Properties

Label 4026.2.a.z.1.4
Level $4026$
Weight $2$
Character 4026.1
Self dual yes
Analytic conductor $32.148$
Analytic rank $0$
Dimension $7$
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] = [4026,2,Mod(1,4026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, 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("4026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4026.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: \(32.1477718538\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 18x^{5} - 10x^{4} + 91x^{3} + 90x^{2} - 66x - 56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.70776\) of defining polynomial
Character \(\chi\) \(=\) 4026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.745947 q^{5} +1.00000 q^{6} -1.28209 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.745947 q^{5} +1.00000 q^{6} -1.28209 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.745947 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.02684 q^{13} +1.28209 q^{14} +0.745947 q^{15} +1.00000 q^{16} -3.94118 q^{17} -1.00000 q^{18} +1.02435 q^{19} -0.745947 q^{20} +1.28209 q^{21} -1.00000 q^{22} +2.77474 q^{23} +1.00000 q^{24} -4.44356 q^{25} -1.02684 q^{26} -1.00000 q^{27} -1.28209 q^{28} +4.28967 q^{29} -0.745947 q^{30} -5.11863 q^{31} -1.00000 q^{32} -1.00000 q^{33} +3.94118 q^{34} +0.956374 q^{35} +1.00000 q^{36} -1.64876 q^{37} -1.02435 q^{38} -1.02684 q^{39} +0.745947 q^{40} +5.94995 q^{41} -1.28209 q^{42} +3.78917 q^{43} +1.00000 q^{44} -0.745947 q^{45} -2.77474 q^{46} +6.38880 q^{47} -1.00000 q^{48} -5.35623 q^{49} +4.44356 q^{50} +3.94118 q^{51} +1.02684 q^{52} -0.306449 q^{53} +1.00000 q^{54} -0.745947 q^{55} +1.28209 q^{56} -1.02435 q^{57} -4.28967 q^{58} -6.02604 q^{59} +0.745947 q^{60} -1.00000 q^{61} +5.11863 q^{62} -1.28209 q^{63} +1.00000 q^{64} -0.765971 q^{65} +1.00000 q^{66} -9.15286 q^{67} -3.94118 q^{68} -2.77474 q^{69} -0.956374 q^{70} +7.77380 q^{71} -1.00000 q^{72} -0.861474 q^{73} +1.64876 q^{74} +4.44356 q^{75} +1.02435 q^{76} -1.28209 q^{77} +1.02684 q^{78} -3.78432 q^{79} -0.745947 q^{80} +1.00000 q^{81} -5.94995 q^{82} -10.9806 q^{83} +1.28209 q^{84} +2.93991 q^{85} -3.78917 q^{86} -4.28967 q^{87} -1.00000 q^{88} +9.92651 q^{89} +0.745947 q^{90} -1.31651 q^{91} +2.77474 q^{92} +5.11863 q^{93} -6.38880 q^{94} -0.764114 q^{95} +1.00000 q^{96} +3.30314 q^{97} +5.35623 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 4 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 4 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} + 7 q^{11} - 7 q^{12} - 7 q^{13} + 4 q^{14} - 2 q^{15} + 7 q^{16} - 4 q^{17} - 7 q^{18} - 4 q^{19} + 2 q^{20} + 4 q^{21} - 7 q^{22} - q^{23} + 7 q^{24} + 5 q^{25} + 7 q^{26} - 7 q^{27} - 4 q^{28} + 6 q^{29} + 2 q^{30} + 7 q^{31} - 7 q^{32} - 7 q^{33} + 4 q^{34} + 13 q^{35} + 7 q^{36} - 15 q^{37} + 4 q^{38} + 7 q^{39} - 2 q^{40} + q^{41} - 4 q^{42} - 13 q^{43} + 7 q^{44} + 2 q^{45} + q^{46} + 11 q^{47} - 7 q^{48} + 9 q^{49} - 5 q^{50} + 4 q^{51} - 7 q^{52} + 14 q^{53} + 7 q^{54} + 2 q^{55} + 4 q^{56} + 4 q^{57} - 6 q^{58} + 39 q^{59} - 2 q^{60} - 7 q^{61} - 7 q^{62} - 4 q^{63} + 7 q^{64} - 2 q^{65} + 7 q^{66} - 3 q^{67} - 4 q^{68} + q^{69} - 13 q^{70} + 12 q^{71} - 7 q^{72} - 21 q^{73} + 15 q^{74} - 5 q^{75} - 4 q^{76} - 4 q^{77} - 7 q^{78} + 15 q^{79} + 2 q^{80} + 7 q^{81} - q^{82} + 5 q^{83} + 4 q^{84} - 34 q^{85} + 13 q^{86} - 6 q^{87} - 7 q^{88} - 8 q^{89} - 2 q^{90} + 29 q^{91} - q^{92} - 7 q^{93} - 11 q^{94} + 13 q^{95} + 7 q^{96} - 20 q^{97} - 9 q^{98} + 7 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.745947 −0.333598 −0.166799 0.985991i \(-0.553343\pi\)
−0.166799 + 0.985991i \(0.553343\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.28209 −0.484586 −0.242293 0.970203i \(-0.577900\pi\)
−0.242293 + 0.970203i \(0.577900\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0.745947 0.235889
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 1.02684 0.284795 0.142398 0.989810i \(-0.454519\pi\)
0.142398 + 0.989810i \(0.454519\pi\)
\(14\) 1.28209 0.342654
\(15\) 0.745947 0.192603
\(16\) 1.00000 0.250000
\(17\) −3.94118 −0.955877 −0.477939 0.878393i \(-0.658616\pi\)
−0.477939 + 0.878393i \(0.658616\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.02435 0.235003 0.117502 0.993073i \(-0.462512\pi\)
0.117502 + 0.993073i \(0.462512\pi\)
\(20\) −0.745947 −0.166799
\(21\) 1.28209 0.279776
\(22\) −1.00000 −0.213201
\(23\) 2.77474 0.578574 0.289287 0.957242i \(-0.406582\pi\)
0.289287 + 0.957242i \(0.406582\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.44356 −0.888713
\(26\) −1.02684 −0.201381
\(27\) −1.00000 −0.192450
\(28\) −1.28209 −0.242293
\(29\) 4.28967 0.796571 0.398286 0.917261i \(-0.369605\pi\)
0.398286 + 0.917261i \(0.369605\pi\)
\(30\) −0.745947 −0.136191
\(31\) −5.11863 −0.919333 −0.459666 0.888092i \(-0.652031\pi\)
−0.459666 + 0.888092i \(0.652031\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 3.94118 0.675907
\(35\) 0.956374 0.161657
\(36\) 1.00000 0.166667
\(37\) −1.64876 −0.271055 −0.135527 0.990774i \(-0.543273\pi\)
−0.135527 + 0.990774i \(0.543273\pi\)
\(38\) −1.02435 −0.166172
\(39\) −1.02684 −0.164427
\(40\) 0.745947 0.117945
\(41\) 5.94995 0.929226 0.464613 0.885514i \(-0.346193\pi\)
0.464613 + 0.885514i \(0.346193\pi\)
\(42\) −1.28209 −0.197831
