Properties

Label 3420.2.bj.c.1189.7
Level $3420$
Weight $2$
Character 3420.1189
Analytic conductor $27.309$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3420,2,Mod(1189,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (of order \(6\), degree \(2\), 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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1189.7
Root \(-2.10552 - 1.21562i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.c.2629.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.896156 + 2.04863i) q^{5} -0.663818i q^{7} +1.80905 q^{11} +(-1.99526 + 1.15197i) q^{13} +(-3.77643 - 2.18033i) q^{17} +(-4.21168 + 1.12329i) q^{19} +(1.81374 - 1.04716i) q^{23} +(-3.39381 + 3.67179i) q^{25} +(-0.974621 - 1.68809i) q^{29} -9.52527 q^{31} +(1.35992 - 0.594885i) q^{35} -2.97461i q^{37} +(0.247657 - 0.428954i) q^{41} +(-6.81715 - 3.93588i) q^{43} +(-5.69449 + 3.28772i) q^{47} +6.55935 q^{49} +(1.99575 - 1.15225i) q^{53} +(1.62119 + 3.70609i) q^{55} +(-3.88559 + 6.73003i) q^{59} +(-5.36021 - 9.28415i) q^{61} +(-4.14802 - 3.05522i) q^{65} +(-3.96984 + 2.29199i) q^{67} +(2.95914 - 5.12538i) q^{71} +(4.86313 + 2.80773i) q^{73} -1.20088i q^{77} +(-2.99810 + 5.19286i) q^{79} +6.20090i q^{83} +(1.08242 - 9.69045i) q^{85} +(6.65028 + 11.5186i) q^{89} +(0.764696 + 1.32449i) q^{91} +(-6.07552 - 7.62155i) q^{95} +(-8.80695 - 5.08470i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.896156 + 2.04863i 0.400773 + 0.916177i
\(6\) 0 0
\(7\) 0.663818i 0.250900i −0.992100 0.125450i \(-0.959963\pi\)
0.992100 0.125450i \(-0.0400374\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.80905 0.545450 0.272725 0.962092i \(-0.412075\pi\)
0.272725 + 0.962092i \(0.412075\pi\)
\(12\) 0 0
\(13\) −1.99526 + 1.15197i −0.553386 + 0.319498i −0.750487 0.660886i \(-0.770181\pi\)
0.197100 + 0.980383i \(0.436847\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.77643 2.18033i −0.915920 0.528807i −0.0335887 0.999436i \(-0.510694\pi\)
−0.882331 + 0.470629i \(0.844027\pi\)
\(18\) 0 0
\(19\) −4.21168 + 1.12329i −0.966225 + 0.257699i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.81374 1.04716i 0.378191 0.218349i −0.298840 0.954303i \(-0.596600\pi\)
0.677031 + 0.735955i \(0.263266\pi\)
\(24\) 0 0
\(25\) −3.39381 + 3.67179i −0.678762 + 0.734359i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.974621 1.68809i −0.180983 0.313471i 0.761233 0.648479i \(-0.224594\pi\)
−0.942215 + 0.335008i \(0.891261\pi\)
\(30\) 0 0
\(31\) −9.52527 −1.71079 −0.855394 0.517977i \(-0.826685\pi\)
−0.855394 + 0.517977i \(0.826685\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.35992 0.594885i 0.229869 0.100554i
\(36\) 0 0
\(37\) 2.97461i 0.489023i −0.969646 0.244511i \(-0.921372\pi\)
0.969646 0.244511i \(-0.0786276\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.247657 0.428954i 0.0386775 0.0669914i −0.846039 0.533122i \(-0.821019\pi\)
0.884716 + 0.466130i \(0.154352\pi\)
\(42\) 0 0
\(43\) −6.81715 3.93588i −1.03960 0.600216i −0.119884 0.992788i \(-0.538252\pi\)
−0.919721 + 0.392572i \(0.871585\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.69449 + 3.28772i −0.830627 + 0.479563i −0.854067 0.520163i \(-0.825871\pi\)
0.0234403 + 0.999725i \(0.492538\pi\)
\(48\) 0 0
\(49\) 6.55935 0.937049
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.99575 1.15225i 0.274137 0.158273i −0.356629 0.934246i \(-0.616074\pi\)
0.630766 + 0.775973i \(0.282741\pi\)
\(54\) 0 0
\(55\) 1.62119 + 3.70609i 0.218602 + 0.499729i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.88559 + 6.73003i −0.505860 + 0.876176i 0.494117 + 0.869396i \(0.335492\pi\)
−0.999977 + 0.00678007i \(0.997842\pi\)
\(60\) 0 0
\(61\) −5.36021 9.28415i −0.686304 1.18871i −0.973025 0.230700i \(-0.925899\pi\)
0.286721 0.958014i \(-0.407435\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.14802 3.05522i −0.514499 0.378954i
\(66\) 0 0
\(67\) −3.96984 + 2.29199i −0.484993 + 0.280011i −0.722495 0.691376i \(-0.757005\pi\)
0.237502 + 0.971387i \(0.423671\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.95914 5.12538i 0.351185 0.608270i −0.635272 0.772288i \(-0.719112\pi\)
0.986457 + 0.164018i \(0.0524455\pi\)
\(72\) 0 0
\(73\) 4.86313 + 2.80773i 0.569187 + 0.328620i 0.756824 0.653618i \(-0.226750\pi\)
−0.187638 + 0.982238i \(0.560083\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.20088i 0.136853i
\(78\) 0 0
\(79\) −2.99810 + 5.19286i −0.337312 + 0.584242i −0.983926 0.178575i \(-0.942851\pi\)
0.646614 + 0.762817i \(0.276185\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.20090i 0.680638i 0.940310 + 0.340319i \(0.110535\pi\)
−0.940310 + 0.340319i \(0.889465\pi\)
\(84\) 0 0
