Properties

Label 5625.2.a.v.1.3
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5625,2,Mod(1,5625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.33620000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.936839\) 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)
 
\(f(q)\) \(=\) \(q-0.936839 q^{2} -1.12233 q^{4} -2.74037 q^{7} +2.92512 q^{8} +O(q^{10})\) \(q-0.936839 q^{2} -1.12233 q^{4} -2.74037 q^{7} +2.92512 q^{8} +0.291988 q^{11} +3.55634 q^{13} +2.56728 q^{14} -0.495700 q^{16} -4.37511 q^{17} +2.55634 q^{19} -0.273546 q^{22} +5.13149 q^{23} -3.33172 q^{26} +3.07561 q^{28} -9.40004 q^{29} -8.94362 q^{31} -5.38585 q^{32} +4.09877 q^{34} +0.735053 q^{37} -2.39488 q^{38} +11.2939 q^{41} -0.249982 q^{43} -0.327708 q^{44} -4.80737 q^{46} +2.88442 q^{47} +0.509614 q^{49} -3.99140 q^{52} -4.87769 q^{53} -8.01591 q^{56} +8.80632 q^{58} +5.10135 q^{59} -1.99140 q^{61} +8.37873 q^{62} +6.03708 q^{64} +13.7135 q^{67} +4.91033 q^{68} -12.4081 q^{71} -4.55038 q^{73} -0.688626 q^{74} -2.86907 q^{76} -0.800156 q^{77} +7.54669 q^{79} -10.5806 q^{82} +11.1622 q^{83} +0.234192 q^{86} +0.854102 q^{88} +7.99383 q^{89} -9.74568 q^{91} -5.75924 q^{92} -2.70224 q^{94} +16.5086 q^{97} -0.477426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 10 q^{13} - 8 q^{16} - 18 q^{19} + 30 q^{28} - 26 q^{31} - 20 q^{34} + 40 q^{43} - 30 q^{46} - 16 q^{49} - 40 q^{52} - 10 q^{58} - 24 q^{61} - 34 q^{64} + 40 q^{67} - 40 q^{73} - 44 q^{76} - 42 q^{79} - 60 q^{82} - 20 q^{88} - 40 q^{91} - 10 q^{94} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.936839 −0.662445 −0.331222 0.943553i \(-0.607461\pi\)
−0.331222 + 0.943553i \(0.607461\pi\)
\(3\) 0 0
\(4\) −1.12233 −0.561167
\(5\) 0 0
\(6\) 0 0
\(7\) −2.74037 −1.03576 −0.517881 0.855453i \(-0.673279\pi\)
−0.517881 + 0.855453i \(0.673279\pi\)
\(8\) 2.92512 1.03419
\(9\) 0 0
\(10\) 0 0
\(11\) 0.291988 0.0880378 0.0440189 0.999031i \(-0.485984\pi\)
0.0440189 + 0.999031i \(0.485984\pi\)
\(12\) 0 0
\(13\) 3.55634 0.986352 0.493176 0.869930i \(-0.335836\pi\)
0.493176 + 0.869930i \(0.335836\pi\)
\(14\) 2.56728 0.686135
\(15\) 0 0
\(16\) −0.495700 −0.123925
\(17\) −4.37511 −1.06112 −0.530560 0.847648i \(-0.678018\pi\)
−0.530560 + 0.847648i \(0.678018\pi\)
\(18\) 0 0
\(19\) 2.55634 0.586465 0.293232 0.956041i \(-0.405269\pi\)
0.293232 + 0.956041i \(0.405269\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.273546 −0.0583202
\(23\) 5.13149 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.33172 −0.653404
\(27\) 0 0
\(28\) 3.07561 0.581235
\(29\) −9.40004 −1.74554 −0.872772 0.488128i \(-0.837680\pi\)
−0.872772 + 0.488128i \(0.837680\pi\)
\(30\) 0 0
\(31\) −8.94362 −1.60632 −0.803161 0.595762i \(-0.796850\pi\)
−0.803161 + 0.595762i \(0.796850\pi\)
\(32\) −5.38585 −0.952093
\(33\) 0 0
\(34\) 4.09877 0.702933
\(35\) 0 0
\(36\) 0 0
\(37\) 0.735053 0.120842 0.0604210 0.998173i \(-0.480756\pi\)
0.0604210 + 0.998173i \(0.480756\pi\)
\(38\) −2.39488 −0.388501
\(39\) 0 0
\(40\) 0 0
\(41\) 11.2939 1.76381 0.881905 0.471427i \(-0.156261\pi\)
0.881905 + 0.471427i \(0.156261\pi\)
\(42\) 0 0
\(43\) −0.249982 −0.0381218 −0.0190609 0.999818i \(-0.506068\pi\)
−0.0190609 + 0.999818i \(0.506068\pi\)
\(44\) −0.327708 −0.0494039
\(45\) 0 0
\(46\) −4.80737 −0.708809
\(47\) 2.88442 0.420736 0.210368 0.977622i \(-0.432534\pi\)
0.210368 + 0.977622i \(0.432534\pi\)
\(48\) 0 0
\(49\) 0.509614 0.0728020
\(50\) 0 0
\(51\) 0 0
\(52\) −3.99140 −0.553508
\(53\) −4.87769 −0.670002 −0.335001 0.942218i \(-0.608737\pi\)
−0.335001 + 0.942218i \(0.608737\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −8.01591 −1.07117
\(57\) 0 0
\(58\) 8.80632 1.15633
\(59\) 5.10135 0.664140 0.332070 0.943255i \(-0.392253\pi\)
0.332070 + 0.943255i \(0.392253\pi\)
\(60\) 0 0
\(61\) −1.99140 −0.254973 −0.127486 0.991840i \(-0.540691\pi\)
−0.127486 + 0.991840i \(0.540691\pi\)
\(62\) 8.37873 1.06410
\(63\) 0 0
\(64\) 6.03708 0.754635
\(65\) 0 0
\(66\) 0 0
\(67\) 13.7135 1.67537 0.837686 0.546151i \(-0.183908\pi\)
0.837686 + 0.546151i \(0.183908\pi\)
\(68\) 4.91033 0.595465
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4081 −1.47257 −0.736286 0.676670i \(-0.763422\pi\)
−0.736286 + 0.676670i \(0.763422\pi\)
\(72\) 0 0
\(73\) −4.55038 −0.532581 −0.266291 0.963893i \(-0.585798\pi\)
−0.266291 + 0.963893i \(0.585798\pi\)
