Properties

Label 2695.2.a.j.1.2
Level $2695$
Weight $2$
Character 2695.1
Self dual yes
Analytic conductor $21.520$
Analytic rank $1$
Dimension $4$
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] = [2695,2,Mod(1,2695)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, 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("2695.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2695.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: \(21.5196833447\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.693822\) of defining polynomial
Character \(\chi\) \(=\) 2695.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.74747 q^{2} -2.44129 q^{3} +1.05365 q^{4} +1.00000 q^{5} +4.26608 q^{6} +1.65372 q^{8} +2.95990 q^{9} +O(q^{10})\) \(q-1.74747 q^{2} -2.44129 q^{3} +1.05365 q^{4} +1.00000 q^{5} +4.26608 q^{6} +1.65372 q^{8} +2.95990 q^{9} -1.74747 q^{10} +1.00000 q^{11} -2.57226 q^{12} +1.88258 q^{13} -2.44129 q^{15} -4.99712 q^{16} -2.57226 q^{17} -5.17233 q^{18} +3.06719 q^{19} +1.05365 q^{20} -1.74747 q^{22} -5.83834 q^{23} -4.03722 q^{24} +1.00000 q^{25} -3.28975 q^{26} +0.0978926 q^{27} -2.56286 q^{29} +4.26608 q^{30} +2.84846 q^{31} +5.42487 q^{32} -2.44129 q^{33} +4.49494 q^{34} +3.11869 q^{36} -4.98357 q^{37} -5.35982 q^{38} -4.59593 q^{39} +1.65372 q^{40} +1.57928 q^{41} -10.7818 q^{43} +1.05365 q^{44} +2.95990 q^{45} +10.2023 q^{46} -4.87843 q^{47} +12.1994 q^{48} -1.74747 q^{50} +6.27963 q^{51} +1.98357 q^{52} -1.90499 q^{53} -0.171064 q^{54} +1.00000 q^{55} -7.48791 q^{57} +4.47851 q^{58} +8.36812 q^{59} -2.57226 q^{60} -6.89613 q^{61} -4.97760 q^{62} +0.514463 q^{64} +1.88258 q^{65} +4.26608 q^{66} +14.6142 q^{67} -2.71025 q^{68} +14.2531 q^{69} +4.51573 q^{71} +4.89486 q^{72} +0.100993 q^{73} +8.70864 q^{74} -2.44129 q^{75} +3.23174 q^{76} +8.03124 q^{78} +9.19528 q^{79} -4.99712 q^{80} -9.11869 q^{81} -2.75975 q^{82} +7.62375 q^{83} -2.57226 q^{85} +18.8409 q^{86} +6.25668 q^{87} +1.65372 q^{88} +14.1911 q^{89} -5.17233 q^{90} -6.15154 q^{92} -6.95392 q^{93} +8.52491 q^{94} +3.06719 q^{95} -13.2437 q^{96} +1.79824 q^{97} +2.95990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 4 q^{5} + 7 q^{6} - 9 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 4 q^{5} + 7 q^{6} - 9 q^{8} - q^{9} - 3 q^{10} + 4 q^{11} - 3 q^{12} - 6 q^{13} - 3 q^{15} + 5 q^{16} - 3 q^{17} + q^{18} - 3 q^{19} + 3 q^{20} - 3 q^{22} - 6 q^{23} - 4 q^{24} + 4 q^{25} - 5 q^{26} + 3 q^{27} - 8 q^{29} + 7 q^{30} + 10 q^{31} + 4 q^{32} - 3 q^{33} + 10 q^{34} - 8 q^{36} - 9 q^{37} - 23 q^{38} - 13 q^{39} - 9 q^{40} + 15 q^{41} - 2 q^{43} + 3 q^{44} - q^{45} + 16 q^{46} - 15 q^{47} - q^{48} - 3 q^{50} + q^{51} - 3 q^{52} - 30 q^{53} - 13 q^{54} + 4 q^{55} - 6 q^{57} - q^{58} + 17 q^{59} - 3 q^{60} + 16 q^{62} + 5 q^{64} - 6 q^{65} + 7 q^{66} - 25 q^{67} - 19 q^{68} + 6 q^{69} - 13 q^{71} + 26 q^{72} + 3 q^{73} - 16 q^{74} - 3 q^{75} + 40 q^{76} - 5 q^{78} - 4 q^{79} + 5 q^{80} - 16 q^{81} - 27 q^{82} + 18 q^{83} - 3 q^{85} - 10 q^{86} + 20 q^{87} - 9 q^{88} + 25 q^{89} + q^{90} - 26 q^{92} + 10 q^{93} + 23 q^{94} - 3 q^{95} - 7 q^{96} - 23 q^{97} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.74747 −1.23565 −0.617823 0.786317i \(-0.711985\pi\)
−0.617823 + 0.786317i \(0.711985\pi\)
\(3\) −2.44129 −1.40948 −0.704740 0.709466i \(-0.748936\pi\)
−0.704740 + 0.709466i \(0.748936\pi\)
\(4\) 1.05365 0.526823
\(5\) 1.00000 0.447214
\(6\) 4.26608 1.74162
\(7\) 0 0
\(8\) 1.65372 0.584680
\(9\) 2.95990 0.986634
\(10\) −1.74747 −0.552598
\(11\) 1.00000 0.301511
\(12\) −2.57226 −0.742546
\(13\) 1.88258 0.522134 0.261067 0.965321i \(-0.415926\pi\)
0.261067 + 0.965321i \(0.415926\pi\)
\(14\) 0 0
\(15\) −2.44129 −0.630339
\(16\) −4.99712 −1.24928
\(17\) −2.57226 −0.623864 −0.311932 0.950104i \(-0.600976\pi\)
−0.311932 + 0.950104i \(0.600976\pi\)
\(18\) −5.17233 −1.21913
\(19\) 3.06719 0.703662 0.351831 0.936063i \(-0.385559\pi\)
0.351831 + 0.936063i \(0.385559\pi\)
\(20\) 1.05365 0.235602
\(21\) 0 0
\(22\) −1.74747 −0.372562
\(23\) −5.83834 −1.21738 −0.608689 0.793409i \(-0.708304\pi\)
−0.608689 + 0.793409i \(0.708304\pi\)
\(24\) −4.03722 −0.824094
\(25\) 1.00000 0.200000
\(26\) −3.28975 −0.645174
\(27\) 0.0978926 0.0188394
\(28\) 0 0
\(29\) −2.56286 −0.475911 −0.237955 0.971276i \(-0.576477\pi\)
−0.237955 + 0.971276i \(0.576477\pi\)
\(30\) 4.26608 0.778876
\(31\) 2.84846 0.511599 0.255799 0.966730i \(-0.417661\pi\)
0.255799 + 0.966730i \(0.417661\pi\)
\(32\) 5.42487 0.958990
\(33\) −2.44129 −0.424974
\(34\) 4.49494 0.770875
\(35\) 0 0
\(36\) 3.11869 0.519781
\(37\) −4.98357 −0.819295 −0.409647 0.912244i \(-0.634348\pi\)
−0.409647 + 0.912244i \(0.634348\pi\)
\(38\) −5.35982 −0.869478
\(39\) −4.59593 −0.735938
\(40\) 1.65372 0.261477
\(41\) 1.57928 0.246642 0.123321 0.992367i \(-0.460645\pi\)
0.123321 + 0.992367i \(0.460645\pi\)
\(42\) 0 0
\(43\) −10.7818 −1.64421 −0.822105 0.569335i \(-0.807201\pi\)
−0.822105 + 0.569335i \(0.807201\pi\)
\(44\) 1.05365 0.158843
