Properties

Label 354.6.a.g.1.1
Level $354$
Weight $6$
Character 354.1
Self dual yes
Analytic conductor $56.776$
Analytic rank $0$
Dimension $6$
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] = [354,6,Mod(1,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, 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("354.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 354.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: \(56.7758722138\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 358x^{4} - 404x^{3} + 26492x^{2} - 11664x - 353376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(17.2905\) of defining polynomial
Character \(\chi\) \(=\) 354.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -75.0224 q^{5} +36.0000 q^{6} +17.7979 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -75.0224 q^{5} +36.0000 q^{6} +17.7979 q^{7} -64.0000 q^{8} +81.0000 q^{9} +300.090 q^{10} +485.600 q^{11} -144.000 q^{12} -507.251 q^{13} -71.1916 q^{14} +675.201 q^{15} +256.000 q^{16} -346.426 q^{17} -324.000 q^{18} -1662.38 q^{19} -1200.36 q^{20} -160.181 q^{21} -1942.40 q^{22} -593.797 q^{23} +576.000 q^{24} +2503.36 q^{25} +2029.00 q^{26} -729.000 q^{27} +284.766 q^{28} -6931.65 q^{29} -2700.81 q^{30} -7789.67 q^{31} -1024.00 q^{32} -4370.40 q^{33} +1385.70 q^{34} -1335.24 q^{35} +1296.00 q^{36} +4846.32 q^{37} +6649.53 q^{38} +4565.26 q^{39} +4801.43 q^{40} +14695.2 q^{41} +640.724 q^{42} -15929.6 q^{43} +7769.60 q^{44} -6076.81 q^{45} +2375.19 q^{46} +22466.5 q^{47} -2304.00 q^{48} -16490.2 q^{49} -10013.4 q^{50} +3117.83 q^{51} -8116.01 q^{52} -16983.6 q^{53} +2916.00 q^{54} -36430.9 q^{55} -1139.07 q^{56} +14961.4 q^{57} +27726.6 q^{58} +3481.00 q^{59} +10803.2 q^{60} -5681.13 q^{61} +31158.7 q^{62} +1441.63 q^{63} +4096.00 q^{64} +38055.2 q^{65} +17481.6 q^{66} -60959.7 q^{67} -5542.81 q^{68} +5344.18 q^{69} +5340.96 q^{70} +32944.4 q^{71} -5184.00 q^{72} +74404.5 q^{73} -19385.3 q^{74} -22530.2 q^{75} -26598.1 q^{76} +8642.66 q^{77} -18261.0 q^{78} -36190.1 q^{79} -19205.7 q^{80} +6561.00 q^{81} -58780.7 q^{82} -70280.2 q^{83} -2562.90 q^{84} +25989.7 q^{85} +63718.4 q^{86} +62384.8 q^{87} -31078.4 q^{88} +25275.7 q^{89} +24307.3 q^{90} -9028.00 q^{91} -9500.76 q^{92} +70107.0 q^{93} -89866.1 q^{94} +124716. q^{95} +9216.00 q^{96} +25544.4 q^{97} +65960.9 q^{98} +39333.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{2} - 54 q^{3} + 96 q^{4} + 4 q^{5} + 216 q^{6} - 54 q^{7} - 384 q^{8} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{2} - 54 q^{3} + 96 q^{4} + 4 q^{5} + 216 q^{6} - 54 q^{7} - 384 q^{8} + 486 q^{9} - 16 q^{10} + 436 q^{11} - 864 q^{12} - 536 q^{13} + 216 q^{14} - 36 q^{15} + 1536 q^{16} + 910 q^{17} - 1944 q^{18} + 1462 q^{19} + 64 q^{20} + 486 q^{21} - 1744 q^{22} + 1634 q^{23} + 3456 q^{24} - 1186 q^{25} + 2144 q^{26} - 4374 q^{27} - 864 q^{28} - 1598 q^{29} + 144 q^{30} - 5670 q^{31} - 6144 q^{32} - 3924 q^{33} - 3640 q^{34} - 7242 q^{35} + 7776 q^{36} - 20458 q^{37} - 5848 q^{38} + 4824 q^{39} - 256 q^{40} + 262 q^{41} - 1944 q^{42} - 34028 q^{43} + 6976 q^{44} + 324 q^{45} - 6536 q^{46} - 11194 q^{47} - 13824 q^{48} - 32652 q^{49} + 4744 q^{50} - 8190 q^{51} - 8576 q^{52} - 17164 q^{53} + 17496 q^{54} - 37040 q^{55} + 3456 q^{56} - 13158 q^{57} + 6392 q^{58} + 20886 q^{59} - 576 q^{60} - 43546 q^{61} + 22680 q^{62} - 4374 q^{63} + 24576 q^{64} + 65568 q^{65} + 15696 q^{66} - 52772 q^{67} + 14560 q^{68} - 14706 q^{69} + 28968 q^{70} + 84740 q^{71} - 31104 q^{72} - 36578 q^{73} + 81832 q^{74} + 10674 q^{75} + 23392 q^{76} + 90678 q^{77} - 19296 q^{78} + 85196 q^{79} + 1024 q^{80} + 39366 q^{81} - 1048 q^{82} + 217026 q^{83} + 7776 q^{84} + 26570 q^{85} + 136112 q^{86} + 14382 q^{87} - 27904 q^{88} + 333850 q^{89} - 1296 q^{90} + 214914 q^{91} + 26144 q^{92} + 51030 q^{93} + 44776 q^{94} + 458758 q^{95} + 55296 q^{96} + 173148 q^{97} + 130608 q^{98} + 35316 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) −75.0224 −1.34204 −0.671021 0.741439i \(-0.734144\pi\)
−0.671021 + 0.741439i \(0.734144\pi\)
\(6\) 36.0000 0.408248
\(7\) 17.7979 0.137285 0.0686426 0.997641i \(-0.478133\pi\)
0.0686426 + 0.997641i \(0.478133\pi\)
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 300.090 0.948966
\(11\) 485.600 1.21003 0.605016 0.796213i \(-0.293167\pi\)
0.605016 + 0.796213i \(0.293167\pi\)
\(12\) −144.000 −0.288675
\(13\) −507.251 −0.832462 −0.416231 0.909259i \(-0.636649\pi\)
−0.416231 + 0.909259i \(0.636649\pi\)
\(14\) −71.1916 −0.0970753
\(15\) 675.201 0.774828
\(16\) 256.000 0.250000
\(17\) −346.426 −0.290729 −0.145364 0.989378i \(-0.546435\pi\)
−0.145364 + 0.989378i \(0.546435\pi\)
\(18\) −324.000 −0.235702
\(19\) −1662.38 −1.05644 −0.528222 0.849106i \(-0.677141\pi\)
−0.528222 + 0.849106i \(0.677141\pi\)
\(20\) −1200.36 −0.671021
\(21\) −160.181 −0.0792616
\(22\) −1942.40 −0.855622
\(23\) −593.797 −0.234055 −0.117028 0.993129i \(-0.537337\pi\)
−0.117028 + 0.993129i \(0.537337\pi\)
\(24\) 576.000 0.204124
\(25\) 2503.36 0.801075
\(26\) 2029.00 0.588640
\(27\) −729.000 −0.192450
\(28\) 284.766 0.0686426
\(29\) −6931.65 −1.53053 −0.765265 0.643716i \(-0.777392\pi\)
−0.765265 + 0.643716i \(0.777392\pi\)
\(30\) −2700.81 −0.547886
\(31\) −7789.67 −1.45584 −0.727922 0.685660i \(-0.759514\pi\)
−0.727922 + 0.685660i \(0.759514\pi\)
\(32\) −1024.00 −0.176777
\(33\) −4370.40 −0.698613
\(34\) 1385.70 0.205576
\(35\) −1335.24 −0.184242
\(36\) 1296.00 0.166667
