Properties

Label 315.10.a.b.1.1
Level $315$
Weight $10$
Character 315.1
Self dual yes
Analytic conductor $162.236$
Analytic rank $1$
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] = [315,10,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(162.236288392\)
Analytic rank: \(1\)
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.17157 q^{2} -427.882 q^{4} -625.000 q^{5} +2401.00 q^{7} -8620.20 q^{8} +O(q^{10})\) \(q+9.17157 q^{2} -427.882 q^{4} -625.000 q^{5} +2401.00 q^{7} -8620.20 q^{8} -5732.23 q^{10} -35089.6 q^{11} -77401.4 q^{13} +22020.9 q^{14} +140015. q^{16} +229907. q^{17} +16433.6 q^{19} +267426. q^{20} -321827. q^{22} +2.57284e6 q^{23} +390625. q^{25} -709892. q^{26} -1.02735e6 q^{28} +6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6 q^{32} +2.10861e6 q^{34} -1.50062e6 q^{35} +9.70272e6 q^{37} +150722. q^{38} +5.38762e6 q^{40} -2.98108e7 q^{41} -1.95343e7 q^{43} +1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6 q^{47} +5.76480e6 q^{49} +3.58265e6 q^{50} +3.31187e7 q^{52} +2.74263e7 q^{53} +2.19310e7 q^{55} -2.06971e7 q^{56} +6.07908e7 q^{58} -5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7 q^{62} -1.94308e7 q^{64} +4.83759e7 q^{65} +2.74351e8 q^{67} -9.83733e7 q^{68} -1.37631e7 q^{70} +3.63673e8 q^{71} +2.09245e7 q^{73} +8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7 q^{77} -2.65896e8 q^{79} -8.75093e7 q^{80} -2.73412e8 q^{82} +9.43764e6 q^{83} -1.43692e8 q^{85} -1.79160e8 q^{86} +3.02479e8 q^{88} +6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9 q^{92} -5.44603e7 q^{94} -1.02710e7 q^{95} -1.20731e9 q^{97} +5.28723e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22} + 498908 q^{23} + 781250 q^{25} - 319736 q^{26} - 1728720 q^{28} + 11577554 q^{29} - 3953760 q^{31} + 11398656 q^{32} + 4243944 q^{34} - 3001250 q^{35} - 3205412 q^{37} - 2217520 q^{38} + 12840000 q^{40} - 1058992 q^{41} + 15948180 q^{43} + 10187376 q^{44} - 7156176 q^{46} - 65501290 q^{47} + 11529602 q^{49} + 9375000 q^{50} + 25432656 q^{52} + 25114688 q^{53} + 11603750 q^{55} - 49326144 q^{56} + 134182296 q^{58} + 116159208 q^{59} - 44688544 q^{61} - 12388000 q^{62} + 79055872 q^{64} + 31931250 q^{65} + 118092496 q^{67} - 140439024 q^{68} - 36015000 q^{70} + 294165824 q^{71} - 57419332 q^{73} - 102418064 q^{74} + 39622368 q^{76} - 44576966 q^{77} - 692852854 q^{79} - 70480000 q^{80} + 152932128 q^{82} + 540679928 q^{83} - 233693750 q^{85} + 346989008 q^{86} + 105455296 q^{88} + 779043704 q^{89} - 122667090 q^{91} - 495040608 q^{92} - 937691032 q^{94} + 89547500 q^{95} - 2673039406 q^{97} + 138355224 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.17157 0.405330 0.202665 0.979248i \(-0.435040\pi\)
0.202665 + 0.979248i \(0.435040\pi\)
\(3\) 0 0
\(4\) −427.882 −0.835708
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) −8620.20 −0.744067
\(9\) 0 0
\(10\) −5732.23 −0.181269
\(11\) −35089.6 −0.722622 −0.361311 0.932445i \(-0.617671\pi\)
−0.361311 + 0.932445i \(0.617671\pi\)
\(12\) 0 0
\(13\) −77401.4 −0.751629 −0.375815 0.926695i \(-0.622637\pi\)
−0.375815 + 0.926695i \(0.622637\pi\)
\(14\) 22020.9 0.153200
\(15\) 0 0
\(16\) 140015. 0.534115
\(17\) 229907. 0.667626 0.333813 0.942639i \(-0.391665\pi\)
0.333813 + 0.942639i \(0.391665\pi\)
\(18\) 0 0
\(19\) 16433.6 0.0289295 0.0144647 0.999895i \(-0.495396\pi\)
0.0144647 + 0.999895i \(0.495396\pi\)
\(20\) 267426. 0.373740
\(21\) 0 0
\(22\) −321827. −0.292900
\(23\) 2.57284e6 1.91707 0.958535 0.284975i \(-0.0919852\pi\)
0.958535 + 0.284975i \(0.0919852\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) −709892. −0.304658
\(27\) 0 0
\(28\) −1.02735e6 −0.315868
\(29\) 6.62817e6 1.74022 0.870108 0.492862i \(-0.164049\pi\)
0.870108 + 0.492862i \(0.164049\pi\)
\(30\) 0 0
\(31\) −8.17416e6 −1.58970 −0.794851 0.606805i \(-0.792451\pi\)
−0.794851 + 0.606805i \(0.792451\pi\)
\(32\) 5.69770e6 0.960560
\(33\) 0 0
\(34\) 2.10861e6 0.270609
\(35\) −1.50062e6 −0.169031
\(36\) 0 0
\(37\) 9.70272e6 0.851110 0.425555 0.904933i \(-0.360079\pi\)
0.425555 + 0.904933i \(0.360079\pi\)
\(38\) 150722. 0.0117260
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) 2.35970e7 0.777046
\(47\) −5.93794e6 −0.177499 −0.0887494 0.996054i \(-0.528287\pi\)
