Properties

Label 624.4.q.i.289.2
Level $624$
Weight $4$
Character 624.289
Analytic conductor $36.817$
Analytic rank $0$
Dimension $8$
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] = [624,4,Mod(289,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\), 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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.2
Root \(-2.11303 + 3.65987i\) of defining polynomial
Character \(\chi\) \(=\) 624.289
Dual form 624.4.q.i.529.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 2.59808i) q^{3} -5.85953 q^{5} +(-12.0627 + 20.8932i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\) \(q+(-1.50000 - 2.59808i) q^{3} -5.85953 q^{5} +(-12.0627 + 20.8932i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-16.9446 - 29.3489i) q^{11} +(-40.8020 - 23.0694i) q^{13} +(8.78930 + 15.2235i) q^{15} +(24.6978 - 42.7779i) q^{17} +(-38.4274 + 66.5582i) q^{19} +72.3763 q^{21} +(3.14582 + 5.44871i) q^{23} -90.6659 q^{25} +27.0000 q^{27} +(-50.4977 - 87.4645i) q^{29} +307.580 q^{31} +(-50.8338 + 88.0467i) q^{33} +(70.6819 - 122.425i) q^{35} +(38.0095 + 65.8343i) q^{37} +(1.26677 + 140.611i) q^{39} +(257.209 + 445.499i) q^{41} +(-134.092 + 232.254i) q^{43} +(26.3679 - 45.6705i) q^{45} +460.912 q^{47} +(-119.519 - 207.012i) q^{49} -148.187 q^{51} +67.8057 q^{53} +(99.2874 + 171.971i) q^{55} +230.564 q^{57} +(12.6010 - 21.8256i) q^{59} +(294.416 - 509.944i) q^{61} +(-108.565 - 188.039i) q^{63} +(239.080 + 135.176i) q^{65} +(-502.230 - 869.888i) q^{67} +(9.43745 - 16.3461i) q^{69} +(447.740 - 775.509i) q^{71} +968.599 q^{73} +(135.999 + 235.557i) q^{75} +817.592 q^{77} +119.053 q^{79} +(-40.5000 - 70.1481i) q^{81} -480.784 q^{83} +(-144.718 + 250.658i) q^{85} +(-151.493 + 262.394i) q^{87} +(-542.954 - 940.423i) q^{89} +(974.179 - 574.205i) q^{91} +(-461.370 - 799.116i) q^{93} +(225.167 - 390.000i) q^{95} +(8.32761 - 14.4239i) q^{97} +305.003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9} + 40 q^{11} - 60 q^{13} + 18 q^{15} - 98 q^{17} + 124 q^{19} + 84 q^{21} + 104 q^{23} - 116 q^{25} + 216 q^{27} - 194 q^{29} - 52 q^{31} + 120 q^{33} + 88 q^{35} - 102 q^{37} - 342 q^{39} + 1054 q^{41} + 450 q^{43} + 54 q^{45} + 192 q^{47} - 1070 q^{49} + 588 q^{51} + 524 q^{53} + 204 q^{55} - 744 q^{57} + 308 q^{59} + 928 q^{61} - 126 q^{63} + 2346 q^{65} - 1134 q^{67} + 312 q^{69} + 1064 q^{71} + 1904 q^{73} + 174 q^{75} + 5016 q^{77} + 1492 q^{79} - 324 q^{81} + 808 q^{83} + 1394 q^{85} - 582 q^{87} - 1620 q^{89} - 3278 q^{91} + 78 q^{93} + 2204 q^{95} - 2166 q^{97} - 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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 0
\(3\) −1.50000 2.59808i −0.288675 0.500000i
\(4\) 0 0
\(5\) −5.85953 −0.524093 −0.262046 0.965055i \(-0.584397\pi\)
−0.262046 + 0.965055i \(0.584397\pi\)
\(6\) 0 0
\(7\) −12.0627 + 20.8932i −0.651326 + 1.12813i 0.331476 + 0.943464i \(0.392454\pi\)
−0.982801 + 0.184666i \(0.940880\pi\)
\(8\) 0 0
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −16.9446 29.3489i −0.464453 0.804457i 0.534723 0.845027i \(-0.320416\pi\)
−0.999177 + 0.0405703i \(0.987083\pi\)
\(12\) 0 0
\(13\) −40.8020 23.0694i −0.870495 0.492178i
\(14\) 0 0
\(15\) 8.78930 + 15.2235i 0.151292 + 0.262046i
\(16\) 0 0
\(17\) 24.6978 42.7779i 0.352359 0.610303i −0.634303 0.773084i \(-0.718713\pi\)
0.986662 + 0.162781i \(0.0520463\pi\)
\(18\) 0 0
\(19\) −38.4274 + 66.5582i −0.463992 + 0.803658i −0.999155 0.0410905i \(-0.986917\pi\)
0.535163 + 0.844749i \(0.320250\pi\)
\(20\) 0 0
\(21\) 72.3763 0.752086
\(22\) 0 0
\(23\) 3.14582 + 5.44871i 0.0285195 + 0.0493972i 0.879933 0.475098i \(-0.157587\pi\)
−0.851413 + 0.524495i \(0.824254\pi\)
\(24\) 0 0
\(25\) −90.6659 −0.725327
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −50.4977 87.4645i −0.323351 0.560061i 0.657826 0.753170i \(-0.271476\pi\)
−0.981177 + 0.193109i \(0.938143\pi\)
\(30\) 0 0
\(31\) 307.580 1.78203 0.891016 0.453972i \(-0.149993\pi\)
0.891016 + 0.453972i \(0.149993\pi\)
\(32\) 0 0
\(33\) −50.8338 + 88.0467i −0.268152 + 0.464453i
\(34\) 0 0
\(35\) 70.6819 122.425i 0.341355 0.591244i
\(36\) 0 0
\(37\) 38.0095 + 65.8343i 0.168884 + 0.292516i 0.938028 0.346560i \(-0.112650\pi\)
−0.769144 + 0.639076i \(0.779317\pi\)
\(38\) 0 0
\(39\) 1.26677 + 140.611i 0.00520115 + 0.577327i
\(40\) 0 0
\(41\) 257.209 + 445.499i 0.979740 + 1.69696i 0.663312 + 0.748343i \(0.269150\pi\)
0.316427 + 0.948617i \(0.397517\pi\)
\(42\) 0 0
\(43\) −134.092 + 232.254i −0.475554 + 0.823684i −0.999608 0.0280012i \(-0.991086\pi\)
0.524054 + 0.851685i \(0.324419\pi\)
\(44\) 0 0
\(45\) 26.3679 45.6705i 0.0873488 0.151292i
\(46\) 0 0
\(47\) 460.912 1.43045 0.715223 0.698896i \(-0.246325\pi\)
0.715223 + 0.698896i \(0.246325\pi\)
\(48\) 0 0
\(49\) −119.519 207.012i −0.348451 0.603534i
\(50\) 0 0
\(51\) −148.187 −0.406869
\(52\) 0 0
\(53\) 67.8057 0.175733 0.0878663 0.996132i \(-0.471995\pi\)
0.0878663 + 0.996132i \(0.471995\pi\)
\(54\) 0 0
\(55\) 99.2874 + 171.971i 0.243417 + 0.421610i
\(56\) 0 0
\(57\) 230.564 0.535772
\(58\) 0 0
\(59\) 12.6010 21.8256i 0.0278053 0.0481603i −0.851788 0.523887i \(-0.824481\pi\)
0.879593 + 0.475727i \(0.157815\pi\)
\(60\) 0 0
\(61\) 294.416 509.944i 0.617969 1.07035i −0.371886 0.928278i \(-0.621289\pi\)
0.989856 0.142076i \(-0.0453778\pi\)