\(43\) 3.78917 0.577842 0.288921 0.957353i \(-0.406703\pi\)
0.288921 + 0.957353i \(0.406703\pi\)
\(44\) 1.00000 0.150756
\(45\) −0.745947 −0.111199
\(46\) −2.77474 −0.409113
\(47\) 6.38880 0.931903 0.465951 0.884810i \(-0.345712\pi\)
0.465951 + 0.884810i \(0.345712\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.35623 −0.765176
\(50\) 4.44356 0.628415
\(51\) 3.94118 0.551876
\(52\) 1.02684 0.142398
\(53\) −0.306449 −0.0420939 −0.0210470 0.999778i \(-0.506700\pi\)
−0.0210470 + 0.999778i \(0.506700\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.745947 −0.100583
\(56\) 1.28209 0.171327
\(57\) −1.02435 −0.135679
\(58\) −4.28967 −0.563261
\(59\) −6.02604 −0.784524 −0.392262 0.919854i \(-0.628307\pi\)
−0.392262 + 0.919854i \(0.628307\pi\)
\(60\) 0.745947 0.0963014
\(61\) −1.00000 −0.128037
\(62\) 5.11863 0.650066
\(63\) −1.28209 −0.161529
\(64\) 1.00000 0.125000
\(65\) −0.765971 −0.0950070
\(66\) 1.00000 0.123091
\(67\) −9.15286 −1.11820 −0.559100 0.829100i \(-0.688853\pi\)
−0.559100 + 0.829100i \(0.688853\pi\)
\(68\) −3.94118 −0.477939
\(69\) −2.77474 −0.334040
\(70\) −0.956374 −0.114309
\(71\) 7.77380 0.922580 0.461290 0.887249i \(-0.347387\pi\)
0.461290 + 0.887249i \(0.347387\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.861474 −0.100828 −0.0504140 0.998728i \(-0.516054\pi\)
−0.0504140 + 0.998728i \(0.516054\pi\)
\(74\) 1.64876 0.191664
\(75\) 4.44356 0.513098
\(76\) 1.02435 0.117502
\(77\) −1.28209 −0.146108
\(78\) 1.02684 0.116267
\(79\) −3.78432 −0.425769 −0.212884 0.977077i \(-0.568286\pi\)
−0.212884 + 0.977077i \(0.568286\pi\)
\(80\) −0.745947 −0.0833994
\(81\) 1.00000 0.111111
\(82\) −5.94995 −0.657062
\(83\) −10.9806 −1.20528 −0.602640 0.798013i \(-0.705885\pi\)
−0.602640 + 0.798013i \(0.705885\pi\)
\(84\) 1.28209 0.139888
\(85\) 2.93991 0.318878
\(86\) −3.78917 −0.408596
\(87\) −4.28967 −0.459901
\(88\) −1.00000 −0.106600
\(89\) 9.92651 1.05221 0.526104 0.850420i \(-0.323652\pi\)
0.526104 + 0.850420i \(0.323652\pi\)
\(90\) 0.745947 0.0786297
\(91\) −1.31651 −0.138008
\(92\) 2.77474 0.289287
\(93\) 5.11863 0.530777
\(94\) −6.38880 −0.658955
\(95\) −0.764114 −0.0783965
\(96\) 1.00000 0.102062
\(97\) 3.30314 0.335383 0.167692 0.985839i \(-0.446369\pi\)
0.167692 + 0.985839i \(0.446369\pi\)
\(98\) 5.35623 0.541061
\(99\) 1.00000 0.100504
\(100\) −4.44356 −0.444356
\(101\) 6.19552 0.616477 0.308239 0.951309i \(-0.400260\pi\)
0.308239 + 0.951309i \(0.400260\pi\)
\(102\) −3.94118 −0.390235
\(103\) −1.68344 −0.165875 −0.0829373 0.996555i \(-0.526430\pi\)
−0.0829373 + 0.996555i \(0.526430\pi\)
\(104\) −1.02684 −0.100690
\(105\) −0.956374 −0.0933326
\(106\) 0.306449 0.0297649
\(107\) −16.8399 −1.62798 −0.813989 0.580880i \(-0.802709\pi\)
−0.813989 + 0.580880i \(0.802709\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −1.16123 −0.111226 −0.0556130 0.998452i \(-0.517711\pi\)
−0.0556130 + 0.998452i \(0.517711\pi\)
\(110\) 0.745947 0.0711233
\(111\) 1.64876 0.156493
\(112\) −1.28209 −0.121147
\(113\) 14.0142 1.31834 0.659171 0.751993i \(-0.270908\pi\)
0.659171 + 0.751993i \(0.270908\pi\)
\(114\) 1.02435 0.0959396
\(115\) −2.06981 −0.193011
\(116\) 4.28967 0.398286
\(117\) 1.02684 0.0949317
\(118\) 6.02604 0.554742
\(119\) 5.05297 0.463205
\(120\) −0.745947 −0.0680953
\(121\) 1.00000 0.0909091
\(122\) 1.00000 0.0905357
\(123\) −5.94995 −0.536489
\(124\) −5.11863 −0.459666
\(125\) 7.04440 0.630070
\(126\) 1.28209 0.114218
\(127\) 19.0865 1.69366 0.846828 0.531867i \(-0.178509\pi\)
0.846828 + 0.531867i \(0.178509\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.78917 −0.333617
\(130\) 0.765971 0.0671801
\(131\) 17.6945 1.54597 0.772986 0.634423i \(-0.218762\pi\)
0.772986 + 0.634423i \(0.218762\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −1.31332 −0.113879
\(134\) 9.15286 0.790687
\(135\) 0.745947 0.0642009
\(136\) 3.94118 0.337954
\(137\) −15.6557 −1.33755 −0.668777 0.743463i \(-0.733182\pi\)
−0.668777 + 0.743463i \(0.733182\pi\)
\(138\) 2.77474 0.236202
\(139\) 0.617102 0.0523419 0.0261710 0.999657i \(-0.491669\pi\)
0.0261710 + 0.999657i \(0.491669\pi\)
\(140\) 0.956374 0.0808284
\(141\) −6.38880 −0.538034
\(142\) −7.77380 −0.652362
\(143\) 1.02684 0.0858690
\(144\) 1.00000 0.0833333
\(145\) −3.19986 −0.265734
\(146\) 0.861474 0.0712961
\(147\) 5.35623 0.441775
\(148\) −1.64876 −0.135527
\(149\) −1.52056 −0.124569 −0.0622847 0.998058i \(-0.519839\pi\)
−0.0622847 + 0.998058i \(0.519839\pi\)
\(150\) −4.44356 −0.362815
\(151\) −9.36299 −0.761949 −0.380974 0.924586i \(-0.624411\pi\)
−0.380974 + 0.924586i \(0.624411\pi\)
\(152\) −1.02435 −0.0830861
\(153\) −3.94118 −0.318626
\(154\) 1.28209 0.103314
\(155\) 3.81823 0.306687
\(156\) −1.02684 −0.0822133
\(157\) 15.0841 1.20384 0.601922 0.798555i \(-0.294402\pi\)
0.601922 + 0.798555i \(0.294402\pi\)
\(158\) 3.78432 0.301064
\(159\) 0.306449 0.0243029
\(160\) 0.745947 0.0589723
\(161\) −3.55748 −0.280369
\(162\) −1.00000 −0.0785674
\(163\) 23.0806 1.80781 0.903906 0.427731i \(-0.140687\pi\)
0.903906 + 0.427731i \(0.140687\pi\)
\(164\) 5.94995 0.464613
\(165\) 0.745947 0.0580719
\(166\) 10.9806 0.852262
\(167\) 16.6333 1.28712 0.643561 0.765395i \(-0.277456\pi\)
0.643561 + 0.765395i \(0.277456\pi\)
\(168\) −1.28209 −0.0989157
\(169\) −11.9456 −0.918892
\(170\) −2.93991 −0.225481
\(171\) 1.02435 0.0783343
