Properties

Label 900.3.p.c.401.1
Level $900$
Weight $3$
Character 900.401
Analytic conductor $24.523$
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] = [900,3,Mod(101,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} - 9720 x^{4} + 43740 x^{3} - 72171 x^{2} - 118098 x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 401.1
Root \(2.89597 - 0.783177i\) of defining polynomial
Character \(\chi\) \(=\) 900.401
Dual form 900.3.p.c.101.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.89597 + 0.783177i) q^{3} +(2.35650 + 4.08158i) q^{7} +(7.77327 - 4.53611i) q^{9} +O(q^{10})\) \(q+(-2.89597 + 0.783177i) q^{3} +(2.35650 + 4.08158i) q^{7} +(7.77327 - 4.53611i) q^{9} +(16.6882 - 9.63493i) q^{11} +(-4.68482 + 8.11434i) q^{13} -16.2489i q^{17} -19.0505 q^{19} +(-10.0210 - 9.97458i) q^{21} +(-14.3817 - 8.30327i) q^{23} +(-18.9586 + 19.2243i) q^{27} +(-3.87581 + 2.23770i) q^{29} +(-7.19383 + 12.4601i) q^{31} +(-40.7826 + 40.9723i) q^{33} +55.6092 q^{37} +(7.21211 - 27.1679i) q^{39} +(26.7067 + 15.4191i) q^{41} +(-21.5538 - 37.3322i) q^{43} +(35.9308 - 20.7446i) q^{47} +(13.3938 - 23.1987i) q^{49} +(12.7257 + 47.0562i) q^{51} -30.7037i q^{53} +(55.1696 - 14.9199i) q^{57} +(74.4236 + 42.9685i) q^{59} +(-51.7462 - 89.6270i) q^{61} +(36.8323 + 21.0379i) q^{63} +(61.7879 - 107.020i) q^{67} +(48.1518 + 12.7826i) q^{69} +111.409i q^{71} +90.5781 q^{73} +(78.6515 + 45.4095i) q^{77} +(41.6569 + 72.1519i) q^{79} +(39.8473 - 70.5208i) q^{81} +(-31.0471 + 17.9251i) q^{83} +(9.47171 - 9.51576i) q^{87} -84.9144i q^{89} -44.1591 q^{91} +(11.0747 - 41.7181i) q^{93} +(-52.6259 - 91.1508i) q^{97} +(86.0166 - 150.594i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 6 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 6 q^{7} + 26 q^{9} + 48 q^{11} + 30 q^{13} + 72 q^{19} - 128 q^{21} + 78 q^{23} + 106 q^{27} + 150 q^{29} - 12 q^{31} - 96 q^{33} + 12 q^{37} + 40 q^{39} + 90 q^{41} - 114 q^{43} - 12 q^{47} + 48 q^{49} - 144 q^{51} + 158 q^{57} + 48 q^{59} - 78 q^{61} + 212 q^{63} + 168 q^{67} - 150 q^{69} + 24 q^{73} + 258 q^{77} + 120 q^{79} + 434 q^{81} - 114 q^{83} + 330 q^{87} + 120 q^{91} - 82 q^{93} - 96 q^{97} - 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.89597 + 0.783177i −0.965323 + 0.261059i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.35650 + 4.08158i 0.336643 + 0.583083i 0.983799 0.179274i \(-0.0573750\pi\)
−0.647156 + 0.762358i \(0.724042\pi\)
\(8\) 0 0
\(9\) 7.77327 4.53611i 0.863696 0.504013i
\(10\) 0 0
\(11\) 16.6882 9.63493i 1.51711 0.875902i 0.517310 0.855798i \(-0.326934\pi\)
0.999798 0.0201041i \(-0.00639976\pi\)
\(12\) 0 0
\(13\) −4.68482 + 8.11434i −0.360370 + 0.624180i −0.988022 0.154315i \(-0.950683\pi\)
0.627651 + 0.778495i \(0.284016\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.2489i 0.955816i −0.878410 0.477908i \(-0.841395\pi\)
0.878410 0.477908i \(-0.158605\pi\)
\(18\) 0 0
\(19\) −19.0505 −1.00266 −0.501328 0.865257i \(-0.667155\pi\)
−0.501328 + 0.865257i \(0.667155\pi\)
\(20\) 0 0
\(21\) −10.0210 9.97458i −0.477189 0.474980i
\(22\) 0 0
\(23\) −14.3817 8.30327i −0.625290 0.361012i 0.153635 0.988128i \(-0.450902\pi\)
−0.778926 + 0.627116i \(0.784235\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −18.9586 + 19.2243i −0.702169 + 0.712011i
\(28\) 0 0
\(29\) −3.87581 + 2.23770i −0.133649 + 0.0771621i −0.565334 0.824862i \(-0.691253\pi\)
0.431685 + 0.902024i \(0.357919\pi\)
\(30\) 0 0
\(31\) −7.19383 + 12.4601i −0.232059 + 0.401938i −0.958414 0.285382i \(-0.907880\pi\)
0.726355 + 0.687320i \(0.241213\pi\)
\(32\) 0 0
\(33\) −40.7826 + 40.9723i −1.23584 + 1.24158i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 55.6092 1.50295 0.751476 0.659761i \(-0.229342\pi\)
0.751476 + 0.659761i \(0.229342\pi\)
\(38\) 0 0
\(39\) 7.21211 27.1679i 0.184926 0.696613i
\(40\) 0 0
\(41\) 26.7067 + 15.4191i 0.651384 + 0.376077i 0.788986 0.614411i \(-0.210606\pi\)
−0.137602 + 0.990488i \(0.543940\pi\)
\(42\) 0 0
\(43\) −21.5538 37.3322i −0.501250 0.868191i −0.999999 0.00144406i \(-0.999540\pi\)
0.498749 0.866747i \(-0.333793\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 35.9308 20.7446i 0.764484 0.441375i −0.0664192 0.997792i \(-0.521157\pi\)
0.830903 + 0.556417i \(0.187824\pi\)
\(48\) 0 0
\(49\) 13.3938 23.1987i 0.273343 0.473443i
\(50\) 0 0
\(51\) 12.7257 + 47.0562i 0.249524 + 0.922671i
\(52\) 0 0
\(53\) 30.7037i 0.579316i −0.957130 0.289658i \(-0.906458\pi\)
0.957130 0.289658i \(-0.0935415\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 55.1696 14.9199i 0.967887 0.261753i
\(58\) 0 0
\(59\) 74.4236 + 42.9685i 1.26142 + 0.728279i 0.973348 0.229331i \(-0.0736538\pi\)
0.288068 + 0.957610i \(0.406987\pi\)
\(60\) 0 0
\(61\) −51.7462 89.6270i −0.848298 1.46930i −0.882726 0.469889i \(-0.844294\pi\)
0.0344274 0.999407i \(-0.489039\pi\)
\(62\) 0 0
\(63\) 36.8323 + 21.0379i 0.584639 + 0.333934i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 61.7879 107.020i 0.922207 1.59731i 0.126215 0.992003i \(-0.459717\pi\)
0.795992 0.605307i \(-0.206950\pi\)
\(68\) 0 0
\(69\) 48.1518 + 12.7826i 0.697853 + 0.185255i
\(70\) 0 0
\(71\) 111.409i 1.56914i 0.620043 + 0.784568i \(0.287115\pi\)