\(85\) 1.08242 9.69045i 0.117404 1.05108i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.65028 + 11.5186i 0.704928 + 1.22097i 0.966717 + 0.255847i \(0.0823542\pi\)
−0.261789 + 0.965125i \(0.584312\pi\)
\(90\) 0 0
\(91\) 0.764696 + 1.32449i 0.0801619 + 0.138844i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.07552 7.62155i −0.623335 0.781955i
\(96\) 0 0
\(97\) −8.80695 5.08470i −0.894211 0.516273i −0.0188932 0.999822i \(-0.506014\pi\)
−0.875317 + 0.483549i \(0.839348\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.48770 + 12.9691i 0.745054 + 1.29047i 0.950170 + 0.311733i \(0.100909\pi\)
−0.205116 + 0.978738i \(0.565757\pi\)
\(102\) 0 0
\(103\) 18.1501i 1.78839i −0.447681 0.894193i \(-0.647750\pi\)
0.447681 0.894193i \(-0.352250\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.9698i 1.35051i −0.737584 0.675256i \(-0.764033\pi\)
0.737584 0.675256i \(-0.235967\pi\)
\(108\) 0 0
\(109\) −9.20544 + 15.9443i −0.881721 + 1.52719i −0.0322945 + 0.999478i \(0.510281\pi\)
−0.849426 + 0.527707i \(0.823052\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.8946i 1.02487i −0.858725 0.512437i \(-0.828743\pi\)
0.858725 0.512437i \(-0.171257\pi\)
\(114\) 0 0
\(115\) 3.77065 + 2.77727i 0.351615 + 0.258982i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.44734 + 2.50687i −0.132677 + 0.229804i
\(120\) 0 0
\(121\) −7.72733 −0.702484
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.5635 3.66218i −0.944832 0.327555i
\(126\) 0 0
\(127\) 2.86242 1.65262i 0.253998 0.146646i −0.367595 0.929986i \(-0.619819\pi\)
0.621594 + 0.783340i \(0.286486\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.646627 1.11999i 0.0564961 0.0978541i −0.836394 0.548128i \(-0.815340\pi\)
0.892890 + 0.450274i \(0.148674\pi\)
\(132\) 0 0
\(133\) 0.745658 + 2.79579i 0.0646567 + 0.242426i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.6426 7.29920i 1.08013 0.623613i 0.149197 0.988807i \(-0.452331\pi\)
0.930931 + 0.365195i \(0.118998\pi\)
\(138\) 0 0
\(139\) −1.87915 3.25478i −0.159387 0.276067i 0.775261 0.631641i \(-0.217618\pi\)
−0.934648 + 0.355575i \(0.884285\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.60954 + 2.08397i −0.301844 + 0.174270i
\(144\) 0 0
\(145\) 2.58487 3.50944i 0.214662 0.291443i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.83653 + 6.64507i −0.314301 + 0.544385i −0.979289 0.202469i \(-0.935103\pi\)
0.664988 + 0.746854i \(0.268437\pi\)
\(150\) 0 0
\(151\) 1.62643 0.132357 0.0661785 0.997808i \(-0.478919\pi\)
0.0661785 + 0.997808i \(0.478919\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.53613 19.5138i −0.685638 1.56739i
\(156\) 0 0
\(157\) −2.29893 1.32729i −0.183474 0.105929i 0.405450 0.914117i \(-0.367115\pi\)
−0.588924 + 0.808188i \(0.700448\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.695126 1.20399i −0.0547836 0.0948880i
\(162\) 0 0
\(163\) 19.8052i 1.55127i −0.631184 0.775633i \(-0.717431\pi\)
0.631184 0.775633i \(-0.282569\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.0776925 0.0448558i 0.00601203 0.00347105i −0.496991 0.867756i \(-0.665562\pi\)
0.503003 + 0.864285i \(0.332228\pi\)
\(168\) 0 0
\(169\) −3.84595 + 6.66139i −0.295842 + 0.512414i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.7120 + 7.91662i 1.04250 + 0.601889i 0.920541 0.390646i \(-0.127748\pi\)
0.121962 + 0.992535i \(0.461082\pi\)
\(174\) 0 0
\(175\) 2.43740 + 2.25287i 0.184250 + 0.170301i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.5051 −1.00942 −0.504709 0.863290i \(-0.668400\pi\)
−0.504709 + 0.863290i \(0.668400\pi\)
\(180\) 0 0
\(181\) 4.78591 + 8.28945i 0.355734 + 0.616150i 0.987243 0.159219i \(-0.0508975\pi\)
−0.631509 + 0.775368i \(0.717564\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.09389 2.66571i 0.448032 0.195987i
\(186\) 0 0
\(187\) −6.83177 3.94432i −0.499588 0.288438i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −26.9010 −1.94649 −0.973243 0.229778i \(-0.926200\pi\)
−0.973243 + 0.229778i \(0.926200\pi\)
\(192\) 0 0
\(193\) −19.7692 11.4137i −1.42302 0.821579i −0.426461 0.904506i \(-0.640240\pi\)
−0.996556 + 0.0829272i \(0.973573\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.4172i 0.742192i 0.928594 + 0.371096i \(0.121018\pi\)
−0.928594 + 0.371096i \(0.878982\pi\)
\(198\) 0 0
\(199\) 0.782081 + 1.35460i 0.0554403 + 0.0960254i 0.892414 0.451218i \(-0.149010\pi\)
−0.836973 + 0.547244i \(0.815677\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.12059 + 0.646971i −0.0786498 + 0.0454085i
\(204\) 0 0
\(205\) 1.10071 + 0.122949i 0.0768769 + 0.00858710i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.61915 + 2.03208i −0.527028 + 0.140562i
\(210\) 0 0
\(211\) −10.4253 + 18.0571i −0.717704 + 1.24310i 0.244203 + 0.969724i \(0.421474\pi\)
