Properties

Label 650.6.b.g.599.3
Level $650$
Weight $6$
Character 650.599
Analytic conductor $104.249$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.3
Root \(1.58114 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 650.599
Dual form 650.6.b.g.599.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000i q^{2} -17.8114i q^{3} -16.0000 q^{4} +71.2456 q^{6} -86.7544i q^{7} -64.0000i q^{8} -74.2456 q^{9} +O(q^{10})\) \(q+4.00000i q^{2} -17.8114i q^{3} -16.0000 q^{4} +71.2456 q^{6} -86.7544i q^{7} -64.0000i q^{8} -74.2456 q^{9} -203.285 q^{11} +284.982i q^{12} -169.000i q^{13} +347.018 q^{14} +256.000 q^{16} +406.605i q^{17} -296.982i q^{18} +2326.61 q^{19} -1545.22 q^{21} -813.139i q^{22} +3280.84i q^{23} -1139.93 q^{24} +676.000 q^{26} -3005.75i q^{27} +1388.07i q^{28} +2726.43 q^{29} +4914.62 q^{31} +1024.00i q^{32} +3620.78i q^{33} -1626.42 q^{34} +1187.93 q^{36} -3126.78i q^{37} +9306.43i q^{38} -3010.12 q^{39} +15061.9 q^{41} -6180.87i q^{42} +2794.35i q^{43} +3252.56 q^{44} -13123.4 q^{46} -16679.4i q^{47} -4559.72i q^{48} +9280.67 q^{49} +7242.20 q^{51} +2704.00i q^{52} +17262.9i q^{53} +12023.0 q^{54} -5552.28 q^{56} -41440.1i q^{57} +10905.7i q^{58} -2762.75 q^{59} -16849.0 q^{61} +19658.5i q^{62} +6441.13i q^{63} -4096.00 q^{64} -14483.1 q^{66} -19057.9i q^{67} -6505.68i q^{68} +58436.4 q^{69} +8326.03 q^{71} +4751.72i q^{72} +49498.2i q^{73} +12507.1 q^{74} -37225.7 q^{76} +17635.9i q^{77} -12040.5i q^{78} -95807.1 q^{79} -71578.3 q^{81} +60247.5i q^{82} +34154.5i q^{83} +24723.5 q^{84} -11177.4 q^{86} -48561.6i q^{87} +13010.2i q^{88} +26681.8 q^{89} -14661.5 q^{91} -52493.5i q^{92} -87536.3i q^{93} +66717.7 q^{94} +18238.9 q^{96} -24960.0i q^{97} +37122.7i q^{98} +15093.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{4} + 32 q^{6} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{4} + 32 q^{6} - 44 q^{9} + 768 q^{11} + 2400 q^{14} + 1024 q^{16} + 4816 q^{19} + 2800 q^{21} - 512 q^{24} + 2704 q^{26} - 7056 q^{29} + 1760 q^{31} - 1952 q^{34} + 704 q^{36} - 1352 q^{39} + 32040 q^{41} - 12288 q^{44} - 2656 q^{46} - 38772 q^{49} + 18976 q^{51} - 8576 q^{54} - 38400 q^{56} - 85744 q^{59} - 66384 q^{61} - 16384 q^{64} - 93856 q^{66} + 198328 q^{69} - 139040 q^{71} + 85952 q^{74} - 77056 q^{76} - 70416 q^{79} - 230404 q^{81} - 44800 q^{84} - 23712 q^{86} + 36904 q^{89} - 101400 q^{91} + 306336 q^{94} + 8192 q^{96} + 91552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000i 0.707107i
\(3\) − 17.8114i − 1.14260i −0.820741 0.571301i \(-0.806439\pi\)
0.820741 0.571301i \(-0.193561\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 71.2456 0.807941
\(7\) − 86.7544i − 0.669186i −0.942363 0.334593i \(-0.891401\pi\)
0.942363 0.334593i \(-0.108599\pi\)
\(8\) − 64.0000i − 0.353553i
\(9\) −74.2456 −0.305537
\(10\) 0 0
\(11\) −203.285 −0.506551 −0.253275 0.967394i \(-0.581508\pi\)
−0.253275 + 0.967394i \(0.581508\pi\)
\(12\) 284.982i 0.571301i
\(13\) − 169.000i − 0.277350i
\(14\) 347.018 0.473186
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 406.605i 0.341233i 0.985338 + 0.170616i \(0.0545758\pi\)
−0.985338 + 0.170616i \(0.945424\pi\)
\(18\) − 296.982i − 0.216047i
\(19\) 2326.61 1.47856 0.739281 0.673397i \(-0.235166\pi\)
0.739281 + 0.673397i \(0.235166\pi\)
\(20\) 0 0
\(21\) −1545.22 −0.764612
\(22\) − 813.139i − 0.358185i
\(23\) 3280.84i 1.29320i 0.762829 + 0.646600i \(0.223810\pi\)
−0.762829 + 0.646600i \(0.776190\pi\)
\(24\) −1139.93 −0.403970
\(25\) 0 0
\(26\) 676.000 0.196116
\(27\) − 3005.75i − 0.793494i
\(28\) 1388.07i 0.334593i
\(29\) 2726.43 0.602005 0.301002 0.953623i \(-0.402679\pi\)
0.301002 + 0.953623i \(0.402679\pi\)
\(30\) 0 0
\(31\) 4914.62 0.918514 0.459257 0.888303i \(-0.348116\pi\)
0.459257 + 0.888303i \(0.348116\pi\)
\(32\) 1024.00i 0.176777i
\(33\) 3620.78i 0.578785i
\(34\) −1626.42 −0.241288
\(35\) 0 0
\(36\) 1187.93 0.152769
\(37\) − 3126.78i − 0.375486i −0.982218 0.187743i \(-0.939883\pi\)
0.982218 0.187743i \(-0.0601172\pi\)
\(38\) 9306.43i 1.04550i
\(39\) −3010.12 −0.316901
\(40\) 0 0
\(41\) 15061.9 1.39933 0.699664 0.714472i \(-0.253333\pi\)
0.699664 + 0.714472i \(0.253333\pi\)
\(42\) − 6180.87i − 0.540663i
\(43\) 2794.35i 0.230467i 0.993338 + 0.115234i \(0.0367616\pi\)
−0.993338 + 0.115234i \(0.963238\pi\)
\(44\) 3252.56 0.253275
\(45\) 0 0
\(46\) −13123.4 −0.914431
\(47\) − 16679.4i − 1.10138i −0.834711 0.550689i \(-0.814365\pi\)
0.834711 0.550689i \(-0.185635\pi\)
\(48\) − 4559.72i − 0.285650i
\(49\) 9280.67 0.552191
\(50\) 0 0
\(51\) 7242.20 0.389893
\(52\) 2704.00i 0.138675i
\(53\) 17262.9i 0.844158i 0.906559 + 0.422079i \(0.138700\pi\)
−0.906559 + 0.422079i \(0.861300\pi\)
\(54\) 12023.0 0.561085
\(55\) 0 0
\(56\) −5552.28 −0.236593
\(57\) − 41440.1i − 1.68941i
\(58\) 10905.7i 0.425682i
\(59\) −2762.75 −0.103326 −0.0516632 0.998665i \(-0.516452\pi\)
−0.0516632 + 0.998665i \(0.516452\pi\)
\(60\) 0 0
\(61\) −16849.0 −0.579761 −0.289881 0.957063i \(-0.593616\pi\)
−0.289881 + 0.957063i \(0.593616\pi\)
\(62\) 19658.5i 0.649488i
\(63\) 6441.13i 0.204461i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) −14483.1 −0.409263
