Properties

Label 1500.2.m.c.601.1
Level $1500$
Weight $2$
Character 1500.601
Analytic conductor $11.978$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 601.1
Character \(\chi\) \(=\) 1500.601
Dual form 1500.2.m.c.901.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.809017 - 0.587785i) q^{3} -4.13266 q^{7} +(0.309017 + 0.951057i) q^{9} +O(q^{10})\) \(q+(-0.809017 - 0.587785i) q^{3} -4.13266 q^{7} +(0.309017 + 0.951057i) q^{9} +(-1.16386 + 3.58198i) q^{11} +(-0.215899 - 0.664470i) q^{13} +(4.28173 - 3.11086i) q^{17} +(-4.63966 + 3.37091i) q^{19} +(3.34339 + 2.42912i) q^{21} +(1.68924 - 5.19894i) q^{23} +(0.309017 - 0.951057i) q^{27} +(5.68284 + 4.12883i) q^{29} +(8.16460 - 5.93193i) q^{31} +(3.04701 - 2.21379i) q^{33} +(1.78763 + 5.50175i) q^{37} +(-0.215899 + 0.664470i) q^{39} +(-2.03813 - 6.27271i) q^{41} +4.79668 q^{43} +(-7.82626 - 5.68611i) q^{47} +10.0789 q^{49} -5.29251 q^{51} +(2.74773 + 1.99634i) q^{53} +5.73494 q^{57} +(0.230309 + 0.708820i) q^{59} +(3.64886 - 11.2300i) q^{61} +(-1.27706 - 3.93039i) q^{63} +(3.88133 - 2.81995i) q^{67} +(-4.42248 + 3.21312i) q^{69} +(2.54239 + 1.84715i) q^{71} +(3.32110 - 10.2213i) q^{73} +(4.80982 - 14.8031i) q^{77} +(7.38222 + 5.36349i) q^{79} +(-0.809017 + 0.587785i) q^{81} +(-6.82889 + 4.96148i) q^{83} +(-2.17065 - 6.68058i) q^{87} +(-1.04690 + 3.22202i) q^{89} +(0.892239 + 2.74603i) q^{91} -10.0920 q^{93} +(-8.44215 - 6.13358i) q^{97} -3.76632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{3} + 16 q^{7} - 6 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(877\) \(1001\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(1\)

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
\(3\) −0.809017 0.587785i −0.467086 0.339358i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.13266 −1.56200 −0.780999 0.624532i \(-0.785290\pi\)
−0.780999 + 0.624532i \(0.785290\pi\)
\(8\) 0 0
\(9\) 0.309017 + 0.951057i 0.103006 + 0.317019i
\(10\) 0 0
\(11\) −1.16386 + 3.58198i −0.350916 + 1.08001i 0.607424 + 0.794378i \(0.292203\pi\)
−0.958340 + 0.285630i \(0.907797\pi\)
\(12\) 0 0
\(13\) −0.215899 0.664470i −0.0598797 0.184291i 0.916642 0.399709i \(-0.130889\pi\)
−0.976522 + 0.215418i \(0.930889\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.28173 3.11086i 1.03847 0.754494i 0.0684847 0.997652i \(-0.478184\pi\)
0.969987 + 0.243159i \(0.0781836\pi\)
\(18\) 0 0
\(19\) −4.63966 + 3.37091i −1.06441 + 0.773340i −0.974899 0.222646i \(-0.928531\pi\)
−0.0895120 + 0.995986i \(0.528531\pi\)
\(20\) 0 0
\(21\) 3.34339 + 2.42912i 0.729588 + 0.530077i
\(22\) 0 0
\(23\) 1.68924 5.19894i 0.352231 1.08405i −0.605367 0.795946i \(-0.706974\pi\)
0.957598 0.288108i \(-0.0930263\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.309017 0.951057i 0.0594703 0.183031i
\(28\) 0 0
\(29\) 5.68284 + 4.12883i 1.05528 + 0.766704i 0.973209 0.229923i \(-0.0738474\pi\)
0.0820685 + 0.996627i \(0.473847\pi\)
\(30\) 0 0
\(31\) 8.16460 5.93193i 1.46641 1.06541i 0.484769 0.874642i \(-0.338904\pi\)
0.981636 0.190764i \(-0.0610964\pi\)
\(32\) 0 0
\(33\) 3.04701 2.21379i 0.530417 0.385371i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.78763 + 5.50175i 0.293884 + 0.904482i 0.983594 + 0.180396i \(0.0577381\pi\)
−0.689710 + 0.724086i \(0.742262\pi\)
\(38\) 0 0
\(39\) −0.215899 + 0.664470i −0.0345716 + 0.106400i
\(40\) 0 0
\(41\) −2.03813 6.27271i −0.318302 0.979633i −0.974374 0.224935i \(-0.927783\pi\)
0.656072 0.754698i \(-0.272217\pi\)
\(42\) 0 0
\(43\) 4.79668 0.731488 0.365744 0.930716i \(-0.380815\pi\)
0.365744 + 0.930716i \(0.380815\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.82626 5.68611i −1.14158 0.829405i −0.154239 0.988033i \(-0.549293\pi\)
−0.987338 + 0.158629i \(0.949293\pi\)
\(48\) 0 0
\(49\) 10.0789 1.43984
\(50\) 0 0
\(51\) −5.29251 −0.741099
\(52\) 0 0
\(53\) 2.74773 + 1.99634i 0.377429 + 0.274219i 0.760285 0.649590i \(-0.225059\pi\)
−0.382856 + 0.923808i \(0.625059\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.73494 0.759611
\(58\) 0 0
\(59\) 0.230309 + 0.708820i 0.0299837 + 0.0922804i 0.964928 0.262513i \(-0.0845512\pi\)
−0.934945 + 0.354793i \(0.884551\pi\)
\(60\) 0 0
\(61\) 3.64886 11.2300i 0.467188 1.43786i −0.389021 0.921229i \(-0.627187\pi\)
0.856210 0.516629i \(-0.172813\pi\)
\(62\) 0 0
\(63\) −1.27706 3.93039i −0.160895 0.495183i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.88133 2.81995i 0.474180 0.344512i −0.324888 0.945752i \(-0.605327\pi\)
0.799068 + 0.601241i \(0.205327\pi\)
\(68\) 0 0
\(69\) −4.42248 + 3.21312i −0.532405 + 0.386815i
\(70\) 0 0
\(71\) 2.54239 + 1.84715i 0.301726 + 0.219217i 0.728338 0.685218i \(-0.240293\pi\)
−0.426612 + 0.904435i \(0.640293\pi\)
\(72\) 0 0
\(73\) 3.32110 10.2213i 0.388706 1.19631i −0.545051 0.838403i \(-0.683490\pi\)
0.933756 0.357910i \(-0.116510\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.80982 14.8031i 0.548130 1.68697i
\(78\) 0 0
\(79\) 7.38222 + 5.36349i 0.830564 + 0.603440i 0.919719 0.392578i \(-0.128417\pi\)
−0.0891546 + 0.996018i \(0.528417\pi\)
