Properties

Label 1950.2.e.b.1249.2
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1950,2,Mod(1249,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (of order \(2\), degree \(1\), 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: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.b.1249.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} +8.00000 q^{19} +1.00000 q^{21} -3.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} +7.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +8.00000i q^{38} +1.00000 q^{39} -6.00000 q^{41} +1.00000i q^{42} +12.0000i q^{43} +3.00000 q^{44} +4.00000 q^{46} +3.00000i q^{47} +1.00000i q^{48} +6.00000 q^{49} +1.00000 q^{51} +1.00000i q^{52} -5.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} +8.00000i q^{57} +7.00000i q^{58} +9.00000 q^{59} +5.00000 q^{61} +1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -11.0000i q^{67} +1.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} -4.00000 q^{74} -8.00000 q^{76} +3.00000i q^{77} +1.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -7.00000i q^{83} -1.00000 q^{84} -12.0000 q^{86} +7.00000i q^{87} +3.00000i q^{88} +8.00000 q^{89} -1.00000 q^{91} +4.00000i q^{92} +1.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} +6.00000i q^{97} +6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} + 2 q^{14} + 2 q^{16} + 16 q^{19} + 2 q^{21} + 2 q^{24} + 2 q^{26} + 14 q^{29} + 2 q^{31} + 2 q^{34} + 2 q^{36} + 2 q^{39} - 12 q^{41} + 6 q^{44} + 8 q^{46} + 12 q^{49} + 2 q^{51} + 2 q^{54} - 2 q^{56} + 18 q^{59} + 10 q^{61} - 2 q^{64} + 6 q^{66} + 8 q^{69} + 16 q^{71} - 8 q^{74} - 16 q^{76} + 16 q^{79} + 2 q^{81} - 2 q^{84} - 24 q^{86} + 16 q^{89} - 2 q^{91} - 6 q^{94} - 2 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 1.00000i − 0.277350i
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.00000i − 0.242536i −0.992620 0.121268i \(-0.961304\pi\)
0.992620 0.121268i \(-0.0386960\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) − 3.00000i − 0.639602i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) − 1.00000i − 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.00000i − 0.522233i
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 8.00000i 1.29777i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 1.00000i 0.138675i
\(53\) − 5.00000i − 0.686803i −0.939189 0.343401i \(-0.888421\pi\)
0.939189 0.343401i \(-0.111579\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 8.00000i 1.05963i
\(58\) 7.00000i 0.919145i
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 11.0000i − 1.34386i −0.740613 0.671932i \(-0.765465\pi\)
0.740613 0.671932i \(-0.234535\pi\)
\(68\) 1.00000i 0.121268i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 3.00000i 0.341882i
\(78\) 1.00000i 0.113228i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 7.00000i − 0.768350i −0.923260 0.384175i \(-0.874486\pi\)
0.923260 0.384175i \(-0.125514\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 7.00000i 0.750479i
\(88\) 3.00000i 0.319801i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 4.00000i 0.417029i
\(93\) 1.00000i 0.103695i
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 6.00000i 0.606092i
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 1.00000 0.0995037 0.0497519 0.998762i \(-0.484157\pi\)
0.0497519 + 0.998762i \(0.484157\pi\)
\(102\) 1.00000i 0.0990148i
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 5.00000 0.485643
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) − 1.00000i − 0.0944911i
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) −7.00000 −0.649934
\(117\) 1.00000i 0.0924500i
\(118\) 9.00000i 0.828517i
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 5.00000i 0.452679i
\(123\) − 6.00000i − 0.541002i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 12.0000i 1.06483i 0.846484 + 0.532414i \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 3.00000i 0.261116i
