Properties

Label 1500.2.m.c.301.6
Level $1500$
Weight $2$
Character 1500.301
Analytic conductor $11.978$
Analytic rank $0$
Dimension $24$
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] = [1500,2,Mod(301,1500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1500, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1500.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.m (of order \(5\), degree \(4\), 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: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 301.6
Character \(\chi\) \(=\) 1500.301
Dual form 1500.2.m.c.1201.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.309017 - 0.951057i) q^{3} +4.62675 q^{7} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})\) \(q+(0.309017 - 0.951057i) q^{3} +4.62675 q^{7} +(-0.809017 - 0.587785i) q^{9} +(4.00366 - 2.90883i) q^{11} +(3.04414 + 2.21170i) q^{13} +(-0.831378 - 2.55872i) q^{17} +(1.81426 + 5.58371i) q^{19} +(1.42974 - 4.40030i) q^{21} +(-5.40285 + 3.92540i) q^{23} +(-0.809017 + 0.587785i) q^{27} +(0.370972 - 1.14173i) q^{29} +(1.02048 + 3.14072i) q^{31} +(-1.52926 - 4.70659i) q^{33} +(-1.51873 - 1.10342i) q^{37} +(3.04414 - 2.21170i) q^{39} +(2.45366 + 1.78269i) q^{41} -10.6626 q^{43} +(0.0801015 - 0.246527i) q^{47} +14.4068 q^{49} -2.69040 q^{51} +(3.02731 - 9.31711i) q^{53} +5.87106 q^{57} +(7.78643 + 5.65717i) q^{59} +(-5.07552 + 3.68758i) q^{61} +(-3.74312 - 2.71953i) q^{63} +(0.791247 + 2.43521i) q^{67} +(2.06370 + 6.35143i) q^{69} +(2.68143 - 8.25259i) q^{71} +(-3.94381 + 2.86534i) q^{73} +(18.5239 - 13.4584i) q^{77} +(3.85443 - 11.8627i) q^{79} +(0.309017 + 0.951057i) q^{81} +(-2.74724 - 8.45513i) q^{83} +(-0.971218 - 0.705631i) q^{87} +(-11.7934 + 8.56841i) q^{89} +(14.0845 + 10.2330i) q^{91} +3.30235 q^{93} +(-1.23202 + 3.79176i) q^{97} -4.94880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{3} + 16 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 6 q^{3} + 16 q^{7} - 6 q^{9} - 6 q^{11} - 4 q^{17} - 10 q^{19} - 4 q^{21} - 14 q^{23} - 6 q^{27} - 4 q^{29} + 6 q^{31} + 4 q^{33} + 8 q^{37} - 10 q^{41} + 56 q^{43} - 26 q^{47} + 56 q^{49} + 16 q^{51} + 32 q^{53} + 20 q^{57} + 36 q^{59} - 12 q^{61} - 4 q^{63} - 36 q^{67} - 4 q^{69} + 40 q^{71} - 32 q^{73} + 46 q^{77} - 8 q^{79} - 6 q^{81} + 6 q^{83} - 4 q^{87} - 30 q^{91} - 4 q^{93} - 48 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(877\) \(1001\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(1\)

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
\(3\) 0.309017 0.951057i 0.178411 0.549093i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.62675 1.74875 0.874373 0.485254i \(-0.161273\pi\)
0.874373 + 0.485254i \(0.161273\pi\)
\(8\) 0 0
\(9\) −0.809017 0.587785i −0.269672 0.195928i
\(10\) 0 0
\(11\) 4.00366 2.90883i 1.20715 0.877045i 0.212180 0.977231i \(-0.431944\pi\)
0.994969 + 0.100185i \(0.0319435\pi\)
\(12\) 0 0
\(13\) 3.04414 + 2.21170i 0.844294 + 0.613415i 0.923567 0.383438i \(-0.125260\pi\)
−0.0792730 + 0.996853i \(0.525260\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.831378 2.55872i −0.201639 0.620580i −0.999835 0.0181824i \(-0.994212\pi\)
0.798196 0.602398i \(-0.205788\pi\)
\(18\) 0 0
\(19\) 1.81426 + 5.58371i 0.416219 + 1.28099i 0.911156 + 0.412062i \(0.135191\pi\)
−0.494937 + 0.868929i \(0.664809\pi\)
\(20\) 0 0
\(21\) 1.42974 4.40030i 0.311996 0.960224i
\(22\) 0 0
\(23\) −5.40285 + 3.92540i −1.12657 + 0.818502i −0.985192 0.171453i \(-0.945154\pi\)
−0.141379 + 0.989955i \(0.545154\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.809017 + 0.587785i −0.155695 + 0.113119i
\(28\) 0 0
\(29\) 0.370972 1.14173i 0.0688878 0.212015i −0.910686 0.413099i \(-0.864446\pi\)
0.979574 + 0.201084i \(0.0644464\pi\)
\(30\) 0 0
\(31\) 1.02048 + 3.14072i 0.183284 + 0.564090i 0.999915 0.0130708i \(-0.00416068\pi\)
−0.816631 + 0.577161i \(0.804161\pi\)
\(32\) 0 0
\(33\) −1.52926 4.70659i −0.266210 0.819311i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.51873 1.10342i −0.249677 0.181401i 0.455907 0.890028i \(-0.349315\pi\)
−0.705584 + 0.708627i \(0.749315\pi\)
\(38\) 0 0
\(39\) 3.04414 2.21170i 0.487453 0.354156i
\(40\) 0 0
\(41\) 2.45366 + 1.78269i 0.383198 + 0.278410i 0.762663 0.646797i \(-0.223892\pi\)
−0.379464 + 0.925206i \(0.623892\pi\)
\(42\) 0 0
\(43\) −10.6626 −1.62603 −0.813014 0.582244i \(-0.802175\pi\)
−0.813014 + 0.582244i \(0.802175\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.0801015 0.246527i 0.0116840 0.0359597i −0.945045 0.326942i \(-0.893982\pi\)
0.956729 + 0.290982i \(0.0939820\pi\)
\(48\) 0 0
\(49\) 14.4068 2.05811
\(50\) 0 0
\(51\) −2.69040 −0.376731
\(52\) 0 0
\(53\) 3.02731 9.31711i 0.415833 1.27980i −0.495670 0.868511i \(-0.665078\pi\)
0.911504 0.411292i \(-0.134922\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.87106 0.777641
\(58\) 0 0
\(59\) 7.78643 + 5.65717i 1.01371 + 0.736501i 0.964983 0.262311i \(-0.0844847\pi\)
0.0487233 + 0.998812i \(0.484485\pi\)
\(60\) 0 0
\(61\) −5.07552 + 3.68758i −0.649854 + 0.472147i −0.863222 0.504825i \(-0.831557\pi\)
0.213367 + 0.976972i \(0.431557\pi\)
\(62\) 0 0