\(80\) 0 0
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 0 0
\(83\) −6.82889 + 4.96148i −0.749569 + 0.544593i −0.895693 0.444673i \(-0.853320\pi\)
0.146125 + 0.989266i \(0.453320\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.17065 6.68058i −0.232718 0.716234i
\(88\) 0 0
\(89\) −1.04690 + 3.22202i −0.110971 + 0.341533i −0.991085 0.133228i \(-0.957466\pi\)
0.880114 + 0.474761i \(0.157466\pi\)
\(90\) 0 0
\(91\) 0.892239 + 2.74603i 0.0935321 + 0.287862i
\(92\) 0 0
\(93\) −10.0920 −1.04649
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.44215 6.13358i −0.857170 0.622771i 0.0699435 0.997551i \(-0.477718\pi\)
−0.927114 + 0.374780i \(0.877718\pi\)
\(98\) 0 0
\(99\) −3.76632 −0.378529
\(100\) 0 0
\(101\) 7.10799 0.707272 0.353636 0.935383i \(-0.384945\pi\)
0.353636 + 0.935383i \(0.384945\pi\)
\(102\) 0 0
\(103\) 9.87733 + 7.17630i 0.973243 + 0.707102i 0.956188 0.292752i \(-0.0945712\pi\)
0.0170543 + 0.999855i \(0.494571\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.04303 0.390855 0.195427 0.980718i \(-0.437391\pi\)
0.195427 + 0.980718i \(0.437391\pi\)
\(108\) 0 0
\(109\) −0.239122 0.735942i −0.0229037 0.0704904i 0.938951 0.344050i \(-0.111799\pi\)
−0.961855 + 0.273560i \(0.911799\pi\)
\(110\) 0 0
\(111\) 1.78763 5.50175i 0.169674 0.522203i
\(112\) 0 0
\(113\) 0.352104 + 1.08366i 0.0331231 + 0.101943i 0.966251 0.257602i \(-0.0829322\pi\)
−0.933128 + 0.359544i \(0.882932\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.565232 0.410665i 0.0522557 0.0379660i
\(118\) 0 0
\(119\) −17.6949 + 12.8561i −1.62209 + 1.17852i
\(120\) 0 0
\(121\) −2.57684 1.87218i −0.234258 0.170198i
\(122\) 0 0
\(123\) −2.03813 + 6.27271i −0.183772 + 0.565591i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −0.166210 + 0.511543i −0.0147488 + 0.0453921i −0.958160 0.286233i \(-0.907597\pi\)
0.943411 + 0.331625i \(0.107597\pi\)
\(128\) 0 0
\(129\) −3.88060 2.81942i −0.341668 0.248236i
\(130\) 0 0
\(131\) 5.43855 3.95134i 0.475169 0.345230i −0.324284 0.945960i \(-0.605123\pi\)
0.799452 + 0.600730i \(0.205123\pi\)
\(132\) 0 0
\(133\) 19.1741 13.9308i 1.66261 1.20796i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.0566672 0.174404i −0.00484140 0.0149003i 0.948607 0.316458i \(-0.102494\pi\)
−0.953448 + 0.301557i \(0.902494\pi\)
\(138\) 0 0
\(139\) 2.54014 7.81775i 0.215452 0.663093i −0.783669 0.621178i \(-0.786654\pi\)
0.999121 0.0419144i \(-0.0133457\pi\)
\(140\) 0 0
\(141\) 2.98937 + 9.20032i 0.251750 + 0.774807i
\(142\) 0 0
\(143\) 2.63140 0.220048
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −8.15398 5.92421i −0.672529 0.488621i
\(148\) 0 0
\(149\) 5.53790 0.453682 0.226841 0.973932i \(-0.427160\pi\)
0.226841 + 0.973932i \(0.427160\pi\)
\(150\) 0 0
\(151\) 12.8736 1.04764 0.523821 0.851828i \(-0.324506\pi\)
0.523821 + 0.851828i \(0.324506\pi\)
\(152\) 0 0
\(153\) 4.28173 + 3.11086i 0.346157 + 0.251498i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.6395 1.16836 0.584179 0.811625i \(-0.301417\pi\)
0.584179 + 0.811625i \(0.301417\pi\)
\(158\) 0 0
\(159\) −1.04954 3.23015i −0.0832338 0.256167i
\(160\) 0 0
\(161\) −6.98105 + 21.4855i −0.550184 + 1.69329i
\(162\) 0 0
\(163\) 1.37334 + 4.22672i 0.107569 + 0.331062i 0.990325 0.138769i \(-0.0443146\pi\)
−0.882756 + 0.469832i \(0.844315\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.2267 + 13.9690i −1.48780 + 1.08095i −0.512869 + 0.858467i \(0.671417\pi\)
−0.974936 + 0.222487i \(0.928583\pi\)
\(168\) 0 0
\(169\) 10.1223 7.35429i 0.778639 0.565715i
\(170\) 0 0
\(171\) −4.63966 3.37091i −0.354804 0.257780i
\(172\) 0 0
\(173\) 4.66207 14.3484i 0.354451 1.09089i −0.601877 0.798589i \(-0.705580\pi\)
0.956327 0.292298i \(-0.0944198\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.230309 0.708820i 0.0173111 0.0532781i
\(178\) 0 0
\(179\) −13.5238 9.82561i −1.01082 0.734401i −0.0464356 0.998921i \(-0.514786\pi\)
−0.964380 + 0.264521i \(0.914786\pi\)
\(180\) 0 0
\(181\) −8.40755 + 6.10844i −0.624928 + 0.454037i −0.854639 0.519222i \(-0.826222\pi\)
0.229711 + 0.973259i \(0.426222\pi\)
\(182\) 0 0
\(183\) −9.55283 + 6.94054i −0.706166 + 0.513059i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.15971 + 18.9576i 0.450443 + 1.38632i
\(188\) 0 0
\(189\) −1.27706 + 3.93039i −0.0928926 + 0.285894i
\(190\) 0 0
\(191\) −6.10485 18.7888i −0.441732 1.35951i −0.886029 0.463631i \(-0.846546\pi\)
0.444297 0.895880i \(-0.353454\pi\)
\(192\) 0 0
\(193\) 2.23549 0.160914 0.0804571 0.996758i \(-0.474362\pi\)
0.0804571 + 0.996758i \(0.474362\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.4106 + 13.3761i 1.31170 + 0.953008i 0.999996 + 0.00282909i \(0.000900528\pi\)
0.311706 + 0.950178i \(0.399099\pi\)
\(198\) 0 0
\(199\) 22.8171 1.61746 0.808731 0.588179i \(-0.200155\pi\)
0.808731 + 0.588179i \(0.200155\pi\)
\(200\) 0 0
\(201\) −4.79759 −0.338396
\(202\) 0 0
\(203\) −23.4853 17.0630i −1.64834 1.19759i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.46649 0.379947
\(208\) 0 0
