Properties

Label 5610.2.a.bu.1.2
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.23607 q^{13} +3.23607 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -7.70820 q^{19} -1.00000 q^{20} -3.23607 q^{21} +1.00000 q^{22} -5.23607 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} -8.47214 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -3.23607 q^{35} +1.00000 q^{36} -6.00000 q^{37} -7.70820 q^{38} -1.23607 q^{39} -1.00000 q^{40} -8.94427 q^{41} -3.23607 q^{42} -3.23607 q^{43} +1.00000 q^{44} -1.00000 q^{45} -5.23607 q^{46} +2.47214 q^{47} -1.00000 q^{48} +3.47214 q^{49} +1.00000 q^{50} +1.00000 q^{51} +1.23607 q^{52} +3.23607 q^{53} -1.00000 q^{54} -1.00000 q^{55} +3.23607 q^{56} +7.70820 q^{57} -8.47214 q^{58} -3.23607 q^{59} +1.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +3.23607 q^{63} +1.00000 q^{64} -1.23607 q^{65} -1.00000 q^{66} +6.00000 q^{67} -1.00000 q^{68} +5.23607 q^{69} -3.23607 q^{70} -0.763932 q^{71} +1.00000 q^{72} +1.23607 q^{73} -6.00000 q^{74} -1.00000 q^{75} -7.70820 q^{76} +3.23607 q^{77} -1.23607 q^{78} +3.70820 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.94427 q^{82} +4.94427 q^{83} -3.23607 q^{84} +1.00000 q^{85} -3.23607 q^{86} +8.47214 q^{87} +1.00000 q^{88} -6.94427 q^{89} -1.00000 q^{90} +4.00000 q^{91} -5.23607 q^{92} +4.00000 q^{93} +2.47214 q^{94} +7.70820 q^{95} -1.00000 q^{96} +3.47214 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 3.23607 0.864876
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) −7.70820 −1.76838 −0.884192 0.467124i \(-0.845290\pi\)
−0.884192 + 0.467124i \(0.845290\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.23607 −0.706168
\(22\) 1.00000 0.213201
\(23\) −5.23607 −1.09180 −0.545898 0.837852i \(-0.683811\pi\)
−0.545898 + 0.837852i \(0.683811\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.23607 0.242413
\(27\) −1.00000 −0.192450
\(28\) 3.23607 0.611559
\(29\) −8.47214 −1.57324 −0.786618 0.617440i \(-0.788170\pi\)
−0.786618 + 0.617440i \(0.788170\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) −3.23607 −0.546995
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −7.70820 −1.25044
\(39\) −1.23607 −0.197929
\(40\) −1.00000 −0.158114
\(41\) −8.94427 −1.39686 −0.698430 0.715678i \(-0.746118\pi\)
−0.698430 + 0.715678i \(0.746118\pi\)
\(42\) −3.23607 −0.499336
\(43\) −3.23607 −0.493496 −0.246748 0.969080i \(-0.579362\pi\)
−0.246748 + 0.969080i \(0.579362\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) −5.23607 −0.772016
\(47\) 2.47214 0.360598 0.180299 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.47214 0.496019
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 1.23607 0.171412
\(53\) 3.23607 0.444508 0.222254 0.974989i \(-0.428659\pi\)
0.222254 + 0.974989i \(0.428659\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 3.23607 0.432438
\(57\) 7.70820 1.02098
\(58\) −8.47214 −1.11245
\(59\) −3.23607 −0.421300 −0.210650 0.977562i \(-0.567558\pi\)
−0.210650 + 0.977562i \(0.567558\pi\)
\(60\) 1.00000 0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) 3.23607 0.407706
\(64\) 1.00000 0.125000
\(65\) −1.23607 −0.153315
\(66\) −1.00000 −0.123091
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) −1.00000 −0.121268
\(69\) 5.23607 0.630349
\(70\) −3.23607 −0.386784
\(71\) −0.763932 −0.0906621 −0.0453310 0.998972i \(-0.514434\pi\)
−0.0453310 + 0.998972i \(0.514434\pi\)
\(72\) 1.00000 0.117851
\(73\) 1.23607 0.144671 0.0723354 0.997380i \(-0.476955\pi\)
0.0723354 + 0.997380i \(0.476955\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) −7.70820 −0.884192
\(77\) 3.23607 0.368784
\(78\) −1.23607 −0.139957
\(79\) 3.70820 0.417206 0.208603 0.978000i \(-0.433108\pi\)
0.208603 + 0.978000i \(0.433108\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −8.94427 −0.987730
\(83\) 4.94427 0.542704 0.271352 0.962480i \(-0.412529\pi\)
0.271352 + 0.962480i \(0.412529\pi\)
\(84\) −3.23607 −0.353084
\(85\) 1.00000 0.108465
\(86\) −3.23607 −0.348954
\(87\) 8.47214 0.908308
\(88\) 1.00000 0.106600
\(89\) −6.94427 −0.736091 −0.368046 0.929808i \(-0.619973\pi\)
−0.368046 + 0.929808i \(0.619973\pi\)
\(90\) −1.00000 −0.105409
\(91\) 4.00000 0.419314
\(92\) −5.23607 −0.545898
\(93\) 4.00000 0.414781
\(94\) 2.47214 0.254981
\(95\) 7.70820 0.790845
\(96\) −1.00000 −0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 3.47214 0.350739
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 8.94427 0.889988 0.444994 0.895533i \(-0.353206\pi\)
0.444994 + 0.895533i \(0.353206\pi\)
\(102\) 1.00000 0.0990148
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 1.23607 0.121206
\(105\) 3.23607 0.315808
\(106\) 3.23607 0.314315
\(107\) 0.944272 0.0912862 0.0456431 0.998958i \(-0.485466\pi\)
