Properties

Label 525.4.d.h.274.2
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
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] = [525,4,Mod(274,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("525.274");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (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: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{65})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 33x^{2} + 256 \) 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 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(-3.53113i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.h.274.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.53113i q^{2} -3.00000i q^{3} -4.46887 q^{4} -10.5934 q^{6} -7.00000i q^{7} -12.4689i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-3.53113i q^{2} -3.00000i q^{3} -4.46887 q^{4} -10.5934 q^{6} -7.00000i q^{7} -12.4689i q^{8} -9.00000 q^{9} -2.93774 q^{11} +13.4066i q^{12} +19.0623i q^{13} -24.7179 q^{14} -79.7802 q^{16} +122.498i q^{17} +31.7802i q^{18} -107.436 q^{19} -21.0000 q^{21} +10.3735i q^{22} -210.623i q^{23} -37.4066 q^{24} +67.3113 q^{26} +27.0000i q^{27} +31.2821i q^{28} -95.4942 q^{29} -94.3074 q^{31} +181.963i q^{32} +8.81323i q^{33} +432.556 q^{34} +40.2198 q^{36} +97.1206i q^{37} +379.370i q^{38} +57.1868 q^{39} -491.113 q^{41} +74.1537i q^{42} +43.0039i q^{43} +13.1284 q^{44} -743.735 q^{46} +473.494i q^{47} +239.340i q^{48} -49.0000 q^{49} +367.494 q^{51} -85.1868i q^{52} +183.677i q^{53} +95.3405 q^{54} -87.2821 q^{56} +322.307i q^{57} +337.202i q^{58} +760.615 q^{59} -198.747 q^{61} +333.012i q^{62} +63.0000i q^{63} +4.29373 q^{64} +31.1206 q^{66} -309.992i q^{67} -547.428i q^{68} -631.868 q^{69} +665.693 q^{71} +112.220i q^{72} -621.288i q^{73} +342.945 q^{74} +480.117 q^{76} +20.5642i q^{77} -201.934i q^{78} +24.7626 q^{79} +81.0000 q^{81} +1734.18i q^{82} +406.724i q^{83} +93.8463 q^{84} +151.852 q^{86} +286.483i q^{87} +36.6303i q^{88} -261.751 q^{89} +133.436 q^{91} +941.245i q^{92} +282.922i q^{93} +1671.97 q^{94} +545.889 q^{96} -1004.77i q^{97} +173.025i q^{98} +26.4397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 34 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 34 q^{4} + 6 q^{6} - 36 q^{9} - 44 q^{11} + 14 q^{14} - 174 q^{16} - 204 q^{19} - 84 q^{21} - 198 q^{24} + 108 q^{26} + 392 q^{29} + 300 q^{31} + 924 q^{34} + 306 q^{36} + 132 q^{39} - 352 q^{41} + 504 q^{44} - 1040 q^{46} - 196 q^{49} + 696 q^{51} - 54 q^{54} - 462 q^{56} + 1688 q^{59} - 408 q^{61} + 1678 q^{64} - 456 q^{66} - 1560 q^{69} + 3340 q^{71} + 2436 q^{74} + 824 q^{76} + 1776 q^{79} + 324 q^{81} + 714 q^{84} - 2424 q^{86} - 1176 q^{89} + 308 q^{91} + 2560 q^{94} - 90 q^{96} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.53113i − 1.24844i −0.781248 0.624221i \(-0.785416\pi\)
0.781248 0.624221i \(-0.214584\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −4.46887 −0.558609
\(5\) 0 0
\(6\) −10.5934 −0.720789
\(7\) − 7.00000i − 0.377964i
\(8\) − 12.4689i − 0.551051i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −2.93774 −0.0805239 −0.0402619 0.999189i \(-0.512819\pi\)
−0.0402619 + 0.999189i \(0.512819\pi\)
\(12\) 13.4066i 0.322513i
\(13\) 19.0623i 0.406686i 0.979108 + 0.203343i \(0.0651807\pi\)
−0.979108 + 0.203343i \(0.934819\pi\)
\(14\) −24.7179 −0.471867
\(15\) 0 0
\(16\) −79.7802 −1.24656
\(17\) 122.498i 1.74766i 0.486236 + 0.873828i \(0.338370\pi\)
−0.486236 + 0.873828i \(0.661630\pi\)
\(18\) 31.7802i 0.416148i
\(19\) −107.436 −1.29723 −0.648617 0.761115i \(-0.724652\pi\)
−0.648617 + 0.761115i \(0.724652\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 10.3735i 0.100529i
\(23\) − 210.623i − 1.90947i −0.297455 0.954736i \(-0.596138\pi\)
0.297455 0.954736i \(-0.403862\pi\)
\(24\) −37.4066 −0.318150
\(25\) 0 0
\(26\) 67.3113 0.507724
\(27\) 27.0000i 0.192450i
\(28\) 31.2821i 0.211134i
\(29\) −95.4942 −0.611477 −0.305738 0.952116i \(-0.598903\pi\)
−0.305738 + 0.952116i \(0.598903\pi\)
\(30\) 0 0
\(31\) −94.3074 −0.546391 −0.273195 0.961959i \(-0.588081\pi\)
−0.273195 + 0.961959i \(0.588081\pi\)
\(32\) 181.963i 1.00521i
\(33\) 8.81323i 0.0464905i
\(34\) 432.556 2.18185
\(35\) 0 0
\(36\) 40.2198 0.186203
\(37\) 97.1206i 0.431528i 0.976446 + 0.215764i \(0.0692242\pi\)
−0.976446 + 0.215764i \(0.930776\pi\)
\(38\) 379.370i 1.61952i
\(39\) 57.1868 0.234800
\(40\) 0 0
\(41\) −491.113 −1.87071 −0.935353 0.353716i \(-0.884918\pi\)
−0.935353 + 0.353716i \(0.884918\pi\)
\(42\) 74.1537i 0.272433i
\(43\) 43.0039i 0.152512i 0.997088 + 0.0762562i \(0.0242967\pi\)
−0.997088 + 0.0762562i \(0.975703\pi\)
\(44\) 13.1284 0.0449814
\(45\) 0 0
\(46\) −743.735 −2.38387
\(47\) 473.494i 1.46949i 0.678341 + 0.734747i \(0.262699\pi\)
−0.678341 + 0.734747i \(0.737301\pi\)
\(48\) 239.340i 0.719705i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 367.494 1.00901
\(52\) − 85.1868i − 0.227178i
\(53\) 183.677i 0.476038i 0.971261 + 0.238019i \(0.0764980\pi\)
−0.971261 + 0.238019i \(0.923502\pi\)
\(54\) 95.3405 0.240263