\(63\) −3.74312 2.71953i −0.471589 0.342629i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.791247 + 2.43521i 0.0966662 + 0.297508i 0.987684 0.156460i \(-0.0500082\pi\)
−0.891018 + 0.453968i \(0.850008\pi\)
\(68\) 0 0
\(69\) 2.06370 + 6.35143i 0.248441 + 0.764622i
\(70\) 0 0
\(71\) 2.68143 8.25259i 0.318227 0.979403i −0.656178 0.754606i \(-0.727828\pi\)
0.974406 0.224797i \(-0.0721718\pi\)
\(72\) 0 0
\(73\) −3.94381 + 2.86534i −0.461588 + 0.335363i −0.794154 0.607717i \(-0.792086\pi\)
0.332566 + 0.943080i \(0.392086\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.5239 13.4584i 2.11100 1.53373i
\(78\) 0 0
\(79\) 3.85443 11.8627i 0.433657 1.33466i −0.460800 0.887504i \(-0.652437\pi\)
0.894457 0.447155i \(-0.147563\pi\)
\(80\) 0 0
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 0 0
\(83\) −2.74724 8.45513i −0.301549 0.928071i −0.980943 0.194298i \(-0.937757\pi\)
0.679394 0.733774i \(-0.262243\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.971218 0.705631i −0.104125 0.0756516i
\(88\) 0 0
\(89\) −11.7934 + 8.56841i −1.25010 + 0.908249i −0.998228 0.0595118i \(-0.981046\pi\)
−0.251870 + 0.967761i \(0.581046\pi\)
\(90\) 0 0
\(91\) 14.0845 + 10.2330i 1.47646 + 1.07271i
\(92\) 0 0
\(93\) 3.30235 0.342437
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.23202 + 3.79176i −0.125092 + 0.384995i −0.993918 0.110123i \(-0.964875\pi\)
0.868825 + 0.495119i \(0.164875\pi\)
\(98\) 0 0
\(99\) −4.94880 −0.497373
\(100\) 0 0
\(101\) 9.36896 0.932246 0.466123 0.884720i \(-0.345650\pi\)
0.466123 + 0.884720i \(0.345650\pi\)
\(102\) 0 0
\(103\) 3.22123 9.91391i 0.317397 0.976847i −0.657360 0.753577i \(-0.728327\pi\)
0.974757 0.223270i \(-0.0716731\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.220683 −0.0213342 −0.0106671 0.999943i \(-0.503396\pi\)
−0.0106671 + 0.999943i \(0.503396\pi\)
\(108\) 0 0
\(109\) −5.40941 3.93017i −0.518127 0.376442i 0.297771 0.954637i \(-0.403757\pi\)
−0.815898 + 0.578196i \(0.803757\pi\)
\(110\) 0 0
\(111\) −1.51873 + 1.10342i −0.144151 + 0.104732i
\(112\) 0 0
\(113\) −7.77355 5.64782i −0.731274 0.531302i 0.158692 0.987328i \(-0.449272\pi\)
−0.889966 + 0.456026i \(0.849272\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.16276 3.57861i −0.107497 0.330842i
\(118\) 0 0
\(119\) −3.84658 11.8385i −0.352615 1.08524i
\(120\) 0 0
\(121\) 4.16882 12.8303i 0.378984 1.16639i
\(122\) 0 0
\(123\) 2.45366 1.78269i 0.221239 0.160740i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.9380 + 9.40003i −1.14806 + 0.834118i −0.988222 0.153025i \(-0.951098\pi\)
−0.159842 + 0.987143i \(0.551098\pi\)
\(128\) 0 0
\(129\) −3.29492 + 10.1407i −0.290101 + 0.892840i
\(130\) 0 0
\(131\) −3.80795 11.7197i −0.332702 1.02395i −0.967843 0.251556i \(-0.919058\pi\)
0.635140 0.772397i \(-0.280942\pi\)
\(132\) 0 0
\(133\) 8.39411 + 25.8344i 0.727862 + 2.24013i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.60698 + 3.34717i 0.393601 + 0.285968i 0.766930 0.641731i \(-0.221784\pi\)
−0.373329 + 0.927699i \(0.621784\pi\)
\(138\) 0 0
\(139\) 14.5598 10.5783i 1.23495 0.897242i 0.237697 0.971339i \(-0.423607\pi\)
0.997251 + 0.0740969i \(0.0236074\pi\)
\(140\) 0 0
\(141\) −0.209709 0.152362i −0.0176606 0.0128312i
\(142\) 0 0
\(143\) 18.6212 1.55718
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.45195 13.7017i 0.367190 1.13010i
\(148\) 0 0
\(149\) −1.09001 −0.0892972 −0.0446486 0.999003i \(-0.514217\pi\)
−0.0446486 + 0.999003i \(0.514217\pi\)
\(150\) 0 0
\(151\) 11.3789 0.926004 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(152\) 0 0
\(153\) −0.831378 + 2.55872i −0.0672129 + 0.206860i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.98415 −0.317970 −0.158985 0.987281i \(-0.550822\pi\)
−0.158985 + 0.987281i \(0.550822\pi\)
\(158\) 0 0
\(159\) −7.92560 5.75829i −0.628541 0.456662i
\(160\) 0 0
\(161\) −24.9976 + 18.1618i −1.97009 + 1.43135i
\(162\) 0 0
\(163\) −17.6620 12.8322i −1.38339 1.00509i −0.996554 0.0829441i \(-0.973568\pi\)
−0.386837 0.922148i \(-0.626432\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.78015 14.7118i −0.369899 1.13843i −0.946856 0.321658i \(-0.895760\pi\)
0.576957 0.816775i \(-0.304240\pi\)
\(168\) 0 0
\(169\) 0.357976 + 1.10174i 0.0275366 + 0.0847490i
\(170\) 0 0
\(171\) 1.81426 5.58371i 0.138740 0.426997i
\(172\) 0 0
\(173\) −13.8911 + 10.0925i −1.05612 + 0.767317i −0.973367 0.229253i \(-0.926372\pi\)
−0.0827547 + 0.996570i \(0.526372\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.78643 5.65717i 0.585264 0.425219i
\(178\) 0 0
\(179\) −4.46392 + 13.7385i −0.333649 + 1.02687i 0.633734 + 0.773551i \(0.281521\pi\)
−0.967384 + 0.253316i \(0.918479\pi\)
\(180\) 0 0
\(181\) 3.83071 + 11.7897i 0.284734 + 0.876322i 0.986478 + 0.163893i \(0.0524051\pi\)
−0.701744 + 0.712429i \(0.747595\pi\)
\(182\) 0 0
\(183\) 1.93868 + 5.96664i 0.143311 + 0.441066i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.7714 7.82590i −0.787685 0.572287i
\(188\) 0 0
\(189\) −3.74312 + 2.71953i −0.272272 + 0.197817i
\(190\) 0 0
\(191\) 1.33930 + 0.973056i 0.0969081 + 0.0704078i 0.635184 0.772361i \(-0.280924\pi\)
−0.538276 + 0.842769i \(0.680924\pi\)
\(192\) 0 0
