Properties

Label 9075.2.a.cm.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.35567\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.35567 q^{2} -1.00000 q^{3} +3.54920 q^{4} +2.35567 q^{6} +0.193527 q^{7} -3.64941 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.35567 q^{2} -1.00000 q^{3} +3.54920 q^{4} +2.35567 q^{6} +0.193527 q^{7} -3.64941 q^{8} +1.00000 q^{9} -3.54920 q^{12} +0.973708 q^{13} -0.455887 q^{14} +1.49843 q^{16} -2.67571 q^{17} -2.35567 q^{18} +5.54920 q^{19} -0.193527 q^{21} +4.80040 q^{23} +3.64941 q^{24} -2.29374 q^{26} -1.00000 q^{27} +0.686867 q^{28} +10.1178 q^{29} -2.50533 q^{31} +3.76902 q^{32} +6.30309 q^{34} +3.54920 q^{36} -5.71217 q^{37} -13.0721 q^{38} -0.973708 q^{39} +8.27861 q^{41} +0.455887 q^{42} +5.11353 q^{43} -11.3082 q^{46} +10.8523 q^{47} -1.49843 q^{48} -6.96255 q^{49} +2.67571 q^{51} +3.45589 q^{52} +9.28879 q^{53} +2.35567 q^{54} -0.706260 q^{56} -5.54920 q^{57} -23.8342 q^{58} -11.2180 q^{59} -1.20602 q^{61} +5.90173 q^{62} +0.193527 q^{63} -11.8754 q^{64} -3.25922 q^{67} -9.49662 q^{68} -4.80040 q^{69} +5.99078 q^{71} -3.64941 q^{72} +3.30624 q^{73} +13.4560 q^{74} +19.6952 q^{76} +2.29374 q^{78} -10.1529 q^{79} +1.00000 q^{81} -19.5017 q^{82} +8.31508 q^{83} -0.686867 q^{84} -12.0458 q^{86} -10.1178 q^{87} +7.34270 q^{89} +0.188439 q^{91} +17.0376 q^{92} +2.50533 q^{93} -25.5645 q^{94} -3.76902 q^{96} +15.8523 q^{97} +16.4015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 4 q^{3} + q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 4 q^{3} + q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 4 q^{9} - q^{12} - 7 q^{13} + 3 q^{14} - q^{16} - 10 q^{17} - 3 q^{18} + 9 q^{19} + 6 q^{21} + 3 q^{23} + 3 q^{24} - 4 q^{26} - 4 q^{27} + 7 q^{28} + 15 q^{29} - 13 q^{31} + 6 q^{32} - 3 q^{34} + q^{36} + 3 q^{37} - 15 q^{38} + 7 q^{39} + 22 q^{41} - 3 q^{42} + q^{46} + 2 q^{47} + q^{48} - 12 q^{49} + 10 q^{51} + 9 q^{52} - 10 q^{53} + 3 q^{54} - 8 q^{56} - 9 q^{57} - 39 q^{58} - 21 q^{59} + 11 q^{61} - 10 q^{62} - 6 q^{63} - 3 q^{64} - q^{67} - 3 q^{68} - 3 q^{69} - 13 q^{71} - 3 q^{72} - q^{73} - 11 q^{74} + 19 q^{76} + 4 q^{78} - 4 q^{79} + 4 q^{81} - 25 q^{82} - 3 q^{83} - 7 q^{84} - 15 q^{87} - 10 q^{89} + 12 q^{91} + 24 q^{92} + 13 q^{93} - 35 q^{94} - 6 q^{96} + 22 q^{97} + 11 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.35567 −1.66571 −0.832857 0.553489i \(-0.813296\pi\)
−0.832857 + 0.553489i \(0.813296\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.54920 1.77460
\(5\) 0 0
\(6\) 2.35567 0.961700
\(7\) 0.193527 0.0731464 0.0365732 0.999331i \(-0.488356\pi\)
0.0365732 + 0.999331i \(0.488356\pi\)
\(8\) −3.64941 −1.29026
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −3.54920 −1.02457
\(13\) 0.973708 0.270058 0.135029 0.990842i \(-0.456887\pi\)
0.135029 + 0.990842i \(0.456887\pi\)
\(14\) −0.455887 −0.121841
\(15\) 0 0
\(16\) 1.49843 0.374607
\(17\) −2.67571 −0.648954 −0.324477 0.945894i \(-0.605188\pi\)
−0.324477 + 0.945894i \(0.605188\pi\)
\(18\) −2.35567 −0.555238
\(19\) 5.54920 1.27307 0.636537 0.771246i \(-0.280366\pi\)
0.636537 + 0.771246i \(0.280366\pi\)
\(20\) 0 0
\(21\) −0.193527 −0.0422311
\(22\) 0 0
\(23\) 4.80040 1.00095 0.500476 0.865750i \(-0.333158\pi\)
0.500476 + 0.865750i \(0.333158\pi\)
\(24\) 3.64941 0.744933
\(25\) 0 0
\(26\) −2.29374 −0.449839
\(27\) −1.00000 −0.192450
\(28\) 0.686867 0.129806
\(29\) 10.1178 1.87883 0.939414 0.342785i \(-0.111370\pi\)
0.939414 + 0.342785i \(0.111370\pi\)
\(30\) 0 0
\(31\) −2.50533 −0.449970 −0.224985 0.974362i \(-0.572233\pi\)
−0.224985 + 0.974362i \(0.572233\pi\)
\(32\) 3.76902 0.666275
\(33\) 0 0
\(34\) 6.30309 1.08097
\(35\) 0 0
\(36\) 3.54920 0.591534
\(37\) −5.71217 −0.939076 −0.469538 0.882912i \(-0.655579\pi\)
−0.469538 + 0.882912i \(0.655579\pi\)
\(38\) −13.0721 −2.12058
\(39\) −0.973708 −0.155918
\(40\) 0 0
\(41\) 8.27861 1.29290 0.646451 0.762956i \(-0.276253\pi\)
0.646451 + 0.762956i \(0.276253\pi\)
\(42\) 0.455887 0.0703449
\(43\) 5.11353 0.779807 0.389903 0.920856i \(-0.372508\pi\)
0.389903 + 0.920856i \(0.372508\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −11.3082 −1.66730
\(47\) 10.8523 1.58297 0.791485 0.611189i \(-0.209308\pi\)
0.791485 + 0.611189i \(0.209308\pi\)
\(48\) −1.49843 −0.216279
\(49\) −6.96255 −0.994650
\(50\) 0 0
\(51\) 2.67571 0.374674
\(52\) 3.45589 0.479245
\(53\) 9.28879 1.27591 0.637956 0.770072i \(-0.279780\pi\)
0.637956 + 0.770072i \(0.279780\pi\)
\(54\) 2.35567 0.320567
\(55\) 0 0
\(56\) −0.706260 −0.0943780
\(57\) −5.54920 −0.735010
\(58\) −23.8342 −3.12959
\(59\) −11.2180 −1.46046 −0.730230 0.683201i \(-0.760587\pi\)
−0.730230 + 0.683201i \(0.760587\pi\)
\(60\) 0 0
\(61\) −1.20602 −0.154415 −0.0772077 0.997015i \(-0.524600\pi\)
