Properties

Label 5625.2.a.ba.1.8
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5625,2,Mod(1,5625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5625, 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("5625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.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: \(44.9158511370\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.46980000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.44055\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.44055 q^{2} +3.95630 q^{4} +0.591023 q^{7} +4.77444 q^{8} +O(q^{10})\) \(q+2.44055 q^{2} +3.95630 q^{4} +0.591023 q^{7} +4.77444 q^{8} +4.26423 q^{11} +5.06316 q^{13} +1.44242 q^{14} +3.73968 q^{16} +5.39132 q^{17} -5.73968 q^{19} +10.4071 q^{22} -2.12522 q^{23} +12.3569 q^{26} +2.33826 q^{28} +9.48296 q^{29} -1.38761 q^{31} -0.421994 q^{32} +13.1578 q^{34} -11.3502 q^{37} -14.0080 q^{38} -0.403551 q^{41} +5.30211 q^{43} +16.8705 q^{44} -5.18672 q^{46} -8.60289 q^{47} -6.65069 q^{49} +20.0314 q^{52} -0.337629 q^{53} +2.82180 q^{56} +23.1437 q^{58} -2.73744 q^{59} +9.01478 q^{61} -3.38654 q^{62} -8.50926 q^{64} -5.86133 q^{67} +21.3296 q^{68} +10.7281 q^{71} +5.02136 q^{73} -27.7007 q^{74} -22.7079 q^{76} +2.52026 q^{77} +8.44321 q^{79} -0.984886 q^{82} +11.0136 q^{83} +12.9401 q^{86} +20.3593 q^{88} -13.1464 q^{89} +2.99244 q^{91} -8.40801 q^{92} -20.9958 q^{94} +7.24927 q^{97} -16.2314 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} - 18 q^{16} + 2 q^{19} + 20 q^{22} + 10 q^{28} + 4 q^{31} + 50 q^{34} + 50 q^{43} - 30 q^{46} - 6 q^{49} + 30 q^{52} + 60 q^{58} + 46 q^{61} - 14 q^{64} + 40 q^{67} + 50 q^{73} - 34 q^{76} - 12 q^{79} + 60 q^{82} + 70 q^{88} - 10 q^{91} - 20 q^{94} + 50 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.44055 1.72573 0.862866 0.505434i \(-0.168667\pi\)
0.862866 + 0.505434i \(0.168667\pi\)
\(3\) 0 0
\(4\) 3.95630 1.97815
\(5\) 0 0
\(6\) 0 0
\(7\) 0.591023 0.223386 0.111693 0.993743i \(-0.464373\pi\)
0.111693 + 0.993743i \(0.464373\pi\)
\(8\) 4.77444 1.68802
\(9\) 0 0
\(10\) 0 0
\(11\) 4.26423 1.28571 0.642856 0.765987i \(-0.277749\pi\)
0.642856 + 0.765987i \(0.277749\pi\)
\(12\) 0 0
\(13\) 5.06316 1.40427 0.702134 0.712045i \(-0.252231\pi\)
0.702134 + 0.712045i \(0.252231\pi\)
\(14\) 1.44242 0.385504
\(15\) 0 0
\(16\) 3.73968 0.934920
\(17\) 5.39132 1.30759 0.653793 0.756673i \(-0.273177\pi\)
0.653793 + 0.756673i \(0.273177\pi\)
\(18\) 0 0
\(19\) −5.73968 −1.31677 −0.658387 0.752680i \(-0.728761\pi\)
−0.658387 + 0.752680i \(0.728761\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 10.4071 2.21879
\(23\) −2.12522 −0.443140 −0.221570 0.975145i \(-0.571118\pi\)
−0.221570 + 0.975145i \(0.571118\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 12.3569 2.42339
\(27\) 0 0
\(28\) 2.33826 0.441890
\(29\) 9.48296 1.76094 0.880471 0.474101i \(-0.157227\pi\)
0.880471 + 0.474101i \(0.157227\pi\)
\(30\) 0 0
\(31\) −1.38761 −0.249223 −0.124611 0.992206i \(-0.539768\pi\)
−0.124611 + 0.992206i \(0.539768\pi\)
\(32\) −0.421994 −0.0745986
\(33\) 0 0
\(34\) 13.1578 2.25654
\(35\) 0 0
\(36\) 0 0
\(37\) −11.3502 −1.86595 −0.932977 0.359935i \(-0.882799\pi\)
−0.932977 + 0.359935i \(0.882799\pi\)
\(38\) −14.0080 −2.27240
\(39\) 0 0
\(40\) 0 0
\(41\) −0.403551 −0.0630240 −0.0315120 0.999503i \(-0.510032\pi\)
−0.0315120 + 0.999503i \(0.510032\pi\)
\(42\) 0 0
\(43\) 5.30211 0.808565 0.404282 0.914634i \(-0.367521\pi\)
0.404282 + 0.914634i \(0.367521\pi\)
\(44\) 16.8705 2.54333
\(45\) 0 0
\(46\) −5.18672 −0.764740
\(47\) −8.60289 −1.25486 −0.627430 0.778673i \(-0.715893\pi\)
−0.627430 + 0.778673i \(0.715893\pi\)
\(48\) 0 0
\(49\) −6.65069 −0.950099
\(50\) 0 0
\(51\) 0 0
\(52\) 20.0314 2.77785
\(53\) −0.337629 −0.0463769 −0.0231884 0.999731i \(-0.507382\pi\)
−0.0231884 + 0.999731i \(0.507382\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.82180 0.377079
\(57\) 0 0
\(58\) 23.1437 3.03891
\(59\) −2.73744 −0.356384 −0.178192 0.983996i \(-0.557025\pi\)
−0.178192 + 0.983996i \(0.557025\pi\)
\(60\) 0 0
\(61\) 9.01478 1.15422 0.577112 0.816665i \(-0.304179\pi\)
0.577112 + 0.816665i \(0.304179\pi\)
\(62\) −3.38654 −0.430091
\(63\) 0 0
\(64\) −8.50926 −1.06366
\(65\) 0 0
\(66\) 0 0
\(67\) −5.86133 −0.716075 −0.358038 0.933707i \(-0.616554\pi\)
−0.358038 + 0.933707i \(0.616554\pi\)
