Properties

Label 525.4.a.p.1.2
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
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] = [525,4,Mod(1,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("525.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.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: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.56155 q^{2} +3.00000 q^{3} +22.9309 q^{4} +16.6847 q^{6} +7.00000 q^{7} +83.0388 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.56155 q^{2} +3.00000 q^{3} +22.9309 q^{4} +16.6847 q^{6} +7.00000 q^{7} +83.0388 q^{8} +9.00000 q^{9} -33.6155 q^{11} +68.7926 q^{12} +38.3542 q^{13} +38.9309 q^{14} +278.378 q^{16} -65.7235 q^{17} +50.0540 q^{18} +33.3996 q^{19} +21.0000 q^{21} -186.955 q^{22} -207.447 q^{23} +249.116 q^{24} +213.309 q^{26} +27.0000 q^{27} +160.516 q^{28} -189.170 q^{29} +202.108 q^{31} +883.902 q^{32} -100.847 q^{33} -365.525 q^{34} +206.378 q^{36} +16.5227 q^{37} +185.754 q^{38} +115.062 q^{39} +388.617 q^{41} +116.793 q^{42} -41.8144 q^{43} -770.833 q^{44} -1153.73 q^{46} -368.648 q^{47} +835.133 q^{48} +49.0000 q^{49} -197.170 q^{51} +879.494 q^{52} -458.172 q^{53} +150.162 q^{54} +581.272 q^{56} +100.199 q^{57} -1052.08 q^{58} +256.216 q^{59} -123.511 q^{61} +1124.03 q^{62} +63.0000 q^{63} +2688.85 q^{64} -560.864 q^{66} +336.277 q^{67} -1507.10 q^{68} -622.341 q^{69} -453.312 q^{71} +747.349 q^{72} -22.0436 q^{73} +91.8920 q^{74} +765.882 q^{76} -235.309 q^{77} +639.926 q^{78} +385.417 q^{79} +81.0000 q^{81} +2161.32 q^{82} -23.7501 q^{83} +481.548 q^{84} -232.553 q^{86} -567.511 q^{87} -2791.39 q^{88} -1482.81 q^{89} +268.479 q^{91} -4756.94 q^{92} +606.324 q^{93} -2050.25 q^{94} +2651.71 q^{96} -51.9867 q^{97} +272.516 q^{98} -302.540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{2} + 6 q^{3} + 17 q^{4} + 21 q^{6} + 14 q^{7} + 63 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 7 q^{2} + 6 q^{3} + 17 q^{4} + 21 q^{6} + 14 q^{7} + 63 q^{8} + 18 q^{9} - 26 q^{11} + 51 q^{12} - 14 q^{13} + 49 q^{14} + 297 q^{16} - 16 q^{17} + 63 q^{18} + 174 q^{19} + 42 q^{21} - 176 q^{22} - 184 q^{23} + 189 q^{24} + 138 q^{26} + 54 q^{27} + 119 q^{28} - 32 q^{29} + 330 q^{31} + 1071 q^{32} - 78 q^{33} - 294 q^{34} + 153 q^{36} + 132 q^{37} + 388 q^{38} - 42 q^{39} + 200 q^{41} + 147 q^{42} - 364 q^{43} - 816 q^{44} - 1120 q^{46} - 292 q^{47} + 891 q^{48} + 98 q^{49} - 48 q^{51} + 1190 q^{52} - 34 q^{53} + 189 q^{54} + 441 q^{56} + 522 q^{57} - 826 q^{58} + 364 q^{59} + 792 q^{61} + 1308 q^{62} + 126 q^{63} + 2809 q^{64} - 528 q^{66} + 788 q^{67} - 1802 q^{68} - 552 q^{69} + 454 q^{71} + 567 q^{72} - 778 q^{73} + 258 q^{74} - 68 q^{76} - 182 q^{77} + 414 q^{78} + 408 q^{79} + 162 q^{81} + 1890 q^{82} - 1136 q^{83} + 357 q^{84} - 696 q^{86} - 96 q^{87} - 2944 q^{88} + 36 q^{89} - 98 q^{91} - 4896 q^{92} + 990 q^{93} - 1940 q^{94} + 3213 q^{96} + 498 q^{97} + 343 q^{98} - 234 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.56155 1.96631 0.983153 0.182785i \(-0.0585112\pi\)
0.983153 + 0.182785i \(0.0585112\pi\)
\(3\) 3.00000 0.577350
\(4\) 22.9309 2.86636
\(5\) 0 0
\(6\) 16.6847 1.13525
\(7\) 7.00000 0.377964
\(8\) 83.0388 3.66983
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −33.6155 −0.921406 −0.460703 0.887554i \(-0.652403\pi\)
−0.460703 + 0.887554i \(0.652403\pi\)
\(12\) 68.7926 1.65489
\(13\) 38.3542 0.818272 0.409136 0.912474i \(-0.365830\pi\)
0.409136 + 0.912474i \(0.365830\pi\)
\(14\) 38.9309 0.743194
\(15\) 0 0
\(16\) 278.378 4.34965
\(17\) −65.7235 −0.937664 −0.468832 0.883287i \(-0.655325\pi\)
−0.468832 + 0.883287i \(0.655325\pi\)
\(18\) 50.0540 0.655435
\(19\) 33.3996 0.403284 0.201642 0.979459i \(-0.435372\pi\)
0.201642 + 0.979459i \(0.435372\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) −186.955 −1.81177
\(23\) −207.447 −1.88068 −0.940341 0.340234i \(-0.889494\pi\)
−0.940341 + 0.340234i \(0.889494\pi\)
\(24\) 249.116 2.11878
\(25\) 0 0
\(26\) 213.309 1.60897
\(27\) 27.0000 0.192450
\(28\) 160.516 1.08338
\(29\) −189.170 −1.21131 −0.605656 0.795726i \(-0.707089\pi\)
−0.605656 + 0.795726i \(0.707089\pi\)
\(30\) 0 0
\(31\) 202.108 1.17096 0.585478 0.810688i \(-0.300907\pi\)
0.585478 + 0.810688i \(0.300907\pi\)
\(32\) 883.902 4.88292
\(33\) −100.847 −0.531974
\(34\) −365.525 −1.84373
\(35\) 0 0
\(36\) 206.378 0.955453
\(37\) 16.5227 0.0734141 0.0367070 0.999326i \(-0.488313\pi\)
0.0367070 + 0.999326i \(0.488313\pi\)
\(38\) 185.754 0.792980
\(39\) 115.062 0.472429
\(40\) 0 0
\(41\) 388.617 1.48029 0.740144 0.672448i \(-0.234757\pi\)
0.740144 + 0.672448i \(0.234757\pi\)
\(42\) 116.793 0.429083
\(43\) −41.8144 −0.148294 −0.0741469 0.997247i \(-0.523623\pi\)
−0.0741469 + 0.997247i \(0.523623\pi\)
\(44\) −770.833 −2.64108
\(45\) 0 0
\(46\) −1153.73 −3.69800
\(47\) −368.648 −1.14410 −0.572051 0.820218i \(-0.693852\pi\)
