Properties

Label 225.6.a.s.1.1
Level $225$
Weight $6$
Character 225.1
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,6,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.26209\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.26209 q^{2} -4.31044 q^{4} -131.048 q^{7} +191.069 q^{8} +O(q^{10})\) \(q-5.26209 q^{2} -4.31044 q^{4} -131.048 q^{7} +191.069 q^{8} -290.104 q^{11} -68.3868 q^{13} +689.588 q^{14} -867.486 q^{16} -310.644 q^{17} -2133.35 q^{19} +1526.55 q^{22} +873.145 q^{23} +359.857 q^{26} +564.876 q^{28} +2580.97 q^{29} -9086.30 q^{31} -1549.41 q^{32} +1634.64 q^{34} -3990.64 q^{37} +11225.9 q^{38} -16981.8 q^{41} +18017.7 q^{43} +1250.48 q^{44} -4594.57 q^{46} +24864.7 q^{47} +366.670 q^{49} +294.777 q^{52} -7652.91 q^{53} -25039.2 q^{56} -13581.3 q^{58} +9233.69 q^{59} +3326.17 q^{61} +47812.9 q^{62} +35912.7 q^{64} +32340.7 q^{67} +1339.01 q^{68} +35885.9 q^{71} +26513.6 q^{73} +20999.1 q^{74} +9195.65 q^{76} +38017.7 q^{77} +71705.7 q^{79} +89359.9 q^{82} +39630.1 q^{83} -94810.8 q^{86} -55429.9 q^{88} +117441. q^{89} +8961.98 q^{91} -3763.64 q^{92} -130840. q^{94} -21878.3 q^{97} -1929.45 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 69 q^{4} - 200 q^{7} + 615 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} + 69 q^{4} - 200 q^{7} + 615 q^{8} + 196 q^{11} + 360 q^{13} - 18 q^{14} + 1137 q^{16} + 1490 q^{17} - 3180 q^{19} + 6515 q^{22} + 1560 q^{23} + 4756 q^{26} - 4490 q^{28} + 3920 q^{29} - 1096 q^{31} + 5455 q^{32} + 20113 q^{34} - 2020 q^{37} + 485 q^{38} - 27754 q^{41} + 3000 q^{43} + 36887 q^{44} + 2454 q^{46} + 25760 q^{47} - 11686 q^{49} + 31700 q^{52} - 26980 q^{53} - 54270 q^{56} + 160 q^{58} - 11960 q^{59} - 24396 q^{61} + 129810 q^{62} + 43649 q^{64} + 40060 q^{67} + 133345 q^{68} + 87296 q^{71} + 70290 q^{73} + 41222 q^{74} - 67535 q^{76} + 4500 q^{77} + 65480 q^{79} - 21185 q^{82} + 92580 q^{83} - 248924 q^{86} + 150645 q^{88} + 72810 q^{89} - 20576 q^{91} + 46590 q^{92} - 121652 q^{94} + 126140 q^{97} - 125615 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.26209 −0.930214 −0.465107 0.885254i \(-0.653984\pi\)
−0.465107 + 0.885254i \(0.653984\pi\)
\(3\) 0 0
\(4\) −4.31044 −0.134701
\(5\) 0 0
\(6\) 0 0
\(7\) −131.048 −1.01085 −0.505425 0.862871i \(-0.668664\pi\)
−0.505425 + 0.862871i \(0.668664\pi\)
\(8\) 191.069 1.05552
\(9\) 0 0
\(10\) 0 0
\(11\) −290.104 −0.722891 −0.361445 0.932393i \(-0.617717\pi\)
−0.361445 + 0.932393i \(0.617717\pi\)
\(12\) 0 0
\(13\) −68.3868 −0.112231 −0.0561156 0.998424i \(-0.517872\pi\)
−0.0561156 + 0.998424i \(0.517872\pi\)
\(14\) 689.588 0.940307
\(15\) 0 0
\(16\) −867.486 −0.847154
\(17\) −310.644 −0.260700 −0.130350 0.991468i \(-0.541610\pi\)
−0.130350 + 0.991468i \(0.541610\pi\)
\(18\) 0 0
\(19\) −2133.35 −1.35574 −0.677871 0.735180i \(-0.737097\pi\)
−0.677871 + 0.735180i \(0.737097\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1526.55 0.672443
\(23\) 873.145 0.344165 0.172083 0.985083i \(-0.444950\pi\)
0.172083 + 0.985083i \(0.444950\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 359.857 0.104399
\(27\) 0 0
\(28\) 564.876 0.136163
\(29\) 2580.97 0.569885 0.284943 0.958545i \(-0.408025\pi\)
0.284943 + 0.958545i \(0.408025\pi\)
\(30\) 0 0
\(31\) −9086.30 −1.69818 −0.849088 0.528252i \(-0.822848\pi\)
−0.849088 + 0.528252i \(0.822848\pi\)
\(32\) −1549.41 −0.267480
\(33\) 0 0
\(34\) 1634.64 0.242507
\(35\) 0 0
\(36\) 0 0
\(37\) −3990.64 −0.479224 −0.239612 0.970869i \(-0.577020\pi\)
−0.239612 + 0.970869i \(0.577020\pi\)
\(38\) 11225.9 1.26113
\(39\) 0 0
\(40\) 0 0
\(41\) −16981.8 −1.57770 −0.788851 0.614584i \(-0.789324\pi\)
−0.788851 + 0.614584i \(0.789324\pi\)
\(42\) 0 0
\(43\) 18017.7 1.48603 0.743017 0.669273i \(-0.233394\pi\)
0.743017 + 0.669273i \(0.233394\pi\)
\(44\) 1250.48 0.0973742
\(45\) 0 0
\(46\) −4594.57 −0.320147
\(47\) 24864.7 1.64187 0.820933 0.571024i \(-0.193454\pi\)
0.820933 + 0.571024i \(0.193454\pi\)
\(48\) 0 0
\(49\) 366.670 0.0218165
\(50\) 0 0
\(51\) 0 0
\(52\) 294.777 0.0151177
\(53\) −7652.91 −0.374229 −0.187114 0.982338i \(-0.559913\pi\)
−0.187114 + 0.982338i \(0.559913\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −25039.2 −1.06697
\(57\) 0 0
\(58\) −13581.3 −0.530116
\(59\) 9233.69 0.345339 0.172669 0.984980i \(-0.444761\pi\)
0.172669 + 0.984980i \(0.444761\pi\)
\(60\) 0 0
\(61\) 3326.17 0.114451 0.0572256 0.998361i \(-0.481775\pi\)
0.0572256 + 0.998361i \(0.481775\pi\)
\(62\) 47812.9 1.57967
\(63\) 0 0
\(64\) 35912.7 1.09597
\(65\) 0 0
