Properties

Label 1600.2.j.d.1007.5
Level $1600$
Weight $2$
Character 1600.1007
Analytic conductor $12.776$
Analytic rank $0$
Dimension $18$
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] = [1600,2,Mod(143,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.j (of order \(4\), degree \(2\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1007.5
Root \(-1.37691 + 0.322680i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1007
Dual form 1600.2.j.d.143.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.614566i q^{3} +(2.83610 + 2.83610i) q^{7} +2.62231 q^{9} +O(q^{10})\) \(q-0.614566i q^{3} +(2.83610 + 2.83610i) q^{7} +2.62231 q^{9} +(-1.95928 - 1.95928i) q^{11} +2.05493 q^{13} +(4.06774 + 4.06774i) q^{17} +(-0.683479 - 0.683479i) q^{19} +(1.74297 - 1.74297i) q^{21} +(-4.95014 + 4.95014i) q^{23} -3.45528i q^{27} +(-0.835439 + 0.835439i) q^{29} +2.35978i q^{31} +(-1.20411 + 1.20411i) q^{33} +4.54384 q^{37} -1.26289i q^{39} -5.07255i q^{41} -0.849753 q^{43} +(2.72646 - 2.72646i) q^{47} +9.08690i q^{49} +(2.49989 - 2.49989i) q^{51} +5.17605i q^{53} +(-0.420043 + 0.420043i) q^{57} +(-4.16328 + 4.16328i) q^{59} +(5.55706 + 5.55706i) q^{61} +(7.43712 + 7.43712i) q^{63} -1.73609 q^{67} +(3.04219 + 3.04219i) q^{69} -2.33526 q^{71} +(-4.39686 - 4.39686i) q^{73} -11.1134i q^{77} +14.0993 q^{79} +5.74343 q^{81} +2.75725i q^{83} +(0.513433 + 0.513433i) q^{87} -11.6448 q^{89} +(5.82797 + 5.82797i) q^{91} +1.45024 q^{93} +(3.52933 + 3.52933i) q^{97} +(-5.13783 - 5.13783i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{7} - 10 q^{9} + 2 q^{11} + 6 q^{17} - 2 q^{19} - 16 q^{21} - 2 q^{23} - 14 q^{29} + 8 q^{33} - 8 q^{37} - 44 q^{43} - 38 q^{47} - 8 q^{51} - 24 q^{57} + 10 q^{59} + 14 q^{61} + 6 q^{63} + 12 q^{67} + 32 q^{69} - 24 q^{71} - 14 q^{73} - 16 q^{79} + 2 q^{81} + 24 q^{87} - 12 q^{89} - 16 q^{93} - 18 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.614566i 0.354820i −0.984137 0.177410i \(-0.943228\pi\)
0.984137 0.177410i \(-0.0567718\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.83610 + 2.83610i 1.07194 + 1.07194i 0.997203 + 0.0747413i \(0.0238131\pi\)
0.0747413 + 0.997203i \(0.476187\pi\)
\(8\) 0 0
\(9\) 2.62231 0.874103
\(10\) 0 0
\(11\) −1.95928 1.95928i −0.590745 0.590745i 0.347088 0.937833i \(-0.387171\pi\)
−0.937833 + 0.347088i \(0.887171\pi\)
\(12\) 0 0
\(13\) 2.05493 0.569934 0.284967 0.958537i \(-0.408017\pi\)
0.284967 + 0.958537i \(0.408017\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.06774 + 4.06774i 0.986571 + 0.986571i 0.999911 0.0133401i \(-0.00424641\pi\)
−0.0133401 + 0.999911i \(0.504246\pi\)
\(18\) 0 0
\(19\) −0.683479 0.683479i −0.156801 0.156801i 0.624347 0.781147i \(-0.285365\pi\)
−0.781147 + 0.624347i \(0.785365\pi\)
\(20\) 0 0
\(21\) 1.74297 1.74297i 0.380347 0.380347i
\(22\) 0 0
\(23\) −4.95014 + 4.95014i −1.03218 + 1.03218i −0.0327113 + 0.999465i \(0.510414\pi\)
−0.999465 + 0.0327113i \(0.989586\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.45528i 0.664969i
\(28\) 0 0
\(29\) −0.835439 + 0.835439i −0.155137 + 0.155137i −0.780408 0.625271i \(-0.784989\pi\)
0.625271 + 0.780408i \(0.284989\pi\)
\(30\) 0 0
\(31\) 2.35978i 0.423829i 0.977288 + 0.211915i \(0.0679698\pi\)
−0.977288 + 0.211915i \(0.932030\pi\)
\(32\) 0 0
\(33\) −1.20411 + 1.20411i −0.209608 + 0.209608i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.54384 0.747002 0.373501 0.927630i \(-0.378157\pi\)
0.373501 + 0.927630i \(0.378157\pi\)
\(38\) 0 0
\(39\) 1.26289i 0.202224i
\(40\) 0 0
\(41\) 5.07255i 0.792199i −0.918208 0.396100i \(-0.870364\pi\)
0.918208 0.396100i \(-0.129636\pi\)
\(42\) 0 0
\(43\) −0.849753 −0.129586 −0.0647930 0.997899i \(-0.520639\pi\)
−0.0647930 + 0.997899i \(0.520639\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.72646 2.72646i 0.397696 0.397696i −0.479724 0.877419i \(-0.659263\pi\)
0.877419 + 0.479724i \(0.159263\pi\)
\(48\) 0 0
\(49\) 9.08690i 1.29813i
\(50\) 0 0
\(51\) 2.49989 2.49989i 0.350055 0.350055i
\(52\) 0 0
\(53\) 5.17605i 0.710985i 0.934679 + 0.355492i \(0.115687\pi\)
−0.934679 + 0.355492i \(0.884313\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.420043 + 0.420043i −0.0556360 + 0.0556360i
\(58\) 0 0
\(59\) −4.16328 + 4.16328i −0.542013 + 0.542013i −0.924119 0.382105i \(-0.875199\pi\)
0.382105 + 0.924119i \(0.375199\pi\)
\(60\) 0 0
\(61\) 5.55706 + 5.55706i 0.711509 + 0.711509i 0.966851 0.255342i \(-0.0821880\pi\)
−0.255342 + 0.966851i \(0.582188\pi\)
\(62\) 0 0
\(63\) 7.43712 + 7.43712i 0.936990 + 0.936990i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.73609 −0.212097 −0.106048 0.994361i \(-0.533820\pi\)
−0.106048 + 0.994361i \(0.533820\pi\)
\(68\) 0 0
\(69\) 3.04219 + 3.04219i 0.366237 + 0.366237i
\(70\) 0 0
\(71\) −2.33526 −0.277144 −0.138572 0.990352i \(-0.544251\pi\)
−0.138572 + 0.990352i \(0.544251\pi\)
\(72\) 0 0
