Properties

Label 320.6.c.j.129.1
Level $320$
Weight $6$
Character 320.129
Analytic conductor $51.323$
Analytic rank $0$
Dimension $8$
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] = [320,6,Mod(129,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), 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: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 41x^{6} + 460x^{4} + 969x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{31}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(1.64654i\) of defining polynomial
Character \(\chi\) \(=\) 320.129
Dual form 320.6.c.j.129.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-28.9338i q^{3} +(13.1588 - 54.3309i) q^{5} -146.828i q^{7} -594.165 q^{9} +O(q^{10})\) \(q-28.9338i q^{3} +(13.1588 - 54.3309i) q^{5} -146.828i q^{7} -594.165 q^{9} -191.129 q^{11} -83.9971i q^{13} +(-1572.00 - 380.734i) q^{15} +2000.23i q^{17} +677.481 q^{19} -4248.29 q^{21} +1296.23i q^{23} +(-2778.69 - 1429.86i) q^{25} +10160.5i q^{27} -3266.00 q^{29} +6157.98 q^{31} +5530.08i q^{33} +(-7977.29 - 1932.08i) q^{35} -11369.5i q^{37} -2430.36 q^{39} -10599.8 q^{41} -12926.2i q^{43} +(-7818.50 + 32281.5i) q^{45} -9521.88i q^{47} -4751.45 q^{49} +57874.4 q^{51} -14786.8i q^{53} +(-2515.03 + 10384.2i) q^{55} -19602.1i q^{57} -38226.9 q^{59} +3581.69 q^{61} +87240.0i q^{63} +(-4563.64 - 1105.30i) q^{65} +21780.0i q^{67} +37504.9 q^{69} +51390.1 q^{71} -13305.9i q^{73} +(-41371.3 + 80398.1i) q^{75} +28063.1i q^{77} -15944.3 q^{79} +149601. q^{81} +53305.0i q^{83} +(108675. + 26320.7i) q^{85} +94497.8i q^{87} +51330.4 q^{89} -12333.1 q^{91} -178174. i q^{93} +(8914.84 - 36808.1i) q^{95} -80849.9i q^{97} +113562. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 1000 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 1000 q^{9} + 736 q^{11} - 992 q^{15} - 1376 q^{19} - 1984 q^{21} - 2136 q^{25} - 5872 q^{29} + 4224 q^{31} - 19232 q^{35} - 3008 q^{39} + 23600 q^{41} + 28328 q^{45} - 45000 q^{49} + 124800 q^{51} + 15008 q^{55} - 91680 q^{59} - 123856 q^{61} - 72064 q^{65} + 76736 q^{69} - 125632 q^{71} - 222784 q^{75} + 43264 q^{79} + 409672 q^{81} + 293760 q^{85} - 41904 q^{89} + 487616 q^{91} + 442592 q^{95} - 266848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 28.9338i 1.85610i −0.372451 0.928052i \(-0.621482\pi\)
0.372451 0.928052i \(-0.378518\pi\)
\(4\) 0 0
\(5\) 13.1588 54.3309i 0.235392 0.971901i
\(6\) 0 0
\(7\) 146.828i 1.13257i −0.824211 0.566283i \(-0.808381\pi\)
0.824211 0.566283i \(-0.191619\pi\)
\(8\) 0 0
\(9\) −594.165 −2.44512
\(10\) 0 0
\(11\) −191.129 −0.476260 −0.238130 0.971233i \(-0.576534\pi\)
−0.238130 + 0.971233i \(0.576534\pi\)
\(12\) 0 0
\(13\) 83.9971i 0.137850i −0.997622 0.0689249i \(-0.978043\pi\)
0.997622 0.0689249i \(-0.0219569\pi\)
\(14\) 0 0
\(15\) −1572.00 380.734i −1.80395 0.436912i
\(16\) 0 0
\(17\) 2000.23i 1.67864i 0.543635 + 0.839322i \(0.317048\pi\)
−0.543635 + 0.839322i \(0.682952\pi\)
\(18\) 0 0
\(19\) 677.481 0.430540 0.215270 0.976555i \(-0.430937\pi\)
0.215270 + 0.976555i \(0.430937\pi\)
\(20\) 0 0
\(21\) −4248.29 −2.10216
\(22\) 0 0
\(23\) 1296.23i 0.510931i 0.966818 + 0.255466i \(0.0822288\pi\)
−0.966818 + 0.255466i \(0.917771\pi\)
\(24\) 0 0
\(25\) −2778.69 1429.86i −0.889181 0.457555i
\(26\) 0 0
\(27\) 10160.5i 2.68230i
\(28\) 0 0
\(29\) −3266.00 −0.721143 −0.360571 0.932732i \(-0.617418\pi\)
−0.360571 + 0.932732i \(0.617418\pi\)
\(30\) 0 0
\(31\) 6157.98 1.15089 0.575445 0.817841i \(-0.304829\pi\)
0.575445 + 0.817841i \(0.304829\pi\)
\(32\) 0 0
\(33\) 5530.08i 0.883989i
\(34\) 0 0
\(35\) −7977.29 1932.08i −1.10074 0.266597i
\(36\) 0 0
\(37\) 11369.5i 1.36533i −0.730730 0.682667i \(-0.760820\pi\)
0.730730 0.682667i \(-0.239180\pi\)
\(38\) 0 0
\(39\) −2430.36 −0.255863
\(40\) 0 0
\(41\) −10599.8 −0.984774 −0.492387 0.870376i \(-0.663876\pi\)
−0.492387 + 0.870376i \(0.663876\pi\)
\(42\) 0 0
\(43\) 12926.2i 1.06610i −0.846082 0.533052i \(-0.821045\pi\)
0.846082 0.533052i \(-0.178955\pi\)
\(44\) 0 0
\(45\) −7818.50 + 32281.5i −0.575562 + 2.37642i
\(46\) 0 0
\(47\) 9521.88i 0.628750i −0.949299 0.314375i \(-0.898205\pi\)
0.949299 0.314375i \(-0.101795\pi\)
\(48\) 0 0
\(49\) −4751.45 −0.282706
\(50\) 0 0
\(51\) 57874.4 3.11574
\(52\) 0 0
\(53\) 14786.8i 0.723078i −0.932357 0.361539i \(-0.882251\pi\)
0.932357 0.361539i \(-0.117749\pi\)
\(54\) 0 0
\(55\) −2515.03 + 10384.2i −0.112108 + 0.462878i
\(56\) 0 0
\(57\) 19602.1i 0.799126i
\(58\) 0 0
\(59\) −38226.9 −1.42968 −0.714841 0.699287i \(-0.753501\pi\)
−0.714841 + 0.699287i \(0.753501\pi\)
\(60\) 0 0
\(61\) 3581.69 0.123243 0.0616217 0.998100i \(-0.480373\pi\)
0.0616217 + 0.998100i \(0.480373\pi\)
\(62\) 0 0
\(63\) 87240.0i 2.76926i
\(64\) 0 0
