Properties

Label 192.3.i.b.17.4
Level $192$
Weight $3$
Character 192.17
Analytic conductor $5.232$
Analytic rank $0$
Dimension $20$
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] = [192,3,Mod(17,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.i (of order \(4\), 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: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{23} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.4
Root \(1.21144 - 1.59136i\) of defining polynomial
Character \(\chi\) \(=\) 192.17
Dual form 192.3.i.b.113.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.14944 - 2.77106i) q^{3} +(4.80434 + 4.80434i) q^{5} +7.36187i q^{7} +(-6.35757 + 6.37035i) q^{9} +O(q^{10})\) \(q+(-1.14944 - 2.77106i) q^{3} +(4.80434 + 4.80434i) q^{5} +7.36187i q^{7} +(-6.35757 + 6.37035i) q^{9} +(0.514693 + 0.514693i) q^{11} +(7.12969 + 7.12969i) q^{13} +(7.79081 - 18.8354i) q^{15} -11.1126i q^{17} +(21.1403 + 21.1403i) q^{19} +(20.4002 - 8.46203i) q^{21} +7.80231 q^{23} +21.1633i q^{25} +(24.9603 + 10.2949i) q^{27} +(34.6058 - 34.6058i) q^{29} -24.8644 q^{31} +(0.834637 - 2.01786i) q^{33} +(-35.3689 + 35.3689i) q^{35} +(-18.2760 + 18.2760i) q^{37} +(11.5617 - 27.9520i) q^{39} -64.2448 q^{41} +(-7.24058 + 7.24058i) q^{43} +(-61.1492 + 0.0613789i) q^{45} -23.0508i q^{47} -5.19710 q^{49} +(-30.7938 + 12.7733i) q^{51} +(31.9199 + 31.9199i) q^{53} +4.94552i q^{55} +(34.2816 - 82.8807i) q^{57} +(-17.6272 - 17.6272i) q^{59} +(-12.3933 - 12.3933i) q^{61} +(-46.8976 - 46.8036i) q^{63} +68.5069i q^{65} +(-41.1425 - 41.1425i) q^{67} +(-8.96830 - 21.6207i) q^{69} +25.6785 q^{71} +56.1845i q^{73} +(58.6449 - 24.3260i) q^{75} +(-3.78910 + 3.78910i) q^{77} +35.7013 q^{79} +(-0.162608 - 80.9998i) q^{81} +(94.9424 - 94.9424i) q^{83} +(53.3889 - 53.3889i) q^{85} +(-135.672 - 56.1175i) q^{87} -44.8713 q^{89} +(-52.4878 + 52.4878i) q^{91} +(28.5802 + 68.9008i) q^{93} +203.131i q^{95} -82.3636 q^{97} +(-6.55097 + 0.00657558i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{3} + 92 q^{13} + 116 q^{15} + 52 q^{19} + 48 q^{21} - 18 q^{27} + 80 q^{31} + 60 q^{33} - 116 q^{37} - 172 q^{43} + 60 q^{45} - 364 q^{49} - 128 q^{51} - 244 q^{61} - 296 q^{63} - 356 q^{67} - 20 q^{69} + 146 q^{75} - 384 q^{79} - 188 q^{81} + 48 q^{85} - 136 q^{91} - 132 q^{93} + 472 q^{97} + 452 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.14944 2.77106i −0.383147 0.923687i
\(4\) 0 0
\(5\) 4.80434 + 4.80434i 0.960868 + 0.960868i 0.999263 0.0383950i \(-0.0122245\pi\)
−0.0383950 + 0.999263i \(0.512225\pi\)
\(6\) 0 0
\(7\) 7.36187i 1.05170i 0.850579 + 0.525848i \(0.176252\pi\)
−0.850579 + 0.525848i \(0.823748\pi\)
\(8\) 0 0
\(9\) −6.35757 + 6.37035i −0.706397 + 0.707816i
\(10\) 0 0
\(11\) 0.514693 + 0.514693i 0.0467903 + 0.0467903i 0.730115 0.683325i \(-0.239467\pi\)
−0.683325 + 0.730115i \(0.739467\pi\)
\(12\) 0 0
\(13\) 7.12969 + 7.12969i 0.548438 + 0.548438i 0.925989 0.377551i \(-0.123234\pi\)
−0.377551 + 0.925989i \(0.623234\pi\)
\(14\) 0 0
\(15\) 7.79081 18.8354i 0.519388 1.25569i
\(16\) 0 0
\(17\) 11.1126i 0.653684i −0.945079 0.326842i \(-0.894015\pi\)
0.945079 0.326842i \(-0.105985\pi\)
\(18\) 0 0
\(19\) 21.1403 + 21.1403i 1.11265 + 1.11265i 0.992791 + 0.119858i \(0.0382440\pi\)
0.119858 + 0.992791i \(0.461756\pi\)
\(20\) 0 0
\(21\) 20.4002 8.46203i 0.971438 0.402954i
\(22\) 0 0
\(23\) 7.80231 0.339231 0.169615 0.985510i \(-0.445747\pi\)
0.169615 + 0.985510i \(0.445747\pi\)
\(24\) 0 0
\(25\) 21.1633i 0.846533i
\(26\) 0 0
\(27\) 24.9603 + 10.2949i 0.924455 + 0.381292i
\(28\) 0 0
\(29\) 34.6058 34.6058i 1.19330 1.19330i 0.217169 0.976134i \(-0.430318\pi\)
0.976134 0.217169i \(-0.0696823\pi\)
\(30\) 0 0
\(31\) −24.8644 −0.802078 −0.401039 0.916061i \(-0.631351\pi\)
−0.401039 + 0.916061i \(0.631351\pi\)
\(32\) 0 0
\(33\) 0.834637 2.01786i 0.0252920 0.0611472i
\(34\) 0 0
\(35\) −35.3689 + 35.3689i −1.01054 + 1.01054i
\(36\) 0 0
\(37\) −18.2760 + 18.2760i −0.493946 + 0.493946i −0.909547 0.415601i \(-0.863571\pi\)
0.415601 + 0.909547i \(0.363571\pi\)
\(38\) 0 0
\(39\) 11.5617 27.9520i 0.296453 0.716717i
\(40\) 0 0
\(41\) −64.2448 −1.56695 −0.783473 0.621426i \(-0.786554\pi\)
−0.783473 + 0.621426i \(0.786554\pi\)
\(42\) 0 0
\(43\) −7.24058 + 7.24058i −0.168386 + 0.168386i −0.786269 0.617884i \(-0.787990\pi\)
0.617884 + 0.786269i \(0.287990\pi\)
\(44\) 0 0
\(45\) −61.1492 + 0.0613789i −1.35887 + 0.00136398i
\(46\) 0 0
\(47\) 23.0508i 0.490442i −0.969467 0.245221i \(-0.921139\pi\)
0.969467 0.245221i \(-0.0788606\pi\)
\(48\) 0 0
\(49\) −5.19710 −0.106063
\(50\) 0 0
\(51\) −30.7938 + 12.7733i −0.603800 + 0.250457i
\(52\) 0 0
\(53\) 31.9199 + 31.9199i 0.602263 + 0.602263i 0.940913 0.338650i \(-0.109970\pi\)
−0.338650 + 0.940913i \(0.609970\pi\)
\(54\) 0 0
\(55\) 4.94552i 0.0899185i
\(56\) 0 0
\(57\) 34.2816 82.8807i 0.601432 1.45405i
\(58\) 0 0
\(59\) −17.6272 17.6272i −0.298766 0.298766i 0.541764 0.840530i \(-0.317756\pi\)
−0.840530 + 0.541764i \(0.817756\pi\)
\(60\) 0 0
\(61\) −12.3933 12.3933i −0.203170 0.203170i 0.598187 0.801357i \(-0.295888\pi\)
−0.801357 + 0.598187i \(0.795888\pi\)
\(62\) 0 0
\(63\) −46.8976 46.8036i −0.744407 0.742914i
\(64\) 0 0
\(65\) 68.5069i 1.05395i
\(66\) 0 0
\(67\) −41.1425 41.1425i −0.614067 0.614067i 0.329936 0.944003i \(-0.392973\pi\)
