Properties

Label 124.4.e.c.5.1
Level $124$
Weight $4$
Character 124.5
Analytic conductor $7.316$
Analytic rank $0$
Dimension $8$
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] = [124,4,Mod(5,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.e (of order \(3\), 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: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 29x^{6} - 58x^{5} + 824x^{4} - 1198x^{3} + 1933x^{2} + 129x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 5.1
Root \(-0.0334318 + 0.0579056i\) of defining polynomial
Character \(\chi\) \(=\) 124.5
Dual form 124.4.e.c.25.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.31198 + 5.73652i) q^{3} +(7.27011 + 12.5922i) q^{5} +(-4.03211 + 6.98382i) q^{7} +(-8.43843 - 14.6158i) q^{9} +O(q^{10})\) \(q+(-3.31198 + 5.73652i) q^{3} +(7.27011 + 12.5922i) q^{5} +(-4.03211 + 6.98382i) q^{7} +(-8.43843 - 14.6158i) q^{9} +(-11.2996 - 19.5714i) q^{11} +(-4.70324 - 8.14624i) q^{13} -96.3138 q^{15} +(-22.1864 + 38.4279i) q^{17} +(-13.0694 + 22.6368i) q^{19} +(-26.7085 - 46.2605i) q^{21} -205.597 q^{23} +(-43.2089 + 74.8399i) q^{25} -67.0553 q^{27} +147.971 q^{29} +(170.223 - 28.5490i) q^{31} +149.696 q^{33} -117.255 q^{35} +(17.4333 - 30.1954i) q^{37} +62.3081 q^{39} +(175.754 + 304.415i) q^{41} +(22.1749 - 38.4081i) q^{43} +(122.697 - 212.517i) q^{45} +174.136 q^{47} +(138.984 + 240.728i) q^{49} +(-146.962 - 254.545i) q^{51} +(-44.8332 - 77.6534i) q^{53} +(164.298 - 284.573i) q^{55} +(-86.5709 - 149.945i) q^{57} +(-232.918 + 403.425i) q^{59} +156.265 q^{61} +136.099 q^{63} +(68.3860 - 118.448i) q^{65} +(-244.979 - 424.316i) q^{67} +(680.935 - 1179.41i) q^{69} +(467.320 + 809.423i) q^{71} +(149.091 + 258.233i) q^{73} +(-286.214 - 495.737i) q^{75} +182.244 q^{77} +(-522.561 + 905.102i) q^{79} +(449.923 - 779.290i) q^{81} +(285.827 + 495.067i) q^{83} -645.189 q^{85} +(-490.078 + 848.839i) q^{87} +749.227 q^{89} +75.8558 q^{91} +(-400.004 + 1071.04i) q^{93} -380.062 q^{95} -1215.61 q^{97} +(-190.701 + 330.304i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{5} - 32 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{5} - 32 q^{7} - 4 q^{9} - 80 q^{11} - 28 q^{13} - 304 q^{15} - 8 q^{17} - 56 q^{19} - 76 q^{21} - 624 q^{23} - 48 q^{25} - 432 q^{27} - 216 q^{29} + 528 q^{31} + 584 q^{33} + 592 q^{35} + 96 q^{37} + 128 q^{39} + 552 q^{41} - 112 q^{43} + 524 q^{45} - 304 q^{47} - 4 q^{49} - 232 q^{51} + 1316 q^{53} - 208 q^{55} + 464 q^{57} + 224 q^{59} + 2664 q^{61} - 1856 q^{63} + 504 q^{65} - 272 q^{67} + 496 q^{69} + 1120 q^{71} + 248 q^{73} - 912 q^{75} + 624 q^{77} + 824 q^{79} + 104 q^{81} + 1616 q^{83} - 2232 q^{85} + 456 q^{87} + 184 q^{89} - 6928 q^{91} - 1960 q^{93} + 544 q^{95} + 2520 q^{97} + 1336 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −3.31198 + 5.73652i −0.637391 + 1.10399i 0.348612 + 0.937267i \(0.386653\pi\)
−0.986003 + 0.166726i \(0.946680\pi\)
\(4\) 0 0
\(5\) 7.27011 + 12.5922i 0.650258 + 1.12628i 0.983060 + 0.183283i \(0.0586725\pi\)
−0.332802 + 0.942997i \(0.607994\pi\)
\(6\) 0 0
\(7\) −4.03211 + 6.98382i −0.217713 + 0.377091i −0.954109 0.299461i \(-0.903193\pi\)
0.736395 + 0.676552i \(0.236526\pi\)
\(8\) 0 0
\(9\) −8.43843 14.6158i −0.312535 0.541326i
\(10\) 0 0
\(11\) −11.2996 19.5714i −0.309722 0.536455i 0.668579 0.743641i \(-0.266903\pi\)
−0.978302 + 0.207186i \(0.933569\pi\)
\(12\) 0 0
\(13\) −4.70324 8.14624i −0.100342 0.173797i 0.811484 0.584375i \(-0.198660\pi\)
−0.911825 + 0.410578i \(0.865327\pi\)
\(14\) 0 0
\(15\) −96.3138 −1.65787
\(16\) 0 0
\(17\) −22.1864 + 38.4279i −0.316528 + 0.548243i −0.979761 0.200170i \(-0.935851\pi\)
0.663233 + 0.748413i \(0.269184\pi\)
\(18\) 0 0
\(19\) −13.0694 + 22.6368i −0.157806 + 0.273328i −0.934077 0.357071i \(-0.883775\pi\)
0.776271 + 0.630399i \(0.217109\pi\)
\(20\) 0 0
\(21\) −26.7085 46.2605i −0.277537 0.480708i
\(22\) 0 0
\(23\) −205.597 −1.86391 −0.931957 0.362568i \(-0.881900\pi\)
−0.931957 + 0.362568i \(0.881900\pi\)
\(24\) 0 0
\(25\) −43.2089 + 74.8399i −0.345671 + 0.598720i
\(26\) 0 0
\(27\) −67.0553 −0.477955
\(28\) 0 0
\(29\) 147.971 0.947502 0.473751 0.880659i \(-0.342900\pi\)
0.473751 + 0.880659i \(0.342900\pi\)
\(30\) 0 0
\(31\) 170.223 28.5490i 0.986226 0.165405i
\(32\) 0 0
\(33\) 149.696 0.789657
\(34\) 0 0
\(35\) −117.255 −0.566280
\(36\) 0 0
\(37\) 17.4333 30.1954i 0.0774601 0.134165i −0.824693 0.565580i \(-0.808652\pi\)
0.902153 + 0.431415i \(0.141986\pi\)
\(38\) 0 0
\(39\) 62.3081 0.255828
\(40\) 0 0
\(41\) 175.754 + 304.415i 0.669468 + 1.15955i 0.978053 + 0.208356i \(0.0668114\pi\)
−0.308584 + 0.951197i \(0.599855\pi\)
\(42\) 0 0
\(43\) 22.1749 38.4081i 0.0786428 0.136213i −0.824022 0.566558i \(-0.808275\pi\)
0.902665 + 0.430345i \(0.141608\pi\)
\(44\) 0 0
\(45\) 122.697 212.517i 0.406456 0.704003i
\(46\) 0 0
\(47\) 174.136 0.540434 0.270217 0.962799i \(-0.412905\pi\)
0.270217 + 0.962799i \(0.412905\pi\)
\(48\) 0 0
\(49\) 138.984 + 240.728i 0.405202 + 0.701830i
\(50\) 0 0
\(51\) −146.962 254.545i −0.403505 0.698890i
\(52\) 0 0
\(53\) −44.8332 77.6534i −0.116195 0.201255i 0.802062 0.597241i \(-0.203736\pi\)
−0.918257 + 0.395986i \(0.870403\pi\)
\(54\) 0 0
\(55\) 164.298 284.573i 0.402799 0.697668i
\(56\) 0 0
\(57\) −86.5709 149.945i −0.201168 0.348434i