\(172\) 3.78917 0.288921
\(173\) −19.1561 −1.45641 −0.728205 0.685360i \(-0.759645\pi\)
−0.728205 + 0.685360i \(0.759645\pi\)
\(174\) 4.28967 0.325199
\(175\) 5.69707 0.430658
\(176\) 1.00000 0.0753778
\(177\) 6.02604 0.452945
\(178\) −9.92651 −0.744024
\(179\) 15.0366 1.12389 0.561943 0.827176i \(-0.310054\pi\)
0.561943 + 0.827176i \(0.310054\pi\)
\(180\) −0.745947 −0.0555996
\(181\) 3.34104 0.248337 0.124169 0.992261i \(-0.460374\pi\)
0.124169 + 0.992261i \(0.460374\pi\)
\(182\) 1.31651 0.0975862
\(183\) 1.00000 0.0739221
\(184\) −2.77474 −0.204557
\(185\) 1.22989 0.0904232
\(186\) −5.11863 −0.375316
\(187\) −3.94118 −0.288208
\(188\) 6.38880 0.465951
\(189\) 1.28209 0.0932586
\(190\) 0.764114 0.0554347
\(191\) 23.6842 1.71373 0.856865 0.515541i \(-0.172409\pi\)
0.856865 + 0.515541i \(0.172409\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −25.2226 −1.81556 −0.907781 0.419444i \(-0.862225\pi\)
−0.907781 + 0.419444i \(0.862225\pi\)
\(194\) −3.30314 −0.237152
\(195\) 0.765971 0.0548523
\(196\) −5.35623 −0.382588
\(197\) −19.7129 −1.40449 −0.702243 0.711937i \(-0.747818\pi\)
−0.702243 + 0.711937i \(0.747818\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −13.3077 −0.943361 −0.471681 0.881770i \(-0.656352\pi\)
−0.471681 + 0.881770i \(0.656352\pi\)
\(200\) 4.44356 0.314207
\(201\) 9.15286 0.645593
\(202\) −6.19552 −0.435915
\(203\) −5.49976 −0.386007
\(204\) 3.94118 0.275938
\(205\) −4.43835 −0.309988
\(206\) 1.68344 0.117291
\(207\) 2.77474 0.192858
\(208\) 1.02684 0.0711988
\(209\) 1.02435 0.0708561
\(210\) 0.956374 0.0659961
\(211\) 13.9516 0.960468 0.480234 0.877140i \(-0.340552\pi\)
0.480234 + 0.877140i \(0.340552\pi\)
\(212\) −0.306449 −0.0210470
\(213\) −7.77380 −0.532652
\(214\) 16.8399 1.15115
\(215\) −2.82652 −0.192767
\(216\) 1.00000 0.0680414
\(217\) 6.56256 0.445496
\(218\) 1.16123 0.0786487
\(219\) 0.861474 0.0582130
\(220\) −0.745947 −0.0502917
\(221\) −4.04698 −0.272229
\(222\) −1.64876 −0.110658
\(223\) 2.90024 0.194214 0.0971071 0.995274i \(-0.469041\pi\)
0.0971071 + 0.995274i \(0.469041\pi\)
\(224\) 1.28209 0.0856635
\(225\) −4.44356 −0.296238
\(226\) −14.0142 −0.932208
\(227\) 8.77009 0.582091 0.291046 0.956709i \(-0.405997\pi\)
0.291046 + 0.956709i \(0.405997\pi\)
\(228\) −1.02435 −0.0678395
\(229\) 12.6487 0.835850 0.417925 0.908481i \(-0.362757\pi\)
0.417925 + 0.908481i \(0.362757\pi\)
\(230\) 2.06981 0.136479
\(231\) 1.28209 0.0843556
\(232\) −4.28967 −0.281630
\(233\) −6.13121 −0.401669 −0.200834 0.979625i \(-0.564365\pi\)
−0.200834 + 0.979625i \(0.564365\pi\)
\(234\) −1.02684 −0.0671269
\(235\) −4.76571 −0.310881
\(236\) −6.02604 −0.392262
\(237\) 3.78432 0.245818
\(238\) −5.05297 −0.327535
\(239\) 17.9825 1.16319 0.581597 0.813477i \(-0.302428\pi\)
0.581597 + 0.813477i \(0.302428\pi\)
\(240\) 0.745947 0.0481507
\(241\) 29.8941 1.92565 0.962826 0.270124i \(-0.0870647\pi\)
0.962826 + 0.270124i \(0.0870647\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 3.99547 0.255261
\(246\) 5.94995 0.379355
\(247\) 1.05185 0.0669277
\(248\) 5.11863 0.325033
\(249\) 10.9806 0.695869
\(250\) −7.04440 −0.445527
\(251\) 24.7252 1.56064 0.780320 0.625380i \(-0.215056\pi\)
0.780320 + 0.625380i \(0.215056\pi\)
\(252\) −1.28209 −0.0807643
\(253\) 2.77474 0.174446
\(254\) −19.0865 −1.19760
\(255\) −2.93991 −0.184105
\(256\) 1.00000 0.0625000
\(257\) 26.9857 1.68332 0.841660 0.540008i \(-0.181579\pi\)
0.841660 + 0.540008i \(0.181579\pi\)
\(258\) 3.78917 0.235903
\(259\) 2.11387 0.131349
\(260\) −0.765971 −0.0475035
\(261\) 4.28967 0.265524
\(262\) −17.6945 −1.09317
\(263\) 5.78655 0.356814 0.178407 0.983957i \(-0.442906\pi\)
0.178407 + 0.983957i \(0.442906\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0.228594 0.0140424
\(266\) 1.31332 0.0805247
\(267\) −9.92651 −0.607493
\(268\) −9.15286 −0.559100
\(269\) 11.6113 0.707952 0.353976 0.935254i \(-0.384829\pi\)
0.353976 + 0.935254i \(0.384829\pi\)
\(270\) −0.745947 −0.0453969
\(271\) −25.1036 −1.52494 −0.762468 0.647025i \(-0.776013\pi\)
−0.762468 + 0.647025i \(0.776013\pi\)
\(272\) −3.94118 −0.238969
\(273\) 1.31651 0.0796788
\(274\) 15.6557 0.945794
\(275\) −4.44356 −0.267957
\(276\) −2.77474 −0.167020
\(277\) 2.63951 0.158593 0.0792964 0.996851i \(-0.474733\pi\)
0.0792964 + 0.996851i \(0.474733\pi\)
\(278\) −0.617102 −0.0370113
\(279\) −5.11863 −0.306444
\(280\) −0.956374 −0.0571543
\(281\) 0.322393 0.0192324 0.00961618 0.999954i \(-0.496939\pi\)
0.00961618 + 0.999954i \(0.496939\pi\)
\(282\) 6.38880 0.380448
\(283\) −14.2970 −0.849867 −0.424933 0.905225i \(-0.639702\pi\)
−0.424933 + 0.905225i \(0.639702\pi\)
\(284\) 7.77380 0.461290
\(285\) 0.764114 0.0452622
\(286\) −1.02684 −0.0607185
\(287\) −7.62840 −0.450290
\(288\) −1.00000 −0.0589256
\(289\) −1.46708 −0.0862988
\(290\) 3.19986 0.187903
\(291\) −3.30314 −0.193634
\(292\) −0.861474 −0.0504140
\(293\) 7.05756 0.412307 0.206153 0.978520i \(-0.433905\pi\)
0.206153 + 0.978520i \(0.433905\pi\)
\(294\) −5.35623 −0.312382
\(295\) 4.49511 0.261715
\(296\) 1.64876 0.0958322
\(297\) −1.00000 −0.0580259
\(298\) 1.52056 0.0880838
\(299\) 2.84922 0.164775
\(300\) 4.44356 0.256549
\(301\) −4.85807 −0.280014
\(302\) 9.36299 0.538779
\(303\) −6.19552 −0.355923
\(304\) 1.02435 0.0587508
\(305\) 0.745947 0.0427128
\(306\) 3.94118 0.225302
