Properties

Label 450.4.f.e.143.3
Level $450$
Weight $4$
Character 450.143
Analytic conductor $26.551$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,4,Mod(107,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.f (of order \(4\), degree \(2\), 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: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 71x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.3
Root \(1.97374 - 1.97374i\) of defining polynomial
Character \(\chi\) \(=\) 450.143
Dual form 450.4.f.e.107.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 + 1.41421i) q^{2} +4.00000i q^{4} +(-21.9129 + 21.9129i) q^{7} +(-5.65685 + 5.65685i) q^{8} +O(q^{10})\) \(q+(1.41421 + 1.41421i) q^{2} +4.00000i q^{4} +(-21.9129 + 21.9129i) q^{7} +(-5.65685 + 5.65685i) q^{8} -64.9306i q^{11} +(-13.0871 - 13.0871i) q^{13} -61.9790 q^{14} -16.0000 q^{16} +(5.53365 + 5.53365i) q^{17} -103.826i q^{19} +(91.8258 - 91.8258i) q^{22} +(-19.7990 + 19.7990i) q^{23} -37.0160i q^{26} +(-87.6515 - 87.6515i) q^{28} +184.646 q^{29} +19.8258 q^{31} +(-22.6274 - 22.6274i) q^{32} +15.6515i q^{34} +(254.739 - 254.739i) q^{37} +(146.832 - 146.832i) q^{38} +63.1468i q^{41} +(-104.174 - 104.174i) q^{43} +259.722 q^{44} -56.0000 q^{46} +(-185.567 - 185.567i) q^{47} -617.348i q^{49} +(52.3485 - 52.3485i) q^{52} +(-259.476 + 259.476i) q^{53} -247.916i q^{56} +(261.129 + 261.129i) q^{58} -255.539 q^{59} -894.083 q^{61} +(28.0379 + 28.0379i) q^{62} -64.0000i q^{64} +(100.523 - 100.523i) q^{67} +(-22.1346 + 22.1346i) q^{68} -491.653i q^{71} +(-603.780 - 603.780i) q^{73} +720.510 q^{74} +415.303 q^{76} +(1422.82 + 1422.82i) q^{77} -698.780i q^{79} +(-89.3030 + 89.3030i) q^{82} +(707.101 - 707.101i) q^{83} -294.649i q^{86} +(367.303 + 367.303i) q^{88} +389.648 q^{89} +573.553 q^{91} +(-79.1960 - 79.1960i) q^{92} -524.864i q^{94} +(29.2576 - 29.2576i) q^{97} +(873.063 - 873.063i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} - 288 q^{13} - 128 q^{16} + 368 q^{22} + 32 q^{28} - 208 q^{31} + 1488 q^{37} - 1200 q^{43} - 448 q^{46} + 1152 q^{52} + 256 q^{58} - 3120 q^{61} + 1904 q^{67} - 2264 q^{73} + 1856 q^{76} + 752 q^{82} + 1472 q^{88} - 8976 q^{91} - 3432 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 + 1.41421i 0.500000 + 0.500000i
\(3\) 0 0
\(4\) 4.00000i 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) −21.9129 + 21.9129i −1.18318 + 1.18318i −0.204270 + 0.978915i \(0.565482\pi\)
−0.978915 + 0.204270i \(0.934518\pi\)
\(8\) −5.65685 + 5.65685i −0.250000 + 0.250000i
\(9\) 0 0
\(10\) 0 0
\(11\) 64.9306i 1.77976i −0.456198 0.889878i \(-0.650789\pi\)
0.456198 0.889878i \(-0.349211\pi\)
\(12\) 0 0
\(13\) −13.0871 13.0871i −0.279209 0.279209i 0.553584 0.832793i \(-0.313260\pi\)
−0.832793 + 0.553584i \(0.813260\pi\)
\(14\) −61.9790 −1.18318
\(15\) 0 0
\(16\) −16.0000 −0.250000
\(17\) 5.53365 + 5.53365i 0.0789474 + 0.0789474i 0.745478 0.666530i \(-0.232221\pi\)
−0.666530 + 0.745478i \(0.732221\pi\)
\(18\) 0 0
\(19\) 103.826i 1.25365i −0.779162 0.626823i \(-0.784355\pi\)
0.779162 0.626823i \(-0.215645\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 91.8258 91.8258i 0.889878 0.889878i
\(23\) −19.7990 + 19.7990i −0.179495 + 0.179495i −0.791135 0.611641i \(-0.790510\pi\)
0.611641 + 0.791135i \(0.290510\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 37.0160i 0.279209i
\(27\) 0 0
\(28\) −87.6515 87.6515i −0.591592 0.591592i
\(29\) 184.646 1.18234 0.591171 0.806547i \(-0.298666\pi\)
0.591171 + 0.806547i \(0.298666\pi\)
\(30\) 0 0
\(31\) 19.8258 0.114865 0.0574324 0.998349i \(-0.481709\pi\)
0.0574324 + 0.998349i \(0.481709\pi\)
\(32\) −22.6274 22.6274i −0.125000 0.125000i
\(33\) 0 0
\(34\) 15.6515i 0.0789474i
\(35\) 0 0
\(36\) 0 0
\(37\) 254.739 254.739i 1.13186 1.13186i 0.141991 0.989868i \(-0.454650\pi\)
0.989868 0.141991i \(-0.0453503\pi\)
\(38\) 146.832 146.832i 0.626823 0.626823i
\(39\) 0 0
\(40\) 0 0
\(41\) 63.1468i 0.240533i 0.992742 + 0.120267i \(0.0383750\pi\)
−0.992742 + 0.120267i \(0.961625\pi\)
\(42\) 0 0
\(43\) −104.174 104.174i −0.369452 0.369452i 0.497826 0.867277i \(-0.334132\pi\)
−0.867277 + 0.497826i \(0.834132\pi\)
\(44\) 259.722 0.889878
\(45\) 0 0
\(46\) −56.0000 −0.179495
\(47\) −185.567 185.567i −0.575910 0.575910i 0.357864 0.933774i \(-0.383505\pi\)
−0.933774 + 0.357864i \(0.883505\pi\)
\(48\) 0 0
\(49\) 617.348i 1.79985i
\(50\) 0 0
\(51\) 0 0
\(52\) 52.3485 52.3485i 0.139604 0.139604i
\(53\) −259.476 + 259.476i −0.672486 + 0.672486i −0.958289 0.285802i \(-0.907740\pi\)
0.285802 + 0.958289i \(0.407740\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 247.916i 0.591592i
\(57\) 0 0
\(58\) 261.129 + 261.129i 0.591171 + 0.591171i
\(59\) −255.539 −0.563870 −0.281935 0.959434i \(-0.590976\pi\)
−0.281935 + 0.959434i \(0.590976\pi\)
\(60\) 0 0
\(61\) −894.083 −1.87665 −0.938325 0.345755i \(-0.887623\pi\)
−0.938325 + 0.345755i \(0.887623\pi\)
\(62\) 28.0379 + 28.0379i 0.0574324 + 0.0574324i
\(63\) 0 0
\(64\) 64.0000i 0.125000i
\(65\) 0 0
\(66\) 0 0
\(67\) 100.523 100.523i 0.183296 0.183296i −0.609495 0.792790i \(-0.708628\pi\)
0.792790 + 0.609495i \(0.208628\pi\)
\(68\) −22.1346 + 22.1346i −0.0394737 + 0.0394737i
\(69\) 0 0
