Properties

Label 324.5.k.a.89.2
Level $324$
Weight $5$
Character 324.89
Analytic conductor $33.492$
Analytic rank $0$
Dimension $72$
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] = [324,5,Mod(17,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.k (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 89.2
Character \(\chi\) \(=\) 324.89
Dual form 324.5.k.a.233.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-23.3523 - 27.8302i) q^{5} +(44.9114 - 16.3464i) q^{7} +O(q^{10})\) \(q+(-23.3523 - 27.8302i) q^{5} +(44.9114 - 16.3464i) q^{7} +(93.1464 - 111.008i) q^{11} +(41.5560 + 235.676i) q^{13} +(70.2567 - 40.5627i) q^{17} +(327.274 - 566.855i) q^{19} +(-117.142 + 321.845i) q^{23} +(-120.660 + 684.294i) q^{25} +(-1287.14 - 226.958i) q^{29} +(208.261 + 75.8006i) q^{31} +(-1503.71 - 868.166i) q^{35} +(-577.060 - 999.497i) q^{37} +(2496.16 - 440.141i) q^{41} +(-2375.48 - 1993.27i) q^{43} +(-989.606 - 2718.92i) q^{47} +(-89.4477 + 75.0556i) q^{49} +2001.37i q^{53} -5264.55 q^{55} +(-352.871 - 420.535i) q^{59} +(-2801.19 + 1019.55i) q^{61} +(5588.48 - 6660.09i) q^{65} +(-206.959 - 1173.72i) q^{67} +(3982.84 - 2299.50i) q^{71} +(750.689 - 1300.23i) q^{73} +(2368.76 - 6508.11i) q^{77} +(157.278 - 891.966i) q^{79} +(-2210.32 - 389.740i) q^{83} +(-2769.53 - 1008.02i) q^{85} +(3605.84 + 2081.83i) q^{89} +(5718.79 + 9905.23i) q^{91} +(-23418.3 + 4129.28i) q^{95} +(-12717.1 - 10670.9i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 9 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −23.3523 27.8302i −0.934092 1.11321i −0.993369 0.114966i \(-0.963324\pi\)
0.0592770 0.998242i \(-0.481120\pi\)
\(6\) 0 0
\(7\) 44.9114 16.3464i 0.916558 0.333600i 0.159690 0.987167i \(-0.448951\pi\)
0.756868 + 0.653567i \(0.226728\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 93.1464 111.008i 0.769805 0.917418i −0.228620 0.973516i \(-0.573421\pi\)
0.998425 + 0.0560975i \(0.0178658\pi\)
\(12\) 0 0
\(13\) 41.5560 + 235.676i 0.245894 + 1.39453i 0.818408 + 0.574637i \(0.194857\pi\)
−0.572515 + 0.819895i \(0.694032\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 70.2567 40.5627i 0.243103 0.140355i −0.373499 0.927631i \(-0.621842\pi\)
0.616602 + 0.787275i \(0.288509\pi\)
\(18\) 0 0
\(19\) 327.274 566.855i 0.906576 1.57024i 0.0877883 0.996139i \(-0.472020\pi\)
0.818788 0.574096i \(-0.194647\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −117.142 + 321.845i −0.221440 + 0.608403i −0.999812 0.0194015i \(-0.993824\pi\)
0.778371 + 0.627804i \(0.216046\pi\)
\(24\) 0 0
\(25\) −120.660 + 684.294i −0.193055 + 1.09487i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1287.14 226.958i −1.53049 0.269867i −0.655944 0.754809i \(-0.727729\pi\)
−0.874546 + 0.484942i \(0.838841\pi\)
\(30\) 0 0
\(31\) 208.261 + 75.8006i 0.216712 + 0.0788768i 0.448095 0.893986i \(-0.352103\pi\)
−0.231383 + 0.972863i \(0.574325\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1503.71 868.166i −1.22752 0.708707i
\(36\) 0 0
\(37\) −577.060 999.497i −0.421519 0.730093i 0.574569 0.818456i \(-0.305170\pi\)
−0.996088 + 0.0883635i \(0.971836\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2496.16 440.141i 1.48493 0.261833i 0.628382 0.777905i \(-0.283718\pi\)
0.856545 + 0.516072i \(0.172606\pi\)
\(42\) 0 0
\(43\) −2375.48 1993.27i −1.28474 1.07802i −0.992572 0.121657i \(-0.961179\pi\)
−0.292167 0.956367i \(-0.594376\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −989.606 2718.92i −0.447988 1.23084i −0.934122 0.356954i \(-0.883815\pi\)
0.486134 0.873884i \(-0.338407\pi\)
\(48\) 0 0
\(49\) −89.4477 + 75.0556i −0.0372544 + 0.0312601i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2001.37i 0.712485i 0.934394 + 0.356242i \(0.115942\pi\)
−0.934394 + 0.356242i \(0.884058\pi\)
\(54\) 0 0
\(55\) −5264.55 −1.74035
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −352.871 420.535i −0.101370 0.120809i 0.712974 0.701190i \(-0.247348\pi\)
−0.814344 + 0.580382i \(0.802903\pi\)
\(60\) 0 0
\(61\) −2801.19 + 1019.55i −0.752805 + 0.273998i −0.689785 0.724014i \(-0.742295\pi\)
−0.0630192 + 0.998012i \(0.520073\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5588.48 6660.09i 1.32272 1.57635i
\(66\) 0 0
\(67\) −206.959 1173.72i −0.0461035 0.261466i 0.953040 0.302844i \(-0.0979363\pi\)
−0.999144 + 0.0413782i \(0.986825\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3982.84 2299.50i 0.790090 0.456159i −0.0499044 0.998754i \(-0.515892\pi\)
0.839994 + 0.542595i \(0.182558\pi\)
\(72\) 0 0
\(73\) 750.689 1300.23i 0.140869 0.243992i −0.786955 0.617010i \(-0.788344\pi\)
0.927824 + 0.373018i \(0.121677\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2368.76 6508.11i 0.399521 1.09767i
\(78\) 0 0
\(79\) 157.278 891.966i 0.0252007 0.142920i −0.969611 0.244651i \(-0.921327\pi\)
