Properties

Label 525.4.d.i.274.4
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) 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 274.4
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.i.274.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.56155i q^{2} -3.00000i q^{3} -22.9309 q^{4} +16.6847 q^{6} +7.00000i q^{7} -83.0388i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+5.56155i q^{2} -3.00000i q^{3} -22.9309 q^{4} +16.6847 q^{6} +7.00000i q^{7} -83.0388i q^{8} -9.00000 q^{9} -33.6155 q^{11} +68.7926i q^{12} -38.3542i q^{13} -38.9309 q^{14} +278.378 q^{16} -65.7235i q^{17} -50.0540i q^{18} -33.3996 q^{19} +21.0000 q^{21} -186.955i q^{22} +207.447i q^{23} -249.116 q^{24} +213.309 q^{26} +27.0000i q^{27} -160.516i q^{28} +189.170 q^{29} +202.108 q^{31} +883.902i q^{32} +100.847i q^{33} +365.525 q^{34} +206.378 q^{36} +16.5227i q^{37} -185.754i q^{38} -115.062 q^{39} +388.617 q^{41} +116.793i q^{42} +41.8144i q^{43} +770.833 q^{44} -1153.73 q^{46} -368.648i q^{47} -835.133i q^{48} -49.0000 q^{49} -197.170 q^{51} +879.494i q^{52} +458.172i q^{53} -150.162 q^{54} +581.272 q^{56} +100.199i q^{57} +1052.08i q^{58} -256.216 q^{59} -123.511 q^{61} +1124.03i q^{62} -63.0000i q^{63} -2688.85 q^{64} -560.864 q^{66} +336.277i q^{67} +1507.10i q^{68} +622.341 q^{69} -453.312 q^{71} +747.349i q^{72} +22.0436i q^{73} -91.8920 q^{74} +765.882 q^{76} -235.309i q^{77} -639.926i q^{78} -385.417 q^{79} +81.0000 q^{81} +2161.32i q^{82} +23.7501i q^{83} -481.548 q^{84} -232.553 q^{86} -567.511i q^{87} +2791.39i q^{88} +1482.81 q^{89} +268.479 q^{91} -4756.94i q^{92} -606.324i q^{93} +2050.25 q^{94} +2651.71 q^{96} -51.9867i q^{97} -272.516i q^{98} +302.540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 34 q^{4} + 42 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 34 q^{4} + 42 q^{6} - 36 q^{9} - 52 q^{11} - 98 q^{14} + 594 q^{16} - 348 q^{19} + 84 q^{21} - 378 q^{24} + 276 q^{26} + 64 q^{29} + 660 q^{31} + 588 q^{34} + 306 q^{36} + 84 q^{39} + 400 q^{41} + 1632 q^{44} - 2240 q^{46} - 196 q^{49} - 96 q^{51} - 378 q^{54} + 882 q^{56} - 728 q^{59} + 1584 q^{61} - 5618 q^{64} - 1056 q^{66} + 1104 q^{69} + 908 q^{71} - 516 q^{74} - 136 q^{76} - 816 q^{79} + 324 q^{81} - 714 q^{84} - 1392 q^{86} - 72 q^{89} - 196 q^{91} + 3880 q^{94} + 6426 q^{96} + 468 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.56155i 1.96631i 0.182785 + 0.983153i \(0.441489\pi\)
−0.182785 + 0.983153i \(0.558511\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −22.9309 −2.86636
\(5\) 0 0
\(6\) 16.6847 1.13525
\(7\) 7.00000i 0.377964i
\(8\) − 83.0388i − 3.66983i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −33.6155 −0.921406 −0.460703 0.887554i \(-0.652403\pi\)
−0.460703 + 0.887554i \(0.652403\pi\)
\(12\) 68.7926i 1.65489i
\(13\) − 38.3542i − 0.818272i −0.912474 0.409136i \(-0.865830\pi\)
0.912474 0.409136i \(-0.134170\pi\)
\(14\) −38.9309 −0.743194
\(15\) 0 0
\(16\) 278.378 4.34965
\(17\) − 65.7235i − 0.937664i −0.883287 0.468832i \(-0.844675\pi\)
0.883287 0.468832i \(-0.155325\pi\)
\(18\) − 50.0540i − 0.655435i
\(19\) −33.3996 −0.403284 −0.201642 0.979459i \(-0.564628\pi\)
−0.201642 + 0.979459i \(0.564628\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) − 186.955i − 1.81177i
\(23\) 207.447i 1.88068i 0.340234 + 0.940341i \(0.389494\pi\)
−0.340234 + 0.940341i \(0.610506\pi\)
\(24\) −249.116 −2.11878
\(25\) 0 0
\(26\) 213.309 1.60897
\(27\) 27.0000i 0.192450i
\(28\) − 160.516i − 1.08338i
\(29\) 189.170 1.21131 0.605656 0.795726i \(-0.292911\pi\)
0.605656 + 0.795726i \(0.292911\pi\)
\(30\) 0 0
\(31\) 202.108 1.17096 0.585478 0.810688i \(-0.300907\pi\)
0.585478 + 0.810688i \(0.300907\pi\)
\(32\) 883.902i 4.88292i
\(33\) 100.847i 0.531974i
\(34\) 365.525 1.84373
\(35\) 0 0
\(36\) 206.378 0.955453
\(37\) 16.5227i 0.0734141i 0.999326 + 0.0367070i \(0.0116868\pi\)
−0.999326 + 0.0367070i \(0.988313\pi\)
\(38\) − 185.754i − 0.792980i
\(39\) −115.062 −0.472429
\(40\) 0 0
\(41\) 388.617 1.48029 0.740144 0.672448i \(-0.234757\pi\)
0.740144 + 0.672448i \(0.234757\pi\)
\(42\) 116.793i 0.429083i
\(43\) 41.8144i 0.148294i 0.997247 + 0.0741469i \(0.0236234\pi\)
−0.997247 + 0.0741469i \(0.976377\pi\)
\(44\) 770.833 2.64108
\(45\) 0 0
\(46\) −1153.73 −3.69800
\(47\) − 368.648i − 1.14410i −0.820218 0.572051i \(-0.806148\pi\)
0.820218 0.572051i \(-0.193852\pi\)
\(48\) − 835.133i − 2.51127i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −197.170 −0.541360
\(52\) 879.494i 2.34546i
\(53\) 458.172i 1.18745i 0.804668 + 0.593725i \(0.202343\pi\)
−0.804668 + 0.593725i \(0.797657\pi\)
\(54\) −150.162 −0.378416
\(55\) 0 0
\(56\) 581.272 1.38707
\(57\) 100.199i 0.232836i
\(58\) 1052.08i 2.38181i
\(59\) −256.216 −0.565364 −0.282682 0.959214i \(-0.591224\pi\)
−0.282682 + 0.959214i \(0.591224\pi\)
\(60\) 0 0
\(61\) −123.511 −0.259246 −0.129623 0.991563i \(-0.541377\pi\)
−0.129623 + 0.991563i \(0.541377\pi\)
\(62\) 1124.03i 2.30246i