\(193\) 16.3253 1.17512 0.587560 0.809181i \(-0.300089\pi\)
0.587560 + 0.809181i \(0.300089\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.21264 + 12.9652i −0.300138 + 0.923730i 0.681309 + 0.731996i \(0.261411\pi\)
−0.981447 + 0.191734i \(0.938589\pi\)
\(198\) 0 0
\(199\) −6.07817 −0.430870 −0.215435 0.976518i \(-0.569117\pi\)
−0.215435 + 0.976518i \(0.569117\pi\)
\(200\) 0 0
\(201\) 2.56053 0.180606
\(202\) 0 0
\(203\) 1.71639 5.28252i 0.120467 0.370760i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.67829 0.464173
\(208\) 0 0
\(209\) 23.5057 + 17.0779i 1.62593 + 1.18130i
\(210\) 0 0
\(211\) −13.8200 + 10.0408i −0.951409 + 0.691239i −0.951140 0.308761i \(-0.900086\pi\)
−0.000268984 1.00000i \(0.500086\pi\)
\(212\) 0 0
\(213\) −7.02007 5.10038i −0.481008 0.349472i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.72151 + 14.5313i 0.320517 + 0.986450i
\(218\) 0 0
\(219\) 1.50640 + 4.63622i 0.101793 + 0.313287i
\(220\) 0 0
\(221\) 3.12828 9.62787i 0.210431 0.647640i
\(222\) 0 0
\(223\) −0.634100 + 0.460700i −0.0424624 + 0.0308508i −0.608814 0.793313i \(-0.708354\pi\)
0.566352 + 0.824164i \(0.308354\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.3755 + 14.0771i −1.28600 + 0.934331i −0.999716 0.0238153i \(-0.992419\pi\)
−0.286280 + 0.958146i \(0.592419\pi\)
\(228\) 0 0
\(229\) −1.71005 + 5.26299i −0.113003 + 0.347788i −0.991525 0.129914i \(-0.958530\pi\)
0.878522 + 0.477702i \(0.158530\pi\)
\(230\) 0 0
\(231\) −7.07551 21.7762i −0.465535 1.43277i
\(232\) 0 0
\(233\) 0.280508 + 0.863314i 0.0183767 + 0.0565576i 0.959824 0.280602i \(-0.0905340\pi\)
−0.941448 + 0.337159i \(0.890534\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.0910 7.33156i −0.655482 0.476236i
\(238\) 0 0
\(239\) 9.09227 6.60592i 0.588130 0.427302i −0.253516 0.967331i \(-0.581587\pi\)
0.841646 + 0.540030i \(0.181587\pi\)
\(240\) 0 0
\(241\) 16.3840 + 11.9037i 1.05539 + 0.766783i 0.973229 0.229837i \(-0.0738193\pi\)
0.0821564 + 0.996619i \(0.473819\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.82663 + 21.0102i −0.434368 + 1.33685i
\(248\) 0 0
\(249\) −8.89025 −0.563397
\(250\) 0 0
\(251\) −30.6919 −1.93725 −0.968627 0.248520i \(-0.920056\pi\)
−0.968627 + 0.248520i \(0.920056\pi\)
\(252\) 0 0
\(253\) −10.2129 + 31.4319i −0.642077 + 1.97611i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.77543 0.297883 0.148941 0.988846i \(-0.452413\pi\)
0.148941 + 0.988846i \(0.452413\pi\)
\(258\) 0 0
\(259\) −7.02676 5.10524i −0.436622 0.317224i
\(260\) 0 0
\(261\) −0.971218 + 0.705631i −0.0601169 + 0.0436775i
\(262\) 0 0
\(263\) −0.978638 0.711022i −0.0603454 0.0438435i 0.557204 0.830376i \(-0.311874\pi\)
−0.617549 + 0.786532i \(0.711874\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.50468 + 13.8640i 0.275682 + 0.848461i
\(268\) 0 0
\(269\) −7.88763 24.2756i −0.480917 1.48011i −0.837808 0.545965i \(-0.816163\pi\)
0.356891 0.934146i \(-0.383837\pi\)
\(270\) 0 0
\(271\) −1.81499 + 5.58596i −0.110253 + 0.339323i −0.990927 0.134399i \(-0.957090\pi\)
0.880675 + 0.473722i \(0.157090\pi\)
\(272\) 0 0
\(273\) 14.0845 10.2330i 0.852432 0.619328i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.96855 2.88332i 0.238447 0.173242i −0.462144 0.886805i \(-0.652920\pi\)
0.700591 + 0.713563i \(0.252920\pi\)
\(278\) 0 0
\(279\) 1.02048 3.14072i 0.0610946 0.188030i
\(280\) 0 0
\(281\) −0.400257 1.23186i −0.0238773 0.0734868i 0.938408 0.345530i \(-0.112301\pi\)
−0.962285 + 0.272043i \(0.912301\pi\)
\(282\) 0 0
\(283\) 9.34149 + 28.7501i 0.555294 + 1.70902i 0.695166 + 0.718849i \(0.255331\pi\)
−0.139873 + 0.990169i \(0.544669\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.3525 + 8.24807i 0.670116 + 0.486868i
\(288\) 0 0
\(289\) 7.89744 5.73783i 0.464555 0.337519i
\(290\) 0 0
\(291\) 3.22546 + 2.34344i 0.189080 + 0.137375i
\(292\) 0 0
\(293\) 8.06831 0.471356 0.235678 0.971831i \(-0.424269\pi\)
0.235678 + 0.971831i \(0.424269\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.52926 + 4.70659i −0.0887368 + 0.273104i
\(298\) 0 0
\(299\) −25.1289 −1.45324
\(300\) 0 0
\(301\) −49.3331 −2.84351
\(302\) 0 0
\(303\) 2.89517 8.91041i 0.166323 0.511890i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −14.7750 −0.843255 −0.421628 0.906769i \(-0.638541\pi\)
−0.421628 + 0.906769i \(0.638541\pi\)
\(308\) 0 0
\(309\) −8.43328 6.12713i −0.479752 0.348560i
\(310\) 0 0
\(311\) 7.40552 5.38043i 0.419929 0.305096i −0.357680 0.933844i \(-0.616432\pi\)
0.777609 + 0.628748i \(0.216432\pi\)
\(312\) 0 0
\(313\) 10.7238 + 7.79128i 0.606144 + 0.440389i 0.848054 0.529909i \(-0.177774\pi\)
−0.241910 + 0.970299i \(0.577774\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.244293 0.751858i −0.0137209 0.0422286i 0.943962 0.330055i \(-0.107067\pi\)
−0.957683 + 0.287827i \(0.907067\pi\)
\(318\) 0 0
\(319\) −1.83587 5.65021i −0.102789 0.316351i
\(320\) 0 0
\(321\) −0.0681947 + 0.209882i −0.00380626 + 0.0117145i
\(322\) 0 0
\(323\) 12.7788 9.28434i 0.711032 0.516595i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −5.40941 + 3.93017i −0.299141 + 0.217339i
\(328\) 0 0
\(329\) 0.370610 1.14062i 0.0204324 0.0628844i