\(67\) − 19057.9i − 0.518665i −0.965788 0.259332i \(-0.916497\pi\)
0.965788 0.259332i \(-0.0835025\pi\)
\(68\) − 6505.68i − 0.170616i
\(69\) 58436.4 1.47761
\(70\) 0 0
\(71\) 8326.03 0.196016 0.0980082 0.995186i \(-0.468753\pi\)
0.0980082 + 0.995186i \(0.468753\pi\)
\(72\) 4751.72i 0.108024i
\(73\) 49498.2i 1.08713i 0.839366 + 0.543566i \(0.182926\pi\)
−0.839366 + 0.543566i \(0.817074\pi\)
\(74\) 12507.1 0.265508
\(75\) 0 0
\(76\) −37225.7 −0.739281
\(77\) 17635.9i 0.338977i
\(78\) − 12040.5i − 0.224083i
\(79\) −95807.1 −1.72715 −0.863575 0.504220i \(-0.831780\pi\)
−0.863575 + 0.504220i \(0.831780\pi\)
\(80\) 0 0
\(81\) −71578.3 −1.21218
\(82\) 60247.5i 0.989474i
\(83\) 34154.5i 0.544192i 0.962270 + 0.272096i \(0.0877168\pi\)
−0.962270 + 0.272096i \(0.912283\pi\)
\(84\) 24723.5 0.382306
\(85\) 0 0
\(86\) −11177.4 −0.162965
\(87\) − 48561.6i − 0.687851i
\(88\) 13010.2i 0.179093i
\(89\) 26681.8 0.357059 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(90\) 0 0
\(91\) −14661.5 −0.185599
\(92\) − 52493.5i − 0.646600i
\(93\) − 87536.3i − 1.04950i
\(94\) 66717.7 0.778792
\(95\) 0 0
\(96\) 18238.9 0.201985
\(97\) − 24960.0i − 0.269349i −0.990890 0.134674i \(-0.957001\pi\)
0.990890 0.134674i \(-0.0429988\pi\)
\(98\) 37122.7i 0.390458i
\(99\) 15093.0 0.154770
\(100\) 0 0
\(101\) 99719.9 0.972698 0.486349 0.873765i \(-0.338328\pi\)
0.486349 + 0.873765i \(0.338328\pi\)
\(102\) 28968.8i 0.275696i
\(103\) − 68985.8i − 0.640718i −0.947296 0.320359i \(-0.896197\pi\)
0.947296 0.320359i \(-0.103803\pi\)
\(104\) −10816.0 −0.0980581
\(105\) 0 0
\(106\) −69051.6 −0.596910
\(107\) − 120316.i − 1.01593i −0.861378 0.507964i \(-0.830398\pi\)
0.861378 0.507964i \(-0.169602\pi\)
\(108\) 48092.0i 0.396747i
\(109\) 173293. 1.39706 0.698529 0.715582i \(-0.253838\pi\)
0.698529 + 0.715582i \(0.253838\pi\)
\(110\) 0 0
\(111\) −55692.3 −0.429030
\(112\) − 22209.1i − 0.167296i
\(113\) − 95289.3i − 0.702018i −0.936372 0.351009i \(-0.885839\pi\)
0.936372 0.351009i \(-0.114161\pi\)
\(114\) 165761. 1.19459
\(115\) 0 0
\(116\) −43622.9 −0.301002
\(117\) 12547.5i 0.0847408i
\(118\) − 11051.0i − 0.0730629i
\(119\) 35274.8 0.228348
\(120\) 0 0
\(121\) −119726. −0.743406
\(122\) − 67395.9i − 0.409953i
\(123\) − 268273.i − 1.59887i
\(124\) −78634.0 −0.459257
\(125\) 0 0
\(126\) −25764.5 −0.144576
\(127\) − 70746.9i − 0.389223i −0.980880 0.194611i \(-0.937655\pi\)
0.980880 0.194611i \(-0.0623446\pi\)
\(128\) − 16384.0i − 0.0883883i
\(129\) 49771.2 0.263332
\(130\) 0 0
\(131\) 192442. 0.979764 0.489882 0.871789i \(-0.337040\pi\)
0.489882 + 0.871789i \(0.337040\pi\)
\(132\) − 57932.5i − 0.289393i
\(133\) − 201844.i − 0.989432i
\(134\) 76231.4 0.366752
\(135\) 0 0
\(136\) 26022.7 0.120644
\(137\) − 280120.i − 1.27509i −0.770411 0.637547i \(-0.779949\pi\)
0.770411 0.637547i \(-0.220051\pi\)
\(138\) 233746.i 1.04483i
\(139\) −331722. −1.45626 −0.728128 0.685441i \(-0.759609\pi\)
−0.728128 + 0.685441i \(0.759609\pi\)
\(140\) 0 0
\(141\) −297084. −1.25844
\(142\) 33304.1i 0.138604i
\(143\) 34355.1i 0.140492i
\(144\) −19006.9 −0.0763843
\(145\) 0 0
\(146\) −197993. −0.768718
\(147\) − 165302.i − 0.630933i
\(148\) 50028.5i 0.187743i
\(149\) −205133. −0.756956 −0.378478 0.925610i \(-0.623553\pi\)
−0.378478 + 0.925610i \(0.623553\pi\)
\(150\) 0 0
\(151\) 493417. 1.76105 0.880525 0.474000i \(-0.157190\pi\)
0.880525 + 0.474000i \(0.157190\pi\)
\(152\) − 148903.i − 0.522750i
\(153\) − 30188.6i − 0.104259i
\(154\) −70543.4 −0.239693
\(155\) 0 0
\(156\) 48162.0 0.158450
\(157\) − 299589.i − 0.970011i −0.874511 0.485006i \(-0.838818\pi\)
0.874511 0.485006i \(-0.161182\pi\)
\(158\) − 383229.i − 1.22128i
\(159\) 307476. 0.964536
\(160\) 0 0
\(161\) 284628. 0.865391
\(162\) − 286313.i − 0.857144i
\(163\) − 479059.i − 1.41228i −0.708073 0.706139i \(-0.750435\pi\)
0.708073 0.706139i \(-0.249565\pi\)
\(164\) −240990. −0.699664
\(165\) 0 0
\(166\) −136618. −0.384802
\(167\) − 388300.i − 1.07740i −0.842498 0.538699i \(-0.818916\pi\)
0.842498 0.538699i \(-0.181084\pi\)
\(168\) 98893.9i 0.270331i
\(169\) −28561.0 −0.0769231
\(170\) 0 0
\(171\) −172740. −0.451756
\(172\) − 44709.5i − 0.115234i
\(173\) 20482.9i 0.0520327i 0.999662 + 0.0260163i \(0.00828219\pi\)
−0.999662 + 0.0260163i \(0.991718\pi\)
\(174\) 194246. 0.486384
\(175\) 0 0
\(176\) −52040.9 −0.126638
\(177\) 49208.4i 0.118061i
\(178\) 106727.i 0.252479i
\(179\) −320609. −0.747898 −0.373949 0.927449i \(-0.621997\pi\)
−0.373949 + 0.927449i \(0.621997\pi\)
\(180\) 0 0
\(181\) −685416. −1.55510 −0.777549 0.628822i \(-0.783537\pi\)
−0.777549 + 0.628822i \(0.783537\pi\)
\(182\) − 58646.0i − 0.131238i
\(183\) 300104.i 0.662436i
\(184\) 209974. 0.457215
\(185\) 0 0
\(186\) 350145. 0.742105
\(187\) − 82656.6i − 0.172852i
\(188\) 266871.i 0.550689i
\(189\) −260762. −0.530995
\(190\) 0 0
\(191\) −139964. −0.277609 −0.138804 0.990320i \(-0.544326\pi\)
−0.138804 + 0.990320i \(0.544326\pi\)
\(192\) 72955.4i 0.142825i
\(193\) − 520732.i − 1.00628i −0.864203 0.503142i \(-0.832177\pi\)
0.864203 0.503142i \(-0.167823\pi\)
