Properties

Label 117.5.j.b.109.4
Level $117$
Weight $5$
Character 117.109
Analytic conductor $12.094$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,5,Mod(73,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(4))
 
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("117.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 117.j (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: \(12.0942856808\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 109.4
Root \(-1.66937 + 1.66937i\) of defining polynomial
Character \(\chi\) \(=\) 117.109
Dual form 117.5.j.b.73.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.66937 + 1.66937i) q^{2} +10.4264i q^{4} +(28.4160 - 28.4160i) q^{5} +(-17.1341 - 17.1341i) q^{7} +(-44.1155 - 44.1155i) q^{8} +94.8736i q^{10} +(-54.2346 - 54.2346i) q^{11} +(-168.969 - 3.22849i) q^{13} +57.2062 q^{14} -19.5322 q^{16} -70.4411i q^{17} +(194.123 - 194.123i) q^{19} +(296.276 + 296.276i) q^{20} +181.075 q^{22} -716.329i q^{23} -989.935i q^{25} +(287.462 - 276.683i) q^{26} +(178.647 - 178.647i) q^{28} -725.866 q^{29} +(586.446 - 586.446i) q^{31} +(738.454 - 738.454i) q^{32} +(117.592 + 117.592i) q^{34} -973.762 q^{35} +(575.923 + 575.923i) q^{37} +648.126i q^{38} -2507.17 q^{40} +(1810.07 - 1810.07i) q^{41} +2879.48i q^{43} +(565.472 - 565.472i) q^{44} +(1195.82 + 1195.82i) q^{46} +(-1064.86 - 1064.86i) q^{47} -1813.85i q^{49} +(1652.57 + 1652.57i) q^{50} +(33.6615 - 1761.74i) q^{52} -2549.76 q^{53} -3082.26 q^{55} +1511.75i q^{56} +(1211.74 - 1211.74i) q^{58} +(2264.35 + 2264.35i) q^{59} +6434.68 q^{61} +1957.99i q^{62} +2152.99i q^{64} +(-4893.16 + 4709.68i) q^{65} +(-5472.27 + 5472.27i) q^{67} +734.447 q^{68} +(1625.57 - 1625.57i) q^{70} +(-4643.32 + 4643.32i) q^{71} +(-5605.57 - 5605.57i) q^{73} -1922.86 q^{74} +(2024.00 + 2024.00i) q^{76} +1858.52i q^{77} -6871.05 q^{79} +(-555.027 + 555.027i) q^{80} +6043.36i q^{82} +(2647.22 - 2647.22i) q^{83} +(-2001.65 - 2001.65i) q^{85} +(-4806.92 - 4806.92i) q^{86} +4785.17i q^{88} +(6804.23 + 6804.23i) q^{89} +(2839.81 + 2950.45i) q^{91} +7468.73 q^{92} +3555.29 q^{94} -11032.4i q^{95} +(9498.95 - 9498.95i) q^{97} +(3027.99 + 3027.99i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.66937 + 1.66937i −0.417343 + 0.417343i −0.884287 0.466944i \(-0.845355\pi\)
0.466944 + 0.884287i \(0.345355\pi\)
\(3\) 0 0
\(4\) 10.4264i 0.651650i
\(5\) 28.4160 28.4160i 1.13664 1.13664i 0.147591 0.989049i \(-0.452848\pi\)
0.989049 0.147591i \(-0.0471518\pi\)
\(6\) 0 0
\(7\) −17.1341 17.1341i −0.349675 0.349675i 0.510314 0.859988i \(-0.329529\pi\)
−0.859988 + 0.510314i \(0.829529\pi\)
\(8\) −44.1155 44.1155i −0.689304 0.689304i
\(9\) 0 0
\(10\) 94.8736i 0.948736i
\(11\) −54.2346 54.2346i −0.448220 0.448220i 0.446542 0.894762i \(-0.352655\pi\)
−0.894762 + 0.446542i \(0.852655\pi\)
\(12\) 0 0
\(13\) −168.969 3.22849i −0.999818 0.0191035i
\(14\) 57.2062 0.291868
\(15\) 0 0
\(16\) −19.5322 −0.0762978
\(17\) 70.4411i 0.243741i −0.992546 0.121870i \(-0.961111\pi\)
0.992546 0.121870i \(-0.0388892\pi\)
\(18\) 0 0
\(19\) 194.123 194.123i 0.537736 0.537736i −0.385127 0.922863i \(-0.625843\pi\)
0.922863 + 0.385127i \(0.125843\pi\)
\(20\) 296.276 + 296.276i 0.740691 + 0.740691i
\(21\) 0 0
\(22\) 181.075 0.374123
\(23\) 716.329i 1.35412i −0.735928 0.677059i \(-0.763254\pi\)
0.735928 0.677059i \(-0.236746\pi\)
\(24\) 0 0
\(25\) 989.935i 1.58390i
\(26\) 287.462 276.683i 0.425239 0.409294i
\(27\) 0 0
\(28\) 178.647 178.647i 0.227866 0.227866i
\(29\) −725.866 −0.863099 −0.431550 0.902089i \(-0.642033\pi\)
−0.431550 + 0.902089i \(0.642033\pi\)
\(30\) 0 0
\(31\) 586.446 586.446i 0.610245 0.610245i −0.332765 0.943010i \(-0.607981\pi\)
0.943010 + 0.332765i \(0.107981\pi\)
\(32\) 738.454 738.454i 0.721147 0.721147i
\(33\) 0 0
\(34\) 117.592 + 117.592i 0.101723 + 0.101723i
\(35\) −973.762 −0.794908
\(36\) 0 0
\(37\) 575.923 + 575.923i 0.420689 + 0.420689i 0.885441 0.464752i \(-0.153856\pi\)
−0.464752 + 0.885441i \(0.653856\pi\)
\(38\) 648.126i 0.448840i
\(39\) 0 0
\(40\) −2507.17 −1.56698
\(41\) 1810.07 1810.07i 1.07678 1.07678i 0.0799854 0.996796i \(-0.474513\pi\)
0.996796 0.0799854i \(-0.0254874\pi\)
\(42\) 0 0
\(43\) 2879.48i 1.55732i 0.627448 + 0.778659i \(0.284100\pi\)
−0.627448 + 0.778659i \(0.715900\pi\)
\(44\) 565.472 565.472i 0.292083 0.292083i
\(45\) 0 0
\(46\) 1195.82 + 1195.82i 0.565132 + 0.565132i
\(47\) −1064.86 1064.86i −0.482055 0.482055i 0.423733 0.905787i \(-0.360720\pi\)
−0.905787 + 0.423733i \(0.860720\pi\)
\(48\) 0 0
\(49\) 1813.85i 0.755455i
\(50\) 1652.57 + 1652.57i 0.661028 + 0.661028i
\(51\) 0 0
\(52\) 33.6615 1761.74i 0.0124488 0.651531i
\(53\) −2549.76 −0.907710 −0.453855 0.891076i \(-0.649952\pi\)
−0.453855 + 0.891076i \(0.649952\pi\)
\(54\) 0 0
\(55\) −3082.26 −1.01893
\(56\) 1511.75i 0.482065i
\(57\) 0 0
\(58\) 1211.74 1211.74i 0.360208 0.360208i
\(59\) 2264.35 + 2264.35i 0.650487 + 0.650487i 0.953110 0.302623i \(-0.0978624\pi\)
−0.302623 + 0.953110i \(0.597862\pi\)
\(60\) 0 0
\(61\) 6434.68 1.72929 0.864644 0.502386i \(-0.167544\pi\)
0.864644 + 0.502386i \(0.167544\pi\)
\(62\) 1957.99i 0.509363i
\(63\) 0 0
\(64\) 2152.99i 0.525633i
\(65\) −4893.16 + 4709.68i −1.15815 + 1.11472i
\(66\) 0 0
\(67\) −5472.27 + 5472.27i −1.21904 + 1.21904i −0.251070 + 0.967969i \(0.580783\pi\)
−0.967969 + 0.251070i \(0.919217\pi\)
\(68\) 734.447 0.158834
\(69\) 0 0
\(70\) 1625.57 1625.57i 0.331749 0.331749i
\(71\) −4643.32 + 4643.32i −0.921111 + 0.921111i −0.997108 0.0759967i \(-0.975786\pi\)
0.0759967 + 0.997108i \(0.475786\pi\)
