Properties

Label 384.6.c.d.383.14
Level $384$
Weight $6$
Character 384.383
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,6,Mod(383,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.c (of order \(2\), degree \(1\), 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: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + 18093172325 x^{8} + 74958811500 x^{6} + 79355888475 x^{4} + \cdots + 2870280625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{14}\cdot 41^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.14
Root \(-9.62962i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.6.c.d.383.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.16061 + 14.7095i) q^{3} +48.6693i q^{5} +167.754i q^{7} +(-189.736 + 151.820i) q^{9} +O(q^{10})\) \(q+(5.16061 + 14.7095i) q^{3} +48.6693i q^{5} +167.754i q^{7} +(-189.736 + 151.820i) q^{9} +319.315 q^{11} -152.547 q^{13} +(-715.899 + 251.163i) q^{15} +1323.90i q^{17} +1617.20i q^{19} +(-2467.58 + 865.716i) q^{21} +1222.40 q^{23} +756.298 q^{25} +(-3212.34 - 2007.43i) q^{27} +4549.54i q^{29} +5353.25i q^{31} +(1647.86 + 4696.95i) q^{33} -8164.49 q^{35} -3269.34 q^{37} +(-787.239 - 2243.89i) q^{39} -13077.7i q^{41} -20435.5i q^{43} +(-7388.96 - 9234.33i) q^{45} +20059.0 q^{47} -11334.6 q^{49} +(-19473.8 + 6832.12i) q^{51} +33272.4i q^{53} +15540.9i q^{55} +(-23788.2 + 8345.77i) q^{57} +5883.70 q^{59} +22967.3 q^{61} +(-25468.4 - 31829.1i) q^{63} -7424.38i q^{65} -13938.9i q^{67} +(6308.32 + 17980.8i) q^{69} -65642.7 q^{71} +55673.0 q^{73} +(3902.96 + 11124.7i) q^{75} +53566.6i q^{77} -82204.2i q^{79} +(12950.6 - 57611.3i) q^{81} -19118.5 q^{83} -64433.2 q^{85} +(-66921.3 + 23478.4i) q^{87} -97061.0i q^{89} -25590.5i q^{91} +(-78743.3 + 27626.0i) q^{93} -78708.3 q^{95} +159302. q^{97} +(-60585.7 + 48478.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 948 q^{11} + 852 q^{15} - 1640 q^{21} + 328 q^{23} - 12500 q^{25} + 2030 q^{27} + 2836 q^{33} - 7184 q^{35} - 15056 q^{37} - 12980 q^{39} - 11800 q^{45} + 36640 q^{47} - 33388 q^{49} + 1936 q^{51} + 15404 q^{57} + 62908 q^{59} - 73264 q^{61} + 23608 q^{63} + 84024 q^{69} + 34888 q^{71} + 52568 q^{73} + 115698 q^{75} + 55444 q^{81} - 225172 q^{83} + 30112 q^{85} - 225700 q^{87} + 148016 q^{93} + 418616 q^{95} + 7600 q^{97} + 378260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 5.16061 + 14.7095i 0.331053 + 0.943612i
\(4\) 0 0
\(5\) 48.6693i 0.870623i 0.900280 + 0.435312i \(0.143362\pi\)
−0.900280 + 0.435312i \(0.856638\pi\)
\(6\) 0 0
\(7\) 167.754i 1.29398i 0.762497 + 0.646992i \(0.223973\pi\)
−0.762497 + 0.646992i \(0.776027\pi\)
\(8\) 0 0
\(9\) −189.736 + 151.820i −0.780807 + 0.624772i
\(10\) 0 0
\(11\) 319.315 0.795679 0.397840 0.917455i \(-0.369760\pi\)
0.397840 + 0.917455i \(0.369760\pi\)
\(12\) 0 0
\(13\) −152.547 −0.250349 −0.125175 0.992135i \(-0.539949\pi\)
−0.125175 + 0.992135i \(0.539949\pi\)
\(14\) 0 0
\(15\) −715.899 + 251.163i −0.821530 + 0.288223i
\(16\) 0 0
\(17\) 1323.90i 1.11105i 0.831501 + 0.555523i \(0.187482\pi\)
−0.831501 + 0.555523i \(0.812518\pi\)
\(18\) 0 0
\(19\) 1617.20i 1.02773i 0.857870 + 0.513867i \(0.171788\pi\)
−0.857870 + 0.513867i \(0.828212\pi\)
\(20\) 0 0
\(21\) −2467.58 + 865.716i −1.22102 + 0.428378i
\(22\) 0 0
\(23\) 1222.40 0.481829 0.240914 0.970546i \(-0.422553\pi\)
0.240914 + 0.970546i \(0.422553\pi\)
\(24\) 0 0
\(25\) 756.298 0.242015
\(26\) 0 0
\(27\) −3212.34 2007.43i −0.848031 0.529946i
\(28\) 0 0
\(29\) 4549.54i 1.00455i 0.864707 + 0.502276i \(0.167504\pi\)
−0.864707 + 0.502276i \(0.832496\pi\)
\(30\) 0 0
\(31\) 5353.25i 1.00049i 0.865884 + 0.500245i \(0.166757\pi\)
−0.865884 + 0.500245i \(0.833243\pi\)
\(32\) 0 0
\(33\) 1647.86 + 4696.95i 0.263412 + 0.750813i
\(34\) 0 0
\(35\) −8164.49 −1.12657
\(36\) 0 0
\(37\) −3269.34 −0.392605 −0.196303 0.980543i \(-0.562894\pi\)
−0.196303 + 0.980543i \(0.562894\pi\)
\(38\) 0 0
\(39\) −787.239 2243.89i −0.0828791 0.236233i
\(40\) 0 0
\(41\) 13077.7i 1.21499i −0.794323 0.607495i \(-0.792174\pi\)
0.794323 0.607495i \(-0.207826\pi\)
\(42\) 0 0
\(43\) 20435.5i 1.68544i −0.538350 0.842721i \(-0.680952\pi\)
0.538350 0.842721i \(-0.319048\pi\)
\(44\) 0 0
\(45\) −7388.96 9234.33i −0.543941 0.679789i
\(46\) 0 0
\(47\) 20059.0 1.32454 0.662269 0.749266i \(-0.269594\pi\)
0.662269 + 0.749266i \(0.269594\pi\)
\(48\) 0 0
\(49\) −11334.6 −0.674395
\(50\) 0 0
\(51\) −19473.8 + 6832.12i −1.04840 + 0.367816i
\(52\) 0 0
\(53\) 33272.4i 1.62703i 0.581546 + 0.813513i \(0.302448\pi\)
−0.581546 + 0.813513i \(0.697552\pi\)
\(54\) 0 0
\(55\) 15540.9i 0.692737i
\(56\) 0 0
\(57\) −23788.2 + 8345.77i −0.969783 + 0.340235i
\(58\) 0 0
\(59\) 5883.70 0.220050 0.110025 0.993929i \(-0.464907\pi\)
0.110025 + 0.993929i \(0.464907\pi\)
\(60\) 0 0
\(61\) 22967.3 0.790289 0.395144 0.918619i \(-0.370695\pi\)
0.395144 + 0.918619i \(0.370695\pi\)
\(62\) 0 0
