Properties

Label 1334.2.a.k.1.6
Level $1334$
Weight $2$
Character 1334.1
Self dual yes
Analytic conductor $10.652$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1334,2,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.a (trivial)

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: yes
Analytic conductor: \(10.6520436296\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5x^{9} - 8x^{8} + 60x^{7} + 13x^{6} - 241x^{5} + 6x^{4} + 346x^{3} + 16x^{2} - 64x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.267188\) of defining polynomial
Character \(\chi\) \(=\) 1334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.26719 q^{3} +1.00000 q^{4} +1.36043 q^{5} +1.26719 q^{6} +1.67775 q^{7} +1.00000 q^{8} -1.39423 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.26719 q^{3} +1.00000 q^{4} +1.36043 q^{5} +1.26719 q^{6} +1.67775 q^{7} +1.00000 q^{8} -1.39423 q^{9} +1.36043 q^{10} +4.80785 q^{11} +1.26719 q^{12} -1.92323 q^{13} +1.67775 q^{14} +1.72392 q^{15} +1.00000 q^{16} +3.00120 q^{17} -1.39423 q^{18} -2.45647 q^{19} +1.36043 q^{20} +2.12603 q^{21} +4.80785 q^{22} +1.00000 q^{23} +1.26719 q^{24} -3.14924 q^{25} -1.92323 q^{26} -5.56832 q^{27} +1.67775 q^{28} +1.00000 q^{29} +1.72392 q^{30} -6.93732 q^{31} +1.00000 q^{32} +6.09245 q^{33} +3.00120 q^{34} +2.28246 q^{35} -1.39423 q^{36} +8.33349 q^{37} -2.45647 q^{38} -2.43709 q^{39} +1.36043 q^{40} -1.49177 q^{41} +2.12603 q^{42} +0.840503 q^{43} +4.80785 q^{44} -1.89675 q^{45} +1.00000 q^{46} -0.754099 q^{47} +1.26719 q^{48} -4.18516 q^{49} -3.14924 q^{50} +3.80308 q^{51} -1.92323 q^{52} +4.47440 q^{53} -5.56832 q^{54} +6.54072 q^{55} +1.67775 q^{56} -3.11281 q^{57} +1.00000 q^{58} +10.0339 q^{59} +1.72392 q^{60} +5.19712 q^{61} -6.93732 q^{62} -2.33918 q^{63} +1.00000 q^{64} -2.61641 q^{65} +6.09245 q^{66} -14.7127 q^{67} +3.00120 q^{68} +1.26719 q^{69} +2.28246 q^{70} -5.76111 q^{71} -1.39423 q^{72} -0.364931 q^{73} +8.33349 q^{74} -3.99068 q^{75} -2.45647 q^{76} +8.06636 q^{77} -2.43709 q^{78} -14.4028 q^{79} +1.36043 q^{80} -2.87341 q^{81} -1.49177 q^{82} +4.02625 q^{83} +2.12603 q^{84} +4.08291 q^{85} +0.840503 q^{86} +1.26719 q^{87} +4.80785 q^{88} -5.58835 q^{89} -1.89675 q^{90} -3.22670 q^{91} +1.00000 q^{92} -8.79089 q^{93} -0.754099 q^{94} -3.34185 q^{95} +1.26719 q^{96} +16.2117 q^{97} -4.18516 q^{98} -6.70326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 5 q^{3} + 10 q^{4} - q^{5} + 5 q^{6} + 2 q^{7} + 10 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 5 q^{3} + 10 q^{4} - q^{5} + 5 q^{6} + 2 q^{7} + 10 q^{8} + 11 q^{9} - q^{10} + q^{11} + 5 q^{12} + 9 q^{13} + 2 q^{14} + 5 q^{15} + 10 q^{16} + 11 q^{18} + 20 q^{19} - q^{20} + 14 q^{21} + q^{22} + 10 q^{23} + 5 q^{24} + 23 q^{25} + 9 q^{26} + 23 q^{27} + 2 q^{28} + 10 q^{29} + 5 q^{30} + 21 q^{31} + 10 q^{32} + q^{33} + 2 q^{35} + 11 q^{36} - 8 q^{37} + 20 q^{38} + 23 q^{39} - q^{40} - 4 q^{41} + 14 q^{42} - 3 q^{43} + q^{44} - 18 q^{45} + 10 q^{46} - q^{47} + 5 q^{48} + 18 q^{49} + 23 q^{50} - 6 q^{51} + 9 q^{52} - 13 q^{53} + 23 q^{54} + 13 q^{55} + 2 q^{56} - 22 q^{57} + 10 q^{58} + 14 q^{59} + 5 q^{60} + 12 q^{61} + 21 q^{62} - 26 q^{63} + 10 q^{64} - 25 q^{65} + q^{66} + 6 q^{67} + 5 q^{69} + 2 q^{70} + 8 q^{71} + 11 q^{72} + 16 q^{73} - 8 q^{74} + 8 q^{75} + 20 q^{76} - 16 q^{77} + 23 q^{78} + 7 q^{79} - q^{80} - 2 q^{81} - 4 q^{82} + 18 q^{83} + 14 q^{84} + 8 q^{85} - 3 q^{86} + 5 q^{87} + q^{88} + 2 q^{89} - 18 q^{90} + 8 q^{91} + 10 q^{92} - 41 q^{93} - q^{94} - 16 q^{95} + 5 q^{96} + 6 q^{97} + 18 q^{98} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.26719 0.731612 0.365806 0.930691i \(-0.380793\pi\)
0.365806 + 0.930691i \(0.380793\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.36043 0.608401 0.304201 0.952608i \(-0.401611\pi\)
0.304201 + 0.952608i \(0.401611\pi\)
\(6\) 1.26719 0.517328
\(7\) 1.67775 0.634130 0.317065 0.948404i \(-0.397303\pi\)
0.317065 + 0.948404i \(0.397303\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.39423 −0.464744
\(10\) 1.36043 0.430205
\(11\) 4.80785 1.44962 0.724810 0.688949i \(-0.241927\pi\)
0.724810 + 0.688949i \(0.241927\pi\)
\(12\) 1.26719 0.365806
\(13\) −1.92323 −0.533407 −0.266704 0.963779i \(-0.585935\pi\)
−0.266704 + 0.963779i \(0.585935\pi\)
\(14\) 1.67775 0.448398
\(15\) 1.72392 0.445113
\(16\) 1.00000 0.250000
\(17\) 3.00120 0.727897 0.363948 0.931419i \(-0.381428\pi\)
0.363948 + 0.931419i \(0.381428\pi\)
\(18\) −1.39423 −0.328624
\(19\) −2.45647 −0.563553 −0.281777 0.959480i \(-0.590924\pi\)
−0.281777 + 0.959480i \(0.590924\pi\)
\(20\) 1.36043 0.304201
\(21\) 2.12603 0.463937
\(22\) 4.80785 1.02504
\(23\) 1.00000 0.208514
\(24\) 1.26719 0.258664
\(25\) −3.14924 −0.629848
\(26\) −1.92323 −0.377176
\(27\) −5.56832 −1.07162
\(28\) 1.67775 0.317065
\(29\) 1.00000 0.185695
\(30\) 1.72392 0.314743
\(31\) −6.93732 −1.24598 −0.622990 0.782230i \(-0.714082\pi\)
−0.622990 + 0.782230i \(0.714082\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.09245 1.06056
\(34\) 3.00120 0.514701
\(35\) 2.28246 0.385805
\(36\) −1.39423 −0.232372
\(37\) 8.33349 1.37002 0.685008 0.728535i \(-0.259798\pi\)
0.685008 + 0.728535i \(0.259798\pi\)
\(38\) −2.45647 −0.398492
\(39\) −2.43709 −0.390247
\(40\) 1.36043 0.215102
\(41\) −1.49177 −0.232975 −0.116488 0.993192i \(-0.537164\pi\)
