Properties

Label 1800.2.k.u.901.12
Level $1800$
Weight $2$
Character 1800.901
Analytic conductor $14.373$
Analytic rank $0$
Dimension $12$
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] = [1800,2,Mod(901,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.k (of order \(2\), degree \(1\), 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: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.180227832610816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 901.12
Root \(1.37729 - 0.321037i\) of defining polynomial
Character \(\chi\) \(=\) 1800.901
Dual form 1800.2.k.u.901.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.37729 + 0.321037i) q^{2} +(1.79387 + 0.884323i) q^{4} -4.05705 q^{7} +(2.18678 + 1.79387i) q^{8} +O(q^{10})\) \(q+(1.37729 + 0.321037i) q^{2} +(1.79387 + 0.884323i) q^{4} -4.05705 q^{7} +(2.18678 + 1.79387i) q^{8} +0.985939i q^{11} +4.94567i q^{13} +(-5.58774 - 1.30246i) q^{14} +(2.43594 + 3.17272i) q^{16} -4.52323 q^{17} +2.60492i q^{19} +(-0.316523 + 1.35793i) q^{22} -3.53729 q^{23} +(-1.58774 + 6.81163i) q^{26} +(-7.27782 - 3.58774i) q^{28} -7.59434i q^{29} -3.28415 q^{31} +(2.33645 + 5.15180i) q^{32} +(-6.22982 - 1.45212i) q^{34} +0.945668i q^{37} +(-0.836276 + 3.58774i) q^{38} -0.568295 q^{41} +8.45963i q^{43} +(-0.871889 + 1.76865i) q^{44} +(-4.87189 - 1.13560i) q^{46} -2.60492 q^{47} +9.45963 q^{49} +(-4.37357 + 8.87189i) q^{52} -0.229815i q^{53} +(-8.87189 - 7.27782i) q^{56} +(2.43806 - 10.4596i) q^{58} +9.10003i q^{59} +11.0183i q^{61} +(-4.52323 - 1.05433i) q^{62} +(1.56406 + 7.84562i) q^{64} +8.45963i q^{67} +(-8.11409 - 4.00000i) q^{68} -1.43171 q^{71} +11.9507 q^{73} +(-0.303594 + 1.30246i) q^{74} +(-2.30359 + 4.67289i) q^{76} -4.00000i q^{77} -3.28415 q^{79} +(-0.782708 - 0.182443i) q^{82} -9.89134i q^{83} +(-2.71585 + 11.6514i) q^{86} +(-1.76865 + 2.15604i) q^{88} +12.3510 q^{89} -20.0648i q^{91} +(-6.34545 - 3.12811i) q^{92} +(-3.58774 - 0.836276i) q^{94} -3.23797 q^{97} +(13.0287 + 3.03689i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{4} - 20 q^{14} + 2 q^{16} + 28 q^{26} - 32 q^{31} - 24 q^{34} + 8 q^{41} + 44 q^{44} - 4 q^{46} + 12 q^{49} - 52 q^{56} + 46 q^{64} - 32 q^{71} + 36 q^{74} + 12 q^{76} - 32 q^{79} - 40 q^{86} - 40 q^{89} + 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37729 + 0.321037i 0.973893 + 0.227007i
\(3\) 0 0
\(4\) 1.79387 + 0.884323i 0.896935 + 0.442162i
\(5\) 0 0
\(6\) 0 0
\(7\) −4.05705 −1.53342 −0.766710 0.641994i \(-0.778107\pi\)
−0.766710 + 0.641994i \(0.778107\pi\)
\(8\) 2.18678 + 1.79387i 0.773145 + 0.634229i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.985939i 0.297272i 0.988892 + 0.148636i \(0.0474882\pi\)
−0.988892 + 0.148636i \(0.952512\pi\)
\(12\) 0 0
\(13\) 4.94567i 1.37168i 0.727752 + 0.685841i \(0.240565\pi\)
−0.727752 + 0.685841i \(0.759435\pi\)
\(14\) −5.58774 1.30246i −1.49339 0.348097i
\(15\) 0 0
\(16\) 2.43594 + 3.17272i 0.608986 + 0.793181i
\(17\) −4.52323 −1.09704 −0.548522 0.836136i \(-0.684809\pi\)
−0.548522 + 0.836136i \(0.684809\pi\)
\(18\) 0 0
\(19\) 2.60492i 0.597610i 0.954314 + 0.298805i \(0.0965881\pi\)
−0.954314 + 0.298805i \(0.903412\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.316523 + 1.35793i −0.0674829 + 0.289511i
\(23\) −3.53729 −0.737577 −0.368788 0.929513i \(-0.620227\pi\)
−0.368788 + 0.929513i \(0.620227\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.58774 + 6.81163i −0.311382 + 1.33587i
\(27\) 0 0
\(28\) −7.27782 3.58774i −1.37538 0.678019i
\(29\) 7.59434i 1.41023i −0.709091 0.705117i \(-0.750895\pi\)
0.709091 0.705117i \(-0.249105\pi\)
\(30\) 0 0
\(31\) −3.28415 −0.589850 −0.294925 0.955520i \(-0.595295\pi\)
−0.294925 + 0.955520i \(0.595295\pi\)
\(32\) 2.33645 + 5.15180i 0.413029 + 0.910718i
\(33\) 0 0
\(34\) −6.22982 1.45212i −1.06840 0.249037i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.945668i 0.155467i 0.996974 + 0.0777334i \(0.0247683\pi\)
−0.996974 + 0.0777334i \(0.975232\pi\)
\(38\) −0.836276 + 3.58774i −0.135662 + 0.582009i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.568295 −0.0887527 −0.0443763 0.999015i \(-0.514130\pi\)
−0.0443763 + 0.999015i \(0.514130\pi\)
\(42\) 0 0
\(43\) 8.45963i 1.29008i 0.764148 + 0.645041i \(0.223160\pi\)
−0.764148 + 0.645041i \(0.776840\pi\)
\(44\) −0.871889 + 1.76865i −0.131442 + 0.266634i
\(45\) 0 0
\(46\) −4.87189 1.13560i −0.718321 0.167435i
\(47\) −2.60492 −0.379967 −0.189984 0.981787i \(-0.560843\pi\)
−0.189984 + 0.981787i \(0.560843\pi\)
\(48\) 0 0
\(49\) 9.45963 1.35138
\(50\) 0 0
\(51\) 0 0
\(52\) −4.37357 + 8.87189i −0.606505 + 1.23031i
\(53\) 0.229815i 0.0315675i −0.999875 0.0157838i \(-0.994976\pi\)
0.999875 0.0157838i \(-0.00502434\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −8.87189 7.27782i −1.18556 0.972539i
\(57\) 0 0
\(58\) 2.43806 10.4596i 0.320133 1.37342i
\(59\) 9.10003i 1.18472i 0.805672 + 0.592362i \(0.201804\pi\)
−0.805672 + 0.592362i \(0.798196\pi\)
\(60\) 0 0
\(61\) 11.0183i 1.41075i 0.708832 + 0.705377i \(0.249222\pi\)
−0.708832 + 0.705377i \(0.750778\pi\)
\(62\) −4.52323 1.05433i −0.574451 0.133900i
\(63\) 0 0
\(64\) 1.56406 + 7.84562i 0.195507 + 0.980702i
\(65\) 0 0
\(66\) 0 0
\(67\) 8.45963i 1.03351i 0.856134 + 0.516754i \(0.172860\pi\)
−0.856134 + 0.516754i \(0.827140\pi\)
\(68\) −8.11409 4.00000i −0.983978 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) −1.43171 −0.169912 −0.0849561 0.996385i \(-0.527075\pi\)