\(194\) 99839.9 0.190458
\(195\) 0 0
\(196\) −148491. −0.276095
\(197\) 408213.i 0.749414i 0.927143 + 0.374707i \(0.122257\pi\)
−0.927143 + 0.374707i \(0.877743\pi\)
\(198\) 60371.9i 0.109439i
\(199\) 853536. 1.52788 0.763940 0.645288i \(-0.223262\pi\)
0.763940 + 0.645288i \(0.223262\pi\)
\(200\) 0 0
\(201\) −339447. −0.592627
\(202\) 398879.i 0.687802i
\(203\) − 236530.i − 0.402853i
\(204\) −115875. −0.194946
\(205\) 0 0
\(206\) 275943. 0.453056
\(207\) − 243588.i − 0.395121i
\(208\) − 43264.0i − 0.0693375i
\(209\) −472964. −0.748966
\(210\) 0 0
\(211\) −277058. −0.428415 −0.214207 0.976788i \(-0.568717\pi\)
−0.214207 + 0.976788i \(0.568717\pi\)
\(212\) − 276206.i − 0.422079i
\(213\) − 148298.i − 0.223968i
\(214\) 481263. 0.718370
\(215\) 0 0
\(216\) −192368. −0.280542
\(217\) − 426365.i − 0.614657i
\(218\) 693171.i 0.987869i
\(219\) 881632. 1.24216
\(220\) 0 0
\(221\) 68716.2 0.0946409
\(222\) − 222769.i − 0.303370i
\(223\) − 1.40025e6i − 1.88557i −0.333403 0.942784i \(-0.608197\pi\)
0.333403 0.942784i \(-0.391803\pi\)
\(224\) 88836.6 0.118296
\(225\) 0 0
\(226\) 381157. 0.496402
\(227\) 114539.i 0.147533i 0.997276 + 0.0737666i \(0.0235020\pi\)
−0.997276 + 0.0737666i \(0.976498\pi\)
\(228\) 663042.i 0.844703i
\(229\) 501586. 0.632057 0.316029 0.948750i \(-0.397650\pi\)
0.316029 + 0.948750i \(0.397650\pi\)
\(230\) 0 0
\(231\) 314119. 0.387315
\(232\) − 174492.i − 0.212841i
\(233\) 147822.i 0.178381i 0.996015 + 0.0891904i \(0.0284280\pi\)
−0.996015 + 0.0891904i \(0.971572\pi\)
\(234\) −50190.0 −0.0599208
\(235\) 0 0
\(236\) 44204.0 0.0516632
\(237\) 1.70646e6i 1.97344i
\(238\) 141099.i 0.161466i
\(239\) 442991. 0.501649 0.250824 0.968033i \(-0.419298\pi\)
0.250824 + 0.968033i \(0.419298\pi\)
\(240\) 0 0
\(241\) 807221. 0.895262 0.447631 0.894218i \(-0.352268\pi\)
0.447631 + 0.894218i \(0.352268\pi\)
\(242\) − 478905.i − 0.525668i
\(243\) 544511.i 0.591549i
\(244\) 269584. 0.289881
\(245\) 0 0
\(246\) 1.07309e6 1.13057
\(247\) − 393197.i − 0.410079i
\(248\) − 314536.i − 0.324744i
\(249\) 608338. 0.621794
\(250\) 0 0
\(251\) 1.21143e6 1.21371 0.606853 0.794814i \(-0.292432\pi\)
0.606853 + 0.794814i \(0.292432\pi\)
\(252\) − 103058.i − 0.102231i
\(253\) − 666945.i − 0.655072i
\(254\) 282988. 0.275222
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) − 1.15116e6i − 1.08718i −0.839350 0.543592i \(-0.817064\pi\)
0.839350 0.543592i \(-0.182936\pi\)
\(258\) 199085.i 0.186204i
\(259\) −271262. −0.251270
\(260\) 0 0
\(261\) −202426. −0.183935
\(262\) 769768.i 0.692798i
\(263\) 930449.i 0.829475i 0.909941 + 0.414737i \(0.136127\pi\)
−0.909941 + 0.414737i \(0.863873\pi\)
\(264\) 231730. 0.204632
\(265\) 0 0
\(266\) 807375. 0.699634
\(267\) − 475239.i − 0.407976i
\(268\) 304926.i 0.259332i
\(269\) 2.08547e6 1.75721 0.878605 0.477549i \(-0.158475\pi\)
0.878605 + 0.477549i \(0.158475\pi\)
\(270\) 0 0
\(271\) −1.48375e6 −1.22726 −0.613632 0.789592i \(-0.710292\pi\)
−0.613632 + 0.789592i \(0.710292\pi\)
\(272\) 104091.i 0.0853081i
\(273\) 261142.i 0.212065i
\(274\) 1.12048e6 0.901628
\(275\) 0 0
\(276\) −934982. −0.738806
\(277\) 753280.i 0.589871i 0.955517 + 0.294935i \(0.0952982\pi\)
−0.955517 + 0.294935i \(0.904702\pi\)
\(278\) − 1.32689e6i − 1.02973i
\(279\) −364889. −0.280640
\(280\) 0 0
\(281\) −771041. −0.582521 −0.291261 0.956644i \(-0.594075\pi\)
−0.291261 + 0.956644i \(0.594075\pi\)
\(282\) − 1.18833e6i − 0.889848i
\(283\) 132716.i 0.0985047i 0.998786 + 0.0492523i \(0.0156839\pi\)
−0.998786 + 0.0492523i \(0.984316\pi\)
\(284\) −133217. −0.0980082
\(285\) 0 0
\(286\) −137420. −0.0993428
\(287\) − 1.30668e6i − 0.936410i
\(288\) − 76027.4i − 0.0540119i
\(289\) 1.25453e6 0.883560
\(290\) 0 0
\(291\) −444572. −0.307758
\(292\) − 791971.i − 0.543566i
\(293\) − 435111.i − 0.296095i −0.988980 0.148047i \(-0.952701\pi\)
0.988980 0.148047i \(-0.0472988\pi\)
\(294\) 661206. 0.446137
\(295\) 0 0
\(296\) −200114. −0.132754
\(297\) 611023.i 0.401945i
\(298\) − 820534.i − 0.535249i
\(299\) 554463. 0.358669
\(300\) 0 0
\(301\) 242422. 0.154225
\(302\) 1.97367e6i 1.24525i
\(303\) − 1.77615e6i − 1.11141i
\(304\) 595612. 0.369640
\(305\) 0 0
\(306\) 120754. 0.0737224
\(307\) 707726.i 0.428567i 0.976771 + 0.214284i \(0.0687417\pi\)
−0.976771 + 0.214284i \(0.931258\pi\)
\(308\) − 282174.i − 0.169488i
\(309\) −1.22873e6 −0.732085
\(310\) 0 0
\(311\) 2.19086e6 1.28444 0.642221 0.766519i \(-0.278013\pi\)
0.642221 + 0.766519i \(0.278013\pi\)
\(312\) 192648.i 0.112041i
\(313\) 1.16467e6i 0.671960i 0.941869 + 0.335980i \(0.109067\pi\)
−0.941869 + 0.335980i \(0.890933\pi\)
\(314\) 1.19836e6 0.685902
\(315\) 0 0
\(316\) 1.53291e6 0.863575
\(317\) 2.33420e6i 1.30464i 0.757946 + 0.652318i \(0.226203\pi\)
−0.757946 + 0.652318i \(0.773797\pi\)
\(318\) 1.22990e6i 0.682030i
\(319\) −554242. −0.304946
\(320\) 0 0
\(321\) −2.14299e6 −1.16080
\(322\) 1.13851e6i 0.611924i
\(323\) 946011.i 0.504533i
\(324\) 1.14525e6 0.606092
\(325\) 0 0
\(326\) 1.91624e6 0.998631
\(327\) − 3.08659e6i − 1.59628i
\(328\) − 963960.i − 0.494737i
\(329\) −1.44701e6 −0.737026
\(330\) 0 0
