Properties

Label 261.4.a.f.1.2
Level $261$
Weight $4$
Character 261.1
Self dual yes
Analytic conductor $15.399$
Analytic rank $0$
Dimension $5$
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] = [261,4,Mod(1,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 261.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: \(15.3994985115\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.328194\) of defining polynomial
Character \(\chi\) \(=\) 261.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24125 q^{2} -2.97681 q^{4} -18.3339 q^{5} -16.8583 q^{7} +24.6017 q^{8} +O(q^{10})\) \(q-2.24125 q^{2} -2.97681 q^{4} -18.3339 q^{5} -16.8583 q^{7} +24.6017 q^{8} +41.0908 q^{10} -52.4385 q^{11} -87.5580 q^{13} +37.7836 q^{14} -31.3241 q^{16} -15.4072 q^{17} +67.0156 q^{19} +54.5766 q^{20} +117.528 q^{22} -132.679 q^{23} +211.133 q^{25} +196.239 q^{26} +50.1839 q^{28} +29.0000 q^{29} +90.2221 q^{31} -126.609 q^{32} +34.5314 q^{34} +309.078 q^{35} +11.1247 q^{37} -150.199 q^{38} -451.047 q^{40} +18.8392 q^{41} -147.756 q^{43} +156.100 q^{44} +297.366 q^{46} -21.0963 q^{47} -58.7983 q^{49} -473.201 q^{50} +260.644 q^{52} +290.454 q^{53} +961.404 q^{55} -414.743 q^{56} -64.9962 q^{58} +337.343 q^{59} +84.0147 q^{61} -202.210 q^{62} +534.355 q^{64} +1605.28 q^{65} +330.821 q^{67} +45.8644 q^{68} -692.721 q^{70} -492.420 q^{71} -347.053 q^{73} -24.9333 q^{74} -199.493 q^{76} +884.023 q^{77} -986.297 q^{79} +574.294 q^{80} -42.2234 q^{82} -594.382 q^{83} +282.475 q^{85} +331.157 q^{86} -1290.08 q^{88} -1387.04 q^{89} +1476.08 q^{91} +394.960 q^{92} +47.2820 q^{94} -1228.66 q^{95} -334.003 q^{97} +131.781 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.24125 −0.792400 −0.396200 0.918164i \(-0.629671\pi\)
−0.396200 + 0.918164i \(0.629671\pi\)
\(3\) 0 0
\(4\) −2.97681 −0.372101
\(5\) −18.3339 −1.63984 −0.819918 0.572481i \(-0.805981\pi\)
−0.819918 + 0.572481i \(0.805981\pi\)
\(6\) 0 0
\(7\) −16.8583 −0.910262 −0.455131 0.890425i \(-0.650408\pi\)
−0.455131 + 0.890425i \(0.650408\pi\)
\(8\) 24.6017 1.08725
\(9\) 0 0
\(10\) 41.0908 1.29941
\(11\) −52.4385 −1.43735 −0.718673 0.695348i \(-0.755250\pi\)
−0.718673 + 0.695348i \(0.755250\pi\)
\(12\) 0 0
\(13\) −87.5580 −1.86802 −0.934008 0.357252i \(-0.883714\pi\)
−0.934008 + 0.357252i \(0.883714\pi\)
\(14\) 37.7836 0.721292
\(15\) 0 0
\(16\) −31.3241 −0.489439
\(17\) −15.4072 −0.219812 −0.109906 0.993942i \(-0.535055\pi\)
−0.109906 + 0.993942i \(0.535055\pi\)
\(18\) 0 0
\(19\) 67.0156 0.809181 0.404591 0.914498i \(-0.367414\pi\)
0.404591 + 0.914498i \(0.367414\pi\)
\(20\) 54.5766 0.610185
\(21\) 0 0
\(22\) 117.528 1.13895
\(23\) −132.679 −1.20285 −0.601423 0.798931i \(-0.705399\pi\)
−0.601423 + 0.798931i \(0.705399\pi\)
\(24\) 0 0
\(25\) 211.133 1.68906
\(26\) 196.239 1.48022
\(27\) 0 0
\(28\) 50.1839 0.338710
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 90.2221 0.522721 0.261361 0.965241i \(-0.415829\pi\)
0.261361 + 0.965241i \(0.415829\pi\)
\(32\) −126.609 −0.699422
\(33\) 0 0
\(34\) 34.5314 0.174179
\(35\) 309.078 1.49268
\(36\) 0 0
\(37\) 11.1247 0.0494296 0.0247148 0.999695i \(-0.492132\pi\)
0.0247148 + 0.999695i \(0.492132\pi\)
\(38\) −150.199 −0.641196
\(39\) 0 0
\(40\) −451.047 −1.78292
\(41\) 18.8392 0.0717608 0.0358804 0.999356i \(-0.488576\pi\)
0.0358804 + 0.999356i \(0.488576\pi\)
\(42\) 0 0
\(43\) −147.756 −0.524013 −0.262007 0.965066i \(-0.584384\pi\)
−0.262007 + 0.965066i \(0.584384\pi\)
\(44\) 156.100 0.534839
\(45\) 0 0
\(46\) 297.366 0.953136
\(47\) −21.0963 −0.0654726 −0.0327363 0.999464i \(-0.510422\pi\)
−0.0327363 + 0.999464i \(0.510422\pi\)
\(48\) 0 0
\(49\) −58.7983 −0.171424
\(50\) −473.201 −1.33841
\(51\) 0 0
\(52\) 260.644 0.695092
\(53\) 290.454 0.752772 0.376386 0.926463i \(-0.377167\pi\)
0.376386 + 0.926463i \(0.377167\pi\)
\(54\) 0 0
\(55\) 961.404 2.35701
\(56\) −414.743 −0.989686
\(57\) 0 0
\(58\) −64.9962 −0.147145
\(59\) 337.343 0.744379 0.372190 0.928157i \(-0.378607\pi\)
0.372190 + 0.928157i \(0.378607\pi\)
\(60\) 0 0
\(61\) 84.0147 0.176344 0.0881720 0.996105i \(-0.471897\pi\)
0.0881720 + 0.996105i \(0.471897\pi\)
\(62\) −202.210 −0.414205
\(63\) 0 0
\(64\) 534.355 1.04366
\(65\) 1605.28 3.06324
\(66\) 0 0
\(67\) 330.821 0.603228 0.301614 0.953430i \(-0.402475\pi\)
0.301614 + 0.953430i \(0.402475\pi\)
\(68\) 45.8644 0.0817922
\(69\) 0 0
\(70\) −692.721 −1.18280
\(71\) −492.420 −0.823092 −0.411546 0.911389i \(-0.635011\pi\)
−0.411546 + 0.911389i \(0.635011\pi\)
\(72\) 0 0
