.179045 + 0.983841i \(0.442699\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) − 1683.00i − 0.269029i
\(34\) 7252.00 1.07587
\(35\) 0 0
\(36\) −2592.00 −0.333333
\(37\) 6578.00i 0.789932i 0.918696 + 0.394966i \(0.129244\pi\)
−0.918696 + 0.394966i \(0.870756\pi\)
\(38\) 1032.00i 0.115937i
\(39\) 5643.00 0.594085
\(40\) 0 0
\(41\) 6676.00 0.620236 0.310118 0.950698i \(-0.399632\pi\)
0.310118 + 0.950698i \(0.399632\pi\)
\(42\) − 1764.00i − 0.154303i
\(43\) − 3178.00i − 0.262109i −0.991375 0.131055i \(-0.958164\pi\)
0.991375 0.131055i \(-0.0418364\pi\)
\(44\) 2992.00 0.232986
\(45\) 0 0
\(46\) −11880.0 −0.827793
\(47\) − 22001.0i − 1.45277i −0.687286 0.726387i \(-0.741198\pi\)
0.687286 0.726387i \(-0.258802\pi\)
\(48\) 2304.00i 0.144338i
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) −16317.0 −0.878446
\(52\) 10032.0i 0.514493i
\(53\) − 26168.0i − 1.27962i −0.768533 0.639810i \(-0.779013\pi\)
0.768533 0.639810i \(-0.220987\pi\)
\(54\) 14580.0 0.680414
\(55\) 0 0
\(56\) 3136.00 0.133631
\(57\) − 2322.00i − 0.0946619i
\(58\) 5196.00i 0.202815i
\(59\) −3932.00 −0.147056 −0.0735281 0.997293i \(-0.523426\pi\)
−0.0735281 + 0.997293i \(0.523426\pi\)
\(60\) 0 0
\(61\) −48740.0 −1.67711 −0.838554 0.544819i \(-0.816598\pi\)
−0.838554 + 0.544819i \(0.816598\pi\)
\(62\) − 7664.00i − 0.253207i
\(63\) − 7938.00i − 0.251976i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) −6732.00 −0.190232
\(67\) − 44832.0i − 1.22012i −0.792357 0.610058i \(-0.791146\pi\)
0.792357 0.610058i \(-0.208854\pi\)
\(68\) − 29008.0i − 0.760756i
\(69\) 26730.0 0.675890
\(70\) 0 0
\(71\) 63736.0 1.50051 0.750255 0.661148i \(-0.229931\pi\)
0.750255 + 0.661148i \(0.229931\pi\)
\(72\) 10368.0i 0.235702i
\(73\) − 60470.0i − 1.32811i −0.747685 0.664053i \(-0.768835\pi\)
0.747685 0.664053i \(-0.231165\pi\)
\(74\) 26312.0 0.558566
\(75\) 0 0
\(76\) 4128.00 0.0819796
\(77\) 9163.00i 0.176121i
\(78\) − 22572.0i − 0.420081i
\(79\) 43721.0 0.788174 0.394087 0.919073i \(-0.371061\pi\)
0.394087 + 0.919073i \(0.371061\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) − 26704.0i − 0.438573i
\(83\) − 97276.0i − 1.54992i −0.632008 0.774962i \(-0.717769\pi\)
0.632008 0.774962i \(-0.282231\pi\)
\(84\) −7056.00 −0.109109
\(85\) 0 0
\(86\) −12712.0 −0.185339
\(87\) − 11691.0i − 0.165597i
\(88\) − 11968.0i − 0.164746i
\(89\) −45560.0 −0.609689 −0.304845 0.952402i \(-0.598605\pi\)
−0.304845 + 0.952402i \(0.598605\pi\)
\(90\) 0 0
\(91\) −30723.0 −0.388920
\(92\) 47520.0i 0.585338i
\(93\) 17244.0i 0.206743i
\(94\) −88004.0 −1.02727
\(95\) 0 0
\(96\) 9216.00 0.102062
\(97\) − 57295.0i − 0.618283i −0.951016 0.309142i \(-0.899958\pi\)
0.951016 0.309142i \(-0.100042\pi\)
\(98\) 9604.00i 0.101015i
\(99\) −30294.0 −0.310648
\(100\) 0 0
\(101\) −44970.0 −0.438651 −0.219326 0.975652i \(-0.570386\pi\)
−0.219326 + 0.975652i \(0.570386\pi\)
\(102\) 65268.0i 0.621155i
\(103\) 101405.i 0.941817i 0.882182 + 0.470908i \(0.156074\pi\)
−0.882182 + 0.470908i \(0.843926\pi\)
\(104\) 40128.0 0.363801
\(105\) 0 0
\(106\) −104672. −0.904828
\(107\) − 166002.i − 1.40170i −0.713311 0.700848i \(-0.752805\pi\)
0.713311 0.700848i \(-0.247195\pi\)
\(108\) − 58320.0i − 0.481125i
\(109\) −8289.00 −0.0668245 −0.0334123 0.999442i \(-0.510637\pi\)
−0.0334123 + 0.999442i \(0.510637\pi\)
\(110\) 0 0
\(111\) −59202.0 −0.456067
\(112\) − 12544.0i − 0.0944911i
\(113\) − 263206.i − 1.93910i −0.244898 0.969549i \(-0.578755\pi\)
0.244898 0.969549i \(-0.421245\pi\)
\(114\) −9288.00 −0.0669360
\(115\) 0 0
\(116\) 20784.0 0.143412
\(117\) − 101574.i − 0.685990i
\(118\) 15728.0i 0.103984i
\(119\) 88837.0 0.575078
\(120\) 0 0
\(121\) −126082. −0.782870
\(122\) 194960.i 1.18589i
\(123\) 60084.0i 0.358093i
\(124\) −30656.0 −0.179045
\(125\) 0 0
\(126\) −31752.0 −0.178174
\(127\) 30052.0i 0.165335i 0.996577 + 0.0826674i \(0.0263439\pi\)
−0.996577 + 0.0826674i \(0.973656\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) 28602.0 0.151329
\(130\) 0 0
\(131\) −120050. −0.611201 −0.305600 0.952160i \(-0.598857\pi\)
−0.305600 + 0.952160i \(0.598857\pi\)
\(132\) 26928.0i 0.134515i
\(133\) 12642.0i 0.0619707i
\(134\) −179328. −0.862752
\(135\) 0 0
\(136\) −116032. −0.537936
\(137\) 31776.0i 0.144643i 0.997381 + 0.0723216i \(0.0230408\pi\)
−0.997381 + 0.0723216i \(0.976959\pi\)
\(138\) − 106920.i − 0.477927i
\(139\) 200162. 0.878708 0.439354 0.898314i \(-0.355207\pi\)
0.439354 + 0.898314i \(0.355207\pi\)
\(140\) 0 0
\(141\) 198009. 0.838759
\(142\) − 254944.i − 1.06102i
\(143\) 117249.i 0.479478i
\(144\) 41472.0 0.166667
\(145\) 0 0
\(146\) −241880. −0.939113
\(147\) − 21609.0i − 0.0824786i
\(148\) − 105248.i − 0.394966i