\(63\) − 63.0000i − 0.125988i
\(64\) −2688.85 −5.25166
\(65\) 0 0
\(66\) −560.864 −1.04602
\(67\) 336.277i 0.613175i 0.951843 + 0.306587i \(0.0991872\pi\)
−0.951843 + 0.306587i \(0.900813\pi\)
\(68\) 1507.10i 2.68768i
\(69\) 622.341 1.08581
\(70\) 0 0
\(71\) −453.312 −0.757722 −0.378861 0.925454i \(-0.623684\pi\)
−0.378861 + 0.925454i \(0.623684\pi\)
\(72\) 747.349i 1.22328i
\(73\) 22.0436i 0.0353426i 0.999844 + 0.0176713i \(0.00562524\pi\)
−0.999844 + 0.0176713i \(0.994375\pi\)
\(74\) −91.8920 −0.144355
\(75\) 0 0
\(76\) 765.882 1.15596
\(77\) − 235.309i − 0.348259i
\(78\) − 639.926i − 0.928941i
\(79\) −385.417 −0.548896 −0.274448 0.961602i \(-0.588495\pi\)
−0.274448 + 0.961602i \(0.588495\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 2161.32i 2.91070i
\(83\) 23.7501i 0.0314085i 0.999877 + 0.0157043i \(0.00499903\pi\)
−0.999877 + 0.0157043i \(0.995001\pi\)
\(84\) −481.548 −0.625491
\(85\) 0 0
\(86\) −232.553 −0.291591
\(87\) − 567.511i − 0.699352i
\(88\) 2791.39i 3.38140i
\(89\) 1482.81 1.76604 0.883020 0.469335i \(-0.155506\pi\)
0.883020 + 0.469335i \(0.155506\pi\)
\(90\) 0 0
\(91\) 268.479 0.309278
\(92\) − 4756.94i − 5.39071i
\(93\) − 606.324i − 0.676052i
\(94\) 2050.25 2.24965
\(95\) 0 0
\(96\) 2651.71 2.81915
\(97\) − 51.9867i − 0.0544170i −0.999630 0.0272085i \(-0.991338\pi\)
0.999630 0.0272085i \(-0.00866181\pi\)
\(98\) − 272.516i − 0.280901i
\(99\) 302.540 0.307135
\(100\) 0 0
\(101\) 1429.30 1.40812 0.704062 0.710138i \(-0.251368\pi\)
0.704062 + 0.710138i \(0.251368\pi\)
\(102\) − 1096.57i − 1.06448i
\(103\) 434.212i 0.415381i 0.978195 + 0.207690i \(0.0665946\pi\)
−0.978195 + 0.207690i \(0.933405\pi\)
\(104\) −3184.88 −3.00292
\(105\) 0 0
\(106\) −2548.15 −2.33489
\(107\) − 666.307i − 0.602003i −0.953624 0.301001i \(-0.902679\pi\)
0.953624 0.301001i \(-0.0973209\pi\)
\(108\) − 619.133i − 0.551631i
\(109\) 1199.51 1.05406 0.527029 0.849847i \(-0.323306\pi\)
0.527029 + 0.849847i \(0.323306\pi\)
\(110\) 0 0
\(111\) 49.5682 0.0423856
\(112\) 1948.64i 1.64401i
\(113\) − 81.5171i − 0.0678627i −0.999424 0.0339314i \(-0.989197\pi\)
0.999424 0.0339314i \(-0.0108028\pi\)
\(114\) −557.261 −0.457827
\(115\) 0 0
\(116\) −4337.84 −3.47206
\(117\) 345.187i 0.272757i
\(118\) − 1424.96i − 1.11168i
\(119\) 460.064 0.354404
\(120\) 0 0
\(121\) −200.996 −0.151011
\(122\) − 686.915i − 0.509757i
\(123\) − 1165.85i − 0.854645i
\(124\) −4634.51 −3.35638
\(125\) 0 0
\(126\) 350.378 0.247731
\(127\) 336.985i 0.235453i 0.993046 + 0.117727i \(0.0375607\pi\)
−0.993046 + 0.117727i \(0.962439\pi\)
\(128\) − 7882.95i − 5.44344i
\(129\) 125.443 0.0856175
\(130\) 0 0
\(131\) 2931.15 1.95493 0.977465 0.211097i \(-0.0677035\pi\)
0.977465 + 0.211097i \(0.0677035\pi\)
\(132\) − 2312.50i − 1.52483i
\(133\) − 233.797i − 0.152427i
\(134\) −1870.22 −1.20569
\(135\) 0 0
\(136\) −5457.60 −3.44107
\(137\) 1585.07i 0.988477i 0.869326 + 0.494238i \(0.164553\pi\)
−0.869326 + 0.494238i \(0.835447\pi\)
\(138\) 3461.18i 2.13504i
\(139\) 1298.85 0.792569 0.396284 0.918128i \(-0.370299\pi\)
0.396284 + 0.918128i \(0.370299\pi\)
\(140\) 0 0
\(141\) −1105.94 −0.660548
\(142\) − 2521.12i − 1.48991i
\(143\) 1289.30i 0.753960i
\(144\) −2505.40 −1.44988
\(145\) 0 0
\(146\) −122.597 −0.0694943
\(147\) 147.000i 0.0824786i
\(148\) − 378.881i − 0.210431i
\(149\) 2003.29 1.10145 0.550724 0.834687i \(-0.314352\pi\)
0.550724 + 0.834687i \(0.314352\pi\)
\(150\) 0 0
\(151\) 2740.96 1.47719 0.738596 0.674148i \(-0.235489\pi\)
0.738596 + 0.674148i \(0.235489\pi\)
\(152\) 2773.47i 1.47999i
\(153\) 591.511i 0.312555i
\(154\) 1308.68 0.684783
\(155\) 0 0
\(156\) 2638.48 1.35415
\(157\) − 3644.22i − 1.85249i −0.376928 0.926243i \(-0.623019\pi\)
0.376928 0.926243i \(-0.376981\pi\)
\(158\) − 2143.52i − 1.07930i
\(159\) 1374.52 0.685574
\(160\) 0 0
\(161\) −1452.13 −0.710831
\(162\) 450.486i 0.218478i
\(163\) 2774.27i 1.33311i 0.745454 + 0.666557i \(0.232233\pi\)
−0.745454 + 0.666557i \(0.767767\pi\)
\(164\) −8911.33 −4.24304
\(165\) 0 0
\(166\) −132.087 −0.0617588
\(167\) − 1154.91i − 0.535149i −0.963537 0.267574i \(-0.913778\pi\)
0.963537 0.267574i \(-0.0862221\pi\)
\(168\) − 1743.82i − 0.800823i
\(169\) 725.958 0.330432
\(170\) 0 0
\(171\) 300.597 0.134428
\(172\) − 958.841i − 0.425064i
\(173\) − 3387.46i − 1.48869i −0.667794 0.744346i \(-0.732761\pi\)
0.667794 0.744346i \(-0.267239\pi\)
\(174\) 3156.24 1.37514
\(175\) 0 0
\(176\) −9357.82 −4.00780
\(177\) 768.648i 0.326413i
\(178\) 8246.73i 3.47258i
\(179\) −1603.32 −0.669486 −0.334743 0.942309i \(-0.608650\pi\)
−0.334743 + 0.942309i \(0.608650\pi\)
\(180\) 0 0
\(181\) 544.220 0.223489 0.111745 0.993737i \(-0.464356\pi\)
0.111745 + 0.993737i \(0.464356\pi\)
\(182\) 1493.16i 0.608134i
\(183\) 370.534i 0.149676i
\(184\) 17226.2 6.90179
\(185\) 0 0
\(186\) 3372.10 1.32933
\(187\) 2209.33i 0.863969i
\(188\) 8453.41i 3.27941i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 2993.44 1.13402 0.567010 0.823711i \(-0.308100\pi\)