−0.961907 + 0.273376i \(0.911860\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.95396 17.4930i 0.133259 1.19301i
\(216\) 0 0
\(217\) 6.32305i 0.429236i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0466 0.675810
\(222\) 0 0
\(223\) −19.5472 11.2856i −1.30898 0.755740i −0.327054 0.945006i \(-0.606056\pi\)
−0.981926 + 0.189266i \(0.939389\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.4187i 0.757883i −0.925421 0.378941i \(-0.876288\pi\)
0.925421 0.378941i \(-0.123712\pi\)
\(228\) 0 0
\(229\) 15.8237 1.04566 0.522829 0.852438i \(-0.324877\pi\)
0.522829 + 0.852438i \(0.324877\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.6621 6.73314i −0.764013 0.441103i 0.0667219 0.997772i \(-0.478746\pi\)
−0.830735 + 0.556669i \(0.812079\pi\)
\(234\) 0 0
\(235\) −11.8385 8.71963i −0.772257 0.568806i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.7437 1.27711 0.638557 0.769574i \(-0.279532\pi\)
0.638557 + 0.769574i \(0.279532\pi\)
\(240\) 0 0
\(241\) 2.69793 + 4.67295i 0.173789 + 0.301011i 0.939742 0.341886i \(-0.111066\pi\)
−0.765953 + 0.642897i \(0.777732\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.87820 + 13.4377i 0.375544 + 0.858503i
\(246\) 0 0
\(247\) 7.10942 7.09296i 0.452361 0.451314i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.78646 3.09424i −0.112760 0.195306i 0.804122 0.594464i \(-0.202636\pi\)
−0.916882 + 0.399158i \(0.869303\pi\)
\(252\) 0 0
\(253\) 3.28115 1.89437i 0.206284 0.119098i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.0500 + 9.26649i −1.00117 + 0.578028i −0.908595 0.417678i \(-0.862844\pi\)
−0.0925780 + 0.995705i \(0.529511\pi\)
\(258\) 0 0
\(259\) −1.97460 −0.122696
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.30609 5.37288i −0.573838 0.331306i 0.184843 0.982768i \(-0.440822\pi\)
−0.758681 + 0.651462i \(0.774156\pi\)
\(264\) 0 0
\(265\) 4.14904 + 3.05597i 0.254873 + 0.187727i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.05027 10.4794i 0.368892 0.638939i −0.620501 0.784206i \(-0.713071\pi\)
0.989393 + 0.145267i \(0.0464040\pi\)
\(270\) 0 0
\(271\) 9.34472 16.1855i 0.567651 0.983201i −0.429146 0.903235i \(-0.641185\pi\)
0.996798 0.0799661i \(-0.0254812\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.13958 + 6.64247i −0.370231 + 0.400556i
\(276\) 0 0
\(277\) 5.94922i 0.357454i −0.983899 0.178727i \(-0.942802\pi\)
0.983899 0.178727i \(-0.0571979\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.78943 15.2237i −0.524333 0.908172i −0.999599 0.0283294i \(-0.990981\pi\)
0.475265 0.879843i \(-0.342352\pi\)
\(282\) 0 0
\(283\) 27.2029 + 15.7056i 1.61705 + 0.933603i 0.987679 + 0.156495i \(0.0500196\pi\)
0.629368 + 0.777107i \(0.283314\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.284748 0.164399i −0.0168081 0.00970418i
\(288\) 0 0
\(289\) 1.00764 + 1.74528i 0.0592727 + 0.102663i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.1435i 1.64416i 0.569372 + 0.822080i \(0.307186\pi\)
−0.569372 + 0.822080i \(0.692814\pi\)
\(294\) 0 0
\(295\) −17.2695 1.92899i −1.00547 0.112310i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.41259 + 4.17873i −0.139524 + 0.241662i
\(300\) 0 0
\(301\) −2.61271 + 4.52535i −0.150594 + 0.260837i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 14.2163 19.3012i 0.814020 1.10518i
\(306\) 0 0
\(307\) 23.3588 + 13.4862i 1.33316 + 0.769698i 0.985782 0.168028i \(-0.0537400\pi\)
0.347374 + 0.937727i \(0.387073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.99721 0.283366 0.141683 0.989912i \(-0.454749\pi\)
0.141683 + 0.989912i \(0.454749\pi\)
\(312\) 0 0
\(313\) −22.6484 + 13.0761i −1.28017 + 0.739104i −0.976878 0.213795i \(-0.931417\pi\)
−0.303287 + 0.952899i \(0.598084\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.5314 + 13.0085i −1.26549 + 0.730632i −0.974131 0.225982i \(-0.927441\pi\)
−0.291359 + 0.956614i \(0.594108\pi\)
\(318\) 0 0
\(319\) −1.76314 3.05385i −0.0987169 0.170983i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.3543 + 4.94081i 1.02126 + 0.274914i
\(324\) 0 0
\(325\) 2.54176 11.2357i 0.140992 0.623247i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.18245 + 3.78011i 0.120322 + 0.208404i
\(330\) 0 0
\(331\) 7.97402 0.438292 0.219146 0.975692i \(-0.429673\pi\)
0.219146 + 0.975692i \(0.429673\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.25304 6.07877i −0.450912 0.332119i
\(336\) 0 0
\(337\) −3.42862 1.97951i −0.186769 0.107831i 0.403700 0.914891i \(-0.367724\pi\)
−0.590469 + 0.807060i \(0.701057\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −17.2317 −0.933150
\(342\) 0 0
\(343\) 9.00094i 0.486005i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.16047 2.40205i −0.223345 0.128949i 0.384153 0.923269i \(-0.374494\pi\)