0.0456431 + 0.998958i \(0.485466\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 6.00000 0.569495
\(112\) 3.23607 0.305780
\(113\) −15.2361 −1.43329 −0.716644 0.697439i \(-0.754323\pi\)
−0.716644 + 0.697439i \(0.754323\pi\)
\(114\) 7.70820 0.721939
\(115\) 5.23607 0.488266
\(116\) −8.47214 −0.786618
\(117\) 1.23607 0.114275
\(118\) −3.23607 −0.297904
\(119\) −3.23607 −0.296650
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) 6.00000 0.543214
\(123\) 8.94427 0.806478
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 3.23607 0.288292
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.23607 0.284920
\(130\) −1.23607 −0.108410
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −24.9443 −2.16294
\(134\) 6.00000 0.518321
\(135\) 1.00000 0.0860663
\(136\) −1.00000 −0.0857493
\(137\) −15.8885 −1.35745 −0.678725 0.734393i \(-0.737467\pi\)
−0.678725 + 0.734393i \(0.737467\pi\)
\(138\) 5.23607 0.445724
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −3.23607 −0.273498
\(141\) −2.47214 −0.208191
\(142\) −0.763932 −0.0641078
\(143\) 1.23607 0.103365
\(144\) 1.00000 0.0833333
\(145\) 8.47214 0.703573
\(146\) 1.23607 0.102298
\(147\) −3.47214 −0.286377
\(148\) −6.00000 −0.493197
\(149\) −9.52786 −0.780553 −0.390277 0.920698i \(-0.627621\pi\)
−0.390277 + 0.920698i \(0.627621\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −14.4721 −1.17773 −0.588863 0.808233i \(-0.700424\pi\)
−0.588863 + 0.808233i \(0.700424\pi\)
\(152\) −7.70820 −0.625218
\(153\) −1.00000 −0.0808452
\(154\) 3.23607 0.260770
\(155\) 4.00000 0.321288
\(156\) −1.23607 −0.0989646
\(157\) 19.4164 1.54960 0.774799 0.632208i \(-0.217851\pi\)
0.774799 + 0.632208i \(0.217851\pi\)
\(158\) 3.70820 0.295009
\(159\) −3.23607 −0.256637
\(160\) −1.00000 −0.0790569
\(161\) −16.9443 −1.33540
\(162\) 1.00000 0.0785674
\(163\) −6.47214 −0.506937 −0.253468 0.967344i \(-0.581571\pi\)
−0.253468 + 0.967344i \(0.581571\pi\)
\(164\) −8.94427 −0.698430
\(165\) 1.00000 0.0778499
\(166\) 4.94427 0.383750
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −3.23607 −0.249668
\(169\) −11.4721 −0.882472
\(170\) 1.00000 0.0766965
\(171\) −7.70820 −0.589461
\(172\) −3.23607 −0.246748
\(173\) 13.4164 1.02003 0.510015 0.860165i \(-0.329640\pi\)
0.510015 + 0.860165i \(0.329640\pi\)
\(174\) 8.47214 0.642271
\(175\) 3.23607 0.244624
\(176\) 1.00000 0.0753778
\(177\) 3.23607 0.243238
\(178\) −6.94427 −0.520495
\(179\) −12.7639 −0.954021 −0.477011 0.878898i \(-0.658280\pi\)
−0.477011 + 0.878898i \(0.658280\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 8.47214 0.629729 0.314864 0.949137i \(-0.398041\pi\)
0.314864 + 0.949137i \(0.398041\pi\)
\(182\) 4.00000 0.296500
\(183\) −6.00000 −0.443533
\(184\) −5.23607 −0.386008
\(185\) 6.00000 0.441129
\(186\) 4.00000 0.293294
\(187\) −1.00000 −0.0731272
\(188\) 2.47214 0.180299
\(189\) −3.23607 −0.235389
\(190\) 7.70820 0.559212
\(191\) 1.52786 0.110552 0.0552762 0.998471i \(-0.482396\pi\)
0.0552762 + 0.998471i \(0.482396\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 7.70820 0.554849 0.277424 0.960747i \(-0.410519\pi\)
0.277424 + 0.960747i \(0.410519\pi\)
\(194\) 0 0
\(195\) 1.23607 0.0885167
\(196\) 3.47214 0.248010
\(197\) 14.9443 1.06474 0.532368 0.846513i \(-0.321302\pi\)
0.532368 + 0.846513i \(0.321302\pi\)
\(198\) 1.00000 0.0710669
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.00000 −0.423207
\(202\) 8.94427 0.629317
\(203\) −27.4164 −1.92425
\(204\) 1.00000 0.0700140
\(205\) 8.94427 0.624695
\(206\) 4.00000 0.278693
\(207\) −5.23607 −0.363932
\(208\) 1.23607 0.0857059
\(209\) −7.70820 −0.533188
\(210\) 3.23607 0.223310
\(211\) 20.3607 1.40169 0.700844 0.713315i \(-0.252807\pi\)
0.700844 + 0.713315i \(0.252807\pi\)
\(212\) 3.23607 0.222254
\(213\) 0.763932 0.0523438
\(214\) 0.944272 0.0645491
\(215\) 3.23607 0.220698
\(216\) −1.00000 −0.0680414
\(217\) −12.9443 −0.878714
\(218\) −6.00000 −0.406371
\(219\) −1.23607 −0.0835257
\(220\) −1.00000 −0.0674200
\(221\) −1.23607 −0.0831469
\(222\) 6.00000 0.402694
\(223\) −23.4164 −1.56808 −0.784039 0.620711i \(-0.786844\pi\)
−0.784039 + 0.620711i \(0.786844\pi\)
\(224\) 3.23607 0.216219
\(225\) 1.00000 0.0666667
\(226\) −15.2361 −1.01349
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 7.70820 0.510488
\(229\) 29.4164 1.94389 0.971945 0.235206i \(-0.0755766\pi\)
0.971945 + 0.235206i \(0.0755766\pi\)
\(230\) 5.23607 0.345256
\(231\) −3.23607 −0.212918
\(232\) −8.47214 −0.556223
\(233\) −7.52786 −0.493167 −0.246583 0.969122i \(-0.579308\pi\)
−0.246583 + 0.969122i \(0.579308\pi\)
\(234\) 1.23607 0.0808043
\(235\) −2.47214 −0.161264
\(236\) −3.23607 −0.210650
\(237\) −3.70820 −0.240874
