Properties

Label 75.8.a.c.1.1
Level $75$
Weight $8$
Character 75.1
Self dual yes
Analytic conductor $23.429$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,8,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+22.0000 q^{2} -27.0000 q^{3} +356.000 q^{4} -594.000 q^{6} +420.000 q^{7} +5016.00 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+22.0000 q^{2} -27.0000 q^{3} +356.000 q^{4} -594.000 q^{6} +420.000 q^{7} +5016.00 q^{8} +729.000 q^{9} -2944.00 q^{11} -9612.00 q^{12} +11006.0 q^{13} +9240.00 q^{14} +64784.0 q^{16} +16546.0 q^{17} +16038.0 q^{18} -25364.0 q^{19} -11340.0 q^{21} -64768.0 q^{22} +5880.00 q^{23} -135432. q^{24} +242132. q^{26} -19683.0 q^{27} +149520. q^{28} +163042. q^{29} -201600. q^{31} +783200. q^{32} +79488.0 q^{33} +364012. q^{34} +259524. q^{36} -120530. q^{37} -558008. q^{38} -297162. q^{39} -115910. q^{41} -249480. q^{42} +19148.0 q^{43} -1.04806e6 q^{44} +129360. q^{46} -841016. q^{47} -1.74917e6 q^{48} -647143. q^{49} -446742. q^{51} +3.91814e6 q^{52} -501890. q^{53} -433026. q^{54} +2.10672e6 q^{56} +684828. q^{57} +3.58692e6 q^{58} -1.58618e6 q^{59} -372962. q^{61} -4.43520e6 q^{62} +306180. q^{63} +8.93805e6 q^{64} +1.74874e6 q^{66} -4.56104e6 q^{67} +5.89038e6 q^{68} -158760. q^{69} +1.51283e6 q^{71} +3.65666e6 q^{72} +1.52291e6 q^{73} -2.65166e6 q^{74} -9.02958e6 q^{76} -1.23648e6 q^{77} -6.53756e6 q^{78} +4.23192e6 q^{79} +531441. q^{81} -2.55002e6 q^{82} +1.85420e6 q^{83} -4.03704e6 q^{84} +421256. q^{86} -4.40213e6 q^{87} -1.47671e7 q^{88} -6.88817e6 q^{89} +4.62252e6 q^{91} +2.09328e6 q^{92} +5.44320e6 q^{93} -1.85024e7 q^{94} -2.11464e7 q^{96} -3.70003e6 q^{97} -1.42371e7 q^{98} -2.14618e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.0000 1.94454 0.972272 0.233854i \(-0.0751336\pi\)
0.972272 + 0.233854i \(0.0751336\pi\)
\(3\) −27.0000 −0.577350
\(4\) 356.000 2.78125
\(5\) 0 0
\(6\) −594.000 −1.12268
\(7\) 420.000 0.462814 0.231407 0.972857i \(-0.425667\pi\)
0.231407 + 0.972857i \(0.425667\pi\)
\(8\) 5016.00 3.46372
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −2944.00 −0.666904 −0.333452 0.942767i \(-0.608214\pi\)
−0.333452 + 0.942767i \(0.608214\pi\)
\(12\) −9612.00 −1.60576
\(13\) 11006.0 1.38940 0.694701 0.719299i \(-0.255537\pi\)
0.694701 + 0.719299i \(0.255537\pi\)
\(14\) 9240.00 0.899961
\(15\) 0 0
\(16\) 64784.0 3.95410
\(17\) 16546.0 0.816811 0.408406 0.912801i \(-0.366085\pi\)
0.408406 + 0.912801i \(0.366085\pi\)
\(18\) 16038.0 0.648181
\(19\) −25364.0 −0.848360 −0.424180 0.905578i \(-0.639438\pi\)
−0.424180 + 0.905578i \(0.639438\pi\)
\(20\) 0 0
\(21\) −11340.0 −0.267206
\(22\) −64768.0 −1.29682
\(23\) 5880.00 0.100770 0.0503848 0.998730i \(-0.483955\pi\)
0.0503848 + 0.998730i \(0.483955\pi\)
\(24\) −135432. −1.99978
\(25\) 0 0
\(26\) 242132. 2.70175
\(27\) −19683.0 −0.192450
\(28\) 149520. 1.28720
\(29\) 163042. 1.24139 0.620693 0.784054i \(-0.286852\pi\)
0.620693 + 0.784054i \(0.286852\pi\)
\(30\) 0 0
\(31\) −201600. −1.21542 −0.607708 0.794161i \(-0.707911\pi\)
−0.607708 + 0.794161i \(0.707911\pi\)
\(32\) 783200. 4.22520
\(33\) 79488.0 0.385037
\(34\) 364012. 1.58833
\(35\) 0 0
\(36\) 259524. 0.927083
\(37\) −120530. −0.391191 −0.195596 0.980685i \(-0.562664\pi\)
−0.195596 + 0.980685i \(0.562664\pi\)
\(38\) −558008. −1.64967
\(39\) −297162. −0.802171
\(40\) 0 0
\(41\) −115910. −0.262650 −0.131325 0.991339i \(-0.541923\pi\)
−0.131325 + 0.991339i \(0.541923\pi\)
\(42\) −249480. −0.519593
\(43\) 19148.0 0.0367269 0.0183634 0.999831i \(-0.494154\pi\)
0.0183634 + 0.999831i \(0.494154\pi\)
\(44\) −1.04806e6 −1.85483
\(45\) 0 0
\(46\) 129360. 0.195951
\(47\) −841016. −1.18158 −0.590788 0.806827i \(-0.701183\pi\)
−0.590788 + 0.806827i \(0.701183\pi\)
\(48\) −1.74917e6 −2.28290
\(49\) −647143. −0.785804
\(50\) 0 0
\(51\) −446742. −0.471586
\(52\) 3.91814e6 3.86427
\(53\) −501890. −0.463066 −0.231533 0.972827i \(-0.574374\pi\)
−0.231533 + 0.972827i \(0.574374\pi\)
\(54\) −433026. −0.374228
\(55\) 0 0
\(56\) 2.10672e6 1.60306
\(57\) 684828. 0.489801
\(58\) 3.58692e6 2.41393
\(59\) −1.58618e6 −1.00547 −0.502735 0.864440i \(-0.667673\pi\)
−0.502735 + 0.864440i \(0.667673\pi\)
\(60\) 0 0
\(61\) −372962. −0.210383 −0.105191 0.994452i \(-0.533546\pi\)
−0.105191 + 0.994452i \(0.533546\pi\)
\(62\) −4.43520e6 −2.36343
\(63\) 306180. 0.154271
\(64\) 8.93805e6 4.26199
\(65\) 0 0
\(66\) 1.74874e6 0.748722
\(67\) −4.56104e6 −1.85269 −0.926344 0.376678i \(-0.877066\pi\)
−0.926344 + 0.376678i \(0.877066\pi\)
\(68\) 5.89038e6 2.27176
\(69\) −158760. −0.0581794
\(70\) 0 0
\(71\) 1.51283e6 0.501633 0.250817 0.968035i \(-0.419301\pi\)
0.250817 + 0.968035i \(0.419301\pi\)
\(72\) 3.65666e6 1.15457
\(73\) 1.52291e6 0.458189 0.229094 0.973404i \(-0.426424\pi\)
0.229094 + 0.973404i \(0.426424\pi\)
\(74\) −2.65166e6 −0.760688
\(75\) 0 0
\(76\) −9.02958e6 −2.35950
\(77\) −1.23648e6 −0.308652
\(78\) −6.53756e6 −1.55986
\(79\) 4.23192e6 0.965701 0.482850 0.875703i \(-0.339601\pi\)
