Properties

Label 400.5.p.c.257.1
Level $400$
Weight $5$
Character 400.257
Analytic conductor $41.348$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,5,Mod(193,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, 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("400.193");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(41.3479852335\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.257
Dual form 400.5.p.c.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(9.00000 - 9.00000i) q^{3} +(29.0000 + 29.0000i) q^{7} -81.0000i q^{9} +118.000 q^{11} +(-69.0000 + 69.0000i) q^{13} +(271.000 + 271.000i) q^{17} -280.000i q^{19} +522.000 q^{21} +(269.000 - 269.000i) q^{23} +680.000i q^{29} -202.000 q^{31} +(1062.00 - 1062.00i) q^{33} +(651.000 + 651.000i) q^{37} +1242.00i q^{39} +1682.00 q^{41} +(1089.00 - 1089.00i) q^{43} +(1269.00 + 1269.00i) q^{47} -719.000i q^{49} +4878.00 q^{51} +(611.000 - 611.000i) q^{53} +(-2520.00 - 2520.00i) q^{57} -1160.00i q^{59} -5598.00 q^{61} +(2349.00 - 2349.00i) q^{63} +(-751.000 - 751.000i) q^{67} -4842.00i q^{69} -6442.00 q^{71} +(2951.00 - 2951.00i) q^{73} +(3422.00 + 3422.00i) q^{77} -10560.0i q^{79} +6561.00 q^{81} +(-6231.00 + 6231.00i) q^{83} +(6120.00 + 6120.00i) q^{87} -14480.0i q^{89} -4002.00 q^{91} +(-1818.00 + 1818.00i) q^{93} +(7311.00 + 7311.00i) q^{97} -9558.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} + 58 q^{7} + 236 q^{11} - 138 q^{13} + 542 q^{17} + 1044 q^{21} + 538 q^{23} - 404 q^{31} + 2124 q^{33} + 1302 q^{37} + 3364 q^{41} + 2178 q^{43} + 2538 q^{47} + 9756 q^{51} + 1222 q^{53}+ \cdots + 14622 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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 \(0\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 29.0000 + 29.0000i 0.591837 + 0.591837i 0.938127 0.346291i \(-0.112559\pi\)
−0.346291 + 0.938127i \(0.612559\pi\)
\(8\) 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 −0.881140 0.472856i \(-0.843223\pi\)
0.472856 + 0.881140i \(0.343223\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 271.000 + 271.000i 0.937716 + 0.937716i 0.998171 0.0604547i \(-0.0192551\pi\)
−0.0604547 + 0.998171i \(0.519255\pi\)
\(18\) 0 0
\(19\) 280.000i 0.775623i −0.921739 0.387812i \(-0.873231\pi\)
0.921739 0.387812i \(-0.126769\pi\)
\(20\) 0 0
\(21\) 522.000 1.18367
\(22\) 0 0
\(23\) 269.000 269.000i 0.508507 0.508507i −0.405561 0.914068i \(-0.632924\pi\)
0.914068 + 0.405561i \(0.132924\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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 \(-0.533516\pi\)
−0.105099 + 0.994462i \(0.533516\pi\)
\(32\) 0 0
\(33\) 1062.00 1062.00i 0.975207 0.975207i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 651.000 + 651.000i 0.475530 + 0.475530i 0.903699 0.428169i \(-0.140841\pi\)
−0.428169 + 0.903699i \(0.640841\pi\)
\(38\) 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 \(-0.613270\pi\)
0.937352 + 0.348385i \(0.113270\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1269.00 + 1269.00i 0.574468 + 0.574468i 0.933374 0.358906i \(-0.116850\pi\)
−0.358906 + 0.933374i \(0.616850\pi\)
\(48\) 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 −0.589935 0.807450i \(-0.700847\pi\)
0.807450 + 0.589935i \(0.200847\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2520.00 2520.00i −0.775623 0.775623i
\(58\) 0 0
\(59\) 1160.00i 0.333238i −0.986021 0.166619i \(-0.946715\pi\)
0.986021 0.166619i \(-0.0532849\pi\)
\(60\) 0 0
\(61\) −5598.00 −1.50443 −0.752217 0.658915i \(-0.771016\pi\)
