Properties

Label 1900.2.i.g.201.6
Level $1900$
Weight $2$
Character 1900.201
Analytic conductor $15.172$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,2,Mod(201,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), 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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 201.6
Root \(-0.226426 - 0.392182i\) of defining polynomial
Character \(\chi\) \(=\) 1900.201
Dual form 1900.2.i.g.501.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.226426 - 0.392182i) q^{3} -2.54366 q^{7} +(1.39746 + 2.42048i) q^{9} +2.22377 q^{11} +(-3.51096 - 6.08116i) q^{13} +(-1.27878 + 2.21492i) q^{17} +(-2.70498 + 3.41805i) q^{19} +(-0.575952 + 0.997578i) q^{21} +(4.01448 + 6.95328i) q^{23} +2.62425 q^{27} +(0.941734 + 1.63113i) q^{29} +5.98111 q^{31} +(0.503519 - 0.872121i) q^{33} +2.86105 q^{37} -3.17989 q^{39} +(-3.67524 + 6.36571i) q^{41} +(1.84706 - 3.19919i) q^{43} +(2.36448 + 4.09540i) q^{47} -0.529782 q^{49} +(0.579100 + 1.00303i) q^{51} +(5.14549 + 8.91226i) q^{53} +(0.728020 + 1.83478i) q^{57} +(3.73666 - 6.47208i) q^{59} +(4.17839 + 7.23719i) q^{61} +(-3.55467 - 6.15687i) q^{63} +(-6.17997 - 10.7040i) q^{67} +3.63593 q^{69} +(-4.13931 + 7.16950i) q^{71} +(-6.32134 + 10.9489i) q^{73} -5.65651 q^{77} +(2.13067 - 3.69043i) q^{79} +(-3.59819 + 6.23225i) q^{81} -14.7613 q^{83} +0.852933 q^{87} +(7.19403 + 12.4604i) q^{89} +(8.93069 + 15.4684i) q^{91} +(1.35428 - 2.34568i) q^{93} +(-2.91488 + 5.04871i) q^{97} +(3.10763 + 5.38258i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0.226426 0.392182i 0.130727 0.226426i −0.793230 0.608922i \(-0.791602\pi\)
0.923957 + 0.382496i \(0.124935\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.54366 −0.961414 −0.480707 0.876881i \(-0.659620\pi\)
−0.480707 + 0.876881i \(0.659620\pi\)
\(8\) 0 0
\(9\) 1.39746 + 2.42048i 0.465821 + 0.806825i
\(10\) 0 0
\(11\) 2.22377 0.670491 0.335246 0.942131i \(-0.391181\pi\)
0.335246 + 0.942131i \(0.391181\pi\)
\(12\) 0 0
\(13\) −3.51096 6.08116i −0.973765 1.68661i −0.683953 0.729526i \(-0.739741\pi\)
−0.289812 0.957084i \(-0.593593\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.27878 + 2.21492i −0.310151 + 0.537197i −0.978395 0.206745i \(-0.933713\pi\)
0.668244 + 0.743942i \(0.267046\pi\)
\(18\) 0 0
\(19\) −2.70498 + 3.41805i −0.620565 + 0.784155i
\(20\) 0 0
\(21\) −0.575952 + 0.997578i −0.125683 + 0.217689i
\(22\) 0 0
\(23\) 4.01448 + 6.95328i 0.837076 + 1.44986i 0.892329 + 0.451387i \(0.149070\pi\)
−0.0552521 + 0.998472i \(0.517596\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.62425 0.505036
\(28\) 0 0
\(29\) 0.941734 + 1.63113i 0.174876 + 0.302893i 0.940118 0.340849i \(-0.110714\pi\)
−0.765243 + 0.643742i \(0.777381\pi\)
\(30\) 0 0
\(31\) 5.98111 1.07424 0.537120 0.843506i \(-0.319512\pi\)
0.537120 + 0.843506i \(0.319512\pi\)
\(32\) 0 0
\(33\) 0.503519 0.872121i 0.0876515 0.151817i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.86105 0.470353 0.235177 0.971953i \(-0.424433\pi\)
0.235177 + 0.971953i \(0.424433\pi\)
\(38\) 0 0
\(39\) −3.17989 −0.509190
\(40\) 0 0
\(41\) −3.67524 + 6.36571i −0.573977 + 0.994157i 0.422175 + 0.906514i \(0.361267\pi\)
−0.996152 + 0.0876426i \(0.972067\pi\)
\(42\) 0 0
\(43\) 1.84706 3.19919i 0.281673 0.487873i −0.690124 0.723691i \(-0.742444\pi\)
0.971797 + 0.235819i \(0.0757772\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.36448 + 4.09540i 0.344895 + 0.597376i 0.985335 0.170633i \(-0.0545812\pi\)
−0.640440 + 0.768008i \(0.721248\pi\)
\(48\) 0 0
\(49\) −0.529782 −0.0756832
\(50\) 0 0
\(51\) 0.579100 + 1.00303i 0.0810903 + 0.140453i
\(52\) 0 0
\(53\) 5.14549 + 8.91226i 0.706788 + 1.22419i 0.966042 + 0.258383i \(0.0831898\pi\)
−0.259255 + 0.965809i \(0.583477\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.728020 + 1.83478i 0.0964286 + 0.243023i
\(58\) 0 0
\(59\) 3.73666 6.47208i 0.486472 0.842593i −0.513408 0.858145i \(-0.671617\pi\)
0.999879 + 0.0155515i \(0.00495040\pi\)
\(60\) 0 0
\(61\) 4.17839 + 7.23719i 0.534988 + 0.926627i 0.999164 + 0.0408838i \(0.0130174\pi\)
−0.464176 + 0.885743i \(0.653649\pi\)
\(62\) 0 0
\(63\) −3.55467 6.15687i −0.447847 0.775693i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.17997 10.7040i −0.755004 1.30771i −0.945372 0.325993i \(-0.894302\pi\)
0.190368 0.981713i \(-0.439032\pi\)
\(68\) 0 0
\(69\) 3.63593 0.437715
\(70\) 0 0
\(71\) −4.13931 + 7.16950i −0.491246 + 0.850863i −0.999949 0.0100790i \(-0.996792\pi\)
0.508703 + 0.860942i \(0.330125\pi\)
\(72\) 0 0
\(73\) −6.32134 + 10.9489i −0.739857 + 1.28147i 0.212703 + 0.977117i \(0.431773\pi\)
−0.952559 + 0.304352i \(0.901560\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.65651 −0.644620
\(78\) 0 0
\(79\) 2.13067 3.69043i 0.239719 0.415206i −0.720914 0.693024i \(-0.756278\pi\)
0.960634 + 0.277818i \(0.0896112\pi\)
\(80\) 0 0
\(81\) −3.59819 + 6.23225i −0.399799 + 0.692472i
