Properties

Label 230.5.d.a.91.17
Level $230$
Weight $5$
Character 230.91
Analytic conductor $23.775$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [230,5,Mod(91,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.91");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 230.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.7750915093\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.17
Character \(\chi\) \(=\) 230.91
Dual form 230.5.d.a.91.32

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843 q^{2} +17.4598 q^{3} +8.00000 q^{4} +11.1803i q^{5} +49.3837 q^{6} -69.3646i q^{7} +22.6274 q^{8} +223.844 q^{9} +O(q^{10})\) \(q+2.82843 q^{2} +17.4598 q^{3} +8.00000 q^{4} +11.1803i q^{5} +49.3837 q^{6} -69.3646i q^{7} +22.6274 q^{8} +223.844 q^{9} +31.6228i q^{10} +62.8983i q^{11} +139.678 q^{12} -93.8132 q^{13} -196.193i q^{14} +195.206i q^{15} +64.0000 q^{16} -294.893i q^{17} +633.126 q^{18} +108.705i q^{19} +89.4427i q^{20} -1211.09i q^{21} +177.903i q^{22} +(-94.8175 + 520.433i) q^{23} +395.070 q^{24} -125.000 q^{25} -265.344 q^{26} +2494.02 q^{27} -554.917i q^{28} +365.974 q^{29} +552.127i q^{30} -501.410 q^{31} +181.019 q^{32} +1098.19i q^{33} -834.082i q^{34} +775.520 q^{35} +1790.75 q^{36} +1893.15i q^{37} +307.463i q^{38} -1637.96 q^{39} +252.982i q^{40} -2682.34 q^{41} -3425.48i q^{42} -606.856i q^{43} +503.186i q^{44} +2502.65i q^{45} +(-268.184 + 1472.01i) q^{46} -1816.05 q^{47} +1117.43 q^{48} -2410.45 q^{49} -353.553 q^{50} -5148.76i q^{51} -750.506 q^{52} -866.268i q^{53} +7054.16 q^{54} -703.224 q^{55} -1569.54i q^{56} +1897.96i q^{57} +1035.13 q^{58} +4425.54 q^{59} +1561.65i q^{60} +552.555i q^{61} -1418.20 q^{62} -15526.8i q^{63} +512.000 q^{64} -1048.86i q^{65} +3106.15i q^{66} +6584.19i q^{67} -2359.14i q^{68} +(-1655.49 + 9086.65i) q^{69} +2193.50 q^{70} -7800.85 q^{71} +5065.01 q^{72} +2030.45 q^{73} +5354.65i q^{74} -2182.47 q^{75} +869.636i q^{76} +4362.91 q^{77} -4632.84 q^{78} +11883.7i q^{79} +715.542i q^{80} +25413.7 q^{81} -7586.81 q^{82} -9302.52i q^{83} -9688.73i q^{84} +3297.00 q^{85} -1716.45i q^{86} +6389.82 q^{87} +1423.23i q^{88} -11080.9i q^{89} +7078.57i q^{90} +6507.32i q^{91} +(-758.540 + 4163.47i) q^{92} -8754.51 q^{93} -5136.56 q^{94} -1215.35 q^{95} +3160.56 q^{96} -9252.99i q^{97} -6817.78 q^{98} +14079.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 256 q^{4} + 64 q^{6} + 832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 256 q^{4} + 64 q^{6} + 832 q^{9} - 408 q^{13} + 2048 q^{16} + 1024 q^{18} + 1332 q^{23} + 512 q^{24} - 4000 q^{25} + 2208 q^{27} + 3732 q^{29} - 412 q^{31} + 300 q^{35} + 6656 q^{36} - 9208 q^{39} - 6156 q^{41} + 4480 q^{46} + 5184 q^{47} - 13820 q^{49} - 3264 q^{52} - 3328 q^{54} - 6000 q^{55} + 3200 q^{58} - 30468 q^{59} - 10752 q^{62} + 16384 q^{64} - 1168 q^{69} + 4800 q^{70} - 37644 q^{71} + 8192 q^{72} + 19984 q^{73} + 27528 q^{77} + 15744 q^{78} + 69056 q^{81} - 17408 q^{82} + 3300 q^{85} + 28936 q^{87} + 10656 q^{92} + 40648 q^{93} - 23808 q^{94} - 21600 q^{95} + 4096 q^{96} + 43008 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843 0.707107
\(3\) 17.4598 1.93998 0.969988 0.243154i \(-0.0781819\pi\)
0.969988 + 0.243154i \(0.0781819\pi\)
\(4\) 8.00000 0.500000
\(5\) 11.1803i 0.447214i
\(6\) 49.3837 1.37177
\(7\) 69.3646i 1.41560i −0.706411 0.707802i \(-0.749687\pi\)
0.706411 0.707802i \(-0.250313\pi\)
\(8\) 22.6274 0.353553
\(9\) 223.844 2.76351
\(10\) 31.6228i 0.316228i
\(11\) 62.8983i 0.519821i 0.965633 + 0.259910i \(0.0836930\pi\)
−0.965633 + 0.259910i \(0.916307\pi\)
\(12\) 139.678 0.969988
\(13\) −93.8132 −0.555108 −0.277554 0.960710i \(-0.589524\pi\)
−0.277554 + 0.960710i \(0.589524\pi\)
\(14\) 196.193i 1.00098i
\(15\) 195.206i 0.867583i
\(16\) 64.0000 0.250000
\(17\) 294.893i 1.02039i −0.860059 0.510195i \(-0.829573\pi\)
0.860059 0.510195i \(-0.170427\pi\)
\(18\) 633.126 1.95409
\(19\) 108.705i 0.301121i 0.988601 + 0.150560i \(0.0481078\pi\)
−0.988601 + 0.150560i \(0.951892\pi\)
\(20\) 89.4427i 0.223607i
\(21\) 1211.09i 2.74624i
\(22\) 177.903i 0.367569i
\(23\) −94.8175 + 520.433i −0.179239 + 0.983806i
\(24\) 395.070 0.685885
\(25\) −125.000 −0.200000
\(26\) −265.344 −0.392520
\(27\) 2494.02 3.42116
\(28\) 554.917i 0.707802i
\(29\) 365.974 0.435165 0.217583 0.976042i \(-0.430183\pi\)
0.217583 + 0.976042i \(0.430183\pi\)
\(30\) 552.127i 0.613474i
\(31\) −501.410 −0.521759 −0.260879 0.965371i \(-0.584012\pi\)
−0.260879 + 0.965371i \(0.584012\pi\)
\(32\) 181.019 0.176777
\(33\) 1098.19i 1.00844i
\(34\) 834.082i 0.721525i
\(35\) 775.520 0.633077
\(36\) 1790.75 1.38175
\(37\) 1893.15i 1.38287i 0.722437 + 0.691437i \(0.243022\pi\)
−0.722437 + 0.691437i \(0.756978\pi\)
\(38\) 307.463i 0.212924i
\(39\) −1637.96 −1.07690
\(40\) 252.982i 0.158114i
\(41\) −2682.34 −1.59568 −0.797841 0.602868i \(-0.794025\pi\)
−0.797841 + 0.602868i \(0.794025\pi\)
\(42\) 3425.48i 1.94188i
\(43\) 606.856i 0.328208i −0.986443 0.164104i \(-0.947527\pi\)
0.986443 0.164104i \(-0.0524732\pi\)
\(44\) 503.186i 0.259910i
\(45\) 2502.65i 1.23588i
\(46\) −268.184 + 1472.01i −0.126741 + 0.695656i
\(47\) −1816.05 −0.822113 −0.411056 0.911610i \(-0.634840\pi\)
−0.411056 + 0.911610i \(0.634840\pi\)
\(48\) 1117.43 0.484994
\(49\) −2410.45 −1.00394
\(50\) −353.553 −0.141421
\(51\) 5148.76i 1.97953i
\(52\) −750.506 −0.277554
\(53\) 866.268i 0.308390i −0.988040 0.154195i \(-0.950722\pi\)
