Properties

Label 208.4.w.a.49.1
Level $208$
Weight $4$
Character 208.49
Analytic conductor $12.272$
Analytic rank $0$
Dimension $2$
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] = [208,4,Mod(17,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 208.49
Dual form 208.4.w.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.50000 - 6.06218i) q^{3} -13.8564i q^{5} +(-19.5000 - 11.2583i) q^{7} +(-11.0000 + 19.0526i) q^{9} +O(q^{10})\) \(q+(-3.50000 - 6.06218i) q^{3} -13.8564i q^{5} +(-19.5000 - 11.2583i) q^{7} +(-11.0000 + 19.0526i) q^{9} +(19.5000 - 11.2583i) q^{11} +(-13.0000 - 45.0333i) q^{13} +(-84.0000 + 48.4974i) q^{15} +(-13.5000 + 23.3827i) q^{17} +(76.5000 + 44.1673i) q^{19} +157.617i q^{21} +(28.5000 + 49.3634i) q^{23} -67.0000 q^{25} -35.0000 q^{27} +(34.5000 + 59.7558i) q^{29} -72.7461i q^{31} +(-136.500 - 78.8083i) q^{33} +(-156.000 + 270.200i) q^{35} +(-34.5000 + 19.9186i) q^{37} +(-227.500 + 236.425i) q^{39} +(-340.500 + 196.588i) q^{41} +(-42.5000 + 73.6122i) q^{43} +(264.000 + 152.420i) q^{45} -342.946i q^{47} +(82.0000 + 142.028i) q^{49} +189.000 q^{51} +426.000 q^{53} +(-156.000 - 270.200i) q^{55} -618.342i q^{57} +(16.5000 + 9.52628i) q^{59} +(8.50000 - 14.7224i) q^{61} +(429.000 - 247.683i) q^{63} +(-624.000 + 180.133i) q^{65} +(-142.500 + 82.2724i) q^{67} +(199.500 - 345.544i) q^{69} +(-505.500 - 291.851i) q^{71} -1004.59i q^{73} +(234.500 + 406.166i) q^{75} -507.000 q^{77} +1244.00 q^{79} +(419.500 + 726.595i) q^{81} -426.084i q^{83} +(324.000 + 187.061i) q^{85} +(241.500 - 418.290i) q^{87} +(265.500 - 153.286i) q^{89} +(-253.500 + 1024.51i) q^{91} +(-441.000 + 254.611i) q^{93} +(612.000 - 1060.02i) q^{95} +(1069.50 + 617.476i) q^{97} +495.367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9} + 39 q^{11} - 26 q^{13} - 168 q^{15} - 27 q^{17} + 153 q^{19} + 57 q^{23} - 134 q^{25} - 70 q^{27} + 69 q^{29} - 273 q^{33} - 312 q^{35} - 69 q^{37} - 455 q^{39} - 681 q^{41} - 85 q^{43} + 528 q^{45} + 164 q^{49} + 378 q^{51} + 852 q^{53} - 312 q^{55} + 33 q^{59} + 17 q^{61} + 858 q^{63} - 1248 q^{65} - 285 q^{67} + 399 q^{69} - 1011 q^{71} + 469 q^{75} - 1014 q^{77} + 2488 q^{79} + 839 q^{81} + 648 q^{85} + 483 q^{87} + 531 q^{89} - 507 q^{91} - 882 q^{93} + 1224 q^{95} + 2139 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −3.50000 6.06218i −0.673575 1.16667i −0.976883 0.213774i \(-0.931424\pi\)
0.303308 0.952893i \(-0.401909\pi\)
\(4\) 0 0
\(5\) 13.8564i 1.23935i −0.784857 0.619677i \(-0.787263\pi\)
0.784857 0.619677i \(-0.212737\pi\)
\(6\) 0 0
\(7\) −19.5000 11.2583i −1.05290 0.607893i −0.129441 0.991587i \(-0.541318\pi\)
−0.923460 + 0.383694i \(0.874652\pi\)
\(8\) 0 0
\(9\) −11.0000 + 19.0526i −0.407407 + 0.705650i
\(10\) 0 0
\(11\) 19.5000 11.2583i 0.534497 0.308592i −0.208349 0.978055i \(-0.566809\pi\)
0.742846 + 0.669462i \(0.233475\pi\)
\(12\) 0 0
\(13\) −13.0000 45.0333i −0.277350 0.960769i
\(14\) 0 0
\(15\) −84.0000 + 48.4974i −1.44591 + 0.834799i
\(16\) 0 0
\(17\) −13.5000 + 23.3827i −0.192602 + 0.333596i −0.946112 0.323840i \(-0.895026\pi\)
0.753510 + 0.657437i \(0.228359\pi\)
\(18\) 0 0
\(19\) 76.5000 + 44.1673i 0.923700 + 0.533299i 0.884814 0.465945i \(-0.154286\pi\)
0.0388865 + 0.999244i \(0.487619\pi\)
\(20\) 0 0
\(21\) 157.617i 1.63785i
\(22\) 0 0
\(23\) 28.5000 + 49.3634i 0.258377 + 0.447521i 0.965807 0.259261i \(-0.0834791\pi\)
−0.707431 + 0.706783i \(0.750146\pi\)
\(24\) 0 0
\(25\) −67.0000 −0.536000
\(26\) 0 0
\(27\) −35.0000 −0.249472
\(28\) 0 0
\(29\) 34.5000 + 59.7558i 0.220913 + 0.382633i 0.955086 0.296330i \(-0.0957628\pi\)
−0.734172 + 0.678963i \(0.762430\pi\)
\(30\) 0 0
\(31\) 72.7461i 0.421471i −0.977543 0.210735i \(-0.932414\pi\)
0.977543 0.210735i \(-0.0675858\pi\)
\(32\) 0 0
\(33\) −136.500 78.8083i −0.720048 0.415720i
\(34\) 0 0
\(35\) −156.000 + 270.200i −0.753395 + 1.30492i
\(36\) 0 0
\(37\) −34.5000 + 19.9186i −0.153291 + 0.0885026i −0.574683 0.818376i \(-0.694875\pi\)
0.421393 + 0.906878i \(0.361541\pi\)
\(38\) 0 0
\(39\) −227.500 + 236.425i −0.934081 + 0.970725i
\(40\) 0 0
\(41\) −340.500 + 196.588i −1.29700 + 0.748826i −0.979886 0.199560i \(-0.936049\pi\)
−0.317118 + 0.948386i \(0.602715\pi\)
\(42\) 0 0
\(43\) −42.5000 + 73.6122i −0.150725 + 0.261064i −0.931494 0.363756i \(-0.881494\pi\)
0.780769 + 0.624820i \(0.214828\pi\)
\(44\) 0 0
\(45\) 264.000 + 152.420i 0.874551 + 0.504922i
\(46\) 0 0
\(47\) 342.946i 1.06434i −0.846639 0.532168i \(-0.821377\pi\)
0.846639 0.532168i \(-0.178623\pi\)
\(48\) 0 0
\(49\) 82.0000 + 142.028i 0.239067 + 0.414076i
\(50\) 0 0
\(51\) 189.000 0.518927
\(52\) 0 0
\(53\) 426.000 1.10407 0.552034 0.833822i \(-0.313852\pi\)
0.552034 + 0.833822i \(0.313852\pi\)
\(54\) 0 0
\(55\) −156.000 270.200i −0.382455 0.662432i
\(56\) 0 0
\(57\) 618.342i 1.43687i
\(58\) 0 0
\(59\) 16.5000 + 9.52628i 0.0364088 + 0.0210206i 0.518094 0.855324i \(-0.326642\pi\)
−0.481685 + 0.876344i \(0.659975\pi\)
\(60\) 0 0
\(61\) 8.50000 14.7224i 0.0178412 0.0309019i −0.856967 0.515371i \(-0.827654\pi\)
0.874808 + 0.484469i \(0.160987\pi\)
\(62\) 0 0
\(63\) 429.000 247.683i 0.857919 0.495320i
\(64\) 0 0
\(65\) −624.000 + 180.133i −1.19073 + 0.343735i
\(66\) 0 0
\(67\) −142.500 + 82.2724i −0.259838 + 0.150018i −0.624261 0.781216i \(-0.714600\pi\)
0.364423 + 0.931234i \(0.381266\pi\)