\(65\) −4563.64 1105.30i −0.133976 0.0324487i
\(66\) 0 0
\(67\) 21780.0i 0.592748i 0.955072 + 0.296374i \(0.0957774\pi\)
−0.955072 + 0.296374i \(0.904223\pi\)
\(68\) 0 0
\(69\) 37504.9 0.948341
\(70\) 0 0
\(71\) 51390.1 1.20986 0.604928 0.796280i \(-0.293202\pi\)
0.604928 + 0.796280i \(0.293202\pi\)
\(72\) 0 0
\(73\) 13305.9i 0.292238i −0.989267 0.146119i \(-0.953322\pi\)
0.989267 0.146119i \(-0.0466782\pi\)
\(74\) 0 0
\(75\) −41371.3 + 80398.1i −0.849270 + 1.65041i
\(76\) 0 0
\(77\) 28063.1i 0.539396i
\(78\) 0 0
\(79\) −15944.3 −0.287434 −0.143717 0.989619i \(-0.545906\pi\)
−0.143717 + 0.989619i \(0.545906\pi\)
\(80\) 0 0
\(81\) 149601. 2.53350
\(82\) 0 0
\(83\) 53305.0i 0.849323i 0.905352 + 0.424662i \(0.139607\pi\)
−0.905352 + 0.424662i \(0.860393\pi\)
\(84\) 0 0
\(85\) 108675. + 26320.7i 1.63148 + 0.395139i
\(86\) 0 0
\(87\) 94497.8i 1.33852i
\(88\) 0 0
\(89\) 51330.4 0.686910 0.343455 0.939169i \(-0.388403\pi\)
0.343455 + 0.939169i \(0.388403\pi\)
\(90\) 0 0
\(91\) −12333.1 −0.156124
\(92\) 0 0
\(93\) 178174.i 2.13617i
\(94\) 0 0
\(95\) 8914.84 36808.1i 0.101346 0.418442i
\(96\) 0 0
\(97\) 80849.9i 0.872469i −0.899833 0.436234i \(-0.856312\pi\)
0.899833 0.436234i \(-0.143688\pi\)
\(98\) 0 0
\(99\) 113562. 1.16451
\(100\) 0 0
\(101\) −89471.2 −0.872729 −0.436365 0.899770i \(-0.643734\pi\)
−0.436365 + 0.899770i \(0.643734\pi\)
\(102\) 0 0
\(103\) 44184.6i 0.410372i −0.978723 0.205186i \(-0.934220\pi\)
0.978723 0.205186i \(-0.0657799\pi\)
\(104\) 0 0
\(105\) −55902.4 + 230813.i −0.494832 + 2.04309i
\(106\) 0 0
\(107\) 56446.0i 0.476622i 0.971189 + 0.238311i \(0.0765937\pi\)
−0.971189 + 0.238311i \(0.923406\pi\)
\(108\) 0 0
\(109\) 52651.1 0.424465 0.212232 0.977219i \(-0.431927\pi\)
0.212232 + 0.977219i \(0.431927\pi\)
\(110\) 0 0
\(111\) −328964. −2.53420
\(112\) 0 0
\(113\) 78399.6i 0.577587i −0.957391 0.288794i \(-0.906746\pi\)
0.957391 0.288794i \(-0.0932542\pi\)
\(114\) 0 0
\(115\) 70425.3 + 17056.8i 0.496574 + 0.120269i
\(116\) 0 0
\(117\) 49908.1i 0.337059i
\(118\) 0 0
\(119\) 293690. 1.90118
\(120\) 0 0
\(121\) −124521. −0.773176
\(122\) 0 0
\(123\) 306691.i 1.82784i
\(124\) 0 0
\(125\) −114250. + 132154.i −0.654004 + 0.756491i
\(126\) 0 0
\(127\) 296102.i 1.62904i 0.580133 + 0.814522i \(0.303001\pi\)
−0.580133 + 0.814522i \(0.696999\pi\)
\(128\) 0 0
\(129\) −374004. −1.97880
\(130\) 0 0
\(131\) 199030. 1.01331 0.506653 0.862150i \(-0.330883\pi\)
0.506653 + 0.862150i \(0.330883\pi\)
\(132\) 0 0
\(133\) 99473.2i 0.487615i
\(134\) 0 0
\(135\) 552031. + 133700.i 2.60693 + 0.631391i
\(136\) 0 0
\(137\) 361224.i 1.64428i 0.569288 + 0.822138i \(0.307219\pi\)
−0.569288 + 0.822138i \(0.692781\pi\)
\(138\) 0 0
\(139\) −257315. −1.12961 −0.564805 0.825224i \(-0.691049\pi\)
−0.564805 + 0.825224i \(0.691049\pi\)
\(140\) 0 0
\(141\) −275504. −1.16702
\(142\) 0 0
\(143\) 16054.3i 0.0656524i
\(144\) 0 0
\(145\) −42976.7 + 177445.i −0.169751 + 0.700879i
\(146\) 0 0
\(147\) 137477.i 0.524733i
\(148\) 0 0
\(149\) −478333. −1.76508 −0.882540 0.470238i \(-0.844168\pi\)
−0.882540 + 0.470238i \(0.844168\pi\)
\(150\) 0 0
\(151\) −193483. −0.690558 −0.345279 0.938500i \(-0.612216\pi\)
−0.345279 + 0.938500i \(0.612216\pi\)
\(152\) 0 0
\(153\) 1.18847e6i 4.10449i
\(154\) 0 0
\(155\) 81031.6 334568.i 0.270910 1.11855i
\(156\) 0 0
\(157\) 460294.i 1.49034i 0.666873 + 0.745172i \(0.267633\pi\)
−0.666873 + 0.745172i \(0.732367\pi\)
\(158\) 0 0
\(159\) −427839. −1.34211
\(160\) 0 0
\(161\) 190323. 0.578663
\(162\) 0 0
\(163\) 372211.i 1.09729i −0.836057 0.548643i \(-0.815145\pi\)
0.836057 0.548643i \(-0.184855\pi\)
\(164\) 0 0
\(165\) 300454. + 72769.3i 0.859149 + 0.208084i
\(166\) 0 0
\(167\) 423266.i 1.17442i −0.809436 0.587208i \(-0.800227\pi\)
0.809436 0.587208i \(-0.199773\pi\)
\(168\) 0 0
\(169\) 364237. 0.980997
\(170\) 0 0
\(171\) −402535. −1.05272
\(172\) 0 0
\(173\) 223915.i 0.568810i −0.958704 0.284405i \(-0.908204\pi\)
0.958704 0.284405i \(-0.0917961\pi\)
\(174\) 0 0
\(175\) −209943. + 407990.i −0.518211 + 1.00706i
\(176\) 0 0
\(177\) 1.10605e6i 2.65364i
\(178\) 0 0
\(179\) −29324.8 −0.0684072 −0.0342036 0.999415i \(-0.510889\pi\)
−0.0342036 + 0.999415i \(0.510889\pi\)
\(180\) 0 0
\(181\) 46594.3 0.105715 0.0528574 0.998602i \(-0.483167\pi\)
0.0528574 + 0.998602i \(0.483167\pi\)
\(182\) 0 0
\(183\) 103632.i 0.228752i
\(184\) 0 0
\(185\) −617718. 149610.i −1.32697 0.321388i
\(186\) 0 0
\(187\) 382302.i 0.799472i
\(188\) 0 0
\(189\) 1.49185e6 3.03788
\(190\) 0 0
\(191\) 327452. 0.649477 0.324739 0.945804i \(-0.394724\pi\)
0.324739 + 0.945804i \(0.394724\pi\)
\(192\) 0 0
\(193\) 406622.i 0.785773i −0.919587 0.392887i \(-0.871476\pi\)
0.919587 0.392887i \(-0.128524\pi\)
\(194\) 0 0
\(195\) −31980.6 + 132043.i −0.0602282 + 0.248674i