\(66\) 0 0
\(67\) 32340.7 0.880161 0.440080 0.897958i \(-0.354950\pi\)
0.440080 + 0.897958i \(0.354950\pi\)
\(68\) 1339.01 0.0351165
\(69\) 0 0
\(70\) 0 0
\(71\) 35885.9 0.844847 0.422424 0.906399i \(-0.361179\pi\)
0.422424 + 0.906399i \(0.361179\pi\)
\(72\) 0 0
\(73\) 26513.6 0.582319 0.291159 0.956675i \(-0.405959\pi\)
0.291159 + 0.956675i \(0.405959\pi\)
\(74\) 20999.1 0.445781
\(75\) 0 0
\(76\) 9195.65 0.182620
\(77\) 38017.7 0.730733
\(78\) 0 0
\(79\) 71705.7 1.29266 0.646332 0.763056i \(-0.276302\pi\)
0.646332 + 0.763056i \(0.276302\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 89359.9 1.46760
\(83\) 39630.1 0.631437 0.315719 0.948853i \(-0.397754\pi\)
0.315719 + 0.948853i \(0.397754\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −94810.8 −1.38233
\(87\) 0 0
\(88\) −55429.9 −0.763022
\(89\) 117441. 1.57161 0.785806 0.618473i \(-0.212248\pi\)
0.785806 + 0.618473i \(0.212248\pi\)
\(90\) 0 0
\(91\) 8961.98 0.113449
\(92\) −3763.64 −0.0463594
\(93\) 0 0
\(94\) −130840. −1.52729
\(95\) 0 0
\(96\) 0 0
\(97\) −21878.3 −0.236093 −0.118047 0.993008i \(-0.537663\pi\)
−0.118047 + 0.993008i \(0.537663\pi\)
\(98\) −1929.45 −0.0202940
\(99\) 0 0
\(100\) 0 0
\(101\) 75072.1 0.732276 0.366138 0.930561i \(-0.380680\pi\)
0.366138 + 0.930561i \(0.380680\pi\)
\(102\) 0 0
\(103\) −47928.6 −0.445145 −0.222573 0.974916i \(-0.571445\pi\)
−0.222573 + 0.974916i \(0.571445\pi\)
\(104\) −13066.6 −0.118462
\(105\) 0 0
\(106\) 40270.3 0.348113
\(107\) 92012.3 0.776938 0.388469 0.921462i \(-0.373004\pi\)
0.388469 + 0.921462i \(0.373004\pi\)
\(108\) 0 0
\(109\) −10647.5 −0.0858387 −0.0429194 0.999079i \(-0.513666\pi\)
−0.0429194 + 0.999079i \(0.513666\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 113683. 0.856346
\(113\) 87373.9 0.643703 0.321852 0.946790i \(-0.395695\pi\)
0.321852 + 0.946790i \(0.395695\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −11125.1 −0.0767642
\(117\) 0 0
\(118\) −48588.5 −0.321239
\(119\) 40709.4 0.263528
\(120\) 0 0
\(121\) −76890.5 −0.477429
\(122\) −17502.6 −0.106464
\(123\) 0 0
\(124\) 39165.9 0.228746
\(125\) 0 0
\(126\) 0 0
\(127\) −197379. −1.08591 −0.542953 0.839763i \(-0.682694\pi\)
−0.542953 + 0.839763i \(0.682694\pi\)
\(128\) −139395. −0.752005
\(129\) 0 0
\(130\) 0 0
\(131\) 118490. 0.603258 0.301629 0.953425i \(-0.402470\pi\)
0.301629 + 0.953425i \(0.402470\pi\)
\(132\) 0 0
\(133\) 279571. 1.37045
\(134\) −170179. −0.818738
\(135\) 0 0
\(136\) −59354.3 −0.275173
\(137\) −302570. −1.37728 −0.688642 0.725101i \(-0.741793\pi\)
−0.688642 + 0.725101i \(0.741793\pi\)
\(138\) 0 0
\(139\) 157190. 0.690062 0.345031 0.938591i \(-0.387868\pi\)
0.345031 + 0.938591i \(0.387868\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −188835. −0.785889
\(143\) 19839.3 0.0811309
\(144\) 0 0
\(145\) 0 0
\(146\) −139517. −0.541681
\(147\) 0 0
\(148\) 17201.4 0.0645520
\(149\) −526340. −1.94223 −0.971115 0.238612i \(-0.923308\pi\)
−0.971115 + 0.238612i \(0.923308\pi\)
\(150\) 0 0
\(151\) 1849.08 0.00659954 0.00329977 0.999995i \(-0.498950\pi\)
0.00329977 + 0.999995i \(0.498950\pi\)
\(152\) −407616. −1.43101
\(153\) 0 0
\(154\) −200052. −0.679739
\(155\) 0 0
\(156\) 0 0
\(157\) −343342. −1.11167 −0.555837 0.831292i \(-0.687602\pi\)
−0.555837 + 0.831292i \(0.687602\pi\)
\(158\) −377322. −1.20246
\(159\) 0 0
\(160\) 0 0
\(161\) −114424. −0.347899
\(162\) 0 0
\(163\) 267463. 0.788487 0.394243 0.919006i \(-0.371007\pi\)
0.394243 + 0.919006i \(0.371007\pi\)
\(164\) 73199.1 0.212518
\(165\) 0 0
\(166\) −208537. −0.587372
\(167\) 122968. 0.341193 0.170596 0.985341i \(-0.445431\pi\)
0.170596 + 0.985341i \(0.445431\pi\)
\(168\) 0 0
\(169\) −366616. −0.987404
\(170\) 0 0
\(171\) 0 0
\(172\) −77664.3 −0.200170
\(173\) 288020. 0.731657 0.365829 0.930682i \(-0.380786\pi\)
0.365829 + 0.930682i \(0.380786\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 251662. 0.612400
\(177\) 0 0
\(178\) −617986. −1.46194
\(179\) −246177. −0.574268 −0.287134 0.957890i \(-0.592702\pi\)
−0.287134 + 0.957890i \(0.592702\pi\)
\(180\) 0 0
\(181\) 433120. 0.982678 0.491339 0.870968i \(-0.336508\pi\)
0.491339 + 0.870968i \(0.336508\pi\)
\(182\) −47158.7 −0.105532
\(183\) 0 0
\(184\) 166831. 0.363272
\(185\) 0 0
\(186\) 0 0
\(187\) 90119.1 0.188457
\(188\) −107178. −0.221161
\(189\) 0 0
\(190\) 0 0
\(191\) 701011. 1.39040 0.695202 0.718814i \(-0.255315\pi\)
0.695202 + 0.718814i \(0.255315\pi\)
\(192\) 0 0
\(193\) −215730. −0.416887 −0.208443 0.978034i \(-0.566840\pi\)
−0.208443 + 0.978034i \(0.566840\pi\)
