Properties

Label 2205.4.a.bh.1.1
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
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] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, 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("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(130.099211563\)
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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.43845 q^{2} -5.93087 q^{4} -5.00000 q^{5} -20.0388 q^{8} +O(q^{10})\) \(q+1.43845 q^{2} -5.93087 q^{4} -5.00000 q^{5} -20.0388 q^{8} -7.19224 q^{10} -7.61553 q^{11} -52.3542 q^{13} +18.6222 q^{16} -49.7235 q^{17} -140.600 q^{19} +29.6543 q^{20} -10.9545 q^{22} +23.4470 q^{23} +25.0000 q^{25} -75.3087 q^{26} -157.170 q^{29} -127.892 q^{31} +187.098 q^{32} -71.5246 q^{34} -115.477 q^{37} -202.246 q^{38} +100.194 q^{40} -188.617 q^{41} +322.186 q^{43} +45.1667 q^{44} +33.7272 q^{46} -76.6477 q^{47} +35.9612 q^{50} +310.506 q^{52} +424.172 q^{53} +38.0776 q^{55} -226.081 q^{58} +107.784 q^{59} -915.511 q^{61} -183.966 q^{62} +120.153 q^{64} +261.771 q^{65} -451.723 q^{67} +294.903 q^{68} -907.312 q^{71} -755.956 q^{73} -166.108 q^{74} +833.882 q^{76} +22.5834 q^{79} -93.1109 q^{80} -271.316 q^{82} +1112.25 q^{83} +248.617 q^{85} +463.447 q^{86} +152.606 q^{88} +1518.81 q^{89} -139.061 q^{92} -110.254 q^{94} +703.002 q^{95} +549.987 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{2} + 17 q^{4} - 10 q^{5} + 63 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 7 q^{2} + 17 q^{4} - 10 q^{5} + 63 q^{8} - 35 q^{10} + 26 q^{11} - 14 q^{13} + 297 q^{16} + 16 q^{17} - 174 q^{19} - 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} + 138 q^{26} + 32 q^{29} - 330 q^{31} + 1071 q^{32} + 294 q^{34} - 132 q^{37} - 388 q^{38} - 315 q^{40} + 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 175 q^{50} + 1190 q^{52} - 34 q^{53} - 130 q^{55} + 826 q^{58} + 364 q^{59} - 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} + 68 q^{76} + 408 q^{79} - 1485 q^{80} + 1890 q^{82} + 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} + 36 q^{89} - 4896 q^{92} + 1940 q^{94} + 870 q^{95} + 498 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.43845 0.508568 0.254284 0.967130i \(-0.418160\pi\)
0.254284 + 0.967130i \(0.418160\pi\)
\(3\) 0 0
\(4\) −5.93087 −0.741359
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −20.0388 −0.885599
\(9\) 0 0
\(10\) −7.19224 −0.227438
\(11\) −7.61553 −0.208743 −0.104371 0.994538i \(-0.533283\pi\)
−0.104371 + 0.994538i \(0.533283\pi\)
\(12\) 0 0
\(13\) −52.3542 −1.11696 −0.558478 0.829519i \(-0.688615\pi\)
−0.558478 + 0.829519i \(0.688615\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 18.6222 0.290971
\(17\) −49.7235 −0.709395 −0.354697 0.934981i \(-0.615416\pi\)
−0.354697 + 0.934981i \(0.615416\pi\)
\(18\) 0 0
\(19\) −140.600 −1.69768 −0.848840 0.528649i \(-0.822699\pi\)
−0.848840 + 0.528649i \(0.822699\pi\)
\(20\) 29.6543 0.331546
\(21\) 0 0
\(22\) −10.9545 −0.106160
\(23\) 23.4470 0.212566 0.106283 0.994336i \(-0.466105\pi\)
0.106283 + 0.994336i \(0.466105\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −75.3087 −0.568048
\(27\) 0 0
\(28\) 0 0
\(29\) −157.170 −1.00641 −0.503204 0.864168i \(-0.667845\pi\)
−0.503204 + 0.864168i \(0.667845\pi\)
\(30\) 0 0
\(31\) −127.892 −0.740971 −0.370485 0.928838i \(-0.620809\pi\)
−0.370485 + 0.928838i \(0.620809\pi\)
\(32\) 187.098 1.03358
\(33\) 0 0
\(34\) −71.5246 −0.360776
\(35\) 0 0
\(36\) 0 0
\(37\) −115.477 −0.513090 −0.256545 0.966532i \(-0.582584\pi\)
−0.256545 + 0.966532i \(0.582584\pi\)
\(38\) −202.246 −0.863386
\(39\) 0 0
\(40\) 100.194 0.396052
\(41\) −188.617 −0.718466 −0.359233 0.933248i \(-0.616962\pi\)
−0.359233 + 0.933248i \(0.616962\pi\)
\(42\) 0 0
\(43\) 322.186 1.14262 0.571312 0.820733i \(-0.306435\pi\)
0.571312 + 0.820733i \(0.306435\pi\)
\(44\) 45.1667 0.154753
\(45\) 0 0
\(46\) 33.7272 0.108104
\(47\) −76.6477 −0.237877 −0.118938 0.992902i \(-0.537949\pi\)
−0.118938 + 0.992902i \(0.537949\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 35.9612 0.101714
\(51\) 0 0
\(52\) 310.506 0.828065
\(53\) 424.172 1.09933 0.549666 0.835385i \(-0.314755\pi\)
0.549666 + 0.835385i \(0.314755\pi\)
\(54\) 0 0
\(55\) 38.0776 0.0933525
\(56\) 0 0
\(57\) 0 0
\(58\) −226.081 −0.511827
\(59\) 107.784 0.237835 0.118918 0.992904i \(-0.462058\pi\)
0.118918 + 0.992904i \(0.462058\pi\)
\(60\) 0 0
\(61\) −915.511 −1.92163 −0.960813 0.277197i \(-0.910595\pi\)