\(55\) 0 0
\(56\) −87.2821 −0.208278
\(57\) 322.307i 0.748959i
\(58\) 337.202i 0.763394i
\(59\) 760.615 1.67837 0.839183 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(60\) 0 0
\(61\) −198.747 −0.417163 −0.208582 0.978005i \(-0.566885\pi\)
−0.208582 + 0.978005i \(0.566885\pi\)
\(62\) 333.012i 0.682137i
\(63\) 63.0000i 0.125988i
\(64\) 4.29373 0.00838618
\(65\) 0 0
\(66\) 31.1206 0.0580407
\(67\) − 309.992i − 0.565247i −0.959231 0.282624i \(-0.908795\pi\)
0.959231 0.282624i \(-0.0912048\pi\)
\(68\) − 547.428i − 0.976256i
\(69\) −631.868 −1.10243
\(70\) 0 0
\(71\) 665.693 1.11272 0.556360 0.830941i \(-0.312197\pi\)
0.556360 + 0.830941i \(0.312197\pi\)
\(72\) 112.220i 0.183684i
\(73\) − 621.288i − 0.996113i −0.867145 0.498057i \(-0.834047\pi\)
0.867145 0.498057i \(-0.165953\pi\)
\(74\) 342.945 0.538738
\(75\) 0 0
\(76\) 480.117 0.724647
\(77\) 20.5642i 0.0304352i
\(78\) − 201.934i − 0.293135i
\(79\) 24.7626 0.0352659 0.0176330 0.999845i \(-0.494387\pi\)
0.0176330 + 0.999845i \(0.494387\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1734.18i 2.33547i
\(83\) 406.724i 0.537876i 0.963157 + 0.268938i \(0.0866727\pi\)
−0.963157 + 0.268938i \(0.913327\pi\)
\(84\) 93.8463 0.121898
\(85\) 0 0
\(86\) 151.852 0.190403
\(87\) 286.483i 0.353036i
\(88\) 36.6303i 0.0443728i
\(89\) −261.751 −0.311748 −0.155874 0.987777i \(-0.549819\pi\)
−0.155874 + 0.987777i \(0.549819\pi\)
\(90\) 0 0
\(91\) 133.436 0.153713
\(92\) 941.245i 1.06665i
\(93\) 282.922i 0.315459i
\(94\) 1671.97 1.83458
\(95\) 0 0
\(96\) 545.889 0.580360
\(97\) − 1004.77i − 1.05175i −0.850563 0.525873i \(-0.823739\pi\)
0.850563 0.525873i \(-0.176261\pi\)
\(98\) 173.025i 0.178349i
\(99\) 26.4397 0.0268413
\(100\) 0 0
\(101\) −128.872 −0.126962 −0.0634812 0.997983i \(-0.520220\pi\)
−0.0634812 + 0.997983i \(0.520220\pi\)
\(102\) − 1297.67i − 1.25969i
\(103\) − 806.008i − 0.771051i −0.922697 0.385526i \(-0.874020\pi\)
0.922697 0.385526i \(-0.125980\pi\)
\(104\) 237.685 0.224105
\(105\) 0 0
\(106\) 648.587 0.594305
\(107\) − 769.712i − 0.695429i −0.937600 0.347714i \(-0.886958\pi\)
0.937600 0.347714i \(-0.113042\pi\)
\(108\) − 120.660i − 0.107504i
\(109\) 780.856 0.686169 0.343085 0.939304i \(-0.388528\pi\)
0.343085 + 0.939304i \(0.388528\pi\)
\(110\) 0 0
\(111\) 291.362 0.249143
\(112\) 558.461i 0.471157i
\(113\) 1115.65i 0.928771i 0.885633 + 0.464386i \(0.153725\pi\)
−0.885633 + 0.464386i \(0.846275\pi\)
\(114\) 1138.11 0.935032
\(115\) 0 0
\(116\) 426.751 0.341576
\(117\) − 171.560i − 0.135562i
\(118\) − 2685.83i − 2.09534i
\(119\) 857.486 0.660552
\(120\) 0 0
\(121\) −1322.37 −0.993516
\(122\) 701.802i 0.520804i
\(123\) 1473.34i 1.08005i
\(124\) 421.448 0.305219
\(125\) 0 0
\(126\) 222.461 0.157289
\(127\) − 1875.98i − 1.31076i −0.755299 0.655381i \(-0.772508\pi\)
0.755299 0.655381i \(-0.227492\pi\)
\(128\) 1440.54i 0.994744i
\(129\) 129.012 0.0880530
\(130\) 0 0
\(131\) 364.203 0.242905 0.121452 0.992597i \(-0.461245\pi\)
0.121452 + 0.992597i \(0.461245\pi\)
\(132\) − 39.3852i − 0.0259700i
\(133\) 752.051i 0.490309i
\(134\) −1094.62 −0.705679
\(135\) 0 0
\(136\) 1527.41 0.963048
\(137\) 1603.13i 0.999743i 0.866099 + 0.499872i \(0.166620\pi\)
−0.866099 + 0.499872i \(0.833380\pi\)
\(138\) 2231.21i 1.37633i
\(139\) −2431.12 −1.48349 −0.741746 0.670681i \(-0.766002\pi\)
−0.741746 + 0.670681i \(0.766002\pi\)
\(140\) 0 0
\(141\) 1420.48 0.848413
\(142\) − 2350.65i − 1.38917i
\(143\) − 56.0000i − 0.0327479i
\(144\) 718.021 0.415522
\(145\) 0 0
\(146\) −2193.85 −1.24359
\(147\) 147.000i 0.0824786i
\(148\) − 434.020i − 0.241055i
\(149\) −2341.57 −1.28744 −0.643722 0.765260i \(-0.722611\pi\)
−0.643722 + 0.765260i \(0.722611\pi\)
\(150\) 0 0
\(151\) −2104.07 −1.13395 −0.566976 0.823734i \(-0.691887\pi\)
−0.566976 + 0.823734i \(0.691887\pi\)
\(152\) 1339.60i 0.714843i
\(153\) − 1102.48i − 0.582552i
\(154\) 72.6148 0.0379966
\(155\) 0 0
\(156\) −255.560 −0.131162
\(157\) − 593.467i − 0.301680i −0.988558 0.150840i \(-0.951802\pi\)
0.988558 0.150840i \(-0.0481979\pi\)
\(158\) − 87.4399i − 0.0440275i
\(159\) 551.031 0.274840
\(160\) 0 0
\(161\) −1474.36 −0.721712
\(162\) − 286.021i − 0.138716i
\(163\) − 2178.71i − 1.04693i −0.852047 0.523465i \(-0.824639\pi\)
0.852047 0.523465i \(-0.175361\pi\)
\(164\) 2194.72 1.04499
\(165\) 0 0
\(166\) 1436.19 0.671508
\(167\) 799.502i 0.370463i 0.982695 + 0.185231i \(0.0593035\pi\)
−0.982695 + 0.185231i \(0.940697\pi\)
\(168\) 261.846i 0.120249i
\(169\) 1833.63 0.834606
\(170\) 0 0
\(171\) 966.922 0.432412
\(172\) − 192.179i − 0.0851947i
\(173\) 1444.36i 0.634754i 0.948299 + 0.317377i \(0.102802\pi\)
−0.948299 + 0.317377i \(0.897198\pi\)
\(174\) 1011.61 0.440745
\(175\) 0 0
\(176\) 234.374 0.100378
\(177\) − 2281.84i − 0.969005i
\(178\) 924.276i 0.389199i
\(179\) −3343.49 −1.39611 −0.698056 0.716043i \(-0.745952\pi\)
−0.698056 + 0.716043i \(0.745952\pi\)
\(180\) 0 0
\(181\) 2251.81 0.924729 0.462365 0.886690i \(-0.347001\pi\)
0.462365 + 0.886690i \(0.347001\pi\)
\(182\) − 471.179i − 0.191902i
\(183\) 596.241i 0.240849i
