Properties

Label 325.6.a.j.1.10
Level $325$
Weight $6$
Character 325.1
Self dual yes
Analytic conductor $52.125$
Analytic rank $0$
Dimension $11$
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] = [325,6,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, 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("325.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.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: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-9.60154\) of defining polynomial
Character \(\chi\) \(=\) 325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.60154 q^{2} -25.9222 q^{3} +60.1895 q^{4} -248.893 q^{6} -181.554 q^{7} +270.662 q^{8} +428.963 q^{9} +512.944 q^{11} -1560.25 q^{12} -169.000 q^{13} -1743.19 q^{14} +672.710 q^{16} -1668.15 q^{17} +4118.70 q^{18} -464.071 q^{19} +4706.28 q^{21} +4925.05 q^{22} +2095.04 q^{23} -7016.17 q^{24} -1622.66 q^{26} -4820.57 q^{27} -10927.6 q^{28} +5935.75 q^{29} +8397.76 q^{31} -2202.14 q^{32} -13296.7 q^{33} -16016.8 q^{34} +25819.0 q^{36} +8909.35 q^{37} -4455.79 q^{38} +4380.86 q^{39} +3256.85 q^{41} +45187.5 q^{42} +5951.12 q^{43} +30873.8 q^{44} +20115.6 q^{46} +10741.1 q^{47} -17438.1 q^{48} +16154.7 q^{49} +43242.3 q^{51} -10172.0 q^{52} +39344.1 q^{53} -46284.9 q^{54} -49139.7 q^{56} +12029.8 q^{57} +56992.4 q^{58} -3940.20 q^{59} -41477.5 q^{61} +80631.4 q^{62} -77879.7 q^{63} -42670.7 q^{64} -127668. q^{66} -18589.7 q^{67} -100405. q^{68} -54308.1 q^{69} +70348.8 q^{71} +116104. q^{72} +78602.6 q^{73} +85543.4 q^{74} -27932.2 q^{76} -93126.8 q^{77} +42063.0 q^{78} -71703.3 q^{79} +20722.1 q^{81} +31270.8 q^{82} -208.877 q^{83} +283268. q^{84} +57139.9 q^{86} -153868. q^{87} +138835. q^{88} -43537.5 q^{89} +30682.6 q^{91} +126099. q^{92} -217689. q^{93} +103131. q^{94} +57084.5 q^{96} -15298.0 q^{97} +155110. q^{98} +220034. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.60154 1.69733 0.848664 0.528933i \(-0.177408\pi\)
0.848664 + 0.528933i \(0.177408\pi\)
\(3\) −25.9222 −1.66291 −0.831456 0.555590i \(-0.812492\pi\)
−0.831456 + 0.555590i \(0.812492\pi\)
\(4\) 60.1895 1.88092
\(5\) 0 0
\(6\) −248.893 −2.82251
\(7\) −181.554 −1.40042 −0.700212 0.713935i \(-0.746911\pi\)
−0.700212 + 0.713935i \(0.746911\pi\)
\(8\) 270.662 1.49521
\(9\) 428.963 1.76528
\(10\) 0 0
\(11\) 512.944 1.27817 0.639084 0.769137i \(-0.279313\pi\)
0.639084 + 0.769137i \(0.279313\pi\)
\(12\) −1560.25 −3.12781
\(13\) −169.000 −0.277350
\(14\) −1743.19 −2.37698
\(15\) 0 0
\(16\) 672.710 0.656943
\(17\) −1668.15 −1.39995 −0.699977 0.714165i \(-0.746807\pi\)
−0.699977 + 0.714165i \(0.746807\pi\)
\(18\) 4118.70 2.99626
\(19\) −464.071 −0.294917 −0.147459 0.989068i \(-0.547109\pi\)
−0.147459 + 0.989068i \(0.547109\pi\)
\(20\) 0 0
\(21\) 4706.28 2.32878
\(22\) 4925.05 2.16947
\(23\) 2095.04 0.825795 0.412897 0.910778i \(-0.364517\pi\)
0.412897 + 0.910778i \(0.364517\pi\)
\(24\) −7016.17 −2.48641
\(25\) 0 0
\(26\) −1622.66 −0.470754
\(27\) −4820.57 −1.27259
\(28\) −10927.6 −2.63409
\(29\) 5935.75 1.31063 0.655316 0.755355i \(-0.272535\pi\)
0.655316 + 0.755355i \(0.272535\pi\)
\(30\) 0 0
\(31\) 8397.76 1.56949 0.784746 0.619817i \(-0.212793\pi\)
0.784746 + 0.619817i \(0.212793\pi\)
\(32\) −2202.14 −0.380164
\(33\) −13296.7 −2.12548
\(34\) −16016.8 −2.37618
\(35\) 0 0
\(36\) 25819.0 3.32035
\(37\) 8909.35 1.06990 0.534948 0.844885i \(-0.320331\pi\)
0.534948 + 0.844885i \(0.320331\pi\)
\(38\) −4455.79 −0.500571
\(39\) 4380.86 0.461209
\(40\) 0 0
\(41\) 3256.85 0.302579 0.151289 0.988490i \(-0.451657\pi\)
0.151289 + 0.988490i \(0.451657\pi\)
\(42\) 45187.5 3.95271
\(43\) 5951.12 0.490826 0.245413 0.969419i \(-0.421076\pi\)
0.245413 + 0.969419i \(0.421076\pi\)
\(44\) 30873.8 2.40413
\(45\) 0 0
\(46\) 20115.6 1.40164
\(47\) 10741.1 0.709255 0.354628 0.935008i \(-0.384608\pi\)
0.354628 + 0.935008i \(0.384608\pi\)
\(48\) −17438.1 −1.09244
\(49\) 16154.7 0.961190
\(50\) 0 0
\(51\) 43242.3 2.32800
\(52\) −10172.0 −0.521674
\(53\) 39344.1 1.92393 0.961967 0.273167i \(-0.0880710\pi\)
0.961967 + 0.273167i \(0.0880710\pi\)
\(54\) −46284.9 −2.16001
\(55\) 0 0
\(56\) −49139.7 −2.09393
\(57\) 12029.8 0.490422
\(58\) 56992.4 2.22457
\(59\) −3940.20 −0.147363 −0.0736815 0.997282i \(-0.523475\pi\)
−0.0736815 + 0.997282i \(0.523475\pi\)
\(60\) 0 0
\(61\) −41477.5 −1.42721 −0.713605 0.700548i \(-0.752939\pi\)
−0.713605 + 0.700548i \(0.752939\pi\)
\(62\) 80631.4 2.66394
\(63\) −77879.7 −2.47214
\(64\) −42670.7 −1.30221
\(65\) 0 0
\(66\) −127668. −3.60764
\(67\) −18589.7 −0.505923 −0.252961 0.967476i \(-0.581405\pi\)
−0.252961 + 0.967476i \(0.581405\pi\)
\(68\) −100405. −2.63320
\(69\) −54308.1 −1.37322
\(70\) 0 0
\(71\) 70348.8 1.65619 0.828096 0.560586i \(-0.189424\pi\)
0.828096 + 0.560586i \(0.189424\pi\)
\(72\) 116104. 2.63947
\(73\) 78602.6 1.72635 0.863176 0.504902i \(-0.168471\pi\)
