Properties

Label 5625.2.a.p.1.5
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $6$
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] = [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: \(6\)
Coefficient field: 6.6.44400625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.01887\) 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)
 
\(f(q)\) \(=\) \(q+2.01887 q^{2} +2.07584 q^{4} +1.01887 q^{7} +0.153106 q^{8} +O(q^{10})\) \(q+2.01887 q^{2} +2.07584 q^{4} +1.01887 q^{7} +0.153106 q^{8} -4.75961 q^{11} -0.103837 q^{13} +2.05697 q^{14} -3.84257 q^{16} +5.83776 q^{17} -0.724404 q^{19} -9.60904 q^{22} +9.07152 q^{23} -0.209634 q^{26} +2.11501 q^{28} +3.98847 q^{29} +1.06662 q^{31} -8.06387 q^{32} +11.7857 q^{34} +4.02621 q^{37} -1.46248 q^{38} -7.20977 q^{41} +8.62791 q^{43} -9.88018 q^{44} +18.3142 q^{46} +8.19797 q^{47} -5.96190 q^{49} -0.215549 q^{52} +4.36719 q^{53} +0.155995 q^{56} +8.05221 q^{58} +4.91285 q^{59} +6.96435 q^{61} +2.15338 q^{62} -8.59476 q^{64} -9.91998 q^{67} +12.1183 q^{68} +10.7866 q^{71} +8.63115 q^{73} +8.12840 q^{74} -1.50375 q^{76} -4.84943 q^{77} +2.48291 q^{79} -14.5556 q^{82} -4.24385 q^{83} +17.4186 q^{86} -0.728724 q^{88} +18.3752 q^{89} -0.105797 q^{91} +18.8310 q^{92} +16.5506 q^{94} +6.69876 q^{97} -12.0363 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37} - 9 q^{38} + q^{41} - 2 q^{43} - 9 q^{44} + 19 q^{46} + 14 q^{47} - 14 q^{49} - 13 q^{52} + 23 q^{53} + 35 q^{56} - 22 q^{58} - 9 q^{59} + 11 q^{61} - 23 q^{62} - 23 q^{64} - 8 q^{67} + 50 q^{68} + 8 q^{71} - 13 q^{73} + 22 q^{74} - 26 q^{76} - 13 q^{77} - 5 q^{79} + 13 q^{82} - 20 q^{83} + 37 q^{86} - 28 q^{88} + 4 q^{89} + 34 q^{91} + 61 q^{92} + 41 q^{94} + 7 q^{97} - 41 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.01887 1.42756 0.713778 0.700372i \(-0.246982\pi\)
0.713778 + 0.700372i \(0.246982\pi\)
\(3\) 0 0
\(4\) 2.07584 1.03792
\(5\) 0 0
\(6\) 0 0
\(7\) 1.01887 0.385097 0.192548 0.981287i \(-0.438325\pi\)
0.192548 + 0.981287i \(0.438325\pi\)
\(8\) 0.153106 0.0541311
\(9\) 0 0
\(10\) 0 0
\(11\) −4.75961 −1.43508 −0.717538 0.696519i \(-0.754731\pi\)
−0.717538 + 0.696519i \(0.754731\pi\)
\(12\) 0 0
\(13\) −0.103837 −0.0287992 −0.0143996 0.999896i \(-0.504584\pi\)
−0.0143996 + 0.999896i \(0.504584\pi\)
\(14\) 2.05697 0.549748
\(15\) 0 0
\(16\) −3.84257 −0.960643
\(17\) 5.83776 1.41587 0.707933 0.706280i \(-0.249628\pi\)
0.707933 + 0.706280i \(0.249628\pi\)
\(18\) 0 0
\(19\) −0.724404 −0.166190 −0.0830949 0.996542i \(-0.526480\pi\)
−0.0830949 + 0.996542i \(0.526480\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −9.60904 −2.04865
\(23\) 9.07152 1.89154 0.945771 0.324834i \(-0.105308\pi\)
0.945771 + 0.324834i \(0.105308\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.209634 −0.0411126
\(27\) 0 0
\(28\) 2.11501 0.399699
\(29\) 3.98847 0.740641 0.370320 0.928904i \(-0.379248\pi\)
0.370320 + 0.928904i \(0.379248\pi\)
\(30\) 0 0
\(31\) 1.06662 0.191571 0.0957857 0.995402i \(-0.469464\pi\)
0.0957857 + 0.995402i \(0.469464\pi\)
\(32\) −8.06387 −1.42550
\(33\) 0 0
\(34\) 11.7857 2.02123
\(35\) 0 0
\(36\) 0 0
\(37\) 4.02621 0.661905 0.330953 0.943647i \(-0.392630\pi\)
0.330953 + 0.943647i \(0.392630\pi\)
\(38\) −1.46248 −0.237245
\(39\) 0 0
\(40\) 0 0
\(41\) −7.20977 −1.12598 −0.562988 0.826465i \(-0.690348\pi\)
−0.562988 + 0.826465i \(0.690348\pi\)
\(42\) 0 0
\(43\) 8.62791 1.31574 0.657872 0.753130i \(-0.271457\pi\)
0.657872 + 0.753130i \(0.271457\pi\)
\(44\) −9.88018 −1.48949
\(45\) 0 0
\(46\) 18.3142 2.70028
\(47\) 8.19797 1.19580 0.597899 0.801572i \(-0.296003\pi\)
0.597899 + 0.801572i \(0.296003\pi\)
\(48\) 0 0
\(49\) −5.96190 −0.851700
\(50\) 0 0
\(51\) 0 0
\(52\) −0.215549 −0.0298913
\(53\) 4.36719 0.599880 0.299940 0.953958i \(-0.403033\pi\)
0.299940 + 0.953958i \(0.403033\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.155995 0.0208457
\(57\) 0 0
\(58\) 8.05221 1.05731
\(59\) 4.91285 0.639599 0.319799 0.947485i \(-0.396384\pi\)
0.319799 + 0.947485i \(0.396384\pi\)
\(60\) 0 0
\(61\) 6.96435 0.891694 0.445847 0.895109i \(-0.352903\pi\)
0.445847 + 0.895109i \(0.352903\pi\)
\(62\) 2.15338 0.273479
\(63\) 0 0
\(64\) −8.59476 −1.07435
\(65\) 0 0
\(66\) 0 0
\(67\) −9.91998 −1.21192 −0.605959 0.795496i \(-0.707210\pi\)
−0.605959 + 0.795496i \(0.707210\pi\)
\(68\) 12.1183 1.46955
\(69\) 0 0
\(70\) 0 0
\(71\) 10.7866 1.28013 0.640065 0.768320i \(-0.278907\pi\)