0.567010 + 0.823711i \(0.308100\pi\)
\(192\) 8066.54i 3.03204i
\(193\) 1309.32i 0.488325i 0.969734 + 0.244163i \(0.0785131\pi\)
−0.969734 + 0.244163i \(0.921487\pi\)
\(194\) 289.127 0.107001
\(195\) 0 0
\(196\) 1123.61 0.409480
\(197\) 1141.38i 0.412790i 0.978469 + 0.206395i \(0.0661733\pi\)
−0.978469 + 0.206395i \(0.933827\pi\)
\(198\) 1682.59i 0.603922i
\(199\) −2370.23 −0.844327 −0.422164 0.906520i \(-0.638729\pi\)
−0.422164 + 0.906520i \(0.638729\pi\)
\(200\) 0 0
\(201\) 1008.83 0.354017
\(202\) 7949.12i 2.76880i
\(203\) 1324.19i 0.457833i
\(204\) 4521.29 1.55173
\(205\) 0 0
\(206\) −2414.89 −0.816765
\(207\) − 1867.02i − 0.626894i
\(208\) − 10676.9i − 3.55920i
\(209\) 1122.75 0.371588
\(210\) 0 0
\(211\) −687.159 −0.224199 −0.112099 0.993697i \(-0.535758\pi\)
−0.112099 + 0.993697i \(0.535758\pi\)
\(212\) − 10506.3i − 3.40366i
\(213\) 1359.94i 0.437471i
\(214\) 3705.70 1.18372
\(215\) 0 0
\(216\) 2242.05 0.706260
\(217\) 1414.76i 0.442580i
\(218\) 6671.15i 2.07260i
\(219\) 66.1308 0.0204050
\(220\) 0 0
\(221\) −2520.77 −0.767264
\(222\) 275.676i 0.0833431i
\(223\) 990.496i 0.297437i 0.988880 + 0.148719i \(0.0475149\pi\)
−0.988880 + 0.148719i \(0.952485\pi\)
\(224\) −6187.32 −1.84557
\(225\) 0 0
\(226\) 453.362 0.133439
\(227\) − 1479.25i − 0.432517i −0.976336 0.216258i \(-0.930615\pi\)
0.976336 0.216258i \(-0.0693853\pi\)
\(228\) − 2297.65i − 0.667392i
\(229\) −6704.47 −1.93469 −0.967345 0.253463i \(-0.918430\pi\)
−0.967345 + 0.253463i \(0.918430\pi\)
\(230\) 0 0
\(231\) −705.926 −0.201067
\(232\) − 15708.5i − 4.44531i
\(233\) − 1749.09i − 0.491789i −0.969297 0.245895i \(-0.920918\pi\)
0.969297 0.245895i \(-0.0790817\pi\)
\(234\) −1919.78 −0.536324
\(235\) 0 0
\(236\) 5875.25 1.62054
\(237\) 1156.25i 0.316905i
\(238\) 2558.67i 0.696866i
\(239\) 6320.89 1.71073 0.855365 0.518027i \(-0.173333\pi\)
0.855365 + 0.518027i \(0.173333\pi\)
\(240\) 0 0
\(241\) 3359.62 0.897975 0.448988 0.893538i \(-0.351785\pi\)
0.448988 + 0.893538i \(0.351785\pi\)
\(242\) − 1117.85i − 0.296935i
\(243\) − 243.000i − 0.0641500i
\(244\) 2832.22 0.743092
\(245\) 0 0
\(246\) 6483.95 1.68049
\(247\) 1281.01i 0.329996i
\(248\) − 16782.8i − 4.29721i
\(249\) 71.2502 0.0181337
\(250\) 0 0
\(251\) −1330.50 −0.334582 −0.167291 0.985908i \(-0.553502\pi\)
−0.167291 + 0.985908i \(0.553502\pi\)
\(252\) 1444.64i 0.361127i
\(253\) − 6973.44i − 1.73287i
\(254\) −1874.16 −0.462973
\(255\) 0 0
\(256\) 22330.6 5.45182
\(257\) 2476.95i 0.601197i 0.953751 + 0.300599i \(0.0971865\pi\)
−0.953751 + 0.300599i \(0.902814\pi\)
\(258\) 697.659i 0.168350i
\(259\) −115.659 −0.0277479
\(260\) 0 0
\(261\) −1702.53 −0.403771
\(262\) 16301.8i 3.84399i
\(263\) − 5152.56i − 1.20806i −0.796960 0.604032i \(-0.793560\pi\)
0.796960 0.604032i \(-0.206440\pi\)
\(264\) 8374.18 1.95225
\(265\) 0 0
\(266\) 1300.28 0.299718
\(267\) − 4448.43i − 1.01962i
\(268\) − 7711.11i − 1.75758i
\(269\) 1150.97 0.260876 0.130438 0.991456i \(-0.458362\pi\)
0.130438 + 0.991456i \(0.458362\pi\)
\(270\) 0 0
\(271\) 1838.32 0.412067 0.206034 0.978545i \(-0.433944\pi\)
0.206034 + 0.978545i \(0.433944\pi\)
\(272\) − 18296.0i − 4.07851i
\(273\) − 805.437i − 0.178561i
\(274\) −8815.43 −1.94365
\(275\) 0 0
\(276\) −14270.8 −3.11233
\(277\) − 568.447i − 0.123302i −0.998098 0.0616510i \(-0.980363\pi\)
0.998098 0.0616510i \(-0.0196366\pi\)
\(278\) 7223.62i 1.55843i
\(279\) −1818.97 −0.390319
\(280\) 0 0
\(281\) 6015.00 1.27696 0.638479 0.769640i \(-0.279564\pi\)
0.638479 + 0.769640i \(0.279564\pi\)
\(282\) − 6150.76i − 1.29884i
\(283\) 3985.75i 0.837202i 0.908170 + 0.418601i \(0.137479\pi\)
−0.908170 + 0.418601i \(0.862521\pi\)
\(284\) 10394.8 2.17190
\(285\) 0 0
\(286\) −7170.48 −1.48252
\(287\) 2720.32i 0.559497i
\(288\) − 7955.12i − 1.62764i
\(289\) 593.424 0.120787
\(290\) 0 0
\(291\) −155.960 −0.0314177
\(292\) − 505.479i − 0.101305i
\(293\) − 2490.01i − 0.496478i −0.968699 0.248239i \(-0.920148\pi\)
0.968699 0.248239i \(-0.0798518\pi\)
\(294\) −817.548 −0.162178
\(295\) 0 0
\(296\) 1372.03 0.269417
\(297\) − 907.619i − 0.177325i
\(298\) 11141.4i 2.16578i
\(299\) 7956.45 1.53891
\(300\) 0 0
\(301\) −292.701 −0.0560498
\(302\) 15244.0i 2.90461i
\(303\) − 4287.90i − 0.812981i
\(304\) −9297.72 −1.75415
\(305\) 0 0
\(306\) −3289.72 −0.614578
\(307\) 141.853i 0.0263712i 0.999913 + 0.0131856i \(0.00419723\pi\)
−0.999913 + 0.0131856i \(0.995803\pi\)
\(308\) 5395.83i 0.998234i
\(309\) 1302.64 0.239820
\(310\) 0 0
\(311\) 2091.92 0.381420 0.190710 0.981646i \(-0.438921\pi\)
0.190710 + 0.981646i \(0.438921\pi\)
\(312\) 9554.65i 1.73374i
\(313\) 5521.44i 0.997094i 0.866863 + 0.498547i \(0.166133\pi\)
−0.866863 + 0.498547i \(0.833867\pi\)
\(314\) 20267.5 3.64255
\(315\) 0 0
\(316\) 8837.94 1.57333
\(317\) − 5351.63i − 0.948195i −0.880472 0.474097i \(-0.842774\pi\)
0.880472 0.474097i \(-0.157226\pi\)
\(318\) 7644.45i 1.34805i
\(319\) −6359.06 −1.11611
\(320\) 0 0
\(321\) −1998.92 −0.347567
\(322\) − 8076.09i − 1.39771i
\(323\) 2195.14i 0.378145i
