Properties

Label 4030.2.a.p.1.1
Level $4030$
Weight $2$
Character 4030.1
Self dual yes
Analytic conductor $32.180$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4030,2,Mod(1,4030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4030.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: \(32.1797120146\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 16x^{7} + 46x^{6} + 80x^{5} - 212x^{4} - 133x^{3} + 294x^{2} + 52x - 112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.25457\) of defining polynomial
Character \(\chi\) \(=\) 4030.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.25457 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.25457 q^{6} -1.51784 q^{7} -1.00000 q^{8} +7.59221 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.25457 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.25457 q^{6} -1.51784 q^{7} -1.00000 q^{8} +7.59221 q^{9} +1.00000 q^{10} -2.88520 q^{11} -3.25457 q^{12} -1.00000 q^{13} +1.51784 q^{14} +3.25457 q^{15} +1.00000 q^{16} -1.75666 q^{17} -7.59221 q^{18} -4.82261 q^{19} -1.00000 q^{20} +4.93991 q^{21} +2.88520 q^{22} -1.81249 q^{23} +3.25457 q^{24} +1.00000 q^{25} +1.00000 q^{26} -14.9456 q^{27} -1.51784 q^{28} -1.34372 q^{29} -3.25457 q^{30} +1.00000 q^{31} -1.00000 q^{32} +9.39008 q^{33} +1.75666 q^{34} +1.51784 q^{35} +7.59221 q^{36} +6.10218 q^{37} +4.82261 q^{38} +3.25457 q^{39} +1.00000 q^{40} +8.61726 q^{41} -4.93991 q^{42} -8.63631 q^{43} -2.88520 q^{44} -7.59221 q^{45} +1.81249 q^{46} -8.82339 q^{47} -3.25457 q^{48} -4.69616 q^{49} -1.00000 q^{50} +5.71717 q^{51} -1.00000 q^{52} +5.04595 q^{53} +14.9456 q^{54} +2.88520 q^{55} +1.51784 q^{56} +15.6955 q^{57} +1.34372 q^{58} +0.167892 q^{59} +3.25457 q^{60} -12.5603 q^{61} -1.00000 q^{62} -11.5238 q^{63} +1.00000 q^{64} +1.00000 q^{65} -9.39008 q^{66} -9.70815 q^{67} -1.75666 q^{68} +5.89888 q^{69} -1.51784 q^{70} -3.09566 q^{71} -7.59221 q^{72} -14.9215 q^{73} -6.10218 q^{74} -3.25457 q^{75} -4.82261 q^{76} +4.37927 q^{77} -3.25457 q^{78} +7.76579 q^{79} -1.00000 q^{80} +25.8650 q^{81} -8.61726 q^{82} -9.01972 q^{83} +4.93991 q^{84} +1.75666 q^{85} +8.63631 q^{86} +4.37321 q^{87} +2.88520 q^{88} +10.3332 q^{89} +7.59221 q^{90} +1.51784 q^{91} -1.81249 q^{92} -3.25457 q^{93} +8.82339 q^{94} +4.82261 q^{95} +3.25457 q^{96} -18.0929 q^{97} +4.69616 q^{98} -21.9051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9} + 9 q^{10} + 14 q^{11} - 3 q^{12} - 9 q^{13} + 3 q^{14} + 3 q^{15} + 9 q^{16} + q^{17} - 14 q^{18} + 6 q^{19} - 9 q^{20} + q^{21} - 14 q^{22} - 4 q^{23} + 3 q^{24} + 9 q^{25} + 9 q^{26} - 15 q^{27} - 3 q^{28} + 15 q^{29} - 3 q^{30} + 9 q^{31} - 9 q^{32} + 14 q^{33} - q^{34} + 3 q^{35} + 14 q^{36} - 9 q^{37} - 6 q^{38} + 3 q^{39} + 9 q^{40} + 18 q^{41} - q^{42} - 23 q^{43} + 14 q^{44} - 14 q^{45} + 4 q^{46} + 3 q^{47} - 3 q^{48} + 12 q^{49} - 9 q^{50} - 11 q^{51} - 9 q^{52} - 6 q^{53} + 15 q^{54} - 14 q^{55} + 3 q^{56} + 17 q^{57} - 15 q^{58} + 28 q^{59} + 3 q^{60} - 9 q^{62} + 12 q^{63} + 9 q^{64} + 9 q^{65} - 14 q^{66} - 16 q^{67} + q^{68} - 6 q^{69} - 3 q^{70} + 32 q^{71} - 14 q^{72} - 11 q^{73} + 9 q^{74} - 3 q^{75} + 6 q^{76} - 29 q^{77} - 3 q^{78} - 8 q^{79} - 9 q^{80} + 9 q^{81} - 18 q^{82} + 15 q^{83} + q^{84} - q^{85} + 23 q^{86} - 19 q^{87} - 14 q^{88} + 51 q^{89} + 14 q^{90} + 3 q^{91} - 4 q^{92} - 3 q^{93} - 3 q^{94} - 6 q^{95} + 3 q^{96} - 26 q^{97} - 12 q^{98} + 27 q^{99}+O(q^{100}) \) 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
\(3\) −3.25457 −1.87903 −0.939513 0.342514i \(-0.888721\pi\)
−0.939513 + 0.342514i \(0.888721\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 3.25457 1.32867
\(7\) −1.51784 −0.573689 −0.286845 0.957977i \(-0.592606\pi\)
−0.286845 + 0.957977i \(0.592606\pi\)
\(8\) −1.00000 −0.353553
\(9\) 7.59221 2.53074
\(10\) 1.00000 0.316228
\(11\) −2.88520 −0.869921 −0.434961 0.900449i \(-0.643238\pi\)
−0.434961 + 0.900449i \(0.643238\pi\)
\(12\) −3.25457 −0.939513
\(13\) −1.00000 −0.277350
\(14\) 1.51784 0.405660
\(15\) 3.25457 0.840326
\(16\) 1.00000 0.250000
\(17\) −1.75666 −0.426053 −0.213026 0.977046i \(-0.568332\pi\)
−0.213026 + 0.977046i \(0.568332\pi\)
\(18\) −7.59221 −1.78950
\(19\) −4.82261 −1.10638 −0.553191 0.833054i \(-0.686590\pi\)
−0.553191 + 0.833054i \(0.686590\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.93991 1.07798
\(22\) 2.88520 0.615127
\(23\) −1.81249 −0.377931 −0.188966 0.981984i \(-0.560513\pi\)
−0.188966 + 0.981984i \(0.560513\pi\)
\(24\) 3.25457 0.664336
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) −14.9456 −2.87629
\(28\) −1.51784 −0.286845
\(29\) −1.34372 −0.249522 −0.124761 0.992187i \(-0.539816\pi\)
−0.124761 + 0.992187i \(0.539816\pi\)
\(30\) −3.25457 −0.594200
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 9.39008 1.63460
\(34\) 1.75666 0.301265
\(35\) 1.51784 0.256562
\(36\) 7.59221 1.26537
\(37\) 6.10218 1.00319 0.501596 0.865102i \(-0.332746\pi\)
0.501596 + 0.865102i \(0.332746\pi\)
\(38\) 4.82261 0.782331
\(39\) 3.25457 0.521148
\(40\) 1.00000 0.158114
\(41\) 8.61726 1.34579 0.672895 0.739738i \(-0.265051\pi\)
0.672895 + 0.739738i \(0.265051\pi\)
\(42\) −4.93991 −0.762245
\(43\) −8.63631 −1.31703 −0.658513 0.752570i \(-0.728814\pi\)
−0.658513 + 0.752570i \(0.728814\pi\)
