Properties

Label 2394.2.e.b.1063.2
Level $2394$
Weight $2$
Character 2394.1063
Analytic conductor $19.116$
Analytic rank $0$
Dimension $12$
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] = [2394,2,Mod(1063,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.1063");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.e (of order \(2\), degree \(1\), 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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} + 54x^{8} - 114x^{7} + 120x^{6} + 46x^{5} + 9x^{4} - 4x^{3} + 8x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 798)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1063.2
Root \(-0.301222 + 0.301222i\) of defining polynomial
Character \(\chi\) \(=\) 2394.1063
Dual form 2394.2.e.b.1063.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -0.602444i q^{5} +(-0.307956 - 2.62777i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -0.602444i q^{5} +(-0.307956 - 2.62777i) q^{7} +1.00000i q^{8} -0.602444 q^{10} -1.46816 q^{11} +2.60244 q^{13} +(-2.62777 + 0.307956i) q^{14} +1.00000 q^{16} +1.08663i q^{17} +(4.00633 - 1.71737i) q^{19} +0.602444i q^{20} +1.46816i q^{22} +2.83683 q^{23} +4.63706 q^{25} -2.60244i q^{26} +(0.307956 + 2.62777i) q^{28} -6.19185i q^{29} +7.12381 q^{31} -1.00000i q^{32} +1.08663 q^{34} +(-1.58308 + 0.185526i) q^{35} -5.73972i q^{37} +(-1.71737 - 4.00633i) q^{38} +0.602444 q^{40} -9.86097 q^{41} -12.1295 q^{43} +1.46816 q^{44} -2.83683i q^{46} +0.752759i q^{47} +(-6.81033 + 1.61847i) q^{49} -4.63706i q^{50} -2.60244 q^{52} -4.81882i q^{53} +0.884483i q^{55} +(2.62777 - 0.307956i) q^{56} -6.19185 q^{58} -0.799650 q^{59} +8.69481i q^{61} -7.12381i q^{62} -1.00000 q^{64} -1.56783i q^{65} -5.90487i q^{67} -1.08663i q^{68} +(0.185526 + 1.58308i) q^{70} -4.75712i q^{71} -7.95747i q^{73} -5.73972 q^{74} +(-4.00633 + 1.71737i) q^{76} +(0.452128 + 3.85798i) q^{77} +0.00299468i q^{79} -0.602444i q^{80} +9.86097i q^{82} +15.9162i q^{83} +0.654635 q^{85} +12.1295i q^{86} -1.46816i q^{88} +9.74408 q^{89} +(-0.801438 - 6.83862i) q^{91} -2.83683 q^{92} +0.752759 q^{94} +(-1.03462 - 2.41359i) q^{95} +0.784815 q^{97} +(1.61847 + 6.81033i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 4 q^{10} - 12 q^{11} + 20 q^{13} + 4 q^{14} + 12 q^{16} - 8 q^{19} + 12 q^{23} - 12 q^{25} + 24 q^{31} + 4 q^{34} - 4 q^{35} - 4 q^{40} + 16 q^{41} + 12 q^{44} + 4 q^{49} - 20 q^{52} - 4 q^{56} + 8 q^{58} - 40 q^{59} - 12 q^{64} - 24 q^{70} + 8 q^{76} - 8 q^{77} + 8 q^{85} + 16 q^{89} + 24 q^{91} - 12 q^{92} + 44 q^{95} - 60 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0.602444i 0.269421i −0.990885 0.134711i \(-0.956990\pi\)
0.990885 0.134711i \(-0.0430105\pi\)
\(6\) 0 0
\(7\) −0.307956 2.62777i −0.116396 0.993203i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −0.602444 −0.190509
\(11\) −1.46816 −0.442666 −0.221333 0.975198i \(-0.571041\pi\)
−0.221333 + 0.975198i \(0.571041\pi\)
\(12\) 0 0
\(13\) 2.60244 0.721788 0.360894 0.932607i \(-0.382472\pi\)
0.360894 + 0.932607i \(0.382472\pi\)
\(14\) −2.62777 + 0.307956i −0.702300 + 0.0823047i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.08663i 0.263547i 0.991280 + 0.131774i \(0.0420672\pi\)
−0.991280 + 0.131774i \(0.957933\pi\)
\(18\) 0 0
\(19\) 4.00633 1.71737i 0.919114 0.393991i
\(20\) 0.602444i 0.134711i
\(21\) 0 0
\(22\) 1.46816i 0.313012i
\(23\) 2.83683 0.591520 0.295760 0.955262i \(-0.404427\pi\)
0.295760 + 0.955262i \(0.404427\pi\)
\(24\) 0 0
\(25\) 4.63706 0.927412
\(26\) 2.60244i 0.510381i
\(27\) 0 0
\(28\) 0.307956 + 2.62777i 0.0581982 + 0.496601i
\(29\) 6.19185i 1.14980i −0.818224 0.574899i \(-0.805041\pi\)
0.818224 0.574899i \(-0.194959\pi\)
\(30\) 0 0
\(31\) 7.12381 1.27947 0.639737 0.768594i \(-0.279043\pi\)
0.639737 + 0.768594i \(0.279043\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 1.08663 0.186356
\(35\) −1.58308 + 0.185526i −0.267590 + 0.0313597i
\(36\) 0 0
\(37\) 5.73972i 0.943605i −0.881704 0.471802i \(-0.843604\pi\)
0.881704 0.471802i \(-0.156396\pi\)
\(38\) −1.71737 4.00633i −0.278594 0.649912i
\(39\) 0 0
\(40\) 0.602444 0.0952547
\(41\) −9.86097 −1.54003 −0.770013 0.638029i \(-0.779750\pi\)
−0.770013 + 0.638029i \(0.779750\pi\)
\(42\) 0 0
\(43\) −12.1295 −1.84974 −0.924869 0.380286i \(-0.875826\pi\)
−0.924869 + 0.380286i \(0.875826\pi\)
\(44\) 1.46816 0.221333
\(45\) 0 0
\(46\) 2.83683i 0.418268i
\(47\) 0.752759i 0.109801i 0.998492 + 0.0549006i \(0.0174842\pi\)
−0.998492 + 0.0549006i \(0.982516\pi\)
\(48\) 0 0
\(49\) −6.81033 + 1.61847i −0.972904 + 0.231211i
\(50\) 4.63706i 0.655780i
\(51\) 0 0
\(52\) −2.60244 −0.360894
\(53\) 4.81882i 0.661916i −0.943645 0.330958i \(-0.892628\pi\)
0.943645 0.330958i \(-0.107372\pi\)
\(54\) 0 0
\(55\) 0.884483i 0.119264i
\(56\) 2.62777 0.307956i 0.351150 0.0411524i
\(57\) 0 0
\(58\) −6.19185 −0.813030
\(59\) −0.799650 −0.104106 −0.0520528 0.998644i \(-0.516576\pi\)
−0.0520528 + 0.998644i \(0.516576\pi\)
\(60\) 0 0
\(61\) 8.69481i 1.11326i 0.830762 + 0.556628i \(0.187905\pi\)
