Properties

Label 450.6.f.g.143.2
Level $450$
Weight $6$
Character 450.143
Analytic conductor $72.173$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + 19220200150 x^{4} - 149540021784 x^{2} + \cdots + 498214340649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.2
Root \(-5.20458 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 450.143
Dual form 450.6.f.g.107.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.82843 - 2.82843i) q^{2} +16.0000i q^{4} +(-50.1584 + 50.1584i) q^{7} +(45.2548 - 45.2548i) q^{8} +O(q^{10})\) \(q+(-2.82843 - 2.82843i) q^{2} +16.0000i q^{4} +(-50.1584 + 50.1584i) q^{7} +(45.2548 - 45.2548i) q^{8} +659.741i q^{11} +(517.605 + 517.605i) q^{13} +283.739 q^{14} -256.000 q^{16} +(62.7710 + 62.7710i) q^{17} -38.1009i q^{19} +(1866.03 - 1866.03i) q^{22} +(-455.322 + 455.322i) q^{23} -2928.02i q^{26} +(-802.535 - 802.535i) q^{28} -4598.69 q^{29} +2690.73 q^{31} +(724.077 + 724.077i) q^{32} -355.086i q^{34} +(1486.78 - 1486.78i) q^{37} +(-107.766 + 107.766i) q^{38} +194.091i q^{41} +(9864.36 + 9864.36i) q^{43} -10555.9 q^{44} +2575.69 q^{46} +(-5300.97 - 5300.97i) q^{47} +11775.3i q^{49} +(-8281.68 + 8281.68i) q^{52} +(20592.5 - 20592.5i) q^{53} +4539.82i q^{56} +(13007.1 + 13007.1i) q^{58} -18628.3 q^{59} -44820.8 q^{61} +(-7610.55 - 7610.55i) q^{62} -4096.00i q^{64} +(25712.6 - 25712.6i) q^{67} +(-1004.34 + 1004.34i) q^{68} -61412.8i q^{71} +(27244.9 + 27244.9i) q^{73} -8410.49 q^{74} +609.615 q^{76} +(-33091.6 - 33091.6i) q^{77} -73400.0i q^{79} +(548.974 - 548.974i) q^{82} +(-58926.6 + 58926.6i) q^{83} -55801.3i q^{86} +(29856.5 + 29856.5i) q^{88} -112776. q^{89} -51924.5 q^{91} +(-7285.16 - 7285.16i) q^{92} +29986.8i q^{94} +(-87995.6 + 87995.6i) q^{97} +(33305.5 - 33305.5i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 528 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 528 q^{7} - 192 q^{13} - 4096 q^{16} + 2688 q^{22} + 8448 q^{28} + 13024 q^{31} + 47328 q^{37} + 55440 q^{43} + 44544 q^{46} + 3072 q^{52} + 101184 q^{58} + 28400 q^{61} - 242256 q^{67} + 430944 q^{73} - 7168 q^{76} - 158208 q^{82} + 43008 q^{88} - 185472 q^{91} - 457152 q^{97}+O(q^{100}) \) 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
\(7\) −50.1584 + 50.1584i −0.386900 + 0.386900i −0.873580 0.486680i \(-0.838208\pi\)
0.486680 + 0.873580i \(0.338208\pi\)
\(8\) 45.2548 45.2548i 0.250000 0.250000i
\(9\) 0 0
\(10\) 0 0
\(11\) 659.741i 1.64396i 0.569514 + 0.821981i \(0.307131\pi\)
−0.569514 + 0.821981i \(0.692869\pi\)
\(12\) 0 0
\(13\) 517.605 + 517.605i 0.849454 + 0.849454i 0.990065 0.140610i \(-0.0449065\pi\)
−0.140610 + 0.990065i \(0.544907\pi\)
\(14\) 283.739 0.386900
\(15\) 0 0
\(16\) −256.000 −0.250000
\(17\) 62.7710 + 62.7710i 0.0526789 + 0.0526789i 0.732955 0.680277i \(-0.238140\pi\)
−0.680277 + 0.732955i \(0.738140\pi\)
\(18\) 0 0
\(19\) 38.1009i 0.0242132i −0.999927 0.0121066i \(-0.996146\pi\)
0.999927 0.0121066i \(-0.00385374\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1866.03 1866.03i 0.821981 0.821981i
\(23\) −455.322 + 455.322i −0.179473 + 0.179473i −0.791126 0.611653i \(-0.790505\pi\)
0.611653 + 0.791126i \(0.290505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2928.02i 0.849454i
\(27\) 0 0
\(28\) −802.535 802.535i −0.193450 0.193450i
\(29\) −4598.69 −1.01540 −0.507702 0.861533i \(-0.669505\pi\)
−0.507702 + 0.861533i \(0.669505\pi\)
\(30\) 0 0
\(31\) 2690.73 0.502883 0.251441 0.967873i \(-0.419095\pi\)
0.251441 + 0.967873i \(0.419095\pi\)
\(32\) 724.077 + 724.077i 0.125000 + 0.125000i
\(33\) 0 0
\(34\) 355.086i 0.0526789i
\(35\) 0 0
\(36\) 0 0
\(37\) 1486.78 1486.78i 0.178543 0.178543i −0.612178 0.790720i \(-0.709706\pi\)
0.790720 + 0.612178i \(0.209706\pi\)
\(38\) −107.766 + 107.766i −0.0121066 + 0.0121066i
\(39\) 0 0
\(40\) 0 0
\(41\) 194.091i 0.0180321i 0.999959 + 0.00901606i \(0.00286994\pi\)
−0.999959 + 0.00901606i \(0.997130\pi\)
\(42\) 0 0
\(43\) 9864.36 + 9864.36i 0.813575 + 0.813575i 0.985168 0.171593i \(-0.0548913\pi\)
−0.171593 + 0.985168i \(0.554891\pi\)
\(44\) −10555.9 −0.821981
\(45\) 0 0
\(46\) 2575.69 0.179473
\(47\) −5300.97 5300.97i −0.350034 0.350034i 0.510088 0.860122i \(-0.329613\pi\)
−0.860122 + 0.510088i \(0.829613\pi\)
\(48\) 0 0
\(49\) 11775.3i 0.700617i
\(50\) 0 0
\(51\) 0 0
\(52\) −8281.68 + 8281.68i −0.424727 + 0.424727i
\(53\) 20592.5 20592.5i 1.00697 1.00697i 0.00699943 0.999976i \(-0.497772\pi\)
0.999976 0.00699943i \(-0.00222801\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4539.82i 0.193450i
\(57\) 0 0
\(58\) 13007.1 + 13007.1i 0.507702 + 0.507702i
\(59\) −18628.3 −0.696694 −0.348347 0.937366i \(-0.613257\pi\)
−0.348347 + 0.937366i \(0.613257\pi\)
\(60\) 0 0
\(61\) −44820.8 −1.54225 −0.771126 0.636682i \(-0.780306\pi\)
−0.771126 + 0.636682i \(0.780306\pi\)
\(62\) −7610.55 7610.55i −0.251441 0.251441i
\(63\) 0 0
\(64\) 4096.00i 0.125000i
\(65\) 0 0
\(66\) 0 0
\(67\) 25712.6 25712.6i 0.699776 0.699776i −0.264586 0.964362i \(-0.585235\pi\)
0.964362 + 0.264586i \(0.0852352\pi\)
\(68\) −1004.34 + 1004.34i −0.0263395 + 0.0263395i
\(69\) 0 0
\(70\) 0 0
\(71\) 61412.8i 1.44582i −0.690944 0.722908i \(-0.742805\pi\)
0.690944 0.722908i \(-0.257195\pi\)
\(72\) 0 0
\(73\) 27244.9 + 27244.9i 0.598381 + 0.598381i 0.939882 0.341501i \(-0.110935\pi\)
