Properties

Label 320.5.p.b.193.1
Level $320$
Weight $5$
Character 320.193
Analytic conductor $33.078$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,5,Mod(193,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.193");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 320.193
Dual form 320.5.p.b.257.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-9.00000 - 9.00000i) q^{3} +(15.0000 + 20.0000i) q^{5} +(29.0000 - 29.0000i) q^{7} +81.0000i q^{9} +118.000 q^{11} +(-69.0000 - 69.0000i) q^{13} +(45.0000 - 315.000i) q^{15} +(-271.000 + 271.000i) q^{17} +280.000i q^{19} -522.000 q^{21} +(269.000 + 269.000i) q^{23} +(-175.000 + 600.000i) q^{25} +680.000i q^{29} +202.000 q^{31} +(-1062.00 - 1062.00i) q^{33} +(1015.00 + 145.000i) q^{35} +(651.000 - 651.000i) q^{37} +1242.00i q^{39} +1682.00 q^{41} +(-1089.00 - 1089.00i) q^{43} +(-1620.00 + 1215.00i) q^{45} +(1269.00 - 1269.00i) q^{47} +719.000i q^{49} +4878.00 q^{51} +(611.000 + 611.000i) q^{53} +(1770.00 + 2360.00i) q^{55} +(2520.00 - 2520.00i) q^{57} +1160.00i q^{59} +5598.00 q^{61} +(2349.00 + 2349.00i) q^{63} +(345.000 - 2415.00i) q^{65} +(751.000 - 751.000i) q^{67} -4842.00i q^{69} +6442.00 q^{71} +(-2951.00 - 2951.00i) q^{73} +(6975.00 - 3825.00i) q^{75} +(3422.00 - 3422.00i) q^{77} -10560.0i q^{79} +6561.00 q^{81} +(6231.00 + 6231.00i) q^{83} +(-9485.00 - 1355.00i) q^{85} +(6120.00 - 6120.00i) q^{87} +14480.0i q^{89} -4002.00 q^{91} +(-1818.00 - 1818.00i) q^{93} +(-5600.00 + 4200.00i) q^{95} +(-7311.00 + 7311.00i) q^{97} +9558.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{3} + 30 q^{5} + 58 q^{7} + 236 q^{11} - 138 q^{13} + 90 q^{15} - 542 q^{17} - 1044 q^{21} + 538 q^{23} - 350 q^{25} + 404 q^{31} - 2124 q^{33} + 2030 q^{35} + 1302 q^{37} + 3364 q^{41} - 2178 q^{43}+ \cdots - 14622 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −9.00000 9.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 15.0000 + 20.0000i 0.600000 + 0.800000i
\(6\) 0 0
\(7\) 29.0000 29.0000i 0.591837 0.591837i −0.346291 0.938127i \(-0.612559\pi\)
0.938127 + 0.346291i \(0.112559\pi\)
\(8\) 0 0
\(9\) 81.0000i 1.00000i
\(10\) 0 0
\(11\) 118.000 0.975207 0.487603 0.873065i \(-0.337871\pi\)
0.487603 + 0.873065i \(0.337871\pi\)
\(12\) 0 0
\(13\) −69.0000 69.0000i −0.408284 0.408284i 0.472856 0.881140i \(-0.343223\pi\)
−0.881140 + 0.472856i \(0.843223\pi\)
\(14\) 0 0
\(15\) 45.0000 315.000i 0.200000 1.40000i
\(16\) 0 0
\(17\) −271.000 + 271.000i −0.937716 + 0.937716i −0.998171 0.0604547i \(-0.980745\pi\)
0.0604547 + 0.998171i \(0.480745\pi\)
\(18\) 0 0
\(19\) 280.000i 0.775623i 0.921739 + 0.387812i \(0.126769\pi\)
−0.921739 + 0.387812i \(0.873231\pi\)
\(20\) 0 0
\(21\) −522.000 −1.18367
\(22\) 0 0
\(23\) 269.000 + 269.000i 0.508507 + 0.508507i 0.914068 0.405561i \(-0.132924\pi\)
−0.405561 + 0.914068i \(0.632924\pi\)
\(24\) 0 0
\(25\) −175.000 + 600.000i −0.280000 + 0.960000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 680.000i 0.808561i 0.914635 + 0.404281i \(0.132478\pi\)
−0.914635 + 0.404281i \(0.867522\pi\)
\(30\) 0 0
\(31\) 202.000 0.210198 0.105099 0.994462i \(-0.466484\pi\)
0.105099 + 0.994462i \(0.466484\pi\)
\(32\) 0 0
\(33\) −1062.00 1062.00i −0.975207 0.975207i
\(34\) 0 0
\(35\) 1015.00 + 145.000i 0.828571 + 0.118367i
\(36\) 0 0
\(37\) 651.000 651.000i 0.475530 0.475530i −0.428169 0.903699i \(-0.640841\pi\)
0.903699 + 0.428169i \(0.140841\pi\)
\(38\) 0 0
\(39\) 1242.00i 0.816568i
\(40\) 0 0
\(41\) 1682.00 1.00059 0.500297 0.865854i \(-0.333224\pi\)
0.500297 + 0.865854i \(0.333224\pi\)
\(42\) 0 0
\(43\) −1089.00 1089.00i −0.588967 0.588967i 0.348385 0.937352i \(-0.386730\pi\)
−0.937352 + 0.348385i \(0.886730\pi\)
\(44\) 0 0
\(45\) −1620.00 + 1215.00i −0.800000 + 0.600000i
\(46\) 0 0
\(47\) 1269.00 1269.00i 0.574468 0.574468i −0.358906 0.933374i \(-0.616850\pi\)
0.933374 + 0.358906i \(0.116850\pi\)
\(48\) 0 0
\(49\) 719.000i 0.299459i
\(50\) 0 0
\(51\) 4878.00 1.87543
\(52\) 0 0
\(53\) 611.000 + 611.000i 0.217515 + 0.217515i 0.807450 0.589935i \(-0.200847\pi\)
−0.589935 + 0.807450i \(0.700847\pi\)
\(54\) 0 0
\(55\) 1770.00 + 2360.00i 0.585124 + 0.780165i
\(56\) 0 0
\(57\) 2520.00 2520.00i 0.775623 0.775623i
\(58\) 0 0
\(59\) 1160.00i 0.333238i 0.986021 + 0.166619i \(0.0532849\pi\)
−0.986021 + 0.166619i \(0.946715\pi\)
\(60\) 0 0
\(61\) 5598.00 1.50443 0.752217 0.658915i \(-0.228984\pi\)
0.752217 + 0.658915i \(0.228984\pi\)
\(62\) 0 0
\(63\) 2349.00 + 2349.00i 0.591837 + 0.591837i
\(64\) 0 0
\(65\) 345.000 2415.00i 0.0816568 0.571598i
\(66\) 0 0
\(67\) 751.000 751.000i 0.167298 0.167298i −0.618493 0.785791i \(-0.712256\pi\)
0.785791 + 0.618493i \(0.212256\pi\)
\(68\) 0 0
\(69\) 4842.00i 1.01701i
\(70\) 0 0
\(71\) 6442.00 1.27792 0.638961 0.769240i \(-0.279365\pi\)
0.638961 + 0.769240i \(0.279365\pi\)
\(72\) 0 0
\(73\) −2951.00 2951.00i −0.553762 0.553762i 0.373762 0.927525i \(-0.378068\pi\)
−0.927525 + 0.373762i \(0.878068\pi\)
\(74\) 0 0
\(75\) 6975.00 3825.00i 1.24000 0.680000i
\(76\) 0 0
\(77\) 3422.00 3422.00i 0.577163 0.577163i
\(78\) 0 0
