Properties

Label 2601.2.a.m.1.1
Level $2601$
Weight $2$
Character 2601.1
Self dual yes
Analytic conductor $20.769$
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] = [2601,2,Mod(1,2601)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2601.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2601.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: \(20.7690895657\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 867)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2601.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} +3.82843 q^{4} -2.82843 q^{5} -3.00000 q^{7} -4.41421 q^{8} +O(q^{10})\) \(q-2.41421 q^{2} +3.82843 q^{4} -2.82843 q^{5} -3.00000 q^{7} -4.41421 q^{8} +6.82843 q^{10} -0.828427 q^{11} +6.65685 q^{13} +7.24264 q^{14} +3.00000 q^{16} -3.00000 q^{19} -10.8284 q^{20} +2.00000 q^{22} -2.82843 q^{23} +3.00000 q^{25} -16.0711 q^{26} -11.4853 q^{28} -3.17157 q^{29} +4.65685 q^{31} +1.58579 q^{32} +8.48528 q^{35} -6.65685 q^{37} +7.24264 q^{38} +12.4853 q^{40} +8.82843 q^{41} +3.00000 q^{43} -3.17157 q^{44} +6.82843 q^{46} +9.65685 q^{47} +2.00000 q^{49} -7.24264 q^{50} +25.4853 q^{52} +1.65685 q^{53} +2.34315 q^{55} +13.2426 q^{56} +7.65685 q^{58} +8.82843 q^{59} +4.65685 q^{61} -11.2426 q^{62} -9.82843 q^{64} -18.8284 q^{65} +1.00000 q^{67} -20.4853 q^{70} -5.17157 q^{71} +7.65685 q^{73} +16.0711 q^{74} -11.4853 q^{76} +2.48528 q^{77} -12.0000 q^{79} -8.48528 q^{80} -21.3137 q^{82} -1.17157 q^{83} -7.24264 q^{86} +3.65685 q^{88} -5.31371 q^{89} -19.9706 q^{91} -10.8284 q^{92} -23.3137 q^{94} +8.48528 q^{95} +10.6569 q^{97} -4.82843 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 6 q^{8} + 8 q^{10} + 4 q^{11} + 2 q^{13} + 6 q^{14} + 6 q^{16} - 6 q^{19} - 16 q^{20} + 4 q^{22} + 6 q^{25} - 18 q^{26} - 6 q^{28} - 12 q^{29} - 2 q^{31} + 6 q^{32} - 2 q^{37} + 6 q^{38} + 8 q^{40} + 12 q^{41} + 6 q^{43} - 12 q^{44} + 8 q^{46} + 8 q^{47} + 4 q^{49} - 6 q^{50} + 34 q^{52} - 8 q^{53} + 16 q^{55} + 18 q^{56} + 4 q^{58} + 12 q^{59} - 2 q^{61} - 14 q^{62} - 14 q^{64} - 32 q^{65} + 2 q^{67} - 24 q^{70} - 16 q^{71} + 4 q^{73} + 18 q^{74} - 6 q^{76} - 12 q^{77} - 24 q^{79} - 20 q^{82} - 8 q^{83} - 6 q^{86} - 4 q^{88} + 12 q^{89} - 6 q^{91} - 16 q^{92} - 24 q^{94} + 10 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −4.41421 −1.56066
\(9\) 0 0
\(10\) 6.82843 2.15934
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) 6.65685 1.84628 0.923140 0.384465i \(-0.125614\pi\)
0.923140 + 0.384465i \(0.125614\pi\)
\(14\) 7.24264 1.93568
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 0 0
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −10.8284 −2.42131
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) −16.0711 −3.15180
\(27\) 0 0
\(28\) −11.4853 −2.17051
\(29\) −3.17157 −0.588946 −0.294473 0.955660i \(-0.595144\pi\)
−0.294473 + 0.955660i \(0.595144\pi\)
\(30\) 0 0
\(31\) 4.65685 0.836396 0.418198 0.908356i \(-0.362662\pi\)
0.418198 + 0.908356i \(0.362662\pi\)
\(32\) 1.58579 0.280330
\(33\) 0 0
\(34\) 0 0
\(35\) 8.48528 1.43427
\(36\) 0 0
\(37\) −6.65685 −1.09438 −0.547190 0.837008i \(-0.684303\pi\)
−0.547190 + 0.837008i \(0.684303\pi\)
\(38\) 7.24264 1.17491
\(39\) 0 0
\(40\) 12.4853 1.97410
\(41\) 8.82843 1.37877 0.689384 0.724396i \(-0.257881\pi\)
0.689384 + 0.724396i \(0.257881\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) −3.17157 −0.478133
\(45\) 0 0
\(46\) 6.82843 1.00680
\(47\) 9.65685 1.40860 0.704298 0.709904i \(-0.251262\pi\)
0.704298 + 0.709904i \(0.251262\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −7.24264 −1.02426
\(51\) 0 0
\(52\) 25.4853 3.53417
\(53\) 1.65685 0.227586 0.113793 0.993504i \(-0.463700\pi\)
0.113793 + 0.993504i \(0.463700\pi\)
\(54\) 0 0
\(55\) 2.34315 0.315950
\(56\) 13.2426 1.76962
\(57\) 0 0
\(58\) 7.65685 1.00539
\(59\) 8.82843 1.14936 0.574682 0.818377i \(-0.305126\pi\)
0.574682 + 0.818377i \(0.305126\pi\)
\(60\) 0 0
\(61\) 4.65685 0.596249 0.298125 0.954527i \(-0.403639\pi\)
0.298125 + 0.954527i \(0.403639\pi\)
\(62\) −11.2426 −1.42782
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) −18.8284 −2.33538
\(66\) 0 0
\(67\) 1.00000 0.122169 0.0610847 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −20.4853 −2.44846
\(71\) −5.17157 −0.613753 −0.306876 0.951749i \(-0.599284\pi\)
−0.306876 + 0.951749i \(0.599284\pi\)
\(72\) 0 0
