Properties

Label 900.3.p.c.401.4
Level $900$
Weight $3$
Character 900.401
Analytic conductor $24.523$
Analytic rank $0$
Dimension $12$
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] = [900,3,Mod(101,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} - 9720 x^{4} + 43740 x^{3} - 72171 x^{2} - 118098 x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 401.4
Root \(0.459278 + 2.96464i\) of defining polynomial
Character \(\chi\) \(=\) 900.401
Dual form 900.3.p.c.101.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.459278 - 2.96464i) q^{3} +(-4.13490 - 7.16186i) q^{7} +(-8.57813 + 2.72318i) q^{9} +O(q^{10})\) \(q+(-0.459278 - 2.96464i) q^{3} +(-4.13490 - 7.16186i) q^{7} +(-8.57813 + 2.72318i) q^{9} +(3.29604 - 1.90297i) q^{11} +(9.65850 - 16.7290i) q^{13} -7.87471i q^{17} -12.3401 q^{19} +(-19.3332 + 15.5478i) q^{21} +(-1.04738 - 0.604705i) q^{23} +(12.0130 + 24.1803i) q^{27} +(10.7443 - 6.20324i) q^{29} +(-4.65731 + 8.06669i) q^{31} +(-7.15541 - 8.89757i) q^{33} -37.0571 q^{37} +(-54.0314 - 20.9507i) q^{39} +(40.1391 + 23.1743i) q^{41} +(-30.7083 - 53.1884i) q^{43} +(-79.7091 + 46.0201i) q^{47} +(-9.69479 + 16.7919i) q^{49} +(-23.3456 + 3.61668i) q^{51} +52.1540i q^{53} +(5.66752 + 36.5838i) q^{57} +(-54.4701 - 31.4484i) q^{59} +(11.6631 + 20.2011i) q^{61} +(54.9727 + 50.1752i) q^{63} +(-54.1692 + 93.8237i) q^{67} +(-1.31169 + 3.38283i) q^{69} -134.509i q^{71} +66.7798 q^{73} +(-27.2576 - 15.7372i) q^{77} +(62.2089 + 107.749i) q^{79} +(66.1686 - 46.7196i) q^{81} +(69.8921 - 40.3522i) q^{83} +(-23.3250 - 29.0040i) q^{87} +176.513i q^{89} -159.748 q^{91} +(26.0538 + 10.1024i) q^{93} +(-45.9441 - 79.5774i) q^{97} +(-23.0917 + 25.2997i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 6 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 6 q^{7} + 26 q^{9} + 48 q^{11} + 30 q^{13} + 72 q^{19} - 128 q^{21} + 78 q^{23} + 106 q^{27} + 150 q^{29} - 12 q^{31} - 96 q^{33} + 12 q^{37} + 40 q^{39} + 90 q^{41} - 114 q^{43} - 12 q^{47} + 48 q^{49} - 144 q^{51} + 158 q^{57} + 48 q^{59} - 78 q^{61} + 212 q^{63} + 168 q^{67} - 150 q^{69} + 24 q^{73} + 258 q^{77} + 120 q^{79} + 434 q^{81} - 114 q^{83} + 330 q^{87} + 120 q^{91} - 82 q^{93} - 96 q^{97} - 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) −0.459278 2.96464i −0.153093 0.988212i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.13490 7.16186i −0.590700 1.02312i −0.994138 0.108115i \(-0.965518\pi\)
0.403438 0.915007i \(-0.367815\pi\)
\(8\) 0 0
\(9\) −8.57813 + 2.72318i −0.953125 + 0.302576i
\(10\) 0 0
\(11\) 3.29604 1.90297i 0.299640 0.172997i −0.342641 0.939466i \(-0.611321\pi\)
0.642281 + 0.766469i \(0.277988\pi\)
\(12\) 0 0
\(13\) 9.65850 16.7290i 0.742962 1.28685i −0.208179 0.978091i \(-0.566754\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.87471i 0.463218i −0.972809 0.231609i \(-0.925601\pi\)
0.972809 0.231609i \(-0.0743990\pi\)
\(18\) 0 0
\(19\) −12.3401 −0.649478 −0.324739 0.945804i \(-0.605276\pi\)
−0.324739 + 0.945804i \(0.605276\pi\)
\(20\) 0 0
\(21\) −19.3332 + 15.5478i −0.920630 + 0.740369i
\(22\) 0 0
\(23\) −1.04738 0.604705i −0.0455383 0.0262915i 0.477058 0.878872i \(-0.341703\pi\)
−0.522596 + 0.852580i \(0.675036\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 12.0130 + 24.1803i 0.444925 + 0.895568i
\(28\) 0 0
\(29\) 10.7443 6.20324i 0.370494 0.213905i −0.303180 0.952933i \(-0.598048\pi\)
0.673674 + 0.739028i \(0.264715\pi\)
\(30\) 0 0
\(31\) −4.65731 + 8.06669i −0.150236 + 0.260216i −0.931314 0.364217i \(-0.881337\pi\)
0.781078 + 0.624433i \(0.214670\pi\)
\(32\) 0 0
\(33\) −7.15541 8.89757i −0.216831 0.269623i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −37.0571 −1.00154 −0.500772 0.865579i \(-0.666950\pi\)
−0.500772 + 0.865579i \(0.666950\pi\)
\(38\) 0 0
\(39\) −54.0314 20.9507i −1.38542 0.537197i
\(40\) 0 0
\(41\) 40.1391 + 23.1743i 0.979002 + 0.565227i 0.901969 0.431802i \(-0.142122\pi\)
0.0770331 + 0.997029i \(0.475455\pi\)
\(42\) 0 0
\(43\) −30.7083 53.1884i −0.714148 1.23694i −0.963287 0.268472i \(-0.913481\pi\)
0.249140 0.968468i \(-0.419852\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −79.7091 + 46.0201i −1.69594 + 0.979151i −0.746402 + 0.665496i \(0.768220\pi\)
−0.949537 + 0.313655i \(0.898446\pi\)
\(48\) 0 0
\(49\) −9.69479 + 16.7919i −0.197853 + 0.342691i
\(50\) 0 0
\(51\) −23.3456 + 3.61668i −0.457758 + 0.0709152i
\(52\) 0 0
\(53\) 52.1540i 0.984038i 0.870584 + 0.492019i \(0.163741\pi\)
−0.870584 + 0.492019i \(0.836259\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.66752 + 36.5838i 0.0994303 + 0.641822i
\(58\) 0 0
\(59\) −54.4701 31.4484i −0.923223 0.533023i −0.0385610 0.999256i \(-0.512277\pi\)
−0.884662 + 0.466233i \(0.845611\pi\)
\(60\) 0 0
\(61\) 11.6631 + 20.2011i 0.191199 + 0.331166i 0.945648 0.325193i \(-0.105429\pi\)
−0.754449 + 0.656359i \(0.772096\pi\)
\(62\) 0 0
\(63\) 54.9727 + 50.1752i 0.872583 + 0.796432i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −54.1692 + 93.8237i −0.808495 + 1.40035i 0.105411 + 0.994429i \(0.466384\pi\)
−0.913906 + 0.405926i \(0.866949\pi\)
\(68\) 0 0
\(69\) −1.31169 + 3.38283i −0.0190100 + 0.0490265i
\(70\) 0 0
