Properties

Label 15.10.a.c.1.2
Level 15
Weight 10
Character 15.1
Self dual yes
Analytic conductor 7.726
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.72553754246\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{4729}) \)
Defining polynomial: \(x^{2} - x - 1182\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-33.8839\) of defining polynomial
Character \(\chi\) \(=\) 15.1

$q$-expansion

\(f(q)\) \(=\) \(q+43.8839 q^{2} +81.0000 q^{3} +1413.79 q^{4} -625.000 q^{5} +3554.59 q^{6} -7861.50 q^{7} +39574.2 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+43.8839 q^{2} +81.0000 q^{3} +1413.79 q^{4} -625.000 q^{5} +3554.59 q^{6} -7861.50 q^{7} +39574.2 q^{8} +6561.00 q^{9} -27427.4 q^{10} -49373.3 q^{11} +114517. q^{12} +24250.7 q^{13} -344993. q^{14} -50625.0 q^{15} +1.01281e6 q^{16} +268222. q^{17} +287922. q^{18} -168364. q^{19} -883621. q^{20} -636781. q^{21} -2.16669e6 q^{22} -2.12200e6 q^{23} +3.20551e6 q^{24} +390625. q^{25} +1.06422e6 q^{26} +531441. q^{27} -1.11145e7 q^{28} +389624. q^{29} -2.22162e6 q^{30} +90532.2 q^{31} +2.41838e7 q^{32} -3.99924e6 q^{33} +1.17706e7 q^{34} +4.91344e6 q^{35} +9.27590e6 q^{36} -3.31991e6 q^{37} -7.38848e6 q^{38} +1.96431e6 q^{39} -2.47339e7 q^{40} +2.32694e7 q^{41} -2.79444e7 q^{42} +1.91140e7 q^{43} -6.98036e7 q^{44} -4.10063e6 q^{45} -9.31215e7 q^{46} +6.28153e7 q^{47} +8.20372e7 q^{48} +2.14495e7 q^{49} +1.71421e7 q^{50} +2.17260e7 q^{51} +3.42855e7 q^{52} -180207. q^{53} +2.33217e7 q^{54} +3.08583e7 q^{55} -3.11112e8 q^{56} -1.36375e7 q^{57} +1.70982e7 q^{58} +3.84564e7 q^{59} -7.15733e7 q^{60} -553620. q^{61} +3.97290e6 q^{62} -5.15793e7 q^{63} +5.42724e8 q^{64} -1.51567e7 q^{65} -1.75502e8 q^{66} -2.39163e8 q^{67} +3.79211e8 q^{68} -1.71882e8 q^{69} +2.15621e8 q^{70} +1.28653e8 q^{71} +2.59646e8 q^{72} -2.39376e8 q^{73} -1.45690e8 q^{74} +3.16406e7 q^{75} -2.38033e8 q^{76} +3.88148e8 q^{77} +8.62015e7 q^{78} -5.28027e8 q^{79} -6.33003e8 q^{80} +4.30467e7 q^{81} +1.02115e9 q^{82} +2.12210e8 q^{83} -9.00277e8 q^{84} -1.67639e8 q^{85} +8.38797e8 q^{86} +3.15595e7 q^{87} -1.95391e9 q^{88} -2.07724e8 q^{89} -1.79951e8 q^{90} -1.90647e8 q^{91} -3.00007e9 q^{92} +7.33311e6 q^{93} +2.75658e9 q^{94} +1.05228e8 q^{95} +1.95889e9 q^{96} +1.70780e9 q^{97} +9.41288e8 q^{98} -3.23938e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + 49647q^{8} + 13122q^{9} + O(q^{10}) \) \( 2q + 19q^{2} + 162q^{3} + 1521q^{4} - 1250q^{5} + 1539q^{6} - 11872q^{7} + 49647q^{8} + 13122q^{9} - 11875q^{10} + 35488q^{11} + 123201q^{12} + 143676q^{13} - 245196q^{14} - 101250q^{15} + 707265q^{16} + 385156q^{17} + 124659q^{18} - 403296q^{19} - 950625q^{20} - 961632q^{21} - 4278368q^{22} + 223704q^{23} + 4021407q^{24} + 781250q^{25} - 1907546q^{26} + 1062882q^{27} - 11544484q^{28} - 74572q^{29} - 961875q^{30} - 5027128q^{31} + 26629583q^{32} + 2874528q^{33} + 8860882q^{34} + 7420000q^{35} + 9979281q^{36} + 5373628q^{37} - 1542476q^{38} + 11637756q^{39} - 31029375q^{40} + 14211332q^{41} - 19860876q^{42} + 27748920q^{43} - 60705952q^{44} - 8201250q^{45} - 151491648q^{46} + 95966440q^{47} + 57288465q^{48} - 2819950q^{49} + 7421875q^{50} + 31197636q^{51} + 47088706q^{52} - 64305596q^{53} + 10097379q^{54} - 22180000q^{55} - 351509340q^{56} - 32666976q^{57} + 28649198q^{58} + 187863136q^{59} - 77000625q^{60} + 154080060q^{61} + 131320056q^{62} - 77892192q^{63} + 638301089q^{64} - 89797500q^{65} - 346547808q^{66} + 33592376q^{67} + 391747238q^{68} + 18120024q^{69} + 153247500q^{70} - 228270976q^{71} + 325733967q^{72} - 33122316q^{73} - 362019226q^{74} + 63281250q^{75} - 263218724q^{76} + 47811456q^{77} - 154511226q^{78} - 932406760q^{79} - 442040625q^{80} + 86093442q^{81} + 1246549646q^{82} + 207040152q^{83} - 935103204q^{84} - 240722500q^{85} + 623926708q^{86} - 6040332q^{87} - 1099114848q^{88} + 224518164q^{89} - 77911875q^{90} - 669602528q^{91} - 2748592992q^{92} - 407197368q^{93} + 1931650816q^{94} + 252060000q^{95} + 2156996223q^{96} + 387134596q^{97} + 1545205739q^{98} + 232836768q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 43.8839 1.93941 0.969706 0.244277i \(-0.0785506\pi\)
0.969706 + 0.244277i \(0.0785506\pi\)
\(3\) 81.0000 0.577350
\(4\) 1413.79 2.76132
\(5\) −625.000 −0.447214
\(6\) 3554.59 1.11972
\(7\) −7861.50 −1.23755 −0.618777 0.785567i \(-0.712371\pi\)
−0.618777 + 0.785567i \(0.712371\pi\)
\(8\) 39574.2 3.41591
\(9\) 6561.00 0.333333
\(10\) −27427.4 −0.867331
\(11\) −49373.3 −1.01678 −0.508388 0.861128i \(-0.669758\pi\)
−0.508388 + 0.861128i \(0.669758\pi\)
\(12\) 114517. 1.59425
\(13\) 24250.7 0.235494 0.117747 0.993044i \(-0.462433\pi\)
0.117747 + 0.993044i \(0.462433\pi\)
\(14\) −344993. −2.40013
\(15\) −50625.0 −0.258199
\(16\) 1.01281e6 3.86355
\(17\) 268222. 0.778888 0.389444 0.921050i \(-0.372667\pi\)
0.389444 + 0.921050i \(0.372667\pi\)
\(18\) 287922. 0.646470
\(19\) −168364. −0.296387 −0.148193 0.988958i \(-0.547346\pi\)
−0.148193 + 0.988958i \(0.547346\pi\)
\(20\) −883621. −1.23490
\(21\) −636781. −0.714502
\(22\) −2.16669e6 −1.97195
\(23\) −2.12200e6 −1.58114 −0.790569 0.612373i \(-0.790215\pi\)
−0.790569 + 0.612373i \(0.790215\pi\)
\(24\) 3.20551e6 1.97218
\(25\) 390625. 0.200000
\(26\) 1.06422e6 0.456720
\(27\) 531441. 0.192450
\(28\) −1.11145e7 −3.41728
\(29\) 389624. 0.102295 0.0511475 0.998691i \(-0.483712\pi\)
0.0511475 + 0.998691i \(0.483712\pi\)
\(30\) −2.22162e6 −0.500754
\(31\) 90532.2 0.0176066 0.00880330 0.999961i \(-0.497198\pi\)
0.00880330 + 0.999961i \(0.497198\pi\)
\(32\) 2.41838e7 4.07709
\(33\) −3.99924e6 −0.587036
\(34\) 1.17706e7 1.51058
\(35\) 4.91344e6 0.553451
\(36\) 9.27590e6 0.920438
\(37\) −3.31991e6 −0.291218 −0.145609 0.989342i \(-0.546514\pi\)
−0.145609 + 0.989342i \(0.546514\pi\)