\(70\) 0 0
\(71\) 491.653i 0.821810i −0.911678 0.410905i \(-0.865213\pi\)
0.911678 0.410905i \(-0.134787\pi\)
\(72\) 0 0
\(73\) −603.780 603.780i −0.968043 0.968043i 0.0314620 0.999505i \(-0.489984\pi\)
−0.999505 + 0.0314620i \(0.989984\pi\)
\(74\) 720.510 1.13186
\(75\) 0 0
\(76\) 415.303 0.626823
\(77\) 1422.82 + 1422.82i 2.10578 + 2.10578i
\(78\) 0 0
\(79\) 698.780i 0.995176i −0.867413 0.497588i \(-0.834219\pi\)
0.867413 0.497588i \(-0.165781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −89.3030 + 89.3030i −0.120267 + 0.120267i
\(83\) 707.101 707.101i 0.935114 0.935114i −0.0629055 0.998019i \(-0.520037\pi\)
0.998019 + 0.0629055i \(0.0200367\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 294.649i 0.369452i
\(87\) 0 0
\(88\) 367.303 + 367.303i 0.444939 + 0.444939i
\(89\) 389.648 0.464074 0.232037 0.972707i \(-0.425461\pi\)
0.232037 + 0.972707i \(0.425461\pi\)
\(90\) 0 0
\(91\) 573.553 0.660711
\(92\) −79.1960 79.1960i −0.0897473 0.0897473i
\(93\) 0 0
\(94\) 524.864i 0.575910i
\(95\) 0 0
\(96\) 0 0
\(97\) 29.2576 29.2576i 0.0306253 0.0306253i −0.691628 0.722254i \(-0.743106\pi\)
0.722254 + 0.691628i \(0.243106\pi\)
\(98\) 873.063 873.063i 0.899925 0.899925i
\(99\) 0 0
\(100\) 0 0
\(101\) 386.204i 0.380482i 0.981737 + 0.190241i \(0.0609269\pi\)
−0.981737 + 0.190241i \(0.939073\pi\)
\(102\) 0 0
\(103\) −793.564 793.564i −0.759148 0.759148i 0.217019 0.976167i \(-0.430366\pi\)
−0.976167 + 0.217019i \(0.930366\pi\)
\(104\) 148.064 0.139604
\(105\) 0 0
\(106\) −733.909 −0.672486
\(107\) 501.494 + 501.494i 0.453096 + 0.453096i 0.896381 0.443285i \(-0.146187\pi\)
−0.443285 + 0.896381i \(0.646187\pi\)
\(108\) 0 0
\(109\) 875.295i 0.769157i 0.923092 + 0.384578i \(0.125653\pi\)
−0.923092 + 0.384578i \(0.874347\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 350.606 350.606i 0.295796 0.295796i
\(113\) −943.768 + 943.768i −0.785683 + 0.785683i −0.980783 0.195100i \(-0.937497\pi\)
0.195100 + 0.980783i \(0.437497\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 738.584i 0.591171i
\(117\) 0 0
\(118\) −361.386 361.386i −0.281935 0.281935i
\(119\) −242.516 −0.186819
\(120\) 0 0
\(121\) −2884.98 −2.16753
\(122\) −1264.42 1264.42i −0.938325 0.938325i
\(123\) 0 0
\(124\) 79.3030i 0.0574324i
\(125\) 0 0
\(126\) 0 0
\(127\) −535.822 + 535.822i −0.374382 + 0.374382i −0.869070 0.494688i \(-0.835282\pi\)
0.494688 + 0.869070i \(0.335282\pi\)
\(128\) 90.5097 90.5097i 0.0625000 0.0625000i
\(129\) 0 0
\(130\) 0 0
\(131\) 1852.00i 1.23519i 0.786495 + 0.617597i \(0.211894\pi\)
−0.786495 + 0.617597i \(0.788106\pi\)
\(132\) 0 0
\(133\) 2275.12 + 2275.12i 1.48329 + 1.48329i
\(134\) 284.321 0.183296
\(135\) 0 0
\(136\) −62.6061 −0.0394737
\(137\) −888.796 888.796i −0.554270 0.554270i 0.373401 0.927670i \(-0.378192\pi\)
−0.927670 + 0.373401i \(0.878192\pi\)
\(138\) 0 0
\(139\) 252.348i 0.153985i −0.997032 0.0769925i \(-0.975468\pi\)
0.997032 0.0769925i \(-0.0245318\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 695.303 695.303i 0.410905 0.410905i
\(143\) −849.755 + 849.755i −0.496924 + 0.496924i
\(144\) 0 0
\(145\) 0 0
\(146\) 1707.75i 0.968043i
\(147\) 0 0
\(148\) 1018.95 + 1018.95i 0.565929 + 0.565929i
\(149\) 169.025 0.0929335 0.0464668 0.998920i \(-0.485204\pi\)
0.0464668 + 0.998920i \(0.485204\pi\)
\(150\) 0 0
\(151\) 2783.65 1.50020 0.750100 0.661324i \(-0.230005\pi\)
0.750100 + 0.661324i \(0.230005\pi\)
\(152\) 587.327 + 587.327i 0.313411 + 0.313411i
\(153\) 0 0
\(154\) 4024.33i 2.10578i
\(155\) 0 0
\(156\) 0 0
\(157\) −2018.22 + 2018.22i −1.02593 + 1.02593i −0.0262763 + 0.999655i \(0.508365\pi\)
−0.999655 + 0.0262763i \(0.991635\pi\)
\(158\) 988.225 988.225i 0.497588 0.497588i
\(159\) 0 0
\(160\) 0 0
\(161\) 867.706i 0.424750i
\(162\) 0 0
\(163\) −2132.69 2132.69i −1.02482 1.02482i −0.999684 0.0251326i \(-0.991999\pi\)
−0.0251326 0.999684i \(-0.508001\pi\)
\(164\) −252.587 −0.120267
\(165\) 0 0
\(166\) 1999.98 0.935114
\(167\) −1027.95 1027.95i −0.476319 0.476319i 0.427633 0.903952i \(-0.359347\pi\)
−0.903952 + 0.427633i \(0.859347\pi\)
\(168\) 0 0
\(169\) 1854.45i 0.844085i
\(170\) 0 0
\(171\) 0 0
\(172\) 416.697 416.697i 0.184726 0.184726i
\(173\) −868.268 + 868.268i −0.381579 + 0.381579i −0.871671 0.490092i \(-0.836963\pi\)
0.490092 + 0.871671i \(0.336963\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1038.89i 0.444939i
\(177\) 0 0
\(178\) 551.045 + 551.045i 0.232037 + 0.232037i
\(179\) −3448.21 −1.43984 −0.719920 0.694057i \(-0.755822\pi\)
−0.719920 + 0.694057i \(0.755822\pi\)
\(180\) 0 0
\(181\) 3511.81 1.44216 0.721080 0.692852i \(-0.243646\pi\)
0.721080 + 0.692852i \(0.243646\pi\)
\(182\) 811.126 + 811.126i 0.330355 + 0.330355i
\(183\) 0 0
\(184\) 224.000i 0.0897473i
\(185\) 0 0
\(186\) 0 0
\(187\) 359.303 359.303i 0.140507 0.140507i
\(188\) 742.269 742.269i 0.287955 0.287955i
\(189\) 0 0
\(190\) 0 0
\(191\) 2037.43i 0.771850i 0.922530 + 0.385925i \(0.126118\pi\)
−0.922530 + 0.385925i \(0.873882\pi\)
\(192\) 0 0
\(193\) −1127.78 1127.78i −0.420619 0.420619i 0.464798 0.885417i \(-0.346127\pi\)
−0.885417 + 0.464798i \(0.846127\pi\)
\(194\) 82.7529 0.0306253
\(195\) 0 0
\(196\) 2469.39 0.899925
