Properties

Label 354.5.d.a.235.18
Level $354$
Weight $5$
Character 354.235
Analytic conductor $36.593$
Analytic rank $0$
Dimension $40$
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] = [354,5,Mod(235,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.235");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 354.d (of order \(2\), degree \(1\), 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: \(36.5929669317\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 235.18
Character \(\chi\) \(=\) 354.235
Dual form 354.5.d.a.235.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} -4.63644 q^{5} +14.6969i q^{6} -33.3282 q^{7} -22.6274i q^{8} +27.0000 q^{9} +O(q^{10})\) \(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} -4.63644 q^{5} +14.6969i q^{6} -33.3282 q^{7} -22.6274i q^{8} +27.0000 q^{9} -13.1138i q^{10} -4.16353i q^{11} -41.5692 q^{12} +241.996i q^{13} -94.2665i q^{14} -24.0916 q^{15} +64.0000 q^{16} +291.659 q^{17} +76.3675i q^{18} -369.712 q^{19} +37.0915 q^{20} -173.179 q^{21} +11.7762 q^{22} -496.520i q^{23} -117.576i q^{24} -603.503 q^{25} -684.469 q^{26} +140.296 q^{27} +266.626 q^{28} -329.812 q^{29} -68.1414i q^{30} -1480.23i q^{31} +181.019i q^{32} -21.6343i q^{33} +824.936i q^{34} +154.524 q^{35} -216.000 q^{36} -727.964i q^{37} -1045.70i q^{38} +1257.45i q^{39} +104.911i q^{40} -1746.67 q^{41} -489.823i q^{42} -1687.92i q^{43} +33.3082i q^{44} -125.184 q^{45} +1404.37 q^{46} -2187.33i q^{47} +332.554 q^{48} -1290.23 q^{49} -1706.97i q^{50} +1515.50 q^{51} -1935.97i q^{52} -2224.64 q^{53} +396.817i q^{54} +19.3039i q^{55} +754.132i q^{56} -1921.08 q^{57} -932.848i q^{58} +(-3416.38 - 667.631i) q^{59} +192.733 q^{60} +4206.97i q^{61} +4186.72 q^{62} -899.863 q^{63} -512.000 q^{64} -1122.00i q^{65} +61.1911 q^{66} -279.854i q^{67} -2333.27 q^{68} -2580.00i q^{69} +437.061i q^{70} -2456.24 q^{71} -610.940i q^{72} -2001.19i q^{73} +2058.99 q^{74} -3135.90 q^{75} +2957.70 q^{76} +138.763i q^{77} -3556.61 q^{78} +3622.55 q^{79} -296.732 q^{80} +729.000 q^{81} -4940.33i q^{82} +2780.03i q^{83} +1385.43 q^{84} -1352.26 q^{85} +4774.16 q^{86} -1713.75 q^{87} -94.2099 q^{88} -4088.90i q^{89} -354.073i q^{90} -8065.31i q^{91} +3972.16i q^{92} -7691.49i q^{93} +6186.70 q^{94} +1714.15 q^{95} +940.604i q^{96} -14089.3i q^{97} -3649.32i q^{98} -112.415i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 0.707107i
\(3\) 5.19615 0.577350
\(4\) −8.00000 −0.500000
\(5\) −4.63644 −0.185457 −0.0927287 0.995691i \(-0.529559\pi\)
−0.0927287 + 0.995691i \(0.529559\pi\)
\(6\) 14.6969i 0.408248i
\(7\) −33.3282 −0.680168 −0.340084 0.940395i \(-0.610456\pi\)
−0.340084 + 0.940395i \(0.610456\pi\)
\(8\) 22.6274i 0.353553i
\(9\) 27.0000 0.333333
\(10\) 13.1138i 0.131138i
\(11\) 4.16353i 0.0344093i −0.999852 0.0172047i \(-0.994523\pi\)
0.999852 0.0172047i \(-0.00547669\pi\)
\(12\) −41.5692 −0.288675
\(13\) 241.996i 1.43193i 0.698136 + 0.715965i \(0.254013\pi\)
−0.698136 + 0.715965i \(0.745987\pi\)
\(14\) 94.2665i 0.480952i
\(15\) −24.0916 −0.107074
\(16\) 64.0000 0.250000
\(17\) 291.659 1.00920 0.504600 0.863353i \(-0.331640\pi\)
0.504600 + 0.863353i \(0.331640\pi\)
\(18\) 76.3675i 0.235702i
\(19\) −369.712 −1.02413 −0.512067 0.858946i \(-0.671120\pi\)
−0.512067 + 0.858946i \(0.671120\pi\)
\(20\) 37.0915 0.0927287
\(21\) −173.179 −0.392695
\(22\) 11.7762 0.0243311
\(23\) 496.520i 0.938602i −0.883038 0.469301i \(-0.844506\pi\)
0.883038 0.469301i \(-0.155494\pi\)
\(24\) 117.576i 0.204124i
\(25\) −603.503 −0.965606
\(26\) −684.469 −1.01253
\(27\) 140.296 0.192450
\(28\) 266.626 0.340084
\(29\) −329.812 −0.392166 −0.196083 0.980587i \(-0.562822\pi\)
−0.196083 + 0.980587i \(0.562822\pi\)
\(30\) 68.1414i 0.0757127i
\(31\) 1480.23i 1.54030i −0.637863 0.770150i \(-0.720181\pi\)
0.637863 0.770150i \(-0.279819\pi\)
\(32\) 181.019i 0.176777i
\(33\) 21.6343i 0.0198662i
\(34\) 824.936i 0.713612i
\(35\) 154.524 0.126142
\(36\) −216.000 −0.166667
\(37\) 727.964i 0.531749i −0.964008 0.265874i \(-0.914339\pi\)
0.964008 0.265874i \(-0.0856606\pi\)
\(38\) 1045.70i 0.724172i
\(39\) 1257.45i 0.826726i
\(40\) 104.911i 0.0655691i
\(41\) −1746.67 −1.03907 −0.519533 0.854451i \(-0.673894\pi\)
−0.519533 + 0.854451i \(0.673894\pi\)
\(42\) 489.823i 0.277678i
\(43\) 1687.92i 0.912883i −0.889753 0.456442i \(-0.849124\pi\)
0.889753 0.456442i \(-0.150876\pi\)
\(44\) 33.3082i 0.0172047i
\(45\) −125.184 −0.0618192
\(46\) 1404.37 0.663692
\(47\) 2187.33i 0.990190i −0.868839 0.495095i \(-0.835133\pi\)
0.868839 0.495095i \(-0.164867\pi\)
\(48\) 332.554 0.144338
\(49\) −1290.23 −0.537371
\(50\) 1706.97i 0.682786i
\(51\) 1515.50 0.582662
\(52\) 1935.97i 0.715965i
\(53\) −2224.64 −0.791968 −0.395984 0.918258i \(-0.629596\pi\)
−0.395984 + 0.918258i \(0.629596\pi\)
\(54\) 396.817i 0.136083i
\(55\) 19.3039i 0.00638147i
\(56\) 754.132i 0.240476i
\(57\) −1921.08 −0.591284
\(58\) 932.848i 0.277303i
\(59\) −3416.38 667.631i −0.981435 0.191793i
\(60\) 192.733 0.0535370
\(61\) 4206.97i 1.13060i 0.824885 + 0.565301i \(0.191240\pi\)
−0.824885 + 0.565301i \(0.808760\pi\)
\(62\) 4186.72 1.08916
\(63\) −899.863 −0.226723
\(64\) −512.000 −0.125000
\(65\) 1122.00i 0.265562i
\(66\) 61.1911 0.0140476
\(67\) 279.854i 0.0623422i −0.999514 0.0311711i \(-0.990076\pi\)