0.988040 0.154195i \(-0.0492784\pi\)
\(54\) 7054.16 2.41912
\(55\) −703.224 −0.232471
\(56\) 1569.54i 0.500492i
\(57\) 1897.96i 0.584167i
\(58\) 1035.13 0.307708
\(59\) 4425.54 1.27134 0.635671 0.771960i \(-0.280723\pi\)
0.635671 + 0.771960i \(0.280723\pi\)
\(60\) 1561.65i 0.433792i
\(61\) 552.555i 0.148496i 0.997240 + 0.0742482i \(0.0236557\pi\)
−0.997240 + 0.0742482i \(0.976344\pi\)
\(62\) −1418.20 −0.368939
\(63\) 15526.8i 3.91203i
\(64\) 512.000 0.125000
\(65\) 1048.86i 0.248252i
\(66\) 3106.15i 0.713074i
\(67\) 6584.19i 1.46674i 0.679830 + 0.733370i \(0.262054\pi\)
−0.679830 + 0.733370i \(0.737946\pi\)
\(68\) 2359.14i 0.510195i
\(69\) −1655.49 + 9086.65i −0.347720 + 1.90856i
\(70\) 2193.50 0.447653
\(71\) −7800.85 −1.54748 −0.773740 0.633503i \(-0.781616\pi\)
−0.773740 + 0.633503i \(0.781616\pi\)
\(72\) 5065.01 0.977047
\(73\) 2030.45 0.381019 0.190509 0.981685i \(-0.438986\pi\)
0.190509 + 0.981685i \(0.438986\pi\)
\(74\) 5354.65i 0.977840i
\(75\) −2182.47 −0.387995
\(76\) 869.636i 0.150560i
\(77\) 4362.91 0.735860
\(78\) −4632.84 −0.761480
\(79\) 11883.7i 1.90413i 0.305891 + 0.952066i \(0.401046\pi\)
−0.305891 + 0.952066i \(0.598954\pi\)
\(80\) 715.542i 0.111803i
\(81\) 25413.7 3.87346
\(82\) −7586.81 −1.12832
\(83\) 9302.52i 1.35034i −0.737660 0.675172i \(-0.764069\pi\)
0.737660 0.675172i \(-0.235931\pi\)
\(84\) 9688.73i 1.37312i
\(85\) 3297.00 0.456332
\(86\) 1716.45i 0.232078i
\(87\) 6389.82 0.844210
\(88\) 1423.23i 0.183784i
\(89\) 11080.9i 1.39892i −0.714669 0.699462i \(-0.753423\pi\)
0.714669 0.699462i \(-0.246577\pi\)
\(90\) 7078.57i 0.873897i
\(91\) 6507.32i 0.785813i
\(92\) −758.540 + 4163.47i −0.0896196 + 0.491903i
\(93\) −8754.51 −1.01220
\(94\) −5136.56 −0.581321
\(95\) −1215.35 −0.134665
\(96\) 3160.56 0.342942
\(97\) 9252.99i 0.983419i −0.870759 0.491710i \(-0.836372\pi\)
0.870759 0.491710i \(-0.163628\pi\)
\(98\) −6817.78 −0.709889
\(99\) 14079.4i 1.43653i
\(100\) −1000.00 −0.100000
\(101\) −4885.92 −0.478965 −0.239482 0.970901i \(-0.576978\pi\)
−0.239482 + 0.970901i \(0.576978\pi\)
\(102\) 14562.9i 1.39974i
\(103\) 8790.26i 0.828567i −0.910148 0.414283i \(-0.864032\pi\)
0.910148 0.414283i \(-0.135968\pi\)
\(104\) −2122.75 −0.196260
\(105\) 13540.4 1.22815
\(106\) 2450.18i 0.218065i
\(107\) 22541.8i 1.96889i −0.175698 0.984444i \(-0.556218\pi\)
0.175698 0.984444i \(-0.443782\pi\)
\(108\) 19952.2 1.71058
\(109\) 5212.01i 0.438684i −0.975648 0.219342i \(-0.929609\pi\)
0.975648 0.219342i \(-0.0703911\pi\)
\(110\) −1989.02 −0.164382
\(111\) 33054.1i 2.68274i
\(112\) 4439.33i 0.353901i
\(113\) 13202.4i 1.03395i 0.856002 + 0.516973i \(0.172941\pi\)
−0.856002 + 0.516973i \(0.827059\pi\)
\(114\) 5368.23i 0.413068i
\(115\) −5818.62 1060.09i −0.439971 0.0801582i
\(116\) 2927.79 0.217583
\(117\) −20999.5 −1.53404
\(118\) 12517.3 0.898975
\(119\) −20455.1 −1.44447
\(120\) 4417.01i 0.306737i
\(121\) 10684.8 0.729787
\(122\) 1562.86i 0.105003i
\(123\) −46833.1 −3.09558
\(124\) −4011.28 −0.260879
\(125\) 1397.54i 0.0894427i
\(126\) 43916.6i 2.76622i
\(127\) −16087.6 −0.997430 −0.498715 0.866766i \(-0.666195\pi\)
−0.498715 + 0.866766i \(0.666195\pi\)
\(128\) 1448.15 0.0883883
\(129\) 10595.6i 0.636715i
\(130\) 2966.63i 0.175540i
\(131\) −21055.6 −1.22695 −0.613473 0.789716i \(-0.710228\pi\)
−0.613473 + 0.789716i \(0.710228\pi\)
\(132\) 8785.52i 0.504220i
\(133\) 7540.25 0.426268
\(134\) 18622.9i 1.03714i
\(135\) 27884.0i 1.52999i
\(136\) 6672.66i 0.360762i
\(137\) 34815.7i 1.85496i 0.373875 + 0.927479i \(0.378029\pi\)
−0.373875 + 0.927479i \(0.621971\pi\)
\(138\) −4682.44 + 25700.9i −0.245875 + 1.34955i
\(139\) 13204.2 0.683409 0.341705 0.939807i \(-0.388996\pi\)
0.341705 + 0.939807i \(0.388996\pi\)
\(140\) 6204.16 0.316539
\(141\) −31707.8 −1.59488
\(142\) −22064.1 −1.09423
\(143\) 5900.69i 0.288556i
\(144\) 14326.0 0.690876
\(145\) 4091.71i 0.194612i
\(146\) 5742.98 0.269421
\(147\) −42085.9 −1.94761
\(148\) 15145.2i 0.691437i
\(149\) 3486.31i 0.157034i 0.996913 + 0.0785169i \(0.0250184\pi\)
−0.996913 + 0.0785169i \(0.974982\pi\)
\(150\) −6172.96 −0.274354
\(151\) −33962.4 −1.48951 −0.744757 0.667336i \(-0.767435\pi\)
−0.744757 + 0.667336i \(0.767435\pi\)
\(152\) 2459.70i 0.106462i
\(153\) 66009.9i 2.81985i
\(154\) 12340.2 0.520332
\(155\) 5605.94i 0.233338i
\(156\) −13103.7 −0.538448
\(157\) 1022.17i 0.0414690i 0.999785 + 0.0207345i \(0.00660047\pi\)
−0.999785 + 0.0207345i \(0.993400\pi\)
\(158\) 33612.2i 1.34643i
\(159\) 15124.8i 0.598269i
\(160\) 2023.86i 0.0790569i
\(161\) 36099.6 + 6576.98i 1.39268 + 0.253732i
\(162\) 71880.9 2.73895
\(163\) 11549.3 0.434691 0.217346 0.976095i \(-0.430260\pi\)
0.217346 + 0.976095i \(0.430260\pi\)
\(164\) −21458.7 −0.797841
\(165\) −12278.1 −0.450988
\(166\) 26311.5i 0.954838i
\(167\) 41559.9 1.49019 0.745095 0.666958i \(-0.232404\pi\)
0.745095 + 0.666958i \(0.232404\pi\)
\(168\) 27403.9i 0.970942i
\(169\) −19760.1 −0.691855
\(170\) 9325.32 0.322676
\(171\) 24332.8i 0.832148i
\(172\) 4854.85i 0.164104i
\(173\) 31791.6 1.06223 0.531117 0.847299i \(-0.321773\pi\)
0.531117 + 0.847299i \(0.321773\pi\)
\(174\) 18073.2 0.596946
\(175\) 8670.58i 0.283121i
\(176\) 4025.49i 0.129955i
\(177\) 77269.0 2.46637
\(178\) 31341.5i 0.989189i
\(179\) −28774.4 −0.898050 −0.449025 0.893519i \(-0.648229\pi\)
−0.449025 + 0.893519i \(0.648229\pi\)
\(180\) 20021.2i 0.617939i
\(181\) 55583.7i 1.69664i 0.529481 + 0.848322i \(0.322387\pi\)