\(209\) −6.67464 20.5424i −0.461695 1.42095i
\(210\) 0 0
\(211\) 6.21019 19.1130i 0.427527 1.31579i −0.473027 0.881048i \(-0.656839\pi\)
0.900554 0.434745i \(-0.143161\pi\)
\(212\) 0 0
\(213\) −0.971105 2.98875i −0.0665390 0.204786i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −33.7415 + 24.5146i −2.29052 + 1.66416i
\(218\) 0 0
\(219\) −8.69476 + 6.31711i −0.587537 + 0.426871i
\(220\) 0 0
\(221\) −2.99149 2.17345i −0.201230 0.146202i
\(222\) 0 0
\(223\) −4.14665 + 12.7621i −0.277680 + 0.854612i 0.710817 + 0.703377i \(0.248325\pi\)
−0.988498 + 0.151236i \(0.951675\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.71411 17.5862i 0.379259 1.16724i −0.561301 0.827612i \(-0.689699\pi\)
0.940560 0.339628i \(-0.110301\pi\)
\(228\) 0 0
\(229\) 15.7050 + 11.4104i 1.03782 + 0.754017i 0.969858 0.243671i \(-0.0783516\pi\)
0.0679579 + 0.997688i \(0.478352\pi\)
\(230\) 0 0
\(231\) −12.5923 + 9.14882i −0.828511 + 0.601948i
\(232\) 0 0
\(233\) −9.85646 + 7.16114i −0.645718 + 0.469142i −0.861810 0.507231i \(-0.830669\pi\)
0.216092 + 0.976373i \(0.430669\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.81976 8.67832i −0.183163 0.563717i
\(238\) 0 0
\(239\) 1.39823 4.30330i 0.0904438 0.278357i −0.895596 0.444869i \(-0.853250\pi\)
0.986040 + 0.166512i \(0.0532503\pi\)
\(240\) 0 0
\(241\) 2.78479 + 8.57071i 0.179384 + 0.552088i 0.999807 0.0196702i \(-0.00626163\pi\)
−0.820422 + 0.571758i \(0.806262\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.24157 + 2.35514i 0.206256 + 0.149854i
\(248\) 0 0
\(249\) 8.44098 0.534925
\(250\) 0 0
\(251\) 3.16965 0.200066 0.100033 0.994984i \(-0.468105\pi\)
0.100033 + 0.994984i \(0.468105\pi\)
\(252\) 0 0
\(253\) 16.6565 + 12.1016i 1.04718 + 0.760824i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.25241 −0.202880 −0.101440 0.994842i \(-0.532345\pi\)
−0.101440 + 0.994842i \(0.532345\pi\)
\(258\) 0 0
\(259\) −7.38766 22.7369i −0.459047 1.41280i
\(260\) 0 0
\(261\) −2.17065 + 6.68058i −0.134360 + 0.413518i
\(262\) 0 0
\(263\) 3.41774 + 10.5187i 0.210747 + 0.648612i 0.999428 + 0.0338097i \(0.0107640\pi\)
−0.788682 + 0.614802i \(0.789236\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.74081 1.99132i 0.167735 0.121867i
\(268\) 0 0
\(269\) −20.0279 + 14.5511i −1.22112 + 0.887197i −0.996193 0.0871780i \(-0.972215\pi\)
−0.224929 + 0.974375i \(0.572215\pi\)
\(270\) 0 0
\(271\) −0.202420 0.147067i −0.0122962 0.00893369i 0.581620 0.813460i \(-0.302419\pi\)
−0.593916 + 0.804527i \(0.702419\pi\)
\(272\) 0 0
\(273\) 0.892239 2.74603i 0.0540008 0.166197i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.74121 + 8.43658i −0.164703 + 0.506905i −0.999014 0.0443894i \(-0.985866\pi\)
0.834311 + 0.551294i \(0.185866\pi\)
\(278\) 0 0
\(279\) 8.16460 + 5.93193i 0.488802 + 0.355135i
\(280\) 0 0
\(281\) −15.9683 + 11.6016i −0.952588 + 0.692096i −0.951418 0.307904i \(-0.900373\pi\)
−0.00117050 + 0.999999i \(0.500373\pi\)
\(282\) 0 0
\(283\) −21.3919 + 15.5421i −1.27162 + 0.923882i −0.999266 0.0383159i \(-0.987801\pi\)
−0.272349 + 0.962198i \(0.587801\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.42289 + 25.9230i 0.497187 + 1.53019i
\(288\) 0 0
\(289\) 3.40247 10.4717i 0.200145 0.615983i
\(290\) 0 0
\(291\) 3.22461 + 9.92434i 0.189030 + 0.581775i
\(292\) 0 0
\(293\) −19.1882 −1.12099 −0.560494 0.828158i \(-0.689389\pi\)
−0.560494 + 0.828158i \(0.689389\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.04701 + 2.21379i 0.176806 + 0.128457i
\(298\) 0 0
\(299\) −3.81925 −0.220873
\(300\) 0 0
\(301\) −19.8231 −1.14258
\(302\) 0 0
\(303\) −5.75049 4.17797i −0.330357 0.240018i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9.26979 0.529055 0.264527 0.964378i \(-0.414784\pi\)
0.264527 + 0.964378i \(0.414784\pi\)
\(308\) 0 0
\(309\) −3.77281 11.6115i −0.214627 0.660555i
\(310\) 0 0
\(311\) −7.69255 + 23.6752i −0.436204 + 1.34250i 0.455643 + 0.890163i \(0.349409\pi\)
−0.891847 + 0.452336i \(0.850591\pi\)
\(312\) 0 0
\(313\) −1.86866 5.75115i −0.105623 0.325074i 0.884253 0.467008i \(-0.154668\pi\)
−0.989876 + 0.141934i \(0.954668\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.6389 8.45614i 0.653705 0.474944i −0.210827 0.977523i \(-0.567616\pi\)
0.864531 + 0.502579i \(0.167616\pi\)
\(318\) 0 0
\(319\) −21.4034 + 15.5505i −1.19836 + 0.870659i
\(320\) 0 0
\(321\) −3.27088 2.37644i −0.182563 0.132640i
\(322\) 0 0
\(323\) −9.37934 + 28.8666i −0.521881 + 1.60618i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −0.239122 + 0.735942i −0.0132235 + 0.0406977i
\(328\) 0 0
\(329\) 32.3433 + 23.4988i 1.78314 + 1.29553i
\(330\) 0 0
\(331\) −0.863126 + 0.627098i −0.0474417 + 0.0344684i −0.611253 0.791435i \(-0.709334\pi\)
0.563812 + 0.825903i \(0.309334\pi\)
\(332\) 0 0
\(333\) −4.68007 + 3.40027i −0.256466 + 0.186334i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.51226 13.8873i −0.245799 0.756490i −0.995504 0.0947186i \(-0.969805\pi\)
0.749705 0.661772i \(-0.230195\pi\)
\(338\) 0 0
\(339\) 0.352104 1.08366i 0.0191236 0.0588565i
\(340\) 0 0