\(238\) −3.23607 −0.209763
\(239\) −16.3607 −1.05828 −0.529142 0.848533i \(-0.677486\pi\)
−0.529142 + 0.848533i \(0.677486\pi\)
\(240\) 1.00000 0.0645497
\(241\) 4.18034 0.269279 0.134640 0.990895i \(-0.457012\pi\)
0.134640 + 0.990895i \(0.457012\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) −3.47214 −0.221827
\(246\) 8.94427 0.570266
\(247\) −9.52786 −0.606243
\(248\) −4.00000 −0.254000
\(249\) −4.94427 −0.313331
\(250\) −1.00000 −0.0632456
\(251\) 23.2361 1.46665 0.733324 0.679880i \(-0.237968\pi\)
0.733324 + 0.679880i \(0.237968\pi\)
\(252\) 3.23607 0.203853
\(253\) −5.23607 −0.329189
\(254\) −12.0000 −0.752947
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) −4.47214 −0.278964 −0.139482 0.990225i \(-0.544544\pi\)
−0.139482 + 0.990225i \(0.544544\pi\)
\(258\) 3.23607 0.201469
\(259\) −19.4164 −1.20648
\(260\) −1.23607 −0.0766577
\(261\) −8.47214 −0.524412
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −3.23607 −0.198790
\(266\) −24.9443 −1.52943
\(267\) 6.94427 0.424983
\(268\) 6.00000 0.366508
\(269\) −24.4721 −1.49209 −0.746046 0.665894i \(-0.768050\pi\)
−0.746046 + 0.665894i \(0.768050\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −4.00000 −0.242091
\(274\) −15.8885 −0.959862
\(275\) 1.00000 0.0603023
\(276\) 5.23607 0.315174
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 8.00000 0.479808
\(279\) −4.00000 −0.239474
\(280\) −3.23607 −0.193392
\(281\) −16.4721 −0.982645 −0.491323 0.870978i \(-0.663486\pi\)
−0.491323 + 0.870978i \(0.663486\pi\)
\(282\) −2.47214 −0.147214
\(283\) −3.41641 −0.203084 −0.101542 0.994831i \(-0.532378\pi\)
−0.101542 + 0.994831i \(0.532378\pi\)
\(284\) −0.763932 −0.0453310
\(285\) −7.70820 −0.456595
\(286\) 1.23607 0.0730902
\(287\) −28.9443 −1.70853
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 8.47214 0.497501
\(291\) 0 0
\(292\) 1.23607 0.0723354
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −3.47214 −0.202499
\(295\) 3.23607 0.188411
\(296\) −6.00000 −0.348743
\(297\) −1.00000 −0.0580259
\(298\) −9.52786 −0.551934
\(299\) −6.47214 −0.374293
\(300\) −1.00000 −0.0577350
\(301\) −10.4721 −0.603604
\(302\) −14.4721 −0.832778
\(303\) −8.94427 −0.513835
\(304\) −7.70820 −0.442096
\(305\) −6.00000 −0.343559
\(306\) −1.00000 −0.0571662
\(307\) 10.2918 0.587384 0.293692 0.955900i \(-0.405116\pi\)
0.293692 + 0.955900i \(0.405116\pi\)
\(308\) 3.23607 0.184392
\(309\) −4.00000 −0.227552
\(310\) 4.00000 0.227185
\(311\) 11.2361 0.637139 0.318569 0.947900i \(-0.396798\pi\)
0.318569 + 0.947900i \(0.396798\pi\)
\(312\) −1.23607 −0.0699786
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 19.4164 1.09573
\(315\) −3.23607 −0.182332
\(316\) 3.70820 0.208603
\(317\) −13.0557 −0.733283 −0.366641 0.930362i \(-0.619492\pi\)
−0.366641 + 0.930362i \(0.619492\pi\)
\(318\) −3.23607 −0.181470
\(319\) −8.47214 −0.474349
\(320\) −1.00000 −0.0559017
\(321\) −0.944272 −0.0527041
\(322\) −16.9443 −0.944267
\(323\) 7.70820 0.428896
\(324\) 1.00000 0.0555556
\(325\) 1.23607 0.0685647
\(326\) −6.47214 −0.358458
\(327\) 6.00000 0.331801
\(328\) −8.94427 −0.493865
\(329\) 8.00000 0.441054
\(330\) 1.00000 0.0550482
\(331\) 33.8885 1.86268 0.931341 0.364147i \(-0.118639\pi\)
0.931341 + 0.364147i \(0.118639\pi\)
\(332\) 4.94427 0.271352
\(333\) −6.00000 −0.328798
\(334\) −12.0000 −0.656611
\(335\) −6.00000 −0.327815
\(336\) −3.23607 −0.176542
\(337\) −3.70820 −0.201999 −0.100999 0.994886i \(-0.532204\pi\)
−0.100999 + 0.994886i \(0.532204\pi\)
\(338\) −11.4721 −0.624002
\(339\) 15.2361 0.827510
\(340\) 1.00000 0.0542326
\(341\) −4.00000 −0.216612
\(342\) −7.70820 −0.416812
\(343\) −11.4164 −0.616428
\(344\) −3.23607 −0.174477
\(345\) −5.23607 −0.281900
\(346\) 13.4164 0.721271
\(347\) 12.9443 0.694885 0.347442 0.937701i \(-0.387050\pi\)
0.347442 + 0.937701i \(0.387050\pi\)
\(348\) 8.47214 0.454154
\(349\) 1.70820 0.0914381 0.0457190 0.998954i \(-0.485442\pi\)
0.0457190 + 0.998954i \(0.485442\pi\)
\(350\) 3.23607 0.172975
\(351\) −1.23607 −0.0659764
\(352\) 1.00000 0.0533002
\(353\) −4.47214 −0.238028 −0.119014 0.992893i \(-0.537973\pi\)
−0.119014 + 0.992893i \(0.537973\pi\)
\(354\) 3.23607 0.171995
\(355\) 0.763932 0.0405453
\(356\) −6.94427 −0.368046
\(357\) 3.23607 0.171271
\(358\) −12.7639 −0.674595
\(359\) −7.41641 −0.391423 −0.195712 0.980662i \(-0.562702\pi\)
−0.195712 + 0.980662i \(0.562702\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 40.4164 2.12718
\(362\) 8.47214 0.445286
\(363\) −1.00000 −0.0524864
\(364\) 4.00000 0.209657
\(365\) −1.23607 −0.0646988
\(366\) −6.00000 −0.313625
\(367\) 28.8328 1.50506 0.752530 0.658558i \(-0.228833\pi\)
0.752530 + 0.658558i \(0.228833\pi\)
\(368\) −5.23607 −0.272949
\(369\) −8.94427 −0.465620
