Properties

Label 450.6.f.c.143.1
Level $450$
Weight $6$
Character 450.143
Analytic conductor $72.173$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,6,Mod(107,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.107");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.f (of order \(4\), 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: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 450.143
Dual form 450.6.f.c.107.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.82843 - 2.82843i) q^{2} +16.0000i q^{4} +(33.0000 - 33.0000i) q^{7} +(45.2548 - 45.2548i) q^{8} +O(q^{10})\) \(q+(-2.82843 - 2.82843i) q^{2} +16.0000i q^{4} +(33.0000 - 33.0000i) q^{7} +(45.2548 - 45.2548i) q^{8} -258.801i q^{11} +(213.000 + 213.000i) q^{13} -186.676 q^{14} -256.000 q^{16} +(-462.448 - 462.448i) q^{17} -722.000i q^{19} +(-732.000 + 732.000i) q^{22} +(-810.344 + 810.344i) q^{23} -1204.91i q^{26} +(528.000 + 528.000i) q^{28} +2537.10 q^{29} +2764.00 q^{31} +(724.077 + 724.077i) q^{32} +2616.00i q^{34} +(-4467.00 + 4467.00i) q^{37} +(-2042.12 + 2042.12i) q^{38} -7632.51i q^{41} +(3240.00 + 3240.00i) q^{43} +4140.82 q^{44} +4584.00 q^{46} +(-9927.78 - 9927.78i) q^{47} +14629.0i q^{49} +(-3408.00 + 3408.00i) q^{52} +(7937.98 - 7937.98i) q^{53} -2986.82i q^{56} +(-7176.00 - 7176.00i) q^{58} +11450.9 q^{59} +29750.0 q^{61} +(-7817.77 - 7817.77i) q^{62} -4096.00i q^{64} +(-19866.0 + 19866.0i) q^{67} +(7399.17 - 7399.17i) q^{68} -71284.8i q^{71} +(-45066.0 - 45066.0i) q^{73} +25269.2 q^{74} +11552.0 q^{76} +(-8540.44 - 8540.44i) q^{77} -2992.00i q^{79} +(-21588.0 + 21588.0i) q^{82} +(28603.9 - 28603.9i) q^{83} -18328.2i q^{86} +(-11712.0 - 11712.0i) q^{88} +11705.4 q^{89} +14058.0 q^{91} +(-12965.5 - 12965.5i) q^{92} +56160.0i q^{94} +(-111372. + 111372. i) q^{97} +(41377.1 - 41377.1i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 132 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 132 q^{7} + 852 q^{13} - 1024 q^{16} - 2928 q^{22} + 2112 q^{28} + 11056 q^{31} - 17868 q^{37} + 12960 q^{43} + 18336 q^{46} - 13632 q^{52} - 28704 q^{58} + 119000 q^{61} - 79464 q^{67} - 180264 q^{73} + 46208 q^{76} - 86352 q^{82} - 46848 q^{88} + 56232 q^{91} - 445488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82843 2.82843i −0.500000 0.500000i
\(3\) 0 0
\(4\) 16.0000i 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) 33.0000 33.0000i 0.254548 0.254548i −0.568285 0.822832i \(-0.692393\pi\)
0.822832 + 0.568285i \(0.192393\pi\)
\(8\) 45.2548 45.2548i 0.250000 0.250000i
\(9\) 0 0
\(10\) 0 0
\(11\) 258.801i 0.644888i −0.946589 0.322444i \(-0.895496\pi\)
0.946589 0.322444i \(-0.104504\pi\)
\(12\) 0 0
\(13\) 213.000 + 213.000i 0.349560 + 0.349560i 0.859945 0.510386i \(-0.170497\pi\)
−0.510386 + 0.859945i \(0.670497\pi\)
\(14\) −186.676 −0.254548
\(15\) 0 0
\(16\) −256.000 −0.250000
\(17\) −462.448 462.448i −0.388097 0.388097i 0.485911 0.874008i \(-0.338488\pi\)
−0.874008 + 0.485911i \(0.838488\pi\)
\(18\) 0 0
\(19\) 722.000i 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −732.000 + 732.000i −0.322444 + 0.322444i
\(23\) −810.344 + 810.344i −0.319411 + 0.319411i −0.848541 0.529130i \(-0.822518\pi\)
0.529130 + 0.848541i \(0.322518\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1204.91i 0.349560i
\(27\) 0 0
\(28\) 528.000 + 528.000i 0.127274 + 0.127274i
\(29\) 2537.10 0.560199 0.280100 0.959971i \(-0.409633\pi\)
0.280100 + 0.959971i \(0.409633\pi\)
\(30\) 0 0
\(31\) 2764.00 0.516575 0.258288 0.966068i \(-0.416842\pi\)
0.258288 + 0.966068i \(0.416842\pi\)
\(32\) 724.077 + 724.077i 0.125000 + 0.125000i
\(33\) 0 0
\(34\) 2616.00i 0.388097i
\(35\) 0 0
\(36\) 0 0
\(37\) −4467.00 + 4467.00i −0.536428 + 0.536428i −0.922478 0.386050i \(-0.873839\pi\)
0.386050 + 0.922478i \(0.373839\pi\)
\(38\) −2042.12 + 2042.12i −0.229416 + 0.229416i
\(39\) 0 0
\(40\) 0 0
\(41\) 7632.51i 0.709100i −0.935037 0.354550i \(-0.884634\pi\)
0.935037 0.354550i \(-0.115366\pi\)
\(42\) 0 0
\(43\) 3240.00 + 3240.00i 0.267223 + 0.267223i 0.827980 0.560757i \(-0.189490\pi\)
−0.560757 + 0.827980i \(0.689490\pi\)
\(44\) 4140.82 0.322444
\(45\) 0 0
\(46\) 4584.00 0.319411
\(47\) −9927.78 9927.78i −0.655552 0.655552i 0.298772 0.954325i \(-0.403423\pi\)
−0.954325 + 0.298772i \(0.903423\pi\)
\(48\) 0 0
\(49\) 14629.0i 0.870411i
\(50\) 0 0
\(51\) 0 0
\(52\) −3408.00 + 3408.00i −0.174780 + 0.174780i
\(53\) 7937.98 7937.98i 0.388169 0.388169i −0.485865 0.874034i \(-0.661495\pi\)
0.874034 + 0.485865i \(0.161495\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2986.82i 0.127274i
\(57\) 0 0
\(58\) −7176.00 7176.00i −0.280100 0.280100i
\(59\) 11450.9 0.428262 0.214131 0.976805i \(-0.431308\pi\)
0.214131 + 0.976805i \(0.431308\pi\)
\(60\) 0 0
\(61\) 29750.0 1.02368 0.511838 0.859082i \(-0.328965\pi\)
0.511838 + 0.859082i \(0.328965\pi\)
\(62\) −7817.77 7817.77i −0.258288 0.258288i
\(63\) 0 0
\(64\) 4096.00i 0.125000i
\(65\) 0 0
\(66\) 0 0
\(67\) −19866.0 + 19866.0i −0.540659 + 0.540659i −0.923722 0.383063i \(-0.874869\pi\)
0.383063 + 0.923722i \(0.374869\pi\)
\(68\) 7399.17 7399.17i 0.194049 0.194049i
\(69\) 0 0
\(70\) 0 0
\(71\) 71284.8i 1.67823i −0.543954 0.839115i \(-0.683074\pi\)
0.543954 0.839115i \(-0.316926\pi\)
\(72\) 0 0
\(73\) −45066.0 45066.0i −0.989787 0.989787i 0.0101611 0.999948i \(-0.496766\pi\)
