Properties

Label 91.10.a.a.1.3
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-33.9466\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-35.9466 q^{2} +230.724 q^{3} +780.160 q^{4} +434.116 q^{5} -8293.74 q^{6} +2401.00 q^{7} -9639.45 q^{8} +33550.4 q^{9} +O(q^{10})\) \(q-35.9466 q^{2} +230.724 q^{3} +780.160 q^{4} +434.116 q^{5} -8293.74 q^{6} +2401.00 q^{7} -9639.45 q^{8} +33550.4 q^{9} -15605.0 q^{10} -37127.2 q^{11} +180001. q^{12} +28561.0 q^{13} -86307.9 q^{14} +100161. q^{15} -52936.3 q^{16} -546796. q^{17} -1.20603e6 q^{18} -902942. q^{19} +338680. q^{20} +553968. q^{21} +1.33460e6 q^{22} -1.65189e6 q^{23} -2.22405e6 q^{24} -1.76467e6 q^{25} -1.02667e6 q^{26} +3.19955e6 q^{27} +1.87316e6 q^{28} -2.15158e6 q^{29} -3.60045e6 q^{30} +2.15325e6 q^{31} +6.83828e6 q^{32} -8.56612e6 q^{33} +1.96555e7 q^{34} +1.04231e6 q^{35} +2.61747e7 q^{36} +1.55393e7 q^{37} +3.24577e7 q^{38} +6.58970e6 q^{39} -4.18464e6 q^{40} -2.64886e6 q^{41} -1.99133e7 q^{42} -3.32538e7 q^{43} -2.89651e7 q^{44} +1.45648e7 q^{45} +5.93799e7 q^{46} -4.58283e6 q^{47} -1.22137e7 q^{48} +5.76480e6 q^{49} +6.34339e7 q^{50} -1.26159e8 q^{51} +2.22821e7 q^{52} +1.52908e7 q^{53} -1.15013e8 q^{54} -1.61175e7 q^{55} -2.31443e7 q^{56} -2.08330e8 q^{57} +7.73421e7 q^{58} -1.05493e8 q^{59} +7.81416e7 q^{60} +515476. q^{61} -7.74022e7 q^{62} +8.05546e7 q^{63} -2.18710e8 q^{64} +1.23988e7 q^{65} +3.07923e8 q^{66} +2.84354e8 q^{67} -4.26589e8 q^{68} -3.81130e8 q^{69} -3.74677e7 q^{70} +3.30668e8 q^{71} -3.23408e8 q^{72} -1.87286e8 q^{73} -5.58584e8 q^{74} -4.07151e8 q^{75} -7.04439e8 q^{76} -8.91423e7 q^{77} -2.36877e8 q^{78} +4.60455e8 q^{79} -2.29805e7 q^{80} +7.78381e7 q^{81} +9.52175e7 q^{82} -1.63544e8 q^{83} +4.32183e8 q^{84} -2.37373e8 q^{85} +1.19536e9 q^{86} -4.96421e8 q^{87} +3.57885e8 q^{88} -8.24421e8 q^{89} -5.23555e8 q^{90} +6.85750e7 q^{91} -1.28874e9 q^{92} +4.96806e8 q^{93} +1.64737e8 q^{94} -3.91982e8 q^{95} +1.57775e9 q^{96} -6.27004e8 q^{97} -2.07225e8 q^{98} -1.24563e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −35.9466 −1.58863 −0.794316 0.607505i \(-0.792170\pi\)
−0.794316 + 0.607505i \(0.792170\pi\)
\(3\) 230.724 1.64455 0.822274 0.569092i \(-0.192705\pi\)
0.822274 + 0.569092i \(0.192705\pi\)
\(4\) 780.160 1.52375
\(5\) 434.116 0.310628 0.155314 0.987865i \(-0.450361\pi\)
0.155314 + 0.987865i \(0.450361\pi\)
\(6\) −8293.74 −2.61258
\(7\) 2401.00 0.377964
\(8\) −9639.45 −0.832046
\(9\) 33550.4 1.70454
\(10\) −15605.0 −0.493474
\(11\) −37127.2 −0.764583 −0.382292 0.924042i \(-0.624865\pi\)
−0.382292 + 0.924042i \(0.624865\pi\)
\(12\) 180001. 2.50588
\(13\) 28561.0 0.277350
\(14\) −86307.9 −0.600446
\(15\) 100161. 0.510843
\(16\) −52936.3 −0.201936
\(17\) −546796. −1.58784 −0.793918 0.608025i \(-0.791962\pi\)
−0.793918 + 0.608025i \(0.791962\pi\)
\(18\) −1.20603e6 −2.70788
\(19\) −902942. −1.58953 −0.794765 0.606917i \(-0.792406\pi\)
−0.794765 + 0.606917i \(0.792406\pi\)
\(20\) 338680. 0.473320
\(21\) 553968. 0.621581
\(22\) 1.33460e6 1.21464
\(23\) −1.65189e6 −1.23085 −0.615426 0.788194i \(-0.711016\pi\)
−0.615426 + 0.788194i \(0.711016\pi\)
\(24\) −2.22405e6 −1.36834
\(25\) −1.76467e6 −0.903510
\(26\) −1.02667e6 −0.440607
\(27\) 3.19955e6 1.15865
\(28\) 1.87316e6 0.575923
\(29\) −2.15158e6 −0.564894 −0.282447 0.959283i \(-0.591146\pi\)
−0.282447 + 0.959283i \(0.591146\pi\)
\(30\) −3.60045e6 −0.811542
\(31\) 2.15325e6 0.418762 0.209381 0.977834i \(-0.432855\pi\)
0.209381 + 0.977834i \(0.432855\pi\)
\(32\) 6.83828e6 1.15285
\(33\) −8.56612e6 −1.25739
\(34\) 1.96555e7 2.52249
\(35\) 1.04231e6 0.117407
\(36\) 2.61747e7 2.59729
\(37\) 1.55393e7 1.36308 0.681542 0.731779i \(-0.261310\pi\)
0.681542 + 0.731779i \(0.261310\pi\)
\(38\) 3.24577e7 2.52518
\(39\) 6.58970e6 0.456116
\(40\) −4.18464e6 −0.258457
\(41\) −2.64886e6 −0.146397 −0.0731983 0.997317i \(-0.523321\pi\)
−0.0731983 + 0.997317i \(0.523321\pi\)
\(42\) −1.99133e7 −0.987463
\(43\) −3.32538e7 −1.48331 −0.741656 0.670780i \(-0.765959\pi\)
−0.741656 + 0.670780i \(0.765959\pi\)
\(44\) −2.89651e7 −1.16503
\(45\) 1.45648e7 0.529478
\(46\) 5.93799e7 1.95537
\(47\) −4.58283e6 −0.136991 −0.0684956 0.997651i \(-0.521820\pi\)
−0.0684956 + 0.997651i \(0.521820\pi\)
\(48\) −1.22137e7 −0.332094
\(49\) 5.76480e6 0.142857
\(50\) 6.34339e7 1.43534
\(51\) −1.26159e8 −2.61127
\(52\) 2.22821e7 0.422612
\(53\) 1.52908e7 0.266188 0.133094 0.991103i \(-0.457509\pi\)
0.133094 + 0.991103i \(0.457509\pi\)
\(54\) −1.15013e8 −1.84066
\(55\) −1.61175e7 −0.237501
\(56\) −2.31443e7 −0.314484
\(57\) −2.08330e8 −2.61406
\(58\) 7.73421e7 0.897408
\(59\) −1.05493e8 −1.13342 −0.566710 0.823917i \(-0.691784\pi\)
−0.566710 + 0.823917i \(0.691784\pi\)
\(60\) 7.81416e7 0.778398
\(61\) 515476. 0.00476677 0.00238338 0.999997i \(-0.499241\pi\)
0.00238338 + 0.999997i \(0.499241\pi\)
\(62\) −7.74022e7 −0.665258
\(63\) 8.05546e7 0.644255
\(64\) −2.18710e8 −1.62951
\(65\) 1.23988e7 0.0861528
\(66\) 3.07923e8 1.99754
\(67\) 2.84354e8 1.72394 0.861972 0.506956i \(-0.169229\pi\)
0.861972 + 0.506956i \(0.169229\pi\)
\(68\) −4.26589e8 −2.41946
\(69\) −3.81130e8 −2.02420
\(70\) −3.74677e7 −0.186516
\(71\) 3.30668e8 1.54429 0.772147 0.635444i \(-0.219183\pi\)
0.772147 + 0.635444i \(0.219183\pi\)
\(72\) −3.23408e8 −1.41825
\(73\) −1.87286e8 −0.771885 −0.385943 0.922523i \(-0.626124\pi\)
−0.385943 + 0.922523i \(0.626124\pi\)
\(74\) −5.58584e8 −2.16544