−0.116488 + 0.993192i \(0.537164\pi\)
\(42\) 2.12603 0.328053
\(43\) 0.840503 0.128175 0.0640877 0.997944i \(-0.479586\pi\)
0.0640877 + 0.997944i \(0.479586\pi\)
\(44\) 4.80785 0.724810
\(45\) −1.89675 −0.282751
\(46\) 1.00000 0.147442
\(47\) −0.754099 −0.109997 −0.0549983 0.998486i \(-0.517515\pi\)
−0.0549983 + 0.998486i \(0.517515\pi\)
\(48\) 1.26719 0.182903
\(49\) −4.18516 −0.597879
\(50\) −3.14924 −0.445370
\(51\) 3.80308 0.532538
\(52\) −1.92323 −0.266704
\(53\) 4.47440 0.614606 0.307303 0.951612i \(-0.400573\pi\)
0.307303 + 0.951612i \(0.400573\pi\)
\(54\) −5.56832 −0.757753
\(55\) 6.54072 0.881951
\(56\) 1.67775 0.224199
\(57\) −3.11281 −0.412302
\(58\) 1.00000 0.131306
\(59\) 10.0339 1.30631 0.653153 0.757226i \(-0.273446\pi\)
0.653153 + 0.757226i \(0.273446\pi\)
\(60\) 1.72392 0.222557
\(61\) 5.19712 0.665423 0.332712 0.943029i \(-0.392036\pi\)
0.332712 + 0.943029i \(0.392036\pi\)
\(62\) −6.93732 −0.881041
\(63\) −2.33918 −0.294708
\(64\) 1.00000 0.125000
\(65\) −2.61641 −0.324526
\(66\) 6.09245 0.749928
\(67\) −14.7127 −1.79744 −0.898721 0.438520i \(-0.855503\pi\)
−0.898721 + 0.438520i \(0.855503\pi\)
\(68\) 3.00120 0.363948
\(69\) 1.26719 0.152552
\(70\) 2.28246 0.272806
\(71\) −5.76111 −0.683718 −0.341859 0.939751i \(-0.611057\pi\)
−0.341859 + 0.939751i \(0.611057\pi\)
\(72\) −1.39423 −0.164312
\(73\) −0.364931 −0.0427119 −0.0213560 0.999772i \(-0.506798\pi\)
−0.0213560 + 0.999772i \(0.506798\pi\)
\(74\) 8.33349 0.968748
\(75\) −3.99068 −0.460804
\(76\) −2.45647 −0.281777
\(77\) 8.06636 0.919248
\(78\) −2.43709 −0.275946
\(79\) −14.4028 −1.62044 −0.810222 0.586123i \(-0.800653\pi\)
−0.810222 + 0.586123i \(0.800653\pi\)
\(80\) 1.36043 0.152100
\(81\) −2.87341 −0.319268
\(82\) −1.49177 −0.164738
\(83\) 4.02625 0.441938 0.220969 0.975281i \(-0.429078\pi\)
0.220969 + 0.975281i \(0.429078\pi\)
\(84\) 2.12603 0.231968
\(85\) 4.08291 0.442853
\(86\) 0.840503 0.0906337
\(87\) 1.26719 0.135857
\(88\) 4.80785 0.512518
\(89\) −5.58835 −0.592363 −0.296182 0.955132i \(-0.595713\pi\)
−0.296182 + 0.955132i \(0.595713\pi\)
\(90\) −1.89675 −0.199935
\(91\) −3.22670 −0.338250
\(92\) 1.00000 0.104257
\(93\) −8.79089 −0.911573
\(94\) −0.754099 −0.0777794
\(95\) −3.34185 −0.342867
\(96\) 1.26719 0.129332
\(97\) 16.2117 1.64604 0.823022 0.568010i \(-0.192286\pi\)
0.823022 + 0.568010i \(0.192286\pi\)
\(98\) −4.18516 −0.422765
\(99\) −6.70326 −0.673703
\(100\) −3.14924 −0.314924
\(101\) 5.25982 0.523372 0.261686 0.965153i \(-0.415722\pi\)
0.261686 + 0.965153i \(0.415722\pi\)
\(102\) 3.80308 0.376561
\(103\) 7.07418 0.697039 0.348520 0.937301i \(-0.386684\pi\)
0.348520 + 0.937301i \(0.386684\pi\)
\(104\) −1.92323 −0.188588
\(105\) 2.89230 0.282260
\(106\) 4.47440 0.434592
\(107\) 2.68162 0.259242 0.129621 0.991564i \(-0.458624\pi\)
0.129621 + 0.991564i \(0.458624\pi\)
\(108\) −5.56832 −0.535812
\(109\) −11.8123 −1.13141 −0.565704 0.824608i \(-0.691396\pi\)
−0.565704 + 0.824608i \(0.691396\pi\)
\(110\) 6.54072 0.623633
\(111\) 10.5601 1.00232
\(112\) 1.67775 0.158532
\(113\) −14.8431 −1.39633 −0.698163 0.715939i \(-0.745999\pi\)
−0.698163 + 0.715939i \(0.745999\pi\)
\(114\) −3.11281 −0.291542
\(115\) 1.36043 0.126860
\(116\) 1.00000 0.0928477
\(117\) 2.68143 0.247898
\(118\) 10.0339 0.923697
\(119\) 5.03526 0.461581
\(120\) 1.72392 0.157371
\(121\) 12.1154 1.10140
\(122\) 5.19712 0.470525
\(123\) −1.89035 −0.170447
\(124\) −6.93732 −0.622990
\(125\) −11.0864 −0.991602
\(126\) −2.33918 −0.208390
\(127\) −13.9826 −1.24076 −0.620379 0.784302i \(-0.713021\pi\)
−0.620379 + 0.784302i \(0.713021\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.06508 0.0937747
\(130\) −2.61641 −0.229474
\(131\) −1.69582 −0.148165 −0.0740824 0.997252i \(-0.523603\pi\)
−0.0740824 + 0.997252i \(0.523603\pi\)
\(132\) 6.09245 0.530280
\(133\) −4.12135 −0.357366
\(134\) −14.7127 −1.27098
\(135\) −7.57529 −0.651977
\(136\) 3.00120 0.257350
\(137\) 12.1768 1.04034 0.520168 0.854064i \(-0.325869\pi\)
0.520168 + 0.854064i \(0.325869\pi\)
\(138\) 1.26719 0.107870
\(139\) 4.82859 0.409556 0.204778 0.978808i \(-0.434353\pi\)
0.204778 + 0.978808i \(0.434353\pi\)
\(140\) 2.28246 0.192903
\(141\) −0.955586 −0.0804748
\(142\) −5.76111 −0.483462
\(143\) −9.24659 −0.773238
\(144\) −1.39423 −0.116186
\(145\) 1.36043 0.112977
\(146\) −0.364931 −0.0302019
\(147\) −5.30338 −0.437415
\(148\) 8.33349 0.685008
\(149\) −20.4742 −1.67731 −0.838657 0.544660i \(-0.816659\pi\)
−0.838657 + 0.544660i \(0.816659\pi\)
\(150\) −3.99068 −0.325838
\(151\) 14.9617 1.21757 0.608784 0.793336i \(-0.291658\pi\)
0.608784 + 0.793336i \(0.291658\pi\)
\(152\) −2.45647 −0.199246
\(153\) −4.18437 −0.338286
\(154\) 8.06636 0.650006
\(155\) −9.43772 −0.758056
\(156\) −2.43709 −0.195124
\(157\) −7.35532 −0.587019 −0.293509 0.955956i \(-0.594823\pi\)
−0.293509 + 0.955956i \(0.594823\pi\)
\(158\) −14.4028 −1.14583
\(159\) 5.66991 0.449653
\(160\) 1.36043 0.107551
\(161\) 1.67775 0.132225
\(162\) −2.87341 −0.225757
\(163\) 23.8132 1.86519 0.932596 0.360923i \(-0.117538\pi\)
0.932596 + 0.360923i \(0.117538\pi\)
\(164\) −1.49177 −0.116488
\(165\) 8.28833 0.645246
\(166\) 4.02625 0.312498
\(167\) 13.1045 1.01406 0.507029 0.861929i \(-0.330744\pi\)
0.507029 + 0.861929i \(0.330744\pi\)
\(168\) 2.12603 0.164026
