Properties

Label 245.10.a.c.1.2
Level $245$
Weight $10$
Character 245.1
Self dual yes
Analytic conductor $126.184$
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] = [245,10,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(126.183779860\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.17157 q^{2} -65.7351 q^{3} -427.882 q^{4} -625.000 q^{5} +602.894 q^{6} +8620.20 q^{8} -15361.9 q^{9} +O(q^{10})\) \(q-9.17157 q^{2} -65.7351 q^{3} -427.882 q^{4} -625.000 q^{5} +602.894 q^{6} +8620.20 q^{8} -15361.9 q^{9} +5732.23 q^{10} +35089.6 q^{11} +28126.9 q^{12} +77401.4 q^{13} +41084.4 q^{15} +140015. q^{16} +229907. q^{17} +140893. q^{18} -16433.6 q^{19} +267426. q^{20} -321827. q^{22} -2.57284e6 q^{23} -566649. q^{24} +390625. q^{25} -709892. q^{26} +2.30368e6 q^{27} -6.62817e6 q^{29} -376809. q^{30} +8.17416e6 q^{31} -5.69770e6 q^{32} -2.30662e6 q^{33} -2.10861e6 q^{34} +6.57308e6 q^{36} +9.70272e6 q^{37} +150722. q^{38} -5.08798e6 q^{39} -5.38762e6 q^{40} -2.98108e7 q^{41} -1.95343e7 q^{43} -1.50142e7 q^{44} +9.60119e6 q^{45} +2.35970e7 q^{46} -5.93794e6 q^{47} -9.20389e6 q^{48} -3.58265e6 q^{50} -1.51130e7 q^{51} -3.31187e7 q^{52} -2.74263e7 q^{53} -2.11284e7 q^{54} -2.19310e7 q^{55} +1.08026e6 q^{57} +6.07908e7 q^{58} -5.24915e7 q^{59} -1.75793e7 q^{60} -2.23282e7 q^{61} -7.49699e7 q^{62} -1.94308e7 q^{64} -4.83759e7 q^{65} +2.11553e7 q^{66} +2.74351e8 q^{67} -9.83733e7 q^{68} +1.69126e8 q^{69} -3.63673e8 q^{71} -1.32423e8 q^{72} -2.09245e7 q^{73} -8.89892e7 q^{74} -2.56778e7 q^{75} +7.03163e6 q^{76} +4.66648e7 q^{78} -2.65896e8 q^{79} -8.75093e7 q^{80} +1.50936e8 q^{81} +2.73412e8 q^{82} +9.43764e6 q^{83} -1.43692e8 q^{85} +1.79160e8 q^{86} +4.35704e8 q^{87} +3.02479e8 q^{88} +6.64876e8 q^{89} -8.80580e7 q^{90} +1.10087e9 q^{92} -5.37329e8 q^{93} +5.44603e7 q^{94} +1.02710e7 q^{95} +3.74539e8 q^{96} +1.20731e9 q^{97} -5.39042e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{2} + 174 q^{3} - 720 q^{4} - 1250 q^{5} - 2952 q^{6} + 20544 q^{8} + 22428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 24 q^{2} + 174 q^{3} - 720 q^{4} - 1250 q^{5} - 2952 q^{6} + 20544 q^{8} + 22428 q^{9} + 15000 q^{10} + 18566 q^{11} - 41904 q^{12} + 51090 q^{13} - 108750 q^{15} + 112768 q^{16} + 373910 q^{17} - 419472 q^{18} + 143276 q^{19} + 450000 q^{20} - 76808 q^{22} - 498908 q^{23} + 2291904 q^{24} + 781250 q^{25} - 319736 q^{26} + 6644538 q^{27} - 11577554 q^{29} + 1845000 q^{30} + 3953760 q^{31} - 11398656 q^{32} - 6267894 q^{33} - 4243944 q^{34} - 4466016 q^{36} - 3205412 q^{37} - 2217520 q^{38} - 11395746 q^{39} - 12840000 q^{40} - 1058992 q^{41} + 15948180 q^{43} - 10187376 q^{44} - 14017500 q^{45} - 7156176 q^{46} - 65501290 q^{47} - 15735936 q^{48} - 9375000 q^{50} + 19409466 q^{51} - 25432656 q^{52} - 25114688 q^{53} - 85496472 q^{54} - 11603750 q^{55} + 39368244 q^{57} + 134182296 q^{58} + 116159208 q^{59} + 26190000 q^{60} + 44688544 q^{61} - 12388000 q^{62} + 79055872 q^{64} - 31931250 q^{65} + 79894824 q^{66} + 118092496 q^{67} - 140439024 q^{68} + 666320892 q^{69} - 294165824 q^{71} + 318176640 q^{72} + 57419332 q^{73} + 102418064 q^{74} + 67968750 q^{75} - 39622368 q^{76} + 140199000 q^{78} - 692852854 q^{79} - 70480000 q^{80} + 447773346 q^{81} - 152932128 q^{82} + 540679928 q^{83} - 233693750 q^{85} - 346989008 q^{86} - 750836190 q^{87} + 105455296 q^{88} + 779043704 q^{89} + 262170000 q^{90} + 495040608 q^{92} - 1549107360 q^{93} + 937691032 q^{94} - 89547500 q^{95} - 992180736 q^{96} + 2673039406 q^{97} - 1163466540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.17157 −0.405330 −0.202665 0.979248i \(-0.564960\pi\)
−0.202665 + 0.979248i \(0.564960\pi\)
\(3\) −65.7351 −0.468545 −0.234273 0.972171i \(-0.575271\pi\)
−0.234273 + 0.972171i \(0.575271\pi\)
\(4\) −427.882 −0.835708
\(5\) −625.000 −0.447214
\(6\) 602.894 0.189915
\(7\) 0 0
\(8\) 8620.20 0.744067
\(9\) −15361.9 −0.780465
\(10\) 5732.23 0.181269
\(11\) 35089.6 0.722622 0.361311 0.932445i \(-0.382329\pi\)
0.361311 + 0.932445i \(0.382329\pi\)
\(12\) 28126.9 0.391567
\(13\) 77401.4 0.751629 0.375815 0.926695i \(-0.377363\pi\)
0.375815 + 0.926695i \(0.377363\pi\)
\(14\) 0 0
\(15\) 41084.4 0.209540
\(16\) 140015. 0.534115
\(17\) 229907. 0.667626 0.333813 0.942639i \(-0.391665\pi\)
0.333813 + 0.942639i \(0.391665\pi\)
\(18\) 140893. 0.316346
\(19\) −16433.6 −0.0289295 −0.0144647 0.999895i \(-0.504604\pi\)
−0.0144647 + 0.999895i \(0.504604\pi\)
\(20\) 267426. 0.373740
\(21\) 0 0
\(22\) −321827. −0.292900
\(23\) −2.57284e6 −1.91707 −0.958535 0.284975i \(-0.908015\pi\)
−0.958535 + 0.284975i \(0.908015\pi\)
\(24\) −566649. −0.348629
\(25\) 390625. 0.200000
\(26\) −709892. −0.304658
\(27\) 2.30368e6 0.834228
\(28\) 0 0
\(29\) −6.62817e6 −1.74022 −0.870108 0.492862i \(-0.835951\pi\)
−0.870108 + 0.492862i \(0.835951\pi\)
\(30\) −376809. −0.0849328
\(31\) 8.17416e6 1.58970 0.794851 0.606805i \(-0.207549\pi\)
0.794851 + 0.606805i \(0.207549\pi\)
\(32\) −5.69770e6 −0.960560
\(33\) −2.30662e6 −0.338581
\(34\) −2.10861e6 −0.270609
\(35\) 0 0
\(36\) 6.57308e6 0.652241
\(37\) 9.70272e6 0.851110 0.425555 0.904933i \(-0.360079\pi\)
0.425555 + 0.904933i \(0.360079\pi\)
\(38\) 150722. 0.0117260
\(39\) −5.08798e6 −0.352172
\(40\) −5.38762e6 −0.332757
\(41\) −2.98108e7 −1.64758 −0.823789 0.566896i \(-0.808144\pi\)
−0.823789 + 0.566896i \(0.808144\pi\)
\(42\) 0 0