0.999514 0.0311711i \(-0.00992368\pi\)
\(68\) −2333.27 −0.504600
\(69\) 2580.00i 0.541902i
\(70\) 437.061i 0.0891961i
\(71\) −2456.24 −0.487253 −0.243627 0.969869i \(-0.578337\pi\)
−0.243627 + 0.969869i \(0.578337\pi\)
\(72\) 610.940i 0.117851i
\(73\) 2001.19i 0.375528i −0.982214 0.187764i \(-0.939876\pi\)
0.982214 0.187764i \(-0.0601241\pi\)
\(74\) 2058.99 0.376003
\(75\) −3135.90 −0.557493
\(76\) 2957.70 0.512067
\(77\) 138.763i 0.0234041i
\(78\) −3556.61 −0.584583
\(79\) 3622.55 0.580444 0.290222 0.956959i \(-0.406271\pi\)
0.290222 + 0.956959i \(0.406271\pi\)
\(80\) −296.732 −0.0463644
\(81\) 729.000 0.111111
\(82\) 4940.33i 0.734730i
\(83\) 2780.03i 0.403546i 0.979432 + 0.201773i \(0.0646704\pi\)
−0.979432 + 0.201773i \(0.935330\pi\)
\(84\) 1385.43 0.196348
\(85\) −1352.26 −0.187164
\(86\) 4774.16 0.645506
\(87\) −1713.75 −0.226417
\(88\) −94.2099 −0.0121655
\(89\) 4088.90i 0.516209i −0.966117 0.258105i \(-0.916902\pi\)
0.966117 0.258105i \(-0.0830980\pi\)
\(90\) 354.073i 0.0437127i
\(91\) 8065.31i 0.973954i
\(92\) 3972.16i 0.469301i
\(93\) 7691.49i 0.889293i
\(94\) 6186.70 0.700170
\(95\) 1714.15 0.189933
\(96\) 940.604i 0.102062i
\(97\) 14089.3i 1.49743i −0.662892 0.748715i \(-0.730671\pi\)
0.662892 0.748715i \(-0.269329\pi\)
\(98\) 3649.32i 0.379979i
\(99\) 112.415i 0.0114698i
\(100\) 4828.03 0.482803
\(101\) 4785.01i 0.469073i 0.972107 + 0.234536i \(0.0753572\pi\)
−0.972107 + 0.234536i \(0.924643\pi\)
\(102\) 4286.49i 0.412004i
\(103\) 5397.36i 0.508753i −0.967105 0.254376i \(-0.918130\pi\)
0.967105 0.254376i \(-0.0818702\pi\)
\(104\) 5475.75 0.506264
\(105\) 802.932 0.0728283
\(106\) 6292.22i 0.560006i
\(107\) 9551.89 0.834299 0.417150 0.908838i \(-0.363029\pi\)
0.417150 + 0.908838i \(0.363029\pi\)
\(108\) −1122.37 −0.0962250
\(109\) 6217.45i 0.523310i 0.965161 + 0.261655i \(0.0842683\pi\)
−0.965161 + 0.261655i \(0.915732\pi\)
\(110\) −54.5998 −0.00451238
\(111\) 3782.61i 0.307005i
\(112\) −2133.01 −0.170042
\(113\) 2816.11i 0.220543i 0.993902 + 0.110271i \(0.0351720\pi\)
−0.993902 + 0.110271i \(0.964828\pi\)
\(114\) 5433.64i 0.418101i
\(115\) 2302.09i 0.174071i
\(116\) 2638.49 0.196083
\(117\) 6533.90i 0.477310i
\(118\) 1888.35 9662.97i 0.135618 0.693980i
\(119\) −9720.48 −0.686426
\(120\) 545.131i 0.0378564i
\(121\) 14623.7 0.998816
\(122\) −11899.1 −0.799456
\(123\) −9075.96 −0.599905
\(124\) 11841.8i 0.770150i
\(125\) 5695.88 0.364536
\(126\) 2545.20i 0.160317i
\(127\) −23586.6 −1.46237 −0.731185 0.682179i \(-0.761033\pi\)
−0.731185 + 0.682179i \(0.761033\pi\)
\(128\) 1448.15i 0.0883883i
\(129\) 8770.70i 0.527054i
\(130\) 3173.50 0.187781
\(131\) 14769.0i 0.860612i 0.902683 + 0.430306i \(0.141594\pi\)
−0.902683 + 0.430306i \(0.858406\pi\)
\(132\) 173.075i 0.00993312i
\(133\) 12321.9 0.696583
\(134\) 791.547 0.0440826
\(135\) −650.474 −0.0356913
\(136\) 6599.49i 0.356806i
\(137\) −30624.2 −1.63164 −0.815818 0.578308i \(-0.803713\pi\)
−0.815818 + 0.578308i \(0.803713\pi\)
\(138\) 7297.33 0.383183
\(139\) 25204.3 1.30450 0.652252 0.758002i \(-0.273824\pi\)
0.652252 + 0.758002i \(0.273824\pi\)
\(140\) −1236.19 −0.0630711
\(141\) 11365.7i 0.571687i
\(142\) 6947.31i 0.344540i
\(143\) 1007.56 0.0492718
\(144\) 1728.00 0.0833333
\(145\) 1529.15 0.0727301
\(146\) 5660.22 0.265539
\(147\) −6704.22 −0.310251
\(148\) 5823.71i 0.265874i
\(149\) 20530.3i 0.924747i −0.886685 0.462374i \(-0.846998\pi\)
0.886685 0.462374i \(-0.153002\pi\)
\(150\) 8869.65i 0.394207i
\(151\) 15895.5i 0.697139i −0.937283 0.348570i \(-0.886668\pi\)
0.937283 0.348570i \(-0.113332\pi\)
\(152\) 8365.63i 0.362086i
\(153\) 7874.79 0.336400
\(154\) −392.481 −0.0165492
\(155\) 6862.98i 0.285660i
\(156\) 10059.6i 0.413363i
\(157\) 4402.72i 0.178616i −0.996004 0.0893082i \(-0.971534\pi\)
0.996004 0.0893082i \(-0.0284656\pi\)
\(158\) 10246.1i 0.410436i
\(159\) −11559.6 −0.457243
\(160\) 839.285i 0.0327846i
\(161\) 16548.2i 0.638407i
\(162\) 2061.92i 0.0785674i
\(163\) 16309.9 0.613868 0.306934 0.951731i \(-0.400697\pi\)
0.306934 + 0.951731i \(0.400697\pi\)
\(164\) 13973.4 0.519533
\(165\) 100.306i 0.00368434i
\(166\) −7863.11 −0.285350
\(167\) −29499.3 −1.05774 −0.528870 0.848703i \(-0.677384\pi\)
−0.528870 + 0.848703i \(0.677384\pi\)
\(168\) 3918.59i 0.138839i
\(169\) −30001.2 −1.05043
\(170\) 3824.76i 0.132345i
\(171\) −9982.23 −0.341378
\(172\) 13503.4i 0.456442i
\(173\) 45951.4i 1.53535i 0.640841 + 0.767674i \(0.278586\pi\)
−0.640841 + 0.767674i \(0.721414\pi\)
\(174\) 4847.22i 0.160101i
\(175\) 20113.7 0.656774
\(176\) 266.466i 0.00860233i
\(177\) −17752.0 3469.11i −0.566632 0.110732i
\(178\) 11565.1 0.365015
\(179\) 50111.3i 1.56397i 0.623295 + 0.781987i \(0.285793\pi\)
−0.623295 + 0.781987i \(0.714207\pi\)
\(180\) 1001.47 0.0309096
\(181\) −55894.1 −1.70612 −0.853058 0.521816i \(-0.825255\pi\)
−0.853058 + 0.521816i \(0.825255\pi\)
\(182\) 22812.1 0.688689
\(183\) 21860.0i 0.652753i
\(184\) −11235.0 −0.331846
\(185\) 3375.16i 0.0986168i
\(186\) 21754.8 0.628825
\(187\) 1214.33i 0.0347259i
\(188\) 17498.6i 0.495095i
\(189\) −4675.82 −0.130898
\(190\) 4848.34i 0.134303i
\(191\) 28684.0i 0.786273i 0.919480 + 0.393137i \(0.128610\pi\)
−0.919480 + 0.393137i \(0.871390\pi\)
\(192\) −2660.43 −0.0721688
\(193\) 23152.2 0.621553 0.310777 0.950483i \(-0.399411\pi\)
0.310777 + 0.950483i \(0.399411\pi\)
