Properties

Label 525.4.d.o.274.3
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $8$
CM no
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 40x^{6} + 488x^{4} + 1945x^{2} + 1936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.3
Root \(-3.56826i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.o.274.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56826i q^{2} -3.00000i q^{3} +1.40404 q^{4} -7.70478 q^{6} -7.00000i q^{7} -24.1520i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-2.56826i q^{2} -3.00000i q^{3} +1.40404 q^{4} -7.70478 q^{6} -7.00000i q^{7} -24.1520i q^{8} -9.00000 q^{9} +66.4181 q^{11} -4.21213i q^{12} -34.0329i q^{13} -17.9778 q^{14} -50.7963 q^{16} -6.12934i q^{17} +23.1143i q^{18} +163.424 q^{19} -21.0000 q^{21} -170.579i q^{22} +35.8799i q^{23} -72.4561 q^{24} -87.4053 q^{26} +27.0000i q^{27} -9.82830i q^{28} -27.7256 q^{29} +74.3417 q^{31} -62.7581i q^{32} -199.254i q^{33} -15.7417 q^{34} -12.6364 q^{36} +260.966i q^{37} -419.716i q^{38} -102.099 q^{39} -445.485 q^{41} +53.9334i q^{42} -474.985i q^{43} +93.2538 q^{44} +92.1489 q^{46} +51.0436i q^{47} +152.389i q^{48} -49.0000 q^{49} -18.3880 q^{51} -47.7837i q^{52} -676.667i q^{53} +69.3430 q^{54} -169.064 q^{56} -490.273i q^{57} +71.2066i q^{58} +115.979 q^{59} -390.356 q^{61} -190.929i q^{62} +63.0000i q^{63} -567.550 q^{64} -511.737 q^{66} +713.721i q^{67} -8.60586i q^{68} +107.640 q^{69} +810.524 q^{71} +217.368i q^{72} +350.196i q^{73} +670.227 q^{74} +229.455 q^{76} -464.927i q^{77} +262.216i q^{78} +50.9307 q^{79} +81.0000 q^{81} +1144.12i q^{82} -84.8904i q^{83} -29.4849 q^{84} -1219.88 q^{86} +83.1769i q^{87} -1604.13i q^{88} -1521.82 q^{89} -238.230 q^{91} +50.3769i q^{92} -223.025i q^{93} +131.093 q^{94} -188.274 q^{96} +1232.08i q^{97} +125.845i q^{98} -597.763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 36 q^{6} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} + 36 q^{6} - 72 q^{9} + 114 q^{11} + 84 q^{14} + 432 q^{16} + 24 q^{19} - 168 q^{21} - 558 q^{24} - 162 q^{26} - 756 q^{29} - 186 q^{31} - 1566 q^{34} + 288 q^{36} - 258 q^{39} - 930 q^{41} - 1362 q^{44} + 620 q^{46} - 392 q^{49} + 594 q^{51} - 324 q^{54} - 1302 q^{56} - 462 q^{59} - 2706 q^{61} - 6214 q^{64} - 246 q^{66} - 936 q^{69} - 3450 q^{71} + 3906 q^{74} - 6092 q^{76} - 3258 q^{79} + 648 q^{81} + 672 q^{84} - 9084 q^{86} + 1956 q^{89} - 602 q^{91} - 4960 q^{94} + 4140 q^{96} - 1026 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.56826i − 0.908017i −0.890997 0.454008i \(-0.849994\pi\)
0.890997 0.454008i \(-0.150006\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 1.40404 0.175505
\(5\) 0 0
\(6\) −7.70478 −0.524244
\(7\) − 7.00000i − 0.377964i
\(8\) − 24.1520i − 1.06738i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 66.4181 1.82053 0.910264 0.414029i \(-0.135879\pi\)
0.910264 + 0.414029i \(0.135879\pi\)
\(12\) − 4.21213i − 0.101328i
\(13\) − 34.0329i − 0.726079i −0.931774 0.363040i \(-0.881739\pi\)
0.931774 0.363040i \(-0.118261\pi\)
\(14\) −17.9778 −0.343198
\(15\) 0 0
\(16\) −50.7963 −0.793692
\(17\) − 6.12934i − 0.0874461i −0.999044 0.0437231i \(-0.986078\pi\)
0.999044 0.0437231i \(-0.0139219\pi\)
\(18\) 23.1143i 0.302672i
\(19\) 163.424 1.97327 0.986634 0.162951i \(-0.0521013\pi\)
0.986634 + 0.162951i \(0.0521013\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) − 170.579i − 1.65307i
\(23\) 35.8799i 0.325282i 0.986685 + 0.162641i \(0.0520012\pi\)
−0.986685 + 0.162641i \(0.947999\pi\)
\(24\) −72.4561 −0.616251
\(25\) 0 0
\(26\) −87.4053 −0.659292
\(27\) 27.0000i 0.192450i
\(28\) − 9.82830i − 0.0663348i
\(29\) −27.7256 −0.177535 −0.0887676 0.996052i \(-0.528293\pi\)
−0.0887676 + 0.996052i \(0.528293\pi\)
\(30\) 0 0
\(31\) 74.3417 0.430715 0.215358 0.976535i \(-0.430908\pi\)
0.215358 + 0.976535i \(0.430908\pi\)
\(32\) − 62.7581i − 0.346693i
\(33\) − 199.254i − 1.05108i
\(34\) −15.7417 −0.0794026
\(35\) 0 0
\(36\) −12.6364 −0.0585018
\(37\) 260.966i 1.15953i 0.814785 + 0.579763i \(0.196855\pi\)
−0.814785 + 0.579763i \(0.803145\pi\)
\(38\) − 419.716i − 1.79176i
\(39\) −102.099 −0.419202
\(40\) 0 0
\(41\) −445.485 −1.69690 −0.848452 0.529272i \(-0.822465\pi\)
−0.848452 + 0.529272i \(0.822465\pi\)
\(42\) 53.9334i 0.198146i
\(43\) − 474.985i − 1.68452i −0.539070 0.842261i \(-0.681224\pi\)
0.539070 0.842261i \(-0.318776\pi\)
\(44\) 93.2538 0.319512
\(45\) 0 0
\(46\) 92.1489 0.295361
\(47\) 51.0436i 0.158414i 0.996858 + 0.0792072i \(0.0252389\pi\)
−0.996858 + 0.0792072i \(0.974761\pi\)
\(48\) 152.389i 0.458239i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −18.3880 −0.0504871
\(52\) − 47.7837i − 0.127431i
\(53\) − 676.667i − 1.75372i −0.480742 0.876862i \(-0.659633\pi\)
0.480742 0.876862i \(-0.340367\pi\)
\(54\) 69.3430 0.174748
\(55\) 0 0
\(56\) −169.064 −0.403431
\(57\) − 490.273i − 1.13927i
\(58\) 71.2066i 0.161205i
\(59\) 115.979 0.255917 0.127959 0.991779i \(-0.459158\pi\)
0.127959 + 0.991779i \(0.459158\pi\)
\(60\) 0 0
