Properties

Label 2925.2.a.y.1.2
Level $2925$
Weight $2$
Character 2925.1
Self dual yes
Analytic conductor $23.356$
Analytic rank $1$
Dimension $2$
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] = [2925,2,Mod(1,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, 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("2925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.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: \(23.3562425912\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2925.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} -1.73205 q^{8} +3.46410 q^{11} -1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} -6.92820 q^{17} +2.00000 q^{19} +6.00000 q^{22} +6.92820 q^{23} -1.73205 q^{26} -2.00000 q^{28} -6.92820 q^{29} +2.00000 q^{31} -5.19615 q^{32} -12.0000 q^{34} -2.00000 q^{37} +3.46410 q^{38} -6.92820 q^{41} -8.00000 q^{43} +3.46410 q^{44} +12.0000 q^{46} -10.3923 q^{47} -3.00000 q^{49} -1.00000 q^{52} +3.46410 q^{56} -12.0000 q^{58} -3.46410 q^{59} -10.0000 q^{61} +3.46410 q^{62} +1.00000 q^{64} -14.0000 q^{67} -6.92820 q^{68} -3.46410 q^{71} +10.0000 q^{73} -3.46410 q^{74} +2.00000 q^{76} -6.92820 q^{77} -4.00000 q^{79} -12.0000 q^{82} +10.3923 q^{83} -13.8564 q^{86} -6.00000 q^{88} -6.92820 q^{89} +2.00000 q^{91} +6.92820 q^{92} -18.0000 q^{94} +10.0000 q^{97} -5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 4 q^{7} - 2 q^{13} - 10 q^{16} + 4 q^{19} + 12 q^{22} - 4 q^{28} + 4 q^{31} - 24 q^{34} - 4 q^{37} - 16 q^{43} + 24 q^{46} - 6 q^{49} - 2 q^{52} - 24 q^{58} - 20 q^{61} + 2 q^{64} - 28 q^{67} + 20 q^{73} + 4 q^{76} - 8 q^{79} - 24 q^{82} - 12 q^{88} + 4 q^{91} - 36 q^{94} + 20 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −6.92820 −1.68034 −0.840168 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.73205 −0.339683
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) −12.0000 −2.05798
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 3.46410 0.561951
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) −10.3923 −1.51587 −0.757937 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) −12.0000 −1.57568
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 3.46410 0.439941
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) −6.92820 −0.840168
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −3.46410 −0.402694
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −12.0000 −1.32518
\(83\) 10.3923 1.14070 0.570352 0.821401i \(-0.306807\pi\)
0.570352 + 0.821401i \(0.306807\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −13.8564 −1.49417
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 6.92820 0.722315
\(93\) 0 0
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −5.19615 −0.524891
\(99\) 0 0
\(100\) 0 0
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 1.73205 0.169842
\(105\) 0 0
\(106\) 0 0
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.0000 0.944911
\(113\) −6.92820 −0.651751 −0.325875 0.945413i \(-0.605659\pi\)
−0.325875 + 0.945413i \(0.605659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.92820 −0.643268
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 13.8564 1.27021
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −17.3205 −1.56813
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) 0 0
\(131\) 20.7846 1.81596 0.907980 0.419014i \(-0.137624\pi\)
0.907980 + 0.419014i \(0.137624\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −24.2487 −2.09477
\(135\) 0 0
\(136\) 12.0000 1.02899
\(137\) −6.92820 −0.591916 −0.295958 0.955201i \(-0.595639\pi\)
−0.295958 + 0.955201i \(0.595639\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −3.46410 −0.289683
\(144\) 0 0
\(145\) 0 0
\(146\) 17.3205 1.43346
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) −3.46410 −0.280976
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −6.92820 −0.551178
\(159\) 0 0
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) −6.92820 −0.541002