\(169\) −9.30119 −0.715476
\(170\) 4.08291 0.313145
\(171\) 3.42490 0.261908
\(172\) 0.840503 0.0640877
\(173\) −13.2765 −1.00940 −0.504698 0.863296i \(-0.668396\pi\)
−0.504698 + 0.863296i \(0.668396\pi\)
\(174\) 1.26719 0.0960653
\(175\) −5.28364 −0.399405
\(176\) 4.80785 0.362405
\(177\) 12.7149 0.955708
\(178\) −5.58835 −0.418864
\(179\) −3.06552 −0.229127 −0.114564 0.993416i \(-0.536547\pi\)
−0.114564 + 0.993416i \(0.536547\pi\)
\(180\) −1.89675 −0.141376
\(181\) 1.51768 0.112809 0.0564043 0.998408i \(-0.482036\pi\)
0.0564043 + 0.998408i \(0.482036\pi\)
\(182\) −3.22670 −0.239179
\(183\) 6.58573 0.486831
\(184\) 1.00000 0.0737210
\(185\) 11.3371 0.833520
\(186\) −8.79089 −0.644579
\(187\) 14.4293 1.05517
\(188\) −0.754099 −0.0549983
\(189\) −9.34225 −0.679549
\(190\) −3.34185 −0.242443
\(191\) 14.9770 1.08370 0.541850 0.840475i \(-0.317724\pi\)
0.541850 + 0.840475i \(0.317724\pi\)
\(192\) 1.26719 0.0914514
\(193\) −15.0546 −1.08365 −0.541825 0.840491i \(-0.682267\pi\)
−0.541825 + 0.840491i \(0.682267\pi\)
\(194\) 16.2117 1.16393
\(195\) −3.31549 −0.237427
\(196\) −4.18516 −0.298940
\(197\) −14.4974 −1.03290 −0.516448 0.856318i \(-0.672746\pi\)
−0.516448 + 0.856318i \(0.672746\pi\)
\(198\) −6.70326 −0.476380
\(199\) 0.866870 0.0614508 0.0307254 0.999528i \(-0.490218\pi\)
0.0307254 + 0.999528i \(0.490218\pi\)
\(200\) −3.14924 −0.222685
\(201\) −18.6438 −1.31503
\(202\) 5.25982 0.370080
\(203\) 1.67775 0.117755
\(204\) 3.80308 0.266269
\(205\) −2.02944 −0.141742
\(206\) 7.07418 0.492881
\(207\) −1.39423 −0.0969059
\(208\) −1.92323 −0.133352
\(209\) −11.8103 −0.816938
\(210\) 2.89230 0.199588
\(211\) 18.4414 1.26956 0.634779 0.772694i \(-0.281091\pi\)
0.634779 + 0.772694i \(0.281091\pi\)
\(212\) 4.47440 0.307303
\(213\) −7.30041 −0.500216
\(214\) 2.68162 0.183312
\(215\) 1.14344 0.0779821
\(216\) −5.56832 −0.378876
\(217\) −11.6391 −0.790113
\(218\) −11.8123 −0.800027
\(219\) −0.462436 −0.0312485
\(220\) 6.54072 0.440975
\(221\) −5.77198 −0.388266
\(222\) 10.5601 0.708747
\(223\) −2.35395 −0.157632 −0.0788161 0.996889i \(-0.525114\pi\)
−0.0788161 + 0.996889i \(0.525114\pi\)
\(224\) 1.67775 0.112099
\(225\) 4.39077 0.292718
\(226\) −14.8431 −0.987351
\(227\) −19.0700 −1.26572 −0.632859 0.774267i \(-0.718119\pi\)
−0.632859 + 0.774267i \(0.718119\pi\)
\(228\) −3.11281 −0.206151
\(229\) −8.57773 −0.566832 −0.283416 0.958997i \(-0.591468\pi\)
−0.283416 + 0.958997i \(0.591468\pi\)
\(230\) 1.36043 0.0897039
\(231\) 10.2216 0.672532
\(232\) 1.00000 0.0656532
\(233\) −15.3009 −1.00240 −0.501198 0.865332i \(-0.667107\pi\)
−0.501198 + 0.865332i \(0.667107\pi\)
\(234\) 2.68143 0.175290
\(235\) −1.02590 −0.0669221
\(236\) 10.0339 0.653153
\(237\) −18.2511 −1.18554
\(238\) 5.03526 0.326387
\(239\) −13.7120 −0.886954 −0.443477 0.896286i \(-0.646255\pi\)
−0.443477 + 0.896286i \(0.646255\pi\)
\(240\) 1.72392 0.111278
\(241\) −10.7067 −0.689678 −0.344839 0.938662i \(-0.612066\pi\)
−0.344839 + 0.938662i \(0.612066\pi\)
\(242\) 12.1154 0.778807
\(243\) 13.0638 0.838044
\(244\) 5.19712 0.332712
\(245\) −5.69360 −0.363751
\(246\) −1.89035 −0.120524
\(247\) 4.72436 0.300604
\(248\) −6.93732 −0.440520
\(249\) 5.10202 0.323327
\(250\) −11.0864 −0.701168
\(251\) −9.17063 −0.578845 −0.289422 0.957201i \(-0.593463\pi\)
−0.289422 + 0.957201i \(0.593463\pi\)
\(252\) −2.33918 −0.147354
\(253\) 4.80785 0.302267
\(254\) −13.9826 −0.877349
\(255\) 5.17381 0.323997
\(256\) 1.00000 0.0625000
\(257\) −5.17021 −0.322509 −0.161255 0.986913i \(-0.551554\pi\)
−0.161255 + 0.986913i \(0.551554\pi\)
\(258\) 1.06508 0.0663087
\(259\) 13.9815 0.868769
\(260\) −2.61641 −0.162263
\(261\) −1.39423 −0.0863009
\(262\) −1.69582 −0.104768
\(263\) −10.8593 −0.669613 −0.334806 0.942287i \(-0.608671\pi\)
−0.334806 + 0.942287i \(0.608671\pi\)
\(264\) 6.09245 0.374964
\(265\) 6.08709 0.373927
\(266\) −4.12135 −0.252696
\(267\) −7.08149 −0.433380
\(268\) −14.7127 −0.898721
\(269\) 22.5988 1.37787 0.688935 0.724823i \(-0.258078\pi\)
0.688935 + 0.724823i \(0.258078\pi\)
\(270\) −7.57529 −0.461018
\(271\) −22.9257 −1.39264 −0.696320 0.717732i \(-0.745180\pi\)
−0.696320 + 0.717732i \(0.745180\pi\)
\(272\) 3.00120 0.181974
\(273\) −4.08883 −0.247467
\(274\) 12.1768 0.735628
\(275\) −15.1411 −0.913040
\(276\) 1.26719 0.0762758
\(277\) 6.16710 0.370545 0.185273 0.982687i \(-0.440683\pi\)
0.185273 + 0.982687i \(0.440683\pi\)
\(278\) 4.82859 0.289600
\(279\) 9.67224 0.579062
\(280\) 2.28246 0.136403
\(281\) 24.4453 1.45828 0.729141 0.684364i \(-0.239920\pi\)
0.729141 + 0.684364i \(0.239920\pi\)
\(282\) −0.955586 −0.0569043
\(283\) −12.9213 −0.768089 −0.384044 0.923315i \(-0.625469\pi\)
−0.384044 + 0.923315i \(0.625469\pi\)
\(284\) −5.76111 −0.341859
\(285\) −4.23475 −0.250845
\(286\) −9.24659 −0.546762
\(287\) −2.50281 −0.147736
\(288\) −1.39423 −0.0821560
\(289\) −7.99282 −0.470166
\(290\) 1.36043 0.0798870
\(291\) 20.5432 1.20426
\(292\) −0.364931 −0.0213560
\(293\) 19.3518 1.13055 0.565273 0.824904i \(-0.308771\pi\)
0.565273 + 0.824904i \(0.308771\pi\)
\(294\) −5.30338 −0.309299
\(295\) 13.6504 0.794758
\(296\) 8.33349 0.484374
\(297\) −26.7716 −1.55345
\(298\) −20.4742 −1.18604
\(299\) −1.92323 −0.111223
\(300\) −3.99068 −0.230402
\(301\) 1.41015 0.0812799
\(302\) 14.9617 0.860951
\(303\) 6.66519 0.382905
