Properties

Label 448.6.a.t.1.2
Level $448$
Weight $6$
Character 448.1
Self dual yes
Analytic conductor $71.852$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,6,Mod(1,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

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: yes
Analytic conductor: \(71.8519512762\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.78709\) of defining polynomial
Character \(\chi\) \(=\) 448.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+15.5742 q^{3} -96.7225 q^{5} +49.0000 q^{7} -0.445054 q^{9} +O(q^{10})\) \(q+15.5742 q^{3} -96.7225 q^{5} +49.0000 q^{7} -0.445054 q^{9} +281.445 q^{11} +269.393 q^{13} -1506.37 q^{15} +1719.45 q^{17} -1172.84 q^{19} +763.135 q^{21} -785.341 q^{23} +6230.25 q^{25} -3791.46 q^{27} -6149.46 q^{29} +7006.58 q^{31} +4383.27 q^{33} -4739.40 q^{35} +11499.9 q^{37} +4195.57 q^{39} -13245.3 q^{41} -19824.2 q^{43} +43.0467 q^{45} -6887.96 q^{47} +2401.00 q^{49} +26778.9 q^{51} -8654.97 q^{53} -27222.1 q^{55} -18266.0 q^{57} -47856.6 q^{59} -52762.1 q^{61} -21.8076 q^{63} -26056.4 q^{65} -24040.2 q^{67} -12231.0 q^{69} -12540.9 q^{71} +4079.29 q^{73} +97031.0 q^{75} +13790.8 q^{77} +11919.1 q^{79} -58940.7 q^{81} +81916.2 q^{83} -166309. q^{85} -95772.7 q^{87} -96968.7 q^{89} +13200.2 q^{91} +109122. q^{93} +113440. q^{95} -26410.7 q^{97} -125.258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 82 q^{5} + 98 q^{7} + 222 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 82 q^{5} + 98 q^{7} + 222 q^{9} + 340 q^{11} - 910 q^{13} - 1824 q^{15} + 3216 q^{17} - 674 q^{19} - 294 q^{21} + 1104 q^{23} + 3322 q^{25} - 3348 q^{27} - 8064 q^{29} + 6212 q^{31} + 3120 q^{33} - 4018 q^{35} + 8512 q^{37} + 29640 q^{39} - 1304 q^{41} - 10004 q^{43} + 3318 q^{45} + 12748 q^{47} + 4802 q^{49} - 5508 q^{51} + 11220 q^{53} - 26360 q^{55} - 29028 q^{57} - 12018 q^{59} - 102738 q^{61} + 10878 q^{63} - 43420 q^{65} + 24136 q^{67} - 52992 q^{69} - 89720 q^{71} - 55588 q^{73} + 159774 q^{75} + 16660 q^{77} - 48824 q^{79} - 122562 q^{81} + 35782 q^{83} - 144276 q^{85} - 54468 q^{87} - 18300 q^{89} - 44590 q^{91} + 126264 q^{93} + 120784 q^{95} - 69984 q^{97} + 12900 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 15.5742 0.999084 0.499542 0.866290i \(-0.333502\pi\)
0.499542 + 0.866290i \(0.333502\pi\)
\(4\) 0 0
\(5\) −96.7225 −1.73023 −0.865113 0.501578i \(-0.832753\pi\)
−0.865113 + 0.501578i \(0.832753\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) −0.445054 −0.00183150
\(10\) 0 0
\(11\) 281.445 0.701313 0.350657 0.936504i \(-0.385958\pi\)
0.350657 + 0.936504i \(0.385958\pi\)
\(12\) 0 0
\(13\) 269.393 0.442107 0.221054 0.975262i \(-0.429050\pi\)
0.221054 + 0.975262i \(0.429050\pi\)
\(14\) 0 0
\(15\) −1506.37 −1.72864
\(16\) 0 0
\(17\) 1719.45 1.44300 0.721499 0.692415i \(-0.243453\pi\)
0.721499 + 0.692415i \(0.243453\pi\)
\(18\) 0 0
\(19\) −1172.84 −0.745339 −0.372670 0.927964i \(-0.621558\pi\)
−0.372670 + 0.927964i \(0.621558\pi\)
\(20\) 0 0
\(21\) 763.135 0.377618
\(22\) 0 0
\(23\) −785.341 −0.309555 −0.154778 0.987949i \(-0.549466\pi\)
−0.154778 + 0.987949i \(0.549466\pi\)
\(24\) 0 0
\(25\) 6230.25 1.99368
\(26\) 0 0
\(27\) −3791.46 −1.00091
\(28\) 0 0
\(29\) −6149.46 −1.35782 −0.678909 0.734222i \(-0.737547\pi\)
−0.678909 + 0.734222i \(0.737547\pi\)
\(30\) 0 0
\(31\) 7006.58 1.30949 0.654744 0.755851i \(-0.272776\pi\)
0.654744 + 0.755851i \(0.272776\pi\)
\(32\) 0 0
\(33\) 4383.27 0.700671
\(34\) 0 0
\(35\) −4739.40 −0.653964
\(36\) 0 0
\(37\) 11499.9 1.38099 0.690495 0.723337i \(-0.257393\pi\)
0.690495 + 0.723337i \(0.257393\pi\)
\(38\) 0 0
\(39\) 4195.57 0.441702
\(40\) 0 0
\(41\) −13245.3 −1.23056 −0.615279 0.788310i \(-0.710957\pi\)
−0.615279 + 0.788310i \(0.710957\pi\)
\(42\) 0 0
\(43\) −19824.2 −1.63502 −0.817512 0.575911i \(-0.804647\pi\)
−0.817512 + 0.575911i \(0.804647\pi\)
\(44\) 0 0
\(45\) 43.0467 0.00316890
\(46\) 0 0
\(47\) −6887.96 −0.454827 −0.227413 0.973798i \(-0.573027\pi\)
−0.227413 + 0.973798i \(0.573027\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 26778.9 1.44168
\(52\) 0 0
\(53\) −8654.97 −0.423229 −0.211615 0.977353i \(-0.567872\pi\)
−0.211615 + 0.977353i \(0.567872\pi\)
\(54\) 0 0
\(55\) −27222.1 −1.21343
\(56\) 0 0
\(57\) −18266.0 −0.744656
\(58\) 0 0
\(59\) −47856.6 −1.78983 −0.894915 0.446236i \(-0.852764\pi\)
−0.894915 + 0.446236i \(0.852764\pi\)
