Properties

Label 1134.2.g.j.487.1
Level $1134$
Weight $2$
Character 1134.487
Analytic conductor $9.055$
Analytic rank $0$
Dimension $4$
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] = [1134,2,Mod(163,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.g (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: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 487.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1134.487
Dual form 1134.2.g.j.163.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.62132 - 0.358719i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.62132 - 0.358719i) q^{7} -1.00000 q^{8} +(2.12132 - 3.67423i) q^{11} +6.24264 q^{13} +(-1.00000 - 2.44949i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.12132 - 5.40629i) q^{19} +4.24264 q^{22} +(3.62132 + 6.27231i) q^{23} +(2.50000 - 4.33013i) q^{25} +(3.12132 + 5.40629i) q^{26} +(1.62132 - 2.09077i) q^{28} +4.24264 q^{29} +(-0.378680 + 0.655892i) q^{31} +(0.500000 - 0.866025i) q^{32} +(2.00000 + 3.46410i) q^{37} +(3.12132 - 5.40629i) q^{38} +5.48528 q^{41} -6.48528 q^{43} +(2.12132 + 3.67423i) q^{44} +(-3.62132 + 6.27231i) q^{46} +(6.62132 + 11.4685i) q^{47} +(6.74264 + 1.88064i) q^{49} +5.00000 q^{50} +(-3.12132 + 5.40629i) q^{52} +(2.12132 - 3.67423i) q^{53} +(2.62132 + 0.358719i) q^{56} +(2.12132 + 3.67423i) q^{58} +(-3.12132 - 5.40629i) q^{61} -0.757359 q^{62} +1.00000 q^{64} +(4.12132 - 7.13834i) q^{67} +7.24264 q^{71} +(3.50000 - 6.06218i) q^{73} +(-2.00000 + 3.46410i) q^{74} +6.24264 q^{76} +(-6.87868 + 8.87039i) q^{77} +(-4.62132 - 8.00436i) q^{79} +(2.74264 + 4.75039i) q^{82} -7.75736 q^{83} +(-3.24264 - 5.61642i) q^{86} +(-2.12132 + 3.67423i) q^{88} +(-2.74264 - 4.75039i) q^{89} +(-16.3640 - 2.23936i) q^{91} -7.24264 q^{92} +(-6.62132 + 11.4685i) q^{94} -12.4853 q^{97} +(1.74264 + 6.77962i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 2 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 2 q^{7} - 4 q^{8} + 8 q^{13} - 4 q^{14} - 2 q^{16} - 4 q^{19} + 6 q^{23} + 10 q^{25} + 4 q^{26} - 2 q^{28} - 10 q^{31} + 2 q^{32} + 8 q^{37} + 4 q^{38} - 12 q^{41} + 8 q^{43} - 6 q^{46} + 18 q^{47} + 10 q^{49} + 20 q^{50} - 4 q^{52} + 2 q^{56} - 4 q^{61} - 20 q^{62} + 4 q^{64} + 8 q^{67} + 12 q^{71} + 14 q^{73} - 8 q^{74} + 8 q^{76} - 36 q^{77} - 10 q^{79} - 6 q^{82} - 48 q^{83} + 4 q^{86} + 6 q^{89} - 40 q^{91} - 12 q^{92} - 18 q^{94} - 16 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −2.62132 0.358719i −0.990766 0.135583i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.12132 3.67423i 0.639602 1.10782i −0.345918 0.938265i \(-0.612432\pi\)
0.985520 0.169559i \(-0.0542342\pi\)
\(12\) 0 0
\(13\) 6.24264 1.73140 0.865699 0.500566i \(-0.166875\pi\)
0.865699 + 0.500566i \(0.166875\pi\)
\(14\) −1.00000 2.44949i −0.267261 0.654654i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.125000 0.216506i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −3.12132 5.40629i −0.716080 1.24029i −0.962542 0.271134i \(-0.912601\pi\)
0.246462 0.969153i \(-0.420732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.24264 0.904534
\(23\) 3.62132 + 6.27231i 0.755097 + 1.30787i 0.945326 + 0.326127i \(0.105744\pi\)
−0.190228 + 0.981740i \(0.560923\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 3.12132 + 5.40629i 0.612141 + 1.06026i
\(27\) 0 0
\(28\) 1.62132 2.09077i 0.306401 0.395118i
\(29\) 4.24264 0.787839 0.393919 0.919145i \(-0.371119\pi\)
0.393919 + 0.919145i \(0.371119\pi\)
\(30\) 0 0
\(31\) −0.378680 + 0.655892i −0.0680129 + 0.117802i −0.898027 0.439941i \(-0.854999\pi\)
0.830014 + 0.557743i \(0.188333\pi\)
\(32\) 0.500000 0.866025i 0.0883883 0.153093i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 + 3.46410i 0.328798 + 0.569495i 0.982274 0.187453i \(-0.0600231\pi\)
−0.653476 + 0.756948i \(0.726690\pi\)
\(38\) 3.12132 5.40629i 0.506345 0.877015i
\(39\) 0 0
\(40\) 0 0
\(41\) 5.48528 0.856657 0.428329 0.903623i \(-0.359103\pi\)
0.428329 + 0.903623i \(0.359103\pi\)
\(42\) 0 0
\(43\) −6.48528 −0.988996 −0.494498 0.869179i \(-0.664648\pi\)
−0.494498 + 0.869179i \(0.664648\pi\)
\(44\) 2.12132 + 3.67423i 0.319801 + 0.553912i
\(45\) 0 0
\(46\) −3.62132 + 6.27231i −0.533935 + 0.924802i
\(47\) 6.62132 + 11.4685i 0.965819 + 1.67285i 0.707399 + 0.706815i \(0.249869\pi\)
0.258420 + 0.966033i \(0.416798\pi\)
\(48\) 0 0
\(49\) 6.74264 + 1.88064i 0.963234 + 0.268662i
\(50\) 5.00000 0.707107
\(51\) 0 0
\(52\) −3.12132 + 5.40629i −0.432849 + 0.749717i
\(53\) 2.12132 3.67423i 0.291386 0.504695i −0.682752 0.730650i \(-0.739217\pi\)
0.974138 + 0.225955i \(0.0725503\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.62132 + 0.358719i 0.350289 + 0.0479359i
\(57\) 0 0
\(58\) 2.12132 + 3.67423i 0.278543 + 0.482451i
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −3.12132 5.40629i −0.399644 0.692204i 0.594038 0.804437i \(-0.297533\pi\)
−0.993682 + 0.112233i \(0.964200\pi\)
\(62\) −0.757359 −0.0961847
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.12132 7.13834i 0.503499 0.872087i −0.496492 0.868041i \(-0.665379\pi\)
0.999992 0.00404550i \(-0.00128773\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.24264 0.859543 0.429772 0.902938i \(-0.358594\pi\)
0.429772 + 0.902938i \(0.358594\pi\)
\(72\) 0 0
\(73\) 3.50000 6.06218i 0.409644 0.709524i −0.585206 0.810885i \(-0.698986\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) −2.00000 + 3.46410i −0.232495 + 0.402694i