\(58\) 0 0
\(59\) −232.918 + 403.425i −0.513954 + 0.890194i 0.485915 + 0.874006i \(0.338486\pi\)
−0.999869 + 0.0161882i \(0.994847\pi\)
\(60\) 0 0
\(61\) 156.265 0.327994 0.163997 0.986461i \(-0.447561\pi\)
0.163997 + 0.986461i \(0.447561\pi\)
\(62\) 0 0
\(63\) 136.099 0.272172
\(64\) 0 0
\(65\) 68.3860 118.448i 0.130496 0.226026i
\(66\) 0 0
\(67\) −244.979 424.316i −0.446701 0.773709i 0.551468 0.834196i \(-0.314068\pi\)
−0.998169 + 0.0604873i \(0.980735\pi\)
\(68\) 0 0
\(69\) 680.935 1179.41i 1.18804 2.05775i
\(70\) 0 0
\(71\) 467.320 + 809.423i 0.781137 + 1.35297i 0.931280 + 0.364305i \(0.118693\pi\)
−0.150143 + 0.988664i \(0.547973\pi\)
\(72\) 0 0
\(73\) 149.091 + 258.233i 0.239038 + 0.414026i 0.960439 0.278492i \(-0.0898345\pi\)
−0.721400 + 0.692518i \(0.756501\pi\)
\(74\) 0 0
\(75\) −286.214 495.737i −0.440655 0.763237i
\(76\) 0 0
\(77\) 182.244 0.269723
\(78\) 0 0
\(79\) −522.561 + 905.102i −0.744212 + 1.28901i 0.206351 + 0.978478i \(0.433841\pi\)
−0.950562 + 0.310534i \(0.899492\pi\)
\(80\) 0 0
\(81\) 449.923 779.290i 0.617179 1.06899i
\(82\) 0 0
\(83\) 285.827 + 495.067i 0.377995 + 0.654707i 0.990770 0.135550i \(-0.0432803\pi\)
−0.612775 + 0.790257i \(0.709947\pi\)
\(84\) 0 0
\(85\) −645.189 −0.823300
\(86\) 0 0
\(87\) −490.078 + 848.839i −0.603929 + 1.04604i
\(88\) 0 0
\(89\) 749.227 0.892336 0.446168 0.894949i \(-0.352788\pi\)
0.446168 + 0.894949i \(0.352788\pi\)
\(90\) 0 0
\(91\) 75.8558 0.0873830
\(92\) 0 0
\(93\) −400.004 + 1071.04i −0.446005 + 1.19421i
\(94\) 0 0
\(95\) −380.062 −0.410459
\(96\) 0 0
\(97\) −1215.61 −1.27243 −0.636217 0.771510i \(-0.719502\pi\)
−0.636217 + 0.771510i \(0.719502\pi\)
\(98\) 0 0
\(99\) −190.701 + 330.304i −0.193598 + 0.335321i
\(100\) 0 0
\(101\) 597.214 0.588367 0.294183 0.955749i \(-0.404952\pi\)
0.294183 + 0.955749i \(0.404952\pi\)
\(102\) 0 0
\(103\) 181.986 + 315.209i 0.174093 + 0.301538i 0.939847 0.341595i \(-0.110967\pi\)
−0.765754 + 0.643134i \(0.777634\pi\)
\(104\) 0 0
\(105\) 388.348 672.638i 0.360942 0.625169i
\(106\) 0 0
\(107\) −497.071 + 860.953i −0.449100 + 0.777864i −0.998328 0.0578088i \(-0.981589\pi\)
0.549228 + 0.835673i \(0.314922\pi\)
\(108\) 0 0
\(109\) −2004.47 −1.76141 −0.880706 0.473664i \(-0.842931\pi\)
−0.880706 + 0.473664i \(0.842931\pi\)
\(110\) 0 0
\(111\) 115.478 + 200.013i 0.0987447 + 0.171031i
\(112\) 0 0
\(113\) −72.7745 126.049i −0.0605845 0.104935i 0.834142 0.551549i \(-0.185963\pi\)
−0.894727 + 0.446614i \(0.852630\pi\)
\(114\) 0 0
\(115\) −1494.72 2588.92i −1.21203 2.09929i
\(116\) 0 0
\(117\) −79.3759 + 137.483i −0.0627205 + 0.108635i
\(118\) 0 0
\(119\) −178.916 309.891i −0.137825 0.238720i
\(120\) 0 0
\(121\) 410.140 710.383i 0.308144 0.533721i
\(122\) 0 0
\(123\) −2328.38 −1.70685
\(124\) 0 0
\(125\) 560.995 0.401415
\(126\) 0 0
\(127\) −40.9997 + 71.0136i −0.0286467 + 0.0496176i −0.879993 0.474986i \(-0.842453\pi\)
0.851347 + 0.524604i \(0.175786\pi\)
\(128\) 0 0
\(129\) 146.886 + 254.414i 0.100252 + 0.173642i
\(130\) 0 0
\(131\) 1294.29 2241.78i 0.863226 1.49515i −0.00557139 0.999984i \(-0.501773\pi\)
0.868798 0.495167i \(-0.164893\pi\)
\(132\) 0 0
\(133\) −105.394 182.548i −0.0687130 0.119014i
\(134\) 0 0
\(135\) −487.499 844.373i −0.310794 0.538311i
\(136\) 0 0
\(137\) 1070.86 + 1854.79i 0.667811 + 1.15668i 0.978515 + 0.206175i \(0.0661017\pi\)
−0.310704 + 0.950507i \(0.600565\pi\)
\(138\) 0 0
\(139\) −254.389 −0.155230 −0.0776150 0.996983i \(-0.524730\pi\)
−0.0776150 + 0.996983i \(0.524730\pi\)
\(140\) 0 0
\(141\) −576.737 + 998.937i −0.344468 + 0.596636i
\(142\) 0 0
\(143\) −106.289 + 184.098i −0.0621562 + 0.107658i
\(144\) 0 0
\(145\) 1075.77 + 1863.28i 0.616121 + 1.06715i
\(146\) 0 0
\(147\) −1841.25 −1.03309
\(148\) 0 0
\(149\) −1240.05 + 2147.83i −0.681805 + 1.18092i 0.292625 + 0.956227i \(0.405471\pi\)
−0.974430 + 0.224693i \(0.927862\pi\)
\(150\) 0 0
\(151\) 2779.12 1.49776 0.748880 0.662706i \(-0.230592\pi\)
0.748880 + 0.662706i \(0.230592\pi\)
\(152\) 0 0
\(153\) 748.872 0.395704
\(154\) 0 0
\(155\) 1597.04 + 1935.93i 0.827593 + 1.00321i
\(156\) 0 0
\(157\) 101.582 0.0516377 0.0258188 0.999667i \(-0.491781\pi\)
0.0258188 + 0.999667i \(0.491781\pi\)
\(158\) 0 0
\(159\) 593.947 0.296246
\(160\) 0 0
\(161\) 828.991 1435.86i 0.405799 0.702865i
\(162\) 0 0
\(163\) 2303.54 1.10692 0.553458 0.832877i \(-0.313308\pi\)
0.553458 + 0.832877i \(0.313308\pi\)
\(164\) 0 0
\(165\) 1088.30 + 1885.00i 0.513481 + 0.889375i
\(166\) 0 0
\(167\) 1292.05 2237.90i 0.598695 1.03697i −0.394319 0.918974i \(-0.629019\pi\)
0.993014 0.117997i \(-0.0376472\pi\)
\(168\) 0 0
\(169\) 1054.26 1826.03i 0.479863 0.831147i
\(170\) 0 0
\(171\) 441.140 0.197279
\(172\) 0 0
\(173\) 1547.93 + 2681.09i 0.680270 + 1.17826i 0.974898 + 0.222651i \(0.0714709\pi\)
−0.294628 + 0.955612i \(0.595196\pi\)
\(174\) 0 0
\(175\) −348.446 603.526i −0.150514 0.260699i
\(176\) 0 0
\(177\) −1542.84 2672.27i −0.655179 1.13480i
\(178\) 0 0
\(179\) 621.702 1076.82i 0.259599 0.449638i −0.706536 0.707677i \(-0.749743\pi\)
0.966134 + 0.258039i \(0.0830764\pi\)
\(180\) 0 0
\(181\) −2067.19 3580.48i −0.848911 1.47036i −0.882181 0.470911i \(-0.843925\pi\)
0.0332697 0.999446i \(-0.489408\pi\)
\(182\) 0 0
\(183\) −517.545 + 896.414i −0.209060 + 0.362103i
