Properties

Label 117.5.j.b.73.4
Level $117$
Weight $5$
Character 117.73
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 73.4
Root \(-1.66937 - 1.66937i\) of defining polynomial
Character \(\chi\) \(=\) 117.73
Dual form 117.5.j.b.109.4

$q$-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 \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

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

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 0.466944 0.884287i \(-0.345355\pi\)
−0.884287 + 0.466944i \(0.845355\pi\)
\(3\) 0 0
\(4\) 10.4264i 0.651650i
\(5\) 28.4160 + 28.4160i 1.13664 + 1.13664i 0.989049 + 0.147591i \(0.0471518\pi\)
0.147591 + 0.989049i \(0.452848\pi\)
\(6\) 0 0
\(7\) −17.1341 + 17.1341i −0.349675 + 0.349675i −0.859988 0.510314i \(-0.829529\pi\)
0.510314 + 0.859988i \(0.329529\pi\)
\(8\) −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 −0.894762 0.446542i \(-0.852655\pi\)
0.446542 + 0.894762i \(0.352655\pi\)
\(12\) 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 \(0.0388892\pi\)
−0.992546 + 0.121870i \(0.961111\pi\)
\(18\) 0 0
\(19\) 194.123 + 194.123i 0.537736 + 0.537736i 0.922863 0.385127i \(-0.125843\pi\)
−0.385127 + 0.922863i \(0.625843\pi\)
\(20\) 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 \(0.236746\pi\)
−0.735928 + 0.677059i \(0.763254\pi\)
\(24\) 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 0.943010 0.332765i \(-0.107981\pi\)
−0.332765 + 0.943010i \(0.607981\pi\)
\(32\) 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 −0.464752 0.885441i \(-0.653856\pi\)
0.885441 + 0.464752i \(0.153856\pi\)
\(38\) 648.126i 0.448840i
\(39\) 0 0
\(40\) −2507.17 −1.56698
\(41\) 1810.07 + 1810.07i 1.07678 + 1.07678i 0.996796 + 0.0799854i \(0.0254874\pi\)
0.0799854 + 0.996796i \(0.474513\pi\)
\(42\) 0 0
\(43\) 2879.48i 1.55732i −0.627448 0.778659i \(-0.715900\pi\)
0.627448 0.778659i \(-0.284100\pi\)
\(44\) 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 −0.905787 0.423733i \(-0.860720\pi\)
0.423733 + 0.905787i \(0.360720\pi\)
\(48\) 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 −0.302623 0.953110i \(-0.597862\pi\)
0.953110 + 0.302623i \(0.0978624\pi\)
\(60\) 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 −0.967969 0.251070i \(-0.919217\pi\)
−0.251070 0.967969i \(-0.580783\pi\)
\(68\) 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 0.0759967 0.997108i \(-0.475786\pi\)
−0.997108 + 0.0759967i \(0.975786\pi\)
\(72\) 0 0
\(73\) −5605.57 + 5605.57i −1.05190 + 1.05190i −0.0533222 + 0.998577i \(0.516981\pi\)
−0.998577 + 0.0533222i \(0.983019\pi\)
\(74\) −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 0.872637 0.488370i \(-0.162408\pi\)
−0.488370 + 0.872637i \(0.662408\pi\)
\(84\) 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 −0.132210 0.991222i \(-0.542207\pi\)
0.991222 + 0.132210i \(0.0422073\pi\)
\(90\) 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 0.999954 + 0.00960578i \(0.00305766\pi\)
0.00960578 + 0.999954i \(0.496942\pi\)
\(98\) 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 \(0.300953\pi\)
−0.585361 + 0.810773i \(0.699047\pi\)
\(102\) 0 0
\(103\) 11148.4i 1.05084i −0.850842 0.525421i \(-0.823908\pi\)
0.850842 0.525421i \(-0.176092\pi\)
\(104\) 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 0.785719 0.618583i \(-0.212293\pi\)
−0.618583 + 0.785719i \(0.712293\pi\)
\(110\) 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 \(-0.924721\pi\)
0.972165 0.234297i \(-0.0752789\pi\)
\(128\) 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 −0.485163 0.874424i \(-0.661240\pi\)
0.874424 + 0.485163i \(0.161240\pi\)
