Properties

Label 245.4.a.m.1.3
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
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] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.227497\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.227497 q^{2} -1.80858 q^{3} -7.94824 q^{4} -5.00000 q^{5} +0.411448 q^{6} +3.62818 q^{8} -23.7290 q^{9} +O(q^{10})\) \(q-0.227497 q^{2} -1.80858 q^{3} -7.94824 q^{4} -5.00000 q^{5} +0.411448 q^{6} +3.62818 q^{8} -23.7290 q^{9} +1.13749 q^{10} +17.7019 q^{11} +14.3751 q^{12} -62.3178 q^{13} +9.04291 q^{15} +62.7606 q^{16} +87.1623 q^{17} +5.39829 q^{18} -101.724 q^{19} +39.7412 q^{20} -4.02713 q^{22} +93.6396 q^{23} -6.56187 q^{24} +25.0000 q^{25} +14.1771 q^{26} +91.7477 q^{27} +297.385 q^{29} -2.05724 q^{30} +91.2779 q^{31} -43.3033 q^{32} -32.0153 q^{33} -19.8292 q^{34} +188.604 q^{36} +281.210 q^{37} +23.1418 q^{38} +112.707 q^{39} -18.1409 q^{40} -271.754 q^{41} -7.81066 q^{43} -140.699 q^{44} +118.645 q^{45} -21.3028 q^{46} -92.4524 q^{47} -113.508 q^{48} -5.68743 q^{50} -157.640 q^{51} +495.317 q^{52} +180.304 q^{53} -20.8723 q^{54} -88.5094 q^{55} +183.975 q^{57} -67.6544 q^{58} -99.7011 q^{59} -71.8753 q^{60} +434.218 q^{61} -20.7655 q^{62} -492.233 q^{64} +311.589 q^{65} +7.28340 q^{66} -461.554 q^{67} -692.787 q^{68} -169.355 q^{69} +518.417 q^{71} -86.0932 q^{72} -542.866 q^{73} -63.9744 q^{74} -45.2146 q^{75} +808.524 q^{76} -25.6405 q^{78} +239.701 q^{79} -313.803 q^{80} +474.751 q^{81} +61.8232 q^{82} +299.184 q^{83} -435.812 q^{85} +1.77690 q^{86} -537.846 q^{87} +64.2257 q^{88} +1054.36 q^{89} -26.9914 q^{90} -744.271 q^{92} -165.084 q^{93} +21.0327 q^{94} +508.618 q^{95} +78.3176 q^{96} +288.854 q^{97} -420.049 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9} - 5 q^{10} + 47 q^{11} - 98 q^{12} + q^{13} + 40 q^{15} + 171 q^{16} - 2 q^{17} - 51 q^{18} - 21 q^{19} - 175 q^{20} + 523 q^{22} + 201 q^{23} + 848 q^{24} + 125 q^{25} - 47 q^{26} - 518 q^{27} + 190 q^{29} - 80 q^{30} + 388 q^{31} - 95 q^{32} - 262 q^{33} + 130 q^{34} + 1229 q^{36} - 145 q^{37} + 835 q^{38} + 14 q^{39} - 165 q^{40} - 281 q^{41} + 568 q^{43} + 1091 q^{44} - 405 q^{45} + 337 q^{46} - 473 q^{47} + 70 q^{48} + 25 q^{50} + 732 q^{51} - 379 q^{52} + 351 q^{53} - 774 q^{54} - 235 q^{55} + 954 q^{57} + 1818 q^{58} + 708 q^{59} + 490 q^{60} + 1944 q^{61} + 448 q^{62} - 125 q^{64} - 5 q^{65} + 1482 q^{66} + 1118 q^{67} - 3118 q^{68} + 374 q^{69} + 864 q^{71} - 2219 q^{72} - 1652 q^{73} - 3285 q^{74} - 200 q^{75} + 691 q^{76} - 5574 q^{78} + 218 q^{79} - 855 q^{80} - 455 q^{81} + 1027 q^{82} - 1502 q^{83} + 10 q^{85} - 4264 q^{86} + 390 q^{87} + 2131 q^{88} + 2322 q^{89} + 255 q^{90} - 2957 q^{92} - 2288 q^{93} - 2677 q^{94} + 105 q^{95} + 4592 q^{96} + 598 q^{97} + 35 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.227497 −0.0804324 −0.0402162 0.999191i \(-0.512805\pi\)
−0.0402162 + 0.999191i \(0.512805\pi\)
\(3\) −1.80858 −0.348062 −0.174031 0.984740i \(-0.555679\pi\)
−0.174031 + 0.984740i \(0.555679\pi\)
\(4\) −7.94824 −0.993531
\(5\) −5.00000 −0.447214
\(6\) 0.411448 0.0279955
\(7\) 0 0
\(8\) 3.62818 0.160345
\(9\) −23.7290 −0.878853
\(10\) 1.13749 0.0359705
\(11\) 17.7019 0.485211 0.242605 0.970125i \(-0.421998\pi\)
0.242605 + 0.970125i \(0.421998\pi\)
\(12\) 14.3751 0.345810
\(13\) −62.3178 −1.32953 −0.664763 0.747054i \(-0.731467\pi\)
−0.664763 + 0.747054i \(0.731467\pi\)
\(14\) 0 0
\(15\) 9.04291 0.155658
\(16\) 62.7606 0.980634
\(17\) 87.1623 1.24353 0.621764 0.783205i \(-0.286416\pi\)
0.621764 + 0.783205i \(0.286416\pi\)
\(18\) 5.39829 0.0706883
\(19\) −101.724 −1.22826 −0.614131 0.789204i \(-0.710493\pi\)
−0.614131 + 0.789204i \(0.710493\pi\)
\(20\) 39.7412 0.444320
\(21\) 0 0
\(22\) −4.02713 −0.0390267
\(23\) 93.6396 0.848922 0.424461 0.905446i \(-0.360464\pi\)
0.424461 + 0.905446i \(0.360464\pi\)
\(24\) −6.56187 −0.0558098
\(25\) 25.0000 0.200000
\(26\) 14.1771 0.106937
\(27\) 91.7477 0.653957
\(28\) 0 0
\(29\) 297.385 1.90424 0.952122 0.305718i \(-0.0988963\pi\)
0.952122 + 0.305718i \(0.0988963\pi\)
\(30\) −2.05724 −0.0125200
\(31\) 91.2779 0.528838 0.264419 0.964408i \(-0.414820\pi\)
0.264419 + 0.964408i \(0.414820\pi\)
\(32\) −43.3033 −0.239219
\(33\) −32.0153 −0.168883
\(34\) −19.8292 −0.100020
\(35\) 0 0
\(36\) 188.604 0.873167
\(37\) 281.210 1.24948 0.624738 0.780835i \(-0.285206\pi\)
0.624738 + 0.780835i \(0.285206\pi\)
\(38\) 23.1418 0.0987921
\(39\) 112.707 0.462757
\(40\) −18.1409 −0.0717082
\(41\) −271.754 −1.03514 −0.517570 0.855641i \(-0.673163\pi\)
−0.517570 + 0.855641i \(0.673163\pi\)
\(42\) 0 0
\(43\) −7.81066 −0.0277003 −0.0138502 0.999904i \(-0.504409\pi\)
−0.0138502 + 0.999904i \(0.504409\pi\)
\(44\) −140.699 −0.482072
\(45\) 118.645 0.393035
\(46\) −21.3028 −0.0682809
\(47\) −92.4524 −0.286927 −0.143464 0.989656i \(-0.545824\pi\)
−0.143464 + 0.989656i \(0.545824\pi\)
\(48\) −113.508 −0.341321
\(49\) 0 0
\(50\) −5.68743 −0.0160865
\(51\) −157.640 −0.432825
\(52\) 495.317 1.32092
\(53\) 180.304 0.467295 0.233648 0.972321i \(-0.424934\pi\)
0.233648 + 0.972321i \(0.424934\pi\)