0.863176 + 0.504902i \(0.168471\pi\)
\(74\) 85543.4 1.81596
\(75\) 0 0
\(76\) −27932.2 −0.554716
\(77\) −93126.8 −1.78998
\(78\) 42063.0 0.782823
\(79\) −71703.3 −1.29262 −0.646310 0.763075i \(-0.723689\pi\)
−0.646310 + 0.763075i \(0.723689\pi\)
\(80\) 0 0
\(81\) 20722.1 0.350931
\(82\) 31270.8 0.513575
\(83\) −208.877 −0.00332808 −0.00166404 0.999999i \(-0.500530\pi\)
−0.00166404 + 0.999999i \(0.500530\pi\)
\(84\) 283268. 4.38026
\(85\) 0 0
\(86\) 57139.9 0.833093
\(87\) −153868. −2.17947
\(88\) 138835. 1.91113
\(89\) −43537.5 −0.582624 −0.291312 0.956628i \(-0.594092\pi\)
−0.291312 + 0.956628i \(0.594092\pi\)
\(90\) 0 0
\(91\) 30682.6 0.388408
\(92\) 126099. 1.55325
\(93\) −217689. −2.60993
\(94\) 103131. 1.20384
\(95\) 0 0
\(96\) 57084.5 0.632179
\(97\) −15298.0 −0.165084 −0.0825419 0.996588i \(-0.526304\pi\)
−0.0825419 + 0.996588i \(0.526304\pi\)
\(98\) 155110. 1.63145
\(99\) 220034. 2.25632
\(100\) 0 0
\(101\) 2600.25 0.0253636 0.0126818 0.999920i \(-0.495963\pi\)
0.0126818 + 0.999920i \(0.495963\pi\)
\(102\) 415193. 3.95138
\(103\) 210878. 1.95857 0.979283 0.202497i \(-0.0649056\pi\)
0.979283 + 0.202497i \(0.0649056\pi\)
\(104\) −45741.9 −0.414697
\(105\) 0 0
\(106\) 377764. 3.26555
\(107\) −107005. −0.903531 −0.451765 0.892137i \(-0.649206\pi\)
−0.451765 + 0.892137i \(0.649206\pi\)
\(108\) −290148. −2.39365
\(109\) 151212. 1.21904 0.609522 0.792769i \(-0.291362\pi\)
0.609522 + 0.792769i \(0.291362\pi\)
\(110\) 0 0
\(111\) −230950. −1.77914
\(112\) −122133. −0.920000
\(113\) −176779. −1.30237 −0.651186 0.758918i \(-0.725728\pi\)
−0.651186 + 0.758918i \(0.725728\pi\)
\(114\) 115504. 0.832407
\(115\) 0 0
\(116\) 357270. 2.46520
\(117\) −72494.7 −0.489600
\(118\) −37832.0 −0.250123
\(119\) 302860. 1.96053
\(120\) 0 0
\(121\) 102060. 0.633715
\(122\) −398248. −2.42244
\(123\) −84424.9 −0.503162
\(124\) 505457. 2.95209
\(125\) 0 0
\(126\) −747765. −4.19603
\(127\) 3379.52 0.0185929 0.00929643 0.999957i \(-0.497041\pi\)
0.00929643 + 0.999957i \(0.497041\pi\)
\(128\) −339235. −1.83011
\(129\) −154266. −0.816201
\(130\) 0 0
\(131\) −165563. −0.842915 −0.421458 0.906848i \(-0.638481\pi\)
−0.421458 + 0.906848i \(0.638481\pi\)
\(132\) −800319. −3.99787
\(133\) 84253.8 0.413010
\(134\) −178489. −0.858717
\(135\) 0 0
\(136\) −451507. −2.09323
\(137\) −54105.0 −0.246284 −0.123142 0.992389i \(-0.539297\pi\)
−0.123142 + 0.992389i \(0.539297\pi\)
\(138\) −521441. −2.33081
\(139\) −248178. −1.08950 −0.544749 0.838599i \(-0.683375\pi\)
−0.544749 + 0.838599i \(0.683375\pi\)
\(140\) 0 0
\(141\) −278432. −1.17943
\(142\) 675457. 2.81110
\(143\) −86687.5 −0.354500
\(144\) 288567. 1.15969
\(145\) 0 0
\(146\) 754705. 2.93019
\(147\) −418767. −1.59838
\(148\) 536249. 2.01239
\(149\) 102637. 0.378736 0.189368 0.981906i \(-0.439356\pi\)
0.189368 + 0.981906i \(0.439356\pi\)
\(150\) 0 0
\(151\) 513812. 1.83384 0.916921 0.399068i \(-0.130666\pi\)
0.916921 + 0.399068i \(0.130666\pi\)
\(152\) −125606. −0.440964
\(153\) −715576. −2.47131
\(154\) −894161. −3.03818
\(155\) 0 0
\(156\) 263682. 0.867498
\(157\) 276514. 0.895300 0.447650 0.894209i \(-0.352261\pi\)
0.447650 + 0.894209i \(0.352261\pi\)
\(158\) −688461. −2.19400
\(159\) −1.01989e6 −3.19933
\(160\) 0 0
\(161\) −380362. −1.15646
\(162\) 198964. 0.595644
\(163\) −278424. −0.820801 −0.410401 0.911905i \(-0.634611\pi\)
−0.410401 + 0.911905i \(0.634611\pi\)
\(164\) 196028. 0.569127
\(165\) 0 0
\(166\) −2005.54 −0.00564885
\(167\) 407177. 1.12978 0.564888 0.825167i \(-0.308919\pi\)
0.564888 + 0.825167i \(0.308919\pi\)
\(168\) 1.27381e6 3.48203
\(169\) 28561.0 0.0769231
\(170\) 0 0
\(171\) −199069. −0.520611
\(172\) 358195. 0.923206
\(173\) −49123.8 −0.124789 −0.0623946 0.998052i \(-0.519874\pi\)
−0.0623946 + 0.998052i \(0.519874\pi\)
\(174\) −1.47737e6 −3.69927
\(175\) 0 0
\(176\) 345062. 0.839684
\(177\) 102139. 0.245052
\(178\) −418026. −0.988903
\(179\) 386317. 0.901179 0.450590 0.892731i \(-0.351214\pi\)
0.450590 + 0.892731i \(0.351214\pi\)
\(180\) 0 0
\(181\) −218966. −0.496799 −0.248400 0.968658i \(-0.579905\pi\)
−0.248400 + 0.968658i \(0.579905\pi\)
\(182\) 294600. 0.659256
\(183\) 1.07519e6 2.37333
\(184\) 567047. 1.23474
\(185\) 0 0
\(186\) −2.09015e6 −4.42991
\(187\) −855670. −1.78938
\(188\) 646499. 1.33405
\(189\) 875192. 1.78217
\(190\) 0 0
\(191\) 277244. 0.549894 0.274947 0.961459i \(-0.411340\pi\)
0.274947 + 0.961459i \(0.411340\pi\)
\(192\) 1.10612e6 2.16545
\(193\) −310407. −0.599844 −0.299922 0.953964i \(-0.596961\pi\)
−0.299922 + 0.953964i \(0.596961\pi\)
\(194\) −146884. −0.280201
\(195\) 0 0
\(196\) 972344. 1.80792
\(197\) −516120. −0.947513 −0.473757 0.880656i \(-0.657102\pi\)
−0.473757 + 0.880656i \(0.657102\pi\)
\(198\) 2.11266e6 3.82972
\(199\) 647020. 1.15820 0.579102 0.815255i \(-0.303403\pi\)
0.579102 + 0.815255i \(0.303403\pi\)
