Properties

Label 5625.2.a.p.1.4
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 1.4
Root \(0.141689\) 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)
 
\(f(q)\) \(=\) \(q+0.141689 q^{2} -1.97992 q^{4} -0.858311 q^{7} -0.563913 q^{8} +O(q^{10})\) \(q+0.141689 q^{2} -1.97992 q^{4} -0.858311 q^{7} -0.563913 q^{8} +3.67993 q^{11} +4.58134 q^{13} -0.121614 q^{14} +3.87995 q^{16} +5.30702 q^{17} +6.36870 q^{19} +0.521407 q^{22} -3.42379 q^{23} +0.649128 q^{26} +1.69939 q^{28} -3.73405 q^{29} +1.25290 q^{31} +1.67757 q^{32} +0.751949 q^{34} -7.45067 q^{37} +0.902378 q^{38} +2.53168 q^{41} -3.37972 q^{43} -7.28598 q^{44} -0.485114 q^{46} +8.49937 q^{47} -6.26330 q^{49} -9.07071 q^{52} +2.34827 q^{53} +0.484013 q^{56} -0.529076 q^{58} -13.1264 q^{59} +10.3476 q^{61} +0.177523 q^{62} -7.52220 q^{64} +3.34649 q^{67} -10.5075 q^{68} +4.32289 q^{71} -9.08007 q^{73} -1.05568 q^{74} -12.6095 q^{76} -3.15852 q^{77} -3.24730 q^{79} +0.358712 q^{82} +7.39269 q^{83} -0.478870 q^{86} -2.07516 q^{88} +15.4975 q^{89} -3.93221 q^{91} +6.77883 q^{92} +1.20427 q^{94} -10.0386 q^{97} -0.887444 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.141689 0.100190 0.0500948 0.998744i \(-0.484048\pi\)
0.0500948 + 0.998744i \(0.484048\pi\)
\(3\) 0 0
\(4\) −1.97992 −0.989962
\(5\) 0 0
\(6\) 0 0
\(7\) −0.858311 −0.324411 −0.162205 0.986757i \(-0.551861\pi\)
−0.162205 + 0.986757i \(0.551861\pi\)
\(8\) −0.563913 −0.199374
\(9\) 0 0
\(10\) 0 0
\(11\) 3.67993 1.10954 0.554770 0.832004i \(-0.312806\pi\)
0.554770 + 0.832004i \(0.312806\pi\)
\(12\) 0 0
\(13\) 4.58134 1.27064 0.635318 0.772251i \(-0.280869\pi\)
0.635318 + 0.772251i \(0.280869\pi\)
\(14\) −0.121614 −0.0325026
\(15\) 0 0
\(16\) 3.87995 0.969987
\(17\) 5.30702 1.28714 0.643571 0.765387i \(-0.277452\pi\)
0.643571 + 0.765387i \(0.277452\pi\)
\(18\) 0 0
\(19\) 6.36870 1.46108 0.730540 0.682870i \(-0.239268\pi\)
0.730540 + 0.682870i \(0.239268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.521407 0.111164
\(23\) −3.42379 −0.713909 −0.356954 0.934122i \(-0.616185\pi\)
−0.356954 + 0.934122i \(0.616185\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.649128 0.127304
\(27\) 0 0
\(28\) 1.69939 0.321154
\(29\) −3.73405 −0.693396 −0.346698 0.937977i \(-0.612697\pi\)
−0.346698 + 0.937977i \(0.612697\pi\)
\(30\) 0 0
\(31\) 1.25290 0.225028 0.112514 0.993650i \(-0.464110\pi\)
0.112514 + 0.993650i \(0.464110\pi\)
\(32\) 1.67757 0.296556
\(33\) 0 0
\(34\) 0.751949 0.128958
\(35\) 0 0
\(36\) 0 0
\(37\) −7.45067 −1.22488 −0.612441 0.790516i \(-0.709812\pi\)
−0.612441 + 0.790516i \(0.709812\pi\)
\(38\) 0.902378 0.146385
\(39\) 0 0
\(40\) 0 0
\(41\) 2.53168 0.395381 0.197691 0.980264i \(-0.436656\pi\)
0.197691 + 0.980264i \(0.436656\pi\)
\(42\) 0 0
\(43\) −3.37972 −0.515402 −0.257701 0.966225i \(-0.582965\pi\)
−0.257701 + 0.966225i \(0.582965\pi\)
\(44\) −7.28598 −1.09840
\(45\) 0 0
\(46\) −0.485114 −0.0715262
\(47\) 8.49937 1.23976 0.619880 0.784696i \(-0.287181\pi\)
0.619880 + 0.784696i \(0.287181\pi\)
\(48\) 0 0
\(49\) −6.26330 −0.894758
\(50\) 0 0
\(51\) 0 0
\(52\) −9.07071 −1.25788
\(53\) 2.34827 0.322560 0.161280 0.986909i \(-0.448438\pi\)
0.161280 + 0.986909i \(0.448438\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.484013 0.0646789
\(57\) 0 0
\(58\) −0.529076 −0.0694710
\(59\) −13.1264 −1.70891 −0.854455 0.519525i \(-0.826109\pi\)
−0.854455 + 0.519525i \(0.826109\pi\)
\(60\) 0 0
\(61\) 10.3476 1.32488 0.662439 0.749116i \(-0.269521\pi\)
0.662439 + 0.749116i \(0.269521\pi\)
\(62\) 0.177523 0.0225454
\(63\) 0 0
\(64\) −7.52220 −0.940275
\(65\) 0 0
\(66\) 0 0
\(67\) 3.34649 0.408838 0.204419 0.978883i \(-0.434469\pi\)
0.204419 + 0.978883i \(0.434469\pi\)
\(68\) −10.5075 −1.27422
\(69\) 0 0
\(70\) 0 0
\(71\) 4.32289 0.513032 0.256516 0.966540i \(-0.417425\pi\)
0.256516 + 0.966540i \(0.417425\pi\)
\(72\) 0 0
\(73\) −9.08007 −1.06274 −0.531371 0.847139i \(-0.678323\pi\)
−0.531371 + 0.847139i \(0.678323\pi\)
\(74\) −1.05568 −0.122721
\(75\) 0 0
\(76\) −12.6095 −1.44641
\(77\) −3.15852 −0.359947
\(78\) 0 0
\(79\) −3.24730 −0.365350 −0.182675 0.983173i \(-0.558476\pi\)
−0.182675 + 0.983173i \(0.558476\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.358712 0.0396131
\(83\) 7.39269 0.811453 0.405727 0.913994i \(-0.367019\pi\)
0.405727 + 0.913994i \(0.367019\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.478870 −0.0516379
