Properties

Label 9.24.a.d.1.1
Level $9$
Weight $24$
Character 9.1
Self dual yes
Analytic conductor $30.168$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.1683633611\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 29258 x^{2} + 97377280\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{10}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-159.463\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q-4783.90 q^{2} +1.44971e7 q^{4} -1.42520e8 q^{5} +6.49474e9 q^{7} -2.92224e10 q^{8} +O(q^{10})\) \(q-4783.90 q^{2} +1.44971e7 q^{4} -1.42520e8 q^{5} +6.49474e9 q^{7} -2.92224e10 q^{8} +6.81799e11 q^{10} +6.56843e11 q^{11} -1.03084e13 q^{13} -3.10702e13 q^{14} +1.81865e13 q^{16} -2.36156e14 q^{17} +5.53329e14 q^{19} -2.06612e15 q^{20} -3.14227e15 q^{22} -3.36472e15 q^{23} +8.39088e15 q^{25} +4.93141e16 q^{26} +9.41548e16 q^{28} +8.75953e15 q^{29} -1.61750e17 q^{31} +1.58133e17 q^{32} +1.12975e18 q^{34} -9.25627e17 q^{35} +1.42652e18 q^{37} -2.64707e18 q^{38} +4.16476e18 q^{40} +4.86336e18 q^{41} -3.63780e18 q^{43} +9.52231e18 q^{44} +1.60965e19 q^{46} +1.28817e19 q^{47} +1.48129e19 q^{49} -4.01411e19 q^{50} -1.49441e20 q^{52} -1.00229e19 q^{53} -9.36129e19 q^{55} -1.89792e20 q^{56} -4.19047e19 q^{58} -1.29320e19 q^{59} +1.60482e20 q^{61} +7.73794e20 q^{62} -9.09050e20 q^{64} +1.46914e21 q^{65} +3.50743e20 q^{67} -3.42358e21 q^{68} +4.42811e21 q^{70} +2.05647e21 q^{71} -9.56238e20 q^{73} -6.82431e21 q^{74} +8.02166e21 q^{76} +4.26602e21 q^{77} -4.09106e21 q^{79} -2.59193e21 q^{80} -2.32658e22 q^{82} +1.62083e22 q^{83} +3.36569e22 q^{85} +1.74029e22 q^{86} -1.91945e22 q^{88} +2.59976e21 q^{89} -6.69501e22 q^{91} -4.87787e22 q^{92} -6.16249e22 q^{94} -7.88602e22 q^{95} +2.98925e22 q^{97} -7.08634e22 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 19109968 q^{4} + 8561438480 q^{7} + O(q^{10}) \) \( 4 q + 19109968 q^{4} + 8561438480 q^{7} + 898978809600 q^{10} - 13534569998680 q^{13} + 27399081725056 q^{16} + 823113907401824 q^{19} - 2742157478016000 q^{22} + 24257341946701900 q^{25} + 210193505445412160 q^{28} + 199165518593535632 q^{31} + 2993143687606771200 q^{34} + 6114312513425419640 q^{37} + 14523223786943846400 q^{40} + 8384436518889991520 q^{43} + 27149798727824025600 q^{46} - 15307929032390927772 q^{49} - 333882838377412430560 q^{52} - 425998563364306944000 q^{55} - 519004760872044384000 q^{58} - 187635897281945053672 q^{61} - 1002932785066778037248 q^{64} + 2217868901107775746880 q^{67} + 9884892109681717632000 q^{70} + 5558219032297686657560 q^{73} + 17444623005047704385408 q^{76} - 1562583260526294231664 q^{79} - 41334076267904219328000 q^{82} + 17862754059765861580800 q^{85} - 85611540851539743744000 q^{88} - 149580096340642991861600 q^{91} - 164391971318917439846400 q^{94} + 61688465492693971809080 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4783.90 −1.65172 −0.825861 0.563873i \(-0.809311\pi\)
−0.825861 + 0.563873i \(0.809311\pi\)
\(3\) 0 0
\(4\) 1.44971e7 1.72819
\(5\) −1.42520e8 −1.30533 −0.652663 0.757648i \(-0.726348\pi\)
−0.652663 + 0.757648i \(0.726348\pi\)
\(6\) 0 0
\(7\) 6.49474e9 1.24146 0.620732 0.784023i \(-0.286835\pi\)
0.620732 + 0.784023i \(0.286835\pi\)
\(8\) −2.92224e10 −1.20276
\(9\) 0 0
\(10\) 6.81799e11 2.15604
\(11\) 6.56843e11 0.694138 0.347069 0.937840i \(-0.387177\pi\)
0.347069 + 0.937840i \(0.387177\pi\)
\(12\) 0 0
\(13\) −1.03084e13 −1.59529 −0.797647 0.603125i \(-0.793922\pi\)
−0.797647 + 0.603125i \(0.793922\pi\)
\(14\) −3.10702e13 −2.05056
\(15\) 0 0
\(16\) 1.81865e13 0.258445
\(17\) −2.36156e14 −1.67123 −0.835617 0.549313i \(-0.814890\pi\)
−0.835617 + 0.549313i \(0.814890\pi\)
\(18\) 0 0
\(19\) 5.53329e14 1.08973 0.544863 0.838525i \(-0.316582\pi\)
0.544863 + 0.838525i \(0.316582\pi\)
\(20\) −2.06612e15 −2.25585
\(21\) 0 0
\(22\) −3.14227e15 −1.14652
\(23\) −3.36472e15 −0.736341 −0.368171 0.929758i \(-0.620016\pi\)
−0.368171 + 0.929758i \(0.620016\pi\)
\(24\) 0 0
\(25\) 8.39088e15 0.703878
\(26\) 4.93141e16 2.63498
\(27\) 0 0
\(28\) 9.41548e16 2.14548
\(29\) 8.75953e15 0.133323 0.0666613 0.997776i \(-0.478765\pi\)
0.0666613 + 0.997776i \(0.478765\pi\)
\(30\) 0 0
\(31\) −1.61750e17 −1.14336 −0.571681 0.820476i \(-0.693709\pi\)
−0.571681 + 0.820476i \(0.693709\pi\)
\(32\) 1.58133e17 0.775884
\(33\) 0 0
\(34\) 1.12975e18 2.76041
\(35\) −9.25627e17 −1.62052
\(36\) 0 0
\(37\) 1.42652e18 1.31813 0.659063 0.752088i \(-0.270953\pi\)
0.659063 + 0.752088i \(0.270953\pi\)
\(38\) −2.64707e18 −1.79992
\(39\) 0 0
\(40\) 4.16476e18 1.57000
\(41\) 4.86336e18 1.38014 0.690069 0.723744i \(-0.257580\pi\)
0.690069 + 0.723744i \(0.257580\pi\)
\(42\) 0 0
\(43\) −3.63780e18 −0.596969 −0.298485 0.954414i \(-0.596481\pi\)
−0.298485 + 0.954414i \(0.596481\pi\)
\(44\) 9.52231e18 1.19960
\(45\) 0 0
\(46\) 1.60965e19 1.21623
\(47\) 1.28817e19 0.760062 0.380031 0.924974i \(-0.375913\pi\)
