Properties

Label 25.34.b.c.24.3
Level $25$
Weight $34$
Character 25.24
Analytic conductor $172.457$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 20265063301 x^{10} + 151877130254993985540 x^{8} + 529361026408998369955851761920 x^{6} + 862054500541443268723145801151954288640 x^{4} + 544606882049987247786598582768552479276511789056 x^{2} + 39700341273686227371481766034965422500156905479886340096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{47}\cdot 3^{10}\cdot 5^{36}\cdot 7^{2}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.3
Root \(-63200.4i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.c.24.10

$q$-expansion

\(f(q)\) \(=\) \(q-126401. i q^{2} +8.63211e7i q^{3} -7.38725e9 q^{4} +1.09111e13 q^{6} +7.78650e13i q^{7} -1.52021e14i q^{8} -1.89228e15 q^{9} +O(q^{10})\) \(q-126401. i q^{2} +8.63211e7i q^{3} -7.38725e9 q^{4} +1.09111e13 q^{6} +7.78650e13i q^{7} -1.52021e14i q^{8} -1.89228e15 q^{9} +1.61939e17 q^{11} -6.37676e17i q^{12} +2.28706e18i q^{13} +9.84220e18 q^{14} -8.26715e19 q^{16} +2.46494e20i q^{17} +2.39186e20i q^{18} +1.53881e21 q^{19} -6.72140e21 q^{21} -2.04692e22i q^{22} +2.16284e22i q^{23} +1.31226e22 q^{24} +2.89086e23 q^{26} +3.16521e23i q^{27} -5.75208e23i q^{28} +2.07535e23 q^{29} +1.75532e24 q^{31} +9.14391e24i q^{32} +1.39788e25i q^{33} +3.11571e25 q^{34} +1.39787e25 q^{36} +8.12289e25i q^{37} -1.94506e26i q^{38} -1.97422e26 q^{39} -1.92486e26 q^{41} +8.49590e26i q^{42} -8.01003e26i q^{43} -1.19628e27 q^{44} +2.73385e27 q^{46} -7.53703e27i q^{47} -7.13630e27i q^{48} +1.66804e27 q^{49} -2.12777e28 q^{51} -1.68951e28i q^{52} -1.75257e28i q^{53} +4.00085e28 q^{54} +1.18371e28 q^{56} +1.32832e29i q^{57} -2.62326e28i q^{58} +2.97225e29 q^{59} -3.70944e29 q^{61} -2.21873e29i q^{62} -1.47342e29i q^{63} +4.45655e29 q^{64} +1.76693e30 q^{66} +8.43824e29i q^{67} -1.82091e30i q^{68} -1.86699e30 q^{69} +5.92473e30 q^{71} +2.87666e29i q^{72} -7.49289e30i q^{73} +1.02674e31 q^{74} -1.13675e31 q^{76} +1.26094e31i q^{77} +2.49543e31i q^{78} -6.07295e30 q^{79} -3.78417e31 q^{81} +2.43304e31i q^{82} -3.47153e31i q^{83} +4.96526e31 q^{84} -1.01248e32 q^{86} +1.79147e31i q^{87} -2.46181e31i q^{88} -1.80671e32 q^{89} -1.78082e32 q^{91} -1.59775e32i q^{92} +1.51521e32i q^{93} -9.52688e32 q^{94} -7.89312e32 q^{96} +1.09143e33i q^{97} -2.10841e32i q^{98} -3.06434e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + O(q^{10}) \) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + 301521793111780384q^{11} + 14172220017979782912q^{14} + \)\(12\!\cdots\!32\)\(q^{16} - \)\(34\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!56\)\(q^{21} - \)\(42\!\cdots\!00\)\(q^{24} - \)\(72\!\cdots\!76\)\(q^{26} - \)\(26\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!04\)\(q^{31} - \)\(17\!\cdots\!48\)\(q^{34} + \)\(33\!\cdots\!92\)\(q^{36} - \)\(11\!\cdots\!12\)\(q^{39} + \)\(89\!\cdots\!64\)\(q^{41} - \)\(14\!\cdots\!28\)\(q^{44} + \)\(69\!\cdots\!44\)\(q^{46} + \)\(14\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!76\)\(q^{51} + \)\(36\!\cdots\!00\)\(q^{54} - \)\(78\!\cdots\!00\)\(q^{56} + \)\(35\!\cdots\!00\)\(q^{59} - \)\(56\!\cdots\!16\)\(q^{61} - \)\(38\!\cdots\!44\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{66} - \)\(63\!\cdots\!72\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} - \)\(40\!\cdots\!68\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{76} + \)\(44\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!48\)\(q^{81} + \)\(47\!\cdots\!52\)\(q^{84} - \)\(25\!\cdots\!16\)\(q^{86} + \)\(67\!\cdots\!00\)\(q^{89} + \)\(38\!\cdots\!64\)\(q^{91} - \)\(29\!\cdots\!28\)\(q^{94} + \)\(63\!\cdots\!44\)\(q^{96} - \)\(33\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 126401.i − 1.36381i −0.731439 0.681907i \(-0.761151\pi\)
0.731439 0.681907i \(-0.238849\pi\)
\(3\) 8.63211e7i 1.15775i 0.815415 + 0.578877i \(0.196509\pi\)
−0.815415 + 0.578877i \(0.803491\pi\)
\(4\) −7.38725e9 −0.859989
\(5\) 0 0
\(6\) 1.09111e13 1.57896
\(7\) 7.78650e13i 0.885573i 0.896627 + 0.442787i \(0.146010\pi\)
−0.896627 + 0.442787i \(0.853990\pi\)
\(8\) − 1.52021e14i − 0.190949i
\(9\) −1.89228e15 −0.340396
\(10\) 0 0
\(11\) 1.61939e17 1.06260 0.531302 0.847182i \(-0.321703\pi\)
0.531302 + 0.847182i \(0.321703\pi\)
\(12\) − 6.37676e17i − 0.995656i
\(13\) 2.28706e18i 0.953262i 0.879104 + 0.476631i \(0.158142\pi\)
−0.879104 + 0.476631i \(0.841858\pi\)
\(14\) 9.84220e18 1.20776
\(15\) 0 0
\(16\) −8.26715e19 −1.12041
\(17\) 2.46494e20i 1.22857i 0.789085 + 0.614284i \(0.210555\pi\)
−0.789085 + 0.614284i \(0.789445\pi\)
\(18\) 2.39186e20i 0.464236i
\(19\) 1.53881e21 1.22391 0.611955 0.790893i \(-0.290383\pi\)
0.611955 + 0.790893i \(0.290383\pi\)
\(20\) 0 0
\(21\) −6.72140e21 −1.02528
\(22\) − 2.04692e22i − 1.44920i
\(23\) 2.16284e22i 0.735387i 0.929947 + 0.367693i \(0.119852\pi\)
−0.929947 + 0.367693i \(0.880148\pi\)
\(24\) 1.31226e22 0.221072
\(25\) 0 0
\(26\) 2.89086e23 1.30007
\(27\) 3.16521e23i 0.763660i
\(28\) − 5.75208e23i − 0.761583i
\(29\) 2.07535e23 0.154001 0.0770007 0.997031i \(-0.475466\pi\)
0.0770007 + 0.997031i \(0.475466\pi\)
\(30\) 0 0
\(31\) 1.75532e24 0.433398 0.216699 0.976238i \(-0.430471\pi\)
0.216699 + 0.976238i \(0.430471\pi\)
\(32\) 9.14391e24i 1.33708i
\(33\) 1.39788e25i 1.23024i
\(34\) 3.11571e25 1.67554
\(35\) 0 0
\(36\) 1.39787e25 0.292736
\(37\) 8.12289e25i 1.08239i 0.840898 + 0.541193i \(0.182027\pi\)