−0.0887494 + 0.996054i \(0.528287\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 3.58265e6 0.0810660
\(51\) 0 0
\(52\) 3.31187e7 0.628142
\(53\) 2.74263e7 0.477448 0.238724 0.971088i \(-0.423271\pi\)
0.238724 + 0.971088i \(0.423271\pi\)
\(54\) 0 0
\(55\) 2.19310e7 0.323166
\(56\) −2.06971e7 −0.281231
\(57\) 0 0
\(58\) 6.07908e7 0.705362
\(59\) −5.24915e7 −0.563969 −0.281984 0.959419i \(-0.590993\pi\)
−0.281984 + 0.959419i \(0.590993\pi\)
\(60\) 0 0
\(61\) 2.23282e7 0.206476 0.103238 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(62\) −7.49699e7 −0.644354
\(63\) 0 0
\(64\) −1.94308e7 −0.144771
\(65\) 4.83759e7 0.336139
\(66\) 0 0
\(67\) 2.74351e8 1.66330 0.831649 0.555302i \(-0.187397\pi\)
0.831649 + 0.555302i \(0.187397\pi\)
\(68\) −9.83733e7 −0.557940
\(69\) 0 0
\(70\) −1.37631e7 −0.0685133
\(71\) 3.63673e8 1.69843 0.849216 0.528046i \(-0.177075\pi\)
0.849216 + 0.528046i \(0.177075\pi\)
\(72\) 0 0
\(73\) 2.09245e7 0.0862387 0.0431193 0.999070i \(-0.486270\pi\)
0.0431193 + 0.999070i \(0.486270\pi\)
\(74\) 8.89892e7 0.344980
\(75\) 0 0
\(76\) −7.03163e6 −0.0241766
\(77\) −8.42501e7 −0.273125
\(78\) 0 0
\(79\) −2.65896e8 −0.768051 −0.384025 0.923323i \(-0.625462\pi\)
−0.384025 + 0.923323i \(0.625462\pi\)
\(80\) −8.75093e7 −0.238863
\(81\) 0 0
\(82\) −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
\(87\) 0 0
\(88\) 3.02479e8 0.537679
\(89\) 6.64876e8 1.12327 0.561637 0.827384i \(-0.310172\pi\)
0.561637 + 0.827384i \(0.310172\pi\)
\(90\) 0 0
\(91\) −1.85841e8 −0.284089
\(92\) −1.10087e9 −1.60211
\(93\) 0 0
\(94\) −5.44603e7 −0.0719456
\(95\) −1.02710e7 −0.0129377
\(96\) 0 0
\(97\) −1.20731e9 −1.38467 −0.692336 0.721575i \(-0.743418\pi\)
−0.692336 + 0.721575i \(0.743418\pi\)
\(98\) 5.28723e7 0.0579043
\(99\) 0 0
\(100\) −1.67142e8 −0.167142
\(101\) −1.18204e9 −1.13028 −0.565139 0.824996i \(-0.691177\pi\)
−0.565139 + 0.824996i \(0.691177\pi\)
\(102\) 0 0
\(103\) 1.97811e9 1.73174 0.865870 0.500268i \(-0.166765\pi\)
0.865870 + 0.500268i \(0.166765\pi\)
\(104\) 6.67215e8 0.559263
\(105\) 0 0
\(106\) 2.51542e8 0.193524
\(107\) −1.67828e8 −0.123776 −0.0618881 0.998083i \(-0.519712\pi\)
−0.0618881 + 0.998083i \(0.519712\pi\)
\(108\) 0 0
\(109\) −1.02540e9 −0.695784 −0.347892 0.937535i \(-0.613102\pi\)
−0.347892 + 0.937535i \(0.613102\pi\)
\(110\) 2.01142e8 0.130989
\(111\) 0 0
\(112\) 3.36176e8 0.201876
\(113\) −1.27533e9 −0.735814 −0.367907 0.929863i \(-0.619926\pi\)
−0.367907 + 0.929863i \(0.619926\pi\)
\(114\) 0 0
\(115\) −1.60803e9 −0.857340
\(116\) −2.83608e9 −1.45431
\(117\) 0 0
\(118\) −4.81430e8 −0.228594
\(119\) 5.52008e8 0.252339
\(120\) 0 0
\(121\) −1.12667e9 −0.477818
\(122\) 2.04785e8 0.0836909
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 4.43683e8 0.136247
\(131\) −2.05173e9 −0.608694 −0.304347 0.952561i \(-0.598438\pi\)
−0.304347 + 0.952561i \(0.598438\pi\)
\(132\) 0 0
\(133\) 3.94570e7 0.0109343
\(134\) 2.51623e9 0.674185
\(135\) 0 0
\(136\) −1.98185e9 −0.496759
\(137\) −3.25539e9 −0.789514 −0.394757 0.918786i \(-0.629171\pi\)
−0.394757 + 0.918786i \(0.629171\pi\)
\(138\) 0 0
\(139\) −8.26776e9 −1.87854 −0.939272 0.343173i \(-0.888498\pi\)
−0.939272 + 0.343173i \(0.888498\pi\)
\(140\) 6.42091e8 0.141260
\(141\) 0 0
\(142\) 3.33545e9 0.688425
\(143\) 2.71598e9 0.543143
\(144\) 0 0
\(145\) −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 \(-0.528376\pi\)
−0.0890289 + 0.996029i \(0.528376\pi\)
\(150\) 0 0
\(151\) 1.97304e9 0.308844 0.154422 0.988005i \(-0.450649\pi\)
0.154422 + 0.988005i \(0.450649\pi\)
\(152\) −1.41661e8 −0.0215255
\(153\) 0 0
\(154\) −7.72706e8 −0.110706
\(155\) 5.10885e9 0.710936
\(156\) 0 0
\(157\) −4.61623e9 −0.606372 −0.303186 0.952931i \(-0.598050\pi\)
−0.303186 + 0.952931i \(0.598050\pi\)
\(158\) −2.43868e9 −0.311314
\(159\) 0 0
\(160\) −3.56106e9 −0.429576
\(161\) 6.17740e9 0.724584
\(162\) 0 0
\(163\) 6.26525e9 0.695175 0.347588 0.937648i \(-0.387001\pi\)
0.347588 + 0.937648i \(0.387001\pi\)
\(164\) 1.27555e10 1.37689
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 8.35837e9 0.728188
\(173\) −8.97209e9 −0.761528 −0.380764 0.924672i \(-0.624339\pi\)
−0.380764 + 0.924672i \(0.624339\pi\)
\(174\) 0 0
\(175\) 9.37891e8 0.0755929
\(176\) −4.91306e9 −0.385963
\(177\) 0 0
\(178\) 6.09796e9 0.455297
\(179\) −1.76242e10 −1.28313 −0.641565 0.767069i \(-0.721714\pi\)