−0.607498 + 0.794321i \(0.707827\pi\)
\(348\) 0 0
\(349\) −9.02058 −0.482861 −0.241430 0.970418i \(-0.577617\pi\)
−0.241430 + 0.970418i \(0.577617\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0910i 0.962889i 0.876477 + 0.481444i \(0.159888\pi\)
−0.876477 + 0.481444i \(0.840112\pi\)
\(354\) 0 0
\(355\) 13.1519 + 1.46905i 0.698029 + 0.0779693i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.4849 + 30.2848i −0.922820 + 1.59837i −0.127791 + 0.991801i \(0.540789\pi\)
−0.795030 + 0.606571i \(0.792545\pi\)
\(360\) 0 0
\(361\) 16.4765 9.46183i 0.867182 0.497991i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.39389 + 12.4789i −0.0729595 + 0.653178i
\(366\) 0 0
\(367\) 8.55319 4.93819i 0.446473 0.257771i −0.259866 0.965645i \(-0.583679\pi\)
0.706339 + 0.707873i \(0.250345\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.764883 1.32482i −0.0397107 0.0687810i
\(372\) 0 0
\(373\) 18.9698i 0.982220i −0.871098 0.491110i \(-0.836591\pi\)
0.871098 0.491110i \(-0.163409\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.88925 + 2.24546i 0.200306 + 0.115647i
\(378\) 0 0
\(379\) −8.93773 −0.459101 −0.229550 0.973297i \(-0.573726\pi\)
−0.229550 + 0.973297i \(0.573726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.9643 + 8.06230i 0.713543 + 0.411964i 0.812371 0.583140i \(-0.198176\pi\)
−0.0988287 + 0.995104i \(0.531510\pi\)
\(384\) 0 0
\(385\) 2.46017 1.07618i 0.125382 0.0548471i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.0603 + 17.4249i 0.510075 + 0.883476i 0.999932 + 0.0116730i \(0.00371573\pi\)
−0.489857 + 0.871803i \(0.662951\pi\)
\(390\) 0 0
\(391\) −9.13262 −0.461857
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13.3250 1.48840i −0.670455 0.0748894i
\(396\) 0 0
\(397\) 13.8674 + 8.00633i 0.695983 + 0.401826i 0.805850 0.592120i \(-0.201709\pi\)
−0.109866 + 0.993946i \(0.535042\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.83602 + 11.8403i −0.341375 + 0.591278i −0.984688 0.174324i \(-0.944226\pi\)
0.643313 + 0.765603i \(0.277559\pi\)
\(402\) 0 0
\(403\) 19.0054 10.9728i 0.946727 0.546593i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.38123i 0.266738i
\(408\) 0 0
\(409\) 3.03518 + 5.25709i 0.150080 + 0.259946i 0.931257 0.364364i \(-0.118713\pi\)
−0.781177 + 0.624310i \(0.785380\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.46752 + 2.57932i 0.219832 + 0.126920i
\(414\) 0 0
\(415\) −12.7034 + 5.55698i −0.623585 + 0.272781i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.33985 −0.309722 −0.154861 0.987936i \(-0.549493\pi\)
−0.154861 + 0.987936i \(0.549493\pi\)
\(420\) 0 0
\(421\) 3.52768 6.11011i 0.171928 0.297789i −0.767166 0.641449i \(-0.778334\pi\)
0.939094 + 0.343660i \(0.111667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 20.8222 6.46668i 1.01002 0.313680i
\(426\) 0 0
\(427\) −6.16299 + 3.55820i −0.298248 + 0.172194i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13.0390 22.5843i −0.628069 1.08785i −0.987939 0.154845i \(-0.950512\pi\)
0.359870 0.933002i \(-0.382821\pi\)
\(432\) 0 0
\(433\) −29.6662 + 17.1278i −1.42567 + 0.823109i −0.996775 0.0802463i \(-0.974429\pi\)
−0.428892 + 0.903356i \(0.641096\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.46262 + 6.44766i −0.309149 + 0.308433i
\(438\) 0 0
\(439\) 3.13427 5.42872i 0.149591 0.259099i −0.781485 0.623923i \(-0.785538\pi\)
0.931076 + 0.364825i \(0.118871\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −28.3099 + 16.3447i −1.34504 + 0.776561i −0.987543 0.157352i \(-0.949704\pi\)
−0.357500 + 0.933913i \(0.616371\pi\)
\(444\) 0 0
\(445\) −17.6378 + 23.9465i −0.836110 + 1.13517i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −23.0822 −1.08932 −0.544659 0.838658i \(-0.683341\pi\)
−0.544659 + 0.838658i \(0.683341\pi\)
\(450\) 0 0
\(451\) 0.448025 0.776001i 0.0210967 0.0365405i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.02811 + 2.75353i −0.0950794 + 0.129088i
\(456\) 0 0
\(457\) 19.9166i 0.931661i 0.884874 + 0.465830i \(0.154244\pi\)
−0.884874 + 0.465830i \(0.845756\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.2586 36.8209i 0.990111 1.71492i 0.373562 0.927605i \(-0.378136\pi\)
0.616548 0.787317i \(-0.288530\pi\)
\(462\) 0 0
\(463\) 27.2843i 1.26801i 0.773329 + 0.634005i \(0.218590\pi\)
−0.773329 + 0.634005i \(0.781410\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 39.5912i 1.83206i −0.401107 0.916031i \(-0.631375\pi\)
0.401107 0.916031i \(-0.368625\pi\)
\(468\) 0 0
\(469\) 1.52146 + 2.63525i 0.0702547 + 0.121685i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.3326 7.12022i −0.567053 0.327388i
\(474\) 0 0