0.994812 + 0.101730i \(0.0324379\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2210.32 389.740i −0.320848 0.0565742i 0.0109047 0.999941i \(-0.496529\pi\)
−0.331753 + 0.943366i \(0.607640\pi\)
\(84\) 0 0
\(85\) −2769.53 1008.02i −0.383325 0.139519i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3605.84 + 2081.83i 0.455226 + 0.262825i 0.710035 0.704167i \(-0.248679\pi\)
−0.254809 + 0.966991i \(0.582013\pi\)
\(90\) 0 0
\(91\) 5718.79 + 9905.23i 0.690592 + 1.19614i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −23418.3 + 4129.28i −2.59482 + 0.457538i
\(96\) 0 0
\(97\) −12717.1 10670.9i −1.35159 1.13412i −0.978483 0.206327i \(-0.933849\pi\)
−0.373104 0.927789i \(-0.621707\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1625.30 + 4465.47i 0.159327 + 0.437748i 0.993511 0.113739i \(-0.0362828\pi\)
−0.834183 + 0.551487i \(0.814061\pi\)
\(102\) 0 0
\(103\) 3110.55 2610.06i 0.293199 0.246023i −0.484308 0.874898i \(-0.660929\pi\)
0.777507 + 0.628874i \(0.216484\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13840.5i 1.20888i −0.796650 0.604441i \(-0.793397\pi\)
0.796650 0.604441i \(-0.206603\pi\)
\(108\) 0 0
\(109\) −20651.8 −1.73822 −0.869112 0.494616i \(-0.835309\pi\)
−0.869112 + 0.494616i \(0.835309\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6495.37 7740.88i −0.508682 0.606224i 0.449184 0.893439i \(-0.351715\pi\)
−0.957866 + 0.287215i \(0.907270\pi\)
\(114\) 0 0
\(115\) 11692.5 4255.74i 0.884125 0.321795i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2492.27 2970.17i 0.175995 0.209743i
\(120\) 0 0
\(121\) −1104.05 6261.36i −0.0754078 0.427659i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2197.67 1268.82i 0.140651 0.0812047i
\(126\) 0 0
\(127\) −2913.07 + 5045.59i −0.180611 + 0.312827i −0.942089 0.335364i \(-0.891141\pi\)
0.761478 + 0.648191i \(0.224474\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7585.77 + 20841.7i −0.442035 + 1.21448i 0.496116 + 0.868256i \(0.334759\pi\)
−0.938151 + 0.346226i \(0.887463\pi\)
\(132\) 0 0
\(133\) 5432.28 30808.0i 0.307099 1.74165i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −23386.2 4123.61i −1.24600 0.219703i −0.488515 0.872556i \(-0.662461\pi\)
−0.757485 + 0.652852i \(0.773572\pi\)
\(138\) 0 0
\(139\) 19862.6 + 7229.41i 1.02803 + 0.374174i 0.800332 0.599557i \(-0.204657\pi\)
0.227702 + 0.973731i \(0.426879\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 30032.6 + 17339.3i 1.46866 + 0.847931i
\(144\) 0 0
\(145\) 23741.5 + 41121.4i 1.12920 + 1.95583i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10205.6 1799.53i 0.459692 0.0810561i 0.0609936 0.998138i \(-0.480573\pi\)
0.398699 + 0.917082i \(0.369462\pi\)
\(150\) 0 0
\(151\) 33851.0 + 28404.4i 1.48463 + 1.24575i 0.901065 + 0.433684i \(0.142787\pi\)
0.583564 + 0.812067i \(0.301658\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2753.82 7566.05i −0.114623 0.314924i
\(156\) 0 0
\(157\) 8441.79 7083.50i 0.342480 0.287375i −0.455282 0.890347i \(-0.650462\pi\)
0.797762 + 0.602972i \(0.206017\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16369.3i 0.631509i
\(162\) 0 0
\(163\) −29240.4 −1.10055 −0.550273 0.834985i \(-0.685476\pi\)
−0.550273 + 0.834985i \(0.685476\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21160.9 + 25218.5i 0.758753 + 0.904247i 0.997768 0.0667690i \(-0.0212691\pi\)
−0.239015 + 0.971016i \(0.576825\pi\)
\(168\) 0 0
\(169\) −26977.7 + 9819.07i −0.944563 + 0.343793i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13402.4 15972.4i 0.447807 0.533675i −0.494165 0.869368i \(-0.664526\pi\)
0.941971 + 0.335693i \(0.108971\pi\)
\(174\) 0 0
\(175\) 5766.76 + 32704.9i 0.188302 + 1.06792i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10186.4 5881.11i 0.317917 0.183550i −0.332547 0.943087i \(-0.607908\pi\)
0.650464 + 0.759537i \(0.274575\pi\)
\(180\) 0 0
\(181\) 12812.8 22192.4i 0.391099 0.677403i −0.601496 0.798876i \(-0.705428\pi\)
0.992595 + 0.121473i \(0.0387618\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14340.5 + 39400.2i −0.419007 + 1.15121i
\(186\) 0 0
\(187\) 2041.39 11577.3i 0.0583772 0.331073i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 43133.9 + 7605.67i 1.18237 + 0.208483i 0.730062 0.683381i \(-0.239491\pi\)
0.452304 + 0.891864i \(0.350602\pi\)
\(192\) 0 0
\(193\) 42585.5 + 15499.8i 1.14326 + 0.416114i 0.843091 0.537770i \(-0.180733\pi\)
0.300173 + 0.953885i \(0.402955\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 41059.8 + 23705.9i 1.05800 + 0.610835i 0.924878 0.380265i \(-0.124167\pi\)
0.133120 + 0.991100i \(0.457500\pi\)
\(198\) 0 0
\(199\) −8841.99 15314.8i −0.223277 0.386727i 0.732524 0.680741i \(-0.238342\pi\)
−0.955801 + 0.294014i \(0.905009\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −61517.3 + 10847.2i −1.49281 + 0.263223i
\(204\) 0 0