−0.572051 + 0.820218i \(0.693852\pi\)
\(48\) 835.133 2.51127
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −197.170 −0.541360
\(52\) 879.494 2.34546
\(53\) −458.172 −1.18745 −0.593725 0.804668i \(-0.702343\pi\)
−0.593725 + 0.804668i \(0.702343\pi\)
\(54\) 150.162 0.378416
\(55\) 0 0
\(56\) 581.272 1.38707
\(57\) 100.199 0.232836
\(58\) −1052.08 −2.38181
\(59\) 256.216 0.565364 0.282682 0.959214i \(-0.408776\pi\)
0.282682 + 0.959214i \(0.408776\pi\)
\(60\) 0 0
\(61\) −123.511 −0.259246 −0.129623 0.991563i \(-0.541377\pi\)
−0.129623 + 0.991563i \(0.541377\pi\)
\(62\) 1124.03 2.30246
\(63\) 63.0000 0.125988
\(64\) 2688.85 5.25166
\(65\) 0 0
\(66\) −560.864 −1.04602
\(67\) 336.277 0.613175 0.306587 0.951843i \(-0.400813\pi\)
0.306587 + 0.951843i \(0.400813\pi\)
\(68\) −1507.10 −2.68768
\(69\) −622.341 −1.08581
\(70\) 0 0
\(71\) −453.312 −0.757722 −0.378861 0.925454i \(-0.623684\pi\)
−0.378861 + 0.925454i \(0.623684\pi\)
\(72\) 747.349 1.22328
\(73\) −22.0436 −0.0353426 −0.0176713 0.999844i \(-0.505625\pi\)
−0.0176713 + 0.999844i \(0.505625\pi\)
\(74\) 91.8920 0.144355
\(75\) 0 0
\(76\) 765.882 1.15596
\(77\) −235.309 −0.348259
\(78\) 639.926 0.928941
\(79\) 385.417 0.548896 0.274448 0.961602i \(-0.411505\pi\)
0.274448 + 0.961602i \(0.411505\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 2161.32 2.91070
\(83\) −23.7501 −0.0314085 −0.0157043 0.999877i \(-0.504999\pi\)
−0.0157043 + 0.999877i \(0.504999\pi\)
\(84\) 481.548 0.625491
\(85\) 0 0
\(86\) −232.553 −0.291591
\(87\) −567.511 −0.699352
\(88\) −2791.39 −3.38140
\(89\) −1482.81 −1.76604 −0.883020 0.469335i \(-0.844494\pi\)
−0.883020 + 0.469335i \(0.844494\pi\)
\(90\) 0 0
\(91\) 268.479 0.309278
\(92\) −4756.94 −5.39071
\(93\) 606.324 0.676052
\(94\) −2050.25 −2.24965
\(95\) 0 0
\(96\) 2651.71 2.81915
\(97\) −51.9867 −0.0544170 −0.0272085 0.999630i \(-0.508662\pi\)
−0.0272085 + 0.999630i \(0.508662\pi\)
\(98\) 272.516 0.280901
\(99\) −302.540 −0.307135
\(100\) 0 0
\(101\) 1429.30 1.40812 0.704062 0.710138i \(-0.251368\pi\)
0.704062 + 0.710138i \(0.251368\pi\)
\(102\) −1096.57 −1.06448
\(103\) −434.212 −0.415381 −0.207690 0.978195i \(-0.566595\pi\)
−0.207690 + 0.978195i \(0.566595\pi\)
\(104\) 3184.88 3.00292
\(105\) 0 0
\(106\) −2548.15 −2.33489
\(107\) −666.307 −0.602003 −0.301001 0.953624i \(-0.597321\pi\)
−0.301001 + 0.953624i \(0.597321\pi\)
\(108\) 619.133 0.551631
\(109\) −1199.51 −1.05406 −0.527029 0.849847i \(-0.676694\pi\)
−0.527029 + 0.849847i \(0.676694\pi\)
\(110\) 0 0
\(111\) 49.5682 0.0423856
\(112\) 1948.64 1.64401
\(113\) 81.5171 0.0678627 0.0339314 0.999424i \(-0.489197\pi\)
0.0339314 + 0.999424i \(0.489197\pi\)
\(114\) 557.261 0.457827
\(115\) 0 0
\(116\) −4337.84 −3.47206
\(117\) 345.187 0.272757
\(118\) 1424.96 1.11168
\(119\) −460.064 −0.354404
\(120\) 0 0
\(121\) −200.996 −0.151011
\(122\) −686.915 −0.509757
\(123\) 1165.85 0.854645
\(124\) 4634.51 3.35638
\(125\) 0 0
\(126\) 350.378 0.247731
\(127\) 336.985 0.235453 0.117727 0.993046i \(-0.462439\pi\)
0.117727 + 0.993046i \(0.462439\pi\)
\(128\) 7882.95 5.44344
\(129\) −125.443 −0.0856175
\(130\) 0 0
\(131\) 2931.15 1.95493 0.977465 0.211097i \(-0.0677035\pi\)
0.977465 + 0.211097i \(0.0677035\pi\)
\(132\) −2312.50 −1.52483
\(133\) 233.797 0.152427
\(134\) 1870.22 1.20569
\(135\) 0 0
\(136\) −5457.60 −3.44107
\(137\) 1585.07 0.988477 0.494238 0.869326i \(-0.335447\pi\)
0.494238 + 0.869326i \(0.335447\pi\)
\(138\) −3461.18 −2.13504
\(139\) −1298.85 −0.792569 −0.396284 0.918128i \(-0.629701\pi\)
−0.396284 + 0.918128i \(0.629701\pi\)
\(140\) 0 0
\(141\) −1105.94 −0.660548
\(142\) −2521.12 −1.48991
\(143\) −1289.30 −0.753960
\(144\) 2505.40 1.44988
\(145\) 0 0
\(146\) −122.597 −0.0694943
\(147\) 147.000 0.0824786
\(148\) 378.881 0.210431
\(149\) −2003.29 −1.10145 −0.550724 0.834687i \(-0.685648\pi\)
−0.550724 + 0.834687i \(0.685648\pi\)
\(150\) 0 0
\(151\) 2740.96 1.47719 0.738596 0.674148i \(-0.235489\pi\)
0.738596 + 0.674148i \(0.235489\pi\)
\(152\) 2773.47 1.47999
\(153\) −591.511 −0.312555
\(154\) −1308.68 −0.684783
\(155\) 0 0
\(156\) 2638.48 1.35415
\(157\) −3644.22 −1.85249 −0.926243 0.376928i \(-0.876981\pi\)
−0.926243 + 0.376928i \(0.876981\pi\)
\(158\) 2143.52 1.07930
\(159\) −1374.52 −0.685574
\(160\) 0 0
\(161\) −1452.13 −0.710831
\(162\) 450.486 0.218478
\(163\) −2774.27 −1.33311 −0.666557 0.745454i \(-0.732233\pi\)
−0.666557 + 0.745454i \(0.732233\pi\)
\(164\) 8911.33 4.24304
\(165\) 0 0
\(166\) −132.087 −0.0617588
\(167\) −1154.91 −0.535149 −0.267574 0.963537i \(-0.586222\pi\)
−0.267574 + 0.963537i \(0.586222\pi\)
\(168\) 1743.82 0.800823
\(169\) −725.958 −0.330432
\(170\) 0 0
\(171\) 300.597 0.134428
\(172\) −958.841 −0.425064
\(173\) 3387.46 1.48869 0.744346 0.667794i \(-0.232761\pi\)
0.744346 + 0.667794i \(0.232761\pi\)
\(174\) −3156.24 −1.37514
\(175\) 0 0
\(176\) −9357.82 −4.00780
\(177\) 768.648 0.326413
