Properties

Label 29.12.a.a.1.5
Level $29$
Weight $12$
Character 29.1
Self dual yes
Analytic conductor $22.282$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2819522362\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-20.7036\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q-23.7036 q^{2} -804.018 q^{3} -1486.14 q^{4} +6643.74 q^{5} +19058.1 q^{6} -61145.1 q^{7} +83771.9 q^{8} +469298. q^{9} +O(q^{10})\) \(q-23.7036 q^{2} -804.018 q^{3} -1486.14 q^{4} +6643.74 q^{5} +19058.1 q^{6} -61145.1 q^{7} +83771.9 q^{8} +469298. q^{9} -157481. q^{10} +97159.7 q^{11} +1.19488e6 q^{12} +2.25195e6 q^{13} +1.44936e6 q^{14} -5.34169e6 q^{15} +1.05792e6 q^{16} -5.47274e6 q^{17} -1.11241e7 q^{18} -1.36844e7 q^{19} -9.87353e6 q^{20} +4.91618e7 q^{21} -2.30304e6 q^{22} +3.78724e7 q^{23} -6.73541e7 q^{24} -4.68878e6 q^{25} -5.33794e7 q^{26} -2.34895e8 q^{27} +9.08701e7 q^{28} +2.05111e7 q^{29} +1.26617e8 q^{30} +7.66307e7 q^{31} -1.96641e8 q^{32} -7.81182e7 q^{33} +1.29724e8 q^{34} -4.06232e8 q^{35} -6.97442e8 q^{36} +5.87994e8 q^{37} +3.24370e8 q^{38} -1.81061e9 q^{39} +5.56559e8 q^{40} -5.25000e8 q^{41} -1.16531e9 q^{42} +1.40657e9 q^{43} -1.44393e8 q^{44} +3.11790e9 q^{45} -8.97712e8 q^{46} -1.05859e9 q^{47} -8.50585e8 q^{48} +1.76140e9 q^{49} +1.11141e8 q^{50} +4.40018e9 q^{51} -3.34671e9 q^{52} +1.82460e8 q^{53} +5.56786e9 q^{54} +6.45504e8 q^{55} -5.12224e9 q^{56} +1.10025e10 q^{57} -4.86188e8 q^{58} -4.81226e9 q^{59} +7.93850e9 q^{60} +1.42171e9 q^{61} -1.81643e9 q^{62} -2.86953e10 q^{63} +2.49449e9 q^{64} +1.49614e10 q^{65} +1.85168e9 q^{66} +1.58194e9 q^{67} +8.13325e9 q^{68} -3.04501e10 q^{69} +9.62917e9 q^{70} -7.93869e9 q^{71} +3.93140e10 q^{72} -2.06263e10 q^{73} -1.39376e10 q^{74} +3.76987e9 q^{75} +2.03369e10 q^{76} -5.94084e9 q^{77} +4.29180e10 q^{78} -3.33041e10 q^{79} +7.02853e9 q^{80} +1.05725e11 q^{81} +1.24444e10 q^{82} -4.80488e9 q^{83} -7.30612e10 q^{84} -3.63595e10 q^{85} -3.33408e10 q^{86} -1.64913e10 q^{87} +8.13925e9 q^{88} -3.03829e10 q^{89} -7.39054e10 q^{90} -1.37696e11 q^{91} -5.62836e10 q^{92} -6.16125e10 q^{93} +2.50925e10 q^{94} -9.09157e10 q^{95} +1.58103e11 q^{96} -2.92126e10 q^{97} -4.17514e10 q^{98} +4.55969e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} + O(q^{10}) \) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} - 685834q^{10} - 612246q^{11} + 2578538q^{12} + 1510364q^{13} + 3955400q^{14} - 2462818q^{15} + 3024818q^{16} - 3291098q^{17} - 27885614q^{18} - 44121388q^{19} - 49472662q^{20} - 46916800q^{21} - 43435618q^{22} - 88684076q^{23} - 224700678q^{24} - 44195521q^{25} - 324999762q^{26} - 236304286q^{27} - 391274848q^{28} + 225622639q^{29} - 494910382q^{30} - 292235934q^{31} - 632542514q^{32} - 1079766410q^{33} - 1113307936q^{34} - 1312820120q^{35} - 2236726492q^{36} - 1380429338q^{37} - 1222857284q^{38} - 1186931090q^{39} - 2713154106q^{40} - 1062067494q^{41} + 205598960q^{42} + 74588594q^{43} + 52891466q^{44} + 4527996830q^{45} - 87670324q^{46} - 1821239394q^{47} + 2666035542q^{48} + 4692522003q^{49} + 9494259926q^{50} + 8768158380q^{51} + 3266669866q^{52} + 7818635688q^{53} + 17402728558q^{54} - 191002682q^{55} + 11263587512q^{56} + 15495358340q^{57} - 656356768q^{58} + 1230002712q^{59} + 31834046430q^{60} - 18602654230q^{61} + 22075953162q^{62} - 9964531456q^{63} + 11813658086q^{64} + 32245789334q^{65} + 42677188354q^{66} + 27481284652q^{67} + 29588811820q^{68} - 20565315068q^{69} + 42862666712q^{70} - 20347168516q^{71} + 47061083616q^{72} - 57740010478q^{73} - 2640709564q^{74} - 23544691000q^{75} - 33350650772q^{76} + 871959792q^{77} - 15384525342q^{78} - 120245016462q^{79} - 84319695274q^{80} - 48880047865q^{81} - 111495532412q^{82} - 142463983824q^{83} - 134146226376q^{84} - 181628566552q^{85} + 47870165542q^{86} - 20141948318q^{87} - 180608014462q^{88} - 96700717270q^{89} - 25522461244q^{90} - 355162031176q^{91} - 22429477796q^{92} - 172582115142q^{93} + 172608565078q^{94} - 195922150708q^{95} + 226391047758q^{96} - 303190852014q^{97} - 123776497136q^{98} - 139125462440q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −23.7036 −0.523781 −0.261890 0.965098i \(-0.584346\pi\)
−0.261890 + 0.965098i \(0.584346\pi\)
\(3\) −804.018 −1.91029 −0.955144 0.296141i \(-0.904300\pi\)
−0.955144 + 0.296141i \(0.904300\pi\)
\(4\) −1486.14 −0.725654
\(5\) 6643.74 0.950775 0.475388 0.879776i \(-0.342308\pi\)
0.475388 + 0.879776i \(0.342308\pi\)
\(6\) 19058.1 1.00057
\(7\) −61145.1 −1.37506 −0.687531 0.726155i \(-0.741306\pi\)
−0.687531 + 0.726155i \(0.741306\pi\)
\(8\) 83771.9 0.903864
\(9\) 469298. 2.64920
\(10\) −157481. −0.497998
\(11\) 97159.7 0.181897 0.0909487 0.995856i \(-0.471010\pi\)
0.0909487 + 0.995856i \(0.471010\pi\)
\(12\) 1.19488e6 1.38621
\(13\) 2.25195e6 1.68217 0.841086 0.540901i \(-0.181917\pi\)
0.841086 + 0.540901i \(0.181917\pi\)
\(14\) 1.44936e6 0.720231
\(15\) −5.34169e6 −1.81626
\(16\) 1.05792e6 0.252227
\(17\) −5.47274e6 −0.934837 −0.467418 0.884036i \(-0.654816\pi\)
−0.467418 + 0.884036i \(0.654816\pi\)
\(18\) −1.11241e7 −1.38760
\(19\) −1.36844e7 −1.26789 −0.633944 0.773379i \(-0.718565\pi\)
−0.633944 + 0.773379i \(0.718565\pi\)
\(20\) −9.87353e6 −0.689934
\(21\) 4.91618e7 2.62677
\(22\) −2.30304e6 −0.0952743
\(23\) 3.78724e7 1.22693 0.613464 0.789722i \(-0.289775\pi\)
0.613464 + 0.789722i \(0.289775\pi\)
\(24\) −6.73541e7 −1.72664
\(25\) −4.68878e6 −0.0960263
\(26\) −5.33794e7 −0.881090
\(27\) −2.34895e8 −3.15045
\(28\) 9.08701e7 0.997819
\(29\) 2.05111e7 0.185695
\(30\) 1.26617e8 0.951320
\(31\) 7.66307e7 0.480744 0.240372 0.970681i \(-0.422731\pi\)
0.240372 + 0.970681i \(0.422731\pi\)
\(32\) −1.96641e8 −1.03598
\(33\) −7.81182e7 −0.347476
\(34\) 1.29724e8 0.489649
