Properties

Label 384.6.d.j.193.5
Level $384$
Weight $6$
Character 384.193
Analytic conductor $61.587$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,6,Mod(193,384)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.d (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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 338x^{10} + 43555x^{8} + 2692222x^{6} + 81680965x^{4} + 1098257588x^{2} + 4742525956 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.5
Root \(10.8525i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.6.d.j.193.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000i q^{3} +46.1613i q^{5} +59.9278 q^{7} -81.0000 q^{9} +O(q^{10})\) \(q-9.00000i q^{3} +46.1613i q^{5} +59.9278 q^{7} -81.0000 q^{9} +400.065i q^{11} -351.898i q^{13} +415.451 q^{15} +1662.58 q^{17} -564.314i q^{19} -539.350i q^{21} -3831.57 q^{23} +994.139 q^{25} +729.000i q^{27} +5944.12i q^{29} -2678.95 q^{31} +3600.59 q^{33} +2766.34i q^{35} +771.803i q^{37} -3167.08 q^{39} -5119.70 q^{41} -9518.90i q^{43} -3739.06i q^{45} +8138.39 q^{47} -13215.7 q^{49} -14963.2i q^{51} +11051.9i q^{53} -18467.5 q^{55} -5078.83 q^{57} +32143.3i q^{59} +35454.8i q^{61} -4854.15 q^{63} +16244.0 q^{65} +11362.8i q^{67} +34484.1i q^{69} -10394.7 q^{71} -40098.7 q^{73} -8947.25i q^{75} +23975.0i q^{77} +25754.8 q^{79} +6561.00 q^{81} +2239.75i q^{83} +76746.5i q^{85} +53497.1 q^{87} -15149.4 q^{89} -21088.5i q^{91} +24110.6i q^{93} +26049.5 q^{95} +158426. q^{97} -32405.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 972 q^{9} - 4888 q^{17} - 18660 q^{25} - 720 q^{33} - 25096 q^{41} + 135564 q^{49} - 13104 q^{57} + 254272 q^{65} - 187032 q^{73} + 78732 q^{81} - 575544 q^{89} + 93864 q^{97}+O(q^{100}) \) 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
\(3\) − 9.00000i − 0.577350i
\(4\) 0 0
\(5\) 46.1613i 0.825758i 0.910786 + 0.412879i \(0.135477\pi\)
−0.910786 + 0.412879i \(0.864523\pi\)
\(6\) 0 0
\(7\) 59.9278 0.462257 0.231128 0.972923i \(-0.425758\pi\)
0.231128 + 0.972923i \(0.425758\pi\)
\(8\) 0 0
\(9\) −81.0000 −0.333333
\(10\) 0 0
\(11\) 400.065i 0.996894i 0.866920 + 0.498447i \(0.166096\pi\)
−0.866920 + 0.498447i \(0.833904\pi\)
\(12\) 0 0
\(13\) − 351.898i − 0.577508i −0.957403 0.288754i \(-0.906759\pi\)
0.957403 0.288754i \(-0.0932410\pi\)
\(14\) 0 0
\(15\) 415.451 0.476751
\(16\) 0 0
\(17\) 1662.58 1.39527 0.697636 0.716452i \(-0.254235\pi\)
0.697636 + 0.716452i \(0.254235\pi\)
\(18\) 0 0
\(19\) − 564.314i − 0.358622i −0.983792 0.179311i \(-0.942613\pi\)
0.983792 0.179311i \(-0.0573869\pi\)
\(20\) 0 0
\(21\) − 539.350i − 0.266884i
\(22\) 0 0
\(23\) −3831.57 −1.51028 −0.755139 0.655565i \(-0.772430\pi\)
−0.755139 + 0.655565i \(0.772430\pi\)
\(24\) 0 0
\(25\) 994.139 0.318124
\(26\) 0 0
\(27\) 729.000i 0.192450i
\(28\) 0 0
\(29\) 5944.12i 1.31248i 0.754552 + 0.656240i \(0.227854\pi\)
−0.754552 + 0.656240i \(0.772146\pi\)
\(30\) 0 0
\(31\) −2678.95 −0.500680 −0.250340 0.968158i \(-0.580542\pi\)
−0.250340 + 0.968158i \(0.580542\pi\)
\(32\) 0 0
\(33\) 3600.59 0.575557
\(34\) 0 0
\(35\) 2766.34i 0.381712i
\(36\) 0 0
\(37\) 771.803i 0.0926834i 0.998926 + 0.0463417i \(0.0147563\pi\)
−0.998926 + 0.0463417i \(0.985244\pi\)
\(38\) 0 0
\(39\) −3167.08 −0.333425
\(40\) 0 0
\(41\) −5119.70 −0.475647 −0.237824 0.971308i \(-0.576434\pi\)
−0.237824 + 0.971308i \(0.576434\pi\)
\(42\) 0 0
\(43\) − 9518.90i − 0.785083i −0.919734 0.392541i \(-0.871596\pi\)
0.919734 0.392541i \(-0.128404\pi\)
\(44\) 0 0
\(45\) − 3739.06i − 0.275253i
\(46\) 0 0
\(47\) 8138.39 0.537395 0.268698 0.963225i \(-0.413407\pi\)
0.268698 + 0.963225i \(0.413407\pi\)
\(48\) 0 0
\(49\) −13215.7 −0.786319
\(50\) 0 0
\(51\) − 14963.2i − 0.805561i
\(52\) 0 0
\(53\) 11051.9i 0.540441i 0.962799 + 0.270220i \(0.0870966\pi\)
−0.962799 + 0.270220i \(0.912903\pi\)
\(54\) 0 0
\(55\) −18467.5 −0.823193
\(56\) 0 0
\(57\) −5078.83 −0.207051
\(58\) 0 0
\(59\) 32143.3i 1.20215i 0.799191 + 0.601077i \(0.205261\pi\)
−0.799191 + 0.601077i \(0.794739\pi\)
\(60\) 0 0
\(61\) 35454.8i 1.21997i 0.792412 + 0.609986i \(0.208825\pi\)
−0.792412 + 0.609986i \(0.791175\pi\)
\(62\) 0 0
\(63\) −4854.15 −0.154086
\(64\) 0 0
\(65\) 16244.0 0.476882
\(66\) 0 0
\(67\) 11362.8i 0.309242i 0.987974 + 0.154621i \(0.0494157\pi\)
−0.987974 + 0.154621i \(0.950584\pi\)
\(68\) 0 0
\(69\) 34484.1i 0.871959i
\(70\) 0 0
\(71\) −10394.7 −0.244719 −0.122360 0.992486i \(-0.539046\pi\)
