Properties

Label 245.6.a.e.1.2
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,6,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.73688\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.73688 q^{2} +28.3358 q^{3} -24.5095 q^{4} -25.0000 q^{5} -77.5516 q^{6} +154.660 q^{8} +559.917 q^{9} +O(q^{10})\) \(q-2.73688 q^{2} +28.3358 q^{3} -24.5095 q^{4} -25.0000 q^{5} -77.5516 q^{6} +154.660 q^{8} +559.917 q^{9} +68.4220 q^{10} +63.4296 q^{11} -694.496 q^{12} +868.146 q^{13} -708.395 q^{15} +361.019 q^{16} -1189.74 q^{17} -1532.43 q^{18} -1532.71 q^{19} +612.737 q^{20} -173.599 q^{22} -2337.74 q^{23} +4382.40 q^{24} +625.000 q^{25} -2376.01 q^{26} +8980.10 q^{27} +8169.03 q^{29} +1938.79 q^{30} +2369.72 q^{31} -5937.17 q^{32} +1797.33 q^{33} +3256.18 q^{34} -13723.3 q^{36} +12841.3 q^{37} +4194.85 q^{38} +24599.6 q^{39} -3866.49 q^{40} -1652.93 q^{41} +12971.1 q^{43} -1554.63 q^{44} -13997.9 q^{45} +6398.11 q^{46} +6219.00 q^{47} +10229.8 q^{48} -1710.55 q^{50} -33712.3 q^{51} -21277.8 q^{52} +19944.4 q^{53} -24577.5 q^{54} -1585.74 q^{55} -43430.7 q^{57} -22357.7 q^{58} +30708.0 q^{59} +17362.4 q^{60} -19901.3 q^{61} -6485.64 q^{62} +4696.70 q^{64} -21703.6 q^{65} -4919.07 q^{66} +31668.6 q^{67} +29160.0 q^{68} -66241.7 q^{69} -32747.5 q^{71} +86596.6 q^{72} +54441.9 q^{73} -35145.0 q^{74} +17709.9 q^{75} +37566.1 q^{76} -67326.1 q^{78} +47649.2 q^{79} -9025.48 q^{80} +118398. q^{81} +4523.87 q^{82} +56509.0 q^{83} +29743.6 q^{85} -35500.2 q^{86} +231476. q^{87} +9810.00 q^{88} -2533.69 q^{89} +38310.6 q^{90} +57296.8 q^{92} +67148.0 q^{93} -17020.7 q^{94} +38317.9 q^{95} -168235. q^{96} -58075.1 q^{97} +35515.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 7 q^{2} - 14 q^{3} + 49 q^{4} - 100 q^{5} - 136 q^{6} + 489 q^{8} + 774 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 7 q^{2} - 14 q^{3} + 49 q^{4} - 100 q^{5} - 136 q^{6} + 489 q^{8} + 774 q^{9} - 175 q^{10} + 770 q^{11} - 840 q^{12} - 58 q^{13} + 350 q^{15} - 615 q^{16} - 2006 q^{17} - 1409 q^{18} - 564 q^{19} - 1225 q^{20} - 1736 q^{22} - 6340 q^{23} + 6244 q^{24} + 2500 q^{25} + 8730 q^{26} + 7438 q^{27} + 8066 q^{29} + 3400 q^{30} + 5856 q^{31} - 3495 q^{32} + 8130 q^{33} - 3402 q^{34} - 28759 q^{36} + 29544 q^{37} + 36860 q^{38} + 57466 q^{39} - 12225 q^{40} - 13156 q^{41} - 5692 q^{43} + 44952 q^{44} - 19350 q^{45} + 30928 q^{46} - 39926 q^{47} + 57156 q^{48} + 4375 q^{50} + 9830 q^{51} + 23398 q^{52} + 20300 q^{53} - 77292 q^{54} - 19250 q^{55} - 50876 q^{57} - 22234 q^{58} - 8432 q^{59} + 21000 q^{60} - 30540 q^{61} + 137568 q^{62} + 37121 q^{64} + 1450 q^{65} + 166756 q^{66} + 32792 q^{67} + 72554 q^{68} + 36540 q^{69} - 83920 q^{71} - 71795 q^{72} + 75424 q^{73} + 168762 q^{74} - 8750 q^{75} + 125092 q^{76} - 89204 q^{78} + 129486 q^{79} + 15375 q^{80} + 146308 q^{81} - 247230 q^{82} + 187520 q^{83} + 50150 q^{85} + 36156 q^{86} + 238670 q^{87} + 475556 q^{88} + 30324 q^{89} + 35225 q^{90} + 124464 q^{92} + 8256 q^{93} - 245756 q^{94} + 14100 q^{95} - 172340 q^{96} - 180270 q^{97} - 420668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73688 −0.483816 −0.241908 0.970299i \(-0.577773\pi\)
−0.241908 + 0.970299i \(0.577773\pi\)
\(3\) 28.3358 1.81774 0.908871 0.417077i \(-0.136946\pi\)
0.908871 + 0.417077i \(0.136946\pi\)
\(4\) −24.5095 −0.765922
\(5\) −25.0000 −0.447214
\(6\) −77.5516 −0.879453
\(7\) 0 0
\(8\) 154.660 0.854382
\(9\) 559.917 2.30419
\(10\) 68.4220 0.216369
\(11\) 63.4296 0.158056 0.0790279 0.996872i \(-0.474818\pi\)
0.0790279 + 0.996872i \(0.474818\pi\)
\(12\) −694.496 −1.39225
\(13\) 868.146 1.42474 0.712368 0.701806i \(-0.247623\pi\)
0.712368 + 0.701806i \(0.247623\pi\)
\(14\) 0 0
\(15\) −708.395 −0.812919
\(16\) 361.019 0.352558
\(17\) −1189.74 −0.998460 −0.499230 0.866469i \(-0.666384\pi\)
−0.499230 + 0.866469i \(0.666384\pi\)
\(18\) −1532.43 −1.11480
\(19\) −1532.71 −0.974041 −0.487020 0.873391i \(-0.661916\pi\)
−0.487020 + 0.873391i \(0.661916\pi\)
\(20\) 612.737 0.342531
\(21\) 0 0
\(22\) −173.599 −0.0764699
\(23\) −2337.74 −0.921460 −0.460730 0.887540i \(-0.652412\pi\)
−0.460730 + 0.887540i \(0.652412\pi\)
\(24\) 4382.40 1.55305
\(25\) 625.000 0.200000
\(26\) −2376.01 −0.689310
\(27\) 8980.10 2.37067
\(28\) 0 0
\(29\) 8169.03 1.80375 0.901874 0.432000i \(-0.142192\pi\)
0.901874 + 0.432000i \(0.142192\pi\)
\(30\) 1938.79 0.393303
\(31\) 2369.72 0.442887 0.221444 0.975173i \(-0.428923\pi\)
0.221444 + 0.975173i \(0.428923\pi\)
\(32\) −5937.17 −1.02496
\(33\) 1797.33 0.287305
\(34\) 3256.18 0.483071
\(35\) 0 0
\(36\) −13723.3 −1.76483
\(37\) 12841.3 1.54207 0.771035 0.636793i \(-0.219739\pi\)
0.771035 + 0.636793i \(0.219739\pi\)
\(38\) 4194.85 0.471257
\(39\) 24599.6 2.58980
\(40\) −3866.49 −0.382091
\(41\) −1652.93 −0.153566 −0.0767830 0.997048i \(-0.524465\pi\)
−0.0767830 + 0.997048i \(0.524465\pi\)
\(42\) 0 0
\(43\) 12971.1 1.06980 0.534902 0.844914i \(-0.320348\pi\)
0.534902 + 0.844914i \(0.320348\pi\)
\(44\) −1554.63 −0.121058
\(45\) −13997.9 −1.03046
\(46\) 6398.11 0.445817
\(47\) 6219.00 0.410654 0.205327 0.978693i \(-0.434174\pi\)
0.205327 + 0.978693i \(0.434174\pi\)