\(45\) 2.95990 0.441236
\(46\) 10.2023 1.50425
\(47\) −4.87843 −0.711593 −0.355796 0.934564i \(-0.615790\pi\)
−0.355796 + 0.934564i \(0.615790\pi\)
\(48\) 12.1994 1.76084
\(49\) 0 0
\(50\) −1.74747 −0.247129
\(51\) 6.27963 0.879324
\(52\) 1.98357 0.275072
\(53\) −1.90499 −0.261670 −0.130835 0.991404i \(-0.541766\pi\)
−0.130835 + 0.991404i \(0.541766\pi\)
\(54\) −0.171064 −0.0232789
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −7.48791 −0.991798
\(58\) 4.47851 0.588057
\(59\) 8.36812 1.08944 0.544718 0.838619i \(-0.316637\pi\)
0.544718 + 0.838619i \(0.316637\pi\)
\(60\) −2.57226 −0.332077
\(61\) −6.89613 −0.882959 −0.441479 0.897271i \(-0.645546\pi\)
−0.441479 + 0.897271i \(0.645546\pi\)
\(62\) −4.97760 −0.632155
\(63\) 0 0
\(64\) 0.514463 0.0643078
\(65\) 1.88258 0.233506
\(66\) 4.26608 0.525118
\(67\) 14.6142 1.78540 0.892702 0.450647i \(-0.148807\pi\)
0.892702 + 0.450647i \(0.148807\pi\)
\(68\) −2.71025 −0.328666
\(69\) 14.2531 1.71587
\(70\) 0 0
\(71\) 4.51573 0.535919 0.267959 0.963430i \(-0.413651\pi\)
0.267959 + 0.963430i \(0.413651\pi\)
\(72\) 4.89486 0.576865
\(73\) 0.100993 0.0118203 0.00591017 0.999983i \(-0.498119\pi\)
0.00591017 + 0.999983i \(0.498119\pi\)
\(74\) 8.70864 1.01236
\(75\) −2.44129 −0.281896
\(76\) 3.23174 0.370706
\(77\) 0 0
\(78\) 8.03124 0.909359
\(79\) 9.19528 1.03455 0.517275 0.855819i \(-0.326946\pi\)
0.517275 + 0.855819i \(0.326946\pi\)
\(80\) −4.99712 −0.558695
\(81\) −9.11869 −1.01319
\(82\) −2.75975 −0.304763
\(83\) 7.62375 0.836815 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(84\) 0 0
\(85\) −2.57226 −0.279000
\(86\) 18.8409 2.03166
\(87\) 6.25668 0.670786
\(88\) 1.65372 0.176288
\(89\) 14.1911 1.50426 0.752129 0.659016i \(-0.229027\pi\)
0.752129 + 0.659016i \(0.229027\pi\)
\(90\) −5.17233 −0.545212
\(91\) 0 0
\(92\) −6.15154 −0.641342
\(93\) −6.95392 −0.721088
\(94\) 8.52491 0.879277
\(95\) 3.06719 0.314687
\(96\) −13.2437 −1.35168
\(97\) 1.79824 0.182583 0.0912916 0.995824i \(-0.470900\pi\)
0.0912916 + 0.995824i \(0.470900\pi\)
\(98\) 0 0
\(99\) 2.95990 0.297481
\(100\) 1.05365 0.105365
\(101\) −10.7664 −1.07130 −0.535650 0.844440i \(-0.679933\pi\)
−0.535650 + 0.844440i \(0.679933\pi\)
\(102\) −10.9734 −1.08653
\(103\) 4.63635 0.456833 0.228417 0.973563i \(-0.426645\pi\)
0.228417 + 0.973563i \(0.426645\pi\)
\(104\) 3.11327 0.305281
\(105\) 0 0
\(106\) 3.32890 0.323332
\(107\) −1.63708 −0.158262 −0.0791311 0.996864i \(-0.525215\pi\)
−0.0791311 + 0.996864i \(0.525215\pi\)
\(108\) 0.103144 0.00992505
\(109\) −16.6501 −1.59479 −0.797396 0.603456i \(-0.793790\pi\)
−0.797396 + 0.603456i \(0.793790\pi\)
\(110\) −1.74747 −0.166615
\(111\) 12.1664 1.15478
\(112\) 0 0
\(113\) −15.0135 −1.41236 −0.706178 0.708034i \(-0.749582\pi\)
−0.706178 + 0.708034i \(0.749582\pi\)
\(114\) 13.0849 1.22551
\(115\) −5.83834 −0.544428
\(116\) −2.70034 −0.250721
\(117\) 5.57226 0.515155
\(118\) −14.6230 −1.34616
\(119\) 0 0
\(120\) −4.03722 −0.368546
\(121\) 1.00000 0.0909091
\(122\) 12.0508 1.09103
\(123\) −3.85549 −0.347638
\(124\) 3.00127 0.269522
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 10.5179 0.933311 0.466656 0.884439i \(-0.345459\pi\)
0.466656 + 0.884439i \(0.345459\pi\)
\(128\) −11.7487 −1.03845
\(129\) 26.3215 2.31748
\(130\) −3.28975 −0.288530
\(131\) 20.6852 1.80727 0.903637 0.428300i \(-0.140887\pi\)
0.903637 + 0.428300i \(0.140887\pi\)
\(132\) −2.57226 −0.223886
\(133\) 0 0
\(134\) −25.5378 −2.20613
\(135\) 0.0978926 0.00842525
\(136\) −4.25380 −0.364760
\(137\) −21.6149 −1.84669 −0.923343 0.383977i \(-0.874554\pi\)
−0.923343 + 0.383977i \(0.874554\pi\)
\(138\) −24.9068 −2.12021
\(139\) −16.4798 −1.39780 −0.698898 0.715221i \(-0.746326\pi\)
−0.698898 + 0.715221i \(0.746326\pi\)
\(140\) 0 0
\(141\) 11.9097 1.00298
\(142\) −7.89110 −0.662206
\(143\) 1.88258 0.157429
\(144\) −14.7910 −1.23258
\(145\) −2.56286 −0.212834
\(146\) −0.176482 −0.0146058
\(147\) 0 0
\(148\) −5.25092 −0.431623
\(149\) 8.94452 0.732764 0.366382 0.930465i \(-0.380596\pi\)
0.366382 + 0.930465i \(0.380596\pi\)
\(150\) 4.26608 0.348324
\(151\) 13.9427 1.13464 0.567322 0.823496i \(-0.307979\pi\)
0.567322 + 0.823496i \(0.307979\pi\)
\(152\) 5.07229 0.411417
\(153\) −7.61363 −0.615525
\(154\) 0 0
\(155\) 2.84846 0.228794
\(156\) −4.84248 −0.387709
\(157\) −3.00288 −0.239656 −0.119828 0.992795i \(-0.538234\pi\)
−0.119828 + 0.992795i \(0.538234\pi\)
\(158\) −16.0685 −1.27834
\(159\) 4.65062 0.368818
\(160\) 5.42487 0.428873
\(161\) 0 0
\(162\) 15.9346 1.25194
\(163\) −22.2697 −1.74430 −0.872150 0.489239i \(-0.837275\pi\)
−0.872150 + 0.489239i \(0.837275\pi\)
\(164\) 1.66400 0.129937
\(165\) −2.44129 −0.190054
\(166\) −13.3223 −1.03401
\(167\) −7.35385 −0.569058 −0.284529 0.958667i \(-0.591837\pi\)
−0.284529 + 0.958667i \(0.591837\pi\)
\(168\) 0 0
\(169\) −9.45589 −0.727376
\(170\) 4.49494 0.344746
\(171\) 9.07859 0.694257
\(172\) −11.3602 −0.866208
\(173\) −23.2108 −1.76469 −0.882343 0.470607i \(-0.844035\pi\)
−0.882343 + 0.470607i \(0.844035\pi\)