\(60\) 0 0
\(61\) −52762.1 −1.81550 −0.907752 0.419507i \(-0.862203\pi\)
−0.907752 + 0.419507i \(0.862203\pi\)
\(62\) 0 0
\(63\) −21.8076 −0.000692241 0
\(64\) 0 0
\(65\) −26056.4 −0.764945
\(66\) 0 0
\(67\) −24040.2 −0.654261 −0.327130 0.944979i \(-0.606082\pi\)
−0.327130 + 0.944979i \(0.606082\pi\)
\(68\) 0 0
\(69\) −12231.0 −0.309272
\(70\) 0 0
\(71\) −12540.9 −0.295246 −0.147623 0.989044i \(-0.547162\pi\)
−0.147623 + 0.989044i \(0.547162\pi\)
\(72\) 0 0
\(73\) 4079.29 0.0895936 0.0447968 0.998996i \(-0.485736\pi\)
0.0447968 + 0.998996i \(0.485736\pi\)
\(74\) 0 0
\(75\) 97031.0 1.99185
\(76\) 0 0
\(77\) 13790.8 0.265071
\(78\) 0 0
\(79\) 11919.1 0.214870 0.107435 0.994212i \(-0.465736\pi\)
0.107435 + 0.994212i \(0.465736\pi\)
\(80\) 0 0
\(81\) −58940.7 −0.998165
\(82\) 0 0
\(83\) 81916.2 1.30519 0.652596 0.757706i \(-0.273680\pi\)
0.652596 + 0.757706i \(0.273680\pi\)
\(84\) 0 0
\(85\) −166309. −2.49671
\(86\) 0 0
\(87\) −95772.7 −1.35657
\(88\) 0 0
\(89\) −96968.7 −1.29765 −0.648824 0.760939i \(-0.724739\pi\)
−0.648824 + 0.760939i \(0.724739\pi\)
\(90\) 0 0
\(91\) 13200.2 0.167101
\(92\) 0 0
\(93\) 109122. 1.30829
\(94\) 0 0
\(95\) 113440. 1.28960
\(96\) 0 0
\(97\) −26410.7 −0.285004 −0.142502 0.989795i \(-0.545515\pi\)
−0.142502 + 0.989795i \(0.545515\pi\)
\(98\) 0 0
\(99\) −125.258 −0.00128445
\(100\) 0 0
\(101\) 73137.7 0.713408 0.356704 0.934217i \(-0.383901\pi\)
0.356704 + 0.934217i \(0.383901\pi\)
\(102\) 0 0
\(103\) −87649.5 −0.814060 −0.407030 0.913415i \(-0.633435\pi\)
−0.407030 + 0.913415i \(0.633435\pi\)
\(104\) 0 0
\(105\) −73812.3 −0.653365
\(106\) 0 0
\(107\) −115438. −0.974743 −0.487372 0.873195i \(-0.662044\pi\)
−0.487372 + 0.873195i \(0.662044\pi\)
\(108\) 0 0
\(109\) 117303. 0.945675 0.472837 0.881150i \(-0.343230\pi\)
0.472837 + 0.881150i \(0.343230\pi\)
\(110\) 0 0
\(111\) 179102. 1.37973
\(112\) 0 0
\(113\) 181535. 1.33741 0.668705 0.743528i \(-0.266849\pi\)
0.668705 + 0.743528i \(0.266849\pi\)
\(114\) 0 0
\(115\) 75960.1 0.535601
\(116\) 0 0
\(117\) −119.894 −0.000809718 0
\(118\) 0 0
\(119\) 84252.8 0.545402
\(120\) 0 0
\(121\) −81839.7 −0.508160
\(122\) 0 0
\(123\) −206284. −1.22943
\(124\) 0 0
\(125\) −300347. −1.71929
\(126\) 0 0
\(127\) −106111. −0.583780 −0.291890 0.956452i \(-0.594284\pi\)
−0.291890 + 0.956452i \(0.594284\pi\)
\(128\) 0 0
\(129\) −308745. −1.63353
\(130\) 0 0
\(131\) 300334. 1.52907 0.764534 0.644584i \(-0.222969\pi\)
0.764534 + 0.644584i \(0.222969\pi\)
\(132\) 0 0
\(133\) −57469.1 −0.281712
\(134\) 0 0
\(135\) 366719. 1.73181
\(136\) 0 0
\(137\) −91892.5 −0.418291 −0.209146 0.977885i \(-0.567068\pi\)
−0.209146 + 0.977885i \(0.567068\pi\)
\(138\) 0 0
\(139\) 153133. 0.672252 0.336126 0.941817i \(-0.390883\pi\)
0.336126 + 0.941817i \(0.390883\pi\)
\(140\) 0 0
\(141\) −107274. −0.454410
\(142\) 0 0
\(143\) 75819.3 0.310056
\(144\) 0 0
\(145\) 594791. 2.34933
\(146\) 0 0
\(147\) 37393.6 0.142726
\(148\) 0 0
\(149\) −498602. −1.83987 −0.919937 0.392066i \(-0.871760\pi\)
−0.919937 + 0.392066i \(0.871760\pi\)
\(150\) 0 0
\(151\) −209609. −0.748114 −0.374057 0.927406i \(-0.622034\pi\)
−0.374057 + 0.927406i \(0.622034\pi\)
\(152\) 0 0
\(153\) −765.245 −0.00264285
\(154\) 0 0
\(155\) −677694. −2.26571
\(156\) 0 0
\(157\) 7381.72 0.0239006 0.0119503 0.999929i \(-0.496196\pi\)
0.0119503 + 0.999929i \(0.496196\pi\)
\(158\) 0 0
\(159\) −134794. −0.422842
\(160\) 0 0
\(161\) −38481.7 −0.117001
\(162\) 0 0
\(163\) 388154. 1.14429 0.572144 0.820153i \(-0.306112\pi\)
0.572144 + 0.820153i \(0.306112\pi\)
\(164\) 0 0
\(165\) −423961. −1.21232
\(166\) 0 0
\(167\) 168240. 0.466808 0.233404 0.972380i \(-0.425013\pi\)
0.233404 + 0.972380i \(0.425013\pi\)
\(168\) 0 0
\(169\) −298720. −0.804541
\(170\) 0 0
\(171\) 521.976 0.00136509
\(172\) 0 0
\(173\) −321513. −0.816740 −0.408370 0.912817i \(-0.633903\pi\)
−0.408370 + 0.912817i \(0.633903\pi\)
\(174\) 0 0
\(175\) 305282. 0.753540
\(176\) 0 0
\(177\) −745327. −1.78819
\(178\) 0 0
\(179\) −611906. −1.42742 −0.713710 0.700441i \(-0.752987\pi\)
−0.713710 + 0.700441i \(0.752987\pi\)
\(180\) 0 0
\(181\) −261474. −0.593242 −0.296621 0.954995i \(-0.595860\pi\)
−0.296621 + 0.954995i \(0.595860\pi\)
\(182\) 0 0
\(183\) −821726. −1.81384
\(184\) 0 0
\(185\) −1.11230e6 −2.38943
\(186\) 0 0
\(187\) 483929. 1.01199
\(188\) 0 0
\(189\) −185781. −0.378310
\(190\) 0 0
\(191\) −703301. −1.39495 −0.697474 0.716610i \(-0.745693\pi\)