\(54\) −20.8723 −0.0525994
\(55\) −88.5094 −0.216993
\(56\) 0 0
\(57\) 183.975 0.427511
\(58\) −67.6544 −0.153163
\(59\) −99.7011 −0.220000 −0.110000 0.993932i \(-0.535085\pi\)
−0.110000 + 0.993932i \(0.535085\pi\)
\(60\) −71.8753 −0.154651
\(61\) 434.218 0.911408 0.455704 0.890131i \(-0.349388\pi\)
0.455704 + 0.890131i \(0.349388\pi\)
\(62\) −20.7655 −0.0425358
\(63\) 0 0
\(64\) −492.233 −0.961393
\(65\) 311.589 0.594582
\(66\) 7.28340 0.0135837
\(67\) −461.554 −0.841609 −0.420805 0.907151i \(-0.638252\pi\)
−0.420805 + 0.907151i \(0.638252\pi\)
\(68\) −692.787 −1.23548
\(69\) −169.355 −0.295478
\(70\) 0 0
\(71\) 518.417 0.866546 0.433273 0.901263i \(-0.357359\pi\)
0.433273 + 0.901263i \(0.357359\pi\)
\(72\) −86.0932 −0.140919
\(73\) −542.866 −0.870380 −0.435190 0.900339i \(-0.643319\pi\)
−0.435190 + 0.900339i \(0.643319\pi\)
\(74\) −63.9744 −0.100498
\(75\) −45.2146 −0.0696124
\(76\) 808.524 1.22032
\(77\) 0 0
\(78\) −25.6405 −0.0372207
\(79\) 239.701 0.341373 0.170687 0.985325i \(-0.445401\pi\)
0.170687 + 0.985325i \(0.445401\pi\)
\(80\) −313.803 −0.438553
\(81\) 474.751 0.651235
\(82\) 61.8232 0.0832589
\(83\) 299.184 0.395660 0.197830 0.980236i \(-0.436611\pi\)
0.197830 + 0.980236i \(0.436611\pi\)
\(84\) 0 0
\(85\) −435.812 −0.556122
\(86\) 1.77690 0.00222801
\(87\) −537.846 −0.662795
\(88\) 64.2257 0.0778009
\(89\) 1054.36 1.25575 0.627876 0.778314i \(-0.283925\pi\)
0.627876 + 0.778314i \(0.283925\pi\)
\(90\) −26.9914 −0.0316128
\(91\) 0 0
\(92\) −744.271 −0.843430
\(93\) −165.084 −0.184069
\(94\) 21.0327 0.0230782
\(95\) 508.618 0.549296
\(96\) 78.3176 0.0832631
\(97\) 288.854 0.302358 0.151179 0.988506i \(-0.451693\pi\)
0.151179 + 0.988506i \(0.451693\pi\)
\(98\) 0 0
\(99\) −420.049 −0.426429
\(100\) −198.706 −0.198706
\(101\) 354.839 0.349582 0.174791 0.984606i \(-0.444075\pi\)
0.174791 + 0.984606i \(0.444075\pi\)
\(102\) 35.8627 0.0348131
\(103\) −1064.15 −1.01799 −0.508997 0.860768i \(-0.669984\pi\)
−0.508997 + 0.860768i \(0.669984\pi\)
\(104\) −226.100 −0.213182
\(105\) 0 0
\(106\) −41.0186 −0.0375857
\(107\) −1393.71 −1.25920 −0.629601 0.776918i \(-0.716782\pi\)
−0.629601 + 0.776918i \(0.716782\pi\)
\(108\) −729.233 −0.649727
\(109\) −144.484 −0.126964 −0.0634818 0.997983i \(-0.520220\pi\)
−0.0634818 + 0.997983i \(0.520220\pi\)
\(110\) 20.1357 0.0174533
\(111\) −508.591 −0.434895
\(112\) 0 0
\(113\) −2023.91 −1.68489 −0.842447 0.538780i \(-0.818886\pi\)
−0.842447 + 0.538780i \(0.818886\pi\)
\(114\) −41.8539 −0.0343858
\(115\) −468.198 −0.379650
\(116\) −2363.69 −1.89193
\(117\) 1478.74 1.16846
\(118\) 22.6817 0.0176951
\(119\) 0 0
\(120\) 32.8093 0.0249589
\(121\) −1017.64 −0.764570
\(122\) −98.7834 −0.0733068
\(123\) 491.489 0.360293
\(124\) −725.499 −0.525417
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1606.56 1.12252 0.561258 0.827641i \(-0.310318\pi\)
0.561258 + 0.827641i \(0.310318\pi\)
\(128\) 458.408 0.316546
\(129\) 14.1262 0.00964143
\(130\) −70.8856 −0.0478237
\(131\) 2848.32 1.89969 0.949843 0.312726i \(-0.101242\pi\)
0.949843 + 0.312726i \(0.101242\pi\)
\(132\) 254.466 0.167791
\(133\) 0 0
\(134\) 105.002 0.0676927
\(135\) −458.738 −0.292459
\(136\) 316.241 0.199393
\(137\) 841.717 0.524910 0.262455 0.964944i \(-0.415468\pi\)
0.262455 + 0.964944i \(0.415468\pi\)
\(138\) 38.5278 0.0237660
\(139\) 2429.10 1.48226 0.741128 0.671363i \(-0.234291\pi\)
0.741128 + 0.671363i \(0.234291\pi\)
\(140\) 0 0
\(141\) 167.208 0.0998684
\(142\) −117.938 −0.0696984
\(143\) −1103.14 −0.645100
\(144\) −1489.25 −0.861833
\(145\) −1486.93 −0.851604
\(146\) 123.501 0.0700067
\(147\) 0 0
\(148\) −2235.12 −1.24139
\(149\) −577.947 −0.317767 −0.158884 0.987297i \(-0.550789\pi\)
−0.158884 + 0.987297i \(0.550789\pi\)
\(150\) 10.2862 0.00559909
\(151\) 783.616 0.422316 0.211158 0.977452i \(-0.432276\pi\)
0.211158 + 0.977452i \(0.432276\pi\)
\(152\) −369.072 −0.196945
\(153\) −2068.28 −1.09288
\(154\) 0 0
\(155\) −456.389 −0.236504
\(156\) −895.822 −0.459764
\(157\) −1372.40 −0.697642 −0.348821 0.937189i \(-0.613418\pi\)
−0.348821 + 0.937189i \(0.613418\pi\)
\(158\) −54.5314 −0.0274575
\(159\) −326.095 −0.162648
\(160\) 216.517 0.106982
\(161\) 0 0
\(162\) −108.004 −0.0523804
\(163\) 3754.78 1.80428 0.902139 0.431445i \(-0.141996\pi\)
0.902139 + 0.431445i \(0.141996\pi\)
\(164\) 2159.96 1.02844
\(165\) 160.077 0.0755270
\(166\) −68.0636 −0.0318239
\(167\) −1945.55 −0.901505 −0.450752 0.892649i \(-0.648844\pi\)
−0.450752 + 0.892649i \(0.648844\pi\)
\(168\) 0 0
\(169\) 1686.50 0.767639
\(170\) 99.1459 0.0447303
\(171\) 2413.80 1.07946
\(172\) 62.0810 0.0275211
\(173\) −2067.34 −0.908538 −0.454269 0.890864i \(-0.650100\pi\)
−0.454269 + 0.890864i \(0.650100\pi\)
\(174\) 122.359 0.0533102
\(175\) 0 0
\(176\) 1110.98 0.475814
\(177\) 180.318 0.0765735
\(178\) −239.864 −0.101003
\(179\) 2395.92 1.00044 0.500222 0.865897i \(-0.333252\pi\)
0.500222 + 0.865897i \(0.333252\pi\)
\(180\) −943.021 −0.390492
\(181\) 1319.40 0.541826 0.270913 0.962604i \(-0.412674\pi\)
0.270913 + 0.962604i \(0.412674\pi\)
\(182\) 0 0
\(183\) −785.319 −0.317227
\(184\) 339.742 0.136120
\(185\) −1406.05 −0.558782