−0.960813 + 0.277197i \(0.910595\pi\)
\(62\) −183.966 −0.376834
\(63\) 0 0
\(64\) 120.153 0.234673
\(65\) 261.771 0.499518
\(66\) 0 0
\(67\) −451.723 −0.823684 −0.411842 0.911255i \(-0.635114\pi\)
−0.411842 + 0.911255i \(0.635114\pi\)
\(68\) 294.903 0.525916
\(69\) 0 0
\(70\) 0 0
\(71\) −907.312 −1.51659 −0.758297 0.651909i \(-0.773968\pi\)
−0.758297 + 0.651909i \(0.773968\pi\)
\(72\) 0 0
\(73\) −755.956 −1.21203 −0.606014 0.795454i \(-0.707232\pi\)
−0.606014 + 0.795454i \(0.707232\pi\)
\(74\) −166.108 −0.260941
\(75\) 0 0
\(76\) 833.882 1.25859
\(77\) 0 0
\(78\) 0 0
\(79\) 22.5834 0.0321624 0.0160812 0.999871i \(-0.494881\pi\)
0.0160812 + 0.999871i \(0.494881\pi\)
\(80\) −93.1109 −0.130126
\(81\) 0 0
\(82\) −271.316 −0.365389
\(83\) 1112.25 1.47091 0.735454 0.677575i \(-0.236969\pi\)
0.735454 + 0.677575i \(0.236969\pi\)
\(84\) 0 0
\(85\) 248.617 0.317251
\(86\) 463.447 0.581102
\(87\) 0 0
\(88\) 152.606 0.184862
\(89\) 1518.81 1.80892 0.904458 0.426562i \(-0.140275\pi\)
0.904458 + 0.426562i \(0.140275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −139.061 −0.157588
\(93\) 0 0
\(94\) −110.254 −0.120977
\(95\) 703.002 0.759226
\(96\) 0 0
\(97\) 549.987 0.575698 0.287849 0.957676i \(-0.407060\pi\)
0.287849 + 0.957676i \(0.407060\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −148.272 −0.148272
\(101\) −533.299 −0.525398 −0.262699 0.964878i \(-0.584613\pi\)
−0.262699 + 0.964878i \(0.584613\pi\)
\(102\) 0 0
\(103\) −1357.79 −1.29890 −0.649451 0.760404i \(-0.725001\pi\)
−0.649451 + 0.760404i \(0.725001\pi\)
\(104\) 1049.12 0.989176
\(105\) 0 0
\(106\) 610.149 0.559084
\(107\) −913.693 −0.825515 −0.412757 0.910841i \(-0.635434\pi\)
−0.412757 + 0.910841i \(0.635434\pi\)
\(108\) 0 0
\(109\) −160.489 −0.141028 −0.0705139 0.997511i \(-0.522464\pi\)
−0.0705139 + 0.997511i \(0.522464\pi\)
\(110\) 54.7727 0.0474761
\(111\) 0 0
\(112\) 0 0
\(113\) 1788.48 1.48891 0.744453 0.667675i \(-0.232710\pi\)
0.744453 + 0.667675i \(0.232710\pi\)
\(114\) 0 0
\(115\) −117.235 −0.0950626
\(116\) 932.157 0.746109
\(117\) 0 0
\(118\) 155.042 0.120955
\(119\) 0 0
\(120\) 0 0
\(121\) −1273.00 −0.956427
\(122\) −1316.91 −0.977277
\(123\) 0 0
\(124\) 758.511 0.549325
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −271.015 −0.189360 −0.0946799 0.995508i \(-0.530183\pi\)
−0.0946799 + 0.995508i \(0.530183\pi\)
\(128\) −1323.95 −0.914231
\(129\) 0 0
\(130\) 376.543 0.254039
\(131\) −763.151 −0.508984 −0.254492 0.967075i \(-0.581908\pi\)
−0.254492 + 0.967075i \(0.581908\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −649.780 −0.418899
\(135\) 0 0
\(136\) 996.400 0.628240
\(137\) 240.934 0.150251 0.0751254 0.997174i \(-0.476064\pi\)
0.0751254 + 0.997174i \(0.476064\pi\)
\(138\) 0 0
\(139\) 103.150 0.0629427 0.0314714 0.999505i \(-0.489981\pi\)
0.0314714 + 0.999505i \(0.489981\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1305.12 −0.771291
\(143\) 398.705 0.233156
\(144\) 0 0
\(145\) 785.852 0.450079
\(146\) −1087.40 −0.616398
\(147\) 0 0
\(148\) 684.881 0.380384
\(149\) 3256.71 1.79061 0.895303 0.445458i \(-0.146959\pi\)
0.895303 + 0.445458i \(0.146959\pi\)
\(150\) 0 0
\(151\) 1471.04 0.792793 0.396396 0.918080i \(-0.370261\pi\)
0.396396 + 0.918080i \(0.370261\pi\)
\(152\) 2817.47 1.50346
\(153\) 0 0
\(154\) 0 0
\(155\) 639.460 0.331372
\(156\) 0 0
\(157\) 1394.22 0.708730 0.354365 0.935107i \(-0.384697\pi\)
0.354365 + 0.935107i \(0.384697\pi\)
\(158\) 32.4850 0.0163567
\(159\) 0 0
\(160\) −935.488 −0.462230
\(161\) 0 0
\(162\) 0 0
\(163\) −3674.27 −1.76559 −0.882794 0.469760i \(-0.844341\pi\)
−0.882794 + 0.469760i \(0.844341\pi\)
\(164\) 1118.67 0.532641
\(165\) 0 0
\(166\) 1599.91 0.748056
\(167\) 4041.09 1.87251 0.936254 0.351325i \(-0.114269\pi\)
0.936254 + 0.351325i \(0.114269\pi\)
\(168\) 0 0
\(169\) 543.958 0.247591
\(170\) 357.623 0.161344
\(171\) 0 0
\(172\) −1910.84 −0.847094
\(173\) 59.4582 0.0261302 0.0130651 0.999915i \(-0.495841\pi\)
0.0130651 + 0.999915i \(0.495841\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −141.818 −0.0607381
\(177\) 0 0
\(178\) 2184.73 0.919957
\(179\) 2973.32 1.24155 0.620773 0.783991i \(-0.286819\pi\)
0.620773 + 0.783991i \(0.286819\pi\)
\(180\) 0 0
\(181\) 676.220 0.277696 0.138848 0.990314i \(-0.455660\pi\)
0.138848 + 0.990314i \(0.455660\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −469.849 −0.188249
\(185\) 577.386 0.229461
\(186\) 0 0
\(187\) 378.671 0.148081
\(188\) 454.588 0.176352