\(38\) −7.38848e6 −0.574816
\(39\) 1.96431e6 0.135963
\(40\) −2.47339e7 −1.52764
\(41\) 2.32694e7 1.28605 0.643024 0.765846i \(-0.277679\pi\)
0.643024 + 0.765846i \(0.277679\pi\)
\(42\) −2.79444e7 −1.38571
\(43\) 1.91140e7 0.852597 0.426298 0.904583i \(-0.359817\pi\)
0.426298 + 0.904583i \(0.359817\pi\)
\(44\) −6.98036e7 −2.80764
\(45\) −4.10063e6 −0.149071
\(46\) −9.31215e7 −3.06648
\(47\) 6.28153e7 1.87770 0.938848 0.344332i \(-0.111895\pi\)
0.938848 + 0.344332i \(0.111895\pi\)
\(48\) 8.20372e7 2.23062
\(49\) 2.14495e7 0.531539
\(50\) 1.71421e7 0.387882
\(51\) 2.17260e7 0.449691
\(52\) 3.42855e7 0.650273
\(53\) −180207. −0.00313711 −0.00156856 0.999999i \(-0.500499\pi\)
−0.00156856 + 0.999999i \(0.500499\pi\)
\(54\) 2.33217e7 0.373240
\(55\) 3.08583e7 0.454716
\(56\) −3.11112e8 −4.22738
\(57\) −1.36375e7 −0.171119
\(58\) 1.70982e7 0.198392
\(59\) 3.84564e7 0.413175 0.206588 0.978428i \(-0.433764\pi\)
0.206588 + 0.978428i \(0.433764\pi\)
\(60\) −7.15733e7 −0.712969
\(61\) −553620. −0.00511950 −0.00255975 0.999997i \(-0.500815\pi\)
−0.00255975 + 0.999997i \(0.500815\pi\)
\(62\) 3.97290e6 0.0341464
\(63\) −5.15793e7 −0.412518
\(64\) 5.42724e8 4.04361
\(65\) −1.51567e7 −0.105316
\(66\) −1.75502e8 −1.13850
\(67\) −2.39163e8 −1.44996 −0.724982 0.688768i \(-0.758152\pi\)
−0.724982 + 0.688768i \(0.758152\pi\)
\(68\) 3.79211e8 2.15076
\(69\) −1.71882e8 −0.912871
\(70\) 2.15621e8 1.07337
\(71\) 1.28653e8 0.600838 0.300419 0.953807i \(-0.402873\pi\)
0.300419 + 0.953807i \(0.402873\pi\)
\(72\) 2.59646e8 1.13864
\(73\) −2.39376e8 −0.986569 −0.493284 0.869868i \(-0.664204\pi\)
−0.493284 + 0.869868i \(0.664204\pi\)
\(74\) −1.45690e8 −0.564792
\(75\) 3.16406e7 0.115470
\(76\) −2.38033e8 −0.818418
\(77\) 3.88148e8 1.25831
\(78\) 8.62015e7 0.263687
\(79\) −5.28027e8 −1.52523 −0.762613 0.646855i \(-0.776084\pi\)
−0.762613 + 0.646855i \(0.776084\pi\)
\(80\) −6.33003e8 −1.72783
\(81\) 4.30467e7 0.111111
\(82\) 1.02115e9 2.49418
\(83\) 2.12210e8 0.490812 0.245406 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(84\) −9.00277e8 −1.97297
\(85\) −1.67639e8 −0.348329
\(86\) 8.38797e8 1.65354
\(87\) 3.15595e7 0.0590601
\(88\) −1.95391e9 −3.47322
\(89\) −2.07724e8 −0.350939 −0.175469 0.984485i \(-0.556144\pi\)
−0.175469 + 0.984485i \(0.556144\pi\)
\(90\) −1.79951e8 −0.289110
\(91\) −1.90647e8 −0.291436
\(92\) −3.00007e9 −4.36602
\(93\) 7.33311e6 0.0101652
\(94\) 2.75658e9 3.64162
\(95\) 1.05228e8 0.132548
\(96\) 1.95889e9 2.35391
\(97\) 1.70780e9 1.95868 0.979341 0.202214i \(-0.0648136\pi\)
0.979341 + 0.202214i \(0.0648136\pi\)
\(98\) 9.41288e8 1.03087
\(99\) −3.23938e8 −0.338925
\(100\) 5.52263e8 0.552263
\(101\) −1.81384e8 −0.173441 −0.0867206 0.996233i \(-0.527639\pi\)
−0.0867206 + 0.996233i \(0.527639\pi\)
\(102\) 9.53422e8 0.872136
\(103\) −1.70453e9 −1.49223 −0.746116 0.665816i \(-0.768083\pi\)
−0.746116 + 0.665816i \(0.768083\pi\)
\(104\) 9.59703e8 0.804427
\(105\) 3.97988e8 0.319535
\(106\) −7.90818e6 −0.00608415
\(107\) 1.73298e9 1.27811 0.639053 0.769163i \(-0.279326\pi\)
0.639053 + 0.769163i \(0.279326\pi\)
\(108\) 7.51348e8 0.531415
\(109\) −1.17238e9 −0.795515 −0.397757 0.917491i \(-0.630211\pi\)
−0.397757 + 0.917491i \(0.630211\pi\)
\(110\) 1.35418e9 0.881881
\(111\) −2.68913e8 −0.168135
\(112\) −7.96217e9 −4.78135
\(113\) −1.22180e9 −0.704935 −0.352467 0.935824i \(-0.614657\pi\)
−0.352467 + 0.935824i \(0.614657\pi\)
\(114\) −5.98467e8 −0.331870
\(115\) 1.32625e9 0.707107
\(116\) 5.50848e8 0.282469
\(117\) 1.59109e8 0.0784980
\(118\) 1.68761e9 0.801317
\(119\) −2.10863e9 −0.963916
\(120\) −2.00344e9 −0.881985
\(121\) 7.97750e7 0.0338324
\(122\) −2.42950e7 −0.00992881
\(123\) 1.88482e9 0.742501
\(124\) 1.27994e8 0.0486174
\(125\) −2.44141e8 −0.0894427
\(126\) −2.26350e9 −0.800042
\(127\) −9.50324e8 −0.324157 −0.162078 0.986778i \(-0.551820\pi\)
−0.162078 + 0.986778i \(0.551820\pi\)
\(128\) 1.14347e10 3.76513
\(129\) 1.54823e9 0.492247
\(130\) −6.65135e8 −0.204251
\(131\) −4.03050e9 −1.19574 −0.597872 0.801592i \(-0.703987\pi\)
−0.597872 + 0.801592i \(0.703987\pi\)
\(132\) −5.65410e9 −1.62099
\(133\) 1.32360e9 0.366795
\(134\) −1.04954e10 −2.81207
\(135\) −3.32151e8 −0.0860663
\(136\) 1.06147e10 2.66061
\(137\) 5.51871e9 1.33843 0.669214 0.743070i \(-0.266631\pi\)
0.669214 + 0.743070i \(0.266631\pi\)
\(138\) −7.54284e9 −1.77043
\(139\) −3.97776e9 −0.903798 −0.451899 0.892069i \(-0.649253\pi\)
−0.451899 + 0.892069i \(0.649253\pi\)
\(140\) 6.94658e9 1.52825
\(141\) 5.08804e9 1.08409
\(142\) 5.64579e9 1.16527
\(143\) −1.19734e9 −0.239445
\(144\) 6.64502e9 1.28785
\(145\) −2.43515e8 −0.0457478
\(146\) −1.05047e10 −1.91336
\(147\) 1.73741e9 0.306884
\(148\) −4.69367e9 −0.804145
\(149\) −2.26538e9 −0.376533 −0.188266 0.982118i \(-0.560287\pi\)
−0.188266 + 0.982118i \(0.560287\pi\)
\(150\) 1.38851e9 0.223944
\(151\) −8.49158e9 −1.32921 −0.664603 0.747197i \(-0.731399\pi\)
−0.664603 + 0.747197i \(0.731399\pi\)
\(152\) −6.66288e9 −1.01243
\(153\) 1.75981e9 0.259629
\(154\) 1.70334e10 2.44039
\(155\) −5.65826e7 −0.00787391
\(156\) 2.77713e9 0.375435
\(157\) 7.28379e9 0.956774 0.478387 0.878149i \(-0.341222\pi\)
0.478387 + 0.878149i \(0.341222\pi\)
\(158\) −2.31719e10 −2.95804
\(159\) −1.45968e7 −0.00181121
\(160\) −1.51149e10 −1.82333
\(161\) 1.66821e10 1.95674
\(162\) 1.88906e9 0.215490
\(163\) 1.38419e10 1.53586 0.767929 0.640535i \(-0.221288\pi\)
0.767929 + 0.640535i \(0.221288\pi\)
\(164\) 3.28981e10 3.55119
\(165\) 2.49952e9 0.262530
\(166\) 9.31262e9 0.951887
\(167\) −6.92038e9 −0.688503 −0.344252 0.938877i \(-0.611867\pi\)
−0.344252 + 0.938877i \(0.611867\pi\)
\(168\) −2.52001e10 −2.44068
\(169\) −1.00164e10 −0.944543
\(170\) −7.35665e9 −0.675554
\(171\) −1.10464e9 −0.0987957