\(75\) −4.07151e8 −1.48587
\(76\) −7.04439e8 −2.42205
\(77\) −8.91423e7 −0.288985
\(78\) −2.36877e8 −0.724600
\(79\) 4.60455e8 1.33004 0.665020 0.746825i \(-0.268423\pi\)
0.665020 + 0.746825i \(0.268423\pi\)
\(80\) −2.29805e7 −0.0627271
\(81\) 7.78381e7 0.200914
\(82\) 9.52175e7 0.232570
\(83\) −1.63544e8 −0.378253 −0.189127 0.981953i \(-0.560566\pi\)
−0.189127 + 0.981953i \(0.560566\pi\)
\(84\) 4.32183e8 0.947134
\(85\) −2.37373e8 −0.493227
\(86\) 1.19536e9 2.35644
\(87\) −4.96421e8 −0.928995
\(88\) 3.57885e8 0.636168
\(89\) −8.24421e8 −1.39282 −0.696408 0.717646i \(-0.745220\pi\)
−0.696408 + 0.717646i \(0.745220\pi\)
\(90\) −5.23555e8 −0.841146
\(91\) 6.85750e7 0.104828
\(92\) −1.28874e9 −1.87551
\(93\) 4.96806e8 0.688674
\(94\) 1.64737e8 0.217629
\(95\) −3.91982e8 −0.493753
\(96\) 1.57775e9 1.89591
\(97\) −6.27004e8 −0.719113 −0.359557 0.933123i \(-0.617072\pi\)
−0.359557 + 0.933123i \(0.617072\pi\)
\(98\) −2.07225e8 −0.226947
\(99\) −1.24563e9 −1.30326
\(100\) −1.37672e9 −1.37672
\(101\) 8.53911e8 0.816520 0.408260 0.912866i \(-0.366136\pi\)
0.408260 + 0.912866i \(0.366136\pi\)
\(102\) 4.53499e9 4.14835
\(103\) −6.22026e8 −0.544554 −0.272277 0.962219i \(-0.587777\pi\)
−0.272277 + 0.962219i \(0.587777\pi\)
\(104\) −2.75312e8 −0.230768
\(105\) 2.40486e8 0.193081
\(106\) −5.49652e8 −0.422874
\(107\) 3.72891e8 0.275014 0.137507 0.990501i \(-0.456091\pi\)
0.137507 + 0.990501i \(0.456091\pi\)
\(108\) 2.49616e9 1.76549
\(109\) 1.62826e9 1.10485 0.552427 0.833562i \(-0.313702\pi\)
0.552427 + 0.833562i \(0.313702\pi\)
\(110\) 5.79370e8 0.377302
\(111\) 3.58528e9 2.24166
\(112\) −1.27100e8 −0.0763247
\(113\) −2.54267e9 −1.46702 −0.733511 0.679678i \(-0.762120\pi\)
−0.733511 + 0.679678i \(0.762120\pi\)
\(114\) 7.48877e9 4.15278
\(115\) −7.17113e8 −0.382338
\(116\) −1.67858e9 −0.860757
\(117\) 9.58234e8 0.472754
\(118\) 3.79213e9 1.80059
\(119\) −1.31286e9 −0.600146
\(120\) −9.65496e8 −0.425045
\(121\) −9.79521e8 −0.415413
\(122\) −1.85296e7 −0.00757264
\(123\) −6.11154e8 −0.240756
\(124\) 1.67988e9 0.638089
\(125\) −1.61396e9 −0.591284
\(126\) −2.89567e9 −1.02348
\(127\) −2.88641e9 −0.984559 −0.492279 0.870437i \(-0.663836\pi\)
−0.492279 + 0.870437i \(0.663836\pi\)
\(128\) 4.36068e9 1.43585
\(129\) −7.67243e9 −2.43938
\(130\) −4.45695e8 −0.136865
\(131\) −2.81188e9 −0.834210 −0.417105 0.908858i \(-0.636955\pi\)
−0.417105 + 0.908858i \(0.636955\pi\)
\(132\) −6.68294e9 −1.91595
\(133\) −2.16796e9 −0.600786
\(134\) −1.02216e10 −2.73871
\(135\) 1.38898e9 0.359909
\(136\) 5.27082e9 1.32115
\(137\) 3.31457e9 0.803867 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(138\) 1.37004e10 3.21570
\(139\) 5.32393e9 1.20967 0.604833 0.796352i \(-0.293240\pi\)
0.604833 + 0.796352i \(0.293240\pi\)
\(140\) 8.13171e8 0.178898
\(141\) −1.05737e9 −0.225289
\(142\) −1.18864e10 −2.45332
\(143\) −1.06039e9 −0.212057
\(144\) −1.77604e9 −0.344208
\(145\) −9.34037e8 −0.175472
\(146\) 6.73231e9 1.22624
\(147\) 1.33008e9 0.234935
\(148\) 1.21231e10 2.07700
\(149\) −3.15502e9 −0.524402 −0.262201 0.965013i \(-0.584448\pi\)
−0.262201 + 0.965013i \(0.584448\pi\)
\(150\) 1.46357e10 2.36049
\(151\) −1.66845e9 −0.261166 −0.130583 0.991437i \(-0.541685\pi\)
−0.130583 + 0.991437i \(0.541685\pi\)
\(152\) 8.70386e9 1.32256
\(153\) −1.83453e10 −2.70653
\(154\) 3.20437e9 0.459091
\(155\) 9.34762e8 0.130079
\(156\) 5.14102e9 0.695006
\(157\) 3.85793e9 0.506765 0.253382 0.967366i \(-0.418457\pi\)
0.253382 + 0.967366i \(0.418457\pi\)
\(158\) −1.65518e10 −2.11294
\(159\) 3.52795e9 0.437759
\(160\) 2.96861e9 0.358107
\(161\) −3.96619e9 −0.465219
\(162\) −2.79802e9 −0.319178
\(163\) 1.14839e10 1.27422 0.637111 0.770772i \(-0.280129\pi\)
0.637111 + 0.770772i \(0.280129\pi\)
\(164\) −2.06653e9 −0.223072
\(165\) −3.71869e9 −0.390582
\(166\) 5.87885e9 0.600905
\(167\) −1.49406e10 −1.48643 −0.743213 0.669055i \(-0.766699\pi\)
−0.743213 + 0.669055i \(0.766699\pi\)
\(168\) −5.33994e9 −0.517184
\(169\) 8.15731e8 0.0769231
\(170\) 8.53277e9 0.783556
\(171\) −3.02941e10 −2.70942
\(172\) −2.59433e10 −2.26020
\(173\) 5.46601e9 0.463941 0.231971 0.972723i \(-0.425483\pi\)
0.231971 + 0.972723i \(0.425483\pi\)
\(174\) 1.78447e10 1.47583
\(175\) −4.23697e9 −0.341495
\(176\) 1.96538e9 0.154397
\(177\) −2.43398e10 −1.86396
\(178\) 2.96351e10 2.21267
\(179\) 2.22364e10 1.61892 0.809461 0.587173i \(-0.199759\pi\)
0.809461 + 0.587173i \(0.199759\pi\)
\(180\) 1.13629e10 0.806792
\(181\) −4.21143e9 −0.291659 −0.145830 0.989310i \(-0.546585\pi\)
−0.145830 + 0.989310i \(0.546585\pi\)
\(182\) −2.46504e9 −0.166534
\(183\) 1.18932e8 0.00783918
\(184\) 1.59233e10 1.02413
\(185\) 6.74585e9 0.423413
\(186\) −1.78585e10 −1.09405
\(187\) 2.03010e10 1.21403
\(188\) −3.57534e9 −0.208740
\(189\) 7.68211e9 0.437928
\(190\) 1.40904e10 0.784392
\(191\) 1.11392e10 0.605627 0.302813 0.953050i \(-0.402074\pi\)
0.302813 + 0.953050i \(0.402074\pi\)
\(192\) −5.04615e10 −2.67981
\(193\) 2.91759e10 1.51362 0.756808 0.653637i \(-0.226758\pi\)
0.756808 + 0.653637i \(0.226758\pi\)
\(194\) 2.25387e10 1.14241
\(195\) 2.86070e9 0.141682
\(196\) 4.49747e9 0.217679
\(197\) −2.66926e10 −1.26268 −0.631338 0.775507i \(-0.717494\pi\)
−0.631338 + 0.775507i \(0.717494\pi\)
\(198\) 4.47763e10 2.07040
\(199\) 2.08728e10 0.943501 0.471751 0.881732i \(-0.343622\pi\)
0.471751 + 0.881732i \(0.343622\pi\)
\(200\) 1.70104e10 0.751761
\(201\) 6.56073e10 2.83511
\(202\) −3.06952e10 −1.29715