−0.620043 + 0.784568i \(0.712885\pi\)
\(72\) 0 0
\(73\) 90.5781 1.24080 0.620398 0.784287i \(-0.286971\pi\)
0.620398 + 0.784287i \(0.286971\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 78.6515 + 45.4095i 1.02145 + 0.589733i
\(78\) 0 0
\(79\) 41.6569 + 72.1519i 0.527302 + 0.913315i 0.999494 + 0.0318185i \(0.0101299\pi\)
−0.472191 + 0.881496i \(0.656537\pi\)
\(80\) 0 0
\(81\) 39.8473 70.5208i 0.491942 0.870628i
\(82\) 0 0
\(83\) −31.0471 + 17.9251i −0.374062 + 0.215965i −0.675231 0.737606i \(-0.735956\pi\)
0.301170 + 0.953571i \(0.402623\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.47171 9.51576i 0.108870 0.109377i
\(88\) 0 0
\(89\) 84.9144i 0.954095i −0.878878 0.477047i \(-0.841707\pi\)
0.878878 0.477047i \(-0.158293\pi\)
\(90\) 0 0
\(91\) −44.1591 −0.485265
\(92\) 0 0
\(93\) 11.0747 41.7181i 0.119082 0.448581i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −52.6259 91.1508i −0.542535 0.939699i −0.998758 0.0498329i \(-0.984131\pi\)
0.456222 0.889866i \(-0.349202\pi\)
\(98\) 0 0
\(99\) 86.0166 150.594i 0.868854 1.52116i
\(100\) 0 0
\(101\) 69.4641 40.1051i 0.687763 0.397080i −0.115010 0.993364i \(-0.536690\pi\)
0.802773 + 0.596284i \(0.203357\pi\)
\(102\) 0 0
\(103\) −65.2103 + 112.947i −0.633109 + 1.09658i 0.353803 + 0.935320i \(0.384888\pi\)
−0.986912 + 0.161258i \(0.948445\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 49.8390i 0.465785i 0.972502 + 0.232893i \(0.0748191\pi\)
−0.972502 + 0.232893i \(0.925181\pi\)
\(108\) 0 0
\(109\) 121.431 1.11404 0.557022 0.830498i \(-0.311944\pi\)
0.557022 + 0.830498i \(0.311944\pi\)
\(110\) 0 0
\(111\) −161.043 + 43.5519i −1.45083 + 0.392359i
\(112\) 0 0
\(113\) 24.4862 + 14.1371i 0.216692 + 0.125107i 0.604418 0.796668i \(-0.293406\pi\)
−0.387725 + 0.921775i \(0.626739\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.391246 + 84.3258i 0.00334398 + 0.720733i
\(118\) 0 0
\(119\) 66.3211 38.2905i 0.557320 0.321769i
\(120\) 0 0
\(121\) 125.164 216.790i 1.03441 1.79165i
\(122\) 0 0
\(123\) −89.4178 23.7372i −0.726974 0.192986i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 44.8753 0.353349 0.176674 0.984269i \(-0.443466\pi\)
0.176674 + 0.984269i \(0.443466\pi\)
\(128\) 0 0
\(129\) 91.6567 + 91.2324i 0.710517 + 0.707228i
\(130\) 0 0
\(131\) 63.7326 + 36.7960i 0.486508 + 0.280886i 0.723125 0.690717i \(-0.242705\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(132\) 0 0
\(133\) −44.8925 77.7561i −0.337538 0.584632i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 188.952 109.091i 1.37921 0.796286i 0.387144 0.922019i \(-0.373461\pi\)
0.992064 + 0.125733i \(0.0401282\pi\)
\(138\) 0 0
\(139\) 33.0004 57.1584i 0.237413 0.411212i −0.722558 0.691310i \(-0.757034\pi\)
0.959971 + 0.280099i \(0.0903672\pi\)
\(140\) 0 0
\(141\) −87.8076 + 88.2160i −0.622749 + 0.625645i
\(142\) 0 0
\(143\) 180.551i 1.26260i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −20.6193 + 77.6725i −0.140267 + 0.528384i
\(148\) 0 0
\(149\) −169.401 97.8035i −1.13692 0.656399i −0.191252 0.981541i \(-0.561255\pi\)
−0.945665 + 0.325142i \(0.894588\pi\)
\(150\) 0 0
\(151\) 79.5523 + 137.789i 0.526836 + 0.912507i 0.999511 + 0.0312701i \(0.00995522\pi\)
−0.472675 + 0.881237i \(0.656711\pi\)
\(152\) 0 0
\(153\) −73.7067 126.307i −0.481743 0.825534i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −64.1927 + 111.185i −0.408871 + 0.708185i −0.994764 0.102204i \(-0.967411\pi\)
0.585893 + 0.810389i \(0.300744\pi\)
\(158\) 0 0
\(159\) 24.0465 + 88.9170i 0.151236 + 0.559227i
\(160\) 0 0
\(161\) 78.2667i 0.486129i
\(162\) 0 0
\(163\) −21.6963 −0.133106 −0.0665531 0.997783i \(-0.521200\pi\)
−0.0665531 + 0.997783i \(0.521200\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 219.896 + 126.957i 1.31674 + 0.760223i 0.983204 0.182513i \(-0.0584230\pi\)
0.333541 + 0.942736i \(0.391756\pi\)
\(168\) 0 0
\(169\) 40.6050 + 70.3299i 0.240266 + 0.416153i
\(170\) 0 0
\(171\) −148.084 + 86.4151i −0.865991 + 0.505352i
\(172\) 0 0
\(173\) 51.4018 29.6768i 0.297120 0.171542i −0.344028 0.938959i \(-0.611792\pi\)
0.641148 + 0.767417i \(0.278458\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −249.180 66.1485i −1.40780 0.373720i
\(178\) 0 0
\(179\) 5.43887i 0.0303848i 0.999885 + 0.0151924i \(0.00483607\pi\)
−0.999885 + 0.0151924i \(0.995164\pi\)
\(180\) 0 0
\(181\) 45.4874 0.251312 0.125656 0.992074i \(-0.459896\pi\)
0.125656 + 0.992074i \(0.459896\pi\)
\(182\) 0 0
\(183\) 220.049 + 219.031i 1.20245 + 1.19689i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −156.557 271.164i −0.837201 1.45007i
\(188\) 0 0
\(189\) −123.141 32.0788i −0.651542 0.169729i
\(190\) 0 0
\(191\) 199.987 115.463i 1.04705 0.604516i 0.125230 0.992128i \(-0.460033\pi\)
0.921823 + 0.387612i \(0.126700\pi\)
\(192\) 0 0
\(193\) 36.4441 63.1231i 0.188830 0.327063i −0.756031 0.654536i \(-0.772864\pi\)
0.944860 + 0.327474i \(0.106197\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 39.2251i 0.199112i 0.995032 + 0.0995561i \(0.0317423\pi\)
−0.995032 + 0.0995561i \(0.968258\pi\)
\(198\) 0 0
\(199\) −268.068 −1.34707 −0.673537 0.739154i \(-0.735226\pi\)
−0.673537 + 0.739154i \(0.735226\pi\)