\(194\) 115125. 0.219618
\(195\) 0 0
\(196\) −1580.51 −0.00293871
\(197\) −700484. −1.28598 −0.642988 0.765876i \(-0.722305\pi\)
−0.642988 + 0.765876i \(0.722305\pi\)
\(198\) 0 0
\(199\) 22097.5 0.0395558 0.0197779 0.999804i \(-0.493704\pi\)
0.0197779 + 0.999804i \(0.493704\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −395036. −0.681174
\(203\) −338231. −0.576068
\(204\) 0 0
\(205\) 0 0
\(206\) 252205. 0.414081
\(207\) 0 0
\(208\) 59324.6 0.0950772
\(209\) 618893. 0.980054
\(210\) 0 0
\(211\) 910782. 1.40834 0.704172 0.710030i \(-0.251319\pi\)
0.704172 + 0.710030i \(0.251319\pi\)
\(212\) 32987.4 0.0504090
\(213\) 0 0
\(214\) −484177. −0.722719
\(215\) 0 0
\(216\) 0 0
\(217\) 1.19074e6 1.71660
\(218\) 56028.3 0.0798484
\(219\) 0 0
\(220\) 0 0
\(221\) 21243.9 0.0292587
\(222\) 0 0
\(223\) −132745. −0.178754 −0.0893768 0.995998i \(-0.528488\pi\)
−0.0893768 + 0.995998i \(0.528488\pi\)
\(224\) 203048. 0.270382
\(225\) 0 0
\(226\) −459769. −0.598782
\(227\) −354321. −0.456386 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(228\) 0 0
\(229\) 366643. 0.462013 0.231007 0.972952i \(-0.425798\pi\)
0.231007 + 0.972952i \(0.425798\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 493142. 0.601523
\(233\) 1.02388e6 1.23555 0.617776 0.786355i \(-0.288034\pi\)
0.617776 + 0.786355i \(0.288034\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −39801.2 −0.0465175
\(237\) 0 0
\(238\) −214216. −0.245138
\(239\) −1.19966e6 −1.35852 −0.679258 0.733899i \(-0.737698\pi\)
−0.679258 + 0.733899i \(0.737698\pi\)
\(240\) 0 0
\(241\) −94967.5 −0.105325 −0.0526626 0.998612i \(-0.516771\pi\)
−0.0526626 + 0.998612i \(0.516771\pi\)
\(242\) 404604. 0.444112
\(243\) 0 0
\(244\) −14337.3 −0.0154167
\(245\) 0 0
\(246\) 0 0
\(247\) 145893. 0.152157
\(248\) −1.73611e6 −1.79245
\(249\) 0 0
\(250\) 0 0
\(251\) −418053. −0.418839 −0.209419 0.977826i \(-0.567157\pi\)
−0.209419 + 0.977826i \(0.567157\pi\)
\(252\) 0 0
\(253\) −253303. −0.248794
\(254\) 1.03863e6 1.01012
\(255\) 0 0
\(256\) −415700. −0.396442
\(257\) 2.04586e6 1.93216 0.966079 0.258246i \(-0.0831444\pi\)
0.966079 + 0.258246i \(0.0831444\pi\)
\(258\) 0 0
\(259\) 522967. 0.484423
\(260\) 0 0
\(261\) 0 0
\(262\) −623505. −0.561160
\(263\) 1.64024e6 1.46224 0.731119 0.682250i \(-0.238998\pi\)
0.731119 + 0.682250i \(0.238998\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.47113e6 −1.27481
\(267\) 0 0
\(268\) −139402. −0.118559
\(269\) 720582. 0.607160 0.303580 0.952806i \(-0.401818\pi\)
0.303580 + 0.952806i \(0.401818\pi\)
\(270\) 0 0
\(271\) 1.14186e6 0.944477 0.472238 0.881471i \(-0.343446\pi\)
0.472238 + 0.881471i \(0.343446\pi\)
\(272\) 269479. 0.220853
\(273\) 0 0
\(274\) 1.59215e6 1.28117
\(275\) 0 0
\(276\) 0 0
\(277\) −377028. −0.295239 −0.147620 0.989044i \(-0.547161\pi\)
−0.147620 + 0.989044i \(0.547161\pi\)
\(278\) −827148. −0.641906
\(279\) 0 0
\(280\) 0 0
\(281\) 617249. 0.466331 0.233166 0.972437i \(-0.425092\pi\)
0.233166 + 0.972437i \(0.425092\pi\)
\(282\) 0 0
\(283\) 1.25311e6 0.930087 0.465044 0.885288i \(-0.346039\pi\)
0.465044 + 0.885288i \(0.346039\pi\)
\(284\) −154684. −0.113802
\(285\) 0 0
\(286\) −104396. −0.0754692
\(287\) 2.22544e6 1.59482
\(288\) 0 0
\(289\) −1.32336e6 −0.932036
\(290\) 0 0
\(291\) 0 0
\(292\) −114285. −0.0784390
\(293\) 818972. 0.557314 0.278657 0.960391i \(-0.410111\pi\)
0.278657 + 0.960391i \(0.410111\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −762487. −0.505828
\(297\) 0 0
\(298\) 2.76965e6 1.80669
\(299\) −59711.6 −0.0386261
\(300\) 0 0
\(301\) −2.36119e6 −1.50216
\(302\) −9730.03 −0.00613899
\(303\) 0 0
\(304\) 1.85065e6 1.14852
\(305\) 0 0
\(306\) 0 0
\(307\) 136224. 0.0824915 0.0412458 0.999149i \(-0.486867\pi\)
0.0412458 + 0.999149i \(0.486867\pi\)
\(308\) −163873. −0.0984306
\(309\) 0 0
\(310\) 0 0
\(311\) −2.62886e6 −1.54122 −0.770612 0.637304i \(-0.780050\pi\)
−0.770612 + 0.637304i \(0.780050\pi\)
\(312\) 0 0
\(313\) 218161. 0.125868 0.0629341 0.998018i \(-0.479954\pi\)
0.0629341 + 0.998018i \(0.479954\pi\)
\(314\) 1.80669e6 1.03409
\(315\) 0 0
\(316\) −309083. −0.174123
\(317\) −1.25865e6 −0.703491 −0.351745 0.936096i \(-0.614412\pi\)
−0.351745 + 0.936096i \(0.614412\pi\)
\(318\) 0 0
\(319\) −748750. −0.411965
\(320\) 0 0
\(321\) 0 0
\(322\) 602110. 0.323621
\(323\) 662711. 0.353442
\(324\) 0 0
\(325\) 0 0
\(326\) −1.40741e6 −0.733462
\(327\) 0 0
\(328\) −3.24470e6 −1.66529
\(329\) −3.25847e6 −1.65968
\(330\) 0 0
\(331\) −3.21863e6 −1.61473 −0.807366 0.590051i \(-0.799108\pi\)