\(61\) −390.356 −0.819344 −0.409672 0.912233i \(-0.634357\pi\)
−0.409672 + 0.912233i \(0.634357\pi\)
\(62\) − 190.929i − 0.391097i
\(63\) 63.0000i 0.125988i
\(64\) −567.550 −1.10850
\(65\) 0 0
\(66\) −511.737 −0.954400
\(67\) 713.721i 1.30142i 0.759328 + 0.650708i \(0.225528\pi\)
−0.759328 + 0.650708i \(0.774472\pi\)
\(68\) − 8.60586i − 0.0153473i
\(69\) 107.640 0.187801
\(70\) 0 0
\(71\) 810.524 1.35481 0.677405 0.735610i \(-0.263104\pi\)
0.677405 + 0.735610i \(0.263104\pi\)
\(72\) 217.368i 0.355793i
\(73\) 350.196i 0.561470i 0.959785 + 0.280735i \(0.0905782\pi\)
−0.959785 + 0.280735i \(0.909422\pi\)
\(74\) 670.227 1.05287
\(75\) 0 0
\(76\) 229.455 0.346319
\(77\) − 464.927i − 0.688095i
\(78\) 262.216i 0.380642i
\(79\) 50.9307 0.0725336 0.0362668 0.999342i \(-0.488453\pi\)
0.0362668 + 0.999342i \(0.488453\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1144.12i 1.54082i
\(83\) − 84.8904i − 0.112264i −0.998423 0.0561321i \(-0.982123\pi\)
0.998423 0.0561321i \(-0.0178768\pi\)
\(84\) −29.4849 −0.0382984
\(85\) 0 0
\(86\) −1219.88 −1.52957
\(87\) 83.1769i 0.102500i
\(88\) − 1604.13i − 1.94319i
\(89\) −1521.82 −1.81251 −0.906253 0.422735i \(-0.861070\pi\)
−0.906253 + 0.422735i \(0.861070\pi\)
\(90\) 0 0
\(91\) −238.230 −0.274432
\(92\) 50.3769i 0.0570887i
\(93\) − 223.025i − 0.248674i
\(94\) 131.093 0.143843
\(95\) 0 0
\(96\) −188.274 −0.200163
\(97\) 1232.08i 1.28968i 0.764317 + 0.644840i \(0.223076\pi\)
−0.764317 + 0.644840i \(0.776924\pi\)
\(98\) 125.845i 0.129717i
\(99\) −597.763 −0.606843
\(100\) 0 0
\(101\) −1661.64 −1.63702 −0.818510 0.574492i \(-0.805200\pi\)
−0.818510 + 0.574492i \(0.805200\pi\)
\(102\) 47.2252i 0.0458431i
\(103\) − 193.631i − 0.185233i −0.995702 0.0926167i \(-0.970477\pi\)
0.995702 0.0926167i \(-0.0295231\pi\)
\(104\) −821.963 −0.775001
\(105\) 0 0
\(106\) −1737.86 −1.59241
\(107\) − 921.040i − 0.832153i −0.909330 0.416076i \(-0.863405\pi\)
0.909330 0.416076i \(-0.136595\pi\)
\(108\) 37.9092i 0.0337760i
\(109\) −1304.73 −1.14652 −0.573260 0.819373i \(-0.694322\pi\)
−0.573260 + 0.819373i \(0.694322\pi\)
\(110\) 0 0
\(111\) 782.897 0.669453
\(112\) 355.574i 0.299988i
\(113\) 237.585i 0.197789i 0.995098 + 0.0988945i \(0.0315306\pi\)
−0.995098 + 0.0988945i \(0.968469\pi\)
\(114\) −1259.15 −1.03447
\(115\) 0 0
\(116\) −38.9280 −0.0311584
\(117\) 306.296i 0.242026i
\(118\) − 297.863i − 0.232377i
\(119\) −42.9054 −0.0330515
\(120\) 0 0
\(121\) 3080.36 2.31432
\(122\) 1002.54i 0.743978i
\(123\) 1336.45i 0.979708i
\(124\) 104.379 0.0755928
\(125\) 0 0
\(126\) 161.800 0.114399
\(127\) 1442.31i 1.00775i 0.863776 + 0.503876i \(0.168093\pi\)
−0.863776 + 0.503876i \(0.831907\pi\)
\(128\) 955.550i 0.659840i
\(129\) −1424.95 −0.972559
\(130\) 0 0
\(131\) 644.035 0.429539 0.214770 0.976665i \(-0.431100\pi\)
0.214770 + 0.976665i \(0.431100\pi\)
\(132\) − 279.762i − 0.184471i
\(133\) − 1143.97i − 0.745825i
\(134\) 1833.02 1.18171
\(135\) 0 0
\(136\) −148.036 −0.0933381
\(137\) 355.576i 0.221744i 0.993835 + 0.110872i \(0.0353644\pi\)
−0.993835 + 0.110872i \(0.964636\pi\)
\(138\) − 276.447i − 0.170527i
\(139\) 472.016 0.288028 0.144014 0.989576i \(-0.453999\pi\)
0.144014 + 0.989576i \(0.453999\pi\)
\(140\) 0 0
\(141\) 153.131 0.0914606
\(142\) − 2081.64i − 1.23019i
\(143\) − 2260.40i − 1.32185i
\(144\) 457.167 0.264564
\(145\) 0 0
\(146\) 899.393 0.509824
\(147\) 147.000i 0.0824786i
\(148\) 366.407i 0.203503i
\(149\) 1091.53 0.600143 0.300072 0.953917i \(-0.402989\pi\)
0.300072 + 0.953917i \(0.402989\pi\)
\(150\) 0 0
\(151\) 2914.48 1.57071 0.785354 0.619047i \(-0.212481\pi\)
0.785354 + 0.619047i \(0.212481\pi\)
\(152\) − 3947.03i − 2.10622i
\(153\) 55.1641i 0.0291487i
\(154\) −1194.05 −0.624802
\(155\) 0 0
\(156\) −143.351 −0.0735722
\(157\) 1476.94i 0.750780i 0.926867 + 0.375390i \(0.122491\pi\)
−0.926867 + 0.375390i \(0.877509\pi\)
\(158\) − 130.803i − 0.0658617i
\(159\) −2030.00 −1.01251
\(160\) 0 0
\(161\) 251.159 0.122945
\(162\) − 208.029i − 0.100891i
\(163\) 2764.56i 1.32845i 0.747534 + 0.664224i \(0.231238\pi\)
−0.747534 + 0.664224i \(0.768762\pi\)
\(164\) −625.480 −0.297816
\(165\) 0 0
\(166\) −218.020 −0.101938
\(167\) 255.508i 0.118394i 0.998246 + 0.0591970i \(0.0188540\pi\)
−0.998246 + 0.0591970i \(0.981146\pi\)
\(168\) 507.192i 0.232921i
\(169\) 1038.76 0.472809
\(170\) 0 0
\(171\) −1470.82 −0.657756
\(172\) − 666.899i − 0.295643i
\(173\) − 3850.56i − 1.69221i −0.533016 0.846105i \(-0.678941\pi\)
0.533016 0.846105i \(-0.321059\pi\)
\(174\) 213.620 0.0930717
\(175\) 0 0
\(176\) −3373.79 −1.44494
\(177\) − 347.936i − 0.147754i
\(178\) 3908.44i 1.64579i
\(179\) −952.804 −0.397854 −0.198927 0.980014i \(-0.563746\pi\)
−0.198927 + 0.980014i \(0.563746\pi\)
\(180\) 0 0
\(181\) 1648.83 0.677109 0.338554 0.940947i \(-0.390062\pi\)
0.338554 + 0.940947i \(0.390062\pi\)
\(182\) 611.837i 0.249189i
\(183\) 1171.07i 0.473048i
\(184\) 866.572 0.347199
\(185\) 0 0
\(186\) −572.787 −0.225800
\(187\) − 407.099i − 0.159198i
\(188\) 71.6675i 0.0278026i
