Properties

Label 117.5.j.b.73.8
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 73.8
Root \(4.05053 + 4.05053i\) of defining polynomial
Character \(\chi\) \(=\) 117.73
Dual form 117.5.j.b.109.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.05053 + 4.05053i) q^{2} +16.8136i q^{4} +(-33.5573 - 33.5573i) q^{5} +(23.7132 - 23.7132i) q^{7} +(-3.29565 + 3.29565i) q^{8} -271.850i q^{10} +(-32.2545 + 32.2545i) q^{11} +(72.8001 - 152.516i) q^{13} +192.102 q^{14} +242.320 q^{16} -354.222i q^{17} +(-168.478 - 168.478i) q^{19} +(564.221 - 564.221i) q^{20} -261.296 q^{22} +182.161i q^{23} +1627.19i q^{25} +(912.650 - 322.892i) q^{26} +(398.704 + 398.704i) q^{28} -913.769 q^{29} +(-876.095 - 876.095i) q^{31} +(1034.25 + 1034.25i) q^{32} +(1434.79 - 1434.79i) q^{34} -1591.50 q^{35} +(112.687 - 112.687i) q^{37} -1364.85i q^{38} +221.186 q^{40} +(1133.48 + 1133.48i) q^{41} +411.247i q^{43} +(-542.315 - 542.315i) q^{44} +(-737.848 + 737.848i) q^{46} +(3000.97 - 3000.97i) q^{47} +1276.37i q^{49} +(-6590.99 + 6590.99i) q^{50} +(2564.35 + 1224.03i) q^{52} +4228.78 q^{53} +2164.75 q^{55} +156.301i q^{56} +(-3701.25 - 3701.25i) q^{58} +(-1690.19 + 1690.19i) q^{59} -3760.65 q^{61} -7097.31i q^{62} +4501.45i q^{64} +(-7561.01 + 2675.06i) q^{65} +(-2631.54 - 2631.54i) q^{67} +5955.76 q^{68} +(-6446.43 - 6446.43i) q^{70} +(-297.661 - 297.661i) q^{71} +(2616.29 - 2616.29i) q^{73} +912.887 q^{74} +(2832.73 - 2832.73i) q^{76} +1529.71i q^{77} +7885.52 q^{79} +(-8131.61 - 8131.61i) q^{80} +9182.36i q^{82} +(4078.45 + 4078.45i) q^{83} +(-11886.7 + 11886.7i) q^{85} +(-1665.77 + 1665.77i) q^{86} -212.599i q^{88} +(-1722.86 + 1722.86i) q^{89} +(-1890.32 - 5342.96i) q^{91} -3062.78 q^{92} +24311.0 q^{94} +11307.4i q^{95} +(1560.33 + 1560.33i) q^{97} +(-5169.99 + 5169.99i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.05053 + 4.05053i 1.01263 + 1.01263i 0.999919 + 0.0127141i \(0.00404712\pi\)
0.0127141 + 0.999919i \(0.495953\pi\)
\(3\) 0 0
\(4\) 16.8136i 1.05085i
\(5\) −33.5573 33.5573i −1.34229 1.34229i −0.893772 0.448521i \(-0.851951\pi\)
−0.448521 0.893772i \(-0.648049\pi\)
\(6\) 0 0
\(7\) 23.7132 23.7132i 0.483942 0.483942i −0.422446 0.906388i \(-0.638828\pi\)
0.906388 + 0.422446i \(0.138828\pi\)
\(8\) −3.29565 + 3.29565i −0.0514945 + 0.0514945i
\(9\) 0 0
\(10\) 271.850i 2.71850i
\(11\) −32.2545 + 32.2545i −0.266566 + 0.266566i −0.827715 0.561149i \(-0.810359\pi\)
0.561149 + 0.827715i \(0.310359\pi\)
\(12\) 0 0
\(13\) 72.8001 152.516i 0.430770 0.902462i
\(14\) 192.102 0.980111
\(15\) 0 0
\(16\) 242.320 0.946562
\(17\) 354.222i 1.22568i −0.790206 0.612841i \(-0.790027\pi\)
0.790206 0.612841i \(-0.209973\pi\)
\(18\) 0 0
\(19\) −168.478 168.478i −0.466699 0.466699i 0.434145 0.900843i \(-0.357051\pi\)
−0.900843 + 0.434145i \(0.857051\pi\)
\(20\) 564.221 564.221i 1.41055 1.41055i
\(21\) 0 0
\(22\) −261.296 −0.539867
\(23\) 182.161i 0.344349i 0.985066 + 0.172175i \(0.0550793\pi\)
−0.985066 + 0.172175i \(0.944921\pi\)
\(24\) 0 0
\(25\) 1627.19i 2.60350i
\(26\) 912.650 322.892i 1.35007 0.477651i
\(27\) 0 0
\(28\) 398.704 + 398.704i 0.508551 + 0.508551i
\(29\) −913.769 −1.08653 −0.543264 0.839562i \(-0.682812\pi\)
−0.543264 + 0.839562i \(0.682812\pi\)
\(30\) 0 0
\(31\) −876.095 876.095i −0.911650 0.911650i 0.0847522 0.996402i \(-0.472990\pi\)
−0.996402 + 0.0847522i \(0.972990\pi\)
\(32\) 1034.25 + 1034.25i 1.01001 + 1.01001i
\(33\) 0 0
\(34\) 1434.79 1434.79i 1.24117 1.24117i
\(35\) −1591.50 −1.29918
\(36\) 0 0
\(37\) 112.687 112.687i 0.0823135 0.0823135i −0.664751 0.747065i \(-0.731462\pi\)
0.747065 + 0.664751i \(0.231462\pi\)
\(38\) 1364.85i 0.945189i
\(39\) 0 0
\(40\) 221.186 0.138242
\(41\) 1133.48 + 1133.48i 0.674287 + 0.674287i 0.958701 0.284415i \(-0.0917993\pi\)
−0.284415 + 0.958701i \(0.591799\pi\)
\(42\) 0 0
\(43\) 411.247i 0.222416i 0.993797 + 0.111208i \(0.0354720\pi\)
−0.993797 + 0.111208i \(0.964528\pi\)
\(44\) −542.315 542.315i −0.280121 0.280121i
\(45\) 0 0
\(46\) −737.848 + 737.848i −0.348700 + 0.348700i
\(47\) 3000.97 3000.97i 1.35852 1.35852i 0.482772 0.875746i \(-0.339630\pi\)
0.875746 0.482772i \(-0.160370\pi\)
\(48\) 0 0
\(49\) 1276.37i 0.531600i
\(50\) −6590.99 + 6590.99i −2.63639 + 2.63639i
\(51\) 0 0
\(52\) 2564.35 + 1224.03i 0.948354 + 0.452675i
\(53\) 4228.78 1.50544 0.752720 0.658341i \(-0.228741\pi\)
0.752720 + 0.658341i \(0.228741\pi\)
\(54\) 0 0
\(55\) 2164.75 0.715619
\(56\) 156.301i 0.0498407i
\(57\) 0 0
\(58\) −3701.25 3701.25i −1.10025 1.10025i
\(59\) −1690.19 + 1690.19i −0.485547 + 0.485547i −0.906898 0.421350i \(-0.861556\pi\)
0.421350 + 0.906898i \(0.361556\pi\)
\(60\) 0 0
\(61\) −3760.65 −1.01066 −0.505328 0.862928i \(-0.668628\pi\)
−0.505328 + 0.862928i \(0.668628\pi\)
\(62\) 7097.31i 1.84633i
\(63\) 0 0
\(64\) 4501.45i 1.09899i
\(65\) −7561.01 + 2675.06i −1.78959 + 0.633149i
\(66\) 0 0
\(67\) −2631.54 2631.54i −0.586219 0.586219i 0.350386 0.936605i \(-0.386050\pi\)
−0.936605 + 0.350386i \(0.886050\pi\)
\(68\) 5955.76 1.28801
\(69\) 0 0
\(70\) −6446.43 6446.43i −1.31560 1.31560i
\(71\) −297.661 297.661i −0.0590479 0.0590479i 0.676966 0.736014i \(-0.263294\pi\)
−0.736014 + 0.676966i \(0.763294\pi\)
\(72\) 0 0
\(73\) 2616.29 2616.29i 0.490953 0.490953i −0.417653 0.908606i \(-0.637147\pi\)
0.908606 + 0.417653i \(0.137147\pi\)
\(74\) 912.887 0.166707
\(75\) 0 0
\(76\) 2832.73 2832.73i 0.490431 0.490431i
\(77\) 1529.71i 0.258005i
\(78\) 0 0
\(79\) 7885.52 1.26350 0.631752 0.775171i \(-0.282336\pi\)
