Properties

Label 1425.2.c.g.799.2
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
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] = [1425,2,Mod(799,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, 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("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (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: \(11.3786822880\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.g.799.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} -1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -4.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +5.00000i q^{32} +6.00000 q^{34} -1.00000 q^{36} -10.0000i q^{37} +1.00000i q^{38} -6.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} +4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -1.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -7.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} -10.0000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -6.00000i q^{78} +1.00000 q^{81} -2.00000i q^{82} -16.0000i q^{83} -4.00000 q^{86} +2.00000i q^{87} +2.00000 q^{89} -4.00000i q^{92} -8.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{16} + 2 q^{19} + 6 q^{24} + 12 q^{26} - 4 q^{29} + 16 q^{31} + 12 q^{34} - 2 q^{36} - 12 q^{39} - 4 q^{41} + 8 q^{46} + 14 q^{49} - 12 q^{51} - 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{69} + 20 q^{74} + 2 q^{76} + 2 q^{81} - 8 q^{86} + 4 q^{89} - 24 q^{94} + 10 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) − 6.00000i − 0.832050i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) − 2.00000i − 0.262613i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) − 6.00000i − 0.679366i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) − 8.00000i − 0.829561i
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 6.00000i 0.554700i
\(118\) 12.0000i 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) − 2.00000i − 0.181071i
\(123\) 2.00000i 0.180334i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 18.0000 1.54349
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) − 7.00000i − 0.577350i
\(148\) − 10.0000i − 0.821995i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 3.00000i 0.243332i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 24.0000i 1.85718i 0.371113 + 0.928588i \(0.378976\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 4.00000i 0.304997i
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 2.00000i 0.149906i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 12.0000i 0.875190i
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 7.00000i 0.505181i
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) − 10.0000i − 0.703598i
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000i 0.278019i
\(208\) 6.00000i 0.416025i
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) 10.0000i 0.677285i
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −36.0000 −2.42162
\(222\) − 10.0000i − 0.671156i
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) − 6.00000i − 0.381771i
\(248\) 24.0000i 1.52400i
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 8.00000i 0.494242i
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.00000i − 0.122398i
\(268\) − 4.00000i − 0.244339i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 12.0000i 0.714590i
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) − 5.00000i − 0.294628i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) − 10.0000i − 0.585206i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) 30.0000 1.74371
\(297\) 0 0
\(298\) − 6.00000i − 0.347571i
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.00000i − 0.460348i
\(303\) 10.0000i 0.574485i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) − 18.0000i − 1.01905i
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 6.00000i − 0.333849i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) − 10.0000i − 0.553001i
\(328\) − 6.00000i − 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) − 16.0000i − 0.878114i
\(333\) 10.0000i 0.547997i
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) − 1.00000i − 0.0540738i
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 2.00000i 0.107211i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 22.0000i 1.17094i 0.810693 + 0.585471i \(0.199090\pi\)
−0.810693 + 0.585471i \(0.800910\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 14.0000i 0.735824i
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) − 8.00000i − 0.414781i
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −36.0000 −1.85656
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) − 12.0000i − 0.613973i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) − 4.00000i − 0.203331i
\(388\) 10.0000i 0.507673i
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 21.0000i 1.06066i
\(393\) − 8.00000i − 0.403547i
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) 38.0000 1.89763 0.948815 0.315833i \(-0.102284\pi\)
0.948815 + 0.315833i \(0.102284\pi\)
\(402\) − 4.00000i − 0.199502i
\(403\) − 48.0000i − 2.39105i
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) − 18.0000i − 0.891133i