\(196\) 0 0
\(197\) 364071.i 0.668376i −0.942506 0.334188i \(-0.891538\pi\)
0.942506 0.334188i \(-0.108462\pi\)
\(198\) 0 0
\(199\) −55181.3 −0.0987777 −0.0493889 0.998780i \(-0.515727\pi\)
−0.0493889 + 0.998780i \(0.515727\pi\)
\(200\) 0 0
\(201\) 630177. 1.10020
\(202\) 0 0
\(203\) 479540.i 0.816742i
\(204\) 0 0
\(205\) −139480. + 575895.i −0.231808 + 0.957102i
\(206\) 0 0
\(207\) 770174.i 1.24929i
\(208\) 0 0
\(209\) −129486. −0.205049
\(210\) 0 0
\(211\) 831145. 1.28520 0.642600 0.766202i \(-0.277856\pi\)
0.642600 + 0.766202i \(0.277856\pi\)
\(212\) 0 0
\(213\) 1.48691e6i 2.24562i
\(214\) 0 0
\(215\) −702292. 170093.i −1.03615 0.250952i
\(216\) 0 0
\(217\) 904163.i 1.30346i
\(218\) 0 0
\(219\) −384989. −0.542423
\(220\) 0 0
\(221\) 168014. 0.231401
\(222\) 0 0
\(223\) 458628.i 0.617587i −0.951129 0.308793i \(-0.900075\pi\)
0.951129 0.308793i \(-0.0999252\pi\)
\(224\) 0 0
\(225\) 1.65100e6 + 849572.i 2.17416 + 1.11878i
\(226\) 0 0
\(227\) 22779.7i 0.0293416i −0.999892 0.0146708i \(-0.995330\pi\)
0.999892 0.0146708i \(-0.00467003\pi\)
\(228\) 0 0
\(229\) −1.36560e6 −1.72082 −0.860408 0.509606i \(-0.829791\pi\)
−0.860408 + 0.509606i \(0.829791\pi\)
\(230\) 0 0
\(231\) 811971. 1.00118
\(232\) 0 0
\(233\) 673470.i 0.812696i 0.913718 + 0.406348i \(0.133198\pi\)
−0.913718 + 0.406348i \(0.866802\pi\)
\(234\) 0 0
\(235\) −517332. 125297.i −0.611082 0.148003i
\(236\) 0 0
\(237\) 461330.i 0.533508i
\(238\) 0 0
\(239\) −461339. −0.522426 −0.261213 0.965281i \(-0.584123\pi\)
−0.261213 + 0.965281i \(0.584123\pi\)
\(240\) 0 0
\(241\) −842868. −0.934796 −0.467398 0.884047i \(-0.654809\pi\)
−0.467398 + 0.884047i \(0.654809\pi\)
\(242\) 0 0
\(243\) 1.85951e6i 2.02014i
\(244\) 0 0
\(245\) −62523.4 + 258150.i −0.0665468 + 0.274763i
\(246\) 0 0
\(247\) 56906.4i 0.0593498i
\(248\) 0 0
\(249\) 1.54232e6 1.57643
\(250\) 0 0
\(251\) 1.34039e6 1.34291 0.671457 0.741044i \(-0.265669\pi\)
0.671457 + 0.741044i \(0.265669\pi\)
\(252\) 0 0
\(253\) 247747.i 0.243336i
\(254\) 0 0
\(255\) 761558. 3.14437e6i 0.733419 3.02819i
\(256\) 0 0
\(257\) 337841.i 0.319066i −0.987193 0.159533i \(-0.949001\pi\)
0.987193 0.159533i \(-0.0509988\pi\)
\(258\) 0 0
\(259\) −1.66937e6 −1.54633
\(260\) 0 0
\(261\) 1.94054e6 1.76328
\(262\) 0 0
\(263\) 836410.i 0.745641i −0.927903 0.372821i \(-0.878391\pi\)
0.927903 0.372821i \(-0.121609\pi\)
\(264\) 0 0
\(265\) −803381. 194577.i −0.702760 0.170207i
\(266\) 0 0
\(267\) 1.48518e6i 1.27498i
\(268\) 0 0
\(269\) −36624.0 −0.0308592 −0.0154296 0.999881i \(-0.504912\pi\)
−0.0154296 + 0.999881i \(0.504912\pi\)
\(270\) 0 0
\(271\) 1.92834e6 1.59500 0.797500 0.603319i \(-0.206155\pi\)
0.797500 + 0.603319i \(0.206155\pi\)
\(272\) 0 0
\(273\) 356844.i 0.289782i
\(274\) 0 0
\(275\) 531088. + 273287.i 0.423482 + 0.217915i
\(276\) 0 0
\(277\) 1.42694e6i 1.11739i 0.829373 + 0.558696i \(0.188698\pi\)
−0.829373 + 0.558696i \(0.811302\pi\)
\(278\) 0 0
\(279\) −3.65885e6 −2.81407
\(280\) 0 0
\(281\) −2.36724e6 −1.78845 −0.894223 0.447621i \(-0.852271\pi\)
−0.894223 + 0.447621i \(0.852271\pi\)
\(282\) 0 0
\(283\) 1.35021e6i 1.00215i −0.865403 0.501077i \(-0.832937\pi\)
0.865403 0.501077i \(-0.167063\pi\)
\(284\) 0 0
\(285\) −1.06500e6 257940.i −0.776671 0.188108i
\(286\) 0 0
\(287\) 1.55634e6i 1.11532i
\(288\) 0 0
\(289\) −2.58108e6 −1.81785
\(290\) 0 0
\(291\) −2.33929e6 −1.61939
\(292\) 0 0
\(293\) 506336.i 0.344564i −0.985048 0.172282i \(-0.944886\pi\)
0.985048 0.172282i \(-0.0551140\pi\)
\(294\) 0 0
\(295\) −503021. + 2.07690e6i −0.336536 + 1.38951i
\(296\) 0 0
\(297\) 1.94197e6i 1.27747i
\(298\) 0 0
\(299\) 108880. 0.0704317
\(300\) 0 0
\(301\) −1.89793e6 −1.20743
\(302\) 0 0
\(303\) 2.58874e6i 1.61988i
\(304\) 0 0
\(305\) 47130.8 194596.i 0.0290105 0.119780i
\(306\) 0 0
\(307\) 1.15053e6i 0.696712i 0.937362 + 0.348356i \(0.113260\pi\)
−0.937362 + 0.348356i \(0.886740\pi\)
\(308\) 0 0
\(309\) −1.27843e6 −0.761693
\(310\) 0 0
\(311\) −742231. −0.435149 −0.217574 0.976044i \(-0.569815\pi\)
−0.217574 + 0.976044i \(0.569815\pi\)
\(312\) 0 0
\(313\) 1.67101e6i 0.964090i 0.876146 + 0.482045i \(0.160106\pi\)
−0.876146 + 0.482045i \(0.839894\pi\)
\(314\) 0 0
\(315\) 4.73983e6 + 1.14797e6i 2.69145 + 0.651862i
\(316\) 0 0
\(317\) 646531.i 0.361361i −0.983542 0.180681i \(-0.942170\pi\)
0.983542 0.180681i \(-0.0578300\pi\)
\(318\) 0 0
\(319\) 624227. 0.343452
\(320\) 0 0
\(321\) 1.63320e6 0.884660
\(322\) 0 0
\(323\) 1.35512e6i 0.722723i
\(324\) 0 0
\(325\) −120104. + 233402.i −0.0630738 + 0.122573i
\(326\) 0 0
\(327\) 1.52340e6i 0.787851i
\(328\) 0 0
\(329\) −1.39808e6 −0.712101
\(330\) 0 0
\(331\) −2.73052e6 −1.36986 −0.684928 0.728610i \(-0.740166\pi\)
−0.684928 + 0.728610i \(0.740166\pi\)