−0.0772077 + 0.997015i \(0.524600\pi\)
\(62\) 5.90173 0.749521
\(63\) 0.193527 0.0243821
\(64\) −11.8754 −1.48443
\(65\) 0 0
\(66\) 0 0
\(67\) −3.25922 −0.398176 −0.199088 0.979982i \(-0.563798\pi\)
−0.199088 + 0.979982i \(0.563798\pi\)
\(68\) −9.49662 −1.15163
\(69\) −4.80040 −0.577900
\(70\) 0 0
\(71\) 5.99078 0.710975 0.355488 0.934681i \(-0.384315\pi\)
0.355488 + 0.934681i \(0.384315\pi\)
\(72\) −3.64941 −0.430088
\(73\) 3.30624 0.386966 0.193483 0.981104i \(-0.438022\pi\)
0.193483 + 0.981104i \(0.438022\pi\)
\(74\) 13.4560 1.56423
\(75\) 0 0
\(76\) 19.6952 2.25920
\(77\) 0 0
\(78\) 2.29374 0.259715
\(79\) −10.1529 −1.14229 −0.571147 0.820848i \(-0.693501\pi\)
−0.571147 + 0.820848i \(0.693501\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −19.5017 −2.15360
\(83\) 8.31508 0.912698 0.456349 0.889801i \(-0.349157\pi\)
0.456349 + 0.889801i \(0.349157\pi\)
\(84\) −0.686867 −0.0749433
\(85\) 0 0
\(86\) −12.0458 −1.29893
\(87\) −10.1178 −1.08474
\(88\) 0 0
\(89\) 7.34270 0.778325 0.389163 0.921169i \(-0.372764\pi\)
0.389163 + 0.921169i \(0.372764\pi\)
\(90\) 0 0
\(91\) 0.188439 0.0197538
\(92\) 17.0376 1.77629
\(93\) 2.50533 0.259790
\(94\) −25.5645 −2.63677
\(95\) 0 0
\(96\) −3.76902 −0.384674
\(97\) 15.8523 1.60956 0.804778 0.593576i \(-0.202284\pi\)
0.804778 + 0.593576i \(0.202284\pi\)
\(98\) 16.4015 1.65680
\(99\) 0 0
\(100\) 0 0
\(101\) −13.1591 −1.30938 −0.654692 0.755896i \(-0.727201\pi\)
−0.654692 + 0.755896i \(0.727201\pi\)
\(102\) −6.30309 −0.624099
\(103\) −3.99737 −0.393872 −0.196936 0.980416i \(-0.563099\pi\)
−0.196936 + 0.980416i \(0.563099\pi\)
\(104\) −3.55346 −0.348446
\(105\) 0 0
\(106\) −21.8814 −2.12530
\(107\) 4.88682 0.472426 0.236213 0.971701i \(-0.424094\pi\)
0.236213 + 0.971701i \(0.424094\pi\)
\(108\) −3.54920 −0.341522
\(109\) 7.51977 0.720263 0.360131 0.932902i \(-0.382732\pi\)
0.360131 + 0.932902i \(0.382732\pi\)
\(110\) 0 0
\(111\) 5.71217 0.542176
\(112\) 0.289986 0.0274011
\(113\) −20.2237 −1.90249 −0.951243 0.308442i \(-0.900192\pi\)
−0.951243 + 0.308442i \(0.900192\pi\)
\(114\) 13.0721 1.22432
\(115\) 0 0
\(116\) 35.9101 3.33417
\(117\) 0.973708 0.0900194
\(118\) 26.4260 2.43271
\(119\) −0.517822 −0.0474686
\(120\) 0 0
\(121\) 0 0
\(122\) 2.84100 0.257212
\(123\) −8.27861 −0.746457
\(124\) −8.89190 −0.798517
\(125\) 0 0
\(126\) −0.455887 −0.0406136
\(127\) 7.10021 0.630042 0.315021 0.949085i \(-0.397988\pi\)
0.315021 + 0.949085i \(0.397988\pi\)
\(128\) 20.4366 1.80636
\(129\) −5.11353 −0.450222
\(130\) 0 0
\(131\) 2.50024 0.218447 0.109223 0.994017i \(-0.465164\pi\)
0.109223 + 0.994017i \(0.465164\pi\)
\(132\) 0 0
\(133\) 1.07392 0.0931207
\(134\) 7.67765 0.663248
\(135\) 0 0
\(136\) 9.76476 0.837321
\(137\) 15.2944 1.30669 0.653344 0.757061i \(-0.273366\pi\)
0.653344 + 0.757061i \(0.273366\pi\)
\(138\) 11.3082 0.962616
\(139\) 19.6586 1.66742 0.833711 0.552202i \(-0.186212\pi\)
0.833711 + 0.552202i \(0.186212\pi\)
\(140\) 0 0
\(141\) −10.8523 −0.913928
\(142\) −14.1123 −1.18428
\(143\) 0 0
\(144\) 1.49843 0.124869
\(145\) 0 0
\(146\) −7.78841 −0.644574
\(147\) 6.96255 0.574261
\(148\) −20.2737 −1.66648
\(149\) 10.1329 0.830122 0.415061 0.909794i \(-0.363760\pi\)
0.415061 + 0.909794i \(0.363760\pi\)
\(150\) 0 0
\(151\) −11.9632 −0.973548 −0.486774 0.873528i \(-0.661826\pi\)
−0.486774 + 0.873528i \(0.661826\pi\)
\(152\) −20.2513 −1.64260
\(153\) −2.67571 −0.216318
\(154\) 0 0
\(155\) 0 0
\(156\) −3.45589 −0.276692
\(157\) 2.62605 0.209582 0.104791 0.994494i \(-0.466583\pi\)
0.104791 + 0.994494i \(0.466583\pi\)
\(158\) 23.9170 1.90273
\(159\) −9.28879 −0.736649
\(160\) 0 0
\(161\) 0.929007 0.0732160
\(162\) −2.35567 −0.185079
\(163\) −23.7235 −1.85817 −0.929083 0.369872i \(-0.879402\pi\)
−0.929083 + 0.369872i \(0.879402\pi\)
\(164\) 29.3825 2.29438
\(165\) 0 0
\(166\) −19.5876 −1.52029
\(167\) 3.49175 0.270199 0.135100 0.990832i \(-0.456865\pi\)
0.135100 + 0.990832i \(0.456865\pi\)
\(168\) 0.706260 0.0544892
\(169\) −12.0519 −0.927069
\(170\) 0 0
\(171\) 5.54920 0.424358
\(172\) 18.1490 1.38385
\(173\) −20.3596 −1.54791 −0.773954 0.633241i \(-0.781724\pi\)
−0.773954 + 0.633241i \(0.781724\pi\)
\(174\) 23.8342 1.80687
\(175\) 0 0
\(176\) 0 0
\(177\) 11.2180 0.843197
\(178\) −17.2970 −1.29647
\(179\) 3.85117 0.287850 0.143925 0.989589i \(-0.454028\pi\)
0.143925 + 0.989589i \(0.454028\pi\)
\(180\) 0 0
\(181\) −17.6151 −1.30932 −0.654660 0.755923i \(-0.727188\pi\)
−0.654660 + 0.755923i \(0.727188\pi\)
\(182\) −0.443901 −0.0329041
\(183\) 1.20602 0.0891518
\(184\) −17.5186 −1.29149
\(185\) 0 0
\(186\) −5.90173 −0.432736
\(187\) 0 0
\(188\) 38.5170 2.80914
\(189\) −0.193527 −0.0140770
\(190\) 0 0
\(191\) 0.417099 0.0301802 0.0150901 0.999886i \(-0.495196\pi\)
0.0150901 + 0.999886i \(0.495196\pi\)
\(192\) 11.8754 0.857036