−0.999948 + 0.0101611i \(0.996766\pi\)
\(74\) 25269.2 0.536428
\(75\) 0 0
\(76\) 11552.0 0.229416
\(77\) −8540.44 8540.44i −0.164155 0.164155i
\(78\) 0 0
\(79\) 2992.00i 0.0539379i −0.999636 0.0269689i \(-0.991414\pi\)
0.999636 0.0269689i \(-0.00858552\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −21588.0 + 21588.0i −0.354550 + 0.354550i
\(83\) 28603.9 28603.9i 0.455753 0.455753i −0.441505 0.897259i \(-0.645555\pi\)
0.897259 + 0.441505i \(0.145555\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 18328.2i 0.267223i
\(87\) 0 0
\(88\) −11712.0 11712.0i −0.161222 0.161222i
\(89\) 11705.4 0.156644 0.0783219 0.996928i \(-0.475044\pi\)
0.0783219 + 0.996928i \(0.475044\pi\)
\(90\) 0 0
\(91\) 14058.0 0.177959
\(92\) −12965.5 12965.5i −0.159706 0.159706i
\(93\) 0 0
\(94\) 56160.0i 0.655552i
\(95\) 0 0
\(96\) 0 0
\(97\) −111372. + 111372.i −1.20184 + 1.20184i −0.228234 + 0.973606i \(0.573295\pi\)
−0.973606 + 0.228234i \(0.926705\pi\)
\(98\) 41377.1 41377.1i 0.435206 0.435206i
\(99\) 0 0
\(100\) 0 0
\(101\) 10173.9i 0.0992389i −0.998768 0.0496195i \(-0.984199\pi\)
0.998768 0.0496195i \(-0.0158009\pi\)
\(102\) 0 0
\(103\) −109833. 109833.i −1.02009 1.02009i −0.999794 0.0202991i \(-0.993538\pi\)
−0.0202991 0.999794i \(-0.506462\pi\)
\(104\) 19278.6 0.174780
\(105\) 0 0
\(106\) −44904.0 −0.388169
\(107\) −131352. 131352.i −1.10912 1.10912i −0.993267 0.115851i \(-0.963040\pi\)
−0.115851 0.993267i \(-0.536960\pi\)
\(108\) 0 0
\(109\) 23758.0i 0.191533i −0.995404 0.0957665i \(-0.969470\pi\)
0.995404 0.0957665i \(-0.0305302\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8448.00 + 8448.00i −0.0636369 + 0.0636369i
\(113\) 82336.9 82336.9i 0.606595 0.606595i −0.335460 0.942054i \(-0.608892\pi\)
0.942054 + 0.335460i \(0.108892\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 40593.6i 0.280100i
\(117\) 0 0
\(118\) −32388.0 32388.0i −0.214131 0.214131i
\(119\) −30521.6 −0.197578
\(120\) 0 0
\(121\) 94073.0 0.584119
\(122\) −84145.7 84145.7i −0.511838 0.511838i
\(123\) 0 0
\(124\) 44224.0i 0.258288i
\(125\) 0 0
\(126\) 0 0
\(127\) −227979. + 227979.i −1.25425 + 1.25425i −0.300460 + 0.953795i \(0.597140\pi\)
−0.953795 + 0.300460i \(0.902860\pi\)
\(128\) −11585.2 + 11585.2i −0.0625000 + 0.0625000i
\(129\) 0 0
\(130\) 0 0
\(131\) 46563.0i 0.237062i 0.992950 + 0.118531i \(0.0378186\pi\)
−0.992950 + 0.118531i \(0.962181\pi\)
\(132\) 0 0
\(133\) −23826.0 23826.0i −0.116794 0.116794i
\(134\) 112379. 0.540659
\(135\) 0 0
\(136\) −41856.0 −0.194049
\(137\) −150652. 150652.i −0.685762 0.685762i 0.275531 0.961292i \(-0.411146\pi\)
−0.961292 + 0.275531i \(0.911146\pi\)
\(138\) 0 0
\(139\) 152756.i 0.670596i −0.942112 0.335298i \(-0.891163\pi\)
0.942112 0.335298i \(-0.108837\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −201624. + 201624.i −0.839115 + 0.839115i
\(143\) 55124.6 55124.6i 0.225427 0.225427i
\(144\) 0 0
\(145\) 0 0
\(146\) 254932.i 0.989787i
\(147\) 0 0
\(148\) −71472.0 71472.0i −0.268214 0.268214i
\(149\) −462787. −1.70772 −0.853858 0.520506i \(-0.825743\pi\)
−0.853858 + 0.520506i \(0.825743\pi\)
\(150\) 0 0
\(151\) −179168. −0.639467 −0.319733 0.947508i \(-0.603593\pi\)
−0.319733 + 0.947508i \(0.603593\pi\)
\(152\) −32674.0 32674.0i −0.114708 0.114708i
\(153\) 0 0
\(154\) 48312.0i 0.164155i
\(155\) 0 0
\(156\) 0 0
\(157\) −371961. + 371961.i −1.20434 + 1.20434i −0.231504 + 0.972834i \(0.574365\pi\)
−0.972834 + 0.231504i \(0.925635\pi\)
\(158\) −8462.65 + 8462.65i −0.0269689 + 0.0269689i
\(159\) 0 0
\(160\) 0 0
\(161\) 53482.7i 0.162611i
\(162\) 0 0
\(163\) −189294. 189294.i −0.558043 0.558043i 0.370707 0.928750i \(-0.379115\pi\)
−0.928750 + 0.370707i \(0.879115\pi\)
\(164\) 122120. 0.354550
\(165\) 0 0
\(166\) −161808. −0.455753
\(167\) −258933. 258933.i −0.718448 0.718448i 0.249839 0.968287i \(-0.419622\pi\)
−0.968287 + 0.249839i \(0.919622\pi\)
\(168\) 0 0
\(169\) 280555.i 0.755616i
\(170\) 0 0
\(171\) 0 0
\(172\) −51840.0 + 51840.0i −0.133612 + 0.133612i
\(173\) 160164. 160164.i 0.406864 0.406864i −0.473779 0.880644i \(-0.657111\pi\)
0.880644 + 0.473779i \(0.157111\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 66253.1i 0.161222i
\(177\) 0 0
\(178\) −33108.0 33108.0i −0.0783219 0.0783219i
\(179\) −484420. −1.13003 −0.565015 0.825081i \(-0.691130\pi\)
−0.565015 + 0.825081i \(0.691130\pi\)
\(180\) 0 0
\(181\) −266402. −0.604423 −0.302212 0.953241i \(-0.597725\pi\)
−0.302212 + 0.953241i \(0.597725\pi\)
\(182\) −39762.0 39762.0i −0.0889795 0.0889795i
\(183\) 0 0
\(184\) 73344.0i 0.159706i
\(185\) 0 0
\(186\) 0 0
\(187\) −119682. + 119682.i −0.250279 + 0.250279i
\(188\) 158844. 158844.i 0.327776 0.327776i
\(189\) 0 0
\(190\) 0 0
\(191\) 43283.4i 0.0858496i 0.999078 + 0.0429248i \(0.0136676\pi\)
−0.999078 + 0.0429248i \(0.986332\pi\)
\(192\) 0 0
\(193\) −160494. 160494.i −0.310146 0.310146i 0.534820 0.844966i \(-0.320379\pi\)
−0.844966 + 0.534820i \(0.820379\pi\)
\(194\) 630015. 1.20184
\(195\) 0 0
\(196\) −234064. −0.435206
\(197\) −387345. 387345.i −0.711102 0.711102i 0.255664 0.966766i \(-0.417706\pi\)
−0.966766 + 0.255664i \(0.917706\pi\)
\(198\) 0 0
\(199\) 414340.i 0.741693i −0.928694 0.370846i \(-0.879068\pi\)
0.928694 0.370846i \(-0.120932\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −28776.0 + 28776.0i −0.0496195 + 0.0496195i