\(178\) −8246.73 −3.47258
\(179\) 1603.32 0.669486 0.334743 0.942309i \(-0.391350\pi\)
0.334743 + 0.942309i \(0.391350\pi\)
\(180\) 0 0
\(181\) 544.220 0.223489 0.111745 0.993737i \(-0.464356\pi\)
0.111745 + 0.993737i \(0.464356\pi\)
\(182\) 1493.16 0.608134
\(183\) −370.534 −0.149676
\(184\) −17226.2 −6.90179
\(185\) 0 0
\(186\) 3372.10 1.32933
\(187\) 2209.33 0.863969
\(188\) −8453.41 −3.27941
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 2993.44 1.13402 0.567010 0.823711i \(-0.308100\pi\)
0.567010 + 0.823711i \(0.308100\pi\)
\(192\) 8066.54 3.03204
\(193\) −1309.32 −0.488325 −0.244163 0.969734i \(-0.578513\pi\)
−0.244163 + 0.969734i \(0.578513\pi\)
\(194\) −289.127 −0.107001
\(195\) 0 0
\(196\) 1123.61 0.409480
\(197\) 1141.38 0.412790 0.206395 0.978469i \(-0.433827\pi\)
0.206395 + 0.978469i \(0.433827\pi\)
\(198\) −1682.59 −0.603922
\(199\) 2370.23 0.844327 0.422164 0.906520i \(-0.361271\pi\)
0.422164 + 0.906520i \(0.361271\pi\)
\(200\) 0 0
\(201\) 1008.83 0.354017
\(202\) 7949.12 2.76880
\(203\) −1324.19 −0.457833
\(204\) −4521.29 −1.55173
\(205\) 0 0
\(206\) −2414.89 −0.816765
\(207\) −1867.02 −0.626894
\(208\) 10676.9 3.55920
\(209\) −1122.75 −0.371588
\(210\) 0 0
\(211\) −687.159 −0.224199 −0.112099 0.993697i \(-0.535758\pi\)
−0.112099 + 0.993697i \(0.535758\pi\)
\(212\) −10506.3 −3.40366
\(213\) −1359.94 −0.437471
\(214\) −3705.70 −1.18372
\(215\) 0 0
\(216\) 2242.05 0.706260
\(217\) 1414.76 0.442580
\(218\) −6671.15 −2.07260
\(219\) −66.1308 −0.0204050
\(220\) 0 0
\(221\) −2520.77 −0.767264
\(222\) 275.676 0.0833431
\(223\) −990.496 −0.297437 −0.148719 0.988880i \(-0.547515\pi\)
−0.148719 + 0.988880i \(0.547515\pi\)
\(224\) 6187.32 1.84557
\(225\) 0 0
\(226\) 453.362 0.133439
\(227\) −1479.25 −0.432517 −0.216258 0.976336i \(-0.569385\pi\)
−0.216258 + 0.976336i \(0.569385\pi\)
\(228\) 2297.65 0.667392
\(229\) 6704.47 1.93469 0.967345 0.253463i \(-0.0815696\pi\)
0.967345 + 0.253463i \(0.0815696\pi\)
\(230\) 0 0
\(231\) −705.926 −0.201067
\(232\) −15708.5 −4.44531
\(233\) 1749.09 0.491789 0.245895 0.969297i \(-0.420918\pi\)
0.245895 + 0.969297i \(0.420918\pi\)
\(234\) 1919.78 0.536324
\(235\) 0 0
\(236\) 5875.25 1.62054
\(237\) 1156.25 0.316905
\(238\) −2558.67 −0.696866
\(239\) −6320.89 −1.71073 −0.855365 0.518027i \(-0.826667\pi\)
−0.855365 + 0.518027i \(0.826667\pi\)
\(240\) 0 0
\(241\) 3359.62 0.897975 0.448988 0.893538i \(-0.351785\pi\)
0.448988 + 0.893538i \(0.351785\pi\)
\(242\) −1117.85 −0.296935
\(243\) 243.000 0.0641500
\(244\) −2832.22 −0.743092
\(245\) 0 0
\(246\) 6483.95 1.68049
\(247\) 1281.01 0.329996
\(248\) 16782.8 4.29721
\(249\) −71.2502 −0.0181337
\(250\) 0 0
\(251\) −1330.50 −0.334582 −0.167291 0.985908i \(-0.553502\pi\)
−0.167291 + 0.985908i \(0.553502\pi\)
\(252\) 1444.64 0.361127
\(253\) 6973.44 1.73287
\(254\) 1874.16 0.462973
\(255\) 0 0
\(256\) 22330.6 5.45182
\(257\) 2476.95 0.601197 0.300599 0.953751i \(-0.402814\pi\)
0.300599 + 0.953751i \(0.402814\pi\)
\(258\) −697.659 −0.168350
\(259\) 115.659 0.0277479
\(260\) 0 0
\(261\) −1702.53 −0.403771
\(262\) 16301.8 3.84399
\(263\) 5152.56 1.20806 0.604032 0.796960i \(-0.293560\pi\)
0.604032 + 0.796960i \(0.293560\pi\)
\(264\) −8374.18 −1.95225
\(265\) 0 0
\(266\) 1300.28 0.299718
\(267\) −4448.43 −1.01962
\(268\) 7711.11 1.75758
\(269\) −1150.97 −0.260876 −0.130438 0.991456i \(-0.541638\pi\)
−0.130438 + 0.991456i \(0.541638\pi\)
\(270\) 0 0
\(271\) 1838.32 0.412067 0.206034 0.978545i \(-0.433944\pi\)
0.206034 + 0.978545i \(0.433944\pi\)
\(272\) −18296.0 −4.07851
\(273\) 805.437 0.178561
\(274\) 8815.43 1.94365
\(275\) 0 0
\(276\) −14270.8 −3.11233
\(277\) −568.447 −0.123302 −0.0616510 0.998098i \(-0.519637\pi\)
−0.0616510 + 0.998098i \(0.519637\pi\)
\(278\) −7223.62 −1.55843
\(279\) 1818.97 0.390319
\(280\) 0 0
\(281\) 6015.00 1.27696 0.638479 0.769640i \(-0.279564\pi\)
0.638479 + 0.769640i \(0.279564\pi\)
\(282\) −6150.76 −1.29884
\(283\) −3985.75 −0.837202 −0.418601 0.908170i \(-0.637479\pi\)
−0.418601 + 0.908170i \(0.637479\pi\)
\(284\) −10394.8 −2.17190
\(285\) 0 0
\(286\) −7170.48 −1.48252
\(287\) 2720.32 0.559497
\(288\) 7955.12 1.62764
\(289\) −593.424 −0.120787
\(290\) 0 0
\(291\) −155.960 −0.0314177
\(292\) −505.479 −0.101305
\(293\) 2490.01 0.496478 0.248239 0.968699i \(-0.420148\pi\)
0.248239 + 0.968699i \(0.420148\pi\)
\(294\) 817.548 0.162178
\(295\) 0 0
\(296\) 1372.03 0.269417
\(297\) −907.619 −0.177325
\(298\) −11141.4 −2.16578
\(299\) −7956.45 −1.53891
\(300\) 0 0
\(301\) −292.701 −0.0560498
\(302\) 15244.0 2.90461
\(303\) 4287.90 0.812981
\(304\) 9297.72 1.75415
\(305\) 0 0
\(306\) −3289.72 −0.614578
\(307\) 141.853 0.0263712 0.0131856 0.999913i \(-0.495803\pi\)
0.0131856 + 0.999913i \(0.495803\pi\)
\(308\) −5395.83 −0.998234
\(309\) −1302.64 −0.239820
\(310\) 0 0
\(311\) 2091.92 0.381420 0.190710 0.981646i \(-0.438921\pi\)
