Properties

Label 825.4.c.i.199.1
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-3.37228i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.i.199.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.37228i q^{2} +3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} +4.74456i q^{7} -15.6060i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-3.37228i q^{2} +3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} +4.74456i q^{7} -15.6060i q^{8} -9.00000 q^{9} +11.0000 q^{11} -10.1168i q^{12} -15.0217i q^{13} +16.0000 q^{14} -79.6060 q^{16} -73.1684i q^{17} +30.3505i q^{18} +78.7011 q^{19} -14.2337 q^{21} -37.0951i q^{22} +112.000i q^{23} +46.8179 q^{24} -50.6576 q^{26} -27.0000i q^{27} -16.0000i q^{28} -243.125 q^{29} +278.717 q^{31} +143.606i q^{32} +33.0000i q^{33} -246.745 q^{34} +30.3505 q^{36} -102.380i q^{37} -265.402i q^{38} +45.0652 q^{39} -241.255 q^{41} +48.0000i q^{42} -280.016i q^{43} -37.0951 q^{44} +377.696 q^{46} +169.870i q^{47} -238.818i q^{48} +320.489 q^{49} +219.505 q^{51} +50.6576i q^{52} -409.652i q^{53} -91.0516 q^{54} +74.0435 q^{56} +236.103i q^{57} +819.886i q^{58} -196.000 q^{59} -701.359 q^{61} -939.913i q^{62} -42.7011i q^{63} -152.568 q^{64} +111.285 q^{66} -900.587i q^{67} +246.745i q^{68} -336.000 q^{69} +756.500 q^{71} +140.454i q^{72} -1019.81i q^{73} -345.255 q^{74} -265.402 q^{76} +52.1902i q^{77} -151.973i q^{78} +327.549 q^{79} +81.0000 q^{81} +813.581i q^{82} -756.619i q^{83} +48.0000 q^{84} -944.293 q^{86} -729.375i q^{87} -171.666i q^{88} -508.978 q^{89} +71.2716 q^{91} -377.696i q^{92} +836.152i q^{93} +572.848 q^{94} -430.818 q^{96} -614.358i q^{97} -1080.78i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44} + 224 q^{46} + 1236 q^{49} - 156 q^{51} - 54 q^{54} + 480 q^{56} - 784 q^{59} - 2208 q^{61} - 1426 q^{64} + 66 q^{66} - 1344 q^{69} + 912 q^{71} - 1404 q^{74} - 648 q^{76} + 460 q^{79} + 324 q^{81} + 192 q^{84} - 2904 q^{86} - 1944 q^{89} - 680 q^{91} + 1648 q^{94} - 1482 q^{96} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.37228i − 1.19228i −0.802880 0.596141i \(-0.796700\pi\)
0.802880 0.596141i \(-0.203300\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −3.37228 −0.421535
\(5\) 0 0
\(6\) 10.1168 0.688364
\(7\) 4.74456i 0.256182i 0.991762 + 0.128091i \(0.0408850\pi\)
−0.991762 + 0.128091i \(0.959115\pi\)
\(8\) − 15.6060i − 0.689693i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 10.1168i − 0.243373i
\(13\) − 15.0217i − 0.320483i −0.987078 0.160242i \(-0.948773\pi\)
0.987078 0.160242i \(-0.0512274\pi\)
\(14\) 16.0000 0.305441
\(15\) 0 0
\(16\) −79.6060 −1.24384
\(17\) − 73.1684i − 1.04388i −0.852982 0.521940i \(-0.825209\pi\)
0.852982 0.521940i \(-0.174791\pi\)
\(18\) 30.3505i 0.397427i
\(19\) 78.7011 0.950277 0.475138 0.879911i \(-0.342398\pi\)
0.475138 + 0.879911i \(0.342398\pi\)
\(20\) 0 0
\(21\) −14.2337 −0.147907
\(22\) − 37.0951i − 0.359486i
\(23\) 112.000i 1.01537i 0.861541 + 0.507687i \(0.169499\pi\)
−0.861541 + 0.507687i \(0.830501\pi\)
\(24\) 46.8179 0.398194
\(25\) 0 0
\(26\) −50.6576 −0.382106
\(27\) − 27.0000i − 0.192450i
\(28\) − 16.0000i − 0.107990i
\(29\) −243.125 −1.55680 −0.778399 0.627769i \(-0.783968\pi\)
−0.778399 + 0.627769i \(0.783968\pi\)
\(30\) 0 0
\(31\) 278.717 1.61481 0.807405 0.589998i \(-0.200871\pi\)
0.807405 + 0.589998i \(0.200871\pi\)
\(32\) 143.606i 0.793318i
\(33\) 33.0000i 0.174078i
\(34\) −246.745 −1.24460
\(35\) 0 0
\(36\) 30.3505 0.140512
\(37\) − 102.380i − 0.454898i −0.973790 0.227449i \(-0.926961\pi\)
0.973790 0.227449i \(-0.0730385\pi\)
\(38\) − 265.402i − 1.13300i
\(39\) 45.0652 0.185031
\(40\) 0 0
\(41\) −241.255 −0.918970 −0.459485 0.888186i \(-0.651966\pi\)
−0.459485 + 0.888186i \(0.651966\pi\)
\(42\) 48.0000i 0.176347i
\(43\) − 280.016i − 0.993071i −0.868016 0.496536i \(-0.834605\pi\)
0.868016 0.496536i \(-0.165395\pi\)
\(44\) −37.0951 −0.127098
\(45\) 0 0
\(46\) 377.696 1.21061
\(47\) 169.870i 0.527192i 0.964633 + 0.263596i \(0.0849085\pi\)
−0.964633 + 0.263596i \(0.915091\pi\)
\(48\) − 238.818i − 0.718133i
\(49\) 320.489 0.934371
\(50\) 0 0
\(51\) 219.505 0.602684
\(52\) 50.6576i 0.135095i
\(53\) − 409.652i − 1.06170i −0.847466 0.530849i \(-0.821873\pi\)
0.847466 0.530849i \(-0.178127\pi\)
\(54\) −91.0516 −0.229455
\(55\) 0 0
\(56\) 74.0435 0.176687
\(57\) 236.103i 0.548643i
\(58\) 819.886i 1.85614i
\(59\) −196.000 −0.432492 −0.216246 0.976339i \(-0.569381\pi\)
−0.216246 + 0.976339i \(0.569381\pi\)
\(60\) 0 0
\(61\) −701.359 −1.47213 −0.736064 0.676912i \(-0.763318\pi\)
−0.736064 + 0.676912i \(0.763318\pi\)
\(62\) − 939.913i − 1.92531i
\(63\) − 42.7011i − 0.0853941i
\(64\) −152.568 −0.297984
\(65\) 0 0
\(66\) 111.285 0.207550
\(67\) − 900.587i − 1.64215i −0.570819 0.821076i \(-0.693374\pi\)
0.570819 0.821076i \(-0.306626\pi\)
\(68\) 246.745i 0.440032i
\(69\) −336.000 −0.586227
\(70\) 0 0
\(71\) 756.500 1.26451 0.632254 0.774762i \(-0.282130\pi\)
0.632254 + 0.774762i \(0.282130\pi\)
