Properties

Label 108.4.i.a.49.6
Level $108$
Weight $4$
Character 108.49
Analytic conductor $6.372$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,4,Mod(13,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.i (of order \(9\), degree \(6\), 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: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 49.6
Character \(\chi\) \(=\) 108.49
Dual form 108.4.i.a.97.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.81053 + 4.87052i) q^{3} +(-13.0875 + 4.76347i) q^{5} +(-0.689494 - 3.91032i) q^{7} +(-20.4440 + 17.6365i) q^{9} +O(q^{10})\) \(q+(1.81053 + 4.87052i) q^{3} +(-13.0875 + 4.76347i) q^{5} +(-0.689494 - 3.91032i) q^{7} +(-20.4440 + 17.6365i) q^{9} +(-41.0823 - 14.9527i) q^{11} +(4.04425 + 3.39352i) q^{13} +(-46.8959 - 55.1187i) q^{15} +(-3.06285 + 5.30501i) q^{17} +(-7.42787 - 12.8654i) q^{19} +(17.7969 - 10.4379i) q^{21} +(-21.3706 + 121.199i) q^{23} +(52.8371 - 44.3356i) q^{25} +(-122.913 - 67.6414i) q^{27} +(-108.223 + 90.8101i) q^{29} +(-46.8412 + 265.650i) q^{31} +(-1.55313 - 227.164i) q^{33} +(27.6504 + 47.8920i) q^{35} +(100.261 - 173.657i) q^{37} +(-9.20601 + 25.8417i) q^{39} +(239.406 + 200.886i) q^{41} +(462.032 + 168.166i) q^{43} +(183.550 - 328.202i) q^{45} +(-10.6113 - 60.1798i) q^{47} +(307.499 - 111.921i) q^{49} +(-31.3835 - 5.31279i) q^{51} -380.159 q^{53} +608.892 q^{55} +(49.2130 - 59.4709i) q^{57} +(-154.748 + 56.3238i) q^{59} +(-149.877 - 849.997i) q^{61} +(83.0601 + 67.7821i) q^{63} +(-69.0941 - 25.1482i) q^{65} +(159.359 + 133.718i) q^{67} +(-628.994 + 115.348i) q^{69} +(-451.411 + 781.866i) q^{71} +(353.597 + 612.447i) q^{73} +(311.601 + 177.073i) q^{75} +(-30.1439 + 170.954i) q^{77} +(-847.858 + 711.438i) q^{79} +(106.911 - 721.118i) q^{81} +(652.401 - 547.430i) q^{83} +(14.8148 - 84.0192i) q^{85} +(-638.234 - 362.689i) q^{87} +(214.477 + 371.485i) q^{89} +(10.4813 - 18.1541i) q^{91} +(-1378.66 + 252.826i) q^{93} +(158.497 + 132.994i) q^{95} +(-1166.45 - 424.552i) q^{97} +(1103.60 - 418.853i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.81053 + 4.87052i 0.348437 + 0.937332i
\(4\) 0 0
\(5\) −13.0875 + 4.76347i −1.17058 + 0.426058i −0.852867 0.522128i \(-0.825138\pi\)
−0.317717 + 0.948186i \(0.602916\pi\)
\(6\) 0 0
\(7\) −0.689494 3.91032i −0.0372292 0.211137i 0.960518 0.278216i \(-0.0897432\pi\)
−0.997748 + 0.0670790i \(0.978632\pi\)
\(8\) 0 0
\(9\) −20.4440 + 17.6365i −0.757184 + 0.653202i
\(10\) 0 0
\(11\) −41.0823 14.9527i −1.12607 0.409856i −0.289206 0.957267i \(-0.593391\pi\)
−0.836864 + 0.547411i \(0.815613\pi\)
\(12\) 0 0
\(13\) 4.04425 + 3.39352i 0.0862825 + 0.0723996i 0.684909 0.728629i \(-0.259842\pi\)
−0.598626 + 0.801028i \(0.704287\pi\)
\(14\) 0 0
\(15\) −46.8959 55.1187i −0.807232 0.948772i
\(16\) 0 0
\(17\) −3.06285 + 5.30501i −0.0436970 + 0.0756855i −0.887047 0.461680i \(-0.847247\pi\)
0.843350 + 0.537365i \(0.180580\pi\)
\(18\) 0 0
\(19\) −7.42787 12.8654i −0.0896879 0.155344i 0.817691 0.575657i \(-0.195254\pi\)
−0.907379 + 0.420313i \(0.861920\pi\)
\(20\) 0 0
\(21\) 17.7969 10.4379i 0.184934 0.108464i
\(22\) 0 0
\(23\) −21.3706 + 121.199i −0.193743 + 1.09877i 0.720455 + 0.693502i \(0.243933\pi\)
−0.914198 + 0.405268i \(0.867178\pi\)
\(24\) 0 0
\(25\) 52.8371 44.3356i 0.422697 0.354685i
\(26\) 0 0
\(27\) −122.913 67.6414i −0.876098 0.482133i
\(28\) 0 0
\(29\) −108.223 + 90.8101i −0.692985 + 0.581483i −0.919768 0.392462i \(-0.871623\pi\)
0.226784 + 0.973945i \(0.427179\pi\)
\(30\) 0 0
\(31\) −46.8412 + 265.650i −0.271385 + 1.53910i 0.478831 + 0.877907i \(0.341061\pi\)
−0.750216 + 0.661193i \(0.770050\pi\)
\(32\) 0 0
\(33\) −1.55313 227.164i −0.00819291 1.19831i
\(34\) 0 0
\(35\) 27.6504 + 47.8920i 0.133537 + 0.231292i
\(36\) 0 0
\(37\) 100.261 173.657i 0.445481 0.771595i −0.552605 0.833443i \(-0.686366\pi\)
0.998086 + 0.0618483i \(0.0196995\pi\)
\(38\) 0 0
\(39\) −9.20601 + 25.8417i −0.0377985 + 0.106102i
\(40\) 0 0
\(41\) 239.406 + 200.886i 0.911927 + 0.765197i 0.972485 0.232967i \(-0.0748435\pi\)
−0.0605578 + 0.998165i \(0.519288\pi\)
\(42\) 0 0
\(43\) 462.032 + 168.166i 1.63859 + 0.596397i 0.986791 0.161998i \(-0.0517939\pi\)
0.651795 + 0.758395i \(0.274016\pi\)
\(44\) 0 0
\(45\) 183.550 328.202i 0.608045 1.08723i
\(46\) 0 0
\(47\) −10.6113 60.1798i −0.0329323 0.186769i 0.963904 0.266250i \(-0.0857847\pi\)
−0.996836 + 0.0794815i \(0.974674\pi\)
\(48\) 0 0
\(49\) 307.499 111.921i 0.896500 0.326299i
\(50\) 0 0
\(51\) −31.3835 5.31279i −0.0861681 0.0145870i
\(52\) 0 0
\(53\) −380.159 −0.985262 −0.492631 0.870238i \(-0.663965\pi\)
−0.492631 + 0.870238i \(0.663965\pi\)
\(54\) 0 0
\(55\) 608.892 1.49278
\(56\) 0 0
\(57\) 49.2130 59.4709i 0.114358 0.138195i
\(58\) 0 0
\(59\) −154.748 + 56.3238i −0.341467 + 0.124284i −0.507061 0.861910i \(-0.669268\pi\)
0.165594 + 0.986194i \(0.447046\pi\)
\(60\) 0 0
\(61\) −149.877 849.997i −0.314588 1.78411i −0.574523 0.818488i \(-0.694812\pi\)
0.259935 0.965626i \(-0.416299\pi\)
\(62\) 0 0
\(63\) 83.0601 + 67.7821i 0.166105 + 0.135551i
\(64\) 0 0
\(65\) −69.0941 25.1482i −0.131847 0.0479885i
\(66\) 0 0
\(67\) 159.359 + 133.718i 0.290579 + 0.243825i 0.776410 0.630228i \(-0.217039\pi\)
−0.485831 + 0.874053i \(0.661483\pi\)
\(68\) 0 0
\(69\) −628.994 + 115.348i −1.09742 + 0.201250i
\(70\) 0 0
\(71\) −451.411 + 781.866i −0.754543 + 1.30691i 0.191058 + 0.981579i \(0.438808\pi\)
−0.945601 + 0.325329i \(0.894525\pi\)
\(72\) 0 0
\(73\) 353.597 + 612.447i 0.566923 + 0.981939i 0.996868 + 0.0790840i \(0.0251995\pi\)