\(203\) 83724.3 83724.3i 0.142597 0.142597i
\(204\) 0 0
\(205\) 0 0
\(206\) 621309.i 1.02009i
\(207\) 0 0
\(208\) −54528.0 54528.0i −0.0873899 0.0873899i
\(209\) −186854. −0.295895
\(210\) 0 0
\(211\) 589796. 0.912002 0.456001 0.889979i \(-0.349281\pi\)
0.456001 + 0.889979i \(0.349281\pi\)
\(212\) 127008. + 127008.i 0.194084 + 0.194084i
\(213\) 0 0
\(214\) 743040.i 1.10912i
\(215\) 0 0
\(216\) 0 0
\(217\) 91212.0 91212.0i 0.131493 0.131493i
\(218\) −67197.8 + 67197.8i −0.0957665 + 0.0957665i
\(219\) 0 0
\(220\) 0 0
\(221\) 197003.i 0.271326i
\(222\) 0 0
\(223\) −330381. 330381.i −0.444890 0.444890i 0.448761 0.893652i \(-0.351865\pi\)
−0.893652 + 0.448761i \(0.851865\pi\)
\(224\) 47789.1 0.0636369
\(225\) 0 0
\(226\) −465768. −0.606595
\(227\) 549430. + 549430.i 0.707698 + 0.707698i 0.966051 0.258353i \(-0.0831796\pi\)
−0.258353 + 0.966051i \(0.583180\pi\)
\(228\) 0 0
\(229\) 630494.i 0.794497i −0.917711 0.397248i \(-0.869965\pi\)
0.917711 0.397248i \(-0.130035\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 114816. 114816.i 0.140050 0.140050i
\(233\) 244516. 244516.i 0.295065 0.295065i −0.544012 0.839077i \(-0.683096\pi\)
0.839077 + 0.544012i \(0.183096\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 183214.i 0.214131i
\(237\) 0 0
\(238\) 86328.0 + 86328.0i 0.0987892 + 0.0987892i
\(239\) −792670. −0.897630 −0.448815 0.893625i \(-0.648154\pi\)
−0.448815 + 0.893625i \(0.648154\pi\)
\(240\) 0 0
\(241\) 356612. 0.395506 0.197753 0.980252i \(-0.436636\pi\)
0.197753 + 0.980252i \(0.436636\pi\)
\(242\) −266079. 266079.i −0.292060 0.292060i
\(243\) 0 0
\(244\) 476000.i 0.511838i
\(245\) 0 0
\(246\) 0 0
\(247\) 153786. 153786.i 0.160389 0.160389i
\(248\) 125084. 125084.i 0.129144 0.129144i
\(249\) 0 0
\(250\) 0 0
\(251\) 1.15393e6i 1.15610i −0.816003 0.578048i \(-0.803815\pi\)
0.816003 0.578048i \(-0.196185\pi\)
\(252\) 0 0
\(253\) 209718. + 209718.i 0.205984 + 0.205984i
\(254\) 1.28964e6 1.25425
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 859945. + 859945.i 0.812153 + 0.812153i 0.984956 0.172803i \(-0.0552825\pi\)
−0.172803 + 0.984956i \(0.555282\pi\)
\(258\) 0 0
\(259\) 294822.i 0.273093i
\(260\) 0 0
\(261\) 0 0
\(262\) 131700. 131700.i 0.118531 0.118531i
\(263\) −679047. + 679047.i −0.605356 + 0.605356i −0.941729 0.336373i \(-0.890800\pi\)
0.336373 + 0.941729i \(0.390800\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 134780.i 0.116794i
\(267\) 0 0
\(268\) −317856. 317856.i −0.270329 0.270329i
\(269\) −672535. −0.566675 −0.283338 0.959020i \(-0.591442\pi\)
−0.283338 + 0.959020i \(0.591442\pi\)
\(270\) 0 0
\(271\) 1.57142e6 1.29978 0.649889 0.760029i \(-0.274815\pi\)
0.649889 + 0.760029i \(0.274815\pi\)
\(272\) 118387. + 118387.i 0.0970243 + 0.0970243i
\(273\) 0 0
\(274\) 852216.i 0.685762i
\(275\) 0 0
\(276\) 0 0
\(277\) 1.04804e6 1.04804e6i 0.820689 0.820689i −0.165517 0.986207i \(-0.552929\pi\)
0.986207 + 0.165517i \(0.0529294\pi\)
\(278\) −432059. + 432059.i −0.335298 + 0.335298i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.74425e6i 1.31778i −0.752240 0.658889i \(-0.771027\pi\)
0.752240 0.658889i \(-0.228973\pi\)
\(282\) 0 0
\(283\) 1.43080e6 + 1.43080e6i 1.06197 + 1.06197i 0.997948 + 0.0640246i \(0.0203936\pi\)
0.0640246 + 0.997948i \(0.479606\pi\)
\(284\) 1.14056e6 0.839115
\(285\) 0 0
\(286\) −311832. −0.225427
\(287\) −251873. 251873.i −0.180500 0.180500i
\(288\) 0 0
\(289\) 992141.i 0.698761i
\(290\) 0 0
\(291\) 0 0
\(292\) 721056. 721056.i 0.494894 0.494894i
\(293\) −666018. + 666018.i −0.453228 + 0.453228i −0.896425 0.443196i \(-0.853844\pi\)
0.443196 + 0.896425i \(0.353844\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 404307.i 0.268214i
\(297\) 0 0
\(298\) 1.30896e6 + 1.30896e6i 0.853858 + 0.853858i
\(299\) −345207. −0.223306
\(300\) 0 0
\(301\) 213840. 0.136042
\(302\) 506764. + 506764.i 0.319733 + 0.319733i
\(303\) 0 0
\(304\) 184832.i 0.114708i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.08216e6 1.08216e6i 0.655308 0.655308i −0.298958 0.954266i \(-0.596639\pi\)
0.954266 + 0.298958i \(0.0966391\pi\)
\(308\) 136647. 136647.i 0.0820773 0.0820773i
\(309\) 0 0
\(310\) 0 0
\(311\) 1.79768e6i 1.05393i −0.849887 0.526966i \(-0.823330\pi\)
0.849887 0.526966i \(-0.176670\pi\)
\(312\) 0 0
\(313\) 799266. + 799266.i 0.461138 + 0.461138i 0.899028 0.437891i \(-0.144274\pi\)
−0.437891 + 0.899028i \(0.644274\pi\)
\(314\) 2.10413e6 1.20434
\(315\) 0 0
\(316\) 47872.0 0.0269689
\(317\) 906831. + 906831.i 0.506848 + 0.506848i 0.913558 0.406709i \(-0.133324\pi\)
−0.406709 + 0.913558i \(0.633324\pi\)
\(318\) 0 0
\(319\) 656604.i 0.361266i
\(320\) 0 0
\(321\) 0 0
\(322\) 151272. 151272.i 0.0813053 0.0813053i
\(323\) −333887. + 333887.i −0.178071 + 0.178071i
\(324\) 0 0
\(325\) 0 0
\(326\) 1.07081e6i 0.558043i
\(327\) 0 0
\(328\) −345408. 345408.i −0.177275 0.177275i
\(329\) −655233. −0.333738
\(330\) 0 0
\(331\) 107546. 0.0539541 0.0269770 0.999636i \(-0.491412\pi\)
0.0269770 + 0.999636i \(0.491412\pi\)
\(332\) 457662. + 457662.i 0.227877 + 0.227877i
\(333\) 0 0
\(334\) 1.46474e6i 0.718448i
\(335\) 0 0
\(336\) 0 0
\(337\) 1.31587e6 1.31587e6i 0.631156 0.631156i −0.317202 0.948358i \(-0.602743\pi\)
0.948358 + 0.317202i \(0.102743\pi\)