\(324\) −1857.40 −0.318484
\(325\) 0 0
\(326\) −15429.2 −2.62131
\(327\) − 3598.53i − 0.608561i
\(328\) − 32270.3i − 5.43241i
\(329\) 2580.53 0.432430
\(330\) 0 0
\(331\) −4383.52 −0.727916 −0.363958 0.931415i \(-0.618575\pi\)
−0.363958 + 0.931415i \(0.618575\pi\)
\(332\) − 544.609i − 0.0900281i
\(333\) − 148.705i − 0.0244714i
\(334\) 6423.11 1.05227
\(335\) 0 0
\(336\) 5845.93 0.949172
\(337\) 7124.57i 1.15163i 0.817579 + 0.575817i \(0.195316\pi\)
−0.817579 + 0.575817i \(0.804684\pi\)
\(338\) 4037.46i 0.649730i
\(339\) −244.551 −0.0391806
\(340\) 0 0
\(341\) −6793.97 −1.07893
\(342\) 1671.78i 0.264327i
\(343\) − 343.000i − 0.0539949i
\(344\) 3472.22 0.544214
\(345\) 0 0
\(346\) 18839.5 2.92722
\(347\) − 507.743i − 0.0785506i −0.999228 0.0392753i \(-0.987495\pi\)
0.999228 0.0392753i \(-0.0125049\pi\)
\(348\) 13013.5i 2.00459i
\(349\) −6155.14 −0.944060 −0.472030 0.881582i \(-0.656479\pi\)
−0.472030 + 0.881582i \(0.656479\pi\)
\(350\) 0 0
\(351\) 1035.56 0.157476
\(352\) − 29712.8i − 4.49915i
\(353\) − 6429.56i − 0.969437i −0.874670 0.484718i \(-0.838922\pi\)
0.874670 0.484718i \(-0.161078\pi\)
\(354\) −4274.87 −0.641828
\(355\) 0 0
\(356\) −34002.1 −5.06211
\(357\) − 1380.19i − 0.204615i
\(358\) − 8916.97i − 1.31641i
\(359\) −10075.4 −1.48123 −0.740614 0.671931i \(-0.765465\pi\)
−0.740614 + 0.671931i \(0.765465\pi\)
\(360\) 0 0
\(361\) −5743.46 −0.837362
\(362\) 3026.71i 0.439448i
\(363\) 602.989i 0.0871865i
\(364\) −6156.46 −0.886501
\(365\) 0 0
\(366\) −2060.74 −0.294308
\(367\) − 816.898i − 0.116190i −0.998311 0.0580950i \(-0.981497\pi\)
0.998311 0.0580950i \(-0.0185026\pi\)
\(368\) 57748.6i 8.18031i
\(369\) −3497.56 −0.493430
\(370\) 0 0
\(371\) −3207.21 −0.448814
\(372\) 13903.5i 1.93781i
\(373\) − 3737.85i − 0.518870i −0.965761 0.259435i \(-0.916464\pi\)
0.965761 0.259435i \(-0.0835364\pi\)
\(374\) −12287.3 −1.69883
\(375\) 0 0
\(376\) −30612.1 −4.19866
\(377\) − 7255.47i − 0.991183i
\(378\) − 1051.13i − 0.143028i
\(379\) −1950.47 −0.264351 −0.132176 0.991226i \(-0.542196\pi\)
−0.132176 + 0.991226i \(0.542196\pi\)
\(380\) 0 0
\(381\) 1010.95 0.135939
\(382\) 16648.2i 2.22983i
\(383\) 6762.06i 0.902155i 0.892485 + 0.451077i \(0.148960\pi\)
−0.892485 + 0.451077i \(0.851040\pi\)
\(384\) −23648.8 −3.14277
\(385\) 0 0
\(386\) −7281.84 −0.960197
\(387\) − 376.330i − 0.0494313i
\(388\) 1192.10i 0.155979i
\(389\) 2551.98 0.332624 0.166312 0.986073i \(-0.446814\pi\)
0.166312 + 0.986073i \(0.446814\pi\)
\(390\) 0 0
\(391\) 13634.1 1.76345
\(392\) 4068.90i 0.524262i
\(393\) − 8793.45i − 1.12868i
\(394\) −6347.83 −0.811672
\(395\) 0 0
\(396\) −6937.50 −0.880360
\(397\) 4097.93i 0.518058i 0.965869 + 0.259029i \(0.0834026\pi\)
−0.965869 + 0.259029i \(0.916597\pi\)
\(398\) − 13182.2i − 1.66021i
\(399\) −701.392 −0.0880038
\(400\) 0 0
\(401\) −1046.81 −0.130362 −0.0651811 0.997873i \(-0.520762\pi\)
−0.0651811 + 0.997873i \(0.520762\pi\)
\(402\) 5610.66i 0.696105i
\(403\) − 7751.68i − 0.958161i
\(404\) −32775.1 −4.03619
\(405\) 0 0
\(406\) −7364.57 −0.900240
\(407\) − 555.420i − 0.0676441i
\(408\) 16372.8i 1.98670i
\(409\) −6516.92 −0.787876 −0.393938 0.919137i \(-0.628887\pi\)
−0.393938 + 0.919137i \(0.628887\pi\)
\(410\) 0 0
\(411\) 4755.20 0.570697
\(412\) − 9956.86i − 1.19063i
\(413\) − 1793.51i − 0.213687i
\(414\) 10383.5 1.23267
\(415\) 0 0
\(416\) 33901.3 3.99555
\(417\) − 3896.55i − 0.457590i
\(418\) 6244.21i 0.730656i
\(419\) −12279.1 −1.43168 −0.715838 0.698267i \(-0.753955\pi\)
−0.715838 + 0.698267i \(0.753955\pi\)
\(420\) 0 0
\(421\) 10146.9 1.17465 0.587325 0.809351i \(-0.300181\pi\)
0.587325 + 0.809351i \(0.300181\pi\)
\(422\) − 3821.67i − 0.440844i
\(423\) 3317.83i 0.381367i
\(424\) 38046.1 4.35774
\(425\) 0 0
\(426\) −7563.36 −0.860202
\(427\) − 864.579i − 0.0979858i
\(428\) 15279.0i 1.72556i
\(429\) 3867.89 0.435299
\(430\) 0 0
\(431\) 7059.04 0.788914 0.394457 0.918914i \(-0.370933\pi\)
0.394457 + 0.918914i \(0.370933\pi\)
\(432\) 7516.20i 0.837091i
\(433\) 6468.98i 0.717966i 0.933344 + 0.358983i \(0.116876\pi\)
−0.933344 + 0.358983i \(0.883124\pi\)
\(434\) −7868.24 −0.870248
\(435\) 0 0
\(436\) −27505.8 −3.02131
\(437\) − 6928.65i − 0.758449i
\(438\) 367.790i 0.0401226i
\(439\) −4767.13 −0.518275 −0.259137 0.965840i \(-0.583438\pi\)
−0.259137 + 0.965840i \(0.583438\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) − 14019.4i − 1.50868i
\(443\) 2366.55i 0.253810i 0.991915 + 0.126905i \(0.0405044\pi\)
−0.991915 + 0.126905i \(0.959496\pi\)
\(444\) −1136.64 −0.121492
\(445\) 0 0
\(446\) −5508.70 −0.584853
\(447\) − 6009.86i − 0.635921i
\(448\) − 18821.9i − 1.98494i
\(449\) −1814.17 −0.190681 −0.0953406 0.995445i \(-0.530394\pi\)
−0.0953406 + 0.995445i \(0.530394\pi\)
\(450\) 0 0
\(451\) −13063.6 −1.36395
\(452\) 1869.26i 0.194519i
\(453\) − 8222.87i − 0.852857i
\(454\) 8226.93 0.850460
\(455\) 0 0
\(456\) 8320.40 0.854470
\(457\) − 8284.13i − 0.847955i −0.905673 0.423977i \(-0.860634\pi\)
0.905673 0.423977i \(-0.139366\pi\)