0.380031 + 0.924974i \(0.375913\pi\)
\(48\) 0 0
\(49\) 1.48129e19 0.541234
\(50\) −4.01411e19 −1.16261
\(51\) 0 0
\(52\) −1.49441e20 −2.75697
\(53\) −1.00229e19 −0.148533 −0.0742664 0.997238i \(-0.523662\pi\)
−0.0742664 + 0.997238i \(0.523662\pi\)
\(54\) 0 0
\(55\) −9.36129e19 −0.906077
\(56\) −1.89792e20 −1.49319
\(57\) 0 0
\(58\) −4.19047e19 −0.220212
\(59\) −1.29320e19 −0.0558301 −0.0279150 0.999610i \(-0.508887\pi\)
−0.0279150 + 0.999610i \(0.508887\pi\)
\(60\) 0 0
\(61\) 1.60482e20 0.472207 0.236104 0.971728i \(-0.424130\pi\)
0.236104 + 0.971728i \(0.424130\pi\)
\(62\) 7.73794e20 1.88852
\(63\) 0 0
\(64\) −9.09050e20 −1.53999
\(65\) 1.46914e21 2.08238
\(66\) 0 0
\(67\) 3.50743e20 0.350856 0.175428 0.984492i \(-0.443869\pi\)
0.175428 + 0.984492i \(0.443869\pi\)
\(68\) −3.42358e21 −2.88821
\(69\) 0 0
\(70\) 4.42811e21 2.67664
\(71\) 2.05647e21 1.05597 0.527985 0.849254i \(-0.322948\pi\)
0.527985 + 0.849254i \(0.322948\pi\)
\(72\) 0 0
\(73\) −9.56238e20 −0.356741 −0.178370 0.983963i \(-0.557083\pi\)
−0.178370 + 0.983963i \(0.557083\pi\)
\(74\) −6.82431e21 −2.17718
\(75\) 0 0
\(76\) 8.02166e21 1.88325
\(77\) 4.26602e21 0.861747
\(78\) 0 0
\(79\) −4.09106e21 −0.615353 −0.307677 0.951491i \(-0.599551\pi\)
−0.307677 + 0.951491i \(0.599551\pi\)
\(80\) −2.59193e21 −0.337356
\(81\) 0 0
\(82\) −2.32658e22 −2.27960
\(83\) 1.62083e22 1.38146 0.690730 0.723113i \(-0.257289\pi\)
0.690730 + 0.723113i \(0.257289\pi\)
\(84\) 0 0
\(85\) 3.36569e22 2.18151
\(86\) 1.74029e22 0.986028
\(87\) 0 0
\(88\) −1.91945e22 −0.834884
\(89\) 2.59976e21 0.0992997 0.0496499 0.998767i \(-0.484189\pi\)
0.0496499 + 0.998767i \(0.484189\pi\)
\(90\) 0 0
\(91\) −6.69501e22 −1.98050
\(92\) −4.87787e22 −1.27254
\(93\) 0 0
\(94\) −6.16249e22 −1.25541
\(95\) −7.88602e22 −1.42245
\(96\) 0 0
\(97\) 2.98925e22 0.424314 0.212157 0.977236i \(-0.431951\pi\)
0.212157 + 0.977236i \(0.431951\pi\)
\(98\) −7.08634e22 −0.893968
\(99\) 0 0
\(100\) 1.21643e23 1.21643
\(101\) 1.72257e23 1.53632 0.768159 0.640259i \(-0.221173\pi\)
0.768159 + 0.640259i \(0.221173\pi\)
\(102\) 0 0
\(103\) 1.51134e23 1.07580 0.537902 0.843007i \(-0.319217\pi\)
0.537902 + 0.843007i \(0.319217\pi\)
\(104\) 3.01234e23 1.91876
\(105\) 0 0
\(106\) 4.79487e22 0.245335
\(107\) −3.38525e23 −1.55481 −0.777405 0.629000i \(-0.783464\pi\)
−0.777405 + 0.629000i \(0.783464\pi\)
\(108\) 0 0
\(109\) −1.51292e22 −0.0561579 −0.0280789 0.999606i \(-0.508939\pi\)
−0.0280789 + 0.999606i \(0.508939\pi\)
\(110\) 4.47835e23 1.49659
\(111\) 0 0
\(112\) 1.18116e23 0.320851
\(113\) −2.35991e23 −0.578753 −0.289377 0.957215i \(-0.593448\pi\)
−0.289377 + 0.957215i \(0.593448\pi\)
\(114\) 0 0
\(115\) 4.79539e23 0.961166
\(116\) 1.26988e23 0.230406
\(117\) 0 0
\(118\) 6.18655e22 0.0922158
\(119\) −1.53377e24 −2.07478
\(120\) 0 0
\(121\) −4.63987e23 −0.518173
\(122\) −7.67728e23 −0.779955
\(123\) 0 0
\(124\) −2.34490e24 −1.97595
\(125\) 5.03100e23 0.386536
\(126\) 0 0
\(127\) 1.15026e24 0.736295 0.368147 0.929767i \(-0.379992\pi\)
0.368147 + 0.929767i \(0.379992\pi\)
\(128\) 3.02229e24 1.76775
\(129\) 0 0
\(130\) −7.02822e24 −3.43951
\(131\) 2.28724e24 1.02493 0.512463 0.858709i \(-0.328733\pi\)
0.512463 + 0.858709i \(0.328733\pi\)
\(132\) 0 0
\(133\) 3.59373e24 1.35285
\(134\) −1.67792e24 −0.579517
\(135\) 0 0
\(136\) 6.90105e24 2.01010
\(137\) 6.77439e24 1.81378 0.906888 0.421372i \(-0.138451\pi\)
0.906888 + 0.421372i \(0.138451\pi\)
\(138\) 0 0
\(139\) 2.54141e24 0.575976 0.287988 0.957634i \(-0.407014\pi\)
0.287988 + 0.957634i \(0.407014\pi\)
\(140\) −1.34189e25 −2.80056
\(141\) 0 0
\(142\) −9.83795e24 −1.74417
\(143\) −6.77097e24 −1.10735
\(144\) 0 0
\(145\) −1.24840e24 −0.174030
\(146\) 4.57455e24 0.589237
\(147\) 0 0
\(148\) 2.06803e25 2.27797
\(149\) 5.05631e24 0.515456 0.257728 0.966217i \(-0.417026\pi\)
0.257728 + 0.966217i \(0.417026\pi\)
\(150\) 0 0
\(151\) 1.95057e25 1.70579 0.852897 0.522079i \(-0.174843\pi\)
0.852897 + 0.522079i \(0.174843\pi\)
\(152\) −1.61696e25 −1.31068
\(153\) 0 0
\(154\) −2.04082e25 −1.42337
\(155\) 2.30525e25 1.49246
\(156\) 0 0
\(157\) 1.88837e25 1.05497 0.527487 0.849563i \(-0.323134\pi\)
0.527487 + 0.849563i \(0.323134\pi\)
\(158\) 1.95712e25 1.01639
\(159\) 0 0
\(160\) −2.25370e25 −1.01278
\(161\) −2.18530e25 −0.914141
\(162\) 0 0
\(163\) −3.04668e25 −1.10578 −0.552891 0.833253i \(-0.686475\pi\)
−0.552891 + 0.833253i \(0.686475\pi\)
\(164\) 7.05046e25 2.38514
\(165\) 0 0
\(166\) −7.75387e25 −2.28179
\(167\) 1.11101e24 0.0305127 0.0152563 0.999884i \(-0.495144\pi\)
0.0152563 + 0.999884i \(0.495144\pi\)
\(168\) 0 0
\(169\) 6.45082e25 1.54496
\(170\) −1.61011e26 −3.60324
\(171\) 0 0
\(172\) −5.27376e25 −1.03168
\(173\) −2.97282e25 −0.544049 −0.272025 0.962290i \(-0.587693\pi\)
−0.272025 + 0.962290i \(0.587693\pi\)
\(174\) 0 0