\(341\) 11.7456 + 36.1493i 0.636062 + 1.95760i
\(342\) 0 0
\(343\) −12.7239 −0.687029
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.0618 9.48995i −0.701194 0.509447i 0.179127 0.983826i \(-0.442673\pi\)
−0.880321 + 0.474379i \(0.842673\pi\)
\(348\) 0 0
\(349\) −22.2622 −1.19167 −0.595834 0.803108i \(-0.703178\pi\)
−0.595834 + 0.803108i \(0.703178\pi\)
\(350\) 0 0
\(351\) −0.698665 −0.0372920
\(352\) 0 0
\(353\) 7.70991 + 5.60158i 0.410357 + 0.298142i 0.773746 0.633496i \(-0.218381\pi\)
−0.363389 + 0.931637i \(0.618381\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 21.8721 1.15760
\(358\) 0 0
\(359\) −5.18753 15.9656i −0.273788 0.842632i −0.989538 0.144275i \(-0.953915\pi\)
0.715750 0.698357i \(-0.246085\pi\)
\(360\) 0 0
\(361\) 4.29210 13.2097i 0.225900 0.695248i
\(362\) 0 0
\(363\) 0.984264 + 3.02925i 0.0516604 + 0.158995i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −26.2741 + 19.0893i −1.37150 + 0.996452i −0.373880 + 0.927477i \(0.621973\pi\)
−0.997618 + 0.0689751i \(0.978027\pi\)
\(368\) 0 0
\(369\) 5.33589 3.87675i 0.277775 0.201815i
\(370\) 0 0
\(371\) −11.3554 8.25020i −0.589544 0.428329i
\(372\) 0 0
\(373\) −4.41623 + 13.5918i −0.228664 + 0.703755i 0.769235 + 0.638966i \(0.220637\pi\)
−0.997899 + 0.0647890i \(0.979363\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.51656 4.66749i 0.0781068 0.240388i
\(378\) 0 0
\(379\) 2.60155 + 1.89014i 0.133633 + 0.0970897i 0.652593 0.757708i \(-0.273681\pi\)
−0.518961 + 0.854798i \(0.673681\pi\)
\(380\) 0 0
\(381\) 0.435145 0.316151i 0.0222931 0.0161969i
\(382\) 0 0
\(383\) 9.99929 7.26491i 0.510940 0.371220i −0.302240 0.953232i \(-0.597734\pi\)
0.813180 + 0.582012i \(0.197734\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.48226 + 4.56192i 0.0753474 + 0.231895i
\(388\) 0 0
\(389\) 7.14594 21.9930i 0.362314 1.11509i −0.589332 0.807891i \(-0.700609\pi\)
0.951646 0.307196i \(-0.0993908\pi\)
\(390\) 0 0
\(391\) −8.94031 27.5154i −0.452131 1.39152i
\(392\) 0 0
\(393\) −6.72242 −0.339101
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.72179 1.25095i −0.0864140 0.0627834i 0.543739 0.839254i \(-0.317008\pi\)
−0.630153 + 0.776471i \(0.717008\pi\)
\(398\) 0 0
\(399\) −23.7005 −1.18651
\(400\) 0 0
\(401\) −33.8250 −1.68914 −0.844569 0.535447i \(-0.820143\pi\)
−0.844569 + 0.535447i \(0.820143\pi\)
\(402\) 0 0
\(403\) −5.70432 4.14443i −0.284153 0.206449i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.7877 −1.07998
\(408\) 0 0
\(409\) −10.7434 33.0647i −0.531226 1.63495i −0.751665 0.659545i \(-0.770749\pi\)
0.220439 0.975401i \(-0.429251\pi\)
\(410\) 0 0
\(411\) −0.0566672 + 0.174404i −0.00279518 + 0.00860269i
\(412\) 0 0
\(413\) −0.951790 2.92931i −0.0468345 0.144142i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.65017 + 4.83163i −0.325660 + 0.236606i
\(418\) 0 0
\(419\) 5.49925 3.99544i 0.268656 0.195190i −0.445298 0.895382i \(-0.646902\pi\)
0.713954 + 0.700192i \(0.246902\pi\)
\(420\) 0 0
\(421\) 8.75730 + 6.36255i 0.426805 + 0.310092i 0.780370 0.625318i \(-0.215031\pi\)
−0.353565 + 0.935410i \(0.615031\pi\)
\(422\) 0 0
\(423\) 2.98937 9.20032i 0.145348 0.447335i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −15.0795 + 46.4099i −0.729747 + 2.24593i
\(428\) 0 0
\(429\) −2.12884 1.54670i −0.102782 0.0746752i
\(430\) 0 0
\(431\) 8.40317 6.10526i 0.404767 0.294080i −0.366713 0.930334i \(-0.619517\pi\)
0.771480 + 0.636254i \(0.219517\pi\)
\(432\) 0 0
\(433\) 12.6714 9.20630i 0.608948 0.442427i −0.240096 0.970749i \(-0.577179\pi\)
0.849044 + 0.528323i \(0.177179\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.68768 + 29.8156i 0.463425 + 1.42627i
\(438\) 0 0
\(439\) 6.80071 20.9304i 0.324580 0.998955i −0.647050 0.762448i \(-0.723997\pi\)
0.971630 0.236507i \(-0.0760026\pi\)
\(440\) 0 0
\(441\) 3.11454 + 9.58558i 0.148312 + 0.456456i
\(442\) 0 0
\(443\) 29.4447 1.39896 0.699480 0.714652i \(-0.253415\pi\)
0.699480 + 0.714652i \(0.253415\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.48025 3.25509i −0.211909 0.153961i
\(448\) 0 0
\(449\) 27.3445 1.29047 0.645234 0.763985i \(-0.276760\pi\)
0.645234 + 0.763985i \(0.276760\pi\)
\(450\) 0 0
\(451\) 24.8408 1.16971
\(452\) 0 0
\(453\) −10.4150 7.56693i −0.489339 0.355526i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 33.6394 1.57359 0.786793 0.617217i \(-0.211740\pi\)
0.786793 + 0.617217i \(0.211740\pi\)
\(458\) 0 0
\(459\) −1.63547 5.03347i −0.0763374 0.234942i
\(460\) 0 0
\(461\) 7.86199 24.1967i 0.366169 1.12695i −0.583076 0.812417i \(-0.698151\pi\)
0.949246 0.314536i \(-0.101849\pi\)
\(462\) 0 0
\(463\) 2.42308 + 7.45749i 0.112610 + 0.346579i 0.991441 0.130555i \(-0.0416758\pi\)
−0.878831 + 0.477134i \(0.841676\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.73792 6.34847i 0.404343 0.293772i −0.366965 0.930235i \(-0.619603\pi\)
0.771307 + 0.636463i \(0.219603\pi\)
\(468\) 0 0
\(469\) −16.0402 + 11.6539i −0.740668 + 0.538127i
\(470\) 0 0
\(471\) −11.8436 8.60487i −0.545724 0.396491i
\(472\) 0 0