\(133\) − 8.00000i − 0.693688i
\(134\) 11.0000 0.950255
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 4.00000i 0.340503i
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 8.00000i 0.671345i
\(143\) 3.00000i 0.250873i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) − 4.00000i − 0.328798i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) − 8.00000i − 0.648886i
\(153\) 1.00000i 0.0808452i
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) − 15.0000i − 1.19713i −0.801074 0.598565i \(-0.795738\pi\)
0.801074 0.598565i \(-0.204262\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 5.00000 0.396526
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) 20.0000i 1.54765i 0.633402 + 0.773823i \(0.281658\pi\)
−0.633402 + 0.773823i \(0.718342\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) − 12.0000i − 0.914991i
\(173\) 5.00000i 0.380143i 0.981770 + 0.190071i \(0.0608720\pi\)
−0.981770 + 0.190071i \(0.939128\pi\)
\(174\) −7.00000 −0.530669
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 9.00000i 0.676481i
\(178\) 8.00000i 0.599625i
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) − 1.00000i − 0.0741249i
\(183\) 5.00000i 0.369611i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 3.00000i 0.219382i
\(188\) − 3.00000i − 0.218797i
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 3.00000i 0.213201i
\(199\) 26.0000 1.84309 0.921546 0.388270i \(-0.126927\pi\)
0.921546 + 0.388270i \(0.126927\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) 1.00000i 0.0703598i
\(203\) − 7.00000i − 0.491304i
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 4.00000i 0.278019i
\(208\) − 1.00000i − 0.0693375i
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 5.00000i 0.343401i
\(213\) 8.00000i 0.548151i
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) − 1.00000i − 0.0678844i
\(218\) − 2.00000i − 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) − 4.00000i − 0.268462i
\(223\) − 24.0000i − 1.60716i −0.595198 0.803579i \(-0.702926\pi\)
0.595198 0.803579i \(-0.297074\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) − 3.00000i − 0.199117i −0.995032 0.0995585i \(-0.968257\pi\)
0.995032 0.0995585i \(-0.0317430\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) − 7.00000i − 0.459573i
\(233\) − 26.0000i − 1.70332i −0.524097 0.851658i \(-0.675597\pi\)
0.524097 0.851658i \(-0.324403\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) 8.00000i 0.519656i
\(238\) − 1.00000i − 0.0648204i
\(239\) −3.00000 −0.194054 −0.0970269 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 1.00000i 0.0641500i
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) − 8.00000i − 0.509028i
\(248\) − 1.00000i − 0.0635001i
\(249\) 7.00000 0.443607
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) 12.0000i 0.754434i
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 7.00000i − 0.436648i −0.975876 0.218324i \(-0.929941\pi\)
0.975876 0.218324i \(-0.0700590\pi\)
\(258\) − 12.0000i − 0.747087i
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −7.00000 −0.433289
\(262\) − 10.0000i − 0.617802i
\(263\) − 30.0000i − 1.84988i −0.380114 0.924940i \(-0.624115\pi\)
0.380114 0.924940i \(-0.375885\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) 8.00000i 0.489592i
\(268\) 11.0000i 0.671932i
\(269\) −1.00000 −0.0609711 −0.0304855 0.999535i \(-0.509705\pi\)
−0.0304855 + 0.999535i \(0.509705\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) − 1.00000i − 0.0606339i
\(273\) − 1.00000i − 0.0605228i
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 14.0000i 0.839664i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) − 3.00000i − 0.178647i
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −3.00000 −0.177394
\(287\) 6.00000i 0.354169i
\(288\) − 1.00000i − 0.0589256i
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 0 0