\(72\) 0 0
\(73\) −5605.57 5605.57i −1.05190 1.05190i −0.998577 0.0533222i \(-0.983019\pi\)
−0.0533222 0.998577i \(-0.516981\pi\)
\(74\) −1922.86 −0.351143
\(75\) 0 0
\(76\) 2024.00 + 2024.00i 0.350416 + 0.350416i
\(77\) 1858.52i 0.313462i
\(78\) 0 0
\(79\) −6871.05 −1.10095 −0.550477 0.834850i \(-0.685554\pi\)
−0.550477 + 0.834850i \(0.685554\pi\)
\(80\) −555.027 + 555.027i −0.0867230 + 0.0867230i
\(81\) 0 0
\(82\) 6043.36i 0.898774i
\(83\) 2647.22 2647.22i 0.384267 0.384267i −0.488370 0.872637i \(-0.662408\pi\)
0.872637 + 0.488370i \(0.162408\pi\)
\(84\) 0 0
\(85\) −2001.65 2001.65i −0.277045 0.277045i
\(86\) −4806.92 4806.92i −0.649935 0.649935i
\(87\) 0 0
\(88\) 4785.17i 0.617920i
\(89\) 6804.23 + 6804.23i 0.859012 + 0.859012i 0.991222 0.132210i \(-0.0422073\pi\)
−0.132210 + 0.991222i \(0.542207\pi\)
\(90\) 0 0
\(91\) 2839.81 + 2950.45i 0.342931 + 0.356291i
\(92\) 7468.73 0.882412
\(93\) 0 0
\(94\) 3555.29 0.402364
\(95\) 11032.4i 1.22242i
\(96\) 0 0
\(97\) 9498.95 9498.95i 1.00956 1.00956i 0.00960578 0.999954i \(-0.496942\pi\)
0.999954 0.00960578i \(-0.00305766\pi\)
\(98\) 3027.99 + 3027.99i 0.315284 + 0.315284i
\(99\) 0 0
\(100\) 10321.5 1.03215
\(101\) 16541.4i 1.62155i −0.585361 0.810773i \(-0.699047\pi\)
0.585361 0.810773i \(-0.300953\pi\)
\(102\) 0 0
\(103\) 11148.4i 1.05084i 0.850842 + 0.525421i \(0.176092\pi\)
−0.850842 + 0.525421i \(0.823908\pi\)
\(104\) 7311.73 + 7596.58i 0.676010 + 0.702347i
\(105\) 0 0
\(106\) 4256.49 4256.49i 0.378826 0.378826i
\(107\) −9291.29 −0.811537 −0.405769 0.913976i \(-0.632996\pi\)
−0.405769 + 0.913976i \(0.632996\pi\)
\(108\) 0 0
\(109\) 1985.75 1985.75i 0.167136 0.167136i −0.618583 0.785719i \(-0.712293\pi\)
0.785719 + 0.618583i \(0.212293\pi\)
\(110\) 5145.43 5145.43i 0.425243 0.425243i
\(111\) 0 0
\(112\) 334.666 + 334.666i 0.0266794 + 0.0266794i
\(113\) 1799.10 0.140896 0.0704478 0.997515i \(-0.477557\pi\)
0.0704478 + 0.997515i \(0.477557\pi\)
\(114\) 0 0
\(115\) −20355.2 20355.2i −1.53914 1.53914i
\(116\) 7568.17i 0.562439i
\(117\) 0 0
\(118\) −7560.07 −0.542952
\(119\) −1206.94 + 1206.94i −0.0852300 + 0.0852300i
\(120\) 0 0
\(121\) 8758.21i 0.598198i
\(122\) −10741.9 + 10741.9i −0.721706 + 0.721706i
\(123\) 0 0
\(124\) 6114.52 + 6114.52i 0.397666 + 0.397666i
\(125\) −10370.0 10370.0i −0.663680 0.663680i
\(126\) 0 0
\(127\) 7557.96i 0.468594i 0.972165 + 0.234297i \(0.0752789\pi\)
−0.972165 + 0.234297i \(0.924721\pi\)
\(128\) 8221.12 + 8221.12i 0.501777 + 0.501777i
\(129\) 0 0
\(130\) 306.299 16030.7i 0.0181242 0.948563i
\(131\) −9705.21 −0.565539 −0.282769 0.959188i \(-0.591253\pi\)
−0.282769 + 0.959188i \(0.591253\pi\)
\(132\) 0 0
\(133\) −6652.22 −0.376065
\(134\) 18270.5i 1.01751i
\(135\) 0 0
\(136\) −3107.54 + 3107.54i −0.168012 + 0.168012i
\(137\) 7306.03 + 7306.03i 0.389261 + 0.389261i 0.874424 0.485163i \(-0.161240\pi\)
−0.485163 + 0.874424i \(0.661240\pi\)
\(138\) 0 0
\(139\) 14508.5 0.750921 0.375460 0.926838i \(-0.377485\pi\)
0.375460 + 0.926838i \(0.377485\pi\)
\(140\) 10152.8i 0.518002i
\(141\) 0 0
\(142\) 15502.9i 0.768838i
\(143\) 8988.88 + 9339.07i 0.439576 + 0.456701i
\(144\) 0 0
\(145\) −20626.2 + 20626.2i −0.981032 + 0.981032i
\(146\) 18715.6 0.878005
\(147\) 0 0
\(148\) −6004.81 + 6004.81i −0.274142 + 0.274142i
\(149\) −6209.08 + 6209.08i −0.279675 + 0.279675i −0.832979 0.553304i \(-0.813367\pi\)
0.553304 + 0.832979i \(0.313367\pi\)
\(150\) 0 0
\(151\) 6322.22 + 6322.22i 0.277278 + 0.277278i 0.832021 0.554743i \(-0.187184\pi\)
−0.554743 + 0.832021i \(0.687184\pi\)
\(152\) −17127.6 −0.741327
\(153\) 0 0
\(154\) −3102.56 3102.56i −0.130821 0.130821i
\(155\) 33328.9i 1.38726i
\(156\) 0 0
\(157\) 26942.7 1.09305 0.546526 0.837442i \(-0.315950\pi\)
0.546526 + 0.837442i \(0.315950\pi\)
\(158\) 11470.3 11470.3i 0.459475 0.459475i
\(159\) 0 0
\(160\) 41967.8i 1.63937i
\(161\) −12273.6 + 12273.6i −0.473501 + 0.473501i
\(162\) 0 0
\(163\) 22832.2 + 22832.2i 0.859355 + 0.859355i 0.991262 0.131907i \(-0.0421101\pi\)
−0.131907 + 0.991262i \(0.542110\pi\)
\(164\) 18872.5 + 18872.5i 0.701685 + 0.701685i
\(165\) 0 0
\(166\) 8838.37i 0.320742i
\(167\) −7664.75 7664.75i −0.274831 0.274831i 0.556211 0.831041i \(-0.312255\pi\)
−0.831041 + 0.556211i \(0.812255\pi\)
\(168\) 0 0
\(169\) 28540.2 + 1091.03i 0.999270 + 0.0382000i
\(170\) 6683.00 0.231246
\(171\) 0 0
\(172\) −30022.6 −1.01483
\(173\) 9382.63i 0.313496i 0.987639 + 0.156748i \(0.0501011\pi\)
−0.987639 + 0.156748i \(0.949899\pi\)
\(174\) 0 0
\(175\) −16961.6 + 16961.6i −0.553849 + 0.553849i
\(176\) 1059.32 + 1059.32i 0.0341982 + 0.0341982i
\(177\) 0 0
\(178\) −22717.6 −0.717005
\(179\) 322.023i 0.0100503i −0.999987 0.00502517i \(-0.998400\pi\)
0.999987 0.00502517i \(-0.00159957\pi\)
\(180\) 0 0
\(181\) 28479.3i 0.869306i 0.900598 + 0.434653i \(0.143129\pi\)
−0.900598 + 0.434653i \(0.856871\pi\)
\(182\) −9666.09 184.690i −0.291815 0.00557571i
\(183\) 0 0
\(184\) −31601.2 + 31601.2i −0.933400 + 0.933400i
\(185\) 32730.8 0.956343
\(186\) 0 0
\(187\) −3820.35 + 3820.35i −0.109249 + 0.109249i
\(188\) 11102.6 11102.6i 0.314131 0.314131i
\(189\) 0 0
\(190\) 18417.1 + 18417.1i 0.510170 + 0.510170i
\(191\) −39049.7 −1.07041 −0.535206 0.844722i \(-0.679766\pi\)
−0.535206 + 0.844722i \(0.679766\pi\)
\(192\) 0 0
\(193\) −34849.0 34849.0i −0.935570 0.935570i 0.0624768 0.998046i \(-0.480100\pi\)
−0.998046 + 0.0624768i \(0.980100\pi\)
\(194\) 31714.5i 0.842665i
\(195\) 0 0
\(196\) 18911.9 0.492292
\(197\) −33755.4 + 33755.4i −0.869784 + 0.869784i −0.992448 0.122664i \(-0.960856\pi\)
