Properties

Label 25.10.b.c.24.5
Level $25$
Weight $10$
Character 25.24
Analytic conductor $12.876$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,10,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.8758959041\)
Analytic rank: \(0\)
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: \( x^{6} + 1305x^{4} + 433104x^{2} + 16000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.5
Root \(-22.2334i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.10.b.c.24.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+21.4187i q^{2} +210.171i q^{3} +53.2406 q^{4} -4501.59 q^{6} +9905.49i q^{7} +12106.7i q^{8} -24489.0 q^{9} +O(q^{10})\) \(q+21.4187i q^{2} +210.171i q^{3} +53.2406 q^{4} -4501.59 q^{6} +9905.49i q^{7} +12106.7i q^{8} -24489.0 q^{9} +36453.6 q^{11} +11189.6i q^{12} -164867. i q^{13} -212162. q^{14} -232050. q^{16} +82357.1i q^{17} -524521. i q^{18} +609617. q^{19} -2.08185e6 q^{21} +780787. i q^{22} -1.88578e6i q^{23} -2.54448e6 q^{24} +3.53123e6 q^{26} -1.01008e6i q^{27} +527374. i q^{28} -339235. q^{29} +547314. q^{31} +1.22842e6i q^{32} +7.66150e6i q^{33} -1.76398e6 q^{34} -1.30381e6 q^{36} +5.25687e6i q^{37} +1.30572e7i q^{38} +3.46503e7 q^{39} +2.05812e6 q^{41} -4.45904e7i q^{42} +6.76158e6i q^{43} +1.94081e6 q^{44} +4.03909e7 q^{46} +3.15241e7i q^{47} -4.87703e7i q^{48} -5.77651e7 q^{49} -1.73091e7 q^{51} -8.77761e6i q^{52} +4.89593e7i q^{53} +2.16345e7 q^{54} -1.19923e8 q^{56} +1.28124e8i q^{57} -7.26597e6i q^{58} -8.77960e7 q^{59} +3.84654e7 q^{61} +1.17227e7i q^{62} -2.42575e8i q^{63} -1.45121e8 q^{64} -1.64099e8 q^{66} -1.36116e8i q^{67} +4.38474e6i q^{68} +3.96337e8 q^{69} +3.49218e8 q^{71} -2.96481e8i q^{72} -1.61345e8i q^{73} -1.12595e8 q^{74} +3.24564e7 q^{76} +3.61091e8i q^{77} +7.42163e8i q^{78} +1.26975e8 q^{79} -2.69727e8 q^{81} +4.40822e7i q^{82} +2.87494e8i q^{83} -1.10839e8 q^{84} -1.44824e8 q^{86} -7.12976e7i q^{87} +4.41333e8i q^{88} +5.63133e8 q^{89} +1.63309e9 q^{91} -1.00400e8i q^{92} +1.15030e8i q^{93} -6.75205e8 q^{94} -2.58179e8 q^{96} -4.71704e8i q^{97} -1.23725e9i q^{98} -8.92711e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 682 q^{4} + 6842 q^{6} - 116468 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 682 q^{4} + 6842 q^{6} - 116468 q^{9} - 109398 q^{11} - 545844 q^{14} - 1398494 q^{16} - 1637690 q^{19} - 4750788 q^{21} - 5611470 q^{24} - 2560008 q^{26} - 4350960 q^{29} + 8548132 q^{31} + 13677926 q^{34} + 53591596 q^{36} + 100828184 q^{39} + 11852622 q^{41} + 43991406 q^{44} + 28446492 q^{46} + 12907858 q^{49} - 51269398 q^{51} - 564500990 q^{54} - 339076260 q^{56} - 11341920 q^{59} + 250613852 q^{61} - 335084802 q^{64} - 1113831686 q^{66} + 628549884 q^{69} + 595101192 q^{71} + 503014716 q^{74} + 178829730 q^{76} + 620050340 q^{79} + 2797694726 q^{81} - 76534164 q^{84} + 67894392 q^{86} + 2207720070 q^{89} + 2366375312 q^{91} - 3455782264 q^{94} - 5134360898 q^{96} + 1784469044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.4187i 0.946580i 0.880907 + 0.473290i \(0.156934\pi\)
−0.880907 + 0.473290i \(0.843066\pi\)
\(3\) 210.171i 1.49805i 0.662539 + 0.749027i \(0.269479\pi\)
−0.662539 + 0.749027i \(0.730521\pi\)
\(4\) 53.2406 0.103986
\(5\) 0 0
\(6\) −4501.59 −1.41803
\(7\) 9905.49i 1.55932i 0.626204 + 0.779659i \(0.284608\pi\)
−0.626204 + 0.779659i \(0.715392\pi\)
\(8\) 12106.7i 1.04501i
\(9\) −24489.0 −1.24417
\(10\) 0 0
\(11\) 36453.6 0.750712 0.375356 0.926881i \(-0.377520\pi\)
0.375356 + 0.926881i \(0.377520\pi\)
\(12\) 11189.6i 0.155776i
\(13\) − 164867.i − 1.60099i −0.599341 0.800494i \(-0.704571\pi\)
0.599341 0.800494i \(-0.295429\pi\)
\(14\) −212162. −1.47602
\(15\) 0 0
\(16\) −232050. −0.885201
\(17\) 82357.1i 0.239156i 0.992825 + 0.119578i \(0.0381541\pi\)
−0.992825 + 0.119578i \(0.961846\pi\)
\(18\) − 524521.i − 1.17771i
\(19\) 609617. 1.07316 0.536582 0.843848i \(-0.319715\pi\)
0.536582 + 0.843848i \(0.319715\pi\)
\(20\) 0 0
\(21\) −2.08185e6 −2.33594
\(22\) 780787.i 0.710609i
\(23\) − 1.88578e6i − 1.40513i −0.711620 0.702564i \(-0.752038\pi\)
0.711620 0.702564i \(-0.247962\pi\)
\(24\) −2.54448e6 −1.56548
\(25\) 0 0
\(26\) 3.53123e6 1.51546
\(27\) − 1.01008e6i − 0.365778i
\(28\) 527374.i 0.162147i
\(29\) −339235. −0.0890657 −0.0445328 0.999008i \(-0.514180\pi\)
−0.0445328 + 0.999008i \(0.514180\pi\)
\(30\) 0 0
\(31\) 547314. 0.106441 0.0532205 0.998583i \(-0.483051\pi\)
0.0532205 + 0.998583i \(0.483051\pi\)
\(32\) 1.22842e6i 0.207097i
\(33\) 7.66150e6i 1.12461i
\(34\) −1.76398e6 −0.226380
\(35\) 0 0
\(36\) −1.30381e6 −0.129376
\(37\) 5.25687e6i 0.461126i 0.973057 + 0.230563i \(0.0740567\pi\)
−0.973057 + 0.230563i \(0.925943\pi\)
\(38\) 1.30572e7i 1.01584i
\(39\) 3.46503e7 2.39837
\(40\) 0 0
\(41\) 2.05812e6 0.113748 0.0568739 0.998381i \(-0.481887\pi\)
0.0568739 + 0.998381i \(0.481887\pi\)
\(42\) − 4.45904e7i − 2.21116i
\(43\) 6.76158e6i 0.301606i 0.988564 + 0.150803i \(0.0481859\pi\)
−0.988564 + 0.150803i \(0.951814\pi\)
\(44\) 1.94081e6 0.0780632
\(45\) 0 0
\(46\) 4.03909e7 1.33007
\(47\) 3.15241e7i 0.942330i 0.882045 + 0.471165i \(0.156166\pi\)
−0.882045 + 0.471165i \(0.843834\pi\)
\(48\) − 4.87703e7i − 1.32608i
\(49\) −5.77651e7 −1.43147
\(50\) 0 0
\(51\) −1.73091e7 −0.358268
\(52\) − 8.77761e6i − 0.166480i
\(53\) 4.89593e7i 0.852303i 0.904652 + 0.426151i \(0.140131\pi\)
−0.904652 + 0.426151i \(0.859869\pi\)