\(174\) −10.9333 −0.828855
\(175\) 0 0
\(176\) −4.99712 −0.376672
\(177\) −20.4290 −1.53554
\(178\) −24.7986 −1.85873
\(179\) −18.9160 −1.41385 −0.706923 0.707290i \(-0.749917\pi\)
−0.706923 + 0.707290i \(0.749917\pi\)
\(180\) 3.11869 0.232453
\(181\) 21.8510 1.62417 0.812086 0.583538i \(-0.198332\pi\)
0.812086 + 0.583538i \(0.198332\pi\)
\(182\) 0 0
\(183\) 16.8355 1.24451
\(184\) −9.65499 −0.711776
\(185\) −4.98357 −0.366400
\(186\) 12.1518 0.891010
\(187\) −2.57226 −0.188102
\(188\) −5.14014 −0.374883
\(189\) 0 0
\(190\) −5.35982 −0.388842
\(191\) 3.14756 0.227749 0.113875 0.993495i \(-0.463674\pi\)
0.113875 + 0.993495i \(0.463674\pi\)
\(192\) −1.25595 −0.0906406
\(193\) 21.2390 1.52882 0.764409 0.644731i \(-0.223031\pi\)
0.764409 + 0.644731i \(0.223031\pi\)
\(194\) −3.14236 −0.225608
\(195\) −4.59593 −0.329121
\(196\) 0 0
\(197\) 8.39202 0.597906 0.298953 0.954268i \(-0.403363\pi\)
0.298953 + 0.954268i \(0.403363\pi\)
\(198\) −5.17233 −0.367582
\(199\) 23.4653 1.66341 0.831705 0.555218i \(-0.187365\pi\)
0.831705 + 0.555218i \(0.187365\pi\)
\(200\) 1.65372 0.116936
\(201\) −35.6774 −2.51649
\(202\) 18.8140 1.32375
\(203\) 0 0
\(204\) 6.61650 0.463248
\(205\) 1.57928 0.110302
\(206\) −8.10187 −0.564484
\(207\) −17.2809 −1.20111
\(208\) −9.40749 −0.652292
\(209\) 3.06719 0.212162
\(210\) 0 0
\(211\) 0.283774 0.0195358 0.00976791 0.999952i \(-0.496891\pi\)
0.00976791 + 0.999952i \(0.496891\pi\)
\(212\) −2.00718 −0.137854
\(213\) −11.0242 −0.755367
\(214\) 2.86074 0.195556
\(215\) −10.7818 −0.735313
\(216\) 0.161887 0.0110150
\(217\) 0 0
\(218\) 29.0956 1.97060
\(219\) −0.246553 −0.0166605
\(220\) 1.05365 0.0710368
\(221\) −4.84248 −0.325741
\(222\) −21.2603 −1.42690
\(223\) −27.4540 −1.83845 −0.919226 0.393729i \(-0.871185\pi\)
−0.919226 + 0.393729i \(0.871185\pi\)
\(224\) 0 0
\(225\) 2.95990 0.197327
\(226\) 26.2357 1.74517
\(227\) 2.79586 0.185568 0.0927840 0.995686i \(-0.470423\pi\)
0.0927840 + 0.995686i \(0.470423\pi\)
\(228\) −7.88961 −0.522502
\(229\) −12.8497 −0.849134 −0.424567 0.905396i \(-0.639574\pi\)
−0.424567 + 0.905396i \(0.639574\pi\)
\(230\) 10.2023 0.672720
\(231\) 0 0
\(232\) −4.23826 −0.278255
\(233\) −18.4684 −1.20990 −0.604952 0.796262i \(-0.706808\pi\)
−0.604952 + 0.796262i \(0.706808\pi\)
\(234\) −9.73734 −0.636550
\(235\) −4.87843 −0.318234
\(236\) 8.81704 0.573940
\(237\) −22.4484 −1.45818
\(238\) 0 0
\(239\) −21.0893 −1.36415 −0.682075 0.731282i \(-0.738922\pi\)
−0.682075 + 0.731282i \(0.738922\pi\)
\(240\) 12.1994 0.787470
\(241\) −9.37100 −0.603639 −0.301819 0.953365i \(-0.597594\pi\)
−0.301819 + 0.953365i \(0.597594\pi\)
\(242\) −1.74747 −0.112332
\(243\) 21.9677 1.40923
\(244\) −7.26608 −0.465163
\(245\) 0 0
\(246\) 6.73734 0.429557
\(247\) 5.77424 0.367406
\(248\) 4.71057 0.299121
\(249\) −18.6118 −1.17947
\(250\) −1.74747 −0.110520
\(251\) −14.4075 −0.909393 −0.454696 0.890647i \(-0.650252\pi\)
−0.454696 + 0.890647i \(0.650252\pi\)
\(252\) 0 0
\(253\) −5.83834 −0.367053
\(254\) −18.3797 −1.15324
\(255\) 6.27963 0.393245
\(256\) 19.5016 1.21885
\(257\) 16.3074 1.01723 0.508615 0.860994i \(-0.330158\pi\)
0.508615 + 0.860994i \(0.330158\pi\)
\(258\) −45.9961 −2.86359
\(259\) 0 0
\(260\) 1.98357 0.123016
\(261\) −7.58580 −0.469549
\(262\) −36.1467 −2.23315
\(263\) −26.3192 −1.62291 −0.811456 0.584413i \(-0.801325\pi\)
−0.811456 + 0.584413i \(0.801325\pi\)
\(264\) −4.03722 −0.248474
\(265\) −1.90499 −0.117022
\(266\) 0 0
\(267\) −34.6447 −2.12022
\(268\) 15.3982 0.940592
\(269\) 0.470216 0.0286696 0.0143348 0.999897i \(-0.495437\pi\)
0.0143348 + 0.999897i \(0.495437\pi\)
\(270\) −0.171064 −0.0104106
\(271\) 17.9883 1.09271 0.546355 0.837554i \(-0.316015\pi\)
0.546355 + 0.837554i \(0.316015\pi\)
\(272\) 12.8539 0.779381
\(273\) 0 0
\(274\) 37.7713 2.28185
\(275\) 1.00000 0.0603023
\(276\) 15.0177 0.903959
\(277\) 24.9706 1.50034 0.750168 0.661247i \(-0.229973\pi\)
0.750168 + 0.661247i \(0.229973\pi\)
\(278\) 28.7979 1.72718
\(279\) 8.43116 0.504761
\(280\) 0 0
\(281\) −22.4277 −1.33793 −0.668964 0.743295i \(-0.733262\pi\)
−0.668964 + 0.743295i \(0.733262\pi\)
\(282\) −20.8118 −1.23932
\(283\) −21.2464 −1.26297 −0.631483 0.775390i \(-0.717553\pi\)
−0.631483 + 0.775390i \(0.717553\pi\)
\(284\) 4.75798 0.282334
\(285\) −7.48791 −0.443546
\(286\) −3.28975 −0.194527
\(287\) 0 0
\(288\) 16.0571 0.946172
\(289\) −10.3835 −0.610794
\(290\) 4.47851 0.262987
\(291\) −4.39002 −0.257347
\(292\) 0.106411 0.00622722
\(293\) −16.3043 −0.952510 −0.476255 0.879307i \(-0.658006\pi\)
−0.476255 + 0.879307i \(0.658006\pi\)
\(294\) 0 0
\(295\) 8.36812 0.487211
\(296\) −8.24146 −0.479025
\(297\) 0.0978926 0.00568030
\(298\) −15.6303 −0.905437
\(299\) −10.9911 −0.635634
\(300\) −2.57226 −0.148509
\(301\) 0 0
\(302\) −24.3645 −1.40202
\(303\) 26.2840 1.50998
\(304\) −15.3271 −0.879072
\(305\) −6.89613 −0.394871
\(306\) 13.3046 0.760572
\(307\) 1.80820 0.103199 0.0515996 0.998668i \(-0.483568\pi\)
0.0515996 + 0.998668i \(0.483568\pi\)