−0.697474 + 0.716610i \(0.745693\pi\)
\(192\) 0 0
\(193\) 516411. 0.997934 0.498967 0.866621i \(-0.333713\pi\)
0.498967 + 0.866621i \(0.333713\pi\)
\(194\) 0 0
\(195\) −405806. −0.764244
\(196\) 0 0
\(197\) −142579. −0.261752 −0.130876 0.991399i \(-0.541779\pi\)
−0.130876 + 0.991399i \(0.541779\pi\)
\(198\) 0 0
\(199\) −984837. −1.76292 −0.881458 0.472262i \(-0.843438\pi\)
−0.881458 + 0.472262i \(0.843438\pi\)
\(200\) 0 0
\(201\) −374406. −0.653662
\(202\) 0 0
\(203\) −301323. −0.513207
\(204\) 0 0
\(205\) 1.28112e6 2.12914
\(206\) 0 0
\(207\) 349.519 0.000566950 0
\(208\) 0 0
\(209\) −330089. −0.522716
\(210\) 0 0
\(211\) −242836. −0.375497 −0.187748 0.982217i \(-0.560119\pi\)
−0.187748 + 0.982217i \(0.560119\pi\)
\(212\) 0 0
\(213\) −195315. −0.294976
\(214\) 0 0
\(215\) 1.91745e6 2.82896
\(216\) 0 0
\(217\) 343322. 0.494940
\(218\) 0 0
\(219\) 63531.5 0.0895115
\(220\) 0 0
\(221\) 463206. 0.637960
\(222\) 0 0
\(223\) 797063. 1.07332 0.536661 0.843798i \(-0.319685\pi\)
0.536661 + 0.843798i \(0.319685\pi\)
\(224\) 0 0
\(225\) −2772.79 −0.00365142
\(226\) 0 0
\(227\) −26121.4 −0.0336459 −0.0168229 0.999858i \(-0.505355\pi\)
−0.0168229 + 0.999858i \(0.505355\pi\)
\(228\) 0 0
\(229\) 761592. 0.959695 0.479848 0.877352i \(-0.340692\pi\)
0.479848 + 0.877352i \(0.340692\pi\)
\(230\) 0 0
\(231\) 214780. 0.264829
\(232\) 0 0
\(233\) 657852. 0.793850 0.396925 0.917851i \(-0.370077\pi\)
0.396925 + 0.917851i \(0.370077\pi\)
\(234\) 0 0
\(235\) 666221. 0.786953
\(236\) 0 0
\(237\) 185630. 0.214673
\(238\) 0 0
\(239\) −965388. −1.09322 −0.546609 0.837388i \(-0.684082\pi\)
−0.546609 + 0.837388i \(0.684082\pi\)
\(240\) 0 0
\(241\) 35064.2 0.0388885 0.0194443 0.999811i \(-0.493810\pi\)
0.0194443 + 0.999811i \(0.493810\pi\)
\(242\) 0 0
\(243\) 3371.72 0.00366299
\(244\) 0 0
\(245\) −232231. −0.247175
\(246\) 0 0
\(247\) −315954. −0.329520
\(248\) 0 0
\(249\) 1.27578e6 1.30400
\(250\) 0 0
\(251\) 1.31610e6 1.31857 0.659286 0.751892i \(-0.270859\pi\)
0.659286 + 0.751892i \(0.270859\pi\)
\(252\) 0 0
\(253\) −221030. −0.217095
\(254\) 0 0
\(255\) −2.59013e6 −2.49443
\(256\) 0 0
\(257\) −529802. −0.500358 −0.250179 0.968200i \(-0.580489\pi\)
−0.250179 + 0.968200i \(0.580489\pi\)
\(258\) 0 0
\(259\) 563496. 0.521966
\(260\) 0 0
\(261\) 2736.84 0.00248684
\(262\) 0 0
\(263\) 1.29457e6 1.15408 0.577041 0.816715i \(-0.304207\pi\)
0.577041 + 0.816715i \(0.304207\pi\)
\(264\) 0 0
\(265\) 837130. 0.732282
\(266\) 0 0
\(267\) −1.51021e6 −1.29646
\(268\) 0 0
\(269\) 444336. 0.374396 0.187198 0.982322i \(-0.440059\pi\)
0.187198 + 0.982322i \(0.440059\pi\)
\(270\) 0 0
\(271\) −624757. −0.516759 −0.258379 0.966044i \(-0.583188\pi\)
−0.258379 + 0.966044i \(0.583188\pi\)
\(272\) 0 0
\(273\) 205583. 0.166948
\(274\) 0 0
\(275\) 1.75347e6 1.39819
\(276\) 0 0
\(277\) 2.00202e6 1.56772 0.783860 0.620938i \(-0.213248\pi\)
0.783860 + 0.620938i \(0.213248\pi\)
\(278\) 0 0
\(279\) −3118.30 −0.00239832
\(280\) 0 0
\(281\) 249316. 0.188358 0.0941792 0.995555i \(-0.469977\pi\)
0.0941792 + 0.995555i \(0.469977\pi\)
\(282\) 0 0
\(283\) 645136. 0.478835 0.239417 0.970917i \(-0.423044\pi\)
0.239417 + 0.970917i \(0.423044\pi\)
\(284\) 0 0
\(285\) 1.76673e6 1.28842
\(286\) 0 0
\(287\) −649019. −0.465107
\(288\) 0 0
\(289\) 1.53663e6 1.08225
\(290\) 0 0
\(291\) −411325. −0.284743
\(292\) 0 0
\(293\) −2.00830e6 −1.36666 −0.683328 0.730112i \(-0.739468\pi\)
−0.683328 + 0.730112i \(0.739468\pi\)
\(294\) 0 0
\(295\) 4.62881e6 3.09681
\(296\) 0 0
\(297\) −1.06709e6 −0.701954
\(298\) 0 0
\(299\) −211565. −0.136857
\(300\) 0 0
\(301\) −971385. −0.617981
\(302\) 0 0
\(303\) 1.13906e6 0.712754
\(304\) 0 0
\(305\) 5.10328e6 3.14123
\(306\) 0 0
\(307\) −1.71113e6 −1.03618 −0.518092 0.855325i \(-0.673358\pi\)
−0.518092 + 0.855325i \(0.673358\pi\)
\(308\) 0 0
\(309\) −1.36507e6 −0.813314
\(310\) 0 0
\(311\) 602050. 0.352965 0.176483 0.984304i \(-0.443528\pi\)
0.176483 + 0.984304i \(0.443528\pi\)
\(312\) 0 0
\(313\) −1.34208e6 −0.774318 −0.387159 0.922013i \(-0.626543\pi\)
−0.387159 + 0.922013i \(0.626543\pi\)
\(314\) 0 0
\(315\) 2109.29 0.00119773
\(316\) 0 0
\(317\) −1.01803e6 −0.569001 −0.284500 0.958676i \(-0.591828\pi\)
−0.284500 + 0.958676i \(0.591828\pi\)
\(318\) 0 0
\(319\) −1.73073e6 −0.952256
\(320\) 0 0
\(321\) −1.79786e6 −0.973850
\(322\) 0 0
\(323\) −2.01663e6 −1.07552
\(324\) 0 0
