Properties

Label 160.6.n.d.63.5
Level $160$
Weight $6$
Character 160.63
Analytic conductor $25.661$
Analytic rank $0$
Dimension $16$
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] = [160,6,Mod(63,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.63");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.n (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: \(25.6614111701\)
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: \( x^{16} + 1375 x^{14} + 743087 x^{12} + 198706725 x^{10} + 26872635188 x^{8} + 1612811892960 x^{6} + 24971013117824 x^{4} + \cdots + 177426662425600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{41}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 63.5
Root \(2.29308i\) of defining polynomial
Character \(\chi\) \(=\) 160.63
Dual form 160.6.n.d.127.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.29308 - 3.29308i) q^{3} +(35.3847 - 43.2773i) q^{5} +(-5.96093 - 5.96093i) q^{7} +221.311i q^{9} +O(q^{10})\) \(q+(3.29308 - 3.29308i) q^{3} +(35.3847 - 43.2773i) q^{5} +(-5.96093 - 5.96093i) q^{7} +221.311i q^{9} +634.109i q^{11} +(226.441 + 226.441i) q^{13} +(-25.9911 - 259.040i) q^{15} +(429.196 - 429.196i) q^{17} +540.685 q^{19} -39.2596 q^{21} +(1345.21 - 1345.21i) q^{23} +(-620.850 - 3062.71i) q^{25} +(1529.01 + 1529.01i) q^{27} +1942.50i q^{29} +8018.95i q^{31} +(2088.17 + 2088.17i) q^{33} +(-468.898 + 47.0474i) q^{35} +(11033.1 - 11033.1i) q^{37} +1491.37 q^{39} +13026.2 q^{41} +(-10539.1 + 10539.1i) q^{43} +(9577.76 + 7831.03i) q^{45} +(14464.6 + 14464.6i) q^{47} -16735.9i q^{49} -2826.75i q^{51} +(-6059.07 - 6059.07i) q^{53} +(27442.5 + 22437.7i) q^{55} +(1780.52 - 1780.52i) q^{57} -405.643 q^{59} +10881.1 q^{61} +(1319.22 - 1319.22i) q^{63} +(17812.3 - 1787.22i) q^{65} +(-8886.87 - 8886.87i) q^{67} -8859.76i q^{69} +12438.9i q^{71} +(17069.9 + 17069.9i) q^{73} +(-12130.2 - 8041.22i) q^{75} +(3779.87 - 3779.87i) q^{77} +69057.3 q^{79} -43708.3 q^{81} +(-40609.1 + 40609.1i) q^{83} +(-3387.49 - 33761.4i) q^{85} +(6396.80 + 6396.80i) q^{87} +64770.8i q^{89} -2699.59i q^{91} +(26407.0 + 26407.0i) q^{93} +(19132.0 - 23399.4i) q^{95} +(35342.6 - 35342.6i) q^{97} -140335. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{3} - 42 q^{5} - 86 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{3} - 42 q^{5} - 86 q^{7} + 536 q^{13} - 698 q^{15} - 1828 q^{17} + 2512 q^{19} - 4284 q^{21} - 7642 q^{23} + 9140 q^{25} - 12272 q^{27} + 11876 q^{33} + 10518 q^{35} - 7620 q^{37} + 11244 q^{39} - 21284 q^{41} + 20002 q^{43} + 686 q^{45} + 25298 q^{47} + 12852 q^{53} - 10584 q^{55} + 55848 q^{57} - 142704 q^{59} - 20564 q^{61} - 115282 q^{63} - 38256 q^{65} - 10506 q^{67} + 15432 q^{73} + 256226 q^{75} + 133852 q^{77} - 159344 q^{79} - 236116 q^{81} - 61222 q^{83} + 7056 q^{85} + 162176 q^{87} + 122180 q^{93} + 267512 q^{95} - 17344 q^{97} + 107332 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.29308 3.29308i 0.211251 0.211251i −0.593548 0.804799i \(-0.702273\pi\)
0.804799 + 0.593548i \(0.202273\pi\)
\(4\) 0 0
\(5\) 35.3847 43.2773i 0.632980 0.774168i
\(6\) 0 0
\(7\) −5.96093 5.96093i −0.0459800 0.0459800i 0.683743 0.729723i \(-0.260351\pi\)
−0.729723 + 0.683743i \(0.760351\pi\)
\(8\) 0 0
\(9\) 221.311i 0.910746i
\(10\) 0 0
\(11\) 634.109i 1.58009i 0.613048 + 0.790045i \(0.289943\pi\)
−0.613048 + 0.790045i \(0.710057\pi\)
\(12\) 0 0
\(13\) 226.441 + 226.441i 0.371618 + 0.371618i 0.868066 0.496448i \(-0.165363\pi\)
−0.496448 + 0.868066i \(0.665363\pi\)
\(14\) 0 0
\(15\) −25.9911 259.040i −0.0298260 0.297261i
\(16\) 0 0
\(17\) 429.196 429.196i 0.360192 0.360192i −0.503692 0.863883i \(-0.668025\pi\)
0.863883 + 0.503692i \(0.168025\pi\)
\(18\) 0 0
\(19\) 540.685 0.343606 0.171803 0.985131i \(-0.445041\pi\)
0.171803 + 0.985131i \(0.445041\pi\)
\(20\) 0 0
\(21\) −39.2596 −0.0194266
\(22\) 0 0
\(23\) 1345.21 1345.21i 0.530237 0.530237i −0.390406 0.920643i \(-0.627665\pi\)
0.920643 + 0.390406i \(0.127665\pi\)
\(24\) 0 0
\(25\) −620.850 3062.71i −0.198672 0.980066i
\(26\) 0 0
\(27\) 1529.01 + 1529.01i 0.403647 + 0.403647i
\(28\) 0 0
\(29\) 1942.50i 0.428910i 0.976734 + 0.214455i \(0.0687975\pi\)
−0.976734 + 0.214455i \(0.931202\pi\)
\(30\) 0 0
\(31\) 8018.95i 1.49869i 0.662177 + 0.749347i \(0.269633\pi\)
−0.662177 + 0.749347i \(0.730367\pi\)
\(32\) 0 0
\(33\) 2088.17 + 2088.17i 0.333796 + 0.333796i
\(34\) 0 0
\(35\) −468.898 + 47.0474i −0.0647006 + 0.00649181i
\(36\) 0 0
\(37\) 11033.1 11033.1i 1.32493 1.32493i 0.415205 0.909728i \(-0.363710\pi\)
0.909728 0.415205i \(-0.136290\pi\)
\(38\) 0 0
\(39\) 1491.37 0.157009
\(40\) 0 0
\(41\) 13026.2 1.21021 0.605103 0.796147i \(-0.293132\pi\)
0.605103 + 0.796147i \(0.293132\pi\)
\(42\) 0 0
\(43\) −10539.1 + 10539.1i −0.869226 + 0.869226i −0.992387 0.123161i \(-0.960697\pi\)
0.123161 + 0.992387i \(0.460697\pi\)
\(44\) 0 0
\(45\) 9577.76 + 7831.03i 0.705070 + 0.576484i
\(46\) 0 0
\(47\) 14464.6 + 14464.6i 0.955126 + 0.955126i 0.999035 0.0439100i \(-0.0139815\pi\)
−0.0439100 + 0.999035i \(0.513981\pi\)
\(48\) 0 0
\(49\) 16735.9i 0.995772i
\(50\) 0 0
\(51\) 2826.75i 0.152182i
\(52\) 0 0
\(53\) −6059.07 6059.07i −0.296290 0.296290i 0.543269 0.839559i \(-0.317186\pi\)
−0.839559 + 0.543269i \(0.817186\pi\)
\(54\) 0 0
\(55\) 27442.5 + 22437.7i 1.22326 + 1.00017i
