Properties

Label 2100.2.q.i.1801.1
Level $2100$
Weight $2$
Character 2100.1801
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
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] = [2100,2,Mod(1201,2100)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1201");
 
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.q (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1801.1
Root \(-1.32288 - 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1801
Dual form 2100.2.q.i.1201.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-1.32288 - 2.29129i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(-1.32288 - 2.29129i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(0.177124 + 0.306788i) q^{11} +1.00000 q^{13} +(-1.82288 - 3.15731i) q^{17} +(-0.322876 + 0.559237i) q^{19} +(1.32288 - 2.29129i) q^{21} +(1.00000 - 1.73205i) q^{23} -1.00000 q^{27} -3.29150 q^{29} +(-2.00000 - 3.46410i) q^{31} +(-0.177124 + 0.306788i) q^{33} +(1.14575 - 1.98450i) q^{37} +(0.500000 + 0.866025i) q^{39} -5.64575 q^{41} +3.29150 q^{43} +(6.46863 - 11.2040i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(1.82288 - 3.15731i) q^{51} +(-3.82288 - 6.62141i) q^{53} -0.645751 q^{57} +(0.468627 + 0.811686i) q^{59} +(5.79150 - 10.0312i) q^{61} +2.64575 q^{63} +(-3.96863 - 6.87386i) q^{67} +2.00000 q^{69} -7.29150 q^{71} +(-2.14575 - 3.71655i) q^{73} +(0.468627 - 0.811686i) q^{77} +(0.677124 - 1.17281i) q^{79} +(-0.500000 - 0.866025i) q^{81} -1.64575 q^{83} +(-1.64575 - 2.85052i) q^{87} +(-3.46863 + 6.00784i) q^{89} +(-1.32288 - 2.29129i) q^{91} +(2.00000 - 3.46410i) q^{93} +3.70850 q^{97} -0.354249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{9} + 6 q^{11} + 4 q^{13} - 2 q^{17} + 4 q^{19} + 4 q^{23} - 4 q^{27} + 8 q^{29} - 8 q^{31} - 6 q^{33} - 6 q^{37} + 2 q^{39} - 12 q^{41} - 8 q^{43} + 10 q^{47} - 14 q^{49} + 2 q^{51} - 10 q^{53} + 8 q^{57} - 14 q^{59} + 2 q^{61} + 8 q^{69} - 8 q^{71} + 2 q^{73} - 14 q^{77} + 8 q^{79} - 2 q^{81} + 4 q^{83} + 4 q^{87} + 2 q^{89} + 8 q^{93} + 36 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.32288 2.29129i −0.500000 0.866025i
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0.177124 + 0.306788i 0.0534050 + 0.0925002i 0.891492 0.453036i \(-0.149659\pi\)
−0.838087 + 0.545537i \(0.816326\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.82288 3.15731i −0.442112 0.765761i 0.555734 0.831360i \(-0.312437\pi\)
−0.997846 + 0.0655994i \(0.979104\pi\)
\(18\) 0 0
\(19\) −0.322876 + 0.559237i −0.0740728 + 0.128298i −0.900683 0.434478i \(-0.856933\pi\)
0.826610 + 0.562775i \(0.190266\pi\)
\(20\) 0 0
\(21\) 1.32288 2.29129i 0.288675 0.500000i
\(22\) 0 0
\(23\) 1.00000 1.73205i 0.208514 0.361158i −0.742732 0.669588i \(-0.766471\pi\)
0.951247 + 0.308431i \(0.0998038\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.29150 −0.611217 −0.305608 0.952157i \(-0.598860\pi\)
−0.305608 + 0.952157i \(0.598860\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) −0.177124 + 0.306788i −0.0308334 + 0.0534050i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.14575 1.98450i 0.188360 0.326250i −0.756343 0.654175i \(-0.773016\pi\)
0.944704 + 0.327925i \(0.106349\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.0800641 + 0.138675i
\(40\) 0 0
\(41\) −5.64575 −0.881718 −0.440859 0.897576i \(-0.645326\pi\)
−0.440859 + 0.897576i \(0.645326\pi\)
\(42\) 0 0
\(43\) 3.29150 0.501949 0.250975 0.967994i \(-0.419249\pi\)
0.250975 + 0.967994i \(0.419249\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.46863 11.2040i 0.943546 1.63427i 0.184910 0.982755i \(-0.440801\pi\)
0.758636 0.651515i \(-0.225866\pi\)
\(48\) 0 0
\(49\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) 1.82288 3.15731i 0.255254 0.442112i
\(52\) 0 0
\(53\) −3.82288 6.62141i −0.525112 0.909521i −0.999572 0.0292442i \(-0.990690\pi\)
0.474460 0.880277i \(-0.342643\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.645751 −0.0855319
\(58\) 0 0
\(59\) 0.468627 + 0.811686i 0.0610100 + 0.105672i 0.894917 0.446232i \(-0.147235\pi\)
−0.833907 + 0.551905i \(0.813901\pi\)
\(60\) 0 0
\(61\) 5.79150 10.0312i 0.741526 1.28436i −0.210274 0.977642i \(-0.567436\pi\)
0.951800 0.306718i \(-0.0992309\pi\)
\(62\) 0 0
\(63\) 2.64575 0.333333
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.96863 6.87386i −0.484845 0.839776i 0.515003 0.857188i \(-0.327791\pi\)
−0.999848 + 0.0174120i \(0.994457\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −7.29150 −0.865342 −0.432671 0.901552i \(-0.642429\pi\)
−0.432671 + 0.901552i \(0.642429\pi\)
\(72\) 0 0
\(73\) −2.14575 3.71655i −0.251141 0.434989i 0.712699 0.701470i \(-0.247473\pi\)
−0.963840 + 0.266481i \(0.914139\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.468627 0.811686i 0.0534050 0.0925002i