\(325\) 1.67838e6 0.881420
\(326\) 0 0
\(327\) 1.82689e6 0.944809
\(328\) 0 0
\(329\) −337510. −0.171908
\(330\) 0 0
\(331\) −532371. −0.267082 −0.133541 0.991043i \(-0.542635\pi\)
−0.133541 + 0.991043i \(0.542635\pi\)
\(332\) 0 0
\(333\) −5118.09 −0.00252928
\(334\) 0 0
\(335\) 2.32523e6 1.13202
\(336\) 0 0
\(337\) −234947. −0.112693 −0.0563463 0.998411i \(-0.517945\pi\)
−0.0563463 + 0.998411i \(0.517945\pi\)
\(338\) 0 0
\(339\) 2.82726e6 1.33618
\(340\) 0 0
\(341\) 1.97197e6 0.918361
\(342\) 0 0
\(343\) 117649. 0.0539949
\(344\) 0 0
\(345\) 1.18302e6 0.535110
\(346\) 0 0
\(347\) −1.90054e6 −0.847330 −0.423665 0.905819i \(-0.639257\pi\)
−0.423665 + 0.905819i \(0.639257\pi\)
\(348\) 0 0
\(349\) −341162. −0.149933 −0.0749665 0.997186i \(-0.523885\pi\)
−0.0749665 + 0.997186i \(0.523885\pi\)
\(350\) 0 0
\(351\) −1.02139e6 −0.442511
\(352\) 0 0
\(353\) 673122. 0.287513 0.143756 0.989613i \(-0.454082\pi\)
0.143756 + 0.989613i \(0.454082\pi\)
\(354\) 0 0
\(355\) 1.21299e6 0.510842
\(356\) 0 0
\(357\) 1.31217e6 0.544903
\(358\) 0 0
\(359\) −1.43301e6 −0.586833 −0.293417 0.955985i \(-0.594792\pi\)
−0.293417 + 0.955985i \(0.594792\pi\)
\(360\) 0 0
\(361\) −1.10055e6 −0.444469
\(362\) 0 0
\(363\) −1.27459e6 −0.507694
\(364\) 0 0
\(365\) −394559. −0.155017
\(366\) 0 0
\(367\) 2.04781e6 0.793643 0.396822 0.917896i \(-0.370113\pi\)
0.396822 + 0.917896i \(0.370113\pi\)
\(368\) 0 0
\(369\) 5894.87 0.00225376
\(370\) 0 0
\(371\) −424093. −0.159966
\(372\) 0 0
\(373\) 3.33578e6 1.24144 0.620719 0.784033i \(-0.286841\pi\)
0.620719 + 0.784033i \(0.286841\pi\)
\(374\) 0 0
\(375\) −4.67766e6 −1.71771
\(376\) 0 0
\(377\) −1.65662e6 −0.600301
\(378\) 0 0
\(379\) 2.29135e6 0.819394 0.409697 0.912222i \(-0.365634\pi\)
0.409697 + 0.912222i \(0.365634\pi\)
\(380\) 0 0
\(381\) −1.65258e6 −0.583245
\(382\) 0 0
\(383\) −1.84422e6 −0.642414 −0.321207 0.947009i \(-0.604089\pi\)
−0.321207 + 0.947009i \(0.604089\pi\)
\(384\) 0 0
\(385\) −1.33388e6 −0.458633
\(386\) 0 0
\(387\) 8822.83 0.00299454
\(388\) 0 0
\(389\) −3.66601e6 −1.22834 −0.614172 0.789172i \(-0.710510\pi\)
−0.614172 + 0.789172i \(0.710510\pi\)
\(390\) 0 0
\(391\) −1.35035e6 −0.446688
\(392\) 0 0
\(393\) 4.67746e6 1.52767
\(394\) 0 0
\(395\) −1.15284e6 −0.371773
\(396\) 0 0
\(397\) 4.59382e6 1.46284 0.731421 0.681926i \(-0.238858\pi\)
0.731421 + 0.681926i \(0.238858\pi\)
\(398\) 0 0
\(399\) −895033. −0.281454
\(400\) 0 0
\(401\) −1.10984e6 −0.344666 −0.172333 0.985039i \(-0.555130\pi\)
−0.172333 + 0.985039i \(0.555130\pi\)
\(402\) 0 0
\(403\) 1.88752e6 0.578934
\(404\) 0 0
\(405\) 5.70089e6 1.72705
\(406\) 0 0
\(407\) 3.23660e6 0.968507
\(408\) 0 0
\(409\) 3.60689e6 1.06617 0.533083 0.846063i \(-0.321033\pi\)
0.533083 + 0.846063i \(0.321033\pi\)
\(410\) 0 0
\(411\) −1.43115e6 −0.417908
\(412\) 0 0
\(413\) −2.34497e6 −0.676492
\(414\) 0 0
\(415\) −7.92314e6 −2.25828
\(416\) 0 0
\(417\) 2.38492e6 0.671636
\(418\) 0 0
\(419\) 5.51790e6 1.53546 0.767731 0.640773i \(-0.221386\pi\)
0.767731 + 0.640773i \(0.221386\pi\)
\(420\) 0 0
\(421\) 2.59092e6 0.712440 0.356220 0.934402i \(-0.384065\pi\)
0.356220 + 0.934402i \(0.384065\pi\)
\(422\) 0 0
\(423\) 3065.51 0.000833014 0
\(424\) 0 0
\(425\) 1.07126e7 2.87688
\(426\) 0 0
\(427\) −2.58534e6 −0.686196
\(428\) 0 0
\(429\) 1.18082e6 0.309772
\(430\) 0 0
\(431\) 3.45364e6 0.895537 0.447769 0.894150i \(-0.352219\pi\)
0.447769 + 0.894150i \(0.352219\pi\)
\(432\) 0 0
\(433\) 397522. 0.101892 0.0509461 0.998701i \(-0.483776\pi\)
0.0509461 + 0.998701i \(0.483776\pi\)
\(434\) 0 0
\(435\) 9.26338e6 2.34718
\(436\) 0 0
\(437\) 921077. 0.230724
\(438\) 0 0
\(439\) −1.60903e6 −0.398476 −0.199238 0.979951i \(-0.563847\pi\)
−0.199238 + 0.979951i \(0.563847\pi\)
\(440\) 0 0
\(441\) −1068.57 −0.000261642 0
\(442\) 0 0
\(443\) −7.08182e6 −1.71449 −0.857246 0.514906i \(-0.827827\pi\)
−0.857246 + 0.514906i \(0.827827\pi\)
\(444\) 0 0
\(445\) 9.37906e6 2.24522
\(446\) 0 0
\(447\) −7.76531e6 −1.83819
\(448\) 0 0
\(449\) −6.46156e6 −1.51259 −0.756295 0.654230i \(-0.772993\pi\)
−0.756295 + 0.654230i \(0.772993\pi\)
\(450\) 0 0
\(451\) −3.72782e6 −0.863006
\(452\) 0 0
\(453\) −3.26449e6 −0.747429
\(454\) 0 0
\(455\) −1.27676e6 −0.289122
\(456\) 0 0
\(457\) −3.37073e6 −0.754976 −0.377488 0.926015i \(-0.623212\pi\)
−0.377488 + 0.926015i \(0.623212\pi\)
\(458\) 0 0
\(459\) −6.51920e6 −1.44432
\(460\) 0 0