−0.840898 + 0.541193i \(0.817973\pi\)
\(38\) − 1.94506e26i − 1.66919i
\(39\) −1.97422e26 −1.10364
\(40\) 0 0
\(41\) −1.92486e26 −0.471483 −0.235741 0.971816i \(-0.575752\pi\)
−0.235741 + 0.971816i \(0.575752\pi\)
\(42\) 8.49590e26i 1.39829i
\(43\) − 8.01003e26i − 0.894139i −0.894499 0.447069i \(-0.852468\pi\)
0.894499 0.447069i \(-0.147532\pi\)
\(44\) −1.19628e27 −0.913828
\(45\) 0 0
\(46\) 2.73385e27 1.00293
\(47\) − 7.53703e27i − 1.93904i −0.245021 0.969518i \(-0.578795\pi\)
0.245021 0.969518i \(-0.421205\pi\)
\(48\) − 7.13630e27i − 1.29716i
\(49\) 1.66804e27 0.215760
\(50\) 0 0
\(51\) −2.12777e28 −1.42238
\(52\) − 1.68951e28i − 0.819795i
\(53\) − 1.75257e28i − 0.621047i −0.950566 0.310523i \(-0.899496\pi\)
0.950566 0.310523i \(-0.100504\pi\)
\(54\) 4.00085e28 1.04149
\(55\) 0 0
\(56\) 1.18371e28 0.169100
\(57\) 1.32832e29i 1.41699i
\(58\) − 2.62326e28i − 0.210029i
\(59\) 2.97225e29 1.79484 0.897422 0.441172i \(-0.145437\pi\)
0.897422 + 0.441172i \(0.145437\pi\)
\(60\) 0 0
\(61\) −3.70944e29 −1.29231 −0.646154 0.763207i \(-0.723624\pi\)
−0.646154 + 0.763207i \(0.723624\pi\)
\(62\) − 2.21873e29i − 0.591075i
\(63\) − 1.47342e29i − 0.301445i
\(64\) 4.45655e29 0.703119
\(65\) 0 0
\(66\) 1.76693e30 1.67781
\(67\) 8.43824e29i 0.625198i 0.949885 + 0.312599i \(0.101200\pi\)
−0.949885 + 0.312599i \(0.898800\pi\)
\(68\) − 1.82091e30i − 1.05656i
\(69\) −1.86699e30 −0.851397
\(70\) 0 0
\(71\) 5.92473e30 1.68619 0.843094 0.537767i \(-0.180732\pi\)
0.843094 + 0.537767i \(0.180732\pi\)
\(72\) 2.87666e29i 0.0649982i
\(73\) − 7.49289e30i − 1.34841i −0.738544 0.674205i \(-0.764486\pi\)
0.738544 0.674205i \(-0.235514\pi\)
\(74\) 1.02674e31 1.47617
\(75\) 0 0
\(76\) −1.13675e31 −1.05255
\(77\) 1.26094e31i 0.941015i
\(78\) 2.49543e31i 1.50516i
\(79\) −6.07295e30 −0.296861 −0.148430 0.988923i \(-0.547422\pi\)
−0.148430 + 0.988923i \(0.547422\pi\)
\(80\) 0 0
\(81\) −3.78417e31 −1.22453
\(82\) 2.43304e31i 0.643015i
\(83\) − 3.47153e31i − 0.751160i −0.926790 0.375580i \(-0.877444\pi\)
0.926790 0.375580i \(-0.122556\pi\)
\(84\) 4.96526e31 0.881727
\(85\) 0 0
\(86\) −1.01248e32 −1.21944
\(87\) 1.79147e31i 0.178296i
\(88\) − 2.46181e31i − 0.202904i
\(89\) −1.80671e32 −1.23582 −0.617909 0.786250i \(-0.712020\pi\)
−0.617909 + 0.786250i \(0.712020\pi\)
\(90\) 0 0
\(91\) −1.78082e32 −0.844183
\(92\) − 1.59775e32i − 0.632424i
\(93\) 1.51521e32i 0.501769i
\(94\) −9.52688e32 −2.64448
\(95\) 0 0
\(96\) −7.89312e32 −1.54801
\(97\) 1.09143e33i 1.80411i 0.431622 + 0.902055i \(0.357942\pi\)
−0.431622 + 0.902055i \(0.642058\pi\)
\(98\) − 2.10841e32i − 0.294256i
\(99\) −3.06434e32 −0.361706
\(100\) 0 0
\(101\) −7.13542e32 −0.605504 −0.302752 0.953069i \(-0.597905\pi\)
−0.302752 + 0.953069i \(0.597905\pi\)
\(102\) 2.68952e33i 1.93986i
\(103\) 1.80265e33i 1.10687i 0.832891 + 0.553437i \(0.186684\pi\)
−0.832891 + 0.553437i \(0.813316\pi\)
\(104\) 3.47680e32 0.182025
\(105\) 0 0
\(106\) −2.21527e33 −0.846992
\(107\) 4.02336e33i 1.31752i 0.752354 + 0.658759i \(0.228918\pi\)
−0.752354 + 0.658759i \(0.771082\pi\)
\(108\) − 2.33822e33i − 0.656739i
\(109\) −2.00598e33 −0.483937 −0.241968 0.970284i \(-0.577793\pi\)
−0.241968 + 0.970284i \(0.577793\pi\)
\(110\) 0 0
\(111\) −7.01177e33 −1.25314
\(112\) − 6.43722e33i − 0.992204i
\(113\) 6.69356e33i 0.890970i 0.895289 + 0.445485i \(0.146969\pi\)
−0.895289 + 0.445485i \(0.853031\pi\)
\(114\) 1.67900e34 1.93251
\(115\) 0 0
\(116\) −1.53311e33 −0.132440
\(117\) − 4.32776e33i − 0.324486i
\(118\) − 3.75695e34i − 2.44783i
\(119\) −1.91933e34 −1.08799
\(120\) 0 0
\(121\) 2.99905e33 0.129129
\(122\) 4.68877e34i 1.76247i
\(123\) − 1.66156e34i − 0.545861i
\(124\) −1.29670e34 −0.372718
\(125\) 0 0
\(126\) −1.86242e34 −0.411115
\(127\) − 1.27300e34i − 0.246643i −0.992367 0.123321i \(-0.960645\pi\)
0.992367 0.123321i \(-0.0393546\pi\)
\(128\) 2.22144e34i 0.378155i
\(129\) 6.91435e34 1.03519
\(130\) 0 0
\(131\) 1.12002e35 1.30092 0.650461 0.759540i \(-0.274576\pi\)
0.650461 + 0.759540i \(0.274576\pi\)
\(132\) − 1.03264e35i − 1.05799i
\(133\) 1.19819e35i 1.08386i
\(134\) 1.06660e35 0.852653
\(135\) 0 0
\(136\) 3.74722e34 0.234594
\(137\) 7.78279e34i 0.431763i 0.976420 + 0.215882i \(0.0692626\pi\)
−0.976420 + 0.215882i \(0.930737\pi\)
\(138\) 2.35989e35i 1.16115i
\(139\) −3.58047e35 −1.56385 −0.781925 0.623372i \(-0.785762\pi\)
−0.781925 + 0.623372i \(0.785762\pi\)
\(140\) 0 0
\(141\) 6.50605e35 2.24493
\(142\) − 7.48891e35i − 2.29965i
\(143\) 3.70364e35i 1.01294i
\(144\) 1.56438e35 0.381382
\(145\) 0 0
\(146\) −9.47108e35 −1.83898
\(147\) 1.43987e35i 0.249797i
\(148\) − 6.00058e35i − 0.930840i
\(149\) 1.72179e35 0.239005 0.119502 0.992834i \(-0.461870\pi\)
0.119502 + 0.992834i \(0.461870\pi\)
\(150\) 0 0
\(151\) 1.08017e36 1.20330 0.601651 0.798759i \(-0.294510\pi\)
0.601651 + 0.798759i \(0.294510\pi\)
\(152\) − 2.33930e35i − 0.233705i
\(153\) − 4.66436e35i − 0.418199i
\(154\) 1.59384e36 1.28337
\(155\) 0 0
\(156\) 1.45840e36 0.949121
\(157\) 2.63407e36i 1.54271i 0.636407 + 0.771353i \(0.280420\pi\)
−0.636407 + 0.771353i \(0.719580\pi\)
\(158\) 7.67626e35i 0.404863i
\(159\) 1.51284e36 0.719019
\(160\) 0 0
\(161\) −1.68410e36 −0.651239
\(162\) 4.78323e36i 1.67003i
\(163\) − 5.17457e36i − 1.63222i −0.577896 0.816111i \(-0.696126\pi\)
0.577896 0.816111i \(-0.303874\pi\)
\(164\) 1.42194e36 0.405470
\(165\) 0 0
\(166\) −4.38804e36 −1.02444