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 3.00000i 0.174078i
\(298\) − 10.0000i − 0.579284i
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 13.0000i 0.748066i
\(303\) 1.00000i 0.0574485i
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) − 3.00000i − 0.170941i
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) − 1.00000i − 0.0566139i
\(313\) − 3.00000i − 0.169570i −0.996399 0.0847850i \(-0.972980\pi\)
0.996399 0.0847850i \(-0.0270203\pi\)
\(314\) 15.0000 0.846499
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 20.0000i 1.12331i 0.827371 + 0.561656i \(0.189836\pi\)
−0.827371 + 0.561656i \(0.810164\pi\)
\(318\) 5.00000i 0.280386i
\(319\) −21.0000 −1.17577
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) − 4.00000i − 0.222911i
\(323\) − 8.00000i − 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 2.00000i − 0.110600i
\(328\) 6.00000i 0.331295i
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 7.00000i 0.384175i
\(333\) − 4.00000i − 0.219199i
\(334\) −20.0000 −1.09435
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 13.0000i 0.708155i 0.935216 + 0.354078i \(0.115205\pi\)
−0.935216 + 0.354078i \(0.884795\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) − 8.00000i − 0.432590i
\(343\) − 13.0000i − 0.701934i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −5.00000 −0.268802
\(347\) − 14.0000i − 0.751559i −0.926709 0.375780i \(-0.877375\pi\)
0.926709 0.375780i \(-0.122625\pi\)
\(348\) − 7.00000i − 0.375239i
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 3.00000i − 0.159901i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) −9.00000 −0.478345
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) − 1.00000i − 0.0529256i
\(358\) − 10.0000i − 0.528516i
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 1.00000i 0.0525588i
\(363\) − 2.00000i − 0.104973i
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) −5.00000 −0.261354
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −5.00000 −0.259587
\(372\) − 1.00000i − 0.0518476i
\(373\) − 13.0000i − 0.673114i −0.941663 0.336557i \(-0.890737\pi\)
0.941663 0.336557i \(-0.109263\pi\)
\(374\) −3.00000 −0.155126
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) − 7.00000i − 0.360518i
\(378\) − 1.00000i − 0.0514344i
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) − 10.0000i − 0.511645i
\(383\) 32.0000i 1.63512i 0.575841 + 0.817562i \(0.304675\pi\)
−0.575841 + 0.817562i \(0.695325\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) − 12.0000i − 0.609994i
\(388\) − 6.00000i − 0.304604i
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) − 6.00000i − 0.303046i
\(393\) − 10.0000i − 0.504433i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) 26.0000i 1.30490i 0.757831 + 0.652451i \(0.226259\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(398\) 26.0000i 1.30326i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 11.0000i 0.548630i
\(403\) − 1.00000i − 0.0498135i
\(404\) −1.00000 −0.0497519
\(405\) 0 0
\(406\) 7.00000 0.347404
\(407\) − 12.0000i − 0.594818i
\(408\) − 1.00000i − 0.0495074i
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) − 14.0000i − 0.689730i
\(413\) − 9.00000i − 0.442861i
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 14.0000i 0.685583i
\(418\) − 24.0000i − 1.17388i
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 12.0000i 0.584151i
\(423\) − 3.00000i − 0.145865i
\(424\) −5.00000 −0.242821
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) − 5.00000i − 0.241967i
\(428\) 18.0000i 0.870063i
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 10.0000i 0.480569i 0.970702 + 0.240285i \(0.0772408\pi\)
−0.970702 + 0.240285i \(0.922759\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 32.0000i − 1.53077i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) − 1.00000i − 0.0475651i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) − 10.0000i − 0.472984i