\(71\) 134.509i 1.89449i −0.320510 0.947245i \(-0.603854\pi\)
0.320510 0.947245i \(-0.396146\pi\)
\(72\) 0 0
\(73\) 66.7798 0.914792 0.457396 0.889263i \(-0.348782\pi\)
0.457396 + 0.889263i \(0.348782\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −27.2576 15.7372i −0.353995 0.204379i
\(78\) 0 0
\(79\) 62.2089 + 107.749i 0.787455 + 1.36391i 0.927521 + 0.373770i \(0.121935\pi\)
−0.140066 + 0.990142i \(0.544732\pi\)
\(80\) 0 0
\(81\) 66.1686 46.7196i 0.816896 0.576785i
\(82\) 0 0
\(83\) 69.8921 40.3522i 0.842074 0.486172i −0.0158947 0.999874i \(-0.505060\pi\)
0.857969 + 0.513702i \(0.171726\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −23.3250 29.0040i −0.268103 0.333380i
\(88\) 0 0
\(89\) 176.513i 1.98330i 0.128976 + 0.991648i \(0.458831\pi\)
−0.128976 + 0.991648i \(0.541169\pi\)
\(90\) 0 0
\(91\) −159.748 −1.75547
\(92\) 0 0
\(93\) 26.0538 + 10.1024i 0.280148 + 0.108628i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −45.9441 79.5774i −0.473650 0.820386i 0.525895 0.850550i \(-0.323731\pi\)
−0.999545 + 0.0301635i \(0.990397\pi\)
\(98\) 0 0
\(99\) −23.0917 + 25.2997i −0.233250 + 0.255552i
\(100\) 0 0
\(101\) 35.2716 20.3641i 0.349223 0.201624i −0.315120 0.949052i \(-0.602045\pi\)
0.664343 + 0.747428i \(0.268711\pi\)
\(102\) 0 0
\(103\) −51.6856 + 89.5221i −0.501802 + 0.869147i 0.498196 + 0.867065i \(0.333996\pi\)
−0.999998 + 0.00208213i \(0.999337\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 21.9162i 0.204824i −0.994742 0.102412i \(-0.967344\pi\)
0.994742 0.102412i \(-0.0326560\pi\)
\(108\) 0 0
\(109\) −80.9662 −0.742810 −0.371405 0.928471i \(-0.621124\pi\)
−0.371405 + 0.928471i \(0.621124\pi\)
\(110\) 0 0
\(111\) 17.0195 + 109.861i 0.153329 + 0.989737i
\(112\) 0 0
\(113\) −103.180 59.5708i −0.913094 0.527175i −0.0316683 0.999498i \(-0.510082\pi\)
−0.881425 + 0.472324i \(0.843415\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −37.2957 + 169.805i −0.318767 + 1.45133i
\(118\) 0 0
\(119\) −56.3975 + 32.5611i −0.473929 + 0.273623i
\(120\) 0 0
\(121\) −53.2574 + 92.2445i −0.440144 + 0.762351i
\(122\) 0 0
\(123\) 50.2684 129.641i 0.408686 1.05399i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −136.157 −1.07211 −0.536053 0.844185i \(-0.680085\pi\)
−0.536053 + 0.844185i \(0.680085\pi\)
\(128\) 0 0
\(129\) −143.581 + 115.467i −1.11303 + 0.895095i
\(130\) 0 0
\(131\) −162.827 94.0080i −1.24295 0.717618i −0.273257 0.961941i \(-0.588101\pi\)
−0.969694 + 0.244323i \(0.921434\pi\)
\(132\) 0 0
\(133\) 51.0250 + 88.3779i 0.383647 + 0.664495i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 226.868 130.982i 1.65597 0.956075i 0.681424 0.731889i \(-0.261361\pi\)
0.974546 0.224186i \(-0.0719722\pi\)
\(138\) 0 0
\(139\) 41.3482 71.6171i 0.297469 0.515231i −0.678087 0.734981i \(-0.737191\pi\)
0.975556 + 0.219750i \(0.0705243\pi\)
\(140\) 0 0
\(141\) 173.041 + 215.172i 1.22724 + 1.52605i
\(142\) 0 0
\(143\) 73.5194i 0.514122i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 54.2344 + 21.0294i 0.368941 + 0.143057i
\(148\) 0 0
\(149\) −17.5623 10.1396i −0.117868 0.0680510i 0.439907 0.898043i \(-0.355011\pi\)
−0.557775 + 0.829992i \(0.688345\pi\)
\(150\) 0 0
\(151\) −9.31435 16.1329i −0.0616844 0.106841i 0.833534 0.552468i \(-0.186314\pi\)
−0.895218 + 0.445628i \(0.852981\pi\)
\(152\) 0 0
\(153\) 21.4443 + 67.5502i 0.140159 + 0.441505i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −89.4268 + 154.892i −0.569598 + 0.986572i 0.427008 + 0.904248i \(0.359568\pi\)
−0.996606 + 0.0823242i \(0.973766\pi\)
\(158\) 0 0
\(159\) 154.618 23.9532i 0.972438 0.150649i
\(160\) 0 0
\(161\) 10.0016i 0.0621216i
\(162\) 0 0
\(163\) −4.42383 −0.0271401 −0.0135700 0.999908i \(-0.504320\pi\)
−0.0135700 + 0.999908i \(0.504320\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.9679 8.06435i −0.0836399 0.0482895i 0.457597 0.889160i \(-0.348710\pi\)
−0.541237 + 0.840870i \(0.682044\pi\)
\(168\) 0 0
\(169\) −102.073 176.796i −0.603984 1.04613i
\(170\) 0 0
\(171\) 105.855 33.6043i 0.619034 0.196516i
\(172\) 0 0
\(173\) 180.607 104.274i 1.04397 0.602738i 0.123017 0.992405i \(-0.460743\pi\)
0.920956 + 0.389666i \(0.127410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −68.2160 + 175.928i −0.385401 + 0.993942i
\(178\) 0 0
\(179\) 15.0917i 0.0843110i 0.999111 + 0.0421555i \(0.0134225\pi\)
−0.999111 + 0.0421555i \(0.986578\pi\)
\(180\) 0 0
\(181\) −334.817 −1.84982 −0.924910 0.380187i \(-0.875860\pi\)
−0.924910 + 0.380187i \(0.875860\pi\)
\(182\) 0 0
\(183\) 54.5323 43.8548i 0.297991 0.239644i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −14.9853 25.9554i −0.0801355 0.138799i
\(188\) 0 0
\(189\) 123.504 186.018i 0.653458 0.984225i
\(190\) 0 0
\(191\) 262.232 151.399i 1.37294 0.792667i 0.381643 0.924310i \(-0.375358\pi\)
0.991297 + 0.131642i \(0.0420251\pi\)
\(192\) 0 0
\(193\) −84.4137 + 146.209i −0.437377 + 0.757559i −0.997486 0.0708599i \(-0.977426\pi\)
0.560110 + 0.828418i \(0.310759\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 327.080i 1.66030i −0.557537 0.830152i \(-0.688254\pi\)
0.557537 0.830152i \(-0.311746\pi\)
\(198\) 0 0
\(199\) 165.715 0.832736 0.416368 0.909196i \(-0.363303\pi\)
