Properties

Label 175.4.a.c.1.2
Level $175$
Weight $4$
Character 175.1
Self dual yes
Analytic conductor $10.325$
Analytic rank $1$
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] = [175,4,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, 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("175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(10.3253342510\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +17.2132 q^{6} +7.00000 q^{7} +24.0833 q^{8} +17.3137 q^{9} +O(q^{10})\) \(q-2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +17.2132 q^{6} +7.00000 q^{7} +24.0833 q^{8} +17.3137 q^{9} +38.2548 q^{11} +8.74517 q^{12} -19.3431 q^{13} -18.1005 q^{14} -51.7645 q^{16} +87.2254 q^{17} -44.7696 q^{18} -44.2254 q^{19} -46.5980 q^{21} -98.9188 q^{22} -218.167 q^{23} -160.319 q^{24} +50.0172 q^{26} +64.4802 q^{27} -9.19596 q^{28} -46.9411 q^{29} +194.558 q^{31} -58.8141 q^{32} -254.657 q^{33} -225.546 q^{34} -22.7452 q^{36} -366.853 q^{37} +114.357 q^{38} +128.765 q^{39} -339.362 q^{41} +120.492 q^{42} +226.167 q^{43} -50.2557 q^{44} +564.132 q^{46} -11.6762 q^{47} +344.589 q^{48} +49.0000 q^{49} -580.647 q^{51} +25.4113 q^{52} +209.019 q^{53} -166.732 q^{54} +168.583 q^{56} +294.402 q^{57} +121.380 q^{58} -616.000 q^{59} +320.735 q^{61} -503.087 q^{62} +121.196 q^{63} +566.197 q^{64} +658.488 q^{66} -14.5097 q^{67} -114.589 q^{68} +1452.30 q^{69} -952.000 q^{71} +416.971 q^{72} -824.489 q^{73} +948.603 q^{74} +58.0993 q^{76} +267.784 q^{77} -332.958 q^{78} +156.275 q^{79} -896.706 q^{81} +877.519 q^{82} +1036.53 q^{83} +61.2162 q^{84} -584.818 q^{86} +312.480 q^{87} +921.301 q^{88} -170.225 q^{89} -135.402 q^{91} +286.607 q^{92} -1295.15 q^{93} +30.1921 q^{94} +391.517 q^{96} -1059.87 q^{97} -126.704 q^{98} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 56 q^{14} + 168 q^{16} + 50 q^{17} - 16 q^{18} + 36 q^{19} - 14 q^{21} + 184 q^{22} - 244 q^{23} - 496 q^{24} + 216 q^{26} - 86 q^{27} + 140 q^{28} - 26 q^{29} - 120 q^{31} - 672 q^{32} - 498 q^{33} - 24 q^{34} - 136 q^{36} - 564 q^{37} - 320 q^{38} - 14 q^{39} - 328 q^{41} - 56 q^{42} + 260 q^{43} - 1164 q^{44} + 704 q^{46} + 350 q^{47} + 1368 q^{48} + 98 q^{49} - 754 q^{51} - 628 q^{52} + 56 q^{53} + 648 q^{54} - 336 q^{56} + 668 q^{57} + 8 q^{58} - 1232 q^{59} + 336 q^{61} + 1200 q^{62} + 84 q^{63} + 2128 q^{64} + 1976 q^{66} + 152 q^{67} - 908 q^{68} + 1332 q^{69} - 1904 q^{71} + 800 q^{72} - 676 q^{73} + 2016 q^{74} + 1768 q^{76} - 98 q^{77} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} + 816 q^{82} + 376 q^{83} + 756 q^{84} - 768 q^{86} + 410 q^{87} + 4688 q^{88} - 216 q^{89} - 350 q^{91} - 264 q^{92} - 2760 q^{93} - 1928 q^{94} - 2464 q^{96} - 2742 q^{97} - 392 q^{98} + 940 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58579 −0.914214 −0.457107 0.889412i \(-0.651114\pi\)
−0.457107 + 0.889412i \(0.651114\pi\)
\(3\) −6.65685 −1.28111 −0.640556 0.767911i \(-0.721296\pi\)
−0.640556 + 0.767911i \(0.721296\pi\)
\(4\) −1.31371 −0.164214
\(5\) 0 0
\(6\) 17.2132 1.17121
\(7\) 7.00000 0.377964
\(8\) 24.0833 1.06434
\(9\) 17.3137 0.641248
\(10\) 0 0
\(11\) 38.2548 1.04857 0.524285 0.851543i \(-0.324333\pi\)
0.524285 + 0.851543i \(0.324333\pi\)
\(12\) 8.74517 0.210376
\(13\) −19.3431 −0.412679 −0.206339 0.978480i \(-0.566155\pi\)
−0.206339 + 0.978480i \(0.566155\pi\)
\(14\) −18.1005 −0.345540
\(15\) 0 0
\(16\) −51.7645 −0.808820
\(17\) 87.2254 1.24443 0.622214 0.782847i \(-0.286233\pi\)
0.622214 + 0.782847i \(0.286233\pi\)
\(18\) −44.7696 −0.586238
\(19\) −44.2254 −0.534000 −0.267000 0.963697i \(-0.586032\pi\)
−0.267000 + 0.963697i \(0.586032\pi\)
\(20\) 0 0
\(21\) −46.5980 −0.484215
\(22\) −98.9188 −0.958617
\(23\) −218.167 −1.97786 −0.988932 0.148371i \(-0.952597\pi\)
−0.988932 + 0.148371i \(0.952597\pi\)
\(24\) −160.319 −1.36354
\(25\) 0 0
\(26\) 50.0172 0.377276
\(27\) 64.4802 0.459601
\(28\) −9.19596 −0.0620669
\(29\) −46.9411 −0.300578 −0.150289 0.988642i \(-0.548020\pi\)
−0.150289 + 0.988642i \(0.548020\pi\)
\(30\) 0 0
\(31\) 194.558 1.12722 0.563609 0.826042i \(-0.309413\pi\)
0.563609 + 0.826042i \(0.309413\pi\)
\(32\) −58.8141 −0.324905
\(33\) −254.657 −1.34334
\(34\) −225.546 −1.13767
\(35\) 0 0
\(36\) −22.7452 −0.105302
\(37\) −366.853 −1.63001 −0.815003 0.579457i \(-0.803265\pi\)
−0.815003 + 0.579457i \(0.803265\pi\)
\(38\) 114.357 0.488190
\(39\) 128.765 0.528688
\(40\) 0 0
\(41\) −339.362 −1.29267 −0.646336 0.763053i \(-0.723699\pi\)
−0.646336 + 0.763053i \(0.723699\pi\)
\(42\) 120.492 0.442676
\(43\) 226.167 0.802095 0.401047 0.916057i \(-0.368646\pi\)
0.401047 + 0.916057i \(0.368646\pi\)
\(44\) −50.2557 −0.172189
\(45\) 0 0
\(46\) 564.132 1.80819
\(47\) −11.6762 −0.0362372 −0.0181186 0.999836i \(-0.505768\pi\)
−0.0181186 + 0.999836i \(0.505768\pi\)
\(48\) 344.589 1.03619
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −580.647 −1.59425
\(52\) 25.4113 0.0677674
