Properties

Label 25.6.b.a.24.2
Level $25$
Weight $6$
Character 25.24
Analytic conductor $4.010$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.6.b.a.24.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} -148.000 q^{11} +112.000i q^{12} -286.000i q^{13} -384.000 q^{14} +656.000 q^{16} -1678.00i q^{17} +454.000i q^{18} -1060.00 q^{19} -768.000 q^{21} -296.000i q^{22} -2976.00i q^{23} -480.000 q^{24} +572.000 q^{26} +1880.00i q^{27} +5376.00i q^{28} +3410.00 q^{29} -2448.00 q^{31} +5152.00i q^{32} -592.000i q^{33} +3356.00 q^{34} +6356.00 q^{36} +182.000i q^{37} -2120.00i q^{38} +1144.00 q^{39} -9398.00 q^{41} -1536.00i q^{42} +1244.00i q^{43} -4144.00 q^{44} +5952.00 q^{46} -12088.0i q^{47} +2624.00i q^{48} -20057.0 q^{49} +6712.00 q^{51} -8008.00i q^{52} -23846.0i q^{53} -3760.00 q^{54} -23040.0 q^{56} -4240.00i q^{57} +6820.00i q^{58} +20020.0 q^{59} +32302.0 q^{61} -4896.00i q^{62} +43584.0i q^{63} +10688.0 q^{64} +1184.00 q^{66} +60972.0i q^{67} -46984.0i q^{68} +11904.0 q^{69} -32648.0 q^{71} +27240.0i q^{72} +38774.0i q^{73} -364.000 q^{74} -29680.0 q^{76} -28416.0i q^{77} +2288.00i q^{78} +33360.0 q^{79} +47641.0 q^{81} -18796.0i q^{82} -16716.0i q^{83} -21504.0 q^{84} -2488.00 q^{86} +13640.0i q^{87} -17760.0i q^{88} -101370. q^{89} +54912.0 q^{91} -83328.0i q^{92} -9792.00i q^{93} +24176.0 q^{94} -20608.0 q^{96} -119038. i q^{97} -40114.0i q^{98} -33596.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} + O(q^{10}) \) \( 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} - 296 q^{11} - 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1536 q^{21} - 960 q^{24} + 1144 q^{26} + 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 12712 q^{36} + 2288 q^{39} - 18796 q^{41} - 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 13424 q^{51} - 7520 q^{54} - 46080 q^{56} + 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 2368 q^{66} + 23808 q^{69} - 65296 q^{71} - 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 95282 q^{81} - 43008 q^{84} - 4976 q^{86} - 202740 q^{89} + 109824 q^{91} + 48352 q^{94} - 41216 q^{96} - 67192 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.353553i 0.984251 + 0.176777i \(0.0565670\pi\)
−0.984251 + 0.176777i \(0.943433\pi\)
\(3\) 4.00000i 0.256600i 0.991735 + 0.128300i \(0.0409521\pi\)
−0.991735 + 0.128300i \(0.959048\pi\)
\(4\) 28.0000 0.875000
\(5\) 0 0
\(6\) −8.00000 −0.0907218
\(7\) 192.000i 1.48100i 0.672054 + 0.740502i \(0.265412\pi\)
−0.672054 + 0.740502i \(0.734588\pi\)
\(8\) 120.000i 0.662913i
\(9\) 227.000 0.934156
\(10\) 0 0
\(11\) −148.000 −0.368791 −0.184395 0.982852i \(-0.559033\pi\)
−0.184395 + 0.982852i \(0.559033\pi\)
\(12\) 112.000i 0.224525i
\(13\) − 286.000i − 0.469362i −0.972072 0.234681i \(-0.924595\pi\)
0.972072 0.234681i \(-0.0754045\pi\)
\(14\) −384.000 −0.523614
\(15\) 0 0
\(16\) 656.000 0.640625
\(17\) − 1678.00i − 1.40822i −0.710092 0.704109i \(-0.751347\pi\)
0.710092 0.704109i \(-0.248653\pi\)
\(18\) 454.000i 0.330274i
\(19\) −1060.00 −0.673631 −0.336815 0.941571i \(-0.609350\pi\)
−0.336815 + 0.941571i \(0.609350\pi\)
\(20\) 0 0
\(21\) −768.000 −0.380026
\(22\) − 296.000i − 0.130387i
\(23\) − 2976.00i − 1.17304i −0.809934 0.586521i \(-0.800497\pi\)
0.809934 0.586521i \(-0.199503\pi\)
\(24\) −480.000 −0.170103
\(25\) 0 0
\(26\) 572.000 0.165944
\(27\) 1880.00i 0.496305i
\(28\) 5376.00i 1.29588i
\(29\) 3410.00 0.752938 0.376469 0.926429i \(-0.377138\pi\)
0.376469 + 0.926429i \(0.377138\pi\)
\(30\) 0 0
\(31\) −2448.00 −0.457517 −0.228758 0.973483i \(-0.573467\pi\)
−0.228758 + 0.973483i \(0.573467\pi\)
\(32\) 5152.00i 0.889408i
\(33\) − 592.000i − 0.0946317i
\(34\) 3356.00 0.497880
\(35\) 0 0
\(36\) 6356.00 0.817387
\(37\) 182.000i 0.0218558i 0.999940 + 0.0109279i \(0.00347853\pi\)
−0.999940 + 0.0109279i \(0.996521\pi\)
\(38\) − 2120.00i − 0.238164i
\(39\) 1144.00 0.120438
\(40\) 0 0
\(41\) −9398.00 −0.873124 −0.436562 0.899674i \(-0.643804\pi\)
−0.436562 + 0.899674i \(0.643804\pi\)
\(42\) − 1536.00i − 0.134359i
\(43\) 1244.00i 0.102600i 0.998683 + 0.0513002i \(0.0163365\pi\)
−0.998683 + 0.0513002i \(0.983663\pi\)
\(44\) −4144.00 −0.322692
\(45\) 0 0
\(46\) 5952.00 0.414733
\(47\) − 12088.0i − 0.798196i −0.916908 0.399098i \(-0.869323\pi\)
0.916908 0.399098i \(-0.130677\pi\)
\(48\) 2624.00i 0.164384i
\(49\) −20057.0 −1.19337
\(50\) 0 0
\(51\) 6712.00 0.361349
\(52\) − 8008.00i − 0.410691i
\(53\) − 23846.0i − 1.16607i −0.812446 0.583037i \(-0.801864\pi\)
0.812446 0.583037i \(-0.198136\pi\)
\(54\) −3760.00 −0.175470
\(55\) 0 0
\(56\) −23040.0 −0.981776
\(57\) − 4240.00i − 0.172854i
\(58\) 6820.00i 0.266204i
\(59\) 20020.0 0.748745 0.374373 0.927278i \(-0.377858\pi\)
0.374373 + 0.927278i \(0.377858\pi\)
\(60\) 0 0
\(61\) 32302.0 1.11149 0.555744 0.831353i \(-0.312433\pi\)
0.555744 + 0.831353i \(0.312433\pi\)
\(62\) − 4896.00i − 0.161757i
\(63\) 43584.0i 1.38349i
\(64\) 10688.0 0.326172
\(65\) 0 0
\(66\) 1184.00 0.0334574
\(67\) 60972.0i 1.65937i 0.558231 + 0.829685i \(0.311480\pi\)
−0.558231 + 0.829685i \(0.688520\pi\)
\(68\) − 46984.0i − 1.23219i
\(69\) 11904.0 0.301003
\(70\) 0 0
\(71\) −32648.0 −0.768618 −0.384309 0.923204i \(-0.625560\pi\)