0.416368 + 0.909196i \(0.363303\pi\)
\(200\) 0 0
\(201\) 303.032 + 117.501i 1.50762 + 0.584580i
\(202\) 0 0
\(203\) −88.8535 51.2996i −0.437702 0.252707i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10.6313 + 2.33503i 0.0513589 + 0.0112803i
\(208\) 0 0
\(209\) −40.6734 + 23.4828i −0.194610 + 0.112358i
\(210\) 0 0
\(211\) 94.4460 163.585i 0.447611 0.775285i −0.550619 0.834757i \(-0.685608\pi\)
0.998230 + 0.0594715i \(0.0189415\pi\)
\(212\) 0 0
\(213\) −398.770 + 61.7769i −1.87216 + 0.290032i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 77.0300 0.354977
\(218\) 0 0
\(219\) −30.6705 197.978i −0.140048 0.904009i
\(220\) 0 0
\(221\) −131.736 76.0579i −0.596091 0.344153i
\(222\) 0 0
\(223\) −143.892 249.229i −0.645256 1.11762i −0.984242 0.176825i \(-0.943417\pi\)
0.338986 0.940792i \(-0.389916\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 171.961 99.2817i 0.757537 0.437364i −0.0708736 0.997485i \(-0.522579\pi\)
0.828411 + 0.560121i \(0.189245\pi\)
\(228\) 0 0
\(229\) 191.729 332.085i 0.837245 1.45015i −0.0549441 0.998489i \(-0.517498\pi\)
0.892189 0.451662i \(-0.149169\pi\)
\(230\) 0 0
\(231\) −34.1362 + 88.0366i −0.147776 + 0.381111i
\(232\) 0 0
\(233\) 240.304i 1.03135i 0.856784 + 0.515675i \(0.172459\pi\)
−0.856784 + 0.515675i \(0.827541\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 290.865 233.914i 1.22728 0.986977i
\(238\) 0 0
\(239\) −62.2301 35.9285i −0.260377 0.150329i 0.364130 0.931348i \(-0.381367\pi\)
−0.624506 + 0.781020i \(0.714700\pi\)
\(240\) 0 0
\(241\) −85.1369 147.461i −0.353265 0.611873i 0.633554 0.773698i \(-0.281595\pi\)
−0.986819 + 0.161825i \(0.948262\pi\)
\(242\) 0 0
\(243\) −168.896 174.708i −0.695047 0.718965i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −119.187 + 206.437i −0.482537 + 0.835779i
\(248\) 0 0
\(249\) −151.730 188.672i −0.609356 0.757718i
\(250\) 0 0
\(251\) 78.9688i 0.314617i −0.987550 0.157308i \(-0.949718\pi\)
0.987550 0.157308i \(-0.0502817\pi\)
\(252\) 0 0
\(253\) −4.60295 −0.0181935
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −46.4932 26.8428i −0.180907 0.104447i 0.406812 0.913512i \(-0.366641\pi\)
−0.587719 + 0.809065i \(0.699974\pi\)
\(258\) 0 0
\(259\) 153.227 + 265.398i 0.591612 + 1.02470i
\(260\) 0 0
\(261\) −75.2737 + 82.4710i −0.288405 + 0.315981i
\(262\) 0 0
\(263\) 8.94989 5.16722i 0.0340300 0.0196472i −0.482888 0.875682i \(-0.660412\pi\)
0.516918 + 0.856035i \(0.327079\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 523.298 81.0686i 1.95992 0.303628i
\(268\) 0 0
\(269\) 30.2492i 0.112451i 0.998418 + 0.0562254i \(0.0179065\pi\)
−0.998418 + 0.0562254i \(0.982093\pi\)
\(270\) 0 0
\(271\) −139.993 −0.516579 −0.258290 0.966068i \(-0.583159\pi\)
−0.258290 + 0.966068i \(0.583159\pi\)
\(272\) 0 0
\(273\) 73.3686 + 473.594i 0.268749 + 1.73478i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −27.4980 47.6279i −0.0992708 0.171942i 0.812112 0.583501i \(-0.198318\pi\)
−0.911383 + 0.411559i \(0.864984\pi\)
\(278\) 0 0
\(279\) 17.9839 81.8798i 0.0644584 0.293476i
\(280\) 0 0
\(281\) 427.134 246.606i 1.52005 0.877600i 0.520328 0.853967i \(-0.325810\pi\)
0.999721 0.0236336i \(-0.00752352\pi\)
\(282\) 0 0
\(283\) 121.143 209.825i 0.428066 0.741433i −0.568635 0.822590i \(-0.692528\pi\)
0.996701 + 0.0811572i \(0.0258616\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 383.294i 1.33552i
\(288\) 0 0
\(289\) 226.989 0.785429
\(290\) 0 0
\(291\) −214.817 + 172.756i −0.738203 + 0.593662i
\(292\) 0 0
\(293\) 112.750 + 65.0963i 0.384812 + 0.222172i 0.679910 0.733296i \(-0.262019\pi\)
−0.295098 + 0.955467i \(0.595352\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 85.6098 + 56.8390i 0.288248 + 0.191377i
\(298\) 0 0
\(299\) −20.2322 + 11.6811i −0.0676664 + 0.0390672i
\(300\) 0 0
\(301\) −253.952 + 439.857i −0.843694 + 1.46132i
\(302\) 0 0
\(303\) −76.5714 95.2146i −0.252711 0.314240i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −117.072 −0.381341 −0.190670 0.981654i \(-0.561066\pi\)
−0.190670 + 0.981654i \(0.561066\pi\)
\(308\) 0 0
\(309\) 289.138 + 112.113i 0.935723 + 0.362827i
\(310\) 0 0
\(311\) 16.4771 + 9.51306i 0.0529810 + 0.0305886i 0.526257 0.850326i \(-0.323595\pi\)
−0.473276 + 0.880914i \(0.656928\pi\)
\(312\) 0 0
\(313\) 83.5346 + 144.686i 0.266884 + 0.462256i 0.968055 0.250736i \(-0.0806727\pi\)
−0.701172 + 0.712992i \(0.747339\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 169.663 97.9549i 0.535214 0.309006i −0.207923 0.978145i \(-0.566670\pi\)
0.743137 + 0.669139i \(0.233337\pi\)
\(318\) 0 0
\(319\) 23.6092 40.8923i 0.0740100 0.128189i
\(320\) 0 0
\(321\) −64.9735 + 10.0656i −0.202410 + 0.0313571i
\(322\) 0 0
\(323\) 97.1745i 0.300850i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 37.1860 + 240.035i 0.113719 + 0.734053i
\(328\) 0 0
\(329\) 659.178 + 380.577i 2.00358 + 1.15677i
\(330\) 0 0
\(331\) 40.9636 + 70.9510i 0.123757 + 0.214353i 0.921246 0.388980i \(-0.127172\pi\)
−0.797489 + 0.603333i \(0.793839\pi\)
\(332\) 0 0
\(333\) 317.881 100.913i 0.954596 0.303043i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 263.107 455.714i 0.780732 1.35227i −0.150784 0.988567i \(-0.548180\pi\)