−0.529481 + 0.848322i \(0.677613\pi\)
\(182\) 18405.5i 0.555654i
\(183\) 9647.49i 0.288079i
\(184\) −2145.48 + 11776.1i −0.0633706 + 0.347828i
\(185\) −21166.1 −0.618440
\(186\) −24761.5 −0.715733
\(187\) 18548.2 0.530420
\(188\) −14528.4 −0.411056
\(189\) 172997.i 4.84300i
\(190\) −3437.54 −0.0952227
\(191\) 9119.25i 0.249973i −0.992158 0.124986i \(-0.960111\pi\)
0.992158 0.124986i \(-0.0398887\pi\)
\(192\) 8939.41 0.242497
\(193\) 1202.93 0.0322944 0.0161472 0.999870i \(-0.494860\pi\)
0.0161472 + 0.999870i \(0.494860\pi\)
\(194\) 26171.4i 0.695382i
\(195\) 18312.9i 0.481602i
\(196\) −19283.6 −0.501968
\(197\) −69432.6 −1.78908 −0.894542 0.446984i \(-0.852498\pi\)
−0.894542 + 0.446984i \(0.852498\pi\)
\(198\) 39822.6i 1.01578i
\(199\) 17511.6i 0.442201i −0.975251 0.221101i \(-0.929035\pi\)
0.975251 0.221101i \(-0.0709649\pi\)
\(200\) −2828.43 −0.0707107
\(201\) 114959.i 2.84544i
\(202\) −13819.5 −0.338679
\(203\) 25385.6i 0.616022i
\(204\) 41190.1i 0.989766i
\(205\) 29989.5i 0.713611i
\(206\) 24862.6i 0.585885i
\(207\) −21224.3 + 116496.i −0.495328 + 2.71875i
\(208\) −6004.05 −0.138777
\(209\) −6837.33 −0.156529
\(210\) 38298.1 0.868437
\(211\) 43957.8 0.987350 0.493675 0.869646i \(-0.335653\pi\)
0.493675 + 0.869646i \(0.335653\pi\)
\(212\) 6930.14i 0.154195i
\(213\) −136201. −3.00207
\(214\) 63757.8i 1.39221i
\(215\) 6784.86 0.146779
\(216\) 56433.3 1.20956
\(217\) 34780.1i 0.738604i
\(218\) 14741.8i 0.310197i
\(219\) 35451.2 0.739167
\(220\) −5625.79 −0.116235
\(221\) 27664.8i 0.566426i
\(222\) 93491.0i 1.89698i
\(223\) 33778.7 0.679255 0.339628 0.940560i \(-0.389699\pi\)
0.339628 + 0.940560i \(0.389699\pi\)
\(224\) 12556.3i 0.250246i
\(225\) −27980.5 −0.552701
\(226\) 37342.2i 0.731110i
\(227\) 25376.5i 0.492471i 0.969210 + 0.246236i \(0.0791937\pi\)
−0.969210 + 0.246236i \(0.920806\pi\)
\(228\) 15183.7i 0.292083i
\(229\) 84049.3i 1.60274i −0.598168 0.801370i \(-0.704105\pi\)
0.598168 0.801370i \(-0.295895\pi\)
\(230\) −16457.5 2998.39i −0.311107 0.0566804i
\(231\) 76175.5 1.42755
\(232\) 8281.04 0.153854
\(233\) 32229.2 0.593661 0.296830 0.954930i \(-0.404070\pi\)
0.296830 + 0.954930i \(0.404070\pi\)
\(234\) −59395.6 −1.08473
\(235\) 20304.0i 0.367660i
\(236\) 35404.3 0.635671
\(237\) 207487.i 3.69397i
\(238\) −57855.8 −1.02139
\(239\) 30122.2 0.527340 0.263670 0.964613i \(-0.415067\pi\)
0.263670 + 0.964613i \(0.415067\pi\)
\(240\) 12493.2i 0.216896i
\(241\) 15549.4i 0.267719i −0.991000 0.133859i \(-0.957263\pi\)
0.991000 0.133859i \(-0.0427370\pi\)
\(242\) 30221.2 0.516037
\(243\) 241703. 4.09325
\(244\) 4420.44i 0.0742482i
\(245\) 26949.6i 0.448973i
\(246\) −132464. −2.18891
\(247\) 10197.9i 0.167154i
\(248\) −11345.6 −0.184470
\(249\) 162420.i 2.61964i
\(250\) 3952.85i 0.0632456i
\(251\) 59069.3i 0.937593i −0.883306 0.468796i \(-0.844688\pi\)
0.883306 0.468796i \(-0.155312\pi\)
\(252\) 124215.i 1.95601i
\(253\) −32734.4 5963.86i −0.511402 0.0931722i
\(254\) −45502.5 −0.705290
\(255\) 57564.9 0.885273
\(256\) 4096.00 0.0625000
\(257\) 99291.1 1.50329 0.751647 0.659566i \(-0.229260\pi\)
0.751647 + 0.659566i \(0.229260\pi\)
\(258\) 29968.8i 0.450225i
\(259\) 131318. 1.95760
\(260\) 8390.91i 0.124126i
\(261\) 81921.0 1.20258
\(262\) −59554.3 −0.867581
\(263\) 37218.4i 0.538080i −0.963129 0.269040i \(-0.913294\pi\)
0.963129 0.269040i \(-0.0867063\pi\)
\(264\) 24849.2i 0.356537i
\(265\) 9685.17 0.137916
\(266\) 21327.0 0.301417
\(267\) 193470.i 2.71388i
\(268\) 52673.6i 0.733370i
\(269\) −10708.0 −0.147981 −0.0739905 0.997259i \(-0.523573\pi\)
−0.0739905 + 0.997259i \(0.523573\pi\)
\(270\) 78868.0i 1.08186i
\(271\) −18835.5 −0.256471 −0.128236 0.991744i \(-0.540931\pi\)
−0.128236 + 0.991744i \(0.540931\pi\)
\(272\) 18873.1i 0.255097i
\(273\) 113616.i 1.52446i
\(274\) 98473.7i 1.31165i
\(275\) 7862.29i 0.103964i
\(276\) −13243.9 + 72693.2i −0.173860 + 0.954279i
\(277\) 27259.2 0.355266 0.177633 0.984097i \(-0.443156\pi\)
0.177633 + 0.984097i \(0.443156\pi\)
\(278\) 37347.0 0.483243
\(279\) −112238. −1.44188
\(280\) 17548.0 0.223827
\(281\) 19178.1i 0.242881i −0.992599 0.121440i \(-0.961249\pi\)
0.992599 0.121440i \(-0.0387513\pi\)
\(282\) −89683.1 −1.12775
\(283\) 60935.5i 0.760848i −0.924812 0.380424i \(-0.875778\pi\)
0.924812 0.380424i \(-0.124222\pi\)
\(284\) −62406.8 −0.773740
\(285\) −21219.8 −0.261247
\(286\) 16689.7i 0.204040i
\(287\) 186060.i 2.25885i
\(288\) 40520.1 0.488523
\(289\) −3440.69 −0.0411955
\(290\) 11573.1i 0.137611i
\(291\) 161555.i 1.90781i
\(292\) 16243.6 0.190509
\(293\) 69688.9i 0.811761i 0.913926 + 0.405880i \(0.133035\pi\)
−0.913926 + 0.405880i \(0.866965\pi\)
\(294\) −119037. −1.37717
\(295\) 49479.1i 0.568562i
\(296\) 42837.2i 0.488920i
\(297\) 156870.i 1.77839i
\(298\) 9860.76i 0.111040i
\(299\) 8895.14 48823.5i 0.0994970 0.546118i
\(300\) −17459.8 −0.193998
\(301\) −42094.3 −0.464612
\(302\) −96060.2 −1.05325
\(303\) −85307.1 −0.929180
\(304\) 6957.09i 0.0752801i
\(305\) −6177.75 −0.0664096
\(306\) 186704.i 1.99394i
\(307\) 148352. 1.57405 0.787024 0.616923i \(-0.211621\pi\)
0.787024 + 0.616923i \(0.211621\pi\)
\(308\) 34903.3 0.367930
\(309\) 153476.i 1.60740i
\(310\) 15856.0i 0.164995i
\(311\) 134338. 1.38892 0.694460 0.719531i \(-0.255643\pi\)
0.694460 + 0.719531i \(0.255643\pi\)
\(312\) −37062.8 −0.380740
\(313\) 23016.1i 0.234933i 0.993077 + 0.117466i \(0.0374773\pi\)
−0.993077 + 0.117466i \(0.962523\pi\)
