Properties

Label 252.4.j.b.169.3
Level $252$
Weight $4$
Character 252.169
Analytic conductor $14.868$
Analytic rank $0$
Dimension $18$
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] = [252,4,Mod(85,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.85");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.j (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: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 7 x^{17} + 81 x^{16} - 470 x^{15} + 2687 x^{14} - 12243 x^{13} + 43732 x^{12} - 169303 x^{11} + 166449 x^{10} - 57172 x^{9} - 3758051 x^{8} + \cdots + 5500612092612 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 169.3
Root \(2.54993 - 4.95961i\) of defining polynomial
Character \(\chi\) \(=\) 252.169
Dual form 252.4.j.b.85.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.54993 - 4.95961i) q^{3} +(3.85902 + 6.68401i) q^{5} +(-3.50000 + 6.06218i) q^{7} +(-22.1955 + 15.3741i) q^{9} +O(q^{10})\) \(q+(-1.54993 - 4.95961i) q^{3} +(3.85902 + 6.68401i) q^{5} +(-3.50000 + 6.06218i) q^{7} +(-22.1955 + 15.3741i) q^{9} +(20.9501 - 36.2866i) q^{11} +(-19.8256 - 34.3390i) q^{13} +(27.1689 - 29.4989i) q^{15} -68.0598 q^{17} -104.181 q^{19} +(35.4908 + 7.96270i) q^{21} +(-72.9636 - 126.377i) q^{23} +(32.7160 - 56.6658i) q^{25} +(110.651 + 86.2521i) q^{27} +(-145.615 + 252.213i) q^{29} +(133.567 + 231.344i) q^{31} +(-212.438 - 47.6626i) q^{33} -54.0262 q^{35} -341.523 q^{37} +(-139.580 + 151.550i) q^{39} +(-116.552 - 201.874i) q^{41} +(20.6831 - 35.8241i) q^{43} +(-188.413 - 89.0259i) q^{45} +(-120.893 + 209.392i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(105.488 + 337.550i) q^{51} -505.372 q^{53} +323.387 q^{55} +(161.472 + 516.695i) q^{57} +(-90.6191 - 156.957i) q^{59} +(339.158 - 587.439i) q^{61} +(-15.5162 - 188.362i) q^{63} +(153.015 - 265.029i) q^{65} +(-443.394 - 767.980i) q^{67} +(-513.691 + 557.746i) q^{69} +594.531 q^{71} +704.964 q^{73} +(-331.747 - 74.4308i) q^{75} +(146.651 + 254.006i) q^{77} +(287.818 - 498.516i) q^{79} +(256.276 - 682.469i) q^{81} +(-170.316 + 294.997i) q^{83} +(-262.644 - 454.912i) q^{85} +(1476.57 + 331.283i) q^{87} -1427.89 q^{89} +277.559 q^{91} +(940.358 - 1021.00i) q^{93} +(-402.034 - 696.344i) q^{95} +(76.5670 - 132.618i) q^{97} +(92.8760 + 1127.49i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 11 q^{3} - 6 q^{5} - 63 q^{7} - 109 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 11 q^{3} - 6 q^{5} - 63 q^{7} - 109 q^{9} - 93 q^{11} - 18 q^{13} + 162 q^{15} - 54 q^{17} - 90 q^{19} - 49 q^{21} - 246 q^{23} - 315 q^{25} - 394 q^{27} - 318 q^{29} - 18 q^{31} + 33 q^{33} + 84 q^{35} - 72 q^{37} + 268 q^{39} - 57 q^{41} - 171 q^{43} - 318 q^{45} - 1056 q^{47} - 441 q^{49} + 705 q^{51} + 1512 q^{53} - 1800 q^{55} - 1271 q^{57} - 411 q^{59} - 198 q^{61} + 308 q^{63} - 1326 q^{65} - 441 q^{67} + 642 q^{69} + 3516 q^{71} + 54 q^{73} - 2497 q^{75} - 651 q^{77} + 72 q^{79} + 1163 q^{81} - 558 q^{83} - 1008 q^{85} + 2766 q^{87} + 2784 q^{89} + 252 q^{91} - 2618 q^{93} - 156 q^{95} + 909 q^{97} + 6318 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.54993 4.95961i −0.298284 0.954477i
\(4\) 0 0
\(5\) 3.85902 + 6.68401i 0.345161 + 0.597836i 0.985383 0.170354i \(-0.0544910\pi\)
−0.640222 + 0.768190i \(0.721158\pi\)
\(6\) 0 0
\(7\) −3.50000 + 6.06218i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) −22.1955 + 15.3741i −0.822054 + 0.569410i
\(10\) 0 0
\(11\) 20.9501 36.2866i 0.574244 0.994620i −0.421879 0.906652i \(-0.638629\pi\)
0.996123 0.0879680i \(-0.0280373\pi\)
\(12\) 0 0
\(13\) −19.8256 34.3390i −0.422972 0.732609i 0.573257 0.819376i \(-0.305680\pi\)
−0.996229 + 0.0867670i \(0.972346\pi\)
\(14\) 0 0
\(15\) 27.1689 29.4989i 0.467665 0.507773i
\(16\) 0 0
\(17\) −68.0598 −0.970996 −0.485498 0.874238i \(-0.661362\pi\)
−0.485498 + 0.874238i \(0.661362\pi\)
\(18\) 0 0
\(19\) −104.181 −1.25793 −0.628965 0.777434i \(-0.716521\pi\)
−0.628965 + 0.777434i \(0.716521\pi\)
\(20\) 0 0
\(21\) 35.4908 + 7.96270i 0.368796 + 0.0827430i
\(22\) 0 0
\(23\) −72.9636 126.377i −0.661477 1.14571i −0.980228 0.197873i \(-0.936597\pi\)
0.318751 0.947839i \(-0.396737\pi\)
\(24\) 0 0
\(25\) 32.7160 56.6658i 0.261728 0.453326i
\(26\) 0 0
\(27\) 110.651 + 86.2521i 0.788694 + 0.614786i
\(28\) 0 0
\(29\) −145.615 + 252.213i −0.932417 + 1.61499i −0.153240 + 0.988189i \(0.548971\pi\)
−0.779177 + 0.626804i \(0.784363\pi\)
\(30\) 0 0
\(31\) 133.567 + 231.344i 0.773847 + 1.34034i 0.935440 + 0.353485i \(0.115003\pi\)
−0.161593 + 0.986857i \(0.551663\pi\)
\(32\) 0 0
\(33\) −212.438 47.6626i −1.12063 0.251424i
\(34\) 0 0
\(35\) −54.0262 −0.260917
\(36\) 0 0
\(37\) −341.523 −1.51746 −0.758731 0.651404i \(-0.774180\pi\)
−0.758731 + 0.651404i \(0.774180\pi\)
\(38\) 0 0
\(39\) −139.580 + 151.550i −0.573093 + 0.622242i
\(40\) 0 0
\(41\) −116.552 201.874i −0.443960 0.768962i 0.554019 0.832504i \(-0.313094\pi\)
−0.997979 + 0.0635423i \(0.979760\pi\)
\(42\) 0 0
\(43\) 20.6831 35.8241i 0.0733521 0.127050i −0.827016 0.562178i \(-0.809964\pi\)
0.900368 + 0.435128i \(0.143297\pi\)
\(44\) 0 0
\(45\) −188.413 89.0259i −0.624155 0.294916i
\(46\) 0 0
\(47\) −120.893 + 209.392i −0.375192 + 0.649851i −0.990356 0.138548i \(-0.955756\pi\)
0.615164 + 0.788399i \(0.289090\pi\)
\(48\) 0 0
\(49\) −24.5000 42.4352i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 105.488 + 337.550i 0.289632 + 0.926793i
\(52\) 0 0