\(44\) −2.88520 −0.434961
\(45\) −7.59221 −1.13178
\(46\) 1.81249 0.267238
\(47\) −8.82339 −1.28702 −0.643512 0.765436i \(-0.722523\pi\)
−0.643512 + 0.765436i \(0.722523\pi\)
\(48\) −3.25457 −0.469756
\(49\) −4.69616 −0.670880
\(50\) −1.00000 −0.141421
\(51\) 5.71717 0.800564
\(52\) −1.00000 −0.138675
\(53\) 5.04595 0.693114 0.346557 0.938029i \(-0.387351\pi\)
0.346557 + 0.938029i \(0.387351\pi\)
\(54\) 14.9456 2.03384
\(55\) 2.88520 0.389041
\(56\) 1.51784 0.202830
\(57\) 15.6955 2.07892
\(58\) 1.34372 0.176439
\(59\) 0.167892 0.0218577 0.0109288 0.999940i \(-0.496521\pi\)
0.0109288 + 0.999940i \(0.496521\pi\)
\(60\) 3.25457 0.420163
\(61\) −12.5603 −1.60818 −0.804091 0.594506i \(-0.797348\pi\)
−0.804091 + 0.594506i \(0.797348\pi\)
\(62\) −1.00000 −0.127000
\(63\) −11.5238 −1.45186
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −9.39008 −1.15584
\(67\) −9.70815 −1.18604 −0.593020 0.805188i \(-0.702064\pi\)
−0.593020 + 0.805188i \(0.702064\pi\)
\(68\) −1.75666 −0.213026
\(69\) 5.89888 0.710142
\(70\) −1.51784 −0.181417
\(71\) −3.09566 −0.367387 −0.183694 0.982984i \(-0.558805\pi\)
−0.183694 + 0.982984i \(0.558805\pi\)
\(72\) −7.59221 −0.894750
\(73\) −14.9215 −1.74643 −0.873217 0.487332i \(-0.837970\pi\)
−0.873217 + 0.487332i \(0.837970\pi\)
\(74\) −6.10218 −0.709364
\(75\) −3.25457 −0.375805
\(76\) −4.82261 −0.553191
\(77\) 4.37927 0.499065
\(78\) −3.25457 −0.368507
\(79\) 7.76579 0.873720 0.436860 0.899530i \(-0.356091\pi\)
0.436860 + 0.899530i \(0.356091\pi\)
\(80\) −1.00000 −0.111803
\(81\) 25.8650 2.87389
\(82\) −8.61726 −0.951617
\(83\) −9.01972 −0.990043 −0.495021 0.868881i \(-0.664840\pi\)
−0.495021 + 0.868881i \(0.664840\pi\)
\(84\) 4.93991 0.538988
\(85\) 1.75666 0.190537
\(86\) 8.63631 0.931278
\(87\) 4.37321 0.468858
\(88\) 2.88520 0.307564
\(89\) 10.3332 1.09532 0.547661 0.836701i \(-0.315518\pi\)
0.547661 + 0.836701i \(0.315518\pi\)
\(90\) 7.59221 0.800289
\(91\) 1.51784 0.159113
\(92\) −1.81249 −0.188966
\(93\) −3.25457 −0.337483
\(94\) 8.82339 0.910063
\(95\) 4.82261 0.494789
\(96\) 3.25457 0.332168
\(97\) −18.0929 −1.83706 −0.918529 0.395354i \(-0.870622\pi\)
−0.918529 + 0.395354i \(0.870622\pi\)
\(98\) 4.69616 0.474384
\(99\) −21.9051 −2.20154
\(100\) 1.00000 0.100000
\(101\) −19.0906 −1.89959 −0.949794 0.312877i \(-0.898707\pi\)
−0.949794 + 0.312877i \(0.898707\pi\)
\(102\) −5.71717 −0.566084
\(103\) −10.2723 −1.01216 −0.506080 0.862486i \(-0.668906\pi\)
−0.506080 + 0.862486i \(0.668906\pi\)
\(104\) 1.00000 0.0980581
\(105\) −4.93991 −0.482086
\(106\) −5.04595 −0.490106
\(107\) 0.752780 0.0727740 0.0363870 0.999338i \(-0.488415\pi\)
0.0363870 + 0.999338i \(0.488415\pi\)
\(108\) −14.9456 −1.43815
\(109\) −0.215107 −0.0206036 −0.0103018 0.999947i \(-0.503279\pi\)
−0.0103018 + 0.999947i \(0.503279\pi\)
\(110\) −2.88520 −0.275093
\(111\) −19.8599 −1.88502
\(112\) −1.51784 −0.143422
\(113\) −11.0171 −1.03640 −0.518201 0.855259i \(-0.673398\pi\)
−0.518201 + 0.855259i \(0.673398\pi\)
\(114\) −15.6955 −1.47002
\(115\) 1.81249 0.169016
\(116\) −1.34372 −0.124761
\(117\) −7.59221 −0.701900
\(118\) −0.167892 −0.0154557
\(119\) 2.66633 0.244422
\(120\) −3.25457 −0.297100
\(121\) −2.67561 −0.243237
\(122\) 12.5603 1.13716
\(123\) −28.0455 −2.52877
\(124\) 1.00000 0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 11.5238 1.02662
\(127\) −1.84777 −0.163963 −0.0819814 0.996634i \(-0.526125\pi\)
−0.0819814 + 0.996634i \(0.526125\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 28.1075 2.47472
\(130\) −1.00000 −0.0877058
\(131\) 7.86380 0.687063 0.343532 0.939141i \(-0.388377\pi\)
0.343532 + 0.939141i \(0.388377\pi\)
\(132\) 9.39008 0.817302
\(133\) 7.31995 0.634720
\(134\) 9.70815 0.838656
\(135\) 14.9456 1.28632
\(136\) 1.75666 0.150632
\(137\) 13.3808 1.14320 0.571600 0.820533i \(-0.306323\pi\)
0.571600 + 0.820533i \(0.306323\pi\)
\(138\) −5.89888 −0.502146
\(139\) 4.40179 0.373355 0.186678 0.982421i \(-0.440228\pi\)
0.186678 + 0.982421i \(0.440228\pi\)
\(140\) 1.51784 0.128281
\(141\) 28.7163 2.41835
\(142\) 3.09566 0.259782
\(143\) 2.88520 0.241273
\(144\) 7.59221 0.632684
\(145\) 1.34372 0.111590
\(146\) 14.9215 1.23491
\(147\) 15.2840 1.26060
\(148\) 6.10218 0.501596
\(149\) 0.984443 0.0806487 0.0403244 0.999187i \(-0.487161\pi\)
0.0403244 + 0.999187i \(0.487161\pi\)
\(150\) 3.25457 0.265734
\(151\) 3.33309 0.271243 0.135621 0.990761i \(-0.456697\pi\)
0.135621 + 0.990761i \(0.456697\pi\)
\(152\) 4.82261 0.391165
\(153\) −13.3369 −1.07823
\(154\) −4.37927 −0.352892
\(155\) −1.00000 −0.0803219
\(156\) 3.25457 0.260574
\(157\) −14.2429 −1.13671 −0.568355 0.822783i \(-0.692420\pi\)
−0.568355 + 0.822783i \(0.692420\pi\)
\(158\) −7.76579 −0.617813
\(159\) −16.4224 −1.30238
\(160\) 1.00000 0.0790569
\(161\) 2.75108 0.216815
\(162\) −25.8650 −2.03215
\(163\) 4.45499 0.348941 0.174471 0.984662i \(-0.444179\pi\)
0.174471 + 0.984662i \(0.444179\pi\)
\(164\) 8.61726 0.672895
\(165\) −9.39008 −0.731017
\(166\) 9.01972 0.700066
\(167\) 23.1918 1.79464 0.897318 0.441384i \(-0.145512\pi\)
0.897318 + 0.441384i \(0.145512\pi\)
\(168\) −4.93991 −0.381122
\(169\) 1.00000 0.0769231
\(170\) −1.75666 −0.134730
\(171\) −36.6143 −2.79996
\(172\) −8.63631 −0.658513
