Properties

Label 64.24.a.n.1.2
Level $64$
Weight $24$
Character 64.1
Self dual yes
Analytic conductor $214.531$
Analytic rank $1$
Dimension $6$
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] = [64,24,Mod(1,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(214.530583901\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 389656213x^{4} + 47522643058672215x^{2} - 1756479932541937262634975 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{71}\cdot 3^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-11884.9\) of defining polynomial
Character \(\chi\) \(=\) 64.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-216966. q^{3} -1.90927e8 q^{5} +9.74240e9 q^{7} -4.70690e10 q^{9} +O(q^{10})\) \(q-216966. q^{3} -1.90927e8 q^{5} +9.74240e9 q^{7} -4.70690e10 q^{9} -9.58624e11 q^{11} +2.14517e12 q^{13} +4.14246e13 q^{15} +2.65620e13 q^{17} -7.63669e14 q^{19} -2.11377e15 q^{21} +3.75905e15 q^{23} +2.45321e16 q^{25} +3.06382e16 q^{27} -5.01247e16 q^{29} -1.90008e17 q^{31} +2.07989e17 q^{33} -1.86008e18 q^{35} +1.56774e18 q^{37} -4.65429e17 q^{39} -3.05851e18 q^{41} -3.07525e18 q^{43} +8.98674e18 q^{45} +2.72401e18 q^{47} +6.75455e19 q^{49} -5.76305e18 q^{51} +2.52631e19 q^{53} +1.83027e20 q^{55} +1.65690e20 q^{57} +2.29901e20 q^{59} +6.13479e20 q^{61} -4.58565e20 q^{63} -4.09571e20 q^{65} +1.20452e21 q^{67} -8.15585e20 q^{69} -2.43377e20 q^{71} -3.45063e20 q^{73} -5.32263e21 q^{75} -9.33929e21 q^{77} +3.08236e21 q^{79} -2.21622e21 q^{81} +1.36342e22 q^{83} -5.07140e21 q^{85} +1.08753e22 q^{87} -1.67114e22 q^{89} +2.08991e22 q^{91} +4.12252e22 q^{93} +1.45805e23 q^{95} +7.71584e22 q^{97} +4.51215e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 163121700 q^{5} + 14120212542 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 163121700 q^{5} + 14120212542 q^{9} - 1101295489524 q^{13} + 56435517243468 q^{17} - 14\!\cdots\!84 q^{21}+ \cdots + 25\!\cdots\!16 q^{97}+O(q^{100}) \) 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
\(3\) −216966. −0.707126 −0.353563 0.935411i \(-0.615030\pi\)
−0.353563 + 0.935411i \(0.615030\pi\)
\(4\) 0 0
\(5\) −1.90927e8 −1.74869 −0.874343 0.485308i \(-0.838707\pi\)
−0.874343 + 0.485308i \(0.838707\pi\)
\(6\) 0 0
\(7\) 9.74240e9 1.86225 0.931126 0.364698i \(-0.118828\pi\)
0.931126 + 0.364698i \(0.118828\pi\)
\(8\) 0 0
\(9\) −4.70690e10 −0.499973
\(10\) 0 0
\(11\) −9.58624e11 −1.01305 −0.506527 0.862224i \(-0.669071\pi\)
−0.506527 + 0.862224i \(0.669071\pi\)
\(12\) 0 0
\(13\) 2.14517e12 0.331981 0.165991 0.986127i \(-0.446918\pi\)
0.165991 + 0.986127i \(0.446918\pi\)
\(14\) 0 0
\(15\) 4.14246e13 1.23654
\(16\) 0 0
\(17\) 2.65620e13 0.187974 0.0939872 0.995573i \(-0.470039\pi\)
0.0939872 + 0.995573i \(0.470039\pi\)
\(18\) 0 0
\(19\) −7.63669e14 −1.50397 −0.751985 0.659181i \(-0.770903\pi\)
−0.751985 + 0.659181i \(0.770903\pi\)
\(20\) 0 0
\(21\) −2.11377e15 −1.31685
\(22\) 0 0
\(23\) 3.75905e15 0.822636 0.411318 0.911492i \(-0.365069\pi\)
0.411318 + 0.911492i \(0.365069\pi\)
\(24\) 0 0
\(25\) 2.45321e16 2.05790
\(26\) 0 0
\(27\) 3.06382e16 1.06067
\(28\) 0 0
\(29\) −5.01247e16 −0.762912 −0.381456 0.924387i \(-0.624577\pi\)
−0.381456 + 0.924387i \(0.624577\pi\)
\(30\) 0 0
\(31\) −1.90008e17 −1.34311 −0.671556 0.740954i \(-0.734374\pi\)
−0.671556 + 0.740954i \(0.734374\pi\)
\(32\) 0 0
\(33\) 2.07989e17 0.716357
\(34\) 0 0
\(35\) −1.86008e18 −3.25649
\(36\) 0 0
\(37\) 1.56774e18 1.44862 0.724311 0.689474i \(-0.242158\pi\)
0.724311 + 0.689474i \(0.242158\pi\)
\(38\) 0 0
\(39\) −4.65429e17 −0.234753
\(40\) 0 0
\(41\) −3.05851e18 −0.867951 −0.433975 0.900925i \(-0.642890\pi\)
−0.433975 + 0.900925i \(0.642890\pi\)
\(42\) 0 0
\(43\) −3.07525e18 −0.504653 −0.252327 0.967642i \(-0.581196\pi\)
−0.252327 + 0.967642i \(0.581196\pi\)
\(44\) 0 0
\(45\) 8.98674e18 0.874295
\(46\) 0 0
\(47\) 2.72401e18 0.160725 0.0803626 0.996766i \(-0.474392\pi\)
0.0803626 + 0.996766i \(0.474392\pi\)
\(48\) 0 0
\(49\) 6.75455e19 2.46798
\(50\) 0 0
\(51\) −5.76305e18 −0.132922
\(52\) 0 0
\(53\) 2.52631e19 0.374381 0.187191 0.982324i \(-0.440062\pi\)
0.187191 + 0.982324i \(0.440062\pi\)
\(54\) 0 0
\(55\) 1.83027e20 1.77151
\(56\) 0 0
\(57\) 1.65690e20 1.06350
\(58\) 0 0
\(59\) 2.29901e20 0.992527 0.496263 0.868172i \(-0.334705\pi\)
0.496263 + 0.868172i \(0.334705\pi\)
\(60\) 0 0
\(61\) 6.13479e20 1.80512 0.902561 0.430562i \(-0.141685\pi\)
0.902561 + 0.430562i \(0.141685\pi\)