−0.122360 + 0.992486i \(0.539046\pi\)
\(72\) 0 0
\(73\) −40098.7 −0.880689 −0.440345 0.897829i \(-0.645144\pi\)
−0.440345 + 0.897829i \(0.645144\pi\)
\(74\) 0 0
\(75\) − 8947.25i − 0.183669i
\(76\) 0 0
\(77\) 23975.0i 0.460821i
\(78\) 0 0
\(79\) 25754.8 0.464291 0.232146 0.972681i \(-0.425425\pi\)
0.232146 + 0.972681i \(0.425425\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 2239.75i 0.0356866i 0.999841 + 0.0178433i \(0.00568000\pi\)
−0.999841 + 0.0178433i \(0.994320\pi\)
\(84\) 0 0
\(85\) 76746.5i 1.15216i
\(86\) 0 0
\(87\) 53497.1 0.757761
\(88\) 0 0
\(89\) −15149.4 −0.202732 −0.101366 0.994849i \(-0.532321\pi\)
−0.101366 + 0.994849i \(0.532321\pi\)
\(90\) 0 0
\(91\) − 21088.5i − 0.266957i
\(92\) 0 0
\(93\) 24110.6i 0.289068i
\(94\) 0 0
\(95\) 26049.5 0.296135
\(96\) 0 0
\(97\) 158426. 1.70961 0.854804 0.518951i \(-0.173677\pi\)
0.854804 + 0.518951i \(0.173677\pi\)
\(98\) 0 0
\(99\) − 32405.3i − 0.332298i
\(100\) 0 0
\(101\) 154639.i 1.50840i 0.656647 + 0.754198i \(0.271974\pi\)
−0.656647 + 0.754198i \(0.728026\pi\)
\(102\) 0 0
\(103\) 89671.0 0.832835 0.416417 0.909174i \(-0.363286\pi\)
0.416417 + 0.909174i \(0.363286\pi\)
\(104\) 0 0
\(105\) 24897.1 0.220382
\(106\) 0 0
\(107\) 113338.i 0.957006i 0.878086 + 0.478503i \(0.158820\pi\)
−0.878086 + 0.478503i \(0.841180\pi\)
\(108\) 0 0
\(109\) 221403.i 1.78491i 0.451133 + 0.892457i \(0.351020\pi\)
−0.451133 + 0.892457i \(0.648980\pi\)
\(110\) 0 0
\(111\) 6946.23 0.0535108
\(112\) 0 0
\(113\) −6039.39 −0.0444936 −0.0222468 0.999753i \(-0.507082\pi\)
−0.0222468 + 0.999753i \(0.507082\pi\)
\(114\) 0 0
\(115\) − 176870.i − 1.24712i
\(116\) 0 0
\(117\) 28503.7i 0.192503i
\(118\) 0 0
\(119\) 99634.5 0.644974
\(120\) 0 0
\(121\) 998.858 0.00620212
\(122\) 0 0
\(123\) 46077.3i 0.274615i
\(124\) 0 0
\(125\) 190145.i 1.08845i
\(126\) 0 0
\(127\) −328162. −1.80542 −0.902711 0.430247i \(-0.858427\pi\)
−0.902711 + 0.430247i \(0.858427\pi\)
\(128\) 0 0
\(129\) −85670.1 −0.453268
\(130\) 0 0
\(131\) − 176176.i − 0.896952i −0.893795 0.448476i \(-0.851967\pi\)
0.893795 0.448476i \(-0.148033\pi\)
\(132\) 0 0
\(133\) − 33818.1i − 0.165776i
\(134\) 0 0
\(135\) −33651.6 −0.158917
\(136\) 0 0
\(137\) 50245.9 0.228718 0.114359 0.993440i \(-0.463519\pi\)
0.114359 + 0.993440i \(0.463519\pi\)
\(138\) 0 0
\(139\) 364925.i 1.60201i 0.598655 + 0.801007i \(0.295702\pi\)
−0.598655 + 0.801007i \(0.704298\pi\)
\(140\) 0 0
\(141\) − 73245.5i − 0.310265i
\(142\) 0 0
\(143\) 140782. 0.575715
\(144\) 0 0
\(145\) −274388. −1.08379
\(146\) 0 0
\(147\) 118941.i 0.453981i
\(148\) 0 0
\(149\) 502438.i 1.85403i 0.375023 + 0.927016i \(0.377635\pi\)
−0.375023 + 0.927016i \(0.622365\pi\)
\(150\) 0 0
\(151\) −225684. −0.805485 −0.402743 0.915313i \(-0.631943\pi\)
−0.402743 + 0.915313i \(0.631943\pi\)
\(152\) 0 0
\(153\) −134669. −0.465091
\(154\) 0 0
\(155\) − 123664.i − 0.413441i
\(156\) 0 0
\(157\) 464228.i 1.50308i 0.659688 + 0.751539i \(0.270688\pi\)
−0.659688 + 0.751539i \(0.729312\pi\)
\(158\) 0 0
\(159\) 99467.3 0.312024
\(160\) 0 0
\(161\) −229617. −0.698136
\(162\) 0 0
\(163\) − 143142.i − 0.421987i −0.977487 0.210994i \(-0.932330\pi\)
0.977487 0.210994i \(-0.0676699\pi\)
\(164\) 0 0
\(165\) 166208.i 0.475271i
\(166\) 0 0
\(167\) −170711. −0.473663 −0.236831 0.971551i \(-0.576109\pi\)
−0.236831 + 0.971551i \(0.576109\pi\)
\(168\) 0 0
\(169\) 247461. 0.666484
\(170\) 0 0
\(171\) 45709.5i 0.119541i
\(172\) 0 0
\(173\) − 38779.6i − 0.0985118i −0.998786 0.0492559i \(-0.984315\pi\)
0.998786 0.0492559i \(-0.0156850\pi\)
\(174\) 0 0
\(175\) 59576.6 0.147055
\(176\) 0 0
\(177\) 289289. 0.694064
\(178\) 0 0
\(179\) − 399489.i − 0.931907i −0.884809 0.465954i \(-0.845711\pi\)
0.884809 0.465954i \(-0.154289\pi\)
\(180\) 0 0
\(181\) − 168069.i − 0.381322i −0.981656 0.190661i \(-0.938937\pi\)
0.981656 0.190661i \(-0.0610632\pi\)
\(182\) 0 0
\(183\) 319093. 0.704351
\(184\) 0 0
\(185\) −35627.4 −0.0765340
\(186\) 0 0
\(187\) 665138.i 1.39094i
\(188\) 0 0
\(189\) 43687.4i 0.0889614i
\(190\) 0 0
\(191\) −230781. −0.457738 −0.228869 0.973457i \(-0.573503\pi\)
−0.228869 + 0.973457i \(0.573503\pi\)
\(192\) 0 0
\(193\) 7236.58 0.0139843 0.00699214 0.999976i \(-0.497774\pi\)
0.00699214 + 0.999976i \(0.497774\pi\)
\(194\) 0 0
\(195\) − 146196.i − 0.275328i
\(196\) 0 0
\(197\) − 582795.i − 1.06992i −0.844878 0.534959i \(-0.820327\pi\)
0.844878 0.534959i \(-0.179673\pi\)