\(473\) −5.58265 + 17.1816i −0.256691 + 0.790012i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.04954 + 3.23015i −0.0480551 + 0.147898i
\(478\) 0 0
\(479\) −18.3447 13.3282i −0.838191 0.608981i 0.0836740 0.996493i \(-0.473335\pi\)
−0.921865 + 0.387512i \(0.873335\pi\)
\(480\) 0 0
\(481\) 3.26980 2.37565i 0.149090 0.108320i
\(482\) 0 0
\(483\) 18.2766 13.2787i 0.831615 0.604204i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.54072 + 4.74184i 0.0698166 + 0.214873i 0.979877 0.199602i \(-0.0639650\pi\)
−0.910060 + 0.414476i \(0.863965\pi\)
\(488\) 0 0
\(489\) 1.37334 4.22672i 0.0621048 0.191139i
\(490\) 0 0
\(491\) −8.17443 25.1583i −0.368907 1.13538i −0.947498 0.319761i \(-0.896397\pi\)
0.578592 0.815617i \(-0.303603\pi\)
\(492\) 0 0
\(493\) 37.1766 1.67435
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.5068 7.63365i −0.471295 0.342416i
\(498\) 0 0
\(499\) 38.0875 1.70503 0.852516 0.522702i \(-0.175076\pi\)
0.852516 + 0.522702i \(0.175076\pi\)
\(500\) 0 0
\(501\) 23.7655 1.06176
\(502\) 0 0
\(503\) 1.90603 + 1.38481i 0.0849856 + 0.0617456i 0.629467 0.777027i \(-0.283273\pi\)
−0.544481 + 0.838773i \(0.683273\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.5119 −0.555672
\(508\) 0 0
\(509\) −9.02282 27.7694i −0.399929 1.23086i −0.925056 0.379832i \(-0.875982\pi\)
0.525126 0.851024i \(-0.324018\pi\)
\(510\) 0 0
\(511\) −13.7250 + 42.2412i −0.607158 + 1.86864i
\(512\) 0 0
\(513\) 1.77219 + 5.45425i 0.0782442 + 0.240811i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 29.4762 21.4157i 1.29636 0.941862i
\(518\) 0 0
\(519\) −12.2055 + 8.86778i −0.535760 + 0.389252i
\(520\) 0 0
\(521\) 34.4787 + 25.0503i 1.51054 + 1.09747i 0.965937 + 0.258779i \(0.0833200\pi\)
0.544604 + 0.838693i \(0.316680\pi\)
\(522\) 0 0
\(523\) 6.38893 19.6631i 0.279368 0.859808i −0.708662 0.705548i \(-0.750701\pi\)
0.988030 0.154259i \(-0.0492992\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.5052 50.7978i 0.718978 2.21279i
\(528\) 0 0
\(529\) −5.56809 4.04545i −0.242091 0.175889i
\(530\) 0 0
\(531\) −0.602958 + 0.438075i −0.0261661 + 0.0190108i
\(532\) 0 0
\(533\) −3.72800 + 2.70855i −0.161478 + 0.117320i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.16563 + 15.8982i 0.222913 + 0.686057i
\(538\) 0 0
\(539\) −11.7304 + 36.1023i −0.505262 + 1.55504i
\(540\) 0 0
\(541\) 3.69426 + 11.3698i 0.158829 + 0.488824i 0.998529 0.0542264i \(-0.0172693\pi\)
−0.839700 + 0.543050i \(0.817269\pi\)
\(542\) 0 0
\(543\) 10.3923 0.445976
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −21.5220 15.6366i −0.920214 0.668575i 0.0233630 0.999727i \(-0.492563\pi\)
−0.943577 + 0.331152i \(0.892563\pi\)
\(548\) 0 0
\(549\) 11.8079 0.503951
\(550\) 0 0
\(551\) −40.2844 −1.71617
\(552\) 0 0
\(553\) −30.5082 22.1655i −1.29734 0.942573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −28.3420 −1.20089 −0.600444 0.799667i \(-0.705010\pi\)
−0.600444 + 0.799667i \(0.705010\pi\)
\(558\) 0 0
\(559\) −1.03560 3.18725i −0.0438013 0.134806i
\(560\) 0 0
\(561\) 6.15971 18.9576i 0.260063 0.800393i
\(562\) 0 0
\(563\) 1.25342 + 3.85764i 0.0528255 + 0.162580i 0.973989 0.226596i \(-0.0727597\pi\)
−0.921163 + 0.389176i \(0.872760\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.34339 2.42912i 0.140409 0.102013i
\(568\) 0 0
\(569\) −1.59124 + 1.15610i −0.0667081 + 0.0484663i −0.620640 0.784096i \(-0.713127\pi\)
0.553931 + 0.832562i \(0.313127\pi\)
\(570\) 0 0
\(571\) −4.17273 3.03167i −0.174623 0.126871i 0.497040 0.867727i \(-0.334420\pi\)
−0.671664 + 0.740856i \(0.734420\pi\)
\(572\) 0 0
\(573\) −6.10485 + 18.7888i −0.255034 + 0.784914i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.9338 36.7286i 0.496812 1.52903i −0.317301 0.948325i \(-0.602776\pi\)
0.814113 0.580706i \(-0.197224\pi\)
\(578\) 0 0
\(579\) −1.80855 1.31399i −0.0751608 0.0546075i
\(580\) 0 0
\(581\) 28.2215 20.5041i 1.17083 0.850654i
\(582\) 0 0
\(583\) −10.3488 + 7.51885i −0.428604 + 0.311399i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.39659 + 25.8420i 0.346564 + 1.06662i 0.960741 + 0.277447i \(0.0894882\pi\)
−0.614177 + 0.789168i \(0.710512\pi\)
\(588\) 0 0
\(589\) −17.8850 + 55.0443i −0.736937 + 2.26806i
\(590\) 0 0
\(591\) −7.03223 21.6430i −0.289267 0.890273i
\(592\) 0 0
\(593\) −30.3486 −1.24627 −0.623135 0.782114i \(-0.714141\pi\)
−0.623135 + 0.782114i \(0.714141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.4594 13.4116i −0.755494 0.548899i
\(598\) 0 0
\(599\) −26.9205 −1.09994 −0.549971 0.835184i \(-0.685361\pi\)
−0.549971 + 0.835184i \(0.685361\pi\)
\(600\) 0 0
\(601\) −16.9133 −0.689910 −0.344955 0.938619i \(-0.612106\pi\)
−0.344955 + 0.938619i \(0.612106\pi\)
\(602\) 0 0
\(603\) 3.88133 + 2.81995i 0.158060 + 0.114837i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.9715 0.770028 0.385014 0.922911i \(-0.374197\pi\)
0.385014 + 0.922911i \(0.374197\pi\)
\(608\) 0 0
\(609\) 8.97057 + 27.6086i 0.363506 + 1.11876i
\(610\) 0 0
\(611\) −2.08857 + 6.42795i −0.0844944 + 0.260047i
\(612\) 0 0