−0.752217 + 0.658915i \(0.771016\pi\)
\(62\) 0 0
\(63\) 2349.00 2349.00i 0.591837 0.591837i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −751.000 751.000i −0.167298 0.167298i 0.618493 0.785791i \(-0.287744\pi\)
−0.785791 + 0.618493i \(0.787744\pi\)
\(68\) 0 0
\(69\) 4842.00i 1.01701i
\(70\) 0 0
\(71\) −6442.00 −1.27792 −0.638961 0.769240i \(-0.720635\pi\)
−0.638961 + 0.769240i \(0.720635\pi\)
\(72\) 0 0
\(73\) 2951.00 2951.00i 0.553762 0.553762i −0.373762 0.927525i \(-0.621932\pi\)
0.927525 + 0.373762i \(0.121932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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 \(-0.970887\pi\)
0.0913348 + 0.995820i \(0.470887\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6120.00 + 6120.00i 0.808561 + 0.808561i
\(88\) 0 0
\(89\) 14480.0i 1.82805i −0.405656 0.914026i \(-0.632957\pi\)
0.405656 0.914026i \(-0.367043\pi\)
\(90\) 0 0
\(91\) −4002.00 −0.483275
\(92\) 0 0
\(93\) −1818.00 + 1818.00i −0.210198 + 0.210198i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7311.00 + 7311.00i 0.777022 + 0.777022i 0.979323 0.202301i \(-0.0648420\pi\)
−0.202301 + 0.979323i \(0.564842\pi\)
\(98\) 0 0
\(99\) 9558.00i 0.975207i
\(100\) 0 0
\(101\) −878.000 −0.0860700 −0.0430350 0.999074i \(-0.513703\pi\)
−0.0430350 + 0.999074i \(0.513703\pi\)
\(102\) 0 0
\(103\) 10429.0 10429.0i 0.983033 0.983033i −0.0168252 0.999858i \(-0.505356\pi\)
0.999858 + 0.0168252i \(0.00535587\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4711.00 4711.00i −0.411477 0.411477i 0.470776 0.882253i \(-0.343974\pi\)
−0.882253 + 0.470776i \(0.843974\pi\)
\(108\) 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 \(-0.712704\pi\)
0.784920 + 0.619597i \(0.212704\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 5909.00 + 5909.00i 0.366359 + 0.366359i 0.866147 0.499789i \(-0.166589\pi\)
−0.499789 + 0.866147i \(0.666589\pi\)
\(128\) 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 −0.995798 0.0915804i \(-0.970808\pi\)
−0.0915804 0.995798i \(-0.529192\pi\)
\(138\) 0 0
\(139\) 9400.00i 0.486517i 0.969961 + 0.243259i \(0.0782164\pi\)
−0.969961 + 0.243259i \(0.921784\pi\)
\(140\) 0 0
\(141\) 22842.0 1.14894
\(142\) 0 0
\(143\) −8142.00 + 8142.00i −0.398161 + 0.398161i
\(144\) 0 0
\(145\) 0 0
\(146\) 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 \(-0.363220\pi\)
0.416605 + 0.909088i \(0.363220\pi\)
\(152\) 0 0
\(153\) 21951.0 21951.0i 0.937716 0.937716i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16371.0 + 16371.0i 0.664165 + 0.664165i 0.956359 0.292194i \(-0.0943853\pi\)
−0.292194 + 0.956359i \(0.594385\pi\)
\(158\) 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 \(-0.571245\pi\)
0.975056 + 0.221960i \(0.0712455\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1549.00 + 1549.00i 0.0555416 + 0.0555416i 0.734332 0.678790i \(-0.237496\pi\)
−0.678790 + 0.734332i \(0.737496\pi\)
\(168\) 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 −0.752164 0.658976i \(-0.770990\pi\)
0.658976 + 0.752164i \(0.270990\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10440.0 10440.0i −0.333238 0.333238i
\(178\) 0 0
\(179\) 2600.00i 0.0811460i 0.999177 + 0.0405730i \(0.0129183\pi\)
−0.999177 + 0.0405730i \(0.987082\pi\)
\(180\) 0 0
\(181\) −44398.0 −1.35521 −0.677604 0.735427i \(-0.736982\pi\)
−0.677604 + 0.735427i \(0.736982\pi\)
\(182\) 0 0
\(183\) −50382.0 + 50382.0i −1.50443 + 1.50443i
\(184\) 0 0
\(185\) 0 0
\(186\) 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 \(-0.435525\pi\)
0.201173 + 0.979556i \(0.435525\pi\)
