Properties

Label 9.24.a.d.1.4
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 1.4
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 \(-0.190689\pi\)
0.825861 + 0.563873i \(0.190689\pi\)
\(3\) 0 0
\(4\) 1.44971e7 1.72819
\(5\) 1.42520e8 1.30533 0.652663 0.757648i \(-0.273652\pi\)
0.652663 + 0.757648i \(0.273652\pi\)
\(6\) 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 \(-0.612823\pi\)
−0.347069 + 0.937840i \(0.612823\pi\)
\(12\) 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 \(-0.185110\pi\)
0.835617 + 0.549313i \(0.185110\pi\)
\(18\) 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 \(-0.379984\pi\)
0.368171 + 0.929758i \(0.379984\pi\)
\(24\) 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 \(-0.521235\pi\)
−0.0666613 + 0.997776i \(0.521235\pi\)
\(30\) 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 \(-0.742420\pi\)
−0.690069 + 0.723744i \(0.742420\pi\)
\(42\) 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 \(-0.624087\pi\)
−0.380031 + 0.924974i \(0.624087\pi\)
\(48\) 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 \(-0.476338\pi\)
0.0742664 + 0.997238i \(0.476338\pi\)
\(54\) 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 \(-0.491113\pi\)
0.0279150 + 0.999610i \(0.491113\pi\)
\(60\) 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 \(-0.677052\pi\)
−0.527985 + 0.849254i \(0.677052\pi\)
\(72\) 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 \(-0.742711\pi\)
−0.690730 + 0.723113i \(0.742711\pi\)
\(84\) 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 \(-0.515811\pi\)
−0.0496499 + 0.998767i \(0.515811\pi\)
\(90\) 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 \(-0.778827\pi\)
−0.768159 + 0.640259i \(0.778827\pi\)
\(102\) 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 \(-0.216536\pi\)
0.777405 + 0.629000i \(0.216536\pi\)
\(108\) 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 \(-0.406552\pi\)
0.289377 + 0.957215i \(0.406552\pi\)
\(114\) 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 \(-0.671267\pi\)
−0.512463 + 0.858709i \(0.671267\pi\)
\(132\) 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 \(-0.861549\pi\)
−0.906888 + 0.421372i \(0.861549\pi\)
\(138\) 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 \(-0.582974\pi\)
−0.257728 + 0.966217i \(0.582974\pi\)
\(150\) 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 \(-0.504856\pi\)
−0.0152563 + 0.999884i \(0.504856\pi\)
\(168\) 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 \(-0.412307\pi\)
0.272025 + 0.962290i \(0.412307\pi\)
\(174\) 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 \(-0.429320\pi\)
0.220227 + 0.975449i \(0.429320\pi\)
\(180\) 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 \(-0.631527\pi\)
−0.401545 + 0.915839i \(0.631527\pi\)
\(192\) 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 \(-0.445899\pi\)
0.169146 + 0.985591i \(0.445899\pi\)
\(198\) 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 \(-0.106907\pi\)
0.944128 + 0.329580i \(0.106907\pi\)
\(228\) 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 \(-0.364705\pi\)
0.412359 + 0.911021i \(0.364705\pi\)
\(234\) 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 \(-0.474001\pi\)
0.0815888 + 0.996666i \(0.474001\pi\)
\(240\) 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 \(-0.627758\pi\)
−0.390674 + 0.920529i \(0.627758\pi\)
\(252\) 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 \(-0.555219\pi\)
−0.172608 + 0.984991i \(0.555219\pi\)
\(258\) 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 \(-0.134599\pi\)
0.911921 + 0.410367i \(0.134599\pi\)
\(264\) 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 \(-0.404161\pi\)
0.296559 + 0.955015i \(0.404161\pi\)
\(270\) 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 \(-0.474408\pi\)
0.0803145 + 0.996770i \(0.474408\pi\)
\(282\) 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 \(-0.573861\pi\)
−0.229964 + 0.973199i \(0.573861\pi\)
\(294\) 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 \(-0.546743\pi\)
−0.146320 + 0.989237i \(0.546743\pi\)
\(312\) 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 \(-0.443774\pi\)
0.175723 + 0.984440i \(0.443774\pi\)
\(318\) 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 \(-0.201095\pi\)
0.806991 + 0.590564i \(0.201095\pi\)
\(348\) 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 \(-0.669615\pi\)
−0.508000 + 0.861357i \(0.669615\pi\)
\(354\) 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 \(-0.744320\pi\)
−0.694377 + 0.719612i \(0.744320\pi\)
\(360\) 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 \(-0.563836\pi\)
−0.199206 + 0.979958i \(0.563836\pi\)
\(384\) 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 \(-0.571905\pi\)
−0.223979 + 0.974594i \(0.571905\pi\)
\(390\) 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 \(-0.314868\pi\)
0.549370 + 0.835579i \(0.314868\pi\)
\(402\) 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 \(-0.434773\pi\)
0.203487 + 0.979078i \(0.434773\pi\)
\(420\) 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 \(-0.356916\pi\)
0.434525 + 0.900660i \(0.356916\pi\)
\(432\) 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 \(-0.149725\pi\)
0.891398 + 0.453221i \(0.149725\pi\)