\(75\) 0 0
\(76\) 6.24264 0.716080
\(77\) −6.87868 + 8.87039i −0.783898 + 1.01087i
\(78\) 0 0
\(79\) −4.62132 8.00436i −0.519939 0.900561i −0.999731 0.0231789i \(-0.992621\pi\)
0.479792 0.877382i \(-0.340712\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.74264 + 4.75039i 0.302874 + 0.524593i
\(83\) −7.75736 −0.851481 −0.425740 0.904845i \(-0.639986\pi\)
−0.425740 + 0.904845i \(0.639986\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −3.24264 5.61642i −0.349663 0.605634i
\(87\) 0 0
\(88\) −2.12132 + 3.67423i −0.226134 + 0.391675i
\(89\) −2.74264 4.75039i −0.290719 0.503541i 0.683261 0.730175i \(-0.260561\pi\)
−0.973980 + 0.226634i \(0.927228\pi\)
\(90\) 0 0
\(91\) −16.3640 2.23936i −1.71541 0.234748i
\(92\) −7.24264 −0.755097
\(93\) 0 0
\(94\) −6.62132 + 11.4685i −0.682937 + 1.18288i
\(95\) 0 0
\(96\) 0 0
\(97\) −12.4853 −1.26769 −0.633844 0.773461i \(-0.718524\pi\)
−0.633844 + 0.773461i \(0.718524\pi\)
\(98\) 1.74264 + 6.77962i 0.176033 + 0.684845i
\(99\) 0 0
\(100\) 2.50000 + 4.33013i 0.250000 + 0.433013i
\(101\) 3.87868 6.71807i 0.385943 0.668473i −0.605956 0.795498i \(-0.707209\pi\)
0.991900 + 0.127025i \(0.0405428\pi\)
\(102\) 0 0
\(103\) −0.378680 0.655892i −0.0373124 0.0646270i 0.846766 0.531965i \(-0.178546\pi\)
−0.884079 + 0.467338i \(0.845213\pi\)
\(104\) −6.24264 −0.612141
\(105\) 0 0
\(106\) 4.24264 0.412082
\(107\) −1.24264 2.15232i −0.120131 0.208072i 0.799688 0.600415i \(-0.204998\pi\)
−0.919819 + 0.392343i \(0.871665\pi\)
\(108\) 0 0
\(109\) 1.12132 1.94218i 0.107403 0.186027i −0.807314 0.590122i \(-0.799080\pi\)
0.914717 + 0.404094i \(0.132413\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 + 2.44949i 0.0944911 + 0.231455i
\(113\) 20.4853 1.92709 0.963547 0.267541i \(-0.0862110\pi\)
0.963547 + 0.267541i \(0.0862110\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.12132 + 3.67423i −0.196960 + 0.341144i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.50000 6.06218i −0.318182 0.551107i
\(122\) 3.12132 5.40629i 0.282591 0.489462i
\(123\) 0 0
\(124\) −0.378680 0.655892i −0.0340064 0.0589009i
\(125\) 0 0
\(126\) 0 0
\(127\) 6.75736 0.599619 0.299809 0.953999i \(-0.403077\pi\)
0.299809 + 0.953999i \(0.403077\pi\)
\(128\) 0.500000 + 0.866025i 0.0441942 + 0.0765466i
\(129\) 0 0
\(130\) 0 0
\(131\) −3.87868 6.71807i −0.338882 0.586961i 0.645341 0.763895i \(-0.276715\pi\)
−0.984223 + 0.176934i \(0.943382\pi\)
\(132\) 0 0
\(133\) 6.24264 + 15.2913i 0.541306 + 1.32592i
\(134\) 8.24264 0.712056
\(135\) 0 0
\(136\) 0 0
\(137\) −9.98528 + 17.2950i −0.853100 + 1.47761i 0.0252962 + 0.999680i \(0.491947\pi\)
−0.878396 + 0.477933i \(0.841386\pi\)
\(138\) 0 0
\(139\) −4.72792 −0.401017 −0.200509 0.979692i \(-0.564259\pi\)
−0.200509 + 0.979692i \(0.564259\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.62132 + 6.27231i 0.303894 + 0.526361i
\(143\) 13.2426 22.9369i 1.10741 1.91808i
\(144\) 0 0
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 8.12132 + 14.0665i 0.665324 + 1.15238i 0.979197 + 0.202911i \(0.0650401\pi\)
−0.313873 + 0.949465i \(0.601627\pi\)
\(150\) 0 0
\(151\) 1.37868 2.38794i 0.112195 0.194328i −0.804460 0.594007i \(-0.797545\pi\)
0.916655 + 0.399679i \(0.130878\pi\)
\(152\) 3.12132 + 5.40629i 0.253173 + 0.438508i
\(153\) 0 0
\(154\) −11.1213 1.52192i −0.896182 0.122640i
\(155\) 0 0
\(156\) 0 0
\(157\) −7.00000 + 12.1244i −0.558661 + 0.967629i 0.438948 + 0.898513i \(0.355351\pi\)
−0.997609 + 0.0691164i \(0.977982\pi\)
\(158\) 4.62132 8.00436i 0.367653 0.636793i
\(159\) 0 0
\(160\) 0 0
\(161\) −7.24264 17.7408i −0.570800 1.39817i
\(162\) 0 0
\(163\) 5.87868 + 10.1822i 0.460454 + 0.797529i 0.998984 0.0450772i \(-0.0143534\pi\)
−0.538530 + 0.842606i \(0.681020\pi\)
\(164\) −2.74264 + 4.75039i −0.214164 + 0.370943i
\(165\) 0 0
\(166\) −3.87868 6.71807i −0.301044 0.521423i
\(167\) −24.2132 −1.87367 −0.936837 0.349766i \(-0.886261\pi\)
−0.936837 + 0.349766i \(0.886261\pi\)
\(168\) 0 0
\(169\) 25.9706 1.99774
\(170\) 0 0
\(171\) 0 0
\(172\) 3.24264 5.61642i 0.247249 0.428248i
\(173\) −5.48528 9.50079i −0.417038 0.722331i 0.578602 0.815610i \(-0.303599\pi\)
−0.995640 + 0.0932788i \(0.970265\pi\)
\(174\) 0 0
\(175\) −8.10660 + 10.4539i −0.612801 + 0.790237i
\(176\) −4.24264 −0.319801
\(177\) 0 0
\(178\) 2.74264 4.75039i 0.205570 0.356057i
\(179\) −8.12132 + 14.0665i −0.607016 + 1.05138i 0.384713 + 0.923036i \(0.374300\pi\)
−0.991729 + 0.128346i \(0.959033\pi\)
\(180\) 0 0
\(181\) −20.2426 −1.50462 −0.752312 0.658807i \(-0.771061\pi\)
−0.752312 + 0.658807i \(0.771061\pi\)
\(182\) −6.24264 15.2913i −0.462735 1.13347i
\(183\) 0 0
\(184\) −3.62132 6.27231i −0.266967 0.462401i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −13.2426 −0.965819
\(189\) 0 0
\(190\) 0 0
\(191\) 4.24264 + 7.34847i 0.306987 + 0.531717i 0.977702 0.209999i \(-0.0673460\pi\)
−0.670715 + 0.741715i \(0.734013\pi\)
\(192\) 0 0
\(193\) 2.25736 3.90986i 0.162488 0.281438i −0.773272 0.634074i \(-0.781381\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −6.24264 10.8126i −0.448195 0.776297i
\(195\) 0 0
\(196\) −5.00000 + 4.89898i −0.357143 + 0.349927i
\(197\) −16.9706 −1.20910 −0.604551 0.796566i \(-0.706648\pi\)
−0.604551 + 0.796566i \(0.706648\pi\)
\(198\) 0 0
\(199\) 7.37868 12.7802i 0.523061 0.905968i −0.476579 0.879132i \(-0.658123\pi\)
0.999640 0.0268362i \(-0.00854325\pi\)