\(73\) 7.65685 0.896167 0.448084 0.893992i \(-0.352107\pi\)
0.448084 + 0.893992i \(0.352107\pi\)
\(74\) 16.0711 1.86822
\(75\) 0 0
\(76\) −11.4853 −1.31745
\(77\) 2.48528 0.283224
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −8.48528 −0.948683
\(81\) 0 0
\(82\) −21.3137 −2.35371
\(83\) −1.17157 −0.128597 −0.0642984 0.997931i \(-0.520481\pi\)
−0.0642984 + 0.997931i \(0.520481\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.24264 −0.780994
\(87\) 0 0
\(88\) 3.65685 0.389822
\(89\) −5.31371 −0.563252 −0.281626 0.959524i \(-0.590874\pi\)
−0.281626 + 0.959524i \(0.590874\pi\)
\(90\) 0 0
\(91\) −19.9706 −2.09348
\(92\) −10.8284 −1.12894
\(93\) 0 0
\(94\) −23.3137 −2.40463
\(95\) 8.48528 0.870572
\(96\) 0 0
\(97\) 10.6569 1.08204 0.541020 0.841010i \(-0.318038\pi\)
0.541020 + 0.841010i \(0.318038\pi\)
\(98\) −4.82843 −0.487745
\(99\) 0 0
\(100\) 11.4853 1.14853
\(101\) 11.6569 1.15990 0.579950 0.814652i \(-0.303072\pi\)
0.579950 + 0.814652i \(0.303072\pi\)
\(102\) 0 0
\(103\) −15.0000 −1.47799 −0.738997 0.673709i \(-0.764700\pi\)
−0.738997 + 0.673709i \(0.764700\pi\)
\(104\) −29.3848 −2.88141
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 3.65685 0.350263 0.175132 0.984545i \(-0.443965\pi\)
0.175132 + 0.984545i \(0.443965\pi\)
\(110\) −5.65685 −0.539360
\(111\) 0 0
\(112\) −9.00000 −0.850420
\(113\) 3.65685 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −12.1421 −1.12737
\(117\) 0 0
\(118\) −21.3137 −1.96209
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) −11.2426 −1.01786
\(123\) 0 0
\(124\) 17.8284 1.60104
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) −1.34315 −0.119185 −0.0595925 0.998223i \(-0.518980\pi\)
−0.0595925 + 0.998223i \(0.518980\pi\)
\(128\) 20.5563 1.81694
\(129\) 0 0
\(130\) 45.4558 3.98674
\(131\) −9.17157 −0.801324 −0.400662 0.916226i \(-0.631220\pi\)
−0.400662 + 0.916226i \(0.631220\pi\)
\(132\) 0 0
\(133\) 9.00000 0.780399
\(134\) −2.41421 −0.208556
\(135\) 0 0
\(136\) 0 0
\(137\) −17.7990 −1.52067 −0.760335 0.649531i \(-0.774965\pi\)
−0.760335 + 0.649531i \(0.774965\pi\)
\(138\) 0 0
\(139\) −5.34315 −0.453200 −0.226600 0.973988i \(-0.572761\pi\)
−0.226600 + 0.973988i \(0.572761\pi\)
\(140\) 32.4853 2.74551
\(141\) 0 0
\(142\) 12.4853 1.04774
\(143\) −5.51472 −0.461164
\(144\) 0 0
\(145\) 8.97056 0.744965
\(146\) −18.4853 −1.52985
\(147\) 0 0
\(148\) −25.4853 −2.09488
\(149\) −18.8284 −1.54248 −0.771242 0.636542i \(-0.780364\pi\)
−0.771242 + 0.636542i \(0.780364\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 13.2426 1.07412
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) −13.1716 −1.05797
\(156\) 0 0
\(157\) −17.9706 −1.43421 −0.717104 0.696967i \(-0.754533\pi\)
−0.717104 + 0.696967i \(0.754533\pi\)
\(158\) 28.9706 2.30477
\(159\) 0 0
\(160\) −4.48528 −0.354593
\(161\) 8.48528 0.668734
\(162\) 0 0
\(163\) −1.65685 −0.129775 −0.0648874 0.997893i \(-0.520669\pi\)
−0.0648874 + 0.997893i \(0.520669\pi\)
\(164\) 33.7990 2.63926
\(165\) 0 0
\(166\) 2.82843 0.219529
\(167\) 9.31371 0.720716 0.360358 0.932814i \(-0.382654\pi\)
0.360358 + 0.932814i \(0.382654\pi\)
\(168\) 0 0
\(169\) 31.3137 2.40875
\(170\) 0 0
\(171\) 0 0
\(172\) 11.4853 0.875744
\(173\) 21.3137 1.62045 0.810226 0.586118i \(-0.199345\pi\)
0.810226 + 0.586118i \(0.199345\pi\)
\(174\) 0 0
\(175\) −9.00000 −0.680336
\(176\) −2.48528 −0.187335
\(177\) 0 0
\(178\) 12.8284 0.961531
\(179\) −12.1421 −0.907546 −0.453773 0.891117i \(-0.649922\pi\)
−0.453773 + 0.891117i \(0.649922\pi\)
\(180\) 0 0
\(181\) −19.9706 −1.48440 −0.742200 0.670178i \(-0.766218\pi\)
−0.742200 + 0.670178i \(0.766218\pi\)
\(182\) 48.2132 3.57380
\(183\) 0 0
\(184\) 12.4853 0.920427
\(185\) 18.8284 1.38429
\(186\) 0 0
\(187\) 0 0
\(188\) 36.9706 2.69636
\(189\) 0 0
\(190\) −20.4853 −1.48616
\(191\) −26.1421 −1.89158 −0.945789 0.324781i \(-0.894709\pi\)
−0.945789 + 0.324781i \(0.894709\pi\)
\(192\) 0 0
\(193\) 10.3137 0.742397 0.371198 0.928554i \(-0.378947\pi\)
0.371198 + 0.928554i \(0.378947\pi\)
\(194\) −25.7279 −1.84716
\(195\) 0 0
\(196\) 7.65685 0.546918
\(197\) 2.34315 0.166942 0.0834711 0.996510i \(-0.473399\pi\)
0.0834711 + 0.996510i \(0.473399\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −13.2426 −0.936396
\(201\) 0 0
\(202\) −28.1421 −1.98007
\(203\) 9.51472 0.667802
\(204\) 0 0
\(205\) −24.9706 −1.74402
\(206\) 36.2132 2.52309
\(207\) 0 0