\(475\) 10.1692 19.2766i 0.466593 0.884472i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13.7954 + 23.8943i 0.630326 + 1.09176i 0.987485 + 0.157713i \(0.0504121\pi\)
−0.357159 + 0.934044i \(0.616255\pi\)
\(480\) 0 0
\(481\) 3.42665 + 5.93513i 0.156242 + 0.270618i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.52428 22.5989i 0.114622 1.02616i
\(486\) 0 0
\(487\) 29.9289i 1.35621i −0.734966 0.678104i \(-0.762802\pi\)
0.734966 0.678104i \(-0.237198\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.23550 12.5323i 0.326534 0.565573i −0.655288 0.755379i \(-0.727453\pi\)
0.981822 + 0.189806i \(0.0607860\pi\)
\(492\) 0 0
\(493\) 8.49996i 0.382819i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.40232 1.96433i −0.152615 0.0881122i
\(498\) 0 0
\(499\) −9.06799 + 15.7062i −0.405939 + 0.703107i −0.994430 0.105396i \(-0.966389\pi\)
0.588491 + 0.808504i \(0.299722\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.7022 8.48833i 0.655539 0.378476i −0.135036 0.990841i \(-0.543115\pi\)
0.790575 + 0.612365i \(0.209782\pi\)
\(504\) 0 0
\(505\) −19.8587 + 26.9619i −0.883703 + 1.19979i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.6888 + 28.9058i 0.739717 + 1.28123i 0.952623 + 0.304155i \(0.0983739\pi\)
−0.212906 + 0.977073i \(0.568293\pi\)
\(510\) 0 0
\(511\) 1.86382 3.22824i 0.0824507 0.142809i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 37.1830 16.2654i 1.63848 0.716737i
\(516\) 0 0
\(517\) −10.3016 + 5.94765i −0.453065 + 0.261577i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.09359 −0.0479110 −0.0239555 0.999713i \(-0.507626\pi\)
−0.0239555 + 0.999713i \(0.507626\pi\)
\(522\) 0 0
\(523\) 3.51597 2.02994i 0.153742 0.0887633i −0.421155 0.906989i \(-0.638375\pi\)
0.574898 + 0.818225i \(0.305042\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 35.9715 + 20.7682i 1.56695 + 0.904676i
\(528\) 0 0
\(529\) −9.30690 + 16.1200i −0.404648 + 0.700871i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.14117i 0.0494295i
\(534\) 0 0
\(535\) 28.6190 12.5191i 1.23731 0.541249i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.8662 0.511114
\(540\) 0 0
\(541\) −6.65097 11.5198i −0.285947 0.495275i 0.686891 0.726760i \(-0.258975\pi\)
−0.972838 + 0.231485i \(0.925642\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −40.9135 4.57001i −1.75254 0.195758i
\(546\) 0 0
\(547\) −29.6808 + 17.1362i −1.26906 + 0.732692i −0.974810 0.223036i \(-0.928403\pi\)
−0.294250 + 0.955729i \(0.595070\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00100 + 6.01493i 0.255651 + 0.256244i
\(552\) 0 0
\(553\) 3.44712 + 1.99019i 0.146586 + 0.0846316i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 35.2047 20.3255i 1.49167 0.861217i 0.491717 0.870755i \(-0.336369\pi\)
0.999955 + 0.00953795i \(0.00303607\pi\)
\(558\) 0 0
\(559\) 18.1360 0.767071
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.7137i 0.872980i −0.899709 0.436490i \(-0.856221\pi\)
0.899709 0.436490i \(-0.143779\pi\)
\(564\) 0 0
\(565\) 22.3190 9.76323i 0.938967 0.410742i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.0551 1.42766 0.713831 0.700318i \(-0.246958\pi\)
0.713831 + 0.700318i \(0.246958\pi\)
\(570\) 0 0
\(571\) 18.5413 0.775929 0.387965 0.921674i \(-0.373178\pi\)
0.387965 + 0.921674i \(0.373178\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.31052 + 10.2135i −0.0963553 + 0.425934i
\(576\) 0 0
\(577\) 2.96818i 0.123567i −0.998090 0.0617834i \(-0.980321\pi\)
0.998090 0.0617834i \(-0.0196788\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.11627 0.170772
\(582\) 0 0
\(583\) 3.61042 2.08448i 0.149528 0.0863302i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.9328 + 19.0138i 1.35928 + 0.784781i 0.989527 0.144348i \(-0.0461085\pi\)
0.369754 + 0.929130i \(0.379442\pi\)
\(588\) 0 0
\(589\) 40.1174 10.6996i 1.65301 0.440869i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 38.0060 21.9428i 1.56072 0.901081i 0.563533 0.826093i \(-0.309442\pi\)
0.997184 0.0749876i \(-0.0238917\pi\)
\(594\) 0 0
\(595\) −6.43270 0.718528i −0.263715 0.0294568i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 17.4883 + 30.2906i 0.714552 + 1.23764i 0.963132 + 0.269029i \(0.0867026\pi\)
−0.248581 + 0.968611i \(0.579964\pi\)
\(600\) 0 0
\(601\) 31.9988 1.30526 0.652630 0.757677i \(-0.273666\pi\)
0.652630 + 0.757677i \(0.273666\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.92489 15.8305i −0.281537 0.643600i
\(606\) 0 0
\(607\) 2.10357i 0.0853814i −0.999088 0.0426907i \(-0.986407\pi\)
0.999088 0.0426907i \(-0.0135930\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.57467 13.1197i 0.306438 0.530767i
\(612\) 0 0
\(613\) 30.1598 + 17.4127i 1.21814 + 0.703294i 0.964520 0.264010i \(-0.0850450\pi\)