\(68\) 21.3296 2.58660
\(69\) 0 0
\(70\) 0 0
\(71\) 10.7281 1.27319 0.636596 0.771197i \(-0.280342\pi\)
0.636596 + 0.771197i \(0.280342\pi\)
\(72\) 0 0
\(73\) 5.02136 0.587706 0.293853 0.955851i \(-0.405062\pi\)
0.293853 + 0.955851i \(0.405062\pi\)
\(74\) −27.7007 −3.22014
\(75\) 0 0
\(76\) −22.7079 −2.60477
\(77\) 2.52026 0.287210
\(78\) 0 0
\(79\) 8.44321 0.949936 0.474968 0.880003i \(-0.342460\pi\)
0.474968 + 0.880003i \(0.342460\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.984886 −0.108762
\(83\) 11.0136 1.20890 0.604450 0.796643i \(-0.293393\pi\)
0.604450 + 0.796643i \(0.293393\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.9401 1.39537
\(87\) 0 0
\(88\) 20.3593 2.17031
\(89\) −13.1464 −1.39351 −0.696756 0.717308i \(-0.745374\pi\)
−0.696756 + 0.717308i \(0.745374\pi\)
\(90\) 0 0
\(91\) 2.99244 0.313693
\(92\) −8.40801 −0.876595
\(93\) 0 0
\(94\) −20.9958 −2.16555
\(95\) 0 0
\(96\) 0 0
\(97\) 7.24927 0.736052 0.368026 0.929815i \(-0.380034\pi\)
0.368026 + 0.929815i \(0.380034\pi\)
\(98\) −16.2314 −1.63962
\(99\) 0 0
\(100\) 0 0
\(101\) −10.2203 −1.01696 −0.508478 0.861075i \(-0.669792\pi\)
−0.508478 + 0.861075i \(0.669792\pi\)
\(102\) 0 0
\(103\) 11.6781 1.15068 0.575339 0.817915i \(-0.304870\pi\)
0.575339 + 0.817915i \(0.304870\pi\)
\(104\) 24.1738 2.37043
\(105\) 0 0
\(106\) −0.824000 −0.0800340
\(107\) −1.07495 −0.103920 −0.0519598 0.998649i \(-0.516547\pi\)
−0.0519598 + 0.998649i \(0.516547\pi\)
\(108\) 0 0
\(109\) −0.279773 −0.0267974 −0.0133987 0.999910i \(-0.504265\pi\)
−0.0133987 + 0.999910i \(0.504265\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.21024 0.208848
\(113\) −13.6813 −1.28703 −0.643513 0.765436i \(-0.722524\pi\)
−0.643513 + 0.765436i \(0.722524\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 37.5174 3.48340
\(117\) 0 0
\(118\) −6.68086 −0.615023
\(119\) 3.18639 0.292096
\(120\) 0 0
\(121\) 7.18363 0.653057
\(122\) 22.0011 1.99188
\(123\) 0 0
\(124\) −5.48981 −0.492999
\(125\) 0 0
\(126\) 0 0
\(127\) 14.9264 1.32450 0.662252 0.749281i \(-0.269601\pi\)
0.662252 + 0.749281i \(0.269601\pi\)
\(128\) −19.9233 −1.76099
\(129\) 0 0
\(130\) 0 0
\(131\) 20.2324 1.76771 0.883856 0.467758i \(-0.154938\pi\)
0.883856 + 0.467758i \(0.154938\pi\)
\(132\) 0 0
\(133\) −3.39228 −0.294148
\(134\) −14.3049 −1.23575
\(135\) 0 0
\(136\) 25.7405 2.20723
\(137\) 8.00831 0.684196 0.342098 0.939664i \(-0.388862\pi\)
0.342098 + 0.939664i \(0.388862\pi\)
\(138\) 0 0
\(139\) 10.0009 0.848261 0.424131 0.905601i \(-0.360580\pi\)
0.424131 + 0.905601i \(0.360580\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 26.1825 2.19719
\(143\) 21.5905 1.80548
\(144\) 0 0
\(145\) 0 0
\(146\) 12.2549 1.01422
\(147\) 0 0
\(148\) −44.9046 −3.69113
\(149\) −20.2500 −1.65895 −0.829474 0.558546i \(-0.811360\pi\)
−0.829474 + 0.558546i \(0.811360\pi\)
\(150\) 0 0
\(151\) −0.0600683 −0.00488829 −0.00244414 0.999997i \(-0.500778\pi\)
−0.00244414 + 0.999997i \(0.500778\pi\)
\(152\) −27.4038 −2.22274
\(153\) 0 0
\(154\) 6.15082 0.495647
\(155\) 0 0
\(156\) 0 0
\(157\) 3.69886 0.295201 0.147601 0.989047i \(-0.452845\pi\)
0.147601 + 0.989047i \(0.452845\pi\)
\(158\) 20.6061 1.63933
\(159\) 0 0
\(160\) 0 0
\(161\) −1.25606 −0.0989910
\(162\) 0 0
\(163\) 4.88965 0.382987 0.191493 0.981494i \(-0.438667\pi\)
0.191493 + 0.981494i \(0.438667\pi\)
\(164\) −1.59656 −0.124671
\(165\) 0 0
\(166\) 26.8793 2.08624
\(167\) −16.3419 −1.26457 −0.632286 0.774735i \(-0.717883\pi\)
−0.632286 + 0.774735i \(0.717883\pi\)
\(168\) 0 0
\(169\) 12.6356 0.971968
\(170\) 0 0
\(171\) 0 0
\(172\) 20.9767 1.59946
\(173\) 11.1117 0.844811 0.422405 0.906407i \(-0.361186\pi\)
0.422405 + 0.906407i \(0.361186\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 15.9468 1.20204
\(177\) 0 0
\(178\) −32.0844 −2.40483
\(179\) −18.0527 −1.34932 −0.674659 0.738129i \(-0.735709\pi\)
−0.674659 + 0.738129i \(0.735709\pi\)
\(180\) 0 0
\(181\) −3.11291 −0.231381 −0.115690 0.993285i \(-0.536908\pi\)
−0.115690 + 0.993285i \(0.536908\pi\)
\(182\) 7.30321 0.541350
\(183\) 0 0
\(184\) −10.1467 −0.748028
\(185\) 0 0
\(186\) 0 0
\(187\) 22.9898 1.68118
\(188\) −34.0356 −2.48230
\(189\) 0 0
\(190\) 0 0
\(191\) −15.0422 −1.08842 −0.544208 0.838950i \(-0.683170\pi\)
−0.544208 + 0.838950i \(0.683170\pi\)
\(192\) 0 0
\(193\) −5.39017 −0.387993 −0.193996 0.981002i \(-0.562145\pi\)