−0.0849561 + 0.996385i \(0.527075\pi\)
\(72\) 0 0
\(73\) 11.9507 1.39873 0.699363 0.714767i \(-0.253467\pi\)
0.699363 + 0.714767i \(0.253467\pi\)
\(74\) −0.303594 + 1.30246i −0.0352921 + 0.151408i
\(75\) 0 0
\(76\) −2.30359 + 4.67289i −0.264240 + 0.536018i
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −3.28415 −0.369495 −0.184748 0.982786i \(-0.559147\pi\)
−0.184748 + 0.982786i \(0.559147\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.782708 0.182443i −0.0864356 0.0201475i
\(83\) 9.89134i 1.08572i −0.839825 0.542858i \(-0.817342\pi\)
0.839825 0.542858i \(-0.182658\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.71585 + 11.6514i −0.292858 + 1.25640i
\(87\) 0 0
\(88\) −1.76865 + 2.15604i −0.188538 + 0.229834i
\(89\) 12.3510 1.30920 0.654600 0.755976i \(-0.272837\pi\)
0.654600 + 0.755976i \(0.272837\pi\)
\(90\) 0 0
\(91\) 20.0648i 2.10336i
\(92\) −6.34545 3.12811i −0.661559 0.326128i
\(93\) 0 0
\(94\) −3.58774 0.836276i −0.370047 0.0862553i
\(95\) 0 0
\(96\) 0 0
\(97\) −3.23797 −0.328766 −0.164383 0.986397i \(-0.552563\pi\)
−0.164383 + 0.986397i \(0.552563\pi\)
\(98\) 13.0287 + 3.03689i 1.31610 + 0.306772i
\(99\) 0 0
\(100\) 0 0
\(101\) 4.35637i 0.433475i −0.976230 0.216738i \(-0.930458\pi\)
0.976230 0.216738i \(-0.0695416\pi\)
\(102\) 0 0
\(103\) −15.0754 −1.48542 −0.742711 0.669612i \(-0.766460\pi\)
−0.742711 + 0.669612i \(0.766460\pi\)
\(104\) −8.87189 + 10.8151i −0.869960 + 1.06051i
\(105\) 0 0
\(106\) 0.0737791 0.316523i 0.00716606 0.0307434i
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 4.17034i 0.399446i −0.979852 0.199723i \(-0.935996\pi\)
0.979852 0.199723i \(-0.0640042\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −9.88274 12.8719i −0.933831 1.21628i
\(113\) 1.28526 0.120907 0.0604537 0.998171i \(-0.480745\pi\)
0.0604537 + 0.998171i \(0.480745\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.71585 13.6233i 0.623551 1.26489i
\(117\) 0 0
\(118\) −2.92145 + 12.5334i −0.268941 + 1.15379i
\(119\) 18.3510 1.68223
\(120\) 0 0
\(121\) 10.0279 0.911630
\(122\) −3.53729 + 15.1755i −0.320252 + 1.37392i
\(123\) 0 0
\(124\) −5.89134 2.90425i −0.529058 0.260809i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.15280 −0.102294 −0.0511472 0.998691i \(-0.516288\pi\)
−0.0511472 + 0.998691i \(0.516288\pi\)
\(128\) −0.364570 + 11.3078i −0.0322237 + 0.999481i
\(129\) 0 0
\(130\) 0 0
\(131\) 3.89019i 0.339887i 0.985454 + 0.169944i \(0.0543586\pi\)
−0.985454 + 0.169944i \(0.945641\pi\)
\(132\) 0 0
\(133\) 10.5683i 0.916387i
\(134\) −2.71585 + 11.6514i −0.234614 + 1.00653i
\(135\) 0 0
\(136\) −9.89134 8.11409i −0.848175 0.695778i
\(137\) 17.5135 1.49628 0.748138 0.663544i \(-0.230948\pi\)
0.748138 + 0.663544i \(0.230948\pi\)
\(138\) 0 0
\(139\) 16.8612i 1.43015i −0.699047 0.715076i \(-0.746392\pi\)
0.699047 0.715076i \(-0.253608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.97188 0.459630i −0.165476 0.0385713i
\(143\) −4.87613 −0.407762
\(144\) 0 0
\(145\) 0 0
\(146\) 16.4596 + 3.83662i 1.36221 + 0.317521i
\(147\) 0 0
\(148\) −0.836276 + 1.69641i −0.0687415 + 0.139444i
\(149\) 10.4986i 0.860078i −0.902810 0.430039i \(-0.858500\pi\)
0.902810 0.430039i \(-0.141500\pi\)
\(150\) 0 0
\(151\) 4.71585 0.383771 0.191885 0.981417i \(-0.438540\pi\)
0.191885 + 0.981417i \(0.438540\pi\)
\(152\) −4.67289 + 5.69641i −0.379022 + 0.462040i
\(153\) 0 0
\(154\) 1.28415 5.50917i 0.103480 0.443942i
\(155\) 0 0
\(156\) 0 0
\(157\) 8.94567i 0.713942i 0.934115 + 0.356971i \(0.116191\pi\)
−0.934115 + 0.356971i \(0.883809\pi\)
\(158\) −4.52323 1.05433i −0.359849 0.0838782i
\(159\) 0 0
\(160\) 0 0
\(161\) 14.3510 1.13101
\(162\) 0 0
\(163\) 15.7827i 1.23619i −0.786102 0.618097i \(-0.787904\pi\)
0.786102 0.618097i \(-0.212096\pi\)
\(164\) −1.01945 0.502556i −0.0796054 0.0392430i
\(165\) 0 0
\(166\) 3.17548 13.6233i 0.246465 1.05737i
\(167\) −5.50917 −0.426312 −0.213156 0.977018i \(-0.568374\pi\)
−0.213156 + 0.977018i \(0.568374\pi\)
\(168\) 0 0
\(169\) −11.4596 −0.881510
\(170\) 0 0
\(171\) 0 0
\(172\) −7.48105 + 15.1755i −0.570425 + 1.15712i
\(173\) 10.3385i 0.786020i −0.919534 0.393010i \(-0.871434\pi\)
0.919534 0.393010i \(-0.128566\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.12811 + 2.40169i −0.235790 + 0.181034i
\(177\) 0 0
\(178\) 17.0109 + 3.96511i 1.27502 + 0.297198i
\(179\) 16.1746i 1.20895i −0.796625 0.604474i \(-0.793383\pi\)
0.796625 0.604474i \(-0.206617\pi\)
\(180\) 0 0
\(181\) 8.11409i 0.603116i 0.953448 + 0.301558i \(0.0975067\pi\)
−0.953448 + 0.301558i \(0.902493\pi\)
\(182\) 6.44154 27.6351i 0.477479 2.04845i
\(183\) 0 0
\(184\) −7.73530 6.34545i −0.570254 0.467793i
\(185\) 0 0
\(186\) 0 0
\(187\) 4.45963i 0.326120i
\(188\) −4.67289 2.30359i −0.340806 0.168007i
\(189\) 0 0
\(190\) 0 0
\(191\) 24.9193 1.80309 0.901547 0.432681i \(-0.142432\pi\)
0.901547 + 0.432681i \(0.142432\pi\)
\(192\) 0 0
\(193\) 1.03951 0.0748254 0.0374127 0.999300i \(-0.488088\pi\)
0.0374127 + 0.999300i \(0.488088\pi\)
\(194\) −4.45963 1.03951i −0.320183 0.0746323i
\(195\) 0 0
\(196\) 16.9694 + 8.36537i 1.21210 + 0.597527i
\(197\) 9.66152i 0.688355i 0.938905 + 0.344177i \(0.111842\pi\)
−0.938905 + 0.344177i \(0.888158\pi\)
\(198\) 0 0
\(199\) 23.0668 1.63516 0.817582 0.575813i \(-0.195314\pi\)
0.817582 + 0.575813i \(0.195314\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.39856 6.00000i 0.0984020 0.422159i
\(203\) 30.8106i 2.16248i
\(204\) 0 0
\(205\) 0 0