\(87\) 0 0
\(88\) −2.07516 −0.221213
\(89\) 15.4975 1.64273 0.821367 0.570400i \(-0.193212\pi\)
0.821367 + 0.570400i \(0.193212\pi\)
\(90\) 0 0
\(91\) −3.93221 −0.412208
\(92\) 6.77883 0.706742
\(93\) 0 0
\(94\) 1.20427 0.124211
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0386 −1.01926 −0.509631 0.860393i \(-0.670218\pi\)
−0.509631 + 0.860393i \(0.670218\pi\)
\(98\) −0.887444 −0.0896454
\(99\) 0 0
\(100\) 0 0
\(101\) 0.714616 0.0711070 0.0355535 0.999368i \(-0.488681\pi\)
0.0355535 + 0.999368i \(0.488681\pi\)
\(102\) 0 0
\(103\) 0.780764 0.0769310 0.0384655 0.999260i \(-0.487753\pi\)
0.0384655 + 0.999260i \(0.487753\pi\)
\(104\) −2.58348 −0.253331
\(105\) 0 0
\(106\) 0.332725 0.0323171
\(107\) −11.9601 −1.15623 −0.578113 0.815957i \(-0.696211\pi\)
−0.578113 + 0.815957i \(0.696211\pi\)
\(108\) 0 0
\(109\) 2.43384 0.233119 0.116560 0.993184i \(-0.462813\pi\)
0.116560 + 0.993184i \(0.462813\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.33020 −0.314674
\(113\) 16.0354 1.50848 0.754240 0.656599i \(-0.228006\pi\)
0.754240 + 0.656599i \(0.228006\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.39313 0.686435
\(117\) 0 0
\(118\) −1.85987 −0.171215
\(119\) −4.55507 −0.417563
\(120\) 0 0
\(121\) 2.54188 0.231080
\(122\) 1.46615 0.132739
\(123\) 0 0
\(124\) −2.48065 −0.222769
\(125\) 0 0
\(126\) 0 0
\(127\) −3.83848 −0.340610 −0.170305 0.985391i \(-0.554475\pi\)
−0.170305 + 0.985391i \(0.554475\pi\)
\(128\) −4.42097 −0.390762
\(129\) 0 0
\(130\) 0 0
\(131\) 12.4495 1.08771 0.543857 0.839178i \(-0.316963\pi\)
0.543857 + 0.839178i \(0.316963\pi\)
\(132\) 0 0
\(133\) −5.46632 −0.473990
\(134\) 0.474162 0.0409613
\(135\) 0 0
\(136\) −2.99270 −0.256622
\(137\) 11.8589 1.01317 0.506587 0.862189i \(-0.330907\pi\)
0.506587 + 0.862189i \(0.330907\pi\)
\(138\) 0 0
\(139\) −4.66089 −0.395332 −0.197666 0.980269i \(-0.563336\pi\)
−0.197666 + 0.980269i \(0.563336\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.612508 0.0514005
\(143\) 16.8590 1.40982
\(144\) 0 0
\(145\) 0 0
\(146\) −1.28655 −0.106476
\(147\) 0 0
\(148\) 14.7518 1.21259
\(149\) 11.5480 0.946053 0.473026 0.881048i \(-0.343162\pi\)
0.473026 + 0.881048i \(0.343162\pi\)
\(150\) 0 0
\(151\) 24.4694 1.99129 0.995646 0.0932103i \(-0.0297129\pi\)
0.995646 + 0.0932103i \(0.0297129\pi\)
\(152\) −3.59140 −0.291301
\(153\) 0 0
\(154\) −0.447529 −0.0360629
\(155\) 0 0
\(156\) 0 0
\(157\) −22.3660 −1.78500 −0.892501 0.451045i \(-0.851052\pi\)
−0.892501 + 0.451045i \(0.851052\pi\)
\(158\) −0.460109 −0.0366043
\(159\) 0 0
\(160\) 0 0
\(161\) 2.93867 0.231600
\(162\) 0 0
\(163\) −0.813450 −0.0637143 −0.0318572 0.999492i \(-0.510142\pi\)
−0.0318572 + 0.999492i \(0.510142\pi\)
\(164\) −5.01253 −0.391413
\(165\) 0 0
\(166\) 1.04747 0.0812992
\(167\) 25.1660 1.94740 0.973700 0.227833i \(-0.0731639\pi\)
0.973700 + 0.227833i \(0.0731639\pi\)
\(168\) 0 0
\(169\) 7.98869 0.614515
\(170\) 0 0
\(171\) 0 0
\(172\) 6.69158 0.510229
\(173\) −0.410945 −0.0312436 −0.0156218 0.999878i \(-0.504973\pi\)
−0.0156218 + 0.999878i \(0.504973\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 14.2779 1.07624
\(177\) 0 0
\(178\) 2.19584 0.164585
\(179\) 14.8502 1.10996 0.554978 0.831865i \(-0.312727\pi\)
0.554978 + 0.831865i \(0.312727\pi\)
\(180\) 0 0
\(181\) 0.739223 0.0549460 0.0274730 0.999623i \(-0.491254\pi\)
0.0274730 + 0.999623i \(0.491254\pi\)
\(182\) −0.557153 −0.0412990
\(183\) 0 0
\(184\) 1.93072 0.142334
\(185\) 0 0
\(186\) 0 0
\(187\) 19.5295 1.42814
\(188\) −16.8281 −1.22732
\(189\) 0 0
\(190\) 0 0
\(191\) 2.72506 0.197179 0.0985893 0.995128i \(-0.468567\pi\)
0.0985893 + 0.995128i \(0.468567\pi\)
\(192\) 0 0
\(193\) 14.2040 1.02243 0.511215 0.859453i \(-0.329196\pi\)
0.511215 + 0.859453i \(0.329196\pi\)
\(194\) −1.42236 −0.102120
\(195\) 0 0
\(196\) 12.4009 0.885776
\(197\) 5.54591 0.395130 0.197565 0.980290i \(-0.436697\pi\)
0.197565 + 0.980290i \(0.436697\pi\)
\(198\) 0 0
\(199\) 5.96371 0.422756 0.211378 0.977404i \(-0.432205\pi\)
0.211378 + 0.977404i \(0.432205\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.101254 0.00712418
\(203\) 3.20497 0.224945
\(204\) 0 0
\(205\) 0 0
\(206\) 0.110626 0.00770768
\(207\) 0 0
\(208\) 17.7754 1.23250
\(209\) 23.4364 1.62113
\(210\) 0 0
\(211\) −13.7183 −0.944407 −0.472204 0.881490i \(-0.656541\pi\)
−0.472204 + 0.881490i \(0.656541\pi\)