\(82\) 0 0
\(83\) −14.7613 −1.62026 −0.810132 0.586248i \(-0.800604\pi\)
−0.810132 + 0.586248i \(0.800604\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.852933 0.0914440
\(88\) 0 0
\(89\) 7.19403 + 12.4604i 0.762566 + 1.32080i 0.941524 + 0.336946i \(0.109394\pi\)
−0.178958 + 0.983857i \(0.557273\pi\)
\(90\) 0 0
\(91\) 8.93069 + 15.4684i 0.936191 + 1.62153i
\(92\) 0 0
\(93\) 1.35428 2.34568i 0.140432 0.243236i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.91488 + 5.04871i −0.295961 + 0.512619i −0.975208 0.221291i \(-0.928973\pi\)
0.679247 + 0.733910i \(0.262306\pi\)
\(98\) 0 0
\(99\) 3.10763 + 5.38258i 0.312329 + 0.540969i
\(100\) 0 0
\(101\) 6.47686 + 11.2183i 0.644472 + 1.11626i 0.984423 + 0.175815i \(0.0562560\pi\)
−0.339951 + 0.940443i \(0.610411\pi\)
\(102\) 0 0
\(103\) 16.6116 1.63679 0.818394 0.574658i \(-0.194865\pi\)
0.818394 + 0.574658i \(0.194865\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.59275 0.443998 0.221999 0.975047i \(-0.428742\pi\)
0.221999 + 0.975047i \(0.428742\pi\)
\(108\) 0 0
\(109\) 1.30902 2.26728i 0.125381 0.217166i −0.796501 0.604637i \(-0.793318\pi\)
0.921882 + 0.387471i \(0.126651\pi\)
\(110\) 0 0
\(111\) 0.647816 1.12205i 0.0614880 0.106500i
\(112\) 0 0
\(113\) 0.509357 0.0479163 0.0239581 0.999713i \(-0.492373\pi\)
0.0239581 + 0.999713i \(0.492373\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.81286 16.9964i 0.907200 1.57132i
\(118\) 0 0
\(119\) 3.25279 5.63401i 0.298183 0.516468i
\(120\) 0 0
\(121\) −6.05486 −0.550441
\(122\) 0 0
\(123\) 1.66434 + 2.88273i 0.150069 + 0.259927i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −0.367242 0.636081i −0.0325874 0.0564431i 0.849272 0.527956i \(-0.177041\pi\)
−0.881859 + 0.471513i \(0.843708\pi\)
\(128\) 0 0
\(129\) −0.836444 1.44876i −0.0736448 0.127556i
\(130\) 0 0
\(131\) 2.84274 4.92377i 0.248371 0.430192i −0.714703 0.699428i \(-0.753438\pi\)
0.963074 + 0.269236i \(0.0867713\pi\)
\(132\) 0 0
\(133\) 6.88055 8.69437i 0.596620 0.753898i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.39993 + 16.2812i 0.803091 + 1.39099i 0.917573 + 0.397568i \(0.130146\pi\)
−0.114482 + 0.993425i \(0.536521\pi\)
\(138\) 0 0
\(139\) −2.05362 3.55697i −0.174186 0.301698i 0.765694 0.643206i \(-0.222396\pi\)
−0.939879 + 0.341507i \(0.889063\pi\)
\(140\) 0 0
\(141\) 2.14152 0.180349
\(142\) 0 0
\(143\) −7.80756 13.5231i −0.652901 1.13086i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.119957 + 0.207771i −0.00989385 + 0.0171367i
\(148\) 0 0
\(149\) 2.08175 3.60570i 0.170544 0.295391i −0.768066 0.640370i \(-0.778781\pi\)
0.938610 + 0.344980i \(0.112114\pi\)
\(150\) 0 0
\(151\) 17.7955 1.44818 0.724090 0.689706i \(-0.242260\pi\)
0.724090 + 0.689706i \(0.242260\pi\)
\(152\) 0 0
\(153\) −7.14821 −0.577899
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.67476 8.09693i 0.373087 0.646205i −0.616952 0.787001i \(-0.711633\pi\)
0.990039 + 0.140796i \(0.0449660\pi\)
\(158\) 0 0
\(159\) 4.66030 0.369586
\(160\) 0 0
\(161\) −10.2115 17.6868i −0.804777 1.39391i
\(162\) 0 0
\(163\) −1.55215 −0.121574 −0.0607869 0.998151i \(-0.519361\pi\)
−0.0607869 + 0.998151i \(0.519361\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.05980 7.03177i −0.314157 0.544135i 0.665101 0.746753i \(-0.268388\pi\)
−0.979258 + 0.202618i \(0.935055\pi\)
\(168\) 0 0
\(169\) −18.1537 + 31.4431i −1.39644 + 2.41870i
\(170\) 0 0
\(171\) −12.0534 1.77073i −0.921748 0.135411i
\(172\) 0 0
\(173\) −0.365686 + 0.633386i −0.0278026 + 0.0481555i −0.879592 0.475729i \(-0.842184\pi\)
0.851789 + 0.523884i \(0.175518\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.69216 2.93090i −0.127190 0.220300i
\(178\) 0 0
\(179\) 2.68858 0.200954 0.100477 0.994939i \(-0.467963\pi\)
0.100477 + 0.994939i \(0.467963\pi\)
\(180\) 0 0
\(181\) 2.75090 + 4.76469i 0.204473 + 0.354157i 0.949965 0.312358i \(-0.101119\pi\)
−0.745492 + 0.666515i \(0.767785\pi\)
\(182\) 0 0
\(183\) 3.78439 0.279750
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.84372 + 4.92547i −0.207953 + 0.360186i
\(188\) 0 0
\(189\) −6.67520 −0.485549
\(190\) 0 0
\(191\) 9.39845 0.680048 0.340024 0.940417i \(-0.389565\pi\)
0.340024 + 0.940417i \(0.389565\pi\)
\(192\) 0 0
\(193\) 5.77864 10.0089i 0.415956 0.720457i −0.579572 0.814921i \(-0.696780\pi\)
0.995528 + 0.0944641i \(0.0301138\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.11825 −0.364660 −0.182330 0.983237i \(-0.558364\pi\)
−0.182330 + 0.983237i \(0.558364\pi\)
\(198\) 0 0
\(199\) −6.71897 11.6376i −0.476295 0.824968i 0.523336 0.852127i \(-0.324687\pi\)
−0.999631 + 0.0271588i \(0.991354\pi\)
\(200\) 0 0
\(201\) −5.59723 −0.394798
\(202\) 0 0
\(203\) −2.39545 4.14905i −0.168128 0.291206i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −11.2202 + 19.4339i −0.779855 + 1.35075i
\(208\) 0 0
\(209\) −6.01524 + 7.60096i −0.416083 + 0.525769i