\(138\) 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 0.553304 0.832979i \(-0.313367\pi\)
−0.832979 + 0.553304i \(0.813367\pi\)
\(150\) 0 0
\(151\) 6322.22 6322.22i 0.277278 0.277278i −0.554743 0.832021i \(-0.687184\pi\)
0.832021 + 0.554743i \(0.187184\pi\)
\(152\) −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 −0.131907 0.991262i \(-0.542110\pi\)
0.991262 + 0.131907i \(0.0421101\pi\)
\(164\) 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 −0.831041 0.556211i \(-0.812255\pi\)
0.556211 + 0.831041i \(0.312255\pi\)
\(168\) 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 \(-0.949899\pi\)
0.987639 0.156748i \(-0.0501011\pi\)
\(174\) 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 \(0.00159957\pi\)
−0.999987 + 0.00502517i \(0.998400\pi\)
\(180\) 0 0
\(181\) 28479.3i 0.869306i −0.900598 0.434653i \(-0.856871\pi\)
0.900598 0.434653i \(-0.143129\pi\)
\(182\) −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 −0.998046 0.0624768i \(-0.980100\pi\)
0.0624768 + 0.998046i \(0.480100\pi\)
\(194\) 31714.5i 0.842665i
\(195\) 0 0
\(196\) 18911.9 0.492292
\(197\) −33755.4 33755.4i −0.869784 0.869784i 0.122664 0.992448i \(-0.460856\pi\)
−0.992448 + 0.122664i \(0.960856\pi\)
\(198\) 0 0
\(199\) 12091.0i 0.305322i −0.988279 0.152661i \(-0.951216\pi\)
0.988279 0.152661i \(-0.0487842\pi\)
\(200\) −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 0.781721 0.623628i \(-0.214342\pi\)
−0.623628 + 0.781721i \(0.714342\pi\)
\(224\) −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 0.794452 0.607327i \(-0.207758\pi\)
−0.607327 + 0.794452i \(0.707758\pi\)
\(228\) 0 0
\(229\) 36285.4 36285.4i 0.691928 0.691928i −0.270728 0.962656i \(-0.587264\pi\)
0.962656 + 0.270728i \(0.0872643\pi\)
\(230\) 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 \(-0.993066\pi\)
0.999763 0.0217819i \(-0.00693393\pi\)
\(234\) 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 0.852529 0.522679i \(-0.175067\pi\)
−0.522679 + 0.852529i \(0.675067\pi\)
\(240\) 0 0
\(241\) −27569.6 + 27569.6i −0.474676 + 0.474676i −0.903424 0.428748i \(-0.858955\pi\)
0.428748 + 0.903424i \(0.358955\pi\)
\(242\) 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 \(0.136188\pi\)
−0.909861 + 0.414913i \(0.863812\pi\)
\(252\) 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 \(0.229232\pi\)
−0.751703 + 0.659501i \(0.770768\pi\)
\(258\) 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 −0.165249 0.986252i \(-0.552843\pi\)
0.986252 + 0.165249i \(0.0528428\pi\)
\(272\) 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 \(0.133915\pi\)
−0.912800 + 0.408406i \(0.866085\pi\)
\(278\) −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 −0.806783 0.590848i \(-0.798793\pi\)
0.590848 + 0.806783i \(0.298793\pi\)
\(282\) 0 0
\(283\) 135558.i 1.69259i 0.532716 + 0.846294i \(0.321171\pi\)
−0.532716 + 0.846294i \(0.678829\pi\)
\(284\) −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 −0.483480 0.875356i \(-0.660627\pi\)
0.875356 + 0.483480i \(0.160627\pi\)
\(294\) 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 −0.362112 + 0.932135i \(0.617944\pi\)
−0.932135 + 0.362112i \(0.882056\pi\)
\(308\) −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 \(-0.997745\pi\)
0.999975 0.00708495i \(-0.00225523\pi\)
\(312\) 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 0.999916 + 0.0129239i \(0.00411392\pi\)
0.0129239 + 0.999916i \(0.495886\pi\)
\(318\) 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 0.986935 0.161122i \(-0.0515112\pi\)
−0.161122 + 0.986935i \(0.551511\pi\)
\(332\) 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 \(0.0683979\pi\)