\(37\) 4846.32 0.581980 0.290990 0.956726i \(-0.406015\pi\)
0.290990 + 0.956726i \(0.406015\pi\)
\(38\) 6649.53 0.747019
\(39\) 4565.26 0.480622
\(40\) 4801.43 0.474483
\(41\) 14695.2 1.36526 0.682629 0.730765i \(-0.260836\pi\)
0.682629 + 0.730765i \(0.260836\pi\)
\(42\) 640.724 0.0560464
\(43\) −15929.6 −1.31381 −0.656907 0.753972i \(-0.728135\pi\)
−0.656907 + 0.753972i \(0.728135\pi\)
\(44\) 7769.60 0.605016
\(45\) −6076.81 −0.447347
\(46\) 2375.19 0.165502
\(47\) 22466.5 1.48351 0.741756 0.670669i \(-0.233993\pi\)
0.741756 + 0.670669i \(0.233993\pi\)
\(48\) −2304.00 −0.144338
\(49\) −16490.2 −0.981153
\(50\) −10013.4 −0.566445
\(51\) 3117.83 0.167852
\(52\) −8116.01 −0.416231
\(53\) −16983.6 −0.830501 −0.415250 0.909707i \(-0.636306\pi\)
−0.415250 + 0.909707i \(0.636306\pi\)
\(54\) 2916.00 0.136083
\(55\) −36430.9 −1.62391
\(56\) −1139.07 −0.0485376
\(57\) 14961.4 0.609938
\(58\) 27726.6 1.08225
\(59\) 3481.00 0.130189
\(60\) 10803.2 0.387414
\(61\) −5681.13 −0.195484 −0.0977418 0.995212i \(-0.531162\pi\)
−0.0977418 + 0.995212i \(0.531162\pi\)
\(62\) 31158.7 1.02944
\(63\) 1441.63 0.0457617
\(64\) 4096.00 0.125000
\(65\) 38055.2 1.11720
\(66\) 17481.6 0.493994
\(67\) −60959.7 −1.65904 −0.829518 0.558480i \(-0.811385\pi\)
−0.829518 + 0.558480i \(0.811385\pi\)
\(68\) −5542.81 −0.145364
\(69\) 5344.18 0.135132
\(70\) 5340.96 0.130279
\(71\) 32944.4 0.775596 0.387798 0.921744i \(-0.373236\pi\)
0.387798 + 0.921744i \(0.373236\pi\)
\(72\) −5184.00 −0.117851
\(73\) 74404.5 1.63415 0.817075 0.576532i \(-0.195594\pi\)
0.817075 + 0.576532i \(0.195594\pi\)
\(74\) −19385.3 −0.411522
\(75\) −22530.2 −0.462501
\(76\) −26598.1 −0.528222
\(77\) 8642.66 0.166120
\(78\) −18261.0 −0.339851
\(79\) −36190.1 −0.652412 −0.326206 0.945299i \(-0.605770\pi\)
−0.326206 + 0.945299i \(0.605770\pi\)
\(80\) −19205.7 −0.335510
\(81\) 6561.00 0.111111
\(82\) −58780.7 −0.965383
\(83\) −70280.2 −1.11979 −0.559896 0.828563i \(-0.689159\pi\)
−0.559896 + 0.828563i \(0.689159\pi\)
\(84\) −2562.90 −0.0396308
\(85\) 25989.7 0.390170
\(86\) 63718.4 0.929006
\(87\) 62384.8 0.883651
\(88\) −31078.4 −0.427811
\(89\) 25275.7 0.338243 0.169121 0.985595i \(-0.445907\pi\)
0.169121 + 0.985595i \(0.445907\pi\)
\(90\) 24307.3 0.316322
\(91\) −9028.00 −0.114285
\(92\) −9500.76 −0.117028
\(93\) 70107.0 0.840531
\(94\) −89866.1 −1.04900
\(95\) 124716. 1.41779
\(96\) 9216.00 0.102062
\(97\) 25544.4 0.275656 0.137828 0.990456i \(-0.455988\pi\)
0.137828 + 0.990456i \(0.455988\pi\)
\(98\) 65960.9 0.693780
\(99\) 39333.6 0.403344
\(100\) 40053.7 0.400537
\(101\) 177892. 1.73521 0.867607 0.497250i \(-0.165657\pi\)
0.867607 + 0.497250i \(0.165657\pi\)
\(102\) −12471.3 −0.118689
\(103\) 92651.8 0.860520 0.430260 0.902705i \(-0.358422\pi\)
0.430260 + 0.902705i \(0.358422\pi\)
\(104\) 32464.1 0.294320
\(105\) 12017.2 0.106372
\(106\) 67934.4 0.587253
\(107\) 55810.0 0.471251 0.235626 0.971844i \(-0.424286\pi\)
0.235626 + 0.971844i \(0.424286\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 210675. 1.69842 0.849211 0.528053i \(-0.177078\pi\)
0.849211 + 0.528053i \(0.177078\pi\)
\(110\) 145724. 1.14828
\(111\) −43616.9 −0.336006
\(112\) 4556.26 0.0343213
\(113\) −254472. −1.87476 −0.937378 0.348315i \(-0.886754\pi\)
−0.937378 + 0.348315i \(0.886754\pi\)
\(114\) −59845.7 −0.431292
\(115\) 44548.1 0.314112
\(116\) −110906. −0.765265
\(117\) −41087.3 −0.277487
\(118\) −13924.0 −0.0920575
\(119\) −6165.65 −0.0399127
\(120\) −43212.9 −0.273943
\(121\) 74756.6 0.464179
\(122\) 22724.5 0.138228
\(123\) −132256. −0.788232
\(124\) −124635. −0.727922
\(125\) 46637.0 0.266966
\(126\) −5766.52 −0.0323584
\(127\) 247007. 1.35894 0.679471 0.733703i \(-0.262210\pi\)
0.679471 + 0.733703i \(0.262210\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 143366. 0.758530
\(130\) −152221. −0.789979
\(131\) 269965. 1.37445 0.687225 0.726445i \(-0.258829\pi\)
0.687225 + 0.726445i \(0.258829\pi\)
\(132\) −69926.4 −0.349306
\(133\) −29586.9 −0.145034
\(134\) 243839. 1.17312
\(135\) 54691.3 0.258276
\(136\) 22171.2 0.102788
\(137\) 356377. 1.62221 0.811107 0.584897i \(-0.198865\pi\)
0.811107 + 0.584897i \(0.198865\pi\)
\(138\) −21376.7 −0.0955527
\(139\) −50616.3 −0.222205 −0.111102 0.993809i \(-0.535438\pi\)
−0.111102 + 0.993809i \(0.535438\pi\)
\(140\) −21363.9 −0.0921212
\(141\) −202199. −0.856506
\(142\) −131778. −0.548429
\(143\) −246321. −1.00731
\(144\) 20736.0 0.0833333
\(145\) 520029. 2.05403
\(146\) −297618. −1.15552
\(147\) 148412. 0.566469
\(148\) 77541.1 0.290990
\(149\) 73911.6 0.272739 0.136369 0.990658i \(-0.456457\pi\)
0.136369 + 0.990658i \(0.456457\pi\)
\(150\) 90120.9 0.327037
\(151\) −43522.5 −0.155336 −0.0776679 0.996979i \(-0.524747\pi\)
−0.0776679 + 0.996979i \(0.524747\pi\)
\(152\) 106392. 0.373510
\(153\) −28060.5 −0.0969096
\(154\) −34570.7 −0.117464
\(155\) 584399. 1.95380
\(156\) 73044.1 0.240311
\(157\) −102497. −0.331864 −0.165932 0.986137i \(-0.553063\pi\)
−0.165932 + 0.986137i \(0.553063\pi\)
\(158\) 144760. 0.461325
\(159\) 152852. 0.479490
\(160\) 76822.9 0.237242
\(161\) −10568.3 −0.0321323
\(162\) −26244.0 −0.0785674
\(163\) −90752.2 −0.267540 −0.133770 0.991012i \(-0.542708\pi\)
−0.133770 + 0.991012i \(0.542708\pi\)
\(164\) 235123. 0.682629
\(165\) 327878. 0.937567
\(166\) 281121. 0.791813
\(167\) −59309.4 −0.164563 −0.0822815 0.996609i \(-0.526221\pi\)
−0.0822815 + 0.996609i \(0.526221\pi\)
\(168\) 10251.6 0.0280232
\(169\) −113990. −0.307007
\(170\) −103959. −0.275892