\(198\) 0 0
\(199\) −1.07918e6 −1.93180 −0.965899 0.258920i \(-0.916633\pi\)
−0.965899 + 0.258920i \(0.916633\pi\)
\(200\) 0 0
\(201\) 102265. 0.178541
\(202\) 0 0
\(203\) 356218.i 0.606703i
\(204\) 0 0
\(205\) − 236332.i − 0.392769i
\(206\) 0 0
\(207\) 310357. 0.503426
\(208\) 0 0
\(209\) 225763. 0.357508
\(210\) 0 0
\(211\) − 860927.i − 1.33125i −0.746286 0.665626i \(-0.768165\pi\)
0.746286 0.665626i \(-0.231835\pi\)
\(212\) 0 0
\(213\) 93552.7i 0.141289i
\(214\) 0 0
\(215\) 439404. 0.648288
\(216\) 0 0
\(217\) −160544. −0.231443
\(218\) 0 0
\(219\) 360888.i 0.508466i
\(220\) 0 0
\(221\) − 585057.i − 0.805781i
\(222\) 0 0
\(223\) −442484. −0.595848 −0.297924 0.954590i \(-0.596294\pi\)
−0.297924 + 0.954590i \(0.596294\pi\)
\(224\) 0 0
\(225\) −80525.2 −0.106041
\(226\) 0 0
\(227\) − 1.25053e6i − 1.61076i −0.592761 0.805378i \(-0.701962\pi\)
0.592761 0.805378i \(-0.298038\pi\)
\(228\) 0 0
\(229\) 508471.i 0.640734i 0.947293 + 0.320367i \(0.103806\pi\)
−0.947293 + 0.320367i \(0.896194\pi\)
\(230\) 0 0
\(231\) 215775. 0.266055
\(232\) 0 0
\(233\) 1.28856e6 1.55495 0.777474 0.628915i \(-0.216501\pi\)
0.777474 + 0.628915i \(0.216501\pi\)
\(234\) 0 0
\(235\) 375678.i 0.443758i
\(236\) 0 0
\(237\) − 231793.i − 0.268059i
\(238\) 0 0
\(239\) 1.42040e6 1.60848 0.804240 0.594304i \(-0.202572\pi\)
0.804240 + 0.594304i \(0.202572\pi\)
\(240\) 0 0
\(241\) −196969. −0.218452 −0.109226 0.994017i \(-0.534837\pi\)
−0.109226 + 0.994017i \(0.534837\pi\)
\(242\) 0 0
\(243\) − 59049.0i − 0.0641500i
\(244\) 0 0
\(245\) − 610051.i − 0.649309i
\(246\) 0 0
\(247\) −198581. −0.207107
\(248\) 0 0
\(249\) 20157.8 0.0206037
\(250\) 0 0
\(251\) 1.87291e6i 1.87643i 0.346052 + 0.938215i \(0.387522\pi\)
−0.346052 + 0.938215i \(0.612478\pi\)
\(252\) 0 0
\(253\) − 1.53288e6i − 1.50559i
\(254\) 0 0
\(255\) 690719. 0.665198
\(256\) 0 0
\(257\) −854792. −0.807287 −0.403643 0.914916i \(-0.632256\pi\)
−0.403643 + 0.914916i \(0.632256\pi\)
\(258\) 0 0
\(259\) 46252.4i 0.0428435i
\(260\) 0 0
\(261\) − 481474.i − 0.437493i
\(262\) 0 0
\(263\) 1.94301e6 1.73215 0.866076 0.499913i \(-0.166635\pi\)
0.866076 + 0.499913i \(0.166635\pi\)
\(264\) 0 0
\(265\) −510170. −0.446273
\(266\) 0 0
\(267\) 136345.i 0.117047i
\(268\) 0 0
\(269\) − 223635.i − 0.188434i −0.995552 0.0942168i \(-0.969965\pi\)
0.995552 0.0942168i \(-0.0300347\pi\)
\(270\) 0 0
\(271\) −1.73900e6 −1.43839 −0.719196 0.694807i \(-0.755490\pi\)
−0.719196 + 0.694807i \(0.755490\pi\)
\(272\) 0 0
\(273\) −189796. −0.154128
\(274\) 0 0
\(275\) 397720.i 0.317136i
\(276\) 0 0
\(277\) − 2.35793e6i − 1.84642i −0.384294 0.923211i \(-0.625555\pi\)
0.384294 0.923211i \(-0.374445\pi\)
\(278\) 0 0
\(279\) 216995. 0.166893
\(280\) 0 0
\(281\) −923073. −0.697382 −0.348691 0.937238i \(-0.613374\pi\)
−0.348691 + 0.937238i \(0.613374\pi\)
\(282\) 0 0
\(283\) − 848669.i − 0.629901i −0.949108 0.314951i \(-0.898012\pi\)
0.949108 0.314951i \(-0.101988\pi\)
\(284\) 0 0
\(285\) − 234445.i − 0.170974i
\(286\) 0 0
\(287\) −306813. −0.219871
\(288\) 0 0
\(289\) 1.34430e6 0.946785
\(290\) 0 0
\(291\) − 1.42583e6i − 0.987043i
\(292\) 0 0
\(293\) − 1.09123e6i − 0.742585i −0.928516 0.371293i \(-0.878915\pi\)
0.928516 0.371293i \(-0.121085\pi\)
\(294\) 0 0
\(295\) −1.48377e6 −0.992687
\(296\) 0 0
\(297\) −291648. −0.191852
\(298\) 0 0
\(299\) 1.34832e6i 0.872197i
\(300\) 0 0
\(301\) − 570447.i − 0.362910i
\(302\) 0 0
\(303\) 1.39175e6 0.870873
\(304\) 0 0
\(305\) −1.63664e6 −1.00740
\(306\) 0 0
\(307\) − 1.55616e6i − 0.942340i −0.882042 0.471170i \(-0.843832\pi\)
0.882042 0.471170i \(-0.156168\pi\)
\(308\) 0 0
\(309\) − 807039.i − 0.480837i
\(310\) 0 0
\(311\) −457231. −0.268062 −0.134031 0.990977i \(-0.542792\pi\)
−0.134031 + 0.990977i \(0.542792\pi\)
\(312\) 0 0
\(313\) −1.73593e6 −1.00154 −0.500772 0.865579i \(-0.666951\pi\)
−0.500772 + 0.865579i \(0.666951\pi\)
\(314\) 0 0
\(315\) − 224074.i − 0.127237i
\(316\) 0 0
\(317\) − 450224.i − 0.251640i −0.992053 0.125820i \(-0.959844\pi\)
0.992053 0.125820i \(-0.0401562\pi\)
\(318\) 0 0
\(319\) −2.37804e6 −1.30840
\(320\) 0 0
\(321\) 1.02004e6 0.552528
\(322\) 0 0
\(323\) − 938215.i − 0.500376i
\(324\) 0 0
\(325\) − 349835.i − 0.183719i
\(326\) 0 0
\(327\) 1.99263e6 1.03052
\(328\) 0 0
\(329\) 487716. 0.248415
\(330\) 0 0
\(331\) 2.79853e6i 1.40397i 0.712189 + 0.701987i \(0.247704\pi\)
−0.712189 + 0.701987i \(0.752296\pi\)
\(332\) 0 0
\(333\) − 62516.0i − 0.0308945i