\(172\) 2.70233e10 2.35429
\(173\) 3.26987e9 0.277539 0.138769 0.990325i \(-0.455685\pi\)
0.138769 + 0.990325i \(0.455685\pi\)
\(174\) 1.38495e9 0.114542
\(175\) −3.07090e9 −0.247511
\(176\) −5.00056e10 −3.92836
\(177\) 3.11497e9 0.238547
\(178\) −9.11572e9 −0.680614
\(179\) −1.12431e10 −0.818556 −0.409278 0.912410i \(-0.634219\pi\)
−0.409278 + 0.912410i \(0.634219\pi\)
\(180\) −5.79744e9 −0.411633
\(181\) 3.75039e9 0.259731 0.129865 0.991532i \(-0.458546\pi\)
0.129865 + 0.991532i \(0.458546\pi\)
\(182\) −8.36633e9 −0.565215
\(183\) −4.48432e7 −0.00295574
\(184\) −8.39764e10 −5.40103
\(185\) 2.07494e9 0.130237
\(186\) 3.21805e8 0.0197145
\(187\) −1.32430e10 −0.791954
\(188\) 8.88079e10 5.18491
\(189\) −4.17792e9 −0.238167
\(190\) 4.61780e9 0.257066
\(191\) 1.72426e10 0.937460 0.468730 0.883342i \(-0.344712\pi\)
0.468730 + 0.883342i \(0.344712\pi\)
\(192\) 4.39606e10 2.33458
\(193\) −6.01060e9 −0.311824 −0.155912 0.987771i \(-0.549832\pi\)
−0.155912 + 0.987771i \(0.549832\pi\)
\(194\) 7.49448e10 3.79869
\(195\) −1.22769e9 −0.0608043
\(196\) 3.03252e10 1.46775
\(197\) −1.19077e10 −0.563289 −0.281645 0.959519i \(-0.590880\pi\)
−0.281645 + 0.959519i \(0.590880\pi\)
\(198\) −1.42157e10 −0.657315
\(199\) −6.10946e9 −0.276162 −0.138081 0.990421i \(-0.544093\pi\)
−0.138081 + 0.990421i \(0.544093\pi\)
\(200\) 1.54587e10 0.683183
\(201\) −1.93722e10 −0.837137
\(202\) −7.95982e9 −0.336374
\(203\) −3.06303e9 −0.126596
\(204\) 3.07161e10 1.24174
\(205\) −1.45434e10 −0.575139
\(206\) −7.48012e10 −2.89405
\(207\) −1.39224e10 −0.527046
\(208\) 2.45613e10 0.909842
\(209\) 8.31271e9 0.301359
\(210\) 1.74653e10 0.619710
\(211\) −3.67892e10 −1.27776 −0.638881 0.769306i \(-0.720602\pi\)
−0.638881 + 0.769306i \(0.720602\pi\)
\(212\) −2.54775e8 −0.00866256
\(213\) 1.04209e10 0.346894
\(214\) 7.60499e10 2.47877
\(215\) −1.19463e10 −0.381293
\(216\) 2.10313e10 0.657393
\(217\) −7.11719e8 −0.0217891
\(218\) −5.14485e10 −1.54283
\(219\) −1.93894e10 −0.569596
\(220\) 4.36273e10 1.25561
\(221\) 6.50459e9 0.183423
\(222\) −1.18009e10 −0.326083
\(223\) −5.01307e9 −0.135747 −0.0678737 0.997694i \(-0.521621\pi\)
−0.0678737 + 0.997694i \(0.521621\pi\)
\(224\) −1.90121e11 −5.04562
\(225\) 2.56289e9 0.0666667
\(226\) −5.36175e10 −1.36716
\(227\) −2.73858e10 −0.684555 −0.342278 0.939599i \(-0.611198\pi\)
−0.342278 + 0.939599i \(0.611198\pi\)
\(228\) −1.92806e10 −0.472514
\(229\) 5.36868e10 1.29005 0.645027 0.764159i \(-0.276846\pi\)
0.645027 + 0.764159i \(0.276846\pi\)
\(230\) 5.82009e10 1.37137
\(231\) 3.14400e10 0.726488
\(232\) 1.54191e10 0.349431
\(233\) 7.25874e10 1.61347 0.806733 0.590916i \(-0.201234\pi\)
0.806733 + 0.590916i \(0.201234\pi\)
\(234\) 6.98232e9 0.152240
\(235\) −3.92596e10 −0.839731
\(236\) 5.43694e10 1.14091
\(237\) −4.27702e10 −0.880590
\(238\) −9.25348e10 −1.86943
\(239\) 1.73313e10 0.343589 0.171795 0.985133i \(-0.445043\pi\)
0.171795 + 0.985133i \(0.445043\pi\)
\(240\) −5.12733e10 −0.997563
\(241\) 4.06448e10 0.776119 0.388059 0.921634i \(-0.373146\pi\)
0.388059 + 0.921634i \(0.373146\pi\)
\(242\) 3.50084e9 0.0656149
\(243\) 3.48678e9 0.0641500
\(244\) −7.82704e8 −0.0141365
\(245\) −1.34059e10 −0.237711
\(246\) 8.27132e10 1.44001
\(247\) −4.08296e9 −0.0697973
\(248\) 3.58274e9 0.0601426
\(249\) 1.71890e10 0.283371
\(250\) −1.07138e10 −0.173466
\(251\) 1.00920e11 1.60490 0.802448 0.596723i \(-0.203531\pi\)
0.802448 + 0.596723i \(0.203531\pi\)
\(252\) −7.29224e10 −1.13909
\(253\) 1.04770e11 1.60766
\(254\) −4.17039e10 −0.628673
\(255\) −1.35788e10 −0.201108
\(256\) 2.23924e11 3.25852
\(257\) 5.03660e10 0.720176 0.360088 0.932918i \(-0.382747\pi\)
0.360088 + 0.932918i \(0.382747\pi\)
\(258\) 6.79425e10 0.954669
\(259\) 2.60995e10 0.360398
\(260\) −2.14285e10 −0.290811
\(261\) 2.55632e9 0.0340984
\(262\) −1.76874e11 −2.31904
\(263\) −5.58677e10 −0.720046 −0.360023 0.932943i \(-0.617231\pi\)
−0.360023 + 0.932943i \(0.617231\pi\)
\(264\) −1.58267e11 −2.00526
\(265\) 1.12629e8 0.00140296
\(266\) 5.80845e10 0.711366
\(267\) −1.68256e10 −0.202614
\(268\) −3.38127e11 −4.00381
\(269\) −1.05175e11 −1.22469 −0.612345 0.790590i \(-0.709774\pi\)
−0.612345 + 0.790590i \(0.709774\pi\)
\(270\) −1.45761e10 −0.166918
\(271\) −5.44180e10 −0.612887 −0.306444 0.951889i \(-0.599139\pi\)
−0.306444 + 0.951889i \(0.599139\pi\)
\(272\) 2.71657e11 3.00927
\(273\) −1.54424e10 −0.168261
\(274\) 2.42182e11 2.59576
\(275\) −1.92864e10 −0.203355
\(276\) −2.43005e11 −2.52072
\(277\) 1.64528e11 1.67912 0.839560 0.543268i \(-0.182813\pi\)
0.839560 + 0.543268i \(0.182813\pi\)
\(278\) −1.74559e11 −1.75284
\(279\) 5.93982e8 0.00586887
\(280\) 1.94445e11 1.89054
\(281\) −6.74255e10 −0.645128 −0.322564 0.946548i \(-0.604545\pi\)
−0.322564 + 0.946548i \(0.604545\pi\)
\(282\) 2.23283e11 2.10249
\(283\) 1.41917e11 1.31521 0.657603 0.753364i \(-0.271570\pi\)
0.657603 + 0.753364i \(0.271570\pi\)
\(284\) 1.81889e11 1.65910
\(285\) 8.52345e9 0.0765268
\(286\) −5.25438e10 −0.464381
\(287\) −1.82932e11 −1.59155
\(288\) 1.58670e11 1.35903
\(289\) −4.66446e10 −0.393333
\(290\) −1.06864e10 −0.0887237
\(291\) 1.38332e11 1.13085
\(292\) −3.38428e11 −2.72423
\(293\) 2.79097e10 0.221233 0.110617 0.993863i \(-0.464717\pi\)
0.110617 + 0.993863i \(0.464717\pi\)
\(294\) 7.62443e10 0.595175
\(295\) −2.40352e10 −0.184778
\(296\) −1.31383e11 −0.994776
\(297\) −2.62390e10 −0.195679
\(298\) −9.94136e10 −0.730252
\(299\) −5.14600e10 −0.372349
\(300\) 4.47333e10 0.318849
\(301\) −1.50265e11 −1.05513
\(302\) −3.72643e11 −2.57788
\(303\) −1.46921e10 −0.100136
\(304\) −1.70520e11 −1.14510
\(305\) 3.46012e8 0.00228951
\(306\) 7.72272e10 0.503528
\(307\) 1.59767e11 1.02651 0.513257 0.858235i \(-0.328439\pi\)
0.513257 + 0.858235i \(0.328439\pi\)