\(63\) −25468.4 31829.1i −0.808445 1.01035i
\(64\) 0 0
\(65\) 7424.38i 0.217960i
\(66\) 0 0
\(67\) 13938.9i 0.379352i −0.981847 0.189676i \(-0.939256\pi\)
0.981847 0.189676i \(-0.0607438\pi\)
\(68\) 0 0
\(69\) 6308.32 + 17980.8i 0.159511 + 0.454659i
\(70\) 0 0
\(71\) −65642.7 −1.54540 −0.772700 0.634772i \(-0.781094\pi\)
−0.772700 + 0.634772i \(0.781094\pi\)
\(72\) 0 0
\(73\) 55673.0 1.22275 0.611375 0.791341i \(-0.290617\pi\)
0.611375 + 0.791341i \(0.290617\pi\)
\(74\) 0 0
\(75\) 3902.96 + 11124.7i 0.0801200 + 0.228369i
\(76\) 0 0
\(77\) 53566.6i 1.02960i
\(78\) 0 0
\(79\) 82204.2i 1.48192i −0.671546 0.740962i \(-0.734370\pi\)
0.671546 0.740962i \(-0.265630\pi\)
\(80\) 0 0
\(81\) 12950.6 57611.3i 0.219320 0.975653i
\(82\) 0 0
\(83\) −19118.5 −0.304620 −0.152310 0.988333i \(-0.548671\pi\)
−0.152310 + 0.988333i \(0.548671\pi\)
\(84\) 0 0
\(85\) −64433.2 −0.967302
\(86\) 0 0
\(87\) −66921.3 + 23478.4i −0.947908 + 0.332561i
\(88\) 0 0
\(89\) 97061.0i 1.29888i −0.760412 0.649441i \(-0.775003\pi\)
0.760412 0.649441i \(-0.224997\pi\)
\(90\) 0 0
\(91\) 25590.5i 0.323948i
\(92\) 0 0
\(93\) −78743.3 + 27626.0i −0.944075 + 0.331216i
\(94\) 0 0
\(95\) −78708.3 −0.894770
\(96\) 0 0
\(97\) 159302. 1.71906 0.859532 0.511082i \(-0.170755\pi\)
0.859532 + 0.511082i \(0.170755\pi\)
\(98\) 0 0
\(99\) −60585.7 + 48478.3i −0.621272 + 0.497118i
\(100\) 0 0
\(101\) 77267.6i 0.753692i −0.926276 0.376846i \(-0.877009\pi\)
0.926276 0.376846i \(-0.122991\pi\)
\(102\) 0 0
\(103\) 43795.1i 0.406755i 0.979100 + 0.203377i \(0.0651918\pi\)
−0.979100 + 0.203377i \(0.934808\pi\)
\(104\) 0 0
\(105\) −42133.8 120095.i −0.372956 1.06305i
\(106\) 0 0
\(107\) −220827. −1.86463 −0.932315 0.361647i \(-0.882215\pi\)
−0.932315 + 0.361647i \(0.882215\pi\)
\(108\) 0 0
\(109\) 152973. 1.23325 0.616623 0.787259i \(-0.288500\pi\)
0.616623 + 0.787259i \(0.288500\pi\)
\(110\) 0 0
\(111\) −16871.8 48090.3i −0.129973 0.370467i
\(112\) 0 0
\(113\) 142603.i 1.05059i −0.850922 0.525293i \(-0.823956\pi\)
0.850922 0.525293i \(-0.176044\pi\)
\(114\) 0 0
\(115\) 59493.2i 0.419491i
\(116\) 0 0
\(117\) 28943.8 23159.7i 0.195475 0.156411i
\(118\) 0 0
\(119\) −222090. −1.43768
\(120\) 0 0
\(121\) −59088.7 −0.366894
\(122\) 0 0
\(123\) 192366. 67489.1i 1.14648 0.402227i
\(124\) 0 0
\(125\) 188900.i 1.08133i
\(126\) 0 0
\(127\) 49483.5i 0.272240i 0.990692 + 0.136120i \(0.0434632\pi\)
−0.990692 + 0.136120i \(0.956537\pi\)
\(128\) 0 0
\(129\) 300595. 105460.i 1.59040 0.557972i
\(130\) 0 0
\(131\) −260380. −1.32565 −0.662826 0.748773i \(-0.730643\pi\)
−0.662826 + 0.748773i \(0.730643\pi\)
\(132\) 0 0
\(133\) −271293. −1.32987
\(134\) 0 0
\(135\) 97700.4 156342.i 0.461383 0.738316i
\(136\) 0 0
\(137\) 107894.i 0.491132i 0.969380 + 0.245566i \(0.0789737\pi\)
−0.969380 + 0.245566i \(0.921026\pi\)
\(138\) 0 0
\(139\) 91076.8i 0.399826i −0.979814 0.199913i \(-0.935934\pi\)
0.979814 0.199913i \(-0.0640659\pi\)
\(140\) 0 0
\(141\) 103517. + 295057.i 0.438493 + 1.24985i
\(142\) 0 0
\(143\) −48710.8 −0.199198
\(144\) 0 0
\(145\) −221423. −0.874587
\(146\) 0 0
\(147\) −58493.2 166725.i −0.223261 0.636367i
\(148\) 0 0
\(149\) 396732.i 1.46397i 0.681322 + 0.731984i \(0.261405\pi\)
−0.681322 + 0.731984i \(0.738595\pi\)
\(150\) 0 0
\(151\) 396191.i 1.41404i −0.707192 0.707022i \(-0.750038\pi\)
0.707192 0.707022i \(-0.249962\pi\)
\(152\) 0 0
\(153\) −200994. 251191.i −0.694150 0.867513i
\(154\) 0 0
\(155\) −260539. −0.871050
\(156\) 0 0
\(157\) −92070.4 −0.298106 −0.149053 0.988829i \(-0.547623\pi\)
−0.149053 + 0.988829i \(0.547623\pi\)
\(158\) 0 0
\(159\) −489419. + 171706.i −1.53528 + 0.538633i
\(160\) 0 0
\(161\) 205062.i 0.623478i
\(162\) 0 0
\(163\) 299401.i 0.882640i −0.897350 0.441320i \(-0.854510\pi\)
0.897350 0.441320i \(-0.145490\pi\)
\(164\) 0 0
\(165\) −228598. + 80200.4i −0.653675 + 0.229333i
\(166\) 0 0
\(167\) 61166.1 0.169715 0.0848574 0.996393i \(-0.472957\pi\)
0.0848574 + 0.996393i \(0.472957\pi\)
\(168\) 0 0
\(169\) −348022. −0.937325
\(170\) 0 0
\(171\) −245523. 306842.i −0.642100 0.802463i
\(172\) 0 0
\(173\) 349424.i 0.887641i 0.896116 + 0.443821i \(0.146377\pi\)
−0.896116 + 0.443821i \(0.853623\pi\)
\(174\) 0 0
\(175\) 126872.i 0.313164i
\(176\) 0 0
\(177\) 30363.5 + 86546.0i 0.0728482 + 0.207641i
\(178\) 0 0
\(179\) 172914. 0.403365 0.201682 0.979451i \(-0.435359\pi\)
0.201682 + 0.979451i \(0.435359\pi\)
\(180\) 0 0
\(181\) −370619. −0.840874 −0.420437 0.907322i \(-0.638123\pi\)
−0.420437 + 0.907322i \(0.638123\pi\)
\(182\) 0 0
\(183\) 118525. + 337837.i 0.261628 + 0.745726i
\(184\) 0 0
\(185\) 159117.i 0.341811i
\(186\) 0 0
\(187\) 422741.i 0.884036i
\(188\) 0 0
\(189\) 336756. 538884.i 0.685742 1.09734i
\(190\) 0 0
\(191\) 636934. 1.26331 0.631656 0.775249i \(-0.282375\pi\)
0.631656 + 0.775249i \(0.282375\pi\)
\(192\) 0 0
\(193\) 791811. 1.53013 0.765064 0.643954i \(-0.222707\pi\)
0.765064 + 0.643954i \(0.222707\pi\)