\(307\) −2.22142 −0.126783 −0.0633916 0.997989i \(-0.520192\pi\)
−0.0633916 + 0.997989i \(0.520192\pi\)
\(308\) −1.28209 −0.0730541
\(309\) 1.68344 0.0957677
\(310\) −3.81823 −0.216861
\(311\) 23.8939 1.35490 0.677451 0.735568i \(-0.263085\pi\)
0.677451 + 0.735568i \(0.263085\pi\)
\(312\) 1.02684 0.0581336
\(313\) 25.0026 1.41323 0.706616 0.707597i \(-0.250221\pi\)
0.706616 + 0.707597i \(0.250221\pi\)
\(314\) −15.0841 −0.851247
\(315\) 0.956374 0.0538856
\(316\) −3.78432 −0.212884
\(317\) −1.61195 −0.0905364 −0.0452682 0.998975i \(-0.514414\pi\)
−0.0452682 + 0.998975i \(0.514414\pi\)
\(318\) −0.306449 −0.0171848
\(319\) 4.28967 0.240175
\(320\) −0.745947 −0.0416997
\(321\) 16.8399 0.939914
\(322\) 3.55748 0.198251
\(323\) −4.03717 −0.224634
\(324\) 1.00000 0.0555556
\(325\) −4.56284 −0.253101
\(326\) −23.0806 −1.27832
\(327\) 1.16123 0.0642164
\(328\) −5.94995 −0.328531
\(329\) −8.19105 −0.451587
\(330\) −0.745947 −0.0410630
\(331\) 30.4671 1.67462 0.837312 0.546725i \(-0.184126\pi\)
0.837312 + 0.546725i \(0.184126\pi\)
\(332\) −10.9806 −0.602640
\(333\) −1.64876 −0.0903515
\(334\) −16.6333 −0.910133
\(335\) 6.82755 0.373029
\(336\) 1.28209 0.0699440
\(337\) −0.665639 −0.0362597 −0.0181298 0.999836i \(-0.505771\pi\)
−0.0181298 + 0.999836i \(0.505771\pi\)
\(338\) 11.9456 0.649755
\(339\) −14.0142 −0.761145
\(340\) 2.93991 0.159439
\(341\) −5.11863 −0.277189
\(342\) −1.02435 −0.0553907
\(343\) 15.8419 0.855380
\(344\) −3.78917 −0.204298
\(345\) 2.06981 0.111435
\(346\) 19.1561 1.02984
\(347\) 12.6629 0.679781 0.339891 0.940465i \(-0.389610\pi\)
0.339891 + 0.940465i \(0.389610\pi\)
\(348\) −4.28967 −0.229950
\(349\) 8.77444 0.469685 0.234843 0.972033i \(-0.424543\pi\)
0.234843 + 0.972033i \(0.424543\pi\)
\(350\) −5.69707 −0.304521
\(351\) −1.02684 −0.0548089
\(352\) −1.00000 −0.0533002
\(353\) 15.1214 0.804834 0.402417 0.915457i \(-0.368170\pi\)
0.402417 + 0.915457i \(0.368170\pi\)
\(354\) −6.02604 −0.320281
\(355\) −5.79884 −0.307770
\(356\) 9.92651 0.526104
\(357\) −5.05297 −0.267431
\(358\) −15.0366 −0.794708
\(359\) 32.1499 1.69681 0.848404 0.529349i \(-0.177564\pi\)
0.848404 + 0.529349i \(0.177564\pi\)
\(360\) 0.745947 0.0393149
\(361\) −17.9507 −0.944774
\(362\) −3.34104 −0.175601
\(363\) −1.00000 −0.0524864
\(364\) −1.31651 −0.0690039
\(365\) 0.642614 0.0336360
\(366\) −1.00000 −0.0522708
\(367\) 22.0197 1.14942 0.574709 0.818358i \(-0.305115\pi\)
0.574709 + 0.818358i \(0.305115\pi\)
\(368\) 2.77474 0.144643
\(369\) 5.94995 0.309742
\(370\) −1.22989 −0.0639388
\(371\) 0.392896 0.0203981
\(372\) 5.11863 0.265388
\(373\) −37.3428 −1.93354 −0.966769 0.255653i \(-0.917709\pi\)
−0.966769 + 0.255653i \(0.917709\pi\)
\(374\) 3.94118 0.203794
\(375\) −7.04440 −0.363771
\(376\) −6.38880 −0.329477
\(377\) 4.40482 0.226860
\(378\) −1.28209 −0.0659438
\(379\) −19.1507 −0.983706 −0.491853 0.870678i \(-0.663680\pi\)
−0.491853 + 0.870678i \(0.663680\pi\)
\(380\) −0.764114 −0.0391982
\(381\) −19.0865 −0.977833
\(382\) −23.6842 −1.21179
\(383\) 7.54042 0.385298 0.192649 0.981268i \(-0.438292\pi\)
0.192649 + 0.981268i \(0.438292\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0.956374 0.0487414
\(386\) 25.2226 1.28380
\(387\) 3.78917 0.192614
\(388\) 3.30314 0.167692
\(389\) −2.78201 −0.141053 −0.0705267 0.997510i \(-0.522468\pi\)
−0.0705267 + 0.997510i \(0.522468\pi\)
\(390\) −0.765971 −0.0387864
\(391\) −10.9358 −0.553045
\(392\) 5.35623 0.270531
\(393\) −17.6945 −0.892567
\(394\) 19.7129 0.993122
\(395\) 2.82290 0.142036
\(396\) 1.00000 0.0502519
\(397\) −33.5450 −1.68357 −0.841787 0.539810i \(-0.818496\pi\)
−0.841787 + 0.539810i \(0.818496\pi\)
\(398\) 13.3077 0.667057
\(399\) 1.31332 0.0657482
\(400\) −4.44356 −0.222178
\(401\) 2.67571 0.133619 0.0668093 0.997766i \(-0.478718\pi\)
0.0668093 + 0.997766i \(0.478718\pi\)
\(402\) −9.15286 −0.456503
\(403\) −5.25603 −0.261822
\(404\) 6.19552 0.308239
\(405\) −0.745947 −0.0370664
\(406\) 5.49976 0.272948
\(407\) −1.64876 −0.0817260
\(408\) −3.94118 −0.195118
\(409\) −0.271591 −0.0134293 −0.00671465 0.999977i \(-0.502137\pi\)
−0.00671465 + 0.999977i \(0.502137\pi\)
\(410\) 4.43835 0.219194
\(411\) 15.6557 0.772237
\(412\) −1.68344 −0.0829373
\(413\) 7.72595 0.380169
\(414\) −2.77474 −0.136371
\(415\) 8.19097 0.402079
\(416\) −1.02684 −0.0503451
\(417\) −0.617102 −0.0302196
\(418\) −1.02435 −0.0501028
\(419\) −4.66064 −0.227687 −0.113844 0.993499i \(-0.536316\pi\)
−0.113844 + 0.993499i \(0.536316\pi\)
\(420\) −0.956374 −0.0466663
\(421\) 31.9442 1.55686 0.778432 0.627729i \(-0.216016\pi\)
0.778432 + 0.627729i \(0.216016\pi\)
\(422\) −13.9516 −0.679153
\(423\) 6.38880 0.310634
\(424\) 0.306449 0.0148825
\(425\) 17.5129 0.849500
\(426\) 7.77380 0.376642
\(427\) 1.28209 0.0620449
\(428\) −16.8399 −0.813989
\(429\) −1.02684 −0.0495765
\(430\) 2.82652 0.136307
\(431\) 30.6753 1.47758 0.738789 0.673936i \(-0.235398\pi\)
0.738789 + 0.673936i \(0.235398\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 24.0788 1.15715 0.578577 0.815628i \(-0.303608\pi\)
0.578577 + 0.815628i \(0.303608\pi\)
\(434\) −6.56256 −0.315013
\(435\) 3.19986 0.153422
\(436\) −1.16123 −0.0556130
\(437\) 2.84232 0.135967
\(438\) −0.861474 −0.0411628
\(439\) 8.48940 0.405177 0.202589 0.979264i \(-0.435065\pi\)
0.202589 + 0.979264i \(0.435065\pi\)
\(440\) 0.745947 0.0355616