\(332\) 0 0
\(333\) 6.75538e6i 3.33841i
\(334\) 0 0
\(335\) 1.18332e6 + 286598.i 0.576092 + 0.139528i
\(336\) 0 0
\(337\) 390252.i 0.187185i −0.995611 0.0935923i \(-0.970165\pi\)
0.995611 0.0935923i \(-0.0298350\pi\)
\(338\) 0 0
\(339\) −2.26840e6 −1.07206
\(340\) 0 0
\(341\) −1.17697e6 −0.548123
\(342\) 0 0
\(343\) 1.77009e6i 0.812382i
\(344\) 0 0
\(345\) 493519. 2.03767e6i 0.223232 0.921694i
\(346\) 0 0
\(347\) 2.23128e6i 0.994788i −0.867525 0.497394i \(-0.834290\pi\)
0.867525 0.497394i \(-0.165710\pi\)
\(348\) 0 0
\(349\) −502679. −0.220916 −0.110458 0.993881i \(-0.535232\pi\)
−0.110458 + 0.993881i \(0.535232\pi\)
\(350\) 0 0
\(351\) 853455. 0.369754
\(352\) 0 0
\(353\) 4.57148e6i 1.95263i 0.216350 + 0.976316i \(0.430585\pi\)
−0.216350 + 0.976316i \(0.569415\pi\)
\(354\) 0 0
\(355\) 676232. 2.79207e6i 0.284790 1.17586i
\(356\) 0 0
\(357\) 8.49758e6i 3.52878i
\(358\) 0 0
\(359\) 448308. 0.183586 0.0917931 0.995778i \(-0.470740\pi\)
0.0917931 + 0.995778i \(0.470740\pi\)
\(360\) 0 0
\(361\) −2.01712e6 −0.814636
\(362\) 0 0
\(363\) 3.60286e6i 1.43510i
\(364\) 0 0
\(365\) −722920. 175089.i −0.284026 0.0687903i
\(366\) 0 0
\(367\) 2.56484e6i 0.994018i −0.867745 0.497009i \(-0.834432\pi\)
0.867745 0.497009i \(-0.165568\pi\)
\(368\) 0 0
\(369\) 6.29801e6 2.40789
\(370\) 0 0
\(371\) −2.17112e6 −0.818934
\(372\) 0 0
\(373\) 484728.i 0.180396i −0.995924 0.0901978i \(-0.971250\pi\)
0.995924 0.0901978i \(-0.0287499\pi\)
\(374\) 0 0
\(375\) 3.82370e6 + 3.30568e6i 1.40413 + 1.21390i
\(376\) 0 0
\(377\) 274335.i 0.0994094i
\(378\) 0 0
\(379\) 4.26934e6 1.52673 0.763366 0.645966i \(-0.223545\pi\)
0.763366 + 0.645966i \(0.223545\pi\)
\(380\) 0 0
\(381\) 8.56737e6 3.02367
\(382\) 0 0
\(383\) 1.27791e6i 0.445147i −0.974916 0.222573i \(-0.928554\pi\)
0.974916 0.222573i \(-0.0714457\pi\)
\(384\) 0 0
\(385\) 1.52469e6 + 369276.i 0.524240 + 0.126970i
\(386\) 0 0
\(387\) 7.68029e6i 2.60676i
\(388\) 0 0
\(389\) 3.37341e6 1.13030 0.565151 0.824987i \(-0.308818\pi\)
0.565151 + 0.824987i \(0.308818\pi\)
\(390\) 0 0
\(391\) −2.59276e6 −0.857672
\(392\) 0 0
\(393\) 5.75870e6i 1.88080i
\(394\) 0 0
\(395\) −209808. + 866270.i −0.0676597 + 0.279357i
\(396\) 0 0
\(397\) 1.82804e6i 0.582117i 0.956705 + 0.291058i \(0.0940074\pi\)
−0.956705 + 0.291058i \(0.905993\pi\)
\(398\) 0 0
\(399\) −2.87814e6 −0.905064
\(400\) 0 0
\(401\) 1.34118e6 0.416511 0.208256 0.978074i \(-0.433221\pi\)
0.208256 + 0.978074i \(0.433221\pi\)
\(402\) 0 0
\(403\) 517252.i 0.158650i
\(404\) 0 0
\(405\) 1.96857e6 8.12794e6i 0.596365 2.46231i
\(406\) 0 0
\(407\) 2.17305e6i 0.650254i
\(408\) 0 0
\(409\) 803937. 0.237637 0.118818 0.992916i \(-0.462089\pi\)
0.118818 + 0.992916i \(0.462089\pi\)
\(410\) 0 0
\(411\) 1.04516e7 3.05195
\(412\) 0 0
\(413\) 5.61278e6i 1.61921i
\(414\) 0 0
\(415\) 2.89611e6 + 701430.i 0.825458 + 0.199924i
\(416\) 0 0
\(417\) 7.44511e6i 2.09667i
\(418\) 0 0
\(419\) −3.90295e6 −1.08607 −0.543035 0.839710i \(-0.682725\pi\)
−0.543035 + 0.839710i \(0.682725\pi\)
\(420\) 0 0
\(421\) −3.04306e6 −0.836769 −0.418384 0.908270i \(-0.637403\pi\)
−0.418384 + 0.908270i \(0.637403\pi\)
\(422\) 0 0
\(423\) 5.65756e6i 1.53737i
\(424\) 0 0
\(425\) 2.86005e6 5.55804e6i 0.768072 1.49262i
\(426\) 0 0
\(427\) 525892.i 0.139581i
\(428\) 0 0
\(429\) 464511. 0.121858
\(430\) 0 0
\(431\) 772580. 0.200332 0.100166 0.994971i \(-0.468063\pi\)
0.100166 + 0.994971i \(0.468063\pi\)
\(432\) 0 0
\(433\) 4.83430e6i 1.23912i 0.784948 + 0.619561i \(0.212689\pi\)
−0.784948 + 0.619561i \(0.787311\pi\)
\(434\) 0 0
\(435\) 5.13415e6 + 1.24348e6i 1.30090 + 0.315076i
\(436\) 0 0
\(437\) 878171.i 0.219976i
\(438\) 0 0
\(439\) −4.98655e6 −1.23492 −0.617460 0.786602i \(-0.711838\pi\)
−0.617460 + 0.786602i \(0.711838\pi\)
\(440\) 0 0
\(441\) 2.82314e6 0.691252
\(442\) 0 0
\(443\) 7.63238e6i 1.84778i 0.382655 + 0.923891i \(0.375010\pi\)
−0.382655 + 0.923891i \(0.624990\pi\)
\(444\) 0 0
\(445\) 675447. 2.78883e6i 0.161693 0.667608i
\(446\) 0 0
\(447\) 1.38400e7i 3.27617i
\(448\) 0 0
\(449\) −1.92156e6 −0.449819 −0.224910 0.974380i \(-0.572209\pi\)
−0.224910 + 0.974380i \(0.572209\pi\)
\(450\) 0 0
\(451\) 2.02592e6 0.469009
\(452\) 0 0
\(453\) 5.59820e6i 1.28175i
\(454\) 0 0
\(455\) −162289. + 670070.i −0.0367503 + 0.151737i
\(456\) 0 0
\(457\) 6.64057e6i 1.48735i −0.668539 0.743677i \(-0.733080\pi\)
0.668539 0.743677i \(-0.266920\pi\)
\(458\) 0 0
\(459\) −2.03234e7 −4.50262
\(460\) 0 0
\(461\) −5.78658e6 −1.26815 −0.634074 0.773272i \(-0.718619\pi\)
−0.634074 + 0.773272i \(0.718619\pi\)
\(462\) 0 0
\(463\) 4.55738e6i 0.988013i −0.869458 0.494007i \(-0.835532\pi\)
0.869458 0.494007i \(-0.164468\pi\)
\(464\) 0 0
\(465\) −9.68033e6 2.34455e6i −2.07615 0.502837i
\(466\) 0 0