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 13.8564 1.05348 0.526742 0.850026i \(-0.323414\pi\)
0.526742 + 0.850026i \(0.323414\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −17.3205 −1.30558
\(177\) 0 0
\(178\) −12.0000 −0.899438
\(179\) 13.8564 1.03568 0.517838 0.855479i \(-0.326737\pi\)
0.517838 + 0.855479i \(0.326737\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) −10.3923 −0.757937
\(189\) 0 0
\(190\) 0 0
\(191\) −13.8564 −1.00261 −0.501307 0.865269i \(-0.667147\pi\)
−0.501307 + 0.865269i \(0.667147\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −13.8564 −0.987228 −0.493614 0.869681i \(-0.664324\pi\)
−0.493614 + 0.869681i \(0.664324\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 24.0000 1.68863
\(203\) 13.8564 0.972529
\(204\) 0 0
\(205\) 0 0
\(206\) 6.92820 0.482711
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 6.92820 0.479234
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −17.3205 −1.17309
\(219\) 0 0
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 12.0000 0.787839
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.46410 −0.225494
\(237\) 0 0
\(238\) 24.0000 1.55569
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.73205 0.111340
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) −3.46410 −0.219971
\(249\) 0 0
\(250\) 0 0
\(251\) 6.92820 0.437304 0.218652 0.975803i \(-0.429834\pi\)
0.218652 + 0.975803i \(0.429834\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 27.7128 1.73886
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 27.7128 1.72868 0.864339 0.502910i \(-0.167737\pi\)
0.864339 + 0.502910i \(0.167737\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 36.0000 2.22409
\(263\) 6.92820 0.427211 0.213606 0.976920i \(-0.431479\pi\)
0.213606 + 0.976920i \(0.431479\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.92820 −0.424795
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) 6.92820 0.422420 0.211210 0.977441i \(-0.432260\pi\)
0.211210 + 0.977441i \(0.432260\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 34.6410 2.10042
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −6.92820 −0.415526
\(279\) 0 0
\(280\) 0 0
\(281\) 20.7846 1.23991 0.619953 0.784639i \(-0.287152\pi\)
0.619953 + 0.784639i \(0.287152\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −3.46410 −0.205557
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 13.8564 0.817918
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) 13.8564 0.809500 0.404750 0.914427i \(-0.367359\pi\)
0.404750 + 0.914427i \(0.367359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.46410 0.201347
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) −6.92820 −0.400668
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 24.2487 1.39536
\(303\) 0 0
\(304\) −10.0000 −0.573539
\(305\) 0 0
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) −6.92820 −0.394771
\(309\) 0 0
\(310\) 0 0
\(311\) 20.7846 1.17859 0.589294 0.807919i \(-0.299406\pi\)
0.589294 + 0.807919i \(0.299406\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −3.46410 −0.195491
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) −13.8564 −0.770991
\(324\) 0 0
\(325\) 0 0
\(326\) −24.2487 −1.34301
\(327\) 0 0
\(328\) 12.0000 0.662589
\(329\) 20.7846 1.14589
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 10.3923 0.570352
\(333\) 0 0
\(334\) 30.0000 1.64153
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 1.73205 0.0942111
\(339\) 0 0
\(340\) 0 0
\(341\) 6.92820 0.375183
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 13.8564 0.747087
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) −27.7128 −1.48770 −0.743851 0.668346i \(-0.767003\pi\)
−0.743851 + 0.668346i \(0.767003\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −18.0000 −0.959403
\(353\) 6.92820 0.368751 0.184376 0.982856i \(-0.440974\pi\)
0.184376 + 0.982856i \(0.440974\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.92820 −0.367194