\(308\) 5.48761e11 3.47460
\(309\) −1.38067e11 −0.861540
\(310\) −2.48306e9 −0.0152707
\(311\) −7.84292e10 −0.475396 −0.237698 0.971339i \(-0.576393\pi\)
−0.237698 + 0.971339i \(0.576393\pi\)
\(312\) 7.77359e10 0.464436
\(313\) −1.26229e11 −0.743378 −0.371689 0.928357i \(-0.621221\pi\)
−0.371689 + 0.928357i \(0.621221\pi\)
\(314\) 3.19641e11 1.85558
\(315\) 3.22370e10 0.184484
\(316\) −7.46521e11 −4.21163
\(317\) −9.59499e10 −0.533676 −0.266838 0.963741i \(-0.585979\pi\)
−0.266838 + 0.963741i \(0.585979\pi\)
\(318\) −6.40563e8 −0.00351269
\(319\) −1.92370e10 −0.104011
\(320\) −3.39202e11 −1.80836
\(321\) 1.40372e11 0.737915
\(322\) 7.32074e11 3.79493
\(323\) −4.51591e10 −0.230852
\(324\) 6.08592e10 0.306813
\(325\) 9.47294e9 0.0470988
\(326\) 6.07435e11 2.97866
\(327\) −9.49626e10 −0.459291
\(328\) 9.20867e11 4.39303
\(329\) −4.93822e11 −2.32375
\(330\) 1.09689e11 0.509154
\(331\) 7.73728e10 0.354293 0.177146 0.984185i \(-0.443313\pi\)
0.177146 + 0.984185i \(0.443313\pi\)
\(332\) 3.00022e11 1.35529
\(333\) −2.17819e10 −0.0970727
\(334\) −3.03693e11 −1.33529
\(335\) 1.49477e11 0.648443
\(336\) −6.44936e11 −2.76051
\(337\) 1.73809e11 0.734071 0.367035 0.930207i \(-0.380373\pi\)
0.367035 + 0.930207i \(0.380373\pi\)
\(338\) −4.39558e11 −1.83186
\(339\) −9.89662e10 −0.406994
\(340\) −2.37007e11 −0.961847
\(341\) −4.46987e9 −0.0179020
\(342\) −4.84758e10 −0.191605
\(343\) 1.48614e11 0.579746
\(344\) 7.56421e11 2.91240
\(345\) 1.07426e11 0.408248
\(346\) 1.43495e11 0.538262
\(347\) 4.11335e11 1.52305 0.761523 0.648138i \(-0.224452\pi\)
0.761523 + 0.648138i \(0.224452\pi\)
\(348\) 4.46187e10 0.163084
\(349\) 2.97923e11 1.07495 0.537477 0.843278i \(-0.319377\pi\)
0.537477 + 0.843278i \(0.319377\pi\)
\(350\) −1.34763e11 −0.480025
\(351\) 1.28878e10 0.0453208
\(352\) −1.19404e12 −4.14549
\(353\) 5.69499e9 0.0195212 0.00976061 0.999952i \(-0.496893\pi\)
0.00976061 + 0.999952i \(0.496893\pi\)
\(354\) 1.36697e11 0.462640
\(355\) −8.04082e10 −0.268703
\(356\) −2.93678e11 −0.969052
\(357\) −1.70799e11 −0.556517
\(358\) −4.93392e11 −1.58752
\(359\) 6.19565e11 1.96862 0.984310 0.176446i \(-0.0564602\pi\)
0.984310 + 0.176446i \(0.0564602\pi\)
\(360\) −1.62279e11 −0.509214
\(361\) −2.94341e11 −0.912155
\(362\) 1.64582e11 0.503724
\(363\) 6.46178e9 0.0195331
\(364\) −2.69536e11 −0.804748
\(365\) 1.49610e11 0.441207
\(366\) −1.96789e9 −0.00573240
\(367\) −4.06833e9 −0.0117063 −0.00585314 0.999983i \(-0.501863\pi\)
−0.00585314 + 0.999983i \(0.501863\pi\)
\(368\) −2.14917e12 −6.10880
\(369\) 1.52670e11 0.428683
\(370\) 9.10565e10 0.252583
\(371\) 1.41670e9 0.00388235
\(372\) 1.03675e10 0.0280693
\(373\) −6.75397e11 −1.80663 −0.903315 0.428978i \(-0.858874\pi\)
−0.903315 + 0.428978i \(0.858874\pi\)
\(374\) −5.81155e11 −1.53592
\(375\) −1.97754e10 −0.0516398
\(376\) 2.48586e12 6.41405
\(377\) 9.44867e9 0.0240899
\(378\) −1.83343e11 −0.461904
\(379\) −3.67146e11 −0.914033 −0.457017 0.889458i \(-0.651082\pi\)
−0.457017 + 0.889458i \(0.651082\pi\)
\(380\) 1.48770e11 0.366008
\(381\) −7.69763e10 −0.187152
\(382\) 7.56672e11 1.81812
\(383\) 2.22057e11 0.527314 0.263657 0.964616i \(-0.415071\pi\)
0.263657 + 0.964616i \(0.415071\pi\)
\(384\) 9.26210e11 2.17380
\(385\) −2.42593e11 −0.562735
\(386\) −2.63768e11 −0.604756
\(387\) 1.25407e11 0.284199
\(388\) 2.41448e12 5.40854
\(389\) −7.23043e11 −1.60100 −0.800499 0.599334i \(-0.795432\pi\)
−0.800499 + 0.599334i \(0.795432\pi\)
\(390\) −5.38759e10 −0.117925
\(391\) −5.69168e11 −1.23153
\(392\) 8.48847e11 1.81569
\(393\) −3.26470e11 −0.690363
\(394\) −5.22558e11 −1.09245
\(395\) 3.30017e11 0.682102
\(396\) −4.57982e11 −0.935879
\(397\) 6.06161e11 1.22470 0.612352 0.790585i \(-0.290224\pi\)
0.612352 + 0.790585i \(0.290224\pi\)
\(398\) −2.68107e11 −0.535592
\(399\) 1.07211e11 0.211769
\(400\) 3.95627e11 0.772709
\(401\) 6.72510e11 1.29882 0.649411 0.760438i \(-0.275016\pi\)
0.649411 + 0.760438i \(0.275016\pi\)
\(402\) −8.50126e11 −1.62355
\(403\) 2.19547e9 0.00414625
\(404\) −2.56439e11 −0.478926
\(405\) −2.69042e10 −0.0496904
\(406\) −1.34417e11 −0.245521
\(407\) 1.63915e11 0.296103
\(408\) 8.59789e11 1.53611
\(409\) −3.02482e10 −0.0534497 −0.0267248 0.999643i \(-0.508508\pi\)
−0.0267248 + 0.999643i \(0.508508\pi\)
\(410\) −6.38219e11 −1.11543
\(411\) 4.47015e11 0.772742
\(412\) −2.40985e12 −4.12052
\(413\) −3.02325e11 −0.511326
\(414\) −6.10970e11 −1.02216
\(415\) −1.32632e11 −0.219498
\(416\) 5.86476e11 0.960130
\(417\) −3.22198e11 −0.521808
\(418\) 3.64794e11 0.584459
\(419\) −1.61938e11 −0.256676 −0.128338 0.991731i \(-0.540964\pi\)
−0.128338 + 0.991731i \(0.540964\pi\)
\(420\) 5.62673e11 0.882337
\(421\) −8.06547e11 −1.25130 −0.625648 0.780105i \(-0.715165\pi\)
−0.625648 + 0.780105i \(0.715165\pi\)
\(422\) −1.61445e12 −2.47810
\(423\) 4.12131e11 0.625899
\(424\) −7.13154e9 −0.0107161
\(425\) 1.04774e11 0.155778
\(426\) 4.57309e11 0.672770
\(427\) 4.35228e9 0.00633565
\(428\) 2.45008e12 3.52925
\(429\) −9.69844e10 −0.138243
\(430\) −5.24248e11 −0.739484
\(431\) 8.54653e11 1.19300 0.596502 0.802611i \(-0.296557\pi\)
0.596502 + 0.802611i \(0.296557\pi\)
\(432\) 5.38246e11 0.743540
\(433\) 8.21884e11 1.12361 0.561804 0.827270i \(-0.310107\pi\)
0.561804 + 0.827270i \(0.310107\pi\)
\(434\) −3.12330e10 −0.0422580
\(435\) −1.97247e10 −0.0264125
\(436\) −1.65750e12 −2.19667
\(437\) 3.57269e11 0.468629
\(438\) −8.50883e11 −1.10468
\(439\) 9.97807e11 1.28220 0.641100 0.767457i \(-0.278478\pi\)
0.641100 + 0.767457i \(0.278478\pi\)
\(440\) 1.22119e12 1.55327
\(441\) 1.40730e11 0.177180
\(442\) 2.85447e11 0.355733
\(443\) −2.22586e11 −0.274588 −0.137294 0.990530i \(-0.543840\pi\)
−0.137294 + 0.990530i \(0.543840\pi\)
\(444\) −3.80187e11 −0.464273