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 767.536 0.290769 0.145385 0.989375i \(-0.453558\pi\)
0.145385 + 0.989375i \(0.453558\pi\)
\(192\) 1702.65i 0.639990i
\(193\) − 1186.61i − 0.442561i −0.975210 0.221280i \(-0.928976\pi\)
0.975210 0.221280i \(-0.0710236\pi\)
\(194\) 3164.31 1.17105
\(195\) 0 0
\(196\) −68.7981 −0.0250722
\(197\) − 3614.72i − 1.30730i −0.756797 0.653650i \(-0.773237\pi\)
0.756797 0.653650i \(-0.226763\pi\)
\(198\) 1535.21i 0.551023i
\(199\) 3077.54 1.09629 0.548144 0.836384i \(-0.315335\pi\)
0.548144 + 0.836384i \(0.315335\pi\)
\(200\) 0 0
\(201\) 2141.16 0.751373
\(202\) 4267.52i 1.48644i
\(203\) 194.079i 0.0671020i
\(204\) −25.8176 −0.00886075
\(205\) 0 0
\(206\) −497.295 −0.168195
\(207\) − 322.919i − 0.108427i
\(208\) 1728.75i 0.576284i
\(209\) 10854.3 3.59239
\(210\) 0 0
\(211\) 145.901 0.0476031 0.0238016 0.999717i \(-0.492423\pi\)
0.0238016 + 0.999717i \(0.492423\pi\)
\(212\) − 950.070i − 0.307788i
\(213\) − 2431.57i − 0.782200i
\(214\) −2365.47 −0.755609
\(215\) 0 0
\(216\) 652.105 0.205417
\(217\) − 520.392i − 0.162795i
\(218\) 3350.89i 1.04106i
\(219\) 1050.59 0.324165
\(220\) 0 0
\(221\) −208.599 −0.0634928
\(222\) − 2010.68i − 0.607875i
\(223\) 3357.30i 1.00817i 0.863655 + 0.504084i \(0.168170\pi\)
−0.863655 + 0.504084i \(0.831830\pi\)
\(224\) −439.306 −0.131037
\(225\) 0 0
\(226\) 610.181 0.179596
\(227\) 636.529i 0.186114i 0.995661 + 0.0930570i \(0.0296639\pi\)
−0.995661 + 0.0930570i \(0.970336\pi\)
\(228\) − 688.364i − 0.199948i
\(229\) −3652.17 −1.05389 −0.526947 0.849898i \(-0.676663\pi\)
−0.526947 + 0.849898i \(0.676663\pi\)
\(230\) 0 0
\(231\) −1394.78 −0.397272
\(232\) 669.630i 0.189497i
\(233\) 3387.30i 0.952400i 0.879337 + 0.476200i \(0.157986\pi\)
−0.879337 + 0.476200i \(0.842014\pi\)
\(234\) 786.648 0.219764
\(235\) 0 0
\(236\) 162.839 0.0449149
\(237\) − 152.792i − 0.0418773i
\(238\) 110.192i 0.0300113i
\(239\) 3382.64 0.915502 0.457751 0.889080i \(-0.348655\pi\)
0.457751 + 0.889080i \(0.348655\pi\)
\(240\) 0 0
\(241\) 5809.63 1.55283 0.776413 0.630225i \(-0.217037\pi\)
0.776413 + 0.630225i \(0.217037\pi\)
\(242\) − 7911.17i − 2.10144i
\(243\) − 243.000i − 0.0641500i
\(244\) −548.077 −0.143799
\(245\) 0 0
\(246\) 3432.36 0.889591
\(247\) − 5561.80i − 1.43275i
\(248\) − 1795.50i − 0.459736i
\(249\) −254.671 −0.0648158
\(250\) 0 0
\(251\) −345.006 −0.0867593 −0.0433796 0.999059i \(-0.513813\pi\)
−0.0433796 + 0.999059i \(0.513813\pi\)
\(252\) 88.4547i 0.0221116i
\(253\) 2383.07i 0.592184i
\(254\) 3704.23 0.915056
\(255\) 0 0
\(256\) −2086.30 −0.509350
\(257\) 3142.39i 0.762712i 0.924428 + 0.381356i \(0.124543\pi\)
−0.924428 + 0.381356i \(0.875457\pi\)
\(258\) 3659.65i 0.883100i
\(259\) 1826.76 0.438260
\(260\) 0 0
\(261\) 249.531 0.0591784
\(262\) − 1654.05i − 0.390029i
\(263\) − 4949.02i − 1.16034i −0.814495 0.580170i \(-0.802986\pi\)
0.814495 0.580170i \(-0.197014\pi\)
\(264\) −4812.39 −1.12190
\(265\) 0 0
\(266\) −2938.01 −0.677222
\(267\) 4565.47i 1.04645i
\(268\) 1002.10i 0.228406i
\(269\) 4570.60 1.03596 0.517982 0.855391i \(-0.326683\pi\)
0.517982 + 0.855391i \(0.326683\pi\)
\(270\) 0 0
\(271\) −5338.43 −1.19663 −0.598315 0.801261i \(-0.704163\pi\)
−0.598315 + 0.801261i \(0.704163\pi\)
\(272\) 311.348i 0.0694053i
\(273\) 714.691i 0.158443i
\(274\) 913.213 0.201347
\(275\) 0 0
\(276\) 151.131 0.0329602
\(277\) − 2225.98i − 0.482837i −0.970421 0.241419i \(-0.922387\pi\)
0.970421 0.241419i \(-0.0776127\pi\)
\(278\) − 1212.26i − 0.261534i
\(279\) −669.076 −0.143572
\(280\) 0 0
\(281\) 1991.15 0.422712 0.211356 0.977409i \(-0.432212\pi\)
0.211356 + 0.977409i \(0.432212\pi\)
\(282\) − 393.280i − 0.0830478i
\(283\) − 7350.20i − 1.54390i −0.635682 0.771951i \(-0.719281\pi\)
0.635682 0.771951i \(-0.280719\pi\)
\(284\) 1138.01 0.237777
\(285\) 0 0
\(286\) −5805.29 −1.20026
\(287\) 3118.39i 0.641369i
\(288\) 564.823i 0.115564i
\(289\) 4875.43 0.992353
\(290\) 0 0
\(291\) 3696.25 0.744597
\(292\) 491.690i 0.0985410i
\(293\) 6250.16i 1.24621i 0.782140 + 0.623103i \(0.214128\pi\)
−0.782140 + 0.623103i \(0.785872\pi\)
\(294\) 377.534 0.0748920
\(295\) 0 0
\(296\) 6302.85 1.23765
\(297\) 1793.29i 0.350361i
\(298\) − 2803.32i − 0.544940i
\(299\) 1221.10 0.236180
\(300\) 0 0
\(301\) −3324.89 −0.636690
\(302\) − 7485.13i − 1.42623i
\(303\) 4984.91i 0.945134i
\(304\) −8301.35 −1.56617
\(305\) 0 0
\(306\) 141.676 0.0264675
\(307\) 1259.29i 0.234109i 0.993126 + 0.117054i \(0.0373452\pi\)
−0.993126 + 0.117054i \(0.962655\pi\)
\(308\) − 652.777i − 0.120764i
\(309\) −580.893 −0.106945
\(310\) 0 0
\(311\) −6711.92 −1.22379 −0.611894 0.790940i \(-0.709592\pi\)
−0.611894 + 0.790940i \(0.709592\pi\)
\(312\) 2465.89i 0.447447i
\(313\) 2436.27i 0.439955i 0.975505 + 0.219978i \(0.0705984\pi\)
−0.975505 + 0.219978i \(0.929402\pi\)
\(314\) 3793.16 0.681721
\(315\) 0 0
\(316\) 71.5089 0.0127300
\(317\) − 1191.81i − 0.211164i −0.994411 0.105582i \(-0.966330\pi\)
0.994411 0.105582i \(-0.0336705\pi\)
\(318\) 5213.57i 0.919379i