\(206\) −20.7632 4.83975i −1.44664 0.337202i
\(207\) 0 0
\(208\) −15.6912 + 12.0474i −1.08799 + 0.835335i
\(209\) −2.56829 −0.177653
\(210\) 0 0
\(211\) 6.44154i 0.443454i 0.975109 + 0.221727i \(0.0711694\pi\)
−0.975109 + 0.221727i \(0.928831\pi\)
\(212\) 0.203231 0.412259i 0.0139580 0.0283140i
\(213\) 0 0
\(214\) −1.28415 + 5.50917i −0.0877825 + 0.376599i
\(215\) 0 0
\(216\) 0 0
\(217\) 13.3239 0.904488
\(218\) 1.33883 5.74378i 0.0906772 0.389018i
\(219\) 0 0
\(220\) 0 0
\(221\) 22.3704i 1.50480i
\(222\) 0 0
\(223\) −17.9796 −1.20401 −0.602003 0.798494i \(-0.705630\pi\)
−0.602003 + 0.798494i \(0.705630\pi\)
\(224\) −9.47908 20.9011i −0.633347 1.39651i
\(225\) 0 0
\(226\) 1.77018 + 0.412617i 0.117751 + 0.0274469i
\(227\) 7.02792i 0.466460i 0.972422 + 0.233230i \(0.0749295\pi\)
−0.972422 + 0.233230i \(0.925071\pi\)
\(228\) 0 0
\(229\) 4.17034i 0.275584i 0.990461 + 0.137792i \(0.0440005\pi\)
−0.990461 + 0.137792i \(0.955999\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 13.6233 16.6072i 0.894411 1.09032i
\(233\) 23.9894 1.57160 0.785799 0.618483i \(-0.212252\pi\)
0.785799 + 0.618483i \(0.212252\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.04737 + 16.3243i −0.523839 + 1.06262i
\(237\) 0 0
\(238\) 25.2747 + 5.89134i 1.63831 + 0.381879i
\(239\) 8.91926 0.576939 0.288469 0.957489i \(-0.406854\pi\)
0.288469 + 0.957489i \(0.406854\pi\)
\(240\) 0 0
\(241\) −16.3510 −1.05326 −0.526629 0.850095i \(-0.676544\pi\)
−0.526629 + 0.850095i \(0.676544\pi\)
\(242\) 13.8114 + 3.21933i 0.887830 + 0.206947i
\(243\) 0 0
\(244\) −9.74378 + 19.7655i −0.623781 + 1.26536i
\(245\) 0 0
\(246\) 0 0
\(247\) −12.8831 −0.819731
\(248\) −7.18172 5.89134i −0.456040 0.374100i
\(249\) 0 0
\(250\) 0 0
\(251\) 4.22391i 0.266611i 0.991075 + 0.133305i \(0.0425591\pi\)
−0.991075 + 0.133305i \(0.957441\pi\)
\(252\) 0 0
\(253\) 3.48755i 0.219261i
\(254\) −1.58774 0.370091i −0.0996238 0.0232216i
\(255\) 0 0
\(256\) −4.13235 + 15.4572i −0.258272 + 0.966072i
\(257\) −24.6952 −1.54044 −0.770221 0.637777i \(-0.779854\pi\)
−0.770221 + 0.637777i \(0.779854\pi\)
\(258\) 0 0
\(259\) 3.83662i 0.238396i
\(260\) 0 0
\(261\) 0 0
\(262\) −1.24889 + 5.35793i −0.0771569 + 0.331014i
\(263\) −14.6628 −0.904145 −0.452073 0.891981i \(-0.649315\pi\)
−0.452073 + 0.891981i \(0.649315\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.39281 14.5556i 0.208027 0.892463i
\(267\) 0 0
\(268\) −7.48105 + 15.1755i −0.456978 + 0.926990i
\(269\) 11.5381i 0.703490i 0.936096 + 0.351745i \(0.114412\pi\)
−0.936096 + 0.351745i \(0.885588\pi\)
\(270\) 0 0
\(271\) 5.63511 0.342309 0.171154 0.985244i \(-0.445250\pi\)
0.171154 + 0.985244i \(0.445250\pi\)
\(272\) −11.0183 14.3510i −0.668085 0.870155i
\(273\) 0 0
\(274\) 24.1212 + 5.62246i 1.45721 + 0.339665i
\(275\) 0 0
\(276\) 0 0
\(277\) 17.4053i 1.04578i −0.852399 0.522892i \(-0.824853\pi\)
0.852399 0.522892i \(-0.175147\pi\)
\(278\) 5.41308 23.2229i 0.324655 1.39281i
\(279\) 0 0
\(280\) 0 0
\(281\) −21.7827 −1.29945 −0.649723 0.760171i \(-0.725115\pi\)
−0.649723 + 0.760171i \(0.725115\pi\)
\(282\) 0 0
\(283\) 21.5962i 1.28376i 0.766804 + 0.641881i \(0.221846\pi\)
−0.766804 + 0.641881i \(0.778154\pi\)
\(284\) −2.56829 1.26609i −0.152400 0.0751287i
\(285\) 0 0
\(286\) −6.71585 1.56542i −0.397117 0.0925650i
\(287\) 2.30560 0.136095
\(288\) 0 0
\(289\) 3.45963 0.203508
\(290\) 0 0
\(291\) 0 0
\(292\) 21.4380 + 10.5683i 1.25457 + 0.618463i
\(293\) 32.0125i 1.87019i 0.354398 + 0.935095i \(0.384686\pi\)
−0.354398 + 0.935095i \(0.615314\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.69641 + 2.06797i −0.0986016 + 0.120198i
\(297\) 0 0
\(298\) 3.37043 14.4596i 0.195244 0.837624i
\(299\) 17.4943i 1.01172i
\(300\) 0 0
\(301\) 34.3211i 1.97824i
\(302\) 6.49511 + 1.51396i 0.373752 + 0.0871187i
\(303\) 0 0
\(304\) −8.26470 + 6.34545i −0.474013 + 0.363936i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.13659i 0.0648686i 0.999474 + 0.0324343i \(0.0103260\pi\)
−0.999474 + 0.0324343i \(0.989674\pi\)
\(308\) 3.53729 7.17548i 0.201556 0.408861i
\(309\) 0 0
\(310\) 0 0
\(311\) −5.13659 −0.291269 −0.145635 0.989338i \(-0.546522\pi\)
−0.145635 + 0.989338i \(0.546522\pi\)
\(312\) 0 0
\(313\) 23.0762 1.30434 0.652172 0.758071i \(-0.273858\pi\)
0.652172 + 0.758071i \(0.273858\pi\)
\(314\) −2.87189 + 12.3208i −0.162070 + 0.695303i
\(315\) 0 0
\(316\) −5.89134 2.90425i −0.331414 0.163377i
\(317\) 1.66152i 0.0933203i 0.998911 + 0.0466601i \(0.0148578\pi\)
−0.998911 + 0.0466601i \(0.985142\pi\)
\(318\) 0 0
\(319\) 7.48755 0.419223
\(320\) 0 0
\(321\) 0 0
\(322\) 19.7655 + 4.60719i 1.10149 + 0.256749i
\(323\) 11.7827i 0.655605i
\(324\) 0 0
\(325\) 0 0
\(326\) 5.06682 21.7374i 0.280625 1.20392i
\(327\) 0 0
\(328\) −1.24274 1.01945i −0.0686187 0.0562895i
\(329\) 10.5683 0.582649
\(330\) 0 0
\(331\) 25.9077i 1.42402i 0.702171 + 0.712008i \(0.252214\pi\)
−0.702171 + 0.712008i \(0.747786\pi\)
\(332\) 8.74714 17.7438i 0.480062 0.973816i
\(333\) 0 0
\(334\) −7.58774 1.76865i −0.415183 0.0967760i
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00696 0.436167 0.218083 0.975930i \(-0.430020\pi\)
0.218083 + 0.975930i \(0.430020\pi\)
\(338\) −15.7833 3.67896i −0.858496 0.200109i
\(339\) 0 0
\(340\) 0 0
\(341\) 3.23797i 0.175346i
\(342\) 0 0
\(343\) −9.97884 −0.538806
\(344\) −15.1755 + 18.4994i −0.818207 + 0.997420i
\(345\) 0 0
\(346\) 3.31903 14.2391i 0.178432 0.765499i
\(347\) 23.0279i 1.23620i −0.786098 0.618102i \(-0.787902\pi\)