\(200\) 0 0
\(201\) −95.1203 + 358.317i −0.473235 + 1.78267i
\(202\) 0 0
\(203\) −18.2667 10.5463i −0.0899838 0.0519522i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −149.457 + 0.693436i −0.722015 + 0.00334993i
\(208\) 0 0
\(209\) −317.918 + 183.550i −1.52114 + 0.878229i
\(210\) 0 0
\(211\) 173.445 300.416i 0.822015 1.42377i −0.0821652 0.996619i \(-0.526184\pi\)
0.904180 0.427152i \(-0.140483\pi\)
\(212\) 0 0
\(213\) −87.2527 322.636i −0.409637 1.51472i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −67.8092 −0.312485
\(218\) 0 0
\(219\) −262.311 + 70.9387i −1.19777 + 0.323921i
\(220\) 0 0
\(221\) 131.849 + 76.1229i 0.596601 + 0.344448i
\(222\) 0 0
\(223\) −35.3838 61.2866i −0.158672 0.274828i 0.775718 0.631080i \(-0.217388\pi\)
−0.934390 + 0.356252i \(0.884055\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −232.838 + 134.429i −1.02572 + 0.592199i −0.915755 0.401737i \(-0.868407\pi\)
−0.109963 + 0.993936i \(0.535073\pi\)
\(228\) 0 0
\(229\) −103.497 + 179.261i −0.451950 + 0.782801i −0.998507 0.0546210i \(-0.982605\pi\)
0.546557 + 0.837422i \(0.315938\pi\)
\(230\) 0 0
\(231\) −263.336 69.9063i −1.13998 0.302625i
\(232\) 0 0
\(233\) 47.4893i 0.203817i −0.994794 0.101908i \(-0.967505\pi\)
0.994794 0.101908i \(-0.0324948\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −177.145 176.325i −0.747446 0.743986i
\(238\) 0 0
\(239\) −288.824 166.753i −1.20847 0.697710i −0.246045 0.969258i \(-0.579131\pi\)
−0.962425 + 0.271548i \(0.912464\pi\)
\(240\) 0 0
\(241\) −213.356 369.544i −0.885296 1.53338i −0.845374 0.534176i \(-0.820622\pi\)
−0.0399229 0.999203i \(-0.512711\pi\)
\(242\) 0 0
\(243\) −60.1663 + 235.434i −0.247598 + 0.968863i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 89.2480 154.582i 0.361328 0.625838i
\(248\) 0 0
\(249\) 75.8730 76.2258i 0.304711 0.306128i
\(250\) 0 0
\(251\) 394.340i 1.57107i 0.618814 + 0.785537i \(0.287613\pi\)
−0.618814 + 0.785537i \(0.712387\pi\)
\(252\) 0 0
\(253\) −320.005 −1.26484
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 313.600 + 181.057i 1.22023 + 0.704503i 0.964968 0.262368i \(-0.0845033\pi\)
0.255267 + 0.966871i \(0.417837\pi\)
\(258\) 0 0
\(259\) 131.043 + 226.974i 0.505959 + 0.876346i
\(260\) 0 0
\(261\) −19.9772 + 34.9754i −0.0765412 + 0.134005i
\(262\) 0 0
\(263\) 294.065 169.779i 1.11812 0.645546i 0.177198 0.984175i \(-0.443297\pi\)
0.940920 + 0.338630i \(0.109963\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 66.5031 + 245.910i 0.249075 + 0.921010i
\(268\) 0 0
\(269\) 23.2060i 0.0862678i 0.999069 + 0.0431339i \(0.0137342\pi\)
−0.999069 + 0.0431339i \(0.986266\pi\)
\(270\) 0 0
\(271\) −433.939 −1.60125 −0.800626 0.599165i \(-0.795499\pi\)
−0.800626 + 0.599165i \(0.795499\pi\)
\(272\) 0 0
\(273\) 127.883 34.5844i 0.468438 0.126683i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −170.725 295.704i −0.616336 1.06752i −0.990149 0.140021i \(-0.955283\pi\)
0.373813 0.927504i \(-0.378050\pi\)
\(278\) 0 0
\(279\) 0.600783 + 129.488i 0.00215334 + 0.464113i
\(280\) 0 0
\(281\) 175.833 101.517i 0.625741 0.361272i −0.153360 0.988170i \(-0.549009\pi\)
0.779101 + 0.626899i \(0.215676\pi\)
\(282\) 0 0
\(283\) 78.4126 135.815i 0.277076 0.479910i −0.693581 0.720379i \(-0.743968\pi\)
0.970657 + 0.240469i \(0.0773012\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 145.341i 0.506415i
\(288\) 0 0
\(289\) 24.9744 0.0864167
\(290\) 0 0
\(291\) 223.790 + 222.754i 0.769039 + 0.765479i
\(292\) 0 0
\(293\) −54.9412 31.7203i −0.187513 0.108260i 0.403305 0.915066i \(-0.367861\pi\)
−0.590818 + 0.806805i \(0.701195\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −131.159 + 503.483i −0.441613 + 1.69523i
\(298\) 0 0
\(299\) 134.751 77.7986i 0.450672 0.260196i
\(300\) 0 0
\(301\) 101.583 175.947i 0.337485 0.584541i
\(302\) 0 0
\(303\) −169.756 + 170.546i −0.560252 + 0.562857i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −183.760 −0.598568 −0.299284 0.954164i \(-0.596748\pi\)
−0.299284 + 0.954164i \(0.596748\pi\)
\(308\) 0 0
\(309\) 100.389 378.164i 0.324883 1.22383i
\(310\) 0 0
\(311\) 328.599 + 189.717i 1.05659 + 0.610021i 0.924486 0.381215i \(-0.124494\pi\)
0.132101 + 0.991236i \(0.457828\pi\)
\(312\) 0 0
\(313\) 204.306 + 353.868i 0.652734 + 1.13057i 0.982457 + 0.186491i \(0.0597115\pi\)
−0.329723 + 0.944078i \(0.606955\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −449.330 + 259.421i −1.41744 + 0.818362i −0.996074 0.0885271i \(-0.971784\pi\)
−0.421370 + 0.906889i \(0.638451\pi\)
\(318\) 0 0
\(319\) −43.1202 + 74.6863i −0.135173 + 0.234126i
\(320\) 0 0
\(321\) −39.0328 144.332i −0.121597 0.449633i
\(322\) 0 0
\(323\) 309.549i 0.958355i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −351.660 + 95.1019i −1.07541 + 0.290831i
\(328\) 0 0
\(329\) 169.342 + 97.7696i 0.514717 + 0.297172i
\(330\) 0 0
\(331\) −147.622 255.689i −0.445988 0.772474i 0.552133 0.833756i \(-0.313814\pi\)
−0.998120 + 0.0612827i \(0.980481\pi\)
\(332\) 0 0
\(333\) 432.265 252.250i 1.29809 0.757507i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −53.8810 + 93.3245i −0.159884 + 0.276927i −0.934827 0.355104i \(-0.884445\pi\)