−0.944003 + 0.329936i \(0.892973\pi\)
\(68\) 0 0
\(69\) −8.96830 21.6207i −0.129975 0.313343i
\(70\) 0 0
\(71\) 25.6785 0.361669 0.180834 0.983514i \(-0.442120\pi\)
0.180834 + 0.983514i \(0.442120\pi\)
\(72\) 0 0
\(73\) 56.1845i 0.769650i 0.922990 + 0.384825i \(0.125738\pi\)
−0.922990 + 0.384825i \(0.874262\pi\)
\(74\) 0 0
\(75\) 58.6449 24.3260i 0.781932 0.324347i
\(76\) 0 0
\(77\) −3.78910 + 3.78910i −0.0492091 + 0.0492091i
\(78\) 0 0
\(79\) 35.7013 0.451915 0.225957 0.974137i \(-0.427449\pi\)
0.225957 + 0.974137i \(0.427449\pi\)
\(80\) 0 0
\(81\) −0.162608 80.9998i −0.00200751 0.999998i
\(82\) 0 0
\(83\) 94.9424 94.9424i 1.14388 1.14388i 0.156151 0.987733i \(-0.450091\pi\)
0.987733 0.156151i \(-0.0499086\pi\)
\(84\) 0 0
\(85\) 53.3889 53.3889i 0.628104 0.628104i
\(86\) 0 0
\(87\) −135.672 56.1175i −1.55945 0.645028i
\(88\) 0 0
\(89\) −44.8713 −0.504172 −0.252086 0.967705i \(-0.581117\pi\)
−0.252086 + 0.967705i \(0.581117\pi\)
\(90\) 0 0
\(91\) −52.4878 + 52.4878i −0.576789 + 0.576789i
\(92\) 0 0
\(93\) 28.5802 + 68.9008i 0.307314 + 0.740869i
\(94\) 0 0
\(95\) 203.131i 2.13822i
\(96\) 0 0
\(97\) −82.3636 −0.849109 −0.424554 0.905402i \(-0.639569\pi\)
−0.424554 + 0.905402i \(0.639569\pi\)
\(98\) 0 0
\(99\) −6.55097 + 0.00657558i −0.0661714 + 6.64200e-5i
\(100\) 0 0
\(101\) −36.3420 36.3420i −0.359822 0.359822i 0.503925 0.863747i \(-0.331889\pi\)
−0.863747 + 0.503925i \(0.831889\pi\)
\(102\) 0 0
\(103\) 87.5176i 0.849685i 0.905267 + 0.424843i \(0.139671\pi\)
−0.905267 + 0.424843i \(0.860329\pi\)
\(104\) 0 0
\(105\) 138.664 + 57.3550i 1.32061 + 0.546238i
\(106\) 0 0
\(107\) −104.866 104.866i −0.980058 0.980058i 0.0197471 0.999805i \(-0.493714\pi\)
−0.999805 + 0.0197471i \(0.993714\pi\)
\(108\) 0 0
\(109\) 7.64006 + 7.64006i 0.0700923 + 0.0700923i 0.741284 0.671192i \(-0.234217\pi\)
−0.671192 + 0.741284i \(0.734217\pi\)
\(110\) 0 0
\(111\) 71.6511 + 29.6367i 0.645505 + 0.266998i
\(112\) 0 0
\(113\) 13.1273i 0.116171i 0.998312 + 0.0580853i \(0.0184995\pi\)
−0.998312 + 0.0580853i \(0.981500\pi\)
\(114\) 0 0
\(115\) 37.4849 + 37.4849i 0.325956 + 0.325956i
\(116\) 0 0
\(117\) −90.7461 + 0.0910869i −0.775607 + 0.000778521i
\(118\) 0 0
\(119\) 81.8098 0.687477
\(120\) 0 0
\(121\) 120.470i 0.995621i
\(122\) 0 0
\(123\) 73.8456 + 178.026i 0.600371 + 1.44737i
\(124\) 0 0
\(125\) 18.4327 18.4327i 0.147461 0.147461i
\(126\) 0 0
\(127\) 88.2707 0.695045 0.347523 0.937672i \(-0.387023\pi\)
0.347523 + 0.937672i \(0.387023\pi\)
\(128\) 0 0
\(129\) 28.3867 + 11.7415i 0.220052 + 0.0910192i
\(130\) 0 0
\(131\) 57.0518 57.0518i 0.435510 0.435510i −0.454988 0.890498i \(-0.650356\pi\)
0.890498 + 0.454988i \(0.150356\pi\)
\(132\) 0 0
\(133\) −155.632 + 155.632i −1.17017 + 1.17017i
\(134\) 0 0
\(135\) 70.4575 + 169.378i 0.521907 + 1.25465i
\(136\) 0 0
\(137\) 165.112 1.20520 0.602599 0.798045i \(-0.294132\pi\)
0.602599 + 0.798045i \(0.294132\pi\)
\(138\) 0 0
\(139\) 95.0802 95.0802i 0.684030 0.684030i −0.276875 0.960906i \(-0.589299\pi\)
0.960906 + 0.276875i \(0.0892990\pi\)
\(140\) 0 0
\(141\) −63.8752 + 26.4955i −0.453015 + 0.187912i
\(142\) 0 0
\(143\) 7.33920i 0.0513231i
\(144\) 0 0
\(145\) 332.516 2.29321
\(146\) 0 0
\(147\) 5.97376 + 14.4015i 0.0406378 + 0.0979693i
\(148\) 0 0
\(149\) −131.077 131.077i −0.879709 0.879709i 0.113795 0.993504i \(-0.463699\pi\)
−0.993504 + 0.113795i \(0.963699\pi\)
\(150\) 0 0
\(151\) 123.070i 0.815031i −0.913198 0.407515i \(-0.866395\pi\)
0.913198 0.407515i \(-0.133605\pi\)
\(152\) 0 0
\(153\) 70.7913 + 70.6494i 0.462688 + 0.461761i
\(154\) 0 0
\(155\) −119.457 119.457i −0.770690 0.770690i
\(156\) 0 0
\(157\) 139.181 + 139.181i 0.886503 + 0.886503i 0.994185 0.107683i \(-0.0343430\pi\)
−0.107683 + 0.994185i \(0.534343\pi\)
\(158\) 0 0
\(159\) 51.7620 125.142i 0.325547 0.787058i
\(160\) 0 0
\(161\) 57.4396i 0.356768i
\(162\) 0 0
\(163\) 19.9311 + 19.9311i 0.122277 + 0.122277i 0.765597 0.643320i \(-0.222444\pi\)
−0.643320 + 0.765597i \(0.722444\pi\)
\(164\) 0 0
\(165\) 13.7043 5.68458i 0.0830566 0.0344520i
\(166\) 0 0
\(167\) −60.3220 −0.361210 −0.180605 0.983556i \(-0.557806\pi\)
−0.180605 + 0.983556i \(0.557806\pi\)
\(168\) 0 0
\(169\) 67.3351i 0.398432i
\(170\) 0 0
\(171\) −269.072 + 0.270083i −1.57352 + 0.00157943i
\(172\) 0 0
\(173\) 74.8292 74.8292i 0.432539 0.432539i −0.456952 0.889491i \(-0.651059\pi\)
0.889491 + 0.456952i \(0.151059\pi\)
\(174\) 0 0
\(175\) −155.802 −0.890295
\(176\) 0 0
\(177\) −28.5846 + 69.1074i −0.161495 + 0.390438i
\(178\) 0 0
\(179\) −3.96558 + 3.96558i −0.0221541 + 0.0221541i −0.718097 0.695943i \(-0.754987\pi\)
0.695943 + 0.718097i \(0.254987\pi\)
\(180\) 0 0
\(181\) 158.820 158.820i 0.877457 0.877457i −0.115814 0.993271i \(-0.536948\pi\)
0.993271 + 0.115814i \(0.0369475\pi\)
\(182\) 0 0
\(183\) −20.0973 + 48.5882i −0.109821 + 0.265509i
\(184\) 0 0
\(185\) −175.608 −0.949233
\(186\) 0 0
\(187\) 5.71960 5.71960i 0.0305861 0.0305861i
\(188\) 0 0
\(189\) −75.7896 + 183.754i −0.401003 + 0.972245i
\(190\) 0 0
\(191\) 68.8639i 0.360544i 0.983617 + 0.180272i \(0.0576978\pi\)
−0.983617 + 0.180272i \(0.942302\pi\)
\(192\) 0 0
\(193\) −366.645 −1.89971 −0.949856 0.312686i \(-0.898771\pi\)
−0.949856 + 0.312686i \(0.898771\pi\)
\(194\) 0 0
\(195\) 189.837 78.7446i 0.973522 0.403819i