−0.807366 + 0.590051i \(0.799108\pi\)
\(332\) −170823. −0.0850553
\(333\) 0 0
\(334\) −647067. −0.317382
\(335\) 0 0
\(336\) 0 0
\(337\) 1.63574e6 0.784585 0.392293 0.919840i \(-0.371682\pi\)
0.392293 + 0.919840i \(0.371682\pi\)
\(338\) 1.92917e6 0.918498
\(339\) 0 0
\(340\) 0 0
\(341\) 2.63597e6 1.22760
\(342\) 0 0
\(343\) 2.15448e6 0.988796
\(344\) 3.44262e6 1.56853
\(345\) 0 0
\(346\) −1.51559e6 −0.680598
\(347\) 1.83815e6 0.819514 0.409757 0.912195i \(-0.365614\pi\)
0.409757 + 0.912195i \(0.365614\pi\)
\(348\) 0 0
\(349\) −2.53806e6 −1.11542 −0.557710 0.830036i \(-0.688320\pi\)
−0.557710 + 0.830036i \(0.688320\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 449491. 0.193359
\(353\) 1.88471e6 0.805023 0.402511 0.915415i \(-0.368137\pi\)
0.402511 + 0.915415i \(0.368137\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −506223. −0.211698
\(357\) 0 0
\(358\) 1.29540e6 0.534192
\(359\) −305057. −0.124924 −0.0624619 0.998047i \(-0.519895\pi\)
−0.0624619 + 0.998047i \(0.519895\pi\)
\(360\) 0 0
\(361\) 2.07507e6 0.838039
\(362\) −2.27911e6 −0.914102
\(363\) 0 0
\(364\) −38630.0 −0.0152817
\(365\) 0 0
\(366\) 0 0
\(367\) −727834. −0.282077 −0.141038 0.990004i \(-0.545044\pi\)
−0.141038 + 0.990004i \(0.545044\pi\)
\(368\) −757441. −0.291561
\(369\) 0 0
\(370\) 0 0
\(371\) 1.00290e6 0.378289
\(372\) 0 0
\(373\) −4.77676e6 −1.77771 −0.888855 0.458188i \(-0.848499\pi\)
−0.888855 + 0.458188i \(0.848499\pi\)
\(374\) −474215. −0.175306
\(375\) 0 0
\(376\) 4.75086e6 1.73302
\(377\) −176504. −0.0639590
\(378\) 0 0
\(379\) −701558. −0.250880 −0.125440 0.992101i \(-0.540034\pi\)
−0.125440 + 0.992101i \(0.540034\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3.68878e6 −1.29337
\(383\) −4.01069e6 −1.39708 −0.698541 0.715570i \(-0.746167\pi\)
−0.698541 + 0.715570i \(0.746167\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.13519e6 0.387794
\(387\) 0 0
\(388\) 94305.0 0.0318021
\(389\) 4.45952e6 1.49422 0.747108 0.664702i \(-0.231442\pi\)
0.747108 + 0.664702i \(0.231442\pi\)
\(390\) 0 0
\(391\) −271237. −0.0897237
\(392\) 70059.1 0.0230276
\(393\) 0 0
\(394\) 3.68601e6 1.19623
\(395\) 0 0
\(396\) 0 0
\(397\) 3.36993e6 1.07311 0.536555 0.843865i \(-0.319725\pi\)
0.536555 + 0.843865i \(0.319725\pi\)
\(398\) −116279. −0.0367953
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00679e6 0.933775 0.466888 0.884317i \(-0.345375\pi\)
0.466888 + 0.884317i \(0.345375\pi\)
\(402\) 0 0
\(403\) 621383. 0.190588
\(404\) −323593. −0.0986384
\(405\) 0 0
\(406\) 1.77980e6 0.535867
\(407\) 1.15770e6 0.346426
\(408\) 0 0
\(409\) −998012. −0.295004 −0.147502 0.989062i \(-0.547123\pi\)
−0.147502 + 0.989062i \(0.547123\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 206593. 0.0599616
\(413\) −1.21006e6 −0.349085
\(414\) 0 0
\(415\) 0 0
\(416\) 105959. 0.0300196
\(417\) 0 0
\(418\) −3.25667e6 −0.911660
\(419\) −5.53743e6 −1.54090 −0.770448 0.637503i \(-0.779967\pi\)
−0.770448 + 0.637503i \(0.779967\pi\)
\(420\) 0 0
\(421\) 1.98635e6 0.546198 0.273099 0.961986i \(-0.411951\pi\)
0.273099 + 0.961986i \(0.411951\pi\)
\(422\) −4.79262e6 −1.31006
\(423\) 0 0
\(424\) −1.46223e6 −0.395004
\(425\) 0 0
\(426\) 0 0
\(427\) −435890. −0.115693
\(428\) −396613. −0.104654
\(429\) 0 0
\(430\) 0 0
\(431\) −116512. −0.0302118 −0.0151059 0.999886i \(-0.504809\pi\)
−0.0151059 + 0.999886i \(0.504809\pi\)
\(432\) 0 0
\(433\) 4.56166e6 1.16924 0.584619 0.811308i \(-0.301244\pi\)
0.584619 + 0.811308i \(0.301244\pi\)
\(434\) −6.26580e6 −1.59681
\(435\) 0 0
\(436\) 45895.6 0.0115626
\(437\) −1.86272e6 −0.466599
\(438\) 0 0
\(439\) 2.92172e6 0.723565 0.361782 0.932263i \(-0.382168\pi\)
0.361782 + 0.932263i \(0.382168\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −111787. −0.0272168
\(443\) −1.59752e6 −0.386756 −0.193378 0.981124i \(-0.561944\pi\)
−0.193378 + 0.981124i \(0.561944\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 698514. 0.166279
\(447\) 0 0
\(448\) −4.70630e6 −1.10786
\(449\) −3.11073e6 −0.728193 −0.364096 0.931361i \(-0.618622\pi\)
−0.364096 + 0.931361i \(0.618622\pi\)
\(450\) 0 0
\(451\) 4.92650e6 1.14051
\(452\) −376620. −0.0867076
\(453\) 0 0
\(454\) 1.86447e6 0.424537
\(455\) 0 0
\(456\) 0 0
\(457\) −6.47145e6 −1.44948 −0.724738 0.689025i \(-0.758039\pi\)
−0.724738 + 0.689025i \(0.758039\pi\)
\(458\) −1.92931e6 −0.429771
\(459\) 0 0
\(460\) 0 0
\(461\) 5.47864e6 1.20066 0.600330 0.799752i \(-0.295036\pi\)
0.600330 + 0.799752i \(0.295036\pi\)
\(462\) 0 0