\(73\) −4.39686 4.39686i −0.514613 0.514613i 0.401323 0.915936i \(-0.368550\pi\)
−0.915936 + 0.401323i \(0.868550\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.1134i 1.26649i
\(78\) 0 0
\(79\) 14.0993 1.58629 0.793146 0.609032i \(-0.208442\pi\)
0.793146 + 0.609032i \(0.208442\pi\)
\(80\) 0 0
\(81\) 5.74343 0.638159
\(82\) 0 0
\(83\) 2.75725i 0.302648i 0.988484 + 0.151324i \(0.0483536\pi\)
−0.988484 + 0.151324i \(0.951646\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.513433 + 0.513433i 0.0550458 + 0.0550458i
\(88\) 0 0
\(89\) −11.6448 −1.23435 −0.617173 0.786828i \(-0.711722\pi\)
−0.617173 + 0.786828i \(0.711722\pi\)
\(90\) 0 0
\(91\) 5.82797 + 5.82797i 0.610937 + 0.610937i
\(92\) 0 0
\(93\) 1.45024 0.150383
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.52933 + 3.52933i 0.358349 + 0.358349i 0.863204 0.504855i \(-0.168454\pi\)
−0.504855 + 0.863204i \(0.668454\pi\)
\(98\) 0 0
\(99\) −5.13783 5.13783i −0.516372 0.516372i
\(100\) 0 0
\(101\) 7.39467 7.39467i 0.735797 0.735797i −0.235964 0.971762i \(-0.575825\pi\)
0.971762 + 0.235964i \(0.0758249\pi\)
\(102\) 0 0
\(103\) 3.72605 3.72605i 0.367139 0.367139i −0.499294 0.866433i \(-0.666407\pi\)
0.866433 + 0.499294i \(0.166407\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4605i 1.59130i 0.605758 + 0.795649i \(0.292870\pi\)
−0.605758 + 0.795649i \(0.707130\pi\)
\(108\) 0 0
\(109\) 12.8554 12.8554i 1.23133 1.23133i 0.267870 0.963455i \(-0.413680\pi\)
0.963455 0.267870i \(-0.0863199\pi\)
\(110\) 0 0
\(111\) 2.79249i 0.265051i
\(112\) 0 0
\(113\) −0.863630 + 0.863630i −0.0812435 + 0.0812435i −0.746561 0.665317i \(-0.768296\pi\)
0.665317 + 0.746561i \(0.268296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.38865 0.498181
\(118\) 0 0
\(119\) 23.0730i 2.11510i
\(120\) 0 0
\(121\) 3.32246i 0.302042i
\(122\) 0 0
\(123\) −3.11742 −0.281088
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.7944 11.7944i 1.04659 1.04659i 0.0477265 0.998860i \(-0.484802\pi\)
0.998860 0.0477265i \(-0.0151976\pi\)
\(128\) 0 0
\(129\) 0.522229i 0.0459797i
\(130\) 0 0
\(131\) 15.9756 15.9756i 1.39579 1.39579i 0.584132 0.811659i \(-0.301435\pi\)
0.811659 0.584132i \(-0.198565\pi\)
\(132\) 0 0
\(133\) 3.87683i 0.336163i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.29423 1.29423i 0.110573 0.110573i −0.649655 0.760229i \(-0.725087\pi\)
0.760229 + 0.649655i \(0.225087\pi\)
\(138\) 0 0
\(139\) 8.61413 8.61413i 0.730641 0.730641i −0.240106 0.970747i \(-0.577182\pi\)
0.970747 + 0.240106i \(0.0771821\pi\)
\(140\) 0 0
\(141\) −1.67559 1.67559i −0.141110 0.141110i
\(142\) 0 0
\(143\) −4.02617 4.02617i −0.336685 0.336685i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.58450 0.460602
\(148\) 0 0
\(149\) −0.0806133 0.0806133i −0.00660410 0.00660410i 0.703797 0.710401i \(-0.251486\pi\)
−0.710401 + 0.703797i \(0.751486\pi\)
\(150\) 0 0
\(151\) 3.25198 0.264643 0.132321 0.991207i \(-0.457757\pi\)
0.132321 + 0.991207i \(0.457757\pi\)
\(152\) 0 0
\(153\) 10.6669 + 10.6669i 0.862364 + 0.862364i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.06652i 0.723587i 0.932258 + 0.361793i \(0.117835\pi\)
−0.932258 + 0.361793i \(0.882165\pi\)
\(158\) 0 0
\(159\) 3.18102 0.252271
\(160\) 0 0
\(161\) −28.0782 −2.21287
\(162\) 0 0
\(163\) 3.93313i 0.308067i −0.988066 0.154033i \(-0.950774\pi\)
0.988066 0.154033i \(-0.0492263\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.13216 + 8.13216i 0.629285 + 0.629285i 0.947888 0.318603i \(-0.103214\pi\)
−0.318603 + 0.947888i \(0.603214\pi\)
\(168\) 0 0
\(169\) −8.77728 −0.675175
\(170\) 0 0
\(171\) −1.79229 1.79229i −0.137060 0.137060i
\(172\) 0 0
\(173\) −6.86735 −0.522115 −0.261057 0.965323i \(-0.584071\pi\)
−0.261057 + 0.965323i \(0.584071\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.55861 + 2.55861i 0.192317 + 0.192317i
\(178\) 0 0
\(179\) −15.7117 15.7117i −1.17435 1.17435i −0.981163 0.193183i \(-0.938119\pi\)
−0.193183 0.981163i \(-0.561881\pi\)
\(180\) 0 0
\(181\) −13.9112 + 13.9112i −1.03401 + 1.03401i −0.0346142 + 0.999401i \(0.511020\pi\)
−0.999401 + 0.0346142i \(0.988980\pi\)
\(182\) 0 0
\(183\) 3.41518 3.41518i 0.252458 0.252458i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.9397i 1.16562i
\(188\) 0 0
\(189\) 9.79951 9.79951i 0.712810 0.712810i
\(190\) 0 0
\(191\) 10.3393i 0.748123i 0.927404 + 0.374061i \(0.122035\pi\)
−0.927404 + 0.374061i \(0.877965\pi\)
\(192\) 0 0
\(193\) −13.2080 + 13.2080i −0.950734 + 0.950734i −0.998842 0.0481079i \(-0.984681\pi\)
0.0481079 + 0.998842i \(0.484681\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.2437 −1.08607 −0.543036 0.839709i \(-0.682725\pi\)
−0.543036 + 0.839709i \(0.682725\pi\)
\(198\) 0 0
\(199\) 4.98761i 0.353562i −0.984250 0.176781i \(-0.943432\pi\)
0.984250 0.176781i \(-0.0565684\pi\)
\(200\) 0 0
\(201\) 1.06694i 0.0752561i
\(202\) 0 0
\(203\) −4.73878 −0.332597