\(200\) −2.50000 + 4.33013i −0.176777 + 0.306186i
\(201\) 0 0
\(202\) 7.75736 0.545806
\(203\) −11.1213 1.52192i −0.780564 0.106818i
\(204\) 0 0
\(205\) 0 0
\(206\) 0.378680 0.655892i 0.0263839 0.0456982i
\(207\) 0 0
\(208\) −3.12132 5.40629i −0.216425 0.374858i
\(209\) −26.4853 −1.83203
\(210\) 0 0
\(211\) −6.48528 −0.446465 −0.223233 0.974765i \(-0.571661\pi\)
−0.223233 + 0.974765i \(0.571661\pi\)
\(212\) 2.12132 + 3.67423i 0.145693 + 0.252347i
\(213\) 0 0
\(214\) 1.24264 2.15232i 0.0849452 0.147129i
\(215\) 0 0
\(216\) 0 0
\(217\) 1.22792 1.58346i 0.0833568 0.107493i
\(218\) 2.24264 0.151891
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 11.7279 0.785360 0.392680 0.919675i \(-0.371548\pi\)
0.392680 + 0.919675i \(0.371548\pi\)
\(224\) −1.62132 + 2.09077i −0.108329 + 0.139695i
\(225\) 0 0
\(226\) 10.2426 + 17.7408i 0.681330 + 1.18010i
\(227\) 13.2426 22.9369i 0.878945 1.52238i 0.0264448 0.999650i \(-0.491581\pi\)
0.852500 0.522727i \(-0.175085\pi\)
\(228\) 0 0
\(229\) −12.4853 21.6251i −0.825051 1.42903i −0.901881 0.431985i \(-0.857813\pi\)
0.0768300 0.997044i \(-0.475520\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.24264 −0.278543
\(233\) 10.2426 + 17.7408i 0.671018 + 1.16224i 0.977616 + 0.210398i \(0.0674759\pi\)
−0.306598 + 0.951839i \(0.599191\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.72792 −0.241139 −0.120570 0.992705i \(-0.538472\pi\)
−0.120570 + 0.992705i \(0.538472\pi\)
\(240\) 0 0
\(241\) −4.25736 + 7.37396i −0.274241 + 0.474999i −0.969943 0.243331i \(-0.921760\pi\)
0.695703 + 0.718330i \(0.255093\pi\)
\(242\) 3.50000 6.06218i 0.224989 0.389692i
\(243\) 0 0
\(244\) 6.24264 0.399644
\(245\) 0 0
\(246\) 0 0
\(247\) −19.4853 33.7495i −1.23982 2.14743i
\(248\) 0.378680 0.655892i 0.0240462 0.0416492i
\(249\) 0 0
\(250\) 0 0
\(251\) −18.7279 −1.18210 −0.591048 0.806636i \(-0.701286\pi\)
−0.591048 + 0.806636i \(0.701286\pi\)
\(252\) 0 0
\(253\) 30.7279 1.93185
\(254\) 3.37868 + 5.85204i 0.211997 + 0.367190i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.0312500 + 0.0541266i
\(257\) 2.74264 + 4.75039i 0.171081 + 0.296321i 0.938798 0.344468i \(-0.111941\pi\)
−0.767717 + 0.640789i \(0.778607\pi\)
\(258\) 0 0
\(259\) −4.00000 9.79796i −0.248548 0.608816i
\(260\) 0 0
\(261\) 0 0
\(262\) 3.87868 6.71807i 0.239626 0.415044i
\(263\) 11.4853 19.8931i 0.708213 1.22666i −0.257307 0.966330i \(-0.582835\pi\)
0.965519 0.260331i \(-0.0838316\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.1213 + 13.0519i −0.620578 + 0.800265i
\(267\) 0 0
\(268\) 4.12132 + 7.13834i 0.251750 + 0.436043i
\(269\) −5.48528 + 9.50079i −0.334444 + 0.579273i −0.983378 0.181571i \(-0.941882\pi\)
0.648934 + 0.760844i \(0.275215\pi\)
\(270\) 0 0
\(271\) 6.24264 + 10.8126i 0.379213 + 0.656817i 0.990948 0.134246i \(-0.0428613\pi\)
−0.611735 + 0.791063i \(0.709528\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −19.9706 −1.20647
\(275\) −10.6066 18.3712i −0.639602 1.10782i
\(276\) 0 0
\(277\) 9.60660 16.6391i 0.577205 0.999748i −0.418593 0.908174i \(-0.637477\pi\)
0.995798 0.0915743i \(-0.0291899\pi\)
\(278\) −2.36396 4.09450i −0.141781 0.245572i
\(279\) 0 0
\(280\) 0 0
\(281\) −22.4558 −1.33960 −0.669802 0.742540i \(-0.733621\pi\)
−0.669802 + 0.742540i \(0.733621\pi\)
\(282\) 0 0
\(283\) −12.4853 + 21.6251i −0.742173 + 1.28548i 0.209331 + 0.977845i \(0.432871\pi\)
−0.951504 + 0.307636i \(0.900462\pi\)
\(284\) −3.62132 + 6.27231i −0.214886 + 0.372193i
\(285\) 0 0
\(286\) 26.4853 1.56611
\(287\) −14.3787 1.96768i −0.848747 0.116148i
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 3.50000 + 6.06218i 0.204822 + 0.354762i
\(293\) 10.9706 0.640907 0.320454 0.947264i \(-0.396165\pi\)
0.320454 + 0.947264i \(0.396165\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 3.46410i −0.116248 0.201347i
\(297\) 0 0
\(298\) −8.12132 + 14.0665i −0.470455 + 0.814853i
\(299\) 22.6066 + 39.1558i 1.30737 + 2.26444i
\(300\) 0 0
\(301\) 17.0000 + 2.32640i 0.979864 + 0.134091i
\(302\) 2.75736 0.158668
\(303\) 0 0
\(304\) −3.12132 + 5.40629i −0.179020 + 0.310072i
\(305\) 0 0
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −4.24264 10.3923i −0.241747 0.592157i
\(309\) 0 0
\(310\) 0 0
\(311\) −5.48528 + 9.50079i −0.311042 + 0.538740i −0.978588 0.205828i \(-0.934011\pi\)
0.667546 + 0.744568i \(0.267345\pi\)
\(312\) 0 0
\(313\) 8.98528 + 15.5630i 0.507878 + 0.879671i 0.999958 + 0.00912090i \(0.00290331\pi\)
−0.492080 + 0.870550i \(0.663763\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 9.24264 0.519939
\(317\) −8.48528 14.6969i −0.476581 0.825462i 0.523059 0.852296i \(-0.324791\pi\)
−0.999640 + 0.0268342i \(0.991457\pi\)
\(318\) 0 0
\(319\) 9.00000 15.5885i 0.503903 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) 11.7426 15.1427i 0.654392 0.843870i
\(323\) 0 0
\(324\) 0 0
\(325\) 15.6066 27.0314i 0.865699 1.49943i
\(326\) −5.87868 + 10.1822i −0.325590 + 0.563938i
\(327\) 0 0
\(328\) −5.48528 −0.302874
\(329\) −13.2426 32.4377i −0.730090 1.78835i
\(330\) 0 0
\(331\) 9.24264 + 16.0087i 0.508021 + 0.879919i 0.999957 + 0.00928730i \(0.00295628\pi\)
−0.491935 + 0.870632i \(0.663710\pi\)
\(332\) 3.87868 6.71807i 0.212870 0.368702i
\(333\) 0 0
\(334\) −12.1066 20.9692i −0.662444 1.14739i
\(335\) 0 0
\(336\) 0 0
\(337\) 4.48528 0.244329 0.122164 0.992510i \(-0.461016\pi\)
0.122164 + 0.992510i \(0.461016\pi\)