\(463\) −2.35489e6 −0.510526 −0.255263 0.966872i \(-0.582162\pi\)
−0.255263 + 0.966872i \(0.582162\pi\)
\(464\) −2.23895e6 −0.482781
\(465\) 0 0
\(466\) −5.38776e6 −1.14933
\(467\) −4.56027e6 −0.967606 −0.483803 0.875177i \(-0.660745\pi\)
−0.483803 + 0.875177i \(0.660745\pi\)
\(468\) 0 0
\(469\) −4.23819e6 −0.889710
\(470\) 0 0
\(471\) 0 0
\(472\) 1.76427e6 0.364510
\(473\) −5.22702e6 −1.07424
\(474\) 0 0
\(475\) 0 0
\(476\) −175475. −0.0354975
\(477\) 0 0
\(478\) 6.31274e6 1.26371
\(479\) 1.88004e6 0.374394 0.187197 0.982322i \(-0.440060\pi\)
0.187197 + 0.982322i \(0.440060\pi\)
\(480\) 0 0
\(481\) 272907. 0.0537839
\(482\) 499727. 0.0979751
\(483\) 0 0
\(484\) 331431. 0.0643103
\(485\) 0 0
\(486\) 0 0
\(487\) −1.69396e6 −0.323654 −0.161827 0.986819i \(-0.551739\pi\)
−0.161827 + 0.986819i \(0.551739\pi\)
\(488\) 635528. 0.120805
\(489\) 0 0
\(490\) 0 0
\(491\) −1.48645e6 −0.278258 −0.139129 0.990274i \(-0.544430\pi\)
−0.139129 + 0.990274i \(0.544430\pi\)
\(492\) 0 0
\(493\) −801762. −0.148569
\(494\) −767700. −0.141538
\(495\) 0 0
\(496\) 7.88224e6 1.43862
\(497\) −4.70279e6 −0.854013
\(498\) 0 0
\(499\) 7.09934e6 1.27634 0.638170 0.769896i \(-0.279692\pi\)
0.638170 + 0.769896i \(0.279692\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.19983e6 0.389610
\(503\) 9.24224e6 1.62876 0.814381 0.580331i \(-0.197077\pi\)
0.814381 + 0.580331i \(0.197077\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.33290e6 0.231431
\(507\) 0 0
\(508\) 850790. 0.146273
\(509\) 8.12506e6 1.39006 0.695028 0.718983i \(-0.255392\pi\)
0.695028 + 0.718983i \(0.255392\pi\)
\(510\) 0 0
\(511\) −3.47456e6 −0.588637
\(512\) 6.64807e6 1.12078
\(513\) 0 0
\(514\) −1.07655e7 −1.79732
\(515\) 0 0
\(516\) 0 0
\(517\) −7.21335e6 −1.18689
\(518\) −2.75190e6 −0.450617
\(519\) 0 0
\(520\) 0 0
\(521\) −5.06245e6 −0.817084 −0.408542 0.912740i \(-0.633963\pi\)
−0.408542 + 0.912740i \(0.633963\pi\)
\(522\) 0 0
\(523\) 4.76222e6 0.761299 0.380649 0.924719i \(-0.375700\pi\)
0.380649 + 0.924719i \(0.375700\pi\)
\(524\) −510744. −0.0812596
\(525\) 0 0
\(526\) −8.63109e6 −1.36019
\(527\) 2.82260e6 0.442714
\(528\) 0 0
\(529\) −5.67396e6 −0.881550
\(530\) 0 0
\(531\) 0 0
\(532\) −1.20508e6 −0.184601
\(533\) 1.16133e6 0.177067
\(534\) 0 0
\(535\) 0 0
\(536\) 6.17929e6 0.929023
\(537\) 0 0
\(538\) −3.79177e6 −0.564789
\(539\) −106373. −0.0157709
\(540\) 0 0
\(541\) −2.89920e6 −0.425877 −0.212939 0.977066i \(-0.568303\pi\)
−0.212939 + 0.977066i \(0.568303\pi\)
\(542\) −6.00859e6 −0.878566
\(543\) 0 0
\(544\) 481315. 0.0697320
\(545\) 0 0
\(546\) 0 0
\(547\) 5.74434e6 0.820866 0.410433 0.911891i \(-0.365378\pi\)
0.410433 + 0.911891i \(0.365378\pi\)
\(548\) 1.30421e6 0.185522
\(549\) 0 0
\(550\) 0 0
\(551\) −5.50610e6 −0.772618
\(552\) 0 0
\(553\) −9.39691e6 −1.30669
\(554\) 1.98395e6 0.274636
\(555\) 0 0
\(556\) −677558. −0.0929522
\(557\) −7.29174e6 −0.995848 −0.497924 0.867221i \(-0.665904\pi\)
−0.497924 + 0.867221i \(0.665904\pi\)
\(558\) 0 0
\(559\) −1.23217e6 −0.166779
\(560\) 0 0
\(561\) 0 0
\(562\) −3.24802e6 −0.433788
\(563\) 6.65348e6 0.884663 0.442331 0.896852i \(-0.354151\pi\)
0.442331 + 0.896852i \(0.354151\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.59398e6 −0.865181
\(567\) 0 0
\(568\) 6.85667e6 0.891749
\(569\) −5.78715e6 −0.749349 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(570\) 0 0
\(571\) −1.22059e7 −1.56667 −0.783336 0.621599i \(-0.786483\pi\)
−0.783336 + 0.621599i \(0.786483\pi\)
\(572\) −85516.1 −0.0109284
\(573\) 0 0
\(574\) −1.17105e7 −1.48352
\(575\) 0 0
\(576\) 0 0
\(577\) 1.02981e7 1.28771 0.643853 0.765149i \(-0.277335\pi\)
0.643853 + 0.765149i \(0.277335\pi\)
\(578\) 6.96362e6 0.866993
\(579\) 0 0
\(580\) 0 0
\(581\) −5.19346e6 −0.638288
\(582\) 0 0
\(583\) 2.22014e6 0.270526
\(584\) 5.06591e6 0.614647
\(585\) 0 0
\(586\) −4.30950e6 −0.518422
\(587\) 1.30519e7 1.56343 0.781715 0.623636i \(-0.214345\pi\)
0.781715 + 0.623636i \(0.214345\pi\)
\(588\) 0 0
\(589\) 1.93842e7 2.30229
\(590\) 0 0
\(591\) 0 0
\(592\) 3.46183e6 0.405977
\(593\) −6.43920e6 −0.751961 −0.375980 0.926628i \(-0.622694\pi\)
−0.375980 + 0.926628i \(0.622694\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.26876e6 0.261621
\(597\) 0 0
\(598\) 314208. 0.0359305
\(599\) 1.00760e7 1.14741 0.573707 0.819061i \(-0.305505\pi\)
0.573707 + 0.819061i \(0.305505\pi\)
\(600\) 0 0
\(601\) 1.57050e6 0.177358 0.0886791 0.996060i \(-0.471735\pi\)
0.0886791 + 0.996060i \(0.471735\pi\)
\(602\) 1.24248e7 1.39733
\(603\) 0 0
