Properties

Label 5625.2.a.q.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 \(-2.16056\) 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)
 
\(f(q)\) \(=\) \(q+2.16056 q^{2} +2.66802 q^{4} +3.16056 q^{7} +1.44329 q^{8} +O(q^{10})\) \(q+2.16056 q^{2} +2.66802 q^{4} +3.16056 q^{7} +1.44329 q^{8} -1.53835 q^{11} +5.24144 q^{13} +6.82858 q^{14} -2.21771 q^{16} +1.29068 q^{17} +5.44587 q^{19} -3.32370 q^{22} -6.44244 q^{23} +11.3244 q^{26} +8.43243 q^{28} +2.36361 q^{29} -4.46542 q^{31} -7.67809 q^{32} +2.78860 q^{34} +5.95751 q^{37} +11.7661 q^{38} +8.53219 q^{41} +8.48426 q^{43} -4.10435 q^{44} -13.9193 q^{46} +0.753070 q^{47} +2.98914 q^{49} +13.9843 q^{52} +9.74991 q^{53} +4.56162 q^{56} +5.10672 q^{58} -4.11270 q^{59} -10.6939 q^{61} -9.64781 q^{62} -12.1535 q^{64} +1.89864 q^{67} +3.44357 q^{68} +0.0708774 q^{71} -4.01123 q^{73} +12.8716 q^{74} +14.5297 q^{76} -4.86205 q^{77} +1.61849 q^{79} +18.4343 q^{82} +13.1488 q^{83} +18.3308 q^{86} -2.22029 q^{88} -7.27597 q^{89} +16.5659 q^{91} -17.1885 q^{92} +1.62705 q^{94} -10.1939 q^{97} +6.45821 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 10 q^{4} + 6 q^{7} - 3 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.16056 1.52775 0.763873 0.645366i \(-0.223295\pi\)
0.763873 + 0.645366i \(0.223295\pi\)
\(3\) 0 0
\(4\) 2.66802 1.33401
\(5\) 0 0
\(6\) 0 0
\(7\) 3.16056 1.19458 0.597290 0.802026i \(-0.296244\pi\)
0.597290 + 0.802026i \(0.296244\pi\)
\(8\) 1.44329 0.510282
\(9\) 0 0
\(10\) 0 0
\(11\) −1.53835 −0.463830 −0.231915 0.972736i \(-0.574499\pi\)
−0.231915 + 0.972736i \(0.574499\pi\)
\(12\) 0 0
\(13\) 5.24144 1.45371 0.726857 0.686789i \(-0.240981\pi\)
0.726857 + 0.686789i \(0.240981\pi\)
\(14\) 6.82858 1.82501
\(15\) 0 0
\(16\) −2.21771 −0.554428
\(17\) 1.29068 0.313037 0.156518 0.987675i \(-0.449973\pi\)
0.156518 + 0.987675i \(0.449973\pi\)
\(18\) 0 0
\(19\) 5.44587 1.24937 0.624685 0.780877i \(-0.285228\pi\)
0.624685 + 0.780877i \(0.285228\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.32370 −0.708615
\(23\) −6.44244 −1.34334 −0.671671 0.740850i \(-0.734423\pi\)
−0.671671 + 0.740850i \(0.734423\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 11.3244 2.22090
\(27\) 0 0
\(28\) 8.43243 1.59358
\(29\) 2.36361 0.438912 0.219456 0.975622i \(-0.429572\pi\)
0.219456 + 0.975622i \(0.429572\pi\)
\(30\) 0 0
\(31\) −4.46542 −0.802014 −0.401007 0.916075i \(-0.631340\pi\)
−0.401007 + 0.916075i \(0.631340\pi\)
\(32\) −7.67809 −1.35731
\(33\) 0 0
\(34\) 2.78860 0.478241
\(35\) 0 0
\(36\) 0 0
\(37\) 5.95751 0.979408 0.489704 0.871889i \(-0.337105\pi\)
0.489704 + 0.871889i \(0.337105\pi\)
\(38\) 11.7661 1.90872
\(39\) 0 0
\(40\) 0 0
\(41\) 8.53219 1.33250 0.666252 0.745726i \(-0.267897\pi\)
0.666252 + 0.745726i \(0.267897\pi\)
\(42\) 0 0
\(43\) 8.48426 1.29384 0.646919 0.762559i \(-0.276057\pi\)
0.646919 + 0.762559i \(0.276057\pi\)
\(44\) −4.10435 −0.618754
\(45\) 0 0
\(46\) −13.9193 −2.05228
\(47\) 0.753070 0.109847 0.0549233 0.998491i \(-0.482509\pi\)
0.0549233 + 0.998491i \(0.482509\pi\)
\(48\) 0 0
\(49\) 2.98914 0.427020
\(50\) 0 0
\(51\) 0 0
\(52\) 13.9843 1.93927
\(53\) 9.74991 1.33925 0.669626 0.742698i \(-0.266454\pi\)
0.669626 + 0.742698i \(0.266454\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.56162 0.609572
\(57\) 0 0
\(58\) 5.10672 0.670546
\(59\) −4.11270 −0.535428 −0.267714 0.963498i \(-0.586268\pi\)
−0.267714 + 0.963498i \(0.586268\pi\)
\(60\) 0 0
\(61\) −10.6939 −1.36922 −0.684610 0.728910i \(-0.740027\pi\)
−0.684610 + 0.728910i \(0.740027\pi\)
\(62\) −9.64781 −1.22527
\(63\) 0 0
\(64\) −12.1535 −1.51919
\(65\) 0 0
\(66\) 0 0
\(67\) 1.89864 0.231956 0.115978 0.993252i \(-0.463000\pi\)
0.115978 + 0.993252i \(0.463000\pi\)
\(68\) 3.44357 0.417594
\(69\) 0 0
\(70\) 0 0
\(71\) 0.0708774 0.00841160 0.00420580 0.999991i \(-0.498661\pi\)
0.00420580 + 0.999991i \(0.498661\pi\)
\(72\) 0 0
\(73\) −4.01123 −0.469478 −0.234739 0.972058i \(-0.575424\pi\)
−0.234739 + 0.972058i \(0.575424\pi\)
\(74\) 12.8716 1.49629
\(75\) 0 0
\(76\) 14.5297 1.66667
\(77\) −4.86205 −0.554082
\(78\) 0 0
\(79\) 1.61849 0.182094 0.0910472 0.995847i \(-0.470979\pi\)
0.0910472 + 0.995847i \(0.470979\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 18.4343 2.03573
\(83\) 13.1488 1.44327 0.721636 0.692272i \(-0.243390\pi\)
