Properties

Label 2940.2.bb.c.949.1
Level $2940$
Weight $2$
Character 2940.949
Analytic conductor $23.476$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2940,2,Mod(949,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, 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("2940.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.bb (of order \(6\), 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: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2940.949
Dual form 2940.2.bb.c.1549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{3} +(-1.86603 - 1.23205i) q^{5} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{3} +(-1.86603 - 1.23205i) q^{5} +(0.500000 + 0.866025i) q^{9} +(2.00000 - 3.46410i) q^{11} +6.00000i q^{13} +(1.00000 + 2.00000i) q^{15} +(-1.73205 - 1.00000i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(1.73205 - 1.00000i) q^{23} +(1.96410 + 4.59808i) q^{25} -1.00000i q^{27} -6.00000 q^{29} +(-1.00000 + 1.73205i) q^{31} +(-3.46410 + 2.00000i) q^{33} +(3.46410 - 2.00000i) q^{37} +(3.00000 - 5.19615i) q^{39} -8.00000 q^{41} -4.00000i q^{43} +(0.133975 - 2.23205i) q^{45} +(3.46410 - 2.00000i) q^{47} +(1.00000 + 1.73205i) q^{51} +(-5.19615 - 3.00000i) q^{53} +(-8.00000 + 4.00000i) q^{55} +6.00000i q^{57} +(-2.00000 + 3.46410i) q^{59} +(7.00000 + 12.1244i) q^{61} +(7.39230 - 11.1962i) q^{65} +(3.46410 + 2.00000i) q^{67} -2.00000 q^{69} +(-8.66025 - 5.00000i) q^{73} +(0.598076 - 4.96410i) q^{75} +(-0.500000 + 0.866025i) q^{81} +16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +(5.19615 + 3.00000i) q^{87} +(-4.00000 - 6.92820i) q^{89} +(1.73205 - 1.00000i) q^{93} +(-0.803848 + 13.3923i) q^{95} +10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 2 q^{9} + 8 q^{11} + 4 q^{15} - 12 q^{19} - 6 q^{25} - 24 q^{29} - 4 q^{31} + 12 q^{39} - 32 q^{41} + 4 q^{45} + 4 q^{51} - 32 q^{55} - 8 q^{59} + 28 q^{61} - 12 q^{65} - 8 q^{69} - 8 q^{75} - 2 q^{81} + 8 q^{85} - 16 q^{89} - 24 q^{95} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.866025 0.500000i −0.500000 0.288675i
\(4\) 0 0
\(5\) −1.86603 1.23205i −0.834512 0.550990i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.00000 3.46410i 0.603023 1.04447i −0.389338 0.921095i \(-0.627296\pi\)
0.992361 0.123371i \(-0.0393705\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 1.00000 + 2.00000i 0.258199 + 0.516398i
\(16\) 0 0
\(17\) −1.73205 1.00000i −0.420084 0.242536i 0.275029 0.961436i \(-0.411312\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) 0 0
\(19\) −3.00000 5.19615i −0.688247 1.19208i −0.972404 0.233301i \(-0.925047\pi\)
0.284157 0.958778i \(-0.408286\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.73205 1.00000i 0.361158 0.208514i −0.308431 0.951247i \(-0.599804\pi\)
0.669588 + 0.742732i \(0.266471\pi\)
\(24\) 0 0
\(25\) 1.96410 + 4.59808i 0.392820 + 0.919615i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −1.00000 + 1.73205i −0.179605 + 0.311086i −0.941745 0.336327i \(-0.890815\pi\)
0.762140 + 0.647412i \(0.224149\pi\)
\(32\) 0 0
\(33\) −3.46410 + 2.00000i −0.603023 + 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.46410 2.00000i 0.569495 0.328798i −0.187453 0.982274i \(-0.560023\pi\)
0.756948 + 0.653476i \(0.226690\pi\)
\(38\) 0 0
\(39\) 3.00000 5.19615i 0.480384 0.832050i
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0.133975 2.23205i 0.0199718 0.332734i
\(46\) 0 0
\(47\) 3.46410 2.00000i 0.505291 0.291730i −0.225605 0.974219i \(-0.572436\pi\)
0.730896 + 0.682489i \(0.239102\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.00000 + 1.73205i 0.140028 + 0.242536i
\(52\) 0 0
\(53\) −5.19615 3.00000i −0.713746 0.412082i 0.0987002 0.995117i \(-0.468532\pi\)
−0.812447 + 0.583036i \(0.801865\pi\)
\(54\) 0 0
\(55\) −8.00000 + 4.00000i −1.07872 + 0.539360i
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) 7.00000 + 12.1244i 0.896258 + 1.55236i 0.832240 + 0.554416i \(0.187058\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.39230 11.1962i 0.916903 1.38871i
\(66\) 0 0
\(67\) 3.46410 + 2.00000i 0.423207 + 0.244339i 0.696449 0.717607i \(-0.254762\pi\)
−0.273241 + 0.961946i \(0.588096\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −8.66025 5.00000i −1.01361 0.585206i −0.101361 0.994850i \(-0.532320\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 0 0
\(75\) 0.598076 4.96410i 0.0690599 0.573205i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 0 0