\(167\) − 4.37605e36i − 0.925252i −0.886553 0.462626i \(-0.846907\pi\)
0.886553 0.462626i \(-0.153093\pi\)
\(168\) 1.02179e36i 0.195776i
\(169\) 5.25488e35 0.0912919
\(170\) 0 0
\(171\) −2.91185e36 −0.416614
\(172\) 5.91721e36i 0.768949i
\(173\) 1.16497e37i 1.37580i 0.725805 + 0.687900i \(0.241467\pi\)
−0.725805 + 0.687900i \(0.758533\pi\)
\(174\) 2.26443e36 0.243162
\(175\) 0 0
\(176\) −1.33877e37 −1.19055
\(177\) 2.56568e37i 2.07799i
\(178\) 2.28370e37i 1.68543i
\(179\) −3.52955e36 −0.237490 −0.118745 0.992925i \(-0.537887\pi\)
−0.118745 + 0.992925i \(0.537887\pi\)
\(180\) 0 0
\(181\) −1.74051e37 −0.974947 −0.487473 0.873138i \(-0.662081\pi\)
−0.487473 + 0.873138i \(0.662081\pi\)
\(182\) 2.25097e37i 1.15131i
\(183\) − 3.20203e37i − 1.49618i
\(184\) 3.28797e36 0.140421
\(185\) 0 0
\(186\) 1.91524e37 0.684320
\(187\) 3.99170e37i 1.30548i
\(188\) 5.56779e37i 1.66755i
\(189\) −2.46459e37 −0.676277
\(190\) 0 0
\(191\) −2.24606e36 −0.0518048 −0.0259024 0.999664i \(-0.508246\pi\)
−0.0259024 + 0.999664i \(0.508246\pi\)
\(192\) 3.84694e37i 0.814040i
\(193\) − 2.96630e37i − 0.576129i −0.957611 0.288064i \(-0.906988\pi\)
0.957611 0.288064i \(-0.0930117\pi\)
\(194\) 1.37958e38 2.46047
\(195\) 0 0
\(196\) −1.23222e37 −0.185551
\(197\) − 5.56817e37i − 0.770939i −0.922721 0.385470i \(-0.874039\pi\)
0.922721 0.385470i \(-0.125961\pi\)
\(198\) 3.87335e37i 0.493300i
\(199\) 6.53016e36 0.0765328 0.0382664 0.999268i \(-0.487816\pi\)
0.0382664 + 0.999268i \(0.487816\pi\)
\(200\) 0 0
\(201\) −7.28398e37 −0.723826
\(202\) 9.01924e37i 0.825795i
\(203\) 1.61597e37i 0.136380i
\(204\) 1.57183e38 1.22323
\(205\) 0 0
\(206\) 2.27857e38 1.50957
\(207\) − 4.09270e37i − 0.250322i
\(208\) − 1.89075e38i − 1.06804i
\(209\) 2.49193e38 1.30053
\(210\) 0 0
\(211\) 4.18954e38 1.86856 0.934278 0.356545i \(-0.116045\pi\)
0.934278 + 0.356545i \(0.116045\pi\)
\(212\) 1.29467e38i 0.534093i
\(213\) 5.11429e38i 1.95219i
\(214\) 5.08556e38 1.79685
\(215\) 0 0
\(216\) 4.81177e37 0.145820
\(217\) 1.36678e38i 0.383806i
\(218\) 2.53557e38i 0.660000i
\(219\) 6.46795e38 1.56113
\(220\) 0 0
\(221\) −5.63747e38 −1.17115
\(222\) 8.86294e38i 1.70905i
\(223\) 1.62044e38i 0.290136i 0.989422 + 0.145068i \(0.0463402\pi\)
−0.989422 + 0.145068i \(0.953660\pi\)
\(224\) −7.11990e38 −1.18408
\(225\) 0 0
\(226\) 8.46072e38 1.21512
\(227\) 9.68892e38i 1.29375i 0.762598 + 0.646873i \(0.223924\pi\)
−0.762598 + 0.646873i \(0.776076\pi\)
\(228\) − 9.81259e38i − 1.21859i
\(229\) −4.72860e38 −0.546320 −0.273160 0.961969i \(-0.588069\pi\)
−0.273160 + 0.961969i \(0.588069\pi\)
\(230\) 0 0
\(231\) −1.08846e39 −1.08946
\(232\) − 3.15496e37i − 0.0294065i
\(233\) 2.32086e38i 0.201501i 0.994912 + 0.100750i \(0.0321243\pi\)
−0.994912 + 0.100750i \(0.967876\pi\)
\(234\) −5.47032e38 −0.442539
\(235\) 0 0
\(236\) −2.19567e39 −1.54355
\(237\) − 5.24224e38i − 0.343692i
\(238\) 2.42605e39i 1.48381i
\(239\) −1.78755e39 −1.02022 −0.510109 0.860110i \(-0.670395\pi\)
−0.510109 + 0.860110i \(0.670395\pi\)
\(240\) 0 0
\(241\) −2.68792e39 −1.33701 −0.668506 0.743707i \(-0.733066\pi\)
−0.668506 + 0.743707i \(0.733066\pi\)
\(242\) − 3.79082e38i − 0.176108i
\(243\) − 1.50698e39i − 0.654041i
\(244\) 2.74026e39 1.11137
\(245\) 0 0
\(246\) −2.10023e39 −0.744453
\(247\) 3.51934e39i 1.16671i
\(248\) − 2.66844e38i − 0.0827571i
\(249\) 2.99666e39 0.869659
\(250\) 0 0
\(251\) 2.18494e39 0.555677 0.277838 0.960628i \(-0.410382\pi\)
0.277838 + 0.960628i \(0.410382\pi\)
\(252\) 1.08845e39i 0.259240i
\(253\) 3.50248e39i 0.781426i
\(254\) −1.60909e39 −0.336375
\(255\) 0 0
\(256\) 6.63606e39 1.21885
\(257\) 8.25568e39i 1.42186i 0.703264 + 0.710929i \(0.251725\pi\)
−0.703264 + 0.710929i \(0.748275\pi\)
\(258\) − 8.73980e39i − 1.41181i
\(259\) −6.32489e39 −0.958532
\(260\) 0 0
\(261\) −3.92715e38 −0.0524214
\(262\) − 1.41572e40i − 1.77422i
\(263\) − 2.69949e39i − 0.317697i −0.987303 0.158848i \(-0.949222\pi\)
0.987303 0.158848i \(-0.0507781\pi\)
\(264\) 2.12506e39 0.234912
\(265\) 0 0
\(266\) 1.51452e40 1.47819
\(267\) − 1.55958e40i − 1.43077i
\(268\) − 6.23353e39i − 0.537663i
\(269\) −1.21756e40 −0.987592 −0.493796 0.869578i \(-0.664391\pi\)
−0.493796 + 0.869578i \(0.664391\pi\)
\(270\) 0 0
\(271\) −9.05453e39 −0.649939 −0.324969 0.945725i \(-0.605354\pi\)
−0.324969 + 0.945725i \(0.605354\pi\)
\(272\) − 2.03781e40i − 1.37650i
\(273\) − 1.53722e40i − 0.977357i
\(274\) 9.83751e39 0.588845
\(275\) 0 0
\(276\) 1.37919e40 0.732192
\(277\) − 6.18766e39i − 0.309464i −0.987956 0.154732i \(-0.950549\pi\)
0.987956 0.154732i \(-0.0494515\pi\)
\(278\) 4.52574e40i 2.13280i
\(279\) −3.32155e39 −0.147527
\(280\) 0 0
\(281\) 4.51611e39 0.178283 0.0891415 0.996019i \(-0.471588\pi\)
0.0891415 + 0.996019i \(0.471588\pi\)
\(282\) − 8.22371e40i − 3.06166i
\(283\) − 3.25719e40i − 1.14384i −0.820308 0.571921i \(-0.806198\pi\)
0.820308 0.571921i \(-0.193802\pi\)
\(284\) −4.37674e40 −1.45010
\(285\) 0 0
\(286\) 4.68143e40 1.38146
\(287\) − 1.49879e40i − 0.417533i
\(288\) − 1.73028e40i − 0.455136i
\(289\) −2.05049e40 −0.509382
\(290\) 0 0
\(291\) −9.42136e40 −2.08872
\(292\) 5.53518e40i 1.15962i
\(293\) 6.99032e40i 1.38414i 0.721829 + 0.692072i \(0.243302\pi\)
−0.721829 + 0.692072i \(0.756698\pi\)
\(294\) 1.82001e40 0.340676
\(295\) 0 0
\(296\) 1.23485e40 0.206681
\(297\) 5.12570e40i 0.811469i
\(298\) − 2.17636e40i − 0.325958i
\(299\) −4.94655e40 −0.701016
\(300\) 0 0
\(301\) 6.23701e40 0.791825