\(73\) −347.053 −0.556431 −0.278216 0.960519i \(-0.589743\pi\)
−0.278216 + 0.960519i \(0.589743\pi\)
\(74\) −24.9333 −0.0391681
\(75\) 0 0
\(76\) −199.493 −0.301098
\(77\) 884.023 1.30836
\(78\) 0 0
\(79\) −986.297 −1.40465 −0.702324 0.711858i \(-0.747854\pi\)
−0.702324 + 0.711858i \(0.747854\pi\)
\(80\) 574.294 0.802600
\(81\) 0 0
\(82\) −42.2234 −0.0568633
\(83\) −594.382 −0.786048 −0.393024 0.919528i \(-0.628571\pi\)
−0.393024 + 0.919528i \(0.628571\pi\)
\(84\) 0 0
\(85\) 282.475 0.360455
\(86\) 331.157 0.415228
\(87\) 0 0
\(88\) −1290.08 −1.56276
\(89\) −1387.04 −1.65197 −0.825987 0.563689i \(-0.809382\pi\)
−0.825987 + 0.563689i \(0.809382\pi\)
\(90\) 0 0
\(91\) 1476.08 1.70038
\(92\) 394.960 0.447581
\(93\) 0 0
\(94\) 47.2820 0.0518805
\(95\) −1228.66 −1.32692
\(96\) 0 0
\(97\) −334.003 −0.349617 −0.174808 0.984602i \(-0.555931\pi\)
−0.174808 + 0.984602i \(0.555931\pi\)
\(98\) 131.781 0.135836
\(99\) 0 0
\(100\) −628.502 −0.628502
\(101\) 245.919 0.242276 0.121138 0.992636i \(-0.461346\pi\)
0.121138 + 0.992636i \(0.461346\pi\)
\(102\) 0 0
\(103\) −531.298 −0.508255 −0.254128 0.967171i \(-0.581788\pi\)
−0.254128 + 0.967171i \(0.581788\pi\)
\(104\) −2154.08 −2.03101
\(105\) 0 0
\(106\) −650.979 −0.596497
\(107\) 429.030 0.387625 0.193812 0.981039i \(-0.437915\pi\)
0.193812 + 0.981039i \(0.437915\pi\)
\(108\) 0 0
\(109\) −967.263 −0.849972 −0.424986 0.905200i \(-0.639721\pi\)
−0.424986 + 0.905200i \(0.639721\pi\)
\(110\) −2154.74 −1.86770
\(111\) 0 0
\(112\) 528.070 0.445518
\(113\) 1705.23 1.41960 0.709798 0.704405i \(-0.248786\pi\)
0.709798 + 0.704405i \(0.248786\pi\)
\(114\) 0 0
\(115\) 2432.52 1.97247
\(116\) −86.3275 −0.0690975
\(117\) 0 0
\(118\) −756.070 −0.589846
\(119\) 259.739 0.200086
\(120\) 0 0
\(121\) 1418.80 1.06596
\(122\) −188.298 −0.139735
\(123\) 0 0
\(124\) −268.574 −0.194505
\(125\) −1579.15 −1.12995
\(126\) 0 0
\(127\) −2670.28 −1.86574 −0.932870 0.360213i \(-0.882704\pi\)
−0.932870 + 0.360213i \(0.882704\pi\)
\(128\) −184.749 −0.127576
\(129\) 0 0
\(130\) −3597.83 −2.42731
\(131\) −879.993 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(132\) 0 0
\(133\) −1129.77 −0.736567
\(134\) −741.452 −0.477998
\(135\) 0 0
\(136\) −379.044 −0.238991
\(137\) −2064.15 −1.28724 −0.643620 0.765345i \(-0.722568\pi\)
−0.643620 + 0.765345i \(0.722568\pi\)
\(138\) 0 0
\(139\) 605.130 0.369255 0.184628 0.982809i \(-0.440892\pi\)
0.184628 + 0.982809i \(0.440892\pi\)
\(140\) −920.068 −0.555428
\(141\) 0 0
\(142\) 1103.63 0.652218
\(143\) 4591.41 2.68499
\(144\) 0 0
\(145\) −531.684 −0.304510
\(146\) 777.832 0.440917
\(147\) 0 0
\(148\) −33.1163 −0.0183928
\(149\) 775.322 0.426287 0.213144 0.977021i \(-0.431630\pi\)
0.213144 + 0.977021i \(0.431630\pi\)
\(150\) 0 0
\(151\) 427.925 0.230623 0.115311 0.993329i \(-0.463213\pi\)
0.115311 + 0.993329i \(0.463213\pi\)
\(152\) 1648.70 0.879785
\(153\) 0 0
\(154\) −1981.31 −1.03675
\(155\) −1654.12 −0.857177
\(156\) 0 0
\(157\) −1680.93 −0.854474 −0.427237 0.904140i \(-0.640513\pi\)
−0.427237 + 0.904140i \(0.640513\pi\)
\(158\) 2210.54 1.11304
\(159\) 0 0
\(160\) 2321.24 1.14694
\(161\) 2236.74 1.09490
\(162\) 0 0
\(163\) 2038.68 0.979645 0.489822 0.871822i \(-0.337062\pi\)
0.489822 + 0.871822i \(0.337062\pi\)
\(164\) −56.0808 −0.0267023
\(165\) 0 0
\(166\) 1332.16 0.622865
\(167\) −2543.12 −1.17840 −0.589199 0.807988i \(-0.700557\pi\)
−0.589199 + 0.807988i \(0.700557\pi\)
\(168\) 0 0
\(169\) 5469.40 2.48949
\(170\) −633.095 −0.285625
\(171\) 0 0
\(172\) 439.842 0.194986
\(173\) 306.031 0.134492 0.0672460 0.997736i \(-0.478579\pi\)
0.0672460 + 0.997736i \(0.478579\pi\)
\(174\) 0 0
\(175\) −3559.34 −1.53749
\(176\) 1642.59 0.703493
\(177\) 0 0
\(178\) 3108.69 1.30903
\(179\) 478.797 0.199927 0.0999635 0.994991i \(-0.468127\pi\)
0.0999635 + 0.994991i \(0.468127\pi\)
\(180\) 0 0
\(181\) −478.433 −0.196473 −0.0982367 0.995163i \(-0.531320\pi\)
−0.0982367 + 0.995163i \(0.531320\pi\)
\(182\) −3308.25 −1.34739
\(183\) 0 0
\(184\) −3264.13 −1.30780
\(185\) −203.960 −0.0810565
\(186\) 0 0
\(187\) 807.931 0.315945
\(188\) 62.7998 0.0243625
\(189\) 0 0
\(190\) 2753.73 1.05146
\(191\) −833.106 −0.315610 −0.157805 0.987470i \(-0.550442\pi\)
−0.157805 + 0.987470i \(0.550442\pi\)
\(192\) 0 0
\(193\) −1449.88 −0.540751 −0.270376 0.962755i \(-0.587148\pi\)
−0.270376 + 0.962755i \(0.587148\pi\)
\(194\) 748.582 0.277037
\(195\) 0 0
\(196\) 175.031 0.0637870
\(197\) 1993.27 0.720886 0.360443 0.932781i \(-0.382626\pi\)
0.360443 + 0.932781i \(0.382626\pi\)