\(197\) 257.810 + 257.810i 0.0932396 + 0.0932396i 0.752188 0.658948i \(-0.228998\pi\)
−0.658948 + 0.752188i \(0.728998\pi\)
\(198\) 0 0
\(199\) 513.030i 0.182753i −0.995816 0.0913763i \(-0.970873\pi\)
0.995816 0.0913763i \(-0.0291266\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −546.174 + 546.174i −0.190241 + 0.190241i
\(203\) −4046.12 + 4046.12i −1.39893 + 1.39893i
\(204\) 0 0
\(205\) 0 0
\(206\) 2244.54i 0.759148i
\(207\) 0 0
\(208\) 209.394 + 209.394i 0.0698022 + 0.0698022i
\(209\) −6741.47 −2.23118
\(210\) 0 0
\(211\) −1610.24 −0.525373 −0.262686 0.964881i \(-0.584608\pi\)
−0.262686 + 0.964881i \(0.584608\pi\)
\(212\) −1037.90 1037.90i −0.336243 0.336243i
\(213\) 0 0
\(214\) 1418.44i 0.453096i
\(215\) 0 0
\(216\) 0 0
\(217\) −434.439 + 434.439i −0.135906 + 0.135906i
\(218\) −1237.85 + 1237.85i −0.384578 + 0.384578i
\(219\) 0 0
\(220\) 0 0
\(221\) 144.839i 0.0440856i
\(222\) 0 0
\(223\) 2339.82 + 2339.82i 0.702628 + 0.702628i 0.964974 0.262346i \(-0.0844961\pi\)
−0.262346 + 0.964974i \(0.584496\pi\)
\(224\) 991.664 0.295796
\(225\) 0 0
\(226\) −2669.38 −0.785683
\(227\) −1601.14 1601.14i −0.468155 0.468155i 0.433162 0.901316i \(-0.357398\pi\)
−0.901316 + 0.433162i \(0.857398\pi\)
\(228\) 0 0
\(229\) 6303.36i 1.81894i 0.415766 + 0.909472i \(0.363514\pi\)
−0.415766 + 0.909472i \(0.636486\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1044.52 + 1044.52i −0.295585 + 0.295585i
\(233\) 2681.95 2681.95i 0.754080 0.754080i −0.221158 0.975238i \(-0.570984\pi\)
0.975238 + 0.221158i \(0.0709836\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1022.15i 0.281935i
\(237\) 0 0
\(238\) −342.970 342.970i −0.0934093 0.0934093i
\(239\) 2314.62 0.626444 0.313222 0.949680i \(-0.398592\pi\)
0.313222 + 0.949680i \(0.398592\pi\)
\(240\) 0 0
\(241\) 1476.59 0.394670 0.197335 0.980336i \(-0.436771\pi\)
0.197335 + 0.980336i \(0.436771\pi\)
\(242\) −4079.98 4079.98i −1.08377 1.08377i
\(243\) 0 0
\(244\) 3576.33i 0.938325i
\(245\) 0 0
\(246\) 0 0
\(247\) −1358.78 + 1358.78i −0.350029 + 0.350029i
\(248\) −112.151 + 112.151i −0.0287162 + 0.0287162i
\(249\) 0 0
\(250\) 0 0
\(251\) 4021.54i 1.01130i −0.862738 0.505652i \(-0.831252\pi\)
0.862738 0.505652i \(-0.168748\pi\)
\(252\) 0 0
\(253\) 1285.56 + 1285.56i 0.319457 + 0.319457i
\(254\) −1515.53 −0.374382
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −371.183 371.183i −0.0900924 0.0900924i 0.660624 0.750717i \(-0.270292\pi\)
−0.750717 + 0.660624i \(0.770292\pi\)
\(258\) 0 0
\(259\) 11164.1i 2.67839i
\(260\) 0 0
\(261\) 0 0
\(262\) −2619.13 + 2619.13i −0.617597 + 0.617597i
\(263\) 181.405 181.405i 0.0425320 0.0425320i −0.685521 0.728053i \(-0.740425\pi\)
0.728053 + 0.685521i \(0.240425\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6435.01i 1.48329i
\(267\) 0 0
\(268\) 402.091 + 402.091i 0.0916478 + 0.0916478i
\(269\) 3301.79 0.748377 0.374189 0.927353i \(-0.377921\pi\)
0.374189 + 0.927353i \(0.377921\pi\)
\(270\) 0 0
\(271\) −3894.23 −0.872906 −0.436453 0.899727i \(-0.643765\pi\)
−0.436453 + 0.899727i \(0.643765\pi\)
\(272\) −88.5383 88.5383i −0.0197369 0.0197369i
\(273\) 0 0
\(274\) 2513.89i 0.554270i
\(275\) 0 0
\(276\) 0 0
\(277\) 1417.53 1417.53i 0.307476 0.307476i −0.536454 0.843930i \(-0.680236\pi\)
0.843930 + 0.536454i \(0.180236\pi\)
\(278\) 356.875 356.875i 0.0769925 0.0769925i
\(279\) 0 0
\(280\) 0 0
\(281\) 8119.33i 1.72370i 0.507166 + 0.861848i \(0.330693\pi\)
−0.507166 + 0.861848i \(0.669307\pi\)
\(282\) 0 0
\(283\) −1166.94 1166.94i −0.245114 0.245114i 0.573848 0.818962i \(-0.305450\pi\)
−0.818962 + 0.573848i \(0.805450\pi\)
\(284\) 1966.61 0.410905
\(285\) 0 0
\(286\) −2403.47 −0.496924
\(287\) −1383.73 1383.73i −0.284595 0.284595i
\(288\) 0 0
\(289\) 4851.76i 0.987535i
\(290\) 0 0
\(291\) 0 0
\(292\) 2415.12 2415.12i 0.484021 0.484021i
\(293\) 4826.44 4826.44i 0.962333 0.962333i −0.0369831 0.999316i \(-0.511775\pi\)
0.999316 + 0.0369831i \(0.0117748\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2882.04i 0.565929i
\(297\) 0 0
\(298\) 239.038 + 239.038i 0.0464668 + 0.0464668i
\(299\) 518.224 0.100233
\(300\) 0 0
\(301\) 4565.52 0.874259
\(302\) 3936.68 + 3936.68i 0.750100 + 0.750100i
\(303\) 0 0
\(304\) 1661.21i 0.313411i
\(305\) 0 0
\(306\) 0 0
\(307\) 4161.89 4161.89i 0.773718 0.773718i −0.205036 0.978754i \(-0.565731\pi\)
0.978754 + 0.205036i \(0.0657313\pi\)
\(308\) −5691.27 + 5691.27i −1.05289 + 1.05289i
\(309\) 0 0
\(310\) 0 0
\(311\) 3984.60i 0.726514i −0.931689 0.363257i \(-0.881665\pi\)
0.931689 0.363257i \(-0.118335\pi\)
\(312\) 0 0
\(313\) −2427.14 2427.14i −0.438308 0.438308i 0.453135 0.891442i \(-0.350306\pi\)
−0.891442 + 0.453135i \(0.850306\pi\)
\(314\) −5708.38 −1.02593
\(315\) 0 0
\(316\) 2795.12 0.497588
\(317\) 7132.54 + 7132.54i 1.26373 + 1.26373i 0.949269 + 0.314464i \(0.101825\pi\)
0.314464 + 0.949269i \(0.398175\pi\)
\(318\) 0 0
\(319\) 11989.2i 2.10428i
\(320\) 0 0
\(321\) 0 0
\(322\) 1227.12 1227.12i 0.212375 0.212375i
\(323\) 574.535 574.535i 0.0989721 0.0989721i
\(324\) 0 0
\(325\) 0 0
\(326\) 6032.16i 1.02482i
\(327\) 0 0
\(328\) −357.212 357.212i −0.0601334 0.0601334i
\(329\) 8132.63 1.36282
\(330\) 0 0
\(331\) −1793.22 −0.297777 −0.148889 0.988854i \(-0.547570\pi\)