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12.9808 + 12.9808i −0.902228 + 0.902228i
\(208\) 0 0
\(209\) 2.67825i 0.185258i
\(210\) 0 0
\(211\) −10.3803 + 10.3803i −0.714608 + 0.714608i −0.967496 0.252887i \(-0.918620\pi\)
0.252887 + 0.967496i \(0.418620\pi\)
\(212\) 0 0
\(213\) 1.43517i 0.0983362i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.69257 + 6.69257i −0.454321 + 0.454321i
\(218\) 0 0
\(219\) −2.70216 + 2.70216i −0.182595 + 0.182595i
\(220\) 0 0
\(221\) 8.35890 + 8.35890i 0.562280 + 0.562280i
\(222\) 0 0
\(223\) −1.49853 1.49853i −0.100349 0.100349i 0.655150 0.755499i \(-0.272605\pi\)
−0.755499 + 0.655150i \(0.772605\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.6346 −1.03771 −0.518853 0.854864i \(-0.673641\pi\)
−0.518853 + 0.854864i \(0.673641\pi\)
\(228\) 0 0
\(229\) −9.74097 9.74097i −0.643702 0.643702i 0.307762 0.951463i \(-0.400420\pi\)
−0.951463 + 0.307762i \(0.900420\pi\)
\(230\) 0 0
\(231\) −6.82992 −0.449376
\(232\) 0 0
\(233\) −0.509123 0.509123i −0.0333538 0.0333538i 0.690233 0.723587i \(-0.257508\pi\)
−0.723587 + 0.690233i \(0.757508\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.66493i 0.562848i
\(238\) 0 0
\(239\) −8.19486 −0.530081 −0.265041 0.964237i \(-0.585385\pi\)
−0.265041 + 0.964237i \(0.585385\pi\)
\(240\) 0 0
\(241\) 5.66775 0.365092 0.182546 0.983197i \(-0.441566\pi\)
0.182546 + 0.983197i \(0.441566\pi\)
\(242\) 0 0
\(243\) 13.8956i 0.891400i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.40450 1.40450i −0.0893661 0.0893661i
\(248\) 0 0
\(249\) 1.69451 0.107385
\(250\) 0 0
\(251\) −14.7484 14.7484i −0.930911 0.930911i 0.0668521 0.997763i \(-0.478704\pi\)
−0.997763 + 0.0668521i \(0.978704\pi\)
\(252\) 0 0
\(253\) 19.3974 1.21951
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.61143 3.61143i −0.225275 0.225275i 0.585440 0.810715i \(-0.300922\pi\)
−0.810715 + 0.585440i \(0.800922\pi\)
\(258\) 0 0
\(259\) 12.8868 + 12.8868i 0.800745 + 0.800745i
\(260\) 0 0
\(261\) −2.19078 + 2.19078i −0.135606 + 0.135606i
\(262\) 0 0
\(263\) −6.80041 + 6.80041i −0.419331 + 0.419331i −0.884973 0.465642i \(-0.845823\pi\)
0.465642 + 0.884973i \(0.345823\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.15650i 0.437970i
\(268\) 0 0
\(269\) −1.20010 + 1.20010i −0.0731711 + 0.0731711i −0.742745 0.669574i \(-0.766477\pi\)
0.669574 + 0.742745i \(0.266477\pi\)
\(270\) 0 0
\(271\) 2.79591i 0.169840i 0.996388 + 0.0849199i \(0.0270634\pi\)
−0.996388 + 0.0849199i \(0.972937\pi\)
\(272\) 0 0
\(273\) 3.58167 3.58167i 0.216773 0.216773i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.8115 −0.829852 −0.414926 0.909855i \(-0.636193\pi\)
−0.414926 + 0.909855i \(0.636193\pi\)
\(278\) 0 0
\(279\) 6.18807i 0.370470i
\(280\) 0 0
\(281\) 7.21718i 0.430541i 0.976554 + 0.215270i \(0.0690633\pi\)
−0.976554 + 0.215270i \(0.930937\pi\)
\(282\) 0 0
\(283\) 25.2988 1.50386 0.751930 0.659243i \(-0.229123\pi\)
0.751930 + 0.659243i \(0.229123\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.3862 14.3862i 0.849193 0.849193i
\(288\) 0 0
\(289\) 16.0930i 0.946644i
\(290\) 0 0
\(291\) 2.16901 2.16901i 0.127149 0.127149i
\(292\) 0 0
\(293\) 14.1276i 0.825344i −0.910880 0.412672i \(-0.864596\pi\)
0.910880 0.412672i \(-0.135404\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −6.76985 + 6.76985i −0.392827 + 0.392827i
\(298\) 0 0
\(299\) −10.1722 + 10.1722i −0.588272 + 0.588272i
\(300\) 0 0
\(301\) −2.40998 2.40998i −0.138909 0.138909i
\(302\) 0 0
\(303\) −4.54451 4.54451i −0.261076 0.261076i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −22.6081 −1.29031 −0.645156 0.764051i \(-0.723208\pi\)
−0.645156 + 0.764051i \(0.723208\pi\)
\(308\) 0 0
\(309\) −2.28990 2.28990i −0.130268 0.130268i
\(310\) 0 0
\(311\) 10.7903 0.611859 0.305929 0.952054i \(-0.401033\pi\)
0.305929 + 0.952054i \(0.401033\pi\)
\(312\) 0 0
\(313\) −20.6842 20.6842i −1.16914 1.16914i −0.982412 0.186727i \(-0.940212\pi\)
−0.186727 0.982412i \(-0.559788\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.8207i 1.33791i −0.743305 0.668953i \(-0.766743\pi\)
0.743305 0.668953i \(-0.233257\pi\)
\(318\) 0 0
\(319\) 3.27372 0.183293
\(320\) 0 0
\(321\) 10.1161 0.564624
\(322\) 0 0
\(323\) 5.56042i 0.309390i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −7.90050 7.90050i −0.436899 0.436899i
\(328\) 0 0
\(329\) 15.4650 0.852615
\(330\) 0 0
\(331\) 19.7688 + 19.7688i 1.08659 + 1.08659i 0.995877 + 0.0907155i \(0.0289154\pi\)
0.0907155 + 0.995877i \(0.471085\pi\)
\(332\) 0 0
\(333\) 11.9153 0.652957
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.26955 7.26955i −0.395998 0.395998i 0.480821 0.876819i \(-0.340339\pi\)
−0.876819 + 0.480821i \(0.840339\pi\)
\(338\) 0 0
\(339\) 0.530758 + 0.530758i 0.0288268 + 0.0288268i
\(340\) 0 0
\(341\) 4.62347 4.62347i 0.250375 0.250375i
\(342\) 0 0