\(62\) 0 0
\(63\) −4.58565e20 −0.931075
\(64\) 0 0
\(65\) −4.09571e20 −0.580531
\(66\) 0 0
\(67\) 1.20452e21 1.20491 0.602456 0.798152i \(-0.294189\pi\)
0.602456 + 0.798152i \(0.294189\pi\)
\(68\) 0 0
\(69\) −8.15585e20 −0.581707
\(70\) 0 0
\(71\) −2.43377e20 −0.124971 −0.0624854 0.998046i \(-0.519903\pi\)
−0.0624854 + 0.998046i \(0.519903\pi\)
\(72\) 0 0
\(73\) −3.45063e20 −0.128732 −0.0643659 0.997926i \(-0.520502\pi\)
−0.0643659 + 0.997926i \(0.520502\pi\)
\(74\) 0 0
\(75\) −5.32263e21 −1.45520
\(76\) 0 0
\(77\) −9.33929e21 −1.88656
\(78\) 0 0
\(79\) 3.08236e21 0.463630 0.231815 0.972760i \(-0.425534\pi\)
0.231815 + 0.972760i \(0.425534\pi\)
\(80\) 0 0
\(81\) −2.21622e21 −0.250055
\(82\) 0 0
\(83\) 1.36342e22 1.16207 0.581036 0.813878i \(-0.302648\pi\)
0.581036 + 0.813878i \(0.302648\pi\)
\(84\) 0 0
\(85\) −5.07140e21 −0.328708
\(86\) 0 0
\(87\) 1.08753e22 0.539475
\(88\) 0 0
\(89\) −1.67114e22 −0.638303 −0.319152 0.947704i \(-0.603398\pi\)
−0.319152 + 0.947704i \(0.603398\pi\)
\(90\) 0 0
\(91\) 2.08991e22 0.618232
\(92\) 0 0
\(93\) 4.12252e22 0.949749
\(94\) 0 0
\(95\) 1.45805e23 2.62997
\(96\) 0 0
\(97\) 7.71584e22 1.09524 0.547619 0.836728i \(-0.315534\pi\)
0.547619 + 0.836728i \(0.315534\pi\)
\(98\) 0 0
\(99\) 4.51215e22 0.506499
\(100\) 0 0
\(101\) −1.31460e23 −1.17246 −0.586231 0.810144i \(-0.699389\pi\)
−0.586231 + 0.810144i \(0.699389\pi\)
\(102\) 0 0
\(103\) −2.43701e23 −1.73472 −0.867361 0.497680i \(-0.834185\pi\)
−0.867361 + 0.497680i \(0.834185\pi\)
\(104\) 0 0
\(105\) 4.03575e23 2.30275
\(106\) 0 0
\(107\) 1.04639e23 0.480598 0.240299 0.970699i \(-0.422754\pi\)
0.240299 + 0.970699i \(0.422754\pi\)
\(108\) 0 0
\(109\) −5.91477e22 −0.219550 −0.109775 0.993956i \(-0.535013\pi\)
−0.109775 + 0.993956i \(0.535013\pi\)
\(110\) 0 0
\(111\) −3.40147e23 −1.02436
\(112\) 0 0
\(113\) 1.55264e23 0.380774 0.190387 0.981709i \(-0.439026\pi\)
0.190387 + 0.981709i \(0.439026\pi\)
\(114\) 0 0
\(115\) −7.17703e23 −1.43853
\(116\) 0 0
\(117\) −1.00971e23 −0.165981
\(118\) 0 0
\(119\) 2.58778e23 0.350056
\(120\) 0 0
\(121\) 2.35294e22 0.0262772
\(122\) 0 0
\(123\) 6.63591e23 0.613751
\(124\) 0 0
\(125\) −2.40781e24 −1.84994
\(126\) 0 0
\(127\) 2.80213e24 1.79368 0.896841 0.442353i \(-0.145856\pi\)
0.896841 + 0.442353i \(0.145856\pi\)
\(128\) 0 0
\(129\) 6.67224e23 0.356854
\(130\) 0 0
\(131\) −4.42927e23 −0.198478 −0.0992390 0.995064i \(-0.531641\pi\)
−0.0992390 + 0.995064i \(0.531641\pi\)
\(132\) 0 0
\(133\) −7.43997e24 −2.80077
\(134\) 0 0
\(135\) −5.84966e24 −1.85478
\(136\) 0 0
\(137\) 5.28208e24 1.41423 0.707113 0.707101i \(-0.249997\pi\)
0.707113 + 0.707101i \(0.249997\pi\)
\(138\) 0 0
\(139\) −1.03441e24 −0.234435 −0.117218 0.993106i \(-0.537397\pi\)
−0.117218 + 0.993106i \(0.537397\pi\)
\(140\) 0 0
\(141\) −5.91018e23 −0.113653
\(142\) 0 0
\(143\) −2.05641e24 −0.336315
\(144\) 0 0
\(145\) 9.57014e24 1.33409
\(146\) 0 0
\(147\) −1.46551e25 −1.74517
\(148\) 0 0
\(149\) 9.38525e23 0.0956762 0.0478381 0.998855i \(-0.484767\pi\)
0.0478381 + 0.998855i \(0.484767\pi\)
\(150\) 0 0
\(151\) 6.67720e23 0.0583928 0.0291964 0.999574i \(-0.490705\pi\)
0.0291964 + 0.999574i \(0.490705\pi\)
\(152\) 0 0
\(153\) −1.25025e24 −0.0939820
\(154\) 0 0
\(155\) 3.62776e25 2.34868
\(156\) 0 0
\(157\) 1.43848e25 0.803635 0.401818 0.915720i \(-0.368379\pi\)
0.401818 + 0.915720i \(0.368379\pi\)
\(158\) 0 0
\(159\) −5.48123e24 −0.264735
\(160\) 0 0
\(161\) 3.66221e25 1.53195
\(162\) 0 0
\(163\) −2.39784e25 −0.870289 −0.435145 0.900361i \(-0.643303\pi\)
−0.435145 + 0.900361i \(0.643303\pi\)
\(164\) 0 0
\(165\) −3.97106e25 −1.25268
\(166\) 0 0
\(167\) −1.51560e25 −0.416242 −0.208121 0.978103i \(-0.566735\pi\)
−0.208121 + 0.978103i \(0.566735\pi\)
\(168\) 0 0
\(169\) −3.71521e25 −0.889788
\(170\) 0 0
\(171\) 3.59452e25 0.751943
\(172\) 0 0
\(173\) −5.92919e25 −1.08509 −0.542545 0.840027i \(-0.682539\pi\)
−0.542545 + 0.840027i \(0.682539\pi\)
\(174\) 0 0
\(175\) 2.39002e26 3.83233
\(176\) 0 0
\(177\) −4.98806e25 −0.701842
\(178\) 0 0
\(179\) 6.32487e24 0.0782063 0.0391031 0.999235i \(-0.487550\pi\)
0.0391031 + 0.999235i \(0.487550\pi\)
\(180\) 0 0
\(181\) 2.91917e25 0.317655 0.158828 0.987306i \(-0.449229\pi\)
0.158828 + 0.987306i \(0.449229\pi\)
\(182\) 0 0
\(183\) −1.33104e26 −1.27645
\(184\) 0 0
\(185\) −2.99324e26 −2.53318
\(186\) 0 0
\(187\) −2.54630e25 −0.190428
\(188\) 0 0
\(189\) 2.98490e26 1.97523