\(43\) −1.95343e7 −0.871343 −0.435672 0.900106i \(-0.643489\pi\)
−0.435672 + 0.900106i \(0.643489\pi\)
\(44\) −1.50142e7 −0.603900
\(45\) 9.60119e6 0.349035
\(46\) 2.35970e7 0.777046
\(47\) −5.93794e6 −0.177499 −0.0887494 0.996054i \(-0.528287\pi\)
−0.0887494 + 0.996054i \(0.528287\pi\)
\(48\) −9.20389e6 −0.250257
\(49\) 0 0
\(50\) −3.58265e6 −0.0810660
\(51\) −1.51130e7 −0.312813
\(52\) −3.31187e7 −0.628142
\(53\) −2.74263e7 −0.477448 −0.238724 0.971088i \(-0.576729\pi\)
−0.238724 + 0.971088i \(0.576729\pi\)
\(54\) −2.11284e7 −0.338138
\(55\) −2.19310e7 −0.323166
\(56\) 0 0
\(57\) 1.08026e6 0.0135548
\(58\) 6.07908e7 0.705362
\(59\) −5.24915e7 −0.563969 −0.281984 0.959419i \(-0.590993\pi\)
−0.281984 + 0.959419i \(0.590993\pi\)
\(60\) −1.75793e7 −0.175114
\(61\) −2.23282e7 −0.206476 −0.103238 0.994657i \(-0.532920\pi\)
−0.103238 + 0.994657i \(0.532920\pi\)
\(62\) −7.49699e7 −0.644354
\(63\) 0 0
\(64\) −1.94308e7 −0.144771
\(65\) −4.83759e7 −0.336139
\(66\) 2.11553e7 0.137237
\(67\) 2.74351e8 1.66330 0.831649 0.555302i \(-0.187397\pi\)
0.831649 + 0.555302i \(0.187397\pi\)
\(68\) −9.83733e7 −0.557940
\(69\) 1.69126e8 0.898234
\(70\) 0 0
\(71\) −3.63673e8 −1.69843 −0.849216 0.528046i \(-0.822925\pi\)
−0.849216 + 0.528046i \(0.822925\pi\)
\(72\) −1.32423e8 −0.580719
\(73\) −2.09245e7 −0.0862387 −0.0431193 0.999070i \(-0.513730\pi\)
−0.0431193 + 0.999070i \(0.513730\pi\)
\(74\) −8.89892e7 −0.344980
\(75\) −2.56778e7 −0.0937090
\(76\) 7.03163e6 0.0241766
\(77\) 0 0
\(78\) 4.66648e7 0.142746
\(79\) −2.65896e8 −0.768051 −0.384025 0.923323i \(-0.625462\pi\)
−0.384025 + 0.923323i \(0.625462\pi\)
\(80\) −8.75093e7 −0.238863
\(81\) 1.50936e8 0.389592
\(82\) 2.73412e8 0.667813
\(83\) 9.43764e6 0.0218279 0.0109140 0.999940i \(-0.496526\pi\)
0.0109140 + 0.999940i \(0.496526\pi\)
\(84\) 0 0
\(85\) −1.43692e8 −0.298571
\(86\) 1.79160e8 0.353182
\(87\) 4.35704e8 0.815369
\(88\) 3.02479e8 0.537679
\(89\) 6.64876e8 1.12327 0.561637 0.827384i \(-0.310172\pi\)
0.561637 + 0.827384i \(0.310172\pi\)
\(90\) −8.80580e7 −0.141474
\(91\) 0 0
\(92\) 1.10087e9 1.60211
\(93\) −5.37329e8 −0.744847
\(94\) 5.44603e7 0.0719456
\(95\) 1.02710e7 0.0129377
\(96\) 3.74539e8 0.450066
\(97\) 1.20731e9 1.38467 0.692336 0.721575i \(-0.256582\pi\)
0.692336 + 0.721575i \(0.256582\pi\)
\(98\) 0 0
\(99\) −5.39042e8 −0.563981
\(100\) −1.67142e8 −0.167142
\(101\) −1.18204e9 −1.13028 −0.565139 0.824996i \(-0.691177\pi\)
−0.565139 + 0.824996i \(0.691177\pi\)
\(102\) 1.38610e8 0.126792
\(103\) −1.97811e9 −1.73174 −0.865870 0.500268i \(-0.833235\pi\)
−0.865870 + 0.500268i \(0.833235\pi\)
\(104\) 6.67215e8 0.559263
\(105\) 0 0
\(106\) 2.51542e8 0.193524
\(107\) 1.67828e8 0.123776 0.0618881 0.998083i \(-0.480288\pi\)
0.0618881 + 0.998083i \(0.480288\pi\)
\(108\) −9.85703e8 −0.697171
\(109\) −1.02540e9 −0.695784 −0.347892 0.937535i \(-0.613102\pi\)
−0.347892 + 0.937535i \(0.613102\pi\)
\(110\) 2.01142e8 0.130989
\(111\) −6.37809e8 −0.398783
\(112\) 0 0
\(113\) 1.27533e9 0.735814 0.367907 0.929863i \(-0.380074\pi\)
0.367907 + 0.929863i \(0.380074\pi\)
\(114\) −9.90770e6 −0.00549415
\(115\) 1.60803e9 0.857340
\(116\) 2.83608e9 1.45431
\(117\) −1.18903e9 −0.586621
\(118\) 4.81430e8 0.228594
\(119\) 0 0
\(120\) 3.54156e8 0.155912
\(121\) −1.12667e9 −0.477818
\(122\) 2.04785e8 0.0836909
\(123\) 1.95961e9 0.771965
\(124\) −3.49758e9 −1.32853
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) −2.90339e9 −0.990349 −0.495174 0.868794i \(-0.664896\pi\)
−0.495174 + 0.868794i \(0.664896\pi\)
\(128\) 3.09543e9 1.01924
\(129\) 1.28409e9 0.408264
\(130\) 4.43683e8 0.136247
\(131\) −2.05173e9 −0.608694 −0.304347 0.952561i \(-0.598438\pi\)
−0.304347 + 0.952561i \(0.598438\pi\)
\(132\) 9.86960e8 0.282955
\(133\) 0 0
\(134\) −2.51623e9 −0.674185
\(135\) −1.43980e9 −0.373078
\(136\) 1.98185e9 0.496759
\(137\) 3.25539e9 0.789514 0.394757 0.918786i \(-0.370829\pi\)
0.394757 + 0.918786i \(0.370829\pi\)
\(138\) −1.55115e9 −0.364081
\(139\) 8.26776e9 1.87854 0.939272 0.343173i \(-0.111502\pi\)
0.939272 + 0.343173i \(0.111502\pi\)
\(140\) 0 0
\(141\) 3.90331e8 0.0831662
\(142\) 3.33545e9 0.688425
\(143\) 2.71598e9 0.543143
\(144\) −2.15090e9 −0.416858
\(145\) 4.14261e9 0.778248
\(146\) 1.91910e8 0.0349551
\(147\) 0 0
\(148\) −4.15162e9 −0.711279
\(149\) 1.07127e9 0.178058 0.0890289 0.996029i \(-0.471624\pi\)
0.0890289 + 0.996029i \(0.471624\pi\)
\(150\) 2.35505e8 0.0379831
\(151\) 1.97304e9 0.308844 0.154422 0.988005i \(-0.450649\pi\)
0.154422 + 0.988005i \(0.450649\pi\)
\(152\) −1.41661e8 −0.0215255
\(153\) −3.53182e9 −0.521059
\(154\) 0 0
\(155\) −5.10885e9 −0.710936
\(156\) 2.17706e9 0.294313
\(157\) 4.61623e9 0.606372 0.303186 0.952931i \(-0.401950\pi\)
0.303186 + 0.952931i \(0.401950\pi\)
\(158\) 2.43868e9 0.311314
\(159\) 1.80287e9 0.223706
\(160\) 3.56106e9 0.429576
\(161\) 0 0
\(162\) −1.38432e9 −0.157913
\(163\) 6.26525e9 0.695175 0.347588 0.937648i \(-0.387001\pi\)
0.347588 + 0.937648i \(0.387001\pi\)
\(164\) 1.27555e10 1.37689
\(165\) 1.44163e9 0.151418
\(166\) −8.65580e7 −0.00884751
\(167\) 6.21672e9 0.618496 0.309248 0.950981i \(-0.399923\pi\)
0.309248 + 0.950981i \(0.399923\pi\)
\(168\) 0 0
\(169\) −4.61353e9 −0.435054
\(170\) 1.31788e9 0.121020
\(171\) 2.52451e8 0.0225785
\(172\) 8.35837e9 0.728188
\(173\) −8.97209e9 −0.761528 −0.380764 0.924672i \(-0.624339\pi\)