\(194\) 39850.6 1.05884
\(195\) 5830.09i 0.153322i
\(196\) 10321.8 0.268686
\(197\) 10420.0 0.268495 0.134248 0.990948i \(-0.457138\pi\)
0.134248 + 0.990948i \(0.457138\pi\)
\(198\) 317.958 0.00811036
\(199\) −46042.5 −1.16266 −0.581330 0.813668i \(-0.697468\pi\)
−0.581330 + 0.813668i \(0.697468\pi\)
\(200\) 13655.7i 0.341393i
\(201\) 1454.16i 0.0359933i
\(202\) −13534.1 −0.331685
\(203\) 10992.0 0.266739
\(204\) −12124.0 −0.291331
\(205\) 8098.32 0.192702
\(206\) 15266.0 0.359743
\(207\) 13406.0i 0.312867i
\(208\) 15487.8i 0.357983i
\(209\) 1539.31i 0.0352397i
\(210\) 2271.03i 0.0514974i
\(211\) 10156.2i 0.228121i 0.993474 + 0.114060i \(0.0363857\pi\)
−0.993474 + 0.114060i \(0.963614\pi\)
\(212\) 17797.1 0.395984
\(213\) −12763.0 −0.281316
\(214\) 27016.8i 0.589939i
\(215\) 7825.94i 0.169301i
\(216\) 3174.54i 0.0680414i
\(217\) 49333.4i 1.04766i
\(218\) −17585.6 −0.370036
\(219\) 10398.5i 0.216811i
\(220\) 154.432i 0.00319073i
\(221\) 70580.4i 1.44510i
\(222\) 10698.8 0.217085
\(223\) 25793.8 0.518687 0.259343 0.965785i \(-0.416494\pi\)
0.259343 + 0.965785i \(0.416494\pi\)
\(224\) 6033.06i 0.120238i
\(225\) −16294.6 −0.321869
\(226\) −7965.17 −0.155947
\(227\) 79731.8i 1.54732i 0.633601 + 0.773660i \(0.281576\pi\)
−0.633601 + 0.773660i \(0.718424\pi\)
\(228\) 15368.6 0.295642
\(229\) 4958.81i 0.0945598i 0.998882 + 0.0472799i \(0.0150553\pi\)
−0.998882 + 0.0472799i \(0.984945\pi\)
\(230\) −6511.28 −0.123087
\(231\) 721.034i 0.0135124i
\(232\) 7462.78i 0.138652i
\(233\) 91557.2i 1.68648i 0.537539 + 0.843239i \(0.319354\pi\)
−0.537539 + 0.843239i \(0.680646\pi\)
\(234\) −18480.7 −0.337509
\(235\) 10141.4i 0.183638i
\(236\) 27331.0 + 5341.05i 0.490718 + 0.0958964i
\(237\) 18823.3 0.335120
\(238\) 27493.7i 0.485376i
\(239\) −44494.2 −0.778946 −0.389473 0.921038i \(-0.627343\pi\)
−0.389473 + 0.921038i \(0.627343\pi\)
\(240\) −1541.86 −0.0267685
\(241\) 71057.7 1.22342 0.611712 0.791081i \(-0.290481\pi\)
0.611712 + 0.791081i \(0.290481\pi\)
\(242\) 41362.0i 0.706270i
\(243\) 3788.00 0.0641500
\(244\) 33655.7i 0.565301i
\(245\) 5982.06 0.0996595
\(246\) 25670.7i 0.424197i
\(247\) 89469.0i 1.46649i
\(248\) −33493.7 −0.544578
\(249\) 14445.5i 0.232988i
\(250\) 16110.4i 0.257766i
\(251\) −60466.2 −0.959765 −0.479883 0.877333i \(-0.659321\pi\)
−0.479883 + 0.877333i \(0.659321\pi\)
\(252\) 7198.90 0.113361
\(253\) −2067.28 −0.0322967
\(254\) 66712.9i 1.03405i
\(255\) −7026.54 −0.108059
\(256\) 4096.00 0.0625000
\(257\) 94101.6 1.42472 0.712362 0.701812i \(-0.247625\pi\)
0.712362 + 0.701812i \(0.247625\pi\)
\(258\) 24807.3 0.372683
\(259\) 24261.8i 0.361679i
\(260\) 8976.01i 0.132781i
\(261\) −8904.91 −0.130722
\(262\) −41772.9 −0.608545
\(263\) −98841.3 −1.42898 −0.714491 0.699644i \(-0.753342\pi\)
−0.714491 + 0.699644i \(0.753342\pi\)
\(264\) −489.529 −0.00702378
\(265\) 10314.4 0.146876
\(266\) 34851.5i 0.492559i
\(267\) 21246.5i 0.298034i
\(268\) 2238.83i 0.0311711i
\(269\) 130957.i 1.80977i −0.425658 0.904884i \(-0.639957\pi\)
0.425658 0.904884i \(-0.360043\pi\)
\(270\) 1839.82i 0.0252376i
\(271\) −45179.0 −0.615174 −0.307587 0.951520i \(-0.599521\pi\)
−0.307587 + 0.951520i \(0.599521\pi\)
\(272\) 18666.2 0.252300
\(273\) 41908.6i 0.562313i
\(274\) 86618.3i 1.15374i
\(275\) 2512.70i 0.0332258i
\(276\) 20640.0i 0.270951i
\(277\) −102949. −1.34173 −0.670863 0.741582i \(-0.734076\pi\)
−0.670863 + 0.741582i \(0.734076\pi\)
\(278\) 71288.6i 0.922424i
\(279\) 39966.2i 0.513433i
\(280\) 3496.49i 0.0445980i
\(281\) 13476.5 0.170673 0.0853363 0.996352i \(-0.472804\pi\)
0.0853363 + 0.996352i \(0.472804\pi\)
\(282\) 32147.1 0.404243
\(283\) 36120.2i 0.451000i −0.974243 0.225500i \(-0.927598\pi\)
0.974243 0.225500i \(-0.0724016\pi\)
\(284\) 19649.9 0.243627
\(285\) 8906.97 0.109658
\(286\) 2849.81i 0.0348404i
\(287\) 58213.4 0.706739
\(288\) 4887.52i 0.0589256i
\(289\) 1543.85 0.0184846
\(290\) 4325.09i 0.0514280i
\(291\) 73210.3i 0.864542i
\(292\) 16009.5i 0.187764i
\(293\) 24607.5 0.286637 0.143318 0.989677i \(-0.454223\pi\)
0.143318 + 0.989677i \(0.454223\pi\)
\(294\) 18962.4i 0.219381i
\(295\) 15839.8 + 3095.43i 0.182015 + 0.0355694i
\(296\) −16471.9 −0.188002
\(297\) 584.127i 0.00662208i
\(298\) 58068.5 0.653895
\(299\) 120156. 1.34401
\(300\) 25087.2 0.278746
\(301\) 56255.5i 0.620914i
\(302\) 44959.2 0.492952
\(303\) 24863.7i 0.270819i
\(304\) −23661.6 −0.256033
\(305\) 19505.3i 0.209678i
\(306\) 22273.3i 0.237871i
\(307\) 120412. 1.27760 0.638799 0.769374i \(-0.279432\pi\)
0.638799 + 0.769374i \(0.279432\pi\)
\(308\) 1110.11i 0.0117021i
\(309\) 28045.5i 0.293729i
\(310\) −19411.5 −0.201992
\(311\) 122536. 1.26690 0.633449 0.773784i \(-0.281639\pi\)
0.633449 + 0.773784i \(0.281639\pi\)
\(312\) 28452.8 0.292292
\(313\) 22938.9i 0.234145i −0.993123 0.117072i \(-0.962649\pi\)
0.993123 0.117072i \(-0.0373510\pi\)
\(314\) 12452.8 0.126301
\(315\) 4172.16 0.0420474
\(316\) −28980.4 −0.290222
\(317\) −21501.0 −0.213964 −0.106982 0.994261i \(-0.534119\pi\)
−0.106982 + 0.994261i \(0.534119\pi\)
\(318\) 32695.4i 0.323319i
\(319\) 1373.18i 0.0134942i
\(320\) 2373.86 0.0231822
\(321\) 49633.1 0.481683
\(322\) −46805.2 −0.451422
\(323\) −107830. −1.03356
\(324\) −5832.00 −0.0555556
\(325\) 146046.i 1.38268i
\(326\) 46131.2i 0.434070i
\(327\) 32306.8i 0.302133i