0.631752 + 0.775171i \(0.282336\pi\)
\(80\) −8131.61 8131.61i −1.27056 1.27056i
\(81\) 0 0
\(82\) 9182.36i 1.36561i
\(83\) 4078.45 + 4078.45i 0.592024 + 0.592024i 0.938178 0.346154i \(-0.112512\pi\)
−0.346154 + 0.938178i \(0.612512\pi\)
\(84\) 0 0
\(85\) −11886.7 + 11886.7i −1.64522 + 1.64522i
\(86\) −1665.77 + 1665.77i −0.225226 + 0.225226i
\(87\) 0 0
\(88\) 212.599i 0.0274534i
\(89\) −1722.86 + 1722.86i −0.217505 + 0.217505i −0.807446 0.589941i \(-0.799151\pi\)
0.589941 + 0.807446i \(0.299151\pi\)
\(90\) 0 0
\(91\) −1890.32 5342.96i −0.228272 0.645207i
\(92\) −3062.78 −0.361860
\(93\) 0 0
\(94\) 24311.0 2.75136
\(95\) 11307.4i 1.25289i
\(96\) 0 0
\(97\) 1560.33 + 1560.33i 0.165834 + 0.165834i 0.785146 0.619311i \(-0.212588\pi\)
−0.619311 + 0.785146i \(0.712588\pi\)
\(98\) −5169.99 + 5169.99i −0.538316 + 0.538316i
\(99\) 0 0
\(100\) −27359.0 −2.73590
\(101\) 9982.36i 0.978567i 0.872125 + 0.489283i \(0.162742\pi\)
−0.872125 + 0.489283i \(0.837258\pi\)
\(102\) 0 0
\(103\) 6.56148i 0.000618483i 1.00000 0.000309241i \(9.84346e-5\pi\)
−1.00000 0.000309241i \(0.999902\pi\)
\(104\) 262.716 + 742.563i 0.0242896 + 0.0686541i
\(105\) 0 0
\(106\) 17128.8 + 17128.8i 1.52446 + 1.52446i
\(107\) 6183.10 0.540056 0.270028 0.962853i \(-0.412967\pi\)
0.270028 + 0.962853i \(0.412967\pi\)
\(108\) 0 0
\(109\) 3694.46 + 3694.46i 0.310956 + 0.310956i 0.845280 0.534324i \(-0.179434\pi\)
−0.534324 + 0.845280i \(0.679434\pi\)
\(110\) 8768.39 + 8768.39i 0.724660 + 0.724660i
\(111\) 0 0
\(112\) 5746.17 5746.17i 0.458081 0.458081i
\(113\) 21267.9 1.66559 0.832794 0.553583i \(-0.186740\pi\)
0.832794 + 0.553583i \(0.186740\pi\)
\(114\) 0 0
\(115\) 6112.83 6112.83i 0.462218 0.462218i
\(116\) 15363.8i 1.14178i
\(117\) 0 0
\(118\) −13692.3 −0.983363
\(119\) −8399.72 8399.72i −0.593159 0.593159i
\(120\) 0 0
\(121\) 12560.3i 0.857885i
\(122\) −15232.6 15232.6i −1.02342 1.02342i
\(123\) 0 0
\(124\) 14730.3 14730.3i 0.958009 0.958009i
\(125\) 33630.8 33630.8i 2.15237 2.15237i
\(126\) 0 0
\(127\) 26466.7i 1.64094i −0.571693 0.820468i \(-0.693713\pi\)
0.571693 0.820468i \(-0.306287\pi\)
\(128\) −1685.19 + 1685.19i −0.102856 + 0.102856i
\(129\) 0 0
\(130\) −41461.5 19790.7i −2.45334 1.17105i
\(131\) −21491.7 −1.25236 −0.626178 0.779680i \(-0.715382\pi\)
−0.626178 + 0.779680i \(0.715382\pi\)
\(132\) 0 0
\(133\) −7990.30 −0.451710
\(134\) 21318.3i 1.18725i
\(135\) 0 0
\(136\) 1167.39 + 1167.39i 0.0631159 + 0.0631159i
\(137\) 5030.14 5030.14i 0.268002 0.268002i −0.560292 0.828295i \(-0.689311\pi\)
0.828295 + 0.560292i \(0.189311\pi\)
\(138\) 0 0
\(139\) 12182.9 0.630554 0.315277 0.949000i \(-0.397903\pi\)
0.315277 + 0.949000i \(0.397903\pi\)
\(140\) 26758.9i 1.36525i
\(141\) 0 0
\(142\) 2411.37i 0.119588i
\(143\) 2571.20 + 7267.45i 0.125737 + 0.355394i
\(144\) 0 0
\(145\) 30663.7 + 30663.7i 1.45844 + 1.45844i
\(146\) 21194.7 0.994311
\(147\) 0 0
\(148\) 1894.68 + 1894.68i 0.0864994 + 0.0864994i
\(149\) 10032.3 + 10032.3i 0.451887 + 0.451887i 0.895980 0.444094i \(-0.146474\pi\)
−0.444094 + 0.895980i \(0.646474\pi\)
\(150\) 0 0
\(151\) 10057.5 10057.5i 0.441098 0.441098i −0.451283 0.892381i \(-0.649033\pi\)
0.892381 + 0.451283i \(0.149033\pi\)
\(152\) 1110.49 0.0480649
\(153\) 0 0
\(154\) −6196.14 + 6196.14i −0.261264 + 0.261264i
\(155\) 58798.9i 2.44740i
\(156\) 0 0
\(157\) 8554.44 0.347050 0.173525 0.984829i \(-0.444484\pi\)
0.173525 + 0.984829i \(0.444484\pi\)
\(158\) 31940.6 + 31940.6i 1.27947 + 1.27947i
\(159\) 0 0
\(160\) 69413.7i 2.71147i
\(161\) 4319.61 + 4319.61i 0.166645 + 0.166645i
\(162\) 0 0
\(163\) −1164.17 + 1164.17i −0.0438170 + 0.0438170i −0.728676 0.684859i \(-0.759864\pi\)
0.684859 + 0.728676i \(0.259864\pi\)
\(164\) −19057.8 + 19057.8i −0.708576 + 0.708576i
\(165\) 0 0
\(166\) 33039.8i 1.19901i
\(167\) −7534.77 + 7534.77i −0.270170 + 0.270170i −0.829169 0.558999i \(-0.811186\pi\)
0.558999 + 0.829169i \(0.311186\pi\)
\(168\) 0 0
\(169\) −17961.3 22206.4i −0.628875 0.777506i
\(170\) −96295.3 −3.33202
\(171\) 0 0
\(172\) −6914.56 −0.233726
\(173\) 39158.0i 1.30836i −0.756337 0.654182i \(-0.773013\pi\)
0.756337 0.654182i \(-0.226987\pi\)
\(174\) 0 0
\(175\) 38585.8 + 38585.8i 1.25994 + 1.25994i
\(176\) −7815.90 + 7815.90i −0.252321 + 0.252321i
\(177\) 0 0
\(178\) −13957.0 −0.440505
\(179\) 37765.1i 1.17865i −0.807897 0.589324i \(-0.799394\pi\)
0.807897 0.589324i \(-0.200606\pi\)
\(180\) 0 0
\(181\) 9590.57i 0.292744i 0.989230 + 0.146372i \(0.0467596\pi\)
−0.989230 + 0.146372i \(0.953240\pi\)
\(182\) 13985.0 29298.6i 0.422202 0.884513i
\(183\) 0 0
\(184\) −600.338 600.338i −0.0177321 0.0177321i
\(185\) −7562.97 −0.220978
\(186\) 0 0
\(187\) 11425.2 + 11425.2i 0.326725 + 0.326725i
\(188\) 50457.2 + 50457.2i 1.42760 + 1.42760i
\(189\) 0 0
\(190\) −45800.8 + 45800.8i −1.26872 + 1.26872i
\(191\) −46392.7 −1.27170 −0.635848 0.771814i \(-0.719349\pi\)
−0.635848 + 0.771814i \(0.719349\pi\)
\(192\) 0 0
\(193\) 9357.45 9357.45i 0.251213 0.251213i −0.570255 0.821468i \(-0.693155\pi\)
0.821468 + 0.570255i \(0.193155\pi\)
\(194\) 12640.4i 0.335858i
\(195\) 0 0
\(196\) −21460.5 −0.558633
\(197\) 4463.00 + 4463.00i 0.114999 + 0.114999i 0.762265 0.647266i \(-0.224088\pi\)
−0.647266 + 0.762265i \(0.724088\pi\)
\(198\) 0 0
\(199\) 71326.4i 1.80113i 0.434726 + 0.900563i \(0.356845\pi\)
−0.434726 + 0.900563i \(0.643155\pi\)
\(200\) −5362.65 5362.65i −0.134066 0.134066i
\(201\) 0 0
\(202\) −40433.9 + 40433.9i −0.990929 + 0.990929i
\(203\) −21668.4 + 21668.4i −0.525816 + 0.525816i