\(319\) −1841.48 −0.323208
\(320\) 0 0
\(321\) −2763.12 −0.480443
\(322\) − 645.042i − 0.111636i
\(323\) − 1001.68i − 0.172555i
\(324\) 113.727 0.0195006
\(325\) 0 0
\(326\) 7100.11 1.20625
\(327\) 3914.20i 0.661944i
\(328\) 10759.4i 1.81124i
\(329\) 357.305 0.0598750
\(330\) 0 0
\(331\) 5885.63 0.977351 0.488676 0.872466i \(-0.337480\pi\)
0.488676 + 0.872466i \(0.337480\pi\)
\(332\) − 119.190i − 0.0197030i
\(333\) − 2348.69i − 0.386509i
\(334\) 656.210 0.107504
\(335\) 0 0
\(336\) 1066.72 0.173198
\(337\) − 1209.73i − 0.195544i −0.995209 0.0977720i \(-0.968828\pi\)
0.995209 0.0977720i \(-0.0311716\pi\)
\(338\) − 2667.81i − 0.429319i
\(339\) 712.756 0.114193
\(340\) 0 0
\(341\) 4937.63 0.784129
\(342\) 3777.44i 0.597254i
\(343\) 343.000i 0.0539949i
\(344\) −11471.8 −1.79802
\(345\) 0 0
\(346\) −9889.23 −1.53656
\(347\) 5065.21i 0.783617i 0.920047 + 0.391808i \(0.128150\pi\)
−0.920047 + 0.391808i \(0.871850\pi\)
\(348\) 116.784i 0.0179893i
\(349\) −5481.35 −0.840716 −0.420358 0.907358i \(-0.638096\pi\)
−0.420358 + 0.907358i \(0.638096\pi\)
\(350\) 0 0
\(351\) 918.888 0.139734
\(352\) − 4168.27i − 0.631163i
\(353\) − 602.190i − 0.0907970i −0.998969 0.0453985i \(-0.985544\pi\)
0.998969 0.0453985i \(-0.0144558\pi\)
\(354\) −893.589 −0.134163
\(355\) 0 0
\(356\) −2136.71 −0.318105
\(357\) 128.716i 0.0190823i
\(358\) 2447.05i 0.361259i
\(359\) −2004.84 −0.294740 −0.147370 0.989081i \(-0.547081\pi\)
−0.147370 + 0.989081i \(0.547081\pi\)
\(360\) 0 0
\(361\) 19848.5 2.89379
\(362\) − 4234.63i − 0.614826i
\(363\) − 9241.08i − 1.33617i
\(364\) −334.486 −0.0481643
\(365\) 0 0
\(366\) 3007.61 0.429536
\(367\) 7770.37i 1.10520i 0.833445 + 0.552602i \(0.186365\pi\)
−0.833445 + 0.552602i \(0.813635\pi\)
\(368\) − 1822.57i − 0.258174i
\(369\) 4009.36 0.565635
\(370\) 0 0
\(371\) −4736.67 −0.662846
\(372\) − 313.137i − 0.0436435i
\(373\) 7485.06i 1.03904i 0.854458 + 0.519520i \(0.173889\pi\)
−0.854458 + 0.519520i \(0.826111\pi\)
\(374\) −1045.54 −0.144555
\(375\) 0 0
\(376\) 1232.81 0.169088
\(377\) 943.583i 0.128905i
\(378\) − 485.401i − 0.0660485i
\(379\) 4053.61 0.549393 0.274697 0.961531i \(-0.411423\pi\)
0.274697 + 0.961531i \(0.411423\pi\)
\(380\) 0 0
\(381\) 4326.94 0.581826
\(382\) − 1971.23i − 0.264023i
\(383\) − 4531.33i − 0.604543i −0.953222 0.302272i \(-0.902255\pi\)
0.953222 0.302272i \(-0.0977450\pi\)
\(384\) 2866.65 0.380959
\(385\) 0 0
\(386\) −3047.53 −0.401853
\(387\) 4274.86i 0.561507i
\(388\) 1729.90i 0.226346i
\(389\) 1160.19 0.151219 0.0756093 0.997138i \(-0.475910\pi\)
0.0756093 + 0.997138i \(0.475910\pi\)
\(390\) 0 0
\(391\) 219.920 0.0284446
\(392\) 1183.45i 0.152483i
\(393\) − 1932.11i − 0.247995i
\(394\) −9283.53 −1.18705
\(395\) 0 0
\(396\) −839.285 −0.106504
\(397\) − 6720.74i − 0.849632i −0.905280 0.424816i \(-0.860339\pi\)
0.905280 0.424816i \(-0.139661\pi\)
\(398\) − 7903.93i − 0.995448i
\(399\) −3431.91 −0.430602
\(400\) 0 0
\(401\) 2769.06 0.344839 0.172419 0.985024i \(-0.444842\pi\)
0.172419 + 0.985024i \(0.444842\pi\)
\(402\) − 5499.06i − 0.682260i
\(403\) − 2530.07i − 0.312733i
\(404\) −2333.01 −0.287306
\(405\) 0 0
\(406\) 498.446 0.0609297
\(407\) 17332.8i 2.11095i
\(408\) 444.108i 0.0538888i
\(409\) −14438.9 −1.74561 −0.872806 0.488067i \(-0.837702\pi\)
−0.872806 + 0.488067i \(0.837702\pi\)
\(410\) 0 0
\(411\) 1066.73 0.128024
\(412\) − 271.866i − 0.0325095i
\(413\) − 811.850i − 0.0967277i
\(414\) −829.340 −0.0984537
\(415\) 0 0
\(416\) −2135.84 −0.251726
\(417\) − 1416.05i − 0.166293i
\(418\) − 27876.7i − 3.26195i
\(419\) −6396.35 −0.745780 −0.372890 0.927875i \(-0.621633\pi\)
−0.372890 + 0.927875i \(0.621633\pi\)
\(420\) 0 0
\(421\) −8231.07 −0.952868 −0.476434 0.879210i \(-0.658071\pi\)
−0.476434 + 0.879210i \(0.658071\pi\)
\(422\) − 374.712i − 0.0432244i
\(423\) − 459.393i − 0.0528048i
\(424\) −16342.9 −1.87189
\(425\) 0 0
\(426\) −6244.91 −0.710251
\(427\) 2732.49i 0.309683i
\(428\) − 1293.18i − 0.146047i
\(429\) −6781.20 −0.763169
\(430\) 0 0
\(431\) −5975.72 −0.667843 −0.333922 0.942601i \(-0.608372\pi\)
−0.333922 + 0.942601i \(0.608372\pi\)
\(432\) − 1371.50i − 0.152746i
\(433\) 10912.9i 1.21118i 0.795776 + 0.605591i \(0.207063\pi\)
−0.795776 + 0.605591i \(0.792937\pi\)
\(434\) −1336.50 −0.147821
\(435\) 0 0
\(436\) −1831.90 −0.201221
\(437\) 5863.65i 0.641868i
\(438\) − 2698.18i − 0.294347i
\(439\) −3784.05 −0.411396 −0.205698 0.978616i \(-0.565946\pi\)
−0.205698 + 0.978616i \(0.565946\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 535.737i 0.0576526i
\(443\) − 7869.34i − 0.843981i −0.906600 0.421991i \(-0.861331\pi\)
0.906600 0.421991i \(-0.138669\pi\)
\(444\) 1099.22 0.117493
\(445\) 0 0
\(446\) 8622.42 0.915434
\(447\) − 3274.58i − 0.346493i
\(448\) 3972.85i 0.418972i
\(449\) 15601.8 1.63986 0.819929 0.572465i \(-0.194013\pi\)
0.819929 + 0.572465i \(0.194013\pi\)
\(450\) 0 0
\(451\) −29588.3 −3.08926
\(452\) 333.580i 0.0347130i
\(453\) − 8743.43i − 0.906848i
\(454\) 1634.77 0.168995
\(455\) 0 0