\(198\) 0 0
\(199\) −356.359 −0.126943 −0.0634714 0.997984i \(-0.520217\pi\)
−0.0634714 + 0.997984i \(0.520217\pi\)
\(200\) 5194.23 1.83644
\(201\) 0 0
\(202\) −551.165 −0.191979
\(203\) −488.890 −0.169031
\(204\) 0 0
\(205\) −345.397 −0.117676
\(206\) 1190.77 0.402742
\(207\) 0 0
\(208\) 2742.67 0.914280
\(209\) −3514.20 −1.16307
\(210\) 0 0
\(211\) 4131.66 1.34803 0.674017 0.738716i \(-0.264567\pi\)
0.674017 + 0.738716i \(0.264567\pi\)
\(212\) −864.626 −0.280107
\(213\) 0 0
\(214\) −961.562 −0.307154
\(215\) 2708.95 0.859296
\(216\) 0 0
\(217\) −1520.99 −0.475813
\(218\) 2167.87 0.673518
\(219\) 0 0
\(220\) −2861.92 −0.877048
\(221\) 1349.02 0.410612
\(222\) 0 0
\(223\) 1332.32 0.400086 0.200043 0.979787i \(-0.435892\pi\)
0.200043 + 0.979787i \(0.435892\pi\)
\(224\) 2134.41 0.636657
\(225\) 0 0
\(226\) −3821.84 −1.12489
\(227\) 1329.33 0.388681 0.194340 0.980934i \(-0.437743\pi\)
0.194340 + 0.980934i \(0.437743\pi\)
\(228\) 0 0
\(229\) 5455.47 1.57427 0.787135 0.616780i \(-0.211563\pi\)
0.787135 + 0.616780i \(0.211563\pi\)
\(230\) −5451.89 −1.56299
\(231\) 0 0
\(232\) 713.451 0.201898
\(233\) 591.158 0.166215 0.0831075 0.996541i \(-0.473516\pi\)
0.0831075 + 0.996541i \(0.473516\pi\)
\(234\) 0 0
\(235\) 386.778 0.107364
\(236\) −1004.21 −0.276985
\(237\) 0 0
\(238\) −582.139 −0.158548
\(239\) −6946.01 −1.87992 −0.939959 0.341289i \(-0.889137\pi\)
−0.939959 + 0.341289i \(0.889137\pi\)
\(240\) 0 0
\(241\) 7105.62 1.89923 0.949613 0.313426i \(-0.101477\pi\)
0.949613 + 0.313426i \(0.101477\pi\)
\(242\) −3179.88 −0.844670
\(243\) 0 0
\(244\) −250.096 −0.0656179
\(245\) 1078.00 0.281106
\(246\) 0 0
\(247\) −5867.75 −1.51156
\(248\) 2219.62 0.568331
\(249\) 0 0
\(250\) 3539.27 0.895372
\(251\) 4874.53 1.22581 0.612904 0.790158i \(-0.290001\pi\)
0.612904 + 0.790158i \(0.290001\pi\)
\(252\) 0 0
\(253\) 6957.49 1.72891
\(254\) 5984.75 1.47841
\(255\) 0 0
\(256\) −3860.77 −0.942570
\(257\) −2488.22 −0.603934 −0.301967 0.953318i \(-0.597643\pi\)
−0.301967 + 0.953318i \(0.597643\pi\)
\(258\) 0 0
\(259\) −187.544 −0.0449939
\(260\) −4778.62 −1.13984
\(261\) 0 0
\(262\) 1972.28 0.465068
\(263\) 2812.52 0.659419 0.329709 0.944082i \(-0.393049\pi\)
0.329709 + 0.944082i \(0.393049\pi\)
\(264\) 0 0
\(265\) −5325.16 −1.23442
\(266\) 2532.09 0.583656
\(267\) 0 0
\(268\) −984.793 −0.224462
\(269\) 5554.63 1.25900 0.629501 0.777000i \(-0.283259\pi\)
0.629501 + 0.777000i \(0.283259\pi\)
\(270\) 0 0
\(271\) −3168.41 −0.710211 −0.355105 0.934826i \(-0.615555\pi\)
−0.355105 + 0.934826i \(0.615555\pi\)
\(272\) 482.617 0.107584
\(273\) 0 0
\(274\) 4626.26 1.02001
\(275\) −11071.5 −2.42777
\(276\) 0 0
\(277\) 3965.64 0.860189 0.430095 0.902784i \(-0.358480\pi\)
0.430095 + 0.902784i \(0.358480\pi\)
\(278\) −1356.25 −0.292598
\(279\) 0 0
\(280\) 7603.87 1.62292
\(281\) −1655.16 −0.351383 −0.175692 0.984445i \(-0.556216\pi\)
−0.175692 + 0.984445i \(0.556216\pi\)
\(282\) 0 0
\(283\) 7786.09 1.63546 0.817730 0.575602i \(-0.195232\pi\)
0.817730 + 0.575602i \(0.195232\pi\)
\(284\) 1465.84 0.306274
\(285\) 0 0
\(286\) −10290.5 −2.12758
\(287\) −317.597 −0.0653211
\(288\) 0 0
\(289\) −4675.62 −0.951683
\(290\) 1191.63 0.241294
\(291\) 0 0
\(292\) 1033.11 0.207049
\(293\) −8090.80 −1.61321 −0.806603 0.591094i \(-0.798696\pi\)
−0.806603 + 0.591094i \(0.798696\pi\)
\(294\) 0 0
\(295\) −6184.83 −1.22066
\(296\) 273.688 0.0537426
\(297\) 0 0
\(298\) −1737.69 −0.337790
\(299\) 11617.1 2.24694
\(300\) 0 0
\(301\) 2490.91 0.476989
\(302\) −959.086 −0.182746
\(303\) 0 0
\(304\) −2099.20 −0.396045
\(305\) −1540.32 −0.289175
\(306\) 0 0
\(307\) −6129.49 −1.13951 −0.569753 0.821816i \(-0.692961\pi\)
−0.569753 + 0.821816i \(0.692961\pi\)
\(308\) −2631.57 −0.486843
\(309\) 0 0
\(310\) 3707.30 0.679228
\(311\) 8167.93 1.48926 0.744632 0.667476i \(-0.232625\pi\)
0.744632 + 0.667476i \(0.232625\pi\)
\(312\) 0 0
\(313\) 1877.25 0.339005 0.169502 0.985530i \(-0.445784\pi\)
0.169502 + 0.985530i \(0.445784\pi\)
\(314\) 3767.37 0.677086
\(315\) 0 0
\(316\) 2936.02 0.522671
\(317\) −1222.93 −0.216677 −0.108338 0.994114i \(-0.534553\pi\)
−0.108338 + 0.994114i \(0.534553\pi\)
\(318\) 0 0
\(319\) −1520.72 −0.266908
\(320\) −9796.82 −1.71143
\(321\) 0 0
\(322\) −5013.08 −0.867603
\(323\) −1032.52 −0.177867
\(324\) 0 0
\(325\) −18486.4 −3.15520
\(326\) −4569.20 −0.776271
\(327\) 0 0
\(328\) 463.478 0.0780222
\(329\) 355.648 0.0595972
\(330\) 0 0
\(331\) 3769.03 0.625876 0.312938 0.949774i \(-0.398687\pi\)