\(445\) 1.29827e11 0.156945
\(446\) −2.19993e11 −0.263270
\(447\) −1.83496e11 −0.217391
\(448\) −4.26662e12 −5.00418
\(449\) 1.28886e12 1.49657 0.748287 0.663375i \(-0.230877\pi\)
0.748287 + 0.663375i \(0.230877\pi\)
\(450\) 1.12470e11 0.129294
\(451\) −1.14889e12 −1.30762
\(452\) −1.72738e12 −1.94655
\(453\) −6.87818e11 −0.767417
\(454\) −1.20179e12 −1.32763
\(455\) 1.19154e11 0.130334
\(456\) −5.39694e11 −0.584528
\(457\) 8.77644e11 0.941230 0.470615 0.882339i \(-0.344032\pi\)
0.470615 + 0.882339i \(0.344032\pi\)
\(458\) 2.35599e12 2.50195
\(459\) 1.42544e11 0.149897
\(460\) 1.87504e12 1.95254
\(461\) −3.31142e11 −0.341476 −0.170738 0.985316i \(-0.554615\pi\)
−0.170738 + 0.985316i \(0.554615\pi\)
\(462\) 1.37971e12 1.40896
\(463\) −1.05840e12 −1.07037 −0.535186 0.844735i \(-0.679758\pi\)
−0.535186 + 0.844735i \(0.679758\pi\)
\(464\) 3.94613e11 0.395222
\(465\) −4.58319e9 −0.00454600
\(466\) 3.18542e12 3.12917
\(467\) −1.49420e12 −1.45373 −0.726864 0.686782i \(-0.759023\pi\)
−0.726864 + 0.686782i \(0.759023\pi\)
\(468\) 2.24947e11 0.216758
\(469\) 1.88018e12 1.79441
\(470\) −1.72286e12 −1.62858
\(471\) 5.89987e11 0.552394
\(472\) 1.52188e12 1.41137
\(473\) −9.43722e11 −0.866900
\(474\) −1.87692e12 −1.70783
\(475\) −6.57674e10 −0.0592774
\(476\) −2.98117e12 −2.66168
\(477\) −1.18234e9 −0.00104570
\(478\) 7.60563e11 0.666361
\(479\) 1.04497e12 0.906970 0.453485 0.891264i \(-0.350181\pi\)
0.453485 + 0.891264i \(0.350181\pi\)
\(480\) −1.22431e12 −1.05270
\(481\) −8.05102e10 −0.0685801
\(482\) 1.78365e12 1.50521
\(483\) 1.35125e12 1.12973
\(484\) 1.12785e11 0.0934219
\(485\) −1.06737e12 −0.875950
\(486\) 1.53014e11 0.124413
\(487\) −2.30343e12 −1.85564 −0.927820 0.373027i \(-0.878320\pi\)
−0.927820 + 0.373027i \(0.878320\pi\)
\(488\) −2.19091e10 −0.0174878
\(489\) 1.12119e12 0.886728
\(490\) −5.88305e11 −0.461020
\(491\) −5.37950e11 −0.417711 −0.208855 0.977947i \(-0.566974\pi\)
−0.208855 + 0.977947i \(0.566974\pi\)
\(492\) 2.66475e12 2.05028
\(493\) 1.04506e11 0.0796764
\(494\) −1.79176e11 −0.135366
\(495\) 2.02461e11 0.151572
\(496\) 9.16915e10 0.0680239
\(497\) −1.01141e12 −0.743570
\(498\) 7.54322e11 0.549572
\(499\) −1.01572e12 −0.733364 −0.366682 0.930346i \(-0.619506\pi\)
−0.366682 + 0.930346i \(0.619506\pi\)
\(500\) −3.45164e11 −0.246980
\(501\) −5.60551e11 −0.397508
\(502\) 4.42877e12 3.11255
\(503\) −1.07349e12 −0.747723 −0.373861 0.927485i \(-0.621966\pi\)
−0.373861 + 0.927485i \(0.621966\pi\)
\(504\) −2.04121e12 −1.40913
\(505\) 1.13365e11 0.0775653
\(506\) 4.59772e12 3.11792
\(507\) −8.11329e11 −0.545332
\(508\) −1.34356e12 −0.895099
\(509\) −1.48831e12 −0.982799 −0.491400 0.870934i \(-0.663515\pi\)
−0.491400 + 0.870934i \(0.663515\pi\)
\(510\) −5.95889e11 −0.390031
\(511\) 1.88185e12 1.22093
\(512\) 3.97208e12 2.55449
\(513\) −8.94758e10 −0.0570397
\(514\) 2.21026e12 1.39672
\(515\) 1.06533e12 0.667346
\(516\) 2.18888e12 1.35925
\(517\) −3.10140e12 −1.90919
\(518\) 1.14534e12 0.698960
\(519\) 2.64860e11 0.160237
\(520\) −5.99814e11 −0.359751
\(521\) −9.98544e11 −0.593742 −0.296871 0.954918i \(-0.595943\pi\)
−0.296871 + 0.954918i \(0.595943\pi\)
\(522\) 1.12181e11 0.0661307
\(523\) −1.59973e12 −0.934954 −0.467477 0.884005i \(-0.654837\pi\)
−0.467477 + 0.884005i \(0.654837\pi\)
\(524\) −5.69829e12 −3.30182
\(525\) −2.48743e11 −0.142900
\(526\) −2.45169e12 −1.39646
\(527\) 2.42828e10 0.0137136
\(528\) −4.05045e12 −2.26804
\(529\) 2.70173e12 1.50000
\(530\) 4.94261e9 0.00272092
\(531\) 2.52312e11 0.137725
\(532\) 1.87129e12 1.01284
\(533\) 5.64300e11 0.302857
\(534\) −7.38373e11 −0.392953
\(535\) −1.08311e12 −0.571586
\(536\) −9.46467e12 −4.95295
\(537\) −9.10693e11 −0.472593
\(538\) −4.61548e12 −2.37518
\(539\) −1.05903e12 −0.540456
\(540\) −4.69592e11 −0.237656
\(541\) 5.11810e11 0.256875 0.128437 0.991718i \(-0.459004\pi\)
0.128437 + 0.991718i \(0.459004\pi\)
\(542\) −2.38807e12 −1.18864
\(543\) 3.03782e11 0.149956
\(544\) 6.48665e12 3.17560
\(545\) 7.32736e11 0.355765
\(546\) −6.77673e11 −0.326327
\(547\) 1.40910e12 0.672977 0.336489 0.941688i \(-0.390761\pi\)
0.336489 + 0.941688i \(0.390761\pi\)
\(548\) 7.80231e12 3.69582
\(549\) −3.63230e9 −0.00170650
\(550\) −8.46364e11 −0.394389
\(551\) −6.55988e10 −0.0303189
\(552\) −6.80208e12 −3.11829
\(553\) 4.15108e12 1.88755
\(554\) 7.22013e12 3.25650
\(555\) 1.68070e11 0.0751922
\(556\) −5.62372e12 −2.49567
\(557\) 2.36835e12 1.04255 0.521276 0.853388i \(-0.325456\pi\)
0.521276 + 0.853388i \(0.325456\pi\)
\(558\) 2.60662e10 0.0113821
\(559\) 4.63529e11 0.200781
\(560\) 4.97635e12 2.13828
\(561\) −1.07269e12 −0.457235
\(562\) −2.95889e12 −1.25117
\(563\) −2.36245e12 −0.991005 −0.495502 0.868607i \(-0.665016\pi\)
−0.495502 + 0.868607i \(0.665016\pi\)
\(564\) 7.19344e12 2.99351
\(565\) 7.63628e11 0.315256
\(566\) 6.22784e12 2.55073
\(567\) −3.38412e11 −0.137506
\(568\) 5.09134e12 2.05241
\(569\) 2.25679e11 0.0902582 0.0451291 0.998981i \(-0.485630\pi\)
0.0451291 + 0.998981i \(0.485630\pi\)
\(570\) 3.74042e11 0.148417
\(571\) −9.15389e11 −0.360366 −0.180183 0.983633i \(-0.557669\pi\)
−0.180183 + 0.983633i \(0.557669\pi\)
\(572\) −1.69279e12 −0.661182
\(573\) 1.39665e12 0.541243
\(574\) −8.02777e12 −3.08668
\(575\) −8.28906e11 −0.316228
\(576\) 3.56081e12 1.34787
\(577\) 2.22404e12 0.835318 0.417659 0.908604i \(-0.362851\pi\)
0.417659 + 0.908604i \(0.362851\pi\)
\(578\) −2.04694e12 −0.762835
\(579\) −4.86859e11 −0.180032
\(580\) −3.44280e11 −0.126324
\(581\) −1.66829e12 −0.607407
\(582\) 6.07053e12 2.19318
\(583\) 8.89741e9 0.00318974
\(584\) −9.47310e12 −3.37003
\(585\) −9.94432e10 −0.0351054
\(586\) 1.22478e12 0.429062
\(587\) 5.60186e12 1.94742 0.973712 0.227783i \(-0.0731477\pi\)
0.973712 + 0.227783i \(0.0731477\pi\)