0.482850 + 0.875703i \(0.339601\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) −2.55002e6 −0.510734
\(83\) 1.85420e6 0.355946 0.177973 0.984035i \(-0.443046\pi\)
0.177973 + 0.984035i \(0.443046\pi\)
\(84\) −4.03704e6 −0.743166
\(85\) 0 0
\(86\) 421256. 0.0714170
\(87\) −4.40213e6 −0.716714
\(88\) −1.47671e7 −2.30997
\(89\) −6.88817e6 −1.03571 −0.517856 0.855468i \(-0.673270\pi\)
−0.517856 + 0.855468i \(0.673270\pi\)
\(90\) 0 0
\(91\) 4.62252e6 0.643034
\(92\) 2.09328e6 0.280266
\(93\) 5.44320e6 0.701720
\(94\) −1.85024e7 −2.29763
\(95\) 0 0
\(96\) −2.11464e7 −2.43942
\(97\) −3.70003e6 −0.411628 −0.205814 0.978591i \(-0.565984\pi\)
−0.205814 + 0.978591i \(0.565984\pi\)
\(98\) −1.42371e7 −1.52803
\(99\) −2.14618e6 −0.222301
\(100\) 0 0
\(101\) −1.80025e7 −1.73863 −0.869314 0.494259i \(-0.835439\pi\)
−0.869314 + 0.494259i \(0.835439\pi\)
\(102\) −9.82832e6 −0.917020
\(103\) 5.37207e6 0.484408 0.242204 0.970225i \(-0.422130\pi\)
0.242204 + 0.970225i \(0.422130\pi\)
\(104\) 5.52061e7 4.81250
\(105\) 0 0
\(106\) −1.10416e7 −0.900452
\(107\) 1.15398e7 0.910655 0.455327 0.890324i \(-0.349522\pi\)
0.455327 + 0.890324i \(0.349522\pi\)
\(108\) −7.00715e6 −0.535252
\(109\) −1.57179e6 −0.116253 −0.0581263 0.998309i \(-0.518513\pi\)
−0.0581263 + 0.998309i \(0.518513\pi\)
\(110\) 0 0
\(111\) 3.25431e6 0.225854
\(112\) 2.72093e7 1.83001
\(113\) 2.52050e7 1.64328 0.821640 0.570007i \(-0.193060\pi\)
0.821640 + 0.570007i \(0.193060\pi\)
\(114\) 1.50662e7 0.952439
\(115\) 0 0
\(116\) 5.80430e7 3.45260
\(117\) 8.02337e6 0.463134
\(118\) −3.48959e7 −1.95518
\(119\) 6.94932e6 0.378031
\(120\) 0 0
\(121\) −1.08200e7 −0.555239
\(122\) −8.20516e6 −0.409098
\(123\) 3.12957e6 0.151641
\(124\) −7.17696e7 −3.38037
\(125\) 0 0
\(126\) 6.73596e6 0.299987
\(127\) −3.94080e7 −1.70715 −0.853574 0.520971i \(-0.825570\pi\)
−0.853574 + 0.520971i \(0.825570\pi\)
\(128\) 9.63875e7 4.06243
\(129\) −516996. −0.0212043
\(130\) 0 0
\(131\) 1.41082e7 0.548305 0.274153 0.961686i \(-0.411603\pi\)
0.274153 + 0.961686i \(0.411603\pi\)
\(132\) 2.82977e7 1.07088
\(133\) −1.06529e7 −0.392633
\(134\) −1.00343e8 −3.60263
\(135\) 0 0
\(136\) 8.29947e7 2.82920
\(137\) 8.00512e6 0.265978 0.132989 0.991118i \(-0.457542\pi\)
0.132989 + 0.991118i \(0.457542\pi\)
\(138\) −3.49272e6 −0.113132
\(139\) 4.60716e7 1.45506 0.727532 0.686074i \(-0.240667\pi\)
0.727532 + 0.686074i \(0.240667\pi\)
\(140\) 0 0
\(141\) 2.27074e7 0.682183
\(142\) 3.32823e7 0.975448
\(143\) −3.24017e7 −0.926598
\(144\) 4.72275e7 1.31803
\(145\) 0 0
\(146\) 3.35040e7 0.890968
\(147\) 1.74729e7 0.453684
\(148\) −4.29087e7 −1.08800
\(149\) 7.23525e7 1.79185 0.895925 0.444206i \(-0.146514\pi\)
0.895925 + 0.444206i \(0.146514\pi\)
\(150\) 0 0
\(151\) −3.70062e7 −0.874692 −0.437346 0.899293i \(-0.644082\pi\)
−0.437346 + 0.899293i \(0.644082\pi\)
\(152\) −1.27226e8 −2.93848
\(153\) 1.20620e7 0.272270
\(154\) −2.72026e7 −0.600188
\(155\) 0 0
\(156\) −1.05790e8 −2.23104
\(157\) 7.85080e7 1.61907 0.809534 0.587073i \(-0.199720\pi\)
0.809534 + 0.587073i \(0.199720\pi\)
\(158\) 9.31022e7 1.87785
\(159\) 1.35510e7 0.267351
\(160\) 0 0
\(161\) 2.46960e6 0.0466376
\(162\) 1.16917e7 0.216060
\(163\) 4.68184e7 0.846759 0.423380 0.905952i \(-0.360844\pi\)
0.423380 + 0.905952i \(0.360844\pi\)
\(164\) −4.12640e7 −0.730495
\(165\) 0 0
\(166\) 4.07925e7 0.692153
\(167\) 2.50043e7 0.415438 0.207719 0.978188i \(-0.433396\pi\)
0.207719 + 0.978188i \(0.433396\pi\)
\(168\) −5.68814e7 −0.925525
\(169\) 5.83835e7 0.930437
\(170\) 0 0
\(171\) −1.84904e7 −0.282787
\(172\) 6.81669e6 0.102147
\(173\) −5.30671e7 −0.779227 −0.389613 0.920979i \(-0.627391\pi\)
−0.389613 + 0.920979i \(0.627391\pi\)
\(174\) −9.68469e7 −1.39368
\(175\) 0 0
\(176\) −1.90724e8 −2.63701
\(177\) 4.28268e7 0.580509
\(178\) −1.51540e8 −2.01399
\(179\) 4.22054e7 0.550025 0.275012 0.961441i \(-0.411318\pi\)
0.275012 + 0.961441i \(0.411318\pi\)
\(180\) 0 0
\(181\) −1.00020e8 −1.25376 −0.626879 0.779116i \(-0.715668\pi\)
−0.626879 + 0.779116i \(0.715668\pi\)
\(182\) 1.01695e8 1.25041
\(183\) 1.00700e7 0.121465
\(184\) 2.94941e7 0.349038
\(185\) 0 0
\(186\) 1.19750e8 1.36453
\(187\) −4.87114e7 −0.544735
\(188\) −2.99402e8 −3.28626
\(189\) −8.26686e6 −0.0890685
\(190\) 0 0
\(191\) 6.17610e7 0.641354 0.320677 0.947189i \(-0.396090\pi\)
0.320677 + 0.947189i \(0.396090\pi\)
\(192\) −2.41327e8 −2.46066
\(193\) 7.67419e7 0.768390 0.384195 0.923252i \(-0.374479\pi\)
0.384195 + 0.923252i \(0.374479\pi\)
\(194\) −8.14007e7 −0.800428
\(195\) 0 0
\(196\) −2.30383e8 −2.18552
\(197\) 1.81032e8 1.68703 0.843516 0.537105i \(-0.180482\pi\)
0.843516 + 0.537105i \(0.180482\pi\)
\(198\) −4.72159e7 −0.432275
\(199\) 6.16084e7 0.554185 0.277092 0.960843i \(-0.410629\pi\)
0.277092 + 0.960843i \(0.410629\pi\)
\(200\) 0 0
\(201\) 1.23148e8 1.06965
\(202\) −3.96054e8 −3.38084
\(203\) 6.84776e7 0.574530
\(204\) −1.59040e8 −1.31160
\(205\) 0 0