\(338\) −793529. + 793529.i −0.377808 + 0.377808i
\(339\) 0 0
\(340\) 0 0
\(341\) 715326.i 0.333133i
\(342\) 0 0
\(343\) 1.03739e6 + 1.03739e6i 0.476108 + 0.476108i
\(344\) 293251. 0.133612
\(345\) 0 0
\(346\) −906024. −0.406864
\(347\) 1.78519e6 + 1.78519e6i 0.795905 + 0.795905i 0.982447 0.186542i \(-0.0597279\pi\)
−0.186542 + 0.982447i \(0.559728\pi\)
\(348\) 0 0
\(349\) 1.47933e6i 0.650134i 0.945691 + 0.325067i \(0.105387\pi\)
−0.945691 + 0.325067i \(0.894613\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 187392. 187392.i 0.0806110 0.0806110i
\(353\) −2.22216e6 + 2.22216e6i −0.949156 + 0.949156i −0.998769 0.0496125i \(-0.984201\pi\)
0.0496125 + 0.998769i \(0.484201\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 187287.i 0.0783219i
\(357\) 0 0
\(358\) 1.37015e6 + 1.37015e6i 0.565015 + 0.565015i
\(359\) −33075.6 −0.0135448 −0.00677239 0.999977i \(-0.502156\pi\)
−0.00677239 + 0.999977i \(0.502156\pi\)
\(360\) 0 0
\(361\) 1.95482e6 0.789474
\(362\) 753499. + 753499.i 0.302212 + 0.302212i
\(363\) 0 0
\(364\) 224928.i 0.0889795i
\(365\) 0 0
\(366\) 0 0
\(367\) 2.24097e6 2.24097e6i 0.868501 0.868501i −0.123806 0.992306i \(-0.539510\pi\)
0.992306 + 0.123806i \(0.0395099\pi\)
\(368\) 207448. 207448.i 0.0798528 0.0798528i
\(369\) 0 0
\(370\) 0 0
\(371\) 523907.i 0.197615i
\(372\) 0 0
\(373\) 1.01738e6 + 1.01738e6i 0.378625 + 0.378625i 0.870606 0.491981i \(-0.163727\pi\)
−0.491981 + 0.870606i \(0.663727\pi\)
\(374\) 677024. 0.250279
\(375\) 0 0
\(376\) −898560. −0.327776
\(377\) 540402. + 540402.i 0.195823 + 0.195823i
\(378\) 0 0
\(379\) 2.12654e6i 0.760459i −0.924892 0.380229i \(-0.875845\pi\)
0.924892 0.380229i \(-0.124155\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 122424. 122424.i 0.0429248 0.0429248i
\(383\) 1.07621e6 1.07621e6i 0.374885 0.374885i −0.494368 0.869253i \(-0.664600\pi\)
0.869253 + 0.494368i \(0.164600\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 907891.i 0.310146i
\(387\) 0 0
\(388\) −1.78195e6 1.78195e6i −0.600920 0.600920i
\(389\) −4.19716e6 −1.40631 −0.703156 0.711036i \(-0.748226\pi\)
−0.703156 + 0.711036i \(0.748226\pi\)
\(390\) 0 0
\(391\) 749484. 0.247925
\(392\) 662033. + 662033.i 0.217603 + 0.217603i
\(393\) 0 0
\(394\) 2.19115e6i 0.711102i
\(395\) 0 0
\(396\) 0 0
\(397\) 2.02632e6 2.02632e6i 0.645257 0.645257i −0.306586 0.951843i \(-0.599187\pi\)
0.951843 + 0.306586i \(0.0991868\pi\)
\(398\) −1.17193e6 + 1.17193e6i −0.370846 + 0.370846i
\(399\) 0 0
\(400\) 0 0
\(401\) 4.02761e6i 1.25080i 0.780306 + 0.625398i \(0.215063\pi\)
−0.780306 + 0.625398i \(0.784937\pi\)
\(402\) 0 0
\(403\) 588732. + 588732.i 0.180574 + 0.180574i
\(404\) 162782. 0.0496195
\(405\) 0 0
\(406\) −473616. −0.142597
\(407\) 1.15606e6 + 1.15606e6i 0.345936 + 0.345936i
\(408\) 0 0
\(409\) 5.65537e6i 1.67168i −0.548973 0.835840i \(-0.684981\pi\)
0.548973 0.835840i \(-0.315019\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.75733e6 1.75733e6i 0.510047 0.510047i
\(413\) 377879. 377879.i 0.109013 0.109013i
\(414\) 0 0
\(415\) 0 0
\(416\) 308457.i 0.0873899i
\(417\) 0 0
\(418\) 528504. + 528504.i 0.147947 + 0.147947i
\(419\) −43567.7 −0.0121235 −0.00606177 0.999982i \(-0.501930\pi\)
−0.00606177 + 0.999982i \(0.501930\pi\)
\(420\) 0 0
\(421\) −3.65783e6 −1.00582 −0.502908 0.864340i \(-0.667736\pi\)
−0.502908 + 0.864340i \(0.667736\pi\)
\(422\) −1.66820e6 1.66820e6i −0.456001 0.456001i
\(423\) 0 0
\(424\) 718464.i 0.194084i
\(425\) 0 0
\(426\) 0 0
\(427\) 981750. 981750.i 0.260574 0.260574i
\(428\) 2.10163e6 2.10163e6i 0.554559 0.554559i
\(429\) 0 0
\(430\) 0 0
\(431\) 249959.i 0.0648151i −0.999475 0.0324076i \(-0.989683\pi\)
0.999475 0.0324076i \(-0.0103175\pi\)
\(432\) 0 0
\(433\) −1.11368e6 1.11368e6i −0.285456 0.285456i 0.549824 0.835281i \(-0.314695\pi\)
−0.835281 + 0.549824i \(0.814695\pi\)
\(434\) −515973. −0.131493
\(435\) 0 0
\(436\) 380128. 0.0957665
\(437\) 585069. + 585069.i 0.146556 + 0.146556i
\(438\) 0 0
\(439\) 2.89281e6i 0.716405i 0.933644 + 0.358203i \(0.116610\pi\)
−0.933644 + 0.358203i \(0.883390\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −557208. + 557208.i −0.135663 + 0.135663i
\(443\) 739934. 739934.i 0.179136 0.179136i −0.611843 0.790979i \(-0.709572\pi\)
0.790979 + 0.611843i \(0.209572\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.86892e6i 0.444890i
\(447\) 0 0
\(448\) −135168. 135168.i −0.0318184 0.0318184i
\(449\) −1.73408e6 −0.405932 −0.202966 0.979186i \(-0.565058\pi\)
−0.202966 + 0.979186i \(0.565058\pi\)
\(450\) 0 0
\(451\) −1.97530e6 −0.457290
\(452\) 1.31739e6 + 1.31739e6i 0.303297 + 0.303297i
\(453\) 0 0
\(454\) 3.10805e6i 0.707698i
\(455\) 0 0
\(456\) 0 0
\(457\) −2.99052e6 + 2.99052e6i −0.669817 + 0.669817i −0.957674 0.287856i \(-0.907057\pi\)
0.287856 + 0.957674i \(0.407057\pi\)
\(458\) −1.78331e6 + 1.78331e6i −0.397248 + 0.397248i
\(459\) 0 0
\(460\) 0 0
\(461\) 3.77512e6i 0.827329i 0.910429 + 0.413665i \(0.135751\pi\)
−0.910429 + 0.413665i \(0.864249\pi\)
\(462\) 0 0
\(463\) −1.77997e6 1.77997e6i −0.385887 0.385887i 0.487331 0.873218i \(-0.337971\pi\)
−0.873218 + 0.487331i \(0.837971\pi\)
\(464\) −649497. −0.140050
\(465\) 0 0
\(466\) −1.38319e6 −0.295065
\(467\) −1.10978e6 1.10978e6i −0.235475 0.235475i 0.579498 0.814973i \(-0.303249\pi\)
−0.814973 + 0.579498i \(0.803249\pi\)