\(370\) 6.00000 0.311925
\(371\) 10.4721 0.543686
\(372\) 4.00000 0.207390
\(373\) −15.7082 −0.813340 −0.406670 0.913575i \(-0.633310\pi\)
−0.406670 + 0.913575i \(0.633310\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 1.00000 0.0516398
\(376\) 2.47214 0.127491
\(377\) −10.4721 −0.539342
\(378\) −3.23607 −0.166445
\(379\) −24.9443 −1.28130 −0.640651 0.767833i \(-0.721335\pi\)
−0.640651 + 0.767833i \(0.721335\pi\)
\(380\) 7.70820 0.395423
\(381\) 12.0000 0.614779
\(382\) 1.52786 0.0781723
\(383\) 10.4721 0.535101 0.267551 0.963544i \(-0.413786\pi\)
0.267551 + 0.963544i \(0.413786\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.23607 −0.164925
\(386\) 7.70820 0.392337
\(387\) −3.23607 −0.164499
\(388\) 0 0
\(389\) 25.5967 1.29781 0.648903 0.760871i \(-0.275228\pi\)
0.648903 + 0.760871i \(0.275228\pi\)
\(390\) 1.23607 0.0625907
\(391\) 5.23607 0.264799
\(392\) 3.47214 0.175369
\(393\) 0 0
\(394\) 14.9443 0.752882
\(395\) −3.70820 −0.186580
\(396\) 1.00000 0.0502519
\(397\) −19.8885 −0.998177 −0.499089 0.866551i \(-0.666332\pi\)
−0.499089 + 0.866551i \(0.666332\pi\)
\(398\) −12.0000 −0.601506
\(399\) 24.9443 1.24878
\(400\) 1.00000 0.0500000
\(401\) −20.2918 −1.01332 −0.506662 0.862145i \(-0.669121\pi\)
−0.506662 + 0.862145i \(0.669121\pi\)
\(402\) −6.00000 −0.299253
\(403\) −4.94427 −0.246292
\(404\) 8.94427 0.444994
\(405\) −1.00000 −0.0496904
\(406\) −27.4164 −1.36065
\(407\) −6.00000 −0.297409
\(408\) 1.00000 0.0495074
\(409\) 10.9443 0.541159 0.270580 0.962698i \(-0.412785\pi\)
0.270580 + 0.962698i \(0.412785\pi\)
\(410\) 8.94427 0.441726
\(411\) 15.8885 0.783724
\(412\) 4.00000 0.197066
\(413\) −10.4721 −0.515300
\(414\) −5.23607 −0.257339
\(415\) −4.94427 −0.242705
\(416\) 1.23607 0.0606032
\(417\) −8.00000 −0.391762
\(418\) −7.70820 −0.377021
\(419\) 10.4721 0.511597 0.255799 0.966730i \(-0.417662\pi\)
0.255799 + 0.966730i \(0.417662\pi\)
\(420\) 3.23607 0.157904
\(421\) −6.94427 −0.338443 −0.169222 0.985578i \(-0.554125\pi\)
−0.169222 + 0.985578i \(0.554125\pi\)
\(422\) 20.3607 0.991142
\(423\) 2.47214 0.120199
\(424\) 3.23607 0.157157
\(425\) −1.00000 −0.0485071
\(426\) 0.763932 0.0370126
\(427\) 19.4164 0.939626
\(428\) 0.944272 0.0456431
\(429\) −1.23607 −0.0596779
\(430\) 3.23607 0.156057
\(431\) −5.05573 −0.243526 −0.121763 0.992559i \(-0.538855\pi\)
−0.121763 + 0.992559i \(0.538855\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) −12.9443 −0.621345
\(435\) −8.47214 −0.406208
\(436\) −6.00000 −0.287348
\(437\) 40.3607 1.93071
\(438\) −1.23607 −0.0590616
\(439\) −4.29180 −0.204836 −0.102418 0.994741i \(-0.532658\pi\)
−0.102418 + 0.994741i \(0.532658\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 3.47214 0.165340
\(442\) −1.23607 −0.0587938
\(443\) −3.34752 −0.159046 −0.0795228 0.996833i \(-0.525340\pi\)
−0.0795228 + 0.996833i \(0.525340\pi\)
\(444\) 6.00000 0.284747
\(445\) 6.94427 0.329190
\(446\) −23.4164 −1.10880
\(447\) 9.52786 0.450653
\(448\) 3.23607 0.152890
\(449\) 6.18034 0.291668 0.145834 0.989309i \(-0.453413\pi\)
0.145834 + 0.989309i \(0.453413\pi\)
\(450\) 1.00000 0.0471405
\(451\) −8.94427 −0.421169
\(452\) −15.2361 −0.716644
\(453\) 14.4721 0.679960
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 7.70820 0.360970
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 29.4164 1.37454
\(459\) 1.00000 0.0466760
\(460\) 5.23607 0.244133
\(461\) 7.41641 0.345417 0.172708 0.984973i \(-0.444748\pi\)
0.172708 + 0.984973i \(0.444748\pi\)
\(462\) −3.23607 −0.150556
\(463\) 20.3607 0.946241 0.473121 0.880998i \(-0.343128\pi\)
0.473121 + 0.880998i \(0.343128\pi\)
\(464\) −8.47214 −0.393309
\(465\) −4.00000 −0.185496
\(466\) −7.52786 −0.348722
\(467\) 14.7639 0.683193 0.341597 0.939847i \(-0.389032\pi\)
0.341597 + 0.939847i \(0.389032\pi\)
\(468\) 1.23607 0.0571373
\(469\) 19.4164 0.896566
\(470\) −2.47214 −0.114031
\(471\) −19.4164 −0.894661
\(472\) −3.23607 −0.148952
\(473\) −3.23607 −0.148795
\(474\) −3.70820 −0.170323
\(475\) −7.70820 −0.353677
\(476\) −3.23607 −0.148325
\(477\) 3.23607 0.148169
\(478\) −16.3607 −0.748320
\(479\) −27.5279 −1.25778 −0.628890 0.777494i \(-0.716490\pi\)
−0.628890 + 0.777494i \(0.716490\pi\)
\(480\) 1.00000 0.0456435
\(481\) −7.41641 −0.338159
\(482\) 4.18034 0.190409
\(483\) 16.9443 0.770991
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −20.8328 −0.944025 −0.472012 0.881592i \(-0.656472\pi\)
−0.472012 + 0.881592i \(0.656472\pi\)
\(488\) 6.00000 0.271607
\(489\) 6.47214 0.292680
\(490\) −3.47214 −0.156855
\(491\) 1.05573 0.0476443 0.0238222 0.999716i \(-0.492416\pi\)
0.0238222 + 0.999716i \(0.492416\pi\)
\(492\) 8.94427 0.403239
\(493\) 8.47214 0.381566
\(494\) −9.52786 −0.428679