\(314\) 2891.13i 0.0293230i
\(315\) 173595. 1.74951
\(316\) 95069.5i 0.952066i
\(317\) 111202. 1.10661 0.553305 0.832979i \(-0.313366\pi\)
0.553305 + 0.832979i \(0.313366\pi\)
\(318\) 42779.5i 0.423040i
\(319\) 23019.1i 0.226208i
\(320\) 5724.33i 0.0559017i
\(321\) 393575.i 3.81960i
\(322\) 102105. + 18602.5i 0.984773 + 0.179415i
\(323\) 32056.2 0.307260
\(324\) 203310. 1.93673
\(325\) 11726.7 0.111022
\(326\) 32666.4 0.307373
\(327\) 91000.5i 0.851037i
\(328\) −60694.4 −0.564159
\(329\) 125969.i 1.16379i
\(330\) −34727.8 −0.318896
\(331\) −82063.9 −0.749025 −0.374512 0.927222i \(-0.622190\pi\)
−0.374512 + 0.927222i \(0.622190\pi\)
\(332\) 74420.2i 0.675172i
\(333\) 423771.i 3.82158i
\(334\) 117549. 1.05372
\(335\) −73613.5 −0.655946
\(336\) 77509.8i 0.686559i
\(337\) 3251.27i 0.0286281i 0.999898 + 0.0143141i \(0.00455646\pi\)
−0.999898 + 0.0143141i \(0.995444\pi\)
\(338\) −55890.0 −0.489216
\(339\) 230512.i 2.00583i
\(340\) 26376.0 0.228166
\(341\) 31537.8i 0.271221i
\(342\) 68823.7i 0.588418i
\(343\) 655.389i 0.00557071i
\(344\) 13731.6i 0.116039i
\(345\) −101592. 18509.0i −0.853533 0.155505i
\(346\) 89920.2 0.751113
\(347\) 32147.5 0.266986 0.133493 0.991050i \(-0.457381\pi\)
0.133493 + 0.991050i \(0.457381\pi\)
\(348\) 51118.6 0.422105
\(349\) 198530. 1.62995 0.814975 0.579496i \(-0.196751\pi\)
0.814975 + 0.579496i \(0.196751\pi\)
\(350\) 24524.1i 0.200197i
\(351\) −233972. −1.89911
\(352\) 11385.8i 0.0918922i
\(353\) −31094.6 −0.249537 −0.124768 0.992186i \(-0.539819\pi\)
−0.124768 + 0.992186i \(0.539819\pi\)
\(354\) 218550. 1.74399
\(355\) 87216.1i 0.692054i
\(356\) 88647.1i 0.699462i
\(357\) −357142. −2.80223
\(358\) −81386.3 −0.635017
\(359\) 211521.i 1.64121i −0.571495 0.820606i \(-0.693636\pi\)
0.571495 0.820606i \(-0.306364\pi\)
\(360\) 56628.5i 0.436949i
\(361\) 118504. 0.909326
\(362\) 157215.i 1.19971i
\(363\) 186554. 1.41577
\(364\) 52058.5i 0.392906i
\(365\) 22701.1i 0.170397i
\(366\) 27287.2i 0.203703i
\(367\) 36290.1i 0.269436i 0.990884 + 0.134718i \(0.0430129\pi\)
−0.990884 + 0.134718i \(0.956987\pi\)
\(368\) −6068.32 + 33307.7i −0.0448098 + 0.245951i
\(369\) −600426. −4.40967
\(370\) −59866.8 −0.437303
\(371\) −60088.3 −0.436558
\(372\) −70036.1 −0.506100
\(373\) 133203.i 0.957409i −0.877976 0.478705i \(-0.841107\pi\)
0.877976 0.478705i \(-0.158893\pi\)
\(374\) 52462.4 0.375063
\(375\) 24400.8i 0.173517i
\(376\) −41092.4 −0.290661
\(377\) −34333.2 −0.241564
\(378\) 489309.i 3.42452i
\(379\) 228349.i 1.58972i 0.606792 + 0.794861i \(0.292456\pi\)
−0.606792 + 0.794861i \(0.707544\pi\)
\(380\) −9722.83 −0.0673326
\(381\) −280885. −1.93499
\(382\) 25793.1i 0.176757i
\(383\) 74510.6i 0.507949i −0.967211 0.253975i \(-0.918262\pi\)
0.967211 0.253975i \(-0.0817380\pi\)
\(384\) 25284.5 0.171471
\(385\) 48778.9i 0.329087i
\(386\) 3402.41 0.0228356
\(387\) 135841.i 0.907004i
\(388\) 74023.9i 0.491710i
\(389\) 41483.3i 0.274141i 0.990561 + 0.137070i \(0.0437687\pi\)
−0.990561 + 0.137070i \(0.956231\pi\)
\(390\) 51796.8i 0.340544i
\(391\) 153472. + 27961.0i 1.00387 + 0.182894i
\(392\) −54542.2 −0.354945
\(393\) −367626. −2.38024
\(394\) −196385. −1.26507
\(395\) −132864. −0.851554
\(396\) 112635.i 0.718263i
\(397\) −5511.87 −0.0349718 −0.0174859 0.999847i \(-0.505566\pi\)
−0.0174859 + 0.999847i \(0.505566\pi\)
\(398\) 49530.3i 0.312683i
\(399\) 131651. 0.826949
\(400\) −8000.00 −0.0500000
\(401\) 207005.i 1.28733i 0.765306 + 0.643667i \(0.222588\pi\)
−0.765306 + 0.643667i \(0.777412\pi\)
\(402\) 325152.i 2.01203i
\(403\) 47038.9 0.289632
\(404\) −39087.4 −0.239482
\(405\) 284134.i 1.73226i
\(406\) 71801.4i 0.435593i
\(407\) −119076. −0.718846
\(408\) 116503.i 0.699870i
\(409\) −83511.1 −0.499226 −0.249613 0.968346i \(-0.580303\pi\)
−0.249613 + 0.968346i \(0.580303\pi\)
\(410\) 84823.1i 0.504599i
\(411\) 607875.i 3.59857i
\(412\) 70322.1i 0.414283i
\(413\) 306976.i 1.79972i
\(414\) −60031.5 + 329500.i −0.350250 + 1.92245i
\(415\) 104005. 0.603892
\(416\) −16982.0 −0.0981301
\(417\) 230542. 1.32580
\(418\) −19338.9 −0.110682
\(419\) 188064.i 1.07122i 0.844467 + 0.535608i \(0.179917\pi\)
−0.844467 + 0.535608i \(0.820083\pi\)
\(420\) 108323. 0.614077
\(421\) 117145.i 0.660938i −0.943817 0.330469i \(-0.892793\pi\)
0.943817 0.330469i \(-0.107207\pi\)
\(422\) 124331. 0.698162
\(423\) −406511. −2.27191
\(424\) 19601.4i 0.109032i
\(425\) 36861.6i 0.204078i
\(426\) −385235. −2.12279
\(427\) 38327.8 0.210212
\(428\) 180334.i 0.984444i
\(429\) 103025.i 0.559792i
\(430\) 19190.5 0.103788
\(431\) 202900.i 1.09226i −0.837700 0.546131i \(-0.816100\pi\)
0.837700 0.546131i \(-0.183900\pi\)
\(432\) 159618. 0.855289
\(433\) 95101.1i 0.507236i −0.967305 0.253618i \(-0.918379\pi\)
0.967305 0.253618i \(-0.0816205\pi\)
\(434\) 98373.0i 0.522272i
\(435\) 71440.4i 0.377542i
\(436\) 41696.1i 0.219342i
\(437\) −56573.4 10307.1i −0.296244 0.0539726i
\(438\) 100271. 0.522670
\(439\) −40253.8 −0.208871 −0.104435 0.994532i \(-0.533304\pi\)
−0.104435 + 0.994532i \(0.533304\pi\)
\(440\) −15912.1 −0.0821908
\(441\) −539564. −2.77438
\(442\) 78247.9i 0.400524i
\(443\) 233288. 1.18873 0.594367 0.804194i \(-0.297403\pi\)
0.594367 + 0.804194i \(0.297403\pi\)
\(444\) 264432.i 1.34137i
\(445\) 123888. 0.625618
\(446\) 95540.6 0.480306
\(447\) 60870.1i 0.304642i
\(448\) 35514.7i 0.176951i
\(449\) −279358. −1.38570 −0.692850 0.721082i \(-0.743645\pi\)