\(212\) −4.64940 −0.319322
\(213\) 0 0
\(214\) −1.69462 −0.115842
\(215\) 0 0
\(216\) 0 0
\(217\) −1.07538 −0.0730014
\(218\) 0.344849 0.0233561
\(219\) 0 0
\(220\) 0 0
\(221\) 24.3133 1.63549
\(222\) 0 0
\(223\) 3.24170 0.217081 0.108540 0.994092i \(-0.465382\pi\)
0.108540 + 0.994092i \(0.465382\pi\)
\(224\) −1.43988 −0.0962060
\(225\) 0 0
\(226\) 2.27204 0.151134
\(227\) −13.1799 −0.874779 −0.437390 0.899272i \(-0.644097\pi\)
−0.437390 + 0.899272i \(0.644097\pi\)
\(228\) 0 0
\(229\) 5.83810 0.385793 0.192896 0.981219i \(-0.438212\pi\)
0.192896 + 0.981219i \(0.438212\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.10568 0.138245
\(233\) −9.33453 −0.611526 −0.305763 0.952108i \(-0.598911\pi\)
−0.305763 + 0.952108i \(0.598911\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 25.9893 1.69176
\(237\) 0 0
\(238\) −0.645406 −0.0418354
\(239\) −17.1502 −1.10935 −0.554676 0.832066i \(-0.687158\pi\)
−0.554676 + 0.832066i \(0.687158\pi\)
\(240\) 0 0
\(241\) 7.22073 0.465128 0.232564 0.972581i \(-0.425289\pi\)
0.232564 + 0.972581i \(0.425289\pi\)
\(242\) 0.360157 0.0231518
\(243\) 0 0
\(244\) −20.4875 −1.31158
\(245\) 0 0
\(246\) 0 0
\(247\) 29.1772 1.85650
\(248\) −0.706527 −0.0448645
\(249\) 0 0
\(250\) 0 0
\(251\) 5.75708 0.363383 0.181692 0.983356i \(-0.441843\pi\)
0.181692 + 0.983356i \(0.441843\pi\)
\(252\) 0 0
\(253\) −12.5993 −0.792110
\(254\) −0.543872 −0.0341256
\(255\) 0 0
\(256\) 14.4180 0.901125
\(257\) −26.9602 −1.68173 −0.840867 0.541242i \(-0.817954\pi\)
−0.840867 + 0.541242i \(0.817954\pi\)
\(258\) 0 0
\(259\) 6.39499 0.397365
\(260\) 0 0
\(261\) 0 0
\(262\) 1.76396 0.108978
\(263\) −20.6673 −1.27440 −0.637200 0.770699i \(-0.719907\pi\)
−0.637200 + 0.770699i \(0.719907\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.774520 −0.0474889
\(267\) 0 0
\(268\) −6.62579 −0.404734
\(269\) 13.7435 0.837958 0.418979 0.907996i \(-0.362388\pi\)
0.418979 + 0.907996i \(0.362388\pi\)
\(270\) 0 0
\(271\) 3.34392 0.203128 0.101564 0.994829i \(-0.467615\pi\)
0.101564 + 0.994829i \(0.467615\pi\)
\(272\) 20.5910 1.24851
\(273\) 0 0
\(274\) 1.68028 0.101510
\(275\) 0 0
\(276\) 0 0
\(277\) 8.90947 0.535318 0.267659 0.963514i \(-0.413750\pi\)
0.267659 + 0.963514i \(0.413750\pi\)
\(278\) −0.660400 −0.0396081
\(279\) 0 0
\(280\) 0 0
\(281\) 22.4913 1.34172 0.670859 0.741585i \(-0.265926\pi\)
0.670859 + 0.741585i \(0.265926\pi\)
\(282\) 0 0
\(283\) 1.30694 0.0776894 0.0388447 0.999245i \(-0.487632\pi\)
0.0388447 + 0.999245i \(0.487632\pi\)
\(284\) −8.55899 −0.507883
\(285\) 0 0
\(286\) 2.38874 0.141249
\(287\) −2.17296 −0.128266
\(288\) 0 0
\(289\) 11.1645 0.656733
\(290\) 0 0
\(291\) 0 0
\(292\) 17.9779 1.05207
\(293\) −1.97058 −0.115123 −0.0575613 0.998342i \(-0.518332\pi\)
−0.0575613 + 0.998342i \(0.518332\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.20153 0.244209
\(297\) 0 0
\(298\) 1.63624 0.0947847
\(299\) −15.6855 −0.907118
\(300\) 0 0
\(301\) 2.90085 0.167202
\(302\) 3.46706 0.199507
\(303\) 0 0
\(304\) 24.7102 1.41723
\(305\) 0 0
\(306\) 0 0
\(307\) −15.2544 −0.870617 −0.435308 0.900281i \(-0.643361\pi\)
−0.435308 + 0.900281i \(0.643361\pi\)
\(308\) 6.25363 0.356334
\(309\) 0 0
\(310\) 0 0
\(311\) −17.7452 −1.00624 −0.503120 0.864217i \(-0.667815\pi\)
−0.503120 + 0.864217i \(0.667815\pi\)
\(312\) 0 0
\(313\) −27.3869 −1.54800 −0.773999 0.633187i \(-0.781747\pi\)
−0.773999 + 0.633187i \(0.781747\pi\)
\(314\) −3.16903 −0.178839
\(315\) 0 0
\(316\) 6.42941 0.361683
\(317\) −8.64563 −0.485587 −0.242793 0.970078i \(-0.578064\pi\)
−0.242793 + 0.970078i \(0.578064\pi\)
\(318\) 0 0
\(319\) −13.7410 −0.769350
\(320\) 0 0
\(321\) 0 0
\(322\) 0.416379 0.0232039
\(323\) 33.7988 1.88062
\(324\) 0 0
\(325\) 0 0
\(326\) −0.115257 −0.00638351
\(327\) 0 0
\(328\) −1.42765 −0.0788286
\(329\) −7.29510 −0.402192
\(330\) 0 0
\(331\) 4.96260 0.272769 0.136385 0.990656i \(-0.456452\pi\)
0.136385 + 0.990656i \(0.456452\pi\)
\(332\) −14.6370 −0.803308
\(333\) 0 0
\(334\) 3.56575 0.195109
\(335\) 0 0
\(336\) 0 0
\(337\) −7.30529 −0.397945 −0.198972 0.980005i \(-0.563760\pi\)
−0.198972 + 0.980005i \(0.563760\pi\)
\(338\) 1.13191 0.0615680
\(339\) 0 0
\(340\) 0 0
\(341\) 4.61058 0.249677
\(342\) 0 0
\(343\) 11.3840 0.614680
\(344\) 1.90587 0.102758
\(345\) 0 0
\(346\) −0.0582266 −0.00313028
\(347\) −15.9374 −0.855565 −0.427783 0.903882i \(-0.640705\pi\)