−0.341501 + 0.939882i \(0.610935\pi\)
\(74\) −8410.49 −0.178543
\(75\) 0 0
\(76\) 609.615 0.0121066
\(77\) −33091.6 33091.6i −0.636049 0.636049i
\(78\) 0 0
\(79\) 73400.0i 1.32321i −0.749853 0.661604i \(-0.769876\pi\)
0.749853 0.661604i \(-0.230124\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 548.974 548.974i 0.00901606 0.00901606i
\(83\) −58926.6 + 58926.6i −0.938894 + 0.938894i −0.998238 0.0593440i \(-0.981099\pi\)
0.0593440 + 0.998238i \(0.481099\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 55801.3i 0.813575i
\(87\) 0 0
\(88\) 29856.5 + 29856.5i 0.410991 + 0.410991i
\(89\) −112776. −1.50918 −0.754592 0.656194i \(-0.772165\pi\)
−0.754592 + 0.656194i \(0.772165\pi\)
\(90\) 0 0
\(91\) −51924.5 −0.657308
\(92\) −7285.16 7285.16i −0.0897365 0.0897365i
\(93\) 0 0
\(94\) 29986.8i 0.350034i
\(95\) 0 0
\(96\) 0 0
\(97\) −87995.6 + 87995.6i −0.949580 + 0.949580i −0.998789 0.0492086i \(-0.984330\pi\)
0.0492086 + 0.998789i \(0.484330\pi\)
\(98\) 33305.5 33305.5i 0.350308 0.350308i
\(99\) 0 0
\(100\) 0 0
\(101\) 12987.2i 0.126681i −0.997992 0.0633407i \(-0.979825\pi\)
0.997992 0.0633407i \(-0.0201755\pi\)
\(102\) 0 0
\(103\) 147468. + 147468.i 1.36963 + 1.36963i 0.860962 + 0.508669i \(0.169862\pi\)
0.508669 + 0.860962i \(0.330138\pi\)
\(104\) 46848.3 0.424727
\(105\) 0 0
\(106\) −116489. −1.00697
\(107\) −65928.5 65928.5i −0.556691 0.556691i 0.371673 0.928364i \(-0.378784\pi\)
−0.928364 + 0.371673i \(0.878784\pi\)
\(108\) 0 0
\(109\) 181646.i 1.46440i 0.681091 + 0.732198i \(0.261506\pi\)
−0.681091 + 0.732198i \(0.738494\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12840.6 12840.6i 0.0967250 0.0967250i
\(113\) −38222.8 + 38222.8i −0.281596 + 0.281596i −0.833745 0.552149i \(-0.813808\pi\)
0.552149 + 0.833745i \(0.313808\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 73579.0i 0.507702i
\(117\) 0 0
\(118\) 52688.7 + 52688.7i 0.348347 + 0.348347i
\(119\) −6296.99 −0.0407630
\(120\) 0 0
\(121\) −274208. −1.70261
\(122\) 126772. + 126772.i 0.771126 + 0.771126i
\(123\) 0 0
\(124\) 43051.8i 0.251441i
\(125\) 0 0
\(126\) 0 0
\(127\) 72272.3 72272.3i 0.397615 0.397615i −0.479776 0.877391i \(-0.659282\pi\)
0.877391 + 0.479776i \(0.159282\pi\)
\(128\) −11585.2 + 11585.2i −0.0625000 + 0.0625000i
\(129\) 0 0
\(130\) 0 0
\(131\) 163217.i 0.830973i −0.909599 0.415486i \(-0.863611\pi\)
0.909599 0.415486i \(-0.136389\pi\)
\(132\) 0 0
\(133\) 1911.08 + 1911.08i 0.00936808 + 0.00936808i
\(134\) −145453. −0.699776
\(135\) 0 0
\(136\) 5681.38 0.0263395
\(137\) −30415.8 30415.8i −0.138451 0.138451i 0.634484 0.772936i \(-0.281212\pi\)
−0.772936 + 0.634484i \(0.781212\pi\)
\(138\) 0 0
\(139\) 343921.i 1.50981i 0.655835 + 0.754905i \(0.272317\pi\)
−0.655835 + 0.754905i \(0.727683\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −173702. + 173702.i −0.722908 + 0.722908i
\(143\) −341485. + 341485.i −1.39647 + 1.39647i
\(144\) 0 0
\(145\) 0 0
\(146\) 154120.i 0.598381i
\(147\) 0 0
\(148\) 23788.4 + 23788.4i 0.0892713 + 0.0892713i
\(149\) −174105. −0.642461 −0.321230 0.947001i \(-0.604096\pi\)
−0.321230 + 0.947001i \(0.604096\pi\)
\(150\) 0 0
\(151\) −374613. −1.33703 −0.668513 0.743700i \(-0.733069\pi\)
−0.668513 + 0.743700i \(0.733069\pi\)
\(152\) −1724.25 1724.25i −0.00605329 0.00605329i
\(153\) 0 0
\(154\) 187194.i 0.636049i
\(155\) 0 0
\(156\) 0 0
\(157\) 95123.8 95123.8i 0.307992 0.307992i −0.536138 0.844130i \(-0.680117\pi\)
0.844130 + 0.536138i \(0.180117\pi\)
\(158\) −207606. + 207606.i −0.661604 + 0.661604i
\(159\) 0 0
\(160\) 0 0
\(161\) 45676.5i 0.138876i
\(162\) 0 0
\(163\) 124177. + 124177.i 0.366076 + 0.366076i 0.866044 0.499968i \(-0.166655\pi\)
−0.499968 + 0.866044i \(0.666655\pi\)
\(164\) −3105.46 −0.00901606
\(165\) 0 0
\(166\) 333339. 0.938894
\(167\) −22938.8 22938.8i −0.0636473 0.0636473i 0.674567 0.738214i \(-0.264331\pi\)
−0.738214 + 0.674567i \(0.764331\pi\)
\(168\) 0 0
\(169\) 164537.i 0.443146i
\(170\) 0 0
\(171\) 0 0
\(172\) −157830. + 157830.i −0.406788 + 0.406788i
\(173\) −479359. + 479359.i −1.21771 + 1.21771i −0.249284 + 0.968430i \(0.580195\pi\)
−0.968430 + 0.249284i \(0.919805\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 168894.i 0.410991i
\(177\) 0 0
\(178\) 318979. + 318979.i 0.754592 + 0.754592i
\(179\) 265461. 0.619254 0.309627 0.950858i \(-0.399796\pi\)
0.309627 + 0.950858i \(0.399796\pi\)
\(180\) 0 0
\(181\) −743564. −1.68703 −0.843513 0.537109i \(-0.819516\pi\)
−0.843513 + 0.537109i \(0.819516\pi\)
\(182\) 146865. + 146865.i 0.328654 + 0.328654i
\(183\) 0 0
\(184\) 41211.1i 0.0897365i
\(185\) 0 0
\(186\) 0 0
\(187\) −41412.6 + 41412.6i −0.0866021 + 0.0866021i
\(188\) 84815.5 84815.5i 0.175017 0.175017i
\(189\) 0 0
\(190\) 0 0
\(191\) 539270.i 1.06960i 0.844978 + 0.534801i \(0.179614\pi\)
−0.844978 + 0.534801i \(0.820386\pi\)
\(192\) 0 0
\(193\) 111105. + 111105.i 0.214705 + 0.214705i 0.806263 0.591558i \(-0.201487\pi\)
−0.591558 + 0.806263i \(0.701487\pi\)
\(194\) 497778. 0.949580
\(195\) 0 0
\(196\) −188404. −0.350308
\(197\) 112701. + 112701.i 0.206900 + 0.206900i 0.802949 0.596048i \(-0.203263\pi\)
−0.596048 + 0.802949i \(0.703263\pi\)
\(198\) 0 0
\(199\) 328398.i 0.587853i 0.955828 + 0.293926i \(0.0949620\pi\)
−0.955828 + 0.293926i \(0.905038\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −36733.4 + 36733.4i −0.0633407 + 0.0633407i
