Properties

Label 5610.2.a.bq.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
Analytic rank $0$
Dimension $2$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) 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 \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 5610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +3.37228 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +3.37228 q^{21} -1.00000 q^{22} -7.37228 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -3.37228 q^{28} -0.627719 q^{29} +1.00000 q^{30} +8.11684 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -3.37228 q^{35} +1.00000 q^{36} -2.74456 q^{37} -4.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} -0.744563 q^{41} -3.37228 q^{42} -4.62772 q^{43} +1.00000 q^{44} +1.00000 q^{45} +7.37228 q^{46} -0.744563 q^{47} -1.00000 q^{48} +4.37228 q^{49} -1.00000 q^{50} -1.00000 q^{51} -4.00000 q^{52} +1.00000 q^{54} +1.00000 q^{55} +3.37228 q^{56} -4.00000 q^{57} +0.627719 q^{58} +1.25544 q^{59} -1.00000 q^{60} -11.4891 q^{61} -8.11684 q^{62} -3.37228 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} +7.37228 q^{69} +3.37228 q^{70} -6.74456 q^{71} -1.00000 q^{72} -6.00000 q^{73} +2.74456 q^{74} -1.00000 q^{75} +4.00000 q^{76} -3.37228 q^{77} -4.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +0.744563 q^{82} +9.48913 q^{83} +3.37228 q^{84} +1.00000 q^{85} +4.62772 q^{86} +0.627719 q^{87} -1.00000 q^{88} +8.74456 q^{89} -1.00000 q^{90} +13.4891 q^{91} -7.37228 q^{92} -8.11684 q^{93} +0.744563 q^{94} +4.00000 q^{95} +1.00000 q^{96} +10.8614 q^{97} -4.37228 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} - 8 q^{13} + q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 3.37228 0.901280
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.37228 0.735892
\(22\) −1.00000 −0.213201
\(23\) −7.37228 −1.53723 −0.768613 0.639713i \(-0.779053\pi\)
−0.768613 + 0.639713i \(0.779053\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) −3.37228 −0.637301
\(29\) −0.627719 −0.116564 −0.0582822 0.998300i \(-0.518562\pi\)
−0.0582822 + 0.998300i \(0.518562\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.11684 1.45783 0.728914 0.684605i \(-0.240025\pi\)
0.728914 + 0.684605i \(0.240025\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) −3.37228 −0.570020
\(36\) 1.00000 0.166667
\(37\) −2.74456 −0.451203 −0.225602 0.974220i \(-0.572435\pi\)
−0.225602 + 0.974220i \(0.572435\pi\)
\(38\) −4.00000 −0.648886
\(39\) 4.00000 0.640513
\(40\) −1.00000 −0.158114
\(41\) −0.744563 −0.116281 −0.0581406 0.998308i \(-0.518517\pi\)
−0.0581406 + 0.998308i \(0.518517\pi\)
\(42\) −3.37228 −0.520354
\(43\) −4.62772 −0.705720 −0.352860 0.935676i \(-0.614791\pi\)
−0.352860 + 0.935676i \(0.614791\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 7.37228 1.08698
\(47\) −0.744563 −0.108606 −0.0543028 0.998525i \(-0.517294\pi\)
−0.0543028 + 0.998525i \(0.517294\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.37228 0.624612
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) −4.00000 −0.554700
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) 3.37228 0.450640
\(57\) −4.00000 −0.529813
\(58\) 0.627719 0.0824235
\(59\) 1.25544 0.163444 0.0817220 0.996655i \(-0.473958\pi\)
0.0817220 + 0.996655i \(0.473958\pi\)
\(60\) −1.00000 −0.129099
\(61\) −11.4891 −1.47103 −0.735516 0.677507i \(-0.763060\pi\)
−0.735516 + 0.677507i \(0.763060\pi\)
\(62\) −8.11684 −1.03084
\(63\) −3.37228 −0.424868
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 1.00000 0.121268
\(69\) 7.37228 0.887518
\(70\) 3.37228 0.403065
\(71\) −6.74456 −0.800432 −0.400216 0.916421i \(-0.631065\pi\)
−0.400216 + 0.916421i \(0.631065\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.74456 0.319049
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) −3.37228 −0.384307
\(78\) −4.00000 −0.452911
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 0.744563 0.0822232
\(83\) 9.48913 1.04157 0.520783 0.853689i \(-0.325640\pi\)
0.520783 + 0.853689i \(0.325640\pi\)
\(84\) 3.37228 0.367946
\(85\) 1.00000 0.108465
\(86\) 4.62772 0.499020
\(87\) 0.627719 0.0672985
\(88\) −1.00000 −0.106600
\(89\) 8.74456 0.926922 0.463461 0.886117i \(-0.346607\pi\)
0.463461 + 0.886117i \(0.346607\pi\)
\(90\) −1.00000 −0.105409
\(91\) 13.4891 1.41404
\(92\) −7.37228 −0.768613
\(93\) −8.11684 −0.841678
\(94\) 0.744563 0.0767958
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 10.8614 1.10281 0.551404 0.834238i \(-0.314092\pi\)
0.551404 + 0.834238i \(0.314092\pi\)
\(98\) −4.37228 −0.441667
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 1.00000 0.0990148
\(103\) 12.6277 1.24425 0.622123 0.782919i \(-0.286270\pi\)
0.622123 + 0.782919i \(0.286270\pi\)