\(79\) 10560.0i 1.69204i −0.533154 0.846018i \(-0.678993\pi\)
0.533154 0.846018i \(-0.321007\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 6231.00 + 6231.00i 0.904485 + 0.904485i 0.995820 0.0913348i \(-0.0291134\pi\)
−0.0913348 + 0.995820i \(0.529113\pi\)
\(84\) 0 0
\(85\) −9485.00 1355.00i −1.31280 0.187543i
\(86\) 0 0
\(87\) 6120.00 6120.00i 0.808561 0.808561i
\(88\) 0 0
\(89\) 14480.0i 1.82805i 0.405656 + 0.914026i \(0.367043\pi\)
−0.405656 + 0.914026i \(0.632957\pi\)
\(90\) 0 0
\(91\) −4002.00 −0.483275
\(92\) 0 0
\(93\) −1818.00 1818.00i −0.210198 0.210198i
\(94\) 0 0
\(95\) −5600.00 + 4200.00i −0.620499 + 0.465374i
\(96\) 0 0
\(97\) −7311.00 + 7311.00i −0.777022 + 0.777022i −0.979323 0.202301i \(-0.935158\pi\)
0.202301 + 0.979323i \(0.435158\pi\)
\(98\) 0 0
\(99\) 9558.00i 0.975207i
\(100\) 0 0
\(101\) 878.000 0.0860700 0.0430350 0.999074i \(-0.486297\pi\)
0.0430350 + 0.999074i \(0.486297\pi\)
\(102\) 0 0
\(103\) 10429.0 + 10429.0i 0.983033 + 0.983033i 0.999858 0.0168252i \(-0.00535587\pi\)
−0.0168252 + 0.999858i \(0.505356\pi\)
\(104\) 0 0
\(105\) −7830.00 10440.0i −0.710204 0.946939i
\(106\) 0 0
\(107\) 4711.00 4711.00i 0.411477 0.411477i −0.470776 0.882253i \(-0.656026\pi\)
0.882253 + 0.470776i \(0.156026\pi\)
\(108\) 0 0
\(109\) 22040.0i 1.85506i 0.373745 + 0.927531i \(0.378073\pi\)
−0.373745 + 0.927531i \(0.621927\pi\)
\(110\) 0 0
\(111\) −11718.0 −0.951059
\(112\) 0 0
\(113\) −2111.00 2111.00i −0.165322 0.165322i 0.619597 0.784920i \(-0.287296\pi\)
−0.784920 + 0.619597i \(0.787296\pi\)
\(114\) 0 0
\(115\) −1345.00 + 9415.00i −0.101701 + 0.711909i
\(116\) 0 0
\(117\) 5589.00 5589.00i 0.408284 0.408284i
\(118\) 0 0
\(119\) 15718.0i 1.10995i
\(120\) 0 0
\(121\) −717.000 −0.0489721
\(122\) 0 0
\(123\) −15138.0 15138.0i −1.00059 1.00059i
\(124\) 0 0
\(125\) −14625.0 + 5500.00i −0.936000 + 0.352000i
\(126\) 0 0
\(127\) 5909.00 5909.00i 0.366359 0.366359i −0.499789 0.866147i \(-0.666589\pi\)
0.866147 + 0.499789i \(0.166589\pi\)
\(128\) 0 0
\(129\) 19602.0i 1.17793i
\(130\) 0 0
\(131\) 6358.00 0.370491 0.185246 0.982692i \(-0.440692\pi\)
0.185246 + 0.982692i \(0.440692\pi\)
\(132\) 0 0
\(133\) 8120.00 + 8120.00i 0.459042 + 0.459042i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20409.0 20409.0i 1.08738 1.08738i 0.0915804 0.995798i \(-0.470808\pi\)
0.995798 0.0915804i \(-0.0291919\pi\)
\(138\) 0 0
\(139\) 9400.00i 0.486517i −0.969961 0.243259i \(-0.921784\pi\)
0.969961 0.243259i \(-0.0782164\pi\)
\(140\) 0 0
\(141\) −22842.0 −1.14894
\(142\) 0 0
\(143\) −8142.00 8142.00i −0.398161 0.398161i
\(144\) 0 0
\(145\) −13600.0 + 10200.0i −0.646849 + 0.485137i
\(146\) 0 0
\(147\) 6471.00 6471.00i 0.299459 0.299459i
\(148\) 0 0
\(149\) 13800.0i 0.621594i 0.950476 + 0.310797i \(0.100596\pi\)
−0.950476 + 0.310797i \(0.899404\pi\)
\(150\) 0 0
\(151\) −18998.0 −0.833209 −0.416605 0.909088i \(-0.636780\pi\)
−0.416605 + 0.909088i \(0.636780\pi\)
\(152\) 0 0
\(153\) −21951.0 21951.0i −0.937716 0.937716i
\(154\) 0 0
\(155\) 3030.00 + 4040.00i 0.126119 + 0.168158i
\(156\) 0 0
\(157\) 16371.0 16371.0i 0.664165 0.664165i −0.292194 0.956359i \(-0.594385\pi\)
0.956359 + 0.292194i \(0.0943853\pi\)
\(158\) 0 0
\(159\) 10998.0i 0.435030i
\(160\) 0 0
\(161\) 15602.0 0.601906
\(162\) 0 0
\(163\) −20009.0 20009.0i −0.753096 0.753096i 0.221960 0.975056i \(-0.428755\pi\)
−0.975056 + 0.221960i \(0.928755\pi\)
\(164\) 0 0
\(165\) 5310.00 37170.0i 0.195041 1.36529i
\(166\) 0 0
\(167\) 1549.00 1549.00i 0.0555416 0.0555416i −0.678790 0.734332i \(-0.737496\pi\)
0.734332 + 0.678790i \(0.237496\pi\)
\(168\) 0 0
\(169\) 19039.0i 0.666608i
\(170\) 0 0
\(171\) −22680.0 −0.775623
\(172\) 0 0
\(173\) −2789.00 2789.00i −0.0931872 0.0931872i 0.658976 0.752164i \(-0.270990\pi\)
−0.752164 + 0.658976i \(0.770990\pi\)
\(174\) 0 0
\(175\) 12325.0 + 22475.0i 0.402449 + 0.733878i
\(176\) 0 0
\(177\) 10440.0 10440.0i 0.333238 0.333238i
\(178\) 0 0
\(179\) 2600.00i 0.0811460i −0.999177 0.0405730i \(-0.987082\pi\)
0.999177 0.0405730i \(-0.0129183\pi\)
\(180\) 0 0
\(181\) 44398.0 1.35521 0.677604 0.735427i \(-0.263018\pi\)
0.677604 + 0.735427i \(0.263018\pi\)
\(182\) 0 0
\(183\) −50382.0 50382.0i −1.50443 1.50443i
\(184\) 0 0
\(185\) 22785.0 + 3255.00i 0.665741 + 0.0951059i
\(186\) 0 0
\(187\) −31978.0 + 31978.0i −0.914467 + 0.914467i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14678.0 −0.402346 −0.201173 0.979556i \(-0.564475\pi\)
−0.201173 + 0.979556i \(0.564475\pi\)
\(192\) 0 0
\(193\) 42849.0 + 42849.0i 1.15034 + 1.15034i 0.986484 + 0.163855i \(0.0523930\pi\)
0.163855 + 0.986484i \(0.447607\pi\)
\(194\) 0 0
\(195\) −24840.0 + 18630.0i −0.653254 + 0.489941i
\(196\) 0 0
\(197\) 10971.0 10971.0i 0.282692 0.282692i −0.551490 0.834182i \(-0.685940\pi\)
0.834182 + 0.551490i \(0.185940\pi\)
\(198\) 0 0
\(199\) 38160.0i 0.963612i 0.876278 + 0.481806i \(0.160019\pi\)
−0.876278 + 0.481806i \(0.839981\pi\)
\(200\) 0 0
\(201\) −13518.0 −0.334596
\(202\) 0 0
\(203\) 19720.0 + 19720.0i 0.478536 + 0.478536i
\(204\) 0 0
\(205\) 25230.0 + 33640.0i 0.600357 + 0.800476i