−0.641565 + 0.767069i \(0.721714\pi\)
\(180\) 0 0
\(181\) 1.62250e9 0.112365 0.0561824 0.998421i \(-0.482107\pi\)
0.0561824 + 0.998421i \(0.482107\pi\)
\(182\) −1.70445e9 −0.115150
\(183\) 0 0
\(184\) −2.21784e10 −1.42643
\(185\) −6.06420e9 −0.380628
\(186\) 0 0
\(187\) −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 \(-0.649608\pi\)
−0.452894 + 0.891564i \(0.649608\pi\)
\(192\) 0 0
\(193\) 2.41341e10 1.25206 0.626028 0.779801i \(-0.284680\pi\)
0.626028 + 0.779801i \(0.284680\pi\)
\(194\) −1.10730e10 −0.561249
\(195\) 0 0
\(196\) −2.46666e9 −0.119387
\(197\) 3.89843e9 0.184413 0.0922066 0.995740i \(-0.470608\pi\)
0.0922066 + 0.995740i \(0.470608\pi\)
\(198\) 0 0
\(199\) −1.87489e10 −0.847493 −0.423747 0.905781i \(-0.639285\pi\)
−0.423747 + 0.905781i \(0.639285\pi\)
\(200\) −3.36727e9 −0.148813
\(201\) 0 0
\(202\) −1.08411e10 −0.458135
\(203\) 1.59142e10 0.657740
\(204\) 0 0
\(205\) 1.86317e10 0.736820
\(206\) 1.81424e10 0.701927
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) −1.53925e9 −0.0501702
\(215\) 1.22089e10 0.389677
\(216\) 0 0
\(217\) −1.96262e10 −0.600851
\(218\) −9.40454e9 −0.282022
\(219\) 0 0
\(220\) −9.38388e9 −0.270072
\(221\) −1.77952e10 −0.501807
\(222\) 0 0
\(223\) −2.65324e10 −0.718463 −0.359231 0.933249i \(-0.616961\pi\)
−0.359231 + 0.933249i \(0.616961\pi\)
\(224\) 1.36802e10 0.363058
\(225\) 0 0
\(226\) −1.16967e10 −0.298248
\(227\) −7.78091e10 −1.94498 −0.972488 0.232955i \(-0.925161\pi\)
−0.972488 + 0.232955i \(0.925161\pi\)
\(228\) 0 0
\(229\) 4.84637e10 1.16455 0.582274 0.812993i \(-0.302163\pi\)
0.582274 + 0.812993i \(0.302163\pi\)
\(230\) −1.47481e10 −0.347506
\(231\) 0 0
\(232\) −5.71362e10 −1.29484
\(233\) 2.38429e10 0.529978 0.264989 0.964251i \(-0.414632\pi\)
0.264989 + 0.964251i \(0.414632\pi\)
\(234\) 0 0
\(235\) 3.71121e9 0.0793799
\(236\) 2.24602e10 0.471313
\(237\) 0 0
\(238\) 5.06278e9 0.102280
\(239\) −6.25895e10 −1.24083 −0.620413 0.784275i \(-0.713035\pi\)
−0.620413 + 0.784275i \(0.713035\pi\)
\(240\) 0 0
\(241\) −7.96605e10 −1.52113 −0.760565 0.649262i \(-0.775078\pi\)
−0.760565 + 0.649262i \(0.775078\pi\)
\(242\) −1.03333e10 −0.193674
\(243\) 0 0
\(244\) −9.55384e9 −0.172554
\(245\) −3.60300e9 −0.0638877
\(246\) 0 0
\(247\) −1.27198e9 −0.0217442
\(248\) 7.04629e10 1.18285
\(249\) 0 0
\(250\) −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
\(255\) 0 0
\(256\) −1.84414e10 −0.268358
\(257\) 5.35278e10 0.765385 0.382693 0.923876i \(-0.374997\pi\)
0.382693 + 0.923876i \(0.374997\pi\)
\(258\) 0 0
\(259\) 2.32962e10 0.321689
\(260\) −2.06992e10 −0.280914
\(261\) 0 0
\(262\) −1.88176e10 −0.246722
\(263\) 5.81425e10 0.749364 0.374682 0.927153i \(-0.377752\pi\)
0.374682 + 0.927153i \(0.377752\pi\)
\(264\) 0 0
\(265\) −1.71414e10 −0.213521
\(266\) 3.61883e8 0.00443201
\(267\) 0 0
\(268\) −1.17390e11 −1.39003
\(269\) −4.67380e10 −0.544233 −0.272116 0.962264i \(-0.587724\pi\)
−0.272116 + 0.962264i \(0.587724\pi\)
\(270\) 0 0
\(271\) 2.68147e10 0.302003 0.151001 0.988534i \(-0.451750\pi\)
0.151001 + 0.988534i \(0.451750\pi\)
\(272\) 3.21905e10 0.356589
\(273\) 0 0
\(274\) −2.98570e10 −0.320014
\(275\) −1.37069e10 −0.144524
\(276\) 0 0
\(277\) 1.12549e11 1.14863 0.574316 0.818633i \(-0.305268\pi\)
0.574316 + 0.818633i \(0.305268\pi\)
\(278\) −7.58284e10 −0.761431
\(279\) 0 0
\(280\) 1.29357e10 0.125770
\(281\) 4.60761e10 0.440857 0.220428 0.975403i \(-0.429254\pi\)
0.220428 + 0.975403i \(0.429254\pi\)
\(282\) 0 0
\(283\) 7.94071e10 0.735902 0.367951 0.929845i \(-0.380059\pi\)
0.367951 + 0.929845i \(0.380059\pi\)
\(284\) −1.55609e11 −1.41939
\(285\) 0 0
\(286\) 2.49098e10 0.220152
\(287\) −7.15757e10 −0.622726
\(288\) 0 0
\(289\) −6.57304e10 −0.554276
\(290\) −3.79942e10 −0.315447
\(291\) 0 0
\(292\) −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
\(297\) 0 0
\(298\) −9.82523e9 −0.0721721
\(299\) −1.99142e11 −1.44093
\(300\) 0 0
\(301\) −4.69018e10 −0.329337
\(302\) 1.80958e10 0.125184
\(303\) 0 0
\(304\) 2.30094e9 0.0154517
\(305\) −1.39551e10 −0.0923389
\(306\) 0 0
\(307\) −5.58349e10 −0.358742 −0.179371 0.983781i \(-0.557406\pi\)
−0.179371 + 0.983781i \(0.557406\pi\)
\(308\) 3.60491e10 0.228253
\(309\) 0 0
\(310\) 4.68562e10 0.288164
\(311\) −5.26501e10 −0.319137 −0.159569 0.987187i \(-0.551010\pi\)
