Properties

Label 504.2.bs.a.257.2
Level $504$
Weight $2$
Character 504.257
Analytic conductor $4.024$
Analytic rank $0$
Dimension $48$
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] = [504,2,Mod(257,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bs (of order \(6\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.2
Character \(\chi\) \(=\) 504.257
Dual form 504.2.bs.a.353.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.69760 + 0.343722i) q^{3} +(-0.0262870 - 0.0455305i) q^{5} +(-2.10354 + 1.60471i) q^{7} +(2.76371 - 1.16701i) q^{9} +O(q^{10})\) \(q+(-1.69760 + 0.343722i) q^{3} +(-0.0262870 - 0.0455305i) q^{5} +(-2.10354 + 1.60471i) q^{7} +(2.76371 - 1.16701i) q^{9} +(2.15187 + 1.24238i) q^{11} +(-2.51752 - 1.45349i) q^{13} +(0.0602747 + 0.0682572i) q^{15} +(-2.88651 - 4.99959i) q^{17} +(-2.92807 - 1.69052i) q^{19} +(3.01941 - 3.44720i) q^{21} +(-7.47404 + 4.31514i) q^{23} +(2.49862 - 4.32773i) q^{25} +(-4.29056 + 2.93106i) q^{27} +(-6.23728 + 3.60109i) q^{29} -9.92107i q^{31} +(-4.08005 - 1.36942i) q^{33} +(0.128359 + 0.0535922i) q^{35} +(-0.770828 + 1.33511i) q^{37} +(4.77335 + 1.60212i) q^{39} +(-0.392450 + 0.679743i) q^{41} +(-2.03075 - 3.51736i) q^{43} +(-0.125784 - 0.0951558i) q^{45} +1.31552 q^{47} +(1.84979 - 6.75117i) q^{49} +(6.61862 + 7.49515i) q^{51} +(-0.710864 + 0.410417i) q^{53} -0.130634i q^{55} +(5.55176 + 1.86339i) q^{57} -4.64150 q^{59} +5.62413i q^{61} +(-3.94087 + 6.88981i) q^{63} +0.152832i q^{65} -13.9012 q^{67} +(11.2047 - 9.89439i) q^{69} -3.51362i q^{71} +(6.75829 - 3.90190i) q^{73} +(-2.75412 + 8.20560i) q^{75} +(-6.52021 + 0.839727i) q^{77} +15.0014 q^{79} +(6.27619 - 6.45054i) q^{81} +(3.14086 + 5.44013i) q^{83} +(-0.151756 + 0.262848i) q^{85} +(9.35064 - 8.25712i) q^{87} +(-7.93120 + 13.7372i) q^{89} +(7.62815 - 0.982418i) q^{91} +(3.41009 + 16.8420i) q^{93} +0.177755i q^{95} +(-1.57451 + 0.909044i) q^{97} +(7.39700 + 0.922336i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{9} + 8 q^{15} + 8 q^{21} - 12 q^{23} - 24 q^{25} - 18 q^{27} + 18 q^{29} - 10 q^{39} + 6 q^{41} - 6 q^{43} + 6 q^{45} + 36 q^{47} + 6 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} + 46 q^{63} - 54 q^{75} - 36 q^{77} - 12 q^{79} - 24 q^{87} + 18 q^{89} + 6 q^{91} + 16 q^{93} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −1.69760 + 0.343722i −0.980111 + 0.198448i
\(4\) 0 0
\(5\) −0.0262870 0.0455305i −0.0117559 0.0203618i 0.860088 0.510146i \(-0.170409\pi\)
−0.871844 + 0.489785i \(0.837075\pi\)
\(6\) 0 0
\(7\) −2.10354 + 1.60471i −0.795065 + 0.606525i
\(8\) 0 0
\(9\) 2.76371 1.16701i 0.921237 0.389003i
\(10\) 0 0
\(11\) 2.15187 + 1.24238i 0.648812 + 0.374592i 0.788001 0.615674i \(-0.211116\pi\)
−0.139189 + 0.990266i \(0.544449\pi\)
\(12\) 0 0
\(13\) −2.51752 1.45349i −0.698235 0.403126i 0.108455 0.994101i \(-0.465410\pi\)
−0.806690 + 0.590975i \(0.798743\pi\)
\(14\) 0 0
\(15\) 0.0602747 + 0.0682572i 0.0155629 + 0.0176239i
\(16\) 0 0
\(17\) −2.88651 4.99959i −0.700082 1.21258i −0.968437 0.249258i \(-0.919813\pi\)
0.268355 0.963320i \(-0.413520\pi\)
\(18\) 0 0
\(19\) −2.92807 1.69052i −0.671745 0.387832i 0.124993 0.992158i \(-0.460109\pi\)
−0.796737 + 0.604326i \(0.793443\pi\)
\(20\) 0 0
\(21\) 3.01941 3.44720i 0.658888 0.752241i
\(22\) 0 0
\(23\) −7.47404 + 4.31514i −1.55845 + 0.899769i −0.561039 + 0.827789i \(0.689598\pi\)
−0.997406 + 0.0719794i \(0.977068\pi\)
\(24\) 0 0
\(25\) 2.49862 4.32773i 0.499724 0.865547i
\(26\) 0 0
\(27\) −4.29056 + 2.93106i −0.825718 + 0.564083i
\(28\) 0 0
\(29\) −6.23728 + 3.60109i −1.15823 + 0.668706i −0.950880 0.309560i \(-0.899818\pi\)
−0.207353 + 0.978266i \(0.566485\pi\)
\(30\) 0 0
\(31\) 9.92107i 1.78188i −0.454125 0.890938i \(-0.650048\pi\)
0.454125 0.890938i \(-0.349952\pi\)
\(32\) 0 0
\(33\) −4.08005 1.36942i −0.710245 0.238386i
\(34\) 0 0
\(35\) 0.128359 + 0.0535922i 0.0216967 + 0.00905873i
\(36\) 0 0
\(37\) −0.770828 + 1.33511i −0.126723 + 0.219491i −0.922405 0.386223i \(-0.873779\pi\)
0.795682 + 0.605715i \(0.207113\pi\)
\(38\) 0 0
\(39\) 4.77335 + 1.60212i 0.764348 + 0.256545i
\(40\) 0 0
\(41\) −0.392450 + 0.679743i −0.0612904 + 0.106158i −0.895042 0.445981i \(-0.852855\pi\)
0.833752 + 0.552139i \(0.186188\pi\)
\(42\) 0 0
\(43\) −2.03075 3.51736i −0.309686 0.536392i 0.668608 0.743615i \(-0.266891\pi\)
−0.978294 + 0.207223i \(0.933557\pi\)
\(44\) 0 0
\(45\) −0.125784 0.0951558i −0.0187508 0.0141850i
\(46\) 0 0
\(47\) 1.31552 0.191888 0.0959441 0.995387i \(-0.469413\pi\)
0.0959441 + 0.995387i \(0.469413\pi\)
\(48\) 0 0
\(49\) 1.84979 6.75117i 0.264256 0.964453i
\(50\) 0 0
\(51\) 6.61862 + 7.49515i 0.926792 + 1.04953i
\(52\) 0 0
\(53\) −0.710864 + 0.410417i −0.0976447 + 0.0563752i −0.548027 0.836461i \(-0.684621\pi\)
0.450382 + 0.892836i \(0.351288\pi\)
\(54\) 0 0
\(55\) 0.130634i 0.0176147i
\(56\) 0 0
\(57\) 5.55176 + 1.86339i 0.735349 + 0.246812i
\(58\) 0 0
\(59\) −4.64150 −0.604271 −0.302136 0.953265i \(-0.597700\pi\)
−0.302136 + 0.953265i \(0.597700\pi\)
\(60\) 0 0
\(61\) 5.62413i 0.720097i 0.932934 + 0.360048i \(0.117240\pi\)
−0.932934 + 0.360048i \(0.882760\pi\)
\(62\) 0 0
\(63\) −3.94087 + 6.88981i −0.496503 + 0.868035i
\(64\) 0 0
\(65\) 0.152832i 0.0189565i
\(66\) 0 0
\(67\) −13.9012 −1.69830 −0.849149 0.528153i \(-0.822885\pi\)
−0.849149 + 0.528153i \(0.822885\pi\)
\(68\) 0 0
\(69\) 11.2047 9.89439i 1.34889 1.19114i
\(70\) 0 0
\(71\) 3.51362i 0.416990i −0.978023 0.208495i \(-0.933143\pi\)
0.978023 0.208495i \(-0.0668565\pi\)
\(72\) 0 0