\(193\) 11.6965 0.841935 0.420967 0.907076i \(-0.361691\pi\)
0.420967 + 0.907076i \(0.361691\pi\)
\(194\) −37.3428 −2.68106
\(195\) 0 0
\(196\) −24.7115 −1.76511
\(197\) 21.5958 1.53864 0.769320 0.638863i \(-0.220595\pi\)
0.769320 + 0.638863i \(0.220595\pi\)
\(198\) 0 0
\(199\) 7.76028 0.550111 0.275056 0.961428i \(-0.411304\pi\)
0.275056 + 0.961428i \(0.411304\pi\)
\(200\) 0 0
\(201\) 3.25922 0.229887
\(202\) 30.9986 2.18106
\(203\) 1.95807 0.137429
\(204\) 9.49662 0.664896
\(205\) 0 0
\(206\) 9.41649 0.656078
\(207\) 4.80040 0.333651
\(208\) 1.45903 0.101166
\(209\) 0 0
\(210\) 0 0
\(211\) 12.5057 0.860926 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(212\) 32.9678 2.26424
\(213\) −5.99078 −0.410482
\(214\) −11.5117 −0.786927
\(215\) 0 0
\(216\) 3.64941 0.248311
\(217\) −0.484848 −0.0329137
\(218\) −17.7141 −1.19975
\(219\) −3.30624 −0.223415
\(220\) 0 0
\(221\) −2.60536 −0.175255
\(222\) −13.4560 −0.903109
\(223\) 5.37568 0.359982 0.179991 0.983668i \(-0.442393\pi\)
0.179991 + 0.983668i \(0.442393\pi\)
\(224\) 0.729407 0.0487356
\(225\) 0 0
\(226\) 47.6405 3.16900
\(227\) −21.7529 −1.44379 −0.721895 0.692002i \(-0.756729\pi\)
−0.721895 + 0.692002i \(0.756729\pi\)
\(228\) −19.6952 −1.30435
\(229\) 2.67075 0.176488 0.0882441 0.996099i \(-0.471874\pi\)
0.0882441 + 0.996099i \(0.471874\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −36.9240 −2.42418
\(233\) −5.72699 −0.375188 −0.187594 0.982247i \(-0.560069\pi\)
−0.187594 + 0.982247i \(0.560069\pi\)
\(234\) −2.29374 −0.149946
\(235\) 0 0
\(236\) −39.8150 −2.59173
\(237\) 10.1529 0.659504
\(238\) 1.21982 0.0790691
\(239\) 24.3680 1.57624 0.788118 0.615524i \(-0.211056\pi\)
0.788118 + 0.615524i \(0.211056\pi\)
\(240\) 0 0
\(241\) 16.7082 1.07627 0.538135 0.842859i \(-0.319129\pi\)
0.538135 + 0.842859i \(0.319129\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −4.28042 −0.274026
\(245\) 0 0
\(246\) 19.5017 1.24338
\(247\) 5.40330 0.343804
\(248\) 9.14297 0.580579
\(249\) −8.31508 −0.526947
\(250\) 0 0
\(251\) −16.0984 −1.01612 −0.508061 0.861321i \(-0.669638\pi\)
−0.508061 + 0.861321i \(0.669638\pi\)
\(252\) 0.686867 0.0432685
\(253\) 0 0
\(254\) −16.7258 −1.04947
\(255\) 0 0
\(256\) −24.3912 −1.52445
\(257\) −5.96875 −0.372321 −0.186160 0.982519i \(-0.559604\pi\)
−0.186160 + 0.982519i \(0.559604\pi\)
\(258\) 12.0458 0.749940
\(259\) −1.10546 −0.0686900
\(260\) 0 0
\(261\) 10.1178 0.626276
\(262\) −5.88974 −0.363870
\(263\) −0.451149 −0.0278190 −0.0139095 0.999903i \(-0.504428\pi\)
−0.0139095 + 0.999903i \(0.504428\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.52981 −0.155112
\(267\) −7.34270 −0.449366
\(268\) −11.5676 −0.706604
\(269\) 14.7197 0.897477 0.448738 0.893663i \(-0.351873\pi\)
0.448738 + 0.893663i \(0.351873\pi\)
\(270\) 0 0
\(271\) −4.42787 −0.268974 −0.134487 0.990915i \(-0.542939\pi\)
−0.134487 + 0.990915i \(0.542939\pi\)
\(272\) −4.00935 −0.243103
\(273\) −0.188439 −0.0114048
\(274\) −36.0286 −2.17657
\(275\) 0 0
\(276\) −17.0376 −1.02554
\(277\) −0.0923299 −0.00554757 −0.00277378 0.999996i \(-0.500883\pi\)
−0.00277378 + 0.999996i \(0.500883\pi\)
\(278\) −46.3093 −2.77745
\(279\) −2.50533 −0.149990
\(280\) 0 0
\(281\) −5.18301 −0.309192 −0.154596 0.987978i \(-0.549408\pi\)
−0.154596 + 0.987978i \(0.549408\pi\)
\(282\) 25.5645 1.52234
\(283\) 1.56463 0.0930073 0.0465037 0.998918i \(-0.485192\pi\)
0.0465037 + 0.998918i \(0.485192\pi\)
\(284\) 21.2625 1.26170
\(285\) 0 0
\(286\) 0 0
\(287\) 1.60214 0.0945710
\(288\) 3.76902 0.222092
\(289\) −9.84060 −0.578859
\(290\) 0 0
\(291\) −15.8523 −0.929278
\(292\) 11.7345 0.686709
\(293\) −23.1570 −1.35285 −0.676424 0.736512i \(-0.736471\pi\)
−0.676424 + 0.736512i \(0.736471\pi\)
\(294\) −16.4015 −0.956555
\(295\) 0 0
\(296\) 20.8461 1.21165
\(297\) 0 0
\(298\) −23.8699 −1.38274
\(299\) 4.67419 0.270315
\(300\) 0 0
\(301\) 0.989607 0.0570400
\(302\) 28.1813 1.62165
\(303\) 13.1591 0.755973
\(304\) 8.31508 0.476902
\(305\) 0 0
\(306\) 6.30309 0.360324
\(307\) −7.72480 −0.440878 −0.220439 0.975401i \(-0.570749\pi\)
−0.220439 + 0.975401i \(0.570749\pi\)
\(308\) 0 0
\(309\) 3.99737 0.227402
\(310\) 0 0
\(311\) −19.1209 −1.08424 −0.542122 0.840300i \(-0.682379\pi\)
−0.542122 + 0.840300i \(0.682379\pi\)
\(312\) 3.55346 0.201175
\(313\) −2.90307 −0.164091 −0.0820455 0.996629i \(-0.526145\pi\)
−0.0820455 + 0.996629i \(0.526145\pi\)
\(314\) −6.18612 −0.349103
\(315\) 0 0
\(316\) −36.0348 −2.02712
\(317\) −2.26153 −0.127020 −0.0635102 0.997981i \(-0.520230\pi\)
−0.0635102 + 0.997981i \(0.520230\pi\)
\(318\) 21.8814 1.22705
\(319\) 0 0
\(320\) 0 0
\(321\) −4.88682 −0.272756
\(322\) −2.18844 −0.121957
\(323\) −14.8480 −0.826166
\(324\) 3.54920 0.197178
\(325\) 0 0
\(326\) 55.8848 3.09517
\(327\) −7.51977 −0.415844