−0.429945 + 0.902855i \(0.641467\pi\)
\(74\) 0 0
\(75\) 311.601 + 177.073i 0.479741 + 0.272622i
\(76\) 0 0
\(77\) −30.1439 + 170.954i −0.0446132 + 0.253014i
\(78\) 0 0
\(79\) −847.858 + 711.438i −1.20749 + 1.01320i −0.208104 + 0.978107i \(0.566729\pi\)
−0.999384 + 0.0350955i \(0.988826\pi\)
\(80\) 0 0
\(81\) 106.911 721.118i 0.146654 0.989188i
\(82\) 0 0
\(83\) 652.401 547.430i 0.862775 0.723954i −0.0997890 0.995009i \(-0.531817\pi\)
0.962564 + 0.271054i \(0.0873723\pi\)
\(84\) 0 0
\(85\) 14.8148 84.0192i 0.0189047 0.107214i
\(86\) 0 0
\(87\) −638.234 362.689i −0.786504 0.446947i
\(88\) 0 0
\(89\) 214.477 + 371.485i 0.255444 + 0.442442i 0.965016 0.262191i \(-0.0844450\pi\)
−0.709572 + 0.704633i \(0.751112\pi\)
\(90\) 0 0
\(91\) 10.4813 18.1541i 0.0120740 0.0209128i
\(92\) 0 0
\(93\) −1378.66 + 252.826i −1.53721 + 0.281901i
\(94\) 0 0
\(95\) 158.497 + 132.994i 0.171173 + 0.143631i
\(96\) 0 0
\(97\) −1166.45 424.552i −1.22098 0.444400i −0.350480 0.936570i \(-0.613981\pi\)
−0.870499 + 0.492171i \(0.836204\pi\)
\(98\) 0 0
\(99\) 1103.60 418.853i 1.12036 0.425215i
\(100\) 0 0
\(101\) −142.775 809.715i −0.140659 0.797719i −0.970750 0.240091i \(-0.922823\pi\)
0.830091 0.557628i \(-0.188288\pi\)
\(102\) 0 0
\(103\) −1713.66 + 623.722i −1.63934 + 0.596672i −0.986924 0.161187i \(-0.948468\pi\)
−0.652418 + 0.757859i \(0.726245\pi\)
\(104\) 0 0
\(105\) −183.197 + 221.382i −0.170268 + 0.205759i
\(106\) 0 0
\(107\) −2093.70 −1.89164 −0.945819 0.324695i \(-0.894738\pi\)
−0.945819 + 0.324695i \(0.894738\pi\)
\(108\) 0 0
\(109\) 785.096 0.689895 0.344947 0.938622i \(-0.387897\pi\)
0.344947 + 0.938622i \(0.387897\pi\)
\(110\) 0 0
\(111\) 1027.32 + 173.912i 0.878463 + 0.148711i
\(112\) 0 0
\(113\) 1484.56 540.337i 1.23589 0.449828i 0.360281 0.932844i \(-0.382681\pi\)
0.875612 + 0.483015i \(0.160458\pi\)
\(114\) 0 0
\(115\) −297.638 1687.99i −0.241347 1.36875i
\(116\) 0 0
\(117\) −142.530 + 1.94906i −0.112623 + 0.00154009i
\(118\) 0 0
\(119\) 22.8561 + 8.31893i 0.0176068 + 0.00640836i
\(120\) 0 0
\(121\) 444.562 + 373.032i 0.334006 + 0.280265i
\(122\) 0 0
\(123\) −544.966 + 1529.74i −0.399495 + 1.12140i
\(124\) 0 0
\(125\) 390.150 675.760i 0.279169 0.483534i
\(126\) 0 0
\(127\) −197.197 341.555i −0.137783 0.238646i 0.788874 0.614554i \(-0.210664\pi\)
−0.926657 + 0.375908i \(0.877331\pi\)
\(128\) 0 0
\(129\) 17.4673 + 2554.81i 0.0119218 + 1.74371i
\(130\) 0 0
\(131\) 104.311 591.579i 0.0695704 0.394553i −0.930061 0.367405i \(-0.880246\pi\)
0.999631 0.0271482i \(-0.00864260\pi\)
\(132\) 0 0
\(133\) −45.1865 + 37.9160i −0.0294599 + 0.0247198i
\(134\) 0 0
\(135\) 1930.84 + 299.766i 1.23096 + 0.191109i
\(136\) 0 0
\(137\) 875.032 734.239i 0.545686 0.457885i −0.327791 0.944750i \(-0.606304\pi\)
0.873477 + 0.486865i \(0.161860\pi\)
\(138\) 0 0
\(139\) −359.669 + 2039.78i −0.219473 + 1.24469i 0.653501 + 0.756926i \(0.273300\pi\)
−0.872974 + 0.487767i \(0.837812\pi\)
\(140\) 0 0
\(141\) 273.895 160.640i 0.163589 0.0959455i
\(142\) 0 0
\(143\) −115.404 199.886i −0.0674867 0.116890i
\(144\) 0 0
\(145\) 983.804 1704.00i 0.563451 0.975926i
\(146\) 0 0
\(147\) 1101.85 + 1295.05i 0.618224 + 0.726624i
\(148\) 0 0
\(149\) 2383.51 + 2000.00i 1.31050 + 1.09964i 0.988227 + 0.152997i \(0.0488924\pi\)
0.322276 + 0.946646i \(0.395552\pi\)
\(150\) 0 0
\(151\) −35.5500 12.9391i −0.0191590 0.00697332i 0.332423 0.943130i \(-0.392134\pi\)
−0.351582 + 0.936157i \(0.614356\pi\)
\(152\) 0 0
\(153\) −30.9448 162.473i −0.0163512 0.0858508i
\(154\) 0 0
\(155\) −652.379 3699.82i −0.338067 1.91727i
\(156\) 0 0
\(157\) −2170.17 + 789.878i −1.10318 + 0.401523i −0.828486 0.560010i \(-0.810797\pi\)
−0.274689 + 0.961533i \(0.588575\pi\)
\(158\) 0 0
\(159\) −688.290 1851.57i −0.343301 0.923517i
\(160\) 0 0
\(161\) 488.661 0.239204
\(162\) 0 0
\(163\) 232.306 0.111630 0.0558148 0.998441i \(-0.482224\pi\)
0.0558148 + 0.998441i \(0.482224\pi\)
\(164\) 0 0
\(165\) 1102.42 + 2965.62i 0.520140 + 1.39923i
\(166\) 0 0
\(167\) 1530.58 557.084i 0.709218 0.258134i 0.0378768 0.999282i \(-0.487941\pi\)
0.671342 + 0.741148i \(0.265718\pi\)
\(168\) 0 0
\(169\) −376.665 2136.17i −0.171445 0.972314i
\(170\) 0 0
\(171\) 378.756 + 132.019i 0.169381 + 0.0590396i
\(172\) 0 0
\(173\) −1144.54 416.577i −0.502992 0.183074i 0.0780474 0.996950i \(-0.475131\pi\)
−0.581039 + 0.813876i \(0.697354\pi\)
\(174\) 0 0
\(175\) −209.797 176.041i −0.0906238 0.0760424i
\(176\) 0 0
\(177\) −554.503 651.730i −0.235475 0.276763i
\(178\) 0 0
\(179\) 1158.73 2006.98i 0.483842 0.838039i −0.515986 0.856597i \(-0.672574\pi\)
0.999828 + 0.0185580i \(0.00590754\pi\)
\(180\) 0 0
\(181\) −77.3428 133.962i −0.0317616 0.0550127i 0.849708 0.527254i \(-0.176778\pi\)
−0.881469 + 0.472241i \(0.843445\pi\)
\(182\) 0 0
\(183\) 3868.57 2268.93i 1.56269 0.916524i
\(184\) 0 0
\(185\) −484.957 + 2750.33i −0.192728 + 1.09302i
\(186\) 0 0
\(187\) 205.153 172.144i 0.0802260 0.0673176i
\(188\) 0 0
\(189\) −179.751 + 527.268i −0.0691798 + 0.202926i
\(190\) 0 0
\(191\) 2149.29 1803.47i 0.814227 0.683217i −0.137386 0.990518i \(-0.543870\pi\)
0.951613 + 0.307300i \(0.0994256\pi\)
\(192\) 0 0
\(193\) −475.369 + 2695.95i −0.177294 + 1.00549i 0.758168 + 0.652060i \(0.226095\pi\)
−0.935462 + 0.353427i \(0.885016\pi\)
\(194\) 0 0
\(195\) −2.61214 382.056i −0.000959277 0.140306i
\(196\) 0 0
\(197\) 1571.62 + 2722.12i 0.568391 + 0.984482i 0.996725 + 0.0808620i \(0.0257673\pi\)
−0.428334 + 0.903620i \(0.640899\pi\)
\(198\) 0 0
\(199\) −935.924 + 1621.07i −0.333396 + 0.577459i −0.983175 0.182664i \(-0.941528\pi\)