0.640065 + 0.768320i \(0.278907\pi\)
\(72\) 0 0
\(73\) 8.63115 1.01020 0.505100 0.863061i \(-0.331456\pi\)
0.505100 + 0.863061i \(0.331456\pi\)
\(74\) 8.12840 0.944907
\(75\) 0 0
\(76\) −1.50375 −0.172491
\(77\) −4.84943 −0.552644
\(78\) 0 0
\(79\) 2.48291 0.279350 0.139675 0.990197i \(-0.455394\pi\)
0.139675 + 0.990197i \(0.455394\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.5556 −1.60740
\(83\) −4.24385 −0.465823 −0.232911 0.972498i \(-0.574825\pi\)
−0.232911 + 0.972498i \(0.574825\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 17.4186 1.87830
\(87\) 0 0
\(88\) −0.728724 −0.0776823
\(89\) 18.3752 1.94777 0.973884 0.227047i \(-0.0729072\pi\)
0.973884 + 0.227047i \(0.0729072\pi\)
\(90\) 0 0
\(91\) −0.105797 −0.0110905
\(92\) 18.8310 1.96327
\(93\) 0 0
\(94\) 16.5506 1.70707
\(95\) 0 0
\(96\) 0 0
\(97\) 6.69876 0.680156 0.340078 0.940397i \(-0.389546\pi\)
0.340078 + 0.940397i \(0.389546\pi\)
\(98\) −12.0363 −1.21585
\(99\) 0 0
\(100\) 0 0
\(101\) −5.66147 −0.563337 −0.281669 0.959512i \(-0.590888\pi\)
−0.281669 + 0.959512i \(0.590888\pi\)
\(102\) 0 0
\(103\) 0.594489 0.0585767 0.0292884 0.999571i \(-0.490676\pi\)
0.0292884 + 0.999571i \(0.490676\pi\)
\(104\) −0.0158981 −0.00155893
\(105\) 0 0
\(106\) 8.81680 0.856363
\(107\) −1.38651 −0.134039 −0.0670193 0.997752i \(-0.521349\pi\)
−0.0670193 + 0.997752i \(0.521349\pi\)
\(108\) 0 0
\(109\) −8.85677 −0.848325 −0.424162 0.905586i \(-0.639431\pi\)
−0.424162 + 0.905586i \(0.639431\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.91508 −0.369941
\(113\) −6.59039 −0.619971 −0.309986 0.950741i \(-0.600324\pi\)
−0.309986 + 0.950741i \(0.600324\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.27942 0.768725
\(117\) 0 0
\(118\) 9.91841 0.913064
\(119\) 5.94793 0.545245
\(120\) 0 0
\(121\) 11.6539 1.05945
\(122\) 14.0601 1.27294
\(123\) 0 0
\(124\) 2.21414 0.198836
\(125\) 0 0
\(126\) 0 0
\(127\) −7.29144 −0.647011 −0.323505 0.946226i \(-0.604861\pi\)
−0.323505 + 0.946226i \(0.604861\pi\)
\(128\) −1.22397 −0.108184
\(129\) 0 0
\(130\) 0 0
\(131\) 9.55386 0.834725 0.417362 0.908740i \(-0.362955\pi\)
0.417362 + 0.908740i \(0.362955\pi\)
\(132\) 0 0
\(133\) −0.738074 −0.0639991
\(134\) −20.0271 −1.73008
\(135\) 0 0
\(136\) 0.893796 0.0766424
\(137\) 6.19984 0.529688 0.264844 0.964291i \(-0.414680\pi\)
0.264844 + 0.964291i \(0.414680\pi\)
\(138\) 0 0
\(139\) −0.906531 −0.0768910 −0.0384455 0.999261i \(-0.512241\pi\)
−0.0384455 + 0.999261i \(0.512241\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 21.7767 1.82746
\(143\) 0.494225 0.0413291
\(144\) 0 0
\(145\) 0 0
\(146\) 17.4252 1.44212
\(147\) 0 0
\(148\) 8.35776 0.687004
\(149\) 4.89808 0.401267 0.200633 0.979666i \(-0.435700\pi\)
0.200633 + 0.979666i \(0.435700\pi\)
\(150\) 0 0
\(151\) −10.2626 −0.835161 −0.417581 0.908640i \(-0.637122\pi\)
−0.417581 + 0.908640i \(0.637122\pi\)
\(152\) −0.110911 −0.00899603
\(153\) 0 0
\(154\) −9.79036 −0.788930
\(155\) 0 0
\(156\) 0 0
\(157\) −8.89537 −0.709928 −0.354964 0.934880i \(-0.615507\pi\)
−0.354964 + 0.934880i \(0.615507\pi\)
\(158\) 5.01268 0.398788
\(159\) 0 0
\(160\) 0 0
\(161\) 9.24270 0.728427
\(162\) 0 0
\(163\) 11.7791 0.922607 0.461304 0.887242i \(-0.347382\pi\)
0.461304 + 0.887242i \(0.347382\pi\)
\(164\) −14.9663 −1.16867
\(165\) 0 0
\(166\) −8.56778 −0.664989
\(167\) −17.3762 −1.34461 −0.672306 0.740274i \(-0.734696\pi\)
−0.672306 + 0.740274i \(0.734696\pi\)
\(168\) 0 0
\(169\) −12.9892 −0.999171
\(170\) 0 0
\(171\) 0 0
\(172\) 17.9101 1.36564
\(173\) 12.9596 0.985298 0.492649 0.870228i \(-0.336029\pi\)
0.492649 + 0.870228i \(0.336029\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 18.2892 1.37860
\(177\) 0 0
\(178\) 37.0971 2.78055
\(179\) −1.09897 −0.0821409 −0.0410704 0.999156i \(-0.513077\pi\)
−0.0410704 + 0.999156i \(0.513077\pi\)
\(180\) 0 0
\(181\) −14.9797 −1.11343 −0.556716 0.830703i \(-0.687939\pi\)
−0.556716 + 0.830703i \(0.687939\pi\)
\(182\) −0.213590 −0.0158323
\(183\) 0 0
\(184\) 1.38890 0.102391
\(185\) 0 0
\(186\) 0 0
\(187\) −27.7855 −2.03188
\(188\) 17.0177 1.24114
\(189\) 0 0
\(190\) 0 0
\(191\) 12.7404 0.921862 0.460931 0.887436i \(-0.347516\pi\)
0.460931 + 0.887436i \(0.347516\pi\)
\(192\) 0 0
\(193\) −4.38386 −0.315557 −0.157779 0.987475i \(-0.550433\pi\)
−0.157779 + 0.987475i \(0.550433\pi\)
\(194\) 13.5239 0.970962
\(195\) 0 0
\(196\) −12.3759 −0.883996
\(197\) −3.22165 −0.229533 −0.114767 0.993392i \(-0.536612\pi\)