−0.830762 + 0.556628i \(0.812095\pi\)
\(62\) 7.12381i 0.904725i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.56783i 0.194465i
\(66\) 0 0
\(67\) 5.90487i 0.721395i −0.932683 0.360697i \(-0.882539\pi\)
0.932683 0.360697i \(-0.117461\pi\)
\(68\) 1.08663i 0.131774i
\(69\) 0 0
\(70\) 0.185526 + 1.58308i 0.0221746 + 0.189215i
\(71\) 4.75712i 0.564566i −0.959331 0.282283i \(-0.908908\pi\)
0.959331 0.282283i \(-0.0910917\pi\)
\(72\) 0 0
\(73\) 7.95747i 0.931351i −0.884956 0.465675i \(-0.845811\pi\)
0.884956 0.465675i \(-0.154189\pi\)
\(74\) −5.73972 −0.667229
\(75\) 0 0
\(76\) −4.00633 + 1.71737i −0.459557 + 0.196996i
\(77\) 0.452128 + 3.85798i 0.0515248 + 0.439658i
\(78\) 0 0
\(79\) 0.00299468i 0.000336928i 1.00000 0.000168464i \(5.36238e-5\pi\)
−1.00000 0.000168464i \(0.999946\pi\)
\(80\) 0.602444i 0.0673553i
\(81\) 0 0
\(82\) 9.86097i 1.08896i
\(83\) 15.9162i 1.74703i 0.486801 + 0.873513i \(0.338164\pi\)
−0.486801 + 0.873513i \(0.661836\pi\)
\(84\) 0 0
\(85\) 0.654635 0.0710051
\(86\) 12.1295i 1.30796i
\(87\) 0 0
\(88\) 1.46816i 0.156506i
\(89\) 9.74408 1.03287 0.516435 0.856326i \(-0.327259\pi\)
0.516435 + 0.856326i \(0.327259\pi\)
\(90\) 0 0
\(91\) −0.801438 6.83862i −0.0840136 0.716882i
\(92\) −2.83683 −0.295760
\(93\) 0 0
\(94\) 0.752759 0.0776412
\(95\) −1.03462 2.41359i −0.106150 0.247629i
\(96\) 0 0
\(97\) 0.784815 0.0796859 0.0398430 0.999206i \(-0.487314\pi\)
0.0398430 + 0.999206i \(0.487314\pi\)
\(98\) 1.61847 + 6.81033i 0.163491 + 0.687947i
\(99\) 0 0
\(100\) −4.63706 −0.463706
\(101\) 15.7813i 1.57029i −0.619309 0.785147i \(-0.712587\pi\)
0.619309 0.785147i \(-0.287413\pi\)
\(102\) 0 0
\(103\) −9.32160 −0.918484 −0.459242 0.888311i \(-0.651879\pi\)
−0.459242 + 0.888311i \(0.651879\pi\)
\(104\) 2.60244i 0.255191i
\(105\) 0 0
\(106\) −4.81882 −0.468046
\(107\) 12.6843i 1.22624i −0.789989 0.613120i \(-0.789914\pi\)
0.789989 0.613120i \(-0.210086\pi\)
\(108\) 0 0
\(109\) 17.7254i 1.69779i −0.528562 0.848895i \(-0.677269\pi\)
0.528562 0.848895i \(-0.322731\pi\)
\(110\) 0.884483 0.0843321
\(111\) 0 0
\(112\) −0.307956 2.62777i −0.0290991 0.248301i
\(113\) 9.68177i 0.910785i −0.890291 0.455392i \(-0.849499\pi\)
0.890291 0.455392i \(-0.150501\pi\)
\(114\) 0 0
\(115\) 1.70903i 0.159368i
\(116\) 6.19185i 0.574899i
\(117\) 0 0
\(118\) 0.799650i 0.0736138i
\(119\) 2.85542 0.334635i 0.261756 0.0306759i
\(120\) 0 0
\(121\) −8.84451 −0.804046
\(122\) 8.69481 0.787191
\(123\) 0 0
\(124\) −7.12381 −0.639737
\(125\) 5.80579i 0.519285i
\(126\) 0 0
\(127\) 19.3606i 1.71797i 0.512000 + 0.858986i \(0.328905\pi\)
−0.512000 + 0.858986i \(0.671095\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −1.56783 −0.137507
\(131\) 13.9458i 1.21845i 0.792997 + 0.609225i \(0.208520\pi\)
−0.792997 + 0.609225i \(0.791480\pi\)
\(132\) 0 0
\(133\) −5.74662 9.99882i −0.498295 0.867008i
\(134\) −5.90487 −0.510103
\(135\) 0 0
\(136\) −1.08663 −0.0931780
\(137\) −16.2603 −1.38921 −0.694605 0.719392i \(-0.744421\pi\)
−0.694605 + 0.719392i \(0.744421\pi\)
\(138\) 0 0
\(139\) 6.03418i 0.511813i −0.966702 0.255906i \(-0.917626\pi\)
0.966702 0.255906i \(-0.0823739\pi\)
\(140\) 1.58308 0.185526i 0.133795 0.0156798i
\(141\) 0 0
\(142\) −4.75712 −0.399208
\(143\) −3.82080 −0.319511
\(144\) 0 0
\(145\) −3.73024 −0.309780
\(146\) −7.95747 −0.658565
\(147\) 0 0
\(148\) 5.73972i 0.471802i
\(149\) −0.944612 −0.0773856 −0.0386928 0.999251i \(-0.512319\pi\)
−0.0386928 + 0.999251i \(0.512319\pi\)
\(150\) 0 0
\(151\) 2.62997i 0.214024i 0.994258 + 0.107012i \(0.0341283\pi\)
−0.994258 + 0.107012i \(0.965872\pi\)
\(152\) 1.71737 + 4.00633i 0.139297 + 0.324956i
\(153\) 0 0
\(154\) 3.85798 0.452128i 0.310885 0.0364335i
\(155\) 4.29170i 0.344717i
\(156\) 0 0
\(157\) 7.30560i 0.583050i −0.956563 0.291525i \(-0.905837\pi\)
0.956563 0.291525i \(-0.0941627\pi\)
\(158\) 0.00299468 0.000238244
\(159\) 0 0
\(160\) −0.602444 −0.0476274
\(161\) −0.873619 7.45453i −0.0688508 0.587499i
\(162\) 0 0
\(163\) −6.27456 −0.491461 −0.245731 0.969338i \(-0.579028\pi\)
−0.245731 + 0.969338i \(0.579028\pi\)
\(164\) 9.86097 0.770013
\(165\) 0 0
\(166\) 15.9162 1.23533
\(167\) 17.0317 1.31795 0.658975 0.752165i \(-0.270990\pi\)
0.658975 + 0.752165i \(0.270990\pi\)
\(168\) 0 0
\(169\) −6.22729 −0.479022
\(170\) 0.654635i 0.0502082i
\(171\) 0 0
\(172\) 12.1295 0.924869
\(173\) 12.6805 0.964082 0.482041 0.876149i \(-0.339896\pi\)
0.482041 + 0.876149i \(0.339896\pi\)
\(174\) 0 0
\(175\) −1.42801 12.1851i −0.107947 0.921109i
\(176\) −1.46816 −0.110667
\(177\) 0 0
\(178\) 9.74408i 0.730350i
\(179\) 13.5603i 1.01355i 0.862079 + 0.506774i \(0.169162\pi\)
−0.862079 + 0.506774i \(0.830838\pi\)
\(180\) 0 0
\(181\) −14.1772 −1.05378 −0.526891 0.849933i \(-0.676643\pi\)
−0.526891 + 0.849933i \(0.676643\pi\)
\(182\) −6.83862 + 0.801438i −0.506912 + 0.0594066i
\(183\) 0 0
\(184\) 2.83683i 0.209134i
\(185\) −3.45786 −0.254227
\(186\) 0 0
\(187\) 1.59535i 0.116663i
\(188\) 0.752759i 0.0549006i
\(189\) 0 0
\(190\) −2.41359 + 1.03462i −0.175100 + 0.0750591i
