Properties

Label 175.10.b.c.99.3
Level $175$
Weight $10$
Character 175.99
Analytic conductor $90.131$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,10,Mod(99,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.10.b.c.99.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.17157i q^{2} +65.7351i q^{3} +427.882 q^{4} -602.894 q^{6} -2401.00i q^{7} +8620.20i q^{8} +15361.9 q^{9} +O(q^{10})\) \(q+9.17157i q^{2} +65.7351i q^{3} +427.882 q^{4} -602.894 q^{6} -2401.00i q^{7} +8620.20i q^{8} +15361.9 q^{9} +35089.6 q^{11} +28126.9i q^{12} -77401.4i q^{13} +22020.9 q^{14} +140015. q^{16} +229907. i q^{17} +140893. i q^{18} -16433.6 q^{19} +157830. q^{21} +321827. i q^{22} -2.57284e6i q^{23} -566649. q^{24} +709892. q^{26} +2.30368e6i q^{27} -1.02735e6i q^{28} +6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6i q^{32} +2.30662e6i q^{33} -2.10861e6 q^{34} +6.57308e6 q^{36} -9.70272e6i q^{37} -150722. i q^{38} +5.08798e6 q^{39} +2.98108e7 q^{41} +1.44755e6i q^{42} -1.95343e7i q^{43} +1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6i q^{47} +9.20389e6i q^{48} -5.76480e6 q^{49} -1.51130e7 q^{51} -3.31187e7i q^{52} -2.74263e7i q^{53} -2.11284e7 q^{54} +2.06971e7 q^{56} -1.08026e6i q^{57} +6.07908e7i q^{58} -5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7i q^{62} -3.68839e7i q^{63} +1.94308e7 q^{64} -2.11553e7 q^{66} -2.74351e8i q^{67} +9.83733e7i q^{68} +1.69126e8 q^{69} -3.63673e8 q^{71} +1.32423e8i q^{72} +2.09245e7i q^{73} +8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7i q^{77} +4.66648e7i q^{78} +2.65896e8 q^{79} +1.50936e8 q^{81} +2.73412e8i q^{82} -9.43764e6i q^{83} +6.75326e7 q^{84} +1.79160e8 q^{86} +4.35704e8i q^{87} +3.02479e8i q^{88} +6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9i q^{92} -5.37329e8i q^{93} +5.44603e7 q^{94} -3.74539e8 q^{96} +1.20731e9i q^{97} -5.28723e7i q^{98} +5.39042e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9} + 37132 q^{11} + 115248 q^{14} + 225536 q^{16} + 286552 q^{19} - 835548 q^{21} + 4583808 q^{24} + 639472 q^{26} + 23155108 q^{29} - 7907520 q^{31} - 8487888 q^{34} - 8932032 q^{36} + 22791492 q^{39} + 2117984 q^{41} + 20374752 q^{44} - 14312352 q^{46} - 23059204 q^{49} + 38818932 q^{51} - 170992944 q^{54} + 98652288 q^{56} + 232318416 q^{59} - 89377088 q^{61} - 158111744 q^{64} - 159789648 q^{66} + 1332641784 q^{69} - 588331648 q^{71} - 204836128 q^{74} + 79244736 q^{76} + 1385705708 q^{79} + 895546692 q^{81} - 201223008 q^{84} - 693978016 q^{86} + 1558087408 q^{89} - 245334180 q^{91} + 1875382064 q^{94} + 1984361472 q^{96} + 2326933080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.17157i 0.405330i 0.979248 + 0.202665i \(0.0649603\pi\)
−0.979248 + 0.202665i \(0.935040\pi\)
\(3\) 65.7351i 0.468545i 0.972171 + 0.234273i \(0.0752708\pi\)
−0.972171 + 0.234273i \(0.924729\pi\)
\(4\) 427.882 0.835708
\(5\) 0 0
\(6\) −602.894 −0.189915
\(7\) − 2401.00i − 0.377964i
\(8\) 8620.20i 0.744067i
\(9\) 15361.9 0.780465
\(10\) 0 0
\(11\) 35089.6 0.722622 0.361311 0.932445i \(-0.382329\pi\)
0.361311 + 0.932445i \(0.382329\pi\)
\(12\) 28126.9i 0.391567i
\(13\) − 77401.4i − 0.751629i −0.926695 0.375815i \(-0.877363\pi\)
0.926695 0.375815i \(-0.122637\pi\)
\(14\) 22020.9 0.153200
\(15\) 0 0
\(16\) 140015. 0.534115
\(17\) 229907.i 0.667626i 0.942639 + 0.333813i \(0.108335\pi\)
−0.942639 + 0.333813i \(0.891665\pi\)
\(18\) 140893.i 0.316346i
\(19\) −16433.6 −0.0289295 −0.0144647 0.999895i \(-0.504604\pi\)
−0.0144647 + 0.999895i \(0.504604\pi\)
\(20\) 0 0
\(21\) 157830. 0.177093
\(22\) 321827.i 0.292900i
\(23\) − 2.57284e6i − 1.91707i −0.284975 0.958535i \(-0.591985\pi\)
0.284975 0.958535i \(-0.408015\pi\)
\(24\) −566649. −0.348629
\(25\) 0 0
\(26\) 709892. 0.304658
\(27\) 2.30368e6i 0.834228i
\(28\) − 1.02735e6i − 0.315868i
\(29\) 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 \(0.792451\pi\)
\(32\) 5.69770e6i 0.960560i
\(33\) 2.30662e6i 0.338581i
\(34\) −2.10861e6 −0.270609
\(35\) 0 0
\(36\) 6.57308e6 0.652241
\(37\) − 9.70272e6i − 0.851110i −0.904933 0.425555i \(-0.860079\pi\)
0.904933 0.425555i \(-0.139921\pi\)
\(38\) − 150722.i − 0.0117260i
\(39\) 5.08798e6 0.352172
\(40\) 0 0
\(41\) 2.98108e7 1.64758 0.823789 0.566896i \(-0.191856\pi\)
0.823789 + 0.566896i \(0.191856\pi\)
\(42\) 1.44755e6i 0.0717813i
\(43\) − 1.95343e7i − 0.871343i −0.900106 0.435672i \(-0.856511\pi\)
0.900106 0.435672i \(-0.143489\pi\)
\(44\) 1.50142e7 0.603900
\(45\) 0 0
\(46\) 2.35970e7 0.777046
\(47\) − 5.93794e6i − 0.177499i −0.996054 0.0887494i \(-0.971713\pi\)
0.996054 0.0887494i \(-0.0282870\pi\)
\(48\) 9.20389e6i 0.250257i
\(49\) −5.76480e6 −0.142857
\(50\) 0 0
\(51\) −1.51130e7 −0.312813
\(52\) − 3.31187e7i − 0.628142i
\(53\) − 2.74263e7i − 0.477448i −0.971088 0.238724i \(-0.923271\pi\)
0.971088 0.238724i \(-0.0767291\pi\)
\(54\) −2.11284e7 −0.338138
\(55\) 0 0
\(56\) 2.06971e7 0.281231
\(57\) − 1.08026e6i − 0.0135548i
\(58\) 6.07908e7i 0.705362i