\(328\) −30.2121 −1.66818
\(329\) 2.10021 0.115788
\(330\) 0 0
\(331\) 6.02336 0.331074 0.165537 0.986204i \(-0.447064\pi\)
0.165537 + 0.986204i \(0.447064\pi\)
\(332\) 29.5119 1.61967
\(333\) −5.71217 −0.313025
\(334\) −8.22542 −0.450075
\(335\) 0 0
\(336\) −0.289986 −0.0158201
\(337\) −14.5648 −0.793393 −0.396696 0.917950i \(-0.629843\pi\)
−0.396696 + 0.917950i \(0.629843\pi\)
\(338\) 28.3903 1.54423
\(339\) 20.2237 1.09840
\(340\) 0 0
\(341\) 0 0
\(342\) −13.0721 −0.706859
\(343\) −2.70213 −0.145901
\(344\) −18.6614 −1.00616
\(345\) 0 0
\(346\) 47.9605 2.57837
\(347\) 28.8351 1.54795 0.773976 0.633215i \(-0.218265\pi\)
0.773976 + 0.633215i \(0.218265\pi\)
\(348\) −35.9101 −1.92498
\(349\) 1.63234 0.0873771 0.0436886 0.999045i \(-0.486089\pi\)
0.0436886 + 0.999045i \(0.486089\pi\)
\(350\) 0 0
\(351\) −0.973708 −0.0519727
\(352\) 0 0
\(353\) −24.9297 −1.32687 −0.663437 0.748232i \(-0.730903\pi\)
−0.663437 + 0.748232i \(0.730903\pi\)
\(354\) −26.4260 −1.40452
\(355\) 0 0
\(356\) 26.0607 1.38122
\(357\) 0.517822 0.0274060
\(358\) −9.07211 −0.479476
\(359\) −28.4409 −1.50106 −0.750528 0.660839i \(-0.770201\pi\)
−0.750528 + 0.660839i \(0.770201\pi\)
\(360\) 0 0
\(361\) 11.7936 0.620718
\(362\) 41.4955 2.18095
\(363\) 0 0
\(364\) 0.668808 0.0350550
\(365\) 0 0
\(366\) −2.84100 −0.148501
\(367\) 10.4413 0.545030 0.272515 0.962152i \(-0.412145\pi\)
0.272515 + 0.962152i \(0.412145\pi\)
\(368\) 7.19305 0.374964
\(369\) 8.27861 0.430967
\(370\) 0 0
\(371\) 1.79763 0.0933284
\(372\) 8.89190 0.461024
\(373\) −2.81747 −0.145883 −0.0729416 0.997336i \(-0.523239\pi\)
−0.0729416 + 0.997336i \(0.523239\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −39.6045 −2.04245
\(377\) 9.85178 0.507393
\(378\) 0.455887 0.0234483
\(379\) 8.93319 0.458867 0.229434 0.973324i \(-0.426313\pi\)
0.229434 + 0.973324i \(0.426313\pi\)
\(380\) 0 0
\(381\) −7.10021 −0.363755
\(382\) −0.982550 −0.0502716
\(383\) 8.53279 0.436005 0.218003 0.975948i \(-0.430046\pi\)
0.218003 + 0.975948i \(0.430046\pi\)
\(384\) −20.4366 −1.04290
\(385\) 0 0
\(386\) −27.5532 −1.40242
\(387\) 5.11353 0.259936
\(388\) 56.2630 2.85632
\(389\) 11.7757 0.597054 0.298527 0.954401i \(-0.403505\pi\)
0.298527 + 0.954401i \(0.403505\pi\)
\(390\) 0 0
\(391\) −12.8445 −0.649572
\(392\) 25.4092 1.28336
\(393\) −2.50024 −0.126120
\(394\) −50.8728 −2.56293
\(395\) 0 0
\(396\) 0 0
\(397\) 1.41214 0.0708735 0.0354368 0.999372i \(-0.488718\pi\)
0.0354368 + 0.999372i \(0.488718\pi\)
\(398\) −18.2807 −0.916328
\(399\) −1.07392 −0.0537633
\(400\) 0 0
\(401\) 8.45917 0.422431 0.211215 0.977440i \(-0.432258\pi\)
0.211215 + 0.977440i \(0.432258\pi\)
\(402\) −7.67765 −0.382926
\(403\) −2.43946 −0.121518
\(404\) −46.7044 −2.32363
\(405\) 0 0
\(406\) −4.61257 −0.228918
\(407\) 0 0
\(408\) −9.76476 −0.483428
\(409\) −10.1255 −0.500675 −0.250337 0.968159i \(-0.580542\pi\)
−0.250337 + 0.968159i \(0.580542\pi\)
\(410\) 0 0
\(411\) −15.2944 −0.754416
\(412\) −14.1875 −0.698966
\(413\) −2.17099 −0.106827
\(414\) −11.3082 −0.555767
\(415\) 0 0
\(416\) 3.66993 0.179933
\(417\) −19.6586 −0.962686
\(418\) 0 0
\(419\) 32.8019 1.60248 0.801240 0.598343i \(-0.204174\pi\)
0.801240 + 0.598343i \(0.204174\pi\)
\(420\) 0 0
\(421\) −14.1293 −0.688619 −0.344309 0.938856i \(-0.611887\pi\)
−0.344309 + 0.938856i \(0.611887\pi\)
\(422\) −29.4593 −1.43406
\(423\) 10.8523 0.527657
\(424\) −33.8986 −1.64626
\(425\) 0 0
\(426\) 14.1123 0.683745
\(427\) −0.233398 −0.0112949
\(428\) 17.3443 0.838368
\(429\) 0 0
\(430\) 0 0
\(431\) −19.1057 −0.920291 −0.460145 0.887844i \(-0.652203\pi\)
−0.460145 + 0.887844i \(0.652203\pi\)
\(432\) −1.49843 −0.0720931
\(433\) −15.1200 −0.726622 −0.363311 0.931668i \(-0.618354\pi\)
−0.363311 + 0.931668i \(0.618354\pi\)
\(434\) 1.14214 0.0548247
\(435\) 0 0
\(436\) 26.6892 1.27818
\(437\) 26.6384 1.27429
\(438\) 7.78841 0.372145
\(439\) −37.1642 −1.77375 −0.886876 0.462008i \(-0.847129\pi\)
−0.886876 + 0.462008i \(0.847129\pi\)
\(440\) 0 0
\(441\) −6.96255 −0.331550
\(442\) 6.13737 0.291925
\(443\) 2.76049 0.131155 0.0655775 0.997847i \(-0.479111\pi\)
0.0655775 + 0.997847i \(0.479111\pi\)
\(444\) 20.2737 0.962145
\(445\) 0 0
\(446\) −12.6633 −0.599627
\(447\) −10.1329 −0.479271
\(448\) −2.29822 −0.108581
\(449\) −17.8661 −0.843156 −0.421578 0.906792i \(-0.638524\pi\)
−0.421578 + 0.906792i \(0.638524\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −71.7780 −3.37615
\(453\) 11.9632 0.562078
\(454\) 51.2428 2.40494
\(455\) 0 0
\(456\) 20.2513 0.948356
\(457\) −21.1621 −0.989922 −0.494961 0.868915i \(-0.664818\pi\)
−0.494961 + 0.868915i \(0.664818\pi\)
\(458\) −6.29142 −0.293979
\(459\) 2.67571 0.124891
\(460\) 0 0
\(461\) 34.3476 1.59973 0.799864 0.600181i \(-0.204905\pi\)
0.799864 + 0.600181i \(0.204905\pi\)