\(184\) −2626.23 −1.05222
\(185\) 0 0
\(186\) 999.035 0.393832
\(187\) − 359.868i − 0.140728i
\(188\) − 2115.98i − 0.820873i
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −1001.93 −0.379565 −0.189782 0.981826i \(-0.560778\pi\)
−0.189782 + 0.981826i \(0.560778\pi\)
\(192\) − 12.8812i − 0.00484177i
\(193\) 4054.97i 1.51235i 0.654372 + 0.756173i \(0.272933\pi\)
−0.654372 + 0.756173i \(0.727067\pi\)
\(194\) −3547.99 −1.31304
\(195\) 0 0
\(196\) 218.975 0.0798013
\(197\) − 5140.23i − 1.85902i −0.368802 0.929508i \(-0.620232\pi\)
0.368802 0.929508i \(-0.379768\pi\)
\(198\) − 93.3619i − 0.0335098i
\(199\) −585.631 −0.208614 −0.104307 0.994545i \(-0.533263\pi\)
−0.104307 + 0.994545i \(0.533263\pi\)
\(200\) 0 0
\(201\) −929.977 −0.326346
\(202\) 455.062i 0.158505i
\(203\) 668.459i 0.231116i
\(204\) −1642.28 −0.563642
\(205\) 0 0
\(206\) −2846.12 −0.962614
\(207\) 1895.60i 0.636490i
\(208\) − 1520.79i − 0.506961i
\(209\) 315.619 0.104458
\(210\) 0 0
\(211\) −1055.16 −0.344266 −0.172133 0.985074i \(-0.555066\pi\)
−0.172133 + 0.985074i \(0.555066\pi\)
\(212\) − 820.829i − 0.265919i
\(213\) − 1997.08i − 0.642430i
\(214\) −2717.95 −0.868203
\(215\) 0 0
\(216\) 336.660 0.106050
\(217\) 660.152i 0.206516i
\(218\) − 2757.30i − 0.856643i
\(219\) −1863.86 −0.575106
\(220\) 0 0
\(221\) −2335.09 −0.710747
\(222\) − 1028.84i − 0.311040i
\(223\) − 4675.85i − 1.40412i −0.712119 0.702059i \(-0.752264\pi\)
0.712119 0.702059i \(-0.247736\pi\)
\(224\) 1273.74 0.379935
\(225\) 0 0
\(226\) 3939.49 1.15952
\(227\) 5443.11i 1.59151i 0.605621 + 0.795754i \(0.292925\pi\)
−0.605621 + 0.795754i \(0.707075\pi\)
\(228\) − 1440.35i − 0.418375i
\(229\) 536.303 0.154759 0.0773797 0.997002i \(-0.475345\pi\)
0.0773797 + 0.997002i \(0.475345\pi\)
\(230\) 0 0
\(231\) 61.6926 0.0175717
\(232\) 1190.70i 0.336955i
\(233\) 183.490i 0.0515916i 0.999667 + 0.0257958i \(0.00821196\pi\)
−0.999667 + 0.0257958i \(0.991788\pi\)
\(234\) −605.802 −0.169241
\(235\) 0 0
\(236\) −3399.09 −0.937550
\(237\) − 74.2878i − 0.0203608i
\(238\) − 3027.90i − 0.824661i
\(239\) −643.218 −0.174085 −0.0870425 0.996205i \(-0.527742\pi\)
−0.0870425 + 0.996205i \(0.527742\pi\)
\(240\) 0 0
\(241\) −5755.61 −1.53839 −0.769194 0.639015i \(-0.779342\pi\)
−0.769194 + 0.639015i \(0.779342\pi\)
\(242\) 4669.46i 1.24035i
\(243\) − 243.000i − 0.0641500i
\(244\) 888.175 0.233031
\(245\) 0 0
\(246\) 5202.55 1.34838
\(247\) − 2047.97i − 0.527567i
\(248\) 1175.91i 0.301089i
\(249\) 1220.17 0.310543
\(250\) 0 0
\(251\) −5132.27 −1.29062 −0.645311 0.763920i \(-0.723272\pi\)
−0.645311 + 0.763920i \(0.723272\pi\)
\(252\) − 281.539i − 0.0703781i
\(253\) 618.755i 0.153758i
\(254\) −6624.34 −1.63641
\(255\) 0 0
\(256\) 5121.09 1.25027
\(257\) 5041.74i 1.22372i 0.790967 + 0.611859i \(0.209578\pi\)
−0.790967 + 0.611859i \(0.790422\pi\)
\(258\) − 455.557i − 0.109929i
\(259\) 679.844 0.163102
\(260\) 0 0
\(261\) 859.448 0.203826
\(262\) − 1286.05i − 0.303253i
\(263\) − 7577.00i − 1.77649i −0.459367 0.888246i \(-0.651924\pi\)
0.459367 0.888246i \(-0.348076\pi\)
\(264\) 109.891 0.0256186
\(265\) 0 0
\(266\) 2655.59 0.612122
\(267\) 785.253i 0.179988i
\(268\) 1385.32i 0.315752i
\(269\) −1023.10 −0.231893 −0.115947 0.993255i \(-0.536990\pi\)
−0.115947 + 0.993255i \(0.536990\pi\)
\(270\) 0 0
\(271\) −2251.98 −0.504790 −0.252395 0.967624i \(-0.581218\pi\)
−0.252395 + 0.967624i \(0.581218\pi\)
\(272\) − 9772.91i − 2.17857i
\(273\) − 400.307i − 0.0887462i
\(274\) 5660.87 1.24812
\(275\) 0 0
\(276\) 2823.74 0.615829
\(277\) − 8630.72i − 1.87209i −0.351875 0.936047i \(-0.614456\pi\)
0.351875 0.936047i \(-0.385544\pi\)
\(278\) 8584.61i 1.85205i
\(279\) 848.767 0.182130
\(280\) 0 0
\(281\) −7521.62 −1.59680 −0.798402 0.602124i \(-0.794321\pi\)
−0.798402 + 0.602124i \(0.794321\pi\)
\(282\) − 5015.91i − 1.05919i
\(283\) − 14.8169i − 0.00311226i −0.999999 0.00155613i \(-0.999505\pi\)
0.999999 0.00155613i \(-0.000495333\pi\)
\(284\) −2974.89 −0.621576
\(285\) 0 0
\(286\) −197.743 −0.0408839
\(287\) 3437.79i 0.707060i
\(288\) − 1637.67i − 0.335071i
\(289\) −10092.8 −2.05430
\(290\) 0 0
\(291\) −3014.32 −0.607226
\(292\) 2776.46i 0.556438i
\(293\) − 6913.39i − 1.37844i −0.724550 0.689222i \(-0.757952\pi\)
0.724550 0.689222i \(-0.242048\pi\)
\(294\) 519.076 0.102970
\(295\) 0 0
\(296\) 1210.98 0.237794
\(297\) − 79.3190i − 0.0154968i
\(298\) 8268.39i 1.60730i
\(299\) 4014.94 0.776555
\(300\) 0 0
\(301\) 301.027 0.0576442
\(302\) 7429.74i 1.41567i
\(303\) 386.615i 0.0733018i
\(304\) 8571.25 1.61709
\(305\) 0 0
\(306\) −3893.01 −0.727283
\(307\) − 7644.12i − 1.42108i −0.703655 0.710542i \(-0.748450\pi\)
0.703655 0.710542i \(-0.251550\pi\)
\(308\) − 91.8987i − 0.0170014i
\(309\) −2418.02 −0.445167
\(310\) 0 0
\(311\) 7593.99 1.38462 0.692308 0.721602i \(-0.256594\pi\)
0.692308 + 0.721602i \(0.256594\pi\)
\(312\) − 713.055i − 0.129387i
\(313\) 9127.84i 1.64836i 0.566329 + 0.824179i \(0.308363\pi\)
−0.566329 + 0.824179i \(0.691637\pi\)
\(314\) −2095.61 −0.376631
\(315\) 0 0
\(316\) −110.661 −0.0196999