\(200\) 0 0
\(201\) 481886. 0.841306
\(202\) 24966.4 0.0430504
\(203\) −1.07766e6 −1.83544
\(204\) 2.60273e6 4.37879
\(205\) 0 0
\(206\) 2.02475e6 3.32433
\(207\) 898693. 1.45776
\(208\) −113688. −0.182203
\(209\) −238042. −0.376954
\(210\) 0 0
\(211\) 132130. 0.204312 0.102156 0.994768i \(-0.467426\pi\)
0.102156 + 0.994768i \(0.467426\pi\)
\(212\) 2.36810e6 3.61877
\(213\) −1.82360e6 −2.75410
\(214\) −1.02741e6 −1.53359
\(215\) 0 0
\(216\) −1.30475e6 −1.90279
\(217\) −1.52464e6 −2.19796
\(218\) 1.45186e6 2.06912
\(219\) −2.03756e6 −2.87077
\(220\) 0 0
\(221\) 281918. 0.388278
\(222\) −2.21748e6 −3.01979
\(223\) 919984. 1.23885 0.619424 0.785057i \(-0.287366\pi\)
0.619424 + 0.785057i \(0.287366\pi\)
\(224\) 399807. 0.532391
\(225\) 0 0
\(226\) −1.69735e6 −2.21055
\(227\) 30962.8 0.0398819 0.0199409 0.999801i \(-0.493652\pi\)
0.0199409 + 0.999801i \(0.493652\pi\)
\(228\) 724065. 0.922445
\(229\) 163086. 0.205508 0.102754 0.994707i \(-0.467235\pi\)
0.102754 + 0.994707i \(0.467235\pi\)
\(230\) 0 0
\(231\) 2.41406e6 2.97658
\(232\) 1.60658e6 1.95967
\(233\) −1.19712e6 −1.44460 −0.722298 0.691582i \(-0.756914\pi\)
−0.722298 + 0.691582i \(0.756914\pi\)
\(234\) −696060. −0.831012
\(235\) 0 0
\(236\) −237159. −0.277178
\(237\) 1.85871e6 2.14952
\(238\) 2.90792e6 3.32766
\(239\) −769561. −0.871462 −0.435731 0.900077i \(-0.643510\pi\)
−0.435731 + 0.900077i \(0.643510\pi\)
\(240\) 0 0
\(241\) −850451. −0.943206 −0.471603 0.881811i \(-0.656324\pi\)
−0.471603 + 0.881811i \(0.656324\pi\)
\(242\) 979937. 1.07562
\(243\) 634236. 0.689025
\(244\) −2.49651e6 −2.68447
\(245\) 0 0
\(246\) −810609. −0.854031
\(247\) 78428.0 0.0817954
\(248\) 2.27296e6 2.34672
\(249\) 5414.55 0.00553431
\(250\) 0 0
\(251\) 992979. 0.994846 0.497423 0.867508i \(-0.334280\pi\)
0.497423 + 0.867508i \(0.334280\pi\)
\(252\) −4.68754e6 −4.64990
\(253\) 1.07464e6 1.05550
\(254\) 32448.6 0.0315582
\(255\) 0 0
\(256\) −1.89172e6 −1.80408
\(257\) −1.29318e6 −1.22131 −0.610655 0.791897i \(-0.709094\pi\)
−0.610655 + 0.791897i \(0.709094\pi\)
\(258\) −1.48120e6 −1.38536
\(259\) −1.61752e6 −1.49831
\(260\) 0 0
\(261\) 2.54622e6 2.31363
\(262\) −1.58965e6 −1.43070
\(263\) 1.69375e6 1.50994 0.754972 0.655757i \(-0.227650\pi\)
0.754972 + 0.655757i \(0.227650\pi\)
\(264\) −3.59890e6 −3.17805
\(265\) 0 0
\(266\) 808966. 0.701013
\(267\) 1.12859e6 0.968852
\(268\) −1.11890e6 −0.951601
\(269\) −857080. −0.722172 −0.361086 0.932532i \(-0.617594\pi\)
−0.361086 + 0.932532i \(0.617594\pi\)
\(270\) 0 0
\(271\) 1.66147e6 1.37426 0.687130 0.726534i \(-0.258870\pi\)
0.687130 + 0.726534i \(0.258870\pi\)
\(272\) −1.12218e6 −0.919691
\(273\) −795361. −0.645889
\(274\) −519491. −0.418024
\(275\) 0 0
\(276\) −3.26877e6 −2.58293
\(277\) −2.15375e6 −1.68654 −0.843269 0.537491i \(-0.819372\pi\)
−0.843269 + 0.537491i \(0.819372\pi\)
\(278\) −2.38289e6 −1.84924
\(279\) 3.60233e6 2.77059
\(280\) 0 0
\(281\) −547631. −0.413735 −0.206868 0.978369i \(-0.566327\pi\)
−0.206868 + 0.978369i \(0.566327\pi\)
\(282\) −2.67338e6 −2.00188
\(283\) 609080. 0.452073 0.226036 0.974119i \(-0.427423\pi\)
0.226036 + 0.974119i \(0.427423\pi\)
\(284\) 4.23426e6 3.11517
\(285\) 0 0
\(286\) −832333. −0.601703
\(287\) −591293. −0.423739
\(288\) −944638. −0.671095
\(289\) 1.36288e6 0.959874
\(290\) 0 0
\(291\) 396558. 0.274520
\(292\) 4.73105e6 3.24713
\(293\) −927787. −0.631363 −0.315682 0.948865i \(-0.602233\pi\)
−0.315682 + 0.948865i \(0.602233\pi\)
\(294\) −4.02080e6 −2.71297
\(295\) 0 0
\(296\) 2.41142e6 1.59972
\(297\) −2.47268e6 −1.62659
\(298\) 985469. 0.642839
\(299\) −354061. −0.229034
\(300\) 0 0
\(301\) −1.08045e6 −0.687365
\(302\) 4.93339e6 3.11263
\(303\) −67404.3 −0.0421775
\(304\) −312185. −0.193744
\(305\) 0 0
\(306\) −6.87063e6 −4.19462
\(307\) 2.39073e6 1.44772 0.723859 0.689948i \(-0.242367\pi\)
0.723859 + 0.689948i \(0.242367\pi\)
\(308\) −5.60526e6 −3.36681
\(309\) −5.46643e6 −3.25692
\(310\) 0 0
\(311\) −1.27080e6 −0.745032 −0.372516 0.928026i \(-0.621505\pi\)
−0.372516 + 0.928026i \(0.621505\pi\)
\(312\) 1.18573e6 0.689605
\(313\) 1.69293e6 0.976738 0.488369 0.872637i \(-0.337592\pi\)
0.488369 + 0.872637i \(0.337592\pi\)
\(314\) 2.65496e6 1.51962
\(315\) 0 0
\(316\) −4.31578e6 −2.43132
\(317\) 958135. 0.535523 0.267762 0.963485i \(-0.413716\pi\)
0.267762 + 0.963485i \(0.413716\pi\)
\(318\) −9.79249e6 −5.43032
\(319\) 3.04471e6 1.67521
\(320\) 0 0
\(321\) 2.77380e6 1.50249
\(322\) −3.65205e6 −1.96290
\(323\) 774142. 0.412871
\(324\) 1.24725e6 0.660073
\(325\) 0 0
\(326\) −2.67330e6 −1.39317
\(327\) −3.91975e6 −2.02716
\(328\) 881507. 0.452419
\(329\) −1.95008e6 −0.993259
\(330\) 0 0
\(331\) 2.29139e6 1.14955 0.574777 0.818310i \(-0.305089\pi\)
0.574777 + 0.818310i \(0.305089\pi\)
\(332\) −12572.2 −0.00625986
\(333\) 3.82178e6 1.88866
\(334\) 3.90953e6 1.91760
\(335\) 0 0