\(48\) 10229.8 0.640859
\(49\) 0 0
\(50\) −1710.55 −0.0967633
\(51\) −33712.3 −1.81494
\(52\) −21277.8 −1.09124
\(53\) 19944.4 0.975283 0.487642 0.873044i \(-0.337857\pi\)
0.487642 + 0.873044i \(0.337857\pi\)
\(54\) −24577.5 −1.14697
\(55\) −1585.74 −0.0706847
\(56\) 0 0
\(57\) −43430.7 −1.77056
\(58\) −22357.7 −0.872682
\(59\) 30708.0 1.14847 0.574237 0.818689i \(-0.305299\pi\)
0.574237 + 0.818689i \(0.305299\pi\)
\(60\) 17362.4 0.622632
\(61\) −19901.3 −0.684790 −0.342395 0.939556i \(-0.611238\pi\)
−0.342395 + 0.939556i \(0.611238\pi\)
\(62\) −6485.64 −0.214276
\(63\) 0 0
\(64\) 4696.70 0.143332
\(65\) −21703.6 −0.637161
\(66\) −4919.07 −0.139003
\(67\) 31668.6 0.861869 0.430935 0.902383i \(-0.358184\pi\)
0.430935 + 0.902383i \(0.358184\pi\)
\(68\) 29160.0 0.764743
\(69\) −66241.7 −1.67498
\(70\) 0 0
\(71\) −32747.5 −0.770960 −0.385480 0.922716i \(-0.625964\pi\)
−0.385480 + 0.922716i \(0.625964\pi\)
\(72\) 86596.6 1.96865
\(73\) 54441.9 1.19571 0.597856 0.801604i \(-0.296020\pi\)
0.597856 + 0.801604i \(0.296020\pi\)
\(74\) −35145.0 −0.746078
\(75\) 17709.9 0.363548
\(76\) 37566.1 0.746039
\(77\) 0 0
\(78\) −67326.1 −1.25299
\(79\) 47649.2 0.858989 0.429495 0.903069i \(-0.358692\pi\)
0.429495 + 0.903069i \(0.358692\pi\)
\(80\) −9025.48 −0.157669
\(81\) 118398. 2.00509
\(82\) 4523.87 0.0742977
\(83\) 56509.0 0.900372 0.450186 0.892935i \(-0.351358\pi\)
0.450186 + 0.892935i \(0.351358\pi\)
\(84\) 0 0
\(85\) 29743.6 0.446525
\(86\) −35500.2 −0.517589
\(87\) 231476. 3.27875
\(88\) 9810.00 0.135040
\(89\) −2533.69 −0.0339061 −0.0169531 0.999856i \(-0.505397\pi\)
−0.0169531 + 0.999856i \(0.505397\pi\)
\(90\) 38310.6 0.498555
\(91\) 0 0
\(92\) 57296.8 0.705766
\(93\) 67148.0 0.805055
\(94\) −17020.7 −0.198681
\(95\) 38317.9 0.435604
\(96\) −168235. −1.86310
\(97\) −58075.1 −0.626701 −0.313350 0.949638i \(-0.601451\pi\)
−0.313350 + 0.949638i \(0.601451\pi\)
\(98\) 0 0
\(99\) 35515.3 0.364190
\(100\) −15318.4 −0.153184
\(101\) −98422.7 −0.960045 −0.480023 0.877256i \(-0.659372\pi\)
−0.480023 + 0.877256i \(0.659372\pi\)
\(102\) 92266.5 0.878099
\(103\) −179343. −1.66568 −0.832840 0.553514i \(-0.813287\pi\)
−0.832840 + 0.553514i \(0.813287\pi\)
\(104\) 134267. 1.21727
\(105\) 0 0
\(106\) −54585.3 −0.471858
\(107\) 159436. 1.34626 0.673128 0.739526i \(-0.264950\pi\)
0.673128 + 0.739526i \(0.264950\pi\)
\(108\) −220098. −1.81575
\(109\) −97547.7 −0.786413 −0.393207 0.919450i \(-0.628634\pi\)
−0.393207 + 0.919450i \(0.628634\pi\)
\(110\) 4339.98 0.0341984
\(111\) 363868. 2.80308
\(112\) 0 0
\(113\) 140601. 1.03584 0.517919 0.855430i \(-0.326707\pi\)
0.517919 + 0.855430i \(0.326707\pi\)
\(114\) 118864. 0.856623
\(115\) 58443.5 0.412090
\(116\) −200219. −1.38153
\(117\) 486090. 3.28286
\(118\) −84044.0 −0.555650
\(119\) 0 0
\(120\) −109560. −0.694543
\(121\) −157028. −0.975018
\(122\) 54467.5 0.331312
\(123\) −46837.1 −0.279143
\(124\) −58080.7 −0.339217
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) 233889. 1.28677 0.643385 0.765543i \(-0.277529\pi\)
0.643385 + 0.765543i \(0.277529\pi\)
\(128\) 177135. 0.955609
\(129\) 367546. 1.94463
\(130\) 59400.2 0.308269
\(131\) −181216. −0.922610 −0.461305 0.887242i \(-0.652619\pi\)
−0.461305 + 0.887242i \(0.652619\pi\)
\(132\) −44051.6 −0.220053
\(133\) 0 0
\(134\) −86673.0 −0.416986
\(135\) −224503. −1.06020
\(136\) −184005. −0.853066
\(137\) −203830. −0.927827 −0.463914 0.885880i \(-0.653555\pi\)
−0.463914 + 0.885880i \(0.653555\pi\)
\(138\) 181296. 0.810381
\(139\) −51086.4 −0.224269 −0.112134 0.993693i \(-0.535769\pi\)
−0.112134 + 0.993693i \(0.535769\pi\)
\(140\) 0 0
\(141\) 176220. 0.746463
\(142\) 89625.8 0.373003
\(143\) 55066.1 0.225188
\(144\) 202141. 0.812359
\(145\) −204226. −0.806660
\(146\) −149001. −0.578505
\(147\) 0 0
\(148\) −314733. −1.18110
\(149\) −307059. −1.13307 −0.566533 0.824039i \(-0.691716\pi\)
−0.566533 + 0.824039i \(0.691716\pi\)
\(150\) −48469.8 −0.175891
\(151\) 358221. 1.27852 0.639261 0.768990i \(-0.279240\pi\)
0.639261 + 0.768990i \(0.279240\pi\)
\(152\) −237049. −0.832203
\(153\) −666158. −2.30064
\(154\) 0 0
\(155\) −59243.0 −0.198065
\(156\) −602924. −1.98359
\(157\) −437769. −1.41741 −0.708705 0.705505i \(-0.750720\pi\)
−0.708705 + 0.705505i \(0.750720\pi\)
\(158\) −130410. −0.415593
\(159\) 565140. 1.77281
\(160\) 148429. 0.458374
\(161\) 0 0
\(162\) −324042. −0.970095
\(163\) −430265. −1.26843 −0.634216 0.773156i \(-0.718677\pi\)
−0.634216 + 0.773156i \(0.718677\pi\)
\(164\) 40512.5 0.117620
\(165\) −44933.2 −0.128486
\(166\) −154658. −0.435615
\(167\) −229736. −0.637437 −0.318718 0.947849i \(-0.603252\pi\)
−0.318718 + 0.947849i \(0.603252\pi\)
\(168\) 0 0
\(169\) 382384. 1.02987
\(170\) −81404.6 −0.216036
\(171\) −858193. −2.24437
\(172\) −317914. −0.819387
\(173\) −67029.4 −0.170275 −0.0851374 0.996369i \(-0.527133\pi\)
−0.0851374 + 0.996369i \(0.527133\pi\)
\(174\) −633522. −1.58631
\(175\) 0 0
\(176\) 22899.3 0.0557238
\(177\) 870135. 2.08763
\(178\) 6934.39 0.0164043
\(179\) −234462. −0.546940 −0.273470 0.961881i \(-0.588171\pi\)
−0.273470 + 0.961881i \(0.588171\pi\)
\(180\) 343082. 0.789254