\(604\) −7970.35 −0.000888966 0
\(605\) 0 0
\(606\) 0 0
\(607\) −7.31039e6 −0.805321 −0.402660 0.915349i \(-0.631914\pi\)
−0.402660 + 0.915349i \(0.631914\pi\)
\(608\) 3.30543e6 0.362634
\(609\) 0 0
\(610\) 0 0
\(611\) −1.70041e6 −0.184269
\(612\) 0 0
\(613\) 1.31997e7 1.41878 0.709389 0.704817i \(-0.248971\pi\)
0.709389 + 0.704817i \(0.248971\pi\)
\(614\) −716825. −0.0767348
\(615\) 0 0
\(616\) 7.26399e6 0.771300
\(617\) −1.02423e7 −1.08314 −0.541570 0.840655i \(-0.682170\pi\)
−0.541570 + 0.840655i \(0.682170\pi\)
\(618\) 0 0
\(619\) 1.05614e7 1.10788 0.553942 0.832555i \(-0.313123\pi\)
0.553942 + 0.832555i \(0.313123\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.38333e7 1.43367
\(623\) −1.53905e7 −1.58866
\(624\) 0 0
\(625\) 0 0
\(626\) −1.14798e6 −0.117084
\(627\) 0 0
\(628\) 1.47995e6 0.149744
\(629\) 1.23967e6 0.124933
\(630\) 0 0
\(631\) 1.90535e7 1.90503 0.952513 0.304497i \(-0.0984883\pi\)
0.952513 + 0.304497i \(0.0984883\pi\)
\(632\) 1.37007e7 1.36443
\(633\) 0 0
\(634\) 6.62315e6 0.654397
\(635\) 0 0
\(636\) 0 0
\(637\) −25075.4 −0.00244849
\(638\) 3.93999e6 0.383216
\(639\) 0 0
\(640\) 0 0
\(641\) −8.56937e6 −0.823766 −0.411883 0.911237i \(-0.635129\pi\)
−0.411883 + 0.911237i \(0.635129\pi\)
\(642\) 0 0
\(643\) −1.79513e7 −1.71226 −0.856130 0.516761i \(-0.827138\pi\)
−0.856130 + 0.516761i \(0.827138\pi\)
\(644\) 493218. 0.0468624
\(645\) 0 0
\(646\) −3.48724e6 −0.328777
\(647\) 1.05470e7 0.990534 0.495267 0.868741i \(-0.335070\pi\)
0.495267 + 0.868741i \(0.335070\pi\)
\(648\) 0 0
\(649\) −2.67873e6 −0.249642
\(650\) 0 0
\(651\) 0 0
\(652\) −1.15288e6 −0.106210
\(653\) 1.00324e7 0.920712 0.460356 0.887734i \(-0.347722\pi\)
0.460356 + 0.887734i \(0.347722\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.47315e7 1.33656
\(657\) 0 0
\(658\) 1.71464e7 1.54386
\(659\) 8.99161e6 0.806536 0.403268 0.915082i \(-0.367874\pi\)
0.403268 + 0.915082i \(0.367874\pi\)
\(660\) 0 0
\(661\) 2.39297e6 0.213027 0.106513 0.994311i \(-0.466031\pi\)
0.106513 + 0.994311i \(0.466031\pi\)
\(662\) 1.69367e7 1.50205
\(663\) 0 0
\(664\) 7.57208e6 0.666492
\(665\) 0 0
\(666\) 0 0
\(667\) 2.25356e6 0.196135
\(668\) −530044. −0.0459590
\(669\) 0 0
\(670\) 0 0
\(671\) −964938. −0.0827357
\(672\) 0 0
\(673\) −1.53612e7 −1.30733 −0.653666 0.756783i \(-0.726770\pi\)
−0.653666 + 0.756783i \(0.726770\pi\)
\(674\) −8.60742e6 −0.729833
\(675\) 0 0
\(676\) 1.58028e6 0.133004
\(677\) 1.16026e7 0.972934 0.486467 0.873699i \(-0.338285\pi\)
0.486467 + 0.873699i \(0.338285\pi\)
\(678\) 0 0
\(679\) 2.86711e6 0.238655
\(680\) 0 0
\(681\) 0 0
\(682\) −1.38707e7 −1.14193
\(683\) −1.20315e7 −0.986890 −0.493445 0.869777i \(-0.664263\pi\)
−0.493445 + 0.869777i \(0.664263\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.13371e7 −0.919792
\(687\) 0 0
\(688\) −1.56301e7 −1.25890
\(689\) 523358. 0.0420001
\(690\) 0 0
\(691\) 5.18616e6 0.413191 0.206595 0.978426i \(-0.433762\pi\)
0.206595 + 0.978426i \(0.433762\pi\)
\(692\) −1.24149e6 −0.0985550
\(693\) 0 0
\(694\) −9.67248e6 −0.762323
\(695\) 0 0
\(696\) 0 0
\(697\) 5.27530e6 0.411306
\(698\) 1.33555e7 1.03758
\(699\) 0 0
\(700\) 0 0
\(701\) −6.00859e6 −0.461825 −0.230913 0.972974i \(-0.574171\pi\)
−0.230913 + 0.972974i \(0.574171\pi\)
\(702\) 0 0
\(703\) 8.51342e6 0.649704
\(704\) −1.04184e7 −0.792265
\(705\) 0 0
\(706\) −9.91752e6 −0.748844
\(707\) −9.83807e6 −0.740221
\(708\) 0 0
\(709\) 5.90083e6 0.440857 0.220429 0.975403i \(-0.429254\pi\)
0.220429 + 0.975403i \(0.429254\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.24393e7 1.65886
\(713\) −7.93365e6 −0.584453
\(714\) 0 0
\(715\) 0 0
\(716\) 1.06113e6 0.0773545
\(717\) 0 0
\(718\) 1.60524e6 0.116206
\(719\) 1.36592e7 0.985382 0.492691 0.870204i \(-0.336013\pi\)
0.492691 + 0.870204i \(0.336013\pi\)
\(720\) 0 0
\(721\) 6.28097e6 0.449975
\(722\) −1.09192e7 −0.779556
\(723\) 0 0
\(724\) −1.86693e6 −0.132368
\(725\) 0 0
\(726\) 0 0
\(727\) 1.11594e7 0.783079 0.391539 0.920161i \(-0.371943\pi\)
0.391539 + 0.920161i \(0.371943\pi\)
\(728\) 1.71235e6 0.119747
\(729\) 0 0
\(730\) 0 0
\(731\) −5.59710e6 −0.387409
\(732\) 0 0
\(733\) 1.52510e7 1.04843 0.524215 0.851586i \(-0.324359\pi\)
0.524215 + 0.851586i \(0.324359\pi\)
\(734\) 3.82993e6 0.262392
\(735\) 0 0
\(736\) −1.35286e6 −0.0920573
\(737\) −9.38217e6 −0.636260
\(738\) 0 0
\(739\) −1.11820e7 −0.753196 −0.376598 0.926377i \(-0.622906\pi\)
−0.376598 + 0.926377i \(0.622906\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −5.27735e6 −0.351890