\(317\) − 4929.81i − 0.873456i −0.899593 0.436728i \(-0.856137\pi\)
0.899593 0.436728i \(-0.143863\pi\)
\(318\) − 1945.76i − 0.343122i
\(319\) 280.537 0.0492385
\(320\) 0 0
\(321\) −2309.14 −0.401506
\(322\) 5206.15i 0.901016i
\(323\) − 13160.7i − 2.26712i
\(324\) −361.979 −0.0620677
\(325\) 0 0
\(326\) −7693.30 −1.30703
\(327\) − 2342.57i − 0.396160i
\(328\) 6123.62i 1.03086i
\(329\) 3314.46 0.555417
\(330\) 0 0
\(331\) 1221.67 0.202867 0.101433 0.994842i \(-0.467657\pi\)
0.101433 + 0.994842i \(0.467657\pi\)
\(332\) − 1817.60i − 0.300463i
\(333\) − 874.086i − 0.143843i
\(334\) 2823.14 0.462502
\(335\) 0 0
\(336\) 1675.38 0.272023
\(337\) − 8744.83i − 1.41354i −0.707446 0.706768i \(-0.750153\pi\)
0.707446 0.706768i \(-0.249847\pi\)
\(338\) − 6474.79i − 1.04196i
\(339\) 3346.94 0.536226
\(340\) 0 0
\(341\) 277.051 0.0439975
\(342\) − 3414.33i − 0.539841i
\(343\) 343.000i 0.0539949i
\(344\) 536.210 0.0840421
\(345\) 0 0
\(346\) 5100.21 0.792454
\(347\) 4589.56i 0.710031i 0.934860 + 0.355015i \(0.115524\pi\)
−0.934860 + 0.355015i \(0.884476\pi\)
\(348\) − 1280.25i − 0.197209i
\(349\) 3989.89 0.611960 0.305980 0.952038i \(-0.401016\pi\)
0.305980 + 0.952038i \(0.401016\pi\)
\(350\) 0 0
\(351\) −514.681 −0.0782668
\(352\) − 534.561i − 0.0809437i
\(353\) − 2416.35i − 0.364333i −0.983268 0.182166i \(-0.941689\pi\)
0.983268 0.182166i \(-0.0583109\pi\)
\(354\) −8057.49 −1.20975
\(355\) 0 0
\(356\) 1169.73 0.174145
\(357\) − 2572.46i − 0.381370i
\(358\) 11806.3i 1.74297i
\(359\) 2756.24 0.405206 0.202603 0.979261i \(-0.435060\pi\)
0.202603 + 0.979261i \(0.435060\pi\)
\(360\) 0 0
\(361\) 4683.45 0.682818
\(362\) − 7951.44i − 1.15447i
\(363\) 3967.11i 0.573607i
\(364\) −596.307 −0.0858654
\(365\) 0 0
\(366\) 2105.40 0.300687
\(367\) 11112.8i 1.58061i 0.612711 + 0.790307i \(0.290079\pi\)
−0.612711 + 0.790307i \(0.709921\pi\)
\(368\) 16803.5i 2.38028i
\(369\) 4420.02 0.623569
\(370\) 0 0
\(371\) 1285.74 0.179925
\(372\) − 1264.34i − 0.176218i
\(373\) − 6091.09i − 0.845535i −0.906238 0.422768i \(-0.861059\pi\)
0.906238 0.422768i \(-0.138941\pi\)
\(374\) −1270.74 −0.175691
\(375\) 0 0
\(376\) 5903.94 0.809767
\(377\) − 1820.33i − 0.248679i
\(378\) − 667.383i − 0.0908108i
\(379\) −3984.29 −0.539998 −0.269999 0.962861i \(-0.587023\pi\)
−0.269999 + 0.962861i \(0.587023\pi\)
\(380\) 0 0
\(381\) −5627.95 −0.756768
\(382\) 3537.93i 0.473865i
\(383\) − 318.475i − 0.0424890i −0.999774 0.0212445i \(-0.993237\pi\)
0.999774 0.0212445i \(-0.00676285\pi\)
\(384\) 4321.63 0.574316
\(385\) 0 0
\(386\) 14318.6 1.88808
\(387\) − 387.035i − 0.0508374i
\(388\) 4490.21i 0.587515i
\(389\) −3885.46 −0.506429 −0.253214 0.967410i \(-0.581488\pi\)
−0.253214 + 0.967410i \(0.581488\pi\)
\(390\) 0 0
\(391\) 25800.9 3.33710
\(392\) 610.975i 0.0787216i
\(393\) − 1092.61i − 0.140241i
\(394\) −18150.8 −2.32088
\(395\) 0 0
\(396\) −118.156 −0.0149938
\(397\) 4806.04i 0.607578i 0.952739 + 0.303789i \(0.0982518\pi\)
−0.952739 + 0.303789i \(0.901748\pi\)
\(398\) 2067.94i 0.260443i
\(399\) 2256.15 0.283080
\(400\) 0 0
\(401\) 3618.59 0.450633 0.225316 0.974286i \(-0.427658\pi\)
0.225316 + 0.974286i \(0.427658\pi\)
\(402\) 3283.87i 0.407424i
\(403\) − 1797.71i − 0.222209i
\(404\) 575.911 0.0709223
\(405\) 0 0
\(406\) 2360.42 0.288536
\(407\) − 285.315i − 0.0347483i
\(408\) − 4582.24i − 0.556016i
\(409\) 2109.05 0.254978 0.127489 0.991840i \(-0.459308\pi\)
0.127489 + 0.991840i \(0.459308\pi\)
\(410\) 0 0
\(411\) 4809.40 0.577202
\(412\) 3601.94i 0.430716i
\(413\) − 5324.30i − 0.634363i
\(414\) 6693.62 0.794622
\(415\) 0 0
\(416\) −3468.63 −0.408806
\(417\) 7293.37i 0.856494i
\(418\) − 1114.49i − 0.130410i
\(419\) 6905.91 0.805193 0.402597 0.915377i \(-0.368108\pi\)
0.402597 + 0.915377i \(0.368108\pi\)
\(420\) 0 0
\(421\) −9647.54 −1.11685 −0.558423 0.829556i \(-0.688593\pi\)
−0.558423 + 0.829556i \(0.688593\pi\)
\(422\) 3725.91i 0.429797i
\(423\) − 4261.45i − 0.489831i
\(424\) 2290.25 0.262321
\(425\) 0 0
\(426\) −7051.94 −0.802037
\(427\) 1391.23i 0.157673i
\(428\) 3439.74i 0.388473i
\(429\) −168.000 −0.0189070
\(430\) 0 0
\(431\) −13002.7 −1.45318 −0.726589 0.687073i \(-0.758895\pi\)
−0.726589 + 0.687073i \(0.758895\pi\)
\(432\) − 2154.06i − 0.239902i
\(433\) − 7356.07i − 0.816420i −0.912888 0.408210i \(-0.866153\pi\)
0.912888 0.408210i \(-0.133847\pi\)
\(434\) 2331.08 0.257824
\(435\) 0 0
\(436\) −3489.55 −0.383300
\(437\) 22628.4i 2.47703i
\(438\) 6581.54i 0.717987i
\(439\) −6909.21 −0.751159 −0.375579 0.926790i \(-0.622556\pi\)
−0.375579 + 0.926790i \(0.622556\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 8245.50i 0.887327i
\(443\) 14812.6i 1.58864i 0.607502 + 0.794318i \(0.292172\pi\)
−0.607502 + 0.794318i \(0.707828\pi\)
\(444\) −1302.06 −0.139173
\(445\) 0 0
\(446\) −16511.0 −1.75296
\(447\) 7024.72i 0.743306i
\(448\) − 30.0561i − 0.00316968i
\(449\) 10654.5 1.11986 0.559932 0.828538i \(-0.310827\pi\)
0.559932 + 0.828538i \(0.310827\pi\)
\(450\) 0 0
\(451\) 1442.76 0.150636
\(452\) − 4985.68i − 0.518820i
\(453\) 6312.21i 0.654688i
\(454\) 19220.3 1.98691
\(455\) 0 0