\(208\) 19.9706 1.38471
\(209\) 2.48528 0.171911
\(210\) 0 0
\(211\) −21.9706 −1.51252 −0.756258 0.654274i \(-0.772974\pi\)
−0.756258 + 0.654274i \(0.772974\pi\)
\(212\) 6.34315 0.435649
\(213\) 0 0
\(214\) −14.4853 −0.990193
\(215\) −8.48528 −0.578691
\(216\) 0 0
\(217\) −13.9706 −0.948384
\(218\) −8.82843 −0.597937
\(219\) 0 0
\(220\) 8.97056 0.604795
\(221\) 0 0
\(222\) 0 0
\(223\) 26.6274 1.78310 0.891552 0.452919i \(-0.149617\pi\)
0.891552 + 0.452919i \(0.149617\pi\)
\(224\) −4.75736 −0.317864
\(225\) 0 0
\(226\) −8.82843 −0.587258
\(227\) −26.4853 −1.75789 −0.878945 0.476923i \(-0.841752\pi\)
−0.878945 + 0.476923i \(0.841752\pi\)
\(228\) 0 0
\(229\) 22.9706 1.51794 0.758969 0.651127i \(-0.225704\pi\)
0.758969 + 0.651127i \(0.225704\pi\)
\(230\) −19.3137 −1.27351
\(231\) 0 0
\(232\) 14.0000 0.919145
\(233\) −17.6569 −1.15674 −0.578369 0.815775i \(-0.696311\pi\)
−0.578369 + 0.815775i \(0.696311\pi\)
\(234\) 0 0
\(235\) −27.3137 −1.78175
\(236\) 33.7990 2.20013
\(237\) 0 0
\(238\) 0 0
\(239\) 4.34315 0.280935 0.140467 0.990085i \(-0.455139\pi\)
0.140467 + 0.990085i \(0.455139\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 24.8995 1.60060
\(243\) 0 0
\(244\) 17.8284 1.14135
\(245\) −5.65685 −0.361403
\(246\) 0 0
\(247\) −19.9706 −1.27070
\(248\) −20.5563 −1.30533
\(249\) 0 0
\(250\) −13.6569 −0.863735
\(251\) 4.82843 0.304768 0.152384 0.988321i \(-0.451305\pi\)
0.152384 + 0.988321i \(0.451305\pi\)
\(252\) 0 0
\(253\) 2.34315 0.147312
\(254\) 3.24264 0.203461
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 19.9706 1.24091
\(260\) −72.0833 −4.47041
\(261\) 0 0
\(262\) 22.1421 1.36795
\(263\) −25.3137 −1.56091 −0.780455 0.625212i \(-0.785013\pi\)
−0.780455 + 0.625212i \(0.785013\pi\)
\(264\) 0 0
\(265\) −4.68629 −0.287877
\(266\) −21.7279 −1.33222
\(267\) 0 0
\(268\) 3.82843 0.233858
\(269\) −5.17157 −0.315316 −0.157658 0.987494i \(-0.550394\pi\)
−0.157658 + 0.987494i \(0.550394\pi\)
\(270\) 0 0
\(271\) −24.3137 −1.47695 −0.738476 0.674279i \(-0.764454\pi\)
−0.738476 + 0.674279i \(0.764454\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 42.9706 2.59595
\(275\) −2.48528 −0.149868
\(276\) 0 0
\(277\) −7.97056 −0.478905 −0.239452 0.970908i \(-0.576968\pi\)
−0.239452 + 0.970908i \(0.576968\pi\)
\(278\) 12.8995 0.773660
\(279\) 0 0
\(280\) −37.4558 −2.23841
\(281\) −3.17157 −0.189200 −0.0946001 0.995515i \(-0.530157\pi\)
−0.0946001 + 0.995515i \(0.530157\pi\)
\(282\) 0 0
\(283\) 3.00000 0.178331 0.0891657 0.996017i \(-0.471580\pi\)
0.0891657 + 0.996017i \(0.471580\pi\)
\(284\) −19.7990 −1.17485
\(285\) 0 0
\(286\) 13.3137 0.787256
\(287\) −26.4853 −1.56338
\(288\) 0 0
\(289\) 0 0
\(290\) −21.6569 −1.27173
\(291\) 0 0
\(292\) 29.3137 1.71546
\(293\) −0.686292 −0.0400936 −0.0200468 0.999799i \(-0.506382\pi\)
−0.0200468 + 0.999799i \(0.506382\pi\)
\(294\) 0 0
\(295\) −24.9706 −1.45384
\(296\) 29.3848 1.70796
\(297\) 0 0
\(298\) 45.4558 2.63319
\(299\) −18.8284 −1.08888
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) −12.0711 −0.694612
\(303\) 0 0
\(304\) −9.00000 −0.516185
\(305\) −13.1716 −0.754202
\(306\) 0 0
\(307\) 9.65685 0.551146 0.275573 0.961280i \(-0.411132\pi\)
0.275573 + 0.961280i \(0.411132\pi\)
\(308\) 9.51472 0.542151
\(309\) 0 0
\(310\) 31.7990 1.80606
\(311\) −22.9706 −1.30254 −0.651271 0.758846i \(-0.725764\pi\)
−0.651271 + 0.758846i \(0.725764\pi\)
\(312\) 0 0
\(313\) 13.3431 0.754199 0.377100 0.926173i \(-0.376921\pi\)
0.377100 + 0.926173i \(0.376921\pi\)
\(314\) 43.3848 2.44834
\(315\) 0 0
\(316\) −45.9411 −2.58439
\(317\) −20.4853 −1.15057 −0.575284 0.817954i \(-0.695108\pi\)
−0.575284 + 0.817954i \(0.695108\pi\)
\(318\) 0 0
\(319\) 2.62742 0.147107
\(320\) 27.7990 1.55401
\(321\) 0 0
\(322\) −20.4853 −1.14160
\(323\) 0 0
\(324\) 0 0
\(325\) 19.9706 1.10777
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −38.9706 −2.15179
\(329\) −28.9706 −1.59720
\(330\) 0 0
\(331\) −7.34315 −0.403616 −0.201808 0.979425i \(-0.564682\pi\)
−0.201808 + 0.979425i \(0.564682\pi\)
\(332\) −4.48528 −0.246162
\(333\) 0 0
\(334\) −22.4853 −1.23034
\(335\) −2.82843 −0.154533
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −75.5980 −4.11199
\(339\) 0 0
\(340\) 0 0
\(341\) −3.85786 −0.208915
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) −13.2426 −0.713995
\(345\) 0 0
\(346\) −51.4558 −2.76628