\(743\) 7.71450e6 0.512667 0.256334 0.966588i \(-0.417485\pi\)
0.256334 + 0.966588i \(0.417485\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.51357e7 1.65365
\(747\) 0 0
\(748\) −388453. −0.0253854
\(749\) −1.20581e7 −0.785367
\(750\) 0 0
\(751\) −2.23973e7 −1.44909 −0.724545 0.689228i \(-0.757950\pi\)
−0.724545 + 0.689228i \(0.757950\pi\)
\(752\) −2.15698e7 −1.39091
\(753\) 0 0
\(754\) 928780. 0.0594955
\(755\) 0 0
\(756\) 0 0
\(757\) −2.57267e7 −1.63171 −0.815857 0.578254i \(-0.803734\pi\)
−0.815857 + 0.578254i \(0.803734\pi\)
\(758\) 3.69166e6 0.233372
\(759\) 0 0
\(760\) 0 0
\(761\) 1.48340e7 0.928533 0.464267 0.885696i \(-0.346318\pi\)
0.464267 + 0.885696i \(0.346318\pi\)
\(762\) 0 0
\(763\) 1.39534e6 0.0867700
\(764\) −3.02166e6 −0.187289
\(765\) 0 0
\(766\) 2.11046e7 1.29959
\(767\) −631463. −0.0387578
\(768\) 0 0
\(769\) −5.57112e6 −0.339724 −0.169862 0.985468i \(-0.554332\pi\)
−0.169862 + 0.985468i \(0.554332\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 929892. 0.0561551
\(773\) −1.58230e7 −0.952447 −0.476224 0.879324i \(-0.657995\pi\)
−0.476224 + 0.879324i \(0.657995\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.18026e6 −0.249200
\(777\) 0 0
\(778\) −2.34664e7 −1.38994
\(779\) 3.62281e7 2.13896
\(780\) 0 0
\(781\) −1.04107e7 −0.610732
\(782\) 1.42727e6 0.0834623
\(783\) 0 0
\(784\) −318081. −0.0184819
\(785\) 0 0
\(786\) 0 0
\(787\) 1.31529e7 0.756981 0.378491 0.925605i \(-0.376443\pi\)
0.378491 + 0.925605i \(0.376443\pi\)
\(788\) 3.01939e6 0.173222
\(789\) 0 0
\(790\) 0 0
\(791\) −1.14502e7 −0.650687
\(792\) 0 0
\(793\) −227466. −0.0128450
\(794\) −1.77328e7 −0.998222
\(795\) 0 0
\(796\) −95249.7 −0.00532821
\(797\) −2.58443e7 −1.44118 −0.720590 0.693361i \(-0.756129\pi\)
−0.720590 + 0.693361i \(0.756129\pi\)
\(798\) 0 0
\(799\) −7.72406e6 −0.428034
\(800\) 0 0
\(801\) 0 0
\(802\) −1.58220e7 −0.868611
\(803\) −7.69170e6 −0.420953
\(804\) 0 0
\(805\) 0 0
\(806\) −3.26977e6 −0.177288
\(807\) 0 0
\(808\) 1.43439e7 0.772929
\(809\) −1.78857e7 −0.960804 −0.480402 0.877048i \(-0.659509\pi\)
−0.480402 + 0.877048i \(0.659509\pi\)
\(810\) 0 0
\(811\) −1.41608e7 −0.756026 −0.378013 0.925800i \(-0.623393\pi\)
−0.378013 + 0.925800i \(0.623393\pi\)
\(812\) 1.45793e6 0.0775970
\(813\) 0 0
\(814\) −6.09193e6 −0.322251
\(815\) 0 0
\(816\) 0 0
\(817\) −3.84380e7 −2.01468
\(818\) 5.25162e6 0.274417
\(819\) 0 0
\(820\) 0 0
\(821\) 3.46248e7 1.79279 0.896394 0.443258i \(-0.146177\pi\)
0.896394 + 0.443258i \(0.146177\pi\)
\(822\) 0 0
\(823\) −2.13360e7 −1.09803 −0.549015 0.835813i \(-0.684997\pi\)
−0.549015 + 0.835813i \(0.684997\pi\)
\(824\) −9.15766e6 −0.469858
\(825\) 0 0
\(826\) 6.36744e6 0.324724
\(827\) −1.59813e6 −0.0812548 −0.0406274 0.999174i \(-0.512936\pi\)
−0.0406274 + 0.999174i \(0.512936\pi\)
\(828\) 0 0
\(829\) −2.53923e7 −1.28327 −0.641633 0.767012i \(-0.721743\pi\)
−0.641633 + 0.767012i \(0.721743\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.45595e6 −0.123002
\(833\) −113904. −0.00568755
\(834\) 0 0
\(835\) 0 0
\(836\) −2.66770e6 −0.132014
\(837\) 0 0
\(838\) 2.91384e7 1.43336
\(839\) −1.98528e7 −0.973681 −0.486841 0.873491i \(-0.661851\pi\)
−0.486841 + 0.873491i \(0.661851\pi\)
\(840\) 0 0
\(841\) −1.38498e7 −0.675231
\(842\) −1.04523e7 −0.508081
\(843\) 0 0
\(844\) −3.92587e6 −0.189705
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00764e7 0.482609
\(848\) 6.63879e6 0.317029
\(849\) 0 0
\(850\) 0 0
\(851\) −3.48441e6 −0.164932
\(852\) 0 0
\(853\) −1.59794e7 −0.751948 −0.375974 0.926630i \(-0.622692\pi\)
−0.375974 + 0.926630i \(0.622692\pi\)
\(854\) 2.29369e6 0.107619
\(855\) 0 0
\(856\) 1.75807e7 0.820070
\(857\) −7.00157e6 −0.325644 −0.162822 0.986655i \(-0.552060\pi\)
−0.162822 + 0.986655i \(0.552060\pi\)
\(858\) 0 0
\(859\) −7.28414e6 −0.336818 −0.168409 0.985717i \(-0.553863\pi\)
−0.168409 + 0.985717i \(0.553863\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 613094. 0.0281034
\(863\) 1.76361e7 0.806075 0.403037 0.915184i \(-0.367954\pi\)
0.403037 + 0.915184i \(0.367954\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.40038e7 −1.08764
\(867\) 0 0
\(868\) −5.13263e6 −0.231228
\(869\) −2.08021e7 −0.934455
\(870\) 0 0
\(871\) −2.21167e6 −0.0987816
\(872\) −2.03441e6 −0.0906041
\(873\) 0 0
\(874\) 9.80180e6 0.434037
\(875\) 0 0
\(876\) 0 0
\(877\) −2.69004e7 −1.18102 −0.590512 0.807029i \(-0.701074\pi\)
−0.590512 + 0.807029i \(0.701074\pi\)
\(878\) −1.53743e7 −0.673070
\(879\) 0 0
\(880\) 0 0
\(881\) −2.51911e7 −1.09347 −0.546735 0.837306i \(-0.684130\pi\)
