Properties

Label 980.4.i.w.961.1
Level $980$
Weight $4$
Character 980.961
Analytic conductor $57.822$
Analytic rank $0$
Dimension $4$
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] = [980,4,Mod(361,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), 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: \(57.8218718056\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 22x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(-2.34521 - 4.06202i\) of defining polynomial
Character \(\chi\) \(=\) 980.961
Dual form 980.4.i.w.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.154792 + 0.268108i) q^{3} +(2.50000 - 4.33013i) q^{5} +(13.4521 - 23.2997i) q^{9} +O(q^{10})\) \(q+(0.154792 + 0.268108i) q^{3} +(2.50000 - 4.33013i) q^{5} +(13.4521 - 23.2997i) q^{9} +(-8.30958 - 14.3926i) q^{11} +91.6658 q^{13} +1.54792 q^{15} +(13.8329 + 23.9593i) q^{17} +(39.3096 - 68.0862i) q^{19} +(-18.4644 + 31.9812i) q^{23} +(-12.5000 - 21.6506i) q^{25} +16.6879 q^{27} -209.285 q^{29} +(89.1425 + 154.399i) q^{31} +(2.57252 - 4.45573i) q^{33} +(-68.5700 + 118.767i) q^{37} +(14.1891 + 24.5763i) q^{39} -225.806 q^{41} +502.589 q^{43} +(-67.2604 - 116.498i) q^{45} +(225.904 - 391.277i) q^{47} +(-4.28245 + 7.41742i) q^{51} +(-253.595 - 439.239i) q^{53} -83.0958 q^{55} +24.3393 q^{57} +(-343.118 - 594.298i) q^{59} +(-275.950 + 477.959i) q^{61} +(229.165 - 396.925i) q^{65} +(90.7960 + 157.263i) q^{67} -11.4326 q^{69} +832.801 q^{71} +(-170.781 - 295.802i) q^{73} +(3.86980 - 6.70270i) q^{75} +(-89.6930 + 155.353i) q^{79} +(-360.623 - 624.617i) q^{81} +708.108 q^{83} +138.329 q^{85} +(-32.3957 - 56.1109i) q^{87} +(434.446 - 752.482i) q^{89} +(-27.5971 + 47.7996i) q^{93} +(-196.548 - 340.431i) q^{95} +801.145 q^{97} -447.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} + 10 q^{5} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} + 10 q^{5} - 40 q^{9} - 52 q^{11} + 104 q^{13} + 100 q^{15} - 76 q^{17} + 176 q^{19} - 102 q^{23} - 50 q^{25} - 740 q^{27} - 612 q^{29} + 244 q^{31} + 348 q^{33} + 176 q^{37} - 356 q^{39} + 260 q^{41} + 228 q^{43} + 200 q^{45} + 716 q^{47} + 996 q^{51} - 808 q^{53} - 520 q^{55} + 1936 q^{57} - 1016 q^{59} - 222 q^{61} + 260 q^{65} - 134 q^{67} - 1284 q^{69} + 592 q^{71} + 724 q^{73} + 250 q^{75} - 1128 q^{79} - 2662 q^{81} - 676 q^{83} - 760 q^{85} - 1002 q^{87} - 326 q^{89} - 692 q^{93} - 880 q^{95} + 3880 q^{97} + 3840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0.154792 + 0.268108i 0.0297898 + 0.0515974i 0.880536 0.473979i \(-0.157183\pi\)
−0.850746 + 0.525577i \(0.823850\pi\)
\(4\) 0 0
\(5\) 2.50000 4.33013i 0.223607 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 13.4521 23.2997i 0.498225 0.862951i
\(10\) 0 0
\(11\) −8.30958 14.3926i −0.227767 0.394504i 0.729379 0.684110i \(-0.239809\pi\)
−0.957146 + 0.289606i \(0.906476\pi\)
\(12\) 0 0
\(13\) 91.6658 1.95566 0.977828 0.209410i \(-0.0671544\pi\)
0.977828 + 0.209410i \(0.0671544\pi\)
\(14\) 0 0
\(15\) 1.54792 0.0266448
\(16\) 0 0
\(17\) 13.8329 + 23.9593i 0.197351 + 0.341823i 0.947669 0.319255i \(-0.103433\pi\)
−0.750317 + 0.661078i \(0.770099\pi\)
\(18\) 0 0
\(19\) 39.3096 68.0862i 0.474644 0.822108i −0.524934 0.851143i \(-0.675910\pi\)
0.999578 + 0.0290351i \(0.00924346\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −18.4644 + 31.9812i −0.167395 + 0.289937i −0.937503 0.347976i \(-0.886869\pi\)
0.770108 + 0.637913i \(0.220202\pi\)
\(24\) 0 0
\(25\) −12.5000 21.6506i −0.100000 0.173205i
\(26\) 0 0
\(27\) 16.6879 0.118948
\(28\) 0 0
\(29\) −209.285 −1.34011 −0.670056 0.742311i \(-0.733730\pi\)
−0.670056 + 0.742311i \(0.733730\pi\)
\(30\) 0 0
\(31\) 89.1425 + 154.399i 0.516467 + 0.894546i 0.999817 + 0.0191195i \(0.00608631\pi\)
−0.483351 + 0.875427i \(0.660580\pi\)
\(32\) 0 0
\(33\) 2.57252 4.45573i 0.0135702 0.0235043i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −68.5700 + 118.767i −0.304671 + 0.527706i −0.977188 0.212376i \(-0.931880\pi\)
0.672517 + 0.740082i \(0.265213\pi\)
\(38\) 0 0
\(39\) 14.1891 + 24.5763i 0.0582585 + 0.100907i
\(40\) 0 0
\(41\) −225.806 −0.860120 −0.430060 0.902800i \(-0.641508\pi\)
−0.430060 + 0.902800i \(0.641508\pi\)
\(42\) 0 0
\(43\) 502.589 1.78242 0.891211 0.453588i \(-0.149856\pi\)
0.891211 + 0.453588i \(0.149856\pi\)
\(44\) 0 0
\(45\) −67.2604 116.498i −0.222813 0.385924i
\(46\) 0 0
\(47\) 225.904 391.277i 0.701096 1.21433i −0.266986 0.963700i \(-0.586028\pi\)
0.968082 0.250633i \(-0.0806389\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.28245 + 7.41742i −0.0117581 + 0.0203656i
\(52\) 0 0
\(53\) −253.595 439.239i −0.657243 1.13838i −0.981326 0.192350i \(-0.938389\pi\)
0.324083 0.946029i \(-0.394944\pi\)
\(54\) 0 0
\(55\) −83.0958 −0.203721
\(56\) 0 0
\(57\) 24.3393 0.0565581
\(58\) 0 0
\(59\) −343.118 594.298i −0.757121 1.31137i −0.944313 0.329049i \(-0.893272\pi\)
0.187192 0.982323i \(-0.440061\pi\)
\(60\) 0 0
\(61\) −275.950 + 477.959i −0.579208 + 1.00322i 0.416362 + 0.909199i \(0.363305\pi\)
−0.995570 + 0.0940196i \(0.970028\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 229.165 396.925i 0.437298 0.757422i
