Properties

Label 315.8.a.g.1.3
Level $315$
Weight $8$
Character 315.1
Self dual yes
Analytic conductor $98.401$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,8,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(98.4012830275\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 277x^{2} + 80x + 15348 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-12.8099\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.41438 q^{2} -116.342 q^{4} -125.000 q^{5} +343.000 q^{7} -834.277 q^{8} -426.798 q^{10} +3379.63 q^{11} -14159.0 q^{13} +1171.13 q^{14} +12043.2 q^{16} +19538.0 q^{17} -38098.7 q^{19} +14542.8 q^{20} +11539.4 q^{22} -103511. q^{23} +15625.0 q^{25} -48344.1 q^{26} -39905.3 q^{28} -59794.5 q^{29} +68349.9 q^{31} +147908. q^{32} +66710.3 q^{34} -42875.0 q^{35} +266512. q^{37} -130084. q^{38} +104285. q^{40} -655437. q^{41} -775223. q^{43} -393193. q^{44} -353426. q^{46} +119729. q^{47} +117649. q^{49} +53349.7 q^{50} +1.64728e6 q^{52} -224102. q^{53} -422454. q^{55} -286157. q^{56} -204161. q^{58} +409602. q^{59} +1.42274e6 q^{61} +233373. q^{62} -1.03652e6 q^{64} +1.76987e6 q^{65} +4.22037e6 q^{67} -2.27309e6 q^{68} -146392. q^{70} -2.17619e6 q^{71} +4.43912e6 q^{73} +909973. q^{74} +4.43248e6 q^{76} +1.15921e6 q^{77} +1.63306e6 q^{79} -1.50540e6 q^{80} -2.23791e6 q^{82} +3.49903e6 q^{83} -2.44225e6 q^{85} -2.64691e6 q^{86} -2.81955e6 q^{88} -2.03583e6 q^{89} -4.85652e6 q^{91} +1.20427e7 q^{92} +408799. q^{94} +4.76234e6 q^{95} +1.27049e7 q^{97} +401698. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 11 q^{2} + 141 q^{4} - 500 q^{5} + 1372 q^{7} - 2133 q^{8} + 1375 q^{10} + 2708 q^{11} - 2212 q^{13} - 3773 q^{14} - 9599 q^{16} + 17016 q^{17} + 32668 q^{19} - 17625 q^{20} - 7196 q^{22} - 87696 q^{23}+ \cdots - 1294139 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.41438 0.301791 0.150896 0.988550i \(-0.451784\pi\)
0.150896 + 0.988550i \(0.451784\pi\)
\(3\) 0 0
\(4\) −116.342 −0.908922
\(5\) −125.000 −0.447214
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) −834.277 −0.576096
\(9\) 0 0
\(10\) −426.798 −0.134965
\(11\) 3379.63 0.765588 0.382794 0.923834i \(-0.374962\pi\)
0.382794 + 0.923834i \(0.374962\pi\)
\(12\) 0 0
\(13\) −14159.0 −1.78743 −0.893716 0.448633i \(-0.851911\pi\)
−0.893716 + 0.448633i \(0.851911\pi\)
\(14\) 1171.13 0.114066
\(15\) 0 0
\(16\) 12043.2 0.735061
\(17\) 19538.0 0.964516 0.482258 0.876029i \(-0.339817\pi\)
0.482258 + 0.876029i \(0.339817\pi\)
\(18\) 0 0
\(19\) −38098.7 −1.27430 −0.637152 0.770738i \(-0.719888\pi\)
−0.637152 + 0.770738i \(0.719888\pi\)
\(20\) 14542.8 0.406482
\(21\) 0 0
\(22\) 11539.4 0.231048
\(23\) −103511. −1.77394 −0.886969 0.461828i \(-0.847194\pi\)
−0.886969 + 0.461828i \(0.847194\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) −48344.1 −0.539432
\(27\) 0 0
\(28\) −39905.3 −0.343540
\(29\) −59794.5 −0.455269 −0.227635 0.973747i \(-0.573099\pi\)
−0.227635 + 0.973747i \(0.573099\pi\)
\(30\) 0 0
\(31\) 68349.9 0.412071 0.206035 0.978545i \(-0.433944\pi\)
0.206035 + 0.978545i \(0.433944\pi\)
\(32\) 147908. 0.797931
\(33\) 0 0
\(34\) 66710.3 0.291083
\(35\) −42875.0 −0.169031
\(36\) 0 0
\(37\) 266512. 0.864989 0.432494 0.901637i \(-0.357634\pi\)
0.432494 + 0.901637i \(0.357634\pi\)
\(38\) −130084. −0.384574
\(39\) 0 0
\(40\) 104285. 0.257638
\(41\) −655437. −1.48521 −0.742604 0.669730i \(-0.766410\pi\)
−0.742604 + 0.669730i \(0.766410\pi\)
\(42\) 0 0
\(43\) −775223. −1.48692 −0.743459 0.668782i \(-0.766816\pi\)
−0.743459 + 0.668782i \(0.766816\pi\)
\(44\) −393193. −0.695860
\(45\) 0 0
\(46\) −353426. −0.535360
\(47\) 119729. 0.168211 0.0841057 0.996457i \(-0.473197\pi\)
0.0841057 + 0.996457i \(0.473197\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 53349.7 0.0603583
\(51\) 0 0
\(52\) 1.64728e6 1.62464
\(53\) −224102. −0.206767 −0.103383 0.994642i \(-0.532967\pi\)
−0.103383 + 0.994642i \(0.532967\pi\)
\(54\) 0 0
\(55\) −422454. −0.342382
\(56\) −286157. −0.217744
\(57\) 0 0
\(58\) −204161. −0.137396
\(59\) 409602. 0.259645 0.129823 0.991537i \(-0.458559\pi\)
0.129823 + 0.991537i \(0.458559\pi\)
\(60\) 0 0
\(61\) 1.42274e6 0.802548 0.401274 0.915958i \(-0.368568\pi\)
0.401274 + 0.915958i \(0.368568\pi\)
\(62\) 233373. 0.124359
\(63\) 0 0
\(64\) −1.03652e6 −0.494252
\(65\) 1.76987e6 0.799364
\(66\) 0 0
\(67\) 4.22037e6 1.71431 0.857153 0.515061i \(-0.172231\pi\)
0.857153 + 0.515061i \(0.172231\pi\)
\(68\) −2.27309e6 −0.876670
\(69\) 0 0
\(70\) −146392. −0.0510121
\(71\) −2.17619e6 −0.721595 −0.360798 0.932644i \(-0.617495\pi\)
−0.360798 + 0.932644i \(0.617495\pi\)
\(72\) 0 0
\(73\) 4.43912e6 1.33557 0.667785 0.744354i \(-0.267242\pi\)