0.786098 0.618102i \(-0.212098\pi\)
\(348\) 0 0
\(349\) 21.4380i 1.14755i 0.819012 + 0.573776i \(0.194522\pi\)
−0.819012 + 0.573776i \(0.805478\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.07936 + 2.30359i −0.270731 + 0.122782i
\(353\) 4.52323 0.240747 0.120374 0.992729i \(-0.461591\pi\)
0.120374 + 0.992729i \(0.461591\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 22.1560 + 10.9222i 1.17427 + 0.578878i
\(357\) 0 0
\(358\) 5.19265 22.2772i 0.274440 1.17739i
\(359\) −10.3510 −0.546303 −0.273152 0.961971i \(-0.588066\pi\)
−0.273152 + 0.961971i \(0.588066\pi\)
\(360\) 0 0
\(361\) 12.2144 0.642862
\(362\) −2.60492 + 11.1755i −0.136912 + 0.587370i
\(363\) 0 0
\(364\) 17.7438 35.9937i 0.930027 1.88658i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.485359 −0.0253355 −0.0126678 0.999920i \(-0.504032\pi\)
−0.0126678 + 0.999920i \(0.504032\pi\)
\(368\) −8.61665 11.2229i −0.449174 0.585032i
\(369\) 0 0
\(370\) 0 0
\(371\) 0.932371i 0.0484063i
\(372\) 0 0
\(373\) 30.0823i 1.55760i −0.627272 0.778800i \(-0.715829\pi\)
0.627272 0.778800i \(-0.284171\pi\)
\(374\) 1.43171 6.14222i 0.0740317 0.317606i
\(375\) 0 0
\(376\) −5.69641 4.67289i −0.293770 0.240986i
\(377\) 37.5591 1.93439
\(378\) 0 0
\(379\) 33.6881i 1.73044i 0.501392 + 0.865220i \(0.332821\pi\)
−0.501392 + 0.865220i \(0.667179\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 34.3211 + 8.00000i 1.75602 + 0.409316i
\(383\) 5.17545 0.264453 0.132227 0.991220i \(-0.457787\pi\)
0.132227 + 0.991220i \(0.457787\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.43171 + 0.333720i 0.0728719 + 0.0169859i
\(387\) 0 0
\(388\) −5.80850 2.86341i −0.294882 0.145368i
\(389\) 16.6408i 0.843722i 0.906660 + 0.421861i \(0.138623\pi\)
−0.906660 + 0.421861i \(0.861377\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 20.6862 + 16.9694i 1.04481 + 0.857082i
\(393\) 0 0
\(394\) −3.10170 + 13.3067i −0.156262 + 0.670384i
\(395\) 0 0
\(396\) 0 0
\(397\) 20.4332i 1.02551i −0.858534 0.512757i \(-0.828624\pi\)
0.858534 0.512757i \(-0.171376\pi\)
\(398\) 31.7698 + 7.40530i 1.59247 + 0.371194i
\(399\) 0 0
\(400\) 0 0
\(401\) 4.56829 0.228130 0.114065 0.993473i \(-0.463613\pi\)
0.114065 + 0.993473i \(0.463613\pi\)
\(402\) 0 0
\(403\) 16.2423i 0.809087i
\(404\) 3.85244 7.81477i 0.191666 0.388799i
\(405\) 0 0
\(406\) −9.89134 + 42.4352i −0.490899 + 2.10602i
\(407\) −0.932371 −0.0462159
\(408\) 0 0
\(409\) 19.8913 0.983563 0.491782 0.870719i \(-0.336346\pi\)
0.491782 + 0.870719i \(0.336346\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −27.0433 13.3315i −1.33233 0.656797i
\(413\) 36.9193i 1.81668i
\(414\) 0 0
\(415\) 0 0
\(416\) −25.4791 + 11.5553i −1.24921 + 0.566545i
\(417\) 0 0
\(418\) −3.53729 0.824517i −0.173015 0.0403284i
\(419\) 0.387288i 0.0189203i 0.999955 + 0.00946013i \(0.00301130\pi\)
−0.999955 + 0.00946013i \(0.996989\pi\)
\(420\) 0 0
\(421\) 12.0578i 0.587664i −0.955857 0.293832i \(-0.905069\pi\)
0.955857 0.293832i \(-0.0949306\pi\)
\(422\) −2.06797 + 8.87189i −0.100667 + 0.431877i
\(423\) 0 0
\(424\) 0.412259 0.502556i 0.0200210 0.0244063i
\(425\) 0 0
\(426\) 0 0
\(427\) 44.7019i 2.16328i
\(428\) −3.53729 + 7.17548i −0.170982 + 0.346840i
\(429\) 0 0
\(430\) 0 0
\(431\) −40.4068 −1.94633 −0.973164 0.230113i \(-0.926090\pi\)
−0.973164 + 0.230113i \(0.926090\pi\)
\(432\) 0 0
\(433\) 36.1859 1.73898 0.869491 0.493949i \(-0.164447\pi\)
0.869491 + 0.493949i \(0.164447\pi\)
\(434\) 18.3510 + 4.27748i 0.880875 + 0.205325i
\(435\) 0 0
\(436\) 3.68793 7.48105i 0.176620 0.358277i
\(437\) 9.21438i 0.440783i
\(438\) 0 0
\(439\) −25.4178 −1.21312 −0.606562 0.795036i \(-0.707452\pi\)
−0.606562 + 0.795036i \(0.707452\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.18172 30.8106i 0.341600 1.46551i
\(443\) 7.02792i 0.333907i 0.985965 + 0.166953i \(0.0533929\pi\)
−0.985965 + 0.166953i \(0.946607\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.7632 5.77213i −1.17257 0.273318i
\(447\) 0 0
\(448\) −6.34545 31.8300i −0.299794 1.50383i
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 0.560304i 0.0263837i
\(452\) 2.30560 + 1.13659i 0.108446 + 0.0534607i
\(453\) 0 0
\(454\) −2.25622 + 9.67951i −0.105890 + 0.454282i
\(455\) 0 0
\(456\) 0 0
\(457\) 25.2747 1.18230 0.591149 0.806562i \(-0.298674\pi\)
0.591149 + 0.806562i \(0.298674\pi\)
\(458\) −1.33883 + 5.74378i −0.0625595 + 0.268389i
\(459\) 0 0
\(460\) 0 0
\(461\) 41.0902i 1.91376i 0.290479 + 0.956881i \(0.406185\pi\)
−0.290479 + 0.956881i \(0.593815\pi\)
\(462\) 0 0
\(463\) −13.2106 −0.613951 −0.306975 0.951717i \(-0.599317\pi\)
−0.306975 + 0.951717i \(0.599317\pi\)
\(464\) 24.0947 18.4994i 1.11857 0.858813i
\(465\) 0 0
\(466\) 33.0404 + 7.70148i 1.53057 + 0.356764i
\(467\) 1.89134i 0.0875206i −0.999042 0.0437603i \(-0.986066\pi\)
0.999042 0.0437603i \(-0.0139338\pi\)
\(468\) 0 0
\(469\) 34.3211i 1.58480i
\(470\) 0 0
\(471\) 0 0
\(472\) −16.3243 + 19.8998i −0.751386 + 0.915963i
\(473\) −8.34068 −0.383505
\(474\) 0 0
\(475\) 0 0
\(476\) 32.9193 + 16.2282i 1.50885 + 0.743818i
\(477\) 0 0
\(478\) 12.2844 + 2.86341i 0.561877 + 0.130969i
\(479\) −31.4876 −1.43870 −0.719352 0.694646i \(-0.755561\pi\)
−0.719352 + 0.694646i \(0.755561\pi\)
\(480\) 0 0
\(481\) −4.67696 −0.213251
\(482\) −22.5201 5.24926i −1.02576 0.239097i
\(483\) 0 0
\(484\) 17.9888 + 8.86793i 0.817673 + 0.403088i
\(485\) 0 0
\(486\) 0 0
\(487\) 12.9964 0.588922 0.294461 0.955664i \(-0.404860\pi\)
