Properties

Label 2100.2.bc.f.949.2
Level $2100$
Weight $2$
Character 2100.949
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(949,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bc (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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.2
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2100.949
Dual form 2100.2.bc.f.1549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{3} +(0.358719 + 2.62132i) q^{7} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{3} +(0.358719 + 2.62132i) q^{7} +(0.500000 + 0.866025i) q^{9} +(-2.12132 + 3.67423i) q^{11} -5.24264i q^{13} +(-3.67423 - 2.12132i) q^{17} +(-3.50000 - 6.06218i) q^{19} +(1.00000 - 2.44949i) q^{21} +(3.67423 - 2.12132i) q^{23} -1.00000i q^{27} +10.2426 q^{29} +(-3.74264 + 6.48244i) q^{31} +(3.67423 - 2.12132i) q^{33} +(4.54026 - 2.62132i) q^{37} +(-2.62132 + 4.54026i) q^{39} +4.24264 q^{41} -5.24264i q^{43} +(5.19615 - 3.00000i) q^{47} +(-6.74264 + 1.88064i) q^{49} +(2.12132 + 3.67423i) q^{51} +(7.34847 + 4.24264i) q^{53} +7.00000i q^{57} +(-0.878680 + 1.52192i) q^{59} +(6.24264 + 10.8126i) q^{61} +(-2.09077 + 1.62132i) q^{63} +(2.80821 + 1.62132i) q^{67} -4.24264 q^{69} +12.7279 q^{71} +(-0.655892 - 0.378680i) q^{73} +(-10.3923 - 4.24264i) q^{77} +(5.50000 + 9.52628i) q^{79} +(-0.500000 + 0.866025i) q^{81} -1.75736i q^{83} +(-8.87039 - 5.12132i) q^{87} +(-0.878680 - 1.52192i) q^{89} +(13.7426 - 1.88064i) q^{91} +(6.48244 - 3.74264i) q^{93} -16.4853i q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 28 q^{19} + 8 q^{21} + 48 q^{29} + 4 q^{31} - 4 q^{39} - 20 q^{49} - 24 q^{59} + 16 q^{61} + 44 q^{79} - 4 q^{81} - 24 q^{89} + 76 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.866025 0.500000i −0.500000 0.288675i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.358719 + 2.62132i 0.135583 + 0.990766i
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) −2.12132 + 3.67423i −0.639602 + 1.10782i 0.345918 + 0.938265i \(0.387568\pi\)
−0.985520 + 0.169559i \(0.945766\pi\)
\(12\) 0 0
\(13\) 5.24264i 1.45405i −0.686613 0.727023i \(-0.740903\pi\)
0.686613 0.727023i \(-0.259097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.67423 2.12132i −0.891133 0.514496i −0.0168199 0.999859i \(-0.505354\pi\)
−0.874313 + 0.485363i \(0.838688\pi\)
\(18\) 0 0
\(19\) −3.50000 6.06218i −0.802955 1.39076i −0.917663 0.397360i \(-0.869927\pi\)
0.114708 0.993399i \(-0.463407\pi\)
\(20\) 0 0
\(21\) 1.00000 2.44949i 0.218218 0.534522i
\(22\) 0 0
\(23\) 3.67423 2.12132i 0.766131 0.442326i −0.0653618 0.997862i \(-0.520820\pi\)
0.831493 + 0.555536i \(0.187487\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 10.2426 1.90201 0.951005 0.309175i \(-0.100053\pi\)
0.951005 + 0.309175i \(0.100053\pi\)
\(30\) 0 0
\(31\) −3.74264 + 6.48244i −0.672198 + 1.16428i 0.305081 + 0.952326i \(0.401316\pi\)
−0.977279 + 0.211955i \(0.932017\pi\)
\(32\) 0 0
\(33\) 3.67423 2.12132i 0.639602 0.369274i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.54026 2.62132i 0.746414 0.430942i −0.0779826 0.996955i \(-0.524848\pi\)
0.824397 + 0.566012i \(0.191515\pi\)
\(38\) 0 0
\(39\) −2.62132 + 4.54026i −0.419747 + 0.727023i
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 0 0
\(43\) 5.24264i 0.799495i −0.916625 0.399748i \(-0.869098\pi\)
0.916625 0.399748i \(-0.130902\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.19615 3.00000i 0.757937 0.437595i −0.0706177 0.997503i \(-0.522497\pi\)
0.828554 + 0.559908i \(0.189164\pi\)
\(48\) 0 0
\(49\) −6.74264 + 1.88064i −0.963234 + 0.268662i
\(50\) 0 0
\(51\) 2.12132 + 3.67423i 0.297044 + 0.514496i
\(52\) 0 0
\(53\) 7.34847 + 4.24264i 1.00939 + 0.582772i 0.911013 0.412378i \(-0.135302\pi\)
0.0983769 + 0.995149i \(0.468635\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.00000i 0.927173i
\(58\) 0 0
\(59\) −0.878680 + 1.52192i −0.114394 + 0.198137i −0.917537 0.397649i \(-0.869826\pi\)
0.803143 + 0.595786i \(0.203159\pi\)
\(60\) 0 0
\(61\) 6.24264 + 10.8126i 0.799288 + 1.38441i 0.920080 + 0.391730i \(0.128123\pi\)
−0.120792 + 0.992678i \(0.538543\pi\)
\(62\) 0 0
\(63\) −2.09077 + 1.62132i −0.263412 + 0.204267i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.80821 + 1.62132i 0.343077 + 0.198076i 0.661632 0.749829i \(-0.269864\pi\)
−0.318555 + 0.947904i \(0.603197\pi\)
\(68\) 0 0
\(69\) −4.24264 −0.510754
\(70\) 0 0
\(71\) 12.7279 1.51053 0.755263 0.655422i \(-0.227509\pi\)
0.755263 + 0.655422i \(0.227509\pi\)
\(72\) 0 0
\(73\) −0.655892 0.378680i −0.0767664 0.0443211i 0.461125 0.887335i \(-0.347446\pi\)
−0.537892 + 0.843014i \(0.680779\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.3923 4.24264i −1.18431 0.483494i
\(78\) 0 0
\(79\) 5.50000 + 9.52628i 0.618798 + 1.07179i 0.989705 + 0.143120i \(0.0457135\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 1.75736i 0.192895i −0.995338 0.0964476i \(-0.969252\pi\)