\(74\) −0.688626 −0.0800511
\(75\) 0 0
\(76\) −2.86907 −0.329105
\(77\) −0.800156 −0.0911862
\(78\) 0 0
\(79\) 7.54669 0.849069 0.424534 0.905412i \(-0.360438\pi\)
0.424534 + 0.905412i \(0.360438\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.5806 −1.16843
\(83\) 11.1622 1.22521 0.612605 0.790389i \(-0.290122\pi\)
0.612605 + 0.790389i \(0.290122\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.234192 0.0252536
\(87\) 0 0
\(88\) 0.854102 0.0910476
\(89\) 7.99383 0.847345 0.423672 0.905816i \(-0.360741\pi\)
0.423672 + 0.905816i \(0.360741\pi\)
\(90\) 0 0
\(91\) −9.74568 −1.02163
\(92\) −5.75924 −0.600442
\(93\) 0 0
\(94\) −2.70224 −0.278715
\(95\) 0 0
\(96\) 0 0
\(97\) 16.5086 1.67619 0.838095 0.545524i \(-0.183669\pi\)
0.838095 + 0.545524i \(0.183669\pi\)
\(98\) −0.477426 −0.0482273
\(99\) 0 0
\(100\) 0 0
\(101\) 9.81876 0.977003 0.488501 0.872563i \(-0.337544\pi\)
0.488501 + 0.872563i \(0.337544\pi\)
\(102\) 0 0
\(103\) −4.92642 −0.485415 −0.242707 0.970100i \(-0.578036\pi\)
−0.242707 + 0.970100i \(0.578036\pi\)
\(104\) 10.4027 1.02007
\(105\) 0 0
\(106\) 4.56960 0.443839
\(107\) 4.22786 0.408722 0.204361 0.978896i \(-0.434488\pi\)
0.204361 + 0.978896i \(0.434488\pi\)
\(108\) 0 0
\(109\) −10.5275 −1.00835 −0.504174 0.863602i \(-0.668203\pi\)
−0.504174 + 0.863602i \(0.668203\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.35840 0.128357
\(113\) 11.6955 1.10022 0.550111 0.835092i \(-0.314586\pi\)
0.550111 + 0.835092i \(0.314586\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 10.5500 0.979541
\(117\) 0 0
\(118\) −4.77915 −0.439956
\(119\) 11.9894 1.09907
\(120\) 0 0
\(121\) −10.9147 −0.992249
\(122\) 1.86562 0.168905
\(123\) 0 0
\(124\) 10.0377 0.901414
\(125\) 0 0
\(126\) 0 0
\(127\) −15.7072 −1.39378 −0.696892 0.717176i \(-0.745434\pi\)
−0.696892 + 0.717176i \(0.745434\pi\)
\(128\) 5.11594 0.452190
\(129\) 0 0
\(130\) 0 0
\(131\) 8.84928 0.773165 0.386582 0.922255i \(-0.373655\pi\)
0.386582 + 0.922255i \(0.373655\pi\)
\(132\) 0 0
\(133\) −7.00531 −0.607438
\(134\) −12.8474 −1.10984
\(135\) 0 0
\(136\) −12.7977 −1.09740
\(137\) −8.63561 −0.737790 −0.368895 0.929471i \(-0.620264\pi\)
−0.368895 + 0.929471i \(0.620264\pi\)
\(138\) 0 0
\(139\) −8.01720 −0.680010 −0.340005 0.940424i \(-0.610429\pi\)
−0.340005 + 0.940424i \(0.610429\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 11.6244 0.975498
\(143\) 1.03841 0.0868363
\(144\) 0 0
\(145\) 0 0
\(146\) 4.26297 0.352806
\(147\) 0 0
\(148\) −0.824975 −0.0678125
\(149\) −12.8790 −1.05509 −0.527545 0.849527i \(-0.676887\pi\)
−0.527545 + 0.849527i \(0.676887\pi\)
\(150\) 0 0
\(151\) −23.3541 −1.90053 −0.950263 0.311447i \(-0.899186\pi\)
−0.950263 + 0.311447i \(0.899186\pi\)
\(152\) 7.47761 0.606514
\(153\) 0 0
\(154\) 0.749617 0.0604058
\(155\) 0 0
\(156\) 0 0
\(157\) −9.02953 −0.720635 −0.360317 0.932830i \(-0.617332\pi\)
−0.360317 + 0.932830i \(0.617332\pi\)
\(158\) −7.07003 −0.562461
\(159\) 0 0
\(160\) 0 0
\(161\) −14.0622 −1.10825
\(162\) 0 0
\(163\) −9.22970 −0.722926 −0.361463 0.932386i \(-0.617723\pi\)
−0.361463 + 0.932386i \(0.617723\pi\)
\(164\) −12.6755 −0.989792
\(165\) 0 0
\(166\) −10.4572 −0.811634
\(167\) 20.3209 1.57248 0.786239 0.617922i \(-0.212025\pi\)
0.786239 + 0.617922i \(0.212025\pi\)
\(168\) 0 0
\(169\) −0.352437 −0.0271105
\(170\) 0 0
\(171\) 0 0
\(172\) 0.280563 0.0213927
\(173\) 15.9203 1.21040 0.605199 0.796074i \(-0.293094\pi\)
0.605199 + 0.796074i \(0.293094\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.144739 −0.0109101
\(177\) 0 0
\(178\) −7.48893 −0.561319
\(179\) 5.02991 0.375953 0.187977 0.982173i \(-0.439807\pi\)
0.187977 + 0.982173i \(0.439807\pi\)
\(180\) 0 0
\(181\) −2.83954 −0.211061 −0.105531 0.994416i \(-0.533654\pi\)
−0.105531 + 0.994416i \(0.533654\pi\)
\(182\) 9.13013 0.676770
\(183\) 0 0
\(184\) 15.0102 1.10657
\(185\) 0 0
\(186\) 0 0
\(187\) −1.27748 −0.0934186
\(188\) −3.23729 −0.236103
\(189\) 0 0
\(190\) 0 0
\(191\) −20.7809 −1.50365 −0.751826 0.659362i \(-0.770827\pi\)
−0.751826 + 0.659362i \(0.770827\pi\)
\(192\) 0 0
\(193\) 8.46086 0.609026 0.304513 0.952508i \(-0.401506\pi\)
0.304513 + 0.952508i \(0.401506\pi\)
\(194\) −15.4659 −1.11038
\(195\) 0 0
\(196\) −0.571957 −0.0408541
\(197\) −6.30468 −0.449190 −0.224595 0.974452i \(-0.572106\pi\)
−0.224595 + 0.974452i \(0.572106\pi\)
\(198\) 0 0