\(173\) −19.9078 −1.51356 −0.756780 0.653670i \(-0.773229\pi\)
−0.756780 + 0.653670i \(0.773229\pi\)
\(174\) −4.37321 −0.331532
\(175\) −1.51784 −0.114738
\(176\) −2.88520 −0.217480
\(177\) −0.546416 −0.0410712
\(178\) −10.3332 −0.774509
\(179\) −1.80189 −0.134680 −0.0673399 0.997730i \(-0.521451\pi\)
−0.0673399 + 0.997730i \(0.521451\pi\)
\(180\) −7.59221 −0.565890
\(181\) 15.5429 1.15529 0.577647 0.816287i \(-0.303971\pi\)
0.577647 + 0.816287i \(0.303971\pi\)
\(182\) −1.51784 −0.112510
\(183\) 40.8784 3.02182
\(184\) 1.81249 0.133619
\(185\) −6.10218 −0.448641
\(186\) 3.25457 0.238636
\(187\) 5.06832 0.370632
\(188\) −8.82339 −0.643512
\(189\) 22.6851 1.65010
\(190\) −4.82261 −0.349869
\(191\) −16.3449 −1.18268 −0.591338 0.806424i \(-0.701400\pi\)
−0.591338 + 0.806424i \(0.701400\pi\)
\(192\) −3.25457 −0.234878
\(193\) −8.01480 −0.576918 −0.288459 0.957492i \(-0.593143\pi\)
−0.288459 + 0.957492i \(0.593143\pi\)
\(194\) 18.0929 1.29900
\(195\) −3.25457 −0.233064
\(196\) −4.69616 −0.335440
\(197\) −2.33801 −0.166576 −0.0832881 0.996526i \(-0.526542\pi\)
−0.0832881 + 0.996526i \(0.526542\pi\)
\(198\) 21.9051 1.55672
\(199\) −20.5746 −1.45849 −0.729246 0.684252i \(-0.760129\pi\)
−0.729246 + 0.684252i \(0.760129\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 31.5958 2.22860
\(202\) 19.0906 1.34321
\(203\) 2.03955 0.143148
\(204\) 5.71717 0.400282
\(205\) −8.61726 −0.601856
\(206\) 10.2723 0.715706
\(207\) −13.7608 −0.956444
\(208\) −1.00000 −0.0693375
\(209\) 13.9142 0.962466
\(210\) 4.93991 0.340886
\(211\) −20.5706 −1.41614 −0.708068 0.706144i \(-0.750433\pi\)
−0.708068 + 0.706144i \(0.750433\pi\)
\(212\) 5.04595 0.346557
\(213\) 10.0750 0.690330
\(214\) −0.752780 −0.0514590
\(215\) 8.63631 0.588992
\(216\) 14.9456 1.01692
\(217\) −1.51784 −0.103038
\(218\) 0.215107 0.0145689
\(219\) 48.5631 3.28159
\(220\) 2.88520 0.194520
\(221\) 1.75666 0.118166
\(222\) 19.8599 1.33291
\(223\) 23.6432 1.58326 0.791632 0.610998i \(-0.209232\pi\)
0.791632 + 0.610998i \(0.209232\pi\)
\(224\) 1.51784 0.101415
\(225\) 7.59221 0.506147
\(226\) 11.0171 0.732847
\(227\) 2.65873 0.176466 0.0882331 0.996100i \(-0.471878\pi\)
0.0882331 + 0.996100i \(0.471878\pi\)
\(228\) 15.6955 1.03946
\(229\) 15.0027 0.991407 0.495704 0.868492i \(-0.334910\pi\)
0.495704 + 0.868492i \(0.334910\pi\)
\(230\) −1.81249 −0.119512
\(231\) −14.2526 −0.937755
\(232\) 1.34372 0.0882193
\(233\) 19.9130 1.30454 0.652271 0.757986i \(-0.273816\pi\)
0.652271 + 0.757986i \(0.273816\pi\)
\(234\) 7.59221 0.496318
\(235\) 8.82339 0.575575
\(236\) 0.167892 0.0109288
\(237\) −25.2743 −1.64174
\(238\) −2.66633 −0.172832
\(239\) 15.5385 1.00510 0.502551 0.864547i \(-0.332395\pi\)
0.502551 + 0.864547i \(0.332395\pi\)
\(240\) 3.25457 0.210081
\(241\) 0.0119521 0.000769900 0 0.000384950 1.00000i \(-0.499877\pi\)
0.000384950 1.00000i \(0.499877\pi\)
\(242\) 2.67561 0.171995
\(243\) −39.3424 −2.52382
\(244\) −12.5603 −0.804091
\(245\) 4.69616 0.300027
\(246\) 28.0455 1.78811
\(247\) 4.82261 0.306855
\(248\) −1.00000 −0.0635001
\(249\) 29.3553 1.86032
\(250\) 1.00000 0.0632456
\(251\) −25.9516 −1.63805 −0.819027 0.573755i \(-0.805486\pi\)
−0.819027 + 0.573755i \(0.805486\pi\)
\(252\) −11.5238 −0.725928
\(253\) 5.22941 0.328770
\(254\) 1.84777 0.115939
\(255\) −5.71717 −0.358023
\(256\) 1.00000 0.0625000
\(257\) 21.6042 1.34763 0.673817 0.738898i \(-0.264653\pi\)
0.673817 + 0.738898i \(0.264653\pi\)
\(258\) −28.1075 −1.74989
\(259\) −9.26212 −0.575520
\(260\) 1.00000 0.0620174
\(261\) −10.2018 −0.631474
\(262\) −7.86380 −0.485827
\(263\) 11.1658 0.688512 0.344256 0.938876i \(-0.388131\pi\)
0.344256 + 0.938876i \(0.388131\pi\)
\(264\) −9.39008 −0.577920
\(265\) −5.04595 −0.309970
\(266\) −7.31995 −0.448815
\(267\) −33.6302 −2.05814
\(268\) −9.70815 −0.593020
\(269\) −22.6580 −1.38148 −0.690740 0.723103i \(-0.742715\pi\)
−0.690740 + 0.723103i \(0.742715\pi\)
\(270\) −14.9456 −0.909563
\(271\) 12.6515 0.768524 0.384262 0.923224i \(-0.374456\pi\)
0.384262 + 0.923224i \(0.374456\pi\)
\(272\) −1.75666 −0.106513
\(273\) −4.93991 −0.298977
\(274\) −13.3808 −0.808364
\(275\) −2.88520 −0.173984
\(276\) 5.89888 0.355071
\(277\) 25.7039 1.54440 0.772199 0.635381i \(-0.219157\pi\)
0.772199 + 0.635381i \(0.219157\pi\)
\(278\) −4.40179 −0.264002
\(279\) 7.59221 0.454534
\(280\) −1.51784 −0.0907083
\(281\) −1.64618 −0.0982031 −0.0491016 0.998794i \(-0.515636\pi\)
−0.0491016 + 0.998794i \(0.515636\pi\)
\(282\) −28.7163 −1.71003
\(283\) 16.3504 0.971933 0.485967 0.873977i \(-0.338468\pi\)
0.485967 + 0.873977i \(0.338468\pi\)
\(284\) −3.09566 −0.183694
\(285\) −15.6955 −0.929722
\(286\) −2.88520 −0.170606
\(287\) −13.0796 −0.772065
\(288\) −7.59221 −0.447375
\(289\) −13.9141 −0.818479
\(290\) −1.34372 −0.0789057
\(291\) 58.8846 3.45188
\(292\) −14.9215 −0.873217
\(293\) 13.1221 0.766602 0.383301 0.923624i \(-0.374787\pi\)
0.383301 + 0.923624i \(0.374787\pi\)
\(294\) −15.2840 −0.891380
\(295\) −0.167892 −0.00977506
\(296\) −6.10218 −0.354682
\(297\) 43.1212 2.50215
\(298\) −0.984443 −0.0570273
\(299\) 1.81249 0.104819
\(300\) −3.25457 −0.187903
\(301\) 13.1085 0.755564
\(302\) −3.33309 −0.191798
\(303\) 62.1317 3.56937
\(304\) −4.82261 −0.276596
\(305\) 12.5603 0.719201
\(306\) 13.3369 0.762422