\(613\) 1.68666 + 5.19102i 0.0681237 + 0.209663i 0.979323 0.202302i \(-0.0648424\pi\)
−0.911199 + 0.411966i \(0.864842\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −9.62485 + 6.99287i −0.387482 + 0.281522i −0.764423 0.644715i \(-0.776976\pi\)
0.376941 + 0.926237i \(0.376976\pi\)
\(618\) 0 0
\(619\) 10.6888 7.76586i 0.429619 0.312136i −0.351878 0.936046i \(-0.614457\pi\)
0.781496 + 0.623910i \(0.214457\pi\)
\(620\) 0 0
\(621\) −4.42248 3.21312i −0.177468 0.128938i
\(622\) 0 0
\(623\) 4.32647 13.3155i 0.173336 0.533475i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −6.67464 + 20.5424i −0.266560 + 0.820386i
\(628\) 0 0
\(629\) 24.7693 + 17.9959i 0.987616 + 0.717545i
\(630\) 0 0
\(631\) 9.39932 6.82900i 0.374181 0.271858i −0.384762 0.923016i \(-0.625716\pi\)
0.758943 + 0.651158i \(0.225716\pi\)
\(632\) 0 0
\(633\) −16.2585 + 11.8125i −0.646217 + 0.469504i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.17602 6.69711i −0.0862172 0.265349i
\(638\) 0 0
\(639\) −0.971105 + 2.98875i −0.0384163 + 0.118233i
\(640\) 0 0
\(641\) −10.7784 33.1725i −0.425721 1.31023i −0.902303 0.431103i \(-0.858124\pi\)
0.476582 0.879130i \(-0.341876\pi\)
\(642\) 0 0
\(643\) −2.90629 −0.114613 −0.0573065 0.998357i \(-0.518251\pi\)
−0.0573065 + 0.998357i \(0.518251\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.82163 + 7.13583i 0.386128 + 0.280538i 0.763867 0.645374i \(-0.223298\pi\)
−0.377739 + 0.925912i \(0.623298\pi\)
\(648\) 0 0
\(649\) −2.80702 −0.110185
\(650\) 0 0
\(651\) 41.7068 1.63462
\(652\) 0 0
\(653\) 21.1143 + 15.3404i 0.826266 + 0.600318i 0.918500 0.395420i \(-0.129401\pi\)
−0.0922343 + 0.995737i \(0.529401\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.7473 0.419293
\(658\) 0 0
\(659\) 6.59035 + 20.2830i 0.256724 + 0.790114i 0.993485 + 0.113962i \(0.0363541\pi\)
−0.736761 + 0.676153i \(0.763646\pi\)
\(660\) 0 0
\(661\) −9.79161 + 30.1355i −0.380850 + 1.17213i 0.558597 + 0.829439i \(0.311340\pi\)
−0.939447 + 0.342695i \(0.888660\pi\)
\(662\) 0 0
\(663\) 1.14265 + 3.51671i 0.0443768 + 0.136578i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 31.0652 22.5702i 1.20285 0.873922i
\(668\) 0 0
\(669\) 10.8561 7.88740i 0.419720 0.304945i
\(670\) 0 0
\(671\) 35.9790 + 26.1403i 1.38895 + 1.00913i
\(672\) 0 0
\(673\) −0.613635 + 1.88857i −0.0236539 + 0.0727992i −0.962187 0.272391i \(-0.912186\pi\)
0.938533 + 0.345190i \(0.112186\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.1864 37.5057i 0.468360 1.44146i −0.386348 0.922353i \(-0.626264\pi\)
0.854708 0.519110i \(-0.173736\pi\)
\(678\) 0 0
\(679\) 34.8885 + 25.3480i 1.33890 + 0.972767i
\(680\) 0 0
\(681\) −14.9597 + 10.8689i −0.573259 + 0.416497i
\(682\) 0 0
\(683\) 9.94805 7.22768i 0.380651 0.276559i −0.380962 0.924590i \(-0.624407\pi\)
0.761614 + 0.648031i \(0.224407\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.99878 18.4623i −0.228868 0.704382i
\(688\) 0 0
\(689\) 0.733276 2.25679i 0.0279356 0.0859769i
\(690\) 0 0
\(691\) 4.85334 + 14.9371i 0.184630 + 0.568232i 0.999942 0.0107896i \(-0.00343451\pi\)
−0.815312 + 0.579022i \(0.803435\pi\)
\(692\) 0 0
\(693\) 15.5649 0.591262
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −28.2402 20.5177i −1.06967 0.777164i
\(698\) 0 0
\(699\) 12.1833 0.460813
\(700\) 0 0
\(701\) −26.4222 −0.997953 −0.498976 0.866616i \(-0.666291\pi\)
−0.498976 + 0.866616i \(0.666291\pi\)
\(702\) 0 0
\(703\) −26.8399 19.5003i −1.01229 0.735469i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.3749 −1.10476
\(708\) 0 0
\(709\) 15.0638 + 46.3616i 0.565733 + 1.74115i 0.665763 + 0.746163i \(0.268106\pi\)
−0.100030 + 0.994984i \(0.531894\pi\)
\(710\) 0 0
\(711\) −2.81976 + 8.67832i −0.105749 + 0.325462i
\(712\) 0 0
\(713\) −17.0478 52.4677i −0.638445 1.96493i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.66061 + 2.65959i −0.136708 + 0.0993241i
\(718\) 0 0
\(719\) 27.7117 20.1337i 1.03347 0.750861i 0.0644713 0.997920i \(-0.479464\pi\)
0.969001 + 0.247058i \(0.0794639\pi\)
\(720\) 0 0
\(721\) −40.8197 29.6572i −1.52020 1.10449i
\(722\) 0 0
\(723\) 2.78479 8.57071i 0.103568 0.318748i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.60134 20.3168i 0.244830 0.753510i −0.750834 0.660491i \(-0.770348\pi\)
0.995664 0.0930190i \(-0.0296517\pi\)
\(728\) 0 0
\(729\) −0.809017 0.587785i −0.0299636 0.0217698i
\(730\) 0 0
\(731\) 20.5381 14.9218i 0.759629 0.551903i
\(732\) 0 0
\(733\) 25.3083 18.3875i 0.934782 0.679159i −0.0123772 0.999923i \(-0.503940\pi\)
0.947159 + 0.320765i \(0.103940\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.58370 + 17.1849i 0.205678 + 0.633013i
\(738\) 0 0
\(739\) 0.406330 1.25056i 0.0149471 0.0460025i −0.943305 0.331928i \(-0.892301\pi\)
0.958252 + 0.285925i \(0.0923009\pi\)
\(740\) 0 0
\(741\) −1.23817 3.81070i −0.0454853 0.139989i
\(742\) 0 0
\(743\) −8.76431 −0.321531 −0.160766 0.986993i \(-0.551396\pi\)
−0.160766 + 0.986993i \(0.551396\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −6.82889 4.96148i −0.249856 0.181531i
\(748\) 0 0
\(749\) −16.7085 −0.610515