\(181\) 199344. 0.452278 0.226139 0.974095i \(-0.427390\pi\)
0.226139 + 0.974095i \(0.427390\pi\)
\(182\) 0 0
\(183\) −563920. −1.24477
\(184\) −361554. −0.787279
\(185\) −321032. −0.689634
\(186\) −183776. −0.389499
\(187\) −75464.9 −0.157812
\(188\) −152425. −0.314529
\(189\) 0 0
\(190\) −104871. −0.210752
\(191\) 300904. 0.596821 0.298410 0.954438i \(-0.403544\pi\)
0.298410 + 0.954438i \(0.403544\pi\)
\(192\) 133085. 0.260541
\(193\) −173138. −0.334579 −0.167289 0.985908i \(-0.553501\pi\)
−0.167289 + 0.985908i \(0.553501\pi\)
\(194\) 158944. 0.303208
\(195\) −614990. −1.15819
\(196\) 0 0
\(197\) 100950. 0.185329 0.0926644 0.995697i \(-0.470462\pi\)
0.0926644 + 0.995697i \(0.470462\pi\)
\(198\) −97201.1 −0.176201
\(199\) 809079. 1.44830 0.724149 0.689643i \(-0.242233\pi\)
0.724149 + 0.689643i \(0.242233\pi\)
\(200\) 96662.3 0.170876
\(201\) 897354. 1.56666
\(202\) 269371. 0.464486
\(203\) 0 0
\(204\) 826272. 1.39010
\(205\) 41323.3 0.0686768
\(206\) 490840. 0.805883
\(207\) −1.30894e6 −2.12322
\(208\) 313417. 0.502302
\(209\) −97219.4 −0.153953
\(210\) 0 0
\(211\) 183306. 0.283446 0.141723 0.989906i \(-0.454736\pi\)
0.141723 + 0.989906i \(0.454736\pi\)
\(212\) −488826. −0.746991
\(213\) −927925. −1.40141
\(214\) −436358. −0.651340
\(215\) −324277. −0.478431
\(216\) 1.38886e6 2.02546
\(217\) 0 0
\(218\) 266976. 0.380480
\(219\) 1.54266e6 2.17350
\(220\) 38865.7 0.0541389
\(221\) −1.03287e6 −1.42254
\(222\) −995862. −1.35618
\(223\) 678874. 0.914171 0.457085 0.889423i \(-0.348893\pi\)
0.457085 + 0.889423i \(0.348893\pi\)
\(224\) 0 0
\(225\) 349948. 0.460837
\(226\) −384808. −0.501156
\(227\) 256484. 0.330366 0.165183 0.986263i \(-0.447179\pi\)
0.165183 + 0.986263i \(0.447179\pi\)
\(228\) 1.06446e6 1.35611
\(229\) −321802. −0.405509 −0.202754 0.979230i \(-0.564989\pi\)
−0.202754 + 0.979230i \(0.564989\pi\)
\(230\) −159953. −0.199376
\(231\) 0 0
\(232\) 1.26342e6 1.54109
\(233\) 193589. 0.233609 0.116805 0.993155i \(-0.462735\pi\)
0.116805 + 0.993155i \(0.462735\pi\)
\(234\) −1.33037e6 −1.58830
\(235\) −155475. −0.183650
\(236\) −752637. −0.879641
\(237\) 1.35018e6 1.56142
\(238\) 0 0
\(239\) −482857. −0.546794 −0.273397 0.961901i \(-0.588147\pi\)
−0.273397 + 0.961901i \(0.588147\pi\)
\(240\) −255744. −0.286601
\(241\) 1.06213e6 1.17797 0.588984 0.808145i \(-0.299528\pi\)
0.588984 + 0.808145i \(0.299528\pi\)
\(242\) 429766. 0.471730
\(243\) 1.17275e6 1.27406
\(244\) 487771. 0.524495
\(245\) 0 0
\(246\) 128188. 0.135054
\(247\) −1.33062e6 −1.38775
\(248\) 366500. 0.378395
\(249\) 1.60123e6 1.63664
\(250\) 42763.7 0.0432738
\(251\) −1.43228e6 −1.43497 −0.717484 0.696575i \(-0.754707\pi\)
−0.717484 + 0.696575i \(0.754707\pi\)
\(252\) 0 0
\(253\) −148282. −0.145642
\(254\) −640126. −0.622560
\(255\) 842808. 0.811667
\(256\) −635092. −0.605671
\(257\) 491843. 0.464508 0.232254 0.972655i \(-0.425390\pi\)
0.232254 + 0.972655i \(0.425390\pi\)
\(258\) −1.00593e6 −0.940844
\(259\) 0 0
\(260\) 531945. 0.488016
\(261\) 4.57398e6 4.15617
\(262\) 495966. 0.446374
\(263\) −1.41098e6 −1.25786 −0.628928 0.777464i \(-0.716506\pi\)
−0.628928 + 0.777464i \(0.716506\pi\)
\(264\) 277974. 0.245468
\(265\) −498609. −0.436160
\(266\) 0 0
\(267\) −71794.0 −0.0616325
\(268\) −776181. −0.660124
\(269\) −847237. −0.713878 −0.356939 0.934128i \(-0.616180\pi\)
−0.356939 + 0.934128i \(0.616180\pi\)
\(270\) 614436. 0.512941
\(271\) −1.14845e6 −0.949923 −0.474961 0.880007i \(-0.657538\pi\)
−0.474961 + 0.880007i \(0.657538\pi\)
\(272\) −429520. −0.352015
\(273\) 0 0
\(274\) 557859. 0.448898
\(275\) 39643.5 0.0316111
\(276\) 1.62355e6 1.28290
\(277\) 1.17885e6 0.923124 0.461562 0.887108i \(-0.347289\pi\)
0.461562 + 0.887108i \(0.347289\pi\)
\(278\) 139817. 0.108505
\(279\) 1.32685e6 1.02049
\(280\) 0 0
\(281\) 112265. 0.0848164 0.0424082 0.999100i \(-0.486497\pi\)
0.0424082 + 0.999100i \(0.486497\pi\)
\(282\) −482294. −0.361151
\(283\) −1.67112e6 −1.24034 −0.620172 0.784466i \(-0.712937\pi\)
−0.620172 + 0.784466i \(0.712937\pi\)
\(284\) 802624. 0.590495
\(285\) 1.08577e6 0.791816
\(286\) −150709. −0.108949
\(287\) 0 0
\(288\) −3.32433e6 −2.36169
\(289\) −4368.41 −0.00307665
\(290\) 558941. 0.390275
\(291\) −1.64560e6 −1.13918
\(292\) −1.33434e6 −0.915822
\(293\) −2.28723e6 −1.55647 −0.778235 0.627973i \(-0.783885\pi\)
−0.778235 + 0.627973i \(0.783885\pi\)
\(294\) 0 0
\(295\) −767699. −0.513613
\(296\) 1.98603e6 1.31752
\(297\) 569604. 0.374699
\(298\) 840382. 0.548196
\(299\) −2.02950e6 −1.31284
\(300\) −434060. −0.278450
\(301\) 0 0
\(302\) −980406. −0.618570
\(303\) −2.78888e6 −1.74511
\(304\) −553339. −0.343406
\(305\) 497533. 0.306247
\(306\) 1.82319e6 1.11309
\(307\) −449789. −0.272372 −0.136186 0.990683i \(-0.543485\pi\)
−0.136186 + 0.990683i \(0.543485\pi\)
\(308\) 0 0
\(309\) −5.08183e6 −3.02778
\(310\) 162141. 0.0958272
\(311\) 605586. 0.355038 0.177519 0.984117i \(-0.443193\pi\)
0.177519 + 0.984117i \(0.443193\pi\)
\(312\) 3.80456e6 2.21268
\(313\) 2.03261e6 1.17272 0.586359 0.810052i \(-0.300561\pi\)
0.586359 + 0.810052i \(0.300561\pi\)
\(314\) 1.19812e6 0.685766
\(315\) 0 0
\(316\) −1.16786e6 −0.657918