\(72\) 140.454i 0.229898i
\(73\) − 1019.81i − 1.63507i −0.575877 0.817536i \(-0.695339\pi\)
0.575877 0.817536i \(-0.304661\pi\)
\(74\) −345.255 −0.542367
\(75\) 0 0
\(76\) −265.402 −0.400575
\(77\) 52.1902i 0.0772419i
\(78\) − 151.973i − 0.220609i
\(79\) 327.549 0.466483 0.233241 0.972419i \(-0.425067\pi\)
0.233241 + 0.972419i \(0.425067\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 813.581i 1.09567i
\(83\) − 756.619i − 1.00060i −0.865852 0.500300i \(-0.833223\pi\)
0.865852 0.500300i \(-0.166777\pi\)
\(84\) 48.0000 0.0623480
\(85\) 0 0
\(86\) −944.293 −1.18402
\(87\) − 729.375i − 0.898818i
\(88\) − 171.666i − 0.207950i
\(89\) −508.978 −0.606198 −0.303099 0.952959i \(-0.598021\pi\)
−0.303099 + 0.952959i \(0.598021\pi\)
\(90\) 0 0
\(91\) 71.2716 0.0821022
\(92\) − 377.696i − 0.428016i
\(93\) 836.152i 0.932311i
\(94\) 572.848 0.628561
\(95\) 0 0
\(96\) −430.818 −0.458023
\(97\) − 614.358i − 0.643079i −0.946896 0.321539i \(-0.895800\pi\)
0.946896 0.321539i \(-0.104200\pi\)
\(98\) − 1080.78i − 1.11403i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −1015.92 −1.00087 −0.500434 0.865775i \(-0.666826\pi\)
−0.500434 + 0.865775i \(0.666826\pi\)
\(102\) − 740.234i − 0.718569i
\(103\) 1102.16i 1.05436i 0.849753 + 0.527181i \(0.176751\pi\)
−0.849753 + 0.527181i \(0.823249\pi\)
\(104\) −234.429 −0.221035
\(105\) 0 0
\(106\) −1381.46 −1.26584
\(107\) − 1377.58i − 1.24463i −0.782767 0.622315i \(-0.786192\pi\)
0.782767 0.622315i \(-0.213808\pi\)
\(108\) 91.0516i 0.0811245i
\(109\) −320.217 −0.281388 −0.140694 0.990053i \(-0.544933\pi\)
−0.140694 + 0.990053i \(0.544933\pi\)
\(110\) 0 0
\(111\) 307.141 0.262636
\(112\) − 377.696i − 0.318651i
\(113\) − 1629.45i − 1.35651i −0.734828 0.678254i \(-0.762737\pi\)
0.734828 0.678254i \(-0.237263\pi\)
\(114\) 796.206 0.654136
\(115\) 0 0
\(116\) 819.886 0.656245
\(117\) 135.196i 0.106828i
\(118\) 660.967i 0.515652i
\(119\) 347.152 0.267423
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 2365.18i 1.75519i
\(123\) − 723.766i − 0.530568i
\(124\) −939.913 −0.680699
\(125\) 0 0
\(126\) −144.000 −0.101814
\(127\) − 2291.26i − 1.60091i −0.599390 0.800457i \(-0.704590\pi\)
0.599390 0.800457i \(-0.295410\pi\)
\(128\) 1663.35i 1.14860i
\(129\) 840.049 0.573350
\(130\) 0 0
\(131\) −1147.41 −0.765267 −0.382633 0.923900i \(-0.624983\pi\)
−0.382633 + 0.923900i \(0.624983\pi\)
\(132\) − 111.285i − 0.0733799i
\(133\) 373.402i 0.243444i
\(134\) −3037.03 −1.95791
\(135\) 0 0
\(136\) −1141.86 −0.719956
\(137\) − 1268.60i − 0.791121i −0.918440 0.395561i \(-0.870550\pi\)
0.918440 0.395561i \(-0.129450\pi\)
\(138\) 1133.09i 0.698947i
\(139\) 486.288 0.296737 0.148368 0.988932i \(-0.452598\pi\)
0.148368 + 0.988932i \(0.452598\pi\)
\(140\) 0 0
\(141\) −509.609 −0.304374
\(142\) − 2551.13i − 1.50765i
\(143\) − 165.239i − 0.0966294i
\(144\) 716.454 0.414614
\(145\) 0 0
\(146\) −3439.10 −1.94947
\(147\) 961.467i 0.539459i
\(148\) 345.255i 0.191756i
\(149\) −2354.11 −1.29434 −0.647169 0.762346i \(-0.724047\pi\)
−0.647169 + 0.762346i \(0.724047\pi\)
\(150\) 0 0
\(151\) −570.070 −0.307229 −0.153615 0.988131i \(-0.549091\pi\)
−0.153615 + 0.988131i \(0.549091\pi\)
\(152\) − 1228.21i − 0.655399i
\(153\) 658.516i 0.347960i
\(154\) 176.000 0.0920941
\(155\) 0 0
\(156\) −151.973 −0.0779971
\(157\) 2072.67i 1.05361i 0.849985 + 0.526807i \(0.176611\pi\)
−0.849985 + 0.526807i \(0.823389\pi\)
\(158\) − 1104.59i − 0.556179i
\(159\) 1228.96 0.612972
\(160\) 0 0
\(161\) −531.391 −0.260121
\(162\) − 273.155i − 0.132476i
\(163\) 2676.51i 1.28614i 0.765808 + 0.643069i \(0.222339\pi\)
−0.765808 + 0.643069i \(0.777661\pi\)
\(164\) 813.581 0.387378
\(165\) 0 0
\(166\) −2551.53 −1.19300
\(167\) 1188.12i 0.550536i 0.961368 + 0.275268i \(0.0887665\pi\)
−0.961368 + 0.275268i \(0.911233\pi\)
\(168\) 222.130i 0.102010i
\(169\) 1971.35 0.897290
\(170\) 0 0
\(171\) −708.310 −0.316759
\(172\) 944.293i 0.418615i
\(173\) 807.147i 0.354718i 0.984146 + 0.177359i \(0.0567554\pi\)
−0.984146 + 0.177359i \(0.943245\pi\)
\(174\) −2459.66 −1.07164
\(175\) 0 0
\(176\) −875.666 −0.375033
\(177\) − 588.000i − 0.249699i
\(178\) 1716.42i 0.722758i
\(179\) 1950.39 0.814408 0.407204 0.913337i \(-0.366504\pi\)
0.407204 + 0.913337i \(0.366504\pi\)
\(180\) 0 0
\(181\) 1061.61 0.435959 0.217980 0.975953i \(-0.430053\pi\)
0.217980 + 0.975953i \(0.430053\pi\)
\(182\) − 240.348i − 0.0978889i
\(183\) − 2104.08i − 0.849933i
\(184\) 1747.87 0.700297
\(185\) 0 0
\(186\) 2819.74 1.11158
\(187\) − 804.853i − 0.314742i
\(188\) − 572.848i − 0.222230i
\(189\) 128.103 0.0493023
\(190\) 0 0
\(191\) 2136.41 0.809348 0.404674 0.914461i \(-0.367385\pi\)
0.404674 + 0.914461i \(0.367385\pi\)
\(192\) − 457.704i − 0.172041i
\(193\) 3947.76i 1.47236i 0.676784 + 0.736181i \(0.263373\pi\)
−0.676784 + 0.736181i \(0.736627\pi\)
\(194\) −2071.79 −0.766731
\(195\) 0 0
\(196\) −1080.78 −0.393870
\(197\) − 923.886i − 0.334133i −0.985946 0.167066i \(-0.946571\pi\)
0.985946 0.167066i \(-0.0534294\pi\)