\(186\) 37.5561 0.0148051
\(187\) 1542.94 0.603373
\(188\) 734.834 0.285071
\(189\) 0 0
\(190\) −115.709 −0.0441812
\(191\) 2108.07 0.798611 0.399306 0.916818i \(-0.369251\pi\)
0.399306 + 0.916818i \(0.369251\pi\)
\(192\) 890.244 0.334624
\(193\) 3631.62 1.35445 0.677227 0.735774i \(-0.263181\pi\)
0.677227 + 0.735774i \(0.263181\pi\)
\(194\) −65.7135 −0.0243194
\(195\) −563.534 −0.206951
\(196\) 0 0
\(197\) 3280.46 1.18641 0.593206 0.805051i \(-0.297862\pi\)
0.593206 + 0.805051i \(0.297862\pi\)
\(198\) 95.5599 0.0342987
\(199\) −648.733 −0.231093 −0.115546 0.993302i \(-0.536862\pi\)
−0.115546 + 0.993302i \(0.536862\pi\)
\(200\) 90.7045 0.0320689
\(201\) 834.759 0.292932
\(202\) −80.7249 −0.0281177
\(203\) 0 0
\(204\) 1252.96 0.430024
\(205\) 1358.77 0.462929
\(206\) 242.090 0.0818798
\(207\) −2221.98 −0.746078
\(208\) −3911.10 −1.30378
\(209\) −1800.70 −0.595966
\(210\) 0 0
\(211\) 5072.33 1.65495 0.827474 0.561504i \(-0.189777\pi\)
0.827474 + 0.561504i \(0.189777\pi\)
\(212\) −1433.10 −0.464272
\(213\) −937.600 −0.301612
\(214\) 317.064 0.101281
\(215\) 39.0533 0.0123880
\(216\) 332.877 0.104858
\(217\) 0 0
\(218\) 32.8697 0.0102120
\(219\) 981.819 0.302946
\(220\) 703.495 0.215589
\(221\) −5431.76 −1.65330
\(222\) 115.703 0.0349796
\(223\) −163.717 −0.0491629 −0.0245814 0.999698i \(-0.507825\pi\)
−0.0245814 + 0.999698i \(0.507825\pi\)
\(224\) 0 0
\(225\) −593.226 −0.175771
\(226\) 460.433 0.135520
\(227\) 6326.62 1.84984 0.924918 0.380166i \(-0.124133\pi\)
0.924918 + 0.380166i \(0.124133\pi\)
\(228\) −1462.28 −0.424746
\(229\) 3484.47 1.00550 0.502752 0.864431i \(-0.332321\pi\)
0.502752 + 0.864431i \(0.332321\pi\)
\(230\) 106.514 0.0305361
\(231\) 0 0
\(232\) 1078.97 0.305335
\(233\) −4496.13 −1.26417 −0.632084 0.774900i \(-0.717800\pi\)
−0.632084 + 0.774900i \(0.717800\pi\)
\(234\) −336.409 −0.0939819
\(235\) 462.262 0.128318
\(236\) 792.449 0.218576
\(237\) −433.519 −0.118819
\(238\) 0 0
\(239\) −1529.73 −0.414016 −0.207008 0.978339i \(-0.566373\pi\)
−0.207008 + 0.978339i \(0.566373\pi\)
\(240\) 567.538 0.152644
\(241\) 4572.40 1.22213 0.611067 0.791579i \(-0.290741\pi\)
0.611067 + 0.791579i \(0.290741\pi\)
\(242\) 231.511 0.0614963
\(243\) −3335.81 −0.880627
\(244\) −3451.27 −0.905512
\(245\) 0 0
\(246\) −111.812 −0.0289793
\(247\) 6339.18 1.63301
\(248\) 331.173 0.0847963
\(249\) −541.100 −0.137714
\(250\) 28.4372 0.00719409
\(251\) −537.143 −0.135076 −0.0675382 0.997717i \(-0.521514\pi\)
−0.0675382 + 0.997717i \(0.521514\pi\)
\(252\) 0 0
\(253\) 1657.60 0.411906
\(254\) −365.489 −0.0902866
\(255\) 788.201 0.193565
\(256\) 3833.58 0.935932
\(257\) −878.553 −0.213240 −0.106620 0.994300i \(-0.534003\pi\)
−0.106620 + 0.994300i \(0.534003\pi\)
\(258\) −3.21368 −0.000775484 0
\(259\) 0 0
\(260\) −2476.58 −0.590736
\(261\) −7056.67 −1.67355
\(262\) −647.985 −0.152796
\(263\) 3714.03 0.870786 0.435393 0.900241i \(-0.356609\pi\)
0.435393 + 0.900241i \(0.356609\pi\)
\(264\) −116.157 −0.0270795
\(265\) −901.519 −0.208981
\(266\) 0 0
\(267\) −1906.90 −0.437079
\(268\) 3668.55 0.836165
\(269\) −154.460 −0.0350096 −0.0175048 0.999847i \(-0.505572\pi\)
−0.0175048 + 0.999847i \(0.505572\pi\)
\(270\) 104.362 0.0235232
\(271\) 2406.30 0.539382 0.269691 0.962947i \(-0.413079\pi\)
0.269691 + 0.962947i \(0.413079\pi\)
\(272\) 5470.36 1.21944
\(273\) 0 0
\(274\) −191.488 −0.0422198
\(275\) 442.547 0.0970422
\(276\) 1346.08 0.293566
\(277\) 1449.76 0.314467 0.157234 0.987561i \(-0.449742\pi\)
0.157234 + 0.987561i \(0.449742\pi\)
\(278\) −552.614 −0.119222
\(279\) −2165.94 −0.464771
\(280\) 0 0
\(281\) 2037.07 0.432460 0.216230 0.976342i \(-0.430624\pi\)
0.216230 + 0.976342i \(0.430624\pi\)
\(282\) −38.0393 −0.00803266
\(283\) −7366.22 −1.54727 −0.773633 0.633634i \(-0.781563\pi\)
−0.773633 + 0.633634i \(0.781563\pi\)
\(284\) −4120.51 −0.860940
\(285\) −919.877 −0.191189
\(286\) 250.962 0.0518870
\(287\) 0 0
\(288\) 1027.55 0.210239
\(289\) 2684.27 0.546361
\(290\) 338.272 0.0684966
\(291\) −522.417 −0.105239
\(292\) 4314.84 0.864749
\(293\) −542.935 −0.108255 −0.0541273 0.998534i \(-0.517238\pi\)
−0.0541273 + 0.998534i \(0.517238\pi\)
\(294\) 0 0
\(295\) 498.506 0.0983868
\(296\) 1020.28 0.200346
\(297\) 1624.11 0.317307
\(298\) 131.481 0.0255588
\(299\) −5835.41 −1.12866
\(300\) 359.377 0.0691620
\(301\) 0 0
\(302\) −178.271 −0.0339679
\(303\) −641.756 −0.121676
\(304\) −6384.23 −1.20448
\(305\) −2171.09 −0.407594
\(306\) 470.527 0.0879028
\(307\) 5314.23 0.987945 0.493972 0.869478i \(-0.335544\pi\)
0.493972 + 0.869478i \(0.335544\pi\)
\(308\) 0 0
\(309\) 1924.60 0.354325
\(310\) 103.827 0.0190226
\(311\) −8840.00 −1.61180 −0.805901 0.592051i \(-0.798318\pi\)
−0.805901 + 0.592051i \(0.798318\pi\)
\(312\) 408.921 0.0742006
\(313\) −1896.34 −0.342453 −0.171226 0.985232i \(-0.554773\pi\)
−0.171226 + 0.985232i \(0.554773\pi\)
\(314\) 312.218 0.0561130
\(315\) 0 0
\(316\) −1905.20 −0.339165
\(317\) −6072.81 −1.07597 −0.537985 0.842954i \(-0.680814\pi\)
−0.537985 + 0.842954i \(0.680814\pi\)
\(318\) 74.1856 0.0130821
\(319\) 5264.28 0.923960
\(320\) 2461.17 0.429948