\(194\) 0 0
\(195\) 109209. 38314.4i 0.205670 0.0721564i
\(196\) 0 0
\(197\) 229538.i 0.421395i 0.977551 + 0.210698i \(0.0675735\pi\)
−0.977551 + 0.210698i \(0.932426\pi\)
\(198\) 0 0
\(199\) 42046.2i 0.0752651i 0.999292 + 0.0376325i \(0.0119816\pi\)
−0.999292 + 0.0376325i \(0.988018\pi\)
\(200\) 0 0
\(201\) 205034. 71933.4i 0.357961 0.125586i
\(202\) 0 0
\(203\) −763206. −1.29987
\(204\) 0 0
\(205\) 636484. 1.05780
\(206\) 0 0
\(207\) −231933. + 185584.i −0.376215 + 0.301033i
\(208\) 0 0
\(209\) 516398.i 0.817747i
\(210\) 0 0
\(211\) 80176.3i 0.123977i −0.998077 0.0619883i \(-0.980256\pi\)
0.998077 0.0619883i \(-0.0197442\pi\)
\(212\) 0 0
\(213\) −338757. 965569.i −0.511610 1.45826i
\(214\) 0 0
\(215\) 994581. 1.46739
\(216\) 0 0
\(217\) −898031. −1.29462
\(218\) 0 0
\(219\) 287307. + 818919.i 0.404795 + 1.15380i
\(220\) 0 0
\(221\) 201957.i 0.278150i
\(222\) 0 0
\(223\) 73558.4i 0.0990536i 0.998773 + 0.0495268i \(0.0157713\pi\)
−0.998773 + 0.0495268i \(0.984229\pi\)
\(224\) 0 0
\(225\) −143497. + 114821.i −0.188967 + 0.151204i
\(226\) 0 0
\(227\) 359120. 0.462568 0.231284 0.972886i \(-0.425707\pi\)
0.231284 + 0.972886i \(0.425707\pi\)
\(228\) 0 0
\(229\) −867580. −1.09325 −0.546627 0.837376i \(-0.684088\pi\)
−0.546627 + 0.837376i \(0.684088\pi\)
\(230\) 0 0
\(231\) −787935. + 276436.i −0.971539 + 0.340851i
\(232\) 0 0
\(233\) 520013.i 0.627515i −0.949503 0.313758i \(-0.898412\pi\)
0.949503 0.313758i \(-0.101588\pi\)
\(234\) 0 0
\(235\) 976257.i 1.15317i
\(236\) 0 0
\(237\) 1.20918e6 424224.i 1.39836 0.490596i
\(238\) 0 0
\(239\) −206539. −0.233888 −0.116944 0.993139i \(-0.537310\pi\)
−0.116944 + 0.993139i \(0.537310\pi\)
\(240\) 0 0
\(241\) 1.65009e6 1.83006 0.915030 0.403385i \(-0.132166\pi\)
0.915030 + 0.403385i \(0.132166\pi\)
\(242\) 0 0
\(243\) 914264. 106813.i 0.993244 0.116041i
\(244\) 0 0
\(245\) 551645.i 0.587144i
\(246\) 0 0
\(247\) 246701.i 0.257293i
\(248\) 0 0
\(249\) −98663.3 281223.i −0.100846 0.287443i
\(250\) 0 0
\(251\) 1.56162e6 1.56455 0.782277 0.622931i \(-0.214058\pi\)
0.782277 + 0.622931i \(0.214058\pi\)
\(252\) 0 0
\(253\) 390330. 0.383381
\(254\) 0 0
\(255\) −332515. 947777.i −0.320229 0.912758i
\(256\) 0 0
\(257\) 1.51207e6i 1.42804i 0.700127 + 0.714019i \(0.253127\pi\)
−0.700127 + 0.714019i \(0.746873\pi\)
\(258\) 0 0
\(259\) 548447.i 0.508025i
\(260\) 0 0
\(261\) −690710. 863213.i −0.627616 0.784362i
\(262\) 0 0
\(263\) 387053. 0.345049 0.172524 0.985005i \(-0.444808\pi\)
0.172524 + 0.985005i \(0.444808\pi\)
\(264\) 0 0
\(265\) −1.61935e6 −1.41653
\(266\) 0 0
\(267\) 1.42771e6 500894.i 1.22564 0.429999i
\(268\) 0 0
\(269\) 1.62869e6i 1.37233i 0.727446 + 0.686165i \(0.240707\pi\)
−0.727446 + 0.686165i \(0.759293\pi\)
\(270\) 0 0
\(271\) 549923.i 0.454861i 0.973794 + 0.227431i \(0.0730325\pi\)
−0.973794 + 0.227431i \(0.926968\pi\)
\(272\) 0 0
\(273\) 376423. 132063.i 0.305681 0.107244i
\(274\) 0 0
\(275\) 241498. 0.192567
\(276\) 0 0
\(277\) 1.37805e6 1.07911 0.539553 0.841951i \(-0.318593\pi\)
0.539553 + 0.841951i \(0.318593\pi\)
\(278\) 0 0
\(279\) −812728. 1.01570e6i −0.625078 0.781190i
\(280\) 0 0
\(281\) 2.03707e6i 1.53900i 0.638645 + 0.769502i \(0.279495\pi\)
−0.638645 + 0.769502i \(0.720505\pi\)
\(282\) 0 0
\(283\) 1.15726e6i 0.858947i −0.903079 0.429473i \(-0.858699\pi\)
0.903079 0.429473i \(-0.141301\pi\)
\(284\) 0 0
\(285\) −406183. 1.15776e6i −0.296217 0.844315i
\(286\) 0 0
\(287\) 2.19385e6 1.57218
\(288\) 0 0
\(289\) −332847. −0.234423
\(290\) 0 0
\(291\) 822096. + 2.34325e6i 0.569102 + 1.62213i
\(292\) 0 0
\(293\) 2.48372e6i 1.69018i −0.534621 0.845092i \(-0.679546\pi\)
0.534621 0.845092i \(-0.320454\pi\)
\(294\) 0 0
\(295\) 286356.i 0.191580i
\(296\) 0 0
\(297\) −1.02575e6 641004.i −0.674761 0.421667i
\(298\) 0 0
\(299\) −186474. −0.120626
\(300\) 0 0
\(301\) 3.42814e6 2.18094
\(302\) 0 0
\(303\) 1.13656e6 398748.i 0.711193 0.249512i
\(304\) 0 0
\(305\) 1.11780e6i 0.688044i
\(306\) 0 0
\(307\) 2.54604e6i 1.54177i 0.636976 + 0.770884i \(0.280185\pi\)
−0.636976 + 0.770884i \(0.719815\pi\)
\(308\) 0 0
\(309\) −644202. + 226010.i −0.383819 + 0.134658i
\(310\) 0 0
\(311\) 2.66268e6 1.56106 0.780528 0.625121i \(-0.214950\pi\)
0.780528 + 0.625121i \(0.214950\pi\)
\(312\) 0 0
\(313\) −400195. −0.230893 −0.115446 0.993314i \(-0.536830\pi\)
−0.115446 + 0.993314i \(0.536830\pi\)
\(314\) 0 0
\(315\) 1.54910e6 1.23953e6i 0.879636 0.703851i
\(316\) 0 0
\(317\) 1.54428e6i 0.863136i −0.902080 0.431568i \(-0.857961\pi\)
0.902080 0.431568i \(-0.142039\pi\)
\(318\) 0 0
\(319\) 1.45274e6i 0.799302i
\(320\) 0 0
\(321\) −1.13960e6 3.24824e6i −0.617292 1.75949i
\(322\) 0 0
\(323\) −2.14101e6 −1.14186
\(324\) 0 0
\(325\) −115371. −0.0605884
\(326\) 0 0
\(327\) 789436. + 2.25015e6i 0.408270 + 1.16371i
\(328\) 0 0
\(329\) 3.36498e6i 1.71393i
\(330\) 0 0
\(331\) 337952.i 0.169545i −0.996400 0.0847725i \(-0.972984\pi\)