\(203\) −5.16595e9 −0.213510
\(204\) −9.84241e10 −3.97893
\(205\) −1.14991e9 −0.0454750
\(206\) 2.23597e10 0.865095
\(207\) −5.54217e10 −2.09804
\(208\) −1.51191e9 −0.0560070
\(209\) 3.35237e10 1.21533
\(210\) −8.64468e9 −0.306734
\(211\) −2.72187e10 −0.945358 −0.472679 0.881235i \(-0.656713\pi\)
−0.472679 + 0.881235i \(0.656713\pi\)
\(212\) 1.19293e10 0.405604
\(213\) 7.62931e10 2.53967
\(214\) −1.34042e10 −0.436896
\(215\) −1.44360e10 −0.460759
\(216\) −3.08419e10 −0.964048
\(217\) 5.16996e9 0.158277
\(218\) −5.85305e10 −1.75520
\(219\) −4.32114e10 −1.26940
\(220\) −1.25742e10 −0.361893
\(221\) −1.56171e10 −0.440386
\(222\) −1.28879e11 −3.56117
\(223\) 3.81531e10 1.03314 0.516569 0.856246i \(-0.327209\pi\)
0.516569 + 0.856246i \(0.327209\pi\)
\(224\) 1.64187e10 0.435735
\(225\) −5.92054e10 −1.54007
\(226\) 9.14003e10 2.33056
\(227\) −2.63924e10 −0.659726 −0.329863 0.944029i \(-0.607002\pi\)
−0.329863 + 0.944029i \(0.607002\pi\)
\(228\) −1.62531e11 −3.98317
\(229\) 5.14334e10 1.23591 0.617953 0.786215i \(-0.287962\pi\)
0.617953 + 0.786215i \(0.287962\pi\)
\(230\) 2.57778e10 0.607394
\(231\) −2.05673e10 −0.475250
\(232\) 2.07400e10 0.470017
\(233\) −6.10637e10 −1.35732 −0.678659 0.734454i \(-0.737438\pi\)
−0.678659 + 0.734454i \(0.737438\pi\)
\(234\) −3.44453e10 −0.751032
\(235\) −1.98948e9 −0.0425534
\(236\) −8.23016e10 −1.72705
\(237\) 1.06238e11 2.18732
\(238\) 4.71928e10 0.953410
\(239\) −7.41233e10 −1.46948 −0.734741 0.678348i \(-0.762696\pi\)
−0.734741 + 0.678348i \(0.762696\pi\)
\(240\) −5.30216e9 −0.103158
\(241\) 2.24805e10 0.429269 0.214634 0.976694i \(-0.431144\pi\)
0.214634 + 0.976694i \(0.431144\pi\)
\(242\) 3.52105e10 0.659938
\(243\) −4.50176e10 −0.828236
\(244\) 4.02154e8 0.00726336
\(245\) 2.50259e9 0.0443755
\(246\) 2.19689e10 0.382473
\(247\) −2.57889e10 −0.440856
\(248\) −2.07562e10 −0.348429
\(249\) −3.77334e10 −0.622056
\(250\) 5.80162e10 0.939333
\(251\) −2.53210e10 −0.402669 −0.201335 0.979523i \(-0.564528\pi\)
−0.201335 + 0.979523i \(0.564528\pi\)
\(252\) 6.28455e10 0.981684
\(253\) 6.13300e10 0.941089
\(254\) 1.03757e11 1.56410
\(255\) −5.47677e10 −0.811136
\(256\) −4.47722e10 −0.651522
\(257\) −7.30610e10 −1.04469 −0.522344 0.852735i \(-0.674942\pi\)
−0.522344 + 0.852735i \(0.674942\pi\)
\(258\) 2.75798e11 3.87527
\(259\) 3.73098e10 0.515198
\(260\) 9.67305e9 0.131275
\(261\) −7.21865e10 −0.962883
\(262\) 1.01077e11 1.32525
\(263\) 1.14086e11 1.47038 0.735191 0.677860i \(-0.237093\pi\)
0.735191 + 0.677860i \(0.237093\pi\)
\(264\) 8.25726e10 1.04621
\(265\) 6.63798e9 0.0826855
\(266\) 7.79310e10 0.954427
\(267\) −1.90213e11 −2.29055
\(268\) 2.21842e11 2.62686
\(269\) −1.48238e10 −0.172613 −0.0863067 0.996269i \(-0.527506\pi\)
−0.0863067 + 0.996269i \(0.527506\pi\)
\(270\) −4.99290e10 −0.571763
\(271\) −2.59065e10 −0.291775 −0.145887 0.989301i \(-0.546604\pi\)
−0.145887 + 0.989301i \(0.546604\pi\)
\(272\) 2.89454e10 0.320641
\(273\) 1.58219e10 0.172395
\(274\) −1.19148e11 −1.27705
\(275\) 6.55171e10 0.690809
\(276\) −2.97343e11 −3.08437
\(277\) −3.70344e10 −0.377961 −0.188980 0.981981i \(-0.560518\pi\)
−0.188980 + 0.981981i \(0.560518\pi\)
\(278\) −1.91377e11 −1.92171
\(279\) 7.22426e10 0.713796
\(280\) −1.00473e10 −0.0976876
\(281\) −8.33411e10 −0.797408 −0.398704 0.917080i \(-0.630540\pi\)
−0.398704 + 0.917080i \(0.630540\pi\)
\(282\) 3.80088e10 0.357901
\(283\) −1.02296e11 −0.948026 −0.474013 0.880518i \(-0.657195\pi\)
−0.474013 + 0.880518i \(0.657195\pi\)
\(284\) 2.57974e11 2.35312
\(285\) −9.04396e10 −0.812001
\(286\) 3.81174e10 0.336881
\(287\) −6.35991e9 −0.0553327
\(288\) 2.29427e11 1.96507
\(289\) 1.80399e11 1.52122
\(290\) 3.35755e10 0.278760
\(291\) −1.44665e11 −1.18262
\(292\) −1.46113e11 −1.17616
\(293\) 4.44571e10 0.352400 0.176200 0.984354i \(-0.443619\pi\)
0.176200 + 0.984354i \(0.443619\pi\)
\(294\) −4.78118e10 −0.373226
\(295\) −4.57964e10 −0.352072
\(296\) −1.49790e11 −1.13415
\(297\) −1.18790e11 −0.885883
\(298\) 1.13412e11 0.833081
\(299\) −4.71797e10 −0.341377
\(300\) −3.17643e11 −2.26409
\(301\) −7.98423e10 −0.560639
\(302\) 5.99752e10 0.414897
\(303\) 1.97018e11 1.34281
\(304\) 4.77985e10 0.320984
\(305\) 2.23776e8 0.00148069
\(306\) 6.59450e11 4.29968
\(307\) 4.96403e10 0.318942 0.159471 0.987203i \(-0.449021\pi\)
0.159471 + 0.987203i \(0.449021\pi\)
\(308\) −6.95453e10 −0.440341
\(309\) −1.43516e11 −0.895545
\(310\) −3.36015e10 −0.206648
\(311\) 1.63025e10 0.0988170 0.0494085 0.998779i \(-0.484266\pi\)
0.0494085 + 0.998779i \(0.484266\pi\)
\(312\) −6.35211e10 −0.379509
\(313\) 3.07617e10 0.181159 0.0905797 0.995889i \(-0.471128\pi\)
0.0905797 + 0.995889i \(0.471128\pi\)
\(314\) −1.38680e11 −0.805062
\(315\) 3.49701e10 0.200124
\(316\) 3.59228e11 2.02665
\(317\) 6.00070e9 0.0333761 0.0166880 0.999861i \(-0.494688\pi\)
0.0166880 + 0.999861i \(0.494688\pi\)
\(318\) −1.26818e11 −0.695437
\(319\) 7.98821e10 0.431908
\(320\) −9.49455e10 −0.506173
\(321\) 8.60349e10 0.452274
\(322\) 1.42571e11 0.739061
\(323\) 4.93726e11 2.52391
\(324\) 6.07262e10 0.306142
\(325\) −5.04007e10 −0.250589
\(326\) −4.12808e11 −2.02427
\(327\) 3.75678e11 1.81698
\(328\) 2.55335e10 0.121809
\(329\) −1.10034e10 −0.0517778
\(330\) 1.33674e11 0.620491
\(331\) −2.75688e11 −1.26238 −0.631192 0.775627i \(-0.717434\pi\)
−0.631192 + 0.775627i \(0.717434\pi\)
\(332\) −1.27590e11 −0.576364
\(333\) 5.21349e11 2.32343