0.667785 + 0.744354i \(0.267242\pi\)
\(74\) 909973. 0.261046
\(75\) 0 0
\(76\) 4.43248e6 1.15824
\(77\) 1.15921e6 0.289365
\(78\) 0 0
\(79\) 1.63306e6 0.372656 0.186328 0.982488i \(-0.440341\pi\)
0.186328 + 0.982488i \(0.440341\pi\)
\(80\) −1.50540e6 −0.328729
\(81\) 0 0
\(82\) −2.23791e6 −0.448223
\(83\) 3.49903e6 0.671698 0.335849 0.941916i \(-0.390977\pi\)
0.335849 + 0.941916i \(0.390977\pi\)
\(84\) 0 0
\(85\) −2.44225e6 −0.431345
\(86\) −2.64691e6 −0.448739
\(87\) 0 0
\(88\) −2.81955e6 −0.441053
\(89\) −2.03583e6 −0.306110 −0.153055 0.988218i \(-0.548911\pi\)
−0.153055 + 0.988218i \(0.548911\pi\)
\(90\) 0 0
\(91\) −4.85652e6 −0.675586
\(92\) 1.20427e7 1.61237
\(93\) 0 0
\(94\) 408799. 0.0507648
\(95\) 4.76234e6 0.569886
\(96\) 0 0
\(97\) 1.27049e7 1.41342 0.706709 0.707505i \(-0.250179\pi\)
0.706709 + 0.707505i \(0.250179\pi\)
\(98\) 401698. 0.0431131
\(99\) 0 0
\(100\) −1.81784e6 −0.181784
\(101\) 1.86609e7 1.80222 0.901110 0.433590i \(-0.142753\pi\)
0.901110 + 0.433590i \(0.142753\pi\)
\(102\) 0 0
\(103\) 5.30772e6 0.478606 0.239303 0.970945i \(-0.423081\pi\)
0.239303 + 0.970945i \(0.423081\pi\)
\(104\) 1.18125e7 1.02973
\(105\) 0 0
\(106\) −765171. −0.0624005
\(107\) −1.74818e7 −1.37957 −0.689784 0.724015i \(-0.742295\pi\)
−0.689784 + 0.724015i \(0.742295\pi\)
\(108\) 0 0
\(109\) 467681. 0.0345905 0.0172952 0.999850i \(-0.494494\pi\)
0.0172952 + 0.999850i \(0.494494\pi\)
\(110\) −1.44242e6 −0.103328
\(111\) 0 0
\(112\) 4.13083e6 0.277827
\(113\) 1.45523e7 0.948763 0.474382 0.880319i \(-0.342672\pi\)
0.474382 + 0.880319i \(0.342672\pi\)
\(114\) 0 0
\(115\) 1.29389e7 0.793329
\(116\) 6.95661e6 0.413804
\(117\) 0 0
\(118\) 1.39854e6 0.0783587
\(119\) 6.70154e6 0.364553
\(120\) 0 0
\(121\) −8.06524e6 −0.413874
\(122\) 4.85777e6 0.242202
\(123\) 0 0
\(124\) −7.95196e6 −0.374540
\(125\) −1.95312e6 −0.0894427
\(126\) 0 0
\(127\) 2.04101e6 0.0884161 0.0442081 0.999022i \(-0.485924\pi\)
0.0442081 + 0.999022i \(0.485924\pi\)
\(128\) −2.24713e7 −0.947093
\(129\) 0 0
\(130\) 6.04301e6 0.241241
\(131\) 9.14330e6 0.355348 0.177674 0.984089i \(-0.443143\pi\)
0.177674 + 0.984089i \(0.443143\pi\)
\(132\) 0 0
\(133\) −1.30679e7 −0.481642
\(134\) 1.44099e7 0.517363
\(135\) 0 0
\(136\) −1.63001e7 −0.555654
\(137\) −3.62725e7 −1.20519 −0.602596 0.798047i \(-0.705867\pi\)
−0.602596 + 0.798047i \(0.705867\pi\)
\(138\) 0 0
\(139\) −2.28823e7 −0.722682 −0.361341 0.932434i \(-0.617681\pi\)
−0.361341 + 0.932434i \(0.617681\pi\)
\(140\) 4.98816e6 0.153636
\(141\) 0 0
\(142\) −7.43036e6 −0.217771
\(143\) −4.78521e7 −1.36844
\(144\) 0 0
\(145\) 7.47431e6 0.203603
\(146\) 1.51568e7 0.403064
\(147\) 0 0
\(148\) −3.10065e7 −0.786207
\(149\) 7.18870e7 1.78032 0.890161 0.455646i \(-0.150592\pi\)
0.890161 + 0.455646i \(0.150592\pi\)
\(150\) 0 0
\(151\) 6.39974e7 1.51267 0.756333 0.654187i \(-0.226989\pi\)
0.756333 + 0.654187i \(0.226989\pi\)
\(152\) 3.17849e7 0.734122
\(153\) 0 0
\(154\) 3.95800e6 0.0873280
\(155\) −8.54374e6 −0.184284
\(156\) 0 0
\(157\) 6.00497e6 0.123840 0.0619202 0.998081i \(-0.480278\pi\)
0.0619202 + 0.998081i \(0.480278\pi\)
\(158\) 5.57589e6 0.112464
\(159\) 0 0
\(160\) −1.84885e7 −0.356846
\(161\) −3.55042e7 −0.670486
\(162\) 0 0
\(163\) 9.83028e7 1.77791 0.888954 0.457997i \(-0.151433\pi\)
0.888954 + 0.457997i \(0.151433\pi\)
\(164\) 7.62549e7 1.34994
\(165\) 0 0
\(166\) 1.19470e7 0.202713
\(167\) −5.80791e7 −0.964966 −0.482483 0.875905i \(-0.660265\pi\)
−0.482483 + 0.875905i \(0.660265\pi\)
\(168\) 0 0
\(169\) 1.37728e8 2.19491
\(170\) −8.33878e6 −0.130176
\(171\) 0 0
\(172\) 9.01909e7 1.35149
\(173\) 2.31746e7 0.340291 0.170146 0.985419i \(-0.445576\pi\)
0.170146 + 0.985419i \(0.445576\pi\)
\(174\) 0 0
\(175\) 5.35938e6 0.0755929
\(176\) 4.07017e7 0.562754
\(177\) 0 0
\(178\) −6.95111e6 −0.0923814
\(179\) −3.77902e7 −0.492486 −0.246243 0.969208i \(-0.579196\pi\)
−0.246243 + 0.969208i \(0.579196\pi\)
\(180\) 0 0
\(181\) −9.09325e6 −0.113984 −0.0569920 0.998375i \(-0.518151\pi\)
−0.0569920 + 0.998375i \(0.518151\pi\)
\(182\) −1.65820e7 −0.203886
\(183\) 0 0
\(184\) 8.63567e7 1.02196
\(185\) −3.33140e7 −0.386835
\(186\) 0 0
\(187\) 6.60314e7 0.738422
\(188\) −1.39295e7 −0.152891
\(189\) 0 0
\(190\) 1.62605e7 0.171987
\(191\) 5.34900e7 0.555464 0.277732 0.960659i \(-0.410417\pi\)
0.277732 + 0.960659i \(0.410417\pi\)
\(192\) 0 0
\(193\) −8.24940e7 −0.825984 −0.412992 0.910735i \(-0.635516\pi\)
−0.412992 + 0.910735i \(0.635516\pi\)
\(194\) 4.33794e7 0.426557
\(195\) 0 0
\(196\) −1.36875e7 −0.129846
\(197\) −7.55328e7 −0.703888 −0.351944 0.936021i \(-0.614479\pi\)
−0.351944 + 0.936021i \(0.614479\pi\)
\(198\) 0 0