\(308\) 0 0
\(309\) −11.3187 −0.643897
\(310\) −4.97760 −0.282708
\(311\) −30.3806 −1.72273 −0.861363 0.507990i \(-0.830389\pi\)
−0.861363 + 0.507990i \(0.830389\pi\)
\(312\) −7.60040 −0.430288
\(313\) 17.7832 1.00517 0.502584 0.864528i \(-0.332383\pi\)
0.502584 + 0.864528i \(0.332383\pi\)
\(314\) 5.24743 0.296130
\(315\) 0 0
\(316\) 9.68857 0.545025
\(317\) 5.71699 0.321098 0.160549 0.987028i \(-0.448673\pi\)
0.160549 + 0.987028i \(0.448673\pi\)
\(318\) −8.12682 −0.455729
\(319\) −2.56286 −0.143492
\(320\) 0.514463 0.0287593
\(321\) 3.99658 0.223067
\(322\) 0 0
\(323\) −7.88961 −0.438990
\(324\) −9.60787 −0.533771
\(325\) 1.88258 0.104427
\(326\) 38.9156 2.15534
\(327\) 40.6478 2.24783
\(328\) 2.61170 0.144207
\(329\) 0 0
\(330\) 4.26608 0.234840
\(331\) −8.09032 −0.444684 −0.222342 0.974969i \(-0.571370\pi\)
−0.222342 + 0.974969i \(0.571370\pi\)
\(332\) 8.03273 0.440854
\(333\) −14.7509 −0.808344
\(334\) 12.8506 0.703154
\(335\) 14.6142 0.798457
\(336\) 0 0
\(337\) 11.3367 0.617549 0.308775 0.951135i \(-0.400081\pi\)
0.308775 + 0.951135i \(0.400081\pi\)
\(338\) 16.5239 0.898780
\(339\) 36.6524 1.99069
\(340\) −2.71025 −0.146984
\(341\) 2.84846 0.154253
\(342\) −15.8645 −0.857857
\(343\) 0 0
\(344\) −17.8301 −0.961337
\(345\) 14.2531 0.767360
\(346\) 40.5602 2.18053
\(347\) −0.406764 −0.0218363 −0.0109181 0.999940i \(-0.503475\pi\)
−0.0109181 + 0.999940i \(0.503475\pi\)
\(348\) 6.59233 0.353386
\(349\) 4.87916 0.261175 0.130588 0.991437i \(-0.458314\pi\)
0.130588 + 0.991437i \(0.458314\pi\)
\(350\) 0 0
\(351\) 0.184291 0.00983671
\(352\) 5.42487 0.289146
\(353\) 8.99734 0.478880 0.239440 0.970911i \(-0.423036\pi\)
0.239440 + 0.970911i \(0.423036\pi\)
\(354\) 35.6991 1.89738
\(355\) 4.51573 0.239670
\(356\) 14.9524 0.792477
\(357\) 0 0
\(358\) 33.0551 1.74701
\(359\) −4.69692 −0.247894 −0.123947 0.992289i \(-0.539555\pi\)
−0.123947 + 0.992289i \(0.539555\pi\)
\(360\) 4.89486 0.257982
\(361\) −9.59233 −0.504859
\(362\) −38.1839 −2.00690
\(363\) −2.44129 −0.128135
\(364\) 0 0
\(365\) 0.100993 0.00528621
\(366\) −29.4194 −1.53778
\(367\) 4.59173 0.239686 0.119843 0.992793i \(-0.461761\pi\)
0.119843 + 0.992793i \(0.461761\pi\)
\(368\) 29.1749 1.52085
\(369\) 4.67452 0.243346
\(370\) 8.70864 0.452741
\(371\) 0 0
\(372\) −7.32697 −0.379886
\(373\) −13.0675 −0.676610 −0.338305 0.941036i \(-0.609854\pi\)
−0.338305 + 0.941036i \(0.609854\pi\)
\(374\) 4.49494 0.232428
\(375\) −2.44129 −0.126068
\(376\) −8.06758 −0.416054
\(377\) −4.82479 −0.248489
\(378\) 0 0
\(379\) −10.4099 −0.534719 −0.267359 0.963597i \(-0.586151\pi\)
−0.267359 + 0.963597i \(0.586151\pi\)
\(380\) 3.23174 0.165785
\(381\) −25.6772 −1.31548
\(382\) −5.50026 −0.281418
\(383\) −22.9451 −1.17244 −0.586221 0.810152i \(-0.699385\pi\)
−0.586221 + 0.810152i \(0.699385\pi\)
\(384\) 28.6821 1.46368
\(385\) 0 0
\(386\) −37.1145 −1.88908
\(387\) −31.9131 −1.62223
\(388\) 1.89470 0.0961891
\(389\) 11.6665 0.591517 0.295759 0.955263i \(-0.404428\pi\)
0.295759 + 0.955263i \(0.404428\pi\)
\(390\) 8.03124 0.406678
\(391\) 15.0177 0.759477
\(392\) 0 0
\(393\) −50.4986 −2.54732
\(394\) −14.6648 −0.738801
\(395\) 9.19528 0.462665
\(396\) 3.11869 0.156720
\(397\) −2.41496 −0.121203 −0.0606017 0.998162i \(-0.519302\pi\)
−0.0606017 + 0.998162i \(0.519302\pi\)
\(398\) −41.0048 −2.05539
\(399\) 0 0
\(400\) −4.99712 −0.249856
\(401\) −33.8495 −1.69037 −0.845183 0.534477i \(-0.820509\pi\)
−0.845183 + 0.534477i \(0.820509\pi\)
\(402\) 62.3452 3.10950
\(403\) 5.36246 0.267123
\(404\) −11.3440 −0.564386
\(405\) −9.11869 −0.453111
\(406\) 0 0
\(407\) −4.98357 −0.247027
\(408\) 10.3848 0.514123
\(409\) −9.17056 −0.453455 −0.226728 0.973958i \(-0.572803\pi\)
−0.226728 + 0.973958i \(0.572803\pi\)
\(410\) −2.75975 −0.136294
\(411\) 52.7682 2.60287
\(412\) 4.88507 0.240670
\(413\) 0 0
\(414\) 30.1978 1.48414
\(415\) 7.62375 0.374235
\(416\) 10.2128 0.500721
\(417\) 40.2319 1.97017
\(418\) −5.35982 −0.262158
\(419\) −9.98307 −0.487705 −0.243852 0.969812i \(-0.578411\pi\)
−0.243852 + 0.969812i \(0.578411\pi\)
\(420\) 0 0
\(421\) −16.5141 −0.804846 −0.402423 0.915454i \(-0.631832\pi\)
−0.402423 + 0.915454i \(0.631832\pi\)
\(422\) −0.495886 −0.0241394
\(423\) −14.4397 −0.702081
\(424\) −3.15032 −0.152993
\(425\) −2.57226 −0.124773
\(426\) 19.2645 0.933367
\(427\) 0 0
\(428\) −1.72490 −0.0833761
\(429\) −4.59593 −0.221894
\(430\) 18.8409 0.908588
\(431\) −16.4554 −0.792628 −0.396314 0.918115i \(-0.629711\pi\)
−0.396314 + 0.918115i \(0.629711\pi\)
\(432\) −0.489181 −0.0235357
\(433\) −17.9836 −0.864235 −0.432118 0.901817i \(-0.642234\pi\)
−0.432118 + 0.901817i \(0.642234\pi\)
\(434\) 0 0
\(435\) 6.25668 0.299985
\(436\) −17.5433 −0.840173
\(437\) −17.9073 −0.856622
\(438\) 0.430844 0.0205865
\(439\) 23.0726 1.10120 0.550598 0.834771i \(-0.314400\pi\)
0.550598 + 0.834771i \(0.314400\pi\)
\(440\) 1.65372 0.0788382
\(441\) 0 0
\(442\) 8.46209 0.402500
\(443\) 9.63027 0.457548 0.228774 0.973480i \(-0.426528\pi\)