\(206\) 1.18185e8 0.941952
\(207\) 4.28652e6 0.0335899
\(208\) 7.13013e8 5.49383
\(209\) 7.46716e7 0.565775
\(210\) 0 0
\(211\) −1.69917e8 −1.24523 −0.622613 0.782530i \(-0.713929\pi\)
−0.622613 + 0.782530i \(0.713929\pi\)
\(212\) −1.78673e8 −1.28790
\(213\) −4.08465e7 −0.289618
\(214\) 2.53875e8 1.77081
\(215\) 0 0
\(216\) −9.87299e7 −0.666593
\(217\) −8.46720e7 −0.562511
\(218\) −3.45795e7 −0.226058
\(219\) −4.11186e7 −0.264535
\(220\) 0 0
\(221\) 1.82105e8 1.13488
\(222\) 7.15948e7 0.439184
\(223\) −1.48129e8 −0.894484 −0.447242 0.894413i \(-0.647594\pi\)
−0.447242 + 0.894413i \(0.647594\pi\)
\(224\) 3.28944e8 1.95548
\(225\) 0 0
\(226\) 5.54509e8 3.19543
\(227\) 3.96127e7 0.224773 0.112387 0.993665i \(-0.464151\pi\)
0.112387 + 0.993665i \(0.464151\pi\)
\(228\) 2.43799e8 1.36226
\(229\) −3.71816e7 −0.204599 −0.102300 0.994754i \(-0.532620\pi\)
−0.102300 + 0.994754i \(0.532620\pi\)
\(230\) 0 0
\(231\) 3.33850e7 0.178201
\(232\) 8.17819e8 4.29981
\(233\) 1.79591e8 0.930122 0.465061 0.885279i \(-0.346032\pi\)
0.465061 + 0.885279i \(0.346032\pi\)
\(234\) 1.76514e8 0.900584
\(235\) 0 0
\(236\) −5.64679e8 −2.79646
\(237\) −1.14262e8 −0.557548
\(238\) 1.52885e8 0.735099
\(239\) −3.73328e8 −1.76888 −0.884439 0.466655i \(-0.845459\pi\)
−0.884439 + 0.466655i \(0.845459\pi\)
\(240\) 0 0
\(241\) −2.57022e8 −1.18280 −0.591398 0.806380i \(-0.701424\pi\)
−0.591398 + 0.806380i \(0.701424\pi\)
\(242\) −2.38041e8 −1.07969
\(243\) −1.43489e7 −0.0641500
\(244\) −1.32774e8 −0.585127
\(245\) 0 0
\(246\) 6.88505e7 0.294873
\(247\) −2.79156e8 −1.17871
\(248\) −1.01123e9 −4.20986
\(249\) −5.00635e7 −0.205506
\(250\) 0 0
\(251\) −1.27344e8 −0.508302 −0.254151 0.967165i \(-0.581796\pi\)
−0.254151 + 0.967165i \(0.581796\pi\)
\(252\) 1.09000e8 0.429067
\(253\) −1.73107e7 −0.0672037
\(254\) −8.66976e8 −3.31963
\(255\) 0 0
\(256\) 9.76454e8 3.63757
\(257\) −1.30682e8 −0.480230 −0.240115 0.970744i \(-0.577185\pi\)
−0.240115 + 0.970744i \(0.577185\pi\)
\(258\) −1.13739e7 −0.0412326
\(259\) −5.06226e7 −0.181049
\(260\) 0 0
\(261\) 1.18858e8 0.413795
\(262\) 3.10381e8 1.06620
\(263\) 2.67747e8 0.907568 0.453784 0.891112i \(-0.350074\pi\)
0.453784 + 0.891112i \(0.350074\pi\)
\(264\) 3.98712e8 1.33366
\(265\) 0 0
\(266\) −2.34363e8 −0.763491
\(267\) 1.85981e8 0.597969
\(268\) −1.62373e9 −5.15279
\(269\) 1.49432e8 0.468070 0.234035 0.972228i \(-0.424807\pi\)
0.234035 + 0.972228i \(0.424807\pi\)
\(270\) 0 0
\(271\) −1.53185e8 −0.467545 −0.233773 0.972291i \(-0.575107\pi\)
−0.233773 + 0.972291i \(0.575107\pi\)
\(272\) 1.07192e9 3.22976
\(273\) −1.24808e8 −0.371256
\(274\) 1.76113e8 0.517206
\(275\) 0 0
\(276\) −5.65186e7 −0.161811
\(277\) −6.54462e8 −1.85014 −0.925072 0.379792i \(-0.875996\pi\)
−0.925072 + 0.379792i \(0.875996\pi\)
\(278\) 1.01358e9 2.82943
\(279\) −1.46966e8 −0.405138
\(280\) 0 0
\(281\) 5.51493e8 1.48275 0.741375 0.671091i \(-0.234174\pi\)
0.741375 + 0.671091i \(0.234174\pi\)
\(282\) 4.99564e8 1.32654
\(283\) 2.10200e8 0.551291 0.275646 0.961259i \(-0.411108\pi\)
0.275646 + 0.961259i \(0.411108\pi\)
\(284\) 5.38568e8 1.39517
\(285\) 0 0
\(286\) −7.12837e8 −1.80181
\(287\) −4.86822e7 −0.121558
\(288\) 5.70953e8 1.40840
\(289\) −1.36569e8 −0.332819
\(290\) 0 0
\(291\) 9.99009e7 0.237653
\(292\) 5.42156e8 1.27434
\(293\) −5.42402e8 −1.25975 −0.629875 0.776697i \(-0.716894\pi\)
−0.629875 + 0.776697i \(0.716894\pi\)
\(294\) 3.84403e8 0.882208
\(295\) 0 0
\(296\) −6.04578e8 −1.35498
\(297\) 5.79468e7 0.128346
\(298\) 1.59175e9 3.48433
\(299\) 6.47153e7 0.140010
\(300\) 0 0
\(301\) 8.04216e6 0.0169977
\(302\) −8.14137e8 −1.70088
\(303\) 4.86066e8 1.00380
\(304\) −1.64318e9 −3.35450
\(305\) 0 0
\(306\) 2.65365e8 0.529442
\(307\) −9.26477e8 −1.82747 −0.913736 0.406310i \(-0.866815\pi\)
−0.913736 + 0.406310i \(0.866815\pi\)
\(308\) −4.40187e8 −0.858439
\(309\) −1.45046e8 −0.279673
\(310\) 0 0
\(311\) −2.12976e8 −0.401485 −0.200743 0.979644i \(-0.564335\pi\)
−0.200743 + 0.979644i \(0.564335\pi\)
\(312\) −1.49056e9 −2.77850
\(313\) −3.63896e8 −0.670768 −0.335384 0.942081i \(-0.608866\pi\)
−0.335384 + 0.942081i \(0.608866\pi\)
\(314\) 1.72718e9 3.14835
\(315\) 0 0
\(316\) 1.50656e9 2.68586
\(317\) 3.17049e8 0.559009 0.279505 0.960144i \(-0.409830\pi\)
0.279505 + 0.960144i \(0.409830\pi\)
\(318\) 2.98123e8 0.519876
\(319\) −4.79996e8 −0.827885
\(320\) 0 0
\(321\) −3.11574e8 −0.525767
\(322\) 5.43312e7 0.0906888
\(323\) −4.19673e8 −0.692950
\(324\) 1.89193e8 0.309028
\(325\) 0 0
\(326\) 1.03001e9 1.64656
\(327\) 4.24384e7 0.0671185
\(328\) −5.81405e8 −0.909746
\(329\) −3.53227e8 −0.546850
\(330\) 0 0
\(331\) 2.24556e8 0.340351 0.170176 0.985414i \(-0.445566\pi\)
0.170176 + 0.985414i \(0.445566\pi\)
\(332\) 6.60097e8 0.989975
\(333\) −8.78664e7 −0.130397
\(334\) 5.50094e8 0.807838
\(335\) 0 0
\(336\) −7.34651e8 −1.05656
\(337\) 1.23886e9 1.76327 0.881633 0.471935i \(-0.156444\pi\)
0.881633 + 0.471935i \(0.156444\pi\)
\(338\) 1.28444e9 1.80927