\(85\) 2.00000 + 4.00000i 0.216930 + 0.433861i
\(86\) 0 0
\(87\) 5.19615 + 3.00000i 0.557086 + 0.321634i
\(88\) 0 0
\(89\) −4.00000 6.92820i −0.423999 0.734388i 0.572327 0.820025i \(-0.306041\pi\)
−0.996326 + 0.0856373i \(0.972707\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.73205 1.00000i 0.179605 0.103695i
\(94\) 0 0
\(95\) −0.803848 + 13.3923i −0.0824730 + 1.37402i
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 8.00000 13.8564i 0.796030 1.37876i −0.126153 0.992011i \(-0.540263\pi\)
0.922183 0.386753i \(-0.126403\pi\)
\(102\) 0 0
\(103\) −13.8564 + 8.00000i −1.36531 + 0.788263i −0.990325 0.138767i \(-0.955686\pi\)
−0.374987 + 0.927030i \(0.622353\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.19615 3.00000i 0.502331 0.290021i −0.227345 0.973814i \(-0.573004\pi\)
0.729676 + 0.683793i \(0.239671\pi\)
\(108\) 0 0
\(109\) −7.00000 + 12.1244i −0.670478 + 1.16130i 0.307290 + 0.951616i \(0.400578\pi\)
−0.977769 + 0.209687i \(0.932756\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 0 0
\(115\) −4.46410 0.267949i −0.416280 0.0249864i
\(116\) 0 0
\(117\) −5.19615 + 3.00000i −0.480384 + 0.277350i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 6.92820 + 4.00000i 0.624695 + 0.360668i
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) −2.00000 + 3.46410i −0.176090 + 0.304997i
\(130\) 0 0
\(131\) −6.00000 10.3923i −0.524222 0.907980i −0.999602 0.0281993i \(-0.991023\pi\)
0.475380 0.879781i \(-0.342311\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.23205 + 1.86603i −0.106038 + 0.160602i
\(136\) 0 0
\(137\) 15.5885 + 9.00000i 1.33181 + 0.768922i 0.985577 0.169226i \(-0.0541268\pi\)
0.346235 + 0.938148i \(0.387460\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 20.7846 + 12.0000i 1.73810 + 1.00349i
\(144\) 0 0
\(145\) 11.1962 + 7.39230i 0.929790 + 0.613898i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.0000 + 19.0526i 0.901155 + 1.56085i 0.825997 + 0.563675i \(0.190613\pi\)
0.0751583 + 0.997172i \(0.476054\pi\)
\(150\) 0 0
\(151\) −8.00000 + 13.8564i −0.651031 + 1.12762i 0.331842 + 0.943335i \(0.392330\pi\)
−0.982873 + 0.184284i \(0.941004\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 4.00000 2.00000i 0.321288 0.160644i
\(156\) 0 0
\(157\) −8.66025 5.00000i −0.691164 0.399043i 0.112884 0.993608i \(-0.463991\pi\)
−0.804048 + 0.594565i \(0.797324\pi\)
\(158\) 0 0
\(159\) 3.00000 + 5.19615i 0.237915 + 0.412082i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.92820 + 4.00000i −0.542659 + 0.313304i −0.746156 0.665771i \(-0.768103\pi\)
0.203497 + 0.979076i \(0.434769\pi\)
\(164\) 0 0
\(165\) 8.92820 + 0.535898i 0.695060 + 0.0417196i
\(166\) 0 0
\(167\) 24.0000i 1.85718i 0.371113 + 0.928588i \(0.378976\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 3.00000 5.19615i 0.229416 0.397360i
\(172\) 0 0
\(173\) 1.73205 1.00000i 0.131685 0.0760286i −0.432710 0.901533i \(-0.642443\pi\)
0.564396 + 0.825505i \(0.309109\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.46410 2.00000i 0.260378 0.150329i
\(178\) 0 0
\(179\) 2.00000 3.46410i 0.149487 0.258919i −0.781551 0.623841i \(-0.785571\pi\)
0.931038 + 0.364922i \(0.118904\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 0 0
\(185\) −8.92820 0.535898i −0.656415 0.0394000i
\(186\) 0 0
\(187\) −6.92820 + 4.00000i −0.506640 + 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) −6.92820 4.00000i −0.498703 0.287926i 0.229475 0.973315i \(-0.426299\pi\)
−0.728178 + 0.685388i \(0.759632\pi\)
\(194\) 0 0
\(195\) −12.0000 + 6.00000i −0.859338 + 0.429669i
\(196\) 0 0
\(197\) 10.0000i 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) −9.00000 + 15.5885i −0.637993 + 1.10504i 0.347879 + 0.937539i \(0.386902\pi\)
−0.985873 + 0.167497i \(0.946431\pi\)
\(200\) 0 0
\(201\) −2.00000 3.46410i −0.141069 0.244339i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 14.9282 + 9.85641i 1.04263 + 0.688401i
\(206\) 0 0
\(207\) 1.73205 + 1.00000i 0.120386 + 0.0695048i
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) −24.0000 −1.65223 −0.826114 0.563503i \(-0.809453\pi\)
−0.826114 + 0.563503i \(0.809453\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.92820 + 7.46410i −0.336101 + 0.509048i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 5.00000 + 8.66025i 0.337869 + 0.585206i
\(220\) 0 0
\(221\) 6.00000 10.3923i 0.403604 0.699062i
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) −17.3205 10.0000i −1.14960 0.663723i −0.200812 0.979630i \(-0.564358\pi\)