\(171\) −134653. −0.352148
\(172\) −254874. −0.656907
\(173\) −371287. −0.943179 −0.471589 0.881818i \(-0.656320\pi\)
−0.471589 + 0.881818i \(0.656320\pi\)
\(174\) −249539. −0.624836
\(175\) 44554.5 0.109976
\(176\) 124314. 0.302508
\(177\) −31329.0 −0.0751646
\(178\) −101103. −0.239174
\(179\) 7764.45 0.0181125 0.00905624 0.999959i \(-0.497117\pi\)
0.00905624 + 0.999959i \(0.497117\pi\)
\(180\) −97229.0 −0.223674
\(181\) 207550. 0.470898 0.235449 0.971887i \(-0.424344\pi\)
0.235449 + 0.971887i \(0.424344\pi\)
\(182\) 36112.0 0.0808115
\(183\) 51130.2 0.112863
\(184\) 38003.0 0.0827511
\(185\) −363582. −0.781041
\(186\) −280428. −0.594345
\(187\) −168224. −0.351791
\(188\) 359464. 0.741756
\(189\) −12974.7 −0.0264205
\(190\) −498863. −1.00253
\(191\) 172606. 0.342351 0.171176 0.985241i \(-0.445243\pi\)
0.171176 + 0.985241i \(0.445243\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −912610. −1.76357 −0.881783 0.471655i \(-0.843657\pi\)
−0.881783 + 0.471655i \(0.843657\pi\)
\(194\) −102178. −0.194918
\(195\) −342497. −0.645015
\(196\) −263844. −0.490576
\(197\) 346131. 0.635441 0.317721 0.948184i \(-0.397083\pi\)
0.317721 + 0.948184i \(0.397083\pi\)
\(198\) −157334. −0.285207
\(199\) 219663. 0.393210 0.196605 0.980483i \(-0.437008\pi\)
0.196605 + 0.980483i \(0.437008\pi\)
\(200\) −160215. −0.283223
\(201\) 548637. 0.957845
\(202\) −711568. −1.22698
\(203\) −123369. −0.210119
\(204\) 49885.3 0.0839261
\(205\) −1.10247e6 −1.83223
\(206\) −370607. −0.608479
\(207\) −48097.6 −0.0780185
\(208\) −129856. −0.208115
\(209\) −807253. −1.27833
\(210\) −48068.7 −0.0752166
\(211\) 139453. 0.215636 0.107818 0.994171i \(-0.465614\pi\)
0.107818 + 0.994171i \(0.465614\pi\)
\(212\) −271738. −0.415250
\(213\) −296500. −0.447791
\(214\) −223240. −0.333225
\(215\) 1.19508e6 1.76319
\(216\) 46656.0 0.0680414
\(217\) −138640. −0.199866
\(218\) −842698. −1.20097
\(219\) −669640. −0.943477
\(220\) −582894. −0.811957
\(221\) 175725. 0.242021
\(222\) 174468. 0.237592
\(223\) 524085. 0.705732 0.352866 0.935674i \(-0.385207\pi\)
0.352866 + 0.935674i \(0.385207\pi\)
\(224\) −18225.1 −0.0242688
\(225\) 202772. 0.267025
\(226\) 1.01789e6 1.32565
\(227\) −1.06504e6 −1.37183 −0.685914 0.727683i \(-0.740597\pi\)
−0.685914 + 0.727683i \(0.740597\pi\)
\(228\) 239383. 0.304969
\(229\) 77430.1 0.0975711 0.0487855 0.998809i \(-0.484465\pi\)
0.0487855 + 0.998809i \(0.484465\pi\)
\(230\) −178192. −0.222111
\(231\) −77784.0 −0.0959092
\(232\) 443626. 0.541124
\(233\) 547655. 0.660872 0.330436 0.943828i \(-0.392804\pi\)
0.330436 + 0.943828i \(0.392804\pi\)
\(234\) 164349. 0.196213
\(235\) −1.68549e6 −1.99094
\(236\) 55696.0 0.0650945
\(237\) 325711. 0.376670
\(238\) 24662.6 0.0282226
\(239\) 781054. 0.884477 0.442238 0.896898i \(-0.354185\pi\)
0.442238 + 0.896898i \(0.354185\pi\)
\(240\) 172852. 0.193707
\(241\) 1.10849e6 1.22938 0.614692 0.788767i \(-0.289280\pi\)
0.614692 + 0.788767i \(0.289280\pi\)
\(242\) −299026. −0.328224
\(243\) −59049.0 −0.0641500
\(244\) −90898.1 −0.0977418
\(245\) 1.23714e6 1.31675
\(246\) 529026. 0.557364
\(247\) 843244. 0.879450
\(248\) 498539. 0.514718
\(249\) 632521. 0.646512
\(250\) −186548. −0.188773
\(251\) 1.16262e6 1.16480 0.582402 0.812901i \(-0.302113\pi\)
0.582402 + 0.812901i \(0.302113\pi\)
\(252\) 23066.1 0.0228809
\(253\) −288348. −0.283215
\(254\) −988030. −0.960917
\(255\) −233907. −0.225265
\(256\) 65536.0 0.0625000
\(257\) −1.43559e6 −1.35580 −0.677902 0.735152i \(-0.737111\pi\)
−0.677902 + 0.735152i \(0.737111\pi\)
\(258\) −573465. −0.536362
\(259\) 86254.3 0.0798972
\(260\) 608883. 0.558599
\(261\) −561464. −0.510176
\(262\) −1.07986e6 −0.971882
\(263\) −1.89520e6 −1.68953 −0.844766 0.535135i \(-0.820261\pi\)
−0.844766 + 0.535135i \(0.820261\pi\)
\(264\) 279706. 0.246997
\(265\) 1.27415e6 1.11457
\(266\) 118348. 0.102555
\(267\) −227481. −0.195285
\(268\) −975356. −0.829518
\(269\) 1.77716e6 1.49742 0.748712 0.662895i \(-0.230673\pi\)
0.748712 + 0.662895i \(0.230673\pi\)
\(270\) −218765. −0.182629
\(271\) −760373. −0.628932 −0.314466 0.949269i \(-0.601825\pi\)
−0.314466 + 0.949269i \(0.601825\pi\)
\(272\) −88685.0 −0.0726822
\(273\) 81252.0 0.0659823
\(274\) −1.42551e6 −1.14708
\(275\) 1.21563e6 0.969327
\(276\) 85506.8 0.0675660
\(277\) −1.75250e6 −1.37233 −0.686165 0.727446i \(-0.740707\pi\)
−0.686165 + 0.727446i \(0.740707\pi\)
\(278\) 202465. 0.157122
\(279\) −630963. −0.485281
\(280\) 85455.4 0.0651395
\(281\) 138408. 0.104567 0.0522836 0.998632i \(-0.483350\pi\)
0.0522836 + 0.998632i \(0.483350\pi\)
\(282\) 808795. 0.605642
\(283\) −844853. −0.627068 −0.313534 0.949577i \(-0.601513\pi\)
−0.313534 + 0.949577i \(0.601513\pi\)
\(284\) 527110. 0.387798
\(285\) −1.12244e6 −0.818563
\(286\) 985284. 0.712273
\(287\) 261543. 0.187430
\(288\) −82944.0 −0.0589256
\(289\) −1.29985e6 −0.915477
\(290\) −2.08012e6 −1.45242
\(291\) −229900. −0.159150
\(292\) 1.19047e6 0.817075
\(293\) 2.80754e6 1.91054 0.955271 0.295733i \(-0.0955637\pi\)
0.955271 + 0.295733i \(0.0955637\pi\)
\(294\) −593648. −0.400554
\(295\) −261153. −0.174719
\(296\) −310164. −0.205761
\(297\) −354003. −0.232871
\(298\) −295646. −0.192855
\(299\) 301204. 0.194842
\(300\) −360484. −0.231250
\(301\) −283513. −0.180367
\(302\) 174090. 0.109839
\(303\) −1.60103e6 −1.00183
\(304\) −425570. −0.264111
\(305\) 426212. 0.262347
\(306\) 112242. 0.0685254
\(307\) 712594. 0.431516 0.215758 0.976447i \(-0.430778\pi\)
0.215758 + 0.976447i \(0.430778\pi\)