\(198\) 333.856i 0.119829i
\(199\) 476.152 0.169616 0.0848078 0.996397i \(-0.472972\pi\)
0.0848078 + 0.996397i \(0.472972\pi\)
\(200\) 0 0
\(201\) 2701.76 0.948097
\(202\) 3425.96i 1.19332i
\(203\) − 1153.52i − 0.398824i
\(204\) −740.234 −0.254053
\(205\) 0 0
\(206\) 3716.80 1.25710
\(207\) − 1008.00i − 0.338458i
\(208\) 1195.82i 0.398631i
\(209\) 865.712 0.286519
\(210\) 0 0
\(211\) −4918.24 −1.60467 −0.802336 0.596872i \(-0.796410\pi\)
−0.802336 + 0.596872i \(0.796410\pi\)
\(212\) 1381.46i 0.447543i
\(213\) 2269.50i 0.730064i
\(214\) −4645.57 −1.48395
\(215\) 0 0
\(216\) −421.361 −0.132731
\(217\) 1322.39i 0.413686i
\(218\) 1079.86i 0.335494i
\(219\) 3059.44 0.944010
\(220\) 0 0
\(221\) −1099.12 −0.334546
\(222\) − 1035.77i − 0.313136i
\(223\) 2100.29i 0.630700i 0.948975 + 0.315350i \(0.102122\pi\)
−0.948975 + 0.315350i \(0.897878\pi\)
\(224\) −681.348 −0.203234
\(225\) 0 0
\(226\) −5494.95 −1.61734
\(227\) 2257.16i 0.659970i 0.943986 + 0.329985i \(0.107044\pi\)
−0.943986 + 0.329985i \(0.892956\pi\)
\(228\) − 796.206i − 0.231272i
\(229\) 5311.07 1.53260 0.766301 0.642482i \(-0.222095\pi\)
0.766301 + 0.642482i \(0.222095\pi\)
\(230\) 0 0
\(231\) −156.571 −0.0445956
\(232\) 3794.20i 1.07371i
\(233\) 2466.27i 0.693435i 0.937970 + 0.346718i \(0.112704\pi\)
−0.937970 + 0.346718i \(0.887296\pi\)
\(234\) 455.918 0.127369
\(235\) 0 0
\(236\) 660.967 0.182311
\(237\) 982.646i 0.269324i
\(238\) − 1170.70i − 0.318844i
\(239\) −1429.40 −0.386863 −0.193432 0.981114i \(-0.561962\pi\)
−0.193432 + 0.981114i \(0.561962\pi\)
\(240\) 0 0
\(241\) −978.989 −0.261669 −0.130835 0.991404i \(-0.541766\pi\)
−0.130835 + 0.991404i \(0.541766\pi\)
\(242\) − 408.046i − 0.108389i
\(243\) 243.000i 0.0641500i
\(244\) 2365.18 0.620553
\(245\) 0 0
\(246\) −2440.74 −0.632586
\(247\) − 1182.23i − 0.304548i
\(248\) − 4349.65i − 1.11372i
\(249\) 2269.86 0.577696
\(250\) 0 0
\(251\) −6530.63 −1.64227 −0.821135 0.570734i \(-0.806659\pi\)
−0.821135 + 0.570734i \(0.806659\pi\)
\(252\) 144.000i 0.0359966i
\(253\) 1232.00i 0.306147i
\(254\) −7726.76 −1.90874
\(255\) 0 0
\(256\) 4388.74 1.07147
\(257\) − 8130.26i − 1.97335i −0.162696 0.986676i \(-0.552019\pi\)
0.162696 0.986676i \(-0.447981\pi\)
\(258\) − 2832.88i − 0.683595i
\(259\) 485.750 0.116537
\(260\) 0 0
\(261\) 2188.12 0.518933
\(262\) 3869.40i 0.912414i
\(263\) − 4549.42i − 1.06665i −0.845910 0.533326i \(-0.820942\pi\)
0.845910 0.533326i \(-0.179058\pi\)
\(264\) 514.997 0.120060
\(265\) 0 0
\(266\) 1259.22 0.290254
\(267\) − 1526.93i − 0.349988i
\(268\) 3037.03i 0.692225i
\(269\) 29.1522 0.00660760 0.00330380 0.999995i \(-0.498948\pi\)
0.00330380 + 0.999995i \(0.498948\pi\)
\(270\) 0 0
\(271\) 7711.22 1.72850 0.864250 0.503063i \(-0.167794\pi\)
0.864250 + 0.503063i \(0.167794\pi\)
\(272\) 5824.64i 1.29842i
\(273\) 213.815i 0.0474017i
\(274\) −4278.07 −0.943239
\(275\) 0 0
\(276\) 1133.09 0.247115
\(277\) − 1127.52i − 0.244571i −0.992495 0.122286i \(-0.960978\pi\)
0.992495 0.122286i \(-0.0390224\pi\)
\(278\) − 1639.90i − 0.353794i
\(279\) −2508.46 −0.538270
\(280\) 0 0
\(281\) −1872.47 −0.397517 −0.198758 0.980049i \(-0.563691\pi\)
−0.198758 + 0.980049i \(0.563691\pi\)
\(282\) 1718.54i 0.362900i
\(283\) 2124.48i 0.446245i 0.974790 + 0.223123i \(0.0716251\pi\)
−0.974790 + 0.223123i \(0.928375\pi\)
\(284\) −2551.13 −0.533034
\(285\) 0 0
\(286\) −557.233 −0.115209
\(287\) − 1144.65i − 0.235424i
\(288\) − 1292.45i − 0.264439i
\(289\) −440.621 −0.0896846
\(290\) 0 0
\(291\) 1843.07 0.371282
\(292\) 3439.10i 0.689241i
\(293\) − 3324.19i − 0.662802i −0.943490 0.331401i \(-0.892479\pi\)
0.943490 0.331401i \(-0.107521\pi\)
\(294\) 3242.34 0.643187
\(295\) 0 0
\(296\) −1597.75 −0.313740
\(297\) − 297.000i − 0.0580259i
\(298\) 7938.73i 1.54322i
\(299\) 1682.44 0.325411
\(300\) 0 0
\(301\) 1328.55 0.254407
\(302\) 1922.44i 0.366304i
\(303\) − 3047.75i − 0.577851i
\(304\) −6265.07 −1.18200
\(305\) 0 0
\(306\) 2220.70 0.414866
\(307\) 1698.94i 0.315843i 0.987452 + 0.157921i \(0.0504793\pi\)
−0.987452 + 0.157921i \(0.949521\pi\)
\(308\) − 176.000i − 0.0325602i
\(309\) −3306.49 −0.608736
\(310\) 0 0
\(311\) 6928.83 1.26334 0.631668 0.775239i \(-0.282370\pi\)
0.631668 + 0.775239i \(0.282370\pi\)
\(312\) − 703.287i − 0.127615i
\(313\) − 3560.75i − 0.643020i −0.946906 0.321510i \(-0.895810\pi\)
0.946906 0.321510i \(-0.104190\pi\)
\(314\) 6989.64 1.25620
\(315\) 0 0
\(316\) −1104.59 −0.196639
\(317\) − 332.750i − 0.0589561i −0.999565 0.0294780i \(-0.990615\pi\)
0.999565 0.0294780i \(-0.00938451\pi\)
\(318\) − 4144.39i − 0.730835i
\(319\) −2674.37 −0.469393
\(320\) 0 0
\(321\) 4132.73 0.718587
\(322\) 1792.00i 0.310137i
\(323\) − 5758.43i − 0.991975i
\(324\) −273.155 −0.0468372
\(325\) 0 0
\(326\) 9025.94 1.53344
\(327\) − 960.652i − 0.162459i
\(328\) 3765.02i 0.633807i
\(329\) −805.957 −0.135057
\(330\) 0 0
\(331\) −541.445 −0.0899108 −0.0449554 0.998989i \(-0.514315\pi\)
−0.0449554 + 0.998989i \(0.514315\pi\)
\(332\) 2551.53i 0.421788i