\(444\) 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 \(-0.770016\pi\)
−0.750145 + 0.661273i \(0.770016\pi\)
\(450\) 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 \(-0.805026\pi\)
−0.818197 + 0.574938i \(0.805026\pi\)
\(462\) 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 \(-0.158072\pi\)
0.879208 + 0.476437i \(0.158072\pi\)
\(468\) 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 \(-0.799800\pi\)
−0.808647 + 0.588294i \(0.799800\pi\)
\(480\) 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 \(-0.397234\pi\)
0.317269 + 0.948336i \(0.397234\pi\)
\(492\) 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 \(-0.391035\pi\)
0.335675 + 0.941978i \(0.391035\pi\)
\(504\) 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 \(-0.582463\pi\)
−0.256178 + 0.966630i \(0.582463\pi\)
\(510\) 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 \(-0.262718\pi\)
0.678299 + 0.734786i \(0.262718\pi\)
\(522\) 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 \(-0.498662\pi\)
0.00420205 + 0.999991i \(0.498662\pi\)
\(558\) 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 \(-0.591539\pi\)
−0.283630 + 0.958934i \(0.591539\pi\)
\(564\) 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 \(-0.557347\pi\)
−0.179188 + 0.983815i \(0.557347\pi\)
\(570\) 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 \(-0.481957\pi\)
0.0566545 + 0.998394i \(0.481957\pi\)
\(588\) 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 \(-0.608870\pi\)
−0.335396 + 0.942077i \(0.608870\pi\)
\(594\) 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 \(-0.545075\pi\)
−0.141135 + 0.989990i \(0.545075\pi\)
\(600\) 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 \(-0.361837\pi\)
0.420550 + 0.907269i \(0.361837\pi\)
\(618\) 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 \(-0.616857\pi\)
−0.358927 + 0.933366i \(0.616857\pi\)
\(642\) 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 \(-0.374474\pi\)
0.384208 + 0.923246i \(0.374474\pi\)
\(648\) 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 \(-0.926209\pi\)
−0.973249 + 0.229751i \(0.926209\pi\)
\(654\) 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 \(-0.826539\pi\)
−0.855157 + 0.518369i \(0.826539\pi\)
\(660\) 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 \(-0.450248\pi\)
0.155666 + 0.987810i \(0.450248\pi\)
\(678\) 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 \(-0.683628\pi\)
−0.545415 + 0.838166i \(0.683628\pi\)
\(684\) 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 \(-0.448474\pi\)
0.161167 + 0.986927i \(0.448474\pi\)
\(702\) 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 \(-0.210209\pi\)
0.789753 + 0.613425i \(0.210209\pi\)
\(720\) 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 \(-0.273274\pi\)
0.653561 + 0.756874i \(0.273274\pi\)
\(744\) 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 \(-0.699325\pi\)
−0.586068 + 0.810262i \(0.699325\pi\)
\(762\) 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 \(-0.689110\pi\)
−0.559768 + 0.828649i \(0.689110\pi\)
\(774\) 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 \(-0.680733\pi\)
−0.537769 + 0.843092i \(0.680733\pi\)
\(798\) 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 \(-0.606421\pi\)
−0.328136 + 0.944630i \(0.606421\pi\)
\(810\) 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 \(-0.641175\pi\)
−0.429117 + 0.903249i \(0.641175\pi\)
\(822\) 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 \(-0.532493\pi\)
−0.101904 + 0.994794i \(0.532493\pi\)
\(828\) 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 \(-0.760256\pi\)
−0.729519 + 0.683961i \(0.760256\pi\)
\(840\) 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 \(-0.401785\pi\)
0.303678 + 0.952775i \(0.401785\pi\)
\(858\) 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 \(-0.504173\pi\)
−0.0131100 + 0.999914i \(0.504173\pi\)
\(864\) 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 \(-0.567914\pi\)
−0.211744 + 0.977325i \(0.567914\pi\)
\(882\) 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 \(-0.220963\pi\)
0.768582 + 0.639751i \(0.220963\pi\)
\(888\) 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 \(-0.460077\pi\)
0.125093 + 0.992145i \(0.460077\pi\)
\(912\) 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 \(-0.401436\pi\)
0.304723 + 0.952441i \(0.401436\pi\)
\(930\) 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 \(-0.448321\pi\)
0.161641 + 0.986850i \(0.448321\pi\)
\(942\) 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 \(-0.528483\pi\)
−0.0893634 + 0.995999i \(0.528483\pi\)
\(948\) 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 \(-0.785922\pi\)
−0.782239 + 0.622979i \(0.785922\pi\)
\(954\) 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 \(-0.385391\pi\)
0.352326 + 0.935877i \(0.385391\pi\)
\(972\) 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 \(-0.488559\pi\)
0.0359357 + 0.999354i \(0.488559\pi\)
\(978\) 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 \(-0.453223\pi\)
0.146424 + 0.989222i \(0.453223\pi\)
\(984\) 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.1.4 yes 4
3.2 odd 2 inner 9.24.a.d.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.24.a.d.1.1 4 3.2 odd 2 inner
9.24.a.d.1.4 yes 4 1.1 even 1 trivial