\(54\) 2.16345e7 0.346238
\(55\) 0 0
\(56\) −1.19923e8 −1.62950
\(57\) 1.28124e8i 1.60766i
\(58\) − 7.26597e6i − 0.0843078i
\(59\) −8.77960e7 −0.943281 −0.471640 0.881791i \(-0.656338\pi\)
−0.471640 + 0.881791i \(0.656338\pi\)
\(60\) 0 0
\(61\) 3.84654e7 0.355701 0.177851 0.984057i \(-0.443086\pi\)
0.177851 + 0.984057i \(0.443086\pi\)
\(62\) 1.17227e7i 0.100755i
\(63\) − 2.42575e8i − 1.94005i
\(64\) −1.45121e8 −1.08124
\(65\) 0 0
\(66\) −1.64099e8 −1.06453
\(67\) − 1.36116e8i − 0.825227i −0.910906 0.412613i \(-0.864616\pi\)
0.910906 0.412613i \(-0.135384\pi\)
\(68\) 4.38474e6i 0.0248687i
\(69\) 3.96337e8 2.10496
\(70\) 0 0
\(71\) 3.49218e8 1.63092 0.815462 0.578810i \(-0.196483\pi\)
0.815462 + 0.578810i \(0.196483\pi\)
\(72\) − 2.96481e8i − 1.30017i
\(73\) − 1.61345e8i − 0.664971i −0.943108 0.332486i \(-0.892113\pi\)
0.943108 0.332486i \(-0.107887\pi\)
\(74\) −1.12595e8 −0.436492
\(75\) 0 0
\(76\) 3.24564e7 0.111594
\(77\) 3.61091e8i 1.17060i
\(78\) 7.42163e8i 2.27025i
\(79\) 1.26975e8 0.366772 0.183386 0.983041i \(-0.441294\pi\)
0.183386 + 0.983041i \(0.441294\pi\)
\(80\) 0 0
\(81\) −2.69727e8 −0.696213
\(82\) 4.40822e7i 0.107671i
\(83\) 2.87494e8i 0.664931i 0.943115 + 0.332466i \(0.107881\pi\)
−0.943115 + 0.332466i \(0.892119\pi\)
\(84\) −1.10839e8 −0.242904
\(85\) 0 0
\(86\) −1.44824e8 −0.285494
\(87\) − 7.12976e7i − 0.133425i
\(88\) 4.41333e8i 0.784502i
\(89\) 5.63133e8 0.951385 0.475692 0.879612i \(-0.342198\pi\)
0.475692 + 0.879612i \(0.342198\pi\)
\(90\) 0 0
\(91\) 1.63309e9 2.49645
\(92\) − 1.00400e8i − 0.146113i
\(93\) 1.15030e8i 0.159454i
\(94\) −6.75205e8 −0.891991
\(95\) 0 0
\(96\) −2.58179e8 −0.310242
\(97\) − 4.71704e8i − 0.541000i −0.962720 0.270500i \(-0.912811\pi\)
0.962720 0.270500i \(-0.0871890\pi\)
\(98\) − 1.23725e9i − 1.35500i
\(99\) −8.92711e8 −0.934012
\(100\) 0 0
\(101\) −1.25673e9 −1.20170 −0.600851 0.799361i \(-0.705172\pi\)
−0.600851 + 0.799361i \(0.705172\pi\)
\(102\) − 3.70738e8i − 0.339130i
\(103\) − 1.95398e9i − 1.71062i −0.518119 0.855309i \(-0.673367\pi\)
0.518119 0.855309i \(-0.326633\pi\)
\(104\) 1.99599e9 1.67305
\(105\) 0 0
\(106\) −1.04864e9 −0.806773
\(107\) 2.12306e9i 1.56580i 0.622148 + 0.782899i \(0.286260\pi\)
−0.622148 + 0.782899i \(0.713740\pi\)
\(108\) − 5.37771e7i − 0.0380356i
\(109\) 1.18863e8 0.0806544 0.0403272 0.999187i \(-0.487160\pi\)
0.0403272 + 0.999187i \(0.487160\pi\)
\(110\) 0 0
\(111\) −1.10484e9 −0.690791
\(112\) − 2.29857e9i − 1.38031i
\(113\) 1.85455e9i 1.07001i 0.844850 + 0.535003i \(0.179689\pi\)
−0.844850 + 0.535003i \(0.820311\pi\)
\(114\) −2.74425e9 −1.52178
\(115\) 0 0
\(116\) −1.80611e7 −0.00926154
\(117\) 4.03742e9i 1.99190i
\(118\) − 1.88047e9i − 0.892891i
\(119\) −8.15787e8 −0.372920
\(120\) 0 0
\(121\) −1.02908e9 −0.436432
\(122\) 8.23877e8i 0.336700i
\(123\) 4.32557e8i 0.170400i
\(124\) 2.91393e7 0.0110683
\(125\) 0 0
\(126\) 5.19564e9 1.83642
\(127\) − 5.39208e9i − 1.83924i −0.392805 0.919622i \(-0.628495\pi\)
0.392805 0.919622i \(-0.371505\pi\)
\(128\) − 2.47934e9i − 0.816379i
\(129\) −1.42109e9 −0.451823
\(130\) 0 0
\(131\) 3.83690e9 1.13831 0.569154 0.822231i \(-0.307271\pi\)
0.569154 + 0.822231i \(0.307271\pi\)
\(132\) 4.07903e8i 0.116943i
\(133\) 6.03856e9i 1.67340i
\(134\) 2.91543e9 0.781143
\(135\) 0 0
\(136\) −9.97072e8 −0.249920
\(137\) − 2.55408e9i − 0.619430i −0.950829 0.309715i \(-0.899766\pi\)
0.950829 0.309715i \(-0.100234\pi\)
\(138\) 8.48901e9i 1.99251i
\(139\) 4.09121e9 0.929577 0.464789 0.885422i \(-0.346130\pi\)
0.464789 + 0.885422i \(0.346130\pi\)
\(140\) 0 0
\(141\) −6.62547e9 −1.41166
\(142\) 7.47978e9i 1.54380i
\(143\) − 6.00999e9i − 1.20188i
\(144\) 5.68267e9 1.10134
\(145\) 0 0
\(146\) 3.45580e9 0.629449
\(147\) − 1.21406e10i − 2.14443i
\(148\) 2.79879e8i 0.0479504i
\(149\) 6.13643e9 1.01995 0.509974 0.860190i \(-0.329655\pi\)
0.509974 + 0.860190i \(0.329655\pi\)
\(150\) 0 0
\(151\) −5.83754e9 −0.913763 −0.456882 0.889528i \(-0.651034\pi\)
−0.456882 + 0.889528i \(0.651034\pi\)
\(152\) 7.38046e9i 1.12147i
\(153\) − 2.01684e9i − 0.297550i
\(154\) −7.73408e9 −1.10807
\(155\) 0 0
\(156\) 1.84480e9 0.249396
\(157\) − 1.35031e9i − 0.177371i −0.996060 0.0886857i \(-0.971733\pi\)
0.996060 0.0886857i \(-0.0282667\pi\)
\(158\) 2.71963e9i 0.347179i
\(159\) −1.02898e10 −1.27680
\(160\) 0 0
\(161\) 1.86796e10 2.19104
\(162\) − 5.77720e9i − 0.659022i
\(163\) 4.50806e9i 0.500202i 0.968220 + 0.250101i \(0.0804638\pi\)
−0.968220 + 0.250101i \(0.919536\pi\)
\(164\) 1.09575e8 0.0118281
\(165\) 0 0
\(166\) −6.15773e9 −0.629411
\(167\) 1.81836e10i 1.80908i 0.426393 + 0.904538i \(0.359784\pi\)
−0.426393 + 0.904538i \(0.640216\pi\)
\(168\) − 2.52043e10i − 2.44109i
\(169\) −1.65766e10 −1.56316
\(170\) 0 0
\(171\) −1.49289e10 −1.33520
\(172\) 3.59991e8i 0.0313627i
\(173\) − 9.77143e9i − 0.829375i −0.909964 0.414687i \(-0.863891\pi\)
0.909964 0.414687i \(-0.136109\pi\)
\(174\) 1.52710e9 0.126298
\(175\) 0 0
\(176\) −8.45906e9 −0.664531
\(177\) − 1.84522e10i − 1.41309i
\(178\) 1.20616e10i 0.900562i
\(179\) −6.65866e9 −0.484784 −0.242392 0.970178i \(-0.577932\pi\)
−0.242392 + 0.970178i \(0.577932\pi\)
\(180\) 0 0
\(181\) −7.32873e9 −0.507546 −0.253773 0.967264i \(-0.581672\pi\)
−0.253773 + 0.967264i \(0.581672\pi\)
\(182\) 3.49785e10i 2.36309i
\(183\) 8.08432e9i 0.532860i
\(184\) 2.28306e10 1.46837
\(185\) 0 0
\(186\) −2.46378e9 −0.150936