\(307\) 9.05852 0.516997 0.258498 0.966012i \(-0.416772\pi\)
0.258498 + 0.966012i \(0.416772\pi\)
\(308\) 4.37927 0.249532
\(309\) 33.4319 1.90188
\(310\) 1.00000 0.0567962
\(311\) 5.03422 0.285464 0.142732 0.989761i \(-0.454411\pi\)
0.142732 + 0.989761i \(0.454411\pi\)
\(312\) −3.25457 −0.184254
\(313\) 6.84388 0.386839 0.193419 0.981116i \(-0.438042\pi\)
0.193419 + 0.981116i \(0.438042\pi\)
\(314\) 14.2429 0.803775
\(315\) 11.5238 0.649290
\(316\) 7.76579 0.436860
\(317\) −20.3489 −1.14291 −0.571455 0.820634i \(-0.693621\pi\)
−0.571455 + 0.820634i \(0.693621\pi\)
\(318\) 16.4224 0.920921
\(319\) 3.87689 0.217064
\(320\) −1.00000 −0.0559017
\(321\) −2.44997 −0.136744
\(322\) −2.75108 −0.153311
\(323\) 8.47169 0.471377
\(324\) 25.8650 1.43694
\(325\) −1.00000 −0.0554700
\(326\) −4.45499 −0.246739
\(327\) 0.700081 0.0387146
\(328\) −8.61726 −0.475809
\(329\) 13.3925 0.738352
\(330\) 9.39008 0.516907
\(331\) −11.7549 −0.646108 −0.323054 0.946381i \(-0.604710\pi\)
−0.323054 + 0.946381i \(0.604710\pi\)
\(332\) −9.01972 −0.495021
\(333\) 46.3290 2.53881
\(334\) −23.1918 −1.26900
\(335\) 9.70815 0.530413
\(336\) 4.93991 0.269494
\(337\) −21.0243 −1.14527 −0.572634 0.819811i \(-0.694078\pi\)
−0.572634 + 0.819811i \(0.694078\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 35.8559 1.94742
\(340\) 1.75666 0.0952683
\(341\) −2.88520 −0.156242
\(342\) 36.6143 1.97987
\(343\) 17.7529 0.958566
\(344\) 8.63631 0.465639
\(345\) −5.89888 −0.317585
\(346\) 19.9078 1.07025
\(347\) −13.6654 −0.733599 −0.366800 0.930300i \(-0.619547\pi\)
−0.366800 + 0.930300i \(0.619547\pi\)
\(348\) 4.37321 0.234429
\(349\) −28.5515 −1.52833 −0.764163 0.645023i \(-0.776848\pi\)
−0.764163 + 0.645023i \(0.776848\pi\)
\(350\) 1.51784 0.0811319
\(351\) 14.9456 0.797740
\(352\) 2.88520 0.153782
\(353\) 5.00976 0.266643 0.133321 0.991073i \(-0.457436\pi\)
0.133321 + 0.991073i \(0.457436\pi\)
\(354\) 0.546416 0.0290417
\(355\) 3.09566 0.164300
\(356\) 10.3332 0.547661
\(357\) −8.67775 −0.459275
\(358\) 1.80189 0.0952330
\(359\) 27.6641 1.46006 0.730029 0.683417i \(-0.239507\pi\)
0.730029 + 0.683417i \(0.239507\pi\)
\(360\) 7.59221 0.400144
\(361\) 4.25757 0.224082
\(362\) −15.5429 −0.816916
\(363\) 8.70795 0.457049
\(364\) 1.51784 0.0795564
\(365\) 14.9215 0.781029
\(366\) −40.8784 −2.13675
\(367\) −9.00019 −0.469806 −0.234903 0.972019i \(-0.575477\pi\)
−0.234903 + 0.972019i \(0.575477\pi\)
\(368\) −1.81249 −0.0944828
\(369\) 65.4240 3.40584
\(370\) 6.10218 0.317237
\(371\) −7.65894 −0.397632
\(372\) −3.25457 −0.168741
\(373\) 35.3130 1.82844 0.914218 0.405222i \(-0.132806\pi\)
0.914218 + 0.405222i \(0.132806\pi\)
\(374\) −5.06832 −0.262077
\(375\) 3.25457 0.168065
\(376\) 8.82339 0.455032
\(377\) 1.34372 0.0692049
\(378\) −22.6851 −1.16680
\(379\) −15.8928 −0.816357 −0.408179 0.912902i \(-0.633836\pi\)
−0.408179 + 0.912902i \(0.633836\pi\)
\(380\) 4.82261 0.247395
\(381\) 6.01368 0.308090
\(382\) 16.3449 0.836278
\(383\) −8.98324 −0.459022 −0.229511 0.973306i \(-0.573713\pi\)
−0.229511 + 0.973306i \(0.573713\pi\)
\(384\) 3.25457 0.166084
\(385\) −4.37927 −0.223188
\(386\) 8.01480 0.407942
\(387\) −65.5687 −3.33304
\(388\) −18.0929 −0.918529
\(389\) 34.7293 1.76085 0.880423 0.474188i \(-0.157258\pi\)
0.880423 + 0.474188i \(0.157258\pi\)
\(390\) 3.25457 0.164801
\(391\) 3.18394 0.161019
\(392\) 4.69616 0.237192
\(393\) −25.5933 −1.29101
\(394\) 2.33801 0.117787
\(395\) −7.76579 −0.390739
\(396\) −21.9051 −1.10077
\(397\) 2.31500 0.116187 0.0580933 0.998311i \(-0.481498\pi\)
0.0580933 + 0.998311i \(0.481498\pi\)
\(398\) 20.5746 1.03131
\(399\) −23.8233 −1.19265
\(400\) 1.00000 0.0500000
\(401\) 29.9444 1.49535 0.747677 0.664063i \(-0.231169\pi\)
0.747677 + 0.664063i \(0.231169\pi\)
\(402\) −31.5958 −1.57586
\(403\) −1.00000 −0.0498135
\(404\) −19.0906 −0.949794
\(405\) −25.8650 −1.28524
\(406\) −2.03955 −0.101221
\(407\) −17.6060 −0.872698
\(408\) −5.71717 −0.283042
\(409\) 24.5992 1.21635 0.608176 0.793802i \(-0.291902\pi\)
0.608176 + 0.793802i \(0.291902\pi\)
\(410\) 8.61726 0.425576
\(411\) −43.5487 −2.14810
\(412\) −10.2723 −0.506080
\(413\) −0.254833 −0.0125395
\(414\) 13.7608 0.676308
\(415\) 9.01972 0.442761
\(416\) 1.00000 0.0490290
\(417\) −14.3259 −0.701544
\(418\) −13.9142 −0.680566
\(419\) −14.7647 −0.721304 −0.360652 0.932701i \(-0.617446\pi\)
−0.360652 + 0.932701i \(0.617446\pi\)
\(420\) −4.93991 −0.241043
\(421\) 13.1348 0.640152 0.320076 0.947392i \(-0.396292\pi\)
0.320076 + 0.947392i \(0.396292\pi\)
\(422\) 20.5706 1.00136
\(423\) −66.9890 −3.25712
\(424\) −5.04595 −0.245053
\(425\) −1.75666 −0.0852106
\(426\) −10.0750 −0.488137
\(427\) 19.0645 0.922597
\(428\) 0.752780 0.0363870
\(429\) −9.39008 −0.453358
\(430\) −8.63631 −0.416480
\(431\) 10.4912 0.505345 0.252672 0.967552i \(-0.418691\pi\)
0.252672 + 0.967552i \(0.418691\pi\)
\(432\) −14.9456 −0.719073
\(433\) 28.8205 1.38503 0.692514 0.721405i \(-0.256503\pi\)
0.692514 + 0.721405i \(0.256503\pi\)
\(434\) 1.51784 0.0728586
\(435\) −4.37321 −0.209680
\(436\) −0.215107 −0.0103018
\(437\) 8.74095 0.418136
\(438\) −48.5631 −2.32044
\(439\) −14.3101 −0.682982 −0.341491 0.939885i \(-0.610932\pi\)
−0.341491 + 0.939885i \(0.610932\pi\)
\(440\) −2.88520 −0.137547
\(441\) −35.6542 −1.69782