\(203\) 230663. 230663.i 0.392860 0.392860i
\(204\) 0 0
\(205\) 0 0
\(206\) 834203.i 1.36963i
\(207\) 0 0
\(208\) −132507. 132507.i −0.212364 0.212364i
\(209\) 25136.8 0.0398055
\(210\) 0 0
\(211\) −647753. −1.00162 −0.500811 0.865557i \(-0.666965\pi\)
−0.500811 + 0.865557i \(0.666965\pi\)
\(212\) 329479. + 329479.i 0.503487 + 0.503487i
\(213\) 0 0
\(214\) 372948.i 0.556691i
\(215\) 0 0
\(216\) 0 0
\(217\) −134963. + 134963.i −0.194565 + 0.194565i
\(218\) 513772. 513772.i 0.732198 0.732198i
\(219\) 0 0
\(220\) 0 0
\(221\) 64981.2i 0.0894967i
\(222\) 0 0
\(223\) −360829. 360829.i −0.485892 0.485892i 0.421115 0.907007i \(-0.361639\pi\)
−0.907007 + 0.421115i \(0.861639\pi\)
\(224\) −72637.2 −0.0967250
\(225\) 0 0
\(226\) 216221. 0.281596
\(227\) −1.01792e6 1.01792e6i −1.31114 1.31114i −0.920572 0.390572i \(-0.872277\pi\)
−0.390572 0.920572i \(-0.627723\pi\)
\(228\) 0 0
\(229\) 984516.i 1.24061i −0.784362 0.620303i \(-0.787009\pi\)
0.784362 0.620303i \(-0.212991\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −208113. + 208113.i −0.253851 + 0.253851i
\(233\) −1.03993e6 + 1.03993e6i −1.25491 + 1.25491i −0.301423 + 0.953491i \(0.597461\pi\)
−0.953491 + 0.301423i \(0.902539\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 298052.i 0.348347i
\(237\) 0 0
\(238\) 17810.6 + 17810.6i 0.0203815 + 0.0203815i
\(239\) −810562. −0.917891 −0.458946 0.888464i \(-0.651773\pi\)
−0.458946 + 0.888464i \(0.651773\pi\)
\(240\) 0 0
\(241\) 1.35804e6 1.50616 0.753078 0.657931i \(-0.228568\pi\)
0.753078 + 0.657931i \(0.228568\pi\)
\(242\) 775576. + 775576.i 0.851306 + 0.851306i
\(243\) 0 0
\(244\) 717133.i 0.771126i
\(245\) 0 0
\(246\) 0 0
\(247\) 19721.2 19721.2i 0.0205680 0.0205680i
\(248\) 121769. 121769.i 0.125721 0.125721i
\(249\) 0 0
\(250\) 0 0
\(251\) 176671.i 0.177003i 0.996076 + 0.0885017i \(0.0282079\pi\)
−0.996076 + 0.0885017i \(0.971792\pi\)
\(252\) 0 0
\(253\) −300395. 300395.i −0.295047 0.295047i
\(254\) −408834. −0.397615
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −1.14874e6 1.14874e6i −1.08490 1.08490i −0.996045 0.0888531i \(-0.971680\pi\)
−0.0888531 0.996045i \(-0.528320\pi\)
\(258\) 0 0
\(259\) 149149.i 0.138156i
\(260\) 0 0
\(261\) 0 0
\(262\) −461647. + 461647.i −0.415486 + 0.415486i
\(263\) 837838. 837838.i 0.746914 0.746914i −0.226985 0.973898i \(-0.572887\pi\)
0.973898 + 0.226985i \(0.0728867\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 10810.7i 0.00936808i
\(267\) 0 0
\(268\) 411402. + 411402.i 0.349888 + 0.349888i
\(269\) 240388. 0.202550 0.101275 0.994858i \(-0.467708\pi\)
0.101275 + 0.994858i \(0.467708\pi\)
\(270\) 0 0
\(271\) −26337.3 −0.0217846 −0.0108923 0.999941i \(-0.503467\pi\)
−0.0108923 + 0.999941i \(0.503467\pi\)
\(272\) −16069.4 16069.4i −0.0131697 0.0131697i
\(273\) 0 0
\(274\) 172057.i 0.138451i
\(275\) 0 0
\(276\) 0 0
\(277\) 1.78533e6 1.78533e6i 1.39804 1.39804i 0.592371 0.805665i \(-0.298192\pi\)
0.805665 0.592371i \(-0.201808\pi\)
\(278\) 972756. 972756.i 0.754905 0.754905i
\(279\) 0 0
\(280\) 0 0
\(281\) 2.09342e6i 1.58158i 0.612087 + 0.790790i \(0.290330\pi\)
−0.612087 + 0.790790i \(0.709670\pi\)
\(282\) 0 0
\(283\) −1.00478e6 1.00478e6i −0.745772 0.745772i 0.227910 0.973682i \(-0.426811\pi\)
−0.973682 + 0.227910i \(0.926811\pi\)
\(284\) 982605. 0.722908
\(285\) 0 0
\(286\) 1.93173e6 1.39647
\(287\) −9735.33 9735.33i −0.00697663 0.00697663i
\(288\) 0 0
\(289\) 1.41198e6i 0.994450i
\(290\) 0 0
\(291\) 0 0
\(292\) −435918. + 435918.i −0.299190 + 0.299190i
\(293\) 1.52933e6 1.52933e6i 1.04072 1.04072i 0.0415819 0.999135i \(-0.486760\pi\)
0.999135 0.0415819i \(-0.0132397\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 134568.i 0.0892713i
\(297\) 0 0
\(298\) 492445. + 492445.i 0.321230 + 0.321230i
\(299\) −471354. −0.304908
\(300\) 0 0
\(301\) −989562. −0.629545
\(302\) 1.05956e6 + 1.05956e6i 0.668513 + 0.668513i
\(303\) 0 0
\(304\) 9753.84i 0.00605329i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.95052e6 1.95052e6i 1.18115 1.18115i 0.201702 0.979447i \(-0.435353\pi\)
0.979447 0.201702i \(-0.0646471\pi\)
\(308\) 529465. 529465.i 0.318025 0.318025i
\(309\) 0 0
\(310\) 0 0
\(311\) 1.92784e6i 1.13024i 0.825009 + 0.565120i \(0.191170\pi\)
−0.825009 + 0.565120i \(0.808830\pi\)
\(312\) 0 0
\(313\) −64282.4 64282.4i −0.0370878 0.0370878i 0.688320 0.725407i \(-0.258349\pi\)
−0.725407 + 0.688320i \(0.758349\pi\)
\(314\) −538101. −0.307992
\(315\) 0 0
\(316\) 1.17440e6 0.661604
\(317\) 1.08190e6 + 1.08190e6i 0.604697 + 0.604697i 0.941555 0.336858i \(-0.109364\pi\)
−0.336858 + 0.941555i \(0.609364\pi\)
\(318\) 0 0
\(319\) 3.03395e6i 1.66929i
\(320\) 0 0
\(321\) 0 0
\(322\) −129193. + 129193.i −0.0694381 + 0.0694381i
\(323\) 2391.63 2391.63i 0.00127552 0.00127552i
\(324\) 0 0
\(325\) 0 0
\(326\) 702449.i 0.366076i
\(327\) 0 0
\(328\) 8783.58 + 8783.58i 0.00450803 + 0.00450803i
\(329\) 531777. 0.270857
\(330\) 0 0
\(331\) 3.12047e6 1.56549 0.782744 0.622344i \(-0.213820\pi\)
0.782744 + 0.622344i \(0.213820\pi\)
\(332\) −942826. 942826.i −0.469447 0.469447i
\(333\) 0 0
\(334\) 129762.i 0.0636473i
\(335\) 0 0
\(336\) 0 0
\(337\) 63979.7 63979.7i 0.0306879 0.0306879i −0.691596 0.722284i \(-0.743092\pi\)
0.722284 + 0.691596i \(0.243092\pi\)
