Properties

Label 5070.2.b.v.1351.5
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5070,2,Mod(1351,5070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5070.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not 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: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.5
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.v.1351.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.55496i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.55496i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.85086i q^{11} +1.00000 q^{12} -2.55496 q^{14} +1.00000i q^{15} +1.00000 q^{16} +7.40581 q^{17} +1.00000i q^{18} +2.93900i q^{19} +1.00000i q^{20} -2.55496i q^{21} -3.85086 q^{22} +1.80194 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.55496i q^{28} +8.03684 q^{29} -1.00000 q^{30} -1.13706i q^{31} +1.00000i q^{32} -3.85086i q^{33} +7.40581i q^{34} +2.55496 q^{35} -1.00000 q^{36} -3.44504i q^{37} -2.93900 q^{38} -1.00000 q^{40} +1.55496i q^{41} +2.55496 q^{42} +10.4547 q^{43} -3.85086i q^{44} -1.00000i q^{45} +1.80194i q^{46} -9.71379i q^{47} -1.00000 q^{48} +0.472189 q^{49} -1.00000i q^{50} -7.40581 q^{51} -8.93900 q^{53} -1.00000i q^{54} +3.85086 q^{55} +2.55496 q^{56} -2.93900i q^{57} +8.03684i q^{58} -11.7845i q^{59} -1.00000i q^{60} -10.0586 q^{61} +1.13706 q^{62} +2.55496i q^{63} -1.00000 q^{64} +3.85086 q^{66} -1.36227i q^{67} -7.40581 q^{68} -1.80194 q^{69} +2.55496i q^{70} +12.0761i q^{71} -1.00000i q^{72} +2.29590i q^{73} +3.44504 q^{74} +1.00000 q^{75} -2.93900i q^{76} -9.83877 q^{77} +8.81700 q^{79} -1.00000i q^{80} +1.00000 q^{81} -1.55496 q^{82} +15.9051i q^{83} +2.55496i q^{84} -7.40581i q^{85} +10.4547i q^{86} -8.03684 q^{87} +3.85086 q^{88} -15.6799i q^{89} +1.00000 q^{90} -1.80194 q^{92} +1.13706i q^{93} +9.71379 q^{94} +2.93900 q^{95} -1.00000i q^{96} -1.98254i q^{97} +0.472189i q^{98} +3.85086i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} + 18 q^{17} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 6 q^{27} - 8 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 16 q^{42} + 18 q^{43} - 6 q^{48} - 10 q^{49} - 18 q^{51} - 34 q^{53} - 4 q^{55} + 16 q^{56} + 2 q^{61} - 4 q^{62} - 6 q^{64} - 4 q^{66} - 18 q^{68} - 2 q^{69} + 20 q^{74} + 6 q^{75} + 6 q^{77} - 6 q^{79} + 6 q^{81} - 10 q^{82} + 8 q^{87} - 4 q^{88} + 6 q^{90} - 2 q^{92} + 42 q^{94} - 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) 2.55496i 0.965683i 0.875708 + 0.482842i \(0.160395\pi\)
−0.875708 + 0.482842i \(0.839605\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.85086i 1.16108i 0.814233 + 0.580538i \(0.197158\pi\)
−0.814233 + 0.580538i \(0.802842\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −2.55496 −0.682841
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 7.40581 1.79617 0.898087 0.439818i \(-0.144957\pi\)
0.898087 + 0.439818i \(0.144957\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.93900i 0.674253i 0.941459 + 0.337127i \(0.109455\pi\)
−0.941459 + 0.337127i \(0.890545\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 2.55496i − 0.557538i
\(22\) −3.85086 −0.821005
\(23\) 1.80194 0.375730 0.187865 0.982195i \(-0.439843\pi\)
0.187865 + 0.982195i \(0.439843\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 2.55496i − 0.482842i
\(29\) 8.03684 1.49240 0.746201 0.665720i \(-0.231876\pi\)
0.746201 + 0.665720i \(0.231876\pi\)
\(30\) −1.00000 −0.182574
\(31\) − 1.13706i − 0.204223i −0.994773 0.102111i \(-0.967440\pi\)
0.994773 0.102111i \(-0.0325598\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.85086i − 0.670348i
\(34\) 7.40581i 1.27009i
\(35\) 2.55496 0.431867
\(36\) −1.00000 −0.166667
\(37\) − 3.44504i − 0.566361i −0.959067 0.283181i \(-0.908610\pi\)
0.959067 0.283181i \(-0.0913896\pi\)
\(38\) −2.93900 −0.476769
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 1.55496i 0.242844i 0.992601 + 0.121422i \(0.0387454\pi\)
−0.992601 + 0.121422i \(0.961255\pi\)
\(42\) 2.55496 0.394239
\(43\) 10.4547 1.59433 0.797166 0.603761i \(-0.206332\pi\)
0.797166 + 0.603761i \(0.206332\pi\)
\(44\) − 3.85086i − 0.580538i
\(45\) − 1.00000i − 0.149071i
\(46\) 1.80194i 0.265681i
\(47\) − 9.71379i − 1.41690i −0.705760 0.708451i \(-0.749394\pi\)
0.705760 0.708451i \(-0.250606\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0.472189 0.0674556
\(50\) − 1.00000i − 0.141421i
\(51\) −7.40581 −1.03702
\(52\) 0 0
\(53\) −8.93900 −1.22787 −0.613933 0.789358i \(-0.710414\pi\)
−0.613933 + 0.789358i \(0.710414\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 3.85086 0.519249
\(56\) 2.55496 0.341421
\(57\) − 2.93900i − 0.389280i
\(58\) 8.03684i 1.05529i
\(59\) − 11.7845i − 1.53421i −0.641522 0.767104i \(-0.721697\pi\)
0.641522 0.767104i \(-0.278303\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −10.0586 −1.28787 −0.643936 0.765079i \(-0.722700\pi\)
−0.643936 + 0.765079i \(0.722700\pi\)
\(62\) 1.13706 0.144407
\(63\) 2.55496i 0.321894i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.85086 0.474007
\(67\) − 1.36227i − 0.166428i −0.996532 0.0832140i \(-0.973481\pi\)
0.996532 0.0832140i \(-0.0265185\pi\)
\(68\) −7.40581 −0.898087
\(69\) −1.80194 −0.216928
\(70\) 2.55496i 0.305376i