\(206\) 0 0
\(207\) −21789.0 + 21789.0i −0.508507 + 0.508507i
\(208\) 0 0
\(209\) 33040.0i 0.756393i
\(210\) 0 0
\(211\) −72842.0 −1.63613 −0.818063 0.575128i \(-0.804952\pi\)
−0.818063 + 0.575128i \(0.804952\pi\)
\(212\) 0 0
\(213\) −57978.0 57978.0i −1.27792 1.27792i
\(214\) 0 0
\(215\) 5445.00 38115.0i 0.117793 0.824554i
\(216\) 0 0
\(217\) 5858.00 5858.00i 0.124403 0.124403i
\(218\) 0 0
\(219\) 53118.0i 1.10752i
\(220\) 0 0
\(221\) 37398.0 0.765709
\(222\) 0 0
\(223\) −30891.0 30891.0i −0.621187 0.621187i 0.324648 0.945835i \(-0.394754\pi\)
−0.945835 + 0.324648i \(0.894754\pi\)
\(224\) 0 0
\(225\) −48600.0 14175.0i −0.960000 0.280000i
\(226\) 0 0
\(227\) 54911.0 54911.0i 1.06563 1.06563i 0.0679438 0.997689i \(-0.478356\pi\)
0.997689 0.0679438i \(-0.0216439\pi\)
\(228\) 0 0
\(229\) 50280.0i 0.958792i −0.877599 0.479396i \(-0.840856\pi\)
0.877599 0.479396i \(-0.159144\pi\)
\(230\) 0 0
\(231\) −61596.0 −1.15433
\(232\) 0 0
\(233\) −2391.00 2391.00i −0.0440421 0.0440421i 0.684743 0.728785i \(-0.259915\pi\)
−0.728785 + 0.684743i \(0.759915\pi\)
\(234\) 0 0
\(235\) 44415.0 + 6345.00i 0.804255 + 0.114894i
\(236\) 0 0
\(237\) −95040.0 + 95040.0i −1.69204 + 1.69204i
\(238\) 0 0
\(239\) 17760.0i 0.310919i 0.987842 + 0.155459i \(0.0496858\pi\)
−0.987842 + 0.155459i \(0.950314\pi\)
\(240\) 0 0
\(241\) −28238.0 −0.486183 −0.243092 0.970003i \(-0.578162\pi\)
−0.243092 + 0.970003i \(0.578162\pi\)
\(242\) 0 0
\(243\) −59049.0 59049.0i −1.00000 1.00000i
\(244\) 0 0
\(245\) −14380.0 + 10785.0i −0.239567 + 0.179675i
\(246\) 0 0
\(247\) 19320.0 19320.0i 0.316675 0.316675i
\(248\) 0 0
\(249\) 112158.i 1.80897i
\(250\) 0 0
\(251\) −121002. −1.92064 −0.960318 0.278907i \(-0.910028\pi\)
−0.960318 + 0.278907i \(0.910028\pi\)
\(252\) 0 0
\(253\) 31742.0 + 31742.0i 0.495899 + 0.495899i
\(254\) 0 0
\(255\) 73170.0 + 97560.0i 1.12526 + 1.50035i
\(256\) 0 0
\(257\) −72431.0 + 72431.0i −1.09663 + 1.09663i −0.101823 + 0.994803i \(0.532467\pi\)
−0.994803 + 0.101823i \(0.967533\pi\)
\(258\) 0 0
\(259\) 37758.0i 0.562872i
\(260\) 0 0
\(261\) −55080.0 −0.808561
\(262\) 0 0
\(263\) −14771.0 14771.0i −0.213549 0.213549i 0.592224 0.805773i \(-0.298250\pi\)
−0.805773 + 0.592224i \(0.798250\pi\)
\(264\) 0 0
\(265\) −3055.00 + 21385.0i −0.0435030 + 0.304521i
\(266\) 0 0
\(267\) 130320. 130320.i 1.82805 1.82805i
\(268\) 0 0
\(269\) 89720.0i 1.23989i 0.784644 + 0.619947i \(0.212846\pi\)
−0.784644 + 0.619947i \(0.787154\pi\)
\(270\) 0 0
\(271\) 68202.0 0.928664 0.464332 0.885661i \(-0.346294\pi\)
0.464332 + 0.885661i \(0.346294\pi\)
\(272\) 0 0
\(273\) 36018.0 + 36018.0i 0.483275 + 0.483275i
\(274\) 0 0
\(275\) −20650.0 + 70800.0i −0.273058 + 0.936198i
\(276\) 0 0
\(277\) −18549.0 + 18549.0i −0.241747 + 0.241747i −0.817573 0.575826i \(-0.804681\pi\)
0.575826 + 0.817573i \(0.304681\pi\)
\(278\) 0 0
\(279\) 16362.0i 0.210198i
\(280\) 0 0
\(281\) 2322.00 0.0294069 0.0147035 0.999892i \(-0.495320\pi\)
0.0147035 + 0.999892i \(0.495320\pi\)
\(282\) 0 0
\(283\) 91711.0 + 91711.0i 1.14511 + 1.14511i 0.987501 + 0.157613i \(0.0503797\pi\)
0.157613 + 0.987501i \(0.449620\pi\)
\(284\) 0 0
\(285\) 88200.0 + 12600.0i 1.08587 + 0.155125i
\(286\) 0 0
\(287\) 48778.0 48778.0i 0.592189 0.592189i
\(288\) 0 0
\(289\) 63361.0i 0.758624i
\(290\) 0 0
\(291\) 131598. 1.55404
\(292\) 0 0
\(293\) 4851.00 + 4851.00i 0.0565062 + 0.0565062i 0.734795 0.678289i \(-0.237278\pi\)
−0.678289 + 0.734795i \(0.737278\pi\)
\(294\) 0 0
\(295\) −23200.0 + 17400.0i −0.266590 + 0.199943i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 37122.0i 0.415230i
\(300\) 0 0
\(301\) −63162.0 −0.697145
\(302\) 0 0
\(303\) −7902.00 7902.00i −0.0860700 0.0860700i
\(304\) 0 0
\(305\) 83970.0 + 111960.i 0.902661 + 1.20355i
\(306\) 0 0
\(307\) −42849.0 + 42849.0i −0.454636 + 0.454636i −0.896890 0.442254i \(-0.854179\pi\)
0.442254 + 0.896890i \(0.354179\pi\)
\(308\) 0 0
\(309\) 187722.i 1.96607i
\(310\) 0 0
\(311\) −72278.0 −0.747283 −0.373642 0.927573i \(-0.621891\pi\)
−0.373642 + 0.927573i \(0.621891\pi\)
\(312\) 0 0
\(313\) 18249.0 + 18249.0i 0.186273 + 0.186273i 0.794083 0.607810i \(-0.207952\pi\)
−0.607810 + 0.794083i \(0.707952\pi\)
\(314\) 0 0
\(315\) −11745.0 + 82215.0i −0.118367 + 0.828571i
\(316\) 0 0
\(317\) −25149.0 + 25149.0i −0.250266 + 0.250266i −0.821080 0.570814i \(-0.806628\pi\)
0.570814 + 0.821080i \(0.306628\pi\)
\(318\) 0 0
\(319\) 80240.0i 0.788514i
\(320\) 0 0
\(321\) −84798.0 −0.822954
\(322\) 0 0
\(323\) −75880.0 75880.0i −0.727315 0.727315i
\(324\) 0 0
\(325\) 53475.0 29325.0i 0.506272 0.277633i
\(326\) 0 0
\(327\) 198360. 198360.i 1.85506 1.85506i
\(328\) 0 0
\(329\) 73602.0i 0.679983i
\(330\) 0 0
\(331\) 54038.0 0.493223 0.246611 0.969114i \(-0.420683\pi\)
0.246611 + 0.969114i \(0.420683\pi\)
\(332\) 0 0
\(333\) 52731.0 + 52731.0i 0.475530 + 0.475530i
\(334\) 0 0
\(335\) 26285.0 + 3755.00i 0.234217 + 0.0334596i
\(336\) 0 0
\(337\) 8529.00 8529.00i 0.0750997 0.0750997i −0.668559 0.743659i \(-0.733089\pi\)
0.743659 + 0.668559i \(0.233089\pi\)
\(338\) 0 0
\(339\) 37998.0i 0.330645i