\(175\) 5.44966e25 0.873840
\(176\) 1.19457e25 0.179397
\(177\) 0 0
\(178\) −1.24370e25 −0.164016
\(179\) −3.56213e25 −0.440453 −0.220227 0.975449i \(-0.570680\pi\)
−0.220227 + 0.975449i \(0.570680\pi\)
\(180\) 0 0
\(181\) −2.89955e25 −0.315520 −0.157760 0.987478i \(-0.550427\pi\)
−0.157760 + 0.987478i \(0.550427\pi\)
\(182\) 3.20282e26 3.27124
\(183\) 0 0
\(184\) 9.83252e25 0.885645
\(185\) −2.03306e26 −1.72059
\(186\) 0 0
\(187\) −1.55118e26 −1.16007
\(188\) 1.86748e26 1.31353
\(189\) 0 0
\(190\) 3.77259e26 2.34949
\(191\) 1.36977e26 0.803090 0.401545 0.915839i \(-0.368473\pi\)
0.401545 + 0.915839i \(0.368473\pi\)
\(192\) 0 0
\(193\) 9.06538e25 0.471495 0.235747 0.971814i \(-0.424246\pi\)
0.235747 + 0.971814i \(0.424246\pi\)
\(194\) −1.43003e26 −0.700848
\(195\) 0 0
\(196\) 2.14744e26 0.935354
\(197\) −8.23482e25 −0.338292 −0.169146 0.985591i \(-0.554101\pi\)
−0.169146 + 0.985591i \(0.554101\pi\)
\(198\) 0 0
\(199\) 4.07308e26 1.48975 0.744874 0.667205i \(-0.232510\pi\)
0.744874 + 0.667205i \(0.232510\pi\)
\(200\) −2.45201e26 −0.846600
\(201\) 0 0
\(202\) −8.24061e26 −2.53757
\(203\) 5.68908e25 0.165515
\(204\) 0 0
\(205\) −6.93124e26 −1.80153
\(206\) −7.23008e26 −1.77693
\(207\) 0 0
\(208\) −1.87473e26 −0.412296
\(209\) 3.63450e26 0.756419
\(210\) 0 0
\(211\) −2.71522e26 −0.506474 −0.253237 0.967404i \(-0.581495\pi\)
−0.253237 + 0.967404i \(0.581495\pi\)
\(212\) −1.45303e26 −0.256693
\(213\) 0 0
\(214\) 1.61947e27 2.56812
\(215\) 5.18458e26 0.779240
\(216\) 0 0
\(217\) −1.05052e27 −1.41944
\(218\) 7.23765e25 0.0927572
\(219\) 0 0
\(220\) −1.35712e27 −1.56587
\(221\) 2.43438e27 2.66611
\(222\) 0 0
\(223\) −3.26296e26 −0.322185 −0.161093 0.986939i \(-0.551502\pi\)
−0.161093 + 0.986939i \(0.551502\pi\)
\(224\) 1.02703e27 0.963233
\(225\) 0 0
\(226\) 1.12896e27 0.955940
\(227\) −2.34616e27 −1.88826 −0.944128 0.329580i \(-0.893093\pi\)
−0.944128 + 0.329580i \(0.893093\pi\)
\(228\) 0 0
\(229\) 1.12251e27 0.816739 0.408370 0.912817i \(-0.366097\pi\)
0.408370 + 0.912817i \(0.366097\pi\)
\(230\) −2.29406e27 −1.58758
\(231\) 0 0
\(232\) −2.55974e26 −0.160356
\(233\) −1.38324e27 −0.824719 −0.412359 0.911021i \(-0.635295\pi\)
−0.412359 + 0.911021i \(0.635295\pi\)
\(234\) 0 0
\(235\) −1.83590e27 −0.992129
\(236\) −1.87477e26 −0.0964849
\(237\) 0 0
\(238\) 7.33742e27 3.42696
\(239\) −3.66637e26 −0.163178 −0.0815888 0.996666i \(-0.525999\pi\)
−0.0815888 + 0.996666i \(0.525999\pi\)
\(240\) 0 0
\(241\) 1.56991e27 0.634862 0.317431 0.948281i \(-0.397180\pi\)
0.317431 + 0.948281i \(0.397180\pi\)
\(242\) 2.21967e27 0.855878
\(243\) 0 0
\(244\) 2.32652e27 0.816062
\(245\) −2.11113e27 −0.706487
\(246\) 0 0
\(247\) −5.70391e27 −1.73843
\(248\) 4.72671e27 1.37520
\(249\) 0 0
\(250\) −2.40678e27 −0.638450
\(251\) 3.08385e27 0.781349 0.390674 0.920529i \(-0.372242\pi\)
0.390674 + 0.920529i \(0.372242\pi\)
\(252\) 0 0
\(253\) −2.21009e27 −0.511122
\(254\) −5.50271e27 −1.21615
\(255\) 0 0
\(256\) −6.83268e27 −1.37985
\(257\) 1.78781e27 0.345216 0.172608 0.984991i \(-0.444781\pi\)
0.172608 + 0.984991i \(0.444781\pi\)
\(258\) 0 0
\(259\) 9.26485e27 1.63641
\(260\) 2.12983e28 3.59874
\(261\) 0 0
\(262\) −1.09419e28 −1.69289
\(263\) −1.23162e28 −1.82384 −0.911921 0.410367i \(-0.865401\pi\)
−0.911921 + 0.410367i \(0.865401\pi\)
\(264\) 0 0
\(265\) 1.42846e27 0.193884
\(266\) −1.71920e28 −2.23454
\(267\) 0 0
\(268\) 5.08476e27 0.606345
\(269\) −5.19149e27 −0.593118 −0.296559 0.955015i \(-0.595839\pi\)
−0.296559 + 0.955015i \(0.595839\pi\)
\(270\) 0 0
\(271\) 4.52253e27 0.474498 0.237249 0.971449i \(-0.423754\pi\)
0.237249 + 0.971449i \(0.423754\pi\)
\(272\) −4.29485e27 −0.431923
\(273\) 0 0
\(274\) −3.24080e28 −2.99585
\(275\) 5.51149e27 0.488589
\(276\) 0 0
\(277\) −7.44055e27 −0.606859 −0.303429 0.952854i \(-0.598132\pi\)
−0.303429 + 0.952854i \(0.598132\pi\)
\(278\) −1.21578e28 −0.951353
\(279\) 0 0
\(280\) 2.70490e28 1.94910
\(281\) −2.32245e27 −0.160629 −0.0803145 0.996770i \(-0.525592\pi\)
−0.0803145 + 0.996770i \(0.525592\pi\)
\(282\) 0 0
\(283\) 7.10788e27 0.453102 0.226551 0.973999i \(-0.427255\pi\)
0.226551 + 0.973999i \(0.427255\pi\)
\(284\) 2.98128e28 1.82491
\(285\) 0 0
\(286\) 3.23916e28 1.82904
\(287\) 3.15863e28 1.71339
\(288\) 0 0
\(289\) 3.58023e28 1.79302
\(290\) 5.97224e27 0.287449
\(291\) 0 0
\(292\) −1.38627e28 −0.616515
\(293\) 1.07564e28 0.459928 0.229964 0.973199i \(-0.426139\pi\)
0.229964 + 0.973199i \(0.426139\pi\)
\(294\) 0 0
\(295\) 1.84307e27 0.0728765
\(296\) −4.16862e28 −1.58540
\(297\) 0 0
\(298\) −2.41889e28 −0.851391
\(299\) 3.46847e28 1.17468
\(300\) 0 0
\(301\) −2.36266e28 −0.741116
\(302\) −9.33133e28 −2.81750
\(303\) 0 0
\(304\) 1.00631e28 0.281634
\(305\) −2.28718e28 −0.616384
\(306\) 0 0
\(307\) 4.66353e28 1.16580 0.582899 0.812545i \(-0.301918\pi\)
0.582899 + 0.812545i \(0.301918\pi\)