0.312938 + 0.949774i \(0.398687\pi\)
\(332\) 1769.36 0.292489
\(333\) 0 0
\(334\) 5699.76 0.933763
\(335\) −6065.25 −0.989195
\(336\) 0 0
\(337\) −10900.2 −1.76193 −0.880967 0.473179i \(-0.843107\pi\)
−0.880967 + 0.473179i \(0.843107\pi\)
\(338\) −12258.3 −1.97267
\(339\) 0 0
\(340\) −840.874 −0.134126
\(341\) −4731.11 −0.751332
\(342\) 0 0
\(343\) 6773.63 1.06630
\(344\) −3635.05 −0.569735
\(345\) 0 0
\(346\) −685.892 −0.106572
\(347\) 8542.30 1.32154 0.660771 0.750588i \(-0.270230\pi\)
0.660771 + 0.750588i \(0.270230\pi\)
\(348\) 0 0
\(349\) −993.823 −0.152430 −0.0762151 0.997091i \(-0.524284\pi\)
−0.0762151 + 0.997091i \(0.524284\pi\)
\(350\) 7977.35 1.21831
\(351\) 0 0
\(352\) 6639.19 1.00531
\(353\) −8191.10 −1.23504 −0.617519 0.786556i \(-0.711862\pi\)
−0.617519 + 0.786556i \(0.711862\pi\)
\(354\) 0 0
\(355\) 9027.99 1.34974
\(356\) 4128.95 0.614702
\(357\) 0 0
\(358\) −1073.10 −0.158422
\(359\) 4703.71 0.691510 0.345755 0.938325i \(-0.387623\pi\)
0.345755 + 0.938325i \(0.387623\pi\)
\(360\) 0 0
\(361\) −2367.90 −0.345226
\(362\) 1072.29 0.155686
\(363\) 0 0
\(364\) −4394.00 −0.632715
\(365\) 6362.85 0.912456
\(366\) 0 0
\(367\) 9431.88 1.34153 0.670763 0.741672i \(-0.265967\pi\)
0.670763 + 0.741672i \(0.265967\pi\)
\(368\) 4156.05 0.588720
\(369\) 0 0
\(370\) 457.125 0.0642292
\(371\) −4896.55 −0.685219
\(372\) 0 0
\(373\) −8281.46 −1.14959 −0.574796 0.818297i \(-0.694919\pi\)
−0.574796 + 0.818297i \(0.694919\pi\)
\(374\) −1810.77 −0.250355
\(375\) 0 0
\(376\) −519.006 −0.0711854
\(377\) −2539.18 −0.346882
\(378\) 0 0
\(379\) 6875.50 0.931848 0.465924 0.884825i \(-0.345722\pi\)
0.465924 + 0.884825i \(0.345722\pi\)
\(380\) 3657.49 0.493750
\(381\) 0 0
\(382\) 1867.20 0.250089
\(383\) 4826.61 0.643938 0.321969 0.946750i \(-0.395655\pi\)
0.321969 + 0.946750i \(0.395655\pi\)
\(384\) 0 0
\(385\) −16207.6 −2.14550
\(386\) 3249.55 0.428492
\(387\) 0 0
\(388\) 994.263 0.130093
\(389\) −4970.57 −0.647861 −0.323930 0.946081i \(-0.605004\pi\)
−0.323930 + 0.946081i \(0.605004\pi\)
\(390\) 0 0
\(391\) 2044.21 0.264400
\(392\) −1446.54 −0.186381
\(393\) 0 0
\(394\) −4467.41 −0.571230
\(395\) 18082.7 2.30339
\(396\) 0 0
\(397\) −12288.8 −1.55355 −0.776774 0.629779i \(-0.783145\pi\)
−0.776774 + 0.629779i \(0.783145\pi\)
\(398\) 798.689 0.100590
\(399\) 0 0
\(400\) −6613.54 −0.826693
\(401\) 11971.7 1.49086 0.745432 0.666581i \(-0.232243\pi\)
0.745432 + 0.666581i \(0.232243\pi\)
\(402\) 0 0
\(403\) −7899.66 −0.976452
\(404\) −732.055 −0.0901512
\(405\) 0 0
\(406\) 1095.72 0.133941
\(407\) −583.365 −0.0710475
\(408\) 0 0
\(409\) 11147.7 1.34772 0.673861 0.738858i \(-0.264635\pi\)
0.673861 + 0.738858i \(0.264635\pi\)
\(410\) 774.120 0.0932465
\(411\) 0 0
\(412\) 1581.57 0.189123
\(413\) −5687.03 −0.677580
\(414\) 0 0
\(415\) 10897.4 1.28899
\(416\) 11085.6 1.30653
\(417\) 0 0
\(418\) 7876.19 0.921620
\(419\) −11557.3 −1.34752 −0.673762 0.738949i \(-0.735323\pi\)
−0.673762 + 0.738949i \(0.735323\pi\)
\(420\) 0 0
\(421\) −12874.4 −1.49040 −0.745201 0.666840i \(-0.767647\pi\)
−0.745201 + 0.666840i \(0.767647\pi\)
\(422\) −9260.07 −1.06818
\(423\) 0 0
\(424\) 7145.67 0.818454
\(425\) −3252.97 −0.371275
\(426\) 0 0
\(427\) −1416.34 −0.160519
\(428\) −1277.14 −0.144236
\(429\) 0 0
\(430\) −6071.42 −0.680906
\(431\) 4088.31 0.456907 0.228454 0.973555i \(-0.426633\pi\)
0.228454 + 0.973555i \(0.426633\pi\)
\(432\) 0 0
\(433\) −3865.90 −0.429060 −0.214530 0.976717i \(-0.568822\pi\)
−0.214530 + 0.976717i \(0.568822\pi\)
\(434\) 3408.91 0.377035
\(435\) 0 0
\(436\) 2879.36 0.316276
\(437\) −8891.56 −0.973320
\(438\) 0 0
\(439\) 10662.4 1.15920 0.579600 0.814901i \(-0.303209\pi\)
0.579600 + 0.814901i \(0.303209\pi\)
\(440\) 23652.2 2.56267
\(441\) 0 0
\(442\) −3023.50 −0.325369
\(443\) −10288.9 −1.10347 −0.551736 0.834019i \(-0.686034\pi\)
−0.551736 + 0.834019i \(0.686034\pi\)
\(444\) 0 0
\(445\) 25429.8 2.70897
\(446\) −2986.07 −0.317028
\(447\) 0 0
\(448\) −9008.30 −0.950005
\(449\) −12426.2 −1.30608 −0.653041 0.757323i \(-0.726507\pi\)
−0.653041 + 0.757323i \(0.726507\pi\)
\(450\) 0 0
\(451\) −987.901 −0.103145
\(452\) −5076.14 −0.528234
\(453\) 0 0
\(454\) −2979.35 −0.307991
\(455\) −27062.3 −2.78835
\(456\) 0 0
\(457\) −10657.8 −1.09092 −0.545462 0.838136i \(-0.683646\pi\)
−0.545462 + 0.838136i \(0.683646\pi\)
\(458\) −12227.1 −1.24745
\(459\) 0 0
\(460\) −7241.17 −0.733959
\(461\) −9819.21 −0.992031 −0.496016 0.868314i \(-0.665204\pi\)
−0.496016 + 0.868314i \(0.665204\pi\)
\(462\) 0 0