0.122664 + 0.992448i \(0.460856\pi\)
\(198\) 0 0
\(199\) 12091.0i 0.305322i 0.988279 + 0.152661i \(0.0487842\pi\)
−0.988279 + 0.152661i \(0.951216\pi\)
\(200\) −43671.5 + 43671.5i −1.09179 + 1.09179i
\(201\) 0 0
\(202\) 27613.7 + 27613.7i 0.676740 + 0.676740i
\(203\) 12437.0 + 12437.0i 0.301804 + 0.301804i
\(204\) 0 0
\(205\) 102870.i 2.44782i
\(206\) −18610.8 18610.8i −0.438561 0.438561i
\(207\) 0 0
\(208\) 3300.34 + 63.0596i 0.0762838 + 0.00145755i
\(209\) −21056.3 −0.482048
\(210\) 0 0
\(211\) 51063.6 1.14695 0.573477 0.819221i \(-0.305594\pi\)
0.573477 + 0.819221i \(0.305594\pi\)
\(212\) 26584.8i 0.591509i
\(213\) 0 0
\(214\) 15510.6 15510.6i 0.338689 0.338689i
\(215\) 81823.2 + 81823.2i 1.77011 + 1.77011i
\(216\) 0 0
\(217\) −20096.4 −0.426775
\(218\) 6629.90i 0.139506i
\(219\) 0 0
\(220\) 32136.9i 0.663985i
\(221\) −227.418 + 11902.4i −0.00465630 + 0.243696i
\(222\) 0 0
\(223\) 7861.80 7861.80i 0.158093 0.158093i −0.623628 0.781721i \(-0.714342\pi\)
0.781721 + 0.623628i \(0.214342\pi\)
\(224\) −25305.4 −0.504333
\(225\) 0 0
\(226\) −3003.36 + 3003.36i −0.0588018 + 0.0588018i
\(227\) 9642.38 9642.38i 0.187125 0.187125i −0.607327 0.794452i \(-0.707758\pi\)
0.794452 + 0.607327i \(0.207758\pi\)
\(228\) 0 0
\(229\) 36285.4 + 36285.4i 0.691928 + 0.691928i 0.962656 0.270728i \(-0.0872643\pi\)
−0.270728 + 0.962656i \(0.587264\pi\)
\(230\) 67960.7 1.28470
\(231\) 0 0
\(232\) 32021.9 + 32021.9i 0.594938 + 0.594938i
\(233\) 2365.03i 0.0435637i 0.999763 + 0.0217819i \(0.00693393\pi\)
−0.999763 + 0.0217819i \(0.993066\pi\)
\(234\) 0 0
\(235\) −60518.0 −1.09584
\(236\) −23609.0 + 23609.0i −0.423890 + 0.423890i
\(237\) 0 0
\(238\) 4029.67i 0.0711402i
\(239\) 18841.4 18841.4i 0.329850 0.329850i −0.522679 0.852529i \(-0.675067\pi\)
0.852529 + 0.522679i \(0.175067\pi\)
\(240\) 0 0
\(241\) −27569.6 27569.6i −0.474676 0.474676i 0.428748 0.903424i \(-0.358955\pi\)
−0.903424 + 0.428748i \(0.858955\pi\)
\(242\) 14620.7 + 14620.7i 0.249653 + 0.249653i
\(243\) 0 0
\(244\) 67090.5i 1.12689i
\(245\) −51542.3 51542.3i −0.858680 0.858680i
\(246\) 0 0
\(247\) −33427.5 + 32174.0i −0.547910 + 0.527365i
\(248\) −51742.7 −0.841289
\(249\) 0 0
\(250\) 34622.8 0.553964
\(251\) 52279.9i 0.829826i −0.909861 0.414913i \(-0.863812\pi\)
0.909861 0.414913i \(-0.136188\pi\)
\(252\) 0 0
\(253\) −38849.8 + 38849.8i −0.606943 + 0.606943i
\(254\) −12617.0 12617.0i −0.195564 0.195564i
\(255\) 0 0
\(256\) −61896.1 −0.944459
\(257\) 87118.8i 1.31900i −0.751703 0.659501i \(-0.770768\pi\)
0.751703 0.659501i \(-0.229232\pi\)
\(258\) 0 0
\(259\) 19735.8i 0.294209i
\(260\) −49105.0 51018.1i −0.726406 0.754705i
\(261\) 0 0
\(262\) 16201.6 16201.6i 0.236023 0.236023i
\(263\) 92681.2 1.33992 0.669962 0.742395i \(-0.266310\pi\)
0.669962 + 0.742395i \(0.266310\pi\)
\(264\) 0 0
\(265\) −72453.8 + 72453.8i −1.03174 + 1.03174i
\(266\) 11105.0 11105.0i 0.156948 0.156948i
\(267\) 0 0
\(268\) −57056.0 57056.0i −0.794387 0.794387i
\(269\) 123465. 1.70624 0.853118 0.521717i \(-0.174708\pi\)
0.853118 + 0.521717i \(0.174708\pi\)
\(270\) 0 0
\(271\) 60295.3 + 60295.3i 0.821003 + 0.821003i 0.986252 0.165249i \(-0.0528428\pi\)
−0.165249 + 0.986252i \(0.552843\pi\)
\(272\) 1375.87i 0.0185969i
\(273\) 0 0
\(274\) −24393.0 −0.324910
\(275\) −53688.8 + 53688.8i −0.709934 + 0.709934i
\(276\) 0 0
\(277\) 62673.2i 0.816812i −0.912800 0.408406i \(-0.866085\pi\)
0.912800 0.408406i \(-0.133915\pi\)
\(278\) −24220.1 + 24220.1i −0.313391 + 0.313391i
\(279\) 0 0
\(280\) 42958.0 + 42958.0i 0.547933 + 0.547933i
\(281\) −17050.4 17050.4i −0.215934 0.215934i 0.590848 0.806783i \(-0.298793\pi\)
−0.806783 + 0.590848i \(0.798793\pi\)
\(282\) 0 0
\(283\) 135558.i 1.69259i −0.532716 0.846294i \(-0.678829\pi\)
0.532716 0.846294i \(-0.321171\pi\)
\(284\) −48413.1 48413.1i −0.600242 0.600242i
\(285\) 0 0
\(286\) −30596.2 584.600i −0.374054 0.00714705i
\(287\) −62027.7 −0.753047
\(288\) 0 0
\(289\) 78559.1 0.940590
\(290\) 68865.6i 0.818854i
\(291\) 0 0
\(292\) 58445.9 58445.9i 0.685470 0.685470i
\(293\) 33642.1 + 33642.1i 0.391876 + 0.391876i 0.875356 0.483480i \(-0.160627\pi\)
−0.483480 + 0.875356i \(0.660627\pi\)
\(294\) 0 0
\(295\) 128687. 1.47874
\(296\) 50814.3i 0.579966i
\(297\) 0 0
\(298\) 20730.5i 0.233441i
\(299\) −2312.66 + 121037.i −0.0258684 + 1.35387i
\(300\) 0 0
\(301\) 49337.2 49337.2i 0.544555 0.544555i
\(302\) −21108.2 −0.231440
\(303\) 0 0
\(304\) −3791.65 + 3791.65i −0.0410281 + 0.0410281i
\(305\) 182848. 182848.i 1.96558 1.96558i
\(306\) 0 0
\(307\) −121981. 121981.i −1.29425 1.29425i −0.932135 0.362112i \(-0.882056\pi\)
−0.362112 0.932135i \(-0.617944\pi\)
\(308\) −19377.7 −0.204268
\(309\) 0 0
\(310\) 55638.2 + 55638.2i 0.578962 + 0.578962i
\(311\) 1370.53i 0.0141699i 0.999975 + 0.00708495i \(0.00225523\pi\)
−0.999975 + 0.00708495i \(0.997745\pi\)
\(312\) 0 0
\(313\) 52462.0 0.535495 0.267748 0.963489i \(-0.413721\pi\)
0.267748 + 0.963489i \(0.413721\pi\)
\(314\) −44977.3 + 44977.3i −0.456178 + 0.456178i
\(315\) 0 0
\(316\) 71640.4i 0.717437i
\(317\) 101779. 101779.i 1.01284 1.01284i 0.0129239 0.999916i \(-0.495886\pi\)
0.999916 0.0129239i \(-0.00411392\pi\)
\(318\) 0 0
\(319\) 39367.1 + 39367.1i 0.386858 + 0.386858i
\(320\) 61179.4 + 61179.4i 0.597455 + 0.597455i
\(321\) 0 0
\(322\) 40978.5i 0.395225i
\(323\) −13674.2 13674.2i −0.131068 0.131068i
\(324\) 0 0
\(325\) −3196.00 + 167269.i −0.0302580 + 1.58361i
\(326\) −76230.8 −0.717291
\(327\) 0 0
\(328\) −159704. −1.48446
\(329\) 36490.7i 0.337125i
\(330\) 0 0