\(104\) 4.00000 0.392232
\(105\) 3.37228 0.329101
\(106\) 0 0
\(107\) 4.62772 0.447378 0.223689 0.974661i \(-0.428190\pi\)
0.223689 + 0.974661i \(0.428190\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −18.2337 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 2.74456 0.260502
\(112\) −3.37228 −0.318651
\(113\) −15.4891 −1.45709 −0.728547 0.684996i \(-0.759804\pi\)
−0.728547 + 0.684996i \(0.759804\pi\)
\(114\) 4.00000 0.374634
\(115\) −7.37228 −0.687469
\(116\) −0.627719 −0.0582822
\(117\) −4.00000 −0.369800
\(118\) −1.25544 −0.115572
\(119\) −3.37228 −0.309137
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) 11.4891 1.04018
\(123\) 0.744563 0.0671350
\(124\) 8.11684 0.728914
\(125\) 1.00000 0.0894427
\(126\) 3.37228 0.300427
\(127\) 11.4891 1.01950 0.509748 0.860324i \(-0.329739\pi\)
0.509748 + 0.860324i \(0.329739\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.62772 0.407448
\(130\) 4.00000 0.350823
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −13.4891 −1.16966
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) −16.1168 −1.37695 −0.688477 0.725258i \(-0.741721\pi\)
−0.688477 + 0.725258i \(0.741721\pi\)
\(138\) −7.37228 −0.627570
\(139\) 7.37228 0.625309 0.312654 0.949867i \(-0.398782\pi\)
0.312654 + 0.949867i \(0.398782\pi\)
\(140\) −3.37228 −0.285010
\(141\) 0.744563 0.0627035
\(142\) 6.74456 0.565991
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −0.627719 −0.0521292
\(146\) 6.00000 0.496564
\(147\) −4.37228 −0.360620
\(148\) −2.74456 −0.225602
\(149\) 3.25544 0.266696 0.133348 0.991069i \(-0.457427\pi\)
0.133348 + 0.991069i \(0.457427\pi\)
\(150\) 1.00000 0.0816497
\(151\) −18.2337 −1.48384 −0.741918 0.670490i \(-0.766084\pi\)
−0.741918 + 0.670490i \(0.766084\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) 3.37228 0.271746
\(155\) 8.11684 0.651961
\(156\) 4.00000 0.320256
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 24.8614 1.95935
\(162\) −1.00000 −0.0785674
\(163\) 20.8614 1.63399 0.816996 0.576644i \(-0.195638\pi\)
0.816996 + 0.576644i \(0.195638\pi\)
\(164\) −0.744563 −0.0581406
\(165\) −1.00000 −0.0778499
\(166\) −9.48913 −0.736499
\(167\) −3.48913 −0.269997 −0.134998 0.990846i \(-0.543103\pi\)
−0.134998 + 0.990846i \(0.543103\pi\)
\(168\) −3.37228 −0.260177
\(169\) 3.00000 0.230769
\(170\) −1.00000 −0.0766965
\(171\) 4.00000 0.305888
\(172\) −4.62772 −0.352860
\(173\) 20.2337 1.53834 0.769169 0.639045i \(-0.220670\pi\)
0.769169 + 0.639045i \(0.220670\pi\)
\(174\) −0.627719 −0.0475872
\(175\) −3.37228 −0.254921
\(176\) 1.00000 0.0753778
\(177\) −1.25544 −0.0943645
\(178\) −8.74456 −0.655433
\(179\) 10.7446 0.803086 0.401543 0.915840i \(-0.368474\pi\)
0.401543 + 0.915840i \(0.368474\pi\)
\(180\) 1.00000 0.0745356
\(181\) −8.62772 −0.641293 −0.320647 0.947199i \(-0.603900\pi\)
−0.320647 + 0.947199i \(0.603900\pi\)
\(182\) −13.4891 −0.999880
\(183\) 11.4891 0.849301
\(184\) 7.37228 0.543492
\(185\) −2.74456 −0.201784
\(186\) 8.11684 0.595156
\(187\) 1.00000 0.0731272
\(188\) −0.744563 −0.0543028
\(189\) 3.37228 0.245297
\(190\) −4.00000 −0.290191
\(191\) 13.3723 0.967584 0.483792 0.875183i \(-0.339259\pi\)
0.483792 + 0.875183i \(0.339259\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.4891 1.40286 0.701429 0.712739i \(-0.252546\pi\)
0.701429 + 0.712739i \(0.252546\pi\)
\(194\) −10.8614 −0.779804
\(195\) 4.00000 0.286446
\(196\) 4.37228 0.312306
\(197\) 9.48913 0.676072 0.338036 0.941133i \(-0.390237\pi\)
0.338036 + 0.941133i \(0.390237\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −3.25544 −0.230772 −0.115386 0.993321i \(-0.536810\pi\)
−0.115386 + 0.993321i \(0.536810\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) −2.00000 −0.140720
\(203\) 2.11684 0.148573
\(204\) −1.00000 −0.0700140
\(205\) −0.744563 −0.0520025
\(206\) −12.6277 −0.879815
\(207\) −7.37228 −0.512409
\(208\) −4.00000 −0.277350
\(209\) 4.00000 0.276686
\(210\) −3.37228 −0.232710
\(211\) −15.3723 −1.05827 −0.529136 0.848537i \(-0.677484\pi\)
−0.529136 + 0.848537i \(0.677484\pi\)
\(212\) 0 0
\(213\) 6.74456 0.462130
\(214\) −4.62772 −0.316344
\(215\) −4.62772 −0.315608
\(216\) 1.00000 0.0680414
\(217\) −27.3723 −1.85815
\(218\) 18.2337 1.23494
\(219\) 6.00000 0.405442
\(220\) 1.00000 0.0674200
\(221\) −4.00000 −0.269069
\(222\) −2.74456 −0.184203
\(223\) 14.1168 0.945334 0.472667 0.881241i \(-0.343291\pi\)
0.472667 + 0.881241i \(0.343291\pi\)
\(224\) 3.37228 0.225320
\(225\) 1.00000 0.0666667
\(226\) 15.4891 1.03032
\(227\) −4.86141 −0.322663 −0.161331 0.986900i \(-0.551579\pi\)
−0.161331 + 0.986900i \(0.551579\pi\)
\(228\) −4.00000 −0.264906
\(229\) −12.7446 −0.842184 −0.421092 0.907018i \(-0.638353\pi\)
−0.421092 + 0.907018i \(0.638353\pi\)