\(199\) −1.05546e6 −0.00949412 −0.00474706 0.999989i \(-0.501511\pi\)
−0.00474706 + 0.999989i \(0.501511\pi\)
\(200\) −1.30356e7 −0.115219
\(201\) 0 0
\(202\) 6.37154e7 0.543895
\(203\) −2.05095e7 −0.172076
\(204\) 0 0
\(205\) 8.19296e7 0.664205
\(206\) 1.81226e7 0.144439
\(207\) 0 0
\(208\) −1.70520e8 −1.31387
\(209\) −1.28760e8 −0.975593
\(210\) 0 0
\(211\) 1.14474e7 0.0838916 0.0419458 0.999120i \(-0.486644\pi\)
0.0419458 + 0.999120i \(0.486644\pi\)
\(212\) 2.60725e7 0.187935
\(213\) 0 0
\(214\) −5.96896e7 −0.416342
\(215\) 9.69028e7 0.664970
\(216\) 0 0
\(217\) 2.34440e7 0.155748
\(218\) 1.59684e6 0.0104391
\(219\) 0 0
\(220\) 4.91492e7 0.311198
\(221\) −2.76638e8 −1.72401
\(222\) 0 0
\(223\) 2.81598e8 1.70044 0.850222 0.526425i \(-0.176468\pi\)
0.850222 + 0.526425i \(0.176468\pi\)
\(224\) 5.07323e7 0.301590
\(225\) 0 0
\(226\) 4.96872e7 0.286329
\(227\) 4.00301e7 0.227141 0.113571 0.993530i \(-0.463771\pi\)
0.113571 + 0.993530i \(0.463771\pi\)
\(228\) 0 0
\(229\) −1.24592e8 −0.685591 −0.342796 0.939410i \(-0.611374\pi\)
−0.342796 + 0.939410i \(0.611374\pi\)
\(230\) 4.41782e7 0.239420
\(231\) 0 0
\(232\) 4.98852e7 0.262279
\(233\) 3.21485e8 1.66500 0.832500 0.554025i \(-0.186909\pi\)
0.832500 + 0.554025i \(0.186909\pi\)
\(234\) 0 0
\(235\) −1.49661e7 −0.0752264
\(236\) −4.76540e7 −0.235997
\(237\) 0 0
\(238\) 2.28816e7 0.110019
\(239\) 2.95076e8 1.39811 0.699054 0.715068i \(-0.253605\pi\)
0.699054 + 0.715068i \(0.253605\pi\)
\(240\) 0 0
\(241\) 1.00887e8 0.464273 0.232137 0.972683i \(-0.425428\pi\)
0.232137 + 0.972683i \(0.425428\pi\)
\(242\) −2.75378e7 −0.124904
\(243\) 0 0
\(244\) −1.65524e8 −0.729453
\(245\) −1.47061e7 −0.0638877
\(246\) 0 0
\(247\) 5.39439e8 2.27773
\(248\) −5.70227e7 −0.237393
\(249\) 0 0
\(250\) −6.66871e6 −0.0269931
\(251\) 9.30364e7 0.371360 0.185680 0.982610i \(-0.440551\pi\)
0.185680 + 0.982610i \(0.440551\pi\)
\(252\) 0 0
\(253\) −3.49829e8 −1.35811
\(254\) 6.96877e6 0.0266832
\(255\) 0 0
\(256\) 5.59493e7 0.208428
\(257\) −3.64584e8 −1.33978 −0.669888 0.742462i \(-0.733658\pi\)
−0.669888 + 0.742462i \(0.733658\pi\)
\(258\) 0 0
\(259\) 9.14136e7 0.326935
\(260\) −2.05910e8 −0.726559
\(261\) 0 0
\(262\) 3.12187e7 0.107241
\(263\) −1.11633e8 −0.378396 −0.189198 0.981939i \(-0.560589\pi\)
−0.189198 + 0.981939i \(0.560589\pi\)
\(264\) 0 0
\(265\) 2.80128e7 0.0924690
\(266\) −4.46187e7 −0.145355
\(267\) 0 0
\(268\) −4.91006e8 −1.55817
\(269\) 1.53396e8 0.480487 0.240244 0.970713i \(-0.422773\pi\)
0.240244 + 0.970713i \(0.422773\pi\)
\(270\) 0 0
\(271\) −4.17128e8 −1.27314 −0.636571 0.771218i \(-0.719648\pi\)
−0.636571 + 0.771218i \(0.719648\pi\)
\(272\) 2.35301e8 0.708978
\(273\) 0 0
\(274\) −1.23848e8 −0.363717
\(275\) 5.28068e7 0.153118
\(276\) 0 0
\(277\) −4.05683e8 −1.14685 −0.573426 0.819257i \(-0.694386\pi\)
−0.573426 + 0.819257i \(0.694386\pi\)
\(278\) −7.81288e7 −0.218099
\(279\) 0 0
\(280\) 3.57696e7 0.0973781
\(281\) 1.50774e8 0.405371 0.202686 0.979244i \(-0.435033\pi\)
0.202686 + 0.979244i \(0.435033\pi\)
\(282\) 0 0
\(283\) −3.69597e7 −0.0969338 −0.0484669 0.998825i \(-0.515434\pi\)
−0.0484669 + 0.998825i \(0.515434\pi\)
\(284\) 2.53183e8 0.655874
\(285\) 0 0
\(286\) −1.63385e8 −0.412983
\(287\) −2.24815e8 −0.561356
\(288\) 0 0
\(289\) −2.86042e7 −0.0697088
\(290\) 2.55202e7 0.0614455
\(291\) 0 0
\(292\) −5.16456e8 −1.21393
\(293\) 6.00426e8 1.39451 0.697257 0.716822i \(-0.254404\pi\)
0.697257 + 0.716822i \(0.254404\pi\)
\(294\) 0 0
\(295\) −5.12003e7 −0.116117
\(296\) −2.22345e8 −0.498317
\(297\) 0 0
\(298\) 2.45450e8 0.537286
\(299\) 1.46561e9 3.17079
\(300\) 0 0
\(301\) −2.65901e8 −0.562002
\(302\) 2.18511e8 0.456510
\(303\) 0 0
\(304\) −4.58832e8 −0.936691
\(305\) −1.77842e8 −0.358910
\(306\) 0 0
\(307\) 9.08890e8 1.79278 0.896390 0.443267i \(-0.146181\pi\)
0.896390 + 0.443267i \(0.146181\pi\)
\(308\) −1.34865e8 −0.263010
\(309\) 0 0
\(310\) −2.91716e7 −0.0556153
\(311\) −6.45862e8 −1.21753 −0.608763 0.793352i \(-0.708334\pi\)
−0.608763 + 0.793352i \(0.708334\pi\)
\(312\) 0 0
\(313\) −8.96603e8 −1.65270 −0.826352 0.563154i \(-0.809588\pi\)
−0.826352 + 0.563154i \(0.809588\pi\)
\(314\) 2.05033e7 0.0373740
\(315\) 0 0
\(316\) −1.89994e8 −0.338715
\(317\) −8.47295e8 −1.49392 −0.746959 0.664870i \(-0.768487\pi\)
−0.746959 + 0.664870i \(0.768487\pi\)
\(318\) 0 0
\(319\) −2.02084e8 −0.348549
\(320\) 1.29565e8 0.221036
\(321\) 0 0
\(322\) −1.21225e8 −0.202347
\(323\) −7.44374e8 −1.22909
\(324\) 0 0
\(325\) −2.21234e8 −0.357486
\(326\) 3.35643e8 0.536557
\(327\) 0 0
\(328\) 5.46816e8 0.855623
\(329\) 4.10669e7 0.0635779
\(330\) 0 0
\(331\) 7.42401e8 1.12523 0.562614 0.826720i \(-0.309796\pi\)