\(331\) 755890. 0.379218 0.189609 0.981860i \(-0.439278\pi\)
0.189609 + 0.981860i \(0.439278\pi\)
\(332\) − 546471.i − 0.272096i
\(333\) 232150.i 0.114725i
\(334\) 1.55320e6 0.761835
\(335\) 0 0
\(336\) −395576. −0.191153
\(337\) − 2.81171e6i − 1.34864i −0.738441 0.674319i \(-0.764437\pi\)
0.738441 0.674319i \(-0.235563\pi\)
\(338\) − 114244.i − 0.0543928i
\(339\) −1.69723e6 −0.802126
\(340\) 0 0
\(341\) −999068. −0.465274
\(342\) − 690961.i − 0.319439i
\(343\) − 2.26322e6i − 1.03870i
\(344\) 178838. 0.0814824
\(345\) 0 0
\(346\) −81931.6 −0.0367927
\(347\) − 1.36358e6i − 0.607935i −0.952682 0.303967i \(-0.901689\pi\)
0.952682 0.303967i \(-0.0983114\pi\)
\(348\) 776985.i 0.343926i
\(349\) −488343. −0.214616 −0.107308 0.994226i \(-0.534223\pi\)
−0.107308 + 0.994226i \(0.534223\pi\)
\(350\) 0 0
\(351\) −507972. −0.220076
\(352\) − 208164.i − 0.0895464i
\(353\) 1.45693e6i 0.622301i 0.950361 + 0.311151i \(0.100714\pi\)
−0.950361 + 0.311151i \(0.899286\pi\)
\(354\) −196834. −0.0834817
\(355\) 0 0
\(356\) −426908. −0.178529
\(357\) − 628293.i − 0.260911i
\(358\) − 1.28243e6i − 0.528844i
\(359\) −675187. −0.276495 −0.138248 0.990398i \(-0.544147\pi\)
−0.138248 + 0.990398i \(0.544147\pi\)
\(360\) 0 0
\(361\) 2.93701e6 1.18614
\(362\) − 2.74166e6i − 1.09962i
\(363\) 2.13249e6i 0.849417i
\(364\) 234584. 0.0927994
\(365\) 0 0
\(366\) −1.20042e6 −0.468413
\(367\) 623601.i 0.241680i 0.992672 + 0.120840i \(0.0385589\pi\)
−0.992672 + 0.120840i \(0.961441\pi\)
\(368\) 839896.i 0.323300i
\(369\) −1.11828e6 −0.427547
\(370\) 0 0
\(371\) 1.49763e6 0.564899
\(372\) 1.40058e6i 0.524748i
\(373\) 2.83653e6i 1.05564i 0.849357 + 0.527819i \(0.176990\pi\)
−0.849357 + 0.527819i \(0.823010\pi\)
\(374\) 330626. 0.122225
\(375\) 0 0
\(376\) −1.06748e6 −0.389396
\(377\) − 460767.i − 0.166966i
\(378\) − 1.04305e6i − 0.375470i
\(379\) −5.54522e6 −1.98299 −0.991495 0.130147i \(-0.958455\pi\)
−0.991495 + 0.130147i \(0.958455\pi\)
\(380\) 0 0
\(381\) −1.26010e6 −0.444726
\(382\) − 559856.i − 0.196299i
\(383\) 3.71036e6i 1.29247i 0.763140 + 0.646234i \(0.223657\pi\)
−0.763140 + 0.646234i \(0.776343\pi\)
\(384\) −291822. −0.100993
\(385\) 0 0
\(386\) 2.08293e6 0.711551
\(387\) − 207468.i − 0.0704163i
\(388\) 399360.i 0.134674i
\(389\) −4.96384e6 −1.66320 −0.831599 0.555377i \(-0.812574\pi\)
−0.831599 + 0.555377i \(0.812574\pi\)
\(390\) 0 0
\(391\) −1.33401e6 −0.441282
\(392\) − 593963.i − 0.195229i
\(393\) − 3.42766e6i − 1.11948i
\(394\) −1.63285e6 −0.529916
\(395\) 0 0
\(396\) −241488. −0.0773851
\(397\) 3.75615e6i 1.19610i 0.801459 + 0.598049i \(0.204057\pi\)
−0.801459 + 0.598049i \(0.795943\pi\)
\(398\) 3.41414e6i 1.08037i
\(399\) −3.59512e6 −1.13053
\(400\) 0 0
\(401\) −3.20829e6 −0.996352 −0.498176 0.867076i \(-0.665997\pi\)
−0.498176 + 0.867076i \(0.665997\pi\)
\(402\) − 1.35779e6i − 0.419051i
\(403\) − 830571.i − 0.254750i
\(404\) −1.59552e6 −0.486349
\(405\) 0 0
\(406\) 946121. 0.284860
\(407\) 635627.i 0.190203i
\(408\) − 463501.i − 0.137848i
\(409\) −4.11220e6 −1.21553 −0.607765 0.794117i \(-0.707934\pi\)
−0.607765 + 0.794117i \(0.707934\pi\)
\(410\) 0 0
\(411\) −4.98932e6 −1.45692
\(412\) 1.10377e6i 0.320359i
\(413\) 239681.i 0.0691446i
\(414\) 974352. 0.279393
\(415\) 0 0
\(416\) 173056. 0.0490290
\(417\) 5.90844e6i 1.66392i
\(418\) − 1.89186e6i − 0.529599i
\(419\) 1.21350e6 0.337680 0.168840 0.985643i \(-0.445998\pi\)
0.168840 + 0.985643i \(0.445998\pi\)
\(420\) 0 0
\(421\) 500559. 0.137642 0.0688208 0.997629i \(-0.478076\pi\)
0.0688208 + 0.997629i \(0.478076\pi\)
\(422\) − 1.10823e6i − 0.302935i
\(423\) 1.23837e6i 0.336512i
\(424\) 1.10482e6 0.298455
\(425\) 0 0
\(426\) 593193. 0.158370
\(427\) 1.46172e6i 0.387968i
\(428\) 1.92505e6i 0.507964i
\(429\) 611912. 0.160526
\(430\) 0 0
\(431\) −18753.3 −0.00486279 −0.00243139 0.999997i \(-0.500774\pi\)
−0.00243139 + 0.999997i \(0.500774\pi\)
\(432\) − 769472.i − 0.198373i
\(433\) − 1.10350e6i − 0.282849i −0.989949 0.141424i \(-0.954832\pi\)
0.989949 0.141424i \(-0.0451682\pi\)
\(434\) 1.70546e6 0.434628
\(435\) 0 0
\(436\) −2.77269e6 −0.698529
\(437\) 7.63324e6i 1.91208i
\(438\) 3.52653e6i 0.878338i
\(439\) −3.67024e6 −0.908935 −0.454467 0.890763i \(-0.650170\pi\)
−0.454467 + 0.890763i \(0.650170\pi\)
\(440\) 0 0
\(441\) −689048. −0.168715
\(442\) 274865.i 0.0669212i
\(443\) − 4.19972e6i − 1.01674i −0.861138 0.508371i \(-0.830248\pi\)
0.861138 0.508371i \(-0.169752\pi\)
\(444\) 891077. 0.214515
\(445\) 0 0
\(446\) 5.60099e6 1.33330
\(447\) 3.65371e6i 0.864899i
\(448\) 355346.i 0.0836482i
\(449\) −2.61245e6 −0.611551 −0.305775 0.952104i \(-0.598916\pi\)
−0.305775 + 0.952104i \(0.598916\pi\)
\(450\) 0 0
\(451\) −3.06185e6 −0.708831
\(452\) 1.52463e6i 0.351009i
\(453\) − 8.78844e6i − 2.01218i
\(454\) −458157. −0.104322
\(455\) 0 0
\(456\) −2.65217e6 −0.597295
\(457\) 5.86700e6i 1.31409i 0.753851 + 0.657046i \(0.228194\pi\)
−0.753851 + 0.657046i \(0.771806\pi\)
\(458\) 2.00634e6i 0.446932i
\(459\) 1.22215e6 0.270766
\(460\) 0 0
\(461\) −95542.2 −0.0209384 −0.0104692 0.999945i \(-0.503333\pi\)
−0.0104692 + 0.999945i \(0.503333\pi\)