\(230\) 7.37228 0.486114
\(231\) 3.37228 0.221880
\(232\) 0.627719 0.0412118
\(233\) 16.1168 1.05585 0.527925 0.849291i \(-0.322970\pi\)
0.527925 + 0.849291i \(0.322970\pi\)
\(234\) 4.00000 0.261488
\(235\) −0.744563 −0.0485699
\(236\) 1.25544 0.0817220
\(237\) 0 0
\(238\) 3.37228 0.218593
\(239\) 4.23369 0.273855 0.136927 0.990581i \(-0.456277\pi\)
0.136927 + 0.990581i \(0.456277\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −17.6060 −1.13410 −0.567050 0.823683i \(-0.691915\pi\)
−0.567050 + 0.823683i \(0.691915\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −11.4891 −0.735516
\(245\) 4.37228 0.279335
\(246\) −0.744563 −0.0474716
\(247\) −16.0000 −1.01806
\(248\) −8.11684 −0.515420
\(249\) −9.48913 −0.601349
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −3.37228 −0.212434
\(253\) −7.37228 −0.463491
\(254\) −11.4891 −0.720892
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) 8.11684 0.506315 0.253157 0.967425i \(-0.418531\pi\)
0.253157 + 0.967425i \(0.418531\pi\)
\(258\) −4.62772 −0.288109
\(259\) 9.25544 0.575105
\(260\) −4.00000 −0.248069
\(261\) −0.627719 −0.0388548
\(262\) −8.00000 −0.494242
\(263\) −1.88316 −0.116120 −0.0580602 0.998313i \(-0.518492\pi\)
−0.0580602 + 0.998313i \(0.518492\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 13.4891 0.827071
\(267\) −8.74456 −0.535159
\(268\) 4.00000 0.244339
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 1.00000 0.0608581
\(271\) −13.3723 −0.812308 −0.406154 0.913805i \(-0.633130\pi\)
−0.406154 + 0.913805i \(0.633130\pi\)
\(272\) 1.00000 0.0606339
\(273\) −13.4891 −0.816399
\(274\) 16.1168 0.973654
\(275\) 1.00000 0.0603023
\(276\) 7.37228 0.443759
\(277\) −7.48913 −0.449978 −0.224989 0.974361i \(-0.572235\pi\)
−0.224989 + 0.974361i \(0.572235\pi\)
\(278\) −7.37228 −0.442160
\(279\) 8.11684 0.485943
\(280\) 3.37228 0.201532
\(281\) −14.8614 −0.886557 −0.443279 0.896384i \(-0.646185\pi\)
−0.443279 + 0.896384i \(0.646185\pi\)
\(282\) −0.744563 −0.0443381
\(283\) −13.2554 −0.787954 −0.393977 0.919120i \(-0.628901\pi\)
−0.393977 + 0.919120i \(0.628901\pi\)
\(284\) −6.74456 −0.400216
\(285\) −4.00000 −0.236940
\(286\) 4.00000 0.236525
\(287\) 2.51087 0.148212
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0.627719 0.0368609
\(291\) −10.8614 −0.636707
\(292\) −6.00000 −0.351123
\(293\) 20.1168 1.17524 0.587619 0.809138i \(-0.300065\pi\)
0.587619 + 0.809138i \(0.300065\pi\)
\(294\) 4.37228 0.254997
\(295\) 1.25544 0.0730944
\(296\) 2.74456 0.159524
\(297\) −1.00000 −0.0580259
\(298\) −3.25544 −0.188582
\(299\) 29.4891 1.70540
\(300\) −1.00000 −0.0577350
\(301\) 15.6060 0.899513
\(302\) 18.2337 1.04923
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) −11.4891 −0.657865
\(306\) −1.00000 −0.0571662
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −3.37228 −0.192154
\(309\) −12.6277 −0.718366
\(310\) −8.11684 −0.461006
\(311\) 33.4891 1.89899 0.949497 0.313776i \(-0.101594\pi\)
0.949497 + 0.313776i \(0.101594\pi\)
\(312\) −4.00000 −0.226455
\(313\) −1.37228 −0.0775659 −0.0387830 0.999248i \(-0.512348\pi\)
−0.0387830 + 0.999248i \(0.512348\pi\)
\(314\) −10.0000 −0.564333
\(315\) −3.37228 −0.190007
\(316\) 0 0
\(317\) 16.1168 0.905212 0.452606 0.891711i \(-0.350494\pi\)
0.452606 + 0.891711i \(0.350494\pi\)
\(318\) 0 0
\(319\) −0.627719 −0.0351455
\(320\) 1.00000 0.0559017
\(321\) −4.62772 −0.258294
\(322\) −24.8614 −1.38547
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −20.8614 −1.15541
\(327\) 18.2337 1.00833
\(328\) 0.744563 0.0411116
\(329\) 2.51087 0.138429
\(330\) 1.00000 0.0550482
\(331\) 9.88316 0.543227 0.271614 0.962406i \(-0.412443\pi\)
0.271614 + 0.962406i \(0.412443\pi\)
\(332\) 9.48913 0.520783
\(333\) −2.74456 −0.150401
\(334\) 3.48913 0.190916
\(335\) 4.00000 0.218543
\(336\) 3.37228 0.183973
\(337\) 11.4891 0.625853 0.312926 0.949777i \(-0.398691\pi\)
0.312926 + 0.949777i \(0.398691\pi\)
\(338\) −3.00000 −0.163178
\(339\) 15.4891 0.841254
\(340\) 1.00000 0.0542326
\(341\) 8.11684 0.439552
\(342\) −4.00000 −0.216295
\(343\) 8.86141 0.478471
\(344\) 4.62772 0.249510
\(345\) 7.37228 0.396910
\(346\) −20.2337 −1.08777
\(347\) −5.48913 −0.294672 −0.147336 0.989087i \(-0.547070\pi\)
−0.147336 + 0.989087i \(0.547070\pi\)
\(348\) 0.627719 0.0336493
\(349\) 4.23369 0.226624 0.113312 0.993559i \(-0.463854\pi\)
0.113312 + 0.993559i \(0.463854\pi\)
\(350\) 3.37228 0.180256
\(351\) 4.00000 0.213504
\(352\) −1.00000 −0.0533002
\(353\) 25.3723 1.35043 0.675215 0.737621i \(-0.264051\pi\)
0.675215 + 0.737621i \(0.264051\pi\)
\(354\) 1.25544 0.0667257
\(355\) −6.74456 −0.357964
\(356\) 8.74456 0.463461
\(357\) 3.37228 0.178480
\(358\) −10.7446 −0.567868
\(359\) 13.2554 0.699595 0.349798 0.936825i \(-0.386250\pi\)