\(71\) 12.0761i 1.43317i 0.697502 + 0.716583i \(0.254295\pi\)
−0.697502 + 0.716583i \(0.745705\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.29590i 0.268714i 0.990933 + 0.134357i \(0.0428970\pi\)
−0.990933 + 0.134357i \(0.957103\pi\)
\(74\) 3.44504 0.400478
\(75\) 1.00000 0.115470
\(76\) − 2.93900i − 0.337127i
\(77\) −9.83877 −1.12123
\(78\) 0 0
\(79\) 8.81700 0.991990 0.495995 0.868325i \(-0.334803\pi\)
0.495995 + 0.868325i \(0.334803\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −1.55496 −0.171716
\(83\) 15.9051i 1.74582i 0.487884 + 0.872908i \(0.337769\pi\)
−0.487884 + 0.872908i \(0.662231\pi\)
\(84\) 2.55496i 0.278769i
\(85\) − 7.40581i − 0.803273i
\(86\) 10.4547i 1.12736i
\(87\) −8.03684 −0.861639
\(88\) 3.85086 0.410503
\(89\) − 15.6799i − 1.66207i −0.556220 0.831035i \(-0.687749\pi\)
0.556220 0.831035i \(-0.312251\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.80194 −0.187865
\(93\) 1.13706i 0.117908i
\(94\) 9.71379 1.00190
\(95\) 2.93900 0.301535
\(96\) − 1.00000i − 0.102062i
\(97\) − 1.98254i − 0.201297i −0.994922 0.100648i \(-0.967908\pi\)
0.994922 0.100648i \(-0.0320917\pi\)
\(98\) 0.472189i 0.0476983i
\(99\) 3.85086i 0.387025i
\(100\) 1.00000 0.100000
\(101\) 3.48188 0.346460 0.173230 0.984881i \(-0.444580\pi\)
0.173230 + 0.984881i \(0.444580\pi\)
\(102\) − 7.40581i − 0.733285i
\(103\) −8.65817 −0.853115 −0.426557 0.904460i \(-0.640274\pi\)
−0.426557 + 0.904460i \(0.640274\pi\)
\(104\) 0 0
\(105\) −2.55496 −0.249338
\(106\) − 8.93900i − 0.868233i
\(107\) 13.9366 1.34730 0.673651 0.739049i \(-0.264725\pi\)
0.673651 + 0.739049i \(0.264725\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 5.39373i − 0.516626i −0.966061 0.258313i \(-0.916833\pi\)
0.966061 0.258313i \(-0.0831665\pi\)
\(110\) 3.85086i 0.367165i
\(111\) 3.44504i 0.326989i
\(112\) 2.55496i 0.241421i
\(113\) 9.44265 0.888290 0.444145 0.895955i \(-0.353508\pi\)
0.444145 + 0.895955i \(0.353508\pi\)
\(114\) 2.93900 0.275263
\(115\) − 1.80194i − 0.168032i
\(116\) −8.03684 −0.746201
\(117\) 0 0
\(118\) 11.7845 1.08485
\(119\) 18.9215i 1.73453i
\(120\) 1.00000 0.0912871
\(121\) −3.82908 −0.348099
\(122\) − 10.0586i − 0.910663i
\(123\) − 1.55496i − 0.140206i
\(124\) 1.13706i 0.102111i
\(125\) 1.00000i 0.0894427i
\(126\) −2.55496 −0.227614
\(127\) 0.499336 0.0443089 0.0221545 0.999755i \(-0.492947\pi\)
0.0221545 + 0.999755i \(0.492947\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −10.4547 −0.920488
\(130\) 0 0
\(131\) −1.12498 −0.0982901 −0.0491451 0.998792i \(-0.515650\pi\)
−0.0491451 + 0.998792i \(0.515650\pi\)
\(132\) 3.85086i 0.335174i
\(133\) −7.50902 −0.651115
\(134\) 1.36227 0.117682
\(135\) 1.00000i 0.0860663i
\(136\) − 7.40581i − 0.635043i
\(137\) 14.1468i 1.20864i 0.796742 + 0.604319i \(0.206555\pi\)
−0.796742 + 0.604319i \(0.793445\pi\)
\(138\) − 1.80194i − 0.153391i
\(139\) 7.17629 0.608685 0.304343 0.952563i \(-0.401563\pi\)
0.304343 + 0.952563i \(0.401563\pi\)
\(140\) −2.55496 −0.215933
\(141\) 9.71379i 0.818049i
\(142\) −12.0761 −1.01340
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 8.03684i − 0.667423i
\(146\) −2.29590 −0.190010
\(147\) −0.472189 −0.0389455
\(148\) 3.44504i 0.283181i
\(149\) − 3.25667i − 0.266797i −0.991063 0.133398i \(-0.957411\pi\)
0.991063 0.133398i \(-0.0425890\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 0.137063i 0.0111541i 0.999984 + 0.00557703i \(0.00177523\pi\)
−0.999984 + 0.00557703i \(0.998225\pi\)
\(152\) 2.93900 0.238384
\(153\) 7.40581 0.598725
\(154\) − 9.83877i − 0.792831i
\(155\) −1.13706 −0.0913311
\(156\) 0 0
\(157\) 4.35152 0.347289 0.173645 0.984808i \(-0.444446\pi\)
0.173645 + 0.984808i \(0.444446\pi\)
\(158\) 8.81700i 0.701443i
\(159\) 8.93900 0.708909
\(160\) 1.00000 0.0790569
\(161\) 4.60388i 0.362836i
\(162\) 1.00000i 0.0785674i
\(163\) 11.6189i 0.910066i 0.890475 + 0.455033i \(0.150372\pi\)
−0.890475 + 0.455033i \(0.849628\pi\)
\(164\) − 1.55496i − 0.121422i
\(165\) −3.85086 −0.299789
\(166\) −15.9051 −1.23448
\(167\) 20.8146i 1.61068i 0.592811 + 0.805341i \(0.298018\pi\)
−0.592811 + 0.805341i \(0.701982\pi\)
\(168\) −2.55496 −0.197119
\(169\) 0 0
\(170\) 7.40581 0.568000
\(171\) 2.93900i 0.224751i
\(172\) −10.4547 −0.797166
\(173\) 8.89440 0.676228 0.338114 0.941105i \(-0.390211\pi\)
0.338114 + 0.941105i \(0.390211\pi\)
\(174\) − 8.03684i − 0.609271i
\(175\) − 2.55496i − 0.193137i
\(176\) 3.85086i 0.290269i
\(177\) 11.7845i 0.885776i
\(178\) 15.6799 1.17526
\(179\) −0.188374 −0.0140797 −0.00703985 0.999975i \(-0.502241\pi\)
−0.00703985 + 0.999975i \(0.502241\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −18.1588 −1.34974 −0.674868 0.737939i \(-0.735799\pi\)
−0.674868 + 0.737939i \(0.735799\pi\)
\(182\) 0 0
\(183\) 10.0586 0.743554
\(184\) − 1.80194i − 0.132841i
\(185\) −3.44504 −0.253285
\(186\) −1.13706 −0.0833735
\(187\) 28.5187i 2.08549i
\(188\) 9.71379i 0.708451i
\(189\) − 2.55496i − 0.185846i
\(190\) 2.93900i 0.213218i
\(191\) 0.469205 0.0339505 0.0169752 0.999856i \(-0.494596\pi\)
0.0169752 + 0.999856i \(0.494596\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 17.9312i − 1.29072i −0.763879 0.645359i \(-0.776708\pi\)
0.763879 0.645359i \(-0.223292\pi\)
\(194\) 1.98254 0.142338
\(195\) 0 0