\(339\) −6.80534e8 −0.948748
\(340\) 0 0
\(341\) 5.93510e8 0.810565
\(342\) −4.06788e8 −0.549891
\(343\) −6.17688e8 −0.826494
\(344\) 9.60464e7 0.127212
\(345\) 0 0
\(346\) −1.16748e9 −1.51524
\(347\) 5.83643e8 0.749884 0.374942 0.927048i \(-0.377663\pi\)
0.374942 + 0.927048i \(0.377663\pi\)
\(348\) −1.56716e9 −1.99336
\(349\) −4.69471e8 −0.591180 −0.295590 0.955315i \(-0.595516\pi\)
−0.295590 + 0.955315i \(0.595516\pi\)
\(350\) 0 0
\(351\) −2.16631e8 −0.267390
\(352\) −2.30574e9 −2.81781
\(353\) −6.18559e7 −0.0748461 −0.0374231 0.999300i \(-0.511915\pi\)
−0.0374231 + 0.999300i \(0.511915\pi\)
\(354\) 9.42189e8 1.12882
\(355\) 0 0
\(356\) −2.45219e9 −2.88057
\(357\) −1.87632e8 −0.218257
\(358\) 9.28519e8 1.06955
\(359\) −1.23537e7 −0.0140918 −0.00704592 0.999975i \(-0.502243\pi\)
−0.00704592 + 0.999975i \(0.502243\pi\)
\(360\) 0 0
\(361\) −2.50539e8 −0.280285
\(362\) −2.20045e9 −2.43799
\(363\) 2.92141e8 0.320567
\(364\) 1.64562e9 1.78844
\(365\) 0 0
\(366\) 2.21539e8 0.236193
\(367\) 1.25514e7 0.0132544 0.00662721 0.999978i \(-0.497890\pi\)
0.00662721 + 0.999978i \(0.497890\pi\)
\(368\) 3.80930e8 0.398454
\(369\) −8.44984e7 −0.0875500
\(370\) 0 0
\(371\) −2.10794e8 −0.214313
\(372\) 1.93778e9 1.95166
\(373\) 5.65994e8 0.564717 0.282359 0.959309i \(-0.408883\pi\)
0.282359 + 0.959309i \(0.408883\pi\)
\(374\) −1.07165e9 −1.05926
\(375\) 0 0
\(376\) −4.21854e9 −4.09265
\(377\) 1.79444e9 1.72478
\(378\) −1.81871e8 −0.173198
\(379\) 1.77776e9 1.67740 0.838698 0.544597i \(-0.183317\pi\)
0.838698 + 0.544597i \(0.183317\pi\)
\(380\) 0 0
\(381\) 1.06402e9 0.985623
\(382\) 1.35874e9 1.24714
\(383\) −1.18195e9 −1.07498 −0.537492 0.843269i \(-0.680628\pi\)
−0.537492 + 0.843269i \(0.680628\pi\)
\(384\) −2.60246e9 −2.34544
\(385\) 0 0
\(386\) 1.68832e9 1.49417
\(387\) 1.39589e7 0.0122423
\(388\) −1.31721e9 −1.14484
\(389\) 6.02482e8 0.518944 0.259472 0.965751i \(-0.416451\pi\)
0.259472 + 0.965751i \(0.416451\pi\)
\(390\) 0 0
\(391\) 9.72905e7 0.0823098
\(392\) −3.24607e9 −2.72180
\(393\) −3.80922e8 −0.316564
\(394\) 3.98270e9 3.28051
\(395\) 0 0
\(396\) −7.64039e8 −0.618276
\(397\) 1.86765e8 0.149806 0.0749029 0.997191i \(-0.476135\pi\)
0.0749029 + 0.997191i \(0.476135\pi\)
\(398\) 1.35539e9 1.07764
\(399\) 2.87628e8 0.226687
\(400\) 0 0
\(401\) −9.96333e8 −0.771613 −0.385806 0.922580i \(-0.626077\pi\)
−0.385806 + 0.922580i \(0.626077\pi\)
\(402\) 2.70926e9 2.07998
\(403\) −2.21881e9 −1.68870
\(404\) −6.40887e9 −4.83556
\(405\) 0 0
\(406\) 1.50651e9 1.11720
\(407\) 3.54840e8 0.260887
\(408\) −2.24086e9 −1.63344
\(409\) −2.38644e9 −1.72472 −0.862362 0.506293i \(-0.831015\pi\)
−0.862362 + 0.506293i \(0.831015\pi\)
\(410\) 0 0
\(411\) −2.16138e8 −0.153563
\(412\) 1.91246e9 1.34726
\(413\) −6.66194e8 −0.465345
\(414\) 9.43034e7 0.0653170
\(415\) 0 0
\(416\) 8.61990e9 5.87051
\(417\) −1.24393e9 −0.840081
\(418\) 1.64278e9 1.10017
\(419\) 2.59644e9 1.72437 0.862183 0.506597i \(-0.169097\pi\)
0.862183 + 0.506597i \(0.169097\pi\)
\(420\) 0 0
\(421\) 2.83850e9 1.85396 0.926981 0.375108i \(-0.122394\pi\)
0.926981 + 0.375108i \(0.122394\pi\)
\(422\) −3.73817e9 −2.42139
\(423\) −6.13101e8 −0.393859
\(424\) −2.51748e9 −1.60393
\(425\) 0 0
\(426\) −8.98622e8 −0.563175
\(427\) −1.56644e8 −0.0973680
\(428\) 4.10816e9 2.53276
\(429\) 8.74845e8 0.534971
\(430\) 0 0
\(431\) −1.52808e9 −0.919342 −0.459671 0.888089i \(-0.652033\pi\)
−0.459671 + 0.888089i \(0.652033\pi\)
\(432\) −1.27514e9 −0.760967
\(433\) 2.66061e9 1.57498 0.787488 0.616330i \(-0.211381\pi\)
0.787488 + 0.616330i \(0.211381\pi\)
\(434\) −1.86278e9 −1.09383
\(435\) 0 0
\(436\) −5.59559e8 −0.323328
\(437\) −1.49140e8 −0.0854890
\(438\) −9.04609e8 −0.514400
\(439\) 2.29222e9 1.29310 0.646549 0.762873i \(-0.276212\pi\)
0.646549 + 0.762873i \(0.276212\pi\)
\(440\) 0 0
\(441\) −4.71767e8 −0.261935
\(442\) 4.00632e9 2.20682
\(443\) 9.34043e8 0.510451 0.255225 0.966882i \(-0.417850\pi\)
0.255225 + 0.966882i \(0.417850\pi\)
\(444\) 1.15853e9 0.628157
\(445\) 0 0
\(446\) −3.25884e9 −1.73936
\(447\) −1.95352e9 −1.03452
\(448\) 3.75398e9 1.97251
\(449\) −8.85012e7 −0.0461410 −0.0230705 0.999734i \(-0.507344\pi\)
−0.0230705 + 0.999734i \(0.507344\pi\)
\(450\) 0 0
\(451\) 3.41239e8 0.175162
\(452\) 8.97296e9 4.57037
\(453\) 9.99168e8 0.505004
\(454\) 8.71480e8 0.437081
\(455\) 0 0
\(456\) 3.43510e9 1.69653
\(457\) 1.21064e9 0.593347 0.296674 0.954979i \(-0.404123\pi\)
0.296674 + 0.954979i \(0.404123\pi\)
\(458\) −8.17995e8 −0.397852
\(459\) −3.25675e8 −0.157195
\(460\) 0 0
\(461\) 8.57428e8 0.407610 0.203805 0.979012i \(-0.434669\pi\)
0.203805 + 0.979012i \(0.434669\pi\)
\(462\) 7.34469e8 0.346519
\(463\) 1.94662e9 0.911481 0.455741 0.890113i \(-0.349375\pi\)
0.455741 + 0.890113i \(0.349375\pi\)
\(464\) 1.05625e10 4.90856
\(465\) 0 0
\(466\) 3.95101e9 1.80866
\(467\) 9.59955e8 0.436156 0.218078 0.975931i \(-0.430021\pi\)
0.218078 + 0.975931i \(0.430021\pi\)