\(463\) −19210.5 −1.92827 −0.964135 0.265412i \(-0.914492\pi\)
−0.964135 + 0.265412i \(0.914492\pi\)
\(464\) −908.399 −0.0908865
\(465\) 0 0
\(466\) −1324.93 −0.131709
\(467\) 345.566 0.0342417 0.0171208 0.999853i \(-0.494550\pi\)
0.0171208 + 0.999853i \(0.494550\pi\)
\(468\) 0 0
\(469\) −5577.08 −0.549095
\(470\) −866.865 −0.0850756
\(471\) 0 0
\(472\) 8299.24 0.809329
\(473\) 7748.10 0.753188
\(474\) 0 0
\(475\) 14149.2 1.36676
\(476\) −773.194 −0.0744523
\(477\) 0 0
\(478\) 15567.7 1.48965
\(479\) 253.709 0.0242009 0.0121005 0.999927i \(-0.496148\pi\)
0.0121005 + 0.999927i \(0.496148\pi\)
\(480\) 0 0
\(481\) −974.060 −0.0923354
\(482\) −15925.5 −1.50495
\(483\) 0 0
\(484\) −4223.50 −0.396647
\(485\) 6123.58 0.573314
\(486\) 0 0
\(487\) −13255.1 −1.23336 −0.616680 0.787214i \(-0.711523\pi\)
−0.616680 + 0.787214i \(0.711523\pi\)
\(488\) 2066.91 0.191731
\(489\) 0 0
\(490\) −2416.07 −0.222749
\(491\) 6454.57 0.593260 0.296630 0.954993i \(-0.404137\pi\)
0.296630 + 0.954993i \(0.404137\pi\)
\(492\) 0 0
\(493\) −446.809 −0.0408180
\(494\) 13151.1 1.19776
\(495\) 0 0
\(496\) −2826.12 −0.255840
\(497\) 8301.36 0.749229
\(498\) 0 0
\(499\) 8090.41 0.725805 0.362902 0.931827i \(-0.381786\pi\)
0.362902 + 0.931827i \(0.381786\pi\)
\(500\) 4700.84 0.420455
\(501\) 0 0
\(502\) −10925.0 −0.971330
\(503\) 18897.4 1.67513 0.837567 0.546334i \(-0.183977\pi\)
0.837567 + 0.546334i \(0.183977\pi\)
\(504\) 0 0
\(505\) −4508.66 −0.397293
\(506\) −15593.4 −1.36999
\(507\) 0 0
\(508\) 7948.92 0.694245
\(509\) −4265.15 −0.371413 −0.185707 0.982605i \(-0.559457\pi\)
−0.185707 + 0.982605i \(0.559457\pi\)
\(510\) 0 0
\(511\) 5850.72 0.506498
\(512\) 10130.9 0.874469
\(513\) 0 0
\(514\) 5576.72 0.478558
\(515\) 9740.77 0.833455
\(516\) 0 0
\(517\) 1106.26 0.0941068
\(518\) 420.333 0.0356532
\(519\) 0 0
\(520\) 39492.7 3.33052
\(521\) 3324.96 0.279595 0.139798 0.990180i \(-0.455355\pi\)
0.139798 + 0.990180i \(0.455355\pi\)
\(522\) 0 0
\(523\) 13017.6 1.08838 0.544188 0.838964i \(-0.316838\pi\)
0.544188 + 0.838964i \(0.316838\pi\)
\(524\) 2619.57 0.218390
\(525\) 0 0
\(526\) −6303.54 −0.522524
\(527\) −1390.07 −0.114900
\(528\) 0 0
\(529\) 5436.69 0.446839
\(530\) 11935.0 0.978156
\(531\) 0 0
\(532\) 3363.11 0.274078
\(533\) −1649.52 −0.134050
\(534\) 0 0
\(535\) −7865.80 −0.635641
\(536\) 8138.78 0.655862
\(537\) 0 0
\(538\) −12449.3 −0.997634
\(539\) 3083.29 0.246395
\(540\) 0 0
\(541\) −17906.8 −1.42305 −0.711527 0.702658i \(-0.751996\pi\)
−0.711527 + 0.702658i \(0.751996\pi\)
\(542\) 7101.19 0.562771
\(543\) 0 0
\(544\) 1950.69 0.153741
\(545\) 17733.7 1.39382
\(546\) 0 0
\(547\) 1612.94 0.126078 0.0630389 0.998011i \(-0.479921\pi\)
0.0630389 + 0.998011i \(0.479921\pi\)
\(548\) 6144.58 0.478984
\(549\) 0 0
\(550\) 24813.9 1.92376
\(551\) 1943.45 0.150261
\(552\) 0 0
\(553\) 16627.3 1.27860
\(554\) −8887.99 −0.681614
\(555\) 0 0
\(556\) −1801.36 −0.137400
\(557\) 7803.94 0.593651 0.296826 0.954932i \(-0.404072\pi\)
0.296826 + 0.954932i \(0.404072\pi\)
\(558\) 0 0
\(559\) 12937.2 0.978865
\(560\) −9681.60 −0.730576
\(561\) 0 0
\(562\) 3709.63 0.278436
\(563\) 12329.5 0.922958 0.461479 0.887151i \(-0.347319\pi\)
0.461479 + 0.887151i \(0.347319\pi\)
\(564\) 0 0
\(565\) −31263.5 −2.32790
\(566\) −17450.6 −1.29594
\(567\) 0 0
\(568\) −12114.4 −0.894910
\(569\) −1554.46 −0.114528 −0.0572640 0.998359i \(-0.518238\pi\)
−0.0572640 + 0.998359i \(0.518238\pi\)
\(570\) 0 0
\(571\) 15951.8 1.16911 0.584556 0.811353i \(-0.301269\pi\)
0.584556 + 0.811353i \(0.301269\pi\)
\(572\) −13667.8 −0.999087
\(573\) 0 0
\(574\) 711.813 0.0517605
\(575\) −28012.9 −2.03168
\(576\) 0 0
\(577\) 10491.7 0.756975 0.378488 0.925606i \(-0.376444\pi\)
0.378488 + 0.925606i \(0.376444\pi\)
\(578\) 10479.2 0.754114
\(579\) 0 0
\(580\) 1582.72 0.113309
\(581\) 10020.3 0.715509
\(582\) 0 0
\(583\) −15231.0 −1.08199
\(584\) −8538.11 −0.604982
\(585\) 0 0
\(586\) 18133.5 1.27831
\(587\) −6437.47 −0.452645 −0.226323 0.974052i \(-0.572670\pi\)
−0.226323 + 0.974052i \(0.572670\pi\)
\(588\) 0 0
\(589\) 6046.29 0.422976
\(590\) 13861.7 0.967251
\(591\) 0 0
\(592\) −348.473 −0.0241928
\(593\) −11240.6 −0.778411 −0.389205 0.921151i \(-0.627250\pi\)
−0.389205 + 0.921151i \(0.627250\pi\)
\(594\) 0 0
\(595\) −4762.04 −0.328108
\(596\) −2307.99 −0.158622
\(597\) 0 0
\(598\) −26036.8 −1.78047
\(599\) −15903.5 −1.08481 −0.542405 0.840117i \(-0.682486\pi\)
−0.542405 + 0.840117i \(0.682486\pi\)
\(600\) 0 0
\(601\) 117.190 0.00795385 0.00397692 0.999992i \(-0.498734\pi\)