\(73\) 6.75829 3.90190i 0.790998 0.456683i −0.0493159 0.998783i \(-0.515704\pi\)
0.840314 + 0.542100i \(0.182371\pi\)
\(74\) 0 0
\(75\) −2.75412 + 8.20560i −0.318019 + 0.947501i
\(76\) 0 0
\(77\) −6.52021 + 0.839727i −0.743047 + 0.0956958i
\(78\) 0 0
\(79\) 15.0014 1.68779 0.843895 0.536508i \(-0.180257\pi\)
0.843895 + 0.536508i \(0.180257\pi\)
\(80\) 0 0
\(81\) 6.27619 6.45054i 0.697354 0.716727i
\(82\) 0 0
\(83\) 3.14086 + 5.44013i 0.344754 + 0.597132i 0.985309 0.170781i \(-0.0546290\pi\)
−0.640555 + 0.767912i \(0.721296\pi\)
\(84\) 0 0
\(85\) −0.151756 + 0.262848i −0.0164602 + 0.0285099i
\(86\) 0 0
\(87\) 9.35064 8.25712i 1.00249 0.885256i
\(88\) 0 0
\(89\) −7.93120 + 13.7372i −0.840706 + 1.45615i 0.0485931 + 0.998819i \(0.484526\pi\)
−0.889299 + 0.457326i \(0.848807\pi\)
\(90\) 0 0
\(91\) 7.62815 0.982418i 0.799648 0.102985i
\(92\) 0 0
\(93\) 3.41009 + 16.8420i 0.353610 + 1.74644i
\(94\) 0 0
\(95\) 0.177755i 0.0182373i
\(96\) 0 0
\(97\) −1.57451 + 0.909044i −0.159867 + 0.0922995i −0.577799 0.816179i \(-0.696088\pi\)
0.417932 + 0.908478i \(0.362755\pi\)
\(98\) 0 0
\(99\) 7.39700 + 0.922336i 0.743427 + 0.0926982i
\(100\) 0 0
\(101\) 0.667947 1.15692i 0.0664632 0.115118i −0.830879 0.556453i \(-0.812162\pi\)
0.897342 + 0.441336i \(0.145495\pi\)
\(102\) 0 0
\(103\) −6.62845 + 3.82694i −0.653120 + 0.377079i −0.789651 0.613557i \(-0.789738\pi\)
0.136530 + 0.990636i \(0.456405\pi\)
\(104\) 0 0
\(105\) −0.236324 0.0468583i −0.0230628 0.00457290i
\(106\) 0 0
\(107\) 10.9289 + 6.30978i 1.05653 + 0.609989i 0.924471 0.381252i \(-0.124507\pi\)
0.132062 + 0.991242i \(0.457840\pi\)
\(108\) 0 0
\(109\) 5.12613 + 8.87871i 0.490994 + 0.850426i 0.999946 0.0103684i \(-0.00330042\pi\)
−0.508952 + 0.860795i \(0.669967\pi\)
\(110\) 0 0
\(111\) 0.849651 2.53144i 0.0806453 0.240274i
\(112\) 0 0
\(113\) 13.4379 + 7.75836i 1.26413 + 0.729845i 0.973871 0.227104i \(-0.0729257\pi\)
0.290258 + 0.956949i \(0.406259\pi\)
\(114\) 0 0
\(115\) 0.392940 + 0.226864i 0.0366419 + 0.0211552i
\(116\) 0 0
\(117\) −8.65394 1.07906i −0.800057 0.0997594i
\(118\) 0 0
\(119\) 14.0948 + 5.88482i 1.29207 + 0.539461i
\(120\) 0 0
\(121\) −2.41298 4.17940i −0.219362 0.379946i
\(122\) 0 0
\(123\) 0.432581 1.28883i 0.0390046 0.116210i
\(124\) 0 0
\(125\) −0.525595 −0.0470107
\(126\) 0 0
\(127\) −7.05269 −0.625825 −0.312912 0.949782i \(-0.601305\pi\)
−0.312912 + 0.949782i \(0.601305\pi\)
\(128\) 0 0
\(129\) 4.65640 + 5.27306i 0.409973 + 0.464267i
\(130\) 0 0
\(131\) −1.84958 3.20357i −0.161599 0.279898i 0.773843 0.633377i \(-0.218332\pi\)
−0.935442 + 0.353479i \(0.884998\pi\)
\(132\) 0 0
\(133\) 8.87212 1.14263i 0.769310 0.0990782i
\(134\) 0 0
\(135\) 0.246239 + 0.118302i 0.0211928 + 0.0101818i
\(136\) 0 0
\(137\) −3.54459 2.04647i −0.302835 0.174842i 0.340881 0.940107i \(-0.389275\pi\)
−0.643716 + 0.765265i \(0.722608\pi\)
\(138\) 0 0
\(139\) −12.1602 7.02072i −1.03142 0.595489i −0.114027 0.993478i \(-0.536375\pi\)
−0.917391 + 0.397988i \(0.869708\pi\)
\(140\) 0 0
\(141\) −2.23323 + 0.452173i −0.188072 + 0.0380799i
\(142\) 0 0
\(143\) −3.61158 6.25544i −0.302016 0.523106i
\(144\) 0 0
\(145\) 0.327919 + 0.189324i 0.0272322 + 0.0157225i
\(146\) 0 0
\(147\) −0.819682 + 12.0966i −0.0676062 + 0.997712i
\(148\) 0 0
\(149\) −6.55913 + 3.78692i −0.537345 + 0.310236i −0.744002 0.668177i \(-0.767075\pi\)
0.206657 + 0.978413i \(0.433742\pi\)
\(150\) 0 0
\(151\) 6.34413 10.9883i 0.516278 0.894219i −0.483544 0.875320i \(-0.660651\pi\)
0.999821 0.0188990i \(-0.00601609\pi\)
\(152\) 0 0
\(153\) −13.8120 10.4488i −1.11664 0.844738i
\(154\) 0 0
\(155\) −0.451711 + 0.260795i −0.0362823 + 0.0209476i
\(156\) 0 0
\(157\) 1.09592i 0.0874639i −0.999043 0.0437319i \(-0.986075\pi\)
0.999043 0.0437319i \(-0.0139247\pi\)
\(158\) 0 0
\(159\) 1.06569 0.941066i 0.0845151 0.0746314i
\(160\) 0 0
\(161\) 8.79741 21.0708i 0.693333 1.66061i
\(162\) 0 0
\(163\) 4.43585 7.68311i 0.347442 0.601788i −0.638352 0.769745i \(-0.720384\pi\)
0.985794 + 0.167957i \(0.0537169\pi\)
\(164\) 0 0
\(165\) 0.0449018 + 0.221765i 0.00349560 + 0.0172643i
\(166\) 0 0
\(167\) 4.68302 8.11124i 0.362383 0.627666i −0.625969 0.779848i \(-0.715296\pi\)
0.988353 + 0.152181i \(0.0486298\pi\)
\(168\) 0 0
\(169\) −2.27472 3.93994i −0.174979 0.303072i
\(170\) 0 0
\(171\) −10.0652 1.25503i −0.769703 0.0959746i
\(172\) 0 0
\(173\) −22.9351 −1.74372 −0.871861 0.489753i \(-0.837087\pi\)
−0.871861 + 0.489753i \(0.837087\pi\)
\(174\) 0 0
\(175\) 1.68882 + 13.1131i 0.127663 + 0.991260i
\(176\) 0 0
\(177\) 7.87942 1.59539i 0.592253 0.119916i
\(178\) 0 0
\(179\) −10.5254 + 6.07685i −0.786706 + 0.454205i −0.838802 0.544437i \(-0.816743\pi\)
0.0520958 + 0.998642i \(0.483410\pi\)
\(180\) 0 0
\(181\) 24.1883i 1.79790i 0.438048 + 0.898952i \(0.355670\pi\)
−0.438048 + 0.898952i \(0.644330\pi\)
\(182\) 0 0
\(183\) −1.93314 9.54755i −0.142902 0.705775i
\(184\) 0 0
\(185\) 0.0810510 0.00595899
\(186\) 0 0
\(187\) 14.3446i 1.04898i
\(188\) 0 0
\(189\) 4.32185 13.0507i 0.314368 0.949301i
\(190\) 0 0
\(191\) 12.5335i 0.906889i 0.891284 + 0.453445i \(0.149805\pi\)
−0.891284 + 0.453445i \(0.850195\pi\)
\(192\) 0 0
\(193\) −2.97733 −0.214313 −0.107157 0.994242i \(-0.534175\pi\)
−0.107157 + 0.994242i \(0.534175\pi\)
\(194\) 0 0
\(195\) −0.0525317 0.259448i −0.00376187 0.0185794i
\(196\) 0 0
\(197\) 8.95974i 0.638355i 0.947695 + 0.319178i \(0.103407\pi\)
−0.947695 + 0.319178i \(0.896593\pi\)
\(198\) 0 0
\(199\) −5.95927 + 3.44059i −0.422442 + 0.243897i −0.696121 0.717924i \(-0.745092\pi\)