0.721636 + 0.692272i \(0.243390\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 18.3308 1.97666
\(87\) 0 0
\(88\) −2.22029 −0.236684
\(89\) −7.27597 −0.771251 −0.385626 0.922655i \(-0.626014\pi\)
−0.385626 + 0.922655i \(0.626014\pi\)
\(90\) 0 0
\(91\) 16.5659 1.73658
\(92\) −17.1885 −1.79203
\(93\) 0 0
\(94\) 1.62705 0.167818
\(95\) 0 0
\(96\) 0 0
\(97\) −10.1939 −1.03504 −0.517518 0.855673i \(-0.673144\pi\)
−0.517518 + 0.855673i \(0.673144\pi\)
\(98\) 6.45821 0.652378
\(99\) 0 0
\(100\) 0 0
\(101\) −1.08759 −0.108219 −0.0541096 0.998535i \(-0.517232\pi\)
−0.0541096 + 0.998535i \(0.517232\pi\)
\(102\) 0 0
\(103\) 4.93756 0.486512 0.243256 0.969962i \(-0.421784\pi\)
0.243256 + 0.969962i \(0.421784\pi\)
\(104\) 7.56494 0.741803
\(105\) 0 0
\(106\) 21.0653 2.04604
\(107\) 15.3059 1.47967 0.739837 0.672786i \(-0.234902\pi\)
0.739837 + 0.672786i \(0.234902\pi\)
\(108\) 0 0
\(109\) 6.65900 0.637816 0.318908 0.947786i \(-0.396684\pi\)
0.318908 + 0.947786i \(0.396684\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −7.00922 −0.662309
\(113\) −8.97143 −0.843961 −0.421981 0.906605i \(-0.638665\pi\)
−0.421981 + 0.906605i \(0.638665\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.30616 0.585512
\(117\) 0 0
\(118\) −8.88573 −0.817998
\(119\) 4.07928 0.373947
\(120\) 0 0
\(121\) −8.63348 −0.784861
\(122\) −23.1049 −2.09182
\(123\) 0 0
\(124\) −11.9138 −1.06989
\(125\) 0 0
\(126\) 0 0
\(127\) −10.0185 −0.888995 −0.444497 0.895780i \(-0.646618\pi\)
−0.444497 + 0.895780i \(0.646618\pi\)
\(128\) −10.9023 −0.963635
\(129\) 0 0
\(130\) 0 0
\(131\) 6.37865 0.557305 0.278653 0.960392i \(-0.410112\pi\)
0.278653 + 0.960392i \(0.410112\pi\)
\(132\) 0 0
\(133\) 17.2120 1.49247
\(134\) 4.10214 0.354371
\(135\) 0 0
\(136\) 1.86284 0.159737
\(137\) −9.17732 −0.784071 −0.392036 0.919950i \(-0.628229\pi\)
−0.392036 + 0.919950i \(0.628229\pi\)
\(138\) 0 0
\(139\) −9.26539 −0.785880 −0.392940 0.919564i \(-0.628542\pi\)
−0.392940 + 0.919564i \(0.628542\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.153135 0.0128508
\(143\) −8.06317 −0.674276
\(144\) 0 0
\(145\) 0 0
\(146\) −8.66649 −0.717244
\(147\) 0 0
\(148\) 15.8947 1.30654
\(149\) 10.0817 0.825928 0.412964 0.910747i \(-0.364494\pi\)
0.412964 + 0.910747i \(0.364494\pi\)
\(150\) 0 0
\(151\) 10.7375 0.873804 0.436902 0.899509i \(-0.356076\pi\)
0.436902 + 0.899509i \(0.356076\pi\)
\(152\) 7.86000 0.637530
\(153\) 0 0
\(154\) −10.5048 −0.846497
\(155\) 0 0
\(156\) 0 0
\(157\) −17.4417 −1.39200 −0.696001 0.718041i \(-0.745039\pi\)
−0.696001 + 0.718041i \(0.745039\pi\)
\(158\) 3.49684 0.278194
\(159\) 0 0
\(160\) 0 0
\(161\) −20.3617 −1.60473
\(162\) 0 0
\(163\) 16.2574 1.27338 0.636689 0.771120i \(-0.280303\pi\)
0.636689 + 0.771120i \(0.280303\pi\)
\(164\) 22.7641 1.77757
\(165\) 0 0
\(166\) 28.4089 2.20496
\(167\) −11.3906 −0.881430 −0.440715 0.897647i \(-0.645275\pi\)
−0.440715 + 0.897647i \(0.645275\pi\)
\(168\) 0 0
\(169\) 14.4727 1.11328
\(170\) 0 0
\(171\) 0 0
\(172\) 22.6362 1.72599
\(173\) 9.45845 0.719113 0.359556 0.933123i \(-0.382928\pi\)
0.359556 + 0.933123i \(0.382928\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.41162 0.257161
\(177\) 0 0
\(178\) −15.7202 −1.17828
\(179\) −4.07747 −0.304764 −0.152382 0.988322i \(-0.548694\pi\)
−0.152382 + 0.988322i \(0.548694\pi\)
\(180\) 0 0
\(181\) −11.1202 −0.826558 −0.413279 0.910604i \(-0.635617\pi\)
−0.413279 + 0.910604i \(0.635617\pi\)
\(182\) 35.7916 2.65305
\(183\) 0 0
\(184\) −9.29833 −0.685482
\(185\) 0 0
\(186\) 0 0
\(187\) −1.98552 −0.145196
\(188\) 2.00921 0.146536
\(189\) 0 0
\(190\) 0 0
\(191\) −3.61135 −0.261308 −0.130654 0.991428i \(-0.541708\pi\)
−0.130654 + 0.991428i \(0.541708\pi\)
\(192\) 0 0
\(193\) 17.0562 1.22774 0.613868 0.789409i \(-0.289613\pi\)
0.613868 + 0.789409i \(0.289613\pi\)
\(194\) −22.0246 −1.58127
\(195\) 0 0
\(196\) 7.97508 0.569648
\(197\) −4.40523 −0.313860 −0.156930 0.987610i \(-0.550160\pi\)
−0.156930 + 0.987610i \(0.550160\pi\)
\(198\) 0 0
\(199\) 16.5956 1.17643 0.588214 0.808705i \(-0.299831\pi\)
0.588214 + 0.808705i \(0.299831\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.34980 −0.165332
\(203\) 7.47034 0.524315
\(204\) 0 0
\(205\) 0 0
\(206\) 10.6679 0.743267
\(207\) 0 0
\(208\) −11.6240 −0.805980
\(209\) −8.37767 −0.579495
\(210\) 0 0
\(211\) 18.5454 1.27671 0.638357 0.769740i \(-0.279614\pi\)