\(53\) 209.019 0.541717 0.270859 0.962619i \(-0.412692\pi\)
0.270859 + 0.962619i \(0.412692\pi\)
\(54\) −166.732 −0.420173
\(55\) 0 0
\(56\) 168.583 0.402283
\(57\) 294.402 0.684114
\(58\) 121.380 0.274792
\(59\) −616.000 −1.35926 −0.679630 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 0 0
\(61\) 320.735 0.673212 0.336606 0.941646i \(-0.390721\pi\)
0.336606 + 0.941646i \(0.390721\pi\)
\(62\) −503.087 −1.03052
\(63\) 121.196 0.242369
\(64\) 566.197 1.10585
\(65\) 0 0
\(66\) 658.488 1.22810
\(67\) −14.5097 −0.0264573 −0.0132286 0.999912i \(-0.504211\pi\)
−0.0132286 + 0.999912i \(0.504211\pi\)
\(68\) −114.589 −0.204352
\(69\) 1452.30 2.53387
\(70\) 0 0
\(71\) −952.000 −1.59129 −0.795645 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) 416.971 0.682506
\(73\) −824.489 −1.32191 −0.660953 0.750427i \(-0.729848\pi\)
−0.660953 + 0.750427i \(0.729848\pi\)
\(74\) 948.603 1.49017
\(75\) 0 0
\(76\) 58.0993 0.0876901
\(77\) 267.784 0.396322
\(78\) −332.958 −0.483334
\(79\) 156.275 0.222561 0.111280 0.993789i \(-0.464505\pi\)
0.111280 + 0.993789i \(0.464505\pi\)
\(80\) 0 0
\(81\) −896.706 −1.23005
\(82\) 877.519 1.18178
\(83\) 1036.53 1.37077 0.685384 0.728182i \(-0.259634\pi\)
0.685384 + 0.728182i \(0.259634\pi\)
\(84\) 61.2162 0.0795147
\(85\) 0 0
\(86\) −584.818 −0.733286
\(87\) 312.480 0.385074
\(88\) 921.301 1.11603
\(89\) −170.225 −0.202740 −0.101370 0.994849i \(-0.532323\pi\)
−0.101370 + 0.994849i \(0.532323\pi\)
\(90\) 0 0
\(91\) −135.402 −0.155978
\(92\) 286.607 0.324792
\(93\) −1295.15 −1.44409
\(94\) 30.1921 0.0331285
\(95\) 0 0
\(96\) 391.517 0.416240
\(97\) −1059.87 −1.10942 −0.554710 0.832044i \(-0.687171\pi\)
−0.554710 + 0.832044i \(0.687171\pi\)
\(98\) −126.704 −0.130602
\(99\) 662.333 0.672394
\(100\) 0 0
\(101\) −241.833 −0.238251 −0.119125 0.992879i \(-0.538009\pi\)
−0.119125 + 0.992879i \(0.538009\pi\)
\(102\) 1501.43 1.45749
\(103\) 1679.58 1.60673 0.803367 0.595484i \(-0.203040\pi\)
0.803367 + 0.595484i \(0.203040\pi\)
\(104\) −465.846 −0.439230
\(105\) 0 0
\(106\) −540.479 −0.495245
\(107\) −1506.88 −1.36146 −0.680728 0.732537i \(-0.738336\pi\)
−0.680728 + 0.732537i \(0.738336\pi\)
\(108\) −84.7082 −0.0754727
\(109\) −1252.41 −1.10054 −0.550271 0.834986i \(-0.685476\pi\)
−0.550271 + 0.834986i \(0.685476\pi\)
\(110\) 0 0
\(111\) 2442.09 2.08822
\(112\) −362.352 −0.305705
\(113\) −1370.20 −1.14069 −0.570345 0.821405i \(-0.693190\pi\)
−0.570345 + 0.821405i \(0.693190\pi\)
\(114\) −761.261 −0.625426
\(115\) 0 0
\(116\) 61.6670 0.0493589
\(117\) −334.902 −0.264630
\(118\) 1592.84 1.24265
\(119\) 610.578 0.470349
\(120\) 0 0
\(121\) 132.432 0.0994984
\(122\) −829.352 −0.615459
\(123\) 2259.09 1.65606
\(124\) −255.593 −0.185104
\(125\) 0 0
\(126\) −313.387 −0.221577
\(127\) −1213.49 −0.847873 −0.423936 0.905692i \(-0.639352\pi\)
−0.423936 + 0.905692i \(0.639352\pi\)
\(128\) −993.551 −0.686081
\(129\) −1505.56 −1.02757
\(130\) 0 0
\(131\) −1982.42 −1.32217 −0.661087 0.750309i \(-0.729904\pi\)
−0.661087 + 0.750309i \(0.729904\pi\)
\(132\) 334.545 0.220594
\(133\) −309.578 −0.201833
\(134\) 37.5189 0.0241876
\(135\) 0 0
\(136\) 2100.67 1.32449
\(137\) −2210.95 −1.37879 −0.689394 0.724386i \(-0.742123\pi\)
−0.689394 + 0.724386i \(0.742123\pi\)
\(138\) −3755.34 −2.31649
\(139\) 528.039 0.322213 0.161107 0.986937i \(-0.448494\pi\)
0.161107 + 0.986937i \(0.448494\pi\)
\(140\) 0 0
\(141\) 77.7267 0.0464239
\(142\) 2461.67 1.45478
\(143\) −739.969 −0.432722
\(144\) −896.235 −0.518655
\(145\) 0 0
\(146\) 2131.95 1.20851
\(147\) −326.186 −0.183016
\(148\) 481.938 0.267669
\(149\) −328.372 −0.180545 −0.0902727 0.995917i \(-0.528774\pi\)
−0.0902727 + 0.995917i \(0.528774\pi\)
\(150\) 0 0
\(151\) 1029.43 0.554793 0.277396 0.960756i \(-0.410528\pi\)
0.277396 + 0.960756i \(0.410528\pi\)
\(152\) −1065.09 −0.568358
\(153\) 1510.20 0.797987
\(154\) −692.432 −0.362323
\(155\) 0 0
\(156\) −169.159 −0.0868177
\(157\) −525.098 −0.266926 −0.133463 0.991054i \(-0.542610\pi\)
−0.133463 + 0.991054i \(0.542610\pi\)
\(158\) −404.094 −0.203468
\(159\) −1391.41 −0.694001
\(160\) 0 0
\(161\) −1527.17 −0.747562
\(162\) 2318.69 1.12453
\(163\) −1002.63 −0.481790 −0.240895 0.970551i \(-0.577441\pi\)
−0.240895 + 0.970551i \(0.577441\pi\)
\(164\) 445.823 0.212274
\(165\) 0 0
\(166\) −2680.24 −1.25317
\(167\) 359.422 0.166544 0.0832722 0.996527i \(-0.473463\pi\)
0.0832722 + 0.996527i \(0.473463\pi\)
\(168\) −1122.23 −0.515369
\(169\) −1822.84 −0.829696
\(170\) 0 0
\(171\) −765.706 −0.342427
\(172\) −297.117 −0.131715
\(173\) −3293.65 −1.44747 −0.723733 0.690080i \(-0.757575\pi\)
−0.723733 + 0.690080i \(0.757575\pi\)
\(174\) −808.007 −0.352039
\(175\) 0 0
\(176\) −1980.24 −0.848104
\(177\) 4100.62 1.74137
\(178\) 440.167 0.185348
\(179\) 2978.82 1.24384 0.621921 0.783080i \(-0.286353\pi\)
0.621921 + 0.783080i \(0.286353\pi\)
\(180\) 0 0
\(181\) 1462.31 0.600514 0.300257 0.953858i \(-0.402928\pi\)
0.300257 + 0.953858i \(0.402928\pi\)
