Properties

Label 900.3.f.h.199.7
Level $900$
Weight $3$
Character 900.199
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
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] = [900,3,Mod(199,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.f (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} - 14 x^{13} + 23 x^{12} - 26 x^{11} + 18 x^{10} - 10 x^{9} + 9 x^{8} - 20 x^{7} + 72 x^{6} - 208 x^{5} + 368 x^{4} - 448 x^{3} + 512 x^{2} - 512 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.7
Root \(1.40592 - 0.152947i\) of defining polynomial
Character \(\chi\) \(=\) 900.199
Dual form 900.3.f.h.199.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.305673 - 1.97650i) q^{2} +(-3.81313 + 1.20833i) q^{4} -0.329898 q^{7} +(3.55383 + 7.16731i) q^{8} +O(q^{10})\) \(q+(-0.305673 - 1.97650i) q^{2} +(-3.81313 + 1.20833i) q^{4} -0.329898 q^{7} +(3.55383 + 7.16731i) q^{8} +20.4920i q^{11} -0.416712i q^{13} +(0.100841 + 0.652044i) q^{14} +(13.0799 - 9.21501i) q^{16} -18.5884i q^{17} -12.4503i q^{19} +(40.5024 - 6.26384i) q^{22} -23.2304 q^{23} +(-0.823633 + 0.127378i) q^{26} +(1.25794 - 0.398624i) q^{28} -23.9166 q^{29} -42.0148i q^{31} +(-22.2117 - 23.0357i) q^{32} +(-36.7400 + 5.68197i) q^{34} -50.9523i q^{37} +(-24.6081 + 3.80573i) q^{38} -46.7073 q^{41} +55.5866 q^{43} +(-24.7610 - 78.1385i) q^{44} +(7.10090 + 45.9149i) q^{46} +81.7616 q^{47} -48.8912 q^{49} +(0.503524 + 1.58898i) q^{52} +29.9744i q^{53} +(-1.17240 - 2.36448i) q^{56} +(7.31067 + 47.2713i) q^{58} +24.3311i q^{59} -74.8416 q^{61} +(-83.0424 + 12.8428i) q^{62} +(-38.7406 + 50.9428i) q^{64} -72.8008 q^{67} +(22.4608 + 70.8799i) q^{68} -39.2803i q^{71} -46.5814i q^{73} +(-100.707 + 15.5747i) q^{74} +(15.0441 + 47.4747i) q^{76} -6.76026i q^{77} -101.920i q^{79} +(14.2771 + 92.3170i) q^{82} -5.88913 q^{83} +(-16.9913 - 109.867i) q^{86} +(-146.872 + 72.8250i) q^{88} -61.0100 q^{89} +0.137472i q^{91} +(88.5804 - 28.0699i) q^{92} +(-24.9923 - 161.602i) q^{94} -95.5437i q^{97} +(14.9447 + 96.6335i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} + 44 q^{14} + 80 q^{16} + 132 q^{26} - 64 q^{29} - 248 q^{34} + 32 q^{41} + 80 q^{44} - 152 q^{46} - 32 q^{49} + 344 q^{56} + 272 q^{61} - 32 q^{64} - 216 q^{74} + 240 q^{76} - 428 q^{86} + 256 q^{89} - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.305673 1.97650i −0.152836 0.988251i
\(3\) 0 0
\(4\) −3.81313 + 1.20833i −0.953282 + 0.302082i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.329898 −0.0471283 −0.0235641 0.999722i \(-0.507501\pi\)
−0.0235641 + 0.999722i \(0.507501\pi\)
\(8\) 3.55383 + 7.16731i 0.444229 + 0.895913i
\(9\) 0 0
\(10\) 0 0
\(11\) 20.4920i 1.86291i 0.363861 + 0.931453i \(0.381458\pi\)
−0.363861 + 0.931453i \(0.618542\pi\)
\(12\) 0 0
\(13\) 0.416712i 0.0320548i −0.999872 0.0160274i \(-0.994898\pi\)
0.999872 0.0160274i \(-0.00510189\pi\)
\(14\) 0.100841 + 0.652044i 0.00720292 + 0.0465746i
\(15\) 0 0
\(16\) 13.0799 9.21501i 0.817493 0.575938i
\(17\) 18.5884i 1.09343i −0.837317 0.546717i \(-0.815877\pi\)
0.837317 0.546717i \(-0.184123\pi\)
\(18\) 0 0
\(19\) 12.4503i 0.655281i −0.944803 0.327640i \(-0.893747\pi\)
0.944803 0.327640i \(-0.106253\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 40.5024 6.26384i 1.84102 0.284720i
\(23\) −23.2304 −1.01002 −0.505008 0.863114i \(-0.668511\pi\)
−0.505008 + 0.863114i \(0.668511\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.823633 + 0.127378i −0.0316782 + 0.00489914i
\(27\) 0 0
\(28\) 1.25794 0.398624i 0.0449265 0.0142366i
\(29\) −23.9166 −0.824712 −0.412356 0.911023i \(-0.635294\pi\)
−0.412356 + 0.911023i \(0.635294\pi\)
\(30\) 0 0
\(31\) 42.0148i 1.35532i −0.735377 0.677658i \(-0.762995\pi\)
0.735377 0.677658i \(-0.237005\pi\)
\(32\) −22.2117 23.0357i −0.694115 0.719865i
\(33\) 0 0
\(34\) −36.7400 + 5.68197i −1.08059 + 0.167117i
\(35\) 0 0
\(36\) 0 0
\(37\) 50.9523i 1.37709i −0.725194 0.688545i \(-0.758250\pi\)
0.725194 0.688545i \(-0.241750\pi\)
\(38\) −24.6081 + 3.80573i −0.647582 + 0.100151i
\(39\) 0 0
\(40\) 0 0
\(41\) −46.7073 −1.13920 −0.569601 0.821921i \(-0.692902\pi\)
−0.569601 + 0.821921i \(0.692902\pi\)
\(42\) 0 0
\(43\) 55.5866 1.29271 0.646356 0.763036i \(-0.276292\pi\)
0.646356 + 0.763036i \(0.276292\pi\)
\(44\) −24.7610 78.1385i −0.562750 1.77588i
\(45\) 0 0
\(46\) 7.10090 + 45.9149i 0.154367 + 0.998151i
\(47\) 81.7616 1.73961 0.869804 0.493397i \(-0.164245\pi\)
0.869804 + 0.493397i \(0.164245\pi\)
\(48\) 0 0
\(49\) −48.8912 −0.997779
\(50\) 0 0
\(51\) 0 0
\(52\) 0.503524 + 1.58898i 0.00968316 + 0.0305572i
\(53\) 29.9744i 0.565554i 0.959186 + 0.282777i \(0.0912557\pi\)
−0.959186 + 0.282777i \(0.908744\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.17240 2.36448i −0.0209357 0.0422228i
\(57\) 0 0
\(58\) 7.31067 + 47.2713i 0.126046 + 0.815023i
\(59\) 24.3311i 0.412391i 0.978511 + 0.206196i \(0.0661083\pi\)
−0.978511 + 0.206196i \(0.933892\pi\)
\(60\) 0 0
\(61\) −74.8416 −1.22691 −0.613456 0.789729i \(-0.710221\pi\)
−0.613456 + 0.789729i \(0.710221\pi\)
\(62\) −83.0424 + 12.8428i −1.33939 + 0.207142i
\(63\) 0 0
\(64\) −38.7406 + 50.9428i −0.605321 + 0.795981i
\(65\) 0 0
\(66\) 0 0
\(67\) −72.8008 −1.08658 −0.543290 0.839545i \(-0.682822\pi\)
−0.543290 + 0.839545i \(0.682822\pi\)