0.638357 + 0.769740i \(0.279614\pi\)
\(212\) 26.0129 1.78658
\(213\) 0 0
\(214\) 33.0693 2.26057
\(215\) 0 0
\(216\) 0 0
\(217\) −14.1132 −0.958069
\(218\) 14.3872 0.974422
\(219\) 0 0
\(220\) 0 0
\(221\) 6.76504 0.455066
\(222\) 0 0
\(223\) 0.771557 0.0516673 0.0258336 0.999666i \(-0.491776\pi\)
0.0258336 + 0.999666i \(0.491776\pi\)
\(224\) −24.2671 −1.62141
\(225\) 0 0
\(226\) −19.3833 −1.28936
\(227\) 11.4314 0.758727 0.379363 0.925248i \(-0.376143\pi\)
0.379363 + 0.925248i \(0.376143\pi\)
\(228\) 0 0
\(229\) 10.2248 0.675671 0.337835 0.941205i \(-0.390305\pi\)
0.337835 + 0.941205i \(0.390305\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.41139 0.223969
\(233\) −22.5229 −1.47553 −0.737763 0.675059i \(-0.764118\pi\)
−0.737763 + 0.675059i \(0.764118\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.9728 −0.714266
\(237\) 0 0
\(238\) 8.81353 0.571297
\(239\) 0.589074 0.0381040 0.0190520 0.999818i \(-0.493935\pi\)
0.0190520 + 0.999818i \(0.493935\pi\)
\(240\) 0 0
\(241\) 1.50937 0.0972270 0.0486135 0.998818i \(-0.484520\pi\)
0.0486135 + 0.998818i \(0.484520\pi\)
\(242\) −18.6531 −1.19907
\(243\) 0 0
\(244\) −28.5317 −1.82655
\(245\) 0 0
\(246\) 0 0
\(247\) 28.5442 1.81622
\(248\) −6.44492 −0.409253
\(249\) 0 0
\(250\) 0 0
\(251\) −11.8953 −0.750823 −0.375412 0.926858i \(-0.622499\pi\)
−0.375412 + 0.926858i \(0.622499\pi\)
\(252\) 0 0
\(253\) 9.91073 0.623082
\(254\) −21.6455 −1.35816
\(255\) 0 0
\(256\) 0.752056 0.0470035
\(257\) −18.0492 −1.12588 −0.562939 0.826498i \(-0.690329\pi\)
−0.562939 + 0.826498i \(0.690329\pi\)
\(258\) 0 0
\(259\) 18.8291 1.16998
\(260\) 0 0
\(261\) 0 0
\(262\) 13.7815 0.851421
\(263\) −19.1066 −1.17817 −0.589083 0.808072i \(-0.700511\pi\)
−0.589083 + 0.808072i \(0.700511\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 37.1876 2.28012
\(267\) 0 0
\(268\) 5.06562 0.309432
\(269\) 22.2250 1.35508 0.677542 0.735484i \(-0.263045\pi\)
0.677542 + 0.735484i \(0.263045\pi\)
\(270\) 0 0
\(271\) 20.2107 1.22772 0.613858 0.789417i \(-0.289617\pi\)
0.613858 + 0.789417i \(0.289617\pi\)
\(272\) −2.86237 −0.173556
\(273\) 0 0
\(274\) −19.8281 −1.19786
\(275\) 0 0
\(276\) 0 0
\(277\) 16.0413 0.963826 0.481913 0.876219i \(-0.339942\pi\)
0.481913 + 0.876219i \(0.339942\pi\)
\(278\) −20.0184 −1.20063
\(279\) 0 0
\(280\) 0 0
\(281\) −8.37308 −0.499496 −0.249748 0.968311i \(-0.580348\pi\)
−0.249748 + 0.968311i \(0.580348\pi\)
\(282\) 0 0
\(283\) 1.12995 0.0671685 0.0335843 0.999436i \(-0.489308\pi\)
0.0335843 + 0.999436i \(0.489308\pi\)
\(284\) 0.189102 0.0112211
\(285\) 0 0
\(286\) −17.4210 −1.03012
\(287\) 26.9665 1.59178
\(288\) 0 0
\(289\) −15.3341 −0.902008
\(290\) 0 0
\(291\) 0 0
\(292\) −10.7020 −0.626289
\(293\) −2.37857 −0.138958 −0.0694789 0.997583i \(-0.522134\pi\)
−0.0694789 + 0.997583i \(0.522134\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.59844 0.499774
\(297\) 0 0
\(298\) 21.7822 1.26181
\(299\) −33.7676 −1.95283
\(300\) 0 0
\(301\) 26.8150 1.54559
\(302\) 23.1990 1.33495
\(303\) 0 0
\(304\) −12.0774 −0.692686
\(305\) 0 0
\(306\) 0 0
\(307\) −2.89366 −0.165150 −0.0825748 0.996585i \(-0.526314\pi\)
−0.0825748 + 0.996585i \(0.526314\pi\)
\(308\) −12.9720 −0.739151
\(309\) 0 0
\(310\) 0 0
\(311\) 7.14545 0.405181 0.202591 0.979264i \(-0.435064\pi\)
0.202591 + 0.979264i \(0.435064\pi\)
\(312\) 0 0
\(313\) −23.0318 −1.30184 −0.650918 0.759148i \(-0.725616\pi\)
−0.650918 + 0.759148i \(0.725616\pi\)
\(314\) −37.6839 −2.12663
\(315\) 0 0
\(316\) 4.31816 0.242916
\(317\) −20.7750 −1.16684 −0.583419 0.812171i \(-0.698285\pi\)
−0.583419 + 0.812171i \(0.698285\pi\)
\(318\) 0 0
\(319\) −3.63606 −0.203581
\(320\) 0 0
\(321\) 0 0
\(322\) −43.9927 −2.45162
\(323\) 7.02890 0.391098
\(324\) 0 0
\(325\) 0 0
\(326\) 35.1251 1.94540
\(327\) 0 0
\(328\) 12.3145 0.679953
\(329\) 2.38012 0.131220
\(330\) 0 0
\(331\) −19.2504 −1.05810 −0.529048 0.848592i \(-0.677451\pi\)
−0.529048 + 0.848592i \(0.677451\pi\)
\(332\) 35.0814 1.92534
\(333\) 0 0
\(334\) −24.6100 −1.34660
\(335\) 0 0
\(336\) 0 0
\(337\) −21.7064 −1.18243 −0.591213 0.806516i \(-0.701351\pi\)
−0.591213 + 0.806516i \(0.701351\pi\)
\(338\) 31.2690 1.70081
\(339\) 0 0
\(340\) 0 0
\(341\) 6.86939 0.371998
\(342\) 0 0
\(343\) −12.6766 −0.684470
\(344\) 12.2453 0.660221
\(345\) 0 0
\(346\) 20.4356 1.09862