\(461\) −6.20821e6 −1.36055 −0.680274 0.732958i \(-0.738139\pi\)
−0.680274 + 0.732958i \(0.738139\pi\)
\(462\) 0 0
\(463\) 5.82940e6 1.26378 0.631890 0.775058i \(-0.282279\pi\)
0.631890 + 0.775058i \(0.282279\pi\)
\(464\) 0 0
\(465\) −1.05545e7 −2.26363
\(466\) 0 0
\(467\) −174742. −0.0370770 −0.0185385 0.999828i \(-0.505901\pi\)
−0.0185385 + 0.999828i \(0.505901\pi\)
\(468\) 0 0
\(469\) −1.17797e6 −0.247287
\(470\) 0 0
\(471\) 114964. 0.0238787
\(472\) 0 0
\(473\) −5.57942e6 −1.14666
\(474\) 0 0
\(475\) −7.30707e6 −1.48597
\(476\) 0 0
\(477\) 3851.93 0.000775143 0
\(478\) 0 0
\(479\) −371746. −0.0740299 −0.0370150 0.999315i \(-0.511785\pi\)
−0.0370150 + 0.999315i \(0.511785\pi\)
\(480\) 0 0
\(481\) 3.09800e6 0.610546
\(482\) 0 0
\(483\) −599321. −0.116894
\(484\) 0 0
\(485\) 2.55451e6 0.493121
\(486\) 0 0
\(487\) 8.71307e6 1.66475 0.832375 0.554213i \(-0.186981\pi\)
0.832375 + 0.554213i \(0.186981\pi\)
\(488\) 0 0
\(489\) 6.04519e6 1.14324
\(490\) 0 0
\(491\) 6.88530e6 1.28890 0.644450 0.764646i \(-0.277086\pi\)
0.644450 + 0.764646i \(0.277086\pi\)
\(492\) 0 0
\(493\) −1.05737e7 −1.95933
\(494\) 0 0
\(495\) 12115.3 0.00222239
\(496\) 0 0
\(497\) −614506. −0.111592
\(498\) 0 0
\(499\) −45442.7 −0.00816982 −0.00408491 0.999992i \(-0.501300\pi\)
−0.00408491 + 0.999992i \(0.501300\pi\)
\(500\) 0 0
\(501\) 2.62020e6 0.466381
\(502\) 0 0
\(503\) 623417. 0.109865 0.0549324 0.998490i \(-0.482506\pi\)
0.0549324 + 0.998490i \(0.482506\pi\)
\(504\) 0 0
\(505\) −7.07406e6 −1.23436
\(506\) 0 0
\(507\) −4.65233e6 −0.803804
\(508\) 0 0
\(509\) −7.69092e6 −1.31578 −0.657891 0.753113i \(-0.728551\pi\)
−0.657891 + 0.753113i \(0.728551\pi\)
\(510\) 0 0
\(511\) 199885. 0.0338632
\(512\) 0 0
\(513\) 4.44676e6 0.746020
\(514\) 0 0
\(515\) 8.47768e6 1.40851
\(516\) 0 0
\(517\) −1.93858e6 −0.318976
\(518\) 0 0
\(519\) −5.00730e6 −0.815991
\(520\) 0 0
\(521\) 7.32268e6 1.18189 0.590943 0.806713i \(-0.298756\pi\)
0.590943 + 0.806713i \(0.298756\pi\)
\(522\) 0 0
\(523\) 5.17976e6 0.828047 0.414024 0.910266i \(-0.364123\pi\)
0.414024 + 0.910266i \(0.364123\pi\)
\(524\) 0 0
\(525\) 4.75452e6 0.752850
\(526\) 0 0
\(527\) 1.20474e7 1.88959
\(528\) 0 0
\(529\) −5.81958e6 −0.904175
\(530\) 0 0
\(531\) 21298.8 0.00327807
\(532\) 0 0
\(533\) −3.56819e6 −0.544038
\(534\) 0 0
\(535\) 1.11655e7 1.68653
\(536\) 0 0
\(537\) −9.52993e6 −1.42611
\(538\) 0 0
\(539\) 675750. 0.100188
\(540\) 0 0
\(541\) −9.31804e6 −1.36877 −0.684387 0.729119i \(-0.739930\pi\)
−0.684387 + 0.729119i \(0.739930\pi\)
\(542\) 0 0
\(543\) −4.07224e6 −0.592698
\(544\) 0 0
\(545\) −1.13458e7 −1.63623
\(546\) 0 0
\(547\) 1.26417e7 1.80650 0.903250 0.429114i \(-0.141174\pi\)
0.903250 + 0.429114i \(0.141174\pi\)
\(548\) 0 0
\(549\) 23482.0 0.00332509
\(550\) 0 0
\(551\) 7.21232e6 1.01204
\(552\) 0 0
\(553\) 584035. 0.0812131
\(554\) 0 0
\(555\) −1.73232e7 −2.38724
\(556\) 0 0
\(557\) 3.84693e6 0.525384 0.262692 0.964880i \(-0.415390\pi\)
0.262692 + 0.964880i \(0.415390\pi\)
\(558\) 0 0
\(559\) −5.34050e6 −0.722856
\(560\) 0 0
\(561\) 7.53680e6 1.01107
\(562\) 0 0
\(563\) 3.69193e6 0.490888 0.245444 0.969411i \(-0.421066\pi\)
0.245444 + 0.969411i \(0.421066\pi\)
\(564\) 0 0
\(565\) −1.75585e7 −2.31402
\(566\) 0 0
\(567\) −2.88809e6 −0.377271
\(568\) 0 0
\(569\) −6.64173e6 −0.860005 −0.430002 0.902828i \(-0.641487\pi\)
−0.430002 + 0.902828i \(0.641487\pi\)
\(570\) 0 0
\(571\) 292599. 0.0375562 0.0187781 0.999824i \(-0.494022\pi\)
0.0187781 + 0.999824i \(0.494022\pi\)
\(572\) 0 0
\(573\) −1.09533e7 −1.39367
\(574\) 0 0
\(575\) −4.89287e6 −0.617154
\(576\) 0 0
\(577\) 3.55569e6 0.444615 0.222307 0.974977i \(-0.428641\pi\)
0.222307 + 0.974977i \(0.428641\pi\)
\(578\) 0 0
\(579\) 8.04267e6 0.997020
\(580\) 0 0
\(581\) 4.01389e6 0.493316
\(582\) 0 0
\(583\) −2.43590e6 −0.296816
\(584\) 0 0
\(585\) 11596.5 0.00140099
\(586\) 0 0
\(587\) −2.64810e6 −0.317204 −0.158602 0.987343i \(-0.550699\pi\)
−0.158602 + 0.987343i \(0.550699\pi\)
\(588\) 0 0
\(589\) −8.21758e6 −0.976013
\(590\) 0 0
\(591\) −2.22055e6 −0.261512
\(592\) 0 0
\(593\) 1.05356e6 0.123034 0.0615168 0.998106i \(-0.480406\pi\)
0.0615168 + 0.998106i \(0.480406\pi\)
\(594\) 0 0
\(595\) −8.14914e6 −0.943669
\(596\) 0 0
\(597\) −1.53380e7 −1.76130
\(598\) 0 0
\(599\) −9.19012e6 −1.04654 −0.523268 0.852168i \(-0.675287\pi\)
−0.523268 + 0.852168i \(0.675287\pi\)
\(600\) 0 0
\(601\) 3.46924e6 0.391785 0.195893 0.980625i \(-0.437240\pi\)