\(189\) 0 0
\(190\) 1011.23 0.386118
\(191\) −16.5589 −0.00627308 −0.00313654 0.999995i \(-0.500998\pi\)
−0.00313654 + 0.999995i \(0.500998\pi\)
\(192\) 0 0
\(193\) 2694.68 1.00501 0.502506 0.864574i \(-0.332411\pi\)
0.502506 + 0.864574i \(0.332411\pi\)
\(194\) 791.127 0.292781
\(195\) 0 0
\(196\) 0 0
\(197\) −1027.38 −0.371561 −0.185781 0.982591i \(-0.559481\pi\)
−0.185781 + 0.982591i \(0.559481\pi\)
\(198\) 0 0
\(199\) −2823.77 −1.00589 −0.502944 0.864319i \(-0.667750\pi\)
−0.502944 + 0.864319i \(0.667750\pi\)
\(200\) −500.971 −0.177120
\(201\) 0 0
\(202\) −767.123 −0.267201
\(203\) 0 0
\(204\) 0 0
\(205\) 943.087 0.321308
\(206\) −1953.11 −0.660579
\(207\) 0 0
\(208\) −974.948 −0.325002
\(209\) 1070.75 0.354378
\(210\) 0 0
\(211\) 5151.16 1.68067 0.840333 0.542071i \(-0.182360\pi\)
0.840333 + 0.542071i \(0.182360\pi\)
\(212\) −2515.71 −0.814999
\(213\) 0 0
\(214\) −1314.30 −0.419830
\(215\) −1610.93 −0.510997
\(216\) 0 0
\(217\) 0 0
\(218\) −230.855 −0.0717222
\(219\) 0 0
\(220\) −225.834 −0.0692077
\(221\) 2603.23 0.792363
\(222\) 0 0
\(223\) 114.496 0.0343822 0.0171911 0.999852i \(-0.494528\pi\)
0.0171911 + 0.999852i \(0.494528\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2572.64 0.757209
\(227\) 4744.75 1.38731 0.693657 0.720306i \(-0.255999\pi\)
0.693657 + 0.720306i \(0.255999\pi\)
\(228\) 0 0
\(229\) 5384.47 1.55378 0.776891 0.629635i \(-0.216796\pi\)
0.776891 + 0.629635i \(0.216796\pi\)
\(230\) −168.636 −0.0483458
\(231\) 0 0
\(232\) 3149.51 0.891274
\(233\) 1608.91 0.452373 0.226187 0.974084i \(-0.427374\pi\)
0.226187 + 0.974084i \(0.427374\pi\)
\(234\) 0 0
\(235\) 383.239 0.106382
\(236\) −639.253 −0.176321
\(237\) 0 0
\(238\) 0 0
\(239\) 3113.11 0.842554 0.421277 0.906932i \(-0.361582\pi\)
0.421277 + 0.906932i \(0.361582\pi\)
\(240\) 0 0
\(241\) −7136.38 −1.90745 −0.953724 0.300685i \(-0.902785\pi\)
−0.953724 + 0.300685i \(0.902785\pi\)
\(242\) −1831.15 −0.486408
\(243\) 0 0
\(244\) 5429.78 1.42461
\(245\) 0 0
\(246\) 0 0
\(247\) 7361.01 1.89624
\(248\) 2562.81 0.656203
\(249\) 0 0
\(250\) −179.806 −0.0454877
\(251\) −225.504 −0.0567079 −0.0283539 0.999598i \(-0.509027\pi\)
−0.0283539 + 0.999598i \(0.509027\pi\)
\(252\) 0 0
\(253\) −178.561 −0.0443717
\(254\) −389.841 −0.0963024
\(255\) 0 0
\(256\) −2865.65 −0.699621
\(257\) −4423.05 −1.07355 −0.536775 0.843725i \(-0.680358\pi\)
−0.536775 + 0.843725i \(0.680358\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1552.53 −0.370322
\(261\) 0 0
\(262\) −1097.75 −0.258853
\(263\) −6540.56 −1.53349 −0.766746 0.641950i \(-0.778125\pi\)
−0.766746 + 0.641950i \(0.778125\pi\)
\(264\) 0 0
\(265\) −2120.86 −0.491636
\(266\) 0 0
\(267\) 0 0
\(268\) 2679.11 0.610645
\(269\) 2262.97 0.512920 0.256460 0.966555i \(-0.417444\pi\)
0.256460 + 0.966555i \(0.417444\pi\)
\(270\) 0 0
\(271\) −1615.68 −0.362160 −0.181080 0.983468i \(-0.557959\pi\)
−0.181080 + 0.983468i \(0.557959\pi\)
\(272\) −925.959 −0.206414
\(273\) 0 0
\(274\) 346.571 0.0764127
\(275\) −190.388 −0.0417485
\(276\) 0 0
\(277\) 4691.55 1.01765 0.508823 0.860871i \(-0.330081\pi\)
0.508823 + 0.860871i \(0.330081\pi\)
\(278\) 148.375 0.0320107
\(279\) 0 0
\(280\) 0 0
\(281\) 8119.00 1.72363 0.861813 0.507226i \(-0.169329\pi\)
0.861813 + 0.507226i \(0.169329\pi\)
\(282\) 0 0
\(283\) 3633.75 0.763265 0.381632 0.924314i \(-0.375362\pi\)
0.381632 + 0.924314i \(0.375362\pi\)
\(284\) 5381.15 1.12434
\(285\) 0 0
\(286\) 573.515 0.118576
\(287\) 0 0
\(288\) 0 0
\(289\) −2440.58 −0.496759
\(290\) 1130.41 0.228896
\(291\) 0 0
\(292\) 4483.48 0.898547
\(293\) −7981.99 −1.59151 −0.795756 0.605618i \(-0.792926\pi\)
−0.795756 + 0.605618i \(0.792926\pi\)
\(294\) 0 0
\(295\) −538.920 −0.106363
\(296\) 2314.03 0.454392
\(297\) 0 0
\(298\) 4684.61 0.910645
\(299\) −1227.55 −0.237427
\(300\) 0 0
\(301\) 0 0
\(302\) 2116.02 0.403189
\(303\) 0 0
\(304\) −2618.28 −0.493977
\(305\) 4577.56 0.859377
\(306\) 0 0
\(307\) 7118.15 1.32330 0.661652 0.749811i \(-0.269856\pi\)
0.661652 + 0.749811i \(0.269856\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 919.830 0.168525
\(311\) −9155.92 −1.66940 −0.834702 0.550703i \(-0.814360\pi\)
−0.834702 + 0.550703i \(0.814360\pi\)
\(312\) 0 0
\(313\) 6163.44 1.11303 0.556515 0.830838i \(-0.312138\pi\)
0.556515 + 0.830838i \(0.312138\pi\)
\(314\) 2005.51 0.360438
\(315\) 0 0
\(316\) −133.939 −0.0238438
\(317\) −8658.37 −1.53408 −0.767038 0.641601i \(-0.778270\pi\)
−0.767038 + 0.641601i \(0.778270\pi\)