−0.692850 + 0.721082i \(0.743645\pi\)
\(450\) −79140.8 −0.390819
\(451\) 168715.i 0.829468i
\(452\) 105620.i 0.516973i
\(453\) −592976. −2.88962
\(454\) 71775.7i 0.348230i
\(455\) −72754.0 −0.351426
\(456\) 42945.9i 0.206534i
\(457\) 273520.i 1.30966i 0.755778 + 0.654828i \(0.227259\pi\)
−0.755778 + 0.654828i \(0.772741\pi\)
\(458\) 237727.i 1.13331i
\(459\) 735469.i 3.49091i
\(460\) −46549.0 8480.74i −0.219986 0.0400791i
\(461\) −201691. −0.949042 −0.474521 0.880244i \(-0.657379\pi\)
−0.474521 + 0.880244i \(0.657379\pi\)
\(462\) 215457. 1.00943
\(463\) 179802. 0.838749 0.419375 0.907813i \(-0.362249\pi\)
0.419375 + 0.907813i \(0.362249\pi\)
\(464\) 23422.3 0.108791
\(465\) 97878.4i 0.452669i
\(466\) 91158.1 0.419781
\(467\) 304288.i 1.39525i −0.716465 0.697623i \(-0.754241\pi\)
0.716465 0.697623i \(-0.245759\pi\)
\(468\) −167996. −0.767022
\(469\) 456710. 2.07632
\(470\) 57428.4i 0.259975i
\(471\) 17846.8i 0.0804488i
\(472\) 100139. 0.449487
\(473\) 38170.2 0.170609
\(474\) 586861.i 2.61203i
\(475\) 13588.1i 0.0602241i
\(476\) −163641. −0.722234
\(477\) 193909.i 0.852238i
\(478\) 85198.4 0.372886
\(479\) 146012.i 0.636381i 0.948027 + 0.318191i \(0.103075\pi\)
−0.948027 + 0.318191i \(0.896925\pi\)
\(480\) 35336.1i 0.153369i
\(481\) 177603.i 0.767644i
\(482\) 43980.2i 0.189306i
\(483\) 630292. + 114833.i 2.70176 + 0.492233i
\(484\) 85478.4 0.364893
\(485\) 103452. 0.439798
\(486\) 683638. 2.89437
\(487\) 304198. 1.28262 0.641311 0.767281i \(-0.278391\pi\)
0.641311 + 0.767281i \(0.278391\pi\)
\(488\) 12502.9i 0.0525014i
\(489\) 201648. 0.843290
\(490\) 76225.1i 0.317472i
\(491\) 71274.4 0.295645 0.147823 0.989014i \(-0.452774\pi\)
0.147823 + 0.989014i \(0.452774\pi\)
\(492\) −374665. −1.54779
\(493\) 107923.i 0.444038i
\(494\) 28844.1i 0.118196i
\(495\) −157412. −0.642434
\(496\) −32090.2 −0.130440
\(497\) 541103.i 2.19062i
\(498\) 459393.i 1.85236i
\(499\) 317321. 1.27438 0.637188 0.770708i \(-0.280097\pi\)
0.637188 + 0.770708i \(0.280097\pi\)
\(500\) 11180.3i 0.0447214i
\(501\) 725627. 2.89093
\(502\) 167073.i 0.662978i
\(503\) 374840.i 1.48153i 0.671766 + 0.740764i \(0.265536\pi\)
−0.671766 + 0.740764i \(0.734464\pi\)
\(504\) 351332.i 1.38311i
\(505\) 54626.3i 0.214200i
\(506\) −92586.7 16868.3i −0.361616 0.0658827i
\(507\) −345007. −1.34218
\(508\) −128700. −0.498715
\(509\) −482427. −1.86207 −0.931036 0.364929i \(-0.881093\pi\)
−0.931036 + 0.364929i \(0.881093\pi\)
\(510\) 162818. 0.625983
\(511\) 140841.i 0.539372i
\(512\) 11585.2 0.0441942
\(513\) 271112.i 1.03018i
\(514\) 280838. 1.06299
\(515\) 98278.1 0.370546
\(516\) 84764.6i 0.318357i
\(517\) 114226.i 0.427351i
\(518\) 371423. 1.38423
\(519\) 555074. 2.06071
\(520\) 23733.1i 0.0877702i
\(521\) 114785.i 0.422873i −0.977392 0.211437i \(-0.932186\pi\)
0.977392 0.211437i \(-0.0678142\pi\)
\(522\) 231708. 0.850353
\(523\) 122797.i 0.448937i −0.974481 0.224469i \(-0.927935\pi\)
0.974481 0.224469i \(-0.0720646\pi\)
\(524\) −168445. −0.613473
\(525\) 151386.i 0.549248i
\(526\) 105270.i 0.380480i
\(527\) 147862.i 0.532397i
\(528\) 70284.2i 0.252110i
\(529\) −261860. 98692.3i −0.935747 0.352673i
\(530\) 27393.8 0.0975215
\(531\) 990631. 3.51336
\(532\) 60322.0 0.213134
\(533\) 251639. 0.885775
\(534\) 547215.i 1.91900i
\(535\) 252025. 0.880514
\(536\) 148983.i 0.518571i
\(537\) −502395. −1.74219
\(538\) −30286.9 −0.104638
\(539\) 151613.i 0.521866i
\(540\) 223072.i 0.764994i
\(541\) 171718. 0.586706 0.293353 0.956004i \(-0.405229\pi\)
0.293353 + 0.956004i \(0.405229\pi\)
\(542\) −53274.9 −0.181353
\(543\) 970480.i 3.29145i
\(544\) 53381.3i 0.180381i
\(545\) 58272.0 0.196186
\(546\) 321355.i 1.07795i
\(547\) 283125. 0.946245 0.473122 0.880997i \(-0.343127\pi\)
0.473122 + 0.880997i \(0.343127\pi\)
\(548\) 278526.i 0.927479i
\(549\) 123686.i 0.410371i
\(550\) 22237.9i 0.0735137i
\(551\) 39783.0i 0.131037i
\(552\) −37459.5 + 205607.i −0.122937 + 0.674777i
\(553\) 824308. 2.69550
\(554\) 77100.6 0.251211
\(555\) −369556. −1.19976
\(556\) 105633. 0.341705
\(557\) 265207.i 0.854821i −0.904058 0.427411i \(-0.859426\pi\)
0.904058 0.427411i \(-0.140574\pi\)
\(558\) −317456. −1.01957
\(559\) 56931.1i 0.182191i
\(560\) 49633.3 0.158269
\(561\) 323848. 1.02900
\(562\) 54243.9i 0.171743i
\(563\) 297822.i 0.939594i 0.882774 + 0.469797i \(0.155673\pi\)
−0.882774 + 0.469797i \(0.844327\pi\)
\(564\) −253662. −0.797439
\(565\) −147608. −0.462394
\(566\) 172352.i 0.538001i
\(567\) 1.76281e6i 5.48328i
\(568\) −176513. −0.547117
\(569\) 6160.11i 0.0190267i 0.999955 + 0.00951336i \(0.00302824\pi\)
−0.999955 + 0.00951336i \(0.996972\pi\)
\(570\) −60018.7 −0.184730
\(571\) 111470.i 0.341888i 0.985281 + 0.170944i \(0.0546817\pi\)
−0.985281 + 0.170944i \(0.945318\pi\)
\(572\) 47205.5i 0.144278i
\(573\) 159220.i 0.484941i
\(574\) 526256.i 1.59725i
\(575\) 11852.2 65054.1i 0.0358478 0.196761i
\(576\) 114608. 0.345438
\(577\) −356286. −1.07015 −0.535077 0.844803i \(-0.679718\pi\)
−0.535077 + 0.844803i \(0.679718\pi\)
\(578\) −9731.73 −0.0291296
\(579\) 21002.9 0.0626503
\(580\) 32733.7i 0.0973059i
\(581\) −645266. −1.91155
\(582\) 456947.i 1.34902i
\(583\) 54486.8 0.160308
\(584\) 45943.8 0.134711
\(585\) 234782.i 0.686045i
\(586\) 197110.i 0.574002i
\(587\) 21337.8 0.0619262 0.0309631 0.999521i \(-0.490143\pi\)
0.0309631 + 0.999521i \(0.490143\pi\)
\(588\) −336687. −0.973805
\(589\) 54505.5i 0.157112i
\(590\) 139948.i 0.402034i