0.996400 0.0847725i \(-0.0270163\pi\)
\(332\) 0 0
\(333\) 620313. 496351.i 0.306549 0.245289i
\(334\) 0 0
\(335\) 678398. 0.330273
\(336\) 0 0
\(337\) 1.92048e6 0.921158 0.460579 0.887619i \(-0.347642\pi\)
0.460579 + 0.887619i \(0.347642\pi\)
\(338\) 0 0
\(339\) 2.09761e6 735917.i 0.991345 0.347800i
\(340\) 0 0
\(341\) 1.70937e6i 0.796070i
\(342\) 0 0
\(343\) 918027.i 0.421328i
\(344\) 0 0
\(345\) −875113. + 307021.i −0.395837 + 0.138874i
\(346\) 0 0
\(347\) −1.67762e6 −0.747946 −0.373973 0.927440i \(-0.622005\pi\)
−0.373973 + 0.927440i \(0.622005\pi\)
\(348\) 0 0
\(349\) −3.93176e6 −1.72792 −0.863960 0.503560i \(-0.832023\pi\)
−0.863960 + 0.503560i \(0.832023\pi\)
\(350\) 0 0
\(351\) 490034. + 306229.i 0.212304 + 0.132672i
\(352\) 0 0
\(353\) 3.20933e6i 1.37081i 0.728161 + 0.685406i \(0.240375\pi\)
−0.728161 + 0.685406i \(0.759625\pi\)
\(354\) 0 0
\(355\) 3.19479e6i 1.34546i
\(356\) 0 0
\(357\) −1.14612e6 3.26682e6i −0.475948 1.35661i
\(358\) 0 0
\(359\) 913329. 0.374017 0.187008 0.982358i \(-0.440121\pi\)
0.187008 + 0.982358i \(0.440121\pi\)
\(360\) 0 0
\(361\) −139253. −0.0562389
\(362\) 0 0
\(363\) −304934. 869163.i −0.121462 0.346206i
\(364\) 0 0
\(365\) 2.70957e6i 1.06455i
\(366\) 0 0
\(367\) 205159.i 0.0795106i −0.999209 0.0397553i \(-0.987342\pi\)
0.999209 0.0397553i \(-0.0126578\pi\)
\(368\) 0 0
\(369\) 1.98546e6 + 2.48132e6i 0.759092 + 0.948673i
\(370\) 0 0
\(371\) −5.58160e6 −2.10535
\(372\) 0 0
\(373\) −2.30940e6 −0.859465 −0.429732 0.902956i \(-0.641392\pi\)
−0.429732 + 0.902956i \(0.641392\pi\)
\(374\) 0 0
\(375\) −2.77862e6 + 974840.i −1.02035 + 0.357977i
\(376\) 0 0
\(377\) 694021.i 0.251489i
\(378\) 0 0
\(379\) 1.30015e6i 0.464939i 0.972604 + 0.232470i \(0.0746806\pi\)
−0.972604 + 0.232470i \(0.925319\pi\)
\(380\) 0 0
\(381\) −727876. + 255365.i −0.256889 + 0.0901259i
\(382\) 0 0
\(383\) 421140. 0.146700 0.0733499 0.997306i \(-0.476631\pi\)
0.0733499 + 0.997306i \(0.476631\pi\)
\(384\) 0 0
\(385\) −2.60705e6 −0.896390
\(386\) 0 0
\(387\) 3.10251e6 + 3.87735e6i 1.05302 + 1.31601i
\(388\) 0 0
\(389\) 3.99605e6i 1.33893i 0.742844 + 0.669464i \(0.233476\pi\)
−0.742844 + 0.669464i \(0.766524\pi\)
\(390\) 0 0
\(391\) 1.61833e6i 0.535334i
\(392\) 0 0
\(393\) −1.34372e6 3.83005e6i −0.438862 1.25090i
\(394\) 0 0
\(395\) 4.00082e6 1.29020
\(396\) 0 0
\(397\) 1.38885e6 0.442262 0.221131 0.975244i \(-0.429025\pi\)
0.221131 + 0.975244i \(0.429025\pi\)
\(398\) 0 0
\(399\) −1.40004e6 3.99058e6i −0.440259 1.25488i
\(400\) 0 0
\(401\) 1.31621e6i 0.408758i 0.978892 + 0.204379i \(0.0655174\pi\)
−0.978892 + 0.204379i \(0.934483\pi\)
\(402\) 0 0
\(403\) 816624.i 0.250472i
\(404\) 0 0
\(405\) 2.80390e6 + 630297.i 0.849426 + 0.190945i
\(406\) 0 0
\(407\) −1.04395e6 −0.312388
\(408\) 0 0
\(409\) 2.83849e6 0.839033 0.419516 0.907748i \(-0.362200\pi\)
0.419516 + 0.907748i \(0.362200\pi\)
\(410\) 0 0
\(411\) −1.58707e6 + 556802.i −0.463438 + 0.162591i
\(412\) 0 0
\(413\) 987017.i 0.284741i
\(414\) 0 0
\(415\) 930485.i 0.265210i
\(416\) 0 0
\(417\) 1.33969e6 470012.i 0.377280 0.132364i
\(418\) 0 0
\(419\) 983535. 0.273687 0.136844 0.990593i \(-0.456304\pi\)
0.136844 + 0.990593i \(0.456304\pi\)
\(420\) 0 0
\(421\) −3.50363e6 −0.963414 −0.481707 0.876332i \(-0.659983\pi\)
−0.481707 + 0.876332i \(0.659983\pi\)
\(422\) 0 0
\(423\) −3.80591e6 + 3.04535e6i −1.03421 + 0.827534i
\(424\) 0 0
\(425\) 1.00126e6i 0.268890i
\(426\) 0 0
\(427\) 3.85287e6i 1.02262i
\(428\) 0 0
\(429\) −251377. 716509.i −0.0659452 0.187966i
\(430\) 0 0
\(431\) 4.96042e6 1.28625 0.643125 0.765762i \(-0.277638\pi\)
0.643125 + 0.765762i \(0.277638\pi\)
\(432\) 0 0
\(433\) −3.86224e6 −0.989964 −0.494982 0.868903i \(-0.664825\pi\)
−0.494982 + 0.868903i \(0.664825\pi\)
\(434\) 0 0
\(435\) −1.14268e6 3.25701e6i −0.289535 0.825270i
\(436\) 0 0
\(437\) 1.97687e6i 0.495192i
\(438\) 0 0
\(439\) 7.78526e6i 1.92802i −0.265862 0.964011i \(-0.585657\pi\)
0.265862 0.964011i \(-0.414343\pi\)
\(440\) 0 0
\(441\) 2.15057e6 1.72081e6i 0.526572 0.421343i
\(442\) 0 0
\(443\) −6.74367e6 −1.63263 −0.816314 0.577609i \(-0.803986\pi\)
−0.816314 + 0.577609i \(0.803986\pi\)
\(444\) 0 0
\(445\) 4.72389e6 1.13084
\(446\) 0 0
\(447\) −5.83571e6 + 2.04738e6i −1.38142 + 0.484651i
\(448\) 0 0
\(449\) 3.85833e6i 0.903200i −0.892221 0.451600i \(-0.850853\pi\)
0.892221 0.451600i \(-0.149147\pi\)
\(450\) 0 0
\(451\) 4.17592e6i 0.966743i
\(452\) 0 0
\(453\) 5.82776e6 2.04459e6i 1.33431 0.468124i
\(454\) 0 0
\(455\) 1.24547e6 0.282037
\(456\) 0 0
\(457\) 1.64250e6 0.367888 0.183944 0.982937i \(-0.441114\pi\)
0.183944 + 0.982937i \(0.441114\pi\)
\(458\) 0 0
\(459\) 2.65764e6 4.25281e6i 0.588794 0.942202i
\(460\) 0 0
\(461\) 77799.5i 0.0170500i −0.999964 0.00852500i \(-0.997286\pi\)
0.999964 0.00852500i \(-0.00271363\pi\)
\(462\) 0 0
\(463\) 213758.i 0.0463414i −0.999732 0.0231707i \(-0.992624\pi\)