−0.114767 + 0.993392i \(0.536612\pi\)
\(198\) 0 0
\(199\) −2.70518 −0.191765 −0.0958825 0.995393i \(-0.530567\pi\)
−0.0958825 + 0.995393i \(0.530567\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −11.4298 −0.804196
\(203\) 4.06374 0.285218
\(204\) 0 0
\(205\) 0 0
\(206\) 1.20020 0.0836216
\(207\) 0 0
\(208\) 0.399002 0.0276658
\(209\) 3.44788 0.238495
\(210\) 0 0
\(211\) 26.4435 1.82044 0.910222 0.414122i \(-0.135911\pi\)
0.910222 + 0.414122i \(0.135911\pi\)
\(212\) 9.06559 0.622627
\(213\) 0 0
\(214\) −2.79918 −0.191348
\(215\) 0 0
\(216\) 0 0
\(217\) 1.08675 0.0737736
\(218\) −17.8807 −1.21103
\(219\) 0 0
\(220\) 0 0
\(221\) −0.606177 −0.0407759
\(222\) 0 0
\(223\) −8.03245 −0.537893 −0.268946 0.963155i \(-0.586675\pi\)
−0.268946 + 0.963155i \(0.586675\pi\)
\(224\) −8.21604 −0.548957
\(225\) 0 0
\(226\) −13.3051 −0.885044
\(227\) −16.4576 −1.09233 −0.546166 0.837677i \(-0.683913\pi\)
−0.546166 + 0.837677i \(0.683913\pi\)
\(228\) 0 0
\(229\) 0.409285 0.0270463 0.0135231 0.999909i \(-0.495695\pi\)
0.0135231 + 0.999909i \(0.495695\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.610658 0.0400917
\(233\) 18.4641 1.20962 0.604811 0.796369i \(-0.293249\pi\)
0.604811 + 0.796369i \(0.293249\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.1983 0.663852
\(237\) 0 0
\(238\) 12.0081 0.778369
\(239\) 4.61682 0.298637 0.149319 0.988789i \(-0.452292\pi\)
0.149319 + 0.988789i \(0.452292\pi\)
\(240\) 0 0
\(241\) −29.2022 −1.88108 −0.940540 0.339682i \(-0.889681\pi\)
−0.940540 + 0.339682i \(0.889681\pi\)
\(242\) 23.5277 1.51242
\(243\) 0 0
\(244\) 14.4569 0.925506
\(245\) 0 0
\(246\) 0 0
\(247\) 0.0752201 0.00478614
\(248\) 0.163307 0.0103700
\(249\) 0 0
\(250\) 0 0
\(251\) −0.389664 −0.0245954 −0.0122977 0.999924i \(-0.503915\pi\)
−0.0122977 + 0.999924i \(0.503915\pi\)
\(252\) 0 0
\(253\) −43.1769 −2.71451
\(254\) −14.7205 −0.923645
\(255\) 0 0
\(256\) 14.7185 0.919906
\(257\) −8.03324 −0.501100 −0.250550 0.968104i \(-0.580611\pi\)
−0.250550 + 0.968104i \(0.580611\pi\)
\(258\) 0 0
\(259\) 4.10219 0.254898
\(260\) 0 0
\(261\) 0 0
\(262\) 19.2880 1.19162
\(263\) 24.2131 1.49304 0.746522 0.665360i \(-0.231722\pi\)
0.746522 + 0.665360i \(0.231722\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.49008 −0.0913624
\(267\) 0 0
\(268\) −20.5923 −1.25787
\(269\) −26.4063 −1.61002 −0.805009 0.593263i \(-0.797839\pi\)
−0.805009 + 0.593263i \(0.797839\pi\)
\(270\) 0 0
\(271\) −10.4088 −0.632287 −0.316144 0.948711i \(-0.602388\pi\)
−0.316144 + 0.948711i \(0.602388\pi\)
\(272\) −22.4320 −1.36014
\(273\) 0 0
\(274\) 12.5167 0.756160
\(275\) 0 0
\(276\) 0 0
\(277\) −24.5207 −1.47330 −0.736652 0.676272i \(-0.763595\pi\)
−0.736652 + 0.676272i \(0.763595\pi\)
\(278\) −1.83017 −0.109766
\(279\) 0 0
\(280\) 0 0
\(281\) 14.4228 0.860393 0.430197 0.902735i \(-0.358444\pi\)
0.430197 + 0.902735i \(0.358444\pi\)
\(282\) 0 0
\(283\) 2.14925 0.127760 0.0638798 0.997958i \(-0.479653\pi\)
0.0638798 + 0.997958i \(0.479653\pi\)
\(284\) 22.3912 1.32867
\(285\) 0 0
\(286\) 0.997775 0.0589997
\(287\) −7.34582 −0.433610
\(288\) 0 0
\(289\) 17.0795 1.00468
\(290\) 0 0
\(291\) 0 0
\(292\) 17.9169 1.04851
\(293\) 9.02970 0.527521 0.263760 0.964588i \(-0.415037\pi\)
0.263760 + 0.964588i \(0.415037\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.616437 0.0358297
\(297\) 0 0
\(298\) 9.88860 0.572831
\(299\) −0.941960 −0.0544750
\(300\) 0 0
\(301\) 8.79072 0.506689
\(302\) −20.7189 −1.19224
\(303\) 0 0
\(304\) 2.78358 0.159649
\(305\) 0 0
\(306\) 0 0
\(307\) 5.03454 0.287336 0.143668 0.989626i \(-0.454110\pi\)
0.143668 + 0.989626i \(0.454110\pi\)
\(308\) −10.0666 −0.573599
\(309\) 0 0
\(310\) 0 0
\(311\) 4.89158 0.277376 0.138688 0.990336i \(-0.455711\pi\)
0.138688 + 0.990336i \(0.455711\pi\)
\(312\) 0 0
\(313\) −3.17282 −0.179338 −0.0896691 0.995972i \(-0.528581\pi\)
−0.0896691 + 0.995972i \(0.528581\pi\)
\(314\) −17.9586 −1.01346
\(315\) 0 0
\(316\) 5.15413 0.289942
\(317\) 19.9953 1.12305 0.561524 0.827461i \(-0.310215\pi\)
0.561524 + 0.827461i \(0.310215\pi\)
\(318\) 0 0
\(319\) −18.9836 −1.06288
\(320\) 0 0
\(321\) 0 0
\(322\) 18.6598 1.03987
\(323\) −4.22890 −0.235302
\(324\) 0 0
\(325\) 0 0
\(326\) 23.7804 1.31707
\(327\) 0 0
\(328\) −1.10386 −0.0609503
\(329\) 8.35267 0.460498
\(330\) 0 0
\(331\) 6.01724 0.330738 0.165369 0.986232i \(-0.447119\pi\)
0.165369 + 0.986232i \(0.447119\pi\)
\(332\) −8.80954 −0.483486
\(333\) 0 0