−0.380764 + 0.924672i \(0.624339\pi\)
\(174\) −3.99609e9 −0.330494
\(175\) 0 0
\(176\) 4.91306e9 0.385963
\(177\) 3.45053e9 0.264245
\(178\) −6.09796e9 −0.455297
\(179\) 1.76242e10 1.28313 0.641565 0.767069i \(-0.278286\pi\)
0.641565 + 0.767069i \(0.278286\pi\)
\(180\) −4.10818e9 −0.291691
\(181\) −1.62250e9 −0.112365 −0.0561824 0.998421i \(-0.517893\pi\)
−0.0561824 + 0.998421i \(0.517893\pi\)
\(182\) 0 0
\(183\) 1.46775e9 0.0967433
\(184\) −2.21784e10 −1.42643
\(185\) −6.06420e9 −0.380628
\(186\) 4.92815e9 0.301909
\(187\) 8.06735e9 0.482441
\(188\) 2.54074e9 0.148337
\(189\) 0 0
\(190\) −9.42010e7 −0.00524402
\(191\) 1.66601e10 0.905788 0.452894 0.891564i \(-0.350392\pi\)
0.452894 + 0.891564i \(0.350392\pi\)
\(192\) 1.27728e9 0.0678316
\(193\) 2.41341e10 1.25206 0.626028 0.779801i \(-0.284680\pi\)
0.626028 + 0.779801i \(0.284680\pi\)
\(194\) −1.10730e10 −0.561249
\(195\) 3.17999e9 0.157496
\(196\) 0 0
\(197\) −3.89843e9 −0.184413 −0.0922066 0.995740i \(-0.529392\pi\)
−0.0922066 + 0.995740i \(0.529392\pi\)
\(198\) 4.94387e9 0.228599
\(199\) 1.87489e10 0.847493 0.423747 0.905781i \(-0.360715\pi\)
0.423747 + 0.905781i \(0.360715\pi\)
\(200\) 3.36727e9 0.148813
\(201\) −1.80345e10 −0.779330
\(202\) 1.08411e10 0.458135
\(203\) 0 0
\(204\) 6.46658e9 0.261420
\(205\) 1.86317e10 0.736820
\(206\) 1.81424e10 0.701927
\(207\) 3.95238e10 1.49621
\(208\) 1.08373e10 0.401456
\(209\) −5.76647e8 −0.0209051
\(210\) 0 0
\(211\) −2.20489e9 −0.0765801 −0.0382900 0.999267i \(-0.512191\pi\)
−0.0382900 + 0.999267i \(0.512191\pi\)
\(212\) 1.17352e10 0.399007
\(213\) 2.39060e10 0.795792
\(214\) −1.53925e9 −0.0501702
\(215\) 1.22089e10 0.389677
\(216\) 1.98582e10 0.620722
\(217\) 0 0
\(218\) 9.40454e9 0.282022
\(219\) 1.37547e9 0.0404067
\(220\) 9.38388e9 0.270072
\(221\) 1.77952e10 0.501807
\(222\) 5.84971e9 0.161639
\(223\) 2.65324e10 0.718463 0.359231 0.933249i \(-0.383039\pi\)
0.359231 + 0.933249i \(0.383039\pi\)
\(224\) 0 0
\(225\) −6.00074e9 −0.156093
\(226\) −1.16967e10 −0.298248
\(227\) −7.78091e10 −1.94498 −0.972488 0.232955i \(-0.925161\pi\)
−0.972488 + 0.232955i \(0.925161\pi\)
\(228\) −4.62225e8 −0.0113278
\(229\) −4.84637e10 −1.16455 −0.582274 0.812993i \(-0.697837\pi\)
−0.582274 + 0.812993i \(0.697837\pi\)
\(230\) −1.47481e10 −0.347506
\(231\) 0 0
\(232\) −5.71362e10 −1.29484
\(233\) −2.38429e10 −0.529978 −0.264989 0.964251i \(-0.585368\pi\)
−0.264989 + 0.964251i \(0.585368\pi\)
\(234\) 1.09053e10 0.237775
\(235\) 3.71121e9 0.0793799
\(236\) 2.24602e10 0.471313
\(237\) 1.74787e10 0.359866
\(238\) 0 0
\(239\) 6.25895e10 1.24083 0.620413 0.784275i \(-0.286965\pi\)
0.620413 + 0.784275i \(0.286965\pi\)
\(240\) 5.75243e9 0.111918
\(241\) 7.96605e10 1.52113 0.760565 0.649262i \(-0.224922\pi\)
0.760565 + 0.649262i \(0.224922\pi\)
\(242\) 1.03333e10 0.193674
\(243\) −5.52651e10 −1.01677
\(244\) 9.55384e9 0.172554
\(245\) 0 0
\(246\) −1.79727e10 −0.312901
\(247\) −1.27198e9 −0.0217442
\(248\) 7.04629e10 1.18285
\(249\) −6.20384e8 −0.0102274
\(250\) 2.23915e9 0.0362538
\(251\) 5.44549e10 0.865975 0.432988 0.901400i \(-0.357459\pi\)
0.432988 + 0.901400i \(0.357459\pi\)
\(252\) 0 0
\(253\) −9.02799e10 −1.38532
\(254\) 2.66286e10 0.401418
\(255\) 9.44562e9 0.139894
\(256\) −1.84414e10 −0.268358
\(257\) 5.35278e10 0.765385 0.382693 0.923876i \(-0.374997\pi\)
0.382693 + 0.923876i \(0.374997\pi\)
\(258\) −1.17771e10 −0.165482
\(259\) 0 0
\(260\) 2.06992e10 0.280914
\(261\) 1.01821e11 1.35818
\(262\) 1.88176e10 0.246722
\(263\) −5.81425e10 −0.749364 −0.374682 0.927153i \(-0.622248\pi\)
−0.374682 + 0.927153i \(0.622248\pi\)
\(264\) −1.98835e10 −0.251927
\(265\) 1.71414e10 0.213521
\(266\) 0 0
\(267\) −4.37057e10 −0.526304
\(268\) −1.17390e11 −1.39003
\(269\) −4.67380e10 −0.544233 −0.272116 0.962264i \(-0.587724\pi\)
−0.272116 + 0.962264i \(0.587724\pi\)
\(270\) 1.32052e10 0.151220
\(271\) −2.68147e10 −0.302003 −0.151001 0.988534i \(-0.548250\pi\)
−0.151001 + 0.988534i \(0.548250\pi\)
\(272\) 3.21905e10 0.356589
\(273\) 0 0
\(274\) −2.98570e10 −0.320014
\(275\) 1.37069e10 0.144524
\(276\) −7.23660e10 −0.750661
\(277\) 1.12549e11 1.14863 0.574316 0.818633i \(-0.305268\pi\)
0.574316 + 0.818633i \(0.305268\pi\)
\(278\) −7.58284e10 −0.761431
\(279\) −1.25571e11 −1.24071
\(280\) 0 0
\(281\) −4.60761e10 −0.440857 −0.220428 0.975403i \(-0.570746\pi\)
−0.220428 + 0.975403i \(0.570746\pi\)
\(282\) −3.57995e9 −0.0337098
\(283\) −7.94071e10 −0.735902 −0.367951 0.929845i \(-0.619941\pi\)
−0.367951 + 0.929845i \(0.619941\pi\)
\(284\) 1.55609e11 1.41939
\(285\) −6.75163e8 −0.00606188
\(286\) −2.49098e10 −0.220152
\(287\) 0 0
\(288\) 8.75275e10 0.749684
\(289\) −6.57304e10 −0.554276
\(290\) −3.79942e10 −0.315447
\(291\) −7.93628e10 −0.648782
\(292\) 8.95322e9 0.0720703
\(293\) 1.41265e11 1.11977 0.559887 0.828569i \(-0.310844\pi\)
0.559887 + 0.828569i \(0.310844\pi\)
\(294\) 0 0
\(295\) 3.28072e10 0.252215
\(296\) 8.36393e10 0.633283
\(297\) 8.08351e10 0.602832
\(298\) −9.82523e9 −0.0721721
\(299\) −1.99142e11 −1.44093
\(300\) 1.09871e10 0.0783133
\(301\) 0 0
\(302\) −1.80958e10 −0.125184
\(303\) 7.77013e10 0.529586
\(304\) −2.30094e9 −0.0154517
\(305\) 1.39551e10 0.0923389
\(306\) 3.23923e10 0.211201
\(307\) 5.58349e10 0.358742 0.179371 0.983781i \(-0.442594\pi\)
0.179371 + 0.983781i \(0.442594\pi\)
\(308\) 0 0
\(309\) 1.30031e11 0.811399
\(310\) 4.68562e10 0.288164