\(68\) 22.4608 + 70.8799i 0.330307 + 1.04235i
\(69\) 0 0
\(70\) 0 0
\(71\) 39.2803i 0.553244i −0.960979 0.276622i \(-0.910785\pi\)
0.960979 0.276622i \(-0.0892150\pi\)
\(72\) 0 0
\(73\) 46.5814i 0.638101i −0.947738 0.319051i \(-0.896636\pi\)
0.947738 0.319051i \(-0.103364\pi\)
\(74\) −100.707 + 15.5747i −1.36091 + 0.210469i
\(75\) 0 0
\(76\) 15.0441 + 47.4747i 0.197948 + 0.624667i
\(77\) 6.76026i 0.0877955i
\(78\) 0 0
\(79\) 101.920i 1.29012i −0.764131 0.645062i \(-0.776832\pi\)
0.764131 0.645062i \(-0.223168\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 14.2771 + 92.3170i 0.174112 + 1.12582i
\(83\) −5.88913 −0.0709534 −0.0354767 0.999371i \(-0.511295\pi\)
−0.0354767 + 0.999371i \(0.511295\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −16.9913 109.867i −0.197574 1.27752i
\(87\) 0 0
\(88\) −146.872 + 72.8250i −1.66900 + 0.827557i
\(89\) −61.0100 −0.685506 −0.342753 0.939426i \(-0.611359\pi\)
−0.342753 + 0.939426i \(0.611359\pi\)
\(90\) 0 0
\(91\) 0.137472i 0.00151069i
\(92\) 88.5804 28.0699i 0.962831 0.305108i
\(93\) 0 0
\(94\) −24.9923 161.602i −0.265876 1.71917i
\(95\) 0 0
\(96\) 0 0
\(97\) 95.5437i 0.984987i −0.870316 0.492494i \(-0.836086\pi\)
0.870316 0.492494i \(-0.163914\pi\)
\(98\) 14.9447 + 96.6335i 0.152497 + 0.986057i
\(99\) 0 0
\(100\) 0 0
\(101\) −162.675 −1.61064 −0.805322 0.592838i \(-0.798008\pi\)
−0.805322 + 0.592838i \(0.798008\pi\)
\(102\) 0 0
\(103\) −158.196 −1.53588 −0.767941 0.640521i \(-0.778718\pi\)
−0.767941 + 0.640521i \(0.778718\pi\)
\(104\) 2.98670 1.48092i 0.0287183 0.0142397i
\(105\) 0 0
\(106\) 59.2445 9.16236i 0.558910 0.0864373i
\(107\) −18.1827 −0.169932 −0.0849660 0.996384i \(-0.527078\pi\)
−0.0849660 + 0.996384i \(0.527078\pi\)
\(108\) 0 0
\(109\) 156.842 1.43891 0.719457 0.694537i \(-0.244391\pi\)
0.719457 + 0.694537i \(0.244391\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.31503 + 3.04001i −0.0385270 + 0.0271430i
\(113\) 98.7245i 0.873668i 0.899542 + 0.436834i \(0.143900\pi\)
−0.899542 + 0.436834i \(0.856100\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 91.1972 28.8991i 0.786183 0.249130i
\(117\) 0 0
\(118\) 48.0904 7.43735i 0.407546 0.0630284i
\(119\) 6.13227i 0.0515317i
\(120\) 0 0
\(121\) −298.921 −2.47042
\(122\) 22.8770 + 147.925i 0.187517 + 1.21250i
\(123\) 0 0
\(124\) 50.7676 + 160.208i 0.409416 + 1.29200i
\(125\) 0 0
\(126\) 0 0
\(127\) 27.0938 0.213337 0.106669 0.994295i \(-0.465982\pi\)
0.106669 + 0.994295i \(0.465982\pi\)
\(128\) 112.531 + 60.9990i 0.879145 + 0.476555i
\(129\) 0 0
\(130\) 0 0
\(131\) 4.45811i 0.0340314i 0.999855 + 0.0170157i \(0.00541653\pi\)
−0.999855 + 0.0170157i \(0.994583\pi\)
\(132\) 0 0
\(133\) 4.10734i 0.0308822i
\(134\) 22.2533 + 143.891i 0.166069 + 1.07381i
\(135\) 0 0
\(136\) 133.229 66.0600i 0.979622 0.485735i
\(137\) 181.700i 1.32628i −0.748496 0.663139i \(-0.769224\pi\)
0.748496 0.663139i \(-0.230776\pi\)
\(138\) 0 0
\(139\) 223.419i 1.60733i 0.595083 + 0.803664i \(0.297119\pi\)
−0.595083 + 0.803664i \(0.702881\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −77.6377 + 12.0069i −0.546744 + 0.0845559i
\(143\) 8.53925 0.0597151
\(144\) 0 0
\(145\) 0 0
\(146\) −92.0683 + 14.2387i −0.630604 + 0.0975251i
\(147\) 0 0
\(148\) 61.5670 + 194.288i 0.415994 + 1.31275i
\(149\) 123.867 0.831324 0.415662 0.909519i \(-0.363550\pi\)
0.415662 + 0.909519i \(0.363550\pi\)
\(150\) 0 0
\(151\) 76.0961i 0.503948i 0.967734 + 0.251974i \(0.0810797\pi\)
−0.967734 + 0.251974i \(0.918920\pi\)
\(152\) 89.2353 44.2464i 0.587075 0.291095i
\(153\) 0 0
\(154\) −13.3617 + 2.06643i −0.0867641 + 0.0134184i
\(155\) 0 0
\(156\) 0 0
\(157\) 34.2940i 0.218433i −0.994018 0.109217i \(-0.965166\pi\)
0.994018 0.109217i \(-0.0348342\pi\)
\(158\) −201.445 + 31.1541i −1.27497 + 0.197178i
\(159\) 0 0
\(160\) 0 0
\(161\) 7.66365 0.0476003
\(162\) 0 0
\(163\) 165.538 1.01557 0.507786 0.861483i \(-0.330464\pi\)
0.507786 + 0.861483i \(0.330464\pi\)
\(164\) 178.101 56.4377i 1.08598 0.344132i
\(165\) 0 0
\(166\) 1.80015 + 11.6399i 0.0108443 + 0.0701198i
\(167\) −83.6064 −0.500637 −0.250319 0.968164i \(-0.580535\pi\)
−0.250319 + 0.968164i \(0.580535\pi\)
\(168\) 0 0
\(169\) 168.826 0.998972
\(170\) 0 0
\(171\) 0 0
\(172\) −211.959 + 67.1668i −1.23232 + 0.390505i
\(173\) 192.900i 1.11503i −0.830168 0.557513i \(-0.811756\pi\)
0.830168 0.557513i \(-0.188244\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 188.834 + 268.033i 1.07292 + 1.52291i
\(177\) 0 0
\(178\) 18.6491 + 120.587i 0.104770 + 0.677452i
\(179\) 120.939i 0.675637i −0.941211 0.337819i \(-0.890311\pi\)
0.941211 0.337819i \(-0.109689\pi\)
\(180\) 0 0
\(181\) −107.583 −0.594381 −0.297191 0.954818i \(-0.596050\pi\)
−0.297191 + 0.954818i \(0.596050\pi\)
\(182\) 0.271715 0.0420216i 0.00149294 0.000230888i
\(183\) 0 0
\(184\) −82.5569 166.499i −0.448679 0.904887i
\(185\) 0 0
\(186\) 0 0
\(187\) 380.913 2.03697
\(188\) −311.767 + 98.7947i −1.65834 + 0.525504i
\(189\) 0 0
\(190\) 0 0
\(191\) 279.706i 1.46443i −0.681075 0.732214i \(-0.738487\pi\)
0.681075 0.732214i \(-0.261513\pi\)
\(192\) 0 0
\(193\) 102.534i 0.531263i −0.964075 0.265632i \(-0.914420\pi\)
0.964075 0.265632i \(-0.0855805\pi\)
\(194\) −188.842 + 29.2051i −0.973415 + 0.150542i
\(195\) 0 0
\(196\) 186.428 59.0765i 0.951165 0.301411i