\(78\) 0 0
\(79\) 0.677124 1.17281i 0.0761824 0.131952i −0.825417 0.564523i \(-0.809060\pi\)
0.901600 + 0.432571i \(0.142394\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −1.64575 −0.180645 −0.0903223 0.995913i \(-0.528790\pi\)
−0.0903223 + 0.995913i \(0.528790\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.64575 2.85052i −0.176443 0.305608i
\(88\) 0 0
\(89\) −3.46863 + 6.00784i −0.367674 + 0.636830i −0.989201 0.146563i \(-0.953179\pi\)
0.621528 + 0.783392i \(0.286512\pi\)
\(90\) 0 0
\(91\) −1.32288 2.29129i −0.138675 0.240192i
\(92\) 0 0
\(93\) 2.00000 3.46410i 0.207390 0.359211i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.70850 0.376541 0.188270 0.982117i \(-0.439712\pi\)
0.188270 + 0.982117i \(0.439712\pi\)
\(98\) 0 0
\(99\) −0.354249 −0.0356033
\(100\) 0 0
\(101\) −2.46863 4.27579i −0.245638 0.425457i 0.716673 0.697409i \(-0.245664\pi\)
−0.962311 + 0.271953i \(0.912331\pi\)
\(102\) 0 0
\(103\) −0.0313730 + 0.0543397i −0.00309128 + 0.00535425i −0.867567 0.497320i \(-0.834317\pi\)
0.864476 + 0.502675i \(0.167651\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.11438 14.0545i 0.784447 1.35870i −0.144883 0.989449i \(-0.546280\pi\)
0.929329 0.369252i \(-0.120386\pi\)
\(108\) 0 0
\(109\) 5.79150 + 10.0312i 0.554725 + 0.960812i 0.997925 + 0.0643895i \(0.0205100\pi\)
−0.443200 + 0.896423i \(0.646157\pi\)
\(110\) 0 0
\(111\) 2.29150 0.217500
\(112\) 0 0
\(113\) −8.58301 −0.807421 −0.403711 0.914887i \(-0.632280\pi\)
−0.403711 + 0.914887i \(0.632280\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.500000 + 0.866025i −0.0462250 + 0.0800641i
\(118\) 0 0
\(119\) −4.82288 + 8.35347i −0.442112 + 0.765761i
\(120\) 0 0
\(121\) 5.43725 9.41760i 0.494296 0.856145i
\(122\) 0 0
\(123\) −2.82288 4.88936i −0.254530 0.440859i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.35425 −0.120170 −0.0600851 0.998193i \(-0.519137\pi\)
−0.0600851 + 0.998193i \(0.519137\pi\)
\(128\) 0 0
\(129\) 1.64575 + 2.85052i 0.144900 + 0.250975i
\(130\) 0 0
\(131\) −0.291503 + 0.504897i −0.0254687 + 0.0441131i −0.878479 0.477781i \(-0.841441\pi\)
0.853010 + 0.521894i \(0.174774\pi\)
\(132\) 0 0
\(133\) 1.70850 0.148146
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.645751 1.11847i −0.0551703 0.0955577i 0.837121 0.547017i \(-0.184237\pi\)
−0.892292 + 0.451460i \(0.850903\pi\)
\(138\) 0 0
\(139\) 15.9373 1.35178 0.675890 0.737002i \(-0.263759\pi\)
0.675890 + 0.737002i \(0.263759\pi\)
\(140\) 0 0
\(141\) 12.9373 1.08951
\(142\) 0 0
\(143\) 0.177124 + 0.306788i 0.0148119 + 0.0256549i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) −2.46863 + 4.27579i −0.202238 + 0.350286i −0.949249 0.314525i \(-0.898155\pi\)
0.747011 + 0.664811i \(0.231488\pi\)
\(150\) 0 0
\(151\) 5.67712 + 9.83307i 0.461998 + 0.800204i 0.999060 0.0433387i \(-0.0137995\pi\)
−0.537063 + 0.843542i \(0.680466\pi\)
\(152\) 0 0
\(153\) 3.64575 0.294742
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −8.50000 14.7224i −0.678374 1.17498i −0.975470 0.220131i \(-0.929352\pi\)
0.297097 0.954847i \(-0.403982\pi\)
\(158\) 0 0
\(159\) 3.82288 6.62141i 0.303174 0.525112i
\(160\) 0 0
\(161\) −5.29150 −0.417029
\(162\) 0 0
\(163\) 8.61438 14.9205i 0.674730 1.16867i −0.301818 0.953366i \(-0.597593\pi\)
0.976548 0.215301i \(-0.0690733\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.5203 1.51052 0.755262 0.655424i \(-0.227510\pi\)
0.755262 + 0.655424i \(0.227510\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −0.322876 0.559237i −0.0246909 0.0427659i
\(172\) 0 0
\(173\) −10.2915 + 17.8254i −0.782448 + 1.35524i 0.148063 + 0.988978i \(0.452696\pi\)
−0.930512 + 0.366263i \(0.880637\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.468627 + 0.811686i −0.0352242 + 0.0610100i
\(178\) 0 0
\(179\) 11.6458 + 20.1710i 0.870444 + 1.50765i 0.861538 + 0.507693i \(0.169502\pi\)
0.00890653 + 0.999960i \(0.497165\pi\)
\(180\) 0 0
\(181\) 14.5830 1.08395 0.541973 0.840396i \(-0.317677\pi\)
0.541973 + 0.840396i \(0.317677\pi\)
\(182\) 0 0
\(183\) 11.5830 0.856240
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.645751 1.11847i 0.0472220 0.0817909i
\(188\) 0 0
\(189\) 1.32288 + 2.29129i 0.0962250 + 0.166667i
\(190\) 0 0
\(191\) −6.76013 + 11.7089i −0.489146 + 0.847225i −0.999922 0.0124883i \(-0.996025\pi\)
0.510776 + 0.859714i \(0.329358\pi\)
\(192\) 0 0
\(193\) −8.00000 13.8564i −0.575853 0.997406i −0.995948 0.0899262i \(-0.971337\pi\)
0.420096 0.907480i \(-0.361996\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.70850 −0.620455 −0.310227 0.950662i \(-0.600405\pi\)
−0.310227 + 0.950662i \(0.600405\pi\)
\(198\) 0 0
\(199\) −0.322876 0.559237i −0.0228880 0.0396433i 0.854355 0.519691i \(-0.173953\pi\)
−0.877243 + 0.480047i \(0.840619\pi\)