\(328\) 39522.6i 0.367365i
\(329\) 72899.9i 0.673496i
\(330\) −283.709 −0.00260522
\(331\) −141825. −1.29449 −0.647244 0.762283i \(-0.724079\pi\)
−0.647244 + 0.762283i \(0.724079\pi\)
\(332\) 22240.2i 0.201773i
\(333\) 19655.0i 0.177250i
\(334\) 83436.7i 0.747935i
\(335\) 1297.53i 0.0115618i
\(336\) −11083.4 −0.0981738
\(337\) 133252.i 1.17331i −0.809836 0.586657i \(-0.800444\pi\)
0.809836 0.586657i \(-0.199556\pi\)
\(338\) 84856.3i 0.742763i
\(339\) 14633.0i 0.127331i
\(340\) 10818.1 0.0935818
\(341\) −6162.97 −0.0530007
\(342\) 28234.0i 0.241391i
\(343\) 123022. 1.04567
\(344\) −38193.3 −0.322753
\(345\) 11962.0i 0.100500i
\(346\) −129970. −1.08565
\(347\) 97310.6i 0.808167i −0.914722 0.404083i \(-0.867591\pi\)
0.914722 0.404083i \(-0.132409\pi\)
\(348\) 13710.0 0.113209
\(349\) 102463.i 0.841233i 0.907239 + 0.420616i \(0.138186\pi\)
−0.907239 + 0.420616i \(0.861814\pi\)
\(350\) 56890.2i 0.464410i
\(351\) 33951.1i 0.275575i
\(352\) 753.679 0.00608277
\(353\) 16785.8i 0.134708i −0.997729 0.0673539i \(-0.978544\pi\)
0.997729 0.0673539i \(-0.0214557\pi\)
\(354\) 9812.13 50210.3i 0.0782991 0.400669i
\(355\) 11388.2 0.0903648
\(356\) 32711.2i 0.258105i
\(357\) −50509.1 −0.396308
\(358\) −141736. −1.10590
\(359\) 204196. 1.58438 0.792188 0.610277i \(-0.208942\pi\)
0.792188 + 0.610277i \(0.208942\pi\)
\(360\) 2832.59i 0.0218564i
\(361\) 6366.05 0.0488490
\(362\) 158092.i 1.20641i
\(363\) 75986.8 0.576667
\(364\) 64522.5i 0.486977i
\(365\) 9278.40i 0.0696446i
\(366\) −61829.5 −0.461566
\(367\) 107995.i 0.801807i 0.916120 + 0.400903i \(0.131304\pi\)
−0.916120 + 0.400903i \(0.868696\pi\)
\(368\) 31777.3i 0.234650i
\(369\) −47160.1 −0.346355
\(370\) −9546.39 −0.0697326
\(371\) 74143.2 0.538671
\(372\) 61531.9i 0.444646i
\(373\) −259803. −1.86735 −0.933676 0.358120i \(-0.883418\pi\)
−0.933676 + 0.358120i \(0.883418\pi\)
\(374\) 3434.64 0.0245549
\(375\) 29596.7 0.210465
\(376\) −49493.6 −0.350085
\(377\) 79813.2i 0.561555i
\(378\) 13225.2i 0.0925592i
\(379\) 180363. 1.25565 0.627826 0.778354i \(-0.283945\pi\)
0.627826 + 0.778354i \(0.283945\pi\)
\(380\) −13713.2 −0.0949666
\(381\) −122559. −0.844300
\(382\) −81130.7 −0.555979
\(383\) −53195.1 −0.362639 −0.181319 0.983424i \(-0.558037\pi\)
−0.181319 + 0.983424i \(0.558037\pi\)
\(384\) 7524.83i 0.0510310i
\(385\) 643.367i 0.00434047i
\(386\) 65484.4i 0.439504i
\(387\) 45573.9i 0.304294i
\(388\) 112715.i 0.748715i
\(389\) −184071. −1.21642 −0.608212 0.793774i \(-0.708113\pi\)
−0.608212 + 0.793774i \(0.708113\pi\)
\(390\) 16490.0 0.108415
\(391\) 144815.i 0.947237i
\(392\) 29194.5i 0.189989i
\(393\) 76741.8i 0.496875i
\(394\) 29472.3i 0.189855i
\(395\) −16795.7 −0.107648
\(396\) 899.322i 0.00573489i
\(397\) 182691.i 1.15914i −0.814922 0.579570i \(-0.803220\pi\)
0.814922 0.579570i \(-0.196780\pi\)
\(398\) 130228.i 0.822125i
\(399\) 64026.2 0.402172
\(400\) −38624.2 −0.241401
\(401\) 27488.4i 0.170947i 0.996340 + 0.0854733i \(0.0272402\pi\)
−0.996340 + 0.0854733i \(0.972760\pi\)
\(402\) 4113.00 0.0254511
\(403\) 358210. 2.20560
\(404\) 38280.1i 0.234536i
\(405\) −3379.96 −0.0206064
\(406\) 31090.2i 0.188613i
\(407\) −3030.90 −0.0182971
\(408\) 34291.9i 0.206002i
\(409\) 185149.i 1.10682i −0.832910 0.553409i \(-0.813327\pi\)
0.832910 0.553409i \(-0.186673\pi\)
\(410\) 22905.5i 0.136261i
\(411\) −159128. −0.942026
\(412\) 43178.9i 0.254376i
\(413\) 113862. + 22251.0i 0.667541 + 0.130451i
\(414\) 37918.0 0.221231
\(415\) 12889.4i 0.0748407i
\(416\) −43806.0 −0.253132
\(417\) 130965. 0.753156
\(418\) −4353.82 −0.0249183
\(419\) 143622.i 0.818077i −0.912517 0.409038i \(-0.865864\pi\)
0.912517 0.409038i \(-0.134136\pi\)
\(420\) −6423.46 −0.0364141
\(421\) 240408.i 1.35639i 0.734883 + 0.678194i \(0.237237\pi\)
−0.734883 + 0.678194i \(0.762763\pi\)
\(422\) −28725.9 −0.161306
\(423\) 59057.9i 0.330063i
\(424\) 50337.8i 0.280003i
\(425\) −176017. −0.974489
\(426\) 36099.3i 0.198920i
\(427\) 140211.i 0.768999i
\(428\) −76415.2 −0.417150
\(429\) 5235.43 0.0284471
\(430\) −22135.1 −0.119714
\(431\) 185431.i 0.998223i −0.866538 0.499112i \(-0.833660\pi\)
0.866538 0.499112i \(-0.166340\pi\)
\(432\) 8978.95 0.0481125
\(433\) 5575.97 0.0297402 0.0148701 0.999889i \(-0.495267\pi\)
0.0148701 + 0.999889i \(0.495267\pi\)
\(434\) −139536. −0.740810
\(435\) 7945.70 0.0419908
\(436\) 49739.6i 0.261655i
\(437\) 183570.i 0.961253i
\(438\) 29411.4 0.153309
\(439\) 190431. 0.988120 0.494060 0.869428i \(-0.335512\pi\)
0.494060 + 0.869428i \(0.335512\pi\)
\(440\) 436.798 0.00225619
\(441\) −34836.2 −0.179124
\(442\) −199631. −1.02184
\(443\) 190993.i 0.973218i 0.873620 + 0.486609i \(0.161766\pi\)
−0.873620 + 0.486609i \(0.838234\pi\)
\(444\) 30260.9i 0.153503i
\(445\) 18957.9i 0.0957349i
\(446\) 72955.8i 0.366767i
\(447\) 106679.i 0.533903i
\(448\) 17064.1 0.0850210
\(449\) −166726. −0.827010 −0.413505 0.910502i \(-0.635696\pi\)
−0.413505 + 0.910502i \(0.635696\pi\)
\(450\) 46088.1i 0.227595i
\(451\) 7272.31i 0.0357535i
\(452\) 22528.9i 0.110271i
\(453\) 82595.3i 0.402493i
\(454\) −225516. −1.09412
\(455\) 37394.3i 0.180627i
\(456\) 43469.1i 0.209050i
\(457\) 383054.i 1.83412i −0.398748 0.917060i \(-0.630555\pi\)
0.398748 0.917060i \(-0.369445\pi\)
\(458\) −14025.6 −0.0668639
\(459\) 40918.6 0.194221
\(460\) 18416.7i 0.0870354i