\(190\) 0 0
\(191\) −3.38328e26 −1.98360 −0.991800 0.127796i \(-0.959210\pi\)
−0.991800 + 0.127796i \(0.959210\pi\)
\(192\) 0 0
\(193\) −1.27890e26 −0.665163 −0.332581 0.943075i \(-0.607920\pi\)
−0.332581 + 0.943075i \(0.607920\pi\)
\(194\) 0 0
\(195\) 8.88629e25 0.410509
\(196\) 0 0
\(197\) −3.48404e26 −1.43127 −0.715635 0.698474i \(-0.753863\pi\)
−0.715635 + 0.698474i \(0.753863\pi\)
\(198\) 0 0
\(199\) 1.41137e26 0.516214 0.258107 0.966116i \(-0.416901\pi\)
0.258107 + 0.966116i \(0.416901\pi\)
\(200\) 0 0
\(201\) −2.61341e26 −0.852025
\(202\) 0 0
\(203\) −4.88334e26 −1.42073
\(204\) 0 0
\(205\) 5.83951e26 1.51777
\(206\) 0 0
\(207\) −1.76935e26 −0.411295
\(208\) 0 0
\(209\) 7.32072e26 1.52360
\(210\) 0 0
\(211\) 4.52456e26 0.843972 0.421986 0.906602i \(-0.361333\pi\)
0.421986 + 0.906602i \(0.361333\pi\)
\(212\) 0 0
\(213\) 5.28045e25 0.0883701
\(214\) 0 0
\(215\) 5.87148e26 0.882480
\(216\) 0 0
\(217\) −1.85113e27 −2.50121
\(218\) 0 0
\(219\) 7.48669e25 0.0910296
\(220\) 0 0
\(221\) 5.69801e25 0.0624040
\(222\) 0 0
\(223\) −9.10896e26 −0.899421 −0.449711 0.893174i \(-0.648473\pi\)
−0.449711 + 0.893174i \(0.648473\pi\)
\(224\) 0 0
\(225\) −1.15470e27 −1.02890
\(226\) 0 0
\(227\) −4.31344e26 −0.347158 −0.173579 0.984820i \(-0.555533\pi\)
−0.173579 + 0.984820i \(0.555533\pi\)
\(228\) 0 0
\(229\) −1.66792e27 −1.21357 −0.606787 0.794864i \(-0.707542\pi\)
−0.606787 + 0.794864i \(0.707542\pi\)
\(230\) 0 0
\(231\) 2.02631e27 1.33404
\(232\) 0 0
\(233\) −5.01267e26 −0.298866 −0.149433 0.988772i \(-0.547745\pi\)
−0.149433 + 0.988772i \(0.547745\pi\)
\(234\) 0 0
\(235\) −5.20088e26 −0.281058
\(236\) 0 0
\(237\) −6.68766e26 −0.327845
\(238\) 0 0
\(239\) −2.25295e27 −1.00271 −0.501355 0.865241i \(-0.667165\pi\)
−0.501355 + 0.865241i \(0.667165\pi\)
\(240\) 0 0
\(241\) −6.48722e26 −0.262339 −0.131169 0.991360i \(-0.541873\pi\)
−0.131169 + 0.991360i \(0.541873\pi\)
\(242\) 0 0
\(243\) −2.40354e27 −0.883850
\(244\) 0 0
\(245\) −1.28963e28 −4.31572
\(246\) 0 0
\(247\) −1.63820e27 −0.499290
\(248\) 0 0
\(249\) −2.95817e27 −0.821732
\(250\) 0 0
\(251\) 7.85626e26 0.199053 0.0995264 0.995035i \(-0.468267\pi\)
0.0995264 + 0.995035i \(0.468267\pi\)
\(252\) 0 0
\(253\) −3.60351e27 −0.833374
\(254\) 0 0
\(255\) 1.10032e27 0.232438
\(256\) 0 0
\(257\) 6.31330e27 1.21906 0.609531 0.792762i \(-0.291358\pi\)
0.609531 + 0.792762i \(0.291358\pi\)
\(258\) 0 0
\(259\) 1.52736e28 2.69770
\(260\) 0 0
\(261\) 2.35932e27 0.381435
\(262\) 0 0
\(263\) 2.63348e27 0.389977 0.194989 0.980806i \(-0.437533\pi\)
0.194989 + 0.980806i \(0.437533\pi\)
\(264\) 0 0
\(265\) −4.82340e27 −0.654675
\(266\) 0 0
\(267\) 3.62580e27 0.451361
\(268\) 0 0
\(269\) −8.67658e27 −0.991282 −0.495641 0.868527i \(-0.665067\pi\)
−0.495641 + 0.868527i \(0.665067\pi\)
\(270\) 0 0
\(271\) 1.35097e28 1.41742 0.708711 0.705499i \(-0.249277\pi\)
0.708711 + 0.705499i \(0.249277\pi\)
\(272\) 0 0
\(273\) −4.53439e27 −0.437168
\(274\) 0 0
\(275\) −2.35171e28 −2.08477
\(276\) 0 0
\(277\) 2.14474e27 0.174927 0.0874635 0.996168i \(-0.472124\pi\)
0.0874635 + 0.996168i \(0.472124\pi\)
\(278\) 0 0
\(279\) 8.94348e27 0.671519
\(280\) 0 0
\(281\) 1.26495e28 0.874883 0.437441 0.899247i \(-0.355885\pi\)
0.437441 + 0.899247i \(0.355885\pi\)
\(282\) 0 0
\(283\) 3.33583e27 0.212647 0.106324 0.994332i \(-0.466092\pi\)
0.106324 + 0.994332i \(0.466092\pi\)
\(284\) 0 0
\(285\) −3.16347e28 −1.85972
\(286\) 0 0
\(287\) −2.97972e28 −1.61634
\(288\) 0 0
\(289\) −1.92620e28 −0.964666
\(290\) 0 0
\(291\) −1.67407e28 −0.774472
\(292\) 0 0
\(293\) −2.39590e28 −1.02445 −0.512225 0.858851i \(-0.671179\pi\)
−0.512225 + 0.858851i \(0.671179\pi\)
\(294\) 0 0
\(295\) −4.38942e28 −1.73562
\(296\) 0 0
\(297\) −2.93705e28 −1.07452
\(298\) 0 0
\(299\) 8.06380e27 0.273100
\(300\) 0 0
\(301\) −2.99603e28 −0.939792
\(302\) 0 0
\(303\) 2.85224e28 0.829078
\(304\) 0 0
\(305\) −1.17130e29 −3.15659
\(306\) 0 0
\(307\) −4.06329e28 −1.01575 −0.507874 0.861431i \(-0.669569\pi\)
−0.507874 + 0.861431i \(0.669569\pi\)
\(308\) 0 0
\(309\) 5.28749e28 1.22667
\(310\) 0 0
\(311\) −2.40957e27 −0.0519033 −0.0259517 0.999663i \(-0.508262\pi\)
−0.0259517 + 0.999663i \(0.508262\pi\)
\(312\) 0 0
\(313\) −3.08964e28 −0.618226 −0.309113 0.951025i \(-0.600032\pi\)
−0.309113 + 0.951025i \(0.600032\pi\)
\(314\) 0 0
\(315\) 8.75523e28 1.62816
\(316\) 0 0
\(317\) 4.47714e28 0.774140 0.387070 0.922050i \(-0.373487\pi\)