\(149\) −309642. −1.14260 −0.571300 0.820741i \(-0.693561\pi\)
−0.571300 + 0.820741i \(0.693561\pi\)
\(150\) 0 0
\(151\) −208657. −0.744716 −0.372358 0.928089i \(-0.621451\pi\)
−0.372358 + 0.928089i \(0.621451\pi\)
\(152\) − 16512.0i − 0.0579683i
\(153\) 293706.i 1.01434i
\(154\) 36652.0 0.124536
\(155\) 0 0
\(156\) −90288.0 −0.297042
\(157\) 36010.0i 0.116593i 0.998299 + 0.0582967i \(0.0185669\pi\)
−0.998299 + 0.0582967i \(0.981433\pi\)
\(158\) − 174884.i − 0.557324i
\(159\) 235512. 0.738789
\(160\) 0 0
\(161\) −145530. −0.442474
\(162\) − 26244.0i − 0.0785674i
\(163\) − 175670.i − 0.517879i −0.965893 0.258940i \(-0.916627\pi\)
0.965893 0.258940i \(-0.0833731\pi\)
\(164\) −106816. −0.310118
\(165\) 0 0
\(166\) −389104. −1.09596
\(167\) − 157413.i − 0.436767i −0.975863 0.218383i \(-0.929922\pi\)
0.975863 0.218383i \(-0.0700783\pi\)
\(168\) 28224.0i 0.0771517i
\(169\) −21836.0 −0.0588107
\(170\) 0 0
\(171\) −41796.0 −0.109306
\(172\) 50848.0i 0.131055i
\(173\) 23471.0i 0.0596233i 0.999556 + 0.0298117i \(0.00949076\pi\)
−0.999556 + 0.0298117i \(0.990509\pi\)
\(174\) −46764.0 −0.117095
\(175\) 0 0
\(176\) −47872.0 −0.116493
\(177\) − 35388.0i − 0.0849030i
\(178\) 182240.i 0.431116i
\(179\) −612228. −1.42817 −0.714086 0.700058i \(-0.753158\pi\)
−0.714086 + 0.700058i \(0.753158\pi\)
\(180\) 0 0
\(181\) 528832. 1.19983 0.599917 0.800062i \(-0.295200\pi\)
0.599917 + 0.800062i \(0.295200\pi\)
\(182\) 122892.i 0.275008i
\(183\) − 438660.i − 0.968279i
\(184\) 190080. 0.413897
\(185\) 0 0
\(186\) 68976.0 0.146189
\(187\) − 339031.i − 0.708982i
\(188\) 352016.i 0.726387i
\(189\) 178605. 0.363696
\(190\) 0 0
\(191\) 540369. 1.07178 0.535892 0.844287i \(-0.319976\pi\)
0.535892 + 0.844287i \(0.319976\pi\)
\(192\) − 36864.0i − 0.0721688i
\(193\) − 960320.i − 1.85576i −0.372874 0.927882i \(-0.621628\pi\)
0.372874 0.927882i \(-0.378372\pi\)
\(194\) −229180. −0.437192
\(195\) 0 0
\(196\) 38416.0 0.0714286
\(197\) 761944.i 1.39881i 0.714728 + 0.699403i \(0.246551\pi\)
−0.714728 + 0.699403i \(0.753449\pi\)
\(198\) 121176.i 0.219661i
\(199\) −125084. −0.223908 −0.111954 0.993713i \(-0.535711\pi\)
−0.111954 + 0.993713i \(0.535711\pi\)
\(200\) 0 0
\(201\) 403488. 0.704434
\(202\) 179880.i 0.310173i
\(203\) 63651.0i 0.108409i
\(204\) 261072. 0.439223
\(205\) 0 0
\(206\) 405620. 0.665965
\(207\) − 481140.i − 0.780451i
\(208\) − 160512.i − 0.257246i
\(209\) 48246.0 0.0764004
\(210\) 0 0
\(211\) 627547. 0.970376 0.485188 0.874410i \(-0.338751\pi\)
0.485188 + 0.874410i \(0.338751\pi\)
\(212\) 418688.i 0.639810i
\(213\) 573624.i 0.866320i
\(214\) −664008. −0.991149
\(215\) 0 0
\(216\) −233280. −0.340207
\(217\) − 93884.0i − 0.135345i
\(218\) 33156.0i 0.0472521i
\(219\) 544230. 0.766782
\(220\) 0 0
\(221\) 1.13675e6 1.56561
\(222\) 236808.i 0.322488i
\(223\) 1.22110e6i 1.64433i 0.569248 + 0.822166i \(0.307234\pi\)
−0.569248 + 0.822166i \(0.692766\pi\)
\(224\) −50176.0 −0.0668153
\(225\) 0 0
\(226\) −1.05282e6 −1.37115
\(227\) 390547.i 0.503047i 0.967851 + 0.251524i \(0.0809316\pi\)
−0.967851 + 0.251524i \(0.919068\pi\)
\(228\) 37152.0i 0.0473309i
\(229\) 712124. 0.897360 0.448680 0.893692i \(-0.351894\pi\)
0.448680 + 0.893692i \(0.351894\pi\)
\(230\) 0 0
\(231\) −82467.0 −0.101683
\(232\) − 83136.0i − 0.101407i
\(233\) − 561576.i − 0.677671i −0.940846 0.338835i \(-0.889967\pi\)
0.940846 0.338835i \(-0.110033\pi\)
\(234\) −406296. −0.485068
\(235\) 0 0
\(236\) 62912.0 0.0735281
\(237\) 393489.i 0.455053i
\(238\) − 355348.i − 0.406641i
\(239\) 1.36084e6 1.54103 0.770515 0.637421i \(-0.219999\pi\)
0.770515 + 0.637421i \(0.219999\pi\)
\(240\) 0 0
\(241\) 530050. 0.587860 0.293930 0.955827i \(-0.405037\pi\)
0.293930 + 0.955827i \(0.405037\pi\)
\(242\) 504328.i 0.553573i
\(243\) 944784.i 1.02640i
\(244\) 779840. 0.838554
\(245\) 0 0
\(246\) 240336. 0.253210
\(247\) 161766.i 0.168712i
\(248\) 122624.i 0.126604i
\(249\) 875484. 0.894849
\(250\) 0 0
\(251\) 990330. 0.992192 0.496096 0.868268i \(-0.334766\pi\)
0.496096 + 0.868268i \(0.334766\pi\)
\(252\) 127008.i 0.125988i
\(253\) 555390.i 0.545503i
\(254\) 120208. 0.116909
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) − 1.81643e6i − 1.71548i −0.514083 0.857740i \(-0.671868\pi\)
0.514083 0.857740i \(-0.328132\pi\)
\(258\) − 114408.i − 0.107006i
\(259\) 322322. 0.298566
\(260\) 0 0
\(261\) −210438. −0.191215
\(262\) 480200.i 0.432184i
\(263\) 1.95847e6i 1.74594i 0.487777 + 0.872968i \(0.337808\pi\)
−0.487777 + 0.872968i \(0.662192\pi\)
\(264\) 107712. 0.0951162
\(265\) 0 0
\(266\) 50568.0 0.0438199
\(267\) − 410040.i − 0.352004i
\(268\) 717312.i 0.610058i