\(442\) −1.75666 −0.0835558
\(443\) 32.7324 1.55516 0.777581 0.628782i \(-0.216446\pi\)
0.777581 + 0.628782i \(0.216446\pi\)
\(444\) −19.8599 −0.942511
\(445\) −10.3332 −0.489843
\(446\) −23.6432 −1.11954
\(447\) −3.20394 −0.151541
\(448\) −1.51784 −0.0717112
\(449\) −4.57873 −0.216084 −0.108042 0.994146i \(-0.534458\pi\)
−0.108042 + 0.994146i \(0.534458\pi\)
\(450\) −7.59221 −0.357900
\(451\) −24.8625 −1.17073
\(452\) −11.0171 −0.518201
\(453\) −10.8478 −0.509672
\(454\) −2.65873 −0.124780
\(455\) −1.51784 −0.0711574
\(456\) −15.6955 −0.735009
\(457\) −29.7216 −1.39032 −0.695158 0.718857i \(-0.744666\pi\)
−0.695158 + 0.718857i \(0.744666\pi\)
\(458\) −15.0027 −0.701031
\(459\) 26.2544 1.22545
\(460\) 1.81249 0.0845080
\(461\) −15.4727 −0.720634 −0.360317 0.932830i \(-0.617332\pi\)
−0.360317 + 0.932830i \(0.617332\pi\)
\(462\) 14.2526 0.663093
\(463\) −11.8854 −0.552360 −0.276180 0.961106i \(-0.589069\pi\)
−0.276180 + 0.961106i \(0.589069\pi\)
\(464\) −1.34372 −0.0623804
\(465\) 3.25457 0.150927
\(466\) −19.9130 −0.922450
\(467\) 13.1182 0.607036 0.303518 0.952826i \(-0.401839\pi\)
0.303518 + 0.952826i \(0.401839\pi\)
\(468\) −7.59221 −0.350950
\(469\) 14.7354 0.680418
\(470\) −8.82339 −0.406993
\(471\) 46.3546 2.13591
\(472\) −0.167892 −0.00772786
\(473\) 24.9175 1.14571
\(474\) 25.2743 1.16089
\(475\) −4.82261 −0.221277
\(476\) 2.66633 0.122211
\(477\) 38.3099 1.75409
\(478\) −15.5385 −0.710715
\(479\) −10.9823 −0.501792 −0.250896 0.968014i \(-0.580725\pi\)
−0.250896 + 0.968014i \(0.580725\pi\)
\(480\) −3.25457 −0.148550
\(481\) −6.10218 −0.278235
\(482\) −0.0119521 −0.000544402 0
\(483\) −8.95356 −0.407401
\(484\) −2.67561 −0.121619
\(485\) 18.0929 0.821557
\(486\) 39.3424 1.78461
\(487\) 13.1634 0.596490 0.298245 0.954489i \(-0.403599\pi\)
0.298245 + 0.954489i \(0.403599\pi\)
\(488\) 12.5603 0.568578
\(489\) −14.4991 −0.655670
\(490\) −4.69616 −0.212151
\(491\) −13.7510 −0.620576 −0.310288 0.950643i \(-0.600425\pi\)
−0.310288 + 0.950643i \(0.600425\pi\)
\(492\) −28.0455 −1.26439
\(493\) 2.36045 0.106309
\(494\) −4.82261 −0.216979
\(495\) 21.9051 0.984559
\(496\) 1.00000 0.0449013
\(497\) 4.69871 0.210766
\(498\) −29.3553 −1.31544
\(499\) 5.02756 0.225064 0.112532 0.993648i \(-0.464104\pi\)
0.112532 + 0.993648i \(0.464104\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −75.4793 −3.37217
\(502\) 25.9516 1.15828
\(503\) 30.3408 1.35283 0.676415 0.736521i \(-0.263533\pi\)
0.676415 + 0.736521i \(0.263533\pi\)
\(504\) 11.5238 0.513309
\(505\) 19.0906 0.849521
\(506\) −5.22941 −0.232476
\(507\) −3.25457 −0.144540
\(508\) −1.84777 −0.0819814
\(509\) 22.0452 0.977136 0.488568 0.872526i \(-0.337520\pi\)
0.488568 + 0.872526i \(0.337520\pi\)
\(510\) 5.71717 0.253161
\(511\) 22.6485 1.00191
\(512\) −1.00000 −0.0441942
\(513\) 72.0770 3.18228
\(514\) −21.6042 −0.952922
\(515\) 10.2723 0.452652
\(516\) 28.1075 1.23736
\(517\) 25.4573 1.11961
\(518\) 9.26212 0.406954
\(519\) 64.7912 2.84402
\(520\) −1.00000 −0.0438529
\(521\) −14.1827 −0.621354 −0.310677 0.950516i \(-0.600556\pi\)
−0.310677 + 0.950516i \(0.600556\pi\)
\(522\) 10.2018 0.446519
\(523\) 11.9919 0.524371 0.262186 0.965017i \(-0.415557\pi\)
0.262186 + 0.965017i \(0.415557\pi\)
\(524\) 7.86380 0.343532
\(525\) 4.93991 0.215595
\(526\) −11.1658 −0.486852
\(527\) −1.75666 −0.0765213
\(528\) 9.39008 0.408651
\(529\) −19.7149 −0.857168
\(530\) 5.04595 0.219182
\(531\) 1.27467 0.0553161
\(532\) 7.31995 0.317360
\(533\) −8.61726 −0.373255
\(534\) 33.6302 1.45532
\(535\) −0.752780 −0.0325455
\(536\) 9.70815 0.419328
\(537\) 5.86438 0.253067
\(538\) 22.6580 0.976854
\(539\) 13.5494 0.583613
\(540\) 14.9456 0.643158
\(541\) −17.1915 −0.739122 −0.369561 0.929207i \(-0.620492\pi\)
−0.369561 + 0.929207i \(0.620492\pi\)
\(542\) −12.6515 −0.543429
\(543\) −50.5854 −2.17083
\(544\) 1.75666 0.0753162
\(545\) 0.215107 0.00921419
\(546\) 4.93991 0.211409
\(547\) −32.8323 −1.40381 −0.701905 0.712271i \(-0.747667\pi\)
−0.701905 + 0.712271i \(0.747667\pi\)
\(548\) 13.3808 0.571600
\(549\) −95.3605 −4.06988
\(550\) 2.88520 0.123025
\(551\) 6.48022 0.276067
\(552\) −5.89888 −0.251073
\(553\) −11.7872 −0.501244
\(554\) −25.7039 −1.09205
\(555\) 19.8599 0.843008
\(556\) 4.40179 0.186678
\(557\) 38.9688 1.65116 0.825580 0.564285i \(-0.190848\pi\)
0.825580 + 0.564285i \(0.190848\pi\)
\(558\) −7.59221 −0.321404
\(559\) 8.63631 0.365277
\(560\) 1.51784 0.0641404
\(561\) −16.4952 −0.696428
\(562\) 1.64618 0.0694401
\(563\) −26.7868 −1.12893 −0.564464 0.825458i \(-0.690917\pi\)
−0.564464 + 0.825458i \(0.690917\pi\)
\(564\) 28.7163 1.20918
\(565\) 11.0171 0.463493
\(566\) −16.3504 −0.687261
\(567\) −39.2589 −1.64872
\(568\) 3.09566 0.129891
\(569\) 25.0563 1.05041 0.525207 0.850974i \(-0.323988\pi\)
0.525207 + 0.850974i \(0.323988\pi\)
\(570\) 15.6955 0.657412
\(571\) −42.3338 −1.77161 −0.885807 0.464053i \(-0.846395\pi\)
−0.885807 + 0.464053i \(0.846395\pi\)
\(572\) 2.88520 0.120636
\(573\) 53.1956 2.22228
\(574\) 13.0796 0.545933
\(575\) −1.81249 −0.0755862
\(576\) 7.59221 0.316342
\(577\) 27.1552 1.13048 0.565242 0.824925i \(-0.308783\pi\)
0.565242 + 0.824925i \(0.308783\pi\)
\(578\) 13.9141 0.578752
\(579\) 26.0847 1.08404
\(580\) 1.34372 0.0557948
\(581\) 13.6905 0.567977