\(302\) − 1.36535e41i − 1.64108i
\(303\) − 6.15938e40i − 0.701025i
\(304\) −1.27215e41 −1.37128
\(305\) 0 0
\(306\) −5.89579e40 −0.570346
\(307\) 1.35007e40i 0.123758i 0.998084 + 0.0618788i \(0.0197092\pi\)
−0.998084 + 0.0618788i \(0.980291\pi\)
\(308\) − 9.31485e40i − 0.809262i
\(309\) −1.55607e41 −1.28149
\(310\) 0 0
\(311\) 5.12682e40 0.379579 0.189789 0.981825i \(-0.439220\pi\)
0.189789 + 0.981825i \(0.439220\pi\)
\(312\) 3.00122e40i 0.210740i
\(313\) − 1.36507e39i − 0.00909232i −0.999990 0.00454616i \(-0.998553\pi\)
0.999990 0.00454616i \(-0.00144709\pi\)
\(314\) 3.32949e41 2.10397
\(315\) 0 0
\(316\) 4.48624e40 0.255297
\(317\) 2.88324e41i 1.55741i 0.627390 + 0.778705i \(0.284123\pi\)
−0.627390 + 0.778705i \(0.715877\pi\)
\(318\) − 1.91224e41i − 0.980609i
\(319\) 3.36080e40 0.163643
\(320\) 0 0
\(321\) −3.47301e41 −1.52536
\(322\) 2.12871e41i 0.888169i
\(323\) 3.79307e41i 1.50366i
\(324\) 2.79546e41 1.05308
\(325\) 0 0
\(326\) −6.54070e41 −2.22605
\(327\) − 1.73158e41i − 0.560280i
\(328\) 2.92619e40i 0.0900292i
\(329\) 5.86871e41 1.71716
\(330\) 0 0
\(331\) 1.22306e41 0.323808 0.161904 0.986807i \(-0.448237\pi\)
0.161904 + 0.986807i \(0.448237\pi\)
\(332\) 2.56450e41i 0.645989i
\(333\) − 1.53708e41i − 0.368439i
\(334\) −5.53136e41 −1.26187
\(335\) 0 0
\(336\) 5.55668e41 1.14873
\(337\) 1.95036e40i 0.0383902i 0.999816 + 0.0191951i \(0.00611037\pi\)
−0.999816 + 0.0191951i \(0.993890\pi\)
\(338\) − 6.64222e40i − 0.124505i
\(339\) −5.77796e41 −1.03152
\(340\) 0 0
\(341\) 2.84254e41 0.460531
\(342\) 3.68061e41i 0.568183i
\(343\) 7.31855e41i 1.07664i
\(344\) −1.21769e41 −0.170735
\(345\) 0 0
\(346\) 1.47254e42 1.87634
\(347\) − 8.38690e41i − 1.01898i −0.860477 0.509490i \(-0.829834\pi\)
0.860477 0.509490i \(-0.170166\pi\)
\(348\) − 1.32340e41i − 0.153333i
\(349\) 2.92359e40 0.0323070 0.0161535 0.999870i \(-0.494858\pi\)
0.0161535 + 0.999870i \(0.494858\pi\)
\(350\) 0 0
\(351\) −7.23902e41 −0.727968
\(352\) 1.48075e42i 1.42079i
\(353\) 1.88083e42i 1.72213i 0.508492 + 0.861067i \(0.330203\pi\)
−0.508492 + 0.861067i \(0.669797\pi\)
\(354\) 3.24304e42 2.83399
\(355\) 0 0
\(356\) 1.33466e42 1.06279
\(357\) − 1.65679e42i − 1.25962i
\(358\) 4.46139e41i 0.323892i
\(359\) −1.60953e41 −0.111594 −0.0557968 0.998442i \(-0.517770\pi\)
−0.0557968 + 0.998442i \(0.517770\pi\)
\(360\) 0 0
\(361\) 7.87154e41 0.497956
\(362\) 2.20002e42i 1.32965i
\(363\) 2.58881e41i 0.149500i
\(364\) 1.31554e42 0.725988
\(365\) 0 0
\(366\) −4.04740e42 −2.04051
\(367\) 2.15460e42i 1.03843i 0.854645 + 0.519213i \(0.173775\pi\)
−0.854645 + 0.519213i \(0.826225\pi\)
\(368\) − 1.78806e42i − 0.823933i
\(369\) 3.64238e41 0.160491
\(370\) 0 0
\(371\) 1.36464e42 0.549982
\(372\) − 1.11932e42i − 0.431516i
\(373\) − 2.59028e42i − 0.955325i −0.878543 0.477663i \(-0.841484\pi\)
0.878543 0.477663i \(-0.158516\pi\)
\(374\) 5.04554e42 1.78044
\(375\) 0 0
\(376\) −1.14578e42 −0.370257
\(377\) 4.74646e41i 0.146804i
\(378\) 3.11526e42i 0.922316i
\(379\) −4.01905e42 −1.13914 −0.569568 0.821944i \(-0.692890\pi\)
−0.569568 + 0.821944i \(0.692890\pi\)
\(380\) 0 0
\(381\) 1.09887e42 0.285551
\(382\) 2.83904e41i 0.0706522i
\(383\) − 5.03299e42i − 1.19963i −0.800140 0.599813i \(-0.795242\pi\)
0.800140 0.599813i \(-0.204758\pi\)
\(384\) −1.91757e42 −0.437811
\(385\) 0 0
\(386\) −3.74943e42 −0.785732
\(387\) 1.51572e42i 0.304361i
\(388\) − 8.06268e42i − 1.55151i
\(389\) −4.14807e42 −0.765029 −0.382515 0.923949i \(-0.624942\pi\)
−0.382515 + 0.923949i \(0.624942\pi\)
\(390\) 0 0
\(391\) −5.33128e42 −0.903473
\(392\) − 2.53576e41i − 0.0411991i
\(393\) 9.66818e42i 1.50615i
\(394\) −7.03822e42 −1.05142
\(395\) 0 0
\(396\) 2.26370e42 0.311063
\(397\) − 8.89168e42i − 1.17203i −0.810299 0.586017i \(-0.800695\pi\)
0.810299 0.586017i \(-0.199305\pi\)
\(398\) − 8.25418e41i − 0.104377i
\(399\) −1.03429e43 −1.25485
\(400\) 0 0
\(401\) 1.71192e41 0.0191250 0.00956251 0.999954i \(-0.496956\pi\)
0.00956251 + 0.999954i \(0.496956\pi\)
\(402\) 9.20702e42i 0.987163i
\(403\) 4.01451e42i 0.413142i
\(404\) 5.27111e42 0.520727
\(405\) 0 0
\(406\) 2.04260e42 0.185996
\(407\) 1.31541e43i 1.15015i
\(408\) 3.23464e42i 0.271603i
\(409\) −1.40883e43 −1.13612 −0.568061 0.822986i \(-0.692306\pi\)
−0.568061 + 0.822986i \(0.692306\pi\)
\(410\) 0 0
\(411\) −6.71819e42 −0.499876
\(412\) − 1.33166e43i − 0.951899i
\(413\) 2.31434e43i 1.58947i
\(414\) −5.17321e42 −0.341393
\(415\) 0 0
\(416\) −2.09127e43 −1.27459
\(417\) − 3.09070e43i − 1.81056i
\(418\) − 3.14982e43i − 1.77369i
\(419\) 2.03304e43 1.10056 0.550280 0.834980i \(-0.314521\pi\)
0.550280 + 0.834980i \(0.314521\pi\)
\(420\) 0 0
\(421\) 2.81610e43 1.40927 0.704633 0.709572i \(-0.251111\pi\)
0.704633 + 0.709572i \(0.251111\pi\)
\(422\) − 5.29562e43i − 2.54836i
\(423\) 1.42622e43i 0.660039i
\(424\) −2.66427e42 −0.118588
\(425\) 0 0
\(426\) 6.46451e43 2.66243
\(427\) − 2.88836e43i − 1.14443i
\(428\) − 2.97216e43i − 1.13305i
\(429\) −3.19702e43 −1.17274
\(430\) 0 0
\(431\) −1.08445e43 −0.368412 −0.184206 0.982888i \(-0.558971\pi\)
−0.184206 + 0.982888i \(0.558971\pi\)
\(432\) − 2.61673e43i − 0.855611i
\(433\) − 3.95138e43i − 1.24365i −0.783156 0.621825i \(-0.786392\pi\)
0.783156 0.621825i \(-0.213608\pi\)
\(434\) 1.72762e43 0.523440
\(435\) 0 0
\(436\) 1.48186e43 0.416180
\(437\) 3.32820e43i 0.900047i
\(438\) − 8.17554e43i − 2.12909i
\(439\) −1.22543e43 −0.307342 −0.153671 0.988122i \(-0.549110\pi\)