0.253621 + 0.967304i \(0.418378\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.8782 + 9.16726i −0.639231 + 0.369060i −0.784318 0.620359i \(-0.786987\pi\)
0.145087 + 0.989419i \(0.453654\pi\)
\(618\) 0 0
\(619\) −32.4325 −1.30357 −0.651786 0.758403i \(-0.725980\pi\)
−0.651786 + 0.758403i \(0.725980\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.64628 4.41458i 0.306342 0.176866i
\(624\) 0 0
\(625\) −1.96412 24.9227i −0.0785649 0.996909i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.48562 + 11.2334i −0.258598 + 0.447906i
\(630\) 0 0
\(631\) −7.25551 12.5669i −0.288837 0.500281i 0.684695 0.728830i \(-0.259935\pi\)
−0.973532 + 0.228549i \(0.926602\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5.95078 + 4.38304i 0.236150 + 0.173936i
\(636\) 0 0
\(637\) −13.0876 + 7.55614i −0.518550 + 0.299385i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.8849 + 24.0494i −0.548421 + 0.949892i 0.449962 + 0.893048i \(0.351437\pi\)
−0.998383 + 0.0568449i \(0.981896\pi\)
\(642\) 0 0
\(643\) 9.72615 + 5.61539i 0.383562 + 0.221450i 0.679367 0.733799i \(-0.262255\pi\)
−0.295805 + 0.955248i \(0.595588\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.00223i 0.0787157i 0.999225 + 0.0393578i \(0.0125312\pi\)
−0.999225 + 0.0393578i \(0.987469\pi\)
\(648\) 0 0
\(649\) −7.02923 + 12.1750i −0.275921 + 0.477910i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.23945i 0.126770i 0.997989 + 0.0633848i \(0.0201895\pi\)
−0.997989 + 0.0633848i \(0.979810\pi\)
\(654\) 0 0
\(655\) 2.87393 + 0.321016i 0.112294 + 0.0125431i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.14745 + 5.45154i 0.122607 + 0.212362i 0.920795 0.390047i \(-0.127541\pi\)
−0.798188 + 0.602409i \(0.794208\pi\)
\(660\) 0 0
\(661\) −0.196536 0.340410i −0.00764435 0.0132404i 0.862178 0.506606i \(-0.169100\pi\)
−0.869822 + 0.493365i \(0.835767\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.05933 + 4.03304i −0.196192 + 0.156395i
\(666\) 0 0
\(667\) −3.53542 2.04117i −0.136892 0.0790346i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.69690 16.7955i −0.374345 0.648384i
\(672\) 0 0
\(673\) 43.1041i 1.66154i −0.556616 0.830770i \(-0.687900\pi\)
0.556616 0.830770i \(-0.312100\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.3727i 0.475522i −0.971324 0.237761i \(-0.923587\pi\)
0.971324 0.237761i \(-0.0764134\pi\)
\(678\) 0 0
\(679\) −3.37532 + 5.84622i −0.129533 + 0.224357i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.8415i 1.25664i −0.777954 0.628322i \(-0.783742\pi\)
0.777954 0.628322i \(-0.216258\pi\)
\(684\) 0 0
\(685\) 26.2831 + 19.3588i 1.00423 + 0.739662i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.65470 + 4.59807i −0.101136 + 0.175172i
\(690\) 0 0
\(691\) −16.6704 −0.634171 −0.317085 0.948397i \(-0.602704\pi\)
−0.317085 + 0.948397i \(0.602704\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.98384 6.76648i 0.189048 0.256667i
\(696\) 0 0
\(697\) −1.87052 + 1.07995i −0.0708510 + 0.0409059i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.9399 27.6087i 0.602041 1.04277i −0.390470 0.920616i \(-0.627688\pi\)
0.992512 0.122151i \(-0.0389791\pi\)
\(702\) 0 0
\(703\) 3.34134 + 12.5281i 0.126021 + 0.472506i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.60911 4.97047i 0.323779 0.186934i
\(708\) 0 0
\(709\) −2.99202 5.18233i −0.112368 0.194626i 0.804357 0.594147i \(-0.202510\pi\)
−0.916724 + 0.399520i \(0.869177\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −17.2764 + 9.97451i −0.647004 + 0.373548i
\(714\) 0 0
\(715\) −7.50399 5.52706i −0.280633 0.206700i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.345920 0.599151i 0.0129006 0.0223446i −0.859503 0.511131i \(-0.829227\pi\)
0.872404 + 0.488786i \(0.162560\pi\)
\(720\) 0 0
\(721\) −12.0484 −0.448706
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.50600 + 2.15046i 0.353044 + 0.0798660i
\(726\) 0 0
\(727\) 19.1410 + 11.0511i 0.709902 + 0.409862i 0.811025 0.585012i \(-0.198910\pi\)
−0.101123 + 0.994874i \(0.532244\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.1630 + 29.7272i 0.634796 + 1.09950i
\(732\) 0 0
\(733\) 19.0945i 0.705271i 0.935761 + 0.352635i \(0.114714\pi\)
−0.935761 + 0.352635i \(0.885286\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.18165 + 4.14633i −0.264540 + 0.152732i
\(738\) 0 0
\(739\) 6.05045 10.4797i 0.222569 0.385501i −0.733018 0.680209i \(-0.761889\pi\)
0.955587 + 0.294708i \(0.0952223\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.6722 + 7.89364i 0.501584 + 0.289590i 0.729368 0.684122i \(-0.239814\pi\)
−0.227783 + 0.973712i \(0.573148\pi\)
\(744\) 0 0
\(745\) −17.0514 1.90463i −0.624717 0.0697804i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.27341 −0.338843
\(750\) 0 0