\(582\) −58.8846 −2.44085
\(583\) −14.5586 −0.602955
\(584\) 14.9215 0.617457
\(585\) 7.59221 0.313899
\(586\) −13.1221 −0.542069
\(587\) 14.9418 0.616714 0.308357 0.951271i \(-0.400221\pi\)
0.308357 + 0.951271i \(0.400221\pi\)
\(588\) 15.2840 0.630301
\(589\) −4.82261 −0.198712
\(590\) 0.167892 0.00691201
\(591\) 7.60921 0.313001
\(592\) 6.10218 0.250798
\(593\) 24.7525 1.01646 0.508232 0.861220i \(-0.330299\pi\)
0.508232 + 0.861220i \(0.330299\pi\)
\(594\) −43.1212 −1.76928
\(595\) −2.66633 −0.109309
\(596\) 0.984443 0.0403244
\(597\) 66.9613 2.74054
\(598\) −1.81249 −0.0741184
\(599\) −11.0424 −0.451181 −0.225591 0.974222i \(-0.572431\pi\)
−0.225591 + 0.974222i \(0.572431\pi\)
\(600\) 3.25457 0.132867
\(601\) 1.46009 0.0595581 0.0297791 0.999557i \(-0.490520\pi\)
0.0297791 + 0.999557i \(0.490520\pi\)
\(602\) −13.1085 −0.534264
\(603\) −73.7063 −3.00155
\(604\) 3.33309 0.135621
\(605\) 2.67561 0.108779
\(606\) −62.1317 −2.52393
\(607\) 9.83217 0.399075 0.199538 0.979890i \(-0.436056\pi\)
0.199538 + 0.979890i \(0.436056\pi\)
\(608\) 4.82261 0.195583
\(609\) −6.63784 −0.268979
\(610\) −12.5603 −0.508552
\(611\) 8.82339 0.356956
\(612\) −13.3369 −0.539114
\(613\) −12.0717 −0.487570 −0.243785 0.969829i \(-0.578389\pi\)
−0.243785 + 0.969829i \(0.578389\pi\)
\(614\) −9.05852 −0.365572
\(615\) 28.0455 1.13090
\(616\) −4.37927 −0.176446
\(617\) −35.5836 −1.43254 −0.716271 0.697822i \(-0.754152\pi\)
−0.716271 + 0.697822i \(0.754152\pi\)
\(618\) −33.4319 −1.34483
\(619\) −16.7610 −0.673683 −0.336842 0.941561i \(-0.609359\pi\)
−0.336842 + 0.941561i \(0.609359\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 27.0889 1.08704
\(622\) −5.03422 −0.201854
\(623\) −15.6842 −0.628374
\(624\) 3.25457 0.130287
\(625\) 1.00000 0.0400000
\(626\) −6.84388 −0.273536
\(627\) −45.2847 −1.80850
\(628\) −14.2429 −0.568355
\(629\) −10.7195 −0.427413
\(630\) −11.5238 −0.459117
\(631\) −39.0561 −1.55480 −0.777400 0.629006i \(-0.783462\pi\)
−0.777400 + 0.629006i \(0.783462\pi\)
\(632\) −7.76579 −0.308907
\(633\) 66.9483 2.66096
\(634\) 20.3489 0.808159
\(635\) 1.84777 0.0733264
\(636\) −16.4224 −0.651189
\(637\) 4.69616 0.186069
\(638\) −3.87689 −0.153488
\(639\) −23.5029 −0.929760
\(640\) 1.00000 0.0395285
\(641\) 1.93474 0.0764176 0.0382088 0.999270i \(-0.487835\pi\)
0.0382088 + 0.999270i \(0.487835\pi\)
\(642\) 2.44997 0.0966928
\(643\) −13.9835 −0.551457 −0.275728 0.961236i \(-0.588919\pi\)
−0.275728 + 0.961236i \(0.588919\pi\)
\(644\) 2.75108 0.108408
\(645\) −28.1075 −1.10673
\(646\) −8.47169 −0.333314
\(647\) −7.20166 −0.283126 −0.141563 0.989929i \(-0.545213\pi\)
−0.141563 + 0.989929i \(0.545213\pi\)
\(648\) −25.8650 −1.01607
\(649\) −0.484403 −0.0190145
\(650\) 1.00000 0.0392232
\(651\) 4.93991 0.193610
\(652\) 4.45499 0.174471
\(653\) 17.4178 0.681612 0.340806 0.940134i \(-0.389300\pi\)
0.340806 + 0.940134i \(0.389300\pi\)
\(654\) −0.700081 −0.0273754
\(655\) −7.86380 −0.307264
\(656\) 8.61726 0.336447
\(657\) −113.287 −4.41976
\(658\) −13.3925 −0.522094
\(659\) 33.3649 1.29971 0.649857 0.760057i \(-0.274829\pi\)
0.649857 + 0.760057i \(0.274829\pi\)
\(660\) −9.39008 −0.365509
\(661\) 1.59456 0.0620212 0.0310106 0.999519i \(-0.490127\pi\)
0.0310106 + 0.999519i \(0.490127\pi\)
\(662\) 11.7549 0.456867
\(663\) −5.71717 −0.222037
\(664\) 9.01972 0.350033
\(665\) −7.31995 −0.283855
\(666\) −46.3290 −1.79521
\(667\) 2.43548 0.0943020
\(668\) 23.1918 0.897318
\(669\) −76.9483 −2.97499
\(670\) −9.70815 −0.375059
\(671\) 36.2390 1.39899
\(672\) −4.93991 −0.190561
\(673\) 11.9636 0.461164 0.230582 0.973053i \(-0.425937\pi\)
0.230582 + 0.973053i \(0.425937\pi\)
\(674\) 21.0243 0.809826
\(675\) −14.9456 −0.575258
\(676\) 1.00000 0.0384615
\(677\) −43.8752 −1.68626 −0.843131 0.537708i \(-0.819290\pi\)
−0.843131 + 0.537708i \(0.819290\pi\)
\(678\) −35.8559 −1.37704
\(679\) 27.4621 1.05390
\(680\) −1.75666 −0.0673649
\(681\) −8.65302 −0.331584
\(682\) 2.88520 0.110480
\(683\) 32.5152 1.24416 0.622079 0.782954i \(-0.286288\pi\)
0.622079 + 0.782954i \(0.286288\pi\)
\(684\) −36.6143 −1.39998
\(685\) −13.3808 −0.511254
\(686\) −17.7529 −0.677809
\(687\) −48.8273 −1.86288
\(688\) −8.63631 −0.329256
\(689\) −5.04595 −0.192235
\(690\) 5.89888 0.224567
\(691\) −12.2965 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(692\) −19.9078 −0.756780
\(693\) 33.2484 1.26300
\(694\) 13.6654 0.518733
\(695\) −4.40179 −0.166970
\(696\) −4.37321 −0.165766
\(697\) −15.1376 −0.573378
\(698\) 28.5515 1.08069
\(699\) −64.8081 −2.45127
\(700\) −1.51784 −0.0573689
\(701\) 5.93240 0.224064 0.112032 0.993705i \(-0.464264\pi\)
0.112032 + 0.993705i \(0.464264\pi\)
\(702\) −14.9456 −0.564087
\(703\) −29.4284 −1.10991
\(704\) −2.88520 −0.108740
\(705\) −28.7163 −1.08152
\(706\) −5.00976 −0.188545
\(707\) 28.9765 1.08977
\(708\) −0.546416 −0.0205356
\(709\) 43.2227 1.62326 0.811632 0.584169i \(-0.198580\pi\)
0.811632 + 0.584169i \(0.198580\pi\)
\(710\) −3.09566 −0.116178
\(711\) 58.9595 2.21115
\(712\) −10.3332 −0.387255
\(713\) −1.81249 −0.0678784
\(714\) 8.67775 0.324757
\(715\) −2.88520 −0.107900
\(716\) −1.80189 −0.0673399
\(717\) −50.5711 −1.88861
\(718\) −27.6641 −1.03242
\(719\) 8.93000 0.333033 0.166516 0.986039i \(-0.446748\pi\)
0.166516 + 0.986039i \(0.446748\pi\)