\(184\) 0 0
\(185\) 506.969 0.201476
\(186\) 0 0
\(187\) 1002.78 0.392144
\(188\) 0 0
\(189\) 270.374 468.302i 0.104057 0.180233i
\(190\) 0 0
\(191\) 1679.13 + 2908.34i 0.636113 + 1.10178i 0.986278 + 0.165092i \(0.0527922\pi\)
−0.350165 + 0.936688i \(0.613874\pi\)
\(192\) 0 0
\(193\) 1063.13 1841.40i 0.396508 0.686773i −0.596784 0.802402i \(-0.703555\pi\)
0.993292 + 0.115629i \(0.0368885\pi\)
\(194\) 0 0
\(195\) 452.986 + 784.596i 0.166354 + 0.288134i
\(196\) 0 0
\(197\) −2262.95 3919.54i −0.818418 1.41754i −0.906847 0.421460i \(-0.861518\pi\)
0.0884289 0.996082i \(-0.471815\pi\)
\(198\) 0 0
\(199\) −2314.66 4009.11i −0.824533 1.42813i −0.902276 0.431160i \(-0.858105\pi\)
0.0777425 0.996973i \(-0.475229\pi\)
\(200\) 0 0
\(201\) 3245.46 1.13889
\(202\) 0 0
\(203\) −596.636 + 1033.40i −0.206284 + 0.357294i
\(204\) 0 0
\(205\) −2555.50 + 4426.26i −0.870654 + 1.50802i
\(206\) 0 0
\(207\) 1734.92 + 3004.97i 0.582538 + 1.00898i
\(208\) 0 0
\(209\) 590.712 0.195504
\(210\) 0 0
\(211\) 1624.65 2813.98i 0.530074 0.918115i −0.469311 0.883033i \(-0.655498\pi\)
0.999384 0.0350817i \(-0.0111691\pi\)
\(212\) 0 0
\(213\) −6191.03 −1.99156
\(214\) 0 0
\(215\) 644.856 0.204553
\(216\) 0 0
\(217\) −486.978 + 1303.92i −0.152342 + 0.407907i
\(218\) 0 0
\(219\) −1975.15 −0.609443
\(220\) 0 0
\(221\) 417.391 0.127044
\(222\) 0 0
\(223\) 1385.70 2400.10i 0.416113 0.720729i −0.579431 0.815021i \(-0.696725\pi\)
0.995545 + 0.0942919i \(0.0300587\pi\)
\(224\) 0 0
\(225\) 1458.46 0.432136
\(226\) 0 0
\(227\) 2217.13 + 3840.18i 0.648264 + 1.12283i 0.983537 + 0.180705i \(0.0578380\pi\)
−0.335273 + 0.942121i \(0.608829\pi\)
\(228\) 0 0
\(229\) −1173.98 + 2033.39i −0.338772 + 0.586770i −0.984202 0.177049i \(-0.943345\pi\)
0.645430 + 0.763819i \(0.276678\pi\)
\(230\) 0 0
\(231\) −603.590 + 1045.45i −0.171919 + 0.297772i
\(232\) 0 0
\(233\) −5784.03 −1.62629 −0.813143 0.582064i \(-0.802245\pi\)
−0.813143 + 0.582064i \(0.802245\pi\)
\(234\) 0 0
\(235\) 1265.99 + 2192.76i 0.351422 + 0.608680i
\(236\) 0 0
\(237\) −3461.42 5995.36i −0.948708 1.64321i
\(238\) 0 0
\(239\) 1012.16 + 1753.11i 0.273937 + 0.474473i 0.969866 0.243637i \(-0.0783407\pi\)
−0.695929 + 0.718110i \(0.745007\pi\)
\(240\) 0 0
\(241\) 1111.44 1925.07i 0.297071 0.514542i −0.678394 0.734699i \(-0.737324\pi\)
0.975464 + 0.220157i \(0.0706569\pi\)
\(242\) 0 0
\(243\) 2075.03 + 3594.06i 0.547791 + 0.948802i
\(244\) 0 0
\(245\) −2020.86 + 3500.23i −0.526971 + 0.912741i
\(246\) 0 0
\(247\) 245.873 0.0633382
\(248\) 0 0
\(249\) −3786.61 −0.963723
\(250\) 0 0
\(251\) 1428.08 2473.50i 0.359121 0.622015i −0.628693 0.777653i \(-0.716410\pi\)
0.987814 + 0.155638i \(0.0497433\pi\)
\(252\) 0 0
\(253\) 2323.16 + 4023.83i 0.577296 + 0.999906i
\(254\) 0 0
\(255\) 2136.85 3701.14i 0.524764 0.908918i
\(256\) 0 0
\(257\) 95.4369 + 165.302i 0.0231642 + 0.0401215i 0.877375 0.479805i \(-0.159293\pi\)
−0.854211 + 0.519927i \(0.825959\pi\)
\(258\) 0 0
\(259\) 140.586 + 243.502i 0.0337282 + 0.0584190i
\(260\) 0 0
\(261\) −1248.64 2162.72i −0.296127 0.512907i
\(262\) 0 0
\(263\) 240.843 0.0564678 0.0282339 0.999601i \(-0.491012\pi\)
0.0282339 + 0.999601i \(0.491012\pi\)
\(264\) 0 0
\(265\) 651.885 1129.10i 0.151113 0.261735i
\(266\) 0 0
\(267\) −2481.43 + 4297.95i −0.568767 + 0.985133i
\(268\) 0 0
\(269\) 964.105 + 1669.88i 0.218522 + 0.378492i 0.954356 0.298670i \(-0.0965430\pi\)
−0.735834 + 0.677162i \(0.763210\pi\)
\(270\) 0 0
\(271\) −6571.59 −1.47305 −0.736523 0.676412i \(-0.763534\pi\)
−0.736523 + 0.676412i \(0.763534\pi\)
\(272\) 0 0
\(273\) −251.233 + 435.148i −0.0556971 + 0.0964703i
\(274\) 0 0
\(275\) 1952.97 0.428248
\(276\) 0 0
\(277\) −8626.52 −1.87118 −0.935591 0.353086i \(-0.885132\pi\)
−0.935591 + 0.353086i \(0.885132\pi\)
\(278\) 0 0
\(279\) −1853.68 2247.04i −0.397768 0.482174i
\(280\) 0 0
\(281\) 3100.09 0.658135 0.329067 0.944306i \(-0.393266\pi\)
0.329067 + 0.944306i \(0.393266\pi\)
\(282\) 0 0
\(283\) 720.027 0.151241 0.0756204 0.997137i \(-0.475906\pi\)
0.0756204 + 0.997137i \(0.475906\pi\)
\(284\) 0 0
\(285\) 1258.76 2180.24i 0.261623 0.453144i
\(286\) 0 0
\(287\) −2834.64 −0.583009
\(288\) 0 0
\(289\) 1472.03 + 2549.63i 0.299620 + 0.518956i
\(290\) 0 0
\(291\) 4026.06 6973.35i 0.811038 1.40476i
\(292\) 0 0
\(293\) −1450.44 + 2512.24i −0.289201 + 0.500910i −0.973619 0.228179i \(-0.926723\pi\)
0.684418 + 0.729089i \(0.260056\pi\)
\(294\) 0 0
\(295\) −6773.34 −1.33681
\(296\) 0 0
\(297\) 757.695 + 1312.37i 0.148033 + 0.256401i
\(298\) 0 0
\(299\) 966.973 + 1674.85i 0.187028 + 0.323943i
\(300\) 0 0
\(301\) 178.823 + 309.731i 0.0342432 + 0.0593110i
\(302\) 0 0
\(303\) −1977.96 + 3425.93i −0.375020 + 0.649553i
\(304\) 0 0
\(305\) 1136.06 + 1967.71i 0.213281 + 0.369413i
\(306\) 0 0
\(307\) 1909.53 3307.41i 0.354993 0.614865i −0.632124 0.774867i \(-0.717817\pi\)
0.987117 + 0.160002i \(0.0511501\pi\)
\(308\) 0 0
\(309\) −2410.94 −0.443862
\(310\) 0 0
\(311\) −2936.50 −0.535413 −0.267707 0.963500i \(-0.586266\pi\)
−0.267707 + 0.963500i \(0.586266\pi\)
\(312\) 0 0
\(313\) −2513.05 + 4352.72i −0.453820 + 0.786040i −0.998620 0.0525265i \(-0.983273\pi\)
0.544799 + 0.838567i \(0.316606\pi\)
\(314\) 0 0
\(315\) 989.452 + 1713.78i 0.176982 + 0.306542i
\(316\) 0 0