\(199\) −19.4190 −1.37658 −0.688290 0.725436i \(-0.741638\pi\)
−0.688290 + 0.725436i \(0.741638\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −9.19859 −0.647211
\(203\) 25.7596 1.80797
\(204\) 0 0
\(205\) 0 0
\(206\) 4.61526 0.321561
\(207\) 0 0
\(208\) −1.76288 −0.122234
\(209\) 0.746422 0.0516311
\(210\) 0 0
\(211\) −17.2678 −1.18877 −0.594383 0.804182i \(-0.702604\pi\)
−0.594383 + 0.804182i \(0.702604\pi\)
\(212\) 5.47439 0.375983
\(213\) 0 0
\(214\) −3.96082 −0.270756
\(215\) 0 0
\(216\) 0 0
\(217\) 24.5088 1.66377
\(218\) 9.86253 0.667975
\(219\) 0 0
\(220\) 0 0
\(221\) −15.5594 −1.04664
\(222\) 0 0
\(223\) −1.42443 −0.0953869 −0.0476935 0.998862i \(-0.515187\pi\)
−0.0476935 + 0.998862i \(0.515187\pi\)
\(224\) 14.7592 0.986142
\(225\) 0 0
\(226\) −10.9568 −0.728836
\(227\) −9.26834 −0.615161 −0.307581 0.951522i \(-0.599519\pi\)
−0.307581 + 0.951522i \(0.599519\pi\)
\(228\) 0 0
\(229\) −16.8235 −1.11173 −0.555865 0.831273i \(-0.687613\pi\)
−0.555865 + 0.831273i \(0.687613\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −27.4963 −1.80522
\(233\) −9.87404 −0.646870 −0.323435 0.946250i \(-0.604838\pi\)
−0.323435 + 0.946250i \(0.604838\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −5.72542 −0.372693
\(237\) 0 0
\(238\) −11.2321 −0.728071
\(239\) −20.9884 −1.35763 −0.678815 0.734310i \(-0.737506\pi\)
−0.678815 + 0.734310i \(0.737506\pi\)
\(240\) 0 0
\(241\) 0.599134 0.0385936 0.0192968 0.999814i \(-0.493857\pi\)
0.0192968 + 0.999814i \(0.493857\pi\)
\(242\) 10.2254 0.657311
\(243\) 0 0
\(244\) 2.23502 0.143082
\(245\) 0 0
\(246\) 0 0
\(247\) 9.09122 0.578461
\(248\) −26.1612 −1.66124
\(249\) 0 0
\(250\) 0 0
\(251\) 22.8068 1.43955 0.719776 0.694207i \(-0.244245\pi\)
0.719776 + 0.694207i \(0.244245\pi\)
\(252\) 0 0
\(253\) 1.49833 0.0941995
\(254\) 14.7151 0.923305
\(255\) 0 0
\(256\) −16.8670 −1.05419
\(257\) 4.10576 0.256110 0.128055 0.991767i \(-0.459127\pi\)
0.128055 + 0.991767i \(0.459127\pi\)
\(258\) 0 0
\(259\) −2.01432 −0.125163
\(260\) 0 0
\(261\) 0 0
\(262\) −8.29034 −0.512179
\(263\) 12.8759 0.793964 0.396982 0.917826i \(-0.370058\pi\)
0.396982 + 0.917826i \(0.370058\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.56285 0.402394
\(267\) 0 0
\(268\) −15.3911 −0.940164
\(269\) −7.36300 −0.448930 −0.224465 0.974482i \(-0.572063\pi\)
−0.224465 + 0.974482i \(0.572063\pi\)
\(270\) 0 0
\(271\) −1.31070 −0.0796191 −0.0398096 0.999207i \(-0.512675\pi\)
−0.0398096 + 0.999207i \(0.512675\pi\)
\(272\) 2.16874 0.131499
\(273\) 0 0
\(274\) 8.09017 0.488745
\(275\) 0 0
\(276\) 0 0
\(277\) −11.4437 −0.687583 −0.343791 0.939046i \(-0.611711\pi\)
−0.343791 + 0.939046i \(0.611711\pi\)
\(278\) 7.51082 0.450469
\(279\) 0 0
\(280\) 0 0
\(281\) 3.14013 0.187324 0.0936621 0.995604i \(-0.470143\pi\)
0.0936621 + 0.995604i \(0.470143\pi\)
\(282\) 0 0
\(283\) −18.2903 −1.08725 −0.543624 0.839329i \(-0.682948\pi\)
−0.543624 + 0.839329i \(0.682948\pi\)
\(284\) 13.9260 0.826359
\(285\) 0 0
\(286\) −0.972823 −0.0575242
\(287\) −30.9494 −1.82689
\(288\) 0 0
\(289\) 2.14156 0.125974
\(290\) 0 0
\(291\) 0 0
\(292\) 5.10704 0.298867
\(293\) −12.2342 −0.714727 −0.357364 0.933965i \(-0.616324\pi\)
−0.357364 + 0.933965i \(0.616324\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.15012 0.124973
\(297\) 0 0
\(298\) 12.0656 0.698939
\(299\) 18.2493 1.05539
\(300\) 0 0
\(301\) 0.685041 0.0394851
\(302\) 21.8790 1.25899
\(303\) 0 0
\(304\) −1.26718 −0.0726777
\(305\) 0 0
\(306\) 0 0
\(307\) −34.8868 −1.99110 −0.995549 0.0942495i \(-0.969955\pi\)
−0.995549 + 0.0942495i \(0.969955\pi\)
\(308\) 0.898042 0.0511707
\(309\) 0 0
\(310\) 0 0
\(311\) 9.56937 0.542629 0.271315 0.962491i \(-0.412542\pi\)
0.271315 + 0.962491i \(0.412542\pi\)
\(312\) 0 0
\(313\) 28.6193 1.61766 0.808828 0.588045i \(-0.200102\pi\)
0.808828 + 0.588045i \(0.200102\pi\)
\(314\) 8.45921 0.477381
\(315\) 0 0
\(316\) −8.46990 −0.476469
\(317\) −12.9343 −0.726462 −0.363231 0.931699i \(-0.618326\pi\)
−0.363231 + 0.931699i \(0.618326\pi\)
\(318\) 0 0
\(319\) −2.74470 −0.153674
\(320\) 0 0
\(321\) 0 0
\(322\) 13.1740 0.734157
\(323\) −11.1843 −0.622309
\(324\) 0 0
\(325\) 0 0
\(326\) 8.64674 0.478899
\(327\) 0 0
\(328\) 33.0360 1.82411
\(329\) −7.90438 −0.435783
\(330\) 0 0
\(331\) −14.1230 −0.776269 −0.388135 0.921603i \(-0.626880\pi\)
−0.388135 + 0.921603i \(0.626880\pi\)