0.931516 0.363700i \(-0.118487\pi\)
\(338\) 0 0
\(339\) −129.217 + 333.249i −0.381173 + 0.983036i
\(340\) 0 0
\(341\) 35.4509i 0.103962i
\(342\) 0 0
\(343\) −244.872 −0.713913
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 393.556 + 227.219i 1.13417 + 0.654811i 0.944979 0.327130i \(-0.106082\pi\)
0.189186 + 0.981941i \(0.439415\pi\)
\(348\) 0 0
\(349\) 102.845 + 178.134i 0.294686 + 0.510411i 0.974912 0.222591i \(-0.0714516\pi\)
−0.680226 + 0.733003i \(0.738118\pi\)
\(350\) 0 0
\(351\) 520.541 + 32.5803i 1.48302 + 0.0928213i
\(352\) 0 0
\(353\) −87.3821 + 50.4501i −0.247541 + 0.142918i −0.618638 0.785676i \(-0.712315\pi\)
0.371097 + 0.928594i \(0.378982\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 122.434 + 152.243i 0.342952 + 0.426452i
\(358\) 0 0
\(359\) 430.783i 1.19995i −0.800018 0.599977i \(-0.795177\pi\)
0.800018 0.599977i \(-0.204823\pi\)
\(360\) 0 0
\(361\) −208.722 −0.578178
\(362\) 0 0
\(363\) 297.931 + 115.523i 0.820748 + 0.318245i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −162.310 281.129i −0.442261 0.766018i 0.555596 0.831452i \(-0.312490\pi\)
−0.997857 + 0.0654344i \(0.979157\pi\)
\(368\) 0 0
\(369\) −407.426 89.4861i −1.10414 0.242510i
\(370\) 0 0
\(371\) 373.520 215.652i 1.00679 0.581271i
\(372\) 0 0
\(373\) 68.2485 118.210i 0.182972 0.316917i −0.759919 0.650017i \(-0.774762\pi\)
0.942891 + 0.333101i \(0.108095\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 239.656i 0.635693i
\(378\) 0 0
\(379\) −26.1240 −0.0689286 −0.0344643 0.999406i \(-0.510973\pi\)
−0.0344643 + 0.999406i \(0.510973\pi\)
\(380\) 0 0
\(381\) 62.5341 + 403.657i 0.164131 + 1.05947i
\(382\) 0 0
\(383\) −421.006 243.068i −1.09923 0.634642i −0.163213 0.986591i \(-0.552186\pi\)
−0.936019 + 0.351948i \(0.885519\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 408.262 + 372.633i 1.05494 + 0.962875i
\(388\) 0 0
\(389\) 188.122 108.612i 0.483605 0.279209i −0.238313 0.971188i \(-0.576594\pi\)
0.721918 + 0.691979i \(0.243261\pi\)
\(390\) 0 0
\(391\) −4.76188 + 8.24781i −0.0121787 + 0.0210941i
\(392\) 0 0
\(393\) −203.917 + 525.897i −0.518872 + 1.33816i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −506.995 −1.27707 −0.638533 0.769595i \(-0.720458\pi\)
−0.638533 + 0.769595i \(0.720458\pi\)
\(398\) 0 0
\(399\) 238.574 191.861i 0.597929 0.480853i
\(400\) 0 0
\(401\) −433.469 250.263i −1.08097 0.624098i −0.149812 0.988715i \(-0.547867\pi\)
−0.931158 + 0.364616i \(0.881200\pi\)
\(402\) 0 0
\(403\) 89.9652 + 155.824i 0.223239 + 0.386661i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −122.142 + 70.5186i −0.300103 + 0.173264i
\(408\) 0 0
\(409\) −221.644 + 383.899i −0.541917 + 0.938628i 0.456877 + 0.889530i \(0.348968\pi\)
−0.998794 + 0.0490978i \(0.984365\pi\)
\(410\) 0 0
\(411\) −492.510 612.423i −1.19832 1.49008i
\(412\) 0 0
\(413\) 520.143i 1.25943i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −231.309 89.6901i −0.554698 0.215084i
\(418\) 0 0
\(419\) −79.3438 45.8092i −0.189365 0.109330i 0.402320 0.915499i \(-0.368204\pi\)
−0.591685 + 0.806169i \(0.701537\pi\)
\(420\) 0 0
\(421\) 1.07420 + 1.86057i 0.00255155 + 0.00441941i 0.867298 0.497789i \(-0.165854\pi\)
−0.864747 + 0.502208i \(0.832521\pi\)
\(422\) 0 0
\(423\) 558.434 611.829i 1.32017 1.44640i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 96.4516 167.059i 0.225882 0.391239i
\(428\) 0 0
\(429\) −217.958 + 33.7658i −0.508061 + 0.0787082i
\(430\) 0 0
\(431\) 770.877i 1.78858i −0.447491 0.894289i \(-0.647682\pi\)
0.447491 0.894289i \(-0.352318\pi\)
\(432\) 0 0
\(433\) −304.898 −0.704153 −0.352076 0.935971i \(-0.614524\pi\)
−0.352076 + 0.935971i \(0.614524\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.9248 + 7.46211i 0.0295761 + 0.0170758i
\(438\) 0 0
\(439\) −183.954 318.618i −0.419031 0.725782i 0.576812 0.816877i \(-0.304297\pi\)
−0.995842 + 0.0910950i \(0.970963\pi\)
\(440\) 0 0
\(441\) 37.4358 170.443i 0.0848885 0.386493i
\(442\) 0 0
\(443\) −156.585 + 90.4041i −0.353464 + 0.204073i −0.666210 0.745764i \(-0.732085\pi\)
0.312746 + 0.949837i \(0.398751\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −21.9942 + 56.7227i −0.0492041 + 0.126896i
\(448\) 0 0
\(449\) 284.309i 0.633205i 0.948558 + 0.316603i \(0.102542\pi\)
−0.948558 + 0.316603i \(0.897458\pi\)
\(450\) 0 0
\(451\) 176.400 0.391131
\(452\) 0 0
\(453\) −43.5504 + 35.0232i −0.0961377 + 0.0773138i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 119.510 + 206.998i 0.261510 + 0.452949i 0.966643 0.256126i \(-0.0824461\pi\)
−0.705133 + 0.709075i \(0.749113\pi\)
\(458\) 0 0
\(459\) 190.413 94.5987i 0.414843 0.206097i
\(460\) 0 0
\(461\) −35.0175 + 20.2174i −0.0759599 + 0.0438555i −0.537499 0.843264i \(-0.680631\pi\)
0.461539 + 0.887120i \(0.347297\pi\)
\(462\) 0 0
\(463\) −317.928 + 550.667i −0.686669 + 1.18935i 0.286240 + 0.958158i \(0.407595\pi\)
−0.972909 + 0.231188i \(0.925739\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.8159i 0.0745522i −0.999305 0.0372761i \(-0.988132\pi\)
0.999305 0.0372761i \(-0.0118681\pi\)
\(468\) 0 0
\(469\) 895.936 1.91031
\(470\) 0 0
\(471\) 500.270 + 193.980i 1.06214 + 0.411846i