\(196\) 0 0
\(197\) 246.744 + 246.744i 1.25251 + 1.25251i 0.954593 + 0.297912i \(0.0962901\pi\)
0.297912 + 0.954593i \(0.403710\pi\)
\(198\) 0 0
\(199\) 287.802i 1.44624i −0.690722 0.723120i \(-0.742707\pi\)
0.690722 0.723120i \(-0.257293\pi\)
\(200\) 0 0
\(201\) −66.7176 + 161.299i −0.331928 + 0.802484i
\(202\) 0 0
\(203\) 254.763 + 254.763i 1.25499 + 1.25499i
\(204\) 0 0
\(205\) −308.654 308.654i −1.50563 1.50563i
\(206\) 0 0
\(207\) −49.6037 + 49.7034i −0.239632 + 0.240113i
\(208\) 0 0
\(209\) 21.7616i 0.104122i
\(210\) 0 0
\(211\) −156.146 156.146i −0.740027 0.740027i 0.232556 0.972583i \(-0.425291\pi\)
−0.972583 + 0.232556i \(0.925291\pi\)
\(212\) 0 0
\(213\) −29.5159 71.1567i −0.138572 0.334069i
\(214\) 0 0
\(215\) −69.5724 −0.323593
\(216\) 0 0
\(217\) 183.048i 0.843541i
\(218\) 0 0
\(219\) 155.691 64.5807i 0.710916 0.294889i
\(220\) 0 0
\(221\) 79.2296 79.2296i 0.358505 0.358505i
\(222\) 0 0
\(223\) −45.2998 −0.203138 −0.101569 0.994828i \(-0.532386\pi\)
−0.101569 + 0.994828i \(0.532386\pi\)
\(224\) 0 0
\(225\) −134.818 134.547i −0.599190 0.597988i
\(226\) 0 0
\(227\) −300.757 + 300.757i −1.32492 + 1.32492i −0.415186 + 0.909737i \(0.636283\pi\)
−0.909737 + 0.415186i \(0.863717\pi\)
\(228\) 0 0
\(229\) 65.7088 65.7088i 0.286938 0.286938i −0.548930 0.835868i \(-0.684965\pi\)
0.835868 + 0.548930i \(0.184965\pi\)
\(230\) 0 0
\(231\) 14.8552 + 6.14449i 0.0643082 + 0.0265995i
\(232\) 0 0
\(233\) −42.8218 −0.183785 −0.0918923 0.995769i \(-0.529292\pi\)
−0.0918923 + 0.995769i \(0.529292\pi\)
\(234\) 0 0
\(235\) 110.744 110.744i 0.471250 0.471250i
\(236\) 0 0
\(237\) −41.0365 98.9305i −0.173150 0.417428i
\(238\) 0 0
\(239\) 100.598i 0.420913i −0.977603 0.210456i \(-0.932505\pi\)
0.977603 0.210456i \(-0.0674950\pi\)
\(240\) 0 0
\(241\) −5.23162 −0.0217080 −0.0108540 0.999941i \(-0.503455\pi\)
−0.0108540 + 0.999941i \(0.503455\pi\)
\(242\) 0 0
\(243\) −224.269 + 93.5551i −0.922916 + 0.385001i
\(244\) 0 0
\(245\) −24.9686 24.9686i −0.101913 0.101913i
\(246\) 0 0
\(247\) 301.448i 1.22044i
\(248\) 0 0
\(249\) −372.222 153.961i −1.49487 0.618315i
\(250\) 0 0
\(251\) −17.4381 17.4381i −0.0694747 0.0694747i 0.671516 0.740990i \(-0.265644\pi\)
−0.740990 + 0.671516i \(0.765644\pi\)
\(252\) 0 0
\(253\) 4.01579 + 4.01579i 0.0158727 + 0.0158727i
\(254\) 0 0
\(255\) −209.311 86.5765i −0.820828 0.339516i
\(256\) 0 0
\(257\) 343.676i 1.33726i −0.743595 0.668630i \(-0.766881\pi\)
0.743595 0.668630i \(-0.233119\pi\)
\(258\) 0 0
\(259\) −134.545 134.545i −0.519480 0.519480i
\(260\) 0 0
\(261\) 0.442114 + 440.460i 0.00169392 + 1.68758i
\(262\) 0 0
\(263\) −98.0863 −0.372952 −0.186476 0.982460i \(-0.559707\pi\)
−0.186476 + 0.982460i \(0.559707\pi\)
\(264\) 0 0
\(265\) 306.708i 1.15739i
\(266\) 0 0
\(267\) 51.5769 + 124.341i 0.193172 + 0.465697i
\(268\) 0 0
\(269\) −126.560 + 126.560i −0.470482 + 0.470482i −0.902070 0.431589i \(-0.857953\pi\)
0.431589 + 0.902070i \(0.357953\pi\)
\(270\) 0 0
\(271\) 206.487 0.761945 0.380972 0.924586i \(-0.375589\pi\)
0.380972 + 0.924586i \(0.375589\pi\)
\(272\) 0 0
\(273\) 205.779 + 85.1154i 0.753768 + 0.311778i
\(274\) 0 0
\(275\) −10.8926 + 10.8926i −0.0396095 + 0.0396095i
\(276\) 0 0
\(277\) 183.416 183.416i 0.662153 0.662153i −0.293734 0.955887i \(-0.594898\pi\)
0.955887 + 0.293734i \(0.0948980\pi\)
\(278\) 0 0
\(279\) 158.077 158.395i 0.566585 0.567723i
\(280\) 0 0
\(281\) −109.143 −0.388409 −0.194204 0.980961i \(-0.562213\pi\)
−0.194204 + 0.980961i \(0.562213\pi\)
\(282\) 0 0
\(283\) −60.4623 + 60.4623i −0.213648 + 0.213648i −0.805815 0.592167i \(-0.798272\pi\)
0.592167 + 0.805815i \(0.298272\pi\)
\(284\) 0 0
\(285\) 562.888 233.487i 1.97504 0.819252i
\(286\) 0 0
\(287\) 472.962i 1.64795i
\(288\) 0 0
\(289\) 165.509 0.572697
\(290\) 0 0
\(291\) 94.6721 + 228.235i 0.325334 + 0.784311i
\(292\) 0 0
\(293\) 19.4639 + 19.4639i 0.0664296 + 0.0664296i 0.739541 0.673111i \(-0.235043\pi\)
−0.673111 + 0.739541i \(0.735043\pi\)
\(294\) 0 0
\(295\) 169.374i 0.574149i
\(296\) 0 0
\(297\) 7.54818 + 18.1456i 0.0254147 + 0.0610963i
\(298\) 0 0
\(299\) 55.6280 + 55.6280i 0.186047 + 0.186047i
\(300\) 0 0
\(301\) −53.3042 53.3042i −0.177090 0.177090i
\(302\) 0 0
\(303\) −58.9329 + 142.479i −0.194498 + 0.470227i
\(304\) 0 0
\(305\) 119.084i 0.390438i
\(306\) 0 0
\(307\) 408.201 + 408.201i 1.32964 + 1.32964i 0.905677 + 0.423967i \(0.139363\pi\)
0.423967 + 0.905677i \(0.360637\pi\)
\(308\) 0 0
\(309\) 242.517 100.596i 0.784844 0.325554i
\(310\) 0 0
\(311\) 360.965 1.16066 0.580330 0.814381i \(-0.302924\pi\)
0.580330 + 0.814381i \(0.302924\pi\)
\(312\) 0 0
\(313\) 73.9217i 0.236172i 0.993003 + 0.118086i \(0.0376758\pi\)
−0.993003 + 0.118086i \(0.962324\pi\)
\(314\) 0 0
\(315\) −0.451863 450.172i −0.00143449 1.42912i
\(316\) 0 0
\(317\) −172.709 + 172.709i −0.544825 + 0.544825i −0.924939 0.380115i \(-0.875884\pi\)
0.380115 + 0.924939i \(0.375884\pi\)
\(318\) 0 0
\(319\) 35.6227 0.111670
\(320\) 0 0
\(321\) −170.053 + 411.128i −0.529761 + 1.28077i
\(322\) 0 0
\(323\) 234.925 234.925i 0.727321 0.727321i
\(324\) 0 0
\(325\) −150.888 + 150.888i −0.464271 + 0.464271i
\(326\) 0 0
\(327\) 12.3893 29.9529i 0.0378877 0.0915990i
\(328\) 0 0
\(329\) 169.697 0.515796
\(330\) 0 0
\(331\) −261.507 + 261.507i −0.790051 + 0.790051i −0.981502 0.191451i \(-0.938681\pi\)