\(334\) 0 0
\(335\) −524522. −0.255359
\(336\) 0 0
\(337\) 1.88545e6 0.904359 0.452180 0.891927i \(-0.350647\pi\)
0.452180 + 0.891927i \(0.350647\pi\)
\(338\) 0 0
\(339\) 54354.5i 0.0256884i
\(340\) 0 0
\(341\) − 1.07176e6i − 0.499125i
\(342\) 0 0
\(343\) −1.79919e6 −0.825738
\(344\) 0 0
\(345\) −1.59183e6 −0.720027
\(346\) 0 0
\(347\) − 841512.i − 0.375178i −0.982248 0.187589i \(-0.939933\pi\)
0.982248 0.187589i \(-0.0600672\pi\)
\(348\) 0 0
\(349\) − 3.05425e6i − 1.34227i −0.741333 0.671137i \(-0.765806\pi\)
0.741333 0.671137i \(-0.234194\pi\)
\(350\) 0 0
\(351\) 256533. 0.111142
\(352\) 0 0
\(353\) 3.20066e6 1.36711 0.683555 0.729899i \(-0.260433\pi\)
0.683555 + 0.729899i \(0.260433\pi\)
\(354\) 0 0
\(355\) − 479834.i − 0.202079i
\(356\) 0 0
\(357\) − 896710.i − 0.372376i
\(358\) 0 0
\(359\) 1.95304e6 0.799788 0.399894 0.916561i \(-0.369047\pi\)
0.399894 + 0.916561i \(0.369047\pi\)
\(360\) 0 0
\(361\) 2.15765e6 0.871390
\(362\) 0 0
\(363\) − 8989.72i − 0.00358080i
\(364\) 0 0
\(365\) − 1.85100e6i − 0.727236i
\(366\) 0 0
\(367\) −3.31321e6 −1.28406 −0.642028 0.766682i \(-0.721907\pi\)
−0.642028 + 0.766682i \(0.721907\pi\)
\(368\) 0 0
\(369\) 414696. 0.158549
\(370\) 0 0
\(371\) 662317.i 0.249822i
\(372\) 0 0
\(373\) 3.65917e6i 1.36179i 0.732381 + 0.680895i \(0.238409\pi\)
−0.732381 + 0.680895i \(0.761591\pi\)
\(374\) 0 0
\(375\) 1.71130e6 0.628418
\(376\) 0 0
\(377\) 2.09172e6 0.757968
\(378\) 0 0
\(379\) − 1.76629e6i − 0.631631i −0.948821 0.315816i \(-0.897722\pi\)
0.948821 0.315816i \(-0.102278\pi\)
\(380\) 0 0
\(381\) 2.95346e6i 1.04236i
\(382\) 0 0
\(383\) 3.69691e6 1.28778 0.643890 0.765118i \(-0.277320\pi\)
0.643890 + 0.765118i \(0.277320\pi\)
\(384\) 0 0
\(385\) −1.10672e6 −0.380526
\(386\) 0 0
\(387\) 771031.i 0.261694i
\(388\) 0 0
\(389\) − 669156.i − 0.224209i −0.993696 0.112105i \(-0.964241\pi\)
0.993696 0.112105i \(-0.0357592\pi\)
\(390\) 0 0
\(391\) −6.37027e6 −2.10725
\(392\) 0 0
\(393\) −1.58559e6 −0.517855
\(394\) 0 0
\(395\) 1.18887e6i 0.383392i
\(396\) 0 0
\(397\) 2.19806e6i 0.699945i 0.936760 + 0.349973i \(0.113809\pi\)
−0.936760 + 0.349973i \(0.886191\pi\)
\(398\) 0 0
\(399\) −304363. −0.0957105
\(400\) 0 0
\(401\) −4.54750e6 −1.41225 −0.706125 0.708087i \(-0.749558\pi\)
−0.706125 + 0.708087i \(0.749558\pi\)
\(402\) 0 0
\(403\) 942717.i 0.289147i
\(404\) 0 0
\(405\) 302864.i 0.0917508i
\(406\) 0 0
\(407\) −308771. −0.0923956
\(408\) 0 0
\(409\) −1.46461e6 −0.432926 −0.216463 0.976291i \(-0.569452\pi\)
−0.216463 + 0.976291i \(0.569452\pi\)
\(410\) 0 0
\(411\) − 452213.i − 0.132050i
\(412\) 0 0
\(413\) 1.92628e6i 0.555704i
\(414\) 0 0
\(415\) −103390. −0.0294685
\(416\) 0 0
\(417\) 3.28432e6 0.924923
\(418\) 0 0
\(419\) 868371.i 0.241641i 0.992674 + 0.120820i \(0.0385525\pi\)
−0.992674 + 0.120820i \(0.961447\pi\)
\(420\) 0 0
\(421\) 1.31760e6i 0.362308i 0.983455 + 0.181154i \(0.0579833\pi\)
−0.983455 + 0.181154i \(0.942017\pi\)
\(422\) 0 0
\(423\) −659210. −0.179132
\(424\) 0 0
\(425\) 1.65283e6 0.443870
\(426\) 0 0
\(427\) 2.12473e6i 0.563941i
\(428\) 0 0
\(429\) − 1.26704e6i − 0.332389i
\(430\) 0 0
\(431\) 3.26185e6 0.845805 0.422902 0.906175i \(-0.361011\pi\)
0.422902 + 0.906175i \(0.361011\pi\)
\(432\) 0 0
\(433\) −278010. −0.0712591 −0.0356295 0.999365i \(-0.511344\pi\)
−0.0356295 + 0.999365i \(0.511344\pi\)
\(434\) 0 0
\(435\) 2.46949e6i 0.625727i
\(436\) 0 0
\(437\) 2.16221e6i 0.541619i
\(438\) 0 0
\(439\) −2.77446e6 −0.687096 −0.343548 0.939135i \(-0.611629\pi\)
−0.343548 + 0.939135i \(0.611629\pi\)
\(440\) 0 0
\(441\) 1.07047e6 0.262106
\(442\) 0 0
\(443\) − 5.97809e6i − 1.44728i −0.690176 0.723641i \(-0.742467\pi\)
0.690176 0.723641i \(-0.257533\pi\)
\(444\) 0 0
\(445\) − 699317.i − 0.167407i
\(446\) 0 0
\(447\) 4.52194e6 1.07043
\(448\) 0 0
\(449\) −5.68566e6 −1.33096 −0.665480 0.746416i \(-0.731773\pi\)
−0.665480 + 0.746416i \(0.731773\pi\)
\(450\) 0 0
\(451\) − 2.04821e6i − 0.474170i
\(452\) 0 0
\(453\) 2.03115e6i 0.465047i
\(454\) 0 0
\(455\) 973470. 0.220442
\(456\) 0 0
\(457\) −2.65598e6 −0.594886 −0.297443 0.954740i \(-0.596134\pi\)
−0.297443 + 0.954740i \(0.596134\pi\)
\(458\) 0 0
\(459\) 1.21202e6i 0.268520i
\(460\) 0 0
\(461\) 3.88629e6i 0.851692i 0.904796 + 0.425846i \(0.140024\pi\)
−0.904796 + 0.425846i \(0.859976\pi\)
\(462\) 0 0
\(463\) −3.78966e6 −0.821576 −0.410788 0.911731i \(-0.634746\pi\)
−0.410788 + 0.911731i \(0.634746\pi\)