\(334\) 5.37064e11 2.36138
\(335\) 1.23443e11 0.535506
\(336\) −2.93250e10 −0.125520
\(337\) −1.39653e11 −0.589815 −0.294907 0.955526i \(-0.595289\pi\)
−0.294907 + 0.955526i \(0.595289\pi\)
\(338\) −2.93228e10 −0.122202
\(339\) −5.86654e11 −2.41259
\(340\) −1.85189e11 −0.751555
\(341\) −7.99442e10 −0.320178
\(342\) 1.08897e12 4.30426
\(343\) 1.38413e10 0.0539949
\(344\) 3.20548e11 1.23418
\(345\) −1.65455e11 −0.628773
\(346\) −1.96485e11 −0.737031
\(347\) −4.96372e11 −1.83791 −0.918955 0.394363i \(-0.870965\pi\)
−0.918955 + 0.394363i \(0.870965\pi\)
\(348\) −3.87288e11 −1.41556
\(349\) 4.34232e11 1.56678 0.783388 0.621533i \(-0.213490\pi\)
0.783388 + 0.621533i \(0.213490\pi\)
\(350\) 1.52305e11 0.542509
\(351\) 9.13823e10 0.321351
\(352\) −2.53886e11 −0.881448
\(353\) 5.55167e11 1.90299 0.951496 0.307660i \(-0.0995458\pi\)
0.951496 + 0.307660i \(0.0995458\pi\)
\(354\) 8.74934e11 2.96115
\(355\) 1.43549e11 0.479702
\(356\) −6.43180e11 −2.12230
\(357\) −3.02908e11 −0.986968
\(358\) −7.99324e11 −2.57187
\(359\) 1.82924e11 0.581227 0.290613 0.956841i \(-0.406141\pi\)
0.290613 + 0.956841i \(0.406141\pi\)
\(360\) −1.40397e11 −0.440550
\(361\) 4.92617e11 1.52661
\(362\) 1.51387e11 0.463339
\(363\) −2.25999e11 −0.683166
\(364\) 5.34994e10 0.159732
\(365\) −8.13040e10 −0.239770
\(366\) −4.27522e9 −0.0124536
\(367\) −1.84215e11 −0.530064 −0.265032 0.964240i \(-0.585383\pi\)
−0.265032 + 0.964240i \(0.585383\pi\)
\(368\) 8.74451e10 0.248554
\(369\) −8.88703e10 −0.249539
\(370\) −2.42491e11 −0.672647
\(371\) 3.67132e10 0.100610
\(372\) 3.87588e11 1.04937
\(373\) 4.08286e10 0.109213 0.0546065 0.998508i \(-0.482610\pi\)
0.0546065 + 0.998508i \(0.482610\pi\)
\(374\) −7.29753e11 −1.92865
\(375\) −3.72378e11 −0.972396
\(376\) 4.41759e10 0.113983
\(377\) −6.14513e10 −0.156673
\(378\) −2.76146e11 −0.695706
\(379\) −1.76701e11 −0.439908 −0.219954 0.975510i \(-0.570591\pi\)
−0.219954 + 0.975510i \(0.570591\pi\)
\(380\) −3.05809e11 −0.752357
\(381\) −6.65964e11 −1.61915
\(382\) −4.00418e11 −0.962117
\(383\) −7.60004e11 −1.80477 −0.902384 0.430932i \(-0.858185\pi\)
−0.902384 + 0.430932i \(0.858185\pi\)
\(384\) 1.00611e12 2.36132
\(385\) −3.86982e10 −0.0897670
\(386\) −1.04877e12 −2.40458
\(387\) −1.11568e12 −2.52836
\(388\) −4.89163e11 −1.09575
\(389\) 3.59913e10 0.0796938 0.0398469 0.999206i \(-0.487313\pi\)
0.0398469 + 0.999206i \(0.487313\pi\)
\(390\) −1.02832e11 −0.225081
\(391\) 9.03248e11 1.95439
\(392\) −5.55695e10 −0.118864
\(393\) −6.48766e11 −1.37190
\(394\) 9.59508e11 2.00593
\(395\) 1.99891e11 0.413148
\(396\) −9.71793e11 −1.98584
\(397\) −5.85617e11 −1.18319 −0.591597 0.806234i \(-0.701502\pi\)
−0.591597 + 0.806234i \(0.701502\pi\)
\(398\) −7.50308e11 −1.49888
\(399\) −5.00201e11 −0.988021
\(400\) 9.34151e10 0.182451
\(401\) 9.17734e10 0.177242 0.0886211 0.996065i \(-0.471754\pi\)
0.0886211 + 0.996065i \(0.471754\pi\)
\(402\) −2.35836e12 −4.50394
\(403\) 6.14990e10 0.116144
\(404\) 6.66187e11 1.24417
\(405\) 3.37908e10 0.0624095
\(406\) 1.85698e11 0.339188
\(407\) −5.76929e11 −1.04219
\(408\) 1.21610e12 2.17270
\(409\) 3.83284e11 0.677276 0.338638 0.940917i \(-0.390034\pi\)
0.338638 + 0.940917i \(0.390034\pi\)
\(410\) 4.13355e10 0.0722430
\(411\) 7.64749e11 1.32200
\(412\) −4.85279e11 −0.829764
\(413\) −2.53289e11 −0.428392
\(414\) 1.99222e12 3.33301
\(415\) −7.09971e10 −0.117496
\(416\) 1.95308e11 0.319742
\(417\) 1.22836e12 1.98935
\(418\) −1.20506e12 −1.93071
\(419\) −1.50550e11 −0.238625 −0.119313 0.992857i \(-0.538069\pi\)
−0.119313 + 0.992857i \(0.538069\pi\)
\(420\) 1.87618e11 0.294207
\(421\) 1.08971e12 1.69061 0.845303 0.534287i \(-0.179420\pi\)
0.845303 + 0.534287i \(0.179420\pi\)
\(422\) 9.78421e11 1.50183
\(423\) −1.53756e11 −0.233507
\(424\) −1.47395e11 −0.221480
\(425\) 9.64914e11 1.43463
\(426\) −2.74248e12 −4.03459
\(427\) 1.23766e9 0.00180167
\(428\) 2.90915e11 0.419053
\(429\) −2.44657e11 −0.348738
\(430\) 5.18926e11 0.731976
\(431\) −4.54924e11 −0.635025 −0.317512 0.948254i \(-0.602848\pi\)
−0.317512 + 0.948254i \(0.602848\pi\)
\(432\) −1.69372e11 −0.233973
\(433\) −8.89329e11 −1.21581 −0.607906 0.794009i \(-0.707990\pi\)
−0.607906 + 0.794009i \(0.707990\pi\)
\(434\) −1.85843e11 −0.251444
\(435\) −2.15504e11 −0.288572
\(436\) 1.27030e12 1.68352
\(437\) 1.49156e12 1.95648
\(438\) 1.55330e12 2.01661
\(439\) −1.19486e12 −1.53541 −0.767707 0.640801i \(-0.778602\pi\)
−0.767707 + 0.640801i \(0.778602\pi\)
\(440\) 1.55364e11 0.197612
\(441\) 1.93412e11 0.243506
\(442\) 5.61380e11 0.699612
\(443\) 7.07379e11 0.872640 0.436320 0.899791i \(-0.356282\pi\)
0.436320 + 0.899791i \(0.356282\pi\)
\(444\) 2.79709e12 3.41573
\(445\) −3.57895e11 −0.432648
\(446\) −1.37148e12 −1.64128
\(447\) −7.27938e11 −0.862404
\(448\) −5.25122e11 −0.615898
\(449\) −1.52186e12 −1.76712 −0.883558 0.468322i \(-0.844859\pi\)
−0.883558 + 0.468322i \(0.844859\pi\)
\(450\) 2.12823e12 2.44660
\(451\) 9.83446e10 0.111932
\(452\) −1.98369e12 −2.23537
\(453\) −3.84951e11 −0.429500
\(454\) 9.48719e11 1.04806
\(455\) 2.97695e10 0.0325627
\(456\) 2.00819e12 2.17502
\(457\) 4.30973e11 0.462197 0.231098 0.972930i \(-0.425768\pi\)
0.231098 + 0.972930i \(0.425768\pi\)
\(458\) −1.84886e12 −1.96340
\(459\) −1.74950e12 −1.83974
\(460\) −5.59463e11 −0.582587
\(461\) −3.63737e11 −0.375088 −0.187544 0.982256i \(-0.560053\pi\)
−0.187544 + 0.982256i \(0.560053\pi\)
\(462\) 7.39323e11 0.754997