0.349798 + 0.936825i \(0.386250\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 8.62772 0.453463
\(363\) −1.00000 −0.0524864
\(364\) 13.4891 0.707022
\(365\) −6.00000 −0.314054
\(366\) −11.4891 −0.600546
\(367\) 23.4891 1.22612 0.613061 0.790035i \(-0.289938\pi\)
0.613061 + 0.790035i \(0.289938\pi\)
\(368\) −7.37228 −0.384307
\(369\) −0.744563 −0.0387604
\(370\) 2.74456 0.142683
\(371\) 0 0
\(372\) −8.11684 −0.420839
\(373\) −21.7228 −1.12476 −0.562382 0.826877i \(-0.690115\pi\)
−0.562382 + 0.826877i \(0.690115\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −1.00000 −0.0516398
\(376\) 0.744563 0.0383979
\(377\) 2.51087 0.129317
\(378\) −3.37228 −0.173451
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 4.00000 0.205196
\(381\) −11.4891 −0.588606
\(382\) −13.3723 −0.684185
\(383\) 8.74456 0.446826 0.223413 0.974724i \(-0.428280\pi\)
0.223413 + 0.974724i \(0.428280\pi\)
\(384\) 1.00000 0.0510310
\(385\) −3.37228 −0.171867
\(386\) −19.4891 −0.991970
\(387\) −4.62772 −0.235240
\(388\) 10.8614 0.551404
\(389\) 9.25544 0.469269 0.234635 0.972084i \(-0.424611\pi\)
0.234635 + 0.972084i \(0.424611\pi\)
\(390\) −4.00000 −0.202548
\(391\) −7.37228 −0.372832
\(392\) −4.37228 −0.220834
\(393\) −8.00000 −0.403547
\(394\) −9.48913 −0.478055
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −12.2337 −0.613991 −0.306996 0.951711i \(-0.599324\pi\)
−0.306996 + 0.951711i \(0.599324\pi\)
\(398\) 3.25544 0.163180
\(399\) 13.4891 0.675301
\(400\) 1.00000 0.0500000
\(401\) 1.13859 0.0568586 0.0284293 0.999596i \(-0.490949\pi\)
0.0284293 + 0.999596i \(0.490949\pi\)
\(402\) 4.00000 0.199502
\(403\) −32.4674 −1.61732
\(404\) 2.00000 0.0995037
\(405\) 1.00000 0.0496904
\(406\) −2.11684 −0.105057
\(407\) −2.74456 −0.136043
\(408\) 1.00000 0.0495074
\(409\) −22.2337 −1.09939 −0.549693 0.835367i \(-0.685255\pi\)
−0.549693 + 0.835367i \(0.685255\pi\)
\(410\) 0.744563 0.0367713
\(411\) 16.1168 0.794985
\(412\) 12.6277 0.622123
\(413\) −4.23369 −0.208326
\(414\) 7.37228 0.362328
\(415\) 9.48913 0.465803
\(416\) 4.00000 0.196116
\(417\) −7.37228 −0.361022
\(418\) −4.00000 −0.195646
\(419\) 32.6277 1.59397 0.796984 0.604000i \(-0.206427\pi\)
0.796984 + 0.604000i \(0.206427\pi\)
\(420\) 3.37228 0.164550
\(421\) −36.7446 −1.79082 −0.895410 0.445242i \(-0.853118\pi\)
−0.895410 + 0.445242i \(0.853118\pi\)
\(422\) 15.3723 0.748311
\(423\) −0.744563 −0.0362019
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) −6.74456 −0.326775
\(427\) 38.7446 1.87498
\(428\) 4.62772 0.223689
\(429\) 4.00000 0.193122
\(430\) 4.62772 0.223168
\(431\) 10.8614 0.523176 0.261588 0.965180i \(-0.415754\pi\)
0.261588 + 0.965180i \(0.415754\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 7.25544 0.348674 0.174337 0.984686i \(-0.444222\pi\)
0.174337 + 0.984686i \(0.444222\pi\)
\(434\) 27.3723 1.31391
\(435\) 0.627719 0.0300968
\(436\) −18.2337 −0.873235
\(437\) −29.4891 −1.41066
\(438\) −6.00000 −0.286691
\(439\) 21.2554 1.01447 0.507233 0.861809i \(-0.330668\pi\)
0.507233 + 0.861809i \(0.330668\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 4.37228 0.208204
\(442\) 4.00000 0.190261
\(443\) 4.62772 0.219870 0.109935 0.993939i \(-0.464936\pi\)
0.109935 + 0.993939i \(0.464936\pi\)
\(444\) 2.74456 0.130251
\(445\) 8.74456 0.414532
\(446\) −14.1168 −0.668452
\(447\) −3.25544 −0.153977
\(448\) −3.37228 −0.159325
\(449\) 26.8614 1.26767 0.633834 0.773469i \(-0.281480\pi\)
0.633834 + 0.773469i \(0.281480\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −0.744563 −0.0350601
\(452\) −15.4891 −0.728547
\(453\) 18.2337 0.856693
\(454\) 4.86141 0.228157
\(455\) 13.4891 0.632380
\(456\) 4.00000 0.187317
\(457\) 28.9783 1.35555 0.677773 0.735271i \(-0.262945\pi\)
0.677773 + 0.735271i \(0.262945\pi\)
\(458\) 12.7446 0.595514
\(459\) −1.00000 −0.0466760
\(460\) −7.37228 −0.343734
\(461\) 0.510875 0.0237938 0.0118969 0.999929i \(-0.496213\pi\)
0.0118969 + 0.999929i \(0.496213\pi\)
\(462\) −3.37228 −0.156893
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −0.627719 −0.0291411
\(465\) −8.11684 −0.376410
\(466\) −16.1168 −0.746598
\(467\) −14.9783 −0.693111 −0.346555 0.938030i \(-0.612649\pi\)
−0.346555 + 0.938030i \(0.612649\pi\)
\(468\) −4.00000 −0.184900
\(469\) −13.4891 −0.622870
\(470\) 0.744563 0.0343441
\(471\) −10.0000 −0.460776
\(472\) −1.25544 −0.0577862
\(473\) −4.62772 −0.212783
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −3.37228 −0.154568
\(477\) 0 0
\(478\) −4.23369 −0.193644
\(479\) −15.8832 −0.725720 −0.362860 0.931844i \(-0.618200\pi\)
−0.362860 + 0.931844i \(0.618200\pi\)
\(480\) 1.00000 0.0456435
\(481\) 10.9783 0.500565
\(482\) 17.6060 0.801930
\(483\) −24.8614 −1.13123