\(336\) 3.16596e6 1.52988
\(337\) 1.20806e6 0.579446 0.289723 0.957110i \(-0.406437\pi\)
0.289723 + 0.957110i \(0.406437\pi\)
\(338\) 274229. 0.130564
\(339\) 4.58251e6 2.16573
\(340\) 0 0
\(341\) 4.30758e6 2.00608
\(342\) −1.91137e6 −0.883648
\(343\) 118424. 0.0543505
\(344\) 1.61074e6 0.733889
\(345\) 0 0
\(346\) −471664. −0.211808
\(347\) 4.27369e6 1.90537 0.952684 0.303962i \(-0.0983097\pi\)
0.952684 + 0.303962i \(0.0983097\pi\)
\(348\) −9.26124e6 −4.09941
\(349\) −1.49638e6 −0.657625 −0.328812 0.944395i \(-0.606648\pi\)
−0.328812 + 0.944395i \(0.606648\pi\)
\(350\) 0 0
\(351\) 814677. 0.352954
\(352\) −1.12958e6 −0.485914
\(353\) −730249. −0.311914 −0.155957 0.987764i \(-0.549846\pi\)
−0.155957 + 0.987764i \(0.549846\pi\)
\(354\) 980690. 0.415933
\(355\) 0 0
\(356\) −2.62050e6 −1.09587
\(357\) −7.85080e6 −3.26019
\(358\) 3.70924e6 1.52960
\(359\) 1.04399e6 0.427522 0.213761 0.976886i \(-0.431429\pi\)
0.213761 + 0.976886i \(0.431429\pi\)
\(360\) 0 0
\(361\) −2.26074e6 −0.913024
\(362\) −2.10241e6 −0.843231
\(363\) −2.64564e6 −1.05381
\(364\) 1.84677e6 0.730565
\(365\) 0 0
\(366\) 1.03235e7 4.02831
\(367\) 75157.4 0.0291277 0.0145639 0.999894i \(-0.495364\pi\)
0.0145639 + 0.999894i \(0.495364\pi\)
\(368\) 1.40935e6 0.542500
\(369\) 1.39707e6 0.534136
\(370\) 0 0
\(371\) −7.14307e6 −2.69432
\(372\) −1.31026e7 −4.90907
\(373\) 1.44619e6 0.538212 0.269106 0.963111i \(-0.413272\pi\)
0.269106 + 0.963111i \(0.413272\pi\)
\(374\) −8.21574e6 −3.03716
\(375\) 0 0
\(376\) 2.90720e6 1.06049
\(377\) −1.00314e6 −0.363504
\(378\) 8.40319e6 3.02493
\(379\) 291357. 0.104190 0.0520951 0.998642i \(-0.483410\pi\)
0.0520951 + 0.998642i \(0.483410\pi\)
\(380\) 0 0
\(381\) −87604.8 −0.0309183
\(382\) 2.66197e6 0.933350
\(383\) 3.40227e6 1.18515 0.592573 0.805517i \(-0.298112\pi\)
0.592573 + 0.805517i \(0.298112\pi\)
\(384\) 8.79374e6 3.04331
\(385\) 0 0
\(386\) −2.98038e6 −1.01813
\(387\) 2.55281e6 0.866445
\(388\) −920777. −0.310510
\(389\) 1.64309e6 0.550537 0.275268 0.961367i \(-0.411233\pi\)
0.275268 + 0.961367i \(0.411233\pi\)
\(390\) 0 0
\(391\) −3.49485e6 −1.15608
\(392\) 4.37247e6 1.43718
\(393\) 4.29175e6 1.40169
\(394\) −4.95555e6 −1.60824
\(395\) 0 0
\(396\) 1.32437e7 4.24397
\(397\) 2.47328e6 0.787585 0.393792 0.919199i \(-0.371163\pi\)
0.393792 + 0.919199i \(0.371163\pi\)
\(398\) 6.21239e6 1.96585
\(399\) −2.18405e6 −0.686799
\(400\) 0 0
\(401\) 4.41469e6 1.37100 0.685502 0.728071i \(-0.259583\pi\)
0.685502 + 0.728071i \(0.259583\pi\)
\(402\) 4.62684e6 1.42797
\(403\) −1.41922e6 −0.435299
\(404\) 156508. 0.0477070
\(405\) 0 0
\(406\) −1.03472e7 −3.11535
\(407\) 4.56999e6 1.36751
\(408\) 1.17041e7 3.48086
\(409\) −2.67344e6 −0.790244 −0.395122 0.918629i \(-0.629298\pi\)
−0.395122 + 0.918629i \(0.629298\pi\)
\(410\) 0 0
\(411\) 1.40252e6 0.409549
\(412\) 1.26926e7 3.68391
\(413\) 715358. 0.206371
\(414\) 8.62883e6 2.47429
\(415\) 0 0
\(416\) 372162. 0.105439
\(417\) 6.43334e6 1.81174
\(418\) −2.28557e6 −0.639815
\(419\) 2.27577e6 0.633277 0.316639 0.948546i \(-0.397446\pi\)
0.316639 + 0.948546i \(0.397446\pi\)
\(420\) 0 0
\(421\) 4.79147e6 1.31754 0.658770 0.752345i \(-0.271077\pi\)
0.658770 + 0.752345i \(0.271077\pi\)
\(422\) 1.26865e6 0.346785
\(423\) 4.60751e6 1.25203
\(424\) 1.06490e7 2.87669
\(425\) 0 0
\(426\) −1.75093e7 −4.67462
\(427\) 7.53039e6 1.99870
\(428\) −6.44055e6 −1.69947
\(429\) 2.24714e6 0.589503
\(430\) 0 0
\(431\) −2.09885e6 −0.544238 −0.272119 0.962264i \(-0.587725\pi\)
−0.272119 + 0.962264i \(0.587725\pi\)
\(432\) −3.24285e6 −0.836021
\(433\) 2.53735e6 0.650370 0.325185 0.945650i \(-0.394573\pi\)
0.325185 + 0.945650i \(0.394573\pi\)
\(434\) −1.46389e7 −3.73065
\(435\) 0 0
\(436\) 9.10135e6 2.29292
\(437\) −972246. −0.243541
\(438\) −1.95637e7 −4.87264
\(439\) 3.67042e6 0.908979 0.454490 0.890752i \(-0.349822\pi\)
0.454490 + 0.890752i \(0.349822\pi\)
\(440\) 0 0
\(441\) 6.92977e6 1.69677
\(442\) 2.70685e6 0.659034
\(443\) −2.32270e6 −0.562321 −0.281161 0.959661i \(-0.590719\pi\)
−0.281161 + 0.959661i \(0.590719\pi\)
\(444\) −1.39008e7 −3.34643
\(445\) 0 0
\(446\) 8.83325e6 2.10273
\(447\) −2.66057e6 −0.629805
\(448\) 7.74702e6 1.82364
\(449\) −4.30759e6 −1.00837 −0.504183 0.863597i \(-0.668206\pi\)
−0.504183 + 0.863597i \(0.668206\pi\)
\(450\) 0 0
\(451\) 1.67058e6 0.386747
\(452\) −1.06403e7 −2.44966
\(453\) −1.33192e7 −3.04952
\(454\) 297290. 0.0676926
\(455\) 0 0
\(456\) 3.25600e6 0.733285
\(457\) 1.14361e6 0.256145 0.128073 0.991765i \(-0.459121\pi\)
0.128073 + 0.991765i \(0.459121\pi\)
\(458\) 1.56588e6 0.348814
\(459\) 8.04146e6 1.78157
\(460\) 0 0
\(461\) −3.73288e6 −0.818071 −0.409036 0.912518i \(-0.634135\pi\)
−0.409036 + 0.912518i \(0.634135\pi\)
\(462\) 2.31786e7 5.05223
\(463\) 4.46756e6 0.968540 0.484270 0.874919i \(-0.339085\pi\)
0.484270 + 0.874919i \(0.339085\pi\)