\(304\) −2.45647 −0.140888
\(305\) 7.07030 0.404844
\(306\) −4.18437 −0.239204
\(307\) 26.5296 1.51413 0.757064 0.653341i \(-0.226633\pi\)
0.757064 + 0.653341i \(0.226633\pi\)
\(308\) 8.06636 0.459624
\(309\) 8.96432 0.509962
\(310\) −9.43772 −0.536026
\(311\) 1.44899 0.0821648 0.0410824 0.999156i \(-0.486919\pi\)
0.0410824 + 0.999156i \(0.486919\pi\)
\(312\) −2.43709 −0.137973
\(313\) 11.4897 0.649436 0.324718 0.945811i \(-0.394731\pi\)
0.324718 + 0.945811i \(0.394731\pi\)
\(314\) −7.35532 −0.415085
\(315\) −3.18228 −0.179301
\(316\) −14.4028 −0.810222
\(317\) 12.2314 0.686986 0.343493 0.939155i \(-0.388390\pi\)
0.343493 + 0.939155i \(0.388390\pi\)
\(318\) 5.66991 0.317953
\(319\) 4.80785 0.269188
\(320\) 1.36043 0.0760502
\(321\) 3.39812 0.189664
\(322\) 1.67775 0.0934973
\(323\) −7.37235 −0.410209
\(324\) −2.87341 −0.159634
\(325\) 6.05670 0.335966
\(326\) 23.8132 1.31889
\(327\) −14.9684 −0.827752
\(328\) −1.49177 −0.0823691
\(329\) −1.26519 −0.0697522
\(330\) 8.28833 0.456257
\(331\) −3.46211 −0.190295 −0.0951473 0.995463i \(-0.530332\pi\)
−0.0951473 + 0.995463i \(0.530332\pi\)
\(332\) 4.02625 0.220969
\(333\) −11.6188 −0.636708
\(334\) 13.1045 0.717047
\(335\) −20.0156 −1.09357
\(336\) 2.12603 0.115984
\(337\) 10.6391 0.579548 0.289774 0.957095i \(-0.406420\pi\)
0.289774 + 0.957095i \(0.406420\pi\)
\(338\) −9.30119 −0.505918
\(339\) −18.8091 −1.02157
\(340\) 4.08291 0.221427
\(341\) −33.3536 −1.80620
\(342\) 3.42490 0.185197
\(343\) −18.7659 −1.01326
\(344\) 0.840503 0.0453169
\(345\) 1.72392 0.0928126
\(346\) −13.2765 −0.713750
\(347\) 1.01251 0.0543542 0.0271771 0.999631i \(-0.491348\pi\)
0.0271771 + 0.999631i \(0.491348\pi\)
\(348\) 1.26719 0.0679284
\(349\) 31.6050 1.69178 0.845888 0.533360i \(-0.179071\pi\)
0.845888 + 0.533360i \(0.179071\pi\)
\(350\) −5.28364 −0.282422
\(351\) 10.7092 0.571612
\(352\) 4.80785 0.256259
\(353\) −34.4148 −1.83171 −0.915857 0.401505i \(-0.868487\pi\)
−0.915857 + 0.401505i \(0.868487\pi\)
\(354\) 12.7149 0.675788
\(355\) −7.83757 −0.415975
\(356\) −5.58835 −0.296182
\(357\) 6.38062 0.337698
\(358\) −3.06552 −0.162018
\(359\) 5.12711 0.270598 0.135299 0.990805i \(-0.456800\pi\)
0.135299 + 0.990805i \(0.456800\pi\)
\(360\) −1.89675 −0.0999676
\(361\) −12.9657 −0.682408
\(362\) 1.51768 0.0797677
\(363\) 15.3525 0.805796
\(364\) −3.22670 −0.169125
\(365\) −0.496462 −0.0259860
\(366\) 6.58573 0.344242
\(367\) −22.6211 −1.18081 −0.590405 0.807107i \(-0.701032\pi\)
−0.590405 + 0.807107i \(0.701032\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.07987 0.108274
\(370\) 11.3371 0.589388
\(371\) 7.50692 0.389740
\(372\) −8.79089 −0.455787
\(373\) 6.33703 0.328119 0.164060 0.986450i \(-0.447541\pi\)
0.164060 + 0.986450i \(0.447541\pi\)
\(374\) 14.4293 0.746121
\(375\) −14.0486 −0.725467
\(376\) −0.754099 −0.0388897
\(377\) −1.92323 −0.0990513
\(378\) −9.34225 −0.480514
\(379\) 27.1482 1.39451 0.697255 0.716823i \(-0.254404\pi\)
0.697255 + 0.716823i \(0.254404\pi\)
\(380\) −3.34185 −0.171433
\(381\) −17.7186 −0.907753
\(382\) 14.9770 0.766292
\(383\) −2.29092 −0.117060 −0.0585302 0.998286i \(-0.518641\pi\)
−0.0585302 + 0.998286i \(0.518641\pi\)
\(384\) 1.26719 0.0646659
\(385\) 10.9737 0.559271
\(386\) −15.0546 −0.766257
\(387\) −1.17186 −0.0595688
\(388\) 16.2117 0.823022
\(389\) −32.7508 −1.66053 −0.830267 0.557366i \(-0.811812\pi\)
−0.830267 + 0.557366i \(0.811812\pi\)
\(390\) −3.31549 −0.167886
\(391\) 3.00120 0.151777
\(392\) −4.18516 −0.211382
\(393\) −2.14893 −0.108399
\(394\) −14.4974 −0.730368
\(395\) −19.5940 −0.985880
\(396\) −6.70326 −0.336852
\(397\) 2.74992 0.138015 0.0690073 0.997616i \(-0.478017\pi\)
0.0690073 + 0.997616i \(0.478017\pi\)
\(398\) 0.866870 0.0434523
\(399\) −5.22252 −0.261453
\(400\) −3.14924 −0.157462
\(401\) 12.5672 0.627578 0.313789 0.949493i \(-0.398402\pi\)
0.313789 + 0.949493i \(0.398402\pi\)
\(402\) −18.6438 −0.929867
\(403\) 13.3420 0.664615
\(404\) 5.25982 0.261686
\(405\) −3.90907 −0.194243
\(406\) 1.67775 0.0832653
\(407\) 40.0661 1.98600
\(408\) 3.80308 0.188281
\(409\) −35.5121 −1.75596 −0.877981 0.478696i \(-0.841110\pi\)
−0.877981 + 0.478696i \(0.841110\pi\)
\(410\) −2.02944 −0.100227
\(411\) 15.4303 0.761121
\(412\) 7.07418 0.348520
\(413\) 16.8344 0.828367
\(414\) −1.39423 −0.0685228
\(415\) 5.47742 0.268876
\(416\) −1.92323 −0.0942940
\(417\) 6.11874 0.299636
\(418\) −11.8103 −0.577663
\(419\) 26.3900 1.28924 0.644618 0.764505i \(-0.277016\pi\)
0.644618 + 0.764505i \(0.277016\pi\)
\(420\) 2.89230 0.141130
\(421\) 16.7524 0.816460 0.408230 0.912879i \(-0.366146\pi\)
0.408230 + 0.912879i \(0.366146\pi\)
\(422\) 18.4414 0.897713
\(423\) 1.05139 0.0511203
\(424\) 4.47440 0.217296
\(425\) −9.45148 −0.458464
\(426\) −7.30041 −0.353706
\(427\) 8.71947 0.421965
\(428\) 2.68162 0.129621
\(429\) −11.7172 −0.565710
\(430\) 1.14344 0.0551417
\(431\) 28.3187 1.36406 0.682032 0.731323i \(-0.261097\pi\)
0.682032 + 0.731323i \(0.261097\pi\)
\(432\) −5.56832 −0.267906
\(433\) 21.9376 1.05425 0.527127 0.849787i \(-0.323269\pi\)
0.527127 + 0.849787i \(0.323269\pi\)
\(434\) −11.6391 −0.558694
\(435\) 1.72392 0.0826555
\(436\) −11.8123 −0.565704
\(437\) −2.45647 −0.117509
\(438\) −0.462436 −0.0220961
\(439\) 5.61332 0.267909 0.133955 0.990987i \(-0.457232\pi\)