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −17.3205 −0.910346
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −34.6410 −1.80579
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −41.5692 −2.14949
\(375\) 0 0
\(376\) 18.0000 0.928279
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) −17.3205 −0.885037 −0.442518 0.896759i \(-0.645915\pi\)
−0.442518 + 0.896759i \(0.645915\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −45.0333 −2.29214
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) −20.7846 −1.05382 −0.526911 0.849921i \(-0.676650\pi\)
−0.526911 + 0.849921i \(0.676650\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) 5.19615 0.262445
\(393\) 0 0
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −27.7128 −1.38912
\(399\) 0 0
\(400\) 0 0
\(401\) 34.6410 1.72989 0.864945 0.501867i \(-0.167353\pi\)
0.864945 + 0.501867i \(0.167353\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) 13.8564 0.689382
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) −6.92820 −0.343418
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) 5.19615 0.254762
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) −6.92820 −0.338465 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 13.8564 0.674519
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 6.92820 0.334887
\(429\) 0 0
\(430\) 0 0
\(431\) −38.1051 −1.83546 −0.917729 0.397206i \(-0.869980\pi\)
−0.917729 + 0.397206i \(0.869980\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −6.92820 −0.332564
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 13.8564 0.662842
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.46410 −0.164030
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 6.92820 0.326962 0.163481 0.986546i \(-0.447728\pi\)
0.163481 + 0.986546i \(0.447728\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) −6.92820 −0.325875
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 24.2487 1.13307
\(459\) 0 0
\(460\) 0 0
\(461\) 27.7128 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(462\) 0 0
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 34.6410 1.60817
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) −27.7128 −1.27424
\(474\) 0 0
\(475\) 0 0
\(476\) 13.8564 0.635107
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) 3.46410 0.158279 0.0791394 0.996864i \(-0.474783\pi\)
0.0791394 + 0.996864i \(0.474783\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −17.3205 −0.788928
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 17.3205 0.784063
\(489\) 0 0
\(490\) 0 0
\(491\) −6.92820 −0.312665 −0.156333 0.987704i \(-0.549967\pi\)
−0.156333 + 0.987704i \(0.549967\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) −3.46410 −0.155857
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −6.92820 −0.308913 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 41.5692 1.84798
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) 13.8564 0.614174 0.307087 0.951681i \(-0.400646\pi\)
0.307087 + 0.951681i \(0.400646\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) 48.0000 2.11719
\(515\) 0 0
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 6.92820 0.304408
\(519\) 0 0
\(520\) 0 0
\(521\) −20.7846 −0.910590 −0.455295 0.890341i \(-0.650466\pi\)
−0.455295 + 0.890341i \(0.650466\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 20.7846 0.907980
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −13.8564 −0.603595
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 6.92820 0.300094
\(534\) 0 0
\(535\) 0 0
\(536\) 24.2487 1.04738
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −38.1051 −1.63675
\(543\) 0 0
\(544\) 36.0000 1.54349
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −6.92820 −0.295958
\(549\) 0 0
\(550\) 0 0
\(551\) −13.8564 −0.590303
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −3.46410 −0.147176
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 27.7128 1.17423 0.587115 0.809504i \(-0.300264\pi\)