\(441\) −5.35623 −0.255059
\(442\) 4.04698 0.192495
\(443\) 32.7393 1.55549 0.777745 0.628580i \(-0.216364\pi\)
0.777745 + 0.628580i \(0.216364\pi\)
\(444\) 1.64876 0.0782467
\(445\) −7.40466 −0.351014
\(446\) −2.90024 −0.137330
\(447\) 1.52056 0.0719201
\(448\) −1.28209 −0.0605733
\(449\) −26.1057 −1.23200 −0.616002 0.787745i \(-0.711249\pi\)
−0.616002 + 0.787745i \(0.711249\pi\)
\(450\) 4.44356 0.209472
\(451\) 5.94995 0.280172
\(452\) 14.0142 0.659171
\(453\) 9.36299 0.439911
\(454\) −8.77009 −0.411601
\(455\) 0.982047 0.0460391
\(456\) 1.02435 0.0479698
\(457\) 15.2573 0.713708 0.356854 0.934160i \(-0.383849\pi\)
0.356854 + 0.934160i \(0.383849\pi\)
\(458\) −12.6487 −0.591036
\(459\) 3.94118 0.183959
\(460\) −2.06981 −0.0965054
\(461\) 29.7034 1.38343 0.691713 0.722173i \(-0.256856\pi\)
0.691713 + 0.722173i \(0.256856\pi\)
\(462\) −1.28209 −0.0596484
\(463\) −7.74787 −0.360074 −0.180037 0.983660i \(-0.557622\pi\)
−0.180037 + 0.983660i \(0.557622\pi\)
\(464\) 4.28967 0.199143
\(465\) −3.81823 −0.177066
\(466\) 6.13121 0.284023
\(467\) −18.7407 −0.867215 −0.433607 0.901102i \(-0.642760\pi\)
−0.433607 + 0.901102i \(0.642760\pi\)
\(468\) 1.02684 0.0474659
\(469\) 11.7348 0.541864
\(470\) 4.76571 0.219826
\(471\) −15.0841 −0.695040
\(472\) 6.02604 0.277371
\(473\) 3.78917 0.174226
\(474\) −3.78432 −0.173819
\(475\) −4.55178 −0.208850
\(476\) 5.05297 0.231602
\(477\) −0.306449 −0.0140313
\(478\) −17.9825 −0.822502
\(479\) −8.02669 −0.366749 −0.183374 0.983043i \(-0.558702\pi\)
−0.183374 + 0.983043i \(0.558702\pi\)
\(480\) −0.745947 −0.0340477
\(481\) −1.69302 −0.0771950
\(482\) −29.8941 −1.36164
\(483\) 3.55748 0.161871
\(484\) 1.00000 0.0454545
\(485\) −2.46397 −0.111883
\(486\) 1.00000 0.0453609
\(487\) 6.25479 0.283431 0.141716 0.989907i \(-0.454738\pi\)
0.141716 + 0.989907i \(0.454738\pi\)
\(488\) 1.00000 0.0452679
\(489\) −23.0806 −1.04374
\(490\) −3.99547 −0.180497
\(491\) −4.48630 −0.202464 −0.101232 0.994863i \(-0.532278\pi\)
−0.101232 + 0.994863i \(0.532278\pi\)
\(492\) −5.94995 −0.268245
\(493\) −16.9064 −0.761424
\(494\) −1.05185 −0.0473250
\(495\) −0.745947 −0.0335278
\(496\) −5.11863 −0.229833
\(497\) −9.96674 −0.447069
\(498\) −10.9806 −0.492054
\(499\) 28.6832 1.28404 0.642018 0.766689i \(-0.278097\pi\)
0.642018 + 0.766689i \(0.278097\pi\)
\(500\) 7.04440 0.315035
\(501\) −16.6333 −0.743120
\(502\) −24.7252 −1.10354
\(503\) 42.1944 1.88136 0.940678 0.339299i \(-0.110190\pi\)
0.940678 + 0.339299i \(0.110190\pi\)
\(504\) 1.28209 0.0571090
\(505\) −4.62153 −0.205655
\(506\) −2.77474 −0.123352
\(507\) 11.9456 0.530522
\(508\) 19.0865 0.846828
\(509\) 39.9620 1.77128 0.885642 0.464369i \(-0.153719\pi\)
0.885642 + 0.464369i \(0.153719\pi\)
\(510\) 2.93991 0.130182
\(511\) 1.10449 0.0488598
\(512\) −1.00000 −0.0441942
\(513\) −1.02435 −0.0452264
\(514\) −26.9857 −1.19029
\(515\) 1.25576 0.0553354
\(516\) −3.78917 −0.166809
\(517\) 6.38880 0.280979
\(518\) −2.11387 −0.0928779
\(519\) 19.1561 0.840858
\(520\) 0.765971 0.0335901
\(521\) −23.4648 −1.02801 −0.514007 0.857786i \(-0.671839\pi\)
−0.514007 + 0.857786i \(0.671839\pi\)
\(522\) −4.28967 −0.187754
\(523\) 6.63988 0.290342 0.145171 0.989407i \(-0.453627\pi\)
0.145171 + 0.989407i \(0.453627\pi\)
\(524\) 17.6945 0.772986
\(525\) −5.69707 −0.248640
\(526\) −5.78655 −0.252306
\(527\) 20.1734 0.878769
\(528\) −1.00000 −0.0435194
\(529\) −15.3008 −0.665253
\(530\) −0.228594 −0.00992951
\(531\) −6.02604 −0.261508
\(532\) −1.31332 −0.0569396
\(533\) 6.10967 0.264639
\(534\) 9.92651 0.429562
\(535\) 12.5617 0.543090
\(536\) 9.15286 0.395343
\(537\) −15.0366 −0.648876
\(538\) −11.6113 −0.500598
\(539\) −5.35623 −0.230709
\(540\) 0.745947 0.0321005
\(541\) 9.23968 0.397245 0.198622 0.980076i \(-0.436353\pi\)
0.198622 + 0.980076i \(0.436353\pi\)
\(542\) 25.1036 1.07829
\(543\) −3.34104 −0.143378
\(544\) 3.94118 0.168977
\(545\) 0.866219 0.0371048
\(546\) −1.31651 −0.0563414
\(547\) −9.44136 −0.403684 −0.201842 0.979418i \(-0.564693\pi\)
−0.201842 + 0.979418i \(0.564693\pi\)
\(548\) −15.6557 −0.668777
\(549\) −1.00000 −0.0426790
\(550\) 4.44356 0.189474
\(551\) 4.39414 0.187197
\(552\) 2.77474 0.118101
\(553\) 4.85185 0.206322
\(554\) −2.63951 −0.112142
\(555\) −1.22989 −0.0522058
\(556\) 0.617102 0.0261710
\(557\) −29.5229 −1.25092 −0.625462 0.780255i \(-0.715089\pi\)
−0.625462 + 0.780255i \(0.715089\pi\)
\(558\) 5.11863 0.216689
\(559\) 3.89088 0.164567
\(560\) 0.956374 0.0404142
\(561\) 3.94118 0.166397
\(562\) −0.322393 −0.0135993
\(563\) 27.9114 1.17632 0.588162 0.808743i \(-0.299852\pi\)
0.588162 + 0.808743i \(0.299852\pi\)
\(564\) −6.38880 −0.269017
\(565\) −10.4538 −0.439796
\(566\) 14.2970 0.600947
\(567\) −1.28209 −0.0538429
\(568\) −7.77380 −0.326181
\(569\) −33.2074 −1.39213 −0.696063 0.717980i \(-0.745067\pi\)
−0.696063 + 0.717980i \(0.745067\pi\)
\(570\) −0.764114 −0.0320052
\(571\) 40.3175 1.68724 0.843618 0.536943i \(-0.180421\pi\)
0.843618 + 0.536943i \(0.180421\pi\)
\(572\) 1.02684 0.0429345
\(573\) −23.6842 −0.989423
\(574\) 7.62840 0.318403
\(575\) −12.3297 −0.514186
\(576\) 1.00000 0.0416667
\(577\) −6.24593 −0.260022 −0.130011 0.991513i \(-0.541501\pi\)
−0.130011 + 0.991513i \(0.541501\pi\)
\(578\) 1.46708 0.0610225
\(579\) 25.2226 1.04822
\(580\) −3.19986 −0.132867
\(581\) 14.0782 0.584062
\(582\) 3.30314 0.136920