\(204\) 0 0
\(205\) 76072.9i 1.81018i
\(206\) −26.5775 + 26.5775i −0.000626296 + 0.000626296i
\(207\) 0 0
\(208\) 17640.9 36957.7i 0.407750 0.854236i
\(209\) 10868.4 0.248812
\(210\) 0 0
\(211\) −45146.1 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(212\) 71101.2i 1.58199i
\(213\) 0 0
\(214\) 25044.8 + 25044.8i 0.546878 + 0.546878i
\(215\) 13800.4 13800.4i 0.298548 0.298548i
\(216\) 0 0
\(217\) −41550.0 −0.882371
\(218\) 29929.1i 0.629768i
\(219\) 0 0
\(220\) 36397.3i 0.752010i
\(221\) −54024.5 25787.4i −1.10613 0.527986i
\(222\) 0 0
\(223\) 22521.9 + 22521.9i 0.452892 + 0.452892i 0.896313 0.443421i \(-0.146235\pi\)
−0.443421 + 0.896313i \(0.646235\pi\)
\(224\) 49050.9 0.977577
\(225\) 0 0
\(226\) 86146.3 + 86146.3i 1.68663 + 1.68663i
\(227\) −33678.9 33678.9i −0.653592 0.653592i 0.300264 0.953856i \(-0.402925\pi\)
−0.953856 + 0.300264i \(0.902925\pi\)
\(228\) 0 0
\(229\) 55067.0 55067.0i 1.05008 1.05008i 0.0513970 0.998678i \(-0.483633\pi\)
0.998678 0.0513970i \(-0.0163674\pi\)
\(230\) 49520.4 0.936114
\(231\) 0 0
\(232\) 3011.46 3011.46i 0.0559502 0.0559502i
\(233\) 6668.81i 0.122839i 0.998112 + 0.0614195i \(0.0195627\pi\)
−0.998112 + 0.0614195i \(0.980437\pi\)
\(234\) 0 0
\(235\) −201409. −3.64706
\(236\) −28418.2 28418.2i −0.510239 0.510239i
\(237\) 0 0
\(238\) 68046.7i 1.20130i
\(239\) 24924.8 + 24924.8i 0.436350 + 0.436350i 0.890782 0.454431i \(-0.150158\pi\)
−0.454431 + 0.890782i \(0.650158\pi\)
\(240\) 0 0
\(241\) 4314.21 4314.21i 0.0742792 0.0742792i −0.668991 0.743270i \(-0.733274\pi\)
0.743270 + 0.668991i \(0.233274\pi\)
\(242\) −50875.9 + 50875.9i −0.868723 + 0.868723i
\(243\) 0 0
\(244\) 63230.2i 1.06205i
\(245\) 42831.7 42831.7i 0.713564 0.713564i
\(246\) 0 0
\(247\) −37960.9 + 13430.4i −0.622217 + 0.220138i
\(248\) 5774.61 0.0938900
\(249\) 0 0
\(250\) 272445. 4.35913
\(251\) 100235.i 1.59100i 0.605954 + 0.795500i \(0.292791\pi\)
−0.605954 + 0.795500i \(0.707209\pi\)
\(252\) 0 0
\(253\) −5875.50 5875.50i −0.0917918 0.0917918i
\(254\) 107204. 107204.i 1.66167 1.66167i
\(255\) 0 0
\(256\) 58371.4 0.890676
\(257\) 14954.5i 0.226415i −0.993571 0.113208i \(-0.963887\pi\)
0.993571 0.113208i \(-0.0361125\pi\)
\(258\) 0 0
\(259\) 5344.34i 0.0796700i
\(260\) −44977.4 127128.i −0.665346 1.88059i
\(261\) 0 0
\(262\) −87052.7 87052.7i −1.26818 1.26818i
\(263\) 7574.07 0.109501 0.0547505 0.998500i \(-0.482564\pi\)
0.0547505 + 0.998500i \(0.482564\pi\)
\(264\) 0 0
\(265\) −141907. 141907.i −2.02074 2.02074i
\(266\) −32365.0 32365.0i −0.457417 0.457417i
\(267\) 0 0
\(268\) 44245.7 44245.7i 0.616030 0.616030i
\(269\) −100457. −1.38827 −0.694136 0.719844i \(-0.744213\pi\)
−0.694136 + 0.719844i \(0.744213\pi\)
\(270\) 0 0
\(271\) 66620.4 66620.4i 0.907128 0.907128i −0.0889115 0.996040i \(-0.528339\pi\)
0.996040 + 0.0889115i \(0.0283388\pi\)
\(272\) 85835.0i 1.16018i
\(273\) 0 0
\(274\) 40749.5 0.542776
\(275\) −52484.2 52484.2i −0.694005 0.694005i
\(276\) 0 0
\(277\) 70180.3i 0.914652i 0.889299 + 0.457326i \(0.151193\pi\)
−0.889299 + 0.457326i \(0.848807\pi\)
\(278\) 49347.3 + 49347.3i 0.638519 + 0.638519i
\(279\) 0 0
\(280\) 5245.03 5245.03i 0.0669009 0.0669009i
\(281\) 20515.7 20515.7i 0.259820 0.259820i −0.565161 0.824981i \(-0.691186\pi\)
0.824981 + 0.565161i \(0.191186\pi\)
\(282\) 0 0
\(283\) 30170.2i 0.376708i −0.982101 0.188354i \(-0.939685\pi\)
0.982101 0.188354i \(-0.0603153\pi\)
\(284\) 5004.76 5004.76i 0.0620507 0.0620507i
\(285\) 0 0
\(286\) −19022.3 + 39851.8i −0.232558 + 0.487209i
\(287\) 53756.6 0.652631
\(288\) 0 0
\(289\) −41952.2 −0.502295
\(290\) 248408.i 2.95373i
\(291\) 0 0
\(292\) 43989.3 + 43989.3i 0.515919 + 0.515919i
\(293\) 67660.9 67660.9i 0.788139 0.788139i −0.193050 0.981189i \(-0.561838\pi\)
0.981189 + 0.193050i \(0.0618380\pi\)
\(294\) 0 0
\(295\) 113437. 1.30349
\(296\) 742.755i 0.00847739i
\(297\) 0 0
\(298\) 81272.6i 0.915191i
\(299\) 27782.4 + 13261.3i 0.310762 + 0.148335i
\(300\) 0 0
\(301\) 9751.97 + 9751.97i 0.107636 + 0.107636i
\(302\) 81476.3 0.893341
\(303\) 0 0
\(304\) −40825.6 40825.6i −0.441759 0.441759i
\(305\) 126197. + 126197.i 1.35660 + 1.35660i
\(306\) 0 0
\(307\) −96671.1 + 96671.1i −1.02570 + 1.02570i −0.0260377 + 0.999661i \(0.508289\pi\)
−0.999661 + 0.0260377i \(0.991711\pi\)
\(308\) −25720.0 −0.271125
\(309\) 0 0
\(310\) −238167. + 238167.i −2.47832 + 2.47832i
\(311\) 70642.4i 0.730373i −0.930934 0.365186i \(-0.881005\pi\)
0.930934 0.365186i \(-0.118995\pi\)
\(312\) 0 0
\(313\) 68524.0 0.699446 0.349723 0.936853i \(-0.386276\pi\)
0.349723 + 0.936853i \(0.386276\pi\)
\(314\) 34650.0 + 34650.0i 0.351435 + 0.351435i
\(315\) 0 0
\(316\) 132584.i 1.32775i
\(317\) −140417. 140417.i −1.39734 1.39734i −0.807578 0.589760i \(-0.799222\pi\)
−0.589760 0.807578i \(-0.700778\pi\)
\(318\) 0 0
\(319\) 29473.2 29473.2i 0.289631 0.289631i
\(320\) 151057. 151057.i 1.47516 1.47516i
\(321\) 0 0
\(322\) 34993.4i 0.337501i
\(323\) −59678.7 + 59678.7i −0.572024 + 0.572024i
\(324\) 0 0
\(325\) 248173. + 118460.i 2.34956 + 1.12151i
\(326\) −9431.05 −0.0887411
\(327\) 0 0
\(328\) −7471.08 −0.0694441
\(329\) 142325.i 1.31489i
\(330\) 0 0
\(331\) −73424.6 73424.6i −0.670171 0.670171i 0.287585 0.957755i \(-0.407148\pi\)
−0.957755 + 0.287585i \(0.907148\pi\)
\(332\) −68573.6 + 68573.6i −0.622130 + 0.622130i
\(333\) 0 0
\(334\) −61039.7 −0.547166
\(335\) 176615.i 1.57376i
\(336\) 0 0
\(337\) 14312.2i 0.126022i −0.998013 0.0630111i \(-0.979930\pi\)
0.998013 0.0630111i \(-0.0200703\pi\)