\(472\) 0 0
\(473\) −202.432 116.874i −0.427975 0.247091i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −142.025 447.384i −0.297746 0.937912i
\(478\) 0 0
\(479\) −24.1869 + 13.9643i −0.0504945 + 0.0291530i −0.525035 0.851081i \(-0.675948\pi\)
0.474540 + 0.880234i \(0.342614\pi\)
\(480\) 0 0
\(481\) −357.916 + 619.929i −0.744108 + 1.28883i
\(482\) 0 0
\(483\) 29.6510 4.59350i 0.0613893 0.00951036i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 236.451 0.485527 0.242763 0.970086i \(-0.421946\pi\)
0.242763 + 0.970086i \(0.421946\pi\)
\(488\) 0 0
\(489\) 2.03177 + 13.1150i 0.00415494 + 0.0268201i
\(490\) 0 0
\(491\) 59.4344 + 34.3145i 0.121048 + 0.0698869i 0.559301 0.828964i \(-0.311070\pi\)
−0.438254 + 0.898851i \(0.644403\pi\)
\(492\) 0 0
\(493\) −48.8487 84.6085i −0.0990846 0.171620i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −963.333 + 556.180i −1.93829 + 1.11908i
\(498\) 0 0
\(499\) −148.403 + 257.042i −0.297402 + 0.515115i −0.975541 0.219819i \(-0.929453\pi\)
0.678139 + 0.734934i \(0.262787\pi\)
\(500\) 0 0
\(501\) −17.4927 + 45.1134i −0.0349156 + 0.0900467i
\(502\) 0 0
\(503\) 854.175i 1.69816i −0.528263 0.849081i \(-0.677157\pi\)
0.528263 0.849081i \(-0.322843\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −477.256 + 383.809i −0.941334 + 0.757019i
\(508\) 0 0
\(509\) −567.421 327.601i −1.11478 0.643616i −0.174714 0.984619i \(-0.555900\pi\)
−0.940062 + 0.341003i \(0.889233\pi\)
\(510\) 0 0
\(511\) −276.128 478.268i −0.540368 0.935944i
\(512\) 0 0
\(513\) −148.241 298.387i −0.288969 0.581651i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −175.150 + 303.368i −0.338781 + 0.586786i
\(518\) 0 0
\(519\) −392.083 487.545i −0.755458 0.939392i
\(520\) 0 0
\(521\) 886.683i 1.70189i 0.525258 + 0.850943i \(0.323969\pi\)
−0.525258 + 0.850943i \(0.676031\pi\)
\(522\) 0 0
\(523\) 416.390 0.796158 0.398079 0.917351i \(-0.369677\pi\)
0.398079 + 0.917351i \(0.369677\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 63.5229 + 36.6749i 0.120537 + 0.0695919i
\(528\) 0 0
\(529\) −263.769 456.861i −0.498618 0.863631i
\(530\) 0 0
\(531\) 552.891 + 121.436i 1.04123 + 0.228693i
\(532\) 0 0
\(533\) 775.367 447.658i 1.45472 0.839884i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 44.7413 6.93127i 0.0833172 0.0129074i
\(538\) 0 0
\(539\) 73.7956i 0.136912i
\(540\) 0 0
\(541\) 932.175 1.72306 0.861529 0.507708i \(-0.169507\pi\)
0.861529 + 0.507708i \(0.169507\pi\)
\(542\) 0 0
\(543\) 153.774 + 992.611i 0.283194 + 1.82801i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 320.108 + 554.443i 0.585206 + 1.01361i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(548\) 0 0
\(549\) −155.059 141.527i −0.282439 0.257790i
\(550\) 0 0
\(551\) −132.586 + 76.5485i −0.240628 + 0.138927i
\(552\) 0 0
\(553\) 514.455 891.063i 0.930299 1.61133i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 255.289i 0.458328i 0.973388 + 0.229164i \(0.0735992\pi\)
−0.973388 + 0.229164i \(0.926401\pi\)
\(558\) 0 0
\(559\) −1186.39 −2.12234
\(560\) 0 0
\(561\) −70.0658 + 56.3468i −0.124894 + 0.100440i
\(562\) 0 0
\(563\) −453.510 261.834i −0.805525 0.465070i 0.0398747 0.999205i \(-0.487304\pi\)
−0.845399 + 0.534135i \(0.820637\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −608.199 280.709i −1.07266 0.495077i
\(568\) 0 0
\(569\) −714.948 + 412.775i −1.25650 + 0.725440i −0.972392 0.233353i \(-0.925030\pi\)
−0.284107 + 0.958793i \(0.591697\pi\)
\(570\) 0 0
\(571\) 72.0237 124.749i 0.126136 0.218474i −0.796040 0.605244i \(-0.793076\pi\)
0.922176 + 0.386769i \(0.126409\pi\)
\(572\) 0 0
\(573\) −569.281 707.887i −0.993510 1.23540i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −444.934 −0.771116 −0.385558 0.922684i \(-0.625991\pi\)
−0.385558 + 0.922684i \(0.625991\pi\)
\(578\) 0 0
\(579\) 472.225 + 183.105i 0.815587 + 0.316244i
\(580\) 0 0
\(581\) −577.994 333.705i −0.994826 0.574363i
\(582\) 0 0
\(583\) 99.2476 + 171.902i 0.170236 + 0.294857i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 62.3769 36.0133i 0.106264 0.0613515i −0.445926 0.895070i \(-0.647126\pi\)
0.552190 + 0.833718i \(0.313792\pi\)
\(588\) 0 0
\(589\) 57.4716 99.5437i 0.0975748 0.169005i
\(590\) 0 0
\(591\) −969.673 + 150.221i −1.64073 + 0.254180i
\(592\) 0 0
\(593\) 1018.32i 1.71724i 0.512614 + 0.858619i \(0.328677\pi\)
−0.512614 + 0.858619i \(0.671323\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −76.1090 491.283i −0.127486 0.822920i
\(598\) 0 0
\(599\) −728.467 420.580i −1.21614 0.702138i −0.252048 0.967715i \(-0.581104\pi\)
−0.964090 + 0.265577i \(0.914438\pi\)
\(600\) 0 0
\(601\) 288.843 + 500.290i 0.480603 + 0.832429i 0.999752 0.0222543i \(-0.00708436\pi\)
−0.519149 + 0.854684i \(0.673751\pi\)
\(602\) 0 0
\(603\) 209.171 952.344i 0.346884 1.57934i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −142.103 + 246.130i −0.234108 + 0.405486i −0.959013 0.283362i \(-0.908550\pi\)
0.724905 + 0.688848i \(0.241883\pi\)
\(608\) 0 0
\(609\) −111.276 + 286.979i −0.182719 + 0.471230i
\(610\) 0 0
\(611\) 1777.94i 2.90989i
\(612\) 0 0
\(613\) −272.783 −0.444996 −0.222498 0.974933i \(-0.571421\pi\)