0.273680 + 0.961821i \(0.411759\pi\)
\(200\) 0 0
\(201\) 23.5987 4.77814i 1.66452 0.337024i
\(202\) 0 0
\(203\) 7.34166 17.5841i 0.515284 1.23416i
\(204\) 0 0
\(205\) 0.0412654 0.00288210
\(206\) 0 0
\(207\) −15.6203 + 20.6481i −1.08568 + 1.43514i
\(208\) 0 0
\(209\) −4.20054 7.27555i −0.290557 0.503260i
\(210\) 0 0
\(211\) 5.51329 9.54930i 0.379550 0.657401i −0.611446 0.791286i \(-0.709412\pi\)
0.990997 + 0.133885i \(0.0427453\pi\)
\(212\) 0 0
\(213\) 1.20771 + 5.96473i 0.0827509 + 0.408697i
\(214\) 0 0
\(215\) −0.106765 + 0.184922i −0.00728128 + 0.0126116i
\(216\) 0 0
\(217\) 15.9205 + 20.8694i 1.08075 + 1.41671i
\(218\) 0 0
\(219\) −10.1317 + 8.94685i −0.684638 + 0.604572i
\(220\) 0 0
\(221\) 16.7821i 1.12889i
\(222\) 0 0
\(223\) −5.55863 + 3.20928i −0.372234 + 0.214909i −0.674434 0.738335i \(-0.735612\pi\)
0.302200 + 0.953244i \(0.402279\pi\)
\(224\) 0 0
\(225\) 1.85496 14.8765i 0.123664 0.991767i
\(226\) 0 0
\(227\) 12.1083 20.9722i 0.803658 1.39198i −0.113535 0.993534i \(-0.536218\pi\)
0.917193 0.398443i \(-0.130449\pi\)
\(228\) 0 0
\(229\) −9.51981 + 5.49626i −0.629087 + 0.363203i −0.780398 0.625283i \(-0.784984\pi\)
0.151312 + 0.988486i \(0.451650\pi\)
\(230\) 0 0
\(231\) 10.7801 3.66666i 0.709278 0.241249i
\(232\) 0 0
\(233\) −18.3691 10.6054i −1.20340 0.694783i −0.242090 0.970254i \(-0.577833\pi\)
−0.961309 + 0.275471i \(0.911166\pi\)
\(234\) 0 0
\(235\) −0.0345811 0.0598962i −0.00225582 0.00390720i
\(236\) 0 0
\(237\) −25.4664 + 5.15632i −1.65422 + 0.334939i
\(238\) 0 0
\(239\) −1.14539 0.661293i −0.0740893 0.0427755i 0.462498 0.886620i \(-0.346953\pi\)
−0.536587 + 0.843845i \(0.680287\pi\)
\(240\) 0 0
\(241\) 3.20494 + 1.85037i 0.206448 + 0.119193i 0.599660 0.800255i \(-0.295303\pi\)
−0.393212 + 0.919448i \(0.628636\pi\)
\(242\) 0 0
\(243\) −8.43728 + 13.1077i −0.541252 + 0.840861i
\(244\) 0 0
\(245\) −0.356009 + 0.0932463i −0.0227446 + 0.00595729i
\(246\) 0 0
\(247\) 4.91431 + 8.51184i 0.312690 + 0.541595i
\(248\) 0 0
\(249\) −7.20182 8.15559i −0.456397 0.516840i
\(250\) 0 0
\(251\) 26.6932 1.68486 0.842429 0.538807i \(-0.181125\pi\)
0.842429 + 0.538807i \(0.181125\pi\)
\(252\) 0 0
\(253\) −21.4442 −1.34818
\(254\) 0 0
\(255\) 0.167274 0.498374i 0.0104751 0.0312094i
\(256\) 0 0
\(257\) 0.888958 + 1.53972i 0.0554517 + 0.0960451i 0.892419 0.451208i \(-0.149007\pi\)
−0.836967 + 0.547253i \(0.815673\pi\)
\(258\) 0 0
\(259\) −0.521004 4.04542i −0.0323736 0.251370i
\(260\) 0 0
\(261\) −13.0355 + 17.2313i −0.806879 + 1.06659i
\(262\) 0 0
\(263\) −16.4352 9.48887i −1.01344 0.585109i −0.101241 0.994862i \(-0.532281\pi\)
−0.912196 + 0.409753i \(0.865615\pi\)
\(264\) 0 0
\(265\) 0.0373730 + 0.0215773i 0.00229580 + 0.00132548i
\(266\) 0 0
\(267\) 8.74223 26.0465i 0.535016 1.59402i
\(268\) 0 0
\(269\) 7.17800 + 12.4327i 0.437651 + 0.758033i 0.997508 0.0705560i \(-0.0224773\pi\)
−0.559857 + 0.828589i \(0.689144\pi\)
\(270\) 0 0
\(271\) −11.0202 6.36254i −0.669432 0.386497i 0.126429 0.991976i \(-0.459648\pi\)
−0.795861 + 0.605479i \(0.792982\pi\)
\(272\) 0 0
\(273\) −12.6119 + 4.28972i −0.763307 + 0.259626i
\(274\) 0 0
\(275\) 10.7534 6.20847i 0.648454 0.374385i
\(276\) 0 0
\(277\) 12.4763 21.6095i 0.749625 1.29839i −0.198377 0.980126i \(-0.563567\pi\)
0.948002 0.318264i \(-0.103100\pi\)
\(278\) 0 0
\(279\) −11.5780 27.4189i −0.693154 1.64153i
\(280\) 0 0
\(281\) 9.13275 5.27279i 0.544814 0.314549i −0.202214 0.979341i \(-0.564814\pi\)
0.747028 + 0.664793i \(0.231480\pi\)
\(282\) 0 0
\(283\) 15.3438i 0.912094i −0.889956 0.456047i \(-0.849265\pi\)
0.889956 0.456047i \(-0.150735\pi\)
\(284\) 0 0
\(285\) −0.0610983 0.301757i −0.00361915 0.0178746i
\(286\) 0 0
\(287\) −0.265258 2.05964i −0.0156577 0.121577i
\(288\) 0 0
\(289\) −8.16392 + 14.1403i −0.480230 + 0.831783i
\(290\) 0 0
\(291\) 2.36043 2.08439i 0.138371 0.122189i
\(292\) 0 0
\(293\) 1.55166 2.68756i 0.0906490 0.157009i −0.817135 0.576446i \(-0.804439\pi\)
0.907784 + 0.419437i \(0.137773\pi\)
\(294\) 0 0
\(295\) 0.122011 + 0.211329i 0.00710376 + 0.0123041i
\(296\) 0 0
\(297\) −12.8742 + 0.976755i −0.747037 + 0.0566771i
\(298\) 0 0
\(299\) 25.0881 1.45088
\(300\) 0 0
\(301\) 9.91612 + 4.14015i 0.571555 + 0.238634i
\(302\) 0 0
\(303\) −0.736250 + 2.19358i −0.0422965 + 0.126018i
\(304\) 0 0
\(305\) 0.256069 0.147842i 0.0146625 0.00846539i
\(306\) 0 0
\(307\) 22.1272i 1.26286i 0.775431 + 0.631432i \(0.217533\pi\)
−0.775431 + 0.631432i \(0.782467\pi\)
\(308\) 0 0
\(309\) 9.93707 8.77496i 0.565300 0.499190i
\(310\) 0 0
\(311\) −8.33250 −0.472493 −0.236246 0.971693i \(-0.575917\pi\)
−0.236246 + 0.971693i \(0.575917\pi\)
\(312\) 0 0
\(313\) 23.6015i 1.33403i 0.745043 + 0.667017i \(0.232429\pi\)
−0.745043 + 0.667017i \(0.767571\pi\)
\(314\) 0 0
\(315\) 0.417290 0.00168300i 0.0235116 9.48263e-5i
\(316\) 0 0
\(317\) 17.0871i 0.959708i −0.877348 0.479854i \(-0.840690\pi\)
0.877348 0.479854i \(-0.159310\pi\)
\(318\) 0 0
\(319\) −17.8957 −1.00197
\(320\) 0 0
\(321\) −20.7217 6.95501i −1.15657 0.388191i
\(322\) 0 0
\(323\) 19.5188i 1.08606i
\(324\) 0 0
\(325\) −12.5806 + 7.26344i −0.697849 + 0.402903i
\(326\) 0 0
\(327\) −11.7539 13.3106i −0.649994 0.736076i
\(328\) 0 0
\(329\) −2.76725 + 2.11103i −0.152564 + 0.116385i
\(330\) 0 0
\(331\) 19.3681 1.06457 0.532283 0.846567i \(-0.321334\pi\)
0.532283 + 0.846567i \(0.321334\pi\)
\(332\) 0 0
\(333\) −0.572258 + 4.58943i −0.0313595 + 0.251499i
\(334\) 0 0
\(335\) 0.365420 + 0.632927i 0.0199651 + 0.0345805i
\(336\) 0 0
\(337\) 4.58170 7.93573i 0.249581 0.432287i −0.713829 0.700320i \(-0.753040\pi\)