\(331\) 90476.9 90476.9i 0.825813 0.825813i −0.161122 0.986935i \(-0.551511\pi\)
0.986935 + 0.161122i \(0.0515112\pi\)
\(332\) 27600.9 + 27600.9i 0.250408 + 0.250408i
\(333\) 0 0
\(334\) 25590.6 0.229397
\(335\) 311000.i 2.77122i
\(336\) 0 0
\(337\) 48432.3i 0.426457i −0.977002 0.213228i \(-0.931602\pi\)
0.977002 0.213228i \(-0.0683979\pi\)
\(338\) −49465.4 + 45822.8i −0.432981 + 0.401096i
\(339\) 0 0
\(340\) 20870.0 20870.0i 0.180537 0.180537i
\(341\) −63611.3 −0.547048
\(342\) 0 0
\(343\) −72217.5 + 72217.5i −0.613838 + 0.613838i
\(344\) 127030. 127030.i 1.07347 1.07347i
\(345\) 0 0
\(346\) −15663.1 15663.1i −0.130835 0.130835i
\(347\) −50065.9 −0.415799 −0.207899 0.978150i \(-0.566663\pi\)
−0.207899 + 0.978150i \(0.566663\pi\)
\(348\) 0 0
\(349\) −36678.7 36678.7i −0.301137 0.301137i 0.540322 0.841458i \(-0.318302\pi\)
−0.841458 + 0.540322i \(0.818302\pi\)
\(350\) 56630.5i 0.462290i
\(351\) 0 0
\(352\) −80099.5 −0.646465
\(353\) 91471.6 91471.6i 0.734069 0.734069i −0.237354 0.971423i \(-0.576280\pi\)
0.971423 + 0.237354i \(0.0762803\pi\)
\(354\) 0 0
\(355\) 263889.i 2.09394i
\(356\) −70943.6 + 70943.6i −0.559775 + 0.559775i
\(357\) 0 0
\(358\) 537.576 + 537.576i 0.00419444 + 0.00419444i
\(359\) −19904.6 19904.6i −0.154442 0.154442i 0.625657 0.780098i \(-0.284831\pi\)
−0.780098 + 0.625657i \(0.784831\pi\)
\(360\) 0 0
\(361\) 54953.8i 0.421680i
\(362\) −47542.6 47542.6i −0.362799 0.362799i
\(363\) 0 0
\(364\) −30762.5 + 29609.0i −0.232177 + 0.223471i
\(365\) −318576. −2.39126
\(366\) 0 0
\(367\) 13539.5 0.100524 0.0502620 0.998736i \(-0.483994\pi\)
0.0502620 + 0.998736i \(0.483994\pi\)
\(368\) 13991.5i 0.103316i
\(369\) 0 0
\(370\) −54639.9 + 54639.9i −0.399123 + 0.399123i
\(371\) 43687.7 + 43687.7i 0.317403 + 0.317403i
\(372\) 0 0
\(373\) −76501.0 −0.549857 −0.274928 0.961465i \(-0.588654\pi\)
−0.274928 + 0.961465i \(0.588654\pi\)
\(374\) 12755.1i 0.0911890i
\(375\) 0 0
\(376\) 93953.5i 0.664565i
\(377\) 122649. + 2343.45i 0.862942 + 0.0164882i
\(378\) 0 0
\(379\) 9873.34 9873.34i 0.0687362 0.0687362i −0.671903 0.740639i \(-0.734523\pi\)
0.740639 + 0.671903i \(0.234523\pi\)
\(380\) 115028. 0.796592
\(381\) 0 0
\(382\) 65188.4 65188.4i 0.446729 0.446729i
\(383\) −92064.1 + 92064.1i −0.627614 + 0.627614i −0.947467 0.319853i \(-0.896366\pi\)
0.319853 + 0.947467i \(0.396366\pi\)
\(384\) 0 0
\(385\) 52811.6 + 52811.6i 0.356294 + 0.356294i
\(386\) 116352. 0.780906
\(387\) 0 0
\(388\) 99039.8 + 99039.8i 0.657880 + 0.657880i
\(389\) 35489.2i 0.234529i −0.993101 0.117265i \(-0.962587\pi\)
0.993101 0.117265i \(-0.0374125\pi\)
\(390\) 0 0
\(391\) −50459.0 −0.330054
\(392\) −80018.7 + 80018.7i −0.520738 + 0.520738i
\(393\) 0 0
\(394\) 112701.i 0.725996i
\(395\) −195248. + 195248.i −1.25139 + 1.25139i
\(396\) 0 0
\(397\) 45764.3 + 45764.3i 0.290366 + 0.290366i 0.837225 0.546859i \(-0.184177\pi\)
−0.546859 + 0.837225i \(0.684177\pi\)
\(398\) −20184.4 20184.4i −0.127424 0.127424i
\(399\) 0 0
\(400\) 19335.6i 0.120848i
\(401\) −53143.4 53143.4i −0.330492 0.330492i 0.522281 0.852773i \(-0.325081\pi\)
−0.852773 + 0.522281i \(0.825081\pi\)
\(402\) 0 0
\(403\) −100985. + 97197.9i −0.621792 + 0.598476i
\(404\) 172467. 1.05668
\(405\) 0 0
\(406\) −41524.1 −0.251911
\(407\) 62470.0i 0.377123i
\(408\) 0 0
\(409\) 75294.0 75294.0i 0.450105 0.450105i −0.445284 0.895389i \(-0.646897\pi\)
0.895389 + 0.445284i \(0.146897\pi\)
\(410\) 171728. + 171728.i 1.02158 + 1.02158i
\(411\) 0 0
\(412\) −116237. −0.684781
\(413\) 77594.9i 0.454918i
\(414\) 0 0
\(415\) 150447.i 0.873546i
\(416\) −127160. + 122392.i −0.734791 + 0.707239i
\(417\) 0 0
\(418\) 35150.8 35150.8i 0.201179 0.201179i
\(419\) 264545. 1.50686 0.753429 0.657530i \(-0.228399\pi\)
0.753429 + 0.657530i \(0.228399\pi\)
\(420\) 0 0
\(421\) 30687.4 30687.4i 0.173140 0.173140i −0.615218 0.788357i \(-0.710932\pi\)
0.788357 + 0.615218i \(0.210932\pi\)
\(422\) −85244.1 + 85244.1i −0.478673 + 0.478673i
\(423\) 0 0
\(424\) 112484. + 112484.i 0.625688 + 0.625688i
\(425\) −69732.1 −0.386060
\(426\) 0 0
\(427\) −110252. 110252.i −0.604688 0.604688i
\(428\) 96874.7i 0.528838i
\(429\) 0 0
\(430\) −273187. −1.47748
\(431\) −138755. + 138755.i −0.746956 + 0.746956i −0.973906 0.226950i \(-0.927124\pi\)
0.226950 + 0.973906i \(0.427124\pi\)
\(432\) 0 0
\(433\) 192478.i 1.02661i −0.858206 0.513305i \(-0.828421\pi\)
0.858206 0.513305i \(-0.171579\pi\)
\(434\) 33548.3 33548.3i 0.178111 0.178111i
\(435\) 0 0
\(436\) 20704.2 + 20704.2i 0.108914 + 0.108914i
\(437\) −139056. 139056.i −0.728158 0.728158i
\(438\) 0 0
\(439\) 221370.i 1.14865i 0.818626 + 0.574327i \(0.194736\pi\)
−0.818626 + 0.574327i \(0.805264\pi\)
\(440\) 135975. + 135975.i 0.702352 + 0.702352i
\(441\) 0 0
\(442\) −19489.8 20249.1i −0.0997616 0.103648i
\(443\) −354252. −1.80511 −0.902557 0.430570i \(-0.858313\pi\)
−0.902557 + 0.430570i \(0.858313\pi\)
\(444\) 0 0
\(445\) 386698. 1.95277
\(446\) 26248.5i 0.131958i
\(447\) 0 0
\(448\) 36889.5 36889.5i 0.183801 0.183801i
\(449\) 187721. + 187721.i 0.931152 + 0.931152i 0.997778 0.0666259i \(-0.0212234\pi\)
−0.0666259 + 0.997778i \(0.521223\pi\)
\(450\) 0 0
\(451\) −196337. −0.965270
\(452\) 18758.1i 0.0918147i
\(453\) 0 0
\(454\) 32193.4i 0.156191i
\(455\) 164536. + 3143.78i 0.794763 + 0.0151855i
\(456\) 0 0
\(457\) −88792.4 + 88792.4i −0.425151 + 0.425151i −0.886973 0.461822i \(-0.847196\pi\)
0.461822 + 0.886973i \(0.347196\pi\)
\(458\) −121148. −0.577543
\(459\) 0 0
\(460\) 212231. 212231.i 1.00298 1.00298i
\(461\) 56596.4 56596.4i 0.266310 0.266310i −0.561302 0.827611i \(-0.689699\pi\)