−0.384309 + 0.923204i \(0.625560\pi\)
\(72\) 27240.0i 0.619264i
\(73\) 38774.0i 0.851596i 0.904818 + 0.425798i \(0.140007\pi\)
−0.904818 + 0.425798i \(0.859993\pi\)
\(74\) −364.000 −0.00772720
\(75\) 0 0
\(76\) −29680.0 −0.589427
\(77\) − 28416.0i − 0.546180i
\(78\) 2288.00i 0.0425814i
\(79\) 33360.0 0.601393 0.300696 0.953720i \(-0.402781\pi\)
0.300696 + 0.953720i \(0.402781\pi\)
\(80\) 0 0
\(81\) 47641.0 0.806805
\(82\) − 18796.0i − 0.308696i
\(83\) − 16716.0i − 0.266340i −0.991093 0.133170i \(-0.957484\pi\)
0.991093 0.133170i \(-0.0425157\pi\)
\(84\) −21504.0 −0.332522
\(85\) 0 0
\(86\) −2488.00 −0.0362747
\(87\) 13640.0i 0.193204i
\(88\) − 17760.0i − 0.244476i
\(89\) −101370. −1.35655 −0.678273 0.734810i \(-0.737271\pi\)
−0.678273 + 0.734810i \(0.737271\pi\)
\(90\) 0 0
\(91\) 54912.0 0.695126
\(92\) − 83328.0i − 1.02641i
\(93\) − 9792.00i − 0.117399i
\(94\) 24176.0 0.282205
\(95\) 0 0
\(96\) −20608.0 −0.228222
\(97\) − 119038.i − 1.28457i −0.766468 0.642283i \(-0.777987\pi\)
0.766468 0.642283i \(-0.222013\pi\)
\(98\) − 40114.0i − 0.421921i
\(99\) −33596.0 −0.344508
\(100\) 0 0
\(101\) −89898.0 −0.876893 −0.438446 0.898757i \(-0.644471\pi\)
−0.438446 + 0.898757i \(0.644471\pi\)
\(102\) 13424.0i 0.127756i
\(103\) 19504.0i 0.181147i 0.995890 + 0.0905734i \(0.0288700\pi\)
−0.995890 + 0.0905734i \(0.971130\pi\)
\(104\) 34320.0 0.311146
\(105\) 0 0
\(106\) 47692.0 0.412269
\(107\) 158292.i 1.33659i 0.743895 + 0.668297i \(0.232976\pi\)
−0.743895 + 0.668297i \(0.767024\pi\)
\(108\) 52640.0i 0.434267i
\(109\) −36830.0 −0.296917 −0.148459 0.988919i \(-0.547431\pi\)
−0.148459 + 0.988919i \(0.547431\pi\)
\(110\) 0 0
\(111\) −728.000 −0.00560821
\(112\) 125952.i 0.948768i
\(113\) − 11186.0i − 0.0824098i −0.999151 0.0412049i \(-0.986880\pi\)
0.999151 0.0412049i \(-0.0131196\pi\)
\(114\) 8480.00 0.0611130
\(115\) 0 0
\(116\) 95480.0 0.658821
\(117\) − 64922.0i − 0.438457i
\(118\) 40040.0i 0.264721i
\(119\) 322176. 2.08557
\(120\) 0 0
\(121\) −139147. −0.863993
\(122\) 64604.0i 0.392970i
\(123\) − 37592.0i − 0.224044i
\(124\) −68544.0 −0.400327
\(125\) 0 0
\(126\) −87168.0 −0.489137
\(127\) 70552.0i 0.388150i 0.980987 + 0.194075i \(0.0621706\pi\)
−0.980987 + 0.194075i \(0.937829\pi\)
\(128\) 186240.i 1.00473i
\(129\) −4976.00 −0.0263273
\(130\) 0 0
\(131\) 76452.0 0.389234 0.194617 0.980879i \(-0.437654\pi\)
0.194617 + 0.980879i \(0.437654\pi\)
\(132\) − 16576.0i − 0.0828028i
\(133\) − 203520.i − 0.997650i
\(134\) −121944. −0.586676
\(135\) 0 0
\(136\) 201360. 0.933525
\(137\) − 144918.i − 0.659661i −0.944040 0.329831i \(-0.893008\pi\)
0.944040 0.329831i \(-0.106992\pi\)
\(138\) 23808.0i 0.106420i
\(139\) −112220. −0.492644 −0.246322 0.969188i \(-0.579222\pi\)
−0.246322 + 0.969188i \(0.579222\pi\)
\(140\) 0 0
\(141\) 48352.0 0.204817
\(142\) − 65296.0i − 0.271748i
\(143\) 42328.0i 0.173096i
\(144\) 148912. 0.598444
\(145\) 0 0
\(146\) −77548.0 −0.301085
\(147\) − 80228.0i − 0.306219i
\(148\) 5096.00i 0.0191238i
\(149\) −403750. −1.48986 −0.744932 0.667140i \(-0.767518\pi\)
−0.744932 + 0.667140i \(0.767518\pi\)
\(150\) 0 0
\(151\) −446648. −1.59413 −0.797064 0.603895i \(-0.793615\pi\)
−0.797064 + 0.603895i \(0.793615\pi\)
\(152\) − 127200.i − 0.446558i
\(153\) − 380906.i − 1.31550i
\(154\) 56832.0 0.193104
\(155\) 0 0
\(156\) 32032.0 0.105383
\(157\) − 262258.i − 0.849141i −0.905395 0.424570i \(-0.860425\pi\)
0.905395 0.424570i \(-0.139575\pi\)
\(158\) 66720.0i 0.212625i
\(159\) 95384.0 0.299215
\(160\) 0 0
\(161\) 571392. 1.73728
\(162\) 95282.0i 0.285248i
\(163\) 154564.i 0.455658i 0.973701 + 0.227829i \(0.0731628\pi\)
−0.973701 + 0.227829i \(0.926837\pi\)
\(164\) −263144. −0.763983
\(165\) 0 0
\(166\) 33432.0 0.0941656
\(167\) 396672.i 1.10063i 0.834958 + 0.550314i \(0.185492\pi\)
−0.834958 + 0.550314i \(0.814508\pi\)
\(168\) − 92160.0i − 0.251924i
\(169\) 289497. 0.779700
\(170\) 0 0
\(171\) −240620. −0.629276
\(172\) 34832.0i 0.0897754i
\(173\) 573474.i 1.45680i 0.685155 + 0.728398i \(0.259735\pi\)
−0.685155 + 0.728398i \(0.740265\pi\)
\(174\) −27280.0 −0.0683079
\(175\) 0 0
\(176\) −97088.0 −0.236257
\(177\) 80080.0i 0.192128i
\(178\) − 202740.i − 0.479611i
\(179\) 594460. 1.38672 0.693362 0.720589i \(-0.256129\pi\)
0.693362 + 0.720589i \(0.256129\pi\)
\(180\) 0 0
\(181\) −107098. −0.242988 −0.121494 0.992592i \(-0.538769\pi\)
−0.121494 + 0.992592i \(0.538769\pi\)
\(182\) 109824.i 0.245764i
\(183\) 129208.i 0.285208i
\(184\) 357120. 0.777624
\(185\) 0 0
\(186\) 19584.0 0.0415068
\(187\) 248344.i 0.519337i
\(188\) − 338464.i − 0.698422i
\(189\) −360960. −0.735029
\(190\) 0 0
\(191\) 469552. 0.931323 0.465661 0.884963i \(-0.345816\pi\)
0.465661 + 0.884963i \(0.345816\pi\)
\(192\) 42752.0i 0.0836957i
\(193\) − 52706.0i − 0.101851i −0.998702 0.0509257i \(-0.983783\pi\)
0.998702 0.0509257i \(-0.0162172\pi\)
\(194\) 238076. 0.454163
\(195\) 0 0
\(196\) −561596. −1.04420
\(197\) 455862.i 0.836889i 0.908242 + 0.418444i \(0.137425\pi\)
−0.908242 + 0.418444i \(0.862575\pi\)
\(198\) − 67192.0i − 0.121802i
\(199\) −865000. −1.54840 −0.774200 0.632940i \(-0.781848\pi\)
−0.774200 + 0.632940i \(0.781848\pi\)