0.195893 + 0.980625i \(0.437240\pi\)
\(602\) 0 0
\(603\) 10699.2 0.00119828
\(604\) 0 0
\(605\) 7.91574e6 0.879231
\(606\) 0 0
\(607\) 744150. 0.0819764 0.0409882 0.999160i \(-0.486949\pi\)
0.0409882 + 0.999160i \(0.486949\pi\)
\(608\) 0 0
\(609\) −4.69286e6 −0.512737
\(610\) 0 0
\(611\) −1.85557e6 −0.201082
\(612\) 0 0
\(613\) −1.14578e7 −1.23154 −0.615772 0.787925i \(-0.711156\pi\)
−0.615772 + 0.787925i \(0.711156\pi\)
\(614\) 0 0
\(615\) 1.99524e7 2.12719
\(616\) 0 0
\(617\) −3.90052e6 −0.412486 −0.206243 0.978501i \(-0.566124\pi\)
−0.206243 + 0.978501i \(0.566124\pi\)
\(618\) 0 0
\(619\) 2.65029e6 0.278014 0.139007 0.990291i \(-0.455609\pi\)
0.139007 + 0.990291i \(0.455609\pi\)
\(620\) 0 0
\(621\) 2.97758e6 0.309838
\(622\) 0 0
\(623\) −4.75147e6 −0.490464
\(624\) 0 0
\(625\) 9.58083e6 0.981077
\(626\) 0 0
\(627\) −5.14087e6 −0.522237
\(628\) 0 0
\(629\) 1.97735e7 1.99277
\(630\) 0 0
\(631\) 8.97599e6 0.897447 0.448723 0.893671i \(-0.351879\pi\)
0.448723 + 0.893671i \(0.351879\pi\)
\(632\) 0 0
\(633\) −3.78197e6 −0.375153
\(634\) 0 0
\(635\) 1.02633e7 1.01007
\(636\) 0 0
\(637\) 646812. 0.0631582
\(638\) 0 0
\(639\) 5581.39 0.000540742 0
\(640\) 0 0
\(641\) −2.87152e6 −0.276036 −0.138018 0.990430i \(-0.544073\pi\)
−0.138018 + 0.990430i \(0.544073\pi\)
\(642\) 0 0
\(643\) 1.38761e7 1.32354 0.661772 0.749705i \(-0.269805\pi\)
0.661772 + 0.749705i \(0.269805\pi\)
\(644\) 0 0
\(645\) 2.98626e7 2.82637
\(646\) 0 0
\(647\) 7.24853e6 0.680752 0.340376 0.940289i \(-0.389446\pi\)
0.340376 + 0.940289i \(0.389446\pi\)
\(648\) 0 0
\(649\) −1.34690e7 −1.25523
\(650\) 0 0
\(651\) 5.34696e6 0.494487
\(652\) 0 0
\(653\) 1.23719e7 1.13541 0.567706 0.823231i \(-0.307831\pi\)
0.567706 + 0.823231i \(0.307831\pi\)
\(654\) 0 0
\(655\) −2.90491e7 −2.64563
\(656\) 0 0
\(657\) −1815.50 −0.000164090 0
\(658\) 0 0
\(659\) −572294. −0.0513341 −0.0256670 0.999671i \(-0.508171\pi\)
−0.0256670 + 0.999671i \(0.508171\pi\)
\(660\) 0 0
\(661\) −6.73679e6 −0.599721 −0.299861 0.953983i \(-0.596940\pi\)
−0.299861 + 0.953983i \(0.596940\pi\)
\(662\) 0 0
\(663\) 7.21405e6 0.637376
\(664\) 0 0
\(665\) 5.55855e6 0.487425
\(666\) 0 0
\(667\) 4.82942e6 0.420320
\(668\) 0 0
\(669\) 1.24136e7 1.07234
\(670\) 0 0
\(671\) −1.48496e7 −1.27324
\(672\) 0 0
\(673\) −1.22608e7 −1.04348 −0.521738 0.853106i \(-0.674716\pi\)
−0.521738 + 0.853106i \(0.674716\pi\)
\(674\) 0 0
\(675\) −2.36217e7 −1.99550
\(676\) 0 0
\(677\) 462085. 0.0387481 0.0193741 0.999812i \(-0.493833\pi\)
0.0193741 + 0.999812i \(0.493833\pi\)
\(678\) 0 0
\(679\) −1.29413e6 −0.107721
\(680\) 0 0
\(681\) −406819. −0.0336151
\(682\) 0 0
\(683\) −1.15378e7 −0.946392 −0.473196 0.880957i \(-0.656900\pi\)
−0.473196 + 0.880957i \(0.656900\pi\)
\(684\) 0 0
\(685\) 8.88807e6 0.723738
\(686\) 0 0
\(687\) 1.18612e7 0.958816
\(688\) 0 0
\(689\) −2.33159e6 −0.187113
\(690\) 0 0
\(691\) 3.09921e6 0.246920 0.123460 0.992350i \(-0.460601\pi\)
0.123460 + 0.992350i \(0.460601\pi\)
\(692\) 0 0
\(693\) −6137.65 −0.000485477 0
\(694\) 0 0
\(695\) −1.48114e7 −1.16315
\(696\) 0 0
\(697\) −2.27746e7 −1.77569
\(698\) 0 0
\(699\) 1.02455e7 0.793122
\(700\) 0 0
\(701\) −5.83889e6 −0.448782 −0.224391 0.974499i \(-0.572039\pi\)
−0.224391 + 0.974499i \(0.572039\pi\)
\(702\) 0 0
\(703\) −1.34876e7 −1.02931
\(704\) 0 0
\(705\) 1.03758e7 0.786232
\(706\) 0 0
\(707\) 3.58375e6 0.269643
\(708\) 0 0
\(709\) −75435.7 −0.00563587 −0.00281794 0.999996i \(-0.500897\pi\)
−0.00281794 + 0.999996i \(0.500897\pi\)
\(710\) 0 0
\(711\) −5304.63 −0.000393533 0
\(712\) 0 0
\(713\) −5.50255e6 −0.405359
\(714\) 0 0
\(715\) −7.33343e6 −0.536466
\(716\) 0 0
\(717\) −1.50351e7 −1.09222
\(718\) 0 0
\(719\) 1.30869e7 0.944091 0.472045 0.881574i \(-0.343516\pi\)
0.472045 + 0.881574i \(0.343516\pi\)
\(720\) 0 0
\(721\) −4.29482e6 −0.307686
\(722\) 0 0
\(723\) 546096. 0.0388529
\(724\) 0 0
\(725\) −3.83126e7 −2.70705
\(726\) 0 0
\(727\) 3.58160e6 0.251328 0.125664 0.992073i \(-0.459894\pi\)
0.125664 + 0.992073i \(0.459894\pi\)
\(728\) 0 0
\(729\) 1.43751e7 1.00182
\(730\) 0 0
\(731\) −3.40866e7 −2.35934
\(732\) 0 0
\(733\) 9.23826e6 0.635083 0.317541 0.948244i \(-0.397143\pi\)
0.317541 + 0.948244i \(0.397143\pi\)
\(734\) 0 0
\(735\) −3.61680e6 −0.246949
\(736\) 0 0
\(737\) −6.76599e6 −0.458842
\(738\) 0 0
\(739\) −1.90444e7 −1.28279 −0.641397 0.767210i \(-0.721645\pi\)