0.191451 + 0.981502i \(0.438681\pi\)
\(332\) 0 0
\(333\) −0.233489 232.615i −0.000701168 0.698544i
\(334\) 0 0
\(335\) 395.325i 1.18007i
\(336\) 0 0
\(337\) 18.2211 0.0540684 0.0270342 0.999635i \(-0.491394\pi\)
0.0270342 + 0.999635i \(0.491394\pi\)
\(338\) 0 0
\(339\) 36.3765 15.0890i 0.107305 0.0445104i
\(340\) 0 0
\(341\) −12.7975 12.7975i −0.0375294 0.0375294i
\(342\) 0 0
\(343\) 322.471i 0.940149i
\(344\) 0 0
\(345\) 60.7863 146.960i 0.176192 0.425970i
\(346\) 0 0
\(347\) 173.710 + 173.710i 0.500605 + 0.500605i 0.911626 0.411021i \(-0.134828\pi\)
−0.411021 + 0.911626i \(0.634828\pi\)
\(348\) 0 0
\(349\) 387.899 + 387.899i 1.11146 + 1.11146i 0.992953 + 0.118506i \(0.0378106\pi\)
0.118506 + 0.992953i \(0.462189\pi\)
\(350\) 0 0
\(351\) 104.560 + 251.358i 0.297891 + 0.716121i
\(352\) 0 0
\(353\) 676.812i 1.91732i 0.284561 + 0.958658i \(0.408152\pi\)
−0.284561 + 0.958658i \(0.591848\pi\)
\(354\) 0 0
\(355\) 123.368 + 123.368i 0.347516 + 0.347516i
\(356\) 0 0
\(357\) −94.0355 226.700i −0.263405 0.635014i
\(358\) 0 0
\(359\) −240.896 −0.671020 −0.335510 0.942037i \(-0.608909\pi\)
−0.335510 + 0.942037i \(0.608909\pi\)
\(360\) 0 0
\(361\) 532.827i 1.47598i
\(362\) 0 0
\(363\) −333.830 + 138.473i −0.919643 + 0.381469i
\(364\) 0 0
\(365\) −269.929 + 269.929i −0.739532 + 0.739532i
\(366\) 0 0
\(367\) −666.702 −1.81663 −0.908313 0.418291i \(-0.862629\pi\)
−0.908313 + 0.418291i \(0.862629\pi\)
\(368\) 0 0
\(369\) 408.441 409.261i 1.10689 1.10911i
\(370\) 0 0
\(371\) −234.990 + 234.990i −0.633397 + 0.633397i
\(372\) 0 0
\(373\) −358.513 + 358.513i −0.961160 + 0.961160i −0.999273 0.0381137i \(-0.987865\pi\)
0.0381137 + 0.999273i \(0.487865\pi\)
\(374\) 0 0
\(375\) −72.2653 29.8908i −0.192708 0.0797088i
\(376\) 0 0
\(377\) 493.457 1.30890
\(378\) 0 0
\(379\) 140.959 140.959i 0.371925 0.371925i −0.496253 0.868178i \(-0.665291\pi\)
0.868178 + 0.496253i \(0.165291\pi\)
\(380\) 0 0
\(381\) −101.462 244.604i −0.266305 0.642004i
\(382\) 0 0
\(383\) 69.4683i 0.181379i 0.995879 + 0.0906897i \(0.0289071\pi\)
−0.995879 + 0.0906897i \(0.971093\pi\)
\(384\) 0 0
\(385\) −36.4083 −0.0945669
\(386\) 0 0
\(387\) −0.0925037 92.1575i −0.000239028 0.238133i
\(388\) 0 0
\(389\) −265.362 265.362i −0.682165 0.682165i 0.278322 0.960488i \(-0.410222\pi\)
−0.960488 + 0.278322i \(0.910222\pi\)
\(390\) 0 0
\(391\) 86.7042i 0.221750i
\(392\) 0 0
\(393\) −223.672 92.5164i −0.569139 0.235411i
\(394\) 0 0
\(395\) 171.521 + 171.521i 0.434230 + 0.434230i
\(396\) 0 0
\(397\) 259.123 + 259.123i 0.652703 + 0.652703i 0.953643 0.300940i \(-0.0973004\pi\)
−0.300940 + 0.953643i \(0.597300\pi\)
\(398\) 0 0
\(399\) 610.157 + 252.377i 1.52922 + 0.632523i
\(400\) 0 0
\(401\) 664.163i 1.65627i −0.560531 0.828133i \(-0.689403\pi\)
0.560531 0.828133i \(-0.310597\pi\)
\(402\) 0 0
\(403\) −177.275 177.275i −0.439889 0.439889i
\(404\) 0 0
\(405\) 388.369 389.932i 0.958937 0.962795i
\(406\) 0 0
\(407\) −18.8131 −0.0462237
\(408\) 0 0
\(409\) 530.421i 1.29687i −0.761269 0.648437i \(-0.775423\pi\)
0.761269 0.648437i \(-0.224577\pi\)
\(410\) 0 0
\(411\) −189.787 457.536i −0.461768 1.11323i
\(412\) 0 0
\(413\) 129.769 129.769i 0.314211 0.314211i
\(414\) 0 0
\(415\) 912.271 2.19824
\(416\) 0 0
\(417\) −372.762 154.184i −0.893914 0.369746i
\(418\) 0 0
\(419\) 404.149 404.149i 0.964556 0.964556i −0.0348367 0.999393i \(-0.511091\pi\)
0.999393 + 0.0348367i \(0.0110911\pi\)
\(420\) 0 0
\(421\) −264.630 + 264.630i −0.628575 + 0.628575i −0.947710 0.319134i \(-0.896608\pi\)
0.319134 + 0.947710i \(0.396608\pi\)
\(422\) 0 0
\(423\) 146.842 + 146.547i 0.347143 + 0.346447i
\(424\) 0 0
\(425\) 235.180 0.553366
\(426\) 0 0
\(427\) 91.2382 91.2382i 0.213673 0.213673i
\(428\) 0 0
\(429\) 20.3374 8.43598i 0.0474065 0.0196643i
\(430\) 0 0
\(431\) 766.652i 1.77877i 0.457155 + 0.889387i \(0.348868\pi\)
−0.457155 + 0.889387i \(0.651132\pi\)
\(432\) 0 0
\(433\) 151.222 0.349243 0.174622 0.984636i \(-0.444130\pi\)
0.174622 + 0.984636i \(0.444130\pi\)
\(434\) 0 0
\(435\) −382.207 921.422i −0.878638 2.11821i
\(436\) 0 0
\(437\) 164.943 + 164.943i 0.377445 + 0.377445i
\(438\) 0 0
\(439\) 565.007i 1.28703i 0.765433 + 0.643516i \(0.222525\pi\)
−0.765433 + 0.643516i \(0.777475\pi\)
\(440\) 0 0
\(441\) 33.0409 33.1073i 0.0749228 0.0750733i
\(442\) 0 0
\(443\) 100.963 + 100.963i 0.227907 + 0.227907i 0.811818 0.583911i \(-0.198478\pi\)
−0.583911 + 0.811818i \(0.698478\pi\)
\(444\) 0 0
\(445\) −215.577 215.577i −0.484442 0.484442i
\(446\) 0 0
\(447\) −212.557 + 513.887i −0.475518 + 1.14963i
\(448\) 0 0
\(449\) 131.725i 0.293375i −0.989183 0.146687i \(-0.953139\pi\)
0.989183 0.146687i \(-0.0468611\pi\)
\(450\) 0 0
\(451\) −33.0663 33.0663i −0.0733178 0.0733178i
\(452\) 0 0
\(453\) −341.034 + 141.461i −0.752833 + 0.312277i
\(454\) 0 0
\(455\) −504.339 −1.10844
\(456\) 0 0
\(457\) 137.963i 0.301888i −0.988542 0.150944i \(-0.951769\pi\)
0.988542 0.150944i \(-0.0482313\pi\)
\(458\) 0 0
\(459\) 114.403 277.374i 0.249245 0.604302i
\(460\) 0 0
\(461\) −303.536 + 303.536i −0.658430 + 0.658430i −0.955008 0.296579i \(-0.904154\pi\)
0.296579 + 0.955008i \(0.404154\pi\)
\(462\) 0 0
\(463\) −280.379 −0.605570 −0.302785 0.953059i \(-0.597916\pi\)
−0.302785 + 0.953059i \(0.597916\pi\)
\(464\) 0 0
\(465\) −193.714 + 468.332i −0.416589 + 1.00716i