0.190710 + 0.981646i \(0.438921\pi\)
\(312\) 9554.65 1.73374
\(313\) −5521.44 −0.997094 −0.498547 0.866863i \(-0.666133\pi\)
−0.498547 + 0.866863i \(0.666133\pi\)
\(314\) −20267.5 −3.64255
\(315\) 0 0
\(316\) 8837.94 1.57333
\(317\) −5351.63 −0.948195 −0.474097 0.880472i \(-0.657226\pi\)
−0.474097 + 0.880472i \(0.657226\pi\)
\(318\) −7644.45 −1.34805
\(319\) 6359.06 1.11611
\(320\) 0 0
\(321\) −1998.92 −0.347567
\(322\) −8076.09 −1.39771
\(323\) −2195.14 −0.378145
\(324\) 1857.40 0.318484
\(325\) 0 0
\(326\) −15429.2 −2.62131
\(327\) −3598.53 −0.608561
\(328\) 32270.3 5.43241
\(329\) −2580.53 −0.432430
\(330\) 0 0
\(331\) −4383.52 −0.727916 −0.363958 0.931415i \(-0.618575\pi\)
−0.363958 + 0.931415i \(0.618575\pi\)
\(332\) −544.609 −0.0900281
\(333\) 148.705 0.0244714
\(334\) −6423.11 −1.05227
\(335\) 0 0
\(336\) 5845.93 0.949172
\(337\) 7124.57 1.15163 0.575817 0.817579i \(-0.304684\pi\)
0.575817 + 0.817579i \(0.304684\pi\)
\(338\) −4037.46 −0.649730
\(339\) 244.551 0.0391806
\(340\) 0 0
\(341\) −6793.97 −1.07893
\(342\) 1671.78 0.264327
\(343\) 343.000 0.0539949
\(344\) −3472.22 −0.544214
\(345\) 0 0
\(346\) 18839.5 2.92722
\(347\) −507.743 −0.0785506 −0.0392753 0.999228i \(-0.512505\pi\)
−0.0392753 + 0.999228i \(0.512505\pi\)
\(348\) −13013.5 −2.00459
\(349\) 6155.14 0.944060 0.472030 0.881582i \(-0.343521\pi\)
0.472030 + 0.881582i \(0.343521\pi\)
\(350\) 0 0
\(351\) 1035.56 0.157476
\(352\) −29712.8 −4.49915
\(353\) 6429.56 0.969437 0.484718 0.874670i \(-0.338922\pi\)
0.484718 + 0.874670i \(0.338922\pi\)
\(354\) 4274.87 0.641828
\(355\) 0 0
\(356\) −34002.1 −5.06211
\(357\) −1380.19 −0.204615
\(358\) 8916.97 1.31641
\(359\) 10075.4 1.48123 0.740614 0.671931i \(-0.234535\pi\)
0.740614 + 0.671931i \(0.234535\pi\)
\(360\) 0 0
\(361\) −5743.46 −0.837362
\(362\) 3026.71 0.439448
\(363\) −602.989 −0.0871865
\(364\) 6156.46 0.886501
\(365\) 0 0
\(366\) −2060.74 −0.294308
\(367\) −816.898 −0.116190 −0.0580950 0.998311i \(-0.518503\pi\)
−0.0580950 + 0.998311i \(0.518503\pi\)
\(368\) −57748.6 −8.18031
\(369\) 3497.56 0.493430
\(370\) 0 0
\(371\) −3207.21 −0.448814
\(372\) 13903.5 1.93781
\(373\) 3737.85 0.518870 0.259435 0.965761i \(-0.416464\pi\)
0.259435 + 0.965761i \(0.416464\pi\)
\(374\) 12287.3 1.69883
\(375\) 0 0
\(376\) −30612.1 −4.19866
\(377\) −7255.47 −0.991183
\(378\) 1051.13 0.143028
\(379\) 1950.47 0.264351 0.132176 0.991226i \(-0.457804\pi\)
0.132176 + 0.991226i \(0.457804\pi\)
\(380\) 0 0
\(381\) 1010.95 0.135939
\(382\) 16648.2 2.22983
\(383\) −6762.06 −0.902155 −0.451077 0.892485i \(-0.648960\pi\)
−0.451077 + 0.892485i \(0.648960\pi\)
\(384\) 23648.8 3.14277
\(385\) 0 0
\(386\) −7281.84 −0.960197
\(387\) −376.330 −0.0494313
\(388\) −1192.10 −0.155979
\(389\) −2551.98 −0.332624 −0.166312 0.986073i \(-0.553186\pi\)
−0.166312 + 0.986073i \(0.553186\pi\)
\(390\) 0 0
\(391\) 13634.1 1.76345
\(392\) 4068.90 0.524262
\(393\) 8793.45 1.12868
\(394\) 6347.83 0.811672
\(395\) 0 0
\(396\) −6937.50 −0.880360
\(397\) 4097.93 0.518058 0.259029 0.965869i \(-0.416597\pi\)
0.259029 + 0.965869i \(0.416597\pi\)
\(398\) 13182.2 1.66021
\(399\) 701.392 0.0880038
\(400\) 0 0
\(401\) −1046.81 −0.130362 −0.0651811 0.997873i \(-0.520762\pi\)
−0.0651811 + 0.997873i \(0.520762\pi\)
\(402\) 5610.66 0.696105
\(403\) 7751.68 0.958161
\(404\) 32775.1 4.03619
\(405\) 0 0
\(406\) −7364.57 −0.900240
\(407\) −555.420 −0.0676441
\(408\) −16372.8 −1.98670
\(409\) 6516.92 0.787876 0.393938 0.919137i \(-0.371113\pi\)
0.393938 + 0.919137i \(0.371113\pi\)
\(410\) 0 0
\(411\) 4755.20 0.570697
\(412\) −9956.86 −1.19063
\(413\) 1793.51 0.213687
\(414\) −10383.5 −1.23267
\(415\) 0 0
\(416\) 33901.3 3.99555
\(417\) −3896.55 −0.457590
\(418\) −6244.21 −0.730656
\(419\) 12279.1 1.43168 0.715838 0.698267i \(-0.246045\pi\)
0.715838 + 0.698267i \(0.246045\pi\)
\(420\) 0 0
\(421\) 10146.9 1.17465 0.587325 0.809351i \(-0.300181\pi\)
0.587325 + 0.809351i \(0.300181\pi\)
\(422\) −3821.67 −0.440844
\(423\) −3317.83 −0.381367
\(424\) −38046.1 −4.35774
\(425\) 0 0
\(426\) −7563.36 −0.860202
\(427\) −864.579 −0.0979858
\(428\) −15279.0 −1.72556
\(429\) −3867.89 −0.435299
\(430\) 0 0
\(431\) 7059.04 0.788914 0.394457 0.918914i \(-0.370933\pi\)
0.394457 + 0.918914i \(0.370933\pi\)
\(432\) 7516.20 0.837091
\(433\) −6468.98 −0.717966 −0.358983 0.933344i \(-0.616876\pi\)
−0.358983 + 0.933344i \(0.616876\pi\)
\(434\) 7868.24 0.870248
\(435\) 0 0
\(436\) −27505.8 −3.02131
\(437\) −6928.65 −0.758449
\(438\) −367.790 −0.0401226
\(439\) 4767.13 0.518275 0.259137 0.965840i \(-0.416562\pi\)
0.259137 + 0.965840i \(0.416562\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) −14019.4 −1.50868
\(443\) −2366.55 −0.253810 −0.126905 0.991915i \(-0.540504\pi\)
−0.126905 + 0.991915i \(0.540504\pi\)
\(444\) 1136.64 0.121492
\(445\) 0 0
\(446\) −5508.70 −0.584853
\(447\) −6009.86 −0.635921
\(448\) 18821.9 1.98494