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) − 8.00000i − 0.394132i
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 30.0000 1.47087
\(417\) 4.00000i 0.195881i
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) − 12.0000i − 0.583460i
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000i 0.193347i
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) − 4.00000i − 0.191346i
\(438\) − 10.0000i − 0.477818i
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 36.0000i − 1.71235i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 6.00000i − 0.282216i
\(453\) 8.00000i 0.375873i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) − 6.00000i − 0.280668i −0.990104 0.140334i \(-0.955182\pi\)
0.990104 0.140334i \(-0.0448177\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) − 32.0000i − 1.48078i −0.672176 0.740392i \(-0.734640\pi\)
0.672176 0.740392i \(-0.265360\pi\)
\(468\) 6.00000i 0.277350i
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 36.0000i 1.65703i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 12.0000i 0.548867i
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) − 6.00000i − 0.273293i
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 12.0000i 0.540453i
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) − 16.0000i − 0.716977i
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) − 24.0000i − 1.07117i
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) − 8.00000i − 0.354943i
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 11.0000i − 0.486136i
\(513\) 1.00000i 0.0441511i
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) − 48.0000i − 2.09091i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) − 4.00000i − 0.172613i
\(538\) 6.00000i 0.258678i
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) − 14.0000i − 0.600798i
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) − 12.0000i − 0.510754i
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) − 10.0000i − 0.421825i
\(563\) − 20.0000i − 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 18.0000i 0.749350i 0.927156 + 0.374675i \(0.122246\pi\)
−0.927156 + 0.374675i \(0.877754\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000i 0.414513i
\(583\) 0 0
\(584\) 30.0000 1.24141
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 10.0000i 0.410997i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) − 8.00000i − 0.327418i
\(598\) − 24.0000i − 0.981433i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) 24.0000i 0.974130i 0.873366 + 0.487065i \(0.161933\pi\)
−0.873366 + 0.487065i \(0.838067\pi\)
\(608\) 5.00000i 0.202777i
\(609\) 0 0
\(610\) 0 0
\(611\) 72.0000 2.91281
\(612\) 6.00000i 0.242536i
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 4.00000i 0.160385i
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 42.0000i − 1.66410i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.0000 1.50091 0.750455 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) 4.00000i 0.157867i
\(643\) − 20.0000i − 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 36.0000i 1.41531i 0.706560 + 0.707653i \(0.250246\pi\)
−0.706560 + 0.707653i \(0.749754\pi\)
\(648\) 3.00000i 0.117851i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 0 0
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) 12.0000i 0.466393i
\(663\) 36.0000i 1.39812i
\(664\) 48.0000 1.86276
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 8.00000i 0.309761i
\(668\) 24.0000i 0.928588i
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 46.0000i 1.77317i 0.462566 + 0.886585i \(0.346929\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) 6.00000i 0.228914i
\(688\) − 4.00000i − 0.152499i
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 22.0000i 0.836315i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 12.0000i 0.454532i
\(698\) 2.00000i 0.0757011i
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 6.00000i 0.226455i
\(703\) − 10.0000i − 0.377157i
\(704\) 0 0
\(705\) 0 0
\(706\) −22.0000 −0.827981
\(707\) 0 0
\(708\) − 12.0000i − 0.450988i
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000i 0.224860i
\(713\) − 32.0000i − 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) − 12.0000i − 0.448148i
\(718\) 20.0000i 0.746393i
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000i 0.0372161i
\(723\) 6.00000i 0.223142i
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 2.00000i 0.0739221i
\(733\) − 46.0000i − 1.69905i −0.527549 0.849524i \(-0.676889\pi\)
0.527549 0.849524i \(-0.323111\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) 2.00000i 0.0736210i
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 24.0000 0.879883
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 16.0000i 0.585409i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) 24.0000i 0.874609i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) − 12.0000i − 0.435860i
\(759\) 0 0
\(760\) 0 0
\(761\) 50.0000 1.81250 0.906249 0.422744i \(-0.138933\pi\)
0.906249 + 0.422744i \(0.138933\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) − 72.0000i − 2.59977i
\(768\) 17.0000i 0.613435i