\(311\) −5.26501e10 −0.319137 −0.159569 0.987187i \(-0.551010\pi\)
−0.159569 + 0.987187i \(0.551010\pi\)
\(312\) −4.38594e10 −0.262040
\(313\) 2.51256e11 1.47968 0.739838 0.672785i \(-0.234902\pi\)
0.739838 + 0.672785i \(0.234902\pi\)
\(314\) −4.23381e10 −0.245781
\(315\) 0 0
\(316\) 1.13772e11 0.641866
\(317\) −1.16999e11 −0.650749 −0.325375 0.945585i \(-0.605490\pi\)
−0.325375 + 0.945585i \(0.605490\pi\)
\(318\) −1.65351e10 −0.0906747
\(319\) −2.32580e11 −1.25752
\(320\) 1.21442e10 0.0647434
\(321\) −1.10322e10 −0.0579948
\(322\) 0 0
\(323\) −3.77820e9 −0.0193141
\(324\) −6.45828e10 −0.325585
\(325\) 3.02349e10 0.150326
\(326\) −5.74622e10 −0.281775
\(327\) 6.74048e10 0.326006
\(328\) −2.56975e11 −1.22591
\(329\) 0 0
\(330\) −1.32221e10 −0.0613743
\(331\) −2.51419e11 −1.15126 −0.575629 0.817711i \(-0.695243\pi\)
−0.575629 + 0.817711i \(0.695243\pi\)
\(332\) −4.03820e9 −0.0182417
\(333\) −1.49052e11 −0.664262
\(334\) −5.70171e10 −0.250695
\(335\) −1.71469e11 −0.743849
\(336\) 0 0
\(337\) 6.11427e10 0.258232 0.129116 0.991630i \(-0.458786\pi\)
0.129116 + 0.991630i \(0.458786\pi\)
\(338\) 4.23133e10 0.176340
\(339\) −8.38336e10 −0.344762
\(340\) 6.14833e10 0.249518
\(341\) 2.86828e11 1.14875
\(342\) −2.31537e9 −0.00915173
\(343\) 0 0
\(344\) −1.68389e11 −0.648338
\(345\) −1.05704e11 −0.401702
\(346\) 8.22882e10 0.308670
\(347\) −1.68668e11 −0.624524 −0.312262 0.949996i \(-0.601087\pi\)
−0.312262 + 0.949996i \(0.601087\pi\)
\(348\) −1.86430e11 −0.681410
\(349\) 3.31182e11 1.19496 0.597479 0.801885i \(-0.296169\pi\)
0.597479 + 0.801885i \(0.296169\pi\)
\(350\) 0 0
\(351\) 1.78308e11 0.627030
\(352\) −1.99930e11 −0.694122
\(353\) −3.78560e11 −1.29762 −0.648811 0.760949i \(-0.724734\pi\)
−0.648811 + 0.760949i \(0.724734\pi\)
\(354\) −3.16468e10 −0.107106
\(355\) 2.27295e11 0.759562
\(356\) −2.84489e11 −0.938728
\(357\) 0 0
\(358\) −1.61641e11 −0.520091
\(359\) −1.60137e11 −0.508822 −0.254411 0.967096i \(-0.581882\pi\)
−0.254411 + 0.967096i \(0.581882\pi\)
\(360\) 8.27641e10 0.259705
\(361\) −3.22418e11 −0.999163
\(362\) 1.48809e10 0.0455449
\(363\) 7.40617e10 0.223879
\(364\) 0 0
\(365\) 1.30778e10 0.0385671
\(366\) −1.34615e10 −0.0392130
\(367\) −5.13837e11 −1.47852 −0.739261 0.673419i \(-0.764825\pi\)
−0.739261 + 0.673419i \(0.764825\pi\)
\(368\) −3.60236e11 −1.02394
\(369\) 4.57950e11 1.28588
\(370\) 5.56182e10 0.154280
\(371\) 0 0
\(372\) 2.29914e11 0.622474
\(373\) −6.70900e10 −0.179460 −0.0897301 0.995966i \(-0.528600\pi\)
−0.0897301 + 0.995966i \(0.528600\pi\)
\(374\) −7.39903e10 −0.195548
\(375\) 1.60486e10 0.0419080
\(376\) −5.11862e10 −0.132071
\(377\) −5.13030e11 −1.30800
\(378\) 0 0
\(379\) 4.15471e11 1.03434 0.517171 0.855882i \(-0.326985\pi\)
0.517171 + 0.855882i \(0.326985\pi\)
\(380\) −4.39477e9 −0.0108121
\(381\) 1.90854e11 0.464023
\(382\) −1.52799e11 −0.367143
\(383\) 3.51976e11 0.835831 0.417915 0.908486i \(-0.362761\pi\)
0.417915 + 0.908486i \(0.362761\pi\)
\(384\) −2.03478e11 −0.477560
\(385\) 0 0
\(386\) −2.21348e11 −0.507496
\(387\) 3.00084e11 0.680053
\(388\) −5.16588e11 −1.15718
\(389\) −2.60061e11 −0.575840 −0.287920 0.957654i \(-0.592964\pi\)
−0.287920 + 0.957654i \(0.592964\pi\)
\(390\) −2.91655e10 −0.0638379
\(391\) −5.91516e11 −1.27989
\(392\) 0 0
\(393\) 1.34871e11 0.285201
\(394\) 3.57548e10 0.0747482
\(395\) 1.66185e11 0.343483
\(396\) 2.30647e11 0.471323
\(397\) −7.34338e11 −1.48367 −0.741837 0.670580i \(-0.766045\pi\)
−0.741837 + 0.670580i \(0.766045\pi\)
\(398\) −1.71957e11 −0.343514
\(399\) 0 0
\(400\) 5.46933e10 0.106823
\(401\) 8.08296e11 1.56106 0.780532 0.625116i \(-0.214948\pi\)
0.780532 + 0.625116i \(0.214948\pi\)
\(402\) 1.65405e11 0.315886
\(403\) 6.32691e11 1.19487
\(404\) 5.05773e11 0.944581
\(405\) −9.43349e10 −0.174231
\(406\) 0 0
\(407\) 3.40464e11 0.615030
\(408\) −1.30277e11 −0.232754
\(409\) 9.11153e11 1.61004 0.805020 0.593248i \(-0.202155\pi\)
0.805020 + 0.593248i \(0.202155\pi\)
\(410\) −1.70882e11 −0.298655
\(411\) −2.13993e11 −0.369923
\(412\) 8.46398e11 1.44723
\(413\) 0 0
\(414\) −3.62495e11 −0.606458
\(415\) −5.89853e9 −0.00976174
\(416\) −4.41010e11 −0.721985
\(417\) −5.43482e11 −0.880183
\(418\) 5.28876e9 0.00847345
\(419\) −4.94109e11 −0.783177 −0.391589 0.920140i \(-0.628074\pi\)
−0.391589 + 0.920140i \(0.628074\pi\)
\(420\) 0 0
\(421\) −1.15145e10 −0.0178639 −0.00893197 0.999960i \(-0.502843\pi\)
−0.00893197 + 0.999960i \(0.502843\pi\)
\(422\) 2.02223e10 0.0310402
\(423\) 9.12181e10 0.138532
\(424\) −2.36420e11 −0.355253
\(425\) 8.98076e10 0.133525
\(426\) −2.19256e11 −0.322558
\(427\) 0 0
\(428\) −7.18106e10 −0.103441
\(429\) −1.78535e11 −0.254487
\(430\) −1.11975e11 −0.157948
\(431\) −9.42534e11 −1.31568 −0.657839 0.753159i \(-0.728529\pi\)
−0.657839 + 0.753159i \(0.728529\pi\)
\(432\) 3.22549e11 0.445574
\(433\) −1.01849e12 −1.39239 −0.696196 0.717852i \(-0.745125\pi\)
−0.696196 + 0.717852i \(0.745125\pi\)
\(434\) 0 0
\(435\) −2.72315e11 −0.364644
\(436\) 4.38751e11 0.581472
\(437\) 4.22810e10 0.0554598
\(438\) −1.26152e10 −0.0163781
\(439\) 7.89357e11 1.01434 0.507169 0.861847i \(-0.330692\pi\)
0.507169 + 0.861847i \(0.330692\pi\)
\(440\) −1.89049e11 −0.240457
\(441\) 0 0
\(442\) −1.63210e11 −0.203397
\(443\) 1.06770e12 1.31714 0.658572 0.752518i \(-0.271161\pi\)
0.658572 + 0.752518i \(0.271161\pi\)
\(444\) 2.72907e11 0.333266
\(445\) −4.15547e11 −0.502343