\(205\) −70540.3 59190.4i −1.67853 1.40846i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −32440.8 89130.4i −0.742676 2.04049i
\(210\) 0 0
\(211\) 46294.0 38845.3i 1.03982 0.872515i 0.0478364 0.998855i \(-0.484767\pi\)
0.991987 + 0.126340i \(0.0403229\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 112658.i 2.43716i
\(216\) 0 0
\(217\) 10592.3 0.224943
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12479.2 + 14872.2i 0.255508 + 0.304502i
\(222\) 0 0
\(223\) −14524.0 + 5286.30i −0.292063 + 0.106302i −0.483896 0.875125i \(-0.660779\pi\)
0.191834 + 0.981427i \(0.438557\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21204.1 + 25270.1i −0.411499 + 0.490405i −0.931490 0.363766i \(-0.881491\pi\)
0.519991 + 0.854172i \(0.325935\pi\)
\(228\) 0 0
\(229\) 11374.1 + 64505.7i 0.216893 + 1.23006i 0.877591 + 0.479410i \(0.159149\pi\)
−0.660698 + 0.750652i \(0.729740\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10438.5 + 6026.66i −0.192276 + 0.111011i −0.593048 0.805167i \(-0.702075\pi\)
0.400772 + 0.916178i \(0.368742\pi\)
\(234\) 0 0
\(235\) −52558.5 + 91034.0i −0.951716 + 1.64842i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2021.25 5553.34i 0.0353854 0.0972207i −0.920742 0.390173i \(-0.872415\pi\)
0.956127 + 0.292952i \(0.0946376\pi\)
\(240\) 0 0
\(241\) 2803.01 15896.7i 0.0482604 0.273698i −0.951123 0.308813i \(-0.900068\pi\)
0.999383 + 0.0351143i \(0.0111795\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4177.62 + 736.628i 0.0695980 + 0.0122720i
\(246\) 0 0
\(247\) 147194. + 53574.3i 2.41266 + 0.878138i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −39754.0 22952.0i −0.631006 0.364311i 0.150136 0.988665i \(-0.452029\pi\)
−0.781141 + 0.624354i \(0.785362\pi\)
\(252\) 0 0
\(253\) 24815.9 + 42982.4i 0.387694 + 0.671505i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −86522.3 + 15256.2i −1.30997 + 0.230983i −0.784660 0.619926i \(-0.787163\pi\)
−0.525312 + 0.850910i \(0.676051\pi\)
\(258\) 0 0
\(259\) −42254.7 35455.9i −0.629906 0.528554i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5894.23 16194.3i −0.0852149 0.234126i 0.889765 0.456419i \(-0.150868\pi\)
−0.974980 + 0.222293i \(0.928646\pi\)
\(264\) 0 0
\(265\) 55698.5 46736.6i 0.793144 0.665526i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21865.7i 0.302176i −0.988520 0.151088i \(-0.951722\pi\)
0.988520 0.151088i \(-0.0482776\pi\)
\(270\) 0 0
\(271\) 57620.5 0.784582 0.392291 0.919841i \(-0.371683\pi\)
0.392291 + 0.919841i \(0.371683\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 64722.8 + 77133.7i 0.855839 + 1.01995i
\(276\) 0 0
\(277\) 89309.9 32506.1i 1.16396 0.423649i 0.313453 0.949604i \(-0.398514\pi\)
0.850512 + 0.525955i \(0.176292\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 58640.0 69884.4i 0.742645 0.885049i −0.253974 0.967211i \(-0.581738\pi\)
0.996619 + 0.0821616i \(0.0261824\pi\)
\(282\) 0 0
\(283\) −22694.5 128707.i −0.283366 1.60705i −0.711065 0.703127i \(-0.751787\pi\)
0.427699 0.903921i \(-0.359324\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 104911. 60570.6i 1.27367 0.735356i
\(288\) 0 0
\(289\) −38469.8 + 66631.7i −0.460601 + 0.797784i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 40038.7 110006.i 0.466386 1.28138i −0.454220 0.890890i \(-0.650082\pi\)
0.920605 0.390494i \(-0.127696\pi\)
\(294\) 0 0
\(295\) −3463.22 + 19640.9i −0.0397957 + 0.225693i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −80719.1 14232.9i −0.902888 0.159203i
\(300\) 0 0
\(301\) −139269. 50689.7i −1.53717 0.559483i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 93788.4 + 54148.8i 1.00821 + 0.582088i
\(306\) 0 0
\(307\) −26734.0 46304.7i −0.283653 0.491302i 0.688628 0.725114i \(-0.258213\pi\)
−0.972282 + 0.233812i \(0.924880\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −140065. + 24697.2i −1.44813 + 0.255345i −0.841768 0.539840i \(-0.818485\pi\)
−0.606367 + 0.795185i \(0.707374\pi\)
\(312\) 0 0
\(313\) 1973.39 + 1655.87i 0.0201430 + 0.0169019i 0.652804 0.757527i \(-0.273593\pi\)
−0.632661 + 0.774429i \(0.718037\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −31672.4 87019.3i −0.315183 0.865959i −0.991589 0.129429i \(-0.958686\pi\)
0.676406 0.736529i \(-0.263537\pi\)
\(318\) 0 0
\(319\) −145087. + 121742.i −1.42576 + 1.19636i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 53100.5i 0.508972i
\(324\) 0 0
\(325\) −166286. −1.57430
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −88889.1 105934.i −0.821215 0.978686i
\(330\) 0 0
\(331\) 40376.6 14695.9i 0.368531 0.134134i −0.151115 0.988516i \(-0.548286\pi\)
0.519645 + 0.854382i \(0.326064\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27831.9 + 33168.8i −0.248001 + 0.295556i
\(336\) 0 0
\(337\) 5958.86 + 33794.4i 0.0524691 + 0.297567i 0.999738 0.0228716i \(-0.00728088\pi\)