0.562614 + 0.826720i \(0.309796\pi\)
\(332\) −4.07084e8 −0.610521
\(333\) 0 0
\(334\) −1.98304e8 −0.291219
\(335\) −5.27546e8 −0.766661
\(336\) 0 0
\(337\) −7.24512e8 −1.03120 −0.515598 0.856831i \(-0.672430\pi\)
−0.515598 + 0.856831i \(0.672430\pi\)
\(338\) 4.70254e8 0.662406
\(339\) 0 0
\(340\) 2.84137e8 0.392059
\(341\) 2.30998e8 0.315477
\(342\) 0 0
\(343\) 4.03536e7 0.0539949
\(344\) 6.46750e8 0.856608
\(345\) 0 0
\(346\) 7.91269e7 0.102697
\(347\) −5.09846e8 −0.655068 −0.327534 0.944839i \(-0.606218\pi\)
−0.327534 + 0.944839i \(0.606218\pi\)
\(348\) 0 0
\(349\) −8.76052e8 −1.10317 −0.551583 0.834120i \(-0.685976\pi\)
−0.551583 + 0.834120i \(0.685976\pi\)
\(350\) 1.82989e7 0.0228133
\(351\) 0 0
\(352\) 4.99874e8 0.610887
\(353\) −1.26670e9 −1.53272 −0.766358 0.642413i \(-0.777933\pi\)
−0.766358 + 0.642413i \(0.777933\pi\)
\(354\) 0 0
\(355\) 2.72024e8 0.322707
\(356\) 2.36853e8 0.278230
\(357\) 0 0
\(358\) −1.29030e8 −0.148628
\(359\) −1.66558e9 −1.89992 −0.949961 0.312369i \(-0.898878\pi\)
−0.949961 + 0.312369i \(0.898878\pi\)
\(360\) 0 0
\(361\) 5.57643e8 0.623851
\(362\) −3.10478e7 −0.0343994
\(363\) 0 0
\(364\) 5.65018e8 0.614055
\(365\) −5.54890e8 −0.597285
\(366\) 0 0
\(367\) 1.67575e9 1.76961 0.884807 0.465957i \(-0.154290\pi\)
0.884807 + 0.465957i \(0.154290\pi\)
\(368\) −1.24661e9 −1.30395
\(369\) 0 0
\(370\) −1.13747e8 −0.116743
\(371\) −7.68672e7 −0.0781506
\(372\) 0 0
\(373\) 1.47694e9 1.47360 0.736802 0.676109i \(-0.236335\pi\)
0.736802 + 0.676109i \(0.236335\pi\)
\(374\) 2.25456e8 0.222850
\(375\) 0 0
\(376\) −9.98868e7 −0.0969060
\(377\) 8.46628e8 0.813763
\(378\) 0 0
\(379\) −1.64279e9 −1.55005 −0.775024 0.631932i \(-0.782262\pi\)
−0.775024 + 0.631932i \(0.782262\pi\)
\(380\) −5.54061e8 −0.517982
\(381\) 0 0
\(382\) 1.82635e8 0.167634
\(383\) 1.89784e9 1.72609 0.863044 0.505129i \(-0.168555\pi\)
0.863044 + 0.505129i \(0.168555\pi\)
\(384\) 0 0
\(385\) −1.44902e8 −0.129408
\(386\) −2.81666e8 −0.249275
\(387\) 0 0
\(388\) −1.47811e9 −1.28469
\(389\) −1.05432e9 −0.908131 −0.454065 0.890968i \(-0.650027\pi\)
−0.454065 + 0.890968i \(0.650027\pi\)
\(390\) 0 0
\(391\) −2.02240e9 −1.71099
\(392\) −9.81518e7 −0.0822995
\(393\) 0 0
\(394\) −2.57898e8 −0.212428
\(395\) −2.04133e8 −0.166657
\(396\) 0 0
\(397\) 2.96848e8 0.238104 0.119052 0.992888i \(-0.462014\pi\)
0.119052 + 0.992888i \(0.462014\pi\)
\(398\) −3.60373e6 −0.00286525
\(399\) 0 0
\(400\) 1.88176e8 0.147012
\(401\) 7.79941e8 0.604028 0.302014 0.953304i \(-0.402341\pi\)
0.302014 + 0.953304i \(0.402341\pi\)
\(402\) 0 0
\(403\) −9.67763e8 −0.736549
\(404\) −2.17105e9 −1.63808
\(405\) 0 0
\(406\) −7.00273e7 −0.0519310
\(407\) 9.00713e8 0.662225
\(408\) 0 0
\(409\) 8.75152e8 0.632488 0.316244 0.948678i \(-0.397578\pi\)
0.316244 + 0.948678i \(0.397578\pi\)
\(410\) 2.79739e8 0.200452
\(411\) 0 0
\(412\) −6.17511e8 −0.435015
\(413\) 1.40494e8 0.0981367
\(414\) 0 0
\(415\) −4.37378e8 −0.300392
\(416\) −2.09422e9 −1.42625
\(417\) 0 0
\(418\) −4.39635e8 −0.294426
\(419\) −1.07058e9 −0.711000 −0.355500 0.934676i \(-0.615689\pi\)
−0.355500 + 0.934676i \(0.615689\pi\)
\(420\) 0 0
\(421\) 2.54184e9 1.66020 0.830100 0.557614i \(-0.188283\pi\)
0.830100 + 0.557614i \(0.188283\pi\)
\(422\) 3.90858e7 0.0253178
\(423\) 0 0
\(424\) 1.86963e8 0.119118
\(425\) 3.05282e8 0.192903
\(426\) 0 0
\(427\) 4.87999e8 0.303335
\(428\) 2.03387e9 1.25392
\(429\) 0 0
\(430\) 3.30863e8 0.200682
\(431\) −8.91747e8 −0.536502 −0.268251 0.963349i \(-0.586446\pi\)
−0.268251 + 0.963349i \(0.586446\pi\)
\(432\) 0 0
\(433\) −1.03966e9 −0.615440 −0.307720 0.951477i \(-0.599566\pi\)
−0.307720 + 0.951477i \(0.599566\pi\)
\(434\) 8.00468e7 0.0470035
\(435\) 0 0
\(436\) −5.44109e7 −0.0314400
\(437\) 3.94363e9 2.26054
\(438\) 0 0
\(439\) −1.55931e9 −0.879641 −0.439821 0.898086i \(-0.644958\pi\)
−0.439821 + 0.898086i \(0.644958\pi\)
\(440\) 3.52444e8 0.197245
\(441\) 0 0
\(442\) −9.44548e8 −0.520291
\(443\) 2.79970e8 0.153002 0.0765012 0.997069i \(-0.475625\pi\)
0.0765012 + 0.997069i \(0.475625\pi\)
\(444\) 0 0
\(445\) 2.54479e8 0.136897
\(446\) 9.61482e8 0.513179
\(447\) 0 0
\(448\) −3.55527e8 −0.186810
\(449\) −6.18452e8 −0.322436 −0.161218 0.986919i \(-0.551542\pi\)
−0.161218 + 0.986919i \(0.551542\pi\)
\(450\) 0 0
\(451\) −2.21514e9 −1.13706
\(452\) −1.69305e9 −0.862352
\(453\) 0 0
\(454\) 1.36678e8 0.0685494
\(455\) 6.07065e8 0.302131
\(456\) 0 0
\(457\) 9.07329e8 0.444691 0.222345 0.974968i \(-0.428629\pi\)
0.222345 + 0.974968i \(0.428629\pi\)
\(458\) −4.25404e8 −0.206906
\(459\) 0 0
\(460\) −1.50533e9 −0.721075
\(461\) 2.41729e9 1.14915 0.574573 0.818453i \(-0.305168\pi\)