\(333\) 921.423i 0.151633i
\(334\) 4006.67 0.656393
\(335\) 0 0
\(336\) 1133.09 0.183973
\(337\) − 816.531i − 0.131986i −0.997820 0.0659930i \(-0.978978\pi\)
0.997820 0.0659930i \(-0.0210215\pi\)
\(338\) − 6647.94i − 1.06982i
\(339\) 4888.34 0.783180
\(340\) 0 0
\(341\) 3065.89 0.486883
\(342\) 2388.62i 0.377666i
\(343\) 3147.97i 0.495552i
\(344\) −4369.92 −0.684914
\(345\) 0 0
\(346\) 2721.93 0.422924
\(347\) − 6260.53i − 0.968539i −0.874919 0.484269i \(-0.839086\pi\)
0.874919 0.484269i \(-0.160914\pi\)
\(348\) 2459.66i 0.378884i
\(349\) 12768.5 1.95840 0.979198 0.202906i \(-0.0650386\pi\)
0.979198 + 0.202906i \(0.0650386\pi\)
\(350\) 0 0
\(351\) −405.587 −0.0616771
\(352\) 1579.67i 0.239194i
\(353\) − 2649.28i − 0.399453i −0.979852 0.199727i \(-0.935995\pi\)
0.979852 0.199727i \(-0.0640054\pi\)
\(354\) −1982.90 −0.297712
\(355\) 0 0
\(356\) 1716.42 0.255534
\(357\) 1041.46i 0.154397i
\(358\) − 6577.27i − 0.971004i
\(359\) 3203.91 0.471020 0.235510 0.971872i \(-0.424324\pi\)
0.235510 + 0.971872i \(0.424324\pi\)
\(360\) 0 0
\(361\) −665.143 −0.0969737
\(362\) − 3580.04i − 0.519786i
\(363\) 363.000i 0.0524864i
\(364\) −240.348 −0.0346089
\(365\) 0 0
\(366\) −7095.54 −1.01336
\(367\) 8429.40i 1.19894i 0.800397 + 0.599470i \(0.204622\pi\)
−0.800397 + 0.599470i \(0.795378\pi\)
\(368\) − 8915.87i − 1.26297i
\(369\) 2171.30 0.306323
\(370\) 0 0
\(371\) 1943.62 0.271988
\(372\) − 2819.74i − 0.393002i
\(373\) − 9388.53i − 1.30327i −0.758533 0.651635i \(-0.774083\pi\)
0.758533 0.651635i \(-0.225917\pi\)
\(374\) −2714.19 −0.375261
\(375\) 0 0
\(376\) 2650.98 0.363600
\(377\) 3652.16i 0.498928i
\(378\) − 432.000i − 0.0587822i
\(379\) 14264.5 1.93329 0.966647 0.256112i \(-0.0824415\pi\)
0.966647 + 0.256112i \(0.0824415\pi\)
\(380\) 0 0
\(381\) 6873.77 0.924288
\(382\) − 7204.58i − 0.964970i
\(383\) 13462.2i 1.79605i 0.439942 + 0.898026i \(0.354999\pi\)
−0.439942 + 0.898026i \(0.645001\pi\)
\(384\) −4990.05 −0.663144
\(385\) 0 0
\(386\) 13313.0 1.75547
\(387\) 2520.15i 0.331024i
\(388\) 2071.79i 0.271080i
\(389\) 941.881 0.122764 0.0613821 0.998114i \(-0.480449\pi\)
0.0613821 + 0.998114i \(0.480449\pi\)
\(390\) 0 0
\(391\) 8194.87 1.05993
\(392\) − 5001.54i − 0.644429i
\(393\) − 3442.24i − 0.441827i
\(394\) −3115.60 −0.398380
\(395\) 0 0
\(396\) 333.856 0.0423659
\(397\) 847.839i 0.107183i 0.998563 + 0.0535917i \(0.0170669\pi\)
−0.998563 + 0.0535917i \(0.982933\pi\)
\(398\) − 1605.72i − 0.202230i
\(399\) −1120.21 −0.140553
\(400\) 0 0
\(401\) 12203.6 1.51975 0.759875 0.650069i \(-0.225260\pi\)
0.759875 + 0.650069i \(0.225260\pi\)
\(402\) − 9111.10i − 1.13040i
\(403\) − 4186.82i − 0.517520i
\(404\) 3425.96 0.421901
\(405\) 0 0
\(406\) −3890.00 −0.475511
\(407\) − 1126.18i − 0.137157i
\(408\) − 3425.59i − 0.415667i
\(409\) −8759.53 −1.05900 −0.529500 0.848310i \(-0.677620\pi\)
−0.529500 + 0.848310i \(0.677620\pi\)
\(410\) 0 0
\(411\) 3805.79 0.456754
\(412\) − 3716.80i − 0.444451i
\(413\) − 929.934i − 0.110797i
\(414\) −3399.26 −0.403537
\(415\) 0 0
\(416\) 2157.21 0.254245
\(417\) 1458.86i 0.171321i
\(418\) − 2919.42i − 0.341612i
\(419\) 11188.4 1.30451 0.652256 0.757999i \(-0.273823\pi\)
0.652256 + 0.757999i \(0.273823\pi\)
\(420\) 0 0
\(421\) −14082.3 −1.63023 −0.815116 0.579298i \(-0.803327\pi\)
−0.815116 + 0.579298i \(0.803327\pi\)
\(422\) 16585.7i 1.91322i
\(423\) − 1528.83i − 0.175731i
\(424\) −6393.02 −0.732246
\(425\) 0 0
\(426\) 7653.39 0.870441
\(427\) − 3327.64i − 0.377133i
\(428\) 4645.57i 0.524655i
\(429\) 495.718 0.0557890
\(430\) 0 0
\(431\) −5616.05 −0.627647 −0.313823 0.949481i \(-0.601610\pi\)
−0.313823 + 0.949481i \(0.601610\pi\)
\(432\) 2149.36i 0.239378i
\(433\) 7195.75i 0.798627i 0.916814 + 0.399314i \(0.130752\pi\)
−0.916814 + 0.399314i \(0.869248\pi\)
\(434\) 4459.48 0.493230
\(435\) 0 0
\(436\) 1079.86 0.118615
\(437\) 8814.52i 0.964887i
\(438\) − 10317.3i − 1.12553i
\(439\) −101.959 −0.0110848 −0.00554240 0.999985i \(-0.501764\pi\)
−0.00554240 + 0.999985i \(0.501764\pi\)
\(440\) 0 0
\(441\) −2884.40 −0.311457
\(442\) 3706.53i 0.398873i
\(443\) 4953.74i 0.531285i 0.964072 + 0.265642i \(0.0855841\pi\)
−0.964072 + 0.265642i \(0.914416\pi\)
\(444\) −1035.77 −0.110710
\(445\) 0 0
\(446\) 7082.78 0.751972
\(447\) − 7062.34i − 0.747287i
\(448\) − 723.869i − 0.0763383i
\(449\) 11602.0 1.21945 0.609723 0.792615i \(-0.291281\pi\)
0.609723 + 0.792615i \(0.291281\pi\)
\(450\) 0 0
\(451\) −2653.81 −0.277080
\(452\) 5494.95i 0.571816i
\(453\) − 1710.21i − 0.177379i
\(454\) 7611.79 0.786870
\(455\) 0 0
\(456\) 3684.62 0.378395
\(457\) 3530.68i 0.361397i 0.983539 + 0.180698i \(0.0578358\pi\)
−0.983539 + 0.180698i \(0.942164\pi\)
\(458\) − 17910.4i − 1.82729i
\(459\) −1975.55 −0.200895
\(460\) 0 0
\(461\) 11566.3 1.16854 0.584271 0.811559i \(-0.301381\pi\)
0.584271 + 0.811559i \(0.301381\pi\)
\(462\) 528.000i 0.0531705i
\(463\) 10888.5i 1.09294i 0.837479 + 0.546470i \(0.184029\pi\)
−0.837479 + 0.546470i \(0.815971\pi\)