\(448\) 1.00000i 0.0472456i
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) − 2.00000i − 0.0940721i
\(453\) 13.0000i 0.610793i
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) − 6.00000i − 0.280668i −0.990104 0.140334i \(-0.955182\pi\)
0.990104 0.140334i \(-0.0448177\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) − 3.00000i − 0.139573i
\(463\) 11.0000i 0.511213i 0.966781 + 0.255607i \(0.0822752\pi\)
−0.966781 + 0.255607i \(0.917725\pi\)
\(464\) 7.00000 0.324967
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 24.0000i 1.11059i 0.831654 + 0.555294i \(0.187394\pi\)
−0.831654 + 0.555294i \(0.812606\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) −11.0000 −0.507933
\(470\) 0 0
\(471\) 15.0000 0.691164
\(472\) − 9.00000i − 0.414259i
\(473\) − 36.0000i − 1.65528i
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) 5.00000i 0.228934i
\(478\) − 3.00000i − 0.137217i
\(479\) −23.0000 −1.05090 −0.525448 0.850825i \(-0.676102\pi\)
−0.525448 + 0.850825i \(0.676102\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) − 10.0000i − 0.455488i
\(483\) − 4.00000i − 0.182006i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 25.0000i − 1.13286i −0.824110 0.566429i \(-0.808325\pi\)
0.824110 0.566429i \(-0.191675\pi\)
\(488\) − 5.00000i − 0.226339i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 6.00000i 0.270501i
\(493\) − 7.00000i − 0.315264i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) − 8.00000i − 0.358849i
\(498\) 7.00000i 0.313678i
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 18.0000i 0.803379i
\(503\) 22.0000i 0.980932i 0.871460 + 0.490466i \(0.163173\pi\)
−0.871460 + 0.490466i \(0.836827\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) − 1.00000i − 0.0444116i
\(508\) − 12.0000i − 0.532414i
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) 7.00000 0.308757
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) − 9.00000i − 0.395820i
\(518\) 4.00000i 0.175750i
\(519\) −5.00000 −0.219476
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) − 7.00000i − 0.306382i
\(523\) 2.00000i 0.0874539i 0.999044 + 0.0437269i \(0.0139232\pi\)
−0.999044 + 0.0437269i \(0.986077\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) − 1.00000i − 0.0435607i
\(528\) − 3.00000i − 0.130558i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) 8.00000i 0.346844i
\(533\) 6.00000i 0.259889i
\(534\) −8.00000 −0.346194
\(535\) 0 0
\(536\) −11.0000 −0.475128
\(537\) − 10.0000i − 0.431532i
\(538\) − 1.00000i − 0.0431131i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 1.00000i 0.0429537i
\(543\) 1.00000i 0.0429141i
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 1.00000 0.0427960
\(547\) − 6.00000i − 0.256541i −0.991739 0.128271i \(-0.959057\pi\)
0.991739 0.128271i \(-0.0409426\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 56.0000 2.38568
\(552\) − 4.00000i − 0.170251i
\(553\) − 8.00000i − 0.340195i
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) − 18.0000i − 0.759284i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) − 1.00000i − 0.0419961i
\(568\) − 8.00000i − 0.335673i
\(569\) −3.00000 −0.125767 −0.0628833 0.998021i \(-0.520030\pi\)
−0.0628833 + 0.998021i \(0.520030\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) − 3.00000i − 0.125436i
\(573\) − 10.0000i − 0.417756i
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) 16.0000i 0.665512i
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) −7.00000 −0.290409
\(582\) − 6.00000i − 0.248708i
\(583\) 15.0000i 0.621237i
\(584\) 0 0
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 39.0000i − 1.60970i −0.593477 0.804851i \(-0.702245\pi\)
0.593477 0.804851i \(-0.297755\pi\)
\(588\) − 6.00000i − 0.247436i
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 4.00000i 0.164399i
\(593\) − 34.0000i − 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 26.0000i 1.06411i
\(598\) − 4.00000i − 0.163572i