\(462\) 1.25648e6i 0.273873i
\(463\) − 3.43498e6i − 0.744682i −0.928096 0.372341i \(-0.878555\pi\)
0.928096 0.372341i \(-0.121445\pi\)
\(464\) 697967. 0.150501
\(465\) 0 0
\(466\) −591286. −0.126134
\(467\) − 4.27512e6i − 0.907102i −0.891230 0.453551i \(-0.850157\pi\)
0.891230 0.453551i \(-0.149843\pi\)
\(468\) − 200760.i − 0.0423704i
\(469\) −1.65335e6 −0.347083
\(470\) 0 0
\(471\) −5.33610e6 −1.10834
\(472\) 176816.i 0.0365314i
\(473\) − 568048.i − 0.116743i
\(474\) −6.82583e6 −1.39544
\(475\) 0 0
\(476\) −564397. −0.114174
\(477\) − 1.28169e6i − 0.257922i
\(478\) 1.77196e6i 0.354719i
\(479\) 331105. 0.0659367 0.0329683 0.999456i \(-0.489504\pi\)
0.0329683 + 0.999456i \(0.489504\pi\)
\(480\) 0 0
\(481\) −528426. −0.104141
\(482\) 3.22888e6i 0.633046i
\(483\) − 5.06962e6i − 0.988797i
\(484\) 1.91562e6 0.371703
\(485\) 0 0
\(486\) −2.17804e6 −0.418288
\(487\) − 9.33331e6i − 1.78326i −0.452769 0.891628i \(-0.649564\pi\)
0.452769 0.891628i \(-0.350436\pi\)
\(488\) 1.07833e6i 0.204977i
\(489\) −8.53271e6 −1.61367
\(490\) 0 0
\(491\) 5.20484e6 0.974325 0.487162 0.873311i \(-0.338032\pi\)
0.487162 + 0.873311i \(0.338032\pi\)
\(492\) 4.29237e6i 0.799437i
\(493\) 1.10858e6i 0.205424i
\(494\) 1.57279e6 0.289970
\(495\) 0 0
\(496\) 1.25814e6 0.229629
\(497\) − 722320.i − 0.131171i
\(498\) 2.43335e6i 0.439675i
\(499\) 9.51149e6 1.71000 0.855002 0.518624i \(-0.173556\pi\)
0.855002 + 0.518624i \(0.173556\pi\)
\(500\) 0 0
\(501\) −6.91616e6 −1.23104
\(502\) 4.84572e6i 0.858220i
\(503\) 2.34504e6i 0.413267i 0.978418 + 0.206633i \(0.0662507\pi\)
−0.978418 + 0.206633i \(0.933749\pi\)
\(504\) 412232. 0.0722879
\(505\) 0 0
\(506\) 2.66778e6 0.463206
\(507\) 508711.i 0.0878924i
\(508\) 1.13195e6i 0.194611i
\(509\) −1.04128e6 −0.178144 −0.0890720 0.996025i \(-0.528390\pi\)
−0.0890720 + 0.996025i \(0.528390\pi\)
\(510\) 0 0
\(511\) 4.29419e6 0.727493
\(512\) 262144.i 0.0441942i
\(513\) − 6.99321e6i − 1.17323i
\(514\) 4.60464e6 0.768755
\(515\) 0 0
\(516\) −796339. −0.131666
\(517\) 3.39067e6i 0.557904i
\(518\) − 1.08505e6i − 0.177674i
\(519\) 364829. 0.0594526
\(520\) 0 0
\(521\) −1.07451e6 −0.173427 −0.0867135 0.996233i \(-0.527636\pi\)
−0.0867135 + 0.996233i \(0.527636\pi\)
\(522\) − 809702.i − 0.130062i
\(523\) 2.30085e6i 0.367819i 0.982943 + 0.183910i \(0.0588754\pi\)
−0.982943 + 0.183910i \(0.941125\pi\)
\(524\) −3.07907e6 −0.489882
\(525\) 0 0
\(526\) −3.72180e6 −0.586527
\(527\) 1.99831e6i 0.313427i
\(528\) 926920.i 0.144696i
\(529\) −4.32759e6 −0.672368
\(530\) 0 0
\(531\) 205122. 0.0315701
\(532\) 3.22950e6i 0.494716i
\(533\) − 2.54546e6i − 0.388104i
\(534\) 1.90096e6 0.288482
\(535\) 0 0
\(536\) −1.21970e6 −0.183376
\(537\) 5.71048e6i 0.854549i
\(538\) 8.34189e6i 1.24254i
\(539\) −1.88662e6 −0.279713
\(540\) 0 0
\(541\) −9.85493e6 −1.44764 −0.723819 0.689989i \(-0.757615\pi\)
−0.723819 + 0.689989i \(0.757615\pi\)
\(542\) − 5.93501e6i − 0.867807i
\(543\) 1.22082e7i 1.77686i
\(544\) −416364. −0.0603220
\(545\) 0 0
\(546\) −1.04457e6 −0.149953
\(547\) − 4.90591e6i − 0.701054i −0.936553 0.350527i \(-0.886002\pi\)
0.936553 0.350527i \(-0.113998\pi\)
\(548\) 4.48192e6i 0.637547i
\(549\) 1.25096e6 0.177139
\(550\) 0 0
\(551\) 6.34335e6 0.890101
\(552\) − 3.73993e6i − 0.522415i
\(553\) 8.31169e6i 1.15578i
\(554\) −3.01312e6 −0.417102
\(555\) 0 0
\(556\) 5.30756e6 0.728128
\(557\) 1.82826e6i 0.249689i 0.992176 + 0.124845i \(0.0398432\pi\)
−0.992176 + 0.124845i \(0.960157\pi\)
\(558\) − 1.45956e6i − 0.198443i
\(559\) 472244. 0.0639201
\(560\) 0 0
\(561\) −1.47223e6 −0.197500
\(562\) − 3.08417e6i − 0.411905i
\(563\) − 8.05618e6i − 1.07117i −0.844481 0.535585i \(-0.820091\pi\)
0.844481 0.535585i \(-0.179909\pi\)
\(564\) 4.75334e6 0.629218
\(565\) 0 0
\(566\) −530864. −0.0696533
\(567\) 6.20973e6i 0.811176i
\(568\) − 532866.i − 0.0693022i
\(569\) −8.62680e6 −1.11704 −0.558521 0.829491i \(-0.688631\pi\)
−0.558521 + 0.829491i \(0.688631\pi\)
\(570\) 0 0
\(571\) 8.14626e6 1.04561 0.522803 0.852454i \(-0.324886\pi\)
0.522803 + 0.852454i \(0.324886\pi\)
\(572\) − 549682.i − 0.0702460i
\(573\) 2.49295e6i 0.317196i
\(574\) 5.22674e6 0.662142
\(575\) 0 0
\(576\) 304110. 0.0381922
\(577\) − 6.35136e6i − 0.794195i −0.917777 0.397097i \(-0.870018\pi\)
0.917777 0.397097i \(-0.129982\pi\)
\(578\) 5.01812e6i 0.624772i
\(579\) −9.27496e6 −1.14978
\(580\) 0 0
\(581\) 2.96305e6 0.364165
\(582\) − 1.77829e6i − 0.217618i
\(583\) − 3.50928e6i − 0.427609i
\(584\) 3.16788e6 0.384359
\(585\) 0 0
\(586\) 1.74044e6 0.209371
\(587\) 1.42583e7i 1.70794i 0.520324 + 0.853969i \(0.325811\pi\)
−0.520324 + 0.853969i \(0.674189\pi\)
\(588\) 2.64482e6i 0.315467i
\(589\) 1.14344e7 1.35808
\(590\) 0 0
\(591\) 7.27085e6 0.856281
\(592\) − 800456.i − 0.0938714i
\(593\) 5.76780e6i 0.673556i 0.941584 + 0.336778i \(0.109337\pi\)
−0.941584 + 0.336778i \(0.890663\pi\)
\(594\) −2.44409e6 −0.284218
\(595\) 0 0
\(596\) 3.28214e6 0.378478
\(597\) − 1.52027e7i − 1.74576i
\(598\) 2.21785e6i 0.253618i
\(599\) 1.08344e7 1.23378 0.616891 0.787049i \(-0.288392\pi\)
0.616891 + 0.787049i \(0.288392\pi\)
\(600\) 0 0