\(308\) 138283. 0.0830598
\(309\) −833866. −0.496821
\(310\) −2.33760e6 −1.38155
\(311\) 2.03477e6 1.19293 0.596464 0.802640i \(-0.296572\pi\)
0.596464 + 0.802640i \(0.296572\pi\)
\(312\) −292176. −0.169926
\(313\) 1.04329e6 0.601927 0.300964 0.953636i \(-0.402692\pi\)
0.300964 + 0.953636i \(0.402692\pi\)
\(314\) 409986. 0.234663
\(315\) −108155. −0.0614141
\(316\) −579042. −0.326206
\(317\) 97044.0 0.0542401 0.0271200 0.999632i \(-0.491366\pi\)
0.0271200 + 0.999632i \(0.491366\pi\)
\(318\) −611410. −0.339051
\(319\) −3.36601e6 −1.85199
\(320\) −307292. −0.167755
\(321\) −502290. −0.272077
\(322\) 42273.4 0.0227210
\(323\) 575892. 0.307139
\(324\) 104976. 0.0555556
\(325\) −1.26983e6 −0.666864
\(326\) 363009. 0.189179
\(327\) −1.89607e6 −0.980585
\(328\) −940491. −0.482692
\(329\) 399857. 0.203664
\(330\) −1.31151e6 −0.662960
\(331\) 3.17003e6 1.59035 0.795177 0.606377i \(-0.207378\pi\)
0.795177 + 0.606377i \(0.207378\pi\)
\(332\) −1.12448e6 −0.559896
\(333\) 392552. 0.193993
\(334\) 237237. 0.116364
\(335\) 4.57334e6 2.22650
\(336\) −41006.4 −0.0198154
\(337\) −3.75855e6 −1.80279 −0.901397 0.432994i \(-0.857457\pi\)
−0.901397 + 0.432994i \(0.857457\pi\)
\(338\) 455958. 0.217087
\(339\) 2.29025e6 1.08239
\(340\) 415835. 0.195085
\(341\) −3.78266e6 −1.76162
\(342\) 538612. 0.249006
\(343\) −592621. −0.271983
\(344\) 1.01949e6 0.464503
\(345\) −400933. −0.181353
\(346\) 1.48515e6 0.666928
\(347\) −2.43095e6 −1.08381 −0.541905 0.840440i \(-0.682297\pi\)
−0.541905 + 0.840440i \(0.682297\pi\)
\(348\) 998157. 0.441826
\(349\) 1.14487e6 0.503146 0.251573 0.967838i \(-0.419052\pi\)
0.251573 + 0.967838i \(0.419052\pi\)
\(350\) −178218. −0.0777646
\(351\) 369786. 0.160207
\(352\) −497255. −0.213906
\(353\) −2.72502e6 −1.16395 −0.581974 0.813207i \(-0.697720\pi\)
−0.581974 + 0.813207i \(0.697720\pi\)
\(354\) 125316. 0.0531494
\(355\) −2.47157e6 −1.04088
\(356\) 404412. 0.169121
\(357\) 55490.9 0.0230436
\(358\) −31057.8 −0.0128075
\(359\) 3.82400e6 1.56596 0.782982 0.622045i \(-0.213698\pi\)
0.782982 + 0.622045i \(0.213698\pi\)
\(360\) 388916. 0.158161
\(361\) 287413. 0.116075
\(362\) −830201. −0.332975
\(363\) −672809. −0.267994
\(364\) −144448. −0.0571423
\(365\) −5.58200e6 −2.19310
\(366\) −204521. −0.0798059
\(367\) 2.89599e6 1.12236 0.561179 0.827695i \(-0.310348\pi\)
0.561179 + 0.827695i \(0.310348\pi\)
\(368\) −152012. −0.0585139
\(369\) 1.19031e6 0.455086
\(370\) 1.45433e6 0.552279
\(371\) −302272. −0.114015
\(372\) 1.12171e6 0.420266
\(373\) 799129. 0.297403 0.148701 0.988882i \(-0.452491\pi\)
0.148701 + 0.988882i \(0.452491\pi\)
\(374\) 672898. 0.248754
\(375\) −419733. −0.154133
\(376\) −1.43786e6 −0.524501
\(377\) 3.51608e6 1.27411
\(378\) 51898.7 0.0186821
\(379\) −281568. −0.100690 −0.0503449 0.998732i \(-0.516032\pi\)
−0.0503449 + 0.998732i \(0.516032\pi\)
\(380\) 1.99545e6 0.708896
\(381\) −2.22307e6 −0.784585
\(382\) −690423. −0.242079
\(383\) 2.61292e6 0.910183 0.455092 0.890445i \(-0.349606\pi\)
0.455092 + 0.890445i \(0.349606\pi\)
\(384\) 147456. 0.0510310
\(385\) −648393. −0.222939
\(386\) 3.65044e6 1.24703
\(387\) −1.29030e6 −0.437938
\(388\) 408711. 0.137828
\(389\) −1.44947e6 −0.485664 −0.242832 0.970068i \(-0.578076\pi\)
−0.242832 + 0.970068i \(0.578076\pi\)
\(390\) 1.36999e6 0.456094
\(391\) 205707. 0.0680466
\(392\) 1.05538e6 0.346890
\(393\) −2.42968e6 −0.793539
\(394\) −1.38453e6 −0.449325
\(395\) 2.71507e6 0.875564
\(396\) 629338. 0.201672
\(397\) 4.38919e6 1.39768 0.698841 0.715277i \(-0.253700\pi\)
0.698841 + 0.715277i \(0.253700\pi\)
\(398\) −878653. −0.278042
\(399\) 266282. 0.0837355
\(400\) 640860. 0.200269
\(401\) −3.73715e6 −1.16059 −0.580296 0.814406i \(-0.697063\pi\)
−0.580296 + 0.814406i \(0.697063\pi\)
\(402\) −2.19455e6 −0.677299
\(403\) 3.95131e6 1.21193
\(404\) 2.84627e6 0.867607
\(405\) −492222. −0.149116
\(406\) 493475. 0.148577
\(407\) 2.35337e6 0.704214
\(408\) −199541. −0.0593447
\(409\) 3.02555e6 0.894327 0.447163 0.894452i \(-0.352434\pi\)
0.447163 + 0.894452i \(0.352434\pi\)
\(410\) 4.40987e6 1.29558
\(411\) −3.20739e6 −0.936586
\(412\) 1.48243e6 0.430260
\(413\) 61954.5 0.0178730
\(414\) 192390. 0.0551674
\(415\) 5.27259e6 1.50281
\(416\) 519425. 0.147160
\(417\) 455547. 0.128290
\(418\) 3.22901e6 0.903918
\(419\) −2.98553e6 −0.830781 −0.415391 0.909643i \(-0.636355\pi\)
−0.415391 + 0.909643i \(0.636355\pi\)
\(420\) 192275. 0.0531862
\(421\) 1.71749e6 0.472268 0.236134 0.971721i \(-0.424120\pi\)
0.236134 + 0.971721i \(0.424120\pi\)
\(422\) −557810. −0.152477
\(423\) 1.81979e6 0.494504
\(424\) 1.08695e6 0.293626
\(425\) −867228. −0.232895
\(426\) 1.18600e6 0.316636
\(427\) −101112. −0.0268370
\(428\) 892960. 0.235626
\(429\) 2.21689e6 0.581569
\(430\) −4.78031e6 −1.24676
\(431\) −367341. −0.0952525 −0.0476263 0.998865i \(-0.515166\pi\)
−0.0476263 + 0.998865i \(0.515166\pi\)
\(432\) −186624. −0.0481125
\(433\) 684609. 0.175478 0.0877390 0.996143i \(-0.472036\pi\)
0.0877390 + 0.996143i \(0.472036\pi\)
\(434\) 554559. 0.141326
\(435\) −4.68026e6 −1.18590
\(436\) 3.37079e6 0.849211
\(437\) 987118. 0.247267
\(438\) 2.67856e6 0.667139
\(439\) 3.84247e6 0.951590 0.475795 0.879556i \(-0.342160\pi\)
0.475795 + 0.879556i \(0.342160\pi\)
\(440\) 2.33158e6 0.574140
\(441\) −1.33571e6 −0.327051
\(442\) −702899. −0.171134
\(443\) 3.26654e6 0.790823 0.395412 0.918504i \(-0.370602\pi\)
0.395412 + 0.918504i \(0.370602\pi\)
\(444\) −697870. −0.168003
\(445\) −1.89625e6 −0.453936