0.00397692 + 0.999992i \(0.498734\pi\)
\(602\) −5582.75 −0.377966
\(603\) 0 0
\(604\) −1273.85 −0.0858151
\(605\) −26012.1 −1.74801
\(606\) 0 0
\(607\) −22047.8 −1.47429 −0.737143 0.675737i \(-0.763825\pi\)
−0.737143 + 0.675737i \(0.763825\pi\)
\(608\) −8484.78 −0.565959
\(609\) 0 0
\(610\) 3452.24 0.229143
\(611\) 1847.15 0.122304
\(612\) 0 0
\(613\) −12719.2 −0.838050 −0.419025 0.907975i \(-0.637628\pi\)
−0.419025 + 0.907975i \(0.637628\pi\)
\(614\) 13737.7 0.902945
\(615\) 0 0
\(616\) 21748.5 1.42252
\(617\) −12736.9 −0.831064 −0.415532 0.909578i \(-0.636405\pi\)
−0.415532 + 0.909578i \(0.636405\pi\)
\(618\) 0 0
\(619\) 28083.4 1.82353 0.911767 0.410709i \(-0.134719\pi\)
0.911767 + 0.410709i \(0.134719\pi\)
\(620\) 4924.02 0.318957
\(621\) 0 0
\(622\) −18306.4 −1.18009
\(623\) 23383.1 1.50373
\(624\) 0 0
\(625\) 2560.44 0.163868
\(626\) −4207.38 −0.268628
\(627\) 0 0
\(628\) 5003.80 0.317951
\(629\) −171.401 −0.0108652
\(630\) 0 0
\(631\) 281.496 0.0177594 0.00887969 0.999961i \(-0.497173\pi\)
0.00887969 + 0.999961i \(0.497173\pi\)
\(632\) −24264.6 −1.52721
\(633\) 0 0
\(634\) 2740.89 0.171695
\(635\) 48956.7 3.05951
\(636\) 0 0
\(637\) 5148.26 0.320222
\(638\) 3408.30 0.211498
\(639\) 0 0
\(640\) 3387.18 0.209203
\(641\) 8440.98 0.520123 0.260061 0.965592i \(-0.416257\pi\)
0.260061 + 0.965592i \(0.416257\pi\)
\(642\) 0 0
\(643\) −1173.61 −0.0719792 −0.0359896 0.999352i \(-0.511458\pi\)
−0.0359896 + 0.999352i \(0.511458\pi\)
\(644\) −6658.35 −0.407416
\(645\) 0 0
\(646\) 2314.14 0.140942
\(647\) 10845.7 0.659025 0.329513 0.944151i \(-0.393116\pi\)
0.329513 + 0.944151i \(0.393116\pi\)
\(648\) 0 0
\(649\) −17689.8 −1.06993
\(650\) 41432.5 2.50018
\(651\) 0 0
\(652\) −6068.78 −0.364527
\(653\) 5282.40 0.316564 0.158282 0.987394i \(-0.449404\pi\)
0.158282 + 0.987394i \(0.449404\pi\)
\(654\) 0 0
\(655\) 16133.7 0.962437
\(656\) −590.122 −0.0351225
\(657\) 0 0
\(658\) −797.094 −0.0472249
\(659\) 19243.7 1.13752 0.568761 0.822503i \(-0.307423\pi\)
0.568761 + 0.822503i \(0.307423\pi\)
\(660\) 0 0
\(661\) 29196.9 1.71804 0.859021 0.511940i \(-0.171073\pi\)
0.859021 + 0.511940i \(0.171073\pi\)
\(662\) −8447.34 −0.495944
\(663\) 0 0
\(664\) −14622.8 −0.854633
\(665\) 20713.1 1.20785
\(666\) 0 0
\(667\) −3847.69 −0.223363
\(668\) 7570.39 0.438484
\(669\) 0 0
\(670\) 13593.7 0.783838
\(671\) −4405.61 −0.253467
\(672\) 0 0
\(673\) 19924.5 1.14121 0.570605 0.821224i \(-0.306709\pi\)
0.570605 + 0.821224i \(0.306709\pi\)
\(674\) 24430.0 1.39616
\(675\) 0 0
\(676\) −16281.4 −0.926341
\(677\) 4980.43 0.282738 0.141369 0.989957i \(-0.454850\pi\)
0.141369 + 0.989957i \(0.454850\pi\)
\(678\) 0 0
\(679\) 5630.71 0.318243
\(680\) 6949.37 0.391906
\(681\) 0 0
\(682\) 10603.6 0.595355
\(683\) −29295.8 −1.64125 −0.820624 0.571468i \(-0.806374\pi\)
−0.820624 + 0.571468i \(0.806374\pi\)
\(684\) 0 0
\(685\) 37843.9 2.11086
\(686\) −15181.4 −0.844938
\(687\) 0 0
\(688\) 4628.32 0.256472
\(689\) −25431.5 −1.40619
\(690\) 0 0
\(691\) −32759.8 −1.80353 −0.901766 0.432225i \(-0.857729\pi\)
−0.901766 + 0.432225i \(0.857729\pi\)
\(692\) −910.998 −0.0500447
\(693\) 0 0
\(694\) −19145.4 −1.04719
\(695\) −11094.4 −0.605518
\(696\) 0 0
\(697\) −290.260 −0.0157739
\(698\) 2227.40 0.120786
\(699\) 0 0
\(700\) 10595.5 0.572102
\(701\) −27958.5 −1.50639 −0.753195 0.657797i \(-0.771488\pi\)
−0.753195 + 0.657797i \(0.771488\pi\)
\(702\) 0 0
\(703\) 745.532 0.0399975
\(704\) −28020.8 −1.50010
\(705\) 0 0
\(706\) 18358.3 0.978644
\(707\) −4145.77 −0.220534
\(708\) 0 0
\(709\) −31863.5 −1.68781 −0.843906 0.536492i \(-0.819749\pi\)
−0.843906 + 0.536492i \(0.819749\pi\)
\(710\) −20234.0 −1.06953
\(711\) 0 0
\(712\) −34123.6 −1.79612
\(713\) −11970.6 −0.628753
\(714\) 0 0
\(715\) −84178.6 −4.40294
\(716\) −1425.29 −0.0743931
\(717\) 0 0
\(718\) −10542.2 −0.547953
\(719\) −7944.76 −0.412085 −0.206043 0.978543i \(-0.566059\pi\)
−0.206043 + 0.978543i \(0.566059\pi\)
\(720\) 0 0
\(721\) 8956.76 0.462645
\(722\) 5307.06 0.273557
\(723\) 0 0
\(724\) 1424.21 0.0731080
\(725\) 6122.85 0.313651
\(726\) 0 0
\(727\) 28640.3 1.46109 0.730543 0.682866i \(-0.239267\pi\)
0.730543 + 0.682866i \(0.239267\pi\)
\(728\) 36314.1 1.84875
\(729\) 0 0
\(730\) −14260.7 −0.723031
\(731\) 2276.51 0.115184
\(732\) 0 0
\(733\) 11852.7 0.597258 0.298629 0.954369i \(-0.403471\pi\)
0.298629 + 0.954369i \(0.403471\pi\)
\(734\) −21139.2 −1.06303
\(735\) 0 0
\(736\) 16798.3 0.841297
\(737\) −17347.8 −0.867047
\(738\) 0 0