\(583\) −0.306449 −0.0126918
\(584\) 0.861474 0.0356480
\(585\) −0.765971 −0.0316690
\(586\) −7.05756 −0.291545
\(587\) 10.0478 0.414718 0.207359 0.978265i \(-0.433513\pi\)
0.207359 + 0.978265i \(0.433513\pi\)
\(588\) 5.35623 0.220887
\(589\) −5.24329 −0.216046
\(590\) −4.49511 −0.185061
\(591\) 19.7129 0.810881
\(592\) −1.64876 −0.0677636
\(593\) −13.6042 −0.558656 −0.279328 0.960196i \(-0.590112\pi\)
−0.279328 + 0.960196i \(0.590112\pi\)
\(594\) 1.00000 0.0410305
\(595\) −3.76925 −0.154524
\(596\) −1.52056 −0.0622847
\(597\) 13.3077 0.544650
\(598\) −2.84922 −0.116513
\(599\) −19.2922 −0.788258 −0.394129 0.919055i \(-0.628954\pi\)
−0.394129 + 0.919055i \(0.628954\pi\)
\(600\) −4.44356 −0.181408
\(601\) 39.0394 1.59245 0.796224 0.605001i \(-0.206827\pi\)
0.796224 + 0.605001i \(0.206827\pi\)
\(602\) 4.85807 0.198000
\(603\) −9.15286 −0.372733
\(604\) −9.36299 −0.380974
\(605\) −0.745947 −0.0303271
\(606\) 6.19552 0.251676
\(607\) −41.7549 −1.69478 −0.847389 0.530972i \(-0.821827\pi\)
−0.847389 + 0.530972i \(0.821827\pi\)
\(608\) −1.02435 −0.0415431
\(609\) 5.49976 0.222861
\(610\) −0.745947 −0.0302025
\(611\) 6.56030 0.265401
\(612\) −3.94118 −0.159313
\(613\) 2.89232 0.116820 0.0584098 0.998293i \(-0.481397\pi\)
0.0584098 + 0.998293i \(0.481397\pi\)
\(614\) 2.22142 0.0896492
\(615\) 4.43835 0.178972
\(616\) 1.28209 0.0516570
\(617\) 22.2130 0.894260 0.447130 0.894469i \(-0.352446\pi\)
0.447130 + 0.894469i \(0.352446\pi\)
\(618\) −1.68344 −0.0677180
\(619\) 25.9433 1.04275 0.521374 0.853328i \(-0.325420\pi\)
0.521374 + 0.853328i \(0.325420\pi\)
\(620\) 3.81823 0.153344
\(621\) −2.77474 −0.111347
\(622\) −23.8939 −0.958060
\(623\) −12.7267 −0.509885
\(624\) −1.02684 −0.0411066
\(625\) 16.9631 0.678523
\(626\) −25.0026 −0.999307
\(627\) −1.02435 −0.0409088
\(628\) 15.0841 0.601922
\(629\) 6.49806 0.259095
\(630\) −0.956374 −0.0381029
\(631\) −31.2154 −1.24267 −0.621333 0.783547i \(-0.713408\pi\)
−0.621333 + 0.783547i \(0.713408\pi\)
\(632\) 3.78432 0.150532
\(633\) −13.9516 −0.554526
\(634\) 1.61195 0.0640189
\(635\) −14.2375 −0.565000
\(636\) 0.306449 0.0121515
\(637\) −5.50002 −0.217919
\(638\) −4.28967 −0.169830
\(639\) 7.77380 0.307527
\(640\) 0.745947 0.0294862
\(641\) −7.75435 −0.306278 −0.153139 0.988205i \(-0.548938\pi\)
−0.153139 + 0.988205i \(0.548938\pi\)
\(642\) −16.8399 −0.664620
\(643\) −33.1226 −1.30623 −0.653114 0.757260i \(-0.726538\pi\)
−0.653114 + 0.757260i \(0.726538\pi\)
\(644\) −3.55748 −0.140184
\(645\) 2.82652 0.111294
\(646\) 4.03717 0.158840
\(647\) −34.8551 −1.37030 −0.685148 0.728404i \(-0.740262\pi\)
−0.685148 + 0.728404i \(0.740262\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −6.02604 −0.236543
\(650\) 4.56284 0.178969
\(651\) −6.56256 −0.257207
\(652\) 23.0806 0.903906
\(653\) −41.5559 −1.62621 −0.813105 0.582117i \(-0.802224\pi\)
−0.813105 + 0.582117i \(0.802224\pi\)
\(654\) −1.16123 −0.0454079
\(655\) −13.1991 −0.515733
\(656\) 5.94995 0.232307
\(657\) −0.861474 −0.0336093
\(658\) 8.19105 0.319320
\(659\) −21.6449 −0.843165 −0.421582 0.906790i \(-0.638525\pi\)
−0.421582 + 0.906790i \(0.638525\pi\)
\(660\) 0.745947 0.0290360
\(661\) 28.8759 1.12314 0.561570 0.827429i \(-0.310197\pi\)
0.561570 + 0.827429i \(0.310197\pi\)
\(662\) −30.4671 −1.18414
\(663\) 4.04698 0.157172
\(664\) 10.9806 0.426131
\(665\) 0.979666 0.0379898
\(666\) 1.64876 0.0638882
\(667\) 11.9027 0.460875
\(668\) 16.6333 0.643561
\(669\) −2.90024 −0.112130
\(670\) −6.82755 −0.263771
\(671\) −1.00000 −0.0386046
\(672\) −1.28209 −0.0494579
\(673\) −17.4235 −0.671626 −0.335813 0.941929i \(-0.609011\pi\)
−0.335813 + 0.941929i \(0.609011\pi\)
\(674\) 0.665639 0.0256395
\(675\) 4.44356 0.171033
\(676\) −11.9456 −0.459446
\(677\) −48.9322 −1.88062 −0.940308 0.340324i \(-0.889463\pi\)
−0.940308 + 0.340324i \(0.889463\pi\)
\(678\) 14.0142 0.538210
\(679\) −4.23494 −0.162522
\(680\) −2.93991 −0.112741
\(681\) −8.77009 −0.336070
\(682\) 5.11863 0.196002
\(683\) 31.4190 1.20221 0.601107 0.799168i \(-0.294726\pi\)
0.601107 + 0.799168i \(0.294726\pi\)
\(684\) 1.02435 0.0391672
\(685\) 11.6783 0.446205
\(686\) −15.8419 −0.604845
\(687\) −12.6487 −0.482579
\(688\) 3.78917 0.144461
\(689\) −0.314675 −0.0119882
\(690\) −2.06981 −0.0787963
\(691\) −21.0351 −0.800215 −0.400107 0.916468i \(-0.631027\pi\)
−0.400107 + 0.916468i \(0.631027\pi\)
\(692\) −19.1561 −0.728205
\(693\) −1.28209 −0.0487027
\(694\) −12.6629 −0.480678
\(695\) −0.460325 −0.0174611
\(696\) 4.28967 0.162599
\(697\) −23.4498 −0.888226
\(698\) −8.77444 −0.332117
\(699\) 6.13121 0.231904
\(700\) 5.69707 0.215329
\(701\) −1.43256 −0.0541071 −0.0270535 0.999634i \(-0.508612\pi\)
−0.0270535 + 0.999634i \(0.508612\pi\)
\(702\) 1.02684 0.0387557
\(703\) −1.68891 −0.0636986
\(704\) 1.00000 0.0376889
\(705\) 4.76571 0.179487
\(706\) −15.1214 −0.569103
\(707\) −7.94324 −0.298736
\(708\) 6.02604 0.226473
\(709\) 14.8380 0.557254 0.278627 0.960399i \(-0.410121\pi\)
0.278627 + 0.960399i \(0.410121\pi\)
\(710\) 5.79884 0.217627
\(711\) −3.78432 −0.141923
\(712\) −9.92651 −0.372012
\(713\) −14.2029 −0.531902
\(714\) 5.05297 0.189103
\(715\) −0.765971 −0.0286457
\(716\) 15.0366 0.561943
\(717\) −17.9825 −0.671570
\(718\) −32.1499 −1.19983
\(719\) 33.7422 1.25837 0.629187 0.777254i \(-0.283388\pi\)
0.629187 + 0.777254i \(0.283388\pi\)