\(334\) −35.0803 −1.91951
\(335\) 0 0
\(336\) 0 0
\(337\) 22.8136 1.24274 0.621369 0.783518i \(-0.286577\pi\)
0.621369 + 0.783518i \(0.286577\pi\)
\(338\) −26.2235 −1.42637
\(339\) 0 0
\(340\) 0 0
\(341\) −5.07672 −0.274920
\(342\) 0 0
\(343\) −13.2065 −0.713084
\(344\) 1.32098 0.0712227
\(345\) 0 0
\(346\) 26.1637 1.40657
\(347\) −9.79068 −0.525591 −0.262796 0.964852i \(-0.584644\pi\)
−0.262796 + 0.964852i \(0.584644\pi\)
\(348\) 0 0
\(349\) −1.28648 −0.0688639 −0.0344320 0.999407i \(-0.510962\pi\)
−0.0344320 + 0.999407i \(0.510962\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 38.3809 2.04571
\(353\) −1.16422 −0.0619654 −0.0309827 0.999520i \(-0.509864\pi\)
−0.0309827 + 0.999520i \(0.509864\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 38.1439 2.02162
\(357\) 0 0
\(358\) −2.21868 −0.117261
\(359\) −19.0504 −1.00544 −0.502720 0.864449i \(-0.667667\pi\)
−0.502720 + 0.864449i \(0.667667\pi\)
\(360\) 0 0
\(361\) −18.4752 −0.972381
\(362\) −30.2421 −1.58949
\(363\) 0 0
\(364\) −0.219617 −0.0115110
\(365\) 0 0
\(366\) 0 0
\(367\) 16.7694 0.875357 0.437679 0.899131i \(-0.355801\pi\)
0.437679 + 0.899131i \(0.355801\pi\)
\(368\) −34.8580 −1.81710
\(369\) 0 0
\(370\) 0 0
\(371\) 4.44960 0.231012
\(372\) 0 0
\(373\) −17.0520 −0.882920 −0.441460 0.897281i \(-0.645539\pi\)
−0.441460 + 0.897281i \(0.645539\pi\)
\(374\) −56.0953 −2.90062
\(375\) 0 0
\(376\) 1.25516 0.0647298
\(377\) −0.414152 −0.0213299
\(378\) 0 0
\(379\) 1.66611 0.0855825 0.0427913 0.999084i \(-0.486375\pi\)
0.0427913 + 0.999084i \(0.486375\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 25.7212 1.31601
\(383\) −3.05361 −0.156032 −0.0780162 0.996952i \(-0.524859\pi\)
−0.0780162 + 0.996952i \(0.524859\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.85045 −0.450476
\(387\) 0 0
\(388\) 13.9055 0.705947
\(389\) −28.2725 −1.43347 −0.716737 0.697343i \(-0.754365\pi\)
−0.716737 + 0.697343i \(0.754365\pi\)
\(390\) 0 0
\(391\) 52.9574 2.67817
\(392\) −0.912802 −0.0461035
\(393\) 0 0
\(394\) −6.50410 −0.327672
\(395\) 0 0
\(396\) 0 0
\(397\) −20.4783 −1.02778 −0.513888 0.857858i \(-0.671795\pi\)
−0.513888 + 0.857858i \(0.671795\pi\)
\(398\) −5.46140 −0.273755
\(399\) 0 0
\(400\) 0 0
\(401\) −25.5952 −1.27816 −0.639081 0.769139i \(-0.720685\pi\)
−0.639081 + 0.769139i \(0.720685\pi\)
\(402\) 0 0
\(403\) −0.110755 −0.00551711
\(404\) −11.7523 −0.584698
\(405\) 0 0
\(406\) 8.20416 0.407165
\(407\) −19.1632 −0.949885
\(408\) 0 0
\(409\) −12.0402 −0.595349 −0.297675 0.954667i \(-0.596211\pi\)
−0.297675 + 0.954667i \(0.596211\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.23406 0.0607979
\(413\) 5.00556 0.246307
\(414\) 0 0
\(415\) 0 0
\(416\) 0.837329 0.0410534
\(417\) 0 0
\(418\) 6.96083 0.340465
\(419\) 34.6045 1.69054 0.845270 0.534340i \(-0.179440\pi\)
0.845270 + 0.534340i \(0.179440\pi\)
\(420\) 0 0
\(421\) −14.9525 −0.728738 −0.364369 0.931255i \(-0.618715\pi\)
−0.364369 + 0.931255i \(0.618715\pi\)
\(422\) 53.3859 2.59879
\(423\) 0 0
\(424\) 0.668643 0.0324722
\(425\) 0 0
\(426\) 0 0
\(427\) 7.09577 0.343389
\(428\) −2.87816 −0.139121
\(429\) 0 0
\(430\) 0 0
\(431\) −4.81107 −0.231741 −0.115871 0.993264i \(-0.536966\pi\)
−0.115871 + 0.993264i \(0.536966\pi\)
\(432\) 0 0
\(433\) 10.3435 0.497077 0.248538 0.968622i \(-0.420050\pi\)
0.248538 + 0.968622i \(0.420050\pi\)
\(434\) 2.19401 0.105316
\(435\) 0 0
\(436\) −18.3852 −0.880492
\(437\) −6.57145 −0.314355
\(438\) 0 0
\(439\) 30.7640 1.46829 0.734143 0.678995i \(-0.237584\pi\)
0.734143 + 0.678995i \(0.237584\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.22379 −0.0582099
\(443\) 17.7545 0.843543 0.421772 0.906702i \(-0.361408\pi\)
0.421772 + 0.906702i \(0.361408\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.2165 −0.767873
\(447\) 0 0
\(448\) −8.75695 −0.413727
\(449\) 37.2184 1.75645 0.878223 0.478251i \(-0.158729\pi\)
0.878223 + 0.478251i \(0.158729\pi\)
\(450\) 0 0
\(451\) 34.3157 1.61586
\(452\) −13.6806 −0.643480
\(453\) 0 0
\(454\) −33.2258 −1.55937
\(455\) 0 0
\(456\) 0 0
\(457\) −27.6987 −1.29569 −0.647846 0.761772i \(-0.724330\pi\)
−0.647846 + 0.761772i \(0.724330\pi\)
\(458\) 0.826293 0.0386101
\(459\) 0 0
\(460\) 0 0
\(461\) −9.81742 −0.457243 −0.228621 0.973515i \(-0.573422\pi\)
−0.228621 + 0.973515i \(0.573422\pi\)
\(462\) 0 0
\(463\) −3.72732 −0.173223 −0.0866115 0.996242i \(-0.527604\pi\)
−0.0866115 + 0.996242i \(0.527604\pi\)
\(464\) −15.3260 −0.711492