\(332\) −12.5277 −0.687547
\(333\) 0 0
\(334\) −19.0374 −1.04168
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0852 1.42095 0.710475 0.703722i \(-0.248480\pi\)
0.710475 + 0.703722i \(0.248480\pi\)
\(338\) 0.330176 0.0179592
\(339\) 0 0
\(340\) 0 0
\(341\) −2.61143 −0.141417
\(342\) 0 0
\(343\) 17.7860 0.960356
\(344\) −0.731227 −0.0394251
\(345\) 0 0
\(346\) −14.9147 −0.801822
\(347\) −33.3393 −1.78975 −0.894874 0.446318i \(-0.852735\pi\)
−0.894874 + 0.446318i \(0.852735\pi\)
\(348\) 0 0
\(349\) −2.42476 −0.129794 −0.0648972 0.997892i \(-0.520672\pi\)
−0.0648972 + 0.997892i \(0.520672\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.57261 −0.0838202
\(353\) −25.6561 −1.36554 −0.682768 0.730635i \(-0.739224\pi\)
−0.682768 + 0.730635i \(0.739224\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.97175 −0.475502
\(357\) 0 0
\(358\) −4.71222 −0.249048
\(359\) 0.933761 0.0492820 0.0246410 0.999696i \(-0.492156\pi\)
0.0246410 + 0.999696i \(0.492156\pi\)
\(360\) 0 0
\(361\) −12.4651 −0.656059
\(362\) 2.66019 0.139816
\(363\) 0 0
\(364\) 10.9379 0.573302
\(365\) 0 0
\(366\) 0 0
\(367\) 18.1358 0.946683 0.473342 0.880879i \(-0.343048\pi\)
0.473342 + 0.880879i \(0.343048\pi\)
\(368\) −2.54368 −0.132599
\(369\) 0 0
\(370\) 0 0
\(371\) 13.3667 0.693962
\(372\) 0 0
\(373\) 23.8975 1.23736 0.618682 0.785641i \(-0.287667\pi\)
0.618682 + 0.785641i \(0.287667\pi\)
\(374\) 1.19679 0.0618847
\(375\) 0 0
\(376\) 8.43729 0.435120
\(377\) −33.4298 −1.72172
\(378\) 0 0
\(379\) 0.812689 0.0417450 0.0208725 0.999782i \(-0.493356\pi\)
0.0208725 + 0.999782i \(0.493356\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 19.4683 0.996086
\(383\) −23.3827 −1.19480 −0.597400 0.801943i \(-0.703800\pi\)
−0.597400 + 0.801943i \(0.703800\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.92646 −0.403446
\(387\) 0 0
\(388\) −18.5281 −0.940622
\(389\) 11.5129 0.583727 0.291863 0.956460i \(-0.405725\pi\)
0.291863 + 0.956460i \(0.405725\pi\)
\(390\) 0 0
\(391\) −22.4508 −1.13539
\(392\) 1.49068 0.0752909
\(393\) 0 0
\(394\) 5.90647 0.297563
\(395\) 0 0
\(396\) 0 0
\(397\) −4.78381 −0.240093 −0.120046 0.992768i \(-0.538304\pi\)
−0.120046 + 0.992768i \(0.538304\pi\)
\(398\) 18.1925 0.911908
\(399\) 0 0
\(400\) 0 0
\(401\) −29.7663 −1.48646 −0.743228 0.669038i \(-0.766706\pi\)
−0.743228 + 0.669038i \(0.766706\pi\)
\(402\) 0 0
\(403\) −31.8066 −1.58440
\(404\) −11.0199 −0.548261
\(405\) 0 0
\(406\) −24.1326 −1.19768
\(407\) 0.214627 0.0106387
\(408\) 0 0
\(409\) −5.98832 −0.296103 −0.148052 0.988980i \(-0.547300\pi\)
−0.148052 + 0.988980i \(0.547300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5.52909 0.272399
\(413\) −13.9796 −0.687890
\(414\) 0 0
\(415\) 0 0
\(416\) −19.1539 −0.939099
\(417\) 0 0
\(418\) −0.699277 −0.0342028
\(419\) 4.25686 0.207961 0.103981 0.994579i \(-0.466842\pi\)
0.103981 + 0.994579i \(0.466842\pi\)
\(420\) 0 0
\(421\) −23.2688 −1.13405 −0.567026 0.823700i \(-0.691906\pi\)
−0.567026 + 0.823700i \(0.691906\pi\)
\(422\) 16.1772 0.787492
\(423\) 0 0
\(424\) −14.2678 −0.692907
\(425\) 0 0
\(426\) 0 0
\(427\) 5.45717 0.264091
\(428\) −4.74507 −0.229361
\(429\) 0 0
\(430\) 0 0
\(431\) 12.5171 0.602929 0.301465 0.953477i \(-0.402525\pi\)
0.301465 + 0.953477i \(0.402525\pi\)
\(432\) 0 0
\(433\) 18.9565 0.910990 0.455495 0.890238i \(-0.349462\pi\)
0.455495 + 0.890238i \(0.349462\pi\)
\(434\) −22.9608 −1.10215
\(435\) 0 0
\(436\) 11.8153 0.565851
\(437\) 13.1178 0.627511
\(438\) 0 0
\(439\) −7.29914 −0.348369 −0.174184 0.984713i \(-0.555729\pi\)
−0.174184 + 0.984713i \(0.555729\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 14.5766 0.693339
\(443\) 5.75210 0.273291 0.136645 0.990620i \(-0.456368\pi\)
0.136645 + 0.990620i \(0.456368\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.33446 0.0631886
\(447\) 0 0
\(448\) −16.5438 −0.781621
\(449\) −4.75061 −0.224195 −0.112098 0.993697i \(-0.535757\pi\)
−0.112098 + 0.993697i \(0.535757\pi\)
\(450\) 0 0
\(451\) 3.29769 0.155282
\(452\) −13.1263 −0.617408
\(453\) 0 0
\(454\) 8.68294 0.407510
\(455\) 0 0
\(456\) 0 0
\(457\) −4.72678 −0.221110 −0.110555 0.993870i \(-0.535263\pi\)
−0.110555 + 0.993870i \(0.535263\pi\)
\(458\) 15.7609 0.736460
\(459\) 0 0
\(460\) 0 0
\(461\) 7.71028 0.359103 0.179552 0.983749i \(-0.442535\pi\)
0.179552 + 0.983749i \(0.442535\pi\)
\(462\) 0 0