\(187\) 3.00221e9i 0.179537i
\(188\) 1.67836e9i 0.0979887i
\(189\) 1.00053e10 0.570364
\(190\) 0 0
\(191\) 8.06956e9 0.438732 0.219366 0.975643i \(-0.429601\pi\)
0.219366 + 0.975643i \(0.429601\pi\)
\(192\) − 3.05003e10i − 1.61975i
\(193\) − 3.12603e9i − 0.162176i −0.996707 0.0810878i \(-0.974161\pi\)
0.996707 0.0810878i \(-0.0258394\pi\)
\(194\) 1.01033e10 0.512100
\(195\) 0 0
\(196\) −3.07545e9 −0.148853
\(197\) − 2.40133e10i − 1.13594i −0.823051 0.567968i \(-0.807730\pi\)
0.823051 0.567968i \(-0.192270\pi\)
\(198\) − 1.91207e10i − 0.884117i
\(199\) −9.73605e9 −0.440093 −0.220046 0.975489i \(-0.570621\pi\)
−0.220046 + 0.975489i \(0.570621\pi\)
\(200\) 0 0
\(201\) 2.86077e10 1.23623
\(202\) − 2.69175e10i − 1.13751i
\(203\) − 3.36029e9i − 0.138882i
\(204\) −9.21546e8 −0.0372547
\(205\) 0 0
\(206\) 4.18517e10 1.61924
\(207\) 4.61808e10i 1.74822i
\(208\) 3.82574e10i 1.41720i
\(209\) 2.22227e10 0.805637
\(210\) 0 0
\(211\) −2.39305e10 −0.831151 −0.415576 0.909559i \(-0.636420\pi\)
−0.415576 + 0.909559i \(0.636420\pi\)
\(212\) 2.60662e9i 0.0886272i
\(213\) 7.33956e10i 2.44321i
\(214\) −4.54732e10 −1.48215
\(215\) 0 0
\(216\) 1.22287e10 0.382242
\(217\) 5.42141e9i 0.165975i
\(218\) 2.54589e9i 0.0763459i
\(219\) 3.39101e10 0.996163
\(220\) 0 0
\(221\) 1.35779e10 0.382885
\(222\) − 2.36643e10i − 0.653890i
\(223\) − 2.02289e10i − 0.547773i −0.961762 0.273886i \(-0.911691\pi\)
0.961762 0.273886i \(-0.0883092\pi\)
\(224\) −1.21681e10 −0.322930
\(225\) 0 0
\(226\) −3.97220e10 −1.01285
\(227\) 4.42152e10i 1.10524i 0.833434 + 0.552618i \(0.186371\pi\)
−0.833434 + 0.552618i \(0.813629\pi\)
\(228\) 6.82140e9i 0.167173i
\(229\) 4.89483e10 1.17619 0.588096 0.808791i \(-0.299878\pi\)
0.588096 + 0.808791i \(0.299878\pi\)
\(230\) 0 0
\(231\) −7.58909e10 −1.75362
\(232\) − 4.10702e9i − 0.0930746i
\(233\) − 6.68132e10i − 1.48512i −0.669781 0.742559i \(-0.733612\pi\)
0.669781 0.742559i \(-0.266388\pi\)
\(234\) −8.64761e10 −1.88549
\(235\) 0 0
\(236\) −4.67431e9 −0.0980876
\(237\) 2.66865e10i 0.549444i
\(238\) − 1.74731e10i − 0.352999i
\(239\) 2.21153e10 0.438431 0.219216 0.975676i \(-0.429650\pi\)
0.219216 + 0.975676i \(0.429650\pi\)
\(240\) 0 0
\(241\) 9.97637e10 1.90500 0.952502 0.304532i \(-0.0985002\pi\)
0.952502 + 0.304532i \(0.0985002\pi\)
\(242\) − 2.20416e10i − 0.413118i
\(243\) − 7.65703e10i − 1.40874i
\(244\) 2.04792e9 0.0369878
\(245\) 0 0
\(246\) −9.26480e9 −0.161298
\(247\) − 1.00506e11i − 1.71812i
\(248\) 6.62616e9i 0.111232i
\(249\) −6.04229e10 −0.996104
\(250\) 0 0
\(251\) −4.97276e10 −0.790799 −0.395399 0.918509i \(-0.629394\pi\)
−0.395399 + 0.918509i \(0.629394\pi\)
\(252\) − 1.29149e10i − 0.201738i
\(253\) − 6.87435e10i − 1.05485i
\(254\) 1.15491e11 1.74099
\(255\) 0 0
\(256\) −2.11977e10 −0.308467
\(257\) 5.81308e10i 0.831204i 0.909547 + 0.415602i \(0.136429\pi\)
−0.909547 + 0.415602i \(0.863571\pi\)
\(258\) − 3.04379e10i − 0.427686i
\(259\) −5.20718e10 −0.719041
\(260\) 0 0
\(261\) 8.30753e9 0.110813
\(262\) 8.21814e10i 1.07750i
\(263\) 3.23476e10i 0.416909i 0.978032 + 0.208454i \(0.0668433\pi\)
−0.978032 + 0.208454i \(0.933157\pi\)
\(264\) −9.27555e10 −1.17523
\(265\) 0 0
\(266\) −1.29338e11 −1.58401
\(267\) 1.18354e11i 1.42523i
\(268\) − 7.24691e9i − 0.0858116i
\(269\) −6.24220e9 −0.0726863 −0.0363431 0.999339i \(-0.511571\pi\)
−0.0363431 + 0.999339i \(0.511571\pi\)
\(270\) 0 0
\(271\) 8.05816e10 0.907557 0.453779 0.891114i \(-0.350076\pi\)
0.453779 + 0.891114i \(0.350076\pi\)
\(272\) − 1.91110e10i − 0.211701i
\(273\) 3.43228e11i 3.73982i
\(274\) 5.47050e10 0.586340
\(275\) 0 0
\(276\) 2.11012e10 0.218885
\(277\) − 7.07976e10i − 0.722536i −0.932462 0.361268i \(-0.882344\pi\)
0.932462 0.361268i \(-0.117656\pi\)
\(278\) 8.76284e10i 0.879920i
\(279\) −1.34031e10 −0.132430
\(280\) 0 0
\(281\) 8.15619e10 0.780385 0.390193 0.920733i \(-0.372408\pi\)
0.390193 + 0.920733i \(0.372408\pi\)
\(282\) − 1.41909e11i − 1.33625i
\(283\) − 1.89301e11i − 1.75434i −0.480184 0.877168i \(-0.659430\pi\)
0.480184 0.877168i \(-0.340570\pi\)
\(284\) 1.85926e10 0.169593
\(285\) 0 0
\(286\) 1.28726e11 1.13768
\(287\) 2.03867e10i 0.177369i
\(288\) − 3.00828e10i − 0.257663i
\(289\) 1.11805e11 0.942805
\(290\) 0 0
\(291\) 9.91387e10 0.810447
\(292\) − 8.59011e9i − 0.0691474i
\(293\) 5.39489e10i 0.427640i 0.976873 + 0.213820i \(0.0685906\pi\)
−0.976873 + 0.213820i \(0.931409\pi\)
\(294\) 2.60035e11 2.02987
\(295\) 0 0
\(296\) −6.36433e10 −0.481881
\(297\) − 3.68209e10i − 0.274594i
\(298\) 1.31434e11i 0.965462i
\(299\) −3.10903e11 −2.24959
\(300\) 0 0
\(301\) −6.69768e10 −0.470300
\(302\) − 1.25032e11i − 0.864950i
\(303\) − 2.64129e11i − 1.80022i
\(304\) −1.41462e11 −0.949966
\(305\) 0 0
\(306\) 4.31980e10 0.281655
\(307\) − 1.30037e11i − 0.835496i −0.908563 0.417748i \(-0.862819\pi\)
0.908563 0.417748i \(-0.137181\pi\)
\(308\) 1.92247e10i 0.121725i
\(309\) 4.10671e11 2.56260
\(310\) 0 0
\(311\) 2.09580e9 0.0127036 0.00635182 0.999980i \(-0.497978\pi\)
0.00635182 + 0.999980i \(0.497978\pi\)
\(312\) 4.19500e11i 2.50632i
\(313\) 2.43788e11i 1.43570i 0.696200 + 0.717848i \(0.254873\pi\)
−0.696200 + 0.717848i \(0.745127\pi\)
\(314\) 2.89217e10 0.167896
\(315\) 0 0
\(316\) 6.76022e9 0.0381390
\(317\) − 3.46483e11i − 1.92715i −0.267445 0.963573i \(-0.586179\pi\)
0.267445 0.963573i \(-0.413821\pi\)
\(318\) − 2.20395e11i − 1.20859i
\(319\) −1.23664e10 −0.0668626