\(446\) −2.09634e6 −0.499028
\(447\) −665204. −0.157466
\(448\) 72900.2 0.0171606
\(449\) −300699. −0.0703909 −0.0351954 0.999380i \(-0.511205\pi\)
−0.0351954 + 0.999380i \(0.511205\pi\)
\(450\) −811088. −0.188815
\(451\) 7.13598e6 1.65201
\(452\) −4.07156e6 −0.937378
\(453\) 391703. 0.0896832
\(454\) 4.26014e6 0.970028
\(455\) 677302. 0.153375
\(456\) −957532. −0.215646
\(457\) 5.51670e6 1.23563 0.617815 0.786323i \(-0.288018\pi\)
0.617815 + 0.786323i \(0.288018\pi\)
\(458\) −309720. −0.0689932
\(459\) 252544. 0.0559508
\(460\) 712770. 0.157056
\(461\) 2.95614e6 0.647848 0.323924 0.946083i \(-0.394998\pi\)
0.323924 + 0.946083i \(0.394998\pi\)
\(462\) 311136. 0.0678180
\(463\) 6.61613e6 1.43434 0.717169 0.696899i \(-0.245437\pi\)
0.717169 + 0.696899i \(0.245437\pi\)
\(464\) −1.77450e6 −0.382632
\(465\) −5.25959e6 −1.12803
\(466\) −2.19062e6 −0.467307
\(467\) 7.18933e6 1.52544 0.762721 0.646727i \(-0.223863\pi\)
0.762721 + 0.646727i \(0.223863\pi\)
\(468\) −657397. −0.138744
\(469\) −1.08496e6 −0.227761
\(470\) 6.74197e6 1.40780
\(471\) 922469. 0.191602
\(472\) −222784. −0.0460287
\(473\) −7.73542e6 −1.58976
\(474\) −1.30284e6 −0.266346
\(475\) −4.16154e6 −0.846291
\(476\) −98650.4 −0.0199564
\(477\) −1.37567e6 −0.276834
\(478\) −3.12422e6 −0.625420
\(479\) 890884. 0.177412 0.0887059 0.996058i \(-0.471727\pi\)
0.0887059 + 0.996058i \(0.471727\pi\)
\(480\) −691406. −0.136972
\(481\) −2.45830e6 −0.484476
\(482\) −4.43394e6 −0.869305
\(483\) 95115.1 0.0185516
\(484\) 1.19610e6 0.232090
\(485\) −1.91641e6 −0.369941
\(486\) 236196. 0.0453609
\(487\) −2.18101e6 −0.416712 −0.208356 0.978053i \(-0.566811\pi\)
−0.208356 + 0.978053i \(0.566811\pi\)
\(488\) 363593. 0.0691139
\(489\) 816770. 0.154464
\(490\) −4.94855e6 −0.931081
\(491\) 8.22986e6 1.54060 0.770298 0.637684i \(-0.220107\pi\)
0.770298 + 0.637684i \(0.220107\pi\)
\(492\) −2.11610e6 −0.394116
\(493\) 2.40130e6 0.444969
\(494\) −3.37298e6 −0.621865
\(495\) −2.95090e6 −0.541305
\(496\) −1.99415e6 −0.363961
\(497\) 586341. 0.106478
\(498\) −2.53009e6 −0.457153
\(499\) −7.67960e6 −1.38066 −0.690331 0.723494i \(-0.742535\pi\)
−0.690331 + 0.723494i \(0.742535\pi\)
\(500\) 746192. 0.133483
\(501\) 533784. 0.0950104
\(502\) −4.65047e6 −0.823641
\(503\) −732941. −0.129166 −0.0645831 0.997912i \(-0.520572\pi\)
−0.0645831 + 0.997912i \(0.520572\pi\)
\(504\) −92264.3 −0.0161792
\(505\) −1.33459e7 −2.32873
\(506\) 1.15339e6 0.200263
\(507\) 1.02591e6 0.177251
\(508\) 3.95212e6 0.679471
\(509\) −3.43769e6 −0.588128 −0.294064 0.955786i \(-0.595008\pi\)
−0.294064 + 0.955786i \(0.595008\pi\)
\(510\) 935629. 0.159286
\(511\) 1.32424e6 0.224345
\(512\) −262144. −0.0441942
\(513\) 1.21188e6 0.203313
\(514\) 5.74235e6 0.958698
\(515\) −6.95096e6 −1.15485
\(516\) 2.29386e6 0.379265
\(517\) 1.09097e7 1.79510
\(518\) −345017. −0.0564958
\(519\) 3.34158e6 0.544545
\(520\) −2.43553e6 −0.394989
\(521\) 1.80746e6 0.291726 0.145863 0.989305i \(-0.453404\pi\)
0.145863 + 0.989305i \(0.453404\pi\)
\(522\) 2.24585e6 0.360749
\(523\) 7.89318e6 1.26182 0.630911 0.775855i \(-0.282681\pi\)
0.630911 + 0.775855i \(0.282681\pi\)
\(524\) 4.31944e6 0.687225
\(525\) −400991. −0.0634945
\(526\) 7.58082e6 1.19468
\(527\) 2.69854e6 0.423255
\(528\) −1.11882e6 −0.174653
\(529\) −6.08375e6 −0.945218
\(530\) −5.09660e6 −0.788117
\(531\) 281961. 0.0433963
\(532\) −473390. −0.0725171
\(533\) −7.45414e6 −1.13653
\(534\) 909926. 0.138087
\(535\) −4.18700e6 −0.632439
\(536\) 3.90142e6 0.586558
\(537\) −69880.0 −0.0104572
\(538\) −7.10862e6 −1.05884
\(539\) −8.00766e6 −1.18723
\(540\) 875061. 0.129138
\(541\) −1.31054e7 −1.92512 −0.962559 0.271071i \(-0.912622\pi\)
−0.962559 + 0.271071i \(0.912622\pi\)
\(542\) 3.04149e6 0.444722
\(543\) −1.86795e6 −0.271873
\(544\) 354740. 0.0513941
\(545\) −1.58053e7 −2.27935
\(546\) −325008. −0.0466565
\(547\) −1.07610e7 −1.53774 −0.768872 0.639403i \(-0.779181\pi\)
−0.768872 + 0.639403i \(0.779181\pi\)
\(548\) 5.70203e6 0.811107
\(549\) −460172. −0.0651612
\(550\) −4.86253e6 −0.685418
\(551\) 1.15230e7 1.61692
\(552\) −342027. −0.0477764
\(553\) −644108. −0.0895665
\(554\) 7.00999e6 0.970383
\(555\) 3.27224e6 0.450934
\(556\) −809860. −0.111102
\(557\) 3.89845e6 0.532420 0.266210 0.963915i \(-0.414229\pi\)
0.266210 + 0.963915i \(0.414229\pi\)
\(558\) 2.52385e6 0.343146
\(559\) 8.08030e6 1.09370
\(560\) −341822. −0.0460606
\(561\) 1.51402e6 0.203107
\(562\) −553632. −0.0739402
\(563\) −5.32082e6 −0.707469 −0.353735 0.935346i \(-0.615088\pi\)
−0.353735 + 0.935346i \(0.615088\pi\)
\(564\) −3.23518e6 −0.428253
\(565\) 1.90911e7 2.51600
\(566\) 3.37941e6 0.443404
\(567\) 116772. 0.0152539
\(568\) −2.10844e6 −0.274215
\(569\) 7.86449e6 1.01833 0.509166 0.860668i \(-0.329954\pi\)
0.509166 + 0.860668i \(0.329954\pi\)
\(570\) 4.48977e6 0.578811
\(571\) −4.16368e6 −0.534425 −0.267213 0.963638i \(-0.586103\pi\)
−0.267213 + 0.963638i \(0.586103\pi\)
\(572\) −3.94114e6 −0.503653
\(573\) −1.55345e6 −0.197656
\(574\) −1.04617e6 −0.132533
\(575\) −1.48649e6 −0.187496
\(576\) 331776. 0.0416667
\(577\) −7.27913e6 −0.910206 −0.455103 0.890439i \(-0.650398\pi\)
−0.455103 + 0.890439i \(0.650398\pi\)
\(578\) 5.19938e6 0.647340
\(579\) 8.21349e6 1.01820
\(580\) 8.32046e6 1.02702
\(581\) −1.25084e6 −0.153731
\(582\) 919600. 0.112536
\(583\) −8.24724e6 −1.00493
\(584\) −4.76189e6 −0.577759
\(585\) 3.08247e6 0.372399
\(586\) −1.12301e7 −1.35096
\(587\) −1.22184e7 −1.46359 −0.731797 0.681523i \(-0.761318\pi\)