\(338\) 17194.8 162700.i 0.150509 1.42415i
\(339\) 0 0
\(340\) −199859. 199859.i −1.72889 1.72889i
\(341\) 56516.0 0.486030
\(342\) 0 0
\(343\) 87202.1 + 87202.1i 0.741206 + 0.741206i
\(344\) −1355.33 1355.33i −0.0114532 0.0114532i
\(345\) 0 0
\(346\) 158611. 158611.i 1.32489 1.32489i
\(347\) −216160. −1.79521 −0.897607 0.440797i \(-0.854696\pi\)
−0.897607 + 0.440797i \(0.854696\pi\)
\(348\) 0 0
\(349\) −20673.5 + 20673.5i −0.169732 + 0.169732i −0.786862 0.617130i \(-0.788295\pi\)
0.617130 + 0.786862i \(0.288295\pi\)
\(350\) 312586.i 2.55172i
\(351\) 0 0
\(352\) −66718.7 −0.538471
\(353\) 70477.9 + 70477.9i 0.565592 + 0.565592i 0.930890 0.365298i \(-0.119033\pi\)
−0.365298 + 0.930890i \(0.619033\pi\)
\(354\) 0 0
\(355\) 19977.4i 0.158519i
\(356\) −28967.5 28967.5i −0.228565 0.228565i
\(357\) 0 0
\(358\) 152969. 152969.i 1.19354 1.19354i
\(359\) 90432.4 90432.4i 0.701674 0.701674i −0.263096 0.964770i \(-0.584744\pi\)
0.964770 + 0.263096i \(0.0847436\pi\)
\(360\) 0 0
\(361\) 73551.2i 0.564385i
\(362\) −38846.9 + 38846.9i −0.296442 + 0.296442i
\(363\) 0 0
\(364\) 89834.5 31783.1i 0.678017 0.239880i
\(365\) −175591. −1.31801
\(366\) 0 0
\(367\) −156339. −1.16074 −0.580369 0.814354i \(-0.697091\pi\)
−0.580369 + 0.814354i \(0.697091\pi\)
\(368\) 44141.2i 0.325948i
\(369\) 0 0
\(370\) −30634.1 30634.1i −0.223770 0.223770i
\(371\) 100278. 100278.i 0.728546 0.728546i
\(372\) 0 0
\(373\) 12154.0 0.0873580 0.0436790 0.999046i \(-0.486092\pi\)
0.0436790 + 0.999046i \(0.486092\pi\)
\(374\) 92556.7i 0.661705i
\(375\) 0 0
\(376\) 19780.3i 0.139913i
\(377\) −66522.5 + 139364.i −0.468043 + 0.980549i
\(378\) 0 0
\(379\) 56764.9 + 56764.9i 0.395186 + 0.395186i 0.876531 0.481345i \(-0.159852\pi\)
−0.481345 + 0.876531i \(0.659852\pi\)
\(380\) −190118. −1.31661
\(381\) 0 0
\(382\) −187915. 187915.i −1.28776 1.28776i
\(383\) −14158.7 14158.7i −0.0965218 0.0965218i 0.657197 0.753719i \(-0.271742\pi\)
−0.753719 + 0.657197i \(0.771742\pi\)
\(384\) 0 0
\(385\) 51333.0 51333.0i 0.346318 0.346318i
\(386\) 75805.3 0.508774
\(387\) 0 0
\(388\) −26234.9 + 26234.9i −0.174267 + 0.174267i
\(389\) 14487.6i 0.0957405i 0.998854 + 0.0478703i \(0.0152434\pi\)
−0.998854 + 0.0478703i \(0.984757\pi\)
\(390\) 0 0
\(391\) 64525.4 0.422063
\(392\) −4206.48 4206.48i −0.0273745 0.0273745i
\(393\) 0 0
\(394\) 36155.0i 0.232904i
\(395\) −264617. 264617.i −1.69599 1.69599i
\(396\) 0 0
\(397\) 120415. 120415.i 0.764013 0.764013i −0.213032 0.977045i \(-0.568334\pi\)
0.977045 + 0.213032i \(0.0683339\pi\)
\(398\) −288910. + 288910.i −1.82388 + 1.82388i
\(399\) 0 0
\(400\) 394300.i 2.46438i
\(401\) −31598.4 + 31598.4i −0.196506 + 0.196506i −0.798500 0.601994i \(-0.794373\pi\)
0.601994 + 0.798500i \(0.294373\pi\)
\(402\) 0 0
\(403\) −197398. + 69838.8i −1.21544 + 0.430018i
\(404\) −167840. −1.02833
\(405\) 0 0
\(406\) −175537. −1.06492
\(407\) 7269.34i 0.0438840i
\(408\) 0 0
\(409\) 16690.9 + 16690.9i 0.0997776 + 0.0997776i 0.755233 0.655456i \(-0.227523\pi\)
−0.655456 + 0.755233i \(0.727523\pi\)
\(410\) 308136. 308136.i 1.83305 1.83305i
\(411\) 0 0
\(412\) −110.322 −0.000649934
\(413\) 80159.5i 0.469954i
\(414\) 0 0
\(415\) 273724.i 1.58934i
\(416\) 233034. 82446.6i 1.34658 0.476416i
\(417\) 0 0
\(418\) 44022.6 + 44022.6i 0.251955 + 0.251955i
\(419\) 45247.7 0.257732 0.128866 0.991662i \(-0.458866\pi\)
0.128866 + 0.991662i \(0.458866\pi\)
\(420\) 0 0
\(421\) 95387.7 + 95387.7i 0.538181 + 0.538181i 0.922994 0.384814i \(-0.125734\pi\)
−0.384814 + 0.922994i \(0.625734\pi\)
\(422\) −182866. 182866.i −1.02685 1.02685i
\(423\) 0 0
\(424\) −13936.6 + 13936.6i −0.0775219 + 0.0775219i
\(425\) 576386. 3.19107
\(426\) 0 0
\(427\) −89176.8 + 89176.8i −0.489098 + 0.489098i
\(428\) 103960.i 0.567519i
\(429\) 0 0
\(430\) 111798. 0.604639
\(431\) 120110. + 120110.i 0.646582 + 0.646582i 0.952165 0.305583i \(-0.0988514\pi\)
−0.305583 + 0.952165i \(0.598851\pi\)
\(432\) 0 0
\(433\) 104909.i 0.559549i 0.960066 + 0.279774i \(0.0902596\pi\)
−0.960066 + 0.279774i \(0.909740\pi\)
\(434\) −168300. 168300.i −0.893518 0.893518i
\(435\) 0 0
\(436\) −62117.4 + 62117.4i −0.326768 + 0.326768i
\(437\) 30690.1 30690.1i 0.160707 0.160707i
\(438\) 0 0
\(439\) 189251.i 0.981997i 0.871160 + 0.490998i \(0.163368\pi\)
−0.871160 + 0.490998i \(0.836632\pi\)
\(440\) −7134.25 + 7134.25i −0.0368505 + 0.0368505i
\(441\) 0 0
\(442\) −114376. 323281.i −0.585448 1.65476i
\(443\) 154643. 0.787995 0.393998 0.919111i \(-0.371092\pi\)
0.393998 + 0.919111i \(0.371092\pi\)
\(444\) 0 0
\(445\) 115629. 0.583910
\(446\) 182451.i 0.917227i
\(447\) 0 0
\(448\) 106744. + 106744.i 0.531846 + 0.531846i
\(449\) −54131.2 + 54131.2i −0.268507 + 0.268507i −0.828498 0.559992i \(-0.810804\pi\)
0.559992 + 0.828498i \(0.310804\pi\)
\(450\) 0 0
\(451\) −73119.3 −0.359484
\(452\) 357591.i 1.75029i
\(453\) 0 0
\(454\) 272835.i 1.32370i
\(455\) −115861. + 242729.i −0.559649 + 1.17246i
\(456\) 0 0
\(457\) −44160.2 44160.2i −0.211446 0.211446i 0.593436 0.804881i \(-0.297771\pi\)
−0.804881 + 0.593436i \(0.797771\pi\)
\(458\) 446101. 2.12668
\(459\) 0 0
\(460\) 102779. + 102779.i 0.485723 + 0.485723i
\(461\) 145494. + 145494.i 0.684610 + 0.684610i 0.961035 0.276425i \(-0.0891499\pi\)
−0.276425 + 0.961035i \(0.589150\pi\)
\(462\) 0 0
\(463\) −168589. + 168589.i −0.786443 + 0.786443i −0.980909 0.194467i \(-0.937702\pi\)
0.194467 + 0.980909i \(0.437702\pi\)
\(464\) −221424. −1.02847
\(465\) 0 0
\(466\) −27012.2 + 27012.2i −0.124391 + 0.124391i