0.999732 0.0231707i \(-0.00737613\pi\)
\(464\) 0 0
\(465\) −1.34454e6 3.83238e6i −0.288364 0.821933i
\(466\) 0 0
\(467\) 2.85122e6 0.604977 0.302489 0.953153i \(-0.402183\pi\)
0.302489 + 0.953153i \(0.402183\pi\)
\(468\) 0 0
\(469\) 2.33832e6 0.490875
\(470\) 0 0
\(471\) −475140. 1.35431e6i −0.0986891 0.281297i
\(472\) 0 0
\(473\) 6.52537e6i 1.34107i
\(474\) 0 0
\(475\) 1.22309e6i 0.248728i
\(476\) 0 0
\(477\) −5.05141e6 6.31298e6i −1.01652 1.27039i
\(478\) 0 0
\(479\) −8.52181e6 −1.69704 −0.848522 0.529161i \(-0.822507\pi\)
−0.848522 + 0.529161i \(0.822507\pi\)
\(480\) 0 0
\(481\) 498730. 0.0982886
\(482\) 0 0
\(483\) −3.01636e6 + 1.05825e6i −0.588322 + 0.206405i
\(484\) 0 0
\(485\) 7.75312e6i 1.49666i
\(486\) 0 0
\(487\) 6.25067e6i 1.19427i −0.802139 0.597137i \(-0.796305\pi\)
0.802139 0.597137i \(-0.203695\pi\)
\(488\) 0 0
\(489\) 4.40402e6 1.54509e6i 0.832870 0.292201i
\(490\) 0 0
\(491\) −5.91380e6 −1.10704 −0.553519 0.832837i \(-0.686715\pi\)
−0.553519 + 0.832837i \(0.686715\pi\)
\(492\) 0 0
\(493\) −6.02313e6 −1.11610
\(494\) 0 0
\(495\) −2.35941e6 2.94866e6i −0.432803 0.540894i
\(496\) 0 0
\(497\) 1.10119e7i 1.99972i
\(498\) 0 0
\(499\) 916915.i 0.164846i 0.996597 + 0.0824228i \(0.0262658\pi\)
−0.996597 + 0.0824228i \(0.973734\pi\)
\(500\) 0 0
\(501\) 315655. + 899720.i 0.0561847 + 0.160145i
\(502\) 0 0
\(503\) 4.30667e6 0.758964 0.379482 0.925199i \(-0.376102\pi\)
0.379482 + 0.925199i \(0.376102\pi\)
\(504\) 0 0
\(505\) 3.76056e6 0.656181
\(506\) 0 0
\(507\) −1.79601e6 5.11922e6i −0.310305 0.884471i
\(508\) 0 0
\(509\) 3.12915e6i 0.535342i 0.963510 + 0.267671i \(0.0862540\pi\)
−0.963510 + 0.267671i \(0.913746\pi\)
\(510\) 0 0
\(511\) 9.33939e6i 1.58222i
\(512\) 0 0
\(513\) 3.24643e6 5.19501e6i 0.544644 0.871551i
\(514\) 0 0
\(515\) −2.13148e6 −0.354130
\(516\) 0 0
\(517\) 6.40514e6 1.05391
\(518\) 0 0
\(519\) −5.13984e6 + 1.80324e6i −0.837589 + 0.293857i
\(520\) 0 0
\(521\) 181277.i 0.0292583i −0.999893 0.0146292i \(-0.995343\pi\)
0.999893 0.0146292i \(-0.00465677\pi\)
\(522\) 0 0
\(523\) 7.82852e6i 1.25148i 0.780030 + 0.625742i \(0.215204\pi\)
−0.780030 + 0.625742i \(0.784796\pi\)
\(524\) 0 0
\(525\) −1.86622e6 + 654739.i −0.295505 + 0.103674i
\(526\) 0 0
\(527\) −7.08715e6 −1.11159
\(528\) 0 0
\(529\) −4.94209e6 −0.767841
\(530\) 0 0
\(531\) −1.11635e6 + 893261.i −0.171816 + 0.137481i
\(532\) 0 0
\(533\) 1.99498e6i 0.304172i
\(534\) 0 0
\(535\) 1.07475e7i 1.62339i
\(536\) 0 0
\(537\) 892343. + 2.54347e6i 0.133535 + 0.380620i
\(538\) 0 0
\(539\) −3.61930e6 −0.536602
\(540\) 0 0
\(541\) −1.25741e7 −1.84706 −0.923532 0.383520i \(-0.874712\pi\)
−0.923532 + 0.383520i \(0.874712\pi\)
\(542\) 0 0
\(543\) −1.91262e6 5.45160e6i −0.278374 0.793459i
\(544\) 0 0
\(545\) 7.44511e6i 1.07369i
\(546\) 0 0
\(547\) 3.40198e6i 0.486143i −0.970008 0.243071i \(-0.921845\pi\)
0.970008 0.243071i \(-0.0781549\pi\)
\(548\) 0 0
\(549\) −4.35773e6 + 3.48689e6i −0.617063 + 0.493750i
\(550\) 0 0
\(551\) −7.35754e6 −1.03241
\(552\) 0 0
\(553\) 1.37901e7 1.91759
\(554\) 0 0
\(555\) 2.34052e6 821140.i 0.322537 0.113158i
\(556\) 0 0
\(557\) 7.75680e6i 1.05936i 0.848197 + 0.529681i \(0.177689\pi\)
−0.848197 + 0.529681i \(0.822311\pi\)
\(558\) 0 0
\(559\) 3.11738e6i 0.421950i
\(560\) 0 0
\(561\) −6.21829e6 + 2.18160e6i −0.834187 + 0.292663i
\(562\) 0 0
\(563\) −3.17845e6 −0.422614 −0.211307 0.977420i \(-0.567772\pi\)
−0.211307 + 0.977420i \(0.567772\pi\)
\(564\) 0 0
\(565\) 6.94037e6 0.914664
\(566\) 0 0
\(567\) 9.66456e6 + 2.17252e6i 1.26248 + 0.283796i
\(568\) 0 0
\(569\) 1.26620e7i 1.63954i 0.572691 + 0.819771i \(0.305900\pi\)
−0.572691 + 0.819771i \(0.694100\pi\)
\(570\) 0 0
\(571\) 3.92129e6i 0.503314i −0.967816 0.251657i \(-0.919025\pi\)
0.967816 0.251657i \(-0.0809755\pi\)
\(572\) 0 0
\(573\) 3.28697e6 + 9.36895e6i 0.418224 + 1.19208i
\(574\) 0 0
\(575\) 924496. 0.116610
\(576\) 0 0
\(577\) −1.12254e6 −0.140366 −0.0701828 0.997534i \(-0.522358\pi\)
−0.0701828 + 0.997534i \(0.522358\pi\)
\(578\) 0 0
\(579\) 4.08623e6 + 1.16471e7i 0.506554 + 1.44385i
\(580\) 0 0
\(581\) 3.20722e6i 0.394174i
\(582\) 0 0
\(583\) 1.06244e7i 1.29459i
\(584\) 0 0
\(585\) 1.12717e6 + 1.40867e6i 0.136175 + 0.170185i
\(586\) 0 0
\(587\) −3.81465e6 −0.456940 −0.228470 0.973551i \(-0.573372\pi\)
−0.228470 + 0.973551i \(0.573372\pi\)
\(588\) 0 0
\(589\) −8.65730e6 −1.02824
\(590\) 0 0
\(591\) −3.37638e6 + 1.18456e6i −0.397633 + 0.139504i
\(592\) 0 0
\(593\) 6.15809e6i 0.719133i −0.933119 0.359567i \(-0.882925\pi\)
0.933119 0.359567i \(-0.117075\pi\)
\(594\) 0 0
\(595\) 1.08090e7i 1.25167i
\(596\) 0 0
\(597\) −618476. + 216984.i −0.0710210 + 0.0249168i
\(598\) 0 0
\(599\) −8.31895e6 −0.947331 −0.473665 0.880705i \(-0.657069\pi\)
−0.473665 + 0.880705i \(0.657069\pi\)
\(600\) 0 0
\(601\) −1.13436e7 −1.28104 −0.640521 0.767941i \(-0.721282\pi\)
−0.640521 + 0.767941i \(0.721282\pi\)
\(602\) 0 0