\(446\) −2.43344e11 −0.291215
\(447\) −7.04200e10 −0.0834281
\(448\) 0 0
\(449\) −3.31695e9 −0.00385150 −0.00192575 0.999998i \(-0.500613\pi\)
−0.00192575 + 0.999998i \(0.500613\pi\)
\(450\) 5.50362e10 0.0632692
\(451\) −1.04605e12 −1.19058
\(452\) −5.45689e11 −0.614926
\(453\) −1.29698e11 −0.144707
\(454\) 7.13632e11 0.788357
\(455\) 0 0
\(456\) 9.31207e9 0.0100857
\(457\) −7.03146e11 −0.754089 −0.377045 0.926195i \(-0.623060\pi\)
−0.377045 + 0.926195i \(0.623060\pi\)
\(458\) 4.44489e11 0.472026
\(459\) 5.29633e11 0.556952
\(460\) −6.88046e11 −0.716485
\(461\) 1.86192e12 1.92003 0.960015 0.279950i \(-0.0903178\pi\)
0.960015 + 0.279950i \(0.0903178\pi\)
\(462\) 0 0
\(463\) −1.06950e11 −0.108160 −0.0540799 0.998537i \(-0.517223\pi\)
−0.0540799 + 0.998537i \(0.517223\pi\)
\(464\) −9.28043e11 −0.929474
\(465\) 3.35831e11 0.333106
\(466\) 2.18677e11 0.214816
\(467\) −4.13997e11 −0.402783 −0.201392 0.979511i \(-0.564546\pi\)
−0.201392 + 0.979511i \(0.564546\pi\)
\(468\) 5.08766e11 0.490243
\(469\) 0 0
\(470\) −3.40377e10 −0.0321751
\(471\) −3.03448e11 −0.284113
\(472\) −4.52487e11 −0.419631
\(473\) −6.85450e11 −0.629652
\(474\) −1.60307e11 −0.145865
\(475\) −6.41936e9 −0.00578589
\(476\) 0 0
\(477\) 4.21320e11 0.372631
\(478\) −5.74044e11 −0.502944
\(479\) 8.54131e10 0.0741336 0.0370668 0.999313i \(-0.488199\pi\)
0.0370668 + 0.999313i \(0.488199\pi\)
\(480\) −2.34087e11 −0.201276
\(481\) 7.51004e11 0.639719
\(482\) −7.30612e11 −0.616560
\(483\) 0 0
\(484\) 4.82082e11 0.399316
\(485\) −7.54571e11 −0.619244
\(486\) 5.06868e11 0.412127
\(487\) 4.94789e11 0.398602 0.199301 0.979938i \(-0.436133\pi\)
0.199301 + 0.979938i \(0.436133\pi\)
\(488\) −1.92474e11 −0.153632
\(489\) −4.11847e11 −0.325721
\(490\) 0 0
\(491\) 1.11163e12 0.863168 0.431584 0.902073i \(-0.357955\pi\)
0.431584 + 0.902073i \(0.357955\pi\)
\(492\) −8.38484e11 −0.645137
\(493\) −1.52387e12 −1.16181
\(494\) 1.16661e10 0.00881359
\(495\) 3.36902e11 0.252220
\(496\) 1.14450e12 0.849083
\(497\) 0 0
\(498\) 5.68990e9 0.00414546
\(499\) 8.18377e11 0.590882 0.295441 0.955361i \(-0.404533\pi\)
0.295441 + 0.955361i \(0.404533\pi\)
\(500\) 1.04463e11 0.0747480
\(501\) −4.08656e11 −0.289793
\(502\) −4.99437e11 −0.351006
\(503\) −3.13384e11 −0.218284 −0.109142 0.994026i \(-0.534810\pi\)
−0.109142 + 0.994026i \(0.534810\pi\)
\(504\) 0 0
\(505\) 7.38773e11 0.505475
\(506\) 8.28009e11 0.561510
\(507\) 3.03270e11 0.203842
\(508\) 1.24231e12 0.827642
\(509\) 1.02554e12 0.677211 0.338606 0.940928i \(-0.390045\pi\)
0.338606 + 0.940928i \(0.390045\pi\)
\(510\) −8.66311e10 −0.0567033
\(511\) 0 0
\(512\) −1.41572e12 −0.910467
\(513\) −3.78577e10 −0.0241338
\(514\) −4.90934e11 −0.310234
\(515\) 1.23632e12 0.774458
\(516\) −5.49438e11 −0.341189
\(517\) −2.08360e11 −0.128265
\(518\) 0 0
\(519\) 5.89781e11 0.356810
\(520\) −4.17010e11 −0.250110
\(521\) 3.18952e12 1.89651 0.948255 0.317510i \(-0.102847\pi\)
0.948255 + 0.317510i \(0.102847\pi\)
\(522\) −9.33862e11 −0.550510
\(523\) −9.43708e11 −0.551544 −0.275772 0.961223i \(-0.588933\pi\)
−0.275772 + 0.961223i \(0.588933\pi\)
\(524\) 8.77898e11 0.508690
\(525\) 0 0
\(526\) 5.33258e11 0.303740
\(527\) 1.87930e12 1.06133
\(528\) −3.22961e11 −0.180841
\(529\) 4.81837e12 2.67516
\(530\) −1.57214e11 −0.0865465
\(531\) 8.06370e11 0.440158
\(532\) 0 0
\(533\) −2.30740e12 −1.23837
\(534\) 4.00850e11 0.213327
\(535\) −1.04892e11 −0.0553544
\(536\) 2.36496e12 1.23761
\(537\) −1.15853e12 −0.601204
\(538\) 4.28661e11 0.220594
\(539\) 0 0
\(540\) 6.16065e11 0.311784
\(541\) −8.78618e11 −0.440973 −0.220487 0.975390i \(-0.570765\pi\)
−0.220487 + 0.975390i \(0.570765\pi\)
\(542\) 2.45933e11 0.122411
\(543\) 1.06655e11 0.0526480
\(544\) −1.30994e12 −0.641295
\(545\) 6.40875e11 0.311164
\(546\) 0 0
\(547\) 4.56216e11 0.217885 0.108943 0.994048i \(-0.465254\pi\)
0.108943 + 0.994048i \(0.465254\pi\)
\(548\) −1.39292e12 −0.659803
\(549\) 3.43004e11 0.161147
\(550\) −1.25713e11 −0.0585801
\(551\) 1.08925e11 0.0503435
\(552\) 1.45790e12 0.668347
\(553\) 0 0
\(554\) −1.03225e12 −0.465575
\(555\) 3.98630e11 0.178341
\(556\) −3.53763e12 −1.56991
\(557\) 1.63980e12 0.721842 0.360921 0.932596i \(-0.382462\pi\)
0.360921 + 0.932596i \(0.382462\pi\)
\(558\) 1.15168e12 0.502896
\(559\) −1.51198e12 −0.654927
\(560\) 0 0
\(561\) −5.30308e11 −0.226045
\(562\) 4.22590e11 0.178692
\(563\) −4.36151e11 −0.182957 −0.0914786 0.995807i \(-0.529159\pi\)
−0.0914786 + 0.995807i \(0.529159\pi\)
\(564\) −1.67016e11 −0.0695026
\(565\) −7.97079e11 −0.329066
\(566\) 7.28288e11 0.298283
\(567\) 0 0
\(568\) −3.13493e12 −1.26375
\(569\) 1.76284e12 0.705029 0.352514 0.935806i \(-0.385327\pi\)
0.352514 + 0.935806i \(0.385327\pi\)
\(570\) 6.19231e9 0.00245706
\(571\) 2.37232e11 0.0933922 0.0466961 0.998909i \(-0.485131\pi\)
0.0466961 + 0.998909i \(0.485131\pi\)
\(572\) −1.16212e12 −0.453909
\(573\) −1.09515e12 −0.424403
\(574\) 0 0
\(575\) −1.00502e12 −0.383414
\(576\) 2.98494e11 0.112988
\(577\) 3.72080e12 1.39748 0.698739 0.715377i \(-0.253745\pi\)
0.698739 + 0.715377i \(0.253745\pi\)
\(578\) 6.02851e11 0.224665
\(579\) −1.58646e12 −0.586644
\(580\) −1.77255e12 −0.650388
\(581\) 0 0
\(582\) 7.27882e11 0.262971
\(583\) −9.62377e11 −0.345014
\(584\) −1.80373e11 −0.0641674
\(585\) 7.43145e11 0.262345
\(586\) −1.29562e12 −0.453878
\(587\) 6.46176e11 0.224636 0.112318 0.993672i \(-0.464172\pi\)