0.587115 + 0.809504i \(0.300264\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 36.0000 1.51857
\(563\) 13.8564 0.583978 0.291989 0.956422i \(-0.405683\pi\)
0.291989 + 0.956422i \(0.405683\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.92820 0.291214
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 13.8564 0.580891 0.290445 0.956892i \(-0.406197\pi\)
0.290445 + 0.956892i \(0.406197\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −3.46410 −0.144841
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 53.6936 2.23336
\(579\) 0 0
\(580\) 0 0
\(581\) −20.7846 −0.862291
\(582\) 0 0
\(583\) 0 0
\(584\) −17.3205 −0.716728
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) −45.0333 −1.85872 −0.929362 0.369170i \(-0.879642\pi\)
−0.929362 + 0.369170i \(0.879642\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) 20.7846 0.853522 0.426761 0.904365i \(-0.359655\pi\)
0.426761 + 0.904365i \(0.359655\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13.8564 −0.567581
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) −20.7846 −0.849236 −0.424618 0.905373i \(-0.639592\pi\)
−0.424618 + 0.905373i \(0.639592\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 27.7128 1.12949
\(603\) 0 0
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) −10.3923 −0.421464
\(609\) 0 0
\(610\) 0 0
\(611\) 10.3923 0.420428
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −45.0333 −1.81740
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) −6.92820 −0.278919 −0.139459 0.990228i \(-0.544536\pi\)
−0.139459 + 0.990228i \(0.544536\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 36.0000 1.44347
\(623\) 13.8564 0.555145
\(624\) 0 0
\(625\) 0 0
\(626\) −24.2487 −0.969173
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 6.92820 0.275589
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) −41.5692 −1.64574
\(639\) 0 0
\(640\) 0 0
\(641\) −34.6410 −1.36824 −0.684119 0.729370i \(-0.739813\pi\)
−0.684119 + 0.729370i \(0.739813\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −13.8564 −0.544752 −0.272376 0.962191i \(-0.587809\pi\)
−0.272376 + 0.962191i \(0.587809\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) −14.0000 −0.548282
\(653\) −34.6410 −1.35561 −0.677804 0.735243i \(-0.737068\pi\)
−0.677804 + 0.735243i \(0.737068\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 34.6410 1.35250
\(657\) 0 0
\(658\) 36.0000 1.40343
\(659\) −13.8564 −0.539769 −0.269884 0.962893i \(-0.586986\pi\)
−0.269884 + 0.962893i \(0.586986\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −17.3205 −0.673181
\(663\) 0 0
\(664\) −18.0000 −0.698535
\(665\) 0 0
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) 17.3205 0.670151
\(669\) 0 0
\(670\) 0 0
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −3.46410 −0.133432
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −41.5692 −1.59763 −0.798817 0.601574i \(-0.794541\pi\)
−0.798817 + 0.601574i \(0.794541\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) −17.3205 −0.662751 −0.331375 0.943499i \(-0.607513\pi\)
−0.331375 + 0.943499i \(0.607513\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) 40.0000 1.52499
\(689\) 0 0
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 13.8564 0.526742
\(693\) 0 0
\(694\) −48.0000 −1.82206
\(695\) 0 0
\(696\) 0 0
\(697\) 48.0000 1.81813
\(698\) 45.0333 1.70454
\(699\) 0 0
\(700\) 0 0
\(701\) 20.7846 0.785024 0.392512 0.919747i \(-0.371606\pi\)
0.392512 + 0.919747i \(0.371606\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 3.46410 0.130558
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) −27.7128 −1.04225
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) 13.8564 0.518927
\(714\) 0 0
\(715\) 0 0
\(716\) 13.8564 0.517838
\(717\) 0 0
\(718\) 18.0000 0.671754
\(719\) −34.6410 −1.29189 −0.645946 0.763383i \(-0.723537\pi\)