\(461\) −108184. −0.509052 −0.254526 0.967066i \(-0.581919\pi\)
−0.254526 + 0.967066i \(0.581919\pi\)
\(462\) −2039.39 −0.00955470
\(463\) 199196.i 0.929221i 0.885515 + 0.464610i \(0.153806\pi\)
−0.885515 + 0.464610i \(0.846194\pi\)
\(464\) −21107.9 −0.0980415
\(465\) 35661.1i 0.164926i
\(466\) −258963. −1.19252
\(467\) 253351.i 1.16169i −0.814015 0.580844i \(-0.802723\pi\)
0.814015 0.580844i \(-0.197277\pi\)
\(468\) 52271.2i 0.238655i
\(469\) 9327.05i 0.0424032i
\(470\) −28684.3 −0.129852
\(471\) 22877.2i 0.103124i
\(472\) −15106.8 + 77303.8i −0.0678090 + 0.346990i
\(473\) −7027.71 −0.0314117
\(474\) 53240.4i 0.236965i
\(475\) 223123. 0.988909
\(476\) 77763.8 0.343213
\(477\) −60065.2 −0.263989
\(478\) 125849.i 0.550798i
\(479\) 7178.05 0.0312849 0.0156425 0.999878i \(-0.495021\pi\)
0.0156425 + 0.999878i \(0.495021\pi\)
\(480\) 4361.05i 0.0189282i
\(481\) 176165. 0.761427
\(482\) 200981.i 0.865091i
\(483\) 85986.7i 0.368585i
\(484\) −116989. −0.499408
\(485\) 65324.3i 0.277710i
\(486\) 10714.1i 0.0453609i
\(487\) 183775. 0.774869 0.387435 0.921897i \(-0.373361\pi\)
0.387435 + 0.921897i \(0.373361\pi\)
\(488\) 95192.8 0.399728
\(489\) 84748.5 0.354417
\(490\) 16919.8i 0.0704699i
\(491\) −693.629 −0.00287716 −0.00143858 0.999999i \(-0.500458\pi\)
−0.00143858 + 0.999999i \(0.500458\pi\)
\(492\) 72607.7 0.299952
\(493\) −96192.5 −0.395774
\(494\) 253056. 1.03696
\(495\) 521.206i 0.00212716i
\(496\) 94734.6i 0.385075i
\(497\) 81862.3 0.331414
\(498\) −40857.9 −0.164747
\(499\) 192263. 0.772138 0.386069 0.922470i \(-0.373833\pi\)
0.386069 + 0.922470i \(0.373833\pi\)
\(500\) −45567.0 −0.182268
\(501\) −153283. −0.610687
\(502\) 171024.i 0.678656i
\(503\) 240765.i 0.951607i 0.879552 + 0.475803i \(0.157843\pi\)
−0.879552 + 0.475803i \(0.842157\pi\)
\(504\) 20361.6i 0.0801586i
\(505\) 22185.4i 0.0869931i
\(506\) 5847.14i 0.0228372i
\(507\) −155891. −0.606464
\(508\) 188693. 0.731185
\(509\) 51894.8i 0.200304i 0.994972 + 0.100152i \(0.0319328\pi\)
−0.994972 + 0.100152i \(0.968067\pi\)
\(510\) 19874.0i 0.0764093i
\(511\) 66696.2i 0.255423i
\(512\) 11585.2i 0.0441942i
\(513\) −51869.2 −0.197095
\(514\) 266160.i 1.00743i
\(515\) 25024.5i 0.0943520i
\(516\) 70165.6i 0.263527i
\(517\) −9107.01 −0.0340718
\(518\) −68622.6 −0.255745
\(519\) 238771.i 0.886433i
\(520\) −25388.0 −0.0938905
\(521\) −62938.8 −0.231869 −0.115935 0.993257i \(-0.536986\pi\)
−0.115935 + 0.993257i \(0.536986\pi\)
\(522\) 25186.9i 0.0924344i
\(523\) −249247. −0.911225 −0.455613 0.890178i \(-0.650580\pi\)
−0.455613 + 0.890178i \(0.650580\pi\)
\(524\) 118152.i 0.430306i
\(525\) 104514. 0.379189
\(526\) 279565.i 1.01044i
\(527\) 431722.i 1.55447i
\(528\) 1384.60i 0.00496656i
\(529\) 33308.6 0.119027
\(530\) 29173.5i 0.103857i
\(531\) −92242.2 18026.0i −0.327145 0.0639310i
\(532\) −98574.8 −0.348291
\(533\) 422687.i 1.48787i
\(534\) 60094.2 0.210742
\(535\) −44286.8 −0.154727
\(536\) −6332.38 −0.0220413
\(537\) 260386.i 0.902960i
\(538\) 370401. 1.27970
\(539\) 5371.90i 0.0184906i
\(540\) 5203.79 0.0178457
\(541\) 488861.i 1.67029i 0.550032 + 0.835144i \(0.314616\pi\)
−0.550032 + 0.835144i \(0.685384\pi\)
\(542\) 127785.i 0.434994i
\(543\) −290434. −0.985026
\(544\) 52795.9i 0.178403i
\(545\) 28826.8i 0.0970518i
\(546\) 118535. 0.397615
\(547\) 253690. 0.847869 0.423934 0.905693i \(-0.360649\pi\)
0.423934 + 0.905693i \(0.360649\pi\)
\(548\) 244994. 0.815818
\(549\) 113588.i 0.376867i
\(550\) −7107.00 −0.0234942
\(551\) 121935. 0.401630
\(552\) −58378.6 −0.191591
\(553\) −120733. −0.394800
\(554\) 291184.i 0.948743i
\(555\) 17537.8i 0.0569364i
\(556\) −201635. −0.652252
\(557\) −56303.7 −0.181479 −0.0907396 0.995875i \(-0.528923\pi\)
−0.0907396 + 0.995875i \(0.528923\pi\)
\(558\) 113041. 0.363052
\(559\) 408471. 1.30719
\(560\) 9889.56 0.0315356
\(561\) 6309.84i 0.0200490i
\(562\) 38117.2i 0.120684i
\(563\) 368407.i 1.16228i −0.813803 0.581141i \(-0.802607\pi\)
0.813803 0.581141i \(-0.197393\pi\)
\(564\) 90925.6i 0.285843i
\(565\) 13056.7i 0.0409013i
\(566\) 102163. 0.318905
\(567\) −24296.3 −0.0755743
\(568\) 55578.5i 0.172270i
\(569\) 373276.i 1.15294i 0.817119 + 0.576469i \(0.195570\pi\)
−0.817119 + 0.576469i \(0.804430\pi\)
\(570\) 25192.7i 0.0775399i
\(571\) 216523.i 0.664097i −0.943262 0.332049i \(-0.892260\pi\)
0.943262 0.332049i \(-0.107740\pi\)
\(572\) −8060.47 −0.0246359
\(573\) 149047.i 0.453955i
\(574\) 164652.i 0.499740i
\(575\) 299652.i 0.906319i
\(576\) −13824.0 −0.0416667
\(577\) −30989.5 −0.0930815 −0.0465407 0.998916i \(-0.514820\pi\)
−0.0465407 + 0.998916i \(0.514820\pi\)
\(578\) 4366.66i 0.0130706i
\(579\) 120303. 0.358854
\(580\) −12233.2 −0.0363651
\(581\) 92653.6i 0.274479i
\(582\) 207070. 0.611324
\(583\) 9262.34i 0.0272511i
\(584\) −45281.8 −0.132769
\(585\) 30294.0i 0.0885208i
\(586\) 69600.5i 0.202683i
\(587\) 83561.5i 0.242510i 0.992621 + 0.121255i \(0.0386919\pi\)
−0.992621 + 0.121255i \(0.961308\pi\)
\(588\) 53633.8 0.155126
\(589\) 547258.i 1.57747i
\(590\) −8755.20 + 44801.8i −0.0251514 + 0.128704i
\(591\) 54144.1 0.155016
\(592\) 46589.7i 0.132937i
\(593\) −351929. −1.00080 −0.500398 0.865795i \(-0.666813\pi\)
−0.500398 + 0.865795i \(0.666813\pi\)
\(594\) 1652.16 0.00468252
\(595\) 45068.4 0.127303
\(596\) 164243.i 0.462374i
\(597\) −239244. −0.671262