\(191\) −21.4891 −1.55490 −0.777449 0.628947i \(-0.783486\pi\)
−0.777449 + 0.628947i \(0.783486\pi\)
\(192\) 0 0
\(193\) 9.87657i 0.710931i 0.934689 + 0.355466i \(0.115678\pi\)
−0.934689 + 0.355466i \(0.884322\pi\)
\(194\) 0.784815i 0.0563464i
\(195\) 0 0
\(196\) 6.81033 1.61847i 0.486452 0.115605i
\(197\) 18.6902 1.33162 0.665811 0.746120i \(-0.268086\pi\)
0.665811 + 0.746120i \(0.268086\pi\)
\(198\) 0 0
\(199\) 14.0550i 0.996329i −0.867083 0.498165i \(-0.834008\pi\)
0.867083 0.498165i \(-0.165992\pi\)
\(200\) 4.63706i 0.327890i
\(201\) 0 0
\(202\) −15.7813 −1.11037
\(203\) −16.2707 + 1.90682i −1.14198 + 0.133832i
\(204\) 0 0
\(205\) 5.94068i 0.414915i
\(206\) 9.32160i 0.649467i
\(207\) 0 0
\(208\) 2.60244 0.180447
\(209\) −5.88192 + 2.52137i −0.406861 + 0.174407i
\(210\) 0 0
\(211\) 11.9385i 0.821877i −0.911663 0.410939i \(-0.865201\pi\)
0.911663 0.410939i \(-0.134799\pi\)
\(212\) 4.81882i 0.330958i
\(213\) 0 0
\(214\) −12.6843 −0.867083
\(215\) 7.30737i 0.498358i
\(216\) 0 0
\(217\) −2.19382 18.7197i −0.148926 1.27078i
\(218\) −17.7254 −1.20052
\(219\) 0 0
\(220\) 0.884483i 0.0596318i
\(221\) 2.82790i 0.190225i
\(222\) 0 0
\(223\) 12.8928 0.863364 0.431682 0.902026i \(-0.357920\pi\)
0.431682 + 0.902026i \(0.357920\pi\)
\(224\) −2.62777 + 0.307956i −0.175575 + 0.0205762i
\(225\) 0 0
\(226\) −9.68177 −0.644022
\(227\) 2.30063 0.152698 0.0763491 0.997081i \(-0.475674\pi\)
0.0763491 + 0.997081i \(0.475674\pi\)
\(228\) 0 0
\(229\) 4.77358i 0.315447i 0.987483 + 0.157724i \(0.0504155\pi\)
−0.987483 + 0.157724i \(0.949585\pi\)
\(230\) −1.70903 −0.112690
\(231\) 0 0
\(232\) 6.19185 0.406515
\(233\) 24.6124 1.61241 0.806205 0.591636i \(-0.201518\pi\)
0.806205 + 0.591636i \(0.201518\pi\)
\(234\) 0 0
\(235\) 0.453495 0.0295828
\(236\) 0.799650 0.0520528
\(237\) 0 0
\(238\) −0.334635 2.85542i −0.0216912 0.185089i
\(239\) 14.1110 0.912762 0.456381 0.889785i \(-0.349145\pi\)
0.456381 + 0.889785i \(0.349145\pi\)
\(240\) 0 0
\(241\) 2.19261 0.141239 0.0706193 0.997503i \(-0.477502\pi\)
0.0706193 + 0.997503i \(0.477502\pi\)
\(242\) 8.84451i 0.568547i
\(243\) 0 0
\(244\) 8.69481i 0.556628i
\(245\) 0.975040 + 4.10284i 0.0622930 + 0.262121i
\(246\) 0 0
\(247\) 10.4262 4.46935i 0.663406 0.284378i
\(248\) 7.12381i 0.452363i
\(249\) 0 0
\(250\) −5.80579 −0.367190
\(251\) 15.5451i 0.981198i −0.871385 0.490599i \(-0.836778\pi\)
0.871385 0.490599i \(-0.163222\pi\)
\(252\) 0 0
\(253\) −4.16492 −0.261846
\(254\) 19.3606 1.21479
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.27531 0.329065 0.164532 0.986372i \(-0.447389\pi\)
0.164532 + 0.986372i \(0.447389\pi\)
\(258\) 0 0
\(259\) −15.0827 + 1.76758i −0.937191 + 0.109832i
\(260\) 1.56783i 0.0972325i
\(261\) 0 0
\(262\) 13.9458 0.861575
\(263\) −26.3265 −1.62336 −0.811681 0.584101i \(-0.801447\pi\)
−0.811681 + 0.584101i \(0.801447\pi\)
\(264\) 0 0
\(265\) −2.90307 −0.178334
\(266\) −9.99882 + 5.74662i −0.613067 + 0.352348i
\(267\) 0 0
\(268\) 5.90487i 0.360697i
\(269\) 11.7926 0.719004 0.359502 0.933144i \(-0.382946\pi\)
0.359502 + 0.933144i \(0.382946\pi\)
\(270\) 0 0
\(271\) 7.03439i 0.427309i 0.976909 + 0.213654i \(0.0685366\pi\)
−0.976909 + 0.213654i \(0.931463\pi\)
\(272\) 1.08663i 0.0658868i
\(273\) 0 0
\(274\) 16.2603i 0.982319i
\(275\) −6.80794 −0.410534
\(276\) 0 0
\(277\) −1.79468 −0.107832 −0.0539158 0.998545i \(-0.517170\pi\)
−0.0539158 + 0.998545i \(0.517170\pi\)
\(278\) −6.03418 −0.361906
\(279\) 0 0
\(280\) −0.185526 1.58308i −0.0110873 0.0946073i
\(281\) 11.6228i 0.693357i −0.937984 0.346678i \(-0.887310\pi\)
0.937984 0.346678i \(-0.112690\pi\)
\(282\) 0 0
\(283\) 3.48005i 0.206867i −0.994636 0.103434i \(-0.967017\pi\)
0.994636 0.103434i \(-0.0329830\pi\)
\(284\) 4.75712i 0.282283i
\(285\) 0 0
\(286\) 3.82080i 0.225929i
\(287\) 3.03675 + 25.9123i 0.179253 + 1.52956i
\(288\) 0 0
\(289\) 15.8192 0.930543
\(290\) 3.73024i 0.219047i
\(291\) 0 0
\(292\) 7.95747i 0.465675i
\(293\) 7.09787 0.414662 0.207331 0.978271i \(-0.433522\pi\)
0.207331 + 0.978271i \(0.433522\pi\)
\(294\) 0 0
\(295\) 0.481744i 0.0280482i
\(296\) 5.73972 0.333615
\(297\) 0 0
\(298\) 0.944612i 0.0547199i
\(299\) 7.38269 0.426952
\(300\) 0 0
\(301\) 3.73537 + 31.8736i 0.215303 + 1.83717i
\(302\) 2.62997 0.151338
\(303\) 0 0
\(304\) 4.00633 1.71737i 0.229779 0.0984978i
\(305\) 5.23813 0.299935
\(306\) 0 0
\(307\) −3.10840 −0.177405 −0.0887027 0.996058i \(-0.528272\pi\)
−0.0887027 + 0.996058i \(0.528272\pi\)
\(308\) −0.452128 3.85798i −0.0257624 0.219829i
\(309\) 0 0
\(310\) −4.29170 −0.243752
\(311\) 1.56565i 0.0887797i −0.999014 0.0443899i \(-0.985866\pi\)
0.999014 0.0443899i \(-0.0141344\pi\)
\(312\) 0 0
\(313\) 15.4848i 0.875253i 0.899157 + 0.437626i \(0.144181\pi\)
−0.899157 + 0.437626i \(0.855819\pi\)
\(314\) −7.30560 −0.412279
\(315\) 0 0
\(316\) 0.00299468i 0.000168464i
\(317\) 23.4743i 1.31845i −0.751946 0.659225i \(-0.770885\pi\)
0.751946 0.659225i \(-0.229115\pi\)
\(318\) 0 0
\(319\) 9.09062i 0.508977i
\(320\) 0.602444i 0.0336776i
\(321\) 0 0