\(464\) 19354.2 1.93641
\(465\) 0 0
\(466\) 8316.94 0.826770
\(467\) − 10688.0i − 1.05906i −0.848292 0.529529i \(-0.822369\pi\)
0.848292 0.529529i \(-0.177631\pi\)
\(468\) − 455.918i − 0.0450317i
\(469\) 4272.89 0.420690
\(470\) 0 0
\(471\) −6218.02 −0.608304
\(472\) 3058.77i 0.298287i
\(473\) − 3080.18i − 0.299422i
\(474\) 3313.76 0.321110
\(475\) 0 0
\(476\) −1170.70 −0.112728
\(477\) 3686.87i 0.353900i
\(478\) 4820.35i 0.461250i
\(479\) −2341.90 −0.223391 −0.111696 0.993742i \(-0.535628\pi\)
−0.111696 + 0.993742i \(0.535628\pi\)
\(480\) 0 0
\(481\) −1537.93 −0.145787
\(482\) 3301.43i 0.311983i
\(483\) − 1594.17i − 0.150181i
\(484\) −408.046 −0.0383214
\(485\) 0 0
\(486\) 819.464 0.0764849
\(487\) − 6748.91i − 0.627972i −0.949428 0.313986i \(-0.898335\pi\)
0.949428 0.313986i \(-0.101665\pi\)
\(488\) 10945.4i 1.01532i
\(489\) −8029.53 −0.742552
\(490\) 0 0
\(491\) 7361.40 0.676609 0.338305 0.941037i \(-0.390147\pi\)
0.338305 + 0.941037i \(0.390147\pi\)
\(492\) 2440.74i 0.223653i
\(493\) 17789.1i 1.62511i
\(494\) −3986.80 −0.363107
\(495\) 0 0
\(496\) −22187.6 −2.00857
\(497\) 3589.26i 0.323944i
\(498\) − 7654.60i − 0.688777i
\(499\) −10381.7 −0.931359 −0.465680 0.884953i \(-0.654190\pi\)
−0.465680 + 0.884953i \(0.654190\pi\)
\(500\) 0 0
\(501\) −3564.36 −0.317852
\(502\) 22023.1i 1.95805i
\(503\) 19149.0i 1.69744i 0.528840 + 0.848721i \(0.322627\pi\)
−0.528840 + 0.848721i \(0.677373\pi\)
\(504\) −666.391 −0.0588957
\(505\) 0 0
\(506\) 4154.65 0.365013
\(507\) 5914.04i 0.518051i
\(508\) 7726.76i 0.674841i
\(509\) −16073.2 −1.39967 −0.699836 0.714303i \(-0.746744\pi\)
−0.699836 + 0.714303i \(0.746744\pi\)
\(510\) 0 0
\(511\) 4838.58 0.418877
\(512\) − 1493.27i − 0.128894i
\(513\) − 2124.93i − 0.182881i
\(514\) −27417.5 −2.35279
\(515\) 0 0
\(516\) −2832.88 −0.241687
\(517\) 1868.56i 0.158954i
\(518\) − 1638.09i − 0.138945i
\(519\) −2421.44 −0.204797
\(520\) 0 0
\(521\) −18955.3 −1.59395 −0.796975 0.604012i \(-0.793568\pi\)
−0.796975 + 0.604012i \(0.793568\pi\)
\(522\) − 7378.97i − 0.618714i
\(523\) − 4442.19i − 0.371402i −0.982606 0.185701i \(-0.940544\pi\)
0.982606 0.185701i \(-0.0594556\pi\)
\(524\) 3869.40 0.322587
\(525\) 0 0
\(526\) −15341.9 −1.27175
\(527\) − 20393.3i − 1.68567i
\(528\) − 2627.00i − 0.216525i
\(529\) −377.000 −0.0309855
\(530\) 0 0
\(531\) 1764.00 0.144164
\(532\) − 1259.22i − 0.102620i
\(533\) 3624.08i 0.294515i
\(534\) −5149.25 −0.417285
\(535\) 0 0
\(536\) −14054.5 −1.13258
\(537\) 5851.17i 0.470199i
\(538\) − 98.3096i − 0.00787812i
\(539\) 3525.38 0.281723
\(540\) 0 0
\(541\) 2180.90 0.173316 0.0866580 0.996238i \(-0.472381\pi\)
0.0866580 + 0.996238i \(0.472381\pi\)
\(542\) − 26004.4i − 2.06086i
\(543\) 3184.82i 0.251701i
\(544\) 10507.4 0.828129
\(545\) 0 0
\(546\) 721.044 0.0565162
\(547\) − 8225.04i − 0.642920i −0.946923 0.321460i \(-0.895826\pi\)
0.946923 0.321460i \(-0.104174\pi\)
\(548\) 4278.07i 0.333485i
\(549\) 6312.23 0.490709
\(550\) 0 0
\(551\) −19134.2 −1.47939
\(552\) 5243.61i 0.404316i
\(553\) 1554.08i 0.119505i
\(554\) −3802.32 −0.291598
\(555\) 0 0
\(556\) −1639.90 −0.125085
\(557\) 25181.9i 1.91561i 0.287423 + 0.957804i \(0.407201\pi\)
−0.287423 + 0.957804i \(0.592799\pi\)
\(558\) 8459.22i 0.641769i
\(559\) −4206.33 −0.318263
\(560\) 0 0
\(561\) 2414.56 0.181716
\(562\) 6314.50i 0.473952i
\(563\) − 4504.50i − 0.337197i −0.985685 0.168599i \(-0.946076\pi\)
0.985685 0.168599i \(-0.0539242\pi\)
\(564\) 1718.54 0.128304
\(565\) 0 0
\(566\) 7164.36 0.532050
\(567\) 384.310i 0.0284647i
\(568\) − 11805.9i − 0.872122i
\(569\) 13447.0 0.990732 0.495366 0.868684i \(-0.335034\pi\)
0.495366 + 0.868684i \(0.335034\pi\)
\(570\) 0 0
\(571\) −2605.52 −0.190959 −0.0954795 0.995431i \(-0.530438\pi\)
−0.0954795 + 0.995431i \(0.530438\pi\)
\(572\) 557.233i 0.0407327i
\(573\) 6409.24i 0.467277i
\(574\) −3860.09 −0.280691
\(575\) 0 0
\(576\) 1373.11 0.0993281
\(577\) − 6339.65i − 0.457406i −0.973496 0.228703i \(-0.926552\pi\)
0.973496 0.228703i \(-0.0734484\pi\)
\(578\) 1485.90i 0.106929i
\(579\) −11843.3 −0.850069
\(580\) 0 0
\(581\) 3589.83 0.256336
\(582\) − 6215.37i − 0.442672i
\(583\) − 4506.17i − 0.320114i
\(584\) −15915.2 −1.12770
\(585\) 0 0
\(586\) −11210.1 −0.790247
\(587\) 13370.6i 0.940140i 0.882629 + 0.470070i \(0.155771\pi\)
−0.882629 + 0.470070i \(0.844229\pi\)
\(588\) − 3242.34i − 0.227401i
\(589\) 21935.3 1.53452
\(590\) 0 0
\(591\) 2771.66 0.192912
\(592\) 8150.09i 0.565822i
\(593\) 14319.3i 0.991608i 0.868434 + 0.495804i \(0.165127\pi\)
−0.868434 + 0.495804i \(0.834873\pi\)
\(594\) −1001.57 −0.0691832
\(595\) 0 0
\(596\) 7938.73 0.545609
\(597\) 1428.46i 0.0979276i
\(598\) − 5673.65i − 0.387981i
\(599\) 5788.63 0.394853 0.197427 0.980318i \(-0.436742\pi\)
0.197427 + 0.980318i \(0.436742\pi\)
\(600\) 0 0
\(601\) 23968.1 1.62675 0.813375 0.581739i \(-0.197628\pi\)
0.813375 + 0.581739i \(0.197628\pi\)
\(602\) − 4480.26i − 0.303325i
\(603\) 8105.28i 0.547384i
\(604\) 1922.44 0.129508
\(605\) 0 0