\(330\) 0 0
\(331\) −1.04253 3.20858i −0.0573026 0.176359i 0.918308 0.395866i \(-0.129555\pi\)
−0.975611 + 0.219506i \(0.929555\pi\)
\(332\) 0 0
\(333\) 0.580102 + 1.78537i 0.0317894 + 0.0978376i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.6157 + 10.6189i 0.796166 + 0.578449i 0.909787 0.415076i \(-0.136245\pi\)
−0.113621 + 0.993524i \(0.536245\pi\)
\(338\) 0 0
\(339\) −7.77355 + 5.64782i −0.422201 + 0.306747i
\(340\) 0 0
\(341\) 13.2215 + 9.60597i 0.715983 + 0.520192i
\(342\) 0 0
\(343\) 34.2694 1.85037
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.943191 + 2.90284i −0.0506332 + 0.155833i −0.973176 0.230062i \(-0.926107\pi\)
0.922543 + 0.385895i \(0.126107\pi\)
\(348\) 0 0
\(349\) −0.628744 −0.0336559 −0.0168280 0.999858i \(-0.505357\pi\)
−0.0168280 + 0.999858i \(0.505357\pi\)
\(350\) 0 0
\(351\) −3.76277 −0.200842
\(352\) 0 0
\(353\) 5.83721 17.9651i 0.310683 0.956184i −0.666812 0.745226i \(-0.732342\pi\)
0.977495 0.210958i \(-0.0676584\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −12.4478 −0.658807
\(358\) 0 0
\(359\) −23.4087 17.0074i −1.23546 0.897617i −0.238176 0.971222i \(-0.576550\pi\)
−0.997287 + 0.0736053i \(0.976550\pi\)
\(360\) 0 0
\(361\) −12.5149 + 9.09264i −0.658681 + 0.478560i
\(362\) 0 0
\(363\) −10.9141 7.92957i −0.572843 0.416195i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.66342 5.11949i −0.0868299 0.267235i 0.898209 0.439570i \(-0.144869\pi\)
−0.985038 + 0.172334i \(0.944869\pi\)
\(368\) 0 0
\(369\) −0.937217 2.88446i −0.0487895 0.150159i
\(370\) 0 0
\(371\) 14.0066 43.1079i 0.727187 2.23805i
\(372\) 0 0
\(373\) −8.89714 + 6.46415i −0.460677 + 0.334701i −0.793797 0.608183i \(-0.791899\pi\)
0.333120 + 0.942884i \(0.391899\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.65447 2.65513i 0.188215 0.136746i
\(378\) 0 0
\(379\) 2.18405 6.72183i 0.112187 0.345277i −0.879163 0.476522i \(-0.841897\pi\)
0.991350 + 0.131245i \(0.0418974\pi\)
\(380\) 0 0
\(381\) 4.94189 + 15.2096i 0.253181 + 0.779210i
\(382\) 0 0
\(383\) −2.50346 7.70484i −0.127921 0.393699i 0.866501 0.499175i \(-0.166363\pi\)
−0.994422 + 0.105476i \(0.966363\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.62621 + 6.26731i 0.438495 + 0.318585i
\(388\) 0 0
\(389\) −26.7325 + 19.4223i −1.35539 + 0.984750i −0.356669 + 0.934231i \(0.616088\pi\)
−0.998723 + 0.0505192i \(0.983912\pi\)
\(390\) 0 0
\(391\) 14.5358 + 10.5609i 0.735107 + 0.534086i
\(392\) 0 0
\(393\) −12.3228 −0.621603
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.82155 11.7615i 0.191798 0.590294i −0.808201 0.588907i \(-0.799559\pi\)
0.999999 0.00138688i \(-0.000441458\pi\)
\(398\) 0 0
\(399\) 27.1639 1.35990
\(400\) 0 0
\(401\) −14.7983 −0.738993 −0.369496 0.929232i \(-0.620470\pi\)
−0.369496 + 0.929232i \(0.620470\pi\)
\(402\) 0 0
\(403\) −3.83984 + 11.8178i −0.191276 + 0.588687i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.29012 −0.460494
\(408\) 0 0
\(409\) 19.1618 + 13.9219i 0.947491 + 0.688392i 0.950212 0.311604i \(-0.100866\pi\)
−0.00272132 + 0.999996i \(0.500866\pi\)
\(410\) 0 0
\(411\) 4.60698 3.34717i 0.227246 0.165104i
\(412\) 0 0
\(413\) 36.0258 + 26.1743i 1.77272 + 1.28795i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.56136 17.1161i −0.272341 0.838179i
\(418\) 0 0
\(419\) 0.477059 + 1.46824i 0.0233058 + 0.0717280i 0.962033 0.272933i \(-0.0879938\pi\)
−0.938727 + 0.344661i \(0.887994\pi\)
\(420\) 0 0
\(421\) −5.43760 + 16.7352i −0.265013 + 0.815625i 0.726678 + 0.686978i \(0.241063\pi\)
−0.991691 + 0.128647i \(0.958937\pi\)
\(422\) 0 0
\(423\) −0.209709 + 0.152362i −0.0101964 + 0.00740810i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −23.4832 + 17.0615i −1.13643 + 0.825665i
\(428\) 0 0
\(429\) 5.75426 17.7098i 0.277818 0.855037i
\(430\) 0 0
\(431\) 4.83527 + 14.8814i 0.232907 + 0.716814i 0.997392 + 0.0721724i \(0.0229932\pi\)
−0.764485 + 0.644641i \(0.777007\pi\)
\(432\) 0 0
\(433\) 0.0707586 + 0.217772i 0.00340044 + 0.0104655i 0.952742 0.303780i \(-0.0982486\pi\)
−0.949342 + 0.314245i \(0.898249\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −31.7204 23.0462i −1.51739 1.10245i
\(438\) 0 0
\(439\) −15.7111 + 11.4147i −0.749848 + 0.544796i −0.895780 0.444498i \(-0.853382\pi\)
0.145932 + 0.989295i \(0.453382\pi\)
\(440\) 0 0
\(441\) −11.6553 8.46810i −0.555016 0.403243i
\(442\) 0 0
\(443\) −12.7980 −0.608051 −0.304026 0.952664i \(-0.598331\pi\)
−0.304026 + 0.952664i \(0.598331\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.336832 + 1.03666i −0.0159316 + 0.0490325i
\(448\) 0 0
\(449\) −21.1499 −0.998124 −0.499062 0.866566i \(-0.666322\pi\)
−0.499062 + 0.866566i \(0.666322\pi\)
\(450\) 0 0
\(451\) 15.0092 0.706755
\(452\) 0 0
\(453\) 3.51628 10.8220i 0.165209 0.508462i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.97118 −0.419654 −0.209827 0.977739i \(-0.567290\pi\)
−0.209827 + 0.977739i \(0.567290\pi\)
\(458\) 0 0
\(459\) 2.17658 + 1.58137i 0.101594 + 0.0738123i
\(460\) 0 0
\(461\) −22.2764 + 16.1847i −1.03751 + 0.753797i −0.969799 0.243907i \(-0.921571\pi\)
−0.0677146 + 0.997705i \(0.521571\pi\)
\(462\) 0 0