0.387070 + 0.922050i \(0.373487\pi\)
\(318\) 0 0
\(319\) 4.80507e28 0.772871
\(320\) 0 0
\(321\) −2.27032e28 −0.339843
\(322\) 0 0
\(323\) −2.02846e28 −0.282708
\(324\) 0 0
\(325\) 5.26256e28 0.683185
\(326\) 0 0
\(327\) 1.28330e28 0.155249
\(328\) 0 0
\(329\) 2.65384e28 0.299311
\(330\) 0 0
\(331\) 5.57485e28 0.586423 0.293212 0.956048i \(-0.405276\pi\)
0.293212 + 0.956048i \(0.405276\pi\)
\(332\) 0 0
\(333\) −7.37921e28 −0.724271
\(334\) 0 0
\(335\) −2.29976e29 −2.10701
\(336\) 0 0
\(337\) 6.21767e28 0.531966 0.265983 0.963978i \(-0.414303\pi\)
0.265983 + 0.963978i \(0.414303\pi\)
\(338\) 0 0
\(339\) −3.36869e28 −0.269255
\(340\) 0 0
\(341\) 1.82146e29 1.36064
\(342\) 0 0
\(343\) 3.91418e29 2.73375
\(344\) 0 0
\(345\) 1.55717e29 1.01722
\(346\) 0 0
\(347\) −1.40504e29 −0.858817 −0.429409 0.903110i \(-0.641278\pi\)
−0.429409 + 0.903110i \(0.641278\pi\)
\(348\) 0 0
\(349\) 9.62640e28 0.550772 0.275386 0.961334i \(-0.411194\pi\)
0.275386 + 0.961334i \(0.411194\pi\)
\(350\) 0 0
\(351\) 6.57242e28 0.352122
\(352\) 0 0
\(353\) −2.84480e29 −1.42772 −0.713860 0.700288i \(-0.753055\pi\)
−0.713860 + 0.700288i \(0.753055\pi\)
\(354\) 0 0
\(355\) 4.64672e28 0.218535
\(356\) 0 0
\(357\) −5.61459e28 −0.247533
\(358\) 0 0
\(359\) −1.00468e29 −0.415376 −0.207688 0.978195i \(-0.566594\pi\)
−0.207688 + 0.978195i \(0.566594\pi\)
\(360\) 0 0
\(361\) 3.25361e29 1.26192
\(362\) 0 0
\(363\) −5.10508e27 −0.0185813
\(364\) 0 0
\(365\) 6.58818e28 0.225112
\(366\) 0 0
\(367\) −5.45272e29 −1.74966 −0.874829 0.484431i \(-0.839027\pi\)
−0.874829 + 0.484431i \(0.839027\pi\)
\(368\) 0 0
\(369\) 1.43961e29 0.433952
\(370\) 0 0
\(371\) 2.46123e29 0.697192
\(372\) 0 0
\(373\) −1.93840e29 −0.516170 −0.258085 0.966122i \(-0.583091\pi\)
−0.258085 + 0.966122i \(0.583091\pi\)
\(374\) 0 0
\(375\) 5.22413e29 1.30814
\(376\) 0 0
\(377\) −1.07526e29 −0.253272
\(378\) 0 0
\(379\) 2.74346e29 0.608063 0.304031 0.952662i \(-0.401667\pi\)
0.304031 + 0.952662i \(0.401667\pi\)
\(380\) 0 0
\(381\) −6.07967e29 −1.26836
\(382\) 0 0
\(383\) −4.18842e29 −0.822744 −0.411372 0.911468i \(-0.634950\pi\)
−0.411372 + 0.911468i \(0.634950\pi\)
\(384\) 0 0
\(385\) 1.78312e30 3.29900
\(386\) 0 0
\(387\) 1.44749e29 0.252313
\(388\) 0 0
\(389\) −4.88162e29 −0.801944 −0.400972 0.916090i \(-0.631328\pi\)
−0.400972 + 0.916090i \(0.631328\pi\)
\(390\) 0 0
\(391\) 9.98479e28 0.154634
\(392\) 0 0
\(393\) 9.61000e28 0.140349
\(394\) 0 0
\(395\) −5.88505e29 −0.810743
\(396\) 0 0
\(397\) 1.29652e30 1.68535 0.842673 0.538425i \(-0.180981\pi\)
0.842673 + 0.538425i \(0.180981\pi\)
\(398\) 0 0
\(399\) 1.61422e30 1.98050
\(400\) 0 0
\(401\) 1.12391e30 1.30188 0.650939 0.759130i \(-0.274375\pi\)
0.650939 + 0.759130i \(0.274375\pi\)
\(402\) 0 0
\(403\) −4.07599e29 −0.445888
\(404\) 0 0
\(405\) 4.23136e29 0.437267
\(406\) 0 0
\(407\) −1.50288e30 −1.46753
\(408\) 0 0
\(409\) 2.82215e29 0.260472 0.130236 0.991483i \(-0.458426\pi\)
0.130236 + 0.991483i \(0.458426\pi\)
\(410\) 0 0
\(411\) −1.14603e30 −1.00004
\(412\) 0 0
\(413\) 2.23978e30 1.84833
\(414\) 0 0
\(415\) −2.60314e30 −2.03210
\(416\) 0 0
\(417\) 2.24432e29 0.165775
\(418\) 0 0
\(419\) −2.72724e30 −1.90661 −0.953305 0.302009i \(-0.902343\pi\)
−0.953305 + 0.302009i \(0.902343\pi\)
\(420\) 0 0
\(421\) 1.64535e30 1.08897 0.544484 0.838771i \(-0.316726\pi\)
0.544484 + 0.838771i \(0.316726\pi\)
\(422\) 0 0
\(423\) −1.28217e29 −0.0803582
\(424\) 0 0
\(425\) 6.51623e29 0.386833
\(426\) 0 0
\(427\) 5.97675e30 3.36159
\(428\) 0 0
\(429\) 4.46171e29 0.237817
\(430\) 0 0
\(431\) 2.50630e30 1.26632 0.633162 0.774020i \(-0.281757\pi\)
0.633162 + 0.774020i \(0.281757\pi\)
\(432\) 0 0
\(433\) 2.04414e30 0.979262 0.489631 0.871930i \(-0.337131\pi\)
0.489631 + 0.871930i \(0.337131\pi\)
\(434\) 0 0
\(435\) −2.07639e30 −0.943373
\(436\) 0 0
\(437\) −2.87067e30 −1.23722
\(438\) 0 0
\(439\) 4.46273e30 1.82498 0.912492 0.409095i \(-0.134156\pi\)
0.912492 + 0.409095i \(0.134156\pi\)
\(440\) 0 0
\(441\) −3.17930e30 −1.23392
\(442\) 0 0
\(443\) 1.66470e30 0.613330 0.306665 0.951817i \(-0.400787\pi\)
0.306665 + 0.951817i \(0.400787\pi\)
\(444\) 0 0
\(445\) 3.19065e30 1.11619
\(446\) 0 0
\(447\) −2.03628e29 −0.0676551
\(448\) 0 0
\(449\) −3.50605e30 −1.10659 −0.553293 0.832986i \(-0.686629\pi\)
−0.553293 + 0.832986i \(0.686629\pi\)
\(450\) 0 0
\(451\) 2.93196e30 0.879281
\(452\) 0 0
\(453\) −1.44872e29 −0.0412911
\(454\) 0 0