\(601\) 1.23326e7 1.39273 0.696366 0.717687i \(-0.254799\pi\)
0.696366 + 0.717687i \(0.254799\pi\)
\(602\) 969687.i 0.109054i
\(603\) 1.41496e6i 0.158471i
\(604\) −7.89467e6 −0.880525
\(605\) 0 0
\(606\) 7.10460e6 0.785883
\(607\) − 4.32462e6i − 0.476405i −0.971215 0.238203i \(-0.923442\pi\)
0.971215 0.238203i \(-0.0765582\pi\)
\(608\) 2.38245e6i 0.261375i
\(609\) −4.21293e6 −0.460300
\(610\) 0 0
\(611\) −2.81882e6 −0.305467
\(612\) 483018.i 0.0521296i
\(613\) − 1.19846e7i − 1.28816i −0.764957 0.644082i \(-0.777240\pi\)
0.764957 0.644082i \(-0.222760\pi\)
\(614\) −2.83090e6 −0.303043
\(615\) 0 0
\(616\) 1.12869e6 0.119846
\(617\) 8.12346e6i 0.859069i 0.903050 + 0.429535i \(0.141322\pi\)
−0.903050 + 0.429535i \(0.858678\pi\)
\(618\) − 4.91493e6i − 0.517662i
\(619\) 6.07174e6 0.636923 0.318461 0.947936i \(-0.396834\pi\)
0.318461 + 0.947936i \(0.396834\pi\)
\(620\) 0 0
\(621\) 9.86140e6 1.02615
\(622\) 8.76346e6i 0.908238i
\(623\) − 2.31476e6i − 0.238939i
\(624\) −770592. −0.0792251
\(625\) 0 0
\(626\) −4.65869e6 −0.475147
\(627\) 8.42414e6i 0.855770i
\(628\) 4.79343e6i 0.485006i
\(629\) 1.27137e6 0.128128
\(630\) 0 0
\(631\) 7.56443e6 0.756315 0.378158 0.925741i \(-0.376558\pi\)
0.378158 + 0.925741i \(0.376558\pi\)
\(632\) 6.13166e6i 0.610640i
\(633\) 4.93479e6i 0.489507i
\(634\) −9.33679e6 −0.922517
\(635\) 0 0
\(636\) −4.91962e6 −0.482268
\(637\) − 1.56843e6i − 0.153150i
\(638\) − 2.21697e6i − 0.215629i
\(639\) −618171. −0.0598903
\(640\) 0 0
\(641\) 1.04819e7 1.00761 0.503806 0.863817i \(-0.331933\pi\)
0.503806 + 0.863817i \(0.331933\pi\)
\(642\) − 8.57196e6i − 0.820810i
\(643\) 52467.4i 0.00500452i 0.999997 + 0.00250226i \(0.000796494\pi\)
−0.999997 + 0.00250226i \(0.999204\pi\)
\(644\) −4.55404e6 −0.432696
\(645\) 0 0
\(646\) −3.78404e6 −0.356759
\(647\) 1.90086e7i 1.78521i 0.450839 + 0.892605i \(0.351125\pi\)
−0.450839 + 0.892605i \(0.648875\pi\)
\(648\) 4.58101e6i 0.428572i
\(649\) 561625. 0.0523401
\(650\) 0 0
\(651\) −7.59416e6 −0.702307
\(652\) 7.66495e6i 0.706139i
\(653\) − 1.41889e7i − 1.30216i −0.759007 0.651082i \(-0.774315\pi\)
0.759007 0.651082i \(-0.225685\pi\)
\(654\) 1.23463e7 1.12874
\(655\) 0 0
\(656\) 3.85584e6 0.349832
\(657\) − 3.67502e6i − 0.332159i
\(658\) − 5.78806e6i − 0.521156i
\(659\) 1.56797e7 1.40645 0.703226 0.710966i \(-0.251742\pi\)
0.703226 + 0.710966i \(0.251742\pi\)
\(660\) 0 0
\(661\) 1.59619e7 1.42095 0.710477 0.703721i \(-0.248479\pi\)
0.710477 + 0.703721i \(0.248479\pi\)
\(662\) 3.02356e6i 0.268148i
\(663\) − 1.22393e6i − 0.108137i
\(664\) 2.18589e6 0.192401
\(665\) 0 0
\(666\) −928599. −0.0811227
\(667\) 8.94500e6i 0.778513i
\(668\) 6.21280e6i 0.538699i
\(669\) −2.49403e7 −2.15445
\(670\) 0 0
\(671\) 3.42514e6 0.293678
\(672\) − 1.58230e6i − 0.135166i
\(673\) 2.19109e7i 1.86475i 0.361487 + 0.932377i \(0.382269\pi\)
−0.361487 + 0.932377i \(0.617731\pi\)
\(674\) 1.12468e7 0.953631
\(675\) 0 0
\(676\) 456976. 0.0384615
\(677\) − 9.11801e6i − 0.764590i −0.924040 0.382295i \(-0.875134\pi\)
0.924040 0.382295i \(-0.124866\pi\)
\(678\) − 6.78894e6i − 0.567189i
\(679\) −2.16539e6 −0.180244
\(680\) 0 0
\(681\) 2.04010e6 0.168572
\(682\) − 3.99627e6i − 0.328999i
\(683\) 1.57797e7i 1.29434i 0.762346 + 0.647170i \(0.224047\pi\)
−0.762346 + 0.647170i \(0.775953\pi\)
\(684\) 2.76385e6 0.225878
\(685\) 0 0
\(686\) 9.05288e6 0.734474
\(687\) − 8.93394e6i − 0.722189i
\(688\) 715352.i 0.0576168i
\(689\) 2.91743e6 0.234127
\(690\) 0 0
\(691\) −1.32527e7 −1.05587 −0.527935 0.849285i \(-0.677033\pi\)
−0.527935 + 0.849285i \(0.677033\pi\)
\(692\) − 327726.i − 0.0260163i
\(693\) − 1.30938e6i − 0.103570i
\(694\) 5.45432e6 0.429875
\(695\) 0 0
\(696\) −3.10794e6 −0.243192
\(697\) 6.12424e6i 0.477496i
\(698\) − 1.95337e6i − 0.151756i
\(699\) 2.63291e6 0.203818
\(700\) 0 0
\(701\) 5.84764e6 0.449455 0.224727 0.974422i \(-0.427851\pi\)
0.224727 + 0.974422i \(0.427851\pi\)
\(702\) − 2.03189e6i − 0.155617i
\(703\) − 7.27480e6i − 0.555179i
\(704\) 832654. 0.0633188
\(705\) 0 0
\(706\) −5.82770e6 −0.440033
\(707\) − 8.65114e6i − 0.650916i
\(708\) − 787335.i − 0.0590305i
\(709\) 8.45478e6 0.631665 0.315833 0.948815i \(-0.397716\pi\)
0.315833 + 0.948815i \(0.397716\pi\)
\(710\) 0 0
\(711\) 7.11325e6 0.527709
\(712\) − 1.70763e6i − 0.126239i
\(713\) 1.61241e7i 1.18782i
\(714\) 2.51317e6 0.184492
\(715\) 0 0
\(716\) 5.12974e6 0.373949
\(717\) − 7.89028e6i − 0.573184i
\(718\) − 2.70075e6i − 0.195512i
\(719\) 9.16187e6 0.660940 0.330470 0.943817i \(-0.392793\pi\)
0.330470 + 0.943817i \(0.392793\pi\)
\(720\) 0 0
\(721\) −5.98482e6 −0.428759
\(722\) 1.17480e7i 0.838730i
\(723\) − 1.43777e7i − 1.02293i
\(724\) 1.09667e7 0.777549
\(725\) 0 0
\(726\) −8.52997e6 −0.600628
\(727\) − 2.08394e7i − 1.46234i −0.682195 0.731170i \(-0.738974\pi\)
0.682195 0.731170i \(-0.261026\pi\)
\(728\) 938336.i 0.0656191i
\(729\) −7.69503e6 −0.536280
\(730\) 0 0
\(731\) −1.13619e6 −0.0786429
\(732\) − 4.80166e6i − 0.331218i
\(733\) 1.13361e7i 0.779298i 0.920964 + 0.389649i \(0.127404\pi\)
−0.920964 + 0.389649i \(0.872596\pi\)
\(734\) −2.49440e6 −0.170894
\(735\) 0 0