−0.947269 + 0.320438i \(0.896170\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 27813.2 16057.9i 0.239189 0.138096i
\(342\) 0 0
\(343\) −60166.6 + 104212.i −0.511408 + 0.885784i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15708.7 + 43159.3i −0.130461 + 0.358439i −0.987674 0.156523i \(-0.949972\pi\)
0.857213 + 0.514962i \(0.172194\pi\)
\(348\) 0 0
\(349\) 25659.0 145519.i 0.210663 1.19473i −0.677612 0.735420i \(-0.736985\pi\)
0.888275 0.459312i \(-0.151904\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 150996. + 26624.7i 1.21176 + 0.213666i 0.742775 0.669541i \(-0.233509\pi\)
0.468984 + 0.883207i \(0.344620\pi\)
\(354\) 0 0
\(355\) −157004. 57144.8i −1.24582 0.453440i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1793.35 + 1035.39i 0.0139147 + 0.00803368i 0.506941 0.861981i \(-0.330776\pi\)
−0.493026 + 0.870014i \(0.664109\pi\)
\(360\) 0 0
\(361\) −149056. 258172.i −1.14376 1.98105i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −53716.0 + 9471.58i −0.403198 + 0.0710946i
\(366\) 0 0
\(367\) −61804.0 51859.7i −0.458864 0.385033i 0.383849 0.923396i \(-0.374598\pi\)
−0.842713 + 0.538363i \(0.819043\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 32715.2 + 89884.2i 0.237685 + 0.653034i
\(372\) 0 0
\(373\) 94740.5 79496.7i 0.680954 0.571388i −0.235331 0.971915i \(-0.575617\pi\)
0.916285 + 0.400527i \(0.131173\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 312780.i 2.20068i
\(378\) 0 0
\(379\) 253183. 1.76261 0.881306 0.472546i \(-0.156665\pi\)
0.881306 + 0.472546i \(0.156665\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 62618.3 + 74625.6i 0.426878 + 0.508733i 0.936019 0.351950i \(-0.114481\pi\)
−0.509141 + 0.860683i \(0.670037\pi\)
\(384\) 0 0
\(385\) −236438. + 86056.4i −1.59513 + 0.580579i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −27505.9 + 32780.3i −0.181772 + 0.216627i −0.849234 0.528016i \(-0.822936\pi\)
0.667462 + 0.744644i \(0.267381\pi\)
\(390\) 0 0
\(391\) 4824.90 + 27363.4i 0.0315598 + 0.178985i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −28496.4 + 16452.4i −0.182640 + 0.105447i
\(396\) 0 0
\(397\) −88498.8 + 153284.i −0.561509 + 0.972562i 0.435856 + 0.900016i \(0.356446\pi\)
−0.997365 + 0.0725454i \(0.976888\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 66578.7 182923.i 0.414044 1.13758i −0.540977 0.841037i \(-0.681945\pi\)
0.955021 0.296539i \(-0.0958324\pi\)
\(402\) 0 0
\(403\) −9209.90 + 52232.0i −0.0567081 + 0.321608i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −164703. 29041.5i −0.994288 0.175320i
\(408\) 0 0
\(409\) 12003.2 + 4368.82i 0.0717548 + 0.0261166i 0.377648 0.925949i \(-0.376733\pi\)
−0.305893 + 0.952066i \(0.598955\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −22722.1 13118.6i −0.133214 0.0769110i
\(414\) 0 0
\(415\) 40769.6 + 70615.0i 0.236723 + 0.410016i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 193351. 34093.0i 1.10133 0.194195i 0.406702 0.913561i \(-0.366679\pi\)
0.694631 + 0.719366i \(0.255568\pi\)
\(420\) 0 0
\(421\) −14748.7 12375.7i −0.0832129 0.0698239i 0.600232 0.799826i \(-0.295075\pi\)
−0.683445 + 0.730002i \(0.739519\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 19279.7 + 52970.5i 0.106739 + 0.293262i
\(426\) 0 0
\(427\) −109139. + 91578.6i −0.598583 + 0.502271i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 37209.8i 0.200310i 0.994972 + 0.100155i \(0.0319339\pi\)
−0.994972 + 0.100155i \(0.968066\pi\)
\(432\) 0 0
\(433\) 120422. 0.642289 0.321145 0.947030i \(-0.395932\pi\)
0.321145 + 0.947030i \(0.395932\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 144102. + 171734.i 0.754583 + 0.899277i
\(438\) 0 0
\(439\) 72962.8 26556.3i 0.378593 0.137797i −0.145712 0.989327i \(-0.546547\pi\)
0.524305 + 0.851530i \(0.324325\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 144700. 172446.i 0.737327 0.878712i −0.258864 0.965914i \(-0.583348\pi\)
0.996191 + 0.0872021i \(0.0277926\pi\)
\(444\) 0 0
\(445\) −26266.9 148967.i −0.132644 0.752264i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −236745. + 136685.i −1.17432 + 0.677995i −0.954694 0.297588i \(-0.903818\pi\)
−0.219628 + 0.975584i \(0.570484\pi\)
\(450\) 0 0
\(451\) 183650. 318090.i 0.902894 1.56386i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 142118. 390465.i 0.686476 1.88608i
\(456\) 0 0
\(457\) −35070.4 + 198894.i −0.167922 + 0.952334i 0.778078 + 0.628168i \(0.216195\pi\)
−0.946000 + 0.324166i \(0.894916\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −96757.1 17060.9i −0.455282 0.0802786i −0.0586958 0.998276i \(-0.518694\pi\)
−0.396586 + 0.917997i \(0.629805\pi\)
\(462\) 0 0
\(463\) 368771. + 134222.i 1.72026 + 0.626124i 0.997862 0.0653505i \(-0.0208165\pi\)