−0.153671 + 0.988122i \(0.549110\pi\)
\(440\) 0 0
\(441\) −3.15639e42 −0.0734436
\(442\) 7.12581e43i 1.59723i
\(443\) 5.64830e43i 1.21971i 0.792513 + 0.609855i \(0.208772\pi\)
−0.792513 + 0.609855i \(0.791228\pi\)
\(444\) 5.17977e43 1.07768
\(445\) 0 0
\(446\) 2.04825e43 0.395692
\(447\) 1.48627e43i 0.276709i
\(448\) 3.47009e43i 0.622664i
\(449\) −2.63282e43 −0.455362 −0.227681 0.973736i \(-0.573114\pi\)
−0.227681 + 0.973736i \(0.573114\pi\)
\(450\) 0 0
\(451\) −3.11710e43 −0.501000
\(452\) − 4.94470e43i − 0.766224i
\(453\) 9.32419e43i 1.39313i
\(454\) 1.22469e44 1.76443
\(455\) 0 0
\(456\) 2.01931e43 0.270573
\(457\) − 4.92089e43i − 0.635955i −0.948098 0.317977i \(-0.896996\pi\)
0.948098 0.317977i \(-0.103004\pi\)
\(458\) 5.97700e43i 0.745079i
\(459\) −7.80205e43 −0.938209
\(460\) 0 0
\(461\) −1.35047e44 −1.51154 −0.755771 0.654836i \(-0.772737\pi\)
−0.755771 + 0.654836i \(0.772737\pi\)
\(462\) 1.37582e44i 1.48583i
\(463\) − 9.98444e42i − 0.104049i −0.998646 0.0520244i \(-0.983433\pi\)
0.998646 0.0520244i \(-0.0165674\pi\)
\(464\) −1.71573e43 −0.172544
\(465\) 0 0
\(466\) 2.93359e43 0.274810
\(467\) 1.53367e44i 1.38677i 0.720568 + 0.693384i \(0.243881\pi\)
−0.720568 + 0.693384i \(0.756119\pi\)
\(468\) 3.19702e43i 0.279054i
\(469\) −6.57043e43 −0.553659
\(470\) 0 0
\(471\) −2.27376e44 −1.78608
\(472\) − 4.51843e43i − 0.342724i
\(473\) − 1.29714e44i − 0.950116i
\(474\) −6.62624e43 −0.468732
\(475\) 0 0
\(476\) 1.41785e44 0.935658
\(477\) 3.31636e43i 0.211401i
\(478\) 2.25948e44i 1.39139i
\(479\) −5.80916e43 −0.345603 −0.172802 0.984957i \(-0.555282\pi\)
−0.172802 + 0.984957i \(0.555282\pi\)
\(480\) 0 0
\(481\) −1.85775e44 −1.03180
\(482\) 3.39755e44i 1.82344i
\(483\) − 1.45373e44i − 0.753975i
\(484\) −2.21547e43 −0.111050
\(485\) 0 0
\(486\) −1.90484e44 −0.891990
\(487\) 6.21415e43i 0.281290i 0.990060 + 0.140645i \(0.0449176\pi\)
−0.990060 + 0.140645i \(0.955082\pi\)
\(488\) 5.63912e43i 0.246765i
\(489\) 4.46675e44 1.88971
\(490\) 0 0
\(491\) 1.52856e44 0.604557 0.302278 0.953220i \(-0.402253\pi\)
0.302278 + 0.953220i \(0.402253\pi\)
\(492\) 1.22744e44i 0.469435i
\(493\) 5.11562e43i 0.189201i
\(494\) 4.44848e44 1.59117
\(495\) 0 0
\(496\) −1.45115e44 −0.485583
\(497\) 4.61329e44i 1.49324i
\(498\) − 3.78781e44i − 1.18605i
\(499\) −9.99203e43 −0.302688 −0.151344 0.988481i \(-0.548360\pi\)
−0.151344 + 0.988481i \(0.548360\pi\)
\(500\) 0 0
\(501\) 3.77745e44 1.07122
\(502\) − 2.76178e44i − 0.757840i
\(503\) − 9.03153e43i − 0.239822i −0.992785 0.119911i \(-0.961739\pi\)
0.992785 0.119911i \(-0.0382610\pi\)
\(504\) −2.23991e43 −0.0575607
\(505\) 0 0
\(506\) 4.42717e44 1.06572
\(507\) 4.53607e43i 0.105694i
\(508\) 9.40399e43i 0.212110i
\(509\) 5.47217e44 1.19486 0.597429 0.801922i \(-0.296189\pi\)
0.597429 + 0.801922i \(0.296189\pi\)
\(510\) 0 0
\(511\) 5.83434e44 1.19412
\(512\) − 6.47984e44i − 1.28413i
\(513\) 4.87064e44i 0.934651i
\(514\) 1.04353e45 1.93915
\(515\) 0 0
\(516\) −5.10780e44 −0.890254
\(517\) − 1.22054e45i − 2.06043i
\(518\) 7.99471e44i 1.30726i
\(519\) −1.00562e45 −1.59284
\(520\) 0 0
\(521\) 1.19671e45 1.77897 0.889484 0.456966i \(-0.151064\pi\)
0.889484 + 0.456966i \(0.151064\pi\)
\(522\) 4.96395e43i 0.0714931i
\(523\) 6.44705e43i 0.0899670i 0.998988 + 0.0449835i \(0.0143235\pi\)
−0.998988 + 0.0449835i \(0.985676\pi\)
\(524\) −8.27390e44 −1.11878
\(525\) 0 0
\(526\) −3.41218e44 −0.433279
\(527\) 4.32675e44i 0.532460i
\(528\) − 1.15564e45i − 1.37837i
\(529\) 3.97216e44 0.459206
\(530\) 0 0
\(531\) −5.62432e44 −0.610957
\(532\) − 8.85134e44i − 0.932110i
\(533\) − 4.40227e44i − 0.449446i
\(534\) −1.97132e45 −1.95131
\(535\) 0 0
\(536\) 1.28279e44 0.119381
\(537\) − 3.04675e44i − 0.274955i
\(538\) 1.53901e45i 1.34689i
\(539\) 2.70120e44 0.229267
\(540\) 0 0
\(541\) −3.00892e44 −0.240246 −0.120123 0.992759i \(-0.538329\pi\)
−0.120123 + 0.992759i \(0.538329\pi\)
\(542\) 1.14450e45i 0.886396i
\(543\) − 1.50243e45i − 1.12875i
\(544\) −2.25392e45 −1.64269
\(545\) 0 0
\(546\) −1.94306e45 −1.33293
\(547\) 9.15780e44i 0.609537i 0.952426 + 0.304769i \(0.0985791\pi\)
−0.952426 + 0.304769i \(0.901421\pi\)
\(548\) − 5.74934e44i − 0.371312i
\(549\) 7.01930e44 0.439896
\(550\) 0 0
\(551\) 3.19357e44 0.188484
\(552\) 2.83821e44i 0.162574i
\(553\) − 4.72870e44i − 0.262892i
\(554\) −7.82126e44 −0.422052
\(555\) 0 0
\(556\) 2.64498e45 1.34489
\(557\) − 3.51921e45i − 1.73714i −0.495568 0.868569i \(-0.665040\pi\)
0.495568 0.868569i \(-0.334960\pi\)
\(558\) 4.19847e44i 0.201199i
\(559\) 1.83194e45 0.852348
\(560\) 0 0
\(561\) −3.44568e45 −1.51143
\(562\) − 5.70840e44i − 0.243145i
\(563\) − 1.25910e45i − 0.520802i −0.965501 0.260401i \(-0.916145\pi\)
0.965501 0.260401i \(-0.0838547\pi\)
\(564\) −4.80618e45 −1.93061
\(565\) 0 0
\(566\) −4.11711e45 −1.55999
\(567\) − 2.94655e45i − 1.08441i
\(568\) − 9.00681e44i − 0.321976i
\(569\) 1.60105e45 0.555972 0.277986 0.960585i \(-0.410333\pi\)
0.277986 + 0.960585i \(0.410333\pi\)
\(570\) 0 0
\(571\) 5.75140e44 0.188485 0.0942427 0.995549i \(-0.469957\pi\)
0.0942427 + 0.995549i \(0.469957\pi\)
\(572\) − 2.73597e45i − 0.871118i
\(573\) − 1.93882e44i − 0.0599773i
\(574\) −1.89449e45 −0.569437
\(575\) 0 0
\(576\) −8.43303e44 −0.239339
\(577\) 3.64572e45i 1.00550i 0.864432 + 0.502750i \(0.167679\pi\)
−0.864432 + 0.502750i \(0.832321\pi\)
\(578\) 2.59184e45i 0.694702i
\(579\) 2.56054e45 0.667016