\(347\) −1.71494 −0.0920626 −0.0460313 0.998940i \(-0.514657\pi\)
−0.0460313 + 0.998940i \(0.514657\pi\)
\(348\) 0 0
\(349\) −15.5553 −0.832654 −0.416327 0.909215i \(-0.636683\pi\)
−0.416327 + 0.909215i \(0.636683\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.8116 0.629560
\(353\) 15.5536 0.827833 0.413916 0.910315i \(-0.364161\pi\)
0.413916 + 0.910315i \(0.364161\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −19.4124 −1.02886
\(357\) 0 0
\(358\) −8.80961 −0.465602
\(359\) 24.5371 1.29502 0.647508 0.762059i \(-0.275811\pi\)
0.647508 + 0.762059i \(0.275811\pi\)
\(360\) 0 0
\(361\) 10.6575 0.560924
\(362\) −24.0259 −1.26277
\(363\) 0 0
\(364\) 44.1981 2.31661
\(365\) 0 0
\(366\) 0 0
\(367\) −28.1068 −1.46716 −0.733581 0.679602i \(-0.762153\pi\)
−0.733581 + 0.679602i \(0.762153\pi\)
\(368\) 14.2875 0.744786
\(369\) 0 0
\(370\) 0 0
\(371\) 30.8152 1.59984
\(372\) 0 0
\(373\) 9.70825 0.502674 0.251337 0.967900i \(-0.419130\pi\)
0.251337 + 0.967900i \(0.419130\pi\)
\(374\) −4.28984 −0.221823
\(375\) 0 0
\(376\) 1.08690 0.0560527
\(377\) 12.3887 0.638052
\(378\) 0 0
\(379\) −22.2665 −1.14375 −0.571875 0.820340i \(-0.693784\pi\)
−0.571875 + 0.820340i \(0.693784\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −7.80253 −0.399212
\(383\) −13.6299 −0.696458 −0.348229 0.937410i \(-0.613217\pi\)
−0.348229 + 0.937410i \(0.613217\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 36.8510 1.87567
\(387\) 0 0
\(388\) −27.1976 −1.38075
\(389\) −19.2304 −0.975021 −0.487510 0.873117i \(-0.662095\pi\)
−0.487510 + 0.873117i \(0.662095\pi\)
\(390\) 0 0
\(391\) −8.31515 −0.420515
\(392\) 4.31421 0.217900
\(393\) 0 0
\(394\) −9.51777 −0.479498
\(395\) 0 0
\(396\) 0 0
\(397\) −22.2281 −1.11560 −0.557799 0.829976i \(-0.688354\pi\)
−0.557799 + 0.829976i \(0.688354\pi\)
\(398\) 35.8557 1.79729
\(399\) 0 0
\(400\) 0 0
\(401\) −4.71728 −0.235570 −0.117785 0.993039i \(-0.537579\pi\)
−0.117785 + 0.993039i \(0.537579\pi\)
\(402\) 0 0
\(403\) −23.4052 −1.16590
\(404\) −2.90171 −0.144365
\(405\) 0 0
\(406\) 16.1401 0.801020
\(407\) −9.16474 −0.454279
\(408\) 0 0
\(409\) −12.4807 −0.617130 −0.308565 0.951203i \(-0.599849\pi\)
−0.308565 + 0.951203i \(0.599849\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 13.1735 0.649012
\(413\) −12.9984 −0.639611
\(414\) 0 0
\(415\) 0 0
\(416\) −40.2442 −1.97314
\(417\) 0 0
\(418\) −18.1005 −0.885322
\(419\) −34.9484 −1.70734 −0.853669 0.520815i \(-0.825628\pi\)
−0.853669 + 0.520815i \(0.825628\pi\)
\(420\) 0 0
\(421\) −39.5601 −1.92804 −0.964020 0.265829i \(-0.914354\pi\)
−0.964020 + 0.265829i \(0.914354\pi\)
\(422\) 40.0683 1.95050
\(423\) 0 0
\(424\) 14.0720 0.683396
\(425\) 0 0
\(426\) 0 0
\(427\) −33.7989 −1.63564
\(428\) 40.8364 1.97390
\(429\) 0 0
\(430\) 0 0
\(431\) 14.0584 0.677170 0.338585 0.940936i \(-0.390052\pi\)
0.338585 + 0.940936i \(0.390052\pi\)
\(432\) 0 0
\(433\) 0.549678 0.0264158 0.0132079 0.999913i \(-0.495796\pi\)
0.0132079 + 0.999913i \(0.495796\pi\)
\(434\) −30.4925 −1.46369
\(435\) 0 0
\(436\) 17.7663 0.850853
\(437\) −35.0847 −1.67833
\(438\) 0 0
\(439\) −2.97377 −0.141930 −0.0709651 0.997479i \(-0.522608\pi\)
−0.0709651 + 0.997479i \(0.522608\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 14.6163 0.695225
\(443\) −36.6893 −1.74316 −0.871580 0.490253i \(-0.836905\pi\)
−0.871580 + 0.490253i \(0.836905\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.66700 0.0789345
\(447\) 0 0
\(448\) −38.4120 −1.81480
\(449\) 17.8432 0.842071 0.421036 0.907044i \(-0.361667\pi\)
0.421036 + 0.907044i \(0.361667\pi\)
\(450\) 0 0
\(451\) −13.1255 −0.618056
\(452\) −23.9359 −1.12585
\(453\) 0 0
\(454\) 24.6982 1.15914
\(455\) 0 0
\(456\) 0 0
\(457\) 24.1784 1.13102 0.565509 0.824742i \(-0.308680\pi\)
0.565509 + 0.824742i \(0.308680\pi\)
\(458\) 22.0912 1.03225
\(459\) 0 0
\(460\) 0 0
\(461\) 37.3874 1.74130 0.870651 0.491901i \(-0.163698\pi\)
0.870651 + 0.491901i \(0.163698\pi\)
\(462\) 0 0
\(463\) −20.4314 −0.949530 −0.474765 0.880113i \(-0.657467\pi\)
−0.474765 + 0.880113i \(0.657467\pi\)
\(464\) −5.24181 −0.243345
\(465\) 0 0
\(466\) −48.6622 −2.25423
\(467\) 31.2124 1.44434 0.722169 0.691716i \(-0.243145\pi\)
0.722169 + 0.691716i \(0.243145\pi\)
\(468\) 0 0
\(469\) 6.00078 0.277090
\(470\) 0 0
\(471\) 0 0
\(472\) −5.93584 −0.273219