−0.148889 + 0.988854i \(0.547570\pi\)
\(332\) 2828.41 + 2828.41i 0.467557 + 0.467557i
\(333\) 0 0
\(334\) 2907.48i 0.476319i
\(335\) 0 0
\(336\) 0 0
\(337\) −4438.46 + 4438.46i −0.717443 + 0.717443i −0.968081 0.250638i \(-0.919360\pi\)
0.250638 + 0.968081i \(0.419360\pi\)
\(338\) 2622.59 2622.59i 0.422042 0.422042i
\(339\) 0 0
\(340\) 0 0
\(341\) 1287.30i 0.204431i
\(342\) 0 0
\(343\) 6011.77 + 6011.77i 0.946370 + 0.946370i
\(344\) 1178.60 0.184726
\(345\) 0 0
\(346\) −2455.83 −0.381579
\(347\) 4669.58 + 4669.58i 0.722410 + 0.722410i 0.969096 0.246685i \(-0.0793414\pi\)
−0.246685 + 0.969096i \(0.579341\pi\)
\(348\) 0 0
\(349\) 1244.02i 0.190805i 0.995439 + 0.0954026i \(0.0304138\pi\)
−0.995439 + 0.0954026i \(0.969586\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1469.21 + 1469.21i −0.222470 + 0.222470i
\(353\) 5341.19 5341.19i 0.805334 0.805334i −0.178590 0.983924i \(-0.557153\pi\)
0.983924 + 0.178590i \(0.0571535\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1558.59i 0.232037i
\(357\) 0 0
\(358\) −4876.51 4876.51i −0.719920 0.719920i
\(359\) 1040.64 0.152988 0.0764940 0.997070i \(-0.475627\pi\)
0.0764940 + 0.997070i \(0.475627\pi\)
\(360\) 0 0
\(361\) −3920.79 −0.571627
\(362\) 4966.45 + 4966.45i 0.721080 + 0.721080i
\(363\) 0 0
\(364\) 2294.21i 0.330355i
\(365\) 0 0
\(366\) 0 0
\(367\) 1447.50 1447.50i 0.205883 0.205883i −0.596632 0.802515i \(-0.703495\pi\)
0.802515 + 0.596632i \(0.203495\pi\)
\(368\) 316.784 316.784i 0.0448736 0.0448736i
\(369\) 0 0
\(370\) 0 0
\(371\) 11371.7i 1.59135i
\(372\) 0 0
\(373\) −106.398 106.398i −0.0147696 0.0147696i 0.699683 0.714453i \(-0.253324\pi\)
−0.714453 + 0.699683i \(0.753324\pi\)
\(374\) 1016.26 0.140507
\(375\) 0 0
\(376\) 2099.45 0.287955
\(377\) −2416.48 2416.48i −0.330120 0.330120i
\(378\) 0 0
\(379\) 9384.86i 1.27195i −0.771711 0.635974i \(-0.780599\pi\)
0.771711 0.635974i \(-0.219401\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2881.36 + 2881.36i −0.385925 + 0.385925i
\(383\) −8015.85 + 8015.85i −1.06943 + 1.06943i −0.0720253 + 0.997403i \(0.522946\pi\)
−0.997403 + 0.0720253i \(0.977054\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3189.84i 0.420619i
\(387\) 0 0
\(388\) 117.030 + 117.030i 0.0153127 + 0.0153127i
\(389\) 8892.72 1.15907 0.579536 0.814947i \(-0.303234\pi\)
0.579536 + 0.814947i \(0.303234\pi\)
\(390\) 0 0
\(391\) −219.121 −0.0283413
\(392\) 3492.25 + 3492.25i 0.449962 + 0.449962i
\(393\) 0 0
\(394\) 729.197i 0.0932396i
\(395\) 0 0
\(396\) 0 0
\(397\) −3248.62 + 3248.62i −0.410689 + 0.410689i −0.881978 0.471290i \(-0.843789\pi\)
0.471290 + 0.881978i \(0.343789\pi\)
\(398\) 725.534 725.534i 0.0913763 0.0913763i
\(399\) 0 0
\(400\) 0 0
\(401\) 12012.4i 1.49594i 0.663731 + 0.747971i \(0.268972\pi\)
−0.663731 + 0.747971i \(0.731028\pi\)
\(402\) 0 0
\(403\) −259.462 259.462i −0.0320713 0.0320713i
\(404\) −1544.81 −0.190241
\(405\) 0 0
\(406\) −11444.2 −1.39893
\(407\) −16540.3 16540.3i −2.01443 2.01443i
\(408\) 0 0
\(409\) 1710.61i 0.206807i 0.994639 + 0.103403i \(0.0329733\pi\)
−0.994639 + 0.103403i \(0.967027\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3174.26 3174.26i 0.379574 0.379574i
\(413\) 5599.59 5599.59i 0.667162 0.667162i
\(414\) 0 0
\(415\) 0 0
\(416\) 592.256i 0.0698022i
\(417\) 0 0
\(418\) −9533.88 9533.88i −1.11559 1.11559i
\(419\) −2905.88 −0.338811 −0.169405 0.985546i \(-0.554185\pi\)
−0.169405 + 0.985546i \(0.554185\pi\)
\(420\) 0 0
\(421\) −9062.85 −1.04916 −0.524580 0.851361i \(-0.675778\pi\)
−0.524580 + 0.851361i \(0.675778\pi\)
\(422\) −2277.23 2277.23i −0.262686 0.262686i
\(423\) 0 0
\(424\) 2935.64i 0.336243i
\(425\) 0 0
\(426\) 0 0
\(427\) 19591.9 19591.9i 2.22042 2.22042i
\(428\) −2005.98 + 2005.98i −0.226548 + 0.226548i
\(429\) 0 0
\(430\) 0 0
\(431\) 10334.6i 1.15498i −0.816396 0.577492i \(-0.804031\pi\)
0.816396 0.577492i \(-0.195969\pi\)
\(432\) 0 0
\(433\) 3105.17 + 3105.17i 0.344631 + 0.344631i 0.858105 0.513474i \(-0.171642\pi\)
−0.513474 + 0.858105i \(0.671642\pi\)
\(434\) −1228.78 −0.135906
\(435\) 0 0
\(436\) −3501.18 −0.384578
\(437\) 2055.65 + 2055.65i 0.225023 + 0.225023i
\(438\) 0 0
\(439\) 4940.17i 0.537087i −0.963268 0.268544i \(-0.913458\pi\)
0.963268 0.268544i \(-0.0865423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 204.833 204.833i 0.0220428 0.0220428i
\(443\) 11835.5 11835.5i 1.26935 1.26935i 0.322931 0.946423i \(-0.395332\pi\)
0.946423 0.322931i \(-0.104668\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 6618.02i 0.702628i
\(447\) 0 0
\(448\) 1402.42 + 1402.42i 0.147898 + 0.147898i
\(449\) 3384.05 0.355687 0.177843 0.984059i \(-0.443088\pi\)
0.177843 + 0.984059i \(0.443088\pi\)
\(450\) 0 0
\(451\) 4100.16 0.428091
\(452\) −3775.07 3775.07i −0.392842 0.392842i
\(453\) 0 0
\(454\) 4528.70i 0.468155i
\(455\) 0 0
\(456\) 0 0
\(457\) 3509.05 3509.05i 0.359182 0.359182i −0.504329 0.863511i \(-0.668260\pi\)
0.863511 + 0.504329i \(0.168260\pi\)
\(458\) −8914.30 + 8914.30i −0.909472 + 0.909472i
\(459\) 0 0
\(460\) 0 0
\(461\) 8838.05i 0.892904i −0.894807 0.446452i \(-0.852687\pi\)
0.894807 0.446452i \(-0.147313\pi\)
\(462\) 0 0
\(463\) −6343.49 6343.49i −0.636732 0.636732i 0.313016 0.949748i \(-0.398661\pi\)