\(343\) −5.91866 + 5.91866i −0.319578 + 0.319578i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.4667i 1.25976i −0.776692 0.629880i \(-0.783104\pi\)
0.776692 0.629880i \(-0.216896\pi\)
\(348\) 0 0
\(349\) −23.2089 + 23.2089i −1.24234 + 1.24234i −0.283315 + 0.959027i \(0.591434\pi\)
−0.959027 + 0.283315i \(0.908566\pi\)
\(350\) 0 0
\(351\) 7.10035i 0.378988i
\(352\) 0 0
\(353\) 13.3220 13.3220i 0.709059 0.709059i −0.257278 0.966337i \(-0.582826\pi\)
0.966337 + 0.257278i \(0.0828256\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 14.1799 0.750479
\(358\) 0 0
\(359\) 26.9902i 1.42449i −0.701932 0.712244i \(-0.747679\pi\)
0.701932 0.712244i \(-0.252321\pi\)
\(360\) 0 0
\(361\) 18.0657i 0.950827i
\(362\) 0 0
\(363\) −2.04187 −0.107170
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −19.4758 + 19.4758i −1.01663 + 1.01663i −0.0167684 + 0.999859i \(0.505338\pi\)
−0.999859 + 0.0167684i \(0.994662\pi\)
\(368\) 0 0
\(369\) 13.3018i 0.692464i
\(370\) 0 0
\(371\) −14.6798 + 14.6798i −0.762136 + 0.762136i
\(372\) 0 0
\(373\) 4.87069i 0.252195i 0.992018 + 0.126097i \(0.0402452\pi\)
−0.992018 + 0.126097i \(0.959755\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.71677 + 1.71677i −0.0884180 + 0.0884180i
\(378\) 0 0
\(379\) −2.54450 + 2.54450i −0.130702 + 0.130702i −0.769432 0.638729i \(-0.779460\pi\)
0.638729 + 0.769432i \(0.279460\pi\)
\(380\) 0 0
\(381\) −7.24846 7.24846i −0.371350 0.371350i
\(382\) 0 0
\(383\) 0.193238 + 0.193238i 0.00987399 + 0.00987399i 0.712027 0.702153i \(-0.247778\pi\)
−0.702153 + 0.712027i \(0.747778\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.22831 −0.113272
\(388\) 0 0
\(389\) 2.01528 + 2.01528i 0.102179 + 0.102179i 0.756348 0.654169i \(-0.226982\pi\)
−0.654169 + 0.756348i \(0.726982\pi\)
\(390\) 0 0
\(391\) −40.2718 −2.03663
\(392\) 0 0
\(393\) −9.81803 9.81803i −0.495254 0.495254i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 21.5509i 1.08161i 0.841149 + 0.540804i \(0.181880\pi\)
−0.841149 + 0.540804i \(0.818120\pi\)
\(398\) 0 0
\(399\) −2.38257 −0.119277
\(400\) 0 0
\(401\) −10.3965 −0.519176 −0.259588 0.965719i \(-0.583587\pi\)
−0.259588 + 0.965719i \(0.583587\pi\)
\(402\) 0 0
\(403\) 4.84917i 0.241555i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.90264 8.90264i −0.441288 0.441288i
\(408\) 0 0
\(409\) −0.330732 −0.0163536 −0.00817682 0.999967i \(-0.502603\pi\)
−0.00817682 + 0.999967i \(0.502603\pi\)
\(410\) 0 0
\(411\) −0.795389 0.795389i −0.0392337 0.0392337i
\(412\) 0 0
\(413\) −23.6150 −1.16202
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.29395 5.29395i −0.259246 0.259246i
\(418\) 0 0
\(419\) 6.71354 + 6.71354i 0.327978 + 0.327978i 0.851817 0.523839i \(-0.175501\pi\)
−0.523839 + 0.851817i \(0.675501\pi\)
\(420\) 0 0
\(421\) 2.99831 2.99831i 0.146129 0.146129i −0.630258 0.776386i \(-0.717051\pi\)
0.776386 + 0.630258i \(0.217051\pi\)
\(422\) 0 0
\(423\) 7.14963 7.14963i 0.347627 0.347627i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 31.5208i 1.52540i
\(428\) 0 0
\(429\) −2.47435 + 2.47435i −0.119463 + 0.119463i
\(430\) 0 0
\(431\) 19.9548i 0.961191i 0.876942 + 0.480596i \(0.159580\pi\)
−0.876942 + 0.480596i \(0.840420\pi\)
\(432\) 0 0
\(433\) 16.1910 16.1910i 0.778092 0.778092i −0.201414 0.979506i \(-0.564554\pi\)
0.979506 + 0.201414i \(0.0645537\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.76664 0.323692
\(438\) 0 0
\(439\) 29.3734i 1.40191i −0.713204 0.700957i \(-0.752757\pi\)
0.713204 0.700957i \(-0.247243\pi\)
\(440\) 0 0
\(441\) 23.8287i 1.13470i
\(442\) 0 0
\(443\) −19.8713 −0.944115 −0.472057 0.881568i \(-0.656489\pi\)
−0.472057 + 0.881568i \(0.656489\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.0495422 + 0.0495422i −0.00234326 + 0.00234326i
\(448\) 0 0
\(449\) 16.7577i 0.790844i −0.918500 0.395422i \(-0.870598\pi\)
0.918500 0.395422i \(-0.129402\pi\)
\(450\) 0 0
\(451\) −9.93854 + 9.93854i −0.467987 + 0.467987i
\(452\) 0 0
\(453\) 1.99856i 0.0939005i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00267 5.00267i 0.234015 0.234015i −0.580351 0.814366i \(-0.697085\pi\)
0.814366 + 0.580351i \(0.197085\pi\)
\(458\) 0 0
\(459\) 14.0552 14.0552i 0.656039 0.656039i
\(460\) 0 0
\(461\) 2.71518 + 2.71518i 0.126459 + 0.126459i 0.767503 0.641045i \(-0.221499\pi\)
−0.641045 + 0.767503i \(0.721499\pi\)
\(462\) 0 0
\(463\) 9.18551 + 9.18551i 0.426887 + 0.426887i 0.887566 0.460680i \(-0.152394\pi\)
−0.460680 + 0.887566i \(0.652394\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.06405 −0.0492385 −0.0246193 0.999697i \(-0.507837\pi\)
−0.0246193 + 0.999697i \(0.507837\pi\)
\(468\) 0 0
\(469\) −4.92371 4.92371i −0.227356 0.227356i
\(470\) 0 0
\(471\) 5.57197 0.256743
\(472\) 0 0
\(473\) 1.66490 + 1.66490i 0.0765523 + 0.0765523i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 13.5732i 0.621474i
\(478\) 0 0
\(479\) −15.8658 −0.724926 −0.362463 0.931998i \(-0.618064\pi\)