\(182\) 350.121 0.142597
\(183\) −2135.09 −0.862460
\(184\) −5254.16 −2.10512
\(185\) 0 0
\(186\) 3348.97 1.32021
\(187\) 3336.79 1.30487
\(188\) 15.3391 0.00595064
\(189\) 451.362 0.173713
\(190\) 0 0
\(191\) −374.923 −0.142034 −0.0710169 0.997475i \(-0.522624\pi\)
−0.0710169 + 0.997475i \(0.522624\pi\)
\(192\) −3769.09 −1.41672
\(193\) −733.028 −0.273391 −0.136696 0.990613i \(-0.543648\pi\)
−0.136696 + 0.990613i \(0.543648\pi\)
\(194\) 2740.61 1.01425
\(195\) 0 0
\(196\) −64.3717 −0.0234591
\(197\) 2093.24 0.757043 0.378521 0.925593i \(-0.376433\pi\)
0.378521 + 0.925593i \(0.376433\pi\)
\(198\) −1712.65 −0.614711
\(199\) 2865.04 1.02059 0.510295 0.860000i \(-0.329536\pi\)
0.510295 + 0.860000i \(0.329536\pi\)
\(200\) 0 0
\(201\) 96.5887 0.0338948
\(202\) 625.330 0.217812
\(203\) −328.588 −0.113608
\(204\) 762.801 0.261798
\(205\) 0 0
\(206\) −4343.03 −1.46890
\(207\) −3777.27 −1.26830
\(208\) 1001.29 0.333783
\(209\) −1691.84 −0.559936
\(210\) 0 0
\(211\) 5643.65 1.84135 0.920674 0.390331i \(-0.127640\pi\)
0.920674 + 0.390331i \(0.127640\pi\)
\(212\) −274.590 −0.0889573
\(213\) 6337.33 2.03862
\(214\) 3896.47 1.24466
\(215\) 0 0
\(216\) 1552.89 0.489172
\(217\) 1361.91 0.426048
\(218\) 3238.46 1.00613
\(219\) 5488.51 1.69351
\(220\) 0 0
\(221\) −1687.21 −0.513549
\(222\) −6314.71 −1.90908
\(223\) 6369.16 1.91260 0.956302 0.292381i \(-0.0944477\pi\)
0.956302 + 0.292381i \(0.0944477\pi\)
\(224\) −411.699 −0.122803
\(225\) 0 0
\(226\) 3543.05 1.04283
\(227\) 1015.67 0.296972 0.148486 0.988914i \(-0.452560\pi\)
0.148486 + 0.988914i \(0.452560\pi\)
\(228\) −386.758 −0.112341
\(229\) 4108.35 1.18554 0.592768 0.805373i \(-0.298035\pi\)
0.592768 + 0.805373i \(0.298035\pi\)
\(230\) 0 0
\(231\) −1782.60 −0.507733
\(232\) −1130.50 −0.319917
\(233\) −608.431 −0.171071 −0.0855357 0.996335i \(-0.527260\pi\)
−0.0855357 + 0.996335i \(0.527260\pi\)
\(234\) 865.984 0.241928
\(235\) 0 0
\(236\) 809.244 0.223209
\(237\) −1040.30 −0.285126
\(238\) −1578.82 −0.430000
\(239\) −5054.44 −1.36797 −0.683985 0.729496i \(-0.739755\pi\)
−0.683985 + 0.729496i \(0.739755\pi\)
\(240\) 0 0
\(241\) 4.86782 0.00130109 0.000650547 1.00000i \(-0.499793\pi\)
0.000650547 1.00000i \(0.499793\pi\)
\(242\) −342.442 −0.0909628
\(243\) 4228.27 1.11623
\(244\) −421.352 −0.110551
\(245\) 0 0
\(246\) −5841.52 −1.51399
\(247\) 855.458 0.220370
\(248\) 4685.60 1.19974
\(249\) −6900.02 −1.75611
\(250\) 0 0
\(251\) −547.921 −0.137787 −0.0688934 0.997624i \(-0.521947\pi\)
−0.0688934 + 0.997624i \(0.521947\pi\)
\(252\) −159.216 −0.0398003
\(253\) −8345.92 −2.07393
\(254\) 3137.83 0.775137
\(255\) 0 0
\(256\) −1960.46 −0.478629
\(257\) 1774.61 0.430729 0.215364 0.976534i \(-0.430906\pi\)
0.215364 + 0.976534i \(0.430906\pi\)
\(258\) 3893.05 0.939421
\(259\) −2567.97 −0.616084
\(260\) 0 0
\(261\) −812.725 −0.192745
\(262\) 5126.11 1.20875
\(263\) 1199.09 0.281138 0.140569 0.990071i \(-0.455107\pi\)
0.140569 + 0.990071i \(0.455107\pi\)
\(264\) −6132.97 −1.42977
\(265\) 0 0
\(266\) 800.502 0.184519
\(267\) 1133.17 0.259733
\(268\) 19.0615 0.00434464
\(269\) 3250.29 0.736706 0.368353 0.929686i \(-0.379922\pi\)
0.368353 + 0.929686i \(0.379922\pi\)
\(270\) 0 0
\(271\) −896.143 −0.200874 −0.100437 0.994943i \(-0.532024\pi\)
−0.100437 + 0.994943i \(0.532024\pi\)
\(272\) −4515.18 −1.00652
\(273\) 901.352 0.199825
\(274\) 5717.04 1.26051
\(275\) 0 0
\(276\) −1907.90 −0.416095
\(277\) 386.562 0.0838492 0.0419246 0.999121i \(-0.486651\pi\)
0.0419246 + 0.999121i \(0.486651\pi\)
\(278\) −1365.40 −0.294572
\(279\) 3368.53 0.722826
\(280\) 0 0
\(281\) −3335.10 −0.708025 −0.354013 0.935241i \(-0.615183\pi\)
−0.354013 + 0.935241i \(0.615183\pi\)
\(282\) −200.985 −0.0424414
\(283\) −5412.26 −1.13684 −0.568419 0.822739i \(-0.692445\pi\)
−0.568419 + 0.822739i \(0.692445\pi\)
\(284\) 1250.65 0.261311
\(285\) 0 0
\(286\) 1913.40 0.395601
\(287\) −2375.54 −0.488584
\(288\) −1018.29 −0.208345
\(289\) 2695.27 0.548600
\(290\) 0 0
\(291\) 7055.42 1.42129
\(292\) 1083.14 0.217075
\(293\) −282.211 −0.0562695 −0.0281347 0.999604i \(-0.508957\pi\)
−0.0281347 + 0.999604i \(0.508957\pi\)
\(294\) 843.447 0.167316
\(295\) 0 0
\(296\) −8835.01 −1.73488
\(297\) 2466.68 0.481924
\(298\) 849.099 0.165057
\(299\) 4220.03 0.816222
\(300\) 0 0
\(301\) 1583.17 0.303163
\(302\) −2661.88 −0.507199
\(303\) 1609.85 0.305226
\(304\) 2289.31 0.431910
\(305\) 0 0
\(306\) −3905.04 −0.729531
\(307\) −1919.67 −0.356878 −0.178439 0.983951i \(-0.557105\pi\)
−0.178439 + 0.983951i \(0.557105\pi\)
\(308\) −351.790 −0.0650815
\(309\) −11180.7 −2.05841
\(310\) 0 0
\(311\) 1213.31 0.221223 0.110612 0.993864i \(-0.464719\pi\)
0.110612 + 0.993864i \(0.464719\pi\)
\(312\) 3101.07 0.562703
\(313\) 1434.00 0.258960 0.129480 0.991582i \(-0.458669\pi\)
0.129480 + 0.991582i \(0.458669\pi\)
\(314\) 1357.79 0.244028
\(315\) 0 0
\(316\) −205.300 −0.0365475
\(317\) −6496.95 −1.15112 −0.575560 0.817760i \(-0.695216\pi\)