\(197\) 38.9632i 0.197783i 0.995098 + 0.0988913i \(0.0315296\pi\)
−0.995098 + 0.0988913i \(0.968470\pi\)
\(198\) 0 0
\(199\) 147.646i 0.741940i 0.928645 + 0.370970i \(0.120975\pi\)
−0.928645 + 0.370970i \(0.879025\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 49.7254 + 321.528i 0.246165 + 1.59172i
\(203\) 7.89005 0.0388672
\(204\) 0 0
\(205\) 0 0
\(206\) 48.3562 + 312.674i 0.234739 + 1.51784i
\(207\) 0 0
\(208\) −3.84001 5.45055i −0.0184616 0.0262046i
\(209\) 255.132 1.22073
\(210\) 0 0
\(211\) 233.336i 1.10586i −0.833229 0.552928i \(-0.813510\pi\)
0.833229 0.552928i \(-0.186490\pi\)
\(212\) −36.2189 114.296i −0.170844 0.539133i
\(213\) 0 0
\(214\) 5.55797 + 35.9382i 0.0259718 + 0.167936i
\(215\) 0 0
\(216\) 0 0
\(217\) 13.8606i 0.0638737i
\(218\) −47.9422 309.998i −0.219918 1.42201i
\(219\) 0 0
\(220\) 0 0
\(221\) −7.74600 −0.0350498
\(222\) 0 0
\(223\) −82.7105 −0.370899 −0.185450 0.982654i \(-0.559374\pi\)
−0.185450 + 0.982654i \(0.559374\pi\)
\(224\) 7.32758 + 7.59941i 0.0327124 + 0.0339260i
\(225\) 0 0
\(226\) 195.129 30.1774i 0.863403 0.133528i
\(227\) −361.534 −1.59266 −0.796330 0.604862i \(-0.793228\pi\)
−0.796330 + 0.604862i \(0.793228\pi\)
\(228\) 0 0
\(229\) −121.818 −0.531955 −0.265977 0.963979i \(-0.585695\pi\)
−0.265977 + 0.963979i \(0.585695\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −84.9957 171.418i −0.366361 0.738870i
\(233\) 136.615i 0.586329i −0.956062 0.293164i \(-0.905292\pi\)
0.956062 0.293164i \(-0.0947083\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −29.3999 92.7775i −0.124576 0.393125i
\(237\) 0 0
\(238\) 12.1204 1.87447i 0.0509262 0.00787592i
\(239\) 56.4632i 0.236248i −0.992999 0.118124i \(-0.962312\pi\)
0.992999 0.118124i \(-0.0376880\pi\)
\(240\) 0 0
\(241\) −2.24158 −0.00930117 −0.00465059 0.999989i \(-0.501480\pi\)
−0.00465059 + 0.999989i \(0.501480\pi\)
\(242\) 91.3720 + 590.818i 0.377570 + 2.44140i
\(243\) 0 0
\(244\) 285.381 90.4331i 1.16959 0.370627i
\(245\) 0 0
\(246\) 0 0
\(247\) −5.18820 −0.0210049
\(248\) 301.133 149.314i 1.21425 0.602071i
\(249\) 0 0
\(250\) 0 0
\(251\) 395.809i 1.57693i 0.615081 + 0.788464i \(0.289123\pi\)
−0.615081 + 0.788464i \(0.710877\pi\)
\(252\) 0 0
\(253\) 476.036i 1.88157i
\(254\) −8.28184 53.5510i −0.0326057 0.210831i
\(255\) 0 0
\(256\) 86.1671 241.063i 0.336590 0.941651i
\(257\) 109.778i 0.427151i 0.976927 + 0.213576i \(0.0685110\pi\)
−0.976927 + 0.213576i \(0.931489\pi\)
\(258\) 0 0
\(259\) 16.8091i 0.0648998i
\(260\) 0 0
\(261\) 0 0
\(262\) 8.81147 1.36272i 0.0336316 0.00520124i
\(263\) 327.702 1.24601 0.623007 0.782216i \(-0.285911\pi\)
0.623007 + 0.782216i \(0.285911\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.11816 1.25550i 0.0305194 0.00471993i
\(267\) 0 0
\(268\) 277.599 87.9672i 1.03582 0.328236i
\(269\) −130.032 −0.483392 −0.241696 0.970352i \(-0.577704\pi\)
−0.241696 + 0.970352i \(0.577704\pi\)
\(270\) 0 0
\(271\) 329.669i 1.21649i 0.793750 + 0.608245i \(0.208126\pi\)
−0.793750 + 0.608245i \(0.791874\pi\)
\(272\) −171.292 243.134i −0.629751 0.893875i
\(273\) 0 0
\(274\) −359.131 + 55.5408i −1.31070 + 0.202704i
\(275\) 0 0
\(276\) 0 0
\(277\) 304.124i 1.09792i −0.835849 0.548960i \(-0.815024\pi\)
0.835849 0.548960i \(-0.184976\pi\)
\(278\) 441.588 68.2930i 1.58844 0.245658i
\(279\) 0 0
\(280\) 0 0
\(281\) −240.099 −0.854446 −0.427223 0.904146i \(-0.640508\pi\)
−0.427223 + 0.904146i \(0.640508\pi\)
\(282\) 0 0
\(283\) 86.6730 0.306265 0.153133 0.988206i \(-0.451064\pi\)
0.153133 + 0.988206i \(0.451064\pi\)
\(284\) 47.4635 + 149.781i 0.167125 + 0.527398i
\(285\) 0 0
\(286\) −2.61022 16.8779i −0.00912664 0.0590135i
\(287\) 15.4086 0.0536886
\(288\) 0 0
\(289\) −56.5280 −0.195599
\(290\) 0 0
\(291\) 0 0
\(292\) 56.2856 + 177.621i 0.192759 + 0.608290i
\(293\) 390.339i 1.33222i −0.745855 0.666108i \(-0.767959\pi\)
0.745855 0.666108i \(-0.232041\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 365.191 181.076i 1.23375 0.611743i
\(297\) 0 0
\(298\) −37.8629 244.824i −0.127057 0.821557i
\(299\) 9.68038i 0.0323759i
\(300\) 0 0
\(301\) −18.3379 −0.0609233
\(302\) 150.404 23.2605i 0.498027 0.0770216i
\(303\) 0 0
\(304\) −114.730 162.849i −0.377401 0.535687i
\(305\) 0 0
\(306\) 0 0
\(307\) −60.2318 −0.196195 −0.0980973 0.995177i \(-0.531276\pi\)
−0.0980973 + 0.995177i \(0.531276\pi\)
\(308\) 8.16860 + 25.7777i 0.0265214 + 0.0836939i
\(309\) 0 0
\(310\) 0 0
\(311\) 106.594i 0.342747i −0.985206 0.171373i \(-0.945180\pi\)
0.985206 0.171373i \(-0.0548205\pi\)
\(312\) 0 0
\(313\) 46.2243i 0.147682i −0.997270 0.0738408i \(-0.976474\pi\)
0.997270 0.0738408i \(-0.0235257\pi\)
\(314\) −67.7823 + 10.4828i −0.215867 + 0.0333846i
\(315\) 0 0
\(316\) 123.152 + 388.633i 0.389723 + 1.22985i
\(317\) 8.36780i 0.0263969i −0.999913 0.0131984i \(-0.995799\pi\)
0.999913 0.0131984i \(-0.00420131\pi\)
\(318\) 0 0
\(319\) 490.099i 1.53636i
\(320\) 0 0
\(321\) 0 0
\(322\) −2.34257 15.1472i −0.00727507 0.0470411i
\(323\) −231.432 −0.716506
\(324\) 0 0
\(325\) 0 0
\(326\) −50.6006 327.187i −0.155217 1.00364i
\(327\) 0 0
\(328\) −165.990 334.765i −0.506066 1.02063i
\(329\) −26.9730 −0.0819847
\(330\) 0 0
\(331\) 111.072i 0.335564i −0.985824 0.167782i \(-0.946339\pi\)
0.985824 0.167782i \(-0.0536605\pi\)