\(35\) −4.06232e8 −1.30738
\(36\) −6.97442e8 −1.92240
\(37\) 5.87994e8 1.39400 0.697001 0.717070i \(-0.254517\pi\)
0.697001 + 0.717070i \(0.254517\pi\)
\(38\) 3.24370e8 0.664095
\(39\) −1.81061e9 −3.21344
\(40\) 5.56559e8 0.859372
\(41\) −5.25000e8 −0.707699 −0.353849 0.935302i \(-0.615127\pi\)
−0.353849 + 0.935302i \(0.615127\pi\)
\(42\) −1.16531e9 −1.37585
\(43\) 1.40657e9 1.45910 0.729550 0.683927i \(-0.239729\pi\)
0.729550 + 0.683927i \(0.239729\pi\)
\(44\) −1.44393e8 −0.131994
\(45\) 3.11790e9 2.51880
\(46\) −8.97712e8 −0.642642
\(47\) −1.05859e9 −0.673273 −0.336636 0.941635i \(-0.609289\pi\)
−0.336636 + 0.941635i \(0.609289\pi\)
\(48\) −8.50585e8 −0.481826
\(49\) 1.76140e9 0.890796
\(50\) 1.11141e8 0.0502967
\(51\) 4.40018e9 1.78581
\(52\) −3.34671e9 −1.22067
\(53\) 1.82460e8 0.0599307 0.0299654 0.999551i \(-0.490460\pi\)
0.0299654 + 0.999551i \(0.490460\pi\)
\(54\) 5.56786e9 1.65015
\(55\) 6.45504e8 0.172943
\(56\) −5.12224e9 −1.24287
\(57\) 1.10025e10 2.42203
\(58\) −4.86188e8 −0.0972636
\(59\) −4.81226e9 −0.876321 −0.438160 0.898897i \(-0.644370\pi\)
−0.438160 + 0.898897i \(0.644370\pi\)
\(60\) 7.93850e9 1.31797
\(61\) 1.42171e9 0.215524 0.107762 0.994177i \(-0.465632\pi\)
0.107762 + 0.994177i \(0.465632\pi\)
\(62\) −1.81643e9 −0.251804
\(63\) −2.86953e10 −3.64282
\(64\) 2.49449e9 0.290397
\(65\) 1.49614e10 1.59937
\(66\) 1.85168e9 0.182001
\(67\) 1.58194e9 0.143146 0.0715728 0.997435i \(-0.477198\pi\)
0.0715728 + 0.997435i \(0.477198\pi\)
\(68\) 8.13325e9 0.678368
\(69\) −3.04501e10 −2.34379
\(70\) 9.62917e9 0.684778
\(71\) −7.93869e9 −0.522189 −0.261095 0.965313i \(-0.584083\pi\)
−0.261095 + 0.965313i \(0.584083\pi\)
\(72\) 3.93140e10 2.39452
\(73\) −2.06263e10 −1.16452 −0.582258 0.813004i \(-0.697831\pi\)
−0.582258 + 0.813004i \(0.697831\pi\)
\(74\) −1.39376e10 −0.730151
\(75\) 3.76987e9 0.183438
\(76\) 2.03369e10 0.920048
\(77\) −5.94084e9 −0.250120
\(78\) 4.29180e10 1.68314
\(79\) −3.33041e10 −1.21772 −0.608862 0.793276i \(-0.708374\pi\)
−0.608862 + 0.793276i \(0.708374\pi\)
\(80\) 7.02853e9 0.239811
\(81\) 1.05725e11 3.36907
\(82\) 1.24444e10 0.370679
\(83\) −4.80488e9 −0.133891 −0.0669457 0.997757i \(-0.521325\pi\)
−0.0669457 + 0.997757i \(0.521325\pi\)
\(84\) −7.30612e10 −1.90612
\(85\) −3.63595e10 −0.888820
\(86\) −3.33408e10 −0.764249
\(87\) −1.64913e10 −0.354732
\(88\) 8.13925e9 0.164410
\(89\) −3.03829e10 −0.576745 −0.288373 0.957518i \(-0.593114\pi\)
−0.288373 + 0.957518i \(0.593114\pi\)
\(90\) −7.39054e10 −1.31930
\(91\) −1.37696e11 −2.31309
\(92\) −5.62836e10 −0.890325
\(93\) −6.16125e10 −0.918360
\(94\) 2.50925e10 0.352647
\(95\) −9.09157e10 −1.20548
\(96\) 1.58103e11 1.97901
\(97\) −2.92126e10 −0.345402 −0.172701 0.984974i \(-0.555249\pi\)
−0.172701 + 0.984974i \(0.555249\pi\)
\(98\) −4.17514e10 −0.466582
\(99\) 4.55969e10 0.481883
\(100\) 6.96818e9 0.0696818
\(101\) −1.67043e9 −0.0158147 −0.00790734 0.999969i \(-0.502517\pi\)
−0.00790734 + 0.999969i \(0.502517\pi\)
\(102\) −1.04300e11 −0.935372
\(103\) 1.11501e11 0.947706 0.473853 0.880604i \(-0.342863\pi\)
0.473853 + 0.880604i \(0.342863\pi\)
\(104\) 1.88650e11 1.52046
\(105\) 3.26618e11 2.49746
\(106\) −4.32495e9 −0.0313906
\(107\) 1.37541e11 0.948031 0.474015 0.880517i \(-0.342804\pi\)
0.474015 + 0.880517i \(0.342804\pi\)
\(108\) 3.49087e11 2.28614
\(109\) −2.80903e11 −1.74868 −0.874340 0.485314i \(-0.838705\pi\)
−0.874340 + 0.485314i \(0.838705\pi\)
\(110\) −1.53008e10 −0.0905845
\(111\) −4.72758e11 −2.66295
\(112\) −6.46864e10 −0.346828
\(113\) 1.69957e10 0.0867775 0.0433887 0.999058i \(-0.486185\pi\)
0.0433887 + 0.999058i \(0.486185\pi\)
\(114\) −2.60799e11 −1.26861
\(115\) 2.51614e11 1.16653
\(116\) −3.04824e10 −0.134751
\(117\) 1.05684e12 4.45642
\(118\) 1.14068e11 0.459000
\(119\) 3.34631e11 1.28546
\(120\) −4.47483e11 −1.64165
\(121\) −2.75872e11 −0.966913
\(122\) −3.36996e10 −0.112887
\(123\) 4.22110e11 1.35191
\(124\) −1.13884e11 −0.348854
\(125\) −3.55553e11 −1.04207
\(126\) 6.80182e11 1.90804
\(127\) −3.22954e11 −0.867402 −0.433701 0.901057i \(-0.642792\pi\)
−0.433701 + 0.901057i \(0.642792\pi\)
\(128\) 3.43593e11 0.883871
\(129\) −1.13091e12 −2.78730
\(130\) −3.54639e11 −0.837718
\(131\) −1.19707e11 −0.271099 −0.135549 0.990771i \(-0.543280\pi\)
−0.135549 + 0.990771i \(0.543280\pi\)
\(132\) 1.16094e11 0.252148
\(133\) 8.36734e11 1.74343
\(134\) −3.74976e10 −0.0749769
\(135\) −1.56058e12 −2.99537
\(136\) −4.58462e11 −0.844965
\(137\) 1.15117e10 0.0203787 0.0101894 0.999948i \(-0.496757\pi\)
0.0101894 + 0.999948i \(0.496757\pi\)
\(138\) 7.21777e11 1.22763
\(139\) −1.29449e11 −0.211600 −0.105800 0.994387i \(-0.533740\pi\)
−0.105800 + 0.994387i \(0.533740\pi\)
\(140\) 6.03718e11 0.948702
\(141\) 8.51128e11 1.28614
\(142\) 1.88176e11 0.273513
\(143\) 2.18799e11 0.305983
\(144\) 4.96479e11 0.668201
\(145\) 1.36271e11 0.176555
\(146\) 4.88917e11 0.609951
\(147\) −1.41619e12 −1.70168
\(148\) −8.73841e11 −1.01156
\(149\) 1.04551e12 1.16628 0.583139 0.812372i \(-0.301824\pi\)
0.583139 + 0.812372i \(0.301824\pi\)
\(150\) −8.93595e10 −0.0960812
\(151\) −4.17782e11 −0.433088 −0.216544 0.976273i \(-0.569479\pi\)
−0.216544 + 0.976273i \(0.569479\pi\)
\(152\) −1.14637e12 −1.14600
\(153\) −2.56835e12 −2.47657
\(154\) 1.40819e11 0.131008
\(155\) 5.09115e11 0.457079
\(156\) 2.69082e12 2.33184
\(157\) −1.93394e11 −0.161806 −0.0809030 0.996722i \(-0.525780\pi\)
−0.0809030 + 0.996722i \(0.525780\pi\)
\(158\) 7.89427e11 0.637820
\(159\) −1.46701e11 −0.114485
\(160\) −1.30643e12 −0.984980
\(161\) −2.31571e12 −1.68710
\(162\) −2.50607e12 −1.76466
\(163\) −6.42171e11 −0.437138 −0.218569 0.975821i \(-0.570139\pi\)
−0.218569 + 0.975821i \(0.570139\pi\)
\(164\) 7.80223e11 0.513544