\(192\) 0 0
\(193\) −42849.0 + 42849.0i −1.15034 + 1.15034i −0.163855 + 0.986484i \(0.552393\pi\)
−0.986484 + 0.163855i \(0.947607\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10971.0 + 10971.0i 0.282692 + 0.282692i 0.834182 0.551490i \(-0.185940\pi\)
−0.551490 + 0.834182i \(0.685940\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 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 −0.945835 0.324648i \(-0.894754\pi\)
0.324648 + 0.945835i \(0.394754\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −54911.0 54911.0i −1.06563 1.06563i −0.997689 0.0679438i \(-0.978356\pi\)
−0.0679438 0.997689i \(-0.521644\pi\)
\(228\) 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 \(-0.740085\pi\)
0.728785 + 0.684743i \(0.240085\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 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
\(255\) 0 0
\(256\) 0 0
\(257\) 72431.0 + 72431.0i 1.09663 + 1.09663i 0.994803 + 0.101823i \(0.0324674\pi\)
0.101823 + 0.994803i \(0.467533\pi\)
\(258\) 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 −0.805773 0.592224i \(-0.798250\pi\)
0.592224 + 0.805773i \(0.298250\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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 \(-0.653706\pi\)
−0.464332 + 0.885661i \(0.653706\pi\)
\(272\) 0 0
\(273\) −36018.0 + 36018.0i −0.483275 + 0.483275i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18549.0 18549.0i −0.241747 0.241747i 0.575826 0.817573i \(-0.304681\pi\)
−0.817573 + 0.575826i \(0.804681\pi\)
\(278\) 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 −0.157613 + 0.987501i \(0.550380\pi\)
−0.987501 + 0.157613i \(0.949620\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 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 −0.678289 0.734795i \(-0.737278\pi\)
0.734795 + 0.678289i \(0.237278\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 42849.0 + 42849.0i 0.454636 + 0.454636i 0.896890 0.442254i \(-0.145821\pi\)
−0.442254 + 0.896890i \(0.645821\pi\)
\(308\) 0 0
\(309\) 187722.i 1.96607i
\(310\) 0 0
\(311\) 72278.0 0.747283 0.373642 0.927573i \(-0.378109\pi\)
0.373642 + 0.927573i \(0.378109\pi\)
\(312\) 0 0
\(313\) −18249.0 + 18249.0i −0.186273 + 0.186273i −0.794083 0.607810i \(-0.792048\pi\)
0.607810 + 0.794083i \(0.292048\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25149.0 25149.0i −0.250266 0.250266i 0.570814 0.821080i \(-0.306628\pi\)
−0.821080 + 0.570814i \(0.806628\pi\)
\(318\) 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
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) −8529.00 8529.00i −0.0750997 0.0750997i 0.668559 0.743659i \(-0.266911\pi\)
−0.743659 + 0.668559i \(0.766911\pi\)
\(338\) 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
\(345\) 0 0
\(346\) 0 0
\(347\) −56551.0 56551.0i −0.469658 0.469658i 0.432146 0.901804i \(-0.357756\pi\)
−0.901804 + 0.432146i \(0.857756\pi\)
\(348\) 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 \(-0.668720\pi\)
0.862782 + 0.505576i \(0.168720\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) −14971.0 14971.0i −0.111152 0.111152i 0.649343 0.760496i \(-0.275044\pi\)
−0.760496 + 0.649343i \(0.775044\pi\)
\(368\) 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 −0.655729 0.754996i \(-0.727639\pi\)
0.754996 + 0.655729i \(0.227639\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −46920.0 46920.0i −0.330123 0.330123i
\(378\) 0 0
\(379\) 251080.i 1.74797i −0.485954 0.873984i \(-0.661528\pi\)
0.485954 0.873984i \(-0.338472\pi\)