\(495\) −1.00000 −0.0449467
\(496\) −4.00000 −0.179605
\(497\) −2.47214 −0.110890
\(498\) −4.94427 −0.221558
\(499\) −22.8328 −1.02214 −0.511069 0.859540i \(-0.670750\pi\)
−0.511069 + 0.859540i \(0.670750\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 23.2361 1.03708
\(503\) 18.4721 0.823632 0.411816 0.911267i \(-0.364895\pi\)
0.411816 + 0.911267i \(0.364895\pi\)
\(504\) 3.23607 0.144146
\(505\) −8.94427 −0.398015
\(506\) −5.23607 −0.232772
\(507\) 11.4721 0.509495
\(508\) −12.0000 −0.532414
\(509\) 34.7639 1.54088 0.770442 0.637510i \(-0.220036\pi\)
0.770442 + 0.637510i \(0.220036\pi\)
\(510\) −1.00000 −0.0442807
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 7.70820 0.340326
\(514\) −4.47214 −0.197257
\(515\) −4.00000 −0.176261
\(516\) 3.23607 0.142460
\(517\) 2.47214 0.108724
\(518\) −19.4164 −0.853108
\(519\) −13.4164 −0.588915
\(520\) −1.23607 −0.0542052
\(521\) −35.1246 −1.53884 −0.769419 0.638745i \(-0.779454\pi\)
−0.769419 + 0.638745i \(0.779454\pi\)
\(522\) −8.47214 −0.370815
\(523\) 21.1246 0.923715 0.461857 0.886954i \(-0.347183\pi\)
0.461857 + 0.886954i \(0.347183\pi\)
\(524\) 0 0
\(525\) −3.23607 −0.141234
\(526\) −12.0000 −0.523225
\(527\) 4.00000 0.174243
\(528\) −1.00000 −0.0435194
\(529\) 4.41641 0.192018
\(530\) −3.23607 −0.140566
\(531\) −3.23607 −0.140433
\(532\) −24.9443 −1.08147
\(533\) −11.0557 −0.478877
\(534\) 6.94427 0.300508
\(535\) −0.944272 −0.0408244
\(536\) 6.00000 0.259161
\(537\) 12.7639 0.550804
\(538\) −24.4721 −1.05507
\(539\) 3.47214 0.149555
\(540\) 1.00000 0.0430331
\(541\) −45.7771 −1.96811 −0.984055 0.177862i \(-0.943082\pi\)
−0.984055 + 0.177862i \(0.943082\pi\)
\(542\) 20.0000 0.859074
\(543\) −8.47214 −0.363574
\(544\) −1.00000 −0.0428746
\(545\) 6.00000 0.257012
\(546\) −4.00000 −0.171184
\(547\) 4.58359 0.195980 0.0979901 0.995187i \(-0.468759\pi\)
0.0979901 + 0.995187i \(0.468759\pi\)
\(548\) −15.8885 −0.678725
\(549\) 6.00000 0.256074
\(550\) 1.00000 0.0426401
\(551\) 65.3050 2.78208
\(552\) 5.23607 0.222862
\(553\) 12.0000 0.510292
\(554\) 14.0000 0.594803
\(555\) −6.00000 −0.254686
\(556\) 8.00000 0.339276
\(557\) −20.8328 −0.882715 −0.441357 0.897331i \(-0.645503\pi\)
−0.441357 + 0.897331i \(0.645503\pi\)
\(558\) −4.00000 −0.169334
\(559\) −4.00000 −0.169182
\(560\) −3.23607 −0.136749
\(561\) 1.00000 0.0422200
\(562\) −16.4721 −0.694835
\(563\) −35.7771 −1.50782 −0.753912 0.656975i \(-0.771836\pi\)
−0.753912 + 0.656975i \(0.771836\pi\)
\(564\) −2.47214 −0.104096
\(565\) 15.2361 0.640986
\(566\) −3.41641 −0.143602
\(567\) 3.23607 0.135902
\(568\) −0.763932 −0.0320539
\(569\) 19.3050 0.809306 0.404653 0.914470i \(-0.367392\pi\)
0.404653 + 0.914470i \(0.367392\pi\)
\(570\) −7.70820 −0.322861
\(571\) −25.8885 −1.08340 −0.541701 0.840571i \(-0.682219\pi\)
−0.541701 + 0.840571i \(0.682219\pi\)
\(572\) 1.23607 0.0516826
\(573\) −1.52786 −0.0638274
\(574\) −28.9443 −1.20811
\(575\) −5.23607 −0.218359
\(576\) 1.00000 0.0416667
\(577\) −15.8885 −0.661449 −0.330724 0.943727i \(-0.607293\pi\)
−0.330724 + 0.943727i \(0.607293\pi\)
\(578\) 1.00000 0.0415945
\(579\) −7.70820 −0.320342
\(580\) 8.47214 0.351786
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 3.23607 0.134024
\(584\) 1.23607 0.0511489
\(585\) −1.23607 −0.0511051
\(586\) −18.0000 −0.743573
\(587\) −2.18034 −0.0899923 −0.0449961 0.998987i \(-0.514328\pi\)
−0.0449961 + 0.998987i \(0.514328\pi\)
\(588\) −3.47214 −0.143188
\(589\) 30.8328 1.27044
\(590\) 3.23607 0.133227
\(591\) −14.9443 −0.614725
\(592\) −6.00000 −0.246598
\(593\) 13.4164 0.550946 0.275473 0.961309i \(-0.411166\pi\)
0.275473 + 0.961309i \(0.411166\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 3.23607 0.132666
\(596\) −9.52786 −0.390277
\(597\) 12.0000 0.491127
\(598\) −6.47214 −0.264665
\(599\) 4.58359 0.187280 0.0936402 0.995606i \(-0.470150\pi\)
0.0936402 + 0.995606i \(0.470150\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −1.12461 −0.0458739 −0.0229369 0.999737i \(-0.507302\pi\)
−0.0229369 + 0.999737i \(0.507302\pi\)
\(602\) −10.4721 −0.426812
\(603\) 6.00000 0.244339
\(604\) −14.4721 −0.588863
\(605\) −1.00000 −0.0406558
\(606\) −8.94427 −0.363336
\(607\) 29.1246 1.18213 0.591066 0.806623i \(-0.298707\pi\)
0.591066 + 0.806623i \(0.298707\pi\)
\(608\) −7.70820 −0.312609
\(609\) 27.4164 1.11097
\(610\) −6.00000 −0.242933
\(611\) 3.05573 0.123622
\(612\) −1.00000 −0.0404226
\(613\) 7.12461 0.287760 0.143880 0.989595i \(-0.454042\pi\)
0.143880 + 0.989595i \(0.454042\pi\)
\(614\) 10.2918 0.415343
\(615\) −8.94427 −0.360668
\(616\) 3.23607 0.130385
\(617\) −0.763932 −0.0307547 −0.0153774 0.999882i \(-0.504895\pi\)
−0.0153774 + 0.999882i \(0.504895\pi\)
\(618\) −4.00000 −0.160904