\(347\) −15.7990 −0.848134 −0.424067 0.905631i \(-0.639398\pi\)
−0.424067 + 0.905631i \(0.639398\pi\)
\(348\) 0 0
\(349\) −32.6274 −1.74651 −0.873253 0.487267i \(-0.837994\pi\)
−0.873253 + 0.487267i \(0.837994\pi\)
\(350\) 21.7279 1.16141
\(351\) 0 0
\(352\) −1.31371 −0.0700209
\(353\) −11.3137 −0.602168 −0.301084 0.953598i \(-0.597348\pi\)
−0.301084 + 0.953598i \(0.597348\pi\)
\(354\) 0 0
\(355\) 14.6274 0.776343
\(356\) −20.3431 −1.07818
\(357\) 0 0
\(358\) 29.3137 1.54928
\(359\) −18.3431 −0.968114 −0.484057 0.875036i \(-0.660837\pi\)
−0.484057 + 0.875036i \(0.660837\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 48.2132 2.53403
\(363\) 0 0
\(364\) −76.4558 −4.00738
\(365\) −21.6569 −1.13357
\(366\) 0 0
\(367\) 17.3431 0.905305 0.452652 0.891687i \(-0.350478\pi\)
0.452652 + 0.891687i \(0.350478\pi\)
\(368\) −8.48528 −0.442326
\(369\) 0 0
\(370\) −45.4558 −2.36314
\(371\) −4.97056 −0.258059
\(372\) 0 0
\(373\) 31.6569 1.63913 0.819565 0.572986i \(-0.194215\pi\)
0.819565 + 0.572986i \(0.194215\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −42.6274 −2.19834
\(377\) −21.1127 −1.08736
\(378\) 0 0
\(379\) −3.00000 −0.154100 −0.0770498 0.997027i \(-0.524550\pi\)
−0.0770498 + 0.997027i \(0.524550\pi\)
\(380\) 32.4853 1.66646
\(381\) 0 0
\(382\) 63.1127 3.22913
\(383\) 28.1421 1.43800 0.718998 0.695012i \(-0.244601\pi\)
0.718998 + 0.695012i \(0.244601\pi\)
\(384\) 0 0
\(385\) −7.02944 −0.358253
\(386\) −24.8995 −1.26735
\(387\) 0 0
\(388\) 40.7990 2.07125
\(389\) −0.142136 −0.00720656 −0.00360328 0.999994i \(-0.501147\pi\)
−0.00360328 + 0.999994i \(0.501147\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −8.82843 −0.445903
\(393\) 0 0
\(394\) −5.65685 −0.284988
\(395\) 33.9411 1.70776
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 19.3137 0.968109
\(399\) 0 0
\(400\) 9.00000 0.450000
\(401\) 31.7990 1.58797 0.793983 0.607940i \(-0.208004\pi\)
0.793983 + 0.607940i \(0.208004\pi\)
\(402\) 0 0
\(403\) 31.0000 1.54422
\(404\) 44.6274 2.22030
\(405\) 0 0
\(406\) −22.9706 −1.14001
\(407\) 5.51472 0.273354
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 60.2843 2.97723
\(411\) 0 0
\(412\) −57.4264 −2.82920
\(413\) −26.4853 −1.30326
\(414\) 0 0
\(415\) 3.31371 0.162664
\(416\) 10.5563 0.517568
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 22.1421 1.08171 0.540857 0.841115i \(-0.318100\pi\)
0.540857 + 0.841115i \(0.318100\pi\)
\(420\) 0 0
\(421\) 13.6863 0.667029 0.333515 0.942745i \(-0.391765\pi\)
0.333515 + 0.942745i \(0.391765\pi\)
\(422\) 53.0416 2.58203
\(423\) 0 0
\(424\) −7.31371 −0.355185
\(425\) 0 0
\(426\) 0 0
\(427\) −13.9706 −0.676083
\(428\) 22.9706 1.11032
\(429\) 0 0
\(430\) 20.4853 0.987888
\(431\) −40.9706 −1.97348 −0.986741 0.162301i \(-0.948108\pi\)
−0.986741 + 0.162301i \(0.948108\pi\)
\(432\) 0 0
\(433\) 2.02944 0.0975285 0.0487643 0.998810i \(-0.484472\pi\)
0.0487643 + 0.998810i \(0.484472\pi\)
\(434\) 33.7279 1.61899
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 8.48528 0.405906
\(438\) 0 0
\(439\) −22.3137 −1.06498 −0.532488 0.846438i \(-0.678743\pi\)
−0.532488 + 0.846438i \(0.678743\pi\)
\(440\) −10.3431 −0.493090
\(441\) 0 0
\(442\) 0 0
\(443\) −26.8284 −1.27466 −0.637329 0.770592i \(-0.719961\pi\)
−0.637329 + 0.770592i \(0.719961\pi\)
\(444\) 0 0
\(445\) 15.0294 0.712464
\(446\) −64.2843 −3.04395
\(447\) 0 0
\(448\) 29.4853 1.39305
\(449\) 1.02944 0.0485821 0.0242911 0.999705i \(-0.492267\pi\)
0.0242911 + 0.999705i \(0.492267\pi\)
\(450\) 0 0
\(451\) −7.31371 −0.344389
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 63.9411 3.00091
\(455\) 56.4853 2.64807
\(456\) 0 0
\(457\) 2.68629 0.125659 0.0628297 0.998024i \(-0.479988\pi\)
0.0628297 + 0.998024i \(0.479988\pi\)
\(458\) −55.4558 −2.59128
\(459\) 0 0
\(460\) 30.6274 1.42801
\(461\) −16.6274 −0.774416 −0.387208 0.921992i \(-0.626560\pi\)
−0.387208 + 0.921992i \(0.626560\pi\)
\(462\) 0 0
\(463\) −10.3137 −0.479319 −0.239659 0.970857i \(-0.577036\pi\)
−0.239659 + 0.970857i \(0.577036\pi\)
\(464\) −9.51472 −0.441710
\(465\) 0 0
\(466\) 42.6274 1.97468
\(467\) 4.97056 0.230010 0.115005 0.993365i \(-0.463312\pi\)
0.115005 + 0.993365i \(0.463312\pi\)
\(468\) 0 0
\(469\) −3.00000 −0.138527
\(470\) 65.9411 3.04164
\(471\) 0 0
\(472\) −38.9706 −1.79377
\(473\) −2.48528 −0.114273
\(474\) 0 0
\(475\) −9.00000 −0.412948
\(476\) 0 0
\(477\) 0 0
\(478\) −10.4853 −0.479586