\(603\) 2.11620e6 + 2.64472e6i 0.237009 + 0.296201i
\(604\) 0 0
\(605\) 2.87581e6i 0.319427i
\(606\) 0 0
\(607\) 1.37972e7i 1.51991i 0.649974 + 0.759956i \(0.274780\pi\)
−0.649974 + 0.759956i \(0.725220\pi\)
\(608\) 0 0
\(609\) −3.93861e6 1.12263e7i −0.430328 1.22658i
\(610\) 0 0
\(611\) −3.05995e6 −0.331597
\(612\) 0 0
\(613\) −1.47858e6 −0.158926 −0.0794629 0.996838i \(-0.525321\pi\)
−0.0794629 + 0.996838i \(0.525321\pi\)
\(614\) 0 0
\(615\) 3.28465e6 + 9.36234e6i 0.350188 + 0.998152i
\(616\) 0 0
\(617\) 1.64855e6i 0.174337i 0.996194 + 0.0871684i \(0.0277818\pi\)
−0.996194 + 0.0871684i \(0.972218\pi\)
\(618\) 0 0
\(619\) 3.87090e6i 0.406056i 0.979173 + 0.203028i \(0.0650782\pi\)
−0.979173 + 0.203028i \(0.934922\pi\)
\(620\) 0 0
\(621\) −3.92675e6 2.45388e6i −0.408606 0.255343i
\(622\) 0 0
\(623\) 1.62824e7 1.68073
\(624\) 0 0
\(625\) −6.83021e6 −0.699413
\(626\) 0 0
\(627\) −7.59594e6 + 2.66493e6i −0.771636 + 0.270718i
\(628\) 0 0
\(629\) 4.32828e6i 0.436203i
\(630\) 0 0
\(631\) 1.47448e7i 1.47423i −0.675765 0.737117i \(-0.736187\pi\)
0.675765 0.737117i \(-0.263813\pi\)
\(632\) 0 0
\(633\) 1.17935e6 413759.i 0.116986 0.0410429i
\(634\) 0 0
\(635\) −2.40833e6 −0.237018
\(636\) 0 0
\(637\) 1.72906e6 0.168834
\(638\) 0 0
\(639\) 1.24548e7 9.96585e6i 1.20666 0.965522i
\(640\) 0 0
\(641\) 1.09176e7i 1.04950i 0.851256 + 0.524750i \(0.175841\pi\)
−0.851256 + 0.524750i \(0.824159\pi\)
\(642\) 0 0
\(643\) 1.82933e7i 1.74487i −0.488726 0.872437i \(-0.662538\pi\)
0.488726 0.872437i \(-0.337462\pi\)
\(644\) 0 0
\(645\) 5.13265e6 + 1.46298e7i 0.485783 + 1.38464i
\(646\) 0 0
\(647\) 381744. 0.0358518 0.0179259 0.999839i \(-0.494294\pi\)
0.0179259 + 0.999839i \(0.494294\pi\)
\(648\) 0 0
\(649\) 1.87876e6 0.175089
\(650\) 0 0
\(651\) −4.63439e6 1.32095e7i −0.428588 1.22162i
\(652\) 0 0
\(653\) 1.41687e7i 1.30031i −0.759800 0.650157i \(-0.774703\pi\)
0.759800 0.650157i \(-0.225297\pi\)
\(654\) 0 0
\(655\) 1.26725e7i 1.15414i
\(656\) 0 0
\(657\) −1.05632e7 + 8.45225e6i −0.954731 + 0.763939i
\(658\) 0 0
\(659\) 2.91811e6 0.261751 0.130876 0.991399i \(-0.458221\pi\)
0.130876 + 0.991399i \(0.458221\pi\)
\(660\) 0 0
\(661\) 1.08812e7 0.968662 0.484331 0.874885i \(-0.339063\pi\)
0.484331 + 0.874885i \(0.339063\pi\)
\(662\) 0 0
\(663\) 2.97068e6 1.04222e6i 0.262465 0.0920824i
\(664\) 0 0
\(665\) 1.32037e7i 1.15782i
\(666\) 0 0
\(667\) 5.56134e6i 0.484022i
\(668\) 0 0
\(669\) −1.08200e6 + 379607.i −0.0934682 + 0.0327920i
\(670\) 0 0
\(671\) 7.33382e6 0.628816
\(672\) 0 0
\(673\) 1.31612e7 1.12010 0.560049 0.828459i \(-0.310782\pi\)
0.560049 + 0.828459i \(0.310782\pi\)
\(674\) 0 0
\(675\) −2.42949e6 1.51822e6i −0.205237 0.128255i
\(676\) 0 0
\(677\) 5.27738e6i 0.442534i −0.975213 0.221267i \(-0.928981\pi\)
0.975213 0.221267i \(-0.0710193\pi\)
\(678\) 0 0
\(679\) 2.67236e7i 2.22444i
\(680\) 0 0
\(681\) 1.85328e6 + 5.28246e6i 0.153135 + 0.436484i
\(682\) 0 0
\(683\) 7.46685e6 0.612471 0.306236 0.951956i \(-0.400930\pi\)
0.306236 + 0.951956i \(0.400930\pi\)
\(684\) 0 0
\(685\) −5.25115e6 −0.427591
\(686\) 0 0
\(687\) −4.47724e6 1.27616e7i −0.361925 1.03161i
\(688\) 0 0
\(689\) 5.07562e6i 0.407325i
\(690\) 0 0
\(691\) 4.56640e6i 0.363814i 0.983316 + 0.181907i \(0.0582269\pi\)
−0.983316 + 0.181907i \(0.941773\pi\)
\(692\) 0 0
\(693\) −8.13246e6 1.01635e7i −0.643263 0.803916i
\(694\) 0 0
\(695\) 4.43264e6 0.348097
\(696\) 0 0
\(697\) 1.73136e7 1.34991
\(698\) 0 0
\(699\) 7.64911e6 2.68359e6i 0.592131 0.207741i
\(700\) 0 0
\(701\) 967522.i 0.0743645i 0.999309 + 0.0371822i \(0.0118382\pi\)
−0.999309 + 0.0371822i \(0.988162\pi\)
\(702\) 0 0
\(703\) 5.28720e6i 0.403494i
\(704\) 0 0
\(705\) −1.43602e7 + 5.03808e6i −1.08815 + 0.381762i
\(706\) 0 0
\(707\) 1.29620e7 0.975265
\(708\) 0 0
\(709\) 2.59701e7 1.94025 0.970125 0.242606i \(-0.0780023\pi\)
0.970125 + 0.242606i \(0.0780023\pi\)
\(710\) 0 0
\(711\) 1.24802e7 + 1.55971e7i 0.925865 + 1.15710i
\(712\) 0 0
\(713\) 6.54379e6i 0.482065i
\(714\) 0 0
\(715\) 2.37072e6i 0.173426i
\(716\) 0 0
\(717\) −1.06587e6 3.03808e6i −0.0774295 0.220700i
\(718\) 0 0
\(719\) −1.05306e7 −0.759677 −0.379838 0.925053i \(-0.624020\pi\)
−0.379838 + 0.925053i \(0.624020\pi\)
\(720\) 0 0
\(721\) −7.34683e6 −0.526334
\(722\) 0 0
\(723\) 8.51549e6 + 2.42720e7i 0.605848 + 1.72687i
\(724\) 0 0
\(725\) 3.44081e6i 0.243117i
\(726\) 0 0
\(727\) 2.45606e7i 1.72347i −0.507358 0.861735i \(-0.669378\pi\)
0.507358 0.861735i \(-0.330622\pi\)
\(728\) 0 0
\(729\) 6.28933e6 + 1.28971e7i 0.438314 + 0.898822i
\(730\) 0 0
\(731\) 2.70545e7 1.87260
\(732\) 0 0
\(733\) 1.82797e7 1.25664 0.628318 0.777957i \(-0.283744\pi\)
0.628318 + 0.777957i \(0.283744\pi\)
\(734\) 0 0
\(735\) 8.11440e6 2.84683e6i 0.554036 0.194376i
\(736\) 0 0
\(737\) 4.45091e6i 0.301843i
\(738\) 0 0
\(739\) 2.01860e7i 1.35969i 0.733356 + 0.679844i \(0.237953\pi\)
−0.733356 + 0.679844i \(0.762047\pi\)