0.995338 0.0964476i \(-0.0307480\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.87039 5.12132i −0.951005 0.549063i
\(88\) 0 0
\(89\) −0.878680 1.52192i −0.0931399 0.161323i 0.815691 0.578488i \(-0.196357\pi\)
−0.908831 + 0.417165i \(0.863024\pi\)
\(90\) 0 0
\(91\) 13.7426 1.88064i 1.44062 0.197144i
\(92\) 0 0
\(93\) 6.48244 3.74264i 0.672198 0.388094i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.4853i 1.67383i −0.547335 0.836913i \(-0.684358\pi\)
0.547335 0.836913i \(-0.315642\pi\)
\(98\) 0 0
\(99\) −4.24264 −0.426401
\(100\) 0 0
\(101\) 8.12132 14.0665i 0.808102 1.39967i −0.106076 0.994358i \(-0.533829\pi\)
0.914177 0.405315i \(-0.132838\pi\)
\(102\) 0 0
\(103\) −7.58410 + 4.37868i −0.747283 + 0.431444i −0.824711 0.565554i \(-0.808662\pi\)
0.0774283 + 0.996998i \(0.475329\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.0227 6.36396i 1.06561 0.615227i 0.138628 0.990345i \(-0.455731\pi\)
0.926977 + 0.375117i \(0.122398\pi\)
\(108\) 0 0
\(109\) 3.74264 6.48244i 0.358480 0.620906i −0.629227 0.777221i \(-0.716628\pi\)
0.987707 + 0.156316i \(0.0499618\pi\)
\(110\) 0 0
\(111\) −5.24264 −0.497609
\(112\) 0 0
\(113\) 18.0000i 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.54026 2.62132i 0.419747 0.242341i
\(118\) 0 0
\(119\) 4.24264 10.3923i 0.388922 0.952661i
\(120\) 0 0
\(121\) −3.50000 6.06218i −0.318182 0.551107i
\(122\) 0 0
\(123\) −3.67423 2.12132i −0.331295 0.191273i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.2426i 0.997623i 0.866710 + 0.498812i \(0.166230\pi\)
−0.866710 + 0.498812i \(0.833770\pi\)
\(128\) 0 0
\(129\) −2.62132 + 4.54026i −0.230794 + 0.399748i
\(130\) 0 0
\(131\) 1.24264 + 2.15232i 0.108570 + 0.188049i 0.915191 0.403020i \(-0.132039\pi\)
−0.806621 + 0.591069i \(0.798706\pi\)
\(132\) 0 0
\(133\) 14.6354 11.3492i 1.26905 0.984104i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.67423 2.12132i −0.313911 0.181237i 0.334764 0.942302i \(-0.391343\pi\)
−0.648675 + 0.761065i \(0.724677\pi\)
\(138\) 0 0
\(139\) −1.48528 −0.125980 −0.0629900 0.998014i \(-0.520064\pi\)
−0.0629900 + 0.998014i \(0.520064\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 19.2627 + 11.1213i 1.61083 + 0.930012i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.77962 + 1.74264i 0.559173 + 0.143731i
\(148\) 0 0
\(149\) −6.00000 10.3923i −0.491539 0.851371i 0.508413 0.861113i \(-0.330232\pi\)
−0.999953 + 0.00974235i \(0.996899\pi\)
\(150\) 0 0
\(151\) −2.75736 + 4.77589i −0.224391 + 0.388656i −0.956136 0.292922i \(-0.905373\pi\)
0.731746 + 0.681578i \(0.238706\pi\)
\(152\) 0 0
\(153\) 4.24264i 0.342997i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.08052 + 5.24264i 0.724704 + 0.418408i 0.816482 0.577371i \(-0.195921\pi\)
−0.0917773 + 0.995780i \(0.529255\pi\)
\(158\) 0 0
\(159\) −4.24264 7.34847i −0.336463 0.582772i
\(160\) 0 0
\(161\) 6.87868 + 8.87039i 0.542116 + 0.699084i
\(162\) 0 0
\(163\) 6.92820 4.00000i 0.542659 0.313304i −0.203497 0.979076i \(-0.565231\pi\)
0.746156 + 0.665771i \(0.231897\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.72792i 0.520622i 0.965525 + 0.260311i \(0.0838251\pi\)
−0.965525 + 0.260311i \(0.916175\pi\)
\(168\) 0 0
\(169\) −14.4853 −1.11425
\(170\) 0 0
\(171\) 3.50000 6.06218i 0.267652 0.463586i
\(172\) 0 0
\(173\) −3.04384 + 1.75736i −0.231419 + 0.133610i −0.611226 0.791456i \(-0.709323\pi\)
0.379808 + 0.925065i \(0.375990\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.52192 0.878680i 0.114394 0.0660456i
\(178\) 0 0
\(179\) −3.00000 + 5.19615i −0.224231 + 0.388379i −0.956088 0.293079i \(-0.905320\pi\)
0.731858 + 0.681457i \(0.238654\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 0 0
\(183\) 12.4853i 0.922939i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.5885 9.00000i 1.13994 0.658145i
\(188\) 0 0
\(189\) 2.62132 0.358719i 0.190673 0.0260930i
\(190\) 0 0
\(191\) −3.00000 5.19615i −0.217072 0.375980i 0.736839 0.676068i \(-0.236317\pi\)
−0.953912 + 0.300088i \(0.902984\pi\)
\(192\) 0 0
\(193\) −13.2005 7.62132i −0.950194 0.548595i −0.0570527 0.998371i \(-0.518170\pi\)
−0.893141 + 0.449777i \(0.851504\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.75736i 0.552689i 0.961059 + 0.276344i \(0.0891231\pi\)
−0.961059 + 0.276344i \(0.910877\pi\)
\(198\) 0 0
\(199\) −3.24264 + 5.61642i −0.229865 + 0.398137i −0.957768 0.287543i \(-0.907162\pi\)
0.727903 + 0.685680i \(0.240495\pi\)
\(200\) 0 0
\(201\) −1.62132 2.80821i −0.114359 0.198076i
\(202\) 0 0
\(203\) 3.67423 + 26.8492i 0.257881 + 1.88445i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.67423 + 2.12132i 0.255377 + 0.147442i
\(208\) 0 0
\(209\) 29.6985 2.05429
\(210\) 0 0
\(211\) 5.51472 0.379649 0.189824 0.981818i \(-0.439208\pi\)
0.189824 + 0.981818i \(0.439208\pi\)
\(212\) 0 0
\(213\) −11.0227 6.36396i −0.755263 0.436051i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −18.3351 7.48528i −1.24467 0.508134i