\(308\) 6.18449e28 1.48926
\(309\) 0 0
\(310\) −1.10281e29 −2.46513
\(311\) 1.35856e28 0.292639 0.146320 0.989237i \(-0.453257\pi\)
0.146320 + 0.989237i \(0.453257\pi\)
\(312\) 0 0
\(313\) 2.25524e28 0.451267 0.225633 0.974212i \(-0.427555\pi\)
0.225633 + 0.974212i \(0.427555\pi\)
\(314\) −9.03379e28 −1.74252
\(315\) 0 0
\(316\) −5.93085e28 −1.06345
\(317\) −2.03254e28 −0.351445 −0.175723 0.984440i \(-0.556226\pi\)
−0.175723 + 0.984440i \(0.556226\pi\)
\(318\) 0 0
\(319\) 5.75363e27 0.0925442
\(320\) 1.29557e29 2.01019
\(321\) 0 0
\(322\) 1.04543e29 1.50991
\(323\) −1.30672e29 −1.82119
\(324\) 0 0
\(325\) −8.64962e28 −1.12289
\(326\) 1.45750e29 1.82645
\(327\) 0 0
\(328\) −1.42119e29 −1.65998
\(329\) 8.36635e28 0.943590
\(330\) 0 0
\(331\) −7.51986e28 −0.791020 −0.395510 0.918462i \(-0.629432\pi\)
−0.395510 + 0.918462i \(0.629432\pi\)
\(332\) 2.34973e29 2.38742
\(333\) 0 0
\(334\) −5.31498e27 −0.0503984
\(335\) −4.99878e28 −0.457982
\(336\) 0 0
\(337\) 1.54497e29 1.32183 0.660916 0.750460i \(-0.270168\pi\)
0.660916 + 0.750460i \(0.270168\pi\)
\(338\) −3.08601e29 −2.55185
\(339\) 0 0
\(340\) 4.87927e29 3.77005
\(341\) −1.06244e29 −0.793651
\(342\) 0 0
\(343\) −8.15470e28 −0.569542
\(344\) 1.06305e29 0.718013
\(345\) 0 0
\(346\) 1.42217e29 0.898618
\(347\) −2.64051e29 −1.61398 −0.806991 0.590564i \(-0.798905\pi\)
−0.806991 + 0.590564i \(0.798905\pi\)
\(348\) 0 0
\(349\) −1.13409e29 −0.648864 −0.324432 0.945909i \(-0.605173\pi\)
−0.324432 + 0.945909i \(0.605173\pi\)
\(350\) −2.60706e29 −1.44334
\(351\) 0 0
\(352\) 1.03868e29 0.538571
\(353\) 2.02443e29 1.01600 0.508000 0.861357i \(-0.330385\pi\)
0.508000 + 0.861357i \(0.330385\pi\)
\(354\) 0 0
\(355\) −2.93087e29 −1.37839
\(356\) 3.76889e28 0.171609
\(357\) 0 0
\(358\) 1.70409e29 0.727506
\(359\) 3.35901e29 1.38875 0.694377 0.719612i \(-0.255680\pi\)
0.694377 + 0.719612i \(0.255680\pi\)
\(360\) 0 0
\(361\) 4.83433e28 0.187501
\(362\) 1.38711e29 0.521151
\(363\) 0 0
\(364\) −9.70581e29 −3.42268
\(365\) 1.36283e29 0.465664
\(366\) 0 0
\(367\) 4.32640e29 1.38825 0.694125 0.719855i \(-0.255792\pi\)
0.694125 + 0.719855i \(0.255792\pi\)
\(368\) −6.11924e28 −0.190304
\(369\) 0 0
\(370\) 9.72598e29 2.84193
\(371\) −6.50964e28 −0.184398
\(372\) 0 0
\(373\) −6.52581e29 −1.73773 −0.868866 0.495048i \(-0.835151\pi\)
−0.868866 + 0.495048i \(0.835151\pi\)
\(374\) 7.42068e29 1.91611
\(375\) 0 0
\(376\) −3.76435e29 −0.914175
\(377\) −9.02963e28 −0.212689
\(378\) 0 0
\(379\) −3.54426e29 −0.785551 −0.392775 0.919634i \(-0.628485\pi\)
−0.392775 + 0.919634i \(0.628485\pi\)
\(380\) −1.14324e30 −2.45826
\(381\) 0 0
\(382\) −6.55285e29 −1.32648
\(383\) 2.02823e29 0.398411 0.199206 0.979958i \(-0.436164\pi\)
0.199206 + 0.979958i \(0.436164\pi\)
\(384\) 0 0
\(385\) −6.07992e29 −1.12486
\(386\) −4.33679e29 −0.778779
\(387\) 0 0
\(388\) 4.33354e29 0.733293
\(389\) 2.72682e29 0.447957 0.223979 0.974594i \(-0.428095\pi\)
0.223979 + 0.974594i \(0.428095\pi\)
\(390\) 0 0
\(391\) 7.94601e29 1.23060
\(392\) −4.32868e29 −0.650977
\(393\) 0 0
\(394\) 3.93945e29 0.558765
\(395\) 5.83056e29 0.803237
\(396\) 0 0
\(397\) 1.06349e29 0.138243 0.0691213 0.997608i \(-0.477980\pi\)
0.0691213 + 0.997608i \(0.477980\pi\)
\(398\) −1.94852e30 −2.46065
\(399\) 0 0
\(400\) 1.52601e29 0.181914
\(401\) −9.48539e29 −1.09874 −0.549370 0.835579i \(-0.685132\pi\)
−0.549370 + 0.835579i \(0.685132\pi\)
\(402\) 0 0
\(403\) 1.66737e30 1.82400
\(404\) 2.49723e30 2.65505
\(405\) 0 0
\(406\) −2.72160e29 −0.273385
\(407\) 9.36997e29 0.914961
\(408\) 0 0
\(409\) 8.99433e29 0.830139 0.415069 0.909790i \(-0.363757\pi\)
0.415069 + 0.909790i \(0.363757\pi\)
\(410\) 3.31584e30 2.97563
\(411\) 0 0
\(412\) 2.19100e30 1.85919
\(413\) −8.39901e28 −0.0693111
\(414\) 0 0
\(415\) −2.30999e30 −1.80326
\(416\) −1.63009e30 −1.23776
\(417\) 0 0
\(418\) −1.73871e30 −1.24940
\(419\) −5.82139e29 −0.406973 −0.203487 0.979078i \(-0.565227\pi\)
−0.203487 + 0.979078i \(0.565227\pi\)
\(420\) 0 0
\(421\) −8.99599e29 −0.595394 −0.297697 0.954660i \(-0.596219\pi\)
−0.297697 + 0.954660i \(0.596219\pi\)
\(422\) 1.29894e30 0.836555
\(423\) 0 0
\(424\) 2.92894e29 0.178650
\(425\) −1.98156e30 −1.17635
\(426\) 0 0
\(427\) 1.04229e30 0.586228
\(428\) −4.90763e30 −2.68700
\(429\) 0 0
\(430\) −2.48025e30 −1.28709
\(431\) −1.72002e30 −0.869051 −0.434525 0.900660i \(-0.643084\pi\)
−0.434525 + 0.900660i \(0.643084\pi\)
\(432\) 0 0
\(433\) 4.04694e30 1.93872 0.969362 0.245635i \(-0.0789965\pi\)
0.969362 + 0.245635i \(0.0789965\pi\)
\(434\) 5.02559e30 2.34453
\(435\) 0 0
\(436\) −2.19329e29 −0.0970513
\(437\) −1.86180e30 −0.802410
\(438\) 0 0
\(439\) 1.61226e30 0.659317 0.329659 0.944100i \(-0.393066\pi\)
0.329659 + 0.944100i \(0.393066\pi\)
\(440\) 2.73559e30 1.08980
\(441\) 0 0
\(442\) −1.16458e31 −4.40367
\(443\) −4.83888e30 −1.78280 −0.891398 0.453221i \(-0.850275\pi\)