0.649779 + 0.760123i \(0.274861\pi\)
\(200\) 0 0
\(201\) −362.752 + 1018.26i −0.127296 + 0.357326i
\(202\) 0 0
\(203\) 429.716 + 360.574i 0.148572 + 0.124667i
\(204\) 0 0
\(205\) −4090.15 1488.69i −1.39350 0.507194i
\(206\) 0 0
\(207\) −1700.62 2854.69i −0.571020 0.958524i
\(208\) 0 0
\(209\) 112.780 + 639.608i 0.0373262 + 0.211687i
\(210\) 0 0
\(211\) 238.052 86.6438i 0.0776691 0.0282692i −0.302893 0.953024i \(-0.597953\pi\)
0.380562 + 0.924755i \(0.375730\pi\)
\(212\) 0 0
\(213\) −4625.39 783.013i −1.48792 0.251883i
\(214\) 0 0
\(215\) −6847.91 −2.17220
\(216\) 0 0
\(217\) 1071.07 0.335065
\(218\) 0 0
\(219\) −2342.74 + 2831.05i −0.722866 + 0.873539i
\(220\) 0 0
\(221\) −30.3896 + 11.0609i −0.00924988 + 0.00336668i
\(222\) 0 0
\(223\) 600.687 + 3406.66i 0.180381 + 1.02299i 0.931748 + 0.363106i \(0.118284\pi\)
−0.751367 + 0.659885i \(0.770605\pi\)
\(224\) 0 0
\(225\) −298.277 + 1838.25i −0.0883784 + 0.544668i
\(226\) 0 0
\(227\) −1925.67 700.886i −0.563044 0.204931i 0.0447883 0.998997i \(-0.485739\pi\)
−0.607833 + 0.794065i \(0.707961\pi\)
\(228\) 0 0
\(229\) −693.662 582.052i −0.200168 0.167961i 0.537194 0.843459i \(-0.319484\pi\)
−0.737362 + 0.675498i \(0.763929\pi\)
\(230\) 0 0
\(231\) −887.213 + 162.702i −0.252703 + 0.0463419i
\(232\) 0 0
\(233\) −696.488 + 1206.35i −0.195830 + 0.339188i −0.947172 0.320725i \(-0.896073\pi\)
0.751342 + 0.659913i \(0.229407\pi\)
\(234\) 0 0
\(235\) 425.540 + 737.057i 0.118124 + 0.204597i
\(236\) 0 0
\(237\) −5000.15 2841.43i −1.37044 0.778780i
\(238\) 0 0
\(239\) −43.4332 + 246.322i −0.0117551 + 0.0666663i −0.990121 0.140216i \(-0.955220\pi\)
0.978366 + 0.206882i \(0.0663316\pi\)
\(240\) 0 0
\(241\) 3775.11 3167.70i 1.00903 0.846677i 0.0208208 0.999783i \(-0.493372\pi\)
0.988210 + 0.153106i \(0.0489276\pi\)
\(242\) 0 0
\(243\) 3705.79 784.894i 0.978297 0.207206i
\(244\) 0 0
\(245\) −3491.28 + 2929.53i −0.910406 + 0.763921i
\(246\) 0 0
\(247\) 13.6191 77.2377i 0.00350835 0.0198968i
\(248\) 0 0
\(249\) 3847.46 + 2186.40i 0.979208 + 0.556455i
\(250\) 0 0
\(251\) 557.386 + 965.420i 0.140167 + 0.242776i 0.927559 0.373676i \(-0.121903\pi\)
−0.787392 + 0.616452i \(0.788569\pi\)
\(252\) 0 0
\(253\) 2690.21 4659.57i 0.668505 1.15788i
\(254\) 0 0
\(255\) 436.040 79.9632i 0.107082 0.0196372i
\(256\) 0 0
\(257\) −2365.04 1984.51i −0.574037 0.481674i 0.308946 0.951080i \(-0.400024\pi\)
−0.882983 + 0.469406i \(0.844468\pi\)
\(258\) 0 0
\(259\) −748.183 272.316i −0.179497 0.0653317i
\(260\) 0 0
\(261\) 610.944 3765.19i 0.144891 0.892949i
\(262\) 0 0
\(263\) 187.667 + 1064.32i 0.0440003 + 0.249538i 0.998872 0.0474799i \(-0.0151190\pi\)
−0.954872 + 0.297018i \(0.904008\pi\)
\(264\) 0 0
\(265\) 4975.34 1810.88i 1.15333 0.419778i
\(266\) 0 0
\(267\) −1421.01 + 1717.20i −0.325709 + 0.393599i
\(268\) 0 0
\(269\) −5507.29 −1.24827 −0.624137 0.781315i \(-0.714549\pi\)
−0.624137 + 0.781315i \(0.714549\pi\)
\(270\) 0 0
\(271\) 4075.64 0.913570 0.456785 0.889577i \(-0.349001\pi\)
0.456785 + 0.889577i \(0.349001\pi\)
\(272\) 0 0
\(273\) 107.397 + 18.1807i 0.0238093 + 0.00403058i
\(274\) 0 0
\(275\) −2833.61 + 1031.35i −0.621356 + 0.226155i
\(276\) 0 0
\(277\) 615.311 + 3489.60i 0.133467 + 0.756931i 0.975915 + 0.218152i \(0.0700029\pi\)
−0.842447 + 0.538779i \(0.818886\pi\)
\(278\) 0 0
\(279\) −3727.50 6257.04i −0.799855 1.34265i
\(280\) 0 0
\(281\) 6900.33 + 2511.51i 1.46491 + 0.533183i 0.946713 0.322080i \(-0.104382\pi\)
0.518195 + 0.855262i \(0.326604\pi\)
\(282\) 0 0
\(283\) −102.360 85.8899i −0.0215005 0.0180411i 0.631974 0.774990i \(-0.282245\pi\)
−0.653475 + 0.756948i \(0.726689\pi\)
\(284\) 0 0
\(285\) −360.789 + 1012.75i −0.0749871 + 0.210492i
\(286\) 0 0
\(287\) 620.458 1074.66i 0.127611 0.221029i
\(288\) 0 0
\(289\) 2437.74 + 4222.29i 0.496181 + 0.859411i
\(290\) 0 0
\(291\) −44.0982 6449.88i −0.00888343 1.29931i
\(292\) 0 0
\(293\) 710.916 4031.81i 0.141748 0.803893i −0.828173 0.560473i \(-0.810619\pi\)
0.969921 0.243420i \(-0.0782694\pi\)
\(294\) 0 0
\(295\) 1756.98 1474.28i 0.346763 0.290969i
\(296\) 0 0
\(297\) 4038.13 + 4616.75i 0.788942 + 0.901989i
\(298\) 0 0
\(299\) −497.719 + 417.636i −0.0962671 + 0.0807777i
\(300\) 0 0
\(301\) 339.013 1922.64i 0.0649183 0.368170i
\(302\) 0 0
\(303\) 3685.23 2161.40i 0.698717 0.409799i
\(304\) 0 0
\(305\) 6010.46 + 10410.4i 1.12839 + 1.95442i
\(306\) 0 0
\(307\) −4191.33 + 7259.59i −0.779191 + 1.34960i 0.153217 + 0.988192i \(0.451037\pi\)
−0.932409 + 0.361406i \(0.882297\pi\)
\(308\) 0 0
\(309\) −6140.49 7217.16i −1.13049 1.32871i
\(310\) 0 0
\(311\) 1493.12 + 1252.88i 0.272242 + 0.228438i 0.768679 0.639635i \(-0.220914\pi\)
−0.496437 + 0.868073i \(0.665359\pi\)
\(312\) 0 0
\(313\) −1191.53 433.681i −0.215173 0.0783165i 0.232185 0.972672i \(-0.425413\pi\)
−0.447358 + 0.894355i \(0.647635\pi\)
\(314\) 0 0
\(315\) −1409.93 491.446i −0.252192 0.0879042i
\(316\) 0 0
\(317\) 1155.38 + 6552.49i 0.204709 + 1.16096i 0.897898 + 0.440204i \(0.145094\pi\)
−0.693189 + 0.720756i \(0.743795\pi\)
\(318\) 0 0
\(319\) 5803.91 2112.45i 1.01867 0.370767i
\(320\) 0 0
\(321\) −3790.70 10197.4i −0.659116 1.77309i
\(322\) 0 0
\(323\) 91.0017 0.0156764
\(324\) 0 0
\(325\) 364.140 0.0621504
\(326\) 0 0
\(327\) 1421.44 + 3823.83i 0.240385 + 0.646661i
\(328\) 0 0
\(329\) −228.005 + 82.9872i −0.0382077 + 0.0139065i
\(330\) 0 0
\(331\) 988.638 + 5606.85i 0.164171 + 0.931058i 0.949915 + 0.312507i \(0.101169\pi\)
−0.785745 + 0.618551i \(0.787720\pi\)
\(332\) 0 0