−0.645946 + 0.763383i \(0.723537\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −25.9808 −0.966904
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) −3.46410 −0.128388
\(729\) 0 0
\(730\) 0 0
\(731\) 55.4256 2.04999
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −13.8564 −0.511449
\(735\) 0 0
\(736\) −36.0000 −1.32698
\(737\) −48.4974 −1.78643
\(738\) 0 0
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.2487 −0.889599 −0.444799 0.895630i \(-0.646725\pi\)
−0.444799 + 0.895630i \(0.646725\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −45.0333 −1.64879
\(747\) 0 0
\(748\) −24.0000 −0.877527
\(749\) −13.8564 −0.506302
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 51.9615 1.89484
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 3.46410 0.125822
\(759\) 0 0
\(760\) 0 0
\(761\) −34.6410 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) −13.8564 −0.501307
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) 3.46410 0.125081
\(768\) 0 0
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −26.0000 −0.935760
\(773\) 13.8564 0.498380 0.249190 0.968455i \(-0.419836\pi\)
0.249190 + 0.968455i \(0.419836\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −17.3205 −0.621770
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) −13.8564 −0.496457
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) −83.1384 −2.97302
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −13.8564 −0.493614
\(789\) 0 0
\(790\) 0 0
\(791\) 13.8564 0.492677
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) −3.46410 −0.122936
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 34.6410 1.22705 0.613524 0.789676i \(-0.289751\pi\)
0.613524 + 0.789676i \(0.289751\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) 60.0000 2.11867
\(803\) 34.6410 1.22245
\(804\) 0 0
\(805\) 0 0
\(806\) −3.46410 −0.122018
\(807\) 0 0
\(808\) −24.0000 −0.844317
\(809\) −6.92820 −0.243583 −0.121791 0.992556i \(-0.538864\pi\)
−0.121791 + 0.992556i \(0.538864\pi\)
\(810\) 0 0
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 13.8564 0.486265
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −17.3205 −0.605597
\(819\) 0 0
\(820\) 0 0
\(821\) −27.7128 −0.967184 −0.483592 0.875294i \(-0.660668\pi\)
−0.483592 + 0.875294i \(0.660668\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −6.92820 −0.241355
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 10.3923 0.361376 0.180688 0.983540i \(-0.442168\pi\)
0.180688 + 0.983540i \(0.442168\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 20.7846 0.720144
\(834\) 0 0
\(835\) 0 0
\(836\) 6.92820 0.239617
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −24.2487 −0.837158 −0.418579 0.908180i \(-0.637472\pi\)
−0.418579 + 0.908180i \(0.637472\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) −17.3205 −0.596904
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −13.8564 −0.474991
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 34.6410 1.18539
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −66.0000 −2.24797
\(863\) −31.1769 −1.06127 −0.530637 0.847599i \(-0.678047\pi\)
−0.530637 + 0.847599i \(0.678047\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −3.46410 −0.117715
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −13.8564 −0.470046
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 17.3205 0.586546
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −48.4974 −1.63671
\(879\) 0 0
\(880\) 0 0
\(881\) 13.8564 0.466834 0.233417 0.972377i \(-0.425009\pi\)
0.233417 + 0.972377i \(0.425009\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 6.92820 0.233021
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 27.7128 0.930505 0.465253 0.885178i \(-0.345963\pi\)
0.465253 + 0.885178i \(0.345963\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) −20.7846 −0.695530
\(894\) 0 0
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) −13.8564 −0.462137
\(900\) 0 0
\(901\) 0 0
\(902\) −41.5692 −1.38410