\(218\) 0 0
\(219\) 0.378680 + 0.655892i 0.0255888 + 0.0443211i
\(220\) 0 0
\(221\) −11.1213 + 19.2627i −0.748101 + 1.29575i
\(222\) 0 0
\(223\) 24.4853i 1.63966i −0.572610 0.819828i \(-0.694069\pi\)
0.572610 0.819828i \(-0.305931\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.5673 + 13.6066i 1.56422 + 0.903102i 0.996822 + 0.0796568i \(0.0253824\pi\)
0.567396 + 0.823445i \(0.307951\pi\)
\(228\) 0 0
\(229\) −3.50000 6.06218i −0.231287 0.400600i 0.726900 0.686743i \(-0.240960\pi\)
−0.958187 + 0.286143i \(0.907627\pi\)
\(230\) 0 0
\(231\) 6.87868 + 8.87039i 0.452584 + 0.583629i
\(232\) 0 0
\(233\) −2.15232 + 1.24264i −0.141003 + 0.0814081i −0.568842 0.822447i \(-0.692608\pi\)
0.427839 + 0.903855i \(0.359275\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.0000i 0.714527i
\(238\) 0 0
\(239\) 22.9706 1.48584 0.742921 0.669379i \(-0.233440\pi\)
0.742921 + 0.669379i \(0.233440\pi\)
\(240\) 0 0
\(241\) 2.00000 3.46410i 0.128831 0.223142i −0.794393 0.607404i \(-0.792211\pi\)
0.923224 + 0.384262i \(0.125544\pi\)
\(242\) 0 0
\(243\) 0.866025 0.500000i 0.0555556 0.0320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −31.7818 + 18.3492i −2.02223 + 1.16753i
\(248\) 0 0
\(249\) −0.878680 + 1.52192i −0.0556841 + 0.0964476i
\(250\) 0 0
\(251\) 6.72792 0.424663 0.212331 0.977198i \(-0.431894\pi\)
0.212331 + 0.977198i \(0.431894\pi\)
\(252\) 0 0
\(253\) 18.0000i 1.13165i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.52192 + 0.878680i −0.0949346 + 0.0548105i −0.546716 0.837318i \(-0.684122\pi\)
0.451781 + 0.892129i \(0.350789\pi\)
\(258\) 0 0
\(259\) 8.50000 + 10.9612i 0.528164 + 0.681093i
\(260\) 0 0
\(261\) 5.12132 + 8.87039i 0.317002 + 0.549063i
\(262\) 0 0
\(263\) −7.34847 4.24264i −0.453126 0.261612i 0.256023 0.966671i \(-0.417588\pi\)
−0.709150 + 0.705058i \(0.750921\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.75736i 0.107549i
\(268\) 0 0
\(269\) 7.24264 12.5446i 0.441592 0.764859i −0.556216 0.831038i \(-0.687747\pi\)
0.997808 + 0.0661785i \(0.0210807\pi\)
\(270\) 0 0
\(271\) −13.7279 23.7775i −0.833912 1.44438i −0.894913 0.446240i \(-0.852763\pi\)
0.0610014 0.998138i \(-0.480571\pi\)
\(272\) 0 0
\(273\) −12.8418 5.24264i −0.777221 0.317299i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6.69258 3.86396i −0.402118 0.232163i 0.285279 0.958444i \(-0.407914\pi\)
−0.687397 + 0.726281i \(0.741247\pi\)
\(278\) 0 0
\(279\) −7.48528 −0.448132
\(280\) 0 0
\(281\) −4.97056 −0.296519 −0.148259 0.988948i \(-0.547367\pi\)
−0.148259 + 0.988948i \(0.547367\pi\)
\(282\) 0 0
\(283\) 1.49642 + 0.863961i 0.0889532 + 0.0513572i 0.543817 0.839204i \(-0.316979\pi\)
−0.454864 + 0.890561i \(0.650312\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.52192 + 11.1213i 0.0898360 + 0.656471i
\(288\) 0 0
\(289\) 0.500000 + 0.866025i 0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) −8.24264 + 14.2767i −0.483192 + 0.836913i
\(292\) 0 0
\(293\) 28.9706i 1.69248i 0.532803 + 0.846239i \(0.321139\pi\)
−0.532803 + 0.846239i \(0.678861\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.67423 + 2.12132i 0.213201 + 0.123091i
\(298\) 0 0
\(299\) −11.1213 19.2627i −0.643163 1.11399i
\(300\) 0 0
\(301\) 13.7426 1.88064i 0.792113 0.108398i
\(302\) 0 0
\(303\) −14.0665 + 8.12132i −0.808102 + 0.466558i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.24264i 0.299213i 0.988746 + 0.149607i \(0.0478007\pi\)
−0.988746 + 0.149607i \(0.952199\pi\)
\(308\) 0 0
\(309\) 8.75736 0.498189
\(310\) 0 0
\(311\) 10.6066 18.3712i 0.601445 1.04173i −0.391157 0.920324i \(-0.627925\pi\)
0.992602 0.121410i \(-0.0387415\pi\)
\(312\) 0 0
\(313\) −1.49642 + 0.863961i −0.0845829 + 0.0488340i −0.541695 0.840575i \(-0.682217\pi\)
0.457112 + 0.889409i \(0.348884\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.630399 + 0.363961i −0.0354067 + 0.0204421i −0.517599 0.855623i \(-0.673174\pi\)
0.482192 + 0.876065i \(0.339841\pi\)
\(318\) 0 0
\(319\) −21.7279 + 37.6339i −1.21653 + 2.10709i
\(320\) 0 0
\(321\) −12.7279 −0.710403
\(322\) 0 0
\(323\) 29.6985i 1.65247i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −6.48244 + 3.74264i −0.358480 + 0.206969i
\(328\) 0 0
\(329\) 9.72792 + 12.5446i 0.536318 + 0.691607i
\(330\) 0 0
\(331\) −8.50000 14.7224i −0.467202 0.809218i 0.532096 0.846684i \(-0.321405\pi\)
−0.999298 + 0.0374662i \(0.988071\pi\)
\(332\) 0 0
\(333\) 4.54026 + 2.62132i 0.248805 + 0.143647i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11.7279i 0.638861i −0.947610 0.319430i \(-0.896508\pi\)
0.947610 0.319430i \(-0.103492\pi\)
\(338\) 0 0
\(339\) −9.00000 + 15.5885i −0.488813 + 0.846649i
\(340\) 0 0
\(341\) −15.8787 27.5027i −0.859879 1.48935i
\(342\) 0 0
\(343\) −7.34847 17.0000i −0.396780 0.917914i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.7846 12.0000i −1.11578 0.644194i −0.175457 0.984487i \(-0.556140\pi\)