\(606\) −10277.9 −0.688961
\(607\) 23526.6i 1.57317i 0.617482 + 0.786585i \(0.288153\pi\)
−0.617482 + 0.786585i \(0.711847\pi\)
\(608\) 11301.9i 0.753872i
\(609\) 3460.56 0.230261
\(610\) 0 0
\(611\) 2551.74 0.168956
\(612\) − 2220.70i − 0.146677i
\(613\) 1228.07i 0.0809159i 0.999181 + 0.0404579i \(0.0128817\pi\)
−0.999181 + 0.0404579i \(0.987118\pi\)
\(614\) 5729.31 0.376573
\(615\) 0 0
\(616\) 814.478 0.0532732
\(617\) 9844.90i 0.642368i 0.947017 + 0.321184i \(0.104081\pi\)
−0.947017 + 0.321184i \(0.895919\pi\)
\(618\) 11150.4i 0.725785i
\(619\) 6551.68 0.425419 0.212709 0.977115i \(-0.431771\pi\)
0.212709 + 0.977115i \(0.431771\pi\)
\(620\) 0 0
\(621\) 3024.00 0.195409
\(622\) − 23365.9i − 1.50625i
\(623\) − 2414.88i − 0.155297i
\(624\) −3587.46 −0.230150
\(625\) 0 0
\(626\) −12007.8 −0.766661
\(627\) 2597.14i 0.165422i
\(628\) − 6989.64i − 0.444135i
\(629\) −7491.01 −0.474859
\(630\) 0 0
\(631\) −26440.5 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(632\) − 5111.72i − 0.321730i
\(633\) − 14754.7i − 0.926458i
\(634\) −1122.13 −0.0702923
\(635\) 0 0
\(636\) −4144.39 −0.258389
\(637\) − 4814.31i − 0.299450i
\(638\) 9018.74i 0.559648i
\(639\) −6808.50 −0.421502
\(640\) 0 0
\(641\) −27927.2 −1.72084 −0.860421 0.509584i \(-0.829799\pi\)
−0.860421 + 0.509584i \(0.829799\pi\)
\(642\) − 13936.7i − 0.856758i
\(643\) − 16737.7i − 1.02655i −0.858225 0.513274i \(-0.828432\pi\)
0.858225 0.513274i \(-0.171568\pi\)
\(644\) 1792.00 0.109650
\(645\) 0 0
\(646\) −19419.1 −1.18271
\(647\) − 7818.70i − 0.475092i −0.971376 0.237546i \(-0.923657\pi\)
0.971376 0.237546i \(-0.0763431\pi\)
\(648\) − 1264.08i − 0.0766325i
\(649\) −2156.00 −0.130401
\(650\) 0 0
\(651\) −3967.17 −0.238842
\(652\) − 9025.94i − 0.542152i
\(653\) 19747.6i 1.18344i 0.806144 + 0.591719i \(0.201550\pi\)
−0.806144 + 0.591719i \(0.798450\pi\)
\(654\) −3239.59 −0.193697
\(655\) 0 0
\(656\) 19205.4 1.14305
\(657\) 9178.33i 0.545024i
\(658\) 2717.91i 0.161026i
\(659\) −7867.72 −0.465072 −0.232536 0.972588i \(-0.574702\pi\)
−0.232536 + 0.972588i \(0.574702\pi\)
\(660\) 0 0
\(661\) 4227.41 0.248755 0.124378 0.992235i \(-0.460307\pi\)
0.124378 + 0.992235i \(0.460307\pi\)
\(662\) 1825.90i 0.107199i
\(663\) − 3297.35i − 0.193150i
\(664\) −11807.8 −0.690106
\(665\) 0 0
\(666\) 3107.30 0.180789
\(667\) − 27230.0i − 1.58073i
\(668\) − 4006.67i − 0.232070i
\(669\) −6300.88 −0.364135
\(670\) 0 0
\(671\) −7714.94 −0.443863
\(672\) − 2044.04i − 0.117337i
\(673\) 29397.6i 1.68379i 0.539638 + 0.841897i \(0.318561\pi\)
−0.539638 + 0.841897i \(0.681439\pi\)
\(674\) −2753.57 −0.157364
\(675\) 0 0
\(676\) −6647.94 −0.378239
\(677\) − 5737.14i − 0.325696i −0.986651 0.162848i \(-0.947932\pi\)
0.986651 0.162848i \(-0.0520680\pi\)
\(678\) − 16484.8i − 0.933771i
\(679\) 2914.86 0.164745
\(680\) 0 0
\(681\) −6771.49 −0.381034
\(682\) − 10339.0i − 0.580502i
\(683\) 32097.6i 1.79821i 0.437729 + 0.899107i \(0.355783\pi\)
−0.437729 + 0.899107i \(0.644217\pi\)
\(684\) 2388.62 0.133525
\(685\) 0 0
\(686\) 10615.8 0.590837
\(687\) 15933.2i 0.884848i
\(688\) 22291.0i 1.23523i
\(689\) −6153.69 −0.340257
\(690\) 0 0
\(691\) −16456.2 −0.905965 −0.452983 0.891519i \(-0.649640\pi\)
−0.452983 + 0.891519i \(0.649640\pi\)
\(692\) − 2721.93i − 0.149526i
\(693\) − 469.712i − 0.0257473i
\(694\) −21112.3 −1.15477
\(695\) 0 0
\(696\) −11382.6 −0.619909
\(697\) 17652.3i 0.959294i
\(698\) − 43058.9i − 2.33496i
\(699\) −7398.80 −0.400355
\(700\) 0 0
\(701\) 27238.1 1.46758 0.733788 0.679379i \(-0.237751\pi\)
0.733788 + 0.679379i \(0.237751\pi\)
\(702\) 1367.75i 0.0735364i
\(703\) − 8057.44i − 0.432279i
\(704\) −1678.25 −0.0898457
\(705\) 0 0
\(706\) −8934.12 −0.476261
\(707\) − 4820.09i − 0.256405i
\(708\) 1982.90i 0.105257i
\(709\) −28761.4 −1.52349 −0.761747 0.647875i \(-0.775658\pi\)
−0.761747 + 0.647875i \(0.775658\pi\)
\(710\) 0 0
\(711\) −2947.94 −0.155494
\(712\) 7943.10i 0.418090i
\(713\) 31216.3i 1.63964i
\(714\) 3512.09 0.184085
\(715\) 0 0
\(716\) −6577.27 −0.343302
\(717\) − 4288.21i − 0.223356i
\(718\) − 10804.5i − 0.561588i
\(719\) 27272.0 1.41456 0.707282 0.706931i \(-0.249921\pi\)
0.707282 + 0.706931i \(0.249921\pi\)
\(720\) 0 0
\(721\) −5229.28 −0.270109
\(722\) 2243.05i 0.115620i
\(723\) − 2936.97i − 0.151075i
\(724\) −3580.04 −0.183772
\(725\) 0 0
\(726\) 1224.14 0.0625785
\(727\) − 3979.75i − 0.203027i −0.994834 0.101514i \(-0.967631\pi\)
0.994834 0.101514i \(-0.0323685\pi\)
\(728\) − 1112.26i − 0.0566253i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −20488.3 −1.03665
\(732\) 7095.54i 0.358277i
\(733\) 9342.48i 0.470767i 0.971903 + 0.235384i \(0.0756346\pi\)
−0.971903 + 0.235384i \(0.924365\pi\)
\(734\) 28426.3 1.42947
\(735\) 0 0
\(736\) −16083.9 −0.805515
\(737\) − 9906.45i − 0.495127i
\(738\) − 7322.23i − 0.365224i
\(739\) 28928.0 1.43997 0.719983 0.693992i \(-0.244150\pi\)
0.719983 + 0.693992i \(0.244150\pi\)
\(740\) 0 0
\(741\) 3546.68 0.175831
\(742\) − 6554.43i − 0.324287i