−0.193996 + 0.981002i \(0.562145\pi\)
\(194\) 17.6922 1.27023
\(195\) 0 0
\(196\) −26.3121 −1.87944
\(197\) −12.5879 −0.896848 −0.448424 0.893821i \(-0.648015\pi\)
−0.448424 + 0.893821i \(0.648015\pi\)
\(198\) 0 0
\(199\) 16.6365 1.17933 0.589666 0.807648i \(-0.299260\pi\)
0.589666 + 0.807648i \(0.299260\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −24.9431 −1.75499
\(203\) 5.60465 0.393369
\(204\) 0 0
\(205\) 0 0
\(206\) 28.5010 1.98576
\(207\) 0 0
\(208\) 18.9346 1.31288
\(209\) −24.4753 −1.69299
\(210\) 0 0
\(211\) −23.3539 −1.60775 −0.803874 0.594799i \(-0.797232\pi\)
−0.803874 + 0.594799i \(0.797232\pi\)
\(212\) −1.33576 −0.0917403
\(213\) 0 0
\(214\) −2.62348 −0.179337
\(215\) 0 0
\(216\) 0 0
\(217\) −0.820111 −0.0556728
\(218\) −0.682800 −0.0462451
\(219\) 0 0
\(220\) 0 0
\(221\) 27.2971 1.83620
\(222\) 0 0
\(223\) 0.829977 0.0555794 0.0277897 0.999614i \(-0.491153\pi\)
0.0277897 + 0.999614i \(0.491153\pi\)
\(224\) −0.249408 −0.0166643
\(225\) 0 0
\(226\) −33.3898 −2.22106
\(227\) 22.7061 1.50706 0.753530 0.657413i \(-0.228349\pi\)
0.753530 + 0.657413i \(0.228349\pi\)
\(228\) 0 0
\(229\) −27.7726 −1.83527 −0.917633 0.397429i \(-0.869903\pi\)
−0.917633 + 0.397429i \(0.869903\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 45.2758 2.97250
\(233\) 1.14087 0.0747411 0.0373706 0.999301i \(-0.488102\pi\)
0.0373706 + 0.999301i \(0.488102\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.8301 −0.704981
\(237\) 0 0
\(238\) 7.77656 0.504079
\(239\) −7.82335 −0.506051 −0.253025 0.967460i \(-0.581426\pi\)
−0.253025 + 0.967460i \(0.581426\pi\)
\(240\) 0 0
\(241\) −0.865353 −0.0557423 −0.0278711 0.999612i \(-0.508873\pi\)
−0.0278711 + 0.999612i \(0.508873\pi\)
\(242\) 17.5320 1.12700
\(243\) 0 0
\(244\) 35.6651 2.28323
\(245\) 0 0
\(246\) 0 0
\(247\) −29.0609 −1.84910
\(248\) −6.62508 −0.420693
\(249\) 0 0
\(250\) 0 0
\(251\) 6.09930 0.384984 0.192492 0.981299i \(-0.438343\pi\)
0.192492 + 0.981299i \(0.438343\pi\)
\(252\) 0 0
\(253\) −9.06243 −0.569750
\(254\) 36.4287 2.28574
\(255\) 0 0
\(256\) −31.6054 −1.97533
\(257\) −18.2063 −1.13568 −0.567839 0.823140i \(-0.692220\pi\)
−0.567839 + 0.823140i \(0.692220\pi\)
\(258\) 0 0
\(259\) −6.70820 −0.416828
\(260\) 0 0
\(261\) 0 0
\(262\) 49.3782 3.05060
\(263\) 11.7262 0.723071 0.361536 0.932358i \(-0.382253\pi\)
0.361536 + 0.932358i \(0.382253\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.27904 −0.507621
\(267\) 0 0
\(268\) −23.1891 −1.41650
\(269\) −12.0486 −0.734617 −0.367309 0.930099i \(-0.619721\pi\)
−0.367309 + 0.930099i \(0.619721\pi\)
\(270\) 0 0
\(271\) −19.0195 −1.15535 −0.577675 0.816267i \(-0.696040\pi\)
−0.577675 + 0.816267i \(0.696040\pi\)
\(272\) 20.1618 1.22249
\(273\) 0 0
\(274\) 19.5447 1.18074
\(275\) 0 0
\(276\) 0 0
\(277\) −7.22640 −0.434192 −0.217096 0.976150i \(-0.569658\pi\)
−0.217096 + 0.976150i \(0.569658\pi\)
\(278\) 24.4076 1.46387
\(279\) 0 0
\(280\) 0 0
\(281\) −19.2675 −1.14940 −0.574700 0.818364i \(-0.694881\pi\)
−0.574700 + 0.818364i \(0.694881\pi\)
\(282\) 0 0
\(283\) 23.2436 1.38169 0.690843 0.723004i \(-0.257239\pi\)
0.690843 + 0.723004i \(0.257239\pi\)
\(284\) 42.4436 2.51856
\(285\) 0 0
\(286\) 52.6926 3.11578
\(287\) −0.238508 −0.0140787
\(288\) 0 0
\(289\) 12.0663 0.709783
\(290\) 0 0
\(291\) 0 0
\(292\) 19.8660 1.16257
\(293\) −16.9450 −0.989936 −0.494968 0.868911i \(-0.664820\pi\)
−0.494968 + 0.868911i \(0.664820\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −54.1907 −3.14977
\(297\) 0 0
\(298\) −49.4213 −2.86290
\(299\) −10.7603 −0.622286
\(300\) 0 0
\(301\) 3.13367 0.180622
\(302\) −0.146600 −0.00843587
\(303\) 0 0
\(304\) −21.4646 −1.23108
\(305\) 0 0
\(306\) 0 0
\(307\) 7.48318 0.427088 0.213544 0.976933i \(-0.431499\pi\)
0.213544 + 0.976933i \(0.431499\pi\)
\(308\) 9.97087 0.568143
\(309\) 0 0
\(310\) 0 0
\(311\) 29.3081 1.66191 0.830955 0.556339i \(-0.187794\pi\)
0.830955 + 0.556339i \(0.187794\pi\)
\(312\) 0 0
\(313\) −20.6261 −1.16585 −0.582927 0.812524i \(-0.698093\pi\)
−0.582927 + 0.812524i \(0.698093\pi\)
\(314\) 9.02727 0.509438
\(315\) 0 0
\(316\) 33.4039 1.87911
\(317\) 29.7154 1.66898 0.834490 0.551023i \(-0.185762\pi\)
0.834490 + 0.551023i \(0.185762\pi\)
\(318\) 0 0
\(319\) 40.4375 2.26406
\(320\) 0 0
\(321\) 0 0
\(322\) −3.06547 −0.170832
\(323\) −30.9445 −1.72180