\(59\) −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 \(0.467080\pi\)
\(62\) − 7.49699e7i − 0.644354i
\(63\) − 3.68839e7i − 0.294988i
\(64\) 1.94308e7 0.144771
\(65\) 0 0
\(66\) −2.11553e7 −0.137237
\(67\) − 2.74351e8i − 1.66330i −0.555302 0.831649i \(-0.687397\pi\)
0.555302 0.831649i \(-0.312603\pi\)
\(68\) 9.83733e7i 0.557940i
\(69\) 1.69126e8 0.898234
\(70\) 0 0
\(71\) −3.63673e8 −1.69843 −0.849216 0.528046i \(-0.822925\pi\)
−0.849216 + 0.528046i \(0.822925\pi\)
\(72\) 1.32423e8i 0.580719i
\(73\) 2.09245e7i 0.0862387i 0.999070 + 0.0431193i \(0.0137296\pi\)
−0.999070 + 0.0431193i \(0.986270\pi\)
\(74\) 8.89892e7 0.344980
\(75\) 0 0
\(76\) −7.03163e6 −0.0241766
\(77\) − 8.42501e7i − 0.273125i
\(78\) 4.66648e7i 0.142746i
\(79\) 2.65896e8 0.768051 0.384025 0.923323i \(-0.374538\pi\)
0.384025 + 0.923323i \(0.374538\pi\)
\(80\) 0 0
\(81\) 1.50936e8 0.389592
\(82\) 2.73412e8i 0.667813i
\(83\) − 9.43764e6i − 0.0218279i −0.999940 0.0109140i \(-0.996526\pi\)
0.999940 0.0109140i \(-0.00347409\pi\)
\(84\) 6.75326e7 0.147998
\(85\) 0 0
\(86\) 1.79160e8 0.353182
\(87\) 4.35704e8i 0.815369i
\(88\) 3.02479e8i 0.537679i
\(89\) 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
\(92\) − 1.10087e9i − 1.60211i
\(93\) − 5.37329e8i − 0.744847i
\(94\) 5.44603e7 0.0719456
\(95\) 0 0
\(96\) −3.74539e8 −0.450066
\(97\) 1.20731e9i 1.38467i 0.721575 + 0.692336i \(0.243418\pi\)
−0.721575 + 0.692336i \(0.756582\pi\)
\(98\) − 5.28723e7i − 0.0579043i
\(99\) 5.39042e8 0.563981
\(100\) 0 0
\(101\) 1.18204e9 1.13028 0.565139 0.824996i \(-0.308823\pi\)
0.565139 + 0.824996i \(0.308823\pi\)
\(102\) − 1.38610e8i − 0.126792i
\(103\) 1.97811e9i 1.73174i 0.500268 + 0.865870i \(0.333235\pi\)
−0.500268 + 0.865870i \(0.666765\pi\)
\(104\) 6.67215e8 0.559263
\(105\) 0 0
\(106\) 2.51542e8 0.193524
\(107\) − 1.67828e8i − 0.123776i −0.998083 0.0618881i \(-0.980288\pi\)
0.998083 0.0618881i \(-0.0197122\pi\)
\(108\) 9.85703e8i 0.697171i
\(109\) 1.02540e9 0.695784 0.347892 0.937535i \(-0.386898\pi\)
0.347892 + 0.937535i \(0.386898\pi\)
\(110\) 0 0
\(111\) 6.37809e8 0.398783
\(112\) − 3.36176e8i − 0.201876i
\(113\) 1.27533e9i 0.735814i 0.929863 + 0.367907i \(0.119926\pi\)
−0.929863 + 0.367907i \(0.880074\pi\)
\(114\) 9.90770e6 0.00549415
\(115\) 0 0
\(116\) 2.83608e9 1.45431
\(117\) − 1.18903e9i − 0.586621i
\(118\) − 4.81430e8i − 0.228594i
\(119\) 5.52008e8 0.252339
\(120\) 0 0
\(121\) −1.12667e9 −0.477818
\(122\) 2.04785e8i 0.0836909i
\(123\) 1.95961e9i 0.771965i
\(124\) −3.49758e9 −1.32853
\(125\) 0 0
\(126\) 3.38284e8 0.119568
\(127\) 2.90339e9i 0.990349i 0.868794 + 0.495174i \(0.164896\pi\)
−0.868794 + 0.495174i \(0.835104\pi\)
\(128\) 3.09543e9i 1.01924i
\(129\) 1.28409e9 0.408264
\(130\) 0 0
\(131\) 2.05173e9 0.608694 0.304347 0.952561i \(-0.401562\pi\)
0.304347 + 0.952561i \(0.401562\pi\)
\(132\) 9.86960e8i 0.282955i
\(133\) 3.94570e7i 0.0109343i
\(134\) 2.51623e9 0.674185
\(135\) 0 0
\(136\) −1.98185e9 −0.496759
\(137\) − 3.25539e9i − 0.789514i −0.918786 0.394757i \(-0.870829\pi\)
0.918786 0.394757i \(-0.129171\pi\)
\(138\) 1.55115e9i 0.364081i
\(139\) 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
\(142\) − 3.33545e9i − 0.688425i
\(143\) − 2.71598e9i − 0.543143i
\(144\) 2.15090e9 0.416858
\(145\) 0 0
\(146\) −1.91910e8 −0.0349551
\(147\) − 3.78950e8i − 0.0669350i
\(148\) − 4.15162e9i − 0.711279i
\(149\) −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 \(0.450649\pi\)
\(152\) − 1.41661e8i − 0.0215255i
\(153\) 3.53182e9i 0.521059i
\(154\) 7.72706e8 0.110706
\(155\) 0 0
\(156\) 2.17706e9 0.294313
\(157\) 4.61623e9i 0.606372i 0.952931 + 0.303186i \(0.0980503\pi\)
−0.952931 + 0.303186i \(0.901950\pi\)
\(158\) 2.43868e9i 0.311314i
\(159\) 1.80287e9 0.223706
\(160\) 0 0
\(161\) −6.17740e9 −0.724584
\(162\) 1.38432e9i 0.157913i
\(163\) 6.26525e9i 0.695175i 0.937648 + 0.347588i \(0.112999\pi\)
−0.937648 + 0.347588i \(0.887001\pi\)
\(164\) 1.27555e10 1.37689
\(165\) 0 0
\(166\) 8.65580e7 0.00884751
\(167\) 6.21672e9i 0.618496i 0.950981 + 0.309248i \(0.100077\pi\)
−0.950981 + 0.309248i \(0.899923\pi\)
\(168\) 1.36052e9i 0.131769i
\(169\) 4.61353e9 0.435054
\(170\) 0 0
\(171\) −2.52451e8 −0.0225785
\(172\) − 8.35837e9i − 0.728188i
\(173\) 8.97209e9i 0.761528i 0.924672 + 0.380764i \(0.124339\pi\)
−0.924672 + 0.380764i \(0.875661\pi\)
\(174\) −3.99609e9 −0.330494
\(175\) 0 0
\(176\) 4.91306e9 0.385963
\(177\) − 3.45053e9i − 0.264245i
\(178\) 6.09796e9i 0.455297i
\(179\) −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 \(0.482107\pi\)
\(182\) − 1.70445e9i − 0.115150i
\(183\) 1.46775e9i 0.0967433i
\(184\) 2.21784e10 1.42643
\(185\) 0 0
\(186\) 4.92815e9 0.301909
\(187\) 8.06735e9i 0.482441i
\(188\) − 2.54074e9i − 0.148337i
\(189\) 5.53113e9 0.315309
\(190\) 0 0
\(191\) 1.66601e10 0.905788 0.452894 0.891564i \(-0.350392\pi\)
0.452894 + 0.891564i \(0.350392\pi\)
\(192\) 1.27728e9i 0.0678316i
\(193\) 2.41341e10i 1.25206i 0.779801 + 0.626028i \(0.215320\pi\)