\(318\) 0 0
\(319\) 1196.94 0.210080
\(320\) −600.763 −0.104949
\(321\) 0 0
\(322\) 0 0
\(323\) 6991.14 1.20433
\(324\) 0 0
\(325\) −1308.85 −0.223391
\(326\) −5285.24 −0.897922
\(327\) 0 0
\(328\) 3779.67 0.636272
\(329\) 0 0
\(330\) 0 0
\(331\) −128.477 −0.0213346 −0.0106673 0.999943i \(-0.503396\pi\)
−0.0106673 + 0.999943i \(0.503396\pi\)
\(332\) −6596.61 −1.09047
\(333\) 0 0
\(334\) 5812.89 0.952297
\(335\) 2258.62 0.368363
\(336\) 0 0
\(337\) 7784.57 1.25832 0.629158 0.777277i \(-0.283400\pi\)
0.629158 + 0.777277i \(0.283400\pi\)
\(338\) 782.455 0.125917
\(339\) 0 0
\(340\) −1474.52 −0.235197
\(341\) 973.965 0.154672
\(342\) 0 0
\(343\) 0 0
\(344\) −6456.22 −1.01191
\(345\) 0 0
\(346\) 85.5274 0.0132890
\(347\) −3740.26 −0.578638 −0.289319 0.957233i \(-0.593429\pi\)
−0.289319 + 0.957233i \(0.593429\pi\)
\(348\) 0 0
\(349\) −5676.86 −0.870703 −0.435351 0.900261i \(-0.643376\pi\)
−0.435351 + 0.900261i \(0.643376\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1424.85 −0.215752
\(353\) 909.564 0.137142 0.0685711 0.997646i \(-0.478156\pi\)
0.0685711 + 0.997646i \(0.478156\pi\)
\(354\) 0 0
\(355\) 4536.56 0.678241
\(356\) −9007.87 −1.34106
\(357\) 0 0
\(358\) 4276.97 0.631410
\(359\) −2678.57 −0.393788 −0.196894 0.980425i \(-0.563085\pi\)
−0.196894 + 0.980425i \(0.563085\pi\)
\(360\) 0 0
\(361\) 12909.5 1.88212
\(362\) 972.706 0.141227
\(363\) 0 0
\(364\) 0 0
\(365\) 3779.78 0.542035
\(366\) 0 0
\(367\) 716.898 0.101967 0.0509833 0.998700i \(-0.483764\pi\)
0.0509833 + 0.998700i \(0.483764\pi\)
\(368\) 436.633 0.0618508
\(369\) 0 0
\(370\) 830.540 0.116697
\(371\) 0 0
\(372\) 0 0
\(373\) −2006.15 −0.278484 −0.139242 0.990258i \(-0.544467\pi\)
−0.139242 + 0.990258i \(0.544467\pi\)
\(374\) 544.698 0.0753092
\(375\) 0 0
\(376\) 1535.93 0.210664
\(377\) 8228.53 1.12411
\(378\) 0 0
\(379\) 7277.53 0.986336 0.493168 0.869934i \(-0.335839\pi\)
0.493168 + 0.869934i \(0.335839\pi\)
\(380\) −4169.41 −0.562859
\(381\) 0 0
\(382\) −23.8191 −0.00319029
\(383\) 5953.94 0.794339 0.397170 0.917745i \(-0.369992\pi\)
0.397170 + 0.917745i \(0.369992\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3876.16 0.511117
\(387\) 0 0
\(388\) −3261.90 −0.426799
\(389\) −1867.98 −0.243472 −0.121736 0.992563i \(-0.538846\pi\)
−0.121736 + 0.992563i \(0.538846\pi\)
\(390\) 0 0
\(391\) −1165.86 −0.150794
\(392\) 0 0
\(393\) 0 0
\(394\) −1477.83 −0.188964
\(395\) −112.917 −0.0143834
\(396\) 0 0
\(397\) 2160.07 0.273075 0.136538 0.990635i \(-0.456403\pi\)
0.136538 + 0.990635i \(0.456403\pi\)
\(398\) −4061.85 −0.511563
\(399\) 0 0
\(400\) 465.554 0.0581943
\(401\) −1954.81 −0.243438 −0.121719 0.992565i \(-0.538841\pi\)
−0.121719 + 0.992565i \(0.538841\pi\)
\(402\) 0 0
\(403\) 6695.68 0.827632
\(404\) 3162.93 0.389509
\(405\) 0 0
\(406\) 0 0
\(407\) 879.420 0.107104
\(408\) 0 0
\(409\) −14895.1 −1.80077 −0.900384 0.435096i \(-0.856714\pi\)
−0.900384 + 0.435096i \(0.856714\pi\)
\(410\) 1356.58 0.163407
\(411\) 0 0
\(412\) 8052.86 0.962952
\(413\) 0 0
\(414\) 0 0
\(415\) −5561.25 −0.657810
\(416\) −9795.34 −1.15446
\(417\) 0 0
\(418\) 1540.21 0.180225
\(419\) 12608.9 1.47013 0.735067 0.677994i \(-0.237151\pi\)
0.735067 + 0.677994i \(0.237151\pi\)
\(420\) 0 0
\(421\) −7862.86 −0.910243 −0.455122 0.890429i \(-0.650404\pi\)
−0.455122 + 0.890429i \(0.650404\pi\)
\(422\) 7409.67 0.854732
\(423\) 0 0
\(424\) −8499.91 −0.973567
\(425\) −1243.09 −0.141879
\(426\) 0 0
\(427\) 0 0
\(428\) 5419.00 0.612002
\(429\) 0 0
\(430\) −2317.23 −0.259877
\(431\) −14291.0 −1.59715 −0.798575 0.601896i \(-0.794412\pi\)
−0.798575 + 0.601896i \(0.794412\pi\)
\(432\) 0 0
\(433\) 13759.0 1.52705 0.763527 0.645776i \(-0.223466\pi\)
0.763527 + 0.645776i \(0.223466\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 951.838 0.104552
\(437\) −3296.65 −0.360870
\(438\) 0 0
\(439\) 6093.13 0.662436 0.331218 0.943554i \(-0.392541\pi\)
0.331218 + 0.943554i \(0.392541\pi\)
\(440\) −763.031 −0.0826729
\(441\) 0 0
\(442\) 3744.61 0.402970
\(443\) −13449.5 −1.44244 −0.721222 0.692704i \(-0.756419\pi\)
−0.721222 + 0.692704i \(0.756419\pi\)
\(444\) 0 0
\(445\) −7594.05 −0.808972
\(446\) 164.697 0.0174857
\(447\) 0 0
\(448\) 0 0
\(449\) −6893.83 −0.724588 −0.362294 0.932064i \(-0.618006\pi\)
−0.362294 + 0.932064i \(0.618006\pi\)
\(450\) 0 0
\(451\) 1436.42 0.149974
\(452\) −10607.3 −1.10381
\(453\) 0 0
\(454\) 6825.07 0.705543
\(455\) 0 0
\(456\) 0 0
\(457\) −11820.1 −1.20990 −0.604948 0.796265i \(-0.706806\pi\)