\(324\) 0 0
\(325\) 0 0
\(326\) 11.9334 0.660932
\(327\) 0 0
\(328\) −1.92673 −0.106386
\(329\) −5.08451 −0.280318
\(330\) 0 0
\(331\) 33.8462 1.86036 0.930178 0.367109i \(-0.119652\pi\)
0.930178 + 0.367109i \(0.119652\pi\)
\(332\) 43.5731 2.39138
\(333\) 0 0
\(334\) −39.8832 −2.18231
\(335\) 0 0
\(336\) 0 0
\(337\) −5.39168 −0.293704 −0.146852 0.989159i \(-0.546914\pi\)
−0.146852 + 0.989159i \(0.546914\pi\)
\(338\) 30.8378 1.67735
\(339\) 0 0
\(340\) 0 0
\(341\) −5.91710 −0.320429
\(342\) 0 0
\(343\) −8.06787 −0.435624
\(344\) 25.3146 1.36487
\(345\) 0 0
\(346\) 27.1188 1.45792
\(347\) −20.1190 −1.08004 −0.540022 0.841651i \(-0.681584\pi\)
−0.540022 + 0.841651i \(0.681584\pi\)
\(348\) 0 0
\(349\) −15.9106 −0.851672 −0.425836 0.904800i \(-0.640020\pi\)
−0.425836 + 0.904800i \(0.640020\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.79948 −0.0959124
\(353\) −7.21261 −0.383888 −0.191944 0.981406i \(-0.561479\pi\)
−0.191944 + 0.981406i \(0.561479\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −52.0109 −2.75657
\(357\) 0 0
\(358\) −44.0584 −2.32856
\(359\) 3.37048 0.177887 0.0889436 0.996037i \(-0.471651\pi\)
0.0889436 + 0.996037i \(0.471651\pi\)
\(360\) 0 0
\(361\) 13.9439 0.733892
\(362\) −7.59722 −0.399301
\(363\) 0 0
\(364\) 11.8390 0.620532
\(365\) 0 0
\(366\) 0 0
\(367\) −5.37420 −0.280531 −0.140266 0.990114i \(-0.544796\pi\)
−0.140266 + 0.990114i \(0.544796\pi\)
\(368\) −7.94765 −0.414300
\(369\) 0 0
\(370\) 0 0
\(371\) −0.199546 −0.0103599
\(372\) 0 0
\(373\) −16.0783 −0.832505 −0.416252 0.909249i \(-0.636657\pi\)
−0.416252 + 0.909249i \(0.636657\pi\)
\(374\) 56.1078 2.90127
\(375\) 0 0
\(376\) −41.0740 −2.11823
\(377\) 48.0137 2.47283
\(378\) 0 0
\(379\) −27.6162 −1.41855 −0.709274 0.704933i \(-0.750977\pi\)
−0.709274 + 0.704933i \(0.750977\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −36.7113 −1.87831
\(383\) −23.2577 −1.18841 −0.594207 0.804312i \(-0.702534\pi\)
−0.594207 + 0.804312i \(0.702534\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −13.1550 −0.669571
\(387\) 0 0
\(388\) 28.6803 1.45602
\(389\) 12.8126 0.649624 0.324812 0.945779i \(-0.394699\pi\)
0.324812 + 0.945779i \(0.394699\pi\)
\(390\) 0 0
\(391\) −11.4578 −0.579443
\(392\) −31.7533 −1.60379
\(393\) 0 0
\(394\) −30.7214 −1.54772
\(395\) 0 0
\(396\) 0 0
\(397\) −10.8608 −0.545085 −0.272543 0.962144i \(-0.587865\pi\)
−0.272543 + 0.962144i \(0.587865\pi\)
\(398\) 40.6023 2.03521
\(399\) 0 0
\(400\) 0 0
\(401\) −23.0220 −1.14966 −0.574831 0.818272i \(-0.694932\pi\)
−0.574831 + 0.818272i \(0.694932\pi\)
\(402\) 0 0
\(403\) −7.02570 −0.349975
\(404\) −40.4345 −2.01169
\(405\) 0 0
\(406\) 13.6784 0.678849
\(407\) −48.3996 −2.39908
\(408\) 0 0
\(409\) −20.3469 −1.00609 −0.503044 0.864261i \(-0.667787\pi\)
−0.503044 + 0.864261i \(0.667787\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 46.2020 2.27621
\(413\) −1.61789 −0.0796111
\(414\) 0 0
\(415\) 0 0
\(416\) −2.13662 −0.104756
\(417\) 0 0
\(418\) −59.7332 −2.92165
\(419\) −21.6227 −1.05634 −0.528169 0.849140i \(-0.677121\pi\)
−0.528169 + 0.849140i \(0.677121\pi\)
\(420\) 0 0
\(421\) −5.15561 −0.251269 −0.125635 0.992077i \(-0.540097\pi\)
−0.125635 + 0.992077i \(0.540097\pi\)
\(422\) −56.9964 −2.77454
\(423\) 0 0
\(424\) −1.61199 −0.0782850
\(425\) 0 0
\(426\) 0 0
\(427\) 5.32794 0.257837
\(428\) −4.25283 −0.205568
\(429\) 0 0
\(430\) 0 0
\(431\) 1.21065 0.0583150 0.0291575 0.999575i \(-0.490718\pi\)
0.0291575 + 0.999575i \(0.490718\pi\)
\(432\) 0 0
\(433\) −0.0648649 −0.00311721 −0.00155860 0.999999i \(-0.500496\pi\)
−0.00155860 + 0.999999i \(0.500496\pi\)
\(434\) −2.00152 −0.0960762
\(435\) 0 0
\(436\) −1.10686 −0.0530092
\(437\) 12.1981 0.583514
\(438\) 0 0
\(439\) −11.7704 −0.561770 −0.280885 0.959741i \(-0.590628\pi\)
−0.280885 + 0.959741i \(0.590628\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 66.6200 3.16879
\(443\) −19.1683 −0.910716 −0.455358 0.890309i \(-0.650489\pi\)
−0.455358 + 0.890309i \(0.650489\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.02560 0.0959151
\(447\) 0 0
\(448\) −5.02917 −0.237606
\(449\) 0.536369 0.0253128 0.0126564 0.999920i \(-0.495971\pi\)
0.0126564 + 0.999920i \(0.495971\pi\)
\(450\) 0 0
\(451\) −1.72083 −0.0810308
\(452\) −54.1271 −2.54593
\(453\) 0 0
\(454\) 55.4155 2.60078
\(455\) 0 0
\(456\) 0 0