\(736\) −3.35958e6 −0.228608
\(737\) 3.87417e6i 0.262730i
\(738\) − 4.47311e6i − 0.302321i
\(739\) −6.74529e6 −0.454349 −0.227174 0.973854i \(-0.572949\pi\)
−0.227174 + 0.973854i \(0.572949\pi\)
\(740\) 0 0
\(741\) −7.00338e6 −0.468557
\(742\) 5.99053e6i 0.399444i
\(743\) − 1.63250e7i − 1.08488i −0.840094 0.542440i \(-0.817501\pi\)
0.840094 0.542440i \(-0.182499\pi\)
\(744\) −5.60232e6 −0.371053
\(745\) 0 0
\(746\) −1.13461e7 −0.746449
\(747\) − 2.53582e6i − 0.166271i
\(748\) 1.32251e6i 0.0864258i
\(749\) −1.04379e7 −0.679845
\(750\) 0 0
\(751\) −2.54751e7 −1.64822 −0.824110 0.566430i \(-0.808324\pi\)
−0.824110 + 0.566430i \(0.808324\pi\)
\(752\) − 4.26993e6i − 0.275344i
\(753\) − 2.15772e7i − 1.38678i
\(754\) 1.84307e6 0.118063
\(755\) 0 0
\(756\) 4.17220e6 0.265497
\(757\) 2.49077e6i 0.157977i 0.996876 + 0.0789886i \(0.0251691\pi\)
−0.996876 + 0.0789886i \(0.974831\pi\)
\(758\) − 2.21809e7i − 1.40219i
\(759\) −1.18792e7 −0.748486
\(760\) 0 0
\(761\) 6.96708e6 0.436103 0.218052 0.975937i \(-0.430030\pi\)
0.218052 + 0.975937i \(0.430030\pi\)
\(762\) − 5.04040e6i − 0.314469i
\(763\) − 1.50339e7i − 0.934891i
\(764\) 2.23942e6 0.138804
\(765\) 0 0
\(766\) −1.48415e7 −0.913913
\(767\) 466905.i 0.0286576i
\(768\) − 1.16729e6i − 0.0714126i
\(769\) −6.80390e6 −0.414899 −0.207449 0.978246i \(-0.566516\pi\)
−0.207449 + 0.978246i \(0.566516\pi\)
\(770\) 0 0
\(771\) −2.05038e7 −1.24222
\(772\) 8.33171e6i 0.503142i
\(773\) 2.04878e7i 1.23324i 0.787262 + 0.616619i \(0.211498\pi\)
−0.787262 + 0.616619i \(0.788502\pi\)
\(774\) 829871. 0.0497918
\(775\) 0 0
\(776\) −1.59744e6 −0.0952291
\(777\) 4.83156e6i 0.287101i
\(778\) − 1.98554e7i − 1.17606i
\(779\) 3.50431e7 2.06899
\(780\) 0 0
\(781\) −1.69256e6 −0.0992922
\(782\) − 5.33603e6i − 0.312034i
\(783\) − 8.19498e6i − 0.477687i
\(784\) 2.37585e6 0.138048
\(785\) 0 0
\(786\) 1.37106e7 0.791591
\(787\) 1.89888e7i 1.09285i 0.837508 + 0.546425i \(0.184012\pi\)
−0.837508 + 0.546425i \(0.815988\pi\)
\(788\) − 6.53142e6i − 0.374707i
\(789\) 1.65726e7 0.947759
\(790\) 0 0
\(791\) −8.26677e6 −0.469780
\(792\) − 965951.i − 0.0547195i
\(793\) 2.84748e6i 0.160797i
\(794\) −1.50246e7 −0.845769
\(795\) 0 0
\(796\) −1.36566e7 −0.763940
\(797\) − 2.20925e7i − 1.23197i −0.787759 0.615983i \(-0.788759\pi\)
0.787759 0.615983i \(-0.211241\pi\)
\(798\) − 1.43805e7i − 0.799403i
\(799\) 6.78194e6 0.375826
\(800\) 0 0
\(801\) −1.98100e6 −0.109095
\(802\) − 1.28332e7i − 0.704527i
\(803\) − 1.00622e7i − 0.550688i
\(804\) 5.43115e6 0.296314
\(805\) 0 0
\(806\) 3.32229e6 0.180135
\(807\) − 3.71452e7i − 2.00779i
\(808\) − 6.38207e6i − 0.343901i
\(809\) 3.46398e7 1.86082 0.930410 0.366521i \(-0.119451\pi\)
0.930410 + 0.366521i \(0.119451\pi\)
\(810\) 0 0
\(811\) 8.39750e6 0.448330 0.224165 0.974551i \(-0.428035\pi\)
0.224165 + 0.974551i \(0.428035\pi\)
\(812\) 3.78448e6i 0.201427i
\(813\) 2.64277e7i 1.40227i
\(814\) −2.54251e6 −0.134494
\(815\) 0 0
\(816\) 1.85400e6 0.0974732
\(817\) 6.50135e6i 0.340760i
\(818\) − 1.64488e7i − 0.859509i
\(819\) 1.08855e6 0.0567073
\(820\) 0 0
\(821\) −1.98105e7 −1.02574 −0.512869 0.858467i \(-0.671418\pi\)
−0.512869 + 0.858467i \(0.671418\pi\)
\(822\) − 1.99573e7i − 1.03020i
\(823\) 336504.i 0.0173177i 0.999963 + 0.00865886i \(0.00275624\pi\)
−0.999963 + 0.00865886i \(0.997244\pi\)
\(824\) −4.41509e6 −0.226528
\(825\) 0 0
\(826\) −958724. −0.0488926
\(827\) 1.02845e7i 0.522903i 0.965217 + 0.261451i \(0.0842011\pi\)
−0.965217 + 0.261451i \(0.915799\pi\)
\(828\) 3.89741e6i 0.197560i
\(829\) −3.43440e7 −1.73566 −0.867829 0.496863i \(-0.834485\pi\)
−0.867829 + 0.496863i \(0.834485\pi\)
\(830\) 0 0
\(831\) 1.34170e7 0.673987
\(832\) 692224.i 0.0346688i
\(833\) 3.77357e6i 0.188425i
\(834\) −2.36337e7 −1.17657
\(835\) 0 0
\(836\) 7.56742e6 0.374483
\(837\) − 1.47721e7i − 0.728836i
\(838\) 4.85401e6i 0.238776i
\(839\) 1.00761e7 0.494185 0.247092 0.968992i \(-0.420525\pi\)
0.247092 + 0.968992i \(0.420525\pi\)
\(840\) 0 0
\(841\) −1.30777e7 −0.637590
\(842\) 2.00224e6i 0.0973273i
\(843\) 1.37333e7i 0.665590i
\(844\) 4.43293e6 0.214207
\(845\) 0 0
\(846\) −4.95349e6 −0.237950
\(847\) 1.03868e7i 0.497477i
\(848\) 4.41930e6i 0.211040i
\(849\) 2.36385e6 0.112552
\(850\) 0 0
\(851\) 1.02585e7 0.485578
\(852\) 2.37277e6i 0.111984i
\(853\) 3.71613e7i 1.74871i 0.485286 + 0.874355i \(0.338715\pi\)
−0.485286 + 0.874355i \(0.661285\pi\)
\(854\) −5.84690e6 −0.274335
\(855\) 0 0
\(856\) −7.70021e6 −0.359185
\(857\) − 3.89491e7i − 1.81153i −0.423783 0.905764i \(-0.639298\pi\)
0.423783 0.905764i \(-0.360702\pi\)
\(858\) 2.44765e6i 0.113509i
\(859\) 2.13532e7 0.987371 0.493685 0.869641i \(-0.335649\pi\)
0.493685 + 0.869641i \(0.335649\pi\)
\(860\) 0 0
\(861\) −2.32739e7 −1.06994
\(862\) − 75013.4i − 0.00343851i
\(863\) − 6.49502e6i − 0.296861i −0.988923 0.148431i \(-0.952578\pi\)
0.988923 0.148431i \(-0.0474222\pi\)
\(864\) 3.07789e6 0.140271
\(865\) 0 0
\(866\) 4.41402e6 0.200004
\(867\) − 2.23449e7i − 1.00956i
\(868\) 6.82185e6i 0.307328i
\(869\) 1.94761e7 0.874889
\(870\) 0 0
\(871\) −3.22078e6 −0.143852
\(872\) − 1.10907e7i − 0.493934i