\(56\) 0 0
\(57\) 1780.52 1780.52i 0.0725870 0.0725870i
\(58\) 0 0
\(59\) −405.643 −0.0151710 −0.00758550 0.999971i \(-0.502415\pi\)
−0.00758550 + 0.999971i \(0.502415\pi\)
\(60\) 0 0
\(61\) 10881.1 0.374412 0.187206 0.982321i \(-0.440057\pi\)
0.187206 + 0.982321i \(0.440057\pi\)
\(62\) 0 0
\(63\) 1319.22 1319.22i 0.0418761 0.0418761i
\(64\) 0 0
\(65\) 17812.3 1787.22i 0.522921 0.0524679i
\(66\) 0 0
\(67\) −8886.87 8886.87i −0.241859 0.241859i 0.575760 0.817619i \(-0.304706\pi\)
−0.817619 + 0.575760i \(0.804706\pi\)
\(68\) 0 0
\(69\) 8859.76i 0.224026i
\(70\) 0 0
\(71\) 12438.9i 0.292844i 0.989222 + 0.146422i \(0.0467758\pi\)
−0.989222 + 0.146422i \(0.953224\pi\)
\(72\) 0 0
\(73\) 17069.9 + 17069.9i 0.374907 + 0.374907i 0.869261 0.494354i \(-0.164595\pi\)
−0.494354 + 0.869261i \(0.664595\pi\)
\(74\) 0 0
\(75\) −12130.2 8041.22i −0.249010 0.165070i
\(76\) 0 0
\(77\) 3779.87 3779.87i 0.0726525 0.0726525i
\(78\) 0 0
\(79\) 69057.3 1.24492 0.622460 0.782651i \(-0.286133\pi\)
0.622460 + 0.782651i \(0.286133\pi\)
\(80\) 0 0
\(81\) −43708.3 −0.740204
\(82\) 0 0
\(83\) −40609.1 + 40609.1i −0.647035 + 0.647035i −0.952275 0.305240i \(-0.901263\pi\)
0.305240 + 0.952275i \(0.401263\pi\)
\(84\) 0 0
\(85\) −3387.49 33761.4i −0.0508546 0.506843i
\(86\) 0 0
\(87\) 6396.80 + 6396.80i 0.0906076 + 0.0906076i
\(88\) 0 0
\(89\) 64770.8i 0.866771i 0.901209 + 0.433386i \(0.142681\pi\)
−0.901209 + 0.433386i \(0.857319\pi\)
\(90\) 0 0
\(91\) 2699.59i 0.0341739i
\(92\) 0 0
\(93\) 26407.0 + 26407.0i 0.316601 + 0.316601i
\(94\) 0 0
\(95\) 19132.0 23399.4i 0.217496 0.266008i
\(96\) 0 0
\(97\) 35342.6 35342.6i 0.381390 0.381390i −0.490213 0.871603i \(-0.663081\pi\)
0.871603 + 0.490213i \(0.163081\pi\)
\(98\) 0 0
\(99\) −140335. −1.43906
\(100\) 0 0
\(101\) −104115. −1.01557 −0.507784 0.861484i \(-0.669535\pi\)
−0.507784 + 0.861484i \(0.669535\pi\)
\(102\) 0 0
\(103\) 43067.5 43067.5i 0.399997 0.399997i −0.478235 0.878232i \(-0.658723\pi\)
0.878232 + 0.478235i \(0.158723\pi\)
\(104\) 0 0
\(105\) −1389.19 + 1699.05i −0.0122967 + 0.0150395i
\(106\) 0 0
\(107\) −139966. 139966.i −1.18185 1.18185i −0.979265 0.202585i \(-0.935066\pi\)
−0.202585 0.979265i \(-0.564934\pi\)
\(108\) 0 0
\(109\) 96865.6i 0.780914i 0.920621 + 0.390457i \(0.127683\pi\)
−0.920621 + 0.390457i \(0.872317\pi\)
\(110\) 0 0
\(111\) 72665.8i 0.559787i
\(112\) 0 0
\(113\) −58802.8 58802.8i −0.433213 0.433213i 0.456507 0.889720i \(-0.349100\pi\)
−0.889720 + 0.456507i \(0.849100\pi\)
\(114\) 0 0
\(115\) −10617.3 105817.i −0.0748630 0.746123i
\(116\) 0 0
\(117\) −50113.9 + 50113.9i −0.338449 + 0.338449i
\(118\) 0 0
\(119\) −5116.81 −0.0331232
\(120\) 0 0
\(121\) −241043. −1.49669
\(122\) 0 0
\(123\) 42896.4 42896.4i 0.255657 0.255657i
\(124\) 0 0
\(125\) −154514. 81504.1i −0.884491 0.466557i
\(126\) 0 0
\(127\) 156941. + 156941.i 0.863428 + 0.863428i 0.991735 0.128306i \(-0.0409540\pi\)
−0.128306 + 0.991735i \(0.540954\pi\)
\(128\) 0 0
\(129\) 69412.2i 0.367250i
\(130\) 0 0
\(131\) 328683.i 1.67340i −0.547663 0.836699i \(-0.684482\pi\)
0.547663 0.836699i \(-0.315518\pi\)
\(132\) 0 0
\(133\) −3222.98 3222.98i −0.0157990 0.0157990i
\(134\) 0 0
\(135\) 120275. 12067.9i 0.567991 0.0569900i
\(136\) 0 0
\(137\) −68941.8 + 68941.8i −0.313820 + 0.313820i −0.846388 0.532567i \(-0.821227\pi\)
0.532567 + 0.846388i \(0.321227\pi\)
\(138\) 0 0
\(139\) −357947. −1.57138 −0.785692 0.618618i \(-0.787693\pi\)
−0.785692 + 0.618618i \(0.787693\pi\)
\(140\) 0 0
\(141\) 95265.8 0.403542
\(142\) 0 0
\(143\) −143588. + 143588.i −0.587190 + 0.587190i
\(144\) 0 0
\(145\) 84066.2 + 68734.7i 0.332048 + 0.271492i
\(146\) 0 0
\(147\) −55112.7 55112.7i −0.210358 0.210358i
\(148\) 0 0
\(149\) 181781.i 0.670784i 0.942079 + 0.335392i \(0.108869\pi\)
−0.942079 + 0.335392i \(0.891131\pi\)
\(150\) 0 0
\(151\) 281792.i 1.00574i −0.864361 0.502871i \(-0.832277\pi\)
0.864361 0.502871i \(-0.167723\pi\)
\(152\) 0 0
\(153\) 94986.0 + 94986.0i 0.328043 + 0.328043i
\(154\) 0 0
\(155\) 347039. + 283748.i 1.16024 + 0.948644i
\(156\) 0 0
\(157\) 18154.8 18154.8i 0.0587817 0.0587817i −0.677105 0.735887i \(-0.736766\pi\)
0.735887 + 0.677105i \(0.236766\pi\)
\(158\) 0 0
\(159\) −39906.0 −0.125183
\(160\) 0 0
\(161\) −16037.4 −0.0487606
\(162\) 0 0
\(163\) −288172. + 288172.i −0.849537 + 0.849537i −0.990075 0.140538i \(-0.955117\pi\)
0.140538 + 0.990075i \(0.455117\pi\)
\(164\) 0 0
\(165\) 164259. 16481.2i 0.469700 0.0471279i
\(166\) 0 0
\(167\) 31141.0 + 31141.0i 0.0864055 + 0.0864055i 0.748989 0.662583i \(-0.230540\pi\)
−0.662583 + 0.748989i \(0.730540\pi\)
\(168\) 0 0
\(169\) 268742.i 0.723800i
\(170\) 0 0
\(171\) 119660.i 0.312937i
\(172\) 0 0
\(173\) 159322. + 159322.i 0.404726 + 0.404726i 0.879895 0.475169i \(-0.157613\pi\)
−0.475169 + 0.879895i \(0.657613\pi\)
\(174\) 0 0
\(175\) −14555.7 + 21957.4i −0.0359285 + 0.0541983i
\(176\) 0 0
\(177\) −1335.81 + 1335.81i −0.00320489 + 0.00320489i
\(178\) 0 0
\(179\) 657790. 1.53446 0.767229 0.641373i \(-0.221635\pi\)
0.767229 + 0.641373i \(0.221635\pi\)
\(180\) 0 0
\(181\) −662330. −1.50272 −0.751360 0.659892i \(-0.770602\pi\)
−0.751360 + 0.659892i \(0.770602\pi\)
\(182\) 0 0
\(183\) 35832.4 35832.4i 0.0790949 0.0790949i
\(184\) 0 0
\(185\) −87080.4 867887.i −0.187064 1.86438i