\(269\) −218034. −0.183715 −0.0918573 0.995772i \(-0.529280\pi\)
−0.0918573 + 0.995772i \(0.529280\pi\)
\(270\) 0 0
\(271\) 1.26265e6 1.04438 0.522192 0.852828i \(-0.325114\pi\)
0.522192 + 0.852828i \(0.325114\pi\)
\(272\) 464128.i 0.380378i
\(273\) − 276507.i − 0.224543i
\(274\) 127104. 0.102278
\(275\) 0 0
\(276\) −427680. −0.337945
\(277\) − 1.10264e6i − 0.863443i −0.902007 0.431721i \(-0.857906\pi\)
0.902007 0.431721i \(-0.142094\pi\)
\(278\) − 800648.i − 0.621340i
\(279\) 310392. 0.238726
\(280\) 0 0
\(281\) −998213. −0.754149 −0.377075 0.926183i \(-0.623070\pi\)
−0.377075 + 0.926183i \(0.623070\pi\)
\(282\) − 792036.i − 0.593092i
\(283\) 386371.i 0.286773i 0.989667 + 0.143387i \(0.0457992\pi\)
−0.989667 + 0.143387i \(0.954201\pi\)
\(284\) −1.01978e6 −0.750255
\(285\) 0 0
\(286\) 468996. 0.339042
\(287\) − 327124.i − 0.234427i
\(288\) − 165888.i − 0.117851i
\(289\) −1.86711e6 −1.31500
\(290\) 0 0
\(291\) 515655. 0.356966
\(292\) 967520.i 0.664053i
\(293\) 783571.i 0.533224i 0.963804 + 0.266612i \(0.0859041\pi\)
−0.963804 + 0.266612i \(0.914096\pi\)
\(294\) −86436.0 −0.0583212
\(295\) 0 0
\(296\) −420992. −0.279283
\(297\) − 681615.i − 0.448382i
\(298\) 1.23857e6i 0.807940i
\(299\) −1.86219e6 −1.20461
\(300\) 0 0
\(301\) −155722. −0.0990681
\(302\) 834628.i 0.526594i
\(303\) − 404730.i − 0.253255i
\(304\) −66048.0 −0.0409898
\(305\) 0 0
\(306\) 1.17482e6 0.717248
\(307\) 2.81773e6i 1.70629i 0.521670 + 0.853147i \(0.325309\pi\)
−0.521670 + 0.853147i \(0.674691\pi\)
\(308\) − 146608.i − 0.0880604i
\(309\) −912645. −0.543758
\(310\) 0 0
\(311\) −847398. −0.496806 −0.248403 0.968657i \(-0.579906\pi\)
−0.248403 + 0.968657i \(0.579906\pi\)
\(312\) 361152.i 0.210041i
\(313\) − 364955.i − 0.210561i −0.994443 0.105281i \(-0.966426\pi\)
0.994443 0.105281i \(-0.0335741\pi\)
\(314\) 144040. 0.0824440
\(315\) 0 0
\(316\) −699536. −0.394087
\(317\) 1.93744e6i 1.08288i 0.840740 + 0.541439i \(0.182120\pi\)
−0.840740 + 0.541439i \(0.817880\pi\)
\(318\) − 942048.i − 0.522402i
\(319\) 242913. 0.133652
\(320\) 0 0
\(321\) 1.49402e6 0.809270
\(322\) 582120.i 0.312876i
\(323\) − 467754.i − 0.249466i
\(324\) −104976. −0.0555556
\(325\) 0 0
\(326\) −702680. −0.366196
\(327\) − 74601.0i − 0.0385812i
\(328\) 427264.i 0.219286i
\(329\) −1.07805e6 −0.549097
\(330\) 0 0
\(331\) −61460.0 −0.0308335 −0.0154167 0.999881i \(-0.504907\pi\)
−0.0154167 + 0.999881i \(0.504907\pi\)
\(332\) 1.55642e6i 0.774962i
\(333\) 1.06564e6i 0.526621i
\(334\) −629652. −0.308841
\(335\) 0 0
\(336\) 112896. 0.0545545
\(337\) − 3.74116e6i − 1.79445i −0.441574 0.897225i \(-0.645580\pi\)
0.441574 0.897225i \(-0.354420\pi\)
\(338\) 87344.0i 0.0415854i
\(339\) 2.36885e6 1.11954
\(340\) 0 0
\(341\) −358292. −0.166860
\(342\) 167184.i 0.0772911i
\(343\) 117649.i 0.0539949i
\(344\) 203392. 0.0926697
\(345\) 0 0
\(346\) 93884.0 0.0421601
\(347\) 211334.i 0.0942206i 0.998890 + 0.0471103i \(0.0150012\pi\)
−0.998890 + 0.0471103i \(0.984999\pi\)
\(348\) 187056.i 0.0827987i
\(349\) −3.39558e6 −1.49228 −0.746140 0.665789i \(-0.768095\pi\)
−0.746140 + 0.665789i \(0.768095\pi\)
\(350\) 0 0
\(351\) 2.28542e6 0.990142
\(352\) 191488.i 0.0823730i
\(353\) 3.88094e6i 1.65768i 0.559489 + 0.828838i \(0.310997\pi\)
−0.559489 + 0.828838i \(0.689003\pi\)
\(354\) −141552. −0.0600355
\(355\) 0 0
\(356\) 728960. 0.304845
\(357\) 799533.i 0.332021i
\(358\) 2.44891e6i 1.00987i
\(359\) −3.24210e6 −1.32767 −0.663836 0.747878i \(-0.731073\pi\)
−0.663836 + 0.747878i \(0.731073\pi\)
\(360\) 0 0
\(361\) −2.40954e6 −0.973117
\(362\) − 2.11533e6i − 0.848411i
\(363\) − 1.13474e6i − 0.451990i
\(364\) 491568. 0.194460
\(365\) 0 0
\(366\) −1.75464e6 −0.684676
\(367\) 1.44430e6i 0.559749i 0.960037 + 0.279874i \(0.0902928\pi\)
−0.960037 + 0.279874i \(0.909707\pi\)
\(368\) − 760320.i − 0.292669i
\(369\) 1.08151e6 0.413490
\(370\) 0 0
\(371\) −1.28223e6 −0.483651
\(372\) − 275904.i − 0.103371i
\(373\) − 3.43542e6i − 1.27852i −0.768991 0.639260i \(-0.779241\pi\)
0.768991 0.639260i \(-0.220759\pi\)
\(374\) −1.35612e6 −0.501326
\(375\) 0 0
\(376\) 1.40806e6 0.513633
\(377\) 814473.i 0.295137i
\(378\) − 714420.i − 0.257172i
\(379\) 1.68635e6 0.603044 0.301522 0.953459i \(-0.402505\pi\)
0.301522 + 0.953459i \(0.402505\pi\)
\(380\) 0 0
\(381\) −270468. −0.0954560
\(382\) − 2.16148e6i − 0.757865i
\(383\) − 2.64354e6i − 0.920850i −0.887699 0.460425i \(-0.847697\pi\)
0.887699 0.460425i \(-0.152303\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) −3.84128e6 −1.31222
\(387\) − 514836.i − 0.174740i
\(388\) 916720.i 0.309142i
\(389\) 452099. 0.151481 0.0757407 0.997128i \(-0.475868\pi\)
0.0757407 + 0.997128i \(0.475868\pi\)
\(390\) 0 0