\(456\) −11841.1 −1.21603
\(457\) 8840.78i 0.904933i 0.891781 + 0.452466i \(0.149456\pi\)
−0.891781 + 0.452466i \(0.850544\pi\)
\(458\) 9379.71i 0.956954i
\(459\) 165.492 0.0168290
\(460\) 0 0
\(461\) −10523.7 −1.06321 −0.531605 0.846993i \(-0.678411\pi\)
−0.531605 + 0.846993i \(0.678411\pi\)
\(462\) 3582.16i 0.360729i
\(463\) 10781.6i 1.08221i 0.840954 + 0.541107i \(0.181994\pi\)
−0.840954 + 0.541107i \(0.818006\pi\)
\(464\) 1408.36 0.140908
\(465\) 0 0
\(466\) 8699.46 0.864795
\(467\) 5592.55i 0.554160i 0.960847 + 0.277080i \(0.0893666\pi\)
−0.960847 + 0.277080i \(0.910633\pi\)
\(468\) 430.053i 0.0424769i
\(469\) 4996.05 0.491889
\(470\) 0 0
\(471\) 4430.82 0.433463
\(472\) − 2801.12i − 0.273161i
\(473\) − 31547.6i − 3.06672i
\(474\) −392.410 −0.0380253
\(475\) 0 0
\(476\) −60.2410 −0.00580072
\(477\) 6090.01i 0.584575i
\(478\) − 8687.51i − 0.831292i
\(479\) 7835.28 0.747397 0.373698 0.927550i \(-0.378089\pi\)
0.373698 + 0.927550i \(0.378089\pi\)
\(480\) 0 0
\(481\) 8881.42 0.841908
\(482\) − 14920.6i − 1.40999i
\(483\) − 753.478i − 0.0709823i
\(484\) 4324.96 0.406176
\(485\) 0 0
\(486\) −624.087 −0.0582493
\(487\) − 5979.24i − 0.556356i −0.960530 0.278178i \(-0.910269\pi\)
0.960530 0.278178i \(-0.0897305\pi\)
\(488\) 9427.89i 0.874550i
\(489\) 8293.68 0.766980
\(490\) 0 0
\(491\) −1792.66 −0.164769 −0.0823846 0.996601i \(-0.526254\pi\)
−0.0823846 + 0.996601i \(0.526254\pi\)
\(492\) 1876.44i 0.171944i
\(493\) 169.940i 0.0155248i
\(494\) −14284.2 −1.30096
\(495\) 0 0
\(496\) −3776.29 −0.341855
\(497\) − 5673.67i − 0.512070i
\(498\) 654.061i 0.0588538i
\(499\) 20514.4 1.84038 0.920191 0.391470i \(-0.128033\pi\)
0.920191 + 0.391470i \(0.128033\pi\)
\(500\) 0 0
\(501\) 766.523 0.0683548
\(502\) 886.065i 0.0787789i
\(503\) 16294.5i 1.44441i 0.691679 + 0.722205i \(0.256871\pi\)
−0.691679 + 0.722205i \(0.743129\pi\)
\(504\) 1521.58 0.134477
\(505\) 0 0
\(506\) 6120.35 0.537713
\(507\) − 3116.28i − 0.272976i
\(508\) 2025.07i 0.176866i
\(509\) −12183.8 −1.06097 −0.530487 0.847693i \(-0.677991\pi\)
−0.530487 + 0.847693i \(0.677991\pi\)
\(510\) 0 0
\(511\) 2451.37 0.212216
\(512\) 13002.5i 1.12234i
\(513\) 4412.46i 0.379756i
\(514\) 8070.48 0.692555
\(515\) 0 0
\(516\) −2000.70 −0.170689
\(517\) 3390.22i 0.288398i
\(518\) − 4691.59i − 0.397947i
\(519\) −11551.7 −0.976998
\(520\) 0 0
\(521\) −11251.3 −0.946116 −0.473058 0.881031i \(-0.656850\pi\)
−0.473058 + 0.881031i \(0.656850\pi\)
\(522\) − 640.859i − 0.0537350i
\(523\) − 13493.7i − 1.12818i −0.825713 0.564090i \(-0.809227\pi\)
0.825713 0.564090i \(-0.190773\pi\)
\(524\) 904.254 0.0753865
\(525\) 0 0
\(526\) −12710.4 −1.05361
\(527\) − 455.666i − 0.0376644i
\(528\) 10121.4i 0.834236i
\(529\) 10879.6 0.894192
\(530\) 0 0
\(531\) −1043.81 −0.0853058
\(532\) − 1606.18i − 0.130896i
\(533\) 15161.1i 1.23209i
\(534\) 11725.3 0.950195
\(535\) 0 0
\(536\) 17237.8 1.38910
\(537\) 2858.41i 0.229701i
\(538\) − 11738.5i − 0.940673i
\(539\) −3254.49 −0.260075
\(540\) 0 0
\(541\) −18810.8 −1.49490 −0.747449 0.664319i \(-0.768722\pi\)
−0.747449 + 0.664319i \(0.768722\pi\)
\(542\) 13710.5i 1.08656i
\(543\) − 4946.49i − 0.390929i
\(544\) −384.666 −0.0303169
\(545\) 0 0
\(546\) 1835.51 0.143869
\(547\) − 9822.68i − 0.767801i −0.923374 0.383901i \(-0.874581\pi\)
0.923374 0.383901i \(-0.125419\pi\)
\(548\) 499.245i 0.0389173i
\(549\) 3513.20 0.273115
\(550\) 0 0
\(551\) −4531.04 −0.350324
\(552\) − 2599.72i − 0.200455i
\(553\) − 356.515i − 0.0274151i
\(554\) −5716.89 −0.438424
\(555\) 0 0
\(556\) 662.731 0.0505504
\(557\) − 23627.9i − 1.79739i −0.438571 0.898696i \(-0.644515\pi\)
0.438571 0.898696i \(-0.355485\pi\)
\(558\) 1718.36i 0.130366i
\(559\) −16165.1 −1.22310
\(560\) 0 0
\(561\) −1221.30 −0.0919131
\(562\) − 5113.79i − 0.383830i
\(563\) − 966.441i − 0.0723457i −0.999346 0.0361729i \(-0.988483\pi\)
0.999346 0.0361729i \(-0.0115167\pi\)
\(564\) 215.002 0.0160518
\(565\) 0 0
\(566\) −18877.2 −1.40189
\(567\) − 567.000i − 0.0419961i
\(568\) − 19575.8i − 1.44610i
\(569\) −16451.2 −1.21208 −0.606038 0.795435i \(-0.707242\pi\)
−0.606038 + 0.795435i \(0.707242\pi\)
\(570\) 0 0
\(571\) 37.8467 0.00277379 0.00138690 0.999999i \(-0.499559\pi\)
0.00138690 + 0.999999i \(0.499559\pi\)
\(572\) − 3173.70i − 0.231991i
\(573\) − 2302.61i − 0.167876i
\(574\) 8008.85 0.582374
\(575\) 0 0
\(576\) 5107.95 0.369498
\(577\) 23779.1i 1.71566i 0.513933 + 0.857830i \(0.328188\pi\)
−0.513933 + 0.857830i \(0.671812\pi\)
\(578\) − 12521.4i − 0.901073i
\(579\) −3559.84 −0.255513
\(580\) 0 0
\(581\) −594.233 −0.0424319
\(582\) − 9492.92i − 0.676107i
\(583\) − 44942.9i − 3.19270i
\(584\) 8457.93 0.599301
\(585\) 0 0
\(586\) 16052.0 1.13158
\(587\) 12556.0i 0.882863i 0.897295 + 0.441432i \(0.145529\pi\)
−0.897295 + 0.441432i \(0.854471\pi\)
\(588\) 206.394i 0.0144754i
\(589\) 12149.2 0.849917
\(590\) 0 0
\(591\) −10844.2 −0.754770
\(592\) − 13256.1i − 0.920307i
\(593\) 26489.2i 1.83437i 0.398461 + 0.917185i \(0.369544\pi\)
−0.398461 + 0.917185i \(0.630456\pi\)