\(333\) 1012.96 + 5318.48i 0.166697 + 0.875228i
\(334\) 0 0
\(335\) −2722.58 990.936i −0.444030 0.161614i
\(336\) 0 0
\(337\) 4026.87 + 3378.95i 0.650913 + 0.546181i 0.907348 0.420381i \(-0.138103\pi\)
−0.256435 + 0.966561i \(0.582548\pi\)
\(338\) 0 0
\(339\) 5319.57 + 6252.30i 0.852269 + 1.00171i
\(340\) 0 0
\(341\) 5896.53 10213.1i 0.936407 1.62190i
\(342\) 0 0
\(343\) −1330.63 2304.72i −0.209467 0.362808i
\(344\) 0 0
\(345\) 7682.52 4505.81i 1.19888 0.703145i
\(346\) 0 0
\(347\) −1204.18 + 6829.25i −0.186293 + 1.05652i 0.737989 + 0.674813i \(0.235776\pi\)
−0.924282 + 0.381710i \(0.875335\pi\)
\(348\) 0 0
\(349\) −362.713 + 304.352i −0.0556320 + 0.0466808i −0.670180 0.742199i \(-0.733783\pi\)
0.614548 + 0.788880i \(0.289339\pi\)
\(350\) 0 0
\(351\) −267.548 690.667i −0.0406856 0.105029i
\(352\) 0 0
\(353\) 1133.07 950.758i 0.170842 0.143353i −0.553358 0.832944i \(-0.686654\pi\)
0.724200 + 0.689590i \(0.242209\pi\)
\(354\) 0 0
\(355\) 2183.45 12383.0i 0.326438 1.85132i
\(356\) 0 0
\(357\) 0.864085 + 126.383i 0.000128101 + 0.0187364i
\(358\) 0 0
\(359\) 1718.27 + 2976.14i 0.252610 + 0.437534i 0.964244 0.265017i \(-0.0853777\pi\)
−0.711634 + 0.702551i \(0.752044\pi\)
\(360\) 0 0
\(361\) 3319.15 5748.94i 0.483912 0.838160i
\(362\) 0 0
\(363\) −1011.97 + 2840.64i −0.146321 + 0.410729i
\(364\) 0 0
\(365\) −7545.08 6331.07i −1.08199 0.907900i
\(366\) 0 0
\(367\) 8040.63 + 2926.55i 1.14364 + 0.416253i 0.843228 0.537556i \(-0.180652\pi\)
0.300416 + 0.953808i \(0.402874\pi\)
\(368\) 0 0
\(369\) −8437.33 + 115.378i −1.19032 + 0.0162774i
\(370\) 0 0
\(371\) 262.117 + 1486.54i 0.0366805 + 0.208025i
\(372\) 0 0
\(373\) −8918.15 + 3245.94i −1.23797 + 0.450586i −0.876321 0.481727i \(-0.840010\pi\)
−0.361653 + 0.932313i \(0.617787\pi\)
\(374\) 0 0
\(375\) 3997.68 + 676.751i 0.550505 + 0.0931927i
\(376\) 0 0
\(377\) −745.848 −0.101892
\(378\) 0 0
\(379\) −11784.3 −1.59714 −0.798572 0.601899i \(-0.794411\pi\)
−0.798572 + 0.601899i \(0.794411\pi\)
\(380\) 0 0
\(381\) 1306.52 1578.85i 0.175682 0.212301i
\(382\) 0 0
\(383\) −8550.92 + 3112.28i −1.14081 + 0.415222i −0.842207 0.539155i \(-0.818744\pi\)
−0.298606 + 0.954376i \(0.596522\pi\)
\(384\) 0 0
\(385\) −419.827 2380.96i −0.0555750 0.315182i
\(386\) 0 0
\(387\) −12411.6 + 4710.63i −1.63028 + 0.618746i
\(388\) 0 0
\(389\) −8567.80 3118.43i −1.11672 0.406454i −0.283267 0.959041i \(-0.591418\pi\)
−0.833455 + 0.552587i \(0.813641\pi\)
\(390\) 0 0
\(391\) −577.506 484.585i −0.0746949 0.0626765i
\(392\) 0 0
\(393\) 3070.16 563.021i 0.394068 0.0722663i
\(394\) 0 0
\(395\) 7707.46 13349.7i 0.981783 1.70050i
\(396\) 0 0
\(397\) −4029.27 6978.90i −0.509378 0.882269i −0.999941 0.0108631i \(-0.996542\pi\)
0.490563 0.871406i \(-0.336791\pi\)
\(398\) 0 0
\(399\) −266.482 151.434i −0.0334356 0.0190004i
\(400\) 0 0
\(401\) 2031.54 11521.4i 0.252993 1.43479i −0.548180 0.836361i \(-0.684679\pi\)
0.801173 0.598433i \(-0.204210\pi\)
\(402\) 0 0
\(403\) −1090.93 + 915.396i −0.134846 + 0.113149i
\(404\) 0 0
\(405\) 2035.82 + 9946.92i 0.249780 + 1.22041i
\(406\) 0 0
\(407\) −6715.58 + 5635.04i −0.817885 + 0.686287i
\(408\) 0 0
\(409\) −489.847 + 2778.06i −0.0592210 + 0.335859i −0.999995 0.00317545i \(-0.998989\pi\)
0.940774 + 0.339034i \(0.110100\pi\)
\(410\) 0 0
\(411\) 5160.40 + 2932.50i 0.619328 + 0.351945i
\(412\) 0 0
\(413\) 326.942 + 566.281i 0.0389535 + 0.0674694i
\(414\) 0 0
\(415\) −5930.65 + 10272.2i −0.701504 + 1.21504i
\(416\) 0 0
\(417\) −10586.0 + 1941.32i −1.24316 + 0.227978i
\(418\) 0 0
\(419\) 7349.28 + 6166.78i 0.856887 + 0.719014i 0.961295 0.275521i \(-0.0888503\pi\)
−0.104408 + 0.994535i \(0.533295\pi\)
\(420\) 0 0
\(421\) 11950.5 + 4349.63i 1.38345 + 0.503535i 0.923222 0.384266i \(-0.125545\pi\)
0.460228 + 0.887801i \(0.347768\pi\)
\(422\) 0 0
\(423\) 1278.29 + 1043.17i 0.146933 + 0.119907i
\(424\) 0 0
\(425\) 73.3687 + 416.094i 0.00837389 + 0.0474907i
\(426\) 0 0
\(427\) −3220.42 + 1172.14i −0.364981 + 0.132842i
\(428\) 0 0
\(429\) 764.607 923.979i 0.0860502 0.103986i
\(430\) 0 0
\(431\) −7867.05 −0.879216 −0.439608 0.898190i \(-0.644883\pi\)
−0.439608 + 0.898190i \(0.644883\pi\)
\(432\) 0 0
\(433\) −17294.1 −1.91940 −0.959702 0.281018i \(-0.909328\pi\)
−0.959702 + 0.281018i \(0.909328\pi\)
\(434\) 0 0
\(435\) 10080.6 + 1706.50i 1.11109 + 0.188093i
\(436\) 0 0
\(437\) 1718.02 625.306i 0.188064 0.0684496i
\(438\) 0 0
\(439\) 578.944 + 3283.35i 0.0629418 + 0.356961i 0.999970 + 0.00770790i \(0.00245352\pi\)
−0.937028 + 0.349253i \(0.886435\pi\)
\(440\) 0 0
\(441\) −4312.62 + 7711.30i −0.465676 + 0.832664i
\(442\) 0 0
\(443\) 12292.3 + 4474.02i 1.31834 + 0.479836i 0.902925 0.429797i \(-0.141415\pi\)
0.415413 + 0.909633i \(0.363637\pi\)
\(444\) 0 0
\(445\) −4576.53 3840.17i −0.487525 0.409082i
\(446\) 0 0
\(447\) −5425.64 + 15230.0i −0.574103 + 1.61153i
\(448\) 0 0
\(449\) 9483.66 16426.2i 0.996796 1.72650i 0.429132 0.903242i \(-0.358820\pi\)
0.567665 0.823260i \(-0.307847\pi\)
\(450\) 0 0
\(451\) −6831.56 11832.6i −0.713272 1.23542i
\(452\) 0 0
\(453\) −1.34398 196.574i −0.000139395 0.0203881i
\(454\) 0 0
\(455\) −50.6974 + 287.519i −0.00522359 + 0.0296244i
\(456\) 0 0
\(457\) −10486.3 + 8799.07i −1.07337 + 0.900664i −0.995353 0.0962885i \(-0.969303\pi\)
−0.0780156 + 0.996952i \(0.524858\pi\)
\(458\) 0 0
\(459\) 735.302 444.880i 0.0747733 0.0452401i
\(460\) 0 0
\(461\) −7991.70 + 6705.83i −0.807398 + 0.677488i −0.949985 0.312295i \(-0.898902\pi\)
0.142587 + 0.989782i \(0.454458\pi\)
\(462\) 0 0