\(165\) −5.18997e11 −0.330372
\(166\) 1.13893e11 0.0701297
\(167\) −3.12229e11 −0.186008 −0.0930042 0.995666i \(-0.529647\pi\)
−0.0930042 + 0.995666i \(0.529647\pi\)
\(168\) 4.11837e12 2.37424
\(169\) 3.27912e12 1.82970
\(170\) 8.61851e11 0.465547
\(171\) −6.42207e12 −3.35889
\(172\) −2.09036e12 −1.05880
\(173\) 9.22108e11 0.452406 0.226203 0.974080i \(-0.427369\pi\)
0.226203 + 0.974080i \(0.427369\pi\)
\(174\) 3.90904e11 0.185802
\(175\) 2.86696e11 0.132042
\(176\) 1.02787e11 0.0458794
\(177\) 3.86915e12 1.67403
\(178\) 7.20184e11 0.302088
\(179\) −3.84140e11 −0.156242 −0.0781210 0.996944i \(-0.524892\pi\)
−0.0781210 + 0.996944i \(0.524892\pi\)
\(180\) −4.63363e12 −1.82777
\(181\) −3.67123e11 −0.140469 −0.0702344 0.997531i \(-0.522375\pi\)
−0.0702344 + 0.997531i \(0.522375\pi\)
\(182\) 3.26389e12 1.21155
\(183\) −1.14308e12 −0.411713
\(184\) 3.17264e12 1.10898
\(185\) 3.90648e12 1.32538
\(186\) 1.46044e12 0.481019
\(187\) −5.31730e11 −0.170044
\(188\) 1.57322e12 0.488563
\(189\) 1.43627e13 4.33207
\(190\) 2.15503e12 0.631406
\(191\) −2.26967e11 −0.0646070 −0.0323035 0.999478i \(-0.510284\pi\)
−0.0323035 + 0.999478i \(0.510284\pi\)
\(192\) −2.00562e12 −0.554742
\(193\) 1.51356e12 0.406850 0.203425 0.979091i \(-0.434793\pi\)
0.203425 + 0.979091i \(0.434793\pi\)
\(194\) 6.92443e11 0.180915
\(195\) −1.20292e13 −3.05526
\(196\) −2.61768e12 −0.646410
\(197\) −7.38522e12 −1.77337 −0.886684 0.462375i \(-0.846997\pi\)
−0.886684 + 0.462375i \(0.846997\pi\)
\(198\) −1.08081e12 −0.252401
\(199\) 3.16497e12 0.718915 0.359458 0.933161i \(-0.382962\pi\)
0.359458 + 0.933161i \(0.382962\pi\)
\(200\) −3.92788e11 −0.0867947
\(201\) −1.27191e12 −0.273449
\(202\) 3.95952e10 0.00828342
\(203\) −1.25416e12 −0.255343
\(204\) −6.53928e12 −1.29588
\(205\) −3.48797e12 −0.672862
\(206\) −2.64297e12 −0.496390
\(207\) 1.77734e13 3.25038
\(208\) 2.38238e12 0.424289
\(209\) −1.32957e12 −0.230625
\(210\) −7.74203e12 −1.30812
\(211\) 4.52814e12 0.745360 0.372680 0.927960i \(-0.378439\pi\)
0.372680 + 0.927960i \(0.378439\pi\)
\(212\) −2.71160e11 −0.0434890
\(213\) 6.38285e12 0.997532
\(214\) −3.26023e12 −0.496560
\(215\) 9.34490e12 1.38728
\(216\) −1.96776e13 −2.84758
\(217\) −4.68559e12 −0.661053
\(218\) 6.65841e12 0.915925
\(219\) 1.65839e13 2.22456
\(220\) −9.59309e11 −0.125497
\(221\) −1.23243e13 −1.57256
\(222\) 1.12061e13 1.39480
\(223\) −1.19134e13 −1.44664 −0.723319 0.690514i \(-0.757384\pi\)
−0.723319 + 0.690514i \(0.757384\pi\)
\(224\) 1.20236e13 1.42453
\(225\) −2.20044e12 −0.254393
\(226\) −4.02859e11 −0.0454524
\(227\) 1.51576e13 1.66913 0.834564 0.550912i \(-0.185720\pi\)
0.834564 + 0.550912i \(0.185720\pi\)
\(228\) −1.63513e13 −1.75756
\(229\) −9.21835e12 −0.967293 −0.483646 0.875264i \(-0.660688\pi\)
−0.483646 + 0.875264i \(0.660688\pi\)
\(230\) −5.96417e12 −0.611008
\(231\) 4.77654e12 0.477802
\(232\) 1.71826e12 0.167843
\(233\) 1.81932e11 0.0173561 0.00867804 0.999962i \(-0.497238\pi\)
0.00867804 + 0.999962i \(0.497238\pi\)
\(234\) −2.50509e13 −2.33419
\(235\) −7.03302e12 −0.640131
\(236\) 7.15169e12 0.635905
\(237\) 2.67771e13 2.32620
\(238\) −7.93197e12 −0.673298
\(239\) 6.71911e12 0.557344 0.278672 0.960386i \(-0.410106\pi\)
0.278672 + 0.960386i \(0.410106\pi\)
\(240\) −5.65107e12 −0.458109
\(241\) 7.28547e12 0.577250 0.288625 0.957442i \(-0.406802\pi\)
0.288625 + 0.957442i \(0.406802\pi\)
\(242\) 6.53915e12 0.506451
\(243\) −4.33939e13 −3.28545
\(244\) −2.11285e12 −0.156396
\(245\) 1.17023e13 0.846947
\(246\) −1.00055e13 −0.708104
\(247\) −3.08166e13 −2.13281
\(248\) 6.41950e12 0.434527
\(249\) 3.86321e12 0.255771
\(250\) 8.42788e12 0.545819
\(251\) −1.88972e13 −1.19727 −0.598634 0.801023i \(-0.704290\pi\)
−0.598634 + 0.801023i \(0.704290\pi\)
\(252\) 4.26452e13 2.64343
\(253\) 3.67967e12 0.223175
\(254\) 7.65518e12 0.454328
\(255\) 2.92337e13 1.69790
\(256\) −1.32531e13 −0.753352
\(257\) −3.10860e13 −1.72955 −0.864773 0.502163i \(-0.832538\pi\)
−0.864773 + 0.502163i \(0.832538\pi\)
\(258\) 2.68066e13 1.45994
\(259\) −3.59529e13 −1.91684
\(260\) −2.22347e13 −1.16059
\(261\) 9.62585e12 0.491945
\(262\) 2.83749e12 0.141996
\(263\) −1.64154e12 −0.0804440 −0.0402220 0.999191i \(-0.512807\pi\)
−0.0402220 + 0.999191i \(0.512807\pi\)
\(264\) −6.54410e12 −0.314071
\(265\) 1.21222e12 0.0569807
\(266\) −1.98336e13 −0.913173
\(267\) 2.44284e13 1.10175
\(268\) −2.35098e12 −0.103874
\(269\) −4.31521e13 −1.86794 −0.933972 0.357346i \(-0.883682\pi\)
−0.933972 + 0.357346i \(0.883682\pi\)
\(270\) 3.69914e13 1.56892
\(271\) −2.74015e12 −0.113879 −0.0569395 0.998378i \(-0.518134\pi\)
−0.0569395 + 0.998378i \(0.518134\pi\)
\(272\) −5.78970e12 −0.235791
\(273\) 1.10710e14 4.41867
\(274\) −2.72869e11 −0.0106740
\(275\) −4.55561e11 −0.0174669
\(276\) 4.52530e13 1.70078
\(277\) 1.99734e13 0.735892 0.367946 0.929847i \(-0.380061\pi\)
0.367946 + 0.929847i \(0.380061\pi\)
\(278\) 3.06840e12 0.110832
\(279\) 3.59627e13 1.27359
\(280\) −3.40308e13 −1.18169
\(281\) −5.50393e13 −1.87408 −0.937040 0.349221i \(-0.886446\pi\)
−0.937040 + 0.349221i \(0.886446\pi\)
\(282\) −2.01748e13 −0.673658
\(283\) 7.72019e12 0.252815 0.126407 0.991978i \(-0.459655\pi\)
0.126407 + 0.991978i \(0.459655\pi\)
\(284\) 1.17980e13 0.378928
\(285\) 7.30979e13 2.30281
\(286\) −5.18632e12 −0.160268
\(287\) 3.21012e13 0.973130
\(288\) −9.22834e13 −2.74451
\(289\) −4.32102e12 −0.126081
\(290\) −3.23011e12 −0.0924759
\(291\) 2.34874e13 0.659818
\(292\) 3.06535e13 0.845035
\(293\) 1.62367e13 0.439264 0.219632 0.975583i \(-0.429514\pi\)
0.219632 + 0.975583i \(0.429514\pi\)
\(294\) 3.35689e13 0.891306
\(295\) −3.19714e13 −0.833184
\(296\) 4.92573e13 1.25999
\(297\) −2.28223e13 −0.573059
\(298\) −2.47823e13 −0.610874