\(463\) 1.70614e12 1.72544 0.862721 0.505681i \(-0.168759\pi\)
0.862721 + 0.505681i \(0.168759\pi\)
\(464\) 1.13897e11 0.114072
\(465\) 2.15672e11 0.213922
\(466\) 2.19503e12 2.15628
\(467\) −8.23780e11 −0.801466 −0.400733 0.916195i \(-0.631245\pi\)
−0.400733 + 0.916195i \(0.631245\pi\)
\(468\) 7.47576e11 0.720359
\(469\) 6.82735e11 0.651590
\(470\) 7.15151e10 0.0676017
\(471\) 8.90117e11 0.833399
\(472\) 1.01690e12 0.943057
\(473\) 1.23462e12 1.13412
\(474\) −3.81889e12 −3.47484
\(475\) 1.59339e12 1.43616
\(476\) −1.02424e12 −0.914472
\(477\) 5.13012e11 0.453728
\(478\) 2.66448e12 2.33447
\(479\) 1.33355e12 1.15744 0.578722 0.815525i \(-0.303552\pi\)
0.578722 + 0.815525i \(0.303552\pi\)
\(480\) 6.84929e11 0.588925
\(481\) 4.43817e11 0.378052
\(482\) −8.08098e11 −0.681950
\(483\) −9.15094e11 −0.765074
\(484\) −7.64183e11 −0.632985
\(485\) −2.72193e11 −0.223377
\(486\) 1.61823e12 1.31576
\(487\) −3.46476e11 −0.279121 −0.139560 0.990214i \(-0.544569\pi\)
−0.139560 + 0.990214i \(0.544569\pi\)
\(488\) −4.96890e9 −0.00396617
\(489\) 2.64961e12 2.09552
\(490\) −8.99598e10 −0.0704963
\(491\) 2.15900e12 1.67643 0.838216 0.545338i \(-0.183599\pi\)
0.838216 + 0.545338i \(0.183599\pi\)
\(492\) −4.76798e11 −0.366853
\(493\) 1.17648e12 0.896958
\(494\) 9.27025e11 0.700358
\(495\) −5.40750e11 −0.404830
\(496\) −1.13985e11 −0.0845632
\(497\) 7.93935e11 0.583689
\(498\) 1.35639e12 0.988218
\(499\) −1.08273e12 −0.781749 −0.390875 0.920444i \(-0.627827\pi\)
−0.390875 + 0.920444i \(0.627827\pi\)
\(500\) −1.25914e12 −0.900969
\(501\) −3.44715e12 −2.44450
\(502\) 9.10203e11 0.639693
\(503\) −2.04294e12 −1.42298 −0.711492 0.702694i \(-0.751980\pi\)
−0.711492 + 0.702694i \(0.751980\pi\)
\(504\) −7.76502e11 −0.536050
\(505\) 3.70697e11 0.253634
\(506\) −2.20461e12 −1.49504
\(507\) 1.88208e11 0.126504
\(508\) −2.25186e12 −1.50022
\(509\) −2.96107e12 −1.95533 −0.977663 0.210178i \(-0.932596\pi\)
−0.977663 + 0.210178i \(0.932596\pi\)
\(510\) 1.96871e12 1.28860
\(511\) −4.49674e11 −0.291745
\(512\) −6.23255e11 −0.400822
\(513\) −2.88901e12 −1.84171
\(514\) 2.62629e12 1.65962
\(515\) −2.70032e11 −0.169154
\(516\) −5.98572e12 −3.71700
\(517\) 1.70147e11 0.104741
\(518\) −1.34116e12 −0.818459
\(519\) 1.26114e12 0.762974
\(520\) −1.19518e11 −0.0716831
\(521\) −5.60548e11 −0.333306 −0.166653 0.986016i \(-0.553296\pi\)
−0.166653 + 0.986016i \(0.553296\pi\)
\(522\) 2.59486e12 1.52967
\(523\) 2.85263e12 1.66720 0.833599 0.552370i \(-0.186276\pi\)
0.833599 + 0.552370i \(0.186276\pi\)
\(524\) −2.19371e12 −1.27113
\(525\) −9.77569e11 −0.561604
\(526\) −4.10099e12 −2.33589
\(527\) −1.17739e12 −0.664925
\(528\) 4.53459e11 0.253913
\(529\) 9.27591e11 0.514998
\(530\) −2.38613e11 −0.131357
\(531\) −3.53935e12 −1.93196
\(532\) −1.69136e12 −0.915448
\(533\) −7.56540e10 −0.0406031
\(534\) 6.83753e12 3.63884
\(535\) 1.61878e11 0.0854273
\(536\) −2.74102e12 −1.43440
\(537\) 5.13047e12 2.66240
\(538\) 5.32866e11 0.274219
\(539\) −2.14031e11 −0.109226
\(540\) 1.08362e12 0.548411
\(541\) −3.07722e12 −1.54444 −0.772220 0.635355i \(-0.780854\pi\)
−0.772220 + 0.635355i \(0.780854\pi\)
\(542\) 9.31253e11 0.463522
\(543\) −9.71676e11 −0.479648
\(544\) −3.73915e12 −1.83053
\(545\) 7.06855e11 0.343199
\(546\) −5.68743e11 −0.273873
\(547\) −2.99830e12 −1.43196 −0.715982 0.698119i \(-0.754021\pi\)
−0.715982 + 0.698119i \(0.754021\pi\)
\(548\) 2.58589e12 1.22489
\(549\) 1.72944e10 0.00812514
\(550\) −2.35512e12 −1.09744
\(551\) 1.94275e12 0.897915
\(552\) 3.67389e12 1.68422
\(553\) 1.10555e12 0.502708
\(554\) 1.33126e12 0.600440
\(555\) 1.55643e12 0.696323
\(556\) 4.15351e12 1.84323
\(557\) 2.77410e12 1.22116 0.610581 0.791954i \(-0.290936\pi\)
0.610581 + 0.791954i \(0.290936\pi\)
\(558\) −2.59688e12 −1.13396
\(559\) −9.49761e11 −0.411397
\(560\) −5.51763e10 −0.0237086
\(561\) 4.68392e12 1.99653
\(562\) 2.99583e12 1.26679
\(563\) −6.92492e11 −0.290487 −0.145244 0.989396i \(-0.546397\pi\)
−0.145244 + 0.989396i \(0.546397\pi\)
\(564\) −8.24915e11 −0.343284
\(565\) −1.10381e12 −0.455699
\(566\) 3.67720e12 1.50606
\(567\) 1.86889e11 0.0759383
\(568\) −3.18746e12 −1.28492
\(569\) 4.26105e11 0.170417 0.0852083 0.996363i \(-0.472844\pi\)
0.0852083 + 0.996363i \(0.472844\pi\)
\(570\) 3.25100e12 1.28997
\(571\) −3.57448e12 −1.40718 −0.703591 0.710605i \(-0.748421\pi\)
−0.703591 + 0.710605i \(0.748421\pi\)
\(572\) −8.27273e11 −0.323122
\(573\) 2.57008e12 0.995982
\(574\) 2.28617e11 0.0879033
\(575\) 2.91504e12 1.11209
\(576\) −7.33780e12 −2.77757
\(577\) −3.13230e12 −1.17645 −0.588224 0.808698i \(-0.700173\pi\)
−0.588224 + 0.808698i \(0.700173\pi\)
\(578\) −6.48472e12 −2.41666
\(579\) 6.73156e12 2.48922
\(580\) −7.28698e11 −0.267375
\(581\) −3.92669e11 −0.142966
\(582\) 5.20021e12 1.87874
\(583\) −5.67703e11 −0.203523
\(584\) 1.80533e12 0.642244
\(585\) 4.15985e11 0.146851
\(586\) −1.59808e12 −0.559834
\(587\) 7.18927e11 0.249927 0.124963 0.992161i \(-0.460119\pi\)
0.124963 + 0.992161i \(0.460119\pi\)
\(588\) 1.03767e12 0.357983
\(589\) −1.94426e12 −0.665635
\(590\) 1.64623e12 0.559313
\(591\) −6.15861e12 −2.07653
\(592\) −8.22592e11 −0.275256
\(593\) 4.09447e12 1.35973 0.679864 0.733339i \(-0.262039\pi\)
0.679864 + 0.733339i \(0.262039\pi\)
\(594\) 4.27010e12 1.40734
\(595\) −5.69933e11 −0.186422
\(596\) −2.46142e12 −0.799057
\(597\) 4.81586e12 1.55163
\(598\) 1.69595e12 0.542322
\(599\) −9.86003e11 −0.312938 −0.156469 0.987683i \(-0.550011\pi\)