0.827611 + 0.561302i \(0.189699\pi\)
\(462\) 0 0
\(463\) 171067. + 171067.i 0.798003 + 0.798003i 0.982780 0.184777i \(-0.0591563\pi\)
−0.184777 + 0.982780i \(0.559156\pi\)
\(464\) 14177.8 0.0658526
\(465\) 0 0
\(466\) −3948.11 3948.11i −0.0181810 0.0181810i
\(467\) 68140.5i 0.312443i −0.987722 0.156222i \(-0.950069\pi\)
0.987722 0.156222i \(-0.0499314\pi\)
\(468\) 0 0
\(469\) 187524. 0.852534
\(470\) 101027. 101027.i 0.457343 0.457343i
\(471\) 0 0
\(472\) 199785.i 0.896767i
\(473\) 156168. 156168.i 0.698021 0.698021i
\(474\) 0 0
\(475\) −192169. 192169.i −0.851718 0.851718i
\(476\) −12584.1 12584.1i −0.0555401 0.0555401i
\(477\) 0 0
\(478\) 62906.5i 0.275321i
\(479\) 113713. + 113713.i 0.495610 + 0.495610i 0.910068 0.414458i \(-0.136029\pi\)
−0.414458 + 0.910068i \(0.636029\pi\)
\(480\) 0 0
\(481\) −95453.9 99172.6i −0.412576 0.428649i
\(482\) 92047.9 0.396205
\(483\) 0 0
\(484\) 91316.6 0.389816
\(485\) 539844.i 2.29501i
\(486\) 0 0
\(487\) 30988.8 30988.8i 0.130661 0.130661i −0.638752 0.769413i \(-0.720549\pi\)
0.769413 + 0.638752i \(0.220549\pi\)
\(488\) −283869. 283869.i −1.19201 1.19201i
\(489\) 0 0
\(490\) 172086. 0.716728
\(491\) 229476.i 0.951861i −0.879483 0.475931i \(-0.842111\pi\)
0.879483 0.475931i \(-0.157889\pi\)
\(492\) 0 0
\(493\) 51130.8i 0.210372i
\(494\) 2092.47 109513.i 0.00857442 0.448759i
\(495\) 0 0
\(496\) −11454.6 + 11454.6i −0.0465604 + 0.0465604i
\(497\) 159118. 0.644179
\(498\) 0 0
\(499\) 29259.2 29259.2i 0.117506 0.117506i −0.645909 0.763415i \(-0.723521\pi\)
0.763415 + 0.645909i \(0.223521\pi\)
\(500\) 108122. 108122.i 0.432487 0.432487i
\(501\) 0 0
\(502\) 87274.5 + 87274.5i 0.346322 + 0.346322i
\(503\) −432230. −1.70836 −0.854180 0.519978i \(-0.825940\pi\)
−0.854180 + 0.519978i \(0.825940\pi\)
\(504\) 0 0
\(505\) −470040. 470040.i −1.84311 1.84311i
\(506\) 129710.i 0.506607i
\(507\) 0 0
\(508\) −78802.3 −0.305360
\(509\) 150630. 150630.i 0.581402 0.581402i −0.353886 0.935289i \(-0.615140\pi\)
0.935289 + 0.353886i \(0.115140\pi\)
\(510\) 0 0
\(511\) 192092.i 0.735645i
\(512\) −28210.4 + 28210.4i −0.107614 + 0.107614i
\(513\) 0 0
\(514\) 145434. + 145434.i 0.550476 + 0.550476i
\(515\) 316792. + 316792.i 1.19443 + 1.19443i
\(516\) 0 0
\(517\) 115504.i 0.432133i
\(518\) 32946.4 + 32946.4i 0.122786 + 0.122786i
\(519\) 0 0
\(520\) 423634. + 8094.37i 1.56669 + 0.0299348i
\(521\) 152305. 0.561096 0.280548 0.959840i \(-0.409484\pi\)
0.280548 + 0.959840i \(0.409484\pi\)
\(522\) 0 0
\(523\) 78780.8 0.288016 0.144008 0.989577i \(-0.454001\pi\)
0.144008 + 0.989577i \(0.454001\pi\)
\(524\) 101190.i 0.368533i
\(525\) 0 0
\(526\) −154719. + 154719.i −0.559208 + 0.559208i
\(527\) −41309.9 41309.9i −0.148742 0.148742i
\(528\) 0 0
\(529\) −233286. −0.833638
\(530\) 241905.i 0.861177i
\(531\) 0 0
\(532\) 69358.7i 0.245063i
\(533\) −311690. + 300002.i −1.09716 + 1.05601i
\(534\) 0 0
\(535\) −264021. + 264021.i −0.922425 + 0.922425i
\(536\) 482823. 1.68058
\(537\) 0 0
\(538\) −206109. + 206109.i −0.712086 + 0.712086i
\(539\) −98373.3 + 98373.3i −0.338610 + 0.338610i
\(540\) 0 0
\(541\) −35408.4 35408.4i −0.120979 0.120979i 0.644025 0.765004i \(-0.277263\pi\)
−0.765004 + 0.644025i \(0.777263\pi\)
\(542\) −201310. −0.685279
\(543\) 0 0
\(544\) −52017.5 52017.5i −0.175773 0.175773i
\(545\) 112854.i 0.379947i
\(546\) 0 0
\(547\) 10939.9 0.0365628 0.0182814 0.999833i \(-0.494181\pi\)
0.0182814 + 0.999833i \(0.494181\pi\)
\(548\) −76175.6 + 76175.6i −0.253662 + 0.253662i
\(549\) 0 0
\(550\) 179253.i 0.592572i
\(551\) −140907. + 140907.i −0.464120 + 0.464120i
\(552\) 0 0
\(553\) 117729. + 117729.i 0.384976 + 0.384976i
\(554\) 104625. + 104625.i 0.340891 + 0.340891i
\(555\) 0 0
\(556\) 151272.i 0.489338i
\(557\) −32717.3 32717.3i −0.105455 0.105455i 0.652411 0.757866i \(-0.273758\pi\)
−0.757866 + 0.652411i \(0.773758\pi\)
\(558\) 0 0
\(559\) 9296.38 486543.i 0.0297502 1.55703i
\(560\) 19019.7 0.0606497
\(561\) 0 0
\(562\) 56926.8 0.180237
\(563\) 180795.i 0.570388i −0.958470 0.285194i \(-0.907942\pi\)
0.958470 0.285194i \(-0.0920580\pi\)
\(564\) 0 0
\(565\) 51123.1 51123.1i 0.160148 0.160148i
\(566\) 226296. + 226296.i 0.706390 + 0.706390i
\(567\) 0 0
\(568\) 409685. 1.26985
\(569\) 500901.i 1.54713i 0.633715 + 0.773566i \(0.281529\pi\)
−0.633715 + 0.773566i \(0.718471\pi\)
\(570\) 0 0
\(571\) 246396.i 0.755722i −0.925862 0.377861i \(-0.876660\pi\)
0.925862 0.377861i \(-0.123340\pi\)
\(572\) −97372.9 + 93721.7i −0.297609 + 0.286449i
\(573\) 0 0
\(574\) 103547. 103547.i 0.314279 0.314279i
\(575\) −709119. −2.14478
\(576\) 0 0
\(577\) −184815. + 184815.i −0.555119 + 0.555119i −0.927914 0.372794i \(-0.878400\pi\)
0.372794 + 0.927914i \(0.378400\pi\)
\(578\) −131144. + 131144.i −0.392549 + 0.392549i
\(579\) 0 0
\(580\) −215057. 215057.i −0.639290 0.639290i
\(581\) −90715.2 −0.268737
\(582\) 0 0
\(583\) 138285. + 138285.i 0.406854 + 0.406854i
\(584\) 494585.i 1.45016i
\(585\) 0 0
\(586\) −112322. −0.327093
\(587\) 402989. 402989.i 1.16954 1.16954i 0.187227 0.982317i \(-0.440050\pi\)
0.982317 0.187227i \(-0.0599501\pi\)
\(588\) 0 0
\(589\) 227685.i 0.656302i
\(590\) −214827. + 214827.i −0.617141 + 0.617141i
\(591\) 0 0
\(592\) −11249.1 11249.1i −0.0320976 0.0320976i
\(593\) −180607. 180607.i −0.513601 0.513601i 0.402027 0.915628i \(-0.368306\pi\)
−0.915628 + 0.402027i \(0.868306\pi\)
\(594\) 0 0
\(595\) 68592.9i 0.193751i
\(596\) −64738.3 64738.3i −0.182251 0.182251i
\(597\) 0 0
\(598\) −198196. 205917.i −0.554233 0.575825i