0.133955 + 0.990987i \(0.457232\pi\)
\(440\) 6.54072 0.311817
\(441\) 5.83508 0.277861
\(442\) −5.77198 −0.274545
\(443\) 16.4683 0.782435 0.391217 0.920298i \(-0.372054\pi\)
0.391217 + 0.920298i \(0.372054\pi\)
\(444\) 10.5601 0.501160
\(445\) −7.60253 −0.360395
\(446\) −2.35395 −0.111463
\(447\) −25.9447 −1.22714
\(448\) 1.67775 0.0792662
\(449\) −11.0487 −0.521420 −0.260710 0.965417i \(-0.583957\pi\)
−0.260710 + 0.965417i \(0.583957\pi\)
\(450\) 4.39077 0.206983
\(451\) −7.17219 −0.337725
\(452\) −14.8431 −0.698163
\(453\) 18.9593 0.890787
\(454\) −19.0700 −0.894997
\(455\) −4.38968 −0.205791
\(456\) −3.11281 −0.145771
\(457\) −0.859524 −0.0402068 −0.0201034 0.999798i \(-0.506400\pi\)
−0.0201034 + 0.999798i \(0.506400\pi\)
\(458\) −8.57773 −0.400811
\(459\) −16.7116 −0.780032
\(460\) 1.36043 0.0634302
\(461\) −12.9533 −0.603297 −0.301649 0.953419i \(-0.597537\pi\)
−0.301649 + 0.953419i \(0.597537\pi\)
\(462\) 10.2216 0.475552
\(463\) −32.4852 −1.50971 −0.754856 0.655890i \(-0.772294\pi\)
−0.754856 + 0.655890i \(0.772294\pi\)
\(464\) 1.00000 0.0464238
\(465\) −11.9594 −0.554602
\(466\) −15.3009 −0.708802
\(467\) 6.04140 0.279563 0.139781 0.990182i \(-0.455360\pi\)
0.139781 + 0.990182i \(0.455360\pi\)
\(468\) 2.68143 0.123949
\(469\) −24.6842 −1.13981
\(470\) −1.02590 −0.0473211
\(471\) −9.32058 −0.429470
\(472\) 10.0339 0.461849
\(473\) 4.04101 0.185806
\(474\) −18.2511 −0.838300
\(475\) 7.73602 0.354953
\(476\) 5.03526 0.230791
\(477\) −6.23836 −0.285635
\(478\) −13.7120 −0.627171
\(479\) 15.0832 0.689168 0.344584 0.938755i \(-0.388020\pi\)
0.344584 + 0.938755i \(0.388020\pi\)
\(480\) 1.72392 0.0786857
\(481\) −16.0272 −0.730777
\(482\) −10.7067 −0.487676
\(483\) 2.12603 0.0967375
\(484\) 12.1154 0.550700
\(485\) 22.0548 1.00146
\(486\) 13.0638 0.592586
\(487\) 28.7409 1.30238 0.651188 0.758917i \(-0.274271\pi\)
0.651188 + 0.758917i \(0.274271\pi\)
\(488\) 5.19712 0.235263
\(489\) 30.1758 1.36460
\(490\) −5.69360 −0.257211
\(491\) −31.7016 −1.43067 −0.715336 0.698781i \(-0.753726\pi\)
−0.715336 + 0.698781i \(0.753726\pi\)
\(492\) −1.89035 −0.0852236
\(493\) 3.00120 0.135167
\(494\) 4.72436 0.212559
\(495\) −9.11930 −0.409882
\(496\) −6.93732 −0.311495
\(497\) −9.66570 −0.433566
\(498\) 5.10202 0.228627
\(499\) −7.15558 −0.320328 −0.160164 0.987090i \(-0.551202\pi\)
−0.160164 + 0.987090i \(0.551202\pi\)
\(500\) −11.0864 −0.495801
\(501\) 16.6059 0.741896
\(502\) −9.17063 −0.409305
\(503\) 6.70427 0.298929 0.149464 0.988767i \(-0.452245\pi\)
0.149464 + 0.988767i \(0.452245\pi\)
\(504\) −2.33918 −0.104195
\(505\) 7.15561 0.318420
\(506\) 4.80785 0.213735
\(507\) −11.7864 −0.523451
\(508\) −13.9826 −0.620379
\(509\) −8.99690 −0.398781 −0.199390 0.979920i \(-0.563896\pi\)
−0.199390 + 0.979920i \(0.563896\pi\)
\(510\) 5.17381 0.229100
\(511\) −0.612263 −0.0270849
\(512\) 1.00000 0.0441942
\(513\) 13.6784 0.603917
\(514\) −5.17021 −0.228048
\(515\) 9.62390 0.424080
\(516\) 1.06508 0.0468873
\(517\) −3.62559 −0.159453
\(518\) 13.9815 0.614312
\(519\) −16.8239 −0.738485
\(520\) −2.61641 −0.114737
\(521\) 37.9054 1.66067 0.830334 0.557266i \(-0.188150\pi\)
0.830334 + 0.557266i \(0.188150\pi\)
\(522\) −1.39423 −0.0610239
\(523\) 34.2143 1.49609 0.748044 0.663649i \(-0.230993\pi\)
0.748044 + 0.663649i \(0.230993\pi\)
\(524\) −1.69582 −0.0740824
\(525\) −6.69536 −0.292210
\(526\) −10.8593 −0.473488
\(527\) −20.8203 −0.906945
\(528\) 6.09245 0.265140
\(529\) 1.00000 0.0434783
\(530\) 6.08709 0.264406
\(531\) −13.9896 −0.607098
\(532\) −4.12135 −0.178683
\(533\) 2.86901 0.124271
\(534\) −7.08149 −0.306446
\(535\) 3.64815 0.157723
\(536\) −14.7127 −0.635492
\(537\) −3.88459 −0.167632
\(538\) 22.5988 0.974301
\(539\) −20.1216 −0.866698
\(540\) −7.57529 −0.325989
\(541\) −6.07269 −0.261085 −0.130543 0.991443i \(-0.541672\pi\)
−0.130543 + 0.991443i \(0.541672\pi\)
\(542\) −22.9257 −0.984745
\(543\) 1.92319 0.0825321
\(544\) 3.00120 0.128675
\(545\) −16.0697 −0.688351
\(546\) −4.08883 −0.174986
\(547\) 44.4162 1.89910 0.949549 0.313619i \(-0.101542\pi\)
0.949549 + 0.313619i \(0.101542\pi\)
\(548\) 12.1768 0.520168
\(549\) −7.24600 −0.309252
\(550\) −15.1411 −0.645617
\(551\) −2.45647 −0.104649
\(552\) 1.26719 0.0539351
\(553\) −24.1643 −1.02757
\(554\) 6.16710 0.262015
\(555\) 14.3662 0.609813
\(556\) 4.82859 0.204778
\(557\) 3.21542 0.136242 0.0681209 0.997677i \(-0.478300\pi\)
0.0681209 + 0.997677i \(0.478300\pi\)
\(558\) 9.67224 0.409459
\(559\) −1.61648 −0.0683697
\(560\) 2.28246 0.0964514
\(561\) 18.2846 0.771978
\(562\) 24.4453 1.03116
\(563\) 11.9237 0.502525 0.251262 0.967919i \(-0.419154\pi\)
0.251262 + 0.967919i \(0.419154\pi\)
\(564\) −0.955586 −0.0402374
\(565\) −20.1930 −0.849526
\(566\) −12.9213 −0.543121
\(567\) −4.82087 −0.202457
\(568\) −5.76111 −0.241731
\(569\) −10.8574 −0.455167 −0.227584 0.973759i \(-0.573082\pi\)
−0.227584 + 0.973759i \(0.573082\pi\)
\(570\) −4.23475 −0.177374
\(571\) −1.83362 −0.0767345 −0.0383673 0.999264i \(-0.512216\pi\)
−0.0383673 + 0.999264i \(0.512216\pi\)
\(572\) −9.24659 −0.386619
\(573\) 18.9787 0.792847
\(574\) −2.50281 −0.104465
\(575\) −3.14924 −0.131332
\(576\) −1.39423 −0.0580931
\(577\) 15.0644 0.627140 0.313570 0.949565i \(-0.398475\pi\)
0.313570 + 0.949565i \(0.398475\pi\)
\(578\) −7.99282 −0.332458