\(599\) 26.0000 1.06233 0.531166 0.847268i \(-0.321754\pi\)
0.531166 + 0.847268i \(0.321754\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 12.0000i 0.489083i
\(603\) 11.0000i 0.447955i
\(604\) −13.0000 −0.528962
\(605\) 0 0
\(606\) −1.00000 −0.0406222
\(607\) − 22.0000i − 0.892952i −0.894795 0.446476i \(-0.852679\pi\)
0.894795 0.446476i \(-0.147321\pi\)
\(608\) 8.00000i 0.324443i
\(609\) 7.00000 0.283654
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) − 1.00000i − 0.0404226i
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) − 34.0000i − 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) − 14.0000i − 0.563163i
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 18.0000i 0.721734i
\(623\) − 8.00000i − 0.320513i
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 3.00000 0.119904
\(627\) − 24.0000i − 0.958468i
\(628\) 15.0000i 0.598565i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 12.0000i 0.476957i
\(634\) −20.0000 −0.794301
\(635\) 0 0
\(636\) −5.00000 −0.198263
\(637\) − 6.00000i − 0.237729i
\(638\) − 21.0000i − 0.831398i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 18.0000i 0.710403i
\(643\) − 20.0000i − 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −27.0000 −1.05984
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) − 4.00000i − 0.156652i
\(653\) 21.0000i 0.821794i 0.911682 + 0.410897i \(0.134784\pi\)
−0.911682 + 0.410897i \(0.865216\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 3.00000i 0.116952i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 28.0000i 1.08825i
\(663\) − 1.00000i − 0.0388368i
\(664\) −7.00000 −0.271653
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) − 28.0000i − 1.08416i
\(668\) − 20.0000i − 0.773823i
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 1.00000i 0.0385758i
\(673\) 43.0000i 1.65753i 0.559598 + 0.828764i \(0.310955\pi\)
−0.559598 + 0.828764i \(0.689045\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) − 2.00000i − 0.0768095i
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) − 3.00000i − 0.114876i
\(683\) − 27.0000i − 1.03313i −0.856249 0.516563i \(-0.827211\pi\)
0.856249 0.516563i \(-0.172789\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) − 14.0000i − 0.534133i
\(688\) 12.0000i 0.457496i
\(689\) −5.00000 −0.190485
\(690\) 0 0
\(691\) −43.0000 −1.63580 −0.817899 0.575362i \(-0.804861\pi\)
−0.817899 + 0.575362i \(0.804861\pi\)
\(692\) − 5.00000i − 0.190071i
\(693\) − 3.00000i − 0.113961i
\(694\) 14.0000 0.531433
\(695\) 0 0
\(696\) 7.00000 0.265334
\(697\) 6.00000i 0.227266i
\(698\) 28.0000i 1.05982i
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) − 1.00000i − 0.0377426i
\(703\) 32.0000i 1.20690i
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) − 1.00000i − 0.0376089i
\(708\) − 9.00000i − 0.338241i
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) − 8.00000i − 0.299813i
\(713\) − 4.00000i − 0.149801i
\(714\) 1.00000 0.0374241
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) − 3.00000i − 0.112037i
\(718\) 15.0000i 0.559795i
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) 45.0000i 1.67473i
\(723\) − 10.0000i − 0.371904i
\(724\) −1.00000 −0.0371647
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 14.0000i 0.519231i 0.965712 + 0.259616i \(0.0835959\pi\)
−0.965712 + 0.259616i \(0.916404\pi\)
\(728\) 1.00000i 0.0370625i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) − 5.00000i − 0.184805i
\(733\) − 28.0000i − 1.03420i −0.855924 0.517102i \(-0.827011\pi\)
0.855924 0.517102i \(-0.172989\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 33.0000i 1.21557i
\(738\) 6.00000i 0.220863i
\(739\) 47.0000 1.72892 0.864461 0.502699i \(-0.167660\pi\)
0.864461 + 0.502699i \(0.167660\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) − 5.00000i − 0.183556i
\(743\) 29.0000i 1.06391i 0.846774 + 0.531953i \(0.178542\pi\)