\(484\) 1.00000 0.0454545
\(485\) 10.8614 0.493191
\(486\) 1.00000 0.0453609
\(487\) −15.4891 −0.701879 −0.350940 0.936398i \(-0.614138\pi\)
−0.350940 + 0.936398i \(0.614138\pi\)
\(488\) 11.4891 0.520088
\(489\) −20.8614 −0.943385
\(490\) −4.37228 −0.197520
\(491\) 41.4891 1.87238 0.936189 0.351497i \(-0.114327\pi\)
0.936189 + 0.351497i \(0.114327\pi\)
\(492\) 0.744563 0.0335675
\(493\) −0.627719 −0.0282710
\(494\) 16.0000 0.719874
\(495\) 1.00000 0.0449467
\(496\) 8.11684 0.364457
\(497\) 22.7446 1.02023
\(498\) 9.48913 0.425218
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 1.00000 0.0447214
\(501\) 3.48913 0.155883
\(502\) −12.0000 −0.535586
\(503\) −23.7228 −1.05775 −0.528874 0.848700i \(-0.677386\pi\)
−0.528874 + 0.848700i \(0.677386\pi\)
\(504\) 3.37228 0.150213
\(505\) 2.00000 0.0889988
\(506\) 7.37228 0.327738
\(507\) −3.00000 −0.133235
\(508\) 11.4891 0.509748
\(509\) 9.25544 0.410240 0.205120 0.978737i \(-0.434242\pi\)
0.205120 + 0.978737i \(0.434242\pi\)
\(510\) 1.00000 0.0442807
\(511\) 20.2337 0.895086
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −8.11684 −0.358019
\(515\) 12.6277 0.556444
\(516\) 4.62772 0.203724
\(517\) −0.744563 −0.0327458
\(518\) −9.25544 −0.406661
\(519\) −20.2337 −0.888160
\(520\) 4.00000 0.175412
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 0.627719 0.0274745
\(523\) −14.1168 −0.617286 −0.308643 0.951178i \(-0.599875\pi\)
−0.308643 + 0.951178i \(0.599875\pi\)
\(524\) 8.00000 0.349482
\(525\) 3.37228 0.147178
\(526\) 1.88316 0.0821095
\(527\) 8.11684 0.353575
\(528\) −1.00000 −0.0435194
\(529\) 31.3505 1.36307
\(530\) 0 0
\(531\) 1.25544 0.0544813
\(532\) −13.4891 −0.584828
\(533\) 2.97825 0.129002
\(534\) 8.74456 0.378414
\(535\) 4.62772 0.200074
\(536\) −4.00000 −0.172774
\(537\) −10.7446 −0.463662
\(538\) −14.0000 −0.603583
\(539\) 4.37228 0.188327
\(540\) −1.00000 −0.0430331
\(541\) 2.23369 0.0960337 0.0480169 0.998847i \(-0.484710\pi\)
0.0480169 + 0.998847i \(0.484710\pi\)
\(542\) 13.3723 0.574389
\(543\) 8.62772 0.370251
\(544\) −1.00000 −0.0428746
\(545\) −18.2337 −0.781045
\(546\) 13.4891 0.577281
\(547\) 5.25544 0.224706 0.112353 0.993668i \(-0.464161\pi\)
0.112353 + 0.993668i \(0.464161\pi\)
\(548\) −16.1168 −0.688477
\(549\) −11.4891 −0.490344
\(550\) −1.00000 −0.0426401
\(551\) −2.51087 −0.106967
\(552\) −7.37228 −0.313785
\(553\) 0 0
\(554\) 7.48913 0.318182
\(555\) 2.74456 0.116500
\(556\) 7.37228 0.312654
\(557\) −1.13859 −0.0482437 −0.0241219 0.999709i \(-0.507679\pi\)
−0.0241219 + 0.999709i \(0.507679\pi\)
\(558\) −8.11684 −0.343613
\(559\) 18.5109 0.782927
\(560\) −3.37228 −0.142505
\(561\) −1.00000 −0.0422200
\(562\) 14.8614 0.626891
\(563\) 33.4891 1.41140 0.705699 0.708512i \(-0.250633\pi\)
0.705699 + 0.708512i \(0.250633\pi\)
\(564\) 0.744563 0.0313517
\(565\) −15.4891 −0.651632
\(566\) 13.2554 0.557168
\(567\) −3.37228 −0.141623
\(568\) 6.74456 0.282996
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 4.00000 0.167542
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −4.00000 −0.167248
\(573\) −13.3723 −0.558635
\(574\) −2.51087 −0.104802
\(575\) −7.37228 −0.307445
\(576\) 1.00000 0.0416667
\(577\) 38.4674 1.60142 0.800709 0.599054i \(-0.204456\pi\)
0.800709 + 0.599054i \(0.204456\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −19.4891 −0.809940
\(580\) −0.627719 −0.0260646
\(581\) −32.0000 −1.32758
\(582\) 10.8614 0.450220
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) −4.00000 −0.165380
\(586\) −20.1168 −0.831019
\(587\) 22.3505 0.922505 0.461253 0.887269i \(-0.347400\pi\)
0.461253 + 0.887269i \(0.347400\pi\)
\(588\) −4.37228 −0.180310
\(589\) 32.4674 1.33779
\(590\) −1.25544 −0.0516855
\(591\) −9.48913 −0.390330
\(592\) −2.74456 −0.112801
\(593\) 40.9783 1.68278 0.841388 0.540432i \(-0.181739\pi\)
0.841388 + 0.540432i \(0.181739\pi\)
\(594\) 1.00000 0.0410305
\(595\) −3.37228 −0.138250
\(596\) 3.25544 0.133348
\(597\) 3.25544 0.133236
\(598\) −29.4891 −1.20590
\(599\) 44.1168 1.80257 0.901283 0.433231i \(-0.142627\pi\)
0.901283 + 0.433231i \(0.142627\pi\)
\(600\) 1.00000 0.0408248
\(601\) 36.9783 1.50837 0.754187 0.656660i \(-0.228031\pi\)
0.754187 + 0.656660i \(0.228031\pi\)
\(602\) −15.6060 −0.636052
\(603\) 4.00000 0.162893
\(604\) −18.2337 −0.741918
\(605\) 1.00000 0.0406558
\(606\) 2.00000 0.0812444
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −4.00000 −0.162221
\(609\) −2.11684 −0.0857788
\(610\) 11.4891 0.465181
\(611\) 2.97825 0.120487
\(612\) 1.00000 0.0404226
\(613\) −5.25544 −0.212265 −0.106133 0.994352i \(-0.533847\pi\)
−0.106133 + 0.994352i \(0.533847\pi\)
\(614\) 12.0000 0.484281
\(615\) 0.744563 0.0300237
\(616\) 3.37228 0.135873
\(617\) 26.4674 1.06554 0.532768 0.846261i \(-0.321152\pi\)