\(598\) 339853.i 0.950361i
\(599\) −189155. −0.527186 −0.263593 0.964634i \(-0.584908\pi\)
−0.263593 + 0.964634i \(0.584908\pi\)
\(600\) 70957.2i 0.197103i
\(601\) 551712.i 1.52744i 0.645549 + 0.763719i \(0.276629\pi\)
−0.645549 + 0.763719i \(0.723371\pi\)
\(602\) −159114. −0.439053
\(603\) 7556.06i 0.0207807i
\(604\) 127164.i 0.348570i
\(605\) −67801.7 −0.185238
\(606\) −70325.0 −0.191498
\(607\) 237501. 0.644598 0.322299 0.946638i \(-0.395544\pi\)
0.322299 + 0.946638i \(0.395544\pi\)
\(608\) 66925.0i 0.181043i
\(609\) 57116.3 0.154002
\(610\) 55169.4 0.148265
\(611\) 529326. 1.41788
\(612\) −62998.3 −0.168200
\(613\) 496603.i 1.32157i 0.750577 + 0.660783i \(0.229776\pi\)
−0.750577 + 0.660783i \(0.770224\pi\)
\(614\) 340577.i 0.903398i
\(615\) 42080.1 0.111257
\(616\) 3139.85 0.00827461
\(617\) 125936. 0.330811 0.165405 0.986226i \(-0.447107\pi\)
0.165405 + 0.986226i \(0.447107\pi\)
\(618\) 79324.7 0.207698
\(619\) 275923. 0.720122 0.360061 0.932929i \(-0.382756\pi\)
0.360061 + 0.932929i \(0.382756\pi\)
\(620\) 54903.9i 0.142830i
\(621\) 69659.9i 0.180634i
\(622\) 346583.i 0.895833i
\(623\) 136276.i 0.351109i
\(624\) 80476.8i 0.206681i
\(625\) 350781. 0.898000
\(626\) 64881.1 0.165565
\(627\) 7998.48i 0.0203457i
\(628\) 35221.7i 0.0893082i
\(629\) 212317.i 0.536641i
\(630\) 11800.6i 0.0297320i
\(631\) 462752. 1.16222 0.581112 0.813824i \(-0.302618\pi\)
0.581112 + 0.813824i \(0.302618\pi\)
\(632\) 81969.0i 0.205218i
\(633\) 52772.9i 0.131705i
\(634\) 60814.1i 0.151295i
\(635\) 109358. 0.271208
\(636\) 92476.4 0.228621
\(637\) 312230.i 0.769478i
\(638\) −3883.94 −0.00954182
\(639\) −66318.6 −0.162418
\(640\) 6714.28i 0.0163923i
\(641\) −351453. −0.855364 −0.427682 0.903929i \(-0.640670\pi\)
−0.427682 + 0.903929i \(0.640670\pi\)
\(642\) 140384.i 0.340601i
\(643\) −3965.89 −0.00959220 −0.00479610 0.999988i \(-0.501527\pi\)
−0.00479610 + 0.999988i \(0.501527\pi\)
\(644\) 132385.i 0.319204i
\(645\) 40664.8i 0.0977460i
\(646\) 304989.i 0.730834i
\(647\) −132862. −0.317390 −0.158695 0.987328i \(-0.550729\pi\)
−0.158695 + 0.987328i \(0.550729\pi\)
\(648\) 16495.4i 0.0392837i
\(649\) −2779.70 + 14224.2i −0.00659947 + 0.0337705i
\(650\) 413079. 0.977703
\(651\) 256344.i 0.604869i
\(652\) −130479. −0.306934
\(653\) −363151. −0.851650 −0.425825 0.904805i \(-0.640016\pi\)
−0.425825 + 0.904805i \(0.640016\pi\)
\(654\) −91377.4 −0.213640
\(655\) 68475.4i 0.159607i
\(656\) −111787. −0.259766
\(657\) 54032.2i 0.125176i
\(658\) −206192. −0.476234
\(659\) 281539.i 0.648287i −0.946008 0.324144i \(-0.894924\pi\)
0.946008 0.324144i \(-0.105076\pi\)
\(660\) 802.450i 0.00184217i
\(661\) 543997. 1.24507 0.622534 0.782592i \(-0.286103\pi\)
0.622534 + 0.782592i \(0.286103\pi\)
\(662\) 401143.i 0.915341i
\(663\) 366746.i 0.834332i
\(664\) 62904.9 0.142675
\(665\) −57129.5 −0.129187
\(666\) 55592.8 0.125334
\(667\) 163758.i 0.368088i
\(668\) 235994. 0.528870
\(669\) 134028. 0.299464
\(670\) −3669.96 −0.00817545
\(671\) 17515.8 0.0389032
\(672\) 31348.7i 0.0694194i
\(673\) 19850.8i 0.0438276i 0.999760 + 0.0219138i \(0.00697594\pi\)
−0.999760 + 0.0219138i \(0.993024\pi\)
\(674\) 376894. 0.829658
\(675\) −84669.2 −0.185831
\(676\) 240010. 0.525213
\(677\) −649848. −1.41786 −0.708932 0.705277i \(-0.750822\pi\)
−0.708932 + 0.705277i \(0.750822\pi\)
\(678\) −41388.2 −0.0900363
\(679\) 469572.i 1.01850i
\(680\) 30598.1i 0.0661724i
\(681\) 414299.i 0.893345i
\(682\) 17431.5i 0.0374771i
\(683\) 691295.i 1.48191i 0.671554 + 0.740955i \(0.265627\pi\)
−0.671554 + 0.740955i \(0.734373\pi\)
\(684\) 79857.8 0.170689
\(685\) 141987. 0.302599
\(686\) 347959.i 0.739401i
\(687\) 25766.7i 0.0545941i
\(688\) 108027.i 0.228221i
\(689\) 538354.i 1.13404i
\(690\) −33833.6 −0.0710641
\(691\) 115514.i 0.241925i −0.992657 0.120962i \(-0.961402\pi\)
0.992657 0.120962i \(-0.0385980\pi\)
\(692\) 367611.i 0.767674i
\(693\) 3746.60i 0.00780138i
\(694\) 275236. 0.571460
\(695\) −116858. −0.241930
\(696\) 38777.8i 0.0800506i
\(697\) −509431. −1.04862
\(698\) −289809. −0.594841
\(699\) 475745.i 0.973688i
\(700\) −160910. −0.328387
\(701\) 824249.i 1.67734i −0.544636 0.838672i \(-0.683332\pi\)
0.544636 0.838672i \(-0.316668\pi\)
\(702\) −96028.3 −0.194861
\(703\) 269137.i 0.544581i
\(704\) 2131.73i 0.00430117i
\(705\) 52696.4i 0.106024i
\(706\) 47477.4 0.0952528
\(707\) 159476.i 0.319048i
\(708\) 142016. + 27752.9i 0.283316 + 0.0553658i
\(709\) −902813. −1.79599 −0.897997 0.440001i \(-0.854978\pi\)
−0.897997 + 0.440001i \(0.854978\pi\)
\(710\) 32210.7i 0.0638975i
\(711\) 97808.9 0.193481
\(712\) −92521.1 −0.182508
\(713\) −734963. −1.44573
\(714\) 142861.i 0.280232i
\(715\) −4671.48 −0.00913782
\(716\) 400890.i 0.781987i
\(717\) −231199. −0.449725
\(718\) 577553.i 1.12032i
\(719\) 420448.i 0.813307i 0.913583 + 0.406653i \(0.133304\pi\)
−0.913583 + 0.406653i \(0.866696\pi\)
\(720\) −8011.76 −0.0154548
\(721\) 179885.i 0.346038i
\(722\) 18005.9i 0.0345414i
\(723\) 369227. 0.706344
\(724\) 447153. 0.853058
\(725\) 199042. 0.378678
\(726\) 214923.i 0.407765i
\(727\) −518451. −0.980931 −0.490466 0.871460i \(-0.663173\pi\)
−0.490466 + 0.871460i \(0.663173\pi\)
\(728\) −182497. −0.344345
\(729\) 19683.0 0.0370370
\(730\) −26243.3 −0.0492461
\(731\) 492297.i 0.921282i
\(732\) 174880.i 0.326376i