\(473\) −13.0518 −0.600121
\(474\) 0 0
\(475\) 0 0
\(476\) 10.8836 0.498849
\(477\) 0 0
\(478\) 1.27273 0.0582133
\(479\) −11.0970 −0.507033 −0.253516 0.967331i \(-0.581587\pi\)
−0.253516 + 0.967331i \(0.581587\pi\)
\(480\) 0 0
\(481\) 31.2259 1.42378
\(482\) 3.26108 0.148538
\(483\) 0 0
\(484\) −23.0343 −1.04701
\(485\) 0 0
\(486\) 0 0
\(487\) −7.13702 −0.323409 −0.161705 0.986839i \(-0.551699\pi\)
−0.161705 + 0.986839i \(0.551699\pi\)
\(488\) −15.4345 −0.698688
\(489\) 0 0
\(490\) 0 0
\(491\) −7.23348 −0.326442 −0.163221 0.986590i \(-0.552188\pi\)
−0.163221 + 0.986590i \(0.552188\pi\)
\(492\) 0 0
\(493\) 3.05067 0.137395
\(494\) 61.6715 2.77473
\(495\) 0 0
\(496\) 9.90303 0.444659
\(497\) 0.224012 0.0100483
\(498\) 0 0
\(499\) 16.0169 0.717014 0.358507 0.933527i \(-0.383286\pi\)
0.358507 + 0.933527i \(0.383286\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −25.7005 −1.14707
\(503\) −33.7963 −1.50690 −0.753452 0.657503i \(-0.771613\pi\)
−0.753452 + 0.657503i \(0.771613\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 21.4127 0.951912
\(507\) 0 0
\(508\) −26.7294 −1.18593
\(509\) 36.4270 1.61460 0.807300 0.590142i \(-0.200928\pi\)
0.807300 + 0.590142i \(0.200928\pi\)
\(510\) 0 0
\(511\) −12.6777 −0.560829
\(512\) 23.4294 1.03544
\(513\) 0 0
\(514\) −38.9964 −1.72006
\(515\) 0 0
\(516\) 0 0
\(517\) −1.15849 −0.0509502
\(518\) 40.6813 1.78743
\(519\) 0 0
\(520\) 0 0
\(521\) −25.4993 −1.11714 −0.558572 0.829456i \(-0.688651\pi\)
−0.558572 + 0.829456i \(0.688651\pi\)
\(522\) 0 0
\(523\) 7.03174 0.307477 0.153738 0.988112i \(-0.450869\pi\)
0.153738 + 0.988112i \(0.450869\pi\)
\(524\) 17.0184 0.743450
\(525\) 0 0
\(526\) −41.2811 −1.79994
\(527\) −5.76345 −0.251060
\(528\) 0 0
\(529\) 18.5050 0.804565
\(530\) 0 0
\(531\) 0 0
\(532\) 45.9220 1.99097
\(533\) 44.7210 1.93708
\(534\) 0 0
\(535\) 0 0
\(536\) 2.74030 0.118363
\(537\) 0 0
\(538\) 48.0185 2.07022
\(539\) −4.59834 −0.198065
\(540\) 0 0
\(541\) −27.1476 −1.16717 −0.583584 0.812053i \(-0.698350\pi\)
−0.583584 + 0.812053i \(0.698350\pi\)
\(542\) 43.6665 1.87564
\(543\) 0 0
\(544\) −9.90999 −0.424887
\(545\) 0 0
\(546\) 0 0
\(547\) −44.0769 −1.88459 −0.942296 0.334782i \(-0.891337\pi\)
−0.942296 + 0.334782i \(0.891337\pi\)
\(548\) −24.4853 −1.04596
\(549\) 0 0
\(550\) 0 0
\(551\) 12.8719 0.548363
\(552\) 0 0
\(553\) 5.11533 0.217526
\(554\) 34.6581 1.47248
\(555\) 0 0
\(556\) −24.7202 −1.04837
\(557\) −9.85667 −0.417641 −0.208820 0.977954i \(-0.566962\pi\)
−0.208820 + 0.977954i \(0.566962\pi\)
\(558\) 0 0
\(559\) 44.4697 1.88087
\(560\) 0 0
\(561\) 0 0
\(562\) −18.0905 −0.763103
\(563\) −12.3917 −0.522249 −0.261125 0.965305i \(-0.584093\pi\)
−0.261125 + 0.965305i \(0.584093\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.44132 0.102616
\(567\) 0 0
\(568\) 0.102297 0.00429228
\(569\) 11.7158 0.491150 0.245575 0.969378i \(-0.421023\pi\)
0.245575 + 0.969378i \(0.421023\pi\)
\(570\) 0 0
\(571\) 28.9797 1.21276 0.606382 0.795173i \(-0.292620\pi\)
0.606382 + 0.795173i \(0.292620\pi\)
\(572\) −21.5127 −0.899491
\(573\) 0 0
\(574\) 58.2628 2.43184
\(575\) 0 0
\(576\) 0 0
\(577\) 32.8024 1.36558 0.682790 0.730614i \(-0.260766\pi\)
0.682790 + 0.730614i \(0.260766\pi\)
\(578\) −33.1303 −1.37804
\(579\) 0 0
\(580\) 0 0
\(581\) 41.5577 1.72410
\(582\) 0 0
\(583\) −14.9988 −0.621186
\(584\) −5.78938 −0.239566
\(585\) 0 0
\(586\) −5.13905 −0.212292
\(587\) 18.6395 0.769334 0.384667 0.923055i \(-0.374316\pi\)
0.384667 + 0.923055i \(0.374316\pi\)
\(588\) 0 0
\(589\) −24.3181 −1.00201
\(590\) 0 0
\(591\) 0 0
\(592\) −13.2120 −0.543012
\(593\) 1.55882 0.0640129 0.0320064 0.999488i \(-0.489810\pi\)
0.0320064 + 0.999488i \(0.489810\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 26.8983 1.10180
\(597\) 0 0
\(598\) −72.9570 −2.98343
\(599\) 31.5470 1.28898 0.644489 0.764614i \(-0.277070\pi\)
0.644489 + 0.764614i \(0.277070\pi\)
\(600\) 0 0
\(601\) −32.3999 −1.32162 −0.660809 0.750554i \(-0.729787\pi\)
−0.660809 + 0.750554i \(0.729787\pi\)
\(602\) 57.9354 2.36127
\(603\) 0 0
\(604\) 28.6478 1.16566
\(605\) 0 0
\(606\) 0 0
\(607\) −24.8410 −1.00827 −0.504133 0.863626i \(-0.668188\pi\)
−0.504133 + 0.863626i \(0.668188\pi\)
\(608\) −41.8139 −1.69578
\(609\) 0 0
\(610\) 0 0
\(611\) 3.94717 0.159685
\(612\) 0 0
\(613\) 3.94558 0.159360 0.0796802 0.996820i \(-0.474610\pi\)