0.532768 + 0.846261i \(0.321152\pi\)
\(618\) 12.6277 0.507961
\(619\) −16.2337 −0.652487 −0.326244 0.945286i \(-0.605783\pi\)
−0.326244 + 0.945286i \(0.605783\pi\)
\(620\) 8.11684 0.325980
\(621\) 7.37228 0.295839
\(622\) −33.4891 −1.34279
\(623\) −29.4891 −1.18146
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 1.37228 0.0548474
\(627\) −4.00000 −0.159745
\(628\) 10.0000 0.399043
\(629\) −2.74456 −0.109433
\(630\) 3.37228 0.134355
\(631\) −0.233688 −0.00930297 −0.00465148 0.999989i \(-0.501481\pi\)
−0.00465148 + 0.999989i \(0.501481\pi\)
\(632\) 0 0
\(633\) 15.3723 0.610993
\(634\) −16.1168 −0.640082
\(635\) 11.4891 0.455932
\(636\) 0 0
\(637\) −17.4891 −0.692944
\(638\) 0.627719 0.0248516
\(639\) −6.74456 −0.266811
\(640\) −1.00000 −0.0395285
\(641\) −36.1168 −1.42653 −0.713265 0.700895i \(-0.752784\pi\)
−0.713265 + 0.700895i \(0.752784\pi\)
\(642\) 4.62772 0.182641
\(643\) 4.62772 0.182499 0.0912497 0.995828i \(-0.470914\pi\)
0.0912497 + 0.995828i \(0.470914\pi\)
\(644\) 24.8614 0.979677
\(645\) 4.62772 0.182216
\(646\) −4.00000 −0.157378
\(647\) 12.9783 0.510228 0.255114 0.966911i \(-0.417887\pi\)
0.255114 + 0.966911i \(0.417887\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 1.25544 0.0492802
\(650\) 4.00000 0.156893
\(651\) 27.3723 1.07280
\(652\) 20.8614 0.816996
\(653\) 22.6277 0.885491 0.442746 0.896647i \(-0.354005\pi\)
0.442746 + 0.896647i \(0.354005\pi\)
\(654\) −18.2337 −0.712994
\(655\) 8.00000 0.312586
\(656\) −0.744563 −0.0290703
\(657\) −6.00000 −0.234082
\(658\) −2.51087 −0.0978841
\(659\) 8.62772 0.336088 0.168044 0.985779i \(-0.446255\pi\)
0.168044 + 0.985779i \(0.446255\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 28.7446 1.11803 0.559017 0.829156i \(-0.311179\pi\)
0.559017 + 0.829156i \(0.311179\pi\)
\(662\) −9.88316 −0.384120
\(663\) 4.00000 0.155347
\(664\) −9.48913 −0.368249
\(665\) −13.4891 −0.523086
\(666\) 2.74456 0.106350
\(667\) 4.62772 0.179186
\(668\) −3.48913 −0.134998
\(669\) −14.1168 −0.545789
\(670\) −4.00000 −0.154533
\(671\) −11.4891 −0.443533
\(672\) −3.37228 −0.130089
\(673\) 10.2337 0.394480 0.197240 0.980355i \(-0.436802\pi\)
0.197240 + 0.980355i \(0.436802\pi\)
\(674\) −11.4891 −0.442545
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) −34.7446 −1.33534 −0.667671 0.744456i \(-0.732709\pi\)
−0.667671 + 0.744456i \(0.732709\pi\)
\(678\) −15.4891 −0.594856
\(679\) −36.6277 −1.40564
\(680\) −1.00000 −0.0383482
\(681\) 4.86141 0.186290
\(682\) −8.11684 −0.310810
\(683\) 1.02175 0.0390962 0.0195481 0.999809i \(-0.493777\pi\)
0.0195481 + 0.999809i \(0.493777\pi\)
\(684\) 4.00000 0.152944
\(685\) −16.1168 −0.615793
\(686\) −8.86141 −0.338330
\(687\) 12.7446 0.486235
\(688\) −4.62772 −0.176430
\(689\) 0 0
\(690\) −7.37228 −0.280658
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 20.2337 0.769169
\(693\) −3.37228 −0.128102
\(694\) 5.48913 0.208364
\(695\) 7.37228 0.279647
\(696\) −0.627719 −0.0237936
\(697\) −0.744563 −0.0282023
\(698\) −4.23369 −0.160247
\(699\) −16.1168 −0.609595
\(700\) −3.37228 −0.127460
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) −4.00000 −0.150970
\(703\) −10.9783 −0.414053
\(704\) 1.00000 0.0376889
\(705\) 0.744563 0.0280419
\(706\) −25.3723 −0.954898
\(707\) −6.74456 −0.253655
\(708\) −1.25544 −0.0471822
\(709\) −22.7446 −0.854190 −0.427095 0.904207i \(-0.640463\pi\)
−0.427095 + 0.904207i \(0.640463\pi\)
\(710\) 6.74456 0.253119
\(711\) 0 0
\(712\) −8.74456 −0.327716
\(713\) −59.8397 −2.24101
\(714\) −3.37228 −0.126204
\(715\) −4.00000 −0.149592
\(716\) 10.7446 0.401543
\(717\) −4.23369 −0.158110
\(718\) −13.2554 −0.494689
\(719\) 12.2337 0.456240 0.228120 0.973633i \(-0.426742\pi\)
0.228120 + 0.973633i \(0.426742\pi\)
\(720\) 1.00000 0.0372678
\(721\) −42.5842 −1.58592
\(722\) 3.00000 0.111648
\(723\) 17.6060 0.654773
\(724\) −8.62772 −0.320647
\(725\) −0.627719 −0.0233129
\(726\) 1.00000 0.0371135
\(727\) 6.11684 0.226861 0.113431 0.993546i \(-0.463816\pi\)
0.113431 + 0.993546i \(0.463816\pi\)
\(728\) −13.4891 −0.499940
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −4.62772 −0.171162
\(732\) 11.4891 0.424650
\(733\) 9.48913 0.350489 0.175244 0.984525i \(-0.443928\pi\)
0.175244 + 0.984525i \(0.443928\pi\)
\(734\) −23.4891 −0.866999
\(735\) −4.37228 −0.161274
\(736\) 7.37228 0.271746
\(737\) 4.00000 0.147342
\(738\) 0.744563 0.0274077
\(739\) 36.2337 1.33288 0.666439 0.745560i \(-0.267818\pi\)
0.666439 + 0.745560i \(0.267818\pi\)
\(740\) −2.74456 −0.100892
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) −45.2119 −1.65867 −0.829333 0.558755i \(-0.811279\pi\)
−0.829333 + 0.558755i \(0.811279\pi\)
\(744\) 8.11684 0.297578
\(745\) 3.25544 0.119270