\(186\) 0 0
\(187\) 272157. + 272157.i 0.569135 + 0.569135i
\(188\) 0 0
\(189\) 18228.7i 0.0371193i
\(190\) 0 0
\(191\) 414623.i 0.822375i −0.911551 0.411187i \(-0.865114\pi\)
0.911551 0.411187i \(-0.134886\pi\)
\(192\) 0 0
\(193\) 688506. + 688506.i 1.33050 + 1.33050i 0.904923 + 0.425575i \(0.139928\pi\)
0.425575 + 0.904923i \(0.360072\pi\)
\(194\) 0 0
\(195\) 52771.8 64542.7i 0.0993837 0.121552i
\(196\) 0 0
\(197\) 683608. 683608.i 1.25499 1.25499i 0.301540 0.953453i \(-0.402499\pi\)
0.953453 0.301540i \(-0.0975007\pi\)
\(198\) 0 0
\(199\) −808934. −1.44804 −0.724020 0.689779i \(-0.757708\pi\)
−0.724020 + 0.689779i \(0.757708\pi\)
\(200\) 0 0
\(201\) −58530.3 −0.102186
\(202\) 0 0
\(203\) 11579.1 11579.1i 0.0197213 0.0197213i
\(204\) 0 0
\(205\) 460929. 563740.i 0.766036 0.936903i
\(206\) 0 0
\(207\) 297710. + 297710.i 0.482912 + 0.482912i
\(208\) 0 0
\(209\) 342853.i 0.542928i
\(210\) 0 0
\(211\) 308765.i 0.477444i 0.971088 + 0.238722i \(0.0767284\pi\)
−0.971088 + 0.238722i \(0.923272\pi\)
\(212\) 0 0
\(213\) 40962.3 + 40962.3i 0.0618636 + 0.0618636i
\(214\) 0 0
\(215\) 83181.3 + 829027.i 0.122724 + 1.22313i
\(216\) 0 0
\(217\) 47800.4 47800.4i 0.0689099 0.0689099i
\(218\) 0 0
\(219\) 112425. 0.158399
\(220\) 0 0
\(221\) 194375. 0.267707
\(222\) 0 0
\(223\) −410489. + 410489.i −0.552763 + 0.552763i −0.927237 0.374474i \(-0.877823\pi\)
0.374474 + 0.927237i \(0.377823\pi\)
\(224\) 0 0
\(225\) 677812. 137401.i 0.892591 0.180940i
\(226\) 0 0
\(227\) 811943. + 811943.i 1.04583 + 1.04583i 0.998898 + 0.0469317i \(0.0149443\pi\)
0.0469317 + 0.998898i \(0.485056\pi\)
\(228\) 0 0
\(229\) 681576.i 0.858866i 0.903099 + 0.429433i \(0.141287\pi\)
−0.903099 + 0.429433i \(0.858713\pi\)
\(230\) 0 0
\(231\) 24894.8i 0.0306958i
\(232\) 0 0
\(233\) −130959. 130959.i −0.158032 0.158032i 0.623662 0.781694i \(-0.285644\pi\)
−0.781694 + 0.623662i \(0.785644\pi\)
\(234\) 0 0
\(235\) 1.13781e6 114163.i 1.34400 0.134852i
\(236\) 0 0
\(237\) 227411. 227411.i 0.262991 0.262991i
\(238\) 0 0
\(239\) −697425. −0.789773 −0.394887 0.918730i \(-0.629216\pi\)
−0.394887 + 0.918730i \(0.629216\pi\)
\(240\) 0 0
\(241\) −82389.9 −0.0913758 −0.0456879 0.998956i \(-0.514548\pi\)
−0.0456879 + 0.998956i \(0.514548\pi\)
\(242\) 0 0
\(243\) −515485. + 515485.i −0.560016 + 0.560016i
\(244\) 0 0
\(245\) −724286. 592196.i −0.770895 0.630304i
\(246\) 0 0
\(247\) 122433. + 122433.i 0.127690 + 0.127690i
\(248\) 0 0
\(249\) 267458.i 0.273374i
\(250\) 0 0
\(251\) 863770.i 0.865394i −0.901540 0.432697i \(-0.857562\pi\)
0.901540 0.432697i \(-0.142438\pi\)
\(252\) 0 0
\(253\) 853009. + 853009.i 0.837823 + 0.837823i
\(254\) 0 0
\(255\) −122334. 100024.i −0.117814 0.0963280i
\(256\) 0 0
\(257\) 993849. 993849.i 0.938615 0.938615i −0.0596069 0.998222i \(-0.518985\pi\)
0.998222 + 0.0596069i \(0.0189847\pi\)
\(258\) 0 0
\(259\) −131535. −0.121841
\(260\) 0 0
\(261\) −429897. −0.390628
\(262\) 0 0
\(263\) 180655. 180655.i 0.161050 0.161050i −0.621982 0.783032i \(-0.713672\pi\)
0.783032 + 0.621982i \(0.213672\pi\)
\(264\) 0 0
\(265\) −476619. + 47822.1i −0.416924 + 0.0418325i
\(266\) 0 0
\(267\) 213295. + 213295.i 0.183106 + 0.183106i
\(268\) 0 0
\(269\) 72265.9i 0.0608909i 0.999536 + 0.0304455i \(0.00969259\pi\)
−0.999536 + 0.0304455i \(0.990307\pi\)
\(270\) 0 0
\(271\) 2.05138e6i 1.69677i −0.529379 0.848385i \(-0.677575\pi\)
0.529379 0.848385i \(-0.322425\pi\)
\(272\) 0 0
\(273\) −8889.97 8889.97i −0.00721928 0.00721928i
\(274\) 0 0
\(275\) 1.94209e6 393687.i 1.54859 0.313920i
\(276\) 0 0
\(277\) −958531. + 958531.i −0.750597 + 0.750597i −0.974591 0.223994i \(-0.928090\pi\)
0.223994 + 0.974591i \(0.428090\pi\)
\(278\) 0 0
\(279\) −1.77468e6 −1.36493
\(280\) 0 0
\(281\) −2.20230e6 −1.66384 −0.831920 0.554896i \(-0.812758\pi\)
−0.831920 + 0.554896i \(0.812758\pi\)
\(282\) 0 0
\(283\) −872678. + 872678.i −0.647721 + 0.647721i −0.952442 0.304721i \(-0.901437\pi\)
0.304721 + 0.952442i \(0.401437\pi\)
\(284\) 0 0
\(285\) −14053.0 140059.i −0.0102484 0.102141i
\(286\) 0 0
\(287\) −77648.4 77648.4i −0.0556452 0.0556452i
\(288\) 0 0
\(289\) 1.05144e6i 0.740524i
\(290\) 0 0
\(291\) 232772.i 0.161138i
\(292\) 0 0
\(293\) −811890. 811890.i −0.552495 0.552495i 0.374665 0.927160i \(-0.377758\pi\)
−0.927160 + 0.374665i \(0.877758\pi\)
\(294\) 0 0
\(295\) −14353.6 + 17555.2i −0.00960295 + 0.0117449i
\(296\) 0 0
\(297\) −969560. + 969560.i −0.637799 + 0.637799i
\(298\) 0 0
\(299\) 609221. 0.394091
\(300\) 0 0
\(301\) 125646. 0.0799339
\(302\) 0 0
\(303\) −342858. + 342858.i −0.214540 + 0.214540i
\(304\) 0 0
\(305\) 385025. 470906.i 0.236995 0.289858i
\(306\) 0 0
\(307\) −1.50108e6 1.50108e6i −0.908989 0.908989i 0.0872016 0.996191i \(-0.472208\pi\)
−0.996191 + 0.0872016i \(0.972208\pi\)
\(308\) 0 0
\(309\) 283649.i 0.168999i
\(310\) 0 0
\(311\) 2.99788e6i 1.75757i −0.477213 0.878787i \(-0.658353\pi\)
0.477213 0.878787i \(-0.341647\pi\)
\(312\) 0 0
\(313\) 186130. + 186130.i 0.107388 + 0.107388i 0.758759 0.651371i \(-0.225806\pi\)
−0.651371 + 0.758759i \(0.725806\pi\)
\(314\) 0 0
\(315\) −10412.1 103772.i −0.00591239 0.0589258i
\(316\) 0 0
\(317\) −269555. + 269555.i −0.150661 + 0.150661i −0.778413 0.627752i \(-0.783975\pi\)
0.627752 + 0.778413i \(0.283975\pi\)
\(318\) 0 0
\(319\) −1.23176e6 −0.677717