\(467\) 164176.i 0.752793i −0.926459 0.376397i \(-0.877163\pi\)
0.926459 0.376397i \(-0.122837\pi\)
\(468\) 0 0
\(469\) −124804. −0.567392
\(470\) −815813. 815813.i −3.69313 3.69313i
\(471\) 0 0
\(472\) 11140.6i 0.0500061i
\(473\) −13264.6 13264.6i −0.0592885 0.0592885i
\(474\) 0 0
\(475\) 274146. 274146.i 1.21505 1.21505i
\(476\) 141230. 141230.i 0.623322 0.623322i
\(477\) 0 0
\(478\) 201917.i 0.883726i
\(479\) −10906.2 + 10906.2i −0.0475337 + 0.0475337i −0.730474 0.682940i \(-0.760701\pi\)
0.682940 + 0.730474i \(0.260701\pi\)
\(480\) 0 0
\(481\) −8982.97 25390.3i −0.0388267 0.109743i
\(482\) 34949.7 0.150435
\(483\) 0 0
\(484\) −211184. −0.901510
\(485\) 104721.i 0.445196i
\(486\) 0 0
\(487\) 225502. + 225502.i 0.950808 + 0.950808i 0.998846 0.0480379i \(-0.0152968\pi\)
−0.0480379 + 0.998846i \(0.515297\pi\)
\(488\) 12393.8 12393.8i 0.0520432 0.0520432i
\(489\) 0 0
\(490\) 346982. 1.44516
\(491\) 311585.i 1.29245i 0.763147 + 0.646225i \(0.223653\pi\)
−0.763147 + 0.646225i \(0.776347\pi\)
\(492\) 0 0
\(493\) 323677.i 1.33174i
\(494\) −208162. 99361.4i −0.852997 0.407159i
\(495\) 0 0
\(496\) −212295. 212295.i −0.862933 0.862933i
\(497\) −14116.9 −0.0571516
\(498\) 0 0
\(499\) 1928.12 + 1928.12i 0.00774341 + 0.00774341i 0.710968 0.703224i \(-0.248257\pi\)
−0.703224 + 0.710968i \(0.748257\pi\)
\(500\) 565456. + 565456.i 2.26183 + 2.26183i
\(501\) 0 0
\(502\) −406003. + 406003.i −1.61110 + 1.61110i
\(503\) 185299. 0.732382 0.366191 0.930540i \(-0.380662\pi\)
0.366191 + 0.930540i \(0.380662\pi\)
\(504\) 0 0
\(505\) 334981. 334981.i 1.31352 1.31352i
\(506\) 47597.8i 0.185903i
\(507\) 0 0
\(508\) 445001. 1.72438
\(509\) −103850. 103850.i −0.400841 0.400841i 0.477688 0.878529i \(-0.341475\pi\)
−0.878529 + 0.477688i \(0.841475\pi\)
\(510\) 0 0
\(511\) 124081.i 0.475186i
\(512\) 263398. + 263398.i 1.00478 + 1.00478i
\(513\) 0 0
\(514\) 60573.7 60573.7i 0.229276 0.229276i
\(515\) 220.186 220.186i 0.000830185 0.000830185i
\(516\) 0 0
\(517\) 193589.i 0.724269i
\(518\) 21647.4 21647.4i 0.0806765 0.0806765i
\(519\) 0 0
\(520\) 16102.4 33734.5i 0.0595503 0.124758i
\(521\) −200788. −0.739712 −0.369856 0.929089i \(-0.620593\pi\)
−0.369856 + 0.929089i \(0.620593\pi\)
\(522\) 0 0
\(523\) −9411.84 −0.0344089 −0.0172045 0.999852i \(-0.505477\pi\)
−0.0172045 + 0.999852i \(0.505477\pi\)
\(524\) 361353.i 1.31604i
\(525\) 0 0
\(526\) 30679.0 + 30679.0i 0.110884 + 0.110884i
\(527\) −310332. + 310332.i −1.11739 + 1.11739i
\(528\) 0 0
\(529\) 246658. 0.881424
\(530\) 1.14959e6i 4.09254i
\(531\) 0 0
\(532\) 134346.i 0.474681i
\(533\) 255390. 90356.1i 0.898980 0.318056i
\(534\) 0 0
\(535\) −207488. 207488.i −0.724913 0.724913i
\(536\) 17345.3 0.0603742
\(537\) 0 0
\(538\) −406904. 406904.i −1.40581 1.40581i
\(539\) −41168.7 41168.7i −0.141707 0.141707i
\(540\) 0 0
\(541\) −16197.1 + 16197.1i −0.0553406 + 0.0553406i −0.734235 0.678895i \(-0.762459\pi\)
0.678895 + 0.734235i \(0.262459\pi\)
\(542\) 539696. 1.83718
\(543\) 0 0
\(544\) 366356. 366356.i 1.23796 1.23796i
\(545\) 247953.i 0.834788i
\(546\) 0 0
\(547\) −211628. −0.707292 −0.353646 0.935379i \(-0.615058\pi\)
−0.353646 + 0.935379i \(0.615058\pi\)
\(548\) 84574.9 + 84574.9i 0.281631 + 0.281631i
\(549\) 0 0
\(550\) 425178.i 1.40555i
\(551\) 153950. + 153950.i 0.507081 + 0.507081i
\(552\) 0 0
\(553\) 186991. 186991.i 0.611462 0.611462i
\(554\) −284268. + 284268.i −0.926207 + 0.926207i
\(555\) 0 0
\(556\) 204839.i 0.662619i
\(557\) 143135. 143135.i 0.461354 0.461354i −0.437745 0.899099i \(-0.644223\pi\)
0.899099 + 0.437745i \(0.144223\pi\)
\(558\) 0 0
\(559\) 62721.8 + 29938.8i 0.200722 + 0.0958101i
\(560\) −385652. −1.22976
\(561\) 0 0
\(562\) 166199. 0.526206
\(563\) 230941.i 0.728592i 0.931283 + 0.364296i \(0.118690\pi\)
−0.931283 + 0.364296i \(0.881310\pi\)
\(564\) 0 0
\(565\) −713694. 713694.i −2.23571 2.23571i
\(566\) 122205. 122205.i 0.381467 0.381467i
\(567\) 0 0
\(568\) 1961.97 0.00608129
\(569\) 243018.i 0.750609i 0.926902 + 0.375305i \(0.122462\pi\)
−0.926902 + 0.375305i \(0.877538\pi\)
\(570\) 0 0
\(571\) 229921.i 0.705189i −0.935776 0.352595i \(-0.885299\pi\)
0.935776 0.352595i \(-0.114701\pi\)
\(572\) −122192. + 43231.2i −0.373467 + 0.132131i
\(573\) 0 0
\(574\) 217743. + 217743.i 0.660876 + 0.660876i
\(575\) −296410. −0.896515
\(576\) 0 0
\(577\) −45748.5 45748.5i −0.137412 0.137412i 0.635055 0.772467i \(-0.280977\pi\)
−0.772467 + 0.635055i \(0.780977\pi\)
\(578\) −169929. 169929.i −0.508641 0.508641i
\(579\) 0 0
\(580\) −515568. + 515568.i −1.53260 + 1.53260i
\(581\) 193426. 0.573010
\(582\) 0 0
\(583\) −136397. + 136397.i −0.401299 + 0.401299i
\(584\) 17244.7i 0.0505628i
\(585\) 0 0
\(586\) 548126. 1.59619
\(587\) 183355. + 183355.i 0.532127 + 0.532127i 0.921205 0.389078i \(-0.127206\pi\)
−0.389078 + 0.921205i \(0.627206\pi\)
\(588\) 0 0
\(589\) 295206.i 0.850931i
\(590\) 459479. + 459479.i 1.31996 + 1.31996i
\(591\) 0 0
\(592\) 27306.4 27306.4i 0.0779149 0.0779149i
\(593\) 83397.8 83397.8i 0.237162 0.237162i −0.578512 0.815674i \(-0.696366\pi\)
0.815674 + 0.578512i \(0.196366\pi\)
\(594\) 0 0
\(595\) 563745.i 1.59239i
\(596\) −168680. + 168680.i −0.474866 + 0.474866i
\(597\) 0 0
\(598\) 58818.3 + 166249.i 0.164479 + 0.464897i
\(599\) 217649. 0.606601 0.303300 0.952895i \(-0.401911\pi\)
0.303300 + 0.952895i \(0.401911\pi\)
\(600\) 0 0
\(601\) −441379. −1.22197 −0.610987 0.791640i \(-0.709227\pi\)
−0.610987 + 0.791640i \(0.709227\pi\)
\(602\) 79001.4i 0.217993i
\(603\) 0 0