\(619\) −9.88854 −0.397454 −0.198727 0.980055i \(-0.563681\pi\)
−0.198727 + 0.980055i \(0.563681\pi\)
\(620\) 4.00000 0.160644
\(621\) 5.23607 0.210116
\(622\) 11.2361 0.450525
\(623\) −22.4721 −0.900327
\(624\) −1.23607 −0.0494823
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.70820 0.307836
\(628\) 19.4164 0.774799
\(629\) 6.00000 0.239236
\(630\) −3.23607 −0.128928
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 3.70820 0.147504
\(633\) −20.3607 −0.809264
\(634\) −13.0557 −0.518509
\(635\) 12.0000 0.476205
\(636\) −3.23607 −0.128318
\(637\) 4.29180 0.170047
\(638\) −8.47214 −0.335415
\(639\) −0.763932 −0.0302207
\(640\) −1.00000 −0.0395285
\(641\) 8.65248 0.341752 0.170876 0.985293i \(-0.445340\pi\)
0.170876 + 0.985293i \(0.445340\pi\)
\(642\) −0.944272 −0.0372674
\(643\) −36.9443 −1.45694 −0.728470 0.685078i \(-0.759768\pi\)
−0.728470 + 0.685078i \(0.759768\pi\)
\(644\) −16.9443 −0.667698
\(645\) −3.23607 −0.127420
\(646\) 7.70820 0.303275
\(647\) −14.8328 −0.583138 −0.291569 0.956550i \(-0.594177\pi\)
−0.291569 + 0.956550i \(0.594177\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.23607 −0.127027
\(650\) 1.23607 0.0484826
\(651\) 12.9443 0.507326
\(652\) −6.47214 −0.253468
\(653\) 11.8885 0.465235 0.232617 0.972568i \(-0.425271\pi\)
0.232617 + 0.972568i \(0.425271\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) −8.94427 −0.349215
\(657\) 1.23607 0.0482236
\(658\) 8.00000 0.311872
\(659\) 44.8328 1.74644 0.873219 0.487328i \(-0.162028\pi\)
0.873219 + 0.487328i \(0.162028\pi\)
\(660\) 1.00000 0.0389249
\(661\) 1.41641 0.0550919 0.0275459 0.999621i \(-0.491231\pi\)
0.0275459 + 0.999621i \(0.491231\pi\)
\(662\) 33.8885 1.31712
\(663\) 1.23607 0.0480049
\(664\) 4.94427 0.191875
\(665\) 24.9443 0.967297
\(666\) −6.00000 −0.232495
\(667\) 44.3607 1.71765
\(668\) −12.0000 −0.464294
\(669\) 23.4164 0.905331
\(670\) −6.00000 −0.231800
\(671\) 6.00000 0.231627
\(672\) −3.23607 −0.124834
\(673\) −45.2361 −1.74372 −0.871861 0.489753i \(-0.837087\pi\)
−0.871861 + 0.489753i \(0.837087\pi\)
\(674\) −3.70820 −0.142835
\(675\) −1.00000 −0.0384900
\(676\) −11.4721 −0.441236
\(677\) −39.8885 −1.53304 −0.766521 0.642220i \(-0.778014\pi\)
−0.766521 + 0.642220i \(0.778014\pi\)
\(678\) 15.2361 0.585138
\(679\) 0 0
\(680\) 1.00000 0.0383482
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −23.7771 −0.909805 −0.454902 0.890541i \(-0.650326\pi\)
−0.454902 + 0.890541i \(0.650326\pi\)
\(684\) −7.70820 −0.294731
\(685\) 15.8885 0.607070
\(686\) −11.4164 −0.435880
\(687\) −29.4164 −1.12231
\(688\) −3.23607 −0.123374
\(689\) 4.00000 0.152388
\(690\) −5.23607 −0.199334
\(691\) 42.8328 1.62944 0.814719 0.579857i \(-0.196891\pi\)
0.814719 + 0.579857i \(0.196891\pi\)
\(692\) 13.4164 0.510015
\(693\) 3.23607 0.122928
\(694\) 12.9443 0.491358
\(695\) −8.00000 −0.303457
\(696\) 8.47214 0.321135
\(697\) 8.94427 0.338788
\(698\) 1.70820 0.0646565
\(699\) 7.52786 0.284730
\(700\) 3.23607 0.122312
\(701\) 26.4721 0.999839 0.499919 0.866072i \(-0.333363\pi\)
0.499919 + 0.866072i \(0.333363\pi\)
\(702\) −1.23607 −0.0466524
\(703\) 46.2492 1.74432
\(704\) 1.00000 0.0376889
\(705\) 2.47214 0.0931060
\(706\) −4.47214 −0.168311
\(707\) 28.9443 1.08856
\(708\) 3.23607 0.121619
\(709\) −24.8328 −0.932616 −0.466308 0.884622i \(-0.654416\pi\)
−0.466308 + 0.884622i \(0.654416\pi\)
\(710\) 0.763932 0.0286699
\(711\) 3.70820 0.139069
\(712\) −6.94427 −0.260248
\(713\) 20.9443 0.784369
\(714\) 3.23607 0.121107
\(715\) −1.23607 −0.0462263
\(716\) −12.7639 −0.477011
\(717\) 16.3607 0.611001
\(718\) −7.41641 −0.276778
\(719\) 48.5410 1.81027 0.905137 0.425119i \(-0.139768\pi\)
0.905137 + 0.425119i \(0.139768\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 12.9443 0.482070
\(722\) 40.4164 1.50414
\(723\) −4.18034 −0.155469
\(724\) 8.47214 0.314864
\(725\) −8.47214 −0.314647
\(726\) −1.00000 −0.0371135
\(727\) −18.8328 −0.698470 −0.349235 0.937035i \(-0.613559\pi\)
−0.349235 + 0.937035i \(0.613559\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) −1.23607 −0.0457489
\(731\) 3.23607 0.119690
\(732\) −6.00000 −0.221766
\(733\) 3.34752 0.123644 0.0618218 0.998087i \(-0.480309\pi\)
0.0618218 + 0.998087i \(0.480309\pi\)
\(734\) 28.8328 1.06424
\(735\) 3.47214 0.128072
\(736\) −5.23607 −0.193004
\(737\) 6.00000 0.221013
\(738\) −8.94427 −0.329243
\(739\) 28.2918 1.04073 0.520365 0.853944i \(-0.325796\pi\)
0.520365 + 0.853944i \(0.325796\pi\)
\(740\) 6.00000 0.220564
\(741\) 9.52786 0.350015
\(742\) 10.4721 0.384444
\(743\) −22.4721 −0.824423 −0.412211 0.911088i \(-0.635243\pi\)
−0.412211 + 0.911088i \(0.635243\pi\)
\(744\) 4.00000 0.146647
\(745\) 9.52786 0.349074
\(746\) −15.7082 −0.575118
\(747\) 4.94427 0.180901