\(317\) −2.61940e6 −1.46404 −0.732021 0.681282i \(-0.761423\pi\)
−0.732021 + 0.681282i \(0.761423\pi\)
\(318\) −1.54672e6 −0.857716
\(319\) 518158. 0.285093
\(320\) −117418. −0.0641000
\(321\) 4.51775e6 2.44715
\(322\) 0 0
\(323\) 1.82354e6 0.972541
\(324\) −2.90189e6 −1.53574
\(325\) 542591. 0.284947
\(326\) 1.17758e6 0.613688
\(327\) −2.76409e6 −1.42950
\(328\) −255642. −0.131204
\(329\) 0 0
\(330\) 122977. 0.0621639
\(331\) 1.79427e6 0.900155 0.450078 0.892989i \(-0.351396\pi\)
0.450078 + 0.892989i \(0.351396\pi\)
\(332\) −1.38501e6 −0.689615
\(333\) 7.19005e6 3.55322
\(334\) 628758. 0.308402
\(335\) −791714. −0.385440
\(336\) 0 0
\(337\) −1.10766e6 −0.531292 −0.265646 0.964071i \(-0.585585\pi\)
−0.265646 + 0.964071i \(0.585585\pi\)
\(338\) −1.04654e6 −0.498269
\(339\) 3.98404e6 1.88289
\(340\) −729000. −0.342003
\(341\) 150311. 0.0700009
\(342\) 2.34877e6 1.08586
\(343\) 0 0
\(344\) 2.00610e6 0.914022
\(345\) 1.65604e6 0.749072
\(346\) 183451. 0.0823817
\(347\) −2.46825e6 −1.10044 −0.550218 0.835021i \(-0.685455\pi\)
−0.550218 + 0.835021i \(0.685455\pi\)
\(348\) −5.67336e6 −2.51126
\(349\) −2.41856e6 −1.06290 −0.531451 0.847089i \(-0.678353\pi\)
−0.531451 + 0.847089i \(0.678353\pi\)
\(350\) 0 0
\(351\) 7.79604e6 3.37758
\(352\) −376593. −0.162000
\(353\) 1.84853e6 0.789570 0.394785 0.918774i \(-0.370819\pi\)
0.394785 + 0.918774i \(0.370819\pi\)
\(354\) −2.38145e6 −1.01003
\(355\) 818687. 0.344784
\(356\) 62099.4 0.0259694
\(357\) 0 0
\(358\) 641693. 0.264618
\(359\) −552362. −0.226197 −0.113099 0.993584i \(-0.536078\pi\)
−0.113099 + 0.993584i \(0.536078\pi\)
\(360\) −2.16491e6 −0.880409
\(361\) −126886. −0.0512444
\(362\) −545579. −0.218820
\(363\) −4.44950e6 −1.77233
\(364\) 0 0
\(365\) −1.36105e6 −0.534739
\(366\) 1.54338e6 0.602241
\(367\) −4.18145e6 −1.62055 −0.810273 0.586053i \(-0.800681\pi\)
−0.810273 + 0.586053i \(0.800681\pi\)
\(368\) −843969. −0.324868
\(369\) −925505. −0.353845
\(370\) 878625. 0.333656
\(371\) 0 0
\(372\) −1.64576e6 −0.616609
\(373\) 2.29821e6 0.855300 0.427650 0.903944i \(-0.359342\pi\)
0.427650 + 0.903944i \(0.359342\pi\)
\(374\) 206538. 0.0763522
\(375\) −442747. −0.162584
\(376\) 961829. 0.350855
\(377\) 7.09191e6 2.56986
\(378\) 0 0
\(379\) 3.55567e6 1.27152 0.635760 0.771886i \(-0.280687\pi\)
0.635760 + 0.771886i \(0.280687\pi\)
\(380\) −939151. −0.333639
\(381\) 6.62744e6 2.33902
\(382\) −823536. −0.288752
\(383\) 1.57080e6 0.547172 0.273586 0.961848i \(-0.411790\pi\)
0.273586 + 0.961848i \(0.411790\pi\)
\(384\) 5.01927e6 1.73705
\(385\) 0 0
\(386\) 473857. 0.161875
\(387\) 7.26273e6 2.46503
\(388\) 1.42339e6 0.480004
\(389\) −1.33556e6 −0.447495 −0.223747 0.974647i \(-0.571829\pi\)
−0.223747 + 0.974647i \(0.571829\pi\)
\(390\) 1.68315e6 0.560353
\(391\) 2.78131e6 0.920042
\(392\) 0 0
\(393\) −5.13490e6 −1.67707
\(394\) −276289. −0.0896651
\(395\) −1.19123e6 −0.384152
\(396\) −870463. −0.278941
\(397\) 1.47124e6 0.468496 0.234248 0.972177i \(-0.424737\pi\)
0.234248 + 0.972177i \(0.424737\pi\)
\(398\) −2.21435e6 −0.700710
\(399\) 0 0
\(400\) 225637. 0.0705116
\(401\) 994719. 0.308915 0.154458 0.987999i \(-0.450637\pi\)
0.154458 + 0.987999i \(0.450637\pi\)
\(402\) −2.45595e6 −0.757974
\(403\) 2.05726e6 0.630997
\(404\) 2.41229e6 0.735320
\(405\) −2.95996e6 −0.896703
\(406\) 0 0
\(407\) 814517. 0.243733
\(408\) −5.21393e6 −1.55065
\(409\) 995359. 0.294220 0.147110 0.989120i \(-0.453003\pi\)
0.147110 + 0.989120i \(0.453003\pi\)
\(410\) −113097. −0.0332270
\(411\) −5.77569e6 −1.68655
\(412\) 4.39561e6 1.27578
\(413\) 0 0
\(414\) 3.58241e6 1.02725
\(415\) −1.41272e6 −0.402659
\(416\) −5.15433e6 −1.46029
\(417\) −1.44757e6 −0.407662
\(418\) 266078. 0.0744848
\(419\) −62520.7 −0.0173976 −0.00869879 0.999962i \(-0.502769\pi\)
−0.00869879 + 0.999962i \(0.502769\pi\)
\(420\) 0 0
\(421\) 3.26482e6 0.897747 0.448874 0.893595i \(-0.351825\pi\)
0.448874 + 0.893595i \(0.351825\pi\)
\(422\) −501686. −0.137136
\(423\) 3.48213e6 0.946224
\(424\) 3.08459e6 0.833264
\(425\) −743589. −0.199692
\(426\) 2.53962e6 0.678023
\(427\) 0 0
\(428\) −3.90770e6 −1.03113
\(429\) 1.56034e6 0.409333
\(430\) 887506. 0.231473
\(431\) −2.05017e6 −0.531615 −0.265807 0.964026i \(-0.585639\pi\)
−0.265807 + 0.964026i \(0.585639\pi\)
\(432\) 3.24199e6 0.835800
\(433\) −627607. −0.160867 −0.0804337 0.996760i \(-0.525631\pi\)
−0.0804337 + 0.996760i \(0.525631\pi\)
\(434\) 0 0
\(435\) −5.78690e6 −1.46630
\(436\) 2.39085e6 0.602331
\(437\) 3.58309e6 0.897540
\(438\) −4.22206e6 −1.05157
\(439\) −6.13266e6 −1.51876 −0.759378 0.650650i \(-0.774496\pi\)
−0.759378 + 0.650650i \(0.774496\pi\)
\(440\) −245250. −0.0603917
\(441\) 0 0
\(442\) 2.82684e6 0.688249
\(443\) −2.08811e6 −0.505526 −0.252763 0.967528i \(-0.581339\pi\)
−0.252763 + 0.967528i \(0.581339\pi\)
\(444\) −8.91822e6 −2.14694
\(445\) 63342.2 0.0151633
\(446\) −1.85800e6 −0.442291
\(447\) −8.70075e6 −2.05962
\(448\) 0 0
\(449\) −6.77740e6 −1.58653 −0.793263 0.608880i \(-0.791619\pi\)
−0.793263 + 0.608880i \(0.791619\pi\)
\(450\) −957766. −0.222961
\(451\) −104845. −0.0242720
\(452\) −3.44606e6 −0.793371
\(453\) 1.01505e7 2.32402
\(454\) −701965. −0.159836