\(317\) −471.163 + 816.078i −0.0834799 + 0.144591i −0.904742 0.425959i \(-0.859937\pi\)
0.821263 + 0.570550i \(0.193270\pi\)
\(318\) 0 0
\(319\) −1672.01 2896.01i −0.293463 0.508292i
\(320\) 0 0
\(321\) −3292.58 5702.92i −0.572505 0.991607i
\(322\) 0 0
\(323\) −579.923 1004.46i −0.0999002 0.173032i
\(324\) 0 0
\(325\) 812.886 0.138741
\(326\) 0 0
\(327\) 6638.78 11498.7i 1.12271 1.94459i
\(328\) 0 0
\(329\) −702.137 + 1216.14i −0.117660 + 0.203793i
\(330\) 0 0
\(331\) 2540.45 + 4400.19i 0.421860 + 0.730683i 0.996121 0.0879889i \(-0.0280440\pi\)
−0.574261 + 0.818672i \(0.694711\pi\)
\(332\) 0 0
\(333\) −588.440 −0.0968358
\(334\) 0 0
\(335\) 3562.05 6169.65i 0.580942 1.00622i
\(336\) 0 0
\(337\) 8615.39 1.39261 0.696306 0.717745i \(-0.254826\pi\)
0.696306 + 0.717745i \(0.254826\pi\)
\(338\) 0 0
\(339\) 964.111 0.154464
\(340\) 0 0
\(341\) −2482.19 3008.92i −0.394189 0.477836i
\(342\) 0 0
\(343\) −5007.62 −0.788298
\(344\) 0 0
\(345\) 19801.9 3.09014
\(346\) 0 0
\(347\) 5082.62 8803.36i 0.786310 1.36193i −0.141904 0.989880i \(-0.545322\pi\)
0.928213 0.372048i \(-0.121344\pi\)
\(348\) 0 0
\(349\) 9573.66 1.46838 0.734192 0.678941i \(-0.237561\pi\)
0.734192 + 0.678941i \(0.237561\pi\)
\(350\) 0 0
\(351\) 315.377 + 546.249i 0.0479589 + 0.0830672i
\(352\) 0 0
\(353\) 1408.79 2440.10i 0.212415 0.367913i −0.740055 0.672546i \(-0.765201\pi\)
0.952470 + 0.304633i \(0.0985340\pi\)
\(354\) 0 0
\(355\) −6794.94 + 11769.2i −1.01588 + 1.75956i
\(356\) 0 0
\(357\) 2370.26 0.351393
\(358\) 0 0
\(359\) 2690.56 + 4660.18i 0.395549 + 0.685111i 0.993171 0.116667i \(-0.0372210\pi\)
−0.597622 + 0.801778i \(0.703888\pi\)
\(360\) 0 0
\(361\) 3087.88 + 5348.37i 0.450194 + 0.779760i
\(362\) 0 0
\(363\) 2716.75 + 4705.55i 0.392816 + 0.680378i
\(364\) 0 0
\(365\) −2167.82 + 3754.77i −0.310873 + 0.538448i
\(366\) 0 0
\(367\) 569.058 + 985.638i 0.0809390 + 0.140190i 0.903654 0.428264i \(-0.140875\pi\)
−0.822715 + 0.568455i \(0.807541\pi\)
\(368\) 0 0
\(369\) 2966.18 5137.58i 0.418464 0.724801i
\(370\) 0 0
\(371\) 723.090 0.101189
\(372\) 0 0
\(373\) −6628.38 −0.920119 −0.460060 0.887888i \(-0.652172\pi\)
−0.460060 + 0.887888i \(0.652172\pi\)
\(374\) 0 0
\(375\) −1858.00 + 3218.16i −0.255858 + 0.443160i
\(376\) 0 0
\(377\) −695.943 1205.41i −0.0950740 0.164673i
\(378\) 0 0
\(379\) −767.668 + 1329.64i −0.104043 + 0.180209i −0.913347 0.407182i \(-0.866511\pi\)
0.809304 + 0.587391i \(0.199845\pi\)
\(380\) 0 0
\(381\) −271.580 470.391i −0.0365183 0.0632516i
\(382\) 0 0
\(383\) 4413.24 + 7643.95i 0.588788 + 1.01981i 0.994391 + 0.105762i \(0.0337281\pi\)
−0.405603 + 0.914049i \(0.632939\pi\)
\(384\) 0 0
\(385\) 1324.94 + 2294.86i 0.175390 + 0.303784i
\(386\) 0 0
\(387\) −748.486 −0.0983144
\(388\) 0 0
\(389\) 6508.83 11273.6i 0.848357 1.46940i −0.0343162 0.999411i \(-0.510925\pi\)
0.882673 0.469987i \(-0.155741\pi\)
\(390\) 0 0
\(391\) 4561.46 7900.68i 0.589982 1.02188i
\(392\) 0 0
\(393\) 8573.33 + 14849.4i 1.10043 + 1.90599i
\(394\) 0 0
\(395\) −15196.3 −1.93572
\(396\) 0 0
\(397\) 444.201 769.379i 0.0561557 0.0972646i −0.836581 0.547843i \(-0.815449\pi\)
0.892737 + 0.450579i \(0.148782\pi\)
\(398\) 0 0
\(399\) 1396.25 0.175188
\(400\) 0 0
\(401\) −8784.47 −1.09395 −0.546977 0.837148i \(-0.684221\pi\)
−0.546977 + 0.837148i \(0.684221\pi\)
\(402\) 0 0
\(403\) −1033.17 1252.41i −0.127707 0.154806i
\(404\) 0 0
\(405\) 13084.0 1.60530
\(406\) 0 0
\(407\) −787.956 −0.0959645
\(408\) 0 0
\(409\) 7158.51 12398.9i 0.865442 1.49899i −0.00116619 0.999999i \(-0.500371\pi\)
0.866608 0.498990i \(-0.166295\pi\)
\(410\) 0 0
\(411\) −14186.7 −1.70263
\(412\) 0 0
\(413\) −1878.30 3253.31i −0.223789 0.387615i
\(414\) 0 0
\(415\) −4155.98 + 7198.38i −0.491589 + 0.851456i
\(416\) 0 0
\(417\) 842.530 1459.31i 0.0989422 0.171373i
\(418\) 0 0
\(419\) 710.070 0.0827904 0.0413952 0.999143i \(-0.486820\pi\)
0.0413952 + 0.999143i \(0.486820\pi\)
\(420\) 0 0
\(421\) −5497.80 9522.48i −0.636453 1.10237i −0.986205 0.165526i \(-0.947068\pi\)
0.349753 0.936842i \(-0.386266\pi\)
\(422\) 0 0
\(423\) −1469.44 2545.14i −0.168904 0.292551i
\(424\) 0 0
\(425\) −1917.29 3320.85i −0.218829 0.379023i
\(426\) 0 0
\(427\) −630.076 + 1091.32i −0.0714087 + 0.123683i
\(428\) 0 0
\(429\) −704.054 1219.46i −0.0792356 0.137240i
\(430\) 0 0
\(431\) −1401.22 + 2426.98i −0.156600 + 0.271238i −0.933640 0.358212i \(-0.883387\pi\)
0.777041 + 0.629450i \(0.216720\pi\)
\(432\) 0 0
\(433\) −7065.63 −0.784186 −0.392093 0.919926i \(-0.628249\pi\)
−0.392093 + 0.919926i \(0.628249\pi\)
\(434\) 0 0
\(435\) −14251.7 −1.57084
\(436\) 0 0
\(437\) 2687.03 4654.07i 0.294137 0.509461i
\(438\) 0 0
\(439\) 1193.21 + 2066.70i 0.129724 + 0.224688i 0.923570 0.383431i \(-0.125258\pi\)
−0.793846 + 0.608119i \(0.791924\pi\)
\(440\) 0 0
\(441\) 2345.62 4062.73i 0.253279 0.438692i
\(442\) 0 0
\(443\) 1971.10 + 3414.05i 0.211399 + 0.366154i 0.952153 0.305623i \(-0.0988646\pi\)
−0.740753 + 0.671777i \(0.765531\pi\)
\(444\) 0 0
\(445\) 5446.96 + 9434.41i 0.580249 + 1.00502i
\(446\) 0 0
\(447\) −8214.05 14227.1i −0.869152 1.50542i
\(448\) 0 0
\(449\) −9890.55 −1.03956 −0.519782 0.854299i \(-0.673987\pi\)
−0.519782 + 0.854299i \(0.673987\pi\)
\(450\) 0 0
\(451\) 3971.89 6879.52i 0.414699 0.718279i
\(452\) 0 0
\(453\) −9204.40 + 15942.5i −0.954659 + 1.65352i