0.228774 + 0.973480i \(0.426528\pi\)
\(444\) 12.8190 0.608364
\(445\) 14.1911 0.672724
\(446\) 47.9749 2.27168
\(447\) −21.8362 −1.03282
\(448\) 0 0
\(449\) −5.45196 −0.257294 −0.128647 0.991690i \(-0.541063\pi\)
−0.128647 + 0.991690i \(0.541063\pi\)
\(450\) −5.17233 −0.243826
\(451\) 1.57928 0.0743655
\(452\) −15.8190 −0.744061
\(453\) −34.0383 −1.59926
\(454\) −4.88568 −0.229296
\(455\) 0 0
\(456\) −12.3829 −0.579884
\(457\) 26.3853 1.23425 0.617125 0.786865i \(-0.288297\pi\)
0.617125 + 0.786865i \(0.288297\pi\)
\(458\) 22.4545 1.04923
\(459\) −0.251805 −0.0117532
\(460\) −6.15154 −0.286817
\(461\) 14.6685 0.683182 0.341591 0.939849i \(-0.389034\pi\)
0.341591 + 0.939849i \(0.389034\pi\)
\(462\) 0 0
\(463\) 29.7719 1.38362 0.691809 0.722081i \(-0.256814\pi\)
0.691809 + 0.722081i \(0.256814\pi\)
\(464\) 12.8069 0.594546
\(465\) −6.95392 −0.322480
\(466\) 32.2729 1.49501
\(467\) 7.01013 0.324390 0.162195 0.986759i \(-0.448143\pi\)
0.162195 + 0.986759i \(0.448143\pi\)
\(468\) 5.87119 0.271396
\(469\) 0 0
\(470\) 8.52491 0.393225
\(471\) 7.33090 0.337790
\(472\) 13.8386 0.636971
\(473\) −10.7818 −0.495748
\(474\) 39.2278 1.80179
\(475\) 3.06719 0.140732
\(476\) 0 0
\(477\) −5.63857 −0.258172
\(478\) 36.8528 1.68561
\(479\) 13.9532 0.637538 0.318769 0.947832i \(-0.396731\pi\)
0.318769 + 0.947832i \(0.396731\pi\)
\(480\) −13.2437 −0.604488
\(481\) −9.38199 −0.427782
\(482\) 16.3755 0.745885
\(483\) 0 0
\(484\) 1.05365 0.0478930
\(485\) 1.79824 0.0816537
\(486\) −38.3879 −1.74131
\(487\) 6.72132 0.304572 0.152286 0.988336i \(-0.451336\pi\)
0.152286 + 0.988336i \(0.451336\pi\)
\(488\) −11.4043 −0.516248
\(489\) 54.3669 2.45855
\(490\) 0 0
\(491\) −25.9471 −1.17098 −0.585489 0.810681i \(-0.699097\pi\)
−0.585489 + 0.810681i \(0.699097\pi\)
\(492\) −4.06232 −0.183143
\(493\) 6.59233 0.296903
\(494\) −10.0903 −0.453984
\(495\) 2.95990 0.133038
\(496\) −14.2341 −0.639130
\(497\) 0 0
\(498\) 32.5235 1.45741
\(499\) 20.8050 0.931359 0.465679 0.884953i \(-0.345810\pi\)
0.465679 + 0.884953i \(0.345810\pi\)
\(500\) 1.05365 0.0471205
\(501\) 17.9529 0.802075
\(502\) 25.1766 1.12369
\(503\) 30.1238 1.34315 0.671576 0.740936i \(-0.265618\pi\)
0.671576 + 0.740936i \(0.265618\pi\)
\(504\) 0 0
\(505\) −10.7664 −0.479100
\(506\) 10.2023 0.453548
\(507\) 23.0846 1.02522
\(508\) 11.0821 0.491690
\(509\) −10.5596 −0.468045 −0.234022 0.972231i \(-0.575189\pi\)
−0.234022 + 0.972231i \(0.575189\pi\)
\(510\) −10.9734 −0.485912
\(511\) 0 0
\(512\) −10.5810 −0.467618
\(513\) 0.300255 0.0132566
\(514\) −28.4968 −1.25694
\(515\) 4.63635 0.204302
\(516\) 27.7336 1.22090
\(517\) −4.87843 −0.214553
\(518\) 0 0
\(519\) 56.6644 2.48729
\(520\) 3.11327 0.136526
\(521\) −14.9118 −0.653299 −0.326650 0.945146i \(-0.605920\pi\)
−0.326650 + 0.945146i \(0.605920\pi\)
\(522\) 13.2560 0.580197
\(523\) 6.86179 0.300045 0.150022 0.988683i \(-0.452065\pi\)
0.150022 + 0.988683i \(0.452065\pi\)
\(524\) 21.7949 0.952113
\(525\) 0 0
\(526\) 45.9920 2.00535
\(527\) −7.32697 −0.319168
\(528\) 12.1994 0.530912
\(529\) 11.0862 0.482007
\(530\) 3.32890 0.144598
\(531\) 24.7688 1.07487
\(532\) 0 0
\(533\) 2.97313 0.128780
\(534\) 60.5405 2.61984
\(535\) −1.63708 −0.0707770
\(536\) 24.1678 1.04389
\(537\) 46.1794 1.99279
\(538\) −0.821688 −0.0354255
\(539\) 0 0
\(540\) 0.103144 0.00443862
\(541\) −13.7677 −0.591920 −0.295960 0.955200i \(-0.595639\pi\)
−0.295960 + 0.955200i \(0.595639\pi\)
\(542\) −31.4339 −1.35020
\(543\) −53.3447 −2.28924
\(544\) −13.9541 −0.598279
\(545\) −16.6501 −0.713213
\(546\) 0 0
\(547\) −8.87135 −0.379312 −0.189656 0.981851i \(-0.560737\pi\)
−0.189656 + 0.981851i \(0.560737\pi\)
\(548\) −22.7744 −0.972876
\(549\) −20.4119 −0.871157
\(550\) −1.74747 −0.0745123
\(551\) −7.86078 −0.334880
\(552\) 23.5706 1.00323
\(553\) 0 0
\(554\) −43.6353 −1.85389
\(555\) 12.1664 0.516433
\(556\) −17.3639 −0.736392
\(557\) 12.5634 0.532328 0.266164 0.963928i \(-0.414244\pi\)
0.266164 + 0.963928i \(0.414244\pi\)
\(558\) −14.7332 −0.623706
\(559\) −20.2976 −0.858499
\(560\) 0 0
\(561\) 6.27963 0.265126
\(562\) 39.1918 1.65321
\(563\) −14.4847 −0.610455 −0.305228 0.952279i \(-0.598733\pi\)
−0.305228 + 0.952279i \(0.598733\pi\)
\(564\) 12.5486 0.528391
\(565\) −15.0135 −0.631625
\(566\) 37.1274 1.56058
\(567\) 0 0
\(568\) 7.46777 0.313341
\(569\) −34.4444 −1.44399 −0.721993 0.691901i \(-0.756774\pi\)
−0.721993 + 0.691901i \(0.756774\pi\)
\(570\) 13.0849 0.548066
\(571\) 45.9789 1.92416 0.962078 0.272775i \(-0.0879415\pi\)
0.962078 + 0.272775i \(0.0879415\pi\)
\(572\) 1.98357 0.0829374
\(573\) −7.68410 −0.321008
\(574\) 0 0
\(575\) −5.83834 −0.243475
\(576\) 1.52276 0.0634483
\(577\) −40.3967 −1.68174 −0.840868 0.541240i \(-0.817955\pi\)
−0.840868 + 0.541240i \(0.817955\pi\)
\(578\) 18.1448 0.754726
\(579\) −51.8506 −2.15484
\(580\) −2.70034 −0.112126
\(581\) 0 0
\(582\) 7.67142 0.317991
\(583\) −1.90499 −0.0788964
\(584\) 0.167014 0.00691111
\(585\) 5.57226 0.230384
\(586\) 28.4913 1.17697