\(200\) 0 0
\(201\) −243888. −0.425795
\(202\) − 179796.i − 0.310028i
\(203\) 654720.i 1.11510i
\(204\) 187936. 0.316180
\(205\) 0 0
\(206\) −39008.0 −0.0640451
\(207\) − 675552.i − 1.09580i
\(208\) − 187616.i − 0.300685i
\(209\) 156880. 0.248429
\(210\) 0 0
\(211\) 1.10565e6 1.70967 0.854835 0.518900i \(-0.173658\pi\)
0.854835 + 0.518900i \(0.173658\pi\)
\(212\) − 667688.i − 1.02031i
\(213\) − 130592.i − 0.197228i
\(214\) −316584. −0.472557
\(215\) 0 0
\(216\) −225600. −0.329007
\(217\) − 470016.i − 0.677584i
\(218\) − 73660.0i − 0.104976i
\(219\) −155096. −0.218520
\(220\) 0 0
\(221\) −479908. −0.660963
\(222\) − 1456.00i − 0.00198280i
\(223\) − 1.12158e6i − 1.51031i −0.655545 0.755156i \(-0.727561\pi\)
0.655545 0.755156i \(-0.272439\pi\)
\(224\) −989184. −1.31722
\(225\) 0 0
\(226\) 22372.0 0.0291363
\(227\) − 23348.0i − 0.0300736i −0.999887 0.0150368i \(-0.995213\pi\)
0.999887 0.0150368i \(-0.00478654\pi\)
\(228\) − 118720.i − 0.151247i
\(229\) 596010. 0.751043 0.375522 0.926814i \(-0.377464\pi\)
0.375522 + 0.926814i \(0.377464\pi\)
\(230\) 0 0
\(231\) 113664. 0.140150
\(232\) 409200.i 0.499132i
\(233\) 485334.i 0.585667i 0.956163 + 0.292834i \(0.0945982\pi\)
−0.956163 + 0.292834i \(0.905402\pi\)
\(234\) 129844. 0.155018
\(235\) 0 0
\(236\) 560560. 0.655152
\(237\) 133440.i 0.154317i
\(238\) 644352.i 0.737362i
\(239\) 48880.0 0.0553524 0.0276762 0.999617i \(-0.491189\pi\)
0.0276762 + 0.999617i \(0.491189\pi\)
\(240\) 0 0
\(241\) −110798. −0.122882 −0.0614411 0.998111i \(-0.519570\pi\)
−0.0614411 + 0.998111i \(0.519570\pi\)
\(242\) − 278294.i − 0.305468i
\(243\) 647404.i 0.703331i
\(244\) 904456. 0.972552
\(245\) 0 0
\(246\) 75184.0 0.0792114
\(247\) 303160.i 0.316176i
\(248\) − 293760.i − 0.303294i
\(249\) 66864.0 0.0683430
\(250\) 0 0
\(251\) −1.64375e6 −1.64684 −0.823419 0.567434i \(-0.807936\pi\)
−0.823419 + 0.567434i \(0.807936\pi\)
\(252\) 1.22035e6i 1.21055i
\(253\) 440448.i 0.432607i
\(254\) −141104. −0.137232
\(255\) 0 0
\(256\) −30464.0 −0.0290527
\(257\) 1.30624e6i 1.23365i 0.787102 + 0.616823i \(0.211581\pi\)
−0.787102 + 0.616823i \(0.788419\pi\)
\(258\) − 9952.00i − 0.00930810i
\(259\) −34944.0 −0.0323685
\(260\) 0 0
\(261\) 774070. 0.703362
\(262\) 152904.i 0.137615i
\(263\) − 2.12834e6i − 1.89736i −0.316231 0.948682i \(-0.602417\pi\)
0.316231 0.948682i \(-0.397583\pi\)
\(264\) 71040.0 0.0627326
\(265\) 0 0
\(266\) 407040. 0.352722
\(267\) − 405480.i − 0.348090i
\(268\) 1.70722e6i 1.45195i
\(269\) 1.44109e6 1.21426 0.607128 0.794604i \(-0.292321\pi\)
0.607128 + 0.794604i \(0.292321\pi\)
\(270\) 0 0
\(271\) −93248.0 −0.0771288 −0.0385644 0.999256i \(-0.512278\pi\)
−0.0385644 + 0.999256i \(0.512278\pi\)
\(272\) − 1.10077e6i − 0.902139i
\(273\) 219648.i 0.178370i
\(274\) 289836. 0.233225
\(275\) 0 0
\(276\) 333312. 0.263377
\(277\) − 110298.i − 0.0863711i −0.999067 0.0431855i \(-0.986249\pi\)
0.999067 0.0431855i \(-0.0137507\pi\)
\(278\) − 224440.i − 0.174176i
\(279\) −555696. −0.427392
\(280\) 0 0
\(281\) −192198. −0.145205 −0.0726027 0.997361i \(-0.523131\pi\)
−0.0726027 + 0.997361i \(0.523131\pi\)
\(282\) 96704.0i 0.0724139i
\(283\) 331884.i 0.246332i 0.992386 + 0.123166i \(0.0393047\pi\)
−0.992386 + 0.123166i \(0.960695\pi\)
\(284\) −914144. −0.672541
\(285\) 0 0
\(286\) −84656.0 −0.0611988
\(287\) − 1.80442e6i − 1.29310i
\(288\) 1.16950e6i 0.830846i
\(289\) −1.39583e6 −0.983076
\(290\) 0 0
\(291\) 476152. 0.329620
\(292\) 1.08567e6i 0.745146i
\(293\) − 2.19481e6i − 1.49358i −0.665063 0.746788i \(-0.731595\pi\)
0.665063 0.746788i \(-0.268405\pi\)
\(294\) 160456. 0.108265
\(295\) 0 0
\(296\) −21840.0 −0.0144885
\(297\) − 278240.i − 0.183033i
\(298\) − 807500.i − 0.526747i
\(299\) −851136. −0.550581
\(300\) 0 0
\(301\) −238848. −0.151952
\(302\) − 893296.i − 0.563609i
\(303\) − 359592.i − 0.225011i
\(304\) −695360. −0.431545
\(305\) 0 0
\(306\) 761812. 0.465098
\(307\) − 2.37751e6i − 1.43971i −0.694123 0.719857i \(-0.744207\pi\)
0.694123 0.719857i \(-0.255793\pi\)
\(308\) − 795648.i − 0.477908i
\(309\) −78016.0 −0.0464823
\(310\) 0 0
\(311\) −2.37305e6 −1.39125 −0.695626 0.718405i \(-0.744873\pi\)
−0.695626 + 0.718405i \(0.744873\pi\)
\(312\) 137280.i 0.0798400i
\(313\) 1.42941e6i 0.824702i 0.911025 + 0.412351i \(0.135292\pi\)
−0.911025 + 0.412351i \(0.864708\pi\)
\(314\) 524516. 0.300217
\(315\) 0 0
\(316\) 934080. 0.526219
\(317\) 2.12462e6i 1.18750i 0.804650 + 0.593750i \(0.202353\pi\)
−0.804650 + 0.593750i \(0.797647\pi\)
\(318\) 190768.i 0.105788i
\(319\) −504680. −0.277677
\(320\) 0 0
\(321\) −633168. −0.342970
\(322\) 1.14278e6i 0.614221i
\(323\) 1.77868e6i 0.948618i
\(324\) 1.33395e6 0.705954
\(325\) 0 0
\(326\) −309128. −0.161100
\(327\) − 147320.i − 0.0761890i
\(328\) − 1.12776e6i − 0.578805i
\(329\) 2.32090e6 1.18213
\(330\) 0 0
\(331\) 3.09985e6 1.55515 0.777573 0.628793i \(-0.216451\pi\)
0.777573 + 0.628793i \(0.216451\pi\)
\(332\) − 468048.i − 0.233048i
\(333\) 41314.0i 0.0204168i
\(334\) −793344. −0.389131
\(335\) 0 0
\(336\) −503808. −0.243454
\(337\) 2.40008e6i 1.15120i 0.817731 + 0.575601i \(0.195232\pi\)
−0.817731 + 0.575601i \(0.804768\pi\)
\(338\) 578994.i 0.275665i