\(594\) 4605.63 0.318133
\(595\) 0 0
\(596\) 1532.55 0.105328
\(597\) − 9232.63i − 0.632942i
\(598\) − 3136.09i − 0.214456i
\(599\) −8910.78 −0.607821 −0.303910 0.952701i \(-0.598292\pi\)
−0.303910 + 0.952701i \(0.598292\pi\)
\(600\) 0 0
\(601\) 2943.50 0.199780 0.0998902 0.994998i \(-0.468151\pi\)
0.0998902 + 0.994998i \(0.468151\pi\)
\(602\) 8539.18i 0.578125i
\(603\) − 6423.49i − 0.433806i
\(604\) 4092.05 0.275668
\(605\) 0 0
\(606\) 12802.5 0.858198
\(607\) − 20445.4i − 1.36714i −0.729886 0.683569i \(-0.760427\pi\)
0.729886 0.683569i \(-0.239573\pi\)
\(608\) − 10256.2i − 0.684117i
\(609\) 582.238 0.0387413
\(610\) 0 0
\(611\) 1737.16 0.115021
\(612\) 77.4528i 0.00511576i
\(613\) 19174.7i 1.26339i 0.775215 + 0.631697i \(0.217641\pi\)
−0.775215 + 0.631697i \(0.782359\pi\)
\(614\) 3234.18 0.212575
\(615\) 0 0
\(616\) −11228.9 −0.734458
\(617\) 21445.6i 1.39930i 0.714486 + 0.699650i \(0.246661\pi\)
−0.714486 + 0.699650i \(0.753339\pi\)
\(618\) 1491.88i 0.0971074i
\(619\) 17031.2 1.10588 0.552941 0.833220i \(-0.313505\pi\)
0.552941 + 0.833220i \(0.313505\pi\)
\(620\) 0 0
\(621\) −968.757 −0.0626005
\(622\) 17237.9i 1.11122i
\(623\) 10652.8i 0.685063i
\(624\) 5186.24 0.332717
\(625\) 0 0
\(626\) 6256.97 0.399487
\(627\) − 32563.0i − 2.07407i
\(628\) 2073.69i 0.131766i
\(629\) 1599.55 0.101396
\(630\) 0 0
\(631\) 25190.1 1.58923 0.794614 0.607115i \(-0.207673\pi\)
0.794614 + 0.607115i \(0.207673\pi\)
\(632\) − 1230.08i − 0.0774208i
\(633\) − 437.704i − 0.0274837i
\(634\) −3060.88 −0.191740
\(635\) 0 0
\(636\) −2850.21 −0.177702
\(637\) 1667.61i 0.103726i
\(638\) 4729.40i 0.293478i
\(639\) −7294.72 −0.451603
\(640\) 0 0
\(641\) 11755.1 0.724335 0.362168 0.932113i \(-0.382037\pi\)
0.362168 + 0.932113i \(0.382037\pi\)
\(642\) 7096.41i 0.436251i
\(643\) 24044.2i 1.47467i 0.675530 + 0.737333i \(0.263915\pi\)
−0.675530 + 0.737333i \(0.736085\pi\)
\(644\) 352.639 0.0215775
\(645\) 0 0
\(646\) −2572.58 −0.156683
\(647\) 5138.50i 0.312234i 0.987739 + 0.156117i \(0.0498976\pi\)
−0.987739 + 0.156117i \(0.950102\pi\)
\(648\) − 1956.31i − 0.118598i
\(649\) 7703.07 0.465905
\(650\) 0 0
\(651\) −1561.18 −0.0939897
\(652\) 3881.56i 0.233150i
\(653\) 6665.57i 0.399455i 0.979851 + 0.199727i \(0.0640057\pi\)
−0.979851 + 0.199727i \(0.935994\pi\)
\(654\) 10052.7 0.601056
\(655\) 0 0
\(656\) 22629.0 1.34682
\(657\) − 3151.76i − 0.187157i
\(658\) − 917.653i − 0.0543675i
\(659\) −8996.31 −0.531785 −0.265893 0.964003i \(-0.585667\pi\)
−0.265893 + 0.964003i \(0.585667\pi\)
\(660\) 0 0
\(661\) 24565.6 1.44553 0.722763 0.691096i \(-0.242872\pi\)
0.722763 + 0.691096i \(0.242872\pi\)
\(662\) − 15115.8i − 0.887451i
\(663\) 625.798i 0.0366576i
\(664\) −2050.27 −0.119828
\(665\) 0 0
\(666\) −6032.05 −0.350957
\(667\) − 994.793i − 0.0577489i
\(668\) 358.744i 0.0207788i
\(669\) 10071.9 0.582066
\(670\) 0 0
\(671\) −25926.7 −1.49164
\(672\) 1317.92i 0.0756545i
\(673\) − 22046.9i − 1.26277i −0.775468 0.631387i \(-0.782486\pi\)
0.775468 0.631387i \(-0.217514\pi\)
\(674\) −3106.91 −0.177557
\(675\) 0 0
\(676\) 1458.47 0.0829805
\(677\) − 25109.8i − 1.42548i −0.701430 0.712738i \(-0.747455\pi\)
0.701430 0.712738i \(-0.252545\pi\)
\(678\) − 1830.54i − 0.103690i
\(679\) 8624.58 0.487453
\(680\) 0 0
\(681\) 1909.59 0.107453
\(682\) − 12681.1i − 0.712002i
\(683\) 20540.1i 1.15073i 0.817898 + 0.575363i \(0.195139\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(684\) −2065.09 −0.115440
\(685\) 0 0
\(686\) 880.913 0.0490283
\(687\) 10956.5i 0.608466i
\(688\) 24127.5i 1.33699i
\(689\) −23029.0 −1.27334
\(690\) 0 0
\(691\) −1635.59 −0.0900446 −0.0450223 0.998986i \(-0.514336\pi\)
−0.0450223 + 0.998986i \(0.514336\pi\)
\(692\) − 5406.35i − 0.296992i
\(693\) 4184.34i 0.229365i
\(694\) 13008.8 0.711537
\(695\) 0 0
\(696\) 2008.89 0.109406
\(697\) 2730.53i 0.148388i
\(698\) 14077.5i 0.763385i
\(699\) 10161.9 0.549868
\(700\) 0 0
\(701\) 16895.6 0.910325 0.455163 0.890408i \(-0.349581\pi\)
0.455163 + 0.890408i \(0.349581\pi\)
\(702\) − 2359.94i − 0.126881i
\(703\) 42648.1i 2.28806i
\(704\) −37695.5 −2.01805
\(705\) 0 0
\(706\) −1546.58 −0.0824452
\(707\) 11631.5i 0.618736i
\(708\) − 488.517i − 0.0259316i
\(709\) 3527.13 0.186833 0.0934163 0.995627i \(-0.470221\pi\)
0.0934163 + 0.995627i \(0.470221\pi\)
\(710\) 0 0
\(711\) −458.376 −0.0241779
\(712\) 36755.1i 1.93463i
\(713\) 2667.37i 0.140104i
\(714\) 330.577 0.0173271
\(715\) 0 0
\(716\) −1337.78 −0.0698256
\(717\) − 10147.9i − 0.528566i
\(718\) 5148.95i 0.267628i
\(719\) 35693.3 1.85137 0.925687 0.378291i \(-0.123488\pi\)
0.925687 + 0.378291i \(0.123488\pi\)
\(720\) 0 0
\(721\) −1355.42 −0.0700116
\(722\) − 50976.1i − 2.62761i
\(723\) − 17428.9i − 0.896524i
\(724\) 2315.03 0.118836
\(725\) 0 0
\(726\) −23733.5 −1.21327
\(727\) 18829.9i 0.960611i 0.877101 + 0.480305i \(0.159474\pi\)
−0.877101 + 0.480305i \(0.840526\pi\)
\(728\) 5753.74i 0.292923i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −2911.34 −0.147305
\(732\) 1644.23i 0.0830225i
