Properties

Label 1725.2.b.j.1174.2
Level $1725$
Weight $2$
Character 1725.1174
Analytic conductor $13.774$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1725,2,Mod(1174,1725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1725.1174");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1725.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(13.7741943487\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1174.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1725.1174
Dual form 1725.2.b.j.1174.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +2.00000 q^{4} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +2.00000 q^{4} +3.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +2.00000i q^{12} +4.00000 q^{16} +3.00000i q^{17} +8.00000 q^{19} -3.00000 q^{21} +1.00000i q^{23} -1.00000i q^{27} +6.00000i q^{28} -9.00000 q^{29} -5.00000 q^{31} -4.00000i q^{33} -2.00000 q^{36} +9.00000i q^{37} +7.00000 q^{41} +4.00000i q^{43} -8.00000 q^{44} +2.00000i q^{47} +4.00000i q^{48} -2.00000 q^{49} -3.00000 q^{51} +13.0000i q^{53} +8.00000i q^{57} +3.00000 q^{59} -14.0000 q^{61} -3.00000i q^{63} +8.00000 q^{64} -13.0000i q^{67} +6.00000i q^{68} -1.00000 q^{69} -13.0000 q^{71} -4.00000i q^{73} +16.0000 q^{76} -12.0000i q^{77} +1.00000 q^{81} -1.00000i q^{83} -6.00000 q^{84} -9.00000i q^{87} +8.00000 q^{89} +2.00000i q^{92} -5.00000i q^{93} -10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 2 q^{9} - 8 q^{11} + 8 q^{16} + 16 q^{19} - 6 q^{21} - 18 q^{29} - 10 q^{31} - 4 q^{36} + 14 q^{41} - 16 q^{44} - 4 q^{49} - 6 q^{51} + 6 q^{59} - 28 q^{61} + 16 q^{64} - 2 q^{69} - 26 q^{71} + 32 q^{76} + 2 q^{81} - 12 q^{84} + 16 q^{89} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 6.00000i 1.13389i
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 9.00000i 1.47959i 0.672832 + 0.739795i \(0.265078\pi\)
−0.672832 + 0.739795i \(0.734922\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −8.00000 −1.20605
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 4.00000i 0.577350i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 13.0000i 1.78569i 0.450367 + 0.892844i \(0.351293\pi\)
−0.450367 + 0.892844i \(0.648707\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) − 3.00000i − 0.377964i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 13.0000i − 1.58820i −0.607785 0.794101i \(-0.707942\pi\)
0.607785 0.794101i \(-0.292058\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −13.0000 −1.54282 −0.771408 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 16.0000 1.83533
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 1.00000i − 0.109764i −0.998493 0.0548821i \(-0.982522\pi\)
0.998493 0.0548821i \(-0.0174783\pi\)
\(84\) −6.00000 −0.654654
\(85\) 0 0
\(86\) 0 0
\(87\) − 9.00000i − 0.964901i
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000i 0.208514i
\(93\) − 5.00000i − 0.518476i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 11.0000i − 1.06341i −0.846930 0.531705i \(-0.821551\pi\)
0.846930 0.531705i \(-0.178449\pi\)
\(108\) − 2.00000i − 0.192450i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −9.00000 −0.854242
\(112\) 12.0000i 1.13389i
\(113\) 1.00000i 0.0940721i 0.998893 + 0.0470360i \(0.0149776\pi\)
−0.998893 + 0.0470360i \(0.985022\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −18.0000 −1.67126
\(117\) 0 0
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 7.00000i 0.631169i
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) 0 0
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) − 8.00000i − 0.696311i
\(133\) 24.0000i 2.08106i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2.00000i − 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) 0 0
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 18.0000i 1.47959i