−0.977002 + 0.213228i \(0.931602\pi\)
\(338\) −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 −0.841458 0.540322i \(-0.818302\pi\)
0.540322 + 0.841458i \(0.318302\pi\)
\(350\) 56630.5i 0.462290i
\(351\) 0 0
\(352\) −80099.5 −0.646465
\(353\) 91471.6 + 91471.6i 0.734069 + 0.734069i 0.971423 0.237354i \(-0.0762803\pi\)
−0.237354 + 0.971423i \(0.576280\pi\)
\(354\) 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 −0.780098 0.625657i \(-0.784831\pi\)
0.625657 + 0.780098i \(0.284831\pi\)
\(360\) 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 0.740639 0.671903i \(-0.234523\pi\)
−0.671903 + 0.740639i \(0.734523\pi\)
\(380\) 115028. 0.796592
\(381\) 0 0
\(382\) 65188.4 + 65188.4i 0.446729 + 0.446729i
\(383\) −92064.1 92064.1i −0.627614 0.627614i 0.319853 0.947467i \(-0.396366\pi\)
−0.947467 + 0.319853i \(0.896366\pi\)
\(384\) 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 \(0.0374125\pi\)
−0.993101 + 0.117265i \(0.962587\pi\)
\(390\) 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 −0.546859 0.837225i \(-0.684177\pi\)
0.837225 + 0.546859i \(0.184177\pi\)
\(398\) −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 −0.852773 0.522281i \(-0.825081\pi\)
0.522281 + 0.852773i \(0.325081\pi\)
\(402\) 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 0.895389 0.445284i \(-0.146897\pi\)
−0.445284 + 0.895389i \(0.646897\pi\)
\(410\) 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 0.788357 0.615218i \(-0.210932\pi\)
−0.615218 + 0.788357i \(0.710932\pi\)
\(422\) −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 0.226950 0.973906i \(-0.427124\pi\)
−0.973906 + 0.226950i \(0.927124\pi\)
\(432\) 0 0
\(433\) 192478.i 1.02661i 0.858206 + 0.513305i \(0.171579\pi\)
−0.858206 + 0.513305i \(0.828421\pi\)
\(434\) 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 \(-0.805264\pi\)
0.818626 0.574327i \(-0.194736\pi\)
\(440\) 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 −0.0666259 0.997778i \(-0.521223\pi\)
0.997778 + 0.0666259i \(0.0212234\pi\)
\(450\) 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 0.461822 0.886973i \(-0.347196\pi\)
−0.886973 + 0.461822i \(0.847196\pi\)
\(458\) −121148. −0.577543
\(459\) 0 0
\(460\) 212231. + 212231.i 1.00298 + 1.00298i
\(461\) 56596.4 + 56596.4i 0.266310 + 0.266310i 0.827611 0.561302i \(-0.189699\pi\)
−0.561302 + 0.827611i \(0.689699\pi\)
\(462\) 0 0
\(463\) 171067. 171067.i 0.798003 0.798003i −0.184777 0.982780i \(-0.559156\pi\)
0.982780 + 0.184777i \(0.0591563\pi\)
\(464\) 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 \(0.0499314\pi\)
−0.987722 + 0.156222i \(0.950069\pi\)
\(468\) 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 −0.414458 0.910068i \(-0.636029\pi\)
0.910068 + 0.414458i \(0.136029\pi\)
\(480\) 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 0.769413 0.638752i \(-0.220549\pi\)
−0.638752 + 0.769413i \(0.720549\pi\)
\(488\) −283869. + 283869.i −1.19201 + 1.19201i
\(489\) 0 0
\(490\) 172086. 0.716728
\(491\) 229476.i 0.951861i 0.879483 + 0.475931i \(0.157889\pi\)
−0.879483 + 0.475931i \(0.842111\pi\)
\(492\) 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 0.763415 0.645909i \(-0.223521\pi\)
−0.645909 + 0.763415i \(0.723521\pi\)
\(500\) 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 0.935289 0.353886i \(-0.115140\pi\)
−0.353886 + 0.935289i \(0.615140\pi\)
\(510\) 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 −0.765004 0.644025i \(-0.777263\pi\)
0.644025 + 0.765004i \(0.277263\pi\)
\(542\) −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 −0.757866 0.652411i \(-0.773758\pi\)
0.652411 + 0.757866i \(0.273758\pi\)