\(340\) 0 0
\(341\) 23836.0 0.204986
\(342\) 0 0
\(343\) 90480.0 + 90480.0i 0.769067 + 0.769067i
\(344\) 0 0
\(345\) 96840.0 72630.0i 0.813611 0.610208i
\(346\) 0 0
\(347\) 56551.0 56551.0i 0.469658 0.469658i −0.432146 0.901804i \(-0.642244\pi\)
0.901804 + 0.432146i \(0.142244\pi\)
\(348\) 0 0
\(349\) 22520.0i 0.184892i −0.995718 0.0924459i \(-0.970531\pi\)
0.995718 0.0924459i \(-0.0294685\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −44511.0 44511.0i −0.357205 0.357205i 0.505576 0.862782i \(-0.331280\pi\)
−0.862782 + 0.505576i \(0.831280\pi\)
\(354\) 0 0
\(355\) 96630.0 + 128840.i 0.766753 + 1.02234i
\(356\) 0 0
\(357\) 141462. 141462.i 1.10995 1.10995i
\(358\) 0 0
\(359\) 9680.00i 0.0751080i 0.999295 + 0.0375540i \(0.0119566\pi\)
−0.999295 + 0.0375540i \(0.988043\pi\)
\(360\) 0 0
\(361\) 51921.0 0.398409
\(362\) 0 0
\(363\) 6453.00 + 6453.00i 0.0489721 + 0.0489721i
\(364\) 0 0
\(365\) 14755.0 103285.i 0.110752 0.775267i
\(366\) 0 0
\(367\) −14971.0 + 14971.0i −0.111152 + 0.111152i −0.760496 0.649343i \(-0.775044\pi\)
0.649343 + 0.760496i \(0.275044\pi\)
\(368\) 0 0
\(369\) 136242.i 1.00059i
\(370\) 0 0
\(371\) 35438.0 0.257467
\(372\) 0 0
\(373\) 13811.0 + 13811.0i 0.0992676 + 0.0992676i 0.754996 0.655729i \(-0.227639\pi\)
−0.655729 + 0.754996i \(0.727639\pi\)
\(374\) 0 0
\(375\) 181125. + 82125.0i 1.28800 + 0.584000i
\(376\) 0 0
\(377\) 46920.0 46920.0i 0.330123 0.330123i
\(378\) 0 0
\(379\) 251080.i 1.74797i 0.485954 + 0.873984i \(0.338472\pi\)
−0.485954 + 0.873984i \(0.661528\pi\)
\(380\) 0 0
\(381\) −106362. −0.732717
\(382\) 0 0
\(383\) −86091.0 86091.0i −0.586895 0.586895i 0.349894 0.936789i \(-0.386217\pi\)
−0.936789 + 0.349894i \(0.886217\pi\)
\(384\) 0 0
\(385\) 119770. + 17110.0i 0.808028 + 0.115433i
\(386\) 0 0
\(387\) 88209.0 88209.0i 0.588967 0.588967i
\(388\) 0 0
\(389\) 75000.0i 0.495635i −0.968807 0.247818i \(-0.920287\pi\)
0.968807 0.247818i \(-0.0797134\pi\)
\(390\) 0 0
\(391\) −145798. −0.953670
\(392\) 0 0
\(393\) −57222.0 57222.0i −0.370491 0.370491i
\(394\) 0 0
\(395\) 211200. 158400.i 1.35363 1.01522i
\(396\) 0 0
\(397\) −29149.0 + 29149.0i −0.184945 + 0.184945i −0.793507 0.608562i \(-0.791747\pi\)
0.608562 + 0.793507i \(0.291747\pi\)
\(398\) 0 0
\(399\) 146160.i 0.918085i
\(400\) 0 0
\(401\) −45918.0 −0.285558 −0.142779 0.989755i \(-0.545604\pi\)
−0.142779 + 0.989755i \(0.545604\pi\)
\(402\) 0 0
\(403\) −13938.0 13938.0i −0.0858204 0.0858204i
\(404\) 0 0
\(405\) 98415.0 + 131220.i 0.600000 + 0.800000i
\(406\) 0 0
\(407\) 76818.0 76818.0i 0.463740 0.463740i
\(408\) 0 0
\(409\) 78720.0i 0.470585i −0.971925 0.235293i \(-0.924395\pi\)
0.971925 0.235293i \(-0.0756049\pi\)
\(410\) 0 0
\(411\) −367362. −2.17476
\(412\) 0 0
\(413\) 33640.0 + 33640.0i 0.197222 + 0.197222i
\(414\) 0 0
\(415\) −31155.0 + 218085.i −0.180897 + 1.26628i
\(416\) 0 0
\(417\) −84600.0 + 84600.0i −0.486517 + 0.486517i
\(418\) 0 0
\(419\) 14760.0i 0.0840733i −0.999116 0.0420367i \(-0.986615\pi\)
0.999116 0.0420367i \(-0.0133846\pi\)
\(420\) 0 0
\(421\) −221282. −1.24848 −0.624240 0.781232i \(-0.714591\pi\)
−0.624240 + 0.781232i \(0.714591\pi\)
\(422\) 0 0
\(423\) 102789. + 102789.i 0.574468 + 0.574468i
\(424\) 0 0
\(425\) −115175. 210025.i −0.637647 1.16277i
\(426\) 0 0
\(427\) 162342. 162342.i 0.890379 0.890379i
\(428\) 0 0
\(429\) 146556.i 0.796323i
\(430\) 0 0
\(431\) 212522. 1.14406 0.572031 0.820232i \(-0.306156\pi\)
0.572031 + 0.820232i \(0.306156\pi\)
\(432\) 0 0
\(433\) 145409. + 145409.i 0.775560 + 0.775560i 0.979072 0.203512i \(-0.0652357\pi\)
−0.203512 + 0.979072i \(0.565236\pi\)
\(434\) 0 0
\(435\) 214200. + 30600.0i 1.13199 + 0.161712i
\(436\) 0 0
\(437\) −75320.0 + 75320.0i −0.394410 + 0.394410i
\(438\) 0 0
\(439\) 299440.i 1.55375i 0.629656 + 0.776874i \(0.283196\pi\)
−0.629656 + 0.776874i \(0.716804\pi\)
\(440\) 0 0
\(441\) −58239.0 −0.299459
\(442\) 0 0
\(443\) −240609. 240609.i −1.22604 1.22604i −0.965450 0.260590i \(-0.916083\pi\)
−0.260590 0.965450i \(-0.583917\pi\)
\(444\) 0 0
\(445\) −289600. + 217200.i −1.46244 + 1.09683i
\(446\) 0 0
\(447\) 124200. 124200.i 0.621594 0.621594i
\(448\) 0 0
\(449\) 82480.0i 0.409125i 0.978854 + 0.204562i \(0.0655772\pi\)
−0.978854 + 0.204562i \(0.934423\pi\)
\(450\) 0 0
\(451\) 198476. 0.975787
\(452\) 0 0
\(453\) 170982. + 170982.i 0.833209 + 0.833209i
\(454\) 0 0
\(455\) −60030.0 80040.0i −0.289965 0.386620i
\(456\) 0 0
\(457\) −188151. + 188151.i −0.900895 + 0.900895i −0.995514 0.0946187i \(-0.969837\pi\)
0.0946187 + 0.995514i \(0.469837\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 326158. 1.53471 0.767355 0.641223i \(-0.221573\pi\)
0.767355 + 0.641223i \(0.221573\pi\)
\(462\) 0 0
\(463\) −218731. 218731.i −1.02035 1.02035i −0.999789 0.0205595i \(-0.993455\pi\)
−0.0205595 0.999789i \(-0.506545\pi\)
\(464\) 0 0
\(465\) 9090.00 63630.0i 0.0420395 0.294277i
\(466\) 0 0
\(467\) −59249.0 + 59249.0i −0.271673 + 0.271673i −0.829774 0.558100i \(-0.811530\pi\)
0.558100 + 0.829774i \(0.311530\pi\)
\(468\) 0 0
\(469\) 43558.0i 0.198026i
\(470\) 0 0