0.774943 + 0.632032i \(0.217779\pi\)
\(338\) 0 0
\(339\) −81.9833 21.7636i −0.241839 0.0641995i
\(340\) 0 0
\(341\) 277.248i 0.813045i
\(342\) 0 0
\(343\) 357.187 1.04136
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −382.308 220.726i −1.10175 0.636097i −0.165071 0.986282i \(-0.552785\pi\)
−0.936681 + 0.350185i \(0.886119\pi\)
\(348\) 0 0
\(349\) −94.7382 164.091i −0.271456 0.470176i 0.697779 0.716313i \(-0.254172\pi\)
−0.969235 + 0.246138i \(0.920839\pi\)
\(350\) 0 0
\(351\) −67.1751 243.898i −0.191382 0.694867i
\(352\) 0 0
\(353\) 435.490 251.430i 1.23368 0.712267i 0.265887 0.964004i \(-0.414335\pi\)
0.967796 + 0.251737i \(0.0810018\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −162.076 + 162.829i −0.453993 + 0.456104i
\(358\) 0 0
\(359\) 2.93084i 0.00816391i −0.999992 0.00408196i \(-0.998701\pi\)
0.999992 0.00408196i \(-0.00129933\pi\)
\(360\) 0 0
\(361\) 1.92045 0.00531981
\(362\) 0 0
\(363\) −192.685 + 725.842i −0.530813 + 1.99956i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 27.3696 + 47.4055i 0.0745765 + 0.129170i 0.900902 0.434023i \(-0.142906\pi\)
−0.826326 + 0.563193i \(0.809573\pi\)
\(368\) 0 0
\(369\) 277.542 1.28771i 0.752145 0.00348973i
\(370\) 0 0
\(371\) 125.320 72.3534i 0.337789 0.195023i
\(372\) 0 0
\(373\) 338.735 586.707i 0.908138 1.57294i 0.0914894 0.995806i \(-0.470837\pi\)
0.816649 0.577135i \(-0.195829\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 41.9329i 0.111228i
\(378\) 0 0
\(379\) −431.714 −1.13909 −0.569543 0.821961i \(-0.692880\pi\)
−0.569543 + 0.821961i \(0.692880\pi\)
\(380\) 0 0
\(381\) −129.957 + 35.1453i −0.341095 + 0.0922449i
\(382\) 0 0
\(383\) −506.226 292.269i −1.32174 0.763106i −0.337732 0.941242i \(-0.609660\pi\)
−0.984006 + 0.178137i \(0.942993\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −336.886 192.423i −0.870507 0.497217i
\(388\) 0 0
\(389\) −49.9565 + 28.8424i −0.128423 + 0.0741450i −0.562835 0.826569i \(-0.690289\pi\)
0.434412 + 0.900714i \(0.356956\pi\)
\(390\) 0 0
\(391\) −134.919 + 233.686i −0.345061 + 0.597662i
\(392\) 0 0
\(393\) −213.385 56.6462i −0.542965 0.144138i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −25.9853 −0.0654542 −0.0327271 0.999464i \(-0.510419\pi\)
−0.0327271 + 0.999464i \(0.510419\pi\)
\(398\) 0 0
\(399\) 190.904 + 190.020i 0.478456 + 0.476242i
\(400\) 0 0
\(401\) −229.069 132.253i −0.571244 0.329808i 0.186402 0.982474i \(-0.440317\pi\)
−0.757646 + 0.652666i \(0.773651\pi\)
\(402\) 0 0
\(403\) −67.4036 116.746i −0.167255 0.289693i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 928.017 535.791i 2.28014 1.31644i
\(408\) 0 0
\(409\) −154.678 + 267.911i −0.378187 + 0.655039i −0.990798 0.135345i \(-0.956786\pi\)
0.612612 + 0.790384i \(0.290119\pi\)
\(410\) 0 0
\(411\) −461.760 + 463.907i −1.12350 + 1.12873i
\(412\) 0 0
\(413\) 405.021i 0.980681i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −50.8030 + 191.374i −0.121830 + 0.458931i
\(418\) 0 0
\(419\) −58.4329 33.7363i −0.139458 0.0805161i 0.428648 0.903472i \(-0.358990\pi\)
−0.568106 + 0.822956i \(0.692324\pi\)
\(420\) 0 0
\(421\) −101.577 175.936i −0.241275 0.417901i 0.719803 0.694179i \(-0.244232\pi\)
−0.961078 + 0.276278i \(0.910899\pi\)
\(422\) 0 0
\(423\) 185.199 324.240i 0.437824 0.766524i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 243.880 422.413i 0.571148 0.989257i
\(428\) 0 0
\(429\) −141.404 522.871i −0.329613 1.21881i
\(430\) 0 0
\(431\) 319.947i 0.742337i 0.928566 + 0.371168i \(0.121043\pi\)
−0.928566 + 0.371168i \(0.878957\pi\)
\(432\) 0 0
\(433\) 454.101 1.04873 0.524366 0.851493i \(-0.324303\pi\)
0.524366 + 0.851493i \(0.324303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 273.978 + 158.181i 0.626951 + 0.361971i
\(438\) 0 0
\(439\) −45.7154 79.1813i −0.104135 0.180367i 0.809249 0.587465i \(-0.199874\pi\)
−0.913385 + 0.407098i \(0.866541\pi\)
\(440\) 0 0
\(441\) −1.11856 241.086i −0.00253643 0.546679i
\(442\) 0 0
\(443\) 188.030 108.559i 0.424448 0.245055i −0.272531 0.962147i \(-0.587861\pi\)
0.696978 + 0.717092i \(0.254527\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 567.176 + 150.565i 1.26885 + 0.336835i
\(448\) 0 0
\(449\) 832.270i 1.85361i 0.375545 + 0.926804i \(0.377456\pi\)
−0.375545 + 0.926804i \(0.622544\pi\)
\(450\) 0 0
\(451\) 594.249 1.31763
\(452\) 0 0
\(453\) −338.294 336.728i −0.746785 0.743329i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 400.398 + 693.510i 0.876145 + 1.51753i 0.855538 + 0.517739i \(0.173226\pi\)
0.0206062 + 0.999788i \(0.493440\pi\)
\(458\) 0 0
\(459\) 312.373 + 308.055i 0.680551 + 0.671144i
\(460\) 0 0
\(461\) 504.549 291.301i 1.09447 0.631890i 0.159704 0.987165i \(-0.448946\pi\)
0.934762 + 0.355275i \(0.115613\pi\)
\(462\) 0 0
\(463\) −367.120 + 635.871i −0.792916 + 1.37337i 0.131237 + 0.991351i \(0.458105\pi\)
−0.924154 + 0.382021i \(0.875228\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 670.573i 1.43592i −0.696086 0.717958i \(-0.745077\pi\)
0.696086 0.717958i \(-0.254923\pi\)
\(468\) 0 0
\(469\) 582.413 1.24182
\(470\) 0 0
\(471\) 98.8225 372.263i 0.209814 0.790367i
\(472\) 0 0