−0.604948 + 0.796265i \(0.706806\pi\)
\(458\) 7745.28 0.790203
\(459\) 0 0
\(460\) 695.304 0.0704755
\(461\) 8443.38 0.853031 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(462\) 0 0
\(463\) −1269.74 −0.127451 −0.0637257 0.997967i \(-0.520298\pi\)
−0.0637257 + 0.997967i \(0.520298\pi\)
\(464\) −2926.86 −0.292836
\(465\) 0 0
\(466\) 2314.33 0.230063
\(467\) −16481.8 −1.63316 −0.816579 0.577233i \(-0.804132\pi\)
−0.816579 + 0.577233i \(0.804132\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 551.268 0.0541024
\(471\) 0 0
\(472\) −2159.87 −0.210627
\(473\) −2453.61 −0.238514
\(474\) 0 0
\(475\) −3515.01 −0.339536
\(476\) 0 0
\(477\) 0 0
\(478\) 4478.05 0.428496
\(479\) −1400.21 −0.133564 −0.0667822 0.997768i \(-0.521273\pi\)
−0.0667822 + 0.997768i \(0.521273\pi\)
\(480\) 0 0
\(481\) 6045.72 0.573100
\(482\) −10265.3 −0.970066
\(483\) 0 0
\(484\) 7550.02 0.709055
\(485\) −2749.93 −0.257460
\(486\) 0 0
\(487\) −14165.9 −1.31811 −0.659055 0.752094i \(-0.729044\pi\)
−0.659055 + 0.752094i \(0.729044\pi\)
\(488\) 18345.8 1.70179
\(489\) 0 0
\(490\) 0 0
\(491\) −4739.28 −0.435603 −0.217801 0.975993i \(-0.569888\pi\)
−0.217801 + 0.975993i \(0.569888\pi\)
\(492\) 0 0
\(493\) 7815.06 0.713940
\(494\) 10588.4 0.964364
\(495\) 0 0
\(496\) −2381.63 −0.215601
\(497\) 0 0
\(498\) 0 0
\(499\) −11370.0 −1.02003 −0.510013 0.860167i \(-0.670360\pi\)
−0.510013 + 0.860167i \(0.670360\pi\)
\(500\) 741.359 0.0663091
\(501\) 0 0
\(502\) −324.375 −0.0288398
\(503\) 9212.48 0.816629 0.408314 0.912841i \(-0.366117\pi\)
0.408314 + 0.912841i \(0.366117\pi\)
\(504\) 0 0
\(505\) 2666.50 0.234965
\(506\) −256.851 −0.0225660
\(507\) 0 0
\(508\) 1607.36 0.140384
\(509\) −15938.3 −1.38792 −0.693960 0.720014i \(-0.744136\pi\)
−0.693960 + 0.720014i \(0.744136\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 6469.49 0.558426
\(513\) 0 0
\(514\) −6362.33 −0.545973
\(515\) 6788.94 0.580886
\(516\) 0 0
\(517\) 583.713 0.0496550
\(518\) 0 0
\(519\) 0 0
\(520\) −5245.58 −0.442373
\(521\) 6442.99 0.541790 0.270895 0.962609i \(-0.412680\pi\)
0.270895 + 0.962609i \(0.412680\pi\)
\(522\) 0 0
\(523\) −986.655 −0.0824922 −0.0412461 0.999149i \(-0.513133\pi\)
−0.0412461 + 0.999149i \(0.513133\pi\)
\(524\) 4526.15 0.377339
\(525\) 0 0
\(526\) −9408.26 −0.779885
\(527\) 6359.24 0.525641
\(528\) 0 0
\(529\) −11617.2 −0.954815
\(530\) −3050.75 −0.250030
\(531\) 0 0
\(532\) 0 0
\(533\) 9874.91 0.802495
\(534\) 0 0
\(535\) 4568.47 0.369181
\(536\) 9052.01 0.729454
\(537\) 0 0
\(538\) 3255.16 0.260855
\(539\) 0 0
\(540\) 0 0
\(541\) −12681.1 −1.00777 −0.503885 0.863771i \(-0.668097\pi\)
−0.503885 + 0.863771i \(0.668097\pi\)
\(542\) −2324.06 −0.184183
\(543\) 0 0
\(544\) −9303.14 −0.733215
\(545\) 802.443 0.0630695
\(546\) 0 0
\(547\) −14826.4 −1.15892 −0.579462 0.814999i \(-0.696737\pi\)
−0.579462 + 0.814999i \(0.696737\pi\)
\(548\) −1428.95 −0.111390
\(549\) 0 0
\(550\) −273.863 −0.0212320
\(551\) 22098.2 1.70856
\(552\) 0 0
\(553\) 0 0
\(554\) 6748.55 0.517542
\(555\) 0 0
\(556\) −611.767 −0.0466632
\(557\) −1926.46 −0.146547 −0.0732737 0.997312i \(-0.523345\pi\)
−0.0732737 + 0.997312i \(0.523345\pi\)
\(558\) 0 0
\(559\) −16867.8 −1.27626
\(560\) 0 0
\(561\) 0 0
\(562\) 11678.8 0.876581
\(563\) −18624.8 −1.39422 −0.697108 0.716966i \(-0.745530\pi\)
−0.697108 + 0.716966i \(0.745530\pi\)
\(564\) 0 0
\(565\) −8942.41 −0.665859
\(566\) 5226.96 0.388172
\(567\) 0 0
\(568\) 18181.5 1.34309
\(569\) 20093.9 1.48045 0.740227 0.672357i \(-0.234718\pi\)
0.740227 + 0.672357i \(0.234718\pi\)
\(570\) 0 0
\(571\) 4535.25 0.332389 0.166195 0.986093i \(-0.446852\pi\)
0.166195 + 0.986093i \(0.446852\pi\)
\(572\) −2364.66 −0.172852
\(573\) 0 0
\(574\) 0 0
\(575\) 586.174 0.0425133
\(576\) 0 0
\(577\) −10034.6 −0.723994 −0.361997 0.932179i \(-0.617905\pi\)
−0.361997 + 0.932179i \(0.617905\pi\)
\(578\) −3510.64 −0.252636
\(579\) 0 0
\(580\) −4660.79 −0.333670
\(581\) 0 0
\(582\) 0 0
\(583\) −3230.30 −0.229477
\(584\) 15148.5 1.07337
\(585\) 0 0
\(586\) −11481.7 −0.809391
\(587\) −11192.6 −0.786999 −0.393499 0.919325i \(-0.628736\pi\)
−0.393499 + 0.919325i \(0.628736\pi\)
\(588\) 0 0
\(589\) 17981.7 1.25793
\(590\) −775.209 −0.0540929
\(591\) 0 0
\(592\) −2150.44 −0.149295
\(593\) 20317.6 1.40699 0.703493 0.710703i \(-0.251623\pi\)
0.703493 + 0.710703i \(0.251623\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −19315.1 −1.32748
\(597\) 0 0
\(598\) −1765.76 −0.120748
\(599\) 26376.5 1.79919 0.899594 0.436727i \(-0.143862\pi\)