\(196\) −0.472189 −0.0337278
\(197\) 8.17390i 0.582366i 0.956667 + 0.291183i \(0.0940489\pi\)
−0.956667 + 0.291183i \(0.905951\pi\)
\(198\) −3.85086 −0.273668
\(199\) 23.7536 1.68385 0.841924 0.539595i \(-0.181423\pi\)
0.841924 + 0.539595i \(0.181423\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 1.36227i 0.0960873i
\(202\) 3.48188i 0.244984i
\(203\) 20.5338i 1.44119i
\(204\) 7.40581 0.518511
\(205\) 1.55496 0.108603
\(206\) − 8.65817i − 0.603243i
\(207\) 1.80194 0.125243
\(208\) 0 0
\(209\) −11.3177 −0.782859
\(210\) − 2.55496i − 0.176309i
\(211\) 24.2392 1.66870 0.834348 0.551238i \(-0.185844\pi\)
0.834348 + 0.551238i \(0.185844\pi\)
\(212\) 8.93900 0.613933
\(213\) − 12.0761i − 0.827438i
\(214\) 13.9366i 0.952687i
\(215\) − 10.4547i − 0.713007i
\(216\) 1.00000i 0.0680414i
\(217\) 2.90515 0.197214
\(218\) 5.39373 0.365310
\(219\) − 2.29590i − 0.155142i
\(220\) −3.85086 −0.259625
\(221\) 0 0
\(222\) −3.44504 −0.231216
\(223\) 22.5579i 1.51059i 0.655384 + 0.755296i \(0.272507\pi\)
−0.655384 + 0.755296i \(0.727493\pi\)
\(224\) −2.55496 −0.170710
\(225\) −1.00000 −0.0666667
\(226\) 9.44265i 0.628116i
\(227\) 23.7439i 1.57594i 0.615714 + 0.787970i \(0.288868\pi\)
−0.615714 + 0.787970i \(0.711132\pi\)
\(228\) 2.93900i 0.194640i
\(229\) − 3.11529i − 0.205864i −0.994688 0.102932i \(-0.967178\pi\)
0.994688 0.102932i \(-0.0328225\pi\)
\(230\) 1.80194 0.118816
\(231\) 9.83877 0.647344
\(232\) − 8.03684i − 0.527644i
\(233\) 12.5603 0.822855 0.411427 0.911443i \(-0.365030\pi\)
0.411427 + 0.911443i \(0.365030\pi\)
\(234\) 0 0
\(235\) −9.71379 −0.633658
\(236\) 11.7845i 0.767104i
\(237\) −8.81700 −0.572726
\(238\) −18.9215 −1.22650
\(239\) 8.52111i 0.551185i 0.961275 + 0.275592i \(0.0888740\pi\)
−0.961275 + 0.275592i \(0.911126\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 11.6136i 0.748095i 0.927409 + 0.374048i \(0.122030\pi\)
−0.927409 + 0.374048i \(0.877970\pi\)
\(242\) − 3.82908i − 0.246143i
\(243\) −1.00000 −0.0641500
\(244\) 10.0586 0.643936
\(245\) − 0.472189i − 0.0301670i
\(246\) 1.55496 0.0991405
\(247\) 0 0
\(248\) −1.13706 −0.0722036
\(249\) − 15.9051i − 1.00795i
\(250\) −1.00000 −0.0632456
\(251\) −22.7463 −1.43573 −0.717867 0.696180i \(-0.754882\pi\)
−0.717867 + 0.696180i \(0.754882\pi\)
\(252\) − 2.55496i − 0.160947i
\(253\) 6.93900i 0.436251i
\(254\) 0.499336i 0.0313311i
\(255\) 7.40581i 0.463770i
\(256\) 1.00000 0.0625000
\(257\) −7.39373 −0.461208 −0.230604 0.973048i \(-0.574070\pi\)
−0.230604 + 0.973048i \(0.574070\pi\)
\(258\) − 10.4547i − 0.650883i
\(259\) 8.80194 0.546926
\(260\) 0 0
\(261\) 8.03684 0.497468
\(262\) − 1.12498i − 0.0695016i
\(263\) −11.7735 −0.725983 −0.362991 0.931792i \(-0.618245\pi\)
−0.362991 + 0.931792i \(0.618245\pi\)
\(264\) −3.85086 −0.237004
\(265\) 8.93900i 0.549118i
\(266\) − 7.50902i − 0.460408i
\(267\) 15.6799i 0.959597i
\(268\) 1.36227i 0.0832140i
\(269\) −22.8810 −1.39508 −0.697539 0.716547i \(-0.745722\pi\)
−0.697539 + 0.716547i \(0.745722\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 2.13036i 0.129410i 0.997904 + 0.0647050i \(0.0206107\pi\)
−0.997904 + 0.0647050i \(0.979389\pi\)
\(272\) 7.40581 0.449043
\(273\) 0 0
\(274\) −14.1468 −0.854637
\(275\) − 3.85086i − 0.232215i
\(276\) 1.80194 0.108464
\(277\) 14.7084 0.883743 0.441872 0.897078i \(-0.354315\pi\)
0.441872 + 0.897078i \(0.354315\pi\)
\(278\) 7.17629i 0.430405i
\(279\) − 1.13706i − 0.0680742i
\(280\) − 2.55496i − 0.152688i
\(281\) − 12.5942i − 0.751306i −0.926760 0.375653i \(-0.877418\pi\)
0.926760 0.375653i \(-0.122582\pi\)
\(282\) −9.71379 −0.578448
\(283\) 1.62565 0.0966346 0.0483173 0.998832i \(-0.484614\pi\)
0.0483173 + 0.998832i \(0.484614\pi\)
\(284\) − 12.0761i − 0.716583i
\(285\) −2.93900 −0.174091
\(286\) 0 0
\(287\) −3.97285 −0.234510
\(288\) 1.00000i 0.0589256i
\(289\) 37.8461 2.22624
\(290\) 8.03684 0.471939
\(291\) 1.98254i 0.116219i
\(292\) − 2.29590i − 0.134357i
\(293\) − 13.0664i − 0.763346i −0.924297 0.381673i \(-0.875348\pi\)
0.924297 0.381673i \(-0.124652\pi\)
\(294\) − 0.472189i − 0.0275386i
\(295\) −11.7845 −0.686119
\(296\) −3.44504 −0.200239
\(297\) − 3.85086i − 0.223449i
\(298\) 3.25667 0.188654
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 26.7114i 1.53962i
\(302\) −0.137063 −0.00788711
\(303\) −3.48188 −0.200029
\(304\) 2.93900i 0.168563i
\(305\) 10.0586i 0.575954i
\(306\) 7.40581i 0.423362i
\(307\) − 26.2489i − 1.49810i −0.662511 0.749052i \(-0.730509\pi\)
0.662511 0.749052i \(-0.269491\pi\)
\(308\) 9.83877 0.560616
\(309\) 8.65817 0.492546
\(310\) − 1.13706i − 0.0645809i
\(311\) −27.1250 −1.53812 −0.769058 0.639179i \(-0.779274\pi\)
−0.769058 + 0.639179i \(0.779274\pi\)
\(312\) 0 0
\(313\) 19.1323 1.08142 0.540710 0.841209i \(-0.318156\pi\)
0.540710 + 0.841209i \(0.318156\pi\)
\(314\) 4.35152i 0.245570i
\(315\) 2.55496 0.143956
\(316\) −8.81700 −0.495995
\(317\) 6.67456i 0.374881i 0.982276 + 0.187440i \(0.0600191\pi\)
−0.982276 + 0.187440i \(0.939981\pi\)
\(318\) 8.93900i 0.501274i
\(319\) 30.9487i 1.73279i
\(320\) 1.00000i 0.0559017i
\(321\) −13.9366 −0.777866
\(322\) −4.60388 −0.256564
\(323\) 21.7657i 1.21108i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −11.6189 −0.643514
\(327\) 5.39373i 0.298274i
\(328\) 1.55496 0.0858582