−0.940319 + 0.340293i \(0.889474\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −5.24264 −0.279831
\(352\) 0 0
\(353\) 1.52192 + 0.878680i 0.0810035 + 0.0467674i 0.539955 0.841694i \(-0.318441\pi\)
−0.458951 + 0.888462i \(0.651775\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.87039 + 6.87868i −0.469471 + 0.364058i
\(358\) 0 0
\(359\) 5.12132 + 8.87039i 0.270293 + 0.468161i 0.968937 0.247309i \(-0.0795461\pi\)
−0.698644 + 0.715470i \(0.746213\pi\)
\(360\) 0 0
\(361\) −15.0000 + 25.9808i −0.789474 + 1.36741i
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.89588 + 5.13604i 0.464361 + 0.268099i 0.713876 0.700272i \(-0.246938\pi\)
−0.249515 + 0.968371i \(0.580271\pi\)
\(368\) 0 0
\(369\) 2.12132 + 3.67423i 0.110432 + 0.191273i
\(370\) 0 0
\(371\) −8.48528 + 20.7846i −0.440534 + 1.07908i
\(372\) 0 0
\(373\) −6.69258 + 3.86396i −0.346528 + 0.200068i −0.663155 0.748482i \(-0.730783\pi\)
0.316627 + 0.948550i \(0.397450\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 53.6985i 2.76561i
\(378\) 0 0
\(379\) −30.4558 −1.56441 −0.782206 0.623020i \(-0.785905\pi\)
−0.782206 + 0.623020i \(0.785905\pi\)
\(380\) 0 0
\(381\) 5.62132 9.73641i 0.287989 0.498812i
\(382\) 0 0
\(383\) −22.0454 + 12.7279i −1.12647 + 0.650366i −0.943044 0.332668i \(-0.892051\pi\)
−0.183424 + 0.983034i \(0.558718\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.54026 2.62132i 0.230794 0.133249i
\(388\) 0 0
\(389\) 10.6066 18.3712i 0.537776 0.931455i −0.461247 0.887272i \(-0.652598\pi\)
0.999023 0.0441839i \(-0.0140687\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 2.48528i 0.125366i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.9326 8.62132i 0.749444 0.432692i −0.0760490 0.997104i \(-0.524231\pi\)
0.825493 + 0.564412i \(0.190897\pi\)
\(398\) 0 0
\(399\) −18.3492 + 2.51104i −0.918611 + 0.125709i
\(400\) 0 0
\(401\) 10.2426 + 17.7408i 0.511493 + 0.885932i 0.999911 + 0.0133223i \(0.00424074\pi\)
−0.488418 + 0.872610i \(0.662426\pi\)
\(402\) 0 0
\(403\) 33.9851 + 19.6213i 1.69292 + 0.977408i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.2426i 1.10253i
\(408\) 0 0
\(409\) 8.50000 14.7224i 0.420298 0.727977i −0.575670 0.817682i \(-0.695259\pi\)
0.995968 + 0.0897044i \(0.0285922\pi\)
\(410\) 0 0
\(411\) 2.12132 + 3.67423i 0.104637 + 0.181237i
\(412\) 0 0
\(413\) −4.30463 1.75736i −0.211817 0.0864740i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.28629 + 0.742641i 0.0629900 + 0.0363673i
\(418\) 0 0
\(419\) −2.48528 −0.121414 −0.0607070 0.998156i \(-0.519336\pi\)
−0.0607070 + 0.998156i \(0.519336\pi\)
\(420\) 0 0
\(421\) 14.5147 0.707404 0.353702 0.935358i \(-0.384923\pi\)
0.353702 + 0.935358i \(0.384923\pi\)
\(422\) 0 0
\(423\) 5.19615 + 3.00000i 0.252646 + 0.145865i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −26.1039 + 20.2426i −1.26325 + 0.979610i
\(428\) 0 0
\(429\) −11.1213 19.2627i −0.536942 0.930012i
\(430\) 0 0
\(431\) 15.7279 27.2416i 0.757587 1.31218i −0.186490 0.982457i \(-0.559711\pi\)
0.944078 0.329723i \(-0.106955\pi\)
\(432\) 0 0
\(433\) 24.7574i 1.18976i 0.803814 + 0.594881i \(0.202801\pi\)
−0.803814 + 0.594881i \(0.797199\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −25.7196 14.8492i −1.23034 0.710336i
\(438\) 0 0
\(439\) −5.00000 8.66025i −0.238637 0.413331i 0.721686 0.692220i \(-0.243367\pi\)
−0.960323 + 0.278889i \(0.910034\pi\)
\(440\) 0 0
\(441\) −5.00000 4.89898i −0.238095 0.233285i
\(442\) 0 0
\(443\) 3.04384 1.75736i 0.144617 0.0834947i −0.425946 0.904749i \(-0.640059\pi\)
0.570563 + 0.821254i \(0.306725\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.0000i 0.567581i
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −9.00000 + 15.5885i −0.423793 + 0.734032i
\(452\) 0 0
\(453\) 4.77589 2.75736i 0.224391 0.129552i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.84489 5.10660i 0.413747 0.238877i −0.278652 0.960392i \(-0.589887\pi\)
0.692398 + 0.721516i \(0.256554\pi\)
\(458\) 0 0
\(459\) −2.12132 + 3.67423i −0.0990148 + 0.171499i
\(460\) 0 0
\(461\) −6.72792 −0.313351 −0.156675 0.987650i \(-0.550078\pi\)
−0.156675 + 0.987650i \(0.550078\pi\)
\(462\) 0 0
\(463\) 35.7279i 1.66042i 0.557453 + 0.830209i \(0.311779\pi\)
−0.557453 + 0.830209i \(0.688221\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.8931 + 11.4853i −0.920542 + 0.531475i −0.883808 0.467850i \(-0.845029\pi\)
−0.0367344 + 0.999325i \(0.511696\pi\)
\(468\) 0 0
\(469\) −3.24264 + 7.94282i −0.149731 + 0.366765i
\(470\) 0 0
\(471\) −5.24264 9.08052i −0.241568 0.418408i
\(472\) 0 0
\(473\) 19.2627 + 11.1213i 0.885700 + 0.511359i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.48528i 0.388514i
\(478\) 0 0
\(479\) 6.00000 10.3923i 0.274147 0.474837i −0.695773 0.718262i \(-0.744938\pi\)
0.969920 + 0.243426i \(0.0782712\pi\)
\(480\) 0 0
\(481\) −13.7426 23.8030i −0.626610 1.08532i