−0.159569 + 0.987187i \(0.551010\pi\)
\(312\) 0 0
\(313\) −2.51256e11 −1.47968 −0.739838 0.672785i \(-0.765098\pi\)
−0.739838 + 0.672785i \(0.765098\pi\)
\(314\) −4.23381e10 −0.245781
\(315\) 0 0
\(316\) 1.13772e11 0.641866
\(317\) 1.16999e11 0.650749 0.325375 0.945585i \(-0.394510\pi\)
0.325375 + 0.945585i \(0.394510\pi\)
\(318\) 0 0
\(319\) −2.32580e11 −1.25752
\(320\) 1.21442e10 0.0647434
\(321\) 0 0
\(322\) 5.66564e10 0.293696
\(323\) 3.77820e9 0.0193141
\(324\) 0 0
\(325\) −3.02349e10 −0.150326
\(326\) 5.74622e10 0.281775
\(327\) 0 0
\(328\) 2.56975e11 1.22591
\(329\) −1.42570e10 −0.0670883
\(330\) 0 0
\(331\) −2.51419e11 −1.15126 −0.575629 0.817711i \(-0.695243\pi\)
−0.575629 + 0.817711i \(0.695243\pi\)
\(332\) −4.03820e9 −0.0182417
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 6.14833e10 0.249518
\(341\) 2.86828e11 1.14875
\(342\) 0 0
\(343\) 1.38413e10 0.0539949
\(344\) 1.68389e11 0.648338
\(345\) 0 0
\(346\) −8.22882e10 −0.308670
\(347\) 1.68668e11 0.624524 0.312262 0.949996i \(-0.398913\pi\)
0.312262 + 0.949996i \(0.398913\pi\)
\(348\) 0 0
\(349\) −3.31182e11 −1.19496 −0.597479 0.801885i \(-0.703831\pi\)
−0.597479 + 0.801885i \(0.703831\pi\)
\(350\) 8.60193e9 0.0306401
\(351\) 0 0
\(352\) −1.99930e11 −0.694122
\(353\) −3.78560e11 −1.29762 −0.648811 0.760949i \(-0.724734\pi\)
−0.648811 + 0.760949i \(0.724734\pi\)
\(354\) 0 0
\(355\) −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 \(-0.418118\pi\)
0.254411 + 0.967096i \(0.418118\pi\)
\(360\) 0 0
\(361\) −3.22418e11 −0.999163
\(362\) 1.48809e10 0.0455449
\(363\) 0 0
\(364\) 7.95179e10 0.237415
\(365\) −1.30778e10 −0.0385671
\(366\) 0 0
\(367\) 5.13837e11 1.47852 0.739261 0.673419i \(-0.235175\pi\)
0.739261 + 0.673419i \(0.235175\pi\)
\(368\) 3.60236e11 1.02394
\(369\) 0 0
\(370\) −5.56182e10 −0.154280
\(371\) 6.58505e10 0.180458
\(372\) 0 0
\(373\) −6.70900e10 −0.179460 −0.0897301 0.995966i \(-0.528600\pi\)
−0.0897301 + 0.995966i \(0.528600\pi\)
\(374\) −7.39903e10 −0.195548
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) −1.52799e11 −0.367143
\(383\) 3.51976e11 0.835831 0.417915 0.908486i \(-0.362761\pi\)
0.417915 + 0.908486i \(0.362761\pi\)
\(384\) 0 0
\(385\) 5.26563e10 0.122145
\(386\) 2.21348e11 0.507496
\(387\) 0 0
\(388\) 5.16588e11 1.15718
\(389\) 2.60061e11 0.575840 0.287920 0.957654i \(-0.407036\pi\)
0.287920 + 0.957654i \(0.407036\pi\)
\(390\) 0 0
\(391\) 5.91516e11 1.27989
\(392\) −4.96937e10 −0.106295
\(393\) 0 0
\(394\) 3.57548e10 0.0747482
\(395\) 1.66185e11 0.343483
\(396\) 0 0
\(397\) 7.34338e11 1.48367 0.741837 0.670580i \(-0.233955\pi\)
0.741837 + 0.670580i \(0.233955\pi\)
\(398\) −1.71957e11 −0.343514
\(399\) 0 0
\(400\) 5.46933e10 0.106823
\(401\) −8.08296e11 −1.56106 −0.780532 0.625116i \(-0.785052\pi\)
−0.780532 + 0.625116i \(0.785052\pi\)
\(402\) 0 0
\(403\) 6.32691e11 1.19487
\(404\) 5.05773e11 0.944581
\(405\) 0 0
\(406\) 1.45959e11 0.266602
\(407\) −3.40464e11 −0.615030
\(408\) 0 0
\(409\) −9.11153e11 −1.61004 −0.805020 0.593248i \(-0.797845\pi\)
−0.805020 + 0.593248i \(0.797845\pi\)
\(410\) 1.70882e11 0.298655
\(411\) 0 0
\(412\) −8.46398e11 −1.44723
\(413\) −1.26032e11 −0.213160
\(414\) 0 0
\(415\) −5.89853e9 −0.00976174
\(416\) −4.41010e11 −0.721985
\(417\) 0 0
\(418\) −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
\(423\) 0 0
\(424\) −2.36420e11 −0.355253
\(425\) 8.98076e10 0.133525
\(426\) 0 0
\(427\) 5.36100e10 0.0780406
\(428\) 7.18106e10 0.103441
\(429\) 0 0
\(430\) 1.11975e11 0.157948
\(431\) 9.42534e11 1.31568 0.657839 0.753159i \(-0.271471\pi\)
0.657839 + 0.753159i \(0.271471\pi\)
\(432\) 0 0
\(433\) 1.01849e12 1.39239 0.696196 0.717852i \(-0.254875\pi\)
0.696196 + 0.717852i \(0.254875\pi\)
\(434\) −1.80003e11 −0.243543
\(435\) 0 0
\(436\) 4.38751e11 0.581472
\(437\) 4.22810e10 0.0554598
\(438\) 0 0
\(439\) −7.89357e11 −1.01434 −0.507169 0.861847i \(-0.669308\pi\)
−0.507169 + 0.861847i \(0.669308\pi\)
\(440\) −1.89049e11 −0.240457
\(441\) 0 0
\(442\) −1.63210e11 −0.203397
\(443\) −1.06770e12 −1.31714 −0.658572 0.752518i \(-0.728839\pi\)
−0.658572 + 0.752518i \(0.728839\pi\)
\(444\) 0 0
\(445\) −4.15547e11 −0.502343
\(446\) −2.43344e11 −0.291215
\(447\) 0 0
\(448\) −4.66533e10 −0.0547182