0.294461 + 0.955664i \(0.404860\pi\)
\(488\) −19.7655 + 24.0947i −0.894741 + 1.09072i
\(489\) 0 0
\(490\) 0 0
\(491\) 14.9085i 0.672812i −0.941717 0.336406i \(-0.890788\pi\)
0.941717 0.336406i \(-0.109212\pi\)
\(492\) 0 0
\(493\) 34.3510i 1.54709i
\(494\) −17.7438 4.13594i −0.798330 0.186085i
\(495\) 0 0
\(496\) −8.00000 10.4197i −0.359211 0.467858i
\(497\) 5.80850 0.260547
\(498\) 0 0
\(499\) 35.6599i 1.59636i −0.602420 0.798179i \(-0.705797\pi\)
0.602420 0.798179i \(-0.294203\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.35603 + 5.81756i −0.0605226 + 0.259650i
\(503\) 25.3090 1.12847 0.564237 0.825613i \(-0.309170\pi\)
0.564237 + 0.825613i \(0.309170\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.11963 4.80338i 0.0497738 0.213536i
\(507\) 0 0
\(508\) −2.06797 1.01945i −0.0917514 0.0452306i
\(509\) 13.7366i 0.608862i 0.952534 + 0.304431i \(0.0984663\pi\)
−0.952534 + 0.304431i \(0.901534\pi\)
\(510\) 0 0
\(511\) −48.4846 −2.14483
\(512\) −10.6538 + 19.9624i −0.470835 + 0.882221i
\(513\) 0 0
\(514\) −34.0125 7.92806i −1.50023 0.349692i
\(515\) 0 0
\(516\) 0 0
\(517\) 2.56829i 0.112953i
\(518\) 1.23170 5.28415i 0.0541176 0.232172i
\(519\) 0 0
\(520\) 0 0
\(521\) 30.9193 1.35460 0.677299 0.735708i \(-0.263150\pi\)
0.677299 + 0.735708i \(0.263150\pi\)
\(522\) 0 0
\(523\) 21.8385i 0.954932i 0.878650 + 0.477466i \(0.158445\pi\)
−0.878650 + 0.477466i \(0.841555\pi\)
\(524\) −3.44018 + 6.97849i −0.150285 + 0.304857i
\(525\) 0 0
\(526\) −20.1949 4.70729i −0.880541 0.205248i
\(527\) 14.8550 0.647092
\(528\) 0 0
\(529\) −10.4876 −0.455981
\(530\) 0 0
\(531\) 0 0
\(532\) 9.34579 18.9582i 0.405191 0.821940i
\(533\) 2.81060i 0.121740i
\(534\) 0 0
\(535\) 0 0
\(536\) −15.1755 + 18.4994i −0.655481 + 0.799052i
\(537\) 0 0
\(538\) −3.70415 + 15.8913i −0.159697 + 0.685124i
\(539\) 9.32662i 0.401726i
\(540\) 0 0
\(541\) 24.3423i 1.04656i 0.852162 + 0.523278i \(0.175291\pi\)
−0.852162 + 0.523278i \(0.824709\pi\)
\(542\) 7.76120 + 1.80908i 0.333372 + 0.0777066i
\(543\) 0 0
\(544\) −10.5683 23.3028i −0.453112 0.999098i
\(545\) 0 0
\(546\) 0 0
\(547\) 33.3789i 1.42718i 0.700564 + 0.713589i \(0.252932\pi\)
−0.700564 + 0.713589i \(0.747068\pi\)
\(548\) 31.4169 + 15.4876i 1.34206 + 0.661596i
\(549\) 0 0
\(550\) 0 0
\(551\) 19.7827 0.842770
\(552\) 0 0
\(553\) 13.3239 0.566592
\(554\) 5.58774 23.9722i 0.237400 1.01848i
\(555\) 0 0
\(556\) 14.9108 30.2469i 0.632358 1.28275i
\(557\) 30.5808i 1.29575i −0.761747 0.647875i \(-0.775658\pi\)
0.761747 0.647875i \(-0.224342\pi\)
\(558\) 0 0
\(559\) −41.8385 −1.76958
\(560\) 0 0
\(561\) 0 0
\(562\) −30.0011 6.99304i −1.26552 0.294984i
\(563\) 7.02792i 0.296192i −0.988973 0.148096i \(-0.952686\pi\)
0.988973 0.148096i \(-0.0473144\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.93318 + 29.7443i −0.291423 + 1.25025i
\(567\) 0 0
\(568\) −3.13083 2.56829i −0.131367 0.107763i
\(569\) −24.3510 −1.02085 −0.510423 0.859924i \(-0.670511\pi\)
−0.510423 + 0.859924i \(0.670511\pi\)
\(570\) 0 0
\(571\) 20.0992i 0.841125i 0.907263 + 0.420563i \(0.138167\pi\)
−0.907263 + 0.420563i \(0.861833\pi\)
\(572\) −8.74714 4.31207i −0.365736 0.180297i
\(573\) 0 0
\(574\) 3.17548 + 0.740182i 0.132542 + 0.0308946i
\(575\) 0 0
\(576\) 0 0
\(577\) −4.76899 −0.198536 −0.0992678 0.995061i \(-0.531650\pi\)
−0.0992678 + 0.995061i \(0.531650\pi\)
\(578\) 4.76492 + 1.11067i 0.198195 + 0.0461977i
\(579\) 0 0
\(580\) 0 0
\(581\) 40.1296i 1.66486i
\(582\) 0 0
\(583\) 0.226584 0.00938413
\(584\) 26.1336 + 21.4380i 1.08142 + 0.887112i
\(585\) 0 0
\(586\) −10.2772 + 44.0906i −0.424547 + 1.82136i
\(587\) 8.21733i 0.339165i 0.985516 + 0.169583i \(0.0542420\pi\)
−0.985516 + 0.169583i \(0.945758\pi\)
\(588\) 0 0
\(589\) 8.55495i 0.352501i
\(590\) 0 0
\(591\) 0 0
\(592\) −3.00034 + 2.30359i −0.123313 + 0.0946771i
\(593\) 7.76120 0.318714 0.159357 0.987221i \(-0.449058\pi\)
0.159357 + 0.987221i \(0.449058\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.28415 18.8331i 0.380293 0.771434i
\(597\) 0 0
\(598\) 5.61631 24.0947i 0.229668 0.985307i
\(599\) 5.64903 0.230813 0.115407 0.993318i \(-0.463183\pi\)
0.115407 + 0.993318i \(0.463183\pi\)
\(600\) 0 0
\(601\) −37.7299 −1.53903 −0.769516 0.638627i \(-0.779503\pi\)
−0.769516 + 0.638627i \(0.779503\pi\)
\(602\) 11.0183 47.2702i 0.449074 1.92659i
\(603\) 0 0
\(604\) 8.45963 + 4.17034i 0.344217 + 0.169689i
\(605\) 0 0
\(606\) 0 0
\(607\) −0.113292 −0.00459837 −0.00229919 0.999997i \(-0.500732\pi\)
−0.00229919 + 0.999997i \(0.500732\pi\)
\(608\) −13.4200 + 6.08627i −0.544254 + 0.246831i
\(609\) 0 0
\(610\) 0 0
\(611\) 12.8831i 0.521194i
\(612\) 0 0
\(613\) 0.703366i 0.0284087i −0.999899 0.0142044i \(-0.995478\pi\)
0.999899 0.0142044i \(-0.00452154\pi\)
\(614\) −0.364887 + 1.56542i −0.0147256 + 0.0631750i
\(615\) 0 0
\(616\) 7.17548 8.74714i 0.289108 0.352432i
\(617\) −24.4809 −0.985564 −0.492782 0.870153i \(-0.664020\pi\)
−0.492782 + 0.870153i \(0.664020\pi\)
\(618\) 0 0
\(619\) 39.4966i 1.58750i 0.608243 + 0.793751i \(0.291874\pi\)
−0.608243 + 0.793751i \(0.708126\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −7.07459 1.64903i −0.283665 0.0661202i
\(623\) −50.1084 −2.00755
\(624\) 0 0
\(625\) 0 0
\(626\) 31.7827 + 7.40831i 1.27029 + 0.296095i
\(627\) 0 0
\(628\) −7.91086 + 16.0474i −0.315678 + 0.640360i
\(629\) 4.27748i 0.170554i
\(630\) 0 0
\(631\) 17.3400 0.690294 0.345147 0.938549i \(-0.387829\pi\)