0.574573 + 0.818453i \(0.305168\pi\)
\(462\) 0 0
\(463\) −3.14336e8 −0.147184 −0.0735921 0.997288i \(-0.523446\pi\)
−0.0735921 + 0.997288i \(0.523446\pi\)
\(464\) −7.20120e8 −0.334651
\(465\) 0 0
\(466\) 1.09767e9 0.502483
\(467\) −4.74332e7 −0.0215513 −0.0107757 0.999942i \(-0.503430\pi\)
−0.0107757 + 0.999942i \(0.503430\pi\)
\(468\) 0 0
\(469\) 1.44759e9 0.647947
\(470\) −5.10999e7 −0.0227027
\(471\) 0 0
\(472\) −3.41722e8 −0.149581
\(473\) −2.61997e9 −1.13837
\(474\) 0 0
\(475\) −5.95293e8 −0.254861
\(476\) −7.79671e8 −0.331350
\(477\) 0 0
\(478\) 1.00750e9 0.421937
\(479\) 2.08794e9 0.868048 0.434024 0.900901i \(-0.357093\pi\)
0.434024 + 0.900901i \(0.357093\pi\)
\(480\) 0 0
\(481\) −3.77353e9 −1.54611
\(482\) 3.44465e8 0.140114
\(483\) 0 0
\(484\) 9.38326e8 0.376179
\(485\) −1.58811e9 −0.632099
\(486\) 0 0
\(487\) −2.19273e9 −0.860267 −0.430134 0.902765i \(-0.641534\pi\)
−0.430134 + 0.902765i \(0.641534\pi\)
\(488\) −1.18696e9 −0.462345
\(489\) 0 0
\(490\) −5.02123e7 −0.0192808
\(491\) −2.19027e9 −0.835050 −0.417525 0.908666i \(-0.637102\pi\)
−0.417525 + 0.908666i \(0.637102\pi\)
\(492\) 0 0
\(493\) −1.16827e9 −0.439115
\(494\) 1.84185e9 0.687400
\(495\) 0 0
\(496\) 8.23154e8 0.302897
\(497\) −7.46435e8 −0.272737
\(498\) 0 0
\(499\) 6.56737e8 0.236614 0.118307 0.992977i \(-0.462253\pi\)
0.118307 + 0.992977i \(0.462253\pi\)
\(500\) 2.27230e8 0.0812964
\(501\) 0 0
\(502\) 3.17662e8 0.112073
\(503\) 2.58040e9 0.904065 0.452033 0.892001i \(-0.350699\pi\)
0.452033 + 0.892001i \(0.350699\pi\)
\(504\) 0 0
\(505\) −2.33261e9 −0.805977
\(506\) −1.19445e9 −0.409865
\(507\) 0 0
\(508\) −2.37455e8 −0.0803633
\(509\) 2.07943e9 0.698928 0.349464 0.936950i \(-0.386364\pi\)
0.349464 + 0.936950i \(0.386364\pi\)
\(510\) 0 0
\(511\) 1.52262e9 0.504798
\(512\) 3.06735e9 1.00999
\(513\) 0 0
\(514\) −1.24483e9 −0.404333
\(515\) −6.63465e8 −0.214039
\(516\) 0 0
\(517\) 4.04639e8 0.128781
\(518\) 3.12121e8 0.0986662
\(519\) 0 0
\(520\) −1.47656e9 −0.460511
\(521\) 2.82646e9 0.875609 0.437804 0.899070i \(-0.355756\pi\)
0.437804 + 0.899070i \(0.355756\pi\)
\(522\) 0 0
\(523\) 4.81610e9 1.47211 0.736055 0.676922i \(-0.236687\pi\)
0.736055 + 0.676922i \(0.236687\pi\)
\(524\) −1.06375e9 −0.322983
\(525\) 0 0
\(526\) −3.81157e8 −0.114197
\(527\) 1.33542e9 0.397449
\(528\) 0 0
\(529\) 7.30968e9 2.14686
\(530\) 9.56464e7 0.0279064
\(531\) 0 0
\(532\) 1.52034e9 0.437775
\(533\) 9.28030e9 2.65471
\(534\) 0 0
\(535\) 2.18523e9 0.616962
\(536\) −3.52096e9 −0.987606
\(537\) 0 0
\(538\) 5.23754e8 0.145007
\(539\) 3.97611e8 0.109370
\(540\) 0 0
\(541\) 1.56108e9 0.423871 0.211936 0.977284i \(-0.432023\pi\)
0.211936 + 0.977284i \(0.432023\pi\)
\(542\) −1.42423e9 −0.384223
\(543\) 0 0
\(544\) 2.88982e9 0.769618
\(545\) −5.84601e7 −0.0154693
\(546\) 0 0
\(547\) −3.82511e8 −0.0999282 −0.0499641 0.998751i \(-0.515911\pi\)
−0.0499641 + 0.998751i \(0.515911\pi\)
\(548\) 4.22002e9 1.09543
\(549\) 0 0
\(550\) 1.80303e8 0.0462096
\(551\) 2.27810e9 0.580152
\(552\) 0 0
\(553\) 5.60140e8 0.140851
\(554\) −1.38516e9 −0.346110
\(555\) 0 0
\(556\) 2.66217e9 0.656861
\(557\) 7.35207e9 1.80267 0.901335 0.433122i \(-0.142588\pi\)
0.901335 + 0.433122i \(0.142588\pi\)
\(558\) 0 0
\(559\) 1.09763e10 2.65776
\(560\) −5.16354e8 −0.124248
\(561\) 0 0
\(562\) 5.14798e8 0.122338
\(563\) −7.91584e9 −1.86947 −0.934733 0.355352i \(-0.884361\pi\)
−0.934733 + 0.355352i \(0.884361\pi\)
\(564\) 0 0
\(565\) −1.81904e9 −0.424300
\(566\) −1.26194e8 −0.0292538
\(567\) 0 0
\(568\) 1.81555e9 0.415708
\(569\) −4.49464e9 −1.02283 −0.511414 0.859335i \(-0.670878\pi\)
−0.511414 + 0.859335i \(0.670878\pi\)
\(570\) 0 0
\(571\) −2.15052e9 −0.483412 −0.241706 0.970350i \(-0.577707\pi\)
−0.241706 + 0.970350i \(0.577707\pi\)
\(572\) 5.56721e9 1.24380
\(573\) 0 0
\(574\) −7.67604e8 −0.169412
\(575\) −1.61736e9 −0.354788
\(576\) 0 0
\(577\) −1.73438e9 −0.375862 −0.187931 0.982182i \(-0.560178\pi\)
−0.187931 + 0.982182i \(0.560178\pi\)
\(578\) −9.76656e7 −0.0210375
\(579\) 0 0
\(580\) −8.69577e8 −0.185059
\(581\) 1.20017e9 0.253878
\(582\) 0 0
\(583\) −7.57385e8 −0.158298
\(584\) −3.70345e9 −0.769418
\(585\) 0 0
\(586\) 2.05008e9 0.420852
\(587\) 2.98775e9 0.609692 0.304846 0.952402i \(-0.401395\pi\)
0.304846 + 0.952402i \(0.401395\pi\)
\(588\) 0 0
\(589\) −2.60405e9 −0.525104
\(590\) −1.74817e8 −0.0350431
\(591\) 0 0
\(592\) 3.20967e9 0.635819
\(593\) −5.56485e9 −1.09588 −0.547938 0.836519i \(-0.684587\pi\)
−0.547938 + 0.836519i \(0.684587\pi\)
\(594\) 0 0
\(595\) −8.37693e8 −0.163033
\(596\) −8.36348e9 −1.61817
\(597\) 0 0
\(598\) 5.00414e9 0.956919
\(599\) 5.39399e9 1.02545 0.512727 0.858552i \(-0.328635\pi\)