−0.948790 + 0.315906i \(0.897691\pi\)
\(228\) 0 0
\(229\) −1.00000 1.73205i −0.0660819 0.114457i 0.831092 0.556136i \(-0.187717\pi\)
−0.897173 + 0.441679i \(0.854383\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.5167 + 13.0000i −1.47512 + 0.851658i −0.999606 0.0280525i \(-0.991069\pi\)
−0.475509 + 0.879711i \(0.657736\pi\)
\(234\) 0 0
\(235\) −8.92820 0.535898i −0.582412 0.0349582i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 5.00000 8.66025i 0.322078 0.557856i −0.658838 0.752285i \(-0.728952\pi\)
0.980917 + 0.194429i \(0.0622852\pi\)
\(242\) 0 0
\(243\) 0.866025 0.500000i 0.0555556 0.0320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 31.1769 18.0000i 1.98374 1.14531i
\(248\) 0 0
\(249\) 8.00000 13.8564i 0.506979 0.878114i
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) 0 0
\(255\) 0.267949 4.46410i 0.0167796 0.279553i
\(256\) 0 0
\(257\) 5.19615 3.00000i 0.324127 0.187135i −0.329104 0.944294i \(-0.606747\pi\)
0.653231 + 0.757159i \(0.273413\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 5.19615i −0.185695 0.321634i
\(262\) 0 0
\(263\) 8.66025 + 5.00000i 0.534014 + 0.308313i 0.742650 0.669680i \(-0.233569\pi\)
−0.208635 + 0.977993i \(0.566902\pi\)
\(264\) 0 0
\(265\) 6.00000 + 12.0000i 0.368577 + 0.737154i
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 0 0
\(269\) 6.00000 10.3923i 0.365826 0.633630i −0.623082 0.782157i \(-0.714120\pi\)
0.988908 + 0.148527i \(0.0474530\pi\)
\(270\) 0 0
\(271\) −7.00000 12.1244i −0.425220 0.736502i 0.571221 0.820796i \(-0.306470\pi\)
−0.996441 + 0.0842940i \(0.973137\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.8564 + 2.39230i 1.19739 + 0.144261i
\(276\) 0 0
\(277\) −24.2487 14.0000i −1.45696 0.841178i −0.458103 0.888899i \(-0.651471\pi\)
−0.998861 + 0.0477206i \(0.984804\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 24.2487 + 14.0000i 1.44144 + 0.832214i 0.997946 0.0640654i \(-0.0204066\pi\)
0.443491 + 0.896279i \(0.353740\pi\)
\(284\) 0 0
\(285\) 7.39230 11.1962i 0.437882 0.663203i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.50000 11.2583i −0.382353 0.662255i
\(290\) 0 0
\(291\) 5.00000 8.66025i 0.293105 0.507673i
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 8.00000 4.00000i 0.465778 0.232889i
\(296\) 0 0
\(297\) −3.46410 2.00000i −0.201008 0.116052i
\(298\) 0 0
\(299\) 6.00000 + 10.3923i 0.346989 + 0.601003i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −13.8564 + 8.00000i −0.796030 + 0.459588i
\(304\) 0 0
\(305\) 1.87564 31.2487i 0.107399 1.78930i
\(306\) 0 0
\(307\) 12.0000i 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 6.00000 10.3923i 0.340229 0.589294i −0.644246 0.764818i \(-0.722829\pi\)
0.984475 + 0.175525i \(0.0561621\pi\)
\(312\) 0 0
\(313\) −12.1244 + 7.00000i −0.685309 + 0.395663i −0.801852 0.597522i \(-0.796152\pi\)
0.116543 + 0.993186i \(0.462819\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.1244 + 7.00000i −0.680972 + 0.393159i −0.800221 0.599705i \(-0.795285\pi\)
0.119249 + 0.992864i \(0.461951\pi\)
\(318\) 0 0
\(319\) −12.0000 + 20.7846i −0.671871 + 1.16371i
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) −27.5885 + 11.7846i −1.53033 + 0.653693i
\(326\) 0 0
\(327\) 12.1244 7.00000i 0.670478 0.387101i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 + 6.92820i 0.219860 + 0.380808i 0.954765 0.297361i \(-0.0961066\pi\)
−0.734905 + 0.678170i \(0.762773\pi\)
\(332\) 0 0
\(333\) 3.46410 + 2.00000i 0.189832 + 0.109599i
\(334\) 0 0
\(335\) −4.00000 8.00000i −0.218543 0.437087i
\(336\) 0 0
\(337\) 8.00000i 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 0 0
\(339\) 9.00000 15.5885i 0.488813 0.846649i
\(340\) 0 0
\(341\) 4.00000 + 6.92820i 0.216612 + 0.375183i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 3.73205 + 2.46410i 0.200927 + 0.132663i
\(346\) 0 0
\(347\) 15.5885 + 9.00000i 0.836832 + 0.483145i 0.856186 0.516667i \(-0.172828\pi\)
−0.0193540 + 0.999813i \(0.506161\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) −1.73205 1.00000i −0.0921878 0.0532246i 0.453197 0.891410i \(-0.350283\pi\)
−0.545385 + 0.838186i \(0.683617\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −8.50000 + 14.7224i −0.447368 + 0.774865i
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 10.0000 + 20.0000i 0.523424 + 1.04685i
\(366\) 0 0
\(367\) 13.8564 + 8.00000i 0.723299 + 0.417597i 0.815966 0.578101i \(-0.196206\pi\)