−0.779801 + 0.626028i \(0.784680\pi\)
\(194\) −1.10730e10 −0.561249
\(195\) 0 0
\(196\) −2.46666e9 −0.119387
\(197\) 3.89843e9i 0.184413i 0.995740 + 0.0922066i \(0.0293920\pi\)
−0.995740 + 0.0922066i \(0.970608\pi\)
\(198\) 4.94387e9i 0.228599i
\(199\) 1.87489e10 0.847493 0.423747 0.905781i \(-0.360715\pi\)
0.423747 + 0.905781i \(0.360715\pi\)
\(200\) 0 0
\(201\) 1.80345e10 0.779330
\(202\) 1.08411e10i 0.458135i
\(203\) − 1.59142e10i − 0.657740i
\(204\) −6.46658e9 −0.261420
\(205\) 0 0
\(206\) −1.81424e10 −0.701927
\(207\) − 3.95238e10i − 1.49621i
\(208\) − 1.08373e10i − 0.401456i
\(209\) −5.76647e8 −0.0209051
\(210\) 0 0
\(211\) −2.20489e9 −0.0765801 −0.0382900 0.999267i \(-0.512191\pi\)
−0.0382900 + 0.999267i \(0.512191\pi\)
\(212\) − 1.17352e10i − 0.399007i
\(213\) − 2.39060e10i − 0.795792i
\(214\) 1.53925e9 0.0501702
\(215\) 0 0
\(216\) −1.98582e10 −0.620722
\(217\) 1.96262e10i 0.600851i
\(218\) 9.40454e9i 0.282022i
\(219\) −1.37547e9 −0.0404067
\(220\) 0 0
\(221\) 1.77952e10 0.501807
\(222\) 5.84971e9i 0.161639i
\(223\) − 2.65324e10i − 0.718463i −0.933249 0.359231i \(-0.883039\pi\)
0.933249 0.359231i \(-0.116961\pi\)
\(224\) 1.36802e10 0.363058
\(225\) 0 0
\(226\) −1.16967e10 −0.298248
\(227\) − 7.78091e10i − 1.94498i −0.232955 0.972488i \(-0.574839\pi\)
0.232955 0.972488i \(-0.425161\pi\)
\(228\) − 4.62225e8i − 0.0113278i
\(229\) −4.84637e10 −1.16455 −0.582274 0.812993i \(-0.697837\pi\)
−0.582274 + 0.812993i \(0.697837\pi\)
\(230\) 0 0
\(231\) 5.53818e9 0.127972
\(232\) 5.71362e10i 1.29484i
\(233\) − 2.38429e10i − 0.529978i −0.964251 0.264989i \(-0.914632\pi\)
0.964251 0.264989i \(-0.0853683\pi\)
\(234\) 1.09053e10 0.237775
\(235\) 0 0
\(236\) −2.24602e10 −0.471313
\(237\) 1.74787e10i 0.359866i
\(238\) 5.06278e9i 0.102280i
\(239\) −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 \(0.775078\pi\)
\(242\) − 1.03333e10i − 0.193674i
\(243\) 5.52651e10i 1.01677i
\(244\) 9.55384e9 0.172554
\(245\) 0 0
\(246\) −1.79727e10 −0.312901
\(247\) 1.27198e9i 0.0217442i
\(248\) − 7.04629e10i − 1.18285i
\(249\) 6.20384e8 0.0102274
\(250\) 0 0
\(251\) −5.44549e10 −0.865975 −0.432988 0.901400i \(-0.642541\pi\)
−0.432988 + 0.901400i \(0.642541\pi\)
\(252\) − 1.57820e10i − 0.246524i
\(253\) − 9.02799e10i − 1.38532i
\(254\) −2.66286e10 −0.401418
\(255\) 0 0
\(256\) −1.84414e10 −0.268358
\(257\) 5.35278e10i 0.765385i 0.923876 + 0.382693i \(0.125003\pi\)
−0.923876 + 0.382693i \(0.874997\pi\)
\(258\) 1.17771e10i 0.165482i
\(259\) −2.32962e10 −0.321689
\(260\) 0 0
\(261\) 1.01821e11 1.35818
\(262\) 1.88176e10i 0.246722i
\(263\) − 5.81425e10i − 0.749364i −0.927153 0.374682i \(-0.877752\pi\)
0.927153 0.374682i \(-0.122248\pi\)
\(264\) −1.98835e10 −0.251927
\(265\) 0 0
\(266\) −3.61883e8 −0.00443201
\(267\) 4.37057e10i 0.526304i
\(268\) − 1.17390e11i − 1.39003i
\(269\) −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 \(0.451750\pi\)
\(272\) 3.21905e10i 0.356589i
\(273\) − 1.22163e10i − 0.133109i
\(274\) 2.98570e10 0.320014
\(275\) 0 0
\(276\) 7.23660e10 0.750661
\(277\) − 1.12549e11i − 1.14863i −0.818633 0.574316i \(-0.805268\pi\)
0.818633 0.574316i \(-0.194732\pi\)
\(278\) 7.58284e10i 0.761431i
\(279\) −1.25571e11 −1.24071
\(280\) 0 0
\(281\) −4.60761e10 −0.440857 −0.220428 0.975403i \(-0.570746\pi\)
−0.220428 + 0.975403i \(0.570746\pi\)
\(282\) 3.57995e9i 0.0337098i
\(283\) 7.94071e10i 0.735902i 0.929845 + 0.367951i \(0.119941\pi\)
−0.929845 + 0.367951i \(0.880059\pi\)
\(284\) −1.55609e11 −1.41939
\(285\) 0 0
\(286\) 2.49098e10 0.220152
\(287\) − 7.15757e10i − 0.622726i
\(288\) 8.75275e10i 0.749684i
\(289\) 6.57304e10 0.554276
\(290\) 0 0
\(291\) −7.93628e10 −0.648782
\(292\) 8.95322e9i 0.0720703i
\(293\) − 1.41265e11i − 1.11977i −0.828569 0.559887i \(-0.810844\pi\)
0.828569 0.559887i \(-0.189156\pi\)
\(294\) 3.47556e9 0.0271308
\(295\) 0 0
\(296\) 8.36393e10 0.633283
\(297\) 8.08351e10i 0.602832i
\(298\) − 9.82523e9i − 0.0721721i
\(299\) −1.99142e11 −1.44093
\(300\) 0 0
\(301\) −4.69018e10 −0.329337
\(302\) 1.80958e10i 0.125184i
\(303\) 7.77013e10i 0.529586i
\(304\) −2.30094e9 −0.0154517
\(305\) 0 0
\(306\) −3.23923e10 −0.211201
\(307\) 5.58349e10i 0.358742i 0.983781 + 0.179371i \(0.0574063\pi\)
−0.983781 + 0.179371i \(0.942594\pi\)
\(308\) − 3.60491e10i − 0.228253i
\(309\) −1.30031e11 −0.811399
\(310\) 0 0
\(311\) 5.26501e10 0.319137 0.159569 0.987187i \(-0.448990\pi\)
0.159569 + 0.987187i \(0.448990\pi\)
\(312\) 4.38594e10i 0.262040i
\(313\) − 2.51256e11i − 1.47968i −0.672785 0.739838i \(-0.734902\pi\)
0.672785 0.739838i \(-0.265098\pi\)
\(314\) −4.23381e10 −0.245781
\(315\) 0 0
\(316\) 1.13772e11 0.641866
\(317\) 1.16999e11i 0.650749i 0.945585 + 0.325375i \(0.105490\pi\)
−0.945585 + 0.325375i \(0.894510\pi\)
\(318\) 1.65351e10i 0.0906747i
\(319\) 2.32580e11 1.25752
\(320\) 0 0
\(321\) 1.10322e10 0.0579948
\(322\) − 5.66564e10i − 0.293696i
\(323\) − 3.77820e9i − 0.0193141i
\(324\) 6.45828e10 0.325585
\(325\) 0 0
\(326\) −5.74622e10 −0.281775