0.512727 + 0.858552i \(0.328635\pi\)
\(600\) 0 0
\(601\) −5.88616e8 −0.110604 −0.0553021 0.998470i \(-0.517612\pi\)
−0.0553021 + 0.998470i \(0.517612\pi\)
\(602\) −9.07888e8 −0.169607
\(603\) 0 0
\(604\) −7.44558e9 −1.37489
\(605\) 1.00815e9 0.185090
\(606\) 0 0
\(607\) 5.38304e9 0.976938 0.488469 0.872581i \(-0.337556\pi\)
0.488469 + 0.872581i \(0.337556\pi\)
\(608\) −5.63509e9 −1.01681
\(609\) 0 0
\(610\) −6.07222e8 −0.108316
\(611\) −1.69523e9 −0.300667
\(612\) 0 0
\(613\) 8.87502e9 1.55617 0.778086 0.628158i \(-0.216191\pi\)
0.778086 + 0.628158i \(0.216191\pi\)
\(614\) 3.10330e9 0.541046
\(615\) 0 0
\(616\) −9.67106e8 −0.166702
\(617\) −5.94996e6 −0.00101980 −0.000509901 1.00000i \(-0.500162\pi\)
−0.000509901 1.00000i \(0.500162\pi\)
\(618\) 0 0
\(619\) 8.21425e9 1.39204 0.696019 0.718024i \(-0.254953\pi\)
0.696019 + 0.718024i \(0.254953\pi\)
\(620\) 9.93996e8 0.167500
\(621\) 0 0
\(622\) −2.20522e9 −0.367439
\(623\) −6.98291e8 −0.115699
\(624\) 0 0
\(625\) 2.44141e8 0.0400000
\(626\) −3.06134e9 −0.498772
\(627\) 0 0
\(628\) −6.98631e8 −0.112561
\(629\) 5.20711e9 0.834295
\(630\) 0 0
\(631\) 2.53051e9 0.400964 0.200482 0.979697i \(-0.435749\pi\)
0.200482 + 0.979697i \(0.435749\pi\)
\(632\) −1.36243e9 −0.214686
\(633\) 0 0
\(634\) −2.89299e9 −0.450852
\(635\) −2.55126e8 −0.0395409
\(636\) 0 0
\(637\) −1.66579e9 −0.255347
\(638\) −6.89990e8 −0.105189
\(639\) 0 0
\(640\) 2.80891e9 0.423553
\(641\) −6.96698e9 −1.04482 −0.522410 0.852694i \(-0.674967\pi\)
−0.522410 + 0.852694i \(0.674967\pi\)
\(642\) 0 0
\(643\) 1.21823e10 1.80714 0.903572 0.428437i \(-0.140936\pi\)
0.903572 + 0.428437i \(0.140936\pi\)
\(644\) 4.13063e9 0.609419
\(645\) 0 0
\(646\) −2.54158e9 −0.370928
\(647\) 9.95454e9 1.44496 0.722481 0.691391i \(-0.243002\pi\)
0.722481 + 0.691391i \(0.243002\pi\)
\(648\) 0 0
\(649\) 1.38431e9 0.198781
\(650\) −7.55376e8 −0.107886
\(651\) 0 0
\(652\) −1.14367e10 −1.61598
\(653\) −2.38576e9 −0.335297 −0.167649 0.985847i \(-0.553617\pi\)
−0.167649 + 0.985847i \(0.553617\pi\)
\(654\) 0 0
\(655\) −1.14291e9 −0.158916
\(656\) −7.89358e9 −1.09172
\(657\) 0 0
\(658\) 1.40218e8 0.0191873
\(659\) −1.98573e8 −0.0270284 −0.0135142 0.999909i \(-0.504302\pi\)
−0.0135142 + 0.999909i \(0.504302\pi\)
\(660\) 0 0
\(661\) 1.68735e9 0.227247 0.113624 0.993524i \(-0.463754\pi\)
0.113624 + 0.993524i \(0.463754\pi\)
\(662\) 2.53484e9 0.339584
\(663\) 0 0
\(664\) −2.91916e9 −0.386963
\(665\) 1.63348e9 0.215397
\(666\) 0 0
\(667\) 6.18938e9 0.807620
\(668\) 6.75704e9 0.877079
\(669\) 0 0
\(670\) −1.80124e9 −0.231372
\(671\) 4.80834e9 0.614421
\(672\) 0 0
\(673\) −5.02352e9 −0.635266 −0.317633 0.948214i \(-0.602888\pi\)
−0.317633 + 0.948214i \(0.602888\pi\)
\(674\) −2.47376e9 −0.311206
\(675\) 0 0
\(676\) −1.60235e10 −1.99501
\(677\) 1.03907e10 1.28702 0.643510 0.765438i \(-0.277477\pi\)
0.643510 + 0.765438i \(0.277477\pi\)
\(678\) 0 0
\(679\) 4.35778e9 0.534222
\(680\) 2.03751e9 0.248496
\(681\) 0 0
\(682\) 7.88714e8 0.0952082
\(683\) −3.61608e9 −0.434276 −0.217138 0.976141i \(-0.569672\pi\)
−0.217138 + 0.976141i \(0.569672\pi\)
\(684\) 0 0
\(685\) 4.53407e9 0.538978
\(686\) 1.37783e8 0.0162952
\(687\) 0 0
\(688\) −9.33619e9 −1.09297
\(689\) 3.17306e9 0.369582
\(690\) 0 0
\(691\) 1.65856e10 1.91231 0.956153 0.292869i \(-0.0946100\pi\)
0.956153 + 0.292869i \(0.0946100\pi\)
\(692\) −2.69618e9 −0.309298
\(693\) 0 0
\(694\) −1.74081e9 −0.197694
\(695\) 2.86028e9 0.323193
\(696\) 0 0
\(697\) −1.28059e10 −1.43251
\(698\) −2.99117e9 −0.332926
\(699\) 0 0
\(700\) −6.23520e8 −0.0687080
\(701\) −1.42971e10 −1.56760 −0.783798 0.621015i \(-0.786720\pi\)
−0.783798 + 0.621015i \(0.786720\pi\)
\(702\) 0 0
\(703\) −1.01538e10 −1.10226
\(704\) −3.50306e9 −0.378394
\(705\) 0 0
\(706\) −4.32499e9 −0.462561
\(707\) 6.40069e9 0.681175
\(708\) 0 0
\(709\) 4.66391e9 0.491460 0.245730 0.969338i \(-0.420972\pi\)
0.245730 + 0.969338i \(0.420972\pi\)
\(710\) 9.28795e8 0.0973903
\(711\) 0 0
\(712\) 1.69845e9 0.176349
\(713\) −7.07496e9 −0.730989
\(714\) 0 0
\(715\) 5.98151e9 0.611984
\(716\) 4.39659e9 0.447631
\(717\) 0 0
\(718\) −5.68694e9 −0.573380
\(719\) −4.82173e9 −0.483784 −0.241892 0.970303i \(-0.577768\pi\)
−0.241892 + 0.970303i \(0.577768\pi\)
\(720\) 0 0
\(721\) 1.82055e9 0.180896
\(722\) 1.90400e9 0.188273
\(723\) 0 0
\(724\) 1.05793e9 0.103603
\(725\) −9.34289e8 −0.0910539
\(726\) 0 0
\(727\) 2.17621e9 0.210054 0.105027 0.994469i \(-0.466507\pi\)
0.105027 + 0.994469i \(0.466507\pi\)
\(728\) 4.05168e9 0.389203
\(729\) 0 0
\(730\) −1.89461e9 −0.180256
\(731\) −1.51463e10 −1.43416
\(732\) 0 0
\(733\) 4.65228e9 0.436317 0.218158 0.975913i \(-0.429995\pi\)