\(62\) 0 0
\(63\) −108.565 188.039i −0.217109 0.376043i
\(64\) 0 0
\(65\) 239.080 + 135.176i 0.456220 + 0.257947i
\(66\) 0 0
\(67\) −502.230 869.888i −0.915778 1.58617i −0.805758 0.592245i \(-0.798242\pi\)
−0.110021 0.993929i \(-0.535092\pi\)
\(68\) 0 0
\(69\) 9.43745 16.3461i 0.0164657 0.0285195i
\(70\) 0 0
\(71\) 447.740 775.509i 0.748408 1.29628i −0.200177 0.979760i \(-0.564152\pi\)
0.948585 0.316522i \(-0.102515\pi\)
\(72\) 0 0
\(73\) 968.599 1.55296 0.776479 0.630143i \(-0.217004\pi\)
0.776479 + 0.630143i \(0.217004\pi\)
\(74\) 0 0
\(75\) 135.999 + 235.557i 0.209384 + 0.362663i
\(76\) 0 0
\(77\) 817.592 1.21004
\(78\) 0 0
\(79\) 119.053 0.169551 0.0847755 0.996400i \(-0.472983\pi\)
0.0847755 + 0.996400i \(0.472983\pi\)
\(80\) 0 0
\(81\) −40.5000 70.1481i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −480.784 −0.635818 −0.317909 0.948121i \(-0.602981\pi\)
−0.317909 + 0.948121i \(0.602981\pi\)
\(84\) 0 0
\(85\) −144.718 + 250.658i −0.184669 + 0.319856i
\(86\) 0 0
\(87\) −151.493 + 262.394i −0.186687 + 0.323351i
\(88\) 0 0
\(89\) −542.954 940.423i −0.646663 1.12005i −0.983915 0.178638i \(-0.942831\pi\)
0.337252 0.941414i \(-0.390503\pi\)
\(90\) 0 0
\(91\) 974.179 574.205i 1.12222 0.661462i
\(92\) 0 0
\(93\) −461.370 799.116i −0.514428 0.891016i
\(94\) 0 0
\(95\) 225.167 390.000i 0.243175 0.421191i
\(96\) 0 0
\(97\) 8.32761 14.4239i 0.00871692 0.0150981i −0.861634 0.507530i \(-0.830559\pi\)
0.870351 + 0.492432i \(0.163892\pi\)
\(98\) 0 0
\(99\) 305.003 0.309636
\(100\) 0 0
\(101\) 479.002 + 829.655i 0.471906 + 0.817364i 0.999483 0.0321423i \(-0.0102330\pi\)
−0.527578 + 0.849507i \(0.676900\pi\)
\(102\) 0 0
\(103\) 2.70560 0.00258826 0.00129413 0.999999i \(-0.499588\pi\)
0.00129413 + 0.999999i \(0.499588\pi\)
\(104\) 0 0
\(105\) −424.092 −0.394163
\(106\) 0 0
\(107\) 675.642 + 1170.25i 0.610437 + 1.05731i 0.991167 + 0.132621i \(0.0423395\pi\)
−0.380730 + 0.924686i \(0.624327\pi\)
\(108\) 0 0
\(109\) 448.455 0.394075 0.197037 0.980396i \(-0.436868\pi\)
0.197037 + 0.980396i \(0.436868\pi\)
\(110\) 0 0
\(111\) 114.028 197.503i 0.0975054 0.168884i
\(112\) 0 0
\(113\) −699.423 + 1211.44i −0.582267 + 1.00852i 0.412943 + 0.910757i \(0.364501\pi\)
−0.995210 + 0.0977596i \(0.968832\pi\)
\(114\) 0 0
\(115\) −18.4330 31.9269i −0.0149468 0.0258887i
\(116\) 0 0
\(117\) 363.417 214.207i 0.287162 0.169260i
\(118\) 0 0
\(119\) 595.846 + 1032.04i 0.459001 + 0.795013i
\(120\) 0 0
\(121\) 91.2613 158.069i 0.0685659 0.118760i
\(122\) 0 0
\(123\) 771.628 1336.50i 0.565653 0.979740i
\(124\) 0 0
\(125\) 1263.70 0.904231
\(126\) 0 0
\(127\) −59.7522 103.494i −0.0417492 0.0723118i 0.844396 0.535720i \(-0.179960\pi\)
−0.886145 + 0.463408i \(0.846626\pi\)
\(128\) 0 0
\(129\) 804.552 0.549123
\(130\) 0 0
\(131\) 2251.70 1.50177 0.750886 0.660432i \(-0.229627\pi\)
0.750886 + 0.660432i \(0.229627\pi\)
\(132\) 0 0
\(133\) −927.078 1605.75i −0.604420 1.04689i
\(134\) 0 0
\(135\) −158.207 −0.100862
\(136\) 0 0
\(137\) −565.310 + 979.146i −0.352538 + 0.610614i −0.986693 0.162592i \(-0.948015\pi\)
0.634155 + 0.773206i \(0.281348\pi\)
\(138\) 0 0
\(139\) −297.644 + 515.534i −0.181624 + 0.314583i −0.942434 0.334393i \(-0.891469\pi\)
0.760809 + 0.648975i \(0.224802\pi\)
\(140\) 0 0
\(141\) −691.368 1197.49i −0.412934 0.715223i
\(142\) 0 0
\(143\) 14.3099 + 1588.40i 0.00836821 + 0.928869i
\(144\) 0 0
\(145\) 295.893 + 512.501i 0.169466 + 0.293524i
\(146\) 0 0
\(147\) −358.556 + 621.037i −0.201178 + 0.348451i
\(148\) 0 0
\(149\) 396.587 686.910i 0.218052 0.377677i −0.736161 0.676807i \(-0.763363\pi\)
0.954212 + 0.299130i \(0.0966965\pi\)
\(150\) 0 0
\(151\) 134.213 0.0723317 0.0361659 0.999346i \(-0.488486\pi\)
0.0361659 + 0.999346i \(0.488486\pi\)
\(152\) 0 0
\(153\) 222.280 + 385.001i 0.117453 + 0.203434i
\(154\) 0 0
\(155\) −1802.28 −0.933950
\(156\) 0 0
\(157\) 1509.07 0.767114 0.383557 0.923517i \(-0.374699\pi\)
0.383557 + 0.923517i \(0.374699\pi\)
\(158\) 0 0
\(159\) −101.709 176.164i −0.0507297 0.0878663i
\(160\) 0 0
\(161\) −151.788 −0.0743019
\(162\) 0 0
\(163\) 587.540 1017.65i 0.282329 0.489009i −0.689629 0.724163i \(-0.742226\pi\)
0.971958 + 0.235155i \(0.0755596\pi\)
\(164\) 0 0
\(165\) 297.862 515.913i 0.140537 0.243417i
\(166\) 0 0
\(167\) 737.007 + 1276.53i 0.341505 + 0.591504i 0.984712 0.174188i \(-0.0557301\pi\)
−0.643208 + 0.765692i \(0.722397\pi\)
\(168\) 0 0
\(169\) 1132.60 + 1882.56i 0.515522 + 0.856877i
\(170\) 0 0
\(171\) −345.847 599.024i −0.154664 0.267886i
\(172\) 0 0
\(173\) 1164.15 2016.37i 0.511612 0.886139i −0.488297 0.872678i \(-0.662382\pi\)
0.999909 0.0134612i \(-0.00428497\pi\)
\(174\) 0 0
\(175\) 1093.68 1894.30i 0.472424 0.818263i
\(176\) 0 0
\(177\) −75.6062 −0.0321069
\(178\) 0 0
\(179\) −1066.93 1847.97i −0.445508 0.771642i 0.552580 0.833460i \(-0.313643\pi\)
−0.998087 + 0.0618183i \(0.980310\pi\)
\(180\) 0 0
\(181\) −2485.41 −1.02066 −0.510329 0.859979i \(-0.670476\pi\)
−0.510329 + 0.859979i \(0.670476\pi\)
\(182\) 0 0
\(183\) −1766.50 −0.713570
\(184\) 0 0
\(185\) −222.718 385.758i −0.0885110 0.153305i
\(186\) 0 0
\(187\) −1673.98 −0.654617
\(188\) 0 0
\(189\) −325.694 + 564.118i −0.125348 + 0.217109i
\(190\) 0 0
\(191\) −1162.53 + 2013.57i −0.440408 + 0.762809i −0.997720 0.0674941i \(-0.978500\pi\)
0.557311 + 0.830304i \(0.311833\pi\)