\(579\) −19.0770 −0.792811
\(580\) 1.36043 0.0564886
\(581\) 6.75504 0.280246
\(582\) 20.5432 0.851544
\(583\) 21.5122 0.890945
\(584\) −0.364931 −0.0151009
\(585\) 3.64789 0.150822
\(586\) 19.3518 0.799417
\(587\) 18.8131 0.776499 0.388250 0.921554i \(-0.373080\pi\)
0.388250 + 0.921554i \(0.373080\pi\)
\(588\) −5.30338 −0.218708
\(589\) 17.0413 0.702176
\(590\) 13.6504 0.561979
\(591\) −18.3709 −0.755679
\(592\) 8.33349 0.342504
\(593\) 39.1535 1.60784 0.803921 0.594736i \(-0.202744\pi\)
0.803921 + 0.594736i \(0.202744\pi\)
\(594\) −26.7716 −1.09845
\(595\) 6.85010 0.280827
\(596\) −20.4742 −0.838657
\(597\) 1.09849 0.0449581
\(598\) −1.92323 −0.0786466
\(599\) 10.7095 0.437577 0.218789 0.975772i \(-0.429790\pi\)
0.218789 + 0.975772i \(0.429790\pi\)
\(600\) −3.99068 −0.162919
\(601\) 23.5653 0.961247 0.480623 0.876927i \(-0.340410\pi\)
0.480623 + 0.876927i \(0.340410\pi\)
\(602\) 1.41015 0.0574736
\(603\) 20.5129 0.835352
\(604\) 14.9617 0.608784
\(605\) 16.4821 0.670093
\(606\) 6.66519 0.270755
\(607\) 30.9235 1.25514 0.627572 0.778558i \(-0.284049\pi\)
0.627572 + 0.778558i \(0.284049\pi\)
\(608\) −2.45647 −0.0996231
\(609\) 2.12603 0.0861509
\(610\) 7.07030 0.286268
\(611\) 1.45030 0.0586730
\(612\) −4.18437 −0.169143
\(613\) −14.7372 −0.595229 −0.297614 0.954686i \(-0.596191\pi\)
−0.297614 + 0.954686i \(0.596191\pi\)
\(614\) 26.5296 1.07065
\(615\) −2.57168 −0.103700
\(616\) 8.06636 0.325003
\(617\) −31.0254 −1.24904 −0.624518 0.781011i \(-0.714704\pi\)
−0.624518 + 0.781011i \(0.714704\pi\)
\(618\) 8.96432 0.360598
\(619\) 38.0968 1.53124 0.765620 0.643293i \(-0.222432\pi\)
0.765620 + 0.643293i \(0.222432\pi\)
\(620\) −9.43772 −0.379028
\(621\) −5.56832 −0.223449
\(622\) 1.44899 0.0580993
\(623\) −9.37585 −0.375635
\(624\) −2.43709 −0.0975618
\(625\) 0.663903 0.0265561
\(626\) 11.4897 0.459221
\(627\) −14.9659 −0.597682
\(628\) −7.35532 −0.293509
\(629\) 25.0104 0.997231
\(630\) −3.18228 −0.126785
\(631\) −19.6621 −0.782736 −0.391368 0.920234i \(-0.627998\pi\)
−0.391368 + 0.920234i \(0.627998\pi\)
\(632\) −14.4028 −0.572913
\(633\) 23.3687 0.928824
\(634\) 12.2314 0.485772
\(635\) −19.0224 −0.754879
\(636\) 5.66991 0.224826
\(637\) 8.04901 0.318913
\(638\) 4.80785 0.190344
\(639\) 8.03233 0.317754
\(640\) 1.36043 0.0537756
\(641\) −22.7454 −0.898390 −0.449195 0.893434i \(-0.648289\pi\)
−0.449195 + 0.893434i \(0.648289\pi\)
\(642\) 3.39812 0.134113
\(643\) −36.1762 −1.42665 −0.713325 0.700833i \(-0.752812\pi\)
−0.713325 + 0.700833i \(0.752812\pi\)
\(644\) 1.67775 0.0661126
\(645\) 1.44896 0.0570526
\(646\) −7.37235 −0.290061
\(647\) 35.9675 1.41403 0.707013 0.707200i \(-0.250042\pi\)
0.707013 + 0.707200i \(0.250042\pi\)
\(648\) −2.87341 −0.112878
\(649\) 48.2416 1.89365
\(650\) 6.05670 0.237564
\(651\) −14.7489 −0.578056
\(652\) 23.8132 0.932596
\(653\) −6.58068 −0.257522 −0.128761 0.991676i \(-0.541100\pi\)
−0.128761 + 0.991676i \(0.541100\pi\)
\(654\) −14.9684 −0.585309
\(655\) −2.30704 −0.0901437
\(656\) −1.49177 −0.0582438
\(657\) 0.508799 0.0198501
\(658\) −1.26519 −0.0493222
\(659\) 16.7980 0.654356 0.327178 0.944963i \(-0.393902\pi\)
0.327178 + 0.944963i \(0.393902\pi\)
\(660\) 8.28833 0.322623
\(661\) 2.79814 0.108835 0.0544175 0.998518i \(-0.482670\pi\)
0.0544175 + 0.998518i \(0.482670\pi\)
\(662\) −3.46211 −0.134559
\(663\) −7.31419 −0.284060
\(664\) 4.02625 0.156249
\(665\) −5.60679 −0.217422
\(666\) −11.6188 −0.450220
\(667\) 1.00000 0.0387202
\(668\) 13.1045 0.507029
\(669\) −2.98290 −0.115326
\(670\) −20.0156 −0.773268
\(671\) 24.9870 0.964611
\(672\) 2.12603 0.0820132
\(673\) 43.8643 1.69085 0.845423 0.534097i \(-0.179348\pi\)
0.845423 + 0.534097i \(0.179348\pi\)
\(674\) 10.6391 0.409802
\(675\) 17.5360 0.674960
\(676\) −9.30119 −0.357738
\(677\) 16.9798 0.652587 0.326293 0.945269i \(-0.394200\pi\)
0.326293 + 0.945269i \(0.394200\pi\)
\(678\) −18.8091 −0.722358
\(679\) 27.1991 1.04381
\(680\) 4.08291 0.156572
\(681\) −24.1652 −0.926013
\(682\) −33.3536 −1.27717
\(683\) 27.1983 1.04072 0.520358 0.853948i \(-0.325799\pi\)
0.520358 + 0.853948i \(0.325799\pi\)
\(684\) 3.42490 0.130954
\(685\) 16.5657 0.632941
\(686\) −18.7659 −0.716485
\(687\) −10.8696 −0.414701
\(688\) 0.840503 0.0320439
\(689\) −8.60529 −0.327835
\(690\) 1.72392 0.0656284
\(691\) −0.269591 −0.0102557 −0.00512786 0.999987i \(-0.501632\pi\)
−0.00512786 + 0.999987i \(0.501632\pi\)
\(692\) −13.2765 −0.504698
\(693\) −11.2464 −0.427215
\(694\) 1.01251 0.0384342
\(695\) 6.56895 0.249174
\(696\) 1.26719 0.0480327
\(697\) −4.47709 −0.169582
\(698\) 31.6050 1.19627
\(699\) −19.3892 −0.733365
\(700\) −5.28364 −0.199703
\(701\) −8.53751 −0.322457 −0.161229 0.986917i \(-0.551546\pi\)
−0.161229 + 0.986917i \(0.551546\pi\)
\(702\) 10.7092 0.404191
\(703\) −20.4710 −0.772078
\(704\) 4.80785 0.181203
\(705\) −1.30000 −0.0489610
\(706\) −34.4148 −1.29522
\(707\) 8.82467 0.331886
\(708\) 12.7149 0.477854
\(709\) −0.578454 −0.0217243 −0.0108621 0.999941i \(-0.503458\pi\)
−0.0108621 + 0.999941i \(0.503458\pi\)
\(710\) −7.83757 −0.294139
\(711\) 20.0809 0.753092
\(712\) −5.58835 −0.209432
\(713\) −6.93732 −0.259805
\(714\) 6.38062 0.238789
\(715\) −12.5793 −0.470439
\(716\) −3.06552 −0.114564
\(717\) −17.3757 −0.648906
\(718\) 5.12711 0.191342
\(719\) −8.24661 −0.307546 −0.153773 0.988106i \(-0.549143\pi\)