\(464\) 0 0
\(465\) −1.11297e6 −0.238700
\(466\) 0 0
\(467\) − 2.31109e6i − 0.490371i −0.969476 0.245185i \(-0.921151\pi\)
0.969476 0.245185i \(-0.0788488\pi\)
\(468\) 0 0
\(469\) 680948.i 0.142949i
\(470\) 0 0
\(471\) 4.17805e6 0.867803
\(472\) 0 0
\(473\) 3.80818e6 0.782645
\(474\) 0 0
\(475\) − 561007.i − 0.114086i
\(476\) 0 0
\(477\) − 895206.i − 0.180147i
\(478\) 0 0
\(479\) 4.62388e6 0.920804 0.460402 0.887710i \(-0.347705\pi\)
0.460402 + 0.887710i \(0.347705\pi\)
\(480\) 0 0
\(481\) 271596. 0.0535254
\(482\) 0 0
\(483\) 2.06656e6i 0.403069i
\(484\) 0 0
\(485\) 7.31313e6i 1.41172i
\(486\) 0 0
\(487\) 7.55618e6 1.44371 0.721855 0.692044i \(-0.243290\pi\)
0.721855 + 0.692044i \(0.243290\pi\)
\(488\) 0 0
\(489\) −1.28828e6 −0.243634
\(490\) 0 0
\(491\) − 4.40919e6i − 0.825382i −0.910871 0.412691i \(-0.864589\pi\)
0.910871 0.412691i \(-0.135411\pi\)
\(492\) 0 0
\(493\) 9.88255e6i 1.83127i
\(494\) 0 0
\(495\) 1.49587e6 0.274398
\(496\) 0 0
\(497\) −622934. −0.113123
\(498\) 0 0
\(499\) 8.45546e6i 1.52015i 0.649837 + 0.760074i \(0.274837\pi\)
−0.649837 + 0.760074i \(0.725163\pi\)
\(500\) 0 0
\(501\) 1.53639e6i 0.273469i
\(502\) 0 0
\(503\) −1.17007e6 −0.206201 −0.103100 0.994671i \(-0.532876\pi\)
−0.103100 + 0.994671i \(0.532876\pi\)
\(504\) 0 0
\(505\) −7.13833e6 −1.24557
\(506\) 0 0
\(507\) − 2.22715e6i − 0.384795i
\(508\) 0 0
\(509\) 9.48209e6i 1.62222i 0.584893 + 0.811110i \(0.301136\pi\)
−0.584893 + 0.811110i \(0.698864\pi\)
\(510\) 0 0
\(511\) −2.40302e6 −0.407105
\(512\) 0 0
\(513\) 411385. 0.0690169
\(514\) 0 0
\(515\) 4.13932e6i 0.687720i
\(516\) 0 0
\(517\) 3.25589e6i 0.535726i
\(518\) 0 0
\(519\) −349016. −0.0568758
\(520\) 0 0
\(521\) 1.08606e7 1.75290 0.876451 0.481491i \(-0.159905\pi\)
0.876451 + 0.481491i \(0.159905\pi\)
\(522\) 0 0
\(523\) − 7.05571e6i − 1.12794i −0.825795 0.563971i \(-0.809273\pi\)
0.825795 0.563971i \(-0.190727\pi\)
\(524\) 0 0
\(525\) − 536189.i − 0.0849023i
\(526\) 0 0
\(527\) −4.45396e6 −0.698585
\(528\) 0 0
\(529\) 8.24455e6 1.28094
\(530\) 0 0
\(531\) − 2.60360e6i − 0.400718i
\(532\) 0 0
\(533\) 1.80161e6i 0.274690i
\(534\) 0 0
\(535\) −5.23181e6 −0.790255
\(536\) 0 0
\(537\) −3.59541e6 −0.538037
\(538\) 0 0
\(539\) − 5.28712e6i − 0.783876i
\(540\) 0 0
\(541\) 2.11215e6i 0.310264i 0.987894 + 0.155132i \(0.0495804\pi\)
−0.987894 + 0.155132i \(0.950420\pi\)
\(542\) 0 0
\(543\) −1.51262e6 −0.220157
\(544\) 0 0
\(545\) −1.02202e7 −1.47391
\(546\) 0 0
\(547\) 9.37947e6i 1.34032i 0.742215 + 0.670162i \(0.233775\pi\)
−0.742215 + 0.670162i \(0.766225\pi\)
\(548\) 0 0
\(549\) − 2.87184e6i − 0.406657i
\(550\) 0 0
\(551\) 3.35435e6 0.470685
\(552\) 0 0
\(553\) 1.54343e6 0.214622
\(554\) 0 0
\(555\) 320646.i 0.0441869i
\(556\) 0 0
\(557\) − 3.61461e6i − 0.493655i −0.969059 0.246827i \(-0.920612\pi\)
0.969059 0.246827i \(-0.0793881\pi\)
\(558\) 0 0
\(559\) −3.34968e6 −0.453392
\(560\) 0 0
\(561\) 5.98625e6 0.803059
\(562\) 0 0
\(563\) 1.04180e7i 1.38520i 0.721321 + 0.692601i \(0.243535\pi\)
−0.721321 + 0.692601i \(0.756465\pi\)
\(564\) 0 0
\(565\) − 278786.i − 0.0367409i
\(566\) 0 0
\(567\) 393186. 0.0513619
\(568\) 0 0
\(569\) 7.80034e6 1.01003 0.505014 0.863111i \(-0.331487\pi\)
0.505014 + 0.863111i \(0.331487\pi\)
\(570\) 0 0
\(571\) − 1.13367e7i − 1.45511i −0.686049 0.727555i \(-0.740657\pi\)
0.686049 0.727555i \(-0.259343\pi\)
\(572\) 0 0
\(573\) 2.07703e6i 0.264275i
\(574\) 0 0
\(575\) −3.80911e6 −0.480456
\(576\) 0 0
\(577\) 3.41197e6 0.426644 0.213322 0.976982i \(-0.431572\pi\)
0.213322 + 0.976982i \(0.431572\pi\)
\(578\) 0 0
\(579\) − 65129.2i − 0.00807383i
\(580\) 0 0
\(581\) 134224.i 0.0164964i
\(582\) 0 0
\(583\) −4.42149e6 −0.538762
\(584\) 0 0
\(585\) −1.31577e6 −0.158961
\(586\) 0 0
\(587\) 9.10881e6i 1.09111i 0.838077 + 0.545553i \(0.183680\pi\)
−0.838077 + 0.545553i \(0.816320\pi\)
\(588\) 0 0
\(589\) 1.51177e6i 0.179555i
\(590\) 0 0
\(591\) −5.24516e6 −0.617717
\(592\) 0 0
\(593\) −5.87052e6 −0.685551 −0.342776 0.939417i \(-0.611367\pi\)
−0.342776 + 0.939417i \(0.611367\pi\)
\(594\) 0 0
\(595\) 4.59925e6i 0.532592i
\(596\) 0 0
\(597\) 9.71263e6i 1.11532i
\(598\) 0 0
\(599\) 9.18159e6 1.04556 0.522782 0.852466i \(-0.324894\pi\)
0.522782 + 0.852466i \(0.324894\pi\)
\(600\) 0 0
\(601\) 1.01852e7 1.15022 0.575112 0.818075i \(-0.304959\pi\)
0.575112 + 0.818075i \(0.304959\pi\)
\(602\) 0 0
\(603\) − 920388.i − 0.103081i
\(604\) 0 0