\(720\) −7.59221 −0.282945
\(721\) 15.5917 0.580666
\(722\) −4.25757 −0.158450
\(723\) −0.0388988 −0.00144666
\(724\) 15.5429 0.577647
\(725\) −1.34372 −0.0499044
\(726\) −8.70795 −0.323182
\(727\) 2.37993 0.0882668 0.0441334 0.999026i \(-0.485947\pi\)
0.0441334 + 0.999026i \(0.485947\pi\)
\(728\) −1.51784 −0.0562549
\(729\) 50.4475 1.86843
\(730\) −14.9215 −0.552271
\(731\) 15.1711 0.561122
\(732\) 40.8784 1.51091
\(733\) 30.2668 1.11793 0.558964 0.829192i \(-0.311199\pi\)
0.558964 + 0.829192i \(0.311199\pi\)
\(734\) 9.00019 0.332203
\(735\) −15.2840 −0.563758
\(736\) 1.81249 0.0668094
\(737\) 28.0100 1.03176
\(738\) −65.4240 −2.40829
\(739\) 26.4573 0.973249 0.486624 0.873611i \(-0.338228\pi\)
0.486624 + 0.873611i \(0.338228\pi\)
\(740\) −6.10218 −0.224320
\(741\) −15.6955 −0.576589
\(742\) 7.65894 0.281168
\(743\) −25.1284 −0.921873 −0.460936 0.887433i \(-0.652486\pi\)
−0.460936 + 0.887433i \(0.652486\pi\)
\(744\) 3.25457 0.119318
\(745\) −0.984443 −0.0360672
\(746\) −35.3130 −1.29290
\(747\) −68.4796 −2.50554
\(748\) 5.06832 0.185316
\(749\) −1.14260 −0.0417497
\(750\) −3.25457 −0.118840
\(751\) −16.2502 −0.592977 −0.296489 0.955036i \(-0.595816\pi\)
−0.296489 + 0.955036i \(0.595816\pi\)
\(752\) −8.82339 −0.321756
\(753\) 84.4614 3.07794
\(754\) −1.34372 −0.0489352
\(755\) −3.33309 −0.121303
\(756\) 22.6851 0.825049
\(757\) −7.05496 −0.256417 −0.128209 0.991747i \(-0.540923\pi\)
−0.128209 + 0.991747i \(0.540923\pi\)
\(758\) 15.8928 0.577252
\(759\) −17.0195 −0.617768
\(760\) −4.82261 −0.174934
\(761\) −19.2886 −0.699210 −0.349605 0.936897i \(-0.613684\pi\)
−0.349605 + 0.936897i \(0.613684\pi\)
\(762\) −6.01368 −0.217853
\(763\) 0.326499 0.0118200
\(764\) −16.3449 −0.591338
\(765\) 13.3369 0.482198
\(766\) 8.98324 0.324578
\(767\) −0.167892 −0.00606223
\(768\) −3.25457 −0.117439
\(769\) −8.35377 −0.301245 −0.150622 0.988591i \(-0.548128\pi\)
−0.150622 + 0.988591i \(0.548128\pi\)
\(770\) 4.37927 0.157818
\(771\) −70.3124 −2.53224
\(772\) −8.01480 −0.288459
\(773\) −26.6413 −0.958221 −0.479111 0.877755i \(-0.659041\pi\)
−0.479111 + 0.877755i \(0.659041\pi\)
\(774\) 65.5687 2.35682
\(775\) 1.00000 0.0359211
\(776\) 18.0929 0.649498
\(777\) 30.1442 1.08142
\(778\) −34.7293 −1.24511
\(779\) −41.5577 −1.48896
\(780\) −3.25457 −0.116532
\(781\) 8.93160 0.319598
\(782\) −3.18394 −0.113857
\(783\) 20.0827 0.717697
\(784\) −4.69616 −0.167720
\(785\) 14.2429 0.508352
\(786\) 25.5933 0.912881
\(787\) 10.3053 0.367346 0.183673 0.982987i \(-0.441201\pi\)
0.183673 + 0.982987i \(0.441201\pi\)
\(788\) −2.33801 −0.0832881
\(789\) −36.3398 −1.29373
\(790\) 7.76579 0.276294
\(791\) 16.7222 0.594573
\(792\) 21.9051 0.778362
\(793\) 12.5603 0.446030
\(794\) −2.31500 −0.0821563
\(795\) 16.4224 0.582442
\(796\) −20.5746 −0.729246
\(797\) −24.5349 −0.869072 −0.434536 0.900655i \(-0.643088\pi\)
−0.434536 + 0.900655i \(0.643088\pi\)
\(798\) 23.8233 0.843334
\(799\) 15.4997 0.548340
\(800\) −1.00000 −0.0353553
\(801\) 78.4521 2.77197
\(802\) −29.9444 −1.05737
\(803\) 43.0516 1.51926
\(804\) 31.5958 1.11430
\(805\) −2.75108 −0.0969627
\(806\) 1.00000 0.0352235
\(807\) 73.7419 2.59584
\(808\) 19.0906 0.671606
\(809\) 25.6199 0.900748 0.450374 0.892840i \(-0.351291\pi\)
0.450374 + 0.892840i \(0.351291\pi\)
\(810\) 25.8650 0.908803
\(811\) 48.0866 1.68855 0.844274 0.535912i \(-0.180032\pi\)
0.844274 + 0.535912i \(0.180032\pi\)
\(812\) 2.03955 0.0715740
\(813\) −41.1752 −1.44408
\(814\) 17.6060 0.617090
\(815\) −4.45499 −0.156051
\(816\) 5.71717 0.200141
\(817\) 41.6496 1.45713
\(818\) −24.5992 −0.860090
\(819\) 11.5238 0.402673
\(820\) −8.61726 −0.300928
\(821\) −19.2169 −0.670674 −0.335337 0.942098i \(-0.608850\pi\)
−0.335337 + 0.942098i \(0.608850\pi\)
\(822\) 43.5487 1.51894
\(823\) −24.4122 −0.850957 −0.425478 0.904969i \(-0.639894\pi\)
−0.425478 + 0.904969i \(0.639894\pi\)
\(824\) 10.2723 0.357853
\(825\) 9.39008 0.326921
\(826\) 0.254833 0.00886679
\(827\) 14.6459 0.509286 0.254643 0.967035i \(-0.418042\pi\)
0.254643 + 0.967035i \(0.418042\pi\)
\(828\) −13.7608 −0.478222
\(829\) −22.5559 −0.783400 −0.391700 0.920093i \(-0.628113\pi\)
−0.391700 + 0.920093i \(0.628113\pi\)
\(830\) −9.01972 −0.313079
\(831\) −83.6551 −2.90196
\(832\) −1.00000 −0.0346688
\(833\) 8.24957 0.285831
\(834\) 14.3259 0.496066
\(835\) −23.1918 −0.802586
\(836\) 13.9142 0.481233
\(837\) −14.9456 −0.516597
\(838\) 14.7647 0.510039
\(839\) −25.8859 −0.893679 −0.446839 0.894614i \(-0.647450\pi\)
−0.446839 + 0.894614i \(0.647450\pi\)
\(840\) 4.93991 0.170443
\(841\) −27.1944 −0.937739
\(842\) −13.1348 −0.452656
\(843\) 5.35762 0.184526
\(844\) −20.5706 −0.708068
\(845\) −1.00000 −0.0344010
\(846\) 66.9890 2.30313
\(847\) 4.06115 0.139543
\(848\) 5.04595 0.173279
\(849\) −53.2136 −1.82629
\(850\) 1.75666 0.0602530
\(851\) −11.0602 −0.379137
\(852\) 10.0750 0.345165
\(853\) −31.2726 −1.07075 −0.535376 0.844614i \(-0.679830\pi\)
−0.535376 + 0.844614i \(0.679830\pi\)
\(854\) −19.0645 −0.652375
\(855\) 36.6143 1.25218
\(856\) −0.752780 −0.0257295
\(857\) −48.4815 −1.65610 −0.828048 0.560657i \(-0.810549\pi\)
−0.828048 + 0.560657i \(0.810549\pi\)
\(858\) 9.39008 0.320572
\(859\) −51.7545 −1.76584 −0.882920 0.469523i \(-0.844426\pi\)
−0.882920 + 0.469523i \(0.844426\pi\)