\(454\) 0 0
\(455\) 551.480 + 955.191i 0.0568215 + 0.0984177i
\(456\) 0 0
\(457\) 18623.9 1.90633 0.953163 0.302458i \(-0.0978073\pi\)
0.953163 + 0.302458i \(0.0978073\pi\)
\(458\) 0 0
\(459\) 1487.71 2576.79i 0.151286 0.262036i
\(460\) 0 0
\(461\) −16497.4 −1.66673 −0.833363 0.552726i \(-0.813588\pi\)
−0.833363 + 0.552726i \(0.813588\pi\)
\(462\) 0 0
\(463\) 5453.00 0.547349 0.273674 0.961822i \(-0.411761\pi\)
0.273674 + 0.961822i \(0.411761\pi\)
\(464\) 0 0
\(465\) −16394.8 + 2749.66i −1.63504 + 0.274221i
\(466\) 0 0
\(467\) 9875.95 0.978596 0.489298 0.872117i \(-0.337253\pi\)
0.489298 + 0.872117i \(0.337253\pi\)
\(468\) 0 0
\(469\) 3951.13 0.389011
\(470\) 0 0
\(471\) −336.437 + 582.727i −0.0329134 + 0.0570077i
\(472\) 0 0
\(473\) −1002.27 −0.0974298
\(474\) 0 0
\(475\) −1129.42 1956.22i −0.109098 0.188963i
\(476\) 0 0
\(477\) −756.644 + 1310.55i −0.0726297 + 0.125798i
\(478\) 0 0
\(479\) −4012.39 + 6949.67i −0.382737 + 0.662919i −0.991452 0.130469i \(-0.958352\pi\)
0.608716 + 0.793388i \(0.291685\pi\)
\(480\) 0 0
\(481\) −327.972 −0.0310899
\(482\) 0 0
\(483\) 5491.21 + 9511.05i 0.517306 + 0.896000i
\(484\) 0 0
\(485\) −8837.58 15307.1i −0.827410 1.43312i
\(486\) 0 0
\(487\) 9581.32 + 16595.3i 0.891522 + 1.54416i 0.838051 + 0.545592i \(0.183695\pi\)
0.0534712 + 0.998569i \(0.482971\pi\)
\(488\) 0 0
\(489\) −7629.28 + 13214.3i −0.705538 + 1.22203i
\(490\) 0 0
\(491\) −9062.63 15696.9i −0.832975 1.44275i −0.895669 0.444722i \(-0.853302\pi\)
0.0626937 0.998033i \(-0.480031\pi\)
\(492\) 0 0
\(493\) −3282.94 + 5686.22i −0.299911 + 0.519461i
\(494\) 0 0
\(495\) −5545.67 −0.503554
\(496\) 0 0
\(497\) −7537.15 −0.680256
\(498\) 0 0
\(499\) 2476.37 4289.19i 0.222159 0.384791i −0.733304 0.679901i \(-0.762023\pi\)
0.955463 + 0.295110i \(0.0953562\pi\)
\(500\) 0 0
\(501\) 8558.51 + 14823.8i 0.763206 + 1.32191i
\(502\) 0 0
\(503\) −4865.54 + 8427.36i −0.431300 + 0.747033i −0.996986 0.0775880i \(-0.975278\pi\)
0.565686 + 0.824621i \(0.308611\pi\)
\(504\) 0 0
\(505\) 4341.81 + 7520.24i 0.382590 + 0.662666i
\(506\) 0 0
\(507\) 6983.37 + 12095.6i 0.611721 + 1.05953i
\(508\) 0 0
\(509\) 1189.71 + 2060.63i 0.103601 + 0.179442i 0.913166 0.407588i \(-0.133630\pi\)
−0.809565 + 0.587031i \(0.800297\pi\)
\(510\) 0 0
\(511\) −2404.61 −0.208167
\(512\) 0 0
\(513\) 876.369 1517.92i 0.0754243 0.130639i
\(514\) 0 0
\(515\) −2646.11 + 4583.20i −0.226411 + 0.392156i
\(516\) 0 0
\(517\) −1967.67 3408.10i −0.167385 0.289919i
\(518\) 0 0
\(519\) −20506.8 −1.73439
\(520\) 0 0
\(521\) 10280.5 17806.3i 0.864482 1.49733i −0.00307815 0.999995i \(-0.500980\pi\)
0.867560 0.497332i \(-0.165687\pi\)
\(522\) 0 0
\(523\) 563.267 0.0470936 0.0235468 0.999723i \(-0.492504\pi\)
0.0235468 + 0.999723i \(0.492504\pi\)
\(524\) 0 0
\(525\) 4616.18 0.383746
\(526\) 0 0
\(527\) −2679.55 + 7174.72i −0.221486 + 0.593047i
\(528\) 0 0
\(529\) 30103.3 2.47418
\(530\) 0 0
\(531\) 7861.83 0.642513
\(532\) 0 0
\(533\) 1653.23 2863.47i 0.134351 0.232703i
\(534\) 0 0
\(535\) −14455.0 −1.16812
\(536\) 0 0
\(537\) 4118.13 + 7132.80i 0.330932 + 0.573190i
\(538\) 0 0
\(539\) 3140.92 5440.24i 0.251000 0.434745i
\(540\) 0 0
\(541\) −1301.29 + 2253.90i −0.103414 + 0.179118i −0.913089 0.407760i \(-0.866310\pi\)
0.809675 + 0.586878i \(0.199643\pi\)
\(542\) 0 0
\(543\) 27386.0 2.16435
\(544\) 0 0
\(545\) −14572.7 25240.7i −1.14537 1.98384i
\(546\) 0 0
\(547\) −11531.9 19973.8i −0.901401 1.56127i −0.825677 0.564144i \(-0.809206\pi\)
−0.0757244 0.997129i \(-0.524127\pi\)
\(548\) 0 0
\(549\) −1318.63 2283.93i −0.102509 0.177551i
\(550\) 0 0
\(551\) −1933.89 + 3349.59i −0.149522 + 0.258979i
\(552\) 0 0
\(553\) −4214.05 7298.94i −0.324050 0.561271i
\(554\) 0 0
\(555\) −1679.07 + 2908.24i −0.128419 + 0.222428i
\(556\) 0 0
\(557\) 1134.19 0.0862788 0.0431394 0.999069i \(-0.486264\pi\)
0.0431394 + 0.999069i \(0.486264\pi\)
\(558\) 0 0
\(559\) −417.175 −0.0315646
\(560\) 0 0
\(561\) −3321.20 + 5752.49i −0.249949 + 0.432924i
\(562\) 0 0
\(563\) −4721.71 8178.25i −0.353457 0.612206i 0.633395 0.773828i \(-0.281661\pi\)
−0.986853 + 0.161622i \(0.948327\pi\)
\(564\) 0 0
\(565\) 1058.16 1832.78i 0.0787911 0.136470i
\(566\) 0 0
\(567\) 3628.28 + 6284.37i 0.268736 + 0.465465i
\(568\) 0 0
\(569\) −3430.12 5941.15i −0.252721 0.437726i 0.711553 0.702632i \(-0.247992\pi\)
−0.964274 + 0.264907i \(0.914659\pi\)
\(570\) 0 0
\(571\) −6211.21 10758.1i −0.455221 0.788465i 0.543480 0.839422i \(-0.317106\pi\)
−0.998701 + 0.0509568i \(0.983773\pi\)
\(572\) 0 0
\(573\) −22245.0 −1.62181
\(574\) 0 0
\(575\) 8883.63 15386.9i 0.644301 1.11596i
\(576\) 0 0
\(577\) 11258.7 19500.6i 0.812313 1.40697i −0.0989284 0.995095i \(-0.531541\pi\)
0.911241 0.411873i \(-0.135125\pi\)
\(578\) 0 0
\(579\) 7042.16 + 12197.4i 0.505462 + 0.875485i
\(580\) 0 0
\(581\) −4609.94 −0.329178
\(582\) 0 0
\(583\) −1013.19 + 1754.90i −0.0719762 + 0.124666i
\(584\) 0 0
\(585\) −2308.28 −0.163138
\(586\) 0 0
\(587\) 14384.0 1.01140 0.505698 0.862711i \(-0.331235\pi\)
0.505698 + 0.862711i \(0.331235\pi\)
\(588\) 0 0
\(589\) −1578.45 + 4226.43i −0.110423 + 0.295665i
\(590\) 0 0
\(591\) 29979.4 2.08661
\(592\) 0 0
\(593\) −2115.81 −0.146520 −0.0732598 0.997313i \(-0.523340\pi\)
−0.0732598 + 0.997313i \(0.523340\pi\)