0.899594 + 0.436727i \(0.143862\pi\)
\(600\) 0 0
\(601\) −9266.13 −0.628907 −0.314454 0.949273i \(-0.601821\pi\)
−0.314454 + 0.949273i \(0.601821\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8724.56 −0.587744
\(605\) 6365.02 0.427727
\(606\) 0 0
\(607\) −11338.0 −0.758149 −0.379075 0.925366i \(-0.623758\pi\)
−0.379075 + 0.925366i \(0.623758\pi\)
\(608\) −26306.0 −1.75469
\(609\) 0 0
\(610\) 6584.57 0.437052
\(611\) 4012.83 0.265698
\(612\) 0 0
\(613\) 25712.5 1.69416 0.847078 0.531469i \(-0.178360\pi\)
0.847078 + 0.531469i \(0.178360\pi\)
\(614\) 10239.1 0.672990
\(615\) 0 0
\(616\) 0 0
\(617\) −663.465 −0.0432903 −0.0216451 0.999766i \(-0.506890\pi\)
−0.0216451 + 0.999766i \(0.506890\pi\)
\(618\) 0 0
\(619\) 12768.7 0.829108 0.414554 0.910025i \(-0.363938\pi\)
0.414554 + 0.910025i \(0.363938\pi\)
\(620\) −3792.56 −0.245666
\(621\) 0 0
\(622\) −13170.3 −0.849005
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 8865.78 0.566051
\(627\) 0 0
\(628\) −8268.92 −0.525423
\(629\) 5741.93 0.363984
\(630\) 0 0
\(631\) −14937.6 −0.942400 −0.471200 0.882026i \(-0.656179\pi\)
−0.471200 + 0.882026i \(0.656179\pi\)
\(632\) −452.544 −0.0284829
\(633\) 0 0
\(634\) −12454.6 −0.780182
\(635\) 1355.08 0.0846843
\(636\) 0 0
\(637\) 0 0
\(638\) 1721.73 0.106840
\(639\) 0 0
\(640\) 6619.74 0.408856
\(641\) −10903.0 −0.671829 −0.335914 0.941893i \(-0.609045\pi\)
−0.335914 + 0.941893i \(0.609045\pi\)
\(642\) 0 0
\(643\) 7623.47 0.467559 0.233779 0.972290i \(-0.424891\pi\)
0.233779 + 0.972290i \(0.424891\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10056.4 0.612482
\(647\) 5384.84 0.327202 0.163601 0.986527i \(-0.447689\pi\)
0.163601 + 0.986527i \(0.447689\pi\)
\(648\) 0 0
\(649\) −820.833 −0.0496464
\(650\) −1882.72 −0.113610
\(651\) 0 0
\(652\) 21791.6 1.30893
\(653\) −297.318 −0.0178177 −0.00890883 0.999960i \(-0.502836\pi\)
−0.00890883 + 0.999960i \(0.502836\pi\)
\(654\) 0 0
\(655\) 3815.76 0.227624
\(656\) −3512.47 −0.209053
\(657\) 0 0
\(658\) 0 0
\(659\) −10324.7 −0.610309 −0.305155 0.952303i \(-0.598708\pi\)
−0.305155 + 0.952303i \(0.598708\pi\)
\(660\) 0 0
\(661\) −4272.98 −0.251437 −0.125718 0.992066i \(-0.540124\pi\)
−0.125718 + 0.992066i \(0.540124\pi\)
\(662\) −184.808 −0.0108501
\(663\) 0 0
\(664\) −22288.2 −1.30263
\(665\) 0 0
\(666\) 0 0
\(667\) −3685.17 −0.213928
\(668\) −23967.2 −1.38820
\(669\) 0 0
\(670\) 3248.90 0.187337
\(671\) 6972.10 0.401125
\(672\) 0 0
\(673\) −23033.4 −1.31927 −0.659637 0.751584i \(-0.729290\pi\)
−0.659637 + 0.751584i \(0.729290\pi\)
\(674\) 11197.7 0.639940
\(675\) 0 0
\(676\) −3226.15 −0.183554
\(677\) 5113.50 0.290292 0.145146 0.989410i \(-0.453635\pi\)
0.145146 + 0.989410i \(0.453635\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −4982.00 −0.280957
\(681\) 0 0
\(682\) 1401.00 0.0786613
\(683\) −1341.02 −0.0751286 −0.0375643 0.999294i \(-0.511960\pi\)
−0.0375643 + 0.999294i \(0.511960\pi\)
\(684\) 0 0
\(685\) −1204.67 −0.0671942
\(686\) 0 0
\(687\) 0 0
\(688\) 5999.80 0.332471
\(689\) −22207.2 −1.22790
\(690\) 0 0
\(691\) 16809.3 0.925404 0.462702 0.886514i \(-0.346880\pi\)
0.462702 + 0.886514i \(0.346880\pi\)
\(692\) −352.639 −0.0193718
\(693\) 0 0
\(694\) −5380.16 −0.294277
\(695\) −515.748 −0.0281489
\(696\) 0 0
\(697\) 9378.71 0.509676
\(698\) −8165.86 −0.442812
\(699\) 0 0
\(700\) 0 0
\(701\) −13467.0 −0.725592 −0.362796 0.931869i \(-0.618178\pi\)
−0.362796 + 0.931869i \(0.618178\pi\)
\(702\) 0 0
\(703\) 16236.1 0.871064
\(704\) −915.025 −0.0489863
\(705\) 0 0
\(706\) 1308.36 0.0697461
\(707\) 0 0
\(708\) 0 0
\(709\) 35514.3 1.88119 0.940597 0.339525i \(-0.110266\pi\)
0.940597 + 0.339525i \(0.110266\pi\)
\(710\) 6525.61 0.344932
\(711\) 0 0
\(712\) −30435.2 −1.60198
\(713\) −2998.68 −0.157506
\(714\) 0 0
\(715\) −1993.52 −0.104271
\(716\) −17634.4 −0.920431
\(717\) 0 0
\(718\) −3852.99 −0.200268
\(719\) 4993.61 0.259013 0.129506 0.991579i \(-0.458661\pi\)
0.129506 + 0.991579i \(0.458661\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18569.6 0.957186
\(723\) 0 0
\(724\) −4010.57 −0.205872
\(725\) −3929.26 −0.201281
\(726\) 0 0
\(727\) 4223.35 0.215454 0.107727 0.994181i \(-0.465643\pi\)
0.107727 + 0.994181i \(0.465643\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 5437.02 0.275662
\(731\) −16020.2 −0.810572
\(732\) 0 0
\(733\) −19030.9 −0.958968 −0.479484 0.877551i \(-0.659176\pi\)
−0.479484 + 0.877551i \(0.659176\pi\)
\(734\) 1031.22 0.0518570
\(735\) 0 0