0.218158 + 0.975913i \(0.429995\pi\)
\(734\) 5.72166e9 0.534055
\(735\) 0 0
\(736\) −1.53100e10 −1.41548
\(737\) 1.42633e10 1.31245
\(738\) 0 0
\(739\) 4.05338e8 0.0369456 0.0184728 0.999829i \(-0.494120\pi\)
0.0184728 + 0.999829i \(0.494120\pi\)
\(740\) 3.87581e9 0.351603
\(741\) 0 0
\(742\) −2.62454e8 −0.0235852
\(743\) −1.08140e10 −0.967223 −0.483611 0.875283i \(-0.660675\pi\)
−0.483611 + 0.875283i \(0.660675\pi\)
\(744\) 0 0
\(745\) −8.98588e9 −0.796184
\(746\) 5.04282e9 0.444721
\(747\) 0 0
\(748\) −7.68222e9 −0.671168
\(749\) −5.99626e9 −0.521428
\(750\) 0 0
\(751\) −1.42070e10 −1.22394 −0.611972 0.790880i \(-0.709623\pi\)
−0.611972 + 0.790880i \(0.709623\pi\)
\(752\) 1.44192e9 0.123646
\(753\) 0 0
\(754\) 2.89071e9 0.245587
\(755\) −7.99967e9 −0.676485
\(756\) 0 0
\(757\) 3.78427e9 0.317064 0.158532 0.987354i \(-0.449324\pi\)
0.158532 + 0.987354i \(0.449324\pi\)
\(758\) −5.60911e9 −0.467791
\(759\) 0 0
\(760\) −3.97311e9 −0.328309
\(761\) −1.01511e10 −0.834966 −0.417483 0.908685i \(-0.637088\pi\)
−0.417483 + 0.908685i \(0.637088\pi\)
\(762\) 0 0
\(763\) 1.60414e8 0.0130740
\(764\) −6.22314e9 −0.504874
\(765\) 0 0
\(766\) 6.47993e9 0.520919
\(767\) −5.79954e9 −0.464098
\(768\) 0 0
\(769\) −1.31076e10 −1.03939 −0.519696 0.854351i \(-0.673955\pi\)
−0.519696 + 0.854351i \(0.673955\pi\)
\(770\) −4.94750e8 −0.0390543
\(771\) 0 0
\(772\) 9.59751e9 0.750755
\(773\) 1.40370e10 1.09307 0.546533 0.837438i \(-0.315947\pi\)
0.546533 + 0.837438i \(0.315947\pi\)
\(774\) 0 0
\(775\) 1.06797e9 0.0824142
\(776\) −1.05994e10 −0.814265
\(777\) 0 0
\(778\) −3.59985e9 −0.274066
\(779\) 2.49713e10 1.89261
\(780\) 0 0
\(781\) −7.35474e9 −0.552445
\(782\) −6.90524e9 −0.516363
\(783\) 0 0
\(784\) 1.41687e9 0.105009
\(785\) −7.50622e8 −0.0553831
\(786\) 0 0
\(787\) −3.87288e9 −0.283219 −0.141610 0.989923i \(-0.545228\pi\)
−0.141610 + 0.989923i \(0.545228\pi\)
\(788\) 8.78764e9 0.639780
\(789\) 0 0
\(790\) −6.96987e8 −0.0502956
\(791\) 4.99145e9 0.358599
\(792\) 0 0
\(793\) −2.01445e10 −1.43450
\(794\) 1.01355e9 0.0718578
\(795\) 0 0
\(796\) 1.22794e8 0.00862942
\(797\) 8.56688e9 0.599403 0.299701 0.954033i \(-0.403113\pi\)
0.299701 + 0.954033i \(0.403113\pi\)
\(798\) 0 0
\(799\) 2.33926e9 0.162243
\(800\) 2.31106e9 0.159586
\(801\) 0 0
\(802\) 2.66302e9 0.182290
\(803\) 1.50026e10 1.02250
\(804\) 0 0
\(805\) 4.43803e9 0.299850
\(806\) −3.30431e9 −0.222284
\(807\) 0 0
\(808\) −1.55684e10 −1.03825
\(809\) 1.49503e10 0.992731 0.496365 0.868114i \(-0.334668\pi\)
0.496365 + 0.868114i \(0.334668\pi\)
\(810\) 0 0
\(811\) −8.61940e9 −0.567419 −0.283710 0.958910i \(-0.591565\pi\)
−0.283710 + 0.958910i \(0.591565\pi\)
\(812\) 2.38612e9 0.156403
\(813\) 0 0
\(814\) 3.07538e9 0.199854
\(815\) −1.22878e10 −0.795104
\(816\) 0 0
\(817\) 2.95350e10 1.89478
\(818\) 2.98810e9 0.190879
\(819\) 0 0
\(820\) −9.53186e9 −0.603711
\(821\) −2.53165e10 −1.59663 −0.798313 0.602243i \(-0.794274\pi\)
−0.798313 + 0.602243i \(0.794274\pi\)
\(822\) 0 0
\(823\) 1.51790e9 0.0949171 0.0474586 0.998873i \(-0.484888\pi\)
0.0474586 + 0.998873i \(0.484888\pi\)
\(824\) −4.42811e9 −0.275723
\(825\) 0 0
\(826\) 4.79699e8 0.0296168
\(827\) −1.68322e10 −1.03484 −0.517418 0.855733i \(-0.673107\pi\)
−0.517418 + 0.855733i \(0.673107\pi\)
\(828\) 0 0
\(829\) −1.54259e10 −0.940396 −0.470198 0.882561i \(-0.655818\pi\)
−0.470198 + 0.882561i \(0.655818\pi\)
\(830\) −1.49338e9 −0.0906559
\(831\) 0 0
\(832\) 1.46761e10 0.883442
\(833\) 2.29863e9 0.137788
\(834\) 0 0
\(835\) 7.25988e9 0.431546
\(836\) 1.49802e10 0.886737
\(837\) 0 0
\(838\) −3.65536e9 −0.214574
\(839\) 1.06044e9 0.0619894 0.0309947 0.999520i \(-0.490132\pi\)
0.0309947 + 0.999520i \(0.490132\pi\)
\(840\) 0 0
\(841\) −1.36745e10 −0.792730
\(842\) 8.67880e9 0.501034
\(843\) 0 0
\(844\) −1.33181e9 −0.0762509
\(845\) −1.72159e10 −0.981595
\(846\) 0 0
\(847\) −2.76638e9 −0.156430
\(848\) −2.69892e9 −0.151986
\(849\) 0 0
\(850\) 1.04235e9 0.0582165
\(851\) −2.75869e10 −1.53444
\(852\) 0 0
\(853\) 3.03756e10 1.67573 0.837864 0.545878i \(-0.183804\pi\)
0.837864 + 0.545878i \(0.183804\pi\)
\(854\) 1.66622e9 0.0915438
\(855\) 0 0
\(856\) 1.45847e10 0.794764
\(857\) 1.27509e10 0.692001 0.346001 0.938234i \(-0.387539\pi\)
0.346001 + 0.938234i \(0.387539\pi\)
\(858\) 0 0
\(859\) 1.08205e10 0.582467 0.291233 0.956652i \(-0.405934\pi\)
0.291233 + 0.956652i \(0.405934\pi\)
\(860\) −1.12739e10 −0.604405
\(861\) 0 0
\(862\) −3.04476e9 −0.161912
\(863\) −6.05794e9 −0.320839 −0.160419 0.987049i \(-0.551285\pi\)
−0.160419 + 0.987049i \(0.551285\pi\)
\(864\) 0 0
\(865\) −2.89682e9 −0.152183
\(866\) −3.54981e9 −0.185734
\(867\) 0 0
\(868\) −2.72752e9 −0.141563