\(580\) 0 0
\(581\) 2.70311e45 0.665207
\(582\) 1.19087e46i 2.84862i
\(583\) − 2.83810e45i − 0.659927i
\(584\) −1.13907e45 −0.257478
\(585\) 0 0
\(586\) 8.83582e45 1.88771
\(587\) − 2.48055e45i − 0.515251i −0.966245 0.257626i \(-0.917060\pi\)
0.966245 0.257626i \(-0.0829401\pi\)
\(588\) − 1.06367e45i − 0.214822i
\(589\) 2.70109e45 0.530441
\(590\) 0 0
\(591\) 4.80651e45 0.892558
\(592\) − 6.71532e45i − 1.21271i
\(593\) − 2.97630e45i − 0.522726i −0.965241 0.261363i \(-0.915828\pi\)
0.965241 0.261363i \(-0.0841720\pi\)
\(594\) 6.47893e45 1.10669
\(595\) 0 0
\(596\) −1.27193e45 −0.205541
\(597\) 5.63691e44i 0.0886062i
\(598\) 6.25249e45i 0.956056i
\(599\) 3.09884e45 0.460953 0.230476 0.973078i \(-0.425972\pi\)
0.230476 + 0.973078i \(0.425972\pi\)
\(600\) 0 0
\(601\) −5.90335e45 −0.831132 −0.415566 0.909563i \(-0.636417\pi\)
−0.415566 + 0.909563i \(0.636417\pi\)
\(602\) − 7.88364e45i − 1.07990i
\(603\) − 1.59675e45i − 0.212815i
\(604\) −7.97952e45 −1.03483
\(605\) 0 0
\(606\) −7.78551e45 −0.956068
\(607\) 6.02204e45i 0.719664i 0.933017 + 0.359832i \(0.117166\pi\)
−0.933017 + 0.359832i \(0.882834\pi\)
\(608\) 1.40707e46i 1.63646i
\(609\) −1.39493e45 −0.157894
\(610\) 0 0
\(611\) 1.72376e46 1.84841
\(612\) 3.44568e45i 0.359647i
\(613\) − 7.92968e45i − 0.805671i −0.915272 0.402836i \(-0.868025\pi\)
0.915272 0.402836i \(-0.131975\pi\)
\(614\) 1.70650e45 0.168782
\(615\) 0 0
\(616\) 1.91688e45 0.179686
\(617\) − 7.56919e45i − 0.690787i −0.938458 0.345394i \(-0.887745\pi\)
0.938458 0.345394i \(-0.112255\pi\)
\(618\) 1.96689e46i 1.74771i
\(619\) 4.72940e45 0.409176 0.204588 0.978848i \(-0.434415\pi\)
0.204588 + 0.978848i \(0.434415\pi\)
\(620\) 0 0
\(621\) −6.84585e45 −0.561585
\(622\) − 6.48035e45i − 0.517675i
\(623\) − 1.40680e46i − 1.09441i
\(624\) 1.63211e46 1.23653
\(625\) 0 0
\(626\) −1.72547e44 −0.0124002
\(627\) 2.15106e46i 1.50570i
\(628\) − 1.94585e46i − 1.32671i
\(629\) −2.00225e46 −1.32979
\(630\) 0 0
\(631\) −9.12534e45 −0.575128 −0.287564 0.957761i \(-0.592845\pi\)
−0.287564 + 0.957761i \(0.592845\pi\)
\(632\) 9.23213e44i 0.0566853i
\(633\) 3.61646e46i 2.16333i
\(634\) 3.64444e46 2.12402
\(635\) 0 0
\(636\) −1.11757e46 −0.618349
\(637\) 3.81490e45i 0.205675i
\(638\) − 4.24808e45i − 0.223178i
\(639\) −1.12112e46 −0.573971
\(640\) 0 0
\(641\) −1.89987e46 −0.923774 −0.461887 0.886939i \(-0.652827\pi\)
−0.461887 + 0.886939i \(0.652827\pi\)
\(642\) 4.38992e46i 2.08031i
\(643\) 1.26235e46i 0.583040i 0.956565 + 0.291520i \(0.0941611\pi\)
−0.956565 + 0.291520i \(0.905839\pi\)
\(644\) 1.24408e46 0.560058
\(645\) 0 0
\(646\) 4.79447e46 2.05071
\(647\) 1.12690e46i 0.469855i 0.972013 + 0.234927i \(0.0754852\pi\)
−0.972013 + 0.234927i \(0.924515\pi\)
\(648\) 5.75272e45i 0.233822i
\(649\) 4.81323e46 1.90721
\(650\) 0 0
\(651\) −1.17982e46 −0.444353
\(652\) 3.82258e46i 1.40369i
\(653\) 3.57189e45i 0.127888i 0.997953 + 0.0639440i \(0.0203679\pi\)
−0.997953 + 0.0639440i \(0.979632\pi\)
\(654\) −2.18873e46 −0.764118
\(655\) 0 0
\(656\) 1.59131e46 0.528253
\(657\) 1.41786e46i 0.458993i
\(658\) − 7.41810e46i − 2.34189i
\(659\) −4.26159e46 −1.31209 −0.656043 0.754723i \(-0.727771\pi\)
−0.656043 + 0.754723i \(0.727771\pi\)
\(660\) 0 0
\(661\) −2.93053e46 −0.858268 −0.429134 0.903241i \(-0.641181\pi\)
−0.429134 + 0.903241i \(0.641181\pi\)
\(662\) − 1.54596e46i − 0.441613i
\(663\) − 4.86633e46i − 1.35590i
\(664\) −5.27744e45 −0.143433
\(665\) 0 0
\(666\) −1.94288e46 −0.502483
\(667\) 4.48866e45i 0.113251i
\(668\) 3.23269e46i 0.795707i
\(669\) −1.39878e46 −0.335907
\(670\) 0 0
\(671\) −6.00703e46 −1.37321
\(672\) − 6.14598e46i − 1.37088i
\(673\) 7.94628e44i 0.0172948i 0.999963 + 0.00864740i \(0.00275259\pi\)
−0.999963 + 0.00864740i \(0.997247\pi\)
\(674\) 2.46527e45 0.0523571
\(675\) 0 0
\(676\) −3.88191e45 −0.0785100
\(677\) − 5.41689e46i − 1.06915i −0.845122 0.534573i \(-0.820473\pi\)
0.845122 0.534573i \(-0.179527\pi\)
\(678\) 7.30339e46i 1.40681i
\(679\) −8.49843e46 −1.59767
\(680\) 0 0
\(681\) −8.36359e46 −1.49784
\(682\) − 3.59299e46i − 0.628079i
\(683\) − 4.80553e46i − 0.819973i −0.912092 0.409986i \(-0.865533\pi\)
0.912092 0.409986i \(-0.134467\pi\)
\(684\) 2.15106e46 0.358283
\(685\) 0 0
\(686\) 9.25072e46 1.46834
\(687\) − 4.08178e46i − 0.632505i
\(688\) 6.62202e46i 1.00180i
\(689\) 4.00824e46 0.592020
\(690\) 0 0
\(691\) 9.58471e46 1.34956 0.674779 0.738020i \(-0.264239\pi\)
0.674779 + 0.738020i \(0.264239\pi\)
\(692\) − 8.60595e46i − 1.18317i
\(693\) − 2.38605e46i − 0.320317i
\(694\) −1.06011e47 −1.38970
\(695\) 0 0
\(696\) 2.72340e45 0.0340455
\(697\) − 4.74467e46i − 0.579249i
\(698\) − 3.69544e45i − 0.0440607i
\(699\) −2.00340e46 −0.233288
\(700\) 0 0
\(701\) 3.14053e46 0.348863 0.174432 0.984669i \(-0.444191\pi\)
0.174432 + 0.984669i \(0.444191\pi\)
\(702\) 9.15018e46i 0.992813i
\(703\) 1.24996e47i 1.32474i
\(704\) 7.21688e46 0.747138
\(705\) 0 0
\(706\) 2.37738e47 2.34867
\(707\) − 5.55600e46i − 0.536218i
\(708\) − 1.89533e47i − 1.78705i
\(709\) 1.07691e47 0.992009 0.496005 0.868320i \(-0.334800\pi\)
0.496005 + 0.868320i \(0.334800\pi\)
\(710\) 0 0
\(711\) 1.14917e46 0.101050
\(712\) 2.74658e46i 0.235978i
\(713\) 3.79647e46i 0.318715i
\(714\) −2.09419e47 −1.71789
\(715\) 0 0
\(716\) 2.60737e46 0.204239
\(717\) − 1.54303e47i − 1.18116i
\(718\) 2.03445e46i 0.152193i
\(719\) −2.02046e47 −1.47715 −0.738573 0.674174i \(-0.764500\pi\)
−0.738573 + 0.674174i \(0.764500\pi\)