\(455\) −3.99020e30 −1.08109
\(456\) 0 0
\(457\) −5.37923e30 −1.38575 −0.692873 0.721059i \(-0.743655\pi\)
−0.692873 + 0.721059i \(0.743655\pi\)
\(458\) 0 0
\(459\) 8.13813e29 0.199379
\(460\) 0 0
\(461\) −2.22741e30 −0.519086 −0.259543 0.965731i \(-0.583572\pi\)
−0.259543 + 0.965731i \(0.583572\pi\)
\(462\) 0 0
\(463\) 2.60053e29 0.0576608 0.0288304 0.999584i \(-0.490822\pi\)
0.0288304 + 0.999584i \(0.490822\pi\)
\(464\) 0 0
\(465\) −7.87100e30 −1.66081
\(466\) 0 0
\(467\) 8.02144e30 1.61105 0.805524 0.592563i \(-0.201884\pi\)
0.805524 + 0.592563i \(0.201884\pi\)
\(468\) 0 0
\(469\) 1.17350e31 2.24385
\(470\) 0 0
\(471\) −3.12102e30 −0.568272
\(472\) 0 0
\(473\) 2.94801e30 0.511241
\(474\) 0 0
\(475\) −1.87344e31 −3.09502
\(476\) 0 0
\(477\) −1.18911e30 −0.187180
\(478\) 0 0
\(479\) 9.99745e30 1.49979 0.749896 0.661556i \(-0.230104\pi\)
0.749896 + 0.661556i \(0.230104\pi\)
\(480\) 0 0
\(481\) 3.36308e30 0.480915
\(482\) 0 0
\(483\) −7.94575e30 −1.08329
\(484\) 0 0
\(485\) −1.47316e31 −1.91523
\(486\) 0 0
\(487\) −5.15003e30 −0.638597 −0.319298 0.947654i \(-0.603447\pi\)
−0.319298 + 0.947654i \(0.603447\pi\)
\(488\) 0 0
\(489\) 5.20250e30 0.615404
\(490\) 0 0
\(491\) 6.37446e30 0.719460 0.359730 0.933056i \(-0.382869\pi\)
0.359730 + 0.933056i \(0.382869\pi\)
\(492\) 0 0
\(493\) −1.33141e30 −0.143408
\(494\) 0 0
\(495\) −8.61490e30 −0.885708
\(496\) 0 0
\(497\) −2.37108e30 −0.232727
\(498\) 0 0
\(499\) −9.09248e30 −0.852170 −0.426085 0.904683i \(-0.640108\pi\)
−0.426085 + 0.904683i \(0.640108\pi\)
\(500\) 0 0
\(501\) 3.28834e30 0.294335
\(502\) 0 0
\(503\) 2.41376e30 0.206377 0.103189 0.994662i \(-0.467095\pi\)
0.103189 + 0.994662i \(0.467095\pi\)
\(504\) 0 0
\(505\) 2.50993e31 2.05027
\(506\) 0 0
\(507\) 8.06075e30 0.629193
\(508\) 0 0
\(509\) −2.40043e31 −1.79075 −0.895376 0.445312i \(-0.853093\pi\)
−0.895376 + 0.445312i \(0.853093\pi\)
\(510\) 0 0
\(511\) −3.36174e30 −0.239731
\(512\) 0 0
\(513\) −2.33975e31 −1.59522
\(514\) 0 0
\(515\) 4.65291e31 3.03348
\(516\) 0 0
\(517\) −2.61131e30 −0.162823
\(518\) 0 0
\(519\) 1.28643e31 0.767295
\(520\) 0 0
\(521\) −3.16451e31 −1.80581 −0.902907 0.429835i \(-0.858572\pi\)
−0.902907 + 0.429835i \(0.858572\pi\)
\(522\) 0 0
\(523\) −2.09227e31 −1.14248 −0.571239 0.820784i \(-0.693537\pi\)
−0.571239 + 0.820784i \(0.693537\pi\)
\(524\) 0 0
\(525\) −5.18552e31 −2.70994
\(526\) 0 0
\(527\) −5.04699e30 −0.252471
\(528\) 0 0
\(529\) −6.75004e30 −0.323270
\(530\) 0 0
\(531\) −1.08212e31 −0.496236
\(532\) 0 0
\(533\) −6.56102e30 −0.288143
\(534\) 0 0
\(535\) −1.99785e31 −0.840415
\(536\) 0 0
\(537\) −1.37228e30 −0.0553017
\(538\) 0 0
\(539\) −6.47508e31 −2.50020
\(540\) 0 0
\(541\) −1.40632e31 −0.520375 −0.260187 0.965558i \(-0.583784\pi\)
−0.260187 + 0.965558i \(0.583784\pi\)
\(542\) 0 0
\(543\) −6.33360e30 −0.224622
\(544\) 0 0
\(545\) 1.12929e31 0.383924
\(546\) 0 0
\(547\) 2.95880e31 0.964410 0.482205 0.876058i \(-0.339836\pi\)
0.482205 + 0.876058i \(0.339836\pi\)
\(548\) 0 0
\(549\) −2.88758e31 −0.902511
\(550\) 0 0
\(551\) 3.82787e31 1.14740
\(552\) 0 0
\(553\) 3.00295e31 0.863395
\(554\) 0 0
\(555\) 6.49431e31 1.79128
\(556\) 0 0
\(557\) 7.87490e30 0.208406 0.104203 0.994556i \(-0.466771\pi\)
0.104203 + 0.994556i \(0.466771\pi\)
\(558\) 0 0
\(559\) −6.59694e30 −0.167535
\(560\) 0 0
\(561\) 5.52460e30 0.134657
\(562\) 0 0
\(563\) −4.45515e31 −1.04235 −0.521177 0.853448i \(-0.674507\pi\)
−0.521177 + 0.853448i \(0.674507\pi\)
\(564\) 0 0
\(565\) −2.96440e31 −0.665854
\(566\) 0 0
\(567\) −2.15913e31 −0.465665
\(568\) 0 0
\(569\) −5.59955e31 −1.15974 −0.579872 0.814708i \(-0.696897\pi\)
−0.579872 + 0.814708i \(0.696897\pi\)
\(570\) 0 0
\(571\) 9.51176e29 0.0189211 0.00946053 0.999955i \(-0.496989\pi\)
0.00946053 + 0.999955i \(0.496989\pi\)
\(572\) 0 0
\(573\) 7.34056e31 1.40266
\(574\) 0 0
\(575\) 9.22174e31 1.69290
\(576\) 0 0
\(577\) −1.28399e31 −0.226485 −0.113243 0.993567i \(-0.536124\pi\)
−0.113243 + 0.993567i \(0.536124\pi\)
\(578\) 0 0
\(579\) 2.77478e31 0.470354
\(580\) 0 0
\(581\) 1.32830e32 2.16407
\(582\) 0 0
\(583\) −2.42178e31 −0.379268
\(584\) 0 0
\(585\) 1.92781e31 0.290250
\(586\) 0 0
\(587\) 7.40632e31 1.07217 0.536086 0.844163i \(-0.319902\pi\)
0.536086 + 0.844163i \(0.319902\pi\)
\(588\) 0 0
\(589\) 1.45103e32 2.02000
\(590\) 0 0
\(591\) 7.55918e31 1.01209
\(592\) 0 0
\(593\) −8.21493e31 −1.05797 −0.528987 0.848630i \(-0.677428\pi\)