\(200\) 0 0
\(201\) 3.96863 6.87386i 0.279925 0.484845i
\(202\) 0 0
\(203\) 4.35425 + 7.54178i 0.305608 + 0.529329i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 + 1.73205i 0.0695048 + 0.120386i
\(208\) 0 0
\(209\) −0.228757 −0.0158234
\(210\) 0 0
\(211\) −2.06275 −0.142005 −0.0710026 0.997476i \(-0.522620\pi\)
−0.0710026 + 0.997476i \(0.522620\pi\)
\(212\) 0 0
\(213\) −3.64575 6.31463i −0.249803 0.432671i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −5.29150 + 9.16515i −0.359211 + 0.622171i
\(218\) 0 0
\(219\) 2.14575 3.71655i 0.144996 0.251141i
\(220\) 0 0
\(221\) −1.82288 3.15731i −0.122620 0.212384i
\(222\) 0 0
\(223\) 13.9373 0.933308 0.466654 0.884440i \(-0.345459\pi\)
0.466654 + 0.884440i \(0.345459\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.35425 + 7.54178i 0.289002 + 0.500566i 0.973572 0.228382i \(-0.0733434\pi\)
−0.684570 + 0.728947i \(0.740010\pi\)
\(228\) 0 0
\(229\) 10.1458 17.5730i 0.670450 1.16125i −0.307326 0.951604i \(-0.599434\pi\)
0.977777 0.209650i \(-0.0672323\pi\)
\(230\) 0 0
\(231\) 0.937254 0.0616668
\(232\) 0 0
\(233\) −12.1144 + 20.9827i −0.793639 + 1.37462i 0.130060 + 0.991506i \(0.458483\pi\)
−0.923700 + 0.383118i \(0.874850\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.35425 0.0879679
\(238\) 0 0
\(239\) −9.41699 −0.609135 −0.304567 0.952491i \(-0.598512\pi\)
−0.304567 + 0.952491i \(0.598512\pi\)
\(240\) 0 0
\(241\) 0.145751 + 0.252449i 0.00938867 + 0.0162616i 0.870682 0.491847i \(-0.163678\pi\)
−0.861293 + 0.508109i \(0.830345\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.322876 + 0.559237i −0.0205441 + 0.0355834i
\(248\) 0 0
\(249\) −0.822876 1.42526i −0.0521476 0.0903223i
\(250\) 0 0
\(251\) −28.8118 −1.81858 −0.909291 0.416161i \(-0.863375\pi\)
−0.909291 + 0.416161i \(0.863375\pi\)
\(252\) 0 0
\(253\) 0.708497 0.0445428
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.82288 15.2817i 0.550356 0.953244i −0.447893 0.894087i \(-0.647825\pi\)
0.998249 0.0591571i \(-0.0188413\pi\)
\(258\) 0 0
\(259\) −6.06275 −0.376721
\(260\) 0 0
\(261\) 1.64575 2.85052i 0.101869 0.176443i
\(262\) 0 0
\(263\) −12.0000 20.7846i −0.739952 1.28163i −0.952517 0.304487i \(-0.901515\pi\)
0.212565 0.977147i \(-0.431818\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.93725 −0.424553
\(268\) 0 0
\(269\) −10.1144 17.5186i −0.616685 1.06813i −0.990086 0.140459i \(-0.955142\pi\)
0.373402 0.927670i \(-0.378191\pi\)
\(270\) 0 0
\(271\) 1.35425 2.34563i 0.0822647 0.142487i −0.821958 0.569549i \(-0.807118\pi\)
0.904222 + 0.427062i \(0.140451\pi\)
\(272\) 0 0
\(273\) 1.32288 2.29129i 0.0800641 0.138675i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12.5000 + 21.6506i 0.751052 + 1.30086i 0.947313 + 0.320309i \(0.103787\pi\)
−0.196261 + 0.980552i \(0.562880\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −22.2288 −1.32606 −0.663028 0.748594i \(-0.730729\pi\)
−0.663028 + 0.748594i \(0.730729\pi\)
\(282\) 0 0
\(283\) 3.26013 + 5.64671i 0.193795 + 0.335662i 0.946505 0.322690i \(-0.104587\pi\)
−0.752710 + 0.658352i \(0.771254\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.46863 + 12.9360i 0.440859 + 0.763590i
\(288\) 0 0
\(289\) 1.85425 3.21165i 0.109073 0.188921i
\(290\) 0 0
\(291\) 1.85425 + 3.21165i 0.108698 + 0.188270i
\(292\) 0 0
\(293\) −9.87451 −0.576875 −0.288437 0.957499i \(-0.593136\pi\)
−0.288437 + 0.957499i \(0.593136\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.177124 0.306788i −0.0102778 0.0178017i
\(298\) 0 0
\(299\) 1.00000 1.73205i 0.0578315 0.100167i
\(300\) 0 0
\(301\) −4.35425 7.54178i −0.250975 0.434701i
\(302\) 0 0
\(303\) 2.46863 4.27579i 0.141819 0.245638i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.87451 −0.106984 −0.0534919 0.998568i \(-0.517035\pi\)
−0.0534919 + 0.998568i \(0.517035\pi\)
\(308\) 0 0
\(309\) −0.0627461 −0.00356950
\(310\) 0 0
\(311\) 6.64575 + 11.5108i 0.376846 + 0.652716i 0.990601 0.136780i \(-0.0436753\pi\)
−0.613755 + 0.789496i \(0.710342\pi\)
\(312\) 0 0
\(313\) −9.64575 + 16.7069i −0.545210 + 0.944332i 0.453384 + 0.891316i \(0.350217\pi\)
−0.998594 + 0.0530161i \(0.983117\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.29150 + 7.43310i −0.241035 + 0.417485i −0.961009 0.276516i \(-0.910820\pi\)
0.719975 + 0.694000i \(0.244153\pi\)
\(318\) 0 0
\(319\) −0.583005 1.00979i −0.0326420 0.0565376i
\(320\) 0 0
\(321\) 16.2288 0.905801
\(322\) 0 0
\(323\) 2.35425 0.130994
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −5.79150 + 10.0312i −0.320271 + 0.554725i
\(328\) 0 0
\(329\) −34.2288 −1.88709
\(330\) 0 0
\(331\) 1.67712 2.90486i 0.0921831 0.159666i −0.816246 0.577704i \(-0.803949\pi\)
0.908429 + 0.418038i \(0.137282\pi\)
\(332\) 0 0