\(471\) −294678. −1.32833
\(472\) 0 0
\(473\) −128502. 128502.i −0.574365 0.574365i
\(474\) 0 0
\(475\) −168000. 49000.0i −0.744598 0.217175i
\(476\) 0 0
\(477\) −49491.0 + 49491.0i −0.217515 + 0.217515i
\(478\) 0 0
\(479\) 273440.i 1.19177i −0.803071 0.595883i \(-0.796802\pi\)
0.803071 0.595883i \(-0.203198\pi\)
\(480\) 0 0
\(481\) −89838.0 −0.388302
\(482\) 0 0
\(483\) −140418. 140418.i −0.601906 0.601906i
\(484\) 0 0
\(485\) −255885. 36555.0i −1.08783 0.155404i
\(486\) 0 0
\(487\) −123651. + 123651.i −0.521362 + 0.521362i −0.917983 0.396620i \(-0.870183\pi\)
0.396620 + 0.917983i \(0.370183\pi\)
\(488\) 0 0
\(489\) 360162.i 1.50619i
\(490\) 0 0
\(491\) −198442. −0.823134 −0.411567 0.911379i \(-0.635018\pi\)
−0.411567 + 0.911379i \(0.635018\pi\)
\(492\) 0 0
\(493\) −184280. 184280.i −0.758201 0.758201i
\(494\) 0 0
\(495\) −191160. + 143370.i −0.780165 + 0.585124i
\(496\) 0 0
\(497\) 186818. 186818.i 0.756321 0.756321i
\(498\) 0 0
\(499\) 269240.i 1.08128i 0.841254 + 0.540640i \(0.181818\pi\)
−0.841254 + 0.540640i \(0.818182\pi\)
\(500\) 0 0
\(501\) −27882.0 −0.111083
\(502\) 0 0
\(503\) 109869. + 109869.i 0.434249 + 0.434249i 0.890071 0.455822i \(-0.150655\pi\)
−0.455822 + 0.890071i \(0.650655\pi\)
\(504\) 0 0
\(505\) 13170.0 + 17560.0i 0.0516420 + 0.0688560i
\(506\) 0 0
\(507\) −171351. + 171351.i −0.666608 + 0.666608i
\(508\) 0 0
\(509\) 211000.i 0.814417i −0.913335 0.407209i \(-0.866502\pi\)
0.913335 0.407209i \(-0.133498\pi\)
\(510\) 0 0
\(511\) −171158. −0.655474
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −52145.0 + 365015.i −0.196607 + 1.37625i
\(516\) 0 0
\(517\) 149742. 149742.i 0.560225 0.560225i
\(518\) 0 0
\(519\) 50202.0i 0.186374i
\(520\) 0 0
\(521\) 297282. 1.09520 0.547600 0.836740i \(-0.315542\pi\)
0.547600 + 0.836740i \(0.315542\pi\)
\(522\) 0 0
\(523\) 25071.0 + 25071.0i 0.0916576 + 0.0916576i 0.751449 0.659791i \(-0.229355\pi\)
−0.659791 + 0.751449i \(0.729355\pi\)
\(524\) 0 0
\(525\) 91350.0 313200.i 0.331429 1.13633i
\(526\) 0 0
\(527\) −54742.0 + 54742.0i −0.197106 + 0.197106i
\(528\) 0 0
\(529\) 135119.i 0.482842i
\(530\) 0 0
\(531\) −93960.0 −0.333238
\(532\) 0 0
\(533\) −116058. 116058.i −0.408527 0.408527i
\(534\) 0 0
\(535\) 164885. + 23555.0i 0.576068 + 0.0822954i
\(536\) 0 0
\(537\) −23400.0 + 23400.0i −0.0811460 + 0.0811460i
\(538\) 0 0
\(539\) 84842.0i 0.292034i
\(540\) 0 0
\(541\) 142478. 0.486803 0.243402 0.969926i \(-0.421737\pi\)
0.243402 + 0.969926i \(0.421737\pi\)
\(542\) 0 0
\(543\) −399582. 399582.i −1.35521 1.35521i
\(544\) 0 0
\(545\) −440800. + 330600.i −1.48405 + 1.11304i
\(546\) 0 0
\(547\) −291009. + 291009.i −0.972594 + 0.972594i −0.999634 0.0270399i \(-0.991392\pi\)
0.0270399 + 0.999634i \(0.491392\pi\)
\(548\) 0 0
\(549\) 453438.i 1.50443i
\(550\) 0 0
\(551\) −190400. −0.627139
\(552\) 0 0
\(553\) −306240. 306240.i −1.00141 1.00141i
\(554\) 0 0
\(555\) −175770. 234360.i −0.570636 0.760847i
\(556\) 0 0
\(557\) 83091.0 83091.0i 0.267820 0.267820i −0.560401 0.828221i \(-0.689353\pi\)
0.828221 + 0.560401i \(0.189353\pi\)
\(558\) 0 0
\(559\) 150282.i 0.480932i
\(560\) 0 0
\(561\) 575604. 1.82893
\(562\) 0 0
\(563\) −43449.0 43449.0i −0.137076 0.137076i 0.635239 0.772316i \(-0.280902\pi\)
−0.772316 + 0.635239i \(0.780902\pi\)
\(564\) 0 0
\(565\) 10555.0 73885.0i 0.0330645 0.231451i
\(566\) 0 0
\(567\) 190269. 190269.i 0.591837 0.591837i
\(568\) 0 0
\(569\) 270560.i 0.835678i 0.908521 + 0.417839i \(0.137212\pi\)
−0.908521 + 0.417839i \(0.862788\pi\)
\(570\) 0 0
\(571\) −57482.0 −0.176303 −0.0881515 0.996107i \(-0.528096\pi\)
−0.0881515 + 0.996107i \(0.528096\pi\)
\(572\) 0 0
\(573\) 132102. + 132102.i 0.402346 + 0.402346i
\(574\) 0 0
\(575\) −208475. + 114325.i −0.630548 + 0.345784i
\(576\) 0 0
\(577\) 195889. 195889.i 0.588381 0.588381i −0.348812 0.937193i \(-0.613415\pi\)
0.937193 + 0.348812i \(0.113415\pi\)
\(578\) 0 0
\(579\) 771282.i 2.30068i
\(580\) 0 0
\(581\) 361398. 1.07062
\(582\) 0 0
\(583\) 72098.0 + 72098.0i 0.212122 + 0.212122i
\(584\) 0 0
\(585\) 195615. + 27945.0i 0.571598 + 0.0816568i
\(586\) 0 0
\(587\) 404631. 404631.i 1.17431 1.17431i 0.193139 0.981171i \(-0.438133\pi\)
0.981171 0.193139i \(-0.0618669\pi\)
\(588\) 0 0
\(589\) 56560.0i 0.163034i
\(590\) 0 0
\(591\) −197478. −0.565384
\(592\) 0 0
\(593\) −210991. 210991.i −0.600005 0.600005i 0.340309 0.940314i \(-0.389468\pi\)
−0.940314 + 0.340309i \(0.889468\pi\)
\(594\) 0 0
\(595\) −314360. + 235770.i −0.887960 + 0.665970i
\(596\) 0 0
\(597\) 343440. 343440.i 0.963612 0.963612i
\(598\) 0 0
\(599\) 300560.i 0.837679i −0.908060 0.418839i \(-0.862437\pi\)
0.908060 0.418839i \(-0.137563\pi\)
\(600\) 0 0
\(601\) 367442. 1.01728 0.508639 0.860980i \(-0.330149\pi\)
0.508639 + 0.860980i \(0.330149\pi\)
\(602\) 0 0
\(603\) 60831.0 + 60831.0i 0.167298 + 0.167298i
\(604\) 0 0
\(605\) −10755.0 14340.0i −0.0293832 0.0391777i
\(606\) 0 0
\(607\) 146469. 146469.i 0.397529 0.397529i −0.479832 0.877360i \(-0.659302\pi\)
0.877360 + 0.479832i \(0.159302\pi\)
\(608\) 0 0
\(609\) 354960.i 0.957072i
\(610\) 0 0
\(611\) −175122. −0.469092