\(467\) 1.02351e6i 0.217171i 0.994087 + 0.108585i \(0.0346321\pi\)
−0.994087 + 0.108585i \(0.965368\pi\)
\(468\) 0 0
\(469\) 3.19791e6 0.671326
\(470\) 0 0
\(471\) 1.33181e7 2.76623
\(472\) 0 0
\(473\) 2.47057e6i 0.507743i
\(474\) 0 0
\(475\) −1.88251e6 968703.i −0.382828 0.196996i
\(476\) 0 0
\(477\) 8.78581e6i 1.76801i
\(478\) 0 0
\(479\) −5.42421e6 −1.08018 −0.540092 0.841606i \(-0.681610\pi\)
−0.540092 + 0.841606i \(0.681610\pi\)
\(480\) 0 0
\(481\) −955009. −0.188211
\(482\) 0 0
\(483\) 5.50676e6i 1.07406i
\(484\) 0 0
\(485\) −4.39265e6 1.06389e6i −0.847953 0.205372i
\(486\) 0 0
\(487\) 8.76364e6i 1.67441i −0.546888 0.837206i \(-0.684188\pi\)
0.546888 0.837206i \(-0.315812\pi\)
\(488\) 0 0
\(489\) −1.07695e7 −2.03668
\(490\) 0 0
\(491\) −2.44420e6 −0.457545 −0.228772 0.973480i \(-0.573471\pi\)
−0.228772 + 0.973480i \(0.573471\pi\)
\(492\) 0 0
\(493\) 6.53277e6i 1.21054i
\(494\) 0 0
\(495\) 1.49434e6 6.16992e6i 0.274117 1.13179i
\(496\) 0 0
\(497\) 7.54550e6i 1.37024i
\(498\) 0 0
\(499\) 5.77888e6 1.03894 0.519472 0.854487i \(-0.326129\pi\)
0.519472 + 0.854487i \(0.326129\pi\)
\(500\) 0 0
\(501\) −1.22467e7 −2.17984
\(502\) 0 0
\(503\) 5.58777e6i 0.984733i 0.870388 + 0.492367i \(0.163868\pi\)
−0.870388 + 0.492367i \(0.836132\pi\)
\(504\) 0 0
\(505\) −1.17733e6 + 4.86105e6i −0.205433 + 0.848206i
\(506\) 0 0
\(507\) 1.05388e7i 1.82083i
\(508\) 0 0
\(509\) −9.32211e6 −1.59485 −0.797425 0.603418i \(-0.793805\pi\)
−0.797425 + 0.603418i \(0.793805\pi\)
\(510\) 0 0
\(511\) −1.95367e6 −0.330978
\(512\) 0 0
\(513\) 6.88357e6i 1.15484i
\(514\) 0 0
\(515\) −2.40059e6 581416.i −0.398841 0.0965983i
\(516\) 0 0
\(517\) 1.81990e6i 0.299449i
\(518\) 0 0
\(519\) −6.47871e6 −1.05577
\(520\) 0 0
\(521\) 7.51696e6 1.21324 0.606622 0.794991i \(-0.292524\pi\)
0.606622 + 0.794991i \(0.292524\pi\)
\(522\) 0 0
\(523\) 618460.i 0.0988683i 0.998777 + 0.0494342i \(0.0157418\pi\)
−0.998777 + 0.0494342i \(0.984258\pi\)
\(524\) 0 0
\(525\) 1.18047e7 + 6.07446e6i 1.86920 + 0.961854i
\(526\) 0 0
\(527\) 1.23174e7i 1.93193i
\(528\) 0 0
\(529\) 4.75613e6 0.738949
\(530\) 0 0
\(531\) 2.27131e7 3.49575
\(532\) 0 0
\(533\) 890350.i 0.135751i
\(534\) 0 0
\(535\) 3.06676e6 + 742762.i 0.463229 + 0.112193i
\(536\) 0 0
\(537\) 848477.i 0.126971i
\(538\) 0 0
\(539\) 908138. 0.134642
\(540\) 0 0
\(541\) 7.78044e6 1.14291 0.571454 0.820634i \(-0.306380\pi\)
0.571454 + 0.820634i \(0.306380\pi\)
\(542\) 0 0
\(543\) 1.34815e6i 0.196218i
\(544\) 0 0
\(545\) 692826. 2.86058e6i 0.0999155 0.412537i
\(546\) 0 0
\(547\) 6.38809e6i 0.912857i 0.889760 + 0.456428i \(0.150872\pi\)
−0.889760 + 0.456428i \(0.849128\pi\)
\(548\) 0 0
\(549\) −2.12811e6 −0.301345
\(550\) 0 0
\(551\) −2.21265e6 −0.310481
\(552\) 0 0
\(553\) 2.34107e6i 0.325538i
\(554\) 0 0
\(555\) −4.32878e6 + 1.78729e7i −0.596530 + 2.46299i
\(556\) 0 0
\(557\) 8.07422e6i 1.10271i −0.834270 0.551356i \(-0.814110\pi\)
0.834270 0.551356i \(-0.185890\pi\)
\(558\) 0 0
\(559\) −1.08576e6 −0.146962
\(560\) 0 0
\(561\) −1.10615e7 −1.48390
\(562\) 0 0
\(563\) 5.02445e6i 0.668064i −0.942562 0.334032i \(-0.891591\pi\)
0.942562 0.334032i \(-0.108409\pi\)
\(564\) 0 0
\(565\) −4.25952e6 1.03165e6i −0.561357 0.135959i
\(566\) 0 0
\(567\) 2.19656e7i 2.86936i
\(568\) 0 0
\(569\) −8.36647e6 −1.08333 −0.541666 0.840594i \(-0.682206\pi\)
−0.541666 + 0.840594i \(0.682206\pi\)
\(570\) 0 0
\(571\) 831558. 0.106734 0.0533670 0.998575i \(-0.483005\pi\)
0.0533670 + 0.998575i \(0.483005\pi\)
\(572\) 0 0
\(573\) 9.47443e6i 1.20550i
\(574\) 0 0
\(575\) 1.85343e6 3.60182e6i 0.233779 0.454310i
\(576\) 0 0
\(577\) 5.63868e6i 0.705080i 0.935797 + 0.352540i \(0.114682\pi\)
−0.935797 + 0.352540i \(0.885318\pi\)
\(578\) 0 0
\(579\) −1.17651e7 −1.45848
\(580\) 0 0
\(581\) 7.82667e6 0.961915
\(582\) 0 0
\(583\) 2.82619e6i 0.344373i
\(584\) 0 0
\(585\) 2.71155e6 + 656731.i 0.327588 + 0.0793411i
\(586\) 0 0
\(587\) 4.13074e6i 0.494803i −0.968913 0.247402i \(-0.920423\pi\)
0.968913 0.247402i \(-0.0795767\pi\)
\(588\) 0 0
\(589\) 4.17191e6 0.495504
\(590\) 0 0
\(591\) −1.05340e7 −1.24057
\(592\) 0 0
\(593\) 2.07710e6i 0.242561i −0.992618 0.121280i \(-0.961300\pi\)
0.992618 0.121280i \(-0.0387000\pi\)
\(594\) 0 0
\(595\) 3.86461e6 1.59565e7i 0.447521 1.84775i
\(596\) 0 0
\(597\) 1.59660e6i 0.183342i
\(598\) 0 0
\(599\) 1.44039e7 1.64026 0.820128 0.572180i \(-0.193902\pi\)
0.820128 + 0.572180i \(0.193902\pi\)
\(600\) 0 0
\(601\) 1.16092e7 1.31104 0.655522 0.755176i \(-0.272449\pi\)
0.655522 + 0.755176i \(0.272449\pi\)
\(602\) 0 0
\(603\) 1.29409e7i 1.44934i
\(604\) 0 0
\(605\) −1.63854e6 + 6.76533e6i −0.181999 + 0.751450i
\(606\) 0 0
\(607\) 1.75662e7i 1.93511i −0.252654 0.967557i \(-0.581304\pi\)
0.252654 0.967557i \(-0.418696\pi\)