−0.891398 + 0.453221i \(0.850275\pi\)
\(444\) 0 0
\(445\) −3.70516e29 −0.129619
\(446\) 1.56097e30 0.532161
\(447\) 0 0
\(448\) −5.90404e30 −1.91184
\(449\) 4.75344e30 1.50029 0.750145 0.661273i \(-0.229984\pi\)
0.750145 + 0.661273i \(0.229984\pi\)
\(450\) 0 0
\(451\) 3.19447e30 0.958005
\(452\) −3.42118e30 −1.00019
\(453\) 0 0
\(454\) 1.12238e31 3.11887
\(455\) 9.54169e30 2.58520
\(456\) 0 0
\(457\) −6.88472e30 −1.77358 −0.886788 0.462176i \(-0.847069\pi\)
−0.886788 + 0.462176i \(0.847069\pi\)
\(458\) −5.36999e30 −1.34903
\(459\) 0 0
\(460\) 6.95191e30 1.66108
\(461\) 7.02180e30 1.63639 0.818197 0.574938i \(-0.194974\pi\)
0.818197 + 0.574938i \(0.194974\pi\)
\(462\) 0 0
\(463\) 4.31929e29 0.0957703 0.0478852 0.998853i \(-0.484752\pi\)
0.0478852 + 0.998853i \(0.484752\pi\)
\(464\) 1.59305e29 0.0344566
\(465\) 0 0
\(466\) 6.61730e30 1.36221
\(467\) −8.75520e30 −1.75842 −0.879208 0.476437i \(-0.841928\pi\)
−0.879208 + 0.476437i \(0.841928\pi\)
\(468\) 0 0
\(469\) 2.27799e30 0.435576
\(470\) 8.78275e30 1.63872
\(471\) 0 0
\(472\) 3.77904e29 0.0671504
\(473\) −2.38947e30 −0.414379
\(474\) 0 0
\(475\) 4.64292e30 0.767034
\(476\) −2.22353e31 −3.58560
\(477\) 0 0
\(478\) 1.75395e30 0.269524
\(479\) 1.07807e31 1.61729 0.808647 0.588294i \(-0.200200\pi\)
0.808647 + 0.588294i \(0.200200\pi\)
\(480\) 0 0
\(481\) −1.47050e31 −2.10280
\(482\) −7.51030e30 −1.04862
\(483\) 0 0
\(484\) −6.72647e30 −0.895500
\(485\) −4.26026e30 −0.553868
\(486\) 0 0
\(487\) 9.49272e30 1.17708 0.588542 0.808466i \(-0.299702\pi\)
0.588542 + 0.808466i \(0.299702\pi\)
\(488\) −4.68966e30 −0.567954
\(489\) 0 0
\(490\) 1.00994e31 1.16692
\(491\) −5.62204e30 −0.634537 −0.317269 0.948336i \(-0.602766\pi\)
−0.317269 + 0.948336i \(0.602766\pi\)
\(492\) 0 0
\(493\) −2.06862e30 −0.222813
\(494\) 2.72869e31 2.87141
\(495\) 0 0
\(496\) −2.94166e30 −0.295497
\(497\) 1.33562e31 1.31095
\(498\) 0 0
\(499\) −1.40826e29 −0.0131986 −0.00659928 0.999978i \(-0.502101\pi\)
−0.00659928 + 0.999978i \(0.502101\pi\)
\(500\) 7.29349e30 0.668006
\(501\) 0 0
\(502\) −1.47528e31 −1.29057
\(503\) −7.85202e30 −0.671351 −0.335675 0.941978i \(-0.608965\pi\)
−0.335675 + 0.941978i \(0.608965\pi\)
\(504\) 0 0
\(505\) −2.45500e31 −2.00540
\(506\) 1.05729e31 0.844232
\(507\) 0 0
\(508\) 1.66754e31 1.27246
\(509\) 6.86794e30 0.512356 0.256178 0.966630i \(-0.417537\pi\)
0.256178 + 0.966630i \(0.417537\pi\)
\(510\) 0 0
\(511\) −6.21052e30 −0.442881
\(512\) 7.33402e30 0.511373
\(513\) 0 0
\(514\) −8.55272e30 −0.570202
\(515\) −2.15395e31 −1.40428
\(516\) 0 0
\(517\) 8.46127e30 0.527588
\(518\) −4.43221e31 −2.70289
\(519\) 0 0
\(520\) −4.29318e31 −2.50461
\(521\) −2.37730e31 −1.35660 −0.678299 0.734786i \(-0.737282\pi\)
−0.678299 + 0.734786i \(0.737282\pi\)
\(522\) 0 0
\(523\) −1.46642e31 −0.800733 −0.400366 0.916355i \(-0.631117\pi\)
−0.400366 + 0.916355i \(0.631117\pi\)
\(524\) 3.31584e31 1.77127
\(525\) 0 0
\(526\) 5.89197e31 3.01248
\(527\) 3.81982e31 1.91083
\(528\) 0 0
\(529\) −9.55911e30 −0.457802
\(530\) −6.83363e30 −0.320243
\(531\) 0 0
\(532\) 5.20986e31 2.33799
\(533\) −5.01332e31 −2.20172
\(534\) 0 0
\(535\) 4.82464e31 2.02954
\(536\) −1.02496e31 −0.421997
\(537\) 0 0
\(538\) 2.48356e31 0.979666
\(539\) 9.72975e30 0.375691
\(540\) 0 0
\(541\) −4.35565e31 −1.61170 −0.805851 0.592119i \(-0.798292\pi\)
−0.805851 + 0.592119i \(0.798292\pi\)
\(542\) −2.16353e31 −0.783740
\(543\) 0 0
\(544\) −3.73441e31 −1.29668
\(545\) 2.15620e30 0.0733044
\(546\) 0 0
\(547\) 2.29429e31 0.747816 0.373908 0.927466i \(-0.378018\pi\)
0.373908 + 0.927466i \(0.378018\pi\)
\(548\) 9.82089e31 3.13455
\(549\) 0 0
\(550\) −2.63664e31 −0.807013
\(551\) 4.84690e30 0.145285
\(552\) 0 0
\(553\) −2.65704e31 −0.763939
\(554\) 3.55948e31 1.00236
\(555\) 0 0
\(556\) 3.68430e31 0.995395
\(557\) −3.17560e29 −0.00840410 −0.00420205 0.999991i \(-0.501338\pi\)
−0.00420205 + 0.999991i \(0.501338\pi\)
\(558\) 0 0
\(559\) 3.74998e31 0.952341
\(560\) −1.68339e31 −0.418815
\(561\) 0 0
\(562\) 1.11104e31 0.265315
\(563\) 2.42453e31 0.567260 0.283630 0.958934i \(-0.408461\pi\)
0.283630 + 0.958934i \(0.408461\pi\)
\(564\) 0 0
\(565\) 3.36333e31 0.755462
\(566\) −3.40034e31 −0.748399
\(567\) 0 0
\(568\) −6.00949e31 −1.27008
\(569\) 1.73034e31 0.358376 0.179188 0.983815i \(-0.442653\pi\)
0.179188 + 0.983815i \(0.442653\pi\)
\(570\) 0 0
\(571\) 1.56501e31 0.311317 0.155658 0.987811i \(-0.450250\pi\)
0.155658 + 0.987811i \(0.450250\pi\)
\(572\) −9.81593e31 −1.91372
\(573\) 0 0
\(574\) −1.51106e32 −2.83005
\(575\) −2.82330e31 −0.518295
\(576\) 0 0
\(577\) 2.29233e30 0.0404348 0.0202174 0.999796i \(-0.493564\pi\)
0.0202174 + 0.999796i \(0.493564\pi\)
\(578\) −1.71275e32 −2.96158
\(579\) 0 0
\(580\) −1.80982e31 −0.300756
\(581\) 1.05268e32 1.71503
\(582\) 0 0
\(583\) −6.58350e30 −0.103102
\(584\) 2.79435e31 0.429075