\(320\) 0 0
\(321\) −921835. −0.499334
\(322\) 0 0
\(323\) 232060. 232060.i 0.123764 0.123764i
\(324\) 0 0
\(325\) 552936. 834108.i 0.290380 0.438040i
\(326\) 0 0
\(327\) 318986. + 318986.i 0.164969 + 0.164969i
\(328\) 0 0
\(329\) 172444.i 0.0878333i
\(330\) 0 0
\(331\) 656419.i 0.329315i 0.986351 + 0.164657i \(0.0526518\pi\)
−0.986351 + 0.164657i \(0.947348\pi\)
\(332\) 0 0
\(333\) 2.44175e6 + 2.44175e6i 1.20668 + 1.20668i
\(334\) 0 0
\(335\) −699059. + 70140.8i −0.340331 + 0.0341475i
\(336\) 0 0
\(337\) 1.29668e6 1.29668e6i 0.621954 0.621954i −0.324077 0.946031i \(-0.605054\pi\)
0.946031 + 0.324077i \(0.105054\pi\)
\(338\) 0 0
\(339\) −387284. −0.183033
\(340\) 0 0
\(341\) −5.08489e6 −2.36807
\(342\) 0 0
\(343\) −199947. + 199947.i −0.0917655 + 0.0917655i
\(344\) 0 0
\(345\) −383426. 313500.i −0.173434 0.141804i
\(346\) 0 0
\(347\) −2.79696e6 2.79696e6i −1.24699 1.24699i −0.957042 0.289948i \(-0.906362\pi\)
−0.289948 0.957042i \(-0.593638\pi\)
\(348\) 0 0
\(349\) 2.63211e6i 1.15675i 0.815770 + 0.578376i \(0.196313\pi\)
−0.815770 + 0.578376i \(0.803687\pi\)
\(350\) 0 0
\(351\) 692462.i 0.300005i
\(352\) 0 0
\(353\) 324347. + 324347.i 0.138539 + 0.138539i 0.772975 0.634436i \(-0.218768\pi\)
−0.634436 + 0.772975i \(0.718768\pi\)
\(354\) 0 0
\(355\) 538323. + 440147.i 0.226711 + 0.185365i
\(356\) 0 0
\(357\) −16850.1 + 16850.1i −0.00699731 + 0.00699731i
\(358\) 0 0
\(359\) 2.87581e6 1.17767 0.588835 0.808253i \(-0.299587\pi\)
0.588835 + 0.808253i \(0.299587\pi\)
\(360\) 0 0
\(361\) −2.18376e6 −0.881935
\(362\) 0 0
\(363\) −793772. + 793772.i −0.316176 + 0.316176i
\(364\) 0 0
\(365\) 1.34275e6 134727.i 0.527550 0.0529323i
\(366\) 0 0
\(367\) 1.93322e6 + 1.93322e6i 0.749232 + 0.749232i 0.974335 0.225103i \(-0.0722719\pi\)
−0.225103 + 0.974335i \(0.572272\pi\)
\(368\) 0 0
\(369\) 2.88285e6i 1.10219i
\(370\) 0 0
\(371\) 72235.4i 0.0272468i
\(372\) 0 0
\(373\) −2.22939e6 2.22939e6i −0.829685 0.829685i 0.157788 0.987473i \(-0.449564\pi\)
−0.987473 + 0.157788i \(0.949564\pi\)
\(374\) 0 0
\(375\) −777227. + 240428.i −0.285410 + 0.0882890i
\(376\) 0 0
\(377\) −439862. + 439862.i −0.159391 + 0.159391i
\(378\) 0 0
\(379\) −1.42092e6 −0.508127 −0.254064 0.967187i \(-0.581767\pi\)
−0.254064 + 0.967187i \(0.581767\pi\)
\(380\) 0 0
\(381\) 1.03364e6 0.364800
\(382\) 0 0
\(383\) 3.39383e6 3.39383e6i 1.18220 1.18220i 0.203032 0.979172i \(-0.434920\pi\)
0.979172 0.203032i \(-0.0650797\pi\)
\(384\) 0 0
\(385\) −29833.2 297332.i −0.0102576 0.102233i
\(386\) 0 0
\(387\) −2.33242e6 2.33242e6i −0.791644 0.791644i
\(388\) 0 0
\(389\) 3.31303e6i 1.11007i −0.831826 0.555036i \(-0.812704\pi\)
0.831826 0.555036i \(-0.187296\pi\)
\(390\) 0 0
\(391\) 1.15472e6i 0.381974i
\(392\) 0 0
\(393\) −1.08238e6 1.08238e6i −0.353507 0.353507i
\(394\) 0 0
\(395\) 2.44357e6 2.98861e6i 0.788010 0.963778i
\(396\) 0 0
\(397\) 2.40606e6 2.40606e6i 0.766179 0.766179i −0.211253 0.977431i \(-0.567754\pi\)
0.977431 + 0.211253i \(0.0677544\pi\)
\(398\) 0 0
\(399\) −21227.1 −0.00667510
\(400\) 0 0
\(401\) 5.13661e6 1.59520 0.797601 0.603185i \(-0.206102\pi\)
0.797601 + 0.603185i \(0.206102\pi\)
\(402\) 0 0
\(403\) −1.81582e6 + 1.81582e6i −0.556942 + 0.556942i
\(404\) 0 0
\(405\) −1.54660e6 + 1.89158e6i −0.468535 + 0.573043i
\(406\) 0 0
\(407\) 6.99619e6 + 6.99619e6i 2.09351 + 2.09351i
\(408\) 0 0
\(409\) 1.51107e6i 0.446658i −0.974743 0.223329i \(-0.928308\pi\)
0.974743 0.223329i \(-0.0716924\pi\)
\(410\) 0 0
\(411\) 454061.i 0.132590i
\(412\) 0 0
\(413\) 2418.01 + 2418.01i 0.000697562 + 0.000697562i
\(414\) 0 0
\(415\) 320513. + 3.19439e6i 0.0913534 + 0.910474i
\(416\) 0 0
\(417\) −1.17875e6 + 1.17875e6i −0.331956 + 0.331956i
\(418\) 0 0
\(419\) −1.07646e6 −0.299544 −0.149772 0.988721i \(-0.547854\pi\)
−0.149772 + 0.988721i \(0.547854\pi\)
\(420\) 0 0
\(421\) 3.18241e6 0.875086 0.437543 0.899197i \(-0.355849\pi\)
0.437543 + 0.899197i \(0.355849\pi\)
\(422\) 0 0
\(423\) −3.20117e6 + 3.20117e6i −0.869877 + 0.869877i
\(424\) 0 0
\(425\) −1.58097e6 1.04804e6i −0.424572 0.281452i
\(426\) 0 0
\(427\) −64861.6 64861.6i −0.0172154 0.0172154i
\(428\) 0 0
\(429\) 945694.i 0.248089i
\(430\) 0 0
\(431\) 1.46800e6i 0.380657i 0.981720 + 0.190329i \(0.0609553\pi\)
−0.981720 + 0.190329i \(0.939045\pi\)
\(432\) 0 0
\(433\) 4.49367e6 + 4.49367e6i 1.15181 + 1.15181i 0.986189 + 0.165623i \(0.0529634\pi\)
0.165623 + 0.986189i \(0.447037\pi\)
\(434\) 0 0
\(435\) 503185. 50487.6i 0.127498 0.0127927i
\(436\) 0 0
\(437\) 727335. 727335.i 0.182193 0.182193i
\(438\) 0 0
\(439\) −25222.7 −0.00624641 −0.00312320 0.999995i \(-0.500994\pi\)
−0.00312320 + 0.999995i \(0.500994\pi\)
\(440\) 0 0
\(441\) 3.70385e6 0.906895
\(442\) 0 0
\(443\) −3.20085e6 + 3.20085e6i −0.774919 + 0.774919i −0.978962 0.204043i \(-0.934592\pi\)
0.204043 + 0.978962i \(0.434592\pi\)
\(444\) 0 0
\(445\) 2.80311e6 + 2.29189e6i 0.671027 + 0.548649i
\(446\) 0 0
\(447\) 598618. + 598618.i 0.141704 + 0.141704i
\(448\) 0 0
\(449\) 4.85115e6i 1.13561i −0.823164 0.567804i \(-0.807793\pi\)
0.823164 0.567804i \(-0.192207\pi\)
\(450\) 0 0
\(451\) 8.26005e6i 1.91224i
\(452\) 0 0
\(453\) −927963. 927963.i −0.212464 0.212464i
\(454\) 0 0
\(455\) −116831. 95524.3i −0.0264564 0.0216314i
\(456\) 0 0
\(457\) 1.87370e6 1.87370e6i 0.419671 0.419671i −0.465419 0.885090i \(-0.654097\pi\)