\(329\) 24.8183 1.36828
\(330\) − 3.85086i − 0.211983i
\(331\) − 18.1360i − 0.996845i −0.866934 0.498422i \(-0.833913\pi\)
0.866934 0.498422i \(-0.166087\pi\)
\(332\) − 15.9051i − 0.872908i
\(333\) − 3.44504i − 0.188787i
\(334\) −20.8146 −1.13892
\(335\) −1.36227 −0.0744289
\(336\) − 2.55496i − 0.139384i
\(337\) −10.8780 −0.592563 −0.296281 0.955101i \(-0.595747\pi\)
−0.296281 + 0.955101i \(0.595747\pi\)
\(338\) 0 0
\(339\) −9.44265 −0.512854
\(340\) 7.40581i 0.401637i
\(341\) 4.37867 0.237118
\(342\) −2.93900 −0.158923
\(343\) 19.0911i 1.03082i
\(344\) − 10.4547i − 0.563681i
\(345\) 1.80194i 0.0970131i
\(346\) 8.89440i 0.478166i
\(347\) −7.42865 −0.398791 −0.199395 0.979919i \(-0.563898\pi\)
−0.199395 + 0.979919i \(0.563898\pi\)
\(348\) 8.03684 0.430820
\(349\) 28.7275i 1.53775i 0.639399 + 0.768875i \(0.279183\pi\)
−0.639399 + 0.768875i \(0.720817\pi\)
\(350\) 2.55496 0.136568
\(351\) 0 0
\(352\) −3.85086 −0.205251
\(353\) 31.2131i 1.66131i 0.556790 + 0.830654i \(0.312033\pi\)
−0.556790 + 0.830654i \(0.687967\pi\)
\(354\) −11.7845 −0.626338
\(355\) 12.0761 0.640931
\(356\) 15.6799i 0.831035i
\(357\) − 18.9215i − 1.00143i
\(358\) − 0.188374i − 0.00995585i
\(359\) 0.0247542i 0.00130648i 1.00000 0.000653238i \(0.000207932\pi\)
−1.00000 0.000653238i \(0.999792\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 10.3623 0.545383
\(362\) − 18.1588i − 0.954407i
\(363\) 3.82908 0.200975
\(364\) 0 0
\(365\) 2.29590 0.120173
\(366\) 10.0586i 0.525772i
\(367\) 17.5284 0.914975 0.457488 0.889216i \(-0.348749\pi\)
0.457488 + 0.889216i \(0.348749\pi\)
\(368\) 1.80194 0.0939325
\(369\) 1.55496i 0.0809479i
\(370\) − 3.44504i − 0.179099i
\(371\) − 22.8388i − 1.18573i
\(372\) − 1.13706i − 0.0589540i
\(373\) −35.3056 −1.82805 −0.914027 0.405653i \(-0.867044\pi\)
−0.914027 + 0.405653i \(0.867044\pi\)
\(374\) −28.5187 −1.47467
\(375\) − 1.00000i − 0.0516398i
\(376\) −9.71379 −0.500951
\(377\) 0 0
\(378\) 2.55496 0.131413
\(379\) − 18.2567i − 0.937782i −0.883256 0.468891i \(-0.844654\pi\)
0.883256 0.468891i \(-0.155346\pi\)
\(380\) −2.93900 −0.150768
\(381\) −0.499336 −0.0255818
\(382\) 0.469205i 0.0240066i
\(383\) 2.83148i 0.144682i 0.997380 + 0.0723409i \(0.0230469\pi\)
−0.997380 + 0.0723409i \(0.976953\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 9.83877i 0.501430i
\(386\) 17.9312 0.912676
\(387\) 10.4547 0.531444
\(388\) 1.98254i 0.100648i
\(389\) −28.8025 −1.46035 −0.730173 0.683262i \(-0.760561\pi\)
−0.730173 + 0.683262i \(0.760561\pi\)
\(390\) 0 0
\(391\) 13.3448 0.674876
\(392\) − 0.472189i − 0.0238491i
\(393\) 1.12498 0.0567478
\(394\) −8.17390 −0.411795
\(395\) − 8.81700i − 0.443632i
\(396\) − 3.85086i − 0.193513i
\(397\) 21.4306i 1.07557i 0.843082 + 0.537785i \(0.180739\pi\)
−0.843082 + 0.537785i \(0.819261\pi\)
\(398\) 23.7536i 1.19066i
\(399\) 7.50902 0.375921
\(400\) −1.00000 −0.0500000
\(401\) 19.5284i 0.975202i 0.873067 + 0.487601i \(0.162128\pi\)
−0.873067 + 0.487601i \(0.837872\pi\)
\(402\) −1.36227 −0.0679440
\(403\) 0 0
\(404\) −3.48188 −0.173230
\(405\) − 1.00000i − 0.0496904i
\(406\) −20.5338 −1.01907
\(407\) 13.2664 0.657589
\(408\) 7.40581i 0.366642i
\(409\) − 30.2707i − 1.49679i −0.663254 0.748394i \(-0.730825\pi\)
0.663254 0.748394i \(-0.269175\pi\)
\(410\) 1.55496i 0.0767939i
\(411\) − 14.1468i − 0.697808i
\(412\) 8.65817 0.426557
\(413\) 30.1089 1.48156
\(414\) 1.80194i 0.0885604i
\(415\) 15.9051 0.780753
\(416\) 0 0
\(417\) −7.17629 −0.351425
\(418\) − 11.3177i − 0.553565i
\(419\) −24.4873 −1.19628 −0.598140 0.801391i \(-0.704093\pi\)
−0.598140 + 0.801391i \(0.704093\pi\)
\(420\) 2.55496 0.124669
\(421\) 8.90408i 0.433959i 0.976176 + 0.216979i \(0.0696204\pi\)
−0.976176 + 0.216979i \(0.930380\pi\)
\(422\) 24.2392i 1.17995i
\(423\) − 9.71379i − 0.472301i
\(424\) 8.93900i 0.434116i
\(425\) −7.40581 −0.359235
\(426\) 12.0761 0.585087
\(427\) − 25.6993i − 1.24368i
\(428\) −13.9366 −0.673651
\(429\) 0 0
\(430\) 10.4547 0.504172
\(431\) − 1.48619i − 0.0715872i −0.999359 0.0357936i \(-0.988604\pi\)
0.999359 0.0357936i \(-0.0113959\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −18.5719 −0.892511 −0.446255 0.894906i \(-0.647243\pi\)
−0.446255 + 0.894906i \(0.647243\pi\)
\(434\) 2.90515i 0.139452i
\(435\) 8.03684i 0.385337i
\(436\) 5.39373i 0.258313i
\(437\) 5.29590i 0.253337i
\(438\) 2.29590 0.109702
\(439\) −0.532123 −0.0253968 −0.0126984 0.999919i \(-0.504042\pi\)
−0.0126984 + 0.999919i \(0.504042\pi\)
\(440\) − 3.85086i − 0.183582i
\(441\) 0.472189 0.0224852
\(442\) 0 0
\(443\) 25.5230 1.21264 0.606318 0.795222i \(-0.292646\pi\)
0.606318 + 0.795222i \(0.292646\pi\)
\(444\) − 3.44504i − 0.163494i
\(445\) −15.6799 −0.743300
\(446\) −22.5579 −1.06815
\(447\) 3.25667i 0.154035i
\(448\) − 2.55496i − 0.120710i
\(449\) − 40.7730i − 1.92420i −0.272703 0.962098i \(-0.587918\pi\)
0.272703 0.962098i \(-0.412082\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −5.98792 −0.281960
\(452\) −9.44265 −0.444145
\(453\) − 0.137063i − 0.00643980i
\(454\) −23.7439 −1.11436
\(455\) 0 0
\(456\) −2.93900 −0.137631
\(457\) 15.1511i 0.708737i 0.935106 + 0.354368i \(0.115304\pi\)
−0.935106 + 0.354368i \(0.884696\pi\)
\(458\) 3.11529 0.145568
\(459\) −7.40581 −0.345674