\(210\) 0 0
\(211\) 2.54063 4.40050i 0.174904 0.302943i −0.765224 0.643764i \(-0.777372\pi\)
0.940128 + 0.340821i \(0.110705\pi\)
\(212\) 0 0
\(213\) 1.87450 + 3.24673i 0.128438 + 0.222462i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −15.2139 −1.03279
\(218\) 0 0
\(219\) 2.86263 + 4.95823i 0.193439 + 0.335046i
\(220\) 0 0
\(221\) 17.9590 1.20806
\(222\) 0 0
\(223\) −11.8464 + 20.5186i −0.793295 + 1.37403i 0.130621 + 0.991432i \(0.458303\pi\)
−0.923916 + 0.382595i \(0.875030\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.0306 −1.26310 −0.631551 0.775334i \(-0.717582\pi\)
−0.631551 + 0.775334i \(0.717582\pi\)
\(228\) 0 0
\(229\) −27.9233 −1.84523 −0.922613 0.385727i \(-0.873951\pi\)
−0.922613 + 0.385727i \(0.873951\pi\)
\(230\) 0 0
\(231\) −1.28078 + 2.21838i −0.0842694 + 0.145959i
\(232\) 0 0
\(233\) 7.08296 12.2680i 0.464020 0.803706i −0.535137 0.844765i \(-0.679740\pi\)
0.999157 + 0.0410592i \(0.0130732\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.964881 1.67122i −0.0626757 0.108558i
\(238\) 0 0
\(239\) 17.0237 1.10117 0.550585 0.834779i \(-0.314405\pi\)
0.550585 + 0.834779i \(0.314405\pi\)
\(240\) 0 0
\(241\) −8.59549 14.8878i −0.553684 0.959009i −0.998005 0.0631409i \(-0.979888\pi\)
0.444321 0.895868i \(-0.353445\pi\)
\(242\) 0 0
\(243\) 5.56582 + 9.64028i 0.357047 + 0.618424i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 30.2828 + 4.44876i 1.92685 + 0.283068i
\(248\) 0 0
\(249\) −3.34235 + 5.78911i −0.211813 + 0.366870i
\(250\) 0 0
\(251\) −11.8853 20.5860i −0.750195 1.29938i −0.947728 0.319079i \(-0.896626\pi\)
0.197534 0.980296i \(-0.436707\pi\)
\(252\) 0 0
\(253\) 8.92727 + 15.4625i 0.561252 + 0.972118i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.716409 + 1.24086i 0.0446884 + 0.0774026i 0.887504 0.460799i \(-0.152437\pi\)
−0.842816 + 0.538202i \(0.819104\pi\)
\(258\) 0 0
\(259\) −7.27754 −0.452204
\(260\) 0 0
\(261\) −2.63208 + 4.55889i −0.162921 + 0.282188i
\(262\) 0 0
\(263\) 3.56505 6.17484i 0.219830 0.380757i −0.734926 0.678148i \(-0.762783\pi\)
0.954756 + 0.297390i \(0.0961163\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.51567 0.398753
\(268\) 0 0
\(269\) 9.13452 15.8214i 0.556941 0.964651i −0.440808 0.897601i \(-0.645308\pi\)
0.997750 0.0670494i \(-0.0213585\pi\)
\(270\) 0 0
\(271\) 2.90265 5.02754i 0.176324 0.305401i −0.764295 0.644867i \(-0.776913\pi\)
0.940619 + 0.339465i \(0.110246\pi\)
\(272\) 0 0
\(273\) 8.08857 0.489543
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.72209 0.343807 0.171904 0.985114i \(-0.445008\pi\)
0.171904 + 0.985114i \(0.445008\pi\)
\(278\) 0 0
\(279\) 8.35838 + 14.4771i 0.500403 + 0.866724i
\(280\) 0 0
\(281\) 3.07155 + 5.32009i 0.183233 + 0.317370i 0.942980 0.332850i \(-0.108010\pi\)
−0.759746 + 0.650220i \(0.774677\pi\)
\(282\) 0 0
\(283\) −8.22632 + 14.2484i −0.489004 + 0.846980i −0.999920 0.0126510i \(-0.995973\pi\)
0.510916 + 0.859631i \(0.329306\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.34858 16.1922i 0.551829 0.955796i
\(288\) 0 0
\(289\) 5.22942 + 9.05763i 0.307613 + 0.532801i
\(290\) 0 0
\(291\) 1.32001 + 2.28632i 0.0773803 + 0.134027i
\(292\) 0 0
\(293\) −19.5684 −1.14320 −0.571599 0.820533i \(-0.693677\pi\)
−0.571599 + 0.820533i \(0.693677\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.83571 0.338622
\(298\) 0 0
\(299\) 28.1893 48.8253i 1.63023 2.82364i
\(300\) 0 0
\(301\) −4.69829 + 8.13767i −0.270805 + 0.469047i
\(302\) 0 0
\(303\) 5.86613 0.337000
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −14.3994 + 24.9405i −0.821817 + 1.42343i 0.0825100 + 0.996590i \(0.473706\pi\)
−0.904327 + 0.426839i \(0.859627\pi\)
\(308\) 0 0
\(309\) 3.76130 6.51476i 0.213973 0.370612i
\(310\) 0 0
\(311\) −19.4309 −1.10182 −0.550911 0.834564i \(-0.685720\pi\)
−0.550911 + 0.834564i \(0.685720\pi\)
\(312\) 0 0
\(313\) −4.81885 8.34649i −0.272377 0.471772i 0.697093 0.716981i \(-0.254477\pi\)
−0.969470 + 0.245209i \(0.921143\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.115348 + 0.199789i 0.00647860 + 0.0112213i 0.869247 0.494379i \(-0.164604\pi\)
−0.862768 + 0.505600i \(0.831271\pi\)
\(318\) 0 0
\(319\) 2.09420 + 3.62726i 0.117253 + 0.203087i
\(320\) 0 0
\(321\) 1.03992 1.80119i 0.0580427 0.100533i
\(322\) 0 0
\(323\) −4.11163 10.3623i −0.228777 0.576572i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −0.592791 1.02674i −0.0327814 0.0567791i
\(328\) 0 0
\(329\) −6.01444 10.4173i −0.331587 0.574325i
\(330\) 0 0
\(331\) −12.1135 −0.665820 −0.332910 0.942959i \(-0.608031\pi\)
−0.332910 + 0.942959i \(0.608031\pi\)
\(332\) 0 0
\(333\) 3.99821 + 6.92510i 0.219100 + 0.379493i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.57090 9.64909i 0.303466 0.525619i −0.673452 0.739231i \(-0.735189\pi\)
0.976919 + 0.213611i \(0.0685226\pi\)
\(338\) 0 0
\(339\) 0.115332 0.199761i 0.00626396 0.0108495i
\(340\) 0 0