\(66\) 0 0
\(67\) 90.7960 + 157.263i 0.165560 + 0.286758i 0.936854 0.349721i \(-0.113724\pi\)
−0.771294 + 0.636479i \(0.780390\pi\)
\(68\) 0 0
\(69\) −11.4326 −0.0199466
\(70\) 0 0
\(71\) 832.801 1.39205 0.696023 0.718020i \(-0.254951\pi\)
0.696023 + 0.718020i \(0.254951\pi\)
\(72\) 0 0
\(73\) −170.781 295.802i −0.273814 0.474260i 0.696021 0.718021i \(-0.254952\pi\)
−0.969835 + 0.243761i \(0.921619\pi\)
\(74\) 0 0
\(75\) 3.86980 6.70270i 0.00595795 0.0103195i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −89.6930 + 155.353i −0.127737 + 0.221247i −0.922800 0.385280i \(-0.874105\pi\)
0.795062 + 0.606528i \(0.207438\pi\)
\(80\) 0 0
\(81\) −360.623 624.617i −0.494682 0.856814i
\(82\) 0 0
\(83\) 708.108 0.936445 0.468222 0.883611i \(-0.344895\pi\)
0.468222 + 0.883611i \(0.344895\pi\)
\(84\) 0 0
\(85\) 138.329 0.176516
\(86\) 0 0
\(87\) −32.3957 56.1109i −0.0399216 0.0691463i
\(88\) 0 0
\(89\) 434.446 752.482i 0.517429 0.896213i −0.482366 0.875970i \(-0.660223\pi\)
0.999795 0.0202433i \(-0.00644407\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −27.5971 + 47.7996i −0.0307708 + 0.0532966i
\(94\) 0 0
\(95\) −196.548 340.431i −0.212267 0.367658i
\(96\) 0 0
\(97\) 801.145 0.838597 0.419299 0.907848i \(-0.362276\pi\)
0.419299 + 0.907848i \(0.362276\pi\)
\(98\) 0 0
\(99\) −447.125 −0.453916
\(100\) 0 0
\(101\) 471.777 + 817.142i 0.464788 + 0.805037i 0.999192 0.0401926i \(-0.0127972\pi\)
−0.534404 + 0.845229i \(0.679464\pi\)
\(102\) 0 0
\(103\) 84.5848 146.505i 0.0809164 0.140151i −0.822727 0.568436i \(-0.807549\pi\)
0.903644 + 0.428285i \(0.140882\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −787.533 + 1364.05i −0.711530 + 1.23241i 0.252753 + 0.967531i \(0.418664\pi\)
−0.964283 + 0.264875i \(0.914669\pi\)
\(108\) 0 0
\(109\) −458.525 794.188i −0.402924 0.697884i 0.591154 0.806559i \(-0.298673\pi\)
−0.994077 + 0.108675i \(0.965339\pi\)
\(110\) 0 0
\(111\) −42.4564 −0.0363043
\(112\) 0 0
\(113\) −90.9042 −0.0756774 −0.0378387 0.999284i \(-0.512047\pi\)
−0.0378387 + 0.999284i \(0.512047\pi\)
\(114\) 0 0
\(115\) 92.3219 + 159.906i 0.0748614 + 0.129664i
\(116\) 0 0
\(117\) 1233.10 2135.78i 0.974357 1.68764i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 527.402 913.486i 0.396245 0.686316i
\(122\) 0 0
\(123\) −34.9530 60.5403i −0.0256228 0.0443800i
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −300.762 −0.210144 −0.105072 0.994465i \(-0.533507\pi\)
−0.105072 + 0.994465i \(0.533507\pi\)
\(128\) 0 0
\(129\) 77.7969 + 134.748i 0.0530979 + 0.0919683i
\(130\) 0 0
\(131\) 460.472 797.560i 0.307111 0.531933i −0.670618 0.741803i \(-0.733971\pi\)
0.977729 + 0.209871i \(0.0673043\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 41.7197 72.2606i 0.0265975 0.0460682i
\(136\) 0 0
\(137\) −1383.27 2395.90i −0.862635 1.49413i −0.869377 0.494150i \(-0.835480\pi\)
0.00674241 0.999977i \(-0.497854\pi\)
\(138\) 0 0
\(139\) 2735.74 1.66937 0.834686 0.550726i \(-0.185649\pi\)
0.834686 + 0.550726i \(0.185649\pi\)
\(140\) 0 0
\(141\) 139.873 0.0835419
\(142\) 0 0
\(143\) −761.705 1319.31i −0.445433 0.771513i
\(144\) 0 0
\(145\) −523.212 + 906.231i −0.299658 + 0.519023i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1334.14 + 2310.80i −0.733538 + 1.27052i 0.221824 + 0.975087i \(0.428799\pi\)
−0.955362 + 0.295438i \(0.904534\pi\)
\(150\) 0 0
\(151\) −1153.28 1997.54i −0.621540 1.07654i −0.989199 0.146579i \(-0.953174\pi\)
0.367659 0.929961i \(-0.380159\pi\)
\(152\) 0 0
\(153\) 744.326 0.393302
\(154\) 0 0
\(155\) 891.425 0.461942
\(156\) 0 0
\(157\) −990.808 1716.13i −0.503663 0.872370i −0.999991 0.00423500i \(-0.998652\pi\)
0.496328 0.868135i \(-0.334681\pi\)
\(158\) 0 0
\(159\) 78.5089 135.981i 0.0391582 0.0678241i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 541.580 938.044i 0.260244 0.450756i −0.706062 0.708150i \(-0.749530\pi\)
0.966307 + 0.257393i \(0.0828635\pi\)
\(164\) 0 0
\(165\) −12.8626 22.2786i −0.00606879 0.0105115i
\(166\) 0 0
\(167\) −1203.77 −0.557789 −0.278895 0.960322i \(-0.589968\pi\)
−0.278895 + 0.960322i \(0.589968\pi\)
\(168\) 0 0
\(169\) 6205.62 2.82459
\(170\) 0 0
\(171\) −1057.59 1831.80i −0.472959 0.819189i
\(172\) 0 0
\(173\) 1341.90 2324.23i 0.589725 1.02143i −0.404543 0.914519i \(-0.632569\pi\)
0.994268 0.106915i \(-0.0340974\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 106.224 183.985i 0.0451089 0.0781309i
\(178\) 0 0
\(179\) −31.1425 53.9404i −0.0130039 0.0225234i 0.859450 0.511219i \(-0.170806\pi\)
−0.872454 + 0.488696i \(0.837473\pi\)
\(180\) 0 0
\(181\) −680.223 −0.279340 −0.139670 0.990198i \(-0.544604\pi\)
−0.139670 + 0.990198i \(0.544604\pi\)
\(182\) 0 0
\(183\) −170.859 −0.0690179
\(184\) 0 0
\(185\) 342.850 + 593.833i 0.136253 + 0.235997i
\(186\) 0 0
\(187\) 229.891 398.184i 0.0899002 0.155712i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 972.059 1683.65i 0.368250 0.637827i −0.621042 0.783777i \(-0.713290\pi\)
0.989292 + 0.145950i \(0.0466238\pi\)
\(192\) 0 0
\(193\) −340.435 589.651i −0.126969 0.219917i 0.795532 0.605912i \(-0.207192\pi\)
−0.922501 + 0.385995i \(0.873858\pi\)