0.885090 + 0.465419i \(0.154097\pi\)
\(458\) 0 0
\(459\) 1.31249e6 0.290780
\(460\) 0 0
\(461\) 2.47636e6 0.542702 0.271351 0.962480i \(-0.412530\pi\)
0.271351 + 0.962480i \(0.412530\pi\)
\(462\) 0 0
\(463\) 5.78796e6 5.78796e6i 1.25480 1.25480i 0.301252 0.953544i \(-0.402595\pi\)
0.953544 0.301252i \(-0.0974046\pi\)
\(464\) 0 0
\(465\) 2.07723e6 208421.i 0.445504 0.0447001i
\(466\) 0 0
\(467\) −1.55806e6 1.55806e6i −0.330591 0.330591i 0.522220 0.852811i \(-0.325104\pi\)
−0.852811 + 0.522220i \(0.825104\pi\)
\(468\) 0 0
\(469\) 105948.i 0.0222413i
\(470\) 0 0
\(471\) 119570.i 0.0248354i
\(472\) 0 0
\(473\) −6.68294e6 6.68294e6i −1.37346 1.37346i
\(474\) 0 0
\(475\) −335684. 1.65596e6i −0.0682649 0.336756i
\(476\) 0 0
\(477\) 1.34094e6 1.34094e6i 0.269845 0.269845i
\(478\) 0 0
\(479\) 1.81391e6 0.361224 0.180612 0.983554i \(-0.442192\pi\)
0.180612 + 0.983554i \(0.442192\pi\)
\(480\) 0 0
\(481\) 4.99670e6 0.984737
\(482\) 0 0
\(483\) −52812.3 + 52812.3i −0.0103007 + 0.0103007i
\(484\) 0 0
\(485\) −278946. 2.78012e6i −0.0538476 0.536673i
\(486\) 0 0
\(487\) −898602. 898602.i −0.171690 0.171690i 0.616031 0.787722i \(-0.288739\pi\)
−0.787722 + 0.616031i \(0.788739\pi\)
\(488\) 0 0
\(489\) 1.89794e6i 0.358931i
\(490\) 0 0
\(491\) 1.86137e6i 0.348441i 0.984707 + 0.174220i \(0.0557405\pi\)
−0.984707 + 0.174220i \(0.944260\pi\)
\(492\) 0 0
\(493\) 833714. + 833714.i 0.154490 + 0.154490i
\(494\) 0 0
\(495\) −4.96572e6 + 6.07334e6i −0.910897 + 1.11408i
\(496\) 0 0
\(497\) 74147.5 74147.5i 0.0134650 0.0134650i
\(498\) 0 0
\(499\) 3.04071e6 0.546668 0.273334 0.961919i \(-0.411874\pi\)
0.273334 + 0.961919i \(0.411874\pi\)
\(500\) 0 0
\(501\) 205099. 0.0365065
\(502\) 0 0
\(503\) 7.23935e6 7.23935e6i 1.27579 1.27579i 0.332791 0.943001i \(-0.392010\pi\)
0.943001 0.332791i \(-0.107990\pi\)
\(504\) 0 0
\(505\) −3.68407e6 + 4.50581e6i −0.642835 + 0.786221i
\(506\) 0 0
\(507\) −884988. 884988.i −0.152904 0.152904i
\(508\) 0 0
\(509\) 8.84454e6i 1.51315i −0.653910 0.756573i \(-0.726872\pi\)
0.653910 0.756573i \(-0.273128\pi\)
\(510\) 0 0
\(511\) 203505.i 0.0344764i
\(512\) 0 0
\(513\) 826714. + 826714.i 0.138695 + 0.138695i
\(514\) 0 0
\(515\) −339916. 3.38777e6i −0.0564747 0.562855i
\(516\) 0 0
\(517\) −9.17210e6 + 9.17210e6i −1.50918 + 1.50918i
\(518\) 0 0
\(519\) 1.04932e6 0.170998
\(520\) 0 0
\(521\) −9.43074e6 −1.52213 −0.761065 0.648676i \(-0.775323\pi\)
−0.761065 + 0.648676i \(0.775323\pi\)
\(522\) 0 0
\(523\) −3.70150e6 + 3.70150e6i −0.591729 + 0.591729i −0.938098 0.346369i \(-0.887415\pi\)
0.346369 + 0.938098i \(0.387415\pi\)
\(524\) 0 0
\(525\) 24374.3 + 120241.i 0.00385953 + 0.0190394i
\(526\) 0 0
\(527\) 3.44170e6 + 3.44170e6i 0.539817 + 0.539817i
\(528\) 0 0
\(529\) 2.81716e6i 0.437696i
\(530\) 0 0
\(531\) 89773.5i 0.0138169i
\(532\) 0 0
\(533\) 2.94967e6 + 2.94967e6i 0.449734 + 0.449734i
\(534\) 0 0
\(535\) −1.10100e7 + 1.10470e6i −1.66304 + 0.166863i
\(536\) 0 0
\(537\) 2.16615e6 2.16615e6i 0.324156 0.324156i
\(538\) 0 0
\(539\) 1.06124e7 1.57341
\(540\) 0 0
\(541\) 4.64526e6 0.682365 0.341183 0.939997i \(-0.389173\pi\)
0.341183 + 0.939997i \(0.389173\pi\)
\(542\) 0 0
\(543\) −2.18110e6 + 2.18110e6i −0.317451 + 0.317451i
\(544\) 0 0
\(545\) 4.19208e6 + 3.42756e6i 0.604559 + 0.494303i
\(546\) 0 0
\(547\) −8.51024e6 8.51024e6i −1.21611 1.21611i −0.968983 0.247128i \(-0.920513\pi\)
−0.247128 0.968983i \(-0.579487\pi\)
\(548\) 0 0
\(549\) 2.40812e6i 0.340994i
\(550\) 0 0
\(551\) 1.05028e6i 0.147376i
\(552\) 0 0
\(553\) −411645. 411645.i −0.0572414 0.0572414i
\(554\) 0 0
\(555\) −3.14478e6 2.57125e6i −0.433369 0.354334i
\(556\) 0 0
\(557\) −1.60437e6 + 1.60437e6i −0.219113 + 0.219113i −0.808124 0.589012i \(-0.799517\pi\)
0.589012 + 0.808124i \(0.299517\pi\)
\(558\) 0 0
\(559\) −4.77297e6 −0.646040
\(560\) 0 0
\(561\) 1.79247e6 0.240461
\(562\) 0 0
\(563\) 9.82070e6 9.82070e6i 1.30578 1.30578i 0.381355 0.924429i \(-0.375458\pi\)
0.924429 0.381355i \(-0.124542\pi\)
\(564\) 0 0
\(565\) −4.62554e6 + 464109.i −0.609595 + 0.0611644i
\(566\) 0 0
\(567\) 260542. + 260542.i 0.0340346 + 0.0340346i
\(568\) 0 0
\(569\) 7.73484e6i 1.00155i −0.865579 0.500773i \(-0.833049\pi\)
0.865579 0.500773i \(-0.166951\pi\)
\(570\) 0 0
\(571\) 8.13215e6i 1.04380i 0.853008 + 0.521898i \(0.174776\pi\)
−0.853008 + 0.521898i \(0.825224\pi\)
\(572\) 0 0
\(573\) −1.36538e6 1.36538e6i −0.173727 0.173727i
\(574\) 0 0
\(575\) −4.95516e6 3.28481e6i −0.625011 0.414324i
\(576\) 0 0
\(577\) −5.27077e6 + 5.27077e6i −0.659074 + 0.659074i −0.955161 0.296087i \(-0.904318\pi\)
0.296087 + 0.955161i \(0.404318\pi\)
\(578\) 0 0
\(579\) 4.53460e6 0.562138
\(580\) 0 0
\(581\) 484135. 0.0595013
\(582\) 0 0
\(583\) 3.84211e6 3.84211e6i 0.468165 0.468165i
\(584\) 0 0
\(585\) 395531. + 3.94206e6i 0.0477849 + 0.476249i
\(586\) 0 0
\(587\) −4.02268e6 4.02268e6i −0.481859 0.481859i 0.423866 0.905725i \(-0.360673\pi\)
−0.905725 + 0.423866i \(0.860673\pi\)
\(588\) 0 0
\(589\) 4.33573e6i 0.514960i
\(590\) 0 0
\(591\) 4.50235e6i 0.530237i
\(592\) 0 0
\(593\) −5.59217e6 5.59217e6i −0.653046 0.653046i 0.300679 0.953725i \(-0.402787\pi\)
−0.953725 + 0.300679i \(0.902787\pi\)
\(594\) 0 0
\(595\) −181057. + 221442.i −0.0209663 + 0.0256429i
\(596\) 0 0