−0.641397 + 0.767210i \(0.721645\pi\)
\(740\) 0 0
\(741\) −4.92073e6 −0.329218
\(742\) 0 0
\(743\) −9.30669e6 −0.618477 −0.309238 0.950985i \(-0.600074\pi\)
−0.309238 + 0.950985i \(0.600074\pi\)
\(744\) 0 0
\(745\) 4.82260e7 3.18340
\(746\) 0 0
\(747\) −36457.1 −0.00239046
\(748\) 0 0
\(749\) −5.65647e6 −0.368418
\(750\) 0 0
\(751\) −2.56243e7 −1.65788 −0.828939 0.559339i \(-0.811055\pi\)
−0.828939 + 0.559339i \(0.811055\pi\)
\(752\) 0 0
\(753\) 2.04971e7 1.31736
\(754\) 0 0
\(755\) 2.02739e7 1.29441
\(756\) 0 0
\(757\) −5.60956e6 −0.355786 −0.177893 0.984050i \(-0.556928\pi\)
−0.177893 + 0.984050i \(0.556928\pi\)
\(758\) 0 0
\(759\) −3.44236e6 −0.216896
\(760\) 0 0
\(761\) −2.83985e7 −1.77760 −0.888799 0.458298i \(-0.848459\pi\)
−0.888799 + 0.458298i \(0.848459\pi\)
\(762\) 0 0
\(763\) 5.74783e6 0.357432
\(764\) 0 0
\(765\) 74016.5 0.00457272
\(766\) 0 0
\(767\) −1.28922e7 −0.791297
\(768\) 0 0
\(769\) −2.41205e6 −0.147086 −0.0735429 0.997292i \(-0.523431\pi\)
−0.0735429 + 0.997292i \(0.523431\pi\)
\(770\) 0 0
\(771\) −8.25123e6 −0.499899
\(772\) 0 0
\(773\) −5.93891e6 −0.357485 −0.178743 0.983896i \(-0.557203\pi\)
−0.178743 + 0.983896i \(0.557203\pi\)
\(774\) 0 0
\(775\) 4.36527e7 2.61070
\(776\) 0 0
\(777\) 8.77599e6 0.521487
\(778\) 0 0
\(779\) 1.55346e7 0.917183
\(780\) 0 0
\(781\) −3.52958e6 −0.207060
\(782\) 0 0
\(783\) 2.33154e7 1.35906
\(784\) 0 0
\(785\) −713979. −0.0413534
\(786\) 0 0
\(787\) 2.62637e7 1.51154 0.755769 0.654839i \(-0.227263\pi\)
0.755769 + 0.654839i \(0.227263\pi\)
\(788\) 0 0
\(789\) 2.01619e7 1.15302
\(790\) 0 0
\(791\) 8.89522e6 0.505494
\(792\) 0 0
\(793\) −1.42137e7 −0.802648
\(794\) 0 0
\(795\) 1.30376e7 0.731611
\(796\) 0 0
\(797\) 9.33689e6 0.520663 0.260331 0.965519i \(-0.416168\pi\)
0.260331 + 0.965519i \(0.416168\pi\)
\(798\) 0 0
\(799\) −1.18435e7 −0.656315
\(800\) 0 0
\(801\) 43156.3 0.00237664
\(802\) 0 0
\(803\) 1.14809e6 0.0628332
\(804\) 0 0
\(805\) 3.72205e6 0.202438
\(806\) 0 0
\(807\) 6.92017e6 0.374053
\(808\) 0 0
\(809\) −3.43772e7 −1.84671 −0.923357 0.383942i \(-0.874566\pi\)
−0.923357 + 0.383942i \(0.874566\pi\)
\(810\) 0 0
\(811\) −2.02626e7 −1.08179 −0.540896 0.841090i \(-0.681915\pi\)
−0.540896 + 0.841090i \(0.681915\pi\)
\(812\) 0 0
\(813\) −9.73008e6 −0.516285
\(814\) 0 0
\(815\) −3.75433e7 −1.97988
\(816\) 0 0
\(817\) 2.32506e7 1.21865
\(818\) 0 0
\(819\) −5874.82 −0.000306045 0
\(820\) 0 0
\(821\) −1.73641e7 −0.899073 −0.449537 0.893262i \(-0.648411\pi\)
−0.449537 + 0.893262i \(0.648411\pi\)
\(822\) 0 0
\(823\) 2.65094e7 1.36427 0.682136 0.731226i \(-0.261051\pi\)
0.682136 + 0.731226i \(0.261051\pi\)
\(824\) 0 0
\(825\) 2.73089e7 1.39691
\(826\) 0 0
\(827\) −2.30816e6 −0.117355 −0.0586775 0.998277i \(-0.518688\pi\)
−0.0586775 + 0.998277i \(0.518688\pi\)
\(828\) 0 0
\(829\) 6.14314e6 0.310459 0.155230 0.987878i \(-0.450388\pi\)
0.155230 + 0.987878i \(0.450388\pi\)
\(830\) 0 0
\(831\) 3.11798e7 1.56628
\(832\) 0 0
\(833\) 4.12839e6 0.206143
\(834\) 0 0
\(835\) −1.62726e7 −0.807684
\(836\) 0 0
\(837\) −2.65651e7 −1.31068
\(838\) 0 0
\(839\) −1.04628e7 −0.513150 −0.256575 0.966524i \(-0.582594\pi\)
−0.256575 + 0.966524i \(0.582594\pi\)
\(840\) 0 0
\(841\) 1.73047e7 0.843671
\(842\) 0 0
\(843\) 3.88290e6 0.188186
\(844\) 0 0
\(845\) 2.88930e7 1.39204
\(846\) 0 0
\(847\) −4.01014e6 −0.192066
\(848\) 0 0
\(849\) 1.00475e7 0.478396
\(850\) 0 0
\(851\) −9.03136e6 −0.427493
\(852\) 0 0
\(853\) 3.45134e7 1.62411 0.812054 0.583583i \(-0.198350\pi\)
0.812054 + 0.583583i \(0.198350\pi\)
\(854\) 0 0
\(855\) −50486.8 −0.00236191
\(856\) 0 0
\(857\) −1.35665e7 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(858\) 0 0
\(859\) 3.12783e7 1.44630 0.723152 0.690688i \(-0.242692\pi\)
0.723152 + 0.690688i \(0.242692\pi\)
\(860\) 0 0
\(861\) −1.01079e7 −0.464681
\(862\) 0 0
\(863\) 1.98071e7 0.905305 0.452653 0.891687i \(-0.350478\pi\)
0.452653 + 0.891687i \(0.350478\pi\)
\(864\) 0 0
\(865\) 3.10976e7 1.41314
\(866\) 0 0
\(867\) 2.39318e7 1.08125
\(868\) 0 0
\(869\) 3.35457e6 0.150691
\(870\) 0 0
\(871\) −6.47626e6 −0.289254
\(872\) 0 0
\(873\) 11754.2 0.000521984 0
\(874\) 0 0
\(875\) −1.47170e7 −0.649830
\(876\) 0 0
\(877\) 1.74647e6 0.0766765 0.0383383 0.999265i \(-0.487794\pi\)
0.0383383 + 0.999265i \(0.487794\pi\)
\(878\) 0 0
\(879\) −3.12776e7 −1.36540
\(880\) 0 0
\(881\) 1.17415e7 0.509663 0.254832 0.966985i \(-0.417980\pi\)