\(605\) 46108.5i 0.00512145i
\(606\) 0 0
\(607\) 4.74894e6 0.523148 0.261574 0.965183i \(-0.415758\pi\)
0.261574 + 0.965183i \(0.415758\pi\)
\(608\) 0 0
\(609\) 3.20596e6 0.350280
\(610\) 0 0
\(611\) − 2.86388e6i − 0.310350i
\(612\) 0 0
\(613\) 1.16623e6i 0.125352i 0.998034 + 0.0626760i \(0.0199635\pi\)
−0.998034 + 0.0626760i \(0.980037\pi\)
\(614\) 0 0
\(615\) −2.12699e6 −0.226766
\(616\) 0 0
\(617\) −1.29443e6 −0.136888 −0.0684440 0.997655i \(-0.521803\pi\)
−0.0684440 + 0.997655i \(0.521803\pi\)
\(618\) 0 0
\(619\) 4.23949e6i 0.444721i 0.974965 + 0.222360i \(0.0713762\pi\)
−0.974965 + 0.222360i \(0.928624\pi\)
\(620\) 0 0
\(621\) − 2.79321e6i − 0.290653i
\(622\) 0 0
\(623\) −907873. −0.0937141
\(624\) 0 0
\(625\) −5.67063e6 −0.580672
\(626\) 0 0
\(627\) − 2.03186e6i − 0.206408i
\(628\) 0 0
\(629\) 1.28318e6i 0.129319i
\(630\) 0 0
\(631\) −1.24132e6 −0.124111 −0.0620553 0.998073i \(-0.519766\pi\)
−0.0620553 + 0.998073i \(0.519766\pi\)
\(632\) 0 0
\(633\) −7.74834e6 −0.768599
\(634\) 0 0
\(635\) − 1.51484e7i − 1.49084i
\(636\) 0 0
\(637\) 4.65056e6i 0.454105i
\(638\) 0 0
\(639\) 841974. 0.0815730
\(640\) 0 0
\(641\) −1.09152e7 −1.04927 −0.524635 0.851327i \(-0.675798\pi\)
−0.524635 + 0.851327i \(0.675798\pi\)
\(642\) 0 0
\(643\) − 1.48054e7i − 1.41219i −0.708118 0.706095i \(-0.750455\pi\)
0.708118 0.706095i \(-0.249545\pi\)
\(644\) 0 0
\(645\) − 3.95464e6i − 0.374289i
\(646\) 0 0
\(647\) −7.50071e6 −0.704436 −0.352218 0.935918i \(-0.614572\pi\)
−0.352218 + 0.935918i \(0.614572\pi\)
\(648\) 0 0
\(649\) −1.28594e7 −1.19842
\(650\) 0 0
\(651\) 1.44489e6i 0.133624i
\(652\) 0 0
\(653\) − 1.84442e7i − 1.69269i −0.532638 0.846343i \(-0.678799\pi\)
0.532638 0.846343i \(-0.321201\pi\)
\(654\) 0 0
\(655\) 8.13252e6 0.740665
\(656\) 0 0
\(657\) 3.24799e6 0.293563
\(658\) 0 0
\(659\) 1.50202e7i 1.34729i 0.739055 + 0.673645i \(0.235272\pi\)
−0.739055 + 0.673645i \(0.764728\pi\)
\(660\) 0 0
\(661\) − 1.62211e7i − 1.44403i −0.691876 0.722017i \(-0.743215\pi\)
0.691876 0.722017i \(-0.256785\pi\)
\(662\) 0 0
\(663\) −5.26551e6 −0.465218
\(664\) 0 0
\(665\) 1.56109e6 0.136890
\(666\) 0 0
\(667\) − 2.27753e7i − 1.98221i
\(668\) 0 0
\(669\) 3.98236e6i 0.344013i
\(670\) 0 0
\(671\) −1.41842e7 −1.21618
\(672\) 0 0
\(673\) 8.38719e6 0.713804 0.356902 0.934142i \(-0.383833\pi\)
0.356902 + 0.934142i \(0.383833\pi\)
\(674\) 0 0
\(675\) 724727.i 0.0612231i
\(676\) 0 0
\(677\) − 5.22589e6i − 0.438216i −0.975701 0.219108i \(-0.929685\pi\)
0.975701 0.219108i \(-0.0703147\pi\)
\(678\) 0 0
\(679\) 9.49411e6 0.790278
\(680\) 0 0
\(681\) −1.12548e7 −0.929970
\(682\) 0 0
\(683\) 1.61671e7i 1.32611i 0.748570 + 0.663056i \(0.230741\pi\)
−0.748570 + 0.663056i \(0.769259\pi\)
\(684\) 0 0
\(685\) 2.31942e6i 0.188865i
\(686\) 0 0
\(687\) 4.57624e6 0.369928
\(688\) 0 0
\(689\) 3.88915e6 0.312109
\(690\) 0 0
\(691\) 2.38372e7i 1.89915i 0.313533 + 0.949577i \(0.398487\pi\)
−0.313533 + 0.949577i \(0.601513\pi\)
\(692\) 0 0
\(693\) − 1.94198e6i − 0.153607i
\(694\) 0 0
\(695\) −1.68454e7 −1.32287
\(696\) 0 0
\(697\) −8.51189e6 −0.663658
\(698\) 0 0
\(699\) − 1.15971e7i − 0.897750i
\(700\) 0 0
\(701\) − 9.30715e6i − 0.715355i −0.933845 0.357678i \(-0.883569\pi\)
0.933845 0.357678i \(-0.116431\pi\)
\(702\) 0 0
\(703\) 435539. 0.0332383
\(704\) 0 0
\(705\) 3.38110e6 0.256204
\(706\) 0 0
\(707\) 9.26717e6i 0.697266i
\(708\) 0 0
\(709\) − 5.17752e6i − 0.386818i −0.981118 0.193409i \(-0.938046\pi\)
0.981118 0.193409i \(-0.0619544\pi\)
\(710\) 0 0
\(711\) −2.08614e6 −0.154764
\(712\) 0 0
\(713\) 1.02646e7 0.756166
\(714\) 0 0
\(715\) 6.49868e6i 0.475401i
\(716\) 0 0
\(717\) − 1.27836e7i − 0.928657i
\(718\) 0 0
\(719\) 3.77696e6 0.272471 0.136236 0.990676i \(-0.456500\pi\)
0.136236 + 0.990676i \(0.456500\pi\)
\(720\) 0 0
\(721\) 5.37378e6 0.384983
\(722\) 0 0
\(723\) 1.77272e6i 0.126123i
\(724\) 0 0
\(725\) 5.90928e6i 0.417532i
\(726\) 0 0
\(727\) 2.50517e7 1.75793 0.878966 0.476885i \(-0.158234\pi\)
0.878966 + 0.476885i \(0.158234\pi\)
\(728\) 0 0
\(729\) −531441. −0.0370370
\(730\) 0 0
\(731\) − 1.58259e7i − 1.09540i
\(732\) 0 0
\(733\) − 2.58312e7i − 1.77576i −0.460074 0.887881i \(-0.652177\pi\)
0.460074 0.887881i \(-0.347823\pi\)
\(734\) 0 0
\(735\) −5.49046e6 −0.374879
\(736\) 0 0
\(737\) −4.54587e6 −0.308282
\(738\) 0 0
\(739\) − 1.40532e7i − 0.946593i −0.880903 0.473297i \(-0.843064\pi\)
0.880903 0.473297i \(-0.156936\pi\)