\(558\) 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 \(0.0920580\pi\)
−0.958470 + 0.285194i \(0.907942\pi\)
\(564\) 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 \(-0.718471\pi\)
0.633715 0.773566i \(-0.281529\pi\)
\(570\) 0 0
\(571\) 246396.i 0.755722i 0.925862 + 0.377861i \(0.123340\pi\)
−0.925862 + 0.377861i \(0.876660\pi\)
\(572\) −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 0.372794 0.927914i \(-0.378400\pi\)
−0.927914 + 0.372794i \(0.878400\pi\)
\(578\) −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 0.982317 + 0.187227i \(0.0599501\pi\)
0.187227 + 0.982317i \(0.440050\pi\)
\(588\) 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 −0.915628 0.402027i \(-0.868306\pi\)
0.402027 + 0.915628i \(0.368306\pi\)
\(594\) 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 0.916118 0.400909i \(-0.131306\pi\)
−0.400909 + 0.916118i \(0.631306\pi\)
\(614\) 407265. 1.08029
\(615\) 0 0
\(616\) 81989.4 + 81989.4i 0.216071 + 0.216071i
\(617\) −234942. 234942.i −0.617149 0.617149i 0.327650 0.944799i \(-0.393743\pi\)
−0.944799 + 0.327650i \(0.893743\pi\)
\(618\) 0 0
\(619\) 518107. 518107.i 1.35219 1.35219i 0.468984 0.883206i \(-0.344620\pi\)
0.883206 0.468984i \(-0.155380\pi\)
\(620\) 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 −0.989231 0.146365i \(-0.953243\pi\)
0.146365 + 0.989231i \(0.453243\pi\)
\(632\) 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 \(-0.763929\pi\)
0.737362 0.675497i \(-0.236071\pi\)
\(642\) 0 0
\(643\) 263449. + 263449.i 0.637197 + 0.637197i 0.949863 0.312666i \(-0.101222\pi\)
−0.312666 + 0.949863i \(0.601222\pi\)
\(644\) −127970. + 127970.i −0.308557 + 0.308557i
\(645\) 0 0
\(646\) 45654.7 0.109401
\(647\) 396185.i 0.946433i 0.880946 + 0.473216i \(0.156907\pi\)
−0.880946 + 0.473216i \(0.843093\pi\)
\(648\) 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 −0.554597 0.832119i \(-0.687128\pi\)
0.832119 + 0.554597i \(0.187128\pi\)
\(662\) 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 \(0.0496062\pi\)
−0.987881 + 0.155213i \(0.950394\pi\)
\(674\) 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 0.0192116 0.999815i \(-0.493884\pi\)
0.999815 0.0192116i \(-0.00611561\pi\)
\(684\) 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 0.448371 0.893847i \(-0.352004\pi\)
−0.893847 + 0.448371i \(0.852004\pi\)
\(692\) −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 \(0.0183257\pi\)
−0.998343 + 0.0575400i \(0.981674\pi\)
\(702\) 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 −0.364668 0.931138i \(-0.618818\pi\)
0.931138 + 0.364668i \(0.118818\pi\)
\(710\) −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 \(-0.834392\pi\)
0.867684 0.497117i \(-0.165608\pi\)
\(720\) 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 \(-0.998396\pi\)
0.999987 0.00503779i \(-0.00160358\pi\)
\(728\) 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 0.993338 0.115238i \(-0.0367631\pi\)
−0.115238 + 0.993338i \(0.536763\pi\)
\(734\) −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 −0.263040 0.964785i \(-0.584725\pi\)
0.964785 + 0.263040i \(0.0847252\pi\)
\(740\) 341265.i 0.623201i
\(741\) 0 0
\(742\) −145862. −0.264932
\(743\) −77467.3 77467.3i −0.140327 0.140327i 0.633454 0.773781i \(-0.281637\pi\)
−0.773781 + 0.633454i \(0.781637\pi\)
\(744\) 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 \(0.0820954\pi\)
−0.966925 + 0.255061i \(0.917905\pi\)
\(752\) 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 −0.990977 0.134034i \(-0.957207\pi\)
0.134034 + 0.990977i \(0.457207\pi\)
\(762\) 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 0.927073 0.374882i \(-0.122317\pi\)