\(587\) −21.4627 −0.885860 −0.442930 0.896556i \(-0.646061\pi\)
−0.442930 + 0.896556i \(0.646061\pi\)
\(588\) 0 0
\(589\) 8.73678 0.359993
\(590\) −14.6230 −0.602020
\(591\) −20.4874 −0.842737
\(592\) 24.9035 1.02353
\(593\) −11.6017 −0.476424 −0.238212 0.971213i \(-0.576561\pi\)
−0.238212 + 0.971213i \(0.576561\pi\)
\(594\) −0.171064 −0.00701885
\(595\) 0 0
\(596\) 9.42436 0.386037
\(597\) −57.2856 −2.34454
\(598\) 19.2067 0.785419
\(599\) −13.7763 −0.562886 −0.281443 0.959578i \(-0.590813\pi\)
−0.281443 + 0.959578i \(0.590813\pi\)
\(600\) −4.03722 −0.164819
\(601\) −28.9935 −1.18267 −0.591335 0.806426i \(-0.701399\pi\)
−0.591335 + 0.806426i \(0.701399\pi\)
\(602\) 0 0
\(603\) 43.2565 1.76154
\(604\) 14.6907 0.597757
\(605\) 1.00000 0.0406558
\(606\) −45.9305 −1.86580
\(607\) 41.4893 1.68400 0.842000 0.539478i \(-0.181378\pi\)
0.842000 + 0.539478i \(0.181378\pi\)
\(608\) 16.6391 0.674805
\(609\) 0 0
\(610\) 12.0508 0.487921
\(611\) −9.18405 −0.371547
\(612\) −8.02207 −0.324273
\(613\) 34.2229 1.38225 0.691124 0.722736i \(-0.257116\pi\)
0.691124 + 0.722736i \(0.257116\pi\)
\(614\) −3.15977 −0.127518
\(615\) −3.85549 −0.155468
\(616\) 0 0
\(617\) 30.4992 1.22785 0.613926 0.789363i \(-0.289589\pi\)
0.613926 + 0.789363i \(0.289589\pi\)
\(618\) 19.7790 0.795629
\(619\) 38.5147 1.54804 0.774018 0.633164i \(-0.218244\pi\)
0.774018 + 0.633164i \(0.218244\pi\)
\(620\) 3.00127 0.120534
\(621\) −0.571530 −0.0229347
\(622\) 53.0891 2.12868
\(623\) 0 0
\(624\) 22.9664 0.919393
\(625\) 1.00000 0.0400000
\(626\) −31.0756 −1.24203
\(627\) −7.48791 −0.299038
\(628\) −3.16397 −0.126256
\(629\) 12.8190 0.511128
\(630\) 0 0
\(631\) 2.18173 0.0868534 0.0434267 0.999057i \(-0.486172\pi\)
0.0434267 + 0.999057i \(0.486172\pi\)
\(632\) 15.2065 0.604880
\(633\) −0.692775 −0.0275353
\(634\) −9.99026 −0.396764
\(635\) 10.5179 0.417389
\(636\) 4.90011 0.194302
\(637\) 0 0
\(638\) 4.47851 0.177306
\(639\) 13.3661 0.528756
\(640\) −11.7487 −0.464410
\(641\) −41.0820 −1.62264 −0.811321 0.584601i \(-0.801251\pi\)
−0.811321 + 0.584601i \(0.801251\pi\)
\(642\) −6.98390 −0.275632
\(643\) −36.7462 −1.44913 −0.724565 0.689206i \(-0.757959\pi\)
−0.724565 + 0.689206i \(0.757959\pi\)
\(644\) 0 0
\(645\) 26.3215 1.03641
\(646\) 13.7868 0.542436
\(647\) −38.3584 −1.50802 −0.754012 0.656861i \(-0.771884\pi\)
−0.754012 + 0.656861i \(0.771884\pi\)
\(648\) −15.0798 −0.592390
\(649\) 8.36812 0.328477
\(650\) −3.28975 −0.129035
\(651\) 0 0
\(652\) −23.4644 −0.918937
\(653\) −14.4251 −0.564497 −0.282249 0.959341i \(-0.591080\pi\)
−0.282249 + 0.959341i \(0.591080\pi\)
\(654\) −71.0307 −2.77752
\(655\) 20.6852 0.808237
\(656\) −7.89187 −0.308126
\(657\) 0.298929 0.0116623
\(658\) 0 0
\(659\) 21.0717 0.820835 0.410418 0.911898i \(-0.365383\pi\)
0.410418 + 0.911898i \(0.365383\pi\)
\(660\) −2.57226 −0.100125
\(661\) −22.0164 −0.856340 −0.428170 0.903698i \(-0.640841\pi\)
−0.428170 + 0.903698i \(0.640841\pi\)
\(662\) 14.1376 0.549473
\(663\) 11.8219 0.459125
\(664\) 12.6076 0.489269
\(665\) 0 0
\(666\) 25.7767 0.998827
\(667\) 14.9628 0.579363
\(668\) −7.74835 −0.299793
\(669\) 67.0231 2.59126
\(670\) −25.5378 −0.986611
\(671\) −6.89613 −0.266222
\(672\) 0 0
\(673\) 29.4552 1.13542 0.567708 0.823230i \(-0.307830\pi\)
0.567708 + 0.823230i \(0.307830\pi\)
\(674\) −19.8105 −0.763072
\(675\) 0.0978926 0.00376789
\(676\) −9.96316 −0.383198
\(677\) −25.5337 −0.981342 −0.490671 0.871345i \(-0.663248\pi\)
−0.490671 + 0.871345i \(0.663248\pi\)
\(678\) −64.0490 −2.45979
\(679\) 0 0
\(680\) −4.25380 −0.163126
\(681\) −6.82551 −0.261554
\(682\) −4.97760 −0.190602
\(683\) 17.0369 0.651898 0.325949 0.945387i \(-0.394316\pi\)
0.325949 + 0.945387i \(0.394316\pi\)
\(684\) 9.56562 0.365751
\(685\) −21.6149 −0.825863
\(686\) 0 0
\(687\) 31.3699 1.19684
\(688\) 53.8780 2.05408
\(689\) −3.58629 −0.136627
\(690\) −24.9068 −0.948186
\(691\) −43.4197 −1.65176 −0.825881 0.563845i \(-0.809322\pi\)
−0.825881 + 0.563845i \(0.809322\pi\)
\(692\) −24.4560 −0.929677
\(693\) 0 0
\(694\) 0.710808 0.0269819
\(695\) −16.4798 −0.625114
\(696\) 10.3468 0.392195
\(697\) −4.06232 −0.153871
\(698\) −8.52618 −0.322721
\(699\) 45.0867 1.70534
\(700\) 0 0
\(701\) −19.0563 −0.719746 −0.359873 0.933001i \(-0.617180\pi\)
−0.359873 + 0.933001i \(0.617180\pi\)
\(702\) −0.322042 −0.0121547
\(703\) −15.2856 −0.576507
\(704\) 0.514463 0.0193895
\(705\) 11.9097 0.448544
\(706\) −15.7226 −0.591727
\(707\) 0 0
\(708\) −21.5249 −0.808957
\(709\) 18.9528 0.711788 0.355894 0.934526i \(-0.384176\pi\)
0.355894 + 0.934526i \(0.384176\pi\)
\(710\) −7.89110 −0.296148
\(711\) 27.2171 1.02072
\(712\) 23.4682 0.879509
\(713\) −16.6303 −0.622809
\(714\) 0 0
\(715\) 1.88258 0.0704046
\(716\) −19.9307 −0.744847
\(717\) 51.4850 1.92274
\(718\) 8.20772 0.306310
\(719\) −33.2346 −1.23944 −0.619721 0.784822i \(-0.712754\pi\)
−0.619721 + 0.784822i \(0.712754\pi\)
\(720\) −14.7910 −0.551228
\(721\) 0 0
\(722\) 16.7623 0.623828
\(723\) 22.8773 0.850817
\(724\) 23.0232 0.855651