\(321\) 2520.63 0.438280
\(322\) 0 0
\(323\) −8866.46 −1.52738
\(324\) −3773.43 −0.647022
\(325\) −1557.94 −0.265905
\(326\) −854.203 −0.145122
\(327\) 261.311 0.0441912
\(328\) −985.971 −0.165979
\(329\) 0 0
\(330\) −36.4170 −0.00607482
\(331\) 3341.78 0.554927 0.277463 0.960736i \(-0.410506\pi\)
0.277463 + 0.960736i \(0.410506\pi\)
\(332\) −2377.99 −0.393100
\(333\) −6672.83 −1.09810
\(334\) 442.608 0.0725102
\(335\) 2307.77 0.376379
\(336\) 0 0
\(337\) −8568.87 −1.38509 −0.692547 0.721373i \(-0.743511\pi\)
−0.692547 + 0.721373i \(0.743511\pi\)
\(338\) −383.675 −0.0617431
\(339\) 3660.40 0.586447
\(340\) 3463.94 0.552525
\(341\) 1615.79 0.256598
\(342\) −549.133 −0.0868237
\(343\) 0 0
\(344\) −28.3385 −0.00444160
\(345\) 846.775 0.132142
\(346\) 470.315 0.0730760
\(347\) −6246.73 −0.966403 −0.483202 0.875509i \(-0.660526\pi\)
−0.483202 + 0.875509i \(0.660526\pi\)
\(348\) 4274.93 0.658507
\(349\) 7091.01 1.08760 0.543801 0.839214i \(-0.316985\pi\)
0.543801 + 0.839214i \(0.316985\pi\)
\(350\) 0 0
\(351\) −5717.51 −0.869453
\(352\) −766.550 −0.116072
\(353\) 7289.22 1.09905 0.549527 0.835476i \(-0.314808\pi\)
0.549527 + 0.835476i \(0.314808\pi\)
\(354\) −41.0218 −0.00615899
\(355\) −2592.09 −0.387531
\(356\) −8380.31 −1.24763
\(357\) 0 0
\(358\) −545.065 −0.0804681
\(359\) −8668.21 −1.27435 −0.637174 0.770720i \(-0.719897\pi\)
−0.637174 + 0.770720i \(0.719897\pi\)
\(360\) 430.466 0.0630210
\(361\) 3488.68 0.508628
\(362\) −300.161 −0.0435804
\(363\) 1840.49 0.266118
\(364\) 0 0
\(365\) 2714.33 0.389246
\(366\) 178.658 0.0255153
\(367\) −3777.62 −0.537302 −0.268651 0.963238i \(-0.586578\pi\)
−0.268651 + 0.963238i \(0.586578\pi\)
\(368\) 5876.88 0.832482
\(369\) 6448.45 0.909737
\(370\) 319.872 0.0449442
\(371\) 0 0
\(372\) 1312.12 0.182878
\(373\) −1742.13 −0.241834 −0.120917 0.992663i \(-0.538584\pi\)
−0.120917 + 0.992663i \(0.538584\pi\)
\(374\) −351.014 −0.0485308
\(375\) 226.073 0.0311316
\(376\) −335.434 −0.0460072
\(377\) −18532.4 −2.53174
\(378\) 0 0
\(379\) −3471.30 −0.470471 −0.235236 0.971938i \(-0.575586\pi\)
−0.235236 + 0.971938i \(0.575586\pi\)
\(380\) −4042.62 −0.545742
\(381\) −2905.60 −0.390705
\(382\) −479.580 −0.0642342
\(383\) 14445.1 1.92718 0.963591 0.267382i \(-0.0861586\pi\)
0.963591 + 0.267382i \(0.0861586\pi\)
\(384\) −829.069 −0.110178
\(385\) 0 0
\(386\) −826.183 −0.108942
\(387\) 185.339 0.0243445
\(388\) −2295.88 −0.300402
\(389\) −7729.56 −1.00747 −0.503733 0.863860i \(-0.668040\pi\)
−0.503733 + 0.863860i \(0.668040\pi\)
\(390\) 128.202 0.0166456
\(391\) 8161.85 1.05566
\(392\) 0 0
\(393\) −5151.43 −0.661209
\(394\) −746.296 −0.0954260
\(395\) −1198.51 −0.152667
\(396\) 3338.65 0.423670
\(397\) −10069.6 −1.27300 −0.636499 0.771278i \(-0.719618\pi\)
−0.636499 + 0.771278i \(0.719618\pi\)
\(398\) 147.585 0.0185874
\(399\) 0 0
\(400\) 1569.01 0.196127
\(401\) 1503.17 0.187194 0.0935972 0.995610i \(-0.470163\pi\)
0.0935972 + 0.995610i \(0.470163\pi\)
\(402\) −189.905 −0.0235612
\(403\) −5688.23 −0.703104
\(404\) −2820.35 −0.347321
\(405\) −2373.75 −0.291241
\(406\) 0 0
\(407\) 4977.94 0.606259
\(408\) −571.948 −0.0694010
\(409\) −613.381 −0.0741559 −0.0370779 0.999312i \(-0.511805\pi\)
−0.0370779 + 0.999312i \(0.511805\pi\)
\(410\) −309.116 −0.0372345
\(411\) −1522.32 −0.182701
\(412\) 8458.10 1.01141
\(413\) 0 0
\(414\) 505.494 0.0600089
\(415\) −1495.92 −0.176944
\(416\) 2698.57 0.318048
\(417\) −4393.23 −0.515917
\(418\) 409.654 0.0479350
\(419\) 12334.4 1.43813 0.719065 0.694943i \(-0.244571\pi\)
0.719065 + 0.694943i \(0.244571\pi\)
\(420\) 0 0
\(421\) −2432.22 −0.281566 −0.140783 0.990040i \(-0.544962\pi\)
−0.140783 + 0.990040i \(0.544962\pi\)
\(422\) −1153.94 −0.133111
\(423\) 2193.81 0.252167
\(424\) 654.175 0.0749282
\(425\) 2179.06 0.248705
\(426\) 213.301 0.0242594
\(427\) 0 0
\(428\) 11077.5 1.25106
\(429\) 1995.12 0.224535
\(430\) −8.88452 −0.000996394 0
\(431\) −1310.48 −0.146458 −0.0732292 0.997315i \(-0.523330\pi\)
−0.0732292 + 0.997315i \(0.523330\pi\)
\(432\) 5758.13 0.641292
\(433\) 4954.86 0.549920 0.274960 0.961456i \(-0.411335\pi\)
0.274960 + 0.961456i \(0.411335\pi\)
\(434\) 0 0
\(435\) 2689.23 0.296411
\(436\) 1148.39 0.126142
\(437\) −9525.36 −1.04270
\(438\) −223.361 −0.0243667
\(439\) −8195.34 −0.890984 −0.445492 0.895286i \(-0.646971\pi\)
−0.445492 + 0.895286i \(0.646971\pi\)
\(440\) −321.128 −0.0347936
\(441\) 0 0
\(442\) 1235.71 0.132979
\(443\) 2956.70 0.317104 0.158552 0.987351i \(-0.449317\pi\)
0.158552 + 0.987351i \(0.449317\pi\)
\(444\) 4042.41 0.432081
\(445\) −5271.80 −0.561589
\(446\) 37.2452 0.00395429
\(447\) 1045.27 0.110603
\(448\) 0 0
\(449\) −10453.5 −1.09873 −0.549364 0.835583i \(-0.685130\pi\)
−0.549364 + 0.835583i \(0.685130\pi\)
\(450\) 134.957 0.0141377
\(451\) −4810.55 −0.502262
\(452\) 16086.5 1.67399
\(453\) −1417.23 −0.146992
\(454\) −1439.29 −0.148787
\(455\) 0 0
\(456\) 667.497 0.0685491
\(457\) 2245.19 0.229816 0.114908 0.993376i \(-0.463343\pi\)
0.114908 + 0.993376i \(0.463343\pi\)
\(458\) −792.707 −0.0808751
\(459\) 7996.94 0.813214
\(460\) 3721.35 0.377194