\(604\) 169103. + 169103.i 0.463529 + 0.463529i
\(605\) 421490. 421490.i 1.15153 1.15153i
\(606\) 0 0
\(607\) −283335. −0.768993 −0.384497 0.923126i \(-0.625625\pi\)
−0.384497 + 0.923126i \(0.625625\pi\)
\(608\) 348499.i 0.942745i
\(609\) 0 0
\(610\) 1.02233e6i 2.74747i
\(611\) −239225. 676166.i −0.640802 1.81122i
\(612\) 0 0
\(613\) 482403. + 482403.i 1.28377 + 1.28377i 0.938503 + 0.345272i \(0.112213\pi\)
0.345272 + 0.938503i \(0.387787\pi\)
\(614\) −783139. −2.07731
\(615\) 0 0
\(616\) −5041.39 5041.39i −0.0132858 0.0132858i
\(617\) −136398. 136398.i −0.358293 0.358293i 0.504890 0.863184i \(-0.331533\pi\)
−0.863184 + 0.504890i \(0.831533\pi\)
\(618\) 0 0
\(619\) 130340. 130340.i 0.340170 0.340170i −0.516261 0.856431i \(-0.672677\pi\)
0.856431 + 0.516261i \(0.172677\pi\)
\(620\) −988623. −2.57186
\(621\) 0 0
\(622\) 286139. 286139.i 0.739600 0.739600i
\(623\) 81708.7i 0.210519i
\(624\) 0 0
\(625\) −1.24013e6 −3.17473
\(626\) 277559. + 277559.i 0.708282 + 0.708282i
\(627\) 0 0
\(628\) 143831.i 0.364698i
\(629\) −39916.3 39916.3i −0.100890 0.100890i
\(630\) 0 0
\(631\) −145847. + 145847.i −0.366302 + 0.366302i −0.866127 0.499825i \(-0.833398\pi\)
0.499825 + 0.866127i \(0.333398\pi\)
\(632\) −25987.9 + 25987.9i −0.0650635 + 0.0650635i
\(633\) 0 0
\(634\) 1.13753e6i 2.82998i
\(635\) −888150. + 888150.i −2.20262 + 2.20262i
\(636\) 0 0
\(637\) 194667. + 92920.0i 0.479749 + 0.228997i
\(638\) 238764. 0.586580
\(639\) 0 0
\(640\) 113101. 0.276126
\(641\) 207974.i 0.506165i 0.967445 + 0.253083i \(0.0814445\pi\)
−0.967445 + 0.253083i \(0.918556\pi\)
\(642\) 0 0
\(643\) −30795.0 30795.0i −0.0744831 0.0744831i 0.668884 0.743367i \(-0.266772\pi\)
−0.743367 + 0.668884i \(0.766772\pi\)
\(644\) −72628.3 + 72628.3i −0.175119 + 0.175119i
\(645\) 0 0
\(646\) −483461. −1.15850
\(647\) 216754.i 0.517795i 0.965905 + 0.258898i \(0.0833592\pi\)
−0.965905 + 0.258898i \(0.916641\pi\)
\(648\) 0 0
\(649\) 109032.i 0.258861i
\(650\) 525407. + 1.48506e6i 1.24357 + 3.51492i
\(651\) 0 0
\(652\) −19574.0 19574.0i −0.0460452 0.0460452i
\(653\) −622399. −1.45963 −0.729815 0.683645i \(-0.760394\pi\)
−0.729815 + 0.683645i \(0.760394\pi\)
\(654\) 0 0
\(655\) 721203. + 721203.i 1.68103 + 1.68103i
\(656\) 274664. + 274664.i 0.638254 + 0.638254i
\(657\) 0 0
\(658\) 576491. 576491.i 1.33150 1.33150i
\(659\) −127164. −0.292814 −0.146407 0.989224i \(-0.546771\pi\)
−0.146407 + 0.989224i \(0.546771\pi\)
\(660\) 0 0
\(661\) 512043. 512043.i 1.17194 1.17194i 0.190187 0.981748i \(-0.439091\pi\)
0.981748 0.190187i \(-0.0609095\pi\)
\(662\) 594817.i 1.35727i
\(663\) 0 0
\(664\) −26882.3 −0.0609720
\(665\) 268133. + 268133.i 0.606328 + 0.606328i
\(666\) 0 0
\(667\) 166453.i 0.374145i
\(668\) −126687. 126687.i −0.283909 0.283909i
\(669\) 0 0
\(670\) −715384. + 715384.i −1.59364 + 1.59364i
\(671\) 121298. 121298.i 0.269406 0.269406i
\(672\) 0 0
\(673\) 70363.7i 0.155353i 0.996979 + 0.0776763i \(0.0247501\pi\)
−0.996979 + 0.0776763i \(0.975250\pi\)
\(674\) 57972.1 57972.1i 0.127614 0.127614i
\(675\) 0 0
\(676\) 373370. 301995.i 0.817044 0.660855i
\(677\) −289817. −0.632334 −0.316167 0.948704i \(-0.602396\pi\)
−0.316167 + 0.948704i \(0.602396\pi\)
\(678\) 0 0
\(679\) 74000.9 0.160508
\(680\) 78349.1i 0.169440i
\(681\) 0 0
\(682\) 228920. + 228920.i 0.492170 + 0.492170i
\(683\) −126812. + 126812.i −0.271844 + 0.271844i −0.829842 0.557998i \(-0.811570\pi\)
0.557998 + 0.829842i \(0.311570\pi\)
\(684\) 0 0
\(685\) −337596. −0.719476
\(686\) 706430.i 1.50114i
\(687\) 0 0
\(688\) 99653.4i 0.210531i
\(689\) 307856. 644957.i 0.648498 1.35860i
\(690\) 0 0
\(691\) 512067. + 512067.i 1.07243 + 1.07243i 0.997163 + 0.0752718i \(0.0239824\pi\)
0.0752718 + 0.997163i \(0.476018\pi\)
\(692\) 658388. 1.37490
\(693\) 0 0
\(694\) −875563. 875563.i −1.81789 1.81789i
\(695\) −408827. 408827.i −0.846388 0.846388i
\(696\) 0 0
\(697\) 401502. 401502.i 0.826461 0.826461i
\(698\) −167478. −0.343753
\(699\) 0 0
\(700\) −648768. + 648768.i −1.32402 + 1.32402i
\(701\) 718609.i 1.46237i 0.682181 + 0.731184i \(0.261032\pi\)
−0.682181 + 0.731184i \(0.738968\pi\)
\(702\) 0 0
\(703\) −37970.7 −0.0768312
\(704\) −145192. 145192.i −0.292952 0.292952i
\(705\) 0 0
\(706\) 570946.i 1.14547i
\(707\) 236713. + 236713.i 0.473570 + 0.473570i
\(708\) 0 0
\(709\) −360756. + 360756.i −0.717664 + 0.717664i −0.968126 0.250462i \(-0.919417\pi\)
0.250462 + 0.968126i \(0.419417\pi\)
\(710\) −80919.1 + 80919.1i −0.160522 + 0.160522i
\(711\) 0 0
\(712\) 11355.9i 0.0224006i
\(713\) 159590. 159590.i 0.313926 0.313926i
\(714\) 0 0
\(715\) 157594. 330159.i 0.308267 0.645819i
\(716\) 634968. 1.23858
\(717\) 0 0
\(718\) 732599. 1.42108
\(719\) 756213.i 1.46280i −0.681946 0.731402i \(-0.738866\pi\)
0.681946 0.731402i \(-0.261134\pi\)
\(720\) 0 0
\(721\) 155.593 + 155.593i 0.000299310 + 0.000299310i
\(722\) 297921. 297921.i 0.571515 0.571515i
\(723\) 0 0
\(724\) −161252. −0.307630
\(725\) 1.48688e6i 2.82878i
\(726\) 0 0
\(727\) 2488.50i 0.00470835i −0.999997 0.00235417i \(-0.999251\pi\)
0.999997 0.00235417i \(-0.000749357\pi\)
\(728\) 23838.3 + 11378.7i 0.0449794 + 0.0214699i
\(729\) 0 0
\(730\) −711239. 711239.i −1.33466 1.33466i
\(731\) 145673. 0.272611
\(732\) 0 0
\(733\) 269312. + 269312.i 0.501243 + 0.501243i 0.911824 0.410581i \(-0.134674\pi\)
−0.410581 + 0.911824i \(0.634674\pi\)
\(734\) −633254. 633254.i −1.17540 1.17540i
\(735\) 0 0
\(736\) −188401. + 188401.i −0.347798 + 0.347798i
\(737\) 169758. 0.312532
\(738\) 0 0