\(740\) 0 0
\(741\) 1.78723e6i 0.119573i
\(742\) 0 0
\(743\) 3.60964e6 0.239878 0.119939 0.992781i \(-0.461730\pi\)
0.119939 + 0.992781i \(0.461730\pi\)
\(744\) 0 0
\(745\) −2.31932e7 −1.53098
\(746\) 0 0
\(747\) − 181420.i − 0.0118955i
\(748\) 0 0
\(749\) 6.79207e6i 0.442382i
\(750\) 0 0
\(751\) 2.91748e7 1.88759 0.943795 0.330531i \(-0.107228\pi\)
0.943795 + 0.330531i \(0.107228\pi\)
\(752\) 0 0
\(753\) 1.68562e7 1.08336
\(754\) 0 0
\(755\) − 1.04178e7i − 0.665136i
\(756\) 0 0
\(757\) − 1.12305e7i − 0.712294i −0.934430 0.356147i \(-0.884090\pi\)
0.934430 0.356147i \(-0.115910\pi\)
\(758\) 0 0
\(759\) −1.37959e7 −0.869251
\(760\) 0 0
\(761\) 1.69614e7 1.06170 0.530849 0.847466i \(-0.321873\pi\)
0.530849 + 0.847466i \(0.321873\pi\)
\(762\) 0 0
\(763\) 1.32682e7i 0.825089i
\(764\) 0 0
\(765\) − 6.21647e6i − 0.384052i
\(766\) 0 0
\(767\) 1.13111e7 0.694254
\(768\) 0 0
\(769\) 2.61093e7 1.59214 0.796068 0.605208i \(-0.206910\pi\)
0.796068 + 0.605208i \(0.206910\pi\)
\(770\) 0 0
\(771\) 7.69313e6i 0.466087i
\(772\) 0 0
\(773\) − 2.28744e7i − 1.37690i −0.725286 0.688448i \(-0.758292\pi\)
0.725286 0.688448i \(-0.241708\pi\)
\(774\) 0 0
\(775\) −2.66325e6 −0.159279
\(776\) 0 0
\(777\) 416272. 0.0247357
\(778\) 0 0
\(779\) 2.88912e6i 0.170578i
\(780\) 0 0
\(781\) − 4.15857e6i − 0.243959i
\(782\) 0 0
\(783\) −4.33327e6 −0.252587
\(784\) 0 0
\(785\) −2.14293e7 −1.24118
\(786\) 0 0
\(787\) 1.44464e6i 0.0831423i 0.999136 + 0.0415712i \(0.0132363\pi\)
−0.999136 + 0.0415712i \(0.986764\pi\)
\(788\) 0 0
\(789\) − 1.74871e7i − 1.00006i
\(790\) 0 0
\(791\) −361927. −0.0205674
\(792\) 0 0
\(793\) 1.24765e7 0.704544
\(794\) 0 0
\(795\) 4.59153e6i 0.257656i
\(796\) 0 0
\(797\) − 5.37136e6i − 0.299529i −0.988722 0.149764i \(-0.952148\pi\)
0.988722 0.149764i \(-0.0478515\pi\)
\(798\) 0 0
\(799\) 1.35307e7 0.749813
\(800\) 0 0
\(801\) 1.22710e6 0.0675772
\(802\) 0 0
\(803\) − 1.60421e7i − 0.877954i
\(804\) 0 0
\(805\) − 1.05994e7i − 0.576491i
\(806\) 0 0
\(807\) −2.01271e6 −0.108792
\(808\) 0 0
\(809\) 8.86137e6 0.476025 0.238012 0.971262i \(-0.423504\pi\)
0.238012 + 0.971262i \(0.423504\pi\)
\(810\) 0 0
\(811\) 2.91251e7i 1.55495i 0.628915 + 0.777474i \(0.283499\pi\)
−0.628915 + 0.777474i \(0.716501\pi\)
\(812\) 0 0
\(813\) 1.56510e7i 0.830456i
\(814\) 0 0
\(815\) 6.60763e6 0.348459
\(816\) 0 0
\(817\) −5.37165e6 −0.281548
\(818\) 0 0
\(819\) 1.70817e6i 0.0889857i
\(820\) 0 0
\(821\) − 2.62586e7i − 1.35961i −0.733394 0.679803i \(-0.762065\pi\)
0.733394 0.679803i \(-0.237935\pi\)
\(822\) 0 0
\(823\) 1.88150e7 0.968287 0.484143 0.874989i \(-0.339131\pi\)
0.484143 + 0.874989i \(0.339131\pi\)
\(824\) 0 0
\(825\) 3.57948e6 0.183099
\(826\) 0 0
\(827\) − 1.26962e7i − 0.645520i −0.946481 0.322760i \(-0.895389\pi\)
0.946481 0.322760i \(-0.104611\pi\)
\(828\) 0 0
\(829\) − 2.01838e7i − 1.02004i −0.860164 0.510018i \(-0.829639\pi\)
0.860164 0.510018i \(-0.170361\pi\)
\(830\) 0 0
\(831\) −2.12213e7 −1.06603
\(832\) 0 0
\(833\) −2.19720e7 −1.09713
\(834\) 0 0
\(835\) − 7.88021e6i − 0.391130i
\(836\) 0 0
\(837\) − 1.95296e6i − 0.0963560i
\(838\) 0 0
\(839\) −1.27185e7 −0.623781 −0.311890 0.950118i \(-0.600962\pi\)
−0.311890 + 0.950118i \(0.600962\pi\)
\(840\) 0 0
\(841\) −1.48215e7 −0.722605
\(842\) 0 0
\(843\) 8.30766e6i 0.402633i
\(844\) 0 0
\(845\) 1.14231e7i 0.550354i
\(846\) 0 0
\(847\) 59859.3 0.00286697
\(848\) 0 0
\(849\) −7.63802e6 −0.363674
\(850\) 0 0
\(851\) − 2.95721e6i − 0.139978i
\(852\) 0 0
\(853\) 1.95957e7i 0.922121i 0.887369 + 0.461060i \(0.152531\pi\)
−0.887369 + 0.461060i \(0.847469\pi\)
\(854\) 0 0
\(855\) −2.11001e6 −0.0987117
\(856\) 0 0
\(857\) 3.68287e7 1.71291 0.856454 0.516224i \(-0.172663\pi\)
0.856454 + 0.516224i \(0.172663\pi\)
\(858\) 0 0
\(859\) − 1.70593e7i − 0.788823i −0.918934 0.394411i \(-0.870949\pi\)
0.918934 0.394411i \(-0.129051\pi\)
\(860\) 0 0
\(861\) 2.76131e6i 0.126943i
\(862\) 0 0
\(863\) −3.56411e7 −1.62901 −0.814506 0.580155i \(-0.802992\pi\)
−0.814506 + 0.580155i \(0.802992\pi\)
\(864\) 0 0
\(865\) 1.79012e6 0.0813468
\(866\) 0 0
\(867\) − 1.20987e7i − 0.546626i
\(868\) 0 0
\(869\) 1.03036e7i 0.462849i
\(870\) 0 0
\(871\) 3.99855e6 0.178590
\(872\) 0 0
\(873\) −1.28325e7 −0.569869
\(874\) 0 0
\(875\) 1.13949e7i 0.503144i
\(876\) 0 0
\(877\) − 6.78596e6i − 0.297929i −0.988843 0.148964i \(-0.952406\pi\)
0.988843 0.148964i \(-0.0475939\pi\)
\(878\) 0 0
\(879\) −9.82105e6 −0.428732