\(341\) 13.3006 0.720268
\(342\) 0 0
\(343\) 19.1532 1.03418
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.84478 4.92730i 0.152716 0.264511i −0.779509 0.626391i \(-0.784532\pi\)
0.932225 + 0.361880i \(0.117865\pi\)
\(348\) 0 0
\(349\) 0.369374 0.0197722 0.00988608 0.999951i \(-0.496853\pi\)
0.00988608 + 0.999951i \(0.496853\pi\)
\(350\) 0 0
\(351\) −9.21362 15.9585i −0.491787 0.851799i
\(352\) 0 0
\(353\) 1.87124 0.0995962 0.0497981 0.998759i \(-0.484142\pi\)
0.0497981 + 0.998759i \(0.484142\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.47304 2.55137i −0.0779613 0.135033i
\(358\) 0 0
\(359\) 8.47802 14.6844i 0.447453 0.775011i −0.550767 0.834659i \(-0.685665\pi\)
0.998219 + 0.0596485i \(0.0189980\pi\)
\(360\) 0 0
\(361\) −4.36618 18.4915i −0.229799 0.973238i
\(362\) 0 0
\(363\) −1.37098 + 2.37460i −0.0719577 + 0.124634i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.89237 3.27769i −0.0987812 0.171094i 0.812399 0.583102i \(-0.198161\pi\)
−0.911180 + 0.412008i \(0.864828\pi\)
\(368\) 0 0
\(369\) −20.5441 −1.06948
\(370\) 0 0
\(371\) −13.0884 22.6698i −0.679516 1.17696i
\(372\) 0 0
\(373\) −4.51203 −0.233624 −0.116812 0.993154i \(-0.537268\pi\)
−0.116812 + 0.993154i \(0.537268\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.61278 11.4537i 0.340575 0.589894i
\(378\) 0 0
\(379\) 4.64242 0.238465 0.119233 0.992866i \(-0.461957\pi\)
0.119233 + 0.992866i \(0.461957\pi\)
\(380\) 0 0
\(381\) −0.332612 −0.0170403
\(382\) 0 0
\(383\) −6.99088 + 12.1086i −0.357218 + 0.618719i −0.987495 0.157651i \(-0.949608\pi\)
0.630277 + 0.776370i \(0.282941\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.3248 0.524837
\(388\) 0 0
\(389\) −3.93460 6.81492i −0.199492 0.345530i 0.748872 0.662715i \(-0.230596\pi\)
−0.948364 + 0.317185i \(0.897263\pi\)
\(390\) 0 0
\(391\) −20.5346 −1.03848
\(392\) 0 0
\(393\) −1.28734 2.22974i −0.0649378 0.112476i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.48528 + 14.6969i −0.425864 + 0.737618i −0.996501 0.0835846i \(-0.973363\pi\)
0.570637 + 0.821203i \(0.306696\pi\)
\(398\) 0 0
\(399\) −1.85184 4.66706i −0.0927078 0.233645i
\(400\) 0 0
\(401\) −7.16651 + 12.4128i −0.357878 + 0.619864i −0.987606 0.156952i \(-0.949833\pi\)
0.629728 + 0.776816i \(0.283166\pi\)
\(402\) 0 0
\(403\) −20.9994 36.3721i −1.04606 1.81182i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.36231 0.315368
\(408\) 0 0
\(409\) 0.122530 + 0.212228i 0.00605872 + 0.0104940i 0.869039 0.494744i \(-0.164738\pi\)
−0.862980 + 0.505238i \(0.831405\pi\)
\(410\) 0 0
\(411\) 8.51357 0.419943
\(412\) 0 0
\(413\) −9.50480 + 16.4628i −0.467701 + 0.810081i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.85997 −0.0910832
\(418\) 0 0
\(419\) −25.4690 −1.24424 −0.622122 0.782920i \(-0.713729\pi\)
−0.622122 + 0.782920i \(0.713729\pi\)
\(420\) 0 0
\(421\) −3.02703 + 5.24297i −0.147529 + 0.255527i −0.930313 0.366765i \(-0.880465\pi\)
0.782785 + 0.622292i \(0.213798\pi\)
\(422\) 0 0
\(423\) −6.60855 + 11.4463i −0.321318 + 0.556540i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −10.6284 18.4090i −0.514345 0.890872i
\(428\) 0 0
\(429\) −7.07134 −0.341408
\(430\) 0 0
\(431\) −11.1005 19.2267i −0.534694 0.926117i −0.999178 0.0405356i \(-0.987094\pi\)
0.464484 0.885581i \(-0.346240\pi\)
\(432\) 0 0
\(433\) −12.1298 21.0094i −0.582920 1.00965i −0.995131 0.0985593i \(-0.968577\pi\)
0.412211 0.911089i \(-0.364757\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −34.6258 5.08677i −1.65637 0.243333i
\(438\) 0 0
\(439\) 11.9487 20.6957i 0.570279 0.987753i −0.426258 0.904602i \(-0.640168\pi\)
0.996537 0.0831509i \(-0.0264983\pi\)
\(440\) 0 0
\(441\) −0.740350 1.28232i −0.0352548 0.0610631i
\(442\) 0 0
\(443\) 10.3356 + 17.9017i 0.491057 + 0.850536i 0.999947 0.0102958i \(-0.00327733\pi\)
−0.508890 + 0.860832i \(0.669944\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.942727 1.63285i −0.0445895 0.0772312i
\(448\) 0 0
\(449\) −6.64143 −0.313429 −0.156714 0.987644i \(-0.550090\pi\)
−0.156714 + 0.987644i \(0.550090\pi\)
\(450\) 0 0
\(451\) −8.17289 + 14.1559i −0.384846 + 0.666573i
\(452\) 0 0
\(453\) 4.02937 6.97908i 0.189316 0.327906i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −14.3433 −0.670952 −0.335476 0.942049i \(-0.608897\pi\)
−0.335476 + 0.942049i \(0.608897\pi\)
\(458\) 0 0
\(459\) −3.35584 + 5.81249i −0.156637 + 0.271304i
\(460\) 0 0
\(461\) 13.3682 23.1544i 0.622618 1.07841i −0.366378 0.930466i \(-0.619402\pi\)
0.988996 0.147940i \(-0.0472643\pi\)
\(462\) 0 0
\(463\) 12.0950 0.562101 0.281051 0.959693i \(-0.409317\pi\)
0.281051 + 0.959693i \(0.409317\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.3169 1.26408 0.632038 0.774938i \(-0.282219\pi\)
0.632038 + 0.774938i \(0.282219\pi\)
\(468\) 0 0
\(469\) 15.7198 + 27.2274i 0.725871 + 1.25725i
\(470\) 0 0
\(471\) −2.11698 3.66671i −0.0975452 0.168953i
\(472\) 0 0