\(739\) −710849. + 710849.i −1.30163 + 1.30163i −0.374342 + 0.927291i \(0.622131\pi\)
−0.927291 + 0.374342i \(0.877869\pi\)
\(740\) 127161.i 0.232215i
\(741\) 0 0
\(742\) 812357. 1.47550
\(743\) 516575. + 516575.i 0.935741 + 0.935741i 0.998057 0.0623154i \(-0.0198485\pi\)
−0.0623154 + 0.998057i \(0.519848\pi\)
\(744\) 0 0
\(745\) 673317.i 1.21313i
\(746\) 49230.3 + 49230.3i 0.0884616 + 0.0884616i
\(747\) 0 0
\(748\) −192100. + 192100.i −0.343340 + 0.343340i
\(749\) 146621. 146621.i 0.261356 0.261356i
\(750\) 0 0
\(751\) 628582.i 1.11450i −0.830343 0.557252i \(-0.811856\pi\)
0.830343 0.557252i \(-0.188144\pi\)
\(752\) 727194. 727194.i 1.28592 1.28592i
\(753\) 0 0
\(754\) −833952. + 295049.i −1.46689 + 0.518981i
\(755\) −675004. −1.18417
\(756\) 0 0
\(757\) 874918. 1.52678 0.763389 0.645940i \(-0.223534\pi\)
0.763389 + 0.645940i \(0.223534\pi\)
\(758\) 459856.i 0.800357i
\(759\) 0 0
\(760\) −37265.1 37265.1i −0.0645171 0.0645171i
\(761\) −176909. + 176909.i −0.305479 + 0.305479i −0.843153 0.537674i \(-0.819303\pi\)
0.537674 + 0.843153i \(0.319303\pi\)
\(762\) 0 0
\(763\) 175215. 0.300969
\(764\) 780031.i 1.33636i
\(765\) 0 0
\(766\) 114700.i 0.195482i
\(767\) 134735. + 380827.i 0.229029 + 0.647347i
\(768\) 0 0
\(769\) 64761.1 + 64761.1i 0.109512 + 0.109512i 0.759740 0.650228i \(-0.225326\pi\)
−0.650228 + 0.759740i \(0.725326\pi\)
\(770\) 415852. 0.701387
\(771\) 0 0
\(772\) 157333. + 157333.i 0.263988 + 0.263988i
\(773\) −65728.0 65728.0i −0.110000 0.110000i 0.649965 0.759964i \(-0.274784\pi\)
−0.759964 + 0.649965i \(0.774784\pi\)
\(774\) 0 0
\(775\) 1.42557e6 1.42557e6i 2.37348 2.37348i
\(776\) −10284.6 −0.0170791
\(777\) 0 0
\(778\) −58682.3 + 58682.3i −0.0969501 + 0.0969501i
\(779\) 381932.i 0.629377i
\(780\) 0 0
\(781\) 19201.8 0.0314803
\(782\) 261362. + 261362.i 0.427395 + 0.427395i
\(783\) 0 0
\(784\) 309290.i 0.503193i
\(785\) −287064. 287064.i −0.465843 0.465843i
\(786\) 0 0
\(787\) 853121. 853121.i 1.37740 1.37740i 0.528420 0.848983i \(-0.322785\pi\)
0.848983 0.528420i \(-0.177215\pi\)
\(788\) −75039.2 + 75039.2i −0.120847 + 0.120847i
\(789\) 0 0
\(790\) 2.14368e6i 3.43484i
\(791\) 504329. 504329.i 0.806048 0.806048i
\(792\) 0 0
\(793\) −273775. + 573559.i −0.435360 + 0.912078i
\(794\) 975493. 1.54733
\(795\) 0 0
\(796\) −1.19926e6 −1.89272
\(797\) 264563.i 0.416498i 0.978076 + 0.208249i \(0.0667765\pi\)
−0.978076 + 0.208249i \(0.933224\pi\)
\(798\) 0 0
\(799\) −1.06301e6 1.06301e6i −1.66511 1.66511i
\(800\) −1.68293e6 + 1.68293e6i −2.62958 + 2.62958i
\(801\) 0 0
\(802\) −255980. −0.397977
\(803\) 168774.i 0.261743i
\(804\) 0 0
\(805\) 289909.i 0.447373i
\(806\) −1.08245e6 516685.i −1.66625 0.795345i
\(807\) 0 0
\(808\) −32898.4 32898.4i −0.0503908 0.0503908i
\(809\) 903066. 1.37982 0.689910 0.723895i \(-0.257650\pi\)
0.689910 + 0.723895i \(0.257650\pi\)
\(810\) 0 0
\(811\) 158801. + 158801.i 0.241442 + 0.241442i 0.817446 0.576005i \(-0.195389\pi\)
−0.576005 + 0.817446i \(0.695389\pi\)
\(812\) −364324. 364324.i −0.552555 0.552555i
\(813\) 0 0
\(814\) −29444.7 + 29444.7i −0.0444384 + 0.0444384i
\(815\) 78133.2 0.117631
\(816\) 0 0
\(817\) 69286.2 69286.2i 0.103801 0.103801i
\(818\) 135214.i 0.202076i
\(819\) 0 0
\(820\) 1.27906e6 1.90223
\(821\) −566290. 566290.i −0.840142 0.840142i 0.148735 0.988877i \(-0.452480\pi\)
−0.988877 + 0.148735i \(0.952480\pi\)
\(822\) 0 0
\(823\) 664034.i 0.980371i 0.871618 + 0.490186i \(0.163071\pi\)
−0.871618 + 0.490186i \(0.836929\pi\)
\(824\) −21.6244 21.6244i −3.18485e−5 3.18485e-5i
\(825\) 0 0
\(826\) −324689. + 324689.i −0.475891 + 0.475891i
\(827\) 694781. 694781.i 1.01587 1.01587i 0.0159954 0.999872i \(-0.494908\pi\)
0.999872 0.0159954i \(-0.00509171\pi\)
\(828\) 0 0
\(829\) 1.07123e6i 1.55874i 0.626564 + 0.779370i \(0.284461\pi\)
−0.626564 + 0.779370i \(0.715539\pi\)
\(830\) 1.10873e6 1.10873e6i 1.60942 1.60942i
\(831\) 0 0
\(832\) 686543. + 327706.i 0.991794 + 0.473410i
\(833\) 452119. 0.651573
\(834\) 0 0
\(835\) 505694. 0.725295
\(836\) 182737.i 0.261465i
\(837\) 0 0
\(838\) 183277. + 183277.i 0.260988 + 0.260988i
\(839\) −710646. + 710646.i −1.00955 + 1.00955i −0.00959972 + 0.999954i \(0.503056\pi\)
−0.999954 + 0.00959972i \(0.996944\pi\)
\(840\) 0 0
\(841\) 127693. 0.180541
\(842\) 772742.i 1.08996i
\(843\) 0 0
\(844\) 759070.i 1.06561i
\(845\) −142453. + 1.34792e6i −0.199507 + 1.88778i
\(846\) 0 0
\(847\) 297844. + 297844.i 0.415167 + 0.415167i
\(848\) 1.02472e6 1.42499
\(849\) 0 0
\(850\) 2.33467e6 + 2.33467e6i 3.23138 + 3.23138i
\(851\) 20527.2 + 20527.2i 0.0283446 + 0.0283446i
\(852\) 0 0
\(853\) −719087. + 719087.i −0.988288 + 0.988288i −0.999932 0.0116445i \(-0.996293\pi\)
0.0116445 + 0.999932i \(0.496293\pi\)
\(854\) −722427. −0.990555
\(855\) 0 0
\(856\) −20377.3 + 20377.3i −0.0278099 + 0.0278099i
\(857\) 625779.i 0.852038i 0.904714 + 0.426019i \(0.140084\pi\)
−0.904714 + 0.426019i \(0.859916\pi\)
\(858\) 0 0
\(859\) −1.38660e6 −1.87917 −0.939585 0.342315i \(-0.888789\pi\)
−0.939585 + 0.342315i \(0.888789\pi\)
\(860\) 232034. + 232034.i 0.313729 + 0.313729i
\(861\) 0 0
\(862\) 973017.i 1.30950i
\(863\) 251871. + 251871.i 0.338187 + 0.338187i 0.855685 0.517497i \(-0.173136\pi\)
−0.517497 + 0.855685i \(0.673136\pi\)
\(864\) 0 0
\(865\) −1.31404e6 + 1.31404e6i −1.75621 + 1.75621i
\(866\) −424938. + 424938.i −0.566618 + 0.566618i
\(867\) 0 0
\(868\) 698606.i 0.927242i
\(869\) −254343. + 254343.i −0.336807 + 0.336807i
\(870\) 0 0
\(871\) −592928. + 209776.i −0.781566 + 0.276515i