−0.949748 + 0.313016i \(0.898661\pi\)
\(464\) −2954.33 −0.295585
\(465\) 0 0
\(466\) 7585.71 0.754080
\(467\) −226.322 226.322i −0.0224260 0.0224260i 0.695805 0.718231i \(-0.255048\pi\)
−0.718231 + 0.695805i \(0.755048\pi\)
\(468\) 0 0
\(469\) 4405.48i 0.433745i
\(470\) 0 0
\(471\) 0 0
\(472\) 1445.55 1445.55i 0.140967 0.140967i
\(473\) −6764.10 + 6764.10i −0.657534 + 0.657534i
\(474\) 0 0
\(475\) 0 0
\(476\) 970.065i 0.0934093i
\(477\) 0 0
\(478\) 3273.36 + 3273.36i 0.313222 + 0.313222i
\(479\) 15546.2 1.48293 0.741465 0.670991i \(-0.234131\pi\)
0.741465 + 0.670991i \(0.234131\pi\)
\(480\) 0 0
\(481\) −6667.59 −0.632050
\(482\) 2088.21 + 2088.21i 0.197335 + 0.197335i
\(483\) 0 0
\(484\) 11539.9i 1.08377i
\(485\) 0 0
\(486\) 0 0
\(487\) 3464.88 3464.88i 0.322399 0.322399i −0.527288 0.849687i \(-0.676791\pi\)
0.849687 + 0.527288i \(0.176791\pi\)
\(488\) 5057.70 5057.70i 0.469162 0.469162i
\(489\) 0 0
\(490\) 0 0
\(491\) 2614.21i 0.240280i 0.992757 + 0.120140i \(0.0383344\pi\)
−0.992757 + 0.120140i \(0.961666\pi\)
\(492\) 0 0
\(493\) 1021.77 + 1021.77i 0.0933428 + 0.0933428i
\(494\) −3843.21 −0.350029
\(495\) 0 0
\(496\) −317.212 −0.0287162
\(497\) 10773.5 + 10773.5i 0.972353 + 0.972353i
\(498\) 0 0
\(499\) 3762.23i 0.337517i 0.985657 + 0.168758i \(0.0539757\pi\)
−0.985657 + 0.168758i \(0.946024\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5687.31 5687.31i 0.505652 0.505652i
\(503\) 7204.69 7204.69i 0.638651 0.638651i −0.311572 0.950223i \(-0.600855\pi\)
0.950223 + 0.311572i \(0.100855\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3636.11i 0.319457i
\(507\) 0 0
\(508\) −2143.29 2143.29i −0.187191 0.187191i
\(509\) 15219.2 1.32530 0.662652 0.748928i \(-0.269431\pi\)
0.662652 + 0.748928i \(0.269431\pi\)
\(510\) 0 0
\(511\) 26461.1 2.29075
\(512\) 362.039 + 362.039i 0.0312500 + 0.0312500i
\(513\) 0 0
\(514\) 1049.86i 0.0900924i
\(515\) 0 0
\(516\) 0 0
\(517\) −12049.0 + 12049.0i −1.02498 + 1.02498i
\(518\) −15788.4 + 15788.4i −1.33920 + 1.33920i
\(519\) 0 0
\(520\) 0 0
\(521\) 4041.49i 0.339848i −0.985457 0.169924i \(-0.945648\pi\)
0.985457 0.169924i \(-0.0543523\pi\)
\(522\) 0 0
\(523\) −14873.0 14873.0i −1.24350 1.24350i −0.958540 0.284958i \(-0.908020\pi\)
−0.284958 0.958540i \(-0.591980\pi\)
\(524\) −7408.01 −0.617597
\(525\) 0 0
\(526\) 513.091 0.0425320
\(527\) 109.709 + 109.709i 0.00906829 + 0.00906829i
\(528\) 0 0
\(529\) 11383.0i 0.935563i
\(530\) 0 0
\(531\) 0 0
\(532\) −9100.48 + 9100.48i −0.741647 + 0.741647i
\(533\) 826.410 826.410i 0.0671590 0.0671590i
\(534\) 0 0
\(535\) 0 0
\(536\) 1137.28i 0.0916478i
\(537\) 0 0
\(538\) 4669.43 + 4669.43i 0.374189 + 0.374189i
\(539\) −40084.8 −3.20329
\(540\) 0 0
\(541\) −8356.96 −0.664129 −0.332065 0.943257i \(-0.607745\pi\)
−0.332065 + 0.943257i \(0.607745\pi\)
\(542\) −5507.27 5507.27i −0.436453 0.436453i
\(543\) 0 0
\(544\) 250.424i 0.0197369i
\(545\) 0 0
\(546\) 0 0
\(547\) −5395.94 + 5395.94i −0.421780 + 0.421780i −0.885816 0.464036i \(-0.846401\pi\)
0.464036 + 0.885816i \(0.346401\pi\)
\(548\) 3555.18 3555.18i 0.277135 0.277135i
\(549\) 0 0
\(550\) 0 0
\(551\) 19171.0i 1.48224i
\(552\) 0 0
\(553\) 15312.3 + 15312.3i 1.17748 + 1.17748i
\(554\) 4009.37 0.307476
\(555\) 0 0
\(556\) 1009.39 0.0769925
\(557\) −4120.65 4120.65i −0.313461 0.313461i 0.532788 0.846249i \(-0.321144\pi\)
−0.846249 + 0.532788i \(0.821144\pi\)
\(558\) 0 0
\(559\) 2726.68i 0.206308i
\(560\) 0 0
\(561\) 0 0
\(562\) −11482.5 + 11482.5i −0.861848 + 0.861848i
\(563\) −10494.4 + 10494.4i −0.785590 + 0.785590i −0.980768 0.195177i \(-0.937472\pi\)
0.195177 + 0.980768i \(0.437472\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3300.60i 0.245114i
\(567\) 0 0
\(568\) 2781.21 + 2781.21i 0.205453 + 0.205453i
\(569\) −25399.7 −1.87137 −0.935687 0.352832i \(-0.885219\pi\)
−0.935687 + 0.352832i \(0.885219\pi\)
\(570\) 0 0
\(571\) 15700.6 1.15070 0.575351 0.817907i \(-0.304865\pi\)
0.575351 + 0.817907i \(0.304865\pi\)
\(572\) −3399.02 3399.02i −0.248462 0.248462i
\(573\) 0 0
\(574\) 3913.77i 0.284595i
\(575\) 0 0
\(576\) 0 0
\(577\) 5627.77 5627.77i 0.406044 0.406044i −0.474313 0.880356i \(-0.657303\pi\)
0.880356 + 0.474313i \(0.157303\pi\)
\(578\) 6861.42 6861.42i 0.493767 0.493767i
\(579\) 0 0
\(580\) 0 0
\(581\) 30989.3i 2.21282i
\(582\) 0 0
\(583\) 16847.9 + 16847.9i 1.19686 + 1.19686i
\(584\) 6830.99 0.484021
\(585\) 0 0
\(586\) 13651.2 0.962333
\(587\) −10563.0 10563.0i −0.742731 0.742731i 0.230372 0.973103i \(-0.426006\pi\)
−0.973103 + 0.230372i \(0.926006\pi\)
\(588\) 0 0
\(589\) 2058.42i 0.144000i
\(590\) 0 0
\(591\) 0 0
\(592\) −4075.82 + 4075.82i −0.282965 + 0.282965i
\(593\) −4080.58 + 4080.58i −0.282579 + 0.282579i −0.834137 0.551558i \(-0.814034\pi\)
0.551558 + 0.834137i \(0.314034\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 676.101i 0.0464668i
\(597\) 0 0
\(598\) 732.879 + 732.879i 0.0501165 + 0.0501165i
\(599\) 10431.0 0.711515 0.355758 0.934578i \(-0.384223\pi\)
0.355758 + 0.934578i \(0.384223\pi\)
\(600\) 0 0
\(601\) −13534.1 −0.918584 −0.459292 0.888285i \(-0.651897\pi\)
−0.459292 + 0.888285i \(0.651897\pi\)
\(602\) 6456.61 + 6456.61i 0.437129 + 0.437129i