0.254832 + 0.966985i \(0.417980\pi\)
\(882\) 0 0
\(883\) −4.26514e7 −1.84091 −0.920453 0.390854i \(-0.872180\pi\)
−0.920453 + 0.390854i \(0.872180\pi\)
\(884\) 0 0
\(885\) 7.20899e7 3.09397
\(886\) 0 0
\(887\) 4.28737e7 1.82971 0.914854 0.403785i \(-0.132306\pi\)
0.914854 + 0.403785i \(0.132306\pi\)
\(888\) 0 0
\(889\) −5.19942e6 −0.220648
\(890\) 0 0
\(891\) −1.65886e7 −0.700026
\(892\) 0 0
\(893\) 8.07846e6 0.339000
\(894\) 0 0
\(895\) 5.91851e7 2.46976
\(896\) 0 0
\(897\) −3.29495e6 −0.136731
\(898\) 0 0
\(899\) −4.30866e7 −1.77805
\(900\) 0 0
\(901\) −1.48817e7 −0.610719
\(902\) 0 0
\(903\) −1.51285e7 −0.617415
\(904\) 0 0
\(905\) 2.52904e7 1.02644
\(906\) 0 0
\(907\) −1.99065e7 −0.803482 −0.401741 0.915753i \(-0.631595\pi\)
−0.401741 + 0.915753i \(0.631595\pi\)
\(908\) 0 0
\(909\) −32550.2 −0.00130660
\(910\) 0 0
\(911\) 1.12063e7 0.447371 0.223685 0.974661i \(-0.428191\pi\)
0.223685 + 0.974661i \(0.428191\pi\)
\(912\) 0 0
\(913\) 2.30549e7 0.915348
\(914\) 0 0
\(915\) 7.94794e7 3.13835
\(916\) 0 0
\(917\) 1.47164e7 0.577933
\(918\) 0 0
\(919\) −3.38620e7 −1.32259 −0.661293 0.750128i \(-0.729992\pi\)
−0.661293 + 0.750128i \(0.729992\pi\)
\(920\) 0 0
\(921\) −2.66494e7 −1.03523
\(922\) 0 0
\(923\) −3.37844e6 −0.130530
\(924\) 0 0
\(925\) 7.16474e7 2.75325
\(926\) 0 0
\(927\) 39008.7 0.00149095
\(928\) 0 0
\(929\) −2.29019e7 −0.870627 −0.435313 0.900279i \(-0.643362\pi\)
−0.435313 + 0.900279i \(0.643362\pi\)
\(930\) 0 0
\(931\) −2.81598e6 −0.106477
\(932\) 0 0
\(933\) 9.37644e6 0.352642
\(934\) 0 0
\(935\) −4.68069e7 −1.75098
\(936\) 0 0
\(937\) 2.97781e7 1.10802 0.554011 0.832510i \(-0.313097\pi\)
0.554011 + 0.832510i \(0.313097\pi\)
\(938\) 0 0
\(939\) −2.09019e7 −0.773608
\(940\) 0 0
\(941\) 1.30322e7 0.479782 0.239891 0.970800i \(-0.422888\pi\)
0.239891 + 0.970800i \(0.422888\pi\)
\(942\) 0 0
\(943\) 1.04021e7 0.380926
\(944\) 0 0
\(945\) 1.79692e7 0.654561
\(946\) 0 0
\(947\) 4.66683e7 1.69101 0.845506 0.533966i \(-0.179299\pi\)
0.845506 + 0.533966i \(0.179299\pi\)
\(948\) 0 0
\(949\) 1.09893e6 0.0396100
\(950\) 0 0
\(951\) −1.58550e7 −0.568480
\(952\) 0 0
\(953\) 2.00948e7 0.716724 0.358362 0.933583i \(-0.383335\pi\)
0.358362 + 0.933583i \(0.383335\pi\)
\(954\) 0 0
\(955\) 6.80251e7 2.41357
\(956\) 0 0
\(957\) −2.69548e7 −0.951383
\(958\) 0 0
\(959\) −4.50273e6 −0.158099
\(960\) 0 0
\(961\) 2.04630e7 0.714760
\(962\) 0 0
\(963\) 51376.2 0.00178524
\(964\) 0 0
\(965\) −4.99485e7 −1.72665
\(966\) 0 0
\(967\) −7.33633e6 −0.252297 −0.126149 0.992011i \(-0.540262\pi\)
−0.126149 + 0.992011i \(0.540262\pi\)
\(968\) 0 0
\(969\) −3.14074e7 −1.07454
\(970\) 0 0
\(971\) 4.01630e7 1.36703 0.683514 0.729937i \(-0.260451\pi\)
0.683514 + 0.729937i \(0.260451\pi\)
\(972\) 0 0
\(973\) 7.50352e6 0.254087
\(974\) 0 0
\(975\) 2.61394e7 0.880613
\(976\) 0 0
\(977\) 2.65602e7 0.890216 0.445108 0.895477i \(-0.353165\pi\)
0.445108 + 0.895477i \(0.353165\pi\)
\(978\) 0 0
\(979\) −2.72914e7 −0.910057
\(980\) 0 0
\(981\) −52206.0 −0.00173200
\(982\) 0 0
\(983\) −2.77362e7 −0.915510 −0.457755 0.889078i \(-0.651346\pi\)
−0.457755 + 0.889078i \(0.651346\pi\)
\(984\) 0 0
\(985\) 1.37906e7 0.452890
\(986\) 0 0
\(987\) −5.25644e6 −0.171751
\(988\) 0 0
\(989\) 1.55687e7 0.506131
\(990\) 0 0
\(991\) 2.35437e7 0.761536 0.380768 0.924671i \(-0.375660\pi\)
0.380768 + 0.924671i \(0.375660\pi\)
\(992\) 0 0
\(993\) −8.29123e6 −0.266837
\(994\) 0 0
\(995\) 9.52559e7 3.05024
\(996\) 0 0
\(997\) 4.05582e7 1.29223 0.646117 0.763239i \(-0.276392\pi\)
0.646117 + 0.763239i \(0.276392\pi\)
\(998\) 0 0
\(999\) −4.36015e7 −1.38225
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 448.6.a.t.1.2 2
4.3 odd 2 448.6.a.v.1.1 2
8.3 odd 2 56.6.a.e.1.2 2
8.5 even 2 112.6.a.i.1.1 2
24.5 odd 2 1008.6.a.bd.1.1 2
24.11 even 2 504.6.a.i.1.1 2
56.3 even 6 392.6.i.i.177.2 4
56.11 odd 6 392.6.i.j.177.1 4
56.13 odd 2 784.6.a.u.1.2 2
56.19 even 6 392.6.i.i.361.2 4
56.27 even 2 392.6.a.d.1.1 2
56.51 odd 6 392.6.i.j.361.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.e.1.2 2 8.3 odd 2
112.6.a.i.1.1 2 8.5 even 2
392.6.a.d.1.1 2 56.27 even 2
392.6.i.i.177.2 4 56.3 even 6
392.6.i.i.361.2 4 56.19 even 6
392.6.i.j.177.1 4 56.11 odd 6
392.6.i.j.361.1 4 56.51 odd 6
448.6.a.t.1.2 2 1.1 even 1 trivial
448.6.a.v.1.1 2 4.3 odd 2
504.6.a.i.1.1 2 24.11 even 2
784.6.a.u.1.2 2 56.13 odd 2
1008.6.a.bd.1.1 2 24.5 odd 2