\(464\) 3.99304e6 0.861011
\(465\) 0 0
\(466\) −1.14942e7 −2.45195
\(467\) −5.73689e6 −1.21726 −0.608632 0.793453i \(-0.708281\pi\)
−0.608632 + 0.793453i \(0.708281\pi\)
\(468\) −4.36342e6 −0.920900
\(469\) 3.37502e6 0.708507
\(470\) 0 0
\(471\) −7.16787e6 −1.48881
\(472\) −1.06646e6 −0.220339
\(473\) 3.05259e6 0.627359
\(474\) 1.78465e7 3.64843
\(475\) 0 0
\(476\) 1.82290e7 3.68761
\(477\) 1.68772e7 3.39628
\(478\) −7.38897e6 −1.47916
\(479\) −8.44282e6 −1.68131 −0.840657 0.541567i \(-0.817831\pi\)
−0.840657 + 0.541567i \(0.817831\pi\)
\(480\) 0 0
\(481\) −1.50568e6 −0.296736
\(482\) −8.16563e6 −1.60093
\(483\) 9.85982e6 1.92310
\(484\) 6.14297e6 1.19197
\(485\) 0 0
\(486\) 6.08964e6 1.16950
\(487\) −2.30080e6 −0.439599 −0.219799 0.975545i \(-0.570540\pi\)
−0.219799 + 0.975545i \(0.570540\pi\)
\(488\) −1.12264e7 −2.13398
\(489\) 7.21738e6 1.36492
\(490\) 0 0
\(491\) −1.83374e6 −0.343269 −0.171635 0.985161i \(-0.554905\pi\)
−0.171635 + 0.985161i \(0.554905\pi\)
\(492\) −5.08149e6 −0.946408
\(493\) −9.90176e6 −1.83483
\(494\) 753029. 0.138834
\(495\) 0 0
\(496\) 5.64926e6 1.03107
\(497\) −1.27721e7 −2.31937
\(498\) 51988.0 0.00939354
\(499\) −1.65708e6 −0.297916 −0.148958 0.988844i \(-0.547592\pi\)
−0.148958 + 0.988844i \(0.547592\pi\)
\(500\) 0 0
\(501\) −1.05550e7 −1.87872
\(502\) 9.53412e6 1.68858
\(503\) 674839. 0.118927 0.0594634 0.998230i \(-0.481061\pi\)
0.0594634 + 0.998230i \(0.481061\pi\)
\(504\) −2.10791e7 −3.69637
\(505\) 0 0
\(506\) 1.03182e7 1.79154
\(507\) −740365. −0.127916
\(508\) 203412. 0.0349717
\(509\) 214603. 0.0367148 0.0183574 0.999831i \(-0.494156\pi\)
0.0183574 + 0.999831i \(0.494156\pi\)
\(510\) 0 0
\(511\) −1.42706e7 −2.41763
\(512\) −7.30787e6 −1.23202
\(513\) 2.23709e6 0.375310
\(514\) −1.24165e7 −2.07296
\(515\) 0 0
\(516\) −9.28522e6 −1.53521
\(517\) 5.50956e6 0.906548
\(518\) −1.55307e7 −2.54312
\(519\) 1.27340e6 0.207513
\(520\) 0 0
\(521\) 9.75675e6 1.57475 0.787373 0.616476i \(-0.211440\pi\)
0.787373 + 0.616476i \(0.211440\pi\)
\(522\) 2.44476e7 3.92699
\(523\) −1.41439e6 −0.226107 −0.113054 0.993589i \(-0.536063\pi\)
−0.113054 + 0.993589i \(0.536063\pi\)
\(524\) −9.96512e6 −1.58546
\(525\) 0 0
\(526\) 1.62626e7 2.56287
\(527\) −1.40088e7 −2.19722
\(528\) −8.94479e6 −1.39632
\(529\) −2.04716e6 −0.318063
\(530\) 0 0
\(531\) −1.69020e6 −0.260137
\(532\) 5.07119e6 0.776839
\(533\) −550408. −0.0839203
\(534\) 1.08362e7 1.64446
\(535\) 0 0
\(536\) −5.03152e6 −0.756462
\(537\) −1.00142e7 −1.49858
\(538\) −8.22929e6 −1.22576
\(539\) 8.28647e6 1.22856
\(540\) 0 0
\(541\) 6.38226e6 0.937522 0.468761 0.883325i \(-0.344700\pi\)
0.468761 + 0.883325i \(0.344700\pi\)
\(542\) 1.59527e7 2.33257
\(543\) 5.67610e6 0.826134
\(544\) 3.67352e6 0.532212
\(545\) 0 0
\(546\) −7.63669e6 −1.09628
\(547\) −6.98550e6 −0.998227 −0.499113 0.866537i \(-0.666341\pi\)
−0.499113 + 0.866537i \(0.666341\pi\)
\(548\) −3.25655e6 −0.463241
\(549\) −1.77923e7 −2.51943
\(550\) 0 0
\(551\) −2.75461e6 −0.386528
\(552\) −1.46991e7 −2.05326
\(553\) 1.30180e7 1.81022
\(554\) −2.06793e7 −2.86261
\(555\) 0 0
\(556\) −1.49377e7 −2.04926
\(557\) −8.50622e6 −1.16171 −0.580856 0.814006i \(-0.697282\pi\)
−0.580856 + 0.814006i \(0.697282\pi\)
\(558\) 3.45879e7 4.70260
\(559\) −1.00574e6 −0.136131
\(560\) 0 0
\(561\) 2.21809e7 2.97558
\(562\) −5.25810e6 −0.702244
\(563\) 5.22994e6 0.695386 0.347693 0.937608i \(-0.386965\pi\)
0.347693 + 0.937608i \(0.386965\pi\)
\(564\) −1.67587e7 −2.21841
\(565\) 0 0
\(566\) 5.84810e6 0.767315
\(567\) −3.76217e6 −0.491452
\(568\) 1.90408e7 2.47636
\(569\) 1.12412e7 1.45556 0.727781 0.685809i \(-0.240552\pi\)
0.727781 + 0.685809i \(0.240552\pi\)
\(570\) 0 0
\(571\) −3.42903e6 −0.440130 −0.220065 0.975485i \(-0.570627\pi\)
−0.220065 + 0.975485i \(0.570627\pi\)
\(572\) −5.21768e6 −0.666787
\(573\) −7.18679e6 −0.914426
\(574\) −5.67732e6 −0.719224
\(575\) 0 0
\(576\) −1.83041e7 −2.29876
\(577\) −1.39888e7 −1.74920 −0.874600 0.484845i \(-0.838876\pi\)
−0.874600 + 0.484845i \(0.838876\pi\)
\(578\) 1.30858e7 1.62922
\(579\) 8.04645e6 0.997488
\(580\) 0 0
\(581\) 37922.3 0.00466073
\(582\) 3.80756e6 0.465950
\(583\) 2.01813e7 2.45911
\(584\) 2.12747e7 2.58126
\(585\) 0 0
\(586\) −8.90818e6 −1.07163
\(587\) 2.19504e6 0.262934 0.131467 0.991321i \(-0.458031\pi\)
0.131467 + 0.991321i \(0.458031\pi\)
\(588\) −2.52053e7 −3.00642
\(589\) −3.89716e6 −0.462871
\(590\) 0 0
\(591\) 1.33790e7 1.57563
\(592\) 5.99340e6 0.702861
\(593\) 1.41425e6 0.165154 0.0825772 0.996585i \(-0.473685\pi\)
0.0825772 + 0.996585i \(0.473685\pi\)
\(594\) −2.37416e7 −2.76085
\(595\) 0 0
\(596\) 6.17765e6 0.712373
\(597\) −1.67722e7 −1.92599
\(598\) −3.39953e6 −0.388746
\(599\) −9.66661e6 −1.10080 −0.550398 0.834902i \(-0.685524\pi\)
−0.550398 + 0.834902i \(0.685524\pi\)
\(600\) 0 0
\(601\) −8.23263e6 −0.929721 −0.464860 0.885384i \(-0.653895\pi\)