\(463\) −23.7859 17.2815i −1.10543 0.803140i −0.123489 0.992346i \(-0.539409\pi\)
−0.981937 + 0.189206i \(0.939409\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.09785 + 12.6119i 0.189626 + 0.583609i 0.999997 0.00230768i \(-0.000734558\pi\)
−0.810371 + 0.585917i \(0.800735\pi\)
\(468\) 0 0
\(469\) 3.66090 + 11.2671i 0.169045 + 0.520266i
\(470\) 0 0
\(471\) −1.23117 + 3.78915i −0.0567293 + 0.174595i
\(472\) 0 0
\(473\) −42.6893 + 31.0156i −1.96286 + 1.42610i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −7.92560 + 5.75829i −0.362888 + 0.263654i
\(478\) 0 0
\(479\) −5.58057 + 17.1752i −0.254983 + 0.784757i 0.738850 + 0.673870i \(0.235369\pi\)
−0.993833 + 0.110887i \(0.964631\pi\)
\(480\) 0 0
\(481\) −2.18279 6.71793i −0.0995266 0.306311i
\(482\) 0 0
\(483\) 9.54824 + 29.3865i 0.434460 + 1.33713i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −11.3725 8.26258i −0.515336 0.374413i 0.299508 0.954094i \(-0.403177\pi\)
−0.814844 + 0.579681i \(0.803177\pi\)
\(488\) 0 0
\(489\) −17.6620 + 12.8322i −0.798701 + 0.580290i
\(490\) 0 0
\(491\) 22.0310 + 16.0065i 0.994247 + 0.722363i 0.960847 0.277079i \(-0.0893665\pi\)
0.0334000 + 0.999442i \(0.489366\pi\)
\(492\) 0 0
\(493\) −3.22980 −0.145463
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.4063 38.1827i 0.556499 1.71273i
\(498\) 0 0
\(499\) −8.17654 −0.366032 −0.183016 0.983110i \(-0.558586\pi\)
−0.183016 + 0.983110i \(0.558586\pi\)
\(500\) 0 0
\(501\) −15.4689 −0.691099
\(502\) 0 0
\(503\) 3.07311 9.45805i 0.137023 0.421714i −0.858876 0.512183i \(-0.828837\pi\)
0.995899 + 0.0904697i \(0.0288368\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.15844 0.0514479
\(508\) 0 0
\(509\) −13.9758 10.1540i −0.619464 0.450067i 0.233270 0.972412i \(-0.425057\pi\)
−0.852734 + 0.522345i \(0.825057\pi\)
\(510\) 0 0
\(511\) −18.2470 + 13.2572i −0.807200 + 0.586465i
\(512\) 0 0
\(513\) −4.74979 3.45092i −0.209708 0.152362i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.396406 1.22001i −0.0174339 0.0536561i
\(518\) 0 0
\(519\) 5.30593 + 16.3300i 0.232905 + 0.716807i
\(520\) 0 0
\(521\) −3.43786 + 10.5807i −0.150615 + 0.463547i −0.997690 0.0679269i \(-0.978362\pi\)
0.847075 + 0.531474i \(0.178362\pi\)
\(522\) 0 0
\(523\) 3.03609 2.20585i 0.132759 0.0964551i −0.519424 0.854517i \(-0.673853\pi\)
0.652183 + 0.758062i \(0.273853\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.18781 5.22225i 0.313106 0.227485i
\(528\) 0 0
\(529\) 6.67462 20.5424i 0.290201 0.893146i
\(530\) 0 0
\(531\) −2.97415 9.15350i −0.129067 0.397228i
\(532\) 0 0
\(533\) 3.52653 + 10.8535i 0.152751 + 0.470119i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 11.6867 + 8.49089i 0.504318 + 0.366409i
\(538\) 0 0
\(539\) 57.6799 41.9069i 2.48445 1.80506i
\(540\) 0 0
\(541\) −15.2752 11.0981i −0.656731 0.477143i 0.208826 0.977953i \(-0.433036\pi\)
−0.865557 + 0.500810i \(0.833036\pi\)
\(542\) 0 0
\(543\) 12.3964 0.531982
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.0479926 0.147706i 0.00205201 0.00631545i −0.950025 0.312173i \(-0.898943\pi\)
0.952077 + 0.305858i \(0.0989431\pi\)
\(548\) 0 0
\(549\) 6.27369 0.267755
\(550\) 0 0
\(551\) 7.04815 0.300261
\(552\) 0 0
\(553\) 17.8335 54.8858i 0.758356 2.33398i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19.0371 −0.806626 −0.403313 0.915062i \(-0.632141\pi\)
−0.403313 + 0.915062i \(0.632141\pi\)
\(558\) 0 0
\(559\) −32.4584 23.5824i −1.37285 0.997430i
\(560\) 0 0
\(561\) −10.7714 + 7.82590i −0.454770 + 0.330410i
\(562\) 0 0
\(563\) 25.1369 + 18.2630i 1.05939 + 0.769694i 0.973976 0.226652i \(-0.0727779\pi\)
0.0854166 + 0.996345i \(0.472778\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.42974 + 4.40030i 0.0600436 + 0.184795i
\(568\) 0 0
\(569\) 12.1573 + 37.4162i 0.509659 + 1.56857i 0.792794 + 0.609490i \(0.208626\pi\)
−0.283135 + 0.959080i \(0.591374\pi\)
\(570\) 0 0
\(571\) 8.78632 27.0415i 0.367696 1.13165i −0.580580 0.814203i \(-0.697174\pi\)
0.948276 0.317448i \(-0.102826\pi\)
\(572\) 0 0
\(573\) 1.33930 0.973056i 0.0559499 0.0406500i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.1701 13.2014i 0.756433 0.549581i −0.141381 0.989955i \(-0.545154\pi\)
0.897814 + 0.440375i \(0.145154\pi\)
\(578\) 0 0
\(579\) 5.04479 15.5263i 0.209654 0.645250i
\(580\) 0 0
\(581\) −12.7108 39.1198i −0.527332 1.62296i
\(582\) 0 0
\(583\) −14.9815 46.1085i −0.620472 1.90962i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.0114 + 9.45335i 0.537039 + 0.390182i 0.822984 0.568064i \(-0.192308\pi\)
−0.285945 + 0.958246i \(0.592308\pi\)
\(588\) 0 0
\(589\) −15.6854 + 11.3961i −0.646307 + 0.469570i
\(590\) 0 0
\(591\) 11.0288 + 8.01291i 0.453665 + 0.329607i
\(592\) 0 0
\(593\) 20.4648 0.840389 0.420194 0.907434i \(-0.361962\pi\)
0.420194 + 0.907434i \(0.361962\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.87826 + 5.78068i −0.0768720 + 0.236588i
\(598\) 0 0
\(599\) 18.5688 0.758699 0.379349 0.925253i \(-0.376148\pi\)
0.379349 + 0.925253i \(0.376148\pi\)
\(600\) 0 0
\(601\) 47.2047 1.92552 0.962761 0.270355i \(-0.0871409\pi\)
0.962761 + 0.270355i \(0.0871409\pi\)
\(602\) 0 0