−0.222498 + 0.974933i \(0.571421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −68.4752 39.5342i −0.110981 0.0640748i 0.443482 0.896283i \(-0.353743\pi\)
−0.554463 + 0.832208i \(0.687076\pi\)
\(618\) 0 0
\(619\) 171.368 + 296.818i 0.276847 + 0.479513i 0.970599 0.240701i \(-0.0773772\pi\)
−0.693753 + 0.720213i \(0.744044\pi\)
\(620\) 0 0
\(621\) 2.03981 32.5903i 0.00328471 0.0524804i
\(622\) 0 0
\(623\) 1264.16 729.865i 2.02915 1.17153i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 88.2984 + 109.797i 0.140827 + 0.175114i
\(628\) 0 0
\(629\) 291.814i 0.463933i
\(630\) 0 0
\(631\) −608.714 −0.964681 −0.482341 0.875984i \(-0.660213\pi\)
−0.482341 + 0.875984i \(0.660213\pi\)
\(632\) 0 0
\(633\) −528.347 204.867i −0.834672 0.323644i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 187.274 + 324.368i 0.293994 + 0.509213i
\(638\) 0 0
\(639\) 366.292 + 1153.83i 0.573227 + 1.80569i
\(640\) 0 0
\(641\) 145.345 83.9152i 0.226748 0.130913i −0.382323 0.924029i \(-0.624876\pi\)
0.609071 + 0.793116i \(0.291542\pi\)
\(642\) 0 0
\(643\) −106.469 + 184.409i −0.165581 + 0.286795i −0.936862 0.349700i \(-0.886283\pi\)
0.771280 + 0.636496i \(0.219617\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 242.229i 0.374388i 0.982323 + 0.187194i \(0.0599393\pi\)
−0.982323 + 0.187194i \(0.940061\pi\)
\(648\) 0 0
\(649\) −239.381 −0.368846
\(650\) 0 0
\(651\) −35.3782 228.366i −0.0543443 0.350792i
\(652\) 0 0
\(653\) 278.702 + 160.909i 0.426802 + 0.246414i 0.697983 0.716114i \(-0.254081\pi\)
−0.271181 + 0.962528i \(0.587414\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −572.846 + 181.854i −0.871912 + 0.276794i
\(658\) 0 0
\(659\) 411.471 237.563i 0.624387 0.360490i −0.154188 0.988042i \(-0.549276\pi\)
0.778575 + 0.627551i \(0.215943\pi\)
\(660\) 0 0
\(661\) 579.528 1003.77i 0.876745 1.51857i 0.0218529 0.999761i \(-0.493043\pi\)
0.854892 0.518806i \(-0.173623\pi\)
\(662\) 0 0
\(663\) −164.980 + 425.481i −0.248839 + 0.641751i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −15.0045 −0.0224956
\(668\) 0 0
\(669\) −672.785 + 541.053i −1.00566 + 0.808749i
\(670\) 0 0
\(671\) 76.8842 + 44.3891i 0.114582 + 0.0661537i
\(672\) 0 0
\(673\) 72.8616 + 126.200i 0.108264 + 0.187518i 0.915067 0.403302i \(-0.132138\pi\)
−0.806803 + 0.590820i \(0.798804\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 341.713 197.288i 0.504745 0.291415i −0.225926 0.974145i \(-0.572541\pi\)
0.730671 + 0.682730i \(0.239207\pi\)
\(678\) 0 0
\(679\) −379.948 + 658.089i −0.559570 + 0.969204i
\(680\) 0 0
\(681\) −373.312 464.204i −0.548182 0.681650i
\(682\) 0 0
\(683\) 510.506i 0.747447i 0.927540 + 0.373723i \(0.121919\pi\)
−0.927540 + 0.373723i \(0.878081\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1072.57 415.888i −1.56123 0.605368i
\(688\) 0 0
\(689\) 872.486 + 503.730i 1.26631 + 0.731103i
\(690\) 0 0
\(691\) −420.136 727.697i −0.608011 1.05311i −0.991568 0.129589i \(-0.958634\pi\)
0.383556 0.923517i \(-0.374699\pi\)
\(692\) 0 0
\(693\) 276.674 + 60.7682i 0.399242 + 0.0876886i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 182.491 316.083i 0.261823 0.453491i
\(698\) 0 0
\(699\) 712.415 110.366i 1.01919 0.157892i
\(700\) 0 0
\(701\) 596.790i 0.851340i 0.904878 + 0.425670i \(0.139962\pi\)
−0.904878 + 0.425670i \(0.860038\pi\)
\(702\) 0 0
\(703\) 457.288 0.650480
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −291.689 168.407i −0.412573 0.238199i
\(708\) 0 0
\(709\) −660.644 1144.27i −0.931797 1.61392i −0.780248 0.625470i \(-0.784907\pi\)
−0.151549 0.988450i \(-0.548426\pi\)
\(710\) 0 0
\(711\) −827.057 754.879i −1.16323 1.06171i
\(712\) 0 0
\(713\) 9.75594 5.63260i 0.0136830 0.00789986i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −77.9342 + 200.991i −0.108695 + 0.280322i
\(718\) 0 0
\(719\) 833.256i 1.15891i 0.815004 + 0.579455i \(0.196734\pi\)
−0.815004 + 0.579455i \(0.803266\pi\)
\(720\) 0 0
\(721\) 854.859 1.18566
\(722\) 0 0
\(723\) −398.068 + 320.126i −0.550578 + 0.442774i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −136.886 237.094i −0.188289 0.326126i 0.756391 0.654120i \(-0.226961\pi\)
−0.944680 + 0.327994i \(0.893628\pi\)
\(728\) 0 0
\(729\) −440.376 + 580.956i −0.604083 + 0.796922i
\(730\) 0 0
\(731\) −418.843 + 241.819i −0.572973 + 0.330806i
\(732\) 0 0
\(733\) 269.002 465.925i 0.366988 0.635642i −0.622105 0.782934i \(-0.713722\pi\)
0.989093 + 0.147292i \(0.0470557\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 412.329i 0.559470i
\(738\) 0 0
\(739\) −51.4149 −0.0695736 −0.0347868 0.999395i \(-0.511075\pi\)
−0.0347868 + 0.999395i \(0.511075\pi\)
\(740\) 0 0
\(741\) 666.752 + 258.533i 0.899800 + 0.348897i
\(742\) 0 0
\(743\) −722.188 416.955i −0.971989 0.561178i −0.0721467 0.997394i \(-0.522985\pi\)
−0.899842 + 0.436216i \(0.856318\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −489.657 + 536.476i −0.655498 + 0.718174i
\(748\) 0 0
\(749\) −156.961 + 90.6212i −0.209560 + 0.120990i
\(750\) 0 0
\(751\) 492.855 853.650i 0.656265 1.13668i −0.325310 0.945607i \(-0.605469\pi\)
0.981575 0.191077i \(-0.0611979\pi\)
\(752\) 0 0