\(192\) 0 0
\(193\) −1675.06 2901.29i −0.624732 1.08207i −0.988593 0.150614i \(-0.951875\pi\)
0.363860 0.931453i \(-0.381458\pi\)
\(194\) 0 0
\(195\) −7.42266 823.914i −0.00272588 0.302573i
\(196\) 0 0
\(197\) 1929.65 + 3342.25i 0.697878 + 1.20876i 0.969201 + 0.246272i \(0.0792056\pi\)
−0.271323 + 0.962488i \(0.587461\pi\)
\(198\) 0 0
\(199\) −2041.80 + 3536.50i −0.727333 + 1.25978i 0.230673 + 0.973031i \(0.425907\pi\)
−0.958006 + 0.286747i \(0.907426\pi\)
\(200\) 0 0
\(201\) −1506.69 + 2609.66i −0.528725 + 0.915778i
\(202\) 0 0
\(203\) 2436.56 0.842428
\(204\) 0 0
\(205\) −1507.13 2610.42i −0.513474 0.889364i
\(206\) 0 0
\(207\) −56.6247 −0.0190130
\(208\) 0 0
\(209\) 2604.55 0.862011
\(210\) 0 0
\(211\) 1513.65 + 2621.72i 0.493857 + 0.855386i 0.999975 0.00707871i \(-0.00225324\pi\)
−0.506118 + 0.862464i \(0.668920\pi\)
\(212\) 0 0
\(213\) −2686.44 −0.864188
\(214\) 0 0
\(215\) 785.716 1360.90i 0.249234 0.431687i
\(216\) 0 0
\(217\) −3710.25 + 6426.34i −1.16068 + 2.01036i
\(218\) 0 0
\(219\) −1452.90 2516.49i −0.448300 0.776479i
\(220\) 0 0
\(221\) −1994.58 + 1175.66i −0.607104 + 0.357843i
\(222\) 0 0
\(223\) −862.379 1493.68i −0.258965 0.448540i 0.707000 0.707213i \(-0.250048\pi\)
−0.965965 + 0.258673i \(0.916715\pi\)
\(224\) 0 0
\(225\) 407.996 706.671i 0.120888 0.209384i
\(226\) 0 0
\(227\) −961.637 + 1665.60i −0.281172 + 0.487005i −0.971674 0.236326i \(-0.924057\pi\)
0.690502 + 0.723331i \(0.257390\pi\)
\(228\) 0 0
\(229\) 373.993 0.107922 0.0539610 0.998543i \(-0.482815\pi\)
0.0539610 + 0.998543i \(0.482815\pi\)
\(230\) 0 0
\(231\) −1226.39 2124.17i −0.349309 0.605021i
\(232\) 0 0
\(233\) 3094.49 0.870073 0.435036 0.900413i \(-0.356735\pi\)
0.435036 + 0.900413i \(0.356735\pi\)
\(234\) 0 0
\(235\) −2700.73 −0.749686
\(236\) 0 0
\(237\) −178.580 309.309i −0.0489452 0.0847755i
\(238\) 0 0
\(239\) 1221.18 0.330510 0.165255 0.986251i \(-0.447155\pi\)
0.165255 + 0.986251i \(0.447155\pi\)
\(240\) 0 0
\(241\) −72.7003 + 125.921i −0.0194317 + 0.0336567i −0.875578 0.483077i \(-0.839519\pi\)
0.856146 + 0.516734i \(0.172852\pi\)
\(242\) 0 0
\(243\) −121.500 + 210.444i −0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 700.323 + 1212.99i 0.182620 + 0.316308i
\(246\) 0 0
\(247\) 3103.38 1829.21i 0.799446 0.471213i
\(248\) 0 0
\(249\) 721.176 + 1249.11i 0.183545 + 0.317909i
\(250\) 0 0
\(251\) −492.835 + 853.615i −0.123934 + 0.214660i −0.921316 0.388815i \(-0.872884\pi\)
0.797382 + 0.603475i \(0.206218\pi\)
\(252\) 0 0
\(253\) 106.609 184.652i 0.0264919 0.0458854i
\(254\) 0 0
\(255\) 868.306 0.213237
\(256\) 0 0
\(257\) −1464.66 2536.86i −0.355498 0.615740i 0.631705 0.775209i \(-0.282355\pi\)
−0.987203 + 0.159469i \(0.949022\pi\)
\(258\) 0 0
\(259\) −1833.99 −0.439995
\(260\) 0 0
\(261\) 908.958 0.215567
\(262\) 0 0
\(263\) 1119.00 + 1938.17i 0.262360 + 0.454420i 0.966869 0.255275i \(-0.0821660\pi\)
−0.704509 + 0.709695i \(0.748833\pi\)
\(264\) 0 0
\(265\) −397.310 −0.0921002
\(266\) 0 0
\(267\) −1628.86 + 2821.27i −0.373351 + 0.646663i
\(268\) 0 0
\(269\) −962.992 + 1667.95i −0.218270 + 0.378055i −0.954279 0.298917i \(-0.903375\pi\)
0.736009 + 0.676972i \(0.236708\pi\)
\(270\) 0 0
\(271\) −1781.14 3085.03i −0.399250 0.691521i 0.594384 0.804182i \(-0.297396\pi\)
−0.993634 + 0.112660i \(0.964063\pi\)
\(272\) 0 0
\(273\) −2953.10 1669.68i −0.654687 0.370160i
\(274\) 0 0
\(275\) 1536.30 + 2660.94i 0.336881 + 0.583494i
\(276\) 0 0
\(277\) 718.712 1244.85i 0.155896 0.270020i −0.777489 0.628897i \(-0.783507\pi\)
0.933385 + 0.358877i \(0.116840\pi\)
\(278\) 0 0
\(279\) −1384.11 + 2397.35i −0.297005 + 0.514428i
\(280\) 0 0
\(281\) −3913.51 −0.830820 −0.415410 0.909634i \(-0.636362\pi\)
−0.415410 + 0.909634i \(0.636362\pi\)
\(282\) 0 0
\(283\) −1606.16 2781.94i −0.337371 0.584344i 0.646566 0.762858i \(-0.276204\pi\)
−0.983937 + 0.178514i \(0.942871\pi\)
\(284\) 0 0
\(285\) −1351.00 −0.280794
\(286\) 0 0
\(287\) −12410.6 −2.55252
\(288\) 0 0
\(289\) 1236.54 + 2141.74i 0.251686 + 0.435934i
\(290\) 0 0
\(291\) −49.9657 −0.0100654
\(292\) 0 0
\(293\) 2450.89 4245.06i 0.488677 0.846413i −0.511238 0.859439i \(-0.670813\pi\)
0.999915 + 0.0130260i \(0.00414642\pi\)
\(294\) 0 0
\(295\) −73.8362 + 127.888i −0.0145726 + 0.0252404i
\(296\) 0 0
\(297\) −457.504 792.420i −0.0893841 0.154818i
\(298\) 0 0
\(299\) −2.65667 294.890i −0.000513844 0.0570366i
\(300\) 0 0
\(301\) −3235.03 5603.23i −0.619481 1.07297i
\(302\) 0 0
\(303\) 1437.01 2488.97i 0.272455 0.471906i
\(304\) 0 0
\(305\) −1725.14 + 2988.03i −0.323873 + 0.560965i
\(306\) 0 0
\(307\) −5800.63 −1.07837 −0.539185 0.842188i \(-0.681267\pi\)
−0.539185 + 0.842188i \(0.681267\pi\)
\(308\) 0 0
\(309\) −4.05840 7.02935i −0.000747165 0.00129413i
\(310\) 0 0
\(311\) 4913.51 0.895884 0.447942 0.894063i \(-0.352157\pi\)
0.447942 + 0.894063i \(0.352157\pi\)
\(312\) 0 0
\(313\) −8104.97 −1.46364 −0.731822 0.681496i \(-0.761330\pi\)
−0.731822 + 0.681496i \(0.761330\pi\)
\(314\) 0 0
\(315\) 636.137 + 1101.82i 0.113785 + 0.197081i
\(316\) 0 0
\(317\) 5149.92 0.912455 0.456227 0.889863i \(-0.349200\pi\)
0.456227 + 0.889863i \(0.349200\pi\)
\(318\) 0 0
\(319\) −1711.33 + 2964.10i −0.300363 + 0.520244i
\(320\) 0 0
\(321\) 2026.93 3510.74i 0.352436 0.610437i
\(322\) 0 0
\(323\) 1898.15 + 3287.69i 0.326984 + 0.566352i
\(324\) 0 0