\(473\) 4.10742 7.11427i 0.188860 0.327114i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −14.3813 + 24.9091i −0.658473 + 1.14051i
\(478\) 0 0
\(479\) −3.69624 6.40208i −0.168886 0.292519i 0.769143 0.639077i \(-0.220684\pi\)
−0.938028 + 0.346559i \(0.887350\pi\)
\(480\) 0 0
\(481\) −10.0450 17.3985i −0.458013 0.793302i
\(482\) 0 0
\(483\) −9.24858 −0.420825
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −12.9424 −0.586477 −0.293239 0.956039i \(-0.594733\pi\)
−0.293239 + 0.956039i \(0.594733\pi\)
\(488\) 0 0
\(489\) −0.351447 + 0.608725i −0.0158930 + 0.0275275i
\(490\) 0 0
\(491\) 5.59974 9.69903i 0.252713 0.437711i −0.711559 0.702626i \(-0.752011\pi\)
0.964272 + 0.264915i \(0.0853439\pi\)
\(492\) 0 0
\(493\) −4.81710 −0.216951
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.5290 18.2368i 0.472291 0.818032i
\(498\) 0 0
\(499\) 12.9699 22.4645i 0.580611 1.00565i −0.414796 0.909914i \(-0.636147\pi\)
0.995407 0.0957335i \(-0.0305196\pi\)
\(500\) 0 0
\(501\) −3.67698 −0.164275
\(502\) 0 0
\(503\) −13.6300 23.6079i −0.607733 1.05262i −0.991613 0.129241i \(-0.958746\pi\)
0.383881 0.923383i \(-0.374587\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.22093 + 14.2391i 0.365104 + 0.632379i
\(508\) 0 0
\(509\) −12.6203 21.8591i −0.559386 0.968885i −0.997548 0.0699892i \(-0.977704\pi\)
0.438161 0.898896i \(-0.355630\pi\)
\(510\) 0 0
\(511\) 16.0794 27.8503i 0.711309 1.23202i
\(512\) 0 0
\(513\) −7.09853 + 8.96981i −0.313408 + 0.396027i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.25806 + 9.10722i 0.231249 + 0.400535i
\(518\) 0 0
\(519\) 0.165602 + 0.286831i 0.00726911 + 0.0125905i
\(520\) 0 0
\(521\) −24.8142 −1.08713 −0.543565 0.839367i \(-0.682926\pi\)
−0.543565 + 0.839367i \(0.682926\pi\)
\(522\) 0 0
\(523\) 3.08063 + 5.33581i 0.134706 + 0.233318i 0.925485 0.378784i \(-0.123658\pi\)
−0.790779 + 0.612102i \(0.790324\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.64855 + 13.2477i −0.333176 + 0.577078i
\(528\) 0 0
\(529\) −20.7321 + 35.9090i −0.901394 + 1.56126i
\(530\) 0 0
\(531\) 20.8874 0.906434
\(532\) 0 0
\(533\) 51.6145 2.23567
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.608765 1.05441i 0.0262702 0.0455013i
\(538\) 0 0
\(539\) −1.17811 −0.0507449
\(540\) 0 0
\(541\) −3.27394 5.67063i −0.140758 0.243799i 0.787025 0.616922i \(-0.211620\pi\)
−0.927782 + 0.373122i \(0.878287\pi\)
\(542\) 0 0
\(543\) 2.49150 0.106921
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −11.2770 19.5323i −0.482169 0.835141i 0.517621 0.855610i \(-0.326818\pi\)
−0.999791 + 0.0204683i \(0.993484\pi\)
\(548\) 0 0
\(549\) −11.6783 + 20.2274i −0.498417 + 0.863284i
\(550\) 0 0
\(551\) −8.12266 1.19328i −0.346037 0.0508353i
\(552\) 0 0
\(553\) −5.41971 + 9.38722i −0.230470 + 0.399185i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.12281 + 14.0691i 0.344175 + 0.596128i 0.985203 0.171389i \(-0.0548254\pi\)
−0.641029 + 0.767517i \(0.721492\pi\)
\(558\) 0 0
\(559\) −25.9397 −1.09713
\(560\) 0 0
\(561\) 1.28779 + 2.23051i 0.0543703 + 0.0941722i
\(562\) 0 0
\(563\) 32.4581 1.36795 0.683974 0.729506i \(-0.260250\pi\)
0.683974 + 0.729506i \(0.260250\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.15258 15.8527i 0.384372 0.665752i
\(568\) 0 0
\(569\) 25.5422 1.07078 0.535392 0.844604i \(-0.320164\pi\)
0.535392 + 0.844604i \(0.320164\pi\)
\(570\) 0 0
\(571\) 25.9974 1.08796 0.543979 0.839099i \(-0.316917\pi\)
0.543979 + 0.839099i \(0.316917\pi\)
\(572\) 0 0
\(573\) 2.12806 3.68590i 0.0889008 0.153981i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 44.5844 1.85607 0.928037 0.372489i \(-0.121496\pi\)
0.928037 + 0.372489i \(0.121496\pi\)
\(578\) 0 0
\(579\) −2.61687 4.53256i −0.108754 0.188367i
\(580\) 0 0
\(581\) 37.5478 1.55774
\(582\) 0 0
\(583\) 11.4424 + 19.8188i 0.473895 + 0.820810i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.28884 12.6246i 0.300842 0.521074i −0.675485 0.737374i \(-0.736065\pi\)
0.976327 + 0.216300i \(0.0693988\pi\)
\(588\) 0 0
\(589\) −16.1788 + 20.4438i −0.666635 + 0.842371i
\(590\) 0 0
\(591\) −1.15891 + 2.00728i −0.0476710 + 0.0825686i
\(592\) 0 0
\(593\) 11.8657 + 20.5520i 0.487265 + 0.843968i 0.999893 0.0146433i \(-0.00466126\pi\)
−0.512628 + 0.858611i \(0.671328\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.08541 −0.249059
\(598\) 0 0
\(599\) 9.86173 + 17.0810i 0.402939 + 0.697911i 0.994079 0.108658i \(-0.0346552\pi\)
−0.591140 + 0.806569i \(0.701322\pi\)
\(600\) 0 0
\(601\) −45.0351 −1.83702 −0.918509 0.395400i \(-0.870606\pi\)
−0.918509 + 0.395400i \(0.870606\pi\)
\(602\) 0 0
\(603\) 17.2726 29.9170i 0.703393 1.21831i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.24160 0.293928 0.146964 0.989142i \(-0.453050\pi\)
0.146964 + 0.989142i \(0.453050\pi\)
\(608\) 0 0
\(609\) −2.16957 −0.0879156
\(610\) 0 0
\(611\) 16.6032 28.7576i 0.671693 1.16341i
\(612\) 0 0