−0.464860 + 0.885384i \(0.653895\pi\)
\(602\) −1.03740e7 −1.16668
\(603\) −7.97427e6 −0.893095
\(604\) 3.09261e7 3.44931
\(605\) 0 0
\(606\) −647185. −0.0715891
\(607\) 7.61203e6 0.838549 0.419275 0.907859i \(-0.362284\pi\)
0.419275 + 0.907859i \(0.362284\pi\)
\(608\) 1.02195e6 0.112117
\(609\) 2.79353e7 3.05218
\(610\) 0 0
\(611\) −1.81524e6 −0.196712
\(612\) −4.30702e7 −4.64834
\(613\) 1.04544e7 1.12369 0.561844 0.827243i \(-0.310092\pi\)
0.561844 + 0.827243i \(0.310092\pi\)
\(614\) 2.29546e7 2.45725
\(615\) 0 0
\(616\) −2.52059e7 −2.67640
\(617\) −8.78600e6 −0.929134 −0.464567 0.885538i \(-0.653790\pi\)
−0.464567 + 0.885538i \(0.653790\pi\)
\(618\) −5.24861e7 −5.52807
\(619\) 8.05994e6 0.845484 0.422742 0.906250i \(-0.361068\pi\)
0.422742 + 0.906250i \(0.361068\pi\)
\(620\) 0 0
\(621\) −1.00993e7 −1.05090
\(622\) −1.22016e7 −1.26456
\(623\) 7.90438e6 0.815921
\(624\) 2.94705e6 0.302988
\(625\) 0 0
\(626\) 1.62547e7 1.65784
\(627\) 6.17059e6 0.626842
\(628\) 1.66433e7 1.68399
\(629\) −1.48622e7 −1.49781
\(630\) 0 0
\(631\) 1.42505e6 0.142481 0.0712404 0.997459i \(-0.477304\pi\)
0.0712404 + 0.997459i \(0.477304\pi\)
\(632\) −1.94074e7 −1.93274
\(633\) −3.42510e6 −0.339754
\(634\) 9.19956e6 0.908959
\(635\) 0 0
\(636\) −6.13865e7 −6.01769
\(637\) −2.73015e6 −0.266586
\(638\) 2.92339e7 2.84338
\(639\) 3.01770e7 2.92364
\(640\) 0 0
\(641\) 1.11250e6 0.106944 0.0534718 0.998569i \(-0.482971\pi\)
0.0534718 + 0.998569i \(0.482971\pi\)
\(642\) 2.66327e7 2.55022
\(643\) −9.07113e6 −0.865235 −0.432617 0.901578i \(-0.642410\pi\)
−0.432617 + 0.901578i \(0.642410\pi\)
\(644\) −2.28938e7 −2.17522
\(645\) 0 0
\(646\) 7.43295e6 0.700777
\(647\) 2.19842e6 0.206467 0.103234 0.994657i \(-0.467081\pi\)
0.103234 + 0.994657i \(0.467081\pi\)
\(648\) 5.60869e6 0.524716
\(649\) −2.02110e6 −0.188355
\(650\) 0 0
\(651\) 3.95222e7 3.65501
\(652\) −1.67582e7 −1.54386
\(653\) −1.42857e7 −1.31105 −0.655525 0.755173i \(-0.727553\pi\)
−0.655525 + 0.755173i \(0.727553\pi\)
\(654\) −3.76356e7 −3.44076
\(655\) 0 0
\(656\) 2.19092e6 0.198777
\(657\) 3.37176e7 3.04749
\(658\) −1.87237e7 −1.68589
\(659\) 1.54265e7 1.38374 0.691869 0.722023i \(-0.256787\pi\)
0.691869 + 0.722023i \(0.256787\pi\)
\(660\) 0 0
\(661\) 1.65514e7 1.47344 0.736719 0.676199i \(-0.236374\pi\)
0.736719 + 0.676199i \(0.236374\pi\)
\(662\) 2.20009e7 1.95117
\(663\) −7.30795e6 −0.645672
\(664\) −56535.0 −0.00497619
\(665\) 0 0
\(666\) 3.66949e7 3.20568
\(667\) 1.24356e7 1.08231
\(668\) 2.45078e7 2.12502
\(669\) −2.38480e7 −2.06010
\(670\) 0 0
\(671\) −2.12756e7 −1.82422
\(672\) −1.03639e7 −0.885320
\(673\) 8.91603e6 0.758811 0.379406 0.925230i \(-0.376129\pi\)
0.379406 + 0.925230i \(0.376129\pi\)
\(674\) 1.15992e7 0.983510
\(675\) 0 0
\(676\) 1.71907e6 0.144686
\(677\) −8.15442e6 −0.683788 −0.341894 0.939739i \(-0.611068\pi\)
−0.341894 + 0.939739i \(0.611068\pi\)
\(678\) 4.39992e7 3.67596
\(679\) 2.77740e6 0.231187
\(680\) 0 0
\(681\) −802625. −0.0663200
\(682\) 4.13594e7 3.40497
\(683\) 2.02314e6 0.165948 0.0829742 0.996552i \(-0.473558\pi\)
0.0829742 + 0.996552i \(0.473558\pi\)
\(684\) −1.19819e7 −0.979229
\(685\) 0 0
\(686\) 1.13705e6 0.0922507
\(687\) −4.22756e6 −0.341742
\(688\) 4.00338e6 0.322445
\(689\) −6.64915e6 −0.533603
\(690\) 0 0
\(691\) −5.61163e6 −0.447089 −0.223545 0.974694i \(-0.571763\pi\)
−0.223545 + 0.974694i \(0.571763\pi\)
\(692\) −2.95674e6 −0.234719
\(693\) −3.99479e7 −3.15981
\(694\) 4.10339e7 3.23403
\(695\) 0 0
\(696\) −4.16463e7 −3.25877
\(697\) −5.43293e6 −0.423597
\(698\) −1.43675e7 −1.11620
\(699\) 3.10319e7 2.40224
\(700\) 0 0
\(701\) −5.09127e6 −0.391319 −0.195660 0.980672i \(-0.562685\pi\)
−0.195660 + 0.980672i \(0.562685\pi\)
\(702\) 7.82215e6 0.599078
\(703\) −4.13457e6 −0.315531
\(704\) −2.18877e7 −1.66444
\(705\) 0 0
\(706\) −7.01152e6 −0.529420
\(707\) −472085. −0.0355199
\(708\) 6.14769e6 0.460923
\(709\) 1.54906e7 1.15732 0.578658 0.815570i \(-0.303577\pi\)
0.578658 + 0.815570i \(0.303577\pi\)
\(710\) 0 0
\(711\) −3.07580e7 −2.28184
\(712\) −1.17839e7 −0.871146
\(713\) 1.75936e7 1.29608
\(714\) −7.53797e7 −5.53362
\(715\) 0 0
\(716\) 2.32522e7 1.69505
\(717\) 1.99488e7 1.44916
\(718\) 1.00239e7 0.725645
\(719\) −1.12604e7 −0.812328 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(720\) 0 0
\(721\) −3.82857e7 −2.74282
\(722\) −2.17065e7 −1.54970
\(723\) 2.20456e7 1.56847
\(724\) −1.31795e7 −0.934440
\(725\) 0 0
\(726\) −2.54022e7 −1.78867
\(727\) −8.14136e6 −0.571296 −0.285648 0.958335i \(-0.592209\pi\)
−0.285648 + 0.958335i \(0.592209\pi\)
\(728\) 8.30461e6 0.580752
\(729\) −2.14763e7 −1.49672
\(730\) 0 0
\(731\) −9.92740e6 −0.687135
\(732\) 6.47151e7 4.46404
\(733\) 1.30341e7 0.896025 0.448013 0.894027i \(-0.352132\pi\)
0.448013 + 0.894027i \(0.352132\pi\)
\(734\) 721626. 0.0494393
\(735\) 0 0