\(320\) 0 0
\(321\) −4.46207e11 −2.34565
\(322\) 4.00092e11i 2.07400i
\(323\) 5.02063e10i 0.256653i
\(324\) −1.43604e10 −0.0723961
\(325\) 0 0
\(326\) −9.65566e10 −0.473481
\(327\) 2.49816e10i 0.120825i
\(328\) 2.49170e10i 0.118868i
\(329\) −3.12262e11 −1.46939
\(330\) 0 0
\(331\) −1.43303e11 −0.656189 −0.328094 0.944645i \(-0.606406\pi\)
−0.328094 + 0.944645i \(0.606406\pi\)
\(332\) 1.53063e10i 0.0691433i
\(333\) − 1.28735e11i − 0.573718i
\(334\) −3.89470e11 −1.71244
\(335\) 0 0
\(336\) 4.83094e11 2.06778
\(337\) 3.34820e11i 1.41409i 0.707170 + 0.707044i \(0.249972\pi\)
−0.707170 + 0.707044i \(0.750028\pi\)
\(338\) − 3.55048e11i − 1.47966i
\(339\) −3.89773e11 −1.60293
\(340\) 0 0
\(341\) 1.99515e10 0.0799064
\(342\) − 3.19757e11i − 1.26387i
\(343\) − 1.72469e11i − 0.672804i
\(344\) −8.18604e10 −0.315182
\(345\) 0 0
\(346\) 2.09291e11 0.785070
\(347\) − 9.22101e10i − 0.341425i −0.985321 0.170713i \(-0.945393\pi\)
0.985321 0.170713i \(-0.0546070\pi\)
\(348\) − 3.79592e9i − 0.0138743i
\(349\) 4.71738e11 1.70211 0.851053 0.525080i \(-0.175965\pi\)
0.851053 + 0.525080i \(0.175965\pi\)
\(350\) 0 0
\(351\) −1.66528e11 −0.585606
\(352\) 4.47805e10i 0.155470i
\(353\) 4.01029e11i 1.37464i 0.726353 + 0.687322i \(0.241214\pi\)
−0.726353 + 0.687322i \(0.758786\pi\)
\(354\) 3.95222e11 1.33760
\(355\) 0 0
\(356\) 2.99816e10 0.0989303
\(357\) − 1.71455e11i − 0.558654i
\(358\) − 1.42620e11i − 0.458887i
\(359\) 2.36110e10 0.0750220 0.0375110 0.999296i \(-0.488057\pi\)
0.0375110 + 0.999296i \(0.488057\pi\)
\(360\) 0 0
\(361\) 4.89457e10 0.151681
\(362\) − 1.56972e11i − 0.480433i
\(363\) − 2.16284e11i − 0.653799i
\(364\) 8.69465e10 0.259595
\(365\) 0 0
\(366\) −1.73155e11 −0.504395
\(367\) 1.23035e11i 0.354022i 0.984209 + 0.177011i \(0.0566427\pi\)
−0.984209 + 0.177011i \(0.943357\pi\)
\(368\) 4.37596e11i 1.24382i
\(369\) −5.04012e10 −0.141521
\(370\) 0 0
\(371\) −4.84966e11 −1.32901
\(372\) 6.12424e9i 0.0165809i
\(373\) 2.39248e11i 0.639968i 0.947423 + 0.319984i \(0.103678\pi\)
−0.947423 + 0.319984i \(0.896322\pi\)
\(374\) −6.43034e10 −0.169946
\(375\) 0 0
\(376\) −3.81653e11 −0.984745
\(377\) 5.59287e10i 0.142593i
\(378\) 2.14300e11i 0.539896i
\(379\) 5.90629e11 1.47041 0.735205 0.677845i \(-0.237086\pi\)
0.735205 + 0.677845i \(0.237086\pi\)
\(380\) 0 0
\(381\) 1.13326e12 2.75529
\(382\) 1.72839e11i 0.415295i
\(383\) 5.30525e10i 0.125983i 0.998014 + 0.0629914i \(0.0200641\pi\)
−0.998014 + 0.0629914i \(0.979936\pi\)
\(384\) 5.21087e11 1.22298
\(385\) 0 0
\(386\) 6.69554e10 0.153512
\(387\) − 1.65584e11i − 0.375249i
\(388\) − 2.51138e10i − 0.0562562i
\(389\) −3.10395e11 −0.687293 −0.343646 0.939099i \(-0.611662\pi\)
−0.343646 + 0.939099i \(0.611662\pi\)
\(390\) 0 0
\(391\) 1.55307e11 0.336045
\(392\) − 6.99345e11i − 1.49591i
\(393\) 8.06407e11i 1.70525i
\(394\) 5.14333e11 1.07525
\(395\) 0 0
\(396\) −4.75285e10 −0.0971238
\(397\) − 2.09351e11i − 0.422979i −0.977380 0.211489i \(-0.932169\pi\)
0.977380 0.211489i \(-0.0678314\pi\)
\(398\) − 2.08533e11i − 0.416583i
\(399\) −1.26913e12 −2.50685
\(400\) 0 0
\(401\) 4.43868e11 0.857244 0.428622 0.903484i \(-0.358999\pi\)
0.428622 + 0.903484i \(0.358999\pi\)
\(402\) 6.12739e11i 1.17020i
\(403\) − 9.02338e10i − 0.170411i
\(404\) −6.69092e10 −0.124960
\(405\) 0 0
\(406\) 7.19730e10 0.131463
\(407\) 1.91632e11i 0.346172i
\(408\) − 2.09556e11i − 0.374394i
\(409\) −4.56031e11 −0.805823 −0.402912 0.915239i \(-0.632002\pi\)
−0.402912 + 0.915239i \(0.632002\pi\)
\(410\) 0 0
\(411\) 5.36794e11 0.927940
\(412\) − 1.04031e11i − 0.177880i
\(413\) − 8.69663e11i − 1.47087i
\(414\) −9.89132e11 −1.65483
\(415\) 0 0
\(416\) 2.02526e11 0.331559
\(417\) 8.59856e11i 1.39256i
\(418\) 4.75982e11i 0.762600i
\(419\) 1.12392e12 1.78144 0.890720 0.454552i \(-0.150201\pi\)
0.890720 + 0.454552i \(0.150201\pi\)
\(420\) 0 0
\(421\) −1.02822e12 −1.59521 −0.797605 0.603180i \(-0.793900\pi\)
−0.797605 + 0.603180i \(0.793900\pi\)
\(422\) − 5.12559e11i − 0.786751i
\(423\) − 7.71994e11i − 1.17242i
\(424\) −5.92736e11 −0.890666
\(425\) 0 0
\(426\) −1.57204e12 −2.31270
\(427\) 3.81018e11i 0.554652i
\(428\) 1.13033e11i 0.162820i
\(429\) 1.26313e12 1.80048
\(430\) 0 0
\(431\) −1.37714e12 −1.92234 −0.961171 0.275953i \(-0.911007\pi\)
−0.961171 + 0.275953i \(0.911007\pi\)
\(432\) 2.34389e11i 0.323787i
\(433\) − 1.94790e11i − 0.266299i −0.991096 0.133150i \(-0.957491\pi\)
0.991096 0.133150i \(-0.0425091\pi\)
\(434\) −1.16119e11 −0.157109
\(435\) 0 0
\(436\) 6.32835e9 0.00838689
\(437\) − 1.14961e12i − 1.50793i
\(438\) 7.26309e11i 0.942948i
\(439\) 5.32533e11 0.684315 0.342157 0.939643i \(-0.388842\pi\)
0.342157 + 0.939643i \(0.388842\pi\)
\(440\) 0 0
\(441\) 1.41461e12 1.78099
\(442\) 2.90821e11i 0.362432i
\(443\) − 7.26633e11i − 0.896393i −0.893935 0.448196i \(-0.852067\pi\)
0.893935 0.448196i \(-0.147933\pi\)
\(444\) −5.88225e10 −0.0718323
\(445\) 0 0
\(446\) 4.33276e11 0.518511
\(447\) 1.28970e12i 1.52794i
\(448\) − 1.43749e12i − 1.68599i
\(449\) −9.34830e11 −1.08549 −0.542743 0.839899i \(-0.682614\pi\)
−0.542743 + 0.839899i \(0.682614\pi\)
\(450\) 0 0
\(451\) 7.50258e10 0.0853918
\(452\) 9.87374e10i 0.111265i
\(453\) − 1.22688e12i − 1.36887i
\(454\) −9.47031e11 −1.04620
\(455\) 0 0
\(456\) −1.55116e12 −1.68002
\(457\) − 1.09190e12i − 1.17101i −0.810670 0.585503i \(-0.800897\pi\)
0.810670 0.585503i \(-0.199103\pi\)