\(333\) 1.14575 + 1.98450i 0.0627868 + 0.108750i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −19.2915 −1.05088 −0.525438 0.850832i \(-0.676098\pi\)
−0.525438 + 0.850832i \(0.676098\pi\)
\(338\) 0 0
\(339\) −4.29150 7.43310i −0.233082 0.403711i
\(340\) 0 0
\(341\) 0.708497 1.22715i 0.0383673 0.0664541i
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.76013 + 9.97684i 0.309220 + 0.535585i 0.978192 0.207703i \(-0.0665987\pi\)
−0.668972 + 0.743288i \(0.733265\pi\)
\(348\) 0 0
\(349\) 4.70850 0.252040 0.126020 0.992028i \(-0.459780\pi\)
0.126020 + 0.992028i \(0.459780\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 8.46863 + 14.6681i 0.450740 + 0.780704i 0.998432 0.0559759i \(-0.0178270\pi\)
−0.547693 + 0.836680i \(0.684494\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −9.64575 −0.510507
\(358\) 0 0
\(359\) 12.4686 21.5963i 0.658069 1.13981i −0.323046 0.946383i \(-0.604707\pi\)
0.981115 0.193426i \(-0.0619598\pi\)
\(360\) 0 0
\(361\) 9.29150 + 16.0934i 0.489026 + 0.847019i
\(362\) 0 0
\(363\) 10.8745 0.570764
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.93725 + 10.2836i 0.309922 + 0.536801i 0.978345 0.206981i \(-0.0663637\pi\)
−0.668423 + 0.743781i \(0.733030\pi\)
\(368\) 0 0
\(369\) 2.82288 4.88936i 0.146953 0.254530i
\(370\) 0 0
\(371\) −10.1144 + 17.5186i −0.525112 + 0.909521i
\(372\) 0 0
\(373\) −11.7915 + 20.4235i −0.610541 + 1.05749i 0.380608 + 0.924736i \(0.375715\pi\)
−0.991149 + 0.132752i \(0.957619\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.29150 −0.169521
\(378\) 0 0
\(379\) −15.3542 −0.788695 −0.394347 0.918961i \(-0.629029\pi\)
−0.394347 + 0.918961i \(0.629029\pi\)
\(380\) 0 0
\(381\) −0.677124 1.17281i −0.0346901 0.0600851i
\(382\) 0 0
\(383\) −4.64575 + 8.04668i −0.237387 + 0.411166i −0.959964 0.280125i \(-0.909624\pi\)
0.722577 + 0.691291i \(0.242958\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.64575 + 2.85052i −0.0836582 + 0.144900i
\(388\) 0 0
\(389\) 0.239870 + 0.415468i 0.0121619 + 0.0210651i 0.872042 0.489430i \(-0.162795\pi\)
−0.859880 + 0.510496i \(0.829462\pi\)
\(390\) 0 0
\(391\) −7.29150 −0.368747
\(392\) 0 0
\(393\) −0.583005 −0.0294087
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −10.2288 + 17.7167i −0.513367 + 0.889177i 0.486513 + 0.873673i \(0.338269\pi\)
−0.999880 + 0.0155038i \(0.995065\pi\)
\(398\) 0 0
\(399\) 0.854249 + 1.47960i 0.0427659 + 0.0740728i
\(400\) 0 0
\(401\) −17.7601 + 30.7614i −0.886899 + 1.53615i −0.0433765 + 0.999059i \(0.513812\pi\)
−0.843522 + 0.537095i \(0.819522\pi\)
\(402\) 0 0
\(403\) −2.00000 3.46410i −0.0996271 0.172559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.811762 0.0402375
\(408\) 0 0
\(409\) −3.79150 6.56708i −0.187478 0.324721i 0.756931 0.653495i \(-0.226698\pi\)
−0.944409 + 0.328774i \(0.893365\pi\)
\(410\) 0 0
\(411\) 0.645751 1.11847i 0.0318526 0.0551703i
\(412\) 0 0
\(413\) 1.23987 2.14752i 0.0610100 0.105672i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7.96863 + 13.8021i 0.390225 + 0.675890i
\(418\) 0 0
\(419\) 31.6458 1.54600 0.772998 0.634408i \(-0.218756\pi\)
0.772998 + 0.634408i \(0.218756\pi\)
\(420\) 0 0
\(421\) 28.8745 1.40726 0.703629 0.710568i \(-0.251562\pi\)
0.703629 + 0.710568i \(0.251562\pi\)
\(422\) 0 0
\(423\) 6.46863 + 11.2040i 0.314515 + 0.544757i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −30.6458 −1.48305
\(428\) 0 0
\(429\) −0.177124 + 0.306788i −0.00855164 + 0.0148119i
\(430\) 0 0
\(431\) 13.9373 + 24.1400i 0.671334 + 1.16278i 0.977526 + 0.210815i \(0.0676117\pi\)
−0.306192 + 0.951970i \(0.599055\pi\)
\(432\) 0 0
\(433\) −3.29150 −0.158180 −0.0790898 0.996867i \(-0.525201\pi\)
−0.0790898 + 0.996867i \(0.525201\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.645751 + 1.11847i 0.0308905 + 0.0535039i
\(438\) 0 0
\(439\) 19.9059 34.4780i 0.950056 1.64555i 0.204759 0.978812i \(-0.434359\pi\)
0.745297 0.666733i \(-0.232308\pi\)
\(440\) 0 0
\(441\) −3.50000 6.06218i −0.166667 0.288675i
\(442\) 0 0
\(443\) 3.35425 5.80973i 0.159365 0.276029i −0.775275 0.631624i \(-0.782389\pi\)
0.934640 + 0.355596i \(0.115722\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.93725 −0.233524
\(448\) 0 0
\(449\) −32.1033 −1.51505 −0.757523 0.652808i \(-0.773591\pi\)
−0.757523 + 0.652808i \(0.773591\pi\)
\(450\) 0 0
\(451\) −1.00000 1.73205i −0.0470882 0.0815591i
\(452\) 0 0
\(453\) −5.67712 + 9.83307i −0.266735 + 0.461998i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.08301 + 7.07197i −0.190995 + 0.330813i −0.945580 0.325389i \(-0.894505\pi\)
0.754585 + 0.656202i \(0.227838\pi\)
\(458\) 0 0
\(459\) 1.82288 + 3.15731i 0.0850845 + 0.147371i
\(460\) 0 0
\(461\) 10.7085 0.498744 0.249372 0.968408i \(-0.419776\pi\)
0.249372 + 0.968408i \(0.419776\pi\)
\(462\) 0 0