\(460\) 1.80194i 0.0840158i
\(461\) 19.5743i 0.911668i 0.890065 + 0.455834i \(0.150659\pi\)
−0.890065 + 0.455834i \(0.849341\pi\)
\(462\) 9.83877i 0.457741i
\(463\) 31.2905i 1.45419i 0.686535 + 0.727097i \(0.259131\pi\)
−0.686535 + 0.727097i \(0.740869\pi\)
\(464\) 8.03684 0.373101
\(465\) 1.13706 0.0527300
\(466\) 12.5603i 0.581846i
\(467\) 32.3032 1.49481 0.747407 0.664367i \(-0.231299\pi\)
0.747407 + 0.664367i \(0.231299\pi\)
\(468\) 0 0
\(469\) 3.48055 0.160717
\(470\) − 9.71379i − 0.448064i
\(471\) −4.35152 −0.200507
\(472\) −11.7845 −0.542425
\(473\) 40.2597i 1.85114i
\(474\) − 8.81700i − 0.404978i
\(475\) − 2.93900i − 0.134851i
\(476\) − 18.9215i − 0.867267i
\(477\) −8.93900 −0.409289
\(478\) −8.52111 −0.389746
\(479\) − 9.32842i − 0.426226i −0.977028 0.213113i \(-0.931640\pi\)
0.977028 0.213113i \(-0.0683603\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −11.6136 −0.528983
\(483\) − 4.60388i − 0.209484i
\(484\) 3.82908 0.174049
\(485\) −1.98254 −0.0900226
\(486\) − 1.00000i − 0.0453609i
\(487\) − 10.6571i − 0.482920i −0.970411 0.241460i \(-0.922374\pi\)
0.970411 0.241460i \(-0.0776262\pi\)
\(488\) 10.0586i 0.455332i
\(489\) − 11.6189i − 0.525427i
\(490\) 0.472189 0.0213313
\(491\) −40.8611 −1.84404 −0.922019 0.387146i \(-0.873461\pi\)
−0.922019 + 0.387146i \(0.873461\pi\)
\(492\) 1.55496i 0.0701029i
\(493\) 59.5193 2.68061
\(494\) 0 0
\(495\) 3.85086 0.173083
\(496\) − 1.13706i − 0.0510557i
\(497\) −30.8538 −1.38398
\(498\) 15.9051 0.712727
\(499\) − 19.0556i − 0.853047i −0.904477 0.426523i \(-0.859738\pi\)
0.904477 0.426523i \(-0.140262\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 20.8146i − 0.929928i
\(502\) − 22.7463i − 1.01522i
\(503\) −16.9825 −0.757214 −0.378607 0.925558i \(-0.623597\pi\)
−0.378607 + 0.925558i \(0.623597\pi\)
\(504\) 2.55496 0.113807
\(505\) − 3.48188i − 0.154942i
\(506\) −6.93900 −0.308476
\(507\) 0 0
\(508\) −0.499336 −0.0221545
\(509\) 43.0116i 1.90646i 0.302252 + 0.953228i \(0.402262\pi\)
−0.302252 + 0.953228i \(0.597738\pi\)
\(510\) −7.40581 −0.327935
\(511\) −5.86592 −0.259493
\(512\) 1.00000i 0.0441942i
\(513\) − 2.93900i − 0.129760i
\(514\) − 7.39373i − 0.326124i
\(515\) 8.65817i 0.381525i
\(516\) 10.4547 0.460244
\(517\) 37.4064 1.64513
\(518\) 8.80194i 0.386735i
\(519\) −8.89440 −0.390421
\(520\) 0 0
\(521\) −25.1021 −1.09975 −0.549873 0.835249i \(-0.685324\pi\)
−0.549873 + 0.835249i \(0.685324\pi\)
\(522\) 8.03684i 0.351763i
\(523\) −3.86725 −0.169103 −0.0845515 0.996419i \(-0.526946\pi\)
−0.0845515 + 0.996419i \(0.526946\pi\)
\(524\) 1.12498 0.0491451
\(525\) 2.55496i 0.111508i
\(526\) − 11.7735i − 0.513347i
\(527\) − 8.42088i − 0.366819i
\(528\) − 3.85086i − 0.167587i
\(529\) −19.7530 −0.858827
\(530\) −8.93900 −0.388285
\(531\) − 11.7845i − 0.511403i
\(532\) 7.50902 0.325558
\(533\) 0 0
\(534\) −15.6799 −0.678537
\(535\) − 13.9366i − 0.602532i
\(536\) −1.36227 −0.0588412
\(537\) 0.188374 0.00812892
\(538\) − 22.8810i − 0.986469i
\(539\) 1.81833i 0.0783211i
\(540\) − 1.00000i − 0.0430331i
\(541\) − 42.6152i − 1.83217i −0.400983 0.916086i \(-0.631331\pi\)
0.400983 0.916086i \(-0.368669\pi\)
\(542\) −2.13036 −0.0915067
\(543\) 18.1588 0.779270
\(544\) 7.40581i 0.317522i
\(545\) −5.39373 −0.231042
\(546\) 0 0
\(547\) −29.7928 −1.27385 −0.636925 0.770926i \(-0.719794\pi\)
−0.636925 + 0.770926i \(0.719794\pi\)
\(548\) − 14.1468i − 0.604319i
\(549\) −10.0586 −0.429291
\(550\) 3.85086 0.164201
\(551\) 23.6203i 1.00626i
\(552\) 1.80194i 0.0766956i
\(553\) 22.5271i 0.957949i
\(554\) 14.7084i 0.624901i
\(555\) 3.44504 0.146234
\(556\) −7.17629 −0.304343
\(557\) 37.7700i 1.60037i 0.599756 + 0.800183i \(0.295264\pi\)
−0.599756 + 0.800183i \(0.704736\pi\)
\(558\) 1.13706 0.0481357
\(559\) 0 0
\(560\) 2.55496 0.107967
\(561\) − 28.5187i − 1.20406i
\(562\) 12.5942 0.531254
\(563\) −25.0237 −1.05462 −0.527311 0.849672i \(-0.676800\pi\)
−0.527311 + 0.849672i \(0.676800\pi\)
\(564\) − 9.71379i − 0.409024i
\(565\) − 9.44265i − 0.397255i
\(566\) 1.62565i 0.0683310i
\(567\) 2.55496i 0.107298i
\(568\) 12.0761 0.506700
\(569\) −3.81402 −0.159892 −0.0799460 0.996799i \(-0.525475\pi\)
−0.0799460 + 0.996799i \(0.525475\pi\)
\(570\) − 2.93900i − 0.123101i
\(571\) 4.76702 0.199494 0.0997468 0.995013i \(-0.468197\pi\)
0.0997468 + 0.995013i \(0.468197\pi\)
\(572\) 0 0
\(573\) −0.469205 −0.0196013
\(574\) − 3.97285i − 0.165824i
\(575\) −1.80194 −0.0751460
\(576\) −1.00000 −0.0416667
\(577\) − 20.9879i − 0.873738i −0.899525 0.436869i \(-0.856087\pi\)
0.899525 0.436869i \(-0.143913\pi\)
\(578\) 37.8461i 1.57419i
\(579\) 17.9312i 0.745197i
\(580\) 8.03684i 0.333711i
\(581\) −40.6370 −1.68591
\(582\) −1.98254 −0.0821790
\(583\) − 34.4228i − 1.42565i
\(584\) 2.29590 0.0950049
\(585\) 0 0
\(586\) 13.0664 0.539767
\(587\) − 13.9312i − 0.575003i −0.957780 0.287502i \(-0.907175\pi\)
0.957780 0.287502i \(-0.0928247\pi\)
\(588\) 0.472189 0.0194727
\(589\) 3.34183 0.137698
\(590\) − 11.7845i − 0.485159i
\(591\) − 8.17390i − 0.336229i
\(592\) − 3.44504i − 0.141590i
\(593\) 9.75600i 0.400631i 0.979731 + 0.200316i \(0.0641967\pi\)
−0.979731 + 0.200316i \(0.935803\pi\)
\(594\) 3.85086 0.158002
\(595\) 18.9215 0.775708
\(596\) 3.25667i 0.133398i
\(597\) −23.7536 −0.972171