\(68\) 0 0
\(69\) 199.500 345.544i 0.348072 0.602879i
\(70\) 0 0
\(71\) −505.500 291.851i −0.844955 0.487835i 0.0139904 0.999902i \(-0.495547\pi\)
−0.858945 + 0.512067i \(0.828880\pi\)
\(72\) 0 0
\(73\) 1004.59i 1.61066i −0.592826 0.805331i \(-0.701988\pi\)
0.592826 0.805331i \(-0.298012\pi\)
\(74\) 0 0
\(75\) 234.500 + 406.166i 0.361036 + 0.625333i
\(76\) 0 0
\(77\) −507.000 −0.750364
\(78\) 0 0
\(79\) 1244.00 1.77166 0.885829 0.464012i \(-0.153591\pi\)
0.885829 + 0.464012i \(0.153591\pi\)
\(80\) 0 0
\(81\) 419.500 + 726.595i 0.575446 + 0.996701i
\(82\) 0 0
\(83\) 426.084i 0.563480i −0.959491 0.281740i \(-0.909088\pi\)
0.959491 0.281740i \(-0.0909116\pi\)
\(84\) 0 0
\(85\) 324.000 + 187.061i 0.413444 + 0.238702i
\(86\) 0 0
\(87\) 241.500 418.290i 0.297604 0.515465i
\(88\) 0 0
\(89\) 265.500 153.286i 0.316213 0.182566i −0.333490 0.942753i \(-0.608226\pi\)
0.649703 + 0.760188i \(0.274893\pi\)
\(90\) 0 0
\(91\) −253.500 + 1024.51i −0.292022 + 1.18019i
\(92\) 0 0
\(93\) −441.000 + 254.611i −0.491716 + 0.283892i
\(94\) 0 0
\(95\) 612.000 1060.02i 0.660946 1.14479i
\(96\) 0 0
\(97\) 1069.50 + 617.476i 1.11950 + 0.646342i 0.941273 0.337647i \(-0.109631\pi\)
0.178225 + 0.983990i \(0.442965\pi\)
\(98\) 0 0
\(99\) 495.367i 0.502891i
\(100\) 0 0
\(101\) −979.500 1696.54i −0.964989 1.67141i −0.709645 0.704560i \(-0.751144\pi\)
−0.255345 0.966850i \(-0.582189\pi\)
\(102\) 0 0
\(103\) −1856.00 −1.77551 −0.887753 0.460320i \(-0.847735\pi\)
−0.887753 + 0.460320i \(0.847735\pi\)
\(104\) 0 0
\(105\) 2184.00 2.02987
\(106\) 0 0
\(107\) −127.500 220.836i −0.115195 0.199524i 0.802663 0.596433i \(-0.203416\pi\)
−0.917858 + 0.396909i \(0.870083\pi\)
\(108\) 0 0
\(109\) 609.682i 0.535752i 0.963453 + 0.267876i \(0.0863217\pi\)
−0.963453 + 0.267876i \(0.913678\pi\)
\(110\) 0 0
\(111\) 241.500 + 139.430i 0.206506 + 0.119226i
\(112\) 0 0
\(113\) −205.500 + 355.936i −0.171078 + 0.296316i −0.938797 0.344471i \(-0.888058\pi\)
0.767719 + 0.640787i \(0.221392\pi\)
\(114\) 0 0
\(115\) 684.000 394.908i 0.554638 0.320220i
\(116\) 0 0
\(117\) 1001.00 + 247.683i 0.790961 + 0.195712i
\(118\) 0 0
\(119\) 526.500 303.975i 0.405581 0.234162i
\(120\) 0 0
\(121\) −412.000 + 713.605i −0.309542 + 0.536142i
\(122\) 0 0
\(123\) 2383.50 + 1376.11i 1.74726 + 1.00878i
\(124\) 0 0
\(125\) 803.672i 0.575061i
\(126\) 0 0
\(127\) −1121.50 1942.49i −0.783599 1.35723i −0.929833 0.367983i \(-0.880049\pi\)
0.146234 0.989250i \(-0.453285\pi\)
\(128\) 0 0
\(129\) 595.000 0.406099
\(130\) 0 0
\(131\) 372.000 0.248105 0.124053 0.992276i \(-0.460411\pi\)
0.124053 + 0.992276i \(0.460411\pi\)
\(132\) 0 0
\(133\) −994.500 1722.52i −0.648377 1.12302i
\(134\) 0 0
\(135\) 484.974i 0.309185i
\(136\) 0 0
\(137\) −1030.50 594.959i −0.642639 0.371028i 0.142991 0.989724i \(-0.454328\pi\)
−0.785630 + 0.618696i \(0.787661\pi\)
\(138\) 0 0
\(139\) −1272.50 + 2204.03i −0.776490 + 1.34492i 0.157464 + 0.987525i \(0.449668\pi\)
−0.933953 + 0.357395i \(0.883665\pi\)
\(140\) 0 0
\(141\) −2079.00 + 1200.31i −1.24173 + 0.716911i
\(142\) 0 0
\(143\) −760.500 731.791i −0.444729 0.427940i
\(144\) 0 0
\(145\) 828.000 478.046i 0.474218 0.273790i
\(146\) 0 0
\(147\) 574.000 994.197i 0.322059 0.557823i
\(148\) 0 0
\(149\) 1129.50 + 652.117i 0.621022 + 0.358547i 0.777267 0.629171i \(-0.216606\pi\)
−0.156245 + 0.987718i \(0.549939\pi\)
\(150\) 0 0
\(151\) 86.6025i 0.0466729i −0.999728 0.0233365i \(-0.992571\pi\)
0.999728 0.0233365i \(-0.00742890\pi\)
\(152\) 0 0
\(153\) −297.000 514.419i −0.156935 0.271819i
\(154\) 0 0
\(155\) −1008.00 −0.522352
\(156\) 0 0
\(157\) −1534.00 −0.779787 −0.389893 0.920860i \(-0.627488\pi\)
−0.389893 + 0.920860i \(0.627488\pi\)
\(158\) 0 0
\(159\) −1491.00 2582.49i −0.743673 1.28808i
\(160\) 0 0
\(161\) 1283.45i 0.628261i
\(162\) 0 0
\(163\) 1414.50 + 816.662i 0.679707 + 0.392429i 0.799745 0.600340i \(-0.204968\pi\)
−0.120038 + 0.992769i \(0.538302\pi\)
\(164\) 0 0
\(165\) −1092.00 + 1891.40i −0.515225 + 0.892395i
\(166\) 0 0
\(167\) −1408.50 + 813.198i −0.652653 + 0.376809i −0.789472 0.613787i \(-0.789645\pi\)
0.136819 + 0.990596i \(0.456312\pi\)
\(168\) 0 0
\(169\) −1859.00 + 1170.87i −0.846154 + 0.532939i
\(170\) 0 0
\(171\) −1683.00 + 971.681i −0.752645 + 0.434540i
\(172\) 0 0
\(173\) 436.500 756.040i 0.191829 0.332258i −0.754027 0.656843i \(-0.771891\pi\)
0.945857 + 0.324585i \(0.105225\pi\)
\(174\) 0 0
\(175\) 1306.50 + 754.308i 0.564355 + 0.325830i
\(176\) 0 0
\(177\) 133.368i 0.0566359i
\(178\) 0 0
\(179\) −643.500 1114.57i −0.268701 0.465403i 0.699826 0.714314i \(-0.253261\pi\)
−0.968527 + 0.248910i \(0.919928\pi\)
\(180\) 0 0
\(181\) 2.00000 0.000821319 0.000410660 1.00000i \(-0.499869\pi\)
0.000410660 1.00000i \(0.499869\pi\)
\(182\) 0 0
\(183\) −119.000 −0.0480696
\(184\) 0 0
\(185\) 276.000 + 478.046i 0.109686 + 0.189982i
\(186\) 0 0
\(187\) 607.950i 0.237742i
\(188\) 0 0
\(189\) 682.500 + 394.042i 0.262670 + 0.151652i
\(190\) 0 0
\(191\) −1420.50 + 2460.38i −0.538135 + 0.932077i 0.460870 + 0.887468i \(0.347538\pi\)
−0.999005 + 0.0446092i \(0.985796\pi\)
\(192\) 0 0
\(193\) −3676.50 + 2122.63i −1.37119 + 0.791659i −0.991078 0.133281i \(-0.957449\pi\)
−0.380115 + 0.924939i \(0.624115\pi\)
\(194\) 0 0
\(195\) 3276.00 + 3152.33i 1.20307 + 1.15766i
\(196\) 0 0
\(197\) 2383.50 1376.11i 0.862017 0.497686i −0.00267023 0.999996i \(-0.500850\pi\)