−0.156469 + 0.987683i \(0.550011\pi\)
\(600\) 3.92471e12 1.23631
\(601\) 3.62685e12 1.13395 0.566976 0.823735i \(-0.308113\pi\)
0.566976 + 0.823735i \(0.308113\pi\)
\(602\) 2.87006e12 0.890650
\(603\) 9.54021e12 2.93853
\(604\) −1.30166e12 −0.397952
\(605\) −4.25226e11 −0.129039
\(606\) −7.08212e12 −2.13322
\(607\) 3.76025e12 1.12426 0.562131 0.827048i \(-0.309982\pi\)
0.562131 + 0.827048i \(0.309982\pi\)
\(608\) −6.17457e12 −1.83249
\(609\) −1.19191e12 −0.351127
\(610\) −8.04401e9 −0.00235228
\(611\) −1.30890e11 −0.0379945
\(612\) −1.43122e13 −4.12407
\(613\) −2.13878e12 −0.611780 −0.305890 0.952067i \(-0.598954\pi\)
−0.305890 + 0.952067i \(0.598954\pi\)
\(614\) −1.78440e12 −0.506681
\(615\) −2.65312e11 −0.0747858
\(616\) 8.59283e11 0.240449
\(617\) 2.69179e12 0.747752 0.373876 0.927479i \(-0.378028\pi\)
0.373876 + 0.927479i \(0.378028\pi\)
\(618\) 5.15892e12 1.42269
\(619\) −7.17858e11 −0.196531 −0.0982654 0.995160i \(-0.531329\pi\)
−0.0982654 + 0.995160i \(0.531329\pi\)
\(620\) 7.29264e11 0.198208
\(621\) −5.28530e12 −1.42613
\(622\) −5.86019e11 −0.156984
\(623\) −1.97943e12 −0.526435
\(624\) −3.48835e11 −0.0921062
\(625\) 2.74597e12 0.719840
\(626\) −1.10578e12 −0.287796
\(627\) 7.73471e12 1.99867
\(628\) 3.00981e12 0.772183
\(629\) −8.49682e12 −2.16435
\(630\) −1.25706e12 −0.317923
\(631\) −4.59827e12 −1.15468 −0.577341 0.816503i \(-0.695910\pi\)
−0.577341 + 0.816503i \(0.695910\pi\)
\(632\) −4.43853e12 −1.10665
\(633\) −6.28000e12 −1.55469
\(634\) −2.15705e11 −0.0530223
\(635\) −1.25304e12 −0.305832
\(636\) 2.75236e12 0.667035
\(637\) 1.64648e11 0.0396214
\(638\) −2.87149e12 −0.686143
\(639\) 1.10941e13 2.63231
\(640\) 1.89304e12 0.446016
\(641\) −5.40351e12 −1.26420 −0.632099 0.774888i \(-0.717806\pi\)
−0.632099 + 0.774888i \(0.717806\pi\)
\(642\) −3.09266e12 −0.718497
\(643\) 2.49434e12 0.575449 0.287724 0.957713i \(-0.407101\pi\)
0.287724 + 0.957713i \(0.407101\pi\)
\(644\) −3.09426e12 −0.708877
\(645\) −3.33073e12 −0.757741
\(646\) −1.77478e13 −4.00957
\(647\) 7.00912e12 1.57251 0.786257 0.617900i \(-0.212016\pi\)
0.786257 + 0.617900i \(0.212016\pi\)
\(648\) −7.50316e11 −0.167169
\(649\) 3.91667e12 0.866594
\(650\) 1.81173e12 0.398093
\(651\) 1.19283e12 0.260294
\(652\) 8.95928e12 1.94160
\(653\) 6.91176e12 1.48758 0.743788 0.668416i \(-0.233027\pi\)
0.743788 + 0.668416i \(0.233027\pi\)
\(654\) −1.35044e13 −2.88652
\(655\) −1.22068e12 −0.259129
\(656\) 1.40221e11 0.0295628
\(657\) −6.28353e12 −1.31571
\(658\) 3.95534e11 0.0822559
\(659\) 2.27667e12 0.470236 0.235118 0.971967i \(-0.424452\pi\)
0.235118 + 0.971967i \(0.424452\pi\)
\(660\) −2.90118e12 −0.595150
\(661\) 2.29883e12 0.468383 0.234191 0.972190i \(-0.424756\pi\)
0.234191 + 0.972190i \(0.424756\pi\)
\(662\) 9.91004e12 2.00546
\(663\) −3.60323e12 −0.724237
\(664\) 1.57647e12 0.314724
\(665\) −9.41149e11 −0.186621
\(666\) −1.87407e13 −3.69108
\(667\) 3.55418e12 0.695301
\(668\) −1.16560e13 −2.26494
\(669\) 8.80283e12 1.69905
\(670\) −4.43736e12 −0.850722
\(671\) −1.91382e10 −0.00364459
\(672\) 3.78819e12 0.716588
\(673\) −5.78650e12 −1.08730 −0.543649 0.839313i \(-0.682958\pi\)
−0.543649 + 0.839313i \(0.682958\pi\)
\(674\) 5.02005e12 0.936998
\(675\) −5.64614e12 −1.04685
\(676\) 6.36400e11 0.117212
\(677\) 8.97926e12 1.64283 0.821413 0.570334i \(-0.193186\pi\)
0.821413 + 0.570334i \(0.193186\pi\)
\(678\) 2.10882e13 3.83271
\(679\) −1.50544e12 −0.271799
\(680\) 2.28815e12 0.410387
\(681\) −6.08936e12 −1.08495
\(682\) 2.87372e12 0.508645
\(683\) 1.01954e13 1.79272 0.896358 0.443331i \(-0.146203\pi\)
0.896358 + 0.443331i \(0.146203\pi\)
\(684\) −2.36342e13 −4.12847
\(685\) 1.43891e12 0.249704
\(686\) −4.97548e11 −0.0857780
\(687\) 1.18669e13 2.03251
\(688\) 1.76033e12 0.299534
\(689\) 4.36720e11 0.0738272
\(690\) 5.94755e12 0.998889
\(691\) 2.79110e12 0.465720 0.232860 0.972510i \(-0.425192\pi\)
0.232860 + 0.972510i \(0.425192\pi\)
\(692\) 4.26436e12 0.706930
\(693\) −2.99076e12 −0.492587
\(694\) 1.78429e13 2.91976
\(695\) 2.31120e12 0.375757
\(696\) 4.78522e12 0.772966
\(697\) 1.44839e12 0.232454
\(698\) −1.56092e13 −2.48903
\(699\) −1.40888e13 −2.23217
\(700\) −3.30551e12 −0.520352
\(701\) −2.22264e12 −0.347647 −0.173824 0.984777i \(-0.555612\pi\)
−0.173824 + 0.984777i \(0.555612\pi\)
\(702\) −3.28488e12 −0.510509
\(703\) −1.40311e13 −2.16666
\(704\) 8.12007e12 1.24590
\(705\) −4.59020e11 −0.0699811
\(706\) −1.99564e13 −3.02315
\(707\) 2.05024e12 0.308615
\(708\) −1.89889e13 −2.84021
\(709\) −6.86568e12 −1.02041 −0.510206 0.860052i \(-0.670431\pi\)
−0.510206 + 0.860052i \(0.670431\pi\)
\(710\) −5.16009e12 −0.762069
\(711\) 1.54485e13 2.26711
\(712\) 7.94696e12 1.15889
\(713\) −3.55694e12 −0.515434
\(714\) 1.08885e13 1.56793
\(715\) −4.60332e11 −0.0658710
\(716\) 1.73480e13 2.46683
\(717\) −1.71020e13 −2.41663
\(718\) −6.57550e12 −0.923355
\(719\) 1.08604e13 1.51553 0.757766 0.652526i \(-0.226291\pi\)
0.757766 + 0.652526i \(0.226291\pi\)
\(720\) −7.71007e11 −0.106921
\(721\) −1.49348e12 −0.205822
\(722\) −1.77079e13 −2.42521
\(723\) 5.18679e12 0.705953
\(724\) −3.28559e12 −0.444416
\(725\) 3.79683e12 0.510387
\(726\) 8.12389e12 1.08530
\(727\) 1.18416e13 1.57220 0.786098 0.618102i \(-0.212098\pi\)
0.786098 + 0.618102i \(0.212098\pi\)
\(728\) −6.61025e11 −0.0872221
\(729\) −1.19187e13 −1.56299
\(730\) 2.92260e12 0.380905
\(731\) 1.81830e13 2.35526
\(732\) 9.27864e10 0.0119449