\(873\) 1.85317e6i 0.0822961i
\(874\) −3.05330e7 −1.35204
\(875\) 0 0
\(876\) −1.41061e7 −0.621079
\(877\) 8.33527e6i 0.365949i 0.983118 + 0.182975i \(0.0585726\pi\)
−0.983118 + 0.182975i \(0.941427\pi\)
\(878\) − 1.46809e7i − 0.642714i
\(879\) −7.74993e6 −0.338318
\(880\) 0 0
\(881\) −4.91817e6 −0.213483 −0.106742 0.994287i \(-0.534042\pi\)
−0.106742 + 0.994287i \(0.534042\pi\)
\(882\) − 2.75619e6i − 0.119299i
\(883\) − 1.93128e7i − 0.833571i −0.909005 0.416786i \(-0.863157\pi\)
0.909005 0.416786i \(-0.136843\pi\)
\(884\) −1.09946e6 −0.0473204
\(885\) 0 0
\(886\) 1.67989e7 0.718945
\(887\) − 1.36071e7i − 0.580708i −0.956919 0.290354i \(-0.906227\pi\)
0.956919 0.290354i \(-0.0937731\pi\)
\(888\) 3.56431e6i 0.151685i
\(889\) −6.13761e6 −0.260462
\(890\) 0 0
\(891\) 1.45508e7 0.614033
\(892\) 2.24039e7i 0.942784i
\(893\) − 3.88065e7i − 1.62845i
\(894\) −1.46148e7 −0.611576
\(895\) 0 0
\(896\) −1.42138e6 −0.0591482
\(897\) − 9.87575e6i − 0.409816i
\(898\) − 1.04498e7i − 0.432432i
\(899\) 1.33994e7 0.552950
\(900\) 0 0
\(901\) −7.01918e6 −0.288054
\(902\) − 1.22474e7i − 0.501219i
\(903\) − 4.31787e6i − 0.176218i
\(904\) −6.09852e6 −0.248201
\(905\) 0 0
\(906\) 3.51538e7 1.42282
\(907\) 6.88886e6i 0.278054i 0.990289 + 0.139027i \(0.0443975\pi\)
−0.990289 + 0.139027i \(0.955603\pi\)
\(908\) − 1.83263e6i − 0.0737666i
\(909\) −7.40376e6 −0.297196
\(910\) 0 0
\(911\) −4.68632e7 −1.87084 −0.935418 0.353544i \(-0.884976\pi\)
−0.935418 + 0.353544i \(0.884976\pi\)
\(912\) − 1.06087e7i − 0.422351i
\(913\) − 6.94308e6i − 0.275661i
\(914\) −2.34680e7 −0.929203
\(915\) 0 0
\(916\) −8.02537e6 −0.316029
\(917\) − 1.66952e7i − 0.655644i
\(918\) 4.88861e6i 0.191460i
\(919\) 1.51786e7 0.592849 0.296425 0.955056i \(-0.404206\pi\)
0.296425 + 0.955056i \(0.404206\pi\)
\(920\) 0 0
\(921\) 1.26056e7 0.489681
\(922\) − 382169.i − 0.0148057i
\(923\) − 1.40710e6i − 0.0543651i
\(924\) −5.02590e6 −0.193657
\(925\) 0 0
\(926\) 1.37399e7 0.526570
\(927\) 5.12189e6i 0.195763i
\(928\) 2.79187e6i 0.106420i
\(929\) 4.61722e7 1.75526 0.877629 0.479341i \(-0.159124\pi\)
0.877629 + 0.479341i \(0.159124\pi\)
\(930\) 0 0
\(931\) 2.15925e7 0.816447
\(932\) − 2.36515e6i − 0.0891904i
\(933\) − 3.90223e7i − 1.46761i
\(934\) 1.71005e7 0.641418
\(935\) 0 0
\(936\) 803040. 0.0299604
\(937\) 9.28215e6i 0.345382i 0.984976 + 0.172691i \(0.0552462\pi\)
−0.984976 + 0.172691i \(0.944754\pi\)
\(938\) − 6.61341e6i − 0.245425i
\(939\) 2.07445e7 0.767782
\(940\) 0 0
\(941\) −4.88066e7 −1.79682 −0.898410 0.439157i \(-0.855277\pi\)
−0.898410 + 0.439157i \(0.855277\pi\)
\(942\) − 2.13444e7i − 0.783712i
\(943\) 4.94157e7i 1.80961i
\(944\) −707264. −0.0258316
\(945\) 0 0
\(946\) 2.27219e6 0.0825500
\(947\) 4.19678e6i 0.152069i 0.997105 + 0.0760347i \(0.0242260\pi\)
−0.997105 + 0.0760347i \(0.975774\pi\)
\(948\) − 2.73033e7i − 0.986722i
\(949\) 8.36520e6 0.301516
\(950\) 0 0
\(951\) 4.15753e7 1.49068
\(952\) − 2.25759e6i − 0.0807332i
\(953\) 2.79666e7i 0.997487i 0.866750 + 0.498743i \(0.166205\pi\)
−0.866750 + 0.498743i \(0.833795\pi\)
\(954\) 5.12677e6 0.182378
\(955\) 0 0
\(956\) −7.08785e6 −0.250824
\(957\) 9.87183e6i 0.348432i
\(958\) 1.32442e6i 0.0466243i
\(959\) −2.43016e7 −0.853275
\(960\) 0 0
\(961\) −4.47563e6 −0.156331
\(962\) − 2.11371e6i − 0.0736388i
\(963\) 8.93291e6i 0.310404i
\(964\) −1.29155e7 −0.447631
\(965\) 0 0
\(966\) 2.02785e7 0.699185
\(967\) − 5.45774e7i − 1.87692i −0.345383 0.938462i \(-0.612251\pi\)
0.345383 0.938462i \(-0.387749\pi\)
\(968\) 7.66248e6i 0.262834i
\(969\) 1.68498e7 0.576480
\(970\) 0 0
\(971\) 2.64787e7 0.901258 0.450629 0.892711i \(-0.351200\pi\)
0.450629 + 0.892711i \(0.351200\pi\)
\(972\) − 8.71217e6i − 0.295775i
\(973\) 2.87784e7i 0.974506i
\(974\) 3.73333e7 1.26095
\(975\) 0 0
\(976\) −4.31334e6 −0.144940
\(977\) − 1.85214e7i − 0.620780i −0.950609 0.310390i \(-0.899540\pi\)
0.950609 0.310390i \(-0.100460\pi\)
\(978\) − 3.41308e7i − 1.14104i
\(979\) −5.42400e6 −0.180868
\(980\) 0 0
\(981\) −1.28662e7 −0.426853
\(982\) 2.08194e7i 0.688952i
\(983\) 2.22953e7i 0.735919i 0.929842 + 0.367959i \(0.119943\pi\)
−0.929842 + 0.367959i \(0.880057\pi\)
\(984\) −1.71695e7 −0.565287
\(985\) 0 0
\(986\) −4.43433e6 −0.145256
\(987\) 2.57733e7i 0.842127i
\(988\) 6.29115e6i 0.205040i
\(989\) −9.16781e6 −0.298040
\(990\) 0 0
\(991\) 4.55658e7 1.47385 0.736927 0.675972i \(-0.236276\pi\)
0.736927 + 0.675972i \(0.236276\pi\)
\(992\) 5.03257e6i 0.162372i
\(993\) − 1.34635e7i − 0.433295i
\(994\) 2.88928e6 0.0927521
\(995\) 0 0
\(996\) −9.73341e6 −0.310897
\(997\) − 3.81521e7i − 1.21557i −0.794101 0.607786i \(-0.792058\pi\)
0.794101 0.607786i \(-0.207942\pi\)
\(998\) 3.80460e7i 1.20916i
\(999\) −9.39833e6 −0.297946
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 650.6.b.g.599.3 4
5.2 odd 4 130.6.a.a.1.1 2
5.3 odd 4 650.6.a.h.1.2 2
5.4 even 2 inner 650.6.b.g.599.2 4
20.7 even 4 1040.6.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.6.a.a.1.1 2 5.2 odd 4
650.6.a.h.1.2 2 5.3 odd 4
650.6.b.g.599.2 4 5.4 even 2 inner
650.6.b.g.599.3 4 1.1 even 1 trivial
1040.6.a.e.1.2 2 20.7 even 4