\(860\) 8.63631 0.294496
\(861\) 42.5685 1.45073
\(862\) −10.4912 −0.357333
\(863\) 10.5718 0.359868 0.179934 0.983679i \(-0.442412\pi\)
0.179934 + 0.983679i \(0.442412\pi\)
\(864\) 14.9456 0.508461
\(865\) 19.9078 0.676885
\(866\) −28.8205 −0.979362
\(867\) 45.2845 1.53794
\(868\) −1.51784 −0.0515188
\(869\) −22.4059 −0.760067
\(870\) 4.37321 0.148266
\(871\) 9.70815 0.328948
\(872\) 0.215107 0.00728446
\(873\) −137.365 −4.64911
\(874\) −8.74095 −0.295667
\(875\) 1.51784 0.0513123
\(876\) 48.5631 1.64080
\(877\) −18.4891 −0.624332 −0.312166 0.950028i \(-0.601055\pi\)
−0.312166 + 0.950028i \(0.601055\pi\)
\(878\) 14.3101 0.482942
\(879\) −42.7068 −1.44046
\(880\) 2.88520 0.0972601
\(881\) 17.8443 0.601190 0.300595 0.953752i \(-0.402815\pi\)
0.300595 + 0.953752i \(0.402815\pi\)
\(882\) 35.6542 1.20054
\(883\) 6.91818 0.232815 0.116408 0.993202i \(-0.462862\pi\)
0.116408 + 0.993202i \(0.462862\pi\)
\(884\) 1.75666 0.0590829
\(885\) 0.546416 0.0183676
\(886\) −32.7324 −1.09967
\(887\) 38.3628 1.28810 0.644048 0.764985i \(-0.277254\pi\)
0.644048 + 0.764985i \(0.277254\pi\)
\(888\) 19.8599 0.666456
\(889\) 2.80461 0.0940638
\(890\) 10.3332 0.346371
\(891\) −74.6257 −2.50006
\(892\) 23.6432 0.791632
\(893\) 42.5518 1.42394
\(894\) 3.20394 0.107156
\(895\) 1.80189 0.0602306
\(896\) 1.51784 0.0507075
\(897\) −5.89888 −0.196958
\(898\) 4.57873 0.152794
\(899\) −1.34372 −0.0448154
\(900\) 7.59221 0.253074
\(901\) −8.86402 −0.295303
\(902\) 24.8625 0.827832
\(903\) −42.6626 −1.41972
\(904\) 11.0171 0.366423
\(905\) −15.5429 −0.516663
\(906\) 10.8478 0.360392
\(907\) −29.8056 −0.989679 −0.494839 0.868984i \(-0.664773\pi\)
−0.494839 + 0.868984i \(0.664773\pi\)
\(908\) 2.65873 0.0882331
\(909\) −144.940 −4.80735
\(910\) 1.51784 0.0503159
\(911\) −16.4615 −0.545395 −0.272697 0.962100i \(-0.587916\pi\)
−0.272697 + 0.962100i \(0.587916\pi\)
\(912\) 15.6955 0.519730
\(913\) 26.0237 0.861259
\(914\) 29.7216 0.983102
\(915\) −40.8784 −1.35140
\(916\) 15.0027 0.495704
\(917\) −11.9360 −0.394161
\(918\) −26.2544 −0.866525
\(919\) −13.1870 −0.435001 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(920\) −1.81249 −0.0597562
\(921\) −29.4816 −0.971450
\(922\) 15.4727 0.509566
\(923\) 3.09566 0.101895
\(924\) −14.2526 −0.468877
\(925\) 6.10218 0.200638
\(926\) 11.8854 0.390577
\(927\) −77.9895 −2.56151
\(928\) 1.34372 0.0441096
\(929\) −14.6966 −0.482181 −0.241090 0.970503i \(-0.577505\pi\)
−0.241090 + 0.970503i \(0.577505\pi\)
\(930\) −3.25457 −0.106721
\(931\) 22.6478 0.742250
\(932\) 19.9130 0.652271
\(933\) −16.3842 −0.536394
\(934\) −13.1182 −0.429240
\(935\) −5.06832 −0.165752
\(936\) 7.59221 0.248159
\(937\) 41.0941 1.34249 0.671243 0.741237i \(-0.265761\pi\)
0.671243 + 0.741237i \(0.265761\pi\)
\(938\) −14.7354 −0.481128
\(939\) −22.2739 −0.726880
\(940\) 8.82339 0.287787
\(941\) −8.51154 −0.277468 −0.138734 0.990330i \(-0.544303\pi\)
−0.138734 + 0.990330i \(0.544303\pi\)
\(942\) −46.3546 −1.51031
\(943\) −15.6187 −0.508616
\(944\) 0.167892 0.00546442
\(945\) −22.6851 −0.737946
\(946\) −24.9175 −0.810138
\(947\) 32.9643 1.07120 0.535598 0.844473i \(-0.320086\pi\)
0.535598 + 0.844473i \(0.320086\pi\)
\(948\) −25.2743 −0.820871
\(949\) 14.9215 0.484373
\(950\) 4.82261 0.156466
\(951\) 66.2270 2.14756
\(952\) −2.66633 −0.0864162
\(953\) −16.8597 −0.546138 −0.273069 0.961994i \(-0.588039\pi\)
−0.273069 + 0.961994i \(0.588039\pi\)
\(954\) −38.3099 −1.24033
\(955\) 16.3449 0.528909
\(956\) 15.5385 0.502551
\(957\) −12.6176 −0.407869
\(958\) 10.9823 0.354821
\(959\) −20.3099 −0.655841
\(960\) 3.25457 0.105041
\(961\) 1.00000 0.0322581
\(962\) 6.10218 0.196742
\(963\) 5.71526 0.184172
\(964\) 0.0119521 0.000384950 0
\(965\) 8.01480 0.258005
\(966\) 8.95356 0.288076
\(967\) 4.85450 0.156110 0.0780551 0.996949i \(-0.475129\pi\)
0.0780551 + 0.996949i \(0.475129\pi\)
\(968\) 2.67561 0.0859973
\(969\) −27.5717 −0.885730
\(970\) −18.0929 −0.580929
\(971\) 17.6846 0.567525 0.283762 0.958895i \(-0.408417\pi\)
0.283762 + 0.958895i \(0.408417\pi\)
\(972\) −39.3424 −1.26191
\(973\) −6.68122 −0.214190
\(974\) −13.1634 −0.421782
\(975\) 3.25457 0.104230
\(976\) −12.5603 −0.402046
\(977\) 3.05627 0.0977787 0.0488893 0.998804i \(-0.484432\pi\)
0.0488893 + 0.998804i \(0.484432\pi\)
\(978\) 14.4991 0.463629
\(979\) −29.8135 −0.952843
\(980\) 4.69616 0.150013
\(981\) −1.63314 −0.0521421
\(982\) 13.7510 0.438813
\(983\) 48.6466 1.55159 0.775793 0.630987i \(-0.217350\pi\)
0.775793 + 0.630987i \(0.217350\pi\)
\(984\) 28.0455 0.894056
\(985\) 2.33801 0.0744952
\(986\) −2.36045 −0.0751721
\(987\) −43.5868 −1.38738
\(988\) 4.82261 0.153428
\(989\) 15.6533 0.497745
\(990\) −21.9051 −0.696188
\(991\) −23.7061 −0.753049 −0.376524 0.926407i \(-0.622881\pi\)
−0.376524 + 0.926407i \(0.622881\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 38.2571 1.21405
\(994\) −4.69871 −0.149034
\(995\) 20.5746 0.652257
\(996\) 29.3553 0.930158
\(997\) 21.3632 0.676578 0.338289 0.941042i \(-0.390152\pi\)
0.338289 + 0.941042i \(0.390152\pi\)
\(998\) −5.02756 −0.159144
\(999\) −91.2010 −2.88547
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 4030.2.a.p.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4030.2.a.p.1.1 9 1.1 even 1 trivial