\(458\) − 37287.3i − 3.80419i
\(459\) 1774.53 0.180453
\(460\) 0 0
\(461\) 1384.62 0.139888 0.0699439 0.997551i \(-0.477718\pi\)
0.0699439 + 0.997551i \(0.477718\pi\)
\(462\) − 3926.05i − 0.395360i
\(463\) − 13210.3i − 1.32599i −0.748624 0.662994i \(-0.769285\pi\)
0.748624 0.662994i \(-0.230715\pi\)
\(464\) 52660.9 5.26879
\(465\) 0 0
\(466\) 9727.67 0.967008
\(467\) 4574.24i 0.453256i 0.973981 + 0.226628i \(0.0727701\pi\)
−0.973981 + 0.226628i \(0.927230\pi\)
\(468\) − 7915.45i − 0.781820i
\(469\) −2353.94 −0.231758
\(470\) 0 0
\(471\) −10932.7 −1.06953
\(472\) 21275.9i 2.07479i
\(473\) − 1405.61i − 0.136639i
\(474\) −6430.55 −0.623132
\(475\) 0 0
\(476\) −10549.7 −1.01585
\(477\) − 4123.55i − 0.395816i
\(478\) 35154.0i 3.36382i
\(479\) 11031.8 1.05231 0.526154 0.850389i \(-0.323634\pi\)
0.526154 + 0.850389i \(0.323634\pi\)
\(480\) 0 0
\(481\) 633.716 0.0600726
\(482\) 18684.7i 1.76569i
\(483\) 4356.39i 0.410398i
\(484\) 4609.02 0.432853
\(485\) 0 0
\(486\) 1351.46 0.126139
\(487\) 5194.06i 0.483296i 0.970364 + 0.241648i \(0.0776880\pi\)
−0.970364 + 0.241648i \(0.922312\pi\)
\(488\) 10256.2i 0.951389i
\(489\) 8322.81 0.769673
\(490\) 0 0
\(491\) 11954.7 1.09880 0.549398 0.835561i \(-0.314857\pi\)
0.549398 + 0.835561i \(0.314857\pi\)
\(492\) 26734.0i 2.44972i
\(493\) − 12432.9i − 1.13580i
\(494\) −7124.43 −0.648873
\(495\) 0 0
\(496\) 56262.4 5.09326
\(497\) − 3173.19i − 0.286392i
\(498\) 396.262i 0.0356564i
\(499\) −2566.05 −0.230205 −0.115102 0.993354i \(-0.536720\pi\)
−0.115102 + 0.993354i \(0.536720\pi\)
\(500\) 0 0
\(501\) −3464.74 −0.308968
\(502\) − 7399.62i − 0.657891i
\(503\) 21103.5i 1.87069i 0.353731 + 0.935347i \(0.384913\pi\)
−0.353731 + 0.935347i \(0.615087\pi\)
\(504\) −5231.45 −0.462355
\(505\) 0 0
\(506\) 38783.1 3.40735
\(507\) − 2177.87i − 0.190775i
\(508\) − 7727.36i − 0.674894i
\(509\) 781.732 0.0680740 0.0340370 0.999421i \(-0.489164\pi\)
0.0340370 + 0.999421i \(0.489164\pi\)
\(510\) 0 0
\(511\) −154.305 −0.0133582
\(512\) 61129.5i 5.27650i
\(513\) − 901.790i − 0.0776121i
\(514\) −13775.7 −1.18214
\(515\) 0 0
\(516\) −2876.52 −0.245411
\(517\) 12392.3i 1.05418i
\(518\) − 643.244i − 0.0545609i
\(519\) −10162.4 −0.859497
\(520\) 0 0
\(521\) 14013.0 1.17835 0.589176 0.808005i \(-0.299453\pi\)
0.589176 + 0.808005i \(0.299453\pi\)
\(522\) − 9468.73i − 0.793937i
\(523\) − 10310.7i − 0.862052i −0.902339 0.431026i \(-0.858152\pi\)
0.902339 0.431026i \(-0.141848\pi\)
\(524\) −67213.8 −5.60353
\(525\) 0 0
\(526\) 28656.3 2.37542
\(527\) − 13283.2i − 1.09796i
\(528\) 28073.5i 2.31390i
\(529\) −30867.2 −2.53696
\(530\) 0 0
\(531\) 2305.94 0.188455
\(532\) 5361.18i 0.436911i
\(533\) − 14905.1i − 1.21128i
\(534\) 24740.2 2.00489
\(535\) 0 0
\(536\) 27924.0 2.25025
\(537\) 4809.97i 0.386528i
\(538\) 6401.16i 0.512962i
\(539\) 1647.16 0.131629
\(540\) 0 0
\(541\) −17562.9 −1.39572 −0.697862 0.716232i \(-0.745865\pi\)
−0.697862 + 0.716232i \(0.745865\pi\)
\(542\) 10223.9i 0.810250i
\(543\) − 1632.66i − 0.129031i
\(544\) 58093.1 4.57853
\(545\) 0 0
\(546\) 4479.48 0.351107
\(547\) 19889.6i 1.55469i 0.629072 + 0.777347i \(0.283435\pi\)
−0.629072 + 0.777347i \(0.716565\pi\)
\(548\) − 36346.9i − 2.83333i
\(549\) 1111.60 0.0864153
\(550\) 0 0
\(551\) −6318.22 −0.488503
\(552\) − 51678.5i − 3.98475i
\(553\) − 2697.92i − 0.207463i
\(554\) 3161.45 0.242450
\(555\) 0 0
\(556\) −29783.8 −2.27179
\(557\) − 5579.54i − 0.424439i −0.977222 0.212219i \(-0.931931\pi\)
0.977222 0.212219i \(-0.0680692\pi\)
\(558\) − 10116.3i − 0.767486i
\(559\) 1603.76 0.121345
\(560\) 0 0
\(561\) 6627.99 0.498813
\(562\) 33452.8i 2.51089i
\(563\) − 24463.2i − 1.83126i −0.402022 0.915630i \(-0.631693\pi\)
0.402022 0.915630i \(-0.368307\pi\)
\(564\) 25360.2 1.89337
\(565\) 0 0
\(566\) −22167.0 −1.64620
\(567\) 567.000i 0.0419961i
\(568\) 37642.5i 2.78071i
\(569\) 8582.14 0.632306 0.316153 0.948708i \(-0.397609\pi\)
0.316153 + 0.948708i \(0.397609\pi\)
\(570\) 0 0
\(571\) 17580.8 1.28850 0.644248 0.764816i \(-0.277170\pi\)
0.644248 + 0.764816i \(0.277170\pi\)
\(572\) − 29564.7i − 2.16112i
\(573\) − 8980.32i − 0.654727i
\(574\) −15129.2 −1.10014
\(575\) 0 0
\(576\) 24199.6 1.75055
\(577\) 8692.57i 0.627169i 0.949560 + 0.313585i \(0.101530\pi\)
−0.949560 + 0.313585i \(0.898470\pi\)
\(578\) 3300.36i 0.237503i
\(579\) 3927.95 0.281935
\(580\) 0 0
\(581\) −166.250 −0.0118713
\(582\) − 867.381i − 0.0617768i
\(583\) − 15401.7i − 1.09412i
\(584\) 1830.47 0.129701
\(585\) 0 0
\(586\) 13848.3 0.976227
\(587\) − 3584.61i − 0.252049i −0.992027 0.126024i \(-0.959778\pi\)
0.992027 0.126024i \(-0.0402217\pi\)
\(588\) − 3370.84i − 0.236413i
\(589\) −6750.33 −0.472228
\(590\) 0 0
\(591\) 3424.13 0.238325
\(592\) 4599.56i 0.319326i
\(593\) − 21853.6i − 1.51335i −0.653790 0.756676i \(-0.726822\pi\)
0.653790 0.756676i \(-0.273178\pi\)
\(594\) 5047.77 0.348674
\(595\) 0 0
\(596\) −45937.1 −3.15714
\(597\) 7110.69i 0.487472i
\(598\) 44250.2i 3.02596i