\(597\) −2.66388e6 + 2.66388e6i −0.305900 + 0.305900i
\(598\) 0 0
\(599\) −1.15230e7 −1.31220 −0.656099 0.754675i \(-0.727795\pi\)
−0.656099 + 0.754675i \(0.727795\pi\)
\(600\) 0 0
\(601\) 2.44067e6 0.275628 0.137814 0.990458i \(-0.455992\pi\)
0.137814 + 0.990458i \(0.455992\pi\)
\(602\) 0 0
\(603\) 1.96676e6 1.96676e6i 0.220272 0.220272i
\(604\) 0 0
\(605\) −8.52922e6 + 1.04317e7i −0.947373 + 1.15869i
\(606\) 0 0
\(607\) −3.35790e6 3.35790e6i −0.369910 0.369910i 0.497535 0.867444i \(-0.334239\pi\)
−0.867444 + 0.497535i \(0.834239\pi\)
\(608\) 0 0
\(609\) 76261.7i 0.00833227i
\(610\) 0 0
\(611\) 6.55073e6i 0.709883i
\(612\) 0 0
\(613\) −1.00846e7 1.00846e7i −1.08395 1.08395i −0.996137 0.0878091i \(-0.972013\pi\)
−0.0878091 0.996137i \(-0.527987\pi\)
\(614\) 0 0
\(615\) −338566. 3.37431e6i −0.0360957 0.359748i
\(616\) 0 0
\(617\) −8.65775e6 + 8.65775e6i −0.915571 + 0.915571i −0.996703 0.0811322i \(-0.974146\pi\)
0.0811322 + 0.996703i \(0.474146\pi\)
\(618\) 0 0
\(619\) −691475. −0.0725354 −0.0362677 0.999342i \(-0.511547\pi\)
−0.0362677 + 0.999342i \(0.511547\pi\)
\(620\) 0 0
\(621\) 4.11369e6 0.428057
\(622\) 0 0
\(623\) 386094. 386094.i 0.0398541 0.0398541i
\(624\) 0 0
\(625\) −8.99471e6 + 3.80296e6i −0.921059 + 0.389424i
\(626\) 0 0
\(627\) 1.12904e6 + 1.12904e6i 0.114694 + 0.114694i
\(628\) 0 0
\(629\) 9.47074e6i 0.954459i
\(630\) 0 0
\(631\) 1.44466e7i 1.44441i −0.691677 0.722207i \(-0.743128\pi\)
0.691677 0.722207i \(-0.256872\pi\)
\(632\) 0 0
\(633\) 1.01679e6 + 1.01679e6i 0.100860 + 0.100860i
\(634\) 0 0
\(635\) 1.23453e7 1.23868e6i 1.21497 0.121906i
\(636\) 0 0
\(637\) 3.78970e6 3.78970e6i 0.370047 0.370047i
\(638\) 0 0
\(639\) −2.75287e6 −0.266707
\(640\) 0 0
\(641\) 1.13910e7 1.09501 0.547504 0.836803i \(-0.315578\pi\)
0.547504 + 0.836803i \(0.315578\pi\)
\(642\) 0 0
\(643\) −7.53859e6 + 7.53859e6i −0.719056 + 0.719056i −0.968412 0.249356i \(-0.919781\pi\)
0.249356 + 0.968412i \(0.419781\pi\)
\(644\) 0 0
\(645\) 3.00397e6 + 2.45613e6i 0.284313 + 0.232462i
\(646\) 0 0
\(647\) −8.23733e6 8.23733e6i −0.773616 0.773616i 0.205120 0.978737i \(-0.434241\pi\)
−0.978737 + 0.205120i \(0.934241\pi\)
\(648\) 0 0
\(649\) 257222.i 0.0239716i
\(650\) 0 0
\(651\) 314820.i 0.0291146i
\(652\) 0 0
\(653\) −7.48193e6 7.48193e6i −0.686642 0.686642i 0.274846 0.961488i \(-0.411373\pi\)
−0.961488 + 0.274846i \(0.911373\pi\)
\(654\) 0 0
\(655\) −1.42245e7 1.16303e7i −1.29549 1.05923i
\(656\) 0 0
\(657\) −3.77776e6 + 3.77776e6i −0.341445 + 0.341445i
\(658\) 0 0
\(659\) 1.09855e7 0.985390 0.492695 0.870202i \(-0.336012\pi\)
0.492695 + 0.870202i \(0.336012\pi\)
\(660\) 0 0
\(661\) 6.62528e6 0.589795 0.294897 0.955529i \(-0.404715\pi\)
0.294897 + 0.955529i \(0.404715\pi\)
\(662\) 0 0
\(663\) 640092. 640092.i 0.0565534 0.0565534i
\(664\) 0 0
\(665\) −253526. + 25437.8i −0.0222315 + 0.00223062i
\(666\) 0 0
\(667\) 2.61307e6 + 2.61307e6i 0.227424 + 0.227424i
\(668\) 0 0
\(669\) 2.70354e6i 0.233544i
\(670\) 0 0
\(671\) 6.89982e6i 0.591605i
\(672\) 0 0
\(673\) −5.68585e6 5.68585e6i −0.483902 0.483902i 0.422473 0.906375i \(-0.361162\pi\)
−0.906375 + 0.422473i \(0.861162\pi\)
\(674\) 0 0
\(675\) 3.73363e6 5.63220e6i 0.315407 0.475794i
\(676\) 0 0
\(677\) 2.53436e6 2.53436e6i 0.212519 0.212519i −0.592818 0.805337i \(-0.701985\pi\)
0.805337 + 0.592818i \(0.201985\pi\)
\(678\) 0 0
\(679\) −421350. −0.0350726
\(680\) 0 0
\(681\) 5.34758e6 0.441865
\(682\) 0 0
\(683\) 1.05560e7 1.05560e7i 0.865864 0.865864i −0.126148 0.992011i \(-0.540261\pi\)
0.992011 + 0.126148i \(0.0402614\pi\)
\(684\) 0 0
\(685\) 544132. + 5.42310e6i 0.0443076 + 0.441592i
\(686\) 0 0
\(687\) 2.24448e6 + 2.24448e6i 0.181436 + 0.181436i
\(688\) 0 0
\(689\) 2.74404e6i 0.220213i
\(690\) 0 0
\(691\) 2.17073e7i 1.72946i 0.502239 + 0.864729i \(0.332510\pi\)
−0.502239 + 0.864729i \(0.667490\pi\)
\(692\) 0 0
\(693\) 836529. + 836529.i 0.0661680 + 0.0661680i
\(694\) 0 0
\(695\) −1.26658e7 + 1.54910e7i −0.994654 + 1.21651i
\(696\) 0 0
\(697\) 5.59081e6 5.59081e6i 0.435906 0.435906i
\(698\) 0 0
\(699\) −862514. −0.0667687
\(700\) 0 0
\(701\) −2.03600e6 −0.156488 −0.0782441 0.996934i \(-0.524931\pi\)
−0.0782441 + 0.996934i \(0.524931\pi\)
\(702\) 0 0
\(703\) 5.96544e6 5.96544e6i 0.455254 0.455254i
\(704\) 0 0
\(705\) 3.37095e6 4.12285e6i 0.255434 0.312410i
\(706\) 0 0
\(707\) 620621. + 620621.i 0.0466958 + 0.0466958i
\(708\) 0 0
\(709\) 5.59025e6i 0.417653i 0.977953 + 0.208827i \(0.0669644\pi\)
−0.977953 + 0.208827i \(0.933036\pi\)
\(710\) 0 0
\(711\) 1.52832e7i 1.13381i
\(712\) 0 0
\(713\) 1.07872e7 + 1.07872e7i 0.794664 + 0.794664i
\(714\) 0 0
\(715\) 1.13329e6 + 1.12949e7i 0.0829040 + 0.826263i
\(716\) 0 0
\(717\) −2.29667e6 + 2.29667e6i −0.166840 + 0.166840i
\(718\) 0 0
\(719\) 1.30592e7 0.942093 0.471046 0.882108i \(-0.343876\pi\)
0.471046 + 0.882108i \(0.343876\pi\)
\(720\) 0 0
\(721\) −513444. −0.0367837
\(722\) 0 0
\(723\) −271316. + 271316.i −0.0193032 + 0.0193032i
\(724\) 0 0
\(725\) 5.94931e6 1.20600e6i 0.420360 0.0852125i
\(726\) 0 0
\(727\) 1.15104e7 + 1.15104e7i 0.807707 + 0.807707i 0.984286 0.176579i \(-0.0565032\pi\)
−0.176579 + 0.984286i \(0.556503\pi\)
\(728\) 0 0
\(729\) 7.22606e6i 0.503597i
\(730\) 0 0
\(731\) 9.04669e6i 0.626176i
\(732\) 0 0