−0.427783 + 0.903882i \(0.640705\pi\)
\(348\) 0 0
\(349\) 16.5844 0.887743 0.443871 0.896091i \(-0.353605\pi\)
0.443871 + 0.896091i \(0.353605\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.17336 0.329041
\(353\) −12.9691 −0.690277 −0.345138 0.938552i \(-0.612168\pi\)
−0.345138 + 0.938552i \(0.612168\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −30.6839 −1.62624
\(357\) 0 0
\(358\) 2.10412 0.111206
\(359\) 13.4920 0.712082 0.356041 0.934470i \(-0.384126\pi\)
0.356041 + 0.934470i \(0.384126\pi\)
\(360\) 0 0
\(361\) 21.5603 1.13475
\(362\) 0.104740 0.00550502
\(363\) 0 0
\(364\) 7.78549 0.408070
\(365\) 0 0
\(366\) 0 0
\(367\) 4.57463 0.238794 0.119397 0.992847i \(-0.461904\pi\)
0.119397 + 0.992847i \(0.461904\pi\)
\(368\) −13.2841 −0.692482
\(369\) 0 0
\(370\) 0 0
\(371\) −2.01555 −0.104642
\(372\) 0 0
\(373\) −20.7463 −1.07420 −0.537101 0.843518i \(-0.680481\pi\)
−0.537101 + 0.843518i \(0.680481\pi\)
\(374\) 2.76712 0.143084
\(375\) 0 0
\(376\) −4.79291 −0.247175
\(377\) −17.1070 −0.881053
\(378\) 0 0
\(379\) −5.78163 −0.296982 −0.148491 0.988914i \(-0.547442\pi\)
−0.148491 + 0.988914i \(0.547442\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.386113 0.0197553
\(383\) −25.8124 −1.31895 −0.659476 0.751726i \(-0.729222\pi\)
−0.659476 + 0.751726i \(0.729222\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.01256 0.102437
\(387\) 0 0
\(388\) 19.8756 1.00903
\(389\) 15.7046 0.796254 0.398127 0.917330i \(-0.369660\pi\)
0.398127 + 0.917330i \(0.369660\pi\)
\(390\) 0 0
\(391\) −18.1701 −0.918901
\(392\) 3.53196 0.178391
\(393\) 0 0
\(394\) 0.785797 0.0395879
\(395\) 0 0
\(396\) 0 0
\(397\) −19.6040 −0.983895 −0.491948 0.870625i \(-0.663715\pi\)
−0.491948 + 0.870625i \(0.663715\pi\)
\(398\) 0.844995 0.0423558
\(399\) 0 0
\(400\) 0 0
\(401\) 14.4239 0.720297 0.360148 0.932895i \(-0.382726\pi\)
0.360148 + 0.932895i \(0.382726\pi\)
\(402\) 0 0
\(403\) 5.73996 0.285928
\(404\) −1.41489 −0.0703932
\(405\) 0 0
\(406\) 0.454111 0.0225372
\(407\) −27.4179 −1.35906
\(408\) 0 0
\(409\) −27.8742 −1.37829 −0.689146 0.724622i \(-0.742014\pi\)
−0.689146 + 0.724622i \(0.742014\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.54585 −0.0761588
\(413\) 11.2665 0.554389
\(414\) 0 0
\(415\) 0 0
\(416\) 7.68554 0.376815
\(417\) 0 0
\(418\) 3.32069 0.162420
\(419\) 36.2881 1.77279 0.886396 0.462928i \(-0.153201\pi\)
0.886396 + 0.462928i \(0.153201\pi\)
\(420\) 0 0
\(421\) −3.37600 −0.164536 −0.0822681 0.996610i \(-0.526216\pi\)
−0.0822681 + 0.996610i \(0.526216\pi\)
\(422\) −1.94374 −0.0946198
\(423\) 0 0
\(424\) −1.32422 −0.0643099
\(425\) 0 0
\(426\) 0 0
\(427\) −8.88148 −0.429805
\(428\) 23.6801 1.14462
\(429\) 0 0
\(430\) 0 0
\(431\) 25.6625 1.23612 0.618059 0.786131i \(-0.287919\pi\)
0.618059 + 0.786131i \(0.287919\pi\)
\(432\) 0 0
\(433\) −39.8070 −1.91300 −0.956501 0.291730i \(-0.905769\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(434\) −0.152370 −0.00731398
\(435\) 0 0
\(436\) −4.81882 −0.230779
\(437\) −21.8051 −1.04308
\(438\) 0 0
\(439\) 33.9180 1.61882 0.809408 0.587247i \(-0.199788\pi\)
0.809408 + 0.587247i \(0.199788\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.44494 0.163859
\(443\) 28.9300 1.37451 0.687253 0.726418i \(-0.258816\pi\)
0.687253 + 0.726418i \(0.258816\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.459315 0.0217492
\(447\) 0 0
\(448\) 6.45638 0.305035
\(449\) 10.2089 0.481788 0.240894 0.970551i \(-0.422559\pi\)
0.240894 + 0.970551i \(0.422559\pi\)
\(450\) 0 0
\(451\) 9.31639 0.438692
\(452\) −31.7488 −1.49334
\(453\) 0 0
\(454\) −1.86745 −0.0876438
\(455\) 0 0
\(456\) 0 0
\(457\) 32.4952 1.52006 0.760030 0.649888i \(-0.225184\pi\)
0.760030 + 0.649888i \(0.225184\pi\)
\(458\) 0.827198 0.0386524
\(459\) 0 0
\(460\) 0 0
\(461\) 0.700568 0.0326287 0.0163144 0.999867i \(-0.494807\pi\)
0.0163144 + 0.999867i \(0.494807\pi\)
\(462\) 0 0
\(463\) 2.17126 0.100907 0.0504535 0.998726i \(-0.483933\pi\)
0.0504535 + 0.998726i \(0.483933\pi\)
\(464\) −14.4879 −0.672585
\(465\) 0 0
\(466\) −1.32261 −0.0612685
\(467\) 3.87710 0.179411 0.0897053 0.995968i \(-0.471407\pi\)
0.0897053 + 0.995968i \(0.471407\pi\)
\(468\) 0 0
\(469\) −2.87232 −0.132632
\(470\) 0 0
\(471\) 0 0
\(472\) 7.40215 0.340711
\(473\) −12.4371 −0.571859
\(474\) 0 0
\(475\) 0 0
\(476\) 9.01870 0.413371
\(477\) 0 0