\(327\) 6.74048e10i 0.326006i
\(328\) 2.56975e11i 1.22591i
\(329\) −1.42570e10 −0.0670883
\(330\) 0 0
\(331\) −2.51419e11 −1.15126 −0.575629 0.817711i \(-0.695243\pi\)
−0.575629 + 0.817711i \(0.695243\pi\)
\(332\) − 4.03820e9i − 0.0182417i
\(333\) − 1.49052e11i − 0.664262i
\(334\) −5.70171e10 −0.250695
\(335\) 0 0
\(336\) 2.20985e10 0.0945882
\(337\) − 6.11427e10i − 0.258232i −0.991630 0.129116i \(-0.958786\pi\)
0.991630 0.129116i \(-0.0412139\pi\)
\(338\) 4.23133e10i 0.176340i
\(339\) −8.38336e10 −0.344762
\(340\) 0 0
\(341\) −2.86828e11 −1.14875
\(342\) − 2.31537e9i − 0.00915173i
\(343\) 1.38413e10i 0.0539949i
\(344\) 1.68389e11 0.648338
\(345\) 0 0
\(346\) −8.22882e10 −0.308670
\(347\) 1.68668e11i 0.624524i 0.949996 + 0.312262i \(0.101087\pi\)
−0.949996 + 0.312262i \(0.898913\pi\)
\(348\) 1.86430e11i 0.681410i
\(349\) 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
\(352\) 1.99930e11i 0.694122i
\(353\) 3.78560e11i 1.29762i 0.760949 + 0.648811i \(0.224734\pi\)
−0.760949 + 0.648811i \(0.775266\pi\)
\(354\) 3.16468e10 0.107106
\(355\) 0 0
\(356\) 2.84489e11 0.938728
\(357\) 3.62863e10i 0.118232i
\(358\) − 1.61641e11i − 0.520091i
\(359\) 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
\(362\) 1.48809e10i 0.0455449i
\(363\) − 7.40617e10i − 0.223879i
\(364\) −7.95179e10 −0.237415
\(365\) 0 0
\(366\) −1.34615e10 −0.0392130
\(367\) − 5.13837e11i − 1.47852i −0.673419 0.739261i \(-0.735175\pi\)
0.673419 0.739261i \(-0.264825\pi\)
\(368\) − 3.60236e11i − 1.02394i
\(369\) 4.57950e11 1.28588
\(370\) 0 0
\(371\) −6.58505e10 −0.180458
\(372\) − 2.29914e11i − 0.622474i
\(373\) − 6.70900e10i − 0.179460i −0.995966 0.0897301i \(-0.971400\pi\)
0.995966 0.0897301i \(-0.0286005\pi\)
\(374\) −7.39903e10 −0.195548
\(375\) 0 0
\(376\) 5.11862e10 0.132071
\(377\) − 5.13030e11i − 1.30800i
\(378\) 5.07292e10i 0.127804i
\(379\) −4.15471e11 −1.03434 −0.517171 0.855882i \(-0.673015\pi\)
−0.517171 + 0.855882i \(0.673015\pi\)
\(380\) 0 0
\(381\) −1.90854e11 −0.464023
\(382\) 1.52799e11i 0.367143i
\(383\) − 3.51976e11i − 0.835831i −0.908486 0.417915i \(-0.862761\pi\)
0.908486 0.417915i \(-0.137239\pi\)
\(384\) −2.03478e11 −0.477560
\(385\) 0 0
\(386\) −2.21348e11 −0.507496
\(387\) − 3.00084e11i − 0.680053i
\(388\) 5.16588e11i 1.15718i
\(389\) 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
\(392\) − 4.96937e10i − 0.106295i
\(393\) 1.34871e11i 0.285201i
\(394\) −3.57548e10 −0.0747482
\(395\) 0 0
\(396\) 2.30647e11 0.471323
\(397\) − 7.34338e11i − 1.48367i −0.670580 0.741837i \(-0.733955\pi\)
0.670580 0.741837i \(-0.266045\pi\)
\(398\) 1.71957e11i 0.343514i
\(399\) −2.59371e9 −0.00512322
\(400\) 0 0
\(401\) 8.08296e11 1.56106 0.780532 0.625116i \(-0.214948\pi\)
0.780532 + 0.625116i \(0.214948\pi\)
\(402\) 1.65405e11i 0.315886i
\(403\) 6.32691e11i 1.19487i
\(404\) 5.05773e11 0.944581
\(405\) 0 0
\(406\) 1.45959e11 0.266602
\(407\) − 3.40464e11i − 0.615030i
\(408\) − 1.30277e11i − 0.232754i
\(409\) 9.11153e11 1.61004 0.805020 0.593248i \(-0.202155\pi\)
0.805020 + 0.593248i \(0.202155\pi\)
\(410\) 0 0
\(411\) 2.13993e11 0.369923
\(412\) 8.46398e11i 1.44723i
\(413\) 1.26032e11i 0.213160i
\(414\) 3.62495e11 0.606458
\(415\) 0 0
\(416\) 4.41010e11 0.721985
\(417\) 5.43482e11i 0.880183i
\(418\) − 5.28876e9i − 0.00847345i
\(419\) −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 \(0.502843\pi\)
\(422\) − 2.02223e10i − 0.0310402i
\(423\) − 9.12181e10i − 0.138532i
\(424\) 2.36420e11 0.355253
\(425\) 0 0
\(426\) 2.19256e11 0.322558
\(427\) − 5.36100e10i − 0.0780406i
\(428\) − 7.18106e10i − 0.103441i
\(429\) 1.78535e11 0.254487
\(430\) 0 0
\(431\) −9.42534e11 −1.31568 −0.657839 0.753159i \(-0.728529\pi\)
−0.657839 + 0.753159i \(0.728529\pi\)
\(432\) 3.22549e11i 0.445574i
\(433\) 1.01849e12i 1.39239i 0.717852 + 0.696196i \(0.245125\pi\)
−0.717852 + 0.696196i \(0.754875\pi\)
\(434\) −1.80003e11 −0.243543
\(435\) 0 0
\(436\) 4.38751e11 0.581472
\(437\) 4.22810e10i 0.0554598i
\(438\) − 1.26152e10i − 0.0163781i
\(439\) 7.89357e11 1.01434 0.507169 0.861847i \(-0.330692\pi\)
0.507169 + 0.861847i \(0.330692\pi\)
\(440\) 0 0
\(441\) −8.85583e10 −0.111495
\(442\) 1.63210e11i 0.203397i
\(443\) 1.06770e12i 1.31714i 0.752518 + 0.658572i \(0.228839\pi\)
−0.752518 + 0.658572i \(0.771161\pi\)
\(444\) 2.72907e11 0.333266
\(445\) 0 0
\(446\) 2.43344e11 0.291215
\(447\) − 7.04200e10i − 0.0834281i
\(448\) − 4.66533e10i − 0.0547182i
\(449\) 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
\(452\) 5.45689e11i 0.614926i
\(453\) 1.29698e11i 0.144707i
\(454\) 7.13632e11 0.788357
\(455\) 0 0
\(456\) 9.31207e9 0.0100857
\(457\) 7.03146e11i 0.754089i 0.926195 + 0.377045i \(0.123060\pi\)
−0.926195 + 0.377045i \(0.876940\pi\)
\(458\) − 4.44489e11i − 0.472026i
\(459\) −5.29633e11 −0.556952
\(460\) 0 0
\(461\) −1.86192e12 −1.92003 −0.960015 0.279950i \(-0.909682\pi\)
−0.960015 + 0.279950i \(0.909682\pi\)
\(462\) 5.07938e10i 0.0518707i