0.864687 + 0.502311i \(0.167517\pi\)
\(198\) 0 0
\(199\) −842.500 + 1459.25i −0.300117 + 0.519818i −0.976162 0.217042i \(-0.930359\pi\)
0.676045 + 0.736860i \(0.263692\pi\)
\(200\) 0 0
\(201\) 997.500 + 575.907i 0.350041 + 0.202096i
\(202\) 0 0
\(203\) 1553.65i 0.537167i
\(204\) 0 0
\(205\) 2724.00 + 4718.11i 0.928061 + 1.60745i
\(206\) 0 0
\(207\) −1254.00 −0.421058
\(208\) 0 0
\(209\) 1989.00 0.658287
\(210\) 0 0
\(211\) 840.500 + 1455.79i 0.274229 + 0.474979i 0.969940 0.243343i \(-0.0782439\pi\)
−0.695711 + 0.718322i \(0.744911\pi\)
\(212\) 0 0
\(213\) 4085.91i 1.31437i
\(214\) 0 0
\(215\) 1020.00 + 588.897i 0.323551 + 0.186802i
\(216\) 0 0
\(217\) −819.000 + 1418.55i −0.256209 + 0.443767i
\(218\) 0 0
\(219\) −6090.00 + 3516.06i −1.87911 + 1.08490i
\(220\) 0 0
\(221\) 1228.50 + 303.975i 0.373927 + 0.0925229i
\(222\) 0 0
\(223\) 3547.50 2048.15i 1.06528 0.615042i 0.138394 0.990377i \(-0.455806\pi\)
0.926889 + 0.375336i \(0.122473\pi\)
\(224\) 0 0
\(225\) 737.000 1276.52i 0.218370 0.378229i
\(226\) 0 0
\(227\) −379.500 219.104i −0.110962 0.0640638i 0.443492 0.896278i \(-0.353739\pi\)
−0.554454 + 0.832215i \(0.687073\pi\)
\(228\) 0 0
\(229\) 180.133i 0.0519805i −0.999662 0.0259903i \(-0.991726\pi\)
0.999662 0.0259903i \(-0.00827389\pi\)
\(230\) 0 0
\(231\) 1774.50 + 3073.52i 0.505427 + 0.875424i
\(232\) 0 0
\(233\) −5778.00 −1.62459 −0.812295 0.583247i \(-0.801782\pi\)
−0.812295 + 0.583247i \(0.801782\pi\)
\(234\) 0 0
\(235\) −4752.00 −1.31909
\(236\) 0 0
\(237\) −4354.00 7541.35i −1.19334 2.06693i
\(238\) 0 0
\(239\) 1860.22i 0.503464i −0.967797 0.251732i \(-0.919000\pi\)
0.967797 0.251732i \(-0.0810001\pi\)
\(240\) 0 0
\(241\) 1783.50 + 1029.70i 0.476703 + 0.275224i 0.719041 0.694967i \(-0.244581\pi\)
−0.242339 + 0.970192i \(0.577915\pi\)
\(242\) 0 0
\(243\) 2464.00 4267.77i 0.650476 1.12666i
\(244\) 0 0
\(245\) 1968.00 1136.23i 0.513187 0.296289i
\(246\) 0 0
\(247\) 994.500 4019.22i 0.256188 1.03537i
\(248\) 0 0
\(249\) −2583.00 + 1491.30i −0.657393 + 0.379546i
\(250\) 0 0
\(251\) 2245.50 3889.32i 0.564680 0.978055i −0.432399 0.901682i \(-0.642333\pi\)
0.997079 0.0763724i \(-0.0243338\pi\)
\(252\) 0 0
\(253\) 1111.50 + 641.725i 0.276203 + 0.159466i
\(254\) 0 0
\(255\) 2618.86i 0.643135i
\(256\) 0 0
\(257\) −2725.50 4720.70i −0.661525 1.14580i −0.980215 0.197936i \(-0.936576\pi\)
0.318690 0.947859i \(-0.396757\pi\)
\(258\) 0 0
\(259\) 897.000 0.215200
\(260\) 0 0
\(261\) −1518.00 −0.360007
\(262\) 0 0
\(263\) −391.500 678.098i −0.0917906 0.158986i 0.816474 0.577382i \(-0.195926\pi\)
−0.908265 + 0.418396i \(0.862592\pi\)
\(264\) 0 0
\(265\) 5902.83i 1.36833i
\(266\) 0 0
\(267\) −1858.50 1073.01i −0.425986 0.245943i
\(268\) 0 0
\(269\) 2542.50 4403.74i 0.576279 0.998144i −0.419623 0.907699i \(-0.637838\pi\)
0.995901 0.0904453i \(-0.0288290\pi\)
\(270\) 0 0
\(271\) 1147.50 662.509i 0.257216 0.148504i −0.365848 0.930675i \(-0.619221\pi\)
0.623064 + 0.782171i \(0.285888\pi\)
\(272\) 0 0
\(273\) 7098.00 2049.02i 1.57359 0.454257i
\(274\) 0 0
\(275\) −1306.50 + 754.308i −0.286491 + 0.165405i
\(276\) 0 0
\(277\) 1710.50 2962.67i 0.371025 0.642635i −0.618698 0.785629i \(-0.712340\pi\)
0.989724 + 0.142994i \(0.0456730\pi\)
\(278\) 0 0
\(279\) 1386.00 + 800.207i 0.297411 + 0.171710i
\(280\) 0 0
\(281\) 810.600i 0.172087i −0.996291 0.0860433i \(-0.972578\pi\)
0.996291 0.0860433i \(-0.0274223\pi\)
\(282\) 0 0
\(283\) 3588.50 + 6215.46i 0.753760 + 1.30555i 0.945988 + 0.324201i \(0.105095\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(284\) 0 0
\(285\) −8568.00 −1.78079
\(286\) 0 0
\(287\) 8853.00 1.82082
\(288\) 0 0
\(289\) 2092.00 + 3623.45i 0.425809 + 0.737523i
\(290\) 0 0
\(291\) 8644.67i 1.74144i
\(292\) 0 0
\(293\) 8065.50 + 4656.62i 1.60816 + 0.928473i 0.989781 + 0.142595i \(0.0455445\pi\)
0.618381 + 0.785878i \(0.287789\pi\)
\(294\) 0 0
\(295\) 132.000 228.631i 0.0260520 0.0451234i
\(296\) 0 0
\(297\) −682.500 + 394.042i −0.133342 + 0.0769852i
\(298\) 0 0
\(299\) 1852.50 1925.17i 0.358304 0.372360i
\(300\) 0 0
\(301\) 1657.50 956.958i 0.317398 0.183250i
\(302\) 0 0
\(303\) −6856.50 + 11875.8i −1.29999 + 2.25164i
\(304\) 0 0
\(305\) −204.000 117.779i −0.0382984 0.0221116i
\(306\) 0 0
\(307\) 4777.00i 0.888070i −0.896009 0.444035i \(-0.853547\pi\)
0.896009 0.444035i \(-0.146453\pi\)
\(308\) 0 0
\(309\) 6496.00 + 11251.4i 1.19594 + 2.07142i
\(310\) 0 0
\(311\) −6192.00 −1.12899 −0.564495 0.825436i \(-0.690929\pi\)
−0.564495 + 0.825436i \(0.690929\pi\)
\(312\) 0 0
\(313\) −770.000 −0.139051 −0.0695255 0.997580i \(-0.522149\pi\)
−0.0695255 + 0.997580i \(0.522149\pi\)
\(314\) 0 0
\(315\) −3432.00 5944.40i −0.613877 1.06327i
\(316\) 0 0
\(317\) 8057.50i 1.42762i −0.700341 0.713808i \(-0.746969\pi\)
0.700341 0.713808i \(-0.253031\pi\)
\(318\) 0 0
\(319\) 1345.50 + 776.825i 0.236155 + 0.136344i
\(320\) 0 0
\(321\) −892.500 + 1545.86i −0.155185 + 0.268789i
\(322\) 0 0
\(323\) −2065.50 + 1192.52i −0.355813 + 0.205429i
\(324\) 0 0
\(325\) 871.000 + 3017.23i 0.148660 + 0.514972i
\(326\) 0 0
\(327\) 3696.00 2133.89i 0.625044 0.360869i
\(328\) 0 0
\(329\) −3861.00 + 6687.45i −0.647002 + 1.12064i
\(330\) 0 0
\(331\) 4570.50 + 2638.78i 0.758965 + 0.438189i 0.828924 0.559361i \(-0.188954\pi\)
−0.0699590 + 0.997550i \(0.522287\pi\)