\(53\) −505.372 −1.30978 −0.654889 0.755725i \(-0.727285\pi\)
−0.654889 + 0.755725i \(0.727285\pi\)
\(54\) 0 0
\(55\) 323.387 0.792826
\(56\) 0 0
\(57\) 161.472 + 516.695i 0.375220 + 1.20067i
\(58\) 0 0
\(59\) −90.6191 156.957i −0.199959 0.346340i 0.748556 0.663072i \(-0.230748\pi\)
−0.948515 + 0.316732i \(0.897414\pi\)
\(60\) 0 0
\(61\) 339.158 587.439i 0.711881 1.23301i −0.252270 0.967657i \(-0.581177\pi\)
0.964150 0.265356i \(-0.0854896\pi\)
\(62\) 0 0
\(63\) −15.5162 188.362i −0.0310295 0.376689i
\(64\) 0 0
\(65\) 153.015 265.029i 0.291987 0.505736i
\(66\) 0 0
\(67\) −443.394 767.980i −0.808495 1.40035i −0.913906 0.405925i \(-0.866949\pi\)
0.105412 0.994429i \(-0.466384\pi\)
\(68\) 0 0
\(69\) −513.691 + 557.746i −0.896248 + 0.973112i
\(70\) 0 0
\(71\) 594.531 0.993773 0.496886 0.867816i \(-0.334477\pi\)
0.496886 + 0.867816i \(0.334477\pi\)
\(72\) 0 0
\(73\) 704.964 1.13027 0.565136 0.824998i \(-0.308824\pi\)
0.565136 + 0.824998i \(0.308824\pi\)
\(74\) 0 0
\(75\) −331.747 74.4308i −0.510759 0.114594i
\(76\) 0 0
\(77\) 146.651 + 254.006i 0.217044 + 0.375931i
\(78\) 0 0
\(79\) 287.818 498.516i 0.409900 0.709967i −0.584978 0.811049i \(-0.698897\pi\)
0.994878 + 0.101082i \(0.0322304\pi\)
\(80\) 0 0
\(81\) 256.276 682.469i 0.351545 0.936171i
\(82\) 0 0
\(83\) −170.316 + 294.997i −0.225237 + 0.390122i −0.956390 0.292091i \(-0.905649\pi\)
0.731154 + 0.682213i \(0.238982\pi\)
\(84\) 0 0
\(85\) −262.644 454.912i −0.335150 0.580496i
\(86\) 0 0
\(87\) 1476.57 + 331.283i 1.81960 + 0.408245i
\(88\) 0 0
\(89\) −1427.89 −1.70063 −0.850317 0.526272i \(-0.823590\pi\)
−0.850317 + 0.526272i \(0.823590\pi\)
\(90\) 0 0
\(91\) 277.559 0.319737
\(92\) 0 0
\(93\) 940.358 1021.00i 1.04850 1.13842i
\(94\) 0 0
\(95\) −402.034 696.344i −0.434188 0.752036i
\(96\) 0 0
\(97\) 76.5670 132.618i 0.0801464 0.138818i −0.823166 0.567800i \(-0.807795\pi\)
0.903313 + 0.428983i \(0.141128\pi\)
\(98\) 0 0
\(99\) 92.8760 + 1127.49i 0.0942868 + 1.14461i
\(100\) 0 0
\(101\) −512.568 + 887.793i −0.504974 + 0.874641i 0.495009 + 0.868888i \(0.335165\pi\)
−0.999983 + 0.00575325i \(0.998169\pi\)
\(102\) 0 0
\(103\) 876.289 + 1517.78i 0.838284 + 1.45195i 0.891328 + 0.453359i \(0.149774\pi\)
−0.0530438 + 0.998592i \(0.516892\pi\)
\(104\) 0 0
\(105\) 83.7367 + 267.949i 0.0778273 + 0.249039i
\(106\) 0 0
\(107\) −247.500 −0.223614 −0.111807 0.993730i \(-0.535664\pi\)
−0.111807 + 0.993730i \(0.535664\pi\)
\(108\) 0 0
\(109\) 1257.29 1.10483 0.552416 0.833568i \(-0.313706\pi\)
0.552416 + 0.833568i \(0.313706\pi\)
\(110\) 0 0
\(111\) 529.336 + 1693.82i 0.452634 + 1.44838i
\(112\) 0 0
\(113\) −249.452 432.064i −0.207668 0.359692i 0.743311 0.668946i \(-0.233254\pi\)
−0.950980 + 0.309254i \(0.899921\pi\)
\(114\) 0 0
\(115\) 563.136 975.380i 0.456632 0.790910i
\(116\) 0 0
\(117\) 967.968 + 457.369i 0.764860 + 0.361400i
\(118\) 0 0
\(119\) 238.209 412.591i 0.183501 0.317833i
\(120\) 0 0
\(121\) −212.311 367.734i −0.159513 0.276284i
\(122\) 0 0
\(123\) −820.569 + 890.943i −0.601531 + 0.653119i
\(124\) 0 0
\(125\) 1469.76 1.05167
\(126\) 0 0
\(127\) 491.102 0.343136 0.171568 0.985172i \(-0.445117\pi\)
0.171568 + 0.985172i \(0.445117\pi\)
\(128\) 0 0
\(129\) −209.731 47.0552i −0.143146 0.0321161i
\(130\) 0 0
\(131\) 109.506 + 189.670i 0.0730349 + 0.126500i 0.900230 0.435415i \(-0.143398\pi\)
−0.827195 + 0.561915i \(0.810065\pi\)
\(132\) 0 0
\(133\) 364.632 631.561i 0.237726 0.411754i
\(134\) 0 0
\(135\) −149.507 + 1072.44i −0.0953151 + 0.683710i
\(136\) 0 0
\(137\) 530.907 919.558i 0.331084 0.573454i −0.651641 0.758528i \(-0.725919\pi\)
0.982725 + 0.185074i \(0.0592524\pi\)
\(138\) 0 0
\(139\) −71.3339 123.554i −0.0435285 0.0753936i 0.843440 0.537223i \(-0.180527\pi\)
−0.886969 + 0.461829i \(0.847193\pi\)
\(140\) 0 0
\(141\) 1225.88 + 275.038i 0.732181 + 0.164272i
\(142\) 0 0
\(143\) −1661.39 −0.971557
\(144\) 0 0
\(145\) −2247.73 −1.28734
\(146\) 0 0
\(147\) −172.489 + 187.282i −0.0967800 + 0.105080i
\(148\) 0 0
\(149\) −1293.63 2240.63i −0.711262 1.23194i −0.964384 0.264508i \(-0.914790\pi\)
0.253121 0.967435i \(-0.418543\pi\)
\(150\) 0 0
\(151\) 879.334 1523.05i 0.473902 0.820822i −0.525652 0.850700i \(-0.676178\pi\)
0.999554 + 0.0298777i \(0.00951178\pi\)
\(152\) 0 0
\(153\) 1510.62 1046.36i 0.798211 0.552894i
\(154\) 0 0
\(155\) −1030.87 + 1785.52i −0.534203 + 0.925267i
\(156\) 0 0
\(157\) −240.519 416.591i −0.122264 0.211768i 0.798396 0.602133i \(-0.205682\pi\)
−0.920660 + 0.390365i \(0.872349\pi\)
\(158\) 0 0
\(159\) 783.290 + 2506.45i 0.390685 + 1.25015i
\(160\) 0 0
\(161\) 1021.49 0.500030
\(162\) 0 0
\(163\) 2967.97 1.42619 0.713097 0.701066i \(-0.247292\pi\)
0.713097 + 0.701066i \(0.247292\pi\)
\(164\) 0 0
\(165\) −501.226 1603.87i −0.236487 0.756735i
\(166\) 0 0
\(167\) −460.641 797.854i −0.213446 0.369699i 0.739345 0.673327i \(-0.235135\pi\)
−0.952791 + 0.303628i \(0.901802\pi\)
\(168\) 0 0
\(169\) 312.390 541.076i 0.142190 0.246279i
\(170\) 0 0
\(171\) 2312.33 1601.68i 1.03409 0.716277i
\(172\) 0 0
\(173\) −524.992 + 909.313i −0.230719 + 0.399617i −0.958020 0.286702i \(-0.907441\pi\)
0.727301 + 0.686319i \(0.240774\pi\)
\(174\) 0 0
\(175\) 229.012 + 396.660i 0.0989239 + 0.171341i
\(176\) 0 0
\(177\) −637.992 + 692.707i −0.270929 + 0.294164i
\(178\) 0 0
\(179\) 1435.40 0.599367 0.299683 0.954039i \(-0.403119\pi\)
0.299683 + 0.954039i \(0.403119\pi\)
\(180\) 0 0
\(181\) 845.711 0.347300 0.173650 0.984807i \(-0.444444\pi\)