\(455\) 0 0
\(456\) −6.71697e6 −1.51273
\(457\) 3.89825e6 0.873131 0.436566 0.899672i \(-0.356195\pi\)
0.436566 + 0.899672i \(0.356195\pi\)
\(458\) 880733. 0.196192
\(459\) −1.06840e7 −2.36702
\(460\) −1.43242e6 −0.315628
\(461\) −4.27807e6 −0.937552 −0.468776 0.883317i \(-0.655305\pi\)
−0.468776 + 0.883317i \(0.655305\pi\)
\(462\) 0 0
\(463\) 7.18974e6 1.55869 0.779347 0.626593i \(-0.215551\pi\)
0.779347 + 0.626593i \(0.215551\pi\)
\(464\) 2.94918e6 0.635925
\(465\) −1.67870e6 −0.360031
\(466\) −529829. −0.113024
\(467\) 5.72641e6 1.21504 0.607519 0.794305i \(-0.292165\pi\)
0.607519 + 0.794305i \(0.292165\pi\)
\(468\) −1.19138e7 −2.51441
\(469\) 0 0
\(470\) 425516. 0.0888529
\(471\) −1.24045e7 −2.57649
\(472\) 4.74928e6 0.981235
\(473\) 822750. 0.169089
\(474\) −3.69527e6 −0.755441
\(475\) −957946. −0.194808
\(476\) 0 0
\(477\) 1.11672e7 2.24723
\(478\) 1.32152e6 0.264548
\(479\) 696853. 0.138772 0.0693861 0.997590i \(-0.477896\pi\)
0.0693861 + 0.997590i \(0.477896\pi\)
\(480\) 4.20586e6 0.833205
\(481\) 1.11481e7 2.19704
\(482\) −2.90691e6 −0.569920
\(483\) 0 0
\(484\) 3.84867e6 0.746788
\(485\) 1.45188e6 0.280269
\(486\) −3.20967e6 −0.616411
\(487\) 7.51712e6 1.43625 0.718123 0.695916i \(-0.245001\pi\)
0.718123 + 0.695916i \(0.245001\pi\)
\(488\) −3.07793e6 −0.585072
\(489\) −1.21919e7 −2.30568
\(490\) 0 0
\(491\) −255540. −0.0478360 −0.0239180 0.999714i \(-0.507614\pi\)
−0.0239180 + 0.999714i \(0.507614\pi\)
\(492\) 1.14795e6 0.213802
\(493\) −9.71905e6 −1.80097
\(494\) 3.64174e6 0.671416
\(495\) −887883. −0.162871
\(496\) 855515. 0.156143
\(497\) 0 0
\(498\) −4.38236e6 −0.791835
\(499\) 8.14998e6 1.46523 0.732614 0.680645i \(-0.238300\pi\)
0.732614 + 0.680645i \(0.238300\pi\)
\(500\) 382961. 0.0685061
\(501\) −6.50974e6 −1.15870
\(502\) 3.91997e6 0.694261
\(503\) −5.64001e6 −0.993939 −0.496969 0.867768i \(-0.665554\pi\)
−0.496969 + 0.867768i \(0.665554\pi\)
\(504\) 0 0
\(505\) 2.46057e6 0.429345
\(506\) 405829. 0.0704640
\(507\) 1.08352e7 1.87204
\(508\) −5.73251e6 −0.985565
\(509\) 7.79954e6 1.33436 0.667182 0.744895i \(-0.267500\pi\)
0.667182 + 0.744895i \(0.267500\pi\)
\(510\) −2.30666e6 −0.392698
\(511\) 0 0
\(512\) −3.93016e6 −0.662575
\(513\) −1.37639e7 −2.30913
\(514\) −1.34611e6 −0.224737
\(515\) 4.48358e6 0.744915
\(516\) −9.00836e6 −1.48943
\(517\) 394469. 0.0649062
\(518\) 0 0
\(519\) −1.89933e6 −0.309516
\(520\) −3.35668e6 −0.544379
\(521\) −6.04835e6 −0.976208 −0.488104 0.872786i \(-0.662311\pi\)
−0.488104 + 0.872786i \(0.662311\pi\)
\(522\) −1.25184e7 −2.01082
\(523\) 7.64614e6 1.22233 0.611165 0.791503i \(-0.290701\pi\)
0.611165 + 0.791503i \(0.290701\pi\)
\(524\) 4.44151e6 0.706647
\(525\) 0 0
\(526\) 3.86167e6 0.608571
\(527\) −2.81936e6 −0.442205
\(528\) 648870. 0.101291
\(529\) −971316. −0.150911
\(530\) 1.36463e6 0.211021
\(531\) 1.71939e7 2.64630
\(532\) 0 0
\(533\) −1.43499e6 −0.218791
\(534\) 196492. 0.0298188
\(535\) −3.98591e6 −0.602064
\(536\) 4.89785e6 0.736365
\(537\) −6.64366e6 −0.994195
\(538\) 2.31878e6 0.345386
\(539\) 0 0
\(540\) 5.50245e6 0.812029
\(541\) −1.17724e7 −1.72930 −0.864651 0.502374i \(-0.832460\pi\)
−0.864651 + 0.502374i \(0.832460\pi\)
\(542\) 3.14316e6 0.459588
\(543\) 5.64856e6 0.822125
\(544\) 7.06371e6 1.02338
\(545\) 2.43869e6 0.351695
\(546\) 0 0
\(547\) 930880. 0.133023 0.0665113 0.997786i \(-0.478813\pi\)
0.0665113 + 0.997786i \(0.478813\pi\)
\(548\) 4.99578e6 0.710643
\(549\) −1.11431e7 −1.57788
\(550\) −108499. −0.0152940
\(551\) −1.25208e7 −1.75692
\(552\) −1.02449e7 −1.43107
\(553\) 0 0
\(554\) −3.22638e6 −0.446622
\(555\) −9.09670e6 −1.25358
\(556\) 1.25210e6 0.171772
\(557\) −4.66844e6 −0.637579 −0.318790 0.947825i \(-0.603276\pi\)
−0.318790 + 0.947825i \(0.603276\pi\)
\(558\) −3.63142e6 −0.493732
\(559\) 1.12608e7 1.52419
\(560\) 0 0
\(561\) −2.13836e6 −0.286862
\(562\) −307256. −0.0410355
\(563\) −1.30772e6 −0.173878 −0.0869388 0.996214i \(-0.527708\pi\)
−0.0869388 + 0.996214i \(0.527708\pi\)
\(564\) −4.31907e6 −0.571732
\(565\) −3.51502e6 −0.463241
\(566\) 4.57366e6 0.600099
\(567\) 0 0
\(568\) −5.06471e6 −0.658694
\(569\) 1.46759e7 1.90031 0.950154 0.311779i \(-0.100925\pi\)
0.950154 + 0.311779i \(0.100925\pi\)
\(570\) −2.97161e6 −0.383094
\(571\) 1.12905e6 0.144918 0.0724590 0.997371i \(-0.476915\pi\)
0.0724590 + 0.997371i \(0.476915\pi\)
\(572\) −1.34964e6 −0.172476
\(573\) 8.52634e6 1.08487
\(574\) 0 0
\(575\) −1.46109e6 −0.184292
\(576\) 2.62977e6 0.330264
\(577\) −8.94245e6 −1.11819 −0.559097 0.829102i \(-0.688852\pi\)
−0.559097 + 0.829102i \(0.688852\pi\)
\(578\) 11955.8 0.00148854
\(579\) −4.90600e6 −0.608178
\(580\) 5.00547e6 0.617839
\(581\) 0 0
\(582\) 4.50382e6 0.551154
\(583\) 1.26506e6 0.154149
\(584\) 8.41997e6 1.02159
\(585\) −1.21522e7 −1.46814
\(586\) 6.25987e6 0.753046
\(587\) −3.97509e6 −0.476158 −0.238079 0.971246i \(-0.576518\pi\)
−0.238079 + 0.971246i \(0.576518\pi\)
\(588\) 0 0
\(589\) −3.63211e6 −0.431390
\(590\) 2.10110e6 0.248494
\(591\) 2.86051e6 0.336880
\(592\) 4.63595e6 0.543669
\(593\) 3.27487e6 0.382434 0.191217 0.981548i \(-0.438757\pi\)
0.191217 + 0.981548i \(0.438757\pi\)