\(325\) 3699.35 + 2091.61i 0.631393 + 0.356990i
\(326\) 0 0
\(327\) −672.682 1165.12i −0.113760 0.197037i
\(328\) 0 0
\(329\) −5559.86 + 9629.96i −0.931687 + 1.61373i
\(330\) 0 0
\(331\) 3030.99 5249.83i 0.503318 0.871772i −0.496675 0.867937i \(-0.665446\pi\)
0.999993 0.00383535i \(-0.00122083\pi\)
\(332\) 0 0
\(333\) −684.170 −0.112589
\(334\) 0 0
\(335\) 2942.83 + 5097.14i 0.479953 + 0.831302i
\(336\) 0 0
\(337\) 3743.50 0.605108 0.302554 0.953132i \(-0.402161\pi\)
0.302554 + 0.953132i \(0.402161\pi\)
\(338\) 0 0
\(339\) 4196.54 0.672344
\(340\) 0 0
\(341\) −5211.82 9027.13i −0.827671 1.43357i
\(342\) 0 0
\(343\) −2508.15 −0.394832
\(344\) 0 0
\(345\) −55.2990 + 95.7807i −0.00862956 + 0.0149468i
\(346\) 0 0
\(347\) −1260.20 + 2182.74i −0.194961 + 0.337682i −0.946888 0.321565i \(-0.895791\pi\)
0.751927 + 0.659246i \(0.229125\pi\)
\(348\) 0 0
\(349\) −5325.37 9223.82i −0.816793 1.41473i −0.908033 0.418898i \(-0.862417\pi\)
0.0912407 0.995829i \(-0.470917\pi\)
\(350\) 0 0
\(351\) −1101.65 622.875i −0.167527 0.0947197i
\(352\) 0 0
\(353\) 4501.41 + 7796.67i 0.678714 + 1.17557i 0.975368 + 0.220582i \(0.0707956\pi\)
−0.296655 + 0.954985i \(0.595871\pi\)
\(354\) 0 0
\(355\) −2623.55 + 4544.12i −0.392235 + 0.679371i
\(356\) 0 0
\(357\) 1787.54 3096.11i 0.265004 0.459001i
\(358\) 0 0
\(359\) 11360.9 1.67021 0.835106 0.550089i \(-0.185406\pi\)
0.835106 + 0.550089i \(0.185406\pi\)
\(360\) 0 0
\(361\) 476.168 + 824.747i 0.0694224 + 0.120243i
\(362\) 0 0
\(363\) −547.568 −0.0791731
\(364\) 0 0
\(365\) −5675.54 −0.813894
\(366\) 0 0
\(367\) −6969.42 12071.4i −0.991283 1.71695i −0.609743 0.792599i \(-0.708727\pi\)
−0.381540 0.924352i \(-0.624606\pi\)
\(368\) 0 0
\(369\) −4629.77 −0.653160
\(370\) 0 0
\(371\) −817.922 + 1416.68i −0.114459 + 0.198249i
\(372\) 0 0
\(373\) −796.535 + 1379.64i −0.110571 + 0.191515i −0.916001 0.401177i \(-0.868601\pi\)
0.805430 + 0.592691i \(0.201935\pi\)
\(374\) 0 0
\(375\) −1895.55 3283.19i −0.261029 0.452116i
\(376\) 0 0
\(377\) 42.6458 + 4733.68i 0.00582592 + 0.646676i
\(378\) 0 0
\(379\) 4568.78 + 7913.36i 0.619215 + 1.07251i 0.989629 + 0.143645i \(0.0458824\pi\)
−0.370414 + 0.928867i \(0.620784\pi\)
\(380\) 0 0
\(381\) −179.257 + 310.482i −0.0241039 + 0.0417492i
\(382\) 0 0
\(383\) 4775.53 8271.47i 0.637124 1.10353i −0.348937 0.937146i \(-0.613457\pi\)
0.986061 0.166385i \(-0.0532093\pi\)
\(384\) 0 0
\(385\) −4790.71 −0.634174
\(386\) 0 0
\(387\) −1206.83 2090.29i −0.158518 0.274561i
\(388\) 0 0
\(389\) 7366.50 0.960145 0.480072 0.877229i \(-0.340610\pi\)
0.480072 + 0.877229i \(0.340610\pi\)
\(390\) 0 0
\(391\) 310.779 0.0401964
\(392\) 0 0
\(393\) −3377.55 5850.10i −0.433524 0.750886i
\(394\) 0 0
\(395\) −697.596 −0.0888604
\(396\) 0 0
\(397\) 5848.24 10129.4i 0.739332 1.28056i −0.213465 0.976951i \(-0.568475\pi\)
0.952797 0.303609i \(-0.0981917\pi\)
\(398\) 0 0
\(399\) −2781.24 + 4817.24i −0.348962 + 0.604420i
\(400\) 0 0
\(401\) 7083.82 + 12269.5i 0.882167 + 1.52796i 0.848927 + 0.528510i \(0.177249\pi\)
0.0332399 + 0.999447i \(0.489417\pi\)
\(402\) 0 0
\(403\) −12549.9 7095.70i −1.55125 0.877077i
\(404\) 0 0
\(405\) 237.311 + 411.035i 0.0291163 + 0.0504308i
\(406\) 0 0
\(407\) 1288.11 2231.07i 0.156878 0.271720i
\(408\) 0 0
\(409\) −1351.85 + 2341.47i −0.163434 + 0.283076i −0.936098 0.351739i \(-0.885590\pi\)
0.772664 + 0.634815i \(0.218924\pi\)
\(410\) 0 0
\(411\) 3391.86 0.407076
\(412\) 0 0
\(413\) 304.006 + 526.553i 0.0362207 + 0.0627361i
\(414\) 0 0
\(415\) 2817.17 0.333228
\(416\) 0 0
\(417\) 1785.86 0.209722
\(418\) 0 0
\(419\) 3571.26 + 6185.61i 0.416390 + 0.721209i 0.995573 0.0939884i \(-0.0299617\pi\)
−0.579183 + 0.815198i \(0.696628\pi\)
\(420\) 0 0
\(421\) −3406.45 −0.394347 −0.197174 0.980369i \(-0.563176\pi\)
−0.197174 + 0.980369i \(0.563176\pi\)
\(422\) 0 0
\(423\) −2074.11 + 3592.46i −0.238408 + 0.412934i
\(424\) 0 0
\(425\) −2239.25 + 3878.49i −0.255575 + 0.442670i
\(426\) 0 0
\(427\) 7102.92 + 12302.6i 0.804999 + 1.39430i
\(428\) 0 0
\(429\) 4105.31 2419.77i 0.462019 0.272326i
\(430\) 0 0
\(431\) 2586.48 + 4479.92i 0.289064 + 0.500673i 0.973587 0.228318i \(-0.0733227\pi\)
−0.684523 + 0.728992i \(0.739989\pi\)
\(432\) 0 0
\(433\) −5477.49 + 9487.28i −0.607924 + 1.05296i 0.383658 + 0.923475i \(0.374664\pi\)
−0.991582 + 0.129480i \(0.958669\pi\)
\(434\) 0 0
\(435\) 887.678 1537.50i 0.0978412 0.169466i
\(436\) 0 0
\(437\) −483.542 −0.0529313
\(438\) 0 0
\(439\) 5916.22 + 10247.2i 0.643202 + 1.11406i 0.984714 + 0.174181i \(0.0557279\pi\)
−0.341511 + 0.939878i \(0.610939\pi\)
\(440\) 0 0
\(441\) 2151.33 0.232300
\(442\) 0 0
\(443\) −13479.8 −1.44570 −0.722852 0.691003i \(-0.757169\pi\)
−0.722852 + 0.691003i \(0.757169\pi\)
\(444\) 0 0
\(445\) 3181.46 + 5510.44i 0.338911 + 0.587011i
\(446\) 0 0
\(447\) −2379.52 −0.251784
\(448\) 0 0
\(449\) 3387.17 5866.75i 0.356014 0.616635i −0.631277 0.775558i \(-0.717469\pi\)
0.987291 + 0.158923i \(0.0508021\pi\)
\(450\) 0 0
\(451\) 8716.61 15097.6i 0.910087 1.57632i
\(452\) 0 0
\(453\) −201.319 348.695i −0.0208804 0.0361659i
\(454\) 0 0
\(455\) −5708.23 + 3364.57i −0.588145 + 0.346667i
\(456\) 0 0
\(457\) 2321.18 + 4020.40i 0.237594 + 0.411524i 0.960023 0.279920i \(-0.0903080\pi\)
−0.722430 + 0.691444i \(0.756975\pi\)
\(458\) 0 0
\(459\) 666.841 1155.00i 0.0678115 0.117453i
\(460\) 0 0