−0.374882 + 0.927073i \(0.622317\pi\)
\(770\) −176324. −0.297393
\(771\) 0 0
\(772\) 363350. + 363350.i 0.609664 + 0.609664i
\(773\) −213799. 213799.i −0.357804 0.357804i 0.505199 0.863003i \(-0.331419\pi\)
−0.863003 + 0.505199i \(0.831419\pi\)
\(774\) 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 0.0388320 0.999246i \(-0.487636\pi\)
0.999246 0.0388320i \(-0.0123637\pi\)
\(788\) −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 \(-0.803596\pi\)
0.815606 0.578607i \(-0.196404\pi\)
\(798\) 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 0.571997 0.820256i \(-0.306169\pi\)
−0.820256 + 0.571997i \(0.806169\pi\)
\(812\) −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 0.939327 0.343024i \(-0.111451\pi\)
−0.343024 + 0.939327i \(0.611451\pi\)
\(822\) 0 0
\(823\) 129326.i 0.190935i −0.995433 0.0954674i \(-0.969565\pi\)
0.995433 0.0954674i \(-0.0304346\pi\)
\(824\) 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 −0.743875 0.668319i \(-0.767014\pi\)
0.668319 + 0.743875i \(0.267014\pi\)
\(828\) 0 0
\(829\) 984088.i 1.43194i 0.698131 + 0.715970i \(0.254015\pi\)
−0.698131 + 0.715970i \(0.745985\pi\)
\(830\) 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 −0.922715 0.385483i \(-0.874035\pi\)
0.385483 + 0.922715i \(0.374035\pi\)
\(840\) 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 −0.880747 0.473586i \(-0.842959\pi\)
0.473586 + 0.880747i \(0.342959\pi\)
\(854\) 368104. 0.504724
\(855\) 0 0
\(856\) 409890. 409890.i 0.559396 0.559396i
\(857\) 1.04442e6i 1.42205i −0.703168 0.711023i \(-0.748232\pi\)
0.703168 0.711023i \(-0.251768\pi\)
\(858\) 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 0.871113 0.491083i \(-0.163399\pi\)
−0.491083 + 0.871113i \(0.663399\pi\)
\(864\) 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 0.761201 0.648516i \(-0.224610\pi\)
−0.648516 + 0.761201i \(0.724610\pi\)
\(878\) −369548. + 369548.i −0.479383 + 0.479383i
\(879\) 0 0
\(880\) 60203.4 0.0777420
\(881\) 346696.i 0.446681i 0.974741 + 0.223340i \(0.0716961\pi\)
−0.974741 + 0.223340i \(0.928304\pi\)
\(882\) 0 0
\(883\) 570534.i 0.731746i 0.930665 + 0.365873i \(0.119230\pi\)
−0.930665 + 0.365873i \(0.880770\pi\)
\(884\) −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 \(-0.855783\pi\)
0.899108 0.437728i \(-0.144217\pi\)
\(908\) 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 0.248966 0.968512i \(-0.419909\pi\)
−0.968512 + 0.248966i \(0.919909\pi\)
\(930\) 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 −0.999821 0.0189381i \(-0.993971\pi\)
−0.0189381 0.999821i \(-0.506029\pi\)
\(942\) 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 −0.670136 0.742239i \(-0.733764\pi\)
0.742239 + 0.670136i \(0.233764\pi\)
\(948\) 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 \(-0.665546\pi\)
0.496948 0.867781i \(-0.334454\pi\)
\(954\) 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 0.339849 0.940480i \(-0.389624\pi\)
−0.940480 + 0.339849i \(0.889624\pi\)
\(968\) −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 0.952164 0.305588i \(-0.0988531\pi\)
−0.305588 + 0.952164i \(0.598853\pi\)
\(978\) 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 −0.961865 0.273526i \(-0.911810\pi\)
0.273526 + 0.961865i \(0.411810\pi\)
\(984\) 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.73.4 20
3.2 odd 2 39.5.g.a.34.7 yes 20
13.5 odd 4 inner 117.5.j.b.109.4 20
39.5 even 4 39.5.g.a.31.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.5.g.a.31.7 20 39.5 even 4
39.5.g.a.34.7 yes 20 3.2 odd 2
117.5.j.b.73.4 20 1.1 even 1 trivial
117.5.j.b.109.4 20 13.5 odd 4 inner