−0.362463 + 0.931998i \(0.618064\pi\)
\(480\) 0 0
\(481\) 9.33725 0.425742
\(482\) 0 0
\(483\) 17.2559i 0.785170i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −13.7947 13.7947i −0.625099 0.625099i 0.321732 0.946831i \(-0.395735\pi\)
−0.946831 + 0.321732i \(0.895735\pi\)
\(488\) 0 0
\(489\) −2.41717 −0.109308
\(490\) 0 0
\(491\) −19.4471 19.4471i −0.877637 0.877637i 0.115652 0.993290i \(-0.463104\pi\)
−0.993290 + 0.115652i \(0.963104\pi\)
\(492\) 0 0
\(493\) −6.79669 −0.306108
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.62302 6.62302i −0.297083 0.297083i
\(498\) 0 0
\(499\) −23.0141 23.0141i −1.03025 1.03025i −0.999528 0.0307258i \(-0.990218\pi\)
−0.0307258 0.999528i \(-0.509782\pi\)
\(500\) 0 0
\(501\) 4.99775 4.99775i 0.223283 0.223283i
\(502\) 0 0
\(503\) 6.63364 6.63364i 0.295780 0.295780i −0.543579 0.839358i \(-0.682931\pi\)
0.839358 + 0.543579i \(0.182931\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 5.39422i 0.239566i
\(508\) 0 0
\(509\) 8.04140 8.04140i 0.356429 0.356429i −0.506066 0.862495i \(-0.668901\pi\)
0.862495 + 0.506066i \(0.168901\pi\)
\(510\) 0 0
\(511\) 24.9398i 1.10327i
\(512\) 0 0
\(513\) −2.36161 + 2.36161i −0.104268 + 0.104268i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −10.6838 −0.469873
\(518\) 0 0
\(519\) 4.22044i 0.185257i
\(520\) 0 0
\(521\) 32.8549i 1.43940i 0.694285 + 0.719700i \(0.255721\pi\)
−0.694285 + 0.719700i \(0.744279\pi\)
\(522\) 0 0
\(523\) −2.46341 −0.107717 −0.0538587 0.998549i \(-0.517152\pi\)
−0.0538587 + 0.998549i \(0.517152\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.59896 + 9.59896i −0.418137 + 0.418137i
\(528\) 0 0
\(529\) 26.0078i 1.13078i
\(530\) 0 0
\(531\) −10.9174 + 10.9174i −0.473775 + 0.473775i
\(532\) 0 0
\(533\) 10.4237i 0.451501i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −9.65586 + 9.65586i −0.416681 + 0.416681i
\(538\) 0 0
\(539\) 17.8038 17.8038i 0.766863 0.766863i
\(540\) 0 0
\(541\) −18.0772 18.0772i −0.777198 0.777198i 0.202156 0.979353i \(-0.435205\pi\)
−0.979353 + 0.202156i \(0.935205\pi\)
\(542\) 0 0
\(543\) 8.54938 + 8.54938i 0.366889 + 0.366889i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 43.6742 1.86738 0.933688 0.358089i \(-0.116572\pi\)
0.933688 + 0.358089i \(0.116572\pi\)
\(548\) 0 0
\(549\) 14.5723 + 14.5723i 0.621932 + 0.621932i
\(550\) 0 0
\(551\) 1.14201 0.0486513
\(552\) 0 0
\(553\) 39.9869 + 39.9869i 1.70042 + 1.70042i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.18948i 0.219885i 0.993938 + 0.109943i \(0.0350667\pi\)
−0.993938 + 0.109943i \(0.964933\pi\)
\(558\) 0 0
\(559\) −1.74618 −0.0738555
\(560\) 0 0
\(561\) −9.79597 −0.413586
\(562\) 0 0
\(563\) 11.3756i 0.479423i −0.970844 0.239711i \(-0.922947\pi\)
0.970844 0.239711i \(-0.0770528\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 16.2889 + 16.2889i 0.684071 + 0.684071i
\(568\) 0 0
\(569\) −7.51787 −0.315165 −0.157583 0.987506i \(-0.550370\pi\)
−0.157583 + 0.987506i \(0.550370\pi\)
\(570\) 0 0
\(571\) 7.76889 + 7.76889i 0.325118 + 0.325118i 0.850726 0.525609i \(-0.176162\pi\)
−0.525609 + 0.850726i \(0.676162\pi\)
\(572\) 0 0
\(573\) 6.35416 0.265449
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.84819 + 9.84819i 0.409986 + 0.409986i 0.881733 0.471748i \(-0.156377\pi\)
−0.471748 + 0.881733i \(0.656377\pi\)
\(578\) 0 0
\(579\) 8.11720 + 8.11720i 0.337339 + 0.337339i
\(580\) 0 0
\(581\) −7.81984 + 7.81984i −0.324421 + 0.324421i
\(582\) 0 0
\(583\) 10.1413 10.1413i 0.420010 0.420010i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0447i 1.36390i −0.731398 0.681951i \(-0.761132\pi\)
0.731398 0.681951i \(-0.238868\pi\)
\(588\) 0 0
\(589\) 1.61286 1.61286i 0.0664567 0.0664567i
\(590\) 0 0
\(591\) 9.36829i 0.385360i
\(592\) 0 0
\(593\) −18.5424 + 18.5424i −0.761445 + 0.761445i −0.976584 0.215139i \(-0.930980\pi\)
0.215139 + 0.976584i \(0.430980\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.06521 −0.125451
\(598\) 0 0
\(599\) 28.3117i 1.15678i 0.815759 + 0.578392i \(0.196319\pi\)
−0.815759 + 0.578392i \(0.803681\pi\)
\(600\) 0 0
\(601\) 41.7630i 1.70355i −0.523909 0.851774i \(-0.675527\pi\)
0.523909 0.851774i \(-0.324473\pi\)
\(602\) 0 0
\(603\) −4.55255 −0.185394
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.01973 4.01973i 0.163156 0.163156i −0.620807 0.783963i \(-0.713195\pi\)
0.783963 + 0.620807i \(0.213195\pi\)
\(608\) 0 0
\(609\) 2.91229i 0.118012i
\(610\) 0 0
\(611\) 5.60268 5.60268i 0.226660 0.226660i
\(612\) 0 0
\(613\) 21.5230i 0.869305i 0.900598 + 0.434652i \(0.143129\pi\)
−0.900598 + 0.434652i \(0.856871\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.4655 26.4655i 1.06546 1.06546i 0.0677580 0.997702i \(-0.478415\pi\)
0.997702 0.0677580i \(-0.0215846\pi\)
\(618\) 0 0
\(619\) −21.7935 + 21.7935i −0.875955 + 0.875955i −0.993113 0.117158i \(-0.962622\pi\)
0.117158 + 0.993113i \(0.462622\pi\)