\(456\) 4018.81 0.412715
\(457\) − 5855.16i − 0.599328i −0.954045 0.299664i \(-0.903125\pi\)
0.954045 0.299664i \(-0.0968745\pi\)
\(458\) − 1893.76i − 0.193208i
\(459\) −3307.45 −0.336336
\(460\) 0 0
\(461\) 3204.74 0.323774 0.161887 0.986809i \(-0.448242\pi\)
0.161887 + 0.986809i \(0.448242\pi\)
\(462\) − 217.844i − 0.0219373i
\(463\) − 371.658i − 0.0373054i −0.999826 0.0186527i \(-0.994062\pi\)
0.999826 0.0186527i \(-0.00593769\pi\)
\(464\) 7618.54 0.762245
\(465\) 0 0
\(466\) 647.927 0.0644091
\(467\) − 19752.3i − 1.95723i −0.205703 0.978614i \(-0.565948\pi\)
0.205703 0.978614i \(-0.434052\pi\)
\(468\) 766.681i 0.0757262i
\(469\) −2169.95 −0.213643
\(470\) 0 0
\(471\) −1780.40 −0.174175
\(472\) − 9484.01i − 0.924866i
\(473\) − 126.334i − 0.0122809i
\(474\) −262.320 −0.0254193
\(475\) 0 0
\(476\) −3832.00 −0.368990
\(477\) − 1653.09i − 0.158679i
\(478\) 2271.28i 0.217335i
\(479\) 20762.0 1.98046 0.990232 0.139433i \(-0.0445279\pi\)
0.990232 + 0.139433i \(0.0445279\pi\)
\(480\) 0 0
\(481\) −1851.34 −0.175496
\(482\) 20323.8i 1.92059i
\(483\) 4423.07i 0.416681i
\(484\) 5909.50 0.554987
\(485\) 0 0
\(486\) −858.064 −0.0800876
\(487\) 17647.6i 1.64207i 0.570878 + 0.821035i \(0.306603\pi\)
−0.570878 + 0.821035i \(0.693397\pi\)
\(488\) 2478.15i 0.229878i
\(489\) −6536.12 −0.604445
\(490\) 0 0
\(491\) −5637.46 −0.518157 −0.259078 0.965856i \(-0.583419\pi\)
−0.259078 + 0.965856i \(0.583419\pi\)
\(492\) − 6584.16i − 0.603327i
\(493\) − 11697.9i − 1.06865i
\(494\) −7231.64 −0.658638
\(495\) 0 0
\(496\) 7523.86 0.681112
\(497\) − 4659.85i − 0.420569i
\(498\) − 4308.58i − 0.387695i
\(499\) 17474.1 1.56764 0.783818 0.620991i \(-0.213270\pi\)
0.783818 + 0.620991i \(0.213270\pi\)
\(500\) 0 0
\(501\) 2398.51 0.213887
\(502\) 18122.7i 1.61127i
\(503\) − 7444.81i − 0.659936i −0.943992 0.329968i \(-0.892962\pi\)
0.943992 0.329968i \(-0.107038\pi\)
\(504\) 785.539 0.0694260
\(505\) 0 0
\(506\) 2184.90 0.191958
\(507\) − 5500.89i − 0.481860i
\(508\) 8383.53i 0.732203i
\(509\) 3384.48 0.294724 0.147362 0.989083i \(-0.452922\pi\)
0.147362 + 0.989083i \(0.452922\pi\)
\(510\) 0 0
\(511\) −4349.02 −0.376495
\(512\) − 6558.89i − 0.566142i
\(513\) − 2900.77i − 0.249653i
\(514\) 17803.0 1.52774
\(515\) 0 0
\(516\) −576.536 −0.0491872
\(517\) − 1391.00i − 0.118329i
\(518\) − 2400.62i − 0.203624i
\(519\) 4333.07 0.366476
\(520\) 0 0
\(521\) 2973.12 0.250009 0.125005 0.992156i \(-0.460105\pi\)
0.125005 + 0.992156i \(0.460105\pi\)
\(522\) − 3034.82i − 0.254465i
\(523\) − 2689.02i − 0.224823i −0.993662 0.112412i \(-0.964142\pi\)
0.993662 0.112412i \(-0.0358575\pi\)
\(524\) −1627.57 −0.135689
\(525\) 0 0
\(526\) −26755.3 −2.21785
\(527\) − 11552.5i − 0.954903i
\(528\) − 703.121i − 0.0579534i
\(529\) −32194.9 −2.64608
\(530\) 0 0
\(531\) −6845.53 −0.559455
\(532\) − 3360.82i − 0.273891i
\(533\) − 9361.72i − 0.760790i
\(534\) 2772.83 0.224704
\(535\) 0 0
\(536\) −3865.25 −0.311480
\(537\) 10030.5i 0.806046i
\(538\) 3612.69i 0.289506i
\(539\) 143.949 0.0115034
\(540\) 0 0
\(541\) −14429.5 −1.14671 −0.573356 0.819306i \(-0.694359\pi\)
−0.573356 + 0.819306i \(0.694359\pi\)
\(542\) 7952.03i 0.630201i
\(543\) − 6755.44i − 0.533893i
\(544\) −22290.1 −1.75677
\(545\) 0 0
\(546\) −1413.54 −0.110795
\(547\) 13811.2i 1.07957i 0.841804 + 0.539784i \(0.181494\pi\)
−0.841804 + 0.539784i \(0.818506\pi\)
\(548\) − 7164.19i − 0.558466i
\(549\) 1788.72 0.139054
\(550\) 0 0
\(551\) 10259.5 0.793229
\(552\) 7878.68i 0.607498i
\(553\) − 173.338i − 0.0133293i
\(554\) −30476.2 −2.33720
\(555\) 0 0
\(556\) 10864.4 0.828692
\(557\) − 6033.26i − 0.458954i −0.973314 0.229477i \(-0.926298\pi\)
0.973314 0.229477i \(-0.0737016\pi\)
\(558\) − 2997.10i − 0.227379i
\(559\) −819.751 −0.0620246
\(560\) 0 0
\(561\) −1079.60 −0.0812493
\(562\) 26559.8i 1.99352i
\(563\) 6958.47i 0.520896i 0.965488 + 0.260448i \(0.0838703\pi\)
−0.965488 + 0.260448i \(0.916130\pi\)
\(564\) −6347.95 −0.473931
\(565\) 0 0
\(566\) −52.3202 −0.00388548
\(567\) − 567.000i − 0.0419961i
\(568\) − 8300.44i − 0.613166i
\(569\) 13396.4 0.987009 0.493505 0.869743i \(-0.335716\pi\)
0.493505 + 0.869743i \(0.335716\pi\)
\(570\) 0 0
\(571\) −8055.84 −0.590414 −0.295207 0.955433i \(-0.595389\pi\)
−0.295207 + 0.955433i \(0.595389\pi\)
\(572\) 250.257i 0.0182933i
\(573\) 3005.78i 0.219142i
\(574\) 12139.3 0.882724
\(575\) 0 0
\(576\) −38.6435 −0.00279539
\(577\) − 21456.9i − 1.54812i −0.633114 0.774059i \(-0.718223\pi\)
0.633114 0.774059i \(-0.281777\pi\)
\(578\) 35638.9i 2.56468i
\(579\) 12164.9 0.873153
\(580\) 0 0
\(581\) 2847.07 0.203298
\(582\) 10644.0i 0.758087i
\(583\) − 539.596i − 0.0383324i
\(584\) −7746.76 −0.548910
\(585\) 0 0
\(586\) −24412.1 −1.72091
\(587\) 20156.3i 1.41728i 0.705572 + 0.708638i \(0.250690\pi\)
−0.705572 + 0.708638i \(0.749310\pi\)
\(588\) − 656.924i − 0.0460733i
\(589\) 10132.0 0.708797
\(590\) 0 0
\(591\) −15420.7 −1.07330
\(592\) − 7748.30i − 0.537928i
\(593\) − 599.307i − 0.0415018i −0.999785 0.0207509i \(-0.993394\pi\)
0.999785 0.0207509i \(-0.00660570\pi\)
\(594\) −280.086 −0.0193469