\(479\) 12.1421 0.554788 0.277394 0.960756i \(-0.410529\pi\)
0.277394 + 0.960756i \(0.410529\pi\)
\(480\) 0 0
\(481\) −44.3137 −2.02053
\(482\) 14.4853 0.659786
\(483\) 0 0
\(484\) −39.4853 −1.79479
\(485\) −30.1421 −1.36868
\(486\) 0 0
\(487\) 33.9706 1.53935 0.769677 0.638434i \(-0.220417\pi\)
0.769677 + 0.638434i \(0.220417\pi\)
\(488\) −20.5563 −0.930542
\(489\) 0 0
\(490\) 13.6569 0.616954
\(491\) 10.9706 0.495095 0.247547 0.968876i \(-0.420375\pi\)
0.247547 + 0.968876i \(0.420375\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 48.2132 2.16921
\(495\) 0 0
\(496\) 13.9706 0.627297
\(497\) 15.5147 0.695930
\(498\) 0 0
\(499\) 23.9706 1.07307 0.536535 0.843878i \(-0.319733\pi\)
0.536535 + 0.843878i \(0.319733\pi\)
\(500\) 21.6569 0.968524
\(501\) 0 0
\(502\) −11.6569 −0.520271
\(503\) −24.1421 −1.07644 −0.538222 0.842803i \(-0.680904\pi\)
−0.538222 + 0.842803i \(0.680904\pi\)
\(504\) 0 0
\(505\) −32.9706 −1.46717
\(506\) −5.65685 −0.251478
\(507\) 0 0
\(508\) −5.14214 −0.228145
\(509\) −11.6569 −0.516681 −0.258340 0.966054i \(-0.583176\pi\)
−0.258340 + 0.966054i \(0.583176\pi\)
\(510\) 0 0
\(511\) −22.9706 −1.01616
\(512\) 31.2426 1.38074
\(513\) 0 0
\(514\) 0 0
\(515\) 42.4264 1.86953
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) −48.2132 −2.11837
\(519\) 0 0
\(520\) 83.1127 3.64473
\(521\) −18.1421 −0.794821 −0.397411 0.917641i \(-0.630091\pi\)
−0.397411 + 0.917641i \(0.630091\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −35.1127 −1.53391
\(525\) 0 0
\(526\) 61.1127 2.66464
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 11.3137 0.491436
\(531\) 0 0
\(532\) 34.4558 1.49385
\(533\) 58.7696 2.54559
\(534\) 0 0
\(535\) −16.9706 −0.733701
\(536\) −4.41421 −0.190665
\(537\) 0 0
\(538\) 12.4853 0.538279
\(539\) −1.65685 −0.0713658
\(540\) 0 0
\(541\) −27.9411 −1.20128 −0.600641 0.799519i \(-0.705088\pi\)
−0.600641 + 0.799519i \(0.705088\pi\)
\(542\) 58.6985 2.52132
\(543\) 0 0
\(544\) 0 0
\(545\) −10.3431 −0.443052
\(546\) 0 0
\(547\) −5.00000 −0.213785 −0.106892 0.994271i \(-0.534090\pi\)
−0.106892 + 0.994271i \(0.534090\pi\)
\(548\) −68.1421 −2.91089
\(549\) 0 0
\(550\) 6.00000 0.255841
\(551\) 9.51472 0.405341
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 19.2426 0.817541
\(555\) 0 0
\(556\) −20.4558 −0.867521
\(557\) −21.6569 −0.917630 −0.458815 0.888532i \(-0.651726\pi\)
−0.458815 + 0.888532i \(0.651726\pi\)
\(558\) 0 0
\(559\) 19.9706 0.844665
\(560\) 25.4558 1.07571
\(561\) 0 0
\(562\) 7.65685 0.322985
\(563\) −4.62742 −0.195022 −0.0975112 0.995234i \(-0.531088\pi\)
−0.0975112 + 0.995234i \(0.531088\pi\)
\(564\) 0 0
\(565\) −10.3431 −0.435139
\(566\) −7.24264 −0.304431
\(567\) 0 0
\(568\) 22.8284 0.957860
\(569\) 16.3431 0.685140 0.342570 0.939492i \(-0.388703\pi\)
0.342570 + 0.939492i \(0.388703\pi\)
\(570\) 0 0
\(571\) 30.9411 1.29485 0.647423 0.762131i \(-0.275847\pi\)
0.647423 + 0.762131i \(0.275847\pi\)
\(572\) −21.1127 −0.882766
\(573\) 0 0
\(574\) 63.9411 2.66885
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −13.9706 −0.581602 −0.290801 0.956784i \(-0.593922\pi\)
−0.290801 + 0.956784i \(0.593922\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 34.3431 1.42602
\(581\) 3.51472 0.145815
\(582\) 0 0
\(583\) −1.37258 −0.0568466
\(584\) −33.7990 −1.39861
\(585\) 0 0
\(586\) 1.65685 0.0684440
\(587\) 34.8284 1.43752 0.718762 0.695257i \(-0.244709\pi\)
0.718762 + 0.695257i \(0.244709\pi\)
\(588\) 0 0
\(589\) −13.9706 −0.575647
\(590\) 60.2843 2.48186
\(591\) 0 0
\(592\) −19.9706 −0.820785
\(593\) 32.8284 1.34810 0.674051 0.738685i \(-0.264553\pi\)
0.674051 + 0.738685i \(0.264553\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −72.0833 −2.95265
\(597\) 0 0
\(598\) 45.4558 1.85883
\(599\) −20.8284 −0.851026 −0.425513 0.904952i \(-0.639906\pi\)
−0.425513 + 0.904952i \(0.639906\pi\)
\(600\) 0 0
\(601\) 16.6569 0.679447 0.339724 0.940525i \(-0.389666\pi\)
0.339724 + 0.940525i \(0.389666\pi\)
\(602\) 21.7279 0.885564
\(603\) 0 0
\(604\) 19.1421 0.778882
\(605\) 29.1716 1.18599
\(606\) 0 0
\(607\) −15.0000 −0.608831 −0.304416 0.952539i \(-0.598461\pi\)
−0.304416 + 0.952539i \(0.598461\pi\)
\(608\) −4.75736 −0.192936
\(609\) 0 0
\(610\) 31.7990 1.28750
\(611\) 64.2843 2.60066
\(612\) 0 0
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) −23.3137 −0.940865
\(615\) 0 0