\(380\) 0 0
\(381\) 106362. 0.732717
\(382\) 0 0
\(383\) −86091.0 + 86091.0i −0.586895 + 0.586895i −0.936789 0.349894i \(-0.886217\pi\)
0.349894 + 0.936789i \(0.386217\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) −29149.0 29149.0i −0.184945 0.184945i 0.608562 0.793507i \(-0.291747\pi\)
−0.793507 + 0.608562i \(0.791747\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 0 0
\(407\) 76818.0 + 76818.0i 0.463740 + 0.463740i
\(408\) 0 0
\(409\) 78720.0i 0.470585i 0.971925 + 0.235293i \(0.0756049\pi\)
−0.971925 + 0.235293i \(0.924395\pi\)
\(410\) 0 0
\(411\) −367362. −2.17476
\(412\) 0 0
\(413\) 33640.0 33640.0i 0.197222 0.197222i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 84600.0 + 84600.0i 0.486517 + 0.486517i
\(418\) 0 0
\(419\) 14760.0i 0.0840733i 0.999116 + 0.0420367i \(0.0133846\pi\)
−0.999116 + 0.0420367i \(0.986615\pi\)
\(420\) 0 0
\(421\) 221282. 1.24848 0.624240 0.781232i \(-0.285409\pi\)
0.624240 + 0.781232i \(0.285409\pi\)
\(422\) 0 0
\(423\) 102789. 102789.i 0.574468 0.574468i
\(424\) 0 0
\(425\) 0 0
\(426\) 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 \(-0.693844\pi\)
−0.572031 + 0.820232i \(0.693844\pi\)
\(432\) 0 0
\(433\) −145409. + 145409.i −0.775560 + 0.775560i −0.979072 0.203512i \(-0.934764\pi\)
0.203512 + 0.979072i \(0.434764\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 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 0.260590 0.965450i \(-0.416083\pi\)
0.965450 0.260590i \(-0.0839170\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 124200. + 124200.i 0.621594 + 0.621594i
\(448\) 0 0
\(449\) 82480.0i 0.409125i −0.978854 0.204562i \(-0.934423\pi\)
0.978854 0.204562i \(-0.0655772\pi\)
\(450\) 0 0
\(451\) 198476. 0.975787
\(452\) 0 0
\(453\) 170982. 170982.i 0.833209 0.833209i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 188151. + 188151.i 0.900895 + 0.900895i 0.995514 0.0946187i \(-0.0301632\pi\)
−0.0946187 + 0.995514i \(0.530163\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −326158. −1.53471 −0.767355 0.641223i \(-0.778427\pi\)
−0.767355 + 0.641223i \(0.778427\pi\)
\(462\) 0 0
\(463\) −218731. + 218731.i −1.02035 + 1.02035i −0.0205595 + 0.999789i \(0.506545\pi\)
−0.999789 + 0.0205595i \(0.993455\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 59249.0 + 59249.0i 0.271673 + 0.271673i 0.829774 0.558100i \(-0.188470\pi\)
−0.558100 + 0.829774i \(0.688470\pi\)
\(468\) 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
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) −123651. 123651.i −0.521362 0.521362i 0.396620 0.917983i \(-0.370183\pi\)
−0.917983 + 0.396620i \(0.870183\pi\)
\(488\) 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
\(495\) 0 0
\(496\) 0 0
\(497\) −186818. 186818.i −0.756321 0.756321i
\(498\) 0 0
\(499\) 269240.i 1.08128i −0.841254 0.540640i \(-0.818182\pi\)
0.841254 0.540640i \(-0.181818\pi\)
\(500\) 0 0
\(501\) 27882.0 0.111083
\(502\) 0 0
\(503\) 109869. 109869.i 0.434249 0.434249i −0.455822 0.890071i \(-0.650655\pi\)
0.890071 + 0.455822i \(0.150655\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 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 \(-0.770645\pi\)
0.659791 + 0.751449i \(0.270645\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 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
\(535\) 0 0
\(536\) 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 \(-0.578263\pi\)
−0.243402 + 0.969926i \(0.578263\pi\)
\(542\) 0 0
\(543\) −399582. + 399582.i −1.35521 + 1.35521i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 291009. + 291009.i 0.972594 + 0.972594i 0.999634 0.0270399i \(-0.00860813\pi\)