\(463\) 3.82956 0.177975 0.0889874 0.996033i \(-0.471637\pi\)
0.0889874 + 0.996033i \(0.471637\pi\)
\(464\) 4.65961 0.216317
\(465\) 0 0
\(466\) 9.25038 0.428516
\(467\) −19.8882 −0.920318 −0.460159 0.887837i \(-0.652208\pi\)
−0.460159 + 0.887837i \(0.652208\pi\)
\(468\) 0 0
\(469\) −37.5801 −1.73529
\(470\) 0 0
\(471\) 0 0
\(472\) 14.9221 0.686845
\(473\) −0.0729917 −0.00335616
\(474\) 0 0
\(475\) 0 0
\(476\) −13.4561 −0.616760
\(477\) 0 0
\(478\) 19.6628 0.899354
\(479\) −27.8212 −1.27118 −0.635591 0.772026i \(-0.719244\pi\)
−0.635591 + 0.772026i \(0.719244\pi\)
\(480\) 0 0
\(481\) 2.61410 0.119193
\(482\) −0.561292 −0.0255661
\(483\) 0 0
\(484\) 12.2500 0.556817
\(485\) 0 0
\(486\) 0 0
\(487\) −31.7141 −1.43710 −0.718551 0.695474i \(-0.755195\pi\)
−0.718551 + 0.695474i \(0.755195\pi\)
\(488\) −5.82509 −0.263690
\(489\) 0 0
\(490\) 0 0
\(491\) −4.90133 −0.221194 −0.110597 0.993865i \(-0.535276\pi\)
−0.110597 + 0.993865i \(0.535276\pi\)
\(492\) 0 0
\(493\) 41.1262 1.85223
\(494\) −8.51701 −0.383198
\(495\) 0 0
\(496\) 4.43336 0.199064
\(497\) 34.0028 1.52523
\(498\) 0 0
\(499\) 3.13604 0.140389 0.0701943 0.997533i \(-0.477638\pi\)
0.0701943 + 0.997533i \(0.477638\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −21.3663 −0.953624
\(503\) −19.2987 −0.860485 −0.430243 0.902713i \(-0.641572\pi\)
−0.430243 + 0.902713i \(0.641572\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.40370 −0.0624020
\(507\) 0 0
\(508\) 17.6287 0.782145
\(509\) 41.7597 1.85097 0.925484 0.378787i \(-0.123659\pi\)
0.925484 + 0.378787i \(0.123659\pi\)
\(510\) 0 0
\(511\) 12.4697 0.551627
\(512\) 5.56974 0.246150
\(513\) 0 0
\(514\) −3.84644 −0.169659
\(515\) 0 0
\(516\) 0 0
\(517\) 0.842218 0.0370407
\(518\) 1.88709 0.0829139
\(519\) 0 0
\(520\) 0 0
\(521\) 34.0368 1.49118 0.745589 0.666406i \(-0.232168\pi\)
0.745589 + 0.666406i \(0.232168\pi\)
\(522\) 0 0
\(523\) 8.00170 0.349890 0.174945 0.984578i \(-0.444025\pi\)
0.174945 + 0.984578i \(0.444025\pi\)
\(524\) −9.93184 −0.433874
\(525\) 0 0
\(526\) −12.0627 −0.525957
\(527\) 39.1293 1.70450
\(528\) 0 0
\(529\) 3.33216 0.144876
\(530\) 0 0
\(531\) 0 0
\(532\) 7.86230 0.340874
\(533\) 40.1649 1.73974
\(534\) 0 0
\(535\) 0 0
\(536\) 40.1137 1.73265
\(537\) 0 0
\(538\) 6.89795 0.297392
\(539\) 0.148801 0.00640933
\(540\) 0 0
\(541\) 9.69559 0.416846 0.208423 0.978039i \(-0.433167\pi\)
0.208423 + 0.978039i \(0.433167\pi\)
\(542\) 1.22791 0.0527433
\(543\) 0 0
\(544\) 23.5637 1.01028
\(545\) 0 0
\(546\) 0 0
\(547\) 14.5504 0.622129 0.311065 0.950389i \(-0.399314\pi\)
0.311065 + 0.950389i \(0.399314\pi\)
\(548\) 9.69203 0.414023
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0297 −1.02370
\(552\) 0 0
\(553\) −20.6807 −0.879433
\(554\) 10.7209 0.455486
\(555\) 0 0
\(556\) 8.99797 0.381599
\(557\) −17.8610 −0.756795 −0.378397 0.925643i \(-0.623525\pi\)
−0.378397 + 0.925643i \(0.623525\pi\)
\(558\) 0 0
\(559\) −0.889020 −0.0376015
\(560\) 0 0
\(561\) 0 0
\(562\) −2.94179 −0.124092
\(563\) −21.6519 −0.912519 −0.456260 0.889847i \(-0.650811\pi\)
−0.456260 + 0.889847i \(0.650811\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 17.1351 0.720242
\(567\) 0 0
\(568\) −36.2953 −1.52292
\(569\) 32.7634 1.37351 0.686757 0.726887i \(-0.259034\pi\)
0.686757 + 0.726887i \(0.259034\pi\)
\(570\) 0 0
\(571\) −3.39726 −0.142171 −0.0710854 0.997470i \(-0.522646\pi\)
−0.0710854 + 0.997470i \(0.522646\pi\)
\(572\) −1.16544 −0.0487296
\(573\) 0 0
\(574\) 28.9946 1.21021
\(575\) 0 0
\(576\) 0 0
\(577\) −12.5670 −0.523170 −0.261585 0.965180i \(-0.584245\pi\)
−0.261585 + 0.965180i \(0.584245\pi\)
\(578\) −2.00630 −0.0834510
\(579\) 0 0
\(580\) 0 0
\(581\) −30.5885 −1.26902
\(582\) 0 0
\(583\) −1.42423 −0.0589855
\(584\) −13.3104 −0.550789
\(585\) 0 0
\(586\) 11.4614 0.473467
\(587\) 36.3462 1.50017 0.750084 0.661343i \(-0.230013\pi\)
0.750084 + 0.661343i \(0.230013\pi\)
\(588\) 0 0
\(589\) −22.8629 −0.942051
\(590\) 0 0
\(591\) 0 0
\(592\) −0.364366 −0.0149754
\(593\) −31.7555 −1.30404 −0.652022 0.758200i \(-0.726079\pi\)
−0.652022 + 0.758200i \(0.726079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.4545 0.592081
\(597\) 0 0
\(598\) −17.0967 −0.699135
\(599\) −23.6306 −0.965519 −0.482759 0.875753i \(-0.660365\pi\)
−0.482759 + 0.875753i \(0.660365\pi\)
\(600\) 0 0
\(601\) 12.9840 0.529628 0.264814 0.964300i \(-0.414689\pi\)