\(616\) −10.9706 −0.442017
\(617\) −46.6274 −1.87715 −0.938575 0.345076i \(-0.887853\pi\)
−0.938575 + 0.345076i \(0.887853\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −50.4264 −2.02517
\(621\) 0 0
\(622\) 55.4558 2.22358
\(623\) 15.9411 0.638668
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) −32.2132 −1.28750
\(627\) 0 0
\(628\) −68.7990 −2.74538
\(629\) 0 0
\(630\) 0 0
\(631\) −33.9706 −1.35235 −0.676173 0.736743i \(-0.736363\pi\)
−0.676173 + 0.736743i \(0.736363\pi\)
\(632\) 52.9706 2.10706
\(633\) 0 0
\(634\) 49.4558 1.96414
\(635\) 3.79899 0.150758
\(636\) 0 0
\(637\) 13.3137 0.527508
\(638\) −6.34315 −0.251128
\(639\) 0 0
\(640\) −58.1421 −2.29827
\(641\) 17.7990 0.703018 0.351509 0.936185i \(-0.385669\pi\)
0.351509 + 0.936185i \(0.385669\pi\)
\(642\) 0 0
\(643\) −24.6569 −0.972371 −0.486186 0.873856i \(-0.661612\pi\)
−0.486186 + 0.873856i \(0.661612\pi\)
\(644\) 32.4853 1.28010
\(645\) 0 0
\(646\) 0 0
\(647\) 1.02944 0.0404714 0.0202357 0.999795i \(-0.493558\pi\)
0.0202357 + 0.999795i \(0.493558\pi\)
\(648\) 0 0
\(649\) −7.31371 −0.287088
\(650\) −48.2132 −1.89108
\(651\) 0 0
\(652\) −6.34315 −0.248417
\(653\) 41.2548 1.61443 0.807213 0.590260i \(-0.200975\pi\)
0.807213 + 0.590260i \(0.200975\pi\)
\(654\) 0 0
\(655\) 25.9411 1.01360
\(656\) 26.4853 1.03408
\(657\) 0 0
\(658\) 69.9411 2.72659
\(659\) −14.2843 −0.556436 −0.278218 0.960518i \(-0.589744\pi\)
−0.278218 + 0.960518i \(0.589744\pi\)
\(660\) 0 0
\(661\) −4.02944 −0.156727 −0.0783635 0.996925i \(-0.524969\pi\)
−0.0783635 + 0.996925i \(0.524969\pi\)
\(662\) 17.7279 0.689015
\(663\) 0 0
\(664\) 5.17157 0.200696
\(665\) −25.4558 −0.987135
\(666\) 0 0
\(667\) 8.97056 0.347342
\(668\) 35.6569 1.37961
\(669\) 0 0
\(670\) 6.82843 0.263805
\(671\) −3.85786 −0.148931
\(672\) 0 0
\(673\) −18.9411 −0.730127 −0.365063 0.930983i \(-0.618953\pi\)
−0.365063 + 0.930983i \(0.618953\pi\)
\(674\) −4.82843 −0.185984
\(675\) 0 0
\(676\) 119.882 4.61086
\(677\) 26.4853 1.01791 0.508956 0.860793i \(-0.330032\pi\)
0.508956 + 0.860793i \(0.330032\pi\)
\(678\) 0 0
\(679\) −31.9706 −1.22692
\(680\) 0 0
\(681\) 0 0
\(682\) 9.31371 0.356640
\(683\) 0.343146 0.0131301 0.00656505 0.999978i \(-0.497910\pi\)
0.00656505 + 0.999978i \(0.497910\pi\)
\(684\) 0 0
\(685\) 50.3431 1.92351
\(686\) −36.2132 −1.38263
\(687\) 0 0
\(688\) 9.00000 0.343122
\(689\) 11.0294 0.420188
\(690\) 0 0
\(691\) −39.2843 −1.49444 −0.747222 0.664574i \(-0.768613\pi\)
−0.747222 + 0.664574i \(0.768613\pi\)
\(692\) 81.5980 3.10189
\(693\) 0 0
\(694\) 38.1421 1.44786
\(695\) 15.1127 0.573257
\(696\) 0 0
\(697\) 0 0
\(698\) 78.7696 2.98147
\(699\) 0 0
\(700\) −34.4558 −1.30231
\(701\) 26.4853 1.00034 0.500168 0.865929i \(-0.333272\pi\)
0.500168 + 0.865929i \(0.333272\pi\)
\(702\) 0 0
\(703\) 19.9706 0.753204
\(704\) 8.14214 0.306868
\(705\) 0 0
\(706\) 27.3137 1.02796
\(707\) −34.9706 −1.31520
\(708\) 0 0
\(709\) −18.0294 −0.677110 −0.338555 0.940947i \(-0.609938\pi\)
−0.338555 + 0.940947i \(0.609938\pi\)
\(710\) −35.3137 −1.32530
\(711\) 0 0
\(712\) 23.4558 0.879045
\(713\) −13.1716 −0.493279
\(714\) 0 0
\(715\) 15.5980 0.583331
\(716\) −46.4853 −1.73724
\(717\) 0 0
\(718\) 44.2843 1.65267
\(719\) 4.62742 0.172574 0.0862868 0.996270i \(-0.472500\pi\)
0.0862868 + 0.996270i \(0.472500\pi\)
\(720\) 0 0
\(721\) 45.0000 1.67589
\(722\) 24.1421 0.898477
\(723\) 0 0
\(724\) −76.4558 −2.84146
\(725\) −9.51472 −0.353368
\(726\) 0 0
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 88.1543 3.26722
\(729\) 0 0
\(730\) 52.2843 1.93513
\(731\) 0 0
\(732\) 0 0
\(733\) −21.6274 −0.798827 −0.399413 0.916771i \(-0.630786\pi\)
−0.399413 + 0.916771i \(0.630786\pi\)
\(734\) −41.8701 −1.54545
\(735\) 0 0
\(736\) −4.48528 −0.165330
\(737\) −0.828427 −0.0305155
\(738\) 0 0
\(739\) −20.0294 −0.736795 −0.368397 0.929668i \(-0.620093\pi\)
−0.368397 + 0.929668i \(0.620093\pi\)
\(740\) 72.0833 2.64983
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 20.8284 0.764121 0.382060 0.924137i \(-0.375215\pi\)
0.382060 + 0.924137i \(0.375215\pi\)
\(744\) 0 0
\(745\) 53.2548 1.95111
\(746\) −76.4264 −2.79817
\(747\) 0 0
\(748\) 0 0
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) −7.00000 −0.255434 −0.127717 0.991811i \(-0.540765\pi\)
−0.127717 + 0.991811i \(0.540765\pi\)
\(752\) 28.9706 1.05645
\(753\) 0 0
\(754\) 50.9706 1.85624
\(755\) −14.1421 −0.514685