\(594\) 0 0
\(595\) 2601.47 4505.88i 0.179244 0.310459i
\(596\) 0 0
\(597\) 30664.5 2.10220
\(598\) 0 0
\(599\) 7643.57 + 13239.1i 0.521382 + 0.903061i 0.999691 + 0.0248686i \(0.00791675\pi\)
−0.478308 + 0.878192i \(0.658750\pi\)
\(600\) 0 0
\(601\) −5175.74 + 8964.65i −0.351286 + 0.608446i −0.986475 0.163911i \(-0.947589\pi\)
0.635189 + 0.772357i \(0.280922\pi\)
\(602\) 0 0
\(603\) −4134.48 + 7161.13i −0.279219 + 0.483621i
\(604\) 0 0
\(605\) 11927.0 0.801492
\(606\) 0 0
\(607\) 1517.64 + 2628.63i 0.101481 + 0.175771i 0.912295 0.409533i \(-0.134309\pi\)
−0.810814 + 0.585304i \(0.800975\pi\)
\(608\) 0 0
\(609\) −3952.09 6845.23i −0.262967 0.455472i
\(610\) 0 0
\(611\) −819.005 1418.56i −0.0542281 0.0939259i
\(612\) 0 0
\(613\) −14234.5 + 24654.9i −0.937892 + 1.62448i −0.168498 + 0.985702i \(0.553892\pi\)
−0.769394 + 0.638774i \(0.779442\pi\)
\(614\) 0 0
\(615\) −16927.6 29319.4i −1.10989 1.92239i
\(616\) 0 0
\(617\) 470.158 814.338i 0.0306773 0.0531346i −0.850279 0.526332i \(-0.823567\pi\)
0.880956 + 0.473197i \(0.156900\pi\)
\(618\) 0 0
\(619\) −25961.5 −1.68575 −0.842876 0.538108i \(-0.819139\pi\)
−0.842876 + 0.538108i \(0.819139\pi\)
\(620\) 0 0
\(621\) 13786.4 0.890868
\(622\) 0 0
\(623\) −3020.96 + 5232.46i −0.194274 + 0.336492i
\(624\) 0 0
\(625\) 9479.60 + 16419.1i 0.606694 + 1.05083i
\(626\) 0 0
\(627\) −1956.43 + 3388.63i −0.124613 + 0.215836i
\(628\) 0 0
\(629\) 773.564 + 1339.85i 0.0490366 + 0.0849339i
\(630\) 0 0
\(631\) 9698.27 + 16797.9i 0.611857 + 1.05977i 0.990927 + 0.134400i \(0.0429107\pi\)
−0.379070 + 0.925368i \(0.623756\pi\)
\(632\) 0 0
\(633\) 10761.6 + 18639.7i 0.675729 + 1.17040i
\(634\) 0 0
\(635\) −1192.29 −0.0745111
\(636\) 0 0
\(637\) 1307.35 2264.40i 0.0813173 0.140846i
\(638\) 0 0
\(639\) 7886.90 13660.5i 0.488265 0.845699i
\(640\) 0 0
\(641\) −681.241 1179.94i −0.0419773 0.0727067i 0.844273 0.535913i \(-0.180032\pi\)
−0.886251 + 0.463206i \(0.846699\pi\)
\(642\) 0 0
\(643\) 11817.0 0.724752 0.362376 0.932032i \(-0.381966\pi\)
0.362376 + 0.932032i \(0.381966\pi\)
\(644\) 0 0
\(645\) −2135.75 + 3699.23i −0.130380 + 0.225825i
\(646\) 0 0
\(647\) 9822.78 0.596868 0.298434 0.954430i \(-0.403536\pi\)
0.298434 + 0.954430i \(0.403536\pi\)
\(648\) 0 0
\(649\) 10527.5 0.636732
\(650\) 0 0
\(651\) −5867.11 7112.12i −0.353226 0.428181i
\(652\) 0 0
\(653\) 30236.8 1.81203 0.906016 0.423243i \(-0.139108\pi\)
0.906016 + 0.423243i \(0.139108\pi\)
\(654\) 0 0
\(655\) 37638.5 2.24528
\(656\) 0 0
\(657\) 2516.19 4358.17i 0.149415 0.258795i
\(658\) 0 0
\(659\) −4483.62 −0.265034 −0.132517 0.991181i \(-0.542306\pi\)
−0.132517 + 0.991181i \(0.542306\pi\)
\(660\) 0 0
\(661\) −5593.01 9687.37i −0.329112 0.570038i 0.653224 0.757164i \(-0.273416\pi\)
−0.982336 + 0.187127i \(0.940083\pi\)
\(662\) 0 0
\(663\) −1382.39 + 2394.37i −0.0809767 + 0.140256i
\(664\) 0 0
\(665\) 1532.45 2654.29i 0.0893624 0.154780i
\(666\) 0 0
\(667\) −30422.5 −1.76606
\(668\) 0 0
\(669\) 9178.81 + 15898.2i 0.530453 + 0.918772i
\(670\) 0 0
\(671\) −1765.72 3058.32i −0.101587 0.175954i
\(672\) 0 0
\(673\) −2325.43 4027.77i −0.133193 0.230697i 0.791713 0.610894i \(-0.209190\pi\)
−0.924906 + 0.380197i \(0.875856\pi\)
\(674\) 0 0
\(675\) 2897.38 5018.41i 0.165215 0.286161i
\(676\) 0 0
\(677\) −8173.67 14157.2i −0.464017 0.803702i 0.535139 0.844764i \(-0.320259\pi\)
−0.999157 + 0.0410621i \(0.986926\pi\)
\(678\) 0 0
\(679\) 4901.46 8489.57i 0.277026 0.479823i
\(680\) 0 0
\(681\) −29372.3 −1.65279
\(682\) 0 0
\(683\) 6320.40 0.354090 0.177045 0.984203i \(-0.443346\pi\)
0.177045 + 0.984203i \(0.443346\pi\)
\(684\) 0 0
\(685\) −15570.6 + 26969.0i −0.868498 + 1.50428i
\(686\) 0 0
\(687\) −7776.39 13469.1i −0.431860 0.748004i
\(688\) 0 0
\(689\) −421.722 + 730.445i −0.0233184 + 0.0403886i
\(690\) 0 0
\(691\) 3839.67 + 6650.51i 0.211386 + 0.366132i 0.952149 0.305635i \(-0.0988688\pi\)
−0.740762 + 0.671767i \(0.765535\pi\)
\(692\) 0 0
\(693\) −1537.86 2663.65i −0.0842977 0.146008i
\(694\) 0 0
\(695\) −1849.43 3203.31i −0.100940 0.174832i
\(696\) 0 0
\(697\) −15597.4 −0.847623
\(698\) 0 0
\(699\) 19156.6 33180.2i 1.03658 1.79541i
\(700\) 0 0
\(701\) 6090.54 10549.1i 0.328155 0.568381i −0.653991 0.756502i \(-0.726907\pi\)
0.982146 + 0.188121i \(0.0602399\pi\)
\(702\) 0 0
\(703\) 455.685 + 789.270i 0.0244473 + 0.0423440i
\(704\) 0 0
\(705\) −16771.7 −0.895972
\(706\) 0 0
\(707\) −2408.03 + 4170.84i −0.128095 + 0.221868i
\(708\) 0 0
\(709\) 1886.15 0.0999094 0.0499547 0.998751i \(-0.484092\pi\)
0.0499547 + 0.998751i \(0.484092\pi\)
\(710\) 0 0
\(711\) 17638.4 0.930367
\(712\) 0 0
\(713\) −34997.5 + 5869.61i −1.83824 + 0.308301i
\(714\) 0 0
\(715\) −3090.93 −0.161670
\(716\) 0 0
\(717\) −13409.0 −0.698420
\(718\) 0 0
\(719\) −9816.14 + 17002.1i −0.509152 + 0.881877i 0.490792 + 0.871277i \(0.336708\pi\)
−0.999944 + 0.0106002i \(0.996626\pi\)
\(720\) 0 0
\(721\) −2935.15 −0.151610
\(722\) 0 0
\(723\) 7362.13 + 12751.6i 0.378701 + 0.655929i
\(724\) 0 0
\(725\) −6393.67 + 11074.2i −0.327524 + 0.567288i
\(726\) 0 0
\(727\) −15442.3 + 26746.9i −0.787790 + 1.36449i 0.139528 + 0.990218i \(0.455441\pi\)
−0.927318 + 0.374274i \(0.877892\pi\)
\(728\) 0 0
\(729\) −3193.96 −0.162270
\(730\) 0 0
\(731\) 983.961 + 1704.27i 0.0497854 + 0.0862308i
\(732\) 0 0