0.0796802 + 0.996820i \(0.474610\pi\)
\(614\) −6.25192 −0.252307
\(615\) 0 0
\(616\) −7.01737 −0.282738
\(617\) −17.4821 −0.703801 −0.351900 0.936037i \(-0.614464\pi\)
−0.351900 + 0.936037i \(0.614464\pi\)
\(618\) 0 0
\(619\) −14.3869 −0.578260 −0.289130 0.957290i \(-0.593366\pi\)
−0.289130 + 0.957290i \(0.593366\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 15.4382 0.619014
\(623\) −22.9961 −0.921321
\(624\) 0 0
\(625\) 0 0
\(626\) −49.7617 −1.98888
\(627\) 0 0
\(628\) −46.5349 −1.85694
\(629\) 7.68926 0.306591
\(630\) 0 0
\(631\) −17.6758 −0.703664 −0.351832 0.936063i \(-0.614441\pi\)
−0.351832 + 0.936063i \(0.614441\pi\)
\(632\) 2.33596 0.0929194
\(633\) 0 0
\(634\) −44.8855 −1.78263
\(635\) 0 0
\(636\) 0 0
\(637\) 15.6674 0.620764
\(638\) −7.85594 −0.311019
\(639\) 0 0
\(640\) 0 0
\(641\) 2.74127 0.108274 0.0541369 0.998534i \(-0.482759\pi\)
0.0541369 + 0.998534i \(0.482759\pi\)
\(642\) 0 0
\(643\) −20.9276 −0.825303 −0.412651 0.910889i \(-0.635397\pi\)
−0.412651 + 0.910889i \(0.635397\pi\)
\(644\) −54.3254 −2.14072
\(645\) 0 0
\(646\) 15.1864 0.597499
\(647\) −1.84667 −0.0726002 −0.0363001 0.999341i \(-0.511557\pi\)
−0.0363001 + 0.999341i \(0.511557\pi\)
\(648\) 0 0
\(649\) 6.32678 0.248348
\(650\) 0 0
\(651\) 0 0
\(652\) 43.3751 1.69870
\(653\) −18.4133 −0.720567 −0.360284 0.932843i \(-0.617320\pi\)
−0.360284 + 0.932843i \(0.617320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −18.9220 −0.738778
\(657\) 0 0
\(658\) 5.14240 0.200472
\(659\) 13.3800 0.521210 0.260605 0.965446i \(-0.416078\pi\)
0.260605 + 0.965446i \(0.416078\pi\)
\(660\) 0 0
\(661\) 39.1834 1.52406 0.762030 0.647542i \(-0.224203\pi\)
0.762030 + 0.647542i \(0.224203\pi\)
\(662\) −41.5915 −1.61650
\(663\) 0 0
\(664\) 18.9776 0.736476
\(665\) 0 0
\(666\) 0 0
\(667\) −15.2274 −0.589608
\(668\) −30.3903 −1.17584
\(669\) 0 0
\(670\) 0 0
\(671\) 16.4510 0.635086
\(672\) 0 0
\(673\) −19.3911 −0.747473 −0.373736 0.927535i \(-0.621923\pi\)
−0.373736 + 0.927535i \(0.621923\pi\)
\(674\) −46.8981 −1.80645
\(675\) 0 0
\(676\) 38.6133 1.48513
\(677\) −45.6836 −1.75576 −0.877882 0.478876i \(-0.841044\pi\)
−0.877882 + 0.478876i \(0.841044\pi\)
\(678\) 0 0
\(679\) −32.2185 −1.23643
\(680\) 0 0
\(681\) 0 0
\(682\) 14.8417 0.568319
\(683\) 11.5893 0.443451 0.221725 0.975109i \(-0.428831\pi\)
0.221725 + 0.975109i \(0.428831\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −27.3885 −1.04570
\(687\) 0 0
\(688\) −18.8157 −0.717340
\(689\) 51.1035 1.94689
\(690\) 0 0
\(691\) −21.6424 −0.823317 −0.411658 0.911338i \(-0.635050\pi\)
−0.411658 + 0.911338i \(0.635050\pi\)
\(692\) 25.2353 0.959303
\(693\) 0 0
\(694\) −3.70522 −0.140648
\(695\) 0 0
\(696\) 0 0
\(697\) 11.0124 0.417123
\(698\) −33.6081 −1.27208
\(699\) 0 0
\(700\) 0 0
\(701\) 8.61904 0.325537 0.162768 0.986664i \(-0.447958\pi\)
0.162768 + 0.986664i \(0.447958\pi\)
\(702\) 0 0
\(703\) 32.4438 1.22364
\(704\) 18.6964 0.704648
\(705\) 0 0
\(706\) 33.6044 1.26472
\(707\) −3.43739 −0.129276
\(708\) 0 0
\(709\) 16.0001 0.600895 0.300447 0.953798i \(-0.402864\pi\)
0.300447 + 0.953798i \(0.402864\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.5014 −0.393555
\(713\) 28.7682 1.07738
\(714\) 0 0
\(715\) 0 0
\(716\) −10.8788 −0.406558
\(717\) 0 0
\(718\) 53.0138 1.97846
\(719\) 48.5838 1.81187 0.905935 0.423416i \(-0.139169\pi\)
0.905935 + 0.423416i \(0.139169\pi\)
\(720\) 0 0
\(721\) 15.6055 0.581177
\(722\) 23.0263 0.856949
\(723\) 0 0
\(724\) −29.6689 −1.10264
\(725\) 0 0
\(726\) 0 0
\(727\) −25.4328 −0.943251 −0.471625 0.881799i \(-0.656333\pi\)
−0.471625 + 0.881799i \(0.656333\pi\)
\(728\) 23.9094 0.886143
\(729\) 0 0
\(730\) 0 0
\(731\) 10.9505 0.405019
\(732\) 0 0
\(733\) 18.6190 0.687708 0.343854 0.939023i \(-0.388267\pi\)
0.343854 + 0.939023i \(0.388267\pi\)
\(734\) −60.7264 −2.24145
\(735\) 0 0
\(736\) 49.4656 1.82333
\(737\) −2.92078 −0.107588
\(738\) 0 0
\(739\) 46.8872 1.72478 0.862388 0.506249i \(-0.168968\pi\)
0.862388 + 0.506249i \(0.168968\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 66.5780 2.44416
\(743\) −42.7061 −1.56674 −0.783368 0.621558i \(-0.786500\pi\)
−0.783368 + 0.621558i \(0.786500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 20.9753 0.767959
\(747\) 0 0
\(748\) −5.29742 −0.193693
\(749\) 48.3751 1.76759
\(750\) 0 0