\(338\) 12.9853 + 22.4912i 0.706306 + 1.22336i
\(339\) 0 0
\(340\) 0 0
\(341\) 1.60660 + 2.78272i 0.0870024 + 0.150693i
\(342\) 0 0
\(343\) −17.0000 7.34847i −0.917914 0.396780i
\(344\) 6.48528 0.349663
\(345\) 0 0
\(346\) 5.48528 9.50079i 0.294891 0.510765i
\(347\) −3.36396 + 5.82655i −0.180587 + 0.312786i −0.942081 0.335387i \(-0.891133\pi\)
0.761494 + 0.648172i \(0.224466\pi\)
\(348\) 0 0
\(349\) 24.9706 1.33664 0.668322 0.743872i \(-0.267013\pi\)
0.668322 + 0.743872i \(0.267013\pi\)
\(350\) −13.1066 1.79360i −0.700577 0.0958718i
\(351\) 0 0
\(352\) −2.12132 3.67423i −0.113067 0.195837i
\(353\) 10.5000 18.1865i 0.558859 0.967972i −0.438733 0.898617i \(-0.644573\pi\)
0.997592 0.0693543i \(-0.0220939\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.48528 0.290719
\(357\) 0 0
\(358\) −16.2426 −0.858450
\(359\) 5.37868 + 9.31615i 0.283876 + 0.491687i 0.972336 0.233587i \(-0.0750463\pi\)
−0.688460 + 0.725274i \(0.741713\pi\)
\(360\) 0 0
\(361\) −9.98528 + 17.2950i −0.525541 + 0.910264i
\(362\) −10.1213 17.5306i −0.531965 0.921390i
\(363\) 0 0
\(364\) 10.1213 13.0519i 0.530501 0.684107i
\(365\) 0 0
\(366\) 0 0
\(367\) −5.86396 + 10.1567i −0.306096 + 0.530174i −0.977505 0.210913i \(-0.932356\pi\)
0.671409 + 0.741087i \(0.265690\pi\)
\(368\) 3.62132 6.27231i 0.188774 0.326967i
\(369\) 0 0
\(370\) 0 0
\(371\) −6.87868 + 8.87039i −0.357123 + 0.460528i
\(372\) 0 0
\(373\) 3.60660 + 6.24682i 0.186743 + 0.323448i 0.944162 0.329480i \(-0.106874\pi\)
−0.757420 + 0.652928i \(0.773540\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.62132 11.4685i −0.341469 0.591441i
\(377\) 26.4853 1.36406
\(378\) 0 0
\(379\) 1.27208 0.0653423 0.0326711 0.999466i \(-0.489599\pi\)
0.0326711 + 0.999466i \(0.489599\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.24264 + 7.34847i −0.217072 + 0.375980i
\(383\) −12.1066 20.9692i −0.618618 1.07148i −0.989738 0.142894i \(-0.954359\pi\)
0.371120 0.928585i \(-0.378974\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.51472 0.229793
\(387\) 0 0
\(388\) 6.24264 10.8126i 0.316922 0.548925i
\(389\) 5.12132 8.87039i 0.259661 0.449746i −0.706490 0.707723i \(-0.749722\pi\)
0.966151 + 0.257977i \(0.0830558\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.74264 1.88064i −0.340555 0.0949865i
\(393\) 0 0
\(394\) −8.48528 14.6969i −0.427482 0.740421i
\(395\) 0 0
\(396\) 0 0
\(397\) 10.1213 + 17.5306i 0.507975 + 0.879838i 0.999957 + 0.00923278i \(0.00293893\pi\)
−0.491983 + 0.870605i \(0.663728\pi\)
\(398\) 14.7574 0.739720
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 14.7426 + 25.5350i 0.736212 + 1.27516i 0.954189 + 0.299203i \(0.0967209\pi\)
−0.217977 + 0.975954i \(0.569946\pi\)
\(402\) 0 0
\(403\) −2.36396 + 4.09450i −0.117757 + 0.203962i
\(404\) 3.87868 + 6.71807i 0.192972 + 0.334236i
\(405\) 0 0
\(406\) −4.24264 10.3923i −0.210559 0.515761i
\(407\) 16.9706 0.841200
\(408\) 0 0
\(409\) −17.5000 + 30.3109i −0.865319 + 1.49878i 0.00141047 + 0.999999i \(0.499551\pi\)
−0.866730 + 0.498778i \(0.833782\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.757359 0.0373124
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 3.12132 5.40629i 0.153035 0.265065i
\(417\) 0 0
\(418\) −13.2426 22.9369i −0.647719 1.12188i
\(419\) 7.75736 0.378972 0.189486 0.981883i \(-0.439318\pi\)
0.189486 + 0.981883i \(0.439318\pi\)
\(420\) 0 0
\(421\) −14.2426 −0.694144 −0.347072 0.937839i \(-0.612824\pi\)
−0.347072 + 0.937839i \(0.612824\pi\)
\(422\) −3.24264 5.61642i −0.157849 0.273403i
\(423\) 0 0
\(424\) −2.12132 + 3.67423i −0.103020 + 0.178437i
\(425\) 0 0
\(426\) 0 0
\(427\) 6.24264 + 15.2913i 0.302103 + 0.739997i
\(428\) 2.48528 0.120131
\(429\) 0 0
\(430\) 0 0
\(431\) −18.6213 + 32.2531i −0.896957 + 1.55358i −0.0655943 + 0.997846i \(0.520894\pi\)
−0.831363 + 0.555729i \(0.812439\pi\)
\(432\) 0 0
\(433\) 3.97056 0.190813 0.0954065 0.995438i \(-0.469585\pi\)
0.0954065 + 0.995438i \(0.469585\pi\)
\(434\) 1.98528 + 0.271680i 0.0952966 + 0.0130410i
\(435\) 0 0
\(436\) 1.12132 + 1.94218i 0.0537015 + 0.0930137i
\(437\) 22.6066 39.1558i 1.08142 1.87308i
\(438\) 0 0
\(439\) −5.86396 10.1567i −0.279872 0.484752i 0.691481 0.722395i \(-0.256959\pi\)
−0.971353 + 0.237643i \(0.923625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.75736 + 8.23999i 0.226029 + 0.391494i 0.956628 0.291313i \(-0.0940923\pi\)
−0.730599 + 0.682807i \(0.760759\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 5.86396 + 10.1567i 0.277667 + 0.480933i
\(447\) 0 0
\(448\) −2.62132 0.358719i −0.123846 0.0169479i
\(449\) 4.97056 0.234575 0.117288 0.993098i \(-0.462580\pi\)
0.117288 + 0.993098i \(0.462580\pi\)
\(450\) 0 0
\(451\) 11.6360 20.1542i 0.547920 0.949025i
\(452\) −10.2426 + 17.7408i −0.481773 + 0.834456i
\(453\) 0 0
\(454\) 26.4853 1.24302
\(455\) 0 0
\(456\) 0 0
\(457\) −5.75736 9.97204i −0.269318 0.466472i 0.699368 0.714762i \(-0.253465\pi\)
−0.968686 + 0.248290i \(0.920132\pi\)
\(458\) 12.4853 21.6251i 0.583399 1.01048i
\(459\) 0 0
\(460\) 0 0
\(461\) −34.2426 −1.59484 −0.797419 0.603425i \(-0.793802\pi\)
−0.797419 + 0.603425i \(0.793802\pi\)
\(462\) 0 0
\(463\) −8.75736 −0.406989 −0.203495 0.979076i \(-0.565230\pi\)
−0.203495 + 0.979076i \(0.565230\pi\)
\(464\) −2.12132 3.67423i −0.0984798 0.170572i
\(465\) 0 0
\(466\) −10.2426 + 17.7408i −0.474481 + 0.821825i