−0.731797 + 0.681523i \(0.761318\pi\)
\(588\) 2.37459e6 0.283234
\(589\) 1.29494e7 1.53802
\(590\) 1.04461e6 0.123545
\(591\) −3.11518e6 −0.366872
\(592\) 1.24066e6 0.145495
\(593\) −1.31282e6 −0.153309 −0.0766545 0.997058i \(-0.524424\pi\)
−0.0766545 + 0.997058i \(0.524424\pi\)
\(594\) 1.41601e6 0.164665
\(595\) 462562. 0.0535645
\(596\) 1.18259e6 0.136369
\(597\) −1.97697e6 −0.227020
\(598\) −1.20482e6 −0.137774
\(599\) −1.51035e6 −0.171993 −0.0859964 0.996295i \(-0.527407\pi\)
−0.0859964 + 0.996295i \(0.527407\pi\)
\(600\) 1.44193e6 0.163519
\(601\) 7.32636e6 0.827374 0.413687 0.910419i \(-0.364241\pi\)
0.413687 + 0.910419i \(0.364241\pi\)
\(602\) 1.13405e6 0.127539
\(603\) −4.93774e6 −0.553012
\(604\) −696360. −0.0776679
\(605\) −5.60842e6 −0.622948
\(606\) 6.40411e6 0.708398
\(607\) −6.67575e6 −0.735408 −0.367704 0.929943i \(-0.619856\pi\)
−0.367704 + 0.929943i \(0.619856\pi\)
\(608\) 1.70228e6 0.186755
\(609\) 1.11032e6 0.121312
\(610\) −1.70485e6 −0.185507
\(611\) −1.13962e7 −1.23497
\(612\) −448968. −0.0484548
\(613\) −5.04738e6 −0.542519 −0.271260 0.962506i \(-0.587440\pi\)
−0.271260 + 0.962506i \(0.587440\pi\)
\(614\) −2.85038e6 −0.305128
\(615\) 9.92220e6 1.05784
\(616\) −553131. −0.0587321
\(617\) 8.09696e6 0.856267 0.428134 0.903715i \(-0.359171\pi\)
0.428134 + 0.903715i \(0.359171\pi\)
\(618\) 3.33546e6 0.351306
\(619\) 2.22019e6 0.232897 0.116448 0.993197i \(-0.462849\pi\)
0.116448 + 0.993197i \(0.462849\pi\)
\(620\) 9.35039e6 0.976901
\(621\) 432878. 0.0450440
\(622\) −8.13908e6 −0.843528
\(623\) 449855. 0.0464357
\(624\) 1.16871e6 0.120156
\(625\) −1.13218e7 −1.15935
\(626\) −4.17316e6 −0.425627
\(627\) 7.26527e6 0.738046
\(628\) −1.63994e6 −0.165932
\(629\) −1.67889e6 −0.169198
\(630\) 432618. 0.0434263
\(631\) 1.40361e7 1.40338 0.701688 0.712485i \(-0.252430\pi\)
0.701688 + 0.712485i \(0.252430\pi\)
\(632\) 2.31617e6 0.230662
\(633\) −1.25507e6 −0.124497
\(634\) −388176. −0.0383535
\(635\) −1.85311e7 −1.82376
\(636\) 2.44564e6 0.239745
\(637\) 8.36469e6 0.816772
\(638\) 1.34640e7 1.30955
\(639\) 2.66850e6 0.258532
\(640\) 1.22917e6 0.118621
\(641\) 376115. 0.0361556 0.0180778 0.999837i \(-0.494245\pi\)
0.0180778 + 0.999837i \(0.494245\pi\)
\(642\) 2.00916e6 0.192388
\(643\) 1.51655e7 1.44653 0.723266 0.690570i \(-0.242640\pi\)
0.723266 + 0.690570i \(0.242640\pi\)
\(644\) −169094. −0.0160662
\(645\) −1.07557e7 −1.01798
\(646\) −2.30357e6 −0.217180
\(647\) 1.74275e7 1.63672 0.818362 0.574703i \(-0.194882\pi\)
0.818362 + 0.574703i \(0.194882\pi\)
\(648\) −419904. −0.0392837
\(649\) 1.69037e6 0.157533
\(650\) 5.07932e6 0.471544
\(651\) 1.24776e6 0.115393
\(652\) −1.45204e6 −0.133770
\(653\) 5.89333e6 0.540851 0.270426 0.962741i \(-0.412836\pi\)
0.270426 + 0.962741i \(0.412836\pi\)
\(654\) 7.58428e6 0.693378
\(655\) −2.02534e7 −1.84457
\(656\) 3.76196e6 0.341315
\(657\) 6.02676e6 0.544717
\(658\) −1.59943e6 −0.144012
\(659\) −2.41916e6 −0.216996 −0.108498 0.994097i \(-0.534604\pi\)
−0.108498 + 0.994097i \(0.534604\pi\)
\(660\) 5.24605e6 0.468784
\(661\) −2.95209e6 −0.262800 −0.131400 0.991329i \(-0.541947\pi\)
−0.131400 + 0.991329i \(0.541947\pi\)
\(662\) −1.26801e7 −1.12455
\(663\) −1.58152e6 −0.139731
\(664\) 4.49793e6 0.395906
\(665\) 2.21968e6 0.194642
\(666\) −1.57021e6 −0.137174
\(667\) 4.11600e6 0.358229
\(668\) −948950. −0.0822815
\(669\) −4.71677e6 −0.407455
\(670\) −1.82934e7 −1.57437
\(671\) −2.75876e6 −0.236542
\(672\) 164025. 0.0140116
\(673\) 2.03480e7 1.73174 0.865872 0.500265i \(-0.166764\pi\)
0.865872 + 0.500265i \(0.166764\pi\)
\(674\) 1.50342e7 1.27477
\(675\) −1.82495e6 −0.154167
\(676\) −1.82383e6 −0.153504
\(677\) −1.67007e7 −1.40044 −0.700218 0.713929i \(-0.746914\pi\)
−0.700218 + 0.713929i \(0.746914\pi\)
\(678\) −9.16101e6 −0.765366
\(679\) 454638. 0.0378435
\(680\) −1.66334e6 −0.137946
\(681\) 9.58532e6 0.792025
\(682\) 1.51307e7 1.24565
\(683\) −9.23369e6 −0.757397 −0.378698 0.925520i \(-0.623628\pi\)
−0.378698 + 0.925520i \(0.623628\pi\)
\(684\) −2.15445e6 −0.176074
\(685\) −2.67363e7 −2.17708
\(686\) 2.37048e6 0.192321
\(687\) −696871. −0.0563327
\(688\) −4.07798e6 −0.328453
\(689\) 8.61495e6 0.691360
\(690\) 1.60373e6 0.128236
\(691\) −1.56683e7 −1.24832 −0.624159 0.781297i \(-0.714558\pi\)
−0.624159 + 0.781297i \(0.714558\pi\)
\(692\) −5.94059e6 −0.471589
\(693\) 700056. 0.0553732
\(694\) 9.72381e6 0.766369
\(695\) 3.79735e6 0.298208
\(696\) −3.99263e6 −0.312418
\(697\) −5.09078e6 −0.396920
\(698\) −4.57950e6 −0.355778
\(699\) −4.92889e6 −0.381554
\(700\) 712872. 0.0549879
\(701\) −1.63457e7 −1.25634 −0.628171 0.778075i \(-0.716196\pi\)
−0.628171 + 0.778075i \(0.716196\pi\)
\(702\) −1.47914e6 −0.113284
\(703\) −8.05643e6 −0.614829
\(704\) 1.98902e6 0.151254
\(705\) 1.51694e7 1.14947
\(706\) 1.09001e7 0.823036
\(707\) 3.16611e6 0.238219
\(708\) −501264. −0.0375823
\(709\) −1.23736e7 −0.924444 −0.462222 0.886764i \(-0.652948\pi\)
−0.462222 + 0.886764i \(0.652948\pi\)
\(710\) 9.88627e6 0.736015
\(711\) −2.93140e6 −0.217471
\(712\) −1.61765e6 −0.119587
\(713\) 4.62548e6 0.340748
\(714\) −221963. −0.0162943
\(715\) 1.84796e7 1.35185
\(716\) 124231. 0.00905624
\(717\) −7.02949e6 −0.510653
\(718\) −1.52960e7 −1.10730
\(719\) −2.02502e7 −1.46086 −0.730428 0.682990i \(-0.760679\pi\)
−0.730428 + 0.682990i \(0.760679\pi\)
\(720\) −1.55566e6 −0.111837
\(721\) 1.64901e6 0.118137
\(722\) −1.14965e6 −0.0820773
\(723\) −9.97637e6 −0.709785
\(724\) 3.32081e6 0.235449