\(720\) −0.745947 −0.0277998
\(721\) 2.15833 0.0803805
\(722\) 17.9507 0.668056
\(723\) −29.8941 −1.11178
\(724\) 3.34104 0.124169
\(725\) −19.0614 −0.707923
\(726\) 1.00000 0.0371135
\(727\) 27.2283 1.00984 0.504922 0.863165i \(-0.331521\pi\)
0.504922 + 0.863165i \(0.331521\pi\)
\(728\) 1.31651 0.0487931
\(729\) 1.00000 0.0370370
\(730\) −0.642614 −0.0237842
\(731\) −14.9338 −0.552346
\(732\) 1.00000 0.0369611
\(733\) 27.1243 1.00186 0.500930 0.865488i \(-0.332992\pi\)
0.500930 + 0.865488i \(0.332992\pi\)
\(734\) −22.0197 −0.812762
\(735\) −3.99547 −0.147375
\(736\) −2.77474 −0.102278
\(737\) −9.15286 −0.337150
\(738\) −5.94995 −0.219021
\(739\) 28.2274 1.03836 0.519181 0.854664i \(-0.326237\pi\)
0.519181 + 0.854664i \(0.326237\pi\)
\(740\) 1.22989 0.0452116
\(741\) −1.05185 −0.0386407
\(742\) −0.392896 −0.0144237
\(743\) −51.4169 −1.88630 −0.943152 0.332362i \(-0.892154\pi\)
−0.943152 + 0.332362i \(0.892154\pi\)
\(744\) −5.11863 −0.187658
\(745\) 1.13426 0.0415560
\(746\) 37.3428 1.36722
\(747\) −10.9806 −0.401760
\(748\) −3.94118 −0.144104
\(749\) 21.5904 0.788896
\(750\) 7.04440 0.257225
\(751\) −23.3658 −0.852631 −0.426315 0.904575i \(-0.640189\pi\)
−0.426315 + 0.904575i \(0.640189\pi\)
\(752\) 6.38880 0.232976
\(753\) −24.7252 −0.901036
\(754\) −4.40482 −0.160414
\(755\) 6.98429 0.254184
\(756\) 1.28209 0.0466293
\(757\) 1.01078 0.0367372 0.0183686 0.999831i \(-0.494153\pi\)
0.0183686 + 0.999831i \(0.494153\pi\)
\(758\) 19.1507 0.695585
\(759\) −2.77474 −0.100717
\(760\) 0.764114 0.0277173
\(761\) −41.7770 −1.51441 −0.757207 0.653175i \(-0.773437\pi\)
−0.757207 + 0.653175i \(0.773437\pi\)
\(762\) 19.0865 0.691432
\(763\) 1.48881 0.0538986
\(764\) 23.6842 0.856865
\(765\) 2.93991 0.106293
\(766\) −7.54042 −0.272447
\(767\) −6.18780 −0.223429
\(768\) −1.00000 −0.0360844
\(769\) 8.66749 0.312558 0.156279 0.987713i \(-0.450050\pi\)
0.156279 + 0.987713i \(0.450050\pi\)
\(770\) −0.956374 −0.0344653
\(771\) −26.9857 −0.971865
\(772\) −25.2226 −0.907781
\(773\) −2.50838 −0.0902201 −0.0451101 0.998982i \(-0.514364\pi\)
−0.0451101 + 0.998982i \(0.514364\pi\)
\(774\) −3.78917 −0.136199
\(775\) 22.7449 0.817023
\(776\) −3.30314 −0.118576
\(777\) −2.11387 −0.0758345
\(778\) 2.78201 0.0997398
\(779\) 6.09486 0.218371
\(780\) 0.765971 0.0274262
\(781\) 7.77380 0.278168
\(782\) 10.9358 0.391062
\(783\) −4.28967 −0.153300
\(784\) −5.35623 −0.191294
\(785\) −11.2520 −0.401600
\(786\) 17.6945 0.631140
\(787\) −49.1920 −1.75350 −0.876752 0.480943i \(-0.840295\pi\)
−0.876752 + 0.480943i \(0.840295\pi\)
\(788\) −19.7129 −0.702243
\(789\) −5.78655 −0.206007
\(790\) −2.82290 −0.100434
\(791\) −17.9675 −0.638850
\(792\) −1.00000 −0.0355335
\(793\) −1.02684 −0.0364643
\(794\) 33.5450 1.19047
\(795\) −0.228594 −0.00810741
\(796\) −13.3077 −0.471681
\(797\) −20.4915 −0.725847 −0.362923 0.931819i \(-0.618221\pi\)
−0.362923 + 0.931819i \(0.618221\pi\)
\(798\) −1.31332 −0.0464910
\(799\) −25.1794 −0.890785
\(800\) 4.44356 0.157104
\(801\) 9.92651 0.350736
\(802\) −2.67571 −0.0944826
\(803\) −0.861474 −0.0304008
\(804\) 9.15286 0.322797
\(805\) 2.65369 0.0935303
\(806\) 5.25603 0.185136
\(807\) −11.6113 −0.408736
\(808\) −6.19552 −0.217958
\(809\) −7.49309 −0.263443 −0.131721 0.991287i \(-0.542050\pi\)
−0.131721 + 0.991287i \(0.542050\pi\)
\(810\) 0.745947 0.0262099
\(811\) −9.87970 −0.346923 −0.173462 0.984841i \(-0.555495\pi\)
−0.173462 + 0.984841i \(0.555495\pi\)
\(812\) −5.49976 −0.193004
\(813\) 25.1036 0.880423
\(814\) 1.64876 0.0577890
\(815\) −17.2169 −0.603082
\(816\) 3.94118 0.137969
\(817\) 3.88145 0.135795
\(818\) 0.271591 0.00949595
\(819\) −1.31651 −0.0460026
\(820\) −4.43835 −0.154994
\(821\) −21.0894 −0.736027 −0.368013 0.929820i \(-0.619962\pi\)
−0.368013 + 0.929820i \(0.619962\pi\)
\(822\) −15.6557 −0.546054
\(823\) 10.0204 0.349290 0.174645 0.984631i \(-0.444122\pi\)
0.174645 + 0.984631i \(0.444122\pi\)
\(824\) 1.68344 0.0586455
\(825\) 4.44356 0.154705
\(826\) −7.72595 −0.268820
\(827\) 29.9687 1.04211 0.521056 0.853522i \(-0.325538\pi\)
0.521056 + 0.853522i \(0.325538\pi\)
\(828\) 2.77474 0.0964289
\(829\) −6.81946 −0.236850 −0.118425 0.992963i \(-0.537784\pi\)
−0.118425 + 0.992963i \(0.537784\pi\)
\(830\) −8.19097 −0.284313
\(831\) −2.63951 −0.0915636
\(832\) 1.02684 0.0355994
\(833\) 21.1099 0.731415
\(834\) 0.617102 0.0213685
\(835\) −12.4076 −0.429381
\(836\) 1.02435 0.0354280
\(837\) 5.11863 0.176926
\(838\) 4.66064 0.160999
\(839\) 20.0108 0.690850 0.345425 0.938446i \(-0.387735\pi\)
0.345425 + 0.938446i \(0.387735\pi\)
\(840\) 0.956374 0.0329981
\(841\) −10.5988 −0.365475
\(842\) −31.9442 −1.10087
\(843\) −0.322393 −0.0111038
\(844\) 13.9516 0.480234
\(845\) 8.91078 0.306540
\(846\) −6.38880 −0.219652
\(847\) −1.28209 −0.0440533
\(848\) −0.306449 −0.0105235
\(849\) 14.2970 0.490671
\(850\) −17.5129 −0.600687
\(851\) −4.57488 −0.156825
\(852\) −7.77380 −0.266326
\(853\) −21.0335 −0.720175 −0.360087 0.932919i \(-0.617253\pi\)
−0.360087 + 0.932919i \(0.617253\pi\)
\(854\) −1.28209 −0.0438724
\(855\) −0.764114 −0.0261322
\(856\) 16.8399 0.575577
\(857\) −8.50611 −0.290563 −0.145282 0.989390i \(-0.546409\pi\)
−0.145282 + 0.989390i \(0.546409\pi\)
\(858\) 1.02684 0.0350559
\(859\) 42.1631 1.43859 0.719293 0.694707i \(-0.244466\pi\)
0.719293 + 0.694707i \(0.244466\pi\)