\(194\) 0 0
\(195\) 141.891 0.0521080
\(196\) 0 0
\(197\) 323.259 0.116910 0.0584549 0.998290i \(-0.481383\pi\)
0.0584549 + 0.998290i \(0.481383\pi\)
\(198\) 0 0
\(199\) 1415.28 + 2451.34i 0.504154 + 0.873221i 0.999988 + 0.00480368i \(0.00152906\pi\)
−0.495834 + 0.868417i \(0.665138\pi\)
\(200\) 0 0
\(201\) −28.1090 + 48.6862i −0.00986397 + 0.0170849i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −564.514 + 977.768i −0.192329 + 0.333123i
\(206\) 0 0
\(207\) 496.768 + 860.428i 0.166801 + 0.288908i
\(208\) 0 0
\(209\) −1306.59 −0.432432
\(210\) 0 0
\(211\) 2488.78 0.812013 0.406006 0.913870i \(-0.366921\pi\)
0.406006 + 0.913870i \(0.366921\pi\)
\(212\) 0 0
\(213\) 128.911 + 223.280i 0.0414687 + 0.0718259i
\(214\) 0 0
\(215\) 1256.47 2176.28i 0.398562 0.690329i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 52.8712 91.5755i 0.0163137 0.0282562i
\(220\) 0 0
\(221\) 1268.01 + 2196.25i 0.385951 + 0.668487i
\(222\) 0 0
\(223\) −1994.33 −0.598881 −0.299440 0.954115i \(-0.596800\pi\)
−0.299440 + 0.954115i \(0.596800\pi\)
\(224\) 0 0
\(225\) −672.604 −0.199290
\(226\) 0 0
\(227\) 889.192 + 1540.13i 0.259990 + 0.450316i 0.966239 0.257648i \(-0.0829474\pi\)
−0.706249 + 0.707964i \(0.749614\pi\)
\(228\) 0 0
\(229\) −505.826 + 876.116i −0.145965 + 0.252818i −0.929732 0.368236i \(-0.879962\pi\)
0.783768 + 0.621054i \(0.213295\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1642.61 + 2845.09i −0.461851 + 0.799949i −0.999053 0.0435046i \(-0.986148\pi\)
0.537203 + 0.843453i \(0.319481\pi\)
\(234\) 0 0
\(235\) −1129.52 1956.39i −0.313540 0.543067i
\(236\) 0 0
\(237\) −55.5350 −0.0152211
\(238\) 0 0
\(239\) −3355.78 −0.908231 −0.454116 0.890943i \(-0.650045\pi\)
−0.454116 + 0.890943i \(0.650045\pi\)
\(240\) 0 0
\(241\) −1446.38 2505.21i −0.386596 0.669604i 0.605393 0.795926i \(-0.293016\pi\)
−0.991989 + 0.126323i \(0.959682\pi\)
\(242\) 0 0
\(243\) 336.930 583.579i 0.0889467 0.154060i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3603.35 6241.18i 0.928240 1.60776i
\(248\) 0 0
\(249\) 109.609 + 189.849i 0.0278965 + 0.0483181i
\(250\) 0 0
\(251\) 6497.64 1.63397 0.816987 0.576656i \(-0.195643\pi\)
0.816987 + 0.576656i \(0.195643\pi\)
\(252\) 0 0
\(253\) 613.725 0.152508
\(254\) 0 0
\(255\) 21.4123 + 37.0871i 0.00525838 + 0.00910778i
\(256\) 0 0
\(257\) −1129.37 + 1956.12i −0.274116 + 0.474783i −0.969912 0.243457i \(-0.921719\pi\)
0.695795 + 0.718240i \(0.255052\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2815.32 + 4876.27i −0.667677 + 1.15645i
\(262\) 0 0
\(263\) −67.4499 116.827i −0.0158142 0.0273910i 0.858010 0.513633i \(-0.171701\pi\)
−0.873824 + 0.486242i \(0.838367\pi\)
\(264\) 0 0
\(265\) −2535.95 −0.587856
\(266\) 0 0
\(267\) 268.995 0.0616563
\(268\) 0 0
\(269\) −3265.28 5655.63i −0.740102 1.28190i −0.952448 0.304700i \(-0.901444\pi\)
0.212346 0.977195i \(-0.431890\pi\)
\(270\) 0 0
\(271\) 1073.74 1859.78i 0.240683 0.416876i −0.720226 0.693740i \(-0.755962\pi\)
0.960909 + 0.276864i \(0.0892951\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −207.740 + 359.816i −0.0455533 + 0.0789007i
\(276\) 0 0
\(277\) −3914.02 6779.28i −0.848991 1.47050i −0.882109 0.471045i \(-0.843877\pi\)
0.0331177 0.999451i \(-0.489456\pi\)
\(278\) 0 0
\(279\) 4796.61 1.02927
\(280\) 0 0
\(281\) −5934.49 −1.25987 −0.629933 0.776650i \(-0.716917\pi\)
−0.629933 + 0.776650i \(0.716917\pi\)
\(282\) 0 0
\(283\) 2619.14 + 4536.48i 0.550147 + 0.952882i 0.998263 + 0.0589072i \(0.0187616\pi\)
−0.448117 + 0.893975i \(0.647905\pi\)
\(284\) 0 0
\(285\) 60.8481 105.392i 0.0126468 0.0219049i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2073.80 3591.93i 0.422105 0.731107i
\(290\) 0 0
\(291\) 124.011 + 214.793i 0.0249816 + 0.0432694i
\(292\) 0 0
\(293\) 3564.95 0.710808 0.355404 0.934713i \(-0.384343\pi\)
0.355404 + 0.934713i \(0.384343\pi\)
\(294\) 0 0
\(295\) −3431.18 −0.677190
\(296\) 0 0
\(297\) −138.669 240.182i −0.0270923 0.0469252i
\(298\) 0 0
\(299\) −1692.55 + 2931.59i −0.327367 + 0.567017i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −146.055 + 252.974i −0.0276919 + 0.0479637i
\(304\) 0 0
\(305\) 1379.75 + 2389.79i 0.259030 + 0.448653i
\(306\) 0 0
\(307\) −5809.11 −1.07995 −0.539973 0.841682i \(-0.681566\pi\)
−0.539973 + 0.841682i \(0.681566\pi\)
\(308\) 0 0
\(309\) 52.3723 0.00964192
\(310\) 0 0
\(311\) 697.216 + 1207.61i 0.127124 + 0.220185i 0.922561 0.385851i \(-0.126092\pi\)
−0.795437 + 0.606036i \(0.792759\pi\)
\(312\) 0 0
\(313\) −2971.29 + 5146.43i −0.536573 + 0.929372i 0.462512 + 0.886613i \(0.346948\pi\)
−0.999085 + 0.0427590i \(0.986385\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −721.455 + 1249.60i −0.127826 + 0.221402i −0.922834 0.385197i \(-0.874133\pi\)
0.795008 + 0.606599i \(0.207467\pi\)
\(318\) 0 0
\(319\) 1739.07 + 3012.16i 0.305233 + 0.528679i
\(320\) 0 0
\(321\) −487.616 −0.0847852
\(322\) 0 0
\(323\) 2175.06 0.374687
\(324\) 0 0
\(325\) −1145.82 1984.62i −0.195566 0.338730i
\(326\) 0 0
\(327\) 141.952 245.868i 0.0240060 0.0415796i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2945.46 + 5101.68i −0.489115 + 0.847171i −0.999922 0.0125241i \(-0.996013\pi\)