0.722400 + 0.691475i \(0.243039\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −158722. 91638.0i −0.727784 0.420186i 0.0898270 0.995957i \(-0.471369\pi\)
−0.817611 + 0.575771i \(0.804702\pi\)
\(468\) 0 0
\(469\) −28480.9 49330.4i −0.129482 0.224269i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −442536. + 78031.0i −1.97800 + 0.348774i
\(474\) 0 0
\(475\) 348407. + 292348.i 1.54419 + 1.29573i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15394.6 + 42296.4i 0.0670962 + 0.184345i 0.968709 0.248198i \(-0.0798384\pi\)
−0.901613 + 0.432544i \(0.857616\pi\)
\(480\) 0 0
\(481\) 211577. 177534.i 0.914488 0.767347i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 603109.i 2.56397i
\(486\) 0 0
\(487\) −89812.6 −0.378686 −0.189343 0.981911i \(-0.560636\pi\)
−0.189343 + 0.981911i \(0.560636\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 193660. + 230794.i 0.803296 + 0.957331i 0.999731 0.0231855i \(-0.00738084\pi\)
−0.196435 + 0.980517i \(0.562936\pi\)
\(492\) 0 0
\(493\) −99636.4 + 36264.7i −0.409944 + 0.149207i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 141286. 168379.i 0.571989 0.681670i
\(498\) 0 0
\(499\) 46801.6 + 265425.i 0.187958 + 1.06596i 0.922096 + 0.386961i \(0.126475\pi\)
−0.734138 + 0.679000i \(0.762414\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −330182. + 190631.i −1.30502 + 0.753454i −0.981261 0.192685i \(-0.938280\pi\)
−0.323760 + 0.946139i \(0.604947\pi\)
\(504\) 0 0
\(505\) 86320.4 149511.i 0.338478 0.586261i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 83468.2 229327.i 0.322170 0.885156i −0.667858 0.744289i \(-0.732789\pi\)
0.990028 0.140867i \(-0.0449891\pi\)
\(510\) 0 0
\(511\) 12460.4 70666.2i 0.0477187 0.270626i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −145277. 25616.3i −0.547750 0.0965831i
\(516\) 0 0
\(517\) −393999. 143404.i −1.47406 0.536513i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −210341. 121440.i −0.774904 0.447391i 0.0597174 0.998215i \(-0.480980\pi\)
−0.834621 + 0.550824i \(0.814313\pi\)
\(522\) 0 0
\(523\) −25165.4 43587.7i −0.0920027 0.159353i 0.816351 0.577556i \(-0.195993\pi\)
−0.908354 + 0.418203i \(0.862660\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17706.4 3122.11i 0.0637542 0.0112416i
\(528\) 0 0
\(529\) 124509. + 104475.i 0.444927 + 0.373338i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 207461. + 569995.i 0.730268 + 2.00639i
\(534\) 0 0
\(535\) −385184. + 323207.i −1.34574 + 1.12921i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 16920.5i 0.0582420i
\(540\) 0 0
\(541\) 223428. 0.763383 0.381692 0.924290i \(-0.375342\pi\)
0.381692 + 0.924290i \(0.375342\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 482268. + 574745.i 1.62366 + 1.93500i
\(546\) 0 0
\(547\) −32840.0 + 11952.8i −0.109756 + 0.0399480i −0.396314 0.918115i \(-0.629711\pi\)
0.286558 + 0.958063i \(0.407489\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −549900. + 655346.i −1.81126 + 2.15858i
\(552\) 0 0
\(553\) −7516.87 42630.3i −0.0245803 0.139402i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 342072. 197495.i 1.10257 0.636571i 0.165677 0.986180i \(-0.447019\pi\)
0.936896 + 0.349609i \(0.113686\pi\)
\(558\) 0 0
\(559\) 371049. 642676.i 1.18743 2.05669i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 88564.9 243330.i 0.279412 0.767678i −0.718018 0.696025i \(-0.754950\pi\)
0.997430 0.0716530i \(-0.0228274\pi\)
\(564\) 0 0
\(565\) −63748.3 + 361535.i −0.199697 + 1.13254i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 433798. + 76490.4i 1.33987 + 0.236256i 0.797213 0.603698i \(-0.206307\pi\)
0.542659 + 0.839953i \(0.317418\pi\)
\(570\) 0 0
\(571\) −320339. 116594.i −0.982511 0.357605i −0.199695 0.979858i \(-0.563995\pi\)
−0.782816 + 0.622253i \(0.786217\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −206102. 118993.i −0.623372 0.359904i
\(576\) 0 0
\(577\) 315660. + 546740.i 0.948132 + 1.64221i 0.749356 + 0.662167i \(0.230363\pi\)
0.198775 + 0.980045i \(0.436304\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −105639. + 18627.1i −0.312949 + 0.0551814i
\(582\) 0 0
\(583\) 222167. + 186420.i 0.653646 + 0.548474i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22503.9 + 61828.9i 0.0653102 + 0.179438i 0.968055 0.250738i \(-0.0806734\pi\)
−0.902745 + 0.430177i \(0.858451\pi\)
\(588\) 0 0
\(589\) 111126. 93246.0i 0.320321 0.268782i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 660484.i 1.87825i −0.343579 0.939124i \(-0.611639\pi\)
0.343579 0.939124i \(-0.388361\pi\)
\(594\) 0 0
\(595\) −140861. −0.397883
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 249715. + 297598.i 0.695969 + 0.829424i 0.992064 0.125735i \(-0.0401288\pi\)
−0.296094 + 0.955159i \(0.595684\pi\)
\(600\) 0 0
\(601\) 308372. 112238.i 0.853740 0.310736i 0.122176 0.992508i \(-0.461013\pi\)