\(322\) −7.45453 + 0.873619i −0.415425 + 0.0486849i
\(323\) 1.86615 + 4.35340i 0.103835 + 0.242230i
\(324\) 0 0
\(325\) 12.0677 0.669395
\(326\) 6.27456i 0.347516i
\(327\) 0 0
\(328\) 9.86097i 0.544481i
\(329\) 1.97808 0.231817i 0.109055 0.0127805i
\(330\) 0 0
\(331\) 21.3550i 1.17378i 0.809668 + 0.586888i \(0.199647\pi\)
−0.809668 + 0.586888i \(0.800353\pi\)
\(332\) 15.9162i 0.873513i
\(333\) 0 0
\(334\) 17.0317i 0.931932i
\(335\) −3.55735 −0.194359
\(336\) 0 0
\(337\) 16.1349i 0.878924i 0.898261 + 0.439462i \(0.144831\pi\)
−0.898261 + 0.439462i \(0.855169\pi\)
\(338\) 6.22729i 0.338720i
\(339\) 0 0
\(340\) −0.654635 −0.0355026
\(341\) −10.4589 −0.566380
\(342\) 0 0
\(343\) 6.35025 + 17.3975i 0.342882 + 0.939379i
\(344\) 12.1295i 0.653981i
\(345\) 0 0
\(346\) 12.6805i 0.681709i
\(347\) 18.8578 1.01234 0.506169 0.862434i \(-0.331061\pi\)
0.506169 + 0.862434i \(0.331061\pi\)
\(348\) 0 0
\(349\) 16.0310i 0.858120i 0.903276 + 0.429060i \(0.141155\pi\)
−0.903276 + 0.429060i \(0.858845\pi\)
\(350\) −12.1851 + 1.42801i −0.651322 + 0.0763304i
\(351\) 0 0
\(352\) 1.46816i 0.0782531i
\(353\) 11.0403i 0.587615i −0.955865 0.293808i \(-0.905078\pi\)
0.955865 0.293808i \(-0.0949225\pi\)
\(354\) 0 0
\(355\) −2.86590 −0.152106
\(356\) −9.74408 −0.516435
\(357\) 0 0
\(358\) 13.5603 0.716686
\(359\) 7.24265 0.382253 0.191126 0.981565i \(-0.438786\pi\)
0.191126 + 0.981565i \(0.438786\pi\)
\(360\) 0 0
\(361\) 13.1013 13.7607i 0.689542 0.724246i
\(362\) 14.1772i 0.745137i
\(363\) 0 0
\(364\) 0.801438 + 6.83862i 0.0420068 + 0.358441i
\(365\) −4.79393 −0.250926
\(366\) 0 0
\(367\) 4.37658i 0.228456i −0.993455 0.114228i \(-0.963561\pi\)
0.993455 0.114228i \(-0.0364394\pi\)
\(368\) 2.83683 0.147880
\(369\) 0 0
\(370\) 3.45786i 0.179766i
\(371\) −12.6627 + 1.48399i −0.657417 + 0.0770447i
\(372\) 0 0
\(373\) 4.14709i 0.214728i 0.994220 + 0.107364i \(0.0342410\pi\)
−0.994220 + 0.107364i \(0.965759\pi\)
\(374\) −1.59535 −0.0824935
\(375\) 0 0
\(376\) −0.752759 −0.0388206
\(377\) 16.1139i 0.829911i
\(378\) 0 0
\(379\) 23.2482i 1.19418i −0.802174 0.597091i \(-0.796323\pi\)
0.802174 0.597091i \(-0.203677\pi\)
\(380\) 1.03462 + 2.41359i 0.0530748 + 0.123814i
\(381\) 0 0
\(382\) 21.4891i 1.09948i
\(383\) −17.7789 −0.908457 −0.454228 0.890885i \(-0.650085\pi\)
−0.454228 + 0.890885i \(0.650085\pi\)
\(384\) 0 0
\(385\) 2.32422 0.272382i 0.118453 0.0138819i
\(386\) 9.87657 0.502704
\(387\) 0 0
\(388\) −0.784815 −0.0398430
\(389\) 4.77920 0.242315 0.121157 0.992633i \(-0.461339\pi\)
0.121157 + 0.992633i \(0.461339\pi\)
\(390\) 0 0
\(391\) 3.08259i 0.155893i
\(392\) −1.61847 6.81033i −0.0817453 0.343973i
\(393\) 0 0
\(394\) 18.6902i 0.941600i
\(395\) 0.00180413 9.07755e−5
\(396\) 0 0
\(397\) 33.0306i 1.65776i 0.559427 + 0.828880i \(0.311021\pi\)
−0.559427 + 0.828880i \(0.688979\pi\)
\(398\) −14.0550 −0.704511
\(399\) 0 0
\(400\) 4.63706 0.231853
\(401\) 6.66991i 0.333080i 0.986035 + 0.166540i \(0.0532594\pi\)
−0.986035 + 0.166540i \(0.946741\pi\)
\(402\) 0 0
\(403\) 18.5393 0.923509
\(404\) 15.7813i 0.785147i
\(405\) 0 0
\(406\) 1.90682 + 16.2707i 0.0946338 + 0.807504i
\(407\) 8.42682i 0.417702i
\(408\) 0 0
\(409\) −5.36786 −0.265423 −0.132712 0.991155i \(-0.542368\pi\)
−0.132712 + 0.991155i \(0.542368\pi\)
\(410\) 5.94068 0.293389
\(411\) 0 0
\(412\) 9.32160 0.459242
\(413\) 0.246257 + 2.10130i 0.0121175 + 0.103398i
\(414\) 0 0
\(415\) 9.58859 0.470685
\(416\) 2.60244i 0.127595i
\(417\) 0 0
\(418\) 2.52137 + 5.88192i 0.123324 + 0.287694i
\(419\) 18.5084i 0.904193i −0.891969 0.452096i \(-0.850676\pi\)
0.891969 0.452096i \(-0.149324\pi\)
\(420\) 0 0
\(421\) 30.7310i 1.49774i 0.662719 + 0.748868i \(0.269403\pi\)
−0.662719 + 0.748868i \(0.730597\pi\)
\(422\) −11.9385 −0.581155
\(423\) 0 0
\(424\) 4.81882 0.234023
\(425\) 5.03878i 0.244417i
\(426\) 0 0
\(427\) 22.8479 2.67762i 1.10569 0.129579i
\(428\) 12.6843i 0.613120i
\(429\) 0 0
\(430\) 7.30737 0.352393
\(431\) 3.63136i 0.174917i 0.996168 + 0.0874583i \(0.0278744\pi\)
−0.996168 + 0.0874583i \(0.972126\pi\)
\(432\) 0 0
\(433\) 15.3433 0.737352 0.368676 0.929558i \(-0.379811\pi\)
0.368676 + 0.929558i \(0.379811\pi\)
\(434\) −18.7197 + 2.19382i −0.898575 + 0.105307i
\(435\) 0 0
\(436\) 17.7254i 0.848895i
\(437\) 11.3653 4.87188i 0.543674 0.233054i
\(438\) 0 0
\(439\) 38.5479 1.83979 0.919895 0.392164i \(-0.128273\pi\)
0.919895 + 0.392164i \(0.128273\pi\)
\(440\) −0.884483 −0.0421661
\(441\) 0 0
\(442\) 2.82790 0.134509
\(443\) −19.0232 −0.903817 −0.451909 0.892064i \(-0.649257\pi\)
−0.451909 + 0.892064i \(0.649257\pi\)
\(444\) 0 0
\(445\) 5.87026i 0.278277i
\(446\) 12.8928i 0.610491i
\(447\) 0 0
\(448\) 0.307956 + 2.62777i 0.0145496 + 0.124150i
\(449\) 27.5946i 1.30227i −0.758962 0.651134i \(-0.774293\pi\)
0.758962 0.651134i \(-0.225707\pi\)
\(450\) 0 0
\(451\) 14.4775 0.681718
\(452\) 9.68177i 0.455392i
\(453\) 0 0
\(454\) 2.30063i 0.107974i
\(455\) −4.11988 + 0.482822i −0.193143 + 0.0226350i
\(456\) 0 0
\(457\) 30.7393 1.43792 0.718962 0.695050i \(-0.244618\pi\)