\(748\) −1.00000 −0.0365636
\(749\) 3.05573 0.111654
\(750\) 1.00000 0.0365148
\(751\) 7.41641 0.270629 0.135314 0.990803i \(-0.456796\pi\)
0.135314 + 0.990803i \(0.456796\pi\)
\(752\) 2.47214 0.0901495
\(753\) −23.2361 −0.846769
\(754\) −10.4721 −0.381373
\(755\) 14.4721 0.526695
\(756\) −3.23607 −0.117695
\(757\) 21.8885 0.795553 0.397776 0.917482i \(-0.369782\pi\)
0.397776 + 0.917482i \(0.369782\pi\)
\(758\) −24.9443 −0.906017
\(759\) 5.23607 0.190057
\(760\) 7.70820 0.279606
\(761\) −36.4721 −1.32211 −0.661057 0.750336i \(-0.729892\pi\)
−0.661057 + 0.750336i \(0.729892\pi\)
\(762\) 12.0000 0.434714
\(763\) −19.4164 −0.702921
\(764\) 1.52786 0.0552762
\(765\) 1.00000 0.0361551
\(766\) 10.4721 0.378374
\(767\) −4.00000 −0.144432
\(768\) −1.00000 −0.0360844
\(769\) 24.4721 0.882488 0.441244 0.897387i \(-0.354537\pi\)
0.441244 + 0.897387i \(0.354537\pi\)
\(770\) −3.23607 −0.116620
\(771\) 4.47214 0.161060
\(772\) 7.70820 0.277424
\(773\) −31.2361 −1.12348 −0.561742 0.827313i \(-0.689868\pi\)
−0.561742 + 0.827313i \(0.689868\pi\)
\(774\) −3.23607 −0.116318
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 19.4164 0.696560
\(778\) 25.5967 0.917688
\(779\) 68.9443 2.47018
\(780\) 1.23607 0.0442583
\(781\) −0.763932 −0.0273356
\(782\) 5.23607 0.187241
\(783\) 8.47214 0.302769
\(784\) 3.47214 0.124005
\(785\) −19.4164 −0.693001
\(786\) 0 0
\(787\) 19.0557 0.679263 0.339632 0.940559i \(-0.389698\pi\)
0.339632 + 0.940559i \(0.389698\pi\)
\(788\) 14.9443 0.532368
\(789\) 12.0000 0.427211
\(790\) −3.70820 −0.131932
\(791\) −49.3050 −1.75308
\(792\) 1.00000 0.0355335
\(793\) 7.41641 0.263364
\(794\) −19.8885 −0.705818
\(795\) 3.23607 0.114772
\(796\) −12.0000 −0.425329
\(797\) 35.0132 1.24023 0.620115 0.784511i \(-0.287086\pi\)
0.620115 + 0.784511i \(0.287086\pi\)
\(798\) 24.9443 0.883018
\(799\) −2.47214 −0.0874579
\(800\) 1.00000 0.0353553
\(801\) −6.94427 −0.245364
\(802\) −20.2918 −0.716528
\(803\) 1.23607 0.0436199
\(804\) −6.00000 −0.211604
\(805\) 16.9443 0.597207
\(806\) −4.94427 −0.174155
\(807\) 24.4721 0.861460
\(808\) 8.94427 0.314658
\(809\) −16.3607 −0.575211 −0.287605 0.957749i \(-0.592859\pi\)
−0.287605 + 0.957749i \(0.592859\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −30.2492 −1.06219 −0.531097 0.847311i \(-0.678220\pi\)
−0.531097 + 0.847311i \(0.678220\pi\)
\(812\) −27.4164 −0.962127
\(813\) −20.0000 −0.701431
\(814\) −6.00000 −0.210300
\(815\) 6.47214 0.226709
\(816\) 1.00000 0.0350070
\(817\) 24.9443 0.872690
\(818\) 10.9443 0.382657
\(819\) 4.00000 0.139771
\(820\) 8.94427 0.312348
\(821\) −10.3607 −0.361590 −0.180795 0.983521i \(-0.557867\pi\)
−0.180795 + 0.983521i \(0.557867\pi\)
\(822\) 15.8885 0.554177
\(823\) 37.4164 1.30425 0.652127 0.758110i \(-0.273877\pi\)
0.652127 + 0.758110i \(0.273877\pi\)
\(824\) 4.00000 0.139347
\(825\) −1.00000 −0.0348155
\(826\) −10.4721 −0.364372
\(827\) −32.9443 −1.14558 −0.572792 0.819701i \(-0.694140\pi\)
−0.572792 + 0.819701i \(0.694140\pi\)
\(828\) −5.23607 −0.181966
\(829\) −49.7771 −1.72883 −0.864415 0.502779i \(-0.832311\pi\)
−0.864415 + 0.502779i \(0.832311\pi\)
\(830\) −4.94427 −0.171618
\(831\) −14.0000 −0.485655
\(832\) 1.23607 0.0428529
\(833\) −3.47214 −0.120302
\(834\) −8.00000 −0.277017
\(835\) 12.0000 0.415277
\(836\) −7.70820 −0.266594
\(837\) 4.00000 0.138260
\(838\) 10.4721 0.361754
\(839\) 14.2918 0.493408 0.246704 0.969091i \(-0.420653\pi\)
0.246704 + 0.969091i \(0.420653\pi\)
\(840\) 3.23607 0.111655
\(841\) 42.7771 1.47507
\(842\) −6.94427 −0.239315
\(843\) 16.4721 0.567330
\(844\) 20.3607 0.700844
\(845\) 11.4721 0.394653
\(846\) 2.47214 0.0849938
\(847\) 3.23607 0.111193
\(848\) 3.23607 0.111127
\(849\) 3.41641 0.117251
\(850\) −1.00000 −0.0342997
\(851\) 31.4164 1.07694
\(852\) 0.763932 0.0261719
\(853\) −14.3607 −0.491700 −0.245850 0.969308i \(-0.579067\pi\)
−0.245850 + 0.969308i \(0.579067\pi\)
\(854\) 19.4164 0.664416
\(855\) 7.70820 0.263615
\(856\) 0.944272 0.0322745
\(857\) −19.8885 −0.679380 −0.339690 0.940538i \(-0.610322\pi\)
−0.339690 + 0.940538i \(0.610322\pi\)
\(858\) −1.23607 −0.0421987
\(859\) 12.9443 0.441653 0.220826 0.975313i \(-0.429125\pi\)
0.220826 + 0.975313i \(0.429125\pi\)
\(860\) 3.23607 0.110349
\(861\) 28.9443 0.986418
\(862\) −5.05573 −0.172199
\(863\) −52.9443 −1.80224 −0.901122 0.433566i \(-0.857255\pi\)
−0.901122 + 0.433566i \(0.857255\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −13.4164 −0.456172
\(866\) −6.00000 −0.203888
\(867\) −1.00000 −0.0339618
\(868\) −12.9443 −0.439357
\(869\) 3.70820 0.125792
\(870\) −8.47214 −0.287232
\(871\) 7.41641 0.251295
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) 40.3607 1.36522
\(875\) −3.23607 −0.109399