\(466\) 0 0
\(467\) 65.6355 65.6355i 0.140547 0.140547i −0.633333 0.773880i \(-0.718313\pi\)
0.773880 + 0.633333i \(0.218313\pi\)
\(468\) 0 0
\(469\) 302.886 302.886i 0.645812 0.645812i
\(470\) 0 0
\(471\) 225.699 545.659i 0.479190 1.15851i
\(472\) 0 0
\(473\) −7.45335 −0.0157576
\(474\) 0 0
\(475\) −447.400 + 447.400i −0.941894 + 0.941894i
\(476\) 0 0
\(477\) −406.274 + 0.407800i −0.851728 + 0.000854927i
\(478\) 0 0
\(479\) 373.272i 0.779273i −0.920969 0.389636i \(-0.872601\pi\)
0.920969 0.389636i \(-0.127399\pi\)
\(480\) 0 0
\(481\) −260.604 −0.541797
\(482\) 0 0
\(483\) 159.169 66.0234i 0.329542 0.136694i
\(484\) 0 0
\(485\) −395.702 395.702i −0.815881 0.815881i
\(486\) 0 0
\(487\) 0.0470526i 9.66171e-5i 1.00000 4.83086e-5i \(1.53771e-5\pi\)
−1.00000 4.83086e-5i \(0.999985\pi\)
\(488\) 0 0
\(489\) 32.3206 78.1398i 0.0660954 0.159795i
\(490\) 0 0
\(491\) −273.442 273.442i −0.556908 0.556908i 0.371518 0.928426i \(-0.378837\pi\)
−0.928426 + 0.371518i \(0.878837\pi\)
\(492\) 0 0
\(493\) −384.562 384.562i −0.780044 0.780044i
\(494\) 0 0
\(495\) −31.5047 31.4415i −0.0636458 0.0635182i
\(496\) 0 0
\(497\) 189.042i 0.380365i
\(498\) 0 0
\(499\) 46.2637 + 46.2637i 0.0927129 + 0.0927129i 0.751942 0.659229i \(-0.229117\pi\)
−0.659229 + 0.751942i \(0.729117\pi\)
\(500\) 0 0
\(501\) 69.3366 + 167.156i 0.138396 + 0.333645i
\(502\) 0 0
\(503\) −864.426 −1.71854 −0.859270 0.511522i \(-0.829082\pi\)
−0.859270 + 0.511522i \(0.829082\pi\)
\(504\) 0 0
\(505\) 349.198i 0.691482i
\(506\) 0 0
\(507\) −186.590 + 77.3977i −0.368027 + 0.152658i
\(508\) 0 0
\(509\) −171.041 + 171.041i −0.336033 + 0.336033i −0.854872 0.518839i \(-0.826365\pi\)
0.518839 + 0.854872i \(0.326365\pi\)
\(510\) 0 0
\(511\) −413.623 −0.809438
\(512\) 0 0
\(513\) 310.031 + 745.306i 0.604349 + 1.45284i
\(514\) 0 0
\(515\) −420.464 + 420.464i −0.816435 + 0.816435i
\(516\) 0 0
\(517\) 11.8641 11.8641i 0.0229479 0.0229479i
\(518\) 0 0
\(519\) −293.368 121.345i −0.565257 0.233805i
\(520\) 0 0
\(521\) 351.572 0.674802 0.337401 0.941361i \(-0.390452\pi\)
0.337401 + 0.941361i \(0.390452\pi\)
\(522\) 0 0
\(523\) 287.638 287.638i 0.549977 0.549977i −0.376457 0.926434i \(-0.622858\pi\)
0.926434 + 0.376457i \(0.122858\pi\)
\(524\) 0 0
\(525\) 179.085 + 431.736i 0.341114 + 0.822354i
\(526\) 0 0
\(527\) 276.309i 0.524306i
\(528\) 0 0
\(529\) −468.124 −0.884922
\(530\) 0 0
\(531\) 224.357 0.225200i 0.422519 0.000424106i
\(532\) 0 0
\(533\) −458.045 458.045i −0.859372 0.859372i
\(534\) 0 0
\(535\) 1007.63i 1.88341i
\(536\) 0 0
\(537\) 15.5471 + 6.43067i 0.0289517 + 0.0119752i
\(538\) 0 0
\(539\) −2.67491 2.67491i −0.00496273 0.00496273i
\(540\) 0 0
\(541\) 419.846 + 419.846i 0.776056 + 0.776056i 0.979158 0.203102i \(-0.0651021\pi\)
−0.203102 + 0.979158i \(0.565102\pi\)
\(542\) 0 0
\(543\) −622.654 257.545i −1.14669 0.474301i
\(544\) 0 0
\(545\) 73.4108i 0.134699i
\(546\) 0 0
\(547\) −517.346 517.346i −0.945789 0.945789i 0.0528155 0.998604i \(-0.483180\pi\)
−0.998604 + 0.0528155i \(0.983180\pi\)
\(548\) 0 0
\(549\) 157.742 0.158334i 0.287325 0.000288404i
\(550\) 0 0
\(551\) 1463.16 2.65546
\(552\) 0 0
\(553\) 262.828i 0.475277i
\(554\) 0 0
\(555\) 201.851 + 486.621i 0.363696 + 0.876794i
\(556\) 0 0
\(557\) 31.8976 31.8976i 0.0572667 0.0572667i −0.677893 0.735160i \(-0.737107\pi\)
0.735160 + 0.677893i \(0.237107\pi\)
\(558\) 0 0
\(559\) −103.246 −0.184698
\(560\) 0 0
\(561\) −22.4237 9.27502i −0.0399709 0.0165330i
\(562\) 0 0
\(563\) 32.9214 32.9214i 0.0584750 0.0584750i −0.677265 0.735740i \(-0.736835\pi\)
0.735740 + 0.677265i \(0.236835\pi\)
\(564\) 0 0
\(565\) −63.0679 + 63.0679i −0.111625 + 0.111625i
\(566\) 0 0
\(567\) 596.310 1.19710i 1.05169 0.00211129i
\(568\) 0 0
\(569\) 647.095 1.13725 0.568624 0.822597i \(-0.307476\pi\)
0.568624 + 0.822597i \(0.307476\pi\)
\(570\) 0 0
\(571\) −451.861 + 451.861i −0.791350 + 0.791350i −0.981714 0.190363i \(-0.939033\pi\)
0.190363 + 0.981714i \(0.439033\pi\)
\(572\) 0 0
\(573\) 190.826 79.1550i 0.333030 0.138141i
\(574\) 0 0
\(575\) 165.123i 0.287170i
\(576\) 0 0
\(577\) 532.176 0.922315 0.461157 0.887318i \(-0.347434\pi\)
0.461157 + 0.887318i \(0.347434\pi\)
\(578\) 0 0
\(579\) 421.436 + 1015.99i 0.727869 + 1.75474i
\(580\) 0 0
\(581\) 698.953 + 698.953i 1.20302 + 1.20302i
\(582\) 0 0
\(583\) 32.8579i 0.0563601i
\(584\) 0 0
\(585\) −436.412 435.537i −0.746004 0.744508i
\(586\) 0 0
\(587\) −532.393 532.393i −0.906973 0.906973i 0.0890534 0.996027i \(-0.471616\pi\)
−0.996027 + 0.0890534i \(0.971616\pi\)
\(588\) 0 0
\(589\) −525.642 525.642i −0.892431 0.892431i
\(590\) 0 0
\(591\) 400.124 967.359i 0.677029 1.63682i
\(592\) 0 0
\(593\) 254.750i 0.429595i 0.976659 + 0.214798i \(0.0689092\pi\)
−0.976659 + 0.214798i \(0.931091\pi\)
\(594\) 0 0
\(595\) 393.042 + 393.042i 0.660574 + 0.660574i
\(596\) 0 0
\(597\) −797.517 + 330.811i −1.33587 + 0.554123i
\(598\) 0 0
\(599\) −624.772 −1.04303 −0.521513 0.853244i \(-0.674632\pi\)
−0.521513 + 0.853244i \(0.674632\pi\)
\(600\) 0 0
\(601\) 386.910i 0.643777i 0.946778 + 0.321889i \(0.104318\pi\)
−0.946778 + 0.321889i \(0.895682\pi\)
\(602\) 0 0
\(603\) 523.658 0.525625i 0.868422 0.000871684i
\(604\) 0 0
\(605\) 578.779 578.779i 0.956660 0.956660i
\(606\) 0 0
\(607\) 951.141 1.56695 0.783477 0.621421i \(-0.213444\pi\)