\(332\) 22.4560 7.11600i 0.0676386 0.0214337i
\(333\) 0 0
\(334\) 25.5562 + 165.248i 0.0765156 + 0.494755i
\(335\) 0 0
\(336\) 0 0
\(337\) 231.853i 0.687990i 0.938972 + 0.343995i \(0.111780\pi\)
−0.938972 + 0.343995i \(0.888220\pi\)
\(338\) −51.6057 333.686i −0.152679 0.987236i
\(339\) 0 0
\(340\) 0 0
\(341\) 860.966 2.52483
\(342\) 0 0
\(343\) 32.2941 0.0941518
\(344\) 197.546 + 398.406i 0.574260 + 1.15816i
\(345\) 0 0
\(346\) −381.266 + 58.9642i −1.10193 + 0.170417i
\(347\) 402.088 1.15875 0.579377 0.815059i \(-0.303296\pi\)
0.579377 + 0.815059i \(0.303296\pi\)
\(348\) 0 0
\(349\) 163.284 0.467864 0.233932 0.972253i \(-0.424841\pi\)
0.233932 + 0.972253i \(0.424841\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 472.046 455.161i 1.34104 1.29307i
\(353\) 175.851i 0.498161i −0.968483 0.249081i \(-0.919872\pi\)
0.968483 0.249081i \(-0.0801285\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 232.639 73.7201i 0.653481 0.207079i
\(357\) 0 0
\(358\) −239.037 + 36.9678i −0.667700 + 0.103262i
\(359\) 345.628i 0.962753i 0.876514 + 0.481377i \(0.159863\pi\)
−0.876514 + 0.481377i \(0.840137\pi\)
\(360\) 0 0
\(361\) 205.989 0.570607
\(362\) 32.8852 + 212.638i 0.0908431 + 0.587398i
\(363\) 0 0
\(364\) −0.166112 0.524200i −0.000456351 0.00144011i
\(365\) 0 0
\(366\) 0 0
\(367\) −728.998 −1.98637 −0.993185 0.116546i \(-0.962818\pi\)
−0.993185 + 0.116546i \(0.962818\pi\)
\(368\) −303.851 + 214.068i −0.825682 + 0.581707i
\(369\) 0 0
\(370\) 0 0
\(371\) 9.88848i 0.0266536i
\(372\) 0 0
\(373\) 46.6749i 0.125134i −0.998041 0.0625668i \(-0.980071\pi\)
0.998041 0.0625668i \(-0.0199287\pi\)
\(374\) −116.435 752.875i −0.311323 2.01303i
\(375\) 0 0
\(376\) 290.567 + 586.010i 0.772784 + 1.55854i
\(377\) 9.96635i 0.0264360i
\(378\) 0 0
\(379\) 117.629i 0.310368i −0.987886 0.155184i \(-0.950403\pi\)
0.987886 0.155184i \(-0.0495970\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −552.839 + 85.4985i −1.44722 + 0.223818i
\(383\) 251.669 0.657100 0.328550 0.944487i \(-0.393440\pi\)
0.328550 + 0.944487i \(0.393440\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −202.658 + 31.3418i −0.525022 + 0.0811964i
\(387\) 0 0
\(388\) 115.448 + 364.321i 0.297547 + 0.938970i
\(389\) 356.890 0.917454 0.458727 0.888577i \(-0.348306\pi\)
0.458727 + 0.888577i \(0.348306\pi\)
\(390\) 0 0
\(391\) 431.815i 1.10439i
\(392\) −173.751 350.418i −0.443242 0.893923i
\(393\) 0 0
\(394\) 77.0108 11.9100i 0.195459 0.0302284i
\(395\) 0 0
\(396\) 0 0
\(397\) 103.819i 0.261508i 0.991415 + 0.130754i \(0.0417398\pi\)
−0.991415 + 0.130754i \(0.958260\pi\)
\(398\) 291.823 45.1314i 0.733224 0.113396i
\(399\) 0 0
\(400\) 0 0
\(401\) 121.598 0.303237 0.151618 0.988439i \(-0.451551\pi\)
0.151618 + 0.988439i \(0.451551\pi\)
\(402\) 0 0
\(403\) −17.5081 −0.0434444
\(404\) 620.301 196.565i 1.53540 0.486546i
\(405\) 0 0
\(406\) −2.41177 15.5947i −0.00594033 0.0384106i
\(407\) 1044.11 2.56539
\(408\) 0 0
\(409\) −182.788 −0.446915 −0.223457 0.974714i \(-0.571734\pi\)
−0.223457 + 0.974714i \(0.571734\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 603.221 191.152i 1.46413 0.463962i
\(413\) 8.02677i 0.0194353i
\(414\) 0 0
\(415\) 0 0
\(416\) −9.59924 + 9.25587i −0.0230751 + 0.0222497i
\(417\) 0 0
\(418\) −77.9869 504.269i −0.186572 1.20638i
\(419\) 168.020i 0.401003i 0.979693 + 0.200502i \(0.0642572\pi\)
−0.979693 + 0.200502i \(0.935743\pi\)
\(420\) 0 0
\(421\) 625.291 1.48525 0.742626 0.669706i \(-0.233580\pi\)
0.742626 + 0.669706i \(0.233580\pi\)
\(422\) −461.189 + 71.3244i −1.09286 + 0.169015i
\(423\) 0 0
\(424\) −214.836 + 106.524i −0.506688 + 0.251236i
\(425\) 0 0
\(426\) 0 0
\(427\) 24.6901 0.0578222
\(428\) 69.3331 21.9707i 0.161993 0.0513334i
\(429\) 0 0
\(430\) 0 0
\(431\) 133.413i 0.309544i −0.987950 0.154772i \(-0.950536\pi\)
0.987950 0.154772i \(-0.0494643\pi\)
\(432\) 0 0
\(433\) 706.716i 1.63214i 0.577954 + 0.816069i \(0.303851\pi\)
−0.577954 + 0.816069i \(0.696149\pi\)
\(434\) 27.3955 4.23681i 0.0631233 0.00976223i
\(435\) 0 0
\(436\) −598.057 + 189.516i −1.37169 + 0.434670i
\(437\) 289.226i 0.661844i
\(438\) 0 0
\(439\) 507.488i 1.15601i 0.816033 + 0.578005i \(0.196169\pi\)
−0.816033 + 0.578005i \(0.803831\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.36774 + 15.3100i 0.00535689 + 0.0346380i
\(443\) −412.172 −0.930410 −0.465205 0.885203i \(-0.654019\pi\)
−0.465205 + 0.885203i \(0.654019\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 25.2824 + 163.478i 0.0566869 + 0.366542i
\(447\) 0 0
\(448\) 12.7804 16.8059i 0.0285277 0.0375132i
\(449\) −808.617 −1.80093 −0.900465 0.434929i \(-0.856773\pi\)
−0.900465 + 0.434929i \(0.856773\pi\)
\(450\) 0 0
\(451\) 957.124i 2.12223i
\(452\) −119.291 376.449i −0.263919 0.832852i
\(453\) 0 0
\(454\) 110.511 + 714.573i 0.243417 + 1.57395i
\(455\) 0 0
\(456\) 0 0
\(457\) 472.873i 1.03473i −0.855764 0.517367i \(-0.826912\pi\)
0.855764 0.517367i \(-0.173088\pi\)
\(458\) 37.2364 + 240.773i 0.0813021 + 0.525705i
\(459\) 0 0
\(460\) 0 0
\(461\) −433.776 −0.940946 −0.470473 0.882414i \(-0.655917\pi\)
−0.470473 + 0.882414i \(0.655917\pi\)
\(462\) 0 0
\(463\) 530.624 1.14606 0.573028 0.819536i \(-0.305769\pi\)
0.573028 + 0.819536i \(0.305769\pi\)
\(464\) −312.827 + 220.392i −0.674196 + 0.474983i