−0.575560 + 0.817760i \(0.695216\pi\)
\(318\) 3597.89 0.634465
\(319\) −1795.72 −0.315176
\(320\) 0 0
\(321\) 10031.1 1.74418
\(322\) 3948.92 0.683432
\(323\) −3857.58 −0.664524
\(324\) 1178.01 0.201991
\(325\) 0 0
\(326\) 2592.58 0.440459
\(327\) 8337.11 1.40992
\(328\) −8172.96 −1.37584
\(329\) −81.7333 −0.0136964
\(330\) 0 0
\(331\) −9683.88 −1.60808 −0.804039 0.594576i \(-0.797320\pi\)
−0.804039 + 0.594576i \(0.797320\pi\)
\(332\) −1361.70 −0.225099
\(333\) −6351.58 −1.04524
\(334\) −929.389 −0.152257
\(335\) 0 0
\(336\) 2412.12 0.391643
\(337\) −29.1319 −0.00470895 −0.00235447 0.999997i \(-0.500749\pi\)
−0.00235447 + 0.999997i \(0.500749\pi\)
\(338\) 4713.48 0.758520
\(339\) 9121.24 1.46135
\(340\) 0 0
\(341\) 7442.80 1.18197
\(342\) 1979.95 0.313051
\(343\) 343.000 0.0539949
\(344\) 5446.83 0.853701
\(345\) 0 0
\(346\) 8516.68 1.32329
\(347\) 7848.58 1.21422 0.607110 0.794618i \(-0.292329\pi\)
0.607110 + 0.794618i \(0.292329\pi\)
\(348\) −410.508 −0.0632343
\(349\) −10269.6 −1.57513 −0.787567 0.616229i \(-0.788659\pi\)
−0.787567 + 0.616229i \(0.788659\pi\)
\(350\) 0 0
\(351\) −1247.25 −0.189668
\(352\) −2249.93 −0.340686
\(353\) −2799.93 −0.422168 −0.211084 0.977468i \(-0.567699\pi\)
−0.211084 + 0.977468i \(0.567699\pi\)
\(354\) −10603.3 −1.59198
\(355\) 0 0
\(356\) 223.627 0.0332927
\(357\) −4064.53 −0.602570
\(358\) −7702.60 −1.13714
\(359\) −3163.29 −0.465048 −0.232524 0.972591i \(-0.574698\pi\)
−0.232524 + 0.972591i \(0.574698\pi\)
\(360\) 0 0
\(361\) −4903.11 −0.714844
\(362\) −3781.23 −0.548998
\(363\) −881.583 −0.127469
\(364\) 177.879 0.0256137
\(365\) 0 0
\(366\) 5520.88 0.788472
\(367\) −3182.85 −0.452706 −0.226353 0.974045i \(-0.572680\pi\)
−0.226353 + 0.974045i \(0.572680\pi\)
\(368\) 11293.3 1.59974
\(369\) −5875.62 −0.828923
\(370\) 0 0
\(371\) 1463.14 0.204750
\(372\) 1701.45 0.237139
\(373\) 2615.14 0.363021 0.181510 0.983389i \(-0.441901\pi\)
0.181510 + 0.983389i \(0.441901\pi\)
\(374\) −8628.23 −1.19293
\(375\) 0 0
\(376\) −281.201 −0.0385687
\(377\) 907.989 0.124042
\(378\) −1167.12 −0.158811
\(379\) −672.434 −0.0911362 −0.0455681 0.998961i \(-0.514510\pi\)
−0.0455681 + 0.998961i \(0.514510\pi\)
\(380\) 0 0
\(381\) 8078.03 1.08622
\(382\) 969.470 0.129849
\(383\) −1169.86 −0.156075 −0.0780377 0.996950i \(-0.524865\pi\)
−0.0780377 + 0.996950i \(0.524865\pi\)
\(384\) 6613.92 0.878946
\(385\) 0 0
\(386\) 1895.45 0.249938
\(387\) 3915.78 0.514342
\(388\) 1392.36 0.182182
\(389\) −1122.22 −0.146269 −0.0731347 0.997322i \(-0.523300\pi\)
−0.0731347 + 0.997322i \(0.523300\pi\)
\(390\) 0 0
\(391\) −19029.7 −2.46131
\(392\) 1180.08 0.152049
\(393\) 13196.7 1.69385
\(394\) −5412.68 −0.692099
\(395\) 0 0
\(396\) −870.113 −0.110416
\(397\) 1985.93 0.251060 0.125530 0.992090i \(-0.459937\pi\)
0.125530 + 0.992090i \(0.459937\pi\)
\(398\) −7408.38 −0.933037
\(399\) 2060.81 0.258571
\(400\) 0 0
\(401\) −4172.38 −0.519597 −0.259799 0.965663i \(-0.583656\pi\)
−0.259799 + 0.965663i \(0.583656\pi\)
\(402\) −249.758 −0.0309870
\(403\) −3763.37 −0.465178
\(404\) 317.699 0.0391240
\(405\) 0 0
\(406\) 849.658 0.103862
\(407\) −14033.9 −1.70918
\(408\) −13983.9 −1.69682
\(409\) −11700.8 −1.41459 −0.707295 0.706919i \(-0.750085\pi\)
−0.707295 + 0.706919i \(0.750085\pi\)
\(410\) 0 0
\(411\) 14718.0 1.76638
\(412\) −2206.47 −0.263848
\(413\) −4312.00 −0.513752
\(414\) 9767.22 1.15950
\(415\) 0 0
\(416\) 1137.65 0.134082
\(417\) −3515.08 −0.412791
\(418\) 4374.72 0.511901
\(419\) 2733.20 0.318677 0.159339 0.987224i \(-0.449064\pi\)
0.159339 + 0.987224i \(0.449064\pi\)
\(420\) 0 0
\(421\) 13549.4 1.56854 0.784272 0.620417i \(-0.213037\pi\)
0.784272 + 0.620417i \(0.213037\pi\)
\(422\) −14593.3 −1.68339
\(423\) −202.158 −0.0232370
\(424\) 5033.87 0.576571
\(425\) 0 0
\(426\) −16387.0 −1.86374
\(427\) 2245.15 0.254450
\(428\) 1979.60 0.223569
\(429\) 4925.86 0.554366
\(430\) 0 0
\(431\) −6429.25 −0.718530 −0.359265 0.933236i \(-0.616973\pi\)
−0.359265 + 0.933236i \(0.616973\pi\)
\(432\) −3337.79 −0.371735
\(433\) −8022.03 −0.890333 −0.445166 0.895448i \(-0.646855\pi\)
−0.445166 + 0.895448i \(0.646855\pi\)
\(434\) −3521.61 −0.389499
\(435\) 0 0
\(436\) 1645.30 0.180724
\(437\) 9648.50 1.05618
\(438\) −14192.1 −1.54823
\(439\) −5569.88 −0.605549 −0.302774 0.953062i \(-0.597913\pi\)
−0.302774 + 0.953062i \(0.597913\pi\)
\(440\) 0 0
\(441\) 848.372 0.0916069
\(442\) 4362.77 0.469493
\(443\) 5486.21 0.588392 0.294196 0.955745i \(-0.404948\pi\)
0.294196 + 0.955745i \(0.404948\pi\)
\(444\) −3208.19 −0.342914
\(445\) 0 0
\(446\) −16469.3 −1.74853
\(447\) 2185.92 0.231299
\(448\) 3963.38 0.417973
\(449\) −7232.67 −0.760203 −0.380101 0.924945i \(-0.624111\pi\)
−0.380101 + 0.924945i \(0.624111\pi\)
\(450\) 0 0
\(451\) −12982.3 −1.35546
\(452\) 1800.05 0.187317
\(453\) −6852.76 −0.710752
\(454\) −2626.32 −0.271496
\(455\) 0 0
\(456\) 7090.16 0.728130
\(457\) 2900.51 0.296893 0.148446 0.988920i \(-0.452573\pi\)