\(449\) 3.31695e9 0.00385150 0.00192575 0.999998i \(-0.499387\pi\)
0.00192575 + 0.999998i \(0.499387\pi\)
\(450\) 0 0
\(451\) 1.04605e12 1.19058
\(452\) 5.45689e11 0.614926
\(453\) 0 0
\(454\) −7.13632e11 −0.788357
\(455\) 1.16150e11 0.127049
\(456\) 0 0
\(457\) −7.03146e11 −0.754089 −0.377045 0.926195i \(-0.623060\pi\)
−0.377045 + 0.926195i \(0.623060\pi\)
\(458\) 4.44489e11 0.472026
\(459\) 0 0
\(460\) 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
\(465\) 0 0
\(466\) 2.18677e11 0.214816
\(467\) −4.13997e11 −0.402783 −0.201392 0.979511i \(-0.564546\pi\)
−0.201392 + 0.979511i \(0.564546\pi\)
\(468\) 0 0
\(469\) 6.58717e11 0.628667
\(470\) 3.40377e10 0.0321751
\(471\) 0 0
\(472\) 4.52487e11 0.419631
\(473\) 6.85450e11 0.629652
\(474\) 0 0
\(475\) 6.41936e9 0.00578589
\(476\) −2.36194e11 −0.210881
\(477\) 0 0
\(478\) −5.74044e11 −0.502944
\(479\) 8.54131e10 0.0741336 0.0370668 0.999313i \(-0.488199\pi\)
0.0370668 + 0.999313i \(0.488199\pi\)
\(480\) 0 0
\(481\) −7.51004e11 −0.639719
\(482\) −7.30612e11 −0.616560
\(483\) 0 0
\(484\) 4.82082e11 0.399316
\(485\) 7.54571e11 0.619244
\(486\) 0 0
\(487\) 4.94789e11 0.398602 0.199301 0.979938i \(-0.436133\pi\)
0.199301 + 0.979938i \(0.436133\pi\)
\(488\) −1.92474e11 −0.153632
\(489\) 0 0
\(490\) −3.30452e10 −0.0258956
\(491\) −1.11163e12 −0.863168 −0.431584 0.902073i \(-0.642045\pi\)
−0.431584 + 0.902073i \(0.642045\pi\)
\(492\) 0 0
\(493\) 1.52387e12 1.16181
\(494\) −1.16661e10 −0.00881359
\(495\) 0 0
\(496\) −1.14450e12 −0.849083
\(497\) 8.73178e11 0.641947
\(498\) 0 0
\(499\) 8.18377e11 0.590882 0.295441 0.955361i \(-0.404533\pi\)
0.295441 + 0.955361i \(0.404533\pi\)
\(500\) 1.04463e11 0.0747480
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 1.24231e12 0.827642
\(509\) 1.02554e12 0.677211 0.338606 0.940928i \(-0.390045\pi\)
0.338606 + 0.940928i \(0.390045\pi\)
\(510\) 0 0
\(511\) 5.02397e10 0.0325951
\(512\) 1.41572e12 0.910467
\(513\) 0 0
\(514\) 4.90934e11 0.310234
\(515\) −1.23632e12 −0.774458
\(516\) 0 0
\(517\) 2.08360e11 0.128265
\(518\) 2.13663e11 0.130390
\(519\) 0 0
\(520\) −4.17010e11 −0.250110
\(521\) 3.18952e12 1.89651 0.948255 0.317510i \(-0.102847\pi\)
0.948255 + 0.317510i \(0.102847\pi\)
\(522\) 0 0
\(523\) 9.43708e11 0.551544 0.275772 0.961223i \(-0.411067\pi\)
0.275772 + 0.961223i \(0.411067\pi\)
\(524\) 8.77898e11 0.508690
\(525\) 0 0
\(526\) 5.33258e11 0.303740
\(527\) −1.87930e12 −1.06133
\(528\) 0 0
\(529\) 4.81837e12 2.67516
\(530\) −1.57214e11 −0.0865465
\(531\) 0 0
\(532\) −1.68829e10 −0.00913789
\(533\) 2.30740e12 1.23837
\(534\) 0 0
\(535\) 1.04892e11 0.0553544
\(536\) −2.36496e12 −1.23761
\(537\) 0 0
\(538\) −4.28661e11 −0.220594
\(539\) −2.02284e11 −0.103232
\(540\) 0 0
\(541\) −8.78618e11 −0.440973 −0.220487 0.975390i \(-0.570765\pi\)
−0.220487 + 0.975390i \(0.570765\pi\)
\(542\) 2.45933e11 0.122411
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) −1.25713e11 −0.0585801
\(551\) 1.08925e11 0.0503435
\(552\) 0 0
\(553\) −6.38416e11 −0.290296
\(554\) 1.03225e12 0.465575
\(555\) 0 0
\(556\) 3.53763e12 1.56991
\(557\) −1.63980e12 −0.721842 −0.360921 0.932596i \(-0.617538\pi\)
−0.360921 + 0.932596i \(0.617538\pi\)
\(558\) 0 0
\(559\) 1.51198e12 0.654927
\(560\) −2.10110e11 −0.0902818
\(561\) 0 0
\(562\) 4.22590e11 0.178692
\(563\) −4.36151e11 −0.182957 −0.0914786 0.995807i \(-0.529159\pi\)
−0.0914786 + 0.995807i \(0.529159\pi\)
\(564\) 0 0
\(565\) 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 \(-0.614673\pi\)
−0.352514 + 0.935806i \(0.614673\pi\)
\(570\) 0 0
\(571\) 2.37232e11 0.0933922 0.0466961 0.998909i \(-0.485131\pi\)
0.0466961 + 0.998909i \(0.485131\pi\)
\(572\) −1.16212e12 −0.453909
\(573\) 0 0
\(574\) −6.56462e11 −0.252410
\(575\) 1.00502e12 0.383414
\(576\) 0 0
\(577\) −3.72080e12 −1.39748 −0.698739 0.715377i \(-0.746255\pi\)
−0.698739 + 0.715377i \(0.746255\pi\)
\(578\) −6.02851e11 −0.224665
\(579\) 0 0
\(580\) 1.77255e12 0.650388
\(581\) 2.26598e10 0.00825017
\(582\) 0 0
\(583\) −9.62377e11 −0.345014
\(584\) −1.80373e11 −0.0641674
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) 1.35853e12 0.454590
\(593\) 5.05774e12 1.67962 0.839808 0.542883i \(-0.182667\pi\)
0.839808 + 0.542883i \(0.182667\pi\)
\(594\) 0 0
\(595\) −3.45005e11 −0.112849
\(596\) 4.58377e11 0.148804