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) −13.3640 + 17.2335i −0.617090 + 0.795768i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −13.7574 + 23.8284i −0.632564 + 1.09563i
\(474\) 0 0
\(475\) −31.2132 −1.43216
\(476\) 0 0
\(477\) 0 0
\(478\) −1.86396 3.22848i −0.0852556 0.147667i
\(479\) −6.62132 + 11.4685i −0.302536 + 0.524007i −0.976710 0.214565i \(-0.931167\pi\)
0.674174 + 0.738573i \(0.264500\pi\)
\(480\) 0 0
\(481\) 12.4853 + 21.6251i 0.569280 + 0.986022i
\(482\) −8.51472 −0.387835
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 1.37868 2.38794i 0.0624739 0.108208i −0.833097 0.553127i \(-0.813434\pi\)
0.895571 + 0.444919i \(0.146768\pi\)
\(488\) 3.12132 + 5.40629i 0.141296 + 0.244731i
\(489\) 0 0
\(490\) 0 0
\(491\) −6.72792 −0.303627 −0.151813 0.988409i \(-0.548511\pi\)
−0.151813 + 0.988409i \(0.548511\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 19.4853 33.7495i 0.876684 1.51846i
\(495\) 0 0
\(496\) 0.757359 0.0340064
\(497\) −18.9853 2.59808i −0.851606 0.116540i
\(498\) 0 0
\(499\) 5.36396 + 9.29065i 0.240124 + 0.415907i 0.960749 0.277418i \(-0.0894786\pi\)
−0.720626 + 0.693325i \(0.756145\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −9.36396 16.2189i −0.417934 0.723883i
\(503\) 24.2132 1.07961 0.539807 0.841789i \(-0.318497\pi\)
0.539807 + 0.841789i \(0.318497\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 15.3640 + 26.6112i 0.683011 + 1.18301i
\(507\) 0 0
\(508\) −3.37868 + 5.85204i −0.149905 + 0.259643i
\(509\) 3.87868 + 6.71807i 0.171919 + 0.297773i 0.939091 0.343669i \(-0.111670\pi\)
−0.767171 + 0.641442i \(0.778336\pi\)
\(510\) 0 0
\(511\) −11.3492 + 14.6354i −0.502061 + 0.647432i
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −2.74264 + 4.75039i −0.120973 + 0.209531i
\(515\) 0 0
\(516\) 0 0
\(517\) 56.1838 2.47096
\(518\) 6.48528 8.36308i 0.284947 0.367453i
\(519\) 0 0
\(520\) 0 0
\(521\) 8.22792 14.2512i 0.360472 0.624355i −0.627567 0.778563i \(-0.715949\pi\)
0.988039 + 0.154207i \(0.0492824\pi\)
\(522\) 0 0
\(523\) 10.1213 + 17.5306i 0.442574 + 0.766561i 0.997880 0.0650852i \(-0.0207319\pi\)
−0.555305 + 0.831647i \(0.687399\pi\)
\(524\) 7.75736 0.338882
\(525\) 0 0
\(526\) 22.9706 1.00156
\(527\) 0 0
\(528\) 0 0
\(529\) −14.7279 + 25.5095i −0.640344 + 1.10911i
\(530\) 0 0
\(531\) 0 0
\(532\) −16.3640 2.23936i −0.709468 0.0970884i
\(533\) 34.2426 1.48321
\(534\) 0 0
\(535\) 0 0
\(536\) −4.12132 + 7.13834i −0.178014 + 0.308329i
\(537\) 0 0
\(538\) −10.9706 −0.472975
\(539\) 21.2132 20.7846i 0.913717 0.895257i
\(540\) 0 0
\(541\) 7.48528 + 12.9649i 0.321817 + 0.557404i 0.980863 0.194699i \(-0.0623730\pi\)
−0.659046 + 0.752103i \(0.729040\pi\)
\(542\) −6.24264 + 10.8126i −0.268144 + 0.464440i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −6.48528 −0.277291 −0.138645 0.990342i \(-0.544275\pi\)
−0.138645 + 0.990342i \(0.544275\pi\)
\(548\) −9.98528 17.2950i −0.426550 0.738806i
\(549\) 0 0
\(550\) 10.6066 18.3712i 0.452267 0.783349i
\(551\) −13.2426 22.9369i −0.564155 0.977146i
\(552\) 0 0
\(553\) 9.24264 + 22.6398i 0.393037 + 0.962740i
\(554\) 19.2132 0.816291
\(555\) 0 0
\(556\) 2.36396 4.09450i 0.100254 0.173646i
\(557\) −20.4853 + 35.4815i −0.867989 + 1.50340i −0.00394110 + 0.999992i \(0.501254\pi\)
−0.864048 + 0.503409i \(0.832079\pi\)
\(558\) 0 0
\(559\) −40.4853 −1.71234
\(560\) 0 0
\(561\) 0 0
\(562\) −11.2279 19.4473i −0.473621 0.820336i
\(563\) 9.36396 16.2189i 0.394644 0.683543i −0.598412 0.801189i \(-0.704201\pi\)
0.993056 + 0.117645i \(0.0375346\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −24.9706 −1.04959
\(567\) 0 0
\(568\) −7.24264 −0.303894
\(569\) 12.4706 + 21.5996i 0.522793 + 0.905504i 0.999648 + 0.0265224i \(0.00844333\pi\)
−0.476855 + 0.878982i \(0.658223\pi\)
\(570\) 0 0
\(571\) 10.4853 18.1610i 0.438795 0.760016i −0.558801 0.829301i \(-0.688739\pi\)
0.997597 + 0.0692856i \(0.0220720\pi\)
\(572\) 13.2426 + 22.9369i 0.553703 + 0.959041i
\(573\) 0 0
\(574\) −5.48528 13.4361i −0.228951 0.560814i
\(575\) 36.2132 1.51019
\(576\) 0 0
\(577\) −17.9706 + 31.1259i −0.748124 + 1.29579i 0.200596 + 0.979674i \(0.435712\pi\)
−0.948721 + 0.316115i \(0.897621\pi\)
\(578\) −8.50000 + 14.7224i −0.353553 + 0.612372i
\(579\) 0 0
\(580\) 0 0
\(581\) 20.3345 + 2.78272i 0.843618 + 0.115447i
\(582\) 0 0
\(583\) −9.00000 15.5885i −0.372742 0.645608i
\(584\) −3.50000 + 6.06218i −0.144831 + 0.250855i
\(585\) 0 0
\(586\) 5.48528 + 9.50079i 0.226595 + 0.392474i
\(587\) 3.21320 0.132623 0.0663115 0.997799i \(-0.478877\pi\)
0.0663115 + 0.997799i \(0.478877\pi\)
\(588\) 0 0
\(589\) 4.72792 0.194811
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000 3.46410i 0.0821995 0.142374i
\(593\) −2.74264 4.75039i −0.112627 0.195075i 0.804202 0.594356i \(-0.202593\pi\)
−0.916829 + 0.399281i \(0.869260\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −16.2426 −0.665324
\(597\) 0 0
\(598\) −22.6066 + 39.1558i −0.924453 + 1.60120i
\(599\) 11.4853 19.8931i 0.469276 0.812810i −0.530107 0.847931i \(-0.677848\pi\)
0.999383 + 0.0351210i \(0.0111817\pi\)
\(600\) 0 0
\(601\) −17.9706 −0.733035 −0.366517 0.930411i \(-0.619450\pi\)
−0.366517 + 0.930411i \(0.619450\pi\)
\(602\) 6.48528 + 15.8856i 0.264320 + 0.647450i
\(603\) 0 0
\(604\) 1.37868 + 2.38794i 0.0560977 + 0.0971640i
\(605\) 0 0
\(606\) 0 0
\(607\) 19.4853 + 33.7495i 0.790883 + 1.36985i 0.925421 + 0.378941i \(0.123712\pi\)
−0.134538 + 0.990909i \(0.542955\pi\)
\(608\) −6.24264 −0.253173
\(609\) 0 0