\(338\) 465381. 465381.i 0.221573 0.221573i
\(339\) 0 0
\(340\) 0 0
\(341\) 1.77519e6i 0.826720i
\(342\) 0 0
\(343\) −1.43364e6 1.43364e6i −0.657969 0.657969i
\(344\) 892820. 0.406788
\(345\) 0 0
\(346\) 2.71166e6 1.21771
\(347\) −1.23074e6 1.23074e6i −0.548711 0.548711i 0.377357 0.926068i \(-0.376833\pi\)
−0.926068 + 0.377357i \(0.876833\pi\)
\(348\) 0 0
\(349\) 847301.i 0.372370i 0.982515 + 0.186185i \(0.0596123\pi\)
−0.982515 + 0.186185i \(0.940388\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −477704. + 477704.i −0.205495 + 0.205495i
\(353\) 690970. 690970.i 0.295136 0.295136i −0.543969 0.839105i \(-0.683079\pi\)
0.839105 + 0.543969i \(0.183079\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.80442e6i 0.754592i
\(357\) 0 0
\(358\) −750838. 750838.i −0.309627 0.309627i
\(359\) −2.53332e6 −1.03742 −0.518709 0.854951i \(-0.673587\pi\)
−0.518709 + 0.854951i \(0.673587\pi\)
\(360\) 0 0
\(361\) 2.47465e6 0.999414
\(362\) 2.10312e6 + 2.10312e6i 0.843513 + 0.843513i
\(363\) 0 0
\(364\) 830792.i 0.328654i
\(365\) 0 0
\(366\) 0 0
\(367\) −1.44127e6 + 1.44127e6i −0.558573 + 0.558573i −0.928901 0.370328i \(-0.879245\pi\)
0.370328 + 0.928901i \(0.379245\pi\)
\(368\) 116562. 116562.i 0.0448683 0.0448683i
\(369\) 0 0
\(370\) 0 0
\(371\) 2.06577e6i 0.779198i
\(372\) 0 0
\(373\) −3.55217e6 3.55217e6i −1.32197 1.32197i −0.912179 0.409793i \(-0.865601\pi\)
−0.409793 0.912179i \(-0.634399\pi\)
\(374\) 234265. 0.0866021
\(375\) 0 0
\(376\) −479789. −0.175017
\(377\) −2.38030e6 2.38030e6i −0.862540 0.862540i
\(378\) 0 0
\(379\) 2.00856e6i 0.718267i −0.933286 0.359133i \(-0.883072\pi\)
0.933286 0.359133i \(-0.116928\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.52528e6 1.52528e6i 0.534801 0.534801i
\(383\) −575760. + 575760.i −0.200560 + 0.200560i −0.800240 0.599680i \(-0.795295\pi\)
0.599680 + 0.800240i \(0.295295\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 628507.i 0.214705i
\(387\) 0 0
\(388\) −1.40793e6 1.40793e6i −0.474790 0.474790i
\(389\) 4.43267e6 1.48522 0.742612 0.669722i \(-0.233587\pi\)
0.742612 + 0.669722i \(0.233587\pi\)
\(390\) 0 0
\(391\) −57162.1 −0.0189089
\(392\) 532888. + 532888.i 0.175154 + 0.175154i
\(393\) 0 0
\(394\) 637531.i 0.206900i
\(395\) 0 0
\(396\) 0 0
\(397\) 337038. 337038.i 0.107325 0.107325i −0.651405 0.758730i \(-0.725820\pi\)
0.758730 + 0.651405i \(0.225820\pi\)
\(398\) 928851. 928851.i 0.293926 0.293926i
\(399\) 0 0
\(400\) 0 0
\(401\) 3.35110e6i 1.04070i 0.853952 + 0.520352i \(0.174199\pi\)
−0.853952 + 0.520352i \(0.825801\pi\)
\(402\) 0 0
\(403\) 1.39274e6 + 1.39274e6i 0.427176 + 0.427176i
\(404\) 207795. 0.0633407
\(405\) 0 0
\(406\) −1.30483e6 −0.392860
\(407\) 980889. + 980889.i 0.293517 + 0.293517i
\(408\) 0 0
\(409\) 3.51691e6i 1.03957i 0.854298 + 0.519784i \(0.173987\pi\)
−0.854298 + 0.519784i \(0.826013\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.35948e6 + 2.35948e6i −0.684816 + 0.684816i
\(413\) 934364. 934364.i 0.269551 0.269551i
\(414\) 0 0
\(415\) 0 0
\(416\) 749572.i 0.212364i
\(417\) 0 0
\(418\) −71097.5 71097.5i −0.0199028 0.0199028i
\(419\) 6.15249e6 1.71205 0.856024 0.516937i \(-0.172928\pi\)
0.856024 + 0.516937i \(0.172928\pi\)
\(420\) 0 0
\(421\) −2.64403e6 −0.727044 −0.363522 0.931586i \(-0.618426\pi\)
−0.363522 + 0.931586i \(0.618426\pi\)
\(422\) 1.83212e6 + 1.83212e6i 0.500811 + 0.500811i
\(423\) 0 0
\(424\) 1.86382e6i 0.503487i
\(425\) 0 0
\(426\) 0 0
\(427\) 2.24814e6 2.24814e6i 0.596698 0.596698i
\(428\) 1.05486e6 1.05486e6i 0.278345 0.278345i
\(429\) 0 0
\(430\) 0 0
\(431\) 823737.i 0.213597i 0.994281 + 0.106799i \(0.0340600\pi\)
−0.994281 + 0.106799i \(0.965940\pi\)
\(432\) 0 0
\(433\) −2.17513e6 2.17513e6i −0.557525 0.557525i 0.371077 0.928602i \(-0.378989\pi\)
−0.928602 + 0.371077i \(0.878989\pi\)
\(434\) 763466. 0.194565
\(435\) 0 0
\(436\) −2.90633e6 −0.732198
\(437\) 17348.2 + 17348.2i 0.00434561 + 0.00434561i
\(438\) 0 0
\(439\) 3.59369e6i 0.889978i −0.895536 0.444989i \(-0.853207\pi\)
0.895536 0.444989i \(-0.146793\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 183795. 183795.i 0.0447483 0.0447483i
\(443\) −1.31093e6 + 1.31093e6i −0.317373 + 0.317373i −0.847757 0.530384i \(-0.822048\pi\)
0.530384 + 0.847757i \(0.322048\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.04116e6i 0.485892i
\(447\) 0 0
\(448\) 205449. + 205449.i 0.0483625 + 0.0483625i
\(449\) −1.25573e6 −0.293954 −0.146977 0.989140i \(-0.546954\pi\)
−0.146977 + 0.989140i \(0.546954\pi\)
\(450\) 0 0
\(451\) −128050. −0.0296441
\(452\) −611564. 611564.i −0.140798 0.140798i
\(453\) 0 0
\(454\) 5.75824e6i 1.31114i
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00853e6 + 6.00853e6i −1.34579 + 1.34579i −0.455614 + 0.890178i \(0.650580\pi\)
−0.890178 + 0.455614i \(0.849420\pi\)
\(458\) −2.78463e6 + 2.78463e6i −0.620303 + 0.620303i
\(459\) 0 0
\(460\) 0 0
\(461\) 4.47946e6i 0.981687i −0.871248 0.490844i \(-0.836689\pi\)
0.871248 0.490844i \(-0.163311\pi\)
\(462\) 0 0
\(463\) −78368.4 78368.4i −0.0169898 0.0169898i 0.698561 0.715551i \(-0.253824\pi\)
−0.715551 + 0.698561i \(0.753824\pi\)
\(464\) 1.17726e6 0.253851
\(465\) 0 0
\(466\) 5.88272e6 1.25491
\(467\) 2.36197e6 + 2.36197e6i 0.501167 + 0.501167i 0.911801 0.410633i \(-0.134692\pi\)
−0.410633 + 0.911801i \(0.634692\pi\)