0.718962 + 0.695050i \(0.244618\pi\)
\(458\) 4.77358 0.223055
\(459\) 0 0
\(460\) 1.70903i 0.0796840i
\(461\) 20.6499i 0.961760i −0.876786 0.480880i \(-0.840317\pi\)
0.876786 0.480880i \(-0.159683\pi\)
\(462\) 0 0
\(463\) −38.0039 −1.76619 −0.883095 0.469193i \(-0.844545\pi\)
−0.883095 + 0.469193i \(0.844545\pi\)
\(464\) 6.19185i 0.287450i
\(465\) 0 0
\(466\) 24.6124i 1.14015i
\(467\) 16.0408i 0.742279i 0.928577 + 0.371139i \(0.121033\pi\)
−0.928577 + 0.371139i \(0.878967\pi\)
\(468\) 0 0
\(469\) −15.5166 + 1.81844i −0.716491 + 0.0839678i
\(470\) 0.453495i 0.0209182i
\(471\) 0 0
\(472\) 0.799650i 0.0368069i
\(473\) 17.8081 0.818817
\(474\) 0 0
\(475\) 18.5776 7.96354i 0.852398 0.365392i
\(476\) −2.85542 + 0.334635i −0.130878 + 0.0153380i
\(477\) 0 0
\(478\) 14.1110i 0.645420i
\(479\) 19.4280i 0.887690i −0.896104 0.443845i \(-0.853614\pi\)
0.896104 0.443845i \(-0.146386\pi\)
\(480\) 0 0
\(481\) 14.9373i 0.681083i
\(482\) 2.19261i 0.0998708i
\(483\) 0 0
\(484\) 8.84451 0.402023
\(485\) 0.472807i 0.0214691i
\(486\) 0 0
\(487\) 1.60382i 0.0726760i 0.999340 + 0.0363380i \(0.0115693\pi\)
−0.999340 + 0.0363380i \(0.988431\pi\)
\(488\) −8.69481 −0.393596
\(489\) 0 0
\(490\) 4.10284 0.975040i 0.185347 0.0440478i
\(491\) 35.6125 1.60717 0.803586 0.595189i \(-0.202923\pi\)
0.803586 + 0.595189i \(0.202923\pi\)
\(492\) 0 0
\(493\) 6.72827 0.303026
\(494\) −4.46935 10.4262i −0.201086 0.469099i
\(495\) 0 0
\(496\) 7.12381 0.319869
\(497\) −12.5006 + 1.46498i −0.560728 + 0.0657135i
\(498\) 0 0
\(499\) 2.89620 0.129652 0.0648258 0.997897i \(-0.479351\pi\)
0.0648258 + 0.997897i \(0.479351\pi\)
\(500\) 5.80579i 0.259643i
\(501\) 0 0
\(502\) −15.5451 −0.693812
\(503\) 34.7741i 1.55050i 0.631654 + 0.775250i \(0.282376\pi\)
−0.631654 + 0.775250i \(0.717624\pi\)
\(504\) 0 0
\(505\) −9.50732 −0.423070
\(506\) 4.16492i 0.185153i
\(507\) 0 0
\(508\) 19.3606i 0.858986i
\(509\) 37.1924 1.64852 0.824262 0.566208i \(-0.191590\pi\)
0.824262 + 0.566208i \(0.191590\pi\)
\(510\) 0 0
\(511\) −20.9104 + 2.45055i −0.925020 + 0.108406i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 5.27531i 0.232684i
\(515\) 5.61574i 0.247459i
\(516\) 0 0
\(517\) 1.10517i 0.0486053i
\(518\) 1.76758 + 15.0827i 0.0776631 + 0.662694i
\(519\) 0 0
\(520\) 1.56783 0.0687537
\(521\) −16.9681 −0.743384 −0.371692 0.928356i \(-0.621222\pi\)
−0.371692 + 0.928356i \(0.621222\pi\)
\(522\) 0 0
\(523\) 1.64279 0.0718340 0.0359170 0.999355i \(-0.488565\pi\)
0.0359170 + 0.999355i \(0.488565\pi\)
\(524\) 13.9458i 0.609225i
\(525\) 0 0
\(526\) 26.3265i 1.14789i
\(527\) 7.74096i 0.337202i
\(528\) 0 0
\(529\) −14.9524 −0.650104
\(530\) 2.90307i 0.126101i
\(531\) 0 0
\(532\) 5.74662 + 9.99882i 0.249147 + 0.433504i
\(533\) −25.6626 −1.11157
\(534\) 0 0
\(535\) −7.64160 −0.330375
\(536\) 5.90487 0.255052
\(537\) 0 0
\(538\) 11.7926i 0.508413i
\(539\) 9.99864 2.37618i 0.430672 0.102349i
\(540\) 0 0
\(541\) 20.5601 0.883947 0.441974 0.897028i \(-0.354278\pi\)
0.441974 + 0.897028i \(0.354278\pi\)
\(542\) 7.03439 0.302153
\(543\) 0 0
\(544\) 1.08663 0.0465890
\(545\) −10.6786 −0.457420
\(546\) 0 0
\(547\) 22.9504i 0.981287i 0.871360 + 0.490644i \(0.163238\pi\)
−0.871360 + 0.490644i \(0.836762\pi\)
\(548\) 16.2603 0.694605
\(549\) 0 0
\(550\) 6.80794i 0.290292i
\(551\) −10.6337 24.8066i −0.453010 1.05680i
\(552\) 0 0
\(553\) 0.00786933 0.000922231i 0.000334638 3.92172e-5i
\(554\) 1.79468i 0.0762484i
\(555\) 0 0
\(556\) 6.03418i 0.255906i
\(557\) 9.90274 0.419593 0.209796 0.977745i \(-0.432720\pi\)
0.209796 + 0.977745i \(0.432720\pi\)
\(558\) 0 0
\(559\) −31.5665 −1.33512
\(560\) −1.58308 + 0.185526i −0.0668974 + 0.00783991i
\(561\) 0 0
\(562\) −11.6228 −0.490277
\(563\) −11.3791 −0.479571 −0.239786 0.970826i \(-0.577077\pi\)
−0.239786 + 0.970826i \(0.577077\pi\)
\(564\) 0 0
\(565\) −5.83272 −0.245385
\(566\) −3.48005 −0.146277
\(567\) 0 0
\(568\) 4.75712 0.199604
\(569\) 34.7575i 1.45711i −0.684987 0.728555i \(-0.740192\pi\)
0.684987 0.728555i \(-0.259808\pi\)
\(570\) 0 0
\(571\) 30.9139 1.29371 0.646854 0.762614i \(-0.276084\pi\)
0.646854 + 0.762614i \(0.276084\pi\)
\(572\) 3.82080 0.159756
\(573\) 0 0
\(574\) 25.9123 3.03675i 1.08156 0.126751i
\(575\) 13.1546 0.548583
\(576\) 0 0
\(577\) 39.8992i 1.66102i −0.557001 0.830512i \(-0.688048\pi\)
0.557001 0.830512i \(-0.311952\pi\)
\(578\) 15.8192i 0.657993i
\(579\) 0 0
\(580\) 3.73024 0.154890
\(581\) 41.8240 4.90148i 1.73515 0.203348i
\(582\) 0 0
\(583\) 7.07480i 0.293008i
\(584\) 7.95747 0.329282
\(585\) 0 0
\(586\) 7.09787i 0.293210i
\(587\) 28.8180i 1.18945i 0.803930 + 0.594724i \(0.202739\pi\)
−0.803930 + 0.594724i \(0.797261\pi\)
\(588\) 0 0
\(589\) 28.5403 12.2342i 1.17598 0.504102i
\(590\) 0.481744 0.0198331
\(591\) 0 0
\(592\) 5.73972i 0.235901i
\(593\) 41.1324i 1.68911i 0.535472 + 0.844553i \(0.320134\pi\)
−0.535472 + 0.844553i \(0.679866\pi\)
\(594\) 0 0
\(595\) −0.201599 1.72023i −0.00826474 0.0705225i
\(596\) 0.944612 0.0386928
\(597\) 0 0
\(598\) 7.38269i 0.301901i