\(610\) 0 0
\(611\) 41.3345 + 71.5935i 1.67222 + 2.89636i
\(612\) 0 0
\(613\) 18.9706 32.8580i 0.766214 1.32712i −0.173389 0.984853i \(-0.555472\pi\)
0.939602 0.342268i \(-0.111195\pi\)
\(614\) −14.0000 24.2487i −0.564994 0.978598i
\(615\) 0 0
\(616\) 6.87868 8.87039i 0.277150 0.357398i
\(617\) 28.4558 1.14559 0.572795 0.819699i \(-0.305859\pi\)
0.572795 + 0.819699i \(0.305859\pi\)
\(618\) 0 0
\(619\) 0.757359 1.31178i 0.0304408 0.0527251i −0.850404 0.526131i \(-0.823642\pi\)
0.880844 + 0.473406i \(0.156976\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −10.9706 −0.439879
\(623\) 5.48528 + 13.4361i 0.219763 + 0.538308i
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) −8.98528 + 15.5630i −0.359124 + 0.622021i
\(627\) 0 0
\(628\) −7.00000 12.1244i −0.279330 0.483814i
\(629\) 0 0
\(630\) 0 0
\(631\) −24.4853 −0.974744 −0.487372 0.873195i \(-0.662044\pi\)
−0.487372 + 0.873195i \(0.662044\pi\)
\(632\) 4.62132 + 8.00436i 0.183826 + 0.318396i
\(633\) 0 0
\(634\) 8.48528 14.6969i 0.336994 0.583690i
\(635\) 0 0
\(636\) 0 0
\(637\) 42.0919 + 11.7401i 1.66774 + 0.465161i
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) 0 0
\(641\) −2.22792 + 3.85887i −0.0879976 + 0.152416i −0.906665 0.421852i \(-0.861380\pi\)
0.818667 + 0.574269i \(0.194713\pi\)
\(642\) 0 0
\(643\) −46.7279 −1.84277 −0.921385 0.388652i \(-0.872941\pi\)
−0.921385 + 0.388652i \(0.872941\pi\)
\(644\) 18.9853 + 2.59808i 0.748125 + 0.102379i
\(645\) 0 0
\(646\) 0 0
\(647\) 1.13604 1.96768i 0.0446623 0.0773574i −0.842830 0.538180i \(-0.819112\pi\)
0.887492 + 0.460822i \(0.152445\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 31.2132 1.22428
\(651\) 0 0
\(652\) −11.7574 −0.460454
\(653\) 12.0000 + 20.7846i 0.469596 + 0.813365i 0.999396 0.0347583i \(-0.0110661\pi\)
−0.529799 + 0.848123i \(0.677733\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.74264 4.75039i −0.107082 0.185472i
\(657\) 0 0
\(658\) 21.4706 27.6873i 0.837010 1.07936i
\(659\) 39.2132 1.52753 0.763765 0.645495i \(-0.223349\pi\)
0.763765 + 0.645495i \(0.223349\pi\)
\(660\) 0 0
\(661\) 21.0919 36.5322i 0.820379 1.42094i −0.0850210 0.996379i \(-0.527096\pi\)
0.905400 0.424559i \(-0.139571\pi\)
\(662\) −9.24264 + 16.0087i −0.359225 + 0.622197i
\(663\) 0 0
\(664\) 7.75736 0.301044
\(665\) 0 0
\(666\) 0 0
\(667\) 15.3640 + 26.6112i 0.594895 + 1.03039i
\(668\) 12.1066 20.9692i 0.468418 0.811325i
\(669\) 0 0
\(670\) 0 0
\(671\) −26.4853 −1.02245
\(672\) 0 0
\(673\) −27.4853 −1.05948 −0.529740 0.848160i \(-0.677710\pi\)
−0.529740 + 0.848160i \(0.677710\pi\)
\(674\) 2.24264 + 3.88437i 0.0863833 + 0.149620i
\(675\) 0 0
\(676\) −12.9853 + 22.4912i −0.499434 + 0.865045i
\(677\) 17.1213 + 29.6550i 0.658026 + 1.13973i 0.981126 + 0.193369i \(0.0619416\pi\)
−0.323100 + 0.946365i \(0.604725\pi\)
\(678\) 0 0
\(679\) 32.7279 + 4.47871i 1.25598 + 0.171877i
\(680\) 0 0
\(681\) 0 0
\(682\) −1.60660 + 2.78272i −0.0615200 + 0.106556i
\(683\) 20.8492 36.1119i 0.797774 1.38179i −0.123289 0.992371i \(-0.539344\pi\)
0.921063 0.389414i \(-0.127323\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.13604 18.3967i −0.0815543 0.702388i
\(687\) 0 0
\(688\) 3.24264 + 5.61642i 0.123625 + 0.214124i
\(689\) 13.2426 22.9369i 0.504504 0.873827i
\(690\) 0 0
\(691\) −3.12132 5.40629i −0.118741 0.205665i 0.800528 0.599295i \(-0.204552\pi\)
−0.919269 + 0.393630i \(0.871219\pi\)
\(692\) 10.9706 0.417038
\(693\) 0 0
\(694\) −6.72792 −0.255388
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 12.4853 + 21.6251i 0.472575 + 0.818524i
\(699\) 0 0
\(700\) −5.00000 12.2474i −0.188982 0.462910i
\(701\) −13.7574 −0.519608 −0.259804 0.965661i \(-0.583658\pi\)
−0.259804 + 0.965661i \(0.583658\pi\)
\(702\) 0 0
\(703\) 12.4853 21.6251i 0.470891 0.815608i
\(704\) 2.12132 3.67423i 0.0799503 0.138478i
\(705\) 0 0
\(706\) 21.0000 0.790345
\(707\) −12.5772 + 16.2189i −0.473013 + 0.609973i
\(708\) 0 0
\(709\) −12.8492 22.2555i −0.482563 0.835824i 0.517236 0.855843i \(-0.326961\pi\)
−0.999800 + 0.0200183i \(0.993628\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.74264 + 4.75039i 0.102785 + 0.178029i
\(713\) −5.48528 −0.205425
\(714\) 0 0
\(715\) 0 0
\(716\) −8.12132 14.0665i −0.303508 0.525691i
\(717\) 0 0
\(718\) −5.37868 + 9.31615i −0.200731 + 0.347675i
\(719\) −1.13604 1.96768i −0.0423671 0.0733820i 0.844064 0.536242i \(-0.180157\pi\)
−0.886431 + 0.462860i \(0.846823\pi\)
\(720\) 0 0
\(721\) 0.757359 + 1.85514i 0.0282055 + 0.0690892i
\(722\) −19.9706 −0.743227
\(723\) 0 0
\(724\) 10.1213 17.5306i 0.376156 0.651521i
\(725\) 10.6066 18.3712i 0.393919 0.682288i
\(726\) 0 0
\(727\) −25.7279 −0.954196 −0.477098 0.878850i \(-0.658311\pi\)
−0.477098 + 0.878850i \(0.658311\pi\)
\(728\) 16.3640 + 2.23936i 0.606489 + 0.0829961i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 19.4853 + 33.7495i 0.719705 + 1.24657i 0.961116 + 0.276143i \(0.0890564\pi\)
−0.241411 + 0.970423i \(0.577610\pi\)
\(734\) −11.7279 −0.432886
\(735\) 0 0
\(736\) 7.24264 0.266967
\(737\) −17.4853 30.2854i −0.644079 1.11558i
\(738\) 0 0
\(739\) −5.24264 + 9.08052i −0.192854 + 0.334032i −0.946195 0.323598i \(-0.895108\pi\)
0.753341 + 0.657630i \(0.228441\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11.1213 1.52192i −0.408277 0.0558714i
\(743\) −34.7574 −1.27512 −0.637562 0.770399i \(-0.720057\pi\)