\(585\) 0 0
\(586\) −5.14576e31 −0.759674
\(587\) −7.82714e30 −0.113309 −0.0566545 0.998394i \(-0.518043\pi\)
−0.0566545 + 0.998394i \(0.518043\pi\)
\(588\) 0 0
\(589\) −8.95008e31 −1.24595
\(590\) −8.81704e30 −0.120372
\(591\) 0 0
\(592\) 2.59433e31 0.340664
\(593\) 5.20855e31 0.670792 0.335396 0.942077i \(-0.391130\pi\)
0.335396 + 0.942077i \(0.391130\pi\)
\(594\) 0 0
\(595\) 2.18593e32 2.70826
\(596\) 7.33018e31 0.890805
\(597\) 0 0
\(598\) −1.65928e32 −1.94025
\(599\) 2.46078e31 0.282270 0.141135 0.989990i \(-0.454925\pi\)
0.141135 + 0.989990i \(0.454925\pi\)
\(600\) 0 0
\(601\) −6.01876e31 −0.664432 −0.332216 0.943203i \(-0.607796\pi\)
−0.332216 + 0.943203i \(0.607796\pi\)
\(602\) 1.13027e32 1.22412
\(603\) 0 0
\(604\) 2.82776e32 2.94793
\(605\) 6.61273e31 0.676385
\(606\) 0 0
\(607\) 1.78920e32 1.76193 0.880964 0.473183i \(-0.156895\pi\)
0.880964 + 0.473183i \(0.156895\pi\)
\(608\) 8.74994e31 0.845501
\(609\) 0 0
\(610\) 1.09416e32 1.01810
\(611\) −1.32789e32 −1.21252
\(612\) 0 0
\(613\) −1.26160e32 −1.10950 −0.554748 0.832018i \(-0.687185\pi\)
−0.554748 + 0.832018i \(0.687185\pi\)
\(614\) −2.23099e32 −1.92558
\(615\) 0 0
\(616\) −1.24663e32 −1.03648
\(617\) −1.03069e32 −0.841101 −0.420550 0.907269i \(-0.638163\pi\)
−0.420550 + 0.907269i \(0.638163\pi\)
\(618\) 0 0
\(619\) 1.21348e32 0.954096 0.477048 0.878877i \(-0.341707\pi\)
0.477048 + 0.878877i \(0.341707\pi\)
\(620\) 3.34194e32 2.57925
\(621\) 0 0
\(622\) −6.49919e31 −0.483359
\(623\) 1.68848e31 0.123277
\(624\) 0 0
\(625\) −1.71729e32 −1.20843
\(626\) −1.07889e32 −0.745368
\(627\) 0 0
\(628\) 2.73759e32 1.82319
\(629\) −3.36881e32 −2.20290
\(630\) 0 0
\(631\) 3.50812e31 0.221175 0.110588 0.993866i \(-0.464727\pi\)
0.110588 + 0.993866i \(0.464727\pi\)
\(632\) 1.19550e32 0.740125
\(633\) 0 0
\(634\) 9.72346e31 0.580490
\(635\) −1.63934e32 −0.961105
\(636\) 0 0
\(637\) −1.52697e32 −0.863427
\(638\) −2.75248e31 −0.152857
\(639\) 0 0
\(640\) −4.30736e32 −2.30750
\(641\) 1.36428e32 0.717855 0.358927 0.933366i \(-0.383143\pi\)
0.358927 + 0.933366i \(0.383143\pi\)
\(642\) 0 0
\(643\) 8.01130e31 0.406702 0.203351 0.979106i \(-0.434817\pi\)
0.203351 + 0.979106i \(0.434817\pi\)
\(644\) −3.16805e32 −1.57981
\(645\) 0 0
\(646\) 6.25123e32 3.00809
\(647\) −1.62553e32 −0.768416 −0.384208 0.923246i \(-0.625526\pi\)
−0.384208 + 0.923246i \(0.625526\pi\)
\(648\) 0 0
\(649\) −8.49431e30 −0.0387538
\(650\) 4.13789e32 1.85471
\(651\) 0 0
\(652\) −4.41680e32 −1.91100
\(653\) 4.57884e32 1.94650 0.973249 0.229751i \(-0.0737912\pi\)
0.973249 + 0.229751i \(0.0737912\pi\)
\(654\) 0 0
\(655\) −3.25977e32 −1.33786
\(656\) 8.84474e31 0.356690
\(657\) 0 0
\(658\) −4.00238e32 −1.55855
\(659\) 4.46949e32 1.71031 0.855157 0.518369i \(-0.173461\pi\)
0.855157 + 0.518369i \(0.173461\pi\)
\(660\) 0 0
\(661\) −1.68932e32 −0.624301 −0.312151 0.950033i \(-0.601049\pi\)
−0.312151 + 0.950033i \(0.601049\pi\)
\(662\) 3.59742e32 1.30655
\(663\) 0 0
\(664\) −4.73644e32 −1.66157
\(665\) −5.12176e32 −1.76592
\(666\) 0 0
\(667\) −2.94734e31 −0.0981709
\(668\) 1.61065e31 0.0527316
\(669\) 0 0
\(670\) 2.39136e32 0.756459
\(671\) 1.05411e32 0.327777
\(672\) 0 0
\(673\) 5.31206e32 1.59621 0.798104 0.602519i \(-0.205836\pi\)
0.798104 + 0.602519i \(0.205836\pi\)
\(674\) −7.39098e32 −2.18330
\(675\) 0 0
\(676\) 9.35181e32 2.66998
\(677\) −1.10916e32 −0.311333 −0.155666 0.987810i \(-0.549752\pi\)
−0.155666 + 0.987810i \(0.549752\pi\)
\(678\) 0 0
\(679\) 1.94144e32 0.526770
\(680\) −9.83534e32 −2.62384
\(681\) 0 0
\(682\) 5.08261e32 1.31089
\(683\) 4.30126e32 1.09083 0.545415 0.838166i \(-0.316372\pi\)
0.545415 + 0.838166i \(0.316372\pi\)
\(684\) 0 0
\(685\) −9.65483e32 −2.36757
\(686\) 3.90113e32 0.940725
\(687\) 0 0
\(688\) −6.61588e31 −0.154284
\(689\) 1.03320e32 0.236954
\(690\) 0 0
\(691\) −1.48977e32 −0.330461 −0.165231 0.986255i \(-0.552837\pi\)
−0.165231 + 0.986255i \(0.552837\pi\)
\(692\) −4.30972e32 −0.940219
\(693\) 0 0
\(694\) 1.26319e33 2.66585
\(695\) −3.62200e32 −0.751837
\(696\) 0 0
\(697\) −1.14851e33 −2.30653
\(698\) 5.42535e32 1.07174
\(699\) 0 0
\(700\) 7.90042e32 1.51016
\(701\) −1.71421e32 −0.322334 −0.161167 0.986927i \(-0.551526\pi\)
−0.161167 + 0.986927i \(0.551526\pi\)
\(702\) 0 0
\(703\) 7.89333e32 1.43640
\(704\) −5.97103e32 −1.06897
\(705\) 0 0
\(706\) −9.68467e32 −1.67815
\(707\) 1.11876e33 1.90728
\(708\) 0 0
\(709\) 1.24223e32 0.205008 0.102504 0.994733i \(-0.467315\pi\)
0.102504 + 0.994733i \(0.467315\pi\)
\(710\) 1.40210e33 2.27671
\(711\) 0 0
\(712\) −7.59711e31 −0.119434
\(713\) 5.44243e32 0.841905
\(714\) 0 0
\(715\) 9.64995e32 1.44546
\(716\) −5.16405e32 −0.761186
\(717\) 0 0
\(718\) −1.60692e33 −2.29384
\(719\) −1.12436e33 −1.57951 −0.789753 0.613425i \(-0.789791\pi\)
−0.789753 + 0.613425i \(0.789791\pi\)
\(720\) 0 0
\(721\) 9.81574e32 1.33557