0.510807 + 0.859695i \(0.329347\pi\)
\(332\) 0 0
\(333\) 1844.82 + 3195.32i 0.303590 + 0.525833i
\(334\) 0 0
\(335\) 907.960 0.148081
\(336\) 0 0
\(337\) −1326.63 −0.214439 −0.107220 0.994235i \(-0.534195\pi\)
−0.107220 + 0.994235i \(0.534195\pi\)
\(338\) 0 0
\(339\) −14.0712 24.3721i −0.00225441 0.00390475i
\(340\) 0 0
\(341\) 1481.47 2565.99i 0.235268 0.407496i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −28.5814 + 49.5044i −0.00446021 + 0.00772530i
\(346\) 0 0
\(347\) −1102.49 1909.57i −0.170562 0.295421i 0.768055 0.640384i \(-0.221225\pi\)
−0.938616 + 0.344963i \(0.887891\pi\)
\(348\) 0 0
\(349\) −1792.52 −0.274933 −0.137466 0.990506i \(-0.543896\pi\)
−0.137466 + 0.990506i \(0.543896\pi\)
\(350\) 0 0
\(351\) 1529.71 0.232620
\(352\) 0 0
\(353\) 3070.75 + 5318.70i 0.463002 + 0.801943i 0.999109 0.0422071i \(-0.0134389\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(354\) 0 0
\(355\) 2082.00 3606.13i 0.311271 0.539137i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −67.6039 + 117.093i −0.00993872 + 0.0172144i −0.870952 0.491368i \(-0.836497\pi\)
0.861013 + 0.508582i \(0.169830\pi\)
\(360\) 0 0
\(361\) 339.013 + 587.188i 0.0494260 + 0.0856084i
\(362\) 0 0
\(363\) 326.550 0.0472161
\(364\) 0 0
\(365\) −1707.81 −0.244907
\(366\) 0 0
\(367\) 4566.68 + 7909.72i 0.649533 + 1.12502i 0.983234 + 0.182346i \(0.0583690\pi\)
−0.333701 + 0.942679i \(0.608298\pi\)
\(368\) 0 0
\(369\) −3037.56 + 5261.20i −0.428534 + 0.742242i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3775.01 + 6538.50i −0.524028 + 0.907643i 0.475581 + 0.879672i \(0.342238\pi\)
−0.999609 + 0.0279712i \(0.991095\pi\)
\(374\) 0 0
\(375\) −19.3490 33.5135i −0.00266448 0.00461501i
\(376\) 0 0
\(377\) −19184.3 −2.62080
\(378\) 0 0
\(379\) −8444.21 −1.14446 −0.572229 0.820094i \(-0.693921\pi\)
−0.572229 + 0.820094i \(0.693921\pi\)
\(380\) 0 0
\(381\) −46.5555 80.6366i −0.00626014 0.0108429i
\(382\) 0 0
\(383\) −6313.53 + 10935.3i −0.842314 + 1.45893i 0.0456204 + 0.998959i \(0.485474\pi\)
−0.887934 + 0.459971i \(0.847860\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6760.87 11710.2i 0.888048 1.53814i
\(388\) 0 0
\(389\) 7180.32 + 12436.7i 0.935878 + 1.62099i 0.773060 + 0.634333i \(0.218725\pi\)
0.162818 + 0.986656i \(0.447942\pi\)
\(390\) 0 0
\(391\) −1021.66 −0.132143
\(392\) 0 0
\(393\) 285.109 0.0365951
\(394\) 0 0
\(395\) 448.465 + 776.764i 0.0571259 + 0.0989449i
\(396\) 0 0
\(397\) −711.358 + 1232.11i −0.0899296 + 0.155763i −0.907481 0.420093i \(-0.861998\pi\)
0.817552 + 0.575855i \(0.195331\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2004.64 3472.14i 0.249643 0.432395i −0.713783 0.700366i \(-0.753020\pi\)
0.963427 + 0.267971i \(0.0863533\pi\)
\(402\) 0 0
\(403\) 8171.32 + 14153.1i 1.01003 + 1.74942i
\(404\) 0 0
\(405\) −3606.23 −0.442457
\(406\) 0 0
\(407\) 2279.15 0.277576
\(408\) 0 0
\(409\) 84.2472 + 145.920i 0.0101852 + 0.0176413i 0.871073 0.491153i \(-0.163425\pi\)
−0.860888 + 0.508795i \(0.830091\pi\)
\(410\) 0 0
\(411\) 428.239 741.732i 0.0513953 0.0890193i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1770.27 3066.20i 0.209395 0.362684i
\(416\) 0 0
\(417\) 423.472 + 733.474i 0.0497302 + 0.0861352i
\(418\) 0 0
\(419\) 2861.16 0.333596 0.166798 0.985991i \(-0.446657\pi\)
0.166798 + 0.985991i \(0.446657\pi\)
\(420\) 0 0
\(421\) 1135.73 0.131477 0.0657387 0.997837i \(-0.479060\pi\)
0.0657387 + 0.997837i \(0.479060\pi\)
\(422\) 0 0
\(423\) −6077.76 10527.0i −0.698607 1.21002i
\(424\) 0 0
\(425\) 345.823 598.983i 0.0394703 0.0683645i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 235.812 408.438i 0.0265387 0.0459664i
\(430\) 0 0
\(431\) 8150.59 + 14117.2i 0.910905 + 1.57773i 0.812788 + 0.582560i \(0.197949\pi\)
0.0981175 + 0.995175i \(0.468718\pi\)
\(432\) 0 0
\(433\) 5065.77 0.562230 0.281115 0.959674i \(-0.409296\pi\)
0.281115 + 0.959674i \(0.409296\pi\)
\(434\) 0 0
\(435\) −323.957 −0.0357070
\(436\) 0 0
\(437\) 1451.65 + 2514.34i 0.158906 + 0.275234i
\(438\) 0 0
\(439\) −5026.07 + 8705.41i −0.546427 + 0.946439i 0.452089 + 0.891973i \(0.350679\pi\)
−0.998516 + 0.0544661i \(0.982654\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8646.73 14976.6i 0.927356 1.60623i 0.139628 0.990204i \(-0.455409\pi\)
0.787728 0.616024i \(-0.211257\pi\)
\(444\) 0 0
\(445\) −2172.23 3762.41i −0.231401 0.400799i
\(446\) 0 0
\(447\) −826.058 −0.0874076
\(448\) 0 0
\(449\) 10414.5 1.09463 0.547315 0.836927i \(-0.315650\pi\)
0.547315 + 0.836927i \(0.315650\pi\)
\(450\) 0 0
\(451\) 1876.35 + 3249.94i 0.195907 + 0.339321i
\(452\) 0 0
\(453\) 357.037 618.407i 0.0370311 0.0641397i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −849.474 + 1471.33i −0.0869513 + 0.150604i −0.906221 0.422804i \(-0.861046\pi\)
0.819270 + 0.573408i \(0.194379\pi\)
\(458\) 0 0
\(459\) 230.842 + 399.830i 0.0234745 + 0.0406590i
\(460\) 0 0
\(461\) 3640.68 0.367817 0.183908 0.982943i \(-0.441125\pi\)
0.183908 + 0.982943i \(0.441125\pi\)
\(462\) 0 0
\(463\) 2701.96 0.271211 0.135605 0.990763i \(-0.456702\pi\)