−0.637562 + 0.770399i \(0.720057\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.60660 + 6.24682i −0.132047 + 0.228712i
\(747\) 0 0
\(748\) 0 0
\(749\) 2.48528 + 6.08767i 0.0908102 + 0.222439i
\(750\) 0 0
\(751\) 8.62132 + 14.9326i 0.314596 + 0.544897i 0.979352 0.202164i \(-0.0647975\pi\)
−0.664755 + 0.747061i \(0.731464\pi\)
\(752\) 6.62132 11.4685i 0.241455 0.418212i
\(753\) 0 0
\(754\) 13.2426 + 22.9369i 0.482269 + 0.835314i
\(755\) 0 0
\(756\) 0 0
\(757\) 12.9706 0.471423 0.235712 0.971823i \(-0.424258\pi\)
0.235712 + 0.971823i \(0.424258\pi\)
\(758\) 0.636039 + 1.10165i 0.0231020 + 0.0400138i
\(759\) 0 0
\(760\) 0 0
\(761\) 10.5000 + 18.1865i 0.380625 + 0.659261i 0.991152 0.132734i \(-0.0423756\pi\)
−0.610527 + 0.791995i \(0.709042\pi\)
\(762\) 0 0
\(763\) −3.63604 + 4.68885i −0.131633 + 0.169748i
\(764\) −8.48528 −0.306987
\(765\) 0 0
\(766\) 12.1066 20.9692i 0.437429 0.757650i
\(767\) 0 0
\(768\) 0 0
\(769\) −12.4853 −0.450231 −0.225115 0.974332i \(-0.572276\pi\)
−0.225115 + 0.974332i \(0.572276\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.25736 + 3.90986i 0.0812441 + 0.140719i
\(773\) −22.6066 + 39.1558i −0.813103 + 1.40834i 0.0975792 + 0.995228i \(0.468890\pi\)
−0.910682 + 0.413108i \(0.864443\pi\)
\(774\) 0 0
\(775\) 1.89340 + 3.27946i 0.0680129 + 0.117802i
\(776\) 12.4853 0.448195
\(777\) 0 0
\(778\) 10.2426 0.367216
\(779\) −17.1213 29.6550i −0.613435 1.06250i
\(780\) 0 0
\(781\) 15.3640 26.6112i 0.549766 0.952222i
\(782\) 0 0
\(783\) 0 0
\(784\) −1.74264 6.77962i −0.0622372 0.242129i
\(785\) 0 0
\(786\) 0 0
\(787\) 15.6066 27.0314i 0.556315 0.963566i −0.441485 0.897269i \(-0.645548\pi\)
0.997800 0.0662975i \(-0.0211186\pi\)
\(788\) 8.48528 14.6969i 0.302276 0.523557i
\(789\) 0 0
\(790\) 0 0
\(791\) −53.6985 7.34847i −1.90930 0.261281i
\(792\) 0 0
\(793\) −19.4853 33.7495i −0.691943 1.19848i
\(794\) −10.1213 + 17.5306i −0.359192 + 0.622139i
\(795\) 0 0
\(796\) 7.37868 + 12.7802i 0.261530 + 0.452984i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.50000 4.33013i −0.0883883 0.153093i
\(801\) 0 0
\(802\) −14.7426 + 25.5350i −0.520581 + 0.901672i
\(803\) −14.8492 25.7196i −0.524018 0.907626i
\(804\) 0 0
\(805\) 0 0
\(806\) −4.72792 −0.166534
\(807\) 0 0
\(808\) −3.87868 + 6.71807i −0.136451 + 0.236341i
\(809\) −4.50000 + 7.79423i −0.158212 + 0.274030i −0.934224 0.356687i \(-0.883906\pi\)
0.776012 + 0.630718i \(0.217239\pi\)
\(810\) 0 0
\(811\) −23.4558 −0.823646 −0.411823 0.911264i \(-0.635108\pi\)
−0.411823 + 0.911264i \(0.635108\pi\)
\(812\) 6.87868 8.87039i 0.241394 0.311290i
\(813\) 0 0
\(814\) 8.48528 + 14.6969i 0.297409 + 0.515127i
\(815\) 0 0
\(816\) 0 0
\(817\) 20.2426 + 35.0613i 0.708200 + 1.22664i
\(818\) −35.0000 −1.22375
\(819\) 0 0
\(820\) 0 0
\(821\) −16.0919 27.8720i −0.561611 0.972738i −0.997356 0.0726682i \(-0.976849\pi\)
0.435746 0.900070i \(-0.356485\pi\)
\(822\) 0 0
\(823\) −20.3492 + 35.2459i −0.709330 + 1.22860i 0.255776 + 0.966736i \(0.417669\pi\)
−0.965106 + 0.261860i \(0.915664\pi\)
\(824\) 0.378680 + 0.655892i 0.0131919 + 0.0228491i
\(825\) 0 0
\(826\) 0 0
\(827\) −16.9706 −0.590124 −0.295062 0.955478i \(-0.595340\pi\)
−0.295062 + 0.955478i \(0.595340\pi\)
\(828\) 0 0
\(829\) 24.9706 43.2503i 0.867263 1.50214i 0.00248151 0.999997i \(-0.499210\pi\)
0.864782 0.502148i \(-0.167457\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.24264 0.216425
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 13.2426 22.9369i 0.458006 0.793290i
\(837\) 0 0
\(838\) 3.87868 + 6.71807i 0.133987 + 0.232072i
\(839\) −15.5147 −0.535628 −0.267814 0.963471i \(-0.586301\pi\)
−0.267814 + 0.963471i \(0.586301\pi\)
\(840\) 0 0
\(841\) −11.0000 −0.379310
\(842\) −7.12132 12.3345i −0.245417 0.425075i
\(843\) 0 0
\(844\) 3.24264 5.61642i 0.111616 0.193325i
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000 + 17.1464i 0.240523 + 0.589158i
\(848\) −4.24264 −0.145693
\(849\) 0 0
\(850\) 0 0
\(851\) −14.4853 + 25.0892i −0.496549 + 0.860048i
\(852\) 0 0
\(853\) 28.1838 0.964994 0.482497 0.875898i \(-0.339730\pi\)
0.482497 + 0.875898i \(0.339730\pi\)
\(854\) −10.1213 + 13.0519i −0.346344 + 0.446628i
\(855\) 0 0
\(856\) 1.24264 + 2.15232i 0.0424726 + 0.0735647i
\(857\) −2.74264 + 4.75039i −0.0936868 + 0.162270i −0.909060 0.416666i \(-0.863199\pi\)
0.815373 + 0.578936i \(0.196532\pi\)
\(858\) 0 0
\(859\) −21.8492 37.8440i −0.745487 1.29122i −0.949967 0.312350i \(-0.898884\pi\)
0.204481 0.978871i \(-0.434449\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −37.2426 −1.26849
\(863\) −0.106602 0.184640i −0.00362876 0.00628520i 0.864205 0.503139i \(-0.167822\pi\)
−0.867834 + 0.496854i \(0.834488\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.98528 + 3.43861i 0.0674626 + 0.116849i
\(867\) 0 0
\(868\) 0.757359 + 1.85514i 0.0257065 + 0.0629677i
\(869\) −39.2132 −1.33022
\(870\) 0 0
\(871\) 25.7279 44.5621i 0.871757 1.50993i
\(872\) −1.12132 + 1.94218i −0.0379727 + 0.0657706i
\(873\) 0 0
\(874\) 45.2132 1.52936
\(875\) 0 0
\(876\) 0 0
\(877\) −12.8492 22.2555i −0.433888 0.751516i 0.563316 0.826241i \(-0.309525\pi\)
−0.997204 + 0.0747253i \(0.976192\pi\)
\(878\) 5.86396 10.1567i 0.197899 0.342771i
\(879\) 0 0
\(880\) 0 0
\(881\) 36.5147 1.23021 0.615106 0.788444i \(-0.289113\pi\)
0.615106 + 0.788444i \(0.289113\pi\)
\(882\) 0 0
\(883\) 49.6985 1.67249 0.836244 0.548358i \(-0.184747\pi\)
0.836244 + 0.548358i \(0.184747\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.75736 + 8.23999i −0.159827 + 0.276828i