\(468\) 0 0
\(469\) 1.31116e6i 0.275247i
\(470\) 0 0
\(471\) 0 0
\(472\) 518208. 518208.i 0.107065 0.107065i
\(473\) 838516. 838516.i 0.172329 0.172329i
\(474\) 0 0
\(475\) 0 0
\(476\) 488345.i 0.0987892i
\(477\) 0 0
\(478\) 2.24201e6 + 2.24201e6i 0.448815 + 0.448815i
\(479\) 521344. 0.103821 0.0519106 0.998652i \(-0.483469\pi\)
0.0519106 + 0.998652i \(0.483469\pi\)
\(480\) 0 0
\(481\) −1.90294e6 −0.375027
\(482\) −1.00865e6 1.00865e6i −0.197753 0.197753i
\(483\) 0 0
\(484\) 1.50517e6i 0.292060i
\(485\) 0 0
\(486\) 0 0
\(487\) −2.98473e6 + 2.98473e6i −0.570272 + 0.570272i −0.932204 0.361932i \(-0.882117\pi\)
0.361932 + 0.932204i \(0.382117\pi\)
\(488\) 1.34633e6 1.34633e6i 0.255919 0.255919i
\(489\) 0 0
\(490\) 0 0
\(491\) 2.65071e6i 0.496202i −0.968734 0.248101i \(-0.920193\pi\)
0.968734 0.248101i \(-0.0798066\pi\)
\(492\) 0 0
\(493\) −1.17328e6 1.17328e6i −0.217412 0.217412i
\(494\) −869945. −0.160389
\(495\) 0 0
\(496\) −707584. −0.129144
\(497\) −2.35240e6 2.35240e6i −0.427189 0.427189i
\(498\) 0 0
\(499\) 1.33298e6i 0.239648i −0.992795 0.119824i \(-0.961767\pi\)
0.992795 0.119824i \(-0.0382330\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.26380e6 + 3.26380e6i −0.578048 + 0.578048i
\(503\) −3.60041e6 + 3.60041e6i −0.634500 + 0.634500i −0.949193 0.314693i \(-0.898098\pi\)
0.314693 + 0.949193i \(0.398098\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.18634e6i 0.205984i
\(507\) 0 0
\(508\) −3.64766e6 3.64766e6i −0.627127 0.627127i
\(509\) 1.32877e6 0.227329 0.113665 0.993519i \(-0.463741\pi\)
0.113665 + 0.993519i \(0.463741\pi\)
\(510\) 0 0
\(511\) −2.97436e6 −0.503896
\(512\) −185364. 185364.i −0.0312500 0.0312500i
\(513\) 0 0
\(514\) 4.86458e6i 0.812153i
\(515\) 0 0
\(516\) 0 0
\(517\) −2.56932e6 + 2.56932e6i −0.422758 + 0.422758i
\(518\) 833883. 833883.i 0.136546 0.136546i
\(519\) 0 0
\(520\) 0 0
\(521\) 7.53594e6i 1.21631i 0.793820 + 0.608153i \(0.208089\pi\)
−0.793820 + 0.608153i \(0.791911\pi\)
\(522\) 0 0
\(523\) −1.14775e6 1.14775e6i −0.183481 0.183481i 0.609390 0.792871i \(-0.291415\pi\)
−0.792871 + 0.609390i \(0.791415\pi\)
\(524\) −745008. −0.118531
\(525\) 0 0
\(526\) 3.84127e6 0.605356
\(527\) −1.27821e6 1.27821e6i −0.200481 0.200481i
\(528\) 0 0
\(529\) 5.12303e6i 0.795953i
\(530\) 0 0
\(531\) 0 0
\(532\) 381216. 381216.i 0.0583972 0.0583972i
\(533\) 1.62572e6 1.62572e6i 0.247873 0.247873i
\(534\) 0 0
\(535\) 0 0
\(536\) 1.79807e6i 0.270329i
\(537\) 0 0
\(538\) 1.90222e6 + 1.90222e6i 0.283338 + 0.283338i
\(539\) 3.78600e6 0.561318
\(540\) 0 0
\(541\) 8642.00 0.00126947 0.000634733 1.00000i \(-0.499798\pi\)
0.000634733 1.00000i \(0.499798\pi\)
\(542\) −4.44465e6 4.44465e6i −0.649889 0.649889i
\(543\) 0 0
\(544\) 669696.i 0.0970243i
\(545\) 0 0
\(546\) 0 0
\(547\) −7.56553e6 + 7.56553e6i −1.08111 + 1.08111i −0.0847073 + 0.996406i \(0.526996\pi\)
−0.996406 + 0.0847073i \(0.973004\pi\)
\(548\) 2.41043e6 2.41043e6i 0.342881 0.342881i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.83179e6i 0.257037i
\(552\) 0 0
\(553\) −98736.0 98736.0i −0.0137298 0.0137298i
\(554\) −5.92862e6 −0.820689
\(555\) 0 0
\(556\) 2.44410e6 0.335298
\(557\) −7.92605e6 7.92605e6i −1.08248 1.08248i −0.996278 0.0861989i \(-0.972528\pi\)
−0.0861989 0.996278i \(-0.527472\pi\)
\(558\) 0 0
\(559\) 1.38024e6i 0.186821i
\(560\) 0 0
\(561\) 0 0
\(562\) −4.93348e6 + 4.93348e6i −0.658889 + 0.658889i
\(563\) 7.04595e6 7.04595e6i 0.936847 0.936847i −0.0612744 0.998121i \(-0.519516\pi\)
0.998121 + 0.0612744i \(0.0195165\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.09384e6i 1.06197i
\(567\) 0 0
\(568\) −3.22598e6 3.22598e6i −0.419557 0.419557i
\(569\) 4.20382e6 0.544332 0.272166 0.962250i \(-0.412260\pi\)
0.272166 + 0.962250i \(0.412260\pi\)
\(570\) 0 0
\(571\) 1.26883e7 1.62860 0.814298 0.580448i \(-0.197122\pi\)
0.814298 + 0.580448i \(0.197122\pi\)
\(572\) 881994. + 881994.i 0.112713 + 0.112713i
\(573\) 0 0
\(574\) 1.42481e6i 0.180500i
\(575\) 0 0
\(576\) 0 0
\(577\) −3.48634e6 + 3.48634e6i −0.435943 + 0.435943i −0.890644 0.454701i \(-0.849746\pi\)
0.454701 + 0.890644i \(0.349746\pi\)
\(578\) −2.80620e6 + 2.80620e6i −0.349381 + 0.349381i
\(579\) 0 0
\(580\) 0 0
\(581\) 1.88786e6i 0.232022i
\(582\) 0 0
\(583\) −2.05436e6 2.05436e6i −0.250325 0.250325i
\(584\) −4.07891e6 −0.494894
\(585\) 0 0
\(586\) 3.76757e6 0.453228
\(587\) −7.17632e6 7.17632e6i −0.859621 0.859621i 0.131673 0.991293i \(-0.457965\pi\)
−0.991293 + 0.131673i \(0.957965\pi\)
\(588\) 0 0
\(589\) 1.99561e6i 0.237021i
\(590\) 0 0
\(591\) 0 0
\(592\) 1.14355e6 1.14355e6i 0.134107 0.134107i
\(593\) 993240. 993240.i 0.115989 0.115989i −0.646730 0.762719i \(-0.723864\pi\)
0.762719 + 0.646730i \(0.223864\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.40460e6i 0.853858i
\(597\) 0 0
\(598\) 976392. + 976392.i 0.111653 + 0.111653i
\(599\) 1.15789e7 1.31856 0.659278 0.751899i \(-0.270862\pi\)
0.659278 + 0.751899i \(0.270862\pi\)
\(600\) 0 0
\(601\) −1.10155e7 −1.24399 −0.621996 0.783020i \(-0.713678\pi\)
−0.621996 + 0.783020i \(0.713678\pi\)
\(602\) −604831. 604831.i −0.0680210 0.0680210i
\(603\) 0 0
\(604\) 2.86669e6i 0.319733i
\(605\) 0 0
\(606\) 0 0
\(607\) −6.27601e6 + 6.27601e6i −0.691372 + 0.691372i −0.962534 0.271162i \(-0.912592\pi\)
0.271162 + 0.962534i \(0.412592\pi\)