\(612\) 0 0
\(613\) −160989. 160989.i −0.428425 0.428425i 0.459666 0.888092i \(-0.347969\pi\)
−0.888092 + 0.459666i \(0.847969\pi\)
\(614\) 0 0
\(615\) 75690.0 529830.i 0.200119 1.40083i
\(616\) 0 0
\(617\) 320409. 320409.i 0.841656 0.841656i −0.147419 0.989074i \(-0.547096\pi\)
0.989074 + 0.147419i \(0.0470965\pi\)
\(618\) 0 0
\(619\) 341160.i 0.890383i 0.895435 + 0.445191i \(0.146864\pi\)
−0.895435 + 0.445191i \(0.853136\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 419920. + 419920.i 1.08191 + 1.08191i
\(624\) 0 0
\(625\) −329375. 210000.i −0.843200 0.537600i
\(626\) 0 0
\(627\) 297360. 297360.i 0.756393 0.756393i
\(628\) 0 0
\(629\) 352842.i 0.891824i
\(630\) 0 0
\(631\) −390998. −0.982010 −0.491005 0.871157i \(-0.663370\pi\)
−0.491005 + 0.871157i \(0.663370\pi\)
\(632\) 0 0
\(633\) 655578. + 655578.i 1.63613 + 1.63613i
\(634\) 0 0
\(635\) 206815. + 29545.0i 0.512902 + 0.0732717i
\(636\) 0 0
\(637\) 49611.0 49611.0i 0.122264 0.122264i
\(638\) 0 0
\(639\) 521802.i 1.27792i
\(640\) 0 0
\(641\) −585038. −1.42386 −0.711931 0.702249i \(-0.752179\pi\)
−0.711931 + 0.702249i \(0.752179\pi\)
\(642\) 0 0
\(643\) 31911.0 + 31911.0i 0.0771824 + 0.0771824i 0.744644 0.667462i \(-0.232619\pi\)
−0.667462 + 0.744644i \(0.732619\pi\)
\(644\) 0 0
\(645\) −392040. + 294030.i −0.942347 + 0.706760i
\(646\) 0 0
\(647\) −280931. + 280931.i −0.671106 + 0.671106i −0.957971 0.286865i \(-0.907387\pi\)
0.286865 + 0.957971i \(0.407387\pi\)
\(648\) 0 0
\(649\) 136880.i 0.324975i
\(650\) 0 0
\(651\) −105444. −0.248805
\(652\) 0 0
\(653\) −523989. 523989.i −1.22884 1.22884i −0.964402 0.264439i \(-0.914813\pi\)
−0.264439 0.964402i \(-0.585187\pi\)
\(654\) 0 0
\(655\) 95370.0 + 127160.i 0.222295 + 0.296393i
\(656\) 0 0
\(657\) 239031. 239031.i 0.553762 0.553762i
\(658\) 0 0
\(659\) 404360.i 0.931102i −0.885021 0.465551i \(-0.845856\pi\)
0.885021 0.465551i \(-0.154144\pi\)
\(660\) 0 0
\(661\) 5278.00 0.0120800 0.00603999 0.999982i \(-0.498077\pi\)
0.00603999 + 0.999982i \(0.498077\pi\)
\(662\) 0 0
\(663\) −336582. 336582.i −0.765709 0.765709i
\(664\) 0 0
\(665\) −40600.0 + 284200.i −0.0918085 + 0.642659i
\(666\) 0 0
\(667\) −182920. + 182920.i −0.411159 + 0.411159i
\(668\) 0 0
\(669\) 556038.i 1.24237i
\(670\) 0 0
\(671\) 660564. 1.46713
\(672\) 0 0
\(673\) −332111. 332111.i −0.733252 0.733252i 0.238011 0.971263i \(-0.423505\pi\)
−0.971263 + 0.238011i \(0.923505\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −578309. + 578309.i −1.26178 + 1.26178i −0.311546 + 0.950231i \(0.600847\pi\)
−0.950231 + 0.311546i \(0.899153\pi\)
\(678\) 0 0
\(679\) 424038.i 0.919740i
\(680\) 0 0
\(681\) −988398. −2.13127
\(682\) 0 0
\(683\) 349311. + 349311.i 0.748809 + 0.748809i 0.974255 0.225447i \(-0.0723842\pi\)
−0.225447 + 0.974255i \(0.572384\pi\)
\(684\) 0 0
\(685\) 714315. + 102045.i 1.52233 + 0.217476i
\(686\) 0 0
\(687\) −452520. + 452520.i −0.958792 + 0.958792i
\(688\) 0 0
\(689\) 84318.0i 0.177616i
\(690\) 0 0
\(691\) −282762. −0.592195 −0.296098 0.955158i \(-0.595685\pi\)
−0.296098 + 0.955158i \(0.595685\pi\)
\(692\) 0 0
\(693\) 277182. + 277182.i 0.577163 + 0.577163i
\(694\) 0 0
\(695\) 188000. 141000.i 0.389214 0.291910i
\(696\) 0 0
\(697\) −455822. + 455822.i −0.938274 + 0.938274i
\(698\) 0 0
\(699\) 43038.0i 0.0880841i
\(700\) 0 0
\(701\) −270242. −0.549942 −0.274971 0.961453i \(-0.588668\pi\)
−0.274971 + 0.961453i \(0.588668\pi\)
\(702\) 0 0
\(703\) 182280. + 182280.i 0.368832 + 0.368832i
\(704\) 0 0
\(705\) −342630. 456840.i −0.689362 0.919149i
\(706\) 0 0
\(707\) 25462.0 25462.0i 0.0509394 0.0509394i
\(708\) 0 0
\(709\) 297800.i 0.592423i −0.955122 0.296212i \(-0.904277\pi\)
0.955122 0.296212i \(-0.0957234\pi\)
\(710\) 0 0
\(711\) 855360. 1.69204
\(712\) 0 0
\(713\) 54338.0 + 54338.0i 0.106887 + 0.106887i
\(714\) 0 0
\(715\) 40710.0 284970.i 0.0796323 0.557426i
\(716\) 0 0
\(717\) 159840. 159840.i 0.310919 0.310919i
\(718\) 0 0
\(719\) 913760.i 1.76756i −0.467902 0.883780i \(-0.654990\pi\)
0.467902 0.883780i \(-0.345010\pi\)
\(720\) 0 0
\(721\) 604882. 1.16359
\(722\) 0 0
\(723\) 254142. + 254142.i 0.486183 + 0.486183i
\(724\) 0 0
\(725\) −408000. 119000.i −0.776219 0.226397i
\(726\) 0 0
\(727\) −417651. + 417651.i −0.790214 + 0.790214i −0.981529 0.191315i \(-0.938725\pi\)
0.191315 + 0.981529i \(0.438725\pi\)
\(728\) 0 0
\(729\) 531441.i 1.00000i
\(730\) 0 0
\(731\) 590238. 1.10457
\(732\) 0 0
\(733\) −394549. 394549.i −0.734333 0.734333i 0.237142 0.971475i \(-0.423789\pi\)
−0.971475 + 0.237142i \(0.923789\pi\)
\(734\) 0 0
\(735\) 226485. + 32355.0i 0.419242 + 0.0598917i
\(736\) 0 0
\(737\) 88618.0 88618.0i 0.163150 0.163150i
\(738\) 0 0
\(739\) 109880.i 0.201201i 0.994927 + 0.100600i \(0.0320764\pi\)
−0.994927 + 0.100600i \(0.967924\pi\)
\(740\) 0 0
\(741\) −347760. −0.633349
\(742\) 0 0
\(743\) −466451. 466451.i −0.844945 0.844945i 0.144552 0.989497i \(-0.453826\pi\)
−0.989497 + 0.144552i \(0.953826\pi\)
\(744\) 0 0
\(745\) −276000. + 207000.i −0.497275 + 0.372956i
\(746\) 0 0
\(747\) −504711. + 504711.i −0.904485 + 0.904485i