0.963410 + 0.268033i \(0.0863738\pi\)
\(338\) 0 0
\(339\) −25.4789 8.55171i −1.38382 0.464465i
\(340\) 0 0
\(341\) 12.3257 21.3488i 0.667476 1.15610i
\(342\) 0 0
\(343\) 6.94258 + 17.1698i 0.374864 + 0.927080i
\(344\) 0 0
\(345\) −0.745035 0.250063i −0.0401113 0.0134629i
\(346\) 0 0
\(347\) 12.8514i 0.689900i −0.938621 0.344950i \(-0.887896\pi\)
0.938621 0.344950i \(-0.112104\pi\)
\(348\) 0 0
\(349\) −17.3111 + 9.99458i −0.926643 + 0.534998i −0.885748 0.464166i \(-0.846354\pi\)
−0.0408947 + 0.999163i \(0.513021\pi\)
\(350\) 0 0
\(351\) 15.0618 1.14273i 0.803942 0.0609944i
\(352\) 0 0
\(353\) −3.52912 + 6.11262i −0.187836 + 0.325342i −0.944529 0.328429i \(-0.893481\pi\)
0.756692 + 0.653771i \(0.226814\pi\)
\(354\) 0 0
\(355\) −0.159977 + 0.0923626i −0.00849068 + 0.00490210i
\(356\) 0 0
\(357\) −25.9501 5.14539i −1.37343 0.272323i
\(358\) 0 0
\(359\) −24.3683 14.0690i −1.28611 0.742536i −0.308151 0.951337i \(-0.599710\pi\)
−0.977958 + 0.208802i \(0.933044\pi\)
\(360\) 0 0
\(361\) −3.78428 6.55457i −0.199173 0.344978i
\(362\) 0 0
\(363\) 5.53283 + 6.26557i 0.290398 + 0.328857i
\(364\) 0 0
\(365\) −0.355310 0.205139i −0.0185978 0.0107374i
\(366\) 0 0
\(367\) 31.2547 + 18.0449i 1.63148 + 0.941936i 0.983637 + 0.180163i \(0.0576624\pi\)
0.647844 + 0.761773i \(0.275671\pi\)
\(368\) 0 0
\(369\) −0.291352 + 2.33661i −0.0151672 + 0.121639i
\(370\) 0 0
\(371\) 0.836731 2.00406i 0.0434409 0.104046i
\(372\) 0 0
\(373\) 3.59811 + 6.23211i 0.186303 + 0.322686i 0.944015 0.329903i \(-0.107016\pi\)
−0.757712 + 0.652589i \(0.773683\pi\)
\(374\) 0 0
\(375\) 0.892252 0.180659i 0.0460757 0.00932917i
\(376\) 0 0
\(377\) 20.9366 1.07829
\(378\) 0 0
\(379\) −28.1915 −1.44810 −0.724051 0.689746i \(-0.757722\pi\)
−0.724051 + 0.689746i \(0.757722\pi\)
\(380\) 0 0
\(381\) 11.9727 2.42416i 0.613378 0.124194i
\(382\) 0 0
\(383\) 2.20827 + 3.82483i 0.112837 + 0.195440i 0.916913 0.399087i \(-0.130673\pi\)
−0.804076 + 0.594527i \(0.797340\pi\)
\(384\) 0 0
\(385\) 0.209630 + 0.274794i 0.0106837 + 0.0140048i
\(386\) 0 0
\(387\) −9.71718 7.35106i −0.493952 0.373675i
\(388\) 0 0
\(389\) 10.6573 + 6.15300i 0.540347 + 0.311970i 0.745220 0.666819i \(-0.232345\pi\)
−0.204873 + 0.978789i \(0.565678\pi\)
\(390\) 0 0
\(391\) 43.1478 + 24.9114i 2.18208 + 1.25982i
\(392\) 0 0
\(393\) 4.24100 + 4.80265i 0.213930 + 0.242262i
\(394\) 0 0
\(395\) −0.394342 0.683021i −0.0198415 0.0343665i
\(396\) 0 0
\(397\) −0.201000 0.116048i −0.0100879 0.00582426i 0.494948 0.868923i \(-0.335187\pi\)
−0.505035 + 0.863099i \(0.668521\pi\)
\(398\) 0 0
\(399\) −14.6686 + 4.98927i −0.734348 + 0.249776i
\(400\) 0 0
\(401\) 30.2036 17.4381i 1.50830 0.870816i 0.508344 0.861154i \(-0.330258\pi\)
0.999953 0.00966170i \(-0.00307546\pi\)
\(402\) 0 0
\(403\) −14.4202 + 24.9765i −0.718321 + 1.24417i
\(404\) 0 0
\(405\) −0.458678 0.116192i −0.0227919 0.00577363i
\(406\) 0 0
\(407\) −3.31744 + 1.91532i −0.164439 + 0.0949390i
\(408\) 0 0
\(409\) 6.67860i 0.330235i 0.986274 + 0.165118i \(0.0528004\pi\)
−0.986274 + 0.165118i \(0.947200\pi\)
\(410\) 0 0
\(411\) 6.72073 + 2.25574i 0.331509 + 0.111267i
\(412\) 0 0
\(413\) 9.76359 7.44827i 0.480435 0.366505i
\(414\) 0 0
\(415\) 0.165128 0.286009i 0.00810580 0.0140397i
\(416\) 0 0
\(417\) 23.0564 + 7.73864i 1.12908 + 0.378963i
\(418\) 0 0
\(419\) 14.8635 25.7443i 0.726128 1.25769i −0.232380 0.972625i \(-0.574651\pi\)
0.958508 0.285066i \(-0.0920154\pi\)
\(420\) 0 0
\(421\) −15.2147 26.3526i −0.741518 1.28435i −0.951804 0.306707i \(-0.900773\pi\)
0.210286 0.977640i \(-0.432560\pi\)
\(422\) 0 0
\(423\) 3.63571 1.53522i 0.176774 0.0746450i
\(424\) 0 0
\(425\) −28.8492 −1.39939
\(426\) 0 0
\(427\) −9.02512 11.8306i −0.436756 0.572523i
\(428\) 0 0
\(429\) 8.28116 + 9.37787i 0.399818 + 0.452768i
\(430\) 0 0
\(431\) −21.8782 + 12.6314i −1.05384 + 0.608432i −0.923721 0.383066i \(-0.874868\pi\)
−0.130115 + 0.991499i \(0.541535\pi\)
\(432\) 0 0
\(433\) 20.7303i 0.996234i −0.867110 0.498117i \(-0.834025\pi\)
0.867110 0.498117i \(-0.165975\pi\)
\(434\) 0 0
\(435\) −0.621751 0.208684i −0.0298107 0.0100056i
\(436\) 0 0
\(437\) 29.1793 1.39584
\(438\) 0 0
\(439\) 36.0015i 1.71826i −0.511761 0.859128i \(-0.671007\pi\)
0.511761 0.859128i \(-0.328993\pi\)
\(440\) 0 0
\(441\) −2.76638 20.8170i −0.131732 0.991285i
\(442\) 0 0
\(443\) 2.35336i 0.111811i −0.998436 0.0559057i \(-0.982195\pi\)
0.998436 0.0559057i \(-0.0178046\pi\)
\(444\) 0 0
\(445\) 0.833951 0.0395331
\(446\) 0 0
\(447\) 9.83315 8.68320i 0.465092 0.410701i
\(448\) 0 0
\(449\) 31.3789i 1.48086i 0.672132 + 0.740431i \(0.265379\pi\)
−0.672132 + 0.740431i \(0.734621\pi\)
\(450\) 0 0
\(451\) −1.68900 + 0.975145i −0.0795319 + 0.0459178i
\(452\) 0 0
\(453\) −6.99287 + 20.8345i −0.328554 + 0.978889i
\(454\) 0 0
\(455\) −0.245251 0.321488i −0.0114976 0.0150716i
\(456\) 0 0
\(457\) 17.9821 0.841168 0.420584 0.907254i \(-0.361825\pi\)
0.420584 + 0.907254i \(0.361825\pi\)
\(458\) 0 0
\(459\) 27.0389 + 12.9905i 1.26207 + 0.606342i
\(460\) 0 0
\(461\) −2.53192 4.38541i −0.117923 0.204249i 0.801021 0.598636i \(-0.204290\pi\)
−0.918945 + 0.394387i \(0.870957\pi\)
\(462\) 0 0
\(463\) −12.8461 + 22.2501i −0.597008 + 1.03405i 0.396252 + 0.918142i \(0.370311\pi\)
−0.993260 + 0.115906i \(0.963023\pi\)
\(464\) 0 0
\(465\) 0.677184 0.597990i 0.0314037 0.0277311i
\(466\) 0 0
\(467\) −19.1382 + 33.1484i −0.885612 + 1.53392i −0.0406007 + 0.999175i \(0.512927\pi\)
−0.845011 + 0.534749i \(0.820406\pi\)
\(468\) 0 0