\(722\) −2.31269e32 −0.309699
\(723\) 0 0
\(724\) −4.20350e32 −0.545277
\(725\) 7.35002e31 0.0938429
\(726\) 0 0
\(727\) 5.20007e32 0.643225 0.321613 0.946871i \(-0.395775\pi\)
0.321613 + 0.946871i \(0.395775\pi\)
\(728\) 1.95644e33 2.38208
\(729\) 0 0
\(730\) −6.51962e32 −0.769147
\(731\) 8.59091e32 0.997675
\(732\) 0 0
\(733\) −1.16237e33 −1.30812 −0.654060 0.756443i \(-0.726935\pi\)
−0.654060 + 0.756443i \(0.726935\pi\)
\(734\) −2.06971e33 −2.29300
\(735\) 0 0
\(736\) −5.32073e32 −0.571316
\(737\) 2.30383e32 0.243543
\(738\) 0 0
\(739\) −1.47548e33 −1.51190 −0.755950 0.654629i \(-0.772825\pi\)
−0.755950 + 0.654629i \(0.772825\pi\)
\(740\) −2.94735e33 −2.97349
\(741\) 0 0
\(742\) 3.11415e32 0.304575
\(743\) −1.35734e33 −1.30712 −0.653561 0.756874i \(-0.726726\pi\)
−0.653561 + 0.756874i \(0.726726\pi\)
\(744\) 0 0
\(745\) −7.20623e32 −0.672839
\(746\) 3.12188e33 2.87025
\(747\) 0 0
\(748\) −2.24876e33 −2.00481
\(749\) −2.19863e33 −1.93024
\(750\) 0 0
\(751\) 1.72188e33 1.46603 0.733016 0.680211i \(-0.238112\pi\)
0.733016 + 0.680211i \(0.238112\pi\)
\(752\) 2.34273e32 0.196434
\(753\) 0 0
\(754\) 4.31968e32 0.351303
\(755\) −2.77994e33 −2.22662
\(756\) 0 0
\(757\) 2.32468e33 1.80618 0.903091 0.429450i \(-0.141293\pi\)
0.903091 + 0.429450i \(0.141293\pi\)
\(758\) 1.69554e33 1.29751
\(759\) 0 0
\(760\) 2.30448e33 1.71087
\(761\) 1.60288e33 1.17214 0.586068 0.810262i \(-0.300675\pi\)
0.586068 + 0.810262i \(0.300675\pi\)
\(762\) 0 0
\(763\) −9.82601e31 −0.0697180
\(764\) 1.98577e33 1.38789
\(765\) 0 0
\(766\) −9.70285e32 −0.658065
\(767\) 1.33308e32 0.0890654
\(768\) 0 0
\(769\) 1.94080e33 1.25842 0.629212 0.777234i \(-0.283378\pi\)
0.629212 + 0.777234i \(0.283378\pi\)
\(770\) 2.90857e33 1.85796
\(771\) 0 0
\(772\) 1.31422e33 0.814831
\(773\) 1.83275e33 1.11954 0.559768 0.828649i \(-0.310890\pi\)
0.559768 + 0.828649i \(0.310890\pi\)
\(774\) 0 0
\(775\) −1.35722e33 −0.804788
\(776\) −8.73529e32 −0.510349
\(777\) 0 0
\(778\) −1.30448e33 −0.739901
\(779\) 2.69104e33 1.50397
\(780\) 0 0
\(781\) 1.35078e33 0.732989
\(782\) −3.80129e33 −2.03261
\(783\) 0 0
\(784\) 2.69394e32 0.139879
\(785\) −2.69130e33 −1.37709
\(786\) 0 0
\(787\) 8.71269e32 0.432955 0.216477 0.976288i \(-0.430543\pi\)
0.216477 + 0.976288i \(0.430543\pi\)
\(788\) −1.19381e33 −0.584633
\(789\) 0 0
\(790\) −2.78928e33 −1.32672
\(791\) −1.53270e33 −0.718502
\(792\) 0 0
\(793\) −1.65430e33 −0.753309
\(794\) −5.08762e32 −0.228338
\(795\) 0 0
\(796\) 5.90478e33 2.57456
\(797\) 2.50263e33 1.07554 0.537769 0.843092i \(-0.319267\pi\)
0.537769 + 0.843092i \(0.319267\pi\)
\(798\) 0 0
\(799\) −3.04210e33 −1.27024
\(800\) 1.32687e33 0.546128
\(801\) 0 0
\(802\) 4.53772e33 1.81481
\(803\) −6.28098e32 −0.247627
\(804\) 0 0
\(805\) 3.11448e33 1.19325
\(806\) −7.97654e33 −3.01274
\(807\) 0 0
\(808\) −5.03376e33 −1.84783
\(809\) 1.81340e33 0.656273 0.328136 0.944630i \(-0.393579\pi\)
0.328136 + 0.944630i \(0.393579\pi\)
\(810\) 0 0
\(811\) 1.10505e33 0.388726 0.194363 0.980930i \(-0.437736\pi\)
0.194363 + 0.980930i \(0.437736\pi\)
\(812\) 8.24752e32 0.286041
\(813\) 0 0
\(814\) −4.48250e33 −1.51126
\(815\) 4.34212e33 1.44341
\(816\) 0 0
\(817\) −2.01290e33 −0.650532
\(818\) −4.30280e33 −1.37116
\(819\) 0 0
\(820\) −1.00483e34 −3.11338
\(821\) 2.80900e33 0.858234 0.429117 0.903249i \(-0.358825\pi\)
0.429117 + 0.903249i \(0.358825\pi\)
\(822\) 0 0
\(823\) −5.09759e33 −1.51449 −0.757245 0.653131i \(-0.773455\pi\)
−0.757245 + 0.653131i \(0.773455\pi\)
\(824\) −4.41648e33 −1.29394
\(825\) 0 0
\(826\) 4.01800e32 0.114483
\(827\) 7.25325e32 0.203807 0.101904 0.994794i \(-0.467507\pi\)
0.101904 + 0.994794i \(0.467507\pi\)
\(828\) 0 0
\(829\) 2.48272e33 0.678500 0.339250 0.940696i \(-0.389827\pi\)
0.339250 + 0.940696i \(0.389827\pi\)
\(830\) 1.10508e34 2.97848
\(831\) 0 0
\(832\) 9.37081e33 2.45674
\(833\) −3.49816e33 −0.904528
\(834\) 0 0
\(835\) −1.58341e32 −0.0398290
\(836\) 5.26897e33 1.30723
\(837\) 0 0
\(838\) 2.78490e33 0.672207
\(839\) 6.12814e33 1.45904 0.729519 0.683961i \(-0.239744\pi\)
0.729519 + 0.683961i \(0.239744\pi\)
\(840\) 0 0
\(841\) −4.23999e33 −0.982225
\(842\) 4.30359e33 0.983427
\(843\) 0 0
\(844\) −3.93628e33 −0.875283
\(845\) −9.19368e33 −2.01668
\(846\) 0 0
\(847\) −3.01348e33 −0.643293
\(848\) −1.82282e32 −0.0383876
\(849\) 0 0
\(850\) 9.47959e33 1.94300
\(851\) −4.79983e33 −0.970591
\(852\) 0 0
\(853\) 5.01234e33 0.986567 0.493283 0.869869i \(-0.335797\pi\)
0.493283 + 0.869869i \(0.335797\pi\)
\(854\) −4.98620e33 −0.968286
\(855\) 0 0
\(856\) 9.89250e33 1.87007
\(857\) −3.25628e33 −0.607355 −0.303678 0.952775i \(-0.598215\pi\)
−0.303678 + 0.952775i \(0.598215\pi\)
\(858\) 0 0
\(859\) −5.33067e33 −0.967968 −0.483984 0.875077i \(-0.660811\pi\)
−0.483984 + 0.875077i \(0.660811\pi\)
\(860\) 7.51613e33 1.34667
\(861\) 0 0
\(862\) 8.22841e33 1.43543