\(746\) 21.7228 0.795329
\(747\) 9.48913 0.347189
\(748\) 1.00000 0.0365636
\(749\) −15.6060 −0.570230
\(750\) 1.00000 0.0365148
\(751\) −0.116844 −0.00426370 −0.00213185 0.999998i \(-0.500679\pi\)
−0.00213185 + 0.999998i \(0.500679\pi\)
\(752\) −0.744563 −0.0271514
\(753\) −12.0000 −0.437304
\(754\) −2.51087 −0.0914407
\(755\) −18.2337 −0.663592
\(756\) 3.37228 0.122649
\(757\) −14.6277 −0.531653 −0.265827 0.964021i \(-0.585645\pi\)
−0.265827 + 0.964021i \(0.585645\pi\)
\(758\) 0 0
\(759\) 7.37228 0.267597
\(760\) −4.00000 −0.145095
\(761\) −39.0951 −1.41720 −0.708598 0.705612i \(-0.750672\pi\)
−0.708598 + 0.705612i \(0.750672\pi\)
\(762\) 11.4891 0.416207
\(763\) 61.4891 2.22606
\(764\) 13.3723 0.483792
\(765\) 1.00000 0.0361551
\(766\) −8.74456 −0.315954
\(767\) −5.02175 −0.181325
\(768\) −1.00000 −0.0360844
\(769\) 24.9783 0.900739 0.450369 0.892842i \(-0.351292\pi\)
0.450369 + 0.892842i \(0.351292\pi\)
\(770\) 3.37228 0.121529
\(771\) −8.11684 −0.292321
\(772\) 19.4891 0.701429
\(773\) −33.4891 −1.20452 −0.602260 0.798300i \(-0.705733\pi\)
−0.602260 + 0.798300i \(0.705733\pi\)
\(774\) 4.62772 0.166340
\(775\) 8.11684 0.291566
\(776\) −10.8614 −0.389902
\(777\) −9.25544 −0.332037
\(778\) −9.25544 −0.331824
\(779\) −2.97825 −0.106707
\(780\) 4.00000 0.143223
\(781\) −6.74456 −0.241339
\(782\) 7.37228 0.263632
\(783\) 0.627719 0.0224328
\(784\) 4.37228 0.156153
\(785\) 10.0000 0.356915
\(786\) 8.00000 0.285351
\(787\) −9.48913 −0.338251 −0.169125 0.985595i \(-0.554094\pi\)
−0.169125 + 0.985595i \(0.554094\pi\)
\(788\) 9.48913 0.338036
\(789\) 1.88316 0.0670421
\(790\) 0 0
\(791\) 52.2337 1.85722
\(792\) −1.00000 −0.0355335
\(793\) 45.9565 1.63196
\(794\) 12.2337 0.434157
\(795\) 0 0
\(796\) −3.25544 −0.115386
\(797\) −5.48913 −0.194435 −0.0972174 0.995263i \(-0.530994\pi\)
−0.0972174 + 0.995263i \(0.530994\pi\)
\(798\) −13.4891 −0.477510
\(799\) −0.744563 −0.0263407
\(800\) −1.00000 −0.0353553
\(801\) 8.74456 0.308974
\(802\) −1.13859 −0.0402051
\(803\) −6.00000 −0.211735
\(804\) −4.00000 −0.141069
\(805\) 24.8614 0.876249
\(806\) 32.4674 1.14361
\(807\) −14.0000 −0.492823
\(808\) −2.00000 −0.0703598
\(809\) 40.9783 1.44072 0.720359 0.693601i \(-0.243977\pi\)
0.720359 + 0.693601i \(0.243977\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −9.48913 −0.333208 −0.166604 0.986024i \(-0.553280\pi\)
−0.166604 + 0.986024i \(0.553280\pi\)
\(812\) 2.11684 0.0742867
\(813\) 13.3723 0.468986
\(814\) 2.74456 0.0961969
\(815\) 20.8614 0.730743
\(816\) −1.00000 −0.0350070
\(817\) −18.5109 −0.647614
\(818\) 22.2337 0.777383
\(819\) 13.4891 0.471348
\(820\) −0.744563 −0.0260013
\(821\) −27.3723 −0.955299 −0.477650 0.878550i \(-0.658511\pi\)
−0.477650 + 0.878550i \(0.658511\pi\)
\(822\) −16.1168 −0.562139
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) −12.6277 −0.439907
\(825\) −1.00000 −0.0348155
\(826\) 4.23369 0.147309
\(827\) 23.8397 0.828986 0.414493 0.910052i \(-0.363959\pi\)
0.414493 + 0.910052i \(0.363959\pi\)
\(828\) −7.37228 −0.256204
\(829\) 24.9783 0.867531 0.433765 0.901026i \(-0.357185\pi\)
0.433765 + 0.901026i \(0.357185\pi\)
\(830\) −9.48913 −0.329372
\(831\) 7.48913 0.259795
\(832\) −4.00000 −0.138675
\(833\) 4.37228 0.151491
\(834\) 7.37228 0.255281
\(835\) −3.48913 −0.120746
\(836\) 4.00000 0.138343
\(837\) −8.11684 −0.280559
\(838\) −32.6277 −1.12711
\(839\) −13.7228 −0.473764 −0.236882 0.971538i \(-0.576125\pi\)
−0.236882 + 0.971538i \(0.576125\pi\)
\(840\) −3.37228 −0.116355
\(841\) −28.6060 −0.986413
\(842\) 36.7446 1.26630
\(843\) 14.8614 0.511854
\(844\) −15.3723 −0.529136
\(845\) 3.00000 0.103203
\(846\) 0.744563 0.0255986
\(847\) −3.37228 −0.115873
\(848\) 0 0
\(849\) 13.2554 0.454925
\(850\) −1.00000 −0.0342997
\(851\) 20.2337 0.693602
\(852\) 6.74456 0.231065
\(853\) −4.11684 −0.140958 −0.0704790 0.997513i \(-0.522453\pi\)
−0.0704790 + 0.997513i \(0.522453\pi\)
\(854\) −38.7446 −1.32581
\(855\) 4.00000 0.136797
\(856\) −4.62772 −0.158172
\(857\) 18.8614 0.644293 0.322147 0.946690i \(-0.395596\pi\)
0.322147 + 0.946690i \(0.395596\pi\)
\(858\) −4.00000 −0.136558
\(859\) −39.8397 −1.35931 −0.679656 0.733531i \(-0.737871\pi\)
−0.679656 + 0.733531i \(0.737871\pi\)
\(860\) −4.62772 −0.157804
\(861\) −2.51087 −0.0855704
\(862\) −10.8614 −0.369941
\(863\) 19.7228 0.671372 0.335686 0.941974i \(-0.391032\pi\)
0.335686 + 0.941974i \(0.391032\pi\)
\(864\) 1.00000 0.0340207
\(865\) 20.2337 0.687966
\(866\) −7.25544 −0.246550
\(867\) −1.00000 −0.0339618
\(868\) −27.3723 −0.929076
\(869\) 0 0
\(870\) −0.627719 −0.0212817
\(871\) −16.0000 −0.542139
\(872\) 18.2337 0.617471
\(873\) 10.8614 0.367603
\(874\) 29.4891 0.997485
\(875\) −3.37228 −0.114004
\(876\) 6.00000 0.202721