\(462\) 0 0
\(463\) −10.4516 −0.485727 −0.242864 0.970060i \(-0.578087\pi\)
−0.242864 + 0.970060i \(0.578087\pi\)
\(464\) 15.1608 0.703822
\(465\) 0 0
\(466\) 13.4909 0.624955
\(467\) −29.6434 −1.37173 −0.685866 0.727728i \(-0.740576\pi\)
−0.685866 + 0.727728i \(0.740576\pi\)
\(468\) 3.45589 0.159748
\(469\) −0.630746 −0.0291252
\(470\) 0 0
\(471\) −2.62605 −0.121002
\(472\) 40.9392 1.88438
\(473\) 0 0
\(474\) −23.9170 −1.09854
\(475\) 0 0
\(476\) −1.83785 −0.0842378
\(477\) 9.28879 0.425304
\(478\) −57.4031 −2.62556
\(479\) 13.2186 0.603974 0.301987 0.953312i \(-0.402350\pi\)
0.301987 + 0.953312i \(0.402350\pi\)
\(480\) 0 0
\(481\) −5.56199 −0.253605
\(482\) −39.3591 −1.79276
\(483\) −0.929007 −0.0422713
\(484\) 0 0
\(485\) 0 0
\(486\) 2.35567 0.106856
\(487\) 42.4976 1.92575 0.962875 0.269949i \(-0.0870070\pi\)
0.962875 + 0.269949i \(0.0870070\pi\)
\(488\) 4.40128 0.199236
\(489\) 23.7235 1.07281
\(490\) 0 0
\(491\) 15.7457 0.710592 0.355296 0.934754i \(-0.384380\pi\)
0.355296 + 0.934754i \(0.384380\pi\)
\(492\) −29.3825 −1.32466
\(493\) −27.0722 −1.21927
\(494\) −12.7284 −0.572679
\(495\) 0 0
\(496\) −3.75405 −0.168562
\(497\) 1.15938 0.0520052
\(498\) 19.5876 0.877742
\(499\) −33.5279 −1.50092 −0.750458 0.660918i \(-0.770167\pi\)
−0.750458 + 0.660918i \(0.770167\pi\)
\(500\) 0 0
\(501\) −3.49175 −0.156000
\(502\) 37.9226 1.69257
\(503\) −22.8163 −1.01733 −0.508664 0.860965i \(-0.669861\pi\)
−0.508664 + 0.860965i \(0.669861\pi\)
\(504\) −0.706260 −0.0314593
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0519 0.535243
\(508\) 25.2001 1.11807
\(509\) 39.7166 1.76041 0.880204 0.474596i \(-0.157406\pi\)
0.880204 + 0.474596i \(0.157406\pi\)
\(510\) 0 0
\(511\) 0.639846 0.0283051
\(512\) 16.5844 0.732933
\(513\) −5.54920 −0.245003
\(514\) 14.0604 0.620179
\(515\) 0 0
\(516\) −18.1490 −0.798963
\(517\) 0 0
\(518\) 2.60410 0.114418
\(519\) 20.3596 0.893686
\(520\) 0 0
\(521\) 25.5004 1.11719 0.558596 0.829440i \(-0.311340\pi\)
0.558596 + 0.829440i \(0.311340\pi\)
\(522\) −23.8342 −1.04320
\(523\) −15.4304 −0.674725 −0.337363 0.941375i \(-0.609535\pi\)
−0.337363 + 0.941375i \(0.609535\pi\)
\(524\) 8.87385 0.387656
\(525\) 0 0
\(526\) 1.06276 0.0463385
\(527\) 6.70351 0.292010
\(528\) 0 0
\(529\) 0.0438407 0.00190612
\(530\) 0 0
\(531\) −11.2180 −0.486820
\(532\) 3.81156 0.165252
\(533\) 8.06095 0.349159
\(534\) 17.2970 0.748515
\(535\) 0 0
\(536\) 11.8942 0.513752
\(537\) −3.85117 −0.166190
\(538\) −34.6749 −1.49494
\(539\) 0 0
\(540\) 0 0
\(541\) −27.6007 −1.18665 −0.593324 0.804964i \(-0.702185\pi\)
−0.593324 + 0.804964i \(0.702185\pi\)
\(542\) 10.4306 0.448033
\(543\) 17.6151 0.755937
\(544\) −10.0848 −0.432382
\(545\) 0 0
\(546\) 0.443901 0.0189972
\(547\) 30.7685 1.31557 0.657784 0.753207i \(-0.271494\pi\)
0.657784 + 0.753207i \(0.271494\pi\)
\(548\) 54.2828 2.31885
\(549\) −1.20602 −0.0514718
\(550\) 0 0
\(551\) 56.1457 2.39189
\(552\) 17.5186 0.745643
\(553\) −1.96487 −0.0835546
\(554\) 0.217499 0.00924065
\(555\) 0 0
\(556\) 69.7723 2.95901
\(557\) −13.9359 −0.590482 −0.295241 0.955423i \(-0.595400\pi\)
−0.295241 + 0.955423i \(0.595400\pi\)
\(558\) 5.90173 0.249840
\(559\) 4.97909 0.210593
\(560\) 0 0
\(561\) 0 0
\(562\) 12.2095 0.515026
\(563\) 5.83988 0.246122 0.123061 0.992399i \(-0.460729\pi\)
0.123061 + 0.992399i \(0.460729\pi\)
\(564\) −38.5170 −1.62186
\(565\) 0 0
\(566\) −3.68575 −0.154924
\(567\) 0.193527 0.00812737
\(568\) −21.8628 −0.917345
\(569\) 5.45394 0.228641 0.114321 0.993444i \(-0.463531\pi\)
0.114321 + 0.993444i \(0.463531\pi\)
\(570\) 0 0
\(571\) −40.2894 −1.68606 −0.843030 0.537866i \(-0.819230\pi\)
−0.843030 + 0.537866i \(0.819230\pi\)
\(572\) 0 0
\(573\) −0.417099 −0.0174246
\(574\) −3.77411 −0.157528
\(575\) 0 0
\(576\) −11.8754 −0.494810
\(577\) 23.8236 0.991789 0.495894 0.868383i \(-0.334840\pi\)
0.495894 + 0.868383i \(0.334840\pi\)
\(578\) 23.1812 0.964213
\(579\) −11.6965 −0.486091
\(580\) 0 0
\(581\) 1.60919 0.0667606
\(582\) 37.3428 1.54791
\(583\) 0 0
\(584\) −12.0658 −0.499287
\(585\) 0 0
\(586\) 54.5504 2.25346
\(587\) −19.6080 −0.809307 −0.404654 0.914470i \(-0.632608\pi\)
−0.404654 + 0.914470i \(0.632608\pi\)
\(588\) 24.7115 1.01908
\(589\) −13.9026 −0.572845
\(590\) 0 0
\(591\) −21.5958 −0.888334
\(592\) −8.55928 −0.351784
\(593\) 3.31095 0.135964 0.0679822 0.997687i \(-0.478344\pi\)
0.0679822 + 0.997687i \(0.478344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 35.9638 1.47313
\(597\) −7.76028 −0.317607
\(598\) −11.0109 −0.450268
\(599\) 32.0710 1.31039 0.655193 0.755462i \(-0.272587\pi\)
0.655193 + 0.755462i \(0.272587\pi\)
\(600\) 0 0
\(601\) 29.2905 1.19478 0.597392 0.801950i \(-0.296204\pi\)
0.597392 + 0.801950i \(0.296204\pi\)
\(602\) −2.33119 −0.0950123
\(603\) −3.25922 −0.132725