\(449\) 1814.17 0.190681 0.0953406 0.995445i \(-0.469606\pi\)
0.0953406 + 0.995445i \(0.469606\pi\)
\(450\) 0 0
\(451\) −13063.6 −1.36395
\(452\) 1869.26 0.194519
\(453\) 8222.87 0.852857
\(454\) −8226.93 −0.850460
\(455\) 0 0
\(456\) 8320.40 0.854470
\(457\) −8284.13 −0.847955 −0.423977 0.905673i \(-0.639366\pi\)
−0.423977 + 0.905673i \(0.639366\pi\)
\(458\) 37287.3 3.80419
\(459\) −1774.53 −0.180453
\(460\) 0 0
\(461\) 1384.62 0.139888 0.0699439 0.997551i \(-0.477718\pi\)
0.0699439 + 0.997551i \(0.477718\pi\)
\(462\) −3926.05 −0.395360
\(463\) 13210.3 1.32599 0.662994 0.748624i \(-0.269285\pi\)
0.662994 + 0.748624i \(0.269285\pi\)
\(464\) −52660.9 −5.26879
\(465\) 0 0
\(466\) 9727.67 0.967008
\(467\) 4574.24 0.453256 0.226628 0.973981i \(-0.427230\pi\)
0.226628 + 0.973981i \(0.427230\pi\)
\(468\) 7915.45 0.781820
\(469\) 2353.94 0.231758
\(470\) 0 0
\(471\) −10932.7 −1.06953
\(472\) 21275.9 2.07479
\(473\) 1405.61 0.136639
\(474\) 6430.55 0.623132
\(475\) 0 0
\(476\) −10549.7 −1.01585
\(477\) −4123.55 −0.395816
\(478\) −35154.0 −3.36382
\(479\) −11031.8 −1.05231 −0.526154 0.850389i \(-0.676366\pi\)
−0.526154 + 0.850389i \(0.676366\pi\)
\(480\) 0 0
\(481\) 633.716 0.0600726
\(482\) 18684.7 1.76569
\(483\) −4356.39 −0.410398
\(484\) −4609.02 −0.432853
\(485\) 0 0
\(486\) 1351.46 0.126139
\(487\) 5194.06 0.483296 0.241648 0.970364i \(-0.422312\pi\)
0.241648 + 0.970364i \(0.422312\pi\)
\(488\) −10256.2 −0.951389
\(489\) −8322.81 −0.769673
\(490\) 0 0
\(491\) 11954.7 1.09880 0.549398 0.835561i \(-0.314857\pi\)
0.549398 + 0.835561i \(0.314857\pi\)
\(492\) 26734.0 2.44972
\(493\) 12432.9 1.13580
\(494\) 7124.43 0.648873
\(495\) 0 0
\(496\) 56262.4 5.09326
\(497\) −3173.19 −0.286392
\(498\) −396.262 −0.0356564
\(499\) 2566.05 0.230205 0.115102 0.993354i \(-0.463280\pi\)
0.115102 + 0.993354i \(0.463280\pi\)
\(500\) 0 0
\(501\) −3464.74 −0.308968
\(502\) −7399.62 −0.657891
\(503\) −21103.5 −1.87069 −0.935347 0.353731i \(-0.884913\pi\)
−0.935347 + 0.353731i \(0.884913\pi\)
\(504\) 5231.45 0.462355
\(505\) 0 0
\(506\) 38783.1 3.40735
\(507\) −2177.87 −0.190775
\(508\) 7727.36 0.674894
\(509\) −781.732 −0.0680740 −0.0340370 0.999421i \(-0.510836\pi\)
−0.0340370 + 0.999421i \(0.510836\pi\)
\(510\) 0 0
\(511\) −154.305 −0.0133582
\(512\) 61129.5 5.27650
\(513\) 901.790 0.0776121
\(514\) 13775.7 1.18214
\(515\) 0 0
\(516\) −2876.52 −0.245411
\(517\) 12392.3 1.05418
\(518\) 643.244 0.0545609
\(519\) 10162.4 0.859497
\(520\) 0 0
\(521\) 14013.0 1.17835 0.589176 0.808005i \(-0.299453\pi\)
0.589176 + 0.808005i \(0.299453\pi\)
\(522\) −9468.73 −0.793937
\(523\) 10310.7 0.862052 0.431026 0.902339i \(-0.358152\pi\)
0.431026 + 0.902339i \(0.358152\pi\)
\(524\) 67213.8 5.60353
\(525\) 0 0
\(526\) 28656.3 2.37542
\(527\) −13283.2 −1.09796
\(528\) −28073.5 −2.31390
\(529\) 30867.2 2.53696
\(530\) 0 0
\(531\) 2305.94 0.188455
\(532\) 5361.18 0.436911
\(533\) 14905.1 1.21128
\(534\) −24740.2 −2.00489
\(535\) 0 0
\(536\) 27924.0 2.25025
\(537\) 4809.97 0.386528
\(538\) −6401.16 −0.512962
\(539\) −1647.16 −0.131629
\(540\) 0 0
\(541\) −17562.9 −1.39572 −0.697862 0.716232i \(-0.745865\pi\)
−0.697862 + 0.716232i \(0.745865\pi\)
\(542\) 10223.9 0.810250
\(543\) 1632.66 0.129031
\(544\) −58093.1 −4.57853
\(545\) 0 0
\(546\) 4479.48 0.351107
\(547\) 19889.6 1.55469 0.777347 0.629072i \(-0.216565\pi\)
0.777347 + 0.629072i \(0.216565\pi\)
\(548\) 36346.9 2.83333
\(549\) −1111.60 −0.0864153
\(550\) 0 0
\(551\) −6318.22 −0.488503
\(552\) −51678.5 −3.98475
\(553\) 2697.92 0.207463
\(554\) −3161.45 −0.242450
\(555\) 0 0
\(556\) −29783.8 −2.27179
\(557\) −5579.54 −0.424439 −0.212219 0.977222i \(-0.568069\pi\)
−0.212219 + 0.977222i \(0.568069\pi\)
\(558\) 10116.3 0.767486
\(559\) −1603.76 −0.121345
\(560\) 0 0
\(561\) 6627.99 0.498813
\(562\) 33452.8 2.51089
\(563\) 24463.2 1.83126 0.915630 0.402022i \(-0.131693\pi\)
0.915630 + 0.402022i \(0.131693\pi\)
\(564\) −25360.2 −1.89337
\(565\) 0 0
\(566\) −22167.0 −1.64620
\(567\) 567.000 0.0419961
\(568\) −37642.5 −2.78071
\(569\) −8582.14 −0.632306 −0.316153 0.948708i \(-0.602391\pi\)
−0.316153 + 0.948708i \(0.602391\pi\)
\(570\) 0 0
\(571\) 17580.8 1.28850 0.644248 0.764816i \(-0.277170\pi\)
0.644248 + 0.764816i \(0.277170\pi\)
\(572\) −29564.7 −2.16112
\(573\) 8980.32 0.654727
\(574\) 15129.2 1.10014
\(575\) 0 0
\(576\) 24199.6 1.75055
\(577\) 8692.57 0.627169 0.313585 0.949560i \(-0.398470\pi\)
0.313585 + 0.949560i \(0.398470\pi\)
\(578\) −3300.36 −0.237503
\(579\) −3927.95 −0.281935
\(580\) 0 0
\(581\) −166.250 −0.0118713
\(582\) −867.381 −0.0617768
\(583\) 15401.7 1.09412
\(584\) −1830.47 −0.129701
\(585\) 0 0
\(586\) 13848.3 0.976227
\(587\) −3584.61 −0.252049 −0.126024 0.992027i \(-0.540222\pi\)
−0.126024 + 0.992027i \(0.540222\pi\)
\(588\) 3370.84 0.236413
\(589\) 6750.33 0.472228
\(590\) 0 0
\(591\) 3424.13 0.238325