−0.546735 + 0.837306i \(0.684130\pi\)
\(882\) 0 0
\(883\) 3.22126e7 1.39035 0.695175 0.718840i \(-0.255327\pi\)
0.695175 + 0.718840i \(0.255327\pi\)
\(884\) −91570.7 −0.00394117
\(885\) 0 0
\(886\) 8.40629e6 0.359766
\(887\) 8.96139e6 0.382443 0.191221 0.981547i \(-0.438755\pi\)
0.191221 + 0.981547i \(0.438755\pi\)
\(888\) 0 0
\(889\) 2.58662e7 1.09769
\(890\) 0 0
\(891\) 0 0
\(892\) 572187. 0.0240783
\(893\) −5.30449e7 −2.22595
\(894\) 0 0
\(895\) 0 0
\(896\) 1.82674e7 0.760164
\(897\) 0 0
\(898\) 1.63689e7 0.677376
\(899\) −2.34514e7 −0.967765
\(900\) 0 0
\(901\) 2.37733e6 0.0975613
\(902\) −2.59237e7 −1.06092
\(903\) 0 0
\(904\) 1.66944e7 0.679439
\(905\) 0 0
\(906\) 0 0
\(907\) 5.81689e6 0.234786 0.117393 0.993086i \(-0.462546\pi\)
0.117393 + 0.993086i \(0.462546\pi\)
\(908\) 1.52728e6 0.0614757
\(909\) 0 0
\(910\) 0 0
\(911\) 1.96435e7 0.784192 0.392096 0.919924i \(-0.371750\pi\)
0.392096 + 0.919924i \(0.371750\pi\)
\(912\) 0 0
\(913\) −1.14969e7 −0.456460
\(914\) 3.40533e7 1.34832
\(915\) 0 0
\(916\) −1.58039e6 −0.0622337
\(917\) −1.55279e7 −0.609803
\(918\) 0 0
\(919\) −89962.4 −0.00351376 −0.00175688 0.999998i \(-0.500559\pi\)
−0.00175688 + 0.999998i \(0.500559\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.88291e7 −1.11687
\(923\) −2.45412e6 −0.0948183
\(924\) 0 0
\(925\) 0 0
\(926\) 1.23916e7 0.474899
\(927\) 0 0
\(928\) −3.99898e6 −0.152433
\(929\) −3.65192e7 −1.38830 −0.694149 0.719832i \(-0.744219\pi\)
−0.694149 + 0.719832i \(0.744219\pi\)
\(930\) 0 0
\(931\) −782234. −0.0295776
\(932\) −4.41338e6 −0.166430
\(933\) 0 0
\(934\) 2.39966e7 0.900081
\(935\) 0 0
\(936\) 0 0
\(937\) 3.58659e7 1.33454 0.667272 0.744814i \(-0.267462\pi\)
0.667272 + 0.744814i \(0.267462\pi\)
\(938\) 2.23017e7 0.827621
\(939\) 0 0
\(940\) 0 0
\(941\) 3.19693e7 1.17695 0.588476 0.808515i \(-0.299728\pi\)
0.588476 + 0.808515i \(0.299728\pi\)
\(942\) 0 0
\(943\) −1.48276e7 −0.542990
\(944\) −8.01010e6 −0.292555
\(945\) 0 0
\(946\) 2.75050e7 0.999273
\(947\) 4.71846e7 1.70972 0.854861 0.518858i \(-0.173643\pi\)
0.854861 + 0.518858i \(0.173643\pi\)
\(948\) 0 0
\(949\) −1.81318e6 −0.0653544
\(950\) 0 0
\(951\) 0 0
\(952\) 7.77829e6 0.278158
\(953\) 1.65226e6 0.0589315 0.0294657 0.999566i \(-0.490619\pi\)
0.0294657 + 0.999566i \(0.490619\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 5.17108e6 0.182994
\(957\) 0 0
\(958\) −9.89295e6 −0.348267
\(959\) 3.96512e7 1.39223
\(960\) 0 0
\(961\) 5.39316e7 1.88380
\(962\) −1.43606e6 −0.0500306
\(963\) 0 0
\(964\) 409351. 0.0141874
\(965\) 0 0
\(966\) 0 0
\(967\) 3.23040e7 1.11094 0.555470 0.831537i \(-0.312538\pi\)
0.555470 + 0.831537i \(0.312538\pi\)
\(968\) −1.46914e7 −0.503934
\(969\) 0 0
\(970\) 0 0
\(971\) −1.15927e7 −0.394582 −0.197291 0.980345i \(-0.563214\pi\)
−0.197291 + 0.980345i \(0.563214\pi\)
\(972\) 0 0
\(973\) −2.05995e7 −0.697549
\(974\) 8.91377e6 0.301068
\(975\) 0 0
\(976\) −2.88541e6 −0.0969579
\(977\) 2.58947e7 0.867909 0.433954 0.900935i \(-0.357118\pi\)
0.433954 + 0.900935i \(0.357118\pi\)
\(978\) 0 0
\(979\) −3.40702e7 −1.13610
\(980\) 0 0
\(981\) 0 0
\(982\) 7.82184e6 0.258839
\(983\) 3.46040e7 1.14220 0.571101 0.820880i \(-0.306517\pi\)
0.571101 + 0.820880i \(0.306517\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.21894e6 0.138201
\(987\) 0 0
\(988\) −628861. −0.0204957
\(989\) 1.57321e7 0.511441
\(990\) 0 0
\(991\) −3.71464e7 −1.20152 −0.600762 0.799428i \(-0.705136\pi\)
−0.600762 + 0.799428i \(0.705136\pi\)
\(992\) 1.40784e7 0.454228
\(993\) 0 0
\(994\) 2.47465e7 0.794415
\(995\) 0 0
\(996\) 0 0
\(997\) 2.47350e7 0.788086 0.394043 0.919092i \(-0.371076\pi\)
0.394043 + 0.919092i \(0.371076\pi\)
\(998\) −3.73573e7 −1.18727
\(999\) 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 225.6.a.s.1.1 2
3.2 odd 2 25.6.a.b.1.2 2
5.2 odd 4 225.6.b.i.199.2 4
5.3 odd 4 225.6.b.i.199.3 4
5.4 even 2 225.6.a.l.1.2 2
12.11 even 2 400.6.a.w.1.2 2
15.2 even 4 25.6.b.b.24.3 4
15.8 even 4 25.6.b.b.24.2 4
15.14 odd 2 25.6.a.d.1.1 yes 2
60.23 odd 4 400.6.c.n.49.4 4
60.47 odd 4 400.6.c.n.49.1 4
60.59 even 2 400.6.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.a.b.1.2 2 3.2 odd 2
25.6.a.d.1.1 yes 2 15.14 odd 2
25.6.b.b.24.2 4 15.8 even 4
25.6.b.b.24.3 4 15.2 even 4
225.6.a.l.1.2 2 5.4 even 2
225.6.a.s.1.1 2 1.1 even 1 trivial
225.6.b.i.199.2 4 5.2 odd 4
225.6.b.i.199.3 4 5.3 odd 4
400.6.a.o.1.1 2 60.59 even 2
400.6.a.w.1.2 2 12.11 even 2
400.6.c.n.49.1 4 60.47 odd 4
400.6.c.n.49.4 4 60.23 odd 4