0.173650 + 0.984807i \(0.444444\pi\)
\(182\) 0 0
\(183\) −3439.14 771.604i −1.38923 0.311686i
\(184\) 0 0
\(185\) −1317.94 2282.75i −0.523769 0.907194i
\(186\) 0 0
\(187\) −1425.86 + 2469.66i −0.557589 + 0.965772i
\(188\) 0 0
\(189\) −910.153 + 368.902i −0.350285 + 0.141977i
\(190\) 0 0
\(191\) −1289.40 + 2233.30i −0.488468 + 0.846052i −0.999912 0.0132651i \(-0.995777\pi\)
0.511444 + 0.859317i \(0.329111\pi\)
\(192\) 0 0
\(193\) −2276.19 3942.47i −0.848930 1.47039i −0.882164 0.470942i \(-0.843914\pi\)
0.0332344 0.999448i \(-0.489419\pi\)
\(194\) 0 0
\(195\) −1551.60 348.117i −0.569808 0.127842i
\(196\) 0 0
\(197\) −2802.02 −1.01338 −0.506690 0.862129i \(-0.669131\pi\)
−0.506690 + 0.862129i \(0.669131\pi\)
\(198\) 0 0
\(199\) 346.315 0.123365 0.0616825 0.998096i \(-0.480353\pi\)
0.0616825 + 0.998096i \(0.480353\pi\)
\(200\) 0 0
\(201\) −3121.65 + 3389.37i −1.09545 + 1.18939i
\(202\) 0 0
\(203\) −1019.31 1765.49i −0.352420 0.610410i
\(204\) 0 0
\(205\) 899.552 1558.07i 0.306475 0.530831i
\(206\) 0 0
\(207\) 3562.38 + 1683.24i 1.19615 + 0.565185i
\(208\) 0 0
\(209\) −2182.59 + 3780.36i −0.722359 + 1.25116i
\(210\) 0 0
\(211\) −861.334 1491.87i −0.281027 0.486753i 0.690611 0.723226i \(-0.257342\pi\)
−0.971638 + 0.236473i \(0.924008\pi\)
\(212\) 0 0
\(213\) −921.480 2948.64i −0.296426 0.948534i
\(214\) 0 0
\(215\) 319.265 0.101273
\(216\) 0 0
\(217\) −1869.93 −0.584973
\(218\) 0 0
\(219\) −1092.64 3496.35i −0.337141 1.07882i
\(220\) 0 0
\(221\) 1349.33 + 2337.10i 0.410704 + 0.711360i
\(222\) 0 0
\(223\) −2103.45 + 3643.29i −0.631648 + 1.09405i 0.355566 + 0.934651i \(0.384288\pi\)
−0.987215 + 0.159396i \(0.949045\pi\)
\(224\) 0 0
\(225\) 145.037 + 1760.70i 0.0429739 + 0.521689i
\(226\) 0 0
\(227\) 1044.04 1808.33i 0.305266 0.528736i −0.672055 0.740502i \(-0.734588\pi\)
0.977321 + 0.211765i \(0.0679213\pi\)
\(228\) 0 0
\(229\) 2036.30 + 3526.97i 0.587609 + 1.01777i 0.994545 + 0.104311i \(0.0332638\pi\)
−0.406936 + 0.913457i \(0.633403\pi\)
\(230\) 0 0
\(231\) 1032.47 1121.02i 0.294077 0.319298i
\(232\) 0 0
\(233\) 3431.73 0.964894 0.482447 0.875925i \(-0.339748\pi\)
0.482447 + 0.875925i \(0.339748\pi\)
\(234\) 0 0
\(235\) −1866.11 −0.518006
\(236\) 0 0
\(237\) −2918.54 654.803i −0.799914 0.179468i
\(238\) 0 0
\(239\) 766.959 + 1328.41i 0.207575 + 0.359530i 0.950950 0.309344i \(-0.100110\pi\)
−0.743375 + 0.668875i \(0.766776\pi\)
\(240\) 0 0
\(241\) −2131.97 + 3692.67i −0.569842 + 0.986996i 0.426739 + 0.904375i \(0.359662\pi\)
−0.996581 + 0.0826208i \(0.973671\pi\)
\(242\) 0 0
\(243\) −3781.99 213.254i −0.998414 0.0562973i
\(244\) 0 0
\(245\) 189.092 327.517i 0.0493087 0.0854052i
\(246\) 0 0
\(247\) 2065.44 + 3577.45i 0.532069 + 0.921570i
\(248\) 0 0
\(249\) 1727.05 + 387.480i 0.439547 + 0.0986165i
\(250\) 0 0
\(251\) 221.139 0.0556102 0.0278051 0.999613i \(-0.491148\pi\)
0.0278051 + 0.999613i \(0.491148\pi\)
\(252\) 0 0
\(253\) −6114.37 −1.51940
\(254\) 0 0
\(255\) −1849.11 + 2007.69i −0.454101 + 0.493045i
\(256\) 0 0
\(257\) 1192.69 + 2065.80i 0.289487 + 0.501406i 0.973687 0.227888i \(-0.0731821\pi\)
−0.684201 + 0.729294i \(0.739849\pi\)
\(258\) 0 0
\(259\) 1195.33 2070.38i 0.286773 0.496706i
\(260\) 0 0
\(261\) −645.543 7836.68i −0.153096 1.85854i
\(262\) 0 0
\(263\) 1103.18 1910.77i 0.258651 0.447996i −0.707230 0.706984i \(-0.750055\pi\)
0.965881 + 0.258987i \(0.0833888\pi\)
\(264\) 0 0
\(265\) −1950.24 3377.92i −0.452084 0.783033i
\(266\) 0 0
\(267\) 2213.13 + 7081.79i 0.507271 + 1.62322i
\(268\) 0 0
\(269\) 4084.22 0.925722 0.462861 0.886431i \(-0.346823\pi\)
0.462861 + 0.886431i \(0.346823\pi\)
\(270\) 0 0
\(271\) 2734.83 0.613022 0.306511 0.951867i \(-0.400838\pi\)
0.306511 + 0.951867i \(0.400838\pi\)
\(272\) 0 0
\(273\) −430.195 1376.58i −0.0953722 0.305181i
\(274\) 0 0
\(275\) −1370.81 2374.30i −0.300591 0.520640i
\(276\) 0 0
\(277\) 72.9088 126.282i 0.0158147 0.0273918i −0.858010 0.513633i \(-0.828299\pi\)
0.873824 + 0.486242i \(0.161632\pi\)
\(278\) 0 0
\(279\) −6521.27 3081.32i −1.39935 0.661197i
\(280\) 0 0
\(281\) −3716.57 + 6437.30i −0.789012 + 1.36661i 0.137561 + 0.990493i \(0.456074\pi\)
−0.926573 + 0.376115i \(0.877260\pi\)
\(282\) 0 0
\(283\) −2721.28 4713.39i −0.571601 0.990043i −0.996402 0.0847557i \(-0.972989\pi\)
0.424800 0.905287i \(-0.360344\pi\)
\(284\) 0 0
\(285\) −2830.47 + 3073.22i −0.588290 + 0.638743i
\(286\) 0 0
\(287\) 1631.73 0.335602
\(288\) 0 0
\(289\) −280.864 −0.0571675
\(290\) 0 0
\(291\) −776.406 174.194i −0.156405 0.0350909i
\(292\) 0 0
\(293\) −1390.00 2407.55i −0.277149 0.480035i 0.693526 0.720431i \(-0.256056\pi\)
−0.970675 + 0.240396i \(0.922723\pi\)
\(294\) 0 0
\(295\) 699.401 1211.40i 0.138036 0.239086i
\(296\) 0 0
\(297\) 5447.94 2208.15i 1.06438 0.431413i
\(298\) 0 0
\(299\) −2893.10 + 5010.99i −0.559572 + 0.969208i
\(300\) 0 0
\(301\) 144.782 + 250.769i 0.0277245 + 0.0480202i
\(302\) 0 0
\(303\) 5197.55 + 1166.12i 0.985450 + 0.221095i
\(304\) 0 0
\(305\) 5235.26 0.982853
\(306\) 0 0
\(307\) 2464.61 0.458184 0.229092 0.973405i \(-0.426424\pi\)
0.229092 + 0.973405i \(0.426424\pi\)
\(308\) 0 0
\(309\) 6169.40 6698.49i 1.13581 1.23322i
\(310\) 0 0
\(311\) −1694.23 2934.49i −0.308910 0.535047i 0.669214 0.743069i \(-0.266631\pi\)
−0.978124 + 0.208022i \(0.933297\pi\)
\(312\) 0 0
\(313\) −5288.19 + 9159.42i −0.954972 + 1.65406i −0.220540 + 0.975378i \(0.570782\pi\)
−0.734432 + 0.678682i \(0.762551\pi\)
\(314\) 0 0