\(739\) −24052.5 −1.19727 −0.598636 0.801021i \(-0.704290\pi\)
−0.598636 + 0.801021i \(0.704290\pi\)
\(740\) 607.151 0.0301612
\(741\) 0 0
\(742\) 10974.4 0.542968
\(743\) 12530.4 0.618704 0.309352 0.950948i \(-0.399888\pi\)
0.309352 + 0.950948i \(0.399888\pi\)
\(744\) 0 0
\(745\) −14214.7 −0.699042
\(746\) 18560.8 0.910937
\(747\) 0 0
\(748\) −2405.06 −0.117564
\(749\) −7232.71 −0.352840
\(750\) 0 0
\(751\) 30921.6 1.50246 0.751229 0.660042i \(-0.229461\pi\)
0.751229 + 0.660042i \(0.229461\pi\)
\(752\) 660.823 0.0320449
\(753\) 0 0
\(754\) 5690.93 0.274869
\(755\) −7845.54 −0.378184
\(756\) 0 0
\(757\) −6257.40 −0.300435 −0.150217 0.988653i \(-0.547997\pi\)
−0.150217 + 0.988653i \(0.547997\pi\)
\(758\) −15409.7 −0.738397
\(759\) 0 0
\(760\) −30227.2 −1.44270
\(761\) 20094.8 0.957211 0.478605 0.878030i \(-0.341142\pi\)
0.478605 + 0.878030i \(0.341142\pi\)
\(762\) 0 0
\(763\) 16306.4 0.773697
\(764\) 2480.00 0.117439
\(765\) 0 0
\(766\) −10817.6 −0.510257
\(767\) −29537.1 −1.39051
\(768\) 0 0
\(769\) −26647.8 −1.24960 −0.624801 0.780784i \(-0.714820\pi\)
−0.624801 + 0.780784i \(0.714820\pi\)
\(770\) 36325.3 1.70009
\(771\) 0 0
\(772\) 4316.03 0.201214
\(773\) 18513.2 0.861414 0.430707 0.902492i \(-0.358264\pi\)
0.430707 + 0.902492i \(0.358264\pi\)
\(774\) 0 0
\(775\) 19048.8 0.882909
\(776\) −8217.05 −0.380122
\(777\) 0 0
\(778\) 11140.3 0.513365
\(779\) 1262.52 0.0580675
\(780\) 0 0
\(781\) 25821.8 1.18307
\(782\) −4581.58 −0.209510
\(783\) 0 0
\(784\) 1841.80 0.0839014
\(785\) 30818.0 1.40120
\(786\) 0 0
\(787\) −29524.5 −1.33727 −0.668636 0.743590i \(-0.733122\pi\)
−0.668636 + 0.743590i \(0.733122\pi\)
\(788\) −5933.59 −0.268243
\(789\) 0 0
\(790\) −40527.8 −1.82521
\(791\) −28747.2 −1.29220
\(792\) 0 0
\(793\) −7356.16 −0.329413
\(794\) 27542.3 1.23103
\(795\) 0 0
\(796\) 1060.81 0.0472356
\(797\) −38789.4 −1.72395 −0.861976 0.506949i \(-0.830773\pi\)
−0.861976 + 0.506949i \(0.830773\pi\)
\(798\) 0 0
\(799\) 325.035 0.0143916
\(800\) −26731.3 −1.18137
\(801\) 0 0
\(802\) −26831.5 −1.18136
\(803\) 18199.0 0.799785
\(804\) 0 0
\(805\) −41008.2 −1.79546
\(806\) 17705.1 0.773741
\(807\) 0 0
\(808\) 6050.04 0.263415
\(809\) 11552.0 0.502037 0.251018 0.967982i \(-0.419235\pi\)
0.251018 + 0.967982i \(0.419235\pi\)
\(810\) 0 0
\(811\) 26939.1 1.16641 0.583205 0.812325i \(-0.301798\pi\)
0.583205 + 0.812325i \(0.301798\pi\)
\(812\) 1455.33 0.0628968
\(813\) 0 0
\(814\) 1307.47 0.0562981
\(815\) −37377.1 −1.60646
\(816\) 0 0
\(817\) −9901.96 −0.424022
\(818\) −24984.8 −1.06794
\(819\) 0 0
\(820\) 1028.18 0.0437874
\(821\) 7558.67 0.321315 0.160657 0.987010i \(-0.448639\pi\)
0.160657 + 0.987010i \(0.448639\pi\)
\(822\) 0 0
\(823\) −3201.90 −0.135615 −0.0678076 0.997698i \(-0.521600\pi\)
−0.0678076 + 0.997698i \(0.521600\pi\)
\(824\) −13070.8 −0.552603
\(825\) 0 0
\(826\) 12746.0 0.536915
\(827\) 11479.1 0.482670 0.241335 0.970442i \(-0.422415\pi\)
0.241335 + 0.970442i \(0.422415\pi\)
\(828\) 0 0
\(829\) −1667.94 −0.0698794 −0.0349397 0.999389i \(-0.511124\pi\)
−0.0349397 + 0.999389i \(0.511124\pi\)
\(830\) −24423.7 −1.02140
\(831\) 0 0
\(832\) −46787.0 −1.94958
\(833\) 905.917 0.0376809
\(834\) 0 0
\(835\) 46625.3 1.93238
\(836\) 10461.1 0.432781
\(837\) 0 0
\(838\) 25902.8 1.06778
\(839\) −15210.3 −0.625884 −0.312942 0.949772i \(-0.601315\pi\)
−0.312942 + 0.949772i \(0.601315\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 28854.7 1.18100
\(843\) 0 0
\(844\) −12299.2 −0.501606
\(845\) −100276. −4.08235
\(846\) 0 0
\(847\) −23918.5 −0.970306
\(848\) −9098.20 −0.368436
\(849\) 0 0
\(850\) 7290.70 0.294199
\(851\) −1476.02 −0.0594563
\(852\) 0 0
\(853\) −18281.2 −0.733805 −0.366902 0.930259i \(-0.619582\pi\)
−0.366902 + 0.930259i \(0.619582\pi\)
\(854\) 3174.38 0.127195
\(855\) 0 0
\(856\) 10554.9 0.421447
\(857\) −585.415 −0.0233342 −0.0116671 0.999932i \(-0.503714\pi\)
−0.0116671 + 0.999932i \(0.503714\pi\)
\(858\) 0 0
\(859\) −935.611 −0.0371626 −0.0185813 0.999827i \(-0.505915\pi\)
−0.0185813 + 0.999827i \(0.505915\pi\)
\(860\) −8064.02 −0.319745
\(861\) 0 0
\(862\) −9162.92 −0.362054
\(863\) 12110.7 0.477696 0.238848 0.971057i \(-0.423230\pi\)
0.238848 + 0.971057i \(0.423230\pi\)
\(864\) 0 0
\(865\) −5610.75 −0.220545
\(866\) 8664.43 0.339988
\(867\) 0 0
\(868\) 4527.70 0.177051
\(869\) 51720.0 2.01896
\(870\) 0 0
\(871\) −28966.1 −1.12684
\(872\) −23796.4 −0.924136
\(873\) 0 0
\(874\) 19928.2 0.771260
\(875\) 26621.8 1.02855
\(876\) 0 0