\(332\) 0 0
\(333\) 876.418i 0.144226i
\(334\) 0 0
\(335\) 1140.00 + 1974.54i 0.185925 + 0.322031i
\(336\) 0 0
\(337\) 8278.00 1.33808 0.669038 0.743228i \(-0.266706\pi\)
0.669038 + 0.743228i \(0.266706\pi\)
\(338\) 0 0
\(339\) 2877.00 0.460936
\(340\) 0 0
\(341\) −819.000 1418.55i −0.130063 0.225275i
\(342\) 0 0
\(343\) 4030.48i 0.634477i
\(344\) 0 0
\(345\) −4788.00 2764.35i −0.747180 0.431385i
\(346\) 0 0
\(347\) 3433.50 5947.00i 0.531181 0.920033i −0.468156 0.883646i \(-0.655082\pi\)
0.999338 0.0363875i \(-0.0115851\pi\)
\(348\) 0 0
\(349\) 10525.5 6076.90i 1.61438 0.932060i 0.626036 0.779794i \(-0.284676\pi\)
0.988340 0.152266i \(-0.0486571\pi\)
\(350\) 0 0
\(351\) 455.000 + 1576.17i 0.0691912 + 0.239685i
\(352\) 0 0
\(353\) 5029.50 2903.78i 0.758338 0.437827i −0.0703608 0.997522i \(-0.522415\pi\)
0.828699 + 0.559695i \(0.189082\pi\)
\(354\) 0 0
\(355\) −4044.00 + 7004.41i −0.604601 + 1.04720i
\(356\) 0 0
\(357\) −3685.50 2127.82i −0.546379 0.315452i
\(358\) 0 0
\(359\) 1340.61i 0.197088i 0.995133 + 0.0985439i \(0.0314185\pi\)
−0.995133 + 0.0985439i \(0.968581\pi\)
\(360\) 0 0
\(361\) 472.000 + 817.528i 0.0688147 + 0.119191i
\(362\) 0 0
\(363\) 5768.00 0.833999
\(364\) 0 0
\(365\) −13920.0 −1.99618
\(366\) 0 0
\(367\) 1832.50 + 3173.98i 0.260642 + 0.451446i 0.966413 0.256995i \(-0.0827324\pi\)
−0.705770 + 0.708441i \(0.749399\pi\)
\(368\) 0 0
\(369\) 8649.86i 1.22031i
\(370\) 0 0
\(371\) −8307.00 4796.05i −1.16247 0.671155i
\(372\) 0 0
\(373\) −2685.50 + 4651.42i −0.372788 + 0.645688i −0.989993 0.141114i \(-0.954931\pi\)
0.617205 + 0.786802i \(0.288265\pi\)
\(374\) 0 0
\(375\) −4872.00 + 2812.85i −0.670904 + 0.387347i
\(376\) 0 0
\(377\) 2242.50 2330.47i 0.306352 0.318370i
\(378\) 0 0
\(379\) 9967.50 5754.74i 1.35091 0.779950i 0.362536 0.931970i \(-0.381911\pi\)
0.988377 + 0.152020i \(0.0485778\pi\)
\(380\) 0 0
\(381\) −7850.50 + 13597.5i −1.05563 + 1.82840i
\(382\) 0 0
\(383\) −2095.50 1209.84i −0.279569 0.161409i 0.353659 0.935374i \(-0.384937\pi\)
−0.633228 + 0.773965i \(0.718271\pi\)
\(384\) 0 0
\(385\) 7025.20i 0.929967i
\(386\) 0 0
\(387\) −935.000 1619.47i −0.122813 0.212719i
\(388\) 0 0
\(389\) −9858.00 −1.28489 −0.642443 0.766334i \(-0.722079\pi\)
−0.642443 + 0.766334i \(0.722079\pi\)
\(390\) 0 0
\(391\) −1539.00 −0.199055
\(392\) 0 0
\(393\) −1302.00 2255.13i −0.167118 0.289456i
\(394\) 0 0
\(395\) 17237.4i 2.19571i
\(396\) 0 0
\(397\) −7552.50 4360.44i −0.954784 0.551245i −0.0602200 0.998185i \(-0.519180\pi\)
−0.894564 + 0.446941i \(0.852514\pi\)
\(398\) 0 0
\(399\) −6961.50 + 12057.7i −0.873461 + 1.51288i
\(400\) 0 0
\(401\) −6568.50 + 3792.33i −0.817993 + 0.472269i −0.849724 0.527228i \(-0.823231\pi\)
0.0317308 + 0.999496i \(0.489898\pi\)
\(402\) 0 0
\(403\) −3276.00 + 945.700i −0.404936 + 0.116895i
\(404\) 0 0
\(405\) 10068.0 5812.76i 1.23527 0.713181i
\(406\) 0 0
\(407\) −448.500 + 776.825i −0.0546224 + 0.0946088i
\(408\) 0 0
\(409\) 3727.50 + 2152.07i 0.450643 + 0.260179i 0.708102 0.706110i \(-0.249552\pi\)
−0.257459 + 0.966289i \(0.582885\pi\)
\(410\) 0 0
\(411\) 8329.43i 0.999661i
\(412\) 0 0
\(413\) −214.500 371.525i −0.0255565 0.0442652i
\(414\) 0 0
\(415\) −5904.00 −0.698352
\(416\) 0 0
\(417\) 17815.0 2.09210
\(418\) 0 0
\(419\) 2698.50 + 4673.94i 0.314631 + 0.544957i 0.979359 0.202129i \(-0.0647859\pi\)
−0.664728 + 0.747085i \(0.731453\pi\)
\(420\) 0 0
\(421\) 7260.76i 0.840541i −0.907399 0.420270i \(-0.861935\pi\)
0.907399 0.420270i \(-0.138065\pi\)
\(422\) 0 0
\(423\) 6534.00 + 3772.41i 0.751050 + 0.433619i
\(424\) 0 0
\(425\) 904.500 1566.64i 0.103235 0.178808i
\(426\) 0 0
\(427\) −331.500 + 191.392i −0.0375700 + 0.0216911i
\(428\) 0 0
\(429\) −1774.50 + 7171.56i −0.199706 + 0.807100i
\(430\) 0 0
\(431\) 421.500 243.353i 0.0471066 0.0271970i −0.476262 0.879304i \(-0.658009\pi\)
0.523368 + 0.852107i \(0.324675\pi\)
\(432\) 0 0
\(433\) −6069.50 + 10512.7i −0.673629 + 1.16676i 0.303238 + 0.952915i \(0.401932\pi\)
−0.976867 + 0.213846i \(0.931401\pi\)
\(434\) 0 0
\(435\) −5796.00 3346.32i −0.638844 0.368836i
\(436\) 0 0
\(437\) 5035.07i 0.551167i
\(438\) 0 0
\(439\) 230.500 + 399.238i 0.0250596 + 0.0434045i 0.878283 0.478141i \(-0.158689\pi\)
−0.853224 + 0.521545i \(0.825356\pi\)
\(440\) 0 0
\(441\) −3608.00 −0.389591
\(442\) 0 0
\(443\) −12156.0 −1.30372 −0.651861 0.758338i \(-0.726012\pi\)
−0.651861 + 0.758338i \(0.726012\pi\)
\(444\) 0 0
\(445\) −2124.00 3678.88i −0.226263 0.391900i
\(446\) 0 0
\(447\) 9129.64i 0.966034i
\(448\) 0 0
\(449\) −256.500 148.090i −0.0269599 0.0155653i 0.486459 0.873703i \(-0.338288\pi\)
−0.513419 + 0.858138i \(0.671621\pi\)
\(450\) 0 0
\(451\) −4426.50 + 7666.92i −0.462164 + 0.800491i
\(452\) 0 0
\(453\) −525.000 + 303.109i −0.0544518 + 0.0314377i
\(454\) 0 0
\(455\) 14196.0 + 3512.60i 1.46268 + 0.361919i
\(456\) 0 0
\(457\) 529.500 305.707i 0.0541990 0.0312918i −0.472656 0.881247i \(-0.656705\pi\)
0.526855 + 0.849955i \(0.323371\pi\)
\(458\) 0 0
\(459\) 472.500 818.394i 0.0480488 0.0832230i
\(460\) 0 0
\(461\) −11368.5 6563.61i −1.14855 0.663119i −0.200020 0.979792i \(-0.564101\pi\)
−0.948535 + 0.316673i \(0.897434\pi\)
\(462\) 0 0
\(463\) 834.848i 0.0837985i −0.999122 0.0418992i \(-0.986659\pi\)
0.999122 0.0418992i \(-0.0133408\pi\)
\(464\) 0 0
\(465\) 3528.00 + 6110.68i 0.351843 + 0.609410i