\(468\) 2.85632e9 1.28809
\(469\) −1.91564e9 −0.857450
\(470\) 0 0
\(471\) −2.11972e9 −0.934769
\(472\) −7.95626e9 −3.48267
\(473\) −5.63717e7 −0.0244933
\(474\) −2.51376e9 −1.08418
\(475\) 0 0
\(476\) 2.47396e9 1.05140
\(477\) −3.65878e8 −0.154355
\(478\) −8.21322e9 −3.43966
\(479\) 3.03579e8 0.126211 0.0631054 0.998007i \(-0.479900\pi\)
0.0631054 + 0.998007i \(0.479900\pi\)
\(480\) 0 0
\(481\) −1.32655e9 −0.543522
\(482\) −5.65447e9 −2.30000
\(483\) −6.66792e7 −0.0269262
\(484\) −3.85193e9 −1.54426
\(485\) 0 0
\(486\) −3.15676e8 −0.124743
\(487\) −4.36059e8 −0.171078 −0.0855389 0.996335i \(-0.527261\pi\)
−0.0855389 + 0.996335i \(0.527261\pi\)
\(488\) −1.87078e9 −0.728707
\(489\) −1.26410e9 −0.488877
\(490\) 0 0
\(491\) 8.34813e8 0.318276 0.159138 0.987256i \(-0.449128\pi\)
0.159138 + 0.987256i \(0.449128\pi\)
\(492\) 1.11413e9 0.421752
\(493\) 2.69769e9 1.01398
\(494\) −6.14144e9 −2.29206
\(495\) 0 0
\(496\) −1.30605e10 −4.80587
\(497\) 6.35389e8 0.232163
\(498\) −1.10140e9 −0.399615
\(499\) −4.50230e9 −1.62212 −0.811059 0.584964i \(-0.801109\pi\)
−0.811059 + 0.584964i \(0.801109\pi\)
\(500\) 0 0
\(501\) −6.75116e8 −0.239854
\(502\) −2.80158e9 −0.988416
\(503\) −2.41700e9 −0.846815 −0.423407 0.905939i \(-0.639166\pi\)
−0.423407 + 0.905939i \(0.639166\pi\)
\(504\) 1.53580e9 0.534352
\(505\) 0 0
\(506\) −3.80836e8 −0.130681
\(507\) −1.57636e9 −0.537188
\(508\) −1.40292e10 −4.74801
\(509\) 3.51209e9 1.18047 0.590233 0.807233i \(-0.299036\pi\)
0.590233 + 0.807233i \(0.299036\pi\)
\(510\) 0 0
\(511\) 6.39622e8 0.212056
\(512\) 9.14439e9 3.01099
\(513\) 4.99240e8 0.163267
\(514\) −2.87500e9 −0.933829
\(515\) 0 0
\(516\) −1.84051e8 −0.0589744
\(517\) 2.47595e9 0.787998
\(518\) −1.11370e9 −0.352057
\(519\) 1.43281e9 0.449887
\(520\) 0 0
\(521\) 9.90013e8 0.306697 0.153348 0.988172i \(-0.450994\pi\)
0.153348 + 0.988172i \(0.450994\pi\)
\(522\) 2.61487e9 0.804642
\(523\) −4.45926e8 −0.136303 −0.0681517 0.997675i \(-0.521710\pi\)
−0.0681517 + 0.997675i \(0.521710\pi\)
\(524\) 5.02252e9 1.52497
\(525\) 0 0
\(526\) 5.89043e9 1.76481
\(527\) −3.33567e9 −0.992765
\(528\) 5.14955e9 1.52248
\(529\) −3.37025e9 −0.989845
\(530\) 0 0
\(531\) −1.15632e9 −0.335157
\(532\) −3.79243e9 −1.09201
\(533\) −1.27571e9 −0.364926
\(534\) 4.09158e9 1.16278
\(535\) 0 0
\(536\) −2.28782e10 −6.41719
\(537\) −1.13955e9 −0.317557
\(538\) 3.28751e9 0.910183
\(539\) 1.90519e9 0.524056
\(540\) 0 0
\(541\) 2.84753e9 0.773175 0.386588 0.922253i \(-0.373654\pi\)
0.386588 + 0.922253i \(0.373654\pi\)
\(542\) −3.37007e9 −0.909162
\(543\) 2.70055e9 0.723858
\(544\) 1.29588e10 3.45120
\(545\) 0 0
\(546\) −2.74578e9 −0.721923
\(547\) −4.34116e9 −1.13410 −0.567048 0.823684i \(-0.691915\pi\)
−0.567048 + 0.823684i \(0.691915\pi\)
\(548\) 2.84982e9 0.739752
\(549\) −2.71889e8 −0.0701276
\(550\) 0 0
\(551\) −4.13540e9 −1.05314
\(552\) −7.96340e8 −0.201517
\(553\) 1.77741e9 0.446940
\(554\) −1.43982e10 −3.59769
\(555\) 0 0
\(556\) 1.64015e10 4.04690
\(557\) −3.00124e9 −0.735880 −0.367940 0.929849i \(-0.619937\pi\)
−0.367940 + 0.929849i \(0.619937\pi\)
\(558\) −3.23326e9 −0.787809
\(559\) 2.10743e8 0.0510284
\(560\) 0 0
\(561\) 1.31521e9 0.314503
\(562\) 1.21328e10 2.88327
\(563\) 7.43886e9 1.75682 0.878409 0.477909i \(-0.158605\pi\)
0.878409 + 0.477909i \(0.158605\pi\)
\(564\) 8.08385e9 1.89732
\(565\) 0 0
\(566\) 4.62441e9 1.07201
\(567\) 2.23205e8 0.0514237
\(568\) 7.58837e9 1.73752
\(569\) −7.20171e8 −0.163886 −0.0819431 0.996637i \(-0.526113\pi\)
−0.0819431 + 0.996637i \(0.526113\pi\)
\(570\) 0 0
\(571\) −2.30186e9 −0.517431 −0.258716 0.965954i \(-0.583299\pi\)
−0.258716 + 0.965954i \(0.583299\pi\)
\(572\) −1.15350e10 −2.57710
\(573\) −1.66755e9 −0.370286
\(574\) −1.07101e9 −0.236375
\(575\) 0 0
\(576\) 6.51584e9 1.42066
\(577\) 7.37257e9 1.59773 0.798865 0.601511i \(-0.205434\pi\)
0.798865 + 0.601511i \(0.205434\pi\)
\(578\) −3.00451e9 −0.647181
\(579\) −2.07203e9 −0.443630
\(580\) 0 0
\(581\) 7.78766e8 0.164737
\(582\) 2.19782e9 0.462127
\(583\) 1.47756e9 0.308821
\(584\) 7.63892e9 1.58704
\(585\) 0 0
\(586\) −1.19328e10 −2.44964
\(587\) −7.36500e8 −0.150293 −0.0751467 0.997172i \(-0.523942\pi\)
−0.0751467 + 0.997172i \(0.523942\pi\)
\(588\) 6.22034e9 1.26181
\(589\) 5.11338e9 1.03111
\(590\) 0 0
\(591\) −4.88786e9 −0.974008
\(592\) −7.80842e9 −1.54681
\(593\) 5.08539e9 1.00146 0.500729 0.865604i \(-0.333065\pi\)
0.500729 + 0.865604i \(0.333065\pi\)
\(594\) 1.27483e9 0.249574
\(595\) 0 0
\(596\) 2.57575e10 4.98358
\(597\) −1.66343e9 −0.319959
\(598\) 1.42374e9 0.272255
\(599\) 4.86023e9 0.923980 0.461990 0.886885i \(-0.347136\pi\)
0.461990 + 0.886885i \(0.347136\pi\)
\(600\) 0 0
\(601\) 6.78466e9 1.27488 0.637438 0.770502i \(-0.279994\pi\)
0.637438 + 0.770502i \(0.279994\pi\)
\(602\) 1.76928e8 0.0330528
\(603\) −3.32500e9 −0.617563
\(604\) −1.31742e10 −2.43274
\(605\) 0 0
\(606\) 1.06935e10 1.95193
\(607\) −2.85250e9 −0.517685 −0.258842 0.965920i \(-0.583341\pi\)