\(478\) −2.43000 −0.111146
\(479\) 21.9994 1.00518 0.502588 0.864526i \(-0.332381\pi\)
0.502588 + 0.864526i \(0.332381\pi\)
\(480\) 0 0
\(481\) −34.1341 −1.55638
\(482\) 1.02310 0.0466010
\(483\) 0 0
\(484\) −5.03272 −0.228760
\(485\) 0 0
\(486\) 0 0
\(487\) 28.3997 1.28692 0.643458 0.765482i \(-0.277499\pi\)
0.643458 + 0.765482i \(0.277499\pi\)
\(488\) −5.83517 −0.264146
\(489\) 0 0
\(490\) 0 0
\(491\) 14.0468 0.633925 0.316963 0.948438i \(-0.397337\pi\)
0.316963 + 0.948438i \(0.397337\pi\)
\(492\) 0 0
\(493\) −19.8167 −0.892498
\(494\) 4.13410 0.186002
\(495\) 0 0
\(496\) 4.86119 0.218274
\(497\) −3.71038 −0.166433
\(498\) 0 0
\(499\) 13.0842 0.585731 0.292866 0.956154i \(-0.405391\pi\)
0.292866 + 0.956154i \(0.405391\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.815717 0.0364072
\(503\) −12.3044 −0.548625 −0.274312 0.961641i \(-0.588450\pi\)
−0.274312 + 0.961641i \(0.588450\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.78519 −0.0793612
\(507\) 0 0
\(508\) 7.59990 0.337191
\(509\) −0.0783897 −0.00347456 −0.00173728 0.999998i \(-0.500553\pi\)
−0.00173728 + 0.999998i \(0.500553\pi\)
\(510\) 0 0
\(511\) 7.79352 0.344765
\(512\) 10.8848 0.481045
\(513\) 0 0
\(514\) −3.81998 −0.168492
\(515\) 0 0
\(516\) 0 0
\(517\) 31.2771 1.37556
\(518\) 0.906103 0.0398119
\(519\) 0 0
\(520\) 0 0
\(521\) −18.1194 −0.793824 −0.396912 0.917857i \(-0.629918\pi\)
−0.396912 + 0.917857i \(0.629918\pi\)
\(522\) 0 0
\(523\) 11.6303 0.508558 0.254279 0.967131i \(-0.418162\pi\)
0.254279 + 0.967131i \(0.418162\pi\)
\(524\) −24.6490 −1.07680
\(525\) 0 0
\(526\) −2.92834 −0.127682
\(527\) 6.64917 0.289642
\(528\) 0 0
\(529\) −11.2777 −0.490335
\(530\) 0 0
\(531\) 0 0
\(532\) 10.8229 0.469232
\(533\) 11.5985 0.502386
\(534\) 0 0
\(535\) 0 0
\(536\) −1.88713 −0.0815115
\(537\) 0 0
\(538\) 1.94732 0.0839547
\(539\) −23.0485 −0.992770
\(540\) 0 0
\(541\) 29.2716 1.25848 0.629242 0.777209i \(-0.283365\pi\)
0.629242 + 0.777209i \(0.283365\pi\)
\(542\) 0.473798 0.0203514
\(543\) 0 0
\(544\) 8.90292 0.381710
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0675 0.515971 0.257985 0.966149i \(-0.416941\pi\)
0.257985 + 0.966149i \(0.416941\pi\)
\(548\) −23.4797 −1.00300
\(549\) 0 0
\(550\) 0 0
\(551\) −23.7810 −1.01311
\(552\) 0 0
\(553\) 2.78719 0.118524
\(554\) 1.26238 0.0536333
\(555\) 0 0
\(556\) 9.22821 0.391363
\(557\) 41.4154 1.75483 0.877413 0.479737i \(-0.159268\pi\)
0.877413 + 0.479737i \(0.159268\pi\)
\(558\) 0 0
\(559\) −15.4836 −0.654888
\(560\) 0 0
\(561\) 0 0
\(562\) 3.18678 0.134426
\(563\) 32.1467 1.35482 0.677411 0.735604i \(-0.263102\pi\)
0.677411 + 0.735604i \(0.263102\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.185179 0.00778367
\(567\) 0 0
\(568\) −2.43773 −0.102285
\(569\) −21.1080 −0.884893 −0.442446 0.896795i \(-0.645889\pi\)
−0.442446 + 0.896795i \(0.645889\pi\)
\(570\) 0 0
\(571\) −24.7493 −1.03573 −0.517863 0.855464i \(-0.673272\pi\)
−0.517863 + 0.855464i \(0.673272\pi\)
\(572\) −33.3796 −1.39567
\(573\) 0 0
\(574\) −0.307886 −0.0128509
\(575\) 0 0
\(576\) 0 0
\(577\) −18.7710 −0.781448 −0.390724 0.920508i \(-0.627775\pi\)
−0.390724 + 0.920508i \(0.627775\pi\)
\(578\) 1.58189 0.0657979
\(579\) 0 0
\(580\) 0 0
\(581\) −6.34522 −0.263244
\(582\) 0 0
\(583\) 8.64147 0.357893
\(584\) 5.12037 0.211883
\(585\) 0 0
\(586\) −0.279211 −0.0115341
\(587\) −14.7572 −0.609095 −0.304547 0.952497i \(-0.598505\pi\)
−0.304547 + 0.952497i \(0.598505\pi\)
\(588\) 0 0
\(589\) 7.97935 0.328783
\(590\) 0 0
\(591\) 0 0
\(592\) −28.9082 −1.18812
\(593\) −8.01859 −0.329284 −0.164642 0.986353i \(-0.552647\pi\)
−0.164642 + 0.986353i \(0.552647\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −22.8643 −0.936556
\(597\) 0 0
\(598\) −2.22247 −0.0908838
\(599\) 1.28951 0.0526878 0.0263439 0.999653i \(-0.491614\pi\)
0.0263439 + 0.999653i \(0.491614\pi\)
\(600\) 0 0
\(601\) −16.8813 −0.688603 −0.344302 0.938859i \(-0.611884\pi\)
−0.344302 + 0.938859i \(0.611884\pi\)
\(602\) 0.411020 0.0167519
\(603\) 0 0
\(604\) −48.4476 −1.97130
\(605\) 0 0
\(606\) 0 0
\(607\) −0.499318 −0.0202667 −0.0101334 0.999949i \(-0.503226\pi\)
−0.0101334 + 0.999949i \(0.503226\pi\)
\(608\) 10.6840 0.433292
\(609\) 0 0
\(610\) 0 0
\(611\) 38.9385 1.57528
\(612\) 0 0
\(613\) 27.4241 1.10765 0.553825 0.832633i \(-0.313168\pi\)
0.553825 + 0.832633i \(0.313168\pi\)
\(614\) −2.16139 −0.0872267