\(339\) 44744.0 0.0211464
\(340\) 0 0
\(341\) 362304. 0.168728
\(342\) − 481240.i − 0.222483i
\(343\) − 624000.i − 0.286384i
\(344\) −149280. −0.0680151
\(345\) 0 0
\(346\) −1.14695e6 −0.515055
\(347\) 1.77741e6i 0.792436i 0.918156 + 0.396218i \(0.129678\pi\)
−0.918156 + 0.396218i \(0.870322\pi\)
\(348\) 381920.i 0.169054i
\(349\) 2.14805e6 0.944019 0.472010 0.881593i \(-0.343529\pi\)
0.472010 + 0.881593i \(0.343529\pi\)
\(350\) 0 0
\(351\) 537680. 0.232946
\(352\) − 762496.i − 0.328005i
\(353\) 661854.i 0.282700i 0.989960 + 0.141350i \(0.0451443\pi\)
−0.989960 + 0.141350i \(0.954856\pi\)
\(354\) −160160. −0.0679275
\(355\) 0 0
\(356\) −2.83836e6 −1.18698
\(357\) 1.28870e6i 0.535159i
\(358\) 1.18892e6i 0.490281i
\(359\) 259320. 0.106194 0.0530970 0.998589i \(-0.483091\pi\)
0.0530970 + 0.998589i \(0.483091\pi\)
\(360\) 0 0
\(361\) −1.35250e6 −0.546222
\(362\) − 214196.i − 0.0859093i
\(363\) − 556588.i − 0.221701i
\(364\) 1.53754e6 0.608236
\(365\) 0 0
\(366\) −258416. −0.100836
\(367\) − 1.49993e6i − 0.581307i −0.956828 0.290653i \(-0.906127\pi\)
0.956828 0.290653i \(-0.0938726\pi\)
\(368\) − 1.95226e6i − 0.751480i
\(369\) −2.13335e6 −0.815634
\(370\) 0 0
\(371\) 4.57843e6 1.72696
\(372\) − 274176.i − 0.102724i
\(373\) 2.23807e6i 0.832918i 0.909154 + 0.416459i \(0.136729\pi\)
−0.909154 + 0.416459i \(0.863271\pi\)
\(374\) −496688. −0.183614
\(375\) 0 0
\(376\) 1.45056e6 0.529135
\(377\) − 975260.i − 0.353400i
\(378\) − 721920.i − 0.259872i
\(379\) −3.15934e6 −1.12979 −0.564896 0.825162i \(-0.691084\pi\)
−0.564896 + 0.825162i \(0.691084\pi\)
\(380\) 0 0
\(381\) −282208. −0.0995994
\(382\) 939104.i 0.329272i
\(383\) − 342216.i − 0.119207i −0.998222 0.0596037i \(-0.981016\pi\)
0.998222 0.0596037i \(-0.0189837\pi\)
\(384\) −744960. −0.257813
\(385\) 0 0
\(386\) 105412. 0.0360099
\(387\) 282388.i 0.0958449i
\(388\) − 3.33306e6i − 1.12399i
\(389\) −88470.0 −0.0296430 −0.0148215 0.999890i \(-0.504718\pi\)
−0.0148215 + 0.999890i \(0.504718\pi\)
\(390\) 0 0
\(391\) −4.99373e6 −1.65190
\(392\) − 2.40684e6i − 0.791101i
\(393\) 305808.i 0.0998775i
\(394\) −911724. −0.295885
\(395\) 0 0
\(396\) −940688. −0.301445
\(397\) − 5.45674e6i − 1.73763i −0.495138 0.868814i \(-0.664883\pi\)
0.495138 0.868814i \(-0.335117\pi\)
\(398\) − 1.73000e6i − 0.547442i
\(399\) 814080. 0.255997
\(400\) 0 0
\(401\) 4.04680e6 1.25676 0.628378 0.777908i \(-0.283719\pi\)
0.628378 + 0.777908i \(0.283719\pi\)
\(402\) − 487776.i − 0.150541i
\(403\) 700128.i 0.214741i
\(404\) −2.51714e6 −0.767281
\(405\) 0 0
\(406\) −1.30944e6 −0.394249
\(407\) − 26936.0i − 0.00806022i
\(408\) 805440.i 0.239543i
\(409\) 2.71207e6 0.801664 0.400832 0.916151i \(-0.368721\pi\)
0.400832 + 0.916151i \(0.368721\pi\)
\(410\) 0 0
\(411\) 579672. 0.169269
\(412\) 546112.i 0.158503i
\(413\) 3.84384e6i 1.10889i
\(414\) 1.35110e6 0.387425
\(415\) 0 0
\(416\) 1.47347e6 0.417454
\(417\) − 448880.i − 0.126413i
\(418\) 313760.i 0.0878328i
\(419\) −3.71746e6 −1.03445 −0.517227 0.855848i \(-0.673036\pi\)
−0.517227 + 0.855848i \(0.673036\pi\)
\(420\) 0 0
\(421\) 3.55250e6 0.976853 0.488426 0.872605i \(-0.337571\pi\)
0.488426 + 0.872605i \(0.337571\pi\)
\(422\) 2.21130e6i 0.604460i
\(423\) − 2.74398e6i − 0.745640i
\(424\) 2.86152e6 0.773005
\(425\) 0 0
\(426\) 261184. 0.0697305
\(427\) 6.20198e6i 1.64612i
\(428\) 4.43218e6i 1.16952i
\(429\) −169312. −0.0444165
\(430\) 0 0
\(431\) −4.06205e6 −1.05330 −0.526650 0.850082i \(-0.676552\pi\)
−0.526650 + 0.850082i \(0.676552\pi\)
\(432\) 1.23328e6i 0.317945i
\(433\) − 7.26287e6i − 1.86161i −0.365518 0.930804i \(-0.619108\pi\)
0.365518 0.930804i \(-0.380892\pi\)
\(434\) 940032. 0.239562
\(435\) 0 0
\(436\) −1.03124e6 −0.259803
\(437\) 3.15456e6i 0.790197i
\(438\) − 310192.i − 0.0772583i
\(439\) 5.41028e6 1.33986 0.669928 0.742426i \(-0.266325\pi\)
0.669928 + 0.742426i \(0.266325\pi\)
\(440\) 0 0
\(441\) −4.55294e6 −1.11480
\(442\) − 959816.i − 0.233686i
\(443\) 6.51524e6i 1.57733i 0.614826 + 0.788663i \(0.289226\pi\)
−0.614826 + 0.788663i \(0.710774\pi\)
\(444\) −20384.0 −0.00490718
\(445\) 0 0
\(446\) 2.24315e6 0.533976
\(447\) − 1.61500e6i − 0.382299i
\(448\) 2.05210e6i 0.483062i
\(449\) 509950. 0.119375 0.0596873 0.998217i \(-0.480990\pi\)
0.0596873 + 0.998217i \(0.480990\pi\)
\(450\) 0 0
\(451\) 1.39090e6 0.322000
\(452\) − 313208.i − 0.0721085i
\(453\) − 1.78659e6i − 0.409053i
\(454\) 46696.0 0.0106326
\(455\) 0 0
\(456\) 508800. 0.114587
\(457\) 1.22084e6i 0.273444i 0.990609 + 0.136722i \(0.0436568\pi\)
−0.990609 + 0.136722i \(0.956343\pi\)
\(458\) 1.19202e6i 0.265534i
\(459\) 3.15464e6 0.698905
\(460\) 0 0
\(461\) −4.07210e6 −0.892413 −0.446207 0.894930i \(-0.647225\pi\)
−0.446207 + 0.894930i \(0.647225\pi\)
\(462\) 227328.i 0.0495505i
\(463\) − 2.02294e6i − 0.438561i −0.975662 0.219280i \(-0.929629\pi\)
0.975662 0.219280i \(-0.0703709\pi\)
\(464\) 2.23696e6 0.482351
\(465\) 0 0
\(466\) −970668. −0.207065
\(467\) 3.25097e6i 0.689797i 0.938640 + 0.344898i \(0.112087\pi\)
−0.938640 + 0.344898i \(0.887913\pi\)
\(468\) − 1.81782e6i − 0.383650i
\(469\) −1.17066e7 −2.45753
\(470\) 0 0
\(471\) 1.04903e6 0.217890
\(472\) 2.40240e6i 0.496353i