−0.528987 + 0.848630i \(0.677428\pi\)
\(594\) 0 0
\(595\) −4.94076e31 −0.612137
\(596\) 0 0
\(597\) −3.06218e31 −0.365028
\(598\) 0 0
\(599\) −1.31000e32 −1.50267 −0.751333 0.659924i \(-0.770589\pi\)
−0.751333 + 0.659924i \(0.770589\pi\)
\(600\) 0 0
\(601\) 1.05699e31 0.116685 0.0583424 0.998297i \(-0.481418\pi\)
0.0583424 + 0.998297i \(0.481418\pi\)
\(602\) 0 0
\(603\) −5.66958e31 −0.602423
\(604\) 0 0
\(605\) −4.49239e30 −0.0459506
\(606\) 0 0
\(607\) 1.29237e31 0.127267 0.0636337 0.997973i \(-0.479731\pi\)
0.0636337 + 0.997973i \(0.479731\pi\)
\(608\) 0 0
\(609\) 1.05952e32 1.00464
\(610\) 0 0
\(611\) 5.84348e30 0.0533578
\(612\) 0 0
\(613\) −8.92595e31 −0.784981 −0.392490 0.919756i \(-0.628386\pi\)
−0.392490 + 0.919756i \(0.628386\pi\)
\(614\) 0 0
\(615\) −1.26697e32 −1.07326
\(616\) 0 0
\(617\) −1.18347e32 −0.965777 −0.482889 0.875682i \(-0.660412\pi\)
−0.482889 + 0.875682i \(0.660412\pi\)
\(618\) 0 0
\(619\) 2.35060e32 1.84815 0.924073 0.382217i \(-0.124839\pi\)
0.924073 + 0.382217i \(0.124839\pi\)
\(620\) 0 0
\(621\) 1.15170e32 0.872545
\(622\) 0 0
\(623\) −1.62809e32 −1.18868
\(624\) 0 0
\(625\) 1.67271e32 1.17706
\(626\) 0 0
\(627\) −1.58835e32 −1.07738
\(628\) 0 0
\(629\) 4.16424e31 0.272304
\(630\) 0 0
\(631\) 1.77460e32 1.11883 0.559415 0.828888i \(-0.311026\pi\)
0.559415 + 0.828888i \(0.311026\pi\)
\(632\) 0 0
\(633\) −9.81674e31 −0.596795
\(634\) 0 0
\(635\) −5.35002e32 −3.13659
\(636\) 0 0
\(637\) 1.44897e32 0.819323
\(638\) 0 0
\(639\) 1.14555e31 0.0624820
\(640\) 0 0
\(641\) 1.82879e32 0.962270 0.481135 0.876646i \(-0.340225\pi\)
0.481135 + 0.876646i \(0.340225\pi\)
\(642\) 0 0
\(643\) −1.51276e32 −0.767971 −0.383986 0.923339i \(-0.625449\pi\)
−0.383986 + 0.923339i \(0.625449\pi\)
\(644\) 0 0
\(645\) −1.27391e32 −0.624025
\(646\) 0 0
\(647\) 1.45858e32 0.689495 0.344747 0.938695i \(-0.387965\pi\)
0.344747 + 0.938695i \(0.387965\pi\)
\(648\) 0 0
\(649\) −2.20388e32 −1.00548
\(650\) 0 0
\(651\) 4.01632e32 1.76867
\(652\) 0 0
\(653\) 2.02823e32 0.862216 0.431108 0.902300i \(-0.358123\pi\)
0.431108 + 0.902300i \(0.358123\pi\)
\(654\) 0 0
\(655\) 8.45666e31 0.347076
\(656\) 0 0
\(657\) 1.62418e31 0.0643624
\(658\) 0 0
\(659\) 3.26709e32 1.25020 0.625099 0.780545i \(-0.285058\pi\)
0.625099 + 0.780545i \(0.285058\pi\)
\(660\) 0 0
\(661\) 5.26703e31 0.194648 0.0973238 0.995253i \(-0.468972\pi\)
0.0973238 + 0.995253i \(0.468972\pi\)
\(662\) 0 0
\(663\) −1.23627e31 −0.0441275
\(664\) 0 0
\(665\) 1.42049e33 4.89767
\(666\) 0 0
\(667\) −1.88421e32 −0.627599
\(668\) 0 0
\(669\) 1.97633e32 0.636004
\(670\) 0 0
\(671\) −5.88095e32 −1.82868
\(672\) 0 0
\(673\) 2.50043e32 0.751349 0.375675 0.926752i \(-0.377411\pi\)
0.375675 + 0.926752i \(0.377411\pi\)
\(674\) 0 0
\(675\) 7.51620e32 2.18276
\(676\) 0 0
\(677\) 1.88330e31 0.0528627 0.0264313 0.999651i \(-0.491586\pi\)
0.0264313 + 0.999651i \(0.491586\pi\)
\(678\) 0 0
\(679\) 7.51708e32 2.03961
\(680\) 0 0
\(681\) 9.35870e31 0.245484
\(682\) 0 0
\(683\) 1.97248e32 0.500236 0.250118 0.968215i \(-0.419531\pi\)
0.250118 + 0.968215i \(0.419531\pi\)
\(684\) 0 0
\(685\) −1.00849e33 −2.47304
\(686\) 0 0
\(687\) 3.61881e32 0.858150
\(688\) 0 0
\(689\) 5.41937e31 0.124288
\(690\) 0 0
\(691\) −5.10523e32 −1.13245 −0.566223 0.824252i \(-0.691596\pi\)
−0.566223 + 0.824252i \(0.691596\pi\)
\(692\) 0 0
\(693\) 4.39591e32 0.943228
\(694\) 0 0
\(695\) 1.97497e32 0.409954
\(696\) 0 0
\(697\) −8.12402e31 −0.163153
\(698\) 0 0
\(699\) 1.08758e32 0.211336
\(700\) 0 0
\(701\) −5.10061e32 −0.959103 −0.479551 0.877514i \(-0.659201\pi\)
−0.479551 + 0.877514i \(0.659201\pi\)
\(702\) 0 0
\(703\) −1.19724e33 −2.17868
\(704\) 0 0
\(705\) 1.12841e32 0.198743
\(706\) 0 0
\(707\) −1.28074e33 −2.18342
\(708\) 0 0
\(709\) 2.56920e32 0.424000 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(710\) 0 0
\(711\) −1.45083e32 −0.231802
\(712\) 0 0
\(713\) −7.14248e32 −1.10489
\(714\) 0 0
\(715\) 3.92624e32 0.588109
\(716\) 0 0
\(717\) 4.88813e32 0.709043
\(718\) 0 0
\(719\) −6.23870e32 −0.876419 −0.438209 0.898873i \(-0.644387\pi\)
−0.438209 + 0.898873i \(0.644387\pi\)
\(720\) 0 0
\(721\) −2.37423e33 −3.23049
\(722\) 0 0
\(723\) 1.40750e32 0.185507
\(724\) 0 0
\(725\) −1.22966e33 −1.57000
\(726\) 0 0
\(727\) 1.55385e32 0.192205 0.0961024 0.995371i \(-0.469362\pi\)
0.0961024 + 0.995371i \(0.469362\pi\)
\(728\) 0 0
\(729\) 7.30127e32 0.875048
\(730\) 0 0