\(733\) −130264. −0.242448 −0.121224 0.992625i \(-0.538682\pi\)
−0.121224 + 0.992625i \(0.538682\pi\)
\(734\) −305455. −0.566963
\(735\) 31083.7 0.0575384
\(736\) 89879.8 0.165923
\(737\) −1165.18 −0.00214515
\(738\) 133389.i 0.244910i
\(739\) 387178.i 0.708960i −0.935064 0.354480i \(-0.884658\pi\)
0.935064 0.354480i \(-0.115342\pi\)
\(740\) 27001.3i 0.0493084i
\(741\) 464894.i 0.846677i
\(742\) 209709.i 0.380898i
\(743\) −430015. −0.778943 −0.389472 0.921038i \(-0.627342\pi\)
−0.389472 + 0.921038i \(0.627342\pi\)
\(744\) −174039. −0.314412
\(745\) 95187.5i 0.171501i
\(746\) 734833.i 1.32042i
\(747\) 75060.8i 0.134515i
\(748\) 9714.64i 0.0173629i
\(749\) −318348. −0.567464
\(750\) 83712.0i 0.148821i
\(751\) 582052.i 1.03200i −0.856587 0.516002i \(-0.827420\pi\)
0.856587 0.516002i \(-0.172580\pi\)
\(752\) 139989.i 0.247548i
\(753\) −314191. −0.554121
\(754\) 225746. 0.397079
\(755\) 73698.3i 0.129290i
\(756\) 37406.6 0.0654492
\(757\) 254818. 0.444671 0.222335 0.974970i \(-0.428632\pi\)
0.222335 + 0.974970i \(0.428632\pi\)
\(758\) 510144.i 0.887880i
\(759\) −10741.9 −0.0186465
\(760\) 38786.7i 0.0671515i
\(761\) 959024. 1.65600 0.828000 0.560728i \(-0.189479\pi\)
0.828000 + 0.560728i \(0.189479\pi\)
\(762\) 346651.i 0.597010i
\(763\) 207217.i 0.355939i
\(764\) 229472.i 0.393137i
\(765\) −36511.0 −0.0623879
\(766\) 150459.i 0.256424i
\(767\) 161564. 826751.i 0.274634 1.40535i
\(768\) 21283.4 0.0360844
\(769\) 799674.i 1.35226i −0.736782 0.676130i \(-0.763656\pi\)
0.736782 0.676130i \(-0.236344\pi\)
\(770\) 1819.72 0.00306918
\(771\) 488966. 0.822565
\(772\) −185218. −0.310777
\(773\) 906357.i 1.51684i −0.651765 0.758421i \(-0.725971\pi\)
0.651765 0.758421i \(-0.274029\pi\)
\(774\) 128902. 0.215169
\(775\) 893323.i 1.48732i
\(776\) −318805. −0.529422
\(777\) 126068.i 0.208815i
\(778\) 520630.i 0.860142i
\(779\) 645765. 1.06414
\(780\) 46640.7i 0.0766612i
\(781\) 10226.6i 0.0167661i
\(782\) 409597. 0.669798
\(783\) −46271.3 −0.0754724
\(784\) −82574.6 −0.134343
\(785\) 20412.9i 0.0331258i
\(786\) −217059. −0.351343
\(787\) −1.16350e6 −1.87852 −0.939260 0.343206i \(-0.888487\pi\)
−0.939260 + 0.343206i \(0.888487\pi\)
\(788\) −83360.3 −0.134248
\(789\) −513594. −0.825023
\(790\) 47505.5i 0.0761185i
\(791\) 93856.1i 0.150006i
\(792\) −2543.67 −0.00405518
\(793\) −1.01807e6 −1.61894
\(794\) 516728. 0.819636
\(795\) 53595.1 0.0847991
\(796\) 368340. 0.581330
\(797\) 1.02880e6i 1.61963i 0.586685 + 0.809815i \(0.300433\pi\)
−0.586685 + 0.809815i \(0.699567\pi\)
\(798\) 181094.i 0.284379i
\(799\) 637954.i 0.999300i
\(800\) 109246.i 0.170697i
\(801\) 110400.i 0.172070i
\(802\) −77748.9 −0.120878
\(803\) −8332.02 −0.0129217
\(804\) 11633.3i 0.0179966i
\(805\) 76724.5i 0.118397i
\(806\) 1.01317e6i 1.55960i
\(807\) 680471.i 1.04487i
\(808\) 108272. 0.165842
\(809\) 218393.i 0.333688i 0.985983 + 0.166844i \(0.0533577\pi\)
−0.985983 + 0.166844i \(0.946642\pi\)
\(810\) 9559.98i 0.0145709i
\(811\) 435909.i 0.662757i 0.943498 + 0.331378i \(0.107514\pi\)
−0.943498 + 0.331378i \(0.892486\pi\)
\(812\) −87936.3 −0.133369
\(813\) −234757. −0.355171
\(814\) 8572.68i 0.0129380i
\(815\) −75619.6 −0.113846
\(816\) 96992.2 0.145665
\(817\) 624045.i 0.934914i
\(818\) 523682. 0.782638
\(819\) 217763.i 0.324651i
\(820\) −64786.6 −0.0963512
\(821\) 110620.i 0.164115i 0.996628 + 0.0820573i \(0.0261490\pi\)
−0.996628 + 0.0820573i \(0.973851\pi\)
\(822\) 450082.i 0.666113i
\(823\) 722280.i 1.06637i −0.846000 0.533183i \(-0.820996\pi\)
0.846000 0.533183i \(-0.179004\pi\)
\(824\) −122128. −0.179871
\(825\) 13056.4i 0.0191829i
\(826\) −62935.3 + 322050.i −0.0922431 + 0.472023i
\(827\) 280361. 0.409927 0.204964 0.978770i \(-0.434292\pi\)
0.204964 + 0.978770i \(0.434292\pi\)
\(828\) 107248.i 0.156434i
\(829\) −442043. −0.643214 −0.321607 0.946873i \(-0.604223\pi\)
−0.321607 + 0.946873i \(0.604223\pi\)
\(830\) 36456.8 0.0529204
\(831\) −534940. −0.774645
\(832\) 123902.i 0.178991i
\(833\) −376306. −0.542315
\(834\) 370426.i 0.532561i
\(835\) 136772. 0.196166
\(836\) 12314.5i 0.0176199i
\(837\) 207670.i 0.296431i
\(838\) 406225. 0.578468
\(839\) 1.15312e6i 1.63814i −0.573696 0.819068i \(-0.694491\pi\)
0.573696 0.819068i \(-0.305509\pi\)
\(840\) 18168.3i 0.0257487i
\(841\) −598505. −0.846206
\(842\) −679975. −0.959111
\(843\) 70025.8 0.0985378
\(844\) 81249.2i 0.114060i
\(845\) 139099. 0.194809
\(846\) 167041. 0.233390
\(847\) −487381. −0.679363
\(848\) −142377. −0.197992
\(849\) 187686.i 0.260385i
\(850\) 497851.i 0.689068i
\(851\) −361449. −0.499100
\(852\) 102104. 0.140658
\(853\) 46970.7 0.0645549 0.0322775 0.999479i \(-0.489724\pi\)
0.0322775 + 0.999479i \(0.489724\pi\)
\(854\) 396576. 0.543765
\(855\) 46282.0 0.0633111
\(856\) 216135.i 0.294969i
\(857\) 545280.i 0.742435i 0.928546 + 0.371217i \(0.121059\pi\)
−0.928546 + 0.371217i \(0.878941\pi\)
\(858\) 14808.0i 0.0201151i
\(859\) 303823.i 0.411751i −0.978578 0.205875i \(-0.933996\pi\)
0.978578 0.205875i \(-0.0660042\pi\)
\(860\) 62607.5i 0.0846505i
\(861\) 302486. 0.408036
\(862\) 524478. 0.705851
\(863\) 14279.4i 0.0191730i −0.999954 0.00958648i \(-0.996948\pi\)
0.999954 0.00958648i \(-0.00305152\pi\)
\(864\) 25396.3i 0.0340207i
\(865\) 213051.i 0.284742i
\(866\) 15771.2i 0.0210295i
\(867\) 8022.07 0.0106721
\(868\) 394667.i 0.523832i