\(463\) − 1.06950e11i − 0.108160i −0.998537 0.0540799i \(-0.982777\pi\)
0.998537 0.0540799i \(-0.0172226\pi\)
\(464\) 9.28043e11 0.929474
\(465\) 0 0
\(466\) 2.18677e11 0.214816
\(467\) − 4.13997e11i − 0.402783i −0.979511 0.201392i \(-0.935454\pi\)
0.979511 0.201392i \(-0.0645464\pi\)
\(468\) − 5.08766e11i − 0.490243i
\(469\) −6.58717e11 −0.628667
\(470\) 0 0
\(471\) −3.03448e11 −0.284113
\(472\) − 4.52487e11i − 0.419631i
\(473\) − 6.85450e11i − 0.629652i
\(474\) −1.60307e11 −0.145865
\(475\) 0 0
\(476\) 2.36194e11 0.210881
\(477\) − 4.21320e11i − 0.372631i
\(478\) − 5.74044e11i − 0.502944i
\(479\) 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
\(482\) − 7.30612e11i − 0.616560i
\(483\) − 4.06072e11i − 0.339501i
\(484\) −4.82082e11 −0.399316
\(485\) 0 0
\(486\) −5.06868e11 −0.412127
\(487\) − 4.94789e11i − 0.398602i −0.979938 0.199301i \(-0.936133\pi\)
0.979938 0.199301i \(-0.0638672\pi\)
\(488\) 1.92474e11i 0.153632i
\(489\) −4.11847e11 −0.325721
\(490\) 0 0
\(491\) 1.11163e12 0.863168 0.431584 0.902073i \(-0.357955\pi\)
0.431584 + 0.902073i \(0.357955\pi\)
\(492\) 8.38484e11i 0.645137i
\(493\) 1.52387e12i 1.16181i
\(494\) −1.16661e10 −0.00881359
\(495\) 0 0
\(496\) −1.14450e12 −0.849083
\(497\) 8.73178e11i 0.641947i
\(498\) 5.68990e9i 0.00414546i
\(499\) −8.18377e11 −0.590882 −0.295441 0.955361i \(-0.595467\pi\)
−0.295441 + 0.955361i \(0.595467\pi\)
\(500\) 0 0
\(501\) −4.08656e11 −0.289793
\(502\) − 4.99437e11i − 0.351006i
\(503\) 3.13384e11i 0.218284i 0.994026 + 0.109142i \(0.0348102\pi\)
−0.994026 + 0.109142i \(0.965190\pi\)
\(504\) 3.17947e11 0.219491
\(505\) 0 0
\(506\) 8.28009e11 0.561510
\(507\) 3.03270e11i 0.203842i
\(508\) 1.24231e12i 0.827642i
\(509\) 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
\(512\) 1.41572e12i 0.910467i
\(513\) − 3.78577e10i − 0.0241338i
\(514\) −4.90934e11 −0.310234
\(515\) 0 0
\(516\) 5.49438e11 0.341189
\(517\) − 2.08360e11i − 0.128265i
\(518\) − 2.13663e11i − 0.130390i
\(519\) −5.89781e11 −0.356810
\(520\) 0 0
\(521\) −3.18952e12 −1.89651 −0.948255 0.317510i \(-0.897153\pi\)
−0.948255 + 0.317510i \(0.897153\pi\)
\(522\) 9.33862e11i 0.550510i
\(523\) 9.43708e11i 0.551544i 0.961223 + 0.275772i \(0.0889335\pi\)
−0.961223 + 0.275772i \(0.911067\pi\)
\(524\) 8.77898e11 0.508690
\(525\) 0 0
\(526\) 5.33258e11 0.303740
\(527\) − 1.87930e12i − 1.06133i
\(528\) 3.22961e11i 0.180841i
\(529\) −4.81837e12 −2.67516
\(530\) 0 0
\(531\) −8.06370e11 −0.440158
\(532\) 1.68829e10i 0.00913789i
\(533\) − 2.30740e12i − 1.23837i
\(534\) −4.00850e11 −0.213327
\(535\) 0 0
\(536\) 2.36496e12 1.23761
\(537\) − 1.15853e12i − 0.601204i
\(538\) − 4.28661e11i − 0.220594i
\(539\) −2.02284e11 −0.103232
\(540\) 0 0
\(541\) −8.78618e11 −0.440973 −0.220487 0.975390i \(-0.570765\pi\)
−0.220487 + 0.975390i \(0.570765\pi\)
\(542\) 2.45933e11i 0.122411i
\(543\) 1.06655e11i 0.0526480i
\(544\) −1.30994e12 −0.641295
\(545\) 0 0
\(546\) 1.12042e11 0.0539529
\(547\) − 4.56216e11i − 0.217885i −0.994048 0.108943i \(-0.965254\pi\)
0.994048 0.108943i \(-0.0347465\pi\)
\(548\) − 1.39292e12i − 0.659803i
\(549\) 3.43004e11 0.161147
\(550\) 0 0
\(551\) −1.08925e11 −0.0503435
\(552\) 1.45790e12i 0.668347i
\(553\) − 6.38416e11i − 0.290296i
\(554\) 1.03225e12 0.465575
\(555\) 0 0
\(556\) 3.53763e12 1.56991
\(557\) − 1.63980e12i − 0.721842i −0.932596 0.360921i \(-0.882462\pi\)
0.932596 0.360921i \(-0.117538\pi\)
\(558\) − 1.15168e12i − 0.502896i
\(559\) −1.51198e12 −0.654927
\(560\) 0 0
\(561\) −5.30308e11 −0.226045
\(562\) − 4.22590e11i − 0.178692i
\(563\) 4.36151e11i 0.182957i 0.995807 + 0.0914786i \(0.0291593\pi\)
−0.995807 + 0.0914786i \(0.970841\pi\)
\(564\) 1.67016e11 0.0695026
\(565\) 0 0
\(566\) −7.28288e11 −0.298283
\(567\) − 3.62397e11i − 0.147252i
\(568\) − 3.13493e12i − 1.26375i
\(569\) −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 \(0.485131\pi\)
\(572\) − 1.16212e12i − 0.453909i
\(573\) 1.09515e12i 0.424403i
\(574\) 6.56462e11 0.252410
\(575\) 0 0
\(576\) 2.98494e11 0.112988
\(577\) 3.72080e12i 1.39748i 0.715377 + 0.698739i \(0.246255\pi\)
−0.715377 + 0.698739i \(0.753745\pi\)
\(578\) 6.02851e11i 0.224665i
\(579\) −1.58646e12 −0.586644
\(580\) 0 0
\(581\) −2.26598e10 −0.00825017
\(582\) − 7.27882e11i − 0.262971i
\(583\) − 9.62377e11i − 0.345014i
\(584\) −1.80373e11 −0.0641674
\(585\) 0 0
\(586\) 1.29562e12 0.453878
\(587\) 6.46176e11i 0.224636i 0.993672 + 0.112318i \(0.0358275\pi\)
−0.993672 + 0.112318i \(0.964172\pi\)
\(588\) − 1.62146e11i − 0.0559381i
\(589\) 1.34331e11 0.0459892
\(590\) 0 0
\(591\) −2.56264e11 −0.0864059
\(592\) − 1.35853e12i − 0.454590i
\(593\) − 5.05774e12i − 1.67962i −0.542883 0.839808i \(-0.682667\pi\)
0.542883 0.839808i \(-0.317333\pi\)
\(594\) −7.41385e11 −0.244346
\(595\) 0 0
\(596\) −4.58377e11 −0.148804
\(597\) 1.23246e12i 0.397089i
\(598\) − 1.82644e12i − 0.584051i
\(599\) 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 \(0.949976\pi\)
\(602\) − 4.30163e11i − 0.133490i
\(603\) − 4.21455e12i − 1.29815i