\(466\) 0 0
\(467\) −14496.0 −1.43639 −0.718196 0.695841i \(-0.755032\pi\)
−0.718196 + 0.695841i \(0.755032\pi\)
\(468\) 0 0
\(469\) 3705.00 0.364778
\(470\) 0 0
\(471\) 5369.00 + 9299.38i 0.525245 + 0.909751i
\(472\) 0 0
\(473\) 1913.92i 0.186051i
\(474\) 0 0
\(475\) −5125.50 2959.21i −0.495103 0.285848i
\(476\) 0 0
\(477\) −4686.00 + 8116.39i −0.449805 + 0.779086i
\(478\) 0 0
\(479\) 7705.50 4448.77i 0.735017 0.424362i −0.0852376 0.996361i \(-0.527165\pi\)
0.820255 + 0.571998i \(0.193832\pi\)
\(480\) 0 0
\(481\) 1345.50 + 1294.71i 0.127546 + 0.122731i
\(482\) 0 0
\(483\) −7780.50 + 4492.07i −0.732971 + 0.423181i
\(484\) 0 0
\(485\) 8556.00 14819.4i 0.801047 1.38745i
\(486\) 0 0
\(487\) −4117.50 2377.24i −0.383125 0.221197i 0.296052 0.955172i \(-0.404330\pi\)
−0.679177 + 0.733975i \(0.737663\pi\)
\(488\) 0 0
\(489\) 11433.3i 1.05732i
\(490\) 0 0
\(491\) −817.500 1415.95i −0.0751390 0.130145i 0.826008 0.563659i \(-0.190607\pi\)
−0.901147 + 0.433514i \(0.857273\pi\)
\(492\) 0 0
\(493\) −1863.00 −0.170193
\(494\) 0 0
\(495\) 6864.00 0.623260
\(496\) 0 0
\(497\) 6571.50 + 11382.2i 0.593103 + 1.02728i
\(498\) 0 0
\(499\) 14434.9i 1.29498i −0.762074 0.647490i \(-0.775819\pi\)
0.762074 0.647490i \(-0.224181\pi\)
\(500\) 0 0
\(501\) 9859.50 + 5692.38i 0.879222 + 0.507619i
\(502\) 0 0
\(503\) 6343.50 10987.3i 0.562312 0.973952i −0.434983 0.900439i \(-0.643246\pi\)
0.997294 0.0735133i \(-0.0234211\pi\)
\(504\) 0 0
\(505\) −23508.0 + 13572.4i −2.07147 + 1.19596i
\(506\) 0 0
\(507\) 13604.5 + 7171.56i 1.19171 + 0.628205i
\(508\) 0 0
\(509\) −4978.50 + 2874.34i −0.433533 + 0.250300i −0.700850 0.713308i \(-0.747196\pi\)
0.267318 + 0.963608i \(0.413863\pi\)
\(510\) 0 0
\(511\) −11310.0 + 19589.5i −0.979109 + 1.69587i
\(512\) 0 0
\(513\) −2677.50 1545.86i −0.230438 0.133043i
\(514\) 0 0
\(515\) 25717.5i 2.20048i
\(516\) 0 0
\(517\) −3861.00 6687.45i −0.328446 0.568885i
\(518\) 0 0
\(519\) −6111.00 −0.516846
\(520\) 0 0
\(521\) 6054.00 0.509080 0.254540 0.967062i \(-0.418076\pi\)
0.254540 + 0.967062i \(0.418076\pi\)
\(522\) 0 0
\(523\) −7401.50 12819.8i −0.618824 1.07183i −0.989701 0.143153i \(-0.954276\pi\)
0.370877 0.928682i \(-0.379057\pi\)
\(524\) 0 0
\(525\) 10560.3i 0.877885i
\(526\) 0 0
\(527\) 1701.00 + 982.073i 0.140601 + 0.0811760i
\(528\) 0 0
\(529\) 4459.00 7723.21i 0.366483 0.634767i
\(530\) 0 0
\(531\) −363.000 + 209.578i −0.0296664 + 0.0171279i
\(532\) 0 0
\(533\) 13279.5 + 12778.2i 1.07917 + 1.03843i
\(534\) 0 0
\(535\) −3060.00 + 1766.69i −0.247281 + 0.142768i
\(536\) 0 0
\(537\) −4504.50 + 7802.02i −0.361980 + 0.626969i
\(538\) 0 0
\(539\) 3198.00 + 1846.37i 0.255561 + 0.147548i
\(540\) 0 0
\(541\) 21470.5i 1.70626i 0.521695 + 0.853132i \(0.325300\pi\)
−0.521695 + 0.853132i \(0.674700\pi\)
\(542\) 0 0
\(543\) −7.00000 12.1244i −0.000553221 0.000958206i
\(544\) 0 0
\(545\) 8448.00 0.663986
\(546\) 0 0
\(547\) 13516.0 1.05649 0.528247 0.849091i \(-0.322849\pi\)
0.528247 + 0.849091i \(0.322849\pi\)
\(548\) 0 0
\(549\) 187.000 + 323.894i 0.0145373 + 0.0251793i
\(550\) 0 0
\(551\) 6095.09i 0.471251i
\(552\) 0 0
\(553\) −24258.0 14005.4i −1.86538 1.07698i
\(554\) 0 0
\(555\) 1932.00 3346.32i 0.147764 0.255934i
\(556\) 0 0
\(557\) 2503.50 1445.40i 0.190443 0.109952i −0.401747 0.915751i \(-0.631597\pi\)
0.592190 + 0.805798i \(0.298264\pi\)
\(558\) 0 0
\(559\) 3867.50 + 956.958i 0.292626 + 0.0724061i
\(560\) 0 0
\(561\) 3685.50 2127.82i 0.277365 0.160137i
\(562\) 0 0
\(563\) 5791.50 10031.2i 0.433539 0.750912i −0.563636 0.826023i \(-0.690598\pi\)
0.997175 + 0.0751113i \(0.0239312\pi\)
\(564\) 0 0
\(565\) 4932.00 + 2847.49i 0.367240 + 0.212026i
\(566\) 0 0
\(567\) 18891.5i 1.39924i
\(568\) 0 0
\(569\) −6439.50 11153.5i −0.474443 0.821759i 0.525129 0.851023i \(-0.324017\pi\)
−0.999572 + 0.0292638i \(0.990684\pi\)
\(570\) 0 0
\(571\) 11636.0 0.852805 0.426402 0.904534i \(-0.359781\pi\)
0.426402 + 0.904534i \(0.359781\pi\)
\(572\) 0 0
\(573\) 19887.0 1.44990
\(574\) 0 0
\(575\) −1909.50 3307.35i −0.138490 0.239871i
\(576\) 0 0
\(577\) 12311.4i 0.888269i −0.895960 0.444134i \(-0.853511\pi\)
0.895960 0.444134i \(-0.146489\pi\)
\(578\) 0 0
\(579\) 25735.5 + 14858.4i 1.84720 + 1.06648i
\(580\) 0 0
\(581\) −4797.00 + 8308.65i −0.342535 + 0.593289i
\(582\) 0 0
\(583\) 8307.00 4796.05i 0.590121 0.340707i
\(584\) 0 0
\(585\) 3432.00 13870.3i 0.242557 0.980282i
\(586\) 0 0
\(587\) 13549.5 7822.81i 0.952722 0.550054i 0.0587964 0.998270i \(-0.481274\pi\)
0.893925 + 0.448216i \(0.147940\pi\)
\(588\) 0 0
\(589\) 3213.00 5565.08i 0.224770 0.389313i
\(590\) 0 0
\(591\) −16684.5 9632.80i −1.16127 0.670458i
\(592\) 0 0
\(593\) 25821.4i 1.78813i −0.447942 0.894063i \(-0.647843\pi\)
0.447942 0.894063i \(-0.352157\pi\)
\(594\) 0 0
\(595\) −4212.00 7295.40i −0.290210 0.502659i
\(596\) 0 0
\(597\) 11795.0 0.808605
\(598\) 0 0
\(599\) −1668.00 −0.113777 −0.0568887 0.998381i \(-0.518118\pi\)
−0.0568887 + 0.998381i \(0.518118\pi\)
\(600\) 0 0
\(601\) −6849.50 11863.7i −0.464887 0.805207i 0.534310 0.845289i \(-0.320572\pi\)
−0.999196 + 0.0400813i \(0.987238\pi\)
\(602\) 0 0
\(603\) 3619.99i 0.244473i
\(604\) 0 0
\(605\) 9888.00 + 5708.84i 0.664470 + 0.383632i
\(606\) 0 0
\(607\) −11586.5 + 20068.4i −0.774764 + 1.34193i 0.160164 + 0.987090i \(0.448798\pi\)
−0.934927 + 0.354839i \(0.884536\pi\)