\(731\) −8.16849e31 −0.0948619
\(732\) 0 0
\(733\) −6.32529e32 −0.711844 −0.355922 0.934516i \(-0.615833\pi\)
−0.355922 + 0.934516i \(0.615833\pi\)
\(734\) 0 0
\(735\) 2.79805e33 3.05176
\(736\) 0 0
\(737\) −1.15469e33 −1.22064
\(738\) 0 0
\(739\) 1.11960e33 1.14723 0.573615 0.819125i \(-0.305541\pi\)
0.573615 + 0.819125i \(0.305541\pi\)
\(740\) 0 0
\(741\) 3.55434e32 0.353061
\(742\) 0 0
\(743\) −9.97216e32 −0.960325 −0.480162 0.877180i \(-0.659422\pi\)
−0.480162 + 0.877180i \(0.659422\pi\)
\(744\) 0 0
\(745\) −1.79190e32 −0.167308
\(746\) 0 0
\(747\) −6.41750e32 −0.581004
\(748\) 0 0
\(749\) 1.01944e33 0.894994
\(750\) 0 0
\(751\) −4.68302e32 −0.398719 −0.199359 0.979926i \(-0.563886\pi\)
−0.199359 + 0.979926i \(0.563886\pi\)
\(752\) 0 0
\(753\) −1.70454e32 −0.140755
\(754\) 0 0
\(755\) −1.27486e32 −0.102111
\(756\) 0 0
\(757\) −1.06644e33 −0.828576 −0.414288 0.910146i \(-0.635969\pi\)
−0.414288 + 0.910146i \(0.635969\pi\)
\(758\) 0 0
\(759\) 7.81839e32 0.589301
\(760\) 0 0
\(761\) 1.10705e33 0.809550 0.404775 0.914416i \(-0.367350\pi\)
0.404775 + 0.914416i \(0.367350\pi\)
\(762\) 0 0
\(763\) −5.76240e32 −0.408857
\(764\) 0 0
\(765\) 2.38706e32 0.164345
\(766\) 0 0
\(767\) 4.93177e32 0.329500
\(768\) 0 0
\(769\) −9.85970e32 −0.639309 −0.319655 0.947534i \(-0.603567\pi\)
−0.319655 + 0.947534i \(0.603567\pi\)
\(770\) 0 0
\(771\) −1.36977e33 −0.862030
\(772\) 0 0
\(773\) 2.87456e33 1.75593 0.877965 0.478725i \(-0.158901\pi\)
0.877965 + 0.478725i \(0.158901\pi\)
\(774\) 0 0
\(775\) −4.66129e33 −2.76399
\(776\) 0 0
\(777\) −3.31384e33 −1.90761
\(778\) 0 0
\(779\) 2.33569e33 1.30537
\(780\) 0 0
\(781\) 2.33307e32 0.126602
\(782\) 0 0
\(783\) −1.53573e33 −0.809198
\(784\) 0 0
\(785\) −2.74645e33 −1.40531
\(786\) 0 0
\(787\) −7.82813e32 −0.388999 −0.194500 0.980903i \(-0.562308\pi\)
−0.194500 + 0.980903i \(0.562308\pi\)
\(788\) 0 0
\(789\) −5.71375e32 −0.275763
\(790\) 0 0
\(791\) 1.51264e33 0.709097
\(792\) 0 0
\(793\) 1.31602e33 0.599266
\(794\) 0 0
\(795\) 1.04651e33 0.462938
\(796\) 0 0
\(797\) 1.46880e33 0.631235 0.315617 0.948887i \(-0.397788\pi\)
0.315617 + 0.948887i \(0.397788\pi\)
\(798\) 0 0
\(799\) 7.23554e31 0.0302122
\(800\) 0 0
\(801\) 7.86588e32 0.319134
\(802\) 0 0
\(803\) 3.30786e32 0.130412
\(804\) 0 0
\(805\) −6.99214e33 −2.67891
\(806\) 0 0
\(807\) 1.88252e33 0.700962
\(808\) 0 0
\(809\) −3.87756e33 −1.40330 −0.701650 0.712522i \(-0.747553\pi\)
−0.701650 + 0.712522i \(0.747553\pi\)
\(810\) 0 0
\(811\) −3.25547e33 −1.14518 −0.572591 0.819841i \(-0.694062\pi\)
−0.572591 + 0.819841i \(0.694062\pi\)
\(812\) 0 0
\(813\) −2.93115e33 −1.00230
\(814\) 0 0
\(815\) 4.57813e33 1.52186
\(816\) 0 0
\(817\) 2.34847e33 0.758983
\(818\) 0 0
\(819\) −9.83700e32 −0.309099
\(820\) 0 0
\(821\) −3.90085e33 −1.19183 −0.595914 0.803049i \(-0.703210\pi\)
−0.595914 + 0.803049i \(0.703210\pi\)
\(822\) 0 0
\(823\) 1.74515e33 0.518482 0.259241 0.965813i \(-0.416528\pi\)
0.259241 + 0.965813i \(0.416528\pi\)
\(824\) 0 0
\(825\) 5.10240e33 1.47419
\(826\) 0 0
\(827\) 4.60301e33 1.29339 0.646694 0.762749i \(-0.276151\pi\)
0.646694 + 0.762749i \(0.276151\pi\)
\(828\) 0 0
\(829\) −7.40688e31 −0.0202422 −0.0101211 0.999949i \(-0.503222\pi\)
−0.0101211 + 0.999949i \(0.503222\pi\)
\(830\) 0 0
\(831\) −4.65335e32 −0.123695
\(832\) 0 0
\(833\) 1.79415e33 0.463917
\(834\) 0 0
\(835\) 2.89369e33 0.727876
\(836\) 0 0
\(837\) −5.82150e33 −1.42460
\(838\) 0 0
\(839\) 2.97493e33 0.708294 0.354147 0.935190i \(-0.384771\pi\)
0.354147 + 0.935190i \(0.384771\pi\)
\(840\) 0 0
\(841\) −1.80424e33 −0.417965
\(842\) 0 0
\(843\) −2.74450e33 −0.618652
\(844\) 0 0
\(845\) 7.09334e33 1.55596
\(846\) 0 0
\(847\) 2.29233e32 0.0489348
\(848\) 0 0
\(849\) −7.23761e32 −0.150368
\(850\) 0 0
\(851\) 5.89322e33 1.19169
\(852\) 0 0
\(853\) 5.32152e32 0.104742 0.0523711 0.998628i \(-0.483322\pi\)
0.0523711 + 0.998628i \(0.483322\pi\)
\(854\) 0 0
\(855\) −6.86290e33 −1.31491
\(856\) 0 0
\(857\) 6.39595e33 1.19296 0.596480 0.802628i \(-0.296566\pi\)
0.596480 + 0.802628i \(0.296566\pi\)
\(858\) 0 0
\(859\) 5.22319e32 0.0948450 0.0474225 0.998875i \(-0.484899\pi\)
0.0474225 + 0.998875i \(0.484899\pi\)
\(860\) 0 0
\(861\) 6.46497e33 1.14296
\(862\) 0 0
\(863\) −7.61664e33 −1.31111 −0.655555 0.755147i \(-0.727565\pi\)
−0.655555 + 0.755147i \(0.727565\pi\)
\(864\) 0 0
\(865\) 1.13204e34 1.89748
\(866\) 0 0
\(867\) 4.17920e33 0.682140
\(868\) 0 0