\(604\) −42.4596 −1.72766
\(605\) 0 0
\(606\) −30.9986 −1.25923
\(607\) −39.0806 −1.58623 −0.793117 0.609070i \(-0.791543\pi\)
−0.793117 + 0.609070i \(0.791543\pi\)
\(608\) 20.9151 0.848217
\(609\) −1.95807 −0.0793449
\(610\) 0 0
\(611\) 10.5670 0.427494
\(612\) −9.49662 −0.383878
\(613\) −12.7324 −0.514256 −0.257128 0.966377i \(-0.582776\pi\)
−0.257128 + 0.966377i \(0.582776\pi\)
\(614\) 18.1971 0.734376
\(615\) 0 0
\(616\) 0 0
\(617\) 8.97789 0.361436 0.180718 0.983535i \(-0.442158\pi\)
0.180718 + 0.983535i \(0.442158\pi\)
\(618\) −9.41649 −0.378787
\(619\) −22.2121 −0.892780 −0.446390 0.894839i \(-0.647290\pi\)
−0.446390 + 0.894839i \(0.647290\pi\)
\(620\) 0 0
\(621\) −4.80040 −0.192633
\(622\) 45.0425 1.80604
\(623\) 1.42101 0.0569316
\(624\) −1.45903 −0.0584080
\(625\) 0 0
\(626\) 6.83868 0.273329
\(627\) 0 0
\(628\) 9.32038 0.371924
\(629\) 15.2841 0.609417
\(630\) 0 0
\(631\) 26.7672 1.06559 0.532794 0.846245i \(-0.321142\pi\)
0.532794 + 0.846245i \(0.321142\pi\)
\(632\) 37.0522 1.47386
\(633\) −12.5057 −0.497056
\(634\) 5.32744 0.211580
\(635\) 0 0
\(636\) −32.9678 −1.30726
\(637\) −6.77949 −0.268613
\(638\) 0 0
\(639\) 5.99078 0.236992
\(640\) 0 0
\(641\) 15.0726 0.595333 0.297666 0.954670i \(-0.403792\pi\)
0.297666 + 0.954670i \(0.403792\pi\)
\(642\) 11.5117 0.454332
\(643\) 39.2872 1.54934 0.774669 0.632367i \(-0.217917\pi\)
0.774669 + 0.632367i \(0.217917\pi\)
\(644\) 3.29723 0.129929
\(645\) 0 0
\(646\) 34.9771 1.37616
\(647\) −23.8136 −0.936208 −0.468104 0.883673i \(-0.655063\pi\)
−0.468104 + 0.883673i \(0.655063\pi\)
\(648\) −3.64941 −0.143363
\(649\) 0 0
\(650\) 0 0
\(651\) 0.484848 0.0190027
\(652\) −84.1994 −3.29750
\(653\) −6.90354 −0.270156 −0.135078 0.990835i \(-0.543129\pi\)
−0.135078 + 0.990835i \(0.543129\pi\)
\(654\) 17.7141 0.692677
\(655\) 0 0
\(656\) 12.4049 0.484330
\(657\) 3.30624 0.128989
\(658\) −4.94742 −0.192870
\(659\) 20.3718 0.793571 0.396786 0.917911i \(-0.370126\pi\)
0.396786 + 0.917911i \(0.370126\pi\)
\(660\) 0 0
\(661\) −19.7451 −0.767994 −0.383997 0.923334i \(-0.625453\pi\)
−0.383997 + 0.923334i \(0.625453\pi\)
\(662\) −14.1891 −0.551474
\(663\) 2.60536 0.101184
\(664\) −30.3452 −1.17762
\(665\) 0 0
\(666\) 13.4560 0.521410
\(667\) 48.5695 1.88062
\(668\) 12.3929 0.479496
\(669\) −5.37568 −0.207836
\(670\) 0 0
\(671\) 0 0
\(672\) −0.729407 −0.0281375
\(673\) 36.6132 1.41134 0.705669 0.708542i \(-0.250647\pi\)
0.705669 + 0.708542i \(0.250647\pi\)
\(674\) 34.3098 1.32157
\(675\) 0 0
\(676\) −42.7746 −1.64518
\(677\) 7.37138 0.283305 0.141653 0.989916i \(-0.454758\pi\)
0.141653 + 0.989916i \(0.454758\pi\)
\(678\) −47.6405 −1.82962
\(679\) 3.06785 0.117733
\(680\) 0 0
\(681\) 21.7529 0.833573
\(682\) 0 0
\(683\) −24.5651 −0.939959 −0.469979 0.882677i \(-0.655739\pi\)
−0.469979 + 0.882677i \(0.655739\pi\)
\(684\) 19.6952 0.753066
\(685\) 0 0
\(686\) 6.36534 0.243030
\(687\) −2.67075 −0.101896
\(688\) 7.66226 0.292121
\(689\) 9.04457 0.344571
\(690\) 0 0
\(691\) 29.8435 1.13530 0.567651 0.823269i \(-0.307852\pi\)
0.567651 + 0.823269i \(0.307852\pi\)
\(692\) −72.2602 −2.74692
\(693\) 0 0
\(694\) −67.9262 −2.57844
\(695\) 0 0
\(696\) 36.9240 1.39960
\(697\) −22.1511 −0.839033
\(698\) −3.84526 −0.145545
\(699\) 5.72699 0.216615
\(700\) 0 0
\(701\) 14.4189 0.544595 0.272297 0.962213i \(-0.412217\pi\)
0.272297 + 0.962213i \(0.412217\pi\)
\(702\) 2.29374 0.0865716
\(703\) −31.6980 −1.19551
\(704\) 0 0
\(705\) 0 0
\(706\) 58.7263 2.21019
\(707\) −2.54665 −0.0957766
\(708\) 39.8150 1.49634
\(709\) −44.5088 −1.67156 −0.835781 0.549063i \(-0.814985\pi\)
−0.835781 + 0.549063i \(0.814985\pi\)
\(710\) 0 0
\(711\) −10.1529 −0.380765
\(712\) −26.7966 −1.00424
\(713\) −12.0266 −0.450398
\(714\) −1.21982 −0.0456506
\(715\) 0 0
\(716\) 13.6686 0.510819
\(717\) −24.3680 −0.910040
\(718\) 66.9976 2.50033
\(719\) 3.59345 0.134013 0.0670066 0.997753i \(-0.478655\pi\)
0.0670066 + 0.997753i \(0.478655\pi\)
\(720\) 0 0
\(721\) −0.773598 −0.0288103
\(722\) −27.7820 −1.03394
\(723\) −16.7082 −0.621385
\(724\) −62.5196 −2.32352
\(725\) 0 0
\(726\) 0 0
\(727\) 39.0846 1.44957 0.724784 0.688976i \(-0.241940\pi\)
0.724784 + 0.688976i \(0.241940\pi\)
\(728\) −0.687692 −0.0254875
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −13.6823 −0.506059
\(732\) 4.28042 0.158209
\(733\) 37.7065 1.39272 0.696360 0.717693i \(-0.254802\pi\)
0.696360 + 0.717693i \(0.254802\pi\)
\(734\) −24.5962 −0.907863
\(735\) 0 0
\(736\) 18.0928 0.666910
\(737\) 0 0
\(738\) −19.5017 −0.717868
\(739\) −14.9324 −0.549298 −0.274649 0.961545i \(-0.588562\pi\)
−0.274649 + 0.961545i \(0.588562\pi\)
\(740\) 0 0
\(741\) −5.40330 −0.198495
\(742\) −4.23463 −0.155458
\(743\) 13.3680 0.490423 0.245211 0.969470i \(-0.421143\pi\)
0.245211 + 0.969470i \(0.421143\pi\)