\(880\) 0 0
\(881\) 6.27597e6 0.272421 0.136211 0.990680i \(-0.456508\pi\)
0.136211 + 0.990680i \(0.456508\pi\)
\(882\) 0 0
\(883\) − 2.46975e7i − 1.06599i −0.846120 0.532993i \(-0.821067\pi\)
0.846120 0.532993i \(-0.178933\pi\)
\(884\) 0 0
\(885\) 1.33540e7i 0.573128i
\(886\) 0 0
\(887\) −1.04054e6 −0.0444070 −0.0222035 0.999753i \(-0.507068\pi\)
−0.0222035 + 0.999753i \(0.507068\pi\)
\(888\) 0 0
\(889\) −1.96660e7 −0.834569
\(890\) 0 0
\(891\) 2.62483e6i 0.110766i
\(892\) 0 0
\(893\) − 4.59261e6i − 0.192722i
\(894\) 0 0
\(895\) 1.84409e7 0.769530
\(896\) 0 0
\(897\) 1.21349e7 0.503563
\(898\) 0 0
\(899\) − 1.59240e7i − 0.657133i
\(900\) 0 0
\(901\) 1.83746e7i 0.754062i
\(902\) 0 0
\(903\) −5.13402e6 −0.209526
\(904\) 0 0
\(905\) 7.75829e6 0.314880
\(906\) 0 0
\(907\) 2.41754e7i 0.975789i 0.872903 + 0.487894i \(0.162235\pi\)
−0.872903 + 0.487894i \(0.837765\pi\)
\(908\) 0 0
\(909\) − 1.25258e7i − 0.502799i
\(910\) 0 0
\(911\) −3.64416e7 −1.45479 −0.727396 0.686217i \(-0.759270\pi\)
−0.727396 + 0.686217i \(0.759270\pi\)
\(912\) 0 0
\(913\) −896047. −0.0355758
\(914\) 0 0
\(915\) 1.47297e7i 0.581624i
\(916\) 0 0
\(917\) − 1.05579e7i − 0.414622i
\(918\) 0 0
\(919\) 3.29796e7 1.28812 0.644060 0.764975i \(-0.277249\pi\)
0.644060 + 0.764975i \(0.277249\pi\)
\(920\) 0 0
\(921\) −1.40054e7 −0.544060
\(922\) 0 0
\(923\) 3.65789e6i 0.141327i
\(924\) 0 0
\(925\) 767279.i 0.0294849i
\(926\) 0 0
\(927\) −7.26335e6 −0.277612
\(928\) 0 0
\(929\) −3.83111e7 −1.45642 −0.728209 0.685356i \(-0.759647\pi\)
−0.728209 + 0.685356i \(0.759647\pi\)
\(930\) 0 0
\(931\) 7.45779e6i 0.281991i
\(932\) 0 0
\(933\) 4.11508e6i 0.154765i
\(934\) 0 0
\(935\) −3.07036e7 −1.14858
\(936\) 0 0
\(937\) 1.48708e7 0.553332 0.276666 0.960966i \(-0.410770\pi\)
0.276666 + 0.960966i \(0.410770\pi\)
\(938\) 0 0
\(939\) 1.56233e7i 0.578242i
\(940\) 0 0
\(941\) − 2.67852e7i − 0.986099i −0.870001 0.493049i \(-0.835882\pi\)
0.870001 0.493049i \(-0.164118\pi\)
\(942\) 0 0
\(943\) 1.96165e7 0.718359
\(944\) 0 0
\(945\) −2.01666e6 −0.0734605
\(946\) 0 0
\(947\) − 1.24319e7i − 0.450467i −0.974305 0.225233i \(-0.927686\pi\)
0.974305 0.225233i \(-0.0723145\pi\)
\(948\) 0 0
\(949\) 1.41106e7i 0.508605i
\(950\) 0 0
\(951\) −4.05201e6 −0.145285
\(952\) 0 0
\(953\) −3.05330e7 −1.08902 −0.544511 0.838753i \(-0.683285\pi\)
−0.544511 + 0.838753i \(0.683285\pi\)
\(954\) 0 0
\(955\) − 1.06532e7i − 0.377981i
\(956\) 0 0
\(957\) 2.14023e7i 0.755407i
\(958\) 0 0
\(959\) 3.01113e6 0.105726
\(960\) 0 0
\(961\) −2.14524e7 −0.749319
\(962\) 0 0
\(963\) − 9.18035e6i − 0.319002i
\(964\) 0 0
\(965\) 334050.i 0.0115476i
\(966\) 0 0
\(967\) 3.23903e7 1.11391 0.556954 0.830543i \(-0.311970\pi\)
0.556954 + 0.830543i \(0.311970\pi\)
\(968\) 0 0
\(969\) −8.44394e6 −0.288892
\(970\) 0 0
\(971\) 5.00157e7i 1.70239i 0.524853 + 0.851193i \(0.324120\pi\)
−0.524853 + 0.851193i \(0.675880\pi\)
\(972\) 0 0
\(973\) 2.18691e7i 0.740542i
\(974\) 0 0
\(975\) −3.14852e6 −0.106070
\(976\) 0 0
\(977\) 621156. 0.0208192 0.0104096 0.999946i \(-0.496686\pi\)
0.0104096 + 0.999946i \(0.496686\pi\)
\(978\) 0 0
\(979\) − 6.06076e6i − 0.202102i
\(980\) 0 0
\(981\) − 1.79336e7i − 0.594971i
\(982\) 0 0
\(983\) 6.73840e6 0.222420 0.111210 0.993797i \(-0.464527\pi\)
0.111210 + 0.993797i \(0.464527\pi\)
\(984\) 0 0
\(985\) 2.69026e7 0.883493
\(986\) 0 0
\(987\) − 4.38944e6i − 0.143422i
\(988\) 0 0
\(989\) 3.64723e7i 1.18569i
\(990\) 0 0
\(991\) −1.73933e7 −0.562597 −0.281298 0.959620i \(-0.590765\pi\)
−0.281298 + 0.959620i \(0.590765\pi\)
\(992\) 0 0
\(993\) 2.51867e7 0.810585
\(994\) 0 0
\(995\) − 4.98163e7i − 1.59520i
\(996\) 0 0
\(997\) 9.67436e6i 0.308237i 0.988052 + 0.154118i \(0.0492537\pi\)
−0.988052 + 0.154118i \(0.950746\pi\)
\(998\) 0 0
\(999\) −562644. −0.0178369
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.d.j.193.5 yes 12
4.3 odd 2 inner 384.6.d.j.193.11 yes 12
8.3 odd 2 inner 384.6.d.j.193.2 12
8.5 even 2 inner 384.6.d.j.193.8 yes 12
16.3 odd 4 768.6.a.be.1.5 6
16.5 even 4 768.6.a.be.1.2 6
16.11 odd 4 768.6.a.bf.1.2 6
16.13 even 4 768.6.a.bf.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.d.j.193.2 12 8.3 odd 2 inner
384.6.d.j.193.5 yes 12 1.1 even 1 trivial
384.6.d.j.193.8 yes 12 8.5 even 2 inner
384.6.d.j.193.11 yes 12 4.3 odd 2 inner
768.6.a.be.1.2 6 16.5 even 4
768.6.a.be.1.5 6 16.3 odd 4
768.6.a.bf.1.2 6 16.11 odd 4
768.6.a.bf.1.5 6 16.13 even 4