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) − 3.00000i − 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.00000i 0.239426i 0.992809 + 0.119713i \(0.0381975\pi\)
−0.992809 + 0.119713i \(0.961803\pi\)
\(158\) 0 0
\(159\) −13.0000 −1.03097
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) − 2.00000i − 0.156652i −0.996928 0.0783260i \(-0.975042\pi\)
0.996928 0.0783260i \(-0.0249575\pi\)
\(164\) 14.0000 1.09322
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0000i 1.70241i 0.524832 + 0.851206i \(0.324128\pi\)
−0.524832 + 0.851206i \(0.675872\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 8.00000i 0.609994i
\(173\) − 24.0000i − 1.82469i −0.409426 0.912343i \(-0.634271\pi\)
0.409426 0.912343i \(-0.365729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −16.0000 −1.20605
\(177\) 3.00000i 0.225494i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) − 14.0000i − 1.03491i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 12.0000i − 0.877527i
\(188\) 4.00000i 0.291730i
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) 8.00000i 0.577350i
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −4.00000 −0.285714
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 13.0000 0.916949
\(202\) 0 0
\(203\) − 27.0000i − 1.89503i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.00000i − 0.0695048i
\(208\) 0 0
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) 26.0000i 1.78569i
\(213\) − 13.0000i − 0.890745i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 15.0000i − 1.01827i
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 16.0000i 1.05963i
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) − 8.00000i − 0.524097i −0.965055 0.262049i \(-0.915602\pi\)
0.965055 0.262049i \(-0.0843981\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) −13.0000 −0.840900 −0.420450 0.907316i \(-0.638128\pi\)
−0.420450 + 0.907316i \(0.638128\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) −28.0000 −1.79252
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.00000 0.0633724
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) − 6.00000i − 0.377964i
\(253\) − 4.00000i − 0.251478i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −27.0000 −1.67770
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) − 23.0000i − 1.41824i −0.705087 0.709120i \(-0.749092\pi\)
0.705087 0.709120i \(-0.250908\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) − 26.0000i − 1.58820i
\(269\) 17.0000 1.03651 0.518254 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) 0 0
\(271\) −23.0000 −1.39715 −0.698575 0.715537i \(-0.746182\pi\)
−0.698575 + 0.715537i \(0.746182\pi\)
\(272\) 12.0000i 0.727607i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) − 5.00000i − 0.297219i −0.988896 0.148610i \(-0.952520\pi\)
0.988896 0.148610i \(-0.0474798\pi\)
\(284\) −26.0000 −1.54282
\(285\) 0 0
\(286\) 0 0
\(287\) 21.0000i 1.23959i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) − 8.00000i − 0.468165i
\(293\) 19.0000i 1.10999i 0.831853 + 0.554996i \(0.187280\pi\)
−0.831853 + 0.554996i \(0.812720\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 9.00000i 0.517036i
\(304\) 32.0000 1.83533
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) − 24.0000i − 1.36753i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 25.0000i 1.41308i 0.707671 + 0.706542i \(0.249746\pi\)
−0.707671 + 0.706542i \(0.750254\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 0 0
\(319\) 36.0000 2.01561
\(320\) 0 0
\(321\) 11.0000 0.613960
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) 16.0000i 0.884802i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) − 2.00000i − 0.109764i
\(333\) − 9.00000i − 0.493197i
\(334\) 0 0