0.135605 + 0.990763i \(0.456702\pi\)
\(464\) 0 0
\(465\) 137.986 + 238.998i 0.0137611 + 0.0238350i
\(466\) 0 0
\(467\) 6223.49 10779.4i 0.616678 1.06812i −0.373409 0.927667i \(-0.621811\pi\)
0.990088 0.140452i \(-0.0448554\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 306.739 531.287i 0.0300080 0.0519754i
\(472\) 0 0
\(473\) −4176.31 7233.58i −0.405977 0.703172i
\(474\) 0 0
\(475\) −1965.48 −0.189858
\(476\) 0 0
\(477\) −13645.5 −1.30982
\(478\) 0 0
\(479\) 111.809 + 193.659i 0.0106653 + 0.0184729i 0.871309 0.490735i \(-0.163272\pi\)
−0.860643 + 0.509208i \(0.829938\pi\)
\(480\) 0 0
\(481\) −6285.52 + 10886.8i −0.595832 + 1.03201i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2002.86 3469.06i 0.187516 0.324787i
\(486\) 0 0
\(487\) 3070.46 + 5318.18i 0.285699 + 0.494846i 0.972779 0.231737i \(-0.0744408\pi\)
−0.687079 + 0.726583i \(0.741107\pi\)
\(488\) 0 0
\(489\) 335.329 0.0310105
\(490\) 0 0
\(491\) −16011.3 −1.47165 −0.735827 0.677170i \(-0.763206\pi\)
−0.735827 + 0.677170i \(0.763206\pi\)
\(492\) 0 0
\(493\) −2895.02 5014.32i −0.264473 0.458081i
\(494\) 0 0
\(495\) −1117.81 + 1936.11i −0.101499 + 0.175801i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9323.69 + 16149.1i −0.836444 + 1.44876i 0.0564050 + 0.998408i \(0.482036\pi\)
−0.892849 + 0.450356i \(0.851297\pi\)
\(500\) 0 0
\(501\) −186.335 322.741i −0.0166164 0.0287805i
\(502\) 0 0
\(503\) 149.768 0.0132760 0.00663800 0.999978i \(-0.497887\pi\)
0.00663800 + 0.999978i \(0.497887\pi\)
\(504\) 0 0
\(505\) 4717.77 0.415719
\(506\) 0 0
\(507\) 960.581 + 1663.78i 0.0841438 + 0.145741i
\(508\) 0 0
\(509\) −6014.68 + 10417.7i −0.523764 + 0.907186i 0.475853 + 0.879525i \(0.342139\pi\)
−0.999617 + 0.0276612i \(0.991194\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 655.994 1136.21i 0.0564577 0.0977877i
\(514\) 0 0
\(515\) −422.924 732.526i −0.0361869 0.0626776i
\(516\) 0 0
\(517\) −7508.68 −0.638745
\(518\) 0 0
\(519\) 830.860 0.0702711
\(520\) 0 0
\(521\) 6052.09 + 10482.5i 0.508919 + 0.881474i 0.999947 + 0.0103300i \(0.00328820\pi\)
−0.491027 + 0.871144i \(0.663378\pi\)
\(522\) 0 0
\(523\) 2081.15 3604.65i 0.174000 0.301377i −0.765815 0.643061i \(-0.777664\pi\)
0.939815 + 0.341684i \(0.110997\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2466.20 + 4271.58i −0.203851 + 0.353080i
\(528\) 0 0
\(529\) 5401.63 + 9355.90i 0.443958 + 0.768957i
\(530\) 0 0
\(531\) −18462.6 −1.50887
\(532\) 0 0
\(533\) −20698.7 −1.68210
\(534\) 0 0
\(535\) 3937.67 + 6820.24i 0.318206 + 0.551149i
\(536\) 0 0
\(537\) 9.64123 16.6991i 0.000774766 0.00134193i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5551.37 + 9615.26i −0.441168 + 0.764126i −0.997776 0.0666491i \(-0.978769\pi\)
0.556608 + 0.830775i \(0.312103\pi\)
\(542\) 0 0
\(543\) −105.293 182.373i −0.00832148 0.0144132i
\(544\) 0 0
\(545\) −4585.25 −0.360386
\(546\) 0 0
\(547\) −17160.3 −1.34135 −0.670677 0.741749i \(-0.733996\pi\)
−0.670677 + 0.741749i \(0.733996\pi\)
\(548\) 0 0
\(549\) 7424.19 + 12859.1i 0.577152 + 0.999657i
\(550\) 0 0
\(551\) −8226.91 + 14249.4i −0.636076 + 1.10172i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −106.141 + 183.841i −0.00811789 + 0.0140606i
\(556\) 0 0
\(557\) −11108.1 19239.7i −0.844997 1.46358i −0.885624 0.464403i \(-0.846269\pi\)
0.0406273 0.999174i \(-0.487064\pi\)
\(558\) 0 0
\(559\) 46070.3 3.48581
\(560\) 0 0
\(561\) 142.342 0.0107124
\(562\) 0 0
\(563\) −4431.18 7675.03i −0.331709 0.574536i 0.651138 0.758959i \(-0.274292\pi\)
−0.982847 + 0.184423i \(0.940958\pi\)
\(564\) 0 0
\(565\) −227.260 + 393.627i −0.0169220 + 0.0293097i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10475.3 + 18143.8i −0.771789 + 1.33678i 0.164793 + 0.986328i \(0.447304\pi\)
−0.936582 + 0.350449i \(0.886029\pi\)
\(570\) 0 0
\(571\) 3895.39 + 6747.02i 0.285494 + 0.494490i 0.972729 0.231945i \(-0.0745090\pi\)
−0.687235 + 0.726435i \(0.741176\pi\)
\(572\) 0 0
\(573\) 601.868 0.0438803
\(574\) 0 0
\(575\) 923.219 0.0669581
\(576\) 0 0
\(577\) −1231.40 2132.84i −0.0888454 0.153885i 0.818178 0.574965i \(-0.194984\pi\)
−0.907023 + 0.421080i \(0.861651\pi\)
\(578\) 0 0
\(579\) 105.393 182.547i 0.00756476 0.0131026i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4214.53 + 7299.78i −0.299396 + 0.518570i
\(584\) 0 0
\(585\) −6165.48 10678.9i −0.435746 0.754734i
\(586\) 0 0
\(587\) −20438.3 −1.43710 −0.718551 0.695475i \(-0.755194\pi\)
−0.718551 + 0.695475i \(0.755194\pi\)
\(588\) 0 0
\(589\) 14016.6 0.980551
\(590\) 0 0
\(591\) 50.0379 + 86.6682i 0.00348271 + 0.00603224i
\(592\) 0 0
\(593\) −1341.34 + 2323.27i −0.0928873 + 0.160886i −0.908725 0.417396i \(-0.862943\pi\)
0.815838 + 0.578281i \(0.196276\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −438.149 + 758.897i −0.0300373 + 0.0520261i
\(598\) 0 0
\(599\) 12842.9 + 22244.6i 0.876039 + 1.51734i 0.855652 + 0.517552i \(0.173157\pi\)
0.0203871 + 0.999792i \(0.493510\pi\)
\(600\) 0 0
\(601\) −6960.19 −0.472400 −0.236200 0.971705i \(-0.575902\pi\)
−0.236200 + 0.971705i \(0.575902\pi\)
\(602\) 0 0
\(603\) 4885.58 0.329944
\(604\) 0 0