\(588\) 2.45634e12 0.847404
\(589\) −1.52424e10 −0.00521837
\(590\) −1.05476e12 −0.358360
\(591\) −9.64527e11 −0.325215
\(592\) −3.36242e12 −1.12513
\(593\) 2.78326e12 0.924289 0.462145 0.886805i \(-0.347080\pi\)
0.462145 + 0.886805i \(0.347080\pi\)
\(594\) −1.15147e12 −0.379501
\(595\) 1.31789e12 0.431076
\(596\) −3.20278e12 −1.03973
\(597\) −4.94867e11 −0.159442
\(598\) −2.25826e12 −0.722137
\(599\) −1.36720e12 −0.433922 −0.216961 0.976180i \(-0.569614\pi\)
−0.216961 + 0.976180i \(0.569614\pi\)
\(600\) 1.25215e12 0.394436
\(601\) 4.74171e12 1.48252 0.741259 0.671219i \(-0.234229\pi\)
0.741259 + 0.671219i \(0.234229\pi\)
\(602\) −6.59420e12 −2.04634
\(603\) −1.56915e12 −0.483321
\(604\) −1.20053e13 −3.67036
\(605\) −4.98594e10 −0.0151303
\(606\) −6.44746e11 −0.194206
\(607\) −5.74836e12 −1.71868 −0.859339 0.511406i \(-0.829125\pi\)
−0.859339 + 0.511406i \(0.829125\pi\)
\(608\) −4.07170e12 −1.20840
\(609\) −2.48105e11 −0.0730900
\(610\) 1.51844e10 0.00444030
\(611\) 1.52332e12 0.442186
\(612\) 2.48800e12 0.716918
\(613\) 2.99054e12 0.855416 0.427708 0.903917i \(-0.359321\pi\)
0.427708 + 0.903917i \(0.359321\pi\)
\(614\) 7.01120e12 1.99083
\(615\) −1.17801e12 −0.332056
\(616\) 1.53606e13 4.29829
\(617\) 6.27174e12 1.74223 0.871114 0.491081i \(-0.163398\pi\)
0.871114 + 0.491081i \(0.163398\pi\)
\(618\) −6.05890e12 −1.67088
\(619\) −2.67240e12 −0.731632 −0.365816 0.930687i \(-0.619210\pi\)
−0.365816 + 0.930687i \(0.619210\pi\)
\(620\) −7.99961e10 −0.0217423
\(621\) −1.12772e12 −0.304290
\(622\) −3.44177e12 −0.921989
\(623\) 1.63302e12 0.434305
\(624\) 1.98946e12 0.525297
\(625\) 1.52588e11 0.0400000
\(626\) −5.53942e12 −1.44172
\(627\) 6.73329e11 0.173990
\(628\) 1.02978e13 2.64195
\(629\) −8.90474e11 −0.226826
\(630\) 1.41469e12 0.357790
\(631\) 2.73608e12 0.687064 0.343532 0.939141i \(-0.388377\pi\)
0.343532 + 0.939141i \(0.388377\pi\)
\(632\) −2.08962e13 −5.21004
\(633\) −2.97993e12 −0.737716
\(634\) −4.21065e12 −1.03502
\(635\) 5.93953e11 0.144967
\(636\) −2.06368e10 −0.00500133
\(637\) 5.20167e11 0.125174
\(638\) −8.44195e11 −0.201720
\(639\) 8.44093e11 0.200279
\(640\) −7.14668e12 −1.68382
\(641\) −2.58542e12 −0.604882 −0.302441 0.953168i \(-0.597802\pi\)
−0.302441 + 0.953168i \(0.597802\pi\)
\(642\) 6.16004e12 1.43112
\(643\) −4.53865e12 −1.04707 −0.523537 0.852003i \(-0.675388\pi\)
−0.523537 + 0.852003i \(0.675388\pi\)
\(644\) 2.35850e13 5.40319
\(645\) −9.67647e11 −0.220140
\(646\) −1.98176e12 −0.447717
\(647\) 2.68597e12 0.602604 0.301302 0.953529i \(-0.402579\pi\)
0.301302 + 0.953529i \(0.402579\pi\)
\(648\) 1.70354e12 0.379546
\(649\) −1.89872e12 −0.420106
\(650\) 4.15709e11 0.0913439
\(651\) −5.76492e10 −0.0125799
\(652\) 1.95696e13 4.24099
\(653\) −6.39120e12 −1.37554 −0.687770 0.725929i \(-0.741410\pi\)
−0.687770 + 0.725929i \(0.741410\pi\)
\(654\) −4.16732e12 −0.890753
\(655\) 2.51906e12 0.534753
\(656\) 2.35674e13 4.96871
\(657\) −1.57054e12 −0.328856
\(658\) −2.16708e13 −4.50670
\(659\) −4.23105e12 −0.873904 −0.436952 0.899485i \(-0.643942\pi\)
−0.436952 + 0.899485i \(0.643942\pi\)
\(660\) 3.53381e12 0.724929
\(661\) −3.77872e12 −0.769907 −0.384953 0.922936i \(-0.625782\pi\)
−0.384953 + 0.922936i \(0.625782\pi\)
\(662\) 3.39542e12 0.687120
\(663\) 5.26872e11 0.105900
\(664\) 8.39806e12 1.67657
\(665\) −8.27248e11 −0.164036
\(666\) −9.55875e11 −0.188264
\(667\) −8.26782e11 −0.161743
\(668\) −9.78399e12 −1.90117
\(669\) −4.06058e11 −0.0783738
\(670\) 6.55962e12 1.25760
\(671\) 2.73340e10 0.00520538
\(672\) −1.53998e13 −2.91309
\(673\) 2.75736e12 0.518114 0.259057 0.965862i \(-0.416588\pi\)
0.259057 + 0.965862i \(0.416588\pi\)
\(674\) 7.62742e12 1.42367
\(675\) 2.07594e11 0.0384900
\(676\) −1.41611e13 −2.60818
\(677\) −2.40567e12 −0.440136 −0.220068 0.975485i \(-0.570628\pi\)
−0.220068 + 0.975485i \(0.570628\pi\)
\(678\) −4.34302e12 −0.789329
\(679\) −1.34259e13 −2.42397
\(680\) −6.63418e12 −1.18986
\(681\) −2.21825e12 −0.395228
\(682\) −1.96155e11 −0.0347193
\(683\) −7.88192e11 −0.138592 −0.0692961 0.997596i \(-0.522075\pi\)
−0.0692961 + 0.997596i \(0.522075\pi\)
\(684\) −1.56173e12 −0.272806
\(685\) −3.44919e12 −0.598563
\(686\) 6.52178e12 1.12436
\(687\) 4.34863e12 0.744813
\(688\) 1.93588e13 3.29405
\(689\) −4.37015e9 −0.000738771 0
\(690\) 4.71428e12 0.791761
\(691\) 3.54288e12 0.591161 0.295580 0.955318i \(-0.404487\pi\)
0.295580 + 0.955318i \(0.404487\pi\)
\(692\) 4.62293e12 0.766372
\(693\) 2.54664e12 0.419438
\(694\) 1.80510e13 2.95381
\(695\) 2.48610e12 0.404191
\(696\) 1.24894e12 0.201744
\(697\) 6.24137e12 1.00169
\(698\) 1.30740e13 2.08478
\(699\) 5.87958e12 0.931535
\(700\) −4.34161e12 −0.683455
\(701\) −3.66998e12 −0.574027 −0.287014 0.957926i \(-0.592663\pi\)
−0.287014 + 0.957926i \(0.592663\pi\)
\(702\) 5.65568e11 0.0878957
\(703\) 5.58955e11 0.0863133
\(704\) −2.67961e13 −4.11144
\(705\) −3.18003e12 −0.484819
\(706\) 2.49918e11 0.0378597
\(707\) 1.42595e12 0.214643
\(708\) 4.40392e12 0.658703
\(709\) 6.68373e12 0.993369 0.496685 0.867931i \(-0.334551\pi\)
0.496685 + 0.867931i \(0.334551\pi\)
\(710\) −3.52862e12 −0.521126
\(711\) −3.46438e12 −0.508409
\(712\) −8.22050e12 −1.19878
\(713\) −1.92109e11 −0.0278385
\(714\) −7.49532e12 −1.07932
\(715\) 7.48337e11 0.107083
\(716\) −1.58954e13 −2.26029
\(717\) 1.40383e12 0.198371
\(718\) 2.71889e13 3.81796
\(719\) −1.23056e13 −1.71721 −0.858604 0.512640i \(-0.828668\pi\)
−0.858604 + 0.512640i \(0.828668\pi\)
\(720\) −4.15314e12 −0.575944
\(721\) 1.34001e13 1.84672
\(722\) −1.29168e13 −1.76904
\(723\) 3.29223e12 0.448092
\(724\) 5.30228e12 0.717198
\(725\) 1.52197e11 0.0204590
\(726\) 2.83568e11 0.0378828
\(727\) −1.07672e13 −1.42954 −0.714770 0.699360i \(-0.753469\pi\)
−0.714770 + 0.699360i \(0.753469\pi\)