\(603\) 0.791247 2.43521i 0.0322221 0.0991693i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0.0786576 0.00319261 0.00159631 0.999999i \(-0.499492\pi\)
0.00159631 + 0.999999i \(0.499492\pi\)
\(608\) 0 0
\(609\) −4.49358 3.26478i −0.182089 0.132295i
\(610\) 0 0
\(611\) 0.789085 0.573304i 0.0319230 0.0231934i
\(612\) 0 0
\(613\) −4.29654 3.12162i −0.173536 0.126081i 0.497627 0.867391i \(-0.334205\pi\)
−0.671163 + 0.741310i \(0.734205\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.40411 + 28.9429i 0.378595 + 1.16520i 0.941021 + 0.338349i \(0.109868\pi\)
−0.562426 + 0.826848i \(0.690132\pi\)
\(618\) 0 0
\(619\) −9.77918 30.0972i −0.393058 1.20971i −0.930463 0.366385i \(-0.880595\pi\)
0.537405 0.843324i \(-0.319405\pi\)
\(620\) 0 0
\(621\) 2.06370 6.35143i 0.0828136 0.254874i
\(622\) 0 0
\(623\) −54.5651 + 39.6439i −2.18610 + 1.58830i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 23.5057 17.0779i 0.938728 0.682026i
\(628\) 0 0
\(629\) −1.56070 + 4.80335i −0.0622293 + 0.191522i
\(630\) 0 0
\(631\) −10.4722 32.2301i −0.416891 1.28306i −0.910548 0.413403i \(-0.864340\pi\)
0.493657 0.869657i \(-0.335660\pi\)
\(632\) 0 0
\(633\) 5.27877 + 16.2464i 0.209812 + 0.645736i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 43.8564 + 31.8635i 1.73765 + 1.26248i
\(638\) 0 0
\(639\) −7.02007 + 5.10038i −0.277710 + 0.201768i
\(640\) 0 0
\(641\) 39.6699 + 28.8219i 1.56687 + 1.13840i 0.930080 + 0.367358i \(0.119737\pi\)
0.636788 + 0.771039i \(0.280263\pi\)
\(642\) 0 0
\(643\) −14.2509 −0.562000 −0.281000 0.959708i \(-0.590666\pi\)
−0.281000 + 0.959708i \(0.590666\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.0820 40.2623i 0.514307 1.58287i −0.270233 0.962795i \(-0.587101\pi\)
0.784539 0.620079i \(-0.212899\pi\)
\(648\) 0 0
\(649\) 47.6300 1.86964
\(650\) 0 0
\(651\) 15.2791 0.598836
\(652\) 0 0
\(653\) 14.5328 44.7274i 0.568713 1.75032i −0.0879407 0.996126i \(-0.528029\pi\)
0.656654 0.754192i \(-0.271971\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.87481 0.190185
\(658\) 0 0
\(659\) 9.05013 + 6.57531i 0.352543 + 0.256138i 0.749935 0.661511i \(-0.230085\pi\)
−0.397392 + 0.917649i \(0.630085\pi\)
\(660\) 0 0
\(661\) 37.7102 27.3981i 1.46676 1.06566i 0.485218 0.874393i \(-0.338740\pi\)
0.981538 0.191268i \(-0.0612599\pi\)
\(662\) 0 0
\(663\) −8.18995 5.95035i −0.318071 0.231092i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.47746 + 7.62483i 0.0959276 + 0.295235i
\(668\) 0 0
\(669\) 0.242204 + 0.745429i 0.00936417 + 0.0288199i
\(670\) 0 0
\(671\) −9.59412 + 29.5277i −0.370377 + 1.13990i
\(672\) 0 0
\(673\) 11.1110 8.07264i 0.428299 0.311178i −0.352669 0.935748i \(-0.614726\pi\)
0.780968 + 0.624570i \(0.214726\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.7888 + 21.6428i −1.14488 + 0.831801i −0.987791 0.155784i \(-0.950210\pi\)
−0.157085 + 0.987585i \(0.550210\pi\)
\(678\) 0 0
\(679\) −5.70024 + 17.5435i −0.218755 + 0.673259i
\(680\) 0 0
\(681\) 7.40078 + 22.7772i 0.283598 + 0.872826i
\(682\) 0 0
\(683\) −9.29850 28.6179i −0.355797 1.09503i −0.955546 0.294843i \(-0.904733\pi\)
0.599748 0.800189i \(-0.295267\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.47697 + 3.25271i 0.170807 + 0.124098i
\(688\) 0 0
\(689\) 29.8222 21.6671i 1.13614 0.825451i
\(690\) 0 0
\(691\) 6.76839 + 4.91752i 0.257482 + 0.187071i 0.709036 0.705172i \(-0.249130\pi\)
−0.451554 + 0.892244i \(0.649130\pi\)
\(692\) 0 0
\(693\) −22.8968 −0.869779
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.52148 7.76033i 0.0955080 0.293943i
\(698\) 0 0
\(699\) 0.907742 0.0343340
\(700\) 0 0
\(701\) −3.42495 −0.129359 −0.0646794 0.997906i \(-0.520602\pi\)
−0.0646794 + 0.997906i \(0.520602\pi\)
\(702\) 0 0
\(703\) 3.40581 10.4820i 0.128453 0.395336i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 43.3478 1.63026
\(708\) 0 0
\(709\) 40.0330 + 29.0857i 1.50347 + 1.09234i 0.968973 + 0.247168i \(0.0794998\pi\)
0.534499 + 0.845169i \(0.320500\pi\)
\(710\) 0 0
\(711\) −10.0910 + 7.33156i −0.378443 + 0.274955i
\(712\) 0 0
\(713\) −17.8421 12.9630i −0.668191 0.485469i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.47294 10.6886i −0.129699 0.399173i
\(718\) 0 0
\(719\) 8.08148 + 24.8722i 0.301388 + 0.927578i 0.981000 + 0.194006i \(0.0621481\pi\)
−0.679612 + 0.733572i \(0.737852\pi\)
\(720\) 0 0
\(721\) 14.9038 45.8692i 0.555046 1.70826i
\(722\) 0 0
\(723\) 16.3840 11.9037i 0.609327 0.442702i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0059 20.3475i 1.03868 0.754647i 0.0686545 0.997640i \(-0.478129\pi\)
0.970028 + 0.242994i \(0.0781294\pi\)
\(728\) 0 0
\(729\) 0.309017 0.951057i 0.0114451 0.0352243i
\(730\) 0 0
\(731\) 8.86463 + 27.2825i 0.327870 + 1.00908i
\(732\) 0 0
\(733\) 5.39704 + 16.6104i 0.199344 + 0.613518i 0.999898 + 0.0142577i \(0.00453852\pi\)
−0.800554 + 0.599260i \(0.795461\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.2515 + 7.44814i 0.377618 + 0.274356i
\(738\) 0 0
\(739\) −16.2358 + 11.7960i −0.597243 + 0.433922i −0.844899 0.534926i \(-0.820340\pi\)
0.247656 + 0.968848i \(0.420340\pi\)
\(740\) 0 0
\(741\) 17.8724 + 12.9850i 0.656557 + 0.477017i
\(742\) 0 0