\(599\) −9090.48 −0.620078 −0.310039 0.950724i \(-0.600342\pi\)
−0.310039 + 0.950724i \(0.600342\pi\)
\(600\) 0 0
\(601\) −19546.1 −1.32663 −0.663314 0.748341i \(-0.730851\pi\)
−0.663314 + 0.748341i \(0.730851\pi\)
\(602\) − 1627.87i − 0.110211i
\(603\) − 3026.49i − 0.204392i
\(604\) −62852.6 −4.23416
\(605\) 0 0
\(606\) 23847.4 1.59857
\(607\) 15726.0i 1.05157i 0.850619 + 0.525783i \(0.176227\pi\)
−0.850619 + 0.525783i \(0.823773\pi\)
\(608\) − 29522.0i − 1.96920i
\(609\) 3972.58 0.264330
\(610\) 0 0
\(611\) −14139.2 −0.936186
\(612\) − 13563.9i − 0.895894i
\(613\) − 13572.5i − 0.894269i −0.894467 0.447135i \(-0.852444\pi\)
0.894467 0.447135i \(-0.147556\pi\)
\(614\) −788.921 −0.0518538
\(615\) 0 0
\(616\) −19539.8 −1.27805
\(617\) − 17378.5i − 1.13393i −0.823743 0.566964i \(-0.808118\pi\)
0.823743 0.566964i \(-0.191882\pi\)
\(618\) 7244.68i 0.471560i
\(619\) 25113.3 1.63068 0.815338 0.578985i \(-0.196551\pi\)
0.815338 + 0.578985i \(0.196551\pi\)
\(620\) 0 0
\(621\) −5601.07 −0.361937
\(622\) 11634.3i 0.749989i
\(623\) 10379.7i 0.667501i
\(624\) −32030.8 −2.05490
\(625\) 0 0
\(626\) −30707.8 −1.96059
\(627\) − 3368.24i − 0.214537i
\(628\) 83565.1i 5.30989i
\(629\) 1085.93 0.0688377
\(630\) 0 0
\(631\) −10814.4 −0.682276 −0.341138 0.940013i \(-0.610812\pi\)
−0.341138 + 0.940013i \(0.610812\pi\)
\(632\) 32004.5i 2.01436i
\(633\) 2061.48i 0.129441i
\(634\) 29763.4 1.86444
\(635\) 0 0
\(636\) −31518.9 −1.96510
\(637\) 1879.35i 0.116896i
\(638\) − 35366.3i − 2.19461i
\(639\) 4079.81 0.252574
\(640\) 0 0
\(641\) −16359.0 −1.00802 −0.504010 0.863698i \(-0.668143\pi\)
−0.504010 + 0.863698i \(0.668143\pi\)
\(642\) − 11117.1i − 0.683422i
\(643\) 8819.47i 0.540911i 0.962732 + 0.270456i \(0.0871744\pi\)
−0.962732 + 0.270456i \(0.912826\pi\)
\(644\) 33298.6 2.03750
\(645\) 0 0
\(646\) −12208.4 −0.743549
\(647\) 13828.8i 0.840290i 0.907457 + 0.420145i \(0.138021\pi\)
−0.907457 + 0.420145i \(0.861979\pi\)
\(648\) − 6726.14i − 0.407759i
\(649\) 8612.83 0.520930
\(650\) 0 0
\(651\) 4244.27 0.255524
\(652\) − 63616.4i − 3.82118i
\(653\) 23988.7i 1.43760i 0.695219 + 0.718798i \(0.255307\pi\)
−0.695219 + 0.718798i \(0.744693\pi\)
\(654\) 20013.4 1.19662
\(655\) 0 0
\(656\) 108182. 6.43874
\(657\) − 198.392i − 0.0117809i
\(658\) 14351.8i 0.850289i
\(659\) −3109.28 −0.183794 −0.0918972 0.995769i \(-0.529293\pi\)
−0.0918972 + 0.995769i \(0.529293\pi\)
\(660\) 0 0
\(661\) 22695.0 1.33545 0.667726 0.744407i \(-0.267268\pi\)
0.667726 + 0.744407i \(0.267268\pi\)
\(662\) − 24379.2i − 1.43131i
\(663\) 7562.31i 0.442980i
\(664\) 1972.18 0.115264
\(665\) 0 0
\(666\) 827.028 0.0481182
\(667\) 39242.8i 2.27809i
\(668\) 26483.2i 1.53393i
\(669\) 2971.49 0.171726
\(670\) 0 0
\(671\) 4151.90 0.238871
\(672\) 18562.0i 1.06554i
\(673\) 22073.4i 1.26429i 0.774850 + 0.632145i \(0.217825\pi\)
−0.774850 + 0.632145i \(0.782175\pi\)
\(674\) −39623.7 −2.26446
\(675\) 0 0
\(676\) −16646.9 −0.947136
\(677\) 2489.50i 0.141328i 0.997500 + 0.0706642i \(0.0225119\pi\)
−0.997500 + 0.0706642i \(0.977488\pi\)
\(678\) − 1360.09i − 0.0770410i
\(679\) 363.907 0.0205677
\(680\) 0 0
\(681\) −4437.75 −0.249714
\(682\) − 37785.0i − 2.12150i
\(683\) 7970.98i 0.446561i 0.974754 + 0.223280i \(0.0716765\pi\)
−0.974754 + 0.223280i \(0.928323\pi\)
\(684\) −6892.94 −0.385319
\(685\) 0 0
\(686\) 1907.61 0.106171
\(687\) 20113.4i 1.11699i
\(688\) 11640.2i 0.645027i
\(689\) 17572.8 0.971656
\(690\) 0 0
\(691\) −23892.7 −1.31537 −0.657687 0.753292i \(-0.728465\pi\)
−0.657687 + 0.753292i \(0.728465\pi\)
\(692\) 77677.4i 4.26712i
\(693\) 2117.78i 0.116086i
\(694\) 2823.84 0.154454
\(695\) 0 0
\(696\) −47125.5 −2.56650
\(697\) − 25541.3i − 1.38801i
\(698\) − 34232.1i − 1.85631i
\(699\) −5247.28 −0.283935
\(700\) 0 0
\(701\) 12197.0 0.657170 0.328585 0.944474i \(-0.393428\pi\)
0.328585 + 0.944474i \(0.393428\pi\)
\(702\) 5759.33i 0.309647i
\(703\) − 551.853i − 0.0296067i
\(704\) 90387.0 4.83891
\(705\) 0 0
\(706\) 35758.4 1.90621
\(707\) 10005.1i 0.532221i
\(708\) − 17625.8i − 0.935617i
\(709\) 8982.28 0.475792 0.237896 0.971291i \(-0.423542\pi\)
0.237896 + 0.971291i \(0.423542\pi\)
\(710\) 0 0
\(711\) 3468.75 0.182965
\(712\) − 123131.i − 6.48107i
\(713\) 41926.7i 2.20220i
\(714\) 7676.02 0.402336
\(715\) 0 0
\(716\) 36765.6 1.91899
\(717\) − 18962.7i − 0.987690i
\(718\) − 56035.0i − 2.91255i
\(719\) 6501.61 0.337231 0.168616 0.985682i \(-0.446070\pi\)
0.168616 + 0.985682i \(0.446070\pi\)
\(720\) 0 0
\(721\) −3039.49 −0.156999
\(722\) − 31942.6i − 1.64651i
\(723\) − 10078.9i − 0.518446i
\(724\) −12479.4 −0.640600
\(725\) 0 0
\(726\) −3353.55 −0.171435
\(727\) 24228.7i 1.23603i 0.786168 + 0.618013i \(0.212062\pi\)
−0.786168 + 0.618013i \(0.787938\pi\)
\(728\) − 22294.2i − 1.13500i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 2748.19 0.139050
\(732\) − 8496.67i − 0.429024i
\(733\) 36719.1i 1.85027i 0.379636 + 0.925136i \(0.376049\pi\)
−0.379636 + 0.925136i \(0.623951\pi\)
\(734\) 4543.22 0.228465