\(457\) 35.1952 1.64636 0.823181 0.567779i \(-0.192197\pi\)
0.823181 + 0.567779i \(0.192197\pi\)
\(458\) −67.7805 −3.16718
\(459\) 0 0
\(460\) 0 0
\(461\) −1.79752 −0.0837188 −0.0418594 0.999124i \(-0.513328\pi\)
−0.0418594 + 0.999124i \(0.513328\pi\)
\(462\) 0 0
\(463\) 15.8682 0.737460 0.368730 0.929537i \(-0.379793\pi\)
0.368730 + 0.929537i \(0.379793\pi\)
\(464\) 35.4632 1.64634
\(465\) 0 0
\(466\) 2.78436 0.128983
\(467\) 41.1577 1.90455 0.952275 0.305240i \(-0.0987368\pi\)
0.952275 + 0.305240i \(0.0987368\pi\)
\(468\) 0 0
\(469\) −3.46418 −0.159961
\(470\) 0 0
\(471\) 0 0
\(472\) −13.0697 −0.601584
\(473\) 22.6094 1.03958
\(474\) 0 0
\(475\) 0 0
\(476\) 12.6063 0.577809
\(477\) 0 0
\(478\) −19.0933 −0.873307
\(479\) −15.1052 −0.690176 −0.345088 0.938570i \(-0.612151\pi\)
−0.345088 + 0.938570i \(0.612151\pi\)
\(480\) 0 0
\(481\) −57.4677 −2.62030
\(482\) −2.11194 −0.0961962
\(483\) 0 0
\(484\) 28.4205 1.29184
\(485\) 0 0
\(486\) 0 0
\(487\) −4.90873 −0.222436 −0.111218 0.993796i \(-0.535475\pi\)
−0.111218 + 0.993796i \(0.535475\pi\)
\(488\) 43.0405 1.94835
\(489\) 0 0
\(490\) 0 0
\(491\) −43.7750 −1.97554 −0.987769 0.155925i \(-0.950164\pi\)
−0.987769 + 0.155925i \(0.950164\pi\)
\(492\) 0 0
\(493\) 51.1257 2.30258
\(494\) −70.9247 −3.19105
\(495\) 0 0
\(496\) −5.18923 −0.233003
\(497\) 6.34056 0.284413
\(498\) 0 0
\(499\) 6.00199 0.268686 0.134343 0.990935i \(-0.457108\pi\)
0.134343 + 0.990935i \(0.457108\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14.8857 0.664379
\(503\) −28.8130 −1.28471 −0.642354 0.766408i \(-0.722042\pi\)
−0.642354 + 0.766408i \(0.722042\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −22.1173 −0.983235
\(507\) 0 0
\(508\) 59.0532 2.62006
\(509\) 12.3745 0.548492 0.274246 0.961660i \(-0.411572\pi\)
0.274246 + 0.961660i \(0.411572\pi\)
\(510\) 0 0
\(511\) 2.96774 0.131285
\(512\) −37.2879 −1.64791
\(513\) 0 0
\(514\) −44.4334 −1.95987
\(515\) 0 0
\(516\) 0 0
\(517\) −36.6847 −1.61339
\(518\) −16.3717 −0.719332
\(519\) 0 0
\(520\) 0 0
\(521\) 16.0357 0.702535 0.351268 0.936275i \(-0.385751\pi\)
0.351268 + 0.936275i \(0.385751\pi\)
\(522\) 0 0
\(523\) 34.7701 1.52039 0.760195 0.649695i \(-0.225103\pi\)
0.760195 + 0.649695i \(0.225103\pi\)
\(524\) 80.0453 3.49680
\(525\) 0 0
\(526\) 28.6185 1.24783
\(527\) −7.48106 −0.325880
\(528\) 0 0
\(529\) −18.4834 −0.803627
\(530\) 0 0
\(531\) 0 0
\(532\) −13.4209 −0.581869
\(533\) −2.04324 −0.0885026
\(534\) 0 0
\(535\) 0 0
\(536\) −27.9846 −1.20875
\(537\) 0 0
\(538\) −29.4053 −1.26775
\(539\) −28.3601 −1.22155
\(540\) 0 0
\(541\) −10.0421 −0.431745 −0.215872 0.976422i \(-0.569260\pi\)
−0.215872 + 0.976422i \(0.569260\pi\)
\(542\) −46.4180 −1.99382
\(543\) 0 0
\(544\) −2.27510 −0.0975442
\(545\) 0 0
\(546\) 0 0
\(547\) 34.1914 1.46192 0.730959 0.682421i \(-0.239073\pi\)
0.730959 + 0.682421i \(0.239073\pi\)
\(548\) 31.6832 1.35344
\(549\) 0 0
\(550\) 0 0
\(551\) −54.4292 −2.31876
\(552\) 0 0
\(553\) 4.99013 0.212202
\(554\) −17.6364 −0.749299
\(555\) 0 0
\(556\) 39.5663 1.67799
\(557\) 1.35501 0.0574135 0.0287068 0.999588i \(-0.490861\pi\)
0.0287068 + 0.999588i \(0.490861\pi\)
\(558\) 0 0
\(559\) 26.8454 1.13544
\(560\) 0 0
\(561\) 0 0
\(562\) −47.0233 −1.98356
\(563\) 14.5306 0.612391 0.306195 0.951969i \(-0.400944\pi\)
0.306195 + 0.951969i \(0.400944\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 56.7271 2.38442
\(567\) 0 0
\(568\) 51.2207 2.14917
\(569\) −19.6012 −0.821727 −0.410863 0.911697i \(-0.634773\pi\)
−0.410863 + 0.911697i \(0.634773\pi\)
\(570\) 0 0
\(571\) 32.8810 1.37603 0.688013 0.725698i \(-0.258483\pi\)
0.688013 + 0.725698i \(0.258483\pi\)
\(572\) 85.4182 3.57151
\(573\) 0 0
\(574\) −0.582090 −0.0242960
\(575\) 0 0
\(576\) 0 0
\(577\) 7.26819 0.302579 0.151289 0.988490i \(-0.451657\pi\)
0.151289 + 0.988490i \(0.451657\pi\)
\(578\) 29.4485 1.22490
\(579\) 0 0
\(580\) 0 0
\(581\) 6.50929 0.270051
\(582\) 0 0
\(583\) −1.43972 −0.0596273
\(584\) 23.9742 0.992060
\(585\) 0 0
\(586\) −41.3551 −1.70836
\(587\) −9.51042 −0.392537 −0.196268 0.980550i \(-0.562882\pi\)
−0.196268 + 0.980550i \(0.562882\pi\)
\(588\) 0 0
\(589\) 7.96446 0.328170
\(590\) 0 0
\(591\) 0 0
\(592\) −42.4460 −1.74452
\(593\) −9.75081 −0.400418 −0.200209 0.979753i \(-0.564162\pi\)
−0.200209 + 0.979753i \(0.564162\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −80.1151 −3.28164