\(720\) 0 0
\(721\) −1.40364e47 −0.980218
\(722\) − 9.94970e46i − 0.679120i
\(723\) − 2.32024e47i − 1.54793i
\(724\) 1.28576e47 0.838444
\(725\) 0 0
\(726\) 3.27228e46 0.203890
\(727\) − 2.34830e47i − 1.43033i −0.698957 0.715164i \(-0.746352\pi\)
0.698957 0.715164i \(-0.253648\pi\)
\(728\) 2.70721e46i 0.161196i
\(729\) −8.02799e46 −0.467308
\(730\) 0 0
\(731\) 1.97443e47 1.09851
\(732\) 2.36542e47i 1.28669i
\(733\) − 7.07756e46i − 0.376416i −0.982129 0.188208i \(-0.939732\pi\)
0.982129 0.188208i \(-0.0602679\pi\)
\(734\) 2.72343e47 1.41622
\(735\) 0 0
\(736\) −1.97768e47 −0.983270
\(737\) 1.36648e47i 0.664338i
\(738\) − 4.60399e46i − 0.218879i
\(739\) 3.25905e47 1.51516 0.757578 0.652744i \(-0.226382\pi\)
0.757578 + 0.652744i \(0.226382\pi\)
\(740\) 0 0
\(741\) −3.03794e47 −1.35076
\(742\) − 1.72492e47i − 0.750074i
\(743\) 1.42154e47i 0.604568i 0.953218 + 0.302284i \(0.0977490\pi\)
−0.953218 + 0.302284i \(0.902251\pi\)
\(744\) 2.30343e46 0.0958124
\(745\) 0 0
\(746\) −3.27414e47 −1.30289
\(747\) 6.56910e46i 0.255691i
\(748\) − 2.94877e47i − 1.12270i
\(749\) −3.13279e47 −1.16676
\(750\) 0 0
\(751\) −8.07948e45 −0.0287955 −0.0143978 0.999896i \(-0.504583\pi\)
−0.0143978 + 0.999896i \(0.504583\pi\)
\(752\) 6.23098e47i 2.17251i
\(753\) 1.88606e47i 0.643338i
\(754\) 5.99956e46 0.200213
\(755\) 0 0
\(756\) 1.82065e47 0.581591
\(757\) 2.98356e47i 0.932509i 0.884651 + 0.466255i \(0.154397\pi\)
−0.884651 + 0.466255i \(0.845603\pi\)
\(758\) 5.08011e47i 1.55357i
\(759\) −3.02338e47 −0.904699
\(760\) 0 0
\(761\) 8.91821e46 0.255524 0.127762 0.991805i \(-0.459221\pi\)
0.127762 + 0.991805i \(0.459221\pi\)
\(762\) − 1.38898e47i − 0.389439i
\(763\) − 1.56195e47i − 0.428562i
\(764\) 1.65922e46 0.0445516
\(765\) 0 0
\(766\) −6.36174e47 −1.63607
\(767\) 6.79771e47i 1.71096i
\(768\) 5.72833e47i 1.41113i
\(769\) 3.11972e47 0.752195 0.376098 0.926580i \(-0.377266\pi\)
0.376098 + 0.926580i \(0.377266\pi\)
\(770\) 0 0
\(771\) −7.12640e47 −1.64616
\(772\) 2.19128e47i 0.495464i
\(773\) − 5.84176e47i − 1.29295i −0.762935 0.646475i \(-0.776242\pi\)
0.762935 0.646475i \(-0.223758\pi\)
\(774\) 1.91589e47 0.415092
\(775\) 0 0
\(776\) 1.65920e47 0.344493
\(777\) − 5.45972e47i − 1.10974i
\(778\) 5.24320e47i 1.04336i
\(779\) −2.96199e47 −0.577053
\(780\) 0 0
\(781\) 9.59444e47 1.79175
\(782\) 6.73879e47i 1.23217i
\(783\) 6.56892e46i 0.117605i
\(784\) −1.37899e47 −0.241739
\(785\) 0 0
\(786\) 1.22207e48 2.05411
\(787\) 3.46036e47i 0.569559i 0.958593 + 0.284779i \(0.0919203\pi\)
−0.958593 + 0.284779i \(0.908080\pi\)
\(788\) 4.11335e47i 0.662999i
\(789\) 2.33023e47 0.367815
\(790\) 0 0
\(791\) −5.21194e47 −0.789019
\(792\) 4.65842e46i 0.0690675i
\(793\) − 8.48371e47i − 1.23191i
\(794\) −1.12392e48 −1.59844
\(795\) 0 0
\(796\) −4.82399e46 −0.0658174
\(797\) − 8.43355e47i − 1.12706i −0.826095 0.563532i \(-0.809442\pi\)
0.826095 0.563532i \(-0.190558\pi\)
\(798\) 1.30735e48i 1.71138i
\(799\) 1.85784e48 2.38224
\(800\) 0 0
\(801\) 3.41881e47 0.420667
\(802\) − 2.16388e46i − 0.0260830i
\(803\) − 1.21339e48i − 1.43283i
\(804\) 5.38086e47 0.622482
\(805\) 0 0
\(806\) 5.07438e47 0.563449
\(807\) − 1.05101e48i − 1.14339i
\(808\) 1.08473e47i 0.115620i
\(809\) 1.87301e48 1.95609 0.978045 0.208392i \(-0.0668229\pi\)
0.978045 + 0.208392i \(0.0668229\pi\)
\(810\) 0 0
\(811\) −1.53865e48 −1.54275 −0.771376 0.636380i \(-0.780431\pi\)
−0.771376 + 0.636380i \(0.780431\pi\)
\(812\) − 1.19376e47i − 0.117285i
\(813\) − 7.81597e47i − 0.752470i
\(814\) 1.66269e48 1.56859
\(815\) 0 0
\(816\) 1.75906e48 1.59365
\(817\) − 1.23259e48i − 1.09435i
\(818\) 1.78078e48i 1.54946i
\(819\) 3.36981e47 0.287356
\(820\) 0 0
\(821\) −1.66705e48 −1.36548 −0.682742 0.730660i \(-0.739213\pi\)
−0.682742 + 0.730660i \(0.739213\pi\)
\(822\) 8.49185e47i 0.681738i
\(823\) − 2.01297e48i − 1.58394i −0.610557 0.791972i \(-0.709055\pi\)
0.610557 0.791972i \(-0.290945\pi\)
\(824\) 2.74041e47 0.211357
\(825\) 0 0
\(826\) 2.92535e48 2.16774
\(827\) 1.15395e48i 0.838195i 0.907941 + 0.419098i \(0.137653\pi\)
−0.907941 + 0.419098i \(0.862347\pi\)
\(828\) 3.02338e47i 0.215274i
\(829\) −1.78893e48 −1.24866 −0.624331 0.781160i \(-0.714628\pi\)
−0.624331 + 0.781160i \(0.714628\pi\)
\(830\) 0 0
\(831\) 5.34126e47 0.358284
\(832\) 1.01924e48i 0.670257i
\(833\) 4.11161e47i 0.265076i
\(834\) −3.90667e48 −2.46926
\(835\) 0 0
\(836\) −1.84085e48 −1.11844
\(837\) 5.55594e47i 0.330969i
\(838\) − 2.56978e48i − 1.50096i
\(839\) −2.30591e48 −1.32059 −0.660297 0.751004i \(-0.729570\pi\)
−0.660297 + 0.751004i \(0.729570\pi\)
\(840\) 0 0
\(841\) −1.77300e48 −0.976284
\(842\) − 3.55957e48i − 1.92198i
\(843\) 3.89835e47i 0.206408i
\(844\) −3.09492e48 −1.60694
\(845\) 0 0
\(846\) 1.80275e48 0.900171
\(847\) 2.33521e47i 0.114354i
\(848\) 1.44888e48i 0.695826i
\(849\) 2.81164e48 1.32429
\(850\) 0 0
\(851\) −1.75685e48 −0.795972
\(852\) − 3.77806e48i − 1.67886i
\(853\) − 3.23565e48i − 1.41027i −0.709073 0.705135i \(-0.750886\pi\)
0.709073 0.705135i \(-0.249114\pi\)
\(854\) −3.65091e48 −1.56080
\(855\) 0 0
\(856\) 6.11634e47 0.251579
\(857\) 1.13232e48i 0.456864i 0.973560 + 0.228432i \(0.0733598\pi\)
−0.973560 + 0.228432i \(0.926640\pi\)
\(858\) 4.04107e48i 1.59939i
\(859\) 1.69678e48 0.658777 0.329388 0.944195i \(-0.393157\pi\)
0.329388 + 0.944195i \(0.393157\pi\)
\(860\) 0 0
\(861\) 1.29378e48 0.483400
\(862\) 1.37075e48i 0.502445i