\(728\) −7.54470e12 −0.995522
\(729\) 2.82430e11 0.0370370
\(730\) 6.56546e12 0.855681
\(731\) 5.12681e12 0.664078
\(732\) −6.33990e10 −0.00816174
\(733\) −1.06410e13 −1.36149 −0.680744 0.732521i \(-0.738343\pi\)
−0.680744 + 0.732521i \(0.738343\pi\)
\(734\) −1.78534e11 −0.0227033
\(735\) −1.08588e12 −0.137243
\(736\) −5.13181e13 −6.44644
\(737\) 1.18083e13 1.47429
\(738\) 6.69977e12 0.831393
\(739\) −4.24704e12 −0.523825 −0.261912 0.965092i \(-0.584353\pi\)
−0.261912 + 0.965092i \(0.584353\pi\)
\(740\) 2.93354e12 0.359625
\(741\) −3.30720e11 −0.0402975
\(742\) 6.21701e10 0.00752947
\(743\) 1.10732e13 1.33298 0.666488 0.745516i \(-0.267797\pi\)
0.666488 + 0.745516i \(0.267797\pi\)
\(744\) 2.90202e11 0.0347234
\(745\) 1.41586e12 0.168391
\(746\) −2.96390e13 −3.50380
\(747\) 1.39231e12 0.163604
\(748\) −1.87229e13 −2.18684
\(749\) −1.36238e13 −1.58172
\(750\) −8.67821e11 −0.100151
\(751\) 1.45365e13 1.66755 0.833777 0.552102i \(-0.186174\pi\)
0.833777 + 0.552102i \(0.186174\pi\)
\(752\) 6.36197e13 7.25456
\(753\) 8.17454e12 0.926587
\(754\) 4.14644e11 0.0467202
\(755\) 5.30724e12 0.594439
\(756\) −5.90672e12 −0.657655
\(757\) −5.75556e12 −0.637025 −0.318512 0.947919i \(-0.603183\pi\)
−0.318512 + 0.947919i \(0.603183\pi\)
\(758\) −1.61118e13 −1.77269
\(759\) 8.48638e12 0.928184
\(760\) 4.16430e12 0.452774
\(761\) −1.44303e13 −1.55971 −0.779854 0.625961i \(-0.784707\pi\)
−0.779854 + 0.625961i \(0.784707\pi\)
\(762\) −3.37802e12 −0.362965
\(763\) 9.21664e12 0.984492
\(764\) 2.43775e13 2.58862
\(765\) −1.09988e12 −0.116110
\(766\) 9.74471e12 1.02268
\(767\) 9.32595e11 0.0973003
\(768\) 1.81378e13 1.88131
\(769\) 2.65690e12 0.273973 0.136986 0.990573i \(-0.456258\pi\)
0.136986 + 0.990573i \(0.456258\pi\)
\(770\) −1.06459e13 −1.09137
\(771\) 4.07965e12 0.415794
\(772\) −8.49775e12 −0.861046
\(773\) −4.35569e12 −0.438783 −0.219391 0.975637i \(-0.570407\pi\)
−0.219391 + 0.975637i \(0.570407\pi\)
\(774\) 5.50334e12 0.551179
\(775\) 3.53641e10 0.00352132
\(776\) 6.75848e13 6.69069
\(777\) 2.11406e12 0.208076
\(778\) −3.17299e13 −3.10499
\(779\) −3.91774e12 −0.381168
\(780\) −1.73570e12 −0.167900
\(781\) −6.35203e12 −0.610918
\(782\) −2.49773e13 −2.38844
\(783\) 2.07062e11 0.0196867
\(784\) 2.17242e13 2.05363
\(785\) −4.55237e12 −0.427882
\(786\) −1.43268e13 −1.33890
\(787\) −2.02013e13 −1.87713 −0.938564 0.345106i \(-0.887843\pi\)
−0.938564 + 0.345106i \(0.887843\pi\)
\(788\) −1.68351e13 −1.55542
\(789\) −4.52529e12 −0.415719
\(790\) 1.44824e13 1.32288
\(791\) 9.60521e12 0.872394
\(792\) −1.28196e13 −1.15774
\(793\) −1.34257e10 −0.00120561
\(794\) 2.66007e13 2.37520
\(795\) 9.12298e9 0.000809999 0
\(796\) −8.63752e12 −0.762571
\(797\) −2.07284e12 −0.181971 −0.0909857 0.995852i \(-0.529002\pi\)
−0.0909857 + 0.995852i \(0.529002\pi\)
\(798\) 4.70485e12 0.410707
\(799\) 1.68485e13 1.46251
\(800\) 9.44681e12 0.815418
\(801\) −1.36288e12 −0.116980
\(802\) 2.95124e13 2.51895
\(803\) 1.18188e13 1.00312
\(804\) −2.73883e13 −2.31160
\(805\) −1.04263e13 −0.875082
\(806\) 9.63458e10 0.00804128
\(807\) −8.51916e12 −0.707075
\(808\) −7.17812e12 −0.592460
\(809\) 1.41673e13 1.16284 0.581418 0.813605i \(-0.302498\pi\)
0.581418 + 0.813605i \(0.302498\pi\)
\(810\) −1.18066e12 −0.0963701
\(811\) 2.04580e12 0.166062 0.0830310 0.996547i \(-0.473540\pi\)
0.0830310 + 0.996547i \(0.473540\pi\)
\(812\) −4.33049e12 −0.349570
\(813\) −4.40786e12 −0.353851
\(814\) 7.19322e12 0.574266
\(815\) −8.65117e12 −0.686856
\(816\) 2.20042e13 1.73740
\(817\) −3.21812e12 −0.252699
\(818\) −1.32741e12 −0.103661
\(819\) −1.25084e12 −0.0971455
\(820\) −2.05613e13 −1.58814
\(821\) −1.27265e13 −0.977607 −0.488804 0.872394i \(-0.662567\pi\)
−0.488804 + 0.872394i \(0.662567\pi\)
\(822\) 1.96168e13 1.49866
\(823\) 6.94698e12 0.527833 0.263917 0.964546i \(-0.414986\pi\)
0.263917 + 0.964546i \(0.414986\pi\)
\(824\) −6.74552e13 −5.09733
\(825\) −1.56220e12 −0.117407
\(826\) −1.32672e13 −0.991672
\(827\) 5.09129e11 0.0378489 0.0189244 0.999821i \(-0.493976\pi\)
0.0189244 + 0.999821i \(0.493976\pi\)
\(828\) −1.96834e13 −1.45534
\(829\) −1.87663e13 −1.38001 −0.690006 0.723804i \(-0.742392\pi\)
−0.690006 + 0.723804i \(0.742392\pi\)
\(830\) −5.82039e12 −0.425697
\(831\) 1.33268e13 0.969440
\(832\) 1.31615e13 0.952245
\(833\) 5.75324e12 0.414009
\(834\) −1.41393e13 −1.01200
\(835\) 4.32524e12 0.307908
\(836\) 1.17525e13 0.832147
\(837\) 4.81125e10 0.00338839
\(838\) −7.10645e12 −0.497800
\(839\) 6.68721e12 0.465925 0.232962 0.972486i \(-0.425158\pi\)
0.232962 + 0.972486i \(0.425158\pi\)
\(840\) 1.57501e13 1.09150
\(841\) −1.43553e13 −0.989536
\(842\) −3.53944e13 −2.42678
\(843\) −5.46147e12 −0.372465
\(844\) −5.20124e13 −3.52830
\(845\) 6.26025e12 0.422412
\(846\) 1.80859e13 1.21387
\(847\) −6.27151e11 −0.0418694
\(848\) −1.82515e11 −0.0121204
\(849\) 1.14952e13 0.759335
\(850\) 4.59791e12 0.302117
\(851\) 7.04484e12 0.460456
\(852\) 1.47330e13 0.957884
\(853\) 9.37029e12 0.606014 0.303007 0.952988i \(-0.402009\pi\)
0.303007 + 0.952988i \(0.402009\pi\)
\(854\) 1.90995e11 0.0122874
\(855\) 6.90399e11 0.0441828
\(856\) 6.85813e13 4.36590
\(857\) 8.81996e12 0.558538 0.279269 0.960213i \(-0.409908\pi\)
0.279269 + 0.960213i \(0.409908\pi\)
\(858\) −4.25605e12 −0.268111
\(859\) 1.98932e13 1.24663 0.623313 0.781973i \(-0.285786\pi\)
0.623313 + 0.781973i \(0.285786\pi\)
\(860\) −1.68895e13 −1.05287
\(861\) −1.48175e13 −0.918884
\(862\) 3.75055e13 2.31373
\(863\) −1.20953e13 −0.742278 −0.371139 0.928577i \(-0.621033\pi\)
−0.371139 + 0.928577i \(0.621033\pi\)
\(864\) 1.28523e13 0.784636
\(865\) −2.04367e12 −0.124119
\(866\) 3.60674e13 2.17914
\(867\) −3.77821e12 −0.227091
\(868\) −1.00622e12 −0.0601666
\(869\) 2.60704e13 1.55081