\(465\) 0 0
\(466\) −270.019 + 41.7594i −0.579440 + 0.0896124i
\(467\) −355.266 −0.760741 −0.380370 0.924834i \(-0.624203\pi\)
−0.380370 + 0.924834i \(0.624203\pi\)
\(468\) 0 0
\(469\) 24.0168 0.0512086
\(470\) 0 0
\(471\) 0 0
\(472\) −174.388 + 86.4686i −0.369467 + 0.183196i
\(473\) 1139.08i 2.40820i
\(474\) 0 0
\(475\) 0 0
\(476\) −7.40978 23.3831i −0.0155668 0.0491242i
\(477\) 0 0
\(478\) −111.600 + 17.2593i −0.233472 + 0.0361072i
\(479\) 548.640i 1.14539i 0.819770 + 0.572693i \(0.194101\pi\)
−0.819770 + 0.572693i \(0.805899\pi\)
\(480\) 0 0
\(481\) −21.2324 −0.0441423
\(482\) 0.685191 + 4.43050i 0.00142156 + 0.00919190i
\(483\) 0 0
\(484\) 1139.82 361.194i 2.35501 0.746269i
\(485\) 0 0
\(486\) 0 0
\(487\) −134.618 −0.276422 −0.138211 0.990403i \(-0.544135\pi\)
−0.138211 + 0.990403i \(0.544135\pi\)
\(488\) −265.974 536.412i −0.545029 1.09921i
\(489\) 0 0
\(490\) 0 0
\(491\) 756.810i 1.54136i 0.637220 + 0.770682i \(0.280084\pi\)
−0.637220 + 0.770682i \(0.719916\pi\)
\(492\) 0 0
\(493\) 444.572i 0.901768i
\(494\) 1.58589 + 10.2545i 0.00321031 + 0.0207581i
\(495\) 0 0
\(496\) −387.167 549.549i −0.780579 1.10796i
\(497\) 12.9585i 0.0260734i
\(498\) 0 0
\(499\) 706.956i 1.41675i −0.705838 0.708373i \(-0.749430\pi\)
0.705838 0.708373i \(-0.250570\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 782.318 120.988i 1.55840 0.241012i
\(503\) 100.567 0.199935 0.0999673 0.994991i \(-0.468126\pi\)
0.0999673 + 0.994991i \(0.468126\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −940.887 + 145.511i −1.85946 + 0.287572i
\(507\) 0 0
\(508\) −103.312 + 32.7382i −0.203370 + 0.0644452i
\(509\) 753.185 1.47973 0.739867 0.672753i \(-0.234888\pi\)
0.739867 + 0.672753i \(0.234888\pi\)
\(510\) 0 0
\(511\) 15.3671i 0.0300726i
\(512\) −502.800 96.6232i −0.982031 0.188717i
\(513\) 0 0
\(514\) 216.976 33.5561i 0.422133 0.0652843i
\(515\) 0 0
\(516\) 0 0
\(517\) 1675.46i 3.24073i
\(518\) 33.2231 5.13807i 0.0641373 0.00991906i
\(519\) 0 0
\(520\) 0 0
\(521\) −117.708 −0.225926 −0.112963 0.993599i \(-0.536034\pi\)
−0.112963 + 0.993599i \(0.536034\pi\)
\(522\) 0 0
\(523\) −617.411 −1.18052 −0.590259 0.807214i \(-0.700974\pi\)
−0.590259 + 0.807214i \(0.700974\pi\)
\(524\) −5.38686 16.9994i −0.0102803 0.0324415i
\(525\) 0 0
\(526\) −100.170 647.704i −0.190437 1.23138i
\(527\) −780.988 −1.48195
\(528\) 0 0
\(529\) 10.6508 0.0201338
\(530\) 0 0
\(531\) 0 0
\(532\) −4.96301 15.6618i −0.00932896 0.0294395i
\(533\) 19.4635i 0.0365169i
\(534\) 0 0
\(535\) 0 0
\(536\) −258.722 521.786i −0.482690 0.973481i
\(537\) 0 0
\(538\) 39.7474 + 257.009i 0.0738799 + 0.477713i
\(539\) 1001.88i 1.85877i
\(540\) 0 0
\(541\) 352.762 0.652056 0.326028 0.945360i \(-0.394290\pi\)
0.326028 + 0.945360i \(0.394290\pi\)
\(542\) 651.591 100.771i 1.20220 0.185924i
\(543\) 0 0
\(544\) −428.196 + 412.879i −0.787125 + 0.758969i
\(545\) 0 0
\(546\) 0 0
\(547\) 295.110 0.539507 0.269753 0.962929i \(-0.413058\pi\)
0.269753 + 0.962929i \(0.413058\pi\)
\(548\) 219.553 + 692.846i 0.400644 + 1.26432i
\(549\) 0 0
\(550\) 0 0
\(551\) 297.770i 0.540418i
\(552\) 0 0
\(553\) 33.6231i 0.0608013i
\(554\) −601.102 + 92.9624i −1.08502 + 0.167802i
\(555\) 0 0
\(556\) −269.963 851.924i −0.485545 1.53224i
\(557\) 31.8538i 0.0571882i 0.999591 + 0.0285941i \(0.00910302\pi\)
−0.999591 + 0.0285941i \(0.990897\pi\)
\(558\) 0 0
\(559\) 23.1636i 0.0414376i
\(560\) 0 0
\(561\) 0 0
\(562\) 73.3919 + 474.557i 0.130591 + 0.844408i
\(563\) −906.668 −1.61042 −0.805211 0.592988i \(-0.797948\pi\)
−0.805211 + 0.592988i \(0.797948\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −26.4936 171.310i −0.0468085 0.302667i
\(567\) 0 0
\(568\) 281.534 139.596i 0.495659 0.245767i
\(569\) −465.009 −0.817239 −0.408620 0.912705i \(-0.633990\pi\)
−0.408620 + 0.912705i \(0.633990\pi\)
\(570\) 0 0
\(571\) 265.895i 0.465666i −0.972517 0.232833i \(-0.925200\pi\)
0.972517 0.232833i \(-0.0747995\pi\)
\(572\) −32.5613 + 10.3182i −0.0569253 + 0.0180388i
\(573\) 0 0
\(574\) −4.71000 30.4552i −0.00820557 0.0530578i
\(575\) 0 0
\(576\) 0 0
\(577\) 138.097i 0.239336i −0.992814 0.119668i \(-0.961817\pi\)
0.992814 0.119668i \(-0.0381830\pi\)
\(578\) 17.2791 + 111.728i 0.0298946 + 0.193301i
\(579\) 0 0
\(580\) 0 0
\(581\) 1.94281 0.00334391
\(582\) 0 0
\(583\) −614.234 −1.05358
\(584\) 333.863 165.542i 0.571683 0.283463i
\(585\) 0 0
\(586\) −771.507 + 119.316i −1.31656 + 0.203611i
\(587\) −648.473 −1.10472 −0.552362 0.833604i \(-0.686273\pi\)
−0.552362 + 0.833604i \(0.686273\pi\)
\(588\) 0 0
\(589\) −523.098 −0.888113
\(590\) 0 0
\(591\) 0 0
\(592\) −469.526 666.451i −0.793118 1.12576i
\(593\) 350.392i 0.590880i 0.955361 + 0.295440i \(0.0954662\pi\)
−0.955361 + 0.295440i \(0.904534\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −472.322 + 149.672i −0.792486 + 0.251128i
\(597\) 0 0
\(598\) 19.1333 2.95903i 0.0319955 0.00494821i
\(599\) 276.745i 0.462012i −0.972952 0.231006i \(-0.925798\pi\)
0.972952 0.231006i \(-0.0742017\pi\)
\(600\) 0 0
\(601\) 815.487 1.35688 0.678442 0.734654i \(-0.262656\pi\)
0.678442 + 0.734654i \(0.262656\pi\)
\(602\) 5.60540 + 36.2449i 0.00931130 + 0.0602075i
\(603\) 0 0
\(604\) −91.9490 290.164i −0.152233 0.480405i
\(605\) 0 0
\(606\) 0 0
\(607\) −247.049 −0.407001 −0.203500 0.979075i \(-0.565232\pi\)