\(461\) −12441.1 −1.25692 −0.628459 0.777842i \(-0.716314\pi\)
−0.628459 + 0.777842i \(0.716314\pi\)
\(462\) 0 0
\(463\) 17086.0 1.71502 0.857510 0.514468i \(-0.172010\pi\)
0.857510 + 0.514468i \(0.172010\pi\)
\(464\) 18664.1 1.86737
\(465\) 825.418 0.0823179
\(466\) 1022.86 0.101680
\(467\) −11709.1 −1.16024 −0.580121 0.814531i \(-0.696995\pi\)
−0.580121 + 0.814531i \(0.696995\pi\)
\(468\) −11753.4 −1.16090
\(469\) 0 0
\(470\) −105.163 −0.0103209
\(471\) 2482.11 0.242823
\(472\) −361.734 −0.0352757
\(473\) −138.263 −0.0134405
\(474\) 98.6245 0.00955690
\(475\) −2543.09 −0.245652
\(476\) 0 0
\(477\) −4278.44 −0.410684
\(478\) 348.009 0.0333003
\(479\) 11692.0 1.11528 0.557641 0.830082i \(-0.311707\pi\)
0.557641 + 0.830082i \(0.311707\pi\)
\(480\) −391.588 −0.0372364
\(481\) −17524.4 −1.66121
\(482\) −1040.21 −0.0982992
\(483\) 0 0
\(484\) 8088.48 0.759624
\(485\) −1444.27 −0.135218
\(486\) 758.888 0.0708310
\(487\) 7128.84 0.663324 0.331662 0.943398i \(-0.392391\pi\)
0.331662 + 0.943398i \(0.392391\pi\)
\(488\) 1575.42 0.146139
\(489\) −6790.84 −0.628001
\(490\) 0 0
\(491\) −4330.34 −0.398015 −0.199008 0.979998i \(-0.563772\pi\)
−0.199008 + 0.979998i \(0.563772\pi\)
\(492\) −3906.47 −0.357962
\(493\) 25920.8 2.36798
\(494\) −1442.15 −0.131347
\(495\) 2100.24 0.190705
\(496\) 5728.65 0.518597
\(497\) 0 0
\(498\) 123.099 0.0110767
\(499\) 9635.67 0.864433 0.432216 0.901770i \(-0.357732\pi\)
0.432216 + 0.901770i \(0.357732\pi\)
\(500\) 993.531 0.0888641
\(501\) 3518.69 0.313779
\(502\) 122.199 0.0108645
\(503\) 18763.3 1.66325 0.831626 0.555335i \(-0.187410\pi\)
0.831626 + 0.555335i \(0.187410\pi\)
\(504\) 0 0
\(505\) −1774.19 −0.156338
\(506\) −377.099 −0.0331306
\(507\) −3050.18 −0.267186
\(508\) −12769.4 −1.11525
\(509\) 15804.5 1.37627 0.688137 0.725580i \(-0.258429\pi\)
0.688137 + 0.725580i \(0.258429\pi\)
\(510\) −179.314 −0.0155689
\(511\) 0 0
\(512\) −4539.39 −0.391826
\(513\) −9332.90 −0.803231
\(514\) 199.869 0.0171514
\(515\) 5320.73 0.455261
\(516\) −112.279 −0.00957906
\(517\) −1636.58 −0.139220
\(518\) 0 0
\(519\) 3738.96 0.316228
\(520\) 1130.50 0.0953380
\(521\) 9451.64 0.794787 0.397393 0.917648i \(-0.369915\pi\)
0.397393 + 0.917648i \(0.369915\pi\)
\(522\) 1605.37 0.134608
\(523\) 2704.59 0.226125 0.113062 0.993588i \(-0.463934\pi\)
0.113062 + 0.993588i \(0.463934\pi\)
\(524\) −22639.2 −1.88740
\(525\) 0 0
\(526\) −844.931 −0.0700394
\(527\) 7955.99 0.657625
\(528\) −2009.30 −0.165613
\(529\) −3398.62 −0.279331
\(530\) 205.093 0.0168088
\(531\) 2365.81 0.193347
\(532\) 0 0
\(533\) 16935.1 1.37625
\(534\) 433.814 0.0351553
\(535\) 6968.53 0.563132
\(536\) −1674.60 −0.134947
\(537\) −4333.22 −0.348216
\(538\) 35.1392 0.00281591
\(539\) 0 0
\(540\) 3646.16 0.290567
\(541\) 60.7636 0.00482889 0.00241445 0.999997i \(-0.499231\pi\)
0.00241445 + 0.999997i \(0.499231\pi\)
\(542\) −547.427 −0.0433838
\(543\) −2386.25 −0.188589
\(544\) −3774.42 −0.297476
\(545\) 722.419 0.0567799
\(546\) 0 0
\(547\) 14267.4 1.11523 0.557614 0.830100i \(-0.311717\pi\)
0.557614 + 0.830100i \(0.311717\pi\)
\(548\) −6690.17 −0.521515
\(549\) −10303.6 −0.800994
\(550\) −100.678 −0.00780534
\(551\) −30251.1 −2.33891
\(552\) −614.451 −0.0473782
\(553\) 0 0
\(554\) −329.816 −0.0252934
\(555\) 2542.95 0.194491
\(556\) −19307.1 −1.47267
\(557\) −19601.8 −1.49112 −0.745559 0.666439i \(-0.767818\pi\)
−0.745559 + 0.666439i \(0.767818\pi\)
\(558\) 492.744 0.0373827
\(559\) 486.743 0.0368283
\(560\) 0 0
\(561\) −2790.53 −0.210011
\(562\) −463.427 −0.0347838
\(563\) 7081.18 0.530082 0.265041 0.964237i \(-0.414615\pi\)
0.265041 + 0.964237i \(0.414615\pi\)
\(564\) −1329.01 −0.0992223
\(565\) 10119.5 0.753507
\(566\) 1675.79 0.124450
\(567\) 0 0
\(568\) 1880.91 0.138946
\(569\) −12034.6 −0.886669 −0.443334 0.896356i \(-0.646205\pi\)
−0.443334 + 0.896356i \(0.646205\pi\)
\(570\) 209.270 0.0153778
\(571\) 4179.38 0.306308 0.153154 0.988202i \(-0.451057\pi\)
0.153154 + 0.988202i \(0.451057\pi\)
\(572\) 8768.04 0.640927
\(573\) −3812.62 −0.277966
\(574\) 0 0
\(575\) 2340.99 0.169784
\(576\) 11680.2 0.844923
\(577\) 16691.6 1.20430 0.602151 0.798382i \(-0.294311\pi\)
0.602151 + 0.798382i \(0.294311\pi\)
\(578\) −610.664 −0.0439451
\(579\) −6568.09 −0.471434
\(580\) 11818.5 0.846095
\(581\) 0 0
\(582\) 118.848 0.00846465
\(583\) 3191.72 0.226737
\(584\) −1969.62 −0.139561
\(585\) −7393.70 −0.522550
\(586\) 123.516 0.00870718
\(587\) −8878.89 −0.624311 −0.312156 0.950031i \(-0.601051\pi\)
−0.312156 + 0.950031i \(0.601051\pi\)
\(588\) 0 0
\(589\) −9285.11 −0.649552
\(590\) −113.409 −0.00791349
\(591\) −5932.99 −0.412945
\(592\) 17648.9 1.22528
\(593\) 23777.1 1.64656 0.823278 0.567639i \(-0.192143\pi\)
0.823278 + 0.567639i \(0.192143\pi\)
\(594\) −369.480 −0.0255218
\(595\) 0 0
\(596\) 4593.67 0.315711
\(597\) 1173.29 0.0804346
\(598\) 1327.54 0.0907812
\(599\) 12709.4 0.866934 0.433467 0.901169i \(-0.357290\pi\)
0.433467 + 0.901169i \(0.357290\pi\)
\(600\) −164.047 −0.0111620
\(601\) −13296.4 −0.902446 −0.451223 0.892411i \(-0.649012\pi\)
−0.451223 + 0.892411i \(0.649012\pi\)
\(602\) 0 0
\(603\) 10952.2 0.739651