\(735\) 0 0
\(736\) −183363. −9.18321
\(737\) − 11304.1i − 0.564983i
\(738\) − 19451.8i − 0.970234i
\(739\) −23304.5 −1.16004 −0.580021 0.814602i \(-0.696956\pi\)
−0.580021 + 0.814602i \(0.696956\pi\)
\(740\) 0 0
\(741\) 3843.04 0.190523
\(742\) − 17837.0i − 0.882505i
\(743\) 6875.35i 0.339478i 0.985489 + 0.169739i \(0.0542924\pi\)
−0.985489 + 0.169739i \(0.945708\pi\)
\(744\) −50348.4 −2.48100
\(745\) 0 0
\(746\) 20788.3 1.02026
\(747\) − 213.751i − 0.0104695i
\(748\) − 50661.8i − 2.47644i
\(749\) 4664.15 0.227536
\(750\) 0 0
\(751\) 1182.65 0.0574640 0.0287320 0.999587i \(-0.490853\pi\)
0.0287320 + 0.999587i \(0.490853\pi\)
\(752\) − 102623.i − 4.97645i
\(753\) 3991.49i 0.193171i
\(754\) 40351.7 1.94897
\(755\) 0 0
\(756\) 4333.93 0.208497
\(757\) − 25226.8i − 1.21121i −0.795766 0.605604i \(-0.792932\pi\)
0.795766 0.605604i \(-0.207068\pi\)
\(758\) − 10847.7i − 0.519795i
\(759\) −20920.3 −1.00047
\(760\) 0 0
\(761\) −10909.2 −0.519655 −0.259827 0.965655i \(-0.583666\pi\)
−0.259827 + 0.965655i \(0.583666\pi\)
\(762\) 5622.48i 0.267298i
\(763\) 8396.58i 0.398397i
\(764\) −68642.2 −3.25051
\(765\) 0 0
\(766\) −37607.6 −1.77391
\(767\) 9826.95i 0.462621i
\(768\) − 66991.9i − 3.14761i
\(769\) 12771.6 0.598903 0.299451 0.954112i \(-0.403196\pi\)
0.299451 + 0.954112i \(0.403196\pi\)
\(770\) 0 0
\(771\) 7430.84 0.347101
\(772\) − 30023.8i − 1.39972i
\(773\) 2199.06i 0.102322i 0.998690 + 0.0511610i \(0.0162922\pi\)
−0.998690 + 0.0511610i \(0.983708\pi\)
\(774\) 2092.98 0.0971971
\(775\) 0 0
\(776\) −4316.92 −0.199701
\(777\) 346.977i 0.0160203i
\(778\) 14193.0i 0.654041i
\(779\) −12979.7 −0.596977
\(780\) 0 0
\(781\) 15238.3 0.698170
\(782\) 75827.0i 3.46748i
\(783\) 5107.60i 0.233117i
\(784\) −13640.5 −0.621379
\(785\) 0 0
\(786\) 48905.3 2.21933
\(787\) 19587.7i 0.887201i 0.896225 + 0.443601i \(0.146299\pi\)
−0.896225 + 0.443601i \(0.853701\pi\)
\(788\) − 26172.8i − 1.18321i
\(789\) −15457.7 −0.697476
\(790\) 0 0
\(791\) 570.620 0.0256497
\(792\) − 25122.5i − 1.12713i
\(793\) 4737.17i 0.212134i
\(794\) −22790.9 −1.01866
\(795\) 0 0
\(796\) 54351.4 2.42014
\(797\) − 21699.3i − 0.964401i −0.876061 0.482200i \(-0.839838\pi\)
0.876061 0.482200i \(-0.160162\pi\)
\(798\) − 3900.83i − 0.173042i
\(799\) −24228.8 −1.07278
\(800\) 0 0
\(801\) −13345.3 −0.588680
\(802\) − 5821.89i − 0.256332i
\(803\) − 741.007i − 0.0325649i
\(804\) −23133.3 −1.01474
\(805\) 0 0
\(806\) 43111.4 1.88404
\(807\) − 3452.90i − 0.150617i
\(808\) − 118687.i − 5.16758i
\(809\) 15649.4 0.680103 0.340051 0.940407i \(-0.389556\pi\)
0.340051 + 0.940407i \(0.389556\pi\)
\(810\) 0 0
\(811\) −33267.3 −1.44041 −0.720206 0.693760i \(-0.755953\pi\)
−0.720206 + 0.693760i \(0.755953\pi\)
\(812\) − 30364.9i − 1.31231i
\(813\) − 5514.97i − 0.237907i
\(814\) 3089.00 0.133009
\(815\) 0 0
\(816\) −54887.9 −2.35473
\(817\) − 1396.59i − 0.0598046i
\(818\) − 36244.2i − 1.54920i
\(819\) −2416.31 −0.103093
\(820\) 0 0
\(821\) 5158.58 0.219288 0.109644 0.993971i \(-0.465029\pi\)
0.109644 + 0.993971i \(0.465029\pi\)
\(822\) 26446.3i 1.12217i
\(823\) 26333.6i 1.11535i 0.830060 + 0.557674i \(0.188306\pi\)
−0.830060 + 0.557674i \(0.811694\pi\)
\(824\) 36056.5 1.52438
\(825\) 0 0
\(826\) 9974.71 0.420175
\(827\) − 19572.7i − 0.822988i −0.911413 0.411494i \(-0.865007\pi\)
0.911413 0.411494i \(-0.134993\pi\)
\(828\) 42812.5i 1.79690i
\(829\) −9642.26 −0.403968 −0.201984 0.979389i \(-0.564739\pi\)
−0.201984 + 0.979389i \(0.564739\pi\)
\(830\) 0 0
\(831\) −1705.34 −0.0711885
\(832\) 103128.i 4.29728i
\(833\) 3220.45i 0.133952i
\(834\) 21670.9 0.899761
\(835\) 0 0
\(836\) −25745.5 −1.06511
\(837\) 5456.91i 0.225351i
\(838\) − 68290.7i − 2.81511i
\(839\) 31081.1 1.27895 0.639475 0.768812i \(-0.279152\pi\)
0.639475 + 0.768812i \(0.279152\pi\)
\(840\) 0 0
\(841\) 11396.5 0.467278
\(842\) 56432.3i 2.30972i
\(843\) − 18045.0i − 0.737252i
\(844\) 15757.1 0.642634
\(845\) 0 0
\(846\) −18452.3 −0.749885
\(847\) − 1406.97i − 0.0570770i
\(848\) 127545.i 5.16499i
\(849\) 11957.2 0.483359
\(850\) 0 0
\(851\) −3427.59 −0.138068
\(852\) − 31184.5i − 1.25395i
\(853\) 25780.9i 1.03484i 0.855731 + 0.517421i \(0.173108\pi\)
−0.855731 + 0.517421i \(0.826892\pi\)
\(854\) 4808.40 0.192670
\(855\) 0 0
\(856\) −55329.3 −2.20925
\(857\) − 14452.6i − 0.576069i −0.957620 0.288035i \(-0.906998\pi\)
0.957620 0.288035i \(-0.0930018\pi\)
\(858\) 21511.5i 0.855931i
\(859\) −889.366 −0.0353257 −0.0176628 0.999844i \(-0.505623\pi\)
−0.0176628 + 0.999844i \(0.505623\pi\)
\(860\) 0 0
\(861\) 8160.97 0.323025
\(862\) 39259.2i 1.55125i
\(863\) 41460.3i 1.63537i 0.575665 + 0.817685i \(0.304743\pi\)
−0.575665 + 0.817685i \(0.695257\pi\)
\(864\) −23865.4 −0.939718
\(865\) 0 0
\(866\) −35977.6 −1.41174
\(867\) − 1780.27i − 0.0697361i
\(868\) − 32441.6i − 1.26859i
\(869\) 12956.0 0.505756
\(870\) 0 0
\(871\) 12897.6 0.501744
\(872\) − 99606.0i − 3.86822i
\(873\) 467.880i 0.0181390i
\(874\) 38534.1 1.49134
\(875\) 0 0
\(876\) −1516.44 −0.0584882