\(751\) −2.64893 −0.0966608 −0.0483304 0.998831i \(-0.515390\pi\)
−0.0483304 + 0.998831i \(0.515390\pi\)
\(752\) −1.67009 −0.0609021
\(753\) 0 0
\(754\) 26.7666 0.974781
\(755\) 0 0
\(756\) 0 0
\(757\) −52.9153 −1.92324 −0.961620 0.274383i \(-0.911526\pi\)
−0.961620 + 0.274383i \(0.911526\pi\)
\(758\) −48.1080 −1.74736
\(759\) 0 0
\(760\) 0 0
\(761\) −50.7787 −1.84073 −0.920363 0.391064i \(-0.872107\pi\)
−0.920363 + 0.391064i \(0.872107\pi\)
\(762\) 0 0
\(763\) 21.0462 0.761922
\(764\) −9.63514 −0.348587
\(765\) 0 0
\(766\) −29.4483 −1.06401
\(767\) −21.5565 −0.778358
\(768\) 0 0
\(769\) 24.5805 0.886396 0.443198 0.896424i \(-0.353844\pi\)
0.443198 + 0.896424i \(0.353844\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 45.5064 1.63781
\(773\) −20.0112 −0.719754 −0.359877 0.933000i \(-0.617181\pi\)
−0.359877 + 0.933000i \(0.617181\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −14.7128 −0.528159
\(777\) 0 0
\(778\) −41.5485 −1.48958
\(779\) 46.4653 1.66479
\(780\) 0 0
\(781\) −0.109034 −0.00390155
\(782\) −17.9654 −0.642441
\(783\) 0 0
\(784\) −6.62905 −0.236752
\(785\) 0 0
\(786\) 0 0
\(787\) −8.99272 −0.320556 −0.160278 0.987072i \(-0.551239\pi\)
−0.160278 + 0.987072i \(0.551239\pi\)
\(788\) −11.7532 −0.418692
\(789\) 0 0
\(790\) 0 0
\(791\) −28.3547 −1.00818
\(792\) 0 0
\(793\) −56.0517 −1.99045
\(794\) −48.0252 −1.70435
\(795\) 0 0
\(796\) 44.2773 1.56937
\(797\) −37.7564 −1.33740 −0.668700 0.743533i \(-0.733149\pi\)
−0.668700 + 0.743533i \(0.733149\pi\)
\(798\) 0 0
\(799\) 0.971976 0.0343860
\(800\) 0 0
\(801\) 0 0
\(802\) −10.1920 −0.359891
\(803\) 6.17067 0.217758
\(804\) 0 0
\(805\) 0 0
\(806\) −50.5684 −1.78120
\(807\) 0 0
\(808\) −1.56971 −0.0552223
\(809\) 48.9001 1.71924 0.859618 0.510937i \(-0.170702\pi\)
0.859618 + 0.510937i \(0.170702\pi\)
\(810\) 0 0
\(811\) −18.0551 −0.633999 −0.317000 0.948426i \(-0.602675\pi\)
−0.317000 + 0.948426i \(0.602675\pi\)
\(812\) 19.9310 0.699441
\(813\) 0 0
\(814\) −19.8010 −0.694024
\(815\) 0 0
\(816\) 0 0
\(817\) 46.2042 1.61648
\(818\) −26.9653 −0.942819
\(819\) 0 0
\(820\) 0 0
\(821\) −50.6097 −1.76629 −0.883146 0.469099i \(-0.844579\pi\)
−0.883146 + 0.469099i \(0.844579\pi\)
\(822\) 0 0
\(823\) 26.8070 0.934433 0.467217 0.884143i \(-0.345257\pi\)
0.467217 + 0.884143i \(0.345257\pi\)
\(824\) 7.12635 0.248258
\(825\) 0 0
\(826\) −28.0839 −0.977163
\(827\) −21.3649 −0.742929 −0.371465 0.928447i \(-0.621144\pi\)
−0.371465 + 0.928447i \(0.621144\pi\)
\(828\) 0 0
\(829\) −48.1123 −1.67101 −0.835504 0.549484i \(-0.814824\pi\)
−0.835504 + 0.549484i \(0.814824\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −63.7021 −2.20847
\(833\) 3.85803 0.133673
\(834\) 0 0
\(835\) 0 0
\(836\) −22.3518 −0.773052
\(837\) 0 0
\(838\) −75.5080 −2.60838
\(839\) 41.8540 1.44496 0.722481 0.691391i \(-0.243002\pi\)
0.722481 + 0.691391i \(0.243002\pi\)
\(840\) 0 0
\(841\) −23.4133 −0.807357
\(842\) −85.4719 −2.94556
\(843\) 0 0
\(844\) 49.4794 1.70315
\(845\) 0 0
\(846\) 0 0
\(847\) −27.2866 −0.937579
\(848\) −21.6225 −0.742520
\(849\) 0 0
\(850\) 0 0
\(851\) −38.3809 −1.31568
\(852\) 0 0
\(853\) 58.2023 1.99281 0.996404 0.0847260i \(-0.0270015\pi\)
0.996404 + 0.0847260i \(0.0270015\pi\)
\(854\) −73.0245 −2.49885
\(855\) 0 0
\(856\) 22.0909 0.755051
\(857\) −3.22045 −0.110008 −0.0550042 0.998486i \(-0.517517\pi\)
−0.0550042 + 0.998486i \(0.517517\pi\)
\(858\) 0 0
\(859\) 49.4431 1.68698 0.843488 0.537147i \(-0.180498\pi\)
0.843488 + 0.537147i \(0.180498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30.3740 1.03454
\(863\) 21.4924 0.731611 0.365806 0.930691i \(-0.380793\pi\)
0.365806 + 0.930691i \(0.380793\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.18761 0.0403567
\(867\) 0 0
\(868\) −37.6544 −1.27807
\(869\) −2.48981 −0.0844609
\(870\) 0 0
\(871\) 9.95163 0.337198
\(872\) 9.61090 0.325466
\(873\) 0 0
\(874\) −75.8026 −2.56406
\(875\) 0 0
\(876\) 0 0
\(877\) −15.3793 −0.519321 −0.259661 0.965700i \(-0.583611\pi\)
−0.259661 + 0.965700i \(0.583611\pi\)
\(878\) −6.42501 −0.216833
\(879\) 0 0
\(880\) 0 0
\(881\) −18.9275 −0.637682 −0.318841 0.947808i \(-0.603294\pi\)
−0.318841 + 0.947808i \(0.603294\pi\)
\(882\) 0 0
\(883\) −6.31708 −0.212587 −0.106293 0.994335i \(-0.533898\pi\)
−0.106293 + 0.994335i \(0.533898\pi\)
\(884\) 18.0492 0.607062
\(885\) 0 0
\(886\) −79.2694 −2.66311