\(615\) 0 0
\(616\) 1.78113 0.0717639
\(617\) 28.3205 1.14014 0.570070 0.821596i \(-0.306916\pi\)
0.570070 + 0.821596i \(0.306916\pi\)
\(618\) 0 0
\(619\) −0.371804 −0.0149440 −0.00747202 0.999972i \(-0.502378\pi\)
−0.00747202 + 0.999972i \(0.502378\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −2.51431 −0.100815
\(623\) −13.3017 −0.532921
\(624\) 0 0
\(625\) 0 0
\(626\) −3.88043 −0.155093
\(627\) 0 0
\(628\) 44.2830 1.76709
\(629\) −39.5409 −1.57660
\(630\) 0 0
\(631\) 16.4507 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(632\) 1.83120 0.0728411
\(633\) 0 0
\(634\) −1.22499 −0.0486508
\(635\) 0 0
\(636\) 0 0
\(637\) −28.6943 −1.13691
\(638\) −1.94696 −0.0770809
\(639\) 0 0
\(640\) 0 0
\(641\) 2.02604 0.0800239 0.0400119 0.999199i \(-0.487260\pi\)
0.0400119 + 0.999199i \(0.487260\pi\)
\(642\) 0 0
\(643\) −33.2034 −1.30941 −0.654706 0.755883i \(-0.727208\pi\)
−0.654706 + 0.755883i \(0.727208\pi\)
\(644\) −5.81834 −0.229275
\(645\) 0 0
\(646\) 4.78894 0.188418
\(647\) 40.5797 1.59535 0.797677 0.603085i \(-0.206062\pi\)
0.797677 + 0.603085i \(0.206062\pi\)
\(648\) 0 0
\(649\) −48.3042 −1.89611
\(650\) 0 0
\(651\) 0 0
\(652\) 1.61057 0.0630748
\(653\) 9.29712 0.363824 0.181912 0.983315i \(-0.441771\pi\)
0.181912 + 0.983315i \(0.441771\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.82277 0.383515
\(657\) 0 0
\(658\) −1.03364 −0.0402954
\(659\) 3.47563 0.135391 0.0676957 0.997706i \(-0.478435\pi\)
0.0676957 + 0.997706i \(0.478435\pi\)
\(660\) 0 0
\(661\) 4.07303 0.158422 0.0792112 0.996858i \(-0.474760\pi\)
0.0792112 + 0.996858i \(0.474760\pi\)
\(662\) 0.703148 0.0273286
\(663\) 0 0
\(664\) −4.16884 −0.161782
\(665\) 0 0
\(666\) 0 0
\(667\) 12.7846 0.495021
\(668\) −49.8267 −1.92785
\(669\) 0 0
\(670\) 0 0
\(671\) 38.0785 1.47001
\(672\) 0 0
\(673\) 21.3843 0.824305 0.412153 0.911115i \(-0.364777\pi\)
0.412153 + 0.911115i \(0.364777\pi\)
\(674\) −1.03508 −0.0398699
\(675\) 0 0
\(676\) −15.8170 −0.608346
\(677\) 20.3257 0.781181 0.390590 0.920565i \(-0.372271\pi\)
0.390590 + 0.920565i \(0.372271\pi\)
\(678\) 0 0
\(679\) 8.61621 0.330660
\(680\) 0 0
\(681\) 0 0
\(682\) 0.653271 0.0250150
\(683\) 18.9591 0.725449 0.362725 0.931896i \(-0.381847\pi\)
0.362725 + 0.931896i \(0.381847\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.61300 0.0615845
\(687\) 0 0
\(688\) −13.1131 −0.499933
\(689\) 10.7582 0.409856
\(690\) 0 0
\(691\) 14.8195 0.563762 0.281881 0.959449i \(-0.409042\pi\)
0.281881 + 0.959449i \(0.409042\pi\)
\(692\) 0.813641 0.0309300
\(693\) 0 0
\(694\) −2.25816 −0.0857187
\(695\) 0 0
\(696\) 0 0
\(697\) 13.4357 0.508912
\(698\) 2.34984 0.0889426
\(699\) 0 0
\(700\) 0 0
\(701\) −31.3996 −1.18595 −0.592973 0.805222i \(-0.702046\pi\)
−0.592973 + 0.805222i \(0.702046\pi\)
\(702\) 0 0
\(703\) −47.4511 −1.78965
\(704\) −27.6812 −1.04327
\(705\) 0 0
\(706\) −1.83759 −0.0691586
\(707\) −0.613363 −0.0230679
\(708\) 0 0
\(709\) 25.3324 0.951379 0.475690 0.879613i \(-0.342199\pi\)
0.475690 + 0.879613i \(0.342199\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −8.73926 −0.327518
\(713\) −4.28966 −0.160649
\(714\) 0 0
\(715\) 0 0
\(716\) −29.4023 −1.09881
\(717\) 0 0
\(718\) 1.91168 0.0713432
\(719\) 4.12711 0.153915 0.0769575 0.997034i \(-0.475479\pi\)
0.0769575 + 0.997034i \(0.475479\pi\)
\(720\) 0 0
\(721\) −0.670138 −0.0249572
\(722\) 3.05487 0.113691
\(723\) 0 0
\(724\) −1.46361 −0.0543945
\(725\) 0 0
\(726\) 0 0
\(727\) 23.0039 0.853168 0.426584 0.904448i \(-0.359717\pi\)
0.426584 + 0.904448i \(0.359717\pi\)
\(728\) 2.21743 0.0821834
\(729\) 0 0
\(730\) 0 0
\(731\) −17.9362 −0.663396
\(732\) 0 0
\(733\) −9.35692 −0.345606 −0.172803 0.984956i \(-0.555282\pi\)
−0.172803 + 0.984956i \(0.555282\pi\)
\(734\) 0.648176 0.0239246
\(735\) 0 0
\(736\) −5.74366 −0.211714
\(737\) 12.3148 0.453623
\(738\) 0 0
\(739\) −47.1754 −1.73538 −0.867688 0.497109i \(-0.834395\pi\)
−0.867688 + 0.497109i \(0.834395\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.285582 −0.0104840
\(743\) −2.39450 −0.0878455 −0.0439228 0.999035i \(-0.513986\pi\)
−0.0439228 + 0.999035i \(0.513986\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.93953 −0.107624
\(747\) 0 0
\(748\) −38.6668 −1.41380
\(749\) 10.2655 0.375092
\(750\) 0 0
\(751\) −7.21632 −0.263327 −0.131664 0.991294i \(-0.542032\pi\)
−0.131664 + 0.991294i \(0.542032\pi\)