−0.846774 + 0.531953i \(0.821458\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 13.0000 0.475964
\(747\) 7.00000i 0.256117i
\(748\) − 3.00000i − 0.109691i
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 3.00000i 0.109399i
\(753\) 18.0000i 0.655956i
\(754\) 7.00000 0.254925
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 43.0000i 1.56286i 0.623992 + 0.781431i \(0.285510\pi\)
−0.623992 + 0.781431i \(0.714490\pi\)
\(758\) − 11.0000i − 0.399538i
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) 2.00000i 0.0724049i
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) − 9.00000i − 0.324971i
\(768\) 1.00000i 0.0360844i
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) 6.00000i 0.215945i
\(773\) 46.0000i 1.65451i 0.561830 + 0.827253i \(0.310097\pi\)
−0.561830 + 0.827253i \(0.689903\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 4.00000i 0.143499i
\(778\) − 34.0000i − 1.21896i
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) − 4.00000i − 0.143040i
\(783\) − 7.00000i − 0.250160i
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 10.0000 0.356688
\(787\) 7.00000i 0.249523i 0.992187 + 0.124762i \(0.0398166\pi\)
−0.992187 + 0.124762i \(0.960183\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 30.0000 1.06803
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) − 3.00000i − 0.106600i
\(793\) − 5.00000i − 0.177555i
\(794\) −26.0000 −0.922705
\(795\) 0 0
\(796\) −26.0000 −0.921546
\(797\) − 11.0000i − 0.389640i −0.980839 0.194820i \(-0.937588\pi\)
0.980839 0.194820i \(-0.0624123\pi\)
\(798\) 8.00000i 0.283197i
\(799\) 3.00000 0.106132
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) − 28.0000i − 0.988714i
\(803\) 0 0
\(804\) −11.0000 −0.387940
\(805\) 0 0
\(806\) 1.00000 0.0352235
\(807\) − 1.00000i − 0.0352017i
\(808\) − 1.00000i − 0.0351799i
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) −47.0000 −1.65039 −0.825197 0.564846i \(-0.808936\pi\)
−0.825197 + 0.564846i \(0.808936\pi\)
\(812\) 7.00000i 0.245652i
\(813\) 1.00000i 0.0350715i
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) 96.0000i 3.35861i
\(818\) 2.00000i 0.0699284i
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −52.0000 −1.81481 −0.907406 0.420255i \(-0.861941\pi\)
−0.907406 + 0.420255i \(0.861941\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 9.00000 0.313150
\(827\) 3.00000i 0.104320i 0.998639 + 0.0521601i \(0.0166106\pi\)
−0.998639 + 0.0521601i \(0.983389\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) −47.0000 −1.63238 −0.816189 0.577785i \(-0.803917\pi\)
−0.816189 + 0.577785i \(0.803917\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 1.00000i 0.0346688i
\(833\) − 6.00000i − 0.207888i
\(834\) −14.0000 −0.484780
\(835\) 0 0
\(836\) 24.0000 0.830057
\(837\) − 1.00000i − 0.0345651i
\(838\) − 6.00000i − 0.207267i
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) − 12.0000i − 0.413547i
\(843\) − 18.0000i − 0.619953i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 2.00000i 0.0687208i
\(848\) − 5.00000i − 0.171701i
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) − 8.00000i − 0.274075i
\(853\) − 8.00000i − 0.273915i −0.990577 0.136957i \(-0.956268\pi\)
0.990577 0.136957i \(-0.0437323\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) − 3.00000i − 0.102418i
\(859\) 24.0000 0.818869 0.409435 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 12.0000i 0.408722i
\(863\) 9.00000i 0.306364i 0.988198 + 0.153182i \(0.0489520\pi\)
−0.988198 + 0.153182i \(0.951048\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −10.0000 −0.339814
\(867\) 16.0000i 0.543388i
\(868\) 1.00000i 0.0339422i
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −11.0000 −0.372721
\(872\) 2.00000i 0.0677285i
\(873\) − 6.00000i − 0.203069i
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) − 6.00000i − 0.202031i
\(883\) − 48.0000i − 1.61533i −0.589643 0.807664i \(-0.700731\pi\)