\(595\) 0 0
\(596\) 10464.2 0.719177
\(597\) 1756.89i 0.120444i
\(598\) − 14177.3i − 0.969485i
\(599\) 5493.05 0.374691 0.187346 0.982294i \(-0.440012\pi\)
0.187346 + 0.982294i \(0.440012\pi\)
\(600\) 0 0
\(601\) 24292.8 1.64879 0.824396 0.566014i \(-0.191515\pi\)
0.824396 + 0.566014i \(0.191515\pi\)
\(602\) − 1062.97i − 0.0719655i
\(603\) 2789.93i 0.188416i
\(604\) 9402.82 0.633436
\(605\) 0 0
\(606\) 1365.19 0.0915131
\(607\) 3029.50i 0.202576i 0.994857 + 0.101288i \(0.0322964\pi\)
−0.994857 + 0.101288i \(0.967704\pi\)
\(608\) − 19549.3i − 1.30400i
\(609\) 2005.38 0.133435
\(610\) 0 0
\(611\) −9025.87 −0.597623
\(612\) 4926.85i 0.325419i
\(613\) 19339.6i 1.27426i 0.770757 + 0.637129i \(0.219878\pi\)
−0.770757 + 0.637129i \(0.780122\pi\)
\(614\) −26992.4 −1.77414
\(615\) 0 0
\(616\) 256.412 0.0167713
\(617\) − 5743.91i − 0.374783i −0.982285 0.187391i \(-0.939997\pi\)
0.982285 0.187391i \(-0.0600033\pi\)
\(618\) 8538.35i 0.555765i
\(619\) 8243.35 0.535264 0.267632 0.963521i \(-0.413759\pi\)
0.267632 + 0.963521i \(0.413759\pi\)
\(620\) 0 0
\(621\) 5686.81 0.367478
\(622\) − 26815.4i − 1.72861i
\(623\) 1832.26i 0.117830i
\(624\) −4562.37 −0.292694
\(625\) 0 0
\(626\) 32231.6 2.05788
\(627\) − 946.856i − 0.0603091i
\(628\) 2652.13i 0.168521i
\(629\) −11897.1 −0.754162
\(630\) 0 0
\(631\) −4376.56 −0.276114 −0.138057 0.990424i \(-0.544086\pi\)
−0.138057 + 0.990424i \(0.544086\pi\)
\(632\) − 308.762i − 0.0194334i
\(633\) 3165.48i 0.198762i
\(634\) −17407.8 −1.09046
\(635\) 0 0
\(636\) −2462.49 −0.153528
\(637\) − 934.051i − 0.0580980i
\(638\) − 990.613i − 0.0614714i
\(639\) −5991.23 −0.370907
\(640\) 0 0
\(641\) 11836.6 0.729357 0.364678 0.931133i \(-0.381179\pi\)
0.364678 + 0.931133i \(0.381179\pi\)
\(642\) 8153.86i 0.501257i
\(643\) − 1448.21i − 0.0888209i −0.999013 0.0444104i \(-0.985859\pi\)
0.999013 0.0444104i \(-0.0141409\pi\)
\(644\) 6588.72 0.403155
\(645\) 0 0
\(646\) −46472.0 −2.83037
\(647\) − 8732.95i − 0.530646i −0.964160 0.265323i \(-0.914521\pi\)
0.964160 0.265323i \(-0.0854785\pi\)
\(648\) − 1009.98i − 0.0612279i
\(649\) −2234.49 −0.135149
\(650\) 0 0
\(651\) 1980.46 0.119232
\(652\) 9736.37i 0.584824i
\(653\) 21978.4i 1.31712i 0.752527 + 0.658562i \(0.228835\pi\)
−0.752527 + 0.658562i \(0.771165\pi\)
\(654\) −8271.91 −0.494583
\(655\) 0 0
\(656\) 39181.1 2.33196
\(657\) 5591.59i 0.332038i
\(658\) − 11703.8i − 0.693406i
\(659\) 27761.7 1.64103 0.820516 0.571623i \(-0.193686\pi\)
0.820516 + 0.571623i \(0.193686\pi\)
\(660\) 0 0
\(661\) −8573.72 −0.504507 −0.252254 0.967661i \(-0.581172\pi\)
−0.252254 + 0.967661i \(0.581172\pi\)
\(662\) − 4313.86i − 0.253267i
\(663\) 7005.27i 0.410350i
\(664\) 5071.39 0.296398
\(665\) 0 0
\(666\) −3086.51 −0.179579
\(667\) 20113.2i 1.16760i
\(668\) − 3572.87i − 0.206944i
\(669\) −14027.6 −0.810668
\(670\) 0 0
\(671\) 583.868 0.0335916
\(672\) − 3821.22i − 0.219356i
\(673\) − 27159.2i − 1.55559i −0.628518 0.777795i \(-0.716338\pi\)
0.628518 0.777795i \(-0.283662\pi\)
\(674\) −30879.1 −1.76472
\(675\) 0 0
\(676\) −8194.26 −0.466219
\(677\) − 1392.30i − 0.0790404i −0.999219 0.0395202i \(-0.987417\pi\)
0.999219 0.0395202i \(-0.0125829\pi\)
\(678\) − 11818.5i − 0.669448i
\(679\) −7033.42 −0.397523
\(680\) 0 0
\(681\) 16329.3 0.918857
\(682\) − 978.302i − 0.0549283i
\(683\) 8675.09i 0.486007i 0.970025 + 0.243004i \(0.0781327\pi\)
−0.970025 + 0.243004i \(0.921867\pi\)
\(684\) −4321.05 −0.241549
\(685\) 0 0
\(686\) 1211.18 0.0674096
\(687\) − 1608.91i − 0.0893504i
\(688\) − 3430.86i − 0.190117i
\(689\) −3501.30 −0.193598
\(690\) 0 0
\(691\) −21426.0 −1.17957 −0.589785 0.807561i \(-0.700787\pi\)
−0.589785 + 0.807561i \(0.700787\pi\)
\(692\) − 6454.65i − 0.354579i
\(693\) − 185.078i − 0.0101451i
\(694\) 16206.3 0.886433
\(695\) 0 0
\(696\) 3572.11 0.194541
\(697\) − 60160.4i − 3.26935i
\(698\) − 14088.8i − 0.763997i
\(699\) 550.470 0.0297864
\(700\) 0 0
\(701\) 24840.5 1.33839 0.669197 0.743085i \(-0.266638\pi\)
0.669197 + 0.743085i \(0.266638\pi\)
\(702\) 1817.40i 0.0977116i
\(703\) − 10434.2i − 0.559793i
\(704\) −12.6139 −0.000675288 0
\(705\) 0 0
\(706\) −8532.45 −0.454848
\(707\) 902.101i 0.0479873i
\(708\) 10197.3i 0.541295i
\(709\) −12525.0 −0.663450 −0.331725 0.943376i \(-0.607631\pi\)
−0.331725 + 0.943376i \(0.607631\pi\)
\(710\) 0 0
\(711\) −222.863 −0.0117553
\(712\) 3263.74i 0.171789i
\(713\) 19863.3i 1.04332i
\(714\) −9083.69 −0.476118
\(715\) 0 0
\(716\) 14941.6 0.779881
\(717\) 1929.65i 0.100508i
\(718\) − 9732.66i − 0.505877i
\(719\) −28085.0 −1.45674 −0.728369 0.685185i \(-0.759721\pi\)
−0.728369 + 0.685185i \(0.759721\pi\)
\(720\) 0 0
\(721\) −5642.05 −0.291430
\(722\) − 16537.9i − 0.852460i
\(723\) 17266.8i 0.888189i
\(724\) −10063.1 −0.516562
\(725\) 0 0
\(726\) 14008.4 0.716115
\(727\) − 14326.2i − 0.730851i −0.930841 0.365426i \(-0.880923\pi\)
0.930841 0.365426i \(-0.119077\pi\)
\(728\) − 1663.79i − 0.0847037i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −5267.89 −0.266539
\(732\) − 2664.53i − 0.134541i
\(733\) − 6727.85i − 0.339016i −0.985529 0.169508i \(-0.945782\pi\)