\(608\) 522784. 522784.i 0.0573539 0.0573539i
\(609\) 0 0
\(610\) 0 0
\(611\) 4.22923e6i 0.458309i
\(612\) 0 0
\(613\) −5.31846e6 5.31846e6i −0.571656 0.571656i 0.360935 0.932591i \(-0.382458\pi\)
−0.932591 + 0.360935i \(0.882458\pi\)
\(614\) −6.12162e6 −0.655308
\(615\) 0 0
\(616\) −772992. −0.0820773
\(617\) 3.55725e6 + 3.55725e6i 0.376185 + 0.376185i 0.869724 0.493539i \(-0.164297\pi\)
−0.493539 + 0.869724i \(0.664297\pi\)
\(618\) 0 0
\(619\) 1.89359e7i 1.98636i 0.116586 + 0.993181i \(0.462805\pi\)
−0.116586 + 0.993181i \(0.537195\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −5.08462e6 + 5.08462e6i −0.526966 + 0.526966i
\(623\) 386280. 386280.i 0.0398733 0.0398733i
\(624\) 0 0
\(625\) 0 0
\(626\) 4.52133e6i 0.461138i
\(627\) 0 0
\(628\) −5.95138e6 5.95138e6i −0.602169 0.602169i
\(629\) 4.13151e6 0.416373
\(630\) 0 0
\(631\) 8.62094e6 0.861948 0.430974 0.902364i \(-0.358170\pi\)
0.430974 + 0.902364i \(0.358170\pi\)
\(632\) −135402. 135402.i −0.0134845 0.0134845i
\(633\) 0 0
\(634\) 5.12981e6i 0.506848i
\(635\) 0 0
\(636\) 0 0
\(637\) −3.11598e6 + 3.11598e6i −0.304261 + 0.304261i
\(638\) −1.85716e6 + 1.85716e6i −0.180633 + 0.180633i
\(639\) 0 0
\(640\) 0 0
\(641\) 2.95492e6i 0.284054i 0.989863 + 0.142027i \(0.0453619\pi\)
−0.989863 + 0.142027i \(0.954638\pi\)
\(642\) 0 0
\(643\) −8.32876e6 8.32876e6i −0.794425 0.794425i 0.187785 0.982210i \(-0.439869\pi\)
−0.982210 + 0.187785i \(0.939869\pi\)
\(644\) −855724. −0.0813053
\(645\) 0 0
\(646\) 1.88875e6 0.178071
\(647\) −2.45184e6 2.45184e6i −0.230267 0.230267i 0.582537 0.812804i \(-0.302060\pi\)
−0.812804 + 0.582537i \(0.802060\pi\)
\(648\) 0 0
\(649\) 2.96350e6i 0.276181i
\(650\) 0 0
\(651\) 0 0
\(652\) 3.02870e6 3.02870e6i 0.279022 0.279022i
\(653\) 1.32744e7 1.32744e7i 1.21824 1.21824i 0.249988 0.968249i \(-0.419573\pi\)
0.968249 0.249988i \(-0.0804265\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.95392e6i 0.177275i
\(657\) 0 0
\(658\) 1.85328e6 + 1.85328e6i 0.166869 + 0.166869i
\(659\) 1.75882e7 1.57764 0.788818 0.614626i \(-0.210693\pi\)
0.788818 + 0.614626i \(0.210693\pi\)
\(660\) 0 0
\(661\) −6.27670e6 −0.558763 −0.279381 0.960180i \(-0.590129\pi\)
−0.279381 + 0.960180i \(0.590129\pi\)
\(662\) −304186. 304186.i −0.0269770 0.0269770i
\(663\) 0 0
\(664\) 2.58893e6i 0.227877i
\(665\) 0 0
\(666\) 0 0
\(667\) −2.05592e6 + 2.05592e6i −0.178934 + 0.178934i
\(668\) 4.14292e6 4.14292e6i 0.359224 0.359224i
\(669\) 0 0
\(670\) 0 0
\(671\) 7.69933e6i 0.660156i
\(672\) 0 0
\(673\) −5.06163e6 5.06163e6i −0.430777 0.430777i 0.458115 0.888893i \(-0.348525\pi\)
−0.888893 + 0.458115i \(0.848525\pi\)
\(674\) −7.44366e6 −0.631156
\(675\) 0 0
\(676\) 4.48888e6 0.377808
\(677\) −4.86767e6 4.86767e6i −0.408178 0.408178i 0.472925 0.881103i \(-0.343198\pi\)
−0.881103 + 0.472925i \(0.843198\pi\)
\(678\) 0 0
\(679\) 7.35055e6i 0.611851i
\(680\) 0 0
\(681\) 0 0
\(682\) −2.02325e6 + 2.02325e6i −0.166567 + 0.166567i
\(683\) −9.89201e6 + 9.89201e6i −0.811395 + 0.811395i −0.984843 0.173448i \(-0.944509\pi\)
0.173448 + 0.984843i \(0.444509\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.86835e6i 0.476108i
\(687\) 0 0
\(688\) −829440. 829440.i −0.0668058 0.0668058i
\(689\) 3.38158e6 0.271376
\(690\) 0 0
\(691\) −3.09191e6 −0.246338 −0.123169 0.992386i \(-0.539306\pi\)
−0.123169 + 0.992386i \(0.539306\pi\)
\(692\) 2.56262e6 + 2.56262e6i 0.203432 + 0.203432i
\(693\) 0 0
\(694\) 1.00986e7i 0.795905i
\(695\) 0 0
\(696\) 0 0
\(697\) −3.52964e6 + 3.52964e6i −0.275200 + 0.275200i
\(698\) 4.18419e6 4.18419e6i 0.325067 0.325067i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.23298e7i 0.947675i 0.880612 + 0.473837i \(0.157132\pi\)
−0.880612 + 0.473837i \(0.842868\pi\)
\(702\) 0 0
\(703\) 3.22517e6 + 3.22517e6i 0.246130 + 0.246130i
\(704\) −1.06005e6 −0.0806110
\(705\) 0 0
\(706\) 1.25704e7 0.949156
\(707\) −335737. 335737.i −0.0252610 0.0252610i
\(708\) 0 0
\(709\) 6.51293e6i 0.486587i 0.969953 + 0.243294i \(0.0782278\pi\)
−0.969953 + 0.243294i \(0.921772\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 529728. 529728.i 0.0391609 0.0391609i
\(713\) −2.23979e6 + 2.23979e6i −0.165000 + 0.165000i
\(714\) 0 0
\(715\) 0 0
\(716\) 7.75073e6i 0.565015i
\(717\) 0 0
\(718\) 93552.0 + 93552.0i 0.00677239 + 0.00677239i
\(719\) 1.02328e7 0.738195 0.369098 0.929391i \(-0.379667\pi\)
0.369098 + 0.929391i \(0.379667\pi\)
\(720\) 0 0
\(721\) −7.24898e6 −0.519324
\(722\) −5.52905e6 5.52905e6i −0.394737 0.394737i
\(723\) 0 0
\(724\) 4.26243e6i 0.302212i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.39935e7 1.39935e7i 0.981949 0.981949i −0.0178905 0.999840i \(-0.505695\pi\)
0.999840 + 0.0178905i \(0.00569504\pi\)
\(728\) 636192. 636192.i 0.0444898 0.0444898i
\(729\) 0 0
\(730\) 0 0
\(731\) 2.99666e6i 0.207417i
\(732\) 0 0
\(733\) 6.89732e6 + 6.89732e6i 0.474155 + 0.474155i 0.903256 0.429101i \(-0.141170\pi\)
−0.429101 + 0.903256i \(0.641170\pi\)
\(734\) −1.26768e7 −0.868501
\(735\) 0 0
\(736\) −1.17350e6 −0.0798528
\(737\) 5.14134e6 + 5.14134e6i 0.348665 + 0.348665i
\(738\) 0 0
\(739\) 9.89996e6i 0.666841i 0.942778 + 0.333421i \(0.108203\pi\)
−0.942778 + 0.333421i \(0.891797\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.48183e6 + 1.48183e6i −0.0988073 + 0.0988073i