\(463\) 33.9373 1.57720 0.788599 0.614908i \(-0.210807\pi\)
0.788599 + 0.614908i \(0.210807\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.3542 19.6661i 0.525412 0.910040i −0.474150 0.880444i \(-0.657245\pi\)
0.999562 0.0295961i \(-0.00942212\pi\)
\(468\) 0 0
\(469\) −10.5000 + 18.1865i −0.484845 + 0.839776i
\(470\) 0 0
\(471\) 8.50000 14.7224i 0.391659 0.678374i
\(472\) 0 0
\(473\) 0.583005 + 1.00979i 0.0268066 + 0.0464304i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.64575 0.350075
\(478\) 0 0
\(479\) −12.8229 22.2099i −0.585892 1.01479i −0.994764 0.102203i \(-0.967411\pi\)
0.408871 0.912592i \(-0.365922\pi\)
\(480\) 0 0
\(481\) 1.14575 1.98450i 0.0522418 0.0904854i
\(482\) 0 0
\(483\) −2.64575 4.58258i −0.120386 0.208514i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 20.2288 + 35.0372i 0.916652 + 1.58769i 0.804464 + 0.594002i \(0.202453\pi\)
0.112189 + 0.993687i \(0.464214\pi\)
\(488\) 0 0
\(489\) 17.2288 0.779111
\(490\) 0 0
\(491\) −34.4575 −1.55505 −0.777523 0.628855i \(-0.783524\pi\)
−0.777523 + 0.628855i \(0.783524\pi\)
\(492\) 0 0
\(493\) 6.00000 + 10.3923i 0.270226 + 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.64575 + 16.7069i 0.432671 + 0.749408i
\(498\) 0 0
\(499\) 9.90588 17.1575i 0.443448 0.768075i −0.554495 0.832187i \(-0.687088\pi\)
0.997943 + 0.0641127i \(0.0204217\pi\)
\(500\) 0 0
\(501\) 9.76013 + 16.9050i 0.436050 + 0.755262i
\(502\) 0 0
\(503\) −16.7085 −0.744995 −0.372498 0.928033i \(-0.621498\pi\)
−0.372498 + 0.928033i \(0.621498\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.00000 10.3923i −0.266469 0.461538i
\(508\) 0 0
\(509\) 12.9373 22.4080i 0.573434 0.993216i −0.422776 0.906234i \(-0.638944\pi\)
0.996210 0.0869822i \(-0.0277223\pi\)
\(510\) 0 0
\(511\) −5.67712 + 9.83307i −0.251141 + 0.434989i
\(512\) 0 0
\(513\) 0.322876 0.559237i 0.0142553 0.0246909i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.58301 0.201560
\(518\) 0 0
\(519\) −20.5830 −0.903494
\(520\) 0 0
\(521\) 7.93725 + 13.7477i 0.347737 + 0.602299i 0.985847 0.167647i \(-0.0536167\pi\)
−0.638110 + 0.769945i \(0.720283\pi\)
\(522\) 0 0
\(523\) −5.06275 + 8.76893i −0.221378 + 0.383439i −0.955227 0.295875i \(-0.904389\pi\)
0.733848 + 0.679313i \(0.237722\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.29150 + 12.6293i −0.317623 + 0.550139i
\(528\) 0 0
\(529\) 9.50000 + 16.4545i 0.413043 + 0.715412i
\(530\) 0 0
\(531\) −0.937254 −0.0406734
\(532\) 0 0
\(533\) −5.64575 −0.244545
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −11.6458 + 20.1710i −0.502551 + 0.870444i
\(538\) 0 0
\(539\) −2.47974 −0.106810
\(540\) 0 0
\(541\) −13.4373 + 23.2740i −0.577713 + 1.00063i 0.418028 + 0.908434i \(0.362721\pi\)
−0.995741 + 0.0921937i \(0.970612\pi\)
\(542\) 0 0
\(543\) 7.29150 + 12.6293i 0.312908 + 0.541973i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.125492 0.00536566 0.00268283 0.999996i \(-0.499146\pi\)
0.00268283 + 0.999996i \(0.499146\pi\)
\(548\) 0 0
\(549\) 5.79150 + 10.0312i 0.247175 + 0.428120i
\(550\) 0 0
\(551\) 1.06275 1.84073i 0.0452745 0.0784177i
\(552\) 0 0
\(553\) −3.58301 −0.152365
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.35425 + 14.4700i 0.353981 + 0.613113i 0.986943 0.161070i \(-0.0514945\pi\)
−0.632962 + 0.774183i \(0.718161\pi\)
\(558\) 0 0
\(559\) 3.29150 0.139216
\(560\) 0 0
\(561\) 1.29150 0.0545273
\(562\) 0 0
\(563\) −2.76013 4.78068i −0.116326 0.201482i 0.801983 0.597346i \(-0.203778\pi\)
−0.918309 + 0.395865i \(0.870445\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.32288 + 2.29129i −0.0555556 + 0.0962250i
\(568\) 0 0
\(569\) −2.46863 + 4.27579i −0.103490 + 0.179250i −0.913120 0.407690i \(-0.866334\pi\)
0.809630 + 0.586940i \(0.199668\pi\)
\(570\) 0 0
\(571\) −2.96863 5.14181i −0.124233 0.215178i 0.797200 0.603716i \(-0.206314\pi\)
−0.921433 + 0.388537i \(0.872980\pi\)
\(572\) 0 0
\(573\) −13.5203 −0.564817
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.29150 + 3.96900i 0.0953965 + 0.165232i 0.909774 0.415104i \(-0.136255\pi\)
−0.814378 + 0.580335i \(0.802921\pi\)
\(578\) 0 0
\(579\) 8.00000 13.8564i 0.332469 0.575853i
\(580\) 0 0
\(581\) 2.17712 + 3.77089i 0.0903223 + 0.156443i
\(582\) 0 0
\(583\) 1.35425 2.34563i 0.0560872 0.0971460i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 0 0
\(589\) 2.58301 0.106431
\(590\) 0 0
\(591\) −4.35425 7.54178i −0.179110 0.310227i
\(592\) 0 0
\(593\) −0.291503 + 0.504897i −0.0119706 + 0.0207336i −0.871949 0.489597i \(-0.837144\pi\)
0.859978 + 0.510331i \(0.170477\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.322876 0.559237i 0.0132144 0.0228880i
\(598\) 0 0
\(599\) 16.4686 + 28.5245i 0.672890 + 1.16548i 0.977081 + 0.212868i \(0.0682806\pi\)
−0.304191 + 0.952611i \(0.598386\pi\)
\(600\) 0 0