\(736\) 4386.87 0.219704
\(737\) 3440.11 0.171938
\(738\) 0 0
\(739\) −27772.5 −1.38245 −0.691224 0.722641i \(-0.742928\pi\)
−0.691224 + 0.722641i \(0.742928\pi\)
\(740\) −3424.40 −0.170113
\(741\) 0 0
\(742\) 0 0
\(743\) −26880.7 −1.32726 −0.663631 0.748060i \(-0.730985\pi\)
−0.663631 + 0.748060i \(0.730985\pi\)
\(744\) 0 0
\(745\) −16283.6 −0.800783
\(746\) −2885.74 −0.141628
\(747\) 0 0
\(748\) −2245.85 −0.109781
\(749\) 0 0
\(750\) 0 0
\(751\) −35166.6 −1.70872 −0.854360 0.519681i \(-0.826051\pi\)
−0.854360 + 0.519681i \(0.826051\pi\)
\(752\) −1427.35 −0.0692154
\(753\) 0 0
\(754\) 11836.3 0.571688
\(755\) −7355.21 −0.354548
\(756\) 0 0
\(757\) 14589.2 0.700467 0.350233 0.936662i \(-0.386102\pi\)
0.350233 + 0.936662i \(0.386102\pi\)
\(758\) 10468.3 0.501619
\(759\) 0 0
\(760\) −14087.3 −0.672370
\(761\) −782.826 −0.0372897 −0.0186448 0.999826i \(-0.505935\pi\)
−0.0186448 + 0.999826i \(0.505935\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 98.2085 0.00465060
\(765\) 0 0
\(766\) 8564.42 0.403975
\(767\) −5642.95 −0.265652
\(768\) 0 0
\(769\) 16548.4 0.776008 0.388004 0.921658i \(-0.373165\pi\)
0.388004 + 0.921658i \(0.373165\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −15981.8 −0.745075
\(773\) 5744.94 0.267310 0.133655 0.991028i \(-0.457329\pi\)
0.133655 + 0.991028i \(0.457329\pi\)
\(774\) 0 0
\(775\) −3197.30 −0.148194
\(776\) −11021.1 −0.509837
\(777\) 0 0
\(778\) −2687.00 −0.123822
\(779\) 26519.7 1.21973
\(780\) 0 0
\(781\) 6909.66 0.316578
\(782\) −1677.03 −0.0766888
\(783\) 0 0
\(784\) 0 0
\(785\) −6971.09 −0.316954
\(786\) 0 0
\(787\) 34744.3 1.57370 0.786848 0.617146i \(-0.211711\pi\)
0.786848 + 0.617146i \(0.211711\pi\)
\(788\) 6093.24 0.275460
\(789\) 0 0
\(790\) −162.425 −0.00731496
\(791\) 0 0
\(792\) 0 0
\(793\) 47930.8 2.14637
\(794\) 3107.15 0.138877
\(795\) 0 0
\(796\) 16747.4 0.745724
\(797\) 21748.7 0.966600 0.483300 0.875455i \(-0.339438\pi\)
0.483300 + 0.875455i \(0.339438\pi\)
\(798\) 0 0
\(799\) 3811.19 0.168749
\(800\) 4677.44 0.206716
\(801\) 0 0
\(802\) −2811.89 −0.123805
\(803\) 5757.01 0.253002
\(804\) 0 0
\(805\) 0 0
\(806\) 9631.38 0.420907
\(807\) 0 0
\(808\) 10686.7 0.465292
\(809\) 42350.6 1.84050 0.920252 0.391325i \(-0.127983\pi\)
0.920252 + 0.391325i \(0.127983\pi\)
\(810\) 0 0
\(811\) 18910.7 0.818796 0.409398 0.912356i \(-0.365739\pi\)
0.409398 + 0.912356i \(0.365739\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1265.00 0.0544696
\(815\) 18371.3 0.789595
\(816\) 0 0
\(817\) −45299.4 −1.93981
\(818\) −21425.8 −0.915813
\(819\) 0 0
\(820\) −5593.33 −0.238204
\(821\) −6593.42 −0.280283 −0.140141 0.990132i \(-0.544756\pi\)
−0.140141 + 0.990132i \(0.544756\pi\)
\(822\) 0 0
\(823\) 26762.4 1.13351 0.566755 0.823886i \(-0.308199\pi\)
0.566755 + 0.823886i \(0.308199\pi\)
\(824\) 27208.5 1.15031
\(825\) 0 0
\(826\) 0 0
\(827\) 24016.7 1.00985 0.504924 0.863164i \(-0.331521\pi\)
0.504924 + 0.863164i \(0.331521\pi\)
\(828\) 0 0
\(829\) 28422.3 1.19077 0.595383 0.803442i \(-0.297000\pi\)
0.595383 + 0.803442i \(0.297000\pi\)
\(830\) −7999.56 −0.334541
\(831\) 0 0
\(832\) −6290.49 −0.262120
\(833\) 0 0
\(834\) 0 0
\(835\) −20205.4 −0.837411
\(836\) −6350.46 −0.262721
\(837\) 0 0
\(838\) 18137.3 0.747663
\(839\) 6637.09 0.273108 0.136554 0.990633i \(-0.456397\pi\)
0.136554 + 0.990633i \(0.456397\pi\)
\(840\) 0 0
\(841\) 313.546 0.0128560
\(842\) −11310.3 −0.462921
\(843\) 0 0
\(844\) −30550.9 −1.24598
\(845\) −2719.79 −0.110726
\(846\) 0 0
\(847\) 0 0
\(848\) 7899.01 0.319874
\(849\) 0 0
\(850\) −1788.11 −0.0721551
\(851\) −2707.59 −0.109066
\(852\) 0 0
\(853\) 1406.88 0.0564720 0.0282360 0.999601i \(-0.491011\pi\)
0.0282360 + 0.999601i \(0.491011\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18309.3 0.731075
\(857\) 27943.4 1.11380 0.556901 0.830579i \(-0.311990\pi\)
0.556901 + 0.830579i \(0.311990\pi\)
\(858\) 0 0
\(859\) −1936.63 −0.0769233 −0.0384616 0.999260i \(-0.512246\pi\)
−0.0384616 + 0.999260i \(0.512246\pi\)
\(860\) 9554.20 0.378832
\(861\) 0 0
\(862\) −20556.8 −0.812259
\(863\) −7947.70 −0.313491 −0.156746 0.987639i \(-0.550100\pi\)
−0.156746 + 0.987639i \(0.550100\pi\)
\(864\) 0 0
\(865\) −297.291 −0.0116858
\(866\) 19791.6 0.776611
\(867\) 0 0
\(868\) 0 0
\(869\) −171.984 −0.00671365
\(870\) 0 0
\(871\) 23649.6 0.920019
\(872\) 3216.00 0.124894
\(873\) 0 0
\(874\) −4742.06 −0.183527
\(875\) 0 0
\(876\) 0 0
\(877\) 38655.0 1.48835 0.744177 0.667983i \(-0.232842\pi\)