\(605\) −2637.01 4567.43i −0.177206 0.306930i
\(606\) 0 0
\(607\) 13746.5 23809.7i 0.919199 1.59210i 0.118565 0.992946i \(-0.462171\pi\)
0.800634 0.599153i \(-0.204496\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20707.7 35866.8i 1.37110 2.37482i
\(612\) 0 0
\(613\) 7280.71 + 12610.6i 0.479715 + 0.830890i 0.999729 0.0232673i \(-0.00740687\pi\)
−0.520015 + 0.854157i \(0.674074\pi\)
\(614\) 0 0
\(615\) −349.530 −0.0229177
\(616\) 0 0
\(617\) −15433.9 −1.00704 −0.503522 0.863983i \(-0.667963\pi\)
−0.503522 + 0.863983i \(0.667963\pi\)
\(618\) 0 0
\(619\) 3792.25 + 6568.37i 0.246241 + 0.426503i 0.962480 0.271353i \(-0.0874710\pi\)
−0.716238 + 0.697856i \(0.754138\pi\)
\(620\) 0 0
\(621\) −308.131 + 533.699i −0.0199112 + 0.0344873i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −312.500 + 541.266i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −202.249 350.306i −0.0128821 0.0223124i
\(628\) 0 0
\(629\) −3794.09 −0.240509
\(630\) 0 0
\(631\) 14755.4 0.930905 0.465453 0.885073i \(-0.345892\pi\)
0.465453 + 0.885073i \(0.345892\pi\)
\(632\) 0 0
\(633\) 385.244 + 667.261i 0.0241897 + 0.0418977i
\(634\) 0 0
\(635\) −751.904 + 1302.34i −0.0469896 + 0.0813884i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 11202.9 19404.0i 0.693552 1.20127i
\(640\) 0 0
\(641\) −14418.3 24973.2i −0.888438 1.53882i −0.841722 0.539911i \(-0.818458\pi\)
−0.0467161 0.998908i \(-0.514876\pi\)
\(642\) 0 0
\(643\) 24838.4 1.52338 0.761689 0.647943i \(-0.224371\pi\)
0.761689 + 0.647943i \(0.224371\pi\)
\(644\) 0 0
\(645\) 777.969 0.0474922
\(646\) 0 0
\(647\) −3594.13 6225.22i −0.218392 0.378267i 0.735924 0.677064i \(-0.236748\pi\)
−0.954317 + 0.298797i \(0.903415\pi\)
\(648\) 0 0
\(649\) −5702.33 + 9876.73i −0.344894 + 0.597374i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3585.82 + 6210.83i −0.214891 + 0.372203i −0.953239 0.302218i \(-0.902273\pi\)
0.738348 + 0.674420i \(0.235606\pi\)
\(654\) 0 0
\(655\) −2302.36 3987.80i −0.137344 0.237887i
\(656\) 0 0
\(657\) −9189.45 −0.545684
\(658\) 0 0
\(659\) 8236.07 0.486847 0.243423 0.969920i \(-0.421730\pi\)
0.243423 + 0.969920i \(0.421730\pi\)
\(660\) 0 0
\(661\) −8487.12 14700.1i −0.499411 0.865005i 0.500589 0.865685i \(-0.333117\pi\)
−1.00000 0.000680243i \(0.999783\pi\)
\(662\) 0 0
\(663\) −392.554 + 679.924i −0.0229948 + 0.0398282i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3864.32 6693.19i 0.224328 0.388548i
\(668\) 0 0
\(669\) −308.707 534.696i −0.0178405 0.0309007i
\(670\) 0 0
\(671\) 9172.10 0.527698
\(672\) 0 0
\(673\) 16220.1 0.929032 0.464516 0.885565i \(-0.346228\pi\)
0.464516 + 0.885565i \(0.346228\pi\)
\(674\) 0 0
\(675\) −208.598 361.303i −0.0118948 0.0206023i
\(676\) 0 0
\(677\) 3200.14 5542.80i 0.181671 0.314664i −0.760779 0.649011i \(-0.775183\pi\)
0.942450 + 0.334348i \(0.108516\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −275.280 + 476.798i −0.0154901 + 0.0268296i
\(682\) 0 0
\(683\) 8278.16 + 14338.2i 0.463770 + 0.803273i 0.999145 0.0413413i \(-0.0131631\pi\)
−0.535375 + 0.844614i \(0.679830\pi\)
\(684\) 0 0
\(685\) −13832.7 −0.771564
\(686\) 0 0
\(687\) −313.192 −0.0173930
\(688\) 0 0
\(689\) −23246.0 40263.2i −1.28534 2.22628i
\(690\) 0 0
\(691\) −3728.61 + 6458.15i −0.205272 + 0.355542i −0.950219 0.311581i \(-0.899141\pi\)
0.744947 + 0.667124i \(0.232475\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6839.36 11846.1i 0.373283 0.646545i
\(696\) 0 0
\(697\) −3123.55 5410.15i −0.169746 0.294009i
\(698\) 0 0
\(699\) −1017.05 −0.0550337
\(700\) 0 0
\(701\) 26830.5 1.44561 0.722807 0.691050i \(-0.242852\pi\)
0.722807 + 0.691050i \(0.242852\pi\)
\(702\) 0 0
\(703\) 5390.91 + 9337.34i 0.289221 + 0.500945i
\(704\) 0 0
\(705\) 349.682 605.667i 0.0186805 0.0323556i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3604.94 + 6243.95i −0.190954 + 0.330742i −0.945567 0.325428i \(-0.894492\pi\)
0.754613 + 0.656171i \(0.227825\pi\)
\(710\) 0 0
\(711\) 2413.11 + 4179.63i 0.127284 + 0.220462i
\(712\) 0 0
\(713\) −6583.84 −0.345816
\(714\) 0 0
\(715\) −7617.05 −0.398408
\(716\) 0 0
\(717\) −519.448 899.710i −0.0270560 0.0468623i
\(718\) 0 0
\(719\) 13975.8 24206.8i 0.724908 1.25558i −0.234104 0.972211i \(-0.575216\pi\)
0.959012 0.283365i \(-0.0914509\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 447.777 775.572i 0.0230332 0.0398947i
\(724\) 0 0
\(725\) 2616.06 + 4531.15i 0.134011 + 0.232114i
\(726\) 0 0
\(727\) 2492.55 0.127158 0.0635788 0.997977i \(-0.479749\pi\)
0.0635788 + 0.997977i \(0.479749\pi\)
\(728\) 0 0
\(729\) −19265.0 −0.978765
\(730\) 0 0
\(731\) 6952.28 + 12041.7i 0.351764 + 0.609272i
\(732\) 0 0
\(733\) −5065.28 + 8773.32i −0.255239 + 0.442087i −0.964960 0.262395i \(-0.915488\pi\)
0.709721 + 0.704483i \(0.248821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1508.95 2613.59i 0.0754180 0.130628i
\(738\) 0 0
\(739\) −3660.25 6339.73i −0.182198 0.315576i 0.760431 0.649419i \(-0.224988\pi\)
−0.942629 + 0.333843i \(0.891655\pi\)
\(740\) 0 0
\(741\) 2231.08 0.110608
\(742\) 0 0
\(743\) 16974.1 0.838116 0.419058 0.907959i \(-0.362360\pi\)
0.419058 + 0.907959i \(0.362360\pi\)