\(597\) 0 0
\(598\) −1.82644e12 −0.584051
\(599\) 4.61588e11 0.146499 0.0732494 0.997314i \(-0.476663\pi\)
0.0732494 + 0.997314i \(0.476663\pi\)
\(600\) 0 0
\(601\) −6.31800e12 −1.97535 −0.987677 0.156509i \(-0.949976\pi\)
−0.987677 + 0.156509i \(0.949976\pi\)
\(602\) −4.30163e11 −0.133490
\(603\) 0 0
\(604\) −8.44227e11 −0.258103
\(605\) 7.04169e11 0.213687
\(606\) 0 0
\(607\) −1.45276e12 −0.434356 −0.217178 0.976132i \(-0.569685\pi\)
−0.217178 + 0.976132i \(0.569685\pi\)
\(608\) 9.36335e10 0.0277885
\(609\) 0 0
\(610\) −1.27990e11 −0.0374277
\(611\) 4.59605e11 0.133413
\(612\) 0 0
\(613\) 1.20124e12 0.343603 0.171802 0.985132i \(-0.445041\pi\)
0.171802 + 0.985132i \(0.445041\pi\)
\(614\) −5.12094e11 −0.145409
\(615\) 0 0
\(616\) 7.26252e11 0.203224
\(617\) −3.13545e12 −0.870997 −0.435498 0.900189i \(-0.643428\pi\)
−0.435498 + 0.900189i \(0.643428\pi\)
\(618\) 0 0
\(619\) −5.17127e12 −1.41576 −0.707880 0.706333i \(-0.750348\pi\)
−0.707880 + 0.706333i \(0.750348\pi\)
\(620\) −2.18599e12 −0.594135
\(621\) 0 0
\(622\) −4.82884e11 −0.129356
\(623\) 1.59637e12 0.424557
\(624\) 0 0
\(625\) 1.52588e11 0.0400000
\(626\) −2.30441e12 −0.599757
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 1.07306e12 0.263768
\(635\) 1.81462e12 0.442897
\(636\) 0 0
\(637\) −4.46204e11 −0.107376
\(638\) −2.13312e12 −0.509710
\(639\) 0 0
\(640\) 1.93465e12 0.455818
\(641\) 5.74174e12 1.34333 0.671665 0.740855i \(-0.265579\pi\)
0.671665 + 0.740855i \(0.265579\pi\)
\(642\) 0 0
\(643\) −5.85135e11 −0.134992 −0.0674958 0.997720i \(-0.521501\pi\)
−0.0674958 + 0.997720i \(0.521501\pi\)
\(644\) −2.64320e12 −0.605541
\(645\) 0 0
\(646\) 3.46520e10 0.00782857
\(647\) 1.80915e12 0.405887 0.202943 0.979190i \(-0.434949\pi\)
0.202943 + 0.979190i \(0.434949\pi\)
\(648\) 0 0
\(649\) 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 \(-0.584536\pi\)
−0.262466 + 0.964941i \(0.584536\pi\)
\(654\) 0 0
\(655\) 1.28233e12 0.272216
\(656\) −4.17396e12 −0.879996
\(657\) 0 0
\(658\) −1.30759e11 −0.0271929
\(659\) −7.42836e12 −1.53429 −0.767147 0.641471i \(-0.778324\pi\)
−0.767147 + 0.641471i \(0.778324\pi\)
\(660\) 0 0
\(661\) −6.34861e12 −1.29352 −0.646759 0.762695i \(-0.723876\pi\)
−0.646759 + 0.762695i \(0.723876\pi\)
\(662\) −2.30591e12 −0.466640
\(663\) 0 0
\(664\) −8.13544e10 −0.0162414
\(665\) −2.46606e10 −0.00488997
\(666\) 0 0
\(667\) 1.70533e13 3.33611
\(668\) −2.66002e12 −0.516882
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 1.97405e12 0.363578
\(677\) 2.32240e12 0.424901 0.212450 0.977172i \(-0.431856\pi\)
0.212450 + 0.977172i \(0.431856\pi\)
\(678\) 0 0
\(679\) −2.89876e12 −0.523357
\(680\) 1.23866e12 0.222157
\(681\) 0 0
\(682\) 2.63066e12 0.465624
\(683\) −3.78639e12 −0.665782 −0.332891 0.942965i \(-0.608024\pi\)
−0.332891 + 0.942965i \(0.608024\pi\)
\(684\) 0 0
\(685\) 2.03462e12 0.353081
\(686\) 1.26946e11 0.0218858
\(687\) 0 0
\(688\) −2.73509e12 −0.465397
\(689\) −2.12283e12 −0.358864
\(690\) 0 0
\(691\) −5.70532e12 −0.951982 −0.475991 0.879450i \(-0.657911\pi\)
−0.475991 + 0.879450i \(0.657911\pi\)
\(692\) 3.83900e12 0.636415
\(693\) 0 0
\(694\) 1.54695e12 0.253138
\(695\) 5.16735e12 0.840111
\(696\) 0 0
\(697\) −6.85372e12 −1.09997
\(698\) −3.03746e12 −0.484352
\(699\) 0 0
\(700\) −4.01307e11 −0.0631736
\(701\) −6.25160e12 −0.977823 −0.488912 0.872333i \(-0.662606\pi\)
−0.488912 + 0.872333i \(0.662606\pi\)
\(702\) 0 0
\(703\) 1.59450e11 0.0246222
\(704\) 6.81818e11 0.104614
\(705\) 0 0
\(706\) −3.47199e12 −0.525965
\(707\) −2.83807e12 −0.427205
\(708\) 0 0
\(709\) −8.01996e12 −1.19197 −0.595983 0.802997i \(-0.703238\pi\)
−0.595983 + 0.802997i \(0.703238\pi\)
\(710\) −2.08466e12 −0.307873
\(711\) 0 0
\(712\) −5.73136e12 −0.835791
\(713\) −2.10308e13 −3.04757
\(714\) 0 0
\(715\) −1.69749e12 −0.242901
\(716\) 7.54107e12 1.07232
\(717\) 0 0
\(718\) 1.46871e12 0.206241
\(719\) −1.35002e12 −0.188391 −0.0941953 0.995554i \(-0.530028\pi\)
−0.0941953 + 0.995554i \(0.530028\pi\)
\(720\) 0 0
\(721\) 4.74944e12 0.654537
\(722\) −2.95708e12 −0.404991
\(723\) 0 0
\(724\) −6.94238e11 −0.0939042
\(725\) 2.58913e12 0.348043
\(726\) 0 0
\(727\) −1.47778e13 −1.96203 −0.981013 0.193941i \(-0.937873\pi\)
−0.981013 + 0.193941i \(0.937873\pi\)
\(728\) 1.60198e12 0.211381
\(729\) 0 0