\(463\) 848.557 4812.41i 0.0851745 0.483048i −0.912144 0.409869i \(-0.865574\pi\)
0.997319 0.0731793i \(-0.0233145\pi\)
\(464\) 0 0
\(465\) 16838.9 9876.07i 1.67933 0.984928i
\(466\) 0 0
\(467\) 3003.39 + 5202.03i 0.297602 + 0.515463i 0.975587 0.219614i \(-0.0704797\pi\)
−0.677984 + 0.735076i \(0.737146\pi\)
\(468\) 0 0
\(469\) 413.003 715.341i 0.0406624 0.0704294i
\(470\) 0 0
\(471\) −7776.28 9139.77i −0.760747 0.894136i
\(472\) 0 0
\(473\) −16466.8 13817.3i −1.60073 1.34317i
\(474\) 0 0
\(475\) −962.864 350.454i −0.0930089 0.0338525i
\(476\) 0 0
\(477\) 7771.96 6704.66i 0.746024 0.643575i
\(478\) 0 0
\(479\) 529.337 + 3002.02i 0.0504928 + 0.286359i 0.999590 0.0286237i \(-0.00911246\pi\)
−0.949097 + 0.314982i \(0.898001\pi\)
\(480\) 0 0
\(481\) 994.788 362.073i 0.0943003 0.0343225i
\(482\) 0 0
\(483\) 884.735 + 2380.03i 0.0833475 + 0.224214i
\(484\) 0 0
\(485\) 17288.3 1.61860
\(486\) 0 0
\(487\) −1795.38 −0.167056 −0.0835281 0.996505i \(-0.526619\pi\)
−0.0835281 + 0.996505i \(0.526619\pi\)
\(488\) 0 0
\(489\) 420.597 + 1131.45i 0.0388959 + 0.104634i
\(490\) 0 0
\(491\) 11759.8 4280.20i 1.08088 0.393407i 0.260643 0.965435i \(-0.416065\pi\)
0.820234 + 0.572028i \(0.193843\pi\)
\(492\) 0 0
\(493\) −150.277 852.262i −0.0137285 0.0778580i
\(494\) 0 0
\(495\) −12448.2 + 10738.7i −1.13031 + 0.975088i
\(496\) 0 0
\(497\) 3368.59 + 1226.07i 0.304028 + 0.110657i
\(498\) 0 0
\(499\) −10407.7 8733.06i −0.933688 0.783458i 0.0427875 0.999084i \(-0.486376\pi\)
−0.976476 + 0.215627i \(0.930821\pi\)
\(500\) 0 0
\(501\) 5484.44 + 6446.08i 0.489075 + 0.574830i
\(502\) 0 0
\(503\) 6402.30 11089.1i 0.567523 0.982979i −0.429287 0.903168i \(-0.641235\pi\)
0.996810 0.0798111i \(-0.0254317\pi\)
\(504\) 0 0
\(505\) 5725.62 + 9917.06i 0.504528 + 0.873868i
\(506\) 0 0
\(507\) 9722.32 5702.16i 0.851644 0.499491i
\(508\) 0 0
\(509\) 14.7863 83.8573i 0.00128761 0.00730237i −0.984157 0.177298i \(-0.943264\pi\)
0.985445 + 0.169996i \(0.0543754\pi\)
\(510\) 0 0
\(511\) 2151.06 1804.95i 0.186218 0.156255i
\(512\) 0 0
\(513\) 42.7457 + 2083.76i 0.00367889 + 0.179338i
\(514\) 0 0
\(515\) 19456.5 16326.0i 1.66477 1.39691i
\(516\) 0 0
\(517\) −463.914 + 2630.99i −0.0394641 + 0.223812i
\(518\) 0 0
\(519\) −43.2698 6328.72i −0.00365960 0.535260i
\(520\) 0 0
\(521\) 8164.42 + 14141.2i 0.686545 + 1.18913i 0.972949 + 0.231021i \(0.0742068\pi\)
−0.286404 + 0.958109i \(0.592460\pi\)
\(522\) 0 0
\(523\) −5508.00 + 9540.13i −0.460512 + 0.797630i −0.998986 0.0450116i \(-0.985668\pi\)
0.538474 + 0.842642i \(0.319001\pi\)
\(524\) 0 0
\(525\) 477.566 1340.55i 0.0397004 0.111441i
\(526\) 0 0
\(527\) −1265.81 1062.14i −0.104629 0.0877940i
\(528\) 0 0
\(529\) −2799.22 1018.83i −0.230067 0.0837374i
\(530\) 0 0
\(531\) 2170.32 3880.70i 0.177371 0.317152i
\(532\) 0 0
\(533\) 286.507 + 1624.86i 0.0232833 + 0.132046i
\(534\) 0 0
\(535\) 27401.3 9973.26i 2.21432 0.805947i
\(536\) 0 0
\(537\) 11873.0 + 2009.93i 0.954110 + 0.161517i
\(538\) 0 0
\(539\) −14306.3 −1.14326
\(540\) 0 0
\(541\) 8618.57 0.684919 0.342460 0.939533i \(-0.388740\pi\)
0.342460 + 0.939533i \(0.388740\pi\)
\(542\) 0 0
\(543\) 512.432 619.242i 0.0404983 0.0489396i
\(544\) 0 0
\(545\) −10275.0 + 3739.78i −0.807580 + 0.293935i
\(546\) 0 0
\(547\) −118.470 671.876i −0.00926034 0.0525180i 0.979827 0.199845i \(-0.0640440\pi\)
−0.989088 + 0.147327i \(0.952933\pi\)
\(548\) 0 0
\(549\) 18055.0 + 14734.0i 1.40359 + 1.14541i
\(550\) 0 0
\(551\) 1972.18 + 717.815i 0.152482 + 0.0554990i
\(552\) 0 0
\(553\) 3366.54 + 2824.86i 0.258878 + 0.217225i
\(554\) 0 0
\(555\) −14273.6 + 2617.56i −1.09167 + 0.200197i
\(556\) 0 0
\(557\) −706.450 + 1223.61i −0.0537401 + 0.0930806i −0.891644 0.452737i \(-0.850448\pi\)
0.837904 + 0.545818i \(0.183781\pi\)
\(558\) 0 0
\(559\) 1297.90 + 2248.02i 0.0982024 + 0.170092i
\(560\) 0 0
\(561\) 1209.87 + 687.530i 0.0910527 + 0.0517425i
\(562\) 0 0
\(563\) 3353.19 19016.9i 0.251013 1.42356i −0.555092 0.831789i \(-0.687317\pi\)
0.806104 0.591774i \(-0.201572\pi\)
\(564\) 0 0
\(565\) −16855.4 + 14143.3i −1.25506 + 1.05312i
\(566\) 0 0
\(567\) −2893.51 + 79.1509i −0.214314 + 0.00586248i
\(568\) 0 0
\(569\) −10023.8 + 8410.99i −0.738525 + 0.619696i −0.932441 0.361322i \(-0.882325\pi\)
0.193916 + 0.981018i \(0.437881\pi\)
\(570\) 0 0
\(571\) 3569.22 20242.1i 0.261589 1.48354i −0.516987 0.855993i \(-0.672946\pi\)
0.778576 0.627551i \(-0.215943\pi\)
\(572\) 0 0
\(573\) 12675.2 + 7202.93i 0.924108 + 0.525143i
\(574\) 0 0
\(575\) 4244.26 + 7351.28i 0.307823 + 0.533164i
\(576\) 0 0
\(577\) 1699.26 2943.21i 0.122602 0.212352i −0.798191 0.602404i \(-0.794210\pi\)
0.920793 + 0.390052i \(0.127543\pi\)
\(578\) 0 0
\(579\) −13991.4 + 2565.81i −1.00425 + 0.184165i
\(580\) 0 0
\(581\) −2590.45 2173.65i −0.184974 0.155212i
\(582\) 0 0
\(583\) 15617.8 + 5684.41i 1.10947 + 0.403815i
\(584\) 0 0
\(585\) 1856.08 704.446i 0.131179 0.0497868i
\(586\) 0 0
\(587\) 2010.20 + 11400.4i 0.141346 + 0.801612i 0.970229 + 0.242190i \(0.0778657\pi\)
−0.828883 + 0.559422i \(0.811023\pi\)
\(588\) 0 0
\(589\) 3765.63 1370.58i 0.263430 0.0958806i
\(590\) 0 0
\(591\) −10412.7 + 12583.1i −0.724739 + 0.875801i
\(592\) 0 0
\(593\) −20424.8 −1.41441 −0.707205 0.707008i \(-0.750044\pi\)
−0.707205 + 0.707008i \(0.750044\pi\)
\(594\) 0 0
\(595\) −338.756 −0.0233406
\(596\) 0 0
\(597\) −9589.96 1623.44i −0.657439 0.111295i
\(598\) 0 0
\(599\) −15491.8 + 5638.56i −1.05672 + 0.384616i −0.811197 0.584773i \(-0.801184\pi\)
−0.245528 + 0.969389i \(0.578961\pi\)
\(600\) 0 0