\(473\) − 184112.i − 0.0378381i
\(474\) −266880. −0.0545595
\(475\) 0 0
\(476\) 9.02093e6 1.82488
\(477\) − 5.41304e6i − 1.08929i
\(478\) 97760.0i 0.0195700i
\(479\) 3.27936e6 0.653056 0.326528 0.945188i \(-0.394121\pi\)
0.326528 + 0.945188i \(0.394121\pi\)
\(480\) 0 0
\(481\) 52052.0 0.0102583
\(482\) − 221596.i − 0.0434455i
\(483\) 2.28557e6i 0.445786i
\(484\) −3.89612e6 −0.755994
\(485\) 0 0
\(486\) −1.29481e6 −0.248665
\(487\) − 8.53197e6i − 1.63015i −0.579357 0.815074i \(-0.696696\pi\)
0.579357 0.815074i \(-0.303304\pi\)
\(488\) 3.87624e6i 0.736819i
\(489\) −618256. −0.116922
\(490\) 0 0
\(491\) 1.51265e6 0.283162 0.141581 0.989927i \(-0.454781\pi\)
0.141581 + 0.989927i \(0.454781\pi\)
\(492\) − 1.05258e6i − 0.196038i
\(493\) − 5.72198e6i − 1.06030i
\(494\) −606320. −0.111785
\(495\) 0 0
\(496\) −1.60589e6 −0.293097
\(497\) − 6.26842e6i − 1.13833i
\(498\) 133728.i 0.0241629i
\(499\) 6.49190e6 1.16713 0.583567 0.812065i \(-0.301657\pi\)
0.583567 + 0.812065i \(0.301657\pi\)
\(500\) 0 0
\(501\) −1.58669e6 −0.282421
\(502\) − 3.28750e6i − 0.582245i
\(503\) − 8.61770e6i − 1.51870i −0.650684 0.759349i \(-0.725518\pi\)
0.650684 0.759349i \(-0.274482\pi\)
\(504\) −5.23008e6 −0.917132
\(505\) 0 0
\(506\) −880896. −0.152950
\(507\) 1.15799e6i 0.200071i
\(508\) 1.97546e6i 0.339632i
\(509\) −2.67323e6 −0.457343 −0.228671 0.973504i \(-0.573438\pi\)
−0.228671 + 0.973504i \(0.573438\pi\)
\(510\) 0 0
\(511\) −7.44461e6 −1.26122
\(512\) 5.89875e6i 0.994455i
\(513\) − 1.99280e6i − 0.334326i
\(514\) −2.61248e6 −0.436160
\(515\) 0 0
\(516\) −139328. −0.0230364
\(517\) 1.78902e6i 0.294367i
\(518\) − 69888.0i − 0.0114440i
\(519\) −2.29390e6 −0.373814
\(520\) 0 0
\(521\) 6.18500e6 0.998264 0.499132 0.866526i \(-0.333652\pi\)
0.499132 + 0.866526i \(0.333652\pi\)
\(522\) 1.54814e6i 0.248676i
\(523\) 6.89452e6i 1.10217i 0.834448 + 0.551087i \(0.185787\pi\)
−0.834448 + 0.551087i \(0.814213\pi\)
\(524\) 2.14066e6 0.340580
\(525\) 0 0
\(526\) 4.25667e6 0.670820
\(527\) 4.10774e6i 0.644283i
\(528\) − 388352.i − 0.0606235i
\(529\) −2.42023e6 −0.376026
\(530\) 0 0
\(531\) 4.54454e6 0.699445
\(532\) − 5.69856e6i − 0.872943i
\(533\) 2.68783e6i 0.409811i
\(534\) 810960. 0.123068
\(535\) 0 0
\(536\) −7.31664e6 −1.10002
\(537\) 2.37784e6i 0.355834i
\(538\) 2.88218e6i 0.429304i
\(539\) 2.96844e6 0.440104
\(540\) 0 0
\(541\) 155502. 0.0228425 0.0114212 0.999935i \(-0.496364\pi\)
0.0114212 + 0.999935i \(0.496364\pi\)
\(542\) − 186496.i − 0.0272691i
\(543\) − 428392.i − 0.0623508i
\(544\) 8.64506e6 1.25248
\(545\) 0 0
\(546\) −439296. −0.0630631
\(547\) 1.26544e7i 1.80831i 0.427201 + 0.904157i \(0.359500\pi\)
−0.427201 + 0.904157i \(0.640500\pi\)
\(548\) − 4.05770e6i − 0.577204i
\(549\) 7.33255e6 1.03830
\(550\) 0 0
\(551\) −3.61460e6 −0.507202
\(552\) 1.42848e6i 0.199538i
\(553\) 6.40512e6i 0.890665i
\(554\) 220596. 0.0305368
\(555\) 0 0
\(556\) −3.14216e6 −0.431064
\(557\) − 7.07786e6i − 0.966638i −0.875444 0.483319i \(-0.839431\pi\)
0.875444 0.483319i \(-0.160569\pi\)
\(558\) − 1.11139e6i − 0.151106i
\(559\) 355784. 0.0481567
\(560\) 0 0
\(561\) −993376. −0.133262
\(562\) − 384396.i − 0.0513379i
\(563\) − 846636.i − 0.112571i −0.998415 0.0562854i \(-0.982074\pi\)
0.998415 0.0562854i \(-0.0179257\pi\)
\(564\) 1.35386e6 0.179215
\(565\) 0 0
\(566\) −663768. −0.0870914
\(567\) 9.14707e6i 1.19488i
\(568\) − 3.91776e6i − 0.509527i
\(569\) −4.96041e6 −0.642299 −0.321149 0.947029i \(-0.604069\pi\)
−0.321149 + 0.947029i \(0.604069\pi\)
\(570\) 0 0
\(571\) 8.96505e6 1.15070 0.575351 0.817907i \(-0.304866\pi\)
0.575351 + 0.817907i \(0.304866\pi\)
\(572\) 1.18518e6i 0.151459i
\(573\) 1.87821e6i 0.238978i
\(574\) 3.60883e6 0.457180
\(575\) 0 0
\(576\) 2.42618e6 0.304696
\(577\) − 2.86080e6i − 0.357724i −0.983874 0.178862i \(-0.942758\pi\)
0.983874 0.178862i \(-0.0572415\pi\)
\(578\) − 2.79165e6i − 0.347570i
\(579\) 210824. 0.0261351
\(580\) 0 0
\(581\) 3.20947e6 0.394451
\(582\) 952304.i 0.116538i
\(583\) 3.52921e6i 0.430037i
\(584\) −4.65288e6 −0.564534
\(585\) 0 0
\(586\) 4.38961e6 0.528059
\(587\) − 6.74027e6i − 0.807387i −0.914894 0.403694i \(-0.867726\pi\)
0.914894 0.403694i \(-0.132274\pi\)
\(588\) − 2.24638e6i − 0.267942i
\(589\) 2.59488e6 0.308197
\(590\) 0 0
\(591\) −1.82345e6 −0.214746
\(592\) 119392.i 0.0140014i
\(593\) 1.78609e6i 0.208578i 0.994547 + 0.104289i \(0.0332566\pi\)
−0.994547 + 0.104289i \(0.966743\pi\)
\(594\) 556480. 0.0647118
\(595\) 0 0
\(596\) −1.13050e7 −1.30363
\(597\) − 3.46000e6i − 0.397320i
\(598\) − 1.70227e6i − 0.194660i
\(599\) −4.94620e6 −0.563254 −0.281627 0.959524i \(-0.590874\pi\)
−0.281627 + 0.959524i \(0.590874\pi\)
\(600\) 0 0
\(601\) −4.58100e6 −0.517337 −0.258669 0.965966i \(-0.583284\pi\)
−0.258669 + 0.965966i \(0.583284\pi\)
\(602\) − 477696.i − 0.0537230i
\(603\) 1.38406e7i 1.55011i
\(604\) −1.25061e7 −1.39486
\(605\) 0 0
\(606\) 719184. 0.0795533
\(607\) 7.07999e6i 0.779940i 0.920828 + 0.389970i \(0.127515\pi\)
−0.920828 + 0.389970i \(0.872485\pi\)
\(608\) − 5.46112e6i − 0.599132i
\(609\) −2.61888e6 −0.286136
\(610\) 0 0
\(611\) −3.45717e6 −0.374643
\(612\) − 1.06654e7i − 1.15106i