\(465\) 0 0
\(466\) 37.2766 1.72680
\(467\) 7.09925 0.328514 0.164257 0.986418i \(-0.447477\pi\)
0.164257 + 0.986418i \(0.447477\pi\)
\(468\) 0 0
\(469\) −10.1072 −0.466706
\(470\) 0 0
\(471\) 0 0
\(472\) 0.752186 0.0346222
\(473\) −41.0655 −1.88819
\(474\) 0 0
\(475\) 0 0
\(476\) 12.3469 0.565920
\(477\) 0 0
\(478\) 9.32076 0.426322
\(479\) 28.4451 1.29969 0.649845 0.760066i \(-0.274834\pi\)
0.649845 + 0.760066i \(0.274834\pi\)
\(480\) 0 0
\(481\) −0.418070 −0.0190624
\(482\) −58.9555 −2.68535
\(483\) 0 0
\(484\) 24.1916 1.09962
\(485\) 0 0
\(486\) 0 0
\(487\) −14.7384 −0.667860 −0.333930 0.942598i \(-0.608375\pi\)
−0.333930 + 0.942598i \(0.608375\pi\)
\(488\) 1.06628 0.0482684
\(489\) 0 0
\(490\) 0 0
\(491\) 28.4014 1.28174 0.640869 0.767650i \(-0.278574\pi\)
0.640869 + 0.767650i \(0.278574\pi\)
\(492\) 0 0
\(493\) 23.2838 1.04865
\(494\) 0.151860 0.00683248
\(495\) 0 0
\(496\) −4.09858 −0.184032
\(497\) 10.9901 0.492974
\(498\) 0 0
\(499\) 26.3842 1.18112 0.590559 0.806995i \(-0.298907\pi\)
0.590559 + 0.806995i \(0.298907\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −0.786682 −0.0351113
\(503\) −16.6592 −0.742795 −0.371398 0.928474i \(-0.621121\pi\)
−0.371398 + 0.928474i \(0.621121\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −87.1686 −3.87512
\(507\) 0 0
\(508\) −15.1358 −0.671544
\(509\) 31.7760 1.40845 0.704223 0.709979i \(-0.251296\pi\)
0.704223 + 0.709979i \(0.251296\pi\)
\(510\) 0 0
\(511\) 8.79403 0.389025
\(512\) 32.1627 1.42140
\(513\) 0 0
\(514\) −16.2181 −0.715349
\(515\) 0 0
\(516\) 0 0
\(517\) −39.0192 −1.71606
\(518\) 8.28179 0.363881
\(519\) 0 0
\(520\) 0 0
\(521\) 0.592363 0.0259519 0.0129759 0.999916i \(-0.495870\pi\)
0.0129759 + 0.999916i \(0.495870\pi\)
\(522\) 0 0
\(523\) −17.2298 −0.753405 −0.376703 0.926334i \(-0.622942\pi\)
−0.376703 + 0.926334i \(0.622942\pi\)
\(524\) 19.8323 0.866376
\(525\) 0 0
\(526\) 48.8832 2.13141
\(527\) 6.22670 0.271240
\(528\) 0 0
\(529\) 59.2924 2.57793
\(530\) 0 0
\(531\) 0 0
\(532\) −1.53212 −0.0664259
\(533\) 0.748642 0.0324273
\(534\) 0 0
\(535\) 0 0
\(536\) −1.51881 −0.0656025
\(537\) 0 0
\(538\) −53.3108 −2.29839
\(539\) 28.3763 1.22226
\(540\) 0 0
\(541\) 14.7990 0.636258 0.318129 0.948047i \(-0.396945\pi\)
0.318129 + 0.948047i \(0.396945\pi\)
\(542\) −21.0139 −0.902626
\(543\) 0 0
\(544\) −47.0750 −2.01832
\(545\) 0 0
\(546\) 0 0
\(547\) 6.50806 0.278264 0.139132 0.990274i \(-0.455569\pi\)
0.139132 + 0.990274i \(0.455569\pi\)
\(548\) 12.8699 0.549773
\(549\) 0 0
\(550\) 0 0
\(551\) −2.88927 −0.123087
\(552\) 0 0
\(553\) 2.52977 0.107577
\(554\) −49.5041 −2.10323
\(555\) 0 0
\(556\) −1.88181 −0.0798066
\(557\) 6.67224 0.282712 0.141356 0.989959i \(-0.454854\pi\)
0.141356 + 0.989959i \(0.454854\pi\)
\(558\) 0 0
\(559\) −0.895898 −0.0378924
\(560\) 0 0
\(561\) 0 0
\(562\) 29.1178 1.22826
\(563\) −20.0663 −0.845694 −0.422847 0.906201i \(-0.638969\pi\)
−0.422847 + 0.906201i \(0.638969\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.33906 0.182384
\(567\) 0 0
\(568\) 1.65149 0.0692949
\(569\) 21.5938 0.905261 0.452631 0.891698i \(-0.350486\pi\)
0.452631 + 0.891698i \(0.350486\pi\)
\(570\) 0 0
\(571\) −26.3338 −1.10203 −0.551017 0.834494i \(-0.685760\pi\)
−0.551017 + 0.834494i \(0.685760\pi\)
\(572\) 1.02593 0.0428963
\(573\) 0 0
\(574\) −14.8303 −0.619003
\(575\) 0 0
\(576\) 0 0
\(577\) −9.18240 −0.382268 −0.191134 0.981564i \(-0.561217\pi\)
−0.191134 + 0.981564i \(0.561217\pi\)
\(578\) 34.4813 1.43423
\(579\) 0 0
\(580\) 0 0
\(581\) −4.32393 −0.179387
\(582\) 0 0
\(583\) −20.7861 −0.860874
\(584\) 1.32148 0.0546832
\(585\) 0 0
\(586\) 18.2298 0.753066
\(587\) −39.9771 −1.65003 −0.825017 0.565108i \(-0.808834\pi\)
−0.825017 + 0.565108i \(0.808834\pi\)
\(588\) 0 0
\(589\) −0.772668 −0.0318372
\(590\) 0 0
\(591\) 0 0
\(592\) −15.4710 −0.635855
\(593\) 41.6331 1.70967 0.854834 0.518902i \(-0.173659\pi\)
0.854834 + 0.518902i \(0.173659\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.1676 0.416482
\(597\) 0 0
\(598\) −1.90170 −0.0777661
\(599\) −39.0726 −1.59646 −0.798232 0.602350i \(-0.794231\pi\)
−0.798232 + 0.602350i \(0.794231\pi\)
\(600\) 0 0
\(601\) 5.46965 0.223112 0.111556 0.993758i \(-0.464417\pi\)
0.111556 + 0.993758i \(0.464417\pi\)
\(602\) 17.7473 0.723327
\(603\) 0 0
\(604\) −21.3036 −0.866829
\(605\) 0 0
\(606\) 0 0
\(607\) 42.3108 1.71734 0.858671 0.512527i \(-0.171291\pi\)
0.858671 + 0.512527i \(0.171291\pi\)