\(608\) 0 0
\(609\) 1.38749e7 1.51596
\(610\) 0 0
\(611\) −799810. −0.0866730
\(612\) 0 0
\(613\) 1.76294e7i 1.89490i −0.319906 0.947449i \(-0.603651\pi\)
0.319906 0.947449i \(-0.396349\pi\)
\(614\) 0 0
\(615\) 1.66628e7 + 4.03569e6i 1.77648 + 0.430259i
\(616\) 0 0
\(617\) 6.44774e6i 0.681859i 0.940089 + 0.340930i \(0.110742\pi\)
−0.940089 + 0.340930i \(0.889258\pi\)
\(618\) 0 0
\(619\) −1.48717e7 −1.56003 −0.780017 0.625758i \(-0.784790\pi\)
−0.780017 + 0.625758i \(0.784790\pi\)
\(620\) 0 0
\(621\) −1.31704e7 −1.37047
\(622\) 0 0
\(623\) 7.53674e6i 0.777971i
\(624\) 0 0
\(625\) 5.67663e6 + 7.94628e6i 0.581287 + 0.813699i
\(626\) 0 0
\(627\) 3.74653e6i 0.380592i
\(628\) 0 0
\(629\) 2.27418e7 2.29191
\(630\) 0 0
\(631\) −5.05549e6 −0.505464 −0.252732 0.967536i \(-0.581329\pi\)
−0.252732 + 0.967536i \(0.581329\pi\)
\(632\) 0 0
\(633\) 2.40482e7i 2.38547i
\(634\) 0 0
\(635\) 1.60875e7 + 3.89635e6i 1.58327 + 0.383464i
\(636\) 0 0
\(637\) 399108.i 0.0389710i
\(638\) 0 0
\(639\) −3.05342e7 −2.95824
\(640\) 0 0
\(641\) 6.66243e6 0.640453 0.320226 0.947341i \(-0.396241\pi\)
0.320226 + 0.947341i \(0.396241\pi\)
\(642\) 0 0
\(643\) 3.29414e6i 0.314206i −0.987582 0.157103i \(-0.949784\pi\)
0.987582 0.157103i \(-0.0502155\pi\)
\(644\) 0 0
\(645\) −4.92145e6 + 2.03200e7i −0.465794 + 1.92320i
\(646\) 0 0
\(647\) 4.79219e6i 0.450063i −0.974351 0.225032i \(-0.927751\pi\)
0.974351 0.225032i \(-0.0722486\pi\)
\(648\) 0 0
\(649\) 7.30627e6 0.680901
\(650\) 0 0
\(651\) −2.61609e7 −2.41936
\(652\) 0 0
\(653\) 1.27962e7i 1.17435i 0.809459 + 0.587177i \(0.199761\pi\)
−0.809459 + 0.587177i \(0.800239\pi\)
\(654\) 0 0
\(655\) 2.61900e6 1.08135e7i 0.238524 0.984832i
\(656\) 0 0
\(657\) 7.90588e6i 0.714556i
\(658\) 0 0
\(659\) −3.06887e6 −0.275274 −0.137637 0.990483i \(-0.543951\pi\)
−0.137637 + 0.990483i \(0.543951\pi\)
\(660\) 0 0
\(661\) 1.79240e6 0.159563 0.0797814 0.996812i \(-0.474578\pi\)
0.0797814 + 0.996812i \(0.474578\pi\)
\(662\) 0 0
\(663\) 4.86128e6i 0.429504i
\(664\) 0 0
\(665\) −5.40447e6 1.30895e6i −0.473913 0.114781i
\(666\) 0 0
\(667\) 4.23349e6i 0.368454i
\(668\) 0 0
\(669\) −1.32698e7 −1.14631
\(670\) 0 0
\(671\) −684564. −0.0586959
\(672\) 0 0
\(673\) 1.37614e7i 1.17119i −0.810605 0.585593i \(-0.800862\pi\)
0.810605 0.585593i \(-0.199138\pi\)
\(674\) 0 0
\(675\) 1.45281e7 2.82330e7i 1.22730 2.38505i
\(676\) 0 0
\(677\) 2.01097e7i 1.68629i −0.537684 0.843147i \(-0.680700\pi\)
0.537684 0.843147i \(-0.319300\pi\)
\(678\) 0 0
\(679\) −1.18710e7 −0.988129
\(680\) 0 0
\(681\) −659104. −0.0544611
\(682\) 0 0
\(683\) 6.08331e6i 0.498986i −0.968377 0.249493i \(-0.919736\pi\)
0.968377 0.249493i \(-0.0802639\pi\)
\(684\) 0 0
\(685\) 1.96256e7 + 4.75327e6i 1.59807 + 0.387049i
\(686\) 0 0
\(687\) 3.95120e7i 3.19401i
\(688\) 0 0
\(689\) −1.24205e6 −0.0996761
\(690\) 0 0
\(691\) −2.23063e7 −1.77719 −0.888593 0.458697i \(-0.848316\pi\)
−0.888593 + 0.458697i \(0.848316\pi\)
\(692\) 0 0
\(693\) 1.66741e7i 1.31889i
\(694\) 0 0
\(695\) −3.38596e6 + 1.39802e7i −0.265901 + 1.09787i
\(696\) 0 0
\(697\) 2.12020e7i 1.65308i
\(698\) 0 0
\(699\) 1.94860e7 1.50845
\(700\) 0 0
\(701\) 2.01349e7 1.54758 0.773792 0.633439i \(-0.218357\pi\)
0.773792 + 0.633439i \(0.218357\pi\)
\(702\) 0 0
\(703\) 7.70265e6i 0.587830i
\(704\) 0 0
\(705\) −3.62530e6 + 1.49684e7i −0.274708 + 1.13423i
\(706\) 0 0
\(707\) 1.31369e7i 0.988424i
\(708\) 0 0
\(709\) −7.09436e6 −0.530027 −0.265013 0.964245i \(-0.585376\pi\)
−0.265013 + 0.964245i \(0.585376\pi\)
\(710\) 0 0
\(711\) 9.47356e6 0.702812
\(712\) 0 0
\(713\) 7.98215e6i 0.588026i
\(714\) 0 0
\(715\) 872243. + 211255.i 0.0638076 + 0.0154540i
\(716\) 0 0
\(717\) 1.33483e7i 0.969677i
\(718\) 0 0
\(719\) 1.57591e7 1.13686 0.568432 0.822730i \(-0.307550\pi\)
0.568432 + 0.822730i \(0.307550\pi\)
\(720\) 0 0
\(721\) −6.48753e6 −0.464774
\(722\) 0 0
\(723\) 2.43874e7i 1.73508i
\(724\) 0 0
\(725\) 9.07521e6 + 4.66992e6i 0.641227 + 0.329963i
\(726\) 0 0
\(727\) 1.93222e7i 1.35588i −0.735118 0.677939i \(-0.762873\pi\)
0.735118 0.677939i \(-0.237127\pi\)
\(728\) 0 0
\(729\) −1.74496e7 −1.21610
\(730\) 0 0
\(731\) 2.58554e7 1.78961
\(732\) 0 0
\(733\) 7.53614e6i 0.518071i 0.965868 + 0.259035i \(0.0834046\pi\)
−0.965868 + 0.259035i \(0.916595\pi\)
\(734\) 0 0
\(735\) 7.46927e6 + 1.80904e6i 0.509988 + 0.123518i
\(736\) 0 0
\(737\) 4.16278e6i 0.282302i
\(738\) 0 0
\(739\) −6.08996e6 −0.410207 −0.205104 0.978740i \(-0.565753\pi\)
−0.205104 + 0.978740i \(0.565753\pi\)
\(740\) 0 0
\(741\) −1.64652e6 −0.110159
\(742\) 0 0
\(743\) 5.29284e6i 0.351736i 0.984414 + 0.175868i \(0.0562731\pi\)
−0.984414 + 0.175868i \(0.943727\pi\)
\(744\) 0 0
\(745\) −6.29429e6 + 2.59882e7i −0.415485 + 1.71548i
\(746\) 0 0