−0.0270399 + 0.999634i \(0.508608\pi\)
\(548\) 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
\(555\) 0 0
\(556\) 0 0
\(557\) 83091.0 + 83091.0i 0.267820 + 0.267820i 0.828221 0.560401i \(-0.189353\pi\)
−0.560401 + 0.828221i \(0.689353\pi\)
\(558\) 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 \(-0.719098\pi\)
0.772316 + 0.635239i \(0.219098\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 190269. + 190269.i 0.591837 + 0.591837i
\(568\) 0 0
\(569\) 270560.i 0.835678i −0.908521 0.417839i \(-0.862788\pi\)
0.908521 0.417839i \(-0.137212\pi\)
\(570\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) −195889. 195889.i −0.588381 0.588381i 0.348812 0.937193i \(-0.386585\pi\)
−0.937193 + 0.348812i \(0.886585\pi\)
\(578\) 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
\(585\) 0 0
\(586\) 0 0
\(587\) −404631. 404631.i −1.17431 1.17431i −0.981171 0.193139i \(-0.938133\pi\)
−0.193139 0.981171i \(-0.561867\pi\)
\(588\) 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 \(-0.610532\pi\)
0.940314 + 0.340309i \(0.110532\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 146469. + 146469.i 0.397529 + 0.397529i 0.877360 0.479832i \(-0.159302\pi\)
−0.479832 + 0.877360i \(0.659302\pi\)
\(608\) 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 −0.888092 0.459666i \(-0.847969\pi\)
0.459666 + 0.888092i \(0.347969\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −320409. 320409.i −0.841656 0.841656i 0.147419 0.989074i \(-0.452904\pi\)
−0.989074 + 0.147419i \(0.952904\pi\)
\(618\) 0 0
\(619\) 341160.i 0.890383i −0.895435 0.445191i \(-0.853136\pi\)
0.895435 0.445191i \(-0.146864\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 419920. 419920.i 1.08191 1.08191i
\(624\) 0 0
\(625\) 0 0
\(626\) 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 \(-0.336630\pi\)
0.491005 + 0.871157i \(0.336630\pi\)
\(632\) 0 0
\(633\) −655578. + 655578.i −1.63613 + 1.63613i
\(634\) 0 0
\(635\) 0 0
\(636\) 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 \(-0.767381\pi\)
0.667462 + 0.744644i \(0.267381\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −280931. 280931.i −0.671106 0.671106i 0.286865 0.957971i \(-0.407387\pi\)
−0.957971 + 0.286865i \(0.907387\pi\)
\(648\) 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 −0.264439 + 0.964402i \(0.585187\pi\)
−0.964402 + 0.264439i \(0.914813\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −239031. 239031.i −0.553762 0.553762i
\(658\) 0 0
\(659\) 404360.i 0.931102i 0.885021 + 0.465551i \(0.154144\pi\)
−0.885021 + 0.465551i \(0.845856\pi\)
\(660\) 0 0
\(661\) −5278.00 −0.0120800 −0.00603999 0.999982i \(-0.501923\pi\)
−0.00603999 + 0.999982i \(0.501923\pi\)
\(662\) 0 0
\(663\) −336582. + 336582.i −0.765709 + 0.765709i
\(664\) 0 0
\(665\) 0 0
\(666\) 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 \(-0.576495\pi\)
0.971263 + 0.238011i \(0.0764953\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −578309. 578309.i −1.26178 1.26178i −0.950231 0.311546i \(-0.899153\pi\)
−0.311546 0.950231i \(-0.600847\pi\)
\(678\) 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 \(-0.927616\pi\)
0.225447 + 0.974255i \(0.427616\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 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 \(-0.411332\pi\)
0.274971 + 0.961453i \(0.411332\pi\)
\(702\) 0 0
\(703\) 182280. 182280.i 0.368832 0.368832i
\(704\) 0 0
\(705\) 0 0
\(706\) 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