\(604\) 8.44227e11 0.258103
\(605\) 0 0
\(606\) −7.12643e11 −0.214657
\(607\) 1.45276e12i 0.434356i 0.976132 + 0.217178i \(0.0696852\pi\)
−0.976132 + 0.217178i \(0.930315\pi\)
\(608\) − 9.36335e10i − 0.0277885i
\(609\) 1.04612e12 0.308181
\(610\) 0 0
\(611\) −4.59605e11 −0.133413
\(612\) 1.51120e12i 0.435453i
\(613\) 1.20124e12i 0.343603i 0.985132 + 0.171802i \(0.0549588\pi\)
−0.985132 + 0.171802i \(0.945041\pi\)
\(614\) −5.12094e11 −0.145409
\(615\) 0 0
\(616\) 7.26252e11 0.203224
\(617\) − 3.13545e12i − 0.870997i −0.900189 0.435498i \(-0.856572\pi\)
0.900189 0.435498i \(-0.143428\pi\)
\(618\) − 1.19259e12i − 0.328884i
\(619\) 5.17127e12 1.41576 0.707880 0.706333i \(-0.249652\pi\)
0.707880 + 0.706333i \(0.249652\pi\)
\(620\) 0 0
\(621\) 5.92700e12 1.59927
\(622\) 4.82884e11i 0.129356i
\(623\) − 1.59637e12i − 0.424557i
\(624\) 7.12394e11 0.188100
\(625\) 0 0
\(626\) 2.30441e12 0.599757
\(627\) − 3.79059e10i − 0.00979497i
\(628\) 1.97520e12i 0.506749i
\(629\) 2.23073e12 0.568223
\(630\) 0 0
\(631\) −6.37331e12 −1.60042 −0.800208 0.599723i \(-0.795278\pi\)
−0.800208 + 0.599723i \(0.795278\pi\)
\(632\) 2.29208e12i 0.571481i
\(633\) − 1.44939e11i − 0.0358812i
\(634\) −1.07306e12 −0.263768
\(635\) 0 0
\(636\) 7.71416e11 0.186953
\(637\) 4.46204e11i 0.107376i
\(638\) 2.13312e12i 0.509710i
\(639\) −5.58670e12 −1.32557
\(640\) 0 0
\(641\) −5.74174e12 −1.34333 −0.671665 0.740855i \(-0.734421\pi\)
−0.671665 + 0.740855i \(0.734421\pi\)
\(642\) 1.01182e11i 0.0235070i
\(643\) − 5.85135e11i − 0.134992i −0.997720 0.0674958i \(-0.978499\pi\)
0.997720 0.0674958i \(-0.0215009\pi\)
\(644\) −2.64320e12 −0.605541
\(645\) 0 0
\(646\) 3.46520e10 0.00782857
\(647\) 1.80915e12i 0.405887i 0.979190 + 0.202943i \(0.0650507\pi\)
−0.979190 + 0.202943i \(0.934949\pi\)
\(648\) 1.30110e12i 0.289883i
\(649\) −1.84191e12 −0.407536
\(650\) 0 0
\(651\) −1.29013e12 −0.281526
\(652\) 2.68079e12i 0.580963i
\(653\) 2.43900e12i 0.524932i 0.964941 + 0.262466i \(0.0845357\pi\)
−0.964941 + 0.262466i \(0.915464\pi\)
\(654\) −6.18208e11 −0.132140
\(655\) 0 0
\(656\) 4.17396e12 0.879996
\(657\) 3.21440e11i 0.0673063i
\(658\) − 1.30759e11i − 0.0271929i
\(659\) −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 \(0.723876\pi\)
\(662\) − 2.30591e12i − 0.466640i
\(663\) 1.16977e12i 0.235119i
\(664\) 8.13544e10 0.0162414
\(665\) 0 0
\(666\) 1.36704e12 0.269245
\(667\) − 1.70533e13i − 3.33611i
\(668\) 2.66002e12i 0.516882i
\(669\) 1.74411e12 0.336632
\(670\) 0 0
\(671\) 7.83487e11 0.149204
\(672\) 8.99267e11i 0.170109i
\(673\) 3.17186e12i 0.596000i 0.954566 + 0.298000i \(0.0963196\pi\)
−0.954566 + 0.298000i \(0.903680\pi\)
\(674\) 5.60774e11 0.104669
\(675\) 0 0
\(676\) 1.97405e12 0.363578
\(677\) 2.32240e12i 0.424901i 0.977172 + 0.212450i \(0.0681444\pi\)
−0.977172 + 0.212450i \(0.931856\pi\)
\(678\) − 7.68886e11i − 0.139742i
\(679\) 2.89876e12 0.523357
\(680\) 0 0
\(681\) 5.11479e12 0.911309
\(682\) − 2.63066e12i − 0.465624i
\(683\) 3.78639e12i 0.665782i 0.942965 + 0.332891i \(0.108024\pi\)
−0.942965 + 0.332891i \(0.891976\pi\)
\(684\) −1.08019e11 −0.0188690
\(685\) 0 0
\(686\) −1.26946e11 −0.0218858
\(687\) − 3.18577e12i − 0.545643i
\(688\) − 2.73509e12i − 0.465397i
\(689\) −2.12283e12 −0.358864
\(690\) 0 0
\(691\) −5.70532e12 −0.951982 −0.475991 0.879450i \(-0.657911\pi\)
−0.475991 + 0.879450i \(0.657911\pi\)
\(692\) 3.83900e12i 0.636415i
\(693\) − 1.29424e12i − 0.213165i
\(694\) −1.54695e12 −0.253138
\(695\) 0 0
\(696\) −3.75585e12 −0.606690
\(697\) 6.85372e12i 1.09997i
\(698\) 3.03746e12i 0.484352i
\(699\) 1.56732e12 0.248318
\(700\) 0 0
\(701\) 6.25160e12 0.977823 0.488912 0.872333i \(-0.337394\pi\)
0.488912 + 0.872333i \(0.337394\pi\)
\(702\) 1.63536e12i 0.254154i
\(703\) 1.59450e11i 0.0246222i
\(704\) 6.81818e11 0.104614
\(705\) 0 0
\(706\) −3.47199e12 −0.525965
\(707\) − 2.83807e12i − 0.427205i
\(708\) − 1.47642e12i − 0.220831i
\(709\) 8.01996e12 1.19197 0.595983 0.802997i \(-0.296762\pi\)
0.595983 + 0.802997i \(0.296762\pi\)
\(710\) 0 0
\(711\) 4.08467e12 0.599437
\(712\) 5.73136e12i 0.835791i
\(713\) 2.10308e13i 3.04757i
\(714\) −3.32802e11 −0.0479230
\(715\) 0 0
\(716\) −7.54107e12 −1.07232
\(717\) − 4.11433e12i − 0.581383i
\(718\) 1.46871e12i 0.206241i
\(719\) −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
\(722\) − 2.95708e12i − 0.404991i
\(723\) − 5.23649e12i − 0.712718i
\(724\) 6.94238e11 0.0939042
\(725\) 0 0
\(726\) 6.79262e11 0.0907450
\(727\) 1.47778e13i 1.96203i 0.193941 + 0.981013i \(0.437873\pi\)
−0.193941 + 0.981013i \(0.562127\pi\)
\(728\) − 1.60198e12i − 0.211381i
\(729\) −6.61984e11 −0.0868108
\(730\) 0 0
\(731\) 4.49108e12 0.581731
\(732\) 6.28022e11i 0.0808491i
\(733\) − 6.70116e12i − 0.857398i −0.903447 0.428699i \(-0.858972\pi\)
0.903447 0.428699i \(-0.141028\pi\)
\(734\) 4.71269e12 0.599290
\(735\) 0 0
\(736\) 1.46593e13 1.84146
\(737\) − 9.62686e12i − 1.20193i
\(738\) 4.20013e12i 0.521205i
\(739\) 1.39054e13 1.71508 0.857540 0.514417i \(-0.171992\pi\)