\(751\) −17.5848 30.4578i −0.641678 1.11142i −0.985058 0.172223i \(-0.944905\pi\)
0.343380 0.939197i \(-0.388428\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.45753 + 3.33196i 0.0530451 + 0.121262i
\(756\) 0 0
\(757\) 12.7465 + 7.35917i 0.463278 + 0.267473i 0.713421 0.700735i \(-0.247145\pi\)
−0.250144 + 0.968209i \(0.580478\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 39.9486 1.44814 0.724068 0.689728i \(-0.242270\pi\)
0.724068 + 0.689728i \(0.242270\pi\)
\(762\) 0 0
\(763\) 10.5841 + 6.11074i 0.383170 + 0.221224i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 17.9042i 0.646485i
\(768\) 0 0
\(769\) 24.9729 + 43.2544i 0.900547 + 1.55979i 0.826786 + 0.562516i \(0.190167\pi\)
0.0737605 + 0.997276i \(0.476500\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.1384 + 11.6269i −0.724327 + 0.418190i −0.816343 0.577567i \(-0.804002\pi\)
0.0920165 + 0.995757i \(0.470669\pi\)
\(774\) 0 0
\(775\) 32.3269 34.9748i 1.16122 1.25633i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.561213 + 2.08481i −0.0201075 + 0.0746960i
\(780\) 0 0
\(781\) 5.35324 9.27208i 0.191554 0.331781i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.658928 5.89912i 0.0235181 0.210549i
\(786\) 0 0
\(787\) 19.5711i 0.697636i 0.937191 + 0.348818i \(0.113417\pi\)
−0.937191 + 0.348818i \(0.886583\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7.23201 −0.257141
\(792\) 0 0
\(793\) 21.3900 + 12.3495i 0.759582 + 0.438545i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 47.6782i 1.68885i 0.535676 + 0.844424i \(0.320057\pi\)
−0.535676 + 0.844424i \(0.679943\pi\)
\(798\) 0 0
\(799\) 28.6732 1.01438
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.79767 + 5.07933i 0.310463 + 0.179246i
\(804\) 0 0
\(805\) 1.84360 2.50303i 0.0649784 0.0882200i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24.4485 −0.859565 −0.429782 0.902933i \(-0.641410\pi\)
−0.429782 + 0.902933i \(0.641410\pi\)
\(810\) 0 0
\(811\) 5.38004 + 9.31850i 0.188919 + 0.327217i 0.944890 0.327388i \(-0.106168\pi\)
−0.755971 + 0.654605i \(0.772835\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 40.5737 17.7486i 1.42123 0.621706i
\(816\) 0 0
\(817\) 33.1327 + 8.91906i 1.15917 + 0.312038i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.60434 6.24289i −0.125792 0.217879i 0.796250 0.604968i \(-0.206814\pi\)
−0.922042 + 0.387089i \(0.873481\pi\)
\(822\) 0 0
\(823\) −6.70433 + 3.87075i −0.233698 + 0.134926i −0.612277 0.790643i \(-0.709746\pi\)
0.378579 + 0.925569i \(0.376413\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −23.0605 + 13.3140i −0.801891 + 0.462972i −0.844132 0.536136i \(-0.819884\pi\)
0.0422411 + 0.999107i \(0.486550\pi\)
\(828\) 0 0
\(829\) 12.1093 0.420572 0.210286 0.977640i \(-0.432561\pi\)
0.210286 + 0.977640i \(0.432561\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −24.7709 14.3015i −0.858262 0.495518i
\(834\) 0 0
\(835\) 0.161518 + 0.118966i 0.00558955 + 0.00411698i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.20591 2.08869i 0.0416325 0.0721096i −0.844458 0.535621i \(-0.820077\pi\)
0.886091 + 0.463512i \(0.153411\pi\)
\(840\) 0 0
\(841\) 12.6002 21.8242i 0.434491 0.752560i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −17.0933 1.90931i −0.588028 0.0656823i
\(846\) 0 0
\(847\) 5.12954i 0.176253i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.11490 5.39517i −0.106777 0.184944i
\(852\) 0 0
\(853\) 23.9717 + 13.8401i 0.820775 + 0.473875i 0.850684 0.525678i \(-0.176188\pi\)
−0.0299085 + 0.999553i \(0.509522\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.2071 8.20245i −0.485304 0.280190i 0.237320 0.971431i \(-0.423731\pi\)
−0.722624 + 0.691241i \(0.757064\pi\)
\(858\) 0 0
\(859\) −17.4715 30.2616i −0.596121 1.03251i −0.993388 0.114808i \(-0.963375\pi\)
0.397267 0.917703i \(-0.369959\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.6243i 0.633979i 0.948429 + 0.316990i \(0.102672\pi\)
−0.948429 + 0.316990i \(0.897328\pi\)
\(864\) 0 0
\(865\) −3.93018 + 35.1854i −0.133630 + 1.19634i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −5.42372 + 9.39416i −0.183987 + 0.318675i
\(870\) 0 0
\(871\) 5.28058 9.14624i 0.178926 0.309908i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.43102 + 7.01228i −0.0821835 + 0.237058i
\(876\) 0 0
\(877\) 26.3087 + 15.1893i 0.888381 + 0.512907i 0.873413 0.486981i \(-0.161902\pi\)
0.0149683 + 0.999888i \(0.495235\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.3772 0.888670 0.444335 0.895861i \(-0.353440\pi\)
0.444335 + 0.895861i \(0.353440\pi\)
\(882\) 0 0
\(883\) −25.2183 + 14.5598i −0.848664 + 0.489977i −0.860200 0.509957i \(-0.829661\pi\)