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) −417651. 417651.i −0.790214 0.790214i 0.191315 0.981529i \(-0.438725\pi\)
−0.981529 + 0.191315i \(0.938725\pi\)
\(728\) 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 −0.971475 0.237142i \(-0.923789\pi\)
0.237142 + 0.971475i \(0.423789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −88618.0 88618.0i −0.163150 0.163150i
\(738\) 0 0
\(739\) 109880.i 0.201201i −0.994927 0.100600i \(-0.967924\pi\)
0.994927 0.100600i \(-0.0320764\pi\)
\(740\) 0 0
\(741\) 347760. 0.633349
\(742\) 0 0
\(743\) −466451. + 466451.i −0.844945 + 0.844945i −0.989497 0.144552i \(-0.953826\pi\)
0.144552 + 0.989497i \(0.453826\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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 \(-0.853687\pi\)
−0.896206 + 0.443638i \(0.853687\pi\)
\(752\) 0 0
\(753\) −1.08902e6 + 1.08902e6i −1.92064 + 1.92064i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −313269. 313269.i −0.546671 0.546671i 0.378806 0.925476i \(-0.376335\pi\)
−0.925476 + 0.378806i \(0.876335\pi\)
\(758\) 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
\(765\) 0 0
\(766\) 0 0
\(767\) 80040.0 + 80040.0i 0.136056 + 0.136056i
\(768\) 0 0
\(769\) 13280.0i 0.0224567i 0.999937 + 0.0112283i \(0.00357417\pi\)
−0.999937 + 0.0112283i \(0.996426\pi\)
\(770\) 0 0
\(771\) 1.30376e6 2.19325
\(772\) 0 0
\(773\) 782211. 782211.i 1.30908 1.30908i 0.386994 0.922082i \(-0.373513\pi\)
0.922082 0.386994i \(-0.126487\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) 201409. + 201409.i 0.325184 + 0.325184i 0.850752 0.525568i \(-0.176147\pi\)
−0.525568 + 0.850752i \(0.676147\pi\)
\(788\) 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
\(795\) 0 0
\(796\) 0 0
\(797\) 36291.0 + 36291.0i 0.0571324 + 0.0571324i 0.735096 0.677963i \(-0.237137\pi\)
−0.677963 + 0.735096i \(0.737137\pi\)
\(798\) 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
\(805\) 0 0
\(806\) 0 0
\(807\) 807480. + 807480.i 1.23989 + 1.23989i
\(808\) 0 0
\(809\) 71600.0i 0.109400i −0.998503 0.0546998i \(-0.982580\pi\)
0.998503 0.0546998i \(-0.0174202\pi\)
\(810\) 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
\(815\) 0 0
\(816\) 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 \(-0.537259\pi\)
−0.116787 + 0.993157i \(0.537259\pi\)
\(822\) 0 0
\(823\) 791309. 791309.i 1.16828 1.16828i 0.185666 0.982613i \(-0.440556\pi\)
0.982613 0.185666i \(-0.0594441\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −889671. 889671.i −1.30082 1.30082i −0.927837 0.372987i \(-0.878334\pi\)
−0.372987 0.927837i \(-0.621666\pi\)
\(828\) 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
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 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 −0.0395127 0.999219i \(-0.512581\pi\)
0.999219 + 0.0395127i \(0.0125806\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −144489. 144489.i −0.196731 0.196731i 0.601866 0.798597i \(-0.294424\pi\)
−0.798597 + 0.601866i \(0.794424\pi\)
\(858\) 0 0
\(859\) 943480.i 1.27863i 0.768943 + 0.639317i \(0.220783\pi\)
−0.768943 + 0.639317i \(0.779217\pi\)
\(860\) 0 0
\(861\) 878004. 1.18438
\(862\) 0 0
\(863\) 438149. 438149.i 0.588302 0.588302i −0.348869 0.937171i \(-0.613434\pi\)
0.937171 + 0.348869i \(0.113434\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) −281469. 281469.i −0.365958 0.365958i 0.500043 0.866001i \(-0.333318\pi\)
−0.866001 + 0.500043i \(0.833318\pi\)
\(878\) 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 \(-0.845969\pi\)