\(744\) 0 0
\(745\) 6670.71 + 11554.0i 0.328048 + 0.568196i
\(746\) 0 0
\(747\) 9525.52 16498.7i 0.466560 0.808106i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −8857.32 + 15341.3i −0.430370 + 0.745423i −0.996905 0.0786146i \(-0.974950\pi\)
0.566535 + 0.824038i \(0.308284\pi\)
\(752\) 0 0
\(753\) 1005.78 + 1742.07i 0.0486757 + 0.0843088i
\(754\) 0 0
\(755\) −11532.8 −0.555923
\(756\) 0 0
\(757\) 30171.9 1.44864 0.724318 0.689467i \(-0.242155\pi\)
0.724318 + 0.689467i \(0.242155\pi\)
\(758\) 0 0
\(759\) 94.9998 + 164.545i 0.00454318 + 0.00786902i
\(760\) 0 0
\(761\) 7211.78 12491.2i 0.343531 0.595013i −0.641555 0.767077i \(-0.721710\pi\)
0.985086 + 0.172064i \(0.0550437\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1860.81 3223.02i 0.0879449 0.152325i
\(766\) 0 0
\(767\) −31452.2 54476.8i −1.48067 2.56459i
\(768\) 0 0
\(769\) 21410.8 1.00402 0.502010 0.864862i \(-0.332594\pi\)
0.502010 + 0.864862i \(0.332594\pi\)
\(770\) 0 0
\(771\) −699.268 −0.0326634
\(772\) 0 0
\(773\) 13566.1 + 23497.1i 0.631227 + 1.09332i 0.987301 + 0.158859i \(0.0507815\pi\)
−0.356075 + 0.934457i \(0.615885\pi\)
\(774\) 0 0
\(775\) 2228.56 3859.98i 0.103293 0.178909i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8876.33 + 15374.3i −0.408251 + 0.707112i
\(780\) 0 0
\(781\) −6920.23 11986.2i −0.317062 0.549167i
\(782\) 0 0
\(783\) −3492.52 −0.159403
\(784\) 0 0
\(785\) −9908.08 −0.450490
\(786\) 0 0
\(787\) 6543.76 + 11334.1i 0.296391 + 0.513364i 0.975308 0.220851i \(-0.0708835\pi\)
−0.678917 + 0.734215i \(0.737550\pi\)
\(788\) 0 0
\(789\) 20.8814 36.1677i 0.000942204 0.00163194i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −25295.1 + 43812.5i −1.13273 + 1.96195i
\(794\) 0 0
\(795\) −392.544 679.907i −0.0175121 0.0303318i
\(796\) 0 0
\(797\) −19398.5 −0.862147 −0.431074 0.902317i \(-0.641865\pi\)
−0.431074 + 0.902317i \(0.641865\pi\)
\(798\) 0 0
\(799\) 12499.6 0.553449
\(800\) 0 0
\(801\) −11688.4 20244.9i −0.515592 0.893032i
\(802\) 0 0
\(803\) −2838.24 + 4915.98i −0.124731 + 0.216041i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1010.88 1750.89i 0.0440949 0.0763747i
\(808\) 0 0
\(809\) 10792.0 + 18692.4i 0.469008 + 0.812346i 0.999372 0.0354238i \(-0.0112781\pi\)
−0.530364 + 0.847770i \(0.677945\pi\)
\(810\) 0 0
\(811\) 26798.2 1.16031 0.580156 0.814505i \(-0.302991\pi\)
0.580156 + 0.814505i \(0.302991\pi\)
\(812\) 0 0
\(813\) 664.827 0.0286796
\(814\) 0 0
\(815\) −2707.90 4690.22i −0.116385 0.201584i
\(816\) 0 0
\(817\) 19756.6 34219.4i 0.846016 1.46534i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1115.88 + 1932.75i −0.0474352 + 0.0821602i −0.888768 0.458357i \(-0.848438\pi\)
0.841333 + 0.540517i \(0.181771\pi\)
\(822\) 0 0
\(823\) −15084.3 26126.7i −0.638888 1.10659i −0.985677 0.168643i \(-0.946061\pi\)
0.346789 0.937943i \(-0.387272\pi\)
\(824\) 0 0
\(825\) −128.626 −0.00542809
\(826\) 0 0
\(827\) −18201.0 −0.765310 −0.382655 0.923891i \(-0.624990\pi\)
−0.382655 + 0.923891i \(0.624990\pi\)
\(828\) 0 0
\(829\) 2428.84 + 4206.87i 0.101758 + 0.176249i 0.912409 0.409280i \(-0.134220\pi\)
−0.810651 + 0.585529i \(0.800887\pi\)
\(830\) 0 0
\(831\) 1211.72 2098.76i 0.0505825 0.0876115i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −3009.43 + 5212.49i −0.124725 + 0.216031i
\(836\) 0 0
\(837\) 1487.60 + 2576.60i 0.0614324 + 0.106404i
\(838\) 0 0
\(839\) 27903.8 1.14821 0.574104 0.818783i \(-0.305351\pi\)
0.574104 + 0.818783i \(0.305351\pi\)
\(840\) 0 0
\(841\) 19411.2 0.795900
\(842\) 0 0
\(843\) −918.613 1591.08i −0.0375311 0.0650057i
\(844\) 0 0
\(845\) 15514.1 26871.1i 0.631597 1.09396i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −810.844 + 1404.42i −0.0327775 + 0.0567723i
\(850\) 0 0
\(851\) −2532.20 4385.91i −0.102001 0.176671i
\(852\) 0 0
\(853\) −14219.1 −0.570755 −0.285377 0.958415i \(-0.592119\pi\)
−0.285377 + 0.958415i \(0.592119\pi\)
\(854\) 0 0
\(855\) −10575.9 −0.423028
\(856\) 0 0
\(857\) −7293.37 12632.5i −0.290708 0.503521i 0.683269 0.730166i \(-0.260557\pi\)
−0.973977 + 0.226646i \(0.927224\pi\)
\(858\) 0 0
\(859\) −13305.2 + 23045.2i −0.528482 + 0.915358i 0.470966 + 0.882151i \(0.343905\pi\)
−0.999448 + 0.0332070i \(0.989428\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2530.39 4382.77i 0.0998095 0.172875i −0.811796 0.583941i \(-0.801510\pi\)
0.911606 + 0.411066i \(0.134843\pi\)
\(864\) 0 0
\(865\) −6709.48 11621.2i −0.263733 0.456799i
\(866\) 0 0
\(867\) 1284.03 0.0502976
\(868\) 0 0
\(869\) 2981.24 0.116377
\(870\) 0 0
\(871\) 8322.89 + 14415.7i 0.323778 + 0.560800i
\(872\) 0 0
\(873\) 10777.1 18666.4i 0.417810 0.723669i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13807.6 23915.5i 0.531643 0.920832i −0.467675 0.883900i \(-0.654908\pi\)
0.999318 0.0369317i \(-0.0117584\pi\)
\(878\) 0 0
\(879\) 551.827 + 955.792i 0.0211748 + 0.0366758i
\(880\) 0 0
\(881\) 40580.4 1.55186 0.775931 0.630818i \(-0.217281\pi\)
0.775931 + 0.630818i \(0.217281\pi\)
\(882\) 0 0
\(883\) 1567.20 0.0597289 0.0298645 0.999554i \(-0.490492\pi\)
0.0298645 + 0.999554i \(0.490492\pi\)
\(884\) 0 0