\(608\) 0 0
\(609\) −9418.50 + 5437.77i −0.626694 + 0.361822i
\(610\) 0 0
\(611\) −15444.0 + 4458.30i −1.02258 + 0.295194i
\(612\) 0 0
\(613\) 14389.5 8307.78i 0.948102 0.547387i 0.0556111 0.998453i \(-0.482289\pi\)
0.892491 + 0.451066i \(0.148956\pi\)
\(614\) 0 0
\(615\) 19068.0 33026.7i 1.25024 2.16547i
\(616\) 0 0
\(617\) 24589.5 + 14196.8i 1.60443 + 0.926321i 0.990585 + 0.136897i \(0.0437130\pi\)
0.613849 + 0.789423i \(0.289620\pi\)
\(618\) 0 0
\(619\) 6245.78i 0.405556i −0.979225 0.202778i \(-0.935003\pi\)
0.979225 0.202778i \(-0.0649969\pi\)
\(620\) 0 0
\(621\) −997.500 1727.72i −0.0644578 0.111644i
\(622\) 0 0
\(623\) −6903.00 −0.443921
\(624\) 0 0
\(625\) −19511.0 −1.24870
\(626\) 0 0
\(627\) −6961.50 12057.7i −0.443406 0.768002i
\(628\) 0 0
\(629\) 1075.60i 0.0681830i
\(630\) 0 0
\(631\) −19381.5 11189.9i −1.22277 0.705964i −0.257259 0.966342i \(-0.582819\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(632\) 0 0
\(633\) 5883.50 10190.5i 0.369428 0.639869i
\(634\) 0 0
\(635\) −26916.0 + 15540.0i −1.68209 + 0.971157i
\(636\) 0 0
\(637\) 5330.00 5539.10i 0.331526 0.344532i
\(638\) 0 0
\(639\) 11121.0 6420.71i 0.688482 0.397495i
\(640\) 0 0
\(641\) −9913.50 + 17170.7i −0.610858 + 1.05804i 0.380239 + 0.924888i \(0.375842\pi\)
−0.991096 + 0.133148i \(0.957491\pi\)
\(642\) 0 0
\(643\) 7318.50 + 4225.34i 0.448855 + 0.259146i 0.707346 0.706867i \(-0.249892\pi\)
−0.258492 + 0.966013i \(0.583225\pi\)
\(644\) 0 0
\(645\) 8244.56i 0.503301i
\(646\) 0 0
\(647\) 1474.50 + 2553.91i 0.0895959 + 0.155185i 0.907340 0.420397i \(-0.138109\pi\)
−0.817744 + 0.575581i \(0.804776\pi\)
\(648\) 0 0
\(649\) 429.000 0.0259472
\(650\) 0 0
\(651\) 11466.0 0.690304
\(652\) 0 0
\(653\) −6019.50 10426.1i −0.360737 0.624815i 0.627345 0.778741i \(-0.284141\pi\)
−0.988082 + 0.153926i \(0.950808\pi\)
\(654\) 0 0
\(655\) 5154.58i 0.307490i
\(656\) 0 0
\(657\) 19140.0 + 11050.5i 1.13656 + 0.656196i
\(658\) 0 0
\(659\) 1681.50 2912.44i 0.0993960 0.172159i −0.812039 0.583603i \(-0.801642\pi\)
0.911435 + 0.411445i \(0.134976\pi\)
\(660\) 0 0
\(661\) 8797.50 5079.24i 0.517675 0.298880i −0.218308 0.975880i \(-0.570054\pi\)
0.735983 + 0.677000i \(0.236720\pi\)
\(662\) 0 0
\(663\) −2457.00 8511.30i −0.143925 0.498569i
\(664\) 0 0
\(665\) −23868.0 + 13780.2i −1.39182 + 0.803569i
\(666\) 0 0
\(667\) −1966.50 + 3406.08i −0.114158 + 0.197727i
\(668\) 0 0
\(669\) −24832.5 14337.1i −1.43510 0.828554i
\(670\) 0 0
\(671\) 382.783i 0.0220226i
\(672\) 0 0
\(673\) 9084.50 + 15734.8i 0.520329 + 0.901237i 0.999721 + 0.0236358i \(0.00752419\pi\)
−0.479391 + 0.877601i \(0.659142\pi\)
\(674\) 0 0
\(675\) 2345.00 0.133717
\(676\) 0 0
\(677\) 9042.00 0.513312 0.256656 0.966503i \(-0.417379\pi\)
0.256656 + 0.966503i \(0.417379\pi\)
\(678\) 0 0
\(679\) −13903.5 24081.6i −0.785813 1.36107i
\(680\) 0 0
\(681\) 3067.46i 0.172607i
\(682\) 0 0
\(683\) 10792.5 + 6231.05i 0.604632 + 0.349084i 0.770862 0.637003i \(-0.219826\pi\)
−0.166230 + 0.986087i \(0.553159\pi\)
\(684\) 0 0
\(685\) −8244.00 + 14279.0i −0.459835 + 0.796458i
\(686\) 0 0
\(687\) −1092.00 + 630.466i −0.0606440 + 0.0350128i
\(688\) 0 0
\(689\) −5538.00 19184.2i −0.306213 1.06075i
\(690\) 0 0
\(691\) 3739.50 2159.00i 0.205872 0.118860i −0.393520 0.919316i \(-0.628743\pi\)
0.599391 + 0.800456i \(0.295409\pi\)
\(692\) 0 0
\(693\) 5577.00 9659.65i 0.305704 0.529494i
\(694\) 0 0
\(695\) 30540.0 + 17632.3i 1.66683 + 0.962346i
\(696\) 0 0
\(697\) 10615.7i 0.576901i
\(698\) 0 0
\(699\) 20223.0 + 35027.3i 1.09428 + 1.89535i
\(700\) 0 0
\(701\) −18270.0 −0.984377 −0.492189 0.870489i \(-0.663803\pi\)
−0.492189 + 0.870489i \(0.663803\pi\)
\(702\) 0 0
\(703\) −3519.00 −0.188793
\(704\) 0 0
\(705\) 16632.0 + 28807.5i 0.888507 + 1.53894i
\(706\) 0 0
\(707\) 44110.1i 2.34644i
\(708\) 0 0
\(709\) 1411.50 + 814.930i 0.0747673 + 0.0431669i 0.536918 0.843635i \(-0.319589\pi\)
−0.462150 + 0.886802i \(0.652922\pi\)
\(710\) 0 0
\(711\) −13684.0 + 23701.4i −0.721786 + 1.25017i
\(712\) 0 0
\(713\) 3591.00 2073.26i 0.188617 0.108898i
\(714\) 0 0
\(715\) −10140.0 + 10537.8i −0.530370 + 0.551177i
\(716\) 0 0
\(717\) −11277.0 + 6510.78i −0.587374 + 0.339121i
\(718\) 0 0
\(719\) 4915.50 8513.90i 0.254961 0.441606i −0.709924 0.704279i \(-0.751271\pi\)
0.964885 + 0.262673i \(0.0846039\pi\)
\(720\) 0 0
\(721\) 36192.0 + 20895.5i 1.86943 + 1.07932i
\(722\) 0 0
\(723\) 14415.9i 0.741537i
\(724\) 0 0
\(725\) −2311.50 4003.64i −0.118410 0.205091i
\(726\) 0 0
\(727\) 15464.0 0.788897 0.394448 0.918918i \(-0.370936\pi\)
0.394448 + 0.918918i \(0.370936\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) −1147.50 1987.53i −0.0580599 0.100563i
\(732\) 0 0
\(733\) 12616.3i 0.635733i 0.948136 + 0.317866i \(0.102966\pi\)
−0.948136 + 0.317866i \(0.897034\pi\)
\(734\) 0 0
\(735\) −13776.0 7953.58i −0.691341 0.399146i
\(736\) 0 0
\(737\) −1852.50 + 3208.62i −0.0925885 + 0.160368i
\(738\) 0 0
\(739\) 14101.5 8141.50i 0.701938 0.405264i −0.106131 0.994352i \(-0.533846\pi\)
0.808069 + 0.589088i \(0.200513\pi\)
\(740\) 0 0
\(741\) −27846.0 + 8038.45i −1.38050 + 0.398515i
\(742\) 0 0
\(743\) −9358.50 + 5403.13i −0.462086 + 0.266786i −0.712921 0.701244i \(-0.752628\pi\)
0.250835 + 0.968030i \(0.419295\pi\)
\(744\) 0 0
\(745\) 9036.00 15650.8i 0.444367 0.769666i
\(746\) 0 0