\(461\) −1230.18 + 2130.74i −0.124285 + 0.215268i −0.921453 0.388489i \(-0.872997\pi\)
0.797168 + 0.603757i \(0.206330\pi\)
\(462\) 0 0
\(463\) −4290.01 −0.430613 −0.215306 0.976547i \(-0.569075\pi\)
−0.215306 + 0.976547i \(0.569075\pi\)
\(464\) 0 0
\(465\) 2703.41 + 4682.45i 0.269608 + 0.466975i
\(466\) 0 0
\(467\) 8798.99 0.871882 0.435941 0.899975i \(-0.356416\pi\)
0.435941 + 0.899975i \(0.356416\pi\)
\(468\) 0 0
\(469\) 24233.0 2.38588
\(470\) 0 0
\(471\) −2263.61 3920.68i −0.221447 0.383557i
\(472\) 0 0
\(473\) 9088.54 0.883491
\(474\) 0 0
\(475\) 3484.05 6034.56i 0.336546 0.582915i
\(476\) 0 0
\(477\) −305.126 + 528.493i −0.0292888 + 0.0507297i
\(478\) 0 0
\(479\) −5486.68 9503.21i −0.523367 0.906499i −0.999630 0.0271958i \(-0.991342\pi\)
0.476263 0.879303i \(-0.341991\pi\)
\(480\) 0 0
\(481\) −32.0994 3563.03i −0.00304284 0.337755i
\(482\) 0 0
\(483\) 227.683 + 394.358i 0.0214491 + 0.0371509i
\(484\) 0 0
\(485\) −48.7959 + 84.5170i −0.00456847 + 0.00791283i
\(486\) 0 0
\(487\) −2604.79 + 4511.62i −0.242370 + 0.419797i −0.961389 0.275193i \(-0.911258\pi\)
0.719019 + 0.694991i \(0.244591\pi\)
\(488\) 0 0
\(489\) −3525.24 −0.326006
\(490\) 0 0
\(491\) −4389.61 7603.03i −0.403463 0.698819i 0.590678 0.806907i \(-0.298860\pi\)
−0.994141 + 0.108089i \(0.965527\pi\)
\(492\) 0 0
\(493\) −4988.73 −0.455742
\(494\) 0 0
\(495\) −1787.17 −0.162278
\(496\) 0 0
\(497\) 10801.9 + 18709.5i 0.974915 + 1.68860i
\(498\) 0 0
\(499\) 15590.1 1.39861 0.699305 0.714823i \(-0.253493\pi\)
0.699305 + 0.714823i \(0.253493\pi\)
\(500\) 0 0
\(501\) 2211.02 3829.60i 0.197168 0.341505i
\(502\) 0 0
\(503\) −32.1955 + 55.7642i −0.00285393 + 0.00494314i −0.867449 0.497526i \(-0.834242\pi\)
0.864595 + 0.502470i \(0.167575\pi\)
\(504\) 0 0
\(505\) −2806.73 4861.39i −0.247322 0.428375i
\(506\) 0 0
\(507\) 3192.13 5766.42i 0.279620 0.505120i
\(508\) 0 0
\(509\) −1607.08 2783.54i −0.139946 0.242393i 0.787530 0.616276i \(-0.211359\pi\)
−0.927476 + 0.373883i \(0.878026\pi\)
\(510\) 0 0
\(511\) −11683.9 + 20237.2i −1.01148 + 1.75194i
\(512\) 0 0
\(513\) −1037.54 + 1797.07i −0.0892954 + 0.154664i
\(514\) 0 0
\(515\) −15.8535 −0.00135649
\(516\) 0 0
\(517\) −7809.97 13527.3i −0.664376 1.15073i
\(518\) 0 0
\(519\) −6984.92 −0.590759
\(520\) 0 0
\(521\) −3053.01 −0.256727 −0.128363 0.991727i \(-0.540972\pi\)
−0.128363 + 0.991727i \(0.540972\pi\)
\(522\) 0 0
\(523\) 2548.01 + 4413.28i 0.213034 + 0.368985i 0.952663 0.304030i \(-0.0983322\pi\)
−0.739629 + 0.673015i \(0.764999\pi\)
\(524\) 0 0
\(525\) −6562.06 −0.545508
\(526\) 0 0
\(527\) 7596.55 13157.6i 0.627915 1.08758i
\(528\) 0 0
\(529\) 6063.71 10502.6i 0.498373 0.863208i
\(530\) 0 0
\(531\) 113.409 + 196.431i 0.00926845 + 0.0160534i
\(532\) 0 0
\(533\) −217.216 24110.9i −0.0176523 1.95940i
\(534\) 0 0
\(535\) −3958.95 6857.09i −0.319925 0.554127i
\(536\) 0 0
\(537\) −3200.78 + 5543.91i −0.257214 + 0.445508i
\(538\) 0 0
\(539\) −4050.39 + 7015.48i −0.323678 + 0.560627i
\(540\) 0 0
\(541\) 7861.99 0.624793 0.312397 0.949952i \(-0.398868\pi\)
0.312397 + 0.949952i \(0.398868\pi\)
\(542\) 0 0
\(543\) 3728.11 + 6457.28i 0.294638 + 0.510329i
\(544\) 0 0
\(545\) −2627.73 −0.206532
\(546\) 0 0
\(547\) 6317.48 0.493814 0.246907 0.969039i \(-0.420586\pi\)
0.246907 + 0.969039i \(0.420586\pi\)
\(548\) 0 0
\(549\) 2649.75 + 4589.49i 0.205990 + 0.356785i
\(550\) 0 0
\(551\) 7761.98 0.600130
\(552\) 0 0
\(553\) −1436.11 + 2487.41i −0.110433 + 0.191275i
\(554\) 0 0
\(555\) −668.153 + 1157.28i −0.0511018 + 0.0885110i
\(556\) 0 0
\(557\) 485.617 + 841.113i 0.0369412 + 0.0639840i 0.883905 0.467667i \(-0.154905\pi\)
−0.846964 + 0.531651i \(0.821572\pi\)
\(558\) 0 0
\(559\) 10829.2 6383.00i 0.819367 0.482955i
\(560\) 0 0
\(561\) 2510.97 + 4349.12i 0.188972 + 0.327309i
\(562\) 0 0
\(563\) 4664.24 8078.71i 0.349155 0.604755i −0.636944 0.770910i \(-0.719802\pi\)
0.986100 + 0.166155i \(0.0531352\pi\)
\(564\) 0 0
\(565\) 4098.29 7098.45i 0.305162 0.528556i
\(566\) 0 0
\(567\) 1954.16 0.144739
\(568\) 0 0
\(569\) −8726.08 15114.0i −0.642911 1.11355i −0.984780 0.173807i \(-0.944393\pi\)
0.341869 0.939748i \(-0.388940\pi\)
\(570\) 0 0
\(571\) 20181.4 1.47910 0.739548 0.673103i \(-0.235039\pi\)
0.739548 + 0.673103i \(0.235039\pi\)
\(572\) 0 0
\(573\) 6975.20 0.508540
\(574\) 0 0
\(575\) −285.218 494.012i −0.0206859 0.0358291i
\(576\) 0 0
\(577\) 6382.72 0.460513 0.230257 0.973130i \(-0.426043\pi\)
0.230257 + 0.973130i \(0.426043\pi\)
\(578\) 0 0
\(579\) −5025.17 + 8703.86i −0.360689 + 0.624732i
\(580\) 0 0
\(581\) 5799.57 10045.1i 0.414125 0.717285i
\(582\) 0 0
\(583\) −1148.94 1990.02i −0.0816197 0.141369i
\(584\) 0 0
\(585\) −2129.46 + 1255.16i −0.150499 + 0.0887082i
\(586\) 0 0
\(587\) −387.763 671.626i −0.0272653 0.0472248i 0.852071 0.523427i \(-0.175347\pi\)
−0.879336 + 0.476202i \(0.842013\pi\)
\(588\) 0 0
\(589\) −11819.5 + 20472.0i −0.826849 + 1.43214i
\(590\) 0 0
\(591\) 5788.95 10026.8i 0.402920 0.697878i
\(592\) 0 0
\(593\) −17843.3 −1.23564 −0.617821 0.786319i \(-0.711984\pi\)
−0.617821 + 0.786319i \(0.711984\pi\)
\(594\) 0 0
\(595\) −3491.38 6047.24i −0.240559 0.416660i
\(596\) 0 0
\(597\) 12250.8 0.839852
\(598\) 0 0
\(599\) 24373.3 1.66255 0.831274 0.555863i \(-0.187612\pi\)
0.831274 + 0.555863i \(0.187612\pi\)
\(600\) 0 0