\(601\) 14.2915 0.582963 0.291481 0.956577i \(-0.405852\pi\)
0.291481 + 0.956577i \(0.405852\pi\)
\(602\) 0 0
\(603\) 7.93725 0.323230
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.32288 14.4156i 0.337815 0.585113i −0.646206 0.763163i \(-0.723645\pi\)
0.984021 + 0.178050i \(0.0569788\pi\)
\(608\) 0 0
\(609\) −4.35425 + 7.54178i −0.176443 + 0.305608i
\(610\) 0 0
\(611\) 6.46863 11.2040i 0.261693 0.453265i
\(612\) 0 0
\(613\) −17.6458 30.5633i −0.712705 1.23444i −0.963838 0.266489i \(-0.914136\pi\)
0.251133 0.967953i \(-0.419197\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −41.8745 −1.68580 −0.842902 0.538067i \(-0.819155\pi\)
−0.842902 + 0.538067i \(0.819155\pi\)
\(618\) 0 0
\(619\) −12.3542 21.3982i −0.496559 0.860066i 0.503433 0.864034i \(-0.332070\pi\)
−0.999992 + 0.00396857i \(0.998737\pi\)
\(620\) 0 0
\(621\) −1.00000 + 1.73205i −0.0401286 + 0.0695048i
\(622\) 0 0
\(623\) 18.3542 0.735347
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −0.114378 0.198109i −0.00456783 0.00791171i
\(628\) 0 0
\(629\) −8.35425 −0.333106
\(630\) 0 0
\(631\) −1.35425 −0.0539118 −0.0269559 0.999637i \(-0.508581\pi\)
−0.0269559 + 0.999637i \(0.508581\pi\)
\(632\) 0 0
\(633\) −1.03137 1.78639i −0.0409934 0.0710026i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.50000 + 6.06218i −0.138675 + 0.240192i
\(638\) 0 0
\(639\) 3.64575 6.31463i 0.144224 0.249803i
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) −27.3542 −1.07875 −0.539373 0.842067i \(-0.681339\pi\)
−0.539373 + 0.842067i \(0.681339\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.3431 + 43.8956i 0.996341 + 1.72571i 0.572183 + 0.820126i \(0.306097\pi\)
0.424159 + 0.905588i \(0.360570\pi\)
\(648\) 0 0
\(649\) −0.166010 + 0.287539i −0.00651648 + 0.0112869i
\(650\) 0 0
\(651\) −10.5830 −0.414781
\(652\) 0 0
\(653\) −8.11438 + 14.0545i −0.317540 + 0.549996i −0.979974 0.199125i \(-0.936190\pi\)
0.662434 + 0.749120i \(0.269523\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.29150 0.167427
\(658\) 0 0
\(659\) 40.1033 1.56220 0.781101 0.624405i \(-0.214659\pi\)
0.781101 + 0.624405i \(0.214659\pi\)
\(660\) 0 0
\(661\) −0.854249 1.47960i −0.0332264 0.0575499i 0.848934 0.528499i \(-0.177245\pi\)
−0.882160 + 0.470949i \(0.843912\pi\)
\(662\) 0 0
\(663\) 1.82288 3.15731i 0.0707946 0.122620i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.29150 + 5.70105i −0.127447 + 0.220746i
\(668\) 0 0
\(669\) 6.96863 + 12.0700i 0.269423 + 0.466654i
\(670\) 0 0
\(671\) 4.10326 0.158405
\(672\) 0 0
\(673\) 2.29150 0.0883309 0.0441655 0.999024i \(-0.485937\pi\)
0.0441655 + 0.999024i \(0.485937\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.3542 33.5225i 0.743844 1.28838i −0.206889 0.978365i \(-0.566334\pi\)
0.950733 0.310011i \(-0.100333\pi\)
\(678\) 0 0
\(679\) −4.90588 8.49723i −0.188270 0.326094i
\(680\) 0 0
\(681\) −4.35425 + 7.54178i −0.166855 + 0.289002i
\(682\) 0 0
\(683\) −10.0516 17.4099i −0.384615 0.666173i 0.607101 0.794625i \(-0.292332\pi\)
−0.991716 + 0.128452i \(0.958999\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 20.2915 0.774169
\(688\) 0 0
\(689\) −3.82288 6.62141i −0.145640 0.252256i
\(690\) 0 0
\(691\) −7.61438 + 13.1885i −0.289665 + 0.501714i −0.973730 0.227707i \(-0.926877\pi\)
0.684065 + 0.729421i \(0.260210\pi\)
\(692\) 0 0
\(693\) 0.468627 + 0.811686i 0.0178017 + 0.0308334i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.2915 + 17.8254i 0.389818 + 0.675185i
\(698\) 0 0
\(699\) −24.2288 −0.916416
\(700\) 0 0
\(701\) −25.0627 −0.946607 −0.473303 0.880899i \(-0.656939\pi\)
−0.473303 + 0.880899i \(0.656939\pi\)
\(702\) 0 0
\(703\) 0.739870 + 1.28149i 0.0279047 + 0.0483324i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.53137 + 11.3127i −0.245638 + 0.425457i
\(708\) 0 0
\(709\) 3.08301 5.33992i 0.115785 0.200545i −0.802308 0.596910i \(-0.796395\pi\)
0.918093 + 0.396365i \(0.129728\pi\)
\(710\) 0 0
\(711\) 0.677124 + 1.17281i 0.0253941 + 0.0439840i
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.70850 8.15536i −0.175842 0.304567i
\(718\) 0 0
\(719\) −1.11438 + 1.93016i −0.0415593 + 0.0719828i −0.886057 0.463577i \(-0.846566\pi\)
0.844498 + 0.535559i \(0.179899\pi\)
\(720\) 0 0
\(721\) 0.166010 0.00618255
\(722\) 0 0
\(723\) −0.145751 + 0.252449i −0.00542055 + 0.00938867i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 27.1033 1.00520 0.502602 0.864518i \(-0.332376\pi\)
0.502602 + 0.864518i \(0.332376\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.00000 10.3923i −0.221918 0.384373i
\(732\) 0 0
\(733\) −16.0203 + 27.7479i −0.591722 + 1.02489i 0.402279 + 0.915517i \(0.368218\pi\)
−0.994001 + 0.109375i \(0.965115\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.40588 2.43506i 0.0517863 0.0896965i