\(613\) − 5.09609e6i − 0.547754i −0.961765 0.273877i \(-0.911694\pi\)
0.961765 0.273877i \(-0.0883061\pi\)
\(614\) 4.75502e6 0.509016
\(615\) 0 0
\(616\) 3.40992e6 0.362070
\(617\) − 1.30003e7i − 1.37480i −0.726279 0.687400i \(-0.758752\pi\)
0.726279 0.687400i \(-0.241248\pi\)
\(618\) − 156032.i − 0.0164340i
\(619\) −4.84406e6 −0.508139 −0.254070 0.967186i \(-0.581769\pi\)
−0.254070 + 0.967186i \(0.581769\pi\)
\(620\) 0 0
\(621\) 5.59488e6 0.582186
\(622\) − 4.74610e6i − 0.491882i
\(623\) − 1.94630e7i − 2.00905i
\(624\) 750464. 0.0771558
\(625\) 0 0
\(626\) −2.85883e6 −0.291576
\(627\) 627520.i 0.0637468i
\(628\) − 7.34322e6i − 0.742998i
\(629\) 305396. 0.0307777
\(630\) 0 0
\(631\) 6.22775e6 0.622670 0.311335 0.950300i \(-0.399224\pi\)
0.311335 + 0.950300i \(0.399224\pi\)
\(632\) 4.00320e6i 0.398671i
\(633\) 4.42261e6i 0.438702i
\(634\) −4.24924e6 −0.419845
\(635\) 0 0
\(636\) 2.67075e6 0.261813
\(637\) 5.73630e6i 0.560123i
\(638\) − 1.00936e6i − 0.0981735i
\(639\) −7.41110e6 −0.718010
\(640\) 0 0
\(641\) 1.53280e6 0.147347 0.0736734 0.997282i \(-0.476528\pi\)
0.0736734 + 0.997282i \(0.476528\pi\)
\(642\) − 1.26634e6i − 0.121258i
\(643\) 1.74382e7i 1.66332i 0.555287 + 0.831659i \(0.312609\pi\)
−0.555287 + 0.831659i \(0.687391\pi\)
\(644\) 1.59990e7 1.52012
\(645\) 0 0
\(646\) −3.55736e6 −0.335387
\(647\) − 4.25469e6i − 0.399583i −0.979838 0.199792i \(-0.935974\pi\)
0.979838 0.199792i \(-0.0640265\pi\)
\(648\) 5.71692e6i 0.534841i
\(649\) −2.96296e6 −0.276130
\(650\) 0 0
\(651\) 1.88006e6 0.173868
\(652\) 4.32779e6i 0.398701i
\(653\) − 3.01085e6i − 0.276316i −0.990410 0.138158i \(-0.955882\pi\)
0.990410 0.138158i \(-0.0441181\pi\)
\(654\) 294640. 0.0269369
\(655\) 0 0
\(656\) −6.16509e6 −0.559345
\(657\) 8.80170e6i 0.795524i
\(658\) 4.64179e6i 0.417947i
\(659\) 8.11462e6 0.727871 0.363936 0.931424i \(-0.381433\pi\)
0.363936 + 0.931424i \(0.381433\pi\)
\(660\) 0 0
\(661\) 2.47370e6 0.220213 0.110107 0.993920i \(-0.464881\pi\)
0.110107 + 0.993920i \(0.464881\pi\)
\(662\) 6.19970e6i 0.549827i
\(663\) − 1.91963e6i − 0.169603i
\(664\) 2.00592e6 0.176560
\(665\) 0 0
\(666\) −82628.0 −0.00721841
\(667\) − 1.01482e7i − 0.883228i
\(668\) 1.11068e7i 0.963049i
\(669\) 4.48630e6 0.387546
\(670\) 0 0
\(671\) −4.78070e6 −0.409907
\(672\) − 3.95674e6i − 0.337998i
\(673\) − 5.77063e6i − 0.491117i −0.969382 0.245559i \(-0.921029\pi\)
0.969382 0.245559i \(-0.0789714\pi\)
\(674\) −4.80016e6 −0.407011
\(675\) 0 0
\(676\) 8.10592e6 0.682237
\(677\) 1.67197e7i 1.40203i 0.713147 + 0.701014i \(0.247269\pi\)
−0.713147 + 0.701014i \(0.752731\pi\)
\(678\) 89488.0i 0.00747637i
\(679\) 2.28553e7 1.90245
\(680\) 0 0
\(681\) 93392.0 0.00771688
\(682\) 724608.i 0.0596544i
\(683\) − 7.14532e6i − 0.586097i −0.956098 0.293049i \(-0.905330\pi\)
0.956098 0.293049i \(-0.0946698\pi\)
\(684\) −6.73736e6 −0.550617
\(685\) 0 0
\(686\) 1.24800e6 0.101252
\(687\) 2.38404e6i 0.192718i
\(688\) 816064.i 0.0657284i
\(689\) −6.81996e6 −0.547310
\(690\) 0 0
\(691\) −8.78395e6 −0.699833 −0.349917 0.936781i \(-0.613790\pi\)
−0.349917 + 0.936781i \(0.613790\pi\)
\(692\) 1.60573e7i 1.27470i
\(693\) − 6.45043e6i − 0.510218i
\(694\) −3.55482e6 −0.280169
\(695\) 0 0
\(696\) −1.63680e6 −0.128077
\(697\) 1.57698e7i 1.22955i
\(698\) 4.29610e6i 0.333761i
\(699\) −1.94134e6 −0.150282
\(700\) 0 0
\(701\) −1.60141e7 −1.23086 −0.615428 0.788193i \(-0.711017\pi\)
−0.615428 + 0.788193i \(0.711017\pi\)
\(702\) 1.07536e6i 0.0823590i
\(703\) − 192920.i − 0.0147228i
\(704\) −1.58182e6 −0.120289
\(705\) 0 0
\(706\) −1.32371e6 −0.0999495
\(707\) − 1.72604e7i − 1.29868i
\(708\) 2.24224e6i 0.168112i
\(709\) 1.91354e7 1.42962 0.714811 0.699318i \(-0.246513\pi\)
0.714811 + 0.699318i \(0.246513\pi\)
\(710\) 0 0
\(711\) 7.57272e6 0.561795
\(712\) − 1.21644e7i − 0.899271i
\(713\) 7.28525e6i 0.536686i
\(714\) −2.57741e6 −0.189207
\(715\) 0 0
\(716\) 1.66449e7 1.21338
\(717\) 195520.i 0.0142034i
\(718\) 518640.i 0.0375452i
\(719\) −1.02934e7 −0.742566 −0.371283 0.928520i \(-0.621082\pi\)
−0.371283 + 0.928520i \(0.621082\pi\)
\(720\) 0 0
\(721\) −3.74477e6 −0.268279
\(722\) − 2.70500e6i − 0.193119i
\(723\) − 443192.i − 0.0315316i
\(724\) −2.99874e6 −0.212615
\(725\) 0 0
\(726\) 1.11318e6 0.0783831
\(727\) − 1.93264e7i − 1.35618i −0.734981 0.678088i \(-0.762809\pi\)
0.734981 0.678088i \(-0.237191\pi\)
\(728\) 6.58944e6i 0.460808i
\(729\) 8.98715e6 0.626330
\(730\) 0 0
\(731\) 2.08743e6 0.144484
\(732\) 3.61782e6i 0.249557i
\(733\) − 5.26197e6i − 0.361733i −0.983508 0.180866i \(-0.942110\pi\)
0.983508 0.180866i \(-0.0578902\pi\)
\(734\) 2.99986e6 0.205523
\(735\) 0 0
\(736\) 1.53324e7 1.04331
\(737\) − 9.02386e6i − 0.611961i
\(738\) − 4.26669e6i − 0.288370i
\(739\) −2.82944e7 −1.90585 −0.952927 0.303199i \(-0.901945\pi\)
−0.952927 + 0.303199i \(0.901945\pi\)
\(740\) 0 0
\(741\) −1.21264e6 −0.0811309
\(742\) 9.15686e6i 0.610572i
\(743\) − 2.09863e7i − 1.39464i −0.716759 0.697321i \(-0.754375\pi\)
0.716759 0.697321i \(-0.245625\pi\)
\(744\) 1.17504e6 0.0778252
\(745\) 0 0
\(746\) −4.47615e6 −0.294481
\(747\) − 3.79453e6i − 0.248804i
\(748\) 6.95363e6i 0.454420i
\(749\) −3.03921e7 −1.97950
\(750\) 0 0