\(736\) −4.61357e6 −0.313937
\(737\) −9.53545e6 −0.646655
\(738\) 1.34140e7 0.906604
\(739\) −1.95666e7 −1.31796 −0.658981 0.752159i \(-0.729012\pi\)
−0.658981 + 0.752159i \(0.729012\pi\)
\(740\) 0 0
\(741\) −2.03303e6 −0.136019
\(742\) −6.85844e7 −4.57315
\(743\) 1.78971e6 0.118935 0.0594675 0.998230i \(-0.481060\pi\)
0.0594675 + 0.998230i \(0.481060\pi\)
\(744\) −5.89201e7 −3.90240
\(745\) 0 0
\(746\) 1.38857e7 0.913523
\(747\) −89600.2 −0.00587500
\(748\) −5.15023e7 −3.36568
\(749\) 1.94271e7 1.26533
\(750\) 0 0
\(751\) −2.13528e7 −1.38151 −0.690755 0.723089i \(-0.742722\pi\)
−0.690755 + 0.723089i \(0.742722\pi\)
\(752\) 7.22562e6 0.465940
\(753\) −2.57402e7 −1.65434
\(754\) −9.63171e6 −0.616985
\(755\) 0 0
\(756\) 5.26774e7 3.35212
\(757\) 7.21618e6 0.457686 0.228843 0.973463i \(-0.426506\pi\)
0.228843 + 0.973463i \(0.426506\pi\)
\(758\) 2.79747e6 0.176845
\(759\) −2.78570e7 −1.75521
\(760\) 0 0
\(761\) −1.18696e7 −0.742974 −0.371487 0.928438i \(-0.621152\pi\)
−0.371487 + 0.928438i \(0.621152\pi\)
\(762\) −841141. −0.0524785
\(763\) −2.74530e7 −1.70718
\(764\) 1.66872e7 1.03431
\(765\) 0 0
\(766\) 3.26670e7 2.01158
\(767\) 665894. 0.0408711
\(768\) 4.90376e7 3.00003
\(769\) −2.02111e7 −1.23247 −0.616233 0.787564i \(-0.711342\pi\)
−0.616233 + 0.787564i \(0.711342\pi\)
\(770\) 0 0
\(771\) 3.35221e7 2.03093
\(772\) −1.86832e7 −1.12826
\(773\) 4.38847e6 0.264158 0.132079 0.991239i \(-0.457835\pi\)
0.132079 + 0.991239i \(0.457835\pi\)
\(774\) 2.45109e7 1.47064
\(775\) 0 0
\(776\) −4.14058e6 −0.246835
\(777\) 4.19299e7 2.49156
\(778\) 1.57762e7 0.934442
\(779\) −1.51141e6 −0.0892357
\(780\) 0 0
\(781\) 3.60850e7 2.11689
\(782\) −3.35559e7 −1.96224
\(783\) −2.86137e7 −1.66790
\(784\) 1.08674e7 0.631447
\(785\) 0 0
\(786\) 4.12074e7 2.37913
\(787\) −4.56154e6 −0.262527 −0.131264 0.991347i \(-0.541903\pi\)
−0.131264 + 0.991347i \(0.541903\pi\)
\(788\) −3.10650e7 −1.78220
\(789\) −4.39059e7 −2.51090
\(790\) 0 0
\(791\) 3.20949e7 1.82387
\(792\) 5.95549e7 3.37368
\(793\) 7.00970e6 0.395837
\(794\) 2.37473e7 1.33679
\(795\) 0 0
\(796\) 3.89438e7 2.17849
\(797\) 1.86077e7 1.03764 0.518820 0.854883i \(-0.326371\pi\)
0.518820 + 0.854883i \(0.326371\pi\)
\(798\) −2.09702e7 −1.16572
\(799\) −1.79177e7 −0.992925
\(800\) 0 0
\(801\) −1.86759e7 −1.02849
\(802\) 4.23878e7 2.32704
\(803\) 4.03187e7 2.20657
\(804\) 2.90044e7 1.58243
\(805\) 0 0
\(806\) −1.36267e7 −0.738845
\(807\) 2.22175e7 1.20091
\(808\) 703789. 0.0379240
\(809\) −1.95576e7 −1.05062 −0.525309 0.850911i \(-0.676050\pi\)
−0.525309 + 0.850911i \(0.676050\pi\)
\(810\) 0 0
\(811\) −1.88885e7 −1.00843 −0.504214 0.863579i \(-0.668218\pi\)
−0.504214 + 0.863579i \(0.668218\pi\)
\(812\) −6.48637e7 −3.45232
\(813\) −4.30690e7 −2.28528
\(814\) 4.38790e7 2.32111
\(815\) 0 0
\(816\) 2.90895e7 1.52937
\(817\) −2.76174e6 −0.144753
\(818\) −2.56691e7 −1.34130
\(819\) 1.31617e7 0.685648
\(820\) 0 0
\(821\) 7.00926e6 0.362923 0.181461 0.983398i \(-0.441917\pi\)
0.181461 + 0.983398i \(0.441917\pi\)
\(822\) 1.34664e7 0.695138
\(823\) −2.33685e7 −1.20263 −0.601313 0.799014i \(-0.705355\pi\)
−0.601313 + 0.799014i \(0.705355\pi\)
\(824\) 5.70767e7 2.92847
\(825\) 0 0
\(826\) 6.86853e6 0.350279
\(827\) 2.84662e7 1.44732 0.723662 0.690155i \(-0.242458\pi\)
0.723662 + 0.690155i \(0.242458\pi\)
\(828\) 5.40918e7 2.74193
\(829\) −1.37621e7 −0.695503 −0.347752 0.937587i \(-0.613055\pi\)
−0.347752 + 0.937587i \(0.613055\pi\)
\(830\) 0 0
\(831\) 5.58301e7 2.80457
\(832\) 7.21135e6 0.361167
\(833\) −2.69486e7 −1.34562
\(834\) 6.17699e7 3.07512
\(835\) 0 0
\(836\) −1.43276e7 −0.709021
\(837\) −4.04820e7 −1.99732
\(838\) 2.18509e7 1.07488
\(839\) −3.18392e7 −1.56155 −0.780776 0.624811i \(-0.785176\pi\)
−0.780776 + 0.624811i \(0.785176\pi\)
\(840\) 0 0
\(841\) 1.47220e7 0.717757
\(842\) 4.60055e7 2.23630
\(843\) 1.41958e7 0.688006
\(844\) 7.95282e6 0.384295
\(845\) 0 0
\(846\) 4.42392e7 2.12511
\(847\) −1.85294e7 −0.887471
\(848\) 2.64672e7 1.26392
\(849\) −1.57887e7 −0.751757
\(850\) 0 0
\(851\) 1.86654e7 0.883514
\(852\) −1.09761e8 −5.18025
\(853\) 2.97955e6 0.140210 0.0701049 0.997540i \(-0.477667\pi\)
0.0701049 + 0.997540i \(0.477667\pi\)
\(854\) 7.23033e7 3.39245
\(855\) 0 0
\(856\) −2.89621e7 −1.35097
\(857\) −1.00599e7 −0.467886 −0.233943 0.972250i \(-0.575163\pi\)
−0.233943 + 0.972250i \(0.575163\pi\)
\(858\) 2.15759e7 1.00058
\(859\) 1.59873e7 0.739252 0.369626 0.929181i \(-0.379486\pi\)
0.369626 + 0.929181i \(0.379486\pi\)
\(860\) 0 0
\(861\) 1.53277e7 0.704641
\(862\) −2.01522e7 −0.923751
\(863\) 9.59580e6 0.438586 0.219293 0.975659i \(-0.429625\pi\)
0.219293 + 0.975659i \(0.429625\pi\)
\(864\) 1.06156e7 0.483794
\(865\) 0 0
\(866\) 2.43624e7 1.10389
\(867\) −3.53290e7 −1.59619
\(868\) −9.17675e7 −4.13418
\(869\) −3.67797e7 −1.65219
\(870\) 0 0
\(871\) 3.14165e6 0.140318
\(872\) 4.09273e7 1.82273