\(473\) −719.386 415.338i −1.52090 0.878092i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −139.276 238.668i −0.291982 0.500353i
\(478\) 0 0
\(479\) 148.723 85.8655i 0.310487 0.179260i −0.336657 0.941627i \(-0.609296\pi\)
0.647144 + 0.762367i \(0.275963\pi\)
\(480\) 0 0
\(481\) −260.519 + 451.232i −0.541619 + 0.938112i
\(482\) 0 0
\(483\) 61.2967 + 226.658i 0.126908 + 0.469271i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −22.0906 −0.0453606 −0.0226803 0.999743i \(-0.507220\pi\)
−0.0226803 + 0.999743i \(0.507220\pi\)
\(488\) 0 0
\(489\) 62.8319 16.9921i 0.128491 0.0347486i
\(490\) 0 0
\(491\) −458.788 264.881i −0.934395 0.539473i −0.0461960 0.998932i \(-0.514710\pi\)
−0.888199 + 0.459459i \(0.848043\pi\)
\(492\) 0 0
\(493\) 36.3601 + 62.9775i 0.0737527 + 0.127743i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −454.724 + 262.535i −0.914937 + 0.528239i
\(498\) 0 0
\(499\) −84.6486 + 146.616i −0.169636 + 0.293819i −0.938292 0.345844i \(-0.887593\pi\)
0.768656 + 0.639663i \(0.220926\pi\)
\(500\) 0 0
\(501\) −736.243 195.446i −1.46955 0.390112i
\(502\) 0 0
\(503\) 22.5687i 0.0448682i 0.999748 + 0.0224341i \(0.00714159\pi\)
−0.999748 + 0.0224341i \(0.992858\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −172.672 171.872i −0.340575 0.338999i
\(508\) 0 0
\(509\) −654.896 378.104i −1.28663 0.742838i −0.308580 0.951198i \(-0.599854\pi\)
−0.978052 + 0.208361i \(0.933187\pi\)
\(510\) 0 0
\(511\) 213.448 + 369.702i 0.417706 + 0.723487i
\(512\) 0 0
\(513\) 361.169 366.232i 0.704034 0.713902i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 399.746 692.380i 0.773203 1.33923i
\(518\) 0 0
\(519\) −125.616 + 126.200i −0.242034 + 0.243160i
\(520\) 0 0
\(521\) 157.921i 0.303112i 0.988449 + 0.151556i \(0.0484283\pi\)
−0.988449 + 0.151556i \(0.951572\pi\)
\(522\) 0 0
\(523\) −535.341 −1.02360 −0.511798 0.859106i \(-0.671021\pi\)
−0.511798 + 0.859106i \(0.671021\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 202.462 + 116.892i 0.384179 + 0.221806i
\(528\) 0 0
\(529\) −126.612 219.298i −0.239341 0.414551i
\(530\) 0 0
\(531\) 773.424 3.58845i 1.45654 0.00675791i
\(532\) 0 0
\(533\) −250.232 + 144.472i −0.469479 + 0.271054i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4.25960 15.7508i −0.00793222 0.0293311i
\(538\) 0 0
\(539\) 516.193i 0.957686i
\(540\) 0 0
\(541\) 446.519 0.825358 0.412679 0.910877i \(-0.364593\pi\)
0.412679 + 0.910877i \(0.364593\pi\)
\(542\) 0 0
\(543\) −131.730 + 35.6247i −0.242597 + 0.0656072i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −104.414 180.850i −0.190885 0.330622i 0.754659 0.656117i \(-0.227802\pi\)
−0.945544 + 0.325495i \(0.894469\pi\)
\(548\) 0 0
\(549\) −808.795 461.968i −1.47322 0.841472i
\(550\) 0 0
\(551\) 73.8360 42.6292i 0.134004 0.0773671i
\(552\) 0 0
\(553\) −196.329 + 340.052i −0.355026 + 0.614923i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 428.915i 0.770044i 0.922907 + 0.385022i \(0.125806\pi\)
−0.922907 + 0.385022i \(0.874194\pi\)
\(558\) 0 0
\(559\) 403.901 0.722543
\(560\) 0 0
\(561\) 665.753 + 662.671i 1.18672 + 1.18123i
\(562\) 0 0
\(563\) 683.687 + 394.727i 1.21436 + 0.701114i 0.963707 0.266962i \(-0.0860199\pi\)
0.250657 + 0.968076i \(0.419353\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 381.737 3.54236i 0.673258 0.00624756i
\(568\) 0 0
\(569\) 153.834 88.8163i 0.270359 0.156092i −0.358692 0.933456i \(-0.616777\pi\)
0.629051 + 0.777364i \(0.283444\pi\)
\(570\) 0 0
\(571\) −9.18137 + 15.9026i −0.0160795 + 0.0278504i −0.873953 0.486010i \(-0.838452\pi\)
0.857874 + 0.513861i \(0.171785\pi\)
\(572\) 0 0
\(573\) −488.729 + 491.001i −0.852929 + 0.856896i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −240.588 −0.416963 −0.208482 0.978026i \(-0.566852\pi\)
−0.208482 + 0.978026i \(0.566852\pi\)
\(578\) 0 0
\(579\) −56.1045 + 211.345i −0.0968989 + 0.365017i
\(580\) 0 0
\(581\) −146.325 84.4809i −0.251851 0.145406i
\(582\) 0 0
\(583\) −295.828 512.389i −0.507424 0.878884i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −143.112 + 82.6260i −0.243803 + 0.140760i −0.616923 0.787023i \(-0.711621\pi\)
0.373120 + 0.927783i \(0.378288\pi\)
\(588\) 0 0
\(589\) 137.046 237.370i 0.232676 0.403006i
\(590\) 0 0
\(591\) −30.7202 113.595i −0.0519801 0.192208i
\(592\) 0 0
\(593\) 901.661i 1.52051i 0.649626 + 0.760254i \(0.274925\pi\)
−0.649626 + 0.760254i \(0.725075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 776.316 209.945i 1.30036 0.351666i
\(598\) 0 0
\(599\) 893.571 + 515.904i 1.49177 + 0.861275i 0.999956 0.00942385i \(-0.00299975\pi\)
0.491817 + 0.870699i \(0.336333\pi\)
\(600\) 0 0
\(601\) −481.489 833.964i −0.801147 1.38763i −0.918862 0.394580i \(-0.870890\pi\)
0.117715 0.993047i \(-0.462443\pi\)
\(602\) 0 0
\(603\) −5.16013 1112.17i −0.00855743 1.84439i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −469.609 + 813.387i −0.773656 + 1.34001i 0.161891 + 0.986809i \(0.448241\pi\)
−0.935547 + 0.353202i \(0.885093\pi\)
\(608\) 0 0
\(609\) 61.1595 + 16.2357i 0.100426 + 0.0266595i
\(610\) 0 0
\(611\) 388.739i 0.636234i
\(612\) 0 0
\(613\) −364.929 −0.595316 −0.297658 0.954673i \(-0.596206\pi\)