\(748\) 0 0
\(749\) 273238.i 0.487054i
\(750\) 0 0
\(751\) 1.01092e6 1.79241 0.896206 0.443638i \(-0.146313\pi\)
0.896206 + 0.443638i \(0.146313\pi\)
\(752\) 0 0
\(753\) 1.08902e6 + 1.08902e6i 1.92064 + 1.92064i
\(754\) 0 0
\(755\) −284970. 379960.i −0.499925 0.666567i
\(756\) 0 0
\(757\) −313269. + 313269.i −0.546671 + 0.546671i −0.925476 0.378806i \(-0.876335\pi\)
0.378806 + 0.925476i \(0.376335\pi\)
\(758\) 0 0
\(759\) 571356.i 0.991798i
\(760\) 0 0
\(761\) 142082. 0.245341 0.122670 0.992447i \(-0.460854\pi\)
0.122670 + 0.992447i \(0.460854\pi\)
\(762\) 0 0
\(763\) 639160. + 639160.i 1.09789 + 1.09789i
\(764\) 0 0
\(765\) 109755. 768285.i 0.187543 1.31280i
\(766\) 0 0
\(767\) 80040.0 80040.0i 0.136056 0.136056i
\(768\) 0 0
\(769\) 13280.0i 0.0224567i −0.999937 0.0112283i \(-0.996426\pi\)
0.999937 0.0112283i \(-0.00357417\pi\)
\(770\) 0 0
\(771\) 1.30376e6 2.19325
\(772\) 0 0
\(773\) 782211. + 782211.i 1.30908 + 1.30908i 0.922082 + 0.386994i \(0.126487\pi\)
0.386994 + 0.922082i \(0.373513\pi\)
\(774\) 0 0
\(775\) −35350.0 + 121200.i −0.0588554 + 0.201790i
\(776\) 0 0
\(777\) −339822. + 339822.i −0.562872 + 0.562872i
\(778\) 0 0
\(779\) 470960.i 0.776085i
\(780\) 0 0
\(781\) 760156. 1.24624
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 572985. + 81855.0i 0.929831 + 0.132833i
\(786\) 0 0
\(787\) −201409. + 201409.i −0.325184 + 0.325184i −0.850752 0.525568i \(-0.823853\pi\)
0.525568 + 0.850752i \(0.323853\pi\)
\(788\) 0 0
\(789\) 265878.i 0.427099i
\(790\) 0 0
\(791\) −122438. −0.195688
\(792\) 0 0
\(793\) −386262. 386262.i −0.614236 0.614236i
\(794\) 0 0
\(795\) 219960. 164970.i 0.348024 0.261018i
\(796\) 0 0
\(797\) 36291.0 36291.0i 0.0571324 0.0571324i −0.677963 0.735096i \(-0.737137\pi\)
0.735096 + 0.677963i \(0.237137\pi\)
\(798\) 0 0
\(799\) 687798.i 1.07738i
\(800\) 0 0
\(801\) −1.17288e6 −1.82805
\(802\) 0 0
\(803\) −348218. 348218.i −0.540033 0.540033i
\(804\) 0 0
\(805\) 234030. + 312040.i 0.361143 + 0.481525i
\(806\) 0 0
\(807\) 807480. 807480.i 1.23989 1.23989i
\(808\) 0 0
\(809\) 71600.0i 0.109400i 0.998503 + 0.0546998i \(0.0174202\pi\)
−0.998503 + 0.0546998i \(0.982580\pi\)
\(810\) 0 0
\(811\) 103318. 0.157085 0.0785424 0.996911i \(-0.474973\pi\)
0.0785424 + 0.996911i \(0.474973\pi\)
\(812\) 0 0
\(813\) −613818. 613818.i −0.928664 0.928664i
\(814\) 0 0
\(815\) 100045. 700315.i 0.150619 1.05433i
\(816\) 0 0
\(817\) 304920. 304920.i 0.456817 0.456817i
\(818\) 0 0
\(819\) 324162.i 0.483275i
\(820\) 0 0
\(821\) 157438. 0.233573 0.116787 0.993157i \(-0.462741\pi\)
0.116787 + 0.993157i \(0.462741\pi\)
\(822\) 0 0
\(823\) 791309. + 791309.i 1.16828 + 1.16828i 0.982613 + 0.185666i \(0.0594441\pi\)
0.185666 + 0.982613i \(0.440556\pi\)
\(824\) 0 0
\(825\) 823050. 451350.i 1.20926 0.663140i
\(826\) 0 0
\(827\) 889671. 889671.i 1.30082 1.30082i 0.372987 0.927837i \(-0.378334\pi\)
0.927837 0.372987i \(-0.121666\pi\)
\(828\) 0 0
\(829\) 618280.i 0.899655i −0.893115 0.449828i \(-0.851485\pi\)
0.893115 0.449828i \(-0.148515\pi\)
\(830\) 0 0
\(831\) 333882. 0.483494
\(832\) 0 0
\(833\) −194849. 194849.i −0.280807 0.280807i
\(834\) 0 0
\(835\) 54215.0 + 7745.00i 0.0777583 + 0.0111083i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 821360.i 1.16684i 0.812172 + 0.583418i \(0.198285\pi\)
−0.812172 + 0.583418i \(0.801715\pi\)
\(840\) 0 0
\(841\) 244881. 0.346229
\(842\) 0 0
\(843\) −20898.0 20898.0i −0.0294069 0.0294069i
\(844\) 0 0
\(845\) 380780. 285585.i 0.533287 0.399965i
\(846\) 0 0
\(847\) −20793.0 + 20793.0i −0.0289835 + 0.0289835i
\(848\) 0 0
\(849\) 1.65080e6i 2.29023i
\(850\) 0 0
\(851\) 350238. 0.483620
\(852\) 0 0
\(853\) 698291. + 698291.i 0.959706 + 0.959706i 0.999219 0.0395127i \(-0.0125806\pi\)
−0.0395127 + 0.999219i \(0.512581\pi\)
\(854\) 0 0
\(855\) −340200. 453600.i −0.465374 0.620499i
\(856\) 0 0
\(857\) 144489. 144489.i 0.196731 0.196731i −0.601866 0.798597i \(-0.705576\pi\)
0.798597 + 0.601866i \(0.205576\pi\)
\(858\) 0 0
\(859\) 943480.i 1.27863i −0.768943 0.639317i \(-0.779217\pi\)
0.768943 0.639317i \(-0.220783\pi\)
\(860\) 0 0
\(861\) −878004. −1.18438
\(862\) 0 0
\(863\) 438149. + 438149.i 0.588302 + 0.588302i 0.937171 0.348869i \(-0.113434\pi\)
−0.348869 + 0.937171i \(0.613434\pi\)
\(864\) 0 0
\(865\) 13945.0 97615.0i 0.0186374 0.130462i
\(866\) 0 0
\(867\) −570249. + 570249.i −0.758624 + 0.758624i
\(868\) 0 0
\(869\) 1.24608e6i 1.65009i
\(870\) 0 0
\(871\) −103638. −0.136610
\(872\) 0 0
\(873\) −592191. 592191.i −0.777022 0.777022i
\(874\) 0 0
\(875\) −264625. + 583625.i −0.345633 + 0.762286i
\(876\) 0 0
\(877\) −281469. + 281469.i −0.365958 + 0.365958i −0.866001 0.500043i \(-0.833318\pi\)
0.500043 + 0.866001i \(0.333318\pi\)
\(878\) 0 0
\(879\) 87318.0i 0.113012i
\(880\) 0 0
\(881\) 876722. 1.12956 0.564781 0.825241i \(-0.308961\pi\)
0.564781 + 0.825241i \(0.308961\pi\)
\(882\) 0 0
\(883\) 327431. + 327431.i 0.419951 + 0.419951i 0.885187 0.465236i \(-0.154031\pi\)
−0.465236 + 0.885187i \(0.654031\pi\)
\(884\) 0 0
\(885\) 365400. + 52200.0i 0.466533 + 0.0666475i
\(886\) 0 0