−0.0926670 + 0.995697i \(0.529539\pi\)
\(368\) 0 0
\(369\) −4.00000 6.92820i −0.208232 0.360668i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −20.7846 + 12.0000i −1.07619 + 0.621336i −0.929865 0.367901i \(-0.880077\pi\)
−0.146321 + 0.989237i \(0.546743\pi\)
\(374\) 0 0
\(375\) −7.23205 + 8.52628i −0.373461 + 0.440295i
\(376\) 0 0
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 8.00000 13.8564i 0.409852 0.709885i
\(382\) 0 0
\(383\) −24.2487 + 14.0000i −1.23905 + 0.715367i −0.968900 0.247451i \(-0.920407\pi\)
−0.270151 + 0.962818i \(0.587074\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.46410 2.00000i 0.176090 0.101666i
\(388\) 0 0
\(389\) 1.00000 1.73205i 0.0507020 0.0878185i −0.839561 0.543266i \(-0.817187\pi\)
0.890263 + 0.455448i \(0.150521\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.66025 + 5.00000i −0.434646 + 0.250943i −0.701324 0.712843i \(-0.747407\pi\)
0.266678 + 0.963786i \(0.414074\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.0000 19.0526i −0.549314 0.951439i −0.998322 0.0579116i \(-0.981556\pi\)
0.449008 0.893528i \(-0.351777\pi\)
\(402\) 0 0
\(403\) −10.3923 6.00000i −0.517678 0.298881i
\(404\) 0 0
\(405\) 2.00000 1.00000i 0.0993808 0.0496904i
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 0 0
\(409\) 5.00000 8.66025i 0.247234 0.428222i −0.715523 0.698589i \(-0.753812\pi\)
0.962757 + 0.270367i \(0.0871450\pi\)
\(410\) 0 0
\(411\) −9.00000 15.5885i −0.443937 0.768922i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 19.7128 29.8564i 0.967664 1.46559i
\(416\) 0 0
\(417\) −8.66025 5.00000i −0.424094 0.244851i
\(418\) 0 0
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 3.46410 + 2.00000i 0.168430 + 0.0972433i
\(424\) 0 0
\(425\) 1.19615 9.92820i 0.0580219 0.481589i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −12.0000 20.7846i −0.579365 1.00349i
\(430\) 0 0
\(431\) 8.00000 13.8564i 0.385346 0.667440i −0.606471 0.795106i \(-0.707415\pi\)
0.991817 + 0.127666i \(0.0407486\pi\)
\(432\) 0 0
\(433\) 10.0000i 0.480569i 0.970702 + 0.240285i \(0.0772408\pi\)
−0.970702 + 0.240285i \(0.922759\pi\)
\(434\) 0 0
\(435\) −6.00000 12.0000i −0.287678 0.575356i
\(436\) 0 0
\(437\) −10.3923 6.00000i −0.497131 0.287019i
\(438\) 0 0
\(439\) 15.0000 + 25.9808i 0.715911 + 1.23999i 0.962607 + 0.270901i \(0.0873217\pi\)
−0.246696 + 0.969093i \(0.579345\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −25.9808 + 15.0000i −1.23438 + 0.712672i −0.967941 0.251179i \(-0.919182\pi\)
−0.266443 + 0.963851i \(0.585848\pi\)
\(444\) 0 0
\(445\) −1.07180 + 17.8564i −0.0508080 + 0.846475i
\(446\) 0 0
\(447\) 22.0000i 1.04056i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −16.0000 + 27.7128i −0.753411 + 1.30495i
\(452\) 0 0
\(453\) 13.8564 8.00000i 0.651031 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.3205 + 10.0000i −0.810219 + 0.467780i −0.847032 0.531542i \(-0.821613\pi\)
0.0368128 + 0.999322i \(0.488279\pi\)
\(458\) 0 0
\(459\) −1.00000 + 1.73205i −0.0466760 + 0.0808452i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 0 0
\(465\) −4.46410 0.267949i −0.207018 0.0124258i
\(466\) 0 0
\(467\) 10.3923 6.00000i 0.480899 0.277647i −0.239892 0.970799i \(-0.577112\pi\)
0.720791 + 0.693153i \(0.243779\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5.00000 + 8.66025i 0.230388 + 0.399043i
\(472\) 0 0
\(473\) −13.8564 8.00000i −0.637118 0.367840i
\(474\) 0 0
\(475\) 18.0000 24.0000i 0.825897 1.10120i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 2.00000 3.46410i 0.0913823 0.158279i −0.816711 0.577047i \(-0.804205\pi\)
0.908093 + 0.418769i \(0.137538\pi\)
\(480\) 0 0
\(481\) 12.0000 + 20.7846i 0.547153 + 0.947697i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.3205 18.6603i 0.559445 0.847318i
\(486\) 0 0
\(487\) −10.3923 6.00000i −0.470920 0.271886i 0.245705 0.969345i \(-0.420981\pi\)
−0.716625 + 0.697459i \(0.754314\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 10.3923 + 6.00000i 0.468046 + 0.270226i
\(494\) 0 0
\(495\) −7.46410 4.92820i −0.335486 0.221506i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −18.0000 31.1769i −0.805791 1.39567i −0.915756 0.401735i \(-0.868407\pi\)
0.109965 0.993935i \(-0.464926\pi\)
\(500\) 0 0
\(501\) 12.0000 20.7846i 0.536120 0.928588i
\(502\) 0 0
\(503\) 36.0000i 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) −32.0000 + 16.0000i −1.42398 + 0.711991i
\(506\) 0 0
\(507\) 19.9186 + 11.5000i 0.884615 + 0.510733i