\(872\) −24351.3 −0.0320250
\(873\) 0 0
\(874\) 248623. 0.325475
\(875\) 1.59499e6i 2.08325i
\(876\) 0 0
\(877\) 248488. + 248488.i 0.323077 + 0.323077i 0.849946 0.526869i \(-0.176634\pi\)
−0.526869 + 0.849946i \(0.676634\pi\)
\(878\) −766569. + 766569.i −0.994403 + 0.994403i
\(879\) 0 0
\(880\) 524562. 0.677378
\(881\) 1.01415e6i 1.30662i 0.757089 + 0.653312i \(0.226621\pi\)
−0.757089 + 0.653312i \(0.773379\pi\)
\(882\) 0 0
\(883\) 1.21924e6i 1.56375i −0.623433 0.781877i \(-0.714263\pi\)
0.623433 0.781877i \(-0.285737\pi\)
\(884\) 433580. 908349.i 0.554836 1.16238i
\(885\) 0 0
\(886\) 626388. + 626388.i 0.797950 + 0.797950i
\(887\) −904300. −1.14938 −0.574692 0.818370i \(-0.694878\pi\)
−0.574692 + 0.818370i \(0.694878\pi\)
\(888\) 0 0
\(889\) −627608. 627608.i −0.794118 0.794118i
\(890\) 468359. + 468359.i 0.591287 + 0.591287i
\(891\) 0 0
\(892\) −378674. + 378674.i −0.475923 + 0.475923i
\(893\) −1.01120e6 −1.26804
\(894\) 0 0
\(895\) −1.26729e6 + 1.26729e6i −1.58209 + 1.58209i
\(896\) 79922.4i 0.0995526i
\(897\) 0 0
\(898\) −438520. −0.543797
\(899\) 800549. + 800549.i 0.990532 + 0.990532i
\(900\) 0 0
\(901\) 1.49793e6i 1.84519i
\(902\) −296172. 296172.i −0.364025 0.364025i
\(903\) 0 0
\(904\) −70091.5 + 70091.5i −0.0857687 + 0.0857687i
\(905\) 321834. 321834.i 0.392948 0.392948i
\(906\) 0 0
\(907\) 1.40623e6i 1.70940i −0.519124 0.854699i \(-0.673742\pi\)
0.519124 0.854699i \(-0.326258\pi\)
\(908\) 566265. 566265.i 0.686828 0.686828i
\(909\) 0 0
\(910\) −1.45248e6 + 513883.i −1.75400 + 0.620557i
\(911\) 320928. 0.386697 0.193348 0.981130i \(-0.438065\pi\)
0.193348 + 0.981130i \(0.438065\pi\)
\(912\) 0 0
\(913\) −263097. −0.315627
\(914\) 357745.i 0.428234i
\(915\) 0 0
\(916\) 925876. + 925876.i 1.10347 + 1.10347i
\(917\) −509635. + 509635.i −0.606067 + 0.606067i
\(918\) 0 0
\(919\) −824931. −0.976757 −0.488378 0.872632i \(-0.662411\pi\)
−0.488378 + 0.872632i \(0.662411\pi\)
\(920\) 40291.5i 0.0476034i
\(921\) 0 0
\(922\) 1.17866e6i 1.38652i
\(923\) −67067.8 + 23728.3i −0.0787246 + 0.0278525i
\(924\) 0 0
\(925\) 183364. + 183364.i 0.214304 + 0.214304i
\(926\) −1.36575e6 −1.59276
\(927\) 0 0
\(928\) −945070. 945070.i −1.09741 1.09741i
\(929\) −678405. 678405.i −0.786064 0.786064i 0.194783 0.980846i \(-0.437600\pi\)
−0.980846 + 0.194783i \(0.937600\pi\)
\(930\) 0 0
\(931\) 215041. 215041.i 0.248097 0.248097i
\(932\) −112127. −0.129086
\(933\) 0 0
\(934\) 665000. 665000.i 0.762303 0.762303i
\(935\) 766802.i 0.877122i
\(936\) 0 0
\(937\) 1.11931e6 1.27488 0.637441 0.770499i \(-0.279993\pi\)
0.637441 + 0.770499i \(0.279993\pi\)
\(938\) −505523. 505523.i −0.574560 0.574560i
\(939\) 0 0
\(940\) 3.38642e6i 3.83252i
\(941\) −498975. 498975.i −0.563507 0.563507i 0.366795 0.930302i \(-0.380455\pi\)
−0.930302 + 0.366795i \(0.880455\pi\)
\(942\) 0 0
\(943\) −206475. + 206475.i −0.232190 + 0.232190i
\(944\) −409567. + 409567.i −0.459601 + 0.459601i
\(945\) 0 0
\(946\) 107457.i 0.120075i
\(947\) 672812. 672812.i 0.750228 0.750228i −0.224293 0.974522i \(-0.572007\pi\)
0.974522 + 0.224293i \(0.0720073\pi\)
\(948\) 0 0
\(949\) −208560. 589492.i −0.231579 0.654554i
\(950\) 2.22087e6 2.46080
\(951\) 0 0
\(952\) 55365.1 0.0610889
\(953\) 226250.i 0.249117i 0.992212 + 0.124558i \(0.0397514\pi\)
−0.992212 + 0.124558i \(0.960249\pi\)
\(954\) 0 0
\(955\) 1.55682e6 + 1.55682e6i 1.70699 + 1.70699i
\(956\) −419076. + 419076.i −0.458540 + 0.458540i
\(957\) 0 0
\(958\) −88351.7 −0.0962684
\(959\) 238561.i 0.259395i
\(960\) 0 0
\(961\) 611566.i 0.662211i
\(962\) 66458.2 139230.i 0.0718123 0.150447i
\(963\) 0 0
\(964\) 72537.6 + 72537.6i 0.0780565 + 0.0780565i
\(965\) −628022. −0.674404
\(966\) 0 0
\(967\) −652291. 652291.i −0.697571 0.697571i 0.266315 0.963886i \(-0.414194\pi\)
−0.963886 + 0.266315i \(0.914194\pi\)
\(968\) −41394.3 41394.3i −0.0441764 0.0441764i
\(969\) 0 0
\(970\) 424177. 424177.i 0.450821 0.450821i
\(971\) −330964. −0.351028 −0.175514 0.984477i \(-0.556159\pi\)
−0.175514 + 0.984477i \(0.556159\pi\)
\(972\) 0 0
\(973\) 288896. 288896.i 0.305151 0.305151i
\(974\) 1.82681e6i 1.92564i
\(975\) 0 0
\(976\) −911280. −0.956648
\(977\) 812569. + 812569.i 0.851278 + 0.851278i 0.990291 0.139013i \(-0.0443930\pi\)
−0.139013 + 0.990291i \(0.544393\pi\)
\(978\) 0 0
\(979\) 111140.i 0.115959i
\(980\) 720156. + 720156.i 0.749850 + 0.749850i
\(981\) 0 0
\(982\) −1.26209e6 + 1.26209e6i −1.30878 + 1.30878i
\(983\) −1.32394e6 + 1.32394e6i −1.37013 + 1.37013i −0.509898 + 0.860235i \(0.670317\pi\)
−0.860235 + 0.509898i \(0.829683\pi\)
\(984\) 0 0
\(985\) 299533.i 0.308725i
\(986\) −1.31107e6 + 1.31107e6i −1.34856 + 1.34856i
\(987\) 0 0
\(988\) −225814. 638260.i −0.231333 0.653858i
\(989\) −74913.1 −0.0765888
\(990\) 0 0
\(991\) −424185. −0.431925 −0.215962 0.976402i \(-0.569289\pi\)
−0.215962 + 0.976402i \(0.569289\pi\)
\(992\) 1.81221e6i 1.84156i
\(993\) 0 0
\(994\) −57181.2 57181.2i −0.0578736 0.0578736i
\(995\) 2.39352e6 2.39352e6i 2.41764 2.41764i
\(996\) 0 0
\(997\) 1.53487e6 1.54412 0.772060 0.635549i \(-0.219226\pi\)
0.772060 + 0.635549i \(0.219226\pi\)
\(998\) 15619.8i 0.0156825i
\(999\) 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.73.8 20
3.2 odd 2 39.5.g.a.34.3 yes 20
13.5 odd 4 inner 117.5.j.b.109.8 20
39.5 even 4 39.5.g.a.31.3 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.5.g.a.31.3 20 39.5 even 4
39.5.g.a.34.3 yes 20 3.2 odd 2
117.5.j.b.73.8 20 1.1 even 1 trivial
117.5.j.b.109.8 20 13.5 odd 4 inner