0.783477 + 0.621421i \(0.213444\pi\)
\(608\) 0 0
\(609\) 413.129 998.800i 0.678373 1.64007i
\(610\) 0 0
\(611\) 164.345 164.345i 0.268977 0.268977i
\(612\) 0 0
\(613\) 387.896 387.896i 0.632783 0.632783i −0.315982 0.948765i \(-0.602334\pi\)
0.948765 + 0.315982i \(0.102334\pi\)
\(614\) 0 0
\(615\) −500.519 + 1210.08i −0.813852 + 1.96761i
\(616\) 0 0
\(617\) −882.945 −1.43103 −0.715514 0.698598i \(-0.753808\pi\)
−0.715514 + 0.698598i \(0.753808\pi\)
\(618\) 0 0
\(619\) 694.731 694.731i 1.12234 1.12234i 0.130955 0.991388i \(-0.458196\pi\)
0.991388 0.130955i \(-0.0418043\pi\)
\(620\) 0 0
\(621\) 194.748 + 80.3239i 0.313604 + 0.129346i
\(622\) 0 0
\(623\) 330.337i 0.530235i
\(624\) 0 0
\(625\) 706.197 1.12991
\(626\) 0 0
\(627\) 60.3027 25.0136i 0.0961765 0.0398942i
\(628\) 0 0
\(629\) 203.094 + 203.094i 0.322885 + 0.322885i
\(630\) 0 0
\(631\) 927.845i 1.47044i 0.677831 + 0.735218i \(0.262920\pi\)
−0.677831 + 0.735218i \(0.737080\pi\)
\(632\) 0 0
\(633\) −253.209 + 612.170i −0.400014 + 0.967093i
\(634\) 0 0
\(635\) 424.082 + 424.082i 0.667846 + 0.667846i
\(636\) 0 0
\(637\) −37.0537 37.0537i −0.0581691 0.0581691i
\(638\) 0 0
\(639\) −163.253 + 163.581i −0.255482 + 0.255995i
\(640\) 0 0
\(641\) 759.287i 1.18453i −0.805741 0.592267i \(-0.798233\pi\)
0.805741 0.592267i \(-0.201767\pi\)
\(642\) 0 0
\(643\) −274.424 274.424i −0.426787 0.426787i 0.460746 0.887532i \(-0.347582\pi\)
−0.887532 + 0.460746i \(0.847582\pi\)
\(644\) 0 0
\(645\) 79.9694 + 192.789i 0.123984 + 0.298898i
\(646\) 0 0
\(647\) 747.683 1.15561 0.577807 0.816173i \(-0.303908\pi\)
0.577807 + 0.816173i \(0.303908\pi\)
\(648\) 0 0
\(649\) 18.1452i 0.0279587i
\(650\) 0 0
\(651\) −507.239 + 210.403i −0.779168 + 0.323200i
\(652\) 0 0
\(653\) 605.127 605.127i 0.926688 0.926688i −0.0708022 0.997490i \(-0.522556\pi\)
0.997490 + 0.0708022i \(0.0225559\pi\)
\(654\) 0 0
\(655\) 548.192 0.836935
\(656\) 0 0
\(657\) −357.914 357.197i −0.544771 0.543678i
\(658\) 0 0
\(659\) −588.767 + 588.767i −0.893425 + 0.893425i −0.994844 0.101418i \(-0.967662\pi\)
0.101418 + 0.994844i \(0.467662\pi\)
\(660\) 0 0
\(661\) 3.60334 3.60334i 0.00545135 0.00545135i −0.704376 0.709827i \(-0.748773\pi\)
0.709827 + 0.704376i \(0.248773\pi\)
\(662\) 0 0
\(663\) −310.620 128.480i −0.468507 0.193786i
\(664\) 0 0
\(665\) −1495.42 −2.24875
\(666\) 0 0
\(667\) 270.005 270.005i 0.404805 0.404805i
\(668\) 0 0
\(669\) 52.0695 + 125.529i 0.0778319 + 0.187636i
\(670\) 0 0
\(671\) 12.7575i 0.0190127i
\(672\) 0 0
\(673\) −460.445 −0.684167 −0.342084 0.939670i \(-0.611133\pi\)
−0.342084 + 0.939670i \(0.611133\pi\)
\(674\) 0 0
\(675\) −217.874 + 528.242i −0.322776 + 0.782581i
\(676\) 0 0
\(677\) 150.713 + 150.713i 0.222618 + 0.222618i 0.809600 0.586982i \(-0.199684\pi\)
−0.586982 + 0.809600i \(0.699684\pi\)
\(678\) 0 0
\(679\) 606.350i 0.893004i
\(680\) 0 0
\(681\) 1179.12 + 487.714i 1.73145 + 0.716174i
\(682\) 0 0
\(683\) 577.893 + 577.893i 0.846109 + 0.846109i 0.989645 0.143536i \(-0.0458473\pi\)
−0.143536 + 0.989645i \(0.545847\pi\)
\(684\) 0 0
\(685\) 793.254 + 793.254i 1.15803 + 1.15803i
\(686\) 0 0
\(687\) −257.612 106.555i −0.374981 0.155102i
\(688\) 0 0
\(689\) 455.158i 0.660607i
\(690\) 0 0
\(691\) 545.023 + 545.023i 0.788745 + 0.788745i 0.981288 0.192544i \(-0.0616737\pi\)
−0.192544 + 0.981288i \(0.561674\pi\)
\(692\) 0 0
\(693\) −0.0484085 48.2274i −6.98536e−5 0.0695922i
\(694\) 0 0
\(695\) 913.595 1.31453
\(696\) 0 0
\(697\) 713.929i 1.02429i
\(698\) 0 0
\(699\) 49.2212 + 118.662i 0.0704165 + 0.169760i
\(700\) 0 0
\(701\) −413.745 + 413.745i −0.590221 + 0.590221i −0.937691 0.347470i \(-0.887041\pi\)
0.347470 + 0.937691i \(0.387041\pi\)
\(702\) 0 0
\(703\) −772.721 −1.09918
\(704\) 0 0
\(705\) −434.171 179.584i −0.615846 0.254730i
\(706\) 0 0
\(707\) 267.545 267.545i 0.378423 0.378423i
\(708\) 0 0
\(709\) 521.959 521.959i 0.736191 0.736191i −0.235648 0.971839i \(-0.575721\pi\)
0.971839 + 0.235648i \(0.0757212\pi\)
\(710\) 0 0
\(711\) −226.973 + 227.429i −0.319231 + 0.319873i
\(712\) 0 0
\(713\) −194.000 −0.272089
\(714\) 0 0
\(715\) −35.2600 + 35.2600i −0.0493147 + 0.0493147i
\(716\) 0 0
\(717\) −278.764 + 115.632i −0.388792 + 0.161272i
\(718\) 0 0
\(719\) 567.983i 0.789963i −0.918689 0.394981i \(-0.870751\pi\)
0.918689 0.394981i \(-0.129249\pi\)
\(720\) 0 0
\(721\) −644.293 −0.893610
\(722\) 0 0
\(723\) 6.01344 + 14.4972i 0.00831735 + 0.0200514i
\(724\) 0 0
\(725\) 732.374 + 732.374i 1.01017 + 1.01017i
\(726\) 0 0
\(727\) 635.396i 0.873998i −0.899462 0.436999i \(-0.856041\pi\)
0.899462 0.436999i \(-0.143959\pi\)
\(728\) 0 0
\(729\) 517.031 + 513.926i 0.709233 + 0.704974i
\(730\) 0 0
\(731\) 80.4619 + 80.4619i 0.110071 + 0.110071i
\(732\) 0 0
\(733\) −637.378 637.378i −0.869547 0.869547i 0.122875 0.992422i \(-0.460788\pi\)
−0.992422 + 0.122875i \(0.960788\pi\)
\(734\) 0 0
\(735\) −40.4897 + 97.8896i −0.0550880 + 0.133183i
\(736\) 0 0
\(737\) 42.3515i 0.0574648i
\(738\) 0 0
\(739\) −397.296 397.296i −0.537613 0.537613i 0.385214 0.922827i \(-0.374128\pi\)
−0.922827 + 0.385214i \(0.874128\pi\)
\(740\) 0 0
\(741\) 835.331 346.497i 1.12730 0.467607i
\(742\) 0 0
\(743\) −1160.78 −1.56229 −0.781145 0.624349i \(-0.785364\pi\)
−0.781145 + 0.624349i \(0.785364\pi\)
\(744\) 0 0
\(745\) 1259.47i 1.69057i