0.264814 + 0.964300i \(0.414689\pi\)
\(602\) −0.641773 −0.0261567
\(603\) 0 0
\(604\) 26.2110 1.06651
\(605\) 0 0
\(606\) 0 0
\(607\) −9.23404 −0.374798 −0.187399 0.982284i \(-0.560006\pi\)
−0.187399 + 0.982284i \(0.560006\pi\)
\(608\) −13.7681 −0.558369
\(609\) 0 0
\(610\) 0 0
\(611\) 10.2580 0.414994
\(612\) 0 0
\(613\) −16.4276 −0.663506 −0.331753 0.943366i \(-0.607640\pi\)
−0.331753 + 0.943366i \(0.607640\pi\)
\(614\) 32.6833 1.31899
\(615\) 0 0
\(616\) −2.34055 −0.0943036
\(617\) −7.92430 −0.319020 −0.159510 0.987196i \(-0.550991\pi\)
−0.159510 + 0.987196i \(0.550991\pi\)
\(618\) 0 0
\(619\) 7.20155 0.289455 0.144727 0.989472i \(-0.453769\pi\)
0.144727 + 0.989472i \(0.453769\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.96495 −0.359462
\(623\) −21.9060 −0.877647
\(624\) 0 0
\(625\) 0 0
\(626\) −26.8116 −1.07161
\(627\) 0 0
\(628\) 10.1341 0.404396
\(629\) −3.21594 −0.128228
\(630\) 0 0
\(631\) 24.2420 0.965057 0.482529 0.875880i \(-0.339718\pi\)
0.482529 + 0.875880i \(0.339718\pi\)
\(632\) 22.0750 0.878096
\(633\) 0 0
\(634\) 12.1173 0.481241
\(635\) 0 0
\(636\) 0 0
\(637\) 1.81236 0.0718084
\(638\) 2.57134 0.101801
\(639\) 0 0
\(640\) 0 0
\(641\) 12.6808 0.500863 0.250432 0.968134i \(-0.419427\pi\)
0.250432 + 0.968134i \(0.419427\pi\)
\(642\) 0 0
\(643\) −25.2886 −0.997287 −0.498643 0.866807i \(-0.666168\pi\)
−0.498643 + 0.866807i \(0.666168\pi\)
\(644\) 15.7824 0.621915
\(645\) 0 0
\(646\) 10.4779 0.412246
\(647\) −9.57188 −0.376309 −0.188155 0.982139i \(-0.560251\pi\)
−0.188155 + 0.982139i \(0.560251\pi\)
\(648\) 0 0
\(649\) 1.48954 0.0584694
\(650\) 0 0
\(651\) 0 0
\(652\) 10.3588 0.405682
\(653\) −12.8318 −0.502147 −0.251074 0.967968i \(-0.580784\pi\)
−0.251074 + 0.967968i \(0.580784\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −5.59839 −0.218580
\(657\) 0 0
\(658\) 7.40513 0.288682
\(659\) −29.7085 −1.15728 −0.578639 0.815584i \(-0.696416\pi\)
−0.578639 + 0.815584i \(0.696416\pi\)
\(660\) 0 0
\(661\) −22.9708 −0.893461 −0.446730 0.894669i \(-0.647412\pi\)
−0.446730 + 0.894669i \(0.647412\pi\)
\(662\) 13.2310 0.514236
\(663\) 0 0
\(664\) 32.6508 1.26710
\(665\) 0 0
\(666\) 0 0
\(667\) −48.2362 −1.86771
\(668\) −22.8068 −0.882423
\(669\) 0 0
\(670\) 0 0
\(671\) −0.581466 −0.0224472
\(672\) 0 0
\(673\) −15.6190 −0.602069 −0.301034 0.953613i \(-0.597332\pi\)
−0.301034 + 0.953613i \(0.597332\pi\)
\(674\) −24.4376 −0.941301
\(675\) 0 0
\(676\) 0.395552 0.0152135
\(677\) −0.209675 −0.00805845 −0.00402923 0.999992i \(-0.501283\pi\)
−0.00402923 + 0.999992i \(0.501283\pi\)
\(678\) 0 0
\(679\) −45.2395 −1.73613
\(680\) 0 0
\(681\) 0 0
\(682\) 2.44649 0.0936810
\(683\) −25.7192 −0.984119 −0.492059 0.870562i \(-0.663756\pi\)
−0.492059 + 0.870562i \(0.663756\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −16.6627 −0.636183
\(687\) 0 0
\(688\) 0.123916 0.00472425
\(689\) −17.3467 −0.660857
\(690\) 0 0
\(691\) −26.7676 −1.01829 −0.509144 0.860682i \(-0.670038\pi\)
−0.509144 + 0.860682i \(0.670038\pi\)
\(692\) −17.8679 −0.679235
\(693\) 0 0
\(694\) 31.2336 1.18561
\(695\) 0 0
\(696\) 0 0
\(697\) −49.4120 −1.87161
\(698\) 2.27161 0.0859816
\(699\) 0 0
\(700\) 0 0
\(701\) −19.6071 −0.740551 −0.370275 0.928922i \(-0.620737\pi\)
−0.370275 + 0.928922i \(0.620737\pi\)
\(702\) 0 0
\(703\) 1.87905 0.0708696
\(704\) 1.76276 0.0664364
\(705\) 0 0
\(706\) 24.0356 0.904593
\(707\) −26.9070 −1.01194
\(708\) 0 0
\(709\) 23.2844 0.874463 0.437232 0.899349i \(-0.355959\pi\)
0.437232 + 0.899349i \(0.355959\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 23.3829 0.876313
\(713\) −45.8941 −1.71875
\(714\) 0 0
\(715\) 0 0
\(716\) −5.64524 −0.210973
\(717\) 0 0
\(718\) −0.874784 −0.0326466
\(719\) −16.2906 −0.607537 −0.303768 0.952746i \(-0.598245\pi\)
−0.303768 + 0.952746i \(0.598245\pi\)
\(720\) 0 0
\(721\) 13.5002 0.502774
\(722\) 11.6778 0.434603
\(723\) 0 0
\(724\) 3.18691 0.118441
\(725\) 0 0
\(726\) 0 0
\(727\) 26.0407 0.965797 0.482899 0.875676i \(-0.339584\pi\)
0.482899 + 0.875676i \(0.339584\pi\)
\(728\) −28.5073 −1.05655
\(729\) 0 0
\(730\) 0 0
\(731\) 1.09370 0.0404518
\(732\) 0 0
\(733\) −24.1779 −0.893031 −0.446516 0.894776i \(-0.647335\pi\)
−0.446516 + 0.894776i \(0.647335\pi\)
\(734\) −16.9904 −0.627126
\(735\) 0 0
\(736\) −27.6374 −1.01873
\(737\) 4.00419 0.147496
\(738\) 0 0