\(877\) −18.6277 −0.629013 −0.314507 0.949255i \(-0.601839\pi\)
−0.314507 + 0.949255i \(0.601839\pi\)
\(878\) −21.2554 −0.717336
\(879\) −20.1168 −0.678524
\(880\) 1.00000 0.0337100
\(881\) −11.8832 −0.400354 −0.200177 0.979760i \(-0.564152\pi\)
−0.200177 + 0.979760i \(0.564152\pi\)
\(882\) −4.37228 −0.147222
\(883\) 1.48913 0.0501131 0.0250565 0.999686i \(-0.492023\pi\)
0.0250565 + 0.999686i \(0.492023\pi\)
\(884\) −4.00000 −0.134535
\(885\) −1.25544 −0.0422011
\(886\) −4.62772 −0.155471
\(887\) −11.7228 −0.393614 −0.196807 0.980442i \(-0.563057\pi\)
−0.196807 + 0.980442i \(0.563057\pi\)
\(888\) −2.74456 −0.0921015
\(889\) −38.7446 −1.29945
\(890\) −8.74456 −0.293118
\(891\) 1.00000 0.0335013
\(892\) 14.1168 0.472667
\(893\) −2.97825 −0.0996634
\(894\) 3.25544 0.108878
\(895\) 10.7446 0.359151
\(896\) 3.37228 0.112660
\(897\) −29.4891 −0.984613
\(898\) −26.8614 −0.896377
\(899\) −5.09509 −0.169931
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0.744563 0.0247912
\(903\) −15.6060 −0.519334
\(904\) 15.4891 0.515161
\(905\) −8.62772 −0.286795
\(906\) −18.2337 −0.605774
\(907\) 19.1386 0.635487 0.317743 0.948177i \(-0.397075\pi\)
0.317743 + 0.948177i \(0.397075\pi\)
\(908\) −4.86141 −0.161331
\(909\) 2.00000 0.0663358
\(910\) −13.4891 −0.447160
\(911\) −40.4674 −1.34074 −0.670372 0.742025i \(-0.733865\pi\)
−0.670372 + 0.742025i \(0.733865\pi\)
\(912\) −4.00000 −0.132453
\(913\) 9.48913 0.314044
\(914\) −28.9783 −0.958515
\(915\) 11.4891 0.379819
\(916\) −12.7446 −0.421092
\(917\) −26.9783 −0.890900
\(918\) 1.00000 0.0330049
\(919\) 27.8832 0.919780 0.459890 0.887976i \(-0.347889\pi\)
0.459890 + 0.887976i \(0.347889\pi\)
\(920\) 7.37228 0.243057
\(921\) 12.0000 0.395413
\(922\) −0.510875 −0.0168248
\(923\) 26.9783 0.888000
\(924\) 3.37228 0.110940
\(925\) −2.74456 −0.0902407
\(926\) −8.00000 −0.262896
\(927\) 12.6277 0.414749
\(928\) 0.627719 0.0206059
\(929\) 25.3723 0.832438 0.416219 0.909265i \(-0.363355\pi\)
0.416219 + 0.909265i \(0.363355\pi\)
\(930\) 8.11684 0.266162
\(931\) 17.4891 0.573183
\(932\) 16.1168 0.527925
\(933\) −33.4891 −1.09638
\(934\) 14.9783 0.490103
\(935\) 1.00000 0.0327035
\(936\) 4.00000 0.130744
\(937\) −34.4674 −1.12600 −0.563000 0.826457i \(-0.690353\pi\)
−0.563000 + 0.826457i \(0.690353\pi\)
\(938\) 13.4891 0.440436
\(939\) 1.37228 0.0447827
\(940\) −0.744563 −0.0242850
\(941\) −14.7446 −0.480659 −0.240330 0.970691i \(-0.577255\pi\)
−0.240330 + 0.970691i \(0.577255\pi\)
\(942\) 10.0000 0.325818
\(943\) 5.48913 0.178751
\(944\) 1.25544 0.0408610
\(945\) 3.37228 0.109700
\(946\) 4.62772 0.150460
\(947\) −34.7446 −1.12905 −0.564523 0.825417i \(-0.690940\pi\)
−0.564523 + 0.825417i \(0.690940\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) −4.00000 −0.129777
\(951\) −16.1168 −0.522624
\(952\) 3.37228 0.109296
\(953\) −7.02175 −0.227457 −0.113728 0.993512i \(-0.536279\pi\)
−0.113728 + 0.993512i \(0.536279\pi\)
\(954\) 0 0
\(955\) 13.3723 0.432717
\(956\) 4.23369 0.136927
\(957\) 0.627719 0.0202913
\(958\) 15.8832 0.513161
\(959\) 54.3505 1.75507
\(960\) −1.00000 −0.0322749
\(961\) 34.8832 1.12526
\(962\) −10.9783 −0.353953
\(963\) 4.62772 0.149126
\(964\) −17.6060 −0.567050
\(965\) 19.4891 0.627377
\(966\) 24.8614 0.799903
\(967\) 26.4674 0.851133 0.425567 0.904927i \(-0.360075\pi\)
0.425567 + 0.904927i \(0.360075\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −4.00000 −0.128499
\(970\) −10.8614 −0.348739
\(971\) 48.4674 1.55539 0.777696 0.628640i \(-0.216388\pi\)
0.777696 + 0.628640i \(0.216388\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −24.8614 −0.797020
\(974\) 15.4891 0.496304
\(975\) 4.00000 0.128103
\(976\) −11.4891 −0.367758
\(977\) −11.4891 −0.367570 −0.183785 0.982966i \(-0.558835\pi\)
−0.183785 + 0.982966i \(0.558835\pi\)
\(978\) 20.8614 0.667074
\(979\) 8.74456 0.279477
\(980\) 4.37228 0.139667
\(981\) −18.2337 −0.582157
\(982\) −41.4891 −1.32397
\(983\) −5.88316 −0.187644 −0.0938218 0.995589i \(-0.529908\pi\)
−0.0938218 + 0.995589i \(0.529908\pi\)
\(984\) −0.744563 −0.0237358
\(985\) 9.48913 0.302349
\(986\) 0.627719 0.0199906
\(987\) −2.51087 −0.0799220
\(988\) −16.0000 −0.509028
\(989\) 34.1168 1.08485
\(990\) −1.00000 −0.0317821
\(991\) 13.3723 0.424785 0.212392 0.977184i \(-0.431875\pi\)
0.212392 + 0.977184i \(0.431875\pi\)
\(992\) −8.11684 −0.257710
\(993\) −9.88316 −0.313632
\(994\) −22.7446 −0.721414
\(995\) −3.25544 −0.103204
\(996\) −9.48913 −0.300674
\(997\) 24.1168 0.763788 0.381894 0.924206i \(-0.375272\pi\)
0.381894 + 0.924206i \(0.375272\pi\)
\(998\) −36.0000 −1.13956
\(999\) 2.74456 0.0868341
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 5610.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bq.1.1 2 1.1 even 1 trivial