\(601\) 1763.50 + 3054.46i 0.119691 + 0.207311i 0.919645 0.392750i \(-0.128476\pi\)
−0.799954 + 0.600061i \(0.795143\pi\)
\(602\) 0 0
\(603\) 9040.14 0.610519
\(604\) 0 0
\(605\) −534.748 + 926.211i −0.0359349 + 0.0622411i
\(606\) 0 0
\(607\) 3995.77 6920.88i 0.267189 0.462784i −0.700946 0.713214i \(-0.747239\pi\)
0.968135 + 0.250430i \(0.0805720\pi\)
\(608\) 0 0
\(609\) −3654.84 6330.36i −0.243188 0.421214i
\(610\) 0 0
\(611\) −18806.1 10633.0i −1.24520 0.704034i
\(612\) 0 0
\(613\) 8166.09 + 14144.1i 0.538051 + 0.931932i 0.999009 + 0.0445098i \(0.0141726\pi\)
−0.460958 + 0.887422i \(0.652494\pi\)
\(614\) 0 0
\(615\) −4521.38 + 7831.26i −0.296455 + 0.513474i
\(616\) 0 0
\(617\) 9676.82 16760.8i 0.631401 1.09362i −0.355865 0.934537i \(-0.615814\pi\)
0.987266 0.159081i \(-0.0508530\pi\)
\(618\) 0 0
\(619\) 9982.52 0.648193 0.324096 0.946024i \(-0.394940\pi\)
0.324096 + 0.946024i \(0.394940\pi\)
\(620\) 0 0
\(621\) 84.9370 + 147.115i 0.00548858 + 0.00950649i
\(622\) 0 0
\(623\) 26198.0 1.68475
\(624\) 0 0
\(625\) 3928.53 0.251426
\(626\) 0 0
\(627\) −3906.82 6766.81i −0.248841 0.431006i
\(628\) 0 0
\(629\) 3755.00 0.238031
\(630\) 0 0
\(631\) −287.887 + 498.636i −0.0181626 + 0.0314586i −0.874964 0.484188i \(-0.839115\pi\)
0.856801 + 0.515647i \(0.172448\pi\)
\(632\) 0 0
\(633\) 4540.95 7865.15i 0.285129 0.493857i
\(634\) 0 0
\(635\) 350.120 + 606.426i 0.0218805 + 0.0378981i
\(636\) 0 0
\(637\) 100.935 + 11203.7i 0.00627814 + 0.696873i
\(638\) 0 0
\(639\) 4029.66 + 6979.58i 0.249469 + 0.432094i
\(640\) 0 0
\(641\) 12260.9 21236.5i 0.755500 1.30856i −0.189625 0.981857i \(-0.560727\pi\)
0.945125 0.326708i \(-0.105939\pi\)
\(642\) 0 0
\(643\) −11333.5 + 19630.2i −0.695099 + 1.20395i 0.275048 + 0.961431i \(0.411306\pi\)
−0.970147 + 0.242517i \(0.922027\pi\)
\(644\) 0 0
\(645\) −4714.30 −0.287791
\(646\) 0 0
\(647\) −1198.73 2076.25i −0.0728389 0.126161i 0.827306 0.561752i \(-0.189873\pi\)
−0.900144 + 0.435592i \(0.856539\pi\)
\(648\) 0 0
\(649\) −854.078 −0.0516572
\(650\) 0 0
\(651\) 22261.5 1.34024
\(652\) 0 0
\(653\) −10001.0 17322.3i −0.599342 1.03809i −0.992918 0.118799i \(-0.962095\pi\)
0.393576 0.919292i \(-0.371238\pi\)
\(654\) 0 0
\(655\) −13193.9 −0.787068
\(656\) 0 0
\(657\) −4358.70 + 7549.48i −0.258826 + 0.448300i
\(658\) 0 0
\(659\) −1758.98 + 3046.64i −0.103976 + 0.180091i −0.913319 0.407244i \(-0.866490\pi\)
0.809343 + 0.587336i \(0.199823\pi\)
\(660\) 0 0
\(661\) −6791.71 11763.6i −0.399647 0.692209i 0.594035 0.804439i \(-0.297534\pi\)
−0.993682 + 0.112230i \(0.964201\pi\)
\(662\) 0 0
\(663\) 6046.32 + 3418.59i 0.354177 + 0.200252i
\(664\) 0 0
\(665\) 5432.25 + 9408.93i 0.316772 + 0.548665i
\(666\) 0 0
\(667\) 317.713 550.295i 0.0184436 0.0319453i
\(668\) 0 0
\(669\) −2587.14 + 4481.05i −0.149513 + 0.258965i
\(670\) 0 0
\(671\) −19955.1 −1.14807
\(672\) 0 0
\(673\) 5447.92 + 9436.07i 0.312038 + 0.540466i 0.978804 0.204801i \(-0.0656549\pi\)
−0.666765 + 0.745268i \(0.732322\pi\)
\(674\) 0 0
\(675\) −2447.98 −0.139589
\(676\) 0 0
\(677\) 1449.03 0.0822609 0.0411305 0.999154i \(-0.486904\pi\)
0.0411305 + 0.999154i \(0.486904\pi\)
\(678\) 0 0
\(679\) 200.907 + 347.982i 0.0113551 + 0.0196676i
\(680\) 0 0
\(681\) 5769.82 0.324670
\(682\) 0 0
\(683\) −7683.21 + 13307.7i −0.430439 + 0.745543i −0.996911 0.0785385i \(-0.974975\pi\)
0.566472 + 0.824081i \(0.308308\pi\)
\(684\) 0 0
\(685\) 3312.45 5737.34i 0.184763 0.320018i
\(686\) 0 0
\(687\) −560.989 971.661i −0.0311544 0.0539610i
\(688\) 0 0
\(689\) −2766.61 1564.24i −0.152974 0.0864918i
\(690\) 0 0
\(691\) −1009.85 1749.12i −0.0555957 0.0962946i 0.836888 0.547374i \(-0.184372\pi\)
−0.892484 + 0.451079i \(0.851039\pi\)
\(692\) 0 0
\(693\) −3679.16 + 6372.50i −0.201674 + 0.349309i
\(694\) 0 0
\(695\) 1744.05 3020.79i 0.0951880 0.164870i
\(696\) 0 0
\(697\) 25410.0 1.38088
\(698\) 0 0
\(699\) −4641.74 8039.73i −0.251168 0.435036i
\(700\) 0 0
\(701\) 28031.6 1.51033 0.755164 0.655536i \(-0.227557\pi\)
0.755164 + 0.655536i \(0.227557\pi\)
\(702\) 0 0
\(703\) −5842.42 −0.313444
\(704\) 0 0
\(705\) 4051.10 + 7016.70i 0.216416 + 0.374843i
\(706\) 0 0
\(707\) −23112.3 −1.22946
\(708\) 0 0
\(709\) −9802.22 + 16977.9i −0.519224 + 0.899323i 0.480526 + 0.876980i \(0.340446\pi\)
−0.999750 + 0.0223423i \(0.992888\pi\)
\(710\) 0 0
\(711\) −535.739 + 927.928i −0.0282585 + 0.0489452i
\(712\) 0 0
\(713\) 967.590 + 1675.91i 0.0508226 + 0.0880274i
\(714\) 0 0
\(715\) −83.8493 9307.26i −0.00438571 0.486813i
\(716\) 0 0
\(717\) −1831.77 3172.73i −0.0954099 0.165255i
\(718\) 0 0
\(719\) −7363.71 + 12754.3i −0.381947 + 0.661552i −0.991341 0.131316i \(-0.958080\pi\)
0.609393 + 0.792868i \(0.291413\pi\)
\(720\) 0 0
\(721\) −32.6369 + 56.5287i −0.00168580 + 0.00291989i
\(722\) 0 0
\(723\) 436.202 0.0224378
\(724\) 0 0
\(725\) 4578.42 + 7930.05i 0.234535 + 0.406227i
\(726\) 0 0
\(727\) 16890.5 0.861668 0.430834 0.902431i \(-0.358219\pi\)
0.430834 + 0.902431i \(0.358219\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 6623.56 + 11472.3i 0.335131 + 0.580465i
\(732\) 0 0
\(733\) 12553.6 0.632578 0.316289 0.948663i \(-0.397563\pi\)
0.316289 + 0.948663i \(0.397563\pi\)
\(734\) 0 0
\(735\) 2100.97 3638.98i 0.105436 0.182620i
\(736\) 0 0
\(737\) −17020.2 + 29479.8i −0.850673 + 1.47341i
\(738\) 0 0
\(739\) −18937.4 32800.5i −0.942656 1.63273i −0.760378 0.649481i \(-0.774986\pi\)