\(597\) 0 0
\(598\) −26.2612 −1.07390
\(599\) 6.63034 0.270908 0.135454 0.990784i \(-0.456751\pi\)
0.135454 + 0.990784i \(0.456751\pi\)
\(600\) 0 0
\(601\) −25.8619 −1.05493 −0.527465 0.849577i \(-0.676857\pi\)
−0.527465 + 0.849577i \(0.676857\pi\)
\(602\) 7.64789 0.311705
\(603\) 0 0
\(604\) −0.237648 −0.00966975
\(605\) 0 0
\(606\) 0 0
\(607\) 11.3597 0.461075 0.230537 0.973063i \(-0.425952\pi\)
0.230537 + 0.973063i \(0.425952\pi\)
\(608\) 2.42211 0.0982295
\(609\) 0 0
\(610\) 0 0
\(611\) −43.5578 −1.76216
\(612\) 0 0
\(613\) 30.2599 1.22218 0.611092 0.791559i \(-0.290730\pi\)
0.611092 + 0.791559i \(0.290730\pi\)
\(614\) 18.2631 0.737039
\(615\) 0 0
\(616\) 12.0328 0.484816
\(617\) −33.1940 −1.33634 −0.668170 0.744009i \(-0.732922\pi\)
−0.668170 + 0.744009i \(0.732922\pi\)
\(618\) 0 0
\(619\) −11.1705 −0.448979 −0.224489 0.974477i \(-0.572071\pi\)
−0.224489 + 0.974477i \(0.572071\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 71.5280 2.86801
\(623\) −7.76980 −0.311291
\(624\) 0 0
\(625\) 0 0
\(626\) −50.3390 −2.01195
\(627\) 0 0
\(628\) 14.6338 0.583952
\(629\) −61.1923 −2.43990
\(630\) 0 0
\(631\) −13.7542 −0.547546 −0.273773 0.961794i \(-0.588272\pi\)
−0.273773 + 0.961794i \(0.588272\pi\)
\(632\) 40.3116 1.60351
\(633\) 0 0
\(634\) 72.5219 2.88021
\(635\) 0 0
\(636\) 0 0
\(637\) −33.6735 −1.33419
\(638\) 98.6898 3.90717
\(639\) 0 0
\(640\) 0 0
\(641\) −26.6242 −1.05159 −0.525797 0.850610i \(-0.676233\pi\)
−0.525797 + 0.850610i \(0.676233\pi\)
\(642\) 0 0
\(643\) −21.6827 −0.855082 −0.427541 0.903996i \(-0.640620\pi\)
−0.427541 + 0.903996i \(0.640620\pi\)
\(644\) −4.96933 −0.195819
\(645\) 0 0
\(646\) −75.5216 −2.97136
\(647\) −13.6216 −0.535520 −0.267760 0.963486i \(-0.586283\pi\)
−0.267760 + 0.963486i \(0.586283\pi\)
\(648\) 0 0
\(649\) −11.6731 −0.458208
\(650\) 0 0
\(651\) 0 0
\(652\) 19.3449 0.757604
\(653\) 30.4783 1.19271 0.596355 0.802721i \(-0.296615\pi\)
0.596355 + 0.802721i \(0.296615\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.50915 −0.0589224
\(657\) 0 0
\(658\) −12.4090 −0.483753
\(659\) −17.7054 −0.689705 −0.344852 0.938657i \(-0.612071\pi\)
−0.344852 + 0.938657i \(0.612071\pi\)
\(660\) 0 0
\(661\) 7.21561 0.280655 0.140327 0.990105i \(-0.455184\pi\)
0.140327 + 0.990105i \(0.455184\pi\)
\(662\) 82.6035 3.21047
\(663\) 0 0
\(664\) 52.5838 2.04065
\(665\) 0 0
\(666\) 0 0
\(667\) −20.1534 −0.780343
\(668\) −64.6533 −2.50151
\(669\) 0 0
\(670\) 0 0
\(671\) 38.4411 1.48400
\(672\) 0 0
\(673\) 41.3984 1.59579 0.797896 0.602796i \(-0.205947\pi\)
0.797896 + 0.602796i \(0.205947\pi\)
\(674\) −13.1587 −0.506853
\(675\) 0 0
\(676\) 49.9901 1.92270
\(677\) 4.67244 0.179576 0.0897882 0.995961i \(-0.471381\pi\)
0.0897882 + 0.995961i \(0.471381\pi\)
\(678\) 0 0
\(679\) 4.28449 0.164423
\(680\) 0 0
\(681\) 0 0
\(682\) −14.4410 −0.552974
\(683\) −0.695173 −0.0266001 −0.0133000 0.999912i \(-0.504234\pi\)
−0.0133000 + 0.999912i \(0.504234\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −19.6901 −0.751770
\(687\) 0 0
\(688\) 19.8282 0.755944
\(689\) −1.70947 −0.0651255
\(690\) 0 0
\(691\) 16.0208 0.609461 0.304731 0.952439i \(-0.401434\pi\)
0.304731 + 0.952439i \(0.401434\pi\)
\(692\) 43.9614 1.67116
\(693\) 0 0
\(694\) −49.1015 −1.86387
\(695\) 0 0
\(696\) 0 0
\(697\) −2.17567 −0.0824094
\(698\) −38.8305 −1.46976
\(699\) 0 0
\(700\) 0 0
\(701\) −33.8563 −1.27873 −0.639367 0.768902i \(-0.720803\pi\)
−0.639367 + 0.768902i \(0.720803\pi\)
\(702\) 0 0
\(703\) 65.1463 2.45704
\(704\) −36.2854 −1.36756
\(705\) 0 0
\(706\) −17.6028 −0.662488
\(707\) −6.04042 −0.227173
\(708\) 0 0
\(709\) −14.4041 −0.540959 −0.270479 0.962726i \(-0.587182\pi\)
−0.270479 + 0.962726i \(0.587182\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −62.7666 −2.35228
\(713\) 2.94899 0.110440
\(714\) 0 0
\(715\) 0 0
\(716\) −71.4216 −2.66915
\(717\) 0 0
\(718\) 8.22584 0.306985
\(719\) −28.5156 −1.06345 −0.531726 0.846916i \(-0.678457\pi\)
−0.531726 + 0.846916i \(0.678457\pi\)
\(720\) 0 0
\(721\) 6.90203 0.257045
\(722\) 34.0309 1.26650
\(723\) 0 0
\(724\) −12.3156 −0.457705
\(725\) 0 0
\(726\) 0 0
\(727\) 47.5108 1.76208 0.881040 0.473042i \(-0.156844\pi\)
0.881040 + 0.473042i \(0.156844\pi\)
\(728\) 14.2872 0.529520
\(729\) 0 0
\(730\) 0 0
\(731\) 28.5854 1.05727
\(732\) 0 0