\(599\) 461474. 1.28615 0.643077 0.765801i \(-0.277657\pi\)
0.643077 + 0.765801i \(0.277657\pi\)
\(600\) 0 0
\(601\) −482801. −1.33666 −0.668328 0.743867i \(-0.732990\pi\)
−0.668328 + 0.743867i \(0.732990\pi\)
\(602\) 164724.i 0.454532i
\(603\) 0 0
\(604\) −65918.0 + 65918.0i −0.180688 + 0.180688i
\(605\) −248873. 248873.i −0.679935 0.679935i
\(606\) 0 0
\(607\) 219694. 0.596266 0.298133 0.954524i \(-0.403636\pi\)
0.298133 + 0.954524i \(0.403636\pi\)
\(608\) 286701.i 0.775573i
\(609\) 0 0
\(610\) 610481.i 1.64064i
\(611\) 176490. + 183366.i 0.472758 + 0.491176i
\(612\) 0 0
\(613\) 193600. 193600.i 0.515209 0.515209i −0.400909 0.916118i \(-0.631306\pi\)
0.916118 + 0.400909i \(0.131306\pi\)
\(614\) 407265. 1.08029
\(615\) 0 0
\(616\) 81989.4 81989.4i 0.216071 0.216071i
\(617\) −234942. + 234942.i −0.617149 + 0.617149i −0.944799 0.327650i \(-0.893743\pi\)
0.327650 + 0.944799i \(0.393743\pi\)
\(618\) 0 0
\(619\) 518107. + 518107.i 1.35219 + 1.35219i 0.883206 + 0.468984i \(0.155380\pi\)
0.468984 + 0.883206i \(0.344620\pi\)
\(620\) 347500. 0.904006
\(621\) 0 0
\(622\) −2287.92 2287.92i −0.00591371 0.00591371i
\(623\) 233168.i 0.600749i
\(624\) 0 0
\(625\) 29362.4 0.0751678
\(626\) −87578.5 + 87578.5i −0.223485 + 0.223485i
\(627\) 0 0
\(628\) 280915.i 0.712288i
\(629\) 40568.7 40568.7i 0.102539 0.102539i
\(630\) 0 0
\(631\) −335596. 335596.i −0.842866 0.842866i 0.146365 0.989231i \(-0.453243\pi\)
−0.989231 + 0.146365i \(0.953243\pi\)
\(632\) 303120. + 303120.i 0.758892 + 0.758892i
\(633\) 0 0
\(634\) 339815.i 0.845403i
\(635\) 214767. + 214767.i 0.532623 + 0.532623i
\(636\) 0 0
\(637\) −5855.99 + 306484.i −0.0144318 + 0.755317i
\(638\) −131437. −0.322905
\(639\) 0 0
\(640\) 467222. 1.14068
\(641\) 555098.i 1.35099i 0.737362 + 0.675497i \(0.236071\pi\)
−0.737362 + 0.675497i \(0.763929\pi\)
\(642\) 0 0
\(643\) 263449. 263449.i 0.637197 0.637197i −0.312666 0.949863i \(-0.601222\pi\)
0.949863 + 0.312666i \(0.101222\pi\)
\(644\) −127970. 127970.i −0.308557 0.308557i
\(645\) 0 0
\(646\) 45654.7 0.109401
\(647\) 396185.i 0.946433i −0.880946 0.473216i \(-0.843093\pi\)
0.880946 0.473216i \(-0.156907\pi\)
\(648\) 0 0
\(649\) 245612.i 0.583123i
\(650\) −273898. 284569.i −0.648279 0.673535i
\(651\) 0 0
\(652\) −238058. + 238058.i −0.559999 + 0.559999i
\(653\) −825533. −1.93601 −0.968006 0.250929i \(-0.919264\pi\)
−0.968006 + 0.250929i \(0.919264\pi\)
\(654\) 0 0
\(655\) −275783. + 275783.i −0.642813 + 0.642813i
\(656\) −35354.7 + 35354.7i −0.0821560 + 0.0821560i
\(657\) 0 0
\(658\) −60916.5 60916.5i −0.140697 0.140697i
\(659\) −123474. −0.284317 −0.142159 0.989844i \(-0.545404\pi\)
−0.142159 + 0.989844i \(0.545404\pi\)
\(660\) 0 0
\(661\) 121255. + 121255.i 0.277522 + 0.277522i 0.832119 0.554597i \(-0.187128\pi\)
−0.554597 + 0.832119i \(0.687128\pi\)
\(662\) 302079.i 0.689294i
\(663\) 0 0
\(664\) −233566. −0.529754
\(665\) −189029. + 189029.i −0.427451 + 0.427451i
\(666\) 0 0
\(667\) 519959.i 1.16874i
\(668\) 79915.8 79915.8i 0.179093 0.179093i
\(669\) 0 0
\(670\) −519174. 519174.i −1.15655 1.15655i
\(671\) −348982. 348982.i −0.775101 0.775101i
\(672\) 0 0
\(673\) 140601.i 0.310425i −0.987881 0.155213i \(-0.950394\pi\)
0.987881 0.155213i \(-0.0496062\pi\)
\(674\) 80851.5 + 80851.5i 0.177979 + 0.177979i
\(675\) 0 0
\(676\) −11375.5 + 297571.i −0.0248930 + 0.651174i
\(677\) 543307. 1.18541 0.592704 0.805420i \(-0.298060\pi\)
0.592704 + 0.805420i \(0.298060\pi\)
\(678\) 0 0
\(679\) −325511. −0.706035
\(680\) 176608.i 0.381937i
\(681\) 0 0
\(682\) 106191. 106191.i 0.228307 0.228307i
\(683\) 475365. + 475365.i 1.01903 + 1.01903i 0.999815 + 0.0192116i \(0.00611561\pi\)
0.0192116 + 0.999815i \(0.493884\pi\)
\(684\) 0 0
\(685\) 415216. 0.884898
\(686\) 241115.i 0.512362i
\(687\) 0 0
\(688\) 56242.7i 0.118820i
\(689\) 430830. + 8231.87i 0.907544 + 0.0173404i
\(690\) 0 0
\(691\) −212706. + 212706.i −0.445476 + 0.445476i −0.893847 0.448371i \(-0.852004\pi\)
0.448371 + 0.893847i \(0.352004\pi\)
\(692\) −97827.1 −0.204290
\(693\) 0 0
\(694\) 83578.6 83578.6i 0.173531 0.173531i
\(695\) 412274. 412274.i 0.853526 0.853526i
\(696\) 0 0
\(697\) −127503. 127503.i −0.262456 0.262456i
\(698\) 122461. 0.251354
\(699\) 0 0
\(700\) −176849. 176849.i −0.360916 0.360916i
\(701\) 56550.4i 0.115080i −0.998343 0.0575400i \(-0.981674\pi\)
0.998343 0.0575400i \(-0.0183257\pi\)
\(702\) 0 0
\(703\) 223600. 0.452439
\(704\) 116767. 116767.i 0.235599 0.235599i
\(705\) 0 0
\(706\) 305400.i 0.612716i
\(707\) −283421. + 283421.i −0.567013 + 0.567013i
\(708\) 0 0
\(709\) 284754. + 284754.i 0.566470 + 0.566470i 0.931138 0.364668i \(-0.118818\pi\)
−0.364668 + 0.931138i \(0.618818\pi\)
\(710\) −440529. 440529.i −0.873892 0.873892i
\(711\) 0 0
\(712\) 600344.i 1.18424i
\(713\) −420088. 420088.i −0.826345 0.826345i
\(714\) 0 0
\(715\) 520807. + 9951.05i 1.01874 + 0.0194651i
\(716\) 3357.54 0.00654931
\(717\) 0 0
\(718\) 66456.3 0.128910
\(719\) 513980.i 0.994234i 0.867684 + 0.497117i \(0.165608\pi\)
−0.867684 + 0.497117i \(0.834392\pi\)
\(720\) 0 0
\(721\) 191017. 191017.i 0.367453 0.367453i
\(722\) −91738.2 91738.2i −0.175985 0.175985i
\(723\) 0 0
\(724\) −296937. −0.566483
\(725\) 718561.i 1.36706i
\(726\) 0 0
\(727\) 5325.23i 0.0100756i 0.999987 + 0.00503779i \(0.00160358\pi\)
−0.999987 + 0.00503779i \(0.998396\pi\)
\(728\) 4880.69 255440.i 0.00920912 0.481977i
\(729\) 0 0
\(730\) 531821. 531821.i 0.997975 0.997975i
\(731\) 202834. 0.379582
\(732\) 0 0
\(733\) 471793. 471793.i 0.878100 0.878100i −0.115238 0.993338i \(-0.536763\pi\)