\(743\) 5.84644 0.214485 0.107243 0.994233i \(-0.465798\pi\)
0.107243 + 0.994233i \(0.465798\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.74724 + 8.45513i −0.100516 + 0.309357i
\(748\) 0 0
\(749\) −1.02104 −0.0373081
\(750\) 0 0
\(751\) −32.7925 −1.19662 −0.598308 0.801266i \(-0.704160\pi\)
−0.598308 + 0.801266i \(0.704160\pi\)
\(752\) 0 0
\(753\) −9.48431 + 29.1897i −0.345627 + 1.06373i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.6371 0.822759 0.411379 0.911464i \(-0.365047\pi\)
0.411379 + 0.911464i \(0.365047\pi\)
\(758\) 0 0
\(759\) 26.7376 + 19.4260i 0.970513 + 0.705119i
\(760\) 0 0
\(761\) 29.3923 21.3547i 1.06547 0.774109i 0.0903767 0.995908i \(-0.471193\pi\)
0.975092 + 0.221799i \(0.0711929\pi\)
\(762\) 0 0
\(763\) −25.0280 18.1839i −0.906073 0.658301i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.1910 + 34.4425i 0.404085 + 1.24365i
\(768\) 0 0
\(769\) 4.51902 + 13.9081i 0.162960 + 0.501539i 0.998880 0.0473119i \(-0.0150655\pi\)
−0.835920 + 0.548851i \(0.815065\pi\)
\(770\) 0 0
\(771\) 1.47569 4.54170i 0.0531456 0.163565i
\(772\) 0 0
\(773\) −24.6953 + 17.9422i −0.888229 + 0.645336i −0.935416 0.353550i \(-0.884975\pi\)
0.0471864 + 0.998886i \(0.484975\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −7.02676 + 5.10524i −0.252084 + 0.183150i
\(778\) 0 0
\(779\) −5.50245 + 16.9348i −0.197146 + 0.606752i
\(780\) 0 0
\(781\) −13.2699 40.8404i −0.474833 1.46138i
\(782\) 0 0
\(783\) 0.370972 + 1.14173i 0.0132575 + 0.0408023i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −39.7644 28.8905i −1.41745 1.02984i −0.992186 0.124766i \(-0.960182\pi\)
−0.425262 0.905070i \(-0.639818\pi\)
\(788\) 0 0
\(789\) −0.978638 + 0.711022i −0.0348404 + 0.0253130i
\(790\) 0 0
\(791\) −35.9663 26.1310i −1.27881 0.929112i
\(792\) 0 0
\(793\) −23.6065 −0.838290
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13.7613 + 42.3529i −0.487450 + 1.50022i 0.340951 + 0.940081i \(0.389251\pi\)
−0.828401 + 0.560136i \(0.810749\pi\)
\(798\) 0 0
\(799\) −0.697388 −0.0246718
\(800\) 0 0
\(801\) 14.5774 0.515069
\(802\) 0 0
\(803\) −7.45487 + 22.9437i −0.263077 + 0.809667i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −25.5249 −0.898519
\(808\) 0 0
\(809\) 24.5060 + 17.8046i 0.861584 + 0.625978i 0.928316 0.371793i \(-0.121257\pi\)
−0.0667312 + 0.997771i \(0.521257\pi\)
\(810\) 0 0
\(811\) −5.05382 + 3.67182i −0.177464 + 0.128935i −0.672971 0.739669i \(-0.734982\pi\)
0.495507 + 0.868604i \(0.334982\pi\)
\(812\) 0 0
\(813\) 4.75170 + 3.45231i 0.166649 + 0.121078i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −19.3447 59.5367i −0.676784 2.08293i
\(818\) 0 0
\(819\) −5.37980 16.5573i −0.187985 0.578559i
\(820\) 0 0
\(821\) −5.95524 + 18.3284i −0.207839 + 0.639664i 0.791745 + 0.610851i \(0.209173\pi\)
−0.999585 + 0.0288127i \(0.990827\pi\)
\(822\) 0 0
\(823\) 6.11414 4.44218i 0.213126 0.154845i −0.476101 0.879391i \(-0.657950\pi\)
0.689227 + 0.724546i \(0.257950\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.3821 18.4412i 0.882621 0.641262i −0.0513225 0.998682i \(-0.516344\pi\)
0.933944 + 0.357420i \(0.116344\pi\)
\(828\) 0 0
\(829\) 8.55625 26.3334i 0.297171 0.914597i −0.685313 0.728249i \(-0.740335\pi\)
0.982484 0.186349i \(-0.0596654\pi\)
\(830\) 0 0
\(831\) −1.51585 4.66531i −0.0525843 0.161838i
\(832\) 0 0
\(833\) −11.9775 36.8629i −0.414996 1.27723i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.67165 1.94107i −0.0923459 0.0670932i
\(838\) 0 0
\(839\) 30.5642 22.2062i 1.05519 0.766643i 0.0819999 0.996632i \(-0.473869\pi\)
0.973193 + 0.229990i \(0.0738693\pi\)
\(840\) 0 0
\(841\) 22.2956 + 16.1987i 0.768812 + 0.558575i
\(842\) 0 0
\(843\) −1.29526 −0.0446110
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 19.2881 59.3626i 0.662747 2.03972i
\(848\) 0 0
\(849\) 30.2297 1.03748
\(850\) 0 0
\(851\) 12.5368 0.429756
\(852\) 0 0
\(853\) −15.2552 + 46.9508i −0.522329 + 1.60756i 0.247208 + 0.968962i \(0.420487\pi\)
−0.769537 + 0.638602i \(0.779513\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.7692 −0.538667 −0.269333 0.963047i \(-0.586803\pi\)
−0.269333 + 0.963047i \(0.586803\pi\)
\(858\) 0 0
\(859\) −7.60223 5.52335i −0.259385 0.188454i 0.450491 0.892781i \(-0.351249\pi\)
−0.709876 + 0.704327i \(0.751249\pi\)
\(860\) 0 0
\(861\) 11.3525 8.24807i 0.386892 0.281093i
\(862\) 0 0
\(863\) 14.4575 + 10.5040i 0.492140 + 0.357560i 0.806007 0.591907i \(-0.201625\pi\)
−0.313867 + 0.949467i \(0.601625\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3.01655 9.28400i −0.102448 0.315301i
\(868\) 0 0
\(869\) −19.0748 58.7061i −0.647068 1.99147i
\(870\) 0 0
\(871\) −2.97728 + 9.16312i −0.100881 + 0.310481i
\(872\) 0 0
\(873\) 3.22546 2.34344i 0.109165 0.0793133i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.3238 10.4069i 0.483681 0.351415i −0.319068 0.947732i \(-0.603370\pi\)
0.802749 + 0.596317i \(0.203370\pi\)
\(878\) 0 0
\(879\) 2.49325 7.67342i 0.0840951 0.258818i
\(880\) 0 0
\(881\) −9.82007 30.2231i −0.330847 1.01824i −0.968732 0.248110i \(-0.920191\pi\)
0.637885 0.770131i \(-0.279809\pi\)
\(882\) 0 0
\(883\) 10.6950 + 32.9158i 0.359916 + 1.10771i 0.953104 + 0.302643i \(0.0978689\pi\)