\(458\) 1.04841e12i 1.11336i
\(459\) 8.31870e10 0.0874779
\(460\) 0 0
\(461\) 9.93683e11 1.02469 0.512347 0.858779i \(-0.328776\pi\)
0.512347 + 0.858779i \(0.328776\pi\)
\(462\) − 1.62548e12i − 1.65994i
\(463\) 6.95734e11i 0.703605i 0.936074 + 0.351802i \(0.114431\pi\)
−0.936074 + 0.351802i \(0.885569\pi\)
\(464\) 7.87197e10 0.0788411
\(465\) 0 0
\(466\) 1.43105e12 1.40578
\(467\) 1.87957e12i 1.82866i 0.404975 + 0.914328i \(0.367280\pi\)
−0.404975 + 0.914328i \(0.632720\pi\)
\(468\) 2.14955e11i 0.207129i
\(469\) 1.34830e12 1.28679
\(470\) 0 0
\(471\) 2.83795e11 0.265712
\(472\) − 1.06292e12i − 0.985739i
\(473\) 2.46484e11i 0.226419i
\(474\) −5.71589e11 −0.520093
\(475\) 0 0
\(476\) −4.34330e10 −0.0387783
\(477\) − 1.19896e12i − 1.06041i
\(478\) 4.73679e11i 0.415010i
\(479\) 2.29011e11 0.198768 0.0993838 0.995049i \(-0.468313\pi\)
0.0993838 + 0.995049i \(0.468313\pi\)
\(480\) 0 0
\(481\) 8.66683e11 0.738256
\(482\) 2.13681e12i 1.80324i
\(483\) 3.92591e12i 3.28230i
\(484\) −5.47890e10 −0.0453826
\(485\) 0 0
\(486\) 1.64003e12 1.33349
\(487\) − 4.81410e11i − 0.387824i −0.981019 0.193912i \(-0.937882\pi\)
0.981019 0.193912i \(-0.0621177\pi\)
\(488\) 4.65689e11i 0.371712i
\(489\) −9.47465e11 −0.749330
\(490\) 0 0
\(491\) −1.70320e12 −1.32251 −0.661257 0.750160i \(-0.729977\pi\)
−0.661257 + 0.750160i \(0.729977\pi\)
\(492\) 2.30296e10i 0.0177192i
\(493\) − 2.79384e10i − 0.0213006i
\(494\) 2.15270e12 1.62634
\(495\) 0 0
\(496\) −1.27004e11 −0.0942217
\(497\) 3.45917e12i 2.54313i
\(498\) − 1.29418e12i − 0.942892i
\(499\) −2.06075e12 −1.48789 −0.743947 0.668238i \(-0.767049\pi\)
−0.743947 + 0.668238i \(0.767049\pi\)
\(500\) 0 0
\(501\) −3.82168e12 −2.71010
\(502\) − 1.06510e12i − 0.748555i
\(503\) 5.97493e11i 0.416176i 0.978110 + 0.208088i \(0.0667240\pi\)
−0.978110 + 0.208088i \(0.933276\pi\)
\(504\) 2.93679e12 2.02738
\(505\) 0 0
\(506\) 1.47239e12 0.998497
\(507\) − 3.48392e12i − 2.34170i
\(508\) − 2.87077e11i − 0.191255i
\(509\) 8.46938e11 0.559270 0.279635 0.960106i \(-0.409787\pi\)
0.279635 + 0.960106i \(0.409787\pi\)
\(510\) 0 0
\(511\) 1.59820e12 1.03690
\(512\) − 1.72345e12i − 1.10837i
\(513\) − 6.15760e11i − 0.392540i
\(514\) −1.24508e12 −0.786801
\(515\) 0 0
\(516\) −7.56597e10 −0.0469830
\(517\) 1.14917e12i 0.707418i
\(518\) − 1.11531e12i − 0.680630i
\(519\) 2.05367e12 1.24245
\(520\) 0 0
\(521\) 7.31013e11 0.434666 0.217333 0.976098i \(-0.430264\pi\)
0.217333 + 0.976098i \(0.430264\pi\)
\(522\) 1.77936e11i 0.104893i
\(523\) − 1.04453e12i − 0.610467i −0.952278 0.305233i \(-0.901266\pi\)
0.952278 0.305233i \(-0.0987345\pi\)
\(524\) 2.04279e11 0.118368
\(525\) 0 0
\(526\) −6.92842e11 −0.394638
\(527\) 4.50751e10i 0.0254559i
\(528\) − 1.77785e12i − 0.995504i
\(529\) −1.75502e12 −0.974386
\(530\) 0 0
\(531\) 2.15003e12 1.17360
\(532\) 3.21496e11i 0.174010i
\(533\) − 3.39315e11i − 0.182109i
\(534\) −2.53500e12 −1.34909
\(535\) 0 0
\(536\) 1.64792e12 0.862371
\(537\) − 1.39946e12i − 0.726233i
\(538\) − 1.33700e11i − 0.0688034i
\(539\) −2.10575e12 −1.07462
\(540\) 0 0
\(541\) −1.88034e12 −0.943733 −0.471866 0.881670i \(-0.656420\pi\)
−0.471866 + 0.881670i \(0.656420\pi\)
\(542\) 1.72595e12i 0.859076i
\(543\) − 1.54029e12i − 0.760331i
\(544\) −1.01169e11 −0.0495284
\(545\) 0 0
\(546\) −7.35148e12 −3.54004
\(547\) − 1.08624e12i − 0.518780i −0.965773 0.259390i \(-0.916479\pi\)
0.965773 0.259390i \(-0.0835215\pi\)
\(548\) − 1.35981e11i − 0.0644117i
\(549\) −9.41977e11 −0.442553
\(550\) 0 0
\(551\) −2.06804e11 −0.0955821
\(552\) 4.79834e12i 2.19971i
\(553\) 1.25775e12i 0.571914i
\(554\) 1.51639e12 0.683938
\(555\) 0 0
\(556\) 2.17819e11 0.0966626
\(557\) 2.11965e12i 0.933072i 0.884502 + 0.466536i \(0.154498\pi\)
−0.884502 + 0.466536i \(0.845502\pi\)
\(558\) − 2.87078e11i − 0.125356i
\(559\) 1.11476e12 0.482868
\(560\) 0 0
\(561\) −6.30978e11 −0.268956
\(562\) 1.74695e12i 0.738697i
\(563\) − 3.04489e12i − 1.27727i −0.769509 0.638636i \(-0.779499\pi\)
0.769509 0.638636i \(-0.220501\pi\)
\(564\) −3.52744e11 −0.146792
\(565\) 0 0
\(566\) 4.05456e12 1.66062
\(567\) − 2.67178e12i − 1.08562i
\(568\) 4.22788e12i 1.70433i
\(569\) −3.25275e12 −1.30090 −0.650452 0.759548i \(-0.725420\pi\)
−0.650452 + 0.759548i \(0.725420\pi\)
\(570\) 0 0
\(571\) −1.35916e12 −0.535066 −0.267533 0.963549i \(-0.586208\pi\)
−0.267533 + 0.963549i \(0.586208\pi\)
\(572\) − 3.19975e11i − 0.124978i
\(573\) 1.69599e12i 0.657245i
\(574\) −4.36655e11 −0.167894
\(575\) 0 0
\(576\) 3.55386e12 1.34524
\(577\) − 2.35782e12i − 0.885564i −0.896629 0.442782i \(-0.853992\pi\)
0.896629 0.442782i \(-0.146008\pi\)
\(578\) 2.39472e12i 0.892440i
\(579\) 6.57002e11 0.242948
\(580\) 0 0
\(581\) −2.84777e12 −1.03684
\(582\) 2.12342e12i 0.767154i
\(583\) 1.78474e12i 0.639834i
\(584\) 1.95336e12 0.694902
\(585\) 0 0
\(586\) −1.15551e12 −0.404796
\(587\) − 2.37065e12i − 0.824130i −0.911154 0.412065i \(-0.864808\pi\)
0.911154 0.412065i \(-0.135192\pi\)
\(588\) − 6.46371e11i − 0.222989i
\(589\) 3.33652e11 0.114229
\(590\) 0 0
\(591\) 5.04690e12 1.70169
\(592\) − 1.21986e12i − 0.408189i
\(593\) − 4.56707e12i − 1.51667i −0.651864 0.758336i \(-0.726013\pi\)
0.651864 0.758336i \(-0.273987\pi\)
\(594\) 7.88655e11 0.259925
\(595\) 0 0
\(596\) 3.26707e11 0.106060
\(597\) − 2.04624e12i − 0.659283i
\(598\) − 6.65912e12i − 2.12942i
\(599\) 5.30493e12 1.68368 0.841839 0.539729i \(-0.181473\pi\)