\(620\) 0 0
\(621\) 17.1041 + 17.1041i 0.686365 + 0.686365i
\(622\) 0 0
\(623\) −33.0258 33.0258i −1.32315 1.32315i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.64596 0.0657334
\(628\) 0 0
\(629\) 18.4831 + 18.4831i 0.736971 + 0.736971i
\(630\) 0 0
\(631\) 42.7412 1.70150 0.850751 0.525570i \(-0.176148\pi\)
0.850751 + 0.525570i \(0.176148\pi\)
\(632\) 0 0
\(633\) 6.37937 + 6.37937i 0.253557 + 0.253557i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 18.6729i 0.739848i
\(638\) 0 0
\(639\) −6.12376 −0.242252
\(640\) 0 0
\(641\) 45.4930 1.79687 0.898433 0.439110i \(-0.144706\pi\)
0.898433 + 0.439110i \(0.144706\pi\)
\(642\) 0 0
\(643\) 31.3531i 1.23645i 0.786002 + 0.618224i \(0.212147\pi\)
−0.786002 + 0.618224i \(0.787853\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0355 24.0355i −0.944932 0.944932i 0.0536292 0.998561i \(-0.482921\pi\)
−0.998561 + 0.0536292i \(0.982921\pi\)
\(648\) 0 0
\(649\) 16.3141 0.640383
\(650\) 0 0
\(651\) 4.11303 + 4.11303i 0.161202 + 0.161202i
\(652\) 0 0
\(653\) −15.4153 −0.603248 −0.301624 0.953427i \(-0.597529\pi\)
−0.301624 + 0.953427i \(0.597529\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −11.5299 11.5299i −0.449825 0.449825i
\(658\) 0 0
\(659\) 30.4355 + 30.4355i 1.18560 + 1.18560i 0.978272 + 0.207327i \(0.0664763\pi\)
0.207327 + 0.978272i \(0.433524\pi\)
\(660\) 0 0
\(661\) −11.2208 + 11.2208i −0.436437 + 0.436437i −0.890811 0.454374i \(-0.849863\pi\)
0.454374 + 0.890811i \(0.349863\pi\)
\(662\) 0 0
\(663\) 5.13709 5.13709i 0.199508 0.199508i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.27109i 0.320258i
\(668\) 0 0
\(669\) −0.920946 + 0.920946i −0.0356058 + 0.0356058i
\(670\) 0 0
\(671\) 21.7757i 0.840640i
\(672\) 0 0
\(673\) 29.2965 29.2965i 1.12930 1.12930i 0.139006 0.990291i \(-0.455609\pi\)
0.990291 0.139006i \(-0.0443908\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.74511 −0.105503 −0.0527516 0.998608i \(-0.516799\pi\)
−0.0527516 + 0.998608i \(0.516799\pi\)
\(678\) 0 0
\(679\) 20.0191i 0.768261i
\(680\) 0 0
\(681\) 9.60850i 0.368199i
\(682\) 0 0
\(683\) −33.0796 −1.26576 −0.632878 0.774251i \(-0.718127\pi\)
−0.632878 + 0.774251i \(0.718127\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.98647 + 5.98647i −0.228398 + 0.228398i
\(688\) 0 0
\(689\) 10.6364i 0.405214i
\(690\) 0 0
\(691\) 30.8216 30.8216i 1.17251 1.17251i 0.190899 0.981610i \(-0.438860\pi\)
0.981610 0.190899i \(-0.0611404\pi\)
\(692\) 0 0
\(693\) 29.1428i 1.10704i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20.6338 20.6338i 0.781561 0.781561i
\(698\) 0 0
\(699\) −0.312890 + 0.312890i −0.0118346 + 0.0118346i
\(700\) 0 0
\(701\) −22.1242 22.1242i −0.835619 0.835619i 0.152660 0.988279i \(-0.451216\pi\)
−0.988279 + 0.152660i \(0.951216\pi\)
\(702\) 0 0
\(703\) −3.10562 3.10562i −0.117131 0.117131i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 41.9440 1.57747
\(708\) 0 0
\(709\) −7.09244 7.09244i −0.266362 0.266362i 0.561270 0.827632i \(-0.310313\pi\)
−0.827632 + 0.561270i \(0.810313\pi\)
\(710\) 0 0
\(711\) 36.9726 1.38658
\(712\) 0 0
\(713\) −11.6812 11.6812i −0.437466 0.437466i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.03628i 0.188083i
\(718\) 0 0
\(719\) −30.2949 −1.12981 −0.564905 0.825156i \(-0.691087\pi\)
−0.564905 + 0.825156i \(0.691087\pi\)
\(720\) 0 0
\(721\) 21.1349 0.787104
\(722\) 0 0
\(723\) 3.48320i 0.129542i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15.9503 + 15.9503i 0.591566 + 0.591566i 0.938054 0.346489i \(-0.112626\pi\)
−0.346489 + 0.938054i \(0.612626\pi\)
\(728\) 0 0
\(729\) 8.69055 0.321872
\(730\) 0 0
\(731\) −3.45657 3.45657i −0.127846 0.127846i
\(732\) 0 0
\(733\) −35.8535 −1.32428 −0.662140 0.749380i \(-0.730352\pi\)
−0.662140 + 0.749380i \(0.730352\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.40147 + 3.40147i 0.125295 + 0.125295i
\(738\) 0 0
\(739\) 21.4532 + 21.4532i 0.789168 + 0.789168i 0.981358 0.192190i \(-0.0615590\pi\)
−0.192190 + 0.981358i \(0.561559\pi\)
\(740\) 0 0
\(741\) −0.863157 + 0.863157i −0.0317089 + 0.0317089i
\(742\) 0 0
\(743\) 13.0311 13.0311i 0.478063 0.478063i −0.426449 0.904512i \(-0.640235\pi\)
0.904512 + 0.426449i \(0.140235\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.23036i 0.264545i
\(748\) 0 0
\(749\) −46.6836 + 46.6836i −1.70578 + 1.70578i
\(750\) 0 0
\(751\) 22.4879i 0.820595i −0.911952 0.410297i \(-0.865425\pi\)
0.911952 0.410297i \(-0.134575\pi\)
\(752\) 0 0
\(753\) −9.06387 + 9.06387i −0.330306 + 0.330306i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −15.8781 −0.577100 −0.288550 0.957465i \(-0.593173\pi\)
−0.288550 + 0.957465i \(0.593173\pi\)
\(758\) 0 0
\(759\) 11.9210i 0.432705i
\(760\) 0 0
\(761\) 19.5227i 0.707696i −0.935303 0.353848i \(-0.884873\pi\)
0.935303 0.353848i \(-0.115127\pi\)
\(762\) 0 0
\(763\) 72.9184 2.63982