\(746\) 0 0
\(747\) 1.21296 + 1208.42i 0.00162377 + 1.61770i
\(748\) 0 0
\(749\) 772.011 772.011i 1.03072 1.03072i
\(750\) 0 0
\(751\) −1220.14 −1.62469 −0.812343 0.583181i \(-0.801808\pi\)
−0.812343 + 0.583181i \(0.801808\pi\)
\(752\) 0 0
\(753\) −28.2781 + 68.3663i −0.0375539 + 0.0907919i
\(754\) 0 0
\(755\) 591.268 591.268i 0.783136 0.783136i
\(756\) 0 0
\(757\) 202.623 202.623i 0.267666 0.267666i −0.560493 0.828159i \(-0.689388\pi\)
0.828159 + 0.560493i \(0.189388\pi\)
\(758\) 0 0
\(759\) 6.51210 15.7439i 0.00857984 0.0207430i
\(760\) 0 0
\(761\) −694.461 −0.912563 −0.456282 0.889835i \(-0.650819\pi\)
−0.456282 + 0.889835i \(0.650819\pi\)
\(762\) 0 0
\(763\) −56.2451 + 56.2451i −0.0737157 + 0.0737157i
\(764\) 0 0
\(765\) 0.682081 + 679.529i 0.000891610 + 0.888273i
\(766\) 0 0
\(767\) 251.353i 0.327709i
\(768\) 0 0
\(769\) 405.268 0.527007 0.263503 0.964658i \(-0.415122\pi\)
0.263503 + 0.964658i \(0.415122\pi\)
\(770\) 0 0
\(771\) −952.347 + 395.035i −1.23521 + 0.512367i
\(772\) 0 0
\(773\) 142.479 + 142.479i 0.184320 + 0.184320i 0.793235 0.608916i \(-0.208395\pi\)
−0.608916 + 0.793235i \(0.708395\pi\)
\(774\) 0 0
\(775\) 526.214i 0.678985i
\(776\) 0 0
\(777\) −218.182 + 527.486i −0.280800 + 0.678875i
\(778\) 0 0
\(779\) −1358.16 1358.16i −1.74346 1.74346i
\(780\) 0 0
\(781\) 13.2165 + 13.2165i 0.0169226 + 0.0169226i
\(782\) 0 0
\(783\) 1220.03 507.507i 1.55815 0.648158i
\(784\) 0 0
\(785\) 1337.34i 1.70362i
\(786\) 0 0
\(787\) 482.883 + 482.883i 0.613574 + 0.613574i 0.943876 0.330301i \(-0.107150\pi\)
−0.330301 + 0.943876i \(0.607150\pi\)
\(788\) 0 0
\(789\) 112.744 + 271.803i 0.142895 + 0.344491i
\(790\) 0 0
\(791\) −96.6413 −0.122176
\(792\) 0 0
\(793\) 176.721i 0.222852i
\(794\) 0 0
\(795\) 849.908 352.543i 1.06907 0.443451i
\(796\) 0 0
\(797\) −552.965 + 552.965i −0.693808 + 0.693808i −0.963068 0.269260i \(-0.913221\pi\)
0.269260 + 0.963068i \(0.413221\pi\)
\(798\) 0 0
\(799\) −256.155 −0.320595
\(800\) 0 0
\(801\) 285.272 285.846i 0.356145 0.356861i
\(802\) 0 0
\(803\) −28.9178 + 28.9178i −0.0360122 + 0.0360122i
\(804\) 0 0
\(805\) −275.959 + 275.959i −0.342806 + 0.342806i
\(806\) 0 0
\(807\) 496.177 + 205.232i 0.614842 + 0.254314i
\(808\) 0 0
\(809\) −930.240 −1.14986 −0.574932 0.818201i \(-0.694972\pi\)
−0.574932 + 0.818201i \(0.694972\pi\)
\(810\) 0 0
\(811\) 611.506 611.506i 0.754015 0.754015i −0.221211 0.975226i \(-0.571001\pi\)
0.975226 + 0.221211i \(0.0710010\pi\)
\(812\) 0 0
\(813\) −237.345 572.188i −0.291937 0.703799i
\(814\) 0 0
\(815\) 191.511i 0.234983i
\(816\) 0 0
\(817\) −306.137 −0.374708
\(818\) 0 0
\(819\) −0.670570 668.061i −0.000818767 0.815703i
\(820\) 0 0
\(821\) −963.577 963.577i −1.17366 1.17366i −0.981330 0.192334i \(-0.938394\pi\)
−0.192334 0.981330i \(-0.561606\pi\)
\(822\) 0 0
\(823\) 1112.81i 1.35214i 0.736836 + 0.676071i \(0.236319\pi\)
−0.736836 + 0.676071i \(0.763681\pi\)
\(824\) 0 0
\(825\) 42.7046 + 17.6637i 0.0517631 + 0.0214105i
\(826\) 0 0
\(827\) −600.156 600.156i −0.725703 0.725703i 0.244058 0.969761i \(-0.421521\pi\)
−0.969761 + 0.244058i \(0.921521\pi\)
\(828\) 0 0
\(829\) −921.578 921.578i −1.11167 1.11167i −0.992924 0.118750i \(-0.962111\pi\)
−0.118750 0.992924i \(-0.537889\pi\)
\(830\) 0 0
\(831\) −719.085 297.432i −0.865324 0.357920i
\(832\) 0 0
\(833\) 57.7535i 0.0693319i
\(834\) 0 0
\(835\) −289.807 289.807i −0.347075 0.347075i
\(836\) 0 0
\(837\) −620.622 255.976i −0.741484 0.305826i
\(838\) 0 0
\(839\) 1230.19 1.46625 0.733127 0.680091i \(-0.238060\pi\)
0.733127 + 0.680091i \(0.238060\pi\)
\(840\) 0 0
\(841\) 1554.12i 1.84794i
\(842\) 0 0
\(843\) 125.453 + 302.442i 0.148818 + 0.358768i
\(844\) 0 0
\(845\) 323.501 323.501i 0.382841 0.382841i
\(846\) 0 0
\(847\) 886.886 1.04709
\(848\) 0 0
\(849\) 237.043 + 98.0470i 0.279202 + 0.115485i
\(850\) 0 0
\(851\) −142.595 + 142.595i −0.167562 + 0.167562i
\(852\) 0 0
\(853\) −1032.73 + 1032.73i −1.21070 + 1.21070i −0.239902 + 0.970797i \(0.577115\pi\)
−0.970797 + 0.239902i \(0.922885\pi\)
\(854\) 0 0
\(855\) −1294.01 1291.42i −1.51346 1.51043i
\(856\) 0 0
\(857\) 609.799 0.711550 0.355775 0.934572i \(-0.384217\pi\)
0.355775 + 0.934572i \(0.384217\pi\)
\(858\) 0 0
\(859\) −889.225 + 889.225i −1.03519 + 1.03519i −0.0358288 + 0.999358i \(0.511407\pi\)
−0.999358 + 0.0358288i \(0.988593\pi\)
\(860\) 0 0
\(861\) −1310.61 + 543.642i −1.52219 + 0.631407i
\(862\) 0 0
\(863\) 1322.86i 1.53286i 0.642329 + 0.766429i \(0.277968\pi\)
−0.642329 + 0.766429i \(0.722032\pi\)
\(864\) 0 0
\(865\) 719.010 0.831225
\(866\) 0 0
\(867\) −190.243 458.637i −0.219427 0.528993i
\(868\) 0 0
\(869\) 18.3752 + 18.3752i 0.0211452 + 0.0211452i
\(870\) 0 0
\(871\) 586.667i 0.673555i
\(872\) 0 0
\(873\) 523.632 524.684i 0.599808 0.601013i
\(874\) 0 0
\(875\) 135.699 + 135.699i 0.155084 + 0.155084i
\(876\) 0 0
\(877\) 853.500 + 853.500i 0.973204 + 0.973204i 0.999650 0.0264461i \(-0.00841905\pi\)
−0.0264461 + 0.999650i \(0.508419\pi\)
\(878\) 0 0
\(879\) 31.5630 76.3081i 0.0359078 0.0868124i
\(880\) 0 0
\(881\) 1075.39i 1.22064i −0.792153 0.610322i \(-0.791040\pi\)
0.792153 0.610322i \(-0.208960\pi\)
\(882\) 0 0
\(883\) −283.702 283.702i −0.321294 0.321294i 0.527970 0.849263i \(-0.322954\pi\)
−0.849263 + 0.527970i \(0.822954\pi\)
\(884\) 0 0