\(743\) 4857.04i 0.239822i 0.992785 + 0.119911i \(0.0382609\pi\)
−0.992785 + 0.119911i \(0.961739\pi\)
\(744\) 13049.0 0.643008
\(745\) 0 0
\(746\) −31660.8 −1.55386
\(747\) 6809.57i 0.333533i
\(748\) 2714.19i 0.132675i
\(749\) 6536.00 0.318852
\(750\) 0 0
\(751\) 14355.4 0.697517 0.348759 0.937213i \(-0.386603\pi\)
0.348759 + 0.937213i \(0.386603\pi\)
\(752\) − 13522.6i − 0.655744i
\(753\) − 19591.9i − 0.948165i
\(754\) 12316.1 0.594863
\(755\) 0 0
\(756\) −432.000 −0.0207827
\(757\) 17714.9i 0.850538i 0.905067 + 0.425269i \(0.139821\pi\)
−0.905067 + 0.425269i \(0.860179\pi\)
\(758\) − 48103.9i − 2.30503i
\(759\) −3696.00 −0.176754
\(760\) 0 0
\(761\) −7945.82 −0.378497 −0.189248 0.981929i \(-0.560605\pi\)
−0.189248 + 0.981929i \(0.560605\pi\)
\(762\) − 23180.3i − 1.10201i
\(763\) − 1519.29i − 0.0720866i
\(764\) −7204.58 −0.341168
\(765\) 0 0
\(766\) 45398.4 2.14140
\(767\) 2944.26i 0.138606i
\(768\) 13166.2i 0.618613i
\(769\) −27308.1 −1.28057 −0.640284 0.768139i \(-0.721183\pi\)
−0.640284 + 0.768139i \(0.721183\pi\)
\(770\) 0 0
\(771\) 24390.8 1.13932
\(772\) − 13313.0i − 0.620653i
\(773\) − 18872.6i − 0.878136i −0.898454 0.439068i \(-0.855309\pi\)
0.898454 0.439068i \(-0.144691\pi\)
\(774\) 8498.64 0.394674
\(775\) 0 0
\(776\) −9587.65 −0.443527
\(777\) 1457.25i 0.0672826i
\(778\) − 3176.29i − 0.146369i
\(779\) −18987.1 −0.873276
\(780\) 0 0
\(781\) 8321.50 0.381263
\(782\) − 27635.4i − 1.26373i
\(783\) 6564.37i 0.299606i
\(784\) −25512.8 −1.16221
\(785\) 0 0
\(786\) −11608.2 −0.526782
\(787\) 14512.1i 0.657307i 0.944450 + 0.328654i \(0.106595\pi\)
−0.944450 + 0.328654i \(0.893405\pi\)
\(788\) 3115.60i 0.140849i
\(789\) 13648.3 0.615832
\(790\) 0 0
\(791\) 7731.01 0.347513
\(792\) 1544.99i 0.0693167i
\(793\) 10535.6i 0.471792i
\(794\) 2859.15 0.127793
\(795\) 0 0
\(796\) −1605.72 −0.0714989
\(797\) − 29108.9i − 1.29371i −0.762612 0.646856i \(-0.776083\pi\)
0.762612 0.646856i \(-0.223917\pi\)
\(798\) 3777.65i 0.167578i
\(799\) 12429.1 0.550325
\(800\) 0 0
\(801\) 4580.80 0.202066
\(802\) − 41154.0i − 1.81197i
\(803\) − 11218.0i − 0.492993i
\(804\) −9111.10 −0.399656
\(805\) 0 0
\(806\) −14119.1 −0.617029
\(807\) 87.4567i 0.00381490i
\(808\) 15854.4i 0.690291i
\(809\) 3000.83 0.130413 0.0652063 0.997872i \(-0.479229\pi\)
0.0652063 + 0.997872i \(0.479229\pi\)
\(810\) 0 0
\(811\) 6239.39 0.270154 0.135077 0.990835i \(-0.456872\pi\)
0.135077 + 0.990835i \(0.456872\pi\)
\(812\) 3890.00i 0.168118i
\(813\) 23133.7i 0.997950i
\(814\) −3797.81 −0.163530
\(815\) 0 0
\(816\) −17473.9 −0.749645
\(817\) − 22037.6i − 0.943693i
\(818\) 29539.6i 1.26263i
\(819\) −641.445 −0.0273674
\(820\) 0 0
\(821\) 14922.4 0.634342 0.317171 0.948368i \(-0.397267\pi\)
0.317171 + 0.948368i \(0.397267\pi\)
\(822\) − 12834.2i − 0.544580i
\(823\) − 25737.8i − 1.09011i −0.838399 0.545057i \(-0.816508\pi\)
0.838399 0.545057i \(-0.183492\pi\)
\(824\) 17200.3 0.727186
\(825\) 0 0
\(826\) −3136.00 −0.132101
\(827\) − 27043.4i − 1.13711i −0.822645 0.568555i \(-0.807503\pi\)
0.822645 0.568555i \(-0.192497\pi\)
\(828\) 3399.26i 0.142672i
\(829\) 9795.41 0.410384 0.205192 0.978722i \(-0.434218\pi\)
0.205192 + 0.978722i \(0.434218\pi\)
\(830\) 0 0
\(831\) 3382.56 0.141203
\(832\) 2291.84i 0.0954990i
\(833\) − 23449.7i − 0.975370i
\(834\) 4919.70 0.204263
\(835\) 0 0
\(836\) −2919.42 −0.120778
\(837\) − 7525.37i − 0.310770i
\(838\) − 37730.5i − 1.55535i
\(839\) −28875.5 −1.18819 −0.594095 0.804395i \(-0.702490\pi\)
−0.594095 + 0.804395i \(0.702490\pi\)
\(840\) 0 0
\(841\) 34720.7 1.42362
\(842\) 47489.3i 1.94369i
\(843\) − 5617.41i − 0.229506i
\(844\) 16585.7 0.676426
\(845\) 0 0
\(846\) −5155.63 −0.209520
\(847\) 574.092i 0.0232893i
\(848\) 32610.7i 1.32059i
\(849\) −6373.45 −0.257640
\(850\) 0 0
\(851\) 11466.6 0.461892
\(852\) − 7653.39i − 0.307747i
\(853\) − 47157.1i − 1.89288i −0.322878 0.946441i \(-0.604650\pi\)
0.322878 0.946441i \(-0.395350\pi\)
\(854\) −11221.7 −0.449649
\(855\) 0 0
\(856\) −21498.4 −0.858412
\(857\) − 5021.31i − 0.200145i −0.994980 0.100073i \(-0.968092\pi\)
0.994980 0.100073i \(-0.0319075\pi\)
\(858\) − 1671.70i − 0.0665162i
\(859\) 22921.1 0.910428 0.455214 0.890382i \(-0.349563\pi\)
0.455214 + 0.890382i \(0.349563\pi\)
\(860\) 0 0
\(861\) 3433.95 0.135922
\(862\) 18938.9i 0.748332i
\(863\) − 19488.1i − 0.768693i −0.923189 0.384347i \(-0.874427\pi\)
0.923189 0.384347i \(-0.125573\pi\)
\(864\) 3877.36 0.152674
\(865\) 0 0
\(866\) 24266.1 0.952188
\(867\) − 1321.86i − 0.0517794i
\(868\) − 4459.48i − 0.174383i
\(869\) 3603.04 0.140650
\(870\) 0 0
\(871\) −13528.4 −0.526282
\(872\) 4997.30i 0.194071i
\(873\) 5529.22i 0.214360i
\(874\) 29725.0 1.15042
\(875\) 0 0
\(876\) −10317.3 −0.397933
\(877\) 8455.67i 0.325573i 0.986661 + 0.162787i \(0.0520482\pi\)
−0.986661 + 0.162787i \(0.947952\pi\)
\(878\) 343.834i 0.0132162i
\(879\) 9972.56 0.382669
\(880\) 0 0
\(881\) −11291.2 −0.431794 −0.215897 0.976416i \(-0.569268\pi\)
−0.215897 + 0.976416i \(0.569268\pi\)