0.993338 + 0.115238i \(0.0367631\pi\)
\(734\) −22602.4 + 22602.4i −0.0419530 + 0.0419530i
\(735\) 0 0
\(736\) −528976. 528976.i −0.976518 0.976518i
\(737\) 593573. 1.09280
\(738\) 0 0
\(739\) 383237. + 383237.i 0.701745 + 0.701745i 0.964785 0.263040i \(-0.0847252\pi\)
−0.263040 + 0.964785i \(0.584725\pi\)
\(740\) 341265.i 0.623201i
\(741\) 0 0
\(742\) −145862. −0.264932
\(743\) −77467.3 + 77467.3i −0.140327 + 0.140327i −0.773781 0.633454i \(-0.781637\pi\)
0.633454 + 0.773781i \(0.281637\pi\)
\(744\) 0 0
\(745\) 352874.i 0.635780i
\(746\) 127709. 127709.i 0.229479 0.229479i
\(747\) 0 0
\(748\) −39832.4 39832.4i −0.0711924 0.0711924i
\(749\) 159198. + 159198.i 0.283774 + 0.283774i
\(750\) 0 0
\(751\) 287709.i 0.510121i −0.966925 0.255061i \(-0.917905\pi\)
0.966925 0.255061i \(-0.0820954\pi\)
\(752\) 20799.1 + 20799.1i 0.0367797 + 0.0367797i
\(753\) 0 0
\(754\) −208659. + 200835.i −0.367024 + 0.353261i
\(755\) 359304. 0.630330
\(756\) 0 0
\(757\) −962044. −1.67882 −0.839408 0.543502i \(-0.817098\pi\)
−0.839408 + 0.543502i \(0.817098\pi\)
\(758\) 32964.5i 0.0573731i
\(759\) 0 0
\(760\) −486698. + 486698.i −0.842622 + 0.842622i
\(761\) −496273. 496273.i −0.856943 0.856943i 0.134034 0.990977i \(-0.457207\pi\)
−0.990977 + 0.134034i \(0.957207\pi\)
\(762\) 0 0
\(763\) −68047.8 −0.116887
\(764\) 407148.i 0.697534i
\(765\) 0 0
\(766\) 307378.i 0.523860i
\(767\) −375294. 389915.i −0.637942 0.662795i
\(768\) 0 0
\(769\) 326544. 326544.i 0.552191 0.552191i −0.374882 0.927073i \(-0.622317\pi\)
0.927073 + 0.374882i \(0.122317\pi\)
\(770\) −176324. −0.297393
\(771\) 0 0
\(772\) 363350. 363350.i 0.609664 0.609664i
\(773\) −213799. + 213799.i −0.357804 + 0.357804i −0.863003 0.505199i \(-0.831419\pi\)
0.505199 + 0.863003i \(0.331419\pi\)
\(774\) 0 0
\(775\) −580544. 580544.i −0.966566 0.966566i
\(776\) −838101. −1.39179
\(777\) 0 0
\(778\) 59244.6 + 59244.6i 0.0978790 + 0.0978790i
\(779\) 702751.i 1.15805i
\(780\) 0 0
\(781\) 503658. 0.825721
\(782\) 84234.8 84234.8i 0.137746 0.137746i
\(783\) 0 0
\(784\) 35428.5i 0.0576395i
\(785\) 765602. 765602.i 1.24241 1.24241i
\(786\) 0 0
\(787\) 642953. + 642953.i 1.03808 + 1.03808i 0.999246 + 0.0388320i \(0.0123637\pi\)
0.0388320 + 0.999246i \(0.487636\pi\)
\(788\) −351948. 351948.i −0.566795 0.566795i
\(789\) 0 0
\(790\) 651882.i 1.04452i
\(791\) −30825.8 30825.8i −0.0492677 0.0492677i
\(792\) 0 0
\(793\) −1.08726e6 20774.3i −1.72897 0.0330354i
\(794\) −152795. −0.242364
\(795\) 0 0
\(796\) −126066. −0.198963
\(797\) 735073.i 1.15721i 0.815606 + 0.578607i \(0.196404\pi\)
−0.815606 + 0.578607i \(0.803596\pi\)
\(798\) 0 0
\(799\) −75009.8 + 75009.8i −0.117496 + 0.117496i
\(800\) −731022. 731022.i −1.14222 1.14222i
\(801\) 0 0
\(802\) 177432. 0.275857
\(803\) 608032.i 0.942965i
\(804\) 0 0
\(805\) 697534.i 1.07640i
\(806\) 6321.36 330840.i 0.00973062 0.509270i
\(807\) 0 0
\(808\) −729731. + 729731.i −1.11774 + 1.11774i
\(809\) 388192. 0.593130 0.296565 0.955013i \(-0.404159\pi\)
0.296565 + 0.955013i \(0.404159\pi\)
\(810\) 0 0
\(811\) −163285. + 163285.i −0.248259 + 0.248259i −0.820256 0.571997i \(-0.806169\pi\)
0.571997 + 0.820256i \(0.306169\pi\)
\(812\) −129674. + 129674.i −0.196671 + 0.196671i
\(813\) 0 0
\(814\) 104286. + 104286.i 0.157389 + 0.157389i
\(815\) 1.29760e6 1.95355
\(816\) 0 0
\(817\) 558972. + 558972.i 0.837426 + 0.837426i
\(818\) 251387.i 0.375696i
\(819\) 0 0
\(820\) 1.07256e6 1.59512
\(821\) 401933. 401933.i 0.596303 0.596303i −0.343024 0.939327i \(-0.611451\pi\)
0.939327 + 0.343024i \(0.111451\pi\)
\(822\) 0 0
\(823\) 129326.i 0.190935i 0.995433 + 0.0954674i \(0.0304346\pi\)
−0.995433 + 0.0954674i \(0.969565\pi\)
\(824\) 491816. 491816.i 0.724350 0.724350i
\(825\) 0 0
\(826\) 129535. + 129535.i 0.189857 + 0.189857i
\(827\) −51675.4 51675.4i −0.0755567 0.0755567i 0.668319 0.743875i \(-0.267014\pi\)
−0.743875 + 0.668319i \(0.767014\pi\)
\(828\) 0 0
\(829\) 984088.i 1.43194i −0.698131 0.715970i \(-0.745985\pi\)
0.698131 0.715970i \(-0.254015\pi\)
\(830\) 251151. + 251151.i 0.364568 + 0.364568i
\(831\) 0 0
\(832\) 6950.92 363789.i 0.0100414 0.525537i
\(833\) −127769. −0.184135
\(834\) 0 0
\(835\) −435603. −0.624767
\(836\) 219542.i 0.314127i
\(837\) 0 0
\(838\) −441624. + 441624.i −0.628876 + 0.628876i
\(839\) −378169. 378169.i −0.537232 0.537232i 0.385483 0.922715i \(-0.374035\pi\)
−0.922715 + 0.385483i \(0.874035\pi\)
\(840\) 0 0
\(841\) −180399. −0.255060
\(842\) 102457.i 0.144517i
\(843\) 0 0
\(844\) 532409.i 0.747413i
\(845\) 841999. 779994.i 1.17923 1.09239i
\(846\) 0 0
\(847\) −150064. + 150064.i −0.209175 + 0.209175i
\(848\) 49802.4 0.0692562
\(849\) 0 0
\(850\) 116409. 116409.i 0.161119 0.161119i
\(851\) 412551. 412551.i 0.569663 0.569663i
\(852\) 0 0
\(853\) −296254. 296254.i −0.407161 0.407161i 0.473586 0.880747i \(-0.342959\pi\)
−0.880747 + 0.473586i \(0.842959\pi\)
\(854\) 368104. 0.504724
\(855\) 0 0
\(856\) 409890. + 409890.i 0.559396 + 0.559396i
\(857\) 1.04442e6i 1.42205i 0.703168 + 0.711023i \(0.251768\pi\)
−0.703168 + 0.711023i \(0.748232\pi\)
\(858\) 0 0
\(859\) 503818. 0.682790 0.341395 0.939920i \(-0.389101\pi\)
0.341395 + 0.939920i \(0.389101\pi\)
\(860\) −853122. + 853122.i −1.15349 + 1.15349i
\(861\) 0 0
\(862\) 463268.i 0.623473i
\(863\) 283034. 283034.i 0.380029 0.380029i −0.491083 0.871113i \(-0.663399\pi\)
0.871113 + 0.491083i \(0.163399\pi\)
\(864\) 0 0
\(865\) 266617. + 266617.i 0.356332 + 0.356332i
\(866\) 321318. + 321318.i 0.428449 + 0.428449i
\(867\) 0 0
\(868\) 209533.i 0.278108i