\(863\) 1.52320e32 0.0262199 0.0131100 0.999914i \(-0.495827\pi\)
0.0131100 + 0.999914i \(0.495827\pi\)
\(864\) 0 0
\(865\) 4.23685e33 0.710162
\(866\) −1.93602e34 −3.20224
\(867\) 0 0
\(868\) −1.52295e34 −2.45307
\(869\) −2.68718e33 −0.427140
\(870\) 0 0
\(871\) −3.61559e33 −0.559719
\(872\) 4.42110e32 0.0675447
\(873\) 0 0
\(874\) 8.90666e33 1.32536
\(875\) 3.26751e33 0.479870
\(876\) 0 0
\(877\) −5.21094e33 −0.745454 −0.372727 0.927941i \(-0.621577\pi\)
−0.372727 + 0.927941i \(0.621577\pi\)
\(878\) −7.71291e33 −1.08901
\(879\) 0 0
\(880\) −1.70249e33 −0.234171
\(881\) 3.11935e33 0.423488 0.211744 0.977325i \(-0.432086\pi\)
0.211744 + 0.977325i \(0.432086\pi\)
\(882\) 0 0
\(883\) −5.95233e33 −0.787298 −0.393649 0.919261i \(-0.628787\pi\)
−0.393649 + 0.919261i \(0.628787\pi\)
\(884\) 3.52915e34 4.60754
\(885\) 0 0
\(886\) 2.31487e34 2.94469
\(887\) −1.22417e34 −1.53716 −0.768582 0.639751i \(-0.779037\pi\)
−0.768582 + 0.639751i \(0.779037\pi\)
\(888\) 0 0
\(889\) 7.47061e33 0.914084
\(890\) 1.77251e33 0.214094
\(891\) 0 0
\(892\) −4.73034e33 −0.556796
\(893\) 7.12783e33 0.828258
\(894\) 0 0
\(895\) 5.07673e33 0.574935
\(896\) 1.96290e34 2.19460
\(897\) 0 0
\(898\) −2.27400e34 −2.47806
\(899\) −1.41685e33 −0.152436
\(900\) 0 0
\(901\) 2.36698e33 0.248233
\(902\) −1.52820e34 −1.58236
\(903\) 0 0
\(904\) 6.89622e33 0.696104
\(905\) 4.13242e33 0.411856
\(906\) 0 0
\(907\) −1.82802e34 −1.77622 −0.888109 0.459632i \(-0.847981\pi\)
−0.888109 + 0.459632i \(0.847981\pi\)
\(908\) −3.40125e34 −3.26326
\(909\) 0 0
\(910\) −4.56465e34 −4.27003
\(911\) −2.70847e33 −0.250185 −0.125093 0.992145i \(-0.539923\pi\)
−0.125093 + 0.992145i \(0.539923\pi\)
\(912\) 0 0
\(913\) 1.06463e34 0.958924
\(914\) 3.29358e34 2.92946
\(915\) 0 0
\(916\) 1.62732e34 1.41148
\(917\) 1.48551e34 1.27241
\(918\) 0 0
\(919\) 3.04477e32 0.0254346 0.0127173 0.999919i \(-0.495952\pi\)
0.0127173 + 0.999919i \(0.495952\pi\)
\(920\) −1.40133e34 −1.15606
\(921\) 0 0
\(922\) −3.35916e34 −2.70287
\(923\) −2.11988e34 −1.68458
\(924\) 0 0
\(925\) 1.19697e34 0.927800
\(926\) −2.06630e33 −0.158186
\(927\) 0 0
\(928\) 1.38517e33 0.103443
\(929\) −8.26259e33 −0.609446 −0.304723 0.952441i \(-0.598564\pi\)
−0.304723 + 0.952441i \(0.598564\pi\)
\(930\) 0 0
\(931\) 8.19640e33 0.589796
\(932\) −2.00530e34 −1.42527
\(933\) 0 0
\(934\) 4.18840e34 2.90442
\(935\) 2.21073e34 1.51427
\(936\) 0 0
\(937\) 2.63092e34 1.75834 0.879169 0.476510i \(-0.158098\pi\)
0.879169 + 0.476510i \(0.158098\pi\)
\(938\) −1.08977e34 −0.719450
\(939\) 0 0
\(940\) −2.66152e34 −1.71459
\(941\) −5.07997e33 −0.323281 −0.161641 0.986850i \(-0.551679\pi\)
−0.161641 + 0.986850i \(0.551679\pi\)
\(942\) 0 0
\(943\) −1.63639e34 −1.01625
\(944\) −2.35188e32 −0.0144290
\(945\) 0 0
\(946\) 1.14310e34 0.684439
\(947\) 3.02144e33 0.178727 0.0893634 0.995999i \(-0.471517\pi\)
0.0893634 + 0.995999i \(0.471517\pi\)
\(948\) 0 0
\(949\) 9.85723e33 0.569107
\(950\) −2.22113e34 −1.26693
\(951\) 0 0
\(952\) 4.48205e34 2.49547
\(953\) 2.84405e34 1.56448 0.782239 0.622979i \(-0.214078\pi\)
0.782239 + 0.622979i \(0.214078\pi\)
\(954\) 0 0
\(955\) −1.95219e34 −1.04830
\(956\) −5.31517e33 −0.282001
\(957\) 0 0
\(958\) −5.15738e34 −2.67132
\(959\) 4.39979e34 2.25174
\(960\) 0 0
\(961\) 6.14965e33 0.307278
\(962\) 7.03474e34 3.47324
\(963\) 0 0
\(964\) 2.27592e34 1.09716
\(965\) −1.29199e34 −0.615455
\(966\) 0 0
\(967\) 1.35564e33 0.0630581 0.0315291 0.999503i \(-0.489962\pi\)
0.0315291 + 0.999503i \(0.489962\pi\)
\(968\) 1.35588e34 0.623240
\(969\) 0 0
\(970\) 2.03807e34 0.914836
\(971\) −1.58853e34 −0.704651 −0.352326 0.935877i \(-0.614609\pi\)
−0.352326 + 0.935877i \(0.614609\pi\)
\(972\) 0 0
\(973\) 1.65058e34 0.715054
\(974\) −4.54122e34 −1.94422
\(975\) 0 0
\(976\) 2.91860e33 0.122040
\(977\) −1.73918e33 −0.0718714 −0.0359357 0.999354i \(-0.511441\pi\)
−0.0359357 + 0.999354i \(0.511441\pi\)
\(978\) 0 0
\(979\) 1.70763e33 0.0689277
\(980\) −3.06052e34 −1.22094
\(981\) 0 0
\(982\) 2.68953e34 1.04808
\(983\) −7.60342e33 −0.292849 −0.146424 0.989222i \(-0.546777\pi\)
−0.146424 + 0.989222i \(0.546777\pi\)
\(984\) 0 0
\(985\) 1.17362e34 0.441582
\(986\) 9.89606e33 0.368026
\(987\) 0 0
\(988\) −8.26901e34 −3.00434
\(989\) 1.22402e34 0.439573
\(990\) 0 0
\(991\) 3.03285e34 1.06415 0.532076 0.846697i \(-0.321412\pi\)
0.532076 + 0.846697i \(0.321412\pi\)
\(992\) −2.55779e34 −0.887117
\(993\) 0 0
\(994\) −6.38949e34 −2.16532
\(995\) −5.80494e34 −1.94461
\(996\) 0 0
\(997\) −1.74476e34 −0.571137 −0.285568 0.958358i \(-0.592182\pi\)
−0.285568 + 0.958358i \(0.592182\pi\)
\(998\) 6.73697e32 0.0218004
\(999\) 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 9.24.a.d.1.1 4
3.2 odd 2 inner 9.24.a.d.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.24.a.d.1.1 4 1.1 even 1 trivial
9.24.a.d.1.4 yes 4 3.2 odd 2 inner