0.731564 + 0.681772i \(0.238791\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −148473. + 176943.i −0.405636 + 0.483418i
\(606\) 0 0
\(607\) −42611.4 241662.i −0.115651 0.655889i −0.986426 0.164207i \(-0.947493\pi\)
0.870775 0.491682i \(-0.163618\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 599660. 346214.i 1.60629 0.927389i
\(612\) 0 0
\(613\) −60158.0 + 104197.i −0.160093 + 0.277289i −0.934902 0.354906i \(-0.884513\pi\)
0.774809 + 0.632196i \(0.217846\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 213630. 586942.i 0.561166 1.54179i −0.256765 0.966474i \(-0.582657\pi\)
0.817931 0.575316i \(-0.195121\pi\)
\(618\) 0 0
\(619\) −45927.9 + 260470.i −0.119866 + 0.679793i 0.864360 + 0.502874i \(0.167724\pi\)
−0.984225 + 0.176919i \(0.943387\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 195974. + 34555.5i 0.504919 + 0.0890309i
\(624\) 0 0
\(625\) 321459. + 117001.i 0.822935 + 0.299524i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −81084.6 46814.2i −0.204945 0.118325i
\(630\) 0 0
\(631\) −129586. 224449.i −0.325461 0.563715i 0.656145 0.754635i \(-0.272186\pi\)
−0.981606 + 0.190921i \(0.938853\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 208447. 36754.8i 0.516949 0.0911520i
\(636\) 0 0
\(637\) −21405.9 17961.7i −0.0527539 0.0442657i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 247047. + 678755.i 0.601261 + 1.65195i 0.748723 + 0.662882i \(0.230667\pi\)
−0.147463 + 0.989068i \(0.547111\pi\)
\(642\) 0 0
\(643\) −514473. + 431694.i −1.24435 + 1.04413i −0.247174 + 0.968971i \(0.579502\pi\)
−0.997172 + 0.0751588i \(0.976054\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 161609.i 0.386062i −0.981193 0.193031i \(-0.938168\pi\)
0.981193 0.193031i \(-0.0618318\pi\)
\(648\) 0 0
\(649\) −79551.2 −0.188868
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −474226. 565160.i −1.11214 1.32540i −0.940330 0.340265i \(-0.889483\pi\)
−0.171809 0.985130i \(-0.554961\pi\)
\(654\) 0 0
\(655\) 757175. 275589.i 1.76487 0.642361i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −202725. + 241599.i −0.466807 + 0.556319i −0.947162 0.320755i \(-0.896063\pi\)
0.480355 + 0.877074i \(0.340508\pi\)
\(660\) 0 0
\(661\) 50722.9 + 287664.i 0.116092 + 0.658388i 0.986204 + 0.165535i \(0.0529351\pi\)
−0.870112 + 0.492854i \(0.835954\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −984248. + 568256.i −2.22567 + 1.28499i
\(666\) 0 0
\(667\) 223824. 387674.i 0.503100 0.871395i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −147743. + 405920.i −0.328142 + 0.901562i
\(672\) 0 0
\(673\) −9336.70 + 52951.1i −0.0206141 + 0.116908i −0.993378 0.114888i \(-0.963349\pi\)
0.972764 + 0.231796i \(0.0744602\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 336350. + 59307.5i 0.733860 + 0.129399i 0.528075 0.849198i \(-0.322914\pi\)
0.205785 + 0.978597i \(0.434025\pi\)
\(678\) 0 0
\(679\) −745572. 271366.i −1.61715 0.588594i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 195201. + 112699.i 0.418446 + 0.241590i 0.694412 0.719577i \(-0.255664\pi\)
−0.275966 + 0.961167i \(0.588998\pi\)
\(684\) 0 0
\(685\) 431360. + 747138.i 0.919304 + 1.59228i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −471674. + 83168.9i −0.993583 + 0.175195i
\(690\) 0 0
\(691\) 35786.7 + 30028.6i 0.0749489 + 0.0628896i 0.679492 0.733682i \(-0.262200\pi\)
−0.604544 + 0.796572i \(0.706644\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −262642. 721604.i −0.543745 1.49393i
\(696\) 0 0
\(697\) 157519. 132174.i 0.324240 0.272070i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 660244.i 1.34360i 0.740735 + 0.671798i \(0.234478\pi\)
−0.740735 + 0.671798i \(0.765522\pi\)
\(702\) 0 0
\(703\) −755426. −1.52856
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 145989. + 173982.i 0.292065 + 0.348070i
\(708\) 0 0
\(709\) 277434. 100978.i 0.551910 0.200879i −0.0509851 0.998699i \(-0.516236\pi\)
0.602895 + 0.797821i \(0.294014\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −48792.1 + 58148.2i −0.0959777 + 0.114382i
\(714\) 0 0
\(715\) −218774. 1.24073e6i −0.427940 2.42697i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 152320. 87942.1i 0.294646 0.170114i −0.345389 0.938459i \(-0.612253\pi\)
0.640035 + 0.768346i \(0.278920\pi\)
\(720\) 0 0
\(721\) 97033.9 168068.i 0.186661 0.323306i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 310612. 853399.i 0.590938 1.62359i
\(726\) 0 0
\(727\) −104736. + 593990.i −0.198166 + 1.12385i 0.709672 + 0.704532i \(0.248843\pi\)
−0.907838 + 0.419322i \(0.862268\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −247746. 43684.3i −0.463630 0.0817505i
\(732\) 0 0
\(733\) −559477. 203633.i −1.04130 0.379001i −0.235925 0.971771i \(-0.575812\pi\)
−0.805372 + 0.592770i \(0.798034\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −149570. 86354.0i −0.275365 0.158982i