\(604\) −6228.37 −0.419584
\(605\) 5088.22 0.341926
\(606\) 145.998 0.00978672
\(607\) −5838.48 −0.390406 −0.195203 0.980763i \(-0.562537\pi\)
−0.195203 + 0.980763i \(0.562537\pi\)
\(608\) 4404.97 0.293824
\(609\) 0 0
\(610\) 493.917 0.0327838
\(611\) 5761.43 0.381477
\(612\) 16439.2 1.08581
\(613\) 4270.31 0.281364 0.140682 0.990055i \(-0.455070\pi\)
0.140682 + 0.990055i \(0.455070\pi\)
\(614\) −1208.97 −0.0794628
\(615\) −2457.44 −0.161128
\(616\) 0 0
\(617\) −2869.54 −0.187234 −0.0936168 0.995608i \(-0.529843\pi\)
−0.0936168 + 0.995608i \(0.529843\pi\)
\(618\) −437.841 −0.0284992
\(619\) −6882.06 −0.446871 −0.223436 0.974719i \(-0.571727\pi\)
−0.223436 + 0.974719i \(0.571727\pi\)
\(620\) 3627.49 0.234974
\(621\) 8591.22 0.555159
\(622\) 2011.07 0.129641
\(623\) 0 0
\(624\) 7073.54 0.453796
\(625\) 625.000 0.0400000
\(626\) 431.413 0.0275443
\(627\) 3256.71 0.207433
\(628\) 10908.2 0.693129
\(629\) 24510.9 1.55376
\(630\) 0 0
\(631\) −28101.9 −1.77293 −0.886467 0.462793i \(-0.846847\pi\)
−0.886467 + 0.462793i \(0.846847\pi\)
\(632\) 869.679 0.0547373
\(633\) −9173.74 −0.576024
\(634\) 1381.55 0.0865429
\(635\) −8032.82 −0.502004
\(636\) 2591.88 0.161595
\(637\) 0 0
\(638\) −1197.61 −0.0743164
\(639\) −12301.5 −0.761567
\(640\) −2292.04 −0.141564
\(641\) 13677.8 0.842809 0.421404 0.906873i \(-0.361537\pi\)
0.421404 + 0.906873i \(0.361537\pi\)
\(642\) −573.437 −0.0352520
\(643\) −17919.1 −1.09901 −0.549503 0.835492i \(-0.685183\pi\)
−0.549503 + 0.835492i \(0.685183\pi\)
\(644\) 0 0
\(645\) −70.6311 −0.00431178
\(646\) 2017.10 0.122851
\(647\) −31488.2 −1.91334 −0.956669 0.291176i \(-0.905953\pi\)
−0.956669 + 0.291176i \(0.905953\pi\)
\(648\) 1722.48 0.104422
\(649\) −1764.90 −0.106746
\(650\) 354.428 0.0213874
\(651\) 0 0
\(652\) −29843.9 −1.79261
\(653\) −1110.46 −0.0665477 −0.0332739 0.999446i \(-0.510593\pi\)
−0.0332739 + 0.999446i \(0.510593\pi\)
\(654\) −59.4475 −0.00355441
\(655\) −14241.6 −0.849566
\(656\) −17055.4 −1.01509
\(657\) 12881.7 0.764936
\(658\) 0 0
\(659\) 12793.3 0.756230 0.378115 0.925759i \(-0.376572\pi\)
0.378115 + 0.925759i \(0.376572\pi\)
\(660\) −1272.33 −0.0750384
\(661\) −2039.60 −0.120017 −0.0600086 0.998198i \(-0.519113\pi\)
−0.0600086 + 0.998198i \(0.519113\pi\)
\(662\) −760.246 −0.0446341
\(663\) 9823.79 0.575452
\(664\) 1085.50 0.0634419
\(665\) 0 0
\(666\) 1518.05 0.0883232
\(667\) 27847.1 1.61656
\(668\) 15463.7 0.895672
\(669\) 296.096 0.0171117
\(670\) −525.012 −0.0302731
\(671\) 7686.48 0.442225
\(672\) 0 0
\(673\) −4039.94 −0.231394 −0.115697 0.993285i \(-0.536910\pi\)
−0.115697 + 0.993285i \(0.536910\pi\)
\(674\) 1949.40 0.111406
\(675\) 2293.69 0.130791
\(676\) −13404.7 −0.762673
\(677\) 1121.55 0.0636701 0.0318351 0.999493i \(-0.489865\pi\)
0.0318351 + 0.999493i \(0.489865\pi\)
\(678\) −832.731 −0.0471694
\(679\) 0 0
\(680\) −1581.20 −0.0891712
\(681\) −11442.2 −0.643858
\(682\) −367.588 −0.0206388
\(683\) −10232.0 −0.573232 −0.286616 0.958046i \(-0.592530\pi\)
−0.286616 + 0.958046i \(0.592530\pi\)
\(684\) −19185.5 −1.07248
\(685\) −4208.59 −0.234747
\(686\) 0 0
\(687\) −6301.95 −0.349977
\(688\) −490.201 −0.0271639
\(689\) −11236.1 −0.621281
\(690\) −192.639 −0.0106285
\(691\) 22514.6 1.23950 0.619750 0.784799i \(-0.287234\pi\)
0.619750 + 0.784799i \(0.287234\pi\)
\(692\) 16431.7 0.902661
\(693\) 0 0
\(694\) 1421.11 0.0777302
\(695\) −12145.5 −0.662885
\(696\) −1951.40 −0.106276
\(697\) −23686.7 −1.28723
\(698\) −1613.19 −0.0874785
\(699\) 8131.62 0.440009
\(700\) 0 0
\(701\) −19172.5 −1.03300 −0.516501 0.856287i \(-0.672766\pi\)
−0.516501 + 0.856287i \(0.672766\pi\)
\(702\) 1300.72 0.0699322
\(703\) −28605.6 −1.53468
\(704\) −8713.45 −0.466478
\(705\) −836.039 −0.0446625
\(706\) −1658.28 −0.0883996
\(707\) 0 0
\(708\) −1433.21 −0.0760781
\(709\) −26333.2 −1.39487 −0.697437 0.716646i \(-0.745676\pi\)
−0.697437 + 0.716646i \(0.745676\pi\)
\(710\) 589.692 0.0311701
\(711\) −5687.88 −0.300017
\(712\) 3825.41 0.201353
\(713\) 8547.23 0.448943
\(714\) 0 0
\(715\) 5515.71 0.288498
\(716\) −19043.3 −0.993971
\(717\) 2766.64 0.144103
\(718\) 1971.99 0.102499
\(719\) 17047.7 0.884246 0.442123 0.896955i \(-0.354226\pi\)
0.442123 + 0.896955i \(0.354226\pi\)
\(720\) 7446.24 0.385423
\(721\) 0 0
\(722\) −793.666 −0.0409102
\(723\) −8269.57 −0.425378
\(724\) −10486.9 −0.538321
\(725\) 7434.64 0.380849
\(726\) −418.707 −0.0214045
\(727\) −16021.2 −0.817325 −0.408662 0.912686i \(-0.634005\pi\)
−0.408662 + 0.912686i \(0.634005\pi\)
\(728\) 0 0
\(729\) −6785.17 −0.344722
\(730\) −617.503 −0.0313080
\(731\) −680.795 −0.0344461
\(732\) 6241.91 0.315174
\(733\) 7544.16 0.380150 0.190075 0.981770i \(-0.439127\pi\)
0.190075 + 0.981770i \(0.439127\pi\)
\(734\) 859.397 0.0432165
\(735\) 0 0
\(736\) −4054.91 −0.203079
\(737\) −8170.38 −0.408358
\(738\) −1467.00 −0.0731723
\(739\) 17543.9 0.873291 0.436645 0.899634i \(-0.356166\pi\)
0.436645 + 0.899634i \(0.356166\pi\)
\(740\) 11175.6 0.555167
\(741\) −11464.9 −0.568387
\(742\) 0 0
\(743\) −26595.9 −1.31320 −0.656600 0.754239i \(-0.728006\pi\)
−0.656600 + 0.754239i \(0.728006\pi\)
\(744\) −598.953 −0.0295144