\(592\) 4599.56 0.319326
\(593\) 21853.6 1.51335 0.756676 0.653790i \(-0.226822\pi\)
0.756676 + 0.653790i \(0.226822\pi\)
\(594\) −5047.77 −0.348674
\(595\) 0 0
\(596\) −45937.1 −3.15714
\(597\) 7110.69 0.487472
\(598\) −44250.2 −3.02596
\(599\) 9090.48 0.620078 0.310039 0.950724i \(-0.399658\pi\)
0.310039 + 0.950724i \(0.399658\pi\)
\(600\) 0 0
\(601\) −19546.1 −1.32663 −0.663314 0.748341i \(-0.730851\pi\)
−0.663314 + 0.748341i \(0.730851\pi\)
\(602\) −1627.87 −0.110211
\(603\) 3026.49 0.204392
\(604\) 62852.6 4.23416
\(605\) 0 0
\(606\) 23847.4 1.59857
\(607\) 15726.0 1.05157 0.525783 0.850619i \(-0.323773\pi\)
0.525783 + 0.850619i \(0.323773\pi\)
\(608\) 29522.0 1.96920
\(609\) −3972.58 −0.264330
\(610\) 0 0
\(611\) −14139.2 −0.936186
\(612\) −13563.9 −0.895894
\(613\) 13572.5 0.894269 0.447135 0.894467i \(-0.352444\pi\)
0.447135 + 0.894467i \(0.352444\pi\)
\(614\) 788.921 0.0518538
\(615\) 0 0
\(616\) −19539.8 −1.27805
\(617\) −17378.5 −1.13393 −0.566964 0.823743i \(-0.691882\pi\)
−0.566964 + 0.823743i \(0.691882\pi\)
\(618\) −7244.68 −0.471560
\(619\) −25113.3 −1.63068 −0.815338 0.578985i \(-0.803449\pi\)
−0.815338 + 0.578985i \(0.803449\pi\)
\(620\) 0 0
\(621\) −5601.07 −0.361937
\(622\) 11634.3 0.749989
\(623\) −10379.7 −0.667501
\(624\) 32030.8 2.05490
\(625\) 0 0
\(626\) −30707.8 −1.96059
\(627\) −3368.24 −0.214537
\(628\) −83565.1 −5.30989
\(629\) −1085.93 −0.0688377
\(630\) 0 0
\(631\) −10814.4 −0.682276 −0.341138 0.940013i \(-0.610812\pi\)
−0.341138 + 0.940013i \(0.610812\pi\)
\(632\) 32004.5 2.01436
\(633\) −2061.48 −0.129441
\(634\) −29763.4 −1.86444
\(635\) 0 0
\(636\) −31518.9 −1.96510
\(637\) 1879.35 0.116896
\(638\) 35366.3 2.19461
\(639\) −4079.81 −0.252574
\(640\) 0 0
\(641\) −16359.0 −1.00802 −0.504010 0.863698i \(-0.668143\pi\)
−0.504010 + 0.863698i \(0.668143\pi\)
\(642\) −11117.1 −0.683422
\(643\) −8819.47 −0.540911 −0.270456 0.962732i \(-0.587174\pi\)
−0.270456 + 0.962732i \(0.587174\pi\)
\(644\) −33298.6 −2.03750
\(645\) 0 0
\(646\) −12208.4 −0.743549
\(647\) 13828.8 0.840290 0.420145 0.907457i \(-0.361979\pi\)
0.420145 + 0.907457i \(0.361979\pi\)
\(648\) 6726.14 0.407759
\(649\) −8612.83 −0.520930
\(650\) 0 0
\(651\) 4244.27 0.255524
\(652\) −63616.4 −3.82118
\(653\) −23988.7 −1.43760 −0.718798 0.695219i \(-0.755307\pi\)
−0.718798 + 0.695219i \(0.755307\pi\)
\(654\) −20013.4 −1.19662
\(655\) 0 0
\(656\) 108182. 6.43874
\(657\) −198.392 −0.0117809
\(658\) −14351.8 −0.850289
\(659\) 3109.28 0.183794 0.0918972 0.995769i \(-0.470707\pi\)
0.0918972 + 0.995769i \(0.470707\pi\)
\(660\) 0 0
\(661\) 22695.0 1.33545 0.667726 0.744407i \(-0.267268\pi\)
0.667726 + 0.744407i \(0.267268\pi\)
\(662\) −24379.2 −1.43131
\(663\) −7562.31 −0.442980
\(664\) −1972.18 −0.115264
\(665\) 0 0
\(666\) 827.028 0.0481182
\(667\) 39242.8 2.27809
\(668\) −26483.2 −1.53393
\(669\) −2971.49 −0.171726
\(670\) 0 0
\(671\) 4151.90 0.238871
\(672\) 18562.0 1.06554
\(673\) −22073.4 −1.26429 −0.632145 0.774850i \(-0.717825\pi\)
−0.632145 + 0.774850i \(0.717825\pi\)
\(674\) 39623.7 2.26446
\(675\) 0 0
\(676\) −16646.9 −0.947136
\(677\) 2489.50 0.141328 0.0706642 0.997500i \(-0.477488\pi\)
0.0706642 + 0.997500i \(0.477488\pi\)
\(678\) 1360.09 0.0770410
\(679\) −363.907 −0.0205677
\(680\) 0 0
\(681\) −4437.75 −0.249714
\(682\) −37785.0 −2.12150
\(683\) −7970.98 −0.446561 −0.223280 0.974754i \(-0.571677\pi\)
−0.223280 + 0.974754i \(0.571677\pi\)
\(684\) 6892.94 0.385319
\(685\) 0 0
\(686\) 1907.61 0.106171
\(687\) 20113.4 1.11699
\(688\) −11640.2 −0.645027
\(689\) −17572.8 −0.971656
\(690\) 0 0
\(691\) −23892.7 −1.31537 −0.657687 0.753292i \(-0.728465\pi\)
−0.657687 + 0.753292i \(0.728465\pi\)
\(692\) 77677.4 4.26712
\(693\) −2117.78 −0.116086
\(694\) −2823.84 −0.154454
\(695\) 0 0
\(696\) −47125.5 −2.56650
\(697\) −25541.3 −1.38801
\(698\) 34232.1 1.85631
\(699\) 5247.28 0.283935
\(700\) 0 0
\(701\) 12197.0 0.657170 0.328585 0.944474i \(-0.393428\pi\)
0.328585 + 0.944474i \(0.393428\pi\)
\(702\) 5759.33 0.309647
\(703\) 551.853 0.0296067
\(704\) −90387.0 −4.83891
\(705\) 0 0
\(706\) 35758.4 1.90621
\(707\) 10005.1 0.532221
\(708\) 17625.8 0.935617
\(709\) −8982.28 −0.475792 −0.237896 0.971291i \(-0.576458\pi\)
−0.237896 + 0.971291i \(0.576458\pi\)
\(710\) 0 0
\(711\) 3468.75 0.182965
\(712\) −123131. −6.48107
\(713\) −41926.7 −2.20220
\(714\) −7676.02 −0.402336
\(715\) 0 0
\(716\) 36765.6 1.91899
\(717\) −18962.7 −0.987690
\(718\) 56035.0 2.91255
\(719\) −6501.61 −0.337231 −0.168616 0.985682i \(-0.553930\pi\)
−0.168616 + 0.985682i \(0.553930\pi\)
\(720\) 0 0
\(721\) −3039.49 −0.156999
\(722\) −31942.6 −1.64651
\(723\) 10078.9 0.518446
\(724\) 12479.4 0.640600
\(725\) 0 0
\(726\) −3353.55 −0.171435
\(727\) 24228.7 1.23603 0.618013 0.786168i \(-0.287938\pi\)
0.618013 + 0.786168i \(0.287938\pi\)
\(728\) 22294.2 1.13500
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 2748.19 0.139050