\(882\) 9727.02i 0.371344i
\(883\) 31818.1i 1.21264i 0.795219 + 0.606322i \(0.207356\pi\)
−0.795219 + 0.606322i \(0.792644\pi\)
\(884\) 3706.53 0.141023
\(885\) 0 0
\(886\) 16705.4 0.633441
\(887\) − 17481.1i − 0.661732i −0.943678 0.330866i \(-0.892659\pi\)
0.943678 0.330866i \(-0.107341\pi\)
\(888\) − 4793.24i − 0.181138i
\(889\) 10871.0 0.410126
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) − 7082.78i − 0.265862i
\(893\) 13368.9i 0.500978i
\(894\) −23816.2 −0.890976
\(895\) 0 0
\(896\) −7891.87 −0.294251
\(897\) 5047.31i 0.187876i
\(898\) − 39125.1i − 1.45392i
\(899\) −67763.1 −2.51393
\(900\) 0 0
\(901\) −29973.6 −1.10829
\(902\) 8949.39i 0.330357i
\(903\) 3985.66i 0.146882i
\(904\) −25429.1 −0.935574
\(905\) 0 0
\(906\) −5767.31 −0.211486
\(907\) − 10607.4i − 0.388326i −0.980969 0.194163i \(-0.937801\pi\)
0.980969 0.194163i \(-0.0621990\pi\)
\(908\) − 7611.79i − 0.278201i
\(909\) 9143.26 0.333623
\(910\) 0 0
\(911\) −41249.2 −1.50016 −0.750080 0.661347i \(-0.769985\pi\)
−0.750080 + 0.661347i \(0.769985\pi\)
\(912\) − 18795.2i − 0.682425i
\(913\) − 8322.81i − 0.301692i
\(914\) 11906.5 0.430887
\(915\) 0 0
\(916\) −17910.4 −0.646045
\(917\) − 5443.97i − 0.196048i
\(918\) 6662.10i 0.239523i
\(919\) 13858.1 0.497429 0.248714 0.968577i \(-0.419992\pi\)
0.248714 + 0.968577i \(0.419992\pi\)
\(920\) 0 0
\(921\) −5096.83 −0.182352
\(922\) − 39004.9i − 1.39323i
\(923\) − 11363.9i − 0.405253i
\(924\) 528.000 0.0187986
\(925\) 0 0
\(926\) 36719.0 1.30309
\(927\) − 9919.47i − 0.351454i
\(928\) − 34914.2i − 1.23504i
\(929\) −20893.7 −0.737890 −0.368945 0.929451i \(-0.620281\pi\)
−0.368945 + 0.929451i \(0.620281\pi\)
\(930\) 0 0
\(931\) 25222.8 0.887911
\(932\) − 8316.94i − 0.292307i
\(933\) 20786.5i 0.729388i
\(934\) −36042.9 −1.26270
\(935\) 0 0
\(936\) 2109.86 0.0736784
\(937\) − 3203.52i − 0.111691i −0.998439 0.0558454i \(-0.982215\pi\)
0.998439 0.0558454i \(-0.0177854\pi\)
\(938\) − 14409.4i − 0.501581i
\(939\) 10682.2 0.371248
\(940\) 0 0
\(941\) 19951.6 0.691182 0.345591 0.938385i \(-0.387678\pi\)
0.345591 + 0.938385i \(0.387678\pi\)
\(942\) 20968.9i 0.725270i
\(943\) − 27020.6i − 0.933099i
\(944\) 15602.8 0.537952
\(945\) 0 0
\(946\) −10387.2 −0.356996
\(947\) 38216.7i 1.31138i 0.755031 + 0.655689i \(0.227622\pi\)
−0.755031 + 0.655689i \(0.772378\pi\)
\(948\) − 3313.76i − 0.113529i
\(949\) −15319.4 −0.524014
\(950\) 0 0
\(951\) 998.249 0.0340383
\(952\) − 5417.65i − 0.184440i
\(953\) − 47661.4i − 1.62004i −0.586399 0.810022i \(-0.699455\pi\)
0.586399 0.810022i \(-0.300545\pi\)
\(954\) 12433.2 0.421948
\(955\) 0 0
\(956\) 4820.35 0.163077
\(957\) − 8023.12i − 0.271004i
\(958\) 7897.56i 0.266345i
\(959\) 6018.94 0.202671
\(960\) 0 0
\(961\) 47892.3 1.60761
\(962\) 5186.34i 0.173819i
\(963\) 12398.2i 0.414876i
\(964\) 3301.43 0.110303
\(965\) 0 0
\(966\) −5376.00 −0.179058
\(967\) − 18933.2i − 0.629628i −0.949153 0.314814i \(-0.898058\pi\)
0.949153 0.314814i \(-0.101942\pi\)
\(968\) − 1888.32i − 0.0626994i
\(969\) 17275.3 0.572717
\(970\) 0 0
\(971\) −40660.3 −1.34382 −0.671911 0.740632i \(-0.734526\pi\)
−0.671911 + 0.740632i \(0.734526\pi\)
\(972\) − 819.464i − 0.0270415i
\(973\) 2307.23i 0.0760188i
\(974\) −22759.2 −0.748720
\(975\) 0 0
\(976\) 55832.3 1.83110
\(977\) 22502.8i 0.736876i 0.929652 + 0.368438i \(0.120107\pi\)
−0.929652 + 0.368438i \(0.879893\pi\)
\(978\) 27077.8i 0.885331i
\(979\) −5598.76 −0.182775
\(980\) 0 0
\(981\) 2881.96 0.0937959
\(982\) − 24824.7i − 0.806709i
\(983\) − 4435.20i − 0.143907i −0.997408 0.0719536i \(-0.977077\pi\)
0.997408 0.0719536i \(-0.0229234\pi\)
\(984\) −11295.1 −0.365929
\(985\) 0 0
\(986\) 59989.8 1.93759
\(987\) − 2417.87i − 0.0779753i
\(988\) 3986.80i 0.128378i
\(989\) 31361.8 1.00834
\(990\) 0 0
\(991\) 7362.76 0.236010 0.118005 0.993013i \(-0.462350\pi\)
0.118005 + 0.993013i \(0.462350\pi\)
\(992\) 40025.5i 1.28106i
\(993\) − 1624.33i − 0.0519101i
\(994\) 12104.0 0.386233
\(995\) 0 0
\(996\) −7654.60 −0.243519
\(997\) 53480.1i 1.69883i 0.527728 + 0.849413i \(0.323044\pi\)
−0.527728 + 0.849413i \(0.676956\pi\)
\(998\) 35010.0i 1.11044i
\(999\) −2764.27 −0.0875452
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 825.4.c.i.199.1 4
5.2 odd 4 33.4.a.d.1.2 2
5.3 odd 4 825.4.a.k.1.1 2
5.4 even 2 inner 825.4.c.i.199.4 4
15.2 even 4 99.4.a.e.1.1 2
15.8 even 4 2475.4.a.o.1.2 2
20.7 even 4 528.4.a.o.1.1 2
35.27 even 4 1617.4.a.j.1.2 2
40.27 even 4 2112.4.a.bh.1.2 2
40.37 odd 4 2112.4.a.ba.1.2 2
55.32 even 4 363.4.a.j.1.1 2
60.47 odd 4 1584.4.a.x.1.2 2
165.32 odd 4 1089.4.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 5.2 odd 4
99.4.a.e.1.1 2 15.2 even 4
363.4.a.j.1.1 2 55.32 even 4
528.4.a.o.1.1 2 20.7 even 4
825.4.a.k.1.1 2 5.3 odd 4
825.4.c.i.199.1 4 1.1 even 1 trivial
825.4.c.i.199.4 4 5.4 even 2 inner
1089.4.a.t.1.2 2 165.32 odd 4
1584.4.a.x.1.2 2 60.47 odd 4
1617.4.a.j.1.2 2 35.27 even 4
2112.4.a.ba.1.2 2 40.37 odd 4
2112.4.a.bh.1.2 2 40.27 even 4
2475.4.a.o.1.2 2 15.8 even 4