0.112318 + 0.993672i \(0.464172\pi\)
\(588\) 0 0
\(589\) −1.34331e11 −0.0459892
\(590\) −3.00894e11 −0.102230
\(591\) 2.56264e11 0.0864059
\(592\) 1.35853e12 0.454590
\(593\) 5.05774e12 1.67962 0.839808 0.542883i \(-0.182667\pi\)
0.839808 + 0.542883i \(0.182667\pi\)
\(594\) −7.41385e11 −0.244346
\(595\) 0 0
\(596\) −4.58377e11 −0.148804
\(597\) −1.23246e12 −0.397089
\(598\) 1.82644e12 0.584051
\(599\) −4.61588e11 −0.146499 −0.0732494 0.997314i \(-0.523337\pi\)
−0.0732494 + 0.997314i \(0.523337\pi\)
\(600\) −2.21347e11 −0.0697258
\(601\) 6.31800e12 1.97535 0.987677 0.156509i \(-0.0500239\pi\)
0.987677 + 0.156509i \(0.0500239\pi\)
\(602\) 0 0
\(603\) −4.21455e12 −1.29815
\(604\) −8.44227e11 −0.258103
\(605\) 7.04169e11 0.213687
\(606\) −7.12643e11 −0.214657
\(607\) 1.45276e12 0.434356 0.217178 0.976132i \(-0.430315\pi\)
0.217178 + 0.976132i \(0.430315\pi\)
\(608\) 9.36335e10 0.0277885
\(609\) 0 0
\(610\) −1.27990e11 −0.0374277
\(611\) −4.59605e11 −0.133413
\(612\) 1.51120e12 0.435453
\(613\) 1.20124e12 0.343603 0.171802 0.985132i \(-0.445041\pi\)
0.171802 + 0.985132i \(0.445041\pi\)
\(614\) −5.12094e11 −0.145409
\(615\) −1.22476e12 −0.345233
\(616\) 0 0
\(617\) 3.13545e12 0.870997 0.435498 0.900189i \(-0.356572\pi\)
0.435498 + 0.900189i \(0.356572\pi\)
\(618\) −1.19259e12 −0.328884
\(619\) 5.17127e12 1.41576 0.707880 0.706333i \(-0.249652\pi\)
0.707880 + 0.706333i \(0.249652\pi\)
\(620\) 2.18599e12 0.594135
\(621\) −5.92700e12 −1.59927
\(622\) 4.82884e11 0.129356
\(623\) 0 0
\(624\) −7.12394e11 −0.188100
\(625\) 1.52588e11 0.0400000
\(626\) −2.30441e12 −0.599757
\(627\) 3.79059e10 0.00979497
\(628\) −1.97520e12 −0.506749
\(629\) 2.23073e12 0.568223
\(630\) 0 0
\(631\) −6.37331e12 −1.60042 −0.800208 0.599723i \(-0.795278\pi\)
−0.800208 + 0.599723i \(0.795278\pi\)
\(632\) −2.29208e12 −0.571481
\(633\) 1.44939e11 0.0358812
\(634\) 1.07306e12 0.263768
\(635\) 1.81462e12 0.442897
\(636\) −7.71416e11 −0.186953
\(637\) 0 0
\(638\) 2.13312e12 0.509710
\(639\) 5.58670e12 1.32557
\(640\) −1.93465e12 −0.455818
\(641\) −5.74174e12 −1.34333 −0.671665 0.740855i \(-0.734421\pi\)
−0.671665 + 0.740855i \(0.734421\pi\)
\(642\) 1.01182e11 0.0235070
\(643\) 5.85135e11 0.134992 0.0674958 0.997720i \(-0.478499\pi\)
0.0674958 + 0.997720i \(0.478499\pi\)
\(644\) 0 0
\(645\) −8.02555e11 −0.182581
\(646\) 3.46520e10 0.00782857
\(647\) 1.80915e12 0.405887 0.202943 0.979190i \(-0.434949\pi\)
0.202943 + 0.979190i \(0.434949\pi\)
\(648\) 1.30110e12 0.289883
\(649\) −1.84191e12 −0.407536
\(650\) −2.77302e11 −0.0609316
\(651\) 0 0
\(652\) −2.68079e12 −0.580963
\(653\) 2.43900e12 0.524932 0.262466 0.964941i \(-0.415464\pi\)
0.262466 + 0.964941i \(0.415464\pi\)
\(654\) −6.18208e11 −0.132140
\(655\) 1.28233e12 0.272216
\(656\) −4.17396e12 −0.879996
\(657\) 3.21440e11 0.0673063
\(658\) 0 0
\(659\) 7.42836e12 1.53429 0.767147 0.641471i \(-0.221676\pi\)
0.767147 + 0.641471i \(0.221676\pi\)
\(660\) −6.16850e11 −0.126541
\(661\) 6.34861e12 1.29352 0.646759 0.762695i \(-0.276124\pi\)
0.646759 + 0.762695i \(0.276124\pi\)
\(662\) 2.30591e12 0.466640
\(663\) −1.16977e12 −0.235119
\(664\) 8.13544e10 0.0162414
\(665\) 0 0
\(666\) 1.36704e12 0.269245
\(667\) 1.70533e13 3.33611
\(668\) −2.66002e12 −0.516882
\(669\) −1.74411e12 −0.336632
\(670\) 1.57264e12 0.301505
\(671\) −7.83487e11 −0.149204
\(672\) 0 0
\(673\) 3.17186e12 0.596000 0.298000 0.954566i \(-0.403680\pi\)
0.298000 + 0.954566i \(0.403680\pi\)
\(674\) −5.60774e11 −0.104669
\(675\) 8.99875e11 0.166846
\(676\) 1.97405e12 0.363578
\(677\) 2.32240e12 0.424901 0.212450 0.977172i \(-0.431856\pi\)
0.212450 + 0.977172i \(0.431856\pi\)
\(678\) 7.68886e11 0.139742
\(679\) 0 0
\(680\) −1.23866e12 −0.222157
\(681\) 5.11479e12 0.911309
\(682\) −2.63066e12 −0.465624
\(683\) 3.78639e12 0.665782 0.332891 0.942965i \(-0.391976\pi\)
0.332891 + 0.942965i \(0.391976\pi\)
\(684\) −1.08019e11 −0.0188690
\(685\) −2.03462e12 −0.353081
\(686\) 0 0
\(687\) 3.18577e12 0.545643
\(688\) −2.73509e12 −0.465397
\(689\) −2.12283e12 −0.358864
\(690\) 9.69470e11 0.162822
\(691\) 5.70532e12 0.951982 0.475991 0.879450i \(-0.342089\pi\)
0.475991 + 0.879450i \(0.342089\pi\)
\(692\) 3.83900e12 0.636415
\(693\) 0 0
\(694\) 1.54695e12 0.253138
\(695\) −5.16735e12 −0.840111
\(696\) 3.75585e12 0.606690
\(697\) −6.85372e12 −1.09997
\(698\) −3.03746e12 −0.484352
\(699\) 1.56732e12 0.248318
\(700\) 0 0
\(701\) 6.25160e12 0.977823 0.488912 0.872333i \(-0.337394\pi\)
0.488912 + 0.872333i \(0.337394\pi\)
\(702\) −1.63536e12 −0.254154
\(703\) −1.59450e11 −0.0246222
\(704\) −6.81818e11 −0.104614
\(705\) −2.43957e11 −0.0371931
\(706\) 3.47199e12 0.525965
\(707\) 0 0
\(708\) −1.47642e12 −0.220831
\(709\) −8.01996e12 −1.19197 −0.595983 0.802997i \(-0.703238\pi\)
−0.595983 + 0.802997i \(0.703238\pi\)
\(710\) −2.08466e12 −0.307873
\(711\) 4.08467e12 0.599437
\(712\) 5.73136e12 0.835791
\(713\) −2.10308e13 −3.04757
\(714\) 0 0
\(715\) −1.69749e12 −0.242901
\(716\) −7.54107e12 −1.07232
\(717\) −4.11433e12 −0.581383
\(718\) 1.46871e12 0.206241
\(719\) −1.35002e12 −0.188391 −0.0941953 0.995554i \(-0.530028\pi\)
−0.0941953 + 0.995554i \(0.530028\pi\)
\(720\) 1.34431e12 0.186425
\(721\) 0 0
\(722\) 2.95708e12 0.404991
\(723\) −5.23649e12 −0.712718
\(724\) 6.94238e11 0.0939042
\(725\) −2.58913e12 −0.348043
\(726\) −6.79262e11 −0.0907450
\(727\) 1.47778e13 1.96203 0.981013 0.193941i \(-0.0621270\pi\)