\(599\) 2.29940i 0.0939511i 0.998896 + 0.0469755i \(0.0149583\pi\)
−0.998896 + 0.0469755i \(0.985042\pi\)
\(600\) 0 0
\(601\) −4.48455 −0.182928 −0.0914642 0.995808i \(-0.529155\pi\)
−0.0914642 + 0.995808i \(0.529155\pi\)
\(602\) 31.8736 3.73537i 1.29907 0.152242i
\(603\) 0 0
\(604\) 2.62997i 0.107012i
\(605\) 5.32832i 0.216627i
\(606\) 0 0
\(607\) −23.0487 −0.935517 −0.467758 0.883856i \(-0.654938\pi\)
−0.467758 + 0.883856i \(0.654938\pi\)
\(608\) −1.71737 4.00633i −0.0696485 0.162478i
\(609\) 0 0
\(610\) 5.23813i 0.212086i
\(611\) 1.95901i 0.0792532i
\(612\) 0 0
\(613\) −29.5731 −1.19445 −0.597224 0.802075i \(-0.703729\pi\)
−0.597224 + 0.802075i \(0.703729\pi\)
\(614\) 3.10840i 0.125445i
\(615\) 0 0
\(616\) −3.85798 + 0.452128i −0.155442 + 0.0182168i
\(617\) −11.9663 −0.481745 −0.240873 0.970557i \(-0.577434\pi\)
−0.240873 + 0.970557i \(0.577434\pi\)
\(618\) 0 0
\(619\) 38.3335i 1.54075i −0.637590 0.770376i \(-0.720069\pi\)
0.637590 0.770376i \(-0.279931\pi\)
\(620\) 4.29170i 0.172359i
\(621\) 0 0
\(622\) −1.56565 −0.0627767
\(623\) −3.00075 25.6052i −0.120222 1.02585i
\(624\) 0 0
\(625\) 19.6876 0.787506
\(626\) 15.4848 0.618897
\(627\) 0 0
\(628\) 7.30560i 0.291525i
\(629\) 6.23697 0.248684
\(630\) 0 0
\(631\) 17.5444 0.698431 0.349215 0.937042i \(-0.386448\pi\)
0.349215 + 0.937042i \(0.386448\pi\)
\(632\) −0.00299468 −0.000119122
\(633\) 0 0
\(634\) −23.4743 −0.932285
\(635\) 11.6636 0.462858
\(636\) 0 0
\(637\) −17.7235 + 4.21199i −0.702230 + 0.166885i
\(638\) 9.09062 0.359901
\(639\) 0 0
\(640\) 0.602444 0.0238137
\(641\) 14.9850i 0.591871i −0.955208 0.295935i \(-0.904369\pi\)
0.955208 0.295935i \(-0.0956313\pi\)
\(642\) 0 0
\(643\) 12.1996i 0.481105i 0.970636 + 0.240553i \(0.0773287\pi\)
−0.970636 + 0.240553i \(0.922671\pi\)
\(644\) 0.873619 + 7.45453i 0.0344254 + 0.293750i
\(645\) 0 0
\(646\) 4.35340 1.86615i 0.171282 0.0734226i
\(647\) 18.9604i 0.745412i 0.927950 + 0.372706i \(0.121570\pi\)
−0.927950 + 0.372706i \(0.878430\pi\)
\(648\) 0 0
\(649\) 1.17401 0.0460841
\(650\) 12.0677i 0.473334i
\(651\) 0 0
\(652\) 6.27456 0.245731
\(653\) −11.9475 −0.467540 −0.233770 0.972292i \(-0.575106\pi\)
−0.233770 + 0.972292i \(0.575106\pi\)
\(654\) 0 0
\(655\) 8.40156 0.328276
\(656\) −9.86097 −0.385006
\(657\) 0 0
\(658\) −0.231817 1.97808i −0.00903716 0.0771135i
\(659\) 14.6240i 0.569671i 0.958576 + 0.284836i \(0.0919390\pi\)
−0.958576 + 0.284836i \(0.908061\pi\)
\(660\) 0 0
\(661\) 38.4941 1.49725 0.748623 0.662996i \(-0.230716\pi\)
0.748623 + 0.662996i \(0.230716\pi\)
\(662\) 21.3550 0.829984
\(663\) 0 0
\(664\) −15.9162 −0.617667
\(665\) −6.02373 + 3.46201i −0.233590 + 0.134251i
\(666\) 0 0
\(667\) 17.5652i 0.680128i
\(668\) −17.0317 −0.658975
\(669\) 0 0
\(670\) 3.55735i 0.137433i
\(671\) 12.7654i 0.492801i
\(672\) 0 0
\(673\) 39.1709i 1.50993i −0.655767 0.754963i \(-0.727655\pi\)
0.655767 0.754963i \(-0.272345\pi\)
\(674\) 16.1349 0.621493
\(675\) 0 0
\(676\) 6.22729 0.239511
\(677\) 13.6204 0.523476 0.261738 0.965139i \(-0.415704\pi\)
0.261738 + 0.965139i \(0.415704\pi\)
\(678\) 0 0
\(679\) −0.241689 2.06231i −0.00927516 0.0791443i
\(680\) 0.654635i 0.0251041i
\(681\) 0 0
\(682\) 10.4589i 0.400491i
\(683\) 44.7869i 1.71372i 0.515545 + 0.856862i \(0.327589\pi\)
−0.515545 + 0.856862i \(0.672411\pi\)
\(684\) 0 0
\(685\) 9.79590i 0.374282i
\(686\) 17.3975 6.35025i 0.664241 0.242454i
\(687\) 0 0
\(688\) −12.1295 −0.462435
\(689\) 12.5407i 0.477763i
\(690\) 0 0
\(691\) 34.8900i 1.32728i 0.748053 + 0.663639i \(0.230989\pi\)
−0.748053 + 0.663639i \(0.769011\pi\)
\(692\) −12.6805 −0.482041
\(693\) 0 0
\(694\) 18.8578i 0.715831i
\(695\) −3.63526 −0.137893
\(696\) 0 0
\(697\) 10.7153i 0.405869i
\(698\) 16.0310 0.606783
\(699\) 0 0
\(700\) 1.42801 + 12.1851i 0.0539737 + 0.460554i
\(701\) 27.2769 1.03024 0.515118 0.857119i \(-0.327748\pi\)
0.515118 + 0.857119i \(0.327748\pi\)
\(702\) 0 0
\(703\) −9.85722 22.9952i −0.371772 0.867281i
\(704\) 1.46816 0.0553333
\(705\) 0 0
\(706\) −11.0403 −0.415507
\(707\) −41.4695 + 4.85993i −1.55962 + 0.182777i
\(708\) 0 0
\(709\) −6.36116 −0.238899 −0.119449 0.992840i \(-0.538113\pi\)
−0.119449 + 0.992840i \(0.538113\pi\)
\(710\) 2.86590i 0.107555i
\(711\) 0 0
\(712\) 9.74408i 0.365175i
\(713\) 20.2090 0.756835
\(714\) 0 0
\(715\) 2.30182i 0.0860831i
\(716\) 13.5603i 0.506774i
\(717\) 0 0
\(718\) 7.24265i 0.270293i
\(719\) 5.38127i 0.200687i 0.994953 + 0.100344i \(0.0319943\pi\)
−0.994953 + 0.100344i \(0.968006\pi\)
\(720\) 0 0
\(721\) 2.87064 + 24.4950i 0.106908 + 0.912241i
\(722\) −13.7607 13.1013i −0.512119 0.487580i
\(723\) 0 0
\(724\) 14.1772 0.526891
\(725\) 28.7120i 1.06634i
\(726\) 0 0
\(727\) 27.6598i 1.02584i −0.858435 0.512922i \(-0.828563\pi\)
0.858435 0.512922i \(-0.171437\pi\)
\(728\) 6.83862 0.801438i 0.253456 0.0297033i
\(729\) 0 0
\(730\) 4.79393i 0.177431i
\(731\) 13.1804i 0.487493i
\(732\) 0 0
\(733\) 3.25771i 0.120326i 0.998189 + 0.0601632i \(0.0191621\pi\)
−0.998189 + 0.0601632i \(0.980838\pi\)
\(734\) −4.37658 −0.161543
\(735\) 0 0
\(736\) 2.83683i 0.104567i