−0.297658 + 0.954673i \(0.596206\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 231.260 + 133.518i 0.374813 + 0.216398i 0.675559 0.737306i \(-0.263902\pi\)
−0.300746 + 0.953704i \(0.597236\pi\)
\(618\) 0 0
\(619\) −453.122 784.830i −0.732022 1.26790i −0.956017 0.293310i \(-0.905243\pi\)
0.223995 0.974590i \(-0.428090\pi\)
\(620\) 0 0
\(621\) 432.280 119.060i 0.696103 0.191722i
\(622\) 0 0
\(623\) 346.585 200.101i 0.556317 0.321190i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 776.928 780.541i 1.23912 1.24488i
\(628\) 0 0
\(629\) 903.586i 1.43654i
\(630\) 0 0
\(631\) 597.193 0.946424 0.473212 0.880949i \(-0.343095\pi\)
0.473212 + 0.880949i \(0.343095\pi\)
\(632\) 0 0
\(633\) −267.013 + 1005.83i −0.421821 + 1.58899i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 125.495 + 217.363i 0.197009 + 0.341230i
\(638\) 0 0
\(639\) 505.362 + 866.009i 0.790864 + 1.35526i
\(640\) 0 0
\(641\) 78.1431 45.1160i 0.121908 0.0703837i −0.437806 0.899070i \(-0.644244\pi\)
0.559714 + 0.828686i \(0.310911\pi\)
\(642\) 0 0
\(643\) −194.434 + 336.770i −0.302386 + 0.523748i −0.976676 0.214719i \(-0.931117\pi\)
0.674290 + 0.738467i \(0.264450\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 607.624i 0.939141i 0.882895 + 0.469570i \(0.155591\pi\)
−0.882895 + 0.469570i \(0.844409\pi\)
\(648\) 0 0
\(649\) 1655.99 2.55161
\(650\) 0 0
\(651\) 196.373 53.1066i 0.301649 0.0815770i
\(652\) 0 0
\(653\) −981.221 566.508i −1.50264 0.867547i −0.999995 0.00305096i \(-0.999029\pi\)
−0.502640 0.864496i \(-0.667638\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 704.088 410.873i 1.07167 0.625377i
\(658\) 0 0
\(659\) 772.309 445.893i 1.17194 0.676621i 0.217805 0.975992i \(-0.430110\pi\)
0.954137 + 0.299372i \(0.0967771\pi\)
\(660\) 0 0
\(661\) −219.766 + 380.645i −0.332474 + 0.575863i −0.982996 0.183625i \(-0.941217\pi\)
0.650522 + 0.759487i \(0.274550\pi\)
\(662\) 0 0
\(663\) −441.448 117.189i −0.665834 0.176755i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 74.3209 0.111426
\(668\) 0 0
\(669\) 150.469 + 149.772i 0.224916 + 0.223875i
\(670\) 0 0
\(671\) −1727.10 997.142i −2.57392 1.48605i
\(672\) 0 0
\(673\) 31.1287 + 53.9165i 0.0462537 + 0.0801137i 0.888225 0.459408i \(-0.151938\pi\)
−0.841972 + 0.539522i \(0.818605\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 116.119 67.0416i 0.171521 0.0990275i −0.411782 0.911282i \(-0.635093\pi\)
0.583303 + 0.812255i \(0.301760\pi\)
\(678\) 0 0
\(679\) 248.026 429.594i 0.365282 0.632687i
\(680\) 0 0
\(681\) 569.010 571.656i 0.835550 0.839436i
\(682\) 0 0
\(683\) 299.004i 0.437780i 0.975750 + 0.218890i \(0.0702436\pi\)
−0.975750 + 0.218890i \(0.929756\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 159.329 600.192i 0.231921 0.873641i
\(688\) 0 0
\(689\) 249.140 + 143.841i 0.361597 + 0.208768i
\(690\) 0 0
\(691\) 188.288 + 326.124i 0.272486 + 0.471960i 0.969498 0.245100i \(-0.0788208\pi\)
−0.697012 + 0.717060i \(0.745487\pi\)
\(692\) 0 0
\(693\) 817.362 3.79231i 1.17945 0.00547231i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 250.544 433.954i 0.359460 0.622603i
\(698\) 0 0
\(699\) 37.1925 + 137.527i 0.0532082 + 0.196749i
\(700\) 0 0
\(701\) 909.992i 1.29813i −0.760731 0.649067i \(-0.775159\pi\)
0.760731 0.649067i \(-0.224841\pi\)
\(702\) 0 0
\(703\) −1059.38 −1.50694
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 327.385 + 189.016i 0.463062 + 0.267349i
\(708\) 0 0
\(709\) 671.694 + 1163.41i 0.947382 + 1.64091i 0.750911 + 0.660404i \(0.229615\pi\)
0.196471 + 0.980510i \(0.437052\pi\)
\(710\) 0 0
\(711\) 651.099 + 371.895i 0.915751 + 0.523059i
\(712\) 0 0
\(713\) 206.919 119.465i 0.290209 0.167552i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 967.023 + 256.710i 1.34871 + 0.358034i
\(718\) 0 0
\(719\) 288.276i 0.400940i 0.979700 + 0.200470i \(0.0642469\pi\)
−0.979700 + 0.200470i \(0.935753\pi\)
\(720\) 0 0
\(721\) −614.673 −0.852528
\(722\) 0 0
\(723\) 907.292 + 903.092i 1.25490 + 1.24909i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 243.149 + 421.147i 0.334456 + 0.579294i 0.983380 0.181559i \(-0.0581143\pi\)
−0.648925 + 0.760853i \(0.724781\pi\)
\(728\) 0 0
\(729\) −10.1466 728.929i −0.0139185 0.999903i
\(730\) 0 0
\(731\) −606.606 + 350.224i −0.829830 + 0.479103i
\(732\) 0 0
\(733\) 629.211 1089.83i 0.858406 1.48680i −0.0150435 0.999887i \(-0.504789\pi\)
0.873449 0.486915i \(-0.161878\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2381.29i 3.23105i
\(738\) 0 0
\(739\) −1192.17 −1.61322 −0.806609 0.591086i \(-0.798700\pi\)
−0.806609 + 0.591086i \(0.798700\pi\)
\(740\) 0 0
\(741\) −137.394 + 517.562i −0.185417 + 0.698464i
\(742\) 0 0
\(743\) 170.987 + 98.7194i 0.230131 + 0.132866i 0.610632 0.791914i \(-0.290915\pi\)
−0.380502 + 0.924780i \(0.624249\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −160.027 + 280.170i −0.214227 + 0.375060i
\(748\) 0 0
\(749\) −203.422 + 117.446i −0.271592 + 0.156803i
\(750\) 0 0
\(751\) −25.0633 + 43.4109i −0.0333733 + 0.0578042i −0.882230 0.470819i \(-0.843958\pi\)
0.848856 + 0.528624i \(0.177292\pi\)
\(752\) 0 0
\(753\) −308.838 1142.00i −0.410143 1.51659i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 282.361 0.373000 0.186500 0.982455i \(-0.440286\pi\)