\(469\) 29.2417 22.3074i 1.35026 1.03006i
\(470\) 0 0
\(471\) 0.376692 + 1.86044i 0.0173570 + 0.0857243i
\(472\) 0 0
\(473\) 10.0918i 0.464024i
\(474\) 0 0
\(475\) −14.6322 + 8.44793i −0.671373 + 0.387617i
\(476\) 0 0
\(477\) −1.48566 + 1.96386i −0.0680238 + 0.0899189i
\(478\) 0 0
\(479\) −4.90031 + 8.48758i −0.223901 + 0.387808i −0.955989 0.293402i \(-0.905213\pi\)
0.732088 + 0.681210i \(0.238546\pi\)
\(480\) 0 0
\(481\) 3.88115 2.24078i 0.176965 0.102171i
\(482\) 0 0
\(483\) −7.69201 + 38.7937i −0.349998 + 1.76517i
\(484\) 0 0
\(485\) 0.0827784 + 0.0477921i 0.00375877 + 0.00217013i
\(486\) 0 0
\(487\) 17.3631 + 30.0737i 0.786796 + 1.36277i 0.927920 + 0.372778i \(0.121595\pi\)
−0.141125 + 0.989992i \(0.545072\pi\)
\(488\) 0 0
\(489\) −4.88945 + 14.5676i −0.221109 + 0.658768i
\(490\) 0 0
\(491\) 23.3641 + 13.4893i 1.05441 + 0.608763i 0.923880 0.382682i \(-0.124999\pi\)
0.130528 + 0.991445i \(0.458333\pi\)
\(492\) 0 0
\(493\) 36.0080 + 20.7892i 1.62172 + 0.936299i
\(494\) 0 0
\(495\) −0.152451 0.361034i −0.00685216 0.0162273i
\(496\) 0 0
\(497\) 5.63835 + 7.39105i 0.252915 + 0.331534i
\(498\) 0 0
\(499\) −2.47273 4.28289i −0.110694 0.191728i 0.805356 0.592791i \(-0.201974\pi\)
−0.916050 + 0.401063i \(0.868641\pi\)
\(500\) 0 0
\(501\) −5.16190 + 15.3793i −0.230617 + 0.687097i
\(502\) 0 0
\(503\) 15.0173 0.669587 0.334794 0.942291i \(-0.391333\pi\)
0.334794 + 0.942291i \(0.391333\pi\)
\(504\) 0 0
\(505\) −0.0702334 −0.00312534
\(506\) 0 0
\(507\) 5.21582 + 5.90657i 0.231643 + 0.262320i
\(508\) 0 0
\(509\) 12.4297 + 21.5289i 0.550938 + 0.954252i 0.998207 + 0.0598530i \(0.0190632\pi\)
−0.447269 + 0.894399i \(0.647603\pi\)
\(510\) 0 0
\(511\) −7.95492 + 19.0529i −0.351905 + 0.842852i
\(512\) 0 0
\(513\) 17.5181 1.32908i 0.773441 0.0586803i
\(514\) 0 0
\(515\) 0.348484 + 0.201197i 0.0153561 + 0.00886582i
\(516\) 0 0
\(517\) 2.83082 + 1.63438i 0.124499 + 0.0718798i
\(518\) 0 0
\(519\) 38.9347 7.88330i 1.70904 0.346038i
\(520\) 0 0
\(521\) −8.60577 14.9056i −0.377025 0.653027i 0.613603 0.789615i \(-0.289720\pi\)
−0.990628 + 0.136588i \(0.956386\pi\)
\(522\) 0 0
\(523\) 23.4139 + 13.5180i 1.02382 + 0.591101i 0.915207 0.402984i \(-0.132027\pi\)
0.108609 + 0.994085i \(0.465360\pi\)
\(524\) 0 0
\(525\) −7.37422 21.6804i −0.321837 0.946211i
\(526\) 0 0
\(527\) −49.6012 + 28.6373i −2.16066 + 1.24746i
\(528\) 0 0
\(529\) 25.7409 44.5845i 1.11917 1.93846i
\(530\) 0 0
\(531\) −12.8277 + 5.41666i −0.556677 + 0.235063i
\(532\) 0 0
\(533\) 1.97600 1.14085i 0.0855902 0.0494155i
\(534\) 0 0
\(535\) 0.663461i 0.0286839i
\(536\) 0 0
\(537\) 15.7792 13.9339i 0.680923 0.601292i
\(538\) 0 0
\(539\) 12.3680 12.2295i 0.532729 0.526761i
\(540\) 0 0
\(541\) 19.4779 33.7367i 0.837419 1.45045i −0.0546263 0.998507i \(-0.517397\pi\)
0.892045 0.451946i \(-0.149270\pi\)
\(542\) 0 0
\(543\) −8.31406 41.0622i −0.356791 1.76215i
\(544\) 0 0
\(545\) 0.269501 0.466790i 0.0115442 0.0199951i
\(546\) 0 0
\(547\) −1.14413 1.98170i −0.0489196 0.0847313i 0.840529 0.541767i \(-0.182244\pi\)
−0.889448 + 0.457036i \(0.848911\pi\)
\(548\) 0 0
\(549\) 6.56341 + 15.5435i 0.280119 + 0.663379i
\(550\) 0 0
\(551\) 24.3509 1.03738
\(552\) 0 0
\(553\) −31.5561 + 24.0730i −1.34190 + 1.02369i
\(554\) 0 0
\(555\) −0.137592 + 0.0278590i −0.00584048 + 0.00118255i
\(556\) 0 0
\(557\) 35.2838 20.3711i 1.49502 0.863151i 0.495038 0.868871i \(-0.335154\pi\)
0.999984 + 0.00572024i \(0.00182082\pi\)
\(558\) 0 0
\(559\) 11.8067i 0.499370i
\(560\) 0 0
\(561\) 4.93056 + 24.3514i 0.208168 + 1.02812i
\(562\) 0 0
\(563\) −23.4287 −0.987403 −0.493701 0.869631i \(-0.664356\pi\)
−0.493701 + 0.869631i \(0.664356\pi\)
\(564\) 0 0
\(565\) 0.815776i 0.0343200i
\(566\) 0 0
\(567\) −2.85096 + 23.6405i −0.119729 + 0.992807i
\(568\) 0 0
\(569\) 6.67098i 0.279662i −0.990175 0.139831i \(-0.955344\pi\)
0.990175 0.139831i \(-0.0446560\pi\)
\(570\) 0 0
\(571\) 2.69555 0.112805 0.0564026 0.998408i \(-0.482037\pi\)
0.0564026 + 0.998408i \(0.482037\pi\)
\(572\) 0 0
\(573\) −4.30803 21.2768i −0.179970 0.888852i
\(574\) 0 0
\(575\) 43.1275i 1.79854i
\(576\) 0 0
\(577\) −27.9482 + 16.1359i −1.16350 + 0.671747i −0.952140 0.305662i \(-0.901122\pi\)
−0.211359 + 0.977408i \(0.567789\pi\)
\(578\) 0 0
\(579\) 5.05433 1.02338i 0.210051 0.0425300i
\(580\) 0 0
\(581\) −15.3368 6.40337i −0.636277 0.265656i
\(582\) 0 0
\(583\) −2.03958 −0.0844708
\(584\) 0 0
\(585\) 0.178356 + 0.422383i 0.00737411 + 0.0174634i
\(586\) 0 0
\(587\) −3.93933 6.82313i −0.162594 0.281621i 0.773204 0.634157i \(-0.218653\pi\)
−0.935798 + 0.352536i \(0.885319\pi\)
\(588\) 0 0
\(589\) −16.7718 + 29.0495i −0.691068 + 1.19697i
\(590\) 0 0
\(591\) −3.07966 15.2101i −0.126680 0.625659i
\(592\) 0 0
\(593\) 10.7133 18.5559i 0.439941 0.762001i −0.557743 0.830014i \(-0.688333\pi\)
0.997684 + 0.0680127i \(0.0216658\pi\)
\(594\) 0 0
\(595\) −0.102572 0.796438i −0.00420504 0.0326508i
\(596\) 0 0
\(597\) 8.93387 7.88909i 0.365639 0.322879i
\(598\) 0 0
\(599\) 1.55531i 0.0635482i 0.999495 + 0.0317741i \(0.0101157\pi\)
−0.999495 + 0.0317741i \(0.989884\pi\)
\(600\) 0 0
\(601\) −25.4367 + 14.6859i −1.03758 + 0.599050i −0.919148 0.393912i \(-0.871121\pi\)
−0.118436 + 0.992962i \(0.537788\pi\)
\(602\) 0 0
\(603\) −38.4188 + 16.2228i −1.56454 + 0.660643i
\(604\) 0 0
\(605\) −0.126860 + 0.219728i −0.00515759 + 0.00893322i
\(606\) 0 0
\(607\) 29.8433 17.2300i 1.21130 0.699345i 0.248258 0.968694i \(-0.420142\pi\)
0.963043 + 0.269349i \(0.0868085\pi\)
\(608\) 0 0
\(609\) −6.41918 + 32.3743i −0.260118 + 1.31187i