\(752\) 32.9771 1.20255
\(753\) 0 0
\(754\) −2.42388 −0.0882724
\(755\) 0 0
\(756\) 0 0
\(757\) 13.3742 0.486094 0.243047 0.970015i \(-0.421853\pi\)
0.243047 + 0.970015i \(0.421853\pi\)
\(758\) −0.819196 −0.0297545
\(759\) 0 0
\(760\) 0 0
\(761\) −21.7541 −0.788584 −0.394292 0.918985i \(-0.629010\pi\)
−0.394292 + 0.918985i \(0.629010\pi\)
\(762\) 0 0
\(763\) −2.08899 −0.0756265
\(764\) −5.39542 −0.195199
\(765\) 0 0
\(766\) −3.65735 −0.132145
\(767\) −60.1365 −2.17140
\(768\) 0 0
\(769\) 14.2003 0.512077 0.256038 0.966667i \(-0.417583\pi\)
0.256038 + 0.966667i \(0.417583\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −28.1229 −1.01217
\(773\) −22.8365 −0.821373 −0.410687 0.911777i \(-0.634711\pi\)
−0.410687 + 0.911777i \(0.634711\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5.66089 0.203214
\(777\) 0 0
\(778\) 2.22517 0.0797763
\(779\) 16.1235 0.577684
\(780\) 0 0
\(781\) 15.9079 0.569230
\(782\) −2.57451 −0.0920644
\(783\) 0 0
\(784\) −24.3013 −0.867903
\(785\) 0 0
\(786\) 0 0
\(787\) −38.4761 −1.37153 −0.685763 0.727825i \(-0.740531\pi\)
−0.685763 + 0.727825i \(0.740531\pi\)
\(788\) −10.9805 −0.391163
\(789\) 0 0
\(790\) 0 0
\(791\) −13.7633 −0.489367
\(792\) 0 0
\(793\) 47.4060 1.68344
\(794\) −2.77768 −0.0985760
\(795\) 0 0
\(796\) −11.8077 −0.418512
\(797\) 1.86739 0.0661464 0.0330732 0.999453i \(-0.489471\pi\)
0.0330732 + 0.999453i \(0.489471\pi\)
\(798\) 0 0
\(799\) 45.1063 1.59575
\(800\) 0 0
\(801\) 0 0
\(802\) 2.04372 0.0721662
\(803\) −33.4140 −1.17916
\(804\) 0 0
\(805\) 0 0
\(806\) 0.813293 0.0286470
\(807\) 0 0
\(808\) −0.402982 −0.0141768
\(809\) −35.1514 −1.23586 −0.617929 0.786234i \(-0.712028\pi\)
−0.617929 + 0.786234i \(0.712028\pi\)
\(810\) 0 0
\(811\) −6.59158 −0.231462 −0.115731 0.993281i \(-0.536921\pi\)
−0.115731 + 0.993281i \(0.536921\pi\)
\(812\) −6.34561 −0.222687
\(813\) 0 0
\(814\) −3.88483 −0.136163
\(815\) 0 0
\(816\) 0 0
\(817\) −21.5244 −0.753044
\(818\) −3.94949 −0.138091
\(819\) 0 0
\(820\) 0 0
\(821\) −5.25743 −0.183486 −0.0917428 0.995783i \(-0.529244\pi\)
−0.0917428 + 0.995783i \(0.529244\pi\)
\(822\) 0 0
\(823\) −22.1919 −0.773560 −0.386780 0.922172i \(-0.626413\pi\)
−0.386780 + 0.922172i \(0.626413\pi\)
\(824\) −0.440283 −0.0153380
\(825\) 0 0
\(826\) 1.59635 0.0555440
\(827\) 6.56256 0.228203 0.114101 0.993469i \(-0.463601\pi\)
0.114101 + 0.993469i \(0.463601\pi\)
\(828\) 0 0
\(829\) 25.0574 0.870279 0.435140 0.900363i \(-0.356699\pi\)
0.435140 + 0.900363i \(0.356699\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −34.4618 −1.19475
\(833\) −33.2395 −1.15168
\(834\) 0 0
\(835\) 0 0
\(836\) −46.4022 −1.60485
\(837\) 0 0
\(838\) 5.14165 0.177615
\(839\) 42.1485 1.45513 0.727564 0.686040i \(-0.240652\pi\)
0.727564 + 0.686040i \(0.240652\pi\)
\(840\) 0 0
\(841\) −15.0569 −0.519203
\(842\) −0.478344 −0.0164848
\(843\) 0 0
\(844\) 27.1612 0.934927
\(845\) 0 0
\(846\) 0 0
\(847\) −2.18172 −0.0749648
\(848\) 9.11117 0.312879
\(849\) 0 0
\(850\) 0 0
\(851\) 25.5095 0.874454
\(852\) 0 0
\(853\) 43.2473 1.48076 0.740379 0.672189i \(-0.234646\pi\)
0.740379 + 0.672189i \(0.234646\pi\)
\(854\) −1.25841 −0.0430620
\(855\) 0 0
\(856\) 6.74446 0.230521
\(857\) 39.2430 1.34052 0.670258 0.742128i \(-0.266183\pi\)
0.670258 + 0.742128i \(0.266183\pi\)
\(858\) 0 0
\(859\) 52.6092 1.79500 0.897501 0.441013i \(-0.145381\pi\)
0.897501 + 0.441013i \(0.145381\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.63611 0.123846
\(863\) 42.8744 1.45946 0.729731 0.683735i \(-0.239645\pi\)
0.729731 + 0.683735i \(0.239645\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −5.64023 −0.191663
\(867\) 0 0
\(868\) 2.12917 0.0722686
\(869\) −11.9498 −0.405371
\(870\) 0 0
\(871\) 15.3314 0.519484
\(872\) −1.37247 −0.0464778
\(873\) 0 0
\(874\) −3.08955 −0.104506
\(875\) 0 0
\(876\) 0 0
\(877\) −0.390314 −0.0131800 −0.00658999 0.999978i \(-0.502098\pi\)
−0.00658999 + 0.999978i \(0.502098\pi\)
\(878\) 4.80582 0.162188
\(879\) 0 0
\(880\) 0 0
\(881\) −44.7002 −1.50599 −0.752994 0.658028i \(-0.771391\pi\)
−0.752994 + 0.658028i \(0.771391\pi\)
\(882\) 0 0
\(883\) 32.7567 1.10235 0.551175 0.834390i \(-0.314180\pi\)
0.551175 + 0.834390i \(0.314180\pi\)
\(884\) −48.1384 −1.61907
\(885\) 0 0
\(886\) 4.09908 0.137711
\(887\) 15.7130 0.527591 0.263796 0.964579i \(-0.415026\pi\)
0.263796 + 0.964579i \(0.415026\pi\)
\(888\) 0 0