\(603\) 0 0
\(604\) 11134.6i 0.750100i
\(605\) 0 0
\(606\) 0 0
\(607\) −8819.49 + 8819.49i −0.589740 + 0.589740i −0.937561 0.347821i \(-0.886922\pi\)
0.347821 + 0.937561i \(0.386922\pi\)
\(608\) −2349.31 + 2349.31i −0.156706 + 0.156706i
\(609\) 0 0
\(610\) 0 0
\(611\) 4857.08i 0.321598i
\(612\) 0 0
\(613\) −6914.61 6914.61i −0.455593 0.455593i 0.441613 0.897206i \(-0.354406\pi\)
−0.897206 + 0.441613i \(0.854406\pi\)
\(614\) 11771.6 0.773718
\(615\) 0 0
\(616\) −16097.3 −1.05289
\(617\) 14354.7 + 14354.7i 0.936629 + 0.936629i 0.998108 0.0614791i \(-0.0195818\pi\)
−0.0614791 + 0.998108i \(0.519582\pi\)
\(618\) 0 0
\(619\) 15236.3i 0.989334i 0.869083 + 0.494667i \(0.164710\pi\)
−0.869083 + 0.494667i \(0.835290\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5635.08 5635.08i 0.363257 0.363257i
\(623\) −8538.31 + 8538.31i −0.549085 + 0.549085i
\(624\) 0 0
\(625\) 0 0
\(626\) 6865.00i 0.438308i
\(627\) 0 0
\(628\) −8072.86 8072.86i −0.512965 0.512965i
\(629\) 2819.27 0.178715
\(630\) 0 0
\(631\) 2894.27 0.182597 0.0912986 0.995824i \(-0.470898\pi\)
0.0912986 + 0.995824i \(0.470898\pi\)
\(632\) 3952.90 + 3952.90i 0.248794 + 0.248794i
\(633\) 0 0
\(634\) 20173.9i 1.26373i
\(635\) 0 0
\(636\) 0 0
\(637\) −8079.31 + 8079.31i −0.502534 + 0.502534i
\(638\) 16955.3 16955.3i 1.05214 1.05214i
\(639\) 0 0
\(640\) 0 0
\(641\) 345.840i 0.0213102i 0.999943 + 0.0106551i \(0.00339169\pi\)
−0.999943 + 0.0106551i \(0.996608\pi\)
\(642\) 0 0
\(643\) 1161.24 + 1161.24i 0.0712207 + 0.0712207i 0.741820 0.670599i \(-0.233963\pi\)
−0.670599 + 0.741820i \(0.733963\pi\)
\(644\) 3470.82 0.212375
\(645\) 0 0
\(646\) 1625.03 0.0989721
\(647\) −15317.4 15317.4i −0.930738 0.930738i 0.0670144 0.997752i \(-0.478653\pi\)
−0.997752 + 0.0670144i \(0.978653\pi\)
\(648\) 0 0
\(649\) 16592.3i 1.00355i
\(650\) 0 0
\(651\) 0 0
\(652\) 8530.76 8530.76i 0.512408 0.512408i
\(653\) 9594.72 9594.72i 0.574993 0.574993i −0.358527 0.933520i \(-0.616721\pi\)
0.933520 + 0.358527i \(0.116721\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1010.35i 0.0601334i
\(657\) 0 0
\(658\) 11501.3 + 11501.3i 0.681408 + 0.681408i
\(659\) 1554.04 0.0918618 0.0459309 0.998945i \(-0.485375\pi\)
0.0459309 + 0.998945i \(0.485375\pi\)
\(660\) 0 0
\(661\) 2698.57 0.158793 0.0793965 0.996843i \(-0.474701\pi\)
0.0793965 + 0.996843i \(0.474701\pi\)
\(662\) −2536.00 2536.00i −0.148889 0.148889i
\(663\) 0 0
\(664\) 7999.94i 0.467557i
\(665\) 0 0
\(666\) 0 0
\(667\) −3655.80 + 3655.80i −0.212224 + 0.212224i
\(668\) 4111.80 4111.80i 0.238159 0.238159i
\(669\) 0 0
\(670\) 0 0
\(671\) 58053.4i 3.33998i
\(672\) 0 0
\(673\) 8704.87 + 8704.87i 0.498586 + 0.498586i 0.910997 0.412412i \(-0.135314\pi\)
−0.412412 + 0.910997i \(0.635314\pi\)
\(674\) −12553.9 −0.717443
\(675\) 0 0
\(676\) 7417.82 0.422042
\(677\) 8512.59 + 8512.59i 0.483258 + 0.483258i 0.906170 0.422913i \(-0.138992\pi\)
−0.422913 + 0.906170i \(0.638992\pi\)
\(678\) 0 0
\(679\) 1282.24i 0.0724708i
\(680\) 0 0
\(681\) 0 0
\(682\) 1820.52 1820.52i 0.102216 0.102216i
\(683\) −4988.15 + 4988.15i −0.279453 + 0.279453i −0.832891 0.553438i \(-0.813316\pi\)
0.553438 + 0.832891i \(0.313316\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17003.8i 0.946370i
\(687\) 0 0
\(688\) 1666.79 + 1666.79i 0.0923629 + 0.0923629i
\(689\) 6791.59 0.375528
\(690\) 0 0
\(691\) −24712.4 −1.36050 −0.680250 0.732980i \(-0.738129\pi\)
−0.680250 + 0.732980i \(0.738129\pi\)
\(692\) −3473.07 3473.07i −0.190790 0.190790i
\(693\) 0 0
\(694\) 13207.6i 0.722410i
\(695\) 0 0
\(696\) 0 0
\(697\) −349.432 + 349.432i −0.0189895 + 0.0189895i
\(698\) −1759.31 + 1759.31i −0.0954026 + 0.0954026i
\(699\) 0 0
\(700\) 0 0
\(701\) 9481.12i 0.510837i −0.966831 0.255419i \(-0.917787\pi\)
0.966831 0.255419i \(-0.0822133\pi\)
\(702\) 0 0
\(703\) −26448.4 26448.4i −1.41895 1.41895i
\(704\) −4155.56 −0.222470
\(705\) 0 0
\(706\) 15107.2 0.805334
\(707\) −8462.83 8462.83i −0.450180 0.450180i
\(708\) 0 0
\(709\) 33090.6i 1.75281i −0.481571 0.876407i \(-0.659934\pi\)
0.481571 0.876407i \(-0.340066\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2204.18 + 2204.18i −0.116019 + 0.116019i
\(713\) −392.530 + 392.530i −0.0206176 + 0.0206176i
\(714\) 0 0
\(715\) 0 0
\(716\) 13792.8i 0.719920i
\(717\) 0 0
\(718\) 1471.68 + 1471.68i 0.0764940 + 0.0764940i
\(719\) −19996.8 −1.03721 −0.518605 0.855014i \(-0.673549\pi\)
−0.518605 + 0.855014i \(0.673549\pi\)
\(720\) 0 0
\(721\) 34778.6 1.79642
\(722\) −5544.83 5544.83i −0.285813 0.285813i
\(723\) 0 0
\(724\) 14047.2i 0.721080i
\(725\) 0 0
\(726\) 0 0
\(727\) −25556.9 + 25556.9i −1.30379 + 1.30379i −0.377969 + 0.925818i \(0.623377\pi\)
−0.925818 + 0.377969i \(0.876623\pi\)
\(728\) −3244.51 + 3244.51i −0.165178 + 0.165178i
\(729\) 0 0
\(730\) 0 0
\(731\) 1152.93i 0.0583345i
\(732\) 0 0
\(733\) 18981.4 + 18981.4i 0.956474 + 0.956474i 0.999091 0.0426176i \(-0.0135697\pi\)
−0.0426176 + 0.999091i \(0.513570\pi\)
\(734\) 4094.16 0.205883
\(735\) 0 0
\(736\) 896.000 0.0448736
\(737\) −6527.00 6527.00i −0.326221 0.326221i
\(738\) 0 0
\(739\) 4651.19i 0.231525i 0.993277 + 0.115762i \(0.0369311\pi\)
−0.993277 + 0.115762i \(0.963069\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 16082.1 16082.1i 0.795675 0.795675i