0.148446 + 0.988920i \(0.452573\pi\)
\(458\) −10623.3 −1.08383
\(459\) 5624.31 0.571940
\(460\) 0 0
\(461\) 6073.57 0.613611 0.306805 0.951772i \(-0.400740\pi\)
0.306805 + 0.951772i \(0.400740\pi\)
\(462\) 4609.42 0.464176
\(463\) 18922.8 1.89939 0.949693 0.313183i \(-0.101395\pi\)
0.949693 + 0.313183i \(0.101395\pi\)
\(464\) 2429.88 0.243113
\(465\) 0 0
\(466\) 1573.27 0.156396
\(467\) 6776.71 0.671496 0.335748 0.941952i \(-0.391011\pi\)
0.335748 + 0.941952i \(0.391011\pi\)
\(468\) 439.963 0.0434558
\(469\) −101.568 −0.00999991
\(470\) 0 0
\(471\) 3495.50 0.341962
\(472\) −14835.3 −1.44672
\(473\) 8651.96 0.841052
\(474\) 2689.99 0.260666
\(475\) 0 0
\(476\) −802.121 −0.0772377
\(477\) 3618.90 0.347375
\(478\) 13069.7 1.25062
\(479\) 2397.32 0.228677 0.114338 0.993442i \(-0.463525\pi\)
0.114338 + 0.993442i \(0.463525\pi\)
\(480\) 0 0
\(481\) 7096.09 0.672669
\(482\) −12.5871 −0.00118948
\(483\) 10166.1 0.957711
\(484\) −173.977 −0.0163390
\(485\) 0 0
\(486\) −10933.4 −1.02047
\(487\) −5586.17 −0.519781 −0.259890 0.965638i \(-0.583686\pi\)
−0.259890 + 0.965638i \(0.583686\pi\)
\(488\) 7724.35 0.716526
\(489\) 6674.33 0.617227
\(490\) 0 0
\(491\) 537.392 0.0493934 0.0246967 0.999695i \(-0.492138\pi\)
0.0246967 + 0.999695i \(0.492138\pi\)
\(492\) −2967.78 −0.271947
\(493\) −4094.46 −0.374047
\(494\) −2212.03 −0.201466
\(495\) 0 0
\(496\) −10071.2 −0.911716
\(497\) −6664.00 −0.601451
\(498\) 17842.0 1.60546
\(499\) 598.965 0.0537342 0.0268671 0.999639i \(-0.491447\pi\)
0.0268671 + 0.999639i \(0.491447\pi\)
\(500\) 0 0
\(501\) −2392.62 −0.213362
\(502\) 1416.81 0.125966
\(503\) −4426.76 −0.392405 −0.196202 0.980563i \(-0.562861\pi\)
−0.196202 + 0.980563i \(0.562861\pi\)
\(504\) 2918.79 0.257963
\(505\) 0 0
\(506\) 21580.8 1.89601
\(507\) 12134.4 1.06293
\(508\) 1594.17 0.139232
\(509\) 17727.7 1.54374 0.771872 0.635779i \(-0.219321\pi\)
0.771872 + 0.635779i \(0.219321\pi\)
\(510\) 0 0
\(511\) −5771.43 −0.499634
\(512\) 13017.7 1.12365
\(513\) −2851.66 −0.245427
\(514\) −4588.77 −0.393778
\(515\) 0 0
\(516\) 1977.86 0.168741
\(517\) −446.671 −0.0379972
\(518\) 6640.22 0.563233
\(519\) 21925.4 1.85437
\(520\) 0 0
\(521\) 8662.79 0.728453 0.364226 0.931310i \(-0.381333\pi\)
0.364226 + 0.931310i \(0.381333\pi\)
\(522\) 2101.53 0.176210
\(523\) 7770.40 0.649667 0.324833 0.945771i \(-0.394692\pi\)
0.324833 + 0.945771i \(0.394692\pi\)
\(524\) 2604.32 0.217119
\(525\) 0 0
\(526\) −3100.60 −0.257020
\(527\) 16970.4 1.40274
\(528\) 13182.2 1.08652
\(529\) 35429.6 2.91194
\(530\) 0 0
\(531\) −10665.2 −0.871624
\(532\) 406.695 0.0331437
\(533\) 6564.34 0.533458
\(534\) −2930.12 −0.237451
\(535\) 0 0
\(536\) −349.440 −0.0281595
\(537\) −19829.6 −1.59350
\(538\) −8404.56 −0.673506
\(539\) 1874.49 0.149796
\(540\) 0 0
\(541\) 21641.0 1.71981 0.859906 0.510453i \(-0.170522\pi\)
0.859906 + 0.510453i \(0.170522\pi\)
\(542\) 2317.23 0.183642
\(543\) −9734.41 −0.769325
\(544\) −5130.09 −0.404321
\(545\) 0 0
\(546\) −2330.70 −0.182683
\(547\) −7489.29 −0.585409 −0.292705 0.956203i \(-0.594555\pi\)
−0.292705 + 0.956203i \(0.594555\pi\)
\(548\) 2904.54 0.226416
\(549\) 5553.11 0.431696
\(550\) 0 0
\(551\) 2075.99 0.160508
\(552\) 34976.2 2.69689
\(553\) 1093.93 0.0841201
\(554\) −999.566 −0.0766561
\(555\) 0 0
\(556\) −693.689 −0.0529118
\(557\) −25297.9 −1.92443 −0.962214 0.272295i \(-0.912217\pi\)
−0.962214 + 0.272295i \(0.912217\pi\)
\(558\) −8710.29 −0.660817
\(559\) −4374.77 −0.331007
\(560\) 0 0
\(561\) −22212.5 −1.67168
\(562\) 8623.85 0.647286
\(563\) −15661.3 −1.17237 −0.586186 0.810177i \(-0.699371\pi\)
−0.586186 + 0.810177i \(0.699371\pi\)
\(564\) −102.110 −0.00762343
\(565\) 0 0
\(566\) 13994.9 1.03931
\(567\) −6276.94 −0.464915
\(568\) −22927.3 −1.69367
\(569\) −9982.75 −0.735498 −0.367749 0.929925i \(-0.619871\pi\)
−0.367749 + 0.929925i \(0.619871\pi\)
\(570\) 0 0
\(571\) −11583.6 −0.848966 −0.424483 0.905436i \(-0.639544\pi\)
−0.424483 + 0.905436i \(0.639544\pi\)
\(572\) 972.103 0.0710589
\(573\) 2495.81 0.181961
\(574\) 6142.63 0.446670
\(575\) 0 0
\(576\) 9802.97 0.709127
\(577\) 595.378 0.0429565 0.0214783 0.999769i \(-0.493163\pi\)
0.0214783 + 0.999769i \(0.493163\pi\)
\(578\) −6969.39 −0.501537
\(579\) 4879.66 0.350245
\(580\) 0 0
\(581\) 7255.70 0.518102
\(582\) −18243.8 −1.29936
\(583\) 7996.00 0.568028
\(584\) −19856.4 −1.40696
\(585\) 0 0
\(586\) 729.738 0.0514423
\(587\) 15750.3 1.10747 0.553736 0.832693i \(-0.313202\pi\)
0.553736 + 0.832693i \(0.313202\pi\)
\(588\) 428.513 0.0300537
\(589\) −8604.42 −0.601934
\(590\) 0 0
\(591\) −13934.4 −0.969856
\(592\) 18990.0 1.31838
\(593\) 417.878 0.0289379 0.0144690 0.999895i \(-0.495394\pi\)
0.0144690 + 0.999895i \(0.495394\pi\)
\(594\) −6378.31 −0.440581
\(595\) 0 0
\(596\) 431.385 0.0296480
\(597\) −19072.2 −1.30749
\(598\) −10912.1 −0.746201
\(599\) −19997.3 −1.36406 −0.682028 0.731326i \(-0.738902\pi\)