0.841839 + 0.539729i \(0.181473\pi\)
\(600\) 0 0
\(601\) −2.71307e12 −0.848255 −0.424128 0.905602i \(-0.639419\pi\)
−0.424128 + 0.905602i \(0.639419\pi\)
\(602\) − 1.43455e12i − 0.445177i
\(603\) 3.33335e12i 1.02672i
\(604\) −3.10794e11 −0.0950182
\(605\) 0 0
\(606\) 5.65730e12 1.70405
\(607\) − 4.00734e12i − 1.19814i −0.800698 0.599069i \(-0.795538\pi\)
0.800698 0.599069i \(-0.204462\pi\)
\(608\) 7.48868e11i 0.222249i
\(609\) 7.06237e11 0.208052
\(610\) 0 0
\(611\) 5.19728e12 1.50866
\(612\) − 1.07378e11i − 0.0309409i
\(613\) 2.24954e12i 0.643459i 0.946832 + 0.321730i \(0.104264\pi\)
−0.946832 + 0.321730i \(0.895736\pi\)
\(614\) 2.78522e12 0.790864
\(615\) 0 0
\(616\) −4.37162e12 −1.22329
\(617\) 1.19664e12i 0.332414i 0.986091 + 0.166207i \(0.0531519\pi\)
−0.986091 + 0.166207i \(0.946848\pi\)
\(618\) 8.79602e12i 2.42571i
\(619\) 4.36723e12 1.19563 0.597817 0.801633i \(-0.296035\pi\)
0.597817 + 0.801633i \(0.296035\pi\)
\(620\) 0 0
\(621\) −1.90478e12 −0.513965
\(622\) 4.48893e10i 0.0120250i
\(623\) 5.57811e12i 1.48351i
\(624\) −8.04060e12 −2.12304
\(625\) 0 0
\(626\) −5.22161e12 −1.35900
\(627\) 4.67058e12i 1.20689i
\(628\) − 7.18911e10i − 0.0184441i
\(629\) −4.32940e11 −0.110281
\(630\) 0 0
\(631\) 2.78790e12 0.700077 0.350039 0.936735i \(-0.386168\pi\)
0.350039 + 0.936735i \(0.386168\pi\)
\(632\) 1.53725e12i 0.383281i
\(633\) − 5.02950e12i − 1.24511i
\(634\) 7.42120e12 1.82420
\(635\) 0 0
\(636\) −5.47837e11 −0.132768
\(637\) 9.52355e12i 2.29177i
\(638\) − 2.64871e11i − 0.0632909i
\(639\) −8.55199e12 −2.02915
\(640\) 0 0
\(641\) −2.28295e12 −0.534115 −0.267058 0.963681i \(-0.586051\pi\)
−0.267058 + 0.963681i \(0.586051\pi\)
\(642\) − 9.55716e12i − 2.22035i
\(643\) 2.29995e12i 0.530603i 0.964166 + 0.265301i \(0.0854715\pi\)
−0.964166 + 0.265301i \(0.914529\pi\)
\(644\) 9.94512e11 0.227837
\(645\) 0 0
\(646\) −1.07535e12 −0.242943
\(647\) 1.85647e12i 0.416503i 0.978075 + 0.208251i \(0.0667772\pi\)
−0.978075 + 0.208251i \(0.933223\pi\)
\(648\) − 3.26551e12i − 0.727550i
\(649\) −3.20048e12 −0.708132
\(650\) 0 0
\(651\) −1.13942e12 −0.248640
\(652\) 2.40012e11i 0.0520138i
\(653\) − 7.75397e12i − 1.66884i −0.551129 0.834420i \(-0.685803\pi\)
0.551129 0.834420i \(-0.314197\pi\)
\(654\) −5.35073e11 −0.114370
\(655\) 0 0
\(656\) −4.77587e11 −0.100690
\(657\) 3.95117e12i 0.827336i
\(658\) − 6.68824e12i − 1.39090i
\(659\) −7.42166e12 −1.53291 −0.766455 0.642298i \(-0.777981\pi\)
−0.766455 + 0.642298i \(0.777981\pi\)
\(660\) 0 0
\(661\) 5.79783e12 1.18130 0.590648 0.806929i \(-0.298872\pi\)
0.590648 + 0.806929i \(0.298872\pi\)
\(662\) − 3.06936e12i − 0.621135i
\(663\) 2.85369e12i 0.573583i
\(664\) −3.48060e12 −0.694861
\(665\) 0 0
\(666\) 2.75734e12 0.543070
\(667\) 6.39724e11i 0.125149i
\(668\) 9.68108e11i 0.188118i
\(669\) 4.25153e12 0.820593
\(670\) 0 0
\(671\) 1.40220e12 0.267029
\(672\) − 2.55739e12i − 0.483766i
\(673\) 6.32456e12i 1.18840i 0.804318 + 0.594200i \(0.202531\pi\)
−0.804318 + 0.594200i \(0.797469\pi\)
\(674\) −7.17139e12 −1.33855
\(675\) 0 0
\(676\) −8.82546e11 −0.162546
\(677\) − 4.95066e12i − 0.905762i −0.891571 0.452881i \(-0.850396\pi\)
0.891571 0.452881i \(-0.149604\pi\)
\(678\) − 8.34843e12i − 1.51730i
\(679\) 4.67246e12 0.843591
\(680\) 0 0
\(681\) −9.29277e12 −1.65571
\(682\) 4.27336e11i 0.0756379i
\(683\) − 7.32611e12i − 1.28819i −0.764945 0.644095i \(-0.777234\pi\)
0.764945 0.644095i \(-0.222766\pi\)
\(684\) −7.94824e11 −0.138841
\(685\) 0 0
\(686\) 3.69406e12 0.636863
\(687\) 1.02875e13i 1.76200i
\(688\) − 1.56903e12i − 0.266982i
\(689\) 8.07176e12 1.36453
\(690\) 0 0
\(691\) −5.66806e12 −0.945765 −0.472882 0.881126i \(-0.656786\pi\)
−0.472882 + 0.881126i \(0.656786\pi\)
\(692\) − 5.20237e11i − 0.0862430i
\(693\) − 8.84274e12i − 1.45642i
\(694\) 1.97502e12 0.323187
\(695\) 0 0
\(696\) 8.63178e11 0.139431
\(697\) 1.69501e11i 0.0272034i
\(698\) 1.01040e13i 1.61118i
\(699\) 1.40422e13 2.22479
\(700\) 0 0
\(701\) −4.20985e12 −0.658470 −0.329235 0.944248i \(-0.606791\pi\)
−0.329235 + 0.944248i \(0.606791\pi\)
\(702\) − 3.56681e12i − 0.554323i
\(703\) 3.20468e12i 0.494863i
\(704\) −5.29018e12 −0.811696
\(705\) 0 0
\(706\) −8.58952e12 −1.30121
\(707\) − 1.24486e13i − 1.87384i
\(708\) − 9.82407e11i − 0.146941i
\(709\) −1.19654e13 −1.77835 −0.889176 0.457566i \(-0.848721\pi\)
−0.889176 + 0.457566i \(0.848721\pi\)
\(710\) 0 0
\(711\) −3.10949e12 −0.456326
\(712\) 6.81769e12i 0.994208i
\(713\) − 1.03211e12i − 0.149563i
\(714\) 3.67234e12 0.528811
\(715\) 0 0
\(716\) −3.54511e11 −0.0504105
\(717\) 4.64799e12i 0.656794i
\(718\) 5.05715e11i 0.0710143i
\(719\) 4.07118e11 0.0568120 0.0284060 0.999596i \(-0.490957\pi\)
0.0284060 + 0.999596i \(0.490957\pi\)
\(720\) 0 0
\(721\) 1.93551e13 2.66740
\(722\) 1.04835e12i 0.143579i
\(723\) 2.09675e13i 2.85380i
\(724\) −3.90186e11 −0.0527774
\(725\) 0 0
\(726\) 4.63251e12 0.618873
\(727\) 6.44424e12i 0.855592i 0.903875 + 0.427796i \(0.140710\pi\)
−0.903875 + 0.427796i \(0.859290\pi\)
\(728\) 1.97713e13i 2.60882i
\(729\) 1.07838e13 1.41416
\(730\) 0 0
\(731\) −5.56864e11 −0.0721308
\(732\) 4.30414e11i 0.0554098i
\(733\) − 5.46122e12i − 0.698749i −0.936983 0.349375i \(-0.886394\pi\)
0.936983 0.349375i \(-0.113606\pi\)
\(734\) −2.63524e12 −0.335110
\(735\) 0 0
\(736\) 2.31654e12 0.290998
\(737\) − 4.96192e12i − 0.619507i
\(738\) − 1.07953e12i − 0.133961i