\(315\) 1199.14 830.603i 0.214488 0.148569i
\(316\) 0 0
\(317\) 4723.53 8181.40i 0.836909 1.44957i −0.0555582 0.998455i \(-0.517694\pi\)
0.892467 0.451113i \(-0.148973\pi\)
\(318\) 0 0
\(319\) 6101.30 + 10567.8i 1.07087 + 1.85480i
\(320\) 0 0
\(321\) 383.607 + 1227.50i 0.0667005 + 0.213435i
\(322\) 0 0
\(323\) 7090.51 1.22144
\(324\) 0 0
\(325\) −2594.46 −0.442814
\(326\) 0 0
\(327\) −1948.71 6235.68i −0.329554 1.05454i
\(328\) 0 0
\(329\) −846.248 1465.75i −0.141809 0.245621i
\(330\) 0 0
\(331\) −2643.85 + 4579.28i −0.439031 + 0.760423i −0.997615 0.0690245i \(-0.978011\pi\)
0.558584 + 0.829448i \(0.311345\pi\)
\(332\) 0 0
\(333\) 7580.27 5250.60i 1.24744 0.864058i
\(334\) 0 0
\(335\) 3422.13 5927.30i 0.558121 0.966695i
\(336\) 0 0
\(337\) 261.033 + 452.122i 0.0421940 + 0.0730821i 0.886351 0.463014i \(-0.153232\pi\)
−0.844157 + 0.536096i \(0.819899\pi\)
\(338\) 0 0
\(339\) −1756.24 + 1906.86i −0.281374 + 0.305505i
\(340\) 0 0
\(341\) 11192.9 1.77751
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −5710.32 1281.17i −0.891111 0.199929i
\(346\) 0 0
\(347\) −4089.91 7083.93i −0.632731 1.09592i −0.986991 0.160776i \(-0.948600\pi\)
0.354260 0.935147i \(-0.384733\pi\)
\(348\) 0 0
\(349\) −3972.34 + 6880.30i −0.609268 + 1.05528i 0.382093 + 0.924124i \(0.375203\pi\)
−0.991361 + 0.131160i \(0.958130\pi\)
\(350\) 0 0
\(351\) 768.091 5509.63i 0.116802 0.837841i
\(352\) 0 0
\(353\) −3647.98 + 6318.48i −0.550035 + 0.952688i 0.448237 + 0.893915i \(0.352052\pi\)
−0.998271 + 0.0587729i \(0.981281\pi\)
\(354\) 0 0
\(355\) 2294.31 + 3973.85i 0.343011 + 0.594113i
\(356\) 0 0
\(357\) −2415.50 541.940i −0.358100 0.0803431i
\(358\) 0 0
\(359\) 5008.17 0.736271 0.368136 0.929772i \(-0.379996\pi\)
0.368136 + 0.929772i \(0.379996\pi\)
\(360\) 0 0
\(361\) 3994.59 0.582386
\(362\) 0 0
\(363\) −1494.75 + 1622.94i −0.216127 + 0.234662i
\(364\) 0 0
\(365\) 2720.47 + 4711.99i 0.390125 + 0.675717i
\(366\) 0 0
\(367\) 6583.34 11402.7i 0.936370 1.62184i 0.164196 0.986428i \(-0.447497\pi\)
0.772173 0.635412i \(-0.219170\pi\)
\(368\) 0 0
\(369\) 5690.55 + 2688.81i 0.802814 + 0.379333i
\(370\) 0 0
\(371\) 1768.80 3063.66i 0.247525 0.428726i
\(372\) 0 0
\(373\) −4393.91 7610.47i −0.609941 1.05645i −0.991250 0.132001i \(-0.957860\pi\)
0.381309 0.924448i \(-0.375473\pi\)
\(374\) 0 0
\(375\) −2278.02 7289.44i −0.313697 1.00380i
\(376\) 0 0
\(377\) 11547.7 1.57754
\(378\) 0 0
\(379\) −687.994 −0.0932451 −0.0466226 0.998913i \(-0.514846\pi\)
−0.0466226 + 0.998913i \(0.514846\pi\)
\(380\) 0 0
\(381\) −761.173 2435.68i −0.102352 0.327516i
\(382\) 0 0
\(383\) −2311.60 4003.81i −0.308401 0.534165i 0.669612 0.742711i \(-0.266460\pi\)
−0.978013 + 0.208546i \(0.933127\pi\)
\(384\) 0 0
\(385\) −1131.85 + 1960.43i −0.149830 + 0.259513i
\(386\) 0 0
\(387\) 91.6924 + 1113.12i 0.0120439 + 0.146209i
\(388\) 0 0
\(389\) −3383.44 + 5860.30i −0.440996 + 0.763827i −0.997764 0.0668410i \(-0.978708\pi\)
0.556768 + 0.830668i \(0.312041\pi\)
\(390\) 0 0
\(391\) 4965.89 + 8601.17i 0.642291 + 1.11248i
\(392\) 0 0
\(393\) 770.961 837.080i 0.0989564 0.107443i
\(394\) 0 0
\(395\) 4442.78 0.565925
\(396\) 0 0
\(397\) −10230.5 −1.29334 −0.646669 0.762771i \(-0.723839\pi\)
−0.646669 + 0.762771i \(0.723839\pi\)
\(398\) 0 0
\(399\) −3697.45 829.559i −0.463920 0.104085i
\(400\) 0 0
\(401\) 5795.59 + 10038.3i 0.721741 + 1.25009i 0.960302 + 0.278964i \(0.0899910\pi\)
−0.238561 + 0.971128i \(0.576676\pi\)
\(402\) 0 0
\(403\) 5296.08 9173.07i 0.654631 1.13385i
\(404\) 0 0
\(405\) 5550.60 920.704i 0.681016 0.112963i
\(406\) 0 0
\(407\) −7154.94 + 12392.7i −0.871394 + 1.50930i
\(408\) 0 0
\(409\) 2924.72 + 5065.76i 0.353589 + 0.612435i 0.986875 0.161483i \(-0.0516277\pi\)
−0.633286 + 0.773918i \(0.718294\pi\)
\(410\) 0 0
\(411\) −5383.52 1207.84i −0.646106 0.144960i
\(412\) 0 0
\(413\) 1268.67 0.151155
\(414\) 0 0
\(415\) −2629.01 −0.310972
\(416\) 0 0
\(417\) −502.217 + 545.288i −0.0589776 + 0.0640356i
\(418\) 0 0
\(419\) −1540.92 2668.95i −0.179663 0.311185i 0.762102 0.647457i \(-0.224167\pi\)
−0.941765 + 0.336272i \(0.890834\pi\)
\(420\) 0 0
\(421\) −3806.37 + 6592.83i −0.440644 + 0.763218i −0.997737 0.0672317i \(-0.978583\pi\)
0.557093 + 0.830450i \(0.311917\pi\)
\(422\) 0 0
\(423\) −535.942 6506.17i −0.0616038 0.747850i
\(424\) 0 0
\(425\) −2226.64 + 3856.66i −0.254137 + 0.440178i
\(426\) 0 0
\(427\) 2374.10 + 4112.07i 0.269066 + 0.466035i
\(428\) 0 0
\(429\) 2575.04 + 8239.86i 0.289799 + 0.927329i
\(430\) 0 0
\(431\) 4831.62 0.539979 0.269989 0.962863i \(-0.412980\pi\)
0.269989 + 0.962863i \(0.412980\pi\)
\(432\) 0 0
\(433\) 5173.85 0.574225 0.287112 0.957897i \(-0.407305\pi\)
0.287112 + 0.957897i \(0.407305\pi\)
\(434\) 0 0
\(435\) 3483.81 + 11147.8i 0.383991 + 1.22873i
\(436\) 0 0
\(437\) 7601.39 + 13166.0i 0.832091 + 1.44122i
\(438\) 0 0
\(439\) −220.798 + 382.433i −0.0240048 + 0.0415775i −0.877778 0.479067i \(-0.840975\pi\)
0.853773 + 0.520645i \(0.174308\pi\)
\(440\) 0 0
\(441\) 1196.19 + 565.205i 0.129164 + 0.0610307i
\(442\) 0 0
\(443\) 5812.33 10067.2i 0.623368 1.07971i −0.365486 0.930817i \(-0.619097\pi\)
0.988854 0.148888i \(-0.0475695\pi\)
\(444\) 0 0
\(445\) −5510.26 9544.05i −0.586992 1.01670i
\(446\) 0 0
\(447\) −9107.61 + 9888.70i −0.963703 + 1.04635i
\(448\) 0 0
\(449\) −11833.8 −1.24381 −0.621906 0.783092i \(-0.713641\pi\)
−0.621906 + 0.783092i \(0.713641\pi\)
\(450\) 0 0
\(451\) −9767.10 −1.01977