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 7.24264 0.263065
\(759\) 0 0
\(760\) −37.4558 −1.35867
\(761\) 43.1127 1.56283 0.781417 0.624009i \(-0.214497\pi\)
0.781417 + 0.624009i \(0.214497\pi\)
\(762\) 0 0
\(763\) −10.9706 −0.397161
\(764\) −100.083 −3.62089
\(765\) 0 0
\(766\) −67.9411 −2.45481
\(767\) 58.7696 2.12205
\(768\) 0 0
\(769\) −3.37258 −0.121618 −0.0608092 0.998149i \(-0.519368\pi\)
−0.0608092 + 0.998149i \(0.519368\pi\)
\(770\) 16.9706 0.611577
\(771\) 0 0
\(772\) 39.4853 1.42111
\(773\) −34.9706 −1.25780 −0.628902 0.777485i \(-0.716495\pi\)
−0.628902 + 0.777485i \(0.716495\pi\)
\(774\) 0 0
\(775\) 13.9706 0.501837
\(776\) −47.0416 −1.68870
\(777\) 0 0
\(778\) 0.343146 0.0123024
\(779\) −26.4853 −0.948934
\(780\) 0 0
\(781\) 4.28427 0.153303
\(782\) 0 0
\(783\) 0 0
\(784\) 6.00000 0.214286
\(785\) 50.8284 1.81414
\(786\) 0 0
\(787\) 28.5980 1.01941 0.509704 0.860350i \(-0.329755\pi\)
0.509704 + 0.860350i \(0.329755\pi\)
\(788\) 8.97056 0.319563
\(789\) 0 0
\(790\) −81.9411 −2.91533
\(791\) −10.9706 −0.390068
\(792\) 0 0
\(793\) 31.0000 1.10084
\(794\) 53.1127 1.88490
\(795\) 0 0
\(796\) −30.6274 −1.08556
\(797\) −26.6863 −0.945277 −0.472638 0.881256i \(-0.656698\pi\)
−0.472638 + 0.881256i \(0.656698\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.75736 0.168198
\(801\) 0 0
\(802\) −76.7696 −2.71083
\(803\) −6.34315 −0.223845
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) −74.8406 −2.63615
\(807\) 0 0
\(808\) −51.4558 −1.81021
\(809\) 20.4853 0.720224 0.360112 0.932909i \(-0.382738\pi\)
0.360112 + 0.932909i \(0.382738\pi\)
\(810\) 0 0
\(811\) 1.97056 0.0691958 0.0345979 0.999401i \(-0.488985\pi\)
0.0345979 + 0.999401i \(0.488985\pi\)
\(812\) 36.4264 1.27832
\(813\) 0 0
\(814\) −13.3137 −0.466645
\(815\) 4.68629 0.164154
\(816\) 0 0
\(817\) −9.00000 −0.314870
\(818\) −12.0711 −0.422055
\(819\) 0 0
\(820\) −95.5980 −3.33843
\(821\) 24.1421 0.842566 0.421283 0.906929i \(-0.361580\pi\)
0.421283 + 0.906929i \(0.361580\pi\)
\(822\) 0 0
\(823\) 27.2843 0.951070 0.475535 0.879697i \(-0.342255\pi\)
0.475535 + 0.879697i \(0.342255\pi\)
\(824\) 66.2132 2.30665
\(825\) 0 0
\(826\) 63.9411 2.22480
\(827\) −43.4558 −1.51111 −0.755554 0.655087i \(-0.772632\pi\)
−0.755554 + 0.655087i \(0.772632\pi\)
\(828\) 0 0
\(829\) −21.6274 −0.751151 −0.375576 0.926792i \(-0.622555\pi\)
−0.375576 + 0.926792i \(0.622555\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −65.4264 −2.26825
\(833\) 0 0
\(834\) 0 0
\(835\) −26.3431 −0.911642
\(836\) 9.51472 0.329073
\(837\) 0 0
\(838\) −53.4558 −1.84660
\(839\) 34.9706 1.20732 0.603659 0.797243i \(-0.293709\pi\)
0.603659 + 0.797243i \(0.293709\pi\)
\(840\) 0 0
\(841\) −18.9411 −0.653142
\(842\) −33.0416 −1.13869
\(843\) 0 0
\(844\) −84.1127 −2.89528
\(845\) −88.5685 −3.04685
\(846\) 0 0
\(847\) 30.9411 1.06315
\(848\) 4.97056 0.170690
\(849\) 0 0
\(850\) 0 0
\(851\) 18.8284 0.645430
\(852\) 0 0
\(853\) −8.02944 −0.274923 −0.137461 0.990507i \(-0.543894\pi\)
−0.137461 + 0.990507i \(0.543894\pi\)
\(854\) 33.7279 1.15415
\(855\) 0 0
\(856\) −26.4853 −0.905248
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −10.0294 −0.342200 −0.171100 0.985254i \(-0.554732\pi\)
−0.171100 + 0.985254i \(0.554732\pi\)
\(860\) −32.4853 −1.10774
\(861\) 0 0
\(862\) 98.9117 3.36895
\(863\) 23.6569 0.805289 0.402644 0.915357i \(-0.368091\pi\)
0.402644 + 0.915357i \(0.368091\pi\)
\(864\) 0 0
\(865\) −60.2843 −2.04973
\(866\) −4.89949 −0.166492
\(867\) 0 0
\(868\) −53.4853 −1.81541
\(869\) 9.94113 0.337230
\(870\) 0 0
\(871\) 6.65685 0.225559
\(872\) −16.1421 −0.546642
\(873\) 0 0
\(874\) −20.4853 −0.692925
\(875\) −16.9706 −0.573710
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 53.8701 1.81803
\(879\) 0 0
\(880\) 7.02944 0.236962
\(881\) 20.6274 0.694955 0.347478 0.937688i \(-0.387038\pi\)
0.347478 + 0.937688i \(0.387038\pi\)
\(882\) 0 0
\(883\) 33.9411 1.14221 0.571105 0.820877i \(-0.306515\pi\)
0.571105 + 0.820877i \(0.306515\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 64.7696 2.17598
\(887\) 7.31371 0.245570 0.122785 0.992433i \(-0.460817\pi\)
0.122785 + 0.992433i \(0.460817\pi\)
\(888\) 0 0
\(889\) 4.02944 0.135143
\(890\) −36.2843 −1.21625
\(891\) 0 0
\(892\) 101.941 3.41324
\(893\) −28.9706 −0.969463
\(894\) 0 0
\(895\) 34.3431 1.14796
\(896\) −61.6690 −2.06022
\(897\) 0 0