\(608\) 5.84150 0.236904
\(609\) 0 0
\(610\) 0 0
\(611\) −0.851254 −0.0344380
\(612\) 0 0
\(613\) −34.0064 −1.37351 −0.686753 0.726891i \(-0.740965\pi\)
−0.686753 + 0.726891i \(0.740965\pi\)
\(614\) 10.1641 0.410189
\(615\) 0 0
\(616\) −0.742476 −0.0299152
\(617\) −23.3173 −0.938717 −0.469359 0.883008i \(-0.655515\pi\)
−0.469359 + 0.883008i \(0.655515\pi\)
\(618\) 0 0
\(619\) 35.2998 1.41882 0.709409 0.704797i \(-0.248962\pi\)
0.709409 + 0.704797i \(0.248962\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 9.87547 0.395970
\(623\) 18.7219 0.750079
\(624\) 0 0
\(625\) 0 0
\(626\) −6.40551 −0.256016
\(627\) 0 0
\(628\) −18.4653 −0.736847
\(629\) 23.5041 0.937169
\(630\) 0 0
\(631\) −3.50433 −0.139505 −0.0697526 0.997564i \(-0.522221\pi\)
−0.0697526 + 0.997564i \(0.522221\pi\)
\(632\) 0.380149 0.0151215
\(633\) 0 0
\(634\) 40.3679 1.60321
\(635\) 0 0
\(636\) 0 0
\(637\) 0.619067 0.0245283
\(638\) −38.3254 −1.51732
\(639\) 0 0
\(640\) 0 0
\(641\) 9.28841 0.366870 0.183435 0.983032i \(-0.441278\pi\)
0.183435 + 0.983032i \(0.441278\pi\)
\(642\) 0 0
\(643\) −0.0291680 −0.00115028 −0.000575138 1.00000i \(-0.500183\pi\)
−0.000575138 1.00000i \(0.500183\pi\)
\(644\) 19.1863 0.756048
\(645\) 0 0
\(646\) −8.53760 −0.335908
\(647\) 0.844100 0.0331850 0.0165925 0.999862i \(-0.494718\pi\)
0.0165925 + 0.999862i \(0.494718\pi\)
\(648\) 0 0
\(649\) −23.3833 −0.917874
\(650\) 0 0
\(651\) 0 0
\(652\) 24.4514 0.957591
\(653\) −38.0711 −1.48984 −0.744918 0.667156i \(-0.767511\pi\)
−0.744918 + 0.667156i \(0.767511\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 27.7041 1.08166
\(657\) 0 0
\(658\) 16.8630 0.657387
\(659\) −37.8343 −1.47382 −0.736908 0.675993i \(-0.763715\pi\)
−0.736908 + 0.675993i \(0.763715\pi\)
\(660\) 0 0
\(661\) −18.5614 −0.721956 −0.360978 0.932574i \(-0.617557\pi\)
−0.360978 + 0.932574i \(0.617557\pi\)
\(662\) 12.1480 0.472147
\(663\) 0 0
\(664\) −0.649758 −0.0252155
\(665\) 0 0
\(666\) 0 0
\(667\) 36.1815 1.40095
\(668\) −36.0702 −1.39560
\(669\) 0 0
\(670\) 0 0
\(671\) −33.1476 −1.27965
\(672\) 0 0
\(673\) 18.1607 0.700044 0.350022 0.936742i \(-0.386174\pi\)
0.350022 + 0.936742i \(0.386174\pi\)
\(674\) 46.0578 1.77408
\(675\) 0 0
\(676\) −26.9635 −1.03706
\(677\) 5.64312 0.216883 0.108441 0.994103i \(-0.465414\pi\)
0.108441 + 0.994103i \(0.465414\pi\)
\(678\) 0 0
\(679\) 6.82517 0.261926
\(680\) 0 0
\(681\) 0 0
\(682\) −10.2492 −0.392464
\(683\) −30.4468 −1.16502 −0.582508 0.812825i \(-0.697929\pi\)
−0.582508 + 0.812825i \(0.697929\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −26.6622 −1.01797
\(687\) 0 0
\(688\) −33.1534 −1.26396
\(689\) −0.453477 −0.0172761
\(690\) 0 0
\(691\) −9.22470 −0.350924 −0.175462 0.984486i \(-0.556142\pi\)
−0.175462 + 0.984486i \(0.556142\pi\)
\(692\) 26.9020 1.02266
\(693\) 0 0
\(694\) −19.7661 −0.750311
\(695\) 0 0
\(696\) 0 0
\(697\) −42.0889 −1.59423
\(698\) −2.59724 −0.0983071
\(699\) 0 0
\(700\) 0 0
\(701\) 19.9822 0.754717 0.377358 0.926067i \(-0.376832\pi\)
0.377358 + 0.926067i \(0.376832\pi\)
\(702\) 0 0
\(703\) −2.91661 −0.110002
\(704\) 40.9077 1.54177
\(705\) 0 0
\(706\) −2.35042 −0.0884591
\(707\) −5.76830 −0.216939
\(708\) 0 0
\(709\) −26.8259 −1.00747 −0.503734 0.863859i \(-0.668041\pi\)
−0.503734 + 0.863859i \(0.668041\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.81335 0.105435
\(713\) 9.67591 0.362366
\(714\) 0 0
\(715\) 0 0
\(716\) −2.28128 −0.0852556
\(717\) 0 0
\(718\) −38.4602 −1.43532
\(719\) −38.8224 −1.44783 −0.723915 0.689889i \(-0.757659\pi\)
−0.723915 + 0.689889i \(0.757659\pi\)
\(720\) 0 0
\(721\) 0.605707 0.0225577
\(722\) −37.2991 −1.38813
\(723\) 0 0
\(724\) −31.0954 −1.15565
\(725\) 0 0
\(726\) 0 0
\(727\) 29.5764 1.09693 0.548465 0.836174i \(-0.315213\pi\)
0.548465 + 0.836174i \(0.315213\pi\)
\(728\) −0.0161981 −0.000600341 0
\(729\) 0 0
\(730\) 0 0
\(731\) 50.3677 1.86292
\(732\) 0 0
\(733\) −48.8782 −1.80536 −0.902679 0.430315i \(-0.858402\pi\)
−0.902679 + 0.430315i \(0.858402\pi\)
\(734\) 33.8553 1.24962
\(735\) 0 0
\(736\) −73.1515 −2.69640
\(737\) 47.2152 1.73920
\(738\) 0 0
\(739\) 21.7603 0.800466 0.400233 0.916413i \(-0.368929\pi\)
0.400233 + 0.916413i \(0.368929\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.98317 0.329783
\(743\) 9.75724 0.357959 0.178979 0.983853i \(-0.442720\pi\)
0.178979 + 0.983853i \(0.442720\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −34.4258 −1.26042
\(747\) 0 0
\(748\) −57.6782 −2.10892