\(887\) 0.0721586 0.00242285 0.00121142 0.999999i \(-0.499614\pi\)
0.00121142 + 0.999999i \(0.499614\pi\)
\(888\) 0 0
\(889\) −31.6639 −1.06197
\(890\) 0 0
\(891\) 0 0
\(892\) 2.05853 0.0689246
\(893\) 4.10113 0.137239
\(894\) 0 0
\(895\) 0 0
\(896\) −34.4573 −1.15114
\(897\) 0 0
\(898\) 38.5512 1.28647
\(899\) −10.5545 −0.352013
\(900\) 0 0
\(901\) 12.5840 0.419235
\(902\) −28.3584 −0.944233
\(903\) 0 0
\(904\) −12.9484 −0.430658
\(905\) 0 0
\(906\) 0 0
\(907\) 25.4951 0.846550 0.423275 0.906001i \(-0.360880\pi\)
0.423275 + 0.906001i \(0.360880\pi\)
\(908\) 30.4991 1.01215
\(909\) 0 0
\(910\) 0 0
\(911\) −3.55803 −0.117883 −0.0589414 0.998261i \(-0.518773\pi\)
−0.0589414 + 0.998261i \(0.518773\pi\)
\(912\) 0 0
\(913\) −20.2275 −0.669434
\(914\) 52.2389 1.72791
\(915\) 0 0
\(916\) 27.2798 0.901351
\(917\) 20.1601 0.665745
\(918\) 0 0
\(919\) 7.96968 0.262896 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 80.7776 2.66027
\(923\) 0.371499 0.0122280
\(924\) 0 0
\(925\) 0 0
\(926\) −44.1434 −1.45064
\(927\) 0 0
\(928\) −18.1480 −0.595738
\(929\) −48.5424 −1.59263 −0.796313 0.604885i \(-0.793219\pi\)
−0.796313 + 0.604885i \(0.793219\pi\)
\(930\) 0 0
\(931\) 16.2785 0.533505
\(932\) −60.0916 −1.96837
\(933\) 0 0
\(934\) 67.4363 2.20658
\(935\) 0 0
\(936\) 0 0
\(937\) −31.7296 −1.03656 −0.518280 0.855211i \(-0.673427\pi\)
−0.518280 + 0.855211i \(0.673427\pi\)
\(938\) 12.9650 0.423324
\(939\) 0 0
\(940\) 0 0
\(941\) −41.0068 −1.33678 −0.668392 0.743809i \(-0.733017\pi\)
−0.668392 + 0.743809i \(0.733017\pi\)
\(942\) 0 0
\(943\) −54.9681 −1.79001
\(944\) 9.12079 0.296856
\(945\) 0 0
\(946\) −28.1991 −0.916833
\(947\) −9.81789 −0.319038 −0.159519 0.987195i \(-0.550994\pi\)
−0.159519 + 0.987195i \(0.550994\pi\)
\(948\) 0 0
\(949\) −21.0246 −0.682487
\(950\) 0 0
\(951\) 0 0
\(952\) 5.88761 0.190818
\(953\) 53.1118 1.72046 0.860230 0.509906i \(-0.170320\pi\)
0.860230 + 0.509906i \(0.170320\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.57166 0.0508311
\(957\) 0 0
\(958\) −23.9756 −0.774618
\(959\) −29.0055 −0.936635
\(960\) 0 0
\(961\) −11.0600 −0.356774
\(962\) 67.4654 2.17517
\(963\) 0 0
\(964\) 4.02702 0.129702
\(965\) 0 0
\(966\) 0 0
\(967\) 2.44732 0.0787004 0.0393502 0.999225i \(-0.487471\pi\)
0.0393502 + 0.999225i \(0.487471\pi\)
\(968\) −12.4606 −0.400500
\(969\) 0 0
\(970\) 0 0
\(971\) 47.9683 1.53938 0.769688 0.638421i \(-0.220412\pi\)
0.769688 + 0.638421i \(0.220412\pi\)
\(972\) 0 0
\(973\) −29.2838 −0.938796
\(974\) −15.4200 −0.494088
\(975\) 0 0
\(976\) 23.7161 0.759134
\(977\) 23.6950 0.758070 0.379035 0.925382i \(-0.376256\pi\)
0.379035 + 0.925382i \(0.376256\pi\)
\(978\) 0 0
\(979\) 11.1930 0.357730
\(980\) 0 0
\(981\) 0 0
\(982\) −15.6284 −0.498721
\(983\) 12.9505 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 6.59116 0.209905
\(987\) 0 0
\(988\) 76.1565 2.42286
\(989\) −54.6593 −1.73807
\(990\) 0 0
\(991\) 44.4967 1.41348 0.706742 0.707471i \(-0.250164\pi\)
0.706742 + 0.707471i \(0.250164\pi\)
\(992\) 34.2859 1.08858
\(993\) 0 0
\(994\) 0.483992 0.0153513
\(995\) 0 0
\(996\) 0 0
\(997\) 55.5513 1.75933 0.879663 0.475597i \(-0.157768\pi\)
0.879663 + 0.475597i \(0.157768\pi\)
\(998\) 34.6054 1.09542
\(999\) 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.q.1.5 6
3.2 odd 2 1875.2.a.k.1.2 6
5.4 even 2 5625.2.a.p.1.2 6
15.2 even 4 1875.2.b.f.1249.3 12
15.8 even 4 1875.2.b.f.1249.10 12
15.14 odd 2 1875.2.a.j.1.5 6
25.4 even 10 225.2.h.d.91.3 12
25.19 even 10 225.2.h.d.136.3 12
75.8 even 20 375.2.i.d.199.1 24
75.17 even 20 375.2.i.d.199.6 24
75.29 odd 10 75.2.g.c.16.1 12
75.44 odd 10 75.2.g.c.61.1 yes 12
75.47 even 20 375.2.i.d.49.1 24
75.53 even 20 375.2.i.d.49.6 24
75.56 odd 10 375.2.g.c.301.3 12
75.71 odd 10 375.2.g.c.76.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.16.1 12 75.29 odd 10
75.2.g.c.61.1 yes 12 75.44 odd 10
225.2.h.d.91.3 12 25.4 even 10
225.2.h.d.136.3 12 25.19 even 10
375.2.g.c.76.3 12 75.71 odd 10
375.2.g.c.301.3 12 75.56 odd 10
375.2.i.d.49.1 24 75.47 even 20
375.2.i.d.49.6 24 75.53 even 20
375.2.i.d.199.1 24 75.8 even 20
375.2.i.d.199.6 24 75.17 even 20
1875.2.a.j.1.5 6 15.14 odd 2
1875.2.a.k.1.2 6 3.2 odd 2
1875.2.b.f.1249.3 12 15.2 even 4
1875.2.b.f.1249.10 12 15.8 even 4
5625.2.a.p.1.2 6 5.4 even 2
5625.2.a.q.1.5 6 1.1 even 1 trivial