\(591\) −1.21228e6 −3.47078
\(592\) 121162.i 0.345718i
\(593\) −470676. −1.33848 −0.669241 0.743045i \(-0.733381\pi\)
−0.669241 + 0.743045i \(0.733381\pi\)
\(594\) 443695.i 1.25751i
\(595\) 228695.i 0.645986i
\(596\) 27890.4i 0.0785169i
\(597\) 305749.i 0.857859i
\(598\) 25159.2 138094.i 0.0703550 0.386164i
\(599\) −241014. −0.671721 −0.335860 0.941912i \(-0.609027\pi\)
−0.335860 + 0.941912i \(0.609027\pi\)
\(600\) −49383.7 −0.137177
\(601\) 355329. 0.983743 0.491872 0.870668i \(-0.336313\pi\)
0.491872 + 0.870668i \(0.336313\pi\)
\(602\) −119061. −0.328530
\(603\) 1.47383e6i 4.05334i
\(604\) −271699. −0.744757
\(605\) 119460.i 0.326370i
\(606\) −241285. −0.657030
\(607\) −226640. −0.615119 −0.307559 0.951529i \(-0.599512\pi\)
−0.307559 + 0.951529i \(0.599512\pi\)
\(608\) 19677.6i 0.0532311i
\(609\) 443228.i 1.19507i
\(610\) −17473.3 −0.0469587
\(611\) 170369. 0.456361
\(612\) 528079.i 1.40993i
\(613\) 297657.i 0.792127i −0.918223 0.396064i \(-0.870376\pi\)
0.918223 0.396064i \(-0.129624\pi\)
\(614\) 419604. 1.11302
\(615\) 523610.i 1.38439i
\(616\) 98721.5 0.260166
\(617\) 614052.i 1.61300i 0.591234 + 0.806500i \(0.298641\pi\)
−0.591234 + 0.806500i \(0.701359\pi\)
\(618\) 434096.i 1.13660i
\(619\) 622814.i 1.62546i 0.582638 + 0.812732i \(0.302021\pi\)
−0.582638 + 0.812732i \(0.697979\pi\)
\(620\) 44847.5i 0.116669i
\(621\) −236477. + 1.29797e6i −0.613205 + 3.36575i
\(622\) 379965. 0.982115
\(623\) −768621. −1.98032
\(624\) −104829. −0.269224
\(625\) 15625.0 0.0400000
\(626\) 65099.5i 0.166123i
\(627\) −119378. −0.303662
\(628\) 8177.35i 0.0207345i
\(629\) 558277. 1.41107
\(630\) 491002. 1.23709
\(631\) 655660.i 1.64672i −0.567519 0.823360i \(-0.692097\pi\)
0.567519 0.823360i \(-0.307903\pi\)
\(632\) 268897.i 0.673213i
\(633\) 767494. 1.91544
\(634\) 314527. 0.782491
\(635\) 179864.i 0.446064i
\(636\) 120999.i 0.299135i
\(637\) 226132. 0.557292
\(638\) 65107.9i 0.159953i
\(639\) −1.74617e6 −4.27647
\(640\) 16190.9i 0.0395285i
\(641\) 703980.i 1.71334i −0.515863 0.856671i \(-0.672529\pi\)
0.515863 0.856671i \(-0.327471\pi\)
\(642\) 1.11320e6i 2.70086i
\(643\) 303481.i 0.734023i −0.930217 0.367011i \(-0.880381\pi\)
0.930217 0.367011i \(-0.119619\pi\)
\(644\) 288797. + 52615.8i 0.696340 + 0.126866i
\(645\) 118462. 0.284748
\(646\) 90668.5 0.217266
\(647\) −463400. −1.10700 −0.553500 0.832849i \(-0.686708\pi\)
−0.553500 + 0.832849i \(0.686708\pi\)
\(648\) 575047. 1.36947
\(649\) 278359.i 0.660870i
\(650\) 33168.0 0.0785041
\(651\) 607253.i 1.43287i
\(652\) 92394.5 0.217346
\(653\) −580918. −1.36235 −0.681175 0.732121i \(-0.738531\pi\)
−0.681175 + 0.732121i \(0.738531\pi\)
\(654\) 257388.i 0.601774i
\(655\) 235409.i 0.548707i
\(656\) −171670. −0.398920
\(657\) 454504. 1.05295
\(658\) 356295.i 0.822921i
\(659\) 846482.i 1.94916i −0.224046 0.974579i \(-0.571926\pi\)
0.224046 0.974579i \(-0.428074\pi\)
\(660\) −98225.1 −0.225494
\(661\) 14348.8i 0.0328407i −0.999865 0.0164203i \(-0.994773\pi\)
0.999865 0.0164203i \(-0.00522699\pi\)
\(662\) −232112. −0.529640
\(663\) 483022.i 1.09885i
\(664\) 210492.i 0.477419i
\(665\) 84302.5i 0.190633i
\(666\) 1.19861e6i 2.70226i
\(667\) −34700.7 + 190465.i −0.0779986 + 0.428118i
\(668\) 332479. 0.745095
\(669\) 589769. 1.31774
\(670\) −208211. −0.463824
\(671\) −34754.8 −0.0771915
\(672\) 219231.i 0.485471i
\(673\) 201489. 0.444857 0.222428 0.974949i \(-0.428602\pi\)
0.222428 + 0.974949i \(0.428602\pi\)
\(674\) 9195.97i 0.0202431i
\(675\) −311753. −0.684231
\(676\) −158081. −0.345928
\(677\) 127799.i 0.278836i 0.990234 + 0.139418i \(0.0445232\pi\)
−0.990234 + 0.139418i \(0.955477\pi\)
\(678\) 651986.i 1.41833i
\(679\) −641830. −1.39213
\(680\) 74602.6 0.161338
\(681\) 443069.i 0.955382i
\(682\) 89202.5i 0.191782i
\(683\) −840247. −1.80122 −0.900608 0.434632i \(-0.856878\pi\)
−0.900608 + 0.434632i \(0.856878\pi\)
\(684\) 194663.i 0.416074i
\(685\) −389252. −0.829563
\(686\) 1853.72i 0.00393909i
\(687\) 1.46748e6i 3.10928i
\(688\) 38838.8i 0.0820519i
\(689\) 81267.4i 0.171190i
\(690\) −287345. 52351.3i −0.603539 0.109959i
\(691\) 304935. 0.638632 0.319316 0.947648i \(-0.396547\pi\)
0.319316 + 0.947648i \(0.396547\pi\)
\(692\) 254333. 0.531117
\(693\) 976612. 2.03355
\(694\) 90926.8 0.188787
\(695\) 147627.i 0.305630i
\(696\) 144585. 0.298473
\(697\) 791003.i 1.62822i
\(698\) 561526. 1.15255
\(699\) 562716. 1.15169
\(700\) 69364.6i 0.141560i
\(701\) 68580.5i 0.139561i −0.997562 0.0697806i \(-0.977770\pi\)
0.997562 0.0697806i \(-0.0222299\pi\)
\(702\) −661774. −1.34287
\(703\) −205794. −0.416412
\(704\) 32203.9i 0.0649776i
\(705\) 354504.i 0.713251i
\(706\) −87948.7 −0.176449
\(707\) 338910.i 0.678025i
\(708\) 618152. 1.23319
\(709\) 883225.i 1.75703i −0.477716 0.878514i \(-0.658535\pi\)
0.477716 0.878514i \(-0.341465\pi\)
\(710\) 246684.i 0.489356i
\(711\) 2.66009e6i 5.26208i
\(712\) 250732.i 0.494595i
\(713\) 47542.5 260950.i 0.0935196 0.513309i
\(714\) −1.01015e6 −1.98148
\(715\) 65971.7 0.129046
\(716\) −230195. −0.449025
\(717\) 525927. 1.02303
\(718\) 598272.i 1.16051i
\(719\) 612080. 1.18400 0.591998 0.805939i \(-0.298339\pi\)
0.591998 + 0.805939i \(0.298339\pi\)
\(720\) 160170.i 0.308969i
\(721\) −609733. −1.17292
\(722\) 335181. 0.642991
\(723\) 271488.i 0.519367i
\(724\) 444670.i 0.848322i
\(725\) −45746.7 −0.0870330
\(726\) 527655. 1.00110
\(727\) 516461.i 0.977166i 0.872517 + 0.488583i \(0.162486\pi\)