\(744\) −9.14297 −0.335198
\(745\) 0 0
\(746\) 6.63705 0.243000
\(747\) 8.31508 0.304233
\(748\) 0 0
\(749\) 0.945731 0.0345563
\(750\) 0 0
\(751\) 40.4655 1.47661 0.738303 0.674469i \(-0.235627\pi\)
0.738303 + 0.674469i \(0.235627\pi\)
\(752\) 16.2614 0.592991
\(753\) 16.0984 0.586658
\(754\) −23.2076 −0.845171
\(755\) 0 0
\(756\) −0.686867 −0.0249811
\(757\) 8.53757 0.310303 0.155152 0.987891i \(-0.450413\pi\)
0.155152 + 0.987891i \(0.450413\pi\)
\(758\) −21.0437 −0.764341
\(759\) 0 0
\(760\) 0 0
\(761\) 21.2805 0.771416 0.385708 0.922621i \(-0.373957\pi\)
0.385708 + 0.922621i \(0.373957\pi\)
\(762\) 16.7258 0.605911
\(763\) 1.45528 0.0526846
\(764\) 1.48037 0.0535579
\(765\) 0 0
\(766\) −20.1005 −0.726260
\(767\) −10.9231 −0.394409
\(768\) 24.3912 0.880140
\(769\) 24.4717 0.882471 0.441235 0.897391i \(-0.354540\pi\)
0.441235 + 0.897391i \(0.354540\pi\)
\(770\) 0 0
\(771\) 5.96875 0.214959
\(772\) 41.5134 1.49410
\(773\) −13.6472 −0.490855 −0.245427 0.969415i \(-0.578928\pi\)
−0.245427 + 0.969415i \(0.578928\pi\)
\(774\) −12.0458 −0.432978
\(775\) 0 0
\(776\) −57.8516 −2.07675
\(777\) 1.10546 0.0396582
\(778\) −27.7398 −0.994520
\(779\) 45.9397 1.64596
\(780\) 0 0
\(781\) 0 0
\(782\) 30.2574 1.08200
\(783\) −10.1178 −0.361581
\(784\) −10.4329 −0.372603
\(785\) 0 0
\(786\) 5.88974 0.210080
\(787\) −7.45709 −0.265816 −0.132908 0.991128i \(-0.542432\pi\)
−0.132908 + 0.991128i \(0.542432\pi\)
\(788\) 76.6480 2.73047
\(789\) 0.451149 0.0160613
\(790\) 0 0
\(791\) −3.91383 −0.139160
\(792\) 0 0
\(793\) −1.17431 −0.0417011
\(794\) −3.32655 −0.118055
\(795\) 0 0
\(796\) 27.5428 0.976228
\(797\) −12.0347 −0.426289 −0.213145 0.977021i \(-0.568371\pi\)
−0.213145 + 0.977021i \(0.568371\pi\)
\(798\) 2.52981 0.0895542
\(799\) −29.0375 −1.02727
\(800\) 0 0
\(801\) 7.34270 0.259442
\(802\) −19.9270 −0.703648
\(803\) 0 0
\(804\) 11.5676 0.407958
\(805\) 0 0
\(806\) 5.74656 0.202414
\(807\) −14.7197 −0.518159
\(808\) 48.0231 1.68945
\(809\) 50.0338 1.75909 0.879546 0.475813i \(-0.157846\pi\)
0.879546 + 0.475813i \(0.157846\pi\)
\(810\) 0 0
\(811\) 11.2188 0.393944 0.196972 0.980409i \(-0.436889\pi\)
0.196972 + 0.980409i \(0.436889\pi\)
\(812\) 6.94958 0.243882
\(813\) 4.42787 0.155292
\(814\) 0 0
\(815\) 0 0
\(816\) 4.00935 0.140355
\(817\) 28.3760 0.992752
\(818\) 23.8524 0.833981
\(819\) 0.188439 0.00658459
\(820\) 0 0
\(821\) −7.37723 −0.257467 −0.128734 0.991679i \(-0.541091\pi\)
−0.128734 + 0.991679i \(0.541091\pi\)
\(822\) 36.0286 1.25664
\(823\) −35.6380 −1.24226 −0.621132 0.783706i \(-0.713327\pi\)
−0.621132 + 0.783706i \(0.713327\pi\)
\(824\) 14.5880 0.508198
\(825\) 0 0
\(826\) 5.11414 0.177944
\(827\) −50.5291 −1.75707 −0.878535 0.477678i \(-0.841478\pi\)
−0.878535 + 0.477678i \(0.841478\pi\)
\(828\) 17.0376 0.592097
\(829\) −6.14491 −0.213422 −0.106711 0.994290i \(-0.534032\pi\)
−0.106711 + 0.994290i \(0.534032\pi\)
\(830\) 0 0
\(831\) 0.0923299 0.00320289
\(832\) −11.5632 −0.400882
\(833\) 18.6297 0.645482
\(834\) 46.3093 1.60356
\(835\) 0 0
\(836\) 0 0
\(837\) 2.50533 0.0865967
\(838\) −77.2707 −2.66927
\(839\) −37.5230 −1.29544 −0.647719 0.761879i \(-0.724277\pi\)
−0.647719 + 0.761879i \(0.724277\pi\)
\(840\) 0 0
\(841\) 73.3698 2.52999
\(842\) 33.2840 1.14704
\(843\) 5.18301 0.178512
\(844\) 44.3852 1.52780
\(845\) 0 0
\(846\) −25.5645 −0.878924
\(847\) 0 0
\(848\) 13.9186 0.477966
\(849\) −1.56463 −0.0536978
\(850\) 0 0
\(851\) −27.4207 −0.939970
\(852\) −21.2625 −0.728441
\(853\) −27.0122 −0.924880 −0.462440 0.886651i \(-0.653026\pi\)
−0.462440 + 0.886651i \(0.653026\pi\)
\(854\) 0.549810 0.0188141
\(855\) 0 0
\(856\) −17.8340 −0.609554
\(857\) −2.51515 −0.0859159 −0.0429580 0.999077i \(-0.513678\pi\)
−0.0429580 + 0.999077i \(0.513678\pi\)
\(858\) 0 0
\(859\) 6.70885 0.228903 0.114451 0.993429i \(-0.463489\pi\)
0.114451 + 0.993429i \(0.463489\pi\)
\(860\) 0 0
\(861\) −1.60214 −0.0546006
\(862\) 45.0069 1.53294
\(863\) 3.30744 0.112586 0.0562932 0.998414i \(-0.482072\pi\)
0.0562932 + 0.998414i \(0.482072\pi\)
\(864\) −3.76902 −0.128225
\(865\) 0 0
\(866\) 35.6179 1.21034
\(867\) 9.84060 0.334204
\(868\) −1.72082 −0.0584086
\(869\) 0 0
\(870\) 0 0
\(871\) −3.17352 −0.107531
\(872\) −27.4427 −0.929328
\(873\) 15.8523 0.536519
\(874\) −62.7514 −2.12260
\(875\) 0 0
\(876\) −11.7345 −0.396472
\(877\) 32.1004 1.08395 0.541976 0.840394i \(-0.317676\pi\)
0.541976 + 0.840394i \(0.317676\pi\)
\(878\) 87.5468 2.95456
\(879\) 23.1570 0.781067
\(880\) 0 0
\(881\) −35.2547 −1.18776 −0.593881 0.804553i \(-0.702405\pi\)
−0.593881 + 0.804553i \(0.702405\pi\)
\(882\) 16.4015 0.552267
\(883\) 0.455852 0.0153406 0.00767032 0.999971i \(-0.497558\pi\)
0.00767032 + 0.999971i \(0.497558\pi\)
\(884\) −9.24694 −0.311008
\(885\) 0 0
\(886\) −6.50282 −0.218467