\(733\) 5.16125e12 0.660369 0.330184 0.943916i \(-0.392889\pi\)
0.330184 + 0.943916i \(0.392889\pi\)
\(734\) 6.62192e12 0.842077
\(735\) 5.77408e11 0.0729776
\(736\) −1.12961e13 −1.41899
\(737\) −1.05573e13 −1.31810
\(738\) 3.19459e12 0.396425
\(739\) −9.11730e12 −1.12452 −0.562259 0.826961i \(-0.690068\pi\)
−0.562259 + 0.826961i \(0.690068\pi\)
\(740\) 5.26284e12 0.645175
\(741\) −5.95012e12 −0.725010
\(742\) −1.31971e12 −0.159832
\(743\) −1.58640e12 −0.190969 −0.0954843 0.995431i \(-0.530440\pi\)
−0.0954843 + 0.995431i \(0.530440\pi\)
\(744\) −4.78894e12 −0.573008
\(745\) −1.36965e12 −0.162894
\(746\) −1.46765e12 −0.173499
\(747\) −5.48697e12 −0.644748
\(748\) 1.58380e13 1.84988
\(749\) 8.95312e11 0.103946
\(750\) 1.33857e13 1.54478
\(751\) 1.04973e13 1.20420 0.602100 0.798420i \(-0.294331\pi\)
0.602100 + 0.798420i \(0.294331\pi\)
\(752\) 2.42598e11 0.0276635
\(753\) −5.84215e12 −0.662209
\(754\) 2.20897e12 0.248896
\(755\) −7.24302e11 −0.0811257
\(756\) 5.99328e12 0.667292
\(757\) 5.39506e12 0.597125 0.298562 0.954390i \(-0.403493\pi\)
0.298562 + 0.954390i \(0.403493\pi\)
\(758\) 6.35179e12 0.698851
\(759\) 1.41503e13 1.54767
\(760\) 3.77849e12 0.410825
\(761\) 3.08992e12 0.333977 0.166988 0.985959i \(-0.446596\pi\)
0.166988 + 0.985959i \(0.446596\pi\)
\(762\) 2.39392e13 2.57224
\(763\) 3.90945e12 0.417595
\(764\) 8.69038e12 0.922823
\(765\) −7.96398e12 −0.840724
\(766\) 2.73196e13 2.86711
\(767\) −3.01299e12 −0.314354
\(768\) −1.03300e13 −1.07146
\(769\) −1.64177e13 −1.69295 −0.846473 0.532432i \(-0.821278\pi\)
−0.846473 + 0.532432i \(0.821278\pi\)
\(770\) 1.39107e12 0.142607
\(771\) −1.68569e13 −1.71804
\(772\) 2.27618e13 2.30637
\(773\) −1.51411e13 −1.52528 −0.762640 0.646823i \(-0.776097\pi\)
−0.762640 + 0.646823i \(0.776097\pi\)
\(774\) 4.01049e13 4.01664
\(775\) −3.79977e12 −0.378356
\(776\) 6.04397e12 0.598335
\(777\) 8.60825e12 0.847267
\(778\) −1.29377e12 −0.126604
\(779\) 2.39177e12 0.232702
\(780\) 2.23180e12 0.215889
\(781\) −1.22768e13 −1.18074
\(782\) −3.24687e13 −3.10481
\(783\) −6.88408e12 −0.654513
\(784\) −3.05167e11 −0.0288480
\(785\) 1.67479e12 0.157416
\(786\) 2.33210e13 2.17944
\(787\) −1.10748e13 −1.02908 −0.514542 0.857465i \(-0.672038\pi\)
−0.514542 + 0.857465i \(0.672038\pi\)
\(788\) −2.08245e13 −1.92400
\(789\) 2.63223e13 2.41811
\(790\) −7.18540e12 −0.656341
\(791\) −6.10494e12 −0.554482
\(792\) 1.20072e13 1.08437
\(793\) 1.47225e10 0.00132206
\(794\) 2.10509e13 1.87966
\(795\) 1.53154e12 0.135980
\(796\) 1.62841e13 1.43766
\(797\) −1.41284e13 −1.24032 −0.620158 0.784477i \(-0.712931\pi\)
−0.620158 + 0.784477i \(0.712931\pi\)
\(798\) 1.79805e13 1.56960
\(799\) 2.50587e12 0.217520
\(800\) −1.20673e13 −1.04161
\(801\) −2.76597e13 −2.37411
\(802\) −3.29894e12 −0.281573
\(803\) 6.95340e12 0.590171
\(804\) 5.11842e13 4.32000
\(805\) −1.72179e12 −0.144510
\(806\) −2.21068e12 −0.184510
\(807\) −3.42020e12 −0.283871
\(808\) −8.23123e12 −0.679382
\(809\) −1.45772e13 −1.19648 −0.598239 0.801317i \(-0.704133\pi\)
−0.598239 + 0.801317i \(0.704133\pi\)
\(810\) −1.21467e12 −0.0991457
\(811\) 8.28702e12 0.672674 0.336337 0.941742i \(-0.390812\pi\)
0.336337 + 0.941742i \(0.390812\pi\)
\(812\) −4.03026e12 −0.325335
\(813\) −5.97725e12 −0.479838
\(814\) 2.07387e13 1.65566
\(815\) 4.98535e12 0.395810
\(816\) 6.67839e12 0.527310
\(817\) 3.00262e13 2.35777
\(818\) −1.37778e13 −1.07594
\(819\) 2.30072e12 0.178684
\(820\) −8.97116e11 −0.0692925
\(821\) −1.34592e13 −1.03389 −0.516944 0.856019i \(-0.672931\pi\)
−0.516944 + 0.856019i \(0.672931\pi\)
\(822\) −2.74902e13 −2.10017
\(823\) −1.12252e13 −0.852892 −0.426446 0.904513i \(-0.640235\pi\)
−0.426446 + 0.904513i \(0.640235\pi\)
\(824\) 5.99598e12 0.453094
\(825\) 1.51164e13 1.13607
\(826\) 9.10490e12 0.680558
\(827\) 1.18754e13 0.882821 0.441411 0.897305i \(-0.354478\pi\)
0.441411 + 0.897305i \(0.354478\pi\)
\(828\) −4.32378e13 −3.19688
\(829\) 1.79748e13 1.32181 0.660903 0.750471i \(-0.270173\pi\)
0.660903 + 0.750471i \(0.270173\pi\)
\(830\) 2.55211e12 0.186658
\(831\) −8.54472e12 −0.621574
\(832\) −6.24657e12 −0.451946
\(833\) −3.15217e12 −0.226834
\(834\) −4.41553e13 −3.16035
\(835\) −6.48595e12 −0.461726
\(836\) 2.61538e13 1.85186
\(837\) 6.88943e12 0.485198
\(838\) 5.41175e12 0.379088
\(839\) −1.21225e13 −0.844623 −0.422312 0.906451i \(-0.638781\pi\)
−0.422312 + 0.906451i \(0.638781\pi\)
\(840\) −2.31816e12 −0.160652
\(841\) −9.87785e12 −0.680895
\(842\) −3.91715e13 −2.68575
\(843\) −1.92288e13 −1.31138
\(844\) −2.12349e13 −1.44049
\(845\) 3.54122e11 0.0238945
\(846\) 5.52700e12 0.370957
\(847\) −2.35183e12 −0.157011
\(848\) −8.09438e11 −0.0537529
\(849\) −2.36021e13 −1.55907
\(850\) −3.46854e13 −2.27909
\(851\) −2.56692e13 −1.67776
\(852\) 5.95208e13 3.86982
\(853\) −1.99327e13 −1.28913 −0.644564 0.764550i \(-0.722961\pi\)
−0.644564 + 0.764550i \(0.722961\pi\)
\(854\) −4.44896e10 −0.00286219
\(855\) −1.31512e13 −0.841622
\(856\) −3.59447e12 −0.228824
\(857\) −1.82579e13 −1.15621 −0.578107 0.815961i \(-0.696208\pi\)
−0.578107 + 0.815961i \(0.696208\pi\)
\(858\) 8.79459e12 0.554017
\(859\) −6.62180e12 −0.414961 −0.207480 0.978239i \(-0.566526\pi\)
−0.207480 + 0.978239i \(0.566526\pi\)
\(860\) −1.12624e13 −0.702082
\(861\) −1.46738e12 −0.0909974
\(862\) 1.63530e13 1.00882
\(863\) 2.17631e12 0.133559 0.0667794 0.997768i \(-0.478728\pi\)
0.0667794 + 0.997768i \(0.478728\pi\)
\(864\) 2.18794e13 1.33574