\(733\) −7406.60 12828.6i −0.373218 0.646433i 0.616841 0.787088i \(-0.288412\pi\)
−0.990059 + 0.140655i \(0.955079\pi\)
\(734\) 0 0
\(735\) −13386.1 23185.4i −0.671773 1.16355i
\(736\) 0 0
\(737\) −5536.31 + 9589.18i −0.276707 + 0.479270i
\(738\) 0 0
\(739\) 1650.29 + 2858.38i 0.0821473 + 0.142283i 0.904172 0.427168i \(-0.140489\pi\)
−0.822025 + 0.569452i \(0.807156\pi\)
\(740\) 0 0
\(741\) −814.327 + 1410.46i −0.0403712 + 0.0699249i
\(742\) 0 0
\(743\) −3851.25 −0.190160 −0.0950799 0.995470i \(-0.530311\pi\)
−0.0950799 + 0.995470i \(0.530311\pi\)
\(744\) 0 0
\(745\) −36061.2 −1.77340
\(746\) 0 0
\(747\) 4823.86 8355.18i 0.236273 0.409237i
\(748\) 0 0
\(749\) −4008.49 6942.91i −0.195550 0.338703i
\(750\) 0 0
\(751\) 10665.5 18473.2i 0.518230 0.897601i −0.481546 0.876421i \(-0.659924\pi\)
0.999776 0.0211799i \(-0.00674227\pi\)
\(752\) 0 0
\(753\) 9459.51 + 16384.4i 0.457801 + 0.792934i
\(754\) 0 0
\(755\) 20204.5 + 34995.2i 0.973930 + 1.68690i
\(756\) 0 0
\(757\) −11493.4 19907.2i −0.551830 0.955797i −0.998143 0.0609200i \(-0.980597\pi\)
0.446313 0.894877i \(-0.352737\pi\)
\(758\) 0 0
\(759\) −30777.1 −1.47185
\(760\) 0 0
\(761\) −4053.56 + 7020.97i −0.193090 + 0.334441i −0.946273 0.323370i \(-0.895184\pi\)
0.753183 + 0.657811i \(0.228518\pi\)
\(762\) 0 0
\(763\) 8082.26 13998.9i 0.383483 0.664212i
\(764\) 0 0
\(765\) 5444.38 + 9429.94i 0.257310 + 0.445674i
\(766\) 0 0
\(767\) 4381.86 0.206284
\(768\) 0 0
\(769\) −3911.06 + 6774.15i −0.183402 + 0.317662i −0.943037 0.332688i \(-0.892044\pi\)
0.759635 + 0.650350i \(0.225378\pi\)
\(770\) 0 0
\(771\) −1264.34 −0.0590585
\(772\) 0 0
\(773\) −29299.4 −1.36329 −0.681646 0.731682i \(-0.738736\pi\)
−0.681646 + 0.731682i \(0.738736\pi\)
\(774\) 0 0
\(775\) −5218.55 + 13973.1i −0.241878 + 0.647648i
\(776\) 0 0
\(777\) −1862.48 −0.0859922
\(778\) 0 0
\(779\) −9187.98 −0.422585
\(780\) 0 0
\(781\) 10561.0 18292.2i 0.483871 0.838090i
\(782\) 0 0
\(783\) −9922.25 −0.452864
\(784\) 0 0
\(785\) 738.511 + 1279.14i 0.0335778 + 0.0581585i
\(786\) 0 0
\(787\) 12076.9 20917.8i 0.547007 0.947445i −0.451470 0.892286i \(-0.649100\pi\)
0.998478 0.0551585i \(-0.0175664\pi\)
\(788\) 0 0
\(789\) −797.669 + 1381.60i −0.0359921 + 0.0623401i
\(790\) 0 0
\(791\) 1173.74 0.0527603
\(792\) 0 0
\(793\) −734.949 1272.97i −0.0329115 0.0570044i
\(794\) 0 0
\(795\) 4318.06 + 7479.10i 0.192636 + 0.333656i
\(796\) 0 0
\(797\) −2282.82 3953.96i −0.101457 0.175729i 0.810828 0.585285i \(-0.199017\pi\)
−0.912285 + 0.409555i \(0.865684\pi\)
\(798\) 0 0
\(799\) −3863.45 + 6691.70i −0.171063 + 0.296289i
\(800\) 0 0
\(801\) −6322.30 10950.5i −0.278886 0.483044i
\(802\) 0 0
\(803\) 3369.33 5835.85i 0.148071 0.256467i
\(804\) 0 0
\(805\) 24107.4 1.05550
\(806\) 0 0
\(807\) −12772.4 −0.557137
\(808\) 0 0
\(809\) −11967.6 + 20728.4i −0.520096 + 0.900832i 0.479632 + 0.877470i \(0.340770\pi\)
−0.999727 + 0.0233619i \(0.992563\pi\)
\(810\) 0 0
\(811\) 4799.14 + 8312.36i 0.207794 + 0.359909i 0.951019 0.309132i \(-0.100038\pi\)
−0.743225 + 0.669041i \(0.766705\pi\)
\(812\) 0 0
\(813\) 21765.0 37698.0i 0.938906 1.62623i
\(814\) 0 0
\(815\) 16747.0 + 29006.6i 0.719781 + 1.24670i
\(816\) 0 0
\(817\) 579.624 + 1003.94i 0.0248206 + 0.0429906i
\(818\) 0 0
\(819\) −640.104 1108.69i −0.0273102 0.0473027i
\(820\) 0 0
\(821\) 16451.6 0.699348 0.349674 0.936871i \(-0.386292\pi\)
0.349674 + 0.936871i \(0.386292\pi\)
\(822\) 0 0
\(823\) −4406.07 + 7631.54i −0.186617 + 0.323230i −0.944120 0.329601i \(-0.893086\pi\)
0.757503 + 0.652832i \(0.226419\pi\)
\(824\) 0 0
\(825\) −6468.18 + 11203.2i −0.272961 + 0.472783i
\(826\) 0 0
\(827\) 9798.52 + 16971.5i 0.412005 + 0.713613i 0.995109 0.0987844i \(-0.0314954\pi\)
−0.583104 + 0.812397i \(0.698162\pi\)
\(828\) 0 0
\(829\) 12302.4 0.515418 0.257709 0.966223i \(-0.417032\pi\)
0.257709 + 0.966223i \(0.417032\pi\)
\(830\) 0 0
\(831\) 28570.9 49486.2i 1.19267 2.06577i
\(832\) 0 0
\(833\) −12334.2 −0.513031
\(834\) 0 0
\(835\) 37573.5 1.55723
\(836\) 0 0
\(837\) −11414.4 + 1914.36i −0.471372 + 0.0790562i
\(838\) 0 0
\(839\) −8585.46 −0.353281 −0.176641 0.984275i \(-0.556523\pi\)
−0.176641 + 0.984275i \(0.556523\pi\)
\(840\) 0 0
\(841\) −2493.53 −0.102240
\(842\) 0 0
\(843\) −10267.4 + 17783.7i −0.419489 + 0.726577i
\(844\) 0 0
\(845\) 30658.3 1.24814
\(846\) 0 0
\(847\) 3307.46 + 5728.68i 0.134174 + 0.232397i
\(848\) 0 0
\(849\) −2384.71 + 4130.45i −0.0963995 + 0.166969i
\(850\) 0 0
\(851\) −3584.25 + 6208.10i −0.144379 + 0.250072i
\(852\) 0 0
\(853\) 946.881 0.0380077 0.0190039 0.999819i \(-0.493951\pi\)
0.0190039 + 0.999819i \(0.493951\pi\)
\(854\) 0 0
\(855\) 3207.13 + 5554.91i 0.128283 + 0.222192i
\(856\) 0 0
\(857\) −10504.9 18194.9i −0.418715 0.725236i 0.577095 0.816677i \(-0.304186\pi\)
−0.995811 + 0.0914408i \(0.970853\pi\)
\(858\) 0 0
\(859\) −95.4124 165.259i −0.00378979 0.00656411i 0.864124 0.503278i \(-0.167873\pi\)
−0.867914 + 0.496714i \(0.834540\pi\)
\(860\) 0 0
\(861\) 9388.28 16261.0i 0.371605 0.643638i
\(862\) 0 0
\(863\) 14503.1 + 25120.1i 0.572064 + 0.990844i 0.996354 + 0.0853178i \(0.0271905\pi\)
−0.424290 + 0.905527i \(0.639476\pi\)
\(864\) 0 0
\(865\) −22507.2 + 38983.6i −0.884702 + 1.53235i
\(866\) 0 0
\(867\) −19501.4 −0.763899
\(868\) 0 0
\(869\) 23618.9 0.921996