\(733\) 1.05032e7 + 1.05032e7i 0.722038 + 0.722038i 0.969020 0.246982i \(-0.0794388\pi\)
−0.246982 + 0.969020i \(0.579439\pi\)
\(734\) 0 0
\(735\) −4.33527e6 + 434985.i −0.296004 + 0.0296999i
\(736\) 0 0
\(737\) 5.63524e6 5.63524e6i 0.382159 0.382159i
\(738\) 0 0
\(739\) 9.39982e6 0.633153 0.316576 0.948567i \(-0.397467\pi\)
0.316576 + 0.948567i \(0.397467\pi\)
\(740\) 0 0
\(741\) 806364. 0.0539493
\(742\) 0 0
\(743\) −1.05839e7 + 1.05839e7i −0.703354 + 0.703354i −0.965129 0.261775i \(-0.915692\pi\)
0.261775 + 0.965129i \(0.415692\pi\)
\(744\) 0 0
\(745\) 7.86699e6 + 6.43226e6i 0.519299 + 0.424593i
\(746\) 0 0
\(747\) −8.98724e6 8.98724e6i −0.589285 0.589285i
\(748\) 0 0
\(749\) 1.66865e6i 0.108683i
\(750\) 0 0
\(751\) 2.49520e6i 0.161438i 0.996737 + 0.0807189i \(0.0257216\pi\)
−0.996737 + 0.0807189i \(0.974278\pi\)
\(752\) 0 0
\(753\) −2.84446e6 2.84446e6i −0.182815 0.182815i
\(754\) 0 0
\(755\) −1.21952e7 9.97113e6i −0.778613 0.636615i
\(756\) 0 0
\(757\) 1.90732e7 1.90732e7i 1.20972 1.20972i 0.238597 0.971119i \(-0.423312\pi\)
0.971119 0.238597i \(-0.0766876\pi\)
\(758\) 0 0
\(759\) 5.61805e6 0.353982
\(760\) 0 0
\(761\) 2.61451e6 0.163655 0.0818274 0.996647i \(-0.473924\pi\)
0.0818274 + 0.996647i \(0.473924\pi\)
\(762\) 0 0
\(763\) 577409. 577409.i 0.0359064 0.0359064i
\(764\) 0 0
\(765\) 7.47178e6 749690.i 0.461605 0.0463157i
\(766\) 0 0
\(767\) −91854.3 91854.3i −0.00563782 0.00563782i
\(768\) 0 0
\(769\) 1.65153e7i 1.00709i −0.863968 0.503547i \(-0.832028\pi\)
0.863968 0.503547i \(-0.167972\pi\)
\(770\) 0 0
\(771\) 6.54564e6i 0.396567i
\(772\) 0 0
\(773\) −5.55165e6 5.55165e6i −0.334174 0.334174i 0.519995 0.854169i \(-0.325934\pi\)
−0.854169 + 0.519995i \(0.825934\pi\)
\(774\) 0 0
\(775\) 2.45597e7 4.97857e6i 1.46882 0.297749i
\(776\) 0 0
\(777\) −433155. + 433155.i −0.0257390 + 0.0257390i
\(778\) 0 0
\(779\) 7.04309e6 0.415834
\(780\) 0 0
\(781\) −7.88763e6 −0.462721
\(782\) 0 0
\(783\) −2.97011e6 + 2.97011e6i −0.173128 + 0.173128i
\(784\) 0 0
\(785\) −143289. 1.42809e6i −0.00829925 0.0827145i
\(786\) 0 0
\(787\) 7.86188e6 + 7.86188e6i 0.452470 + 0.452470i 0.896174 0.443704i \(-0.146336\pi\)
−0.443704 + 0.896174i \(0.646336\pi\)
\(788\) 0 0
\(789\) 1.18982e6i 0.0680438i
\(790\) 0 0
\(791\) 701038.i 0.0398382i
\(792\) 0 0
\(793\) 2.46393e6 + 2.46393e6i 0.139138 + 0.139138i
\(794\) 0 0
\(795\) −1.41206e6 + 1.72702e6i −0.0792383 + 0.0969126i
\(796\) 0 0
\(797\) −1.64771e6 + 1.64771e6i −0.0918830 + 0.0918830i −0.751554 0.659671i \(-0.770696\pi\)
0.659671 + 0.751554i \(0.270696\pi\)
\(798\) 0 0
\(799\) 1.24163e7 0.688056
\(800\) 0 0
\(801\) −1.43345e7 −0.789409
\(802\) 0 0
\(803\) −1.08242e7 + 1.08242e7i −0.592388 + 0.592388i
\(804\) 0 0
\(805\) −567478. + 694055.i −0.0308645 + 0.0377489i
\(806\) 0 0
\(807\) 237977. + 237977.i 0.0128633 + 0.0128633i
\(808\) 0 0
\(809\) 1.20143e7i 0.645396i 0.946502 + 0.322698i \(0.104590\pi\)
−0.946502 + 0.322698i \(0.895410\pi\)
\(810\) 0 0
\(811\) 1.17390e7i 0.626727i 0.949633 + 0.313364i \(0.101456\pi\)
−0.949633 + 0.313364i \(0.898544\pi\)
\(812\) 0 0
\(813\) −6.75535e6 6.75535e6i −0.358444 0.358444i
\(814\) 0 0
\(815\) 2.27444e6 + 2.26682e7i 0.119944 + 1.19543i
\(816\) 0 0
\(817\) −5.69834e6 + 5.69834e6i −0.298671 + 0.298671i
\(818\) 0 0
\(819\) 597451. 0.0311238
\(820\) 0 0
\(821\) 2.76061e7 1.42938 0.714690 0.699441i \(-0.246568\pi\)
0.714690 + 0.699441i \(0.246568\pi\)
\(822\) 0 0
\(823\) 2.25518e7 2.25518e7i 1.16060 1.16060i 0.176250 0.984345i \(-0.443603\pi\)
0.984345 0.176250i \(-0.0563966\pi\)
\(824\) 0 0
\(825\) 5.09901e6 7.69189e6i 0.260826 0.393458i
\(826\) 0 0
\(827\) −1.00811e7 1.00811e7i −0.512559 0.512559i 0.402750 0.915310i \(-0.368054\pi\)
−0.915310 + 0.402750i \(0.868054\pi\)
\(828\) 0 0
\(829\) 2.93364e7i 1.48259i −0.671180 0.741294i \(-0.734212\pi\)
0.671180 0.741294i \(-0.265788\pi\)
\(830\) 0 0
\(831\) 6.31303e6i 0.317128i
\(832\) 0 0
\(833\) −7.18300e6 7.18300e6i −0.358669 0.358669i
\(834\) 0 0
\(835\) 2.44961e6 245785.i 0.121585 0.0121994i
\(836\) 0 0
\(837\) −1.22611e7 + 1.22611e7i −0.604944 + 0.604944i
\(838\) 0 0
\(839\) −1.26054e7 −0.618230 −0.309115 0.951025i \(-0.600033\pi\)
−0.309115 + 0.951025i \(0.600033\pi\)
\(840\) 0 0
\(841\) 1.67378e7 0.816036
\(842\) 0 0
\(843\) −7.25235e6 + 7.25235e6i −0.351488 + 0.351488i
\(844\) 0 0
\(845\) −1.16304e7 9.50935e6i −0.560343 0.458151i
\(846\) 0 0
\(847\) 1.43684e6 + 1.43684e6i 0.0688176 + 0.0688176i
\(848\) 0 0
\(849\) 5.74759e6i 0.273663i
\(850\) 0 0
\(851\) 2.96837e7i 1.40506i
\(852\) 0 0
\(853\) 1.64758e7 + 1.64758e7i 0.775310 + 0.775310i 0.979029 0.203720i \(-0.0653031\pi\)
−0.203720 + 0.979029i \(0.565303\pi\)
\(854\) 0 0
\(855\) 5.17855e6 + 4.23412e6i 0.242266 + 0.198083i
\(856\) 0 0
\(857\) 1.84182e7 1.84182e7i 0.856635 0.856635i −0.134305 0.990940i \(-0.542880\pi\)
0.990940 + 0.134305i \(0.0428803\pi\)
\(858\) 0 0
\(859\) −4.03573e7 −1.86612 −0.933060 0.359720i \(-0.882872\pi\)
−0.933060 + 0.359720i \(0.882872\pi\)
\(860\) 0 0
\(861\) −511404. −0.0235102
\(862\) 0 0
\(863\) −1.37186e7 + 1.37186e7i −0.627020 + 0.627020i −0.947317 0.320297i \(-0.896217\pi\)
0.320297 + 0.947317i \(0.396217\pi\)
\(864\) 0 0
\(865\) 1.25326e7 1.25747e6i 0.569510 0.0571424i
\(866\) 0 0
\(867\) 3.46247e6 + 3.46247e6i 0.156436 + 0.156436i
\(868\) 0 0
\(869\) 4.37898e7i 1.96709i