\(508\) 0 0
\(509\) −6.00000 10.3923i −0.265945 0.460631i 0.701866 0.712309i \(-0.252351\pi\)
−0.967811 + 0.251679i \(0.919017\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −5.19615 + 3.00000i −0.229416 + 0.132453i
\(514\) 0 0
\(515\) 35.7128 + 2.14359i 1.57369 + 0.0944580i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −22.0000 + 38.1051i −0.963837 + 1.66942i −0.251131 + 0.967953i \(0.580803\pi\)
−0.712706 + 0.701462i \(0.752531\pi\)
\(522\) 0 0
\(523\) 17.3205 10.0000i 0.757373 0.437269i −0.0709788 0.997478i \(-0.522612\pi\)
0.828352 + 0.560208i \(0.189279\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.46410 2.00000i 0.150899 0.0871214i
\(528\) 0 0
\(529\) −9.50000 + 16.4545i −0.413043 + 0.715412i
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 48.0000i 2.07911i
\(534\) 0 0
\(535\) −13.3923 0.803848i −0.579000 0.0347534i
\(536\) 0 0
\(537\) −3.46410 + 2.00000i −0.149487 + 0.0863064i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.00000 8.66025i −0.214967 0.372333i 0.738296 0.674477i \(-0.235631\pi\)
−0.953262 + 0.302144i \(0.902298\pi\)
\(542\) 0 0
\(543\) −8.66025 5.00000i −0.371647 0.214571i
\(544\) 0 0
\(545\) 28.0000 14.0000i 1.19939 0.599694i
\(546\) 0 0
\(547\) 20.0000i 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) −7.00000 + 12.1244i −0.298753 + 0.517455i
\(550\) 0 0
\(551\) 18.0000 + 31.1769i 0.766826 + 1.32818i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 7.46410 + 4.92820i 0.316833 + 0.209191i
\(556\) 0 0
\(557\) 5.19615 + 3.00000i 0.220168 + 0.127114i 0.606028 0.795443i \(-0.292762\pi\)
−0.385860 + 0.922557i \(0.626095\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 34.6410 + 20.0000i 1.45994 + 0.842900i 0.999008 0.0445334i \(-0.0141801\pi\)
0.460937 + 0.887433i \(0.347513\pi\)
\(564\) 0 0
\(565\) 22.1769 33.5885i 0.932990 1.41308i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.00000 8.66025i −0.209611 0.363057i 0.741981 0.670421i \(-0.233886\pi\)
−0.951592 + 0.307364i \(0.900553\pi\)
\(570\) 0 0
\(571\) 4.00000 6.92820i 0.167395 0.289936i −0.770108 0.637913i \(-0.779798\pi\)
0.937503 + 0.347977i \(0.113131\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 + 6.00000i 0.333623 + 0.250217i
\(576\) 0 0
\(577\) −32.9090 19.0000i −1.37002 0.790980i −0.379088 0.925361i \(-0.623762\pi\)
−0.990930 + 0.134380i \(0.957096\pi\)
\(578\) 0 0
\(579\) 4.00000 + 6.92820i 0.166234 + 0.287926i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −20.7846 + 12.0000i −0.860811 + 0.496989i
\(584\) 0 0
\(585\) 13.3923 + 0.803848i 0.553704 + 0.0332350i
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −5.00000 + 8.66025i −0.205673 + 0.356235i
\(592\) 0 0
\(593\) −15.5885 + 9.00000i −0.640141 + 0.369586i −0.784669 0.619915i \(-0.787167\pi\)
0.144528 + 0.989501i \(0.453834\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 15.5885 9.00000i 0.637993 0.368345i
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) −0.669873 + 11.1603i −0.0272342 + 0.453729i
\(606\) 0 0
\(607\) 34.6410 20.0000i 1.40604 0.811775i 0.411033 0.911621i \(-0.365168\pi\)
0.995003 + 0.0998457i \(0.0318349\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 + 20.7846i 0.485468 + 0.840855i
\(612\) 0 0
\(613\) −38.1051 22.0000i −1.53905 0.888572i −0.998895 0.0470071i \(-0.985032\pi\)
−0.540157 0.841564i \(-0.681635\pi\)
\(614\) 0 0
\(615\) −8.00000 16.0000i −0.322591 0.645182i
\(616\) 0 0
\(617\) 34.0000i 1.36879i 0.729112 + 0.684394i \(0.239933\pi\)
−0.729112 + 0.684394i \(0.760067\pi\)
\(618\) 0 0
\(619\) −13.0000 + 22.5167i −0.522514 + 0.905021i 0.477143 + 0.878826i \(0.341672\pi\)
−0.999657 + 0.0261952i \(0.991661\pi\)
\(620\) 0 0
\(621\) −1.00000 1.73205i −0.0401286 0.0695048i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −17.2846 + 18.0622i −0.691384 + 0.722487i
\(626\) 0 0
\(627\) 20.7846 + 12.0000i 0.830057 + 0.479234i
\(628\) 0 0
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 20.7846 + 12.0000i 0.826114 + 0.476957i
\(634\) 0 0
\(635\) 19.7128 29.8564i 0.782279 1.18482i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.00000 + 1.73205i −0.0394976 + 0.0684119i −0.885098 0.465404i \(-0.845909\pi\)
0.845601 + 0.533816i \(0.179242\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) 8.00000 4.00000i 0.315000 0.157500i
\(646\) 0 0
\(647\) 20.7846 + 12.0000i 0.817127 + 0.471769i 0.849425 0.527710i \(-0.176949\pi\)
−0.0322975 + 0.999478i \(0.510282\pi\)
\(648\) 0 0