\(747\) 8118.00 + 4686.93i 0.397620 + 0.229566i
\(748\) 0 0
\(749\) 5741.75i 0.280105i
\(750\) 0 0
\(751\) −6807.50 11790.9i −0.330771 0.572913i 0.651892 0.758312i \(-0.273976\pi\)
−0.982663 + 0.185399i \(0.940642\pi\)
\(752\) 0 0
\(753\) −31437.0 −1.52142
\(754\) 0 0
\(755\) −1200.00 −0.0578443
\(756\) 0 0
\(757\) −2775.50 4807.31i −0.133259 0.230812i 0.791672 0.610947i \(-0.209211\pi\)
−0.924931 + 0.380135i \(0.875878\pi\)
\(758\) 0 0
\(759\) 8984.15i 0.429649i
\(760\) 0 0
\(761\) 8731.50 + 5041.13i 0.415922 + 0.240133i 0.693331 0.720619i \(-0.256142\pi\)
−0.277409 + 0.960752i \(0.589476\pi\)
\(762\) 0 0
\(763\) 6864.00 11888.8i 0.325680 0.564093i
\(764\) 0 0
\(765\) −7128.00 + 4115.35i −0.336880 + 0.194498i
\(766\) 0 0
\(767\) 214.500 866.891i 0.0100980 0.0408105i
\(768\) 0 0
\(769\) 25771.5 14879.2i 1.20851 0.697733i 0.246076 0.969250i \(-0.420859\pi\)
0.962434 + 0.271517i \(0.0875253\pi\)
\(770\) 0 0
\(771\) −19078.5 + 33044.9i −0.891174 + 1.54356i
\(772\) 0 0
\(773\) 24019.5 + 13867.7i 1.11762 + 0.645259i 0.940793 0.338983i \(-0.110083\pi\)
0.176829 + 0.984242i \(0.443416\pi\)
\(774\) 0 0
\(775\) 4873.99i 0.225908i
\(776\) 0 0
\(777\) −3139.50 5437.77i −0.144954 0.251067i
\(778\) 0 0
\(779\) −34731.0 −1.59739
\(780\) 0 0
\(781\) −13143.0 −0.602168
\(782\) 0 0
\(783\) −1207.50 2091.45i −0.0551118 0.0954564i
\(784\) 0 0
\(785\) 21255.7i 0.966432i
\(786\) 0 0
\(787\) 27322.5 + 15774.7i 1.23754 + 0.714493i 0.968590 0.248662i \(-0.0799910\pi\)
0.268947 + 0.963155i \(0.413324\pi\)
\(788\) 0 0
\(789\) −2740.50 + 4746.69i −0.123656 + 0.214178i
\(790\) 0 0
\(791\) 8014.50 4627.17i 0.360256 0.207994i
\(792\) 0 0
\(793\) −773.500 191.392i −0.0346378 0.00857064i
\(794\) 0 0
\(795\) −35784.0 + 20659.9i −1.59639 + 0.921674i
\(796\) 0 0
\(797\) −727.500 + 1260.07i −0.0323330 + 0.0560023i −0.881739 0.471737i \(-0.843627\pi\)
0.849406 + 0.527740i \(0.176960\pi\)
\(798\) 0 0
\(799\) 8019.00 + 4629.77i 0.355059 + 0.204993i
\(800\) 0 0
\(801\) 6744.61i 0.297514i
\(802\) 0 0
\(803\) −11310.0 19589.5i −0.497038 0.860894i
\(804\) 0 0
\(805\) −17784.0 −0.778638
\(806\) 0 0
\(807\) −35595.0 −1.55267
\(808\) 0 0
\(809\) −829.500 1436.74i −0.0360490 0.0624388i 0.847438 0.530894i \(-0.178144\pi\)
−0.883487 + 0.468456i \(0.844811\pi\)
\(810\) 0 0
\(811\) 4402.87i 0.190636i 0.995447 + 0.0953180i \(0.0303868\pi\)
−0.995447 + 0.0953180i \(0.969613\pi\)
\(812\) 0 0
\(813\) −8032.50 4637.57i −0.346509 0.200057i
\(814\) 0 0
\(815\) 11316.0 19599.9i 0.486359 0.842398i
\(816\) 0 0
\(817\) −6502.50 + 3754.22i −0.278450 + 0.160763i
\(818\) 0 0
\(819\) −16731.0 16099.4i −0.713832 0.686885i
\(820\) 0 0
\(821\) −24856.5 + 14350.9i −1.05664 + 0.610049i −0.924500 0.381183i \(-0.875517\pi\)
−0.132136 + 0.991232i \(0.542184\pi\)
\(822\) 0 0
\(823\) 7889.50 13665.0i 0.334156 0.578776i −0.649166 0.760647i \(-0.724882\pi\)
0.983322 + 0.181871i \(0.0582153\pi\)
\(824\) 0 0
\(825\) 9145.50 + 5280.16i 0.385946 + 0.222826i
\(826\) 0 0
\(827\) 7354.29i 0.309231i −0.987975 0.154615i \(-0.950586\pi\)
0.987975 0.154615i \(-0.0494138\pi\)
\(828\) 0 0
\(829\) −8685.50 15043.7i −0.363884 0.630266i 0.624712 0.780855i \(-0.285216\pi\)
−0.988596 + 0.150589i \(0.951883\pi\)
\(830\) 0 0
\(831\) −23947.0 −0.999654
\(832\) 0 0
\(833\) −4428.00 −0.184179
\(834\) 0 0
\(835\) 11268.0 + 19516.7i 0.467000 + 0.808868i
\(836\) 0 0
\(837\) 2546.11i 0.105145i
\(838\) 0 0
\(839\) −25525.5 14737.2i −1.05034 0.606416i −0.127598 0.991826i \(-0.540727\pi\)
−0.922745 + 0.385410i \(0.874060\pi\)
\(840\) 0 0
\(841\) 9814.00 16998.3i 0.402395 0.696968i
\(842\) 0 0
\(843\) −4914.00 + 2837.10i −0.200768 + 0.115913i
\(844\) 0 0
\(845\) 16224.0 + 25759.1i 0.660500 + 1.04868i
\(846\) 0 0
\(847\) 16068.0 9276.86i 0.651834 0.376336i
\(848\) 0 0
\(849\) 25119.5 43508.3i 1.01543 1.75877i
\(850\) 0 0
\(851\) −1966.50 1135.36i −0.0792136 0.0457340i
\(852\) 0 0
\(853\) 2909.85i 0.116801i −0.998293 0.0584005i \(-0.981400\pi\)
0.998293 0.0584005i \(-0.0186000\pi\)
\(854\) 0 0
\(855\) 13464.0 + 23320.3i 0.538549 + 0.932794i
\(856\) 0 0
\(857\) −5346.00 −0.213087 −0.106544 0.994308i \(-0.533978\pi\)
−0.106544 + 0.994308i \(0.533978\pi\)
\(858\) 0 0
\(859\) −24244.0 −0.962974 −0.481487 0.876453i \(-0.659903\pi\)
−0.481487 + 0.876453i \(0.659903\pi\)
\(860\) 0 0
\(861\) −30985.5 53668.5i −1.22646 2.12429i
\(862\) 0 0
\(863\) 32780.8i 1.29301i 0.762908 + 0.646507i \(0.223771\pi\)
−0.762908 + 0.646507i \(0.776229\pi\)
\(864\) 0 0
\(865\) −10476.0 6048.32i −0.411786 0.237745i
\(866\) 0 0
\(867\) 14644.0 25364.2i 0.573629 0.993555i
\(868\) 0 0
\(869\) 24258.0 14005.4i 0.946946 0.546720i
\(870\) 0 0
\(871\) 5557.50 + 5347.71i 0.216198 + 0.208037i
\(872\) 0 0
\(873\) −23529.0 + 13584.5i −0.912183 + 0.526649i
\(874\) 0 0
\(875\) −9048.00 + 15671.6i −0.349575 + 0.605482i
\(876\) 0 0
\(877\) −3934.50 2271.58i −0.151492 0.0874640i 0.422338 0.906439i \(-0.361210\pi\)
−0.573830 + 0.818974i \(0.694543\pi\)
\(878\) 0 0
\(879\) 65192.7i 2.50159i
\(880\) 0 0
\(881\) 10258.5 + 17768.2i 0.392302 + 0.679486i 0.992753 0.120175i \(-0.0383456\pi\)
−0.600451 + 0.799661i \(0.705012\pi\)
\(882\) 0 0
\(883\) 23852.0 0.909042 0.454521 0.890736i \(-0.349811\pi\)
0.454521 + 0.890736i \(0.349811\pi\)
\(884\) 0 0
\(885\) −1848.00 −0.0701919
\(886\) 0 0