\(873\) −6.56226e6 −0.291419
\(874\) −9.33505e6 −0.413369
\(875\) 0 0
\(876\) −1.22639e8 −5.39970
\(877\) 1.09080e7 0.478902 0.239451 0.970908i \(-0.423033\pi\)
0.239451 + 0.970908i \(0.423033\pi\)
\(878\) 3.52416e7 1.54284
\(879\) 2.40503e7 1.04990
\(880\) 0 0
\(881\) −897407. −0.0389538 −0.0194769 0.999810i \(-0.506200\pi\)
−0.0194769 + 0.999810i \(0.506200\pi\)
\(882\) 6.65365e7 2.87997
\(883\) −3.37890e7 −1.45839 −0.729195 0.684306i \(-0.760105\pi\)
−0.729195 + 0.684306i \(0.760105\pi\)
\(884\) 1.69685e7 0.730320
\(885\) 0 0
\(886\) −2.23015e7 −0.954443
\(887\) 2.26350e7 0.965989 0.482994 0.875623i \(-0.339549\pi\)
0.482994 + 0.875623i \(0.339549\pi\)
\(888\) −6.25095e7 −2.66020
\(889\) −613565. −0.0260379
\(890\) 0 0
\(891\) 1.06293e7 0.448549
\(892\) 5.53733e7 2.33017
\(893\) −4.98461e6 −0.209172
\(894\) −2.55456e7 −1.06899
\(895\) 0 0
\(896\) 6.15894e7 2.56293
\(897\) 9.17806e6 0.380864
\(898\) −4.13594e7 −1.71153
\(899\) 4.98470e7 2.05703
\(900\) 0 0
\(901\) −6.56321e7 −2.69342
\(902\) 1.60402e7 0.656436
\(903\) 2.80076e7 1.14303
\(904\) −4.78475e7 −1.94732
\(905\) 0 0
\(906\) −1.27884e8 −5.17604
\(907\) −3.13826e7 −1.26669 −0.633345 0.773870i \(-0.718319\pi\)
−0.633345 + 0.773870i \(0.718319\pi\)
\(908\) 1.86363e6 0.0750146
\(909\) 1.11541e6 0.0447739
\(910\) 0 0
\(911\) −2.54672e7 −1.01668 −0.508341 0.861156i \(-0.669741\pi\)
−0.508341 + 0.861156i \(0.669741\pi\)
\(912\) 8.09254e6 0.322179
\(913\) −107142. −0.00425385
\(914\) 1.09804e7 0.434763
\(915\) 0 0
\(916\) 9.81608e6 0.386544
\(917\) 3.00585e7 1.18044
\(918\) 7.72104e7 3.02391
\(919\) −2.19820e7 −0.858577 −0.429288 0.903167i \(-0.641236\pi\)
−0.429288 + 0.903167i \(0.641236\pi\)
\(920\) 0 0
\(921\) −6.19730e7 −2.40743
\(922\) −3.58413e7 −1.38854
\(923\) −1.18889e7 −0.459345
\(924\) 1.45301e8 5.59871
\(925\) 0 0
\(926\) 4.28954e7 1.64393
\(927\) 9.04588e7 3.45741
\(928\) −1.30714e7 −0.498255
\(929\) 2.75340e7 1.04672 0.523359 0.852112i \(-0.324679\pi\)
0.523359 + 0.852112i \(0.324679\pi\)
\(930\) 0 0
\(931\) −7.49694e6 −0.283472
\(932\) −7.20538e7 −2.71717
\(933\) 3.29419e7 1.23892
\(934\) −5.50830e7 −2.06610
\(935\) 0 0
\(936\) −1.96216e7 −0.732056
\(937\) −117742. −0.00438111 −0.00219055 0.999998i \(-0.500697\pi\)
−0.00219055 + 0.999998i \(0.500697\pi\)
\(938\) 3.24054e7 1.20257
\(939\) −4.38845e7 −1.62423
\(940\) 0 0
\(941\) 1.19282e7 0.439136 0.219568 0.975597i \(-0.429535\pi\)
0.219568 + 0.975597i \(0.429535\pi\)
\(942\) −6.88226e7 −2.52699
\(943\) 6.82323e6 0.249868
\(944\) −2.65061e6 −0.0968091
\(945\) 0 0
\(946\) 2.93096e7 1.06483
\(947\) 4.67856e7 1.69526 0.847632 0.530585i \(-0.178028\pi\)
0.847632 + 0.530585i \(0.178028\pi\)
\(948\) 1.11875e8 4.04307
\(949\) −1.32838e7 −0.478804
\(950\) 0 0
\(951\) −2.48370e7 −0.890529
\(952\) 8.19727e7 2.93141
\(953\) 1.84653e6 0.0658603 0.0329301 0.999458i \(-0.489516\pi\)
0.0329301 + 0.999458i \(0.489516\pi\)
\(954\) 1.62047e8 5.76460
\(955\) 0 0
\(956\) −4.63195e7 −1.63915
\(957\) −7.89257e7 −2.78573
\(958\) −8.10641e7 −2.85374
\(959\) 9.82296e6 0.344902
\(960\) 0 0
\(961\) 4.18932e7 1.46331
\(962\) −1.44568e7 −0.503658
\(963\) −4.59010e7 −1.59498
\(964\) −5.11882e7 −1.77410
\(965\) 0 0
\(966\) 9.46694e7 3.26413
\(967\) −3.47395e7 −1.19470 −0.597349 0.801982i \(-0.703779\pi\)
−0.597349 + 0.801982i \(0.703779\pi\)
\(968\) 2.76239e7 0.947538
\(969\) −2.00675e7 −0.686569
\(970\) 0 0
\(971\) 1.02513e7 0.348923 0.174462 0.984664i \(-0.444182\pi\)
0.174462 + 0.984664i \(0.444182\pi\)
\(972\) 3.81743e7 1.29600
\(973\) 4.50577e7 1.52576
\(974\) −2.20912e7 −0.746143
\(975\) 0 0
\(976\) −2.79023e7 −0.937597
\(977\) 1.54408e6 0.0517528 0.0258764 0.999665i \(-0.491762\pi\)
0.0258764 + 0.999665i \(0.491762\pi\)
\(978\) 6.92980e7 2.31672
\(979\) −2.23323e7 −0.744691
\(980\) 0 0
\(981\) 6.48642e7 2.15195
\(982\) −1.76068e7 −0.582640
\(983\) −3.24347e7 −1.07060 −0.535299 0.844663i \(-0.679801\pi\)
−0.535299 + 0.844663i \(0.679801\pi\)
\(984\) −2.28506e7 −0.752334
\(985\) 0 0
\(986\) −9.50721e7 −3.11430
\(987\) 5.05504e7 1.65170
\(988\) 4.72054e6 0.153851
\(989\) 1.24678e7 0.405322
\(990\) 0 0
\(991\) −3.76595e7 −1.21812 −0.609060 0.793124i \(-0.708453\pi\)
−0.609060 + 0.793124i \(0.708453\pi\)
\(992\) −1.84931e7 −0.596664
\(993\) −5.93980e7 −1.91161
\(994\) −1.22632e8 −3.93674
\(995\) 0 0
\(996\) 325899. 0.0104096
\(997\) −3.72285e7 −1.18614 −0.593072 0.805149i \(-0.702085\pi\)
−0.593072 + 0.805149i \(0.702085\pi\)
\(998\) −1.59106e7 −0.505661
\(999\) −4.29481e7 −1.36154
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 325.6.a.j.1.10 11
5.2 odd 4 325.6.b.i.274.20 22
5.3 odd 4 325.6.b.i.274.3 22
5.4 even 2 325.6.a.k.1.2 yes 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.10 11 1.1 even 1 trivial
325.6.a.k.1.2 yes 11 5.4 even 2
325.6.b.i.274.3 22 5.3 odd 4
325.6.b.i.274.20 22 5.2 odd 4