\(877\) − 21173.0i − 0.815236i −0.913153 0.407618i \(-0.866360\pi\)
0.913153 0.407618i \(-0.133640\pi\)
\(878\) − 26512.6i − 1.01909i
\(879\) −7470.03 −0.286642
\(880\) 0 0
\(881\) 9883.39 0.377957 0.188978 0.981981i \(-0.439482\pi\)
0.188978 + 0.981981i \(0.439482\pi\)
\(882\) 2452.64i 0.0936336i
\(883\) − 45273.9i − 1.72547i −0.505658 0.862734i \(-0.668750\pi\)
0.505658 0.862734i \(-0.331250\pi\)
\(884\) 57803.4 2.19925
\(885\) 0 0
\(886\) −13161.7 −0.499069
\(887\) − 644.388i − 0.0243928i −0.999926 0.0121964i \(-0.996118\pi\)
0.999926 0.0121964i \(-0.00388233\pi\)
\(888\) − 4116.08i − 0.155548i
\(889\) −2358.89 −0.0889930
\(890\) 0 0
\(891\) −2722.86 −0.102378
\(892\) − 22712.9i − 0.852562i
\(893\) 12312.7i 0.461398i
\(894\) 33424.2 1.25042
\(895\) 0 0
\(896\) 55180.6 2.05743
\(897\) − 23869.4i − 0.888489i
\(898\) − 10089.6i − 0.374937i
\(899\) 38232.8 1.41839
\(900\) 0 0
\(901\) 30112.7 1.11343
\(902\) − 72653.8i − 2.68194i
\(903\) 878.103i 0.0323604i
\(904\) −6769.09 −0.249045
\(905\) 0 0
\(906\) 45732.0 1.67698
\(907\) 15065.2i 0.551522i 0.961226 + 0.275761i \(0.0889299\pi\)
−0.961226 + 0.275761i \(0.911070\pi\)
\(908\) 33920.5i 1.23975i
\(909\) −12863.7 −0.469375
\(910\) 0 0
\(911\) 28789.9 1.04704 0.523520 0.852014i \(-0.324619\pi\)
0.523520 + 0.852014i \(0.324619\pi\)
\(912\) 27893.1i 1.01276i
\(913\) − 798.371i − 0.0289400i
\(914\) 46072.6 1.66734
\(915\) 0 0
\(916\) 153739. 5.54552
\(917\) 20518.1i 0.738894i
\(918\) 9869.16i 0.354827i
\(919\) 24163.8 0.867345 0.433673 0.901070i \(-0.357217\pi\)
0.433673 + 0.901070i \(0.357217\pi\)
\(920\) 0 0
\(921\) 425.558 0.0152254
\(922\) 7700.65i 0.275062i
\(923\) 17386.4i 0.620023i
\(924\) 16187.5 0.576331
\(925\) 0 0
\(926\) 73469.5 2.60730
\(927\) − 3907.91i − 0.138460i
\(928\) 167208.i 5.91474i
\(929\) 35115.4 1.24015 0.620075 0.784542i \(-0.287102\pi\)
0.620075 + 0.784542i \(0.287102\pi\)
\(930\) 0 0
\(931\) 1636.58 0.0576120
\(932\) 40108.2i 1.40964i
\(933\) − 6275.75i − 0.220213i
\(934\) −25439.9 −0.891239
\(935\) 0 0
\(936\) 28664.0 1.00097
\(937\) 15512.6i 0.540849i 0.962741 + 0.270424i \(0.0871640\pi\)
−0.962741 + 0.270424i \(0.912836\pi\)
\(938\) − 13091.5i − 0.455708i
\(939\) 16564.3 0.575672
\(940\) 0 0
\(941\) −53283.8 −1.84591 −0.922956 0.384905i \(-0.874234\pi\)
−0.922956 + 0.384905i \(0.874234\pi\)
\(942\) − 60802.5i − 2.10303i
\(943\) 80617.5i 2.78395i
\(944\) −71324.8 −2.45914
\(945\) 0 0
\(946\) 7817.39 0.268674
\(947\) 55509.7i 1.90478i 0.304890 + 0.952388i \(0.401380\pi\)
−0.304890 + 0.952388i \(0.598620\pi\)
\(948\) − 26513.8i − 0.908364i
\(949\) 845.464 0.0289198
\(950\) 0 0
\(951\) −16054.9 −0.547441
\(952\) − 38203.2i − 1.30060i
\(953\) − 28080.6i − 0.954480i −0.878773 0.477240i \(-0.841637\pi\)
0.878773 0.477240i \(-0.158363\pi\)
\(954\) 22933.3 0.778296
\(955\) 0 0
\(956\) −144943. −4.90356
\(957\) 19077.2i 0.644387i
\(958\) 61353.9i 2.06916i
\(959\) −11095.5 −0.373609
\(960\) 0 0
\(961\) 11056.6 0.371140
\(962\) 3524.44i 0.118121i
\(963\) 5996.76i 0.200668i
\(964\) −77039.0 −2.57392
\(965\) 0 0
\(966\) −24228.3 −0.806969
\(967\) − 56609.3i − 1.88256i −0.337634 0.941278i \(-0.609626\pi\)
0.337634 0.941278i \(-0.390374\pi\)
\(968\) 16690.5i 0.554187i
\(969\) 6585.42 0.218322
\(970\) 0 0
\(971\) −6782.17 −0.224151 −0.112075 0.993700i \(-0.535750\pi\)
−0.112075 + 0.993700i \(0.535750\pi\)
\(972\) 5572.20i 0.183877i
\(973\) 9091.95i 0.299563i
\(974\) −28887.0 −0.950309
\(975\) 0 0
\(976\) −34382.8 −1.12763
\(977\) 45655.0i 1.49502i 0.664252 + 0.747509i \(0.268750\pi\)
−0.664252 + 0.747509i \(0.731250\pi\)
\(978\) 46287.7i 1.51341i
\(979\) −49845.5 −1.62724
\(980\) 0 0
\(981\) −10795.6 −0.351353
\(982\) 66486.8i 2.16057i
\(983\) − 10102.3i − 0.327785i −0.986478 0.163893i \(-0.947595\pi\)
0.986478 0.163893i \(-0.0524050\pi\)
\(984\) −96811.0 −3.13640
\(985\) 0 0
\(986\) 69146.4 2.23334
\(987\) − 7741.60i − 0.249664i
\(988\) − 29374.8i − 0.945887i
\(989\) −8674.27 −0.278894
\(990\) 0 0
\(991\) 25416.2 0.814705 0.407353 0.913271i \(-0.366452\pi\)
0.407353 + 0.913271i \(0.366452\pi\)
\(992\) 178644.i 5.71768i
\(993\) 13150.6i 0.420262i
\(994\) 17647.8 0.563135
\(995\) 0 0
\(996\) −1633.83 −0.0519777
\(997\) − 48152.5i − 1.52959i −0.644271 0.764797i \(-0.722839\pi\)
0.644271 0.764797i \(-0.277161\pi\)
\(998\) − 14271.2i − 0.452653i
\(999\) −446.114 −0.0141285
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.i.274.4 4
5.2 odd 4 105.4.a.c.1.1 2
5.3 odd 4 525.4.a.p.1.2 2
5.4 even 2 inner 525.4.d.i.274.1 4
15.2 even 4 315.4.a.m.1.2 2
15.8 even 4 1575.4.a.m.1.1 2
20.7 even 4 1680.4.a.bk.1.2 2
35.27 even 4 735.4.a.k.1.1 2
105.62 odd 4 2205.4.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.c.1.1 2 5.2 odd 4
315.4.a.m.1.2 2 15.2 even 4
525.4.a.p.1.2 2 5.3 odd 4
525.4.d.i.274.1 4 5.4 even 2 inner
525.4.d.i.274.4 4 1.1 even 1 trivial
735.4.a.k.1.1 2 35.27 even 4
1575.4.a.m.1.1 2 15.8 even 4
1680.4.a.bk.1.2 2 20.7 even 4
2205.4.a.bh.1.2 2 105.62 odd 4