\(391\) 5.38461e6 1.78120
\(392\) − 153664.i − 0.0505076i
\(393\) − 1.08045e6i − 0.352877i
\(394\) 3.04778e6 0.989105
\(395\) 0 0
\(396\) 484704. 0.155324
\(397\) − 1.95530e6i − 0.622641i −0.950305 0.311321i \(-0.899229\pi\)
0.950305 0.311321i \(-0.100771\pi\)
\(398\) 500336.i 0.158327i
\(399\) −113778. −0.0357788
\(400\) 0 0
\(401\) −4.76737e6 −1.48053 −0.740266 0.672314i \(-0.765300\pi\)
−0.740266 + 0.672314i \(0.765300\pi\)
\(402\) − 1.61395e6i − 0.498110i
\(403\) − 1.20133e6i − 0.368469i
\(404\) 719520. 0.219326
\(405\) 0 0
\(406\) 254604. 0.0766567
\(407\) − 1.23009e6i − 0.368086i
\(408\) − 1.04429e6i − 0.310577i
\(409\) 4.13199e6 1.22138 0.610690 0.791870i \(-0.290892\pi\)
0.610690 + 0.791870i \(0.290892\pi\)
\(410\) 0 0
\(411\) −285984. −0.0835097
\(412\) − 1.62248e6i − 0.470908i
\(413\) 192668.i 0.0555820i
\(414\) −1.92456e6 −0.551862
\(415\) 0 0
\(416\) −642048. −0.181901
\(417\) 1.80146e6i 0.507322i
\(418\) − 192984.i − 0.0540232i
\(419\) −190512. −0.0530136 −0.0265068 0.999649i \(-0.508438\pi\)
−0.0265068 + 0.999649i \(0.508438\pi\)
\(420\) 0 0
\(421\) −5.19186e6 −1.42764 −0.713818 0.700332i \(-0.753035\pi\)
−0.713818 + 0.700332i \(0.753035\pi\)
\(422\) − 2.51019e6i − 0.686160i
\(423\) − 3.56416e6i − 0.968515i
\(424\) 1.67475e6 0.452414
\(425\) 0 0
\(426\) 2.29450e6 0.612581
\(427\) 2.38826e6i 0.633887i
\(428\) 2.65603e6i 0.700848i
\(429\) −1.05524e6 −0.276827
\(430\) 0 0
\(431\) −4.21781e6 −1.09369 −0.546845 0.837234i \(-0.684171\pi\)
−0.546845 + 0.837234i \(0.684171\pi\)
\(432\) 933120.i 0.240563i
\(433\) 4.86027e6i 1.24578i 0.782310 + 0.622890i \(0.214041\pi\)
−0.782310 + 0.622890i \(0.785959\pi\)
\(434\) −375536. −0.0957034
\(435\) 0 0
\(436\) 132624. 0.0334123
\(437\) 766260.i 0.191943i
\(438\) − 2.17692e6i − 0.542197i
\(439\) 2.03113e6 0.503011 0.251505 0.967856i \(-0.419074\pi\)
0.251505 + 0.967856i \(0.419074\pi\)
\(440\) 0 0
\(441\) −388962. −0.0952381
\(442\) − 4.54700e6i − 1.10706i
\(443\) − 2.84199e6i − 0.688038i −0.938963 0.344019i \(-0.888211\pi\)
0.938963 0.344019i \(-0.111789\pi\)
\(444\) 947232. 0.228034
\(445\) 0 0
\(446\) 4.88440e6 1.16272
\(447\) − 2.78678e6i − 0.659680i
\(448\) 200704.i 0.0472456i
\(449\) 4.59682e6 1.07607 0.538037 0.842921i \(-0.319166\pi\)
0.538037 + 0.842921i \(0.319166\pi\)
\(450\) 0 0
\(451\) −1.24841e6 −0.289012
\(452\) 4.21130e6i 0.969549i
\(453\) − 1.87791e6i − 0.429962i
\(454\) 1.56219e6 0.355708
\(455\) 0 0
\(456\) 148608. 0.0334680
\(457\) 4.93367e6i 1.10504i 0.833498 + 0.552522i \(0.186335\pi\)
−0.833498 + 0.552522i \(0.813665\pi\)
\(458\) − 2.84850e6i − 0.634530i
\(459\) −6.60838e6 −1.46408
\(460\) 0 0
\(461\) −4.75667e6 −1.04244 −0.521220 0.853422i \(-0.674523\pi\)
−0.521220 + 0.853422i \(0.674523\pi\)
\(462\) 329868.i 0.0719011i
\(463\) − 4.08619e6i − 0.885862i −0.896556 0.442931i \(-0.853939\pi\)
0.896556 0.442931i \(-0.146061\pi\)
\(464\) −332544. −0.0717058
\(465\) 0 0
\(466\) −2.24630e6 −0.479186
\(467\) − 4.15932e6i − 0.882531i −0.897377 0.441266i \(-0.854530\pi\)
0.897377 0.441266i \(-0.145470\pi\)
\(468\) 1.62518e6i 0.342995i
\(469\) −2.19677e6 −0.461160
\(470\) 0 0
\(471\) −324090. −0.0673152
\(472\) − 251648.i − 0.0519922i
\(473\) 594286.i 0.122136i
\(474\) 1.57396e6 0.321771
\(475\) 0 0
\(476\) −1.42139e6 −0.287539
\(477\) − 4.23922e6i − 0.853080i
\(478\) − 5.44335e6i − 1.08967i
\(479\) 3.36040e6 0.669195 0.334597 0.942361i \(-0.391400\pi\)
0.334597 + 0.942361i \(0.391400\pi\)
\(480\) 0 0
\(481\) 4.12441e6 0.812828
\(482\) − 2.12020e6i − 0.415680i
\(483\) − 1.30977e6i − 0.255463i
\(484\) 2.01731e6 0.391435
\(485\) 0 0
\(486\) 3.77914e6 0.725775
\(487\) 7.05243e6i 1.34746i 0.738977 + 0.673730i \(0.235309\pi\)
−0.738977 + 0.673730i \(0.764691\pi\)
\(488\) − 3.11936e6i − 0.592947i
\(489\) 1.58103e6 0.298998
\(490\) 0 0
\(491\) −83937.0 −0.0157127 −0.00785633 0.999969i \(-0.502501\pi\)
−0.00785633 + 0.999969i \(0.502501\pi\)
\(492\) − 961344.i − 0.179047i
\(493\) − 2.35509e6i − 0.436405i
\(494\) 647064. 0.119297
\(495\) 0 0
\(496\) 490496. 0.0895223
\(497\) − 3.12306e6i − 0.567140i
\(498\) − 3.50194e6i − 0.632754i
\(499\) 7.10526e6 1.27741 0.638703 0.769454i \(-0.279471\pi\)
0.638703 + 0.769454i \(0.279471\pi\)
\(500\) 0 0
\(501\) 1.41672e6 0.252167
\(502\) − 3.96132e6i − 0.701586i
\(503\) − 2.89147e6i − 0.509564i −0.966999 0.254782i \(-0.917996\pi\)
0.966999 0.254782i \(-0.0820037\pi\)
\(504\) 508032. 0.0890871
\(505\) 0 0
\(506\) 2.22156e6 0.385729
\(507\) − 196524.i − 0.0339544i
\(508\) − 480832.i − 0.0826674i
\(509\) −1.03548e6 −0.177153 −0.0885764 0.996069i \(-0.528232\pi\)
−0.0885764 + 0.996069i \(0.528232\pi\)
\(510\) 0 0
\(511\) −2.96303e6 −0.501977