\(725\) −2.56286 −0.0951821
\(726\) 4.26608 0.158329
\(727\) 9.83280 0.364678 0.182339 0.983236i \(-0.441633\pi\)
0.182339 + 0.983236i \(0.441633\pi\)
\(728\) 0 0
\(729\) −26.2735 −0.973091
\(730\) −0.176482 −0.00653189
\(731\) 27.7336 1.02576
\(732\) 17.7386 0.655638
\(733\) −16.7024 −0.616919 −0.308459 0.951238i \(-0.599813\pi\)
−0.308459 + 0.951238i \(0.599813\pi\)
\(734\) −8.02390 −0.296167
\(735\) 0 0
\(736\) −31.6722 −1.16745
\(737\) 14.6142 0.538320
\(738\) −8.16858 −0.300689
\(739\) −27.7647 −1.02134 −0.510670 0.859777i \(-0.670603\pi\)
−0.510670 + 0.859777i \(0.670603\pi\)
\(740\) −5.25092 −0.193028
\(741\) −14.0966 −0.517852
\(742\) 0 0
\(743\) 16.1302 0.591759 0.295880 0.955225i \(-0.404387\pi\)
0.295880 + 0.955225i \(0.404387\pi\)
\(744\) −11.4999 −0.421606
\(745\) 8.94452 0.327702
\(746\) 22.8351 0.836051
\(747\) 22.5656 0.825630
\(748\) −2.71025 −0.0990965
\(749\) 0 0
\(750\) 4.26608 0.155775
\(751\) −30.1891 −1.10162 −0.550809 0.834631i \(-0.685681\pi\)
−0.550809 + 0.834631i \(0.685681\pi\)
\(752\) 24.3781 0.888979
\(753\) 35.1729 1.28177
\(754\) 8.43116 0.307045
\(755\) 13.9427 0.507429
\(756\) 0 0
\(757\) 13.3060 0.483615 0.241807 0.970324i \(-0.422260\pi\)
0.241807 + 0.970324i \(0.422260\pi\)
\(758\) 18.1909 0.660724
\(759\) 14.2531 0.517354
\(760\) 5.07229 0.183991
\(761\) 8.17689 0.296412 0.148206 0.988957i \(-0.452650\pi\)
0.148206 + 0.988957i \(0.452650\pi\)
\(762\) 44.8701 1.62547
\(763\) 0 0
\(764\) 3.31641 0.119984
\(765\) −7.61363 −0.275271
\(766\) 40.0959 1.44872
\(767\) 15.7537 0.568832
\(768\) −47.6091 −1.71795
\(769\) −54.6122 −1.96936 −0.984682 0.174358i \(-0.944215\pi\)
−0.984682 + 0.174358i \(0.944215\pi\)
\(770\) 0 0
\(771\) −39.8112 −1.43377
\(772\) 22.3784 0.805417
\(773\) −39.6980 −1.42784 −0.713918 0.700229i \(-0.753081\pi\)
−0.713918 + 0.700229i \(0.753081\pi\)
\(774\) 55.7671 2.00451
\(775\) 2.84846 0.102320
\(776\) 2.97379 0.106753
\(777\) 0 0
\(778\) −20.3869 −0.730906
\(779\) 4.84396 0.173553
\(780\) −4.84248 −0.173389
\(781\) 4.51573 0.161586
\(782\) −26.2429 −0.938446
\(783\) −0.250885 −0.00896589
\(784\) 0 0
\(785\) −3.00288 −0.107177
\(786\) 88.2446 3.14758
\(787\) 22.9272 0.817266 0.408633 0.912699i \(-0.366006\pi\)
0.408633 + 0.912699i \(0.366006\pi\)
\(788\) 8.84221 0.314991
\(789\) 64.2529 2.28746
\(790\) −16.0685 −0.571690
\(791\) 0 0
\(792\) 4.89486 0.173931
\(793\) −12.9825 −0.461023
\(794\) 4.22007 0.149765
\(795\) 4.65062 0.164941
\(796\) 24.7241 0.876323
\(797\) 26.4654 0.937451 0.468725 0.883344i \(-0.344713\pi\)
0.468725 + 0.883344i \(0.344713\pi\)
\(798\) 0 0
\(799\) 12.5486 0.443937
\(800\) 5.42487 0.191798
\(801\) 42.0044 1.48415
\(802\) 59.1510 2.08870
\(803\) 0.100993 0.00356396
\(804\) −37.5914 −1.32575
\(805\) 0 0
\(806\) −9.37073 −0.330070
\(807\) −1.14793 −0.0404092
\(808\) −17.8047 −0.626367
\(809\) −24.6624 −0.867084 −0.433542 0.901133i \(-0.642736\pi\)
−0.433542 + 0.901133i \(0.642736\pi\)
\(810\) 15.9346 0.559885
\(811\) −36.8401 −1.29363 −0.646815 0.762647i \(-0.723900\pi\)
−0.646815 + 0.762647i \(0.723900\pi\)
\(812\) 0 0
\(813\) −43.9146 −1.54015
\(814\) 8.70864 0.305238
\(815\) −22.2697 −0.780074
\(816\) −31.3801 −1.09852
\(817\) −33.0699 −1.15697
\(818\) 16.0253 0.560310
\(819\) 0 0
\(820\) 1.66400 0.0581096
\(821\) −8.38656 −0.292693 −0.146346 0.989233i \(-0.546751\pi\)
−0.146346 + 0.989233i \(0.546751\pi\)
\(822\) −92.2108 −3.21622
\(823\) −50.8692 −1.77319 −0.886594 0.462548i \(-0.846935\pi\)
−0.886594 + 0.462548i \(0.846935\pi\)
\(824\) 7.66724 0.267101
\(825\) −2.44129 −0.0849948
\(826\) 0 0
\(827\) 38.3029 1.33192 0.665961 0.745987i \(-0.268022\pi\)
0.665961 + 0.745987i \(0.268022\pi\)
\(828\) −18.2079 −0.632770
\(829\) −3.97040 −0.137898 −0.0689489 0.997620i \(-0.521965\pi\)
−0.0689489 + 0.997620i \(0.521965\pi\)
\(830\) −13.3223 −0.462423
\(831\) −60.9604 −2.11469
\(832\) 0.968518 0.0335773
\(833\) 0 0
\(834\) −70.3040 −2.43443
\(835\) −7.35385 −0.254490
\(836\) 3.23174 0.111772
\(837\) 0.278843 0.00963823
\(838\) 17.4451 0.602631
\(839\) −34.3262 −1.18507 −0.592537 0.805543i \(-0.701874\pi\)
−0.592537 + 0.805543i \(0.701874\pi\)
\(840\) 0 0
\(841\) −22.4318 −0.773509
\(842\) 28.8578 0.994505
\(843\) 54.7526 1.88578
\(844\) 0.298997 0.0102919
\(845\) −9.45589 −0.325292
\(846\) 25.2329 0.867525
\(847\) 0 0
\(848\) 9.51944 0.326899
\(849\) 51.8686 1.78012
\(850\) 4.49494 0.154175
\(851\) 29.0958 0.997391
\(852\) −11.6156 −0.397945
\(853\) 11.6335 0.398322 0.199161 0.979967i \(-0.436178\pi\)
0.199161 + 0.979967i \(0.436178\pi\)
\(854\) 0 0
\(855\) 9.07859 0.310481
\(856\) −2.70727 −0.0925326
\(857\) −48.2990 −1.64986 −0.824932 0.565232i \(-0.808787\pi\)
−0.824932 + 0.565232i \(0.808787\pi\)
\(858\) 8.03124 0.274182
\(859\) 4.26640 0.145568 0.0727838 0.997348i \(-0.476812\pi\)
0.0727838 + 0.997348i \(0.476812\pi\)
\(860\) −11.3602 −0.387380
\(861\) 0 0
\(862\) 28.7553 0.979408
\(863\) −25.2236 −0.858621 −0.429310 0.903157i \(-0.641243\pi\)
−0.429310 + 0.903157i \(0.641243\pi\)