\(468\) 0 0
\(469\) 2.57941e6i 0.541487i
\(470\) 0 0
\(471\) 0 0
\(472\) −843019. + 843019.i −0.174174 + 0.174174i
\(473\) −6.50793e6 + 6.50793e6i −1.33749 + 1.33749i
\(474\) 0 0
\(475\) 0 0
\(476\) 100752.i 0.0203815i
\(477\) 0 0
\(478\) 2.29261e6 + 2.29261e6i 0.458946 + 0.458946i
\(479\) −3.71979e6 −0.740763 −0.370382 0.928880i \(-0.620773\pi\)
−0.370382 + 0.928880i \(0.620773\pi\)
\(480\) 0 0
\(481\) 1.53913e6 0.303328
\(482\) −3.84112e6 3.84112e6i −0.753078 0.753078i
\(483\) 0 0
\(484\) 4.38732e6i 0.851306i
\(485\) 0 0
\(486\) 0 0
\(487\) 2.63171e6 2.63171e6i 0.502823 0.502823i −0.409491 0.912314i \(-0.634294\pi\)
0.912314 + 0.409491i \(0.134294\pi\)
\(488\) −2.02836e6 + 2.02836e6i −0.385563 + 0.385563i
\(489\) 0 0
\(490\) 0 0
\(491\) 5.67262e6i 1.06189i −0.847406 0.530946i \(-0.821837\pi\)
0.847406 0.530946i \(-0.178163\pi\)
\(492\) 0 0
\(493\) −288664. 288664.i −0.0534904 0.0534904i
\(494\) −111560. −0.0205680
\(495\) 0 0
\(496\) −688828. −0.125721
\(497\) 3.08037e6 + 3.08037e6i 0.559387 + 0.559387i
\(498\) 0 0
\(499\) 155951.i 0.0280373i 0.999902 + 0.0140186i \(0.00446242\pi\)
−0.999902 + 0.0140186i \(0.995538\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 499702. 499702.i 0.0885017 0.0885017i
\(503\) 185072. 185072.i 0.0326152 0.0326152i −0.690611 0.723226i \(-0.742658\pi\)
0.723226 + 0.690611i \(0.242658\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.69929e6i 0.295047i
\(507\) 0 0
\(508\) 1.15636e6 + 1.15636e6i 0.198807 + 0.198807i
\(509\) 2.07656e6 0.355263 0.177632 0.984097i \(-0.443156\pi\)
0.177632 + 0.984097i \(0.443156\pi\)
\(510\) 0 0
\(511\) −2.73312e6 −0.463027
\(512\) −185364. 185364.i −0.0312500 0.0312500i
\(513\) 0 0
\(514\) 6.49825e6i 1.08490i
\(515\) 0 0
\(516\) 0 0
\(517\) 3.49727e6 3.49727e6i 0.575444 0.575444i
\(518\) 421857. 421857.i 0.0690781 0.0690781i
\(519\) 0 0
\(520\) 0 0
\(521\) 4.46917e6i 0.721328i −0.932696 0.360664i \(-0.882550\pi\)
0.932696 0.360664i \(-0.117450\pi\)
\(522\) 0 0
\(523\) 4.83883e6 + 4.83883e6i 0.773545 + 0.773545i 0.978724 0.205179i \(-0.0657777\pi\)
−0.205179 + 0.978724i \(0.565778\pi\)
\(524\) 2.61147e6 0.415486
\(525\) 0 0
\(526\) −4.73952e6 −0.746914
\(527\) 168900. + 168900.i 0.0264913 + 0.0264913i
\(528\) 0 0
\(529\) 6.02171e6i 0.935579i
\(530\) 0 0
\(531\) 0 0
\(532\) −30577.3 + 30577.3i −0.00468404 + 0.00468404i
\(533\) −100463. + 100463.i −0.0153175 + 0.0153175i
\(534\) 0 0
\(535\) 0 0
\(536\) 2.32724e6i 0.349888i
\(537\) 0 0
\(538\) −679919. 679919.i −0.101275 0.101275i
\(539\) −7.76863e6 −1.15179
\(540\) 0 0
\(541\) −8.63652e6 −1.26866 −0.634330 0.773062i \(-0.718724\pi\)
−0.634330 + 0.773062i \(0.718724\pi\)
\(542\) 74493.3 + 74493.3i 0.0108923 + 0.0108923i
\(543\) 0 0
\(544\) 90902.1i 0.0131697i
\(545\) 0 0
\(546\) 0 0
\(547\) −6.40411e6 + 6.40411e6i −0.915146 + 0.915146i −0.996671 0.0815255i \(-0.974021\pi\)
0.0815255 + 0.996671i \(0.474021\pi\)
\(548\) 486652. 486652.i 0.0692257 0.0692257i
\(549\) 0 0
\(550\) 0 0
\(551\) 175214.i 0.0245862i
\(552\) 0 0
\(553\) 3.68163e6 + 3.68163e6i 0.511949 + 0.511949i
\(554\) −1.00993e7 −1.39804
\(555\) 0 0
\(556\) −5.50274e6 −0.754905
\(557\) 4.41693e6 + 4.41693e6i 0.603230 + 0.603230i 0.941168 0.337938i \(-0.109730\pi\)
−0.337938 + 0.941168i \(0.609730\pi\)
\(558\) 0 0
\(559\) 1.02117e7i 1.38219i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.92110e6 5.92110e6i 0.790790 0.790790i
\(563\) 6.69896e6 6.69896e6i 0.890711 0.890711i −0.103879 0.994590i \(-0.533126\pi\)
0.994590 + 0.103879i \(0.0331256\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 5.68391e6i 0.745772i
\(567\) 0 0
\(568\) −2.77923e6 2.77923e6i −0.361454 0.361454i
\(569\) −5.55048e6 −0.718704 −0.359352 0.933202i \(-0.617002\pi\)
−0.359352 + 0.933202i \(0.617002\pi\)
\(570\) 0 0
\(571\) −7.99758e6 −1.02652 −0.513261 0.858233i \(-0.671563\pi\)
−0.513261 + 0.858233i \(0.671563\pi\)
\(572\) −5.46377e6 5.46377e6i −0.698236 0.698236i
\(573\) 0 0
\(574\) 55071.3i 0.00697663i
\(575\) 0 0
\(576\) 0 0
\(577\) −1.94056e6 + 1.94056e6i −0.242654 + 0.242654i −0.817947 0.575293i \(-0.804888\pi\)
0.575293 + 0.817947i \(0.304888\pi\)
\(578\) −3.99367e6 + 3.99367e6i −0.497225 + 0.497225i
\(579\) 0 0
\(580\) 0 0
\(581\) 5.91134e6i 0.726516i
\(582\) 0 0
\(583\) 1.35857e7 + 1.35857e7i 1.65543 + 1.65543i
\(584\) 2.46592e6 0.299190
\(585\) 0 0
\(586\) −8.65121e6 −1.04072
\(587\) −8.23325e6 8.23325e6i −0.986225 0.986225i 0.0136812 0.999906i \(-0.495645\pi\)
−0.999906 + 0.0136812i \(0.995645\pi\)
\(588\) 0 0
\(589\) 102520.i 0.0121764i
\(590\) 0 0
\(591\) 0 0
\(592\) −380615. + 380615.i −0.0446356 + 0.0446356i
\(593\) −5.96277e6 + 5.96277e6i −0.696324 + 0.696324i −0.963616 0.267292i \(-0.913871\pi\)
0.267292 + 0.963616i \(0.413871\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.78569e6i 0.321230i
\(597\) 0 0
\(598\) 1.33319e6 + 1.33319e6i 0.152454 + 0.152454i
\(599\) 1.96839e6 0.224153 0.112077 0.993700i \(-0.464250\pi\)
0.112077 + 0.993700i \(0.464250\pi\)
\(600\) 0 0
\(601\) 1.21707e7 1.37445 0.687226 0.726443i \(-0.258828\pi\)
0.687226 + 0.726443i \(0.258828\pi\)
\(602\) 2.79890e6 + 2.79890e6i 0.314772 + 0.314772i
\(603\) 0 0
\(604\) 5.99380e6i 0.668513i
\(605\) 0 0
\(606\) 0 0
\(607\) −2.50153e6 + 2.50153e6i −0.275571 + 0.275571i −0.831338 0.555767i \(-0.812425\pi\)