−0.203500 + 0.979075i \(0.565232\pi\)
\(608\) −286.802 + 276.543i −0.471713 + 0.454840i
\(609\) 0 0
\(610\) 0 0
\(611\) 34.0710i 0.0557627i
\(612\) 0 0
\(613\) 1005.15i 1.63972i 0.572561 + 0.819862i \(0.305950\pi\)
−0.572561 + 0.819862i \(0.694050\pi\)
\(614\) 18.4112 + 119.048i 0.0299857 + 0.193890i
\(615\) 0 0
\(616\) 48.4528 24.0248i 0.0786572 0.0390013i
\(617\) 533.282i 0.864314i 0.901798 + 0.432157i \(0.142247\pi\)
−0.901798 + 0.432157i \(0.857753\pi\)
\(618\) 0 0
\(619\) 1136.85i 1.83659i 0.395900 + 0.918294i \(0.370433\pi\)
−0.395900 + 0.918294i \(0.629567\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −210.684 + 32.5830i −0.338720 + 0.0523842i
\(623\) 20.1271 0.0323067
\(624\) 0 0
\(625\) 0 0
\(626\) −91.3625 + 14.1295i −0.145946 + 0.0225711i
\(627\) 0 0
\(628\) 41.4384 + 130.768i 0.0659847 + 0.208229i
\(629\) −947.121 −1.50576
\(630\) 0 0
\(631\) 936.738i 1.48453i 0.670107 + 0.742265i \(0.266248\pi\)
−0.670107 + 0.742265i \(0.733752\pi\)
\(632\) 730.490 362.206i 1.15584 0.573110i
\(633\) 0 0
\(634\) −16.5390 + 2.55781i −0.0260867 + 0.00403440i
\(635\) 0 0
\(636\) 0 0
\(637\) 20.3735i 0.0319836i
\(638\) −968.682 + 149.810i −1.51831 + 0.234812i
\(639\) 0 0
\(640\) 0 0
\(641\) −214.558 −0.334723 −0.167362 0.985896i \(-0.553525\pi\)
−0.167362 + 0.985896i \(0.553525\pi\)
\(642\) 0 0
\(643\) 786.394 1.22301 0.611504 0.791241i \(-0.290565\pi\)
0.611504 + 0.791241i \(0.290565\pi\)
\(644\) −29.2225 + 9.26020i −0.0453765 + 0.0143792i
\(645\) 0 0
\(646\) 70.7424 + 457.425i 0.109508 + 0.708088i
\(647\) 316.550 0.489258 0.244629 0.969617i \(-0.421334\pi\)
0.244629 + 0.969617i \(0.421334\pi\)
\(648\) 0 0
\(649\) −498.592 −0.768246
\(650\) 0 0
\(651\) 0 0
\(652\) −631.219 + 200.024i −0.968127 + 0.306786i
\(653\) 516.391i 0.790797i 0.918510 + 0.395399i \(0.129394\pi\)
−0.918510 + 0.395399i \(0.870606\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −610.926 + 430.408i −0.931290 + 0.656110i
\(657\) 0 0
\(658\) 8.24491 + 53.3121i 0.0125303 + 0.0810215i
\(659\) 285.118i 0.432653i −0.976321 0.216326i \(-0.930592\pi\)
0.976321 0.216326i \(-0.0694076\pi\)
\(660\) 0 0
\(661\) −391.847 −0.592809 −0.296405 0.955062i \(-0.595788\pi\)
−0.296405 + 0.955062i \(0.595788\pi\)
\(662\) −219.534 + 33.9516i −0.331622 + 0.0512865i
\(663\) 0 0
\(664\) −20.9290 42.2092i −0.0315196 0.0635681i
\(665\) 0 0
\(666\) 0 0
\(667\) 555.593 0.832973
\(668\) 318.802 101.024i 0.477248 0.151233i
\(669\) 0 0
\(670\) 0 0
\(671\) 1533.65i 2.28562i
\(672\) 0 0
\(673\) 1213.59i 1.80325i 0.432517 + 0.901626i \(0.357625\pi\)
−0.432517 + 0.901626i \(0.642375\pi\)
\(674\) 458.257 70.8711i 0.679907 0.105150i
\(675\) 0 0
\(676\) −643.756 + 203.997i −0.952303 + 0.301771i
\(677\) 251.863i 0.372028i −0.982547 0.186014i \(-0.940443\pi\)
0.982547 0.186014i \(-0.0595570\pi\)
\(678\) 0 0
\(679\) 31.5197i 0.0464207i
\(680\) 0 0
\(681\) 0 0
\(682\) −263.174 1701.70i −0.385886 2.49517i
\(683\) 664.793 0.973342 0.486671 0.873585i \(-0.338211\pi\)
0.486671 + 0.873585i \(0.338211\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −9.87143 63.8293i −0.0143898 0.0930457i
\(687\) 0 0
\(688\) 727.067 512.231i 1.05678 0.744522i
\(689\) 12.4907 0.0181287
\(690\) 0 0
\(691\) 654.347i 0.946957i 0.880805 + 0.473479i \(0.157002\pi\)
−0.880805 + 0.473479i \(0.842998\pi\)
\(692\) 233.086 + 735.551i 0.336829 + 1.06293i
\(693\) 0 0
\(694\) −122.907 794.728i −0.177100 1.14514i
\(695\) 0 0
\(696\) 0 0
\(697\) 868.213i 1.24564i
\(698\) −49.9117 322.732i −0.0715067 0.462367i
\(699\) 0 0
\(700\) 0 0
\(701\) 1266.25 1.80635 0.903174 0.429275i \(-0.141231\pi\)
0.903174 + 0.429275i \(0.141231\pi\)
\(702\) 0 0
\(703\) −634.373 −0.902380
\(704\) −1043.92 793.870i −1.48284 1.12766i
\(705\) 0 0
\(706\) −347.570 + 53.7529i −0.492309 + 0.0761372i
\(707\) 53.6661 0.0759068
\(708\) 0 0
\(709\) 493.220 0.695656 0.347828 0.937558i \(-0.386919\pi\)
0.347828 + 0.937558i \(0.386919\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −216.819 437.278i −0.304522 0.614154i
\(713\) 976.020i 1.36889i
\(714\) 0 0
\(715\) 0 0
\(716\) 146.134 + 461.156i 0.204098 + 0.644073i
\(717\) 0 0
\(718\) 683.135 105.649i 0.951442 0.147144i
\(719\) 60.3910i 0.0839930i −0.999118 0.0419965i \(-0.986628\pi\)
0.999118 0.0419965i \(-0.0133718\pi\)
\(720\) 0 0
\(721\) 52.1884 0.0723834
\(722\) −62.9653 407.138i −0.0872096 0.563904i
\(723\) 0 0
\(724\) 410.228 129.995i 0.566613 0.179552i
\(725\) 0 0
\(726\) 0 0
\(727\) 994.690 1.36821 0.684106 0.729383i \(-0.260193\pi\)
0.684106 + 0.729383i \(0.260193\pi\)
\(728\) −0.985307 + 0.488554i −0.00135344 + 0.000671090i
\(729\) 0 0
\(730\) 0 0
\(731\) 1033.27i 1.41350i
\(732\) 0 0
\(733\) 1167.65i 1.59298i −0.604654 0.796488i \(-0.706689\pi\)
0.604654 0.796488i \(-0.293311\pi\)
\(734\) 222.835 + 1440.87i 0.303590 + 1.96303i
\(735\) 0 0
\(736\) 515.986 + 535.127i 0.701067 + 0.727075i
\(737\) 1491.83i 2.02420i
\(738\) 0 0
\(739\) 79.9863i 0.108236i −0.998535 0.0541179i \(-0.982765\pi\)
0.998535 0.0541179i \(-0.0172347\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −19.5446 + 3.02264i −0.0263405 + 0.00407364i
\(743\) 402.122 0.541214 0.270607 0.962690i \(-0.412776\pi\)
0.270607 + 0.962690i \(0.412776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −92.2530 + 14.2672i −0.123664 + 0.0191250i