\(738\) 0 0
\(739\) 25.2601 + 43.7518i 0.929209 + 1.60944i 0.784648 + 0.619941i \(0.212843\pi\)
0.144560 + 0.989496i \(0.453823\pi\)
\(740\) 0 0
\(741\) −0.645751 −0.0237223
\(742\) 0 0
\(743\) −11.4170 −0.418849 −0.209424 0.977825i \(-0.567159\pi\)
−0.209424 + 0.977825i \(0.567159\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.822876 1.42526i 0.0301074 0.0521476i
\(748\) 0 0
\(749\) −42.9373 −1.56889
\(750\) 0 0
\(751\) 16.6771 28.8856i 0.608557 1.05405i −0.382922 0.923781i \(-0.625082\pi\)
0.991478 0.130271i \(-0.0415846\pi\)
\(752\) 0 0
\(753\) −14.4059 24.9517i −0.524979 0.909291i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.0000 0.399802 0.199901 0.979816i \(-0.435938\pi\)
0.199901 + 0.979816i \(0.435938\pi\)
\(758\) 0 0
\(759\) 0.354249 + 0.613577i 0.0128584 + 0.0222714i
\(760\) 0 0
\(761\) 22.1771 38.4119i 0.803920 1.39243i −0.113098 0.993584i \(-0.536077\pi\)
0.917018 0.398847i \(-0.130589\pi\)
\(762\) 0 0
\(763\) 15.3229 26.5400i 0.554725 0.960812i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.468627 + 0.811686i 0.0169211 + 0.0293083i
\(768\) 0 0
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 0 0
\(771\) 17.6458 0.635496
\(772\) 0 0
\(773\) 13.2915 + 23.0216i 0.478062 + 0.828028i 0.999684 0.0251491i \(-0.00800604\pi\)
−0.521622 + 0.853177i \(0.674673\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3.03137 5.25049i −0.108750 0.188360i
\(778\) 0 0
\(779\) 1.82288 3.15731i 0.0653113 0.113122i
\(780\) 0 0
\(781\) −1.29150 2.23695i −0.0462136 0.0800443i
\(782\) 0 0
\(783\) 3.29150 0.117629
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −22.4889 38.9519i −0.801642 1.38849i −0.918535 0.395340i \(-0.870627\pi\)
0.116892 0.993145i \(-0.462707\pi\)
\(788\) 0 0
\(789\) 12.0000 20.7846i 0.427211 0.739952i
\(790\) 0 0
\(791\) 11.3542 + 19.6661i 0.403711 + 0.699247i
\(792\) 0 0
\(793\) 5.79150 10.0312i 0.205662 0.356218i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.87451 −0.137242 −0.0686211 0.997643i \(-0.521860\pi\)
−0.0686211 + 0.997643i \(0.521860\pi\)
\(798\) 0 0
\(799\) −47.1660 −1.66861
\(800\) 0 0
\(801\) −3.46863 6.00784i −0.122558 0.212277i
\(802\) 0 0
\(803\) 0.760130 1.31658i 0.0268244 0.0464612i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.1144 17.5186i 0.356043 0.616685i
\(808\) 0 0
\(809\) 17.3542 + 30.0584i 0.610143 + 1.05680i 0.991216 + 0.132254i \(0.0422213\pi\)
−0.381073 + 0.924545i \(0.624445\pi\)
\(810\) 0 0
\(811\) −7.93725 −0.278715 −0.139357 0.990242i \(-0.544504\pi\)
−0.139357 + 0.990242i \(0.544504\pi\)
\(812\) 0 0
\(813\) 2.70850 0.0949912
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.06275 + 1.84073i −0.0371808 + 0.0643990i
\(818\) 0 0
\(819\) 2.64575 0.0924500
\(820\) 0 0
\(821\) 8.69738 15.0643i 0.303541 0.525748i −0.673395 0.739283i \(-0.735164\pi\)
0.976935 + 0.213535i \(0.0684978\pi\)
\(822\) 0 0
\(823\) 8.26013 + 14.3070i 0.287930 + 0.498709i 0.973315 0.229471i \(-0.0736996\pi\)
−0.685385 + 0.728180i \(0.740366\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −42.5830 −1.48076 −0.740378 0.672191i \(-0.765354\pi\)
−0.740378 + 0.672191i \(0.765354\pi\)
\(828\) 0 0
\(829\) 19.6660 + 34.0625i 0.683029 + 1.18304i 0.974052 + 0.226325i \(0.0726710\pi\)
−0.291023 + 0.956716i \(0.593996\pi\)
\(830\) 0 0
\(831\) −12.5000 + 21.6506i −0.433620 + 0.751052i
\(832\) 0 0
\(833\) 25.5203 0.884225
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.00000 + 3.46410i 0.0691301 + 0.119737i
\(838\) 0 0
\(839\) 19.1660 0.661684 0.330842 0.943686i \(-0.392667\pi\)
0.330842 + 0.943686i \(0.392667\pi\)
\(840\) 0 0
\(841\) −18.1660 −0.626414
\(842\) 0 0
\(843\) −11.1144 19.2507i −0.382800 0.663028i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −28.7712 −0.988592
\(848\) 0 0
\(849\) −3.26013 + 5.64671i −0.111887 + 0.193795i
\(850\) 0 0
\(851\) −2.29150 3.96900i −0.0785517 0.136056i
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −13.5830 23.5265i −0.463987 0.803648i 0.535169 0.844745i \(-0.320248\pi\)
−0.999155 + 0.0410969i \(0.986915\pi\)
\(858\) 0 0
\(859\) 20.5830 35.6508i 0.702283 1.21639i −0.265380 0.964144i \(-0.585497\pi\)
0.967663 0.252246i \(-0.0811692\pi\)
\(860\) 0 0
\(861\) −7.46863 + 12.9360i −0.254530 + 0.440859i
\(862\) 0 0
\(863\) 3.05163 5.28558i 0.103879 0.179923i −0.809401 0.587257i \(-0.800208\pi\)
0.913280 + 0.407333i \(0.133541\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.70850 0.125947
\(868\) 0 0
\(869\) 0.479741 0.0162741
\(870\) 0 0
\(871\) −3.96863 6.87386i −0.134472 0.232912i
\(872\) 0 0
\(873\) −1.85425 + 3.21165i −0.0627568 + 0.108698i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.3745 38.7538i 0.755533 1.30862i −0.189575 0.981866i \(-0.560711\pi\)
0.945109 0.326756i \(-0.105956\pi\)