\(733\) 24.3105 0.897928 0.448964 0.893550i \(-0.351793\pi\)
0.448964 + 0.893550i \(0.351793\pi\)
\(734\) −13.1160 −0.484121
\(735\) 0 0
\(736\) 0.896830 0.0330576
\(737\) −24.9940 −0.920667
\(738\) 0 0
\(739\) 41.7625 1.53626 0.768130 0.640294i \(-0.221187\pi\)
0.768130 + 0.640294i \(0.221187\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.487003 −0.0178784
\(743\) −19.7341 −0.723972 −0.361986 0.932183i \(-0.617901\pi\)
−0.361986 + 0.932183i \(0.617901\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −39.2400 −1.43668
\(747\) 0 0
\(748\) 90.9544 3.32562
\(749\) −0.635321 −0.0232141
\(750\) 0 0
\(751\) 18.2348 0.665397 0.332699 0.943033i \(-0.392041\pi\)
0.332699 + 0.943033i \(0.392041\pi\)
\(752\) −32.1721 −1.17319
\(753\) 0 0
\(754\) 117.180 4.26744
\(755\) 0 0
\(756\) 0 0
\(757\) −15.5166 −0.563962 −0.281981 0.959420i \(-0.590992\pi\)
−0.281981 + 0.959420i \(0.590992\pi\)
\(758\) −67.3988 −2.44803
\(759\) 0 0
\(760\) 0 0
\(761\) 11.7633 0.426419 0.213210 0.977006i \(-0.431608\pi\)
0.213210 + 0.977006i \(0.431608\pi\)
\(762\) 0 0
\(763\) −0.165352 −0.00598615
\(764\) −59.5114 −2.15305
\(765\) 0 0
\(766\) −56.7617 −2.05088
\(767\) −13.8601 −0.500459
\(768\) 0 0
\(769\) −9.10438 −0.328312 −0.164156 0.986434i \(-0.552490\pi\)
−0.164156 + 0.986434i \(0.552490\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −21.3251 −0.767507
\(773\) 25.6603 0.922937 0.461468 0.887157i \(-0.347323\pi\)
0.461468 + 0.887157i \(0.347323\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 34.6112 1.24247
\(777\) 0 0
\(778\) 31.2698 1.12108
\(779\) 2.31625 0.0829883
\(780\) 0 0
\(781\) 45.7471 1.63696
\(782\) −27.9632 −0.999963
\(783\) 0 0
\(784\) −24.8715 −0.888267
\(785\) 0 0
\(786\) 0 0
\(787\) −32.6096 −1.16241 −0.581204 0.813758i \(-0.697418\pi\)
−0.581204 + 0.813758i \(0.697418\pi\)
\(788\) −49.8013 −1.77410
\(789\) 0 0
\(790\) 0 0
\(791\) −8.08594 −0.287503
\(792\) 0 0
\(793\) 45.6433 1.62084
\(794\) −26.5062 −0.940671
\(795\) 0 0
\(796\) 65.8190 2.33289
\(797\) −29.7470 −1.05369 −0.526846 0.849961i \(-0.676626\pi\)
−0.526846 + 0.849961i \(0.676626\pi\)
\(798\) 0 0
\(799\) −46.3809 −1.64084
\(800\) 0 0
\(801\) 0 0
\(802\) −56.1863 −1.98401
\(803\) 21.4122 0.755621
\(804\) 0 0
\(805\) 0 0
\(806\) −17.1466 −0.603963
\(807\) 0 0
\(808\) −48.7961 −1.71664
\(809\) −28.9282 −1.01706 −0.508530 0.861044i \(-0.669811\pi\)
−0.508530 + 0.861044i \(0.669811\pi\)
\(810\) 0 0
\(811\) 9.02896 0.317050 0.158525 0.987355i \(-0.449326\pi\)
0.158525 + 0.987355i \(0.449326\pi\)
\(812\) 22.1736 0.778142
\(813\) 0 0
\(814\) −118.122 −4.14017
\(815\) 0 0
\(816\) 0 0
\(817\) −30.4324 −1.06470
\(818\) −49.6576 −1.73624
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0570 0.769796 0.384898 0.922959i \(-0.374237\pi\)
0.384898 + 0.922959i \(0.374237\pi\)
\(822\) 0 0
\(823\) −17.9828 −0.626840 −0.313420 0.949615i \(-0.601475\pi\)
−0.313420 + 0.949615i \(0.601475\pi\)
\(824\) 55.7564 1.94237
\(825\) 0 0
\(826\) −3.94854 −0.137387
\(827\) 13.4856 0.468939 0.234470 0.972123i \(-0.424665\pi\)
0.234470 + 0.972123i \(0.424665\pi\)
\(828\) 0 0
\(829\) 34.6056 1.20190 0.600952 0.799285i \(-0.294788\pi\)
0.600952 + 0.799285i \(0.294788\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −43.0837 −1.49366
\(833\) −35.8560 −1.24234
\(834\) 0 0
\(835\) 0 0
\(836\) −96.8315 −3.34899
\(837\) 0 0
\(838\) −52.7713 −1.82295
\(839\) 25.5209 0.881080 0.440540 0.897733i \(-0.354787\pi\)
0.440540 + 0.897733i \(0.354787\pi\)
\(840\) 0 0
\(841\) 60.9265 2.10091
\(842\) −12.5825 −0.433623
\(843\) 0 0
\(844\) −92.3949 −3.18036
\(845\) 0 0
\(846\) 0 0
\(847\) 4.24569 0.145884
\(848\) −1.26262 −0.0433587
\(849\) 0 0
\(850\) 0 0
\(851\) 24.1216 0.826878
\(852\) 0 0
\(853\) −3.05949 −0.104755 −0.0523775 0.998627i \(-0.516680\pi\)
−0.0523775 + 0.998627i \(0.516680\pi\)
\(854\) 13.0031 0.444958
\(855\) 0 0
\(856\) −5.13229 −0.175418
\(857\) 29.0297 0.991635 0.495817 0.868427i \(-0.334869\pi\)
0.495817 + 0.868427i \(0.334869\pi\)
\(858\) 0 0
\(859\) −11.5059 −0.392576 −0.196288 0.980546i \(-0.562889\pi\)
−0.196288 + 0.980546i \(0.562889\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.95466 0.100636
\(863\) 10.0121 0.340816 0.170408 0.985374i \(-0.445491\pi\)
0.170408 + 0.985374i \(0.445491\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −0.158306 −0.00537946