−0.258842 + 0.965920i \(0.583341\pi\)
\(608\) −1.98651e10 −3.58449
\(609\) −1.84890e9 −0.331705
\(610\) 0 0
\(611\) −9.25622e9 −1.64168
\(612\) 4.29408e9 0.757252
\(613\) −7.75467e9 −1.35973 −0.679864 0.733339i \(-0.737961\pi\)
−0.679864 + 0.733339i \(0.737961\pi\)
\(614\) −2.03825e10 −3.55360
\(615\) 0 0
\(616\) −6.20218e9 −1.06908
\(617\) 6.34097e9 1.08682 0.543410 0.839468i \(-0.317133\pi\)
0.543410 + 0.839468i \(0.317133\pi\)
\(618\) −3.19101e9 −0.543836
\(619\) 3.19831e9 0.542005 0.271003 0.962579i \(-0.412645\pi\)
0.271003 + 0.962579i \(0.412645\pi\)
\(620\) 0 0
\(621\) −1.15736e8 −0.0193931
\(622\) −4.68548e9 −0.780706
\(623\) −2.89303e9 −0.479342
\(624\) −1.92513e10 −3.17187
\(625\) 0 0
\(626\) −8.00572e9 −1.30434
\(627\) −2.01613e9 −0.326650
\(628\) 2.79488e10 4.50303
\(629\) −1.99429e9 −0.319529
\(630\) 0 0
\(631\) −3.79459e9 −0.601260 −0.300630 0.953741i \(-0.597197\pi\)
−0.300630 + 0.953741i \(0.597197\pi\)
\(632\) 2.12273e10 3.34492
\(633\) 4.58776e9 0.718931
\(634\) 6.97508e9 1.08702
\(635\) 0 0
\(636\) 4.82417e9 0.743571
\(637\) −7.12246e9 −1.09180
\(638\) −1.05599e10 −1.60986
\(639\) 1.10285e9 0.167211
\(640\) 0 0
\(641\) 6.77964e9 1.01672 0.508362 0.861143i \(-0.330251\pi\)
0.508362 + 0.861143i \(0.330251\pi\)
\(642\) −6.85462e9 −1.02238
\(643\) 1.13203e9 0.167927 0.0839635 0.996469i \(-0.473242\pi\)
0.0839635 + 0.996469i \(0.473242\pi\)
\(644\) 8.79178e8 0.129711
\(645\) 0 0
\(646\) −9.23280e9 −1.34747
\(647\) 5.54265e9 0.804549 0.402274 0.915519i \(-0.368220\pi\)
0.402274 + 0.915519i \(0.368220\pi\)
\(648\) 2.66571e9 0.384858
\(649\) 4.66970e9 0.670552
\(650\) 0 0
\(651\) 2.28614e9 0.324766
\(652\) 1.66674e10 2.35505
\(653\) −9.41765e9 −1.32357 −0.661784 0.749695i \(-0.730200\pi\)
−0.661784 + 0.749695i \(0.730200\pi\)
\(654\) 9.33646e8 0.130515
\(655\) 0 0
\(656\) −7.50911e9 −1.03854
\(657\) 1.11020e9 0.152730
\(658\) −7.77099e9 −1.06337
\(659\) 7.46390e9 1.01594 0.507969 0.861376i \(-0.330397\pi\)
0.507969 + 0.861376i \(0.330397\pi\)
\(660\) 0 0
\(661\) 5.58309e9 0.751917 0.375958 0.926637i \(-0.377314\pi\)
0.375958 + 0.926637i \(0.377314\pi\)
\(662\) 4.94024e9 0.661828
\(663\) −4.91684e9 −0.655223
\(664\) 9.30069e9 1.23290
\(665\) 0 0
\(666\) −1.93306e9 −0.253563
\(667\) 9.58687e8 0.125094
\(668\) 8.90152e9 1.15544
\(669\) 3.99948e9 0.516431
\(670\) 0 0
\(671\) 1.09800e9 0.140305
\(672\) −8.88149e9 −1.12900
\(673\) −3.39933e9 −0.429873 −0.214936 0.976628i \(-0.568954\pi\)
−0.214936 + 0.976628i \(0.568954\pi\)
\(674\) 2.72549e10 3.42875
\(675\) 0 0
\(676\) 2.07845e10 2.58778
\(677\) 1.38930e10 1.72082 0.860409 0.509604i \(-0.170208\pi\)
0.860409 + 0.509604i \(0.170208\pi\)
\(678\) −1.49717e10 −1.84488
\(679\) −1.55401e9 −0.190507
\(680\) 0 0
\(681\) −1.06954e9 −0.129773
\(682\) 1.30572e10 1.57618
\(683\) 7.22576e9 0.867782 0.433891 0.900965i \(-0.357140\pi\)
0.433891 + 0.900965i \(0.357140\pi\)
\(684\) −6.58257e9 −0.786500
\(685\) 0 0
\(686\) −1.35891e10 −1.60715
\(687\) 1.00390e9 0.118125
\(688\) 1.24048e9 0.145222
\(689\) −5.52380e9 −0.643385
\(690\) 0 0
\(691\) −5.83386e9 −0.672641 −0.336320 0.941748i \(-0.609182\pi\)
−0.336320 + 0.941748i \(0.609182\pi\)
\(692\) −1.88919e10 −2.16722
\(693\) −9.01394e8 −0.102884
\(694\) 1.28401e10 1.45818
\(695\) 0 0
\(696\) −2.20811e10 −2.48250
\(697\) −1.91785e9 −0.214536
\(698\) −1.03284e10 −1.14958
\(699\) −4.84897e9 −0.537006
\(700\) 0 0
\(701\) 4.37486e9 0.479680 0.239840 0.970812i \(-0.422905\pi\)
0.239840 + 0.970812i \(0.422905\pi\)
\(702\) −4.76588e9 −0.519952
\(703\) 3.05712e9 0.331871
\(704\) −2.63136e10 −2.84234
\(705\) 0 0
\(706\) −1.36083e9 −0.145542
\(707\) −7.56103e9 −0.804661
\(708\) 1.52463e10 1.61454
\(709\) −3.54685e9 −0.373750 −0.186875 0.982384i \(-0.559836\pi\)
−0.186875 + 0.982384i \(0.559836\pi\)
\(710\) 0 0
\(711\) 3.08507e9 0.321900
\(712\) −3.45511e10 −3.58742
\(713\) −1.18541e9 −0.122477
\(714\) −4.12790e9 −0.424409
\(715\) 0 0
\(716\) 1.50251e10 1.52976
\(717\) 1.00799e10 1.02126
\(718\) −2.71782e8 −0.0274022
\(719\) −1.06545e10 −1.06901 −0.534507 0.845164i \(-0.679503\pi\)
−0.534507 + 0.845164i \(0.679503\pi\)
\(720\) 0 0
\(721\) 2.25627e9 0.224191
\(722\) −5.51186e9 −0.545027
\(723\) 6.93958e9 0.682888
\(724\) −3.56073e10 −3.48702
\(725\) 0 0
\(726\) 6.42710e9 0.623357
\(727\) 9.21169e9 0.889138 0.444569 0.895745i \(-0.353357\pi\)
0.444569 + 0.895745i \(0.353357\pi\)
\(728\) 2.31866e10 2.22729
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 3.16823e8 0.0299989
\(732\) 3.58491e9 0.337823
\(733\) 5.81770e8 0.0545616 0.0272808 0.999628i \(-0.491315\pi\)
0.0272808 + 0.999628i \(0.491315\pi\)
\(734\) 2.76131e8 0.0257738
\(735\) 0 0
\(736\) 4.60522e9 0.425773
\(737\) 1.34277e10 1.23557
\(738\) −1.85896e9 −0.170245
\(739\) 1.43208e8 0.0130531 0.00652654 0.999979i \(-0.497923\pi\)
0.00652654 + 0.999979i \(0.497923\pi\)
\(740\) 0 0
\(741\) 7.53722e9 0.680530
\(742\) −4.63746e9 −0.416742