0.985529 0.169508i \(-0.0542179\pi\)
\(734\) 39240.8 1.97331
\(735\) 0 0
\(736\) 38325.5 1.91943
\(737\) 910.677i 0.0455159i
\(738\) − 15607.6i − 0.778490i
\(739\) 3418.51 0.170165 0.0850826 0.996374i \(-0.472885\pi\)
0.0850826 + 0.996374i \(0.472885\pi\)
\(740\) 0 0
\(741\) −6143.91 −0.304591
\(742\) − 4540.11i − 0.224626i
\(743\) − 8095.50i − 0.399724i −0.979824 0.199862i \(-0.935951\pi\)
0.979824 0.199862i \(-0.0640494\pi\)
\(744\) 3527.72 0.173834
\(745\) 0 0
\(746\) −21508.4 −1.05560
\(747\) − 3660.51i − 0.179292i
\(748\) 1608.20i 0.0786119i
\(749\) −5387.99 −0.262847
\(750\) 0 0
\(751\) 13446.8 0.653371 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(752\) − 37775.4i − 1.83182i
\(753\) 15396.8i 0.745141i
\(754\) −6427.84 −0.310462
\(755\) 0 0
\(756\) −844.617 −0.0406328
\(757\) − 2593.24i − 0.124508i −0.998060 0.0622541i \(-0.980171\pi\)
0.998060 0.0622541i \(-0.0198289\pi\)
\(758\) 14069.0i 0.674156i
\(759\) 1856.26 0.0887722
\(760\) 0 0
\(761\) 27079.4 1.28992 0.644959 0.764217i \(-0.276875\pi\)
0.644959 + 0.764217i \(0.276875\pi\)
\(762\) 19873.0i 0.944782i
\(763\) − 5465.99i − 0.259348i
\(764\) 4477.48 0.212028
\(765\) 0 0
\(766\) −1124.58 −0.0530451
\(767\) 14499.0i 0.682568i
\(768\) − 15363.3i − 0.721842i
\(769\) −2138.72 −0.100292 −0.0501458 0.998742i \(-0.515969\pi\)
−0.0501458 + 0.998742i \(0.515969\pi\)
\(770\) 0 0
\(771\) 15125.2 0.706513
\(772\) − 18121.1i − 0.844810i
\(773\) − 25864.0i − 1.20345i −0.798704 0.601724i \(-0.794481\pi\)
0.798704 0.601724i \(-0.205519\pi\)
\(774\) −1366.67 −0.0634676
\(775\) 0 0
\(776\) −12528.4 −0.579566
\(777\) − 2039.53i − 0.0941671i
\(778\) 13720.1i 0.632247i
\(779\) 52763.1 2.42675
\(780\) 0 0
\(781\) −1955.63 −0.0896006
\(782\) − 91106.2i − 4.16618i
\(783\) − 2578.34i − 0.117679i
\(784\) 3909.23 0.178081
\(785\) 0 0
\(786\) −3858.14 −0.175083
\(787\) − 32371.3i − 1.46621i −0.680113 0.733107i \(-0.738069\pi\)
0.680113 0.733107i \(-0.261931\pi\)
\(788\) 22971.0i 1.03846i
\(789\) −22731.0 −1.02566
\(790\) 0 0
\(791\) 7809.52 0.351043
\(792\) − 329.673i − 0.0147909i
\(793\) − 3788.57i − 0.169654i
\(794\) 16970.8 0.758526
\(795\) 0 0
\(796\) 2617.11 0.116534
\(797\) 2024.33i 0.0899691i 0.998988 + 0.0449845i \(0.0143239\pi\)
−0.998988 + 0.0449845i \(0.985676\pi\)
\(798\) − 7966.76i − 0.353409i
\(799\) −58002.1 −2.56817
\(800\) 0 0
\(801\) 2355.76 0.103916
\(802\) − 12777.7i − 0.562589i
\(803\) 1825.18i 0.0802109i
\(804\) 4155.95 0.182300
\(805\) 0 0
\(806\) −6347.95 −0.277416
\(807\) 3069.29i 0.133884i
\(808\) 1606.88i 0.0699628i
\(809\) −12391.7 −0.538526 −0.269263 0.963067i \(-0.586780\pi\)
−0.269263 + 0.963067i \(0.586780\pi\)
\(810\) 0 0
\(811\) 14654.5 0.634511 0.317256 0.948340i \(-0.397239\pi\)
0.317256 + 0.948340i \(0.397239\pi\)
\(812\) − 2987.26i − 0.129104i
\(813\) 6755.94i 0.291441i
\(814\) −1007.49 −0.0433813
\(815\) 0 0
\(816\) −29318.7 −1.25780
\(817\) − 4620.16i − 0.197844i
\(818\) − 7447.33i − 0.318325i
\(819\) −1200.92 −0.0512376
\(820\) 0 0
\(821\) 23887.9 1.01546 0.507731 0.861516i \(-0.330485\pi\)
0.507731 + 0.861516i \(0.330485\pi\)
\(822\) − 16982.6i − 0.720604i
\(823\) 4008.41i 0.169774i 0.996391 + 0.0848871i \(0.0270530\pi\)
−0.996391 + 0.0848871i \(0.972947\pi\)
\(824\) −10050.0 −0.424889
\(825\) 0 0
\(826\) −18800.8 −0.791966
\(827\) − 45110.4i − 1.89679i −0.317096 0.948394i \(-0.602708\pi\)
0.317096 0.948394i \(-0.397292\pi\)
\(828\) − 8471.21i − 0.355549i
\(829\) −16165.4 −0.677260 −0.338630 0.940920i \(-0.609964\pi\)
−0.338630 + 0.940920i \(0.609964\pi\)
\(830\) 0 0
\(831\) −25892.2 −1.08085
\(832\) 81.8481i 0.00341054i
\(833\) − 6002.41i − 0.249665i
\(834\) 25753.8 1.06928
\(835\) 0 0
\(836\) −1410.46 −0.0583514
\(837\) − 2546.30i − 0.105153i
\(838\) − 24385.7i − 1.00524i
\(839\) 25244.4 1.03878 0.519388 0.854538i \(-0.326160\pi\)
0.519388 + 0.854538i \(0.326160\pi\)
\(840\) 0 0
\(841\) −15269.9 −0.626096
\(842\) 34066.7i 1.39432i
\(843\) 22564.9i 0.921916i
\(844\) 4715.37 0.192310
\(845\) 0 0
\(846\) −15047.7 −0.611526
\(847\) 9256.59i 0.375514i
\(848\) − 14653.8i − 0.593412i
\(849\) −44.4506 −0.00179687
\(850\) 0 0
\(851\) 20455.8 0.823990
\(852\) 8924.68i 0.358867i
\(853\) 30168.1i 1.21094i 0.795867 + 0.605472i \(0.207016\pi\)
−0.795867 + 0.605472i \(0.792984\pi\)
\(854\) 4912.61 0.196846
\(855\) 0 0
\(856\) −9597.44 −0.383217
\(857\) − 13393.6i − 0.533857i −0.963716 0.266929i \(-0.913991\pi\)
0.963716 0.266929i \(-0.0860088\pi\)
\(858\) 593.230i 0.0236043i
\(859\) −19060.4 −0.757081 −0.378541 0.925585i \(-0.623574\pi\)
−0.378541 + 0.925585i \(0.623574\pi\)
\(860\) 0 0
\(861\) 10313.4 0.408222
\(862\) 45914.3i 1.81421i
\(863\) 9466.86i 0.373413i 0.982416 + 0.186707i \(0.0597814\pi\)
−0.982416 + 0.186707i \(0.940219\pi\)
\(864\) −4913.00 −0.193453
\(865\) 0 0
\(866\) −25975.2 −1.01925
\(867\) 30278.3i 1.18605i
\(868\) − 2950.13i − 0.115362i
\(869\) −72.7461 −0.00283975
\(870\) 0 0
\(871\) 5909.15 0.229878
\(872\) − 9736.39i − 0.378115i
\(873\) 9042.97i 0.350582i
\(874\) 79903.8 3.09243