\(594\) −1.55894e6 −0.181285
\(595\) 0 0
\(596\) 7.52585e6 0.867841
\(597\) 2.29259e7 2.63263
\(598\) 5.55449e6 0.635172
\(599\) −4.45624e6 −0.507460 −0.253730 0.967275i \(-0.581657\pi\)
−0.253730 + 0.967275i \(0.581657\pi\)
\(600\) 2.73900e6 0.310609
\(601\) 8.18581e6 0.924433 0.462216 0.886767i \(-0.347054\pi\)
0.462216 + 0.886767i \(0.347054\pi\)
\(602\) 0 0
\(603\) 1.77318e7 1.98591
\(604\) −8.77981e6 −0.979248
\(605\) 3.92569e6 0.436041
\(606\) 7.63284e6 0.844315
\(607\) 6.82736e6 0.752109 0.376055 0.926598i \(-0.377281\pi\)
0.376055 + 0.926598i \(0.377281\pi\)
\(608\) 9.09999e6 0.998348
\(609\) 0 0
\(610\) −1.36169e6 −0.148167
\(611\) 5.39900e6 0.585073
\(612\) 1.63272e7 1.76211
\(613\) 6.39746e6 0.687632 0.343816 0.939037i \(-0.388280\pi\)
0.343816 + 0.939037i \(0.388280\pi\)
\(614\) 1.23102e6 0.131778
\(615\) 1.17093e6 0.124837
\(616\) 0 0
\(617\) −1.68341e7 −1.78024 −0.890118 0.455730i \(-0.849378\pi\)
−0.890118 + 0.455730i \(0.849378\pi\)
\(618\) 1.39083e7 1.46489
\(619\) 8.82274e6 0.925501 0.462751 0.886488i \(-0.346863\pi\)
0.462751 + 0.886488i \(0.346863\pi\)
\(620\) 1.45202e6 0.151702
\(621\) −2.09931e7 −2.18448
\(622\) −1.65742e6 −0.171773
\(623\) 0 0
\(624\) 8.88093e6 0.913055
\(625\) 390625. 0.0400000
\(626\) −5.56301e6 −0.567380
\(627\) −2.75479e6 −0.279846
\(628\) 1.07295e7 1.08563
\(629\) −1.52778e7 −1.53970
\(630\) 0 0
\(631\) −375030. −0.0374966 −0.0187483 0.999824i \(-0.505968\pi\)
−0.0187483 + 0.999824i \(0.505968\pi\)
\(632\) 7.36940e6 0.733905
\(633\) 5.19412e6 0.515232
\(634\) 7.16898e6 0.708328
\(635\) −5.84723e6 −0.575461
\(636\) −1.38513e7 −1.35784
\(637\) 0 0
\(638\) −1.41814e6 −0.137932
\(639\) −1.83359e7 −1.77644
\(640\) −4.42838e6 −0.427361
\(641\) 3.25010e6 0.312430 0.156215 0.987723i \(-0.450071\pi\)
0.156215 + 0.987723i \(0.450071\pi\)
\(642\) −1.23645e7 −1.18397
\(643\) −2.71365e6 −0.258837 −0.129418 0.991590i \(-0.541311\pi\)
−0.129418 + 0.991590i \(0.541311\pi\)
\(644\) 0 0
\(645\) −9.18864e6 −0.869665
\(646\) −4.99080e6 −0.470531
\(647\) −1.67261e7 −1.57085 −0.785426 0.618956i \(-0.787556\pi\)
−0.785426 + 0.618956i \(0.787556\pi\)
\(648\) 1.83115e7 1.71311
\(649\) 1.94779e6 0.181523
\(650\) −1.48501e6 −0.137862
\(651\) 0 0
\(652\) 1.05456e7 0.971519
\(653\) −1.17470e7 −1.07806 −0.539031 0.842286i \(-0.681209\pi\)
−0.539031 + 0.842286i \(0.681209\pi\)
\(654\) 7.56499e6 0.691614
\(655\) 4.53040e6 0.412604
\(656\) −596740. −0.0541409
\(657\) 3.04830e7 2.75514
\(658\) 0 0
\(659\) −9.16720e6 −0.822286 −0.411143 0.911571i \(-0.634870\pi\)
−0.411143 + 0.911571i \(0.634870\pi\)
\(660\) 1.10129e6 0.0984106
\(661\) −9.54925e6 −0.850092 −0.425046 0.905172i \(-0.639742\pi\)
−0.425046 + 0.905172i \(0.639742\pi\)
\(662\) −4.91069e6 −0.435510
\(663\) −2.92672e7 −2.58581
\(664\) 8.73965e6 0.769262
\(665\) 0 0
\(666\) −1.96783e7 −1.71910
\(667\) −1.90971e7 −1.66208
\(668\) 5.63070e6 0.488227
\(669\) 1.92364e7 1.66173
\(670\) 2.16683e6 0.186482
\(671\) −1.26233e6 −0.108235
\(672\) 0 0
\(673\) 3.70373e6 0.315211 0.157606 0.987502i \(-0.449623\pi\)
0.157606 + 0.987502i \(0.449623\pi\)
\(674\) 3.03154e6 0.257048
\(675\) 5.61256e6 0.474135
\(676\) −9.37204e6 −0.788801
\(677\) −1.93998e7 −1.62677 −0.813386 0.581724i \(-0.802378\pi\)
−0.813386 + 0.581724i \(0.802378\pi\)
\(678\) −1.09038e7 −0.910972
\(679\) 0 0
\(680\) 4.60013e6 0.381503
\(681\) 7.26767e6 0.600520
\(682\) −411382. −0.0338676
\(683\) 290718. 0.0238463 0.0119231 0.999929i \(-0.496205\pi\)
0.0119231 + 0.999929i \(0.496205\pi\)
\(684\) 2.10339e7 1.71901
\(685\) 5.09576e6 0.414937
\(686\) 0 0
\(687\) −9.11852e6 −0.737110
\(688\) 4.68281e6 0.377168
\(689\) 1.73146e7 1.38952
\(690\) −4.53239e6 −0.362413
\(691\) 2.05452e7 1.63688 0.818438 0.574594i \(-0.194840\pi\)
0.818438 + 0.574594i \(0.194840\pi\)
\(692\) 1.64286e6 0.130417
\(693\) 0 0
\(694\) 6.75529e6 0.532409
\(695\) 1.27716e6 0.100296
\(696\) 3.58000e7 2.80130
\(697\) 1.96656e6 0.153330
\(698\) 6.61930e6 0.514249
\(699\) 5.48549e6 0.424642
\(700\) 0 0
\(701\) −1.57106e7 −1.20753 −0.603764 0.797163i \(-0.706333\pi\)
−0.603764 + 0.797163i \(0.706333\pi\)
\(702\) −2.13368e7 −1.63413
\(703\) −1.96820e7 −1.50204
\(704\) 297910. 0.0226545
\(705\) −4.40551e6 −0.333829
\(706\) −5.05921e6 −0.382007
\(707\) 0 0
\(708\) −2.13266e7 −1.59896
\(709\) −6.48343e6 −0.484383 −0.242192 0.970228i \(-0.577866\pi\)
−0.242192 + 0.970228i \(0.577866\pi\)
\(710\) −2.24065e6 −0.166812
\(711\) 2.66796e7 1.97927
\(712\) −391859. −0.0289688
\(713\) −5.53979e6 −0.408103
\(714\) 0 0
\(715\) −1.37665e6 −0.100707
\(716\) 5.74654e6 0.418913
\(717\) −1.36821e7 −0.993930
\(718\) 1.51175e6 0.109438
\(719\) 4.83478e6 0.348782 0.174391 0.984676i \(-0.444204\pi\)
0.174391 + 0.984676i \(0.444204\pi\)
\(720\) −5.05352e6 −0.363298
\(721\) 0 0
\(722\) 347272. 0.0247929
\(723\) 3.00962e7 2.14124
\(724\) −4.88581e6 −0.346410
\(725\) 5.10565e6 0.360749
\(726\) 1.21778e7 0.857483
\(727\) −8.57824e6 −0.601953 −0.300976 0.953632i \(-0.597313\pi\)
−0.300976 + 0.953632i \(0.597313\pi\)
\(728\) 0 0
\(729\) 4.45997e6 0.310823
\(730\) 3.72502e6 0.258715
\(731\) −1.54322e7 −1.06816
\(732\) 1.38214e7 0.953397