\(865\) 2.37288e12 0.144113
\(866\) 3.19684e13 1.93148
\(867\) 4.16222e13 2.50172
\(868\) 4.03339e12 0.241175
\(869\) −1.70954e13 −1.01693
\(870\) 7.74666e12 0.458435
\(871\) 8.12144e12 0.478136
\(872\) −1.56955e13 −0.919288
\(873\) −2.10363e13 −1.22576
\(874\) −5.36166e13 −3.10812
\(875\) −3.87511e12 −0.223484
\(876\) −3.37118e13 −1.93425
\(877\) −1.95150e12 −0.111396 −0.0556981 0.998448i \(-0.517738\pi\)
−0.0556981 + 0.998448i \(0.517738\pi\)
\(878\) 4.29511e13 2.43921
\(879\) 1.02573e13 0.579539
\(880\) 8.53202e11 0.0479601
\(881\) −1.04264e12 −0.0583102 −0.0291551 0.999575i \(-0.509282\pi\)
−0.0291551 + 0.999575i \(0.509282\pi\)
\(882\) −6.95249e12 −0.386841
\(883\) 1.82714e13 1.01146 0.505730 0.862692i \(-0.331223\pi\)
0.505730 + 0.862692i \(0.331223\pi\)
\(884\) −1.21838e13 −0.671039
\(885\) −1.05663e13 −0.579000
\(886\) −2.54279e13 −1.38630
\(887\) −2.54434e13 −1.38012 −0.690062 0.723750i \(-0.742417\pi\)
−0.690062 + 0.723750i \(0.742417\pi\)
\(888\) −3.45601e13 −1.86516
\(889\) −6.93028e12 −0.372128
\(890\) 1.28651e13 0.687319
\(891\) −2.88991e12 −0.153615
\(892\) 2.97656e13 1.57424
\(893\) 4.13803e12 0.217752
\(894\) 2.61669e13 1.37004
\(895\) 9.65319e12 0.502883
\(896\) 1.04700e13 0.542700
\(897\) −1.08855e13 −0.561411
\(898\) 5.47056e13 2.80730
\(899\) −4.63290e12 −0.236556
\(900\) −4.61897e13 −2.34668
\(901\) −8.36095e12 −0.422663
\(902\) −3.53516e12 −0.177819
\(903\) −1.84215e13 −0.921999
\(904\) 2.45099e13 1.22063
\(905\) −1.82825e12 −0.0905976
\(906\) 1.38377e13 0.682318
\(907\) 2.68870e13 1.31920 0.659598 0.751618i \(-0.270726\pi\)
0.659598 + 0.751618i \(0.270726\pi\)
\(908\) −2.05903e13 −1.00526
\(909\) 2.86491e13 1.39179
\(910\) −1.07011e12 −0.0517301
\(911\) 1.32637e13 0.638015 0.319008 0.947752i \(-0.396650\pi\)
0.319008 + 0.947752i \(0.396650\pi\)
\(912\) 1.10282e13 0.527873
\(913\) 6.07192e12 0.289206
\(914\) −1.54920e13 −0.734260
\(915\) 5.16305e10 0.00243507
\(916\) 4.01263e13 1.88321
\(917\) −6.75131e12 −0.315302
\(918\) 6.28887e13 2.92267
\(919\) −2.84043e13 −1.31360 −0.656801 0.754064i \(-0.728091\pi\)
−0.656801 + 0.754064i \(0.728091\pi\)
\(920\) 6.91257e12 0.318123
\(921\) 1.14532e13 0.524515
\(922\) 1.30751e13 0.595877
\(923\) 9.44422e12 0.428310
\(924\) −1.60457e13 −0.724162
\(925\) −2.74216e13 −1.23156
\(926\) −6.13300e13 −2.74109
\(927\) −2.08692e13 −0.928213
\(928\) −1.47131e13 −0.651236
\(929\) −1.41616e12 −0.0623796 −0.0311898 0.999513i \(-0.509930\pi\)
−0.0311898 + 0.999513i \(0.509930\pi\)
\(930\) −7.75267e12 −0.339843
\(931\) −5.20528e12 −0.227076
\(932\) −4.76394e13 −2.06821
\(933\) 3.76137e12 0.162509
\(934\) 2.96121e13 1.27323
\(935\) 8.81300e12 0.377113
\(936\) −9.23685e12 −0.393353
\(937\) 1.36000e13 0.576381 0.288191 0.957573i \(-0.406946\pi\)
0.288191 + 0.957573i \(0.406946\pi\)
\(938\) −2.45420e13 −1.03514
\(939\) 7.09746e12 0.297926
\(940\) −1.55211e12 −0.0648407
\(941\) 1.71397e13 0.712606 0.356303 0.934370i \(-0.384037\pi\)
0.356303 + 0.934370i \(0.384037\pi\)
\(942\) −3.19967e13 −1.32396
\(943\) 4.37562e12 0.180193
\(944\) 5.58443e12 0.228878
\(945\) 3.33493e12 0.136033
\(946\) −4.43803e13 −1.80169
\(947\) 1.72767e13 0.698050 0.349025 0.937113i \(-0.386513\pi\)
0.349025 + 0.937113i \(0.386513\pi\)
\(948\) 8.28825e13 3.33292
\(949\) −5.34908e12 −0.214082
\(950\) −5.72771e13 −2.28152
\(951\) 1.38450e12 0.0548886
\(952\) 1.26552e13 0.499348
\(953\) 2.53499e13 0.995540 0.497770 0.867309i \(-0.334152\pi\)
0.497770 + 0.867309i \(0.334152\pi\)
\(954\) −1.84411e13 −0.720806
\(955\) 4.83572e12 0.188125
\(956\) −5.78280e13 −2.23912
\(957\) 1.84307e13 0.710294
\(958\) −4.79367e13 −1.83875
\(959\) 7.95828e12 0.303833
\(960\) −2.19062e13 −0.832427
\(961\) −2.18031e13 −0.824638
\(962\) −1.59537e13 −0.600585
\(963\) 1.25107e13 0.468773
\(964\) 1.75384e13 0.654098
\(965\) 1.26657e13 0.470172
\(966\) 3.28945e13 1.21542
\(967\) 9.19232e12 0.338069 0.169035 0.985610i \(-0.445935\pi\)
0.169035 + 0.985610i \(0.445935\pi\)
\(968\) 9.44204e12 0.345642
\(969\) 1.13914e14 4.15070
\(970\) 9.78441e12 0.354864
\(971\) −2.88697e13 −1.04221 −0.521106 0.853492i \(-0.674480\pi\)
−0.521106 + 0.853492i \(0.674480\pi\)
\(972\) −3.51209e13 −1.26202
\(973\) 1.27827e13 0.457211
\(974\) 1.24546e13 0.443420
\(975\) −1.16286e13 −0.412105
\(976\) −2.72874e10 −0.000962582 0
\(977\) −1.03809e13 −0.364511 −0.182255 0.983251i \(-0.558340\pi\)
−0.182255 + 0.983251i \(0.558340\pi\)
\(978\) −9.52445e13 −3.32901
\(979\) 3.06084e13 1.06492
\(980\) 1.95242e12 0.0676172
\(981\) 5.46288e13 1.88327
\(982\) −7.76088e13 −2.66323
\(983\) −7.46537e12 −0.255012 −0.127506 0.991838i \(-0.540697\pi\)
−0.127506 + 0.991838i \(0.540697\pi\)
\(984\) 5.89119e12 0.200320
\(985\) −1.15877e13 −0.392223
\(986\) −4.22904e13 −1.42494
\(987\) −2.53874e12 −0.0851512
\(988\) −2.01195e13 −0.671755
\(989\) 5.49316e13 1.82574
\(990\) 1.94381e13 0.643126
\(991\) 1.19731e13 0.394345 0.197173 0.980369i \(-0.436824\pi\)
0.197173 + 0.980369i \(0.436824\pi\)
\(992\) 1.47245e13 0.482769
\(993\) −6.36077e13 −2.07605
\(994\) −2.85393e13 −0.927266
\(995\) 9.06124e12 0.293078
\(996\) −2.94381e13 −0.947858
\(997\) 3.43494e12 0.110101 0.0550504 0.998484i \(-0.482468\pi\)
0.0550504 + 0.998484i \(0.482468\pi\)
\(998\) 3.89205e13 1.24191
\(999\) 4.97186e13 1.57934
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 91.10.a.a.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.3 12 1.1 even 1 trivial