\(738\) 0 0
\(739\) −224309. 388515.i −0.410731 0.711408i 0.584238 0.811582i \(-0.301393\pi\)
−0.994970 + 0.100174i \(0.968060\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −847422. + 149423.i −1.53505 + 0.270670i −0.876327 0.481717i \(-0.840013\pi\)
−0.658721 + 0.752387i \(0.728902\pi\)
\(744\) 0 0
\(745\) −288406. 242001.i −0.519627 0.436019i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −226242. 621595.i −0.403283 1.10801i
\(750\) 0 0
\(751\) 121600. 102035.i 0.215603 0.180912i −0.528590 0.848878i \(-0.677279\pi\)
0.744193 + 0.667965i \(0.232835\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.60539e6i 2.81635i
\(756\) 0 0
\(757\) 782504. 1.36551 0.682755 0.730648i \(-0.260782\pi\)
0.682755 + 0.730648i \(0.260782\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 388634. + 463156.i 0.671075 + 0.799756i 0.988930 0.148383i \(-0.0474068\pi\)
−0.317855 + 0.948139i \(0.602962\pi\)
\(762\) 0 0
\(763\) −927502. + 337583.i −1.59318 + 0.579871i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 84446.0 100639.i 0.143545 0.171070i
\(768\) 0 0
\(769\) 138375. + 784764.i 0.233994 + 1.32705i 0.844722 + 0.535206i \(0.179766\pi\)
−0.610727 + 0.791841i \(0.709123\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −816640. + 471487.i −1.36670 + 0.789062i −0.990505 0.137480i \(-0.956100\pi\)
−0.376191 + 0.926542i \(0.622766\pi\)
\(774\) 0 0
\(775\) −76998.5 + 133365.i −0.128197 + 0.222044i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 567433. 1.55901e6i 0.935060 2.56906i
\(780\) 0 0
\(781\) 115726. 656316.i 0.189727 1.07600i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −394271. 69520.6i −0.639816 0.112817i
\(786\) 0 0
\(787\) 563750. + 205188.i 0.910201 + 0.331286i 0.754333 0.656492i \(-0.227960\pi\)
0.155868 + 0.987778i \(0.450183\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −418251. 241477.i −0.668473 0.385943i
\(792\) 0 0
\(793\) −356689. 617804.i −0.567209 0.982436i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −217709. + 38387.9i −0.342736 + 0.0604335i −0.342367 0.939566i \(-0.611229\pi\)
−0.000368865 1.00000i \(0.500117\pi\)
\(798\) 0 0
\(799\) −179813. 150881.i −0.281662 0.236343i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −74411.6 204444.i −0.115401 0.317061i
\(804\) 0 0
\(805\) 455562. 382262.i 0.703001 0.589888i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 990055.i 1.51273i 0.654148 + 0.756367i \(0.273027\pi\)
−0.654148 + 0.756367i \(0.726973\pi\)
\(810\) 0 0
\(811\) −453692. −0.689794 −0.344897 0.938640i \(-0.612086\pi\)
−0.344897 + 0.938640i \(0.612086\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 682831. + 813766.i 1.02801 + 1.22514i
\(816\) 0 0
\(817\) −1.90733e6 + 694210.i −2.85747 + 1.04003i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 672797. 801809.i 0.998155 1.18955i 0.0163106 0.999867i \(-0.494808\pi\)
0.981844 0.189688i \(-0.0607476\pi\)
\(822\) 0 0
\(823\) 98469.1 + 558446.i 0.145379 + 0.824483i 0.967062 + 0.254539i \(0.0819238\pi\)
−0.821684 + 0.569944i \(0.806965\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 624640. 360636.i 0.913312 0.527301i 0.0318165 0.999494i \(-0.489871\pi\)
0.881495 + 0.472193i \(0.156537\pi\)
\(828\) 0 0
\(829\) −149992. + 259793.i −0.218252 + 0.378023i −0.954274 0.298935i \(-0.903369\pi\)
0.736022 + 0.676958i \(0.236702\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3239.85 + 8901.40i −0.00466911 + 0.0128283i
\(834\) 0 0
\(835\) 207682. 1.17782e6i 0.297869 1.68930i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.35688e6 + 239255.i 1.92761 + 0.339889i 0.999476 0.0323674i \(-0.0103047\pi\)
0.928130 + 0.372256i \(0.121416\pi\)
\(840\) 0 0
\(841\) 940599. + 342350.i 1.32988 + 0.484037i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 903257. + 521496.i 1.26502 + 0.730361i
\(846\) 0 0
\(847\) −151935. 263159.i −0.211783 0.366818i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 389281. 68640.7i 0.537532 0.0947813i
\(852\) 0 0
\(853\) −15257.8 12802.8i −0.0209697 0.0175957i 0.632243 0.774771i \(-0.282135\pi\)
−0.653212 + 0.757175i \(0.726579\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17076.8 + 46918.1i 0.0232512 + 0.0638820i 0.950775 0.309881i \(-0.100289\pi\)
−0.927524 + 0.373764i \(0.878067\pi\)
\(858\) 0 0
\(859\) −435201. + 365177.i −0.589798 + 0.494899i −0.888148 0.459557i \(-0.848008\pi\)
0.298350 + 0.954456i \(0.403564\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 652821.i 0.876542i −0.898843 0.438271i \(-0.855591\pi\)
0.898843 0.438271i \(-0.144409\pi\)
\(864\) 0 0
\(865\) −757491. −1.01238
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −84365.1 100542.i −0.111718 0.133140i
\(870\) 0 0
\(871\) 268018. 97550.4i 0.353286 0.128586i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 77959.5 92908.5i 0.101825 0.121350i