\(747\) 0 0
\(748\) −1452.47 + 460.267i −1.94180 + 0.615330i
\(749\) 5.99844 0.00800860
\(750\) 0 0
\(751\) 58.7486i 0.0782271i 0.999235 + 0.0391136i \(0.0124534\pi\)
−0.999235 + 0.0391136i \(0.987547\pi\)
\(752\) 1069.43 753.434i 1.42212 1.00191i
\(753\) 0 0
\(754\) 19.6985 3.04644i 0.0261254 0.00404038i
\(755\) 0 0
\(756\) 0 0
\(757\) 1040.91i 1.37504i −0.726164 0.687522i \(-0.758699\pi\)
0.726164 0.687522i \(-0.241301\pi\)
\(758\) −232.495 + 35.9561i −0.306721 + 0.0474355i
\(759\) 0 0
\(760\) 0 0
\(761\) −750.095 −0.985670 −0.492835 0.870123i \(-0.664039\pi\)
−0.492835 + 0.870123i \(0.664039\pi\)
\(762\) 0 0
\(763\) −51.7417 −0.0678135
\(764\) 337.976 + 1066.55i 0.442377 + 1.39601i
\(765\) 0 0
\(766\) −76.9285 497.425i −0.100429 0.649380i
\(767\) 10.1391 0.0132191
\(768\) 0 0
\(769\) −1065.98 −1.38619 −0.693094 0.720847i \(-0.743753\pi\)
−0.693094 + 0.720847i \(0.743753\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 123.894 + 390.975i 0.160485 + 0.506444i
\(773\) 947.271i 1.22545i −0.790297 0.612724i \(-0.790074\pi\)
0.790297 0.612724i \(-0.209926\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 684.791 339.546i 0.882463 0.437560i
\(777\) 0 0
\(778\) −109.092 705.393i −0.140220 0.906675i
\(779\) 581.521i 0.746497i
\(780\) 0 0
\(781\) 804.931 1.03064
\(782\) 853.484 131.994i 1.09141 0.168791i
\(783\) 0 0
\(784\) −639.491 + 450.533i −0.815678 + 0.574659i
\(785\) 0 0
\(786\) 0 0
\(787\) −11.5874 −0.0147236 −0.00736178 0.999973i \(-0.502343\pi\)
−0.00736178 + 0.999973i \(0.502343\pi\)
\(788\) −47.0803 148.572i −0.0597465 0.188543i
\(789\) 0 0
\(790\) 0 0
\(791\) 32.5690i 0.0411744i
\(792\) 0 0
\(793\) 31.1874i 0.0393284i
\(794\) 205.198 31.7345i 0.258436 0.0399679i
\(795\) 0 0
\(796\) −178.405 562.994i −0.224127 0.707278i
\(797\) 780.220i 0.978946i 0.872018 + 0.489473i \(0.162811\pi\)
−0.872018 + 0.489473i \(0.837189\pi\)
\(798\) 0 0
\(799\) 1519.82i 1.90215i
\(800\) 0 0
\(801\) 0 0
\(802\) −37.1692 240.339i −0.0463457 0.299674i
\(803\) 954.545 1.18872
\(804\) 0 0
\(805\) 0 0
\(806\) 5.35175 + 34.6048i 0.00663989 + 0.0429340i
\(807\) 0 0
\(808\) −578.120 1165.94i −0.715495 1.44300i
\(809\) −1061.80 −1.31249 −0.656243 0.754549i \(-0.727855\pi\)
−0.656243 + 0.754549i \(0.727855\pi\)
\(810\) 0 0
\(811\) 309.236i 0.381302i 0.981658 + 0.190651i \(0.0610599\pi\)
−0.981658 + 0.190651i \(0.938940\pi\)
\(812\) −30.0858 + 9.53376i −0.0370514 + 0.0117411i
\(813\) 0 0
\(814\) −319.157 2063.69i −0.392085 2.53525i
\(815\) 0 0
\(816\) 0 0
\(817\) 692.072i 0.847089i
\(818\) 55.8734 + 361.281i 0.0683049 + 0.441664i
\(819\) 0 0
\(820\) 0 0
\(821\) 1156.76 1.40897 0.704483 0.709721i \(-0.251179\pi\)
0.704483 + 0.709721i \(0.251179\pi\)
\(822\) 0 0
\(823\) −1441.89 −1.75200 −0.875998 0.482314i \(-0.839796\pi\)
−0.875998 + 0.482314i \(0.839796\pi\)
\(824\) −562.201 1133.84i −0.682283 1.37602i
\(825\) 0 0
\(826\) −15.8649 + 2.45357i −0.0192069 + 0.00297042i
\(827\) 367.599 0.444497 0.222249 0.974990i \(-0.428660\pi\)
0.222249 + 0.974990i \(0.428660\pi\)
\(828\) 0 0
\(829\) −172.743 −0.208375 −0.104188 0.994558i \(-0.533224\pi\)
−0.104188 + 0.994558i \(0.533224\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 21.2285 + 16.1437i 0.0255150 + 0.0194034i
\(833\) 908.808i 1.09101i
\(834\) 0 0
\(835\) 0 0
\(836\) −972.850 + 308.283i −1.16370 + 0.368759i
\(837\) 0 0
\(838\) 332.093 51.3593i 0.396292 0.0612879i
\(839\) 1083.17i 1.29103i −0.763748 0.645514i \(-0.776643\pi\)
0.763748 0.645514i \(-0.223357\pi\)
\(840\) 0 0
\(841\) −268.994 −0.319850
\(842\) −191.135 1235.89i −0.227001 1.46780i
\(843\) 0 0
\(844\) 281.946 + 889.739i 0.334059 + 1.05419i
\(845\) 0 0
\(846\) 0 0
\(847\) 98.6133 0.116427
\(848\) 276.214 + 392.062i 0.325724 + 0.462337i
\(849\) 0 0
\(850\) 0 0
\(851\) 1183.64i 1.39088i
\(852\) 0 0
\(853\) 1218.00i 1.42790i −0.700196 0.713951i \(-0.746904\pi\)
0.700196 0.713951i \(-0.253096\pi\)
\(854\) −7.54709 48.8000i −0.00883734 0.0571429i
\(855\) 0 0
\(856\) −64.6184 130.321i −0.0754888 0.152244i
\(857\) 207.055i 0.241604i 0.992677 + 0.120802i \(0.0385466\pi\)
−0.992677 + 0.120802i \(0.961453\pi\)
\(858\) 0 0
\(859\) 1186.36i 1.38109i 0.723290 + 0.690544i \(0.242629\pi\)
−0.723290 + 0.690544i \(0.757371\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −263.692 + 40.7809i −0.305907 + 0.0473096i
\(863\) −885.953 −1.02660 −0.513298 0.858210i \(-0.671577\pi\)
−0.513298 + 0.858210i \(0.671577\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1396.83 216.024i 1.61296 0.249450i
\(867\) 0 0
\(868\) −16.7481 52.8522i −0.0192951 0.0608897i
\(869\) 2088.54 2.40338
\(870\) 0 0
\(871\) 30.3370i 0.0348301i
\(872\) 557.389 + 1124.13i 0.639207 + 1.28914i
\(873\) 0 0
\(874\) 571.656 88.4086i 0.654069 0.101154i
\(875\) 0 0
\(876\) 0 0
\(877\) 643.339i 0.733567i 0.930306 + 0.366784i \(0.119541\pi\)
−0.930306 + 0.366784i \(0.880459\pi\)
\(878\) 1003.05 155.125i 1.14243 0.176680i
\(879\) 0 0
\(880\) 0 0
\(881\) −353.918 −0.401723 −0.200861 0.979620i \(-0.564374\pi\)
−0.200861 + 0.979620i \(0.564374\pi\)
\(882\) 0 0
\(883\) 1093.51 1.23840 0.619200 0.785233i \(-0.287457\pi\)
0.619200 + 0.785233i \(0.287457\pi\)
\(884\) 29.5365 9.35971i 0.0334123 0.0105879i