\(743\) −1.76012e10 −1.57428 −0.787139 0.616775i \(-0.788439\pi\)
−0.787139 + 0.616775i \(0.788439\pi\)
\(744\) 2.73031e10 2.43056
\(745\) 0 0
\(746\) 1.24519e10 1.09812
\(747\) 1.35171e9 0.118649
\(748\) −1.73413e10 −1.51504
\(749\) 4.84670e9 0.421463
\(750\) 0 0
\(751\) −2.10398e10 −1.81260 −0.906299 0.422636i \(-0.861105\pi\)
−0.906299 + 0.422636i \(0.861105\pi\)
\(752\) −5.44844e10 −4.67207
\(753\) 3.43830e9 0.293468
\(754\) 3.94777e10 3.35391
\(755\) 0 0
\(756\) −2.94300e9 −0.247722
\(757\) 3.91015e9 0.327610 0.163805 0.986493i \(-0.447623\pi\)
0.163805 + 0.986493i \(0.447623\pi\)
\(758\) 3.91107e10 3.26177
\(759\) 4.67389e8 0.0388001
\(760\) 0 0
\(761\) 1.20242e10 0.989032 0.494516 0.869169i \(-0.335345\pi\)
0.494516 + 0.869169i \(0.335345\pi\)
\(762\) 2.34084e10 1.91659
\(763\) −6.60153e8 −0.0538033
\(764\) 2.19869e10 1.78377
\(765\) 0 0
\(766\) −2.60028e10 −2.09035
\(767\) −1.74575e10 −1.39700
\(768\) −2.63643e10 −2.10015
\(769\) 1.18948e10 0.943223 0.471611 0.881807i \(-0.343673\pi\)
0.471611 + 0.881807i \(0.343673\pi\)
\(770\) 0 0
\(771\) 3.52841e9 0.277261
\(772\) 2.73201e10 2.13709
\(773\) 7.77614e9 0.605531 0.302765 0.953065i \(-0.402090\pi\)
0.302765 + 0.953065i \(0.402090\pi\)
\(774\) 3.07096e8 0.0238057
\(775\) 0 0
\(776\) −1.85594e10 −1.42576
\(777\) 1.36681e9 0.104528
\(778\) 1.32546e10 1.00911
\(779\) 2.93994e9 0.222822
\(780\) 0 0
\(781\) −4.45378e9 −0.334541
\(782\) 2.14039e9 0.160055
\(783\) −3.20916e9 −0.238905
\(784\) −4.19245e10 −3.10715
\(785\) 0 0
\(786\) −8.38028e9 −0.615573
\(787\) −2.44365e10 −1.78701 −0.893507 0.449050i \(-0.851763\pi\)
−0.893507 + 0.449050i \(0.851763\pi\)
\(788\) 6.44473e10 4.69206
\(789\) −7.22916e9 −0.523985
\(790\) 0 0
\(791\) 1.05861e10 0.760532
\(792\) −1.07652e10 −0.769989
\(793\) −4.10482e9 −0.292306
\(794\) 4.10883e9 0.291304
\(795\) 0 0
\(796\) 2.19326e10 1.54133
\(797\) 2.26970e9 0.158805 0.0794024 0.996843i \(-0.474699\pi\)
0.0794024 + 0.996843i \(0.474699\pi\)
\(798\) 6.32781e9 0.440802
\(799\) −1.39155e10 −0.965125
\(800\) 0 0
\(801\) −5.02148e9 −0.345237
\(802\) −2.19193e10 −1.50043
\(803\) −4.48345e9 −0.305568
\(804\) 4.38408e10 2.97496
\(805\) 0 0
\(806\) −4.88138e10 −3.28375
\(807\) −4.03467e9 −0.270240
\(808\) −9.03003e10 −6.02212
\(809\) −1.63200e10 −1.08368 −0.541838 0.840483i \(-0.682271\pi\)
−0.541838 + 0.840483i \(0.682271\pi\)
\(810\) 0 0
\(811\) 6.99393e9 0.460414 0.230207 0.973142i \(-0.426060\pi\)
0.230207 + 0.973142i \(0.426060\pi\)
\(812\) 2.43780e10 1.59791
\(813\) 4.13599e9 0.269937
\(814\) 7.80649e9 0.507306
\(815\) 0 0
\(816\) −2.89417e10 −1.86470
\(817\) −4.85670e8 −0.0311576
\(818\) −5.25017e10 −3.35380
\(819\) 3.36982e9 0.214345
\(820\) 0 0
\(821\) 4.00949e9 0.252865 0.126432 0.991975i \(-0.459647\pi\)
0.126432 + 0.991975i \(0.459647\pi\)
\(822\) −4.75504e9 −0.298609
\(823\) −1.88572e10 −1.17917 −0.589586 0.807706i \(-0.700709\pi\)
−0.589586 + 0.807706i \(0.700709\pi\)
\(824\) 2.69463e10 1.67785
\(825\) 0 0
\(826\) −1.46563e10 −0.904885
\(827\) −1.66386e10 −1.02293 −0.511466 0.859304i \(-0.670897\pi\)
−0.511466 + 0.859304i \(0.670897\pi\)
\(828\) 1.52600e9 0.0934219
\(829\) −1.37224e10 −0.836547 −0.418274 0.908321i \(-0.637365\pi\)
−0.418274 + 0.908321i \(0.637365\pi\)
\(830\) 0 0
\(831\) 1.76705e10 1.06818
\(832\) 9.83722e10 5.92162
\(833\) −1.07076e10 −0.641853
\(834\) −2.73666e10 −1.63357
\(835\) 0 0
\(836\) 2.65831e10 1.57356
\(837\) 3.96809e9 0.233907
\(838\) 5.71217e10 3.35311
\(839\) −6.56954e8 −0.0384033 −0.0192016 0.999816i \(-0.506112\pi\)
−0.0192016 + 0.999816i \(0.506112\pi\)
\(840\) 0 0
\(841\) 9.33282e9 0.541037
\(842\) 6.24469e10 3.60511
\(843\) −1.48903e10 −0.856066
\(844\) −6.04904e10 −3.46328
\(845\) 0 0
\(846\) −1.34882e10 −0.765876
\(847\) −4.54441e9 −0.256972
\(848\) −3.25144e10 −1.83101
\(849\) −5.67541e9 −0.318288
\(850\) 0 0
\(851\) −7.08716e8 −0.0394202
\(852\) −1.45413e10 −0.805501
\(853\) 8.70997e9 0.480502 0.240251 0.970711i \(-0.422770\pi\)
0.240251 + 0.970711i \(0.422770\pi\)
\(854\) −3.44617e9 −0.189336
\(855\) 0 0
\(856\) 5.78835e10 3.15425
\(857\) −1.93825e9 −0.105190 −0.0525952 0.998616i \(-0.516749\pi\)
−0.0525952 + 0.998616i \(0.516749\pi\)
\(858\) 1.92466e10 1.04028
\(859\) −7.95332e8 −0.0428127 −0.0214063 0.999771i \(-0.506814\pi\)
−0.0214063 + 0.999771i \(0.506814\pi\)
\(860\) 0 0
\(861\) 1.31442e9 0.0701815
\(862\) −3.36179e10 −1.78770
\(863\) −2.24172e9 −0.118725 −0.0593626 0.998236i \(-0.518907\pi\)
−0.0593626 + 0.998236i \(0.518907\pi\)
\(864\) −1.54157e10 −0.813141
\(865\) 0 0
\(866\) 5.85335e10 3.06261
\(867\) 3.68735e9 0.192153
\(868\) −3.01432e10 −1.56448
\(869\) −1.24588e10 −0.644030
\(870\) 0 0
\(871\) −5.01989e10 −2.57413
\(872\) −7.88412e9 −0.402666
\(873\) −2.69732e9 −0.137209
\(874\) −3.28109e9 −0.166237
\(875\) 0 0
\(876\) −1.46382e10 −0.735739
\(877\) −1.99169e10 −0.997062 −0.498531 0.866872i \(-0.666127\pi\)
−0.498531 + 0.866872i \(0.666127\pi\)
\(878\) 5.04289e10 2.51449