\(887\) 19378.5 + 33564.5i 0.733558 + 1.27056i 0.955353 + 0.295467i \(0.0954753\pi\)
−0.221794 + 0.975093i \(0.571191\pi\)
\(888\) 0 0
\(889\) 50504.9i 1.90538i
\(890\) 0 0
\(891\) 16360.5 + 9445.74i 0.615149 + 0.355156i
\(892\) 0 0
\(893\) 15147.0 26235.4i 0.567609 0.983128i
\(894\) 0 0
\(895\) −15444.0 + 8916.60i −0.576800 + 0.333016i
\(896\) 0 0
\(897\) −18154.5 4492.07i −0.675765 0.167208i
\(898\) 0 0
\(899\) 4347.00 2509.74i 0.161269 0.0931085i
\(900\) 0 0
\(901\) −5751.00 + 9961.02i −0.212645 + 0.368313i
\(902\) 0 0
\(903\) −11602.5 6698.71i −0.427583 0.246865i
\(904\) 0 0
\(905\) 27.7128i 0.00101791i
\(906\) 0 0
\(907\) −19535.5 33836.5i −0.715177 1.23872i −0.962891 0.269890i \(-0.913013\pi\)
0.247714 0.968833i \(-0.420321\pi\)
\(908\) 0 0
\(909\) 43098.0 1.57257
\(910\) 0 0
\(911\) 53040.0 1.92897 0.964486 0.264134i \(-0.0850860\pi\)
0.964486 + 0.264134i \(0.0850860\pi\)
\(912\) 0 0
\(913\) −4797.00 8308.65i −0.173886 0.301179i
\(914\) 0 0
\(915\) 1648.91i 0.0595753i
\(916\) 0 0
\(917\) −7254.00 4188.10i −0.261230 0.150821i
\(918\) 0 0
\(919\) 183.500 317.831i 0.00658662 0.0114084i −0.862713 0.505693i \(-0.831237\pi\)
0.869300 + 0.494285i \(0.164570\pi\)
\(920\) 0 0
\(921\) −28959.0 + 16719.5i −1.03608 + 0.598182i
\(922\) 0 0
\(923\) −6571.50 + 26558.4i −0.234348 + 0.947108i
\(924\) 0 0
\(925\) 2311.50 1334.55i 0.0821639 0.0474374i
\(926\) 0 0
\(927\) 20416.0 35361.5i 0.723354 1.25289i
\(928\) 0 0
\(929\) −25924.5 14967.5i −0.915560 0.528599i −0.0333441 0.999444i \(-0.510616\pi\)
−0.882216 + 0.470845i \(0.843949\pi\)
\(930\) 0 0
\(931\) 14486.9i 0.509976i
\(932\) 0 0
\(933\) 21672.0 + 37537.0i 0.760460 + 1.31716i
\(934\) 0 0
\(935\) 8424.00 0.294646
\(936\) 0 0
\(937\) 42166.0 1.47012 0.735060 0.678002i \(-0.237154\pi\)
0.735060 + 0.678002i \(0.237154\pi\)
\(938\) 0 0
\(939\) 2695.00 + 4667.88i 0.0936613 + 0.162226i
\(940\) 0 0
\(941\) 35022.1i 1.21327i 0.794981 + 0.606635i \(0.207481\pi\)
−0.794981 + 0.606635i \(0.792519\pi\)
\(942\) 0 0
\(943\) −19408.5 11205.5i −0.670231 0.386958i
\(944\) 0 0
\(945\) 5460.00 9457.00i 0.187951 0.325541i
\(946\) 0 0
\(947\) 2251.50 1299.90i 0.0772586 0.0446053i −0.460873 0.887466i \(-0.652464\pi\)
0.538132 + 0.842861i \(0.319130\pi\)
\(948\) 0 0
\(949\) −45240.0 + 13059.7i −1.54747 + 0.446717i
\(950\) 0 0
\(951\) −48846.0 + 28201.3i −1.66555 + 0.961607i
\(952\) 0 0
\(953\) −5311.50 + 9199.79i −0.180542 + 0.312708i −0.942065 0.335430i \(-0.891118\pi\)
0.761523 + 0.648137i \(0.224452\pi\)
\(954\) 0 0
\(955\) 34092.0 + 19683.0i 1.15517 + 0.666940i
\(956\) 0 0
\(957\) 10875.5i 0.367353i
\(958\) 0 0
\(959\) 13396.5 + 23203.4i 0.451090 + 0.781311i
\(960\) 0 0
\(961\) 24499.0 0.822362
\(962\) 0 0
\(963\) 5610.00 0.187726
\(964\) 0 0
\(965\) 29412.0 + 50943.1i 0.981146 + 1.69939i
\(966\) 0 0
\(967\) 20199.2i 0.671729i 0.941910 + 0.335864i \(0.109028\pi\)
−0.941910 + 0.335864i \(0.890972\pi\)
\(968\) 0 0
\(969\) 14458.5 + 8347.62i 0.479333 + 0.276743i
\(970\) 0 0
\(971\) −1162.50 + 2013.51i −0.0384206 + 0.0665464i −0.884596 0.466358i \(-0.845566\pi\)
0.846176 + 0.532904i \(0.178899\pi\)
\(972\) 0 0
\(973\) 49627.5 28652.5i 1.63513 0.944045i
\(974\) 0 0
\(975\) 15242.5 15840.5i 0.500667 0.520309i
\(976\) 0 0
\(977\) 28525.5 16469.2i 0.934096 0.539300i 0.0459912 0.998942i \(-0.485355\pi\)
0.888105 + 0.459641i \(0.152022\pi\)
\(978\) 0 0
\(979\) 3451.50 5978.17i 0.112677 0.195162i
\(980\) 0 0
\(981\) −11616.0 6706.50i −0.378053 0.218269i
\(982\) 0 0
\(983\) 42702.0i 1.38554i 0.721161 + 0.692768i \(0.243609\pi\)
−0.721161 + 0.692768i \(0.756391\pi\)
\(984\) 0 0
\(985\) −19068.0 33026.7i −0.616809 1.06834i
\(986\) 0 0
\(987\) 54054.0 1.74322
\(988\) 0 0
\(989\) −4845.00 −0.155776
\(990\) 0 0
\(991\) −2421.50 4194.16i −0.0776201 0.134442i 0.824603 0.565712i \(-0.191399\pi\)
−0.902223 + 0.431271i \(0.858065\pi\)
\(992\) 0 0
\(993\) 36942.9i 1.18061i
\(994\) 0 0
\(995\) 20220.0 + 11674.0i 0.644238 + 0.371951i
\(996\) 0 0
\(997\) −5471.50 + 9476.92i −0.173806 + 0.301040i −0.939747 0.341870i \(-0.888940\pi\)
0.765942 + 0.642910i \(0.222273\pi\)
\(998\) 0 0
\(999\) 1207.50 697.150i 0.0382419 0.0220789i
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 208.4.w.a.49.1 2
4.3 odd 2 13.4.e.a.10.1 yes 2
12.11 even 2 117.4.q.c.10.1 2
13.4 even 6 inner 208.4.w.a.17.1 2
52.3 odd 6 169.4.b.b.168.1 2
52.7 even 12 169.4.c.i.22.1 4
52.11 even 12 169.4.a.h.1.2 2
52.15 even 12 169.4.a.h.1.1 2
52.19 even 12 169.4.c.i.22.2 4
52.23 odd 6 169.4.b.b.168.2 2
52.31 even 4 169.4.c.i.146.2 4
52.35 odd 6 169.4.e.b.147.1 2
52.43 odd 6 13.4.e.a.4.1 2
52.47 even 4 169.4.c.i.146.1 4
52.51 odd 2 169.4.e.b.23.1 2
156.11 odd 12 1521.4.a.q.1.1 2
156.95 even 6 117.4.q.c.82.1 2
156.119 odd 12 1521.4.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.e.a.4.1 2 52.43 odd 6
13.4.e.a.10.1 yes 2 4.3 odd 2
117.4.q.c.10.1 2 12.11 even 2
117.4.q.c.82.1 2 156.95 even 6
169.4.a.h.1.1 2 52.15 even 12
169.4.a.h.1.2 2 52.11 even 12
169.4.b.b.168.1 2 52.3 odd 6
169.4.b.b.168.2 2 52.23 odd 6
169.4.c.i.22.1 4 52.7 even 12
169.4.c.i.22.2 4 52.19 even 12
169.4.c.i.146.1 4 52.47 even 4
169.4.c.i.146.2 4 52.31 even 4
169.4.e.b.23.1 2 52.51 odd 2
169.4.e.b.147.1 2 52.35 odd 6
208.4.w.a.17.1 2 13.4 even 6 inner
208.4.w.a.49.1 2 1.1 even 1 trivial
1521.4.a.q.1.1 2 156.11 odd 12
1521.4.a.q.1.2 2 156.119 odd 12