\(885\) 0 0
\(886\) 125.990 + 814.659i 0.142201 + 0.919479i
\(887\) −520.234 −0.586510 −0.293255 0.956034i \(-0.594738\pi\)
−0.293255 + 0.956034i \(0.594738\pi\)
\(888\) 0 0
\(889\) −8.93819 −0.0100542
\(890\) 0 0
\(891\) 0 0
\(892\) 315.386 99.9413i 0.353571 0.112042i
\(893\) 1017.96i 1.13993i
\(894\) 0 0
\(895\) 0 0
\(896\) −37.1236 20.1234i −0.0414326 0.0224592i
\(897\) 0 0
\(898\) 247.172 + 1598.23i 0.275248 + 1.77977i
\(899\) 1004.85i 1.11775i
\(900\) 0 0
\(901\) 557.175 0.618397
\(902\) −1891.76 + 292.567i −2.09729 + 0.324354i
\(903\) 0 0
\(904\) −707.588 + 350.850i −0.782730 + 0.388109i
\(905\) 0 0
\(906\) 0 0
\(907\) −567.834 −0.626057 −0.313029 0.949744i \(-0.601344\pi\)
−0.313029 + 0.949744i \(0.601344\pi\)
\(908\) 1378.58 436.851i 1.51825 0.481114i
\(909\) 0 0
\(910\) 0 0
\(911\) 1180.19i 1.29549i 0.761858 + 0.647744i \(0.224287\pi\)
−0.761858 + 0.647744i \(0.775713\pi\)
\(912\) 0 0
\(913\) 120.680i 0.132180i
\(914\) −934.635 + 144.545i −1.02258 + 0.158145i
\(915\) 0 0
\(916\) 464.506 147.196i 0.507103 0.160694i
\(917\) 1.47072i 0.00160384i
\(918\) 0 0
\(919\) 54.5449i 0.0593524i 0.999560 + 0.0296762i \(0.00944761\pi\)
−0.999560 + 0.0296762i \(0.990552\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 132.594 + 857.360i 0.143811 + 0.929892i
\(923\) −16.3686 −0.0177341
\(924\) 0 0
\(925\) 0 0
\(926\) −162.197 1048.78i −0.175159 1.13259i
\(927\) 0 0
\(928\) 531.228 + 550.936i 0.572444 + 0.593681i
\(929\) 175.428 0.188835 0.0944175 0.995533i \(-0.469901\pi\)
0.0944175 + 0.995533i \(0.469901\pi\)
\(930\) 0 0
\(931\) 608.711i 0.653825i
\(932\) 165.075 + 520.929i 0.177119 + 0.558937i
\(933\) 0 0
\(934\) 108.595 + 702.184i 0.116269 + 0.751803i
\(935\) 0 0
\(936\) 0 0
\(937\) 335.374i 0.357923i 0.983856 + 0.178962i \(0.0572738\pi\)
−0.983856 + 0.178962i \(0.942726\pi\)
\(938\) −7.34130 47.4694i −0.00782654 0.0506070i
\(939\) 0 0
\(940\) 0 0
\(941\) −709.182 −0.753647 −0.376823 0.926285i \(-0.622984\pi\)
−0.376823 + 0.926285i \(0.622984\pi\)
\(942\) 0 0
\(943\) 1085.03 1.15061
\(944\) 224.211 + 318.248i 0.237512 + 0.337127i
\(945\) 0 0
\(946\) 2251.39 348.186i 2.37991 0.368061i
\(947\) −992.486 −1.04803 −0.524016 0.851708i \(-0.675567\pi\)
−0.524016 + 0.851708i \(0.675567\pi\)
\(948\) 0 0
\(949\) −19.4110 −0.0204542
\(950\) 0 0
\(951\) 0 0
\(952\) −43.9518 + 21.7930i −0.0461679 + 0.0228919i
\(953\) 1438.16i 1.50908i −0.656251 0.754542i \(-0.727859\pi\)
0.656251 0.754542i \(-0.272141\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 68.2260 + 215.301i 0.0713661 + 0.225211i
\(957\) 0 0
\(958\) 1084.39 167.704i 1.13193 0.175057i
\(959\) 59.9425i 0.0625052i
\(960\) 0 0
\(961\) −804.245 −0.836883
\(962\) 6.49018 + 41.9660i 0.00674655 + 0.0436237i
\(963\) 0 0
\(964\) 8.54744 2.70857i 0.00886664 0.00280971i
\(965\) 0 0
\(966\) 0 0
\(967\) −1376.32 −1.42329 −0.711646 0.702539i \(-0.752050\pi\)
−0.711646 + 0.702539i \(0.752050\pi\)
\(968\) −1062.31 2142.46i −1.09743 2.21328i
\(969\) 0 0
\(970\) 0 0
\(971\) 652.667i 0.672159i 0.941834 + 0.336080i \(0.109101\pi\)
−0.941834 + 0.336080i \(0.890899\pi\)
\(972\) 0 0
\(973\) 73.7053i 0.0757506i
\(974\) 41.1490 + 266.072i 0.0422474 + 0.273175i
\(975\) 0 0
\(976\) −978.920 + 689.666i −1.00299 + 0.706625i
\(977\) 467.260i 0.478260i 0.970988 + 0.239130i \(0.0768622\pi\)
−0.970988 + 0.239130i \(0.923138\pi\)
\(978\) 0 0
\(979\) 1250.22i 1.27703i
\(980\) 0 0
\(981\) 0 0
\(982\) 1495.84 231.336i 1.52326 0.235577i
\(983\) −1044.27 −1.06233 −0.531165 0.847268i \(-0.678246\pi\)
−0.531165 + 0.847268i \(0.678246\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 878.697 135.894i 0.891174 0.137823i
\(987\) 0 0
\(988\) 19.7833 6.26905i 0.0200236 0.00634519i
\(989\) −1291.30 −1.30566
\(990\) 0 0
\(991\) 1705.99i 1.72148i −0.509044 0.860740i \(-0.670001\pi\)
0.509044 0.860740i \(-0.329999\pi\)
\(992\) −967.839 + 933.219i −0.975644 + 0.940745i
\(993\) 0 0
\(994\) 25.6125 3.96106i 0.0257671 0.00398497i
\(995\) 0 0
\(996\) 0 0
\(997\) 1262.24i 1.26604i −0.774136 0.633020i \(-0.781815\pi\)
0.774136 0.633020i \(-0.218185\pi\)
\(998\) −1397.30 + 216.097i −1.40010 + 0.216530i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 900.3.f.h.199.7 16
3.2 odd 2 300.3.f.c.199.10 16
4.3 odd 2 inner 900.3.f.h.199.9 16
5.2 odd 4 900.3.c.t.451.7 8
5.3 odd 4 900.3.c.n.451.2 8
5.4 even 2 inner 900.3.f.h.199.10 16
12.11 even 2 300.3.f.c.199.8 16
15.2 even 4 300.3.c.e.151.2 yes 8
15.8 even 4 300.3.c.g.151.7 yes 8
15.14 odd 2 300.3.f.c.199.7 16
20.3 even 4 900.3.c.n.451.1 8
20.7 even 4 900.3.c.t.451.8 8
20.19 odd 2 inner 900.3.f.h.199.8 16
60.23 odd 4 300.3.c.g.151.8 yes 8
60.47 odd 4 300.3.c.e.151.1 8
60.59 even 2 300.3.f.c.199.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.c.e.151.1 8 60.47 odd 4
300.3.c.e.151.2 yes 8 15.2 even 4
300.3.c.g.151.7 yes 8 15.8 even 4
300.3.c.g.151.8 yes 8 60.23 odd 4
300.3.f.c.199.7 16 15.14 odd 2
300.3.f.c.199.8 16 12.11 even 2
300.3.f.c.199.9 16 60.59 even 2
300.3.f.c.199.10 16 3.2 odd 2
900.3.c.n.451.1 8 20.3 even 4
900.3.c.n.451.2 8 5.3 odd 4
900.3.c.t.451.7 8 5.2 odd 4
900.3.c.t.451.8 8 20.7 even 4
900.3.f.h.199.7 16 1.1 even 1 trivial
900.3.f.h.199.8 16 20.19 odd 2 inner
900.3.f.h.199.9 16 4.3 odd 2 inner
900.3.f.h.199.10 16 5.4 even 2 inner