\(747\) 3.16720e7i 2.07670i
\(748\) 0 0
\(749\) 8.28785e6 0.539806
\(750\) 0 0
\(751\) 1.58109e7 1.02296 0.511478 0.859296i \(-0.329098\pi\)
0.511478 + 0.859296i \(0.329098\pi\)
\(752\) 0 0
\(753\) 3.87827e7i 2.49259i
\(754\) 0 0
\(755\) −2.54600e6 + 1.05121e7i −0.162552 + 0.671154i
\(756\) 0 0
\(757\) 2.01450e7i 1.27770i 0.769332 + 0.638849i \(0.220589\pi\)
−0.769332 + 0.638849i \(0.779411\pi\)
\(758\) 0 0
\(759\) −7.16826e6 −0.451657
\(760\) 0 0
\(761\) −2.10320e7 −1.31650 −0.658248 0.752801i \(-0.728702\pi\)
−0.658248 + 0.752801i \(0.728702\pi\)
\(762\) 0 0
\(763\) 7.73066e6i 0.480734i
\(764\) 0 0
\(765\) −6.45706e7 1.56388e7i −3.98916 0.966164i
\(766\) 0 0
\(767\) 3.21095e6i 0.197081i
\(768\) 0 0
\(769\) 4.09332e6 0.249609 0.124804 0.992181i \(-0.460170\pi\)
0.124804 + 0.992181i \(0.460170\pi\)
\(770\) 0 0
\(771\) −9.77503e6 −0.592219
\(772\) 0 0
\(773\) 1.09247e7i 0.657599i 0.944400 + 0.328799i \(0.106644\pi\)
−0.944400 + 0.328799i \(0.893356\pi\)
\(774\) 0 0
\(775\) −1.71111e7 8.80504e6i −1.02335 0.526595i
\(776\) 0 0
\(777\) 4.83011e7i 2.87015i
\(778\) 0 0
\(779\) −7.18114e6 −0.423984
\(780\) 0 0
\(781\) −9.82213e6 −0.576206
\(782\) 0 0
\(783\) 3.31843e7i 1.93432i
\(784\) 0 0
\(785\) 2.50082e7 + 6.05692e6i 1.44847 + 0.350815i
\(786\) 0 0
\(787\) 1.38257e7i 0.795700i −0.917450 0.397850i \(-0.869756\pi\)
0.917450 0.397850i \(-0.130244\pi\)
\(788\) 0 0
\(789\) −2.42005e7 −1.38399
\(790\) 0 0
\(791\) −1.15113e7 −0.654156
\(792\) 0 0
\(793\) 300852.i 0.0169891i
\(794\) 0 0
\(795\) −5.62985e6 + 2.32449e7i −0.315921 + 1.30440i
\(796\) 0 0
\(797\) 1.38429e6i 0.0771937i −0.999255 0.0385969i \(-0.987711\pi\)
0.999255 0.0385969i \(-0.0122888\pi\)
\(798\) 0 0
\(799\) 1.90460e7 1.05545
\(800\) 0 0
\(801\) −3.04987e7 −1.67958
\(802\) 0 0
\(803\) 2.54313e6i 0.139181i
\(804\) 0 0
\(805\) 2.50442e6 1.03404e7i 0.136213 0.562403i
\(806\) 0 0
\(807\) 1.05967e6i 0.0572780i
\(808\) 0 0
\(809\) 2.08074e7 1.11776 0.558878 0.829250i \(-0.311232\pi\)
0.558878 + 0.829250i \(0.311232\pi\)
\(810\) 0 0
\(811\) 2.24363e7 1.19784 0.598919 0.800810i \(-0.295597\pi\)
0.598919 + 0.800810i \(0.295597\pi\)
\(812\) 0 0
\(813\) 5.57942e7i 2.96049i
\(814\) 0 0
\(815\) −2.02225e7 4.89785e6i −1.06645 0.258292i
\(816\) 0 0
\(817\) 8.75726e6i 0.459000i
\(818\) 0 0
\(819\) 7.32791e6 0.381742
\(820\) 0 0
\(821\) −1.30693e7 −0.676698 −0.338349 0.941021i \(-0.609869\pi\)
−0.338349 + 0.941021i \(0.609869\pi\)
\(822\) 0 0
\(823\) 2.09593e7i 1.07864i 0.842100 + 0.539321i \(0.181319\pi\)
−0.842100 + 0.539321i \(0.818681\pi\)
\(824\) 0 0
\(825\) 7.90724e6 1.53664e7i 0.404473 0.786026i
\(826\) 0 0
\(827\) 4.54670e6i 0.231171i −0.993298 0.115585i \(-0.963126\pi\)
0.993298 0.115585i \(-0.0368744\pi\)
\(828\) 0 0
\(829\) −2.05898e7 −1.04055 −0.520277 0.853997i \(-0.674171\pi\)
−0.520277 + 0.853997i \(0.674171\pi\)
\(830\) 0 0
\(831\) 4.12867e7 2.07399
\(832\) 0 0
\(833\) 9.50401e6i 0.474564i
\(834\) 0 0
\(835\) −2.29964e7 5.56967e6i −1.14142 0.276448i
\(836\) 0 0
\(837\) 6.25683e7i 3.08703i
\(838\) 0 0
\(839\) −1.09532e7 −0.537200 −0.268600 0.963252i \(-0.586561\pi\)
−0.268600 + 0.963252i \(0.586561\pi\)
\(840\) 0 0
\(841\) −9.84438e6 −0.479953
\(842\) 0 0
\(843\) 6.84932e7i 3.31954i
\(844\) 0 0
\(845\) 4.79293e6 1.97893e7i 0.230919 0.953432i
\(846\) 0 0
\(847\) 1.82831e7i 0.875673i
\(848\) 0 0
\(849\) −3.90666e7 −1.86010
\(850\) 0 0
\(851\) 1.47375e7 0.697592
\(852\) 0 0
\(853\) 1.27066e7i 0.597939i 0.954263 + 0.298970i \(0.0966429\pi\)
−0.954263 + 0.298970i \(0.903357\pi\)
\(854\) 0 0
\(855\) −5.29688e6 + 2.18701e7i −0.247802 + 1.02314i
\(856\) 0 0
\(857\) 7.46258e6i 0.347086i 0.984826 + 0.173543i \(0.0555216\pi\)
−0.984826 + 0.173543i \(0.944478\pi\)
\(858\) 0 0
\(859\) 1.13390e7 0.524315 0.262158 0.965025i \(-0.415566\pi\)
0.262158 + 0.965025i \(0.415566\pi\)
\(860\) 0 0
\(861\) 4.50309e7 2.07015
\(862\) 0 0
\(863\) 1.61275e7i 0.737121i 0.929604 + 0.368561i \(0.120149\pi\)
−0.929604 + 0.368561i \(0.879851\pi\)
\(864\) 0 0
\(865\) −1.21655e7 2.94645e6i −0.552827 0.133893i
\(866\) 0 0
\(867\) 7.46805e7i 3.37411i
\(868\) 0 0
\(869\) 3.04742e6 0.136894
\(870\) 0 0
\(871\) 1.82945e6 0.0817101
\(872\) 0 0
\(873\) 4.80381e7i 2.13329i
\(874\) 0 0
\(875\) 1.94038e7 + 1.67751e7i 0.856776 + 0.740703i
\(876\) 0 0
\(877\) 3.89336e7i 1.70933i −0.519182 0.854663i \(-0.673763\pi\)
0.519182 0.854663i \(-0.326237\pi\)
\(878\) 0 0
\(879\) −1.46502e7 −0.639546
\(880\) 0 0
\(881\) 3.57060e7 1.54989 0.774946 0.632027i \(-0.217777\pi\)
0.774946 + 0.632027i \(0.217777\pi\)
\(882\) 0 0
\(883\) 3.48264e7i 1.50317i −0.659638 0.751583i \(-0.729291\pi\)
0.659638 0.751583i \(-0.270709\pi\)
\(884\) 0 0
\(885\) 6.00927e7 + 1.45543e7i 2.57907 + 0.624645i
\(886\) 0 0