\(335\) 0 0
\(336\) −12.0000 −0.654654
\(337\) − 10.0000i − 0.544735i −0.962193 0.272367i \(-0.912193\pi\)
0.962193 0.272367i \(-0.0878066\pi\)
\(338\) 0 0
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.0000i 0.858925i 0.903085 + 0.429463i \(0.141297\pi\)
−0.903085 + 0.429463i \(0.858703\pi\)
\(348\) − 18.0000i − 0.964901i
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 16.0000 0.847998
\(357\) − 9.00000i − 0.476331i
\(358\) 0 0
\(359\) 2.00000 0.105556 0.0527780 0.998606i \(-0.483192\pi\)
0.0527780 + 0.998606i \(0.483192\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 31.0000i − 1.61819i −0.587680 0.809093i \(-0.699959\pi\)
0.587680 0.809093i \(-0.300041\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −7.00000 −0.364405
\(370\) 0 0
\(371\) −39.0000 −2.02478
\(372\) − 10.0000i − 0.518476i
\(373\) − 6.00000i − 0.310668i −0.987862 0.155334i \(-0.950355\pi\)
0.987862 0.155334i \(-0.0496454\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) − 23.0000i − 1.17525i −0.809135 0.587623i \(-0.800064\pi\)
0.809135 0.587623i \(-0.199936\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) − 20.0000i − 1.01535i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) − 4.00000i − 0.201773i
\(394\) 0 0
\(395\) 0 0
\(396\) 8.00000 0.402015
\(397\) − 8.00000i − 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 0 0
\(399\) −24.0000 −1.20150
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) − 36.0000i − 1.78445i
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 16.0000i 0.788263i
\(413\) 9.00000i 0.442861i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.00000i 0.146911i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) − 2.00000i − 0.0972433i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 42.0000i − 2.03252i
\(428\) − 22.0000i − 1.06341i
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 37.0000i − 1.77811i −0.457804 0.889053i \(-0.651364\pi\)
0.457804 0.889053i \(-0.348636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 32.0000 1.53252
\(437\) 8.00000i 0.382692i
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 6.00000i 0.285069i 0.989790 + 0.142534i \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) −18.0000 −0.854242
\(445\) 0 0
\(446\) 0 0
\(447\) 14.0000i 0.662177i
\(448\) 24.0000i 1.13389i
\(449\) 11.0000 0.519122 0.259561 0.965727i \(-0.416422\pi\)
0.259561 + 0.965727i \(0.416422\pi\)
\(450\) 0 0
\(451\) −28.0000 −1.31847
\(452\) 2.00000i 0.0940721i
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 35.0000i 1.63723i 0.574342 + 0.818615i \(0.305258\pi\)
−0.574342 + 0.818615i \(0.694742\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) − 20.0000i − 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) −36.0000 −1.67126
\(465\) 0 0
\(466\) 0 0
\(467\) 13.0000i 0.601568i 0.953692 + 0.300784i \(0.0972484\pi\)
−0.953692 + 0.300784i \(0.902752\pi\)
\(468\) 0 0
\(469\) 39.0000 1.80085
\(470\) 0 0
\(471\) −3.00000 −0.138233
\(472\) 0 0
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) −18.0000 −0.825029
\(477\) − 13.0000i − 0.595229i
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 3.00000i − 0.136505i
\(484\) 10.0000 0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) 3.00000 0.135388 0.0676941 0.997706i \(-0.478436\pi\)
0.0676941 + 0.997706i \(0.478436\pi\)
\(492\) 14.0000i 0.631169i
\(493\) − 27.0000i − 1.21602i
\(494\) 0 0
\(495\) 0 0
\(496\) −20.0000 −0.898027
\(497\) − 39.0000i − 1.74939i
\(498\) 0 0
\(499\) −21.0000 −0.940089 −0.470045 0.882643i \(-0.655762\pi\)
−0.470045 + 0.882643i \(0.655762\pi\)
\(500\) 0 0
\(501\) −22.0000 −0.982888
\(502\) 0 0
\(503\) 21.0000i 0.936344i 0.883637 + 0.468172i \(0.155087\pi\)
−0.883637 + 0.468172i \(0.844913\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 13.0000i 0.577350i