\(299\) 8.52867e13 2.06391
\(300\) −5.60255e12 −0.133112
\(301\) −8.60049e13 −2.00635
\(302\) 9.90294e12 0.226843
\(303\) 1.34306e12 0.0302106
\(304\) −1.44770e13 −0.319796
\(305\) 9.44545e12 0.204915
\(306\) 6.08791e13 1.29718
\(307\) 4.14278e13 0.867024 0.433512 0.901148i \(-0.357274\pi\)
0.433512 + 0.901148i \(0.357274\pi\)
\(308\) 8.82891e12 0.181501
\(309\) −8.96488e13 −1.81039
\(310\) −1.20679e13 −0.239409
\(311\) −6.13636e13 −1.19599 −0.597996 0.801499i \(-0.704036\pi\)
−0.597996 + 0.801499i \(0.704036\pi\)
\(312\) −1.51678e14 −2.90451
\(313\) −4.27555e13 −0.804448 −0.402224 0.915541i \(-0.631763\pi\)
−0.402224 + 0.915541i \(0.631763\pi\)
\(314\) 4.58413e12 0.0847508
\(315\) −1.90644e14 −3.46350
\(316\) 4.94945e13 0.883645
\(317\) −8.82253e13 −1.54799 −0.773993 0.633194i \(-0.781744\pi\)
−0.773993 + 0.633194i \(0.781744\pi\)
\(318\) 3.47734e12 0.0599650
\(319\) 1.99286e12 0.0337775
\(320\) 1.65728e13 0.276102
\(321\) −1.10586e14 −1.81101
\(322\) 5.48907e13 0.883672
\(323\) 7.48912e13 1.18527
\(324\) −1.57122e14 −2.44478
\(325\) −1.05589e13 −0.161533
\(326\) 1.52218e13 0.228964
\(327\) 2.25851e14 3.34048
\(328\) −4.39803e13 −0.639664
\(329\) 6.47278e13 0.925792
\(330\) 1.23021e13 0.173042
\(331\) −3.28413e13 −0.454325 −0.227163 0.973857i \(-0.572945\pi\)
−0.227163 + 0.973857i \(0.572945\pi\)
\(332\) 7.14071e12 0.0971588
\(333\) 2.75945e14 3.69299
\(334\) 7.40096e12 0.0974277
\(335\) 1.05100e13 0.136099
\(336\) 5.20091e13 0.662541
\(337\) −9.63718e13 −1.20777 −0.603887 0.797070i \(-0.706382\pi\)
−0.603887 + 0.797070i \(0.706382\pi\)
\(338\) −7.77271e13 −0.958364
\(339\) −1.36648e13 −0.165770
\(340\) 5.40352e13 0.644975
\(341\) 7.44542e12 0.0874460
\(342\) 1.52226e14 1.75932
\(343\) 1.32031e13 0.150162
\(344\) 1.17831e14 1.31883
\(345\) −2.02303e14 −2.22842
\(346\) −2.18573e13 −0.236962
\(347\) 5.46073e12 0.0582691 0.0291346 0.999575i \(-0.490725\pi\)
0.0291346 + 0.999575i \(0.490725\pi\)
\(348\) 2.45084e13 0.257412
\(349\) 1.74524e14 1.80433 0.902163 0.431395i \(-0.141979\pi\)
0.902163 + 0.431395i \(0.141979\pi\)
\(350\) −6.79573e12 −0.0691611
\(351\) −5.28972e14 −5.29961
\(352\) −1.91056e13 −0.188441
\(353\) 1.76562e14 1.71449 0.857246 0.514908i \(-0.172174\pi\)
0.857246 + 0.514908i \(0.172174\pi\)
\(354\) −9.17127e13 −0.876822
\(355\) −5.27426e13 −0.496484
\(356\) 4.51532e13 0.418517
\(357\) −2.69050e14 −2.45560
\(358\) 9.10551e12 0.0818366
\(359\) 1.66347e14 1.47230 0.736149 0.676820i \(-0.236642\pi\)
0.736149 + 0.676820i \(0.236642\pi\)
\(360\) 2.61192e14 2.27665
\(361\) 7.07726e13 0.607541
\(362\) 8.70215e12 0.0735748
\(363\) 2.21806e14 1.84708
\(364\) 2.04635e14 1.67850
\(365\) −1.37036e14 −1.10719
\(366\) 2.70951e13 0.215647
\(367\) −1.94799e14 −1.52730 −0.763648 0.645632i \(-0.776594\pi\)
−0.763648 + 0.645632i \(0.776594\pi\)
\(368\) 4.00658e13 0.309465
\(369\) −2.46382e14 −1.87484
\(370\) −9.25977e13 −0.694210
\(371\) −1.11565e13 −0.0824085
\(372\) 9.15648e13 0.666411
\(373\) −1.50997e14 −1.08286 −0.541428 0.840747i \(-0.682116\pi\)
−0.541428 + 0.840747i \(0.682116\pi\)
\(374\) 1.26039e13 0.0890659
\(375\) 2.85871e14 1.99066
\(376\) −8.86803e13 −0.608547
\(377\) 4.61901e13 0.312372
\(378\) −3.40447e14 −2.26905
\(379\) −1.39626e14 −0.917173 −0.458586 0.888650i \(-0.651644\pi\)
−0.458586 + 0.888650i \(0.651644\pi\)
\(380\) 1.35113e14 0.874759
\(381\) 2.59661e14 1.65699
\(382\) 5.37994e12 0.0338399
\(383\) 6.01897e13 0.373189 0.186595 0.982437i \(-0.440255\pi\)
0.186595 + 0.982437i \(0.440255\pi\)
\(384\) −2.76255e14 −1.68845
\(385\) −3.94694e13 −0.237808
\(386\) −3.58768e13 −0.213100
\(387\) 6.60102e14 3.86545
\(388\) 4.34139e13 0.250642
\(389\) −2.30214e14 −1.31042 −0.655208 0.755449i \(-0.727419\pi\)
−0.655208 + 0.755449i \(0.727419\pi\)
\(390\) 2.85136e14 1.60028
\(391\) −2.07266e14 −1.14698
\(392\) 1.47555e14 0.805159
\(393\) 9.62466e13 0.517876
\(394\) 1.75056e14 0.928856
\(395\) −2.21264e14 −1.15778
\(396\) −6.77633e13 −0.349680
\(397\) 2.16304e14 1.10082 0.550411 0.834894i \(-0.314471\pi\)
0.550411 + 0.834894i \(0.314471\pi\)
\(398\) −7.50212e13 −0.376554
\(399\) −6.72749e14 −3.33045
\(400\) −4.96034e12 −0.0242204
\(401\) 2.59235e14 1.24853 0.624265 0.781212i \(-0.285398\pi\)
0.624265 + 0.781212i \(0.285398\pi\)
\(402\) 3.01488e13 0.143228
\(403\) 1.72569e14 0.808694
\(404\) 2.48249e12 0.0114760
\(405\) 7.02410e14 3.20323
\(406\) 2.97280e13 0.133744
\(407\) 5.71293e13 0.253565
\(408\) 3.68611e14 1.61413
\(409\) 1.21534e14 0.525071 0.262536 0.964922i \(-0.415441\pi\)
0.262536 + 0.964922i \(0.415441\pi\)
\(410\) 8.26774e13 0.352432
\(411\) −9.25563e12 −0.0389292
\(412\) −1.65706e14 −0.687706
\(413\) 2.94246e14 1.20500
\(414\) −4.21295e14 −1.70249
\(415\) −3.19224e13 −0.127301
\(416\) −4.42826e14 −1.74269
\(417\) 1.04079e14 0.404217
\(418\) 3.15157e13 0.120797
\(419\) −3.00046e14 −1.13504 −0.567519 0.823360i \(-0.692097\pi\)
−0.567519 + 0.823360i \(0.692097\pi\)
\(420\) −4.85400e14 −1.81229
\(421\) −6.52194e11 −0.00240340 −0.00120170 0.999999i \(-0.500383\pi\)
−0.00120170 + 0.999999i \(0.500383\pi\)
\(422\) −1.07333e14 −0.390405
\(423\) −4.96796e14 −1.78364
\(424\) 1.52850e13 0.0541693
\(425\) 2.56605e13 0.0897689
\(426\) −1.51297e14 −0.522488
\(427\) −8.69303e13 −0.296359
\(428\) −2.04405e14 −0.687942
\(429\) −1.75918e14 −0.584515
\(430\) −2.21508e14 −0.726629
\(431\) −7.67212e12 −0.0248479 −0.0124240 0.999923i \(-0.503955\pi\)
−0.0124240 + 0.999923i \(0.503955\pi\)
\(432\) −2.48499e14 −0.794630
\(433\) −3.59957e14 −1.13649 −0.568247 0.822858i \(-0.692378\pi\)
−0.568247 + 0.822858i \(0.692378\pi\)
\(434\) 1.11065e14 0.346247
\(435\) −1.09564e14 −0.337270
\(436\) 4.17460e14 1.26894
\(437\) −5.18261e14 −1.55561