\(649\) 8.00000 + 13.8564i 0.314027 + 0.543912i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −29.4449 + 17.0000i −1.15227 + 0.665261i −0.949439 0.313953i \(-0.898347\pi\)
−0.202828 + 0.979214i \(0.565013\pi\)
\(654\) 0 0
\(655\) −1.60770 + 26.7846i −0.0628178 + 1.04656i
\(656\) 0 0
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 7.00000 12.1244i 0.272268 0.471583i −0.697174 0.716902i \(-0.745559\pi\)
0.969442 + 0.245319i \(0.0788928\pi\)
\(662\) 0 0
\(663\) −10.3923 + 6.00000i −0.403604 + 0.233021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10.3923 + 6.00000i −0.402392 + 0.232321i
\(668\) 0 0
\(669\) 8.00000 13.8564i 0.309298 0.535720i
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) 12.0000i 0.462566i 0.972887 + 0.231283i \(0.0742923\pi\)
−0.972887 + 0.231283i \(0.925708\pi\)
\(674\) 0 0
\(675\) 4.59808 1.96410i 0.176980 0.0755983i
\(676\) 0 0
\(677\) 32.9090 19.0000i 1.26479 0.730229i 0.290796 0.956785i \(-0.406080\pi\)
0.973998 + 0.226556i \(0.0727465\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.0000 + 17.3205i 0.383201 + 0.663723i
\(682\) 0 0
\(683\) −22.5167 13.0000i −0.861576 0.497431i 0.00296369 0.999996i \(-0.499057\pi\)
−0.864540 + 0.502564i \(0.832390\pi\)
\(684\) 0 0
\(685\) −18.0000 36.0000i −0.687745 1.37549i
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) 18.0000 31.1769i 0.685745 1.18775i
\(690\) 0 0
\(691\) −1.00000 1.73205i −0.0380418 0.0658903i 0.846378 0.532583i \(-0.178779\pi\)
−0.884419 + 0.466693i \(0.845445\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18.6603 12.3205i −0.707824 0.467344i
\(696\) 0 0
\(697\) 13.8564 + 8.00000i 0.524849 + 0.303022i
\(698\) 0 0
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) −20.7846 12.0000i −0.783906 0.452589i
\(704\) 0 0
\(705\) 7.46410 + 4.92820i 0.281114 + 0.185607i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.00000 5.19615i −0.112667 0.195146i 0.804178 0.594389i \(-0.202606\pi\)
−0.916845 + 0.399244i \(0.869273\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.00000i 0.149801i
\(714\) 0 0
\(715\) −24.0000 48.0000i −0.897549 1.79510i
\(716\) 0 0
\(717\) −13.8564 8.00000i −0.517477 0.298765i
\(718\) 0 0
\(719\) −22.0000 38.1051i −0.820462 1.42108i −0.905339 0.424690i \(-0.860383\pi\)
0.0848774 0.996391i \(-0.472950\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −8.66025 + 5.00000i −0.322078 + 0.185952i
\(724\) 0 0
\(725\) −11.7846 27.5885i −0.437669 1.02461i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −4.00000 + 6.92820i −0.147945 + 0.256249i
\(732\) 0 0
\(733\) −29.4449 + 17.0000i −1.08757 + 0.627909i −0.932929 0.360061i \(-0.882756\pi\)
−0.154642 + 0.987971i \(0.549422\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.8564 8.00000i 0.510407 0.294684i
\(738\) 0 0
\(739\) 8.00000 13.8564i 0.294285 0.509716i −0.680534 0.732717i \(-0.738252\pi\)
0.974818 + 0.223001i \(0.0715853\pi\)
\(740\) 0 0
\(741\) −36.0000 −1.32249
\(742\) 0 0
\(743\) 54.0000i 1.98107i −0.137268 0.990534i \(-0.543832\pi\)
0.137268 0.990534i \(-0.456168\pi\)
\(744\) 0 0
\(745\) 2.94744 49.1051i 0.107986 1.79907i
\(746\) 0 0
\(747\) −13.8564 + 8.00000i −0.506979 + 0.292705i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 17.3205i −0.364905 0.632034i 0.623856 0.781540i \(-0.285565\pi\)
−0.988761 + 0.149505i \(0.952232\pi\)
\(752\) 0 0
\(753\) −10.3923 6.00000i −0.378717 0.218652i
\(754\) 0 0
\(755\) 32.0000 16.0000i 1.16460 0.582300i
\(756\) 0 0
\(757\) 8.00000i 0.290765i −0.989376 0.145382i \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) 0 0
\(759\) −4.00000 + 6.92820i −0.145191 + 0.251478i
\(760\) 0 0
\(761\) −12.0000 20.7846i −0.435000 0.753442i 0.562296 0.826936i \(-0.309918\pi\)
−0.997296 + 0.0734946i \(0.976585\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.46410 + 3.73205i −0.0890898 + 0.134933i
\(766\) 0 0
\(767\) −20.7846 12.0000i −0.750489 0.433295i
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) −5.19615 3.00000i −0.186893 0.107903i 0.403634 0.914920i \(-0.367747\pi\)
−0.590527 + 0.807018i \(0.701080\pi\)
\(774\) 0 0
\(775\) −9.92820 1.19615i −0.356632 0.0429671i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000 + 41.5692i 0.859889 + 1.48937i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 10.0000 + 20.0000i 0.356915 + 0.713831i
\(786\) 0 0
\(787\) 3.46410 + 2.00000i 0.123482 + 0.0712923i 0.560469 0.828176i \(-0.310621\pi\)
−0.436987 + 0.899468i \(0.643954\pi\)