0.465236 + 0.885187i \(0.345969\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −477171. 477171.i −0.606494 0.606494i 0.335534 0.942028i \(-0.391083\pi\)
−0.942028 + 0.335534i \(0.891083\pi\)
\(888\) 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
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 0 0
\(907\) 1.11209e6 + 1.11209e6i 1.35184 + 1.35184i 0.883604 + 0.468235i \(0.155110\pi\)
0.468235 + 0.883604i \(0.344890\pi\)
\(908\) 0 0
\(909\) 71118.0i 0.0860700i
\(910\) 0 0
\(911\) 883958. 1.06511 0.532556 0.846395i \(-0.321232\pi\)
0.532556 + 0.846395i \(0.321232\pi\)
\(912\) 0 0
\(913\) −735258. + 735258.i −0.882060 + 0.882060i
\(914\) 0 0
\(915\) 0 0
\(916\) 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
\(925\) 0 0
\(926\) 0 0
\(927\) −844749. 844749.i −0.983033 0.983033i
\(928\) 0 0
\(929\) 1.22744e6i 1.42223i 0.703077 + 0.711113i \(0.251809\pi\)
−0.703077 + 0.711113i \(0.748191\pi\)
\(930\) 0 0
\(931\) −201320. −0.232267
\(932\) 0 0
\(933\) 650502. 650502.i 0.747283 0.747283i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.07047e6 + 1.07047e6i 1.21926 + 1.21926i 0.967892 + 0.251366i \(0.0808799\pi\)
0.251366 + 0.967892i \(0.419120\pi\)
\(938\) 0 0
\(939\) 328482.i 0.372546i
\(940\) 0 0
\(941\) 558642. 0.630891 0.315446 0.948944i \(-0.397846\pi\)
0.315446 + 0.948944i \(0.397846\pi\)
\(942\) 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 \(-0.298300\pi\)
−0.805867 + 0.592097i \(0.798300\pi\)
\(948\) 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 \(-0.586748\pi\)
0.963094 + 0.269166i \(0.0867482\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) −345491. 345491.i −0.369474 0.369474i 0.497811 0.867285i \(-0.334137\pi\)
−0.867285 + 0.497811i \(0.834137\pi\)
\(968\) 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
\(975\) 0 0
\(976\) 0 0
\(977\) 146751. + 146751.i 0.153742 + 0.153742i 0.779787 0.626045i \(-0.215327\pi\)
−0.626045 + 0.779787i \(0.715327\pi\)
\(978\) 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 −0.422953 0.906151i \(-0.639007\pi\)
0.906151 + 0.422953i \(0.139007\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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 \(-0.348265\pi\)
0.458841 + 0.888518i \(0.348265\pi\)
\(992\) 0 0
\(993\) 486342. 486342.i 0.493223 0.493223i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −152149. 152149.i −0.153066 0.153066i 0.626420 0.779486i \(-0.284520\pi\)
−0.779486 + 0.626420i \(0.784520\pi\)
\(998\) 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 400.5.p.c.257.1 2
4.3 odd 2 50.5.c.b.7.1 2
5.2 odd 4 80.5.p.b.33.1 2
5.3 odd 4 inner 400.5.p.c.193.1 2
5.4 even 2 80.5.p.b.17.1 2
12.11 even 2 450.5.g.a.307.1 2
20.3 even 4 50.5.c.b.43.1 2
20.7 even 4 10.5.c.a.3.1 2
20.19 odd 2 10.5.c.a.7.1 yes 2
40.19 odd 2 320.5.p.b.257.1 2
40.27 even 4 320.5.p.b.193.1 2
40.29 even 2 320.5.p.i.257.1 2
40.37 odd 4 320.5.p.i.193.1 2
60.23 odd 4 450.5.g.a.343.1 2
60.47 odd 4 90.5.g.b.73.1 2
60.59 even 2 90.5.g.b.37.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.5.c.a.3.1 2 20.7 even 4
10.5.c.a.7.1 yes 2 20.19 odd 2
50.5.c.b.7.1 2 4.3 odd 2
50.5.c.b.43.1 2 20.3 even 4
80.5.p.b.17.1 2 5.4 even 2
80.5.p.b.33.1 2 5.2 odd 4
90.5.g.b.37.1 2 60.59 even 2
90.5.g.b.73.1 2 60.47 odd 4
320.5.p.b.193.1 2 40.27 even 4
320.5.p.b.257.1 2 40.19 odd 2
320.5.p.i.193.1 2 40.37 odd 4
320.5.p.i.257.1 2 40.29 even 2
400.5.p.c.193.1 2 5.3 odd 4 inner
400.5.p.c.257.1 2 1.1 even 1 trivial
450.5.g.a.307.1 2 12.11 even 2
450.5.g.a.343.1 2 60.23 odd 4