\(745\) 2889.74 0.142110
\(746\) 396.330 0.0194513
\(747\) −7099.36 −0.347727
\(748\) −12263.6 −0.599470
\(749\) 0 0
\(750\) −51.4310 −0.00250399
\(751\) 9616.06 0.467237 0.233619 0.972328i \(-0.424943\pi\)
0.233619 + 0.972328i \(0.424943\pi\)
\(752\) −5802.37 −0.281370
\(753\) 971.468 0.0470149
\(754\) 4216.07 0.203634
\(755\) −3918.08 −0.188866
\(756\) 0 0
\(757\) 8519.83 0.409060 0.204530 0.978860i \(-0.434433\pi\)
0.204530 + 0.978860i \(0.434433\pi\)
\(758\) 789.711 0.0378411
\(759\) −2997.90 −0.143369
\(760\) 1845.36 0.0880765
\(761\) −4228.45 −0.201421 −0.100710 0.994916i \(-0.532112\pi\)
−0.100710 + 0.994916i \(0.532112\pi\)
\(762\) 661.017 0.0314253
\(763\) 0 0
\(764\) −16755.5 −0.793444
\(765\) 10341.4 0.488750
\(766\) −3286.22 −0.155008
\(767\) 6213.15 0.292495
\(768\) −6933.34 −0.325762
\(769\) −25293.4 −1.18609 −0.593046 0.805169i \(-0.702075\pi\)
−0.593046 + 0.805169i \(0.702075\pi\)
\(770\) 0 0
\(771\) 1588.94 0.0742207
\(772\) −28865.0 −1.34569
\(773\) 15296.8 0.711756 0.355878 0.934533i \(-0.384182\pi\)
0.355878 + 0.934533i \(0.384182\pi\)
\(774\) −42.1642 −0.00195809
\(775\) 2281.95 0.105768
\(776\) 1048.02 0.0484814
\(777\) 0 0
\(778\) 1758.45 0.0810329
\(779\) 27643.7 1.27142
\(780\) 4479.11 0.205613
\(781\) 9176.96 0.420458
\(782\) −1856.80 −0.0849092
\(783\) 27284.4 1.24529
\(784\) 0 0
\(785\) 6862.02 0.311995
\(786\) 1171.94 0.0531826
\(787\) 16789.8 0.760474 0.380237 0.924889i \(-0.375842\pi\)
0.380237 + 0.924889i \(0.375842\pi\)
\(788\) −26073.9 −1.17874
\(789\) −6717.12 −0.303087
\(790\) 272.657 0.0122794
\(791\) 0 0
\(792\) −1524.01 −0.0683755
\(793\) −27059.5 −1.21174
\(794\) 2290.81 0.102390
\(795\) 1630.47 0.0727382
\(796\) 5156.29 0.229598
\(797\) −18124.8 −0.805537 −0.402768 0.915302i \(-0.631952\pi\)
−0.402768 + 0.915302i \(0.631952\pi\)
\(798\) 0 0
\(799\) −8058.37 −0.356802
\(800\) −1082.58 −0.0478439
\(801\) −25018.9 −1.10362
\(802\) −341.968 −0.0150565
\(803\) −9609.76 −0.422318
\(804\) −6634.87 −0.291037
\(805\) 0 0
\(806\) 1294.06 0.0565524
\(807\) 279.353 0.0121855
\(808\) 1287.42 0.0560536
\(809\) −10777.9 −0.468395 −0.234197 0.972189i \(-0.575246\pi\)
−0.234197 + 0.972189i \(0.575246\pi\)
\(810\) 540.022 0.0234252
\(811\) 11986.8 0.519007 0.259503 0.965742i \(-0.416441\pi\)
0.259503 + 0.965742i \(0.416441\pi\)
\(812\) 0 0
\(813\) −4352.00 −0.187738
\(814\) −1132.47 −0.0487629
\(815\) −18773.9 −0.806898
\(816\) −9893.59 −0.424442
\(817\) 794.528 0.0340233
\(818\) 139.543 0.00596454
\(819\) 0 0
\(820\) −10799.8 −0.459934
\(821\) 10740.4 0.456570 0.228285 0.973594i \(-0.426688\pi\)
0.228285 + 0.973594i \(0.426688\pi\)
\(822\) 346.322 0.0146951
\(823\) 32400.9 1.37233 0.686163 0.727448i \(-0.259294\pi\)
0.686163 + 0.727448i \(0.259294\pi\)
\(824\) −3860.92 −0.163230
\(825\) −800.383 −0.0337767
\(826\) 0 0
\(827\) 3825.30 0.160845 0.0804225 0.996761i \(-0.474373\pi\)
0.0804225 + 0.996761i \(0.474373\pi\)
\(828\) 17660.8 0.741251
\(829\) 31836.3 1.33380 0.666900 0.745147i \(-0.267621\pi\)
0.666900 + 0.745147i \(0.267621\pi\)
\(830\) 340.318 0.0142321
\(831\) −2622.01 −0.109454
\(832\) 30674.9 1.27820
\(833\) 0 0
\(834\) 999.448 0.0414965
\(835\) 9727.76 0.403165
\(836\) 14312.4 0.592111
\(837\) 8374.53 0.345838
\(838\) −2806.05 −0.115672
\(839\) 6354.36 0.261474 0.130737 0.991417i \(-0.458266\pi\)
0.130737 + 0.991417i \(0.458266\pi\)
\(840\) 0 0
\(841\) 64049.1 2.62615
\(842\) 553.324 0.0226470
\(843\) −3684.20 −0.150523
\(844\) −40316.2 −1.64424
\(845\) −8432.52 −0.343299
\(846\) −499.085 −0.0202824
\(847\) 0 0
\(848\) 11316.0 0.458245
\(849\) 13322.4 0.538544
\(850\) −495.730 −0.0200040
\(851\) 26332.4 1.06071
\(852\) 7452.28 0.299661
\(853\) 16620.2 0.667135 0.333568 0.942726i \(-0.391747\pi\)
0.333568 + 0.942726i \(0.391747\pi\)
\(854\) 0 0
\(855\) −12069.0 −0.482750
\(856\) −5056.62 −0.201906
\(857\) −34440.5 −1.37277 −0.686385 0.727238i \(-0.740804\pi\)
−0.686385 + 0.727238i \(0.740804\pi\)
\(858\) −453.885 −0.0180599
\(859\) 29192.5 1.15953 0.579765 0.814784i \(-0.303144\pi\)
0.579765 + 0.814784i \(0.303144\pi\)
\(860\) −310.405 −0.0123078
\(861\) 0 0
\(862\) 298.130 0.0117800
\(863\) 26062.4 1.02801 0.514006 0.857786i \(-0.328161\pi\)
0.514006 + 0.857786i \(0.328161\pi\)
\(864\) −3972.98 −0.156439
\(865\) 10336.7 0.406311
\(866\) −1127.22 −0.0442314
\(867\) −4854.72 −0.190167
\(868\) 0 0
\(869\) 4243.16 0.165638
\(870\) −611.793 −0.0238411
\(871\) 28763.0 1.11894
\(872\) −524.214 −0.0203579
\(873\) −6854.23 −0.265728
\(874\) 2166.99 0.0838668
\(875\) 0 0
\(876\) −7803.74 −0.300986
\(877\) −29048.5 −1.11847 −0.559236 0.829009i \(-0.688905\pi\)
−0.559236 + 0.829009i \(0.688905\pi\)
\(878\) 1864.42 0.0716640
\(879\) 981.943 0.0376793
\(880\) −5554.90 −0.212791
\(881\) 24440.9 0.934660 0.467330 0.884083i \(-0.345216\pi\)
0.467330 + 0.884083i \(0.345216\pi\)
\(882\) 0 0
\(883\) 38951.9 1.48453 0.742264 0.670108i \(-0.233752\pi\)
0.742264 + 0.670108i \(0.233752\pi\)
\(884\) 43173.0 1.64261
\(885\) −901.589 −0.0342447
\(886\) −672.642 −0.0255055
\(887\) 3143.83 0.119007 0.0595036 0.998228i \(-0.481048\pi\)