\(751\) −1.89668e7 −1.22714 −0.613572 0.789639i \(-0.710268\pi\)
−0.613572 + 0.789639i \(0.710268\pi\)
\(752\) − 7.92973e6i − 0.511345i
\(753\) − 6.57499e6i − 0.422579i
\(754\) 1.95052e6 0.124946
\(755\) 0 0
\(756\) −1.01069e7 −0.643151
\(757\) − 1.08257e7i − 0.686617i −0.939223 0.343309i \(-0.888452\pi\)
0.939223 0.343309i \(-0.111548\pi\)
\(758\) − 6.31868e6i − 0.399442i
\(759\) −1.76179e6 −0.111007
\(760\) 0 0
\(761\) 1.90534e7 1.19264 0.596322 0.802745i \(-0.296628\pi\)
0.596322 + 0.802745i \(0.296628\pi\)
\(762\) − 564416.i − 0.0352137i
\(763\) − 7.07136e6i − 0.439736i
\(764\) 1.31475e7 0.814908
\(765\) 0 0
\(766\) 684432. 0.0421462
\(767\) − 5.72572e6i − 0.351432i
\(768\) − 121856.i − 0.00745494i
\(769\) 1.57826e7 0.962415 0.481208 0.876607i \(-0.340198\pi\)
0.481208 + 0.876607i \(0.340198\pi\)
\(770\) 0 0
\(771\) −5.22497e6 −0.316554
\(772\) − 1.47577e6i − 0.0891199i
\(773\) 2.44049e7i 1.46902i 0.678598 + 0.734510i \(0.262588\pi\)
−0.678598 + 0.734510i \(0.737412\pi\)
\(774\) −564776. −0.0338863
\(775\) 0 0
\(776\) 1.42846e7 0.851555
\(777\) − 139776.i − 0.00830577i
\(778\) − 176940.i − 0.0104804i
\(779\) 9.96188e6 0.588163
\(780\) 0 0
\(781\) 4.83190e6 0.283459
\(782\) − 9.98746e6i − 0.584034i
\(783\) 6.41080e6i 0.373687i
\(784\) −1.31574e7 −0.764504
\(785\) 0 0
\(786\) −611616. −0.0353120
\(787\) 3.37607e7i 1.94301i 0.237019 + 0.971505i \(0.423830\pi\)
−0.237019 + 0.971505i \(0.576170\pi\)
\(788\) 1.27641e7i 0.732278i
\(789\) 8.51334e6 0.486864
\(790\) 0 0
\(791\) 2.14771e6 0.122049
\(792\) − 4.03152e6i − 0.228379i
\(793\) − 9.23837e6i − 0.521690i
\(794\) 1.09135e7 0.614344
\(795\) 0 0
\(796\) −2.42200e7 −1.35485
\(797\) 2.19885e7i 1.22617i 0.790019 + 0.613083i \(0.210071\pi\)
−0.790019 + 0.613083i \(0.789929\pi\)
\(798\) 1.62816e6i 0.0905086i
\(799\) −2.02837e7 −1.12403
\(800\) 0 0
\(801\) −2.30110e7 −1.26723
\(802\) 8.09360e6i 0.444330i
\(803\) − 5.73855e6i − 0.314061i
\(804\) −6.82886e6 −0.372570
\(805\) 0 0
\(806\) −1.40026e6 −0.0759224
\(807\) 5.76436e6i 0.311578i
\(808\) − 1.07878e7i − 0.581303i
\(809\) 2.93597e7 1.57717 0.788587 0.614923i \(-0.210813\pi\)
0.788587 + 0.614923i \(0.210813\pi\)
\(810\) 0 0
\(811\) 3.17703e7 1.69617 0.848083 0.529863i \(-0.177757\pi\)
0.848083 + 0.529863i \(0.177757\pi\)
\(812\) 1.83322e7i 0.975716i
\(813\) − 372992.i − 0.0197912i
\(814\) 53872.0 0.00284972
\(815\) 0 0
\(816\) 4.40307e6 0.231489
\(817\) − 1.31864e6i − 0.0691148i
\(818\) 5.42414e6i 0.283431i
\(819\) 1.24650e7 0.649357
\(820\) 0 0
\(821\) −2.71430e6 −0.140540 −0.0702699 0.997528i \(-0.522386\pi\)
−0.0702699 + 0.997528i \(0.522386\pi\)
\(822\) 1.15934e6i 0.0598457i
\(823\) 1.25866e7i 0.647753i 0.946099 + 0.323877i \(0.104986\pi\)
−0.946099 + 0.323877i \(0.895014\pi\)
\(824\) −2.34048e6 −0.120084
\(825\) 0 0
\(826\) −7.68768e6 −0.392053
\(827\) − 8.72355e6i − 0.443537i −0.975099 0.221768i \(-0.928817\pi\)
0.975099 0.221768i \(-0.0711828\pi\)
\(828\) − 1.89155e7i − 0.958829i
\(829\) 1.06178e7 0.536597 0.268299 0.963336i \(-0.413539\pi\)
0.268299 + 0.963336i \(0.413539\pi\)
\(830\) 0 0
\(831\) 441192. 0.0221628
\(832\) − 3.05677e6i − 0.153093i
\(833\) 3.36556e7i 1.68053i
\(834\) 897760. 0.0446936
\(835\) 0 0
\(836\) 4.39264e6 0.217375
\(837\) − 4.60224e6i − 0.227068i
\(838\) − 7.43492e6i − 0.365735i
\(839\) −1.67765e7 −0.822805 −0.411403 0.911454i \(-0.634961\pi\)
−0.411403 + 0.911454i \(0.634961\pi\)
\(840\) 0 0
\(841\) −8.88305e6 −0.433084
\(842\) 7.10500e6i 0.345370i
\(843\) − 768792.i − 0.0372597i
\(844\) 3.09583e7 1.49596
\(845\) 0 0
\(846\) 5.48795e6 0.263624
\(847\) − 2.67162e7i − 1.27958i
\(848\) − 1.56430e7i − 0.747016i
\(849\) −1.32754e6 −0.0632087
\(850\) 0 0
\(851\) 541632. 0.0256378
\(852\) − 3.65658e6i − 0.172574i
\(853\) 2.20186e7i 1.03613i 0.855340 + 0.518067i \(0.173348\pi\)
−0.855340 + 0.518067i \(0.826652\pi\)
\(854\) −1.24040e7 −0.581991
\(855\) 0 0
\(856\) −1.89950e7 −0.886045
\(857\) 3.16676e7i 1.47287i 0.676510 + 0.736434i \(0.263492\pi\)
−0.676510 + 0.736434i \(0.736508\pi\)
\(858\) − 338624.i − 0.0157036i
\(859\) −1.58064e7 −0.730886 −0.365443 0.930834i \(-0.619082\pi\)
−0.365443 + 0.930834i \(0.619082\pi\)
\(860\) 0 0
\(861\) 7.21766e6 0.331809
\(862\) − 8.12410e6i − 0.372398i
\(863\) 1.44287e7i 0.659476i 0.944072 + 0.329738i \(0.106960\pi\)
−0.944072 + 0.329738i \(0.893040\pi\)
\(864\) −9.68576e6 −0.441417
\(865\) 0 0
\(866\) 1.45257e7 0.658178
\(867\) − 5.58331e6i − 0.252257i
\(868\) − 1.31604e7i − 0.592886i
\(869\) −4.93728e6 −0.221788
\(870\) 0 0
\(871\) 1.74380e7 0.778845
\(872\) − 4.41960e6i − 0.196830i
\(873\) − 2.70216e7i − 1.19999i
\(874\) −6.30912e6 −0.279377
\(875\) 0 0
\(876\) −4.34269e6 −0.191205
\(877\) 247902.i 0.0108838i 0.999985 + 0.00544191i \(0.00173222\pi\)
−0.999985 + 0.00544191i \(0.998268\pi\)
\(878\) 1.08206e7i 0.473711i
\(879\) 8.77922e6 0.383252
\(880\) 0 0
\(881\) 4.10268e7 1.78085 0.890426 0.455128i \(-0.150406\pi\)
0.890426 + 0.455128i \(0.150406\pi\)
\(882\) − 9.10588e6i − 0.394140i
\(883\) − 4.18015e7i − 1.80422i −0.431503 0.902112i \(-0.642016\pi\)
0.431503 0.902112i \(-0.357984\pi\)
\(884\) −1.34374e7 −0.578343
\(885\) 0 0
\(886\) −1.30305e7 −0.557669