0.186500 + 0.982455i \(0.440286\pi\)
\(758\) 0 0
\(759\) 926.726 250.621i 1.22098 0.330199i
\(760\) 0 0
\(761\) 89.9243 + 51.9178i 0.118166 + 0.0682231i 0.557918 0.829896i \(-0.311600\pi\)
−0.439752 + 0.898119i \(0.644934\pi\)
\(762\) 0 0
\(763\) 286.152 + 495.630i 0.375036 + 0.649581i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −697.321 + 402.599i −0.909154 + 0.524900i
\(768\) 0 0
\(769\) 190.567 330.073i 0.247812 0.429223i −0.715106 0.699016i \(-0.753622\pi\)
0.962918 + 0.269793i \(0.0869551\pi\)
\(770\) 0 0
\(771\) −1049.98 278.731i −1.36184 0.361519i
\(772\) 0 0
\(773\) 276.633i 0.357870i 0.983861 + 0.178935i \(0.0572651\pi\)
−0.983861 + 0.178935i \(0.942735\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −557.258 554.678i −0.717191 0.713872i
\(778\) 0 0
\(779\) −508.776 293.742i −0.653114 0.377076i
\(780\) 0 0
\(781\) 1073.41 + 1859.21i 1.37441 + 2.38055i
\(782\) 0 0
\(783\) 30.4616 116.933i 0.0389037 0.149340i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −666.067 + 1153.66i −0.846336 + 1.46590i 0.0381190 + 0.999273i \(0.487863\pi\)
−0.884455 + 0.466625i \(0.845470\pi\)
\(788\) 0 0
\(789\) −718.636 + 721.978i −0.910819 + 0.915055i
\(790\) 0 0
\(791\) 133.257i 0.168466i
\(792\) 0 0
\(793\) 969.686 1.22281
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −571.372 329.882i −0.716903 0.413904i 0.0967084 0.995313i \(-0.469169\pi\)
−0.813612 + 0.581408i \(0.802502\pi\)
\(798\) 0 0
\(799\) −337.077 583.834i −0.421873 0.730706i
\(800\) 0 0
\(801\) −385.182 660.063i −0.480876 0.824048i
\(802\) 0 0
\(803\) 1511.58 872.713i 1.88242 1.08682i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −18.1744 67.2039i −0.0225210 0.0832762i
\(808\) 0 0
\(809\) 654.023i 0.808433i 0.914663 + 0.404217i \(0.132456\pi\)
−0.914663 + 0.404217i \(0.867544\pi\)
\(810\) 0 0
\(811\) −408.003 −0.503086 −0.251543 0.967846i \(-0.580938\pi\)
−0.251543 + 0.967846i \(0.580938\pi\)
\(812\) 0 0
\(813\) 1256.67 339.851i 1.54572 0.418021i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 410.609 + 711.196i 0.502582 + 0.870497i
\(818\) 0 0
\(819\) −343.261 + 200.311i −0.419122 + 0.244580i
\(820\) 0 0
\(821\) 762.425 440.186i 0.928654 0.536158i 0.0422680 0.999106i \(-0.486542\pi\)
0.886385 + 0.462948i \(0.153208\pi\)
\(822\) 0 0
\(823\) 64.8244 112.279i 0.0787660 0.136427i −0.823952 0.566660i \(-0.808235\pi\)
0.902718 + 0.430233i \(0.141569\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 646.300i 0.781499i −0.920497 0.390750i \(-0.872216\pi\)
0.920497 0.390750i \(-0.127784\pi\)
\(828\) 0 0
\(829\) −416.080 −0.501906 −0.250953 0.967999i \(-0.580744\pi\)
−0.250953 + 0.967999i \(0.580744\pi\)
\(830\) 0 0
\(831\) 726.003 + 722.642i 0.873650 + 0.869606i
\(832\) 0 0
\(833\) −376.953 217.634i −0.452524 0.261265i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −103.152 374.522i −0.123240 0.447457i
\(838\) 0 0
\(839\) 240.544 138.878i 0.286703 0.165528i −0.349751 0.936843i \(-0.613734\pi\)
0.636454 + 0.771315i \(0.280400\pi\)
\(840\) 0 0
\(841\) −410.485 + 710.982i −0.488092 + 0.845400i
\(842\) 0 0
\(843\) −429.701 + 431.700i −0.509729 + 0.512099i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1179.79 1.39291
\(848\) 0 0
\(849\) −120.714 + 454.726i −0.142183 + 0.535602i
\(850\) 0 0
\(851\) −799.754 461.738i −0.939781 0.542583i
\(852\) 0 0
\(853\) 658.042 + 1139.76i 0.771444 + 1.33618i 0.936771 + 0.349942i \(0.113799\pi\)
−0.165327 + 0.986239i \(0.552868\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −830.857 + 479.696i −0.969495 + 0.559738i −0.899082 0.437780i \(-0.855765\pi\)
−0.0704129 + 0.997518i \(0.522432\pi\)
\(858\) 0 0
\(859\) 439.753 761.675i 0.511936 0.886699i −0.487968 0.872861i \(-0.662262\pi\)
0.999904 0.0138379i \(-0.00440487\pi\)
\(860\) 0 0
\(861\) −113.828 420.903i −0.132204 0.488854i
\(862\) 0 0
\(863\) 326.481i 0.378310i −0.981947 0.189155i \(-0.939425\pi\)
0.981947 0.189155i \(-0.0605748\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −72.3251 + 19.5594i −0.0834200 + 0.0225599i
\(868\) 0 0
\(869\) 1390.36 + 802.722i 1.59995 + 0.923731i
\(870\) 0 0
\(871\) 578.930 + 1002.74i 0.664673 + 1.15125i
\(872\) 0 0
\(873\) −822.546 469.822i −0.942206 0.538170i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −57.1834 + 99.0445i −0.0652034 + 0.112936i −0.896784 0.442468i \(-0.854103\pi\)
0.831581 + 0.555404i \(0.187436\pi\)
\(878\) 0 0
\(879\) 183.951 + 48.8323i 0.209273 + 0.0555544i
\(880\) 0 0
\(881\) 429.252i 0.487233i 0.969872 + 0.243617i \(0.0783338\pi\)
−0.969872 + 0.243617i \(0.921666\pi\)
\(882\) 0 0
\(883\) −294.993 −0.334080 −0.167040 0.985950i \(-0.553421\pi\)
−0.167040 + 0.985950i \(0.553421\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −347.848 200.830i −0.392162 0.226415i 0.290934 0.956743i \(-0.406034\pi\)
−0.683097 + 0.730328i \(0.739367\pi\)
\(888\) 0 0
\(889\) 105.749 + 183.162i 0.118952 + 0.206032i
\(890\) 0 0
\(891\) −14.4835 1560.79i −0.0162553 1.75173i
\(892\) 0 0
\(893\) −684.498 + 395.195i −0.766515 + 0.442548i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −329.305 + 330.836i −0.367118 + 0.368825i
\(898\) 0 0