\(733\) − 14895.4i − 0.750577i −0.926908 0.375288i \(-0.877544\pi\)
0.926908 0.375288i \(-0.122456\pi\)
\(734\) 19956.3 1.00354
\(735\) 0 0
\(736\) 2251.75 0.112773
\(737\) 47404.0i 2.36926i
\(738\) − 10297.1i − 0.513606i
\(739\) −8866.34 −0.441345 −0.220672 0.975348i \(-0.570825\pi\)
−0.220672 + 0.975348i \(0.570825\pi\)
\(740\) 0 0
\(741\) −16685.4 −0.827198
\(742\) 12165.0i 0.601875i
\(743\) 14225.4i 0.702394i 0.936302 + 0.351197i \(0.114225\pi\)
−0.936302 + 0.351197i \(0.885775\pi\)
\(744\) −5386.51 −0.265429
\(745\) 0 0
\(746\) 19223.6 0.943465
\(747\) 764.013i 0.0374214i
\(748\) − 571.585i − 0.0279401i
\(749\) −6447.28 −0.314524
\(750\) 0 0
\(751\) 34300.2 1.66662 0.833311 0.552804i \(-0.186442\pi\)
0.833311 + 0.552804i \(0.186442\pi\)
\(752\) − 2592.83i − 0.125732i
\(753\) 1035.02i 0.0500905i
\(754\) 2423.37 0.117047
\(755\) 0 0
\(756\) 265.364 0.0127661
\(757\) 2340.98i 0.112397i 0.998420 + 0.0561984i \(0.0178979\pi\)
−0.998420 + 0.0561984i \(0.982102\pi\)
\(758\) − 10410.7i − 0.498858i
\(759\) 7149.22 0.341898
\(760\) 0 0
\(761\) 23894.9 1.13823 0.569113 0.822259i \(-0.307287\pi\)
0.569113 + 0.822259i \(0.307287\pi\)
\(762\) − 11112.7i − 0.528308i
\(763\) 9133.13i 0.433344i
\(764\) 1077.65 0.0510316
\(765\) 0 0
\(766\) −11637.6 −0.548935
\(767\) − 3947.09i − 0.185816i
\(768\) 6258.89i 0.294073i
\(769\) −22951.1 −1.07625 −0.538126 0.842864i \(-0.680868\pi\)
−0.538126 + 0.842864i \(0.680868\pi\)
\(770\) 0 0
\(771\) 9427.17 0.440352
\(772\) − 1666.05i − 0.0776718i
\(773\) 13526.7i 0.629393i 0.949192 + 0.314696i \(0.101903\pi\)
−0.949192 + 0.314696i \(0.898097\pi\)
\(774\) 10979.0 0.509858
\(775\) 0 0
\(776\) 29757.3 1.37658
\(777\) − 5480.28i − 0.253029i
\(778\) − 2979.67i − 0.137309i
\(779\) −72803.1 −3.34845
\(780\) 0 0
\(781\) 53833.5 2.46647
\(782\) − 564.812i − 0.0258282i
\(783\) − 748.592i − 0.0341666i
\(784\) 2489.02 0.113385
\(785\) 0 0
\(786\) −4962.15 −0.225183
\(787\) 1611.55i 0.0729929i 0.999334 + 0.0364965i \(0.0116198\pi\)
−0.999334 + 0.0364965i \(0.988380\pi\)
\(788\) − 5075.22i − 0.229438i
\(789\) −14847.1 −0.669923
\(790\) 0 0
\(791\) 1663.10 0.0747572
\(792\) 14437.2i 0.647731i
\(793\) 13285.0i 0.594909i
\(794\) −17260.6 −0.771481
\(795\) 0 0
\(796\) 4321.00 0.192404
\(797\) 7064.99i 0.313996i 0.987599 + 0.156998i \(0.0501816\pi\)
−0.987599 + 0.156998i \(0.949818\pi\)
\(798\) 8814.03i 0.390994i
\(799\) 312.864 0.0138527
\(800\) 0 0
\(801\) 13696.4 0.604169
\(802\) − 7111.67i − 0.313119i
\(803\) 23259.3i 1.02217i
\(804\) 3006.29 0.131870
\(805\) 0 0
\(806\) −6497.86 −0.283967
\(807\) − 13711.8i − 0.598114i
\(808\) 40131.9i 1.74732i
\(809\) 6352.68 0.276079 0.138040 0.990427i \(-0.455920\pi\)
0.138040 + 0.990427i \(0.455920\pi\)
\(810\) 0 0
\(811\) −16809.3 −0.727810 −0.363905 0.931436i \(-0.618557\pi\)
−0.363905 + 0.931436i \(0.618557\pi\)
\(812\) 272.496i 0.0117768i
\(813\) 16015.3i 0.690874i
\(814\) 44515.2 1.91678
\(815\) 0 0
\(816\) 934.044 0.0400712
\(817\) − 77624.0i − 3.32401i
\(818\) 37082.7i 1.58505i
\(819\) 2144.07 0.0914774
\(820\) 0 0
\(821\) −41992.9 −1.78510 −0.892548 0.450952i \(-0.851084\pi\)
−0.892548 + 0.450952i \(0.851084\pi\)
\(822\) − 2739.64i − 0.116248i
\(823\) − 37046.2i − 1.56908i −0.620080 0.784538i \(-0.712900\pi\)
0.620080 0.784538i \(-0.287100\pi\)
\(824\) −4676.58 −0.197714
\(825\) 0 0
\(826\) −2085.04 −0.0878304
\(827\) 14371.8i 0.604301i 0.953260 + 0.302151i \(0.0977046\pi\)
−0.953260 + 0.302151i \(0.902295\pi\)
\(828\) − 453.392i − 0.0190296i
\(829\) 43252.1 1.81207 0.906037 0.423199i \(-0.139093\pi\)
0.906037 + 0.423199i \(0.139093\pi\)
\(830\) 0 0
\(831\) −6677.93 −0.278766
\(832\) 19315.4i 0.804855i
\(833\) 300.338i 0.0124923i
\(834\) −3636.78 −0.150997
\(835\) 0 0
\(836\) 15239.9 0.630484
\(837\) 2007.23i 0.0828912i
\(838\) 16427.5i 0.677181i
\(839\) 6400.97 0.263392 0.131696 0.991290i \(-0.457958\pi\)
0.131696 + 0.991290i \(0.457958\pi\)
\(840\) 0 0
\(841\) −23620.3 −0.968481
\(842\) 21139.5i 0.865221i
\(843\) − 5973.45i − 0.244053i
\(844\) 204.852 0.00835460
\(845\) 0 0
\(846\) −1179.84 −0.0479477
\(847\) − 21562.5i − 0.874731i
\(848\) 34372.2i 1.39192i
\(849\) −22050.6 −0.891372
\(850\) 0 0
\(851\) −9363.42 −0.377173
\(852\) − 3414.03i − 0.137280i
\(853\) 17772.0i 0.713368i 0.934225 + 0.356684i \(0.116093\pi\)
−0.934225 + 0.356684i \(0.883907\pi\)
\(854\) 7017.75 0.281197
\(855\) 0 0
\(856\) −22245.0 −0.888222
\(857\) − 29094.3i − 1.15968i −0.814731 0.579839i \(-0.803115\pi\)
0.814731 0.579839i \(-0.196885\pi\)
\(858\) 17415.9i 0.692970i
\(859\) 599.045 0.0237941 0.0118971 0.999929i \(-0.496213\pi\)
0.0118971 + 0.999929i \(0.496213\pi\)
\(860\) 0 0
\(861\) 9355.18 0.370295
\(862\) 15347.2i 0.606413i
\(863\) − 19253.4i − 0.759435i −0.925103 0.379717i \(-0.876021\pi\)
0.925103 0.379717i \(-0.123979\pi\)
\(864\) 1694.47 0.0667210
\(865\) 0 0
\(866\) 28027.2 1.09977
\(867\) − 14626.3i − 0.572935i
\(868\) − 730.653i − 0.0285714i
\(869\) 3382.72 0.132049
\(870\) 0 0
\(871\) 24290.0 0.944931