\(887\) −1.80160 −0.0604918 −0.0302459 0.999542i \(-0.509629\pi\)
−0.0302459 + 0.999542i \(0.509629\pi\)
\(888\) −20.8461 −0.699549
\(889\) 1.37408 0.0460853
\(890\) 0 0
\(891\) 0 0
\(892\) 19.0794 0.638824
\(893\) 60.2216 2.01524
\(894\) 23.8699 0.798328
\(895\) 0 0
\(896\) 3.95504 0.132129
\(897\) −4.67419 −0.156067
\(898\) 42.0868 1.40446
\(899\) −25.3484 −0.845416
\(900\) 0 0
\(901\) −24.8541 −0.828009
\(902\) 0 0
\(903\) −0.989607 −0.0329321
\(904\) 73.8047 2.45471
\(905\) 0 0
\(906\) −28.1813 −0.936261
\(907\) 35.3469 1.17367 0.586837 0.809705i \(-0.300373\pi\)
0.586837 + 0.809705i \(0.300373\pi\)
\(908\) −77.2054 −2.56215
\(909\) −13.1591 −0.436461
\(910\) 0 0
\(911\) −1.64877 −0.0546262 −0.0273131 0.999627i \(-0.508695\pi\)
−0.0273131 + 0.999627i \(0.508695\pi\)
\(912\) −8.31508 −0.275340
\(913\) 0 0
\(914\) 49.8511 1.64893
\(915\) 0 0
\(916\) 9.47903 0.313196
\(917\) 0.483864 0.0159786
\(918\) −6.30309 −0.208033
\(919\) 0.318990 0.0105225 0.00526125 0.999986i \(-0.498325\pi\)
0.00526125 + 0.999986i \(0.498325\pi\)
\(920\) 0 0
\(921\) 7.72480 0.254541
\(922\) −80.9118 −2.66469
\(923\) 5.83327 0.192005
\(924\) 0 0
\(925\) 0 0
\(926\) 24.6206 0.809082
\(927\) −3.99737 −0.131291
\(928\) 38.1342 1.25182
\(929\) 56.1509 1.84225 0.921125 0.389267i \(-0.127272\pi\)
0.921125 + 0.389267i \(0.127272\pi\)
\(930\) 0 0
\(931\) −38.6366 −1.26626
\(932\) −20.3262 −0.665808
\(933\) 19.1209 0.625989
\(934\) 69.8301 2.28491
\(935\) 0 0
\(936\) −3.55346 −0.116149
\(937\) 36.8424 1.20359 0.601795 0.798651i \(-0.294453\pi\)
0.601795 + 0.798651i \(0.294453\pi\)
\(938\) 1.48583 0.0485142
\(939\) 2.90307 0.0947380
\(940\) 0 0
\(941\) 15.3105 0.499108 0.249554 0.968361i \(-0.419716\pi\)
0.249554 + 0.968361i \(0.419716\pi\)
\(942\) 6.18612 0.201555
\(943\) 39.7406 1.29413
\(944\) −16.8094 −0.547099
\(945\) 0 0
\(946\) 0 0
\(947\) 22.6654 0.736526 0.368263 0.929722i \(-0.379953\pi\)
0.368263 + 0.929722i \(0.379953\pi\)
\(948\) 36.0348 1.17036
\(949\) 3.21931 0.104503
\(950\) 0 0
\(951\) 2.26153 0.0733353
\(952\) 1.88974 0.0612470
\(953\) 39.3324 1.27410 0.637051 0.770822i \(-0.280154\pi\)
0.637051 + 0.770822i \(0.280154\pi\)
\(954\) −21.8814 −0.708435
\(955\) 0 0
\(956\) 86.4870 2.79719
\(957\) 0 0
\(958\) −31.1388 −1.00605
\(959\) 2.95988 0.0955794
\(960\) 0 0
\(961\) −24.7233 −0.797527
\(962\) 13.1022 0.422433
\(963\) 4.88682 0.157475
\(964\) 59.3008 1.90995
\(965\) 0 0
\(966\) 2.18844 0.0704119
\(967\) 3.06103 0.0984360 0.0492180 0.998788i \(-0.484327\pi\)
0.0492180 + 0.998788i \(0.484327\pi\)
\(968\) 0 0
\(969\) 14.8480 0.476987
\(970\) 0 0
\(971\) 32.0385 1.02816 0.514082 0.857741i \(-0.328133\pi\)
0.514082 + 0.857741i \(0.328133\pi\)
\(972\) −3.54920 −0.113841
\(973\) 3.80447 0.121966
\(974\) −100.110 −3.20775
\(975\) 0 0
\(976\) −1.80714 −0.0578451
\(977\) 53.1962 1.70190 0.850948 0.525250i \(-0.176028\pi\)
0.850948 + 0.525250i \(0.176028\pi\)
\(978\) −55.8848 −1.78700
\(979\) 0 0
\(980\) 0 0
\(981\) 7.51977 0.240088
\(982\) −37.0916 −1.18364
\(983\) 24.5004 0.781442 0.390721 0.920509i \(-0.372226\pi\)
0.390721 + 0.920509i \(0.372226\pi\)
\(984\) 30.2121 0.963126
\(985\) 0 0
\(986\) 63.7734 2.03096
\(987\) −2.10021 −0.0668505
\(988\) 19.1774 0.610115
\(989\) 24.5470 0.780549
\(990\) 0 0
\(991\) −6.34819 −0.201657 −0.100828 0.994904i \(-0.532149\pi\)
−0.100828 + 0.994904i \(0.532149\pi\)
\(992\) −9.44262 −0.299804
\(993\) −6.02336 −0.191146
\(994\) −2.73112 −0.0866258
\(995\) 0 0
\(996\) −29.5119 −0.935120
\(997\) −25.9460 −0.821719 −0.410860 0.911699i \(-0.634771\pi\)
−0.410860 + 0.911699i \(0.634771\pi\)
\(998\) 78.9808 2.50010
\(999\) 5.71217 0.180725
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 9075.2.a.cm.1.1 4
5.4 even 2 1815.2.a.w.1.4 4
11.2 odd 10 825.2.n.g.301.1 8
11.6 odd 10 825.2.n.g.751.1 8
11.10 odd 2 9075.2.a.di.1.4 4
15.14 odd 2 5445.2.a.bf.1.1 4
55.2 even 20 825.2.bx.f.499.4 16
55.13 even 20 825.2.bx.f.499.1 16
55.17 even 20 825.2.bx.f.124.1 16
55.24 odd 10 165.2.m.d.136.2 yes 8
55.28 even 20 825.2.bx.f.124.4 16
55.39 odd 10 165.2.m.d.91.2 8
55.54 odd 2 1815.2.a.p.1.1 4
165.134 even 10 495.2.n.a.136.1 8
165.149 even 10 495.2.n.a.91.1 8
165.164 even 2 5445.2.a.bt.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.d.91.2 8 55.39 odd 10
165.2.m.d.136.2 yes 8 55.24 odd 10
495.2.n.a.91.1 8 165.149 even 10
495.2.n.a.136.1 8 165.134 even 10
825.2.n.g.301.1 8 11.2 odd 10
825.2.n.g.751.1 8 11.6 odd 10
825.2.bx.f.124.1 16 55.17 even 20
825.2.bx.f.124.4 16 55.28 even 20
825.2.bx.f.499.1 16 55.13 even 20
825.2.bx.f.499.4 16 55.2 even 20
1815.2.a.p.1.1 4 55.54 odd 2
1815.2.a.w.1.4 4 5.4 even 2
5445.2.a.bf.1.1 4 15.14 odd 2
5445.2.a.bt.1.4 4 165.164 even 2
9075.2.a.cm.1.1 4 1.1 even 1 trivial
9075.2.a.di.1.4 4 11.10 odd 2