\(610\) 0 0
\(611\) −3.31185 1.91210i −0.133983 0.0773552i
\(612\) 0 0
\(613\) −14.9007 25.8087i −0.601832 1.04240i −0.992544 0.121891i \(-0.961104\pi\)
0.390711 0.920513i \(-0.372229\pi\)
\(614\) 0 0
\(615\) −0.0700522 + 0.0141838i −0.00282478 + 0.000571947i
\(616\) 0 0
\(617\) −10.9008 6.29358i −0.438850 0.253370i 0.264260 0.964451i \(-0.414872\pi\)
−0.703110 + 0.711082i \(0.748206\pi\)
\(618\) 0 0
\(619\) −37.0926 21.4154i −1.49088 0.860759i −0.490933 0.871197i \(-0.663344\pi\)
−0.999946 + 0.0104377i \(0.996678\pi\)
\(620\) 0 0
\(621\) 19.4198 40.4212i 0.779291 1.62205i
\(622\) 0 0
\(623\) −5.36071 41.6242i −0.214772 1.66764i
\(624\) 0 0
\(625\) −12.4793 21.6147i −0.499171 0.864589i
\(626\) 0 0
\(627\) 9.63162 + 10.9072i 0.384650 + 0.435590i
\(628\) 0 0
\(629\) 8.90002 0.354867
\(630\) 0 0
\(631\) 21.2015 0.844018 0.422009 0.906592i \(-0.361325\pi\)
0.422009 + 0.906592i \(0.361325\pi\)
\(632\) 0 0
\(633\) −6.07707 + 18.1060i −0.241542 + 0.719647i
\(634\) 0 0
\(635\) 0.185394 + 0.321112i 0.00735714 + 0.0127429i
\(636\) 0 0
\(637\) −14.4697 + 14.3076i −0.573309 + 0.566886i
\(638\) 0 0
\(639\) −4.10042 9.71063i −0.162210 0.384147i
\(640\) 0 0
\(641\) 10.7877 + 6.22827i 0.426088 + 0.246002i 0.697679 0.716411i \(-0.254216\pi\)
−0.271591 + 0.962413i \(0.587550\pi\)
\(642\) 0 0
\(643\) −4.57692 2.64249i −0.180496 0.104209i 0.407030 0.913415i \(-0.366565\pi\)
−0.587526 + 0.809205i \(0.699898\pi\)
\(644\) 0 0
\(645\) 0.117682 0.350621i 0.00463373 0.0138057i
\(646\) 0 0
\(647\) −17.3799 30.1028i −0.683273 1.18346i −0.973976 0.226650i \(-0.927223\pi\)
0.290704 0.956813i \(-0.406111\pi\)
\(648\) 0 0
\(649\) −9.98788 5.76651i −0.392059 0.226355i
\(650\) 0 0
\(651\) −34.1999 29.9557i −1.34040 1.17406i
\(652\) 0 0
\(653\) −26.3498 + 15.2131i −1.03115 + 0.595333i −0.917313 0.398166i \(-0.869647\pi\)
−0.113834 + 0.993500i \(0.536313\pi\)
\(654\) 0 0
\(655\) −0.0972401 + 0.168425i −0.00379949 + 0.00658090i
\(656\) 0 0
\(657\) 14.1244 18.6707i 0.551045 0.728413i
\(658\) 0 0
\(659\) −33.3127 + 19.2331i −1.29768 + 0.749215i −0.980003 0.198984i \(-0.936236\pi\)
−0.317676 + 0.948199i \(0.602903\pi\)
\(660\) 0 0
\(661\) 43.9551i 1.70965i −0.518914 0.854827i \(-0.673663\pi\)
0.518914 0.854827i \(-0.326337\pi\)
\(662\) 0 0
\(663\) −5.76838 28.4893i −0.224025 1.10643i
\(664\) 0 0
\(665\) −0.285246 0.373915i −0.0110614 0.0144998i
\(666\) 0 0
\(667\) 31.0784 53.8294i 1.20336 2.08428i
\(668\) 0 0
\(669\) 8.33325 7.35871i 0.322182 0.284504i
\(670\) 0 0
\(671\) −6.98732 + 12.1024i −0.269742 + 0.467208i
\(672\) 0 0
\(673\) −4.65619 8.06477i −0.179483 0.310874i 0.762221 0.647317i \(-0.224109\pi\)
−0.941704 + 0.336444i \(0.890776\pi\)
\(674\) 0 0
\(675\) 1.96440 + 25.8920i 0.0756099 + 0.996583i
\(676\) 0 0
\(677\) −5.54128 −0.212969 −0.106484 0.994314i \(-0.533959\pi\)
−0.106484 + 0.994314i \(0.533959\pi\)
\(678\) 0 0
\(679\) 1.85330 4.43885i 0.0711230 0.170348i
\(680\) 0 0
\(681\) −13.3465 + 39.7644i −0.511439 + 1.52378i
\(682\) 0 0
\(683\) −25.8479 + 14.9233i −0.989041 + 0.571023i −0.904988 0.425438i \(-0.860120\pi\)
−0.0840538 + 0.996461i \(0.526787\pi\)
\(684\) 0 0
\(685\) 0.215183i 0.00822170i
\(686\) 0 0
\(687\) 14.2717 12.6026i 0.544498 0.480821i
\(688\) 0 0
\(689\) 2.38615 0.0909052
\(690\) 0 0
\(691\) 32.2898i 1.22836i 0.789165 + 0.614181i \(0.210513\pi\)
−0.789165 + 0.614181i \(0.789487\pi\)
\(692\) 0 0
\(693\) −17.0400 + 9.92990i −0.647296 + 0.377206i
\(694\) 0 0
\(695\) 0.738215i 0.0280021i
\(696\) 0 0
\(697\) 4.53125 0.171633
\(698\) 0 0
\(699\) 34.8287 + 11.6899i 1.31734 + 0.442152i
\(700\) 0 0
\(701\) 27.6043i 1.04260i −0.853373 0.521301i \(-0.825447\pi\)
0.853373 0.521301i \(-0.174553\pi\)
\(702\) 0 0
\(703\) 4.51407 2.60620i 0.170251 0.0982947i
\(704\) 0 0
\(705\) 0.0792926 + 0.0897937i 0.00298633 + 0.00338182i
\(706\) 0 0
\(707\) 0.451467 + 3.50549i 0.0169791 + 0.131838i
\(708\) 0 0
\(709\) −39.4354 −1.48103 −0.740514 0.672041i \(-0.765418\pi\)
−0.740514 + 0.672041i \(0.765418\pi\)
\(710\) 0 0
\(711\) 41.4595 17.5068i 1.55485 0.656555i
\(712\) 0 0
\(713\) 42.8108 + 74.1504i 1.60328 + 2.77696i
\(714\) 0 0
\(715\) −0.189875 + 0.328874i −0.00710094 + 0.0122992i
\(716\) 0 0
\(717\) 2.17172 + 0.728915i 0.0811044 + 0.0272218i
\(718\) 0 0
\(719\) 2.51750 4.36045i 0.0938871 0.162617i −0.815257 0.579100i \(-0.803404\pi\)
0.909144 + 0.416483i \(0.136737\pi\)
\(720\) 0 0
\(721\) 7.80209 18.6869i 0.290565 0.695936i
\(722\) 0 0
\(723\) −6.07672 2.03959i −0.225996 0.0758530i
\(724\) 0 0
\(725\) 35.9910i 1.33667i
\(726\) 0 0
\(727\) −15.3848 + 8.88245i −0.570592 + 0.329432i −0.757386 0.652968i \(-0.773524\pi\)
0.186793 + 0.982399i \(0.440190\pi\)
\(728\) 0 0
\(729\) 9.81773 25.1518i 0.363620 0.931548i
\(730\) 0 0
\(731\) −11.7236 + 20.3058i −0.433611 + 0.751037i
\(732\) 0 0
\(733\) 10.6113 6.12644i 0.391938 0.226285i −0.291062 0.956704i \(-0.594008\pi\)
0.682999 + 0.730419i \(0.260675\pi\)
\(734\) 0 0
\(735\) 0.572311 0.280663i 0.0211100 0.0103524i
\(736\) 0 0
\(737\) −29.9135 17.2706i −1.10188 0.636169i
\(738\) 0 0
\(739\) −4.09505 7.09284i −0.150639 0.260914i 0.780824 0.624752i \(-0.214800\pi\)
−0.931463 + 0.363837i \(0.881466\pi\)
\(740\) 0 0
\(741\) −11.2683 12.7606i −0.413950 0.468771i
\(742\) 0 0
\(743\) −20.9393 12.0893i −0.768190 0.443514i 0.0640389 0.997947i \(-0.479602\pi\)
−0.832228 + 0.554433i \(0.812935\pi\)
\(744\) 0 0
\(745\) 0.344840 + 0.199094i 0.0126340 + 0.00729422i
\(746\) 0 0
\(747\) 15.0291 + 11.3695i 0.549886 + 0.415989i
\(748\) 0 0