\(508\) 36.0000i 1.59724i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 0 0
\(513\) − 8.00000i − 0.353209i
\(514\) 0 0
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) − 8.00000i − 0.351840i
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 0 0
\(527\) − 15.0000i − 0.653410i
\(528\) − 16.0000i − 0.696311i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 48.0000i 2.08106i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.00000 0.344584
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) − 4.00000i − 0.170872i
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) −72.0000 −3.06730
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 6.00000 0.254457
\(557\) − 21.0000i − 0.889799i −0.895581 0.444899i \(-0.853239\pi\)
0.895581 0.444899i \(-0.146761\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) − 11.0000i − 0.463595i −0.972764 0.231797i \(-0.925539\pi\)
0.972764 0.231797i \(-0.0744606\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) 0 0
\(567\) 3.00000i 0.125988i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) − 2.00000i − 0.0835512i
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) 20.0000i 0.832611i 0.909225 + 0.416305i \(0.136675\pi\)
−0.909225 + 0.416305i \(0.863325\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) − 52.0000i − 2.15362i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 14.0000i − 0.577842i −0.957353 0.288921i \(-0.906704\pi\)
0.957353 0.288921i \(-0.0932965\pi\)
\(588\) − 4.00000i − 0.164957i
\(589\) −40.0000 −1.64817
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 36.0000i 1.47959i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 28.0000 1.14692
\(597\) 18.0000i 0.736691i
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −41.0000 −1.67242 −0.836212 0.548406i \(-0.815235\pi\)
−0.836212 + 0.548406i \(0.815235\pi\)
\(602\) 0 0
\(603\) 13.0000i 0.529401i
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0000i 0.405887i 0.979190 + 0.202944i \(0.0650509\pi\)
−0.979190 + 0.202944i \(0.934949\pi\)
\(608\) 0 0
\(609\) 27.0000 1.09410
\(610\) 0 0
\(611\) 0 0
\(612\) − 6.00000i − 0.242536i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.0000i 1.32853i 0.747497 + 0.664265i \(0.231255\pi\)
−0.747497 + 0.664265i \(0.768745\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 24.0000i 0.961540i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 32.0000i − 1.27796i
\(628\) 6.00000i 0.239426i
\(629\) −27.0000 −1.07656
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) 7.00000i 0.278225i
\(634\) 0 0
\(635\) 0 0
\(636\) −26.0000 −1.03097
\(637\) 0 0
\(638\) 0 0
\(639\) 13.0000 0.514272
\(640\) 0 0
\(641\) −20.0000 −0.789953 −0.394976 0.918691i \(-0.629247\pi\)
−0.394976 + 0.918691i \(0.629247\pi\)
\(642\) 0 0
\(643\) 13.0000i 0.512670i 0.966588 + 0.256335i \(0.0825150\pi\)
−0.966588 + 0.256335i \(0.917485\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) − 12.0000i − 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 15.0000 0.587896
\(652\) − 4.00000i − 0.156652i
\(653\) 16.0000i 0.626128i 0.949732 + 0.313064i \(0.101356\pi\)
−0.949732 + 0.313064i \(0.898644\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 28.0000 1.09322
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.00000i − 0.348481i
\(668\) 44.0000i 1.70241i
\(669\) −10.0000 −0.386622
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 26.0000 1.00000
\(677\) 45.0000i 1.72949i 0.502211 + 0.864745i \(0.332520\pi\)
−0.502211 + 0.864745i \(0.667480\pi\)
\(678\) 0 0
\(679\) 30.0000 1.15129
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 26.0000i 0.994862i 0.867503 + 0.497431i \(0.165723\pi\)
−0.867503 + 0.497431i \(0.834277\pi\)
\(684\) −16.0000 −0.611775