0.857540 + 0.514417i \(0.171992\pi\)
\(740\) 0 0
\(741\) −8.36137e10 −0.0101882
\(742\) − 6.03953e11i − 0.0731452i
\(743\) − 3.43953e12i − 0.414047i −0.978336 0.207024i \(-0.933622\pi\)
0.978336 0.207024i \(-0.0663777\pi\)
\(744\) 4.63188e12 0.554216
\(745\) 0 0
\(746\) 6.15321e11 0.0727407
\(747\) − 1.44980e11i − 0.0170359i
\(748\) 3.45188e12i 0.403179i
\(749\) −4.02955e11 −0.0467830
\(750\) 0 0
\(751\) 1.00823e13 1.15660 0.578298 0.815826i \(-0.303717\pi\)
0.578298 + 0.815826i \(0.303717\pi\)
\(752\) − 8.31401e11i − 0.0948047i
\(753\) − 3.57960e12i − 0.405748i
\(754\) 4.70529e12 0.530170
\(755\) 0 0
\(756\) 2.36667e12 0.263506
\(757\) 1.02703e13i 1.13672i 0.822780 + 0.568360i \(0.192422\pi\)
−0.822780 + 0.568360i \(0.807578\pi\)
\(758\) − 3.81052e12i − 0.419250i
\(759\) 5.93456e12 0.649083
\(760\) 0 0
\(761\) −3.20840e12 −0.346783 −0.173391 0.984853i \(-0.555473\pi\)
−0.173391 + 0.984853i \(0.555473\pi\)
\(762\) − 1.75043e12i − 0.188083i
\(763\) − 2.46199e12i − 0.262982i
\(764\) 7.12855e12 0.756974
\(765\) 0 0
\(766\) 3.22817e12 0.338787
\(767\) 4.06292e12i 0.423896i
\(768\) − 1.21225e12i − 0.125738i
\(769\) −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
\(772\) 1.03266e13i 1.04635i
\(773\) − 8.54195e11i − 0.0860497i −0.999074 0.0430249i \(-0.986301\pi\)
0.999074 0.0430249i \(-0.0136995\pi\)
\(774\) 2.75224e12 0.275646
\(775\) 0 0
\(776\) −1.04073e13 −1.03029
\(777\) − 1.53138e12i − 0.150726i
\(778\) 2.38517e12i 0.233405i
\(779\) −4.89898e11 −0.0476636
\(780\) 0 0
\(781\) −1.27611e13 −1.22732
\(782\) 5.42513e12i 0.518776i
\(783\) 1.52692e13i 1.45174i
\(784\) −8.07158e11 −0.0763021
\(785\) 0 0
\(786\) −1.23697e12 −0.115600
\(787\) − 4.25016e12i − 0.394929i −0.980310 0.197465i \(-0.936729\pi\)
0.980310 0.197465i \(-0.0632707\pi\)
\(788\) 1.66807e12i 0.154116i
\(789\) 3.82200e12 0.351111
\(790\) 0 0
\(791\) 3.06206e12 0.278112
\(792\) 4.64665e12i 0.419640i
\(793\) − 1.72823e12i − 0.155193i
\(794\) 6.73503e12 0.601378
\(795\) 0 0
\(796\) 8.02231e12 0.708256
\(797\) − 2.02157e13i − 1.77470i −0.461093 0.887352i \(-0.652542\pi\)
0.461093 0.887352i \(-0.347458\pi\)
\(798\) − 2.37884e10i − 0.00207660i
\(799\) 1.36518e12 0.118503
\(800\) 0 0
\(801\) 1.02138e13 0.876676
\(802\) 7.41334e12i 0.632746i
\(803\) 7.34231e11i 0.0623179i
\(804\) 7.71664e12 0.651292
\(805\) 0 0
\(806\) −5.80278e12 −0.484315
\(807\) − 3.07232e12i − 0.254998i
\(808\) 1.01894e13i 0.841002i
\(809\) 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 \(0.623970\pi\)
\(812\) − 6.80942e12i − 0.549678i
\(813\) 1.76267e12i 0.141502i
\(814\) 3.12259e12 0.249290
\(815\) 0 0
\(816\) −2.11604e12 −0.167078
\(817\) 3.21018e11i 0.0252075i
\(818\) 8.35671e12i 0.652597i
\(819\) −2.85487e12 −0.221722
\(820\) 0 0
\(821\) 1.61631e13 1.24160 0.620798 0.783971i \(-0.286809\pi\)
0.620798 + 0.783971i \(0.286809\pi\)
\(822\) 1.96265e12i 0.149941i
\(823\) 5.97042e12i 0.453634i 0.973937 + 0.226817i \(0.0728319\pi\)
−0.973937 + 0.226817i \(0.927168\pi\)
\(824\) −1.70517e13 −1.28853
\(825\) 0 0
\(826\) −1.15591e12 −0.0864003
\(827\) − 7.76424e12i − 0.577197i −0.957450 0.288599i \(-0.906811\pi\)
0.957450 0.288599i \(-0.0931893\pi\)
\(828\) − 1.69115e13i − 1.25039i
\(829\) 4.09585e12 0.301195 0.150598 0.988595i \(-0.451880\pi\)
0.150598 + 0.988595i \(0.451880\pi\)
\(830\) 0 0
\(831\) 7.39839e12 0.538186
\(832\) − 1.50397e12i − 0.108814i
\(833\) − 1.32537e12i − 0.0953751i
\(834\) −4.98458e12 −0.356765
\(835\) 0 0
\(836\) −2.46737e11 −0.0174705
\(837\) − 1.88306e13i − 1.32617i
\(838\) − 4.53176e12i − 0.317445i
\(839\) −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
\(842\) − 1.05606e11i − 0.00724079i
\(843\) − 3.02882e12i − 0.206561i
\(844\) −9.43433e11 −0.0639985
\(845\) 0 0
\(846\) 8.36613e11 0.0561511
\(847\) 2.70513e12i 0.180598i
\(848\) − 3.84009e12i − 0.255012i
\(849\) −5.21983e12 −0.344803
\(850\) 0 0
\(851\) −2.49636e13 −1.63164
\(852\) − 1.02290e13i − 0.665049i
\(853\) 2.72968e13i 1.76539i 0.469945 + 0.882696i \(0.344274\pi\)
−0.469945 + 0.882696i \(0.655726\pi\)
\(854\) 4.91688e11 0.0316322
\(855\) 0 0
\(856\) 1.44671e12 0.0920979
\(857\) − 7.67571e12i − 0.486077i −0.970017 0.243038i \(-0.921856\pi\)
0.970017 0.243038i \(-0.0781441\pi\)
\(858\) 1.63745e12i 0.103151i
\(859\) 1.67172e13 1.04759 0.523797 0.851843i \(-0.324515\pi\)
0.523797 + 0.851843i \(0.324515\pi\)
\(860\) 0 0
\(861\) 4.70503e12 0.291775
\(862\) − 8.64452e12i − 0.533284i
\(863\) − 2.85615e13i − 1.75280i −0.481582 0.876401i \(-0.659938\pi\)
0.481582 0.876401i \(-0.340062\pi\)
\(864\) −1.31257e13 −0.801327
\(865\) 0 0
\(866\) −9.34116e12 −0.564378
\(867\) 4.32079e12i 0.259703i
\(868\) 8.39769e12i 0.502135i
\(869\) 9.33018e12 0.555010
\(870\) 0 0
\(871\) −2.12351e13 −1.25018
\(872\) 8.83916e12i 0.517710i
\(873\) 1.85466e13i 1.08069i
\(874\) −3.87783e11 −0.0224795
\(875\) 0 0
\(876\) −5.88540e11 −0.0337682
\(877\) 8.40714e12i 0.479899i 0.970785 + 0.239950i \(0.0771309\pi\)