\(601\) −3430.04 19452.7i −0.232803 1.32029i −0.847192 0.531286i \(-0.821709\pi\)
0.614390 0.789003i \(-0.289402\pi\)
\(602\) 0 0
\(603\) −5616.24 + 76.8006i −0.379288 + 0.00518667i
\(604\) 0 0
\(605\) −7595.15 2764.41i −0.510391 0.185767i
\(606\) 0 0
\(607\) −3232.28 2712.21i −0.216136 0.181359i 0.528291 0.849063i \(-0.322833\pi\)
−0.744427 + 0.667704i \(0.767277\pi\)
\(608\) 0 0
\(609\) −978.171 + 2745.77i −0.0650862 + 0.182700i
\(610\) 0 0
\(611\) 161.307 279.391i 0.0106805 0.0184991i
\(612\) 0 0
\(613\) 5986.71 + 10369.3i 0.394455 + 0.683216i 0.993031 0.117850i \(-0.0376001\pi\)
−0.598577 + 0.801066i \(0.704267\pi\)
\(614\) 0 0
\(615\) −154.630 22616.5i −0.0101387 1.48290i
\(616\) 0 0
\(617\) −766.653 + 4347.91i −0.0500232 + 0.283695i −0.999550 0.0299897i \(-0.990453\pi\)
0.949527 + 0.313685i \(0.101564\pi\)
\(618\) 0 0
\(619\) 4277.74 3589.45i 0.277766 0.233073i −0.493252 0.869886i \(-0.664192\pi\)
0.771018 + 0.636813i \(0.219748\pi\)
\(620\) 0 0
\(621\) 10824.8 13451.4i 0.699491 0.869220i
\(622\) 0 0
\(623\) 1304.74 1094.81i 0.0839061 0.0704055i
\(624\) 0 0
\(625\) −3384.29 + 19193.3i −0.216594 + 1.22837i
\(626\) 0 0
\(627\) −2911.03 + 1707.33i −0.185415 + 0.108747i
\(628\) 0 0
\(629\) 614.167 + 1063.77i 0.0389324 + 0.0674328i
\(630\) 0 0
\(631\) 6743.30 11679.7i 0.425431 0.736867i −0.571030 0.820929i \(-0.693456\pi\)
0.996461 + 0.0840619i \(0.0267893\pi\)
\(632\) 0 0
\(633\) 853.001 + 1002.57i 0.0535604 + 0.0629517i
\(634\) 0 0
\(635\) 4207.80 + 3530.77i 0.262963 + 0.220652i
\(636\) 0 0
\(637\) 1623.41 + 590.872i 0.100976 + 0.0367523i
\(638\) 0 0
\(639\) −4560.72 23945.7i −0.282347 1.48244i
\(640\) 0 0
\(641\) 1301.91 + 7383.47i 0.0802218 + 0.454960i 0.998286 + 0.0585281i \(0.0186407\pi\)
−0.918064 + 0.396432i \(0.870248\pi\)
\(642\) 0 0
\(643\) −18909.9 + 6882.64i −1.15977 + 0.422122i −0.849016 0.528367i \(-0.822804\pi\)
−0.310756 + 0.950490i \(0.600582\pi\)
\(644\) 0 0
\(645\) −12398.3 33352.9i −0.756875 2.03608i
\(646\) 0 0
\(647\) −13603.1 −0.826575 −0.413288 0.910601i \(-0.635620\pi\)
−0.413288 + 0.910601i \(0.635620\pi\)
\(648\) 0 0
\(649\) 7199.61 0.435454
\(650\) 0 0
\(651\) 1939.21 + 5216.67i 0.116749 + 0.314067i
\(652\) 0 0
\(653\) 21556.3 7845.87i 1.29183 0.470188i 0.397503 0.917601i \(-0.369877\pi\)
0.894327 + 0.447413i \(0.147655\pi\)
\(654\) 0 0
\(655\) 1452.79 + 8239.19i 0.0866645 + 0.491499i
\(656\) 0 0
\(657\) −18030.3 6284.66i −1.07067 0.373193i
\(658\) 0 0
\(659\) 15226.4 + 5541.95i 0.900054 + 0.327593i 0.750274 0.661127i \(-0.229921\pi\)
0.149779 + 0.988719i \(0.452144\pi\)
\(660\) 0 0
\(661\) 8534.64 + 7161.41i 0.502207 + 0.421402i 0.858377 0.513019i \(-0.171473\pi\)
−0.356170 + 0.934421i \(0.615918\pi\)
\(662\) 0 0
\(663\) −108.894 127.987i −0.00637870 0.00749714i
\(664\) 0 0
\(665\) 410.768 711.470i 0.0239532 0.0414882i
\(666\) 0 0
\(667\) −8693.28 15057.2i −0.504655 0.874089i
\(668\) 0 0
\(669\) −15504.7 + 9093.52i −0.896031 + 0.525525i
\(670\) 0 0
\(671\) −6552.46 + 37160.9i −0.376982 + 2.13797i
\(672\) 0 0
\(673\) 972.493 816.019i 0.0557011 0.0467388i −0.614512 0.788907i \(-0.710647\pi\)
0.670213 + 0.742169i \(0.266203\pi\)
\(674\) 0 0
\(675\) −9493.30 + 1875.45i −0.541329 + 0.106942i
\(676\) 0 0
\(677\) −15101.6 + 12671.8i −0.857315 + 0.719372i −0.961388 0.275197i \(-0.911257\pi\)
0.104073 + 0.994570i \(0.466812\pi\)
\(678\) 0 0
\(679\) −855.875 + 4853.91i −0.0483733 + 0.274339i
\(680\) 0 0
\(681\) −72.8008 10648.0i −0.00409653 0.599165i
\(682\) 0 0
\(683\) 4391.39 + 7606.11i 0.246020 + 0.426120i 0.962418 0.271572i \(-0.0875436\pi\)
−0.716398 + 0.697692i \(0.754210\pi\)
\(684\) 0 0
\(685\) −7954.48 + 13777.6i −0.443686 + 0.768487i
\(686\) 0 0
\(687\) 1579.00 4432.32i 0.0876893 0.246148i
\(688\) 0 0
\(689\) −1537.46 1290.08i −0.0850108 0.0713325i
\(690\) 0 0
\(691\) 9374.91 + 3412.19i 0.516119 + 0.187852i 0.586930 0.809638i \(-0.300336\pi\)
−0.0708107 + 0.997490i \(0.522559\pi\)
\(692\) 0 0
\(693\) −2398.77 4026.62i −0.131489 0.220719i
\(694\) 0 0
\(695\) −5009.27 28409.0i −0.273399 1.55053i
\(696\) 0 0
\(697\) −1798.97 + 654.770i −0.0977628 + 0.0355828i
\(698\) 0 0
\(699\) −7136.58 1208.12i −0.386166 0.0653725i
\(700\) 0 0
\(701\) −4822.63 −0.259841 −0.129920 0.991524i \(-0.541472\pi\)
−0.129920 + 0.991524i \(0.541472\pi\)
\(702\) 0 0
\(703\) −2978.90 −0.159817
\(704\) 0 0
\(705\) −2819.40 + 3407.07i −0.150617 + 0.182011i
\(706\) 0 0
\(707\) −3067.80 + 1116.59i −0.163192 + 0.0593969i
\(708\) 0 0
\(709\) 2404.44 + 13636.3i 0.127363 + 0.722314i 0.979876 + 0.199607i \(0.0639666\pi\)
−0.852513 + 0.522707i \(0.824922\pi\)
\(710\) 0 0
\(711\) 4786.34 29497.8i 0.252464 1.55591i
\(712\) 0 0
\(713\) −31195.4 11354.2i −1.63854 0.596379i
\(714\) 0 0
\(715\) 2462.51 + 2066.29i 0.128801 + 0.108077i
\(716\) 0 0
\(717\) −1278.35 + 234.431i −0.0665843 + 0.0122106i
\(718\) 0 0
\(719\) −17247.7 + 29874.0i −0.894621 + 1.54953i −0.0603471 + 0.998177i \(0.519221\pi\)
−0.834273 + 0.551351i \(0.814113\pi\)
\(720\) 0 0
\(721\) 3620.51 + 6270.91i 0.187011 + 0.323913i
\(722\) 0 0
\(723\) 22263.3 + 12651.6i 1.14520 + 0.650784i
\(724\) 0 0
\(725\) −1692.08 + 9596.29i −0.0866792 + 0.491582i
\(726\) 0 0
\(727\) 28116.6 23592.6i 1.43437 1.20358i 0.491295 0.870993i \(-0.336524\pi\)
0.943073 0.332585i \(-0.107921\pi\)
\(728\) 0 0
\(729\) 10532.3 + 16628.0i 0.535095 + 0.844792i
\(730\) 0 0
\(731\) −2307.25 + 1936.02i −0.116740 + 0.0979564i
\(732\) 0 0
\(733\) −2455.96 + 13928.4i −0.123756 + 0.701854i 0.858283 + 0.513176i \(0.171531\pi\)