\(876\) −1.23607 −0.0417629
\(877\) 42.3607 1.43042 0.715209 0.698910i \(-0.246331\pi\)
0.715209 + 0.698910i \(0.246331\pi\)
\(878\) −4.29180 −0.144841
\(879\) 18.0000 0.607125
\(880\) −1.00000 −0.0337100
\(881\) 14.1803 0.477748 0.238874 0.971051i \(-0.423222\pi\)
0.238874 + 0.971051i \(0.423222\pi\)
\(882\) 3.47214 0.116913
\(883\) 33.4164 1.12455 0.562276 0.826950i \(-0.309926\pi\)
0.562276 + 0.826950i \(0.309926\pi\)
\(884\) −1.23607 −0.0415735
\(885\) −3.23607 −0.108779
\(886\) −3.34752 −0.112462
\(887\) −56.7214 −1.90452 −0.952258 0.305293i \(-0.901246\pi\)
−0.952258 + 0.305293i \(0.901246\pi\)
\(888\) 6.00000 0.201347
\(889\) −38.8328 −1.30241
\(890\) 6.94427 0.232773
\(891\) 1.00000 0.0335013
\(892\) −23.4164 −0.784039
\(893\) −19.0557 −0.637676
\(894\) 9.52786 0.318659
\(895\) 12.7639 0.426651
\(896\) 3.23607 0.108109
\(897\) 6.47214 0.216098
\(898\) 6.18034 0.206241
\(899\) 33.8885 1.13025
\(900\) 1.00000 0.0333333
\(901\) −3.23607 −0.107809
\(902\) −8.94427 −0.297812
\(903\) 10.4721 0.348491
\(904\) −15.2361 −0.506744
\(905\) −8.47214 −0.281623
\(906\) 14.4721 0.480805
\(907\) −38.8328 −1.28942 −0.644711 0.764426i \(-0.723022\pi\)
−0.644711 + 0.764426i \(0.723022\pi\)
\(908\) 0 0
\(909\) 8.94427 0.296663
\(910\) −4.00000 −0.132599
\(911\) 31.2361 1.03490 0.517449 0.855714i \(-0.326882\pi\)
0.517449 + 0.855714i \(0.326882\pi\)
\(912\) 7.70820 0.255244
\(913\) 4.94427 0.163632
\(914\) 18.0000 0.595387
\(915\) 6.00000 0.198354
\(916\) 29.4164 0.971945
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −6.47214 −0.213496 −0.106748 0.994286i \(-0.534044\pi\)
−0.106748 + 0.994286i \(0.534044\pi\)
\(920\) 5.23607 0.172628
\(921\) −10.2918 −0.339126
\(922\) 7.41641 0.244246
\(923\) −0.944272 −0.0310811
\(924\) −3.23607 −0.106459
\(925\) −6.00000 −0.197279
\(926\) 20.3607 0.669093
\(927\) 4.00000 0.131377
\(928\) −8.47214 −0.278111
\(929\) 33.0132 1.08313 0.541563 0.840660i \(-0.317833\pi\)
0.541563 + 0.840660i \(0.317833\pi\)
\(930\) −4.00000 −0.131165
\(931\) −26.7639 −0.877152
\(932\) −7.52786 −0.246583
\(933\) −11.2361 −0.367852
\(934\) 14.7639 0.483091
\(935\) 1.00000 0.0327035
\(936\) 1.23607 0.0404021
\(937\) −60.2492 −1.96826 −0.984128 0.177459i \(-0.943212\pi\)
−0.984128 + 0.177459i \(0.943212\pi\)
\(938\) 19.4164 0.633968
\(939\) 0 0
\(940\) −2.47214 −0.0806322
\(941\) 12.4721 0.406580 0.203290 0.979119i \(-0.434837\pi\)
0.203290 + 0.979119i \(0.434837\pi\)
\(942\) −19.4164 −0.632621
\(943\) 46.8328 1.52509
\(944\) −3.23607 −0.105325
\(945\) 3.23607 0.105269
\(946\) −3.23607 −0.105214
\(947\) 44.7214 1.45325 0.726624 0.687035i \(-0.241088\pi\)
0.726624 + 0.687035i \(0.241088\pi\)
\(948\) −3.70820 −0.120437
\(949\) 1.52786 0.0495966
\(950\) −7.70820 −0.250087
\(951\) 13.0557 0.423361
\(952\) −3.23607 −0.104882
\(953\) 43.3050 1.40278 0.701392 0.712775i \(-0.252562\pi\)
0.701392 + 0.712775i \(0.252562\pi\)
\(954\) 3.23607 0.104772
\(955\) −1.52786 −0.0494405
\(956\) −16.3607 −0.529142
\(957\) 8.47214 0.273865
\(958\) −27.5279 −0.889385
\(959\) −51.4164 −1.66032
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) −7.41641 −0.239115
\(963\) 0.944272 0.0304287
\(964\) 4.18034 0.134640
\(965\) −7.70820 −0.248136
\(966\) 16.9443 0.545173
\(967\) −7.41641 −0.238496 −0.119248 0.992865i \(-0.538048\pi\)
−0.119248 + 0.992865i \(0.538048\pi\)
\(968\) 1.00000 0.0321412
\(969\) −7.70820 −0.247623
\(970\) 0 0
\(971\) 48.7639 1.56491 0.782455 0.622708i \(-0.213967\pi\)
0.782455 + 0.622708i \(0.213967\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 25.8885 0.829949
\(974\) −20.8328 −0.667526
\(975\) −1.23607 −0.0395859
\(976\) 6.00000 0.192055
\(977\) −7.52786 −0.240838 −0.120419 0.992723i \(-0.538424\pi\)
−0.120419 + 0.992723i \(0.538424\pi\)
\(978\) 6.47214 0.206956
\(979\) −6.94427 −0.221940
\(980\) −3.47214 −0.110913
\(981\) −6.00000 −0.191565
\(982\) 1.05573 0.0336896
\(983\) 0.652476 0.0208107 0.0104054 0.999946i \(-0.496688\pi\)
0.0104054 + 0.999946i \(0.496688\pi\)
\(984\) 8.94427 0.285133
\(985\) −14.9443 −0.476164
\(986\) 8.47214 0.269808
\(987\) −8.00000 −0.254643
\(988\) −9.52786 −0.303122
\(989\) 16.9443 0.538797
\(990\) −1.00000 −0.0317821
\(991\) 34.2492 1.08796 0.543981 0.839097i \(-0.316916\pi\)
0.543981 + 0.839097i \(0.316916\pi\)
\(992\) −4.00000 −0.127000
\(993\) −33.8885 −1.07542
\(994\) −2.47214 −0.0784114
\(995\) 12.0000 0.380426
\(996\) −4.94427 −0.156665
\(997\) 5.63932 0.178599 0.0892995 0.996005i \(-0.471537\pi\)
0.0892995 + 0.996005i \(0.471537\pi\)
\(998\) −22.8328 −0.722760
\(999\) 6.00000 0.189832
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.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bu.1.2 2 1.1 even 1 trivial