\(749\) −33.1147 + 4.26479i −1.20999 + 0.155832i
\(750\) 0 0
\(751\) 6.15243 + 10.6563i 0.224505 + 0.388855i 0.956171 0.292809i \(-0.0945900\pi\)
−0.731666 + 0.681664i \(0.761257\pi\)
\(752\) 0 0
\(753\) −45.3144 + 9.17504i −1.65135 + 0.334357i
\(754\) 0 0
\(755\) −0.667073 −0.0242773
\(756\) 0 0
\(757\) −29.3788 −1.06779 −0.533895 0.845551i \(-0.679272\pi\)
−0.533895 + 0.845551i \(0.679272\pi\)
\(758\) 0 0
\(759\) 36.4037 7.37084i 1.32137 0.267545i
\(760\) 0 0
\(761\) −12.1690 21.0774i −0.441127 0.764055i 0.556646 0.830750i \(-0.312088\pi\)
−0.997773 + 0.0666951i \(0.978755\pi\)
\(762\) 0 0
\(763\) −25.0308 10.4508i −0.906176 0.378344i
\(764\) 0 0
\(765\) −0.112662 + 0.903537i −0.00407332 + 0.0326674i
\(766\) 0 0
\(767\) 11.6851 + 6.74638i 0.421923 + 0.243597i
\(768\) 0 0
\(769\) 16.0120 + 9.24453i 0.577407 + 0.333366i 0.760102 0.649803i \(-0.225149\pi\)
−0.182695 + 0.983170i \(0.558482\pi\)
\(770\) 0 0
\(771\) −2.03833 2.30828i −0.0734088 0.0831306i
\(772\) 0 0
\(773\) −9.77378 16.9287i −0.351538 0.608882i 0.634981 0.772528i \(-0.281008\pi\)
−0.986519 + 0.163646i \(0.947675\pi\)
\(774\) 0 0
\(775\) −42.9357 24.7890i −1.54230 0.890445i
\(776\) 0 0
\(777\) 2.27496 + 6.68844i 0.0816137 + 0.239947i
\(778\) 0 0
\(779\) 2.29824 1.32689i 0.0823430 0.0475407i
\(780\) 0 0
\(781\) 4.36526 7.56084i 0.156201 0.270548i
\(782\) 0 0
\(783\) 16.2063 33.7326i 0.579168 1.20550i
\(784\) 0 0
\(785\) −0.0498977 + 0.0288084i −0.00178093 + 0.00102822i
\(786\) 0 0
\(787\) 19.3435i 0.689520i −0.938691 0.344760i \(-0.887960\pi\)
0.938691 0.344760i \(-0.112040\pi\)
\(788\) 0 0
\(789\) 31.1620 + 10.4592i 1.10940 + 0.372357i
\(790\) 0 0
\(791\) −40.7171 + 5.24389i −1.44773 + 0.186451i
\(792\) 0 0
\(793\) 8.17463 14.1589i 0.290290 0.502797i
\(794\) 0 0
\(795\) −0.0708611 0.0237838i −0.00251318 0.000843523i
\(796\) 0 0
\(797\) 4.07573 7.05937i 0.144370 0.250056i −0.784768 0.619790i \(-0.787218\pi\)
0.929138 + 0.369734i \(0.120551\pi\)
\(798\) 0 0
\(799\) −3.79726 6.57705i −0.134338 0.232679i
\(800\) 0 0
\(801\) −5.88807 + 47.2215i −0.208045 + 1.66849i
\(802\) 0 0
\(803\) 19.3906 0.684279
\(804\) 0 0
\(805\) −1.19062 + 0.153338i −0.0419638 + 0.00540445i
\(806\) 0 0
\(807\) −16.4588 18.6385i −0.579377 0.656106i
\(808\) 0 0
\(809\) −5.27457 + 3.04527i −0.185444 + 0.107066i −0.589848 0.807514i \(-0.700812\pi\)
0.404404 + 0.914580i \(0.367479\pi\)
\(810\) 0 0
\(811\) 0.909185i 0.0319258i 0.999873 + 0.0159629i \(0.00508137\pi\)
−0.999873 + 0.0159629i \(0.994919\pi\)
\(812\) 0 0
\(813\) 20.8949 + 7.01316i 0.732817 + 0.245962i
\(814\) 0 0
\(815\) −0.466421 −0.0163380
\(816\) 0 0
\(817\) 13.7321i 0.480425i
\(818\) 0 0
\(819\) 19.9355 11.6172i 0.696603 0.405939i
\(820\) 0 0
\(821\) 35.8604i 1.25154i −0.780009 0.625768i \(-0.784786\pi\)
0.780009 0.625768i \(-0.215214\pi\)
\(822\) 0 0
\(823\) −6.26442 −0.218364 −0.109182 0.994022i \(-0.534823\pi\)
−0.109182 + 0.994022i \(0.534823\pi\)
\(824\) 0 0
\(825\) −16.1210 + 14.2357i −0.561261 + 0.495623i
\(826\) 0 0
\(827\) 21.2516i 0.738990i −0.929233 0.369495i \(-0.879531\pi\)
0.929233 0.369495i \(-0.120469\pi\)
\(828\) 0 0
\(829\) 21.3259 12.3125i 0.740678 0.427631i −0.0816377 0.996662i \(-0.526015\pi\)
0.822316 + 0.569031i \(0.192682\pi\)
\(830\) 0 0
\(831\) −13.7521 + 40.9727i −0.477054 + 1.42133i
\(832\) 0 0
\(833\) −39.0925 + 10.2392i −1.35447 + 0.354766i
\(834\) 0 0
\(835\) −0.492411 −0.0170406
\(836\) 0 0
\(837\) 29.0793 + 42.5669i 1.00513 + 1.47133i
\(838\) 0 0
\(839\) 6.92909 + 12.0015i 0.239219 + 0.414339i 0.960490 0.278313i \(-0.0897753\pi\)
−0.721272 + 0.692652i \(0.756442\pi\)
\(840\) 0 0
\(841\) 11.4358 19.8073i 0.394336 0.683011i
\(842\) 0 0
\(843\) −13.6914 + 12.0902i −0.471557 + 0.416410i
\(844\) 0 0
\(845\) −0.119591 + 0.207138i −0.00411407 + 0.00712578i
\(846\) 0 0
\(847\) 11.7825 + 4.91941i 0.404853 + 0.169033i
\(848\) 0 0
\(849\) 5.27401 + 26.0477i 0.181003 + 0.893954i
\(850\) 0 0
\(851\) 13.3049i 0.456087i
\(852\) 0 0
\(853\) −6.86165 + 3.96158i −0.234938 + 0.135642i −0.612848 0.790201i \(-0.709976\pi\)
0.377910 + 0.925842i \(0.376643\pi\)
\(854\) 0 0
\(855\) 0.207441 + 0.491263i 0.00709434 + 0.0168008i
\(856\) 0 0
\(857\) −21.8942 + 37.9219i −0.747892 + 1.29539i 0.200939 + 0.979604i \(0.435601\pi\)
−0.948831 + 0.315784i \(0.897733\pi\)
\(858\) 0 0
\(859\) 43.6923 25.2257i 1.49076 0.860691i 0.490817 0.871263i \(-0.336698\pi\)
0.999944 + 0.0105716i \(0.00336509\pi\)
\(860\) 0 0
\(861\) 1.15825 + 3.40528i 0.0394729 + 0.116051i
\(862\) 0 0
\(863\) −17.5324 10.1223i −0.596810 0.344568i 0.170976 0.985275i \(-0.445308\pi\)
−0.767786 + 0.640707i \(0.778641\pi\)
\(864\) 0 0
\(865\) 0.602895 + 1.04424i 0.0204990 + 0.0355054i
\(866\) 0 0
\(867\) 8.99874 26.8108i 0.305613 0.910541i
\(868\) 0 0
\(869\) 32.2810 + 18.6375i 1.09506 + 0.632233i
\(870\) 0 0
\(871\) 34.9965 + 20.2052i 1.18581 + 0.684629i
\(872\) 0 0
\(873\) −3.29063 + 4.34980i −0.111371 + 0.147218i
\(874\) 0 0
\(875\) 1.10561 0.843429i 0.0373765 0.0285131i
\(876\) 0 0
\(877\) −22.3916 38.7834i −0.756111 1.30962i −0.944820 0.327589i \(-0.893764\pi\)
0.188710 0.982033i \(-0.439569\pi\)
\(878\) 0 0
\(879\) −1.71033 + 5.09574i −0.0576880 + 0.171875i
\(880\) 0 0
\(881\) −8.83016 −0.297496 −0.148748 0.988875i \(-0.547524\pi\)
−0.148748 + 0.988875i \(0.547524\pi\)
\(882\) 0 0
\(883\) 12.5994 0.424005 0.212002 0.977269i \(-0.432002\pi\)
0.212002 + 0.977269i \(0.432002\pi\)
\(884\) 0 0
\(885\) −0.279765 0.316815i −0.00940420 0.0106496i
\(886\) 0 0