\(482\) 0 0
\(483\) −1.52192 11.1213i −0.0692497 0.506038i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −10.6279 6.13604i −0.481598 0.278050i 0.239484 0.970900i \(-0.423022\pi\)
−0.721082 + 0.692850i \(0.756355\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 32.4853 1.46604 0.733020 0.680207i \(-0.238110\pi\)
0.733020 + 0.680207i \(0.238110\pi\)
\(492\) 0 0
\(493\) −37.6339 21.7279i −1.69494 0.978576i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.56575 + 33.3640i 0.204802 + 1.49658i
\(498\) 0 0
\(499\) 1.25736 + 2.17781i 0.0562871 + 0.0974922i 0.892796 0.450461i \(-0.148740\pi\)
−0.836509 + 0.547953i \(0.815407\pi\)
\(500\) 0 0
\(501\) 3.36396 5.82655i 0.150291 0.260311i
\(502\) 0 0
\(503\) 9.51472i 0.424240i −0.977244 0.212120i \(-0.931963\pi\)
0.977244 0.212120i \(-0.0680368\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.5446 + 7.24264i 0.557126 + 0.321657i
\(508\) 0 0
\(509\) 15.7279 + 27.2416i 0.697128 + 1.20746i 0.969458 + 0.245257i \(0.0788724\pi\)
−0.272330 + 0.962204i \(0.587794\pi\)
\(510\) 0 0
\(511\) 0.757359 1.85514i 0.0335036 0.0820667i
\(512\) 0 0
\(513\) −6.06218 + 3.50000i −0.267652 + 0.154529i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 25.4558i 1.11955i
\(518\) 0 0
\(519\) 3.51472 0.154279
\(520\) 0 0
\(521\) −18.7279 + 32.4377i −0.820485 + 1.42112i 0.0848363 + 0.996395i \(0.472963\pi\)
−0.905321 + 0.424727i \(0.860370\pi\)
\(522\) 0 0
\(523\) 2.80821 1.62132i 0.122794 0.0708954i −0.437345 0.899294i \(-0.644081\pi\)
0.560139 + 0.828399i \(0.310748\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 27.5027 15.8787i 1.19804 0.691686i
\(528\) 0 0
\(529\) −2.50000 + 4.33013i −0.108696 + 0.188266i
\(530\) 0 0
\(531\) −1.75736 −0.0762629
\(532\) 0 0
\(533\) 22.2426i 0.963436i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.19615 3.00000i 0.224231 0.129460i
\(538\) 0 0
\(539\) 7.39340 28.7635i 0.318456 1.23893i
\(540\) 0 0
\(541\) −13.4706 23.3317i −0.579145 1.00311i −0.995578 0.0939417i \(-0.970053\pi\)
0.416433 0.909166i \(-0.363280\pi\)
\(542\) 0 0
\(543\) 11.2583 + 6.50000i 0.483141 + 0.278942i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17.4558i 0.746358i 0.927759 + 0.373179i \(0.121732\pi\)
−0.927759 + 0.373179i \(0.878268\pi\)
\(548\) 0 0
\(549\) −6.24264 + 10.8126i −0.266429 + 0.461469i
\(550\) 0 0
\(551\) −35.8492 62.0927i −1.52723 2.64524i
\(552\) 0 0
\(553\) −22.9985 + 17.8345i −0.977995 + 0.758401i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25.9808 15.0000i −1.10084 0.635570i −0.164399 0.986394i \(-0.552568\pi\)
−0.936442 + 0.350824i \(0.885902\pi\)
\(558\) 0 0
\(559\) −27.4853 −1.16250
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) 5.19615 + 3.00000i 0.218992 + 0.126435i 0.605483 0.795858i \(-0.292980\pi\)
−0.386492 + 0.922293i \(0.626313\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.44949 1.00000i −0.102869 0.0419961i
\(568\) 0 0
\(569\) 8.84924 + 15.3273i 0.370980 + 0.642555i 0.989717 0.143043i \(-0.0456887\pi\)
−0.618737 + 0.785598i \(0.712355\pi\)
\(570\) 0 0
\(571\) −19.4706 + 33.7240i −0.814818 + 1.41131i 0.0946410 + 0.995511i \(0.469830\pi\)
−0.909459 + 0.415794i \(0.863504\pi\)
\(572\) 0 0
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.3967 + 10.6213i 0.765863 + 0.442171i 0.831397 0.555679i \(-0.187542\pi\)
−0.0655337 + 0.997850i \(0.520875\pi\)
\(578\) 0 0
\(579\) 7.62132 + 13.2005i 0.316731 + 0.548595i
\(580\) 0 0
\(581\) 4.60660 0.630399i 0.191114 0.0261534i
\(582\) 0 0
\(583\) −31.1769 + 18.0000i −1.29122 + 0.745484i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.78680i 0.115023i −0.998345 0.0575117i \(-0.981683\pi\)
0.998345 0.0575117i \(-0.0183167\pi\)
\(588\) 0 0
\(589\) 52.3970 2.15898
\(590\) 0 0
\(591\) 3.87868 6.71807i 0.159548 0.276344i
\(592\) 0 0
\(593\) −33.9596 + 19.6066i −1.39455 + 0.805147i −0.993815 0.111045i \(-0.964580\pi\)
−0.400740 + 0.916192i \(0.631247\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.61642 3.24264i 0.229865 0.132712i
\(598\) 0 0
\(599\) −7.75736 + 13.4361i −0.316957 + 0.548986i −0.979852 0.199727i \(-0.935994\pi\)
0.662894 + 0.748713i \(0.269328\pi\)
\(600\) 0 0
\(601\) 13.4853 0.550076 0.275038 0.961433i \(-0.411310\pi\)
0.275038 + 0.961433i \(0.411310\pi\)
\(602\) 0 0
\(603\) 3.24264i 0.132051i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17.9764 10.3787i 0.729640 0.421258i −0.0886507 0.996063i \(-0.528255\pi\)
0.818290 + 0.574805i \(0.194922\pi\)
\(608\) 0 0
\(609\) 10.2426 25.0892i 0.415053 1.01667i
\(610\) 0 0
\(611\) −15.7279 27.2416i −0.636284 1.10208i
\(612\) 0 0
\(613\) −39.3659 22.7279i −1.58997 0.917972i −0.993310 0.115482i \(-0.963159\pi\)
−0.596665 0.802491i \(-0.703508\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.51472i 0.383048i 0.981488 + 0.191524i \(0.0613430\pi\)
−0.981488 + 0.191524i \(0.938657\pi\)
\(618\) 0 0