\(512\) − 262144.i − 0.0441942i
\(513\) − 940410.i − 0.157770i
\(514\) −7.26572e6 −1.21303
\(515\) 0 0
\(516\) −457632. −0.0756645
\(517\) 4.11419e6i 0.676952i
\(518\) − 1.28929e6i − 0.211118i
\(519\) −211239. −0.0344236
\(520\) 0 0
\(521\) −7.49715e6 −1.21005 −0.605023 0.796208i \(-0.706836\pi\)
−0.605023 + 0.796208i \(0.706836\pi\)
\(522\) 841752.i 0.135210i
\(523\) − 3.53223e6i − 0.564670i −0.959316 0.282335i \(-0.908891\pi\)
0.959316 0.282335i \(-0.0911089\pi\)
\(524\) 1.92080e6 0.305600
\(525\) 0 0
\(526\) 7.83390e6 1.23456
\(527\) 3.47371e6i 0.544837i
\(528\) − 430848.i − 0.0672573i
\(529\) −2.38456e6 −0.370483
\(530\) 0 0
\(531\) −636984. −0.0980375
\(532\) − 202272.i − 0.0309854i
\(533\) − 4.18585e6i − 0.638213i
\(534\) −1.64016e6 −0.248905
\(535\) 0 0
\(536\) 2.86925e6 0.431376
\(537\) − 5.51005e6i − 0.824556i
\(538\) 872136.i 0.129906i
\(539\) 448987. 0.0665674
\(540\) 0 0
\(541\) −4.99188e6 −0.733281 −0.366641 0.930363i \(-0.619492\pi\)
−0.366641 + 0.930363i \(0.619492\pi\)
\(542\) − 5.05061e6i − 0.738491i
\(543\) 4.75949e6i 0.692725i
\(544\) 1.85651e6 0.268968
\(545\) 0 0
\(546\) −1.10603e6 −0.158776
\(547\) 5.12634e6i 0.732553i 0.930506 + 0.366277i \(0.119368\pi\)
−0.930506 + 0.366277i \(0.880632\pi\)
\(548\) − 508416.i − 0.0723216i
\(549\) −7.89588e6 −1.11807
\(550\) 0 0
\(551\) 335142. 0.0470273
\(552\) 1.71072e6i 0.238963i
\(553\) − 2.14233e6i − 0.297902i
\(554\) −4.41055e6 −0.610546
\(555\) 0 0
\(556\) −3.20259e6 −0.439354
\(557\) 8.86866e6i 1.21121i 0.795765 + 0.605606i \(0.207069\pi\)
−0.795765 + 0.605606i \(0.792931\pi\)
\(558\) − 1.24157e6i − 0.168805i
\(559\) −1.99261e6 −0.269707
\(560\) 0 0
\(561\) 3.05128e6 0.409331
\(562\) 3.99285e6i 0.533264i
\(563\) − 9.07277e6i − 1.20634i −0.797613 0.603169i \(-0.793904\pi\)
0.797613 0.603169i \(-0.206096\pi\)
\(564\) −3.16814e6 −0.419379
\(565\) 0 0
\(566\) 1.54548e6 0.202779
\(567\) − 321489.i − 0.0419961i
\(568\) 4.07910e6i 0.530510i
\(569\) −2.08310e6 −0.269730 −0.134865 0.990864i \(-0.543060\pi\)
−0.134865 + 0.990864i \(0.543060\pi\)
\(570\) 0 0
\(571\) −5.46368e6 −0.701286 −0.350643 0.936509i \(-0.614037\pi\)
−0.350643 + 0.936509i \(0.614037\pi\)
\(572\) − 1.87598e6i − 0.239739i
\(573\) 4.86332e6i 0.618794i
\(574\) −1.30850e6 −0.165765
\(575\) 0 0
\(576\) −663552. −0.0833333
\(577\) 7.66246e6i 0.958140i 0.877777 + 0.479070i \(0.159026\pi\)
−0.877777 + 0.479070i \(0.840974\pi\)
\(578\) 7.46845e6i 0.929845i
\(579\) 8.64288e6 1.07143
\(580\) 0 0
\(581\) −4.76652e6 −0.585816
\(582\) − 2.06262e6i − 0.252413i
\(583\) 4.89342e6i 0.596267i
\(584\) 3.87008e6 0.469556
\(585\) 0 0
\(586\) 3.13428e6 0.377046
\(587\) − 1.57465e7i − 1.88620i −0.332510 0.943100i \(-0.607896\pi\)
0.332510 0.943100i \(-0.392104\pi\)
\(588\) 345744.i 0.0412393i
\(589\) −494328. −0.0587120
\(590\) 0 0
\(591\) −6.85750e6 −0.807601
\(592\) 1.68397e6i 0.197483i
\(593\) 1.62409e7i 1.89658i 0.317398 + 0.948292i \(0.397191\pi\)
−0.317398 + 0.948292i \(0.602809\pi\)
\(594\) −2.72646e6 −0.317054
\(595\) 0 0
\(596\) 4.95427e6 0.571300
\(597\) − 1.12576e6i − 0.129273i
\(598\) 7.44876e6i 0.851787i
\(599\) 1.90793e6 0.217268 0.108634 0.994082i \(-0.465352\pi\)
0.108634 + 0.994082i \(0.465352\pi\)
\(600\) 0 0
\(601\) 3.52970e6 0.398613 0.199306 0.979937i \(-0.436131\pi\)
0.199306 + 0.979937i \(0.436131\pi\)
\(602\) 622888.i 0.0700517i
\(603\) − 7.26278e6i − 0.813411i
\(604\) 3.33851e6 0.372358
\(605\) 0 0
\(606\) −1.61892e6 −0.179079
\(607\) 3.37799e6i 0.372123i 0.982538 + 0.186061i \(0.0595724\pi\)
−0.982538 + 0.186061i \(0.940428\pi\)
\(608\) 264192.i 0.0289842i
\(609\) −572859. −0.0625899
\(610\) 0 0
\(611\) −1.37946e7 −1.49488
\(612\) − 4.69930e6i − 0.507171i
\(613\) − 1.20412e6i − 0.129425i −0.997904 0.0647127i \(-0.979387\pi\)
0.997904 0.0647127i \(-0.0206131\pi\)
\(614\) 1.12709e7 1.20653
\(615\) 0 0
\(616\) −586432. −0.0622681
\(617\) − 5.47330e6i − 0.578810i −0.957207 0.289405i \(-0.906543\pi\)
0.957207 0.289405i \(-0.0934575\pi\)
\(618\) 3.65058e6i 0.384495i
\(619\) −3.22662e6 −0.338471 −0.169236 0.985576i \(-0.554130\pi\)
−0.169236 + 0.985576i \(0.554130\pi\)
\(620\) 0 0
\(621\) 1.08256e7 1.12648
\(622\) 3.38959e6i 0.351295i
\(623\) 2.23244e6i 0.230441i
\(624\) 1.44461e6 0.148521
\(625\) 0 0
\(626\) −1.45982e6 −0.148889
\(627\) 434214.i 0.0441098i
\(628\) − 576160.i − 0.0582967i
\(629\) −1.19259e7 −1.20189
\(630\) 0 0
\(631\) 1.36282e7 1.36259 0.681297 0.732007i \(-0.261416\pi\)
0.681297 + 0.732007i \(0.261416\pi\)
\(632\) 2.79814e6i 0.278662i
\(633\) 5.64792e6i 0.560247i
\(634\) 7.74975e6 0.765711
\(635\) 0 0
\(636\) −3.76819e6 −0.369394
\(637\) 1.50543e6i 0.146998i