0.555767 + 0.831338i \(0.312425\pi\)
\(608\) 27588.0 27588.0i 0.00302665 0.00302665i
\(609\) 0 0
\(610\) 0 0
\(611\) 5.48762e6i 0.594677i
\(612\) 0 0
\(613\) 7.43814e6 + 7.43814e6i 0.799490 + 0.799490i 0.983015 0.183525i \(-0.0587508\pi\)
−0.183525 + 0.983015i \(0.558751\pi\)
\(614\) −1.10338e7 −1.18115
\(615\) 0 0
\(616\) −2.99511e6 −0.318025
\(617\) −3.13725e6 3.13725e6i −0.331769 0.331769i 0.521489 0.853258i \(-0.325377\pi\)
−0.853258 + 0.521489i \(0.825377\pi\)
\(618\) 0 0
\(619\) 6.26468e6i 0.657161i 0.944476 + 0.328581i \(0.106570\pi\)
−0.944476 + 0.328581i \(0.893430\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5.45276e6 5.45276e6i 0.565120 0.565120i
\(623\) 5.65667e6 5.65667e6i 0.583903 0.583903i
\(624\) 0 0
\(625\) 0 0
\(626\) 363636.i 0.0370878i
\(627\) 0 0
\(628\) 1.52198e6 + 1.52198e6i 0.153996 + 0.153996i
\(629\) 186653. 0.0188109
\(630\) 0 0
\(631\) 2.70818e6 0.270772 0.135386 0.990793i \(-0.456773\pi\)
0.135386 + 0.990793i \(0.456773\pi\)
\(632\) −3.32170e6 3.32170e6i −0.330802 0.330802i
\(633\) 0 0
\(634\) 6.12014e6i 0.604697i
\(635\) 0 0
\(636\) 0 0
\(637\) −6.09494e6 + 6.09494e6i −0.595142 + 0.595142i
\(638\) −8.58129e6 + 8.58129e6i −0.834644 + 0.834644i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.42435e7i 1.36921i −0.728914 0.684605i \(-0.759975\pi\)
0.728914 0.684605i \(-0.240025\pi\)
\(642\) 0 0
\(643\) 1.93862e6 + 1.93862e6i 0.184912 + 0.184912i 0.793492 0.608580i \(-0.208261\pi\)
−0.608580 + 0.793492i \(0.708261\pi\)
\(644\) 730824. 0.0694381
\(645\) 0 0
\(646\) −13529.1 −0.00127552
\(647\) −856023. 856023.i −0.0803942 0.0803942i 0.665766 0.746160i \(-0.268105\pi\)
−0.746160 + 0.665766i \(0.768105\pi\)
\(648\) 0 0
\(649\) 1.22898e7i 1.14534i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.98683e6 + 1.98683e6i −0.183038 + 0.183038i
\(653\) −5.94490e6 + 5.94490e6i −0.545584 + 0.545584i −0.925160 0.379576i \(-0.876070\pi\)
0.379576 + 0.925160i \(0.376070\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 49687.4i 0.00450803i
\(657\) 0 0
\(658\) −1.50409e6 1.50409e6i −0.135428 0.135428i
\(659\) −2.00924e7 −1.80226 −0.901130 0.433548i \(-0.857261\pi\)
−0.901130 + 0.433548i \(0.857261\pi\)
\(660\) 0 0
\(661\) −1.49344e7 −1.32949 −0.664745 0.747071i \(-0.731460\pi\)
−0.664745 + 0.747071i \(0.731460\pi\)
\(662\) −8.82601e6 8.82601e6i −0.782744 0.782744i
\(663\) 0 0
\(664\) 5.33343e6i 0.469447i
\(665\) 0 0
\(666\) 0 0
\(667\) 2.09389e6 2.09389e6i 0.182238 0.182238i
\(668\) 367021. 367021.i 0.0318236 0.0318236i
\(669\) 0 0
\(670\) 0 0
\(671\) 2.95702e7i 2.53541i
\(672\) 0 0
\(673\) 5.94402e6 + 5.94402e6i 0.505875 + 0.505875i 0.913257 0.407383i \(-0.133559\pi\)
−0.407383 + 0.913257i \(0.633559\pi\)
\(674\) −361924. −0.0306879
\(675\) 0 0
\(676\) −2.63259e6 −0.221573
\(677\) 1.68563e7 + 1.68563e7i 1.41348 + 1.41348i 0.729469 + 0.684014i \(0.239767\pi\)
0.684014 + 0.729469i \(0.260233\pi\)
\(678\) 0 0
\(679\) 8.82744e6i 0.734785i
\(680\) 0 0
\(681\) 0 0
\(682\) 5.02099e6 5.02099e6i 0.413360 0.413360i
\(683\) 1.34538e7 1.34538e7i 1.10355 1.10355i 0.109574 0.993979i \(-0.465051\pi\)
0.993979 0.109574i \(-0.0349486\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.10990e6i 0.657969i
\(687\) 0 0
\(688\) −2.52528e6 2.52528e6i −0.203394 0.203394i
\(689\) 2.13175e7 1.71076
\(690\) 0 0
\(691\) 2.73492e6 0.217896 0.108948 0.994047i \(-0.465252\pi\)
0.108948 + 0.994047i \(0.465252\pi\)
\(692\) −7.66974e6 7.66974e6i −0.608857 0.608857i
\(693\) 0 0
\(694\) 6.96214e6i 0.548711i
\(695\) 0 0
\(696\) 0 0
\(697\) −12183.3 + 12183.3i −0.000949912 + 0.000949912i
\(698\) 2.39653e6 2.39653e6i 0.186185 0.186185i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.80427e7i 1.38678i 0.720563 + 0.693390i \(0.243884\pi\)
−0.720563 + 0.693390i \(0.756116\pi\)
\(702\) 0 0
\(703\) −56647.6 56647.6i −0.00432308 0.00432308i
\(704\) 2.70230e6 0.205495
\(705\) 0 0
\(706\) −3.90871e6 −0.295136
\(707\) 651418. + 651418.i 0.0490130 + 0.0490130i
\(708\) 0 0
\(709\) 1.11791e7i 0.835200i 0.908631 + 0.417600i \(0.137129\pi\)
−0.908631 + 0.417600i \(0.862871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −5.10366e6 + 5.10366e6i −0.377296 + 0.377296i
\(713\) −1.22515e6 + 1.22515e6i −0.0902539 + 0.0902539i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.24738e6i 0.309627i
\(717\) 0 0
\(718\) 7.16531e6 + 7.16531e6i 0.518709 + 0.518709i
\(719\) −6.75900e6 −0.487596 −0.243798 0.969826i \(-0.578393\pi\)
−0.243798 + 0.969826i \(0.578393\pi\)
\(720\) 0 0
\(721\) −1.47935e7 −1.05982
\(722\) −6.99936e6 6.99936e6i −0.499707 0.499707i
\(723\) 0 0
\(724\) 1.18970e7i 0.843513i
\(725\) 0 0
\(726\) 0 0
\(727\) 389019. 389019.i 0.0272982 0.0272982i −0.693326 0.720624i \(-0.743855\pi\)
0.720624 + 0.693326i \(0.243855\pi\)
\(728\) −2.34984e6 + 2.34984e6i −0.164327 + 0.164327i
\(729\) 0 0
\(730\) 0 0
\(731\) 1.23839e6i 0.0857165i
\(732\) 0 0
\(733\) 268610. + 268610.i 0.0184655 + 0.0184655i 0.716279 0.697814i \(-0.245844\pi\)
−0.697814 + 0.716279i \(0.745844\pi\)
\(734\) 8.15305e6 0.558573
\(735\) 0 0
\(736\) −659377. −0.0448683
\(737\) 1.69637e7 + 1.69637e7i 1.15041 + 1.15041i
\(738\) 0 0
\(739\) 8.34926e6i 0.562389i −0.959651 0.281195i \(-0.909269\pi\)
0.959651 0.281195i \(-0.0907306\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.84288e6 5.84288e6i 0.389599 0.389599i