−0.593188 + 0.805064i \(0.702131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.0735 16.7639i −0.774732 0.562876i 0.128661 0.991689i \(-0.458932\pi\)
−0.903393 + 0.428813i \(0.858932\pi\)
\(888\) 0 0
\(889\) −59.8610 + 43.4916i −2.00767 + 1.45866i
\(890\) 0 0
\(891\) 4.00366 + 2.90883i 0.134128 + 0.0974495i
\(892\) 0 0
\(893\) 1.52186 0.0509271
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.76524 + 23.8990i −0.259274 + 0.797963i
\(898\) 0 0
\(899\) 3.96444 0.132221
\(900\) 0 0
\(901\) −26.3567 −0.878069
\(902\) 0 0
\(903\) −15.2448 + 46.9185i −0.507314 + 1.56135i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 7.01754 0.233013 0.116507 0.993190i \(-0.462830\pi\)
0.116507 + 0.993190i \(0.462830\pi\)
\(908\) 0 0
\(909\) −7.57965 5.50693i −0.251401 0.182653i
\(910\) 0 0
\(911\) 16.9145 12.2891i 0.560403 0.407157i −0.271203 0.962522i \(-0.587422\pi\)
0.831606 + 0.555365i \(0.187422\pi\)
\(912\) 0 0
\(913\) −35.5936 25.8602i −1.17797 0.855849i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −17.6184 54.2240i −0.581812 1.79063i
\(918\) 0 0
\(919\) 18.4658 + 56.8320i 0.609132 + 1.87471i 0.465406 + 0.885097i \(0.345908\pi\)
0.143726 + 0.989618i \(0.454092\pi\)
\(920\) 0 0
\(921\) −4.56573 + 14.0519i −0.150446 + 0.463025i
\(922\) 0 0
\(923\) 26.4149 19.1916i 0.869458 0.631698i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −8.43328 + 6.12713i −0.276985 + 0.201241i
\(928\) 0 0
\(929\) −12.4320 + 38.2618i −0.407881 + 1.25533i 0.510585 + 0.859828i \(0.329429\pi\)
−0.918466 + 0.395501i \(0.870571\pi\)
\(930\) 0 0
\(931\) 26.1376 + 80.4434i 0.856626 + 2.63642i
\(932\) 0 0
\(933\) −2.82866 8.70572i −0.0926061 0.285012i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 39.9780 + 29.0457i 1.30602 + 0.948881i 0.999995 0.00314114i \(-0.000999859\pi\)
0.306028 + 0.952022i \(0.401000\pi\)
\(938\) 0 0
\(939\) 10.7238 7.79128i 0.349957 0.254259i
\(940\) 0 0
\(941\) −18.0130 13.0872i −0.587207 0.426631i 0.254108 0.967176i \(-0.418218\pi\)
−0.841315 + 0.540545i \(0.818218\pi\)
\(942\) 0 0
\(943\) −20.2546 −0.659579
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.0251 + 33.9316i −0.358266 + 1.10263i 0.595825 + 0.803114i \(0.296825\pi\)
−0.954091 + 0.299516i \(0.903175\pi\)
\(948\) 0 0
\(949\) −18.3428 −0.595433
\(950\) 0 0
\(951\) −0.790550 −0.0256354
\(952\) 0 0
\(953\) 0.867341 2.66940i 0.0280959 0.0864704i −0.936025 0.351933i \(-0.885525\pi\)
0.964121 + 0.265462i \(0.0855246\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −5.94099 −0.192045
\(958\) 0 0
\(959\) 21.3153 + 15.4865i 0.688308 + 0.500085i
\(960\) 0 0
\(961\) 16.2568 11.8113i 0.524413 0.381008i
\(962\) 0 0
\(963\) 0.178536 + 0.129714i 0.00575324 + 0.00417998i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −3.56243 10.9640i −0.114560 0.352580i 0.877295 0.479952i \(-0.159346\pi\)
−0.991855 + 0.127372i \(0.959346\pi\)
\(968\) 0 0
\(969\) −4.88107 15.0224i −0.156803 0.482589i
\(970\) 0 0
\(971\) 7.15578 22.0232i 0.229640 0.706759i −0.768148 0.640273i \(-0.778821\pi\)
0.997787 0.0664858i \(-0.0211787\pi\)
\(972\) 0 0
\(973\) 67.3646 48.9433i 2.15961 1.56905i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −6.68304 + 4.85551i −0.213809 + 0.155342i −0.689536 0.724252i \(-0.742185\pi\)
0.475726 + 0.879593i \(0.342185\pi\)
\(978\) 0 0
\(979\) −22.2927 + 68.6100i −0.712479 + 2.19278i
\(980\) 0 0
\(981\) 2.06621 + 6.35914i 0.0659690 + 0.203032i
\(982\) 0 0
\(983\) −6.11506 18.8202i −0.195040 0.600271i −0.999976 0.00690994i \(-0.997800\pi\)
0.804936 0.593361i \(-0.202200\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −0.970269 0.704941i −0.0308840 0.0224385i
\(988\) 0 0
\(989\) 57.6083 41.8549i 1.83184 1.33091i
\(990\) 0 0
\(991\) −17.3036 12.5718i −0.549666 0.399355i 0.277997 0.960582i \(-0.410330\pi\)
−0.827662 + 0.561227i \(0.810330\pi\)
\(992\) 0 0
\(993\) −3.37370 −0.107061
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.43350 + 16.7226i −0.172081 + 0.529610i −0.999488 0.0319923i \(-0.989815\pi\)
0.827407 + 0.561602i \(0.189815\pi\)
\(998\) 0 0
\(999\) 1.87725 0.0593935
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 1500.2.m.c.301.6 24
5.2 odd 4 300.2.o.a.289.2 yes 24
5.3 odd 4 1500.2.o.c.949.6 24
5.4 even 2 1500.2.m.d.301.1 24
15.2 even 4 900.2.w.c.289.3 24
25.3 odd 20 7500.2.d.g.1249.13 24
25.4 even 10 7500.2.a.m.1.1 12
25.9 even 10 1500.2.m.d.1201.1 24
25.12 odd 20 1500.2.o.c.49.6 24
25.13 odd 20 300.2.o.a.109.2 24
25.16 even 5 inner 1500.2.m.c.1201.6 24
25.21 even 5 7500.2.a.n.1.12 12
25.22 odd 20 7500.2.d.g.1249.12 24
75.38 even 20 900.2.w.c.109.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.o.a.109.2 24 25.13 odd 20
300.2.o.a.289.2 yes 24 5.2 odd 4
900.2.w.c.109.3 24 75.38 even 20
900.2.w.c.289.3 24 15.2 even 4
1500.2.m.c.301.6 24 1.1 even 1 trivial
1500.2.m.c.1201.6 24 25.16 even 5 inner
1500.2.m.d.301.1 24 5.4 even 2
1500.2.m.d.1201.1 24 25.9 even 10
1500.2.o.c.49.6 24 25.12 odd 20
1500.2.o.c.949.6 24 5.3 odd 4
7500.2.a.m.1.1 12 25.4 even 10
7500.2.a.n.1.12 12 25.21 even 5
7500.2.d.g.1249.12 24 25.22 odd 20
7500.2.d.g.1249.13 24 25.3 odd 20