\(870\) −8.65597e11 −0.0512246
\(871\) −5.79987e12 −0.341458
\(872\) −4.63959e13 −2.71741
\(873\) 1.12049e13 0.652894
\(874\) 1.56783e13 0.908864
\(875\) 1.91931e12 0.110690
\(876\) −2.74127e13 −1.57283
\(877\) −2.52693e13 −1.44243 −0.721216 0.692710i \(-0.756416\pi\)
−0.721216 + 0.692710i \(0.756416\pi\)
\(878\) 4.37876e13 2.48671
\(879\) 2.26068e12 0.127729
\(880\) 3.12535e13 1.75682
\(881\) 4.23113e12 0.236628 0.118314 0.992976i \(-0.462251\pi\)
0.118314 + 0.992976i \(0.462251\pi\)
\(882\) 6.17579e12 0.343624
\(883\) 5.41348e12 0.299677 0.149839 0.988710i \(-0.452125\pi\)
0.149839 + 0.988710i \(0.452125\pi\)
\(884\) 9.19615e12 0.506490
\(885\) −1.94685e12 −0.106681
\(886\) −9.76794e12 −0.532539
\(887\) −6.28957e12 −0.341165 −0.170583 0.985343i \(-0.554565\pi\)
−0.170583 + 0.985343i \(0.554565\pi\)
\(888\) −1.06420e13 −0.574334
\(889\) 7.47097e12 0.401161
\(890\) 5.69733e12 0.304380
\(891\) −2.12536e12 −0.112975
\(892\) −7.08744e12 −0.374841
\(893\) −1.05759e13 −0.556525
\(894\) −8.05250e12 −0.421611
\(895\) 7.02695e12 0.366069
\(896\) −8.98938e13 −4.65955
\(897\) −4.16826e12 −0.214976
\(898\) 5.65603e13 2.90247
\(899\) 3.52735e10 0.00180107
\(900\) 3.62340e12 0.184088
\(901\) −4.83356e10 −0.00244346
\(902\) −5.04176e13 −2.53602
\(903\) −1.21714e13 −0.609182
\(904\) −4.83519e13 −2.40800
\(905\) −2.34400e12 −0.116155
\(906\) −3.01841e13 −1.48834
\(907\) −1.27094e13 −0.623580 −0.311790 0.950151i \(-0.600928\pi\)
−0.311790 + 0.950151i \(0.600928\pi\)
\(908\) −3.87178e13 −1.89027
\(909\) −1.19006e12 −0.0578138
\(910\) 5.22896e12 0.252772
\(911\) 9.10912e12 0.438171 0.219086 0.975706i \(-0.429693\pi\)
0.219086 + 0.975706i \(0.429693\pi\)
\(912\) −1.38122e13 −0.661127
\(913\) −1.04775e13 −0.499046
\(914\) 3.85144e13 1.82543
\(915\) 2.80270e10 0.00132185
\(916\) 7.59021e13 3.56225
\(917\) 3.16858e13 1.47980
\(918\) 6.25540e12 0.290712
\(919\) −6.78526e12 −0.313795 −0.156898 0.987615i \(-0.550149\pi\)
−0.156898 + 0.987615i \(0.550149\pi\)
\(920\) 5.24852e13 2.41542
\(921\) 1.29411e13 0.592658
\(922\) −1.45318e13 −0.662263
\(923\) 3.11993e12 0.141494
\(924\) 4.44496e13 2.00606
\(925\) −1.29684e12 −0.0582436
\(926\) −4.64466e13 −2.07589
\(927\) −1.11834e13 −0.497410
\(928\) 9.42261e12 0.417066
\(929\) 3.16131e12 0.139250 0.0696252 0.997573i \(-0.477820\pi\)
0.0696252 + 0.997573i \(0.477820\pi\)
\(930\) −2.01128e11 −0.00881657
\(931\) −3.61134e12 −0.157541
\(932\) 1.02624e14 4.45529
\(933\) −6.35276e12 −0.274470
\(934\) −6.55713e13 −2.81938
\(935\) 8.27689e12 0.354173
\(936\) 6.29661e12 0.268142
\(937\) 3.75670e13 1.59213 0.796065 0.605211i \(-0.206911\pi\)
0.796065 + 0.605211i \(0.206911\pi\)
\(938\) 8.25094e13 3.48009
\(939\) −1.02246e13 −0.429190
\(940\) −5.55049e13 −2.31876
\(941\) −3.13713e13 −1.30431 −0.652153 0.758087i \(-0.726134\pi\)
−0.652153 + 0.758087i \(0.726134\pi\)
\(942\) 2.58909e13 1.07132
\(943\) −4.93776e13 −2.03342
\(944\) 3.89488e13 1.59632
\(945\) 2.61120e12 0.106512
\(946\) −4.14142e13 −1.68127
\(947\) −1.78574e13 −0.721513 −0.360757 0.932660i \(-0.617481\pi\)
−0.360757 + 0.932660i \(0.617481\pi\)
\(948\) −6.04682e13 −2.43159
\(949\) −5.80504e12 −0.232331
\(950\) −2.88613e12 −0.114963
\(951\) −7.77194e12 −0.308118
\(952\) −8.34473e13 −3.29265
\(953\) 8.76602e12 0.344258 0.172129 0.985074i \(-0.444935\pi\)
0.172129 + 0.985074i \(0.444935\pi\)
\(954\) −5.18856e10 −0.00202805
\(955\) −1.07766e13 −0.419245
\(956\) 2.45028e13 0.948758
\(957\) −1.55820e12 −0.0600509
\(958\) 4.58572e13 1.75899
\(959\) −4.33853e13 −1.65638
\(960\) −2.74754e13 −1.04406
\(961\) −2.64314e13 −0.999690
\(962\) −3.53310e12 −0.133005
\(963\) 1.13701e13 0.426035
\(964\) 5.74633e13 2.14311
\(965\) 3.75663e12 0.139452
\(966\) 5.92980e13 2.19100
\(967\) 2.92145e13 1.07443 0.537217 0.843444i \(-0.319476\pi\)
0.537217 + 0.843444i \(0.319476\pi\)
\(968\) 3.15703e12 0.115569
\(969\) −3.65789e12 −0.133283
\(970\) −4.68405e13 −1.69883
\(971\) 1.10119e13 0.397536 0.198768 0.980047i \(-0.436306\pi\)
0.198768 + 0.980047i \(0.436306\pi\)
\(972\) 4.92959e12 0.177138
\(973\) 3.12711e13 1.11850
\(974\) −1.01083e14 −3.59885
\(975\) 7.67308e11 0.0271925
\(976\) −5.60709e11 −0.0197794
\(977\) 8.77989e12 0.308293 0.154146 0.988048i \(-0.450737\pi\)
0.154146 + 0.988048i \(0.450737\pi\)
\(978\) 4.92022e13 1.71973
\(979\) 1.02560e13 0.356826
\(980\) −1.89532e13 −0.656396
\(981\) −7.69197e12 −0.265172
\(982\) −2.36073e13 −0.810113
\(983\) −5.68473e13 −1.94186 −0.970932 0.239355i \(-0.923064\pi\)
−0.970932 + 0.239355i \(0.923064\pi\)
\(984\) 7.45902e13 2.53632
\(985\) 7.44234e12 0.251911
\(986\) 4.58612e12 0.154525
\(987\) −3.99996e13 −1.34162
\(988\) −5.77246e12 −0.192732
\(989\) −4.05599e13 −1.34807
\(990\) 8.88479e12 0.293960
\(991\) 4.32479e13 1.42440 0.712202 0.701974i \(-0.247698\pi\)
0.712202 + 0.701974i \(0.247698\pi\)
\(992\) 2.18942e12 0.0717837
\(993\) 6.26720e12 0.204551
\(994\) −4.43844e13 −1.44209
\(995\) 3.81841e12 0.123503
\(996\) 2.43018e13 0.782476
\(997\) 5.30139e13 1.69927 0.849633 0.527374i \(-0.176823\pi\)
0.849633 + 0.527374i \(0.176823\pi\)
\(998\) −4.45735e13 −1.42229
\(999\) −1.76434e12 −0.0560450
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 15.10.a.c.1.2 2
3.2 odd 2 45.10.a.e.1.1 2
4.3 odd 2 240.10.a.m.1.2 2
5.2 odd 4 75.10.b.e.49.4 4
5.3 odd 4 75.10.b.e.49.1 4
5.4 even 2 75.10.a.g.1.1 2
15.2 even 4 225.10.b.g.199.1 4
15.8 even 4 225.10.b.g.199.4 4
15.14 odd 2 225.10.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.c.1.2 2 1.1 even 1 trivial
45.10.a.e.1.1 2 3.2 odd 2
75.10.a.g.1.1 2 5.4 even 2
75.10.b.e.49.1 4 5.3 odd 4
75.10.b.e.49.4 4 5.2 odd 4
225.10.a.j.1.2 2 15.14 odd 2
225.10.b.g.199.1 4 15.2 even 4
225.10.b.g.199.4 4 15.8 even 4
240.10.a.m.1.2 2 4.3 odd 2