\(875\) 0 0
\(876\) 8329.37 0.321259
\(877\) 37740.6i 1.45315i 0.687090 + 0.726573i \(0.258888\pi\)
−0.687090 + 0.726573i \(0.741112\pi\)
\(878\) 24397.3i 0.937778i
\(879\) −20740.2 −0.795845
\(880\) 0 0
\(881\) 25991.5 0.993957 0.496979 0.867763i \(-0.334443\pi\)
0.496979 + 0.867763i \(0.334443\pi\)
\(882\) − 1557.23i − 0.0594496i
\(883\) − 39420.3i − 1.50238i −0.660087 0.751189i \(-0.729481\pi\)
0.660087 0.751189i \(-0.270519\pi\)
\(884\) 10435.2 0.397030
\(885\) 0 0
\(886\) 52305.0 1.98332
\(887\) 46005.2i 1.74149i 0.491735 + 0.870745i \(0.336363\pi\)
−0.491735 + 0.870745i \(0.663637\pi\)
\(888\) − 3632.95i − 0.137290i
\(889\) −13131.9 −0.495421
\(890\) 0 0
\(891\) −237.957 −0.00894710
\(892\) 20895.8i 0.784353i
\(893\) − 50870.2i − 1.90628i
\(894\) 24805.2 0.927975
\(895\) 0 0
\(896\) 10083.8 0.375978
\(897\) − 12044.8i − 0.448345i
\(898\) − 37622.6i − 1.39809i
\(899\) 9005.81 0.334105
\(900\) 0 0
\(901\) −22500.1 −0.831950
\(902\) − 5094.58i − 0.188061i
\(903\) − 903.081i − 0.0332809i
\(904\) 13910.8 0.511801
\(905\) 0 0
\(906\) 22289.2 0.817340
\(907\) − 2838.97i − 0.103932i −0.998649 0.0519661i \(-0.983451\pi\)
0.998649 0.0519661i \(-0.0165488\pi\)
\(908\) − 24324.6i − 0.889030i
\(909\) 1159.84 0.0423208
\(910\) 0 0
\(911\) 39890.9 1.45076 0.725382 0.688347i \(-0.241663\pi\)
0.725382 + 0.688347i \(0.241663\pi\)
\(912\) − 25713.7i − 0.933626i
\(913\) − 1194.85i − 0.0433119i
\(914\) −20675.3 −0.748226
\(915\) 0 0
\(916\) −2396.67 −0.0864500
\(917\) − 2549.42i − 0.0918094i
\(918\) 11679.0i 0.419897i
\(919\) 646.475 0.0232048 0.0116024 0.999933i \(-0.496307\pi\)
0.0116024 + 0.999933i \(0.496307\pi\)
\(920\) 0 0
\(921\) −22932.4 −0.820463
\(922\) − 11316.3i − 0.404213i
\(923\) 12689.6i 0.452528i
\(924\) −275.696 −0.00981574
\(925\) 0 0
\(926\) −1312.37 −0.0465737
\(927\) 7254.07i 0.257017i
\(928\) − 17376.4i − 0.614665i
\(929\) −51188.2 −1.80778 −0.903892 0.427760i \(-0.859303\pi\)
−0.903892 + 0.427760i \(0.859303\pi\)
\(930\) 0 0
\(931\) 5264.35 0.185319
\(932\) − 819.993i − 0.0288195i
\(933\) − 22782.0i − 0.799409i
\(934\) −69747.8 −2.44349
\(935\) 0 0
\(936\) −2139.16 −0.0747017
\(937\) 29786.1i 1.03849i 0.854624 + 0.519247i \(0.173788\pi\)
−0.854624 + 0.519247i \(0.826212\pi\)
\(938\) 7662.36i 0.266722i
\(939\) 27383.5 0.951680
\(940\) 0 0
\(941\) −44817.4 −1.55261 −0.776304 0.630358i \(-0.782908\pi\)
−0.776304 + 0.630358i \(0.782908\pi\)
\(942\) 6286.82i 0.217448i
\(943\) 103439.i 3.57206i
\(944\) −60682.0 −2.09219
\(945\) 0 0
\(946\) −446.103 −0.0153320
\(947\) 54697.1i 1.87689i 0.345425 + 0.938446i \(0.387735\pi\)
−0.345425 + 0.938446i \(0.612265\pi\)
\(948\) 331.983i 0.0113737i
\(949\) 11843.2 0.405105
\(950\) 0 0
\(951\) −14789.4 −0.504290
\(952\) − 10691.9i − 0.363998i
\(953\) 7577.51i 0.257565i 0.991673 + 0.128783i \(0.0411069\pi\)
−0.991673 + 0.128783i \(0.958893\pi\)
\(954\) −5837.29 −0.198102
\(955\) 0 0
\(956\) 2874.46 0.0972454
\(957\) − 841.612i − 0.0284278i
\(958\) − 73313.4i − 2.47249i
\(959\) 11221.9 0.377868
\(960\) 0 0
\(961\) −20897.1 −0.701457
\(962\) 6537.32i 0.219097i
\(963\) 6927.41i 0.231810i
\(964\) 25721.1 0.859357
\(965\) 0 0
\(966\) 15618.4 0.520202
\(967\) 50779.0i 1.68867i 0.535817 + 0.844334i \(0.320004\pi\)
−0.535817 + 0.844334i \(0.679996\pi\)
\(968\) 16488.5i 0.547478i
\(969\) −39482.0 −1.30892
\(970\) 0 0
\(971\) −15313.2 −0.506102 −0.253051 0.967453i \(-0.581434\pi\)
−0.253051 + 0.967453i \(0.581434\pi\)
\(972\) 1085.94i 0.0358348i
\(973\) 17017.9i 0.560707i
\(974\) 62315.9 2.05003
\(975\) 0 0
\(976\) 15856.1 0.520021
\(977\) 46620.4i 1.52663i 0.646025 + 0.763316i \(0.276430\pi\)
−0.646025 + 0.763316i \(0.723570\pi\)
\(978\) 23079.9i 0.754615i
\(979\) 768.957 0.0251031
\(980\) 0 0
\(981\) −7027.70 −0.228723
\(982\) 19906.6i 0.646889i
\(983\) 2824.37i 0.0916414i 0.998950 + 0.0458207i \(0.0145903\pi\)
−0.998950 + 0.0458207i \(0.985410\pi\)
\(984\) 18370.9 0.595165
\(985\) 0 0
\(986\) −41306.6 −1.33415
\(987\) − 9943.38i − 0.320670i
\(988\) 9152.11i 0.294704i
\(989\) 9057.59 0.291218
\(990\) 0 0
\(991\) 16951.4 0.543370 0.271685 0.962386i \(-0.412419\pi\)
0.271685 + 0.962386i \(0.412419\pi\)
\(992\) − 17160.5i − 0.549239i
\(993\) − 3665.00i − 0.117125i
\(994\) −16454.5 −0.525056
\(995\) 0 0
\(996\) −5452.79 −0.173472
\(997\) 23847.8i 0.757540i 0.925491 + 0.378770i \(0.123653\pi\)
−0.925491 + 0.378770i \(0.876347\pi\)
\(998\) − 61703.4i − 1.95710i
\(999\) −2622.26 −0.0830476
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 525.4.d.h.274.2 4
5.2 odd 4 525.4.a.k.1.2 2
5.3 odd 4 105.4.a.f.1.1 2
5.4 even 2 inner 525.4.d.h.274.3 4
15.2 even 4 1575.4.a.w.1.1 2
15.8 even 4 315.4.a.i.1.2 2
20.3 even 4 1680.4.a.bg.1.1 2
35.13 even 4 735.4.a.p.1.1 2
105.83 odd 4 2205.4.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 2 5.3 odd 4
315.4.a.i.1.2 2 15.8 even 4
525.4.a.k.1.2 2 5.2 odd 4
525.4.d.h.274.2 4 1.1 even 1 trivial
525.4.d.h.274.3 4 5.4 even 2 inner
735.4.a.p.1.1 2 35.13 even 4
1575.4.a.w.1.1 2 15.2 even 4
1680.4.a.bg.1.1 2 20.3 even 4
2205.4.a.z.1.2 2 105.83 odd 4