\(750\) 0 0
\(751\) 44.6570 1.62956 0.814779 0.579772i \(-0.196858\pi\)
0.814779 + 0.579772i \(0.196858\pi\)
\(752\) 0 0
\(753\) −2.56430 1.86307i −0.0934482 0.0678941i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.70729 −0.280126 −0.140063 0.990143i \(-0.544731\pi\)
−0.140063 + 0.990143i \(0.544731\pi\)
\(758\) 0 0
\(759\) −6.36221 19.5809i −0.230934 0.710740i
\(760\) 0 0
\(761\) 12.2344 37.6537i 0.443498 1.36495i −0.440625 0.897691i \(-0.645243\pi\)
0.884123 0.467254i \(-0.154757\pi\)
\(762\) 0 0
\(763\) 0.988210 + 3.04140i 0.0357756 + 0.110106i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.421266 0.306068i 0.0152110 0.0110515i
\(768\) 0 0
\(769\) −23.7336 + 17.2435i −0.855855 + 0.621815i −0.926754 0.375668i \(-0.877413\pi\)
0.0708991 + 0.997483i \(0.477413\pi\)
\(770\) 0 0
\(771\) 2.63126 + 1.91172i 0.0947624 + 0.0688489i
\(772\) 0 0
\(773\) 3.37142 10.3762i 0.121262 0.373205i −0.871940 0.489613i \(-0.837138\pi\)
0.993201 + 0.116408i \(0.0371381\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −7.38766 + 22.7369i −0.265031 + 0.815681i
\(778\) 0 0
\(779\) 30.6010 + 22.2329i 1.09639 + 0.796577i
\(780\) 0 0
\(781\) −9.57543 + 6.95696i −0.342636 + 0.248940i
\(782\) 0 0
\(783\) 5.68284 4.12883i 0.203088 0.147552i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −11.6522 35.8619i −0.415358 1.27834i −0.911931 0.410344i \(-0.865409\pi\)
0.496573 0.867995i \(-0.334591\pi\)
\(788\) 0 0
\(789\) 3.41774 10.5187i 0.121675 0.374476i
\(790\) 0 0
\(791\) −1.45512 4.47841i −0.0517383 0.159234i
\(792\) 0 0
\(793\) −8.24980 −0.292959
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.8369 + 28.9432i 1.41110 + 1.02522i 0.993162 + 0.116745i \(0.0372461\pi\)
0.417935 + 0.908477i \(0.362754\pi\)
\(798\) 0 0
\(799\) −51.1986 −1.81128
\(800\) 0 0
\(801\) −3.38783 −0.119703
\(802\) 0 0
\(803\) 32.7472 + 23.7922i 1.15562 + 0.839610i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.7558 0.871447
\(808\) 0 0
\(809\) 8.33675 + 25.6579i 0.293105 + 0.902083i 0.983852 + 0.178987i \(0.0572819\pi\)
−0.690747 + 0.723097i \(0.742718\pi\)
\(810\) 0 0
\(811\) 2.52546 7.77257i 0.0886809 0.272932i −0.896875 0.442285i \(-0.854168\pi\)
0.985555 + 0.169353i \(0.0541679\pi\)
\(812\) 0 0
\(813\) 0.0773177 + 0.237960i 0.00271165 + 0.00834561i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −22.2550 + 16.1692i −0.778604 + 0.565689i
\(818\) 0 0
\(819\) −2.33591 + 1.69714i −0.0816234 + 0.0593029i
\(820\) 0 0
\(821\) 35.4256 + 25.7382i 1.23636 + 0.898269i 0.997350 0.0727478i \(-0.0231768\pi\)
0.239011 + 0.971017i \(0.423177\pi\)
\(822\) 0 0
\(823\) 5.08540 15.6512i 0.177266 0.545568i −0.822464 0.568817i \(-0.807401\pi\)
0.999730 + 0.0232493i \(0.00740114\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.24174 + 3.82167i −0.0431794 + 0.132893i −0.970322 0.241816i \(-0.922257\pi\)
0.927143 + 0.374708i \(0.122257\pi\)
\(828\) 0 0
\(829\) −30.3103 22.0217i −1.05272 0.764847i −0.0799935 0.996795i \(-0.525490\pi\)
−0.972728 + 0.231948i \(0.925490\pi\)
\(830\) 0 0
\(831\) 7.17658 5.21409i 0.248953 0.180875i
\(832\) 0 0
\(833\) 43.1550 31.3539i 1.49523 1.08635i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.11860 9.59806i −0.107795 0.331758i
\(838\) 0 0
\(839\) −3.58335 + 11.0284i −0.123711 + 0.380743i −0.993664 0.112391i \(-0.964149\pi\)
0.869953 + 0.493135i \(0.164149\pi\)
\(840\) 0 0
\(841\) 6.28600 + 19.3463i 0.216759 + 0.667114i
\(842\) 0 0
\(843\) 19.7379 0.679809
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.6492 + 7.73709i 0.365910 + 0.265849i
\(848\) 0 0
\(849\) 26.4418 0.907481
\(850\) 0 0
\(851\) 31.6230 1.08402
\(852\) 0 0
\(853\) −1.06950 0.777036i −0.0366189 0.0266052i 0.569325 0.822112i \(-0.307205\pi\)
−0.605944 + 0.795507i \(0.707205\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.5623 −0.531598 −0.265799 0.964028i \(-0.585636\pi\)
−0.265799 + 0.964028i \(0.585636\pi\)
\(858\) 0 0
\(859\) −8.46169 26.0424i −0.288709 0.888556i −0.985262 0.171050i \(-0.945284\pi\)
0.696553 0.717505i \(-0.254716\pi\)
\(860\) 0 0
\(861\) 8.42289 25.9230i 0.287051 0.883453i
\(862\) 0 0
\(863\) −9.89760 30.4617i −0.336918 1.03693i −0.965769 0.259402i \(-0.916475\pi\)
0.628851 0.777526i \(-0.283525\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.90777 + 6.47188i −0.302524 + 0.219796i
\(868\) 0 0
\(869\) −27.8038 + 20.2006i −0.943178 + 0.685259i
\(870\) 0 0
\(871\) −2.71175 1.97020i −0.0918842 0.0667578i
\(872\) 0 0
\(873\) 3.22461 9.92434i 0.109137 0.335888i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.32606 4.08119i 0.0447778 0.137812i −0.926168 0.377111i \(-0.876918\pi\)
0.970946 + 0.239299i \(0.0769176\pi\)
\(878\) 0 0
\(879\) 15.5236 + 11.2786i 0.523598 + 0.380416i
\(880\) 0 0
\(881\) 43.7520 31.7877i 1.47404 1.07095i 0.494625 0.869106i \(-0.335305\pi\)
0.979417 0.201848i \(-0.0646947\pi\)
\(882\) 0 0
\(883\) −25.6474 + 18.6340i −0.863105 + 0.627083i −0.928728 0.370762i \(-0.879097\pi\)