\(889\) 3.29461 0.110498
\(890\) 0 0
\(891\) 0 0
\(892\) −6.41833 −0.214902
\(893\) 54.1299 1.81139
\(894\) 0 0
\(895\) 0 0
\(896\) 3.79456 0.126767
\(897\) 0 0
\(898\) 1.44649 0.0482701
\(899\) −4.67839 −0.156033
\(900\) 0 0
\(901\) 12.4623 0.415180
\(902\) 1.32003 0.0439523
\(903\) 0 0
\(904\) −9.04256 −0.300751
\(905\) 0 0
\(906\) 0 0
\(907\) −1.50466 −0.0499613 −0.0249806 0.999688i \(-0.507952\pi\)
−0.0249806 + 0.999688i \(0.507952\pi\)
\(908\) 26.0952 0.865998
\(909\) 0 0
\(910\) 0 0
\(911\) −11.4509 −0.379386 −0.189693 0.981843i \(-0.560749\pi\)
−0.189693 + 0.981843i \(0.560749\pi\)
\(912\) 0 0
\(913\) 27.2046 0.900340
\(914\) 4.60422 0.152294
\(915\) 0 0
\(916\) −11.5590 −0.381920
\(917\) −10.6855 −0.352866
\(918\) 0 0
\(919\) −21.8001 −0.719119 −0.359559 0.933122i \(-0.617073\pi\)
−0.359559 + 0.933122i \(0.617073\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.0992632 0.00326906
\(923\) 19.8046 0.651877
\(924\) 0 0
\(925\) 0 0
\(926\) 0.307645 0.0101098
\(927\) 0 0
\(928\) −6.26415 −0.205631
\(929\) 30.7507 1.00890 0.504449 0.863442i \(-0.331696\pi\)
0.504449 + 0.863442i \(0.331696\pi\)
\(930\) 0 0
\(931\) −39.8891 −1.30731
\(932\) 18.4817 0.605387
\(933\) 0 0
\(934\) 0.549344 0.0179751
\(935\) 0 0
\(936\) 0 0
\(937\) −3.81060 −0.124487 −0.0622435 0.998061i \(-0.519826\pi\)
−0.0622435 + 0.998061i \(0.519826\pi\)
\(938\) −0.406978 −0.0132883
\(939\) 0 0
\(940\) 0 0
\(941\) 20.7043 0.674939 0.337470 0.941336i \(-0.390429\pi\)
0.337470 + 0.941336i \(0.390429\pi\)
\(942\) 0 0
\(943\) −8.66792 −0.282266
\(944\) −50.9297 −1.65762
\(945\) 0 0
\(946\) −1.76221 −0.0572944
\(947\) −15.6833 −0.509639 −0.254819 0.966989i \(-0.582016\pi\)
−0.254819 + 0.966989i \(0.582016\pi\)
\(948\) 0 0
\(949\) −41.5989 −1.35036
\(950\) 0 0
\(951\) 0 0
\(952\) 2.56867 0.0832509
\(953\) 7.75207 0.251114 0.125557 0.992086i \(-0.459928\pi\)
0.125557 + 0.992086i \(0.459928\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 33.9560 1.09822
\(957\) 0 0
\(958\) 3.11708 0.100708
\(959\) −10.1786 −0.328685
\(960\) 0 0
\(961\) −29.4302 −0.949363
\(962\) −4.83644 −0.155933
\(963\) 0 0
\(964\) −14.2965 −0.460459
\(965\) 0 0
\(966\) 0 0
\(967\) −48.4273 −1.55732 −0.778659 0.627447i \(-0.784100\pi\)
−0.778659 + 0.627447i \(0.784100\pi\)
\(968\) −1.43340 −0.0460712
\(969\) 0 0
\(970\) 0 0
\(971\) −25.3350 −0.813039 −0.406520 0.913642i \(-0.633258\pi\)
−0.406520 + 0.913642i \(0.633258\pi\)
\(972\) 0 0
\(973\) 4.00049 0.128250
\(974\) 4.02394 0.128936
\(975\) 0 0
\(976\) 40.1483 1.28511
\(977\) 1.16885 0.0373948 0.0186974 0.999825i \(-0.494048\pi\)
0.0186974 + 0.999825i \(0.494048\pi\)
\(978\) 0 0
\(979\) 57.0298 1.82268
\(980\) 0 0
\(981\) 0 0
\(982\) 1.99029 0.0635127
\(983\) 23.9943 0.765299 0.382649 0.923894i \(-0.375012\pi\)
0.382649 + 0.923894i \(0.375012\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.80782 −0.0894190
\(987\) 0 0
\(988\) −57.7686 −1.83786
\(989\) 11.5714 0.367950
\(990\) 0 0
\(991\) −36.6726 −1.16494 −0.582472 0.812851i \(-0.697914\pi\)
−0.582472 + 0.812851i \(0.697914\pi\)
\(992\) 2.10183 0.0667333
\(993\) 0 0
\(994\) −0.525722 −0.0166749
\(995\) 0 0
\(996\) 0 0
\(997\) −57.0203 −1.80585 −0.902925 0.429798i \(-0.858585\pi\)
−0.902925 + 0.429798i \(0.858585\pi\)
\(998\) 1.85390 0.0586842
\(999\) 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.1.4 6
3.2 odd 2 1875.2.a.j.1.3 6
5.4 even 2 5625.2.a.q.1.3 6
15.2 even 4 1875.2.b.f.1249.6 12
15.8 even 4 1875.2.b.f.1249.7 12
15.14 odd 2 1875.2.a.k.1.4 6
25.6 even 5 225.2.h.d.136.2 12
25.21 even 5 225.2.h.d.91.2 12
75.8 even 20 375.2.i.d.199.3 24
75.17 even 20 375.2.i.d.199.4 24
75.29 odd 10 375.2.g.c.76.2 12
75.44 odd 10 375.2.g.c.301.2 12
75.47 even 20 375.2.i.d.49.3 24
75.53 even 20 375.2.i.d.49.4 24
75.56 odd 10 75.2.g.c.61.2 yes 12
75.71 odd 10 75.2.g.c.16.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.16.2 12 75.71 odd 10
75.2.g.c.61.2 yes 12 75.56 odd 10
225.2.h.d.91.2 12 25.21 even 5
225.2.h.d.136.2 12 25.6 even 5
375.2.g.c.76.2 12 75.29 odd 10
375.2.g.c.301.2 12 75.44 odd 10
375.2.i.d.49.3 24 75.47 even 20
375.2.i.d.49.4 24 75.53 even 20
375.2.i.d.199.3 24 75.8 even 20
375.2.i.d.199.4 24 75.17 even 20
1875.2.a.j.1.3 6 3.2 odd 2
1875.2.a.k.1.4 6 15.14 odd 2
1875.2.b.f.1249.6 12 15.2 even 4
1875.2.b.f.1249.7 12 15.8 even 4
5625.2.a.p.1.4 6 1.1 even 1 trivial
5625.2.a.q.1.3 6 5.4 even 2