\(619\) 10.9853 19.0271i 0.441536 0.764762i −0.556268 0.831003i \(-0.687767\pi\)
0.997804 + 0.0662407i \(0.0211005\pi\)
\(620\) 0 0
\(621\) −2.12132 3.67423i −0.0851257 0.147442i
\(622\) 0 0
\(623\) 3.67423 2.84924i 0.147205 0.114152i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −25.7196 14.8492i −1.02714 0.593022i
\(628\) 0 0
\(629\) −22.2426 −0.886872
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) −4.77589 2.75736i −0.189824 0.109595i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.85951 + 35.3492i 0.390648 + 1.40059i
\(638\) 0 0
\(639\) 6.36396 + 11.0227i 0.251754 + 0.436051i
\(640\) 0 0
\(641\) −12.8787 + 22.3065i −0.508677 + 0.881055i 0.491272 + 0.871006i \(0.336532\pi\)
−0.999950 + 0.0100488i \(0.996801\pi\)
\(642\) 0 0
\(643\) 5.72792i 0.225887i 0.993601 + 0.112944i \(0.0360279\pi\)
−0.993601 + 0.112944i \(0.963972\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.52192 + 0.878680i 0.0598328 + 0.0345445i 0.529618 0.848236i \(-0.322335\pi\)
−0.469785 + 0.882781i \(0.655669\pi\)
\(648\) 0 0
\(649\) −3.72792 6.45695i −0.146334 0.253457i
\(650\) 0 0
\(651\) 12.1360 + 15.6500i 0.475649 + 0.613372i
\(652\) 0 0
\(653\) 1.52192 0.878680i 0.0595572 0.0343854i −0.469926 0.882706i \(-0.655719\pi\)
0.529483 + 0.848321i \(0.322386\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.757359i 0.0295474i
\(658\) 0 0
\(659\) −7.02944 −0.273828 −0.136914 0.990583i \(-0.543718\pi\)
−0.136914 + 0.990583i \(0.543718\pi\)
\(660\) 0 0
\(661\) 17.9853 31.1514i 0.699546 1.21165i −0.269077 0.963119i \(-0.586719\pi\)
0.968624 0.248531i \(-0.0799479\pi\)
\(662\) 0 0
\(663\) 19.2627 11.1213i 0.748101 0.431916i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 37.6339 21.7279i 1.45719 0.841309i
\(668\) 0 0
\(669\) −12.2426 + 21.2049i −0.473328 + 0.819828i
\(670\) 0 0
\(671\) −52.9706 −2.04491
\(672\) 0 0
\(673\) 4.27208i 0.164677i 0.996604 + 0.0823383i \(0.0262388\pi\)
−0.996604 + 0.0823383i \(0.973761\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.0227 6.36396i 0.423637 0.244587i −0.272995 0.962015i \(-0.588014\pi\)
0.696632 + 0.717428i \(0.254681\pi\)
\(678\) 0 0
\(679\) 43.2132 5.91359i 1.65837 0.226943i
\(680\) 0 0
\(681\) −13.6066 23.5673i −0.521406 0.903102i
\(682\) 0 0
\(683\) 7.97887 + 4.60660i 0.305303 + 0.176267i 0.644823 0.764332i \(-0.276931\pi\)
−0.339520 + 0.940599i \(0.610265\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.00000i 0.267067i
\(688\) 0 0
\(689\) 22.2426 38.5254i 0.847377 1.46770i
\(690\) 0 0
\(691\) 20.4706 + 35.4561i 0.778737 + 1.34881i 0.932670 + 0.360731i \(0.117473\pi\)
−0.153933 + 0.988081i \(0.549194\pi\)
\(692\) 0 0
\(693\) −1.52192 11.1213i −0.0578129 0.422464i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −15.5885 9.00000i −0.590455 0.340899i
\(698\) 0 0
\(699\) 2.48528 0.0940020
\(700\) 0 0
\(701\) 51.2132 1.93430 0.967148 0.254214i \(-0.0818167\pi\)
0.967148 + 0.254214i \(0.0818167\pi\)
\(702\) 0 0
\(703\) −31.7818 18.3492i −1.19867 0.692055i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 39.7862 + 16.2426i 1.49631 + 0.610867i
\(708\) 0 0
\(709\) −9.75736 16.9002i −0.366445 0.634702i 0.622562 0.782571i \(-0.286092\pi\)
−0.989007 + 0.147869i \(0.952759\pi\)
\(710\) 0 0
\(711\) −5.50000 + 9.52628i −0.206266 + 0.357263i
\(712\) 0 0
\(713\) 31.7574i 1.18932i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −19.8931 11.4853i −0.742921 0.428926i
\(718\) 0 0
\(719\) 4.75736 + 8.23999i 0.177420 + 0.307300i 0.940996 0.338418i \(-0.109892\pi\)
−0.763576 + 0.645718i \(0.776558\pi\)
\(720\) 0 0
\(721\) −14.1985 18.3096i −0.528779 0.681886i
\(722\) 0 0
\(723\) −3.46410 + 2.00000i −0.128831 + 0.0743808i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 9.24264i 0.342791i −0.985202 0.171395i \(-0.945172\pi\)
0.985202 0.171395i \(-0.0548275\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −11.1213 + 19.2627i −0.411337 + 0.712456i
\(732\) 0 0
\(733\) 38.2898 22.1066i 1.41426 0.816526i 0.418478 0.908227i \(-0.362564\pi\)
0.995787 + 0.0917010i \(0.0292304\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.9142 + 6.87868i −0.438866 + 0.253379i
\(738\) 0 0
\(739\) 0.742641 1.28629i 0.0273185 0.0473170i −0.852043 0.523472i \(-0.824637\pi\)
0.879361 + 0.476155i \(0.157970\pi\)
\(740\) 0 0
\(741\) 36.6985 1.34815
\(742\) 0 0
\(743\) 27.2132i 0.998356i −0.866500 0.499178i \(-0.833635\pi\)
0.866500 0.499178i \(-0.166365\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.52192 0.878680i 0.0556841 0.0321492i
\(748\) 0 0
\(749\) 20.6360 + 26.6112i 0.754024 + 0.972351i
\(750\) 0 0
\(751\) 11.4706 + 19.8676i 0.418567 + 0.724979i 0.995796 0.0916035i \(-0.0291992\pi\)
−0.577229 + 0.816582i \(0.695866\pi\)
\(752\) 0 0
\(753\) −5.82655 3.36396i −0.212331 0.122590i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 42.9706i 1.56179i −0.624661 0.780896i \(-0.714763\pi\)