\(613\) −18.4052 + 31.8788i −0.743381 + 1.28757i 0.207567 + 0.978221i \(0.433446\pi\)
−0.950947 + 0.309352i \(0.899888\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.3448 + 31.7741i 0.738533 + 1.27918i 0.953156 + 0.302479i \(0.0978143\pi\)
−0.214623 + 0.976697i \(0.568852\pi\)
\(618\) 0 0
\(619\) 24.1569 0.970948 0.485474 0.874251i \(-0.338647\pi\)
0.485474 + 0.874251i \(0.338647\pi\)
\(620\) 0 0
\(621\) 10.5350 + 18.2471i 0.422754 + 0.732231i
\(622\) 0 0
\(623\) −18.2992 31.6951i −0.733142 1.26984i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.61895 + 4.08013i 0.0646545 + 0.162945i
\(628\) 0 0
\(629\) −3.65866 + 6.33699i −0.145880 + 0.252672i
\(630\) 0 0
\(631\) −7.11258 12.3193i −0.283147 0.490426i 0.689011 0.724751i \(-0.258045\pi\)
−0.972158 + 0.234325i \(0.924712\pi\)
\(632\) 0 0
\(633\) −1.15053 1.99278i −0.0457295 0.0792058i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.86004 + 3.22169i 0.0736976 + 0.127648i
\(638\) 0 0
\(639\) −23.1381 −0.915330
\(640\) 0 0
\(641\) 10.1834 17.6382i 0.402222 0.696668i −0.591772 0.806105i \(-0.701571\pi\)
0.993994 + 0.109437i \(0.0349048\pi\)
\(642\) 0 0
\(643\) 5.77586 10.0041i 0.227778 0.394523i −0.729371 0.684118i \(-0.760187\pi\)
0.957149 + 0.289595i \(0.0935207\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.1233 0.515932 0.257966 0.966154i \(-0.416948\pi\)
0.257966 + 0.966154i \(0.416948\pi\)
\(648\) 0 0
\(649\) 8.30946 14.3924i 0.326175 0.564952i
\(650\) 0 0
\(651\) −3.44483 + 5.96663i −0.135014 + 0.233851i
\(652\) 0 0
\(653\) −19.4406 −0.760769 −0.380385 0.924828i \(-0.624208\pi\)
−0.380385 + 0.924828i \(0.624208\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −35.3353 −1.37856
\(658\) 0 0
\(659\) 13.7848 + 23.8759i 0.536979 + 0.930074i 0.999065 + 0.0432391i \(0.0137677\pi\)
−0.462086 + 0.886835i \(0.652899\pi\)
\(660\) 0 0
\(661\) 3.88006 + 6.72046i 0.150917 + 0.261396i 0.931565 0.363575i \(-0.118444\pi\)
−0.780648 + 0.624971i \(0.785111\pi\)
\(662\) 0 0
\(663\) 4.06640 7.04320i 0.157926 0.273535i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.56114 + 13.0963i −0.292769 + 0.507090i
\(668\) 0 0
\(669\) 5.36468 + 9.29190i 0.207411 + 0.359246i
\(670\) 0 0
\(671\) 9.29178 + 16.0938i 0.358705 + 0.621295i
\(672\) 0 0
\(673\) −24.2617 −0.935220 −0.467610 0.883935i \(-0.654885\pi\)
−0.467610 + 0.883935i \(0.654885\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.88851 0.187881 0.0939403 0.995578i \(-0.470054\pi\)
0.0939403 + 0.995578i \(0.470054\pi\)
\(678\) 0 0
\(679\) 7.41446 12.8422i 0.284541 0.492839i
\(680\) 0 0
\(681\) −4.30902 + 7.46344i −0.165122 + 0.286000i
\(682\) 0 0
\(683\) 0.560031 0.0214290 0.0107145 0.999943i \(-0.496589\pi\)
0.0107145 + 0.999943i \(0.496589\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6.32257 + 10.9510i −0.241221 + 0.417807i
\(688\) 0 0
\(689\) 36.1312 62.5811i 1.37649 2.38415i
\(690\) 0 0
\(691\) −14.5255 −0.552576 −0.276288 0.961075i \(-0.589104\pi\)
−0.276288 + 0.961075i \(0.589104\pi\)
\(692\) 0 0
\(693\) −7.90477 13.6915i −0.300277 0.520095i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −9.39969 16.2807i −0.356039 0.616677i
\(698\) 0 0
\(699\) −3.20754 5.55561i −0.121320 0.210133i
\(700\) 0 0
\(701\) 1.23926 2.14647i 0.0468064 0.0810710i −0.841673 0.539987i \(-0.818429\pi\)
0.888479 + 0.458916i \(0.151762\pi\)
\(702\) 0 0
\(703\) −7.73907 + 9.77921i −0.291885 + 0.368830i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.4749 28.5354i −0.619604 1.07319i
\(708\) 0 0
\(709\) 16.1966 + 28.0533i 0.608276 + 1.05356i 0.991525 + 0.129919i \(0.0414718\pi\)
−0.383249 + 0.923645i \(0.625195\pi\)
\(710\) 0 0
\(711\) 11.9101 0.446665
\(712\) 0 0
\(713\) 24.0110 + 41.5884i 0.899221 + 1.55750i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.85461 6.67637i 0.143953 0.249334i
\(718\) 0 0
\(719\) −8.64373 + 14.9714i −0.322357 + 0.558338i −0.980974 0.194140i \(-0.937808\pi\)
0.658617 + 0.752478i \(0.271142\pi\)
\(720\) 0 0
\(721\) −42.2542 −1.57363
\(722\) 0 0
\(723\) −7.78497 −0.289526
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −9.47034 + 16.4031i −0.351236 + 0.608358i −0.986466 0.163965i \(-0.947572\pi\)
0.635231 + 0.772323i \(0.280905\pi\)
\(728\) 0 0
\(729\) −16.5481 −0.612894
\(730\) 0 0
\(731\) 4.72397 + 8.18216i 0.174722 + 0.302628i
\(732\) 0 0
\(733\) 0.542118 0.0200236 0.0100118 0.999950i \(-0.496813\pi\)
0.0100118 + 0.999950i \(0.496813\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.7428 23.8033i −0.506224 0.876805i
\(738\) 0 0
\(739\) 2.87033 4.97156i 0.105587 0.182882i −0.808391 0.588646i \(-0.799661\pi\)
0.913978 + 0.405764i \(0.132995\pi\)
\(740\) 0 0
\(741\) 8.60154 10.8690i 0.315985 0.399284i
\(742\) 0 0
\(743\) −11.5060 + 19.9290i −0.422115 + 0.731125i −0.996146 0.0877087i \(-0.972046\pi\)
0.574031 + 0.818834i \(0.305379\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −20.6284 35.7294i −0.754752 1.30727i