\(869\) 15082.6i 0.0199727i
\(870\) 22473.8i 0.0296919i
\(871\) 67723.7 0.0892697
\(872\) 140685. 0.185018
\(873\) 380412.i 0.499144i
\(874\) −519213. −0.679709
\(875\) −189834. −0.247946
\(876\) 83187.9i 0.108406i
\(877\) 306520. 0.398529 0.199264 0.979946i \(-0.436145\pi\)
0.199264 + 0.979946i \(0.436145\pi\)
\(878\) 538621.i 0.698706i
\(879\) 127864. 0.165490
\(880\) 1235.45i 0.00159537i
\(881\) 100841.i 0.129923i −0.997888 0.0649613i \(-0.979308\pi\)
0.997888 0.0649613i \(-0.0206924\pi\)
\(882\) 98531.5i 0.126660i
\(883\) 295565. 0.379080 0.189540 0.981873i \(-0.439300\pi\)
0.189540 + 0.981873i \(0.439300\pi\)
\(884\) 564643.i 0.722552i
\(885\) 82306.1 + 16084.3i 0.105086 + 0.0205360i
\(886\) −540210. −0.688169
\(887\) 910360.i 1.15709i 0.815652 + 0.578543i \(0.196379\pi\)
−0.815652 + 0.578543i \(0.803621\pi\)
\(888\) −85590.7 −0.108543
\(889\) 786099. 0.994658
\(890\) −53621.1 −0.0676948
\(891\) 3035.21i 0.00382326i
\(892\) −206350. −0.259343
\(893\) 808682.i 1.01409i
\(894\) 301733. 0.377527
\(895\) 232338.i 0.290051i
\(896\) 48264.5i 0.0601190i
\(897\) 624349. 0.775966
\(898\) 471572.i 0.584784i
\(899\) 488196.i 0.604053i
\(900\) 130357. 0.160934
\(901\) −648835. −0.799254
\(902\) −20569.2 −0.0252816
\(903\) 292312.i 0.358485i
\(904\) 63721.4 0.0779737
\(905\) 259149. 0.316412
\(906\) 233615. 0.284606
\(907\) 968919. 1.17780 0.588902 0.808205i \(-0.299560\pi\)
0.588902 + 0.808205i \(0.299560\pi\)
\(908\) 637855.i 0.773660i
\(909\) 129195.i 0.156358i
\(910\) −105767. −0.127723
\(911\) −1.24637e6 −1.50179 −0.750897 0.660420i \(-0.770378\pi\)
−0.750897 + 0.660420i \(0.770378\pi\)
\(912\) −122949. −0.147821
\(913\) 11574.7 0.0138858
\(914\) 1.08344e6 1.29692
\(915\) 101353.i 0.121058i
\(916\) 39670.5i 0.0472799i
\(917\) 492224.i 0.585361i
\(918\) 115735.i 0.137335i
\(919\) 133235.i 0.157757i 0.996884 + 0.0788784i \(0.0251339\pi\)
−0.996884 + 0.0788784i \(0.974866\pi\)
\(920\) 52090.2 0.0615433
\(921\) 625680. 0.737621
\(922\) 305991.i 0.359954i
\(923\) 594402.i 0.697713i
\(924\) 5768.28i 0.00675619i
\(925\) 439329.i 0.513459i
\(926\) −563412. −0.657058
\(927\) 145729.i 0.169584i
\(928\) 59702.3i 0.0693258i
\(929\) 989613.i 1.14666i −0.819325 0.573329i \(-0.805652\pi\)
0.819325 0.573329i \(-0.194348\pi\)
\(930\) −100865. −0.116620
\(931\) 477013. 0.550340
\(932\) 732457.i 0.843239i
\(933\) 636714. 0.731444
\(934\) 716586. 0.821438
\(935\) 5630.16i 0.00644018i
\(936\) 147845. 0.168755
\(937\) 873861.i 0.995321i −0.867372 0.497661i \(-0.834192\pi\)
0.867372 0.497661i \(-0.165808\pi\)
\(938\) −26380.9 −0.0299836
\(939\) 119194.i 0.135184i
\(940\) 81131.3i 0.0918191i
\(941\) 506935.i 0.572496i 0.958156 + 0.286248i \(0.0924082\pi\)
−0.958156 + 0.286248i \(0.907592\pi\)
\(942\) 64706.5 0.0729199
\(943\) 867257.i 0.975269i
\(944\) −218648. 42728.4i −0.245359 0.0479482i
\(945\) 21679.2 0.0242761
\(946\) 19877.4i 0.0222114i
\(947\) 459717. 0.512614 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(948\) −150587. −0.167560
\(949\) 484281. 0.537731
\(950\) 631086.i 0.699264i
\(951\) −111723. −0.123532
\(952\) 219949.i 0.242688i
\(953\) 1.27134e6 1.39984 0.699918 0.714223i \(-0.253220\pi\)
0.699918 + 0.714223i \(0.253220\pi\)
\(954\) 169890.i 0.186669i
\(955\) 132992.i 0.145820i
\(956\) 355953. 0.389473
\(957\) 7135.25i 0.00779086i
\(958\) 20302.6i 0.0221218i
\(959\) 1.02065e6 1.10979
\(960\) 12334.9 0.0133842
\(961\) −1.26755e6 −1.37252
\(962\) 498269.i 0.538410i
\(963\) 257901. 0.278100
\(964\) −568461. −0.611712
\(965\) −107344. −0.115272
\(966\) −243207. −0.260629
\(967\) 1.36038e6i 1.45481i −0.686208 0.727405i \(-0.740726\pi\)
0.686208 0.727405i \(-0.259274\pi\)
\(968\) 330896.i 0.353135i
\(969\) −560300. −0.596723
\(970\) −184765. −0.196370
\(971\) 669130. 0.709695 0.354848 0.934924i \(-0.384533\pi\)
0.354848 + 0.934924i \(0.384533\pi\)
\(972\) −30304.0 −0.0320750
\(973\) −840016. −0.887282
\(974\) 519794.i 0.547915i
\(975\) 758875.i 0.798291i
\(976\) 269246.i 0.282650i
\(977\) 33771.9i 0.0353807i 0.999844 + 0.0176904i \(0.00563131\pi\)
−0.999844 + 0.0176904i \(0.994369\pi\)
\(978\) 239705.i 0.250611i
\(979\) −17024.2 −0.0177624
\(980\) −47856.5 −0.0498297
\(981\) 167871.i 0.174437i
\(982\) 1961.88i 0.00203446i
\(983\) 192587.i 0.199306i −0.995022 0.0996531i \(-0.968227\pi\)
0.995022 0.0996531i \(-0.0317733\pi\)
\(984\) 205365.i 0.212098i
\(985\) −48311.8 −0.0497945
\(986\) 272073.i 0.279854i
\(987\) 378799.i 0.388843i
\(988\) 715752.i 0.733244i
\(989\) −838087. −0.856834
\(990\) −1474.19 −0.00150413
\(991\) 137261.i 0.139766i 0.997555 + 0.0698828i \(0.0222625\pi\)
−0.997555 + 0.0698828i \(0.977737\pi\)
\(992\) 267950. 0.272289
\(993\) −736946. −0.747373
\(994\) 231542.i 0.234345i
\(995\) 213473. 0.215624
\(996\) 115564.i 0.116494i
\(997\) −514755. −0.517857 −0.258929 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(998\) 543802.i 0.545984i
\(999\) 102131.i 0.102335i
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 354.5.d.a.235.18 yes 40
3.2 odd 2 1062.5.d.b.235.11 40
59.58 odd 2 inner 354.5.d.a.235.17 40
177.176 even 2 1062.5.d.b.235.12 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.5.d.a.235.17 40 59.58 odd 2 inner
354.5.d.a.235.18 yes 40 1.1 even 1 trivial
1062.5.d.b.235.11 40 3.2 odd 2
1062.5.d.b.235.12 40 177.176 even 2