0.0115356 + 0.999933i \(0.496328\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17.3379 + 10.0100i −0.582148 + 0.336104i −0.761987 0.647593i \(-0.775776\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(888\) 0 0
\(889\) −1.09704 1.90012i −0.0367935 0.0637281i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 20.2903 20.2433i 0.678990 0.677418i
\(894\) 0 0
\(895\) −12.1027 27.6670i −0.404547 0.924805i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.28352 + 16.0795i 0.309623 + 0.536283i
\(900\) 0 0
\(901\) −10.0491 −0.334784
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −12.6931 + 17.2332i −0.421934 + 0.572852i
\(906\) 0 0
\(907\) 6.50159 + 3.75370i 0.215882 + 0.124639i 0.604042 0.796952i \(-0.293556\pi\)
−0.388160 + 0.921592i \(0.626889\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −38.3882 −1.27186 −0.635929 0.771748i \(-0.719383\pi\)
−0.635929 + 0.771748i \(0.719383\pi\)
\(912\) 0 0
\(913\) 11.2178i 0.371254i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.743471 0.429243i −0.0245516 0.0141749i
\(918\) 0 0
\(919\) 37.8811 1.24958 0.624791 0.780792i \(-0.285184\pi\)
0.624791 + 0.780792i \(0.285184\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13.6353i 0.448811i
\(924\) 0 0
\(925\) 10.9222 + 10.0953i 0.359118 + 0.331930i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5.96821 10.3372i 0.195811 0.339154i −0.751355 0.659898i \(-0.770600\pi\)
0.947166 + 0.320744i \(0.103933\pi\)
\(930\) 0 0
\(931\) −27.6258 + 7.36802i −0.905401 + 0.241477i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.95815 17.5305i 0.0640383 0.573310i
\(936\) 0 0
\(937\) 11.7299 6.77228i 0.383200 0.221241i −0.296010 0.955185i \(-0.595656\pi\)
0.679210 + 0.733944i \(0.262323\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7.04614 + 12.2043i 0.229698 + 0.397848i 0.957718 0.287707i \(-0.0928929\pi\)
−0.728021 + 0.685555i \(0.759560\pi\)
\(942\) 0 0
\(943\) 1.03735i 0.0337807i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.58806 + 5.53567i 0.311570 + 0.179885i 0.647629 0.761956i \(-0.275761\pi\)
−0.336059 + 0.941841i \(0.609094\pi\)
\(948\) 0 0
\(949\) −12.9376 −0.419973
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.40004 + 3.69507i 0.207318 + 0.119695i 0.600064 0.799952i \(-0.295142\pi\)
−0.392747 + 0.919647i \(0.628475\pi\)
\(954\) 0 0
\(955\) −24.1075 55.1103i −0.780099 1.78333i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.84534 8.39238i −0.156464 0.271004i
\(960\) 0 0
\(961\) 59.7307 1.92680
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.66632 50.7283i 0.182405 1.63300i
\(966\) 0 0
\(967\) −1.31785 0.760861i −0.0423792 0.0244676i 0.478661 0.878000i \(-0.341122\pi\)
−0.521040 + 0.853532i \(0.674456\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27.2897 + 47.2671i −0.875768 + 1.51687i −0.0198254 + 0.999803i \(0.506311\pi\)
−0.855942 + 0.517071i \(0.827022\pi\)
\(972\) 0 0
\(973\) −2.16058 + 1.24741i −0.0692651 + 0.0399902i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.0130i 0.416321i 0.978095 + 0.208161i \(0.0667477\pi\)
−0.978095 + 0.208161i \(0.933252\pi\)
\(978\) 0 0
\(979\) 12.0307 + 20.8378i 0.384503 + 0.665979i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.6166 10.1709i −0.561882 0.324402i 0.192019 0.981391i \(-0.438497\pi\)
−0.753900 + 0.656989i \(0.771830\pi\)
\(984\) 0 0
\(985\) −21.3410 + 9.33541i −0.679980 + 0.297451i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.4860 −0.524225
\(990\) 0 0
\(991\) −16.9449 + 29.3495i −0.538274 + 0.932318i 0.460723 + 0.887544i \(0.347590\pi\)
−0.998997 + 0.0447741i \(0.985743\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.07422 + 2.81614i −0.0657573 + 0.0892775i
\(996\) 0 0
\(997\) −3.05162 + 1.76185i −0.0966458 + 0.0557985i −0.547544 0.836777i \(-0.684437\pi\)
0.450898 + 0.892575i \(0.351104\pi\)
\(998\) 0 0
\(999\) 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 3420.2.bj.c.1189.7 20
3.2 odd 2 380.2.r.a.49.2 20
5.4 even 2 inner 3420.2.bj.c.1189.1 20
15.2 even 4 1900.2.i.g.201.2 20
15.8 even 4 1900.2.i.g.201.9 20
15.14 odd 2 380.2.r.a.49.9 yes 20
19.7 even 3 inner 3420.2.bj.c.2629.1 20
57.26 odd 6 380.2.r.a.349.9 yes 20
95.64 even 6 inner 3420.2.bj.c.2629.7 20
285.83 even 12 1900.2.i.g.501.9 20
285.197 even 12 1900.2.i.g.501.2 20
285.254 odd 6 380.2.r.a.349.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.2 20 3.2 odd 2
380.2.r.a.49.9 yes 20 15.14 odd 2
380.2.r.a.349.2 yes 20 285.254 odd 6
380.2.r.a.349.9 yes 20 57.26 odd 6
1900.2.i.g.201.2 20 15.2 even 4
1900.2.i.g.201.9 20 15.8 even 4
1900.2.i.g.501.2 20 285.197 even 12
1900.2.i.g.501.9 20 285.83 even 12
3420.2.bj.c.1189.1 20 5.4 even 2 inner
3420.2.bj.c.1189.7 20 1.1 even 1 trivial
3420.2.bj.c.2629.1 20 19.7 even 3 inner
3420.2.bj.c.2629.7 20 95.64 even 6 inner