\(452\) 0 0
\(453\) −8916.64 2000.54i −0.924813 0.207491i
\(454\) 0 0
\(455\) 1071.10 + 1855.20i 0.110361 + 0.191150i
\(456\) 0 0
\(457\) −147.660 + 255.754i −0.0151143 + 0.0261787i −0.873484 0.486854i \(-0.838145\pi\)
0.858369 + 0.513032i \(0.171478\pi\)
\(458\) 0 0
\(459\) −7530.86 5870.30i −0.765818 0.596955i
\(460\) 0 0
\(461\) 5995.26 10384.1i 0.605698 1.04910i −0.386242 0.922397i \(-0.626227\pi\)
0.991941 0.126703i \(-0.0404395\pi\)
\(462\) 0 0
\(463\) −5320.43 9215.25i −0.534042 0.924987i −0.999209 0.0397645i \(-0.987339\pi\)
0.465167 0.885223i \(-0.345994\pi\)
\(464\) 0 0
\(465\) 10453.3 + 2345.29i 1.04249 + 0.233893i
\(466\) 0 0
\(467\) 2566.48 0.254309 0.127155 0.991883i \(-0.459416\pi\)
0.127155 + 0.991883i \(0.459416\pi\)
\(468\) 0 0
\(469\) 6207.51 0.611165
\(470\) 0 0
\(471\) −1693.34 + 1838.56i −0.165658 + 0.179865i
\(472\) 0 0
\(473\) −866.624 1501.04i −0.0842440 0.145915i
\(474\) 0 0
\(475\) −3408.37 + 5903.47i −0.329235 + 0.570252i
\(476\) 0 0
\(477\) 11217.0 7769.63i 1.07671 0.745801i
\(478\) 0 0
\(479\) −2200.37 + 3811.15i −0.209890 + 0.363540i −0.951680 0.307092i \(-0.900644\pi\)
0.741790 + 0.670633i \(0.233977\pi\)
\(480\) 0 0
\(481\) 6770.91 + 11727.6i 0.641844 + 1.11171i
\(482\) 0 0
\(483\) −1583.24 5066.20i −0.149151 0.477267i
\(484\) 0 0
\(485\) 1181.89 0.110654
\(486\) 0 0
\(487\) −15244.7 −1.41849 −0.709243 0.704964i \(-0.750963\pi\)
−0.709243 + 0.704964i \(0.750963\pi\)
\(488\) 0 0
\(489\) −4600.14 14720.0i −0.425410 1.36127i
\(490\) 0 0
\(491\) −8684.08 15041.3i −0.798182 1.38249i −0.920799 0.390037i \(-0.872462\pi\)
0.122617 0.992454i \(-0.460871\pi\)
\(492\) 0 0
\(493\) 9910.55 17165.6i 0.905373 1.56815i
\(494\) 0 0
\(495\) −7177.71 + 4971.77i −0.651746 + 0.451443i
\(496\) 0 0
\(497\) −2080.86 + 3604.15i −0.187805 + 0.325288i
\(498\) 0 0
\(499\) −1475.23 2555.17i −0.132345 0.229229i 0.792235 0.610216i \(-0.208917\pi\)
−0.924580 + 0.380987i \(0.875584\pi\)
\(500\) 0 0
\(501\) −3243.08 + 3521.22i −0.289202 + 0.314005i
\(502\) 0 0
\(503\) −13169.6 −1.16740 −0.583701 0.811968i \(-0.698396\pi\)
−0.583701 + 0.811968i \(0.698396\pi\)
\(504\) 0 0
\(505\) −7912.03 −0.697189
\(506\) 0 0
\(507\) −3167.71 710.706i −0.277481 0.0622555i
\(508\) 0 0
\(509\) −4467.78 7738.42i −0.389059 0.673869i 0.603265 0.797541i \(-0.293866\pi\)
−0.992323 + 0.123672i \(0.960533\pi\)
\(510\) 0 0
\(511\) −2467.37 + 4273.62i −0.213601 + 0.369968i
\(512\) 0 0
\(513\) −11527.7 8985.79i −0.992121 0.773358i
\(514\) 0 0
\(515\) −6763.22 + 11714.2i −0.578686 + 1.00231i
\(516\) 0 0
\(517\) 5065.42 + 8773.56i 0.430903 + 0.746346i
\(518\) 0 0
\(519\) 5323.53 + 1194.39i 0.450245 + 0.101017i
\(520\) 0 0
\(521\) −4285.65 −0.360380 −0.180190 0.983632i \(-0.557671\pi\)
−0.180190 + 0.983632i \(0.557671\pi\)
\(522\) 0 0
\(523\) 17587.3 1.47044 0.735220 0.677828i \(-0.237079\pi\)
0.735220 + 0.677828i \(0.237079\pi\)
\(524\) 0 0
\(525\) 1612.33 1750.60i 0.134034 0.145529i
\(526\) 0 0
\(527\) −9090.51 15745.2i −0.751402 1.30147i
\(528\) 0 0
\(529\) −4563.88 + 7904.88i −0.375103 + 0.649698i
\(530\) 0 0
\(531\) 4424.40 + 2090.54i 0.361587 + 0.170851i
\(532\) 0 0
\(533\) −4621.43 + 8004.55i −0.375565 + 0.650499i
\(534\) 0 0
\(535\) −955.107 1654.29i −0.0771829 0.133685i
\(536\) 0 0
\(537\) −2224.76 7119.01i −0.178781 0.572082i
\(538\) 0 0
\(539\) −2053.11 −0.164070
\(540\) 0 0
\(541\) 11196.4 0.889780 0.444890 0.895585i \(-0.353243\pi\)
0.444890 + 0.895585i \(0.353243\pi\)
\(542\) 0 0
\(543\) −1310.79 4194.40i −0.103594 0.331490i
\(544\) 0 0
\(545\) 4851.91 + 8403.76i 0.381345 + 0.660509i
\(546\) 0 0
\(547\) 10695.6 18525.4i 0.836036 1.44806i −0.0571474 0.998366i \(-0.518200\pi\)
0.893184 0.449692i \(-0.148466\pi\)
\(548\) 0 0
\(549\) 1503.56 + 18252.7i 0.116886 + 1.41895i
\(550\) 0 0
\(551\) 15170.3 26275.7i 1.17291 2.03155i
\(552\) 0 0
\(553\) 2014.73 + 3489.61i 0.154928 + 0.268342i
\(554\) 0 0
\(555\) −9278.82 + 10074.6i −0.709665 + 0.770526i
\(556\) 0 0
\(557\) −13798.3 −1.04964 −0.524822 0.851212i \(-0.675868\pi\)
−0.524822 + 0.851212i \(0.675868\pi\)
\(558\) 0 0
\(559\) −1640.22 −0.124103
\(560\) 0 0
\(561\) 14458.5 + 3243.91i 1.08813 + 0.244132i
\(562\) 0 0
\(563\) −1977.64 3425.37i −0.148042 0.256416i 0.782462 0.622698i \(-0.213964\pi\)
−0.930504 + 0.366283i \(0.880630\pi\)
\(564\) 0 0
\(565\) 1925.28 3334.69i 0.143358 0.248303i
\(566\) 0 0
\(567\) 3240.28 + 3942.23i 0.239998 + 0.291990i
\(568\) 0 0
\(569\) −6488.75 + 11238.9i −0.478072 + 0.828044i −0.999684 0.0251383i \(-0.991997\pi\)
0.521612 + 0.853183i \(0.325331\pi\)
\(570\) 0 0
\(571\) −7591.71 13149.2i −0.556398 0.963710i −0.997793 0.0663968i \(-0.978850\pi\)
0.441395 0.897313i \(-0.354484\pi\)
\(572\) 0 0
\(573\) 13074.8 + 2933.45i 0.953239 + 0.213868i
\(574\) 0 0
\(575\) −9548.31 −0.692508
\(576\) 0 0
\(577\) −18148.1 −1.30938 −0.654691 0.755897i \(-0.727201\pi\)
−0.654691 + 0.755897i \(0.727201\pi\)
\(578\) 0 0
\(579\) −16025.2 + 17399.5i −1.15023 + 1.24888i
\(580\) 0 0
\(581\) −1192.21 2064.98i −0.0851315 0.147452i
\(582\) 0 0
\(583\) −10587.6 + 18338.2i −0.752132 + 1.30273i
\(584\) 0 0
\(585\) 678.346 + 8234.90i 0.0479421 + 0.582002i
\(586\) 0 0
\(587\) −12047.6 + 20867.1i −0.847119 + 1.46725i 0.0366501 + 0.999328i \(0.488331\pi\)
−0.883769 + 0.467924i \(0.845002\pi\)
\(588\) 0 0
\(589\) −13915.0 24101.5i −0.973445 1.68606i
\(590\) 0 0
\(591\) 4342.93 + 13896.9i 0.302274 + 0.967247i
\(592\) 0 0