\(685\) 0 0
\(686\) 0 0
\(687\) − 4.00000i − 0.152610i
\(688\) 16.0000i 0.609994i
\(689\) 0 0
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) − 48.0000i − 1.82469i
\(693\) 12.0000i 0.455842i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 21.0000i 0.795432i
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 72.0000i 2.71553i
\(704\) −32.0000 −1.20605
\(705\) 0 0
\(706\) 0 0
\(707\) 27.0000i 1.01544i
\(708\) 6.00000i 0.225494i
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 5.00000i − 0.187251i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 13.0000i − 0.485494i
\(718\) 0 0
\(719\) 7.00000 0.261056 0.130528 0.991445i \(-0.458333\pi\)
0.130528 + 0.991445i \(0.458333\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 20.0000i 0.743808i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 37.0000i 1.37225i 0.727482 + 0.686127i \(0.240691\pi\)
−0.727482 + 0.686127i \(0.759309\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) − 28.0000i − 1.03491i
\(733\) 51.0000i 1.88373i 0.335994 + 0.941864i \(0.390928\pi\)
−0.335994 + 0.941864i \(0.609072\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 52.0000i 1.91544i
\(738\) 0 0
\(739\) −27.0000 −0.993211 −0.496606 0.867976i \(-0.665420\pi\)
−0.496606 + 0.867976i \(0.665420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.00000i 0.0365881i
\(748\) − 24.0000i − 0.877527i
\(749\) 33.0000 1.20579
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 18.0000i 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 6.00000 0.218218
\(757\) − 45.0000i − 1.63555i −0.575536 0.817776i \(-0.695207\pi\)
0.575536 0.817776i \(-0.304793\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −41.0000 −1.48625 −0.743124 0.669153i \(-0.766657\pi\)
−0.743124 + 0.669153i \(0.766657\pi\)
\(762\) 0 0
\(763\) 48.0000i 1.73772i
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 16.0000i 0.577350i
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 8.00000i 0.287926i
\(773\) − 46.0000i − 1.65451i −0.561830 0.827253i \(-0.689903\pi\)
0.561830 0.827253i \(-0.310097\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 27.0000i − 0.968620i
\(778\) 0 0
\(779\) 56.0000 2.00641
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) 9.00000i 0.321634i
\(784\) −8.00000 −0.285714
\(785\) 0 0
\(786\) 0 0
\(787\) 27.0000i 0.962446i 0.876598 + 0.481223i \(0.159807\pi\)
−0.876598 + 0.481223i \(0.840193\pi\)
\(788\) − 4.00000i − 0.142494i
\(789\) 23.0000 0.818822
\(790\) 0 0
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 36.0000 1.27599
\(797\) − 27.0000i − 0.956389i −0.878254 0.478195i \(-0.841291\pi\)
0.878254 0.478195i \(-0.158709\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) 16.0000i 0.564628i
\(804\) 26.0000 0.916949
\(805\) 0 0
\(806\) 0 0
\(807\) 17.0000i 0.598428i
\(808\) 0 0
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) − 54.0000i − 1.89503i
\(813\) − 23.0000i − 0.806645i
\(814\) 0 0
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) 32.0000i 1.11954i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 37.0000i − 1.28662i −0.765607 0.643308i \(-0.777561\pi\)
0.765607 0.643308i \(-0.222439\pi\)
\(828\) − 2.00000i − 0.0695048i
\(829\) 27.0000 0.937749 0.468874 0.883265i \(-0.344660\pi\)
0.468874 + 0.883265i \(0.344660\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) − 6.00000i − 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) −64.0000 −2.21349
\(837\) 5.00000i 0.172825i
\(838\) 0 0
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 24.0000i 0.826604i
\(844\) 14.0000 0.481900
\(845\) 0 0
\(846\) 0 0
\(847\) 15.0000i 0.515406i
\(848\) 52.0000i 1.78569i
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) −9.00000 −0.308516
\(852\) − 26.0000i − 0.890745i