\(887\) 19.8640 + 34.4054i 0.666967 + 1.15522i 0.978748 + 0.205066i \(0.0657409\pi\)
−0.311782 + 0.950154i \(0.600926\pi\)
\(888\) 0 0
\(889\) −17.7132 2.42400i −0.594082 0.0812982i
\(890\) 0 0
\(891\) 0 0
\(892\) −5.86396 + 10.1567i −0.196340 + 0.340071i
\(893\) 41.3345 71.5935i 1.38321 2.39578i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 2.44949i −0.0334077 0.0818317i
\(897\) 0 0
\(898\) 2.48528 + 4.30463i 0.0829349 + 0.143647i
\(899\) −1.60660 + 2.78272i −0.0535832 + 0.0928088i
\(900\) 0 0
\(901\) 0 0
\(902\) 23.2721 0.774875
\(903\) 0 0
\(904\) −20.4853 −0.681330
\(905\) 0 0
\(906\) 0 0
\(907\) 18.9706 32.8580i 0.629907 1.09103i −0.357663 0.933851i \(-0.616426\pi\)
0.987570 0.157180i \(-0.0502404\pi\)
\(908\) 13.2426 + 22.9369i 0.439472 + 0.761189i
\(909\) 0 0
\(910\) 0 0
\(911\) 14.2721 0.472855 0.236428 0.971649i \(-0.424023\pi\)
0.236428 + 0.971649i \(0.424023\pi\)
\(912\) 0 0
\(913\) −16.4558 + 28.5024i −0.544609 + 0.943290i
\(914\) 5.75736 9.97204i 0.190437 0.329846i
\(915\) 0 0
\(916\) 24.9706 0.825051
\(917\) 7.75736 + 19.0016i 0.256171 + 0.627487i
\(918\) 0 0
\(919\) −16.7279 28.9736i −0.551803 0.955751i −0.998145 0.0608884i \(-0.980607\pi\)
0.446341 0.894863i \(-0.352727\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −17.1213 29.6550i −0.563861 0.976635i
\(923\) 45.2132 1.48821
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) −4.37868 7.58410i −0.143892 0.249229i
\(927\) 0 0
\(928\) 2.12132 3.67423i 0.0696358 0.120613i
\(929\) −5.01472 8.68575i −0.164528 0.284970i 0.771960 0.635671i \(-0.219277\pi\)
−0.936487 + 0.350701i \(0.885943\pi\)
\(930\) 0 0
\(931\) −10.8787 42.3227i −0.356534 1.38707i
\(932\) −20.4853 −0.671018
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.4558 0.994949 0.497475 0.867479i \(-0.334261\pi\)
0.497475 + 0.867479i \(0.334261\pi\)
\(938\) −21.6066 2.95680i −0.705481 0.0965428i
\(939\) 0 0
\(940\) 0 0
\(941\) −20.3345 + 35.2204i −0.662887 + 1.14815i 0.316967 + 0.948437i \(0.397335\pi\)
−0.979854 + 0.199717i \(0.935998\pi\)
\(942\) 0 0
\(943\) 19.8640 + 34.4054i 0.646860 + 1.12039i
\(944\) 0 0
\(945\) 0 0
\(946\) −27.5147 −0.894581
\(947\) −10.7574 18.6323i −0.349567 0.605468i 0.636605 0.771190i \(-0.280338\pi\)
−0.986173 + 0.165722i \(0.947005\pi\)
\(948\) 0 0
\(949\) 21.8492 37.8440i 0.709256 1.22847i
\(950\) −15.6066 27.0314i −0.506345 0.877015i
\(951\) 0 0
\(952\) 0 0
\(953\) 6.51472 0.211032 0.105516 0.994418i \(-0.466351\pi\)
0.105516 + 0.994418i \(0.466351\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.86396 3.22848i 0.0602848 0.104416i
\(957\) 0 0
\(958\) −13.2426 −0.427850
\(959\) 32.3787 41.7539i 1.04556 1.34830i
\(960\) 0 0
\(961\) 15.2132 + 26.3500i 0.490748 + 0.850001i
\(962\) −12.4853 + 21.6251i −0.402542 + 0.697223i
\(963\) 0 0
\(964\) −4.25736 7.37396i −0.137120 0.237499i
\(965\) 0 0
\(966\) 0 0
\(967\) −6.69848 −0.215409 −0.107704 0.994183i \(-0.534350\pi\)
−0.107704 + 0.994183i \(0.534350\pi\)
\(968\) 3.50000 + 6.06218i 0.112494 + 0.194846i
\(969\) 0 0
\(970\) 0 0
\(971\) −13.2426 22.9369i −0.424977 0.736081i 0.571442 0.820643i \(-0.306384\pi\)
−0.996418 + 0.0845617i \(0.973051\pi\)
\(972\) 0 0
\(973\) 12.3934 + 1.69600i 0.397314 + 0.0543712i
\(974\) 2.75736 0.0883515
\(975\) 0 0
\(976\) −3.12132 + 5.40629i −0.0999110 + 0.173051i
\(977\) −6.98528 + 12.0989i −0.223479 + 0.387077i −0.955862 0.293816i \(-0.905075\pi\)
0.732383 + 0.680893i \(0.238408\pi\)
\(978\) 0 0
\(979\) −23.2721 −0.743779
\(980\) 0 0
\(981\) 0 0
\(982\) −3.36396 5.82655i −0.107348 0.185933i
\(983\) −7.75736 + 13.4361i −0.247421 + 0.428546i −0.962810 0.270181i \(-0.912917\pi\)
0.715388 + 0.698727i \(0.246250\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 38.9706 1.23982
\(989\) −23.4853 40.6777i −0.746789 1.29348i
\(990\) 0 0
\(991\) −25.1066 + 43.4859i −0.797537 + 1.38138i 0.123678 + 0.992322i \(0.460531\pi\)
−0.921215 + 0.389053i \(0.872802\pi\)
\(992\) 0.378680 + 0.655892i 0.0120231 + 0.0208246i
\(993\) 0 0
\(994\) −7.24264 17.7408i −0.229723 0.562703i
\(995\) 0 0
\(996\) 0 0
\(997\) −7.00000 + 12.1244i −0.221692 + 0.383982i −0.955322 0.295567i \(-0.904491\pi\)
0.733630 + 0.679549i \(0.237825\pi\)
\(998\) −5.36396 + 9.29065i −0.169793 + 0.294090i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1134.2.g.j.487.1 yes 4
3.2 odd 2 1134.2.g.i.487.1 yes 4
7.2 even 3 inner 1134.2.g.j.163.1 yes 4
7.3 odd 6 7938.2.a.bj.1.1 2
7.4 even 3 7938.2.a.bk.1.1 2
9.2 odd 6 1134.2.h.r.109.2 4
9.4 even 3 1134.2.e.r.865.2 4
9.5 odd 6 1134.2.e.s.865.2 4
9.7 even 3 1134.2.h.s.109.2 4
21.2 odd 6 1134.2.g.i.163.1 4
21.11 odd 6 7938.2.a.bq.1.2 2
21.17 even 6 7938.2.a.bp.1.2 2
63.2 odd 6 1134.2.e.s.919.2 4
63.16 even 3 1134.2.e.r.919.2 4
63.23 odd 6 1134.2.h.r.541.1 4
63.58 even 3 1134.2.h.s.541.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.e.r.865.2 4 9.4 even 3
1134.2.e.r.919.2 4 63.16 even 3
1134.2.e.s.865.2 4 9.5 odd 6
1134.2.e.s.919.2 4 63.2 odd 6
1134.2.g.i.163.1 4 21.2 odd 6
1134.2.g.i.487.1 yes 4 3.2 odd 2
1134.2.g.j.163.1 yes 4 7.2 even 3 inner
1134.2.g.j.487.1 yes 4 1.1 even 1 trivial
1134.2.h.r.109.2 4 9.2 odd 6
1134.2.h.r.541.1 4 63.23 odd 6
1134.2.h.s.109.2 4 9.7 even 3
1134.2.h.s.541.1 4 63.58 even 3
7938.2.a.bj.1.1 2 7.3 odd 6
7938.2.a.bk.1.1 2 7.4 even 3
7938.2.a.bp.1.2 2 21.17 even 6
7938.2.a.bq.1.2 2 21.11 odd 6