\(593\) −10618.6 −0.735333 −0.367666 0.929958i \(-0.619843\pi\)
−0.367666 + 0.929958i \(0.619843\pi\)
\(594\) 0 0
\(595\) 3677.01 0.253349
\(596\) 0 0
\(597\) −536.763 1717.59i −0.0367978 0.117749i
\(598\) 0 0
\(599\) −12015.6 20811.6i −0.819606 1.41960i −0.905973 0.423335i \(-0.860859\pi\)
0.0863674 0.996263i \(-0.472474\pi\)
\(600\) 0 0
\(601\) 430.072 744.907i 0.0291897 0.0505580i −0.851062 0.525066i \(-0.824041\pi\)
0.880251 + 0.474508i \(0.157374\pi\)
\(602\) 0 0
\(603\) 21648.3 + 10228.9i 1.46200 + 0.690801i
\(604\) 0 0
\(605\) 1638.63 2838.18i 0.110115 0.190725i
\(606\) 0 0
\(607\) −3506.39 6073.24i −0.234464 0.406104i 0.724653 0.689114i \(-0.242000\pi\)
−0.959117 + 0.283010i \(0.908667\pi\)
\(608\) 0 0
\(609\) −7176.30 + 7791.75i −0.477501 + 0.518453i
\(610\) 0 0
\(611\) 9587.08 0.634782
\(612\) 0 0
\(613\) 10045.8 0.661903 0.330952 0.943648i \(-0.392630\pi\)
0.330952 + 0.943648i \(0.392630\pi\)
\(614\) 0 0
\(615\) −9121.66 2046.53i −0.598083 0.134186i
\(616\) 0 0
\(617\) 9874.66 + 17103.4i 0.644309 + 1.11598i 0.984461 + 0.175605i \(0.0561883\pi\)
−0.340152 + 0.940371i \(0.610478\pi\)
\(618\) 0 0
\(619\) −8995.65 + 15580.9i −0.584112 + 1.01171i 0.410873 + 0.911693i \(0.365224\pi\)
−0.994985 + 0.100020i \(0.968109\pi\)
\(620\) 0 0
\(621\) 2826.78 20276.9i 0.182665 1.31028i
\(622\) 0 0
\(623\) 4997.62 8656.14i 0.321389 0.556663i
\(624\) 0 0
\(625\) 1582.33 + 2740.67i 0.101269 + 0.175403i
\(626\) 0 0
\(627\) 22132.0 + 4965.52i 1.40967 + 0.316274i
\(628\) 0 0
\(629\) 23244.0 1.47345
\(630\) 0 0
\(631\) 14458.5 0.912175 0.456087 0.889935i \(-0.349250\pi\)
0.456087 + 0.889935i \(0.349250\pi\)
\(632\) 0 0
\(633\) −6064.11 + 6584.18i −0.380769 + 0.413424i
\(634\) 0 0
\(635\) 1895.17 + 3282.53i 0.118437 + 0.205139i
\(636\) 0 0
\(637\) −971.455 + 1682.61i −0.0604246 + 0.104658i
\(638\) 0 0
\(639\) −13195.9 + 9140.36i −0.816935 + 0.565864i
\(640\) 0 0
\(641\) −10467.4 + 18130.0i −0.644985 + 1.11715i 0.339320 + 0.940671i \(0.389803\pi\)
−0.984305 + 0.176476i \(0.943530\pi\)
\(642\) 0 0
\(643\) 855.505 + 1481.78i 0.0524694 + 0.0908797i 0.891067 0.453871i \(-0.149957\pi\)
−0.838598 + 0.544751i \(0.816624\pi\)
\(644\) 0 0
\(645\) −494.838 1583.43i −0.0302081 0.0966628i
\(646\) 0 0
\(647\) −20098.7 −1.22127 −0.610636 0.791912i \(-0.709086\pi\)
−0.610636 + 0.791912i \(0.709086\pi\)
\(648\) 0 0
\(649\) −7593.91 −0.459302
\(650\) 0 0
\(651\) 2898.26 + 9274.13i 0.174488 + 0.558344i
\(652\) 0 0
\(653\) −3901.88 6758.26i −0.233832 0.405010i 0.725100 0.688643i \(-0.241793\pi\)
−0.958933 + 0.283634i \(0.908460\pi\)
\(654\) 0 0
\(655\) −845.169 + 1463.88i −0.0504176 + 0.0873258i
\(656\) 0 0
\(657\) −15647.0 + 10838.2i −0.929144 + 0.643588i
\(658\) 0 0
\(659\) −14784.9 + 25608.2i −0.873958 + 1.51374i −0.0160894 + 0.999871i \(0.505122\pi\)
−0.857869 + 0.513869i \(0.828212\pi\)
\(660\) 0 0
\(661\) −10529.2 18237.1i −0.619575 1.07313i −0.989563 0.144099i \(-0.953972\pi\)
0.369989 0.929036i \(-0.379362\pi\)
\(662\) 0 0
\(663\) 9499.76 10314.5i 0.556471 0.604195i
\(664\) 0 0
\(665\) 5628.48 0.328215
\(666\) 0 0
\(667\) 42498.5 2.46709
\(668\) 0 0
\(669\) 21329.5 + 4785.47i 1.23265 + 0.276558i
\(670\) 0 0
\(671\) −14210.8 24613.8i −0.817586 1.41610i
\(672\) 0 0
\(673\) 12119.1 20990.8i 0.694139 1.20228i −0.276331 0.961063i \(-0.589119\pi\)
0.970470 0.241222i \(-0.0775481\pi\)
\(674\) 0 0
\(675\) 8507.59 3448.28i 0.485122 0.196629i
\(676\) 0 0
\(677\) 10950.2 18966.4i 0.621642 1.07672i −0.367538 0.930008i \(-0.619799\pi\)
0.989180 0.146707i \(-0.0468674\pi\)
\(678\) 0 0
\(679\) 535.969 + 928.325i 0.0302925 + 0.0524681i
\(680\) 0 0
\(681\) −10586.8 2375.25i −0.595723 0.133656i
\(682\) 0 0
\(683\) −7529.77 −0.421843 −0.210921 0.977503i \(-0.567646\pi\)
−0.210921 + 0.977503i \(0.567646\pi\)
\(684\) 0 0
\(685\) 8195.12 0.457109
\(686\) 0 0
\(687\) 14336.3 15565.8i 0.796162 0.864443i
\(688\) 0 0
\(689\) 10019.3 + 17354.0i 0.553999 + 0.959555i
\(690\) 0 0
\(691\) 14569.6 25235.3i 0.802105 1.38929i −0.116123 0.993235i \(-0.537047\pi\)
0.918228 0.396052i \(-0.129620\pi\)
\(692\) 0 0
\(693\) −7160.08 3383.17i −0.392481 0.185449i
\(694\) 0 0
\(695\) 550.557 953.593i 0.0300487 0.0520458i
\(696\) 0 0
\(697\) 7932.51 + 13739.5i 0.431084 + 0.746659i
\(698\) 0 0
\(699\) −5318.94 17020.1i −0.287812 0.920970i
\(700\) 0 0
\(701\) −23973.7 −1.29169 −0.645844 0.763470i \(-0.723494\pi\)
−0.645844 + 0.763470i \(0.723494\pi\)
\(702\) 0 0
\(703\) 35580.1 1.90886
\(704\) 0 0
\(705\) 2892.33 + 9255.16i 0.154513 + 0.494425i
\(706\) 0 0
\(707\) −3587.97 6214.55i −0.190862 0.330583i
\(708\) 0 0
\(709\) −5860.85 + 10151.3i −0.310450 + 0.537714i −0.978460 0.206438i \(-0.933813\pi\)
0.668010 + 0.744152i \(0.267146\pi\)
\(710\) 0 0
\(711\) 1275.96 + 15489.7i 0.0673026 + 0.817032i
\(712\) 0 0
\(713\) 19491.0 33759.4i 1.02376 1.77321i
\(714\) 0 0
\(715\) −6411.34 11104.8i −0.335343 0.580832i
\(716\) 0 0
\(717\) 5399.67 5862.75i 0.281247 0.305368i
\(718\) 0 0
\(719\) 28287.7 1.46725 0.733625 0.679554i \(-0.237827\pi\)
0.733625 + 0.679554i \(0.237827\pi\)
\(720\) 0 0
\(721\) −12268.0 −0.633683
\(722\) 0 0
\(723\) 21618.6 + 4850.34i 1.11204 + 0.249497i
\(724\) 0 0
\(725\) 9527.90 + 16502.8i 0.488079 + 0.845378i
\(726\) 0 0
\(727\) 16094.2 27876.0i 0.821048 1.42210i −0.0838538 0.996478i \(-0.526723\pi\)
0.904902 0.425619i \(-0.139944\pi\)
\(728\) 0 0
\(729\) 4804.15 + 19087.7i 0.244076 + 0.969756i
\(730\) 0 0