\(730\) −1.19944e11 −0.0156324
\(731\) −4.49108e12 −0.581731
\(732\) 0 0
\(733\) −6.70116e12 −0.857398 −0.428699 0.903447i \(-0.641028\pi\)
−0.428699 + 0.903447i \(0.641028\pi\)
\(734\) 4.71269e12 0.599290
\(735\) 0 0
\(736\) 1.46593e13 1.84146
\(737\) −9.62686e12 −1.20193
\(738\) 0 0
\(739\) −1.39054e13 −1.71508 −0.857540 0.514417i \(-0.828008\pi\)
−0.857540 + 0.514417i \(0.828008\pi\)
\(740\) 2.59476e12 0.318094
\(741\) 0 0
\(742\) 6.03953e11 0.0731452
\(743\) 3.43953e12 0.414047 0.207024 0.978336i \(-0.433622\pi\)
0.207024 + 0.978336i \(0.433622\pi\)
\(744\) 0 0
\(745\) 6.69544e11 0.0796298
\(746\) −6.15321e11 −0.0727407
\(747\) 0 0
\(748\) 3.45188e12 0.403179
\(749\) −4.02955e11 −0.0467830
\(750\) 0 0
\(751\) 1.00823e13 1.15660 0.578298 0.815826i \(-0.303717\pi\)
0.578298 + 0.815826i \(0.303717\pi\)
\(752\) −8.31401e11 −0.0948047
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) 8.85379e10 0.00962649
\(761\) 3.20840e12 0.346783 0.173391 0.984853i \(-0.444527\pi\)
0.173391 + 0.984853i \(0.444527\pi\)
\(762\) 0 0
\(763\) −2.46199e12 −0.262982
\(764\) 7.12855e12 0.756974
\(765\) 0 0
\(766\) 3.22817e12 0.338787
\(767\) 4.06292e12 0.423896
\(768\) 0 0
\(769\) 1.52349e13 1.57098 0.785491 0.618872i \(-0.212410\pi\)
0.785491 + 0.618872i \(0.212410\pi\)
\(770\) 4.82941e11 0.0495092
\(771\) 0 0
\(772\) −1.03266e13 −1.04635
\(773\) 8.54195e11 0.0860497 0.0430249 0.999074i \(-0.486301\pi\)
0.0430249 + 0.999074i \(0.486301\pi\)
\(774\) 0 0
\(775\) −3.19303e12 −0.317940
\(776\) 1.04073e13 1.03029
\(777\) 0 0
\(778\) 2.38517e12 0.233405
\(779\) −4.89898e11 −0.0476636
\(780\) 0 0
\(781\) −1.27611e13 −1.22732
\(782\) 5.42513e12 0.518776
\(783\) 0 0
\(784\) 8.07158e11 0.0763021
\(785\) 2.88514e12 0.271178
\(786\) 0 0
\(787\) 4.25016e12 0.394929 0.197465 0.980310i \(-0.436729\pi\)
0.197465 + 0.980310i \(0.436729\pi\)
\(788\) −1.66807e12 −0.154116
\(789\) 0 0
\(790\) 1.52418e12 0.139224
\(791\) −3.06206e12 −0.278112
\(792\) 0 0
\(793\) −1.72823e12 −0.155193
\(794\) 6.73503e12 0.601378
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) −7.41334e12 −0.632746
\(803\) −7.34231e11 −0.0623179
\(804\) 0 0
\(805\) −3.86087e12 −0.324044
\(806\) 5.80278e12 0.484315
\(807\) 0 0
\(808\) 1.01894e13 0.841002
\(809\) 6.65781e12 0.546466 0.273233 0.961948i \(-0.411907\pi\)
0.273233 + 0.961948i \(0.411907\pi\)
\(810\) 0 0
\(811\) −9.35525e12 −0.759384 −0.379692 0.925113i \(-0.623970\pi\)
−0.379692 + 0.925113i \(0.623970\pi\)
\(812\) −6.80942e12 −0.549678
\(813\) 0 0
\(814\) −3.12259e12 −0.249290
\(815\) −3.91578e12 −0.310892
\(816\) 0 0
\(817\) −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 \(-0.713191\pi\)
−0.620798 + 0.783971i \(0.713191\pi\)
\(822\) 0 0
\(823\) 5.97042e12 0.453634 0.226817 0.973937i \(-0.427168\pi\)
0.226817 + 0.973937i \(0.427168\pi\)
\(824\) −1.70517e13 −1.28853
\(825\) 0 0
\(826\) −1.15591e12 −0.0864003
\(827\) −7.76424e12 −0.577197 −0.288599 0.957450i \(-0.593189\pi\)
−0.288599 + 0.957450i \(0.593189\pi\)
\(828\) 0 0
\(829\) −4.09585e12 −0.301195 −0.150598 0.988595i \(-0.548120\pi\)
−0.150598 + 0.988595i \(0.548120\pi\)
\(830\) −5.40988e10 −0.00395673
\(831\) 0 0
\(832\) 1.50397e12 0.108814
\(833\) 1.32537e12 0.0953751
\(834\) 0 0
\(835\) −3.88545e12 −0.276600
\(836\) 2.46737e11 0.0174705
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) 9.43433e11 0.0639985
\(845\) 2.88345e12 0.194562
\(846\) 0 0
\(847\) −2.70513e12 −0.180598
\(848\) 3.84009e12 0.255012
\(849\) 0 0
\(850\) 8.23677e11 0.0541217
\(851\) 2.49636e13 1.63164
\(852\) 0 0
\(853\) 2.72968e13 1.76539 0.882696 0.469945i \(-0.155726\pi\)
0.882696 + 0.469945i \(0.155726\pi\)
\(854\) 4.91688e11 0.0316322
\(855\) 0 0
\(856\) 1.44671e12 0.0920979
\(857\) −7.67571e12 −0.486077 −0.243038 0.970017i \(-0.578144\pi\)
−0.243038 + 0.970017i \(0.578144\pi\)
\(858\) 0 0
\(859\) −1.67172e13 −1.04759 −0.523797 0.851843i \(-0.675485\pi\)
−0.523797 + 0.851843i \(0.675485\pi\)
\(860\) −5.22398e12 −0.325656
\(861\) 0 0
\(862\) 8.64452e12 0.533284
\(863\) 2.85615e13 1.75280 0.876401 0.481582i \(-0.159938\pi\)
0.876401 + 0.481582i \(0.159938\pi\)
\(864\) 0 0
\(865\) 5.60756e12 0.340566
\(866\) 9.34116e12 0.564378
\(867\) 0 0
\(868\) 8.39769e12 0.502135
\(869\) 9.33018e12 0.555010
\(870\) 0 0