\(872\) 31511.9i 1.22377i
\(873\) − 11088.7i − 0.429893i
\(874\) 15059.4 0.582827
\(875\) 0 0
\(876\) 1475.07 0.0568927
\(877\) − 42907.7i − 1.65210i −0.563598 0.826049i \(-0.690583\pi\)
0.563598 0.826049i \(-0.309417\pi\)
\(878\) 9718.41i 0.373554i
\(879\) 18750.5 0.719498
\(880\) 0 0
\(881\) −5765.54 −0.220484 −0.110242 0.993905i \(-0.535163\pi\)
−0.110242 + 0.993905i \(0.535163\pi\)
\(882\) − 1132.60i − 0.0432389i
\(883\) − 35426.8i − 1.35018i −0.737736 0.675089i \(-0.764105\pi\)
0.737736 0.675089i \(-0.235895\pi\)
\(884\) −292.883 −0.0111433
\(885\) 0 0
\(886\) −20210.5 −0.766349
\(887\) 24413.1i 0.924138i 0.886844 + 0.462069i \(0.152893\pi\)
−0.886844 + 0.462069i \(0.847107\pi\)
\(888\) − 18908.5i − 0.714560i
\(889\) 10096.2 0.380895
\(890\) 0 0
\(891\) 5379.86 0.202281
\(892\) 4713.80i 0.176939i
\(893\) 8341.77i 0.312594i
\(894\) −8409.97 −0.314621
\(895\) 0 0
\(896\) 6688.85 0.249396
\(897\) − 3663.29i − 0.136359i
\(898\) − 40069.6i − 1.48902i
\(899\) −2061.17 −0.0764671
\(900\) 0 0
\(901\) −4147.53 −0.153356
\(902\) 75990.3i 2.80510i
\(903\) 9974.67i 0.367593i
\(904\) 5738.17 0.211116
\(905\) 0 0
\(906\) −22455.4 −0.823434
\(907\) 23459.2i 0.858818i 0.903110 + 0.429409i \(0.141278\pi\)
−0.903110 + 0.429409i \(0.858722\pi\)
\(908\) 893.714i 0.0326640i
\(909\) 14954.7 0.545674
\(910\) 0 0
\(911\) −45803.8 −1.66580 −0.832902 0.553421i \(-0.813322\pi\)
−0.832902 + 0.553421i \(0.813322\pi\)
\(912\) 24904.1i 0.904228i
\(913\) − 5638.25i − 0.204380i
\(914\) 22705.4 0.821694
\(915\) 0 0
\(916\) −5127.80 −0.184964
\(917\) − 4508.25i − 0.162351i
\(918\) − 425.027i − 0.0152810i
\(919\) −33360.4 −1.19745 −0.598725 0.800955i \(-0.704326\pi\)
−0.598725 + 0.800955i \(0.704326\pi\)
\(920\) 0 0
\(921\) 3777.86 0.135163
\(922\) 27027.7i 0.965412i
\(923\) − 27584.5i − 0.983700i
\(924\) −1958.33 −0.0697233
\(925\) 0 0
\(926\) 27690.0 0.982668
\(927\) 1742.68i 0.0617444i
\(928\) 1740.01i 0.0615501i
\(929\) −11194.3 −0.395341 −0.197670 0.980269i \(-0.563338\pi\)
−0.197670 + 0.980269i \(0.563338\pi\)
\(930\) 0 0
\(931\) −8007.79 −0.281895
\(932\) 4755.91i 0.167151i
\(933\) 20135.8i 0.706554i
\(934\) 14363.1 0.503186
\(935\) 0 0
\(936\) 7397.67 0.258334
\(937\) − 4383.20i − 0.152821i −0.997076 0.0764103i \(-0.975654\pi\)
0.997076 0.0764103i \(-0.0243459\pi\)
\(938\) − 12831.1i − 0.446644i
\(939\) 7308.80 0.254008
\(940\) 0 0
\(941\) −27642.7 −0.957626 −0.478813 0.877917i \(-0.658933\pi\)
−0.478813 + 0.877917i \(0.658933\pi\)
\(942\) − 11379.5i − 0.393592i
\(943\) − 15984.0i − 0.551972i
\(944\) −5891.29 −0.203120
\(945\) 0 0
\(946\) −81022.3 −2.78463
\(947\) − 25657.8i − 0.880428i −0.897893 0.440214i \(-0.854902\pi\)
0.897893 0.440214i \(-0.145098\pi\)
\(948\) − 214.527i − 0.00734969i
\(949\) 11918.2 0.407672
\(950\) 0 0
\(951\) −3575.44 −0.121915
\(952\) 1036.25i 0.0352785i
\(953\) 48210.6i 1.63871i 0.573285 + 0.819356i \(0.305669\pi\)
−0.573285 + 0.819356i \(0.694331\pi\)
\(954\) 15640.7 0.530804
\(955\) 0 0
\(956\) 4749.38 0.160676
\(957\) 5524.45i 0.186604i
\(958\) − 20123.0i − 0.678649i
\(959\) 2489.04 0.0838114
\(960\) 0 0
\(961\) −24264.3 −0.814484
\(962\) − 22809.8i − 0.764467i
\(963\) 8289.36i 0.277384i
\(964\) 8156.97 0.272529
\(965\) 0 0
\(966\) −1935.13 −0.0644531
\(967\) 17711.5i 0.589002i 0.955651 + 0.294501i \(0.0951534\pi\)
−0.955651 + 0.294501i \(0.904847\pi\)
\(968\) − 74396.9i − 2.47026i
\(969\) −3005.05 −0.0996245
\(970\) 0 0
\(971\) 35777.8 1.18245 0.591227 0.806505i \(-0.298644\pi\)
0.591227 + 0.806505i \(0.298644\pi\)
\(972\) − 341.182i − 0.0112587i
\(973\) − 3304.11i − 0.108864i
\(974\) −15356.2 −0.505181
\(975\) 0 0
\(976\) 19828.7 0.650307
\(977\) − 30436.2i − 0.996662i −0.866987 0.498331i \(-0.833946\pi\)
0.866987 0.498331i \(-0.166054\pi\)
\(978\) − 21300.3i − 0.696430i
\(979\) −101077. −3.29972
\(980\) 0 0
\(981\) 11742.6 0.382174
\(982\) 4604.02i 0.149613i
\(983\) − 57841.2i − 1.87675i −0.345617 0.938376i \(-0.612330\pi\)
0.345617 0.938376i \(-0.387670\pi\)
\(984\) 32278.1 1.04572
\(985\) 0 0
\(986\) 436.450 0.0140967
\(987\) − 1071.92i − 0.0345689i
\(988\) − 7809.01i − 0.251455i
\(989\) 17042.4 0.547944
\(990\) 0 0
\(991\) 1521.84 0.0487819 0.0243909 0.999702i \(-0.492235\pi\)
0.0243909 + 0.999702i \(0.492235\pi\)
\(992\) − 4665.54i − 0.149326i
\(993\) − 17656.9i − 0.564274i
\(994\) −14571.5 −0.464968
\(995\) 0 0
\(996\) −357.569 −0.0113755
\(997\) − 46899.5i − 1.48979i −0.667181 0.744896i \(-0.732499\pi\)
0.667181 0.744896i \(-0.267501\pi\)
\(998\) − 52686.3i − 1.67110i
\(999\) −7046.07 −0.223151
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.o.274.3 8
5.2 odd 4 525.4.a.s.1.4 4
5.3 odd 4 525.4.a.v.1.1 yes 4
5.4 even 2 inner 525.4.d.o.274.6 8
15.2 even 4 1575.4.a.bm.1.1 4
15.8 even 4 1575.4.a.bf.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.s.1.4 4 5.2 odd 4
525.4.a.v.1.1 yes 4 5.3 odd 4
525.4.d.o.274.3 8 1.1 even 1 trivial
525.4.d.o.274.6 8 5.4 even 2 inner
1575.4.a.bf.1.4 4 15.8 even 4
1575.4.a.bm.1.1 4 15.2 even 4