−0.982039 + 0.188678i \(0.939580\pi\)
\(734\) 0 0
\(735\) −20589.4 11700.3i −1.03327 0.587175i
\(736\) 0 0
\(737\) −4547.38 7876.28i −0.227279 0.393659i
\(738\) 0 0
\(739\) 17248.4 29875.1i 0.858583 1.48711i −0.0146980 0.999892i \(-0.504679\pi\)
0.873281 0.487217i \(-0.161988\pi\)
\(740\) 0 0
\(741\) 400.845 73.5091i 0.0198724 0.00364430i
\(742\) 0 0
\(743\) 29161.7 + 24469.5i 1.43989 + 1.20821i 0.939564 + 0.342373i \(0.111231\pi\)
0.500326 + 0.865837i \(0.333214\pi\)
\(744\) 0 0
\(745\) −40721.2 14821.3i −2.00256 0.728874i
\(746\) 0 0
\(747\) −3682.95 + 22697.7i −0.180391 + 1.11173i
\(748\) 0 0
\(749\) 1443.59 + 8187.01i 0.0704241 + 0.399395i
\(750\) 0 0
\(751\) −4734.32 + 1723.15i −0.230037 + 0.0837266i −0.454467 0.890764i \(-0.650170\pi\)
0.224430 + 0.974490i \(0.427948\pi\)
\(752\) 0 0
\(753\) −3692.94 + 4462.68i −0.178723 + 0.215975i
\(754\) 0 0
\(755\) 526.896 0.0253983
\(756\) 0 0
\(757\) 39695.9 1.90591 0.952954 0.303115i \(-0.0980266\pi\)
0.952954 + 0.303115i \(0.0980266\pi\)
\(758\) 0 0
\(759\) 27565.3 + 4666.41i 1.31825 + 0.223162i
\(760\) 0 0
\(761\) −6617.71 + 2408.65i −0.315233 + 0.114735i −0.494791 0.869012i \(-0.664755\pi\)
0.179558 + 0.983747i \(0.442533\pi\)
\(762\) 0 0
\(763\) −541.319 3069.97i −0.0256842 0.145662i
\(764\) 0 0
\(765\) 1178.93 + 1978.97i 0.0557179 + 0.0935290i
\(766\) 0 0
\(767\) −816.977 297.355i −0.0384607 0.0139985i
\(768\) 0 0
\(769\) 30249.5 + 25382.4i 1.41850 + 1.19026i 0.952138 + 0.305669i \(0.0988802\pi\)
0.466362 + 0.884594i \(0.345564\pi\)
\(770\) 0 0
\(771\) 5383.60 15112.0i 0.251473 0.705896i
\(772\) 0 0
\(773\) 15979.4 27677.1i 0.743516 1.28781i −0.207369 0.978263i \(-0.566490\pi\)
0.950885 0.309545i \(-0.100177\pi\)
\(774\) 0 0
\(775\) 9302.79 + 16112.9i 0.431182 + 0.746829i
\(776\) 0 0
\(777\) −28.2854 4137.08i −0.00130596 0.191013i
\(778\) 0 0
\(779\) 806.206 4572.22i 0.0370800 0.210291i
\(780\) 0 0
\(781\) 30236.0 25371.0i 1.38531 1.16241i
\(782\) 0 0
\(783\) 19444.6 3841.38i 0.887475 0.175325i
\(784\) 0 0
\(785\) 24639.6 20675.1i 1.12029 0.940033i
\(786\) 0 0
\(787\) 787.039 4463.52i 0.0356479 0.202169i −0.961782 0.273816i \(-0.911714\pi\)
0.997430 + 0.0716463i \(0.0228253\pi\)
\(788\) 0 0
\(789\) −4843.99 + 2841.01i −0.218569 + 0.128191i
\(790\) 0 0
\(791\) −3136.48 5432.55i −0.140987 0.244196i
\(792\) 0 0
\(793\) 2278.35 3946.21i 0.102026 0.176714i
\(794\) 0 0
\(795\) 17827.9 + 20953.9i 0.795335 + 0.934788i
\(796\) 0 0
\(797\) 3143.57 + 2637.77i 0.139713 + 0.117233i 0.709966 0.704236i \(-0.248710\pi\)
−0.570253 + 0.821469i \(0.693155\pi\)
\(798\) 0 0
\(799\) 351.755 + 128.028i 0.0155747 + 0.00566873i
\(800\) 0 0
\(801\) −10936.4 3812.01i −0.482422 0.168153i
\(802\) 0 0
\(803\) −5368.79 30447.9i −0.235941 1.33809i
\(804\) 0 0
\(805\) −6395.36 + 2327.72i −0.280008 + 0.101915i
\(806\) 0 0
\(807\) −9971.12 26823.4i −0.434944 1.17005i
\(808\) 0 0
\(809\) 3418.34 0.148557 0.0742783 0.997238i \(-0.476335\pi\)
0.0742783 + 0.997238i \(0.476335\pi\)
\(810\) 0 0
\(811\) −3687.72 −0.159671 −0.0798357 0.996808i \(-0.525440\pi\)
−0.0798357 + 0.996808i \(0.525440\pi\)
\(812\) 0 0
\(813\) 7379.07 + 19850.5i 0.318322 + 0.856319i
\(814\) 0 0
\(815\) −3040.31 + 1106.58i −0.130672 + 0.0475606i
\(816\) 0 0
\(817\) −1268.38 7193.36i −0.0543147 0.308034i
\(818\) 0 0
\(819\) 105.895 + 555.994i 0.00451804 + 0.0237216i
\(820\) 0 0
\(821\) −3319.42 1208.17i −0.141107 0.0513586i 0.270501 0.962720i \(-0.412811\pi\)
−0.411608 + 0.911361i \(0.635033\pi\)
\(822\) 0 0
\(823\) −3449.91 2894.82i −0.146119 0.122609i 0.566798 0.823857i \(-0.308182\pi\)
−0.712917 + 0.701248i \(0.752626\pi\)
\(824\) 0 0
\(825\) −10153.5 11933.8i −0.428486 0.503616i
\(826\) 0 0
\(827\) −7788.45 + 13490.0i −0.327486 + 0.567223i −0.982012 0.188817i \(-0.939535\pi\)
0.654526 + 0.756039i \(0.272868\pi\)
\(828\) 0 0
\(829\) −9335.06 16168.8i −0.391098 0.677401i 0.601497 0.798875i \(-0.294571\pi\)
−0.992595 + 0.121474i \(0.961238\pi\)
\(830\) 0 0
\(831\) −15882.1 + 9314.91i −0.662991 + 0.388846i
\(832\) 0 0
\(833\) −348.084 + 1974.08i −0.0144783 + 0.0821103i
\(834\) 0 0
\(835\) −17377.8 + 14581.7i −0.720219 + 0.604336i
\(836\) 0 0
\(837\) 23726.3 29483.4i 0.979811 1.21756i
\(838\) 0 0
\(839\) 13863.9 11633.2i 0.570482 0.478691i −0.311324 0.950304i \(-0.600772\pi\)
0.881806 + 0.471613i \(0.156328\pi\)
\(840\) 0 0
\(841\) −769.305 + 4362.95i −0.0315431 + 0.178890i
\(842\) 0 0
\(843\) 260.870 + 38155.4i 0.0106582 + 1.55889i
\(844\) 0 0
\(845\) 15105.2 + 26163.0i 0.614953 + 1.06513i
\(846\) 0 0
\(847\) 1152.15 1995.58i 0.0467395 0.0809552i
\(848\) 0 0
\(849\) 233.003 654.051i 0.00941891 0.0264393i
\(850\) 0 0
\(851\) 18904.4 + 15862.7i 0.761497 + 0.638972i
\(852\) 0 0
\(853\) −35220.9 12819.4i −1.41377 0.514569i −0.481533 0.876428i \(-0.659920\pi\)
−0.932233 + 0.361860i \(0.882142\pi\)
\(854\) 0 0
\(855\) −5585.85 + 76.3850i −0.223429 + 0.00305534i
\(856\) 0 0
\(857\) −3991.51 22637.0i −0.159098 0.902291i −0.954943 0.296790i \(-0.904084\pi\)
0.795844 0.605501i \(-0.207027\pi\)
\(858\) 0 0
\(859\) −29016.9 + 10561.3i −1.15256 + 0.419496i −0.846432 0.532496i \(-0.821254\pi\)
−0.306123 + 0.951992i \(0.599032\pi\)
\(860\) 0 0
\(861\) 6357.53 + 1076.24i 0.251642 + 0.0425995i
\(862\) 0 0
\(863\) −19121.6 −0.754236 −0.377118 0.926165i \(-0.623085\pi\)
−0.377118 + 0.926165i \(0.623085\pi\)
\(864\) 0 0
\(865\) 16963.5 0.666794
\(866\) 0 0
\(867\) −16151.1 + 19517.6i −0.632666 + 0.764537i
\(868\) 0 0
\(869\) 45469.9 16549.7i 1.77498 0.646040i
\(870\) 0 0