\(878\) 0 0
\(879\) −4.93725 8.55157i −0.166529 0.288437i
\(880\) 0 0
\(881\) 44.7085 1.50627 0.753134 0.657867i \(-0.228541\pi\)
0.753134 + 0.657867i \(0.228541\pi\)
\(882\) 0 0
\(883\) 23.3542 0.785933 0.392967 0.919553i \(-0.371449\pi\)
0.392967 + 0.919553i \(0.371449\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20.6458 35.7595i 0.693216 1.20069i −0.277562 0.960708i \(-0.589526\pi\)
0.970778 0.239978i \(-0.0771404\pi\)
\(888\) 0 0
\(889\) 1.79150 + 3.10297i 0.0600851 + 0.104070i
\(890\) 0 0
\(891\) 0.177124 0.306788i 0.00593389 0.0102778i
\(892\) 0 0
\(893\) 4.17712 + 7.23499i 0.139782 + 0.242110i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) 0 0
\(899\) 6.58301 + 11.4021i 0.219556 + 0.380281i
\(900\) 0 0
\(901\) −13.9373 + 24.1400i −0.464317 + 0.804221i
\(902\) 0 0
\(903\) 4.35425 7.54178i 0.144900 0.250975i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −15.4889 26.8275i −0.514300 0.890794i −0.999862 0.0165917i \(-0.994718\pi\)
0.485562 0.874202i \(-0.338615\pi\)
\(908\) 0 0
\(909\) 4.93725 0.163758
\(910\) 0 0
\(911\) 32.3320 1.07121 0.535604 0.844469i \(-0.320084\pi\)
0.535604 + 0.844469i \(0.320084\pi\)
\(912\) 0 0
\(913\) −0.291503 0.504897i −0.00964733 0.0167097i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.54249 0.0509374
\(918\) 0 0
\(919\) −24.4575 + 42.3617i −0.806779 + 1.39738i 0.108305 + 0.994118i \(0.465458\pi\)
−0.915084 + 0.403264i \(0.867876\pi\)
\(920\) 0 0
\(921\) −0.937254 1.62337i −0.0308836 0.0534919i
\(922\) 0 0
\(923\) −7.29150 −0.240003
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −0.0313730 0.0543397i −0.00103043 0.00178475i
\(928\) 0 0
\(929\) −1.17712 + 2.03884i −0.0386202 + 0.0668921i −0.884689 0.466181i \(-0.845630\pi\)
0.846069 + 0.533073i \(0.178963\pi\)
\(930\) 0 0
\(931\) −2.26013 3.91466i −0.0740728 0.128298i
\(932\) 0 0
\(933\) −6.64575 + 11.5108i −0.217572 + 0.376846i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) −19.2915 −0.629554
\(940\) 0 0
\(941\) −6.87451 11.9070i −0.224103 0.388157i 0.731947 0.681361i \(-0.238612\pi\)
−0.956050 + 0.293204i \(0.905278\pi\)
\(942\) 0 0
\(943\) −5.64575 + 9.77873i −0.183851 + 0.318439i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.3542 + 23.1302i −0.433955 + 0.751632i −0.997210 0.0746515i \(-0.976216\pi\)
0.563255 + 0.826283i \(0.309549\pi\)
\(948\) 0 0
\(949\) −2.14575 3.71655i −0.0696540 0.120644i
\(950\) 0 0
\(951\) −8.58301 −0.278323
\(952\) 0 0
\(953\) 47.5203 1.53933 0.769666 0.638447i \(-0.220423\pi\)
0.769666 + 0.638447i \(0.220423\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0.583005 1.00979i 0.0188459 0.0326420i
\(958\) 0 0
\(959\) −1.70850 + 2.95920i −0.0551703 + 0.0955577i
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 8.11438 + 14.0545i 0.261482 + 0.452900i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 29.9373 0.962717 0.481359 0.876524i \(-0.340143\pi\)
0.481359 + 0.876524i \(0.340143\pi\)
\(968\) 0 0
\(969\) 1.17712 + 2.03884i 0.0378147 + 0.0654970i
\(970\) 0 0
\(971\) 26.2804 45.5190i 0.843378 1.46077i −0.0436447 0.999047i \(-0.513897\pi\)
0.887023 0.461726i \(-0.152770\pi\)
\(972\) 0 0
\(973\) −21.0830 36.5168i −0.675890 1.17068i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.3431 21.3789i −0.394892 0.683973i 0.598196 0.801350i \(-0.295885\pi\)
−0.993087 + 0.117377i \(0.962551\pi\)
\(978\) 0 0
\(979\) −2.45751 −0.0785425
\(980\) 0 0
\(981\) −11.5830 −0.369817
\(982\) 0 0
\(983\) 2.82288 + 4.88936i 0.0900358 + 0.155946i 0.907526 0.419996i \(-0.137969\pi\)
−0.817490 + 0.575943i \(0.804635\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −17.1144 29.6430i −0.544757 0.943546i
\(988\) 0 0
\(989\) 3.29150 5.70105i 0.104664 0.181283i
\(990\) 0 0
\(991\) 11.9373 + 20.6759i 0.379199 + 0.656793i 0.990946 0.134261i \(-0.0428661\pi\)
−0.611747 + 0.791054i \(0.709533\pi\)
\(992\) 0 0
\(993\) 3.35425 0.106444
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −24.3745 42.2179i −0.771948 1.33705i −0.936494 0.350684i \(-0.885949\pi\)
0.164545 0.986370i \(-0.447384\pi\)
\(998\) 0 0
\(999\) −1.14575 + 1.98450i −0.0362500 + 0.0627868i
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.q.i.1801.1 yes 4
5.2 odd 4 2100.2.bc.h.1549.4 8
5.3 odd 4 2100.2.bc.h.1549.1 8
5.4 even 2 2100.2.q.g.1801.2 yes 4
7.4 even 3 inner 2100.2.q.i.1201.1 yes 4
35.4 even 6 2100.2.q.g.1201.2 4
35.18 odd 12 2100.2.bc.h.949.4 8
35.32 odd 12 2100.2.bc.h.949.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.q.g.1201.2 4 35.4 even 6
2100.2.q.g.1801.2 yes 4 5.4 even 2
2100.2.q.i.1201.1 yes 4 7.4 even 3 inner
2100.2.q.i.1801.1 yes 4 1.1 even 1 trivial
2100.2.bc.h.949.1 8 35.32 odd 12
2100.2.bc.h.949.4 8 35.18 odd 12
2100.2.bc.h.1549.1 8 5.3 odd 4
2100.2.bc.h.1549.4 8 5.2 odd 4