\(887\) 9.97196e6i 0.425571i 0.977099 + 0.212785i \(0.0682535\pi\)
−0.977099 + 0.212785i \(0.931747\pi\)
\(888\) 0 0
\(889\) 4.34761e7 1.84500
\(890\) 0 0
\(891\) −2.85930e7 −1.20661
\(892\) 0 0
\(893\) 6.45089e6i 0.270702i
\(894\) 0 0
\(895\) −385879. + 1.59324e6i −0.0161025 + 0.0664850i
\(896\) 0 0
\(897\) 3.15030e6i 0.130729i
\(898\) 0 0
\(899\) −2.01120e7 −0.829956
\(900\) 0 0
\(901\) 2.95771e7 1.21379
\(902\) 0 0
\(903\) 5.49143e7i 2.24112i
\(904\) 0 0
\(905\) 613125. 2.53151e6i 0.0248844 0.102744i
\(906\) 0 0
\(907\) 2.17790e7i 0.879064i −0.898227 0.439532i \(-0.855144\pi\)
0.898227 0.439532i \(-0.144856\pi\)
\(908\) 0 0
\(909\) 5.31606e7 2.13393
\(910\) 0 0
\(911\) 4.45704e7 1.77931 0.889653 0.456637i \(-0.150946\pi\)
0.889653 + 0.456637i \(0.150946\pi\)
\(912\) 0 0
\(913\) 1.01881e7i 0.404499i
\(914\) 0 0
\(915\) −5.63041e6 1.36367e6i −0.222325 0.0538465i
\(916\) 0 0
\(917\) 2.92232e7i 1.14764i
\(918\) 0 0
\(919\) −5.05848e7 −1.97575 −0.987873 0.155264i \(-0.950377\pi\)
−0.987873 + 0.155264i \(0.950377\pi\)
\(920\) 0 0
\(921\) 3.32893e7 1.29317
\(922\) 0 0
\(923\) 4.31662e6i 0.166778i
\(924\) 0 0
\(925\) −1.62569e7 + 3.15925e7i −0.624715 + 1.21403i
\(926\) 0 0
\(927\) 2.62529e7i 1.00341i
\(928\) 0 0
\(929\) −9.61614e6 −0.365563 −0.182781 0.983154i \(-0.558510\pi\)
−0.182781 + 0.983154i \(0.558510\pi\)
\(930\) 0 0
\(931\) −3.21902e6 −0.121716
\(932\) 0 0
\(933\) 2.14756e7i 0.807682i
\(934\) 0 0
\(935\) −2.07708e7 5.03064e6i −0.777007 0.188189i
\(936\) 0 0
\(937\) 3.07884e7i 1.14562i −0.819690 0.572808i \(-0.805854\pi\)
0.819690 0.572808i \(-0.194146\pi\)
\(938\) 0 0
\(939\) 4.83486e7 1.78945
\(940\) 0 0
\(941\) −3.94367e7 −1.45186 −0.725932 0.687766i \(-0.758591\pi\)
−0.725932 + 0.687766i \(0.758591\pi\)
\(942\) 0 0
\(943\) 1.37397e7i 0.503152i
\(944\) 0 0
\(945\) 1.96310e7 8.10535e7i 0.715092 2.95252i
\(946\) 0 0
\(947\) 2.02857e7i 0.735045i 0.930015 + 0.367523i \(0.119794\pi\)
−0.930015 + 0.367523i \(0.880206\pi\)
\(948\) 0 0
\(949\) −1.11765e6 −0.0402849
\(950\) 0 0
\(951\) −1.87066e7 −0.670724
\(952\) 0 0
\(953\) 7.23933e6i 0.258206i 0.991631 + 0.129103i \(0.0412098\pi\)
−0.991631 + 0.129103i \(0.958790\pi\)
\(954\) 0 0
\(955\) 4.30888e6 1.77908e7i 0.152882 0.631227i
\(956\) 0 0
\(957\) 1.80613e7i 0.637482i
\(958\) 0 0
\(959\) 5.30377e7 1.86225
\(960\) 0 0
\(961\) 9.29152e6 0.324548
\(962\) 0 0
\(963\) 3.35382e7i 1.16540i
\(964\) 0 0
\(965\) −2.20921e7 5.35066e6i −0.763693 0.184965i
\(966\) 0 0
\(967\) 4.74626e7i 1.63224i −0.577880 0.816122i \(-0.696120\pi\)
0.577880 0.816122i \(-0.303880\pi\)
\(968\) 0 0
\(969\) 3.92088e7 1.34145
\(970\) 0 0
\(971\) 1.13308e7 0.385668 0.192834 0.981231i \(-0.438232\pi\)
0.192834 + 0.981231i \(0.438232\pi\)
\(972\) 0 0
\(973\) 3.77811e7i 1.27936i
\(974\) 0 0
\(975\) 6.75321e6 + 3.47507e6i 0.227509 + 0.117072i
\(976\) 0 0
\(977\) 4.09471e7i 1.37242i −0.727404 0.686210i \(-0.759273\pi\)
0.727404 0.686210i \(-0.240727\pi\)
\(978\) 0 0
\(979\) −9.81072e6 −0.327148
\(980\) 0 0
\(981\) −3.12835e7 −1.03787
\(982\) 0 0
\(983\) 1.17862e7i 0.389036i 0.980899 + 0.194518i \(0.0623143\pi\)
−0.980899 + 0.194518i \(0.937686\pi\)
\(984\) 0 0
\(985\) −1.97803e7 4.79074e6i −0.649595 0.157330i
\(986\) 0 0
\(987\) 4.04517e7i 1.32173i
\(988\) 0 0
\(989\) 1.67553e7 0.544706
\(990\) 0 0
\(991\) −4.78243e7 −1.54691 −0.773454 0.633853i \(-0.781473\pi\)
−0.773454 + 0.633853i \(0.781473\pi\)
\(992\) 0 0
\(993\) 7.90043e7i 2.54260i
\(994\) 0 0
\(995\) −726120. + 2.99805e6i −0.0232515 + 0.0960021i
\(996\) 0 0
\(997\) 5.93264e6i 0.189021i −0.995524 0.0945105i \(-0.969871\pi\)
0.995524 0.0945105i \(-0.0301286\pi\)
\(998\) 0 0
\(999\) 1.15521e8 3.66223
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 320.6.c.j.129.1 8
4.3 odd 2 320.6.c.i.129.8 8
5.4 even 2 inner 320.6.c.j.129.8 8
8.3 odd 2 80.6.c.d.49.1 8
8.5 even 2 40.6.c.a.9.8 yes 8
20.19 odd 2 320.6.c.i.129.1 8
24.5 odd 2 360.6.f.b.289.5 8
24.11 even 2 720.6.f.n.289.5 8
40.3 even 4 400.6.a.ba.1.4 4
40.13 odd 4 200.6.a.j.1.1 4
40.19 odd 2 80.6.c.d.49.8 8
40.27 even 4 400.6.a.z.1.1 4
40.29 even 2 40.6.c.a.9.1 8
40.37 odd 4 200.6.a.k.1.4 4
120.29 odd 2 360.6.f.b.289.6 8
120.59 even 2 720.6.f.n.289.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.c.a.9.1 8 40.29 even 2
40.6.c.a.9.8 yes 8 8.5 even 2
80.6.c.d.49.1 8 8.3 odd 2
80.6.c.d.49.8 8 40.19 odd 2
200.6.a.j.1.1 4 40.13 odd 4
200.6.a.k.1.4 4 40.37 odd 4
320.6.c.i.129.1 8 20.19 odd 2
320.6.c.i.129.8 8 4.3 odd 2
320.6.c.j.129.1 8 1.1 even 1 trivial
320.6.c.j.129.8 8 5.4 even 2 inner
360.6.f.b.289.5 8 24.5 odd 2
360.6.f.b.289.6 8 120.29 odd 2
400.6.a.z.1.1 4 40.27 even 4
400.6.a.ba.1.4 4 40.3 even 4
720.6.f.n.289.5 8 24.11 even 2
720.6.f.n.289.6 8 120.59 even 2