\(438\) −3.93099e14 −1.16518
\(439\) −1.45149e13 −0.0424872 −0.0212436 0.999774i \(-0.506763\pi\)
−0.0212436 + 0.999774i \(0.506763\pi\)
\(440\) 5.40751e13 0.156317
\(441\) 8.26620e14 2.35990
\(442\) 2.92131e14 0.823675
\(443\) 1.97208e14 0.549166 0.274583 0.961563i \(-0.411460\pi\)
0.274583 + 0.961563i \(0.411460\pi\)
\(444\) 7.02584e14 1.93238
\(445\) −2.01856e14 −0.548355
\(446\) 2.82391e14 0.757721
\(447\) −8.40606e14 −2.22793
\(448\) −1.52526e14 −0.399314
\(449\) 3.49615e14 0.904138 0.452069 0.891983i \(-0.350686\pi\)
0.452069 + 0.891983i \(0.350686\pi\)
\(450\) 5.21583e13 0.133246
\(451\) −5.10089e13 −0.128729
\(452\) −2.52579e13 −0.0629704
\(453\) 3.35904e14 0.827323
\(454\) −3.59291e14 −0.874257
\(455\) −9.14816e14 −2.19923
\(456\) 9.21700e14 2.18919
\(457\) 7.08412e14 1.66244 0.831222 0.555940i \(-0.187642\pi\)
0.831222 + 0.555940i \(0.187642\pi\)
\(458\) 2.18508e14 0.506649
\(459\) 1.28552e15 2.94516
\(460\) −3.73934e14 −0.846499
\(461\) 9.39784e13 0.210220 0.105110 0.994461i \(-0.466481\pi\)
0.105110 + 0.994461i \(0.466481\pi\)
\(462\) −1.13221e14 −0.250263
\(463\) −2.32182e14 −0.507146 −0.253573 0.967316i \(-0.581606\pi\)
−0.253573 + 0.967316i \(0.581606\pi\)
\(464\) 2.16991e13 0.0468374
\(465\) −4.09338e14 −0.873154
\(466\) −4.31245e12 −0.00909078
\(467\) 3.22971e14 0.672855 0.336427 0.941709i \(-0.390781\pi\)
0.336427 + 0.941709i \(0.390781\pi\)
\(468\) −1.57061e15 −3.23382
\(469\) −9.67277e13 −0.196834
\(470\) 1.66708e14 0.335288
\(471\) 1.55492e14 0.309096
\(472\) −4.03132e14 −0.792075
\(473\) 1.36662e14 0.265407
\(474\) −6.34714e14 −1.21842
\(475\) 6.41632e13 0.121751
\(476\) −4.97308e14 −0.932798
\(477\) 8.56280e13 0.158769
\(478\) −1.59267e14 −0.291926
\(479\) −1.04417e15 −1.89202 −0.946008 0.324142i \(-0.894924\pi\)
−0.946008 + 0.324142i \(0.894924\pi\)
\(480\) 1.05040e15 1.88160
\(481\) 1.32413e15 2.34495
\(482\) −1.72692e14 −0.302352
\(483\) 1.86187e15 3.22285
\(484\) 4.09984e14 0.701644
\(485\) −1.94081e14 −0.328400
\(486\) 1.02859e15 1.72085
\(487\) −8.32567e14 −1.37724 −0.688620 0.725122i \(-0.741783\pi\)
−0.688620 + 0.725122i \(0.741783\pi\)
\(488\) 1.19099e14 0.194804
\(489\) 5.16317e14 0.835060
\(490\) −2.77386e14 −0.443615
\(491\) −8.03776e14 −1.27112 −0.635560 0.772051i \(-0.719231\pi\)
−0.635560 + 0.772051i \(0.719231\pi\)
\(492\) −6.27314e14 −0.981018
\(493\) −1.12252e14 −0.173595
\(494\) 7.30465e14 1.11712
\(495\) 3.02934e14 0.458162
\(496\) 8.10690e13 0.121257
\(497\) 4.85412e14 0.718042
\(498\) −9.15720e13 −0.133968
\(499\) −4.84727e13 −0.0701365 −0.0350682 0.999385i \(-0.511165\pi\)
−0.0350682 + 0.999385i \(0.511165\pi\)
\(500\) 5.28401e14 0.756185
\(501\) 2.51038e14 0.355330
\(502\) 4.47931e14 0.627106
\(503\) −7.87706e13 −0.109079 −0.0545394 0.998512i \(-0.517369\pi\)
−0.0545394 + 0.998512i \(0.517369\pi\)
\(504\) −2.40386e15 −3.29261
\(505\) −1.10979e13 −0.0150362
\(506\) −8.72214e13 −0.116895
\(507\) −2.63648e15 −3.49526
\(508\) 4.79954e14 0.629433
\(509\) −1.16339e15 −1.50931 −0.754654 0.656123i \(-0.772195\pi\)
−0.754654 + 0.656123i \(0.772195\pi\)
\(510\) −6.92944e14 −0.889328
\(511\) 1.26120e15 1.60128
\(512\) −3.89531e14 −0.489280
\(513\) 3.21440e15 3.99442
\(514\) 7.36849e14 0.905903
\(515\) 7.40784e14 0.901056
\(516\) 1.68069e15 2.02262
\(517\) −1.02853e14 −0.122466
\(518\) 8.52215e14 1.00400
\(519\) −7.41392e14 −0.864226
\(520\) 1.25334e15 1.44561
\(521\) 3.02892e14 0.345684 0.172842 0.984950i \(-0.444705\pi\)
0.172842 + 0.984950i \(0.444705\pi\)
\(522\) −2.28167e14 −0.257671
\(523\) −9.73299e14 −1.08764 −0.543822 0.839200i \(-0.683023\pi\)
−0.543822 + 0.839200i \(0.683023\pi\)
\(524\) 1.77901e14 0.196724
\(525\) −2.30509e14 −0.252239
\(526\) 3.89103e13 0.0421350
\(527\) −4.19380e14 −0.449417
\(528\) −8.26425e13 −0.0876430
\(529\) 4.81506e14 0.505354
\(530\) −2.87339e13 −0.0298454
\(531\) −2.25839e15 −2.32155
\(532\) −1.24350e15 −1.26512
\(533\) −1.18228e15 −1.19047
\(534\) −5.79041e14 −0.577075
\(535\) 9.13789e14 0.901364
\(536\) 1.32522e14 0.129384
\(537\) 3.08856e14 0.298468
\(538\) 1.02286e15 0.978393
\(539\) 1.71137e14 0.162033
\(540\) 2.31924e15 2.17360
\(541\) −6.67127e14 −0.618905 −0.309452 0.950915i \(-0.600146\pi\)
−0.309452 + 0.950915i \(0.600146\pi\)
\(542\) 6.49516e13 0.0596477
\(543\) 2.95174e14 0.268336
\(544\) 1.07617e15 0.968468
\(545\) −1.86625e15 −1.66260
\(546\) −2.62422e15 −2.31442
\(547\) 1.44083e15 1.25801 0.629005 0.777402i \(-0.283463\pi\)
0.629005 + 0.777402i \(0.283463\pi\)
\(548\) −1.71080e13 −0.0147879
\(549\) 6.67204e14 0.570967
\(550\) 1.07984e13 0.00914884
\(551\) −2.80683e14 −0.235441
\(552\) −2.55086e15 −2.11847
\(553\) 2.03638e15 1.67445
\(554\) −4.73443e14 −0.385446
\(555\) −3.14088e15 −2.53186
\(556\) 1.92379e14 0.153548
\(557\) 7.07734e14 0.559328 0.279664 0.960098i \(-0.409777\pi\)
0.279664 + 0.960098i \(0.409777\pi\)
\(558\) −8.52445e14 −0.667081
\(559\) 3.16753e15 2.45446
\(560\) −4.29760e14 −0.329755
\(561\) 4.27520e14 0.324834
\(562\) 1.30463e15 0.981607
\(563\) 1.93114e14 0.143886 0.0719428 0.997409i \(-0.477080\pi\)
0.0719428 + 0.997409i \(0.477080\pi\)
\(564\) −1.26489e15 −0.933296
\(565\) 1.12915e14 0.0825059
\(566\) −1.82996e14 −0.132420
\(567\) −6.46457e15 −4.63268
\(568\) −6.65038e14 −0.471988
\(569\) 1.78850e13 0.0125711 0.00628554 0.999980i \(-0.497999\pi\)
0.00628554 + 0.999980i \(0.497999\pi\)
\(570\) −1.73268e15 −1.20617
\(571\) −2.69139e15 −1.85557 −0.927786 0.373112i \(-0.878291\pi\)
−0.927786 + 0.373112i \(0.878291\pi\)
\(572\) −3.25166e14 −0.222038
\(573\) 1.82486e14 0.123418
\(574\) −7.60914e14 −0.509707
\(575\) −1.77575e14 −0.117817
\(576\) 1.17066e15 0.769321
\(577\) −8.36258e14 −0.544344 −0.272172 0.962249i \(-0.587742\pi\)