\(876\) 0 0
\(877\) 7936.40 + 45009.5i 0.0103187 + 0.0585202i 0.989532 0.144311i \(-0.0460967\pi\)
−0.979214 + 0.202832i \(0.934986\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 840898. 485493.i 1.08341 0.625505i 0.151593 0.988443i \(-0.451560\pi\)
0.931813 + 0.362938i \(0.118226\pi\)
\(882\) 0 0
\(883\) −3610.50 + 6253.58i −0.00463070 + 0.00802061i −0.868331 0.495984i \(-0.834807\pi\)
0.863701 + 0.504005i \(0.168141\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10779.2 + 29615.6i −0.0137006 + 0.0376420i −0.946354 0.323131i \(-0.895265\pi\)
0.932654 + 0.360773i \(0.117487\pi\)
\(888\) 0 0
\(889\) −48352.8 + 274222.i −0.0611812 + 0.346976i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.86511e6 328869.i −2.33884 0.412401i
\(894\) 0 0
\(895\) −401548. 146152.i −0.501293 0.182456i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −250857. 144833.i −0.310390 0.179204i
\(900\) 0 0
\(901\) 81181.0 + 140610.i 0.100001 + 0.173207i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −916827. + 161661.i −1.11941 + 0.197383i
\(906\) 0 0
\(907\) −205539. 172468.i −0.249850 0.209649i 0.509258 0.860614i \(-0.329920\pi\)
−0.759108 + 0.650965i \(0.774365\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −71258.5 195781.i −0.0858618 0.235903i 0.889330 0.457267i \(-0.151172\pi\)
−0.975191 + 0.221363i \(0.928949\pi\)
\(912\) 0 0
\(913\) −249148. + 209060.i −0.298893 + 0.250801i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.06003e6i 1.26061i
\(918\) 0 0
\(919\) 700477. 0.829398 0.414699 0.909959i \(-0.363887\pi\)
0.414699 + 0.909959i \(0.363887\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 707447. + 843102.i 0.830406 + 0.989639i
\(924\) 0 0
\(925\) 753577. 274280.i 0.880733 0.320561i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 260643. 310623.i 0.302006 0.359916i −0.593604 0.804757i \(-0.702295\pi\)
0.895610 + 0.444841i \(0.146740\pi\)
\(930\) 0 0
\(931\) 13271.7 + 75267.6i 0.0153118 + 0.0868378i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −369870. + 213544.i −0.423083 + 0.244267i
\(936\) 0 0
\(937\) −234491. + 406149.i −0.267083 + 0.462601i −0.968107 0.250536i \(-0.919393\pi\)
0.701024 + 0.713137i \(0.252726\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −336340. + 924087.i −0.379839 + 1.04360i 0.591584 + 0.806243i \(0.298503\pi\)
−0.971423 + 0.237355i \(0.923719\pi\)
\(942\) 0 0
\(943\) −150748. + 854936.i −0.169523 + 0.961414i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −720558. 127054.i −0.803469 0.141673i −0.243192 0.969978i \(-0.578194\pi\)
−0.560277 + 0.828305i \(0.689305\pi\)
\(948\) 0 0
\(949\) 337629. + 122887.i 0.374893 + 0.136450i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −408433. 235809.i −0.449713 0.259642i 0.257996 0.966146i \(-0.416938\pi\)
−0.707709 + 0.706504i \(0.750271\pi\)
\(954\) 0 0
\(955\) −795609. 1.37804e6i −0.872354 1.51096i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.11771e6 + 197083.i −1.21532 + 0.214295i
\(960\) 0 0
\(961\) −669831. 562055.i −0.725302 0.608600i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −563105. 1.54712e6i −0.604693 1.66138i
\(966\) 0 0
\(967\) 239642. 201084.i 0.256278 0.215042i −0.505592 0.862773i \(-0.668726\pi\)
0.761870 + 0.647730i \(0.224282\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 299653.i 0.317819i 0.987293 + 0.158909i \(0.0507978\pi\)
−0.987293 + 0.158909i \(0.949202\pi\)
\(972\) 0 0
\(973\) 1.01023e6 1.06708
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −246865. 294202.i −0.258625 0.308217i 0.621070 0.783755i \(-0.286698\pi\)
−0.879695 + 0.475537i \(0.842254\pi\)
\(978\) 0 0
\(979\) 566971. 206361.i 0.591555 0.215309i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −291778. + 347728.i −0.301957 + 0.359859i −0.895592 0.444876i \(-0.853248\pi\)
0.593635 + 0.804734i \(0.297692\pi\)
\(984\) 0 0
\(985\) −299102. 1.69629e6i −0.308281 1.74835i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 919792. 531042.i 0.940366 0.542921i
\(990\) 0 0
\(991\) −931055. + 1.61263e6i −0.948043 + 1.64206i −0.198502 + 0.980100i \(0.563608\pi\)
−0.749541 + 0.661958i \(0.769726\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −219732. + 603710.i −0.221946 + 0.609792i
\(996\) 0 0
\(997\) 206276. 1.16985e6i 0.207519 1.17690i −0.685907 0.727689i \(-0.740594\pi\)
0.893426 0.449210i \(-0.148295\pi\)
\(998\) 0 0
\(999\) 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 324.5.k.a.89.2 72
3.2 odd 2 108.5.k.a.29.10 72
27.13 even 9 108.5.k.a.41.10 yes 72
27.14 odd 18 inner 324.5.k.a.233.2 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.5.k.a.29.10 72 3.2 odd 2
108.5.k.a.41.10 yes 72 27.13 even 9
324.5.k.a.89.2 72 1.1 even 1 trivial
324.5.k.a.233.2 72 27.14 odd 18 inner