\(871\) 190.711 + 1081.58i 0.00741906 + 0.0420756i
\(872\) 0 0
\(873\) 31334.4 11892.5i 1.21479 0.461053i
\(874\) 0 0
\(875\) −2911.44 1059.68i −0.112485 0.0409413i
\(876\) 0 0
\(877\) −12696.7 10653.8i −0.488867 0.410208i 0.364753 0.931104i \(-0.381153\pi\)
−0.853620 + 0.520896i \(0.825598\pi\)
\(878\) 0 0
\(879\) 20924.1 3837.18i 0.802905 0.147241i
\(880\) 0 0
\(881\) −15481.4 + 26814.5i −0.592032 + 1.02543i 0.401926 + 0.915672i \(0.368341\pi\)
−0.993958 + 0.109758i \(0.964993\pi\)
\(882\) 0 0
\(883\) 12180.7 + 21097.6i 0.464229 + 0.804068i 0.999166 0.0408239i \(-0.0129983\pi\)
−0.534938 + 0.844891i \(0.679665\pi\)
\(884\) 0 0
\(885\) 10361.6 + 5888.17i 0.393560 + 0.223648i
\(886\) 0 0
\(887\) 4420.12 25067.7i 0.167320 0.948920i −0.779320 0.626627i \(-0.784435\pi\)
0.946640 0.322293i \(-0.104454\pi\)
\(888\) 0 0
\(889\) −1199.62 + 1006.60i −0.0452576 + 0.0379756i
\(890\) 0 0
\(891\) −15174.8 + 28026.5i −0.570567 + 1.05379i
\(892\) 0 0
\(893\) −695.420 + 583.527i −0.0260597 + 0.0218667i
\(894\) 0 0
\(895\) −5604.74 + 31786.0i −0.209325 + 1.18714i
\(896\) 0 0
\(897\) −2935.24 1668.01i −0.109259 0.0620883i
\(898\) 0 0
\(899\) −19054.4 33003.1i −0.706895 1.22438i
\(900\) 0 0
\(901\) 1164.37 2016.75i 0.0430530 0.0745700i
\(902\) 0 0
\(903\) 9978.05 1829.83i 0.367717 0.0674339i
\(904\) 0 0
\(905\) 1650.35 + 1384.81i 0.0606182 + 0.0508647i
\(906\) 0 0
\(907\) −25805.9 9392.59i −0.944731 0.343854i −0.176699 0.984265i \(-0.556542\pi\)
−0.768033 + 0.640411i \(0.778764\pi\)
\(908\) 0 0
\(909\) 17199.4 + 14035.7i 0.627577 + 0.512141i
\(910\) 0 0
\(911\) 5849.43 + 33173.7i 0.212733 + 1.20647i 0.884797 + 0.465977i \(0.154297\pi\)
−0.672063 + 0.740494i \(0.734592\pi\)
\(912\) 0 0
\(913\) −34987.7 + 12734.5i −1.26826 + 0.461609i
\(914\) 0 0
\(915\) −39822.1 + 48122.5i −1.43877 + 1.73867i
\(916\) 0 0
\(917\) −2385.18 −0.0858949
\(918\) 0 0
\(919\) −16064.8 −0.576635 −0.288318 0.957535i \(-0.593096\pi\)
−0.288318 + 0.957535i \(0.593096\pi\)
\(920\) 0 0
\(921\) −42946.5 7270.24i −1.53652 0.260111i
\(922\) 0 0
\(923\) −4478.90 + 1630.19i −0.159723 + 0.0581346i
\(924\) 0 0
\(925\) −2401.69 13620.7i −0.0853698 0.484156i
\(926\) 0 0
\(927\) 24033.8 42974.3i 0.851536 1.52261i
\(928\) 0 0
\(929\) 34871.8 + 12692.3i 1.23155 + 0.448247i 0.874127 0.485697i \(-0.161434\pi\)
0.357420 + 0.933944i \(0.383656\pi\)
\(930\) 0 0
\(931\) −3723.97 3124.78i −0.131094 0.110001i
\(932\) 0 0
\(933\) −3398.83 + 9540.66i −0.119263 + 0.334777i
\(934\) 0 0
\(935\) −1864.94 + 3230.17i −0.0652301 + 0.112982i
\(936\) 0 0
\(937\) −11227.4 19446.5i −0.391445 0.678003i 0.601195 0.799102i \(-0.294691\pi\)
−0.992640 + 0.121099i \(0.961358\pi\)
\(938\) 0 0
\(939\) −45.0463 6588.55i −0.00156553 0.228977i
\(940\) 0 0
\(941\) 1443.08 8184.09i 0.0499925 0.283521i −0.949555 0.313600i \(-0.898465\pi\)
0.999548 + 0.0300789i \(0.00957587\pi\)
\(942\) 0 0
\(943\) −29463.4 + 24722.7i −1.01746 + 0.853746i
\(944\) 0 0
\(945\) −159.122 7756.87i −0.00547751 0.267017i
\(946\) 0 0
\(947\) 23700.6 19887.2i 0.813270 0.682415i −0.138116 0.990416i \(-0.544105\pi\)
0.951386 + 0.308001i \(0.0996601\pi\)
\(948\) 0 0
\(949\) −648.324 + 3676.83i −0.0221765 + 0.125769i
\(950\) 0 0
\(951\) −29822.2 + 17490.8i −1.01688 + 0.596401i
\(952\) 0 0
\(953\) −26600.7 46073.8i −0.904178 1.56608i −0.822017 0.569463i \(-0.807151\pi\)
−0.0821613 0.996619i \(-0.526182\pi\)
\(954\) 0 0
\(955\) −19538.1 + 33841.0i −0.662031 + 1.14667i
\(956\) 0 0
\(957\) 20796.9 + 24443.4i 0.702475 + 0.825647i
\(958\) 0 0
\(959\) −3474.44 2915.40i −0.116992 0.0981680i
\(960\) 0 0
\(961\) −40381.3 14697.6i −1.35549 0.493357i
\(962\) 0 0
\(963\) 42803.4 36925.4i 1.43232 1.23562i
\(964\) 0 0
\(965\) −6620.68 37547.8i −0.220857 1.25254i
\(966\) 0 0
\(967\) 20742.7 7549.73i 0.689804 0.251068i 0.0267528 0.999642i \(-0.491483\pi\)
0.663051 + 0.748574i \(0.269261\pi\)
\(968\) 0 0
\(969\) 164.761 + 443.226i 0.00546222 + 0.0146940i
\(970\) 0 0
\(971\) 2497.51 0.0825425 0.0412712 0.999148i \(-0.486859\pi\)
0.0412712 + 0.999148i \(0.486859\pi\)
\(972\) 0 0
\(973\) 8224.19 0.270972
\(974\) 0 0
\(975\) 659.287 + 1773.55i 0.0216555 + 0.0582555i
\(976\) 0 0
\(977\) 18538.1 6747.30i 0.607047 0.220947i −0.0201636 0.999797i \(-0.506419\pi\)
0.627211 + 0.778850i \(0.284196\pi\)
\(978\) 0 0
\(979\) −3256.49 18468.5i −0.106310 0.602916i
\(980\) 0 0
\(981\) −16050.5 + 13846.3i −0.522377 + 0.450641i
\(982\) 0 0
\(983\) −9110.97 3316.12i −0.295620 0.107597i 0.189952 0.981793i \(-0.439167\pi\)
−0.485573 + 0.874196i \(0.661389\pi\)
\(984\) 0 0
\(985\) −33535.3 28139.5i −1.08480 0.910252i
\(986\) 0 0
\(987\) −817.002 960.255i −0.0263480 0.0309678i
\(988\) 0 0
\(989\) −30255.4 + 52403.9i −0.972767 + 1.68488i
\(990\) 0 0
\(991\) 16318.5 + 28264.5i 0.523083 + 0.906006i 0.999639 + 0.0268620i \(0.00855148\pi\)
−0.476556 + 0.879144i \(0.658115\pi\)
\(992\) 0 0
\(993\) −25518.3 + 14966.5i −0.815507 + 0.478297i
\(994\) 0 0
\(995\) 4527.02 25674.0i 0.144237 0.818011i
\(996\) 0 0
\(997\) 5862.89 4919.55i 0.186238 0.156273i −0.544902 0.838500i \(-0.683433\pi\)
0.731140 + 0.682227i \(0.238989\pi\)
\(998\) 0 0
\(999\) −24069.8 + 14562.9i −0.762296 + 0.461212i
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 108.4.i.a.49.6 54
3.2 odd 2 324.4.i.a.37.9 54
27.11 odd 18 324.4.i.a.289.9 54
27.16 even 9 inner 108.4.i.a.97.6 yes 54
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.i.a.49.6 54 1.1 even 1 trivial
108.4.i.a.97.6 yes 54 27.16 even 9 inner
324.4.i.a.37.9 54 3.2 odd 2
324.4.i.a.289.9 54 27.11 odd 18