−0.872517 + 0.488583i \(0.837514\pi\)
\(728\) 147244.i 0.277827i
\(729\) 2.16156e6 4.06736
\(730\) 64208.4i 0.120489i
\(731\) −178957. −0.334900
\(732\) 77179.9i 0.144040i
\(733\) 758127.i 1.41102i 0.708698 + 0.705512i \(0.249283\pi\)
−0.708698 + 0.705512i \(0.750717\pi\)
\(734\) 102644.i 0.190520i
\(735\) 470535.i 0.870998i
\(736\) −17163.8 + 94208.5i −0.0316853 + 0.173914i
\(737\) −414135. −0.762441
\(738\) −1.69826e6 −3.11811
\(739\) 641756. 1.17512 0.587558 0.809182i \(-0.300089\pi\)
0.587558 + 0.809182i \(0.300089\pi\)
\(740\) −169329. −0.309220
\(741\) 178053.i 0.324275i
\(742\) −169955. −0.308693
\(743\) 338378.i 0.612950i 0.951879 + 0.306475i \(0.0991495\pi\)
−0.951879 + 0.306475i \(0.900850\pi\)
\(744\) −198092. −0.357866
\(745\) −38978.1 −0.0702276
\(746\) 376756.i 0.676990i
\(747\) 2.08231e6i 3.73168i
\(748\) 148386. 0.265210
\(749\) −1.56360e6 −2.78717
\(750\) 69015.8i 0.122695i
\(751\) 415430.i 0.736577i 0.929712 + 0.368288i \(0.120056\pi\)
−0.929712 + 0.368288i \(0.879944\pi\)
\(752\) −116227. −0.205528
\(753\) 1.03134e6i 1.81891i
\(754\) −97108.9 −0.170811
\(755\) 379711.i 0.666131i
\(756\) 1.38398e6i 2.42150i
\(757\) 163552.i 0.285407i −0.989766 0.142703i \(-0.954421\pi\)
0.989766 0.142703i \(-0.0455795\pi\)
\(758\) 645869.i 1.12410i
\(759\) −571535. 104128.i −0.992108 0.180752i
\(760\) −27500.3 −0.0476113
\(761\) −715636. −1.23573 −0.617864 0.786285i \(-0.712002\pi\)
−0.617864 + 0.786285i \(0.712002\pi\)
\(762\) −794463. −1.36824
\(763\) −361529. −0.621003
\(764\) 72954.0i 0.124986i
\(765\) 738013. 1.26108
\(766\) 210748.i 0.359174i
\(767\) −415174. −0.705732
\(768\) 71515.3 0.121248
\(769\) 896493.i 1.51598i 0.652264 + 0.757991i \(0.273819\pi\)
−0.652264 + 0.757991i \(0.726181\pi\)
\(770\) 137967.i 0.232699i
\(771\) 1.73360e6 2.91635
\(772\) 9623.46 0.0161472
\(773\) 171722.i 0.287387i 0.989622 + 0.143694i \(0.0458980\pi\)
−0.989622 + 0.143694i \(0.954102\pi\)
\(774\) 384216.i 0.641348i
\(775\) 62676.3 0.104352
\(776\) 209371.i 0.347691i
\(777\) 2.29278e6 3.79770
\(778\) 117332.i 0.193847i
\(779\) 291583.i 0.480493i
\(780\) 146503.i 0.240801i
\(781\) 490660.i 0.804412i
\(782\) 434084. + 79085.6i 0.709840 + 0.129325i
\(783\) 912748. 1.48877
\(784\) −154269. −0.250984
\(785\) −11428.2 −0.0185455
\(786\) −1.03980e6 −1.68309
\(787\) 160006.i 0.258337i 0.991623 + 0.129169i \(0.0412308\pi\)
−0.991623 + 0.129169i \(0.958769\pi\)
\(788\) −555460. −0.894542
\(789\) 649826.i 1.04386i
\(790\) −375795. −0.602140
\(791\) 915782. 1.46366
\(792\) 318580.i 0.507889i
\(793\) 51837.0i 0.0824315i
\(794\) −15589.9 −0.0247288
\(795\) 169101. 0.267554
\(796\) 140093.i 0.221101i
\(797\) 442182.i 0.696121i 0.937472 + 0.348060i \(0.113160\pi\)
−0.937472 + 0.348060i \(0.886840\pi\)
\(798\) 372365. 0.584741
\(799\) 535539.i 0.838875i
\(800\) −22627.4 −0.0353553
\(801\) 2.48039e6i 3.86594i
\(802\) 585497.i 0.910282i
\(803\) 127712.i 0.198061i
\(804\) 919669.i 1.42272i
\(805\) −73532.9 + 403606.i −0.113472 + 0.622825i
\(806\) 133046. 0.204801
\(807\) −186960. −0.287079
\(808\) −110556. −0.169340
\(809\) −146806. −0.224309 −0.112154 0.993691i \(-0.535775\pi\)
−0.112154 + 0.993691i \(0.535775\pi\)
\(810\) 803653.i 1.22489i
\(811\) −505788. −0.769001 −0.384500 0.923125i \(-0.625626\pi\)
−0.384500 + 0.923125i \(0.625626\pi\)
\(812\) 203085.i 0.308011i
\(813\) −328864. −0.497548
\(814\) −336798. −0.508301
\(815\) 129125.i 0.194400i
\(816\) 329521.i 0.494883i
\(817\) 65968.0 0.0988301
\(818\) −236205. −0.353006
\(819\) 1.45662e6i 2.17160i
\(820\) 239916.i 0.356805i
\(821\) 79249.8 0.117574 0.0587870 0.998271i \(-0.481277\pi\)
0.0587870 + 0.998271i \(0.481277\pi\)
\(822\) 1.71933e6i 2.54458i
\(823\) 740309. 1.09298 0.546491 0.837465i \(-0.315963\pi\)
0.546491 + 0.837465i \(0.315963\pi\)
\(824\) 198901.i 0.292943i
\(825\) 137274.i 0.201688i
\(826\) 868259.i 1.27259i
\(827\) 274955.i 0.402023i 0.979589 + 0.201011i \(0.0644228\pi\)
−0.979589 + 0.201011i \(0.935577\pi\)
\(828\) −169795. + 931966.i −0.247664 + 1.35938i
\(829\) −885762. −1.28887 −0.644433 0.764661i \(-0.722907\pi\)
−0.644433 + 0.764661i \(0.722907\pi\)
\(830\) 294172. 0.427016
\(831\) 475939. 0.689207
\(832\) −48032.4 −0.0693885
\(833\) 710824.i 1.02441i
\(834\) 652070. 0.937480
\(835\) 464654.i 0.666433i
\(836\) −54698.6 −0.0782643
\(837\) −1.25053e6 −1.78502
\(838\) 531924.i 0.757464i
\(839\) 78938.5i 0.112141i −0.998427 0.0560706i \(-0.982143\pi\)
0.998427 0.0560706i \(-0.0178572\pi\)
\(840\) 306384. 0.434218
\(841\) −573344. −0.810631
\(842\) 331337.i 0.467354i
\(843\) 334845.i 0.471183i
\(844\) 351663. 0.493675
\(845\) 220924.i 0.309407i
\(846\) −1.14979e6 −1.60648
\(847\) 741147.i 1.03309i
\(848\) 55441.1i 0.0770975i
\(849\) 1.06392e6i 1.47603i
\(850\) 104260.i 0.144305i
\(851\) −985260. 179504.i −1.36048 0.247865i
\(852\) −1.08961e6 −1.50104
\(853\) −173285. −0.238156 −0.119078 0.992885i \(-0.537994\pi\)
−0.119078 + 0.992885i \(0.537994\pi\)
\(854\) 108407. 0.148642
\(855\) −272050. −0.372148
\(856\) 510063.i 0.696107i
\(857\) 550193. 0.749124 0.374562 0.927202i \(-0.377793\pi\)
0.374562 + 0.927202i \(0.377793\pi\)
\(858\) 291398.i 0.395833i
\(859\) 175899. 0.238384 0.119192 0.992871i \(-0.461970\pi\)
0.119192 + 0.992871i \(0.461970\pi\)
\(860\) 54278.8 0.0733895
\(861\) 3.24856e6i 4.38212i
\(862\) 573887.i 0.772346i
\(863\) −1.28923e6 −1.73105 −0.865524 0.500867i \(-0.833015\pi\)
−0.865524 + 0.500867i \(0.833015\pi\)