0.345147 + 0.938549i \(0.387829\pi\)
\(632\) −7.18172 5.89134i −0.285674 0.234345i
\(633\) 0 0
\(634\) −0.533409 + 2.28840i −0.0211844 + 0.0908840i
\(635\) 0 0
\(636\) 0 0
\(637\) 46.7842i 1.85366i
\(638\) 10.3126 + 2.40378i 0.408278 + 0.0951666i
\(639\) 0 0
\(640\) 0 0
\(641\) −38.7019 −1.52863 −0.764317 0.644840i \(-0.776924\pi\)
−0.764317 + 0.644840i \(0.776924\pi\)
\(642\) 0 0
\(643\) 1.13659i 0.0448227i −0.999749 0.0224113i \(-0.992866\pi\)
0.999749 0.0224113i \(-0.00713435\pi\)
\(644\) 25.7438 + 12.6909i 1.01445 + 0.500091i
\(645\) 0 0
\(646\) 3.78267 16.2282i 0.148827 0.638490i
\(647\) −6.54868 −0.257455 −0.128728 0.991680i \(-0.541089\pi\)
−0.128728 + 0.991680i \(0.541089\pi\)
\(648\) 0 0
\(649\) −8.97208 −0.352185
\(650\) 0 0
\(651\) 0 0
\(652\) 13.9570 28.3121i 0.546598 1.10879i
\(653\) 8.74226i 0.342111i −0.985261 0.171056i \(-0.945282\pi\)
0.985261 0.171056i \(-0.0547178\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.38433 1.80304i −0.0540492 0.0703969i
\(657\) 0 0
\(658\) 14.5556 + 3.39281i 0.567438 + 0.132266i
\(659\) 35.5336i 1.38419i 0.721804 + 0.692097i \(0.243313\pi\)
−0.721804 + 0.692097i \(0.756687\pi\)
\(660\) 0 0
\(661\) 23.3028i 0.906373i −0.891416 0.453186i \(-0.850287\pi\)
0.891416 0.453186i \(-0.149713\pi\)
\(662\) −8.31732 + 35.6825i −0.323262 + 1.38684i
\(663\) 0 0
\(664\) 17.7438 21.6302i 0.688592 0.839415i
\(665\) 0 0
\(666\) 0 0
\(667\) 26.8634i 1.04016i
\(668\) −9.88274 4.87189i −0.382375 0.188499i
\(669\) 0 0
\(670\) 0 0
\(671\) −10.8634 −0.419377
\(672\) 0 0
\(673\) −14.3634 −0.553670 −0.276835 0.960917i \(-0.589286\pi\)
−0.276835 + 0.960917i \(0.589286\pi\)
\(674\) 11.0279 + 2.57053i 0.424780 + 0.0990130i
\(675\) 0 0
\(676\) −20.5571 10.1340i −0.790658 0.389770i
\(677\) 24.3076i 0.934217i 0.884200 + 0.467109i \(0.154704\pi\)
−0.884200 + 0.467109i \(0.845296\pi\)
\(678\) 0 0
\(679\) 13.1366 0.504136
\(680\) 0 0
\(681\) 0 0
\(682\) 1.03951 4.45963i 0.0398048 0.170768i
\(683\) 38.5933i 1.47673i −0.674401 0.738365i \(-0.735598\pi\)
0.674401 0.738365i \(-0.264402\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.7438 3.20357i −0.524740 0.122313i
\(687\) 0 0
\(688\) −26.8401 + 20.6072i −1.02327 + 0.785642i
\(689\) 1.13659 0.0433006
\(690\) 0 0
\(691\) 13.4090i 0.510102i 0.966928 + 0.255051i \(0.0820923\pi\)
−0.966928 + 0.255051i \(0.917908\pi\)
\(692\) 9.14256 18.5459i 0.347548 0.705009i
\(693\) 0 0
\(694\) 7.39281 31.7162i 0.280627 1.20393i
\(695\) 0 0
\(696\) 0 0
\(697\) 2.57053 0.0973657
\(698\) −6.88240 + 29.5264i −0.260503 + 1.11759i
\(699\) 0 0
\(700\) 0 0
\(701\) 15.6013i 0.589253i 0.955613 + 0.294626i \(0.0951952\pi\)
−0.955613 + 0.294626i \(0.904805\pi\)
\(702\) 0 0
\(703\) −2.46339 −0.0929086
\(704\) −7.73530 + 1.54206i −0.291535 + 0.0581187i
\(705\) 0 0
\(706\) 6.22982 + 1.45212i 0.234462 + 0.0546514i
\(707\) 17.6740i 0.664699i
\(708\) 0 0
\(709\) 15.7873i 0.592906i 0.955047 + 0.296453i \(0.0958038\pi\)
−0.955047 + 0.296453i \(0.904196\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 27.0089 + 22.1560i 1.01220 + 0.830333i
\(713\) 11.6170 0.435060
\(714\) 0 0
\(715\) 0 0
\(716\) 14.3036 29.0152i 0.534550 1.08435i
\(717\) 0 0
\(718\) −14.2563 3.32304i −0.532041 0.124015i
\(719\) −22.5683 −0.841655 −0.420828 0.907141i \(-0.638260\pi\)
−0.420828 + 0.907141i \(0.638260\pi\)
\(720\) 0 0
\(721\) 61.1616 2.27778
\(722\) 16.8228 + 3.92126i 0.626079 + 0.145934i
\(723\) 0 0
\(724\) −7.17548 + 14.5556i −0.266675 + 0.540956i
\(725\) 0 0
\(726\) 0 0
\(727\) 2.79096 0.103511 0.0517554 0.998660i \(-0.483518\pi\)
0.0517554 + 0.998660i \(0.483518\pi\)
\(728\) 35.9937 43.8774i 1.33401 1.62621i
\(729\) 0 0
\(730\) 0 0
\(731\) 38.2649i 1.41528i
\(732\) 0 0
\(733\) 16.4860i 0.608926i 0.952524 + 0.304463i \(0.0984769\pi\)
−0.952524 + 0.304463i \(0.901523\pi\)
\(734\) −0.668481 0.155818i −0.0246741 0.00575135i
\(735\) 0 0
\(736\) −8.26470 18.2234i −0.304641 0.671724i
\(737\) −8.34068 −0.307233
\(738\) 0 0
\(739\) 14.8894i 0.547714i 0.961770 + 0.273857i \(0.0882995\pi\)
−0.961770 + 0.273857i \(0.911701\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.299325 + 1.28415i −0.0109886 + 0.0471425i
\(743\) −41.4301 −1.51992 −0.759961 0.649968i \(-0.774782\pi\)
−0.759961 + 0.649968i \(0.774782\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 9.65751 41.4321i 0.353587 1.51694i
\(747\) 0 0
\(748\) 3.94376 8.00000i 0.144198 0.292509i
\(749\) 16.2282i 0.592965i
\(750\) 0 0
\(751\) 30.5544 1.11494 0.557472 0.830195i \(-0.311771\pi\)
0.557472 + 0.830195i \(0.311771\pi\)
\(752\) −6.34545 8.26470i −0.231395 0.301383i
\(753\) 0 0
\(754\) 51.7299 + 12.0578i 1.88389 + 0.439121i
\(755\) 0 0
\(756\) 0 0
\(757\) 0.433223i 0.0157457i 0.999969 + 0.00787287i \(0.00250604\pi\)
−0.999969 + 0.00787287i \(0.997494\pi\)
\(758\) −10.8151 + 46.3983i −0.392823 + 1.68526i
\(759\) 0 0
\(760\) 0 0
\(761\) 14.9193 0.540823 0.270411 0.962745i \(-0.412840\pi\)
0.270411 + 0.962745i \(0.412840\pi\)
\(762\) 0 0
\(763\) 16.9193i 0.612518i
\(764\) 44.7019 + 22.0367i 1.61726 + 0.797259i
\(765\) 0 0
\(766\) 7.12811 + 1.66151i 0.257549 + 0.0600328i
\(767\) −45.0057 −1.62506
\(768\) 0 0
\(769\) −31.3789 −1.13155 −0.565776 0.824559i \(-0.691423\pi\)
−0.565776 + 0.824559i \(0.691423\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.86474 + 0.919260i 0.0671135 + 0.0330849i
\(773\) 13.4192i 0.482656i 0.970444 + 0.241328i \(0.0775829\pi\)