\(737\) 8.66928i 0.319337i
\(738\) 0 0
\(739\) 5.12780 0.188629 0.0943147 0.995542i \(-0.469934\pi\)
0.0943147 + 0.995542i \(0.469934\pi\)
\(740\) 3.45786 0.127114
\(741\) 0 0
\(742\) 1.48399 + 12.6627i 0.0544788 + 0.464864i
\(743\) 28.9581i 1.06237i 0.847256 + 0.531184i \(0.178253\pi\)
−0.847256 + 0.531184i \(0.821747\pi\)
\(744\) 0 0
\(745\) 0.569075i 0.0208493i
\(746\) 4.14709 0.151836
\(747\) 0 0
\(748\) 1.59535i 0.0583317i
\(749\) −33.3315 + 3.90622i −1.21791 + 0.142730i
\(750\) 0 0
\(751\) 40.2740i 1.46962i 0.678274 + 0.734809i \(0.262728\pi\)
−0.678274 + 0.734809i \(0.737272\pi\)
\(752\) 0.752759i 0.0274503i
\(753\) 0 0
\(754\) −16.1139 −0.586835
\(755\) 1.58441 0.0576625
\(756\) 0 0
\(757\) 25.0198 0.909358 0.454679 0.890655i \(-0.349754\pi\)
0.454679 + 0.890655i \(0.349754\pi\)
\(758\) −23.2482 −0.844414
\(759\) 0 0
\(760\) 2.41359 1.03462i 0.0875500 0.0375295i
\(761\) 24.3619i 0.883119i 0.897232 + 0.441560i \(0.145575\pi\)
−0.897232 + 0.441560i \(0.854425\pi\)
\(762\) 0 0
\(763\) −46.5783 + 5.45866i −1.68625 + 0.197617i
\(764\) 21.4891 0.777449
\(765\) 0 0
\(766\) 17.7789i 0.642376i
\(767\) −2.08105 −0.0751422
\(768\) 0 0
\(769\) 9.31118i 0.335770i −0.985807 0.167885i \(-0.946306\pi\)
0.985807 0.167885i \(-0.0536937\pi\)
\(770\) −0.272382 2.32422i −0.00981596 0.0837589i
\(771\) 0 0
\(772\) 9.87657i 0.355466i
\(773\) 44.0350 1.58383 0.791914 0.610633i \(-0.209085\pi\)
0.791914 + 0.610633i \(0.209085\pi\)
\(774\) 0 0
\(775\) 33.0336 1.18660
\(776\) 0.784815i 0.0281732i
\(777\) 0 0
\(778\) 4.77920i 0.171342i
\(779\) −39.5063 + 16.9349i −1.41546 + 0.606756i
\(780\) 0 0
\(781\) 6.98420i 0.249914i
\(782\) 3.08259 0.110233
\(783\) 0 0
\(784\) −6.81033 + 1.61847i −0.243226 + 0.0578026i
\(785\) −4.40121 −0.157086
\(786\) 0 0
\(787\) −1.21088 −0.0431631 −0.0215816 0.999767i \(-0.506870\pi\)
−0.0215816 + 0.999767i \(0.506870\pi\)
\(788\) −18.6902 −0.665811
\(789\) 0 0
\(790\) 0.00180413i 6.41880e-5i
\(791\) −25.4415 + 2.98156i −0.904594 + 0.106012i
\(792\) 0 0
\(793\) 22.6278i 0.803535i
\(794\) 33.0306 1.17221
\(795\) 0 0
\(796\) 14.0550i 0.498165i
\(797\) 19.0026 0.673107 0.336554 0.941664i \(-0.390739\pi\)
0.336554 + 0.941664i \(0.390739\pi\)
\(798\) 0 0
\(799\) −0.817973 −0.0289378
\(800\) 4.63706i 0.163945i
\(801\) 0 0
\(802\) 6.66991 0.235523
\(803\) 11.6828i 0.412278i
\(804\) 0 0
\(805\) −4.49094 + 0.526306i −0.158285 + 0.0185499i
\(806\) 18.5393i 0.653020i
\(807\) 0 0
\(808\) 15.7813 0.555183
\(809\) 28.0122 0.984858 0.492429 0.870353i \(-0.336109\pi\)
0.492429 + 0.870353i \(0.336109\pi\)
\(810\) 0 0
\(811\) 18.6270 0.654081 0.327041 0.945010i \(-0.393949\pi\)
0.327041 + 0.945010i \(0.393949\pi\)
\(812\) 16.2707 1.90682i 0.570991 0.0669162i
\(813\) 0 0
\(814\) 8.42682 0.295360
\(815\) 3.78007i 0.132410i
\(816\) 0 0
\(817\) −48.5949 + 20.8309i −1.70012 + 0.728781i
\(818\) 5.36786i 0.187683i
\(819\) 0 0
\(820\) 5.94068i 0.207458i
\(821\) −29.2267 −1.02002 −0.510009 0.860169i \(-0.670358\pi\)
−0.510009 + 0.860169i \(0.670358\pi\)
\(822\) 0 0
\(823\) −19.0198 −0.662989 −0.331495 0.943457i \(-0.607553\pi\)
−0.331495 + 0.943457i \(0.607553\pi\)
\(824\) 9.32160i 0.324733i
\(825\) 0 0
\(826\) 2.10130 0.246257i 0.0731134 0.00856838i
\(827\) 10.8091i 0.375870i −0.982181 0.187935i \(-0.939821\pi\)
0.982181 0.187935i \(-0.0601794\pi\)
\(828\) 0 0
\(829\) −45.7073 −1.58748 −0.793739 0.608258i \(-0.791869\pi\)
−0.793739 + 0.608258i \(0.791869\pi\)
\(830\) 9.58859i 0.332825i
\(831\) 0 0
\(832\) −2.60244 −0.0902235
\(833\) −1.75869 7.40032i −0.0609349 0.256406i
\(834\) 0 0
\(835\) 10.2606i 0.355084i
\(836\) 5.88192 2.52137i 0.203430 0.0872033i
\(837\) 0 0
\(838\) −18.5084 −0.639361
\(839\) 31.5684 1.08986 0.544931 0.838481i \(-0.316556\pi\)
0.544931 + 0.838481i \(0.316556\pi\)
\(840\) 0 0
\(841\) −9.33903 −0.322036
\(842\) 30.7310 1.05906
\(843\) 0 0
\(844\) 11.9385i 0.410939i
\(845\) 3.75159i 0.129059i
\(846\) 0 0
\(847\) 2.72372 + 23.2413i 0.0935882 + 0.798581i
\(848\) 4.81882i 0.165479i
\(849\) 0 0
\(850\) 5.03878 0.172829
\(851\) 16.2826i 0.558161i
\(852\) 0 0
\(853\) 28.1651i 0.964355i −0.876074 0.482178i \(-0.839846\pi\)
0.876074 0.482178i \(-0.160154\pi\)
\(854\) −2.67762 22.8479i −0.0916262 0.781840i
\(855\) 0 0
\(856\) 12.6843 0.433542
\(857\) −0.355917 −0.0121579 −0.00607895 0.999982i \(-0.501935\pi\)
−0.00607895 + 0.999982i \(0.501935\pi\)
\(858\) 0 0
\(859\) 28.3548i 0.967454i 0.875219 + 0.483727i \(0.160717\pi\)
−0.875219 + 0.483727i \(0.839283\pi\)
\(860\) 7.30737i 0.249179i
\(861\) 0 0
\(862\) 3.63136 0.123685
\(863\) 26.9169i 0.916264i 0.888884 + 0.458132i \(0.151481\pi\)
−0.888884 + 0.458132i \(0.848519\pi\)
\(864\) 0 0
\(865\) 7.63930i 0.259744i
\(866\) 15.3433i 0.521387i
\(867\) 0 0
\(868\) 2.19382 + 18.7197i 0.0744631 + 0.635389i
\(869\) 0.00439667i 0.000149147i
\(870\) 0 0
\(871\) 15.3671i 0.520694i
\(872\) 17.7254 0.600259
\(873\) 0 0
\(874\) −4.87188 11.3653i −0.164794 0.384436i
\(875\) −15.2563 + 1.78793i −0.515756 + 0.0604430i
\(876\) 0 0