\(863\) 4.68742e48i 1.68561i 0.538223 + 0.842803i \(0.319096\pi\)
−0.538223 + 0.842803i \(0.680904\pi\)
\(864\) −2.89424e48 −1.02107
\(865\) 0 0
\(866\) −4.99457e48 −1.69611
\(867\) − 1.77001e48i − 0.589739i
\(868\) − 1.00967e48i − 0.330069i
\(869\) −9.83446e47 −0.315446
\(870\) 0 0
\(871\) −1.92987e48 −0.595977
\(872\) 3.04950e47i 0.0924073i
\(873\) − 2.06529e48i − 0.614111i
\(874\) 4.20687e48 1.22750
\(875\) 0 0
\(876\) −4.77803e48 −1.34255
\(877\) 6.18989e48i 1.70683i 0.521234 + 0.853414i \(0.325472\pi\)
−0.521234 + 0.853414i \(0.674528\pi\)
\(878\) 1.54895e48i 0.419158i
\(879\) −6.03412e48 −1.60250
\(880\) 0 0
\(881\) 2.27101e48 0.580919 0.290460 0.956887i \(-0.406192\pi\)
0.290460 + 0.956887i \(0.406192\pi\)
\(882\) 3.98971e47i 0.100163i
\(883\) − 6.78750e48i − 1.67247i −0.548372 0.836234i \(-0.684752\pi\)
0.548372 0.836234i \(-0.315248\pi\)
\(884\) 4.16454e48 1.00717
\(885\) 0 0
\(886\) 7.13950e48 1.66346
\(887\) − 6.24043e48i − 1.42717i −0.700570 0.713583i \(-0.747071\pi\)
0.700570 0.713583i \(-0.252929\pi\)
\(888\) 1.06593e48i 0.239285i
\(889\) 9.91224e47 0.218420
\(890\) 0 0
\(891\) −6.12805e48 −1.30119
\(892\) − 1.19706e48i − 0.249514i
\(893\) − 1.15980e49i − 2.37321i
\(894\) 1.87866e48 0.377379
\(895\) 0 0
\(896\) −1.72972e48 −0.334884
\(897\) − 4.26992e48i − 0.811605i
\(898\) 3.32791e48i 0.621030i
\(899\) 3.64290e47 0.0667440
\(900\) 0 0
\(901\) 4.31999e48 0.762998
\(902\) 3.94004e48i 0.683271i
\(903\) 5.38386e48i 0.916739i
\(904\) 1.01756e48 0.170130
\(905\) 0 0
\(906\) 1.17859e49 1.89997
\(907\) − 6.86902e48i − 1.08736i −0.839292 0.543681i \(-0.817030\pi\)
0.839292 0.543681i \(-0.182970\pi\)
\(908\) − 7.15745e48i − 1.11261i
\(909\) 1.35022e48 0.206111
\(910\) 0 0
\(911\) 7.47750e48 1.10079 0.550393 0.834906i \(-0.314478\pi\)
0.550393 + 0.834906i \(0.314478\pi\)
\(912\) − 1.09814e49i − 1.58760i
\(913\) − 5.62176e48i − 0.798186i
\(914\) −6.22004e48 −0.867324
\(915\) 0 0
\(916\) 3.49314e48 0.469829
\(917\) 8.72107e48i 1.15206i
\(918\) 9.86186e48i 1.27954i
\(919\) 1.28656e49 1.63955 0.819777 0.572683i \(-0.194097\pi\)
0.819777 + 0.572683i \(0.194097\pi\)
\(920\) 0 0
\(921\) −1.16539e48 −0.143281
\(922\) 1.70701e49i 2.06146i
\(923\) 1.35502e49i 1.60738i
\(924\) 8.04069e48 0.936927
\(925\) 0 0
\(926\) −1.26204e48 −0.141903
\(927\) − 3.41112e48i − 0.376775i
\(928\) 1.89768e48i 0.205912i
\(929\) −8.12716e48 −0.866322 −0.433161 0.901316i \(-0.642602\pi\)
−0.433161 + 0.901316i \(0.642602\pi\)
\(930\) 0 0
\(931\) 2.56678e48 0.264070
\(932\) − 1.71448e48i − 0.173288i
\(933\) 4.42553e48i 0.439459i
\(934\) 1.93857e49 1.89129
\(935\) 0 0
\(936\) −6.57908e47 −0.0619603
\(937\) 2.32824e48i 0.215439i 0.994181 + 0.107720i \(0.0343549\pi\)
−0.994181 + 0.107720i \(0.965645\pi\)
\(938\) 8.30508e48i 0.755087i
\(939\) 1.17835e47 0.0105267
\(940\) 0 0
\(941\) 3.65122e48 0.314926 0.157463 0.987525i \(-0.449668\pi\)
0.157463 + 0.987525i \(0.449668\pi\)
\(942\) 2.87405e49i 2.43588i
\(943\) − 4.16317e48i − 0.346722i
\(944\) −2.45720e49 −2.01096
\(945\) 0 0
\(946\) −1.63959e49 −1.29578
\(947\) − 1.76048e49i − 1.36728i −0.729821 0.683639i \(-0.760396\pi\)
0.729821 0.683639i \(-0.239604\pi\)
\(948\) 3.87257e48i 0.295571i
\(949\) 1.71367e49 1.28539
\(950\) 0 0
\(951\) −2.48885e49 −1.80310
\(952\) 2.91777e48i 0.207750i
\(953\) 1.48539e49i 1.03946i 0.854330 + 0.519731i \(0.173968\pi\)
−0.854330 + 0.519731i \(0.826032\pi\)
\(954\) 4.19190e48 0.288312
\(955\) 0 0
\(956\) 1.32051e49 0.877377
\(957\) 2.90108e48i 0.189458i
\(958\) 7.34283e48i 0.471338i
\(959\) −6.06007e48 −0.382358
\(960\) 0 0
\(961\) −1.33223e49 −0.812166
\(962\) 2.34822e49i 1.40718i
\(963\) − 7.61332e48i − 0.448477i
\(964\) 1.98563e49 1.14981
\(965\) 0 0
\(966\) −1.83753e49 −1.02828
\(967\) − 7.36558e47i − 0.0405200i −0.999795 0.0202600i \(-0.993551\pi\)
0.999795 0.0202600i \(-0.00644941\pi\)
\(968\) − 4.55917e47i − 0.0246571i
\(969\) −3.27422e49 −1.74087
\(970\) 0 0
\(971\) −1.98790e49 −1.02159 −0.510797 0.859702i \(-0.670649\pi\)
−0.510797 + 0.859702i \(0.670649\pi\)
\(972\) 1.11325e49i 0.562468i
\(973\) − 2.78793e49i − 1.38490i
\(974\) 7.85474e48 0.383627
\(975\) 0 0
\(976\) 3.06665e49 1.44791
\(977\) − 1.42950e49i − 0.663626i −0.943345 0.331813i \(-0.892340\pi\)
0.943345 0.331813i \(-0.107660\pi\)
\(978\) − 5.64601e49i − 2.57721i
\(979\) −2.92577e49 −1.31319
\(980\) 0 0
\(981\) 3.79587e48 0.164730
\(982\) − 1.93211e49i − 0.824503i
\(983\) 1.85942e49i 0.780271i 0.920757 + 0.390136i \(0.127572\pi\)
−0.920757 + 0.390136i \(0.872428\pi\)
\(984\) −2.52592e48 −0.104232
\(985\) 0 0
\(986\) 6.46619e48 0.258036
\(987\) 5.06594e49i 1.98805i
\(988\) − 2.59983e49i − 1.00335i
\(989\) 1.73245e49 0.657538
\(990\) 0 0
\(991\) 3.51455e49 1.29019 0.645095 0.764102i \(-0.276818\pi\)
0.645095 + 0.764102i \(0.276818\pi\)
\(992\) 1.60504e49i 0.579488i
\(993\) 1.05576e49i 0.374890i
\(994\) 5.83124e49 2.03651
\(995\) 0 0
\(996\) −2.21371e49 −0.747897
\(997\) 2.49733e49i 0.829862i 0.909853 + 0.414931i \(0.136194\pi\)
−0.909853 + 0.414931i \(0.863806\pi\)
\(998\) 1.26300e49i 0.412810i
\(999\) −2.57106e49 −0.826575
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 25.34.b.c.24.3 12
5.2 odd 4 25.34.a.c.1.6 6
5.3 odd 4 5.34.a.b.1.1 6
5.4 even 2 inner 25.34.b.c.24.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.1 6 5.3 odd 4
25.34.a.c.1.6 6 5.2 odd 4
25.34.b.c.24.3 12 1.1 even 1 trivial
25.34.b.c.24.10 12 5.4 even 2 inner