0.624661 0.780896i \(-0.285237\pi\)
\(758\) 0 0
\(759\) 9.00000 15.5885i 0.326679 0.565825i
\(760\) 0 0
\(761\) −18.8787 32.6988i −0.684352 1.18533i −0.973640 0.228090i \(-0.926752\pi\)
0.289288 0.957242i \(-0.406581\pi\)
\(762\) 0 0
\(763\) 18.3351 + 7.48528i 0.663776 + 0.270985i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.97887 + 4.60660i 0.288100 + 0.166335i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 1.75736 0.0632897
\(772\) 0 0
\(773\) 41.3081 + 23.8492i 1.48575 + 0.857798i 0.999868 0.0162275i \(-0.00516560\pi\)
0.485881 + 0.874025i \(0.338499\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.88064 13.7426i −0.0674675 0.493014i
\(778\) 0 0
\(779\) −14.8492 25.7196i −0.532029 0.921502i
\(780\) 0 0
\(781\) −27.0000 + 46.7654i −0.966136 + 1.67340i
\(782\) 0 0
\(783\) 10.2426i 0.366042i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.97204 + 5.75736i 0.355465 + 0.205228i 0.667090 0.744978i \(-0.267540\pi\)
−0.311625 + 0.950205i \(0.600873\pi\)
\(788\) 0 0
\(789\) 4.24264 + 7.34847i 0.151042 + 0.261612i
\(790\) 0 0
\(791\) 47.1838 6.45695i 1.67766 0.229583i
\(792\) 0 0
\(793\) 56.6864 32.7279i 2.01299 1.16220i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40.2426i 1.42547i 0.701435 + 0.712734i \(0.252543\pi\)
−0.701435 + 0.712734i \(0.747457\pi\)
\(798\) 0 0
\(799\) −25.4558 −0.900563
\(800\) 0 0
\(801\) 0.878680 1.52192i 0.0310466 0.0537743i
\(802\) 0 0
\(803\) 2.78272 1.60660i 0.0981999 0.0566957i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.5446 + 7.24264i −0.441592 + 0.254953i
\(808\) 0 0
\(809\) −19.9706 + 34.5900i −0.702128 + 1.21612i 0.265591 + 0.964086i \(0.414433\pi\)
−0.967718 + 0.252035i \(0.918900\pi\)
\(810\) 0 0
\(811\) −55.9411 −1.96436 −0.982179 0.187946i \(-0.939817\pi\)
−0.982179 + 0.187946i \(0.939817\pi\)
\(812\) 0 0
\(813\) 27.4558i 0.962918i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −31.7818 + 18.3492i −1.11191 + 0.641959i
\(818\) 0 0
\(819\) 8.50000 + 10.9612i 0.297014 + 0.383014i
\(820\) 0 0
\(821\) 26.8492 + 46.5043i 0.937045 + 1.62301i 0.770946 + 0.636901i \(0.219784\pi\)
0.166099 + 0.986109i \(0.446883\pi\)
\(822\) 0 0
\(823\) 4.35562 + 2.51472i 0.151827 + 0.0876576i 0.573989 0.818863i \(-0.305395\pi\)
−0.422162 + 0.906521i \(0.638729\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.45584i 0.259265i 0.991562 + 0.129633i \(0.0413798\pi\)
−0.991562 + 0.129633i \(0.958620\pi\)
\(828\) 0 0
\(829\) 5.50000 9.52628i 0.191023 0.330861i −0.754567 0.656223i \(-0.772153\pi\)
0.945589 + 0.325362i \(0.105486\pi\)
\(830\) 0 0
\(831\) 3.86396 + 6.69258i 0.134039 + 0.232163i
\(832\) 0 0
\(833\) 28.7635 + 7.39340i 0.996595 + 0.256166i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.48244 + 3.74264i 0.224066 + 0.129365i
\(838\) 0 0
\(839\) −34.2426 −1.18219 −0.591094 0.806603i \(-0.701304\pi\)
−0.591094 + 0.806603i \(0.701304\pi\)
\(840\) 0 0
\(841\) 75.9117 2.61764
\(842\) 0 0
\(843\) 4.30463 + 2.48528i 0.148259 + 0.0855976i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.6354 11.3492i 0.502878 0.389965i
\(848\) 0 0
\(849\) −0.863961 1.49642i −0.0296511 0.0513572i
\(850\) 0 0
\(851\) 11.1213 19.2627i 0.381234 0.660317i
\(852\) 0 0
\(853\) 0.272078i 0.00931577i −0.999989 0.00465789i \(-0.998517\pi\)
0.999989 0.00465789i \(-0.00148266\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.67423 2.12132i −0.125509 0.0724629i 0.435931 0.899980i \(-0.356419\pi\)
−0.561440 + 0.827517i \(0.689753\pi\)
\(858\) 0 0
\(859\) −11.0000 19.0526i −0.375315 0.650065i 0.615059 0.788481i \(-0.289132\pi\)
−0.990374 + 0.138416i \(0.955799\pi\)
\(860\) 0 0
\(861\) 4.24264 10.3923i 0.144589 0.354169i
\(862\) 0 0
\(863\) −33.3292 + 19.2426i −1.13454 + 0.655027i −0.945073 0.326860i \(-0.894009\pi\)
−0.189467 + 0.981887i \(0.560676\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000i 0.0339618i
\(868\) 0 0
\(869\) −46.6690 −1.58314
\(870\) 0 0
\(871\) 8.50000 14.7224i 0.288012 0.498851i
\(872\) 0 0
\(873\) 14.2767 8.24264i 0.483192 0.278971i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.66025 5.00000i 0.292436 0.168838i −0.346604 0.938012i \(-0.612665\pi\)
0.639040 + 0.769174i \(0.279332\pi\)
\(878\) 0 0
\(879\) 14.4853 25.0892i 0.488576 0.846239i
\(880\) 0 0
\(881\) 7.02944 0.236828 0.118414 0.992964i \(-0.462219\pi\)
0.118414 + 0.992964i \(0.462219\pi\)
\(882\) 0 0
\(883\) 19.7279i 0.663897i −0.943297 0.331949i \(-0.892294\pi\)
0.943297 0.331949i \(-0.107706\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23.5673 13.6066i 0.791313 0.456865i −0.0491114 0.998793i \(-0.515639\pi\)
0.840425 + 0.541928i \(0.182306\pi\)
\(888\) 0 0
\(889\) −29.4706 + 4.03295i −0.988411 + 0.135261i
\(890\) 0 0
\(891\) −2.12132 3.67423i −0.0710669 0.123091i
\(892\) 0 0
\(893\) −36.3731 21.0000i −1.21718 0.702738i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 22.2426i 0.742660i