\(877\) 27.5902i 0.931656i −0.884875 0.465828i \(-0.845757\pi\)
0.884875 0.465828i \(-0.154243\pi\)
\(878\) 38.5479i 1.30093i
\(879\) 0 0
\(880\) 0.884483i 0.0298159i
\(881\) 50.3489i 1.69630i 0.529759 + 0.848148i \(0.322282\pi\)
−0.529759 + 0.848148i \(0.677718\pi\)
\(882\) 0 0
\(883\) 15.3680 0.517175 0.258587 0.965988i \(-0.416743\pi\)
0.258587 + 0.965988i \(0.416743\pi\)
\(884\) 2.82790i 0.0951126i
\(885\) 0 0
\(886\) 19.0232i 0.639095i
\(887\) −17.5241 −0.588402 −0.294201 0.955744i \(-0.595054\pi\)
−0.294201 + 0.955744i \(0.595054\pi\)
\(888\) 0 0
\(889\) 50.8750 5.96220i 1.70629 0.199966i
\(890\) −5.87026 −0.196772
\(891\) 0 0
\(892\) −12.8928 −0.431682
\(893\) 1.29276 + 3.01580i 0.0432607 + 0.100920i
\(894\) 0 0
\(895\) 8.16934 0.273071
\(896\) 2.62777 0.307956i 0.0877876 0.0102881i
\(897\) 0 0
\(898\) −27.5946 −0.920843
\(899\) 44.1096i 1.47114i
\(900\) 0 0
\(901\) 5.23629 0.174446
\(902\) 14.4775i 0.482047i
\(903\) 0 0
\(904\) 9.68177 0.322011
\(905\) 8.54097i 0.283911i
\(906\) 0 0
\(907\) 45.6312i 1.51516i 0.652743 + 0.757579i \(0.273618\pi\)
−0.652743 + 0.757579i \(0.726382\pi\)
\(908\) −2.30063 −0.0763491
\(909\) 0 0
\(910\) 0.482822 + 4.11988i 0.0160054 + 0.136573i
\(911\) 36.4617i 1.20803i 0.796973 + 0.604015i \(0.206433\pi\)
−0.796973 + 0.604015i \(0.793567\pi\)
\(912\) 0 0
\(913\) 23.3674i 0.773350i
\(914\) 30.7393i 1.01677i
\(915\) 0 0
\(916\) 4.77358i 0.157724i
\(917\) 36.6463 4.29470i 1.21017 0.141823i
\(918\) 0 0
\(919\) −33.9016 −1.11831 −0.559155 0.829063i \(-0.688874\pi\)
−0.559155 + 0.829063i \(0.688874\pi\)
\(920\) 1.70903 0.0563451
\(921\) 0 0
\(922\) −20.6499 −0.680067
\(923\) 12.3801i 0.407497i
\(924\) 0 0
\(925\) 26.6155i 0.875111i
\(926\) 38.0039i 1.24889i
\(927\) 0 0
\(928\) −6.19185 −0.203258
\(929\) 25.2468i 0.828322i 0.910204 + 0.414161i \(0.135925\pi\)
−0.910204 + 0.414161i \(0.864075\pi\)
\(930\) 0 0
\(931\) −24.5049 + 18.1800i −0.803115 + 0.595824i
\(932\) −24.6124 −0.806205
\(933\) 0 0
\(934\) 16.0408 0.524870
\(935\) −0.961108 −0.0314316
\(936\) 0 0
\(937\) 60.7106i 1.98333i −0.128843 0.991665i \(-0.541126\pi\)
0.128843 0.991665i \(-0.458874\pi\)
\(938\) 1.81844 + 15.5166i 0.0593742 + 0.506636i
\(939\) 0 0
\(940\) −0.453495 −0.0147914
\(941\) −33.8435 −1.10327 −0.551633 0.834087i \(-0.685995\pi\)
−0.551633 + 0.834087i \(0.685995\pi\)
\(942\) 0 0
\(943\) −27.9739 −0.910956
\(944\) −0.799650 −0.0260264
\(945\) 0 0
\(946\) 17.8081i 0.578991i
\(947\) 1.14059 0.0370642 0.0185321 0.999828i \(-0.494101\pi\)
0.0185321 + 0.999828i \(0.494101\pi\)
\(948\) 0 0
\(949\) 20.7089i 0.672238i
\(950\) −7.96354 18.5776i −0.258371 0.602736i
\(951\) 0 0
\(952\) 0.334635 + 2.85542i 0.0108456 + 0.0925446i
\(953\) 37.0477i 1.20009i 0.799965 + 0.600046i \(0.204851\pi\)
−0.799965 + 0.600046i \(0.795149\pi\)
\(954\) 0 0
\(955\) 12.9460i 0.418922i
\(956\) −14.1110 −0.456381
\(957\) 0 0
\(958\) −19.4280 −0.627691
\(959\) 5.00745 + 42.7282i 0.161699 + 1.37977i
\(960\) 0 0
\(961\) 19.7487 0.637055
\(962\) −14.9373 −0.481598
\(963\) 0 0
\(964\) −2.19261 −0.0706193
\(965\) 5.95008 0.191540
\(966\) 0 0
\(967\) −18.4696 −0.593944 −0.296972 0.954886i \(-0.595977\pi\)
−0.296972 + 0.954886i \(0.595977\pi\)
\(968\) 8.84451i 0.284273i
\(969\) 0 0
\(970\) −0.472807 −0.0151809
\(971\) −13.1599 −0.422320 −0.211160 0.977452i \(-0.567724\pi\)
−0.211160 + 0.977452i \(0.567724\pi\)
\(972\) 0 0
\(973\) −15.8564 + 1.85826i −0.508334 + 0.0595732i
\(974\) 1.60382 0.0513897
\(975\) 0 0
\(976\) 8.69481i 0.278314i
\(977\) 15.4551i 0.494452i 0.968958 + 0.247226i \(0.0795190\pi\)
−0.968958 + 0.247226i \(0.920481\pi\)
\(978\) 0 0
\(979\) −14.3059 −0.457217
\(980\) −0.975040 4.10284i −0.0311465 0.131060i
\(981\) 0 0
\(982\) 35.6125i 1.13644i
\(983\) 49.5309 1.57979 0.789895 0.613242i \(-0.210135\pi\)
0.789895 + 0.613242i \(0.210135\pi\)
\(984\) 0 0
\(985\) 11.2598i 0.358767i
\(986\) 6.72827i 0.214272i
\(987\) 0 0
\(988\) −10.4262 + 4.46935i −0.331703 + 0.142189i
\(989\) −34.4095 −1.09416
\(990\) 0 0
\(991\) 16.3846i 0.520474i 0.965545 + 0.260237i \(0.0838007\pi\)
−0.965545 + 0.260237i \(0.916199\pi\)
\(992\) 7.12381i 0.226181i
\(993\) 0 0
\(994\) 1.46498 + 12.5006i 0.0464664 + 0.396495i
\(995\) −8.46732 −0.268432
\(996\) 0 0
\(997\) 5.77042i 0.182751i 0.995817 + 0.0913755i \(0.0291263\pi\)
−0.995817 + 0.0913755i \(0.970874\pi\)
\(998\) 2.89620i 0.0916775i
\(999\) 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 2394.2.e.b.1063.2 12
3.2 odd 2 798.2.e.b.265.11 yes 12
7.6 odd 2 2394.2.e.a.1063.5 12
19.18 odd 2 2394.2.e.a.1063.8 12
21.20 even 2 798.2.e.a.265.8 yes 12
57.56 even 2 798.2.e.a.265.5 12
133.132 even 2 inner 2394.2.e.b.1063.11 12
399.398 odd 2 798.2.e.b.265.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.e.a.265.5 12 57.56 even 2
798.2.e.a.265.8 yes 12 21.20 even 2
798.2.e.b.265.2 yes 12 399.398 odd 2
798.2.e.b.265.11 yes 12 3.2 odd 2
2394.2.e.a.1063.5 12 7.6 odd 2
2394.2.e.a.1063.8 12 19.18 odd 2
2394.2.e.b.1063.2 12 1.1 even 1 trivial
2394.2.e.b.1063.11 12 133.132 even 2 inner