0.589643 0.807664i \(-0.299269\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) − 56.0000i − 1.88030i −0.340766 0.940148i \(-0.610687\pi\)
0.340766 0.940148i \(-0.389313\pi\)
\(888\) 4.00000i 0.134231i
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 24.0000i 0.803579i
\(893\) 24.0000i 0.803129i
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) − 4.00000i − 0.133556i
\(898\) − 36.0000i − 1.20134i
\(899\) 7.00000 0.233463
\(900\) 0 0
\(901\) −5.00000 −0.166574
\(902\) 18.0000i 0.599334i
\(903\) 12.0000i 0.399335i
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) −13.0000 −0.431896
\(907\) 8.00000i 0.265636i 0.991140 + 0.132818i \(0.0424025\pi\)
−0.991140 + 0.132818i \(0.957597\pi\)
\(908\) 3.00000i 0.0995585i
\(909\) −1.00000 −0.0331679
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 8.00000i 0.264906i
\(913\) 21.0000i 0.694999i
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 10.0000i 0.330229i
\(918\) − 1.00000i − 0.0330049i
\(919\) 14.0000 0.461817 0.230909 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 26.0000i 0.856264i
\(923\) − 8.00000i − 0.263323i
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) −11.0000 −0.361482
\(927\) − 14.0000i − 0.459820i
\(928\) 7.00000i 0.229786i
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 48.0000 1.57314
\(932\) 26.0000i 0.851658i
\(933\) 18.0000i 0.589294i
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 21.0000i 0.686040i 0.939328 + 0.343020i \(0.111450\pi\)
−0.939328 + 0.343020i \(0.888550\pi\)
\(938\) − 11.0000i − 0.359163i
\(939\) 3.00000 0.0979013
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) 15.0000i 0.488726i
\(943\) 24.0000i 0.781548i
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) 36.0000 1.17046
\(947\) 47.0000i 1.52729i 0.645634 + 0.763647i \(0.276593\pi\)
−0.645634 + 0.763647i \(0.723407\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 0 0
\(950\) 0 0
\(951\) −20.0000 −0.648544
\(952\) 1.00000i 0.0324102i
\(953\) − 57.0000i − 1.84641i −0.384307 0.923206i \(-0.625559\pi\)
0.384307 0.923206i \(-0.374441\pi\)
\(954\) −5.00000 −0.161881
\(955\) 0 0
\(956\) 3.00000 0.0970269
\(957\) − 21.0000i − 0.678834i
\(958\) − 23.0000i − 0.743096i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 4.00000i 0.128965i
\(963\) 18.0000i 0.580042i
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) 7.00000i 0.225105i 0.993646 + 0.112552i \(0.0359026\pi\)
−0.993646 + 0.112552i \(0.964097\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 14.0000i − 0.448819i
\(974\) 25.0000 0.801052
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 52.0000i 1.66363i 0.555055 + 0.831814i \(0.312697\pi\)
−0.555055 + 0.831814i \(0.687303\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 18.0000i 0.574403i
\(983\) 11.0000i 0.350846i 0.984493 + 0.175423i \(0.0561292\pi\)
−0.984493 + 0.175423i \(0.943871\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 7.00000 0.222925
\(987\) 3.00000i 0.0954911i
\(988\) 8.00000i 0.254514i
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) 28.0000i 0.888553i
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) −7.00000 −0.221803
\(997\) − 17.0000i − 0.538395i −0.963085 0.269198i \(-0.913241\pi\)
0.963085 0.269198i \(-0.0867585\pi\)
\(998\) − 25.0000i − 0.791361i
\(999\) 4.00000 0.126554
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 1950.2.e.b.1249.2 2
3.2 odd 2 5850.2.e.x.5149.1 2
5.2 odd 4 1950.2.a.m.1.1 1
5.3 odd 4 1950.2.a.q.1.1 yes 1
5.4 even 2 inner 1950.2.e.b.1249.1 2
15.2 even 4 5850.2.a.bt.1.1 1
15.8 even 4 5850.2.a.j.1.1 1
15.14 odd 2 5850.2.e.x.5149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.m.1.1 1 5.2 odd 4
1950.2.a.q.1.1 yes 1 5.3 odd 4
1950.2.e.b.1249.1 2 5.4 even 2 inner
1950.2.e.b.1249.2 2 1.1 even 1 trivial
5850.2.a.j.1.1 1 15.8 even 4
5850.2.a.bt.1.1 1 15.2 even 4
5850.2.e.x.5149.1 2 3.2 odd 2
5850.2.e.x.5149.2 2 15.14 odd 2