\(638\) − 971652.i − 0.0945059i
\(639\) 1.03252e7 1.00034
\(640\) 0 0
\(641\) −1.92472e7 −1.85021 −0.925106 0.379710i \(-0.876024\pi\)
−0.925106 + 0.379710i \(0.876024\pi\)
\(642\) − 5.97607e6i − 0.572240i
\(643\) 1.28399e7i 1.22472i 0.790580 + 0.612358i \(0.209779\pi\)
−0.790580 + 0.612358i \(0.790221\pi\)
\(644\) 2.32848e6 0.221237
\(645\) 0 0
\(646\) −1.87102e6 −0.176399
\(647\) 2.00233e7i 1.88050i 0.340481 + 0.940251i \(0.389410\pi\)
−0.340481 + 0.940251i \(0.610590\pi\)
\(648\) 419904.i 0.0392837i
\(649\) 735284. 0.0685241
\(650\) 0 0
\(651\) 844956. 0.0781415
\(652\) 2.81072e6i 0.258940i
\(653\) 7.23655e6i 0.664124i 0.943258 + 0.332062i \(0.107744\pi\)
−0.943258 + 0.332062i \(0.892256\pi\)
\(654\) −298404. −0.0272810
\(655\) 0 0
\(656\) 1.70906e6 0.155059
\(657\) − 9.79614e6i − 0.885404i
\(658\) 4.31220e6i 0.388270i
\(659\) −1.42474e7 −1.27798 −0.638989 0.769216i \(-0.720647\pi\)
−0.638989 + 0.769216i \(0.720647\pi\)
\(660\) 0 0
\(661\) 1.49265e7 1.32878 0.664391 0.747385i \(-0.268691\pi\)
0.664391 + 0.747385i \(0.268691\pi\)
\(662\) 245840.i 0.0218026i
\(663\) 1.02308e7i 0.903908i
\(664\) 6.22566e6 0.547981
\(665\) 0 0
\(666\) 4.26254e6 0.372377
\(667\) 3.85803e6i 0.335777i
\(668\) 2.51861e6i 0.218383i
\(669\) −1.09899e7 −0.949355
\(670\) 0 0
\(671\) 9.11438e6 0.781485
\(672\) − 451584.i − 0.0385758i
\(673\) 1.55062e7i 1.31967i 0.751409 + 0.659837i \(0.229375\pi\)
−0.751409 + 0.659837i \(0.770625\pi\)
\(674\) −1.49646e7 −1.26887
\(675\) 0 0
\(676\) 349376. 0.0294053
\(677\) − 7.80065e6i − 0.654122i −0.945003 0.327061i \(-0.893942\pi\)
0.945003 0.327061i \(-0.106058\pi\)
\(678\) − 9.47542e6i − 0.791633i
\(679\) −2.80745e6 −0.233689
\(680\) 0 0
\(681\) −3.51492e6 −0.290434
\(682\) 1.43317e6i 0.117988i
\(683\) − 1.58547e7i − 1.30049i −0.759727 0.650243i \(-0.774667\pi\)
0.759727 0.650243i \(-0.225333\pi\)
\(684\) 668736. 0.0546531
\(685\) 0 0
\(686\) 470596. 0.0381802
\(687\) 6.40912e6i 0.518091i
\(688\) − 813568.i − 0.0655274i
\(689\) −1.64073e7 −1.31671
\(690\) 0 0
\(691\) 2.03656e7 1.62257 0.811284 0.584652i \(-0.198769\pi\)
0.811284 + 0.584652i \(0.198769\pi\)
\(692\) − 375536.i − 0.0298117i
\(693\) 1.48441e6i 0.117414i
\(694\) 845336. 0.0666240
\(695\) 0 0
\(696\) 748224. 0.0585475
\(697\) 1.21036e7i 0.943696i
\(698\) 1.35823e7i 1.05520i
\(699\) 5.05418e6 0.391253
\(700\) 0 0
\(701\) 2.48036e7 1.90643 0.953213 0.302300i \(-0.0977543\pi\)
0.953213 + 0.302300i \(0.0977543\pi\)
\(702\) − 9.14166e6i − 0.700136i
\(703\) − 1.69712e6i − 0.129517i
\(704\) 765952. 0.0582465
\(705\) 0 0
\(706\) 1.55237e7 1.17215
\(707\) 2.20353e6i 0.165795i
\(708\) 566208.i 0.0424515i
\(709\) −1.81917e7 −1.35912 −0.679560 0.733620i \(-0.737829\pi\)
−0.679560 + 0.733620i \(0.737829\pi\)
\(710\) 0 0
\(711\) 7.08280e6 0.525450
\(712\) − 2.91584e6i − 0.215558i
\(713\) − 5.69052e6i − 0.419207i
\(714\) 3.19813e6 0.234774
\(715\) 0 0
\(716\) 9.79565e6 0.714086
\(717\) 1.22475e7i 0.889715i
\(718\) 1.29684e7i 0.938806i
\(719\) 1.66202e7 1.19899 0.599493 0.800380i \(-0.295369\pi\)
0.599493 + 0.800380i \(0.295369\pi\)
\(720\) 0 0
\(721\) 4.96884e6 0.355973
\(722\) 9.63814e6i 0.688098i
\(723\) 4.77045e6i 0.339401i
\(724\) −8.46131e6 −0.599917
\(725\) 0 0
\(726\) −4.53895e6 −0.319605
\(727\) − 1.57591e7i − 1.10585i −0.833231 0.552925i \(-0.813511\pi\)
0.833231 0.552925i \(-0.186489\pi\)
\(728\) − 1.96627e6i − 0.137504i
\(729\) −6.90873e6 −0.481481
\(730\) 0 0
\(731\) 5.76171e6 0.398803
\(732\) 7.01856e6i 0.484139i
\(733\) 2.15238e6i 0.147965i 0.997260 + 0.0739827i \(0.0235710\pi\)
−0.997260 + 0.0739827i \(0.976429\pi\)
\(734\) 5.77721e6 0.395802
\(735\) 0 0
\(736\) −3.04128e6 −0.206948
\(737\) 8.38358e6i 0.568540i
\(738\) − 4.32605e6i − 0.292382i
\(739\) 2.27267e7 1.53083 0.765413 0.643540i \(-0.222535\pi\)
0.765413 + 0.643540i \(0.222535\pi\)
\(740\) 0 0
\(741\) −1.45589e6 −0.0974057
\(742\) 5.12893e6i 0.341993i
\(743\) − 1.21153e7i − 0.805123i −0.915393 0.402561i \(-0.868120\pi\)
0.915393 0.402561i \(-0.131880\pi\)
\(744\) −1.10362e6 −0.0730947
\(745\) 0 0
\(746\) −1.37417e7 −0.904050
\(747\) − 1.57587e7i − 1.03328i
\(748\) 5.42450e6i 0.354491i
\(749\) −8.13410e6 −0.529791
\(750\) 0 0
\(751\) −2.07590e7 −1.34310 −0.671549 0.740961i \(-0.734371\pi\)
−0.671549 + 0.740961i \(0.734371\pi\)
\(752\) − 5.63226e6i − 0.363193i
\(753\) 8.91297e6i 0.572842i
\(754\) 3.25789e6 0.208693
\(755\) 0 0
\(756\) −2.85768e6 −0.181848
\(757\) − 1.86222e7i − 1.18111i −0.806997 0.590556i \(-0.798909\pi\)
0.806997 0.590556i \(-0.201091\pi\)
\(758\) − 6.74539e6i − 0.426417i
\(759\) −4.99851e6 −0.314946
\(760\) 0 0
\(761\) 2.65336e7 1.66087 0.830434 0.557117i \(-0.188093\pi\)