\(753\) −234.114 + 36.2686i −0.310908 + 0.0481655i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 414.403 0.547428 0.273714 0.961811i \(-0.411748\pi\)
0.273714 + 0.961811i \(0.411748\pi\)
\(758\) 0 0
\(759\) 2.11403 + 13.6461i 0.00278528 + 0.0179790i
\(760\) 0 0
\(761\) −401.140 231.598i −0.527122 0.304334i 0.212721 0.977113i \(-0.431767\pi\)
−0.739844 + 0.672779i \(0.765101\pi\)
\(762\) 0 0
\(763\) 334.787 + 579.869i 0.438778 + 0.759985i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1052.20 + 607.488i −1.37184 + 0.792031i
\(768\) 0 0
\(769\) 105.504 182.738i 0.137196 0.237630i −0.789238 0.614087i \(-0.789524\pi\)
0.926434 + 0.376457i \(0.122858\pi\)
\(770\) 0 0
\(771\) −58.2260 + 150.164i −0.0755201 + 0.194765i
\(772\) 0 0
\(773\) 1401.96i 1.81367i 0.421490 + 0.906833i \(0.361507\pi\)
−0.421490 + 0.906833i \(0.638493\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 716.433 576.155i 0.922051 0.741512i
\(778\) 0 0
\(779\) −495.319 285.973i −0.635840 0.367102i
\(780\) 0 0
\(781\) −255.966 443.347i −0.327742 0.567665i
\(782\) 0 0
\(783\) 279.068 + 185.282i 0.356409 + 0.236631i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −196.231 + 339.881i −0.249340 + 0.431870i −0.963343 0.268273i \(-0.913547\pi\)
0.714003 + 0.700143i \(0.246880\pi\)
\(788\) 0 0
\(789\) −19.4294 24.1600i −0.0246254 0.0306210i
\(790\) 0 0
\(791\) 985.276i 1.24561i
\(792\) 0 0
\(793\) 450.593 0.568213
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 938.996 + 542.129i 1.17816 + 0.680213i 0.955589 0.294703i \(-0.0952208\pi\)
0.222574 + 0.974916i \(0.428554\pi\)
\(798\) 0 0
\(799\) 362.395 + 627.686i 0.453560 + 0.785589i
\(800\) 0 0
\(801\) −480.678 1514.15i −0.600097 1.89033i
\(802\) 0 0
\(803\) 220.109 127.080i 0.274109 0.158257i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 89.6780 13.8928i 0.111125 0.0172154i
\(808\) 0 0
\(809\) 1256.56i 1.55323i −0.629975 0.776616i \(-0.716935\pi\)
0.629975 0.776616i \(-0.283065\pi\)
\(810\) 0 0
\(811\) 1478.32 1.82284 0.911418 0.411482i \(-0.134989\pi\)
0.911418 + 0.411482i \(0.134989\pi\)
\(812\) 0 0
\(813\) 64.2957 + 415.028i 0.0790845 + 0.510490i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 378.943 + 656.349i 0.463823 + 0.803365i
\(818\) 0 0
\(819\) 1370.34 435.022i 1.67318 0.531163i
\(820\) 0 0
\(821\) 757.934 437.593i 0.923184 0.533001i 0.0385346 0.999257i \(-0.487731\pi\)
0.884649 + 0.466257i \(0.154398\pi\)
\(822\) 0 0
\(823\) −242.040 + 419.225i −0.294095 + 0.509387i −0.974774 0.223195i \(-0.928351\pi\)
0.680679 + 0.732582i \(0.261685\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 262.221i 0.317075i 0.987353 + 0.158537i \(0.0506778\pi\)
−0.987353 + 0.158537i \(0.949322\pi\)
\(828\) 0 0
\(829\) 441.413 0.532465 0.266232 0.963909i \(-0.414221\pi\)
0.266232 + 0.963909i \(0.414221\pi\)
\(830\) 0 0
\(831\) −128.570 + 103.396i −0.154718 + 0.124424i
\(832\) 0 0
\(833\) 132.231 + 76.3436i 0.158741 + 0.0916490i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −251.003 15.7101i −0.299885 0.0187696i
\(838\) 0 0
\(839\) 632.454 365.147i 0.753818 0.435217i −0.0732535 0.997313i \(-0.523338\pi\)
0.827072 + 0.562096i \(0.190005\pi\)
\(840\) 0 0
\(841\) −343.540 + 595.028i −0.408489 + 0.707524i
\(842\) 0 0
\(843\) −927.269 1153.03i −1.09996 1.36778i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 880.856 1.03997
\(848\) 0 0
\(849\) −677.694 262.776i −0.798226 0.309512i
\(850\) 0 0
\(851\) 38.8129 + 22.4086i 0.0456086 + 0.0263321i
\(852\) 0 0
\(853\) −296.660 513.830i −0.347784 0.602380i 0.638071 0.769977i \(-0.279733\pi\)
−0.985856 + 0.167597i \(0.946399\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1223.11 + 706.162i −1.42720 + 0.823993i −0.996899 0.0786932i \(-0.974925\pi\)
−0.430299 + 0.902686i \(0.641592\pi\)
\(858\) 0 0
\(859\) −350.028 + 606.266i −0.407483 + 0.705781i −0.994607 0.103716i \(-0.966927\pi\)
0.587124 + 0.809497i \(0.300260\pi\)
\(860\) 0 0
\(861\) −1136.33 + 176.038i −1.31977 + 0.204458i
\(862\) 0 0
\(863\) 1185.65i 1.37387i 0.726719 + 0.686935i \(0.241044\pi\)
−0.726719 + 0.686935i \(0.758956\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −104.251 672.940i −0.120243 0.776170i
\(868\) 0 0
\(869\) 410.087 + 236.764i 0.471906 + 0.272455i
\(870\) 0 0
\(871\) 1046.39 + 1812.39i 1.20136 + 2.08082i
\(872\) 0 0
\(873\) 610.818 + 557.511i 0.699677 + 0.638616i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 747.001 1293.84i 0.851769 1.47531i −0.0278418 0.999612i \(-0.508863\pi\)
0.879611 0.475694i \(-0.157803\pi\)
\(878\) 0 0
\(879\) 141.203 364.160i 0.160641 0.414289i
\(880\) 0 0
\(881\) 89.7969i 0.101926i −0.998701 0.0509630i \(-0.983771\pi\)
0.998701 0.0509630i \(-0.0162291\pi\)
\(882\) 0 0
\(883\) −709.600 −0.803624 −0.401812 0.915722i \(-0.631620\pi\)
−0.401812 + 0.915722i \(0.631620\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −588.161 339.575i −0.663090 0.382835i 0.130363 0.991466i \(-0.458386\pi\)
−0.793453 + 0.608631i \(0.791719\pi\)
\(888\) 0 0
\(889\) 562.997 + 975.140i 0.633293 + 1.09689i
\(890\) 0 0
\(891\) 129.188 279.907i 0.144992 0.314149i
\(892\) 0 0
\(893\) 983.617 567.892i 1.10147 0.635937i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 43.9224 + 54.6164i 0.0489659 + 0.0608878i