\(864\) 451466. 0.604781
\(865\) 355441.i 0.475045i
\(866\) 268987.i 0.358670i
\(867\) −60073.6 −0.0799182
\(868\) 278241.i 0.369302i
\(869\) −747464. −0.989807
\(870\) 202064.i 0.266963i
\(871\) 617684.i 0.814199i
\(872\) 117934.i 0.155098i
\(873\) 2.07123e6i 2.71768i
\(874\) −160014. 29152.9i −0.209476 0.0381644i
\(875\) −96940.0 −0.126615
\(876\) 283610. 0.369584
\(877\) −988741. −1.28553 −0.642767 0.766062i \(-0.722214\pi\)
−0.642767 + 0.766062i \(0.722214\pi\)
\(878\) −113855. −0.147694
\(879\) 1.21675e6i 1.57480i
\(880\) −45006.4 −0.0581177
\(881\) 963126.i 1.24088i −0.784252 0.620442i \(-0.786953\pi\)
0.784252 0.620442i \(-0.213047\pi\)
\(882\) −1.52612e6 −1.96178
\(883\) −821811. −1.05402 −0.527012 0.849858i \(-0.676688\pi\)
−0.527012 + 0.849858i \(0.676688\pi\)
\(884\) 221319.i 0.283213i
\(885\) 863894.i 1.10300i
\(886\) 659838. 0.840562
\(887\) −163425. −0.207717 −0.103858 0.994592i \(-0.533119\pi\)
−0.103858 + 0.994592i \(0.533119\pi\)
\(888\) 747928.i 0.948492i
\(889\) 1.11591e6i 1.41197i
\(890\) 350408. 0.442379
\(891\) 1.59848e6i 2.01350i
\(892\) 270230. 0.339628
\(893\) 197412.i 0.247555i
\(894\) 172167.i 0.215414i
\(895\) 321708.i 0.401620i
\(896\) 100451.i 0.125123i
\(897\) 155307. 852448.i 0.193022 1.05946i
\(898\) −790145. −0.979838
\(899\) −183503. −0.227051
\(900\) −223844. −0.276351
\(901\) −255456. −0.314678
\(902\) 477197.i 0.586523i
\(903\) −734958. −0.901336
\(904\) 298737.i 0.365555i
\(905\) −621445. −0.758762
\(906\) −1.67719e6 −2.04327
\(907\) 778506.i 0.946341i 0.880971 + 0.473170i \(0.156890\pi\)
−0.880971 + 0.473170i \(0.843110\pi\)
\(908\) 203012.i 0.246236i
\(909\) −1.09368e6 −1.32362
\(910\) −205779. −0.248496
\(911\) 631880.i 0.761374i −0.924704 0.380687i \(-0.875688\pi\)
0.924704 0.380687i \(-0.124312\pi\)
\(912\) 121469.i 0.146042i
\(913\) 585113. 0.701937
\(914\) 773632.i 0.926066i
\(915\) −107862. −0.128833
\(916\) 672395.i 0.801370i
\(917\) 1.46051e6i 1.73687i
\(918\) 2.08022e6i 2.46845i
\(919\) 329548.i 0.390201i −0.980783 0.195100i \(-0.937497\pi\)
0.980783 0.195100i \(-0.0625032\pi\)
\(920\) −131660. 23987.1i −0.155553 0.0283402i
\(921\) 2.59020e6 3.05361
\(922\) −570469. −0.671074
\(923\) 731822. 0.859018
\(924\) 609404. 0.713775
\(925\) 236644.i 0.276575i
\(926\) 508556. 0.593085
\(927\) 1.96765e6i 2.28975i
\(928\) 66248.3 0.0769271
\(929\) 392594. 0.454896 0.227448 0.973790i \(-0.426962\pi\)
0.227448 + 0.973790i \(0.426962\pi\)
\(930\) 276842.i 0.320085i
\(931\) 262027.i 0.302306i
\(932\) 257834. 0.296830
\(933\) 2.34551e6 2.69447
\(934\) 860656.i 0.986588i
\(935\) 207376.i 0.237211i
\(936\) −475165. −0.542366
\(937\) 432138.i 0.492202i 0.969244 + 0.246101i \(0.0791496\pi\)
−0.969244 + 0.246101i \(0.920850\pi\)
\(938\) 1.29177e6 1.46818
\(939\) 401857.i 0.455764i
\(940\) 162432.i 0.183830i
\(941\) 35467.3i 0.0400543i 0.999799 + 0.0200272i \(0.00637527\pi\)
−0.999799 + 0.0200272i \(0.993625\pi\)
\(942\) 50478.5i 0.0568859i
\(943\) 254333. 1.39598e6i 0.286009 1.56984i
\(944\) 283235. 0.317836
\(945\) 1.93416e6 2.16586
\(946\) 107962. 0.120639
\(947\) 552193. 0.615731 0.307865 0.951430i \(-0.400385\pi\)
0.307865 + 0.951430i \(0.400385\pi\)
\(948\) 1.65989e6i 1.84699i
\(949\) −190483. −0.211507
\(950\) 38432.9i 0.0425849i
\(951\) 1.94156e6 2.14680
\(952\) −462846. −0.510697
\(953\) 894272.i 0.984654i 0.870410 + 0.492327i \(0.163854\pi\)
−0.870410 + 0.492327i \(0.836146\pi\)
\(954\) 548457.i 0.602623i
\(955\) 101956. 0.111791
\(956\) 240977. 0.263670
\(957\) 401909.i 0.438838i
\(958\) 412984.i 0.449989i
\(959\) 2.41498e6 2.62589
\(960\) 99945.6i 0.108448i
\(961\) −672109. −0.727768
\(962\) 502337.i 0.542806i
\(963\) 5.04585e6i 5.44103i
\(964\) 124395.i 0.133859i
\(965\) 13449.2i 0.0144425i
\(966\) 1.78273e6 + 324796.i 1.91044 + 0.348062i
\(967\) 984833. 1.05320 0.526599 0.850114i \(-0.323467\pi\)
0.526599 + 0.850114i \(0.323467\pi\)
\(968\) 241770. 0.258019
\(969\) 559694. 0.596078
\(970\) 292605. 0.310984
\(971\) 358995.i 0.380758i −0.981711 0.190379i \(-0.939028\pi\)
0.981711 0.190379i \(-0.0609718\pi\)
\(972\) 1.93362e6 2.04663
\(973\) 915901.i 0.967437i
\(974\) 860403. 0.906951
\(975\) 204745. 0.215379
\(976\) 35363.5i 0.0371241i
\(977\) 31385.8i 0.0328810i −0.999865 0.0164405i \(-0.994767\pi\)
0.999865 0.0164405i \(-0.00523340\pi\)
\(978\) 570348. 0.596296
\(979\) 696969. 0.727190
\(980\) 215597.i 0.224487i
\(981\) 1.16668e6i 1.21231i
\(982\) 201594. 0.209053
\(983\) 1.50797e6i 1.56058i 0.625416 + 0.780292i \(0.284929\pi\)
−0.625416 + 0.780292i \(0.715071\pi\)
\(984\) −1.05971e6 −1.09445
\(985\) 776280.i 0.800103i
\(986\) 305252.i 0.313982i
\(987\) 2.19940e6i 2.25772i
\(988\) 81583.4i 0.0835772i
\(989\) 315828. + 57540.6i 0.322893 + 0.0588277i
\(990\) −445230. −0.454270
\(991\) −1.10604e6 −1.12622 −0.563110 0.826382i \(-0.690395\pi\)
−0.563110 + 0.826382i \(0.690395\pi\)
\(992\) −90764.9 −0.0922348
\(993\) −1.43282e6 −1.45309
\(994\) 1.53047e6i 1.54900i
\(995\) 195786. 0.197758
\(996\) 1.29936e6i 1.30982i
\(997\) −960399. −0.966187 −0.483094 0.875569i \(-0.660487\pi\)
−0.483094 + 0.875569i \(0.660487\pi\)
\(998\) 897519. 0.901120
\(999\) 4.72157e6i 4.73103i
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 230.5.d.a.91.17 32
23.22 odd 2 inner 230.5.d.a.91.32 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.5.d.a.91.17 32 1.1 even 1 trivial
230.5.d.a.91.32 yes 32 23.22 odd 2 inner