\(887\) −9.45185 16.3711i −0.317362 0.549687i 0.662575 0.748996i \(-0.269464\pi\)
−0.979937 + 0.199308i \(0.936130\pi\)
\(888\) 0 0
\(889\) 14.8356 11.3175i 0.497571 0.379578i
\(890\) 0 0
\(891\) 21.5195 6.08329i 0.720932 0.203798i
\(892\) 0 0
\(893\) −3.85193 2.22391i −0.128900 0.0744204i
\(894\) 0 0
\(895\) 0.553363 + 0.319484i 0.0184969 + 0.0106792i
\(896\) 0 0
\(897\) −42.5896 + 8.62333i −1.42202 + 0.287925i
\(898\) 0 0
\(899\) 35.7267 + 61.8804i 1.19155 + 2.06383i
\(900\) 0 0
\(901\) 4.10384 + 2.36935i 0.136719 + 0.0789345i
\(902\) 0 0
\(903\) −18.2567 3.61994i −0.607544 0.120464i
\(904\) 0 0
\(905\) 1.10131 0.635839i 0.0366086 0.0211360i
\(906\) 0 0
\(907\) 6.32064 10.9477i 0.209873 0.363511i −0.741801 0.670620i \(-0.766028\pi\)
0.951674 + 0.307109i \(0.0993615\pi\)
\(908\) 0 0
\(909\) 0.495880 3.97689i 0.0164473 0.131905i
\(910\) 0 0
\(911\) −9.36161 + 5.40493i −0.310164 + 0.179073i −0.647000 0.762490i \(-0.723977\pi\)
0.336836 + 0.941563i \(0.390643\pi\)
\(912\) 0 0
\(913\) 15.6086i 0.516568i
\(914\) 0 0
\(915\) −0.383888 + 0.338993i −0.0126909 + 0.0112068i
\(916\) 0 0
\(917\) 9.03150 + 3.77080i 0.298246 + 0.124523i
\(918\) 0 0
\(919\) 7.46198 12.9245i 0.246148 0.426341i −0.716306 0.697787i \(-0.754168\pi\)
0.962454 + 0.271446i \(0.0875017\pi\)
\(920\) 0 0
\(921\) −7.60560 37.5632i −0.250613 1.23775i
\(922\) 0 0
\(923\) −5.10702 + 8.84561i −0.168100 + 0.291157i
\(924\) 0 0
\(925\) 3.85201 + 6.67187i 0.126653 + 0.219370i
\(926\) 0 0
\(927\) −13.8530 + 18.3120i −0.454994 + 0.601445i
\(928\) 0 0
\(929\) 26.5241 0.870227 0.435114 0.900376i \(-0.356708\pi\)
0.435114 + 0.900376i \(0.356708\pi\)
\(930\) 0 0
\(931\) −16.8293 + 16.6408i −0.551558 + 0.545379i
\(932\) 0 0
\(933\) 14.1453 2.86407i 0.463096 0.0937653i
\(934\) 0 0
\(935\) −0.653116 + 0.377077i −0.0213592 + 0.0123317i
\(936\) 0 0
\(937\) 41.3437i 1.35064i −0.737525 0.675320i \(-0.764006\pi\)
0.737525 0.675320i \(-0.235994\pi\)
\(938\) 0 0
\(939\) −8.11235 40.0659i −0.264736 1.30750i
\(940\) 0 0
\(941\) −45.2975 −1.47666 −0.738328 0.674442i \(-0.764384\pi\)
−0.738328 + 0.674442i \(0.764384\pi\)
\(942\) 0 0
\(943\) 6.77391i 0.220589i
\(944\) 0 0
\(945\) −0.707814 + 0.146289i −0.0230252 + 0.00475878i
\(946\) 0 0
\(947\) 15.7268i 0.511053i −0.966802 0.255526i \(-0.917751\pi\)
0.966802 0.255526i \(-0.0822487\pi\)
\(948\) 0 0
\(949\) −22.6855 −0.736403
\(950\) 0 0
\(951\) 5.87322 + 29.0071i 0.190452 + 0.940621i
\(952\) 0 0
\(953\) 40.0501i 1.29735i 0.761065 + 0.648676i \(0.224677\pi\)
−0.761065 + 0.648676i \(0.775323\pi\)
\(954\) 0 0
\(955\) 0.570654 0.329467i 0.0184659 0.0106613i
\(956\) 0 0
\(957\) 30.3798 6.15116i 0.982040 0.198839i
\(958\) 0 0
\(959\) 10.7402 1.38321i 0.346819 0.0446663i
\(960\) 0 0
\(961\) −67.4275 −2.17508
\(962\) 0 0
\(963\) 37.5678 + 4.68434i 1.21060 + 0.150951i
\(964\) 0 0
\(965\) 0.0782652 + 0.135559i 0.00251945 + 0.00436381i
\(966\) 0 0
\(967\) 10.9257 18.9239i 0.351348 0.608553i −0.635138 0.772399i \(-0.719057\pi\)
0.986486 + 0.163846i \(0.0523900\pi\)
\(968\) 0 0
\(969\) −6.70906 33.1352i −0.215526 1.06446i
\(970\) 0 0
\(971\) 16.2295 28.1102i 0.520828 0.902100i −0.478879 0.877881i \(-0.658957\pi\)
0.999707 0.0242194i \(-0.00771003\pi\)
\(972\) 0 0
\(973\) 36.8458 4.74531i 1.18122 0.152128i
\(974\) 0 0
\(975\) 18.8603 16.6547i 0.604014 0.533377i
\(976\) 0 0
\(977\) 54.2114i 1.73438i 0.497980 + 0.867189i \(0.334075\pi\)
−0.497980 + 0.867189i \(0.665925\pi\)
\(978\) 0 0
\(979\) −34.1338 + 19.7071i −1.09092 + 0.629843i
\(980\) 0 0
\(981\) 24.5286 + 18.5560i 0.783139 + 0.592446i
\(982\) 0 0
\(983\) 8.50399 14.7293i 0.271235 0.469793i −0.697943 0.716153i \(-0.745901\pi\)
0.969178 + 0.246360i \(0.0792346\pi\)
\(984\) 0 0
\(985\) 0.407941 0.235525i 0.0129981 0.00750445i
\(986\) 0 0
\(987\) 3.97209 4.53486i 0.126433 0.144346i
\(988\) 0 0
\(989\) 30.3558 + 17.5259i 0.965258 + 0.557292i
\(990\) 0 0
\(991\) 5.21588 + 9.03416i 0.165688 + 0.286980i 0.936899 0.349599i \(-0.113682\pi\)
−0.771212 + 0.636579i \(0.780349\pi\)
\(992\) 0 0
\(993\) −32.8793 + 6.65724i −1.04339 + 0.211261i
\(994\) 0 0
\(995\) 0.313303 + 0.180886i 0.00993237 + 0.00573446i
\(996\) 0 0
\(997\) 15.6171 + 9.01654i 0.494599 + 0.285557i 0.726480 0.687187i \(-0.241155\pi\)
−0.231881 + 0.972744i \(0.574488\pi\)
\(998\) 0 0
\(999\) −0.606022 7.98772i −0.0191737 0.252720i
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 504.2.bs.a.257.2 48
3.2 odd 2 1512.2.bs.a.1097.10 48
4.3 odd 2 1008.2.ca.e.257.23 48
7.3 odd 6 504.2.cx.a.185.6 yes 48
9.2 odd 6 504.2.cx.a.425.6 yes 48
9.7 even 3 1512.2.cx.a.89.10 48
12.11 even 2 3024.2.ca.e.2609.10 48
21.17 even 6 1512.2.cx.a.17.10 48
28.3 even 6 1008.2.df.e.689.19 48
36.7 odd 6 3024.2.df.e.1601.10 48
36.11 even 6 1008.2.df.e.929.19 48
63.38 even 6 inner 504.2.bs.a.353.2 yes 48
63.52 odd 6 1512.2.bs.a.521.10 48
84.59 odd 6 3024.2.df.e.17.10 48
252.115 even 6 3024.2.ca.e.2033.10 48
252.227 odd 6 1008.2.ca.e.353.23 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.2 48 1.1 even 1 trivial
504.2.bs.a.353.2 yes 48 63.38 even 6 inner
504.2.cx.a.185.6 yes 48 7.3 odd 6
504.2.cx.a.425.6 yes 48 9.2 odd 6
1008.2.ca.e.257.23 48 4.3 odd 2
1008.2.ca.e.353.23 48 252.227 odd 6
1008.2.df.e.689.19 48 28.3 even 6
1008.2.df.e.929.19 48 36.11 even 6
1512.2.bs.a.521.10 48 63.52 odd 6
1512.2.bs.a.1097.10 48 3.2 odd 2
1512.2.cx.a.17.10 48 21.17 even 6
1512.2.cx.a.89.10 48 9.7 even 3
3024.2.ca.e.2033.10 48 252.115 even 6
3024.2.ca.e.2609.10 48 12.11 even 2
3024.2.df.e.17.10 48 84.59 odd 6
3024.2.df.e.1601.10 48 36.7 odd 6