−0.682028 + 0.731326i \(0.738902\pi\)
\(600\) 0 0
\(601\) −15992.6 −1.08545 −0.542723 0.839912i \(-0.682607\pi\)
−0.542723 + 0.839912i \(0.682607\pi\)
\(602\) −4093.73 −0.277156
\(603\) −251.216 −0.0169657
\(604\) −1352.37 −0.0911045
\(605\) 0 0
\(606\) −4162.73 −0.279042
\(607\) −14159.2 −0.946793 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(608\) 2601.08 0.173499
\(609\) 2187.36 0.145544
\(610\) 0 0
\(611\) 225.854 0.0149543
\(612\) −1983.96 −0.131040
\(613\) 4629.41 0.305025 0.152512 0.988302i \(-0.451264\pi\)
0.152512 + 0.988302i \(0.451264\pi\)
\(614\) 4963.87 0.326263
\(615\) 0 0
\(616\) 6449.11 0.421821
\(617\) 23165.3 1.51151 0.755753 0.654857i \(-0.227271\pi\)
0.755753 + 0.654857i \(0.227271\pi\)
\(618\) 28910.9 1.88182
\(619\) 12370.6 0.803258 0.401629 0.915803i \(-0.368444\pi\)
0.401629 + 0.915803i \(0.368444\pi\)
\(620\) 0 0
\(621\) −14067.4 −0.909028
\(622\) −3137.36 −0.202245
\(623\) −1191.58 −0.0766285
\(624\) −6665.43 −0.427613
\(625\) 0 0
\(626\) −3708.02 −0.236745
\(627\) 11262.3 0.717341
\(628\) 689.826 0.0438329
\(629\) −31998.9 −2.02842
\(630\) 0 0
\(631\) 13980.2 0.882002 0.441001 0.897507i \(-0.354623\pi\)
0.441001 + 0.897507i \(0.354623\pi\)
\(632\) 3763.61 0.236880
\(633\) −37568.9 −2.35897
\(634\) 16799.7 1.05237
\(635\) 0 0
\(636\) 1827.91 0.113964
\(637\) −947.814 −0.0589541
\(638\) 4643.36 0.288139
\(639\) −16482.7 −1.02041
\(640\) 0 0
\(641\) −16060.9 −0.989655 −0.494828 0.868991i \(-0.664769\pi\)
−0.494828 + 0.868991i \(0.664769\pi\)
\(642\) −25938.3 −1.59455
\(643\) 4502.17 0.276125 0.138063 0.990424i \(-0.455913\pi\)
0.138063 + 0.990424i \(0.455913\pi\)
\(644\) 2006.25 0.122760
\(645\) 0 0
\(646\) 9974.87 0.607517
\(647\) −29414.8 −1.78735 −0.893675 0.448715i \(-0.851882\pi\)
−0.893675 + 0.448715i \(0.851882\pi\)
\(648\) −21595.6 −1.30919
\(649\) −23565.0 −1.42528
\(650\) 0 0
\(651\) −9066.03 −0.545815
\(652\) 1317.16 0.0791164
\(653\) −13013.6 −0.779882 −0.389941 0.920840i \(-0.627505\pi\)
−0.389941 + 0.920840i \(0.627505\pi\)
\(654\) −21558.0 −1.28897
\(655\) 0 0
\(656\) 17566.9 1.04554
\(657\) −14275.0 −0.847671
\(658\) 211.345 0.0125214
\(659\) 23474.2 1.38759 0.693797 0.720171i \(-0.255937\pi\)
0.693797 + 0.720171i \(0.255937\pi\)
\(660\) 0 0
\(661\) −9266.36 −0.545264 −0.272632 0.962118i \(-0.587894\pi\)
−0.272632 + 0.962118i \(0.587894\pi\)
\(662\) 25040.4 1.47013
\(663\) 11231.5 0.657914
\(664\) 24963.0 1.45896
\(665\) 0 0
\(666\) 16423.8 0.955572
\(667\) 10241.0 0.594501
\(668\) −472.176 −0.0273489
\(669\) −42398.6 −2.45026
\(670\) 0 0
\(671\) 12269.7 0.705909
\(672\) 2740.62 0.157324
\(673\) 25067.2 1.43576 0.717882 0.696164i \(-0.245112\pi\)
0.717882 + 0.696164i \(0.245112\pi\)
\(674\) 75.3288 0.00430498
\(675\) 0 0
\(676\) 2394.68 0.136247
\(677\) 22409.6 1.27219 0.636093 0.771613i \(-0.280550\pi\)
0.636093 + 0.771613i \(0.280550\pi\)
\(678\) −23585.6 −1.33599
\(679\) −7419.11 −0.419322
\(680\) 0 0
\(681\) −6761.20 −0.380455
\(682\) −19245.5 −1.08057
\(683\) 8757.53 0.490626 0.245313 0.969444i \(-0.421109\pi\)
0.245313 + 0.969444i \(0.421109\pi\)
\(684\) 1005.91 0.0562311
\(685\) 0 0
\(686\) −886.925 −0.0493629
\(687\) −27348.7 −1.51880
\(688\) −11707.4 −0.648750
\(689\) −4043.09 −0.223555
\(690\) 0 0
\(691\) 8468.42 0.466214 0.233107 0.972451i \(-0.425111\pi\)
0.233107 + 0.972451i \(0.425111\pi\)
\(692\) 4326.90 0.237694
\(693\) 4636.33 0.254141
\(694\) −20294.8 −1.11006
\(695\) 0 0
\(696\) 7525.54 0.409849
\(697\) −29601.0 −1.60864
\(698\) 26555.1 1.44001
\(699\) 4050.23 0.219162
\(700\) 0 0
\(701\) 15996.9 0.861906 0.430953 0.902374i \(-0.358177\pi\)
0.430953 + 0.902374i \(0.358177\pi\)
\(702\) 3225.12 0.173397
\(703\) 16224.2 0.870423
\(704\) 21659.8 1.15956
\(705\) 0 0
\(706\) 7240.03 0.385952
\(707\) −1692.83 −0.0900503
\(708\) −5387.02 −0.285956
\(709\) 19903.0 1.05426 0.527131 0.849784i \(-0.323268\pi\)
0.527131 + 0.849784i \(0.323268\pi\)
\(710\) 0 0
\(711\) 2705.70 0.142717
\(712\) −4099.58 −0.215784
\(713\) −42446.1 −2.22948
\(714\) 10510.0 0.550878
\(715\) 0 0
\(716\) −3913.30 −0.204256
\(717\) 33646.7 1.75252
\(718\) 8179.59 0.425153
\(719\) −11073.1 −0.574347 −0.287174 0.957879i \(-0.592716\pi\)
−0.287174 + 0.957879i \(0.592716\pi\)
\(720\) 0 0
\(721\) 11757.0 0.607288
\(722\) 12678.4 0.653520
\(723\) −32.4043 −0.00166685
\(724\) −1921.06 −0.0986125
\(725\) 0 0
\(726\) 2279.58 0.116533
\(727\) −31652.7 −1.61476 −0.807382 0.590029i \(-0.799116\pi\)
−0.807382 + 0.590029i \(0.799116\pi\)
\(728\) −3260.92 −0.166013
\(729\) −3935.94 −0.199967
\(730\) 0 0
\(731\) 19727.5 0.998149
\(732\) 2804.88 0.141628
\(733\) −16958.3 −0.854528 −0.427264 0.904127i \(-0.640523\pi\)
−0.427264 + 0.904127i \(0.640523\pi\)
\(734\) 8230.16 0.413870
\(735\) 0 0
\(736\) 12831.3 0.642618
\(737\) −555.065 −0.0277423
\(738\) 15193.1 0.757813
\(739\) −11616.6 −0.578245 −0.289123 0.957292i \(-0.593364\pi\)
−0.289123 + 0.957292i \(0.593364\pi\)
\(740\) 0 0