0.830434 + 0.557117i \(0.188093\pi\)
\(762\) 1.08187e6i 0.0674976i
\(763\) 406161.i 0.0252573i
\(764\) −8.64590e6 −0.535892
\(765\) 0 0
\(766\) −1.05742e7 −0.651139
\(767\) 2.46536e6i 0.151319i
\(768\) 589824.i 0.0360844i
\(769\) 2.01595e7 1.22931 0.614657 0.788794i \(-0.289294\pi\)
0.614657 + 0.788794i \(0.289294\pi\)
\(770\) 0 0
\(771\) 1.63479e7 0.990433
\(772\) 1.53651e7i 0.927882i
\(773\) 5.86488e6i 0.353029i 0.984298 + 0.176514i \(0.0564822\pi\)
−0.984298 + 0.176514i \(0.943518\pi\)
\(774\) −2.05934e6 −0.123560
\(775\) 0 0
\(776\) 3.66688e6 0.218596
\(777\) 2.90090e6i 0.172377i
\(778\) − 1.80840e6i − 0.107114i
\(779\) −1.72241e6 −0.101693
\(780\) 0 0
\(781\) −1.19186e7 −0.699196
\(782\) − 2.15384e7i − 1.25950i
\(783\) − 4.73486e6i − 0.275996i
\(784\) −614656. −0.0357143
\(785\) 0 0
\(786\) −4.32180e6 −0.249522
\(787\) − 1.63347e6i − 0.0940100i −0.998895 0.0470050i \(-0.985032\pi\)
0.998895 0.0470050i \(-0.0149677\pi\)
\(788\) − 1.21911e7i − 0.699403i
\(789\) −1.76263e7 −1.00802
\(790\) 0 0
\(791\) −1.28971e7 −0.732910
\(792\) − 1.93882e6i − 0.109831i
\(793\) 3.05600e7i 1.72572i
\(794\) −7.82121e6 −0.440274
\(795\) 0 0
\(796\) 2.00134e6 0.111954
\(797\) 2.07673e7i 1.15807i 0.815303 + 0.579034i \(0.196570\pi\)
−0.815303 + 0.579034i \(0.803430\pi\)
\(798\) 455112.i 0.0252994i
\(799\) 3.98878e7 2.21041
\(800\) 0 0
\(801\) −7.38072e6 −0.406460
\(802\) 1.90695e7i 1.04689i
\(803\) 1.13079e7i 0.618860i
\(804\) −6.45581e6 −0.352217
\(805\) 0 0
\(806\) −4.80533e6 −0.260547
\(807\) − 1.96231e6i − 0.106068i
\(808\) − 2.87808e6i − 0.155087i
\(809\) −3.53936e6 −0.190131 −0.0950656 0.995471i \(-0.530306\pi\)
−0.0950656 + 0.995471i \(0.530306\pi\)
\(810\) 0 0
\(811\) −2.11480e7 −1.12906 −0.564530 0.825412i \(-0.690943\pi\)
−0.564530 + 0.825412i \(0.690943\pi\)
\(812\) − 1.01842e6i − 0.0542045i
\(813\) 1.13639e7i 0.602976i
\(814\) −4.92034e6 −0.260276
\(815\) 0 0
\(816\) −4.17715e6 −0.219611
\(817\) 819924.i 0.0429753i
\(818\) − 1.65280e7i − 0.863646i
\(819\) −4.97713e6 −0.259280
\(820\) 0 0
\(821\) 265389. 0.0137412 0.00687061 0.999976i \(-0.497813\pi\)
0.00687061 + 0.999976i \(0.497813\pi\)
\(822\) 1.14394e6i 0.0590503i
\(823\) − 3.09261e7i − 1.59157i −0.605579 0.795785i \(-0.707058\pi\)
0.605579 0.795785i \(-0.292942\pi\)
\(824\) −6.48992e6 −0.332982
\(825\) 0 0
\(826\) 770672. 0.0393024
\(827\) − 2.84152e7i − 1.44473i −0.691510 0.722367i \(-0.743054\pi\)
0.691510 0.722367i \(-0.256946\pi\)
\(828\) 7.69824e6i 0.390225i
\(829\) 3.33547e7 1.68566 0.842832 0.538177i \(-0.180887\pi\)
0.842832 + 0.538177i \(0.180887\pi\)
\(830\) 0 0
\(831\) 9.92374e6 0.498509
\(832\) 2.56819e6i 0.128623i
\(833\) − 4.35301e6i − 0.217359i
\(834\) 7.20583e6 0.358731
\(835\) 0 0
\(836\) −771936. −0.0382002
\(837\) 6.98382e6i 0.344572i
\(838\) 762048.i 0.0374863i
\(839\) 5.66205e6 0.277695 0.138848 0.990314i \(-0.455660\pi\)
0.138848 + 0.990314i \(0.455660\pi\)
\(840\) 0 0
\(841\) −1.88237e7 −0.917732
\(842\) 2.07674e7i 1.00949i
\(843\) − 8.98392e6i − 0.435408i
\(844\) −1.00408e7 −0.485188
\(845\) 0 0
\(846\) −1.42566e7 −0.684844
\(847\) 6.17802e6i 0.295897i
\(848\) − 6.69901e6i − 0.319905i
\(849\) −3.47734e6 −0.165569
\(850\) 0 0
\(851\) 1.95367e7 0.924754
\(852\) − 9.17798e6i − 0.433160i
\(853\) − 2.19983e7i − 1.03518i −0.855628 0.517592i \(-0.826829\pi\)
0.855628 0.517592i \(-0.173171\pi\)
\(854\) 9.55304e6 0.448226
\(855\) 0 0
\(856\) 1.06241e7 0.495574
\(857\) − 2.17568e7i − 1.01191i −0.862559 0.505956i \(-0.831140\pi\)
0.862559 0.505956i \(-0.168860\pi\)
\(858\) 4.22096e6i 0.195746i
\(859\) 4.09384e7 1.89299 0.946494 0.322721i \(-0.104598\pi\)
0.946494 + 0.322721i \(0.104598\pi\)
\(860\) 0 0
\(861\) 2.94412e6 0.135347
\(862\) 1.68712e7i 0.773355i
\(863\) − 5.65597e6i − 0.258512i −0.991611 0.129256i \(-0.958741\pi\)
0.991611 0.129256i \(-0.0412588\pi\)
\(864\) 3.73248e6 0.170103
\(865\) 0 0
\(866\) 1.94411e7 0.880899
\(867\) − 1.68040e7i − 0.759216i
\(868\) 1.50214e6i 0.0676725i
\(869\) −8.17583e6 −0.367267
\(870\) 0 0
\(871\) −2.81097e7 −1.25548
\(872\) − 530496.i − 0.0236260i
\(873\) − 9.28179e6i − 0.412189i
\(874\) 3.06504e6 0.135724
\(875\) 0 0
\(876\) −8.70768e6 −0.383391
\(877\) 2.61067e7i 1.14618i 0.819493 + 0.573089i \(0.194255\pi\)
−0.819493 + 0.573089i \(0.805745\pi\)
\(878\) − 8.12454e6i − 0.355682i
\(879\) −7.05214e6 −0.307857
\(880\) 0 0
\(881\) 1.44294e6 0.0626339 0.0313170 0.999510i \(-0.490030\pi\)
0.0313170 + 0.999510i \(0.490030\pi\)
\(882\) 1.55585e6i 0.0673435i
\(883\) 1.52432e7i 0.657921i 0.944344 + 0.328960i \(0.106698\pi\)
−0.944344 + 0.328960i \(0.893302\pi\)
\(884\) −1.81880e7 −0.782807
\(885\) 0 0