−0.970444 + 0.241328i \(0.922417\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7.08074 5.80850i −0.254184 0.208513i
\(777\) 0 0
\(778\) −5.34231 + 22.9193i −0.191531 + 0.821695i
\(779\) 1.48036i 0.0530395i
\(780\) 0 0
\(781\) 1.41157i 0.0505101i
\(782\) 22.0367 + 5.13659i 0.788030 + 0.183684i
\(783\) 0 0
\(784\) 23.0431 + 30.0128i 0.822969 + 1.07189i
\(785\) 0 0
\(786\) 0 0
\(787\) 36.4846i 1.30054i 0.759705 + 0.650268i \(0.225343\pi\)
−0.759705 + 0.650268i \(0.774657\pi\)
\(788\) −8.54391 + 17.3315i −0.304364 + 0.617410i
\(789\) 0 0
\(790\) 0 0
\(791\) −5.21438 −0.185402
\(792\) 0 0
\(793\) −54.4931 −1.93511
\(794\) 6.55982 28.1425i 0.232799 0.998741i
\(795\) 0 0
\(796\) 41.3789 + 20.3985i 1.46664 + 0.723007i
\(797\) 26.0683i 0.923388i 0.887039 + 0.461694i \(0.152758\pi\)
−0.887039 + 0.461694i \(0.847242\pi\)
\(798\) 0 0
\(799\) 11.7827 0.416841
\(800\) 0 0
\(801\) 0 0
\(802\) 6.29188 + 1.46659i 0.222174 + 0.0517871i
\(803\) 11.7827i 0.415801i
\(804\) 0 0
\(805\) 0 0
\(806\) 5.21438 22.3704i 0.183669 0.787964i
\(807\) 0 0
\(808\) 7.81477 9.52645i 0.274923 0.335139i
\(809\) −35.6212 −1.25237 −0.626187 0.779673i \(-0.715385\pi\)
−0.626187 + 0.779673i \(0.715385\pi\)
\(810\) 0 0
\(811\) 43.8935i 1.54131i −0.637253 0.770654i \(-0.719929\pi\)
0.637253 0.770654i \(-0.280071\pi\)
\(812\) −27.2465 + 55.2702i −0.956166 + 1.93960i
\(813\) 0 0
\(814\) −1.28415 0.299325i −0.0450093 0.0104913i
\(815\) 0 0
\(816\) 0 0
\(817\) −22.0367 −0.770966
\(818\) 27.3962 + 6.38585i 0.957885 + 0.223276i
\(819\) 0 0
\(820\) 0 0
\(821\) 28.8058i 1.00533i −0.864482 0.502665i \(-0.832353\pi\)
0.864482 0.502665i \(-0.167647\pi\)
\(822\) 0 0
\(823\) 27.9585 0.974571 0.487286 0.873243i \(-0.337987\pi\)
0.487286 + 0.873243i \(0.337987\pi\)
\(824\) −32.9666 27.0433i −1.14845 0.942098i
\(825\) 0 0
\(826\) 11.8524 50.8486i 0.412399 1.76925i
\(827\) 14.8634i 0.516851i −0.966031 0.258426i \(-0.916796\pi\)
0.966031 0.258426i \(-0.0832037\pi\)
\(828\) 0 0
\(829\) 41.7678i 1.45065i −0.688404 0.725327i \(-0.741688\pi\)
0.688404 0.725327i \(-0.258312\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −38.8018 + 7.73530i −1.34521 + 0.268173i
\(833\) −42.7881 −1.48252
\(834\) 0 0
\(835\) 0 0
\(836\) −4.60719 2.27120i −0.159343 0.0785512i
\(837\) 0 0
\(838\) −0.124334 + 0.533409i −0.00429504 + 0.0184263i
\(839\) 21.6490 0.747408 0.373704 0.927548i \(-0.378088\pi\)
0.373704 + 0.927548i \(0.378088\pi\)
\(840\) 0 0
\(841\) −28.6740 −0.988759
\(842\) 3.87101 16.6072i 0.133404 0.572322i
\(843\) 0 0
\(844\) −5.69641 + 11.5553i −0.196078 + 0.397750i
\(845\) 0 0
\(846\) 0 0
\(847\) −40.6838 −1.39791
\(848\) 0.729140 0.559817i 0.0250388 0.0192242i
\(849\) 0 0
\(850\) 0 0
\(851\) 3.34510i 0.114669i
\(852\) 0 0
\(853\) 49.1880i 1.68416i −0.539350 0.842082i \(-0.681330\pi\)
0.539350 0.842082i \(-0.318670\pi\)
\(854\) 14.3510 61.5676i 0.491080 2.10680i
\(855\) 0 0
\(856\) −7.17548 + 8.74714i −0.245253 + 0.298971i
\(857\) 2.65849 0.0908123 0.0454062 0.998969i \(-0.485542\pi\)
0.0454062 + 0.998969i \(0.485542\pi\)
\(858\) 0 0
\(859\) 4.57680i 0.156158i −0.996947 0.0780792i \(-0.975121\pi\)
0.996947 0.0780792i \(-0.0248787\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −55.6520 12.9721i −1.89552 0.441831i
\(863\) 34.0218 1.15812 0.579058 0.815287i \(-0.303421\pi\)
0.579058 + 0.815287i \(0.303421\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 49.8385 + 11.6170i 1.69358 + 0.394761i
\(867\) 0 0
\(868\) 23.9014 + 11.7827i 0.811267 + 0.399930i
\(869\) 3.23797i 0.109841i
\(870\) 0 0
\(871\) −41.8385 −1.41764
\(872\) 7.48105 9.11963i 0.253340 0.308830i
\(873\) 0 0
\(874\) 2.95815 12.6909i 0.100061 0.429276i
\(875\) 0 0
\(876\) 0 0
\(877\) 11.7563i 0.396981i −0.980103 0.198490i \(-0.936396\pi\)
0.980103 0.198490i \(-0.0636039\pi\)
\(878\) −35.0077 8.16004i −1.18145 0.275388i
\(879\) 0 0
\(880\) 0 0
\(881\) −13.2702 −0.447085 −0.223543 0.974694i \(-0.571762\pi\)
−0.223543 + 0.974694i \(0.571762\pi\)
\(882\) 0 0
\(883\) 17.1366i 0.576692i −0.957526 0.288346i \(-0.906895\pi\)
0.957526 0.288346i \(-0.0931054\pi\)
\(884\) 19.7827 40.1296i 0.665363 1.34970i
\(885\) 0 0
\(886\) −2.25622 + 9.67951i −0.0757993 + 0.325189i
\(887\) 32.0883 1.07742 0.538709 0.842492i \(-0.318912\pi\)
0.538709 + 0.842492i \(0.318912\pi\)
\(888\) 0 0
\(889\) 4.67696 0.156860
\(890\) 0 0
\(891\) 0 0
\(892\) −32.2531 15.8998i −1.07992 0.532365i
\(893\) 6.78562i 0.227072i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.47908 45.8764i 0.0494125 1.53262i
\(897\) 0 0
\(898\) 2.75459 + 0.642074i 0.0919217 + 0.0214263i
\(899\) 24.9409i 0.831827i
\(900\) 0 0
\(901\) 1.03951i 0.0346310i
\(902\) 0.179878 0.771702i 0.00598929 0.0256949i
\(903\) 0 0
\(904\) 2.81060 + 2.30560i 0.0934790 + 0.0766830i
\(905\) 0 0
\(906\) 0 0
\(907\) 40.4596i 1.34344i −0.740805 0.671720i \(-0.765556\pi\)
0.740805 0.671720i \(-0.234444\pi\)
\(908\) −6.21496 + 12.6072i −0.206251 + 0.418384i
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 9.75225 0.322752
\(914\) 34.8106 + 8.11409i 1.15143 + 0.268390i
\(915\) 0 0
\(916\) −3.68793 + 7.48105i −0.121853 + 0.247181i
\(917\) 15.7827i 0.521190i
\(918\) 0 0
\(919\) 50.8495 1.67737 0.838685 0.544617i \(-0.183325\pi\)
0.838685 + 0.544617i \(0.183325\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −13.1915 + 56.5933i −0.434438 + 1.86380i
\(923\) 7.08074i 0.233065i
\(924\) 0 0
\(925\) 0 0
\(926\) −18.1949 4.24110i −0.597922 0.139371i
\(927\) 0 0
\(928\) 39.1245 17.7438i 1.28432 0.582468i