−0.153773 + 0.988106i \(0.549143\pi\)
\(720\) −1.89675 −0.0706878
\(721\) 11.8687 0.442014
\(722\) −12.9657 −0.482535
\(723\) −13.5674 −0.504576
\(724\) 1.51768 0.0564043
\(725\) −3.14924 −0.116960
\(726\) 15.3525 0.569784
\(727\) −22.4965 −0.834349 −0.417175 0.908826i \(-0.636980\pi\)
−0.417175 + 0.908826i \(0.636980\pi\)
\(728\) −3.22670 −0.119589
\(729\) 25.1745 0.932391
\(730\) −0.496462 −0.0183749
\(731\) 2.52251 0.0932985
\(732\) 6.58573 0.243416
\(733\) −28.0468 −1.03593 −0.517967 0.855401i \(-0.673311\pi\)
−0.517967 + 0.855401i \(0.673311\pi\)
\(734\) −22.6211 −0.834959
\(735\) −7.21486 −0.266124
\(736\) 1.00000 0.0368605
\(737\) −70.7364 −2.60561
\(738\) 2.07987 0.0765612
\(739\) −8.86610 −0.326145 −0.163072 0.986614i \(-0.552140\pi\)
−0.163072 + 0.986614i \(0.552140\pi\)
\(740\) 11.3371 0.416760
\(741\) 5.98665 0.219925
\(742\) 7.50692 0.275588
\(743\) −17.8425 −0.654576 −0.327288 0.944925i \(-0.606135\pi\)
−0.327288 + 0.944925i \(0.606135\pi\)
\(744\) −8.79089 −0.322290
\(745\) −27.8537 −1.02048
\(746\) 6.33703 0.232015
\(747\) −5.61353 −0.205388
\(748\) 14.4293 0.527587
\(749\) 4.49909 0.164393
\(750\) −14.0486 −0.512983
\(751\) 31.2394 1.13994 0.569972 0.821664i \(-0.306954\pi\)
0.569972 + 0.821664i \(0.306954\pi\)
\(752\) −0.754099 −0.0274992
\(753\) −11.6209 −0.423490
\(754\) −1.92323 −0.0700398
\(755\) 20.3543 0.740770
\(756\) −9.34225 −0.339774
\(757\) 24.1891 0.879166 0.439583 0.898202i \(-0.355126\pi\)
0.439583 + 0.898202i \(0.355126\pi\)
\(758\) 27.1482 0.986068
\(759\) 6.09245 0.221142
\(760\) −3.34185 −0.121222
\(761\) 5.37248 0.194752 0.0973761 0.995248i \(-0.468955\pi\)
0.0973761 + 0.995248i \(0.468955\pi\)
\(762\) −17.7186 −0.641878
\(763\) −19.8180 −0.717460
\(764\) 14.9770 0.541850
\(765\) −5.69253 −0.205814
\(766\) −2.29092 −0.0827742
\(767\) −19.2975 −0.696793
\(768\) 1.26719 0.0457257
\(769\) 25.1668 0.907539 0.453769 0.891119i \(-0.350079\pi\)
0.453769 + 0.891119i \(0.350079\pi\)
\(770\) 10.9737 0.395465
\(771\) −6.55164 −0.235951
\(772\) −15.0546 −0.541825
\(773\) −40.9003 −1.47108 −0.735540 0.677481i \(-0.763072\pi\)
−0.735540 + 0.677481i \(0.763072\pi\)
\(774\) −1.17186 −0.0421215
\(775\) 21.8473 0.784778
\(776\) 16.2117 0.581964
\(777\) 17.7172 0.635601
\(778\) −32.7508 −1.17417
\(779\) 3.66449 0.131294
\(780\) −3.31549 −0.118713
\(781\) −27.6985 −0.991131
\(782\) 3.00120 0.107323
\(783\) −5.56832 −0.198996
\(784\) −4.18516 −0.149470
\(785\) −10.0064 −0.357143
\(786\) −2.14893 −0.0766498
\(787\) 41.2713 1.47116 0.735582 0.677436i \(-0.236909\pi\)
0.735582 + 0.677436i \(0.236909\pi\)
\(788\) −14.4974 −0.516448
\(789\) −13.7608 −0.489896
\(790\) −19.5940 −0.697122
\(791\) −24.9031 −0.885452
\(792\) −6.70326 −0.238190
\(793\) −9.99525 −0.354942
\(794\) 2.74992 0.0975910
\(795\) 7.71349 0.273569
\(796\) 0.866870 0.0307254
\(797\) 0.713375 0.0252691 0.0126345 0.999920i \(-0.495978\pi\)
0.0126345 + 0.999920i \(0.495978\pi\)
\(798\) −5.22252 −0.184875
\(799\) −2.26320 −0.0800662
\(800\) −3.14924 −0.111342
\(801\) 7.79146 0.275298
\(802\) 12.5672 0.443764
\(803\) −1.75453 −0.0619161
\(804\) −18.6438 −0.657515
\(805\) 2.28246 0.0804460
\(806\) 13.3420 0.469954
\(807\) 28.6369 1.00807
\(808\) 5.25982 0.185040
\(809\) 12.0415 0.423357 0.211679 0.977339i \(-0.432107\pi\)
0.211679 + 0.977339i \(0.432107\pi\)
\(810\) −3.90907 −0.137351
\(811\) 43.6002 1.53101 0.765505 0.643430i \(-0.222489\pi\)
0.765505 + 0.643430i \(0.222489\pi\)
\(812\) 1.67775 0.0588775
\(813\) −29.0512 −1.01887
\(814\) 40.0661 1.40432
\(815\) 32.3961 1.13478
\(816\) 3.80308 0.133134
\(817\) −2.06467 −0.0722337
\(818\) −35.5121 −1.24165
\(819\) 4.49877 0.157200
\(820\) −2.02944 −0.0708712
\(821\) 12.0620 0.420966 0.210483 0.977597i \(-0.432496\pi\)
0.210483 + 0.977597i \(0.432496\pi\)
\(822\) 15.4303 0.538194
\(823\) 36.9766 1.28892 0.644461 0.764637i \(-0.277082\pi\)
0.644461 + 0.764637i \(0.277082\pi\)
\(824\) 7.07418 0.246441
\(825\) −19.1866 −0.667991
\(826\) 16.8344 0.585744
\(827\) −1.19387 −0.0415151 −0.0207575 0.999785i \(-0.506608\pi\)
−0.0207575 + 0.999785i \(0.506608\pi\)
\(828\) −1.39423 −0.0484530
\(829\) 28.8452 1.00184 0.500918 0.865495i \(-0.332996\pi\)
0.500918 + 0.865495i \(0.332996\pi\)
\(830\) 5.47742 0.190124
\(831\) 7.81488 0.271095
\(832\) −1.92323 −0.0666759
\(833\) −12.5605 −0.435194
\(834\) 6.11874 0.211875
\(835\) 17.8277 0.616954
\(836\) −11.8103 −0.408469
\(837\) 38.6292 1.33522
\(838\) 26.3900 0.911628
\(839\) −20.7303 −0.715688 −0.357844 0.933781i \(-0.616488\pi\)
−0.357844 + 0.933781i \(0.616488\pi\)
\(840\) 2.89230 0.0997939
\(841\) 1.00000 0.0344828
\(842\) 16.7524 0.577325
\(843\) 30.9767 1.06690
\(844\) 18.4414 0.634779
\(845\) −12.6536 −0.435297
\(846\) 1.05139 0.0361475
\(847\) 20.3266 0.698430
\(848\) 4.47440 0.153652
\(849\) −16.3737 −0.561943
\(850\) −9.45148 −0.324183
\(851\) 8.33349 0.285668
\(852\) −7.30041 −0.250108
\(853\) −29.8055 −1.02052 −0.510261 0.860020i \(-0.670451\pi\)
−0.510261 + 0.860020i \(0.670451\pi\)
\(854\) 8.71947 0.298374
\(855\) 4.65932 0.159345
\(856\) 2.68162 0.0916559
\(857\) 55.0638 1.88094 0.940471 0.339874i \(-0.110384\pi\)
0.940471 + 0.339874i \(0.110384\pi\)
\(858\) −11.7172 −0.400017
\(859\) 42.7439 1.45840 0.729201 0.684299i \(-0.239892\pi\)
0.729201 + 0.684299i \(0.239892\pi\)