\(739\) −10.7908 −0.396947 −0.198473 0.980106i \(-0.563598\pi\)
−0.198473 + 0.980106i \(0.563598\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.5224 −0.459712
\(743\) 23.8051 0.873325 0.436662 0.899625i \(-0.356160\pi\)
0.436662 + 0.899625i \(0.356160\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −22.3881 −0.819686
\(747\) 0 0
\(748\) 1.43376 0.0524234
\(749\) −11.5859 −0.423339
\(750\) 0 0
\(751\) −33.8662 −1.23580 −0.617898 0.786258i \(-0.712016\pi\)
−0.617898 + 0.786258i \(0.712016\pi\)
\(752\) −1.42981 −0.0521398
\(753\) 0 0
\(754\) 31.3183 1.14054
\(755\) 0 0
\(756\) 0 0
\(757\) 26.4066 0.959763 0.479881 0.877333i \(-0.340680\pi\)
0.479881 + 0.877333i \(0.340680\pi\)
\(758\) −0.761359 −0.0276538
\(759\) 0 0
\(760\) 0 0
\(761\) 20.0385 0.726396 0.363198 0.931712i \(-0.381685\pi\)
0.363198 + 0.931712i \(0.381685\pi\)
\(762\) 0 0
\(763\) 28.8491 1.04441
\(764\) 23.3231 0.843799
\(765\) 0 0
\(766\) 21.9058 0.791490
\(767\) 18.1422 0.655075
\(768\) 0 0
\(769\) −16.4049 −0.591577 −0.295788 0.955253i \(-0.595582\pi\)
−0.295788 + 0.955253i \(0.595582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9.49590 −0.341765
\(773\) 11.4461 0.411688 0.205844 0.978585i \(-0.434006\pi\)
0.205844 + 0.978585i \(0.434006\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 48.2896 1.73349
\(777\) 0 0
\(778\) −10.7857 −0.386687
\(779\) 28.8710 1.03441
\(780\) 0 0
\(781\) −3.62303 −0.129642
\(782\) 21.0328 0.752131
\(783\) 0 0
\(784\) −0.252616 −0.00902200
\(785\) 0 0
\(786\) 0 0
\(787\) −6.50658 −0.231934 −0.115967 0.993253i \(-0.536997\pi\)
−0.115967 + 0.993253i \(0.536997\pi\)
\(788\) 7.07595 0.252070
\(789\) 0 0
\(790\) 0 0
\(791\) −32.0500 −1.13957
\(792\) 0 0
\(793\) −7.08210 −0.251493
\(794\) 4.48166 0.159048
\(795\) 0 0
\(796\) 21.7946 0.772491
\(797\) 35.6753 1.26368 0.631841 0.775098i \(-0.282299\pi\)
0.631841 + 0.775098i \(0.282299\pi\)
\(798\) 0 0
\(799\) −12.6197 −0.446452
\(800\) 0 0
\(801\) 0 0
\(802\) 27.8862 0.984696
\(803\) −1.32866 −0.0468873
\(804\) 0 0
\(805\) 0 0
\(806\) 29.7976 1.04958
\(807\) 0 0
\(808\) 28.7211 1.01040
\(809\) 15.8037 0.555630 0.277815 0.960635i \(-0.410390\pi\)
0.277815 + 0.960635i \(0.410390\pi\)
\(810\) 0 0
\(811\) 30.3394 1.06536 0.532681 0.846316i \(-0.321185\pi\)
0.532681 + 0.846316i \(0.321185\pi\)
\(812\) −28.9108 −1.01457
\(813\) 0 0
\(814\) −0.201071 −0.00704753
\(815\) 0 0
\(816\) 0 0
\(817\) −0.639038 −0.0223571
\(818\) 5.61009 0.196152
\(819\) 0 0
\(820\) 0 0
\(821\) 31.4259 1.09677 0.548386 0.836225i \(-0.315242\pi\)
0.548386 + 0.836225i \(0.315242\pi\)
\(822\) 0 0
\(823\) −45.0336 −1.56977 −0.784886 0.619641i \(-0.787278\pi\)
−0.784886 + 0.619641i \(0.787278\pi\)
\(824\) −14.4104 −0.502010
\(825\) 0 0
\(826\) 13.0966 0.455690
\(827\) −11.9965 −0.417161 −0.208580 0.978005i \(-0.566884\pi\)
−0.208580 + 0.978005i \(0.566884\pi\)
\(828\) 0 0
\(829\) −30.9847 −1.07614 −0.538071 0.842899i \(-0.680847\pi\)
−0.538071 + 0.842899i \(0.680847\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 21.4699 0.744335
\(833\) −2.22962 −0.0772516
\(834\) 0 0
\(835\) 0 0
\(836\) −0.837735 −0.0289737
\(837\) 0 0
\(838\) −3.98799 −0.137763
\(839\) −36.2024 −1.24985 −0.624923 0.780687i \(-0.714870\pi\)
−0.624923 + 0.780687i \(0.714870\pi\)
\(840\) 0 0
\(841\) 59.3608 2.04692
\(842\) 21.7991 0.751247
\(843\) 0 0
\(844\) 19.3803 0.667096
\(845\) 0 0
\(846\) 0 0
\(847\) 29.9104 1.02773
\(848\) 2.41787 0.0830300
\(849\) 0 0
\(850\) 0 0
\(851\) 3.77191 0.129300
\(852\) 0 0
\(853\) −32.6123 −1.11662 −0.558312 0.829631i \(-0.688551\pi\)
−0.558312 + 0.829631i \(0.688551\pi\)
\(854\) −5.11249 −0.174946
\(855\) 0 0
\(856\) 12.3670 0.422695
\(857\) 4.85223 0.165749 0.0828744 0.996560i \(-0.473590\pi\)
0.0828744 + 0.996560i \(0.473590\pi\)
\(858\) 0 0
\(859\) −32.3815 −1.10484 −0.552421 0.833565i \(-0.686296\pi\)
−0.552421 + 0.833565i \(0.686296\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −11.7265 −0.399407
\(863\) 21.9498 0.747180 0.373590 0.927594i \(-0.378127\pi\)
0.373590 + 0.927594i \(0.378127\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −17.7592 −0.603481
\(867\) 0 0
\(868\) −27.5071 −0.933650
\(869\) 2.20355 0.0747502
\(870\) 0 0
\(871\) 48.7700 1.65251
\(872\) −30.7941 −1.04282
\(873\) 0 0
\(874\) −12.2893 −0.415691
\(875\) 0 0
\(876\) 0 0
\(877\) −4.27196 −0.144254 −0.0721270 0.997395i \(-0.522979\pi\)