\(887\) − 2.10476e7i − 0.898241i −0.893471 0.449120i \(-0.851737\pi\)
0.893471 0.449120i \(-0.148263\pi\)
\(888\) − 87360.0i − 0.00371775i
\(889\) −1.35460e7 −0.574852
\(890\) 0 0
\(891\) −7.05087e6 −0.297542
\(892\) − 3.14041e7i − 1.32152i
\(893\) 1.28133e7i 0.537690i
\(894\) 3.23000e6 0.135163
\(895\) 0 0
\(896\) −3.57581e7 −1.48800
\(897\) − 3.40454e6i − 0.141279i
\(898\) 1.01990e6i 0.0422053i
\(899\) −8.34768e6 −0.344482
\(900\) 0 0
\(901\) −4.00136e7 −1.64208
\(902\) 2.78181e6i 0.113844i
\(903\) − 955392.i − 0.0389908i
\(904\) 1.34232e6 0.0546305
\(905\) 0 0
\(906\) 3.57318e6 0.144622
\(907\) 7.48309e6i 0.302039i 0.988531 + 0.151019i \(0.0482556\pi\)
−0.988531 + 0.151019i \(0.951744\pi\)
\(908\) − 653744.i − 0.0263144i
\(909\) −2.04068e7 −0.819155
\(910\) 0 0
\(911\) −6.63165e6 −0.264744 −0.132372 0.991200i \(-0.542259\pi\)
−0.132372 + 0.991200i \(0.542259\pi\)
\(912\) − 2.78144e6i − 0.110734i
\(913\) 2.47397e6i 0.0982239i
\(914\) −2.44168e6 −0.0966772
\(915\) 0 0
\(916\) 1.66883e7 0.657163
\(917\) 1.46788e7i 0.576457i
\(918\) 6.30928e6i 0.247100i
\(919\) 1.68976e7 0.659990 0.329995 0.943983i \(-0.392953\pi\)
0.329995 + 0.943983i \(0.392953\pi\)
\(920\) 0 0
\(921\) 9.51003e6 0.369431
\(922\) − 8.14420e6i − 0.315516i
\(923\) 9.33733e6i 0.360760i
\(924\) 3.18259e6 0.122631
\(925\) 0 0
\(926\) 4.04587e6 0.155055
\(927\) 4.42741e6i 0.169219i
\(928\) 1.75683e7i 0.669669i
\(929\) 1.28653e7 0.489081 0.244541 0.969639i \(-0.421363\pi\)
0.244541 + 0.969639i \(0.421363\pi\)
\(930\) 0 0
\(931\) 2.12604e7 0.803892
\(932\) 1.35894e7i 0.512459i
\(933\) − 9.49219e6i − 0.356995i
\(934\) −6.50194e6 −0.243880
\(935\) 0 0
\(936\) 7.79064e6 0.290659
\(937\) 1.06887e7i 0.397718i 0.980028 + 0.198859i \(0.0637236\pi\)
−0.980028 + 0.198859i \(0.936276\pi\)
\(938\) − 2.34132e7i − 0.868870i
\(939\) −5.71766e6 −0.211619
\(940\) 0 0
\(941\) 2.82455e7 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(942\) 2.09806e6i 0.0770356i
\(943\) 2.79684e7i 1.02421i
\(944\) 1.31331e7 0.479665
\(945\) 0 0
\(946\) 368224. 0.0133778
\(947\) − 1.70892e7i − 0.619222i −0.950863 0.309611i \(-0.899801\pi\)
0.950863 0.309611i \(-0.100199\pi\)
\(948\) 3.73632e6i 0.135028i
\(949\) 1.10894e7 0.399706
\(950\) 0 0
\(951\) −8.49849e6 −0.304713
\(952\) 3.86611e7i 1.38255i
\(953\) − 2.22259e7i − 0.792735i −0.918092 0.396367i \(-0.870271\pi\)
0.918092 0.396367i \(-0.129729\pi\)
\(954\) 1.08261e7 0.385124
\(955\) 0 0
\(956\) 1.36864e6 0.0484333
\(957\) − 2.01872e6i − 0.0712519i
\(958\) 6.55872e6i 0.230890i
\(959\) 2.78243e7 0.976961
\(960\) 0 0
\(961\) −2.26364e7 −0.790678
\(962\) 104104.i 0.00362685i
\(963\) 3.59323e7i 1.24859i
\(964\) −3.10234e6 −0.107522
\(965\) 0 0
\(966\) −4.57114e6 −0.157609
\(967\) 2.41551e7i 0.830696i 0.909663 + 0.415348i \(0.136340\pi\)
−0.909663 + 0.415348i \(0.863660\pi\)
\(968\) − 1.66976e7i − 0.572752i
\(969\) −7.11472e6 −0.243416
\(970\) 0 0
\(971\) −5.48313e7 −1.86630 −0.933149 0.359491i \(-0.882950\pi\)
−0.933149 + 0.359491i \(0.882950\pi\)
\(972\) 1.81273e7i 0.615415i
\(973\) − 2.15462e7i − 0.729608i
\(974\) 1.70639e7 0.576344
\(975\) 0 0
\(976\) 2.11901e7 0.712047
\(977\) − 1.56612e7i − 0.524915i −0.964944 0.262457i \(-0.915467\pi\)
0.964944 0.262457i \(-0.0845329\pi\)
\(978\) − 1.23651e6i − 0.0413382i
\(979\) 1.50028e7 0.500281
\(980\) 0 0
\(981\) −8.36041e6 −0.277367
\(982\) 3.02530e6i 0.100113i
\(983\) 1.63420e7i 0.539412i 0.962943 + 0.269706i \(0.0869266\pi\)
−0.962943 + 0.269706i \(0.913073\pi\)
\(984\) 4.51104e6 0.148521
\(985\) 0 0
\(986\) 1.14440e7 0.374873
\(987\) 9.28358e6i 0.303335i
\(988\) 8.48848e6i 0.276654i
\(989\) 3.70214e6 0.120355
\(990\) 0 0
\(991\) 1.37576e7 0.444997 0.222498 0.974933i \(-0.428579\pi\)
0.222498 + 0.974933i \(0.428579\pi\)
\(992\) − 1.26121e7i − 0.406919i
\(993\) 1.23994e7i 0.399050i
\(994\) 1.25368e7 0.402459
\(995\) 0 0
\(996\) 1.87219e6 0.0598001
\(997\) − 1.29097e7i − 0.411320i −0.978624 0.205660i \(-0.934066\pi\)
0.978624 0.205660i \(-0.0659341\pi\)
\(998\) 1.29838e7i 0.412644i
\(999\) −342160. −0.0108471
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 25.6.b.a.24.2 2
3.2 odd 2 225.6.b.e.199.1 2
4.3 odd 2 400.6.c.j.49.1 2
5.2 odd 4 25.6.a.a.1.1 1
5.3 odd 4 5.6.a.a.1.1 1
5.4 even 2 inner 25.6.b.a.24.1 2
15.2 even 4 225.6.a.f.1.1 1
15.8 even 4 45.6.a.b.1.1 1
15.14 odd 2 225.6.b.e.199.2 2
20.3 even 4 80.6.a.e.1.1 1
20.7 even 4 400.6.a.g.1.1 1
20.19 odd 2 400.6.c.j.49.2 2
35.13 even 4 245.6.a.b.1.1 1
40.3 even 4 320.6.a.g.1.1 1
40.13 odd 4 320.6.a.j.1.1 1
55.43 even 4 605.6.a.a.1.1 1
60.23 odd 4 720.6.a.a.1.1 1
65.38 odd 4 845.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 5.3 odd 4
25.6.a.a.1.1 1 5.2 odd 4
25.6.b.a.24.1 2 5.4 even 2 inner
25.6.b.a.24.2 2 1.1 even 1 trivial
45.6.a.b.1.1 1 15.8 even 4
80.6.a.e.1.1 1 20.3 even 4
225.6.a.f.1.1 1 15.2 even 4
225.6.b.e.199.1 2 3.2 odd 2
225.6.b.e.199.2 2 15.14 odd 2
245.6.a.b.1.1 1 35.13 even 4
320.6.a.g.1.1 1 40.3 even 4
320.6.a.j.1.1 1 40.13 odd 4
400.6.a.g.1.1 1 20.7 even 4
400.6.c.j.49.1 2 4.3 odd 2
400.6.c.j.49.2 2 20.19 odd 2
605.6.a.a.1.1 1 55.43 even 4
720.6.a.a.1.1 1 60.23 odd 4
845.6.a.b.1.1 1 65.38 odd 4