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.55524 + 8.55524i −0.308912 + 0.308912i
\(768\) 0 0
\(769\) 8.03843i 0.289873i −0.989441 0.144937i \(-0.953702\pi\)
0.989441 0.144937i \(-0.0462978\pi\)
\(770\) 0 0
\(771\) −2.21946 + 2.21946i −0.0799320 + 0.0799320i
\(772\) 0 0
\(773\) 40.5118i 1.45711i 0.684988 + 0.728554i \(0.259807\pi\)
−0.684988 + 0.728554i \(0.740193\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.91977 7.91977i 0.284120 0.284120i
\(778\) 0 0
\(779\) −3.46698 + 3.46698i −0.124217 + 0.124217i
\(780\) 0 0
\(781\) 4.57542 + 4.57542i 0.163721 + 0.163721i
\(782\) 0 0
\(783\) 2.88668 + 2.88668i 0.103161 + 0.103161i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 15.8333 0.564396 0.282198 0.959356i \(-0.408937\pi\)
0.282198 + 0.959356i \(0.408937\pi\)
\(788\) 0 0
\(789\) 4.17930 + 4.17930i 0.148787 + 0.148787i
\(790\) 0 0
\(791\) −4.89868 −0.174177
\(792\) 0 0
\(793\) 11.4194 + 11.4194i 0.405513 + 0.405513i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.2670i 0.363674i 0.983329 + 0.181837i \(0.0582044\pi\)
−0.983329 + 0.181837i \(0.941796\pi\)
\(798\) 0 0
\(799\) 22.1811 0.784710
\(800\) 0 0
\(801\) −30.5362 −1.07895
\(802\) 0 0
\(803\) 17.2293i 0.608010i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.737538 + 0.737538i 0.0259626 + 0.0259626i
\(808\) 0 0
\(809\) −9.16442 −0.322204 −0.161102 0.986938i \(-0.551505\pi\)
−0.161102 + 0.986938i \(0.551505\pi\)
\(810\) 0 0
\(811\) 22.1702 + 22.1702i 0.778502 + 0.778502i 0.979576 0.201074i \(-0.0644432\pi\)
−0.201074 + 0.979576i \(0.564443\pi\)
\(812\) 0 0
\(813\) 1.71827 0.0602625
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.580788 + 0.580788i 0.0203192 + 0.0203192i
\(818\) 0 0
\(819\) 15.2827 + 15.2827i 0.534022 + 0.534022i
\(820\) 0 0
\(821\) 13.3258 13.3258i 0.465074 0.465074i −0.435240 0.900314i \(-0.643337\pi\)
0.900314 + 0.435240i \(0.143337\pi\)
\(822\) 0 0
\(823\) −34.7796 + 34.7796i −1.21234 + 1.21234i −0.242084 + 0.970255i \(0.577831\pi\)
−0.970255 + 0.242084i \(0.922169\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.5717i 0.576253i 0.957592 + 0.288127i \(0.0930324\pi\)
−0.957592 + 0.288127i \(0.906968\pi\)
\(828\) 0 0
\(829\) −11.9869 + 11.9869i −0.416321 + 0.416321i −0.883933 0.467613i \(-0.845114\pi\)
0.467613 + 0.883933i \(0.345114\pi\)
\(830\) 0 0
\(831\) 8.48807i 0.294448i
\(832\) 0 0
\(833\) −36.9631 + 36.9631i −1.28070 + 1.28070i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.15370 0.281833
\(838\) 0 0
\(839\) 4.44215i 0.153360i −0.997056 0.0766800i \(-0.975568\pi\)
0.997056 0.0766800i \(-0.0244320\pi\)
\(840\) 0 0
\(841\) 27.6041i 0.951865i
\(842\) 0 0
\(843\) 4.43543 0.152764
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9.42281 9.42281i 0.323772 0.323772i
\(848\) 0 0
\(849\) 15.5478i 0.533599i
\(850\) 0 0
\(851\) −22.4926 + 22.4926i −0.771038 + 0.771038i
\(852\) 0 0
\(853\) 35.6748i 1.22148i 0.791830 + 0.610742i \(0.209129\pi\)
−0.791830 + 0.610742i \(0.790871\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.8568 13.8568i 0.473340 0.473340i −0.429654 0.902994i \(-0.641364\pi\)
0.902994 + 0.429654i \(0.141364\pi\)
\(858\) 0 0
\(859\) 19.4217 19.4217i 0.662660 0.662660i −0.293346 0.956006i \(-0.594769\pi\)
0.956006 + 0.293346i \(0.0947690\pi\)
\(860\) 0 0
\(861\) −8.84130 8.84130i −0.301311 0.301311i
\(862\) 0 0
\(863\) 9.22041 + 9.22041i 0.313866 + 0.313866i 0.846405 0.532539i \(-0.178762\pi\)
−0.532539 + 0.846405i \(0.678762\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.89018 0.335888
\(868\) 0 0
\(869\) −27.6244 27.6244i −0.937093 0.937093i
\(870\) 0 0
\(871\) −3.56753 −0.120881
\(872\) 0 0
\(873\) 9.25500 + 9.25500i 0.313234 + 0.313234i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.4267i 0.352084i −0.984383 0.176042i \(-0.943670\pi\)
0.984383 0.176042i \(-0.0563295\pi\)
\(878\) 0 0
\(879\) −8.68236 −0.292849
\(880\) 0 0
\(881\) −12.7405 −0.429239 −0.214619 0.976698i \(-0.568851\pi\)
−0.214619 + 0.976698i \(0.568851\pi\)
\(882\) 0 0
\(883\) 27.9073i 0.939156i −0.882891 0.469578i \(-0.844406\pi\)
0.882891 0.469578i \(-0.155594\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 41.7449 + 41.7449i 1.40166 + 1.40166i 0.794846 + 0.606811i \(0.207552\pi\)
0.606811 + 0.794846i \(0.292448\pi\)
\(888\) 0 0
\(889\) 66.9004 2.24377
\(890\) 0 0
\(891\) −11.2530 11.2530i −0.376989 0.376989i
\(892\) 0 0
\(893\) −3.72696 −0.124718
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.25148 + 6.25148i 0.208731 + 0.208731i
\(898\) 0 0
\(899\) −1.97145 1.97145i −0.0657516 0.0657516i
\(900\) 0 0
\(901\) −21.0548 + 21.0548i −0.701437 + 0.701437i
\(902\) 0 0
\(903\) −1.48109 + 1.48109i −0.0492877 + 0.0492877i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 26.7614i 0.888597i 0.895879 + 0.444298i \(0.146547\pi\)
−0.895879 + 0.444298i \(0.853453\pi\)
\(908\) 0 0
\(909\) 19.3911 19.3911i 0.643163 0.643163i
\(910\) 0 0