0.981013 + 0.193941i \(0.0621270\pi\)
\(728\) 0 0
\(729\) 6.61984e11 0.0868108
\(730\) −1.19944e11 −0.0156324
\(731\) −4.49108e12 −0.581731
\(732\) −6.28022e11 −0.0808491
\(733\) 6.70116e12 0.857398 0.428699 0.903447i \(-0.358972\pi\)
0.428699 + 0.903447i \(0.358972\pi\)
\(734\) 4.71269e12 0.599290
\(735\) 0 0
\(736\) 1.46593e13 1.84146
\(737\) 9.62686e12 1.20193
\(738\) −4.20013e12 −0.521205
\(739\) −1.39054e13 −1.71508 −0.857540 0.514417i \(-0.828008\pi\)
−0.857540 + 0.514417i \(0.828008\pi\)
\(740\) 2.59476e12 0.318094
\(741\) 8.36137e10 0.0101882
\(742\) 0 0
\(743\) −3.43953e12 −0.414047 −0.207024 0.978336i \(-0.566378\pi\)
−0.207024 + 0.978336i \(0.566378\pi\)
\(744\) −4.63188e12 −0.554216
\(745\) −6.69544e11 −0.0796298
\(746\) 6.15321e11 0.0727407
\(747\) −1.44980e11 −0.0170359
\(748\) −3.45188e12 −0.403179
\(749\) 0 0
\(750\) −1.47191e11 −0.0169866
\(751\) 1.00823e13 1.15660 0.578298 0.815826i \(-0.303717\pi\)
0.578298 + 0.815826i \(0.303717\pi\)
\(752\) −8.31401e11 −0.0948047
\(753\) −3.57960e12 −0.405748
\(754\) 4.70529e12 0.530170
\(755\) −1.23315e12 −0.138119
\(756\) 0 0
\(757\) −1.02703e13 −1.13672 −0.568360 0.822780i \(-0.692422\pi\)
−0.568360 + 0.822780i \(0.692422\pi\)
\(758\) −3.81052e12 −0.419250
\(759\) 5.93456e12 0.649083
\(760\) 8.85379e10 0.00962649
\(761\) 3.20840e12 0.346783 0.173391 0.984853i \(-0.444527\pi\)
0.173391 + 0.984853i \(0.444527\pi\)
\(762\) −1.75043e12 −0.188083
\(763\) 0 0
\(764\) −7.12855e12 −0.756974
\(765\) 2.20739e12 0.233025
\(766\) −3.22817e12 −0.338787
\(767\) −4.06292e12 −0.423896
\(768\) 1.21225e12 0.125738
\(769\) −1.52349e13 −1.57098 −0.785491 0.618872i \(-0.787590\pi\)
−0.785491 + 0.618872i \(0.787590\pi\)
\(770\) 0 0
\(771\) −3.51865e12 −0.358618
\(772\) −1.03266e13 −1.04635
\(773\) 8.54195e11 0.0860497 0.0430249 0.999074i \(-0.486301\pi\)
0.0430249 + 0.999074i \(0.486301\pi\)
\(774\) −2.75224e12 −0.275646
\(775\) 3.19303e12 0.317940
\(776\) 1.04073e13 1.03029
\(777\) 0 0
\(778\) 2.38517e12 0.233405
\(779\) 4.89898e11 0.0476636
\(780\) −1.36066e12 −0.131621
\(781\) −1.27611e13 −1.22732
\(782\) 5.42513e12 0.518776
\(783\) −1.52692e13 −1.45174
\(784\) 0 0
\(785\) −2.88514e12 −0.271178
\(786\) −1.23697e12 −0.115600
\(787\) −4.25016e12 −0.394929 −0.197465 0.980310i \(-0.563271\pi\)
−0.197465 + 0.980310i \(0.563271\pi\)
\(788\) 1.66807e12 0.154116
\(789\) 3.82200e12 0.351111
\(790\) −1.52418e12 −0.139224
\(791\) 0 0
\(792\) −4.64665e12 −0.419640
\(793\) −1.72823e12 −0.155193
\(794\) 6.73503e12 0.601378
\(795\) −1.12679e12 −0.100044
\(796\) −8.02231e12 −0.708256
\(797\) −2.02157e13 −1.77470 −0.887352 0.461093i \(-0.847458\pi\)
−0.887352 + 0.461093i \(0.847458\pi\)
\(798\) 0 0
\(799\) −1.36518e12 −0.118503
\(800\) −2.22566e12 −0.192112
\(801\) −1.02138e13 −0.876676
\(802\) −7.41334e12 −0.632746
\(803\) −7.34231e11 −0.0623179
\(804\) 7.71664e12 0.651292
\(805\) 0 0
\(806\) −5.80278e12 −0.484315
\(807\) 3.07232e12 0.254998
\(808\) −1.01894e13 −0.841002
\(809\) −6.65781e12 −0.546466 −0.273233 0.961948i \(-0.588093\pi\)
−0.273233 + 0.961948i \(0.588093\pi\)
\(810\) 8.65199e11 0.0706210
\(811\) 9.35525e12 0.759384 0.379692 0.925113i \(-0.376030\pi\)
0.379692 + 0.925113i \(0.376030\pi\)
\(812\) 0 0
\(813\) 1.76267e12 0.141502
\(814\) −3.12259e12 −0.249290
\(815\) −3.91578e12 −0.310892
\(816\) −2.11604e12 −0.167078
\(817\) 3.21018e11 0.0252075
\(818\) −8.35671e12 −0.652597
\(819\) 0 0
\(820\) −7.97219e12 −0.615766
\(821\) 1.61631e13 1.24160 0.620798 0.783971i \(-0.286809\pi\)
0.620798 + 0.783971i \(0.286809\pi\)
\(822\) 1.96265e12 0.149941
\(823\) 5.97042e12 0.453634 0.226817 0.973937i \(-0.427168\pi\)
0.226817 + 0.973937i \(0.427168\pi\)
\(824\) −1.70517e13 −1.28853
\(825\) −9.01022e11 −0.0677162
\(826\) 0 0
\(827\) 7.76424e12 0.577197 0.288599 0.957450i \(-0.406811\pi\)
0.288599 + 0.957450i \(0.406811\pi\)
\(828\) −1.69115e13 −1.25039
\(829\) 4.09585e12 0.301195 0.150598 0.988595i \(-0.451880\pi\)
0.150598 + 0.988595i \(0.451880\pi\)
\(830\) 5.40988e10 0.00395673
\(831\) −7.39839e12 −0.538186
\(832\) −1.50397e12 −0.108814
\(833\) 0 0
\(834\) 4.98458e12 0.356765
\(835\) −3.88545e12 −0.276600
\(836\) 2.46737e11 0.0174705
\(837\) 1.88306e13 1.32617
\(838\) 4.53176e12 0.317445
\(839\) −4.46741e11 −0.0311263 −0.0155631 0.999879i \(-0.504954\pi\)
−0.0155631 + 0.999879i \(0.504954\pi\)
\(840\) 0 0
\(841\) 2.94256e13 2.02835
\(842\) 1.05606e11 0.00724079
\(843\) 3.02882e12 0.206561
\(844\) 9.43433e11 0.0639985
\(845\) 2.88345e12 0.194562
\(846\) −8.36613e11 −0.0561511
\(847\) 0 0
\(848\) −3.84009e12 −0.255012
\(849\) 5.21983e12 0.344803
\(850\) −8.23677e11 −0.0541217
\(851\) −2.49636e13 −1.63164
\(852\) −1.02290e13 −0.665049
\(853\) −2.72968e13 −1.76539 −0.882696 0.469945i \(-0.844274\pi\)
−0.882696 + 0.469945i \(0.844274\pi\)
\(854\) 0 0
\(855\) −1.57782e11 −0.0100974
\(856\) 1.44671e12 0.0920979
\(857\) −7.67571e12 −0.486077 −0.243038 0.970017i \(-0.578144\pi\)
−0.243038 + 0.970017i \(0.578144\pi\)
\(858\) 1.63745e12 0.103151
\(859\) 1.67172e13 1.04759 0.523797 0.851843i \(-0.324515\pi\)
0.523797 + 0.851843i \(0.324515\pi\)
\(860\) −5.22398e12 −0.325656
\(861\) 0 0
\(862\) 8.64452e12 0.533284
\(863\) −2.85615e13 −1.75280 −0.876401 0.481582i \(-0.840062\pi\)
−0.876401 + 0.481582i \(0.840062\pi\)
\(864\) −1.31257e13 −0.801327
\(865\) 5.60756e12 0.340566