\(929\) −12.6461 −0.414904 −0.207452 0.978245i \(-0.566517\pi\)
−0.207452 + 0.978245i \(0.566517\pi\)
\(930\) 0 0
\(931\) 24.6416i 0.807596i
\(932\) 43.0339 + 21.2144i 1.40962 + 0.694900i
\(933\) 0 0
\(934\) 0.607188 2.60492i 0.0198678 0.0852357i
\(935\) 0 0
\(936\) 0 0
\(937\) −40.7971 −1.33278 −0.666391 0.745603i \(-0.732162\pi\)
−0.666391 + 0.745603i \(0.732162\pi\)
\(938\) 11.0183 47.2702i 0.359762 1.54343i
\(939\) 0 0
\(940\) 0 0
\(941\) 14.1086i 0.459928i −0.973199 0.229964i \(-0.926139\pi\)
0.973199 0.229964i \(-0.0738608\pi\)
\(942\) 0 0
\(943\) 2.01022 0.0654619
\(944\) −28.8719 + 22.1672i −0.939700 + 0.721480i
\(945\) 0 0
\(946\) −11.4876 2.67766i −0.373493 0.0870584i
\(947\) 45.8385i 1.48955i 0.667315 + 0.744776i \(0.267444\pi\)
−0.667315 + 0.744776i \(0.732556\pi\)
\(948\) 0 0
\(949\) 59.1043i 1.91861i
\(950\) 0 0
\(951\) 0 0
\(952\) 40.1296 + 32.9193i 1.30061 + 1.06692i
\(953\) 21.9104 0.709747 0.354873 0.934914i \(-0.384524\pi\)
0.354873 + 0.934914i \(0.384524\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 16.0000 + 7.88751i 0.517477 + 0.255100i
\(957\) 0 0
\(958\) −43.3676 10.1087i −1.40114 0.326596i
\(959\) −71.0529 −2.29442
\(960\) 0 0
\(961\) −20.2144 −0.652077
\(962\) −6.44154 1.50148i −0.207684 0.0484095i
\(963\) 0 0
\(964\) −29.3315 14.4595i −0.944705 0.465710i
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0359 0.451364 0.225682 0.974201i \(-0.427539\pi\)
0.225682 + 0.974201i \(0.427539\pi\)
\(968\) 21.9289 + 17.9888i 0.704822 + 0.578182i
\(969\) 0 0
\(970\) 0 0
\(971\) 21.6494i 0.694762i 0.937724 + 0.347381i \(0.112929\pi\)
−0.937724 + 0.347381i \(0.887071\pi\)
\(972\) 0 0
\(973\) 68.4068i 2.19302i
\(974\) 17.8998 + 4.17231i 0.573547 + 0.133690i
\(975\) 0 0
\(976\) −34.9582 + 26.8401i −1.11898 + 0.859130i
\(977\) −6.60225 −0.211225 −0.105612 0.994407i \(-0.533680\pi\)
−0.105612 + 0.994407i \(0.533680\pi\)
\(978\) 0 0
\(979\) 12.1773i 0.389188i
\(980\) 0 0
\(981\) 0 0
\(982\) 4.78619 20.5334i 0.152733 0.655247i
\(983\) 53.8600 1.71787 0.858934 0.512087i \(-0.171127\pi\)
0.858934 + 0.512087i \(0.171127\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −11.0279 + 47.3113i −0.351201 + 1.50670i
\(987\) 0 0
\(988\) −23.1106 11.3928i −0.735246 0.362454i
\(989\) 29.9242i 0.951534i
\(990\) 0 0
\(991\) 29.7129 0.943861 0.471931 0.881636i \(-0.343557\pi\)
0.471931 + 0.881636i \(0.343557\pi\)
\(992\) −7.67324 16.9193i −0.243626 0.537187i
\(993\) 0 0
\(994\) 8.00000 + 1.86474i 0.253745 + 0.0591460i
\(995\) 0 0
\(996\) 0 0
\(997\) 16.6506i 0.527328i −0.964615 0.263664i \(-0.915069\pi\)
0.964615 0.263664i \(-0.0849311\pi\)
\(998\) 11.4482 49.1142i 0.362385 1.55468i
\(999\) 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 1800.2.k.u.901.12 12
3.2 odd 2 600.2.k.f.301.1 12
4.3 odd 2 7200.2.k.u.3601.11 12
5.2 odd 4 360.2.d.e.109.4 6
5.3 odd 4 360.2.d.f.109.3 6
5.4 even 2 inner 1800.2.k.u.901.1 12
8.3 odd 2 7200.2.k.u.3601.12 12
8.5 even 2 inner 1800.2.k.u.901.11 12
12.11 even 2 2400.2.k.f.1201.12 12
15.2 even 4 120.2.d.b.109.3 yes 6
15.8 even 4 120.2.d.a.109.4 yes 6
15.14 odd 2 600.2.k.f.301.12 12
20.3 even 4 1440.2.d.e.1009.2 6
20.7 even 4 1440.2.d.f.1009.6 6
20.19 odd 2 7200.2.k.u.3601.1 12
24.5 odd 2 600.2.k.f.301.2 12
24.11 even 2 2400.2.k.f.1201.6 12
40.3 even 4 1440.2.d.f.1009.5 6
40.13 odd 4 360.2.d.e.109.3 6
40.19 odd 2 7200.2.k.u.3601.2 12
40.27 even 4 1440.2.d.e.1009.1 6
40.29 even 2 inner 1800.2.k.u.901.2 12
40.37 odd 4 360.2.d.f.109.4 6
60.23 odd 4 480.2.d.a.49.5 6
60.47 odd 4 480.2.d.b.49.1 6
60.59 even 2 2400.2.k.f.1201.1 12
120.29 odd 2 600.2.k.f.301.11 12
120.53 even 4 120.2.d.b.109.4 yes 6
120.59 even 2 2400.2.k.f.1201.7 12
120.77 even 4 120.2.d.a.109.3 6
120.83 odd 4 480.2.d.b.49.2 6
120.107 odd 4 480.2.d.a.49.6 6
240.53 even 4 3840.2.f.l.769.4 12
240.77 even 4 3840.2.f.l.769.3 12
240.83 odd 4 3840.2.f.m.769.3 12
240.107 odd 4 3840.2.f.m.769.4 12
240.173 even 4 3840.2.f.l.769.9 12
240.197 even 4 3840.2.f.l.769.10 12
240.203 odd 4 3840.2.f.m.769.10 12
240.227 odd 4 3840.2.f.m.769.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.d.a.109.3 6 120.77 even 4
120.2.d.a.109.4 yes 6 15.8 even 4
120.2.d.b.109.3 yes 6 15.2 even 4
120.2.d.b.109.4 yes 6 120.53 even 4
360.2.d.e.109.3 6 40.13 odd 4
360.2.d.e.109.4 6 5.2 odd 4
360.2.d.f.109.3 6 5.3 odd 4
360.2.d.f.109.4 6 40.37 odd 4
480.2.d.a.49.5 6 60.23 odd 4
480.2.d.a.49.6 6 120.107 odd 4
480.2.d.b.49.1 6 60.47 odd 4
480.2.d.b.49.2 6 120.83 odd 4
600.2.k.f.301.1 12 3.2 odd 2
600.2.k.f.301.2 12 24.5 odd 2
600.2.k.f.301.11 12 120.29 odd 2
600.2.k.f.301.12 12 15.14 odd 2
1440.2.d.e.1009.1 6 40.27 even 4
1440.2.d.e.1009.2 6 20.3 even 4
1440.2.d.f.1009.5 6 40.3 even 4
1440.2.d.f.1009.6 6 20.7 even 4
1800.2.k.u.901.1 12 5.4 even 2 inner
1800.2.k.u.901.2 12 40.29 even 2 inner
1800.2.k.u.901.11 12 8.5 even 2 inner
1800.2.k.u.901.12 12 1.1 even 1 trivial
2400.2.k.f.1201.1 12 60.59 even 2
2400.2.k.f.1201.6 12 24.11 even 2
2400.2.k.f.1201.7 12 120.59 even 2
2400.2.k.f.1201.12 12 12.11 even 2
3840.2.f.l.769.3 12 240.77 even 4
3840.2.f.l.769.4 12 240.53 even 4
3840.2.f.l.769.9 12 240.173 even 4
3840.2.f.l.769.10 12 240.197 even 4
3840.2.f.m.769.3 12 240.83 odd 4
3840.2.f.m.769.4 12 240.107 odd 4
3840.2.f.m.769.9 12 240.227 odd 4
3840.2.f.m.769.10 12 240.203 odd 4
7200.2.k.u.3601.1 12 20.19 odd 2
7200.2.k.u.3601.2 12 40.19 odd 2
7200.2.k.u.3601.11 12 4.3 odd 2
7200.2.k.u.3601.12 12 8.3 odd 2