\(870\) 0 0
\(871\) −2304.39 + 3991.32i −0.0896455 + 0.155271i
\(872\) 0 0
\(873\) 10257.8 + 17767.0i 0.397680 + 0.688801i
\(874\) 0 0
\(875\) −2261.99 + 3917.88i −0.0873934 + 0.151370i
\(876\) 0 0
\(877\) −13651.2 23644.6i −0.525620 0.910400i −0.999555 0.0298401i \(-0.990500\pi\)
0.473935 0.880560i \(-0.342833\pi\)
\(878\) 0 0
\(879\) −9607.68 16641.0i −0.368668 0.638552i
\(880\) 0 0
\(881\) 5329.69 + 9231.29i 0.203816 + 0.353019i 0.949755 0.312995i \(-0.101332\pi\)
−0.745939 + 0.666014i \(0.767999\pi\)
\(882\) 0 0
\(883\) 30383.9 1.15798 0.578992 0.815333i \(-0.303446\pi\)
0.578992 + 0.815333i \(0.303446\pi\)
\(884\) 0 0
\(885\) 22433.2 38855.4i 0.852071 1.47583i
\(886\) 0 0
\(887\) −21863.4 + 37868.5i −0.827622 + 1.43348i 0.0722761 + 0.997385i \(0.476974\pi\)
−0.899899 + 0.436099i \(0.856360\pi\)
\(888\) 0 0
\(889\) −330.631 572.669i −0.0124736 0.0216048i
\(890\) 0 0
\(891\) −20335.8 −0.764617
\(892\) 0 0
\(893\) −2275.85 + 3941.89i −0.0852838 + 0.147716i
\(894\) 0 0
\(895\) 18079.3 0.675224
\(896\) 0 0
\(897\) −12810.4 −0.476841
\(898\) 0 0
\(899\) 25188.1 4224.43i 0.934451 0.156722i
\(900\) 0 0
\(901\) 3978.74 0.147116
\(902\) 0 0
\(903\) −2369.04 −0.0873053
\(904\) 0 0
\(905\) 30057.4 52060.9i 1.10402 1.91222i
\(906\) 0 0
\(907\) −10279.2 −0.376312 −0.188156 0.982139i \(-0.560251\pi\)
−0.188156 + 0.982139i \(0.560251\pi\)
\(908\) 0 0
\(909\) −5039.55 8728.76i −0.183885 0.318498i
\(910\) 0 0
\(911\) 21156.9 36644.8i 0.769438 1.33271i −0.168430 0.985714i \(-0.553870\pi\)
0.937868 0.346992i \(-0.112797\pi\)
\(912\) 0 0
\(913\) 6459.44 11188.1i 0.234147 0.405555i
\(914\) 0 0
\(915\) −15050.4 −0.543773
\(916\) 0 0
\(917\) 10437.4 + 18078.2i 0.375872 + 0.651029i
\(918\) 0 0
\(919\) −7250.48 12558.2i −0.260252 0.450769i 0.706057 0.708155i \(-0.250472\pi\)
−0.966309 + 0.257386i \(0.917139\pi\)
\(920\) 0 0
\(921\) 12648.7 + 21908.1i 0.452538 + 0.783819i
\(922\) 0 0
\(923\) 4395.84 7613.81i 0.156761 0.271519i
\(924\) 0 0
\(925\) 1506.55 + 2609.42i 0.0535514 + 0.0927537i
\(926\) 0 0
\(927\) 3071.35 5319.74i 0.108820 0.188482i
\(928\) 0 0
\(929\) 6146.06 0.217057 0.108528 0.994093i \(-0.465386\pi\)
0.108528 + 0.994093i \(0.465386\pi\)
\(930\) 0 0
\(931\) −7265.74 −0.255773
\(932\) 0 0
\(933\) 9725.62 16845.3i 0.341267 0.591093i
\(934\) 0 0
\(935\) 7290.35 + 12627.3i 0.254995 + 0.441664i
\(936\) 0 0
\(937\) −9541.44 + 16526.3i −0.332663 + 0.576189i −0.983033 0.183428i \(-0.941280\pi\)
0.650370 + 0.759617i \(0.274614\pi\)
\(938\) 0 0
\(939\) −16646.3 28832.3i −0.578522 1.00203i
\(940\) 0 0
\(941\) −3486.71 6039.17i −0.120790 0.209215i 0.799289 0.600946i \(-0.205210\pi\)
−0.920080 + 0.391732i \(0.871876\pi\)
\(942\) 0 0
\(943\) −36134.6 62587.0i −1.24783 2.16131i
\(944\) 0 0
\(945\) 7862.59 0.270656
\(946\) 0 0
\(947\) −8133.14 + 14087.0i −0.279083 + 0.483386i −0.971157 0.238441i \(-0.923364\pi\)
0.692074 + 0.721826i \(0.256697\pi\)
\(948\) 0 0
\(949\) 1402.42 2429.07i 0.0479710 0.0830883i
\(950\) 0 0
\(951\) −3120.96 5405.67i −0.106419 0.184323i
\(952\) 0 0
\(953\) −42228.7 −1.43539 −0.717693 0.696360i \(-0.754802\pi\)
−0.717693 + 0.696360i \(0.754802\pi\)
\(954\) 0 0
\(955\) −24414.9 + 42287.9i −0.827275 + 1.43288i
\(956\) 0 0
\(957\) 22150.7 0.748202
\(958\) 0 0
\(959\) −17271.4 −0.581565
\(960\) 0 0
\(961\) 28160.9 9719.41i 0.945282 0.326253i
\(962\) 0 0
\(963\) 16778.0 0.561437
\(964\) 0 0
\(965\) 30916.4 1.03133
\(966\) 0 0
\(967\) 8635.30 14956.8i 0.287169 0.497392i −0.685964 0.727636i \(-0.740619\pi\)
0.973133 + 0.230244i \(0.0739525\pi\)
\(968\) 0 0
\(969\) 7682.77 0.254702
\(970\) 0 0
\(971\) 26409.4 + 45742.4i 0.872829 + 1.51178i 0.859058 + 0.511879i \(0.171050\pi\)
0.0137709 + 0.999905i \(0.495616\pi\)
\(972\) 0 0
\(973\) 1025.72 1776.60i 0.0337957 0.0585358i
\(974\) 0 0
\(975\) −2692.26 + 4663.13i −0.0884322 + 0.153169i
\(976\) 0 0
\(977\) 37685.4 1.23405 0.617023 0.786945i \(-0.288339\pi\)
0.617023 + 0.786945i \(0.288339\pi\)
\(978\) 0 0
\(979\) −8465.94 14663.4i −0.276376 0.478698i
\(980\) 0 0
\(981\) 16914.6 + 29297.0i 0.550502 + 0.953497i
\(982\) 0 0
\(983\) −4794.72 8304.70i −0.155573 0.269460i 0.777695 0.628642i \(-0.216389\pi\)
−0.933267 + 0.359182i \(0.883056\pi\)
\(984\) 0 0
\(985\) 32903.7 56991.0i 1.06437 1.84354i
\(986\) 0 0
\(987\) −4650.93 8055.65i −0.149991 0.259791i
\(988\) 0 0
\(989\) −4559.11 + 7896.60i −0.146584 + 0.253890i
\(990\) 0 0
\(991\) 9963.12 0.319363 0.159682 0.987169i \(-0.448953\pi\)
0.159682 + 0.987169i \(0.448953\pi\)
\(992\) 0 0
\(993\) −33655.7 −1.07556
\(994\) 0 0
\(995\) 33655.7 58293.4i 1.07232 1.85731i
\(996\) 0 0
\(997\) 5599.52 + 9698.65i 0.177872 + 0.308084i 0.941151 0.337985i \(-0.109745\pi\)
−0.763279 + 0.646069i \(0.776412\pi\)
\(998\) 0 0
\(999\) −1169.00 + 2024.76i −0.0370224 + 0.0641248i
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 124.4.e.c.5.1 8
3.2 odd 2 1116.4.i.e.253.1 8
4.3 odd 2 496.4.i.e.129.4 8
31.25 even 3 inner 124.4.e.c.25.1 yes 8
93.56 odd 6 1116.4.i.e.397.1 8
124.87 odd 6 496.4.i.e.273.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.4.e.c.5.1 8 1.1 even 1 trivial
124.4.e.c.25.1 yes 8 31.25 even 3 inner
496.4.i.e.129.4 8 4.3 odd 2
496.4.i.e.273.4 8 124.87 odd 6
1116.4.i.e.253.1 8 3.2 odd 2
1116.4.i.e.397.1 8 93.56 odd 6