\(749\) −1.41267 −0.0516178
\(750\) 0 0
\(751\) −17.6413 −0.643741 −0.321871 0.946784i \(-0.604312\pi\)
−0.321871 + 0.946784i \(0.604312\pi\)
\(752\) −31.5013 −1.14873
\(753\) 0 0
\(754\) −0.836118 −0.0304496
\(755\) 0 0
\(756\) 0 0
\(757\) −44.2551 −1.60848 −0.804240 0.594305i \(-0.797427\pi\)
−0.804240 + 0.594305i \(0.797427\pi\)
\(758\) 3.36367 0.122174
\(759\) 0 0
\(760\) 0 0
\(761\) −39.8755 −1.44549 −0.722743 0.691117i \(-0.757119\pi\)
−0.722743 + 0.691117i \(0.757119\pi\)
\(762\) 0 0
\(763\) −9.02390 −0.326687
\(764\) 26.4470 0.956818
\(765\) 0 0
\(766\) −6.16485 −0.222745
\(767\) −0.510137 −0.0184200
\(768\) 0 0
\(769\) 35.2667 1.27175 0.635876 0.771792i \(-0.280639\pi\)
0.635876 + 0.771792i \(0.280639\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9.10018 −0.327523
\(773\) 23.2417 0.835946 0.417973 0.908459i \(-0.362741\pi\)
0.417973 + 0.908459i \(0.362741\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.02562 0.0368176
\(777\) 0 0
\(778\) −57.0786 −2.04637
\(779\) 5.22279 0.187126
\(780\) 0 0
\(781\) −51.3399 −1.83709
\(782\) 106.914 3.82324
\(783\) 0 0
\(784\) 22.9091 0.818180
\(785\) 0 0
\(786\) 0 0
\(787\) 6.83095 0.243497 0.121749 0.992561i \(-0.461150\pi\)
0.121749 + 0.992561i \(0.461150\pi\)
\(788\) −6.68762 −0.238237
\(789\) 0 0
\(790\) 0 0
\(791\) −6.71475 −0.238749
\(792\) 0 0
\(793\) −0.723159 −0.0256801
\(794\) −41.3430 −1.46721
\(795\) 0 0
\(796\) −5.61551 −0.199036
\(797\) −53.9287 −1.91025 −0.955126 0.296198i \(-0.904281\pi\)
−0.955126 + 0.296198i \(0.904281\pi\)
\(798\) 0 0
\(799\) 47.8578 1.69309
\(800\) 0 0
\(801\) 0 0
\(802\) −51.6734 −1.82465
\(803\) −41.0809 −1.44971
\(804\) 0 0
\(805\) 0 0
\(806\) −0.223601 −0.00787599
\(807\) 0 0
\(808\) −0.866804 −0.0304941
\(809\) 18.4104 0.647275 0.323637 0.946181i \(-0.395094\pi\)
0.323637 + 0.946181i \(0.395094\pi\)
\(810\) 0 0
\(811\) −11.7910 −0.414040 −0.207020 0.978337i \(-0.566376\pi\)
−0.207020 + 0.978337i \(0.566376\pi\)
\(812\) 8.43565 0.296033
\(813\) 0 0
\(814\) −38.6880 −1.35601
\(815\) 0 0
\(816\) 0 0
\(817\) −6.25009 −0.218663
\(818\) −24.3076 −0.849895
\(819\) 0 0
\(820\) 0 0
\(821\) 12.3261 0.430185 0.215093 0.976594i \(-0.430995\pi\)
0.215093 + 0.976594i \(0.430995\pi\)
\(822\) 0 0
\(823\) 2.81852 0.0982475 0.0491237 0.998793i \(-0.484357\pi\)
0.0491237 + 0.998793i \(0.484357\pi\)
\(824\) 0.0910197 0.00317082
\(825\) 0 0
\(826\) 10.1056 0.351618
\(827\) 37.4251 1.30140 0.650700 0.759335i \(-0.274476\pi\)
0.650700 + 0.759335i \(0.274476\pi\)
\(828\) 0 0
\(829\) 30.9434 1.07471 0.537355 0.843356i \(-0.319424\pi\)
0.537355 + 0.843356i \(0.319424\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.892455 0.0309403
\(833\) −34.8042 −1.20589
\(834\) 0 0
\(835\) 0 0
\(836\) 7.15724 0.247538
\(837\) 0 0
\(838\) 69.8620 2.41334
\(839\) 26.6975 0.921701 0.460851 0.887478i \(-0.347544\pi\)
0.460851 + 0.887478i \(0.347544\pi\)
\(840\) 0 0
\(841\) −13.0921 −0.451451
\(842\) −30.1871 −1.04032
\(843\) 0 0
\(844\) 54.8923 1.88947
\(845\) 0 0
\(846\) 0 0
\(847\) 11.8738 0.407989
\(848\) −16.7813 −0.576271
\(849\) 0 0
\(850\) 0 0
\(851\) 36.5239 1.25202
\(852\) 0 0
\(853\) 29.2600 1.00184 0.500921 0.865493i \(-0.332995\pi\)
0.500921 + 0.865493i \(0.332995\pi\)
\(854\) 14.3254 0.490207
\(855\) 0 0
\(856\) −0.212282 −0.00725566
\(857\) 33.5284 1.14531 0.572654 0.819797i \(-0.305914\pi\)
0.572654 + 0.819797i \(0.305914\pi\)
\(858\) 0 0
\(859\) −25.8243 −0.881115 −0.440557 0.897725i \(-0.645219\pi\)
−0.440557 + 0.897725i \(0.645219\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −9.71293 −0.330824
\(863\) 28.8886 0.983378 0.491689 0.870771i \(-0.336380\pi\)
0.491689 + 0.870771i \(0.336380\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 20.8822 0.709605
\(867\) 0 0
\(868\) 2.25592 0.0765710
\(869\) −11.8177 −0.400888
\(870\) 0 0
\(871\) 1.03006 0.0349023
\(872\) −1.35602 −0.0459207
\(873\) 0 0
\(874\) −13.2669 −0.448759
\(875\) 0 0
\(876\) 0 0
\(877\) −15.1004 −0.509905 −0.254953 0.966954i \(-0.582060\pi\)
−0.254953 + 0.966954i \(0.582060\pi\)
\(878\) 62.1086 2.09606
\(879\) 0 0
\(880\) 0 0
\(881\) −36.6216 −1.23381 −0.616906 0.787037i \(-0.711614\pi\)
−0.616906 + 0.787037i \(0.711614\pi\)
\(882\) 0 0
\(883\) −34.5459 −1.16256 −0.581281 0.813703i \(-0.697448\pi\)
−0.581281 + 0.813703i \(0.697448\pi\)
\(884\) −1.25832 −0.0423220
\(885\) 0 0
\(886\) 35.8441 1.20421
\(887\) −33.0360 −1.10924 −0.554620 0.832104i \(-0.687136\pi\)