\(871\) −2.12351e13 −1.25018
\(872\) 8.83916e12 0.517710
\(873\) 0 0
\(874\) 3.87783e11 0.0224795
\(875\) −5.86182e11 −0.0338062
\(876\) 0 0
\(877\) −8.40714e12 −0.479899 −0.239950 0.970785i \(-0.577131\pi\)
−0.239950 + 0.970785i \(0.577131\pi\)
\(878\) −7.23964e12 −0.411142
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) −9.79250e12 −0.533878
\(887\) 3.83859e12 0.208217 0.104108 0.994566i \(-0.466801\pi\)
0.104108 + 0.994566i \(0.466801\pi\)
\(888\) 0 0
\(889\) −6.97103e12 −0.374317
\(890\) −3.81122e12 −0.203615
\(891\) 0 0
\(892\) 1.13527e13 0.600425
\(893\) −9.75816e10 −0.00513495
\(894\) 0 0
\(895\) 1.10151e13 0.573833
\(896\) −7.43213e12 −0.385237
\(897\) 0 0
\(898\) 3.04216e10 0.00156113
\(899\) −5.41798e13 −2.76642
\(900\) 0 0
\(901\) 6.30551e12 0.318756
\(902\) 9.59390e12 0.482576
\(903\) 0 0
\(904\) 1.09936e13 0.547495
\(905\) −1.01406e12 −0.0502511
\(906\) 0 0
\(907\) −2.31944e13 −1.13802 −0.569009 0.822331i \(-0.692673\pi\)
−0.569009 + 0.822331i \(0.692673\pi\)
\(908\) 3.32931e13 1.62543
\(909\) 0 0
\(910\) 1.06528e12 0.0514966
\(911\) −1.70743e13 −0.821317 −0.410659 0.911789i \(-0.634701\pi\)
−0.410659 + 0.911789i \(0.634701\pi\)
\(912\) 0 0
\(913\) −3.31163e11 −0.0157733
\(914\) −6.44896e12 −0.305655
\(915\) 0 0
\(916\) −2.07368e13 −0.973221
\(917\) −4.92620e12 −0.230065
\(918\) 0 0
\(919\) 1.49265e13 0.690301 0.345151 0.938547i \(-0.387828\pi\)
0.345151 + 0.938547i \(0.387828\pi\)
\(920\) 1.38615e13 0.637919
\(921\) 0 0
\(922\) 1.70768e13 0.778246
\(923\) −2.81488e13 −1.27659
\(924\) 0 0
\(925\) 3.79012e12 0.170222
\(926\) −9.80898e11 −0.0438404
\(927\) 0 0
\(928\) 3.77653e13 1.67158
\(929\) −2.92103e12 −0.128666 −0.0643332 0.997928i \(-0.520492\pi\)
−0.0643332 + 0.997928i \(0.520492\pi\)
\(930\) 0 0
\(931\) 9.47362e10 0.00413278
\(932\) −1.02020e13 −0.442906
\(933\) 0 0
\(934\) −3.79701e12 −0.163260
\(935\) 5.04210e12 0.215754
\(936\) 0 0
\(937\) −3.53996e13 −1.50027 −0.750135 0.661284i \(-0.770012\pi\)
−0.750135 + 0.661284i \(0.770012\pi\)
\(938\) 6.04147e12 0.254818
\(939\) 0 0
\(940\) −1.58796e12 −0.0663384
\(941\) 4.67286e13 1.94280 0.971402 0.237439i \(-0.0763080\pi\)
0.971402 + 0.237439i \(0.0763080\pi\)
\(942\) 0 0
\(943\) −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 \(-0.466319\pi\)
0.105616 + 0.994407i \(0.466319\pi\)
\(948\) 0 0
\(949\) −1.61958e12 −0.0648195
\(950\) 5.88756e10 0.00234520
\(951\) 0 0
\(952\) −4.75842e12 −0.187757
\(953\) 2.46017e13 0.966156 0.483078 0.875577i \(-0.339519\pi\)
0.483078 + 0.875577i \(0.339519\pi\)
\(954\) 0 0
\(955\) 1.04125e13 0.405081
\(956\) 2.67809e13 1.03697
\(957\) 0 0
\(958\) 7.83373e11 0.0300486
\(959\) −7.81618e12 −0.298408
\(960\) 0 0
\(961\) 4.03773e13 1.52715
\(962\) −6.88788e12 −0.259297
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 6.92060e12 0.250998
\(971\) −5.30324e12 −0.191450 −0.0957248 0.995408i \(-0.530517\pi\)
−0.0957248 + 0.995408i \(0.530517\pi\)
\(972\) 0 0
\(973\) −1.98509e13 −0.710023
\(974\) 4.53799e12 0.161565
\(975\) 0 0
\(976\) 3.12628e12 0.110282
\(977\) 1.45131e13 0.509606 0.254803 0.966993i \(-0.417989\pi\)
0.254803 + 0.966993i \(0.417989\pi\)
\(978\) 0 0
\(979\) −2.33302e13 −0.811702
\(980\) 1.54166e12 0.0533914
\(981\) 0 0
\(982\) −1.01954e13 −0.349868
\(983\) 3.61534e13 1.23498 0.617488 0.786580i \(-0.288150\pi\)
0.617488 + 0.786580i \(0.288150\pi\)
\(984\) 0 0
\(985\) −2.43652e12 −0.0824721
\(986\) 1.39763e13 0.470917
\(987\) 0 0
\(988\) 5.44258e11 0.0181718
\(989\) −5.02586e13 −1.67043
\(990\) 0 0
\(991\) 4.94162e13 1.62756 0.813782 0.581170i \(-0.197405\pi\)
0.813782 + 0.581170i \(0.197405\pi\)
\(992\) −4.65739e13 −1.52700
\(993\) 0 0
\(994\) 8.00841e12 0.260200
\(995\) 1.17180e13 0.379010
\(996\) 0 0
\(997\) −3.03522e13 −0.972886 −0.486443 0.873712i \(-0.661706\pi\)
−0.486443 + 0.873712i \(0.661706\pi\)
\(998\) 7.50580e12 0.239502
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.10.a.b.1.1 2
3.2 odd 2 35.10.a.b.1.2 2
15.2 even 4 175.10.b.c.99.2 4
15.8 even 4 175.10.b.c.99.3 4
15.14 odd 2 175.10.a.c.1.1 2
21.20 even 2 245.10.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 2 3.2 odd 2
175.10.a.c.1.1 2 15.14 odd 2
175.10.b.c.99.2 4 15.2 even 4
175.10.b.c.99.3 4 15.8 even 4
245.10.a.c.1.2 2 21.20 even 2
315.10.a.b.1.1 2 1.1 even 1 trivial