\(879\) 1.46448e10 0.727317
\(880\) 0 0
\(881\) −2.76906e10 −1.36432 −0.682161 0.731202i \(-0.738960\pi\)
−0.682161 + 0.731202i \(0.738960\pi\)
\(882\) −1.03789e10 −0.509343
\(883\) 1.65616e10 0.809544 0.404772 0.914418i \(-0.367351\pi\)
0.404772 + 0.914418i \(0.367351\pi\)
\(884\) 6.48295e10 3.15638
\(885\) 0 0
\(886\) 2.05489e10 0.992593
\(887\) −6.69904e9 −0.322314 −0.161157 0.986929i \(-0.551523\pi\)
−0.161157 + 0.986929i \(0.551523\pi\)
\(888\) 1.63236e10 0.782296
\(889\) −1.65514e10 −0.790092
\(890\) 0 0
\(891\) −1.56456e9 −0.0741005
\(892\) −5.27339e10 −2.48778
\(893\) 2.13315e10 1.00240
\(894\) −4.29774e10 −2.01168
\(895\) 0 0
\(896\) 4.04827e10 1.88015
\(897\) −1.74731e9 −0.0808346
\(898\) −1.94703e9 −0.0897232
\(899\) −3.28693e10 −1.50880
\(900\) 0 0
\(901\) −8.30427e9 −0.378238
\(902\) 7.50726e9 0.340611
\(903\) −2.17138e8 −0.00981362
\(904\) 1.26428e11 5.69186
\(905\) 0 0
\(906\) 2.19817e10 0.982002
\(907\) −3.87814e9 −0.172583 −0.0862916 0.996270i \(-0.527502\pi\)
−0.0862916 + 0.996270i \(0.527502\pi\)
\(908\) 1.41021e10 0.625150
\(909\) −1.31238e10 −0.579543
\(910\) 0 0
\(911\) −7.15870e9 −0.313704 −0.156852 0.987622i \(-0.550135\pi\)
−0.156852 + 0.987622i \(0.550135\pi\)
\(912\) 4.43659e10 1.93672
\(913\) −5.45878e9 −0.237382
\(914\) 2.66341e10 1.15379
\(915\) 0 0
\(916\) −1.32366e10 −0.569041
\(917\) 5.92545e9 0.253763
\(918\) −7.16485e9 −0.305673
\(919\) 2.54160e10 1.08020 0.540098 0.841602i \(-0.318387\pi\)
0.540098 + 0.841602i \(0.318387\pi\)
\(920\) 0 0
\(921\) 2.50149e10 1.05509
\(922\) 1.88634e10 0.792615
\(923\) 1.66502e10 0.696970
\(924\) 1.18850e10 0.495620
\(925\) 0 0
\(926\) 4.28256e10 1.77241
\(927\) 3.91624e9 0.161469
\(928\) 1.27694e11 5.24511
\(929\) −2.34566e10 −0.959865 −0.479932 0.877306i \(-0.659339\pi\)
−0.479932 + 0.877306i \(0.659339\pi\)
\(930\) 0 0
\(931\) 1.64141e10 0.666644
\(932\) 6.39345e10 2.58690
\(933\) 5.75036e9 0.231798
\(934\) 2.11190e10 0.848124
\(935\) 0 0
\(936\) 4.02452e10 1.60417
\(937\) −9.30070e9 −0.369341 −0.184670 0.982801i \(-0.559122\pi\)
−0.184670 + 0.982801i \(0.559122\pi\)
\(938\) −4.21440e10 −1.66735
\(939\) 9.82520e9 0.387268
\(940\) 0 0
\(941\) −1.13117e10 −0.442553 −0.221276 0.975211i \(-0.571022\pi\)
−0.221276 + 0.975211i \(0.571022\pi\)
\(942\) −4.66337e10 −1.81770
\(943\) −6.81551e8 −0.0264672
\(944\) −1.02759e11 −3.97573
\(945\) 0 0
\(946\) −1.24018e9 −0.0476283
\(947\) −4.46270e9 −0.170755 −0.0853774 0.996349i \(-0.527210\pi\)
−0.0853774 + 0.996349i \(0.527210\pi\)
\(948\) −4.06772e10 −1.55068
\(949\) 1.67611e10 0.636608
\(950\) 0 0
\(951\) −8.56032e9 −0.322744
\(952\) 3.48578e10 1.30939
\(953\) −1.90872e10 −0.714359 −0.357179 0.934036i \(-0.616261\pi\)
−0.357179 + 0.934036i \(0.616261\pi\)
\(954\) −8.04931e9 −0.300151
\(955\) 0 0
\(956\) −1.32905e11 −4.91969
\(957\) 1.29599e10 0.477980
\(958\) 6.67873e9 0.245423
\(959\) 3.36215e9 0.123098
\(960\) 0 0
\(961\) 1.31299e10 0.477234
\(962\) −2.91842e10 −1.05690
\(963\) 8.41249e9 0.303552
\(964\) −9.14997e10 −3.28965
\(965\) 0 0
\(966\) −1.46694e9 −0.0523592
\(967\) 1.34102e10 0.476916 0.238458 0.971153i \(-0.423358\pi\)
0.238458 + 0.971153i \(0.423358\pi\)
\(968\) −5.42733e10 −1.92319
\(969\) 1.13312e10 0.400075
\(970\) 0 0
\(971\) 4.64191e10 1.62716 0.813579 0.581455i \(-0.197516\pi\)
0.813579 + 0.581455i \(0.197516\pi\)
\(972\) −5.10821e9 −0.178417
\(973\) 1.93501e10 0.673423
\(974\) −9.59329e9 −0.332668
\(975\) 0 0
\(976\) −2.41620e10 −0.831875
\(977\) −3.92972e10 −1.34813 −0.674064 0.738673i \(-0.735453\pi\)
−0.674064 + 0.738673i \(0.735453\pi\)
\(978\) −2.78101e10 −0.950642
\(979\) 2.02788e10 0.690721
\(980\) 0 0
\(981\) −1.14584e9 −0.0387509
\(982\) 1.83659e10 0.618902
\(983\) 3.69370e10 1.24029 0.620146 0.784486i \(-0.287073\pi\)
0.620146 + 0.784486i \(0.287073\pi\)
\(984\) 1.56979e10 0.525242
\(985\) 0 0
\(986\) 5.93492e10 1.97172
\(987\) 9.53712e9 0.315724
\(988\) −9.93796e10 −3.27829
\(989\) 1.12590e8 0.00370095
\(990\) 0 0
\(991\) −1.02868e10 −0.335753 −0.167877 0.985808i \(-0.553691\pi\)
−0.167877 + 0.985808i \(0.553691\pi\)
\(992\) −1.57893e11 −5.13538
\(993\) −6.06302e9 −0.196502
\(994\) 1.39786e10 0.451451
\(995\) 0 0
\(996\) −1.78226e10 −0.571562
\(997\) 3.24439e10 1.03681 0.518407 0.855134i \(-0.326525\pi\)
0.518407 + 0.855134i \(0.326525\pi\)
\(998\) −9.90506e10 −3.15428
\(999\) 2.37239e9 0.0752848
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 75.8.a.c.1.1 1
3.2 odd 2 225.8.a.a.1.1 1
5.2 odd 4 75.8.b.a.49.2 2
5.3 odd 4 75.8.b.a.49.1 2
5.4 even 2 15.8.a.a.1.1 1
15.2 even 4 225.8.b.a.199.1 2
15.8 even 4 225.8.b.a.199.2 2
15.14 odd 2 45.8.a.g.1.1 1
20.19 odd 2 240.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.8.a.a.1.1 1 5.4 even 2
45.8.a.g.1.1 1 15.14 odd 2
75.8.a.c.1.1 1 1.1 even 1 trivial
75.8.b.a.49.1 2 5.3 odd 4
75.8.b.a.49.2 2 5.2 odd 4
225.8.a.a.1.1 1 3.2 odd 2
225.8.b.a.199.1 2 15.2 even 4
225.8.b.a.199.2 2 15.8 even 4
240.8.a.c.1.1 1 20.19 odd 2