−0.0721270 + 0.997395i \(0.522979\pi\)
\(878\) 6.83812 0.230775
\(879\) 0 0
\(880\) 0 0
\(881\) −29.1356 −0.981604 −0.490802 0.871271i \(-0.663296\pi\)
−0.490802 + 0.871271i \(0.663296\pi\)
\(882\) 0 0
\(883\) −49.1547 −1.65419 −0.827093 0.562065i \(-0.810007\pi\)
−0.827093 + 0.562065i \(0.810007\pi\)
\(884\) 17.4628 0.587338
\(885\) 0 0
\(886\) −5.38879 −0.181040
\(887\) −30.4384 −1.02202 −0.511010 0.859575i \(-0.670729\pi\)
−0.511010 + 0.859575i \(0.670729\pi\)
\(888\) 0 0
\(889\) 43.0434 1.44363
\(890\) 0 0
\(891\) 0 0
\(892\) 1.59869 0.0535280
\(893\) 7.37357 0.246747
\(894\) 0 0
\(895\) 0 0
\(896\) −14.0196 −0.468361
\(897\) 0 0
\(898\) 4.45056 0.148517
\(899\) 84.0704 2.80391
\(900\) 0 0
\(901\) 21.3404 0.710952
\(902\) −3.08940 −0.102866
\(903\) 0 0
\(904\) 34.2108 1.13783
\(905\) 0 0
\(906\) 0 0
\(907\) 14.9403 0.496084 0.248042 0.968749i \(-0.420213\pi\)
0.248042 + 0.968749i \(0.420213\pi\)
\(908\) 10.4022 0.345208
\(909\) 0 0
\(910\) 0 0
\(911\) −11.8295 −0.391928 −0.195964 0.980611i \(-0.562784\pi\)
−0.195964 + 0.980611i \(0.562784\pi\)
\(912\) 0 0
\(913\) 3.25923 0.107865
\(914\) 4.42823 0.146473
\(915\) 0 0
\(916\) 18.8816 0.623866
\(917\) −24.2503 −0.800814
\(918\) 0 0
\(919\) −30.6610 −1.01141 −0.505706 0.862706i \(-0.668768\pi\)
−0.505706 + 0.862706i \(0.668768\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −7.22329 −0.237886
\(923\) −44.1275 −1.45247
\(924\) 0 0
\(925\) 0 0
\(926\) −3.58768 −0.117898
\(927\) 0 0
\(928\) 50.6273 1.66192
\(929\) −6.46400 −0.212077 −0.106039 0.994362i \(-0.533817\pi\)
−0.106039 + 0.994362i \(0.533817\pi\)
\(930\) 0 0
\(931\) 1.30275 0.0426958
\(932\) 11.0820 0.363002
\(933\) 0 0
\(934\) 18.6321 0.609660
\(935\) 0 0
\(936\) 0 0
\(937\) 12.2409 0.399894 0.199947 0.979807i \(-0.435923\pi\)
0.199947 + 0.979807i \(0.435923\pi\)
\(938\) 35.2065 1.14953
\(939\) 0 0
\(940\) 0 0
\(941\) 7.39466 0.241059 0.120530 0.992710i \(-0.461541\pi\)
0.120530 + 0.992710i \(0.461541\pi\)
\(942\) 0 0
\(943\) 57.9545 1.88726
\(944\) −2.52874 −0.0823036
\(945\) 0 0
\(946\) 0.0683815 0.00222327
\(947\) −28.9928 −0.942139 −0.471070 0.882096i \(-0.656132\pi\)
−0.471070 + 0.882096i \(0.656132\pi\)
\(948\) 0 0
\(949\) −16.1827 −0.525312
\(950\) 0 0
\(951\) 0 0
\(952\) 35.0705 1.13664
\(953\) 43.0634 1.39496 0.697481 0.716604i \(-0.254304\pi\)
0.697481 + 0.716604i \(0.254304\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 23.5560 0.761856
\(957\) 0 0
\(958\) 26.0640 0.842089
\(959\) 23.6647 0.764174
\(960\) 0 0
\(961\) 48.9884 1.58027
\(962\) −2.44899 −0.0789586
\(963\) 0 0
\(964\) −0.672428 −0.0216574
\(965\) 0 0
\(966\) 0 0
\(967\) −31.9201 −1.02648 −0.513240 0.858245i \(-0.671555\pi\)
−0.513240 + 0.858245i \(0.671555\pi\)
\(968\) −31.9270 −1.02617
\(969\) 0 0
\(970\) 0 0
\(971\) 11.8218 0.379380 0.189690 0.981844i \(-0.439252\pi\)
0.189690 + 0.981844i \(0.439252\pi\)
\(972\) 0 0
\(973\) 21.9701 0.704328
\(974\) 29.7110 0.952001
\(975\) 0 0
\(976\) 0.987138 0.0315975
\(977\) 15.7868 0.505066 0.252533 0.967588i \(-0.418736\pi\)
0.252533 + 0.967588i \(0.418736\pi\)
\(978\) 0 0
\(979\) 2.33411 0.0745984
\(980\) 0 0
\(981\) 0 0
\(982\) 4.59176 0.146529
\(983\) −25.5427 −0.814685 −0.407342 0.913276i \(-0.633544\pi\)
−0.407342 + 0.913276i \(0.633544\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −38.5286 −1.22700
\(987\) 0 0
\(988\) −10.2034 −0.324613
\(989\) −1.28278 −0.0407899
\(990\) 0 0
\(991\) −1.33158 −0.0422991 −0.0211495 0.999776i \(-0.506733\pi\)
−0.0211495 + 0.999776i \(0.506733\pi\)
\(992\) 48.1690 1.52937
\(993\) 0 0
\(994\) −31.8551 −1.01038
\(995\) 0 0
\(996\) 0 0
\(997\) −51.6975 −1.63728 −0.818639 0.574309i \(-0.805271\pi\)
−0.818639 + 0.574309i \(0.805271\pi\)
\(998\) −2.93797 −0.0929997
\(999\) 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.v.1.3 8
3.2 odd 2 inner 5625.2.a.v.1.6 8
5.4 even 2 5625.2.a.w.1.6 8
15.14 odd 2 5625.2.a.w.1.3 8
25.4 even 10 225.2.h.e.91.2 16
25.19 even 10 225.2.h.e.136.2 yes 16
75.29 odd 10 225.2.h.e.91.3 yes 16
75.44 odd 10 225.2.h.e.136.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.h.e.91.2 16 25.4 even 10
225.2.h.e.91.3 yes 16 75.29 odd 10
225.2.h.e.136.2 yes 16 25.19 even 10
225.2.h.e.136.3 yes 16 75.44 odd 10
5625.2.a.v.1.3 8 1.1 even 1 trivial
5625.2.a.v.1.6 8 3.2 odd 2 inner
5625.2.a.w.1.3 8 15.14 odd 2
5625.2.a.w.1.6 8 5.4 even 2