0.744177 + 0.667983i \(0.232842\pi\)
\(878\) 8764.65 0.336893
\(879\) 0 0
\(880\) 709.088 0.0271629
\(881\) −18879.4 −0.721978 −0.360989 0.932570i \(-0.617561\pi\)
−0.360989 + 0.932570i \(0.617561\pi\)
\(882\) 0 0
\(883\) −35098.1 −1.33765 −0.668825 0.743420i \(-0.733202\pi\)
−0.668825 + 0.743420i \(0.733202\pi\)
\(884\) −15439.4 −0.587425
\(885\) 0 0
\(886\) −19346.3 −0.733581
\(887\) −48816.4 −1.84791 −0.923954 0.382504i \(-0.875062\pi\)
−0.923954 + 0.382504i \(0.875062\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −10923.6 −0.411417
\(891\) 0 0
\(892\) −679.062 −0.0254895
\(893\) 10776.7 0.403839
\(894\) 0 0
\(895\) −14866.6 −0.555236
\(896\) 0 0
\(897\) 0 0
\(898\) −9916.41 −0.368502
\(899\) 20100.8 0.745718
\(900\) 0 0
\(901\) −21091.3 −0.779860
\(902\) 2066.22 0.0762721
\(903\) 0 0
\(904\) −35839.1 −1.31857
\(905\) −3381.10 −0.124190
\(906\) 0 0
\(907\) −6010.83 −0.220051 −0.110026 0.993929i \(-0.535093\pi\)
−0.110026 + 0.993929i \(0.535093\pi\)
\(908\) −28140.5 −1.02850
\(909\) 0 0
\(910\) 0 0
\(911\) −25780.1 −0.937576 −0.468788 0.883311i \(-0.655309\pi\)
−0.468788 + 0.883311i \(0.655309\pi\)
\(912\) 0 0
\(913\) −8470.37 −0.307041
\(914\) −17002.6 −0.615314
\(915\) 0 0
\(916\) −31934.6 −1.15191
\(917\) 0 0
\(918\) 0 0
\(919\) 26731.8 0.959522 0.479761 0.877399i \(-0.340723\pi\)
0.479761 + 0.877399i \(0.340723\pi\)
\(920\) 2349.25 0.0841874
\(921\) 0 0
\(922\) 12145.4 0.433824
\(923\) 47501.6 1.69397
\(924\) 0 0
\(925\) −2886.93 −0.102618
\(926\) −1826.46 −0.0648177
\(927\) 0 0
\(928\) −29406.2 −1.04020
\(929\) −30464.6 −1.07590 −0.537949 0.842977i \(-0.680801\pi\)
−0.537949 + 0.842977i \(0.680801\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −9542.22 −0.335371
\(933\) 0 0
\(934\) −23708.1 −0.830572
\(935\) −1893.35 −0.0662238
\(936\) 0 0
\(937\) 28533.4 0.994819 0.497409 0.867516i \(-0.334285\pi\)
0.497409 + 0.867516i \(0.334285\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2272.94 −0.0788671
\(941\) 34455.8 1.19365 0.596827 0.802370i \(-0.296428\pi\)
0.596827 + 0.802370i \(0.296428\pi\)
\(942\) 0 0
\(943\) −4422.50 −0.152722
\(944\) 2007.17 0.0692033
\(945\) 0 0
\(946\) −3529.39 −0.121301
\(947\) −2477.68 −0.0850198 −0.0425099 0.999096i \(-0.513535\pi\)
−0.0425099 + 0.999096i \(0.513535\pi\)
\(948\) 0 0
\(949\) 39577.5 1.35378
\(950\) −5056.16 −0.172677
\(951\) 0 0
\(952\) 0 0
\(953\) −41690.6 −1.41709 −0.708547 0.705663i \(-0.750649\pi\)
−0.708547 + 0.705663i \(0.750649\pi\)
\(954\) 0 0
\(955\) 82.7943 0.00280541
\(956\) −18463.5 −0.624635
\(957\) 0 0
\(958\) −2014.13 −0.0679266
\(959\) 0 0
\(960\) 0 0
\(961\) −13434.6 −0.450962
\(962\) 8696.44 0.291460
\(963\) 0 0
\(964\) 42325.0 1.41410
\(965\) −13473.4 −0.449455
\(966\) 0 0
\(967\) −20641.3 −0.686430 −0.343215 0.939257i \(-0.611516\pi\)
−0.343215 + 0.939257i \(0.611516\pi\)
\(968\) 25509.5 0.847010
\(969\) 0 0
\(970\) −3955.63 −0.130936
\(971\) 6626.17 0.218995 0.109497 0.993987i \(-0.465076\pi\)
0.109497 + 0.993987i \(0.465076\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −20377.0 −0.670349
\(975\) 0 0
\(976\) −17048.8 −0.559138
\(977\) −41961.0 −1.37405 −0.687027 0.726632i \(-0.741085\pi\)
−0.687027 + 0.726632i \(0.741085\pi\)
\(978\) 0 0
\(979\) −11566.5 −0.377598
\(980\) 0 0
\(981\) 0 0
\(982\) −6817.21 −0.221533
\(983\) −16781.7 −0.544510 −0.272255 0.962225i \(-0.587769\pi\)
−0.272255 + 0.962225i \(0.587769\pi\)
\(984\) 0 0
\(985\) 5136.88 0.166167
\(986\) 11241.6 0.363087
\(987\) 0 0
\(988\) −43657.2 −1.40579
\(989\) 7554.27 0.242884
\(990\) 0 0
\(991\) 50319.8 1.61298 0.806489 0.591250i \(-0.201365\pi\)
0.806489 + 0.591250i \(0.201365\pi\)
\(992\) −23928.3 −0.765851
\(993\) 0 0
\(994\) 0 0
\(995\) 14118.9 0.449847
\(996\) 0 0
\(997\) −12949.5 −0.411348 −0.205674 0.978621i \(-0.565939\pi\)
−0.205674 + 0.978621i \(0.565939\pi\)
\(998\) −16355.2 −0.518753
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2205.4.a.bh.1.1 2
3.2 odd 2 735.4.a.k.1.2 2
7.6 odd 2 315.4.a.m.1.1 2
21.20 even 2 105.4.a.c.1.2 2
35.34 odd 2 1575.4.a.m.1.2 2
84.83 odd 2 1680.4.a.bk.1.1 2
105.62 odd 4 525.4.d.i.274.2 4
105.83 odd 4 525.4.d.i.274.3 4
105.104 even 2 525.4.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.c.1.2 2 21.20 even 2
315.4.a.m.1.1 2 7.6 odd 2
525.4.a.p.1.1 2 105.104 even 2
525.4.d.i.274.2 4 105.62 odd 4
525.4.d.i.274.3 4 105.83 odd 4
735.4.a.k.1.2 2 3.2 odd 2
1575.4.a.m.1.2 2 35.34 odd 2
1680.4.a.bk.1.1 2 84.83 odd 2
2205.4.a.bh.1.1 2 1.1 even 1 trivial