\(886\) −1.13679e7 −0.486517
\(887\) − 3.31500e7i − 1.41473i −0.706847 0.707366i \(-0.749883\pi\)
0.706847 0.707366i \(-0.250117\pi\)
\(888\) − 3.78893e6i − 0.161244i
\(889\) 1.47255e6 0.0624907
\(890\) 0 0
\(891\) −1.22691e6 −0.0517747
\(892\) − 1.95376e7i − 0.822166i
\(893\) 5.67626e6i 0.238195i
\(894\) −1.11471e7 −0.466464
\(895\) 0 0
\(896\) 802816. 0.0334077
\(897\) − 1.67597e7i − 0.695481i
\(898\) − 1.83873e7i − 0.760899i
\(899\) −2.48888e6 −0.102708
\(900\) 0 0
\(901\) 4.74426e7 1.94696
\(902\) 4.99365e6i 0.204363i
\(903\) − 1.40150e6i − 0.0571970i
\(904\) 1.68452e7 0.685575
\(905\) 0 0
\(906\) −7.51165e6 −0.304029
\(907\) 1.16963e7i 0.472096i 0.971741 + 0.236048i \(0.0758523\pi\)
−0.971741 + 0.236048i \(0.924148\pi\)
\(908\) − 6.24875e6i − 0.251524i
\(909\) −7.28514e6 −0.292434
\(910\) 0 0
\(911\) 2.89321e7 1.15501 0.577503 0.816389i \(-0.304027\pi\)
0.577503 + 0.816389i \(0.304027\pi\)
\(912\) − 594432.i − 0.0236655i
\(913\) 1.81906e7i 0.722221i
\(914\) 1.97347e7 0.781384
\(915\) 0 0
\(916\) −1.13940e7 −0.448680
\(917\) 5.88245e6i 0.231012i
\(918\) 2.64335e7i 1.03526i
\(919\) 4.57838e7 1.78823 0.894115 0.447838i \(-0.147806\pi\)
0.894115 + 0.447838i \(0.147806\pi\)
\(920\) 0 0
\(921\) −2.53596e7 −0.985129
\(922\) 1.90267e7i 0.737116i
\(923\) − 3.99625e7i − 1.54400i
\(924\) 1.31947e6 0.0508417
\(925\) 0 0
\(926\) −1.63448e7 −0.626399
\(927\) 1.64276e7i 0.627878i
\(928\) 1.33018e6i 0.0507036i
\(929\) −2.46947e7 −0.938782 −0.469391 0.882990i \(-0.655526\pi\)
−0.469391 + 0.882990i \(0.655526\pi\)
\(930\) 0 0
\(931\) 619458. 0.0234227
\(932\) 8.98522e6i 0.338835i
\(933\) − 7.62658e6i − 0.286831i
\(934\) −1.66373e7 −0.624044
\(935\) 0 0
\(936\) 6.50074e6 0.242534
\(937\) − 1.98926e7i − 0.740187i −0.928994 0.370094i \(-0.879326\pi\)
0.928994 0.370094i \(-0.120674\pi\)
\(938\) 8.78707e6i 0.326090i
\(939\) 3.28460e6 0.121568
\(940\) 0 0
\(941\) −3.73454e7 −1.37488 −0.687438 0.726243i \(-0.741265\pi\)
−0.687438 + 0.726243i \(0.741265\pi\)
\(942\) 1.29636e6i 0.0475991i
\(943\) − 1.98277e7i − 0.726095i
\(944\) −1.00659e6 −0.0367641
\(945\) 0 0
\(946\) 2.37714e6 0.0863630
\(947\) 5.10396e7i 1.84941i 0.380689 + 0.924703i \(0.375687\pi\)
−0.380689 + 0.924703i \(0.624313\pi\)
\(948\) − 6.29582e6i − 0.227526i
\(949\) −3.79147e7 −1.36660
\(950\) 0 0
\(951\) −1.74369e7 −0.625200
\(952\) 5.68557e6i 0.203321i
\(953\) − 254832.i − 0.00908912i −0.999990 0.00454456i \(-0.998553\pi\)
0.999990 0.00454456i \(-0.00144658\pi\)
\(954\) −1.69569e7 −0.603218
\(955\) 0 0
\(956\) −2.17734e7 −0.770515
\(957\) 2.18622e6i 0.0771638i
\(958\) − 1.34416e7i − 0.473192i
\(959\) 1.55702e6 0.0546700
\(960\) 0 0
\(961\) −2.49581e7 −0.871772
\(962\) − 1.64976e7i − 0.574756i
\(963\) − 2.68923e7i − 0.934464i
\(964\) −8.48080e6 −0.293930
\(965\) 0 0
\(966\) −5.23908e6 −0.180639
\(967\) 1.17012e7i 0.402405i 0.979550 + 0.201203i \(0.0644850\pi\)
−0.979550 + 0.201203i \(0.935515\pi\)
\(968\) − 8.06925e6i − 0.276786i
\(969\) 4.20979e6 0.144029
\(970\) 0 0
\(971\) 3.59080e7 1.22220 0.611101 0.791553i \(-0.290727\pi\)
0.611101 + 0.791553i \(0.290727\pi\)
\(972\) − 1.51165e7i − 0.513200i
\(973\) − 9.80794e6i − 0.332120i
\(974\) 2.82097e7 0.952799
\(975\) 0 0
\(976\) −1.24774e7 −0.419277
\(977\) − 5.50592e7i − 1.84541i −0.385504 0.922706i \(-0.625972\pi\)
0.385504 0.922706i \(-0.374028\pi\)
\(978\) − 6.32412e6i − 0.211423i
\(979\) 8.51972e6 0.284098
\(980\) 0 0
\(981\) −1.34282e6 −0.0445497
\(982\) 335748.i 0.0111105i
\(983\) 1.81317e7i 0.598488i 0.954177 + 0.299244i \(0.0967344\pi\)
−0.954177 + 0.299244i \(0.903266\pi\)
\(984\) −3.84538e6 −0.126605
\(985\) 0 0
\(986\) −9.42035e6 −0.308585
\(987\) − 9.70244e6i − 0.317021i
\(988\) − 2.58826e6i − 0.0843558i
\(989\) −9.43866e6 −0.306845
\(990\) 0 0
\(991\) −2.02908e7 −0.656318 −0.328159 0.944623i \(-0.606428\pi\)
−0.328159 + 0.944623i \(0.606428\pi\)
\(992\) − 1.96198e6i − 0.0633018i
\(993\) − 553140.i − 0.0178017i
\(994\) −1.24923e7 −0.401028
\(995\) 0 0
\(996\) −1.40077e7 −0.447425
\(997\) 4.75390e7i 1.51465i 0.653038 + 0.757325i \(0.273494\pi\)
−0.653038 + 0.757325i \(0.726506\pi\)
\(998\) − 2.84211e7i − 0.903262i
\(999\) −2.39768e7 −0.760112
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 350.6.c.g.99.1 2
5.2 odd 4 350.6.a.l.1.1 1
5.3 odd 4 70.6.a.b.1.1 1
5.4 even 2 inner 350.6.c.g.99.2 2
15.8 even 4 630.6.a.i.1.1 1
20.3 even 4 560.6.a.e.1.1 1
35.13 even 4 490.6.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.6.a.b.1.1 1 5.3 odd 4
350.6.a.l.1.1 1 5.2 odd 4
350.6.c.g.99.1 2 1.1 even 1 trivial
350.6.c.g.99.2 2 5.4 even 2 inner
490.6.a.g.1.1 1 35.13 even 4
560.6.a.e.1.1 1 20.3 even 4
630.6.a.i.1.1 1 15.8 even 4