\(748\) 0 0
\(749\) −11.6824 −0.426866
\(750\) 0 0
\(751\) 25.9704 + 44.9821i 0.947674 + 1.64142i 0.750306 + 0.661090i \(0.229906\pi\)
0.197368 + 0.980330i \(0.436761\pi\)
\(752\) 0 0
\(753\) −10.7646 −0.392283
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −6.36406 + 11.0229i −0.231306 + 0.400633i −0.958193 0.286124i \(-0.907633\pi\)
0.726887 + 0.686757i \(0.240966\pi\)
\(758\) 0 0
\(759\) 8.08547 0.293484
\(760\) 0 0
\(761\) −10.6089 −0.384573 −0.192287 0.981339i \(-0.561590\pi\)
−0.192287 + 0.981339i \(0.561590\pi\)
\(762\) 0 0
\(763\) −3.32969 + 5.76720i −0.120543 + 0.208787i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −52.4770 −1.89484
\(768\) 0 0
\(769\) −10.8089 18.7215i −0.389777 0.675114i 0.602642 0.798012i \(-0.294115\pi\)
−0.992419 + 0.122898i \(0.960781\pi\)
\(770\) 0 0
\(771\) 0.648856 0.0233680
\(772\) 0 0
\(773\) −21.6637 37.5227i −0.779190 1.34960i −0.932409 0.361404i \(-0.882298\pi\)
0.153220 0.988192i \(-0.451036\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.64783 + 2.85412i −0.0591154 + 0.102391i
\(778\) 0 0
\(779\) −11.8169 29.7813i −0.423384 1.06703i
\(780\) 0 0
\(781\) −9.20487 + 15.9433i −0.329376 + 0.570496i
\(782\) 0 0
\(783\) 2.47134 + 4.28049i 0.0883185 + 0.152972i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 29.0046 1.03390 0.516950 0.856015i \(-0.327067\pi\)
0.516950 + 0.856015i \(0.327067\pi\)
\(788\) 0 0
\(789\) −1.61444 2.79629i −0.0574756 0.0995507i
\(790\) 0 0
\(791\) −1.29563 −0.0460674
\(792\) 0 0
\(793\) 29.3403 50.8189i 1.04191 1.80463i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.2045 0.503150 0.251575 0.967838i \(-0.419051\pi\)
0.251575 + 0.967838i \(0.419051\pi\)
\(798\) 0 0
\(799\) −12.0946 −0.427878
\(800\) 0 0
\(801\) −20.1068 + 34.8260i −0.710438 + 1.23052i
\(802\) 0 0
\(803\) −14.0572 + 24.3478i −0.496067 + 0.859214i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.13659 7.16478i −0.145615 0.252212i
\(808\) 0 0
\(809\) 14.9781 0.526603 0.263301 0.964714i \(-0.415189\pi\)
0.263301 + 0.964714i \(0.415189\pi\)
\(810\) 0 0
\(811\) 20.5388 + 35.5742i 0.721214 + 1.24918i 0.960513 + 0.278233i \(0.0897489\pi\)
−0.239300 + 0.970946i \(0.576918\pi\)
\(812\) 0 0
\(813\) −1.31447 2.27674i −0.0461006 0.0798486i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5.93877 + 14.9671i 0.207771 + 0.523632i
\(818\) 0 0
\(819\) −24.9606 + 43.2330i −0.872194 + 1.51068i
\(820\) 0 0
\(821\) 21.5219 + 37.2771i 0.751120 + 1.30098i 0.947280 + 0.320407i \(0.103820\pi\)
−0.196160 + 0.980572i \(0.562847\pi\)
\(822\) 0 0
\(823\) −7.18105 12.4379i −0.250316 0.433559i 0.713297 0.700862i \(-0.247201\pi\)
−0.963613 + 0.267302i \(0.913868\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.1891 + 38.4326i 0.771590 + 1.33643i 0.936691 + 0.350157i \(0.113872\pi\)
−0.165101 + 0.986277i \(0.552795\pi\)
\(828\) 0 0
\(829\) 35.6112 1.23683 0.618413 0.785853i \(-0.287776\pi\)
0.618413 + 0.785853i \(0.287776\pi\)
\(830\) 0 0
\(831\) 1.29563 2.24410i 0.0449450 0.0778470i
\(832\) 0 0
\(833\) 0.677477 1.17342i 0.0234732 0.0406567i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 15.6959 0.542530
\(838\) 0 0
\(839\) 21.1147 36.5718i 0.728962 1.26260i −0.228361 0.973577i \(-0.573337\pi\)
0.957322 0.289022i \(-0.0933301\pi\)
\(840\) 0 0
\(841\) 12.7263 22.0426i 0.438837 0.760088i
\(842\) 0 0
\(843\) 2.78192 0.0958144
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 15.4015 0.529202
\(848\) 0 0
\(849\) 3.72531 + 6.45242i 0.127852 + 0.221447i
\(850\) 0 0
\(851\) 11.4856 + 19.8937i 0.393722 + 0.681946i
\(852\) 0 0
\(853\) 9.90917 17.1632i 0.339284 0.587656i −0.645015 0.764170i \(-0.723149\pi\)
0.984298 + 0.176514i \(0.0564821\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.7082 44.5279i 0.878175 1.52104i 0.0248328 0.999692i \(-0.492095\pi\)
0.853342 0.521352i \(-0.174572\pi\)
\(858\) 0 0
\(859\) 10.5243 + 18.2286i 0.359084 + 0.621951i 0.987808 0.155677i \(-0.0497559\pi\)
−0.628724 + 0.777628i \(0.716423\pi\)
\(860\) 0 0
\(861\) −4.23353 7.33268i −0.144278 0.249897i
\(862\) 0 0
\(863\) 38.8358 1.32199 0.660994 0.750391i \(-0.270135\pi\)
0.660994 + 0.750391i \(0.270135\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.73631 0.160854
\(868\) 0 0
\(869\) 4.73812 8.20667i 0.160730 0.278392i
\(870\) 0 0
\(871\) −43.3953 + 75.1628i −1.47039 + 2.54679i
\(872\) 0 0
\(873\) −16.2937 −0.551459
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0379 22.5823i 0.440259 0.762551i −0.557450 0.830211i \(-0.688220\pi\)
0.997708 + 0.0676600i \(0.0215533\pi\)
\(878\) 0 0
\(879\) −4.43080 + 7.67437i −0.149447 + 0.258850i
\(880\) 0 0
\(881\) 8.23096 0.277308 0.138654 0.990341i \(-0.455722\pi\)
0.138654 + 0.990341i \(0.455722\pi\)
\(882\) 0 0
\(883\) 16.4326 + 28.4622i 0.553003 + 0.957828i 0.998056 + 0.0623244i \(0.0198513\pi\)