−0.554620 + 0.832104i \(0.687136\pi\)
\(888\) 0 0
\(889\) −7.42903 −0.249162
\(890\) 0 0
\(891\) 0 0
\(892\) −16.6741 −0.558289
\(893\) −5.93864 −0.198729
\(894\) 0 0
\(895\) 0 0
\(896\) −1.24706 −0.0416614
\(897\) 0 0
\(898\) 75.1392 2.50743
\(899\) 4.25420 0.141886
\(900\) 0 0
\(901\) 25.4947 0.849350
\(902\) 69.2789 2.30674
\(903\) 0 0
\(904\) −1.00903 −0.0335597
\(905\) 0 0
\(906\) 0 0
\(907\) −44.2708 −1.46999 −0.734994 0.678074i \(-0.762815\pi\)
−0.734994 + 0.678074i \(0.762815\pi\)
\(908\) −34.1634 −1.13375
\(909\) 0 0
\(910\) 0 0
\(911\) 38.8844 1.28830 0.644148 0.764901i \(-0.277212\pi\)
0.644148 + 0.764901i \(0.277212\pi\)
\(912\) 0 0
\(913\) 20.1991 0.668492
\(914\) −55.9201 −1.84967
\(915\) 0 0
\(916\) 0.849608 0.0280719
\(917\) 9.73414 0.321450
\(918\) 0 0
\(919\) −28.5303 −0.941127 −0.470563 0.882366i \(-0.655949\pi\)
−0.470563 + 0.882366i \(0.655949\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −19.8201 −0.652740
\(923\) −1.12005 −0.0368668
\(924\) 0 0
\(925\) 0 0
\(926\) −7.52497 −0.247286
\(927\) 0 0
\(928\) −32.1625 −1.05579
\(929\) −36.3886 −1.19387 −0.596936 0.802289i \(-0.703615\pi\)
−0.596936 + 0.802289i \(0.703615\pi\)
\(930\) 0 0
\(931\) 4.31883 0.141544
\(932\) 38.3284 1.25549
\(933\) 0 0
\(934\) 14.3325 0.468973
\(935\) 0 0
\(936\) 0 0
\(937\) −47.3354 −1.54638 −0.773189 0.634175i \(-0.781340\pi\)
−0.773189 + 0.634175i \(0.781340\pi\)
\(938\) −20.4051 −0.666249
\(939\) 0 0
\(940\) 0 0
\(941\) −52.9254 −1.72532 −0.862659 0.505787i \(-0.831202\pi\)
−0.862659 + 0.505787i \(0.831202\pi\)
\(942\) 0 0
\(943\) −65.4035 −2.12983
\(944\) −18.8780 −0.614426
\(945\) 0 0
\(946\) −82.9059 −2.69550
\(947\) −18.9330 −0.615239 −0.307619 0.951509i \(-0.599532\pi\)
−0.307619 + 0.951509i \(0.599532\pi\)
\(948\) 0 0
\(949\) −0.896234 −0.0290930
\(950\) 0 0
\(951\) 0 0
\(952\) 0.910662 0.0295147
\(953\) 11.4499 0.370899 0.185450 0.982654i \(-0.440626\pi\)
0.185450 + 0.982654i \(0.440626\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9.58377 0.309961
\(957\) 0 0
\(958\) 57.4270 1.85538
\(959\) 6.31683 0.203981
\(960\) 0 0
\(961\) −29.8623 −0.963300
\(962\) −0.844030 −0.0272126
\(963\) 0 0
\(964\) −60.6191 −1.95241
\(965\) 0 0
\(966\) 0 0
\(967\) −36.9926 −1.18960 −0.594801 0.803873i \(-0.702769\pi\)
−0.594801 + 0.803873i \(0.702769\pi\)
\(968\) 1.78428 0.0573490
\(969\) 0 0
\(970\) 0 0
\(971\) 30.2897 0.972043 0.486022 0.873947i \(-0.338448\pi\)
0.486022 + 0.873947i \(0.338448\pi\)
\(972\) 0 0
\(973\) −0.923638 −0.0296105
\(974\) −29.7549 −0.953408
\(975\) 0 0
\(976\) −26.7610 −0.856600
\(977\) −16.6410 −0.532394 −0.266197 0.963919i \(-0.585767\pi\)
−0.266197 + 0.963919i \(0.585767\pi\)
\(978\) 0 0
\(979\) −87.4588 −2.79520
\(980\) 0 0
\(981\) 0 0
\(982\) 57.3388 1.82975
\(983\) −26.1897 −0.835320 −0.417660 0.908603i \(-0.637150\pi\)
−0.417660 + 0.908603i \(0.637150\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 47.0069 1.49700
\(987\) 0 0
\(988\) 0.156145 0.00496762
\(989\) 78.2682 2.48878
\(990\) 0 0
\(991\) −24.8373 −0.788982 −0.394491 0.918900i \(-0.629079\pi\)
−0.394491 + 0.918900i \(0.629079\pi\)
\(992\) −8.60112 −0.273086
\(993\) 0 0
\(994\) 22.1876 0.703749
\(995\) 0 0
\(996\) 0 0
\(997\) 25.0863 0.794490 0.397245 0.917713i \(-0.369966\pi\)
0.397245 + 0.917713i \(0.369966\pi\)
\(998\) 53.2662 1.68611
\(999\) 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.p.1.5 6
3.2 odd 2 1875.2.a.j.1.2 6
5.4 even 2 5625.2.a.q.1.2 6
15.2 even 4 1875.2.b.f.1249.4 12
15.8 even 4 1875.2.b.f.1249.9 12
15.14 odd 2 1875.2.a.k.1.5 6
25.6 even 5 225.2.h.d.136.1 12
25.21 even 5 225.2.h.d.91.1 12
75.8 even 20 375.2.i.d.199.2 24
75.17 even 20 375.2.i.d.199.5 24
75.29 odd 10 375.2.g.c.76.1 12
75.44 odd 10 375.2.g.c.301.1 12
75.47 even 20 375.2.i.d.49.2 24
75.53 even 20 375.2.i.d.49.5 24
75.56 odd 10 75.2.g.c.61.3 yes 12
75.71 odd 10 75.2.g.c.16.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.16.3 12 75.71 odd 10
75.2.g.c.61.3 yes 12 75.56 odd 10
225.2.h.d.91.1 12 25.21 even 5
225.2.h.d.136.1 12 25.6 even 5
375.2.g.c.76.1 12 75.29 odd 10
375.2.g.c.301.1 12 75.44 odd 10
375.2.i.d.49.2 24 75.47 even 20
375.2.i.d.49.5 24 75.53 even 20
375.2.i.d.199.2 24 75.8 even 20
375.2.i.d.199.5 24 75.17 even 20
1875.2.a.j.1.2 6 3.2 odd 2
1875.2.a.k.1.5 6 15.14 odd 2
1875.2.b.f.1249.4 12 15.2 even 4
1875.2.b.f.1249.9 12 15.8 even 4
5625.2.a.p.1.5 6 1.1 even 1 trivial
5625.2.a.q.1.2 6 5.4 even 2