\(725\) −1.73524e7 −1.22607
\(726\) 2.69124e6 0.189500
\(727\) 1.15577e7 0.811030 0.405515 0.914088i \(-0.367092\pi\)
0.405515 + 0.914088i \(0.367092\pi\)
\(728\) 577792. 0.0404057
\(729\) 531441. 0.0370370
\(730\) 2.23280e7 1.55075
\(731\) 5.51842e6 0.381963
\(732\) 818083. 0.0564313
\(733\) 1.55295e7 1.06758 0.533788 0.845618i \(-0.320768\pi\)
0.533788 + 0.845618i \(0.320768\pi\)
\(734\) −1.15839e7 −0.793627
\(735\) −1.11342e7 −0.760225
\(736\) 608049. 0.0413755
\(737\) −2.96021e7 −2.00749
\(738\) −4.76123e6 −0.321794
\(739\) −8.05265e6 −0.542410 −0.271205 0.962522i \(-0.587422\pi\)
−0.271205 + 0.962522i \(0.587422\pi\)
\(740\) −5.81732e6 −0.390520
\(741\) −7.58920e6 −0.507751
\(742\) 1.20909e6 0.0806211
\(743\) 1.78280e7 1.18476 0.592381 0.805658i \(-0.298188\pi\)
0.592381 + 0.805658i \(0.298188\pi\)
\(744\) −4.48685e6 −0.297173
\(745\) −5.54502e6 −0.366027
\(746\) −3.19652e6 −0.210295
\(747\) −5.69269e6 −0.373264
\(748\) −2.69159e6 −0.175896
\(749\) 993301. 0.0646958
\(750\) 1.67893e6 0.108988
\(751\) 2.32701e6 0.150556 0.0752782 0.997163i \(-0.476016\pi\)
0.0752782 + 0.997163i \(0.476016\pi\)
\(752\) 5.75143e6 0.370878
\(753\) −1.04636e7 −0.672500
\(754\) −1.40643e7 −0.900930
\(755\) 3.26516e6 0.208467
\(756\) −207595. −0.0132103
\(757\) −3.09474e6 −0.196284 −0.0981418 0.995172i \(-0.531290\pi\)
−0.0981418 + 0.995172i \(0.531290\pi\)
\(758\) 1.12627e6 0.0711984
\(759\) 2.59513e6 0.163514
\(760\) −7.98181e6 −0.501265
\(761\) 3.06304e6 0.191731 0.0958653 0.995394i \(-0.469438\pi\)
0.0958653 + 0.995394i \(0.469438\pi\)
\(762\) 8.89227e6 0.554785
\(763\) 3.74956e6 0.233168
\(764\) 2.76169e6 0.171176
\(765\) 2.10516e6 0.130057
\(766\) −1.04517e7 −0.643597
\(767\) −1.76574e6 −0.108377
\(768\) −589824. −0.0360844
\(769\) 1.40665e7 0.857770 0.428885 0.903359i \(-0.358906\pi\)
0.428885 + 0.903359i \(0.358906\pi\)
\(770\) 2.59357e6 0.157642
\(771\) 1.29203e7 0.782774
\(772\) −1.46018e7 −0.881783
\(773\) −1.90973e7 −1.14954 −0.574768 0.818317i \(-0.694908\pi\)
−0.574768 + 0.818317i \(0.694908\pi\)
\(774\) 5.16119e6 0.309669
\(775\) −1.95003e7 −1.16624
\(776\) −1.63484e6 −0.0974590
\(777\) −776289. −0.0461287
\(778\) 5.79789e6 0.343416
\(779\) −2.44290e7 −1.44232
\(780\) −5.47994e6 −0.322507
\(781\) 1.59978e7 0.938497
\(782\) −822827. −0.0481162
\(783\) 5.05317e6 0.294550
\(784\) −4.22150e6 −0.245288
\(785\) 7.68953e6 0.445375
\(786\) 9.71873e6 0.561117
\(787\) 5.58099e6 0.321199 0.160600 0.987020i \(-0.448657\pi\)
0.160600 + 0.987020i \(0.448657\pi\)
\(788\) 5.53810e6 0.317721
\(789\) 1.70568e7 0.975452
\(790\) −1.08603e7 −0.619117
\(791\) −4.52907e6 −0.257376
\(792\) −2.51735e6 −0.142604
\(793\) 2.88176e6 0.162733
\(794\) −1.75568e7 −0.988310
\(795\) −1.14674e7 −0.643495
\(796\) 3.51461e6 0.196605
\(797\) −2.73946e7 −1.52763 −0.763817 0.645433i \(-0.776677\pi\)
−0.763817 + 0.645433i \(0.776677\pi\)
\(798\) −1.06513e6 −0.0592100
\(799\) −7.78298e6 −0.431300
\(800\) −2.56344e6 −0.141611
\(801\) 2.04733e6 0.112748
\(802\) 1.49486e7 0.820662
\(803\) 3.61308e7 1.97737
\(804\) 8.77820e6 0.478923
\(805\) 792863. 0.0431229
\(806\) −1.58053e7 −0.856967
\(807\) −1.59944e7 −0.864538
\(808\) −1.13851e7 −0.613491
\(809\) 2.18484e7 1.17368 0.586839 0.809704i \(-0.300372\pi\)
0.586839 + 0.809704i \(0.300372\pi\)
\(810\) 1.96889e6 0.105441
\(811\) −2.22347e7 −1.18708 −0.593539 0.804805i \(-0.702270\pi\)
−0.593539 + 0.804805i \(0.702270\pi\)
\(812\) −1.97390e6 −0.105059
\(813\) 6.84336e6 0.363114
\(814\) −9.41350e6 −0.497955
\(815\) 6.80845e6 0.359049
\(816\) 798165. 0.0419631
\(817\) 2.64811e7 1.38797
\(818\) −1.21022e7 −0.632385
\(819\) −731268. −0.0380949
\(820\) −1.76395e7 −0.916117
\(821\) −7.07014e6 −0.366075 −0.183038 0.983106i \(-0.558593\pi\)
−0.183038 + 0.983106i \(0.558593\pi\)
\(822\) 1.28296e7 0.662266
\(823\) 1.64293e7 0.845509 0.422755 0.906244i \(-0.361063\pi\)
0.422755 + 0.906244i \(0.361063\pi\)
\(824\) −5.92971e6 −0.304240
\(825\) −1.09407e7 −0.559641
\(826\) −247818. −0.0126381
\(827\) −2.82840e7 −1.43806 −0.719030 0.694979i \(-0.755414\pi\)
−0.719030 + 0.694979i \(0.755414\pi\)
\(828\) −769562. −0.0390092
\(829\) −5.03085e6 −0.254247 −0.127123 0.991887i \(-0.540574\pi\)
−0.127123 + 0.991887i \(0.540574\pi\)
\(830\) −2.10903e7 −1.06265
\(831\) 1.57725e7 0.792315
\(832\) −2.07770e6 −0.104058
\(833\) 5.71264e6 0.285249
\(834\) −1.82219e6 −0.0907147
\(835\) 4.44953e6 0.220850
\(836\) −1.29160e7 −0.639166
\(837\) 5.67867e6 0.280177
\(838\) 1.19421e7 0.587451
\(839\) 2.05246e7 1.00663 0.503316 0.864102i \(-0.332113\pi\)
0.503316 + 0.864102i \(0.332113\pi\)
\(840\) −769099. −0.0376083
\(841\) 2.75366e7 1.34252
\(842\) −6.86995e6 −0.333944
\(843\) −1.24567e6 −0.0603719
\(844\) 2.23124e6 0.107818
\(845\) 8.55177e6 0.412016
\(846\) −7.27915e6 −0.349667
\(847\) 1.33051e6 0.0637250
\(848\) −4.34780e6 −0.207625
\(849\) 7.60367e6 0.362038
\(850\) 3.46891e6 0.164682
\(851\) −2.87773e6 −0.136215
\(852\) −4.74399e6 −0.223895
\(853\) −3.77826e7 −1.77795 −0.888974 0.457957i \(-0.848581\pi\)
−0.888974 + 0.457957i \(0.848581\pi\)
\(854\) 404449. 0.0189766
\(855\) 1.01020e7 0.472597
\(856\) −3.57184e6 −0.166613
\(857\) 1.71644e6 0.0798319 0.0399159 0.999203i \(-0.487291\pi\)
0.0399159 + 0.999203i \(0.487291\pi\)
\(858\) −8.86756e6 −0.411231
\(859\) 1.12483e7 0.520120 0.260060 0.965592i \(-0.416258\pi\)
0.260060 + 0.965592i \(0.416258\pi\)
\(860\) 1.91212e7 0.881596
\(861\) −2.35389e6 −0.108213
\(862\) 1.46936e6 0.0673537