\(743\) −9737.08 + 9737.08i −0.480779 + 0.480779i −0.905380 0.424601i \(-0.860414\pi\)
0.424601 + 0.905380i \(0.360414\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 300.938i 0.0147696i
\(747\) 0 0
\(748\) 1437.21 + 1437.21i 0.0702536 + 0.0702536i
\(749\) −21978.4 −1.07219
\(750\) 0 0
\(751\) −7800.89 −0.379039 −0.189520 0.981877i \(-0.560693\pi\)
−0.189520 + 0.981877i \(0.560693\pi\)
\(752\) 2969.08 + 2969.08i 0.143978 + 0.143978i
\(753\) 0 0
\(754\) 6834.85i 0.330120i
\(755\) 0 0
\(756\) 0 0
\(757\) −1497.47 + 1497.47i −0.0718974 + 0.0718974i −0.742141 0.670244i \(-0.766190\pi\)
0.670244 + 0.742141i \(0.266190\pi\)
\(758\) 13272.2 13272.2i 0.635974 0.635974i
\(759\) 0 0
\(760\) 0 0
\(761\) 22037.0i 1.04973i −0.851187 0.524863i \(-0.824116\pi\)
0.851187 0.524863i \(-0.175884\pi\)
\(762\) 0 0
\(763\) −19180.2 19180.2i −0.910054 0.910054i
\(764\) −8149.73 −0.385925
\(765\) 0 0
\(766\) −22672.3 −1.06943
\(767\) 3344.27 + 3344.27i 0.157437 + 0.157437i
\(768\) 0 0
\(769\) 6827.73i 0.320174i 0.987103 + 0.160087i \(0.0511775\pi\)
−0.987103 + 0.160087i \(0.948822\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4511.12 4511.12i 0.210309 0.210309i
\(773\) 2717.78 2717.78i 0.126457 0.126457i −0.641045 0.767503i \(-0.721499\pi\)
0.767503 + 0.641045i \(0.221499\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 331.012i 0.0153127i
\(777\) 0 0
\(778\) 12576.2 + 12576.2i 0.579536 + 0.579536i
\(779\) 6556.26 0.301544
\(780\) 0 0
\(781\) −31923.4 −1.46262
\(782\) −309.884 309.884i −0.0141706 0.0141706i
\(783\) 0 0
\(784\) 9877.58i 0.449962i
\(785\) 0 0
\(786\) 0 0
\(787\) −18023.9 + 18023.9i −0.816370 + 0.816370i −0.985580 0.169210i \(-0.945878\pi\)
0.169210 + 0.985580i \(0.445878\pi\)
\(788\) −1031.24 + 1031.24i −0.0466198 + 0.0466198i
\(789\) 0 0
\(790\) 0 0
\(791\) 41361.3i 1.85922i
\(792\) 0 0
\(793\) 11701.0 + 11701.0i 0.523977 + 0.523977i
\(794\) −9188.48 −0.410689
\(795\) 0 0
\(796\) 2052.12 0.0913763
\(797\) 21477.8 + 21477.8i 0.954557 + 0.954557i 0.999011 0.0444541i \(-0.0141549\pi\)
−0.0444541 + 0.999011i \(0.514155\pi\)
\(798\) 0 0
\(799\) 2053.73i 0.0909332i
\(800\) 0 0
\(801\) 0 0
\(802\) −16988.2 + 16988.2i −0.747971 + 0.747971i
\(803\) −39203.8 + 39203.8i −1.72288 + 1.72288i
\(804\) 0 0
\(805\) 0 0
\(806\) 733.870i 0.0320713i
\(807\) 0 0
\(808\) −2184.70 2184.70i −0.0951205 0.0951205i
\(809\) 36819.3 1.60012 0.800060 0.599920i \(-0.204801\pi\)
0.800060 + 0.599920i \(0.204801\pi\)
\(810\) 0 0
\(811\) 38040.8 1.64709 0.823546 0.567249i \(-0.191992\pi\)
0.823546 + 0.567249i \(0.191992\pi\)
\(812\) −16184.5 16184.5i −0.699464 0.699464i
\(813\) 0 0
\(814\) 46783.1i 2.01443i
\(815\) 0 0
\(816\) 0 0
\(817\) −10816.0 + 10816.0i −0.463161 + 0.463161i
\(818\) −2419.16 + 2419.16i −0.103403 + 0.103403i
\(819\) 0 0
\(820\) 0 0
\(821\) 2824.53i 0.120069i −0.998196 0.0600346i \(-0.980879\pi\)
0.998196 0.0600346i \(-0.0191211\pi\)
\(822\) 0 0
\(823\) 14878.3 + 14878.3i 0.630164 + 0.630164i 0.948109 0.317945i \(-0.102993\pi\)
−0.317945 + 0.948109i \(0.602993\pi\)
\(824\) 8978.16 0.379574
\(825\) 0 0
\(826\) 15838.0 0.667162
\(827\) 31885.6 + 31885.6i 1.34072 + 1.34072i 0.895347 + 0.445369i \(0.146928\pi\)
0.445369 + 0.895347i \(0.353072\pi\)
\(828\) 0 0
\(829\) 6395.67i 0.267950i 0.990985 + 0.133975i \(0.0427743\pi\)
−0.990985 + 0.133975i \(0.957226\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −837.576 + 837.576i −0.0349011 + 0.0349011i
\(833\) 3416.19 3416.19i 0.142094 0.142094i
\(834\) 0 0
\(835\) 0 0
\(836\) 26965.9i 1.11559i
\(837\) 0 0
\(838\) −4109.54 4109.54i −0.169405 0.169405i
\(839\) −19006.7 −0.782104 −0.391052 0.920369i \(-0.627889\pi\)
−0.391052 + 0.920369i \(0.627889\pi\)
\(840\) 0 0
\(841\) 9705.12 0.397930
\(842\) −12816.8 12816.8i −0.524580 0.524580i
\(843\) 0 0
\(844\) 6440.97i 0.262686i
\(845\) 0 0
\(846\) 0 0
\(847\) 63218.3 63218.3i 2.56459 2.56459i
\(848\) 4151.62 4151.62i 0.168122 0.168122i
\(849\) 0 0
\(850\) 0 0
\(851\) 10087.1i 0.406325i
\(852\) 0 0
\(853\) −31885.3 31885.3i −1.27987 1.27987i −0.940738 0.339133i \(-0.889866\pi\)
−0.339133 0.940738i \(-0.610134\pi\)
\(854\) 55414.4 2.22042
\(855\) 0 0
\(856\) −5673.76 −0.226548
\(857\) 14082.6 + 14082.6i 0.561322 + 0.561322i 0.929683 0.368361i \(-0.120081\pi\)
−0.368361 + 0.929683i \(0.620081\pi\)
\(858\) 0 0
\(859\) 24821.7i 0.985920i −0.870052 0.492960i \(-0.835915\pi\)
0.870052 0.492960i \(-0.164085\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 14615.3 14615.3i 0.577492 0.577492i
\(863\) 31772.4 31772.4i 1.25324 1.25324i 0.298976 0.954260i \(-0.403355\pi\)
0.954260 0.298976i \(-0.0966452\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8782.76i 0.344631i
\(867\) 0 0
\(868\) −1737.76 1737.76i −0.0679532 0.0679532i
\(869\) −45372.2 −1.77117
\(870\) 0 0
\(871\) −2631.11 −0.102355
\(872\) −4951.42 4951.42i −0.192289 0.192289i
\(873\) 0 0
\(874\) 5814.24i 0.225023i
\(875\) 0 0
\(876\) 0 0
\(877\) 30277.2 30277.2i 1.16578 1.16578i 0.182592 0.983189i \(-0.441551\pi\)
0.983189 0.182592i \(-0.0584488\pi\)
\(878\) 6986.45 6986.45i 0.268544 0.268544i
\(879\) 0 0
\(880\) 0 0