\(741\) −5694.66 −0.282319
\(742\) −3783.36 −0.187185
\(743\) −15928.0 −0.786464 −0.393232 0.919439i \(-0.628643\pi\)
−0.393232 + 0.919439i \(0.628643\pi\)
\(744\) −31191.4 −1.53700
\(745\) 0 0
\(746\) −6762.19 −0.331879
\(747\) 17946.1 0.879003
\(748\) −4383.57 −0.214277
\(749\) −10548.2 −0.514582
\(750\) 0 0
\(751\) 25571.9 1.24252 0.621260 0.783604i \(-0.286621\pi\)
0.621260 + 0.783604i \(0.286621\pi\)
\(752\) 604.412 0.0293094
\(753\) 3647.43 0.176520
\(754\) −2347.87 −0.113401
\(755\) 0 0
\(756\) −592.958 −0.0285260
\(757\) −6202.41 −0.297794 −0.148897 0.988853i \(-0.547572\pi\)
−0.148897 + 0.988853i \(0.547572\pi\)
\(758\) 1738.77 0.0833179
\(759\) 55557.6 2.65693
\(760\) 0 0
\(761\) −29199.1 −1.39089 −0.695444 0.718580i \(-0.744792\pi\)
−0.695444 + 0.718580i \(0.744792\pi\)
\(762\) −20888.1 −0.993037
\(763\) −8766.87 −0.415966
\(764\) 492.539 0.0233239
\(765\) 0 0
\(766\) 3025.00 0.142686
\(767\) 11915.4 0.560938
\(768\) 13050.5 0.613177
\(769\) 21838.2 1.02407 0.512033 0.858966i \(-0.328892\pi\)
0.512033 + 0.858966i \(0.328892\pi\)
\(770\) 0 0
\(771\) −11813.3 −0.551812
\(772\) 962.985 0.0448945
\(773\) 25544.8 1.18859 0.594296 0.804246i \(-0.297431\pi\)
0.594296 + 0.804246i \(0.297431\pi\)
\(774\) −10125.4 −0.470218
\(775\) 0 0
\(776\) −25525.2 −1.18080
\(777\) 17094.6 0.789273
\(778\) 2901.82 0.133721
\(779\) 15008.4 0.690286
\(780\) 0 0
\(781\) −36418.6 −1.66858
\(782\) 49206.6 2.25016
\(783\) −3026.77 −0.138146
\(784\) −2536.46 −0.115546
\(785\) 0 0
\(786\) −34123.8 −1.54854
\(787\) 37223.2 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(788\) −2749.91 −0.124317
\(789\) −7982.19 −0.360169
\(790\) 0 0
\(791\) −9591.42 −0.431140
\(792\) 15951.1 0.715655
\(793\) −6204.03 −0.277820
\(794\) −5135.18 −0.229522
\(795\) 0 0
\(796\) −3763.83 −0.167595
\(797\) 40384.6 1.79485 0.897425 0.441168i \(-0.145436\pi\)
0.897425 + 0.441168i \(0.145436\pi\)
\(798\) −5328.83 −0.236389
\(799\) −1018.46 −0.0450945
\(800\) 0 0
\(801\) −2947.23 −0.130007
\(802\) 10788.9 0.475023
\(803\) −31540.7 −1.38611
\(804\) −126.889 −0.00556598
\(805\) 0 0
\(806\) 9731.28 0.425272
\(807\) −21636.7 −0.943803
\(808\) −5824.14 −0.253580
\(809\) −1955.76 −0.0849948 −0.0424974 0.999097i \(-0.513531\pi\)
−0.0424974 + 0.999097i \(0.513531\pi\)
\(810\) 0 0
\(811\) −34301.8 −1.48520 −0.742600 0.669735i \(-0.766408\pi\)
−0.742600 + 0.669735i \(0.766408\pi\)
\(812\) 431.669 0.0186559
\(813\) 5965.49 0.257342
\(814\) 36288.7 1.56255
\(815\) 0 0
\(816\) 30056.9 1.28946
\(817\) −10002.3 −0.428319
\(818\) 30255.8 1.29324
\(819\) −2344.31 −0.100021
\(820\) 0 0
\(821\) −13665.6 −0.580918 −0.290459 0.956887i \(-0.593808\pi\)
−0.290459 + 0.956887i \(0.593808\pi\)
\(822\) −38057.5 −1.61485
\(823\) 21519.5 0.911449 0.455724 0.890121i \(-0.349380\pi\)
0.455724 + 0.890121i \(0.349380\pi\)
\(824\) 40449.7 1.71011
\(825\) 0 0
\(826\) 11149.9 0.469679
\(827\) −35220.6 −1.48094 −0.740471 0.672088i \(-0.765398\pi\)
−0.740471 + 0.672088i \(0.765398\pi\)
\(828\) 4962.23 0.208272
\(829\) 31365.5 1.31408 0.657039 0.753857i \(-0.271809\pi\)
0.657039 + 0.753857i \(0.271809\pi\)
\(830\) 0 0
\(831\) −2573.28 −0.107420
\(832\) −10952.0 −0.456362
\(833\) 4274.04 0.177775
\(834\) 9089.24 0.377380
\(835\) 0 0
\(836\) 2222.58 0.0919491
\(837\) 12545.2 0.518070
\(838\) −7067.48 −0.291339
\(839\) −28287.1 −1.16398 −0.581990 0.813196i \(-0.697726\pi\)
−0.581990 + 0.813196i \(0.697726\pi\)
\(840\) 0 0
\(841\) −22185.5 −0.909653
\(842\) −35035.8 −1.43398
\(843\) 22201.2 0.907060
\(844\) −7414.11 −0.302374
\(845\) 0 0
\(846\) 522.738 0.0212436
\(847\) 927.026 0.0376068
\(848\) −10819.8 −0.438152
\(849\) 36028.6 1.45642
\(850\) 0 0
\(851\) 80035.0 3.22393
\(852\) −8325.40 −0.334769
\(853\) −9405.41 −0.377533 −0.188766 0.982022i \(-0.560449\pi\)
−0.188766 + 0.982022i \(0.560449\pi\)
\(854\) −5805.47 −0.232622
\(855\) 0 0
\(856\) −36290.6 −1.44905
\(857\) 27966.9 1.11474 0.557369 0.830265i \(-0.311811\pi\)
0.557369 + 0.830265i \(0.311811\pi\)
\(858\) −12737.2 −0.506809
\(859\) 6281.11 0.249486 0.124743 0.992189i \(-0.460189\pi\)
0.124743 + 0.992189i \(0.460189\pi\)
\(860\) 0 0
\(861\) 15813.6 0.625931
\(862\) 16624.7 0.656890
\(863\) 4757.13 0.187642 0.0938208 0.995589i \(-0.470092\pi\)
0.0938208 + 0.995589i \(0.470092\pi\)
\(864\) −3792.35 −0.149327
\(865\) 0 0
\(866\) 20743.3 0.813954
\(867\) −17942.0 −0.702818
\(868\) −1789.15 −0.0699629
\(869\) 5978.28 0.233371
\(870\) 0 0
\(871\) 280.663 0.0109184
\(872\) −30162.1 −1.17135
\(873\) −18350.3 −0.711414
\(874\) −24949.0 −0.965574
\(875\) 0 0
\(876\) −7210.30 −0.278097
\(877\) 30240.5 1.16437 0.582184 0.813057i \(-0.302198\pi\)
0.582184 + 0.813057i \(0.302198\pi\)
\(878\) 14402.5 0.553601
\(879\) 1878.64 0.0720875
\(880\) 0 0
\(881\) 44875.5 1.71611 0.858056 0.513556i \(-0.171672\pi\)
0.858056 + 0.513556i \(0.171672\pi\)
\(882\) −2193.71 −0.0837483
\(883\) −4892.13 −0.186448 −0.0932238 0.995645i \(-0.529717\pi\)