\(731\) −1407.69 + 2438.18i −0.0712245 + 0.123365i
\(732\) 0 0
\(733\) 3176.49 + 5501.85i 0.160063 + 0.277238i 0.934891 0.354934i \(-0.115497\pi\)
−0.774828 + 0.632172i \(0.782163\pi\)
\(734\) 0 0
\(735\) −1917.43 430.195i −0.0962253 0.0215891i
\(736\) 0 0
\(737\) −37156.5 −1.85709
\(738\) 0 0
\(739\) −12283.6 −0.611447 −0.305723 0.952120i \(-0.598898\pi\)
−0.305723 + 0.952120i \(0.598898\pi\)
\(740\) 0 0
\(741\) 14541.5 15788.6i 0.720910 0.782737i
\(742\) 0 0
\(743\) 8697.39 + 15064.3i 0.429443 + 0.743818i 0.996824 0.0796381i \(-0.0253765\pi\)
−0.567381 + 0.823456i \(0.692043\pi\)
\(744\) 0 0
\(745\) 9984.26 17293.2i 0.491000 0.850437i
\(746\) 0 0
\(747\) −755.048 9166.04i −0.0369823 0.448953i
\(748\) 0 0
\(749\) 866.250 1500.39i 0.0422591 0.0731950i
\(750\) 0 0
\(751\) 5029.77 + 8711.82i 0.244393 + 0.423301i 0.961961 0.273188i \(-0.0880780\pi\)
−0.717568 + 0.696489i \(0.754745\pi\)
\(752\) 0 0
\(753\) −342.749 1096.76i −0.0165876 0.0530787i
\(754\) 0 0
\(755\) 13573.5 0.654290
\(756\) 0 0
\(757\) 6358.99 0.305312 0.152656 0.988279i \(-0.451217\pi\)
0.152656 + 0.988279i \(0.451217\pi\)
\(758\) 0 0
\(759\) 9476.83 + 30324.9i 0.453211 + 1.45023i
\(760\) 0 0
\(761\) −15869.3 27486.5i −0.755930 1.30931i −0.944911 0.327329i \(-0.893852\pi\)
0.188980 0.981981i \(-0.439482\pi\)
\(762\) 0 0
\(763\) −4400.52 + 7621.93i −0.208794 + 0.361641i
\(764\) 0 0
\(765\) 12823.4 + 6059.09i 0.606051 + 0.286362i
\(766\) 0 0
\(767\) −3593.16 + 6223.53i −0.169154 + 0.292984i
\(768\) 0 0
\(769\) 13354.6 + 23130.9i 0.626242 + 1.08468i 0.988299 + 0.152527i \(0.0487410\pi\)
−0.362058 + 0.932156i \(0.617926\pi\)
\(770\) 0 0
\(771\) 8396.99 9117.13i 0.392231 0.425870i
\(772\) 0 0
\(773\) −26220.5 −1.22003 −0.610017 0.792388i \(-0.708837\pi\)
−0.610017 + 0.792388i \(0.708837\pi\)
\(774\) 0 0
\(775\) 17479.0 0.810150
\(776\) 0 0
\(777\) −12120.9 2719.45i −0.559635 0.125559i
\(778\) 0 0
\(779\) 12142.5 + 21031.4i 0.558471 + 0.967300i
\(780\) 0 0
\(781\) 12455.5 21573.5i 0.570668 0.988426i
\(782\) 0 0
\(783\) −37866.3 + 15347.9i −1.72827 + 0.700498i
\(784\) 0 0
\(785\) 1856.33 3215.26i 0.0844017 0.146188i
\(786\) 0 0
\(787\) −11280.8 19538.9i −0.510949 0.884989i −0.999919 0.0126892i \(-0.995961\pi\)
0.488971 0.872300i \(-0.337373\pi\)
\(788\) 0 0
\(789\) −11186.5 2509.80i −0.504753 0.113246i
\(790\) 0 0
\(791\) 3492.33 0.156982
\(792\) 0 0
\(793\) −26896.0 −1.20442
\(794\) 0 0
\(795\) −13730.4 + 14908.0i −0.612538 + 0.665070i
\(796\) 0 0
\(797\) −5165.99 8947.76i −0.229597 0.397674i 0.728092 0.685480i \(-0.240408\pi\)
−0.957689 + 0.287806i \(0.907074\pi\)
\(798\) 0 0
\(799\) 8227.93 14251.2i 0.364309 0.631002i
\(800\) 0 0
\(801\) 31692.7 21952.5i 1.39801 0.968357i
\(802\) 0 0
\(803\) 14769.1 25580.7i 0.649052 1.12419i
\(804\) 0 0
\(805\) 3941.95 + 6827.66i 0.172591 + 0.298936i
\(806\) 0 0
\(807\) −6330.24 20256.1i −0.276128 0.883581i
\(808\) 0 0
\(809\) 19586.3 0.851194 0.425597 0.904913i \(-0.360064\pi\)
0.425597 + 0.904913i \(0.360064\pi\)
\(810\) 0 0
\(811\) −5057.68 −0.218988 −0.109494 0.993987i \(-0.534923\pi\)
−0.109494 + 0.993987i \(0.534923\pi\)
\(812\) 0 0
\(813\) −4238.78 13563.7i −0.182854 0.585116i
\(814\) 0 0
\(815\) 11453.5 + 19838.0i 0.492266 + 0.852630i
\(816\) 0 0
\(817\) −2154.77 + 3732.18i −0.0922717 + 0.159819i
\(818\) 0 0
\(819\) −6160.54 + 4267.20i −0.262841 + 0.182061i
\(820\) 0 0
\(821\) −7583.11 + 13134.3i −0.322354 + 0.558333i −0.980973 0.194143i \(-0.937807\pi\)
0.658619 + 0.752476i \(0.271141\pi\)
\(822\) 0 0
\(823\) 14526.0 + 25159.7i 0.615242 + 1.06563i 0.990342 + 0.138646i \(0.0442749\pi\)
−0.375100 + 0.926984i \(0.622392\pi\)
\(824\) 0 0
\(825\) −9650.97 + 10478.7i −0.407277 + 0.442206i
\(826\) 0 0
\(827\) −4623.53 −0.194409 −0.0972044 0.995264i \(-0.530990\pi\)
−0.0972044 + 0.995264i \(0.530990\pi\)
\(828\) 0 0
\(829\) −26368.0 −1.10470 −0.552350 0.833612i \(-0.686269\pi\)
−0.552350 + 0.833612i \(0.686269\pi\)
\(830\) 0 0
\(831\) −739.311 165.872i −0.0308621 0.00692422i
\(832\) 0 0
\(833\) 1667.47 + 2888.13i 0.0693568 + 0.120130i
\(834\) 0 0
\(835\) 3555.24 6157.86i 0.147346 0.255212i
\(836\) 0 0
\(837\) −5174.68 + 37118.8i −0.213695 + 1.53287i
\(838\) 0 0
\(839\) 4781.51 8281.81i 0.196753 0.340787i −0.750721 0.660620i \(-0.770294\pi\)
0.947474 + 0.319833i \(0.103627\pi\)
\(840\) 0 0
\(841\) −30213.1 52330.7i −1.23880 2.14567i
\(842\) 0 0
\(843\) 37686.9 + 8455.42i 1.53975 + 0.345457i
\(844\) 0 0
\(845\) 4822.08 0.196313
\(846\) 0 0
\(847\) 2972.36 0.120580
\(848\) 0 0
\(849\) −19158.8 + 20801.9i −0.774474 + 0.840894i
\(850\) 0 0
\(851\) 24918.8 + 43160.6i 1.00377 + 1.73857i
\(852\) 0 0
\(853\) 4274.68 7403.96i 0.171585 0.297194i −0.767389 0.641182i \(-0.778444\pi\)
0.938974 + 0.343987i \(0.111778\pi\)
\(854\) 0 0
\(855\) 19629.0 + 9274.77i 0.785142 + 0.370983i
\(856\) 0 0
\(857\) −969.021 + 1678.39i −0.0386244 + 0.0668995i −0.884691 0.466177i \(-0.845631\pi\)
0.846067 + 0.533077i \(0.178964\pi\)
\(858\) 0 0
\(859\) 14860.6 + 25739.3i 0.590263 + 1.02237i 0.994197 + 0.107578i \(0.0343095\pi\)
−0.403933 + 0.914788i \(0.632357\pi\)
\(860\) 0 0
\(861\) −2529.06 8092.74i −0.100105 0.320325i
\(862\) 0 0
\(863\) 28125.2 1.10938 0.554688 0.832058i \(-0.312838\pi\)
0.554688 + 0.832058i \(0.312838\pi\)
\(864\) 0 0
\(865\) −8103.81 −0.318541
\(866\) 0 0
\(867\) 435.318 + 1392.97i 0.0170521 + 0.0545650i
\(868\) 0 0
\(869\) −12059.6 20887.9i −0.470765 0.815389i
\(870\) 0 0