\(740\) 0 0
\(741\) 3.62883e6 1.27313e6i 0.242785 0.0851777i
\(742\) 0 0
\(743\) −2.29183e7 −1.52304 −0.761518 0.648143i \(-0.775546\pi\)
−0.761518 + 0.648143i \(0.775546\pi\)
\(744\) 0 0
\(745\) −1.93087e7 −1.27456
\(746\) 0 0
\(747\) 3.62747e6 2.90257e6i 0.237850 0.190318i
\(748\) 0 0
\(749\) 3.70447e7i 2.41280i
\(750\) 0 0
\(751\) 2.29724e7i 1.48630i 0.669126 + 0.743149i \(0.266669\pi\)
−0.669126 + 0.743149i \(0.733331\pi\)
\(752\) 0 0
\(753\) 8.05891e6 + 2.29706e7i 0.517951 + 1.47633i
\(754\) 0 0
\(755\) 1.92824e7 1.23110
\(756\) 0 0
\(757\) −1.71489e7 −1.08767 −0.543835 0.839192i \(-0.683028\pi\)
−0.543835 + 0.839192i \(0.683028\pi\)
\(758\) 0 0
\(759\) 2.01434e6 + 5.74154e6i 0.126920 + 0.361763i
\(760\) 0 0
\(761\) 6.55577e6i 0.410357i 0.978725 + 0.205178i \(0.0657775\pi\)
−0.978725 + 0.205178i \(0.934223\pi\)
\(762\) 0 0
\(763\) 2.56620e7i 1.59580i
\(764\) 0 0
\(765\) 1.22253e7 9.78222e6i 0.755277 0.604343i
\(766\) 0 0
\(767\) −897544. −0.0550893
\(768\) 0 0
\(769\) −2.14557e7 −1.30836 −0.654180 0.756339i \(-0.726986\pi\)
−0.654180 + 0.756339i \(0.726986\pi\)
\(770\) 0 0
\(771\) −2.22417e7 + 7.80322e6i −1.34751 + 0.472757i
\(772\) 0 0
\(773\) 1.13075e7i 0.680640i −0.940310 0.340320i \(-0.889465\pi\)
0.940310 0.340320i \(-0.110535\pi\)
\(774\) 0 0
\(775\) 4.04865e6i 0.242134i
\(776\) 0 0
\(777\) 8.06736e6 2.83032e6i 0.479379 0.168184i
\(778\) 0 0
\(779\) 2.11494e7 1.24869
\(780\) 0 0
\(781\) −2.09607e7 −1.22964
\(782\) 0 0
\(783\) 9.13290e6 1.46147e7i 0.532359 0.851892i
\(784\) 0 0
\(785\) 4.48100e6i 0.259538i
\(786\) 0 0
\(787\) 619355.i 0.0356454i 0.999841 + 0.0178227i \(0.00567344\pi\)
−0.999841 + 0.0178227i \(0.994327\pi\)
\(788\) 0 0
\(789\) 1.99743e6 + 5.69333e6i 0.114230 + 0.325592i
\(790\) 0 0
\(791\) 2.39222e7 1.35944
\(792\) 0 0
\(793\) −3.50361e6 −0.197848
\(794\) 0 0
\(795\) −8.35682e6 2.38197e7i −0.468946 1.33665i
\(796\) 0 0
\(797\) 2.08343e6i 0.116180i −0.998311 0.0580901i \(-0.981499\pi\)
0.998311 0.0580901i \(-0.0185011\pi\)
\(798\) 0 0
\(799\) 2.65560e7i 1.47162i
\(800\) 0 0
\(801\) 1.47358e7 + 1.84160e7i 0.811505 + 1.01418i
\(802\) 0 0
\(803\) 1.77772e7 0.972916
\(804\) 0 0
\(805\) −9.98025e6 −0.542815
\(806\) 0 0
\(807\) −2.39572e7 + 8.40506e6i −1.29495 + 0.454315i
\(808\) 0 0
\(809\) 343117.i 0.0184320i 0.999958 + 0.00921598i \(0.00293358\pi\)
−0.999958 + 0.00921598i \(0.997066\pi\)
\(810\) 0 0
\(811\) 3.02023e6i 0.161246i −0.996745 0.0806228i \(-0.974309\pi\)
0.996745 0.0806228i \(-0.0256909\pi\)
\(812\) 0 0
\(813\) −8.08907e6 + 2.83794e6i −0.429212 + 0.150583i
\(814\) 0 0
\(815\) 1.45716e7 0.768447
\(816\) 0 0
\(817\) 3.30484e7 1.73219
\(818\) 0 0
\(819\) 3.88514e6 + 4.85545e6i 0.202394 + 0.252941i
\(820\) 0 0
\(821\) 4.37269e6i 0.226407i −0.993572 0.113204i \(-0.963889\pi\)
0.993572 0.113204i \(-0.0361113\pi\)
\(822\) 0 0
\(823\) 2.95780e7i 1.52219i −0.648641 0.761095i \(-0.724662\pi\)
0.648641 0.761095i \(-0.275338\pi\)
\(824\) 0 0
\(825\) 1.24628e6 + 3.55230e6i 0.0637499 + 0.181708i
\(826\) 0 0
\(827\) 2.90787e7 1.47846 0.739232 0.673451i \(-0.235189\pi\)
0.739232 + 0.673451i \(0.235189\pi\)
\(828\) 0 0
\(829\) 2.57279e7 1.30023 0.650113 0.759838i \(-0.274722\pi\)
0.650113 + 0.759838i \(0.274722\pi\)
\(830\) 0 0
\(831\) 7.11157e6 + 2.02703e7i 0.357242 + 1.01826i
\(832\) 0 0
\(833\) 1.50058e7i 0.749284i
\(834\) 0 0
\(835\) 2.97691e6i 0.147758i
\(836\) 0 0
\(837\) 1.07463e7 1.71964e7i 0.530206 0.848447i
\(838\) 0 0
\(839\) 1.13952e7 0.558877 0.279439 0.960164i \(-0.409852\pi\)
0.279439 + 0.960164i \(0.409852\pi\)
\(840\) 0 0
\(841\) −187181. −0.00912581
\(842\) 0 0
\(843\) −2.99642e7 + 1.05125e7i −1.45222 + 0.509492i
\(844\) 0 0
\(845\) 1.69380e7i 0.816057i
\(846\) 0 0
\(847\) 9.91239e6i 0.474756i
\(848\) 0 0
\(849\) 1.70227e7 5.97219e6i 0.810513 0.284357i
\(850\) 0 0
\(851\) −3.99644e6 −0.189169
\(852\) 0 0
\(853\) −3.53215e6 −0.166213 −0.0831067 0.996541i \(-0.526484\pi\)
−0.0831067 + 0.996541i \(0.526484\pi\)
\(854\) 0 0
\(855\) 1.49338e7 1.19495e7i 0.698643 0.559027i
\(856\) 0 0
\(857\) 2.78472e7i 1.29518i −0.761990 0.647589i \(-0.775777\pi\)
0.761990 0.647589i \(-0.224223\pi\)
\(858\) 0 0
\(859\) 9.12202e6i 0.421801i 0.977508 + 0.210901i \(0.0676397\pi\)
−0.977508 + 0.210901i \(0.932360\pi\)
\(860\) 0 0
\(861\) 1.13216e7 + 3.22703e7i 0.520475 + 1.48353i
\(862\) 0 0
\(863\) −3.82202e7 −1.74689 −0.873445 0.486923i \(-0.838119\pi\)
−0.873445 + 0.486923i \(0.838119\pi\)
\(864\) 0 0
\(865\) −1.70062e7 −0.772801
\(866\) 0 0
\(867\) −1.71770e6 4.89600e6i −0.0776066 0.221204i
\(868\) 0 0
\(869\) 2.62491e7i 1.17914i
\(870\) 0 0
\(871\) 2.12635e6i 0.0949706i
\(872\) 0 0
\(873\) −3.02254e7 + 2.41852e7i −1.34226 + 1.07402i
\(874\) 0 0
\(875\) −3.16888e7 −1.39922
\(876\) 0 0
\(877\) 1.52280e7 0.668566 0.334283 0.942473i \(-0.391506\pi\)
0.334283 + 0.942473i \(0.391506\pi\)
\(878\) 0 0