\(598\) 0 0
\(599\) −9.88769 −0.404000 −0.202000 0.979386i \(-0.564744\pi\)
−0.202000 + 0.979386i \(0.564744\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) 4.32304 0.176341 0.0881703 0.996105i \(-0.471898\pi\)
0.0881703 + 0.996105i \(0.471898\pi\)
\(602\) −26.7114 −1.08868
\(603\) − 1.36227i − 0.0554760i
\(604\) − 0.137063i − 0.00557703i
\(605\) 3.82908i 0.155674i
\(606\) − 3.48188i − 0.141442i
\(607\) 37.5948 1.52592 0.762962 0.646443i \(-0.223744\pi\)
0.762962 + 0.646443i \(0.223744\pi\)
\(608\) −2.93900 −0.119192
\(609\) − 20.5338i − 0.832071i
\(610\) −10.0586 −0.407261
\(611\) 0 0
\(612\) −7.40581 −0.299362
\(613\) − 33.8595i − 1.36757i −0.729683 0.683786i \(-0.760332\pi\)
0.729683 0.683786i \(-0.239668\pi\)
\(614\) 26.2489 1.05932
\(615\) −1.55496 −0.0627020
\(616\) 9.83877i 0.396415i
\(617\) − 43.1333i − 1.73648i −0.496142 0.868241i \(-0.665250\pi\)
0.496142 0.868241i \(-0.334750\pi\)
\(618\) 8.65817i 0.348283i
\(619\) 14.6606i 0.589258i 0.955612 + 0.294629i \(0.0951960\pi\)
−0.955612 + 0.294629i \(0.904804\pi\)
\(620\) 1.13706 0.0456656
\(621\) −1.80194 −0.0723093
\(622\) − 27.1250i − 1.08761i
\(623\) 40.0616 1.60503
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 19.1323i 0.764680i
\(627\) 11.3177 0.451984
\(628\) −4.35152 −0.173645
\(629\) − 25.5133i − 1.01728i
\(630\) 2.55496i 0.101792i
\(631\) − 10.5265i − 0.419053i −0.977803 0.209526i \(-0.932808\pi\)
0.977803 0.209526i \(-0.0671922\pi\)
\(632\) − 8.81700i − 0.350722i
\(633\) −24.2392 −0.963422
\(634\) −6.67456 −0.265081
\(635\) − 0.499336i − 0.0198155i
\(636\) −8.93900 −0.354454
\(637\) 0 0
\(638\) −30.9487 −1.22527
\(639\) 12.0761i 0.477722i
\(640\) −1.00000 −0.0395285
\(641\) 16.5435 0.653428 0.326714 0.945123i \(-0.394059\pi\)
0.326714 + 0.945123i \(0.394059\pi\)
\(642\) − 13.9366i − 0.550034i
\(643\) 43.4282i 1.71264i 0.516446 + 0.856320i \(0.327255\pi\)
−0.516446 + 0.856320i \(0.672745\pi\)
\(644\) − 4.60388i − 0.181418i
\(645\) 10.4547i 0.411655i
\(646\) −21.7657 −0.856360
\(647\) −28.7687 −1.13101 −0.565507 0.824744i \(-0.691319\pi\)
−0.565507 + 0.824744i \(0.691319\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 45.3803 1.78133
\(650\) 0 0
\(651\) −2.90515 −0.113862
\(652\) − 11.6189i − 0.455033i
\(653\) −30.0978 −1.17782 −0.588910 0.808199i \(-0.700443\pi\)
−0.588910 + 0.808199i \(0.700443\pi\)
\(654\) −5.39373 −0.210912
\(655\) 1.12498i 0.0439567i
\(656\) 1.55496i 0.0607109i
\(657\) 2.29590i 0.0895715i
\(658\) 24.8183i 0.967519i
\(659\) 46.6015 1.81534 0.907668 0.419689i \(-0.137861\pi\)
0.907668 + 0.419689i \(0.137861\pi\)
\(660\) 3.85086 0.149894
\(661\) − 40.9788i − 1.59389i −0.604051 0.796946i \(-0.706448\pi\)
0.604051 0.796946i \(-0.293552\pi\)
\(662\) 18.1360 0.704876
\(663\) 0 0
\(664\) 15.9051 0.617239
\(665\) 7.50902i 0.291187i
\(666\) 3.44504 0.133493
\(667\) 14.4819 0.560741
\(668\) − 20.8146i − 0.805341i
\(669\) − 22.5579i − 0.872140i
\(670\) − 1.36227i − 0.0526292i
\(671\) − 38.7342i − 1.49532i
\(672\) 2.55496 0.0985596
\(673\) 49.2586 1.89878 0.949389 0.314101i \(-0.101703\pi\)
0.949389 + 0.314101i \(0.101703\pi\)
\(674\) − 10.8780i − 0.419005i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −31.1909 −1.19876 −0.599382 0.800463i \(-0.704587\pi\)
−0.599382 + 0.800463i \(0.704587\pi\)
\(678\) − 9.44265i − 0.362643i
\(679\) 5.06531 0.194389
\(680\) −7.40581 −0.284000
\(681\) − 23.7439i − 0.909869i
\(682\) 4.37867i 0.167668i
\(683\) − 27.0164i − 1.03375i −0.856060 0.516877i \(-0.827095\pi\)
0.856060 0.516877i \(-0.172905\pi\)
\(684\) − 2.93900i − 0.112376i
\(685\) 14.1468 0.540520
\(686\) −19.0911 −0.728903
\(687\) 3.11529i 0.118856i
\(688\) 10.4547 0.398583
\(689\) 0 0
\(690\) −1.80194 −0.0685986
\(691\) − 9.03551i − 0.343727i −0.985121 0.171863i \(-0.945021\pi\)
0.985121 0.171863i \(-0.0549788\pi\)
\(692\) −8.89440 −0.338114
\(693\) −9.83877 −0.373744
\(694\) − 7.42865i − 0.281988i
\(695\) − 7.17629i − 0.272212i
\(696\) 8.03684i 0.304635i
\(697\) 11.5157i 0.436189i
\(698\) −28.7275 −1.08735
\(699\) −12.5603 −0.475075
\(700\) 2.55496i 0.0965683i
\(701\) 16.5332 0.624450 0.312225 0.950008i \(-0.398926\pi\)
0.312225 + 0.950008i \(0.398926\pi\)
\(702\) 0 0
\(703\) 10.1250 0.381871
\(704\) − 3.85086i − 0.145135i
\(705\) 9.71379 0.365843
\(706\) −31.2131 −1.17472
\(707\) 8.89605i 0.334570i
\(708\) − 11.7845i − 0.442888i
\(709\) − 28.4868i − 1.06984i −0.844902 0.534922i \(-0.820341\pi\)
0.844902 0.534922i \(-0.179659\pi\)
\(710\) 12.0761i 0.453207i
\(711\) 8.81700 0.330663
\(712\) −15.6799 −0.587631
\(713\) − 2.04892i − 0.0767326i
\(714\) 18.9215 0.708121
\(715\) 0 0
\(716\) 0.188374 0.00703985
\(717\) − 8.52111i − 0.318227i
\(718\) −0.0247542 −0.000923817 0
\(719\) 36.4064 1.35773 0.678865 0.734263i \(-0.262472\pi\)
0.678865 + 0.734263i \(0.262472\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 22.1213i − 0.823839i
\(722\) 10.3623i 0.385644i
\(723\) − 11.6136i − 0.431913i
\(724\) 18.1588 0.674868
\(725\) −8.03684 −0.298481
\(726\) 3.82908i 0.142111i
\(727\) −36.1196 −1.33960 −0.669801 0.742541i \(-0.733621\pi\)
−0.669801 + 0.742541i \(0.733621\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.29590i 0.0849750i
\(731\) 77.4258 2.86370
\(732\) −10.0586 −0.371777