\(739\) 1.88201e12 0.232125 0.116062 0.993242i \(-0.462973\pi\)
0.116062 + 0.993242i \(0.462973\pi\)
\(740\) 0 0
\(741\) 2.11234e13 2.57384
\(742\) − 1.03873e13i − 1.25802i
\(743\) − 1.36704e13i − 1.64563i −0.568312 0.822813i \(-0.692403\pi\)
0.568312 0.822813i \(-0.307597\pi\)
\(744\) −1.39263e12 −0.166632
\(745\) 0 0
\(746\) −5.12437e12 −0.605781
\(747\) − 7.04042e12i − 0.827287i
\(748\) 1.59840e11i 0.0186693i
\(749\) −2.10300e13 −2.44158
\(750\) 0 0
\(751\) 8.52585e12 0.978043 0.489021 0.872272i \(-0.337354\pi\)
0.489021 + 0.872272i \(0.337354\pi\)
\(752\) − 7.31518e12i − 0.834152i
\(753\) − 1.04513e13i − 1.18466i
\(754\) −1.19792e12 −0.134976
\(755\) 0 0
\(756\) 5.32688e11 0.0593096
\(757\) − 7.50792e12i − 0.830975i −0.909599 0.415487i \(-0.863611\pi\)
0.909599 0.415487i \(-0.136389\pi\)
\(758\) 1.26505e13i 1.39186i
\(759\) 1.44479e13 1.58022
\(760\) 0 0
\(761\) −1.61560e13 −1.74624 −0.873118 0.487509i \(-0.837906\pi\)
−0.873118 + 0.487509i \(0.837906\pi\)
\(762\) 2.42729e13i 2.60810i
\(763\) 1.17740e12i 0.125766i
\(764\) 4.29628e11 0.0456218
\(765\) 0 0
\(766\) −1.13631e12 −0.119253
\(767\) 1.44747e13i 1.51018i
\(768\) − 4.45514e12i − 0.462100i
\(769\) −1.11830e12 −0.115317 −0.0576583 0.998336i \(-0.518363\pi\)
−0.0576583 + 0.998336i \(0.518363\pi\)
\(770\) 0 0
\(771\) −1.22174e13 −1.24519
\(772\) − 1.66432e11i − 0.0168639i
\(773\) 1.73165e13i 1.74442i 0.489129 + 0.872212i \(0.337315\pi\)
−0.489129 + 0.872212i \(0.662685\pi\)
\(774\) 3.54659e12 0.355203
\(775\) 0 0
\(776\) 5.71078e12 0.565351
\(777\) − 1.09440e13i − 1.07716i
\(778\) − 6.64825e12i − 0.650578i
\(779\) 1.25466e12 0.122070
\(780\) 0 0
\(781\) 1.27302e13 1.22435
\(782\) 3.32648e12i 0.318093i
\(783\) 3.42654e11i 0.0325783i
\(784\) 1.34044e13 1.26714
\(785\) 0 0
\(786\) −1.72722e13 −1.61416
\(787\) − 3.88515e11i − 0.0361012i −0.999837 0.0180506i \(-0.994254\pi\)
0.999837 0.0180506i \(-0.00574599\pi\)
\(788\) − 1.27848e12i − 0.118121i
\(789\) −6.79853e12 −0.624552
\(790\) 0 0
\(791\) −1.83702e13 −1.66848
\(792\) − 1.08078e13i − 0.976053i
\(793\) − 6.34166e12i − 0.569474i
\(794\) 4.48403e12 0.400383
\(795\) 0 0
\(796\) −5.18353e11 −0.0457633
\(797\) − 1.08432e12i − 0.0951910i −0.998867 0.0475955i \(-0.984844\pi\)
0.998867 0.0475955i \(-0.0151558\pi\)
\(798\) − 2.71831e13i − 2.37294i
\(799\) −2.59624e12 −0.225363
\(800\) 0 0
\(801\) −1.37906e13 −1.18368
\(802\) 9.50706e12i 0.811450i
\(803\) − 5.88161e12i − 0.499202i
\(804\) 1.52309e12 0.128551
\(805\) 0 0
\(806\) 1.93269e12 0.161307
\(807\) − 1.31193e12i − 0.108888i
\(808\) − 1.52149e13i − 1.25579i
\(809\) −4.06803e12 −0.333900 −0.166950 0.985965i \(-0.553392\pi\)
−0.166950 + 0.985965i \(0.553392\pi\)
\(810\) 0 0
\(811\) 4.87468e12 0.395688 0.197844 0.980234i \(-0.436606\pi\)
0.197844 + 0.980234i \(0.436606\pi\)
\(812\) − 1.78904e11i − 0.0144417i
\(813\) 1.69359e13i 1.35957i
\(814\) −4.10449e12 −0.327680
\(815\) 0 0
\(816\) 4.01658e12 0.317140
\(817\) 4.12198e12i 0.323673i
\(818\) − 9.76758e12i − 0.762777i
\(819\) −3.99926e13 −3.10600
\(820\) 0 0
\(821\) −1.03829e12 −0.0797582 −0.0398791 0.999205i \(-0.512697\pi\)
−0.0398791 + 0.999205i \(0.512697\pi\)
\(822\) 1.14974e13i 0.878370i
\(823\) − 1.10923e13i − 0.842794i −0.906876 0.421397i \(-0.861540\pi\)
0.906876 0.421397i \(-0.138460\pi\)
\(824\) 2.36563e13 1.78761
\(825\) 0 0
\(826\) 1.86270e13 1.39230
\(827\) − 1.45526e13i − 1.08184i −0.841073 0.540922i \(-0.818075\pi\)
0.841073 0.540922i \(-0.181925\pi\)
\(828\) 2.45870e12i 0.181789i
\(829\) −1.37985e13 −1.01470 −0.507348 0.861741i \(-0.669374\pi\)
−0.507348 + 0.861741i \(0.669374\pi\)
\(830\) 0 0
\(831\) 1.48796e13 1.08240
\(832\) 2.39256e13i 1.73104i
\(833\) − 4.75736e12i − 0.342345i
\(834\) −1.84170e13 −1.31817
\(835\) 0 0
\(836\) 1.18315e12 0.0837746
\(837\) − 5.52829e11i − 0.0389337i
\(838\) 2.40728e13i 1.68628i
\(839\) −9.20234e11 −0.0641165 −0.0320582 0.999486i \(-0.510206\pi\)
−0.0320582 + 0.999486i \(0.510206\pi\)
\(840\) 0 0
\(841\) −1.43921e13 −0.992067
\(842\) − 2.20232e13i − 1.50999i
\(843\) 1.71420e13i 1.16906i
\(844\) −1.27407e12 −0.0864277
\(845\) 0 0
\(846\) 1.65351e13 1.10979
\(847\) − 1.01936e13i − 0.680536i
\(848\) − 1.13610e13i − 0.754460i
\(849\) 3.97855e13 2.62809
\(850\) 0 0
\(851\) 9.91330e12 0.647941
\(852\) 3.90762e12i 0.254059i
\(853\) 7.46964e12i 0.483091i 0.970390 + 0.241545i \(0.0776543\pi\)
−0.970390 + 0.241545i \(0.922346\pi\)
\(854\) −8.16090e12 −0.525022
\(855\) 0 0
\(856\) −2.57033e13 −1.63628
\(857\) 2.05568e13i 1.30179i 0.759166 + 0.650897i \(0.225607\pi\)
−0.759166 + 0.650897i \(0.774393\pi\)
\(858\) 2.70545e13i 1.70430i
\(859\) 1.05923e12 0.0663775 0.0331888 0.999449i \(-0.489434\pi\)
0.0331888 + 0.999449i \(0.489434\pi\)
\(860\) 0 0
\(861\) −4.28469e12 −0.265709
\(862\) − 2.94965e13i − 1.81965i
\(863\) − 1.53023e13i − 0.939092i −0.882908 0.469546i \(-0.844418\pi\)
0.882908 0.469546i \(-0.155582\pi\)
\(864\) 1.24080e12 0.0757514
\(865\) 0 0
\(866\) 4.17213e12 0.252074
\(867\) 2.34982e13i 1.41237i
\(868\) 2.88639e11i 0.0172590i
\(869\) 4.62869e12 0.275340
\(870\) 0 0
\(871\) −2.24410e13 −1.32118
\(872\) 1.43904e12i 0.0842847i
\(873\) 1.15516e13i 0.673095i
\(874\) 2.46230e13 1.42738
\(875\) 0 0
\(876\) 1.80539e12 0.103587
\(877\) − 1.74558e11i − 0.00996416i −0.999988 0.00498208i \(-0.998414\pi\)
0.999988 0.00498208i \(-0.00158585\pi\)