\(869\) −2.95482e33 −0.469682
\(870\) 0 0
\(871\) 2.58391e33 0.400008
\(872\) 0 0
\(873\) −3.63177e33 −0.547589
\(874\) 0 0
\(875\) −2.34579e34 −3.44506
\(876\) 0 0
\(877\) 5.20154e33 0.744109 0.372055 0.928211i \(-0.378653\pi\)
0.372055 + 0.928211i \(0.378653\pi\)
\(878\) 0 0
\(879\) 5.19828e33 0.724415
\(880\) 0 0
\(881\) −2.52401e33 −0.342664 −0.171332 0.985213i \(-0.554807\pi\)
−0.171332 + 0.985213i \(0.554807\pi\)
\(882\) 0 0
\(883\) −5.17998e33 −0.685141 −0.342570 0.939492i \(-0.611298\pi\)
−0.342570 + 0.939492i \(0.611298\pi\)
\(884\) 0 0
\(885\) 9.52355e33 1.22730
\(886\) 0 0
\(887\) −1.35171e34 −1.69731 −0.848654 0.528948i \(-0.822587\pi\)
−0.848654 + 0.528948i \(0.822587\pi\)
\(888\) 0 0
\(889\) 2.72995e34 3.34029
\(890\) 0 0
\(891\) 2.12452e33 0.253319
\(892\) 0 0
\(893\) −2.08025e33 −0.241726
\(894\) 0 0
\(895\) −1.20759e33 −0.136758
\(896\) 0 0
\(897\) −1.74957e33 −0.193116
\(898\) 0 0
\(899\) 9.52408e33 1.02468
\(900\) 0 0
\(901\) 6.71039e32 0.0703741
\(902\) 0 0
\(903\) 6.50036e33 0.664551
\(904\) 0 0
\(905\) −5.57348e33 −0.555479
\(906\) 0 0
\(907\) 4.97097e32 0.0483011 0.0241505 0.999708i \(-0.492312\pi\)
0.0241505 + 0.999708i \(0.492312\pi\)
\(908\) 0 0
\(909\) 6.18770e33 0.586199
\(910\) 0 0
\(911\) −8.11942e33 −0.750004 −0.375002 0.927024i \(-0.622358\pi\)
−0.375002 + 0.927024i \(0.622358\pi\)
\(912\) 0 0
\(913\) −1.30701e34 −1.17724
\(914\) 0 0
\(915\) 2.54131e34 2.23211
\(916\) 0 0
\(917\) −4.31517e33 −0.369616
\(918\) 0 0
\(919\) 1.62024e34 1.35348 0.676741 0.736221i \(-0.263392\pi\)
0.676741 + 0.736221i \(0.263392\pi\)
\(920\) 0 0
\(921\) 8.81595e33 0.718262
\(922\) 0 0
\(923\) −5.22086e32 −0.0414880
\(924\) 0 0
\(925\) 3.84600e34 2.98112
\(926\) 0 0
\(927\) 1.14708e34 0.867313
\(928\) 0 0
\(929\) −1.55887e34 −1.14982 −0.574910 0.818216i \(-0.694963\pi\)
−0.574910 + 0.818216i \(0.694963\pi\)
\(930\) 0 0
\(931\) −5.15825e34 −3.71177
\(932\) 0 0
\(933\) 5.22795e32 0.0367022
\(934\) 0 0
\(935\) 4.86157e33 0.332999
\(936\) 0 0
\(937\) −4.49867e33 −0.300662 −0.150331 0.988636i \(-0.548034\pi\)
−0.150331 + 0.988636i \(0.548034\pi\)
\(938\) 0 0
\(939\) 6.70345e33 0.437164
\(940\) 0 0
\(941\) −5.17928e32 −0.0329601 −0.0164801 0.999864i \(-0.505246\pi\)
−0.0164801 + 0.999864i \(0.505246\pi\)
\(942\) 0 0
\(943\) −1.14971e34 −0.714007
\(944\) 0 0
\(945\) −5.69897e34 −3.45406
\(946\) 0 0
\(947\) −4.30688e33 −0.254764 −0.127382 0.991854i \(-0.540657\pi\)
−0.127382 + 0.991854i \(0.540657\pi\)
\(948\) 0 0
\(949\) −7.40220e32 −0.0427365
\(950\) 0 0
\(951\) −9.71387e33 −0.547414
\(952\) 0 0
\(953\) 2.71314e34 1.49246 0.746231 0.665687i \(-0.231862\pi\)
0.746231 + 0.665687i \(0.231862\pi\)
\(954\) 0 0
\(955\) 6.45959e34 3.46870
\(956\) 0 0
\(957\) −1.04254e34 −0.546517
\(958\) 0 0
\(959\) 5.14601e34 2.63364
\(960\) 0 0
\(961\) 1.60897e34 0.803948
\(962\) 0 0
\(963\) −4.92528e33 −0.240286
\(964\) 0 0
\(965\) 2.44176e34 1.16316
\(966\) 0 0
\(967\) −4.19669e34 −1.95210 −0.976050 0.217547i \(-0.930194\pi\)
−0.976050 + 0.217547i \(0.930194\pi\)
\(968\) 0 0
\(969\) 4.40107e33 0.199910
\(970\) 0 0
\(971\) −2.05790e34 −0.912857 −0.456428 0.889760i \(-0.650871\pi\)
−0.456428 + 0.889760i \(0.650871\pi\)
\(972\) 0 0
\(973\) −1.00776e34 −0.436577
\(974\) 0 0
\(975\) −1.14180e34 −0.483098
\(976\) 0 0
\(977\) 1.04532e34 0.431979 0.215989 0.976396i \(-0.430702\pi\)
0.215989 + 0.976396i \(0.430702\pi\)
\(978\) 0 0
\(979\) 1.60199e34 0.646635
\(980\) 0 0
\(981\) 2.78402e33 0.109769
\(982\) 0 0
\(983\) −3.14322e34 −1.21063 −0.605313 0.795988i \(-0.706952\pi\)
−0.605313 + 0.795988i \(0.706952\pi\)
\(984\) 0 0
\(985\) 6.65197e34 2.50284
\(986\) 0 0
\(987\) −5.75793e33 −0.211651
\(988\) 0 0
\(989\) −1.15600e34 −0.415146
\(990\) 0 0
\(991\) −1.36431e34 −0.478704 −0.239352 0.970933i \(-0.576935\pi\)
−0.239352 + 0.970933i \(0.576935\pi\)
\(992\) 0 0
\(993\) −1.20955e34 −0.414675
\(994\) 0 0
\(995\) −2.69468e34 −0.902696
\(996\) 0 0
\(997\) −2.61404e34 −0.855693 −0.427847 0.903851i \(-0.640728\pi\)
−0.427847 + 0.903851i \(0.640728\pi\)
\(998\) 0 0
\(999\) 4.80328e34 1.53651
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 64.24.a.n.1.2 6
4.3 odd 2 inner 64.24.a.n.1.5 6
8.3 odd 2 32.24.a.d.1.2 6
8.5 even 2 32.24.a.d.1.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.24.a.d.1.2 6 8.3 odd 2
32.24.a.d.1.5 yes 6 8.5 even 2
64.24.a.n.1.2 6 1.1 even 1 trivial
64.24.a.n.1.5 6 4.3 odd 2 inner