\(788\) 0 0
\(789\) −5.00000 8.66025i −0.178005 0.308313i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −72.7461 + 42.0000i −2.58329 + 1.49146i
\(794\) 0 0
\(795\) 0.803848 13.3923i 0.0285095 0.474976i
\(796\) 0 0
\(797\) 34.0000i 1.20434i −0.798367 0.602171i \(-0.794303\pi\)
0.798367 0.602171i \(-0.205697\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 4.00000 6.92820i 0.141333 0.244796i
\(802\) 0 0
\(803\) −34.6410 + 20.0000i −1.22245 + 0.705785i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.3923 + 6.00000i −0.365826 + 0.211210i
\(808\) 0 0
\(809\) −9.00000 + 15.5885i −0.316423 + 0.548061i −0.979739 0.200279i \(-0.935815\pi\)
0.663316 + 0.748340i \(0.269149\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 0 0
\(813\) 14.0000i 0.491001i
\(814\) 0 0
\(815\) 17.8564 + 1.07180i 0.625483 + 0.0375434i
\(816\) 0 0
\(817\) −20.7846 + 12.0000i −0.727161 + 0.419827i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.00000 8.66025i −0.174501 0.302245i 0.765487 0.643451i \(-0.222498\pi\)
−0.939989 + 0.341206i \(0.889165\pi\)
\(822\) 0 0
\(823\) −20.7846 12.0000i −0.724506 0.418294i 0.0919029 0.995768i \(-0.470705\pi\)
−0.816409 + 0.577474i \(0.804038\pi\)
\(824\) 0 0
\(825\) −16.0000 12.0000i −0.557048 0.417786i
\(826\) 0 0
\(827\) 22.0000i 0.765015i 0.923952 + 0.382507i \(0.124939\pi\)
−0.923952 + 0.382507i \(0.875061\pi\)
\(828\) 0 0
\(829\) −3.00000 + 5.19615i −0.104194 + 0.180470i −0.913409 0.407044i \(-0.866560\pi\)
0.809214 + 0.587513i \(0.199893\pi\)
\(830\) 0 0
\(831\) 14.0000 + 24.2487i 0.485655 + 0.841178i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 29.5692 44.7846i 1.02329 1.54984i
\(836\) 0 0
\(837\) 1.73205 + 1.00000i 0.0598684 + 0.0345651i
\(838\) 0 0
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 5.19615 + 3.00000i 0.178965 + 0.103325i
\(844\) 0 0
\(845\) 42.9186 + 28.3372i 1.47644 + 0.974828i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −14.0000 24.2487i −0.480479 0.832214i
\(850\) 0 0
\(851\) 4.00000 6.92820i 0.137118 0.237496i
\(852\) 0 0
\(853\) 2.00000i 0.0684787i 0.999414 + 0.0342393i \(0.0109009\pi\)
−0.999414 + 0.0342393i \(0.989099\pi\)
\(854\) 0 0
\(855\) −12.0000 + 6.00000i −0.410391 + 0.205196i
\(856\) 0 0
\(857\) −5.19615 3.00000i −0.177497 0.102478i 0.408619 0.912705i \(-0.366010\pi\)
−0.586116 + 0.810227i \(0.699344\pi\)
\(858\) 0 0
\(859\) 23.0000 + 39.8372i 0.784750 + 1.35923i 0.929148 + 0.369707i \(0.120542\pi\)
−0.144399 + 0.989520i \(0.546125\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15.5885 9.00000i 0.530637 0.306364i −0.210639 0.977564i \(-0.567554\pi\)
0.741276 + 0.671200i \(0.234221\pi\)
\(864\) 0 0
\(865\) −4.46410 0.267949i −0.151784 0.00911055i
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −12.0000 + 20.7846i −0.406604 + 0.704260i
\(872\) 0 0
\(873\) −8.66025 + 5.00000i −0.293105 + 0.169224i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38.1051 22.0000i 1.28672 0.742887i 0.308651 0.951175i \(-0.400123\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(878\) 0 0
\(879\) 3.00000 5.19615i 0.101187 0.175262i
\(880\) 0 0
\(881\) −36.0000 −1.21287 −0.606435 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) 24.0000i 0.807664i −0.914833 0.403832i \(-0.867678\pi\)
0.914833 0.403832i \(-0.132322\pi\)
\(884\) 0 0
\(885\) −8.92820 0.535898i −0.300118 0.0180140i
\(886\) 0 0
\(887\) 10.3923 6.00000i 0.348939 0.201460i −0.315279 0.948999i \(-0.602098\pi\)
0.664218 + 0.747539i \(0.268765\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 + 3.46410i 0.0670025 + 0.116052i
\(892\) 0 0
\(893\) −20.7846 12.0000i −0.695530 0.401565i
\(894\) 0 0
\(895\) −8.00000 + 4.00000i −0.267411 + 0.133705i
\(896\) 0 0
\(897\) 12.0000i 0.400668i
\(898\) 0 0
\(899\) 6.00000 10.3923i 0.200111 0.346603i
\(900\) 0 0
\(901\) 6.00000 + 10.3923i 0.199889 + 0.346218i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.6603 12.3205i −0.620288 0.409548i
\(906\) 0 0
\(907\) 20.7846 + 12.0000i 0.690142 + 0.398453i 0.803665 0.595082i \(-0.202880\pi\)
−0.113523 + 0.993535i \(0.536214\pi\)
\(908\) 0 0
\(909\) 16.0000 0.530687
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 55.4256 + 32.0000i 1.83432 + 1.05905i
\(914\) 0 0
\(915\) −17.2487 + 26.1244i −0.570225 + 0.863645i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 18.0000 + 31.1769i 0.593765 + 1.02843i 0.993720 + 0.111897i \(0.0356925\pi\)
−0.399955 + 0.916535i \(0.630974\pi\)