\(733\) − 17.1648i − 0.633996i −0.948426 0.316998i \(-0.897325\pi\)
0.948426 0.316998i \(-0.102675\pi\)
\(734\) 17.5284i 0.646985i
\(735\) 0.472189i 0.0174170i
\(736\) 1.80194i 0.0664203i
\(737\) 5.24591 0.193236
\(738\) −1.55496 −0.0572388
\(739\) 21.5821i 0.793911i 0.917838 + 0.396955i \(0.129933\pi\)
−0.917838 + 0.396955i \(0.870067\pi\)
\(740\) 3.44504 0.126642
\(741\) 0 0
\(742\) 22.8388 0.838438
\(743\) − 16.6698i − 0.611555i −0.952103 0.305777i \(-0.901084\pi\)
0.952103 0.305777i \(-0.0989163\pi\)
\(744\) 1.13706 0.0416868
\(745\) −3.25667 −0.119315
\(746\) − 35.3056i − 1.29263i
\(747\) 15.9051i 0.581939i
\(748\) − 28.5187i − 1.04275i
\(749\) 35.6075i 1.30107i
\(750\) 1.00000 0.0365148
\(751\) −11.8009 −0.430620 −0.215310 0.976546i \(-0.569076\pi\)
−0.215310 + 0.976546i \(0.569076\pi\)
\(752\) − 9.71379i − 0.354226i
\(753\) 22.7463 0.828922
\(754\) 0 0
\(755\) 0.137063 0.00498825
\(756\) 2.55496i 0.0929229i
\(757\) 5.44371 0.197855 0.0989276 0.995095i \(-0.468459\pi\)
0.0989276 + 0.995095i \(0.468459\pi\)
\(758\) 18.2567 0.663112
\(759\) − 6.93900i − 0.251870i
\(760\) − 2.93900i − 0.106609i
\(761\) − 20.4064i − 0.739732i −0.929085 0.369866i \(-0.879404\pi\)
0.929085 0.369866i \(-0.120596\pi\)
\(762\) − 0.499336i − 0.0180890i
\(763\) 13.7808 0.498897
\(764\) −0.469205 −0.0169752
\(765\) − 7.40581i − 0.267758i
\(766\) −2.83148 −0.102305
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 28.4282i 1.02515i 0.858644 + 0.512573i \(0.171308\pi\)
−0.858644 + 0.512573i \(0.828692\pi\)
\(770\) −9.83877 −0.354565
\(771\) 7.39373 0.266279
\(772\) 17.9312i 0.645359i
\(773\) 41.9874i 1.51018i 0.655619 + 0.755092i \(0.272408\pi\)
−0.655619 + 0.755092i \(0.727592\pi\)
\(774\) 10.4547i 0.375788i
\(775\) 1.13706i 0.0408445i
\(776\) −1.98254 −0.0711691
\(777\) −8.80194 −0.315768
\(778\) − 28.8025i − 1.03262i
\(779\) −4.57002 −0.163738
\(780\) 0 0
\(781\) −46.5032 −1.66401
\(782\) 13.3448i 0.477210i
\(783\) −8.03684 −0.287213
\(784\) 0.472189 0.0168639
\(785\) − 4.35152i − 0.155312i
\(786\) 1.12498i 0.0401268i
\(787\) 11.6950i 0.416882i 0.978035 + 0.208441i \(0.0668389\pi\)
−0.978035 + 0.208441i \(0.933161\pi\)
\(788\) − 8.17390i − 0.291183i
\(789\) 11.7735 0.419146
\(790\) 8.81700 0.313695
\(791\) 24.1256i 0.857807i
\(792\) 3.85086 0.136834
\(793\) 0 0
\(794\) −21.4306 −0.760542
\(795\) − 8.93900i − 0.317034i
\(796\) −23.7536 −0.841924
\(797\) −50.5502 −1.79058 −0.895289 0.445485i \(-0.853031\pi\)
−0.895289 + 0.445485i \(0.853031\pi\)
\(798\) 7.50902i 0.265817i
\(799\) − 71.9385i − 2.54500i
\(800\) − 1.00000i − 0.0353553i
\(801\) − 15.6799i − 0.554023i
\(802\) −19.5284 −0.689572
\(803\) −8.84117 −0.311998
\(804\) − 1.36227i − 0.0480437i
\(805\) 4.60388 0.162265
\(806\) 0 0
\(807\) 22.8810 0.805449
\(808\) − 3.48188i − 0.122492i
\(809\) −19.3134 −0.679021 −0.339511 0.940602i \(-0.610261\pi\)
−0.339511 + 0.940602i \(0.610261\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 1.06638i − 0.0374455i −0.999825 0.0187228i \(-0.994040\pi\)
0.999825 0.0187228i \(-0.00595999\pi\)
\(812\) − 20.5338i − 0.720594i
\(813\) − 2.13036i − 0.0747149i
\(814\) 13.2664i 0.464986i
\(815\) 11.6189 0.406994
\(816\) −7.40581 −0.259255
\(817\) 30.7265i 1.07498i
\(818\) 30.2707 1.05839
\(819\) 0 0
\(820\) −1.55496 −0.0543015
\(821\) 3.22760i 0.112644i 0.998413 + 0.0563220i \(0.0179374\pi\)
−0.998413 + 0.0563220i \(0.982063\pi\)
\(822\) 14.1468 0.493425
\(823\) 29.0017 1.01093 0.505467 0.862846i \(-0.331320\pi\)
0.505467 + 0.862846i \(0.331320\pi\)
\(824\) 8.65817i 0.301622i
\(825\) 3.85086i 0.134070i
\(826\) 30.1089i 1.04762i
\(827\) 17.9414i 0.623883i 0.950101 + 0.311942i \(0.100979\pi\)
−0.950101 + 0.311942i \(0.899021\pi\)
\(828\) −1.80194 −0.0626217
\(829\) 18.9903 0.659561 0.329780 0.944058i \(-0.393025\pi\)
0.329780 + 0.944058i \(0.393025\pi\)
\(830\) 15.9051i 0.552076i
\(831\) −14.7084 −0.510229
\(832\) 0 0
\(833\) 3.49694 0.121162
\(834\) − 7.17629i − 0.248495i
\(835\) 20.8146 0.720319
\(836\) 11.3177 0.391430
\(837\) 1.13706i 0.0393027i
\(838\) − 24.4873i − 0.845898i
\(839\) 36.8189i 1.27113i 0.772047 + 0.635565i \(0.219233\pi\)
−0.772047 + 0.635565i \(0.780767\pi\)
\(840\) 2.55496i 0.0881544i
\(841\) 35.5907 1.22727
\(842\) −8.90408 −0.306855
\(843\) 12.5942i 0.433767i
\(844\) −24.2392 −0.834348
\(845\) 0 0
\(846\) 9.71379 0.333967
\(847\) − 9.78315i − 0.336153i
\(848\) −8.93900 −0.306967
\(849\) −1.62565 −0.0557920
\(850\) − 7.40581i − 0.254017i
\(851\) − 6.20775i − 0.212799i
\(852\) 12.0761i 0.413719i
\(853\) − 14.6746i − 0.502447i −0.967929 0.251224i \(-0.919167\pi\)
0.967929 0.251224i \(-0.0808330\pi\)
\(854\) 25.6993 0.879413
\(855\) 2.93900 0.100512
\(856\) − 13.9366i − 0.476343i
\(857\) 14.2784 0.487742 0.243871 0.969808i \(-0.421583\pi\)
0.243871 + 0.969808i \(0.421583\pi\)
\(858\) 0 0
\(859\) 11.3593 0.387574 0.193787 0.981044i \(-0.437923\pi\)
0.193787 + 0.981044i \(0.437923\pi\)
\(860\) 10.4547i 0.356503i
\(861\) 3.97285 0.135394
\(862\) 1.48619 0.0506198
\(863\) 26.7041i 0.909018i 0.890742 + 0.454509i \(0.150185\pi\)
−0.890742 + 0.454509i \(0.849815\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 8.89440i − 0.302419i
\(866\) − 18.5719i − 0.631100i
\(867\) −37.8461 −1.28532