\(733\) −863417. −0.0593555 −0.0296777 0.999560i \(-0.509448\pi\)
−0.0296777 + 0.999560i \(0.509448\pi\)
\(734\) 1.14441e7 0.784047
\(735\) 0 0
\(736\) 1.38796e7 0.944455
\(737\) 2.00872e6 0.136223
\(738\) 2.53299e6 0.171196
\(739\) 1.26850e7 0.854436 0.427218 0.904149i \(-0.359494\pi\)
0.427218 + 0.904149i \(0.359494\pi\)
\(740\) 7.86833e6 0.528206
\(741\) −3.77042e7 −2.52257
\(742\) 0 0
\(743\) 1.16786e7 0.776104 0.388052 0.921637i \(-0.373148\pi\)
0.388052 + 0.921637i \(0.373148\pi\)
\(744\) 1.03851e7 0.687824
\(745\) 7.67646e6 0.506723
\(746\) −6.28993e6 −0.413808
\(747\) 3.16403e7 2.07463
\(748\) 1.84961e6 0.120872
\(749\) 0 0
\(750\) 1.21174e6 0.0786607
\(751\) −6.46655e6 −0.418382 −0.209191 0.977875i \(-0.567083\pi\)
−0.209191 + 0.977875i \(0.567083\pi\)
\(752\) 2.24518e6 0.144779
\(753\) −4.05847e7 −2.60840
\(754\) −1.94097e7 −1.24334
\(755\) −8.95551e6 −0.571772
\(756\) 0 0
\(757\) 3.00203e6 0.190404 0.0952018 0.995458i \(-0.469650\pi\)
0.0952018 + 0.995458i \(0.469650\pi\)
\(758\) −9.73144e6 −0.615183
\(759\) −4.20169e6 −0.264740
\(760\) 5.92622e6 0.372172
\(761\) 2.83373e7 1.77377 0.886885 0.461989i \(-0.152864\pi\)
0.886885 + 0.461989i \(0.152864\pi\)
\(762\) −1.81385e7 −1.13165
\(763\) 0 0
\(764\) −7.37499e6 −0.457118
\(765\) 1.66539e7 1.02888
\(766\) −4.29909e6 −0.264731
\(767\) 2.66590e7 1.63627
\(768\) −1.79958e7 −1.10095
\(769\) 4.74192e6 0.289160 0.144580 0.989493i \(-0.453817\pi\)
0.144580 + 0.989493i \(0.453817\pi\)
\(770\) 0 0
\(771\) 1.39368e7 0.844356
\(772\) 4.24352e6 0.256261
\(773\) 1.37810e7 0.829530 0.414765 0.909928i \(-0.363864\pi\)
0.414765 + 0.909928i \(0.363864\pi\)
\(774\) −1.98772e7 −1.19262
\(775\) 1.48108e6 0.0885774
\(776\) −8.98187e6 −0.535442
\(777\) 0 0
\(778\) 3.65525e6 0.216505
\(779\) 2.53347e6 0.149580
\(780\) 1.50731e7 0.887086
\(781\) −2.07716e6 −0.121855
\(782\) −7.61211e6 −0.445131
\(783\) 7.33588e7 4.27610
\(784\) 0 0
\(785\) 1.09442e7 0.633885
\(786\) 1.40536e7 0.811392
\(787\) −1.22505e7 −0.705046 −0.352523 0.935803i \(-0.614676\pi\)
−0.352523 + 0.935803i \(0.614676\pi\)
\(788\) −2.47424e6 −0.141947
\(789\) −3.99812e7 −2.28646
\(790\) 3.26025e6 0.185859
\(791\) 0 0
\(792\) 5.49279e6 0.311157
\(793\) −1.72772e7 −0.975644
\(794\) −4.02659e6 −0.226666
\(795\) −1.41285e7 −0.792826
\(796\) −1.98301e7 −1.10928
\(797\) 1.42979e7 0.797310 0.398655 0.917101i \(-0.369477\pi\)
0.398655 + 0.917101i \(0.369477\pi\)
\(798\) 0 0
\(799\) −7.39902e6 −0.410022
\(800\) −3.71073e6 −0.204991
\(801\) −1.41865e6 −0.0781260
\(802\) −2.72242e6 −0.149458
\(803\) 3.45323e6 0.188989
\(804\) −2.19937e7 −1.19994
\(805\) 0 0
\(806\) −5.63048e6 −0.305287
\(807\) −2.40071e7 −1.29765
\(808\) −1.52220e7 −0.820245
\(809\) 3.38944e7 1.82078 0.910389 0.413753i \(-0.135782\pi\)
0.910389 + 0.413753i \(0.135782\pi\)
\(810\) 8.10106e6 0.433839
\(811\) 1.94505e7 1.03843 0.519216 0.854643i \(-0.326224\pi\)
0.519216 + 0.854643i \(0.326224\pi\)
\(812\) 0 0
\(813\) −3.25422e7 −1.72671
\(814\) −2.22923e6 −0.117922
\(815\) 1.07566e7 0.567260
\(816\) −1.21708e7 −0.639873
\(817\) −1.98809e7 −1.04203
\(818\) −2.72418e6 −0.142348
\(819\) 0 0
\(820\) −1.01281e6 −0.0526011
\(821\) 5.51562e6 0.285586 0.142793 0.989753i \(-0.454392\pi\)
0.142793 + 0.989753i \(0.454392\pi\)
\(822\) 1.58074e7 0.815981
\(823\) −2.72639e7 −1.40310 −0.701549 0.712621i \(-0.747508\pi\)
−0.701549 + 0.712621i \(0.747508\pi\)
\(824\) −2.77371e7 −1.42313
\(825\) 1.12333e6 0.0574609
\(826\) 0 0
\(827\) −5.97531e6 −0.303806 −0.151903 0.988395i \(-0.548540\pi\)
−0.151903 + 0.988395i \(0.548540\pi\)
\(828\) 3.20815e7 1.62622
\(829\) −6.28083e6 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(830\) 3.86645e6 0.194813
\(831\) 3.34037e7 1.67800
\(832\) 4.07742e6 0.204210
\(833\) 0 0
\(834\) 3.96183e6 0.197234
\(835\) 5.74339e6 0.285070
\(836\) 2.38280e6 0.117916
\(837\) 2.12803e7 1.04994
\(838\) 171112. 0.00841724
\(839\) 1.22748e7 0.602019 0.301010 0.953621i \(-0.402676\pi\)
0.301010 + 0.953621i \(0.402676\pi\)
\(840\) 0 0
\(841\) 4.62220e7 2.25350
\(842\) −8.93542e6 −0.434345
\(843\) 3.18113e6 0.154174
\(844\) −4.49274e6 −0.217098
\(845\) −9.55960e6 −0.460572
\(846\) −9.53016e6 −0.457798
\(847\) 0 0
\(848\) 7.20030e6 0.343844
\(849\) −4.73526e7 −2.25463
\(850\) 2.03511e6 0.0966143
\(851\) −3.00196e7 −1.42096
\(852\) 2.27430e7 1.07337
\(853\) −6.63619e6 −0.312281 −0.156141 0.987735i \(-0.549905\pi\)
−0.156141 + 0.987735i \(0.549905\pi\)
\(854\) 0 0
\(855\) 2.14548e7 1.00371
\(856\) 2.46583e7 1.15022
\(857\) 3.01234e7 1.40105 0.700523 0.713630i \(-0.252950\pi\)
0.700523 + 0.713630i \(0.252950\pi\)
\(858\) −4.27047e6 −0.198042
\(859\) 4.11124e7 1.90103 0.950517 0.310672i \(-0.100554\pi\)
0.950517 + 0.310672i \(0.100554\pi\)
\(860\) 7.94786e6 0.366441
\(861\) 0 0
\(862\) 5.61107e6 0.257204
\(863\) −3.41903e6 −0.156270 −0.0781351 0.996943i \(-0.524897\pi\)
−0.0781351 + 0.996943i \(0.524897\pi\)
\(864\) −5.33164e7 −2.42983
\(865\) 1.67574e6 0.0761492
\(866\) 1.71768e6 0.0778303
\(867\) −123782. −0.00559256
\(868\) 0 0
\(869\) 3.02237e6 0.135768
\(870\) 1.58380e7 0.709420
\(871\) 2.74929e7 1.22794
\(872\) −1.50867e7 −0.671897