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) −43.0000 −1.46714 −0.733571 0.679613i \(-0.762148\pi\)
−0.733571 + 0.679613i \(0.762148\pi\)
\(860\) 0 0
\(861\) −21.0000 −0.715678
\(862\) 0 0
\(863\) 54.0000i 1.83818i 0.394046 + 0.919091i \(0.371075\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000i 0.271694i
\(868\) − 30.0000i − 1.01827i
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.0000i 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 0 0
\(879\) −19.0000 −0.640854
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) − 34.0000i − 1.14419i −0.820187 0.572096i \(-0.806131\pi\)
0.820187 0.572096i \(-0.193869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 46.0000i 1.54453i 0.635301 + 0.772264i \(0.280876\pi\)
−0.635301 + 0.772264i \(0.719124\pi\)
\(888\) 0 0
\(889\) −54.0000 −1.81110
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 20.0000i 0.669650i
\(893\) 16.0000i 0.535420i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 45.0000 1.50083
\(900\) 0 0
\(901\) −39.0000 −1.29928
\(902\) 0 0
\(903\) − 12.0000i − 0.399335i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 53.0000i − 1.75984i −0.475125 0.879918i \(-0.657597\pi\)
0.475125 0.879918i \(-0.342403\pi\)
\(908\) 8.00000i 0.265489i
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 32.0000i 1.05963i
\(913\) 4.00000i 0.132381i
\(914\) 0 0
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) − 12.0000i − 0.396275i
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 24.0000 0.789542
\(925\) 0 0
\(926\) 0 0
\(927\) − 8.00000i − 0.262754i
\(928\) 0 0
\(929\) −57.0000 −1.87011 −0.935055 0.354504i \(-0.884650\pi\)
−0.935055 + 0.354504i \(0.884650\pi\)
\(930\) 0 0
\(931\) −16.0000 −0.524379
\(932\) − 16.0000i − 0.524097i
\(933\) − 8.00000i − 0.261908i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 0 0
\(939\) −25.0000 −0.815844
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) 7.00000i 0.227951i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) − 22.0000i − 0.712650i −0.934362 0.356325i \(-0.884030\pi\)
0.934362 0.356325i \(-0.115970\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −26.0000 −0.840900
\(957\) 36.0000i 1.16371i
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 11.0000i 0.354470i
\(964\) 40.0000 1.28831
\(965\) 0 0
\(966\) 0 0
\(967\) − 10.0000i − 0.321578i −0.986989 0.160789i \(-0.948596\pi\)
0.986989 0.160789i \(-0.0514039\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 2.00000i 0.0641500i
\(973\) 9.00000i 0.288527i
\(974\) 0 0
\(975\) 0 0
\(976\) −56.0000 −1.79252
\(977\) 25.0000i 0.799821i 0.916554 + 0.399910i \(0.130959\pi\)
−0.916554 + 0.399910i \(0.869041\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 0 0
\(983\) 15.0000i 0.478426i 0.970967 + 0.239213i \(0.0768894\pi\)
−0.970967 + 0.239213i \(0.923111\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 6.00000i − 0.190982i
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 0 0
\(993\) 7.00000i 0.222138i
\(994\) 0 0
\(995\) 0 0
\(996\) 2.00000 0.0633724
\(997\) − 8.00000i − 0.253363i −0.991943 0.126681i \(-0.959567\pi\)
0.991943 0.126681i \(-0.0404325\pi\)
\(998\) 0 0
\(999\) 9.00000 0.284747
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 1725.2.b.j.1174.2 2
5.2 odd 4 345.2.a.d.1.1 1
5.3 odd 4 1725.2.a.k.1.1 1
5.4 even 2 inner 1725.2.b.j.1174.1 2
15.2 even 4 1035.2.a.d.1.1 1
15.8 even 4 5175.2.a.o.1.1 1
20.7 even 4 5520.2.a.h.1.1 1
115.22 even 4 7935.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.d.1.1 1 5.2 odd 4
1035.2.a.d.1.1 1 15.2 even 4
1725.2.a.k.1.1 1 5.3 odd 4
1725.2.b.j.1174.1 2 5.4 even 2 inner
1725.2.b.j.1174.2 2 1.1 even 1 trivial
5175.2.a.o.1.1 1 15.8 even 4
5520.2.a.h.1.1 1 20.7 even 4
7935.2.a.i.1.1 1 115.22 even 4