−0.0932238 + 0.995645i \(0.529717\pi\)
\(884\) 2216.51 0.0843317
\(885\) 0 0
\(886\) −14186.2 −0.537916
\(887\) −1761.40 −0.0666765 −0.0333382 0.999444i \(-0.510614\pi\)
−0.0333382 + 0.999444i \(0.510614\pi\)
\(888\) 58813.4 2.22258
\(889\) −8494.43 −0.320466
\(890\) 0 0
\(891\) −34303.3 −1.28979
\(892\) −8367.22 −0.314075
\(893\) 516.384 0.0193507
\(894\) −5652.33 −0.211457
\(895\) 0 0
\(896\) −6954.86 −0.259314
\(897\) −28092.1 −1.04567
\(898\) 18702.2 0.694988
\(899\) −9132.79 −0.338816
\(900\) 0 0
\(901\) 18231.8 0.674128
\(902\) 33569.3 1.23918
\(903\) −10538.9 −0.388386
\(904\) −32999.0 −1.21408
\(905\) 0 0
\(906\) 17719.8 0.649779
\(907\) −23689.1 −0.867238 −0.433619 0.901096i \(-0.642764\pi\)
−0.433619 + 0.901096i \(0.642764\pi\)
\(908\) −1334.30 −0.0487669
\(909\) −4187.03 −0.152778
\(910\) 0 0
\(911\) −13877.3 −0.504692 −0.252346 0.967637i \(-0.581202\pi\)
−0.252346 + 0.967637i \(0.581202\pi\)
\(912\) −15239.6 −0.553325
\(913\) 39652.2 1.43735
\(914\) −7500.09 −0.271423
\(915\) 0 0
\(916\) −5397.18 −0.194681
\(917\) −13876.9 −0.499735
\(918\) −14543.3 −0.522875
\(919\) 14331.6 0.514426 0.257213 0.966355i \(-0.417196\pi\)
0.257213 + 0.966355i \(0.417196\pi\)
\(920\) 0 0
\(921\) 12779.0 0.457201
\(922\) −15705.0 −0.560971
\(923\) 18414.7 0.656692
\(924\) 2341.81 0.0833767
\(925\) 0 0
\(926\) −48930.2 −1.73644
\(927\) 29079.7 1.03032
\(928\) 2760.80 0.0976592
\(929\) 16668.4 0.588668 0.294334 0.955703i \(-0.404902\pi\)
0.294334 + 0.955703i \(0.404902\pi\)
\(930\) 0 0
\(931\) −2167.04 −0.0762857
\(932\) 799.300 0.0280922
\(933\) −8076.82 −0.283412
\(934\) −17523.1 −0.613891
\(935\) 0 0
\(936\) −8065.52 −0.281656
\(937\) −30384.9 −1.05937 −0.529685 0.848194i \(-0.677690\pi\)
−0.529685 + 0.848194i \(0.677690\pi\)
\(938\) 262.632 0.00914206
\(939\) −9545.94 −0.331757
\(940\) 0 0
\(941\) −1196.35 −0.0414452 −0.0207226 0.999785i \(-0.506597\pi\)
−0.0207226 + 0.999785i \(0.506597\pi\)
\(942\) −9038.63 −0.312627
\(943\) 74037.5 2.55673
\(944\) 31886.9 1.09940
\(945\) 0 0
\(946\) −22372.1 −0.768901
\(947\) −1788.41 −0.0613681 −0.0306840 0.999529i \(-0.509769\pi\)
−0.0306840 + 0.999529i \(0.509769\pi\)
\(948\) 1366.65 0.0468215
\(949\) 15948.2 0.545523
\(950\) 0 0
\(951\) 43249.2 1.47471
\(952\) 14704.7 0.500612
\(953\) −8578.60 −0.291593 −0.145796 0.989315i \(-0.546574\pi\)
−0.145796 + 0.989315i \(0.546574\pi\)
\(954\) −9357.70 −0.317575
\(955\) 0 0
\(956\) 6640.06 0.224639
\(957\) 11953.9 0.403776
\(958\) −6198.95 −0.209059
\(959\) −15476.6 −0.521133
\(960\) 0 0
\(961\) 8061.99 0.270618
\(962\) −18349.0 −0.614963
\(963\) −26089.7 −0.873031
\(964\) −6.39489 −0.000213657 0
\(965\) 0 0
\(966\) −26287.4 −0.875552
\(967\) 55459.3 1.84431 0.922156 0.386818i \(-0.126426\pi\)
0.922156 + 0.386818i \(0.126426\pi\)
\(968\) 3189.40 0.105900
\(969\) 25679.3 0.851330
\(970\) 0 0
\(971\) 22047.3 0.728662 0.364331 0.931270i \(-0.381298\pi\)
0.364331 + 0.931270i \(0.381298\pi\)
\(972\) −5554.72 −0.183300
\(973\) 3696.27 0.121785
\(974\) 14444.6 0.475191
\(975\) 0 0
\(976\) −16602.7 −0.544507
\(977\) −14402.3 −0.471617 −0.235809 0.971800i \(-0.575774\pi\)
−0.235809 + 0.971800i \(0.575774\pi\)
\(978\) −17258.4 −0.564277
\(979\) −6511.94 −0.212587
\(980\) 0 0
\(981\) −21683.9 −0.705721
\(982\) −1389.58 −0.0451561
\(983\) −7817.11 −0.253639 −0.126819 0.991926i \(-0.540477\pi\)
−0.126819 + 0.991926i \(0.540477\pi\)
\(984\) 54406.2 1.76261
\(985\) 0 0
\(986\) 10587.4 0.341959
\(987\) 544.087 0.0175466
\(988\) −1123.82 −0.0361878
\(989\) −49342.0 −1.58643
\(990\) 0 0
\(991\) 24501.6 0.785386 0.392693 0.919670i \(-0.371543\pi\)
0.392693 + 0.919670i \(0.371543\pi\)
\(992\) −11442.8 −0.366239
\(993\) 64464.2 2.06013
\(994\) 17231.7 0.549855
\(995\) 0 0
\(996\) 9064.61 0.288377
\(997\) 50696.0 1.61039 0.805195 0.593010i \(-0.202060\pi\)
0.805195 + 0.593010i \(0.202060\pi\)
\(998\) −1548.80 −0.0491245
\(999\) −23654.8 −0.749152
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 175.4.a.c.1.2 2
3.2 odd 2 1575.4.a.z.1.1 2
5.2 odd 4 175.4.b.c.99.2 4
5.3 odd 4 175.4.b.c.99.3 4
5.4 even 2 35.4.a.b.1.1 2
7.6 odd 2 1225.4.a.m.1.2 2
15.14 odd 2 315.4.a.f.1.2 2
20.19 odd 2 560.4.a.r.1.1 2
35.4 even 6 245.4.e.h.226.2 4
35.9 even 6 245.4.e.h.116.2 4
35.19 odd 6 245.4.e.i.116.2 4
35.24 odd 6 245.4.e.i.226.2 4
35.34 odd 2 245.4.a.k.1.1 2
40.19 odd 2 2240.4.a.bo.1.2 2
40.29 even 2 2240.4.a.bn.1.1 2
105.104 even 2 2205.4.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 5.4 even 2
175.4.a.c.1.2 2 1.1 even 1 trivial
175.4.b.c.99.2 4 5.2 odd 4
175.4.b.c.99.3 4 5.3 odd 4
245.4.a.k.1.1 2 35.34 odd 2
245.4.e.h.116.2 4 35.9 even 6
245.4.e.h.226.2 4 35.4 even 6
245.4.e.i.116.2 4 35.19 odd 6
245.4.e.i.226.2 4 35.24 odd 6
315.4.a.f.1.2 2 15.14 odd 2
560.4.a.r.1.1 2 20.19 odd 2
1225.4.a.m.1.2 2 7.6 odd 2
1575.4.a.z.1.1 2 3.2 odd 2
2205.4.a.u.1.2 2 105.104 even 2
2240.4.a.bn.1.1 2 40.29 even 2
2240.4.a.bo.1.2 2 40.19 odd 2