\(869\) 5.51915e9 0.285301
\(870\) 0 0
\(871\) −5.97560e10 −3.06421
\(872\) −3.90175e8 −0.0199274
\(873\) 0 0
\(874\) 1.34651e10 0.682211
\(875\) −6.69922e8 −0.0338062
\(876\) 0 0
\(877\) 5.64172e9 0.282431 0.141216 0.989979i \(-0.454899\pi\)
0.141216 + 0.989979i \(0.454899\pi\)
\(878\) −5.32407e9 −0.265468
\(879\) 0 0
\(880\) −5.08772e9 −0.251671
\(881\) 1.41128e9 0.0695339 0.0347669 0.999395i \(-0.488931\pi\)
0.0347669 + 0.999395i \(0.488931\pi\)
\(882\) 0 0
\(883\) 1.83461e10 0.896773 0.448386 0.893840i \(-0.351999\pi\)
0.448386 + 0.893840i \(0.351999\pi\)
\(884\) 3.21846e10 1.56699
\(885\) 0 0
\(886\) 9.55923e8 0.0461748
\(887\) −6.77821e9 −0.326124 −0.163062 0.986616i \(-0.552137\pi\)
−0.163062 + 0.986616i \(0.552137\pi\)
\(888\) 0 0
\(889\) 7.00065e8 0.0334182
\(890\) 8.68889e8 0.0413142
\(891\) 0 0
\(892\) −3.27617e10 −1.54557
\(893\) −4.56151e9 −0.214353
\(894\) 0 0
\(895\) 4.72378e9 0.220246
\(896\) −7.70764e9 −0.357967
\(897\) 0 0
\(898\) −2.11163e9 −0.0973085
\(899\) −4.08695e9 −0.187603
\(900\) 0 0
\(901\) −4.37852e9 −0.199430
\(902\) −7.56332e9 −0.343155
\(903\) 0 0
\(904\) −1.21407e10 −0.546579
\(905\) 1.13666e9 0.0509752
\(906\) 0 0
\(907\) −3.32813e9 −0.148107 −0.0740535 0.997254i \(-0.523594\pi\)
−0.0740535 + 0.997254i \(0.523594\pi\)
\(908\) −4.65718e9 −0.206454
\(909\) 0 0
\(910\) 2.07275e9 0.0911806
\(911\) −7.48533e9 −0.328017 −0.164009 0.986459i \(-0.552442\pi\)
−0.164009 + 0.986459i \(0.552442\pi\)
\(912\) 0 0
\(913\) 1.18254e10 0.514244
\(914\) 3.09797e9 0.134204
\(915\) 0 0
\(916\) 1.44953e10 0.623149
\(917\) 3.13615e9 0.134309
\(918\) 0 0
\(919\) 3.84326e10 1.63341 0.816705 0.577056i \(-0.195799\pi\)
0.816705 + 0.577056i \(0.195799\pi\)
\(920\) −1.07946e10 −0.457034
\(921\) 0 0
\(922\) 8.25354e9 0.346802
\(923\) 3.08127e10 1.28980
\(924\) 0 0
\(925\) 4.16425e9 0.172998
\(926\) −1.07326e9 −0.0444189
\(927\) 0 0
\(928\) −8.84406e9 −0.363274
\(929\) −2.74203e10 −1.12206 −0.561032 0.827794i \(-0.689596\pi\)
−0.561032 + 0.827794i \(0.689596\pi\)
\(930\) 0 0
\(931\) −4.48228e9 −0.182043
\(932\) −3.74022e10 −1.51336
\(933\) 0 0
\(934\) −1.61955e8 −0.00650400
\(935\) −8.25392e9 −0.330233
\(936\) 0 0
\(937\) −2.33959e9 −0.0929077 −0.0464539 0.998920i \(-0.514792\pi\)
−0.0464539 + 0.998920i \(0.514792\pi\)
\(938\) 4.94261e9 0.195545
\(939\) 0 0
\(940\) 1.74118e9 0.0683750
\(941\) 4.04198e10 1.58136 0.790680 0.612230i \(-0.209727\pi\)
0.790680 + 0.612230i \(0.209727\pi\)
\(942\) 0 0
\(943\) 6.78449e10 2.63467
\(944\) 4.93294e9 0.190855
\(945\) 0 0
\(946\) −8.94557e9 −0.343549
\(947\) −2.68103e10 −1.02583 −0.512917 0.858438i \(-0.671435\pi\)
−0.512917 + 0.858438i \(0.671435\pi\)
\(948\) 0 0
\(949\) −6.28533e10 −2.38724
\(950\) −2.03256e9 −0.0769148
\(951\) 0 0
\(952\) −5.59094e9 −0.210018
\(953\) 6.72263e9 0.251602 0.125801 0.992056i \(-0.459850\pi\)
0.125801 + 0.992056i \(0.459850\pi\)
\(954\) 0 0
\(955\) −6.68625e9 −0.248411
\(956\) −3.43297e10 −1.27077
\(957\) 0 0
\(958\) 7.12902e9 0.261970
\(959\) −1.24415e10 −0.455520
\(960\) 0 0
\(961\) −2.28409e10 −0.830198
\(962\) −1.28843e10 −0.466602
\(963\) 0 0
\(964\) −1.17373e10 −0.421988
\(965\) 1.03117e10 0.369391
\(966\) 0 0
\(967\) 2.39023e10 0.850055 0.425027 0.905180i \(-0.360264\pi\)
0.425027 + 0.905180i \(0.360264\pi\)
\(968\) 6.72864e9 0.238431
\(969\) 0 0
\(970\) −5.42242e9 −0.190762
\(971\) −1.65771e10 −0.581088 −0.290544 0.956862i \(-0.593836\pi\)
−0.290544 + 0.956862i \(0.593836\pi\)
\(972\) 0 0
\(973\) −7.84862e9 −0.273148
\(974\) −7.48681e9 −0.259621
\(975\) 0 0
\(976\) 1.71344e10 0.589921
\(977\) 2.03177e10 0.697018 0.348509 0.937305i \(-0.386688\pi\)
0.348509 + 0.937305i \(0.386688\pi\)
\(978\) 0 0
\(979\) −6.88038e9 −0.234354
\(980\) 1.71094e9 0.0580689
\(981\) 0 0
\(982\) −7.47841e9 −0.252011
\(983\) −4.78793e10 −1.60772 −0.803861 0.594818i \(-0.797224\pi\)
−0.803861 + 0.594818i \(0.797224\pi\)
\(984\) 0 0
\(985\) 9.44160e9 0.314788
\(986\) −3.98891e9 −0.132521
\(987\) 0 0
\(988\) −6.27594e10 −2.07028
\(989\) 8.02440e10 2.63770
\(990\) 0 0
\(991\) −1.99787e10 −0.652092 −0.326046 0.945354i \(-0.605716\pi\)
−0.326046 + 0.945354i \(0.605716\pi\)
\(992\) 1.01095e10 0.328804
\(993\) 0 0
\(994\) −2.54861e9 −0.0823098
\(995\) 1.31932e8 0.00424590
\(996\) 0 0
\(997\) 4.49634e10 1.43690 0.718450 0.695579i \(-0.244852\pi\)
0.718450 + 0.695579i \(0.244852\pi\)
\(998\) 2.24235e9 0.0714080
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.8.a.g.1.3 4
3.2 odd 2 105.8.a.h.1.2 4
15.14 odd 2 525.8.a.i.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.8.a.h.1.2 4 3.2 odd 2
315.8.a.g.1.3 4 1.1 even 1 trivial
525.8.a.i.1.3 4 15.14 odd 2