0.0656228 + 0.997845i \(0.479097\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.66252 + 5.11670i 0.0558218 + 0.171802i 0.975080 0.221853i \(-0.0712105\pi\)
−0.919258 + 0.393655i \(0.871211\pi\)
\(888\) 0 0
\(889\) 0.686891 2.11403i 0.0230376 0.0709024i
\(890\) 0 0
\(891\) −1.16386 3.58198i −0.0389906 0.120001i
\(892\) 0 0
\(893\) 55.4786 1.85652
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.08984 + 2.24490i 0.103167 + 0.0749550i
\(898\) 0 0
\(899\) 70.8900 2.36432
\(900\) 0 0
\(901\) 17.9753 0.598846
\(902\) 0 0
\(903\) 16.0372 + 11.6517i 0.533684 + 0.387744i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 45.7593 1.51941 0.759706 0.650266i \(-0.225343\pi\)
0.759706 + 0.650266i \(0.225343\pi\)
\(908\) 0 0
\(909\) 2.19649 + 6.76010i 0.0728530 + 0.224219i
\(910\) 0 0
\(911\) 0.669870 2.06165i 0.0221938 0.0683054i −0.939346 0.342970i \(-0.888567\pi\)
0.961540 + 0.274665i \(0.0885669\pi\)
\(912\) 0 0
\(913\) −9.82408 30.2354i −0.325130 1.00065i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −22.4757 + 16.3295i −0.742213 + 0.539249i
\(918\) 0 0
\(919\) 15.9053 11.5559i 0.524666 0.381192i −0.293693 0.955900i \(-0.594884\pi\)
0.818359 + 0.574708i \(0.194884\pi\)
\(920\) 0 0
\(921\) −7.49942 5.44865i −0.247114 0.179539i
\(922\) 0 0
\(923\) 0.678477 2.08814i 0.0223324 0.0687319i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −3.77281 + 11.6115i −0.123915 + 0.381372i
\(928\) 0 0
\(929\) −35.2728 25.6272i −1.15726 0.840800i −0.167832 0.985816i \(-0.553677\pi\)
−0.989429 + 0.145016i \(0.953677\pi\)
\(930\) 0 0
\(931\) −46.7626 + 33.9750i −1.53258 + 1.11349i
\(932\) 0 0
\(933\) 20.1393 14.6321i 0.659333 0.479033i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −13.0014 40.0143i −0.424738 1.30721i −0.903245 0.429126i \(-0.858822\pi\)
0.478506 0.878084i \(-0.341178\pi\)
\(938\) 0 0
\(939\) −1.86866 + 5.75115i −0.0609815 + 0.187682i
\(940\) 0 0
\(941\) 15.1162 + 46.5228i 0.492773 + 1.51660i 0.820399 + 0.571792i \(0.193752\pi\)
−0.327625 + 0.944808i \(0.606248\pi\)
\(942\) 0 0
\(943\) −36.0544 −1.17409
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.75486 4.18115i −0.187008 0.135869i 0.490342 0.871530i \(-0.336872\pi\)
−0.677350 + 0.735661i \(0.736872\pi\)
\(948\) 0 0
\(949\) −7.50878 −0.243745
\(950\) 0 0
\(951\) −14.3864 −0.466512
\(952\) 0 0
\(953\) −17.9903 13.0707i −0.582762 0.423402i 0.256957 0.966423i \(-0.417280\pi\)
−0.839719 + 0.543021i \(0.817280\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 26.4560 0.855202
\(958\) 0 0
\(959\) 0.234186 + 0.720751i 0.00756226 + 0.0232743i
\(960\) 0 0
\(961\) 21.8934 67.3809i 0.706238 2.17358i
\(962\) 0 0
\(963\) 1.24937 + 3.84515i 0.0402603 + 0.123908i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 11.2576 8.17914i 0.362021 0.263023i −0.391874 0.920019i \(-0.628173\pi\)
0.753894 + 0.656996i \(0.228173\pi\)
\(968\) 0 0
\(969\) 24.5554 17.8406i 0.788834 0.573122i
\(970\) 0 0
\(971\) −19.6548 14.2801i −0.630754 0.458270i 0.225907 0.974149i \(-0.427465\pi\)
−0.856661 + 0.515879i \(0.827465\pi\)
\(972\) 0 0
\(973\) −10.4975 + 32.3081i −0.336535 + 1.03575i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14.3172 + 44.0637i −0.458047 + 1.40972i 0.409473 + 0.912322i \(0.365713\pi\)
−0.867520 + 0.497402i \(0.834287\pi\)
\(978\) 0 0
\(979\) −10.3228 7.49993i −0.329917 0.239699i
\(980\) 0 0
\(981\) 0.626030 0.454837i 0.0199876 0.0145218i
\(982\) 0 0
\(983\) 33.2168 24.1334i 1.05945 0.769736i 0.0854644 0.996341i \(-0.472763\pi\)
0.973987 + 0.226605i \(0.0727626\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −12.3540 38.0218i −0.393233 1.21025i
\(988\) 0 0
\(989\) 8.10275 24.9377i 0.257652 0.792972i
\(990\) 0 0
\(991\) 6.26579 + 19.2841i 0.199039 + 0.612580i 0.999906 + 0.0137370i \(0.00437276\pi\)
−0.800866 + 0.598843i \(0.795627\pi\)
\(992\) 0 0
\(993\) 1.06688 0.0338565
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −19.7104 14.3204i −0.624234 0.453533i 0.230164 0.973152i \(-0.426074\pi\)
−0.854398 + 0.519619i \(0.826074\pi\)
\(998\) 0 0
\(999\) 5.78488 0.183026
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.601.1 24
5.2 odd 4 1500.2.o.c.649.2 24
5.3 odd 4 300.2.o.a.229.5 yes 24
5.4 even 2 1500.2.m.d.601.6 24
15.8 even 4 900.2.w.c.829.3 24
25.6 even 5 inner 1500.2.m.c.901.1 24
25.8 odd 20 1500.2.o.c.349.2 24
25.9 even 10 7500.2.a.m.1.12 12
25.12 odd 20 7500.2.d.g.1249.1 24
25.13 odd 20 7500.2.d.g.1249.24 24
25.16 even 5 7500.2.a.n.1.1 12
25.17 odd 20 300.2.o.a.169.5 24
25.19 even 10 1500.2.m.d.901.6 24
75.17 even 20 900.2.w.c.469.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.o.a.169.5 24 25.17 odd 20
300.2.o.a.229.5 yes 24 5.3 odd 4
900.2.w.c.469.3 24 75.17 even 20
900.2.w.c.829.3 24 15.8 even 4
1500.2.m.c.601.1 24 1.1 even 1 trivial
1500.2.m.c.901.1 24 25.6 even 5 inner
1500.2.m.d.601.6 24 5.4 even 2
1500.2.m.d.901.6 24 25.19 even 10
1500.2.o.c.349.2 24 25.8 odd 20
1500.2.o.c.649.2 24 5.2 odd 4
7500.2.a.m.1.12 12 25.9 even 10
7500.2.a.n.1.1 12 25.16 even 5
7500.2.d.g.1249.1 24 25.12 odd 20
7500.2.d.g.1249.24 24 25.13 odd 20