\(920\) 0 0
\(921\) −6.00000 + 10.3923i −0.197707 + 0.342438i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 16.0000 + 12.0000i 0.526077 + 0.394558i
\(926\) 0 0
\(927\) −13.8564 8.00000i −0.455104 0.262754i
\(928\) 0 0
\(929\) −12.0000 20.7846i −0.393707 0.681921i 0.599228 0.800578i \(-0.295474\pi\)
−0.992935 + 0.118657i \(0.962141\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −10.3923 + 6.00000i −0.340229 + 0.196431i
\(934\) 0 0
\(935\) 17.8564 + 1.07180i 0.583967 + 0.0350515i
\(936\) 0 0
\(937\) 14.0000i 0.457360i −0.973502 0.228680i \(-0.926559\pi\)
0.973502 0.228680i \(-0.0734410\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 22.0000 38.1051i 0.717180 1.24219i −0.244933 0.969540i \(-0.578766\pi\)
0.962113 0.272651i \(-0.0879006\pi\)
\(942\) 0 0
\(943\) −13.8564 + 8.00000i −0.451227 + 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.19615 + 3.00000i −0.168852 + 0.0974869i −0.582045 0.813157i \(-0.697747\pi\)
0.413192 + 0.910644i \(0.364414\pi\)
\(948\) 0 0
\(949\) 30.0000 51.9615i 0.973841 1.68674i
\(950\) 0 0
\(951\) 14.0000 0.453981
\(952\) 0 0
\(953\) 14.0000i 0.453504i −0.973952 0.226752i \(-0.927189\pi\)
0.973952 0.226752i \(-0.0728108\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.7846 12.0000i 0.671871 0.387905i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 13.5000 + 23.3827i 0.435484 + 0.754280i
\(962\) 0 0
\(963\) 5.19615 + 3.00000i 0.167444 + 0.0966736i
\(964\) 0 0
\(965\) 8.00000 + 16.0000i 0.257529 + 0.515058i
\(966\) 0 0
\(967\) 20.0000i 0.643157i 0.946883 + 0.321578i \(0.104213\pi\)
−0.946883 + 0.321578i \(0.895787\pi\)
\(968\) 0 0
\(969\) 6.00000 10.3923i 0.192748 0.333849i
\(970\) 0 0
\(971\) −12.0000 20.7846i −0.385098 0.667010i 0.606685 0.794943i \(-0.292499\pi\)
−0.991783 + 0.127933i \(0.959166\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 29.7846 + 3.58846i 0.953871 + 0.114923i
\(976\) 0 0
\(977\) −15.5885 9.00000i −0.498719 0.287936i 0.229465 0.973317i \(-0.426302\pi\)
−0.728184 + 0.685381i \(0.759636\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) −6.92820 4.00000i −0.220975 0.127580i 0.385426 0.922739i \(-0.374054\pi\)
−0.606402 + 0.795158i \(0.707388\pi\)
\(984\) 0 0
\(985\) −12.3205 + 18.6603i −0.392564 + 0.594565i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 6.92820i −0.127193 0.220304i
\(990\) 0 0
\(991\) −26.0000 + 45.0333i −0.825917 + 1.43053i 0.0752991 + 0.997161i \(0.476009\pi\)
−0.901216 + 0.433370i \(0.857324\pi\)
\(992\) 0 0
\(993\) 8.00000i 0.253872i
\(994\) 0 0
\(995\) 36.0000 18.0000i 1.14128 0.570638i
\(996\) 0 0
\(997\) 15.5885 + 9.00000i 0.493691 + 0.285033i 0.726105 0.687584i \(-0.241329\pi\)
−0.232413 + 0.972617i \(0.574662\pi\)
\(998\) 0 0
\(999\) −2.00000 3.46410i −0.0632772 0.109599i
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 2940.2.bb.c.949.1 4
5.4 even 2 inner 2940.2.bb.c.949.2 4
7.2 even 3 inner 2940.2.bb.c.1549.2 4
7.3 odd 6 420.2.k.a.169.1 2
7.4 even 3 2940.2.k.d.589.2 2
7.5 odd 6 2940.2.bb.h.1549.1 4
7.6 odd 2 2940.2.bb.h.949.2 4
21.17 even 6 1260.2.k.d.1009.1 2
28.3 even 6 1680.2.t.a.1009.2 2
35.3 even 12 2100.2.a.j.1.1 1
35.4 even 6 2940.2.k.d.589.1 2
35.9 even 6 inner 2940.2.bb.c.1549.1 4
35.17 even 12 2100.2.a.e.1.1 1
35.19 odd 6 2940.2.bb.h.1549.2 4
35.24 odd 6 420.2.k.a.169.2 yes 2
35.34 odd 2 2940.2.bb.h.949.1 4
84.59 odd 6 5040.2.t.o.1009.1 2
105.17 odd 12 6300.2.a.bc.1.1 1
105.38 odd 12 6300.2.a.n.1.1 1
105.59 even 6 1260.2.k.d.1009.2 2
140.3 odd 12 8400.2.a.bh.1.1 1
140.59 even 6 1680.2.t.a.1009.1 2
140.87 odd 12 8400.2.a.cd.1.1 1
420.59 odd 6 5040.2.t.o.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.a.169.1 2 7.3 odd 6
420.2.k.a.169.2 yes 2 35.24 odd 6
1260.2.k.d.1009.1 2 21.17 even 6
1260.2.k.d.1009.2 2 105.59 even 6
1680.2.t.a.1009.1 2 140.59 even 6
1680.2.t.a.1009.2 2 28.3 even 6
2100.2.a.e.1.1 1 35.17 even 12
2100.2.a.j.1.1 1 35.3 even 12
2940.2.k.d.589.1 2 35.4 even 6
2940.2.k.d.589.2 2 7.4 even 3
2940.2.bb.c.949.1 4 1.1 even 1 trivial
2940.2.bb.c.949.2 4 5.4 even 2 inner
2940.2.bb.c.1549.1 4 35.9 even 6 inner
2940.2.bb.c.1549.2 4 7.2 even 3 inner
2940.2.bb.h.949.1 4 35.34 odd 2
2940.2.bb.h.949.2 4 7.6 odd 2
2940.2.bb.h.1549.1 4 7.5 odd 6
2940.2.bb.h.1549.2 4 35.19 odd 6
5040.2.t.o.1009.1 2 84.59 odd 6
5040.2.t.o.1009.2 2 420.59 odd 6
6300.2.a.n.1.1 1 105.38 odd 12
6300.2.a.bc.1.1 1 105.17 odd 12
8400.2.a.bh.1.1 1 140.3 odd 12
8400.2.a.cd.1.1 1 140.87 odd 12