\(878\) 1.14061e13i 0.647759i
\(879\) −1.13385e13 −0.640629
\(880\) 0 0
\(881\) 8.59039e12 0.480420 0.240210 0.970721i \(-0.422784\pi\)
0.240210 + 0.970721i \(0.422784\pi\)
\(882\) 3.02990e13i 1.68585i
\(883\) 1.70227e13i 0.942336i 0.882043 + 0.471168i \(0.156167\pi\)
−0.882043 + 0.471168i \(0.843833\pi\)
\(884\) 7.22898e11 0.0398145
\(885\) 0 0
\(886\) 1.55635e13 0.848508
\(887\) − 2.01246e13i − 1.09162i −0.837909 0.545810i \(-0.816222\pi\)
0.837909 0.545810i \(-0.183778\pi\)
\(888\) − 1.33760e13i − 0.721885i
\(889\) 5.34112e13 2.86797
\(890\) 0 0
\(891\) −9.83253e12 −0.522655
\(892\) − 1.07700e12i − 0.0569604i
\(893\) 1.92177e13i 1.01127i
\(894\) −2.76237e13 −1.44632
\(895\) 0 0
\(896\) 2.45591e13 1.27299
\(897\) − 6.53428e13i − 3.37002i
\(898\) − 2.00228e13i − 1.02750i
\(899\) −1.85668e11 −0.00948023
\(900\) 0 0
\(901\) −4.03214e12 −0.203833
\(902\) 1.60695e12i 0.0808302i
\(903\) − 1.40766e13i − 0.704535i
\(904\) −2.24525e13 −1.11817
\(905\) 0 0
\(906\) 2.62782e13 1.29574
\(907\) − 2.10130e13i − 1.03099i −0.856893 0.515495i \(-0.827608\pi\)
0.856893 0.515495i \(-0.172392\pi\)
\(908\) 2.35404e12i 0.114929i
\(909\) 3.07761e13 1.49512
\(910\) 0 0
\(911\) 1.95345e12 0.0939656 0.0469828 0.998896i \(-0.485039\pi\)
0.0469828 + 0.998896i \(0.485039\pi\)
\(912\) − 2.97312e13i − 1.42310i
\(913\) 1.04802e13i 0.499172i
\(914\) 2.33870e13 1.10845
\(915\) 0 0
\(916\) 2.60604e12 0.122307
\(917\) 3.80064e13i 1.77499i
\(918\) 1.78175e12i 0.0828048i
\(919\) 2.47656e12 0.114533 0.0572663 0.998359i \(-0.481762\pi\)
0.0572663 + 0.998359i \(0.481762\pi\)
\(920\) 0 0
\(921\) 2.73301e13 1.25162
\(922\) 2.12834e13i 0.969954i
\(923\) − 5.75744e13i − 2.61109i
\(924\) −4.04048e12 −0.182351
\(925\) 0 0
\(926\) −1.49017e13 −0.666018
\(927\) 4.78510e13i 2.12830i
\(928\) − 4.16725e11i − 0.0184452i
\(929\) −4.40844e13 −1.94184 −0.970922 0.239395i \(-0.923051\pi\)
−0.970922 + 0.239395i \(0.923051\pi\)
\(930\) 0 0
\(931\) −3.52146e13 −1.53621
\(932\) − 3.55718e12i − 0.154431i
\(933\) 4.40477e11i 0.0190308i
\(934\) −4.02578e13 −1.73097
\(935\) 0 0
\(936\) −4.88798e13 −2.08156
\(937\) 4.92250e12i 0.208621i 0.994545 + 0.104310i \(0.0332635\pi\)
−0.994545 + 0.104310i \(0.966736\pi\)
\(938\) 2.88787e13i 1.21805i
\(939\) −5.12372e13 −2.15075
\(940\) 0 0
\(941\) −1.54680e13 −0.643103 −0.321552 0.946892i \(-0.604204\pi\)
−0.321552 + 0.946892i \(0.604204\pi\)
\(942\) 6.07852e12i 0.251518i
\(943\) − 3.88116e12i − 0.159830i
\(944\) 2.03731e13 0.834993
\(945\) 0 0
\(946\) −5.27936e12 −0.214324
\(947\) 5.49107e12i 0.221862i 0.993828 + 0.110931i \(0.0353832\pi\)
−0.993828 + 0.110931i \(0.964617\pi\)
\(948\) 1.42080e12i 0.0571343i
\(949\) −2.66004e13 −1.06461
\(950\) 0 0
\(951\) 7.28207e13 2.88697
\(952\) − 9.87649e12i − 0.389705i
\(953\) 1.76932e13i 0.694845i 0.937709 + 0.347422i \(0.112943\pi\)
−0.937709 + 0.347422i \(0.887057\pi\)
\(954\) 2.56802e13 1.00376
\(955\) 0 0
\(956\) 1.17743e12 0.0455905
\(957\) − 2.59905e12i − 0.100164i
\(958\) 4.90510e12i 0.188150i
\(959\) 2.52994e13 0.965888
\(960\) 0 0
\(961\) −2.61401e13 −0.988670
\(962\) 1.85632e13i 0.698819i
\(963\) − 5.19916e13i − 1.94812i
\(964\) 5.31148e12 0.198093
\(965\) 0 0
\(966\) −8.40878e13 −3.10696
\(967\) 2.89231e13i 1.06372i 0.846834 + 0.531858i \(0.178506\pi\)
−0.846834 + 0.531858i \(0.821494\pi\)
\(968\) − 1.24588e13i − 0.456076i
\(969\) −1.05519e13 −0.384481
\(970\) 0 0
\(971\) 2.69482e13 0.972843 0.486422 0.873724i \(-0.338302\pi\)
0.486422 + 0.873724i \(0.338302\pi\)
\(972\) − 4.07665e12i − 0.146489i
\(973\) 4.05255e13i 1.44951i
\(974\) 1.03112e13 0.367107
\(975\) 0 0
\(976\) −8.92590e12 −0.314867
\(977\) 3.90598e13i 1.37153i 0.727824 + 0.685764i \(0.240532\pi\)
−0.727824 + 0.685764i \(0.759468\pi\)
\(978\) − 2.02934e13i − 0.709301i
\(979\) 2.05282e13 0.714216
\(980\) 0 0
\(981\) −2.91084e12 −0.100348
\(982\) − 3.64804e13i − 1.25187i
\(983\) − 9.03830e12i − 0.308742i −0.988013 0.154371i \(-0.950665\pi\)
0.988013 0.154371i \(-0.0493351\pi\)
\(984\) −5.23684e12 −0.178070
\(985\) 0 0
\(986\) 5.98404e11 0.0201627
\(987\) − 6.56285e13i − 2.20123i
\(988\) − 5.35098e12i − 0.178660i
\(989\) 1.27509e13 0.423795
\(990\) 0 0
\(991\) 4.89096e13 1.61088 0.805439 0.592678i \(-0.201929\pi\)
0.805439 + 0.592678i \(0.201929\pi\)
\(992\) 6.72333e11i 0.0220436i
\(993\) − 3.01181e13i − 0.983007i
\(994\) −7.40909e13 −2.40728
\(995\) 0 0
\(996\) −3.21695e12 −0.103580
\(997\) 4.01445e13i 1.28676i 0.765547 + 0.643380i \(0.222468\pi\)
−0.765547 + 0.643380i \(0.777532\pi\)
\(998\) − 4.41384e13i − 1.40841i
\(999\) 5.30984e12 0.168670
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 25.10.b.c.24.5 6
3.2 odd 2 225.10.b.m.199.2 6
4.3 odd 2 400.10.c.q.49.2 6
5.2 odd 4 25.10.a.c.1.2 3
5.3 odd 4 25.10.a.d.1.2 yes 3
5.4 even 2 inner 25.10.b.c.24.2 6
15.2 even 4 225.10.a.p.1.2 3
15.8 even 4 225.10.a.m.1.2 3
15.14 odd 2 225.10.b.m.199.5 6
20.3 even 4 400.10.a.u.1.3 3
20.7 even 4 400.10.a.y.1.1 3
20.19 odd 2 400.10.c.q.49.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.10.a.c.1.2 3 5.2 odd 4
25.10.a.d.1.2 yes 3 5.3 odd 4
25.10.b.c.24.2 6 5.4 even 2 inner
25.10.b.c.24.5 6 1.1 even 1 trivial
225.10.a.m.1.2 3 15.8 even 4
225.10.a.p.1.2 3 15.2 even 4
225.10.b.m.199.2 6 3.2 odd 2
225.10.b.m.199.5 6 15.14 odd 2
400.10.a.u.1.3 3 20.3 even 4
400.10.a.y.1.1 3 20.7 even 4
400.10.c.q.49.2 6 4.3 odd 2
400.10.c.q.49.5 6 20.19 odd 2