\(743\) 1.42800e7 1.42800e7i 0.948976 0.948976i −0.0497843 0.998760i \(-0.515853\pi\)
0.998760 + 0.0497843i \(0.0158534\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 5.75514e6i 0.378625i
\(747\) 0 0
\(748\) −1.91491e6 1.91491e6i −0.125140 0.125140i
\(749\) −8.66924e6 −0.564646
\(750\) 0 0
\(751\) 5.78649e6 0.374382 0.187191 0.982324i \(-0.440062\pi\)
0.187191 + 0.982324i \(0.440062\pi\)
\(752\) 2.54151e6 + 2.54151e6i 0.163888 + 0.163888i
\(753\) 0 0
\(754\) 3.05698e6i 0.195823i
\(755\) 0 0
\(756\) 0 0
\(757\) −986385. + 986385.i −0.0625614 + 0.0625614i −0.737695 0.675134i \(-0.764086\pi\)
0.675134 + 0.737695i \(0.264086\pi\)
\(758\) −6.01476e6 + 6.01476e6i −0.380229 + 0.380229i
\(759\) 0 0
\(760\) 0 0
\(761\) 2.59161e7i 1.62221i 0.584900 + 0.811105i \(0.301134\pi\)
−0.584900 + 0.811105i \(0.698866\pi\)
\(762\) 0 0
\(763\) −784014. 784014.i −0.0487542 0.0487542i
\(764\) −692535. −0.0429248
\(765\) 0 0
\(766\) −6.08794e6 −0.374885
\(767\) 2.43904e6 + 2.43904e6i 0.149703 + 0.149703i
\(768\) 0 0
\(769\) 7.56460e6i 0.461286i −0.973038 0.230643i \(-0.925917\pi\)
0.973038 0.230643i \(-0.0740829\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.56790e6 2.56790e6i 0.155073 0.155073i
\(773\) −2.11501e7 + 2.11501e7i −1.27311 + 1.27311i −0.328657 + 0.944449i \(0.606596\pi\)
−0.944449 + 0.328657i \(0.893404\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.00802e7i 0.600920i
\(777\) 0 0
\(778\) 1.18714e7 + 1.18714e7i 0.703156 + 0.703156i
\(779\) −5.51067e6 −0.325358
\(780\) 0 0
\(781\) −1.84486e7 −1.08227
\(782\) −2.11986e6 2.11986e6i −0.123963 0.123963i
\(783\) 0 0
\(784\) 3.74502e6i 0.217603i
\(785\) 0 0
\(786\) 0 0
\(787\) 5.24337e6 5.24337e6i 0.301768 0.301768i −0.539937 0.841705i \(-0.681552\pi\)
0.841705 + 0.539937i \(0.181552\pi\)
\(788\) 6.19751e6 6.19751e6i 0.355551 0.355551i
\(789\) 0 0
\(790\) 0 0
\(791\) 5.43424e6i 0.308814i
\(792\) 0 0
\(793\) 6.33675e6 + 6.33675e6i 0.357836 + 0.357836i
\(794\) −1.14626e7 −0.645257
\(795\) 0 0
\(796\) 6.62944e6 0.370846
\(797\) 1.41641e7 + 1.41641e7i 0.789848 + 0.789848i 0.981469 0.191621i \(-0.0613743\pi\)
−0.191621 + 0.981469i \(0.561374\pi\)
\(798\) 0 0
\(799\) 9.18216e6i 0.508836i
\(800\) 0 0
\(801\) 0 0
\(802\) 1.13918e7 1.13918e7i 0.625398 0.625398i
\(803\) −1.16631e7 + 1.16631e7i −0.638302 + 0.638302i
\(804\) 0 0
\(805\) 0 0
\(806\) 3.33037e6i 0.180574i
\(807\) 0 0
\(808\) −460416. 460416.i −0.0248097 0.0248097i
\(809\) −1.10071e7 −0.591290 −0.295645 0.955298i \(-0.595535\pi\)
−0.295645 + 0.955298i \(0.595535\pi\)
\(810\) 0 0
\(811\) 5.99290e6 0.319952 0.159976 0.987121i \(-0.448858\pi\)
0.159976 + 0.987121i \(0.448858\pi\)
\(812\) 1.33959e6 + 1.33959e6i 0.0712986 + 0.0712986i
\(813\) 0 0
\(814\) 6.53969e6i 0.345936i
\(815\) 0 0
\(816\) 0 0
\(817\) 2.33928e6 2.33928e6i 0.122610 0.122610i
\(818\) −1.59958e7 + 1.59958e7i −0.835840 + 0.835840i
\(819\) 0 0
\(820\) 0 0
\(821\) 3.27886e7i 1.69772i 0.528619 + 0.848859i \(0.322710\pi\)
−0.528619 + 0.848859i \(0.677290\pi\)
\(822\) 0 0
\(823\) 9.94173e6 + 9.94173e6i 0.511637 + 0.511637i 0.915028 0.403391i \(-0.132168\pi\)
−0.403391 + 0.915028i \(0.632168\pi\)
\(824\) −9.94095e6 −0.510047
\(825\) 0 0
\(826\) −2.13761e6 −0.109013
\(827\) 7.37928e6 + 7.37928e6i 0.375189 + 0.375189i 0.869363 0.494174i \(-0.164529\pi\)
−0.494174 + 0.869363i \(0.664529\pi\)
\(828\) 0 0
\(829\) 1.82419e7i 0.921899i −0.887426 0.460950i \(-0.847509\pi\)
0.887426 0.460950i \(-0.152491\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 872448. 872448.i 0.0436949 0.0436949i
\(833\) 6.76515e6 6.76515e6i 0.337804 0.337804i
\(834\) 0 0
\(835\) 0 0
\(836\) 2.98967e6i 0.147947i
\(837\) 0 0
\(838\) 123228. + 123228.i 0.00606177 + 0.00606177i
\(839\) 2.70804e7 1.32816 0.664080 0.747661i \(-0.268823\pi\)
0.664080 + 0.747661i \(0.268823\pi\)
\(840\) 0 0
\(841\) −1.40743e7 −0.686177
\(842\) 1.03459e7 + 1.03459e7i 0.502908 + 0.502908i
\(843\) 0 0
\(844\) 9.43674e6i 0.456001i
\(845\) 0 0
\(846\) 0 0
\(847\) 3.10441e6 3.10441e6i 0.148686 0.148686i
\(848\) −2.03212e6 + 2.03212e6i −0.0970421 + 0.0970421i
\(849\) 0 0
\(850\) 0 0
\(851\) 7.23962e6i 0.342682i
\(852\) 0 0
\(853\) 1.86524e7 + 1.86524e7i 0.877731 + 0.877731i 0.993300 0.115568i \(-0.0368690\pi\)
−0.115568 + 0.993300i \(0.536869\pi\)
\(854\) −5.55362e6 −0.260574
\(855\) 0 0
\(856\) −1.18886e7 −0.554559
\(857\) −2.73159e6 2.73159e6i −0.127047 0.127047i 0.640724 0.767771i \(-0.278634\pi\)
−0.767771 + 0.640724i \(0.778634\pi\)
\(858\) 0 0
\(859\) 1.85975e7i 0.859946i −0.902842 0.429973i \(-0.858523\pi\)
0.902842 0.429973i \(-0.141477\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −706992. + 706992.i −0.0324076 + 0.0324076i
\(863\) −1.12074e7 + 1.12074e7i −0.512245 + 0.512245i −0.915214 0.402969i \(-0.867978\pi\)
0.402969 + 0.915214i \(0.367978\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 6.29991e6i 0.285456i
\(867\) 0 0
\(868\) 1.45939e6 + 1.45939e6i 0.0657465 + 0.0657465i
\(869\) −774333. −0.0347839
\(870\) 0 0
\(871\) −8.46292e6 −0.377985
\(872\) −1.07516e6 1.07516e6i −0.0478833 0.0478833i
\(873\) 0 0
\(874\) 3.30965e6i 0.146556i
\(875\) 0 0
\(876\) 0 0
\(877\) 2.76523e7 2.76523e7i 1.21404 1.21404i 0.244353 0.969686i \(-0.421424\pi\)
0.969686 0.244353i \(-0.0785756\pi\)
\(878\) 8.18211e6 8.18211e6i 0.358203 0.358203i