−0.272172 + 0.962249i \(0.587742\pi\)
\(578\) 1.02424e14 0.0660386
\(579\) −1.21693e15 −0.777200
\(580\) −2.02517e14 −0.128117
\(581\) 2.93795e14 0.184109
\(582\) −5.56737e14 −0.345600
\(583\) 1.77277e13 0.0109012
\(584\) −1.72790e15 −1.05256
\(585\) 7.02136e15 4.23705
\(586\) −3.84868e14 −0.230078
\(587\) 7.39762e14 0.438109 0.219055 0.975713i \(-0.429703\pi\)
0.219055 + 0.975713i \(0.429703\pi\)
\(588\) 2.10466e15 1.23483
\(589\) −1.04865e15 −0.609530
\(590\) 7.57838e14 0.436406
\(591\) 5.93785e15 3.38765
\(592\) 6.22049e14 0.351605
\(593\) −2.54616e15 −1.42588 −0.712942 0.701223i \(-0.752638\pi\)
−0.712942 + 0.701223i \(0.752638\pi\)
\(594\) 5.40971e14 0.300157
\(595\) 2.22320e15 1.22218
\(596\) −1.55377e15 −0.846314
\(597\) −2.54469e15 −1.37334
\(598\) −2.02160e15 −1.08103
\(599\) 3.54163e15 1.87653 0.938265 0.345917i \(-0.112432\pi\)
0.938265 + 0.345917i \(0.112432\pi\)
\(600\) 3.15809e14 0.165803
\(601\) 2.88090e15 1.49871 0.749357 0.662166i \(-0.230363\pi\)
0.749357 + 0.662166i \(0.230363\pi\)
\(602\) 2.03863e15 1.05089
\(603\) 7.42401e14 0.379222
\(604\) 6.20882e14 0.314272
\(605\) −1.83282e15 −0.919317
\(606\) −3.18353e13 −0.0158237
\(607\) 1.11481e15 0.549116 0.274558 0.961571i \(-0.411468\pi\)
0.274558 + 0.961571i \(0.411468\pi\)
\(608\) 2.69092e15 1.31350
\(609\) 1.00836e15 0.487778
\(610\) −2.23891e14 −0.107330
\(611\) −2.38390e15 −1.13256
\(612\) 3.81692e15 1.79713
\(613\) 3.23230e14 0.150827 0.0754136 0.997152i \(-0.475972\pi\)
0.0754136 + 0.997152i \(0.475972\pi\)
\(614\) −9.81990e14 −0.454131
\(615\) 2.80439e15 1.28536
\(616\) −4.97675e14 −0.226075
\(617\) 3.72116e15 1.67537 0.837684 0.546156i \(-0.183909\pi\)
0.837684 + 0.546156i \(0.183909\pi\)
\(618\) 2.12500e15 0.948249
\(619\) −2.61579e15 −1.15692 −0.578461 0.815710i \(-0.696347\pi\)
−0.578461 + 0.815710i \(0.696347\pi\)
\(620\) −7.56616e14 −0.331681
\(621\) −8.89603e15 −3.86538
\(622\) 1.45454e15 0.626438
\(623\) 1.85776e15 0.793060
\(624\) −1.91548e15 −0.810515
\(625\) −2.13326e15 −0.894753
\(626\) 1.01346e15 0.421354
\(627\) 1.06900e15 0.440561
\(628\) 2.87410e14 0.117415
\(629\) −3.21794e15 −1.30316
\(630\) 4.51896e15 1.81412
\(631\) −3.66804e15 −1.45973 −0.729864 0.683592i \(-0.760417\pi\)
−0.729864 + 0.683592i \(0.760417\pi\)
\(632\) −2.78995e15 −1.10066
\(633\) −3.64070e15 −1.42385
\(634\) 2.09126e15 0.810805
\(635\) −2.14562e15 −0.824704
\(636\) 2.18018e14 0.0830765
\(637\) 3.96658e15 1.49847
\(638\) −4.72379e13 −0.0176920
\(639\) −3.72561e15 −1.38338
\(640\) 2.28274e15 0.840363
\(641\) 1.27860e15 0.466676 0.233338 0.972396i \(-0.425035\pi\)
0.233338 + 0.972396i \(0.425035\pi\)
\(642\) 2.62128e15 0.948573
\(643\) −1.13237e15 −0.406281 −0.203140 0.979150i \(-0.565115\pi\)
−0.203140 + 0.979150i \(0.565115\pi\)
\(644\) 3.44147e15 1.22425
\(645\) −7.51347e15 −2.65010
\(646\) −1.77519e15 −0.620821
\(647\) 4.36102e15 1.51222 0.756109 0.654445i \(-0.227098\pi\)
0.756109 + 0.654445i \(0.227098\pi\)
\(648\) 8.85678e15 3.04518
\(649\) −4.67558e14 −0.159400
\(650\) 2.50284e14 0.0846078
\(651\) 3.76730e15 1.26280
\(652\) 9.54355e14 0.317211
\(653\) 2.45898e13 0.00810460 0.00405230 0.999992i \(-0.498710\pi\)
0.00405230 + 0.999992i \(0.498710\pi\)
\(654\) −5.35348e15 −1.74968
\(655\) −7.95302e14 −0.257754
\(656\) −5.55407e14 −0.178501
\(657\) −9.67988e15 −3.08504
\(658\) −1.53428e15 −0.484912
\(659\) −5.68040e15 −1.78037 −0.890183 0.455603i \(-0.849424\pi\)
−0.890183 + 0.455603i \(0.849424\pi\)
\(660\) 7.71302e14 0.239736
\(661\) −1.05643e15 −0.325637 −0.162818 0.986656i \(-0.552058\pi\)
−0.162818 + 0.986656i \(0.552058\pi\)
\(662\) 7.78458e14 0.237967
\(663\) 9.90900e15 3.00404
\(664\) −4.02513e14 −0.121020
\(665\) 5.55905e15 1.65761
\(666\) −6.54088e15 −1.93432
\(667\) 7.76806e14 0.227835
\(668\) 4.64016e14 0.134978
\(669\) 9.57861e15 2.76350
\(670\) −2.49125e14 −0.0712862
\(671\) 1.38132e14 0.0392032
\(672\) −9.66723e15 −2.72127
\(673\) −4.89396e15 −1.36640 −0.683200 0.730231i \(-0.739412\pi\)
−0.683200 + 0.730231i \(0.739412\pi\)
\(674\) 2.28436e15 0.632609
\(675\) 1.10137e15 0.302526
\(676\) −4.87323e15 −1.32773
\(677\) −5.87421e14 −0.158749 −0.0793746 0.996845i \(-0.525292\pi\)
−0.0793746 + 0.996845i \(0.525292\pi\)
\(678\) 3.23906e14 0.0868271
\(679\) 1.78620e15 0.474950
\(680\) −3.04590e15 −0.803372
\(681\) −1.21870e16 −3.18851
\(682\) −1.76483e14 −0.0458025
\(683\) 2.10799e15 0.542694 0.271347 0.962482i \(-0.412531\pi\)
0.271347 + 0.962482i \(0.412531\pi\)
\(684\) 9.54408e15 2.43739
\(685\) 7.64809e13 0.0193756
\(686\) −3.12962e14 −0.0786519
\(687\) 7.41172e15 1.84781
\(688\) 1.48804e15 0.368025
\(689\) 4.10890e14 0.100814
\(690\) 4.79530e15 1.16720
\(691\) −9.03002e14 −0.218052 −0.109026 0.994039i \(-0.534773\pi\)
−0.109026 + 0.994039i \(0.534773\pi\)
\(692\) −1.37038e15 −0.328290
\(693\) −2.78803e15 −0.662619
\(694\) −1.29439e14 −0.0305203
\(695\) −8.60023e14 −0.201184
\(696\) −1.38151e15 −0.320629
\(697\) 2.87319e15 0.661583
\(698\) −4.13685e15 −0.945071
\(699\) −1.46277e14 −0.0331551
\(700\) −4.26070e14 −0.0958168
\(701\) 8.20073e15 1.82980 0.914900 0.403681i \(-0.132269\pi\)
0.914900 + 0.403681i \(0.132269\pi\)
\(702\) 1.25385e16 2.77583
\(703\) −8.04634e15 −1.76744
\(704\) 2.42364e14 0.0528225
\(705\) 5.65468e15 1.22283
\(706\) −4.18515e15 −0.898018
\(707\) 1.02139e14 0.0217462
\(708\) −5.75009e15 −1.21476
\(709\) −3.41692e15 −0.716277 −0.358139 0.933668i \(-0.616588\pi\)
−0.358139 + 0.933668i \(0.616588\pi\)
\(710\) 1.25019e15 0.260049
\(711\) −1.56296e16 −3.22600
\(712\) −2.54523e15 −0.521299
\(713\) 2.90219e15 0.589838
\(714\) 6.37745e15 1.28619
\(715\) 1.45364e15 0.290921
\(716\) 5.70886e14 0.113378
\(717\) −5.40228e15 −1.06469
\(718\) −3.94303e15 −0.771161