−0.445053 + 0.895504i \(0.646815\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.11165 + 12.3177i 0.238786 + 0.413589i 0.960366 0.278741i \(-0.0899173\pi\)
−0.721580 + 0.692331i \(0.756584\pi\)
\(888\) 0 0
\(889\) 0.934139 + 1.61798i 0.0313300 + 0.0542652i
\(890\) 0 0
\(891\) −8.00154 + 13.8591i −0.268062 + 0.464296i
\(892\) 0 0
\(893\) −20.3942 2.99605i −0.682465 0.100259i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −12.7656 22.1107i −0.426231 0.738254i
\(898\) 0 0
\(899\) 5.63262 + 9.75598i 0.187858 + 0.325380i
\(900\) 0 0
\(901\) −26.3199 −0.876843
\(902\) 0 0
\(903\) 2.12763 + 3.68516i 0.0708031 + 0.122635i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.4594 38.9008i 0.745752 1.29168i −0.204090 0.978952i \(-0.565424\pi\)
0.949842 0.312729i \(-0.101243\pi\)
\(908\) 0 0
\(909\) −18.1023 + 31.3542i −0.600417 + 1.03995i
\(910\) 0 0
\(911\) 38.3516 1.27064 0.635322 0.772247i \(-0.280867\pi\)
0.635322 + 0.772247i \(0.280867\pi\)
\(912\) 0 0
\(913\) −32.8257 −1.08637
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.23097 + 12.5244i −0.238788 + 0.413593i
\(918\) 0 0
\(919\) −36.3727 −1.19983 −0.599913 0.800065i \(-0.704798\pi\)
−0.599913 + 0.800065i \(0.704798\pi\)
\(920\) 0 0
\(921\) 6.52081 + 11.2944i 0.214868 + 0.372162i
\(922\) 0 0
\(923\) 58.1318 1.91343
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 23.2140 + 40.2079i 0.762449 + 1.32060i
\(928\) 0 0
\(929\) −1.96542 + 3.40421i −0.0644834 + 0.111689i −0.896465 0.443115i \(-0.853873\pi\)
0.831981 + 0.554804i \(0.187207\pi\)
\(930\) 0 0
\(931\) 1.43305 1.81082i 0.0469663 0.0593473i
\(932\) 0 0
\(933\) −4.39966 + 7.62043i −0.144038 + 0.249482i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 27.0741 + 46.8937i 0.884473 + 1.53195i 0.846317 + 0.532680i \(0.178815\pi\)
0.0381558 + 0.999272i \(0.487852\pi\)
\(938\) 0 0
\(939\) −4.36445 −0.142429
\(940\) 0 0
\(941\) 14.9969 + 25.9755i 0.488886 + 0.846776i 0.999918 0.0127858i \(-0.00406997\pi\)
−0.511032 + 0.859562i \(0.670737\pi\)
\(942\) 0 0
\(943\) −59.0167 −1.92185
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.7095 + 34.1378i −0.640472 + 1.10933i 0.344856 + 0.938656i \(0.387928\pi\)
−0.985328 + 0.170674i \(0.945406\pi\)
\(948\) 0 0
\(949\) 88.7758 2.88179
\(950\) 0 0
\(951\) 0.104471 0.00338772
\(952\) 0 0
\(953\) −0.663142 + 1.14860i −0.0214813 + 0.0372067i −0.876566 0.481281i \(-0.840172\pi\)
0.855085 + 0.518488i \(0.173505\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.89673 0.0613124
\(958\) 0 0
\(959\) −23.9103 41.4138i −0.772102 1.33732i
\(960\) 0 0
\(961\) 4.77372 0.153991
\(962\) 0 0
\(963\) 6.41820 + 11.1167i 0.206824 + 0.358229i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.57052 6.18431i 0.114820 0.198874i −0.802888 0.596130i \(-0.796704\pi\)
0.917708 + 0.397256i \(0.130038\pi\)
\(968\) 0 0
\(969\) −4.99487 0.733782i −0.160458 0.0235725i
\(970\) 0 0
\(971\) −12.5252 + 21.6943i −0.401953 + 0.696202i −0.993962 0.109729i \(-0.965002\pi\)
0.592009 + 0.805931i \(0.298335\pi\)
\(972\) 0 0
\(973\) 5.22371 + 9.04773i 0.167464 + 0.290057i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.00409135 −0.000130894 −6.54469e−5 1.00000i \(-0.500021\pi\)
−6.54469e−5 1.00000i \(0.500021\pi\)
\(978\) 0 0
\(979\) 15.9979 + 27.7091i 0.511294 + 0.885587i
\(980\) 0 0
\(981\) 7.31720 0.233620
\(982\) 0 0
\(983\) −22.1988 + 38.4495i −0.708033 + 1.22635i 0.257553 + 0.966264i \(0.417084\pi\)
−0.965586 + 0.260084i \(0.916250\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −5.44731 −0.173390
\(988\) 0 0
\(989\) 29.6599 0.943128
\(990\) 0 0
\(991\) 23.7636 41.1598i 0.754877 1.30749i −0.190558 0.981676i \(-0.561030\pi\)
0.945435 0.325810i \(-0.105637\pi\)
\(992\) 0 0
\(993\) −2.74282 + 4.75071i −0.0870408 + 0.150759i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −14.8021 25.6381i −0.468788 0.811965i 0.530575 0.847638i \(-0.321976\pi\)
−0.999364 + 0.0356725i \(0.988643\pi\)
\(998\) 0 0
\(999\) 7.50809 0.237545
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 1900.2.i.g.201.6 20
5.2 odd 4 380.2.r.a.49.5 20
5.3 odd 4 380.2.r.a.49.6 yes 20
5.4 even 2 inner 1900.2.i.g.201.5 20
15.2 even 4 3420.2.bj.c.1189.3 20
15.8 even 4 3420.2.bj.c.1189.5 20
19.7 even 3 inner 1900.2.i.g.501.6 20
95.7 odd 12 380.2.r.a.349.6 yes 20
95.64 even 6 inner 1900.2.i.g.501.5 20
95.83 odd 12 380.2.r.a.349.5 yes 20
285.83 even 12 3420.2.bj.c.2629.3 20
285.197 even 12 3420.2.bj.c.2629.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.5 20 5.2 odd 4
380.2.r.a.49.6 yes 20 5.3 odd 4
380.2.r.a.349.5 yes 20 95.83 odd 12
380.2.r.a.349.6 yes 20 95.7 odd 12
1900.2.i.g.201.5 20 5.4 even 2 inner
1900.2.i.g.201.6 20 1.1 even 1 trivial
1900.2.i.g.501.5 20 95.64 even 6 inner
1900.2.i.g.501.6 20 19.7 even 3 inner
3420.2.bj.c.1189.3 20 15.2 even 4
3420.2.bj.c.1189.5 20 15.8 even 4
3420.2.bj.c.2629.3 20 285.83 even 12
3420.2.bj.c.2629.5 20 285.197 even 12