\(887\) −477171. + 477171.i −0.606494 + 0.606494i −0.942028 0.335534i \(-0.891083\pi\)
0.335534 + 0.942028i \(0.391083\pi\)
\(888\) 0 0
\(889\) 342722.i 0.433649i
\(890\) 0 0
\(891\) 774198. 0.975207
\(892\) 0 0
\(893\) 355320. + 355320.i 0.445571 + 0.445571i
\(894\) 0 0
\(895\) 52000.0 39000.0i 0.0649168 0.0486876i
\(896\) 0 0
\(897\) −334098. + 334098.i −0.415230 + 0.415230i
\(898\) 0 0
\(899\) 137360.i 0.169958i
\(900\) 0 0
\(901\) −331162. −0.407935
\(902\) 0 0
\(903\) 568458. + 568458.i 0.697145 + 0.697145i
\(904\) 0 0
\(905\) 665970. + 887960.i 0.813125 + 1.08417i
\(906\) 0 0
\(907\) −1.11209e6 + 1.11209e6i −1.35184 + 1.35184i −0.468235 + 0.883604i \(0.655110\pi\)
−0.883604 + 0.468235i \(0.844890\pi\)
\(908\) 0 0
\(909\) 71118.0i 0.0860700i
\(910\) 0 0
\(911\) −883958. −1.06511 −0.532556 0.846395i \(-0.678768\pi\)
−0.532556 + 0.846395i \(0.678768\pi\)
\(912\) 0 0
\(913\) 735258. + 735258.i 0.882060 + 0.882060i
\(914\) 0 0
\(915\) 251910. 1.76337e6i 0.300887 2.10621i
\(916\) 0 0
\(917\) 184382. 184382.i 0.219270 0.219270i
\(918\) 0 0
\(919\) 1.24040e6i 1.46869i −0.678775 0.734346i \(-0.737489\pi\)
0.678775 0.734346i \(-0.262511\pi\)
\(920\) 0 0
\(921\) 771282. 0.909272
\(922\) 0 0
\(923\) −444498. 444498.i −0.521755 0.521755i
\(924\) 0 0
\(925\) 276675. + 504525.i 0.323360 + 0.589657i
\(926\) 0 0
\(927\) −844749. + 844749.i −0.983033 + 0.983033i
\(928\) 0 0
\(929\) 1.22744e6i 1.42223i −0.703077 0.711113i \(-0.748191\pi\)
0.703077 0.711113i \(-0.251809\pi\)
\(930\) 0 0
\(931\) −201320. −0.232267
\(932\) 0 0
\(933\) 650502. + 650502.i 0.747283 + 0.747283i
\(934\) 0 0
\(935\) −1.11923e6 159890.i −1.28025 0.182893i
\(936\) 0 0
\(937\) −1.07047e6 + 1.07047e6i −1.21926 + 1.21926i −0.251366 + 0.967892i \(0.580880\pi\)
−0.967892 + 0.251366i \(0.919120\pi\)
\(938\) 0 0
\(939\) 328482.i 0.372546i
\(940\) 0 0
\(941\) −558642. −0.630891 −0.315446 0.948944i \(-0.602154\pi\)
−0.315446 + 0.948944i \(0.602154\pi\)
\(942\) 0 0
\(943\) 452458. + 452458.i 0.508809 + 0.508809i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 191711. 191711.i 0.213770 0.213770i −0.592097 0.805867i \(-0.701700\pi\)
0.805867 + 0.592097i \(0.201700\pi\)
\(948\) 0 0
\(949\) 407238.i 0.452185i
\(950\) 0 0
\(951\) 452682. 0.500532
\(952\) 0 0
\(953\) −630231. 630231.i −0.693927 0.693927i 0.269166 0.963094i \(-0.413252\pi\)
−0.963094 + 0.269166i \(0.913252\pi\)
\(954\) 0 0
\(955\) −220170. 293560.i −0.241408 0.321877i
\(956\) 0 0
\(957\) 722160. 722160.i 0.788514 0.788514i
\(958\) 0 0
\(959\) 1.18372e6i 1.28710i
\(960\) 0 0
\(961\) −882717. −0.955817
\(962\) 0 0
\(963\) 381591. + 381591.i 0.411477 + 0.411477i
\(964\) 0 0
\(965\) −214245. + 1.49972e6i −0.230068 + 1.61048i
\(966\) 0 0
\(967\) −345491. + 345491.i −0.369474 + 0.369474i −0.867285 0.497811i \(-0.834137\pi\)
0.497811 + 0.867285i \(0.334137\pi\)
\(968\) 0 0
\(969\) 1.36584e6i 1.45463i
\(970\) 0 0
\(971\) −1.08308e6 −1.14874 −0.574372 0.818595i \(-0.694753\pi\)
−0.574372 + 0.818595i \(0.694753\pi\)
\(972\) 0 0
\(973\) −272600. 272600.i −0.287939 0.287939i
\(974\) 0 0
\(975\) −745200. 217350.i −0.783905 0.228639i
\(976\) 0 0
\(977\) −146751. + 146751.i −0.153742 + 0.153742i −0.779787 0.626045i \(-0.784673\pi\)
0.626045 + 0.779787i \(0.284673\pi\)
\(978\) 0 0
\(979\) 1.70864e6i 1.78273i
\(980\) 0 0
\(981\) −1.78524e6 −1.85506
\(982\) 0 0
\(983\) 466909. + 466909.i 0.483198 + 0.483198i 0.906151 0.422953i \(-0.139007\pi\)
−0.422953 + 0.906151i \(0.639007\pi\)
\(984\) 0 0
\(985\) 383985. + 54855.0i 0.395769 + 0.0565384i
\(986\) 0 0
\(987\) −662418. + 662418.i −0.679983 + 0.679983i
\(988\) 0 0
\(989\) 585882.i 0.598987i
\(990\) 0 0
\(991\) −901238. −0.917682 −0.458841 0.888518i \(-0.651735\pi\)
−0.458841 + 0.888518i \(0.651735\pi\)
\(992\) 0 0
\(993\) −486342. 486342.i −0.493223 0.493223i
\(994\) 0 0
\(995\) −763200. + 572400.i −0.770890 + 0.578167i
\(996\) 0 0
\(997\) −152149. + 152149.i −0.153066 + 0.153066i −0.779486 0.626420i \(-0.784520\pi\)
0.626420 + 0.779486i \(0.284520\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 320.5.p.b.193.1 2
4.3 odd 2 320.5.p.i.193.1 2
5.2 odd 4 inner 320.5.p.b.257.1 2
8.3 odd 2 80.5.p.b.33.1 2
8.5 even 2 10.5.c.a.3.1 2
20.7 even 4 320.5.p.i.257.1 2
24.5 odd 2 90.5.g.b.73.1 2
40.3 even 4 400.5.p.c.257.1 2
40.13 odd 4 50.5.c.b.7.1 2
40.19 odd 2 400.5.p.c.193.1 2
40.27 even 4 80.5.p.b.17.1 2
40.29 even 2 50.5.c.b.43.1 2
40.37 odd 4 10.5.c.a.7.1 yes 2
120.29 odd 2 450.5.g.a.343.1 2
120.53 even 4 450.5.g.a.307.1 2
120.77 even 4 90.5.g.b.37.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.5.c.a.3.1 2 8.5 even 2
10.5.c.a.7.1 yes 2 40.37 odd 4
50.5.c.b.7.1 2 40.13 odd 4
50.5.c.b.43.1 2 40.29 even 2
80.5.p.b.17.1 2 40.27 even 4
80.5.p.b.33.1 2 8.3 odd 2
90.5.g.b.37.1 2 120.77 even 4
90.5.g.b.73.1 2 24.5 odd 2
320.5.p.b.193.1 2 1.1 even 1 trivial
320.5.p.b.257.1 2 5.2 odd 4 inner
320.5.p.i.193.1 2 4.3 odd 2
320.5.p.i.257.1 2 20.7 even 4
400.5.p.c.193.1 2 40.19 odd 2
400.5.p.c.257.1 2 40.3 even 4
450.5.g.a.307.1 2 120.53 even 4
450.5.g.a.343.1 2 120.29 odd 2