\(899\) 64.3906i 0.0716247i
\(900\) 0 0
\(901\) −498.901 −0.553719
\(902\) 0 0
\(903\) −156.383 + 589.094i −0.173182 + 0.652374i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 211.939 + 367.089i 0.233670 + 0.404729i 0.958885 0.283794i \(-0.0915931\pi\)
−0.725215 + 0.688522i \(0.758260\pi\)
\(908\) 0 0
\(909\) 358.041 626.844i 0.393885 0.689598i
\(910\) 0 0
\(911\) −51.7248 + 29.8633i −0.0567780 + 0.0327808i −0.528120 0.849170i \(-0.677103\pi\)
0.471342 + 0.881950i \(0.343770\pi\)
\(912\) 0 0
\(913\) −345.413 + 598.273i −0.378328 + 0.655283i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 346.840i 0.378233i
\(918\) 0 0
\(919\) −522.062 −0.568077 −0.284038 0.958813i \(-0.591674\pi\)
−0.284038 + 0.958813i \(0.591674\pi\)
\(920\) 0 0
\(921\) 532.164 143.917i 0.577811 0.156262i
\(922\) 0 0
\(923\) −904.008 521.929i −0.979423 0.565470i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 5.44595 + 1173.77i 0.00587481 + 1.26621i
\(928\) 0 0
\(929\) −1309.83 + 756.229i −1.40993 + 0.814025i −0.995381 0.0960006i \(-0.969395\pi\)
−0.414552 + 0.910026i \(0.636062\pi\)
\(930\) 0 0
\(931\) −255.158 + 441.947i −0.274069 + 0.474701i
\(932\) 0 0
\(933\) −1100.19 292.062i −1.17920 0.313036i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −272.803 −0.291145 −0.145573 0.989348i \(-0.546502\pi\)
−0.145573 + 0.989348i \(0.546502\pi\)
\(938\) 0 0
\(939\) −868.805 864.783i −0.925245 0.920962i
\(940\) 0 0
\(941\) −1149.68 663.770i −1.22177 0.705388i −0.256473 0.966551i \(-0.582560\pi\)
−0.965295 + 0.261164i \(0.915894\pi\)
\(942\) 0 0
\(943\) −256.059 443.506i −0.271536 0.470314i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.7310 + 11.3917i −0.0208352 + 0.0120292i −0.510381 0.859948i \(-0.670496\pi\)
0.489546 + 0.871977i \(0.337162\pi\)
\(948\) 0 0
\(949\) −424.342 + 734.981i −0.447146 + 0.774480i
\(950\) 0 0
\(951\) 1098.07 1103.18i 1.15465 1.16002i
\(952\) 0 0
\(953\) 563.843i 0.591650i 0.955242 + 0.295825i \(0.0955946\pi\)
−0.955242 + 0.295825i \(0.904405\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 66.3820 250.060i 0.0693647 0.261296i
\(958\) 0 0
\(959\) 890.530 + 514.148i 0.928603 + 0.536129i
\(960\) 0 0
\(961\) 376.998 + 652.979i 0.392297 + 0.679479i
\(962\) 0 0
\(963\) 226.075 + 387.412i 0.234762 + 0.402297i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −637.165 + 1103.60i −0.658909 + 1.14126i 0.321990 + 0.946743i \(0.395648\pi\)
−0.980899 + 0.194520i \(0.937685\pi\)
\(968\) 0 0
\(969\) −242.431 896.443i −0.250187 0.925121i
\(970\) 0 0
\(971\) 119.681i 0.123255i 0.998099 + 0.0616276i \(0.0196291\pi\)
−0.998099 + 0.0616276i \(0.980371\pi\)
\(972\) 0 0
\(973\) 311.062 0.319694
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1307.00 754.597i −1.33777 0.772361i −0.351293 0.936266i \(-0.614258\pi\)
−0.986476 + 0.163905i \(0.947591\pi\)
\(978\) 0 0
\(979\) −818.144 1417.07i −0.835694 1.44746i
\(980\) 0 0
\(981\) 943.914 550.824i 0.962196 0.561492i
\(982\) 0 0
\(983\) −448.144 + 258.736i −0.455894 + 0.263211i −0.710316 0.703883i \(-0.751448\pi\)
0.254422 + 0.967093i \(0.418115\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −566.980 150.513i −0.574448 0.152495i
\(988\) 0 0
\(989\) 715.866i 0.723828i
\(990\) 0 0
\(991\) −201.958 −0.203793 −0.101896 0.994795i \(-0.532491\pi\)
−0.101896 + 0.994795i \(0.532491\pi\)
\(992\) 0 0
\(993\) 627.758 + 624.852i 0.632184 + 0.629257i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −104.672 181.297i −0.104987 0.181842i 0.808746 0.588158i \(-0.200147\pi\)
−0.913733 + 0.406316i \(0.866813\pi\)
\(998\) 0 0
\(999\) −1054.27 + 1069.05i −1.05533 + 1.07012i
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 900.3.p.c.401.1 12
3.2 odd 2 2700.3.p.c.2501.5 12
5.2 odd 4 900.3.u.c.149.8 24
5.3 odd 4 900.3.u.c.149.5 24
5.4 even 2 180.3.o.b.41.6 12
9.2 odd 6 inner 900.3.p.c.101.1 12
9.7 even 3 2700.3.p.c.1601.5 12
15.2 even 4 2700.3.u.c.449.3 24
15.8 even 4 2700.3.u.c.449.10 24
15.14 odd 2 540.3.o.b.341.4 12
20.19 odd 2 720.3.bs.b.401.1 12
45.2 even 12 900.3.u.c.749.5 24
45.4 even 6 1620.3.g.b.161.11 12
45.7 odd 12 2700.3.u.c.2249.10 24
45.14 odd 6 1620.3.g.b.161.5 12
45.29 odd 6 180.3.o.b.101.6 yes 12
45.34 even 6 540.3.o.b.521.4 12
45.38 even 12 900.3.u.c.749.8 24
45.43 odd 12 2700.3.u.c.2249.3 24
60.59 even 2 2160.3.bs.b.881.6 12
180.79 odd 6 2160.3.bs.b.1601.6 12
180.119 even 6 720.3.bs.b.641.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.o.b.41.6 12 5.4 even 2
180.3.o.b.101.6 yes 12 45.29 odd 6
540.3.o.b.341.4 12 15.14 odd 2
540.3.o.b.521.4 12 45.34 even 6
720.3.bs.b.401.1 12 20.19 odd 2
720.3.bs.b.641.1 12 180.119 even 6
900.3.p.c.101.1 12 9.2 odd 6 inner
900.3.p.c.401.1 12 1.1 even 1 trivial
900.3.u.c.149.5 24 5.3 odd 4
900.3.u.c.149.8 24 5.2 odd 4
900.3.u.c.749.5 24 45.2 even 12
900.3.u.c.749.8 24 45.38 even 12
1620.3.g.b.161.5 12 45.14 odd 6
1620.3.g.b.161.11 12 45.4 even 6
2160.3.bs.b.881.6 12 60.59 even 2
2160.3.bs.b.1601.6 12 180.79 odd 6
2700.3.p.c.1601.5 12 9.7 even 3
2700.3.p.c.2501.5 12 3.2 odd 2
2700.3.u.c.449.3 24 15.2 even 4
2700.3.u.c.449.10 24 15.8 even 4
2700.3.u.c.2249.3 24 45.43 odd 12
2700.3.u.c.2249.10 24 45.7 odd 12