0.0595036 + 0.998228i \(0.481048\pi\)
\(888\) −1845.26 −0.0697330
\(889\) 0 0
\(890\) 1199.32 0.0451700
\(891\) 8403.98 0.315986
\(892\) 1301.26 0.0488448
\(893\) 9404.59 0.352422
\(894\) −237.795 −0.00889604
\(895\) −11979.6 −0.447412
\(896\) 0 0
\(897\) 10553.8 0.392845
\(898\) 2378.13 0.0883734
\(899\) 27144.7 1.00704
\(900\) 4715.10 0.174633
\(901\) 15715.7 0.581094
\(902\) 1094.39 0.0403981
\(903\) 0 0
\(904\) −7343.10 −0.270163
\(905\) −6597.02 −0.242312
\(906\) 322.417 0.0118229
\(907\) −12839.5 −0.470044 −0.235022 0.971990i \(-0.575516\pi\)
−0.235022 + 0.971990i \(0.575516\pi\)
\(908\) −50285.6 −1.83787
\(909\) −8419.98 −0.307231
\(910\) 0 0
\(911\) 17451.4 0.634678 0.317339 0.948312i \(-0.397211\pi\)
0.317339 + 0.948312i \(0.397211\pi\)
\(912\) 11546.4 0.419232
\(913\) 5296.13 0.191978
\(914\) −510.775 −0.0184846
\(915\) 3926.60 0.141868
\(916\) −27695.4 −0.998998
\(917\) 0 0
\(918\) −1819.28 −0.0654088
\(919\) −7937.66 −0.284918 −0.142459 0.989801i \(-0.545501\pi\)
−0.142459 + 0.989801i \(0.545501\pi\)
\(920\) −1698.71 −0.0608747
\(921\) −9611.22 −0.343866
\(922\) 2830.31 0.101097
\(923\) −32306.6 −1.15210
\(924\) 0 0
\(925\) 7030.24 0.249895
\(926\) −3887.02 −0.137943
\(927\) 25251.2 0.894668
\(928\) −12877.8 −0.455532
\(929\) −1781.40 −0.0629127 −0.0314563 0.999505i \(-0.510015\pi\)
−0.0314563 + 0.999505i \(0.510015\pi\)
\(930\) −187.780 −0.00662103
\(931\) 0 0
\(932\) 35736.3 1.25599
\(933\) 15987.9 0.561007
\(934\) 2663.79 0.0933210
\(935\) −7714.69 −0.269837
\(936\) 5365.14 0.187356
\(937\) −46363.0 −1.61645 −0.808225 0.588874i \(-0.799571\pi\)
−0.808225 + 0.588874i \(0.799571\pi\)
\(938\) 0 0
\(939\) 3429.70 0.119195
\(940\) −3674.17 −0.127488
\(941\) 16535.3 0.572833 0.286416 0.958105i \(-0.407536\pi\)
0.286416 + 0.958105i \(0.407536\pi\)
\(942\) −564.673 −0.0195308
\(943\) −25446.9 −0.878754
\(944\) −6257.30 −0.215739
\(945\) 0 0
\(946\) 31.4545 0.00108105
\(947\) −2734.31 −0.0938258 −0.0469129 0.998899i \(-0.514938\pi\)
−0.0469129 + 0.998899i \(0.514938\pi\)
\(948\) 3445.72 0.118050
\(949\) 33830.2 1.15719
\(950\) 578.546 0.0197584
\(951\) 10983.2 0.374505
\(952\) 0 0
\(953\) −35524.6 −1.20751 −0.603754 0.797171i \(-0.706329\pi\)
−0.603754 + 0.797171i \(0.706329\pi\)
\(954\) 973.332 0.0330323
\(955\) −10540.4 −0.357150
\(956\) 12158.7 0.411338
\(957\) −9520.89 −0.321595
\(958\) −2659.89 −0.0897048
\(959\) 0 0
\(960\) −4451.22 −0.149649
\(961\) −21459.4 −0.720330
\(962\) 3986.74 0.133615
\(963\) 33071.3 1.10665
\(964\) −36342.6 −1.21423
\(965\) −18158.1 −0.605730
\(966\) 0 0
\(967\) 9328.92 0.310236 0.155118 0.987896i \(-0.450424\pi\)
0.155118 + 0.987896i \(0.450424\pi\)
\(968\) −3692.19 −0.122595
\(969\) 16035.7 0.531622
\(970\) 328.568 0.0108760
\(971\) 49138.1 1.62401 0.812007 0.583648i \(-0.198375\pi\)
0.812007 + 0.583648i \(0.198375\pi\)
\(972\) 26513.9 0.874930
\(973\) 0 0
\(974\) −1621.79 −0.0533527
\(975\) 2817.67 0.0925515
\(976\) 27251.8 0.893758
\(977\) 14771.3 0.483702 0.241851 0.970313i \(-0.422245\pi\)
0.241851 + 0.970313i \(0.422245\pi\)
\(978\) 1544.90 0.0505116
\(979\) 18664.1 0.609304
\(980\) 0 0
\(981\) 3428.46 0.111582
\(982\) 985.141 0.0320133
\(983\) −20651.0 −0.670054 −0.335027 0.942208i \(-0.608746\pi\)
−0.335027 + 0.942208i \(0.608746\pi\)
\(984\) 1783.21 0.0577710
\(985\) −16402.3 −0.530580
\(986\) −5896.91 −0.190462
\(987\) 0 0
\(988\) −50385.4 −1.62244
\(989\) −731.387 −0.0235154
\(990\) −477.799 −0.0153389
\(991\) −45941.4 −1.47263 −0.736315 0.676639i \(-0.763436\pi\)
−0.736315 + 0.676639i \(0.763436\pi\)
\(992\) −3952.63 −0.126508
\(993\) −6043.88 −0.193149
\(994\) 0 0
\(995\) 3243.67 0.103348
\(996\) 4300.79 0.136823
\(997\) 9025.07 0.286687 0.143344 0.989673i \(-0.454215\pi\)
0.143344 + 0.989673i \(0.454215\pi\)
\(998\) −2192.09 −0.0695284
\(999\) 25800.3 0.817103
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 245.4.a.m.1.3 5
3.2 odd 2 2205.4.a.bu.1.3 5
5.4 even 2 1225.4.a.bg.1.3 5
7.2 even 3 35.4.e.c.11.3 10
7.3 odd 6 245.4.e.o.226.3 10
7.4 even 3 35.4.e.c.16.3 yes 10
7.5 odd 6 245.4.e.o.116.3 10
7.6 odd 2 245.4.a.n.1.3 5
21.2 odd 6 315.4.j.g.46.3 10
21.11 odd 6 315.4.j.g.226.3 10
21.20 even 2 2205.4.a.bt.1.3 5
28.11 odd 6 560.4.q.n.401.3 10
28.23 odd 6 560.4.q.n.81.3 10
35.2 odd 12 175.4.k.d.74.5 20
35.4 even 6 175.4.e.d.51.3 10
35.9 even 6 175.4.e.d.151.3 10
35.18 odd 12 175.4.k.d.149.5 20
35.23 odd 12 175.4.k.d.74.6 20
35.32 odd 12 175.4.k.d.149.6 20
35.34 odd 2 1225.4.a.bf.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.e.c.11.3 10 7.2 even 3
35.4.e.c.16.3 yes 10 7.4 even 3
175.4.e.d.51.3 10 35.4 even 6
175.4.e.d.151.3 10 35.9 even 6
175.4.k.d.74.5 20 35.2 odd 12
175.4.k.d.74.6 20 35.23 odd 12
175.4.k.d.149.5 20 35.18 odd 12
175.4.k.d.149.6 20 35.32 odd 12
245.4.a.m.1.3 5 1.1 even 1 trivial
245.4.a.n.1.3 5 7.6 odd 2
245.4.e.o.116.3 10 7.5 odd 6
245.4.e.o.226.3 10 7.3 odd 6
315.4.j.g.46.3 10 21.2 odd 6
315.4.j.g.226.3 10 21.11 odd 6
560.4.q.n.81.3 10 28.23 odd 6
560.4.q.n.401.3 10 28.11 odd 6
1225.4.a.bf.1.3 5 35.34 odd 2
1225.4.a.bg.1.3 5 5.4 even 2
2205.4.a.bt.1.3 5 21.20 even 2
2205.4.a.bu.1.3 5 3.2 odd 2