\(864\) 0.531054 0.0180668
\(865\) −23.2108 −0.789192
\(866\) 31.4257 1.06789
\(867\) 25.3491 0.860902
\(868\) 0 0
\(869\) 9.19528 0.311929
\(870\) −10.9333 −0.370675
\(871\) 27.5124 0.932221
\(872\) −27.5347 −0.932443
\(873\) 5.32260 0.180143
\(874\) 31.2924 1.05848
\(875\) 0 0
\(876\) −0.259780 −0.00877715
\(877\) 1.12831 0.0381003 0.0190502 0.999819i \(-0.493936\pi\)
0.0190502 + 0.999819i \(0.493936\pi\)
\(878\) −40.3187 −1.36069
\(879\) 39.8037 1.34254
\(880\) −4.99712 −0.168453
\(881\) 28.1086 0.947003 0.473501 0.880793i \(-0.342990\pi\)
0.473501 + 0.880793i \(0.342990\pi\)
\(882\) 0 0
\(883\) 17.9702 0.604745 0.302372 0.953190i \(-0.402221\pi\)
0.302372 + 0.953190i \(0.402221\pi\)
\(884\) −5.10226 −0.171608
\(885\) −20.4290 −0.686714
\(886\) −16.8286 −0.565368
\(887\) −5.94398 −0.199579 −0.0997897 0.995009i \(-0.531817\pi\)
−0.0997897 + 0.995009i \(0.531817\pi\)
\(888\) 20.1198 0.675176
\(889\) 0 0
\(890\) −24.7986 −0.831250
\(891\) −9.11869 −0.305488
\(892\) −28.9268 −0.968539
\(893\) −14.9631 −0.500721
\(894\) 38.1580 1.27620
\(895\) −18.9160 −0.632291
\(896\) 0 0
\(897\) 26.8326 0.895914
\(898\) 9.52713 0.317924
\(899\) −7.30020 −0.243475
\(900\) 3.11869 0.103956
\(901\) 4.90011 0.163246
\(902\) −2.75975 −0.0918895
\(903\) 0 0
\(904\) −24.8283 −0.825776
\(905\) 21.8510 0.726352
\(906\) 59.4809 1.97612
\(907\) −8.97383 −0.297971 −0.148985 0.988839i \(-0.547601\pi\)
−0.148985 + 0.988839i \(0.547601\pi\)
\(908\) 2.94585 0.0977615
\(909\) −31.8676 −1.05698
\(910\) 0 0
\(911\) 48.2124 1.59735 0.798673 0.601765i \(-0.205536\pi\)
0.798673 + 0.601765i \(0.205536\pi\)
\(912\) 37.4180 1.23903
\(913\) 7.62375 0.252309
\(914\) −46.1074 −1.52510
\(915\) 16.8355 0.556563
\(916\) −13.5391 −0.447343
\(917\) 0 0
\(918\) 0.440021 0.0145229
\(919\) −12.0349 −0.396996 −0.198498 0.980101i \(-0.563606\pi\)
−0.198498 + 0.980101i \(0.563606\pi\)
\(920\) −9.65499 −0.318316
\(921\) −4.41433 −0.145457
\(922\) −25.6328 −0.844172
\(923\) 8.50124 0.279822
\(924\) 0 0
\(925\) −4.98357 −0.163859
\(926\) −52.0255 −1.70966
\(927\) 13.7231 0.450727
\(928\) −13.9032 −0.456393
\(929\) −42.5633 −1.39646 −0.698229 0.715875i \(-0.746028\pi\)
−0.698229 + 0.715875i \(0.746028\pi\)
\(930\) 12.1518 0.398472
\(931\) 0 0
\(932\) −19.4591 −0.637405
\(933\) 74.1679 2.42815
\(934\) −12.2500 −0.400831
\(935\) −2.57226 −0.0841218
\(936\) 9.21497 0.301201
\(937\) −17.7793 −0.580824 −0.290412 0.956902i \(-0.593792\pi\)
−0.290412 + 0.956902i \(0.593792\pi\)
\(938\) 0 0
\(939\) −43.4141 −1.41676
\(940\) −5.14014 −0.167653
\(941\) 23.1985 0.756250 0.378125 0.925755i \(-0.376569\pi\)
0.378125 + 0.925755i \(0.376569\pi\)
\(942\) −12.8105 −0.417389
\(943\) −9.22038 −0.300257
\(944\) −41.8165 −1.36101
\(945\) 0 0
\(946\) 18.8409 0.612570
\(947\) −53.0306 −1.72326 −0.861632 0.507534i \(-0.830557\pi\)
−0.861632 + 0.507534i \(0.830557\pi\)
\(948\) −23.6526 −0.768201
\(949\) 0.190128 0.00617180
\(950\) −5.35982 −0.173896
\(951\) −13.9568 −0.452582
\(952\) 0 0
\(953\) −41.6025 −1.34764 −0.673818 0.738898i \(-0.735347\pi\)
−0.673818 + 0.738898i \(0.735347\pi\)
\(954\) 9.85322 0.319010
\(955\) 3.14756 0.101853
\(956\) −22.2206 −0.718666
\(957\) 6.25668 0.202250
\(958\) −24.3828 −0.787772
\(959\) 0 0
\(960\) −1.25595 −0.0405357
\(961\) −22.8863 −0.738267
\(962\) 16.3947 0.528587
\(963\) −4.84558 −0.156147
\(964\) −9.87371 −0.318011
\(965\) 21.2390 0.683708
\(966\) 0 0
\(967\) 23.1656 0.744955 0.372478 0.928041i \(-0.378508\pi\)
0.372478 + 0.928041i \(0.378508\pi\)
\(968\) 1.65372 0.0531527
\(969\) 19.2608 0.618747
\(970\) −3.14236 −0.100895
\(971\) 29.2515 0.938724 0.469362 0.883006i \(-0.344484\pi\)
0.469362 + 0.883006i \(0.344484\pi\)
\(972\) 23.1462 0.742414
\(973\) 0 0
\(974\) −11.7453 −0.376344
\(975\) −4.59593 −0.147188
\(976\) 34.4608 1.10306
\(977\) −7.08584 −0.226696 −0.113348 0.993555i \(-0.536158\pi\)
−0.113348 + 0.993555i \(0.536158\pi\)
\(978\) −95.0044 −3.03791
\(979\) 14.1911 0.453551
\(980\) 0 0
\(981\) −49.2827 −1.57348
\(982\) 45.3418 1.44691
\(983\) 19.8510 0.633148 0.316574 0.948568i \(-0.397467\pi\)
0.316574 + 0.948568i \(0.397467\pi\)
\(984\) −6.37591 −0.203257
\(985\) 8.39202 0.267392
\(986\) −11.5199 −0.366868
\(987\) 0 0
\(988\) 6.08401 0.193558
\(989\) 62.9478 2.00162
\(990\) −5.17233 −0.164388
\(991\) −16.0795 −0.510782 −0.255391 0.966838i \(-0.582204\pi\)
−0.255391 + 0.966838i \(0.582204\pi\)
\(992\) 15.4525 0.490618
\(993\) 19.7508 0.626774
\(994\) 0 0
\(995\) 23.4653 0.743900
\(996\) −19.6102 −0.621374
\(997\) 22.8184 0.722665 0.361332 0.932437i \(-0.382322\pi\)
0.361332 + 0.932437i \(0.382322\pi\)
\(998\) −36.3560 −1.15083
\(999\) −0.487855 −0.0154350
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 2695.2.a.j.1.2 4
7.2 even 3 385.2.i.a.221.3 8
7.4 even 3 385.2.i.a.331.3 yes 8
7.6 odd 2 2695.2.a.k.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.i.a.221.3 8 7.2 even 3
385.2.i.a.331.3 yes 8 7.4 even 3
2695.2.a.j.1.2 4 1.1 even 1 trivial
2695.2.a.k.1.2 4 7.6 odd 2