\(898\) −2.48528 −0.0829349
\(899\) −14.7696 −0.492592
\(900\) 0 0
\(901\) 0 0
\(902\) 17.6569 0.587909
\(903\) 0 0
\(904\) −16.1421 −0.536879
\(905\) 56.4853 1.87763
\(906\) 0 0
\(907\) 1.34315 0.0445984 0.0222992 0.999751i \(-0.492901\pi\)
0.0222992 + 0.999751i \(0.492901\pi\)
\(908\) −101.397 −3.36498
\(909\) 0 0
\(910\) −136.368 −4.52054
\(911\) 6.14214 0.203498 0.101749 0.994810i \(-0.467556\pi\)
0.101749 + 0.994810i \(0.467556\pi\)
\(912\) 0 0
\(913\) 0.970563 0.0321209
\(914\) −6.48528 −0.214514
\(915\) 0 0
\(916\) 87.9411 2.90566
\(917\) 27.5147 0.908616
\(918\) 0 0
\(919\) 7.02944 0.231880 0.115940 0.993256i \(-0.463012\pi\)
0.115940 + 0.993256i \(0.463012\pi\)
\(920\) −35.3137 −1.16426
\(921\) 0 0
\(922\) 40.1421 1.32201
\(923\) −34.4264 −1.13316
\(924\) 0 0
\(925\) −19.9706 −0.656628
\(926\) 24.8995 0.818248
\(927\) 0 0
\(928\) −5.02944 −0.165099
\(929\) −19.5980 −0.642989 −0.321494 0.946911i \(-0.604185\pi\)
−0.321494 + 0.946911i \(0.604185\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −67.5980 −2.21425
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −17.9706 −0.587073 −0.293536 0.955948i \(-0.594832\pi\)
−0.293536 + 0.955948i \(0.594832\pi\)
\(938\) 7.24264 0.236481
\(939\) 0 0
\(940\) −104.569 −3.41065
\(941\) 7.02944 0.229153 0.114577 0.993414i \(-0.463449\pi\)
0.114577 + 0.993414i \(0.463449\pi\)
\(942\) 0 0
\(943\) −24.9706 −0.813153
\(944\) 26.4853 0.862022
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 1.02944 0.0334522 0.0167261 0.999860i \(-0.494676\pi\)
0.0167261 + 0.999860i \(0.494676\pi\)
\(948\) 0 0
\(949\) 50.9706 1.65457
\(950\) 21.7279 0.704947
\(951\) 0 0
\(952\) 0 0
\(953\) 31.1127 1.00784 0.503920 0.863751i \(-0.331891\pi\)
0.503920 + 0.863751i \(0.331891\pi\)
\(954\) 0 0
\(955\) 73.9411 2.39268
\(956\) 16.6274 0.537769
\(957\) 0 0
\(958\) −29.3137 −0.947083
\(959\) 53.3970 1.72428
\(960\) 0 0
\(961\) −9.31371 −0.300442
\(962\) 106.983 3.44926
\(963\) 0 0
\(964\) −22.9706 −0.739832
\(965\) −29.1716 −0.939066
\(966\) 0 0
\(967\) −60.2843 −1.93861 −0.969306 0.245858i \(-0.920930\pi\)
−0.969306 + 0.245858i \(0.920930\pi\)
\(968\) 45.5269 1.46329
\(969\) 0 0
\(970\) 72.7696 2.33649
\(971\) 9.37258 0.300781 0.150390 0.988627i \(-0.451947\pi\)
0.150390 + 0.988627i \(0.451947\pi\)
\(972\) 0 0
\(973\) 16.0294 0.513880
\(974\) −82.0122 −2.62784
\(975\) 0 0
\(976\) 13.9706 0.447187
\(977\) 37.1127 1.18734 0.593670 0.804708i \(-0.297678\pi\)
0.593670 + 0.804708i \(0.297678\pi\)
\(978\) 0 0
\(979\) 4.40202 0.140689
\(980\) −21.6569 −0.691803
\(981\) 0 0
\(982\) −26.4853 −0.845179
\(983\) −6.34315 −0.202315 −0.101157 0.994870i \(-0.532255\pi\)
−0.101157 + 0.994870i \(0.532255\pi\)
\(984\) 0 0
\(985\) −6.62742 −0.211167
\(986\) 0 0
\(987\) 0 0
\(988\) −76.4558 −2.43238
\(989\) −8.48528 −0.269816
\(990\) 0 0
\(991\) −40.9706 −1.30147 −0.650736 0.759304i \(-0.725540\pi\)
−0.650736 + 0.759304i \(0.725540\pi\)
\(992\) 7.38478 0.234467
\(993\) 0 0
\(994\) −37.4558 −1.18803
\(995\) 22.6274 0.717337
\(996\) 0 0
\(997\) 32.9411 1.04326 0.521628 0.853173i \(-0.325325\pi\)
0.521628 + 0.853173i \(0.325325\pi\)
\(998\) −57.8701 −1.83184
\(999\) 0 0
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 2601.2.a.m.1.1 2
3.2 odd 2 867.2.a.g.1.2 2
17.16 even 2 2601.2.a.n.1.1 2
51.2 odd 8 867.2.e.b.616.2 4
51.5 even 16 867.2.h.a.688.2 8
51.8 odd 8 867.2.e.c.829.1 4
51.11 even 16 867.2.h.h.733.2 8
51.14 even 16 867.2.h.h.757.2 8
51.20 even 16 867.2.h.h.757.1 8
51.23 even 16 867.2.h.h.733.1 8
51.26 odd 8 867.2.e.b.829.1 4
51.29 even 16 867.2.h.a.688.1 8
51.32 odd 8 867.2.e.c.616.2 4
51.38 odd 4 867.2.d.b.577.2 4
51.41 even 16 867.2.h.a.712.2 8
51.44 even 16 867.2.h.a.712.1 8
51.47 odd 4 867.2.d.b.577.1 4
51.50 odd 2 867.2.a.h.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
867.2.a.g.1.2 2 3.2 odd 2
867.2.a.h.1.2 yes 2 51.50 odd 2
867.2.d.b.577.1 4 51.47 odd 4
867.2.d.b.577.2 4 51.38 odd 4
867.2.e.b.616.2 4 51.2 odd 8
867.2.e.b.829.1 4 51.26 odd 8
867.2.e.c.616.2 4 51.32 odd 8
867.2.e.c.829.1 4 51.8 odd 8
867.2.h.a.688.1 8 51.29 even 16
867.2.h.a.688.2 8 51.5 even 16
867.2.h.a.712.1 8 51.44 even 16
867.2.h.a.712.2 8 51.41 even 16
867.2.h.h.733.1 8 51.23 even 16
867.2.h.h.733.2 8 51.11 even 16
867.2.h.h.757.1 8 51.20 even 16
867.2.h.h.757.2 8 51.14 even 16
2601.2.a.m.1.1 2 1.1 even 1 trivial
2601.2.a.n.1.1 2 17.16 even 2