\(885\) −531.119 919.926i −0.0201733 0.0349412i
\(886\) 0 0
\(887\) −7576.55 + 13123.0i −0.286805 + 0.496760i −0.973045 0.230614i \(-0.925926\pi\)
0.686241 + 0.727375i \(0.259260\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5993.25 + 10380.6i −0.225344 + 0.390307i
\(892\) 0 0
\(893\) −17760.4 30761.9i −0.665542 1.15275i
\(894\) 0 0
\(895\) −311.425 −0.0116310
\(896\) 0 0
\(897\) −1047.97 −0.0390088
\(898\) 0 0
\(899\) −18656.2 32313.5i −0.692123 1.19879i
\(900\) 0 0
\(901\) 7015.90 12151.9i 0.259416 0.449321i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1700.56 + 2945.45i −0.0624624 + 0.108188i
\(906\) 0 0
\(907\) −22828.8 39540.6i −0.835741 1.44755i −0.893425 0.449212i \(-0.851705\pi\)
0.0576841 0.998335i \(-0.481628\pi\)
\(908\) 0 0
\(909\) 25385.5 0.926277
\(910\) 0 0
\(911\) −12137.4 −0.441417 −0.220709 0.975340i \(-0.570837\pi\)
−0.220709 + 0.975340i \(0.570837\pi\)
\(912\) 0 0
\(913\) −5884.08 10191.5i −0.213291 0.369431i
\(914\) 0 0
\(915\) −427.148 + 739.842i −0.0154329 + 0.0267305i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 11352.0 19662.2i 0.407473 0.705763i −0.587133 0.809490i \(-0.699744\pi\)
0.994606 + 0.103727i \(0.0330769\pi\)
\(920\) 0 0
\(921\) −899.204 1557.47i −0.0321713 0.0557224i
\(922\) 0 0
\(923\) 76339.4 2.72236
\(924\) 0 0
\(925\) 3428.50 0.121868
\(926\) 0 0
\(927\) −2275.68 3941.60i −0.0806292 0.139654i
\(928\) 0 0
\(929\) −5669.56 + 9819.97i −0.200229 + 0.346806i −0.948602 0.316472i \(-0.897502\pi\)
0.748373 + 0.663278i \(0.230835\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −215.847 + 373.858i −0.00757398 + 0.0131185i
\(934\) 0 0
\(935\) −1149.46 1990.92i −0.0402046 0.0696364i
\(936\) 0 0
\(937\) −52101.8 −1.81653 −0.908267 0.418391i \(-0.862594\pi\)
−0.908267 + 0.418391i \(0.862594\pi\)
\(938\) 0 0
\(939\) −1839.73 −0.0639375
\(940\) 0 0
\(941\) −22432.9 38855.0i −0.777144 1.34605i −0.933582 0.358365i \(-0.883334\pi\)
0.156438 0.987688i \(-0.449999\pi\)
\(942\) 0 0
\(943\) 4169.36 7221.55i 0.143980 0.249381i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16044.2 27789.4i 0.550547 0.953575i −0.447688 0.894190i \(-0.647753\pi\)
0.998235 0.0593855i \(-0.0189141\pi\)
\(948\) 0 0
\(949\) −15654.8 27114.9i −0.535486 0.927489i
\(950\) 0 0
\(951\) −446.702 −0.0152317
\(952\) 0 0
\(953\) 17825.8 0.605913 0.302957 0.953004i \(-0.402026\pi\)
0.302957 + 0.953004i \(0.402026\pi\)
\(954\) 0 0
\(955\) −4860.29 8418.27i −0.164686 0.285245i
\(956\) 0 0
\(957\) −538.389 + 932.517i −0.0181856 + 0.0314984i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −997.269 + 1727.32i −0.0334755 + 0.0579813i
\(962\) 0 0
\(963\) 21187.9 + 36698.5i 0.709004 + 1.22803i
\(964\) 0 0
\(965\) −3404.35 −0.113565
\(966\) 0 0
\(967\) 34128.6 1.13495 0.567477 0.823389i \(-0.307920\pi\)
0.567477 + 0.823389i \(0.307920\pi\)
\(968\) 0 0
\(969\) 336.683 + 583.152i 0.0111618 + 0.0193328i
\(970\) 0 0
\(971\) 21040.9 36443.9i 0.695401 1.20447i −0.274645 0.961546i \(-0.588560\pi\)
0.970045 0.242924i \(-0.0781065\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 354.729 614.408i 0.0116517 0.0201813i
\(976\) 0 0
\(977\) 400.588 + 693.839i 0.0131177 + 0.0227204i 0.872510 0.488597i \(-0.162491\pi\)
−0.859392 + 0.511317i \(0.829158\pi\)
\(978\) 0 0
\(979\) −14440.3 −0.471412
\(980\) 0 0
\(981\) −24672.4 −0.802987
\(982\) 0 0
\(983\) 4.48329 + 7.76529i 0.000145468 + 0.000251957i 0.866098 0.499874i \(-0.166620\pi\)
−0.865953 + 0.500126i \(0.833287\pi\)
\(984\) 0 0
\(985\) 808.147 1399.75i 0.0261418 0.0452790i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9280.00 + 16073.4i −0.298369 + 0.516790i
\(990\) 0 0
\(991\) 4405.91 + 7631.26i 0.141229 + 0.244617i 0.927960 0.372680i \(-0.121561\pi\)
−0.786730 + 0.617297i \(0.788228\pi\)
\(992\) 0 0
\(993\) −1823.73 −0.0582824
\(994\) 0 0
\(995\) 14152.8 0.450929
\(996\) 0 0
\(997\) 28762.5 + 49818.1i 0.913658 + 1.58250i 0.808854 + 0.588009i \(0.200088\pi\)
0.104804 + 0.994493i \(0.466578\pi\)
\(998\) 0 0
\(999\) −1144.29 + 1981.96i −0.0362399 + 0.0627693i
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 980.4.i.w.961.1 4
7.2 even 3 980.4.a.n.1.2 2
7.3 odd 6 140.4.i.c.81.2 4
7.4 even 3 inner 980.4.i.w.361.1 4
7.5 odd 6 980.4.a.u.1.1 2
7.6 odd 2 140.4.i.c.121.2 yes 4
21.17 even 6 1260.4.s.e.361.1 4
21.20 even 2 1260.4.s.e.541.1 4
28.3 even 6 560.4.q.l.81.1 4
28.27 even 2 560.4.q.l.401.1 4
35.3 even 12 700.4.r.f.249.2 8
35.13 even 4 700.4.r.f.149.3 8
35.17 even 12 700.4.r.f.249.3 8
35.24 odd 6 700.4.i.h.501.1 4
35.27 even 4 700.4.r.f.149.2 8
35.34 odd 2 700.4.i.h.401.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.4.i.c.81.2 4 7.3 odd 6
140.4.i.c.121.2 yes 4 7.6 odd 2
560.4.q.l.81.1 4 28.3 even 6
560.4.q.l.401.1 4 28.27 even 2
700.4.i.h.401.1 4 35.34 odd 2
700.4.i.h.501.1 4 35.24 odd 6
700.4.r.f.149.2 8 35.27 even 4
700.4.r.f.149.3 8 35.13 even 4
700.4.r.f.249.2 8 35.3 even 12
700.4.r.f.249.3 8 35.17 even 12
980.4.a.n.1.2 2 7.2 even 3
980.4.a.u.1.1 2 7.5 odd 6
980.4.i.w.361.1 4 7.4 even 3 inner
980.4.i.w.961.1 4 1.1 even 1 trivial
1260.4.s.e.361.1 4 21.17 even 6
1260.4.s.e.541.1 4 21.20 even 2