\(879\) 3.65342e7 1.28175e7i 1.59488 0.559541i
\(880\) 0 0
\(881\) 2.39351e7i 1.03895i 0.854485 + 0.519477i \(0.173873\pi\)
−0.854485 + 0.519477i \(0.826127\pi\)
\(882\) 0 0
\(883\) 4.20111e7i 1.81327i −0.421919 0.906634i \(-0.638643\pi\)
0.421919 0.906634i \(-0.361357\pi\)
\(884\) 0 0
\(885\) −4.21214e6 + 1.47777e6i −0.180777 + 0.0634233i
\(886\) 0 0
\(887\) 1.42404e7 0.607735 0.303867 0.952714i \(-0.401722\pi\)
0.303867 + 0.952714i \(0.401722\pi\)
\(888\) 0 0
\(889\) −8.30108e6 −0.352274
\(890\) 0 0
\(891\) 4.13533e6 1.83962e7i 0.174508 0.776307i
\(892\) 0 0
\(893\) 3.24395e7i 1.36127i
\(894\) 0 0
\(895\) 8.41562e6i 0.351179i
\(896\) 0 0
\(897\) −962318. 2.74292e6i −0.0399335 0.113824i
\(898\) 0 0
\(899\) −2.43548e7 −1.00505
\(900\) 0 0
\(901\) −4.40493e7 −1.80770
\(902\) 0 0
\(903\) 1.76913e7 + 5.04261e7i 0.722006 + 2.05796i
\(904\) 0 0
\(905\) 1.80378e7i 0.732084i
\(906\) 0 0
\(907\) 9.67370e6i 0.390458i 0.980758 + 0.195229i \(0.0625450\pi\)
−0.980758 + 0.195229i \(0.937455\pi\)
\(908\) 0 0
\(909\) 1.17307e7 + 1.46604e7i 0.470886 + 0.588488i
\(910\) 0 0
\(911\) 915385. 0.0365433 0.0182716 0.999833i \(-0.494184\pi\)
0.0182716 + 0.999833i \(0.494184\pi\)
\(912\) 0 0
\(913\) −6.10484e6 −0.242380
\(914\) 0 0
\(915\) −1.64423e7 + 5.76855e6i −0.649246 + 0.227779i
\(916\) 0 0
\(917\) 4.36799e7i 1.71537i
\(918\) 0 0
\(919\) 2.37044e7i 0.925849i −0.886398 0.462925i \(-0.846800\pi\)
0.886398 0.462925i \(-0.153200\pi\)
\(920\) 0 0
\(921\) −3.74508e7 + 1.31391e7i −1.45483 + 0.510408i
\(922\) 0 0
\(923\) 1.00136e7 0.386890
\(924\) 0 0
\(925\) −2.47260e6 −0.0950166
\(926\) 0 0
\(927\) −6.64896e6 8.30952e6i −0.254129 0.317597i
\(928\) 0 0
\(929\) 726976.i 0.0276364i −0.999905 0.0138182i \(-0.995601\pi\)
0.999905 0.0138182i \(-0.00439860\pi\)
\(930\) 0 0
\(931\) 1.83303e7i 0.693099i
\(932\) 0 0
\(933\) 1.37411e7 + 3.91666e7i 0.516793 + 1.47303i
\(934\) 0 0
\(935\) −2.05745e7 −0.769662
\(936\) 0 0
\(937\) −1.31585e7 −0.489617 −0.244808 0.969571i \(-0.578725\pi\)
−0.244808 + 0.969571i \(0.578725\pi\)
\(938\) 0 0
\(939\) −2.06525e6 5.88665e6i −0.0764379 0.217873i
\(940\) 0 0
\(941\) 2.83098e7i 1.04223i 0.853488 + 0.521113i \(0.174483\pi\)
−0.853488 + 0.521113i \(0.825517\pi\)
\(942\) 0 0
\(943\) 1.59862e7i 0.585417i
\(944\) 0 0
\(945\) 2.62271e7 + 1.63897e7i 0.955369 + 0.597023i
\(946\) 0 0
\(947\) 3.47806e7 1.26026 0.630132 0.776488i \(-0.283001\pi\)
0.630132 + 0.776488i \(0.283001\pi\)
\(948\) 0 0
\(949\) −8.49277e6 −0.306115
\(950\) 0 0
\(951\) 2.27156e7 7.96946e6i 0.814465 0.285744i
\(952\) 0 0
\(953\) 2.91370e7i 1.03923i 0.854399 + 0.519617i \(0.173925\pi\)
−0.854399 + 0.519617i \(0.826075\pi\)
\(954\) 0 0
\(955\) 3.09991e7i 1.09987i
\(956\) 0 0
\(957\) −2.13690e7 + 7.49702e6i −0.754231 + 0.264612i
\(958\) 0 0
\(959\) −1.80998e7 −0.635516
\(960\) 0 0
\(961\) −28086.4 −0.000981042
\(962\) 0 0
\(963\) 4.18989e7 3.35259e7i 1.45592 1.16497i
\(964\) 0 0
\(965\) 3.85369e7i 1.33217i
\(966\) 0 0
\(967\) 1.15601e7i 0.397554i 0.980045 + 0.198777i \(0.0636969\pi\)
−0.980045 + 0.198777i \(0.936303\pi\)
\(968\) 0 0
\(969\) −1.10489e7 3.14931e7i −0.378017 1.07747i
\(970\) 0 0
\(971\) −1.20504e7 −0.410161 −0.205080 0.978745i \(-0.565746\pi\)
−0.205080 + 0.978745i \(0.565746\pi\)
\(972\) 0 0
\(973\) 1.52785e7 0.517368
\(974\) 0 0
\(975\) −595387. 1.69705e6i −0.0200580 0.0571720i
\(976\) 0 0
\(977\) 2.16156e7i 0.724487i −0.932084 0.362243i \(-0.882011\pi\)
0.932084 0.362243i \(-0.117989\pi\)
\(978\) 0 0
\(979\) 3.09931e7i 1.03349i
\(980\) 0 0
\(981\) −2.90246e7 + 2.32244e7i −0.962927 + 0.770497i
\(982\) 0 0
\(983\) −3.88290e7 −1.28166 −0.640829 0.767684i \(-0.721409\pi\)
−0.640829 + 0.767684i \(0.721409\pi\)
\(984\) 0 0
\(985\) −1.11715e7 −0.366876
\(986\) 0 0
\(987\) −4.94971e7 + 1.73654e7i −1.61729 + 0.567403i
\(988\) 0 0
\(989\) 2.49803e7i 0.812094i
\(990\) 0 0
\(991\) 4.70519e7i 1.52192i 0.648797 + 0.760962i \(0.275272\pi\)
−0.648797 + 0.760962i \(0.724728\pi\)
\(992\) 0 0
\(993\) 4.97109e6 1.74404e6i 0.159985 0.0561285i
\(994\) 0 0
\(995\) −2.04636e6 −0.0655275
\(996\) 0 0
\(997\) −5.60514e6 −0.178586 −0.0892932 0.996005i \(-0.528461\pi\)
−0.0892932 + 0.996005i \(0.528461\pi\)
\(998\) 0 0
\(999\) 1.05022e7 + 6.56299e6i 0.332942 + 0.208060i
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 384.6.c.d.383.14 yes 20
3.2 odd 2 384.6.c.a.383.8 yes 20
4.3 odd 2 384.6.c.a.383.7 20
8.3 odd 2 384.6.c.c.383.14 yes 20
8.5 even 2 384.6.c.b.383.7 yes 20
12.11 even 2 inner 384.6.c.d.383.13 yes 20
24.5 odd 2 384.6.c.c.383.13 yes 20
24.11 even 2 384.6.c.b.383.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.c.a.383.7 20 4.3 odd 2
384.6.c.a.383.8 yes 20 3.2 odd 2
384.6.c.b.383.7 yes 20 8.5 even 2
384.6.c.b.383.8 yes 20 24.11 even 2
384.6.c.c.383.13 yes 20 24.5 odd 2
384.6.c.c.383.14 yes 20 8.3 odd 2
384.6.c.d.383.13 yes 20 12.11 even 2 inner
384.6.c.d.383.14 yes 20 1.1 even 1 trivial