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 3.46410 0.114960
\(909\) 0 0
\(910\) 0 0
\(911\) 41.5692 1.37725 0.688625 0.725118i \(-0.258215\pi\)
0.688625 + 0.725118i \(0.258215\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 17.3205 0.572911
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −41.5692 −1.37274
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 48.0000 1.58080
\(923\) 3.46410 0.114022
\(924\) 0 0
\(925\) 0 0
\(926\) −3.46410 −0.113837
\(927\) 0 0
\(928\) 36.0000 1.18176
\(929\) 48.4974 1.59115 0.795574 0.605856i \(-0.207169\pi\)
0.795574 + 0.605856i \(0.207169\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 48.4974 1.58350
\(939\) 0 0
\(940\) 0 0
\(941\) −41.5692 −1.35512 −0.677559 0.735469i \(-0.736962\pi\)
−0.677559 + 0.735469i \(0.736962\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 17.3205 0.563735
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 17.3205 0.562841 0.281420 0.959585i \(-0.409194\pi\)
0.281420 + 0.959585i \(0.409194\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) −24.0000 −0.777844
\(953\) −27.7128 −0.897706 −0.448853 0.893606i \(-0.648167\pi\)
−0.448853 + 0.893606i \(0.648167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.3923 0.336111
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) 13.8564 0.447447
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 3.46410 0.111687
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) −1.73205 −0.0556702
\(969\) 0 0
\(970\) 0 0
\(971\) −55.4256 −1.77869 −0.889346 0.457234i \(-0.848840\pi\)
−0.889346 + 0.457234i \(0.848840\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) −3.46410 −0.110997
\(975\) 0 0
\(976\) 50.0000 1.60046
\(977\) 6.92820 0.221653 0.110826 0.993840i \(-0.464650\pi\)
0.110826 + 0.993840i \(0.464650\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 10.3923 0.331463 0.165732 0.986171i \(-0.447001\pi\)
0.165732 + 0.986171i \(0.447001\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 83.1384 2.64767
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) −55.4256 −1.76243
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −10.3923 −0.329956
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 24.2487 0.767580
\(999\) 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 2925.2.a.y.1.2 2
3.2 odd 2 inner 2925.2.a.y.1.1 2
5.2 odd 4 2925.2.c.s.2224.3 4
5.3 odd 4 2925.2.c.s.2224.2 4
5.4 even 2 117.2.a.b.1.1 2
15.2 even 4 2925.2.c.s.2224.1 4
15.8 even 4 2925.2.c.s.2224.4 4
15.14 odd 2 117.2.a.b.1.2 yes 2
20.19 odd 2 1872.2.a.v.1.1 2
35.34 odd 2 5733.2.a.t.1.1 2
40.19 odd 2 7488.2.a.cj.1.2 2
40.29 even 2 7488.2.a.cq.1.1 2
45.4 even 6 1053.2.e.i.703.2 4
45.14 odd 6 1053.2.e.i.703.1 4
45.29 odd 6 1053.2.e.i.352.1 4
45.34 even 6 1053.2.e.i.352.2 4
60.59 even 2 1872.2.a.v.1.2 2
65.34 odd 4 1521.2.b.i.1351.1 4
65.44 odd 4 1521.2.b.i.1351.4 4
65.64 even 2 1521.2.a.j.1.2 2
105.104 even 2 5733.2.a.t.1.2 2
120.29 odd 2 7488.2.a.cq.1.2 2
120.59 even 2 7488.2.a.cj.1.1 2
195.44 even 4 1521.2.b.i.1351.2 4
195.164 even 4 1521.2.b.i.1351.3 4
195.194 odd 2 1521.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.a.b.1.1 2 5.4 even 2
117.2.a.b.1.2 yes 2 15.14 odd 2
1053.2.e.i.352.1 4 45.29 odd 6
1053.2.e.i.352.2 4 45.34 even 6
1053.2.e.i.703.1 4 45.14 odd 6
1053.2.e.i.703.2 4 45.4 even 6
1521.2.a.j.1.1 2 195.194 odd 2
1521.2.a.j.1.2 2 65.64 even 2
1521.2.b.i.1351.1 4 65.34 odd 4
1521.2.b.i.1351.2 4 195.44 even 4
1521.2.b.i.1351.3 4 195.164 even 4
1521.2.b.i.1351.4 4 65.44 odd 4
1872.2.a.v.1.1 2 20.19 odd 2
1872.2.a.v.1.2 2 60.59 even 2
2925.2.a.y.1.1 2 3.2 odd 2 inner
2925.2.a.y.1.2 2 1.1 even 1 trivial
2925.2.c.s.2224.1 4 15.2 even 4
2925.2.c.s.2224.2 4 5.3 odd 4
2925.2.c.s.2224.3 4 5.2 odd 4
2925.2.c.s.2224.4 4 15.8 even 4
5733.2.a.t.1.1 2 35.34 odd 2
5733.2.a.t.1.2 2 105.104 even 2
7488.2.a.cj.1.1 2 120.59 even 2
7488.2.a.cj.1.2 2 40.19 odd 2
7488.2.a.cq.1.1 2 40.29 even 2
7488.2.a.cq.1.2 2 120.29 odd 2