\(879\) 0 0
\(880\) 0 0
\(881\) 1.14119e7i 0.495357i 0.968842 + 0.247678i \(0.0796677\pi\)
−0.968842 + 0.247678i \(0.920332\pi\)
\(882\) 0 0
\(883\) 1.53482e7 + 1.53482e7i 0.662453 + 0.662453i 0.955958 0.293505i \(-0.0948217\pi\)
−0.293505 + 0.955958i \(0.594822\pi\)
\(884\) 3.15204e6 0.135663
\(885\) 0 0
\(886\) −4.18570e6 −0.179136
\(887\) 1.86085e7 + 1.86085e7i 0.794151 + 0.794151i 0.982166 0.188015i \(-0.0602055\pi\)
−0.188015 + 0.982166i \(0.560206\pi\)
\(888\) 0 0
\(889\) 1.50466e7i 0.638535i
\(890\) 0 0
\(891\) 0 0
\(892\) 5.28610e6 5.28610e6i 0.222445 0.222445i
\(893\) −7.16786e6 + 7.16786e6i −0.300788 + 0.300788i
\(894\) 0 0
\(895\) 0 0
\(896\) 764626.i 0.0318184i
\(897\) 0 0
\(898\) 4.90472e6 + 4.90472e6i 0.202966 + 0.202966i
\(899\) 7.01254e6 0.289385
\(900\) 0 0
\(901\) −7.34180e6 −0.301294
\(902\) 5.58700e6 + 5.58700e6i 0.228645 + 0.228645i
\(903\) 0 0
\(904\) 7.45229e6i 0.303297i
\(905\) 0 0
\(906\) 0 0
\(907\) 2.18773e7 2.18773e7i 0.883030 0.883030i −0.110812 0.993841i \(-0.535345\pi\)
0.993841 + 0.110812i \(0.0353450\pi\)
\(908\) −8.79089e6 + 8.79089e6i −0.353849 + 0.353849i
\(909\) 0 0
\(910\) 0 0
\(911\) 1.41972e7i 0.566771i −0.959006 0.283385i \(-0.908542\pi\)
0.959006 0.283385i \(-0.0914575\pi\)
\(912\) 0 0
\(913\) −7.40272e6 7.40272e6i −0.293910 0.293910i
\(914\) 1.69169e7 0.669817
\(915\) 0 0
\(916\) 1.00879e7 0.397248
\(917\) 1.53658e6 + 1.53658e6i 0.0603436 + 0.0603436i
\(918\) 0 0
\(919\) 1.37905e7i 0.538631i 0.963052 + 0.269315i \(0.0867974\pi\)
−0.963052 + 0.269315i \(0.913203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.06776e7 1.06776e7i 0.413665 0.413665i
\(923\) 1.51837e7 1.51837e7i 0.586641 0.586641i
\(924\) 0 0
\(925\) 0 0
\(926\) 1.00690e7i 0.385887i
\(927\) 0 0
\(928\) 1.83706e6 + 1.83706e6i 0.0700249 + 0.0700249i
\(929\) −4.48053e7 −1.70330 −0.851649 0.524113i \(-0.824397\pi\)
−0.851649 + 0.524113i \(0.824397\pi\)
\(930\) 0 0
\(931\) 1.05621e7 0.399372
\(932\) 3.91226e6 + 3.91226e6i 0.147532 + 0.147532i
\(933\) 0 0
\(934\) 6.27787e6i 0.235475i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.64785e7 2.64785e7i 0.985244 0.985244i −0.0146490 0.999893i \(-0.504663\pi\)
0.999893 + 0.0146490i \(0.00466308\pi\)
\(938\) 3.70851e6 3.70851e6i 0.137623 0.137623i
\(939\) 0 0
\(940\) 0 0
\(941\) 3.46220e7i 1.27461i 0.770610 + 0.637307i \(0.219952\pi\)
−0.770610 + 0.637307i \(0.780048\pi\)
\(942\) 0 0
\(943\) 6.18496e6 + 6.18496e6i 0.226495 + 0.226495i
\(944\) −2.93143e6 −0.107065
\(945\) 0 0
\(946\) −4.74336e6 −0.172329
\(947\) 505689. + 505689.i 0.0183235 + 0.0183235i 0.716209 0.697886i \(-0.245876\pi\)
−0.697886 + 0.716209i \(0.745876\pi\)
\(948\) 0 0
\(949\) 1.91981e7i 0.691979i
\(950\) 0 0
\(951\) 0 0
\(952\) −1.38125e6 + 1.38125e6i −0.0493946 + 0.0493946i
\(953\) −2.13467e7 + 2.13467e7i −0.761374 + 0.761374i −0.976571 0.215197i \(-0.930961\pi\)
0.215197 + 0.976571i \(0.430961\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.26827e7i 0.448815i
\(957\) 0 0
\(958\) −1.47458e6 1.47458e6i −0.0519106 0.0519106i
\(959\) −9.94303e6 −0.349118
\(960\) 0 0
\(961\) −2.09895e7 −0.733150
\(962\) 5.38233e6 + 5.38233e6i 0.187514 + 0.187514i
\(963\) 0 0
\(964\) 5.70579e6i 0.197753i
\(965\) 0 0
\(966\) 0 0
\(967\) 4.35296e6 4.35296e6i 0.149699 0.149699i −0.628285 0.777983i \(-0.716243\pi\)
0.777983 + 0.628285i \(0.216243\pi\)
\(968\) 4.25726e6 4.25726e6i 0.146030 0.146030i
\(969\) 0 0
\(970\) 0 0
\(971\) 5.77443e7i 1.96545i −0.185081 0.982723i \(-0.559255\pi\)
0.185081 0.982723i \(-0.440745\pi\)
\(972\) 0 0
\(973\) −5.04095e6 5.04095e6i −0.170699 0.170699i
\(974\) 1.68842e7 0.570272
\(975\) 0 0
\(976\) −7.61600e6 −0.255919
\(977\) 3.22881e7 + 3.22881e7i 1.08219 + 1.08219i 0.996305 + 0.0858902i \(0.0273734\pi\)
0.0858902 + 0.996305i \(0.472627\pi\)
\(978\) 0 0
\(979\) 3.02938e6i 0.101018i
\(980\) 0 0
\(981\) 0 0
\(982\) −7.49735e6 + 7.49735e6i −0.248101 + 0.248101i
\(983\) −1.46650e7 + 1.46650e7i −0.484059 + 0.484059i −0.906425 0.422366i \(-0.861200\pi\)
0.422366 + 0.906425i \(0.361200\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 6.63705e6i 0.217412i
\(987\) 0 0
\(988\) 2.46058e6 + 2.46058e6i 0.0801945 + 0.0801945i
\(989\) −5.25103e6 −0.170708
\(990\) 0 0
\(991\) 5.45331e7 1.76391 0.881954 0.471335i \(-0.156228\pi\)
0.881954 + 0.471335i \(0.156228\pi\)
\(992\) 2.00135e6 + 2.00135e6i 0.0645719 + 0.0645719i
\(993\) 0 0
\(994\) 1.33072e7i 0.427189i
\(995\) 0 0
\(996\) 0 0
\(997\) −289683. + 289683.i −0.00922965 + 0.00922965i −0.711707 0.702477i \(-0.752077\pi\)
0.702477 + 0.711707i \(0.252077\pi\)
\(998\) −3.77024e6 + 3.77024e6i −0.119824 + 0.119824i
\(999\) 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 450.6.f.c.143.1 yes 4
3.2 odd 2 inner 450.6.f.c.143.2 yes 4
5.2 odd 4 inner 450.6.f.c.107.2 yes 4
5.3 odd 4 450.6.f.a.107.1 4
5.4 even 2 450.6.f.a.143.2 yes 4
15.2 even 4 inner 450.6.f.c.107.1 yes 4
15.8 even 4 450.6.f.a.107.2 yes 4
15.14 odd 2 450.6.f.a.143.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.6.f.a.107.1 4 5.3 odd 4
450.6.f.a.107.2 yes 4 15.8 even 4
450.6.f.a.143.1 yes 4 15.14 odd 2
450.6.f.a.143.2 yes 4 5.4 even 2
450.6.f.c.107.1 yes 4 15.2 even 4 inner
450.6.f.c.107.2 yes 4 5.2 odd 4 inner
450.6.f.c.143.1 yes 4 1.1 even 1 trivial
450.6.f.c.143.2 yes 4 3.2 odd 2 inner