\(881\) 32069.6i 1.22639i 0.789930 + 0.613197i \(0.210117\pi\)
−0.789930 + 0.613197i \(0.789883\pi\)
\(882\) 0 0
\(883\) −7061.36 7061.36i −0.269121 0.269121i 0.559625 0.828746i \(-0.310945\pi\)
−0.828746 + 0.559625i \(0.810945\pi\)
\(884\) 579.356 0.0220428
\(885\) 0 0
\(886\) 33476.0 1.26935
\(887\) −18324.6 18324.6i −0.693664 0.693664i 0.269372 0.963036i \(-0.413184\pi\)
−0.963036 + 0.269372i \(0.913184\pi\)
\(888\) 0 0
\(889\) 23482.8i 0.885926i
\(890\) 0 0
\(891\) 0 0
\(892\) −9359.29 + 9359.29i −0.351314 + 0.351314i
\(893\) −19266.7 + 19266.7i −0.721987 + 0.721987i
\(894\) 0 0
\(895\) 0 0
\(896\) 3966.65i 0.147898i
\(897\) 0 0
\(898\) 4785.77 + 4785.77i 0.177843 + 0.177843i
\(899\) 3660.75 0.135809
\(900\) 0 0
\(901\) −2871.70 −0.106182
\(902\) 5798.50 + 5798.50i 0.214045 + 0.214045i
\(903\) 0 0
\(904\) 10677.5i 0.392842i
\(905\) 0 0
\(906\) 0 0
\(907\) 18445.0 18445.0i 0.675256 0.675256i −0.283667 0.958923i \(-0.591551\pi\)
0.958923 + 0.283667i \(0.0915511\pi\)
\(908\) 6404.54 6404.54i 0.234077 0.234077i
\(909\) 0 0
\(910\) 0 0
\(911\) 7421.18i 0.269895i −0.990853 0.134948i \(-0.956913\pi\)
0.990853 0.134948i \(-0.0430866\pi\)
\(912\) 0 0
\(913\) −45912.5 45912.5i −1.66427 1.66427i
\(914\) 9925.08 0.359182
\(915\) 0 0
\(916\) −25213.5 −0.909472
\(917\) −40582.7 40582.7i −1.46146 1.46146i
\(918\) 0 0
\(919\) 36117.6i 1.29642i −0.761462 0.648209i \(-0.775518\pi\)
0.761462 0.648209i \(-0.224482\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 12498.9 12498.9i 0.446452 0.446452i
\(923\) −6434.33 + 6434.33i −0.229457 + 0.229457i
\(924\) 0 0
\(925\) 0 0
\(926\) 17942.1i 0.636732i
\(927\) 0 0
\(928\) −4178.06 4178.06i −0.147793 0.147793i
\(929\) 11654.8 0.411606 0.205803 0.978593i \(-0.434019\pi\)
0.205803 + 0.978593i \(0.434019\pi\)
\(930\) 0 0
\(931\) −64096.7 −2.25637
\(932\) 10727.8 + 10727.8i 0.377040 + 0.377040i
\(933\) 0 0
\(934\) 640.137i 0.0224260i
\(935\) 0 0
\(936\) 0 0
\(937\) −16788.2 + 16788.2i −0.585323 + 0.585323i −0.936361 0.351038i \(-0.885829\pi\)
0.351038 + 0.936361i \(0.385829\pi\)
\(938\) −6230.30 + 6230.30i −0.216872 + 0.216872i
\(939\) 0 0
\(940\) 0 0
\(941\) 27834.5i 0.964271i 0.876097 + 0.482136i \(0.160139\pi\)
−0.876097 + 0.482136i \(0.839861\pi\)
\(942\) 0 0
\(943\) −1250.24 1250.24i −0.0431744 0.0431744i
\(944\) 4088.62 0.140967
\(945\) 0 0
\(946\) −19131.8 −0.657534
\(947\) 4017.43 + 4017.43i 0.137855 + 0.137855i 0.772667 0.634812i \(-0.218922\pi\)
−0.634812 + 0.772667i \(0.718922\pi\)
\(948\) 0 0
\(949\) 15803.5i 0.540572i
\(950\) 0 0
\(951\) 0 0
\(952\) 1371.88 1371.88i 0.0467047 0.0467047i
\(953\) 14545.6 14545.6i 0.494416 0.494416i −0.415278 0.909694i \(-0.636316\pi\)
0.909694 + 0.415278i \(0.136316\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9258.47i 0.313222i
\(957\) 0 0
\(958\) 21985.6 + 21985.6i 0.741465 + 0.741465i
\(959\) 38952.1 1.31161
\(960\) 0 0
\(961\) −29397.9 −0.986806
\(962\) −9429.40 9429.40i −0.316025 0.316025i
\(963\) 0 0
\(964\) 5906.36i 0.197335i
\(965\) 0 0
\(966\) 0 0
\(967\) 24630.8 24630.8i 0.819105 0.819105i −0.166874 0.985978i \(-0.553367\pi\)
0.985978 + 0.166874i \(0.0533672\pi\)
\(968\) 16319.9 16319.9i 0.541883 0.541883i
\(969\) 0 0
\(970\) 0 0
\(971\) 17813.2i 0.588727i −0.955693 0.294364i \(-0.904892\pi\)
0.955693 0.294364i \(-0.0951077\pi\)
\(972\) 0 0
\(973\) 5529.68 + 5529.68i 0.182193 + 0.182193i
\(974\) 9800.15 0.322399
\(975\) 0 0
\(976\) 14305.3 0.469162
\(977\) 2909.26 + 2909.26i 0.0952665 + 0.0952665i 0.753134 0.657867i \(-0.228541\pi\)
−0.657867 + 0.753134i \(0.728541\pi\)
\(978\) 0 0
\(979\) 25300.1i 0.825939i
\(980\) 0 0
\(981\) 0 0
\(982\) −3697.05 + 3697.05i −0.120140 + 0.120140i
\(983\) 10412.0 10412.0i 0.337834 0.337834i −0.517718 0.855552i \(-0.673218\pi\)
0.855552 + 0.517718i \(0.173218\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2889.99i 0.0933428i
\(987\) 0 0
\(988\) −5435.12 5435.12i −0.175014 0.175014i
\(989\) 4125.09 0.132629
\(990\) 0 0
\(991\) −12712.1 −0.407482 −0.203741 0.979025i \(-0.565310\pi\)
−0.203741 + 0.979025i \(0.565310\pi\)
\(992\) −448.606 448.606i −0.0143581 0.0143581i
\(993\) 0 0
\(994\) 30472.2i 0.972353i
\(995\) 0 0
\(996\) 0 0
\(997\) −7747.71 + 7747.71i −0.246111 + 0.246111i −0.819372 0.573262i \(-0.805678\pi\)
0.573262 + 0.819372i \(0.305678\pi\)
\(998\) −5320.60 + 5320.60i −0.168758 + 0.168758i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 450.4.f.e.143.3 8
3.2 odd 2 inner 450.4.f.e.143.1 8
5.2 odd 4 inner 450.4.f.e.107.1 8
5.3 odd 4 90.4.f.b.17.4 yes 8
5.4 even 2 90.4.f.b.53.1 yes 8
15.2 even 4 inner 450.4.f.e.107.3 8
15.8 even 4 90.4.f.b.17.1 8
15.14 odd 2 90.4.f.b.53.4 yes 8
20.3 even 4 720.4.w.c.17.3 8
20.19 odd 2 720.4.w.c.593.2 8
60.23 odd 4 720.4.w.c.17.2 8
60.59 even 2 720.4.w.c.593.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.f.b.17.1 8 15.8 even 4
90.4.f.b.17.4 yes 8 5.3 odd 4
90.4.f.b.53.1 yes 8 5.4 even 2
90.4.f.b.53.4 yes 8 15.14 odd 2
450.4.f.e.107.1 8 5.2 odd 4 inner
450.4.f.e.107.3 8 15.2 even 4 inner
450.4.f.e.143.1 8 3.2 odd 2 inner
450.4.f.e.143.3 8 1.1 even 1 trivial
720.4.w.c.17.2 8 60.23 odd 4
720.4.w.c.17.3 8 20.3 even 4
720.4.w.c.593.2 8 20.19 odd 2
720.4.w.c.593.3 8 60.59 even 2