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 14.0000i 0.503871i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −30.0000 −1.07694
\(777\) 0 0
\(778\) − 30.0000i − 1.07555i
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 0 0
\(782\) − 24.0000i − 0.858238i
\(783\) − 2.00000i − 0.0714742i
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) 44.0000i 1.56843i 0.620489 + 0.784215i \(0.286934\pi\)
−0.620489 + 0.784215i \(0.713066\pi\)
\(788\) − 2.00000i − 0.0712470i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 38.0000i 1.34183i
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 48.0000 1.69073
\(807\) − 6.00000i − 0.211210i
\(808\) − 30.0000i − 1.05540i
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 4.00000i 0.139942i
\(818\) 14.0000i 0.489499i
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 18.0000i 0.627822i
\(823\) 32.0000i 1.11545i 0.830026 + 0.557725i \(0.188326\pi\)
−0.830026 + 0.557725i \(0.811674\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 4.00000i 0.139010i
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 42.0000i 1.45609i
\(833\) − 42.0000i − 1.45521i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) − 8.00000i − 0.276355i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000i 0.482472i
\(843\) 10.0000i 0.344418i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) − 6.00000i − 0.206041i
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) −40.0000 −1.37118
\(852\) 0 0
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 30.0000i 1.02478i 0.858753 + 0.512390i \(0.171240\pi\)
−0.858753 + 0.512390i \(0.828760\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) 40.0000i 1.36162i 0.732462 + 0.680808i \(0.238371\pi\)
−0.732462 + 0.680808i \(0.761629\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 30.0000i 1.01593i
\(873\) − 10.0000i − 0.338449i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) − 34.0000i − 1.14810i −0.818821 0.574049i \(-0.805372\pi\)
0.818821 0.574049i \(-0.194628\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) 0 0
\(887\) − 40.0000i − 1.34307i −0.740973 0.671534i \(-0.765636\pi\)
0.740973 0.671534i \(-0.234364\pi\)
\(888\) − 30.0000i − 1.00673i
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) − 16.0000i − 0.535720i
\(893\) 12.0000i 0.401565i
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 0 0
\(897\) 24.0000i 0.801337i
\(898\) 2.00000i 0.0667409i
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) 6.00000i 0.198030i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) − 18.0000i − 0.592798i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 32.0000 1.05159
\(927\) 8.00000i 0.262754i
\(928\) − 10.0000i − 0.328266i
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) − 10.0000i − 0.327561i
\(933\) − 4.00000i − 0.130954i
\(934\) 32.0000 1.04707
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 8.00000i 0.260516i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) − 8.00000i − 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) − 38.0000i − 1.23094i −0.788160 0.615470i \(-0.788966\pi\)
0.788160 0.615470i \(-0.211034\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) − 20.0000i − 0.646171i
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 60.0000i − 1.93448i
\(963\) − 4.00000i − 0.128898i
\(964\) −6.00000 −0.193247
\(965\) 0 0
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) − 33.0000i − 1.06066i
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 4.00000i 0.127906i
\(979\) 0 0
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) − 32.0000i − 1.02116i
\(983\) 8.00000i 0.255160i 0.991828 + 0.127580i \(0.0407210\pi\)
−0.991828 + 0.127580i \(0.959279\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) − 6.00000i − 0.190885i
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 40.0000i 1.27000i
\(993\) − 12.0000i − 0.380808i
\(994\) 0 0
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) − 58.0000i − 1.83688i −0.395562 0.918439i \(-0.629450\pi\)
0.395562 0.918439i \(-0.370550\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) 10.0000 0.316386
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 1425.2.c.g.799.2 2
5.2 odd 4 1425.2.a.a.1.1 1
5.3 odd 4 57.2.a.c.1.1 1
5.4 even 2 inner 1425.2.c.g.799.1 2
15.2 even 4 4275.2.a.m.1.1 1
15.8 even 4 171.2.a.a.1.1 1
20.3 even 4 912.2.a.b.1.1 1
35.13 even 4 2793.2.a.i.1.1 1
40.3 even 4 3648.2.a.bf.1.1 1
40.13 odd 4 3648.2.a.o.1.1 1
55.43 even 4 6897.2.a.a.1.1 1
60.23 odd 4 2736.2.a.s.1.1 1
65.38 odd 4 9633.2.a.h.1.1 1
95.18 even 4 1083.2.a.a.1.1 1
105.83 odd 4 8379.2.a.e.1.1 1
285.113 odd 4 3249.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.c.1.1 1 5.3 odd 4
171.2.a.a.1.1 1 15.8 even 4
912.2.a.b.1.1 1 20.3 even 4
1083.2.a.a.1.1 1 95.18 even 4
1425.2.a.a.1.1 1 5.2 odd 4
1425.2.c.g.799.1 2 5.4 even 2 inner
1425.2.c.g.799.2 2 1.1 even 1 trivial
2736.2.a.s.1.1 1 60.23 odd 4
2793.2.a.i.1.1 1 35.13 even 4
3249.2.a.g.1.1 1 285.113 odd 4
3648.2.a.o.1.1 1 40.13 odd 4
3648.2.a.bf.1.1 1 40.3 even 4
4275.2.a.m.1.1 1 15.2 even 4
6897.2.a.a.1.1 1 55.43 even 4
8379.2.a.e.1.1 1 105.83 odd 4
9633.2.a.h.1.1 1 65.38 odd 4