\(885\) −469.346 + 194.685i −0.530334 + 0.219983i
\(886\) 0 0
\(887\) 282.642 0.318650 0.159325 0.987226i \(-0.449068\pi\)
0.159325 + 0.987226i \(0.449068\pi\)
\(888\) 0 0
\(889\) 649.838i 0.730976i
\(890\) 0 0
\(891\) 41.6064 41.7738i 0.0466963 0.0468841i
\(892\) 0 0
\(893\) 487.301 487.301i 0.545690 0.545690i
\(894\) 0 0
\(895\) −38.1040 −0.0425743
\(896\) 0 0
\(897\) 90.2076 218.090i 0.100566 0.243133i
\(898\) 0 0
\(899\) −860.452 + 860.452i −0.957122 + 0.957122i
\(900\) 0 0
\(901\) 354.715 354.715i 0.393690 0.393690i
\(902\) 0 0
\(903\) −86.4392 + 208.979i −0.0957245 + 0.231428i
\(904\) 0 0
\(905\) 1526.05 1.68624
\(906\) 0 0
\(907\) 216.816 216.816i 0.239047 0.239047i −0.577408 0.816456i \(-0.695936\pi\)
0.816456 + 0.577408i \(0.195936\pi\)
\(908\) 0 0
\(909\) 462.558 0.464295i 0.508864 0.000510776i
\(910\) 0 0
\(911\) 1193.35i 1.30994i 0.755657 + 0.654968i \(0.227318\pi\)
−0.755657 + 0.654968i \(0.772682\pi\)
\(912\) 0 0
\(913\) 97.7324 0.107045
\(914\) 0 0
\(915\) −329.988 + 136.880i −0.360643 + 0.149595i
\(916\) 0 0
\(917\) 420.008 + 420.008i 0.458024 + 0.458024i
\(918\) 0 0
\(919\) 345.202i 0.375628i −0.982205 0.187814i \(-0.939860\pi\)
0.982205 0.187814i \(-0.0601402\pi\)
\(920\) 0 0
\(921\) 661.947 1600.35i 0.718727 1.73763i
\(922\) 0 0
\(923\) 183.080 + 183.080i 0.198353 + 0.198353i
\(924\) 0 0
\(925\) −386.781 386.781i −0.418141 0.418141i
\(926\) 0 0
\(927\) −557.517 556.399i −0.601421 0.600215i
\(928\) 0 0
\(929\) 1417.53i 1.52587i 0.646475 + 0.762935i \(0.276242\pi\)
−0.646475 + 0.762935i \(0.723758\pi\)
\(930\) 0 0
\(931\) −109.868 109.868i −0.118011 0.118011i
\(932\) 0 0
\(933\) −414.908 1000.26i −0.444704 1.07209i
\(934\) 0 0
\(935\) 54.9578 0.0587784
\(936\) 0 0
\(937\) 182.650i 0.194931i 0.995239 + 0.0974656i \(0.0310736\pi\)
−0.995239 + 0.0974656i \(0.968926\pi\)
\(938\) 0 0
\(939\) 204.842 84.9687i 0.218149 0.0904885i
\(940\) 0 0
\(941\) −524.733 + 524.733i −0.557634 + 0.557634i −0.928633 0.371000i \(-0.879015\pi\)
0.371000 + 0.928633i \(0.379015\pi\)
\(942\) 0 0
\(943\) −501.258 −0.531556
\(944\) 0 0
\(945\) −1246.94 + 518.699i −1.31951 + 0.548888i
\(946\) 0 0
\(947\) 278.292 278.292i 0.293867 0.293867i −0.544739 0.838606i \(-0.683371\pi\)
0.838606 + 0.544739i \(0.183371\pi\)
\(948\) 0 0
\(949\) −400.578 + 400.578i −0.422105 + 0.422105i
\(950\) 0 0
\(951\) 677.108 + 280.069i 0.711996 + 0.294500i
\(952\) 0 0
\(953\) 545.447 0.572348 0.286174 0.958178i \(-0.407617\pi\)
0.286174 + 0.958178i \(0.407617\pi\)
\(954\) 0 0
\(955\) −330.845 + 330.845i −0.346435 + 0.346435i
\(956\) 0 0
\(957\) −40.9462 98.7128i −0.0427860 0.103148i
\(958\) 0 0
\(959\) 1215.53i 1.26750i
\(960\) 0 0
\(961\) −342.761 −0.356672
\(962\) 0 0
\(963\) 1334.73 1.33974i 1.38601 0.00139122i
\(964\) 0 0
\(965\) −1761.48 1761.48i −1.82537 1.82537i
\(966\) 0 0
\(967\) 216.237i 0.223616i −0.993730 0.111808i \(-0.964336\pi\)
0.993730 0.111808i \(-0.0356642\pi\)
\(968\) 0 0
\(969\) −921.024 380.959i −0.950489 0.393147i
\(970\) 0 0
\(971\) 147.926 + 147.926i 0.152344 + 0.152344i 0.779164 0.626820i \(-0.215644\pi\)
−0.626820 + 0.779164i \(0.715644\pi\)
\(972\) 0 0
\(973\) 699.968 + 699.968i 0.719392 + 0.719392i
\(974\) 0 0
\(975\) 591.557 + 244.683i 0.606725 + 0.250957i
\(976\) 0 0
\(977\) 553.321i 0.566347i −0.959069 0.283174i \(-0.908613\pi\)
0.959069 0.283174i \(-0.0913873\pi\)
\(978\) 0 0
\(979\) −23.0949 23.0949i −0.0235903 0.0235903i
\(980\) 0 0
\(981\) −97.2420 + 0.0976072i −0.0991254 + 9.94977e-5i
\(982\) 0 0
\(983\) −1107.13 −1.12628 −0.563139 0.826362i \(-0.690406\pi\)
−0.563139 + 0.826362i \(0.690406\pi\)
\(984\) 0 0
\(985\) 2370.88i 2.40698i
\(986\) 0 0
\(987\) −195.057 470.241i −0.197626 0.476434i
\(988\) 0 0
\(989\) −56.4932 + 56.4932i −0.0571216 + 0.0571216i
\(990\) 0 0
\(991\) 1635.60 1.65046 0.825228 0.564800i \(-0.191047\pi\)
0.825228 + 0.564800i \(0.191047\pi\)
\(992\) 0 0
\(993\) 1025.24 + 424.065i 1.03247 + 0.427054i
\(994\) 0 0
\(995\) 1382.70 1382.70i 1.38965 1.38965i
\(996\) 0 0
\(997\) 152.140 152.140i 0.152598 0.152598i −0.626679 0.779277i \(-0.715586\pi\)
0.779277 + 0.626679i \(0.215586\pi\)
\(998\) 0 0
\(999\) −644.323 + 268.025i −0.644968 + 0.268293i
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 192.3.i.b.17.4 20
3.2 odd 2 inner 192.3.i.b.17.1 20
4.3 odd 2 48.3.i.b.29.4 yes 20
8.3 odd 2 384.3.i.d.161.4 20
8.5 even 2 384.3.i.c.161.7 20
12.11 even 2 48.3.i.b.29.7 yes 20
16.3 odd 4 384.3.i.d.353.1 20
16.5 even 4 inner 192.3.i.b.113.1 20
16.11 odd 4 48.3.i.b.5.7 yes 20
16.13 even 4 384.3.i.c.353.10 20
24.5 odd 2 384.3.i.c.161.10 20
24.11 even 2 384.3.i.d.161.1 20
48.5 odd 4 inner 192.3.i.b.113.4 20
48.11 even 4 48.3.i.b.5.4 20
48.29 odd 4 384.3.i.c.353.7 20
48.35 even 4 384.3.i.d.353.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.i.b.5.4 20 48.11 even 4
48.3.i.b.5.7 yes 20 16.11 odd 4
48.3.i.b.29.4 yes 20 4.3 odd 2
48.3.i.b.29.7 yes 20 12.11 even 2
192.3.i.b.17.1 20 3.2 odd 2 inner
192.3.i.b.17.4 20 1.1 even 1 trivial
192.3.i.b.113.1 20 16.5 even 4 inner
192.3.i.b.113.4 20 48.5 odd 4 inner
384.3.i.c.161.7 20 8.5 even 2
384.3.i.c.161.10 20 24.5 odd 2
384.3.i.c.353.7 20 48.29 odd 4
384.3.i.c.353.10 20 16.13 even 4
384.3.i.d.161.1 20 24.11 even 2
384.3.i.d.161.4 20 8.3 odd 2
384.3.i.d.353.1 20 16.3 odd 4
384.3.i.d.353.4 20 48.35 even 4