−0.970785 + 0.239950i \(0.922869\pi\)
\(878\) 7.23964e12i 0.411142i
\(879\) 9.28607e12 0.524665
\(880\) 0 0
\(881\) −1.99694e13 −1.11680 −0.558398 0.829573i \(-0.688584\pi\)
−0.558398 + 0.829573i \(0.688584\pi\)
\(882\) − 8.12219e11i − 0.0451923i
\(883\) − 2.36498e13i − 1.30919i −0.755978 0.654597i \(-0.772838\pi\)
0.755978 0.654597i \(-0.227162\pi\)
\(884\) 7.61423e12 0.419364
\(885\) 0 0
\(886\) −9.79250e12 −0.533878
\(887\) 3.83859e12i 0.208217i 0.994566 + 0.104108i \(0.0331989\pi\)
−0.994566 + 0.104108i \(0.966801\pi\)
\(888\) 5.49804e12i 0.296722i
\(889\) 6.97103e12 0.374317
\(890\) 0 0
\(891\) 5.29627e12 0.281527
\(892\) − 1.13527e13i − 0.600425i
\(893\) 9.75816e10i 0.00513495i
\(894\) 6.45862e11 0.0338159
\(895\) 0 0
\(896\) 7.43213e12 0.385237
\(897\) − 1.30906e13i − 0.675139i
\(898\) 3.04216e10i 0.00156113i
\(899\) −5.41798e13 −2.76642
\(900\) 0 0
\(901\) 6.30551e12 0.318756
\(902\) 9.59390e12i 0.482576i
\(903\) − 3.08309e12i − 0.154309i
\(904\) −1.09936e13 −0.547495
\(905\) 0 0
\(906\) −1.18953e12 −0.0586542
\(907\) 2.31944e13i 1.13802i 0.822331 + 0.569009i \(0.192673\pi\)
−0.822331 + 0.569009i \(0.807327\pi\)
\(908\) − 3.32931e13i − 1.62543i
\(909\) 1.81583e13 0.882142
\(910\) 0 0
\(911\) 1.70743e13 0.821317 0.410659 0.911789i \(-0.365299\pi\)
0.410659 + 0.911789i \(0.365299\pi\)
\(912\) − 1.51253e11i − 0.00723980i
\(913\) − 3.31163e11i − 0.0157733i
\(914\) −6.44896e12 −0.305655
\(915\) 0 0
\(916\) −2.07368e13 −0.973221
\(917\) − 4.92620e12i − 0.230065i
\(918\) − 4.85757e12i − 0.225750i
\(919\) −1.49265e13 −0.690301 −0.345151 0.938547i \(-0.612172\pi\)
−0.345151 + 0.938547i \(0.612172\pi\)
\(920\) 0 0
\(921\) −3.67031e12 −0.168087
\(922\) − 1.70768e13i − 0.778246i
\(923\) 2.81488e13i 1.27659i
\(924\) 2.36969e12 0.106947
\(925\) 0 0
\(926\) 9.80898e11 0.0438404
\(927\) 3.03875e13i 1.35156i
\(928\) 3.77653e13i 1.67158i
\(929\) −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
\(932\) − 1.02020e13i − 0.442906i
\(933\) 3.46096e12i 0.149530i
\(934\) 3.79701e12 0.163260
\(935\) 0 0
\(936\) 1.02497e13 0.436485
\(937\) 3.53996e13i 1.50027i 0.661284 + 0.750135i \(0.270012\pi\)
−0.661284 + 0.750135i \(0.729988\pi\)
\(938\) − 6.04147e12i − 0.254818i
\(939\) 1.65163e13 0.693295
\(940\) 0 0
\(941\) −4.67286e13 −1.94280 −0.971402 0.237439i \(-0.923692\pi\)
−0.971402 + 0.237439i \(0.923692\pi\)
\(942\) − 2.78310e12i − 0.115159i
\(943\) − 7.66985e13i − 3.15852i
\(944\) −7.34960e12 −0.301224
\(945\) 0 0
\(946\) 6.28665e12 0.255217
\(947\) 5.22799e12i 0.211232i 0.994407 + 0.105616i \(0.0336815\pi\)
−0.994407 + 0.105616i \(0.966319\pi\)
\(948\) 7.47882e12i 0.300743i
\(949\) 1.61958e12 0.0648195
\(950\) 0 0
\(951\) −7.69091e12 −0.304905
\(952\) 4.75842e12i 0.187757i
\(953\) − 2.46017e13i − 0.966156i −0.875577 0.483078i \(-0.839519\pi\)
0.875577 0.483078i \(-0.160481\pi\)
\(954\) 3.86417e12 0.151039
\(955\) 0 0
\(956\) −2.67809e13 −1.03697
\(957\) 1.52886e13i 0.589204i
\(958\) 7.83373e11i 0.0300486i
\(959\) −7.81618e12 −0.298408
\(960\) 0 0
\(961\) 4.03773e13 1.52715
\(962\) − 6.88788e12i − 0.259297i
\(963\) − 2.57816e12i − 0.0966031i
\(964\) −3.40853e13 −1.27122
\(965\) 0 0
\(966\) 3.72431e12 0.137610
\(967\) − 1.24062e13i − 0.456267i −0.973630 0.228134i \(-0.926738\pi\)
0.973630 0.228134i \(-0.0732623\pi\)
\(968\) − 9.71212e12i − 0.355529i
\(969\) 2.48360e11 0.00904951
\(970\) 0 0
\(971\) 5.30324e12 0.191450 0.0957248 0.995408i \(-0.469483\pi\)
0.0957248 + 0.995408i \(0.469483\pi\)
\(972\) 2.36469e13i 0.849722i
\(973\) − 1.98509e13i − 0.710023i
\(974\) 4.53799e12 0.161565
\(975\) 0 0
\(976\) 3.12628e12 0.110282
\(977\) 1.45131e13i 0.509606i 0.966993 + 0.254803i \(0.0820107\pi\)
−0.966993 + 0.254803i \(0.917989\pi\)
\(978\) − 3.77728e12i − 0.132024i
\(979\) 2.33302e13 0.811702
\(980\) 0 0
\(981\) 1.57521e13 0.543035
\(982\) 1.01954e13i 0.349868i
\(983\) − 3.61534e13i − 1.23498i −0.786580 0.617488i \(-0.788150\pi\)
0.786580 0.617488i \(-0.211850\pi\)
\(984\) −1.68923e13 −0.574394
\(985\) 0 0
\(986\) −1.39763e13 −0.470917
\(987\) − 9.37185e11i − 0.0314339i
\(988\) 5.44258e11i 0.0181718i
\(989\) −5.02586e13 −1.67043
\(990\) 0 0
\(991\) 4.94162e13 1.62756 0.813782 0.581170i \(-0.197405\pi\)
0.813782 + 0.581170i \(0.197405\pi\)
\(992\) − 4.65739e13i − 1.52700i
\(993\) − 1.65271e13i − 0.539417i
\(994\) −8.00841e12 −0.260200
\(995\) 0 0
\(996\) 2.65451e11 0.00854708
\(997\) 3.03522e13i 0.972886i 0.873712 + 0.486443i \(0.161706\pi\)
−0.873712 + 0.486443i \(0.838294\pi\)
\(998\) − 7.50580e12i − 0.239502i
\(999\) 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 175.10.b.c.99.3 4
5.2 odd 4 35.10.a.b.1.2 2
5.3 odd 4 175.10.a.c.1.1 2
5.4 even 2 inner 175.10.b.c.99.2 4
15.2 even 4 315.10.a.b.1.1 2
35.27 even 4 245.10.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 2 5.2 odd 4
175.10.a.c.1.1 2 5.3 odd 4
175.10.b.c.99.2 4 5.4 even 2 inner
175.10.b.c.99.3 4 1.1 even 1 trivial
245.10.a.c.1.2 2 35.27 even 4
315.10.a.b.1.1 2 15.2 even 4