\(732\) −8496.67 −0.429024
\(733\) −36719.1 −1.85027 −0.925136 0.379636i \(-0.876049\pi\)
−0.925136 + 0.379636i \(0.876049\pi\)
\(734\) −4543.22 −0.228465
\(735\) 0 0
\(736\) −183363. −9.18321
\(737\) −11304.1 −0.564983
\(738\) 19451.8 0.970234
\(739\) 23304.5 1.16004 0.580021 0.814602i \(-0.303044\pi\)
0.580021 + 0.814602i \(0.303044\pi\)
\(740\) 0 0
\(741\) 3843.04 0.190523
\(742\) −17837.0 −0.882505
\(743\) −6875.35 −0.339478 −0.169739 0.985489i \(-0.554292\pi\)
−0.169739 + 0.985489i \(0.554292\pi\)
\(744\) 50348.4 2.48100
\(745\) 0 0
\(746\) 20788.3 1.02026
\(747\) −213.751 −0.0104695
\(748\) 50661.8 2.47644
\(749\) −4664.15 −0.227536
\(750\) 0 0
\(751\) 1182.65 0.0574640 0.0287320 0.999587i \(-0.490853\pi\)
0.0287320 + 0.999587i \(0.490853\pi\)
\(752\) −102623. −4.97645
\(753\) −3991.49 −0.193171
\(754\) −40351.7 −1.94897
\(755\) 0 0
\(756\) 4333.93 0.208497
\(757\) −25226.8 −1.21121 −0.605604 0.795766i \(-0.707068\pi\)
−0.605604 + 0.795766i \(0.707068\pi\)
\(758\) 10847.7 0.519795
\(759\) 20920.3 1.00047
\(760\) 0 0
\(761\) −10909.2 −0.519655 −0.259827 0.965655i \(-0.583666\pi\)
−0.259827 + 0.965655i \(0.583666\pi\)
\(762\) 5622.48 0.267298
\(763\) −8396.58 −0.398397
\(764\) 68642.2 3.25051
\(765\) 0 0
\(766\) −37607.6 −1.77391
\(767\) 9826.95 0.462621
\(768\) 66991.9 3.14761
\(769\) −12771.6 −0.598903 −0.299451 0.954112i \(-0.596804\pi\)
−0.299451 + 0.954112i \(0.596804\pi\)
\(770\) 0 0
\(771\) 7430.84 0.347101
\(772\) −30023.8 −1.39972
\(773\) −2199.06 −0.102322 −0.0511610 0.998690i \(-0.516292\pi\)
−0.0511610 + 0.998690i \(0.516292\pi\)
\(774\) −2092.98 −0.0971971
\(775\) 0 0
\(776\) −4316.92 −0.199701
\(777\) 346.977 0.0160203
\(778\) −14193.0 −0.654041
\(779\) 12979.7 0.596977
\(780\) 0 0
\(781\) 15238.3 0.698170
\(782\) 75827.0 3.46748
\(783\) −5107.60 −0.233117
\(784\) 13640.5 0.621379
\(785\) 0 0
\(786\) 48905.3 2.21933
\(787\) 19587.7 0.887201 0.443601 0.896225i \(-0.353701\pi\)
0.443601 + 0.896225i \(0.353701\pi\)
\(788\) 26172.8 1.18321
\(789\) 15457.7 0.697476
\(790\) 0 0
\(791\) 570.620 0.0256497
\(792\) −25122.5 −1.12713
\(793\) −4737.17 −0.212134
\(794\) 22790.9 1.01866
\(795\) 0 0
\(796\) 54351.4 2.42014
\(797\) −21699.3 −0.964401 −0.482200 0.876061i \(-0.660162\pi\)
−0.482200 + 0.876061i \(0.660162\pi\)
\(798\) 3900.83 0.173042
\(799\) 24228.8 1.07278
\(800\) 0 0
\(801\) −13345.3 −0.588680
\(802\) −5821.89 −0.256332
\(803\) 741.007 0.0325649
\(804\) 23133.3 1.01474
\(805\) 0 0
\(806\) 43111.4 1.88404
\(807\) −3452.90 −0.150617
\(808\) 118687. 5.16758
\(809\) −15649.4 −0.680103 −0.340051 0.940407i \(-0.610444\pi\)
−0.340051 + 0.940407i \(0.610444\pi\)
\(810\) 0 0
\(811\) −33267.3 −1.44041 −0.720206 0.693760i \(-0.755953\pi\)
−0.720206 + 0.693760i \(0.755953\pi\)
\(812\) −30364.9 −1.31231
\(813\) 5514.97 0.237907
\(814\) −3089.00 −0.133009
\(815\) 0 0
\(816\) −54887.9 −2.35473
\(817\) −1396.59 −0.0598046
\(818\) 36244.2 1.54920
\(819\) 2416.31 0.103093
\(820\) 0 0
\(821\) 5158.58 0.219288 0.109644 0.993971i \(-0.465029\pi\)
0.109644 + 0.993971i \(0.465029\pi\)
\(822\) 26446.3 1.12217
\(823\) −26333.6 −1.11535 −0.557674 0.830060i \(-0.688306\pi\)
−0.557674 + 0.830060i \(0.688306\pi\)
\(824\) −36056.5 −1.52438
\(825\) 0 0
\(826\) 9974.71 0.420175
\(827\) −19572.7 −0.822988 −0.411494 0.911413i \(-0.634993\pi\)
−0.411494 + 0.911413i \(0.634993\pi\)
\(828\) −42812.5 −1.79690
\(829\) 9642.26 0.403968 0.201984 0.979389i \(-0.435261\pi\)
0.201984 + 0.979389i \(0.435261\pi\)
\(830\) 0 0
\(831\) −1705.34 −0.0711885
\(832\) 103128. 4.29728
\(833\) −3220.45 −0.133952
\(834\) −21670.9 −0.899761
\(835\) 0 0
\(836\) −25745.5 −1.06511
\(837\) 5456.91 0.225351
\(838\) 68290.7 2.81511
\(839\) −31081.1 −1.27895 −0.639475 0.768812i \(-0.720848\pi\)
−0.639475 + 0.768812i \(0.720848\pi\)
\(840\) 0 0
\(841\) 11396.5 0.467278
\(842\) 56432.3 2.30972
\(843\) 18045.0 0.737252
\(844\) −15757.1 −0.642634
\(845\) 0 0
\(846\) −18452.3 −0.749885
\(847\) −1406.97 −0.0570770
\(848\) −127545. −5.16499
\(849\) −11957.2 −0.483359
\(850\) 0 0
\(851\) −3427.59 −0.138068
\(852\) −31184.5 −1.25395
\(853\) −25780.9 −1.03484 −0.517421 0.855731i \(-0.673108\pi\)
−0.517421 + 0.855731i \(0.673108\pi\)
\(854\) −4808.40 −0.192670
\(855\) 0 0
\(856\) −55329.3 −2.20925
\(857\) −14452.6 −0.576069 −0.288035 0.957620i \(-0.593002\pi\)
−0.288035 + 0.957620i \(0.593002\pi\)
\(858\) −21511.5 −0.855931
\(859\) 889.366 0.0353257 0.0176628 0.999844i \(-0.494377\pi\)
0.0176628 + 0.999844i \(0.494377\pi\)
\(860\) 0 0
\(861\) 8160.97 0.323025
\(862\) 39259.2 1.55125
\(863\) −41460.3 −1.63537 −0.817685 0.575665i \(-0.804743\pi\)
−0.817685 + 0.575665i \(0.804743\pi\)
\(864\) 23865.4 0.939718
\(865\) 0 0
\(866\) −35977.6 −1.41174
\(867\) −1780.27 −0.0697361
\(868\) 32441.6 1.26859
\(869\) −12956.0 −0.505756
\(870\) 0 0
\(871\) 12897.6 0.501744
\(872\) −99606.0 −3.86822