\(743\) 9.91181e6 9.91181e6i 0.658690 0.658690i −0.296380 0.955070i \(-0.595780\pi\)
0.955070 + 0.296380i \(0.0957795\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.00941e7i 1.32197i
\(747\) 0 0
\(748\) −662602. 662602.i −0.0433011 0.0433011i
\(749\) 6.61374e6 0.430767
\(750\) 0 0
\(751\) 1.00908e7 0.652872 0.326436 0.945219i \(-0.394152\pi\)
0.326436 + 0.945219i \(0.394152\pi\)
\(752\) 1.35705e6 + 1.35705e6i 0.0875086 + 0.0875086i
\(753\) 0 0
\(754\) 1.34650e7i 0.862540i
\(755\) 0 0
\(756\) 0 0
\(757\) −1.57442e7 + 1.57442e7i −0.998577 + 0.998577i −0.999999 0.00142163i \(-0.999547\pi\)
0.00142163 + 0.999999i \(0.499547\pi\)
\(758\) −5.68105e6 + 5.68105e6i −0.359133 + 0.359133i
\(759\) 0 0
\(760\) 0 0
\(761\) 3.65317e6i 0.228669i −0.993442 0.114335i \(-0.963526\pi\)
0.993442 0.114335i \(-0.0364736\pi\)
\(762\) 0 0
\(763\) −9.11106e6 9.11106e6i −0.566575 0.566575i
\(764\) −8.62831e6 −0.534801
\(765\) 0 0
\(766\) 3.25699e6 0.200560
\(767\) −9.64208e6 9.64208e6i −0.591810 0.591810i
\(768\) 0 0
\(769\) 1.75833e7i 1.07222i 0.844147 + 0.536112i \(0.180107\pi\)
−0.844147 + 0.536112i \(0.819893\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.77768e6 + 1.77768e6i −0.107352 + 0.107352i
\(773\) −1.32600e6 + 1.32600e6i −0.0798171 + 0.0798171i −0.745888 0.666071i \(-0.767975\pi\)
0.666071 + 0.745888i \(0.267975\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 7.96445e6i 0.474790i
\(777\) 0 0
\(778\) −1.25375e7 1.25375e7i −0.742612 0.742612i
\(779\) 7395.07 0.000436615
\(780\) 0 0
\(781\) 4.05166e7 2.37687
\(782\) 161679. + 161679.i 0.00945444 + 0.00945444i
\(783\) 0 0
\(784\) 3.01447e6i 0.175154i
\(785\) 0 0
\(786\) 0 0
\(787\) 5.03092e6 5.03092e6i 0.289541 0.289541i −0.547357 0.836899i \(-0.684366\pi\)
0.836899 + 0.547357i \(0.184366\pi\)
\(788\) −1.80321e6 + 1.80321e6i −0.103450 + 0.103450i
\(789\) 0 0
\(790\) 0 0
\(791\) 3.83439e6i 0.217899i
\(792\) 0 0
\(793\) −2.31995e7 2.31995e7i −1.31007 1.31007i
\(794\) −1.90658e6 −0.107325
\(795\) 0 0
\(796\) −5.25438e6 −0.293926
\(797\) −1.31915e7 1.31915e7i −0.735611 0.735611i 0.236114 0.971725i \(-0.424126\pi\)
−0.971725 + 0.236114i \(0.924126\pi\)
\(798\) 0 0
\(799\) 665495.i 0.0368789i
\(800\) 0 0
\(801\) 0 0
\(802\) 9.47835e6 9.47835e6i 0.520352 0.520352i
\(803\) −1.79746e7 + 1.79746e7i −0.983716 + 0.983716i
\(804\) 0 0
\(805\) 0 0
\(806\) 7.87852e6i 0.427176i
\(807\) 0 0
\(808\) −587734. 587734.i −0.0316703 0.0316703i
\(809\) 4.79366e6 0.257511 0.128756 0.991676i \(-0.458902\pi\)
0.128756 + 0.991676i \(0.458902\pi\)
\(810\) 0 0
\(811\) 2.44630e7 1.30604 0.653021 0.757340i \(-0.273502\pi\)
0.653021 + 0.757340i \(0.273502\pi\)
\(812\) 3.69061e6 + 3.69061e6i 0.196430 + 0.196430i
\(813\) 0 0
\(814\) 5.54874e6i 0.293517i
\(815\) 0 0
\(816\) 0 0
\(817\) 375841. 375841.i 0.0196992 0.0196992i
\(818\) 9.94731e6 9.94731e6i 0.519784 0.519784i
\(819\) 0 0
\(820\) 0 0
\(821\) 1.42838e7i 0.739581i 0.929115 + 0.369790i \(0.120570\pi\)
−0.929115 + 0.369790i \(0.879430\pi\)
\(822\) 0 0
\(823\) 1.56583e6 + 1.56583e6i 0.0805832 + 0.0805832i 0.746250 0.665666i \(-0.231853\pi\)
−0.665666 + 0.746250i \(0.731853\pi\)
\(824\) 1.33472e7 0.684816
\(825\) 0 0
\(826\) −5.28556e6 −0.269551
\(827\) 2.18679e7 + 2.18679e7i 1.11184 + 1.11184i 0.992902 + 0.118939i \(0.0379492\pi\)
0.118939 + 0.992902i \(0.462051\pi\)
\(828\) 0 0
\(829\) 5.85804e6i 0.296050i 0.988984 + 0.148025i \(0.0472917\pi\)
−0.988984 + 0.148025i \(0.952708\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.12011e6 2.12011e6i 0.106182 0.106182i
\(833\) −739145. + 739145.i −0.0369077 + 0.0369077i
\(834\) 0 0
\(835\) 0 0
\(836\) 402188.i 0.0199028i
\(837\) 0 0
\(838\) −1.74019e7 1.74019e7i −0.856024 0.856024i
\(839\) 1.40555e7 0.689354 0.344677 0.938721i \(-0.387988\pi\)
0.344677 + 0.938721i \(0.387988\pi\)
\(840\) 0 0
\(841\) 636798. 0.0310464
\(842\) 7.47844e6 + 7.47844e6i 0.363522 + 0.363522i
\(843\) 0 0
\(844\) 1.03641e7i 0.500811i
\(845\) 0 0
\(846\) 0 0
\(847\) 1.37538e7 1.37538e7i 0.658741 0.658741i
\(848\) −5.27167e6 + 5.27167e6i −0.251744 + 0.251744i
\(849\) 0 0
\(850\) 0 0
\(851\) 1.35393e6i 0.0640872i
\(852\) 0 0
\(853\) −1.46482e7 1.46482e7i −0.689305 0.689305i 0.272773 0.962078i \(-0.412059\pi\)
−0.962078 + 0.272773i \(0.912059\pi\)
\(854\) −1.27174e7 −0.596698
\(855\) 0 0
\(856\) −5.96717e6 −0.278345
\(857\) 2.43742e7 + 2.43742e7i 1.13365 + 1.13365i 0.989565 + 0.144085i \(0.0460238\pi\)
0.144085 + 0.989565i \(0.453976\pi\)
\(858\) 0 0
\(859\) 2.36043e7i 1.09146i −0.837961 0.545731i \(-0.816252\pi\)
0.837961 0.545731i \(-0.183748\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.32988e6 2.32988e6i 0.106799 0.106799i
\(863\) 1.61979e7 1.61979e7i 0.740343 0.740343i −0.232301 0.972644i \(-0.574626\pi\)
0.972644 + 0.232301i \(0.0746255\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.23044e7i 0.557525i
\(867\) 0 0
\(868\) −2.15941e6 2.15941e6i −0.0972827 0.0972827i
\(869\) 4.84250e7 2.17530
\(870\) 0 0
\(871\) 2.66180e7 1.18886
\(872\) 8.22034e6 + 8.22034e6i 0.366099 + 0.366099i
\(873\) 0 0
\(874\) 98136.3i 0.00434561i
\(875\) 0 0
\(876\) 0 0
\(877\) −7.08182e6 + 7.08182e6i −0.310918 + 0.310918i −0.845265 0.534347i \(-0.820557\pi\)
0.534347 + 0.845265i \(0.320557\pi\)