\(719\) 4.14792e15 0.805048 0.402524 0.915409i \(-0.368133\pi\)
0.402524 + 0.915409i \(0.368133\pi\)
\(720\) 3.29848e15 0.635309
\(721\) −6.81773e15 −1.30315
\(722\) −1.67757e15 −0.318218
\(723\) −5.85765e15 −1.10271
\(724\) 5.45596e14 0.101932
\(725\) −9.61723e13 −0.0178316
\(726\) −5.25760e15 −0.967467
\(727\) 8.28661e14 0.151334 0.0756672 0.997133i \(-0.475891\pi\)
0.0756672 + 0.997133i \(0.475891\pi\)
\(728\) −1.15350e16 −2.09072
\(729\) 1.61606e16 2.90708
\(730\) 3.24824e15 0.579926
\(731\) −7.69780e15 −1.36402
\(732\) 1.69877e15 0.298761
\(733\) 2.10145e15 0.366815 0.183408 0.983037i \(-0.441287\pi\)
0.183408 + 0.983037i \(0.441287\pi\)
\(734\) 4.61744e15 0.799969
\(735\) −9.40883e15 −1.61791
\(736\) −7.44727e15 −1.27107
\(737\) 1.53701e14 0.0260378
\(738\) 5.84014e15 0.982004
\(739\) 2.96614e15 0.495049 0.247524 0.968882i \(-0.420383\pi\)
0.247524 + 0.968882i \(0.420383\pi\)
\(740\) −5.80557e15 −0.961769
\(741\) 2.47771e16 4.07428
\(742\) 2.64450e14 0.0431640
\(743\) 7.23073e15 1.17150 0.585751 0.810491i \(-0.300799\pi\)
0.585751 + 0.810491i \(0.300799\pi\)
\(744\) −5.16139e15 −0.830072
\(745\) 6.94607e15 1.10887
\(746\) 3.57918e15 0.567179
\(747\) −2.25492e15 −0.354705
\(748\) 7.90224e14 0.123393
\(749\) −8.40998e15 −1.30360
\(750\) −6.77617e15 −1.04267
\(751\) −2.89034e15 −0.441499 −0.220749 0.975331i \(-0.570850\pi\)
−0.220749 + 0.975331i \(0.570850\pi\)
\(752\) −1.11990e15 −0.169818
\(753\) 1.51937e16 2.28713
\(754\) −1.09487e15 −0.163614
\(755\) −2.77564e15 −0.411770
\(756\) −2.13449e16 −3.14358
\(757\) 1.13817e15 0.166410 0.0832048 0.996532i \(-0.473484\pi\)
0.0832048 + 0.996532i \(0.473484\pi\)
\(758\) 3.30964e15 0.480397
\(759\) −2.95852e15 −0.426329
\(760\) −7.61617e15 −1.08959
\(761\) 9.58101e15 1.36080 0.680402 0.732839i \(-0.261805\pi\)
0.680402 + 0.732839i \(0.261805\pi\)
\(762\) −6.15490e15 −0.867898
\(763\) 1.71758e16 2.40454
\(764\) 3.37305e14 0.0468823
\(765\) −1.70634e16 −2.35466
\(766\) −1.42671e15 −0.195469
\(767\) −1.08370e16 −1.47412
\(768\) 1.06557e16 1.43912
\(769\) 1.45157e16 1.94645 0.973223 0.229864i \(-0.0738280\pi\)
0.973223 + 0.229864i \(0.0738280\pi\)
\(770\) 9.35567e14 0.124559
\(771\) 2.49937e16 3.30393
\(772\) −2.24936e15 −0.295232
\(773\) −4.19629e15 −0.546862 −0.273431 0.961892i \(-0.588159\pi\)
−0.273431 + 0.961892i \(0.588159\pi\)
\(774\) −1.56468e16 −2.02465
\(775\) −3.59305e14 −0.0461640
\(776\) −2.44719e15 −0.312197
\(777\) 2.89068e16 3.66172
\(778\) 5.45690e15 0.686370
\(779\) 7.18432e15 0.897283
\(780\) 1.78771e16 2.21706
\(781\) −7.71320e14 −0.0949848
\(782\) 4.91294e15 0.600765
\(783\) −4.81797e15 −0.585024
\(784\) 1.86341e15 0.224683
\(785\) −1.28486e15 −0.153841
\(786\) −2.28139e15 −0.271254
\(787\) −1.22676e16 −1.44844 −0.724218 0.689571i \(-0.757799\pi\)
−0.724218 + 0.689571i \(0.757799\pi\)
\(788\) 1.09755e16 1.28685
\(789\) 1.31982e15 0.153671
\(790\) 5.24475e15 0.606424
\(791\) −1.03920e15 −0.119324
\(792\) 3.81973e15 0.435557
\(793\) 3.20161e15 0.362549
\(794\) −5.12719e15 −0.576589
\(795\) −9.74643e14 −0.108850
\(796\) −4.70359e15 −0.521684
\(797\) −2.55928e15 −0.281902 −0.140951 0.990017i \(-0.545016\pi\)
−0.140951 + 0.990017i \(0.545016\pi\)
\(798\) 1.59466e16 1.74442
\(799\) 5.79340e15 0.629400
\(800\) 9.22008e14 0.0994809
\(801\) −1.42586e16 −1.52791
\(802\) −6.14480e15 −0.653956
\(803\) −2.00404e15 −0.211822
\(804\) 1.89023e15 0.198430
\(805\) −1.53850e16 −1.60406
\(806\) −4.09050e15 −0.423578
\(807\) 3.46950e16 3.56831
\(808\) −1.39935e14 −0.0142943
\(809\) 1.63176e16 1.65553 0.827767 0.561072i \(-0.189611\pi\)
0.827767 + 0.561072i \(0.189611\pi\)
\(810\) −1.66497e16 −1.67779
\(811\) −9.57950e15 −0.958800 −0.479400 0.877596i \(-0.659146\pi\)
−0.479400 + 0.877596i \(0.659146\pi\)
\(812\) 1.86385e15 0.185290
\(813\) 2.20313e15 0.217542
\(814\) −1.35417e15 −0.132813
\(815\) −4.26642e15 −0.415620
\(816\) 4.65503e15 0.450429
\(817\) −1.92481e16 −1.84998
\(818\) −2.88078e15 −0.275022
\(819\) −6.46204e16 −6.12785
\(820\) 5.18361e15 0.488265
\(821\) −3.68665e15 −0.344941 −0.172470 0.985015i \(-0.555175\pi\)
−0.172470 + 0.985015i \(0.555175\pi\)
\(822\) 2.19392e14 0.0203904
\(823\) −8.96942e15 −0.828066 −0.414033 0.910262i \(-0.635880\pi\)
−0.414033 + 0.910262i \(0.635880\pi\)
\(824\) 9.34064e15 0.856598
\(825\) 3.66279e14 0.0333669
\(826\) −6.97470e15 −0.631153
\(827\) 1.09711e16 0.986208 0.493104 0.869970i \(-0.335862\pi\)
0.493104 + 0.869970i \(0.335862\pi\)
\(828\) −2.64138e16 −2.35865
\(829\) −1.49044e16 −1.32210 −0.661052 0.750340i \(-0.729890\pi\)
−0.661052 + 0.750340i \(0.729890\pi\)
\(830\) 7.56675e14 0.0666776
\(831\) −1.60590e16 −1.40577
\(832\) 5.61748e15 0.488498
\(833\) −9.63966e15 −0.832749
\(834\) −2.46705e15 −0.211721
\(835\) −2.07437e15 −0.176852
\(836\) 1.97593e15 0.167354
\(837\) −1.80002e16 −1.51456
\(838\) 7.11217e15 0.594511
\(839\) 5.79244e15 0.481028 0.240514 0.970646i \(-0.422684\pi\)
0.240514 + 0.970646i \(0.422684\pi\)
\(840\) 2.73614e16 2.25737
\(841\) 4.20707e14 0.0344828
\(842\) 1.54594e13 0.00125885
\(843\) 4.42526e16 3.58004
\(844\) −6.72944e15 −0.540873
\(845\) 2.17857e16 1.73964
\(846\) 1.17759e16 0.934234
\(847\) 1.68682e16 1.32957
\(848\) 1.93027e14 0.0151162
\(849\) −6.20717e15 −0.482949
\(850\) −6.08246e14 −0.0470192
\(851\) 2.22687e16 1.71034
\(852\) −9.48580e15 −0.723863
\(853\) −5.68904e15 −0.431340 −0.215670 0.976466i \(-0.569194\pi\)
−0.215670 + 0.976466i \(0.569194\pi\)
\(854\) 2.06056e15 0.155227
\(855\) −4.26666e16 −3.19355
\(856\) 1.15221e16 0.856891
\(857\) −4.29917e15 −0.317680 −0.158840 0.987304i \(-0.550775\pi\)
−0.158840 + 0.987304i \(0.550775\pi\)
\(858\) 4.16990e15 0.306158
\(859\) −2.27698e16 −1.66110 −0.830552 0.556941i \(-0.811975\pi\)