\(873\) −3.25172e7 −1.44404
\(874\) −9.80647e6 −0.434244
\(875\) 0 0
\(876\) −3.78097e7 −1.66473
\(877\) −2.94070e7 −1.29108 −0.645538 0.763728i \(-0.723367\pi\)
−0.645538 + 0.763728i \(0.723367\pi\)
\(878\) 1.67844e7 0.734798
\(879\) −6.48105e7 −2.82926
\(880\) −572483. −0.0249204
\(881\) −3.32494e7 −1.44326 −0.721628 0.692281i \(-0.756606\pi\)
−0.721628 + 0.692281i \(0.756606\pi\)
\(882\) 0 0
\(883\) 2.53523e7 1.09425 0.547124 0.837051i \(-0.315722\pi\)
0.547124 + 0.837051i \(0.315722\pi\)
\(884\) 2.53151e7 1.08956
\(885\) −2.17534e7 −0.933616
\(886\) 5.71490e6 0.244582
\(887\) −1.95997e6 −0.0836450 −0.0418225 0.999125i \(-0.513316\pi\)
−0.0418225 + 0.999125i \(0.513316\pi\)
\(888\) 5.62757e7 2.39490
\(889\) 0 0
\(890\) −173360. −0.00733624
\(891\) 7.50997e6 0.316916
\(892\) −1.66389e7 −0.700183
\(893\) −9.53196e6 −0.399994
\(894\) 2.38129e7 0.996480
\(895\) 5.86154e6 0.244599
\(896\) 0 0
\(897\) −5.75075e7 −2.38640
\(898\) 1.85489e7 0.767587
\(899\) 1.93583e7 0.798857
\(900\) −8.57706e6 −0.352965
\(901\) −2.37287e7 −0.973782
\(902\) 286947. 0.0117432
\(903\) 0 0
\(904\) 2.17453e7 0.885002
\(905\) −4.98359e6 −0.202265
\(906\) −2.77806e7 −1.12440
\(907\) −8.69678e6 −0.351027 −0.175513 0.984477i \(-0.556159\pi\)
−0.175513 + 0.984477i \(0.556159\pi\)
\(908\) −6.28629e6 −0.253034
\(909\) −5.51086e7 −2.21212
\(910\) 0 0
\(911\) −3.87563e7 −1.54720 −0.773600 0.633674i \(-0.781546\pi\)
−0.773600 + 0.633674i \(0.781546\pi\)
\(912\) −1.56793e7 −0.624223
\(913\) 3.58434e6 0.142309
\(914\) −1.06690e7 −0.422435
\(915\) 1.40980e7 0.556679
\(916\) 7.88721e6 0.310588
\(917\) 0 0
\(918\) 2.92409e7 1.14521
\(919\) −1.97989e7 −0.773309 −0.386655 0.922225i \(-0.626369\pi\)
−0.386655 + 0.922225i \(0.626369\pi\)
\(920\) 9.03885e6 0.352082
\(921\) −1.27451e7 −0.495103
\(922\) 1.17085e7 0.453603
\(923\) −2.84296e7 −1.09841
\(924\) 0 0
\(925\) 8.02580e6 0.308414
\(926\) −1.96774e7 −0.754121
\(927\) −1.00417e8 −3.83804
\(928\) −4.85010e7 −1.84876
\(929\) −4.20864e7 −1.59993 −0.799967 0.600044i \(-0.795150\pi\)
−0.799967 + 0.600044i \(0.795150\pi\)
\(930\) 4.59439e6 0.174189
\(931\) 0 0
\(932\) −4.74476e6 −0.178927
\(933\) 1.71598e7 0.645368
\(934\) −1.56725e7 −0.587856
\(935\) 1.88662e6 0.0705758
\(936\) 7.51785e7 2.80481
\(937\) −2.14089e7 −0.796608 −0.398304 0.917254i \(-0.630401\pi\)
−0.398304 + 0.917254i \(0.630401\pi\)
\(938\) 0 0
\(939\) 5.75956e7 2.13170
\(940\) 3.81062e6 0.140662
\(941\) 2.30140e7 0.847264 0.423632 0.905834i \(-0.360755\pi\)
0.423632 + 0.905834i \(0.360755\pi\)
\(942\) 3.39497e7 1.24655
\(943\) 3.86412e6 0.141505
\(944\) 1.10862e7 0.404904
\(945\) 0 0
\(946\) −2.25177e6 −0.0818079
\(947\) −1.69350e7 −0.613636 −0.306818 0.951768i \(-0.599264\pi\)
−0.306818 + 0.951768i \(0.599264\pi\)
\(948\) −3.30922e7 −1.19593
\(949\) 4.72635e7 1.70357
\(950\) 2.62178e6 0.0942514
\(951\) −7.42228e7 −2.66125
\(952\) 0 0
\(953\) 2.73067e7 0.973952 0.486976 0.873415i \(-0.338100\pi\)
0.486976 + 0.873415i \(0.338100\pi\)
\(954\) −3.05633e7 −1.08725
\(955\) −7.52259e6 −0.266906
\(956\) 1.18346e7 0.418801
\(957\) 1.46824e7 0.518225
\(958\) −1.90720e6 −0.0671402
\(959\) 0 0
\(960\) −3.32712e6 −0.116517
\(961\) −2.30136e7 −0.803851
\(962\) −3.05110e7 −1.06296
\(963\) 8.92711e7 3.10202
\(964\) −2.60322e7 −0.902232
\(965\) 4.32844e6 0.149628
\(966\) 0 0
\(967\) −1.08494e7 −0.373111 −0.186555 0.982444i \(-0.559732\pi\)
−0.186555 + 0.982444i \(0.559732\pi\)
\(968\) −2.42858e7 −0.833038
\(969\) 5.16713e7 1.76783
\(970\) −3.97361e6 −0.135599
\(971\) −5.42053e7 −1.84499 −0.922495 0.386010i \(-0.873853\pi\)
−0.922495 + 0.386010i \(0.873853\pi\)
\(972\) −2.87435e7 −0.975830
\(973\) 0 0
\(974\) −2.05734e7 −0.694879
\(975\) 1.53748e7 0.517960
\(976\) −7.18476e6 −0.241428
\(977\) −4.07229e7 −1.36490 −0.682452 0.730930i \(-0.739086\pi\)
−0.682452 + 0.730930i \(0.739086\pi\)
\(978\) 3.33678e7 1.11553
\(979\) −160711. −0.00535905
\(980\) 0 0
\(981\) −5.46187e7 −1.81204
\(982\) 699382. 0.0231438
\(983\) −1.40230e7 −0.462869 −0.231434 0.972851i \(-0.574342\pi\)
−0.231434 + 0.972851i \(0.574342\pi\)
\(984\) −7.24381e6 −0.238495
\(985\) −2.52376e6 −0.0828815
\(986\) 2.65999e7 0.871339
\(987\) 0 0
\(988\) 3.26128e7 1.06291
\(989\) −3.03230e7 −0.985783
\(990\) 2.43003e6 0.0787995
\(991\) 3.52354e7 1.13971 0.569855 0.821745i \(-0.306999\pi\)
0.569855 + 0.821745i \(0.306999\pi\)
\(992\) −1.40695e7 −0.453939
\(993\) 5.08420e7 1.63625
\(994\) 0 0
\(995\) −2.02270e7 −0.647699
\(996\) −3.92453e7 −1.25354
\(997\) 1.29377e7 0.412210 0.206105 0.978530i \(-0.433921\pi\)
0.206105 + 0.978530i \(0.433921\pi\)
\(998\) −2.23055e7 −0.708901
\(999\) 1.15316e8 3.65574
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 245.6.a.e.1.2 4
7.6 odd 2 35.6.a.d.1.2 4
21.20 even 2 315.6.a.l.1.3 4
28.27 even 2 560.6.a.v.1.4 4
35.13 even 4 175.6.b.f.99.5 8
35.27 even 4 175.6.b.f.99.4 8
35.34 odd 2 175.6.a.f.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.a.d.1.2 4 7.6 odd 2
175.6.a.f.1.3 4 35.34 odd 2
175.6.b.f.99.4 8 35.27 even 4
175.6.b.f.99.5 8 35.13 even 4
245.6.a.e.1.2 4 1.1 even 1 trivial
315.6.a.l.1.3 4 21.20 even 2
560.6.a.v.1.4 4 28.27 even 2