\(878\) −1.01645e7 + 1.01645e7i −0.444989 + 0.444989i
\(879\) 0 0
\(880\) 0 0
\(881\) 2.30452e7i 1.00032i 0.865932 + 0.500162i \(0.166726\pi\)
−0.865932 + 0.500162i \(0.833274\pi\)
\(882\) 0 0
\(883\) 1.99788e7 + 1.99788e7i 0.862317 + 0.862317i 0.991607 0.129290i \(-0.0412699\pi\)
−0.129290 + 0.991607i \(0.541270\pi\)
\(884\) −1.03970e6 −0.0447483
\(885\) 0 0
\(886\) 7.41573e6 0.317373
\(887\) −1.41730e7 1.41730e7i −0.604858 0.604858i 0.336740 0.941598i \(-0.390676\pi\)
−0.941598 + 0.336740i \(0.890676\pi\)
\(888\) 0 0
\(889\) 7.25013e6i 0.307675i
\(890\) 0 0
\(891\) 0 0
\(892\) 5.77327e6 5.77327e6i 0.242946 0.242946i
\(893\) −201972. + 201972.i −0.00847545 + 0.00847545i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.16219e6i 0.0483625i
\(897\) 0 0
\(898\) 3.55173e6 + 3.55173e6i 0.146977 + 0.146977i
\(899\) −1.23739e7 −0.510629
\(900\) 0 0
\(901\) 2.58522e6 0.106093
\(902\) 362181. + 362181.i 0.0148221 + 0.0148221i
\(903\) 0 0
\(904\) 3.45953e6i 0.140798i
\(905\) 0 0
\(906\) 0 0
\(907\) −1.50979e7 + 1.50979e7i −0.609393 + 0.609393i −0.942787 0.333395i \(-0.891806\pi\)
0.333395 + 0.942787i \(0.391806\pi\)
\(908\) 1.62868e7 1.62868e7i 0.655572 0.655572i
\(909\) 0 0
\(910\) 0 0
\(911\) 4.31847e7i 1.72399i 0.506920 + 0.861993i \(0.330784\pi\)
−0.506920 + 0.861993i \(0.669216\pi\)
\(912\) 0 0
\(913\) −3.88763e7 3.88763e7i −1.54351 1.54351i
\(914\) 3.39894e7 1.34579
\(915\) 0 0
\(916\) 1.57523e7 0.620303
\(917\) 8.18670e6 + 8.18670e6i 0.321504 + 0.321504i
\(918\) 0 0
\(919\) 2.65475e7i 1.03690i −0.855109 0.518448i \(-0.826510\pi\)
0.855109 0.518448i \(-0.173490\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.26698e7 + 1.26698e7i −0.490844 + 0.490844i
\(923\) 3.17876e7 3.17876e7i 1.22816 1.22816i
\(924\) 0 0
\(925\) 0 0
\(926\) 443318.i 0.0169898i
\(927\) 0 0
\(928\) −3.32981e6 3.32981e6i −0.126926 0.126926i
\(929\) 9.56387e6 0.363575 0.181788 0.983338i \(-0.441812\pi\)
0.181788 + 0.983338i \(0.441812\pi\)
\(930\) 0 0
\(931\) 448649. 0.0169641
\(932\) −1.66389e7 1.66389e7i −0.627457 0.627457i
\(933\) 0 0
\(934\) 1.33613e7i 0.501167i
\(935\) 0 0
\(936\) 0 0
\(937\) 3.24994e6 3.24994e6i 0.120928 0.120928i −0.644053 0.764981i \(-0.722748\pi\)
0.764981 + 0.644053i \(0.222748\pi\)
\(938\) 7.29567e6 7.29567e6i 0.270744 0.270744i
\(939\) 0 0
\(940\) 0 0
\(941\) 331919.i 0.0122196i 0.999981 + 0.00610982i \(0.00194483\pi\)
−0.999981 + 0.00610982i \(0.998055\pi\)
\(942\) 0 0
\(943\) −88374.2 88374.2i −0.00323628 0.00323628i
\(944\) 4.76883e6 0.174174
\(945\) 0 0
\(946\) 3.68144e7 1.33749
\(947\) −1.33791e6 1.33791e6i −0.0484790 0.0484790i 0.682452 0.730931i \(-0.260914\pi\)
−0.730931 + 0.682452i \(0.760914\pi\)
\(948\) 0 0
\(949\) 2.82042e7i 1.01659i
\(950\) 0 0
\(951\) 0 0
\(952\) −284969. + 284969.i −0.0101907 + 0.0101907i
\(953\) −1.03531e7 + 1.03531e7i −0.369265 + 0.369265i −0.867209 0.497944i \(-0.834089\pi\)
0.497944 + 0.867209i \(0.334089\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.29690e7i 0.458946i
\(957\) 0 0
\(958\) 1.05212e7 + 1.05212e7i 0.370382 + 0.370382i
\(959\) 3.05121e6 0.107134
\(960\) 0 0
\(961\) −2.13891e7 −0.747109
\(962\) −4.35331e6 4.35331e6i −0.151664 0.151664i
\(963\) 0 0
\(964\) 2.17287e7i 0.753078i
\(965\) 0 0
\(966\) 0 0
\(967\) 2.31470e7 2.31470e7i 0.796027 0.796027i −0.186440 0.982466i \(-0.559695\pi\)
0.982466 + 0.186440i \(0.0596949\pi\)
\(968\) −1.24092e7 + 1.24092e7i −0.425653 + 0.425653i
\(969\) 0 0
\(970\) 0 0
\(971\) 2.76465e7i 0.941004i 0.882399 + 0.470502i \(0.155927\pi\)
−0.882399 + 0.470502i \(0.844073\pi\)
\(972\) 0 0
\(973\) −1.72506e7 1.72506e7i −0.584145 0.584145i
\(974\) −1.48872e7 −0.502823
\(975\) 0 0
\(976\) 1.14741e7 0.385563
\(977\) −2.29525e7 2.29525e7i −0.769297 0.769297i 0.208686 0.977983i \(-0.433081\pi\)
−0.977983 + 0.208686i \(0.933081\pi\)
\(978\) 0 0
\(979\) 7.44031e7i 2.48104i
\(980\) 0 0
\(981\) 0 0
\(982\) −1.60446e7 + 1.60446e7i −0.530946 + 0.530946i
\(983\) 8.12183e6 8.12183e6i 0.268084 0.268084i −0.560244 0.828328i \(-0.689293\pi\)
0.828328 + 0.560244i \(0.189293\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.63293e6i 0.0534904i
\(987\) 0 0
\(988\) 315540. + 315540.i 0.0102840 + 0.0102840i
\(989\) −8.98292e6 −0.292030
\(990\) 0 0
\(991\) −3.30343e6 −0.106851 −0.0534257 0.998572i \(-0.517014\pi\)
−0.0534257 + 0.998572i \(0.517014\pi\)
\(992\) 1.94830e6 + 1.94830e6i 0.0628603 + 0.0628603i
\(993\) 0 0
\(994\) 1.74252e7i 0.559387i
\(995\) 0 0
\(996\) 0 0
\(997\) −1.23928e7 + 1.23928e7i −0.394851 + 0.394851i −0.876412 0.481562i \(-0.840070\pi\)
0.481562 + 0.876412i \(0.340070\pi\)
\(998\) 441095. 441095.i 0.0140186 0.0140186i
\(999\) 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.g.143.2 yes 16
3.2 odd 2 inner 450.6.f.g.143.6 yes 16
5.2 odd 4 inner 450.6.f.g.107.6 yes 16
5.3 odd 4 450.6.f.f.107.3 16
5.4 even 2 450.6.f.f.143.7 yes 16
15.2 even 4 inner 450.6.f.g.107.2 yes 16
15.8 even 4 450.6.f.f.107.7 yes 16
15.14 odd 2 450.6.f.f.143.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.6.f.f.107.3 16 5.3 odd 4
450.6.f.f.107.7 yes 16 15.8 even 4
450.6.f.f.143.3 yes 16 15.14 odd 2
450.6.f.f.143.7 yes 16 5.4 even 2
450.6.f.g.107.2 yes 16 15.2 even 4 inner
450.6.f.g.107.6 yes 16 5.2 odd 4 inner
450.6.f.g.143.2 yes 16 1.1 even 1 trivial
450.6.f.g.143.6 yes 16 3.2 odd 2 inner