\(898\) 0 0
\(899\) 115.562i 0.128545i
\(900\) 0 0
\(901\) 410.698 0.455824
\(902\) 0 0
\(903\) 1420.65 + 550.858i 1.57326 + 0.610031i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −125.977 218.198i −0.138894 0.240571i 0.788184 0.615439i \(-0.211021\pi\)
−0.927078 + 0.374868i \(0.877688\pi\)
\(908\) 0 0
\(909\) −247.109 + 270.736i −0.271847 + 0.297840i
\(910\) 0 0
\(911\) −815.790 + 470.997i −0.895488 + 0.517011i −0.875734 0.482794i \(-0.839622\pi\)
−0.0197547 + 0.999805i \(0.506289\pi\)
\(912\) 0 0
\(913\) 153.578 266.005i 0.168213 0.291353i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1554.85i 1.69559i
\(918\) 0 0
\(919\) −795.449 −0.865560 −0.432780 0.901500i \(-0.642467\pi\)
−0.432780 + 0.901500i \(0.642467\pi\)
\(920\) 0 0
\(921\) 53.7684 + 347.075i 0.0583805 + 0.376846i
\(922\) 0 0
\(923\) −2250.20 1299.15i −2.43792 1.40753i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 199.581 908.681i 0.215298 0.980239i
\(928\) 0 0
\(929\) 1414.36 816.581i 1.52245 0.878989i 0.522806 0.852452i \(-0.324885\pi\)
0.999648 0.0265372i \(-0.00844804\pi\)
\(930\) 0 0
\(931\) 119.634 207.213i 0.128501 0.222570i
\(932\) 0 0
\(933\) 20.6352 53.2177i 0.0221170 0.0570394i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1336.80 1.42668 0.713341 0.700817i \(-0.247181\pi\)
0.713341 + 0.700817i \(0.247181\pi\)
\(938\) 0 0
\(939\) 390.576 314.101i 0.415949 0.334506i
\(940\) 0 0
\(941\) −702.449 405.559i −0.746492 0.430987i 0.0779329 0.996959i \(-0.475168\pi\)
−0.824425 + 0.565971i \(0.808501\pi\)
\(942\) 0 0
\(943\) −28.0272 48.5446i −0.0297214 0.0514789i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 453.592 261.881i 0.478978 0.276538i −0.241013 0.970522i \(-0.577480\pi\)
0.719990 + 0.693984i \(0.244146\pi\)
\(948\) 0 0
\(949\) 644.993 1117.16i 0.679656 1.17720i
\(950\) 0 0
\(951\) −368.323 458.000i −0.387301 0.481598i
\(952\) 0 0
\(953\) 784.182i 0.822856i −0.911442 0.411428i \(-0.865030\pi\)
0.911442 0.411428i \(-0.134970\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −132.074 51.2117i −0.138008 0.0535128i
\(958\) 0 0
\(959\) −1876.15 1083.20i −1.95636 1.12951i
\(960\) 0 0
\(961\) 437.119 + 757.112i 0.454858 + 0.787838i
\(962\) 0 0
\(963\) 59.6818 + 188.000i 0.0619748 + 0.195223i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 557.120 964.960i 0.576132 0.997890i −0.419785 0.907623i \(-0.637895\pi\)
0.995918 0.0902669i \(-0.0287720\pi\)
\(968\) 0 0
\(969\) 288.087 44.6301i 0.297303 0.0460579i
\(970\) 0 0
\(971\) 1241.22i 1.27829i 0.769088 + 0.639143i \(0.220711\pi\)
−0.769088 + 0.639143i \(0.779289\pi\)
\(972\) 0 0
\(973\) −683.882 −0.702859
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 825.547 + 476.629i 0.844981 + 0.487850i 0.858954 0.512052i \(-0.171115\pi\)
−0.0139732 + 0.999902i \(0.504448\pi\)
\(978\) 0 0
\(979\) 335.900 + 581.795i 0.343105 + 0.594275i
\(980\) 0 0
\(981\) 694.539 220.486i 0.707991 0.224756i
\(982\) 0 0
\(983\) −741.717 + 428.230i −0.754544 + 0.435636i −0.827333 0.561711i \(-0.810143\pi\)
0.0727894 + 0.997347i \(0.476810\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 825.526 2129.01i 0.836399 2.15706i
\(988\) 0 0
\(989\) 74.2780i 0.0751041i
\(990\) 0 0
\(991\) −995.644 −1.00469 −0.502343 0.864668i \(-0.667529\pi\)
−0.502343 + 0.864668i \(0.667529\pi\)
\(992\) 0 0
\(993\) 191.530 154.028i 0.192880 0.155114i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −692.421 1199.31i −0.694504 1.20292i −0.970348 0.241714i \(-0.922290\pi\)
0.275843 0.961203i \(-0.411043\pi\)
\(998\) 0 0
\(999\) −445.166 896.053i −0.445612 0.896950i
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 900.3.p.c.401.4 12
3.2 odd 2 2700.3.p.c.2501.1 12
5.2 odd 4 900.3.u.c.149.1 24
5.3 odd 4 900.3.u.c.149.12 24
5.4 even 2 180.3.o.b.41.3 12
9.2 odd 6 inner 900.3.p.c.101.4 12
9.7 even 3 2700.3.p.c.1601.1 12
15.2 even 4 2700.3.u.c.449.11 24
15.8 even 4 2700.3.u.c.449.2 24
15.14 odd 2 540.3.o.b.341.6 12
20.19 odd 2 720.3.bs.b.401.4 12
45.2 even 12 900.3.u.c.749.12 24
45.4 even 6 1620.3.g.b.161.7 12
45.7 odd 12 2700.3.u.c.2249.2 24
45.14 odd 6 1620.3.g.b.161.1 12
45.29 odd 6 180.3.o.b.101.3 yes 12
45.34 even 6 540.3.o.b.521.6 12
45.38 even 12 900.3.u.c.749.1 24
45.43 odd 12 2700.3.u.c.2249.11 24
60.59 even 2 2160.3.bs.b.881.4 12
180.79 odd 6 2160.3.bs.b.1601.4 12
180.119 even 6 720.3.bs.b.641.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.o.b.41.3 12 5.4 even 2
180.3.o.b.101.3 yes 12 45.29 odd 6
540.3.o.b.341.6 12 15.14 odd 2
540.3.o.b.521.6 12 45.34 even 6
720.3.bs.b.401.4 12 20.19 odd 2
720.3.bs.b.641.4 12 180.119 even 6
900.3.p.c.101.4 12 9.2 odd 6 inner
900.3.p.c.401.4 12 1.1 even 1 trivial
900.3.u.c.149.1 24 5.2 odd 4
900.3.u.c.149.12 24 5.3 odd 4
900.3.u.c.749.1 24 45.38 even 12
900.3.u.c.749.12 24 45.2 even 12
1620.3.g.b.161.1 12 45.14 odd 6
1620.3.g.b.161.7 12 45.4 even 6
2160.3.bs.b.881.4 12 60.59 even 2
2160.3.bs.b.1601.4 12 180.79 odd 6
2700.3.p.c.1601.1 12 9.7 even 3
2700.3.p.c.2501.1 12 3.2 odd 2
2700.3.u.c.449.2 24 15.8 even 4
2700.3.u.c.449.11 24 15.2 even 4
2700.3.u.c.2249.2 24 45.7 odd 12
2700.3.u.c.2249.11 24 45.43 odd 12