−0.830552 + 0.556941i \(0.811975\pi\)
\(860\) −1.38878e16 −1.00668
\(861\) −2.58099e16 −1.85896
\(862\) 1.81857e14 0.0130149
\(863\) 3.35556e15 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(864\) 4.61900e16 3.26379
\(865\) 6.12625e15 0.430136
\(866\) 8.53227e15 0.595273
\(867\) 3.47418e15 0.240850
\(868\) 6.96344e15 0.479695
\(869\) −3.23582e15 −0.221501
\(870\) 2.59707e15 0.176656
\(871\) 3.56245e15 0.240796
\(872\) −2.35317e16 −1.58057
\(873\) −1.37094e16 −0.915040
\(874\) 1.22846e16 0.814798
\(875\) 2.17403e16 1.43292
\(876\) −2.46460e16 −1.61426
\(877\) −1.29197e15 −0.0840917 −0.0420459 0.999116i \(-0.513388\pi\)
−0.0420459 + 0.999116i \(0.513388\pi\)
\(878\) 3.44054e14 0.0222540
\(879\) −1.30546e16 −0.839121
\(880\) 6.82890e14 0.0436210
\(881\) −1.43080e16 −0.908263 −0.454132 0.890935i \(-0.650050\pi\)
−0.454132 + 0.890935i \(0.650050\pi\)
\(882\) −1.95939e16 −1.23607
\(883\) 2.31801e16 1.45322 0.726609 0.687051i \(-0.241095\pi\)
0.726609 + 0.687051i \(0.241095\pi\)
\(884\) 1.83157e16 1.14113
\(885\) 2.57056e16 1.59162
\(886\) −4.67454e15 −0.287643
\(887\) −1.42457e16 −0.871174 −0.435587 0.900147i \(-0.643459\pi\)
−0.435587 + 0.900147i \(0.643459\pi\)
\(888\) −3.96038e16 −2.40694
\(889\) 1.97471e16 1.19273
\(890\) 4.78472e15 0.287218
\(891\) 1.02722e16 0.612825
\(892\) 1.77050e16 1.04976
\(893\) 1.44862e16 0.853634
\(894\) 1.99254e16 1.16695
\(895\) −2.55213e15 −0.148551
\(896\) −2.10090e16 −1.21538
\(897\) −6.85721e16 −3.94266
\(898\) −8.28713e15 −0.473570
\(899\) 1.57178e15 0.0892719
\(900\) 3.27016e15 0.184601
\(901\) −9.98554e14 −0.0560254
\(902\) 1.20909e15 0.0674255
\(903\) 6.91495e16 3.83272
\(904\) 1.42376e15 0.0784350
\(905\) −2.43907e15 −0.133554
\(906\) −7.96214e15 −0.433336
\(907\) −9.32604e15 −0.504495 −0.252248 0.967663i \(-0.581170\pi\)
−0.252248 + 0.967663i \(0.581170\pi\)
\(908\) −2.25264e16 −1.21121
\(909\) −7.83929e14 −0.0418963
\(910\) 2.16844e16 1.15191
\(911\) 2.12964e16 1.12449 0.562245 0.826970i \(-0.309938\pi\)
0.562245 + 0.826970i \(0.309938\pi\)
\(912\) 1.16397e16 0.610902
\(913\) −4.66840e14 −0.0243545
\(914\) −1.67919e16 −0.870756
\(915\) −7.59431e15 −0.391447
\(916\) 1.36997e16 0.701919
\(917\) 7.31949e15 0.372777
\(918\) −3.04714e16 −1.54262
\(919\) −2.59517e16 −1.30596 −0.652981 0.757374i \(-0.726482\pi\)
−0.652981 + 0.757374i \(0.726482\pi\)
\(920\) 2.10782e16 1.05439
\(921\) −3.33087e16 −1.65627
\(922\) −2.22763e15 −0.110109
\(923\) −1.78775e16 −0.878412
\(924\) −7.09860e15 −0.346719
\(925\) −2.75698e15 −0.133861
\(926\) 5.50355e15 0.265633
\(927\) 5.23272e16 2.51067
\(928\) −4.03334e15 −0.192376
\(929\) −7.00548e15 −0.332163 −0.166082 0.986112i \(-0.553112\pi\)
−0.166082 + 0.986112i \(0.553112\pi\)
\(930\) 9.70278e15 0.457341
\(931\) −2.41036e16 −1.12943
\(932\) −2.70376e14 −0.0125945
\(933\) 4.93374e16 2.28469
\(934\) −7.65559e15 −0.352428
\(935\) −3.53268e15 −0.161674
\(936\) 8.85332e16 4.02800
\(937\) 7.63705e15 0.345428 0.172714 0.984972i \(-0.444746\pi\)
0.172714 + 0.984972i \(0.444746\pi\)
\(938\) 2.29280e15 0.103098
\(939\) 3.43762e16 1.53673
\(940\) 1.04520e16 0.464513
\(941\) −1.44859e16 −0.640033 −0.320017 0.947412i \(-0.603689\pi\)
−0.320017 + 0.947412i \(0.603689\pi\)
\(942\) −3.68572e15 −0.161899
\(943\) −1.98830e16 −0.868296
\(944\) −5.09097e15 −0.221032
\(945\) 9.54219e16 4.11882
\(946\) −3.23938e15 −0.139015
\(947\) −3.99797e16 −1.70575 −0.852873 0.522119i \(-0.825142\pi\)
−0.852873 + 0.522119i \(0.825142\pi\)
\(948\) −3.97945e16 −1.68802
\(949\) −4.64494e16 −1.95892
\(950\) −1.52090e15 −0.0637706
\(951\) 7.09347e16 2.95710
\(952\) 2.80327e16 1.16188
\(953\) 9.97633e15 0.411112 0.205556 0.978645i \(-0.434100\pi\)
0.205556 + 0.978645i \(0.434100\pi\)
\(954\) −2.02969e15 −0.0831600
\(955\) −1.50791e15 −0.0614268
\(956\) −9.98553e15 −0.404439
\(957\) −1.60229e15 −0.0645247
\(958\) 2.47505e16 0.991002
\(959\) −7.03885e14 −0.0280220
\(960\) −1.33248e16 −0.527435
\(961\) −1.95362e16 −0.768885
\(962\) −3.13868e16 −1.22824
\(963\) 6.45479e16 2.51153
\(964\) −1.08272e16 −0.418884
\(965\) 1.00557e16 0.386823
\(966\) −4.41331e16 −1.68807
\(967\) 7.08033e15 0.269283 0.134641 0.990894i \(-0.457012\pi\)
0.134641 + 0.990894i \(0.457012\pi\)
\(968\) −2.31103e16 −0.873958
\(969\) −6.02139e16 −2.26420
\(970\) 4.60042e15 0.172010
\(971\) −1.86946e15 −0.0695043 −0.0347521 0.999396i \(-0.511064\pi\)
−0.0347521 + 0.999396i \(0.511064\pi\)
\(972\) 6.44894e16 2.38410
\(973\) 7.91514e15 0.290963
\(974\) 1.97348e16 0.721372
\(975\) 8.48956e15 0.308574
\(976\) 1.50405e15 0.0543610
\(977\) −2.08248e16 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(978\) −1.22386e16 −0.437388
\(979\) −2.95199e15 −0.104908
\(980\) −1.73912e16 −0.614590
\(981\) −1.31827e17 −4.63261
\(982\) 1.90524e16 0.665789
\(983\) −1.48242e16 −0.515143 −0.257571 0.966259i \(-0.582922\pi\)
−0.257571 + 0.966259i \(0.582922\pi\)
\(984\) 3.53609e16 1.22194
\(985\) −4.90655e16 −1.68608
\(986\) 2.66078e15 0.0909256
\(987\) −5.20423e16 −1.76853
\(988\) 4.57978e16 1.54768
\(989\) 5.32702e16 1.79021
\(990\) −7.18063e15 −0.239977
\(991\) 2.21328e16 0.735584 0.367792 0.929908i \(-0.380114\pi\)
0.367792 + 0.929908i \(0.380114\pi\)
\(992\) −1.50688e16 −0.498039
\(993\) 2.64050e16 0.867892
\(994\) −1.15060e16 −0.376097
\(995\) 2.10273e16 0.683527
\(996\) −5.74126e15 −0.185601
\(997\) −5.35383e16 −1.72124 −0.860619 0.509249i \(-0.829923\pi\)
−0.860619 + 0.509249i \(0.829923\pi\)
\(998\) 1.14898e15 0.0367361
\(999\) −1.38117e17 −4.39174
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 29.12.a.a.1.5 11
3.2 odd 2 261.12.a.a.1.7 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.12.a.a.1.5 11 1.1 even 1 trivial
261.12.a.a.1.7 11 3.2 odd 2