\(868\) −2.90515 −0.0986072
\(869\) 33.9530i 1.15178i
\(870\) −8.03684 −0.272474
\(871\) 0 0
\(872\) −5.39373 −0.182655
\(873\) − 1.98254i − 0.0670989i
\(874\) −5.29590 −0.179136
\(875\) −2.55496 −0.0863733
\(876\) 2.29590i 0.0775712i
\(877\) 12.7041i 0.428987i 0.976725 + 0.214494i \(0.0688101\pi\)
−0.976725 + 0.214494i \(0.931190\pi\)
\(878\) − 0.532123i − 0.0179583i
\(879\) 13.0664i 0.440718i
\(880\) 3.85086 0.129812
\(881\) −52.1342 −1.75645 −0.878223 0.478252i \(-0.841271\pi\)
−0.878223 + 0.478252i \(0.841271\pi\)
\(882\) 0.472189i 0.0158994i
\(883\) −6.17390 −0.207768 −0.103884 0.994589i \(-0.533127\pi\)
−0.103884 + 0.994589i \(0.533127\pi\)
\(884\) 0 0
\(885\) 11.7845 0.396131
\(886\) 25.5230i 0.857463i
\(887\) 0.0978347 0.00328497 0.00164248 0.999999i \(-0.499477\pi\)
0.00164248 + 0.999999i \(0.499477\pi\)
\(888\) 3.44504 0.115608
\(889\) 1.27578i 0.0427884i
\(890\) − 15.6799i − 0.525593i
\(891\) 3.85086i 0.129008i
\(892\) − 22.5579i − 0.755296i
\(893\) 28.5488 0.955351
\(894\) −3.25667 −0.108919
\(895\) 0.188374i 0.00629663i
\(896\) 2.55496 0.0853552
\(897\) 0 0
\(898\) 40.7730 1.36061
\(899\) − 9.13839i − 0.304782i
\(900\) 1.00000 0.0333333
\(901\) −66.2006 −2.20546
\(902\) − 5.98792i − 0.199376i
\(903\) − 26.7114i − 0.888900i
\(904\) − 9.44265i − 0.314058i
\(905\) 18.1588i 0.603620i
\(906\) 0.137063 0.00455362
\(907\) 6.44265 0.213925 0.106962 0.994263i \(-0.465888\pi\)
0.106962 + 0.994263i \(0.465888\pi\)
\(908\) − 23.7439i − 0.787970i
\(909\) 3.48188 0.115487
\(910\) 0 0
\(911\) 32.0965 1.06340 0.531702 0.846931i \(-0.321553\pi\)
0.531702 + 0.846931i \(0.321553\pi\)
\(912\) − 2.93900i − 0.0973201i
\(913\) −61.2484 −2.02703
\(914\) −15.1511 −0.501153
\(915\) − 10.0586i − 0.332527i
\(916\) 3.11529i 0.102932i
\(917\) − 2.87428i − 0.0949171i
\(918\) − 7.40581i − 0.244428i
\(919\) 14.1062 0.465320 0.232660 0.972558i \(-0.425257\pi\)
0.232660 + 0.972558i \(0.425257\pi\)
\(920\) −1.80194 −0.0594081
\(921\) 26.2489i 0.864931i
\(922\) −19.5743 −0.644646
\(923\) 0 0
\(924\) −9.83877 −0.323672
\(925\) 3.44504i 0.113272i
\(926\) −31.2905 −1.02827
\(927\) −8.65817 −0.284372
\(928\) 8.03684i 0.263822i
\(929\) 31.4959i 1.03335i 0.856183 + 0.516673i \(0.172830\pi\)
−0.856183 + 0.516673i \(0.827170\pi\)
\(930\) 1.13706i 0.0372858i
\(931\) 1.38776i 0.0454821i
\(932\) −12.5603 −0.411427
\(933\) 27.1250 0.888032
\(934\) 32.3032i 1.05699i
\(935\) 28.5187 0.932662
\(936\) 0 0
\(937\) −10.0592 −0.328620 −0.164310 0.986409i \(-0.552540\pi\)
−0.164310 + 0.986409i \(0.552540\pi\)
\(938\) 3.48055i 0.113644i
\(939\) −19.1323 −0.624358
\(940\) 9.71379 0.316829
\(941\) − 20.7802i − 0.677414i −0.940892 0.338707i \(-0.890010\pi\)
0.940892 0.338707i \(-0.109990\pi\)
\(942\) − 4.35152i − 0.141780i
\(943\) 2.80194i 0.0912436i
\(944\) − 11.7845i − 0.383552i
\(945\) −2.55496 −0.0831128
\(946\) −40.2597 −1.30895
\(947\) 41.0307i 1.33332i 0.745362 + 0.666660i \(0.232277\pi\)
−0.745362 + 0.666660i \(0.767723\pi\)
\(948\) 8.81700 0.286363
\(949\) 0 0
\(950\) 2.93900 0.0953538
\(951\) − 6.67456i − 0.216438i
\(952\) 18.9215 0.613251
\(953\) 56.3484 1.82530 0.912652 0.408738i \(-0.134031\pi\)
0.912652 + 0.408738i \(0.134031\pi\)
\(954\) − 8.93900i − 0.289411i
\(955\) − 0.469205i − 0.0151831i
\(956\) − 8.52111i − 0.275592i
\(957\) − 30.9487i − 1.00043i
\(958\) 9.32842 0.301388
\(959\) −36.1444 −1.16716
\(960\) − 1.00000i − 0.0322749i
\(961\) 29.7071 0.958293
\(962\) 0 0
\(963\) 13.9366 0.449101
\(964\) − 11.6136i − 0.374048i
\(965\) −17.9312 −0.577227
\(966\) 4.60388 0.148127
\(967\) − 9.63102i − 0.309713i −0.987937 0.154856i \(-0.950509\pi\)
0.987937 0.154856i \(-0.0494915\pi\)
\(968\) 3.82908i 0.123071i
\(969\) − 21.7657i − 0.699215i
\(970\) − 1.98254i − 0.0636556i
\(971\) 4.72050 0.151488 0.0757440 0.997127i \(-0.475867\pi\)
0.0757440 + 0.997127i \(0.475867\pi\)
\(972\) 1.00000 0.0320750
\(973\) 18.3351i 0.587797i
\(974\) 10.6571 0.341476
\(975\) 0 0
\(976\) −10.0586 −0.321968
\(977\) 2.75063i 0.0880004i 0.999032 + 0.0440002i \(0.0140102\pi\)
−0.999032 + 0.0440002i \(0.985990\pi\)
\(978\) 11.6189 0.371533
\(979\) 60.3812 1.92979
\(980\) 0.472189i 0.0150835i
\(981\) − 5.39373i − 0.172209i
\(982\) − 40.8611i − 1.30393i
\(983\) 7.87907i 0.251303i 0.992074 + 0.125652i \(0.0401021\pi\)
−0.992074 + 0.125652i \(0.959898\pi\)
\(984\) −1.55496 −0.0495703
\(985\) 8.17390 0.260442
\(986\) 59.5193i 1.89548i
\(987\) −24.8183 −0.789976
\(988\) 0 0
\(989\) 18.8388 0.599038
\(990\) 3.85086i 0.122388i
\(991\) 13.3609 0.424424 0.212212 0.977224i \(-0.431933\pi\)
0.212212 + 0.977224i \(0.431933\pi\)
\(992\) 1.13706 0.0361018
\(993\) 18.1360i 0.575529i
\(994\) − 30.8538i − 0.978624i
\(995\) − 23.7536i − 0.753040i
\(996\) 15.9051i 0.503974i
\(997\) 47.1554 1.49343 0.746713 0.665147i \(-0.231631\pi\)
0.746713 + 0.665147i \(0.231631\pi\)
\(998\) 19.0556 0.603195
\(999\) 3.44504i 0.108996i
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 5070.2.b.v.1351.5 6
13.5 odd 4 5070.2.a.bs.1.2 yes 3
13.8 odd 4 5070.2.a.bl.1.2 3
13.12 even 2 inner 5070.2.b.v.1351.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bl.1.2 3 13.8 odd 4
5070.2.a.bs.1.2 yes 3 13.5 odd 4
5070.2.b.v.1351.2 6 13.12 even 2 inner
5070.2.b.v.1351.5 6 1.1 even 1 trivial