\(860\) 1.14344 0.0389911
\(861\) −3.17154 −0.108086
\(862\) 28.3187 0.964538
\(863\) 0.0487062 0.00165798 0.000828990 1.00000i \(-0.499736\pi\)
0.000828990 1.00000i \(0.499736\pi\)
\(864\) −5.56832 −0.189438
\(865\) −18.0617 −0.614117
\(866\) 21.9376 0.745470
\(867\) −10.1284 −0.343979
\(868\) −11.6391 −0.395056
\(869\) −69.2465 −2.34903
\(870\) 1.72392 0.0584463
\(871\) 28.2959 0.958769
\(872\) −11.8123 −0.400013
\(873\) −22.6028 −0.764990
\(874\) −2.45647 −0.0830914
\(875\) −18.6003 −0.628804
\(876\) −0.462436 −0.0156243
\(877\) 2.82793 0.0954925 0.0477463 0.998859i \(-0.484796\pi\)
0.0477463 + 0.998859i \(0.484796\pi\)
\(878\) 5.61332 0.189440
\(879\) 24.5224 0.827121
\(880\) 6.54072 0.220488
\(881\) 19.5533 0.658767 0.329383 0.944196i \(-0.393159\pi\)
0.329383 + 0.944196i \(0.393159\pi\)
\(882\) 5.83508 0.196477
\(883\) 41.0246 1.38059 0.690294 0.723529i \(-0.257481\pi\)
0.690294 + 0.723529i \(0.257481\pi\)
\(884\) −5.77198 −0.194133
\(885\) 17.2977 0.581454
\(886\) 16.4683 0.553265
\(887\) 10.3756 0.348380 0.174190 0.984712i \(-0.444269\pi\)
0.174190 + 0.984712i \(0.444269\pi\)
\(888\) 10.5601 0.354374
\(889\) −23.4594 −0.786802
\(890\) −7.60253 −0.254838
\(891\) −13.8149 −0.462817
\(892\) −2.35395 −0.0788161
\(893\) 1.85242 0.0619890
\(894\) −25.9447 −0.867721
\(895\) −4.17041 −0.139401
\(896\) 1.67775 0.0560497
\(897\) −2.43709 −0.0813721
\(898\) −11.0487 −0.368700
\(899\) −6.93732 −0.231373
\(900\) 4.39077 0.146359
\(901\) 13.4285 0.447370
\(902\) −7.17219 −0.238808
\(903\) 1.78693 0.0594653
\(904\) −14.8431 −0.493676
\(905\) 2.06470 0.0686329
\(906\) 18.9593 0.629882
\(907\) −24.1680 −0.802485 −0.401243 0.915972i \(-0.631422\pi\)
−0.401243 + 0.915972i \(0.631422\pi\)
\(908\) −19.0700 −0.632859
\(909\) −7.33342 −0.243234
\(910\) −4.38968 −0.145517
\(911\) 20.8063 0.689344 0.344672 0.938723i \(-0.387990\pi\)
0.344672 + 0.938723i \(0.387990\pi\)
\(912\) −3.11281 −0.103076
\(913\) 19.3576 0.640643
\(914\) −0.859524 −0.0284305
\(915\) 8.95940 0.296189
\(916\) −8.57773 −0.283416
\(917\) −2.84517 −0.0939558
\(918\) −16.7116 −0.551566
\(919\) 7.76601 0.256177 0.128089 0.991763i \(-0.459116\pi\)
0.128089 + 0.991763i \(0.459116\pi\)
\(920\) 1.36043 0.0448519
\(921\) 33.6181 1.10775
\(922\) −12.9533 −0.426596
\(923\) 11.0799 0.364700
\(924\) 10.2216 0.336266
\(925\) −26.2441 −0.862902
\(926\) −32.4852 −1.06753
\(927\) −9.86306 −0.323945
\(928\) 1.00000 0.0328266
\(929\) 47.6223 1.56244 0.781218 0.624258i \(-0.214599\pi\)
0.781218 + 0.624258i \(0.214599\pi\)
\(930\) −11.9594 −0.392163
\(931\) 10.2807 0.336937
\(932\) −15.3009 −0.501198
\(933\) 1.83615 0.0601127
\(934\) 6.04140 0.197681
\(935\) 19.6300 0.641969
\(936\) 2.68143 0.0876452
\(937\) 17.0990 0.558599 0.279300 0.960204i \(-0.409898\pi\)
0.279300 + 0.960204i \(0.409898\pi\)
\(938\) −24.6842 −0.805969
\(939\) 14.5596 0.475135
\(940\) −1.02590 −0.0334611
\(941\) 27.6093 0.900037 0.450019 0.893019i \(-0.351417\pi\)
0.450019 + 0.893019i \(0.351417\pi\)
\(942\) −9.32058 −0.303681
\(943\) −1.49177 −0.0485787
\(944\) 10.0339 0.326576
\(945\) −12.7094 −0.413438
\(946\) 4.04101 0.131385
\(947\) −38.2847 −1.24409 −0.622043 0.782983i \(-0.713697\pi\)
−0.622043 + 0.782983i \(0.713697\pi\)
\(948\) −18.2511 −0.592768
\(949\) 0.701845 0.0227829
\(950\) 7.73602 0.250990
\(951\) 15.4995 0.502607
\(952\) 5.03526 0.163194
\(953\) −3.44005 −0.111434 −0.0557171 0.998447i \(-0.517744\pi\)
−0.0557171 + 0.998447i \(0.517744\pi\)
\(954\) −6.23836 −0.201974
\(955\) 20.3752 0.659325
\(956\) −13.7120 −0.443477
\(957\) 6.09245 0.196941
\(958\) 15.0832 0.487315
\(959\) 20.4296 0.659708
\(960\) 1.72392 0.0556392
\(961\) 17.1264 0.552465
\(962\) −16.0272 −0.516738
\(963\) −3.73880 −0.120481
\(964\) −10.7067 −0.344839
\(965\) −20.4806 −0.659294
\(966\) 2.12603 0.0684037
\(967\) −55.4650 −1.78363 −0.891817 0.452396i \(-0.850569\pi\)
−0.891817 + 0.452396i \(0.850569\pi\)
\(968\) 12.1154 0.389403
\(969\) −9.34216 −0.300113
\(970\) 22.0548 0.708136
\(971\) −54.8260 −1.75945 −0.879725 0.475483i \(-0.842273\pi\)
−0.879725 + 0.475483i \(0.842273\pi\)
\(972\) 13.0638 0.419022
\(973\) 8.10117 0.259712
\(974\) 28.7409 0.920918
\(975\) 7.67499 0.245796
\(976\) 5.19712 0.166356
\(977\) −34.4063 −1.10076 −0.550378 0.834915i \(-0.685516\pi\)
−0.550378 + 0.834915i \(0.685516\pi\)
\(978\) 30.1758 0.964915
\(979\) −26.8679 −0.858702
\(980\) −5.69360 −0.181875
\(981\) 16.4690 0.525816
\(982\) −31.7016 −1.01164
\(983\) −46.9384 −1.49710 −0.748552 0.663076i \(-0.769250\pi\)
−0.748552 + 0.663076i \(0.769250\pi\)
\(984\) −1.89035 −0.0602622
\(985\) −19.7226 −0.628416
\(986\) 3.00120 0.0955775
\(987\) −1.60323 −0.0510315
\(988\) 4.72436 0.150302
\(989\) 0.840503 0.0267264
\(990\) −9.11930 −0.289830
\(991\) 27.4682 0.872555 0.436277 0.899812i \(-0.356297\pi\)
0.436277 + 0.899812i \(0.356297\pi\)
\(992\) −6.93732 −0.220260
\(993\) −4.38714 −0.139222
\(994\) −9.66570 −0.306577
\(995\) 1.17931 0.0373867
\(996\) 5.10202 0.161664
\(997\) 41.7387 1.32188 0.660939 0.750440i \(-0.270158\pi\)
0.660939 + 0.750440i \(0.270158\pi\)
\(998\) −7.15558 −0.226506
\(999\) −46.4035 −1.46814
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 1334.2.a.k.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1334.2.a.k.1.6 10 1.1 even 1 trivial