\(870\) 0 0
\(871\) 4.02470e6i 0.179758i
\(872\) 0 0
\(873\) 7.82172e6 + 7.82172e6i 0.347350 + 0.347350i
\(874\) 0 0
\(875\) 435208. + 1.40689e6i 0.0192166 + 0.0621211i
\(876\) 0 0
\(877\) −7.68375e6 + 7.68375e6i −0.337345 + 0.337345i −0.855367 0.518022i \(-0.826668\pi\)
0.518022 + 0.855367i \(0.326668\pi\)
\(878\) 0 0
\(879\) −5.34723e6 −0.233430
\(880\) 0 0
\(881\) −1.78611e7 −0.775298 −0.387649 0.921807i \(-0.626713\pi\)
−0.387649 + 0.921807i \(0.626713\pi\)
\(882\) 0 0
\(883\) −1.62792e7 + 1.62792e7i −0.702637 + 0.702637i −0.964976 0.262339i \(-0.915506\pi\)
0.262339 + 0.964976i \(0.415506\pi\)
\(884\) 0 0
\(885\) 10543.1 + 105078.i 0.000452491 + 0.00450975i
\(886\) 0 0
\(887\) 5.07996e6 + 5.07996e6i 0.216796 + 0.216796i 0.807147 0.590351i \(-0.201011\pi\)
−0.590351 + 0.807147i \(0.701011\pi\)
\(888\) 0 0
\(889\) 1.87102e6i 0.0794008i
\(890\) 0 0
\(891\) 2.77158e7i 1.16959i
\(892\) 0 0
\(893\) 7.82077e6 + 7.82077e6i 0.328187 + 0.328187i
\(894\) 0 0
\(895\) 2.32757e7 2.84674e7i 0.971281 1.18793i
\(896\) 0 0
\(897\) 2.00621e6 2.00621e6i 0.0832522 0.0832522i
\(898\) 0 0
\(899\) −1.55768e7 −0.642805
\(900\) 0 0
\(901\) −5.20106e6 −0.213442
\(902\) 0 0
\(903\) 413761. 413761.i 0.0168861 0.0168861i
\(904\) 0 0
\(905\) −2.34363e7 + 2.86639e7i −0.951192 + 1.16336i
\(906\) 0 0
\(907\) 1.23719e7 + 1.23719e7i 0.499367 + 0.499367i 0.911241 0.411874i \(-0.135126\pi\)
−0.411874 + 0.911241i \(0.635126\pi\)
\(908\) 0 0
\(909\) 2.30418e7i 0.924925i
\(910\) 0 0
\(911\) 1.71338e7i 0.684003i −0.939699 0.342002i \(-0.888895\pi\)
0.939699 0.342002i \(-0.111105\pi\)
\(912\) 0 0
\(913\) −2.57506e7 2.57506e7i −1.02237 1.02237i
\(914\) 0 0
\(915\) −282812. 2.81865e6i −0.0111672 0.111298i
\(916\) 0 0
\(917\) −1.95926e6 + 1.95926e6i −0.0769428 + 0.0769428i
\(918\) 0 0
\(919\) −1.74717e6 −0.0682411 −0.0341206 0.999418i \(-0.510863\pi\)
−0.0341206 + 0.999418i \(0.510863\pi\)
\(920\) 0 0
\(921\) −9.88636e6 −0.384050
\(922\) 0 0
\(923\) −2.81668e6 + 2.81668e6i −0.108826 + 0.108826i
\(924\) 0 0
\(925\) −4.06411e7 2.69413e7i −1.56175 1.03529i
\(926\) 0 0
\(927\) 9.53132e6 + 9.53132e6i 0.364296 + 0.364296i
\(928\) 0 0
\(929\) 3.80243e7i 1.44551i 0.691102 + 0.722757i \(0.257125\pi\)
−0.691102 + 0.722757i \(0.742875\pi\)
\(930\) 0 0
\(931\) 9.04887e6i 0.342153i
\(932\) 0 0
\(933\) −9.87226e6 9.87226e6i −0.371289 0.371289i
\(934\) 0 0
\(935\) 2.14084e7 2.14804e6i 0.800858 0.0803550i
\(936\) 0 0
\(937\) −1.05384e7 + 1.05384e7i −0.392127 + 0.392127i −0.875445 0.483318i \(-0.839432\pi\)
0.483318 + 0.875445i \(0.339432\pi\)
\(938\) 0 0
\(939\) 1.22588e6 0.0453717
\(940\) 0 0
\(941\) −4.44456e7 −1.63627 −0.818135 0.575026i \(-0.804992\pi\)
−0.818135 + 0.575026i \(0.804992\pi\)
\(942\) 0 0
\(943\) 1.75230e7 1.75230e7i 0.641697 0.641697i
\(944\) 0 0
\(945\) −788887. 645015.i −0.0287366 0.0234958i
\(946\) 0 0
\(947\) −1.69931e7 1.69931e7i −0.615740 0.615740i 0.328696 0.944436i \(-0.393391\pi\)
−0.944436 + 0.328696i \(0.893391\pi\)
\(948\) 0 0
\(949\) 7.73065e6i 0.278644i
\(950\) 0 0
\(951\) 1.77533e6i 0.0636544i
\(952\) 0 0
\(953\) 2.89702e7 + 2.89702e7i 1.03328 + 1.03328i 0.999427 + 0.0338563i \(0.0107788\pi\)
0.0338563 + 0.999427i \(0.489221\pi\)
\(954\) 0 0
\(955\) −1.79438e7 1.46713e7i −0.636656 0.520547i
\(956\) 0 0
\(957\) −4.05627e6 + 4.05627e6i −0.143168 + 0.143168i
\(958\) 0 0
\(959\) 821914. 0.0288589
\(960\) 0 0
\(961\) −3.56744e7 −1.24609
\(962\) 0 0
\(963\) 3.09760e7 3.09760e7i 1.07636 1.07636i
\(964\) 0 0
\(965\) 5.41592e7 5.43413e6i 1.87221 0.187850i
\(966\) 0 0
\(967\) −1.69845e7 1.69845e7i −0.584101 0.584101i 0.351927 0.936028i \(-0.385527\pi\)
−0.936028 + 0.351927i \(0.885527\pi\)
\(968\) 0 0
\(969\) 1.52838e6i 0.0522905i
\(970\) 0 0
\(971\) 4.58022e7i 1.55897i −0.626419 0.779486i \(-0.715480\pi\)
0.626419 0.779486i \(-0.284520\pi\)
\(972\) 0 0
\(973\) 2.13370e6 + 2.13370e6i 0.0722521 + 0.0722521i
\(974\) 0 0
\(975\) −925920. 4.56764e6i −0.0311934 0.153879i
\(976\) 0 0
\(977\) −1.14731e7 + 1.14731e7i −0.384544 + 0.384544i −0.872736 0.488192i \(-0.837657\pi\)
0.488192 + 0.872736i \(0.337657\pi\)
\(978\) 0 0
\(979\) −4.10718e7 −1.36958
\(980\) 0 0
\(981\) −2.14375e7 −0.711215
\(982\) 0 0
\(983\) −2.41644e7 + 2.41644e7i −0.797613 + 0.797613i −0.982719 0.185106i \(-0.940737\pi\)
0.185106 + 0.982719i \(0.440737\pi\)
\(984\) 0 0
\(985\) −5.39547e6 5.37739e7i −0.177190 1.76596i
\(986\) 0 0
\(987\) −567872. 567872.i −0.0185549 0.0185549i
\(988\) 0 0
\(989\) 2.83546e7i 0.921792i
\(990\) 0 0
\(991\) 1.55838e7i 0.504067i −0.967719 0.252033i \(-0.918901\pi\)
0.967719 0.252033i \(-0.0810993\pi\)
\(992\) 0 0
\(993\) 2.16164e6 + 2.16164e6i 0.0695680 + 0.0695680i
\(994\) 0 0
\(995\) −2.86239e7 + 3.50085e7i −0.916581 + 1.12103i
\(996\) 0 0
\(997\) 6.51999e6 6.51999e6i 0.207735 0.207735i −0.595569 0.803304i \(-0.703073\pi\)
0.803304 + 0.595569i \(0.203073\pi\)
\(998\) 0 0
\(999\) 3.37395e7 1.06961
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 160.6.n.d.63.5 yes 16
4.3 odd 2 160.6.n.c.63.4 16
5.2 odd 4 160.6.n.c.127.4 yes 16
20.7 even 4 inner 160.6.n.d.127.5 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.n.c.63.4 16 4.3 odd 2
160.6.n.c.127.4 yes 16 5.2 odd 4
160.6.n.d.63.5 yes 16 1.1 even 1 trivial
160.6.n.d.127.5 yes 16 20.7 even 4 inner