−0.182278 0.983247i \(-0.558347\pi\)
\(740\) 0 0
\(741\) −9407.48 5319.00i −0.466387 0.263695i
\(742\) 0 0
\(743\) 17941.3 + 31075.3i 0.885872 + 1.53438i 0.844711 + 0.535223i \(0.179773\pi\)
0.0411616 + 0.999153i \(0.486894\pi\)
\(744\) 0 0
\(745\) −2323.82 + 4024.97i −0.114279 + 0.197938i
\(746\) 0 0
\(747\) 2163.53 3747.34i 0.105970 0.183545i
\(748\) 0 0
\(749\) −32600.3 −1.59037
\(750\) 0 0
\(751\) −90.8447 157.348i −0.00441407 0.00764540i 0.863810 0.503818i \(-0.168072\pi\)
−0.868224 + 0.496172i \(0.834738\pi\)
\(752\) 0 0
\(753\) 2957.01 0.143107
\(754\) 0 0
\(755\) −786.425 −0.0379085
\(756\) 0 0
\(757\) 245.526 + 425.264i 0.0117884 + 0.0204181i 0.871859 0.489756i \(-0.162914\pi\)
−0.860071 + 0.510174i \(0.829581\pi\)
\(758\) 0 0
\(759\) −639.655 −0.0305903
\(760\) 0 0
\(761\) 4056.50 7026.07i 0.193230 0.334684i −0.753089 0.657919i \(-0.771437\pi\)
0.946319 + 0.323235i \(0.104770\pi\)
\(762\) 0 0
\(763\) −5409.58 + 9369.67i −0.256671 + 0.444567i
\(764\) 0 0
\(765\) −1302.46 2255.93i −0.0615562 0.106619i
\(766\) 0 0
\(767\) −1017.65 + 599.830i −0.0479078 + 0.0282381i
\(768\) 0 0
\(769\) −9932.33 17203.3i −0.465759 0.806719i 0.533476 0.845815i \(-0.320885\pi\)
−0.999235 + 0.0390962i \(0.987552\pi\)
\(770\) 0 0
\(771\) −4393.98 + 7610.59i −0.205247 + 0.355498i
\(772\) 0 0
\(773\) −4523.71 + 7835.29i −0.210487 + 0.364574i −0.951867 0.306511i \(-0.900838\pi\)
0.741380 + 0.671085i \(0.234172\pi\)
\(774\) 0 0
\(775\) −27887.0 −1.29256
\(776\) 0 0
\(777\) 2750.99 + 4764.85i 0.127016 + 0.219997i
\(778\) 0 0
\(779\) −39535.5 −1.81837
\(780\) 0 0
\(781\) −30347.1 −1.39040
\(782\) 0 0
\(783\) −1363.44 2361.54i −0.0622289 0.107784i
\(784\) 0 0
\(785\) −8842.45 −0.402039
\(786\) 0 0
\(787\) −7509.20 + 13006.3i −0.340120 + 0.589105i −0.984455 0.175639i \(-0.943801\pi\)
0.644335 + 0.764743i \(0.277134\pi\)
\(788\) 0 0
\(789\) 3357.00 5814.50i 0.151473 0.262360i
\(790\) 0 0
\(791\) −16873.9 29226.4i −0.758491 1.31375i
\(792\) 0 0
\(793\) −23776.9 + 14014.7i −1.06474 + 0.627587i
\(794\) 0 0
\(795\) 595.965 + 1032.24i 0.0265870 + 0.0460501i
\(796\) 0 0
\(797\) −15970.5 + 27661.8i −0.709794 + 1.22940i 0.255139 + 0.966904i \(0.417879\pi\)
−0.964933 + 0.262495i \(0.915455\pi\)
\(798\) 0 0
\(799\) 11383.5 19716.9i 0.504030 0.873006i
\(800\) 0 0
\(801\) 9773.17 0.431108
\(802\) 0 0
\(803\) −16412.5 28427.3i −0.721277 1.24929i
\(804\) 0 0
\(805\) 889.409 0.0389411
\(806\) 0 0
\(807\) 5777.95 0.252037
\(808\) 0 0
\(809\) −13630.0 23607.9i −0.592344 1.02597i −0.993916 0.110142i \(-0.964869\pi\)
0.401572 0.915827i \(-0.368464\pi\)
\(810\) 0 0
\(811\) 20707.8 0.896607 0.448303 0.893881i \(-0.352028\pi\)
0.448303 + 0.893881i \(0.352028\pi\)
\(812\) 0 0
\(813\) −5343.43 + 9255.09i −0.230507 + 0.399250i
\(814\) 0 0
\(815\) −3442.71 + 5962.95i −0.147967 + 0.256286i
\(816\) 0 0
\(817\) −10305.6 17849.8i −0.441307 0.764366i
\(818\) 0 0
\(819\) 91.6839 + 10176.9i 0.00391171 + 0.434200i
\(820\) 0 0
\(821\) −7829.28 13560.7i −0.332818 0.576458i 0.650245 0.759724i \(-0.274666\pi\)
−0.983063 + 0.183267i \(0.941333\pi\)
\(822\) 0 0
\(823\) −2053.29 + 3556.40i −0.0869662 + 0.150630i −0.906227 0.422791i \(-0.861051\pi\)
0.819261 + 0.573421i \(0.194384\pi\)
\(824\) 0 0
\(825\) 4608.89 7982.83i 0.194498 0.336881i
\(826\) 0 0
\(827\) −16747.3 −0.704184 −0.352092 0.935965i \(-0.614530\pi\)
−0.352092 + 0.935965i \(0.614530\pi\)
\(828\) 0 0
\(829\) 14578.5 + 25250.8i 0.610776 + 1.05790i 0.991110 + 0.133046i \(0.0424758\pi\)
−0.380334 + 0.924849i \(0.624191\pi\)
\(830\) 0 0
\(831\) −4312.27 −0.180013
\(832\) 0 0
\(833\) −11807.4 −0.491119
\(834\) 0 0
\(835\) −4318.52 7479.89i −0.178980 0.310003i
\(836\) 0 0
\(837\) 8304.66 0.342952
\(838\) 0 0
\(839\) 22909.7 39680.8i 0.942707 1.63282i 0.182429 0.983219i \(-0.441604\pi\)
0.760278 0.649598i \(-0.225063\pi\)
\(840\) 0 0
\(841\) 7094.47 12288.0i 0.290888 0.503833i
\(842\) 0 0
\(843\) 5870.26 + 10167.6i 0.239837 + 0.415410i
\(844\) 0 0
\(845\) −6636.51 11030.9i −0.270181 0.449083i
\(846\) 0 0
\(847\) 2201.72 + 3813.49i 0.0893175 + 0.154703i
\(848\) 0 0
\(849\) −4818.47 + 8345.83i −0.194781 + 0.337371i
\(850\) 0 0
\(851\) −239.142 + 414.205i −0.00963298 + 0.0166848i
\(852\) 0 0
\(853\) 17351.1 0.696471 0.348235 0.937407i \(-0.386781\pi\)
0.348235 + 0.937407i \(0.386781\pi\)
\(854\) 0 0
\(855\) 2026.50 + 3510.00i 0.0810583 + 0.140397i
\(856\) 0 0
\(857\) −21768.1 −0.867659 −0.433829 0.900995i \(-0.642838\pi\)
−0.433829 + 0.900995i \(0.642838\pi\)
\(858\) 0 0
\(859\) 29878.4 1.18677 0.593387 0.804918i \(-0.297791\pi\)
0.593387 + 0.804918i \(0.297791\pi\)
\(860\) 0 0
\(861\) 18615.9 + 32243.6i 0.736849 + 1.27626i
\(862\) 0 0
\(863\) 15067.7 0.594335 0.297168 0.954825i \(-0.403958\pi\)
0.297168 + 0.954825i \(0.403958\pi\)
\(864\) 0 0
\(865\) −6821.40 + 11815.0i −0.268132 + 0.464419i
\(866\) 0 0
\(867\) 3709.61 6425.23i 0.145311 0.251686i
\(868\) 0 0
\(869\) −2017.31 3494.08i −0.0787486 0.136397i
\(870\) 0 0
\(871\) 424.138 + 47079.3i 0.0164999 + 1.83148i
\(872\) 0 0
\(873\) 74.9485 + 129.815i 0.00290564 + 0.00503272i
\(874\) 0 0
\(875\) −15243.7 + 26402.8i −0.588949 + 1.02009i
\(876\) 0 0
\(877\) 10559.8 18290.1i 0.406588 0.704232i −0.587917 0.808922i \(-0.700052\pi\)
0.994505 + 0.104690i \(0.0333850\pi\)
\(878\) 0 0
\(879\) −14705.3 −0.564275