\(873\) −467.880 −0.0181390
\(874\) −38534.1 −1.49134
\(875\) 0 0
\(876\) −1516.44 −0.0584882
\(877\) −21173.0 −0.815236 −0.407618 0.913153i \(-0.633640\pi\)
−0.407618 + 0.913153i \(0.633640\pi\)
\(878\) 26512.6 1.01909
\(879\) 7470.03 0.286642
\(880\) 0 0
\(881\) 9883.39 0.377957 0.188978 0.981981i \(-0.439482\pi\)
0.188978 + 0.981981i \(0.439482\pi\)
\(882\) 2452.64 0.0936336
\(883\) 45273.9 1.72547 0.862734 0.505658i \(-0.168750\pi\)
0.862734 + 0.505658i \(0.168750\pi\)
\(884\) −57803.4 −2.19925
\(885\) 0 0
\(886\) −13161.7 −0.499069
\(887\) −644.388 −0.0243928 −0.0121964 0.999926i \(-0.503882\pi\)
−0.0121964 + 0.999926i \(0.503882\pi\)
\(888\) 4116.08 0.155548
\(889\) 2358.89 0.0889930
\(890\) 0 0
\(891\) −2722.86 −0.102378
\(892\) −22712.9 −0.852562
\(893\) −12312.7 −0.461398
\(894\) −33424.2 −1.25042
\(895\) 0 0
\(896\) 55180.6 2.05743
\(897\) −23869.4 −0.888489
\(898\) 10089.6 0.374937
\(899\) −38232.8 −1.41839
\(900\) 0 0
\(901\) 30112.7 1.11343
\(902\) −72653.8 −2.68194
\(903\) −878.103 −0.0323604
\(904\) 6769.09 0.249045
\(905\) 0 0
\(906\) 45732.0 1.67698
\(907\) 15065.2 0.551522 0.275761 0.961226i \(-0.411070\pi\)
0.275761 + 0.961226i \(0.411070\pi\)
\(908\) −33920.5 −1.23975
\(909\) 12863.7 0.469375
\(910\) 0 0
\(911\) 28789.9 1.04704 0.523520 0.852014i \(-0.324619\pi\)
0.523520 + 0.852014i \(0.324619\pi\)
\(912\) 27893.1 1.01276
\(913\) 798.371 0.0289400
\(914\) −46072.6 −1.66734
\(915\) 0 0
\(916\) 153739. 5.54552
\(917\) 20518.1 0.738894
\(918\) −9869.16 −0.354827
\(919\) −24163.8 −0.867345 −0.433673 0.901070i \(-0.642783\pi\)
−0.433673 + 0.901070i \(0.642783\pi\)
\(920\) 0 0
\(921\) 425.558 0.0152254
\(922\) 7700.65 0.275062
\(923\) −17386.4 −0.620023
\(924\) −16187.5 −0.576331
\(925\) 0 0
\(926\) 73469.5 2.60730
\(927\) −3907.91 −0.138460
\(928\) −167208. −5.91474
\(929\) −35115.4 −1.24015 −0.620075 0.784542i \(-0.712898\pi\)
−0.620075 + 0.784542i \(0.712898\pi\)
\(930\) 0 0
\(931\) 1636.58 0.0576120
\(932\) 40108.2 1.40964
\(933\) 6275.75 0.220213
\(934\) 25439.9 0.891239
\(935\) 0 0
\(936\) 28664.0 1.00097
\(937\) 15512.6 0.540849 0.270424 0.962741i \(-0.412836\pi\)
0.270424 + 0.962741i \(0.412836\pi\)
\(938\) 13091.5 0.455708
\(939\) −16564.3 −0.575672
\(940\) 0 0
\(941\) −53283.8 −1.84591 −0.922956 0.384905i \(-0.874234\pi\)
−0.922956 + 0.384905i \(0.874234\pi\)
\(942\) −60802.5 −2.10303
\(943\) −80617.5 −2.78395
\(944\) 71324.8 2.45914
\(945\) 0 0
\(946\) 7817.39 0.268674
\(947\) 55509.7 1.90478 0.952388 0.304890i \(-0.0986196\pi\)
0.952388 + 0.304890i \(0.0986196\pi\)
\(948\) 26513.8 0.908364
\(949\) −845.464 −0.0289198
\(950\) 0 0
\(951\) −16054.9 −0.547441
\(952\) −38203.2 −1.30060
\(953\) 28080.6 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(954\) −22933.3 −0.778296
\(955\) 0 0
\(956\) −144943. −4.90356
\(957\) 19077.2 0.644387
\(958\) −61353.9 −2.06916
\(959\) 11095.5 0.373609
\(960\) 0 0
\(961\) 11056.6 0.371140
\(962\) 3524.44 0.118121
\(963\) −5996.76 −0.200668
\(964\) 77039.0 2.57392
\(965\) 0 0
\(966\) −24228.3 −0.806969
\(967\) −56609.3 −1.88256 −0.941278 0.337634i \(-0.890374\pi\)
−0.941278 + 0.337634i \(0.890374\pi\)
\(968\) −16690.5 −0.554187
\(969\) −6585.42 −0.218322
\(970\) 0 0
\(971\) −6782.17 −0.224151 −0.112075 0.993700i \(-0.535750\pi\)
−0.112075 + 0.993700i \(0.535750\pi\)
\(972\) 5572.20 0.183877
\(973\) −9091.95 −0.299563
\(974\) 28887.0 0.950309
\(975\) 0 0
\(976\) −34382.8 −1.12763
\(977\) 45655.0 1.49502 0.747509 0.664252i \(-0.231250\pi\)
0.747509 + 0.664252i \(0.231250\pi\)
\(978\) −46287.7 −1.51341
\(979\) 49845.5 1.62724
\(980\) 0 0
\(981\) −10795.6 −0.351353
\(982\) 66486.8 2.16057
\(983\) 10102.3 0.327785 0.163893 0.986478i \(-0.447595\pi\)
0.163893 + 0.986478i \(0.447595\pi\)
\(984\) 96811.0 3.13640
\(985\) 0 0
\(986\) 69146.4 2.23334
\(987\) −7741.60 −0.249664
\(988\) 29374.8 0.945887
\(989\) 8674.27 0.278894
\(990\) 0 0
\(991\) 25416.2 0.814705 0.407353 0.913271i \(-0.366452\pi\)
0.407353 + 0.913271i \(0.366452\pi\)
\(992\) 178644. 5.71768
\(993\) −13150.6 −0.420262
\(994\) −17647.8 −0.563135
\(995\) 0 0
\(996\) −1633.83 −0.0519777
\(997\) −48152.5 −1.52959 −0.764797 0.644271i \(-0.777161\pi\)
−0.764797 + 0.644271i \(0.777161\pi\)
\(998\) 14271.2 0.452653
\(999\) 446.114 0.0141285
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 525.4.a.p.1.2 2
3.2 odd 2 1575.4.a.m.1.1 2
5.2 odd 4 525.4.d.i.274.4 4
5.3 odd 4 525.4.d.i.274.1 4
5.4 even 2 105.4.a.c.1.1 2
15.14 odd 2 315.4.a.m.1.2 2
20.19 odd 2 1680.4.a.bk.1.2 2
35.34 odd 2 735.4.a.k.1.1 2
105.104 even 2 2205.4.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.c.1.1 2 5.4 even 2
315.4.a.m.1.2 2 15.14 odd 2
525.4.a.p.1.2 2 1.1 even 1 trivial
525.4.d.i.274.1 4 5.3 odd 4
525.4.d.i.274.4 4 5.2 odd 4
735.4.a.k.1.1 2 35.34 odd 2
1575.4.a.m.1.1 2 3.2 odd 2
1680.4.a.bk.1.2 2 20.19 odd 2
2205.4.a.bh.1.2 2 105.104 even 2