\(860\) −2.82652 −0.0963834
\(861\) 7.62840 0.259975
\(862\) −30.6753 −1.04481
\(863\) −43.3604 −1.47601 −0.738003 0.674798i \(-0.764231\pi\)
−0.738003 + 0.674798i \(0.764231\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.2894 0.485855
\(866\) −24.0788 −0.818232
\(867\) 1.46708 0.0498246
\(868\) 6.56256 0.222748
\(869\) −3.78432 −0.128374
\(870\) −3.19986 −0.108486
\(871\) −9.39856 −0.318458
\(872\) 1.16123 0.0393244
\(873\) 3.30314 0.111794
\(874\) −2.84232 −0.0961429
\(875\) −9.03158 −0.305323
\(876\) 0.861474 0.0291065
\(877\) 15.3868 0.519576 0.259788 0.965666i \(-0.416347\pi\)
0.259788 + 0.965666i \(0.416347\pi\)
\(878\) −8.48940 −0.286503
\(879\) −7.05756 −0.238045
\(880\) −0.745947 −0.0251459
\(881\) −44.2447 −1.49064 −0.745321 0.666705i \(-0.767704\pi\)
−0.745321 + 0.666705i \(0.767704\pi\)
\(882\) 5.35623 0.180354
\(883\) 27.0054 0.908803 0.454401 0.890797i \(-0.349853\pi\)
0.454401 + 0.890797i \(0.349853\pi\)
\(884\) −4.04698 −0.136115
\(885\) −4.49511 −0.151101
\(886\) −32.7393 −1.09990
\(887\) 2.74028 0.0920095 0.0460047 0.998941i \(-0.485351\pi\)
0.0460047 + 0.998941i \(0.485351\pi\)
\(888\) −1.64876 −0.0553288
\(889\) −24.4707 −0.820722
\(890\) 7.40466 0.248205
\(891\) 1.00000 0.0335013
\(892\) 2.90024 0.0971071
\(893\) 6.54440 0.219000
\(894\) −1.52056 −0.0508552
\(895\) −11.2165 −0.374926
\(896\) 1.28209 0.0428318
\(897\) −2.84922 −0.0951329
\(898\) 26.1057 0.871158
\(899\) −21.9572 −0.732314
\(900\) −4.44356 −0.148119
\(901\) 1.20777 0.0402366
\(902\) −5.94995 −0.198112
\(903\) 4.85807 0.161666
\(904\) −14.0142 −0.466104
\(905\) −2.49224 −0.0828448
\(906\) −9.36299 −0.311064
\(907\) 43.7001 1.45104 0.725519 0.688202i \(-0.241600\pi\)
0.725519 + 0.688202i \(0.241600\pi\)
\(908\) 8.77009 0.291046
\(909\) 6.19552 0.205492
\(910\) −0.982047 −0.0325545
\(911\) −10.7348 −0.355661 −0.177830 0.984061i \(-0.556908\pi\)
−0.177830 + 0.984061i \(0.556908\pi\)
\(912\) −1.02435 −0.0339198
\(913\) −10.9806 −0.363406
\(914\) −15.2573 −0.504668
\(915\) −0.745947 −0.0246603
\(916\) 12.6487 0.417925
\(917\) −22.6860 −0.749156
\(918\) −3.94118 −0.130078
\(919\) −31.8104 −1.04933 −0.524665 0.851309i \(-0.675809\pi\)
−0.524665 + 0.851309i \(0.675809\pi\)
\(920\) 2.06981 0.0682396
\(921\) 2.22142 0.0731983
\(922\) −29.7034 −0.978230
\(923\) 7.98247 0.262746
\(924\) 1.28209 0.0421778
\(925\) 7.32637 0.240890
\(926\) 7.74787 0.254611
\(927\) −1.68344 −0.0552915
\(928\) −4.28967 −0.140815
\(929\) 5.77873 0.189594 0.0947969 0.995497i \(-0.469780\pi\)
0.0947969 + 0.995497i \(0.469780\pi\)
\(930\) 3.81823 0.125205
\(931\) −5.48668 −0.179819
\(932\) −6.13121 −0.200834
\(933\) −23.8939 −0.782252
\(934\) 18.7407 0.613214
\(935\) 2.93991 0.0961455
\(936\) −1.02684 −0.0335634
\(937\) −56.1605 −1.83468 −0.917342 0.398099i \(-0.869670\pi\)
−0.917342 + 0.398099i \(0.869670\pi\)
\(938\) −11.7348 −0.383156
\(939\) −25.0026 −0.815930
\(940\) −4.76571 −0.155440
\(941\) −2.36918 −0.0772329 −0.0386165 0.999254i \(-0.512295\pi\)
−0.0386165 + 0.999254i \(0.512295\pi\)
\(942\) 15.0841 0.491468
\(943\) 16.5096 0.537626
\(944\) −6.02604 −0.196131
\(945\) −0.956374 −0.0311109
\(946\) −3.78917 −0.123196
\(947\) 52.4163 1.70330 0.851650 0.524111i \(-0.175602\pi\)
0.851650 + 0.524111i \(0.175602\pi\)
\(948\) 3.78432 0.122909
\(949\) −0.884599 −0.0287153
\(950\) 4.55178 0.147679
\(951\) 1.61195 0.0522712
\(952\) −5.05297 −0.163768
\(953\) 17.5667 0.569041 0.284520 0.958670i \(-0.408166\pi\)
0.284520 + 0.958670i \(0.408166\pi\)
\(954\) 0.306449 0.00992164
\(955\) −17.6672 −0.571696
\(956\) 17.9825 0.581597
\(957\) −4.28967 −0.138665
\(958\) 8.02669 0.259330
\(959\) 20.0720 0.648160
\(960\) 0.745947 0.0240753
\(961\) −4.79965 −0.154827
\(962\) 1.69302 0.0545851
\(963\) −16.8399 −0.542660
\(964\) 29.8941 0.962826
\(965\) 18.8147 0.605668
\(966\) −3.55748 −0.114460
\(967\) −43.8573 −1.41036 −0.705178 0.709031i \(-0.749133\pi\)
−0.705178 + 0.709031i \(0.749133\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 4.03717 0.129693
\(970\) 2.46397 0.0791133
\(971\) 48.5102 1.55677 0.778383 0.627789i \(-0.216040\pi\)
0.778383 + 0.627789i \(0.216040\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −0.791183 −0.0253642
\(974\) −6.25479 −0.200416
\(975\) 4.56284 0.146128
\(976\) −1.00000 −0.0320092
\(977\) 29.9989 0.959751 0.479875 0.877337i \(-0.340682\pi\)
0.479875 + 0.877337i \(0.340682\pi\)
\(978\) 23.0806 0.738036
\(979\) 9.92651 0.317253
\(980\) 3.99547 0.127631
\(981\) −1.16123 −0.0370754
\(982\) 4.48630 0.143164
\(983\) 17.3423 0.553133 0.276566 0.960995i \(-0.410803\pi\)
0.276566 + 0.960995i \(0.410803\pi\)
\(984\) 5.94995 0.189678
\(985\) 14.7048 0.468534
\(986\) 16.9064 0.538408
\(987\) 8.19105 0.260724
\(988\) 1.05185 0.0334639
\(989\) 10.5140 0.334324
\(990\) 0.745947 0.0237078
\(991\) −59.7452 −1.89787 −0.948935 0.315472i \(-0.897837\pi\)
−0.948935 + 0.315472i \(0.897837\pi\)
\(992\) 5.11863 0.162517
\(993\) −30.4671 −0.966845
\(994\) 9.96674 0.316126
\(995\) 9.92688 0.314703
\(996\) 10.9806 0.347934
\(997\) 16.8895 0.534897 0.267449 0.963572i \(-0.413819\pi\)
0.267449 + 0.963572i \(0.413819\pi\)
\(998\) −28.6832 −0.907951
\(999\) 1.64876 0.0521645
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 4026.2.a.z.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4026.2.a.z.1.4 7 1.1 even 1 trivial