\(880\) 0 0
\(881\) −15826.2 27411.9i −0.605221 1.04827i −0.992016 0.126108i \(-0.959751\pi\)
0.386795 0.922166i \(-0.373582\pi\)
\(882\) 0 0
\(883\) −11701.6 −0.445969 −0.222984 0.974822i \(-0.571580\pi\)
−0.222984 + 0.974822i \(0.571580\pi\)
\(884\) 0 0
\(885\) 443.017 0.0168270
\(886\) 0 0
\(887\) 11254.0 + 19492.4i 0.426010 + 0.737871i 0.996514 0.0834239i \(-0.0265855\pi\)
−0.570504 + 0.821295i \(0.693252\pi\)
\(888\) 0 0
\(889\) 2883.10 0.108769
\(890\) 0 0
\(891\) −1372.51 + 2377.26i −0.0516059 + 0.0893841i
\(892\) 0 0
\(893\) −17711.7 + 30677.5i −0.663716 + 1.14959i
\(894\) 0 0
\(895\) 6251.69 + 10828.2i 0.233487 + 0.404412i
\(896\) 0 0
\(897\) −762.163 + 449.238i −0.0283700 + 0.0167220i
\(898\) 0 0
\(899\) −15532.1 26902.3i −0.576222 0.998046i
\(900\) 0 0
\(901\) 1674.65 2900.58i 0.0619210 0.107250i
\(902\) 0 0
\(903\) −9705.08 + 16809.7i −0.357658 + 0.619481i
\(904\) 0 0
\(905\) 14563.3 0.534919
\(906\) 0 0
\(907\) 9359.32 + 16210.8i 0.342636 + 0.593464i 0.984921 0.173002i \(-0.0553468\pi\)
−0.642285 + 0.766466i \(0.722013\pi\)
\(908\) 0 0
\(909\) −8622.03 −0.314604
\(910\) 0 0
\(911\) −18616.7 −0.677057 −0.338529 0.940956i \(-0.609929\pi\)
−0.338529 + 0.940956i \(0.609929\pi\)
\(912\) 0 0
\(913\) 8146.69 + 14110.5i 0.295308 + 0.511488i
\(914\) 0 0
\(915\) 10350.8 0.373977
\(916\) 0 0
\(917\) −27161.7 + 47045.4i −0.978143 + 1.69419i
\(918\) 0 0
\(919\) −27382.2 + 47427.3i −0.982867 + 1.70238i −0.331812 + 0.943346i \(0.607660\pi\)
−0.651056 + 0.759030i \(0.725674\pi\)
\(920\) 0 0
\(921\) 8700.94 + 15070.5i 0.311298 + 0.539185i
\(922\) 0 0
\(923\) −36159.2 + 21313.2i −1.28949 + 0.760056i
\(924\) 0 0
\(925\) −3446.16 5968.93i −0.122496 0.212170i
\(926\) 0 0
\(927\) −12.1752 + 21.0880i −0.000431376 + 0.000747165i
\(928\) 0 0
\(929\) 15916.0 27567.4i 0.562097 0.973581i −0.435216 0.900326i \(-0.643328\pi\)
0.997313 0.0732550i \(-0.0233387\pi\)
\(930\) 0 0
\(931\) 18371.2 0.646713
\(932\) 0 0
\(933\) −7370.27 12765.7i −0.258619 0.447942i
\(934\) 0 0
\(935\) 9808.73 0.343080
\(936\) 0 0
\(937\) −27408.7 −0.955607 −0.477803 0.878467i \(-0.658567\pi\)
−0.477803 + 0.878467i \(0.658567\pi\)
\(938\) 0 0
\(939\) 12157.5 + 21057.3i 0.422517 + 0.731822i
\(940\) 0 0
\(941\) 54837.8 1.89975 0.949874 0.312634i \(-0.101211\pi\)
0.949874 + 0.312634i \(0.101211\pi\)
\(942\) 0 0
\(943\) −1618.27 + 2802.92i −0.0558833 + 0.0967928i
\(944\) 0 0
\(945\) 1908.41 3305.47i 0.0656938 0.113785i
\(946\) 0 0
\(947\) −19853.9 34388.0i −0.681273 1.18000i −0.974593 0.223984i \(-0.928094\pi\)
0.293320 0.956014i \(-0.405240\pi\)
\(948\) 0 0
\(949\) −39520.8 22345.0i −1.35184 0.764332i
\(950\) 0 0
\(951\) −7724.88 13379.9i −0.263403 0.456227i
\(952\) 0 0
\(953\) 8553.32 14814.8i 0.290734 0.503566i −0.683250 0.730185i \(-0.739434\pi\)
0.973983 + 0.226619i \(0.0727673\pi\)
\(954\) 0 0
\(955\) 6811.90 11798.6i 0.230815 0.399783i
\(956\) 0 0
\(957\) 10268.0 0.346829
\(958\) 0 0
\(959\) −13638.4 23622.3i −0.459234 0.795417i
\(960\) 0 0
\(961\) 64814.4 2.17564
\(962\) 0 0
\(963\) −12161.6 −0.406958
\(964\) 0 0
\(965\) 9815.06 + 17000.2i 0.327417 + 0.567104i
\(966\) 0 0
\(967\) 23417.5 0.778756 0.389378 0.921078i \(-0.372690\pi\)
0.389378 + 0.921078i \(0.372690\pi\)
\(968\) 0 0
\(969\) 5694.44 9863.06i 0.188784 0.326984i
\(970\) 0 0
\(971\) 8215.31 14229.3i 0.271516 0.470279i −0.697734 0.716357i \(-0.745808\pi\)
0.969250 + 0.246077i \(0.0791417\pi\)
\(972\) 0 0
\(973\) −7180.78 12437.5i −0.236593 0.409792i
\(974\) 0 0
\(975\) −114.852 12748.6i −0.00377254 0.418751i
\(976\) 0 0
\(977\) 5277.32 + 9140.58i 0.172811 + 0.299318i 0.939402 0.342819i \(-0.111382\pi\)
−0.766591 + 0.642136i \(0.778048\pi\)
\(978\) 0 0
\(979\) −18400.3 + 31870.2i −0.600689 + 1.04042i
\(980\) 0 0
\(981\) −2018.05 + 3495.36i −0.0656791 + 0.113760i
\(982\) 0 0
\(983\) −1534.33 −0.0497839 −0.0248919 0.999690i \(-0.507924\pi\)
−0.0248919 + 0.999690i \(0.507924\pi\)
\(984\) 0 0
\(985\) −11306.9 19584.1i −0.365753 0.633502i
\(986\) 0 0
\(987\) 33359.1 1.07582
\(988\) 0 0
\(989\) −1687.31 −0.0542502
\(990\) 0 0
\(991\) −9009.08 15604.2i −0.288782 0.500185i 0.684737 0.728790i \(-0.259917\pi\)
−0.973519 + 0.228605i \(0.926584\pi\)
\(992\) 0 0
\(993\) −18185.9 −0.581181
\(994\) 0 0
\(995\) 11964.0 20722.2i 0.381190 0.660240i
\(996\) 0 0
\(997\) −24143.8 + 41818.4i −0.766944 + 1.32839i 0.172269 + 0.985050i \(0.444890\pi\)
−0.939213 + 0.343336i \(0.888443\pi\)
\(998\) 0 0
\(999\) 1026.26 + 1777.53i 0.0325018 + 0.0562947i
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 624.4.q.i.289.2 8
4.3 odd 2 39.4.e.c.16.4 8
12.11 even 2 117.4.g.e.55.1 8
13.9 even 3 inner 624.4.q.i.529.2 8
52.3 odd 6 507.4.a.m.1.1 4
52.11 even 12 507.4.b.h.337.2 8
52.15 even 12 507.4.b.h.337.7 8
52.23 odd 6 507.4.a.i.1.4 4
52.35 odd 6 39.4.e.c.22.4 yes 8
156.23 even 6 1521.4.a.bb.1.1 4
156.35 even 6 117.4.g.e.100.1 8
156.107 even 6 1521.4.a.v.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.4 8 4.3 odd 2
39.4.e.c.22.4 yes 8 52.35 odd 6
117.4.g.e.55.1 8 12.11 even 2
117.4.g.e.100.1 8 156.35 even 6
507.4.a.i.1.4 4 52.23 odd 6
507.4.a.m.1.1 4 52.3 odd 6
507.4.b.h.337.2 8 52.11 even 12
507.4.b.h.337.7 8 52.15 even 12
624.4.q.i.289.2 8 1.1 even 1 trivial
624.4.q.i.529.2 8 13.9 even 3 inner
1521.4.a.v.1.4 4 156.107 even 6
1521.4.a.bb.1.1 4 156.23 even 6