\(866\) 9.34116e12 0.564378
\(867\) 4.32079e12 0.259703
\(868\) 0 0
\(869\) −9.33018e12 −0.555010
\(870\) 2.49755e12 0.147801
\(871\) 2.12351e13 1.25018
\(872\) −8.83916e12 −0.517710
\(873\) −1.85466e13 −1.08069
\(874\) −3.87783e11 −0.0224795
\(875\) 0 0
\(876\) −5.88540e11 −0.0337682
\(877\) −8.40714e12 −0.479899 −0.239950 0.970785i \(-0.577131\pi\)
−0.239950 + 0.970785i \(0.577131\pi\)
\(878\) −7.23964e12 −0.411142
\(879\) −9.28607e12 −0.524665
\(880\) −3.07066e12 −0.172608
\(881\) 1.99694e13 1.11680 0.558398 0.829573i \(-0.311416\pi\)
0.558398 + 0.829573i \(0.311416\pi\)
\(882\) 0 0
\(883\) −2.36498e13 −1.30919 −0.654597 0.755978i \(-0.727162\pi\)
−0.654597 + 0.755978i \(0.727162\pi\)
\(884\) −7.61423e12 −0.419364
\(885\) −2.15658e12 −0.118174
\(886\) −9.79250e12 −0.533878
\(887\) 3.83859e12 0.208217 0.104108 0.994566i \(-0.466801\pi\)
0.104108 + 0.994566i \(0.466801\pi\)
\(888\) −5.49804e12 −0.296722
\(889\) 0 0
\(890\) 3.81122e12 0.203615
\(891\) 5.29627e12 0.281527
\(892\) −1.13527e13 −0.600425
\(893\) 9.75816e10 0.00513495
\(894\) 6.45862e11 0.0338159
\(895\) −1.10151e13 −0.573833
\(896\) 0 0
\(897\) 1.30906e13 0.675139
\(898\) 3.04216e10 0.00156113
\(899\) −5.41798e13 −2.76642
\(900\) 2.56761e12 0.130448
\(901\) −6.30551e12 −0.318756
\(902\) 9.59390e12 0.482576
\(903\) 0 0
\(904\) 1.09936e13 0.547495
\(905\) 1.01406e12 0.0502511
\(906\) 1.18953e12 0.0586542
\(907\) −2.31944e13 −1.13802 −0.569009 0.822331i \(-0.692673\pi\)
−0.569009 + 0.822331i \(0.692673\pi\)
\(908\) 3.32931e13 1.62543
\(909\) 1.81583e13 0.882142
\(910\) 0 0
\(911\) 1.70743e13 0.821317 0.410659 0.911789i \(-0.365299\pi\)
0.410659 + 0.911789i \(0.365299\pi\)
\(912\) 1.51253e11 0.00723980
\(913\) 3.31163e11 0.0157733
\(914\) 6.44896e12 0.305655
\(915\) −9.17341e11 −0.0432649
\(916\) 2.07368e13 0.973221
\(917\) 0 0
\(918\) −4.85757e12 −0.225750
\(919\) 1.49265e13 0.690301 0.345151 0.938547i \(-0.387828\pi\)
0.345151 + 0.938547i \(0.387828\pi\)
\(920\) 1.38615e13 0.637919
\(921\) −3.67031e12 −0.168087
\(922\) −1.70768e13 −0.778246
\(923\) −2.81488e13 −1.27659
\(924\) 0 0
\(925\) 3.79012e12 0.170222
\(926\) 9.80898e11 0.0438404
\(927\) 3.03875e13 1.35156
\(928\) 3.77653e13 1.67158
\(929\) −2.92103e12 −0.128666 −0.0643332 0.997928i \(-0.520492\pi\)
−0.0643332 + 0.997928i \(0.520492\pi\)
\(930\) −3.08010e12 −0.135018
\(931\) 0 0
\(932\) 1.02020e13 0.442906
\(933\) 3.46096e12 0.149530
\(934\) 3.79701e12 0.163260
\(935\) −5.04210e12 −0.215754
\(936\) −1.02497e13 −0.436485
\(937\) 3.53996e13 1.50027 0.750135 0.661284i \(-0.229988\pi\)
0.750135 + 0.661284i \(0.229988\pi\)
\(938\) 0 0
\(939\) −1.65163e13 −0.693295
\(940\) −1.58796e12 −0.0663384
\(941\) 4.67286e13 1.94280 0.971402 0.237439i \(-0.0763080\pi\)
0.971402 + 0.237439i \(0.0763080\pi\)
\(942\) 2.78310e12 0.115159
\(943\) 7.66985e13 3.15852
\(944\) −7.34960e12 −0.301224
\(945\) 0 0
\(946\) 6.28665e12 0.255217
\(947\) −5.22799e12 −0.211232 −0.105616 0.994407i \(-0.533681\pi\)
−0.105616 + 0.994407i \(0.533681\pi\)
\(948\) −7.47882e12 −0.300743
\(949\) −1.61958e12 −0.0648195
\(950\) 5.88756e10 0.00234520
\(951\) 7.69091e12 0.304905
\(952\) 0 0
\(953\) −2.46017e13 −0.966156 −0.483078 0.875577i \(-0.660481\pi\)
−0.483078 + 0.875577i \(0.660481\pi\)
\(954\) −3.86417e12 −0.151039
\(955\) −1.04125e13 −0.405081
\(956\) −2.67809e13 −1.03697
\(957\) 1.52886e13 0.589204
\(958\) −7.83373e11 −0.0300486
\(959\) 0 0
\(960\) −7.98302e11 −0.0303352
\(961\) 4.03773e13 1.52715
\(962\) −6.88788e12 −0.259297
\(963\) −2.57816e12 −0.0966031
\(964\) −3.40853e13 −1.27122
\(965\) −1.50838e13 −0.559936
\(966\) 0 0
\(967\) 1.24062e13 0.456267 0.228134 0.973630i \(-0.426738\pi\)
0.228134 + 0.973630i \(0.426738\pi\)
\(968\) −9.71212e12 −0.355529
\(969\) 2.48360e11 0.00904951
\(970\) 6.92060e12 0.250998
\(971\) −5.30324e12 −0.191450 −0.0957248 0.995408i \(-0.530517\pi\)
−0.0957248 + 0.995408i \(0.530517\pi\)
\(972\) 2.36469e13 0.849722
\(973\) 0 0
\(974\) −4.53799e12 −0.161565
\(975\) −1.98749e12 −0.0704344
\(976\) −3.12628e12 −0.110282
\(977\) −1.45131e13 −0.509606 −0.254803 0.966993i \(-0.582011\pi\)
−0.254803 + 0.966993i \(0.582011\pi\)
\(978\) 3.77728e12 0.132024
\(979\) 2.33302e13 0.811702
\(980\) 0 0
\(981\) 1.57521e13 0.543035
\(982\) −1.01954e13 −0.349868
\(983\) 3.61534e13 1.23498 0.617488 0.786580i \(-0.288150\pi\)
0.617488 + 0.786580i \(0.288150\pi\)
\(984\) 1.68923e13 0.574394
\(985\) 2.43652e12 0.0824721
\(986\) 1.39763e13 0.470917
\(987\) 0 0
\(988\) 5.44258e11 0.0181718
\(989\) 5.02586e13 1.67043
\(990\) −3.08992e12 −0.102232
\(991\) 4.94162e13 1.62756 0.813782 0.581170i \(-0.197405\pi\)
0.813782 + 0.581170i \(0.197405\pi\)
\(992\) −4.65739e13 −1.52700
\(993\) 1.65271e13 0.539417
\(994\) 0 0
\(995\) −1.17180e13 −0.379010
\(996\) 2.65451e11 0.00854708
\(997\) 3.03522e13 0.972886 0.486443 0.873712i \(-0.338294\pi\)
0.486443 + 0.873712i \(0.338294\pi\)
\(998\) −7.50580e12 −0.239502
\(999\) 2.23519e13 0.710020
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 245.10.a.c.1.2 2
7.6 odd 2 35.10.a.b.1.2 2
21.20 even 2 315.10.a.b.1.1 2
35.13 even 4 175.10.b.c.99.3 4
35.27 even 4 175.10.b.c.99.2 4
35.34 odd 2 175.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 2 7.6 odd 2
175.10.a.c.1.1 2 35.34 odd 2
175.10.b.c.99.2 4 35.27 even 4
175.10.b.c.99.3 4 35.13 even 4
245.10.a.c.1.2 2 1.1 even 1 trivial
315.10.a.b.1.1 2 21.20 even 2