\(867\) 0 0
\(868\) −3.24460 −0.110129
\(869\) 36.0038 1.22134
\(870\) 0 0
\(871\) −29.6768 −1.00556
\(872\) −1.33576 −0.0452345
\(873\) 0 0
\(874\) 29.7701 1.00699
\(875\) 0 0
\(876\) 0 0
\(877\) 16.6844 0.563392 0.281696 0.959504i \(-0.409103\pi\)
0.281696 + 0.959504i \(0.409103\pi\)
\(878\) −28.7262 −0.969464
\(879\) 0 0
\(880\) 0 0
\(881\) 9.74584 0.328346 0.164173 0.986432i \(-0.447505\pi\)
0.164173 + 0.986432i \(0.447505\pi\)
\(882\) 0 0
\(883\) 19.7999 0.666319 0.333160 0.942870i \(-0.391885\pi\)
0.333160 + 0.942870i \(0.391885\pi\)
\(884\) 107.995 3.63228
\(885\) 0 0
\(886\) −46.7813 −1.57165
\(887\) 16.0655 0.539427 0.269714 0.962941i \(-0.413071\pi\)
0.269714 + 0.962941i \(0.413071\pi\)
\(888\) 0 0
\(889\) 8.82184 0.295875
\(890\) 0 0
\(891\) 0 0
\(892\) 3.28364 0.109944
\(893\) 49.3779 1.65237
\(894\) 0 0
\(895\) 0 0
\(896\) −11.7751 −0.393380
\(897\) 0 0
\(898\) 1.30904 0.0436831
\(899\) −13.1587 −0.438866
\(900\) 0 0
\(901\) −1.82026 −0.0606418
\(902\) −4.19978 −0.139837
\(903\) 0 0
\(904\) −65.3204 −2.17252
\(905\) 0 0
\(906\) 0 0
\(907\) 17.0483 0.566080 0.283040 0.959108i \(-0.408657\pi\)
0.283040 + 0.959108i \(0.408657\pi\)
\(908\) 89.8322 2.98119
\(909\) 0 0
\(910\) 0 0
\(911\) 30.9582 1.02569 0.512846 0.858481i \(-0.328591\pi\)
0.512846 + 0.858481i \(0.328591\pi\)
\(912\) 0 0
\(913\) 46.9645 1.55430
\(914\) 85.8957 2.84118
\(915\) 0 0
\(916\) −109.877 −3.63043
\(917\) 11.9578 0.394882
\(918\) 0 0
\(919\) 17.0914 0.563794 0.281897 0.959445i \(-0.409036\pi\)
0.281897 + 0.959445i \(0.409036\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −4.38694 −0.144476
\(923\) 54.3181 1.78790
\(924\) 0 0
\(925\) 0 0
\(926\) 38.7273 1.27266
\(927\) 0 0
\(928\) −4.00175 −0.131364
\(929\) 3.86068 0.126665 0.0633323 0.997992i \(-0.479827\pi\)
0.0633323 + 0.997992i \(0.479827\pi\)
\(930\) 0 0
\(931\) 38.1729 1.25106
\(932\) 4.51363 0.147849
\(933\) 0 0
\(934\) 100.447 3.28674
\(935\) 0 0
\(936\) 0 0
\(937\) 7.05231 0.230389 0.115194 0.993343i \(-0.463251\pi\)
0.115194 + 0.993343i \(0.463251\pi\)
\(938\) −8.45451 −0.276050
\(939\) 0 0
\(940\) 0 0
\(941\) −11.6549 −0.379937 −0.189969 0.981790i \(-0.560839\pi\)
−0.189969 + 0.981790i \(0.560839\pi\)
\(942\) 0 0
\(943\) 0.857635 0.0279284
\(944\) −10.2371 −0.333191
\(945\) 0 0
\(946\) 55.1794 1.79404
\(947\) −31.9057 −1.03680 −0.518398 0.855140i \(-0.673471\pi\)
−0.518398 + 0.855140i \(0.673471\pi\)
\(948\) 0 0
\(949\) 25.4240 0.825297
\(950\) 0 0
\(951\) 0 0
\(952\) 15.2132 0.493064
\(953\) 26.3431 0.853337 0.426668 0.904408i \(-0.359687\pi\)
0.426668 + 0.904408i \(0.359687\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −30.9515 −1.00104
\(957\) 0 0
\(958\) −36.8651 −1.19106
\(959\) 4.73310 0.152840
\(960\) 0 0
\(961\) −29.0745 −0.937888
\(962\) −140.253 −4.52193
\(963\) 0 0
\(964\) −3.42359 −0.110266
\(965\) 0 0
\(966\) 0 0
\(967\) −2.32875 −0.0748876 −0.0374438 0.999299i \(-0.511922\pi\)
−0.0374438 + 0.999299i \(0.511922\pi\)
\(968\) 34.2978 1.10237
\(969\) 0 0
\(970\) 0 0
\(971\) −43.6099 −1.39951 −0.699755 0.714383i \(-0.746707\pi\)
−0.699755 + 0.714383i \(0.746707\pi\)
\(972\) 0 0
\(973\) 5.91073 0.189489
\(974\) −11.9800 −0.383864
\(975\) 0 0
\(976\) 33.7124 1.07911
\(977\) 36.5836 1.17041 0.585207 0.810884i \(-0.301013\pi\)
0.585207 + 0.810884i \(0.301013\pi\)
\(978\) 0 0
\(979\) −56.0591 −1.79166
\(980\) 0 0
\(981\) 0 0
\(982\) −106.835 −3.40925
\(983\) 26.5522 0.846885 0.423442 0.905923i \(-0.360822\pi\)
0.423442 + 0.905923i \(0.360822\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 124.775 3.97364
\(987\) 0 0
\(988\) −114.974 −3.65780
\(989\) −11.2682 −0.358307
\(990\) 0 0
\(991\) −52.9441 −1.68182 −0.840912 0.541172i \(-0.817981\pi\)
−0.840912 + 0.541172i \(0.817981\pi\)
\(992\) 0.585564 0.0185917
\(993\) 0 0
\(994\) 15.4745 0.490820
\(995\) 0 0
\(996\) 0 0
\(997\) −49.7519 −1.57566 −0.787828 0.615895i \(-0.788795\pi\)
−0.787828 + 0.615895i \(0.788795\pi\)
\(998\) 14.6482 0.463680
\(999\) 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 5625.2.a.ba.1.8 yes 8
3.2 odd 2 inner 5625.2.a.ba.1.1 yes 8
5.4 even 2 5625.2.a.y.1.1 8
15.14 odd 2 5625.2.a.y.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.y.1.1 8 5.4 even 2
5625.2.a.y.1.8 yes 8 15.14 odd 2
5625.2.a.ba.1.1 yes 8 3.2 odd 2 inner
5625.2.a.ba.1.8 yes 8 1.1 even 1 trivial