\(911\) 19.2403i 0.637459i −0.947846 0.318729i \(-0.896744\pi\)
0.947846 0.318729i \(-0.103256\pi\)
\(912\) 0 0
\(913\) 5.40222 5.40222i 0.178787 0.178787i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 90.6165 2.99242
\(918\) 0 0
\(919\) 42.6903i 1.40822i −0.710090 0.704111i \(-0.751346\pi\)
0.710090 0.704111i \(-0.248654\pi\)
\(920\) 0 0
\(921\) 13.8942i 0.457828i
\(922\) 0 0
\(923\) −4.79878 −0.157954
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.77085 9.77085i 0.320917 0.320917i
\(928\) 0 0
\(929\) 5.58037i 0.183086i 0.995801 + 0.0915430i \(0.0291799\pi\)
−0.995801 + 0.0915430i \(0.970820\pi\)
\(930\) 0 0
\(931\) 6.21070 6.21070i 0.203548 0.203548i
\(932\) 0 0
\(933\) 6.63132i 0.217100i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −41.0680 + 41.0680i −1.34163 + 1.34163i −0.447197 + 0.894435i \(0.647578\pi\)
−0.894435 + 0.447197i \(0.852422\pi\)
\(938\) 0 0
\(939\) −12.7118 + 12.7118i −0.414834 + 0.414834i
\(940\) 0 0
\(941\) −31.5476 31.5476i −1.02842 1.02842i −0.999584 0.0288377i \(-0.990819\pi\)
−0.0288377 0.999584i \(-0.509181\pi\)
\(942\) 0 0
\(943\) 25.1098 + 25.1098i 0.817689 + 0.817689i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.7892 −1.13050 −0.565248 0.824921i \(-0.691220\pi\)
−0.565248 + 0.824921i \(0.691220\pi\)
\(948\) 0 0
\(949\) −9.03522 9.03522i −0.293296 0.293296i
\(950\) 0 0
\(951\) −14.6394 −0.474715
\(952\) 0 0
\(953\) −26.7047 26.7047i −0.865050 0.865050i 0.126870 0.991919i \(-0.459507\pi\)
−0.991919 + 0.126870i \(0.959507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.01191i 0.0650360i
\(958\) 0 0
\(959\) 7.34112 0.237057
\(960\) 0 0
\(961\) 25.4314 0.820369
\(962\) 0 0
\(963\) 43.1645i 1.39096i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −12.8711 12.8711i −0.413906 0.413906i 0.469191 0.883097i \(-0.344546\pi\)
−0.883097 + 0.469191i \(0.844546\pi\)
\(968\) 0 0
\(969\) −3.41725 −0.109778
\(970\) 0 0
\(971\) 23.9028 + 23.9028i 0.767078 + 0.767078i 0.977591 0.210513i \(-0.0675134\pi\)
−0.210513 + 0.977591i \(0.567513\pi\)
\(972\) 0 0
\(973\) 48.8610 1.56641
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.71449 2.71449i −0.0868441 0.0868441i 0.662350 0.749194i \(-0.269559\pi\)
−0.749194 + 0.662350i \(0.769559\pi\)
\(978\) 0 0
\(979\) 22.8154 + 22.8154i 0.729183 + 0.729183i
\(980\) 0 0
\(981\) 33.7109 33.7109i 1.07630 1.07630i
\(982\) 0 0
\(983\) −13.7542 + 13.7542i −0.438692 + 0.438692i −0.891572 0.452880i \(-0.850397\pi\)
0.452880 + 0.891572i \(0.350397\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9.50428i 0.302525i
\(988\) 0 0
\(989\) 4.20640 4.20640i 0.133756 0.133756i
\(990\) 0 0
\(991\) 26.5971i 0.844883i −0.906390 0.422442i \(-0.861173\pi\)
0.906390 0.422442i \(-0.138827\pi\)
\(992\) 0 0
\(993\) 12.1492 12.1492i 0.385545 0.385545i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.4590 0.806295 0.403148 0.915135i \(-0.367916\pi\)
0.403148 + 0.915135i \(0.367916\pi\)
\(998\) 0 0
\(999\) 15.7002i 0.496733i
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 1600.2.j.d.1007.5 18
4.3 odd 2 400.2.j.d.307.3 18
5.2 odd 4 320.2.s.b.303.5 18
5.3 odd 4 1600.2.s.d.943.5 18
5.4 even 2 320.2.j.b.47.5 18
16.5 even 4 400.2.s.d.107.7 18
16.11 odd 4 1600.2.s.d.207.5 18
20.3 even 4 400.2.s.d.243.7 18
20.7 even 4 80.2.s.b.3.3 yes 18
20.19 odd 2 80.2.j.b.67.7 yes 18
40.19 odd 2 640.2.j.d.607.5 18
40.27 even 4 640.2.s.d.223.5 18
40.29 even 2 640.2.j.c.607.5 18
40.37 odd 4 640.2.s.c.223.5 18
60.47 odd 4 720.2.z.g.163.7 18
60.59 even 2 720.2.bd.g.307.3 18
80.19 odd 4 640.2.s.c.287.5 18
80.27 even 4 320.2.j.b.143.5 18
80.29 even 4 640.2.s.d.287.5 18
80.37 odd 4 80.2.j.b.43.7 18
80.43 even 4 inner 1600.2.j.d.143.5 18
80.53 odd 4 400.2.j.d.43.3 18
80.59 odd 4 320.2.s.b.207.5 18
80.67 even 4 640.2.j.c.543.5 18
80.69 even 4 80.2.s.b.27.3 yes 18
80.77 odd 4 640.2.j.d.543.5 18
240.149 odd 4 720.2.z.g.667.7 18
240.197 even 4 720.2.bd.g.523.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.7 18 80.37 odd 4
80.2.j.b.67.7 yes 18 20.19 odd 2
80.2.s.b.3.3 yes 18 20.7 even 4
80.2.s.b.27.3 yes 18 80.69 even 4
320.2.j.b.47.5 18 5.4 even 2
320.2.j.b.143.5 18 80.27 even 4
320.2.s.b.207.5 18 80.59 odd 4
320.2.s.b.303.5 18 5.2 odd 4
400.2.j.d.43.3 18 80.53 odd 4
400.2.j.d.307.3 18 4.3 odd 2
400.2.s.d.107.7 18 16.5 even 4
400.2.s.d.243.7 18 20.3 even 4
640.2.j.c.543.5 18 80.67 even 4
640.2.j.c.607.5 18 40.29 even 2
640.2.j.d.543.5 18 80.77 odd 4
640.2.j.d.607.5 18 40.19 odd 2
640.2.s.c.223.5 18 40.37 odd 4
640.2.s.c.287.5 18 80.19 odd 4
640.2.s.d.223.5 18 40.27 even 4
640.2.s.d.287.5 18 80.29 even 4
720.2.z.g.163.7 18 60.47 odd 4
720.2.z.g.667.7 18 240.149 odd 4
720.2.bd.g.307.3 18 60.59 even 2
720.2.bd.g.523.3 18 240.197 even 4
1600.2.j.d.143.5 18 80.43 even 4 inner
1600.2.j.d.1007.5 18 1.1 even 1 trivial
1600.2.s.d.207.5 18 16.11 odd 4
1600.2.s.d.943.5 18 5.3 odd 4