\(869\) 372649. + 372649.i 0.493470 + 0.493470i
\(870\) 0 0
\(871\) 942311. 906977.i 1.24210 1.19553i
\(872\) −175204. −0.230416
\(873\) 0 0
\(874\) 464271. 0.607783
\(875\) 355360.i 0.464144i
\(876\) 0 0
\(877\) 86669.2 86669.2i 0.112685 0.112685i −0.648516 0.761201i \(-0.724610\pi\)
0.761201 + 0.648516i \(0.224610\pi\)
\(878\) −369548. 369548.i −0.479383 0.479383i
\(879\) 0 0
\(880\) 60203.4 0.0777420
\(881\) 346696.i 0.446681i −0.974741 0.223340i \(-0.928304\pi\)
0.974741 0.223340i \(-0.0716961\pi\)
\(882\) 0 0
\(883\) 570534.i 0.731746i −0.930665 0.365873i \(-0.880770\pi\)
0.930665 0.365873i \(-0.119230\pi\)
\(884\) −124099. 2371.16i −0.158805 0.00303428i
\(885\) 0 0
\(886\) 591378. 591378.i 0.753352 0.753352i
\(887\) −988202. −1.25603 −0.628013 0.778203i \(-0.716131\pi\)
−0.628013 + 0.778203i \(0.716131\pi\)
\(888\) 0 0
\(889\) 129499. 129499.i 0.163856 0.163856i
\(890\) −645542. + 645542.i −0.814976 + 0.814976i
\(891\) 0 0
\(892\) 81970.3 + 81970.3i 0.103021 + 0.103021i
\(893\) −413427. −0.518436
\(894\) 0 0
\(895\) −9150.60 9150.60i −0.0114236 0.0114236i
\(896\) 281722.i 0.350918i
\(897\) 0 0
\(898\) −626753. −0.777219
\(899\) −425681. + 425681.i −0.526702 + 0.526702i
\(900\) 0 0
\(901\) 179608.i 0.221246i
\(902\) 327759. 327759.i 0.402848 0.402848i
\(903\) 0 0
\(904\) −79368.0 79368.0i −0.0971200 0.0971200i
\(905\) 809268. + 809268.i 0.988088 + 0.988088i
\(906\) 0 0
\(907\) 720192.i 0.875455i 0.899108 + 0.437728i \(0.144217\pi\)
−0.899108 + 0.437728i \(0.855783\pi\)
\(908\) 100535. + 100535.i 0.121940 + 0.121940i
\(909\) 0 0
\(910\) −279919. + 269423.i −0.338026 + 0.325351i
\(911\) −963155. −1.16054 −0.580269 0.814425i \(-0.697053\pi\)
−0.580269 + 0.814425i \(0.697053\pi\)
\(912\) 0 0
\(913\) −287142. −0.344473
\(914\) 296455.i 0.354867i
\(915\) 0 0
\(916\) −378326. + 378326.i −0.450895 + 0.450895i
\(917\) 166290. + 166290.i 0.197755 + 0.197755i
\(918\) 0 0
\(919\) −1.29069e6 −1.52823 −0.764117 0.645077i \(-0.776825\pi\)
−0.764117 + 0.645077i \(0.776825\pi\)
\(920\) 1.79596e6i 2.12188i
\(921\) 0 0
\(922\) 188961.i 0.222285i
\(923\) 799569. 769587.i 0.938540 0.903347i
\(924\) 0 0
\(925\) 570127. 570127.i 0.666328 0.666328i
\(926\) −571149. −0.666082
\(927\) 0 0
\(928\) −536019. + 536019.i −0.622421 + 0.622421i
\(929\) −620998. + 620998.i −0.719546 + 0.719546i −0.968512 0.248966i \(-0.919909\pi\)
0.248966 + 0.968512i \(0.419909\pi\)
\(930\) 0 0
\(931\) −352109. 352109.i −0.406235 0.406235i
\(932\) −24658.8 −0.0283883
\(933\) 0 0
\(934\) 113752. + 113752.i 0.130396 + 0.130396i
\(935\) 217118.i 0.248354i
\(936\) 0 0
\(937\) 1.00082e6 1.13992 0.569960 0.821672i \(-0.306959\pi\)
0.569960 + 0.821672i \(0.306959\pi\)
\(938\) −313048. + 313048.i −0.355799 + 0.355799i
\(939\) 0 0
\(940\) 630985.i 0.714107i
\(941\) −902091. + 902091.i −1.01876 + 1.01876i −0.0189381 + 0.999821i \(0.506029\pi\)
−0.999821 + 0.0189381i \(0.993971\pi\)
\(942\) 0 0
\(943\) −1.29661e6 1.29661e6i −1.45809 1.45809i
\(944\) −44227.7 44227.7i −0.0496307 0.0496307i
\(945\) 0 0
\(946\) 521403.i 0.582628i
\(947\) 64662.8 + 64662.8i 0.0721032 + 0.0721032i 0.742239 0.670136i \(-0.233764\pi\)
−0.670136 + 0.742239i \(0.733764\pi\)
\(948\) 0 0
\(949\) 929071. + 965266.i 1.03161 + 1.07180i
\(950\) 641603. 0.710917
\(951\) 0 0
\(952\) 106490. 0.117499
\(953\) 1.57625e6i 1.73556i 0.496948 + 0.867781i \(0.334454\pi\)
−0.496948 + 0.867781i \(0.665546\pi\)
\(954\) 0 0
\(955\) −1.10963e6 + 1.10963e6i −1.21667 + 1.21667i
\(956\) 196448. + 196448.i 0.214947 + 0.214947i
\(957\) 0 0
\(958\) −379659. −0.413679
\(959\) 250364.i 0.272229i
\(960\) 0 0
\(961\) 235683.i 0.255201i
\(962\) 324904. + 6207.93i 0.351079 + 0.00670806i
\(963\) 0 0
\(964\) 287452. 287452.i 0.309322 0.309322i
\(965\) −1.98054e6 −2.12681
\(966\) 0 0
\(967\) −561644. + 561644.i −0.600632 + 0.600632i −0.940480 0.339849i \(-0.889624\pi\)
0.339849 + 0.940480i \(0.389624\pi\)
\(968\) −386373. + 386373.i −0.412340 + 0.412340i
\(969\) 0 0
\(970\) 901199. + 901199.i 0.957806 + 0.957806i
\(971\) 70845.6 0.0751405 0.0375703 0.999294i \(-0.488038\pi\)
0.0375703 + 0.999294i \(0.488038\pi\)
\(972\) 0 0
\(973\) −248590. 248590.i −0.262578 0.262578i
\(974\) 103464.i 0.109061i
\(975\) 0 0
\(976\) −125684. −0.131941
\(977\) 617175. 617175.i 0.646576 0.646576i −0.305588 0.952164i \(-0.598853\pi\)
0.952164 + 0.305588i \(0.0988531\pi\)
\(978\) 0 0
\(979\) 738050.i 0.770052i
\(980\) 537400. 537400.i 0.559559 0.559559i
\(981\) 0 0
\(982\) 383080. + 383080.i 0.397252 + 0.397252i
\(983\) −665135. 665135.i −0.688339 0.688339i 0.273526 0.961865i \(-0.411810\pi\)
−0.961865 + 0.273526i \(0.911810\pi\)
\(984\) 0 0
\(985\) 1.91839e6i 1.97726i
\(986\) −85356.3 85356.3i −0.0877974 0.0877974i
\(987\) 0 0
\(988\) −335459. 348528.i −0.343658 0.357046i
\(989\) 2.06265e6 2.10879
\(990\) 0 0
\(991\) 250003. 0.254565 0.127282 0.991867i \(-0.459375\pi\)
0.127282 + 0.991867i \(0.459375\pi\)
\(992\) 866127.i 0.880153i
\(993\) 0 0
\(994\) −265627. + 265627.i −0.268843 + 0.268843i
\(995\) 343579. + 343579.i 0.347041 + 0.347041i
\(996\) 0 0
\(997\) −1.47712e6 −1.48602 −0.743011 0.669279i \(-0.766603\pi\)
−0.743011 + 0.669279i \(0.766603\pi\)
\(998\) 97688.8i 0.0980808i
\(999\) 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 117.5.j.b.109.4 20
3.2 odd 2 39.5.g.a.31.7 20
13.8 odd 4 inner 117.5.j.b.73.4 20
39.8 even 4 39.5.g.a.34.7 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.5.g.a.31.7 20 3.2 odd 2
39.5.g.a.34.7 yes 20 39.8 even 4
117.5.j.b.73.4 20 13.8 odd 4 inner
117.5.j.b.109.4 20 1.1 even 1 trivial