\(898\) 0 0
\(899\) −38.3345 + 66.3973i −1.27853 + 2.21448i
\(900\) 0 0
\(901\) −18.0000 31.1769i −0.599667 1.03865i
\(902\) 0 0
\(903\) −12.8418 5.24264i −0.427348 0.174464i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −48.2618 27.8640i −1.60251 0.925208i −0.990984 0.133981i \(-0.957224\pi\)
−0.611523 0.791227i \(-0.709443\pi\)
\(908\) 0 0
\(909\) 16.2426 0.538734
\(910\) 0 0
\(911\) 8.78680 0.291120 0.145560 0.989349i \(-0.453502\pi\)
0.145560 + 0.989349i \(0.453502\pi\)
\(912\) 0 0
\(913\) 6.45695 + 3.72792i 0.213694 + 0.123376i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.19615 + 4.02944i −0.171592 + 0.133064i
\(918\) 0 0
\(919\) 0.0147186 + 0.0254934i 0.000485523 + 0.000840950i 0.866268 0.499579i \(-0.166512\pi\)
−0.865783 + 0.500420i \(0.833179\pi\)
\(920\) 0 0
\(921\) 2.62132 4.54026i 0.0863754 0.149607i
\(922\) 0 0
\(923\) 66.7279i 2.19638i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −7.58410 4.37868i −0.249094 0.143815i
\(928\) 0 0
\(929\) 5.63604 + 9.76191i 0.184912 + 0.320278i 0.943547 0.331239i \(-0.107467\pi\)
−0.758635 + 0.651516i \(0.774133\pi\)
\(930\) 0 0
\(931\) 35.0000 + 34.2929i 1.14708 + 1.12390i
\(932\) 0 0
\(933\) −18.3712 + 10.6066i −0.601445 + 0.347245i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 40.6985i 1.32956i −0.747039 0.664781i \(-0.768525\pi\)
0.747039 0.664781i \(-0.231475\pi\)
\(938\) 0 0
\(939\) 1.72792 0.0563886
\(940\) 0 0
\(941\) −5.84924 + 10.1312i −0.190680 + 0.330267i −0.945476 0.325693i \(-0.894403\pi\)
0.754796 + 0.655960i \(0.227736\pi\)
\(942\) 0 0
\(943\) 15.5885 9.00000i 0.507630 0.293080i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −33.9596 + 19.6066i −1.10354 + 0.637129i −0.937148 0.348931i \(-0.886545\pi\)
−0.166391 + 0.986060i \(0.553211\pi\)
\(948\) 0 0
\(949\) −1.98528 + 3.43861i −0.0644450 + 0.111622i
\(950\) 0 0
\(951\) 0.727922 0.0236045
\(952\) 0 0
\(953\) 44.4853i 1.44102i −0.693445 0.720510i \(-0.743908\pi\)
0.693445 0.720510i \(-0.256092\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 37.6339 21.7279i 1.21653 0.702364i
\(958\) 0 0
\(959\) 4.24264 10.3923i 0.137002 0.335585i
\(960\) 0 0
\(961\) −12.5147 21.6761i −0.403701 0.699230i
\(962\) 0 0
\(963\) 11.0227 + 6.36396i 0.355202 + 0.205076i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 26.7574i 0.860459i 0.902720 + 0.430229i \(0.141567\pi\)
−0.902720 + 0.430229i \(0.858433\pi\)
\(968\) 0 0
\(969\) 14.8492 25.7196i 0.477026 0.826234i
\(970\) 0 0
\(971\) −9.51472 16.4800i −0.305342 0.528868i 0.671996 0.740555i \(-0.265437\pi\)
−0.977337 + 0.211688i \(0.932104\pi\)
\(972\) 0 0
\(973\) −0.532799 3.89340i −0.0170808 0.124817i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −44.3519 25.6066i −1.41894 0.819228i −0.422738 0.906252i \(-0.638931\pi\)
−0.996206 + 0.0870242i \(0.972264\pi\)
\(978\) 0 0
\(979\) 7.45584 0.238290
\(980\) 0 0
\(981\) 7.48528 0.238987
\(982\) 0 0
\(983\) −15.3273 8.84924i −0.488866 0.282247i 0.235238 0.971938i \(-0.424413\pi\)
−0.724104 + 0.689691i \(0.757746\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.15232 15.7279i −0.0685090 0.500625i
\(988\) 0 0
\(989\) −11.1213 19.2627i −0.353637 0.612518i
\(990\) 0 0
\(991\) 26.4706 45.8484i 0.840865 1.45642i −0.0482991 0.998833i \(-0.515380\pi\)
0.889164 0.457588i \(-0.151287\pi\)
\(992\) 0 0
\(993\) 17.0000i 0.539479i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.235626 0.136039i −0.00746236 0.00430840i 0.496264 0.868172i \(-0.334705\pi\)
−0.503727 + 0.863863i \(0.668038\pi\)
\(998\) 0 0
\(999\) −2.62132 4.54026i −0.0829349 0.143647i
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 2100.2.bc.f.949.2 8
5.2 odd 4 420.2.q.d.361.1 yes 4
5.3 odd 4 2100.2.q.k.1201.2 4
5.4 even 2 inner 2100.2.bc.f.949.3 8
7.2 even 3 inner 2100.2.bc.f.1549.3 8
15.2 even 4 1260.2.s.e.361.1 4
20.7 even 4 1680.2.bg.t.1201.2 4
35.2 odd 12 420.2.q.d.121.1 4
35.9 even 6 inner 2100.2.bc.f.1549.2 8
35.12 even 12 2940.2.q.q.961.1 4
35.17 even 12 2940.2.a.p.1.2 2
35.23 odd 12 2100.2.q.k.1801.2 4
35.27 even 4 2940.2.q.q.361.1 4
35.32 odd 12 2940.2.a.r.1.2 2
105.2 even 12 1260.2.s.e.541.1 4
105.17 odd 12 8820.2.a.bf.1.1 2
105.32 even 12 8820.2.a.bk.1.1 2
140.107 even 12 1680.2.bg.t.961.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.d.121.1 4 35.2 odd 12
420.2.q.d.361.1 yes 4 5.2 odd 4
1260.2.s.e.361.1 4 15.2 even 4
1260.2.s.e.541.1 4 105.2 even 12
1680.2.bg.t.961.2 4 140.107 even 12
1680.2.bg.t.1201.2 4 20.7 even 4
2100.2.q.k.1201.2 4 5.3 odd 4
2100.2.q.k.1801.2 4 35.23 odd 12
2100.2.bc.f.949.2 8 1.1 even 1 trivial
2100.2.bc.f.949.3 8 5.4 even 2 inner
2100.2.bc.f.1549.2 8 35.9 even 6 inner
2100.2.bc.f.1549.3 8 7.2 even 3 inner
2940.2.a.p.1.2 2 35.17 even 12
2940.2.a.r.1.2 2 35.32 odd 12
2940.2.q.q.361.1 4 35.27 even 4
2940.2.q.q.961.1 4 35.12 even 12
8820.2.a.bf.1.1 2 105.17 odd 12
8820.2.a.bk.1.1 2 105.32 even 12