\(871\) −17581.1 + 30451.4i −0.683941 + 1.18462i
\(872\) 0 0
\(873\) 339.437 + 4120.66i 0.0131595 + 0.159752i
\(874\) 0 0
\(875\) −5144.16 + 8909.95i −0.198748 + 0.344241i
\(876\) 0 0
\(877\) 10763.2 + 18642.3i 0.414419 + 0.717796i 0.995367 0.0961452i \(-0.0306513\pi\)
−0.580948 + 0.813941i \(0.697318\pi\)
\(878\) 0 0
\(879\) −9786.10 + 10625.4i −0.375514 + 0.407719i
\(880\) 0 0
\(881\) −41820.1 −1.59927 −0.799634 0.600488i \(-0.794973\pi\)
−0.799634 + 0.600488i \(0.794973\pi\)
\(882\) 0 0
\(883\) −28659.1 −1.09225 −0.546124 0.837704i \(-0.683897\pi\)
−0.546124 + 0.837704i \(0.683897\pi\)
\(884\) 0 0
\(885\) −7092.08 1591.18i −0.269376 0.0604371i
\(886\) 0 0
\(887\) 813.010 + 1408.18i 0.0307759 + 0.0533054i 0.881003 0.473110i \(-0.156869\pi\)
−0.850227 + 0.526416i \(0.823536\pi\)
\(888\) 0 0
\(889\) −1718.86 + 2977.15i −0.0648466 + 0.112318i
\(890\) 0 0
\(891\) −19395.5 23597.2i −0.729262 0.887244i
\(892\) 0 0
\(893\) 12594.7 21814.6i 0.471964 0.817466i
\(894\) 0 0
\(895\) 5539.22 + 9594.22i 0.206878 + 0.358323i
\(896\) 0 0
\(897\) 29336.6 + 6581.96i 1.09200 + 0.245000i
\(898\) 0 0
\(899\) −77797.3 −2.88619
\(900\) 0 0
\(901\) 34395.5 1.27179
\(902\) 0 0
\(903\) 1019.32 1106.73i 0.0375644 0.0407860i
\(904\) 0 0
\(905\) 3263.61 + 5652.74i 0.119874 + 0.207628i
\(906\) 0 0
\(907\) −12888.7 + 22323.8i −0.471843 + 0.817256i −0.999481 0.0322135i \(-0.989744\pi\)
0.527638 + 0.849469i \(0.323078\pi\)
\(908\) 0 0
\(909\) −2272.32 27585.2i −0.0829132 1.00654i
\(910\) 0 0
\(911\) −7520.47 + 13025.8i −0.273506 + 0.473727i −0.969757 0.244072i \(-0.921517\pi\)
0.696251 + 0.717799i \(0.254850\pi\)
\(912\) 0 0
\(913\) 7136.28 + 12360.4i 0.258682 + 0.448050i
\(914\) 0 0
\(915\) −8114.27 25964.9i −0.293169 0.938111i
\(916\) 0 0
\(917\) −1533.08 −0.0552092
\(918\) 0 0
\(919\) −5117.68 −0.183696 −0.0918480 0.995773i \(-0.529277\pi\)
−0.0918480 + 0.995773i \(0.529277\pi\)
\(920\) 0 0
\(921\) −3819.96 12223.5i −0.136669 0.437326i
\(922\) 0 0
\(923\) −11786.9 20415.6i −0.420338 0.728047i
\(924\) 0 0
\(925\) −11173.3 + 19352.7i −0.397162 + 0.687905i
\(926\) 0 0
\(927\) −42784.0 20215.6i −1.51587 0.716255i
\(928\) 0 0
\(929\) 18761.3 32495.5i 0.662581 1.14762i −0.317353 0.948307i \(-0.602794\pi\)
0.979935 0.199317i \(-0.0638725\pi\)
\(930\) 0 0
\(931\) 2552.42 + 4420.93i 0.0898521 + 0.155628i
\(932\) 0 0
\(933\) −11928.0 + 12951.0i −0.418548 + 0.454443i
\(934\) 0 0
\(935\) −22009.6 −0.769831
\(936\) 0 0
\(937\) −1480.05 −0.0516019 −0.0258010 0.999667i \(-0.508214\pi\)
−0.0258010 + 0.999667i \(0.508214\pi\)
\(938\) 0 0
\(939\) 53623.4 + 12030.9i 1.86362 + 0.418120i
\(940\) 0 0
\(941\) 8643.42 + 14970.8i 0.299434 + 0.518635i 0.976007 0.217741i \(-0.0698688\pi\)
−0.676573 + 0.736376i \(0.736536\pi\)
\(942\) 0 0
\(943\) −17008.1 + 29458.9i −0.587339 + 1.01730i
\(944\) 0 0
\(945\) −5978.04 4659.88i −0.205784 0.160408i
\(946\) 0 0
\(947\) 11234.6 19458.9i 0.385508 0.667720i −0.606331 0.795212i \(-0.707359\pi\)
0.991840 + 0.127492i \(0.0406928\pi\)
\(948\) 0 0
\(949\) −13976.3 24207.7i −0.478073 0.828047i
\(950\) 0 0
\(951\) −47897.7 10746.3i −1.63322 0.366428i
\(952\) 0 0
\(953\) 26138.5 0.888468 0.444234 0.895911i \(-0.353476\pi\)
0.444234 + 0.895911i \(0.353476\pi\)
\(954\) 0 0
\(955\) −19903.2 −0.674400
\(956\) 0 0
\(957\) 42955.4 46639.4i 1.45094 1.57538i
\(958\) 0 0
\(959\) 3716.35 + 6436.91i 0.125138 + 0.216745i
\(960\) 0 0
\(961\) −20784.5 + 35999.9i −0.697678 + 1.20841i
\(962\) 0 0
\(963\) 5493.38 3805.08i 0.183823 0.127328i
\(964\) 0 0
\(965\) 17567.7 30428.1i 0.586035 1.01504i
\(966\) 0 0
\(967\) −19505.6 33784.6i −0.648662 1.12352i −0.983443 0.181220i \(-0.941995\pi\)
0.334780 0.942296i \(-0.391338\pi\)
\(968\) 0 0
\(969\) −10989.8 35166.1i −0.364337 1.16584i
\(970\) 0 0
\(971\) 6211.92 0.205304 0.102652 0.994717i \(-0.467267\pi\)
0.102652 + 0.994717i \(0.467267\pi\)
\(972\) 0 0
\(973\) 998.675 0.0329045
\(974\) 0 0
\(975\) 4021.22 + 12867.5i 0.132084 + 0.422656i
\(976\) 0 0
\(977\) 9656.01 + 16724.7i 0.316195 + 0.547667i 0.979691 0.200514i \(-0.0642611\pi\)
−0.663495 + 0.748180i \(0.730928\pi\)
\(978\) 0 0
\(979\) −29914.5 + 51813.4i −0.976579 + 1.69148i
\(980\) 0 0
\(981\) −27906.2 + 19329.7i −0.908232 + 0.629103i
\(982\) 0 0
\(983\) −7520.24 + 13025.4i −0.244006 + 0.422631i −0.961852 0.273571i \(-0.911795\pi\)
0.717845 + 0.696203i \(0.245128\pi\)
\(984\) 0 0
\(985\) −10813.0 18728.7i −0.349779 0.605835i
\(986\) 0 0
\(987\) −5957.90 + 6468.86i −0.192140 + 0.208618i
\(988\) 0 0
\(989\) −6036.45 −0.194083
\(990\) 0 0
\(991\) −16045.6 −0.514333 −0.257166 0.966367i \(-0.582789\pi\)
−0.257166 + 0.966367i \(0.582789\pi\)
\(992\) 0 0
\(993\) 26809.2 + 6014.91i 0.856762 + 0.192223i
\(994\) 0 0
\(995\) 1336.44 + 2314.78i 0.0425808 + 0.0737521i
\(996\) 0 0
\(997\) 7021.12 12160.9i 0.223030 0.386300i −0.732696 0.680556i \(-0.761738\pi\)
0.955727 + 0.294256i \(0.0950718\pi\)
\(998\) 0 0
\(999\) −37789.8 29457.1i −1.19681 0.932915i
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 252.4.j.b.169.3 yes 18
3.2 odd 2 756.4.j.b.505.3 18
9.2 odd 6 2268.4.a.h.1.7 9
9.4 even 3 inner 252.4.j.b.85.3 18
9.5 odd 6 756.4.j.b.253.3 18
9.7 even 3 2268.4.a.i.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.j.b.85.3 18 9.4 even 3 inner
252.4.j.b.169.3 yes 18 1.1 even 1 trivial
756.4.j.b.253.3 18 9.5 odd 6
756.4.j.b.505.3 18 3.2 odd 2
2268.4.a.h.1.7 9 9.2 odd 6
2268.4.a.i.1.3 9 9.7 even 3