\(877\) 5841.41 0.224915 0.112458 0.993657i \(-0.464128\pi\)
0.112458 + 0.993657i \(0.464128\pi\)
\(878\) −23897.1 −0.918551
\(879\) 0 0
\(880\) −30115.1 −1.15361
\(881\) 47826.2 1.82895 0.914476 0.404641i \(-0.132603\pi\)
0.914476 + 0.404641i \(0.132603\pi\)
\(882\) 0 0
\(883\) 17337.5 0.660761 0.330381 0.943848i \(-0.392823\pi\)
0.330381 + 0.943848i \(0.392823\pi\)
\(884\) −4015.79 −0.152789
\(885\) 0 0
\(886\) 23059.9 0.874392
\(887\) −35181.5 −1.33177 −0.665885 0.746055i \(-0.731946\pi\)
−0.665885 + 0.746055i \(0.731946\pi\)
\(888\) 0 0
\(889\) 45016.3 1.69831
\(890\) −56994.6 −2.14659
\(891\) 0 0
\(892\) −3966.08 −0.148872
\(893\) −1413.78 −0.0529792
\(894\) 0 0
\(895\) −8778.22 −0.327848
\(896\) 3114.56 0.116127
\(897\) 0 0
\(898\) 27850.3 1.03494
\(899\) 2616.44 0.0970669
\(900\) 0 0
\(901\) −4475.08 −0.165468
\(902\) 2214.13 0.0817322
\(903\) 0 0
\(904\) 41951.6 1.54346
\(905\) 8771.56 0.322184
\(906\) 0 0
\(907\) −28184.4 −1.03180 −0.515902 0.856648i \(-0.672543\pi\)
−0.515902 + 0.856648i \(0.672543\pi\)
\(908\) −3957.15 −0.144629
\(909\) 0 0
\(910\) 60653.3 2.20949
\(911\) 34136.6 1.24149 0.620745 0.784013i \(-0.286830\pi\)
0.620745 + 0.784013i \(0.286830\pi\)
\(912\) 0 0
\(913\) 31168.5 1.12982
\(914\) 23886.8 0.864448
\(915\) 0 0
\(916\) −16239.9 −0.585788
\(917\) 14835.2 0.534242
\(918\) 0 0
\(919\) 15512.0 0.556794 0.278397 0.960466i \(-0.410197\pi\)
0.278397 + 0.960466i \(0.410197\pi\)
\(920\) 59844.4 2.14458
\(921\) 0 0
\(922\) 22007.3 0.786086
\(923\) 43115.3 1.53755
\(924\) 0 0
\(925\) 2348.80 0.0834897
\(926\) 43055.5 1.52796
\(927\) 0 0
\(928\) −3671.66 −0.129879
\(929\) 3100.72 0.109506 0.0547531 0.998500i \(-0.482563\pi\)
0.0547531 + 0.998500i \(0.482563\pi\)
\(930\) 0 0
\(931\) −3940.40 −0.138713
\(932\) −1759.77 −0.0618488
\(933\) 0 0
\(934\) −774.498 −0.0271331
\(935\) −14812.5 −0.518098
\(936\) 0 0
\(937\) −19638.8 −0.684708 −0.342354 0.939571i \(-0.611224\pi\)
−0.342354 + 0.939571i \(0.611224\pi\)
\(938\) 12499.6 0.435103
\(939\) 0 0
\(940\) −1151.37 −0.0399504
\(941\) 50033.6 1.73332 0.866658 0.498903i \(-0.166264\pi\)
0.866658 + 0.498903i \(0.166264\pi\)
\(942\) 0 0
\(943\) −2499.57 −0.0863172
\(944\) −10567.0 −0.364328
\(945\) 0 0
\(946\) −17365.4 −0.596827
\(947\) −19758.4 −0.677994 −0.338997 0.940787i \(-0.610088\pi\)
−0.338997 + 0.940787i \(0.610088\pi\)
\(948\) 0 0
\(949\) 30387.3 1.03942
\(950\) −31711.8 −1.08302
\(951\) 0 0
\(952\) 6390.03 0.217544
\(953\) −33843.4 −1.15036 −0.575180 0.818027i \(-0.695068\pi\)
−0.575180 + 0.818027i \(0.695068\pi\)
\(954\) 0 0
\(955\) 15274.1 0.517548
\(956\) 20677.0 0.699520
\(957\) 0 0
\(958\) −568.624 −0.0191768
\(959\) 34798.0 1.17173
\(960\) 0 0
\(961\) −21651.0 −0.726762
\(962\) 2183.11 0.0731666
\(963\) 0 0
\(964\) −21152.1 −0.706705
\(965\) 26582.1 0.886743
\(966\) 0 0
\(967\) 35236.9 1.17181 0.585906 0.810379i \(-0.300739\pi\)
0.585906 + 0.810379i \(0.300739\pi\)
\(968\) 34904.9 1.15897
\(969\) 0 0
\(970\) −13724.5 −0.454294
\(971\) 4505.40 0.148903 0.0744517 0.997225i \(-0.476279\pi\)
0.0744517 + 0.997225i \(0.476279\pi\)
\(972\) 0 0
\(973\) −10201.5 −0.336119
\(974\) 29708.0 0.977314
\(975\) 0 0
\(976\) −2631.69 −0.0863096
\(977\) 15167.4 0.496671 0.248336 0.968674i \(-0.420116\pi\)
0.248336 + 0.968674i \(0.420116\pi\)
\(978\) 0 0
\(979\) 72734.2 2.37446
\(980\) −3209.01 −0.104600
\(981\) 0 0
\(982\) −14466.3 −0.470099
\(983\) −25342.5 −0.822278 −0.411139 0.911573i \(-0.634869\pi\)
−0.411139 + 0.911573i \(0.634869\pi\)
\(984\) 0 0
\(985\) −36544.4 −1.18213
\(986\) 1001.41 0.0323442
\(987\) 0 0
\(988\) 17467.2 0.562455
\(989\) 19604.1 0.630307
\(990\) 0 0
\(991\) 40576.1 1.30065 0.650324 0.759657i \(-0.274633\pi\)
0.650324 + 0.759657i \(0.274633\pi\)
\(992\) −11422.9 −0.365603
\(993\) 0 0
\(994\) −18605.4 −0.593689
\(995\) 6533.46 0.208166
\(996\) 0 0
\(997\) 7442.49 0.236415 0.118208 0.992989i \(-0.462285\pi\)
0.118208 + 0.992989i \(0.462285\pi\)
\(998\) −18132.6 −0.575128
\(999\) 0 0
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 261.4.a.f.1.2 5
3.2 odd 2 29.4.a.b.1.4 5
12.11 even 2 464.4.a.l.1.3 5
15.14 odd 2 725.4.a.c.1.2 5
21.20 even 2 1421.4.a.e.1.4 5
24.5 odd 2 1856.4.a.y.1.3 5
24.11 even 2 1856.4.a.bb.1.3 5
87.86 odd 2 841.4.a.b.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.4 5 3.2 odd 2
261.4.a.f.1.2 5 1.1 even 1 trivial
464.4.a.l.1.3 5 12.11 even 2
725.4.a.c.1.2 5 15.14 odd 2
841.4.a.b.1.2 5 87.86 odd 2
1421.4.a.e.1.4 5 21.20 even 2
1856.4.a.y.1.3 5 24.5 odd 2
1856.4.a.bb.1.3 5 24.11 even 2