\(863\) 9.56455e6 0.437157 0.218579 0.975819i \(-0.429858\pi\)
0.218579 + 0.975819i \(0.429858\pi\)
\(864\) 746496. 0.0340207
\(865\) 2.78548e7 1.26578
\(866\) −2.73844e6 −0.124082
\(867\) 1.16986e7 0.528551
\(868\) −2.21824e6 −0.0999328
\(869\) −1.75739e7 −0.789440
\(870\) 1.87210e7 0.838556
\(871\) 3.09219e7 1.38108
\(872\) −1.34832e7 −0.600483
\(873\) 2.06910e6 0.0918853
\(874\) −3.94847e6 −0.174844
\(875\) 830041. 0.0366505
\(876\) −1.07142e7 −0.471738
\(877\) −3.86591e6 −0.169728 −0.0848640 0.996393i \(-0.527046\pi\)
−0.0848640 + 0.996393i \(0.527046\pi\)
\(878\) −1.53699e7 −0.672876
\(879\) −2.52678e7 −1.10305
\(880\) −9.32631e6 −0.405978
\(881\) 2.11609e7 0.918531 0.459265 0.888299i \(-0.348113\pi\)
0.459265 + 0.888299i \(0.348113\pi\)
\(882\) 5.34284e6 0.231260
\(883\) 2.68330e7 1.15816 0.579079 0.815272i \(-0.303412\pi\)
0.579079 + 0.815272i \(0.303412\pi\)
\(884\) 2.81160e6 0.121010
\(885\) 2.35038e6 0.100874
\(886\) −1.30662e7 −0.559196
\(887\) 2.05930e7 0.878840 0.439420 0.898282i \(-0.355184\pi\)
0.439420 + 0.898282i \(0.355184\pi\)
\(888\) 2.79148e6 0.118796
\(889\) 4.39621e6 0.186562
\(890\) 7.58498e6 0.320981
\(891\) 3.18602e6 0.134448
\(892\) 8.38537e6 0.352866
\(893\) −3.73479e7 −1.56725
\(894\) 2.66082e6 0.111345
\(895\) −582507. −0.0243077
\(896\) −291601. −0.0121344
\(897\) −2.71084e6 −0.112492
\(898\) 1.20280e6 0.0497739
\(899\) 5.39952e7 2.22821
\(900\) 3.24435e6 0.133512
\(901\) 5.88356e6 0.241450
\(902\) −2.85439e7 −1.16815
\(903\) 2.55162e6 0.104135
\(904\) 1.62862e7 0.662826
\(905\) −1.55709e7 −0.631965
\(906\) −1.56681e6 −0.0634156
\(907\) 4.32278e7 1.74480 0.872398 0.488796i \(-0.162564\pi\)
0.872398 + 0.488796i \(0.162564\pi\)
\(908\) −1.70406e7 −0.685914
\(909\) 1.44093e7 0.578405
\(910\) −2.70921e6 −0.108452
\(911\) 1.43100e7 0.571271 0.285636 0.958338i \(-0.407795\pi\)
0.285636 + 0.958338i \(0.407795\pi\)
\(912\) 3.83013e6 0.152485
\(913\) −3.41281e7 −1.35499
\(914\) −2.20668e7 −0.873723
\(915\) −3.83591e6 −0.151466
\(916\) 1.23888e6 0.0487855
\(917\) 4.80480e6 0.188692
\(918\) −1.01018e6 −0.0395632
\(919\) 2.96906e7 1.15966 0.579830 0.814737i \(-0.303119\pi\)
0.579830 + 0.814737i \(0.303119\pi\)
\(920\) −2.85108e6 −0.111055
\(921\) −6.41335e6 −0.249136
\(922\) −1.18246e7 −0.458098
\(923\) −1.67111e7 −0.645654
\(924\) −1.24454e6 −0.0479546
\(925\) 1.21321e7 0.466209
\(926\) −2.64645e7 −1.01423
\(927\) 7.50479e6 0.286840
\(928\) 7.09801e6 0.270562
\(929\) 9.42342e6 0.358236 0.179118 0.983828i \(-0.442676\pi\)
0.179118 + 0.983828i \(0.442676\pi\)
\(930\) 2.10384e7 0.797636
\(931\) 2.74131e7 1.03653
\(932\) 8.76248e6 0.330436
\(933\) −1.83129e7 −0.688738
\(934\) −2.87573e7 −1.07865
\(935\) 1.26206e7 0.472118
\(936\) 2.62959e6 0.0981066
\(937\) −1.01301e7 −0.376935 −0.188467 0.982079i \(-0.560352\pi\)
−0.188467 + 0.982079i \(0.560352\pi\)
\(938\) 4.33982e6 0.161051
\(939\) −9.38960e6 −0.347523
\(940\) −2.69679e7 −0.995468
\(941\) −2.81460e6 −0.103620 −0.0518099 0.998657i \(-0.516499\pi\)
−0.0518099 + 0.998657i \(0.516499\pi\)
\(942\) −3.68987e6 −0.135483
\(943\) −8.72595e6 −0.319546
\(944\) 891136. 0.0325472
\(945\) 973391. 0.0354575
\(946\) 3.09417e7 1.12413
\(947\) −1.60727e7 −0.582388 −0.291194 0.956664i \(-0.594053\pi\)
−0.291194 + 0.956664i \(0.594053\pi\)
\(948\) 5.21137e6 0.188335
\(949\) −3.77417e7 −1.36037
\(950\) 1.66461e7 0.598418
\(951\) −873396. −0.0313155
\(952\) 394602. 0.0141113
\(953\) −5.01184e7 −1.78758 −0.893789 0.448488i \(-0.851963\pi\)
−0.893789 + 0.448488i \(0.851963\pi\)
\(954\) 5.50269e6 0.195751
\(955\) −1.29493e7 −0.459449
\(956\) 1.24969e7 0.442238
\(957\) 3.02941e7 1.06925
\(958\) −3.56354e6 −0.125449
\(959\) 6.34276e6 0.222706
\(960\) 2.76563e6 0.0968535
\(961\) 3.20497e7 1.11948
\(962\) 9.83320e6 0.342576
\(963\) 4.52061e6 0.157084
\(964\) 1.77358e7 0.614692
\(965\) 6.84662e7 2.36678
\(966\) −380461. −0.0131180
\(967\) −2.60892e7 −0.897211 −0.448605 0.893730i \(-0.648079\pi\)
−0.448605 + 0.893730i \(0.648079\pi\)
\(968\) −4.78442e6 −0.164112
\(969\) −5.18303e6 −0.177327
\(970\) 7.66562e6 0.261588
\(971\) −3.47981e7 −1.18442 −0.592212 0.805782i \(-0.701745\pi\)
−0.592212 + 0.805782i \(0.701745\pi\)
\(972\) −944784. −0.0320750
\(973\) −900864. −0.0305054
\(974\) 8.72405e6 0.294660
\(975\) 1.14285e7 0.385014
\(976\) −1.45437e6 −0.0488709
\(977\) −5.04875e7 −1.69218 −0.846092 0.533038i \(-0.821050\pi\)
−0.846092 + 0.533038i \(0.821050\pi\)
\(978\) −3.26708e6 −0.109223
\(979\) 1.22739e7 0.409285
\(980\) 1.97942e7 0.658374
\(981\) 1.70646e7 0.566141
\(982\) −3.29194e7 −1.08937
\(983\) −3.69821e7 −1.22070 −0.610348 0.792134i \(-0.708970\pi\)
−0.610348 + 0.792134i \(0.708970\pi\)
\(984\) 8.46442e6 0.278682
\(985\) −2.59676e7 −0.852788
\(986\) −9.60521e6 −0.314640
\(987\) −3.59871e6 −0.117586
\(988\) 1.34919e7 0.439725
\(989\) 9.45895e6 0.307505
\(990\) 1.18036e7 0.382760
\(991\) 4.47136e7 1.44629 0.723146 0.690696i \(-0.242696\pi\)
0.723146 + 0.690696i \(0.242696\pi\)
\(992\) 7.97662e6 0.257359
\(993\) −2.85303e7 −0.918192
\(994\) −2.34536e6 −0.0752912
\(995\) −1.64797e7 −0.527704
\(996\) 1.01203e7 0.323256
\(997\) 3.12814e7 0.996662 0.498331 0.866987i \(-0.333946\pi\)
0.498331 + 0.866987i \(0.333946\pi\)
\(998\) 3.07184e7 0.976276
\(999\) −3.53297e6 −0.112002
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 354.6.a.g.1.1 6
3.2 odd 2 1062.6.a.h.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.6.a.g.1.1 6 1.1 even 1 trivial
1062.6.a.h.1.6 6 3.2 odd 2